LMFDB - Embedded newform 1216.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

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:	yes
Analytic conductor:	$71.7463225670$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.7057.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 22x + 32 $ x^3 - x^2 - 22*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.36181$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33180 q^{3} -18.4427 q^{5} +10.3872 q^{7} -25.2263 q^{9} +O(q^{10})$	$q-1.33180 q^{3} -18.4427 q^{5} +10.3872 q^{7} -25.2263 q^{9} -50.4519 q^{11} +61.8352 q^{13} +24.5619 q^{15} +68.1070 q^{17} +19.0000 q^{19} -13.8337 q^{21} +145.632 q^{23} +215.132 q^{25} +69.5550 q^{27} -42.6097 q^{29} +91.6582 q^{31} +67.1918 q^{33} -191.568 q^{35} +400.965 q^{37} -82.3522 q^{39} -123.355 q^{41} -449.802 q^{43} +465.240 q^{45} -453.075 q^{47} -235.105 q^{49} -90.7050 q^{51} -437.142 q^{53} +930.466 q^{55} -25.3042 q^{57} +159.352 q^{59} +476.816 q^{61} -262.031 q^{63} -1140.41 q^{65} +629.682 q^{67} -193.953 q^{69} +471.459 q^{71} -725.055 q^{73} -286.512 q^{75} -524.055 q^{77} -1057.66 q^{79} +588.477 q^{81} +726.957 q^{83} -1256.07 q^{85} +56.7476 q^{87} -468.065 q^{89} +642.297 q^{91} -122.071 q^{93} -350.410 q^{95} -891.891 q^{97} +1272.71 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9	$ 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9} - 103 q^{11} - 32 q^{13} + 122 q^{15} + 11 q^{17} + 57 q^{19} - 114 q^{21} + 316 q^{23} + 162 q^{25} - 178 q^{27} + 138 q^{29} + 420 q^{31} - 330 q^{33} - 333 q^{35} - 102 q^{37} + 164 q^{39} - 370 q^{41} - 431 q^{43} + 429 q^{45} - 199 q^{47} - 802 q^{49} - 272 q^{51} + 308 q^{53} + 85 q^{55} - 76 q^{57} + 188 q^{59} + 609 q^{61} - 61 q^{63} - 1536 q^{65} + 246 q^{67} + 892 q^{69} - 954 q^{71} - 629 q^{73} + 1074 q^{75} + 71 q^{77} + 452 q^{79} - 361 q^{81} + 780 q^{83} - 1883 q^{85} - 96 q^{87} - 1356 q^{89} + 670 q^{91} - 1092 q^{93} - 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9 - 103 * q^11 - 32 * q^13 + 122 * q^15 + 11 * q^17 + 57 * q^19 - 114 * q^21 + 316 * q^23 + 162 * q^25 - 178 * q^27 + 138 * q^29 + 420 * q^31 - 330 * q^33 - 333 * q^35 - 102 * q^37 + 164 * q^39 - 370 * q^41 - 431 * q^43 + 429 * q^45 - 199 * q^47 - 802 * q^49 - 272 * q^51 + 308 * q^53 + 85 * q^55 - 76 * q^57 + 188 * q^59 + 609 * q^61 - 61 * q^63 - 1536 * q^65 + 246 * q^67 + 892 * q^69 - 954 * q^71 - 629 * q^73 + 1074 * q^75 + 71 * q^77 + 452 * q^79 - 361 * q^81 + 780 * q^83 - 1883 * q^85 - 96 * q^87 - 1356 * q^89 + 670 * q^91 - 1092 * q^93 - 133 * q^95 - 548 * q^97 + 1305 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−1.33180	−0.256305	−0.128153	−	0.991754i	$-0.540905\pi$
$3$	−1.33180	−0.256305	−0.128153	+	0.991754i	$0.540905\pi$
$4$	0	0
$5$	−18.4427	−1.64956	−0.824781	−	0.565453i	$-0.808701\pi$
$5$	−18.4427	−1.64956	−0.824781	+	0.565453i	$0.808701\pi$
$6$	0	0
$7$	10.3872	0.560858	0.280429	−	0.959875i	$-0.409523\pi$
$7$	10.3872	0.560858	0.280429	+	0.959875i	$0.409523\pi$
$8$	0	0
$9$	−25.2263	−0.934308
$10$	0	0
$11$	−50.4519	−1.38289	−0.691446	−	0.722428i	$-0.743026\pi$
$11$	−50.4519	−1.38289	−0.691446	+	0.722428i	$0.743026\pi$
$12$	0	0
$13$	61.8352	1.31923	0.659616	−	0.751603i	$-0.270719\pi$
$13$	61.8352	1.31923	0.659616	+	0.751603i	$0.270719\pi$
$14$	0	0
$15$	24.5619	0.422791
$16$	0	0
$17$	68.1070	0.971669	0.485835	−	0.874051i	$-0.338516\pi$
$17$	68.1070	0.971669	0.485835	+	0.874051i	$0.338516\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−13.8337	−0.143751
$22$	0	0
$23$	145.632	1.32028	0.660138	−	0.751144i	$-0.270498\pi$
$23$	145.632	1.32028	0.660138	+	0.751144i	$0.270498\pi$
$24$	0	0
$25$	215.132	1.72105
$26$	0	0
$27$	69.5550	0.495773
$28$	0	0
$29$	−42.6097	−0.272842	−0.136421	−	0.990651i	$-0.543560\pi$
$29$	−42.6097	−0.272842	−0.136421	+	0.990651i	$0.543560\pi$
$30$	0	0
$31$	91.6582	0.531042	0.265521	−	0.964105i	$-0.414456\pi$
$31$	91.6582	0.531042	0.265521	+	0.964105i	$0.414456\pi$
$32$	0	0
$33$	67.1918	0.354442
$34$	0	0
$35$	−191.568	−0.925169
$36$	0	0
$37$	400.965	1.78158	0.890788	−	0.454419i	$-0.150153\pi$
$37$	400.965	1.78158	0.890788	+	0.454419i	$0.150153\pi$
$38$	0	0
$39$	−82.3522	−0.338126
$40$	0	0
$41$	−123.355	−0.469873	−0.234936	−	0.972011i	$-0.575488\pi$
$41$	−123.355	−0.469873	−0.234936	+	0.972011i	$0.575488\pi$
$42$	0	0
$43$	−449.802	−1.59521	−0.797607	−	0.603178i	$-0.793901\pi$
$43$	−449.802	−1.59521	−0.797607	+	0.603178i	$0.793901\pi$
$44$	0	0
$45$	465.240	1.54120
$46$	0	0
$47$	−453.075	−1.40612	−0.703062	−	0.711129i	$-0.748184\pi$
$47$	−453.075	−1.40612	−0.703062	+	0.711129i	$0.748184\pi$
$48$	0	0
$49$	−235.105	−0.685439
$50$	0	0
$51$	−90.7050	−0.249044
$52$	0	0
$53$	−437.142	−1.13294	−0.566472	−	0.824081i	$-0.691692\pi$
$53$	−437.142	−1.13294	−0.566472	+	0.824081i	$0.691692\pi$
$54$	0	0
$55$	930.466	2.28116
$56$	0	0
$57$	−25.3042	−0.0588004
$58$	0	0
$59$	159.352	0.351624	0.175812	−	0.984424i	$-0.443745\pi$
$59$	159.352	0.351624	0.175812	+	0.984424i	$0.443745\pi$
$60$	0	0
$61$	476.816	1.00082	0.500410	−	0.865789i	$-0.333183\pi$
$61$	476.816	1.00082	0.500410	+	0.865789i	$0.333183\pi$
$62$	0	0
$63$	−262.031	−0.524014
$64$	0	0
$65$	−1140.41	−2.17615
$66$	0	0
$67$	629.682	1.14818	0.574089	−	0.818793i	$-0.305357\pi$
$67$	629.682	1.14818	0.574089	+	0.818793i	$0.305357\pi$
$68$	0	0
$69$	−193.953	−0.338394
$70$	0	0
$71$	471.459	0.788056	0.394028	−	0.919099i	$-0.371081\pi$
$71$	471.459	0.788056	0.394028	+	0.919099i	$0.371081\pi$
$72$	0	0
$73$	−725.055	−1.16248	−0.581241	−	0.813731i	$-0.697433\pi$
$73$	−725.055	−1.16248	−0.581241	+	0.813731i	$0.697433\pi$
$74$	0	0
$75$	−286.512	−0.441115
$76$	0	0
$77$	−524.055	−0.775605
$78$	0	0
$79$	−1057.66	−1.50628	−0.753139	−	0.657861i	$-0.771461\pi$
$79$	−1057.66	−1.50628	−0.753139	+	0.657861i	$0.771461\pi$
$80$	0	0
$81$	588.477	0.807238
$82$	0	0
$83$	726.957	0.961373	0.480686	−	0.876893i	$-0.340388\pi$
$83$	726.957	0.961373	0.480686	+	0.876893i	$0.340388\pi$
$84$	0	0
$85$	−1256.07	−1.60283
$86$	0	0
$87$	56.7476	0.0699309
$88$	0	0
$89$	−468.065	−0.557469	−0.278735	−	0.960368i	$-0.589915\pi$
$89$	−468.065	−0.557469	−0.278735	+	0.960368i	$0.589915\pi$
$90$	0	0
$91$	642.297	0.739901
$92$	0	0
$93$	−122.071	−0.136109
$94$	0	0
$95$	−350.410	−0.378435
$96$	0	0
$97$	−891.891	−0.933585	−0.466793	−	0.884367i	$-0.654591\pi$
$97$	−891.891	−0.933585	−0.466793	+	0.884367i	$0.654591\pi$
$98$	0	0
$99$	1272.71	1.29205
$100$	0	0
$101$	16.6636	0.0164167	0.00820835	−	0.999966i	$-0.497387\pi$
$101$	16.6636	0.0164167	0.00820835	+	0.999966i	$0.497387\pi$
$102$	0	0
$103$	1292.68	1.23662	0.618309	−	0.785935i	$-0.287818\pi$
$103$	1292.68	1.23662	0.618309	+	0.785935i	$0.287818\pi$
$104$	0	0
$105$	255.130	0.237126
$106$	0	0
$107$	176.185	0.159182	0.0795911	−	0.996828i	$-0.474639\pi$
$107$	176.185	0.159182	0.0795911	+	0.996828i	$0.474639\pi$
$108$	0	0
$109$	−1378.74	−1.21155	−0.605776	−	0.795635i	$-0.707137\pi$
$109$	−1378.74	−1.21155	−0.605776	+	0.795635i	$0.707137\pi$
$110$	0	0
$111$	−534.006	−0.456627
$112$	0	0
$113$	−601.795	−0.500992	−0.250496	−	0.968118i	$-0.580594\pi$
$113$	−601.795	−0.500992	−0.250496	+	0.968118i	$0.580594\pi$
$114$	0	0
$115$	−2685.84	−2.17788
$116$	0	0
$117$	−1559.87	−1.23257
$118$	0	0
$119$	707.443	0.544968
$120$	0	0
$121$	1214.39	0.912389
$122$	0	0
$123$	164.284	0.120431
$124$	0	0
$125$	−1662.26	−1.18942
$126$	0	0
$127$	1890.38	1.32082	0.660409	−	0.750906i	$-0.270383\pi$
$127$	1890.38	1.32082	0.660409	+	0.750906i	$0.270383\pi$
$128$	0	0
$129$	599.047	0.408861
$130$	0	0
$131$	−331.964	−0.221403	−0.110701	−	0.993854i	$-0.535310\pi$
$131$	−331.964	−0.221403	−0.110701	+	0.993854i	$0.535310\pi$
$132$	0	0
$133$	197.357	0.128670
$134$	0	0
$135$	−1282.78	−0.817808
$136$	0	0
$137$	−1956.20	−1.21992	−0.609962	−	0.792430i	$-0.708815\pi$
$137$	−1956.20	−1.21992	−0.609962	+	0.792430i	$0.708815\pi$
$138$	0	0
$139$	−547.821	−0.334285	−0.167142	−	0.985933i	$-0.553454\pi$
$139$	−547.821	−0.334285	−0.167142	+	0.985933i	$0.553454\pi$
$140$	0	0
$141$	603.406	0.360397
$142$	0	0
$143$	−3119.70	−1.82435
$144$	0	0
$145$	785.836	0.450070
$146$	0	0
$147$	313.114	0.175682
$148$	0	0
$149$	992.480	0.545686	0.272843	−	0.962059i	$-0.412036\pi$
$149$	992.480	0.545686	0.272843	+	0.962059i	$0.412036\pi$
$150$	0	0
$151$	−148.167	−0.0798520	−0.0399260	−	0.999203i	$-0.512712\pi$
$151$	−148.167	−0.0798520	−0.0399260	+	0.999203i	$0.512712\pi$
$152$	0	0
$153$	−1718.09	−0.907838
$154$	0	0
$155$	−1690.42	−0.875987
$156$	0	0
$157$	−2356.36	−1.19782	−0.598910	−	0.800817i	$-0.704399\pi$
$157$	−2356.36	−1.19782	−0.598910	+	0.800817i	$0.704399\pi$
$158$	0	0
$159$	582.186	0.290380
$160$	0	0
$161$	1512.71	0.740487
$162$	0	0
$163$	2974.86	1.42950	0.714751	−	0.699379i	$-0.246540\pi$
$163$	2974.86	1.42950	0.714751	+	0.699379i	$0.246540\pi$
$164$	0	0
$165$	−1239.20	−0.584674
$166$	0	0
$167$	−3133.09	−1.45177	−0.725885	−	0.687816i	$-0.758570\pi$
$167$	−3133.09	−1.45177	−0.725885	+	0.687816i	$0.758570\pi$
$168$	0	0
$169$	1626.60	0.740372
$170$	0	0
$171$	−479.300	−0.214345
$172$	0	0
$173$	−3415.40	−1.50097	−0.750485	−	0.660887i	$-0.770180\pi$
$173$	−3415.40	−1.50097	−0.750485	+	0.660887i	$0.770180\pi$
$174$	0	0
$175$	2234.62	0.965265
$176$	0	0
$177$	−212.225	−0.0901231
$178$	0	0
$179$	−80.0505	−0.0334260	−0.0167130	−	0.999860i	$-0.505320\pi$
$179$	−80.0505	−0.0334260	−0.0167130	+	0.999860i	$0.505320\pi$
$180$	0	0
$181$	−3379.91	−1.38799	−0.693996	−	0.719978i	$-0.744152\pi$
$181$	−3379.91	−1.38799	−0.693996	+	0.719978i	$0.744152\pi$
$182$	0	0
$183$	−635.023	−0.256515
$184$	0	0
$185$	−7394.87	−2.93882
$186$	0	0
$187$	−3436.13	−1.34371
$188$	0	0
$189$	722.484	0.278058
$190$	0	0
$191$	−3288.78	−1.24590	−0.622952	−	0.782260i	$-0.714067\pi$
$191$	−3288.78	−1.24590	−0.622952	+	0.782260i	$0.714067\pi$
$192$	0	0
$193$	2906.89	1.08416	0.542079	−	0.840327i	$-0.317637\pi$
$193$	2906.89	1.08416	0.542079	+	0.840327i	$0.317637\pi$
$194$	0	0
$195$	1518.79	0.557759
$196$	0	0
$197$	17.1879	0.00621617	0.00310808	−	0.999995i	$-0.499011\pi$
$197$	17.1879	0.00621617	0.00310808	+	0.999995i	$0.499011\pi$
$198$	0	0
$199$	−613.599	−0.218577	−0.109289	−	0.994010i	$-0.534857\pi$
$199$	−613.599	−0.218577	−0.109289	+	0.994010i	$0.534857\pi$
$200$	0	0
$201$	−838.611	−0.294284
$202$	0	0
$203$	−442.597	−0.153026
$204$	0	0
$205$	2274.99	0.775084
$206$	0	0
$207$	−3673.76	−1.23354
$208$	0	0
$209$	−958.585	−0.317257
$210$	0	0
$211$	3991.85	1.30242	0.651209	−	0.758898i	$-0.274262\pi$
$211$	3991.85	1.30242	0.651209	+	0.758898i	$0.274262\pi$
$212$	0	0
$213$	−627.890	−0.201983
$214$	0	0
$215$	8295.54	2.63140
$216$	0	0
$217$	952.075	0.297839
$218$	0	0
$219$	965.628	0.297950
$220$	0	0
$221$	4211.41	1.28186
$222$	0	0
$223$	−1051.16	−0.315653	−0.157827	−	0.987467i	$-0.550449\pi$
$223$	−1051.16	−0.315653	−0.157827	+	0.987467i	$0.550449\pi$
$224$	0	0
$225$	−5426.97	−1.60799
$226$	0	0
$227$	−3430.64	−1.00308	−0.501541	−	0.865134i	$-0.667233\pi$
$227$	−3430.64	−1.00308	−0.501541	+	0.865134i	$0.667233\pi$
$228$	0	0
$229$	5907.75	1.70478	0.852392	−	0.522904i	$-0.175151\pi$
$229$	5907.75	1.70478	0.852392	+	0.522904i	$0.175151\pi$
$230$	0	0
$231$	697.937	0.198792
$232$	0	0
$233$	−2109.67	−0.593171	−0.296586	−	0.955006i	$-0.595848\pi$
$233$	−2109.67	−0.593171	−0.296586	+	0.955006i	$0.595848\pi$
$234$	0	0
$235$	8355.91	2.31949
$236$	0	0
$237$	1408.59	0.386067
$238$	0	0
$239$	2898.33	0.784426	0.392213	−	0.919875i	$-0.371710\pi$
$239$	2898.33	0.784426	0.392213	+	0.919875i	$0.371710\pi$
$240$	0	0
$241$	−734.884	−0.196423	−0.0982117	−	0.995166i	$-0.531312\pi$
$241$	−734.884	−0.196423	−0.0982117	+	0.995166i	$0.531312\pi$
$242$	0	0
$243$	−2661.72	−0.702672
$244$	0	0
$245$	4335.97	1.13067
$246$	0	0
$247$	1174.87	0.302652
$248$	0	0
$249$	−968.162	−0.246405
$250$	0	0
$251$	−2758.13	−0.693593	−0.346796	−	0.937940i	$-0.612730\pi$
$251$	−2758.13	−0.693593	−0.346796	+	0.937940i	$0.612730\pi$
$252$	0	0
$253$	−7347.40	−1.82580
$254$	0	0
$255$	1672.84	0.410813
$256$	0	0
$257$	486.309	0.118036	0.0590178	−	0.998257i	$-0.481203\pi$
$257$	486.309	0.118036	0.0590178	+	0.998257i	$0.481203\pi$
$258$	0	0
$259$	4164.92	0.999211
$260$	0	0
$261$	1074.89	0.254919
$262$	0	0
$263$	3381.74	0.792878	0.396439	−	0.918061i	$-0.370246\pi$
$263$	3381.74	0.792878	0.396439	+	0.918061i	$0.370246\pi$
$264$	0	0
$265$	8062.06	1.86886
$266$	0	0
$267$	623.369	0.142882
$268$	0	0
$269$	2628.18	0.595699	0.297849	−	0.954613i	$-0.403731\pi$
$269$	2628.18	0.595699	0.297849	+	0.954613i	$0.403731\pi$
$270$	0	0
$271$	1104.06	0.247480	0.123740	−	0.992315i	$-0.460511\pi$
$271$	1104.06	0.247480	0.123740	+	0.992315i	$0.460511\pi$
$272$	0	0
$273$	−855.411	−0.189640
$274$	0	0
$275$	−10853.8	−2.38003
$276$	0	0
$277$	549.699	0.119235	0.0596177	−	0.998221i	$-0.481012\pi$
$277$	549.699	0.119235	0.0596177	+	0.998221i	$0.481012\pi$
$278$	0	0
$279$	−2312.20	−0.496157
$280$	0	0
$281$	−182.930	−0.0388352	−0.0194176	−	0.999811i	$-0.506181\pi$
$281$	−182.930	−0.0388352	−0.0194176	+	0.999811i	$0.506181\pi$
$282$	0	0
$283$	−5218.27	−1.09609	−0.548046	−	0.836448i	$-0.684628\pi$
$283$	−5218.27	−1.09609	−0.548046	+	0.836448i	$0.684628\pi$
$284$	0	0
$285$	466.677	0.0969949
$286$	0	0
$287$	−1281.31	−0.263532
$288$	0	0
$289$	−274.434	−0.0558587
$290$	0	0
$291$	1187.82	0.239283
$292$	0	0
$293$	−6160.17	−1.22826	−0.614131	−	0.789204i	$-0.710494\pi$
$293$	−6160.17	−1.22826	−0.614131	+	0.789204i	$0.710494\pi$
$294$	0	0
$295$	−2938.87	−0.580026
$296$	0	0
$297$	−3509.18	−0.685600
$298$	0	0
$299$	9005.19	1.74175
$300$	0	0
$301$	−4672.20	−0.894687
$302$	0	0
$303$	−22.1925	−0.00420768
$304$	0	0
$305$	−8793.75	−1.65091
$306$	0	0
$307$	−6620.89	−1.23086	−0.615430	−	0.788192i	$-0.711018\pi$
$307$	−6620.89	−1.23086	−0.615430	+	0.788192i	$0.711018\pi$
$308$	0	0
$309$	−1721.59	−0.316952
$310$	0	0
$311$	8280.27	1.50975	0.754873	−	0.655870i	$-0.227698\pi$
$311$	8280.27	1.50975	0.754873	+	0.655870i	$0.227698\pi$
$312$	0	0
$313$	3338.77	0.602934	0.301467	−	0.953477i	$-0.402524\pi$
$313$	3338.77	0.602934	0.301467	+	0.953477i	$0.402524\pi$
$314$	0	0
$315$	4832.55	0.864392
$316$	0	0
$317$	−9587.72	−1.69874	−0.849369	−	0.527800i	$-0.823017\pi$
$317$	−9587.72	−1.69874	−0.849369	+	0.527800i	$0.823017\pi$
$318$	0	0
$319$	2149.74	0.377311
$320$	0	0
$321$	−234.644	−0.0407992
$322$	0	0
$323$	1294.03	0.222916
$324$	0	0
$325$	13302.7	2.27047
$326$	0	0
$327$	1836.20	0.310527
$328$	0	0
$329$	−4706.19	−0.788635
$330$	0	0
$331$	−11195.2	−1.85905	−0.929526	−	0.368756i	$-0.879784\pi$
$331$	−11195.2	−1.85905	−0.929526	+	0.368756i	$0.879784\pi$
$332$	0	0
$333$	−10114.9	−1.66454
$334$	0	0
$335$	−11613.0	−1.89399
$336$	0	0
$337$	5991.21	0.968432	0.484216	−	0.874948i	$-0.339105\pi$
$337$	5991.21	0.968432	0.484216	+	0.874948i	$0.339105\pi$
$338$	0	0
$339$	801.470	0.128407
$340$	0	0
$341$	−4624.33	−0.734374
$342$	0	0
$343$	−6004.91	−0.945291
$344$	0	0
$345$	3577.00	0.558201
$346$	0	0
$347$	−6755.35	−1.04509	−0.522545	−	0.852612i	$-0.675017\pi$
$347$	−6755.35	−1.04509	−0.522545	+	0.852612i	$0.675017\pi$
$348$	0	0
$349$	2358.14	0.361685	0.180843	−	0.983512i	$-0.442117\pi$
$349$	2358.14	0.361685	0.180843	+	0.983512i	$0.442117\pi$
$350$	0	0
$351$	4300.95	0.654039
$352$	0	0
$353$	−5950.00	−0.897129	−0.448565	−	0.893750i	$-0.648065\pi$
$353$	−5950.00	−0.897129	−0.448565	+	0.893750i	$0.648065\pi$
$354$	0	0
$355$	−8694.96	−1.29995
$356$	0	0
$357$	−942.173	−0.139678
$358$	0	0
$359$	1892.90	0.278283	0.139141	−	0.990273i	$-0.455566\pi$
$359$	1892.90	0.278283	0.139141	+	0.990273i	$0.455566\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−1617.32	−0.233850
$364$	0	0
$365$	13371.9	1.91759
$366$	0	0
$367$	3643.63	0.518245	0.259122	−	0.965844i	$-0.416567\pi$
$367$	3643.63	0.518245	0.259122	+	0.965844i	$0.416567\pi$
$368$	0	0
$369$	3111.79	0.439006
$370$	0	0
$371$	−4540.69	−0.635421
$372$	0	0
$373$	−4820.77	−0.669195	−0.334598	−	0.942361i	$-0.608600\pi$
$373$	−4820.77	−0.669195	−0.334598	+	0.942361i	$0.608600\pi$
$374$	0	0
$375$	2213.81	0.304854
$376$	0	0
$377$	−2634.78	−0.359942
$378$	0	0
$379$	8138.02	1.10296	0.551480	−	0.834188i	$-0.314063\pi$
$379$	8138.02	1.10296	0.551480	+	0.834188i	$0.314063\pi$
$380$	0	0
$381$	−2517.61	−0.338532
$382$	0	0
$383$	904.971	0.120736	0.0603679	−	0.998176i	$-0.480773\pi$
$383$	904.971	0.120736	0.0603679	+	0.998176i	$0.480773\pi$
$384$	0	0
$385$	9664.96	1.27941
$386$	0	0
$387$	11346.8	1.49042
$388$	0	0
$389$	−8885.27	−1.15810	−0.579050	−	0.815292i	$-0.696576\pi$
$389$	−8885.27	−1.15810	−0.579050	+	0.815292i	$0.696576\pi$
$390$	0	0
$391$	9918.56	1.28287
$392$	0	0
$393$	442.109	0.0567467
$394$	0	0
$395$	19506.1	2.48470
$396$	0	0
$397$	981.841	0.124124	0.0620619	−	0.998072i	$-0.480232\pi$
$397$	981.841	0.124124	0.0620619	+	0.998072i	$0.480232\pi$
$398$	0	0
$399$	−262.841	−0.0329787
$400$	0	0
$401$	−15512.7	−1.93184	−0.965919	−	0.258845i	$-0.916658\pi$
$401$	−15512.7	−1.93184	−0.965919	+	0.258845i	$0.916658\pi$
$402$	0	0
$403$	5667.71	0.700568
$404$	0	0
$405$	−10853.1	−1.33159
$406$	0	0
$407$	−20229.5	−2.46373
$408$	0	0
$409$	−13347.9	−1.61371	−0.806857	−	0.590747i	$-0.798833\pi$
$409$	−13347.9	−1.61371	−0.806857	+	0.590747i	$0.798833\pi$
$410$	0	0
$411$	2605.27	0.312673
$412$	0	0
$413$	1655.22	0.197211
$414$	0	0
$415$	−13407.0	−1.58584
$416$	0	0
$417$	729.588	0.0856789
$418$	0	0
$419$	−1557.12	−0.181552	−0.0907758	−	0.995871i	$-0.528935\pi$
$419$	−1557.12	−0.181552	−0.0907758	+	0.995871i	$0.528935\pi$
$420$	0	0
$421$	−1281.41	−0.148342	−0.0741710	−	0.997246i	$-0.523631\pi$
$421$	−1281.41	−0.148342	−0.0741710	+	0.997246i	$0.523631\pi$
$422$	0	0
$423$	11429.4	1.31375
$424$	0	0
$425$	14652.0	1.67229
$426$	0	0
$427$	4952.79	0.561317
$428$	0	0
$429$	4154.82	0.467591
$430$	0	0
$431$	9742.61	1.08883	0.544414	−	0.838817i	$-0.316752\pi$
$431$	9742.61	1.08883	0.544414	+	0.838817i	$0.316752\pi$
$432$	0	0
$433$	−5262.21	−0.584031	−0.292016	−	0.956414i	$-0.594326\pi$
$433$	−5262.21	−0.584031	−0.292016	+	0.956414i	$0.594326\pi$
$434$	0	0
$435$	−1046.58	−0.115355
$436$	0	0
$437$	2767.01	0.302892
$438$	0	0
$439$	−15015.7	−1.63249	−0.816243	−	0.577708i	$-0.803947\pi$
$439$	−15015.7	−1.63249	−0.816243	+	0.577708i	$0.803947\pi$
$440$	0	0
$441$	5930.84	0.640411
$442$	0	0
$443$	4296.96	0.460846	0.230423	−	0.973091i	$-0.425989\pi$
$443$	4296.96	0.460846	0.230423	+	0.973091i	$0.425989\pi$
$444$	0	0
$445$	8632.36	0.919580
$446$	0	0
$447$	−1321.79	−0.139862
$448$	0	0
$449$	6749.23	0.709389	0.354695	−	0.934982i	$-0.384585\pi$
$449$	6749.23	0.709389	0.354695	+	0.934982i	$0.384585\pi$
$450$	0	0
$451$	6223.48	0.649783
$452$	0	0
$453$	197.329	0.0204665
$454$	0	0
$455$	−11845.7	−1.22051
$456$	0	0
$457$	11399.9	1.16689	0.583443	−	0.812154i	$-0.301705\pi$
$457$	11399.9	1.16689	0.583443	+	0.812154i	$0.301705\pi$
$458$	0	0
$459$	4737.19	0.481727
$460$	0	0
$461$	5799.27	0.585898	0.292949	−	0.956128i	$-0.405363\pi$
$461$	5799.27	0.585898	0.292949	+	0.956128i	$0.405363\pi$
$462$	0	0
$463$	620.160	0.0622490	0.0311245	−	0.999516i	$-0.490091\pi$
$463$	620.160	0.0622490	0.0311245	+	0.999516i	$0.490091\pi$
$464$	0	0
$465$	2251.30	0.224520
$466$	0	0
$467$	−9426.16	−0.934027	−0.467014	−	0.884250i	$-0.654670\pi$
$467$	−9426.16	−0.934027	−0.467014	+	0.884250i	$0.654670\pi$
$468$	0	0
$469$	6540.65	0.643964
$470$	0	0
$471$	3138.20	0.307007
$472$	0	0
$473$	22693.3	2.20601
$474$	0	0
$475$	4087.50	0.394836
$476$	0	0
$477$	11027.5	1.05852
$478$	0	0
$479$	15813.6	1.50844	0.754218	−	0.656625i	$-0.228016\pi$
$479$	15813.6	1.50844	0.754218	+	0.656625i	$0.228016\pi$
$480$	0	0
$481$	24793.8	2.35031
$482$	0	0
$483$	−2014.63	−0.189791
$484$	0	0
$485$	16448.8	1.54001
$486$	0	0
$487$	−6832.36	−0.635736	−0.317868	−	0.948135i	$-0.602967\pi$
$487$	−6832.36	−0.635736	−0.317868	+	0.948135i	$0.602967\pi$
$488$	0	0
$489$	−3961.92	−0.366389
$490$	0	0
$491$	15084.5	1.38646	0.693232	−	0.720714i	$-0.256186\pi$
$491$	15084.5	1.38646	0.693232	+	0.720714i	$0.256186\pi$
$492$	0	0
$493$	−2902.02	−0.265112
$494$	0	0
$495$	−23472.2	−2.13131
$496$	0	0
$497$	4897.16	0.441987
$498$	0	0
$499$	−21771.2	−1.95313	−0.976564	−	0.215226i	$-0.930951\pi$
$499$	−21771.2	−1.95313	−0.976564	+	0.215226i	$0.930951\pi$
$500$	0	0
$501$	4172.65	0.372096
$502$	0	0
$503$	21300.9	1.88819	0.944097	−	0.329668i	$-0.106937\pi$
$503$	21300.9	1.88819	0.944097	+	0.329668i	$0.106937\pi$
$504$	0	0
$505$	−307.320	−0.0270803
$506$	0	0
$507$	−2166.30	−0.189761
$508$	0	0
$509$	15897.9	1.38440	0.692202	−	0.721704i	$-0.256641\pi$
$509$	15897.9	1.38440	0.692202	+	0.721704i	$0.256641\pi$
$510$	0	0
$511$	−7531.31	−0.651987
$512$	0	0
$513$	1321.55	0.113738
$514$	0	0
$515$	−23840.5	−2.03988
$516$	0	0
$517$	22858.5	1.94452
$518$	0	0
$519$	4548.63	0.384707
$520$	0	0
$521$	−13864.6	−1.16587	−0.582937	−	0.812517i	$-0.698097\pi$
$521$	−13864.6	−1.16587	−0.582937	+	0.812517i	$0.698097\pi$
$522$	0	0
$523$	20503.0	1.71421	0.857106	−	0.515141i	$-0.172260\pi$
$523$	20503.0	1.71421	0.857106	+	0.515141i	$0.172260\pi$
$524$	0	0
$525$	−2976.07	−0.247402
$526$	0	0
$527$	6242.57	0.515997
$528$	0	0
$529$	9041.66	0.743130
$530$	0	0
$531$	−4019.86	−0.328525
$532$	0	0
$533$	−7627.67	−0.619871
$534$	0	0
$535$	−3249.33	−0.262581
$536$	0	0
$537$	106.611	0.00856726
$538$	0	0
$539$	11861.5	0.947887
$540$	0	0
$541$	15933.8	1.26626	0.633131	−	0.774045i	$-0.281769\pi$
$541$	15933.8	1.26626	0.633131	+	0.774045i	$0.281769\pi$
$542$	0	0
$543$	4501.37	0.355750
$544$	0	0
$545$	25427.6	1.99853
$546$	0	0
$547$	−20542.0	−1.60569	−0.802847	−	0.596185i	$-0.796682\pi$
$547$	−20542.0	−1.60569	−0.802847	+	0.596185i	$0.796682\pi$
$548$	0	0
$549$	−12028.3	−0.935073
$550$	0	0
$551$	−809.584	−0.0625943
$552$	0	0
$553$	−10986.2	−0.844808
$554$	0	0
$555$	9848.49	0.753235
$556$	0	0
$557$	−12338.5	−0.938599	−0.469299	−	0.883039i	$-0.655493\pi$
$557$	−12338.5	−0.938599	−0.469299	+	0.883039i	$0.655493\pi$
$558$	0	0
$559$	−27813.6	−2.10446
$560$	0	0
$561$	4576.23	0.344401
$562$	0	0
$563$	−7611.22	−0.569759	−0.284880	−	0.958563i	$-0.591954\pi$
$563$	−7611.22	−0.569759	−0.284880	+	0.958563i	$0.591954\pi$
$564$	0	0
$565$	11098.7	0.826417
$566$	0	0
$567$	6112.64	0.452746
$568$	0	0
$569$	9237.35	0.680579	0.340290	−	0.940321i	$-0.389475\pi$
$569$	9237.35	0.680579	0.340290	+	0.940321i	$0.389475\pi$
$570$	0	0
$571$	−7061.48	−0.517537	−0.258769	−	0.965939i	$-0.583317\pi$
$571$	−7061.48	−0.517537	−0.258769	+	0.965939i	$0.583317\pi$
$572$	0	0
$573$	4380.00	0.319332
$574$	0	0
$575$	31330.0	2.27226
$576$	0	0
$577$	−12167.1	−0.877856	−0.438928	−	0.898522i	$-0.644642\pi$
$577$	−12167.1	−0.877856	−0.438928	+	0.898522i	$0.644642\pi$
$578$	0	0
$579$	−3871.40	−0.277876
$580$	0	0
$581$	7551.07	0.539193
$582$	0	0
$583$	22054.6	1.56674
$584$	0	0
$585$	28768.2	2.03320
$586$	0	0
$587$	11396.4	0.801327	0.400663	−	0.916225i	$-0.368780\pi$
$587$	11396.4	0.801327	0.400663	+	0.916225i	$0.368780\pi$
$588$	0	0
$589$	1741.51	0.121829
$590$	0	0
$591$	−22.8908	−0.00159324
$592$	0	0
$593$	−10401.1	−0.720276	−0.360138	−	0.932899i	$-0.617270\pi$
$593$	−10401.1	−0.720276	−0.360138	+	0.932899i	$0.617270\pi$
$594$	0	0
$595$	−13047.1	−0.898958
$596$	0	0
$597$	817.192	0.0560225
$598$	0	0
$599$	2296.79	0.156668	0.0783340	−	0.996927i	$-0.475040\pi$
$599$	2296.79	0.156668	0.0783340	+	0.996927i	$0.475040\pi$
$600$	0	0
$601$	−7376.50	−0.500655	−0.250328	−	0.968161i	$-0.580538\pi$
$601$	−7376.50	−0.500655	−0.250328	+	0.968161i	$0.580538\pi$
$602$	0	0
$603$	−15884.5	−1.07275
$604$	0	0
$605$	−22396.6	−1.50504
$606$	0	0
$607$	−11793.0	−0.788570	−0.394285	−	0.918988i	$-0.629008\pi$
$607$	−11793.0	−0.788570	−0.394285	+	0.918988i	$0.629008\pi$
$608$	0	0
$609$	589.451	0.0392213
$610$	0	0
$611$	−28016.0	−1.85500
$612$	0	0
$613$	−20918.1	−1.37826	−0.689131	−	0.724637i	$-0.742008\pi$
$613$	−20918.1	−1.37826	−0.689131	+	0.724637i	$0.742008\pi$
$614$	0	0
$615$	−3029.83	−0.198658
$616$	0	0
$617$	−3951.13	−0.257806	−0.128903	−	0.991657i	$-0.541146\pi$
$617$	−3951.13	−0.257806	−0.128903	+	0.991657i	$0.541146\pi$
$618$	0	0
$619$	22159.8	1.43890	0.719450	−	0.694544i	$-0.244394\pi$
$619$	22159.8	1.43890	0.719450	+	0.694544i	$0.244394\pi$
$620$	0	0
$621$	10129.4	0.654557
$622$	0	0
$623$	−4861.90	−0.312661
$624$	0	0
$625$	3765.14	0.240969
$626$	0	0
$627$	1276.64	0.0813146
$628$	0	0
$629$	27308.6	1.73110
$630$	0	0
$631$	11645.2	0.734687	0.367344	−	0.930085i	$-0.380267\pi$
$631$	11645.2	0.734687	0.367344	+	0.930085i	$0.380267\pi$
$632$	0	0
$633$	−5316.35	−0.333817
$634$	0	0
$635$	−34863.6	−2.17877
$636$	0	0
$637$	−14537.8	−0.904252
$638$	0	0
$639$	−11893.2	−0.736286
$640$	0	0
$641$	−21970.1	−1.35377	−0.676885	−	0.736089i	$-0.736671\pi$
$641$	−21970.1	−1.35377	−0.676885	+	0.736089i	$0.736671\pi$
$642$	0	0
$643$	14047.2	0.861537	0.430769	−	0.902462i	$-0.358243\pi$
$643$	14047.2	0.861537	0.430769	+	0.902462i	$0.358243\pi$
$644$	0	0
$645$	−11048.0	−0.674442
$646$	0	0
$647$	20700.0	1.25780	0.628902	−	0.777485i	$-0.283505\pi$
$647$	20700.0	1.25780	0.628902	+	0.777485i	$0.283505\pi$
$648$	0	0
$649$	−8039.59	−0.486258
$650$	0	0
$651$	−1267.97	−0.0763377
$652$	0	0
$653$	−17197.8	−1.03063	−0.515315	−	0.857001i	$-0.672325\pi$
$653$	−17197.8	−1.03063	−0.515315	+	0.857001i	$0.672325\pi$
$654$	0	0
$655$	6122.29	0.365218
$656$	0	0
$657$	18290.5	1.08612
$658$	0	0
$659$	23084.0	1.36453	0.682264	−	0.731106i	$-0.260995\pi$
$659$	23084.0	1.36453	0.682264	+	0.731106i	$0.260995\pi$
$660$	0	0
$661$	9283.22	0.546256	0.273128	−	0.961978i	$-0.411942\pi$
$661$	9283.22	0.546256	0.273128	+	0.961978i	$0.411942\pi$
$662$	0	0
$663$	−5608.76	−0.328547
$664$	0	0
$665$	−3639.79	−0.212248
$666$	0	0
$667$	−6205.33	−0.360227
$668$	0	0
$669$	1399.93	0.0809036
$670$	0	0
$671$	−24056.2	−1.38402
$672$	0	0
$673$	11565.0	0.662403	0.331202	−	0.943560i	$-0.392546\pi$
$673$	11565.0	0.662403	0.331202	+	0.943560i	$0.392546\pi$
$674$	0	0
$675$	14963.5	0.853251
$676$	0	0
$677$	−8808.39	−0.500050	−0.250025	−	0.968239i	$-0.580439\pi$
$677$	−8808.39	−0.500050	−0.250025	+	0.968239i	$0.580439\pi$
$678$	0	0
$679$	−9264.27	−0.523608
$680$	0	0
$681$	4568.93	0.257095
$682$	0	0
$683$	−16282.3	−0.912186	−0.456093	−	0.889932i	$-0.650752\pi$
$683$	−16282.3	−0.912186	−0.456093	+	0.889932i	$0.650752\pi$
$684$	0	0
$685$	36077.6	2.01234
$686$	0	0
$687$	−7867.95	−0.436945
$688$	0	0
$689$	−27030.8	−1.49462
$690$	0	0
$691$	21900.4	1.20569	0.602845	−	0.797859i	$-0.294034\pi$
$691$	21900.4	1.20569	0.602845	+	0.797859i	$0.294034\pi$
$692$	0	0
$693$	13220.0	0.724654
$694$	0	0
$695$	10103.3	0.551423
$696$	0	0
$697$	−8401.33	−0.456561
$698$	0	0
$699$	2809.66	0.152033
$700$	0	0
$701$	12286.6	0.661994	0.330997	−	0.943632i	$-0.392615\pi$
$701$	12286.6	0.661994	0.330997	+	0.943632i	$0.392615\pi$
$702$	0	0
$703$	7618.34	0.408722
$704$	0	0
$705$	−11128.4	−0.594496
$706$	0	0
$707$	173.088	0.00920743
$708$	0	0
$709$	12918.9	0.684315	0.342158	−	0.939643i	$-0.388842\pi$
$709$	12918.9	0.684315	0.342158	+	0.939643i	$0.388842\pi$
$710$	0	0
$711$	26680.8	1.40733
$712$	0	0
$713$	13348.4	0.701122
$714$	0	0
$715$	57535.6	3.00938
$716$	0	0
$717$	−3860.00	−0.201052
$718$	0	0
$719$	−15819.9	−0.820558	−0.410279	−	0.911960i	$-0.634569\pi$
$719$	−15819.9	−0.820558	−0.410279	+	0.911960i	$0.634569\pi$
$720$	0	0
$721$	13427.4	0.693567
$722$	0	0
$723$	978.719	0.0503443
$724$	0	0
$725$	−9166.69	−0.469576
$726$	0	0
$727$	1793.81	0.0915112	0.0457556	−	0.998953i	$-0.485430\pi$
$727$	1793.81	0.0915112	0.0457556	+	0.998953i	$0.485430\pi$
$728$	0	0
$729$	−12344.0	−0.627140
$730$	0	0
$731$	−30634.7	−1.55002
$732$	0	0
$733$	−10370.0	−0.522541	−0.261271	−	0.965266i	$-0.584142\pi$
$733$	−10370.0	−0.522541	−0.261271	+	0.965266i	$0.584142\pi$
$734$	0	0
$735$	−5774.65	−0.289797
$736$	0	0
$737$	−31768.6	−1.58780
$738$	0	0
$739$	−29414.1	−1.46416	−0.732081	−	0.681218i	$-0.761451\pi$
$739$	−29414.1	−1.46416	−0.732081	+	0.681218i	$0.761451\pi$
$740$	0	0
$741$	−1564.69	−0.0775714
$742$	0	0
$743$	−14932.0	−0.737286	−0.368643	−	0.929571i	$-0.620177\pi$
$743$	−14932.0	−0.737286	−0.368643	+	0.929571i	$0.620177\pi$
$744$	0	0
$745$	−18304.0	−0.900142
$746$	0	0
$747$	−18338.4	−0.898218
$748$	0	0
$749$	1830.08	0.0892785
$750$	0	0
$751$	21916.9	1.06492	0.532462	−	0.846454i	$-0.321267\pi$
$751$	21916.9	1.06492	0.532462	+	0.846454i	$0.321267\pi$
$752$	0	0
$753$	3673.28	0.177771
$754$	0	0
$755$	2732.59	0.131721
$756$	0	0
$757$	26973.4	1.29506	0.647532	−	0.762038i	$-0.275801\pi$
$757$	26973.4	1.29506	0.647532	+	0.762038i	$0.275801\pi$
$758$	0	0
$759$	9785.27	0.467962
$760$	0	0
$761$	−16145.3	−0.769076	−0.384538	−	0.923109i	$-0.625639\pi$
$761$	−16145.3	−0.769076	−0.384538	+	0.923109i	$0.625639\pi$
$762$	0	0
$763$	−14321.3	−0.679508
$764$	0	0
$765$	31686.1	1.49753
$766$	0	0
$767$	9853.55	0.463874
$768$	0	0
$769$	−23777.1	−1.11499	−0.557494	−	0.830181i	$-0.688237\pi$
$769$	−23777.1	−1.11499	−0.557494	+	0.830181i	$0.688237\pi$
$770$	0	0
$771$	−647.667	−0.0302531
$772$	0	0
$773$	−27962.2	−1.30107	−0.650537	−	0.759475i	$-0.725456\pi$
$773$	−27962.2	−1.30107	−0.650537	+	0.759475i	$0.725456\pi$
$774$	0	0
$775$	19718.6	0.913951
$776$	0	0
$777$	−5546.84	−0.256103
$778$	0	0
$779$	−2343.74	−0.107796
$780$	0	0
$781$	−23786.0	−1.08980
$782$	0	0
$783$	−2963.72	−0.135268
$784$	0	0
$785$	43457.5	1.97588
$786$	0	0
$787$	−19000.4	−0.860600	−0.430300	−	0.902686i	$-0.641592\pi$
$787$	−19000.4	−0.860600	−0.430300	+	0.902686i	$0.641592\pi$
$788$	0	0
$789$	−4503.80	−0.203219
$790$	0	0
$791$	−6250.98	−0.280985
$792$	0	0
$793$	29484.0	1.32031
$794$	0	0
$795$	−10737.1	−0.478999
$796$	0	0
$797$	19991.0	0.888480	0.444240	−	0.895908i	$-0.353474\pi$
$797$	19991.0	0.888480	0.444240	+	0.895908i	$0.353474\pi$
$798$	0	0
$799$	−30857.6	−1.36629
$800$	0	0
$801$	11807.5	0.520848
$802$	0	0
$803$	36580.3	1.60759
$804$	0	0
$805$	−27898.4	−1.22148
$806$	0	0
$807$	−3500.21	−0.152681
$808$	0	0
$809$	−40779.8	−1.77224	−0.886120	−	0.463455i	$-0.846609\pi$
$809$	−40779.8	−1.77224	−0.886120	+	0.463455i	$0.846609\pi$
$810$	0	0
$811$	23160.4	1.00280	0.501400	−	0.865216i	$-0.332819\pi$
$811$	23160.4	1.00280	0.501400	+	0.865216i	$0.332819\pi$
$812$	0	0
$813$	−1470.39	−0.0634304
$814$	0	0
$815$	−54864.3	−2.35805
$816$	0	0
$817$	−8546.24	−0.365967
$818$	0	0
$819$	−16202.8	−0.691295
$820$	0	0
$821$	−29453.3	−1.25204	−0.626021	−	0.779806i	$-0.715318\pi$
$821$	−29453.3	−1.25204	−0.626021	+	0.779806i	$0.715318\pi$
$822$	0	0
$823$	−43227.2	−1.83087	−0.915435	−	0.402465i	$-0.868153\pi$
$823$	−43227.2	−1.83087	−0.915435	+	0.402465i	$0.868153\pi$
$824$	0	0
$825$	14455.1	0.610014
$826$	0	0
$827$	34232.7	1.43940	0.719702	−	0.694283i	$-0.244278\pi$
$827$	34232.7	1.43940	0.719702	+	0.694283i	$0.244278\pi$
$828$	0	0
$829$	−32410.2	−1.35785	−0.678923	−	0.734209i	$-0.737553\pi$
$829$	−32410.2	−1.35785	−0.678923	+	0.734209i	$0.737553\pi$
$830$	0	0
$831$	−732.089	−0.0305606
$832$	0	0
$833$	−16012.3	−0.666020
$834$	0	0
$835$	57782.5	2.39478
$836$	0	0
$837$	6375.29	0.263276
$838$	0	0
$839$	−6363.50	−0.261851	−0.130925	−	0.991392i	$-0.541795\pi$
$839$	−6363.50	−0.261851	−0.130925	+	0.991392i	$0.541795\pi$
$840$	0	0
$841$	−22573.4	−0.925557
$842$	0	0
$843$	243.626	0.00995366
$844$	0	0
$845$	−29998.8	−1.22129
$846$	0	0
$847$	12614.1	0.511720
$848$	0	0
$849$	6949.69	0.280934
$850$	0	0
$851$	58393.4	2.35217
$852$	0	0
$853$	−22798.2	−0.915117	−0.457558	−	0.889180i	$-0.651276\pi$
$853$	−22798.2	−0.915117	−0.457558	+	0.889180i	$0.651276\pi$
$854$	0	0
$855$	8839.56	0.353575
$856$	0	0
$857$	26426.8	1.05335	0.526675	−	0.850067i	$-0.323438\pi$
$857$	26426.8	1.05335	0.526675	+	0.850067i	$0.323438\pi$
$858$	0	0
$859$	−22484.5	−0.893087	−0.446544	−	0.894762i	$-0.647345\pi$
$859$	−22484.5	−0.893087	−0.446544	+	0.894762i	$0.647345\pi$
$860$	0	0
$861$	1706.46	0.0675445
$862$	0	0
$863$	−22062.6	−0.870242	−0.435121	−	0.900372i	$-0.643294\pi$
$863$	−22062.6	−0.870242	−0.435121	+	0.900372i	$0.643294\pi$
$864$	0	0
$865$	62989.0	2.47594
$866$	0	0
$867$	365.491	0.0143169
$868$	0	0
$869$	53360.9	2.08302
$870$	0	0
$871$	38936.5	1.51471
$872$	0	0
$873$	22499.1	0.872256
$874$	0	0
$875$	−17266.3	−0.667095
$876$	0	0
$877$	−17.2794	−0.000665316 0	−0.000332658	−	1.00000i	$-0.500106\pi$
$877$	−17.2794	−0.000665316 0	−0.000332658		1.00000i	$0.500106\pi$
$878$	0	0
$879$	8204.12	0.314810
$880$	0	0
$881$	19338.2	0.739524	0.369762	−	0.929126i	$-0.379439\pi$
$881$	19338.2	0.739524	0.369762	+	0.929126i	$0.379439\pi$
$882$	0	0
$883$	−35899.9	−1.36821	−0.684105	−	0.729384i	$-0.739807\pi$
$883$	−35899.9	−1.36821	−0.684105	+	0.729384i	$0.739807\pi$
$884$	0	0
$885$	3913.99	0.148664
$886$	0	0
$887$	51076.9	1.93348	0.966738	−	0.255767i	$-0.0823280\pi$
$887$	51076.9	1.93348	0.966738	+	0.255767i	$0.0823280\pi$
$888$	0	0
$889$	19635.8	0.740791
$890$	0	0
$891$	−29689.7	−1.11632
$892$	0	0
$893$	−8608.43	−0.322587
$894$	0	0
$895$	1476.34	0.0551383
$896$	0	0
$897$	−11993.1	−0.446420
$898$	0	0
$899$	−3905.53	−0.144891
$900$	0	0
$901$	−29772.4	−1.10085
$902$	0	0
$903$	6222.43	0.229313
$904$	0	0
$905$	62334.5	2.28958
$906$	0	0
$907$	−18999.5	−0.695555	−0.347778	−	0.937577i	$-0.613064\pi$
$907$	−18999.5	−0.695555	−0.347778	+	0.937577i	$0.613064\pi$
$908$	0	0
$909$	−420.360	−0.0153382
$910$	0	0
$911$	17147.0	0.623605	0.311802	−	0.950147i	$-0.399067\pi$
$911$	17147.0	0.623605	0.311802	+	0.950147i	$0.399067\pi$
$912$	0	0
$913$	−36676.3	−1.32947
$914$	0	0
$915$	11711.5	0.423137
$916$	0	0
$917$	−3448.18	−0.124176
$918$	0	0
$919$	−10013.4	−0.359425	−0.179712	−	0.983719i	$-0.557517\pi$
$919$	−10013.4	−0.359425	−0.179712	+	0.983719i	$0.557517\pi$
$920$	0	0
$921$	8817.70	0.315476
$922$	0	0
$923$	29152.8	1.03963
$924$	0	0
$925$	86260.3	3.06619
$926$	0	0
$927$	−32609.6	−1.15538
$928$	0	0
$929$	−22893.7	−0.808524	−0.404262	−	0.914643i	$-0.632472\pi$
$929$	−22893.7	−0.808524	−0.404262	+	0.914643i	$0.632472\pi$
$930$	0	0
$931$	−4467.00	−0.157250
$932$	0	0
$933$	−11027.7	−0.386956
$934$	0	0
$935$	63371.3	2.21654
$936$	0	0
$937$	1003.65	0.0349922	0.0174961	−	0.999847i	$-0.494431\pi$
$937$	1003.65	0.0349922	0.0174961	+	0.999847i	$0.494431\pi$
$938$	0	0
$939$	−4446.58	−0.154535
$940$	0	0
$941$	−14679.7	−0.508550	−0.254275	−	0.967132i	$-0.581837\pi$
$941$	−14679.7	−0.508550	−0.254275	+	0.967132i	$0.581837\pi$
$942$	0	0
$943$	−17964.4	−0.620362
$944$	0	0
$945$	−13324.5	−0.458674
$946$	0	0
$947$	−38570.8	−1.32353	−0.661765	−	0.749711i	$-0.730192\pi$
$947$	−38570.8	−1.32353	−0.661765	+	0.749711i	$0.730192\pi$
$948$	0	0
$949$	−44833.9	−1.53358
$950$	0	0
$951$	12768.9	0.435395
$952$	0	0
$953$	2208.75	0.0750770	0.0375385	−	0.999295i	$-0.488048\pi$
$953$	2208.75	0.0750770	0.0375385	+	0.999295i	$0.488048\pi$
$954$	0	0
$955$	60653.8	2.05520
$956$	0	0
$957$	−2863.02	−0.0967068
$958$	0	0
$959$	−20319.5	−0.684204
$960$	0	0
$961$	−21389.8	−0.717994
$962$	0	0
$963$	−4444.51	−0.148725
$964$	0	0
$965$	−53610.8	−1.78839
$966$	0	0
$967$	5440.71	0.180932	0.0904661	−	0.995900i	$-0.471164\pi$
$967$	5440.71	0.180932	0.0904661	+	0.995900i	$0.471164\pi$
$968$	0	0
$969$	−1723.39	−0.0571346
$970$	0	0
$971$	−25018.4	−0.826857	−0.413428	−	0.910537i	$-0.635669\pi$
$971$	−25018.4	−0.826857	−0.413428	+	0.910537i	$0.635669\pi$
$972$	0	0
$973$	−5690.34	−0.187486
$974$	0	0
$975$	−17716.6	−0.581932
$976$	0	0
$977$	−45055.8	−1.47540	−0.737698	−	0.675131i	$-0.764087\pi$
$977$	−45055.8	−1.47540	−0.737698	+	0.675131i	$0.764087\pi$
$978$	0	0
$979$	23614.7	0.770920
$980$	0	0
$981$	34780.5	1.13196
$982$	0	0
$983$	−13675.5	−0.443725	−0.221862	−	0.975078i	$-0.571214\pi$
$983$	−13675.5	−0.443725	−0.221862	+	0.975078i	$0.571214\pi$
$984$	0	0
$985$	−316.990	−0.0102539
$986$	0	0
$987$	6267.71	0.202131
$988$	0	0
$989$	−65505.5	−2.10612
$990$	0	0
$991$	23018.0	0.737832	0.368916	−	0.929463i	$-0.379729\pi$
$991$	23018.0	0.737832	0.368916	+	0.929463i	$0.379729\pi$
$992$	0	0
$993$	14909.8	0.476485
$994$	0	0
$995$	11316.4	0.360557
$996$	0	0
$997$	16189.1	0.514256	0.257128	−	0.966377i	$-0.417224\pi$
$997$	16189.1	0.514256	0.257128	+	0.966377i	$0.417224\pi$
$998$	0	0
$999$	27889.2	0.883258

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.r.1.2		3
4.3	odd	2		1216.4.a.w.1.2		3
8.3	odd	2		304.4.a.g.1.2		3
8.5	even	2		152.4.a.c.1.2	✓	3
24.5	odd	2		1368.4.a.d.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.c.1.2	✓	3	8.5	even	2
304.4.a.g.1.2		3	8.3	odd	2
1216.4.a.r.1.2		3	1.1	even	1	trivial
1216.4.a.w.1.2		3	4.3	odd	2
1368.4.a.d.1.1		3	24.5	odd	2

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-1.33180 q^{3} -18.4427 q^{5} +10.3872 q^{7} -25.2263 q^{9} +O(q^{10})\)	\(q-1.33180 q^{3} -18.4427 q^{5} +10.3872 q^{7} -25.2263 q^{9} -50.4519 q^{11} +61.8352 q^{13} +24.5619 q^{15} +68.1070 q^{17} +19.0000 q^{19} -13.8337 q^{21} +145.632 q^{23} +215.132 q^{25} +69.5550 q^{27} -42.6097 q^{29} +91.6582 q^{31} +67.1918 q^{33} -191.568 q^{35} +400.965 q^{37} -82.3522 q^{39} -123.355 q^{41} -449.802 q^{43} +465.240 q^{45} -453.075 q^{47} -235.105 q^{49} -90.7050 q^{51} -437.142 q^{53} +930.466 q^{55} -25.3042 q^{57} +159.352 q^{59} +476.816 q^{61} -262.031 q^{63} -1140.41 q^{65} +629.682 q^{67} -193.953 q^{69} +471.459 q^{71} -725.055 q^{73} -286.512 q^{75} -524.055 q^{77} -1057.66 q^{79} +588.477 q^{81} +726.957 q^{83} -1256.07 q^{85} +56.7476 q^{87} -468.065 q^{89} +642.297 q^{91} -122.071 q^{93} -350.410 q^{95} -891.891 q^{97} +1272.71 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9	\( 3 q - 4 q^{3} - 7 q^{5} + 7 q^{7} + 21 q^{9} - 103 q^{11} - 32 q^{13} + 122 q^{15} + 11 q^{17} + 57 q^{19} - 114 q^{21} + 316 q^{23} + 162 q^{25} - 178 q^{27} + 138 q^{29} + 420 q^{31} - 330 q^{33} - 333 q^{35} - 102 q^{37} + 164 q^{39} - 370 q^{41} - 431 q^{43} + 429 q^{45} - 199 q^{47} - 802 q^{49} - 272 q^{51} + 308 q^{53} + 85 q^{55} - 76 q^{57} + 188 q^{59} + 609 q^{61} - 61 q^{63} - 1536 q^{65} + 246 q^{67} + 892 q^{69} - 954 q^{71} - 629 q^{73} + 1074 q^{75} + 71 q^{77} + 452 q^{79} - 361 q^{81} + 780 q^{83} - 1883 q^{85} - 96 q^{87} - 1356 q^{89} + 670 q^{91} - 1092 q^{93} - 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 7 * q^5 + 7 * q^7 + 21 * q^9 - 103 * q^11 - 32 * q^13 + 122 * q^15 + 11 * q^17 + 57 * q^19 - 114 * q^21 + 316 * q^23 + 162 * q^25 - 178 * q^27 + 138 * q^29 + 420 * q^31 - 330 * q^33 - 333 * q^35 - 102 * q^37 + 164 * q^39 - 370 * q^41 - 431 * q^43 + 429 * q^45 - 199 * q^47 - 802 * q^49 - 272 * q^51 + 308 * q^53 + 85 * q^55 - 76 * q^57 + 188 * q^59 + 609 * q^61 - 61 * q^63 - 1536 * q^65 + 246 * q^67 + 892 * q^69 - 954 * q^71 - 629 * q^73 + 1074 * q^75 + 71 * q^77 + 452 * q^79 - 361 * q^81 + 780 * q^83 - 1883 * q^85 - 96 * q^87 - 1356 * q^89 + 670 * q^91 - 1092 * q^93 - 133 * q^95 - 548 * q^97 + 1305 * q^99

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