LMFDB - Embedded newform 471.4.b.a.313.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(313,471)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("471.313");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			313.12
Character	$\chi$	$=$	471.313
Dual form			471.4.b.a.313.29

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64007i q^{2} -3.00000 q^{3} +1.03005 q^{4} -13.5204i q^{5} +7.92020i q^{6} -34.8670i q^{7} -23.8399i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.64007i q^{2} -3.00000 q^{3} +1.03005 q^{4} -13.5204i q^{5} +7.92020i q^{6} -34.8670i q^{7} -23.8399i q^{8} +9.00000 q^{9} -35.6946 q^{10} -26.6926 q^{11} -3.09016 q^{12} -60.4351 q^{13} -92.0511 q^{14} +40.5611i q^{15} -54.6986 q^{16} +133.298 q^{17} -23.7606i q^{18} -24.6574 q^{19} -13.9267i q^{20} +104.601i q^{21} +70.4701i q^{22} -174.093i q^{23} +71.5198i q^{24} -57.8000 q^{25} +159.553i q^{26} -27.0000 q^{27} -35.9148i q^{28} -73.0744i q^{29} +107.084 q^{30} +225.206 q^{31} -46.3116i q^{32} +80.0777 q^{33} -351.914i q^{34} -471.414 q^{35} +9.27047 q^{36} +385.214 q^{37} +65.0972i q^{38} +181.305 q^{39} -322.324 q^{40} +510.719i q^{41} +276.153 q^{42} -83.9424i q^{43} -27.4947 q^{44} -121.683i q^{45} -459.617 q^{46} +239.216 q^{47} +164.096 q^{48} -872.705 q^{49} +152.596i q^{50} -399.893 q^{51} -62.2513 q^{52} +447.340i q^{53} +71.2818i q^{54} +360.893i q^{55} -831.226 q^{56} +73.9723 q^{57} -192.921 q^{58} +123.840i q^{59} +41.7800i q^{60} +203.342i q^{61} -594.558i q^{62} -313.803i q^{63} -559.854 q^{64} +817.104i q^{65} -211.410i q^{66} -340.891 q^{67} +137.304 q^{68} +522.279i q^{69} +1244.56i q^{70} +993.573 q^{71} -214.559i q^{72} -351.464i q^{73} -1016.99i q^{74} +173.400 q^{75} -25.3984 q^{76} +930.688i q^{77} -478.658i q^{78} -814.843i q^{79} +739.544i q^{80} +81.0000 q^{81} +1348.33 q^{82} +628.665i q^{83} +107.744i q^{84} -1802.23i q^{85} -221.614 q^{86} +219.223i q^{87} +636.349i q^{88} -753.633 q^{89} -321.252 q^{90} +2107.19i q^{91} -179.325i q^{92} -675.618 q^{93} -631.546i q^{94} +333.377i q^{95} +138.935i q^{96} -431.519i q^{97} +2304.00i q^{98} -240.233 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) $ 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	$ 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) $ 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/471\mathbb{Z}\right)^\times$.

$n$	$158$	$319$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.64007i		−	0.933404i	−0.884415	−	0.466702i	$-0.845442\pi$
$2$		−	2.64007i		−	0.933404i	0.884415	−	0.466702i	$-0.154558\pi$
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$	1.03005			0.128757
$5$		−	13.5204i		−	1.20930i	−0.796492	−	0.604649i	$-0.793314\pi$
$5$		−	13.5204i		−	1.20930i	0.796492	−	0.604649i	$-0.206686\pi$
$6$			7.92020i			0.538901i
$7$		−	34.8670i		−	1.88264i	−0.337517	−	0.941319i	$-0.609587\pi$
$7$		−	34.8670i		−	1.88264i	0.337517	−	0.941319i	$-0.390413\pi$
$8$		−	23.8399i		−	1.05359i
$9$	9.00000			0.333333
$10$	−35.6946			−1.12876
$11$	−26.6926			−0.731646			−0.365823	−	0.930684i	$-0.619212\pi$
$11$	−26.6926			−0.731646			−0.365823	+	0.930684i	$0.619212\pi$
$12$	−3.09016			−0.0743376
$13$	−60.4351			−1.28936			−0.644680	−	0.764453i	$-0.723009\pi$
$13$	−60.4351			−1.28936			−0.644680	+	0.764453i	$0.723009\pi$
$14$	−92.0511			−1.75726
$15$			40.5611i			0.698188i
$16$	−54.6986			−0.854665
$17$	133.298			1.90173			0.950865	−	0.309605i	$-0.100197\pi$
$17$	133.298			1.90173			0.950865	+	0.309605i	$0.100197\pi$
$18$		−	23.7606i		−	0.311135i
$19$	−24.6574			−0.297726			−0.148863	−	0.988858i	$-0.547561\pi$
$19$	−24.6574			−0.297726			−0.148863	+	0.988858i	$0.547561\pi$
$20$		−	13.9267i		−	0.155705i
$21$			104.601i			1.08694i
$22$			70.4701i			0.682922i
$23$		−	174.093i		−	1.57830i	−0.614201	−	0.789150i	$-0.710521\pi$
$23$		−	174.093i		−	1.57830i	0.614201	−	0.789150i	$-0.289479\pi$
$24$			71.5198i			0.608288i
$25$	−57.8000			−0.462400
$26$			159.553i			1.20349i
$27$	−27.0000			−0.192450
$28$		−	35.9148i		−	0.242402i
$29$		−	73.0744i		−	0.467916i	−0.972247	−	0.233958i	$-0.924832\pi$
$29$		−	73.0744i		−	0.467916i	0.972247	−	0.233958i	$-0.0751679\pi$
$30$	107.084			0.651692
$31$	225.206			1.30478			0.652390	−	0.757884i	$-0.273767\pi$
$31$	225.206			1.30478			0.652390	+	0.757884i	$0.273767\pi$
$32$		−	46.3116i		−	0.255838i
$33$	80.0777			0.422416
$34$		−	351.914i		−	1.77508i
$35$	−471.414			−2.27667
$36$	9.27047			0.0429188
$37$	385.214			1.71159			0.855794	−	0.517317i	$-0.173069\pi$
$37$	385.214			1.71159			0.855794	+	0.517317i	$0.173069\pi$
$38$			65.0972i			0.277899i
$39$	181.305			0.744412
$40$	−322.324			−1.27410
$41$			510.719i			1.94539i	0.232095	+	0.972693i	$0.425442\pi$
$41$			510.719i			1.94539i	−0.232095	+	0.972693i	$0.574558\pi$
$42$	276.153			1.01456
$43$		−	83.9424i		−	0.297700i	−0.988860	−	0.148850i	$-0.952443\pi$
$43$		−	83.9424i		−	0.297700i	0.988860	−	0.148850i	$-0.0475572\pi$
$44$	−27.4947			−0.0942042
$45$		−	121.683i		−	0.403099i
$46$	−459.617			−1.47319
$47$	239.216			0.742409			0.371205	−	0.928551i	$-0.378945\pi$
$47$	239.216			0.742409			0.371205	+	0.928551i	$0.378945\pi$
$48$	164.096			0.493441
$49$	−872.705			−2.54433
$50$			152.596i			0.431606i
$51$	−399.893			−1.09796
$52$	−62.2513			−0.166013
$53$			447.340i			1.15938i	0.814838	+	0.579688i	$0.196826\pi$
$53$			447.340i			1.15938i	−0.814838	+	0.579688i	$0.803174\pi$
$54$			71.2818i			0.179634i
$55$			360.893i			0.884778i
$56$	−831.226			−1.98352
$57$	73.9723			0.171892
$58$	−192.921			−0.436755
$59$			123.840i			0.273264i	0.990622	+	0.136632i	$0.0436277\pi$
$59$			123.840i			0.273264i	−0.990622	+	0.136632i	$0.956372\pi$
$60$			41.7800i			0.0898963i
$61$			203.342i			0.426808i	0.976964	+	0.213404i	$0.0684551\pi$
$61$			203.342i			0.426808i	−0.976964	+	0.213404i	$0.931545\pi$
$62$		−	594.558i		−	1.21789i
$63$		−	313.803i		−	0.627546i
$64$	−559.854			−1.09347
$65$			817.104i			1.55922i
$66$		−	211.410i		−	0.394285i
$67$	−340.891			−0.621589			−0.310795	−	0.950477i	$-0.600595\pi$
$67$	−340.891			−0.621589			−0.310795	+	0.950477i	$0.600595\pi$
$68$	137.304			0.244860
$69$			522.279i			0.911232i
$70$			1244.56i			2.12505i
$71$	993.573			1.66078			0.830390	−	0.557183i	$-0.188118\pi$
$71$	993.573			1.66078			0.830390	+	0.557183i	$0.188118\pi$
$72$		−	214.559i		−	0.351195i
$73$		−	351.464i		−	0.563503i	−0.959487	−	0.281752i	$-0.909085\pi$
$73$		−	351.464i		−	0.563503i	0.959487	−	0.281752i	$-0.0909154\pi$
$74$		−	1016.99i		−	1.59760i
$75$	173.400			0.266967
$76$	−25.3984			−0.0383342
$77$			930.688i			1.37743i
$78$		−	478.658i		−	0.694837i
$79$		−	814.843i		−	1.16047i	−0.814450	−	0.580234i	$-0.802961\pi$
$79$		−	814.843i		−	1.16047i	0.814450	−	0.580234i	$-0.197039\pi$
$80$			739.544i			1.03354i
$81$	81.0000			0.111111
$82$	1348.33			1.81583
$83$			628.665i			0.831385i	0.909505	+	0.415692i	$0.136461\pi$
$83$			628.665i			0.831385i	−0.909505	+	0.415692i	$0.863539\pi$
$84$			107.744i			0.139951i
$85$		−	1802.23i		−	2.29976i
$86$	−221.614			−0.277874
$87$			219.223i			0.270152i
$88$			636.349i			0.770852i
$89$	−753.633			−0.897583			−0.448792	−	0.893636i	$-0.648145\pi$
$89$	−753.633			−0.897583			−0.448792	+	0.893636i	$0.648145\pi$
$90$	−321.252			−0.376254
$91$			2107.19i			2.42740i
$92$		−	179.325i		−	0.203216i
$93$	−675.618			−0.753315
$94$		−	631.546i		−	0.692968i
$95$			333.377i			0.360040i
$96$			138.935i			0.147708i
$97$		−	431.519i		−	0.451692i	−0.974163	−	0.225846i	$-0.927485\pi$
$97$		−	431.519i		−	0.451692i	0.974163	−	0.225846i	$-0.0725146\pi$
$98$			2304.00i			2.37489i
$99$	−240.233			−0.243882
$100$	−59.5370			−0.0595370
$101$	338.641			0.333624			0.166812	−	0.985989i	$-0.446653\pi$
$101$	338.641			0.333624			0.166812	+	0.985989i	$0.446653\pi$
$102$			1055.74i			1.02484i
$103$		−	1106.38i		−	1.05840i	−0.848497	−	0.529200i	$-0.822492\pi$
$103$		−	1106.38i		−	1.05840i	0.848497	−	0.529200i	$-0.177508\pi$
$104$			1440.77i			1.35845i
$105$	1414.24			1.31444
$106$	1181.01			1.08217
$107$			350.081i			0.316295i	0.987415	+	0.158148i	$0.0505522\pi$
$107$			350.081i			0.316295i	−0.987415	+	0.158148i	$0.949448\pi$
$108$	−27.8114			−0.0247792
$109$	588.820			0.517420			0.258710	−	0.965955i	$-0.416703\pi$
$109$	588.820			0.517420			0.258710	+	0.965955i	$0.416703\pi$
$110$	952.781			0.825855
$111$	−1155.64			−0.988186
$112$			1907.17i			1.60903i
$113$	595.421			0.495686			0.247843	−	0.968800i	$-0.420278\pi$
$113$	595.421			0.495686			0.247843	+	0.968800i	$0.420278\pi$
$114$		−	195.292i		−	0.160445i
$115$	−2353.80			−1.90863
$116$		−	75.2704i		−	0.0602473i
$117$	−543.916			−0.429787
$118$	326.945			0.255065
$119$		−	4647.68i		−	3.58027i
$120$	966.973			0.735601
$121$	−618.507			−0.464694
$122$	536.837			0.398384
$123$		−	1532.16i		−	1.12317i
$124$	231.974			0.167999
$125$		−	908.568i		−	0.650118i
$126$	−828.460			−0.585754
$127$	−1119.17			−0.781974			−0.390987	−	0.920396i	$-0.627866\pi$
$127$	−1119.17			−0.781974			−0.390987	+	0.920396i	$0.627866\pi$
$128$			1107.56i			0.764807i
$129$			251.827i			0.171877i
$130$	2157.21			1.45538
$131$			2388.19i			1.59280i	0.604768	+	0.796401i	$0.293266\pi$
$131$			2388.19i			1.59280i	−0.604768	+	0.796401i	$0.706734\pi$
$132$	82.4842			0.0543888
$133$			859.730i			0.560511i
$134$			899.975i			0.580194i
$135$			365.050i			0.232729i
$136$		−	3177.81i		−	2.00364i
$137$			576.428i			0.359471i	0.983715	+	0.179736i	$0.0575242\pi$
$137$			576.428i			0.359471i	−0.983715	+	0.179736i	$0.942476\pi$
$138$	1378.85			0.850548
$139$			331.601i			0.202346i	0.994869	+	0.101173i	$0.0322595\pi$
$139$			331.601i			0.202346i	−0.994869	+	0.101173i	$0.967740\pi$
$140$	−485.581			−0.293136
$141$	−717.648			−0.428630
$142$		−	2623.10i		−	1.55018i
$143$	1613.17			0.943355
$144$	−492.287			−0.284888
$145$	−987.992			−0.565850
$146$	−927.888			−0.525976
$147$	2618.11			1.46897
$148$	396.790			0.220378
$149$			1557.82i			0.856521i	0.903655	+	0.428261i	$0.140873\pi$
$149$			1557.82i			0.856521i	−0.903655	+	0.428261i	$0.859127\pi$
$150$		−	457.787i		−	0.249188i
$151$		−	198.336i		−	0.106890i	−0.998571	−	0.0534448i	$-0.982980\pi$
$151$		−	198.336i		−	0.106890i	0.998571	−	0.0534448i	$-0.0170201\pi$
$152$			587.832i			0.313680i
$153$	1199.68			0.633910
$154$	2457.08			1.28569
$155$		−	3044.86i		−	1.57787i
$156$	186.754			0.0958479
$157$	−1031.41	−	1675.14i	−0.524303	−	0.851532i
$157$	−1031.41	−	1675.14i	−0.524303	−	0.851532i
$158$	−2151.24			−1.08319
$159$		−	1342.02i		−	0.669366i
$160$	−626.150			−0.309384
$161$	−6070.09			−2.97137
$162$		−	213.845i		−	0.103712i
$163$			963.040i			0.462767i	0.972863	+	0.231384i	$0.0743253\pi$
$163$			963.040i			0.462767i	−0.972863	+	0.231384i	$0.925675\pi$
$164$			526.067i			0.250481i
$165$		−	1082.68i		−	0.510827i
$166$	1659.72			0.776018
$167$	3280.99			1.52030			0.760151	−	0.649747i	$-0.225125\pi$
$167$	3280.99			1.52030			0.760151	+	0.649747i	$0.225125\pi$
$168$	2493.68			1.14519
$169$	1455.40			0.662448
$170$	−4758.01			−2.14660
$171$	−221.917			−0.0992422
$172$		−	86.4651i		−	0.0383308i
$173$	1580.01			0.694369			0.347185	−	0.937797i	$-0.387138\pi$
$173$	1580.01			0.694369			0.347185	+	0.937797i	$0.387138\pi$
$174$	578.764			0.252161
$175$			2015.31i			0.870532i
$176$	1460.04			0.625313
$177$		−	371.519i		−	0.157769i
$178$			1989.64i			0.837808i
$179$			2495.05i			1.04184i	0.853606	+	0.520919i	$0.174411\pi$
$179$			2495.05i			1.04184i	−0.853606	+	0.520919i	$0.825589\pi$
$180$		−	125.340i		−	0.0519016i
$181$		−	1545.52i		−	0.634683i	−0.948311	−	0.317341i	$-0.897210\pi$
$181$		−	1545.52i		−	0.634683i	0.948311	−	0.317341i	$-0.102790\pi$
$182$	5563.11			2.26574
$183$		−	610.026i		−	0.246418i
$184$	−4150.37			−1.66287
$185$		−	5208.23i		−	2.06982i
$186$			1783.67i			0.703147i
$187$	−3558.05			−1.39139
$188$	246.405			0.0955901
$189$			941.408i			0.362314i
$190$	880.138			0.336063
$191$		−	2464.28i		−	0.933555i	−0.884375	−	0.466778i	$-0.845415\pi$
$191$		−	2464.28i		−	0.933555i	0.884375	−	0.466778i	$-0.154585\pi$
$192$	1679.56			0.631313
$193$	−275.499			−0.102750			−0.0513752	−	0.998679i	$-0.516360\pi$
$193$	−275.499			−0.102750			−0.0513752	+	0.998679i	$0.516360\pi$
$194$	−1139.24			−0.421611
$195$		−	2451.31i		−	0.900215i
$196$	−898.932			−0.327599
$197$	−1112.23			−0.402251			−0.201125	−	0.979566i	$-0.564460\pi$
$197$	−1112.23			−0.402251			−0.201125	+	0.979566i	$0.564460\pi$
$198$			634.231i			0.227641i
$199$	2812.57			1.00190			0.500950	−	0.865476i	$-0.332984\pi$
$199$	2812.57			1.00190			0.500950	+	0.865476i	$0.332984\pi$
$200$			1377.95i			0.487178i
$201$	1022.67			0.358875
$202$		−	894.035i		−	0.311406i
$203$	−2547.88			−0.880917
$204$	−411.911			−0.141370
$205$	6905.10			2.35255
$206$	−2920.93			−0.987915
$207$		−	1566.84i		−	0.526100i
$208$	3305.71			1.10197
$209$	658.170			0.217830
$210$		−	3733.69i		−	1.22690i
$211$		−	3326.08i		−	1.08520i	−0.839992	−	0.542599i	$-0.817440\pi$
$211$		−	3326.08i		−	1.08520i	0.839992	−	0.542599i	$-0.182560\pi$
$212$			460.784i			0.149277i
$213$	−2980.72			−0.958852
$214$	924.236			0.295231
$215$	−1134.93			−0.360008
$216$			643.678i			0.202763i
$217$		−	7852.24i		−	2.45643i
$218$		−	1554.52i		−	0.482962i
$219$			1054.39i			0.325339i
$220$			371.738i			0.113921i
$221$	−8055.85			−2.45201
$222$			3050.97i			0.922377i
$223$		−	1627.99i		−	0.488871i	−0.969665	−	0.244436i	$-0.921397\pi$
$223$		−	1627.99i		−	0.488871i	0.969665	−	0.244436i	$-0.0786027\pi$
$224$	−1614.75			−0.481651
$225$	−520.200			−0.154133
$226$		−	1571.95i		−	0.462675i
$227$		−	1395.24i		−	0.407952i	−0.978976	−	0.203976i	$-0.934614\pi$
$227$		−	1395.24i		−	0.407952i	0.978976	−	0.203976i	$-0.0653864\pi$
$228$	76.1953			0.0221323
$229$			298.836i			0.0862343i	0.999070	+	0.0431171i	$0.0137289\pi$
$229$			298.836i			0.0862343i	−0.999070	+	0.0431171i	$0.986271\pi$
$230$			6214.18i			1.78153i
$231$		−	2792.06i		−	0.795257i
$232$	−1742.09			−0.492990
$233$	3559.20			1.00073			0.500366	−	0.865814i	$-0.333199\pi$
$233$	3559.20			1.00073			0.500366	+	0.865814i	$0.333199\pi$
$234$			1435.97i			0.401165i
$235$		−	3234.28i		−	0.897794i
$236$			127.561i			0.0351845i
$237$			2444.53i			0.669997i
$238$	−12270.2			−3.34184
$239$	−853.047			−0.230875			−0.115437	−	0.993315i	$-0.536827\pi$
$239$	−853.047			−0.230875			−0.115437	+	0.993315i	$0.536827\pi$
$240$		−	2218.63i		−	0.596717i
$241$			289.249i			0.0773119i	0.999253	+	0.0386560i	$0.0123076\pi$
$241$			289.249i			0.0773119i	−0.999253	+	0.0386560i	$0.987692\pi$
$242$			1632.90i			0.433747i
$243$	−243.000			−0.0641500
$244$			209.453i			0.0549543i
$245$			11799.3i			3.07685i
$246$	−4044.99			−1.04837
$247$	1490.17			0.383876
$248$		−	5368.89i		−	1.37470i
$249$		−	1886.00i		−	0.480000i
$250$	−2398.68			−0.606823
$251$		−	3092.78i		−	0.777748i	−0.921291	−	0.388874i	$-0.872864\pi$
$251$		−	3092.78i		−	0.777748i	0.921291	−	0.388874i	$-0.127136\pi$
$252$		−	323.233i		−	0.0808007i
$253$			4646.99i			1.15476i
$254$			2954.69i			0.729897i
$255$			5406.69i			1.32777i
$256$	−1554.81			−0.379591
$257$	3651.34			0.886242			0.443121	−	0.896462i	$-0.353871\pi$
$257$	3651.34			0.886242			0.443121	+	0.896462i	$0.353871\pi$
$258$	664.841			0.160431
$259$		−	13431.2i		−	3.22230i
$260$			841.659i			0.200760i
$261$		−	657.670i		−	0.155972i
$262$	6304.98			1.48673
$263$	−3867.71			−0.906819			−0.453409	−	0.891302i	$-0.649792\pi$
$263$	−3867.71			−0.906819			−0.453409	+	0.891302i	$0.649792\pi$
$264$		−	1909.05i		−	0.445052i
$265$	6048.20			1.40203
$266$	2269.74			0.523184
$267$	2260.90			0.518220
$268$	−351.136			−0.0800337
$269$		−	4452.11i		−	1.00911i	−0.863380	−	0.504553i	$-0.831657\pi$
$269$		−	4452.11i		−	1.00911i	0.863380	−	0.504553i	$-0.168343\pi$
$270$	963.755			0.217231
$271$			2685.55i			0.601976i	0.953628	+	0.300988i	$0.0973165\pi$
$271$			2685.55i			0.601976i	−0.953628	+	0.300988i	$0.902684\pi$
$272$	−7291.19			−1.62534
$273$		−	6321.56i		−	1.40146i
$274$	1521.81			0.335532
$275$	1542.83			0.338313
$276$			537.975i			0.117327i
$277$	−1994.20			−0.432563			−0.216282	−	0.976331i	$-0.569393\pi$
$277$	−1994.20			−0.432563			−0.216282	+	0.976331i	$0.569393\pi$
$278$	875.449			0.188870
$279$	2026.85			0.434927
$280$			11238.5i			2.39867i
$281$	4312.55			0.915535			0.457767	−	0.889072i	$-0.348649\pi$
$281$	4312.55			0.915535			0.457767	+	0.889072i	$0.348649\pi$
$282$			1894.64i			0.400085i
$283$	1209.55			0.254065			0.127032	−	0.991899i	$-0.459455\pi$
$283$	1209.55			0.254065			0.127032	+	0.991899i	$0.459455\pi$
$284$	1023.43			0.213836
$285$		−	1000.13i		−	0.207869i
$286$		−	4258.87i		−	0.880532i
$287$	17807.2			3.66246
$288$		−	416.805i		−	0.0852793i
$289$	12855.3			2.61658
$290$			2608.36i			0.528167i
$291$			1294.56i			0.260785i
$292$		−	362.026i		−	0.0725547i
$293$			546.739i			0.109013i	0.998513	+	0.0545065i	$0.0173586\pi$
$293$			546.739i			0.109013i	−0.998513	+	0.0545065i	$0.982641\pi$
$294$		−	6911.99i		−	1.37114i
$295$	1674.36			0.330457
$296$		−	9183.47i		−	1.80331i
$297$	720.699			0.140805
$298$	4112.75			0.799481
$299$			10521.3i			2.03500i
$300$	178.611			0.0343737
$301$	−2926.82			−0.560462
$302$	−523.619			−0.0997711
$303$	−1015.92			−0.192618
$304$	1348.73			0.254456
$305$	2749.26			0.516138
$306$		−	3167.23i		−	0.591694i
$307$			4832.95i			0.898472i	0.893413	+	0.449236i	$0.148304\pi$
$307$			4832.95i			0.898472i	−0.893413	+	0.449236i	$0.851696\pi$
$308$			958.658i			0.177353i
$309$			3319.15i			0.611068i
$310$	−8038.64			−1.47279
$311$	5665.59			1.03301			0.516505	−	0.856284i	$-0.327233\pi$
$311$	5665.59			1.03301			0.516505	+	0.856284i	$0.327233\pi$
$312$		−	4322.30i		−	0.784302i
$313$	−9565.64			−1.72742			−0.863710	−	0.503990i	$-0.831865\pi$
$313$	−9565.64			−1.72742			−0.863710	+	0.503990i	$0.831865\pi$
$314$	−4422.47	+	2722.99i	−0.794824	+	0.489386i
$315$	−4242.72			−0.758890
$316$		−	839.331i		−	0.149418i
$317$	−2892.57			−0.512502			−0.256251	−	0.966610i	$-0.582487\pi$
$317$	−2892.57			−0.512502			−0.256251	+	0.966610i	$0.582487\pi$
$318$	−3543.02			−0.624789
$319$			1950.54i			0.342349i
$320$			7569.43i			1.32232i
$321$		−	1050.24i		−	0.182613i
$322$			16025.4i			2.77349i
$323$	−3286.78			−0.566195
$324$	83.4342			0.0143063
$325$	3493.15			0.596200
$326$	2542.49			0.431949
$327$	−1766.46			−0.298733
$328$	12175.5			2.04963
$329$		−	8340.73i		−	1.39769i
$330$	−2858.34			−0.476808
$331$	4257.03			0.706911			0.353456	−	0.935451i	$-0.385007\pi$
$331$	4257.03			0.706911			0.353456	+	0.935451i	$0.385007\pi$
$332$			647.558i			0.107046i
$333$	3466.92			0.570529
$334$		−	8662.02i		−	1.41906i
$335$			4608.97i			0.751686i
$336$		−	5721.52i		−	0.928972i
$337$			5800.84i			0.937662i	0.883288	+	0.468831i	$0.155325\pi$
$337$			5800.84i			0.937662i	−0.883288	+	0.468831i	$0.844675\pi$
$338$		−	3842.35i		−	0.618332i
$339$	−1786.26			−0.286184
$340$		−	1856.39i		−	0.296109i
$341$	−6011.32			−0.954637
$342$			585.875i			0.0926330i
$343$			18469.2i			2.90741i
$344$	−2001.18			−0.313653
$345$	7061.40			1.10195
$346$		−	4171.33i		−	0.648127i
$347$	−870.321			−0.134643			−0.0673217	−	0.997731i	$-0.521445\pi$
$347$	−870.321			−0.134643			−0.0673217	+	0.997731i	$0.521445\pi$
$348$			225.811i			0.0347838i
$349$	1200.97			0.184201			0.0921007	−	0.995750i	$-0.470642\pi$
$349$	1200.97			0.184201			0.0921007	+	0.995750i	$0.470642\pi$
$350$	5320.55			0.812558
$351$	1631.75			0.248137
$352$			1236.18i			0.187183i
$353$	−10936.9			−1.64904			−0.824518	−	0.565835i	$-0.808554\pi$
$353$	−10936.9			−1.64904			−0.824518	+	0.565835i	$0.808554\pi$
$354$	−980.835			−0.147262
$355$		−	13433.5i		−	2.00838i
$356$	−776.281			−0.115570
$357$			13943.0i			2.06707i
$358$	6587.10			0.972456
$359$		−	9674.99i		−	1.42236i	−0.703011	−	0.711179i	$-0.748162\pi$
$359$		−	9674.99i		−	1.42236i	0.703011	−	0.711179i	$-0.251838\pi$
$360$	−2900.92			−0.424700
$361$	−6251.01			−0.911359
$362$	−4080.27			−0.592415
$363$	1855.52			0.268291
$364$			2170.51i			0.312543i
$365$	−4751.92			−0.681443
$366$	−1610.51			−0.230007
$367$			6538.06i			0.929929i	0.885329	+	0.464964i	$0.153933\pi$
$367$			6538.06i			0.929929i	−0.885329	+	0.464964i	$0.846067\pi$
$368$			9522.64i			1.34892i
$369$			4596.47i			0.648462i
$370$	−13750.1			−1.93198
$371$	15597.4			2.18269
$372$	−695.921			−0.0969942
$373$			5406.53i			0.750509i	0.926922	+	0.375254i	$0.122445\pi$
$373$			5406.53i			0.750509i	−0.926922	+	0.375254i	$0.877555\pi$
$374$			9393.50i			1.29873i
$375$			2725.70i			0.375346i
$376$		−	5702.89i		−	0.782192i
$377$			4416.26i			0.603312i
$378$	2485.38			0.338185
$379$		−	1869.39i		−	0.253361i	−0.991944	−	0.126681i	$-0.959568\pi$
$379$		−	1869.39i		−	0.253361i	0.991944	−	0.126681i	$-0.0404324\pi$
$380$			343.396i			0.0463575i
$381$	3357.52			0.451473
$382$	−6505.86			−0.871384
$383$		−	11961.1i		−	1.59578i	−0.602801	−	0.797891i	$-0.705949\pi$
$383$		−	11961.1i		−	1.59578i	0.602801	−	0.797891i	$-0.294051\pi$
$384$		−	3322.68i		−	0.441562i
$385$	12583.2			1.66572
$386$			727.334i			0.0959076i
$387$		−	755.482i		−	0.0992333i
$388$		−	444.487i		−	0.0581583i
$389$	−13332.0			−1.73768			−0.868841	−	0.495092i	$-0.835134\pi$
$389$	−13332.0			−1.73768			−0.868841	+	0.495092i	$0.835134\pi$
$390$	−6471.62			−0.840265
$391$		−	23206.2i		−	3.00150i
$392$			20805.2i			2.68067i
$393$		−	7164.57i		−	0.919605i
$394$			2936.37i			0.375463i
$395$	−11017.0			−1.40335
$396$	−247.453			−0.0314014
$397$		−	13226.3i		−	1.67206i	−0.548683	−	0.836031i	$-0.684870\pi$
$397$		−	13226.3i		−	1.67206i	0.548683	−	0.836031i	$-0.315130\pi$
$398$		−	7425.38i		−	0.935178i
$399$		−	2579.19i		−	0.323611i
$400$	3161.58			0.395197
$401$		−	3720.71i		−	0.463350i	−0.972793	−	0.231675i	$-0.925579\pi$
$401$		−	3720.71i		−	0.463350i	0.972793	−	0.231675i	$-0.0744205\pi$
$402$		−	2699.92i		−	0.334975i
$403$	−13610.3			−1.68233
$404$	348.818			0.0429563
$405$		−	1095.15i		−	0.134366i
$406$			6726.58i			0.822252i
$407$	−10282.3			−1.25228
$408$			9533.42i			1.15680i
$409$		−	5929.59i		−	0.716868i	−0.933555	−	0.358434i	$-0.883311\pi$
$409$		−	5929.59i		−	0.716868i	0.933555	−	0.358434i	$-0.116689\pi$
$410$		−	18229.9i		−	2.19588i
$411$		−	1729.28i		−	0.207541i
$412$		−	1139.63i		−	0.136276i
$413$	4317.91			0.514457
$414$	−4136.55			−0.491064
$415$	8499.77			1.00539
$416$			2798.85i			0.329867i
$417$		−	994.804i		−	0.116824i
$418$		−	1737.61i		−	0.203324i
$419$	10167.0			1.18542			0.592709	−	0.805417i	$-0.298059\pi$
$419$	10167.0			1.18542			0.592709	+	0.805417i	$0.298059\pi$
$420$	1456.74			0.169242
$421$			15179.4i			1.75724i	0.477521	+	0.878620i	$0.341536\pi$
$421$			15179.4i			1.75724i	−0.477521	+	0.878620i	$0.658464\pi$
$422$	−8781.08			−1.01293
$423$	2152.94			0.247470
$424$	10664.6			1.22150
$425$	−7704.60			−0.879360
$426$			7869.29i			0.894996i
$427$	7089.92			0.803525
$428$			360.601i			0.0407251i
$429$	−4839.50			−0.544646
$430$			2996.29i			0.336033i
$431$	−8467.99			−0.946378			−0.473189	−	0.880961i	$-0.656897\pi$
$431$	−8467.99			−0.946378			−0.473189	+	0.880961i	$0.656897\pi$
$432$	1476.86			0.164480
$433$			10760.0i			1.19421i	0.802165	+	0.597103i	$0.203682\pi$
$433$			10760.0i			1.19421i	−0.802165	+	0.597103i	$0.796318\pi$
$434$	−20730.4			−2.29284
$435$	2963.97			0.326694
$436$	606.516			0.0666212
$437$			4292.69i			0.469902i
$438$	2783.66			0.303673
$439$			4512.78i			0.490622i	0.969444	+	0.245311i	$0.0788901\pi$
$439$			4512.78i			0.490622i	−0.969444	+	0.245311i	$0.921110\pi$
$440$	8603.66			0.932190
$441$	−7854.34			−0.848110
$442$			21268.0i			2.28872i
$443$			3199.46i			0.343140i	0.985172	+	0.171570i	$0.0548839\pi$
$443$			3199.46i			0.343140i	−0.985172	+	0.171570i	$0.945116\pi$
$444$	−1190.37			−0.127235
$445$			10189.4i			1.08544i
$446$	−4298.00			−0.456315
$447$		−	4673.46i		−	0.494513i
$448$			19520.4i			2.05860i
$449$			4391.78i			0.461606i	0.973000	+	0.230803i	$0.0741353\pi$
$449$			4391.78i			0.461606i	−0.973000	+	0.230803i	$0.925865\pi$
$450$			1373.36i			0.143869i
$451$		−	13632.4i		−	1.42333i
$452$	613.315			0.0638228
$453$			595.007i			0.0617127i
$454$	−3683.52			−0.380784
$455$	28489.9			2.93545
$456$		−	1763.49i		−	0.181104i
$457$	9477.53			0.970109			0.485055	−	0.874484i	$-0.338800\pi$
$457$	9477.53			0.970109			0.485055	+	0.874484i	$0.338800\pi$
$458$	788.947			0.0804915
$459$	−3599.04			−0.365988
$460$	−2424.54			−0.245749
$461$	−16361.7			−1.65302			−0.826510	−	0.562922i	$-0.809677\pi$
$461$	−16361.7			−1.65302			−0.826510	+	0.562922i	$0.809677\pi$
$462$	−7371.24			−0.742296
$463$			10626.7i			1.06666i	0.845906	+	0.533332i	$0.179060\pi$
$463$			10626.7i			1.06666i	−0.845906	+	0.533332i	$0.820940\pi$
$464$			3997.06i			0.399912i
$465$			9134.59i			0.910982i
$466$		−	9396.52i		−	0.934088i
$467$	18582.2			1.84128			0.920642	−	0.390408i	$-0.127666\pi$
$467$	18582.2			1.84128			0.920642	+	0.390408i	$0.127666\pi$
$468$	−560.262			−0.0553378
$469$			11885.8i			1.17023i
$470$	−8538.72			−0.838004
$471$	3094.23	+	5025.41i	0.302706	+	0.491632i
$472$	2952.33			0.287907
$473$			2240.64i			0.217811i
$474$	6453.72			0.625378
$475$	1425.20			0.137669
$476$		−	4787.36i		−	0.460983i
$477$			4026.06i			0.386459i
$478$			2252.10i			0.215499i
$479$			7318.66i			0.698118i	0.937101	+	0.349059i	$0.113499\pi$
$479$			7318.66i			0.698118i	−0.937101	+	0.349059i	$0.886501\pi$
$480$	1878.45			0.178623
$481$	−23280.4			−2.20685
$482$	763.637			0.0721633
$483$	18210.3			1.71552
$484$	−637.095			−0.0598324
$485$	−5834.29			−0.546230
$486$			641.536i			0.0598779i
$487$	−16218.0			−1.50905			−0.754527	−	0.656269i	$-0.772134\pi$
$487$	−16218.0			−1.50905			−0.754527	+	0.656269i	$0.772134\pi$
$488$	4847.66			0.449679
$489$		−	2889.12i		−	0.267179i
$490$	31150.9			2.87194
$491$		−	2394.72i		−	0.220106i	−0.993926	−	0.110053i	$-0.964898\pi$
$491$		−	2394.72i		−	0.220106i	0.993926	−	0.110053i	$-0.0351021\pi$
$492$		−	1578.20i		−	0.144615i
$493$		−	9740.64i		−	0.889851i
$494$		−	3934.16i		−	0.358312i
$495$			3248.04i			0.294926i
$496$	−12318.4			−1.11515
$497$		−	34642.9i		−	3.12665i
$498$	−4979.15			−0.448034
$499$			11811.8i			1.05966i	0.848105	+	0.529828i	$0.177743\pi$
$499$			11811.8i			1.05966i	−0.848105	+	0.529828i	$0.822257\pi$
$500$		−	935.873i		−	0.0837070i
$501$	−9842.96			−0.877747
$502$	−8165.15			−0.725953
$503$		−	1833.75i		−	0.162550i	−0.996692	−	0.0812750i	$-0.974101\pi$
$503$		−	1833.75i		−	0.162550i	0.996692	−	0.0812750i	$-0.0258992\pi$
$504$	−7481.03			−0.661174
$505$		−	4578.55i		−	0.403451i
$506$	12268.4			1.07786
$507$	−4366.19			−0.382465
$508$	−1152.81			−0.100684
$509$		−	5355.58i		−	0.466369i	−0.972432	−	0.233185i	$-0.925085\pi$
$509$		−	5355.58i		−	0.466369i	0.972432	−	0.233185i	$-0.0749147\pi$
$510$	14274.0			1.23934
$511$	−12254.5			−1.06087
$512$			12965.3i			1.11912i
$513$	665.751			0.0572975
$514$		−	9639.77i		−	0.827222i
$515$	−14958.7			−1.27992
$516$			259.395i			0.0221303i
$517$	−6385.29			−0.543181
$518$	−35459.3			−3.00771
$519$	−4740.03			−0.400894
$520$	19479.7			1.64277
$521$			20886.6i			1.75635i	0.478341	+	0.878174i	$0.341238\pi$
$521$			20886.6i			1.75635i	−0.478341	+	0.878174i	$0.658762\pi$
$522$	−1736.29			−0.145585
$523$	18368.3			1.53573			0.767866	−	0.640611i	$-0.221319\pi$
$523$	18368.3			1.53573			0.767866	+	0.640611i	$0.221319\pi$
$524$			2459.96i			0.205084i
$525$		−	6045.93i		−	0.502602i
$526$			10211.0i			0.846428i
$527$	30019.4			2.48134
$528$	−4380.13			−0.361024
$529$	−18141.4			−1.49103
$530$		−	15967.6i		−	1.30866i
$531$			1114.56i			0.0910879i
$532$			885.567i			0.0721695i
$533$		−	30865.3i		−	2.50830i
$534$		−	5968.92i		−	0.483709i
$535$	4733.21			0.382495
$536$			8126.82i			0.654898i
$537$		−	7485.16i		−	0.601505i
$538$	−11753.9			−0.941905
$539$	23294.7			1.86155
$540$			376.020i			0.0299654i
$541$			13518.7i			1.07433i	0.843477	+	0.537165i	$0.180505\pi$
$541$			13518.7i			1.07433i	−0.843477	+	0.537165i	$0.819495\pi$
$542$	7090.03			0.561887
$543$			4636.56i			0.366434i
$544$		−	6173.23i		−	0.486535i
$545$		−	7961.06i		−	0.625714i
$546$	−16689.3			−1.30813
$547$	−14505.7			−1.13386			−0.566928	−	0.823767i	$-0.691868\pi$
$547$	−14505.7			−1.13386			−0.566928	+	0.823767i	$0.691868\pi$
$548$			593.751i			0.0462843i
$549$			1830.08i			0.142269i
$550$		−	4073.17i		−	0.315783i
$551$			1801.83i			0.139311i
$552$	12451.1			0.960061
$553$	−28411.1			−2.18474
$554$			5264.83i			0.403757i
$555$			15624.7i			1.19501i
$556$			341.567i			0.0260533i
$557$	11684.7			0.888866			0.444433	−	0.895812i	$-0.353405\pi$
$557$	11684.7			0.888866			0.444433	+	0.895812i	$0.353405\pi$
$558$		−	5351.02i		−	0.405962i
$559$			5073.07i			0.383842i
$560$	25785.7			1.94579
$561$	10674.2			0.803322
$562$		−	11385.4i		−	0.854564i
$563$			5644.87i			0.422563i	0.977425	+	0.211282i	$0.0677637\pi$
$563$			5644.87i			0.422563i	−0.977425	+	0.211282i	$0.932236\pi$
$564$	−739.215			−0.0551889
$565$		−	8050.31i		−	0.599432i
$566$		−	3193.29i		−	0.237145i
$567$		−	2824.22i		−	0.209182i
$568$		−	23686.7i		−	1.74977i
$569$		−	3479.18i		−	0.256335i	−0.991753	−	0.128168i	$-0.959090\pi$
$569$		−	3479.18i		−	0.256335i	0.991753	−	0.128168i	$-0.0409096\pi$
$570$	−2640.41			−0.194026
$571$	−15866.9			−1.16289			−0.581446	−	0.813585i	$-0.697513\pi$
$571$	−15866.9			−1.16289			−0.581446	+	0.813585i	$0.697513\pi$
$572$	1661.65			0.121463
$573$			7392.84i			0.538988i
$574$		−	47012.2i		−	3.41856i
$575$			10062.6i			0.729806i
$576$	−5038.69			−0.364488
$577$	−13884.0			−1.00173			−0.500864	−	0.865526i	$-0.666984\pi$
$577$	−13884.0			−1.00173			−0.500864	+	0.865526i	$0.666984\pi$
$578$		−	33938.7i		−	2.44233i
$579$	826.496			0.0593230
$580$	−1017.68			−0.0728569
$581$	21919.6			1.56520
$582$	3417.72			0.243417
$583$		−	11940.7i		−	0.848253i
$584$	−8378.88			−0.593699
$585$			7353.93i			0.519740i
$586$	1443.43			0.101753
$587$		−	6706.20i		−	0.471541i	−0.971809	−	0.235771i	$-0.924239\pi$
$587$		−	6706.20i		−	0.471541i	0.971809	−	0.235771i	$-0.0757614\pi$
$588$	2696.79			0.189139
$589$	−5553.00			−0.388467
$590$		−	4420.41i		−	0.308450i
$591$	3336.70			0.232240
$592$	−21070.6			−1.46283
$593$	8525.26			0.590372			0.295186	−	0.955440i	$-0.404618\pi$
$593$	8525.26			0.590372			0.295186	+	0.955440i	$0.404618\pi$
$594$		−	1902.69i		−	0.131428i
$595$	−62838.3			−4.32961
$596$			1604.64i			0.110283i
$597$	−8437.72			−0.578447
$598$	27777.0			1.89947
$599$		−	24149.3i		−	1.64727i	−0.567123	−	0.823633i	$-0.691943\pi$
$599$		−	24149.3i		−	1.64727i	0.567123	−	0.823633i	$-0.308057\pi$
$600$		−	4133.84i		−	0.281272i
$601$	23781.7			1.61410			0.807050	−	0.590483i	$-0.201063\pi$
$601$	23781.7			1.61410			0.807050	+	0.590483i	$0.201063\pi$
$602$			7726.99i			0.523137i
$603$	−3068.02			−0.207196
$604$		−	204.296i		−	0.0137627i
$605$			8362.44i			0.561953i
$606$			2682.11i			0.179791i
$607$		−	17172.9i		−	1.14831i	−0.818745	−	0.574157i	$-0.805330\pi$
$607$		−	17172.9i		−	1.14831i	0.818745	−	0.574157i	$-0.194670\pi$
$608$			1141.93i			0.0761697i
$609$	7643.65			0.508598
$610$		−	7258.22i		−	0.481765i
$611$	−14457.0			−0.957232
$612$	1235.73			0.0816201
$613$			16080.3i			1.05950i	0.848153	+	0.529752i	$0.177715\pi$
$613$			16080.3i			1.05950i	−0.848153	+	0.529752i	$0.822285\pi$
$614$	12759.3			0.838638
$615$	−20715.3			−1.35825
$616$	22187.5			1.45124
$617$	−10522.3			−0.686565			−0.343283	−	0.939232i	$-0.611539\pi$
$617$	−10522.3			−0.686565			−0.343283	+	0.939232i	$0.611539\pi$
$618$	8762.78			0.570373
$619$	−6007.96			−0.390114			−0.195057	−	0.980792i	$-0.562489\pi$
$619$	−6007.96			−0.390114			−0.195057	+	0.980792i	$0.562489\pi$
$620$		−	3136.37i		−	0.203161i
$621$			4700.51i			0.303744i
$622$		−	14957.5i		−	0.964216i
$623$			26276.9i			1.68982i
$624$	−9917.14			−0.636223
$625$	−19509.2			−1.24859
$626$			25253.9i			1.61238i
$627$	−1974.51			−0.125764
$628$	−1062.41	−	1725.48i	−0.0675074	−	0.109640i
$629$	51348.1			3.25498
$630$			11201.1i			0.708351i
$631$	3463.53			0.218512			0.109256	−	0.994014i	$-0.465153\pi$
$631$	3463.53			0.218512			0.109256	+	0.994014i	$0.465153\pi$
$632$	−19425.8			−1.22265
$633$			9978.25i			0.626540i
$634$			7636.58i			0.478371i
$635$			15131.6i			0.945638i
$636$		−	1382.35i		−	0.0861853i
$637$	52742.0			3.28055
$638$	5149.56			0.319550
$639$	8942.15			0.553593
$640$	14974.6			0.924879
$641$	−9473.79			−0.583763			−0.291882	−	0.956454i	$-0.594281\pi$
$641$	−9473.79			−0.583763			−0.291882	+	0.956454i	$0.594281\pi$
$642$	−2772.71			−0.170452
$643$		−	7086.71i		−	0.434639i	−0.976101	−	0.217319i	$-0.930269\pi$
$643$		−	7086.71i		−	0.434639i	0.976101	−	0.217319i	$-0.0697313\pi$
$644$	−6252.51			−0.382583
$645$	3404.79			0.207851
$646$			8677.31i			0.528489i
$647$	15859.2			0.963663			0.481831	−	0.876264i	$-0.339972\pi$
$647$	15859.2			0.963663			0.481831	+	0.876264i	$0.339972\pi$
$648$		−	1931.03i		−	0.117065i
$649$		−	3305.60i		−	0.199932i
$650$		−	9222.14i		−	0.556495i
$651$			23556.7i			1.41822i
$652$			991.981i			0.0595843i
$653$	−9703.10			−0.581488			−0.290744	−	0.956801i	$-0.593903\pi$
$653$	−9703.10			−0.581488			−0.290744	+	0.956801i	$0.593903\pi$
$654$			4663.57i			0.278838i
$655$	32289.2			1.92617
$656$		−	27935.6i		−	1.66265i
$657$		−	3163.18i		−	0.187834i
$658$	−22020.1			−1.30461
$659$	29009.8			1.71481			0.857407	−	0.514639i	$-0.172074\pi$
$659$	29009.8			1.71481			0.857407	+	0.514639i	$0.172074\pi$
$660$		−	1115.22i		−	0.0657723i
$661$	−7796.78			−0.458789			−0.229395	−	0.973334i	$-0.573675\pi$
$661$	−7796.78			−0.458789			−0.229395	+	0.973334i	$0.573675\pi$
$662$		−	11238.8i		−	0.659834i
$663$	24167.6			1.41567
$664$	14987.3			0.875936
$665$	11623.8			0.677825
$666$		−	9152.91i		−	0.532534i
$667$	−12721.7			−0.738512
$668$	3379.59			0.195749
$669$			4883.97i			0.282250i
$670$	12168.0			0.701627
$671$		−	5427.72i		−	0.312272i
$672$	4844.24			0.278081
$673$		−	11372.7i		−	0.651389i	−0.945475	−	0.325695i	$-0.894402\pi$
$673$		−	11372.7i		−	0.651389i	0.945475	−	0.325695i	$-0.105598\pi$
$674$	15314.6			0.875217
$675$	1560.60			0.0889889
$676$	1499.14			0.0852945
$677$	17330.6			0.983853			0.491926	−	0.870637i	$-0.336293\pi$
$677$	17330.6			0.983853			0.491926	+	0.870637i	$0.336293\pi$
$678$			4715.85i			0.267126i
$679$	−15045.8			−0.850373
$680$	−42965.1			−2.42299
$681$			4185.71i			0.235531i
$682$			15870.3i			0.891062i
$683$			8262.09i			0.462870i	0.972850	+	0.231435i	$0.0743420\pi$
$683$			8262.09i			0.462870i	−0.972850	+	0.231435i	$0.925658\pi$
$684$	−228.586			−0.0127781
$685$	7793.51			0.434708
$686$	48759.9			2.71379
$687$		−	896.509i		−	0.0497874i
$688$			4591.53i			0.254434i
$689$		−	27035.1i		−	1.49485i
$690$		−	18642.6i		−	1.02857i
$691$		−	24536.5i		−	1.35082i	−0.737444	−	0.675408i	$-0.763967\pi$
$691$		−	24536.5i		−	1.35082i	0.737444	−	0.675408i	$-0.236033\pi$
$692$	1627.49			0.0894046
$693$			8376.19i			0.459142i
$694$			2297.70i			0.125677i
$695$	4483.37			0.244696
$696$	5226.27			0.284628
$697$			68077.6i			3.69960i
$698$		−	3170.63i		−	0.171934i
$699$	−10677.6			−0.577773
$700$			2075.87i			0.112087i
$701$			12589.5i			0.678312i	0.940730	+	0.339156i	$0.110142\pi$
$701$			12589.5i			0.678312i	−0.940730	+	0.339156i	$0.889858\pi$
$702$		−	4307.92i		−	0.231612i
$703$	−9498.38			−0.509585
$704$	14943.9			0.800030
$705$			9702.85i			0.518341i
$706$			28874.0i			1.53922i
$707$		−	11807.4i		−	0.628094i
$708$		−	382.684i		−	0.0203138i
$709$	23693.2			1.25503			0.627517	−	0.778603i	$-0.284071\pi$
$709$	23693.2			1.25503			0.627517	+	0.778603i	$0.284071\pi$
$710$	−35465.2			−1.87463
$711$		−	7333.59i		−	0.386823i
$712$			17966.6i			0.945681i
$713$		−	39206.8i		−	2.05933i
$714$	36810.6			1.92941
$715$		−	21810.6i		−	1.14080i
$716$			2570.03i			0.134143i
$717$	2559.14			0.133296
$718$	−25542.6			−1.32763
$719$			4278.41i			0.221916i	0.993825	+	0.110958i	$0.0353920\pi$
$719$			4278.41i			0.221916i	−0.993825	+	0.110958i	$0.964608\pi$
$720$			6655.90i			0.344515i
$721$	−38576.2			−1.99259
$722$			16503.1i			0.850666i
$723$		−	867.747i		−	0.0446361i
$724$		−	1591.97i		−	0.0817195i
$725$			4223.70i			0.216364i
$726$		−	4898.70i		−	0.250424i
$727$	−33553.2			−1.71172			−0.855858	−	0.517211i	$-0.826970\pi$
$727$	−33553.2			−1.71172			−0.855858	+	0.517211i	$0.826970\pi$
$728$	50235.2			2.55747
$729$	729.000			0.0370370
$730$			12545.4i			0.636062i
$731$		−	11189.3i		−	0.566145i
$732$		−	628.359i		−	0.0317279i
$733$	25145.3			1.26707			0.633536	−	0.773713i	$-0.281603\pi$
$733$	25145.3			1.26707			0.633536	+	0.773713i	$0.281603\pi$
$734$	17260.9			0.867999
$735$		−	35397.8i		−	1.77642i
$736$	−8062.53			−0.403789
$737$	9099.25			0.454783
$738$	12135.0			0.605277
$739$	27946.4			1.39110			0.695552	−	0.718476i	$-0.255160\pi$
$739$	27946.4			1.39110			0.695552	+	0.718476i	$0.255160\pi$
$740$		−	5364.75i		−	0.266503i
$741$	−4470.52			−0.221631
$742$		−	41178.2i		−	2.03733i
$743$	−12509.9			−0.617690			−0.308845	−	0.951112i	$-0.599942\pi$
$743$	−12509.9			−0.617690			−0.308845	+	0.951112i	$0.599942\pi$
$744$			16106.7i			0.793682i
$745$	21062.3			1.03579
$746$	14273.6			0.700528
$747$			5657.99i			0.277128i
$748$	−3664.98			−0.179151
$749$	12206.2			0.595469
$750$	7196.04			0.350350
$751$		−	28043.6i		−	1.36261i	−0.731997	−	0.681307i	$-0.761412\pi$
$751$		−	28043.6i		−	1.36261i	0.731997	−	0.681307i	$-0.238588\pi$
$752$	−13084.8			−0.634511
$753$			9278.35i			0.449033i
$754$	11659.2			0.563134
$755$	−2681.57			−0.129261
$756$			969.699i			0.0466503i
$757$			13492.5i			0.647811i	0.946089	+	0.323906i	$0.104996\pi$
$757$			13492.5i			0.647811i	−0.946089	+	0.323906i	$0.895004\pi$
$758$	−4935.31			−0.236489
$759$		−	13941.0i		−	0.666699i
$760$	7947.69			0.379333
$761$			34495.1i			1.64316i	0.570092	+	0.821581i	$0.306908\pi$
$761$			34495.1i			1.64316i	−0.570092	+	0.821581i	$0.693092\pi$
$762$		−	8864.08i		−	0.421406i
$763$		−	20530.4i		−	0.974115i
$764$		−	2538.34i		−	0.120201i
$765$		−	16220.1i		−	0.766586i
$766$	−31578.2			−1.48951
$767$		−	7484.26i		−	0.352335i
$768$	4664.42			0.219157
$769$	−2199.29			−0.103132			−0.0515659	−	0.998670i	$-0.516421\pi$
$769$	−2199.29			−0.103132			−0.0515659	+	0.998670i	$0.516421\pi$
$770$		−	33220.6i		−	1.55479i
$771$	−10954.0			−0.511672
$772$	−283.778			−0.0132298
$773$	−6886.01			−0.320404			−0.160202	−	0.987084i	$-0.551215\pi$
$773$	−6886.01			−0.320404			−0.160202	+	0.987084i	$0.551215\pi$
$774$	−1994.52			−0.0926248
$775$	−13016.9			−0.603330
$776$	−10287.4			−0.475897
$777$			40293.7i			1.86040i
$778$			35197.3i			1.62196i
$779$		−	12593.0i		−	0.579193i
$780$		−	2524.98i		−	0.115909i
$781$	−26521.0			−1.21510
$782$	−61265.8			−2.80161
$783$			1973.01i			0.0900505i
$784$	47735.7			2.17455
$785$	−22648.5	+	13945.0i	−1.02976	+	0.634038i
$786$	−18914.9			−0.858363
$787$		−	40601.7i		−	1.83900i	−0.393089	−	0.919500i	$-0.628594\pi$
$787$		−	40601.7i		−	1.83900i	0.393089	−	0.919500i	$-0.371406\pi$
$788$	−1145.66			−0.0517924
$789$	11603.1			0.523552
$790$			29085.5i			1.30989i
$791$		−	20760.5i		−	0.933198i
$792$			5727.14i			0.256951i
$793$		−	12289.0i		−	0.550309i
$794$	−34918.3			−1.56071
$795$	−18144.6			−0.809463
$796$	2897.10			0.129001
$797$	20901.8			0.928958			0.464479	−	0.885584i	$-0.346242\pi$
$797$	20901.8			0.928958			0.464479	+	0.885584i	$0.346242\pi$
$798$	−6809.23			−0.302060
$799$	31886.9			1.41186
$800$			2676.81i			0.118299i
$801$	−6782.69			−0.299194
$802$	−9822.91			−0.432493
$803$			9381.47i			0.412285i
$804$	1053.41			0.0462075
$805$			82069.8i			3.59327i
$806$			35932.2i			1.57029i
$807$			13356.3i			0.582608i
$808$		−	8073.19i		−	0.351502i
$809$			18539.6i			0.805708i	0.915264	+	0.402854i	$0.131982\pi$
$809$			18539.6i			0.805708i	−0.915264	+	0.402854i	$0.868018\pi$
$810$	−2891.26			−0.125418
$811$			18672.9i			0.808503i	0.914648	+	0.404251i	$0.132468\pi$
$811$			18672.9i			0.808503i	−0.914648	+	0.404251i	$0.867532\pi$
$812$	−2624.45			−0.113424
$813$		−	8056.65i		−	0.347551i
$814$			27146.1i			1.16888i
$815$	13020.6			0.559623
$816$	21873.6			0.938392
$817$			2069.80i			0.0886332i
$818$	−15654.5			−0.669128
$819$			18964.7i			0.809133i
$820$	7112.61			0.302906
$821$	−4746.33			−0.201764			−0.100882	−	0.994898i	$-0.532166\pi$
$821$	−4746.33			−0.201764			−0.100882	+	0.994898i	$0.532166\pi$
$822$	−4565.42			−0.193720
$823$		−	36592.0i		−	1.54984i	−0.632061	−	0.774918i	$-0.717791\pi$
$823$		−	36592.0i		−	1.54984i	0.632061	−	0.774918i	$-0.282209\pi$
$824$	−26376.1			−1.11512
$825$	−4628.49			−0.195325
$826$		−	11399.6i		−	0.480196i
$827$	−28965.3			−1.21792			−0.608962	−	0.793199i	$-0.708414\pi$
$827$	−28965.3			−1.21792			−0.608962	+	0.793199i	$0.708414\pi$
$828$		−	1613.92i		−	0.0677388i
$829$	−31980.0			−1.33982			−0.669909	−	0.742443i	$-0.733667\pi$
$829$	−31980.0			−1.33982			−0.669909	+	0.742443i	$0.733667\pi$
$830$		−	22440.0i		−	0.938437i
$831$	5982.61			0.249741
$832$	33834.8			1.40987
$833$	−116329.			−4.83863
$834$	−2626.35			−0.109044
$835$		−	44360.1i		−	1.83850i
$836$	677.949			0.0280471
$837$	−6080.56			−0.251105
$838$		−	26841.5i		−	1.10647i
$839$		−	25243.0i		−	1.03872i	−0.854556	−	0.519360i	$-0.826170\pi$
$839$		−	25243.0i		−	1.03872i	0.854556	−	0.519360i	$-0.173830\pi$
$840$		−	33715.4i		−	1.38487i
$841$	19049.1			0.781054
$842$	40074.6			1.64022
$843$	−12937.7			−0.528584
$844$		−	3426.04i		−	0.139726i
$845$		−	19677.5i		−	0.801097i
$846$		−	5683.91i		−	0.230989i
$847$			21565.5i			0.874850i
$848$		−	24468.9i		−	0.990878i
$849$	−3628.65			−0.146684
$850$			20340.6i			0.820798i
$851$		−	67063.0i		−	2.70140i
$852$	−3070.30			−0.123458
$853$	2464.08			0.0989081			0.0494540	−	0.998776i	$-0.484252\pi$
$853$	2464.08			0.0989081			0.0494540	+	0.998776i	$0.484252\pi$
$854$		−	18717.9i		−	0.750014i
$855$			3000.39i			0.120013i
$856$	8345.90			0.333244
$857$		−	5210.44i		−	0.207684i	−0.994594	−	0.103842i	$-0.966886\pi$
$857$		−	5210.44i		−	0.207684i	0.994594	−	0.103842i	$-0.0331136\pi$
$858$			12776.6i			0.508375i
$859$			25575.3i			1.01585i	0.861401	+	0.507926i	$0.169588\pi$
$859$			25575.3i			1.01585i	−0.861401	+	0.507926i	$0.830412\pi$
$860$	−1169.04			−0.0463534
$861$	−53421.6			−2.11452
$862$			22356.1i			0.883353i
$863$			14695.2i			0.579642i	0.957081	+	0.289821i	$0.0935958\pi$
$863$			14695.2i			0.579642i	−0.957081	+	0.289821i	$0.906404\pi$
$864$			1250.41i			0.0492360i
$865$		−	21362.3i		−	0.839699i
$866$	28407.0			1.11468
$867$	−38565.8			−1.51068
$868$		−	8088.22i		−	0.316281i
$869$			21750.2i			0.849052i
$870$		−	7825.09i		−	0.304937i
$871$	20601.8			0.801452
$872$		−	14037.4i		−	0.545146i
$873$		−	3883.67i		−	0.150564i
$874$	11333.0			0.438608
$875$	−31679.0			−1.22394
$876$			1086.08i			0.0418895i
$877$		−	30674.1i		−	1.18106i	−0.807015	−	0.590531i	$-0.798918\pi$
$877$		−	30674.1i		−	1.18106i	0.807015	−	0.590531i	$-0.201082\pi$
$878$	11914.0			0.457949
$879$		−	1640.22i		−	0.0629387i
$880$		−	19740.3i		−	0.756189i
$881$			33623.4i			1.28581i	0.765945	+	0.642906i	$0.222271\pi$
$881$			33623.4i			1.28581i	−0.765945	+	0.642906i	$0.777729\pi$
$882$			20736.0i			0.791629i
$883$			34310.5i			1.30763i	0.756654	+	0.653816i	$0.226833\pi$
$883$			34310.5i			1.30763i	−0.756654	+	0.653816i	$0.773167\pi$
$884$	−8297.95			−0.315713
$885$	−5023.07			−0.190789
$886$	8446.78			0.320288
$887$		−	12145.0i		−	0.459740i	−0.973221	−	0.229870i	$-0.926170\pi$
$887$		−	12145.0i		−	0.459740i	0.973221	−	0.229870i	$-0.0738300\pi$
$888$			27550.4i			1.04114i
$889$			39022.2i			1.47217i
$890$	26900.6			1.01316
$891$	−2162.10			−0.0812940
$892$		−	1676.92i		−	0.0629454i
$893$	−5898.45			−0.221035
$894$	−12338.3			−0.461580
$895$	33734.0			1.25989
$896$	38617.2			1.43986
$897$		−	31564.0i		−	1.17491i
$898$	11594.6			0.430865
$899$		−	16456.8i		−	0.610528i
$900$	−535.833			−0.0198457
$901$			59629.4i			2.20482i
$902$	−35990.4			−1.32855
$903$	8780.45			0.323583
$904$		−	14194.8i		−	0.522248i
$905$	−20896.0			−0.767520
$906$	1570.86			0.0576029
$907$	−39801.2			−1.45708			−0.728542	−	0.685001i	$-0.759802\pi$
$907$	−39801.2			−1.45708			−0.728542	+	0.685001i	$0.759802\pi$
$908$		−	1437.17i		−	0.0525265i
$909$	3047.77			0.111208
$910$		−	75215.3i		−	2.73996i
$911$	−6185.57			−0.224958			−0.112479	−	0.993654i	$-0.535879\pi$
$911$	−6185.57			−0.224958			−0.112479	+	0.993654i	$0.535879\pi$
$912$	−4046.18			−0.146910
$913$		−	16780.7i		−	0.608280i
$914$		−	25021.3i		−	0.905504i
$915$	−8247.77			−0.297992
$916$			307.817i			0.0111032i
$917$	83268.9			2.99867
$918$			9501.69i			0.341615i
$919$			7901.94i			0.283636i	0.989893	+	0.141818i	$0.0452947\pi$
$919$			7901.94i			0.283636i	−0.989893	+	0.141818i	$0.954705\pi$
$920$			56114.4i			2.01091i
$921$		−	14498.8i		−	0.518733i
$922$			43196.1i			1.54294i
$923$	−60046.6			−2.14134
$924$		−	2875.97i		−	0.102395i
$925$	−22265.3			−0.791438
$926$	28055.2			0.995629
$927$		−	9957.46i		−	0.352800i
$928$	−3384.19			−0.119711
$929$	−8640.70			−0.305159			−0.152579	−	0.988291i	$-0.548758\pi$
$929$	−8640.70			−0.305159			−0.152579	+	0.988291i	$0.548758\pi$
$930$	24115.9			0.850314
$931$	21518.7			0.757514
$932$	3666.16			0.128851
$933$	−16996.8			−0.596409
$934$		−	49058.1i		−	1.71866i
$935$			48106.1i			1.68261i
$936$			12966.9i			0.452817i
$937$			31381.4i			1.09411i	0.837095	+	0.547057i	$0.184252\pi$
$937$			31381.4i			1.09411i	−0.837095	+	0.547057i	$0.815748\pi$
$938$	31379.4			1.09230
$939$	28696.9			0.997326
$940$		−	3331.48i		−	0.115597i
$941$	−1271.15			−0.0440366			−0.0220183	−	0.999758i	$-0.507009\pi$
$941$	−1271.15			−0.0440366			−0.0220183	+	0.999758i	$0.507009\pi$
$942$	13267.4	−	8168.97i	0.458892	−	0.282547i
$943$	88912.5			3.07040
$944$		−	6773.86i		−	0.233549i
$945$	12728.2			0.438145
$946$	5915.43			0.203306
$947$			14731.7i			0.505509i	0.967530	+	0.252755i	$0.0813365\pi$
$947$			14731.7i			0.505509i	−0.967530	+	0.252755i	$0.918663\pi$
$948$			2517.99i			0.0862665i
$949$			21240.8i			0.726558i
$950$		−	3762.62i		−	0.128501i
$951$	8677.72			0.295893
$952$	−110800.			−3.77212
$953$	51000.6			1.73355			0.866774	−	0.498702i	$-0.166190\pi$
$953$	51000.6			1.73355			0.866774	+	0.498702i	$0.166190\pi$
$954$	10629.1			0.360722
$955$	−33317.9			−1.12895
$956$	−878.684			−0.0297266
$957$		−	5851.63i		−	0.197655i
$958$	19321.8			0.651626
$959$	20098.3			0.676755
$960$		−	22708.3i		−	0.763445i
$961$	20926.7			0.702450
$962$			61461.8i			2.05989i
$963$			3150.73i			0.105432i
$964$			297.942i			0.00995442i
$965$			3724.84i			0.124256i
$966$		−	48076.3i		−	1.60127i
$967$	−30425.8			−1.01182			−0.505909	−	0.862587i	$-0.668843\pi$
$967$	−30425.8			−1.01182			−0.505909	+	0.862587i	$0.668843\pi$
$968$			14745.2i			0.489595i
$969$	9860.33			0.326893
$970$			15402.9i			0.509853i
$971$			43870.6i			1.44992i	0.688789	+	0.724962i	$0.258143\pi$
$971$			43870.6i			1.44992i	−0.688789	+	0.724962i	$0.741857\pi$
$972$	−250.303			−0.00825974
$973$	11561.9			0.380944
$974$			42816.7i			1.40856i
$975$	−10479.4			−0.344216
$976$		−	11122.5i		−	0.364778i
$977$	−28593.8			−0.936333			−0.468166	−	0.883640i	$-0.655085\pi$
$977$	−28593.8			−0.936333			−0.468166	+	0.883640i	$0.655085\pi$
$978$	−7627.46			−0.249386
$979$	20116.4			0.656713
$980$			12153.9i			0.396165i
$981$	5299.38			0.172473
$982$	−6322.22			−0.205448
$983$			41257.1i			1.33865i	0.742968	+	0.669327i	$0.233417\pi$
$983$			41257.1i			1.33865i	−0.742968	+	0.669327i	$0.766583\pi$
$984$	−36526.5			−1.18336
$985$			15037.8i			0.486441i
$986$	−25715.9			−0.830590
$987$			25022.2i			0.806956i
$988$	1534.96			0.0494266
$989$	−14613.8			−0.469860
$990$	8575.03			0.275285
$991$	−58272.1			−1.86788			−0.933942	−	0.357424i	$-0.883655\pi$
$991$	−58272.1			−1.86788			−0.933942	+	0.357424i	$0.883655\pi$
$992$		−	10429.6i		−	0.333812i
$993$	−12771.1			−0.408135
$994$	−91459.4			−2.91843
$995$		−	38027.0i		−	1.21160i
$996$		−	1942.67i		−	0.0618032i
$997$		−	36698.6i		−	1.16575i	−0.812561	−	0.582876i	$-0.801927\pi$
$997$		−	36698.6i		−	1.16575i	0.812561	−	0.582876i	$-0.198073\pi$
$998$	31183.9			0.989087
$999$	−10400.8			−0.329395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.b.a.313.12	✓	40
157.156	even	2	inner	471.4.b.a.313.29	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.b.a.313.12	✓	40	1.1	even	1	trivial
471.4.b.a.313.29	yes	40	157.156	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64007i q^{2} -3.00000 q^{3} +1.03005 q^{4} -13.5204i q^{5} +7.92020i q^{6} -34.8670i q^{7} -23.8399i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.64007i q^{2} -3.00000 q^{3} +1.03005 q^{4} -13.5204i q^{5} +7.92020i q^{6} -34.8670i q^{7} -23.8399i q^{8} +9.00000 q^{9} -35.6946 q^{10} -26.6926 q^{11} -3.09016 q^{12} -60.4351 q^{13} -92.0511 q^{14} +40.5611i q^{15} -54.6986 q^{16} +133.298 q^{17} -23.7606i q^{18} -24.6574 q^{19} -13.9267i q^{20} +104.601i q^{21} +70.4701i q^{22} -174.093i q^{23} +71.5198i q^{24} -57.8000 q^{25} +159.553i q^{26} -27.0000 q^{27} -35.9148i q^{28} -73.0744i q^{29} +107.084 q^{30} +225.206 q^{31} -46.3116i q^{32} +80.0777 q^{33} -351.914i q^{34} -471.414 q^{35} +9.27047 q^{36} +385.214 q^{37} +65.0972i q^{38} +181.305 q^{39} -322.324 q^{40} +510.719i q^{41} +276.153 q^{42} -83.9424i q^{43} -27.4947 q^{44} -121.683i q^{45} -459.617 q^{46} +239.216 q^{47} +164.096 q^{48} -872.705 q^{49} +152.596i q^{50} -399.893 q^{51} -62.2513 q^{52} +447.340i q^{53} +71.2818i q^{54} +360.893i q^{55} -831.226 q^{56} +73.9723 q^{57} -192.921 q^{58} +123.840i q^{59} +41.7800i q^{60} +203.342i q^{61} -594.558i q^{62} -313.803i q^{63} -559.854 q^{64} +817.104i q^{65} -211.410i q^{66} -340.891 q^{67} +137.304 q^{68} +522.279i q^{69} +1244.56i q^{70} +993.573 q^{71} -214.559i q^{72} -351.464i q^{73} -1016.99i q^{74} +173.400 q^{75} -25.3984 q^{76} +930.688i q^{77} -478.658i q^{78} -814.843i q^{79} +739.544i q^{80} +81.0000 q^{81} +1348.33 q^{82} +628.665i q^{83} +107.744i q^{84} -1802.23i q^{85} -221.614 q^{86} +219.223i q^{87} +636.349i q^{88} -753.633 q^{89} -321.252 q^{90} +2107.19i q^{91} -179.325i q^{92} -675.618 q^{93} -631.546i q^{94} +333.377i q^{95} +138.935i q^{96} -431.519i q^{97} +2304.00i q^{98} -240.233 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9}+O(q^{10}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9	\( 40 q - 120 q^{3} - 164 q^{4} + 360 q^{9} - 174 q^{10} + 110 q^{11} + 492 q^{12} - 194 q^{13} - 78 q^{14} + 796 q^{16} - 150 q^{17} + 172 q^{19} - 668 q^{25} - 1080 q^{27} + 522 q^{30} + 66 q^{31} - 330 q^{33} - 400 q^{35} - 1476 q^{36} - 142 q^{37} + 582 q^{39} + 1160 q^{40} + 234 q^{42} - 1182 q^{44} + 132 q^{46} - 244 q^{47} - 2388 q^{48} - 3786 q^{49} + 450 q^{51} + 1596 q^{52} - 256 q^{56} - 516 q^{57} - 1780 q^{58} - 1790 q^{64} - 320 q^{67} + 1646 q^{68} + 712 q^{71} + 2004 q^{75} - 3004 q^{76} + 3240 q^{81} + 4112 q^{82} - 4198 q^{86} + 366 q^{89} - 1566 q^{90} - 198 q^{93} + 990 q^{99}+O(q^{100}) \) 40 * q - 120 * q^3 - 164 * q^4 + 360 * q^9 - 174 * q^10 + 110 * q^11 + 492 * q^12 - 194 * q^13 - 78 * q^14 + 796 * q^16 - 150 * q^17 + 172 * q^19 - 668 * q^25 - 1080 * q^27 + 522 * q^30 + 66 * q^31 - 330 * q^33 - 400 * q^35 - 1476 * q^36 - 142 * q^37 + 582 * q^39 + 1160 * q^40 + 234 * q^42 - 1182 * q^44 + 132 * q^46 - 244 * q^47 - 2388 * q^48 - 3786 * q^49 + 450 * q^51 + 1596 * q^52 - 256 * q^56 - 516 * q^57 - 1780 * q^58 - 1790 * q^64 - 320 * q^67 + 1646 * q^68 + 712 * q^71 + 2004 * q^75 - 3004 * q^76 + 3240 * q^81 + 4112 * q^82 - 4198 * q^86 + 366 * q^89 - 1566 * q^90 - 198 * q^93 + 990 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more