LMFDB - Embedded newform 475.3.d.d.474.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(474,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("475.474");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.d (of order $2$, degree $1$, not 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:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.6
Character	$\chi$	$=$	475.474
Dual form			475.3.d.d.474.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.92865 q^{2} +2.02259 q^{3} +4.57698 q^{4} -5.92346 q^{6} +8.32815i q^{7} -1.68976 q^{8} -4.90912 q^{9} +O(q^{10})$	\(q-2.92865 q^{2} +2.02259 q^{3} +4.57698 q^{4} -5.92346 q^{6} +8.32815i q^{7} -1.68976 q^{8} -4.90912 q^{9} -17.3372 q^{11} +9.25736 q^{12} +5.27368 q^{13} -24.3902i q^{14} -13.3592 q^{16} -19.0798i q^{17} +14.3771 q^{18} +(11.4291 - 15.1781i) q^{19} +16.8445i q^{21} +50.7744 q^{22} +3.32445i q^{23} -3.41769 q^{24} -15.4447 q^{26} -28.1325 q^{27} +38.1178i q^{28} -37.4537i q^{29} +2.86714i q^{31} +45.8834 q^{32} -35.0660 q^{33} +55.8779i q^{34} -22.4689 q^{36} +33.0085 q^{37} +(-33.4719 + 44.4513i) q^{38} +10.6665 q^{39} -20.9092i q^{41} -49.3315i q^{42} -23.9545i q^{43} -79.3518 q^{44} -9.73613i q^{46} +33.8351i q^{47} -27.0202 q^{48} -20.3582 q^{49} -38.5906i q^{51} +24.1375 q^{52} +37.4936 q^{53} +82.3901 q^{54} -14.0726i q^{56} +(23.1165 - 30.6991i) q^{57} +109.689i q^{58} -32.5574i q^{59} +6.58141 q^{61} -8.39685i q^{62} -40.8839i q^{63} -80.9395 q^{64} +102.696 q^{66} -112.029 q^{67} -87.3276i q^{68} +6.72400i q^{69} -110.992i q^{71} +8.29522 q^{72} -134.981i q^{73} -96.6704 q^{74} +(52.3109 - 69.4698i) q^{76} -144.387i q^{77} -31.2384 q^{78} +48.8335i q^{79} -12.7185 q^{81} +61.2355i q^{82} -44.2971i q^{83} +77.0967i q^{84} +70.1541i q^{86} -75.7535i q^{87} +29.2956 q^{88} -173.577i q^{89} +43.9200i q^{91} +15.2159i q^{92} +5.79906i q^{93} -99.0910i q^{94} +92.8034 q^{96} +50.6952 q^{97} +59.6219 q^{98} +85.1102 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) $ 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9	$ 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9} - 8 q^{11} + 72 q^{16} - 78 q^{19} + 88 q^{24} + 60 q^{26} + 8 q^{36} + 64 q^{39} + 104 q^{44} - 468 q^{49} - 196 q^{54} + 444 q^{61} + 436 q^{64} + 184 q^{66} + 184 q^{74} - 702 q^{76} + 804 q^{81} + 380 q^{96} + 360 q^{99}+O(q^{100}) $ 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9 - 8 * q^11 + 72 * q^16 - 78 * q^19 + 88 * q^24 + 60 * q^26 + 8 * q^36 + 64 * q^39 + 104 * q^44 - 468 * q^49 - 196 * q^54 + 444 * q^61 + 436 * q^64 + 184 * q^66 + 184 * q^74 - 702 * q^76 + 804 * q^81 + 380 * q^96 + 360 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−2.92865			−1.46432			−0.732162	−	0.681131i	$-0.761489\pi$
$2$	−2.92865			−1.46432			−0.732162	+	0.681131i	$0.761489\pi$
$3$	2.02259			0.674198			0.337099	−	0.941469i	$-0.390554\pi$
$3$	2.02259			0.674198			0.337099	+	0.941469i	$0.390554\pi$
$4$	4.57698			1.14424
$5$		0			0
$5$		0			0
$6$	−5.92346			−0.987243
$7$			8.32815i			1.18974i	0.803823	+	0.594868i	$0.202796\pi$
$7$			8.32815i			1.18974i	−0.803823	+	0.594868i	$0.797204\pi$
$8$	−1.68976			−0.211220
$9$	−4.90912			−0.545458
$10$		0			0
$11$	−17.3372			−1.57611			−0.788053	−	0.615608i	$-0.788911\pi$
$11$	−17.3372			−1.57611			−0.788053	+	0.615608i	$0.788911\pi$
$12$	9.25736			0.771446
$13$	5.27368			0.405668			0.202834	−	0.979213i	$-0.434985\pi$
$13$	5.27368			0.405668			0.202834	+	0.979213i	$0.434985\pi$
$14$		−	24.3902i		−	1.74216i
$15$		0			0
$16$	−13.3592			−0.834950
$17$		−	19.0798i		−	1.12234i	−0.827701	−	0.561170i	$-0.810351\pi$
$17$		−	19.0798i		−	1.12234i	0.827701	−	0.561170i	$-0.189649\pi$
$18$	14.3771			0.798727
$19$	11.4291	−	15.1781i	0.601534	−	0.798847i
$19$	11.4291	−	15.1781i	0.601534	−	0.798847i
$20$		0			0
$21$			16.8445i			0.802117i
$22$	50.7744			2.30793
$23$			3.32445i			0.144541i	0.997385	+	0.0722706i	$0.0230245\pi$
$23$			3.32445i			0.144541i	−0.997385	+	0.0722706i	$0.976975\pi$
$24$	−3.41769			−0.142404
$25$		0			0
$26$	−15.4447			−0.594029
$27$	−28.1325			−1.04194
$28$			38.1178i			1.36135i
$29$		−	37.4537i		−	1.29151i	−0.763546	−	0.645753i	$-0.776544\pi$
$29$		−	37.4537i		−	1.29151i	0.763546	−	0.645753i	$-0.223456\pi$
$30$		0			0
$31$			2.86714i			0.0924885i	0.998930	+	0.0462443i	$0.0147253\pi$
$31$			2.86714i			0.0924885i	−0.998930	+	0.0462443i	$0.985275\pi$
$32$	45.8834			1.43386
$33$	−35.0660			−1.06261
$34$			55.8779i			1.64347i
$35$		0			0
$36$	−22.4689			−0.624137
$37$	33.0085			0.892123			0.446061	−	0.895002i	$-0.352826\pi$
$37$	33.0085			0.892123			0.446061	+	0.895002i	$0.352826\pi$
$38$	−33.4719	+	44.4513i	−0.880840	+	1.16977i
$39$	10.6665			0.273500
$40$		0			0
$41$		−	20.9092i		−	0.509979i	−0.966944	−	0.254990i	$-0.917928\pi$
$41$		−	20.9092i		−	0.509979i	0.966944	−	0.254990i	$-0.0820721\pi$
$42$		−	49.3315i		−	1.17456i
$43$		−	23.9545i		−	0.557080i	−0.960425	−	0.278540i	$-0.910149\pi$
$43$		−	23.9545i		−	0.557080i	0.960425	−	0.278540i	$-0.0898505\pi$
$44$	−79.3518			−1.80345
$45$		0			0
$46$		−	9.73613i		−	0.211655i
$47$			33.8351i			0.719895i	0.932973	+	0.359947i	$0.117205\pi$
$47$			33.8351i			0.719895i	−0.932973	+	0.359947i	$0.882795\pi$
$48$	−27.0202			−0.562921
$49$	−20.3582			−0.415473
$50$		0			0
$51$		−	38.5906i		−	0.756679i
$52$	24.1375			0.464183
$53$	37.4936			0.707426			0.353713	−	0.935354i	$-0.384919\pi$
$53$	37.4936			0.707426			0.353713	+	0.935354i	$0.384919\pi$
$54$	82.3901			1.52574
$55$		0			0
$56$		−	14.0726i		−	0.251296i
$57$	23.1165	−	30.6991i	0.405553	−	0.538581i
$58$			109.689i			1.89118i
$59$		−	32.5574i		−	0.551821i	−0.961183	−	0.275910i	$-0.911021\pi$
$59$		−	32.5574i		−	0.551821i	0.961183	−	0.275910i	$-0.0889793\pi$
$60$		0			0
$61$	6.58141			0.107892			0.0539460	−	0.998544i	$-0.482820\pi$
$61$	6.58141			0.107892			0.0539460	+	0.998544i	$0.482820\pi$
$62$		−	8.39685i		−	0.135433i
$63$		−	40.8839i		−	0.648951i
$64$	−80.9395			−1.26468
$65$		0			0
$66$	102.696			1.55600
$67$	−112.029			−1.67208			−0.836038	−	0.548671i	$-0.815134\pi$
$67$	−112.029			−1.67208			−0.836038	+	0.548671i	$0.815134\pi$
$68$		−	87.3276i		−	1.28423i
$69$			6.72400i			0.0974493i
$70$		0			0
$71$		−	110.992i		−	1.56327i	−0.623733	−	0.781637i	$-0.714385\pi$
$71$		−	110.992i		−	1.56327i	0.623733	−	0.781637i	$-0.285615\pi$
$72$	8.29522			0.115211
$73$		−	134.981i		−	1.84906i	−0.381109	−	0.924530i	$-0.624458\pi$
$73$		−	134.981i		−	1.84906i	0.381109	−	0.924530i	$-0.375542\pi$
$74$	−96.6704			−1.30636
$75$		0			0
$76$	52.3109	−	69.4698i	0.688301	−	0.914076i
$77$		−	144.387i		−	1.87515i
$78$	−31.2384			−0.400493
$79$			48.8335i			0.618145i	0.951038	+	0.309072i	$0.100019\pi$
$79$			48.8335i			0.618145i	−0.951038	+	0.309072i	$0.899981\pi$
$80$		0			0
$81$	−12.7185			−0.157018
$82$			61.2355i			0.746775i
$83$		−	44.2971i		−	0.533700i	−0.963738	−	0.266850i	$-0.914017\pi$
$83$		−	44.2971i		−	0.533700i	0.963738	−	0.266850i	$-0.0859828\pi$
$84$			77.0967i			0.917818i
$85$		0			0
$86$			70.1541i			0.815746i
$87$		−	75.7535i		−	0.870730i
$88$	29.2956			0.332904
$89$		−	173.577i		−	1.95030i	−0.221538	−	0.975152i	$-0.571108\pi$
$89$		−	173.577i		−	1.95030i	0.221538	−	0.975152i	$-0.428892\pi$
$90$		0			0
$91$			43.9200i			0.482637i
$92$			15.2159i			0.165390i
$93$			5.79906i			0.0623555i
$94$		−	99.0910i		−	1.05416i
$95$		0			0
$96$	92.8034			0.966703
$97$	50.6952			0.522631			0.261315	−	0.965253i	$-0.415844\pi$
$97$	50.6952			0.522631			0.261315	+	0.965253i	$0.415844\pi$
$98$	59.6219			0.608386
$99$	85.1102			0.859699
$100$		0			0
$101$	−100.261			−0.992681			−0.496340	−	0.868128i	$-0.665323\pi$
$101$	−100.261			−0.992681			−0.496340	+	0.868128i	$0.665323\pi$
$102$			113.018i			1.10802i
$103$	31.5721			0.306525			0.153263	−	0.988185i	$-0.451022\pi$
$103$	31.5721			0.306525			0.153263	+	0.988185i	$0.451022\pi$
$104$	−8.91124			−0.0856850
$105$		0			0
$106$	−109.805			−1.03590
$107$	−71.5150			−0.668365			−0.334182	−	0.942508i	$-0.608460\pi$
$107$	−71.5150			−0.668365			−0.334182	+	0.942508i	$0.608460\pi$
$108$	−128.762			−1.19224
$109$			138.250i			1.26835i	0.773191	+	0.634174i	$0.218660\pi$
$109$			138.250i			1.26835i	−0.773191	+	0.634174i	$0.781340\pi$
$110$		0			0
$111$	66.7628			0.601467
$112$		−	111.257i		−	0.993370i
$113$	−115.926			−1.02590			−0.512949	−	0.858419i	$-0.671447\pi$
$113$	−115.926			−1.02590			−0.512949	+	0.858419i	$0.671447\pi$
$114$	−67.7001	+	89.9069i	−0.593860	+	0.788657i
$115$		0			0
$116$		−	171.425i		−	1.47780i
$117$	−25.8891			−0.221275
$118$			95.3492i			0.808044i
$119$	158.899			1.33529
$120$		0			0
$121$	179.577			1.48411
$122$	−19.2746			−0.157989
$123$		−	42.2907i		−	0.343827i
$124$			13.1228i			0.105829i
$125$		0			0
$126$			119.735i			0.950274i
$127$	−114.758			−0.903607			−0.451804	−	0.892117i	$-0.649219\pi$
$127$	−114.758			−0.903607			−0.451804	+	0.892117i	$0.649219\pi$
$128$	53.5097			0.418045
$129$		−	48.4501i		−	0.375582i
$130$		0			0
$131$	−203.387			−1.55257			−0.776286	−	0.630381i	$-0.782899\pi$
$131$	−203.387			−1.55257			−0.776286	+	0.630381i	$0.782899\pi$
$132$	−160.496			−1.21588
$133$	126.406	+	95.1837i	0.950418	+	0.715667i
$134$	328.094			2.44846
$135$		0			0
$136$			32.2402i			0.237060i
$137$			152.757i			1.11501i	0.830172	+	0.557507i	$0.188242\pi$
$137$			152.757i			1.11501i	−0.830172	+	0.557507i	$0.811758\pi$
$138$		−	19.6922i		−	0.142697i
$139$	−155.284			−1.11715			−0.558575	−	0.829454i	$-0.688652\pi$
$139$	−155.284			−1.11715			−0.558575	+	0.829454i	$0.688652\pi$
$140$		0			0
$141$			68.4345i			0.485351i
$142$			325.058i			2.28914i
$143$	−91.4306			−0.639375
$144$	65.5819			0.455430
$145$		0			0
$146$			395.313i			2.70762i
$147$	−41.1763			−0.280111
$148$	151.079			1.02081
$149$	106.711			0.716178			0.358089	−	0.933687i	$-0.383428\pi$
$149$	106.711			0.716178			0.358089	+	0.933687i	$0.383428\pi$
$150$		0			0
$151$			262.585i			1.73897i	0.493955	+	0.869487i	$0.335551\pi$
$151$			262.585i			1.73897i	−0.493955	+	0.869487i	$0.664449\pi$
$152$	−19.3125	+	25.6473i	−0.127056	+	0.168732i
$153$			93.6649i			0.612189i
$154$			422.857i			2.74583i
$155$		0			0
$156$	48.8203			0.312951
$157$			238.486i			1.51902i	0.650496	+	0.759509i	$0.274561\pi$
$157$			238.486i			1.51902i	−0.650496	+	0.759509i	$0.725439\pi$
$158$		−	143.016i		−	0.905164i
$159$	75.8343			0.476945
$160$		0			0
$161$	−27.6865			−0.171966
$162$	37.2479			0.229925
$163$		−	128.660i		−	0.789328i	−0.918826	−	0.394664i	$-0.870861\pi$
$163$		−	128.660i		−	0.789328i	0.918826	−	0.394664i	$-0.129139\pi$
$164$		−	95.7007i		−	0.583541i
$165$		0			0
$166$			129.731i			0.781510i
$167$	194.440			1.16431			0.582155	−	0.813078i	$-0.302210\pi$
$167$	194.440			1.16431			0.582155	+	0.813078i	$0.302210\pi$
$168$		−	28.4631i		−	0.169423i
$169$	−141.188			−0.835434
$170$		0			0
$171$	−56.1070	+	74.5111i	−0.328111	+	0.435737i
$172$		−	109.639i		−	0.637436i
$173$	253.484			1.46523			0.732613	−	0.680646i	$-0.238301\pi$
$173$	253.484			1.46523			0.732613	+	0.680646i	$0.238301\pi$
$174$			221.855i			1.27503i
$175$		0			0
$176$	231.611			1.31597
$177$		−	65.8504i		−	0.372036i
$178$			508.346i			2.85588i
$179$		−	262.044i		−	1.46393i	−0.681341	−	0.731966i	$-0.738603\pi$
$179$		−	262.044i		−	1.46393i	0.681341	−	0.731966i	$-0.261397\pi$
$180$		0			0
$181$		−	266.130i		−	1.47033i	−0.677887	−	0.735166i	$-0.737104\pi$
$181$		−	266.130i		−	1.47033i	0.677887	−	0.735166i	$-0.262896\pi$
$182$		−	128.626i		−	0.706737i
$183$	13.3115			0.0727405
$184$		−	5.61751i		−	0.0305299i
$185$		0			0
$186$		−	16.9834i		−	0.0913087i
$187$			330.789i			1.76893i
$188$			154.862i			0.823735i
$189$		−	234.292i		−	1.23964i
$190$		0			0
$191$	−204.892			−1.07273			−0.536366	−	0.843985i	$-0.680203\pi$
$191$	−204.892			−1.07273			−0.536366	+	0.843985i	$0.680203\pi$
$192$	−163.708			−0.852644
$193$	239.909			1.24305			0.621526	−	0.783394i	$-0.286513\pi$
$193$	239.909			1.24305			0.621526	+	0.783394i	$0.286513\pi$
$194$	−148.468			−0.765301
$195$		0			0
$196$	−93.1788			−0.475402
$197$		−	10.5701i		−	0.0536555i	−0.999640	−	0.0268278i	$-0.991459\pi$
$197$		−	10.5701i		−	0.0536555i	0.999640	−	0.0268278i	$-0.00854057\pi$
$198$	−249.258			−1.25888
$199$	99.3795			0.499394			0.249697	−	0.968324i	$-0.419669\pi$
$199$	99.3795			0.499394			0.249697	+	0.968324i	$0.419669\pi$
$200$		0			0
$201$	−226.589			−1.12731
$202$	293.628			1.45361
$203$	311.920			1.53655
$204$		−	176.628i		−	0.865825i
$205$		0			0
$206$	−92.4635			−0.448852
$207$		−	16.3201i		−	0.0788411i
$208$	−70.4521			−0.338712
$209$	−198.149	+	263.145i	−0.948081	+	1.25907i
$210$		0			0
$211$		−	89.7607i		−	0.425406i	−0.977117	−	0.212703i	$-0.931773\pi$
$211$		−	89.7607i		−	0.425406i	0.977117	−	0.212703i	$-0.0682267\pi$
$212$	171.607			0.809468
$213$		−	224.493i		−	1.05396i
$214$	209.442			0.978702
$215$		0			0
$216$	47.5371			0.220079
$217$	−23.8780			−0.110037
$218$		−	404.885i		−	1.85727i
$219$		−	273.012i		−	1.24663i
$220$		0			0
$221$		−	100.621i		−	0.455297i
$222$	−195.525			−0.880742
$223$	−104.111			−0.466866			−0.233433	−	0.972373i	$-0.574996\pi$
$223$	−104.111			−0.466866			−0.233433	+	0.972373i	$0.574996\pi$
$224$			382.124i			1.70591i
$225$		0			0
$226$	339.508			1.50225
$227$	−254.689			−1.12198			−0.560990	−	0.827823i	$-0.689579\pi$
$227$	−254.689			−1.12198			−0.560990	+	0.827823i	$0.689579\pi$
$228$	105.804	−	140.509i	0.464051	−	0.616268i
$229$	−326.228			−1.42457			−0.712287	−	0.701888i	$-0.752341\pi$
$229$	−326.228			−1.42457			−0.712287	+	0.701888i	$0.752341\pi$
$230$		0			0
$231$		−	292.035i		−	1.26422i
$232$			63.2876i			0.272791i
$233$			66.9003i			0.287126i	0.989641	+	0.143563i	$0.0458559\pi$
$233$			66.9003i			0.287126i	−0.989641	+	0.143563i	$0.954144\pi$
$234$	75.8201			0.324017
$235$		0			0
$236$		−	149.015i		−	0.631417i
$237$			98.7702i			0.416752i
$238$	−465.360			−1.95529
$239$	−246.601			−1.03180			−0.515902	−	0.856647i	$-0.672543\pi$
$239$	−246.601			−1.03180			−0.515902	+	0.856647i	$0.672543\pi$
$240$		0			0
$241$			39.9249i			0.165663i	0.996564	+	0.0828317i	$0.0263964\pi$
$241$			39.9249i			0.165663i	−0.996564	+	0.0828317i	$0.973604\pi$
$242$	−525.918			−2.17322
$243$	227.468			0.936082
$244$	30.1230			0.123455
$245$		0			0
$246$			123.855i			0.503474i
$247$	60.2736	−	80.0444i	0.244023	−	0.324066i
$248$		−	4.84478i		−	0.0195354i
$249$		−	89.5950i		−	0.359819i
$250$		0			0
$251$	135.552			0.540046			0.270023	−	0.962854i	$-0.412969\pi$
$251$	135.552			0.540046			0.270023	+	0.962854i	$0.412969\pi$
$252$		−	187.125i		−	0.742558i
$253$		−	57.6365i		−	0.227812i
$254$	336.086			1.32317
$255$		0			0
$256$	167.047			0.652528
$257$	−315.240			−1.22662			−0.613308	−	0.789844i	$-0.710162\pi$
$257$	−315.240			−1.22662			−0.613308	+	0.789844i	$0.710162\pi$
$258$			141.893i			0.549974i
$259$			274.900i			1.06139i
$260$		0			0
$261$			183.865i			0.704462i
$262$	595.649			2.27347
$263$		−	203.024i		−	0.771956i	−0.922508	−	0.385978i	$-0.873864\pi$
$263$		−	203.024i		−	0.771956i	0.922508	−	0.385978i	$-0.126136\pi$
$264$	59.2530			0.224443
$265$		0			0
$266$	−370.197	−	278.759i	−1.39172	−	1.04797i
$267$		−	351.076i		−	1.31489i
$268$	−512.755			−1.91326
$269$			272.421i			1.01272i	0.862323	+	0.506359i	$0.169009\pi$
$269$			272.421i			1.01272i	−0.862323	+	0.506359i	$0.830991\pi$
$270$		0			0
$271$	225.743			0.833001			0.416501	−	0.909135i	$-0.363256\pi$
$271$	225.743			0.833001			0.416501	+	0.909135i	$0.363256\pi$
$272$			254.890i			0.937097i
$273$			88.8323i			0.325393i
$274$		−	447.371i		−	1.63274i
$275$		0			0
$276$			30.7756i			0.111506i
$277$		−	133.873i		−	0.483296i	−0.970364	−	0.241648i	$-0.922312\pi$
$277$		−	133.873i		−	0.483296i	0.970364	−	0.241648i	$-0.0776879\pi$
$278$	454.772			1.63587
$279$		−	14.0752i		−	0.0504486i
$280$		0			0
$281$		−	272.283i		−	0.968980i	−0.874797	−	0.484490i	$-0.839005\pi$
$281$		−	272.283i		−	0.968980i	0.874797	−	0.484490i	$-0.160995\pi$
$282$		−	200.421i		−	0.710712i
$283$			307.114i			1.08521i	0.839988	+	0.542605i	$0.182562\pi$
$283$			307.114i			1.08521i	−0.839988	+	0.542605i	$0.817438\pi$
$284$		−	508.010i		−	1.78877i
$285$		0			0
$286$	267.768			0.936252
$287$	174.135			0.606741
$288$	−225.247			−0.782108
$289$	−75.0377			−0.259646
$290$		0			0
$291$	102.536			0.352356
$292$		−	617.807i		−	2.11578i
$293$	−272.281			−0.929287			−0.464644	−	0.885498i	$-0.653818\pi$
$293$	−272.281			−0.929287			−0.464644	+	0.885498i	$0.653818\pi$
$294$	120.591			0.410173
$295$		0			0
$296$	−55.7764			−0.188434
$297$	487.737			1.64221
$298$	−312.518			−1.04872
$299$			17.5321i			0.0586357i
$300$		0			0
$301$	199.496			0.662779
$302$		−	769.019i		−	2.54642i
$303$	−202.787			−0.669263
$304$	−152.684	+	202.767i	−0.502251	+	0.666997i
$305$		0			0
$306$		−	274.311i		−	0.896443i
$307$	−41.8126			−0.136198			−0.0680988	−	0.997679i	$-0.521693\pi$
$307$	−41.8126			−0.136198			−0.0680988	+	0.997679i	$0.521693\pi$
$308$		−	660.854i		−	2.14563i
$309$	63.8575			0.206659
$310$		0			0
$311$	399.316			1.28398			0.641988	−	0.766715i	$-0.278110\pi$
$311$	399.316			1.28398			0.641988	+	0.766715i	$0.278110\pi$
$312$	−18.0238			−0.0577686
$313$			273.543i			0.873941i	0.899476	+	0.436970i	$0.143949\pi$
$313$			273.543i			0.873941i	−0.899476	+	0.436970i	$0.856051\pi$
$314$		−	698.441i		−	2.22434i
$315$		0			0
$316$			223.510i			0.707309i
$317$	120.459			0.379996			0.189998	−	0.981784i	$-0.439152\pi$
$317$	120.459			0.379996			0.189998	+	0.981784i	$0.439152\pi$
$318$	−222.092			−0.698402
$319$			649.340i			2.03555i
$320$		0			0
$321$	−144.646			−0.450610
$322$	81.0840			0.251814
$323$	−289.595	−	218.065i	−0.896578	−	0.675125i
$324$	−58.2121			−0.179667
$325$		0			0
$326$			376.801i			1.15583i
$327$			279.623i			0.855117i
$328$			35.3314i			0.107718i
$329$	−281.784			−0.856485
$330$		0			0
$331$		−	55.2508i		−	0.166921i	−0.996511	−	0.0834604i	$-0.973403\pi$
$331$		−	55.2508i		−	0.166921i	0.996511	−	0.0834604i	$-0.0265972\pi$
$332$		−	202.747i		−	0.610683i
$333$	−162.043			−0.486615
$334$	−569.445			−1.70493
$335$		0			0
$336$		−	225.029i		−	0.669728i
$337$	480.908			1.42703			0.713513	−	0.700642i	$-0.247103\pi$
$337$	480.908			1.42703			0.713513	+	0.700642i	$0.247103\pi$
$338$	413.491			1.22335
$339$	−234.472			−0.691658
$340$		0			0
$341$		−	49.7081i		−	0.145772i
$342$	164.318	−	218.217i	0.480461	−	0.638061i
$343$			238.534i			0.695434i
$344$			40.4772i			0.117666i
$345$		0			0
$346$	−742.365			−2.14556
$347$			538.987i			1.55328i	0.629946	+	0.776639i	$0.283077\pi$
$347$			538.987i			1.55328i	−0.629946	+	0.776639i	$0.716923\pi$
$348$		−	346.722i		−	0.996327i
$349$	21.5094			0.0616315			0.0308158	−	0.999525i	$-0.490189\pi$
$349$	21.5094			0.0616315			0.0308158	+	0.999525i	$0.490189\pi$
$350$		0			0
$351$	−148.362			−0.422683
$352$	−795.488			−2.25991
$353$			91.3380i			0.258748i	0.991596	+	0.129374i	$0.0412968\pi$
$353$			91.3380i			0.258748i	−0.991596	+	0.129374i	$0.958703\pi$
$354$			192.853i			0.544781i
$355$		0			0
$356$		−	794.458i		−	2.23162i
$357$	321.389			0.900248
$358$			767.434i			2.14367i
$359$	673.203			1.87522			0.937609	−	0.347691i	$-0.113034\pi$
$359$	673.203			1.87522			0.937609	+	0.347691i	$0.113034\pi$
$360$		0			0
$361$	−99.7493	−	346.945i	−0.276314	−	0.961067i
$362$			779.401i			2.15304i
$363$	363.211			1.00058
$364$			201.021i			0.552255i
$365$		0			0
$366$	−38.9847			−0.106516
$367$		−	595.514i		−	1.62265i	−0.584593	−	0.811327i	$-0.698746\pi$
$367$		−	595.514i		−	1.62265i	0.584593	−	0.811327i	$-0.301254\pi$
$368$		−	44.4120i		−	0.120685i
$369$			102.646i			0.278172i
$370$		0			0
$371$			312.252i			0.841651i
$372$			26.5422i			0.0713499i
$373$	228.595			0.612854			0.306427	−	0.951894i	$-0.400866\pi$
$373$	228.595			0.612854			0.306427	+	0.951894i	$0.400866\pi$
$374$		−	968.765i		−	2.59028i
$375$		0			0
$376$		−	57.1730i		−	0.152056i
$377$		−	197.519i		−	0.523922i
$378$			686.158i			1.81523i
$379$		−	725.490i		−	1.91422i	−0.289723	−	0.957110i	$-0.593563\pi$
$379$		−	725.490i		−	1.91422i	0.289723	−	0.957110i	$-0.406437\pi$
$380$		0			0
$381$	−232.109			−0.609210
$382$	600.056			1.57083
$383$	525.150			1.37115			0.685574	−	0.728003i	$-0.259551\pi$
$383$	525.150			1.37115			0.685574	+	0.728003i	$0.259551\pi$
$384$	108.228			0.281845
$385$		0			0
$386$	−702.609			−1.82023
$387$			117.595i			0.303864i
$388$	232.031			0.598017
$389$	−135.008			−0.347065			−0.173533	−	0.984828i	$-0.555518\pi$
$389$	−135.008			−0.347065			−0.173533	+	0.984828i	$0.555518\pi$
$390$		0			0
$391$	63.4297			0.162224
$392$	34.4003			0.0877560
$393$	−411.369			−1.04674
$394$			30.9562i			0.0785691i
$395$		0			0
$396$	389.547			0.983705
$397$			344.093i			0.866733i	0.901218	+	0.433366i	$0.142674\pi$
$397$			344.093i			0.866733i	−0.901218	+	0.433366i	$0.857326\pi$
$398$	−291.047			−0.731275
$399$	255.667	+	192.518i	0.640769	+	0.482501i
$400$		0			0
$401$		−	66.2162i		−	0.165128i	−0.996586	−	0.0825638i	$-0.973689\pi$
$401$		−	66.2162i		−	0.165128i	0.996586	−	0.0825638i	$-0.0263108\pi$
$402$	663.600			1.65075
$403$			15.1204i			0.0375196i
$404$	−458.891			−1.13587
$405$		0			0
$406$	−913.503			−2.25001
$407$	−572.274			−1.40608
$408$			65.2088i			0.159825i
$409$		−	66.1891i		−	0.161832i	−0.996721	−	0.0809158i	$-0.974216\pi$
$409$		−	66.1891i		−	0.161832i	0.996721	−	0.0809158i	$-0.0257845\pi$
$410$		0			0
$411$			308.965i			0.751739i
$412$	144.505			0.350740
$413$	271.143			0.656521
$414$			47.7958i			0.115449i
$415$		0			0
$416$	241.974			0.581669
$417$	−314.076			−0.753180
$418$	580.308	−	770.659i	1.38830	−	1.84368i
$419$	−448.086			−1.06942			−0.534709	−	0.845036i	$-0.679579\pi$
$419$	−448.086			−1.06942			−0.534709	+	0.845036i	$0.679579\pi$
$420$		0			0
$421$		−	728.066i		−	1.72937i	−0.502312	−	0.864686i	$-0.667517\pi$
$421$		−	728.066i		−	1.72937i	0.502312	−	0.864686i	$-0.332483\pi$
$422$			262.877i			0.622932i
$423$		−	166.100i		−	0.392672i
$424$	−63.3551			−0.149422
$425$		0			0
$426$			657.460i			1.54333i
$427$			54.8110i			0.128363i
$428$	−327.323			−0.764772
$429$	−184.927			−0.431065
$430$		0			0
$431$			623.890i			1.44754i	0.690042	+	0.723770i	$0.257592\pi$
$431$			623.890i			1.44754i	−0.690042	+	0.723770i	$0.742408\pi$
$432$	375.827			0.869971
$433$	−3.90352			−0.00901505			−0.00450752	−	0.999990i	$-0.501435\pi$
$433$	−3.90352			−0.00901505			−0.00450752	+	0.999990i	$0.501435\pi$
$434$	69.9303			0.161130
$435$		0			0
$436$			632.766i			1.45130i
$437$	50.4588	+	37.9956i	0.115466	+	0.0869464i
$438$			799.557i			1.82547i
$439$			748.985i			1.70612i	0.521815	+	0.853059i	$0.325255\pi$
$439$			748.985i			1.70612i	−0.521815	+	0.853059i	$0.674745\pi$
$440$		0			0
$441$	99.9406			0.226623
$442$			294.682i			0.666702i
$443$			77.7877i			0.175593i	0.996138	+	0.0877965i	$0.0279825\pi$
$443$			77.7877i			0.175593i	−0.996138	+	0.0877965i	$0.972017\pi$
$444$	305.572			0.688225
$445$		0			0
$446$	304.905			0.683642
$447$	215.832			0.482846
$448$		−	674.077i		−	1.50464i
$449$		−	112.043i		−	0.249539i	−0.992186	−	0.124770i	$-0.960181\pi$
$449$		−	112.043i		−	0.249539i	0.992186	−	0.124770i	$-0.0398192\pi$
$450$		0			0
$451$			362.505i			0.803781i
$452$	−530.593			−1.17388
$453$			531.103i			1.17241i
$454$	745.895			1.64294
$455$		0			0
$456$	−39.0613	+	51.8740i	−0.0856607	+	0.113759i
$457$		−	563.415i		−	1.23286i	−0.787412	−	0.616428i	$-0.788579\pi$
$457$		−	563.415i		−	1.23286i	0.787412	−	0.616428i	$-0.211421\pi$
$458$	955.406			2.08604
$459$			536.761i			1.16941i
$460$		0			0
$461$	−633.225			−1.37359			−0.686795	−	0.726851i	$-0.740983\pi$
$461$	−633.225			−1.37359			−0.686795	+	0.726851i	$0.740983\pi$
$462$			855.268i			1.85123i
$463$		−	547.595i		−	1.18271i	−0.806411	−	0.591356i	$-0.798593\pi$
$463$		−	547.595i		−	1.18271i	0.806411	−	0.591356i	$-0.201407\pi$
$464$			500.351i			1.07834i
$465$		0			0
$466$		−	195.927i		−	0.420445i
$467$			856.723i			1.83453i	0.398283	+	0.917263i	$0.369606\pi$
$467$			856.723i			1.83453i	−0.398283	+	0.917263i	$0.630394\pi$
$468$	−118.494			−0.253192
$469$		−	932.996i		−	1.98933i
$470$		0			0
$471$			482.360i			1.02412i
$472$			55.0141i			0.116555i
$473$			415.302i			0.878017i
$474$		−	289.263i		−	0.610260i
$475$		0			0
$476$	727.278			1.52790
$477$	−184.060			−0.385871
$478$	722.208			1.51090
$479$	−218.011			−0.455137			−0.227568	−	0.973762i	$-0.573078\pi$
$479$	−218.011			−0.455137			−0.227568	+	0.973762i	$0.573078\pi$
$480$		0			0
$481$	174.076			0.361905
$482$		−	116.926i		−	0.242585i
$483$	−55.9985			−0.115939
$484$	821.920			1.69818
$485$		0			0
$486$	−666.174			−1.37073
$487$	−782.514			−1.60680			−0.803402	−	0.595437i	$-0.796979\pi$
$487$	−782.514			−1.60680			−0.803402	+	0.595437i	$0.796979\pi$
$488$	−11.1210			−0.0227889
$489$		−	260.228i		−	0.532163i
$490$		0			0
$491$	12.9182			0.0263100			0.0131550	−	0.999913i	$-0.495813\pi$
$491$	12.9182			0.0263100			0.0131550	+	0.999913i	$0.495813\pi$
$492$		−	193.563i		−	0.393422i
$493$	−714.607			−1.44951
$494$	−176.520	+	234.422i	−0.357328	+	0.474538i
$495$		0			0
$496$		−	38.3027i		−	0.0772233i
$497$	924.363			1.85988
$498$			262.392i			0.526892i
$499$	−357.210			−0.715851			−0.357926	−	0.933750i	$-0.616516\pi$
$499$	−357.210			−0.715851			−0.357926	+	0.933750i	$0.616516\pi$
$500$		0			0
$501$	393.272			0.784974
$502$	−396.983			−0.790802
$503$			317.657i			0.631526i	0.948838	+	0.315763i	$0.102260\pi$
$503$			317.657i			0.631526i	−0.948838	+	0.315763i	$0.897740\pi$
$504$			69.0839i			0.137071i
$505$		0			0
$506$			168.797i			0.333591i
$507$	−285.566			−0.563247
$508$	−525.245			−1.03395
$509$			386.897i			0.760112i	0.924964	+	0.380056i	$0.124095\pi$
$509$			386.897i			0.760112i	−0.924964	+	0.380056i	$0.875905\pi$
$510$		0			0
$511$	1124.15			2.19989
$512$	−703.261			−1.37356
$513$	−321.530	+	426.998i	−0.626765	+	0.832354i
$514$	923.228			1.79616
$515$		0			0
$516$		−	221.755i		−	0.429758i
$517$		−	586.604i		−	1.13463i
$518$		−	805.086i		−	1.55422i
$519$	512.695			0.987851
$520$		0			0
$521$		−	216.069i		−	0.414719i	−0.978265	−	0.207359i	$-0.933513\pi$
$521$		−	216.069i		−	0.414719i	0.978265	−	0.207359i	$-0.0664870\pi$
$522$		−	538.474i		−	1.03156i
$523$	−208.535			−0.398728			−0.199364	−	0.979926i	$-0.563888\pi$
$523$	−208.535			−0.398728			−0.199364	+	0.979926i	$0.563888\pi$
$524$	−930.897			−1.77652
$525$		0			0
$526$			594.587i			1.13039i
$527$	54.7045			0.103804
$528$	468.454			0.887223
$529$	517.948			0.979108
$530$		0			0
$531$			159.828i			0.300995i
$532$	578.555	+	435.653i	1.08751	+	0.818897i
$533$		−	110.268i		−	0.206882i
$534$			1028.18i			1.92542i
$535$		0			0
$536$	189.302			0.353175
$537$		−	530.008i		−	0.986979i
$538$		−	797.826i		−	1.48295i
$539$	352.953			0.654829
$540$		0			0
$541$	−444.565			−0.821746			−0.410873	−	0.911693i	$-0.634776\pi$
$541$	−444.565			−0.821746			−0.410873	+	0.911693i	$0.634776\pi$
$542$	−661.123			−1.21978
$543$		−	538.273i		−	0.991294i
$544$		−	875.445i		−	1.60927i
$545$		0			0
$546$		−	260.158i		−	0.476481i
$547$	−57.5311			−0.105176			−0.0525879	−	0.998616i	$-0.516747\pi$
$547$	−57.5311			−0.105176			−0.0525879	+	0.998616i	$0.516747\pi$
$548$			699.164i			1.27585i
$549$	−32.3089			−0.0588505
$550$		0			0
$551$	−568.475	−	428.063i	−1.03172	−	0.776885i
$552$		−	11.3619i		−	0.0205832i
$553$	−406.693			−0.735430
$554$			392.067i			0.707702i
$555$		0			0
$556$	−710.731			−1.27829
$557$		−	32.9110i		−	0.0590861i	−0.999564	−	0.0295430i	$-0.990595\pi$
$557$		−	32.9110i		−	0.0590861i	0.999564	−	0.0295430i	$-0.00940521\pi$
$558$			41.2212i			0.0738730i
$559$		−	126.328i		−	0.225989i
$560$		0			0
$561$			669.052i			1.19261i
$562$			797.422i			1.41890i
$563$	−924.040			−1.64128			−0.820640	−	0.571446i	$-0.806383\pi$
$563$	−924.040			−1.64128			−0.820640	+	0.571446i	$0.806383\pi$
$564$			313.223i			0.555360i
$565$		0			0
$566$		−	899.429i		−	1.58910i
$567$		−	105.921i		−	0.186810i
$568$			187.550i			0.330194i
$569$		−	73.0173i		−	0.128326i	−0.997939	−	0.0641628i	$-0.979562\pi$
$569$		−	73.0173i		−	0.128326i	0.997939	−	0.0641628i	$-0.0204377\pi$
$570$		0			0
$571$	709.971			1.24338			0.621690	−	0.783263i	$-0.286446\pi$
$571$	709.971			1.24338			0.621690	+	0.783263i	$0.286446\pi$
$572$	−418.476			−0.731601
$573$	−414.413			−0.723234
$574$	−509.979			−0.888465
$575$		0			0
$576$	397.342			0.689830
$577$		−	152.315i		−	0.263978i	−0.991251	−	0.131989i	$-0.957864\pi$
$577$		−	152.315i		−	0.263978i	0.991251	−	0.131989i	$-0.0421363\pi$
$578$	219.759			0.380206
$579$	485.238			0.838062
$580$		0			0
$581$	368.913			0.634962
$582$	−300.291			−0.515964
$583$	−650.032			−1.11498
$584$			228.086i			0.390558i
$585$		0			0
$586$	797.415			1.36078
$587$			96.2527i			0.163974i	0.996633	+	0.0819870i	$0.0261266\pi$
$587$			96.2527i			0.163974i	−0.996633	+	0.0819870i	$0.973873\pi$
$588$	−188.463			−0.320515
$589$	43.5178	+	32.7690i	0.0738842	+	0.0556350i
$590$		0			0
$591$		−	21.3791i		−	0.0361744i
$592$	−440.968			−0.744878
$593$		−	707.928i		−	1.19381i	−0.802313	−	0.596904i	$-0.796397\pi$
$593$		−	707.928i		−	1.19381i	0.802313	−	0.596904i	$-0.203603\pi$
$594$	−1428.41			−2.40473
$595$		0			0
$596$	488.412			0.819483
$597$	201.004			0.336690
$598$		−	51.3452i		−	0.0858616i
$599$		−	197.129i		−	0.329097i	−0.986369	−	0.164549i	$-0.947383\pi$
$599$		−	197.129i		−	0.329097i	0.986369	−	0.164549i	$-0.0526168\pi$
$600$		0			0
$601$		−	828.260i		−	1.37814i	−0.724697	−	0.689068i	$-0.758020\pi$
$601$		−	828.260i		−	1.37814i	0.724697	−	0.689068i	$-0.241980\pi$
$602$	−584.255			−0.970522
$603$	549.964			0.912047
$604$			1201.85i			1.98981i
$605$		0			0
$606$	593.891			0.980017
$607$	−386.555			−0.636829			−0.318415	−	0.947952i	$-0.603150\pi$
$607$	−386.555			−0.636829			−0.318415	+	0.947952i	$0.603150\pi$
$608$	524.408	−	696.423i	0.862513	−	1.14543i
$609$	630.887			1.03594
$610$		0			0
$611$			178.435i			0.292038i
$612$			428.702i			0.700493i
$613$		−	1061.04i		−	1.73090i	−0.500993	−	0.865451i	$-0.667032\pi$
$613$		−	1061.04i		−	1.73090i	0.500993	−	0.865451i	$-0.332968\pi$
$614$	122.454			0.199437
$615$		0			0
$616$			243.978i			0.396068i
$617$		−	1188.74i		−	1.92665i	−0.268337	−	0.963325i	$-0.586474\pi$
$617$		−	1188.74i		−	1.92665i	0.268337	−	0.963325i	$-0.413526\pi$
$618$	−187.016			−0.302615
$619$	906.151			1.46390			0.731948	−	0.681361i	$-0.238611\pi$
$619$	906.151			1.46390			0.731948	+	0.681361i	$0.238611\pi$
$620$		0			0
$621$		−	93.5250i		−	0.150604i
$622$	−1169.46			−1.88016
$623$	1445.58			2.32035
$624$	−142.496			−0.228359
$625$		0			0
$626$		−	801.112i		−	1.27973i
$627$	−400.775	+	532.235i	−0.639194	+	0.848860i
$628$			1091.54i			1.73813i
$629$		−	629.795i		−	1.00126i
$630$		0			0
$631$	310.563			0.492176			0.246088	−	0.969247i	$-0.420855\pi$
$631$	310.563			0.492176			0.246088	+	0.969247i	$0.420855\pi$
$632$		−	82.5167i		−	0.130564i
$633$		−	181.549i		−	0.286808i
$634$	−352.782			−0.556438
$635$		0			0
$636$	347.091			0.545741
$637$	−107.362			−0.168544
$638$		−	1901.69i		−	2.98070i
$639$			544.875i			0.852700i
$640$		0			0
$641$			673.349i			1.05047i	0.850958	+	0.525233i	$0.176022\pi$
$641$			673.349i			1.05047i	−0.850958	+	0.525233i	$0.823978\pi$
$642$	423.616			0.659839
$643$			454.931i			0.707514i	0.935337	+	0.353757i	$0.115096\pi$
$643$			454.931i			0.707514i	−0.935337	+	0.353757i	$0.884904\pi$
$644$	−126.720			−0.196771
$645$		0			0
$646$	848.121	+	638.637i	1.31288	+	0.988602i
$647$			357.358i			0.552331i	0.961110	+	0.276166i	$0.0890638\pi$
$647$			357.358i			0.552331i	−0.961110	+	0.276166i	$0.910936\pi$
$648$	21.4911			0.0331653
$649$			564.453i			0.869728i
$650$		0			0
$651$	−48.2955			−0.0741866
$652$		−	588.875i		−	0.903183i
$653$			859.520i			1.31626i	0.752903	+	0.658132i	$0.228653\pi$
$653$			859.520i			1.31626i	−0.752903	+	0.658132i	$0.771347\pi$
$654$		−	818.918i		−	1.25217i
$655$		0			0
$656$			279.329i			0.425807i
$657$			662.640i			1.00858i
$658$	825.245			1.25417
$659$			103.485i			0.157033i	0.996913	+	0.0785166i	$0.0250183\pi$
$659$			103.485i			0.157033i	−0.996913	+	0.0785166i	$0.974982\pi$
$660$		0			0
$661$			520.372i			0.787249i	0.919271	+	0.393625i	$0.128779\pi$
$661$			520.372i			0.787249i	−0.919271	+	0.393625i	$0.871221\pi$
$662$			161.810i			0.244426i
$663$		−	203.514i		−	0.306960i
$664$			74.8513i			0.112728i
$665$		0			0
$666$	474.566			0.712562
$667$	124.513			0.186676
$668$	889.945			1.33225
$669$	−210.574			−0.314760
$670$		0			0
$671$	−114.103			−0.170049
$672$			772.881i			1.15012i
$673$	−687.475			−1.02151			−0.510754	−	0.859727i	$-0.670634\pi$
$673$	−687.475			−1.02151			−0.510754	+	0.859727i	$0.670634\pi$
$674$	−1408.41			−2.08963
$675$		0			0
$676$	−646.215			−0.955940
$677$	540.624			0.798558			0.399279	−	0.916830i	$-0.369261\pi$
$677$	540.624			0.798558			0.399279	+	0.916830i	$0.369261\pi$
$678$	686.686			1.01281
$679$			422.197i			0.621793i
$680$		0			0
$681$	−515.133			−0.756436
$682$			145.578i			0.213457i
$683$	405.203			0.593270			0.296635	−	0.954991i	$-0.404136\pi$
$683$	405.203			0.593270			0.296635	+	0.954991i	$0.404136\pi$
$684$	−256.801	+	341.035i	−0.375439	+	0.498590i
$685$		0			0
$686$		−	698.581i		−	1.01834i
$687$	−659.826			−0.960445
$688$			320.012i			0.465134i
$689$	197.729			0.286980
$690$		0			0
$691$	−252.865			−0.365941			−0.182970	−	0.983118i	$-0.558571\pi$
$691$	−252.865			−0.365941			−0.182970	+	0.983118i	$0.558571\pi$
$692$	1160.19			1.67657
$693$			708.811i			1.02282i
$694$		−	1578.50i		−	2.27450i
$695$		0			0
$696$			128.005i			0.183915i
$697$	−398.942			−0.572370
$698$	−62.9935			−0.0902485
$699$			135.312i			0.193579i
$700$		0			0
$701$	−77.0624			−0.109932			−0.0549660	−	0.998488i	$-0.517505\pi$
$701$	−77.0624			−0.109932			−0.0549660	+	0.998488i	$0.517505\pi$
$702$	434.499			0.618944
$703$	377.259	−	501.007i	0.536642	−	0.712670i
$704$	1403.26			1.99327
$705$		0			0
$706$		−	267.497i		−	0.378891i
$707$		−	834.987i		−	1.18103i
$708$		−	301.396i		−	0.425700i
$709$	−876.024			−1.23558			−0.617788	−	0.786344i	$-0.711971\pi$
$709$	−876.024			−1.23558			−0.617788	+	0.786344i	$0.711971\pi$
$710$		0			0
$711$		−	239.729i		−	0.337172i
$712$			293.303i			0.411942i
$713$	−9.53167			−0.0133684
$714$	−941.234			−1.31825
$715$		0			0
$716$		−	1199.37i		−	1.67510i
$717$	−498.774			−0.695640
$718$	−1971.58			−2.74593
$719$	−819.304			−1.13950			−0.569752	−	0.821816i	$-0.692961\pi$
$719$	−819.304			−1.13950			−0.569752	+	0.821816i	$0.692961\pi$
$720$		0			0
$721$			262.937i			0.364684i
$722$	292.131	+	1016.08i	0.404613	+	1.40731i
$723$			80.7517i			0.111690i
$724$		−	1218.07i		−	1.68242i
$725$		0			0
$726$	−1063.72			−1.46518
$727$		−	140.643i		−	0.193457i	−0.995311	−	0.0967286i	$-0.969162\pi$
$727$		−	140.643i		−	0.193457i	0.995311	−	0.0967286i	$-0.0308379\pi$
$728$		−	74.2141i		−	0.101943i
$729$	574.541			0.788123
$730$		0			0
$731$	−457.046			−0.625233
$732$	60.9265			0.0832329
$733$		−	693.708i		−	0.946395i	−0.880956	−	0.473197i	$-0.843100\pi$
$733$		−	693.708i		−	0.946395i	0.880956	−	0.473197i	$-0.156900\pi$
$734$			1744.05i			2.37609i
$735$		0			0
$736$			152.537i			0.207251i
$737$	1942.27			2.63537
$738$		−	300.613i		−	0.407334i
$739$	1424.83			1.92806			0.964028	−	0.265801i	$-0.0856364\pi$
$739$	1424.83			1.92806			0.964028	+	0.265801i	$0.0856364\pi$
$740$		0			0
$741$	121.909	−	161.897i	0.164520	−	0.218485i
$742$		−	914.477i		−	1.23245i
$743$	−1046.28			−1.40818			−0.704092	−	0.710109i	$-0.748646\pi$
$743$	−1046.28			−1.40818			−0.704092	+	0.710109i	$0.748646\pi$
$744$		−	9.79901i		−	0.0131707i
$745$		0			0
$746$	−669.473			−0.897417
$747$			217.460i			0.291111i
$748$			1514.01i			2.02408i
$749$		−	595.588i		−	0.795178i
$750$		0			0
$751$		−	70.9338i		−	0.0944525i	−0.998884	−	0.0472262i	$-0.984962\pi$
$751$		−	70.9338i		−	0.0944525i	0.998884	−	0.0472262i	$-0.0150382\pi$
$752$		−	452.009i		−	0.601076i
$753$	274.166			0.364098
$754$			578.462i			0.767191i
$755$		0			0
$756$		−	1072.35i		−	1.41845i
$757$		−	1422.41i		−	1.87901i	−0.342532	−	0.939506i	$-0.611284\pi$
$757$		−	1422.41i		−	1.87901i	0.342532	−	0.939506i	$-0.388716\pi$
$758$			2124.70i			2.80304i
$759$		−	116.575i		−	0.153590i
$760$		0			0
$761$	110.061			0.144627			0.0723136	−	0.997382i	$-0.476962\pi$
$761$	110.061			0.144627			0.0723136	+	0.997382i	$0.476962\pi$
$762$	679.765			0.892080
$763$	−1151.37			−1.50900
$764$	−937.786			−1.22747
$765$		0			0
$766$	−1537.98			−2.00780
$767$		−	171.697i		−	0.223856i
$768$	337.868			0.439932
$769$	−136.681			−0.177739			−0.0888693	−	0.996043i	$-0.528325\pi$
$769$	−136.681			−0.177739			−0.0888693	+	0.996043i	$0.528325\pi$
$770$		0			0
$771$	−637.603			−0.826981
$772$	1098.06			1.42235
$773$	−177.135			−0.229153			−0.114576	−	0.993414i	$-0.536551\pi$
$773$	−177.135			−0.229153			−0.114576	+	0.993414i	$0.536551\pi$
$774$		−	344.395i		−	0.444955i
$775$		0			0
$776$	−85.6625			−0.110390
$777$			556.011i			0.715587i
$778$	395.392			0.508216
$779$	−317.361	−	238.974i	−0.407396	−	0.306770i
$780$		0			0
$781$			1924.29i			2.46389i
$782$	−185.763			−0.237549
$783$			1053.66i			1.34568i
$784$	271.969			0.346899
$785$		0			0
$786$	1204.75			1.53277
$787$	−210.259			−0.267166			−0.133583	−	0.991038i	$-0.542648\pi$
$787$	−210.259			−0.267166			−0.133583	+	0.991038i	$0.542648\pi$
$788$		−	48.3793i		−	0.0613950i
$789$		−	410.636i		−	0.520451i
$790$		0			0
$791$		−	965.453i		−	1.22055i
$792$	−143.816			−0.181585
$793$	34.7083			0.0437683
$794$		−	1007.73i		−	1.26918i
$795$		0			0
$796$	454.857			0.571429
$797$	474.092			0.594845			0.297423	−	0.954746i	$-0.403873\pi$
$797$	474.092			0.594845			0.297423	+	0.954746i	$0.403873\pi$
$798$	−748.758	−	563.817i	−0.938293	−	0.706537i
$799$	645.565			0.807967
$800$		0			0
$801$			852.110i			1.06381i
$802$			193.924i			0.241800i
$803$			2340.19i			2.91431i
$804$	−1037.09			−1.28992
$805$		0			0
$806$		−	44.2823i		−	0.0549408i
$807$			550.997i			0.682772i
$808$	169.416			0.209674
$809$	1279.92			1.58210			0.791050	−	0.611751i	$-0.209535\pi$
$809$	1279.92			1.58210			0.791050	+	0.611751i	$0.209535\pi$
$810$		0			0
$811$			869.604i			1.07226i	0.844135	+	0.536130i	$0.180114\pi$
$811$			869.604i			1.07226i	−0.844135	+	0.536130i	$0.819886\pi$
$812$	1427.65			1.75819
$813$	456.587			0.561607
$814$	1675.99			2.05896
$815$		0			0
$816$			515.540i			0.631789i
$817$	−363.583	−	273.779i	−0.445022	−	0.335103i
$818$			193.845i			0.236974i
$819$		−	215.609i		−	0.263258i
$820$		0			0
$821$	−1061.20			−1.29257			−0.646287	−	0.763094i	$-0.723679\pi$
$821$	−1061.20			−1.29257			−0.646287	+	0.763094i	$0.723679\pi$
$822$		−	904.849i		−	1.10079i
$823$		−	458.926i		−	0.557626i	−0.960345	−	0.278813i	$-0.910059\pi$
$823$		−	458.926i		−	0.557626i	0.960345	−	0.278813i	$-0.0899409\pi$
$824$	−53.3492			−0.0647441
$825$		0			0
$826$	−794.083			−0.961359
$827$	746.165			0.902255			0.451128	−	0.892459i	$-0.351022\pi$
$827$	746.165			0.902255			0.451128	+	0.892459i	$0.351022\pi$
$828$		−	74.6967i		−	0.0902134i
$829$			200.255i			0.241562i	0.992679	+	0.120781i	$0.0385399\pi$
$829$			200.255i			0.241562i	−0.992679	+	0.120781i	$0.961460\pi$
$830$		0			0
$831$		−	270.771i		−	0.325837i
$832$	−426.849			−0.513040
$833$			388.429i			0.466301i
$834$	919.818			1.10290
$835$		0			0
$836$	−906.923	+	1204.41i	−1.08484	+	1.44068i
$837$		−	80.6599i		−	0.0963678i
$838$	1312.29			1.56597
$839$			247.657i			0.295181i	0.989049	+	0.147591i	$0.0471518\pi$
$839$			247.657i			0.295181i	−0.989049	+	0.147591i	$0.952848\pi$
$840$		0			0
$841$	−561.777			−0.667987
$842$			2132.25i			2.53236i
$843$		−	550.718i		−	0.653284i
$844$		−	410.832i		−	0.486768i
$845$		0			0
$846$			486.449i			0.574999i
$847$			1495.55i			1.76570i
$848$	−500.884			−0.590665
$849$			621.167i			0.731645i
$850$		0			0
$851$			109.735i			0.128948i
$852$		−	1027.50i		−	1.20598i
$853$		−	172.802i		−	0.202581i	−0.994857	−	0.101291i	$-0.967703\pi$
$853$		−	172.802i		−	0.202581i	0.994857	−	0.101291i	$-0.0322972\pi$
$854$		−	160.522i		−	0.187965i
$855$		0			0
$856$	120.843			0.141172
$857$	795.527			0.928270			0.464135	−	0.885765i	$-0.346365\pi$
$857$	795.527			0.928270			0.464135	+	0.885765i	$0.346365\pi$
$858$	541.586			0.631219
$859$	104.671			0.121852			0.0609260	−	0.998142i	$-0.480595\pi$
$859$	104.671			0.121852			0.0609260	+	0.998142i	$0.480595\pi$
$860$		0			0
$861$	352.203			0.409063
$862$		−	1827.15i		−	2.11967i
$863$	−887.350			−1.02822			−0.514108	−	0.857725i	$-0.671877\pi$
$863$	−887.350			−1.02822			−0.514108	+	0.857725i	$0.671877\pi$
$864$	−1290.81			−1.49400
$865$		0			0
$866$	11.4320			0.0132009
$867$	−151.771			−0.175053
$868$	−109.289			−0.125909
$869$		−	846.633i		−	0.974262i
$870$		0			0
$871$	−590.806			−0.678307
$872$		−	233.609i		−	0.267900i
$873$	−248.869			−0.285073
$874$	−147.776	−	111.276i	−0.169080	−	0.127318i
$875$		0			0
$876$		−	1249.57i		−	1.42645i
$877$	−82.4212			−0.0939809			−0.0469904	−	0.998895i	$-0.514963\pi$
$877$	−82.4212			−0.0939809			−0.0469904	+	0.998895i	$0.514963\pi$
$878$		−	2193.51i		−	2.49831i
$879$	−550.714			−0.626523
$880$		0			0
$881$	1311.40			1.48853			0.744267	−	0.667883i	$-0.232799\pi$
$881$	1311.40			1.48853			0.744267	+	0.667883i	$0.232799\pi$
$882$	−292.691			−0.331849
$883$		−	61.4085i		−	0.0695453i	−0.999395	−	0.0347727i	$-0.988929\pi$
$883$		−	61.4085i		−	0.0695453i	0.999395	−	0.0347727i	$-0.0110707\pi$
$884$		−	460.538i		−	0.520971i
$885$		0			0
$886$		−	227.813i		−	0.257125i
$887$	−70.9500			−0.0799887			−0.0399943	−	0.999200i	$-0.512734\pi$
$887$	−70.9500			−0.0799887			−0.0399943	+	0.999200i	$0.512734\pi$
$888$	−112.813			−0.127042
$889$		−	955.723i		−	1.07505i
$890$		0			0
$891$	220.502			0.247477
$892$	−476.514			−0.534208
$893$	513.552	+	386.706i	0.575086	+	0.433041i
$894$	−632.096			−0.707042
$895$		0			0
$896$			445.637i			0.497363i
$897$			35.4602i			0.0395320i
$898$			328.135i			0.365407i
$899$	107.385			0.119449
$900$		0			0
$901$		−	715.369i		−	0.793972i
$902$		−	1061.65i		−	1.17700i
$903$	403.500			0.446844
$904$	195.888			0.216690
$905$		0			0
$906$		−	1555.41i		−	1.71679i
$907$	658.755			0.726301			0.363151	−	0.931730i	$-0.381701\pi$
$907$	658.755			0.726301			0.363151	+	0.931730i	$0.381701\pi$
$908$	−1165.71			−1.28382
$909$	492.192			0.541465
$910$		0			0
$911$			445.396i			0.488909i	0.969661	+	0.244454i	$0.0786089\pi$
$911$			445.396i			0.488909i	−0.969661	+	0.244454i	$0.921391\pi$
$912$	−308.818	+	410.115i	−0.338616	+	0.449688i
$913$			767.986i			0.841168i
$914$			1650.04i			1.80530i
$915$		0			0
$916$	−1493.14			−1.63006
$917$		−	1693.84i		−	1.84715i
$918$		−	1571.98i		−	1.71240i
$919$	−423.867			−0.461226			−0.230613	−	0.973046i	$-0.574073\pi$
$919$	−423.867			−0.461226			−0.230613	+	0.973046i	$0.574073\pi$
$920$		0			0
$921$	−84.5699			−0.0918240
$922$	1854.49			2.01138
$923$		−	585.339i		−	0.634170i
$924$		−	1336.64i		−	1.44658i
$925$		0			0
$926$			1603.71i			1.73187i
$927$	−154.991			−0.167197
$928$		−	1718.50i		−	1.85183i
$929$	−976.138			−1.05074			−0.525370	−	0.850874i	$-0.676073\pi$
$929$	−976.138			−1.05074			−0.525370	+	0.850874i	$0.676073\pi$
$930$		0			0
$931$	−232.676	+	308.998i	−0.249921	+	0.331899i
$932$			306.201i			0.328542i
$933$	807.654			0.865653
$934$		−	2509.04i		−	2.68634i
$935$		0			0
$936$	43.7463			0.0467375
$937$			383.017i			0.408770i	0.978891	+	0.204385i	$0.0655194\pi$
$937$			383.017i			0.408770i	−0.978891	+	0.204385i	$0.934481\pi$
$938$			2732.42i			2.91302i
$939$			553.267i			0.589209i
$940$		0			0
$941$		−	1515.60i		−	1.61062i	−0.592851	−	0.805312i	$-0.701998\pi$
$941$		−	1515.60i		−	1.61062i	0.592851	−	0.805312i	$-0.298002\pi$
$942$		−	1412.66i		−	1.49964i
$943$	69.5114			0.0737130
$944$			434.941i			0.460743i
$945$		0			0
$946$		−	1216.27i		−	1.28570i
$947$		−	33.4812i		−	0.0353550i	−0.999844	−	0.0176775i	$-0.994373\pi$
$947$		−	33.4812i		−	0.0353550i	0.999844	−	0.0176775i	$-0.00562722\pi$
$948$			452.069i			0.476866i
$949$		−	711.849i		−	0.750104i
$950$		0			0
$951$	243.639			0.256193
$952$	−268.501			−0.282039
$953$	−1337.39			−1.40335			−0.701674	−	0.712499i	$-0.747563\pi$
$953$	−1337.39			−1.40335			−0.701674	+	0.712499i	$0.747563\pi$
$954$	539.048			0.565040
$955$		0			0
$956$	−1128.69			−1.18064
$957$			1313.35i			1.37236i
$958$	638.476			0.666468
$959$	−1272.18			−1.32657
$960$		0			0
$961$	952.779			0.991446
$962$	−509.808			−0.529946
$963$	351.076			0.364565
$964$			182.735i			0.189559i
$965$		0			0
$966$	164.000			0.169772
$967$		−	146.380i		−	0.151375i	−0.997132	−	0.0756877i	$-0.975885\pi$
$967$		−	146.380i		−	0.151375i	0.997132	−	0.0756877i	$-0.0241152\pi$
$968$	−303.442			−0.313473
$969$	−585.732	−	441.058i	−0.604471	−	0.455168i
$970$		0			0
$971$			1118.48i			1.15188i	0.817492	+	0.575940i	$0.195364\pi$
$971$			1118.48i			1.15188i	−0.817492	+	0.575940i	$0.804636\pi$
$972$	1041.12			1.07111
$973$		−	1293.23i		−	1.32912i
$974$	2291.71			2.35288
$975$		0			0
$976$	−87.9224			−0.0900844
$977$	−850.328			−0.870346			−0.435173	−	0.900347i	$-0.643313\pi$
$977$	−850.328			−0.870346			−0.435173	+	0.900347i	$0.643313\pi$
$978$			762.115i			0.779258i
$979$			3009.33i			3.07388i
$980$		0			0
$981$		−	678.685i		−	0.691830i
$982$	−37.8328			−0.0385263
$983$	1064.58			1.08299			0.541493	−	0.840705i	$-0.317859\pi$
$983$	1064.58			1.08299			0.541493	+	0.840705i	$0.317859\pi$
$984$			71.4610i			0.0726230i
$985$		0			0
$986$	2092.83			2.12255
$987$	−569.933			−0.577440
$988$	275.871	−	366.361i	0.279222	−	0.370811i
$989$	79.6353			0.0805210
$990$		0			0
$991$		−	1750.89i		−	1.76679i	−0.468631	−	0.883394i	$-0.655253\pi$
$991$		−	1750.89i		−	1.76679i	0.468631	−	0.883394i	$-0.344747\pi$
$992$			131.554i			0.132615i
$993$		−	111.750i		−	0.112538i
$994$	−2707.13			−2.72347
$995$		0			0
$996$		−	410.074i		−	0.411721i
$997$			444.357i			0.445695i	0.974853	+	0.222847i	$0.0715351\pi$
$997$			444.357i			0.445695i	−0.974853	+	0.222847i	$0.928465\pi$
$998$	1046.14			1.04824
$999$	−928.612			−0.929542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.d.d.474.6		28
5.2	odd	4		475.3.c.i.151.3	yes	14
5.3	odd	4		475.3.c.h.151.12	yes	14
5.4	even	2	inner	475.3.d.d.474.23		28
19.18	odd	2	inner	475.3.d.d.474.24		28
95.18	even	4		475.3.c.h.151.3	✓	14
95.37	even	4		475.3.c.i.151.12	yes	14
95.94	odd	2	inner	475.3.d.d.474.5		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.3.c.h.151.3	✓	14	95.18	even	4
475.3.c.h.151.12	yes	14	5.3	odd	4
475.3.c.i.151.3	yes	14	5.2	odd	4
475.3.c.i.151.12	yes	14	95.37	even	4
475.3.d.d.474.5		28	95.94	odd	2	inner
475.3.d.d.474.6		28	1.1	even	1	trivial
475.3.d.d.474.23		28	5.4	even	2	inner
475.3.d.d.474.24		28	19.18	odd	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.92865 q^{2} +2.02259 q^{3} +4.57698 q^{4} -5.92346 q^{6} +8.32815i q^{7} -1.68976 q^{8} -4.90912 q^{9} +O(q^{10})\)	\(q-2.92865 q^{2} +2.02259 q^{3} +4.57698 q^{4} -5.92346 q^{6} +8.32815i q^{7} -1.68976 q^{8} -4.90912 q^{9} -17.3372 q^{11} +9.25736 q^{12} +5.27368 q^{13} -24.3902i q^{14} -13.3592 q^{16} -19.0798i q^{17} +14.3771 q^{18} +(11.4291 - 15.1781i) q^{19} +16.8445i q^{21} +50.7744 q^{22} +3.32445i q^{23} -3.41769 q^{24} -15.4447 q^{26} -28.1325 q^{27} +38.1178i q^{28} -37.4537i q^{29} +2.86714i q^{31} +45.8834 q^{32} -35.0660 q^{33} +55.8779i q^{34} -22.4689 q^{36} +33.0085 q^{37} +(-33.4719 + 44.4513i) q^{38} +10.6665 q^{39} -20.9092i q^{41} -49.3315i q^{42} -23.9545i q^{43} -79.3518 q^{44} -9.73613i q^{46} +33.8351i q^{47} -27.0202 q^{48} -20.3582 q^{49} -38.5906i q^{51} +24.1375 q^{52} +37.4936 q^{53} +82.3901 q^{54} -14.0726i q^{56} +(23.1165 - 30.6991i) q^{57} +109.689i q^{58} -32.5574i q^{59} +6.58141 q^{61} -8.39685i q^{62} -40.8839i q^{63} -80.9395 q^{64} +102.696 q^{66} -112.029 q^{67} -87.3276i q^{68} +6.72400i q^{69} -110.992i q^{71} +8.29522 q^{72} -134.981i q^{73} -96.6704 q^{74} +(52.3109 - 69.4698i) q^{76} -144.387i q^{77} -31.2384 q^{78} +48.8335i q^{79} -12.7185 q^{81} +61.2355i q^{82} -44.2971i q^{83} +77.0967i q^{84} +70.1541i q^{86} -75.7535i q^{87} +29.2956 q^{88} -173.577i q^{89} +43.9200i q^{91} +15.2159i q^{92} +5.79906i q^{93} -99.0910i q^{94} +92.8034 q^{96} +50.6952 q^{97} +59.6219 q^{98} +85.1102 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9}+O(q^{10}) \) 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9	\( 28 q + 56 q^{4} - 8 q^{6} + 72 q^{9} - 8 q^{11} + 72 q^{16} - 78 q^{19} + 88 q^{24} + 60 q^{26} + 8 q^{36} + 64 q^{39} + 104 q^{44} - 468 q^{49} - 196 q^{54} + 444 q^{61} + 436 q^{64} + 184 q^{66} + 184 q^{74} - 702 q^{76} + 804 q^{81} + 380 q^{96} + 360 q^{99}+O(q^{100}) \) 28 * q + 56 * q^4 - 8 * q^6 + 72 * q^9 - 8 * q^11 + 72 * q^16 - 78 * q^19 + 88 * q^24 + 60 * q^26 + 8 * q^36 + 64 * q^39 + 104 * q^44 - 468 * q^49 - 196 * q^54 + 444 * q^61 + 436 * q^64 + 184 * q^66 + 184 * q^74 - 702 * q^76 + 804 * q^81 + 380 * q^96 + 360 * q^99

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