LMFDB - Embedded newform 1386.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,4,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, 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("1386.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1386.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:	$81.7766472680$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -2.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} -2.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} -4.00000 q^{10} -11.0000 q^{11} +26.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +46.0000 q^{17} -48.0000 q^{19} -8.00000 q^{20} -22.0000 q^{22} +128.000 q^{23} -121.000 q^{25} +52.0000 q^{26} -28.0000 q^{28} +146.000 q^{29} -128.000 q^{31} +32.0000 q^{32} +92.0000 q^{34} +14.0000 q^{35} -26.0000 q^{37} -96.0000 q^{38} -16.0000 q^{40} -10.0000 q^{41} +52.0000 q^{43} -44.0000 q^{44} +256.000 q^{46} +544.000 q^{47} +49.0000 q^{49} -242.000 q^{50} +104.000 q^{52} -318.000 q^{53} +22.0000 q^{55} -56.0000 q^{56} +292.000 q^{58} +48.0000 q^{59} +466.000 q^{61} -256.000 q^{62} +64.0000 q^{64} -52.0000 q^{65} +516.000 q^{67} +184.000 q^{68} +28.0000 q^{70} +392.000 q^{71} +754.000 q^{73} -52.0000 q^{74} -192.000 q^{76} +77.0000 q^{77} -32.0000 q^{80} -20.0000 q^{82} -624.000 q^{83} -92.0000 q^{85} +104.000 q^{86} -88.0000 q^{88} +1590.00 q^{89} -182.000 q^{91} +512.000 q^{92} +1088.00 q^{94} +96.0000 q^{95} +1018.00 q^{97} +98.0000 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−2.00000	−0.178885	−0.0894427	−	0.995992i	$-0.528509\pi$
$5$	−2.00000	−0.178885	−0.0894427	+	0.995992i	$0.528509\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−4.00000	−0.126491
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	26.0000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	26.0000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	46.0000	0.656273	0.328136	−	0.944630i	$-0.393579\pi$
$17$	46.0000	0.656273	0.328136	+	0.944630i	$0.393579\pi$
$18$	0	0
$19$	−48.0000	−0.579577	−0.289788	−	0.957091i	$-0.593585\pi$
$19$	−48.0000	−0.579577	−0.289788	+	0.957091i	$0.593585\pi$
$20$	−8.00000	−0.0894427
$21$	0	0
$22$	−22.0000	−0.213201
$23$	128.000	1.16043	0.580214	−	0.814464i	$-0.302969\pi$
$23$	128.000	1.16043	0.580214	+	0.814464i	$0.302969\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	52.0000	0.392232
$27$	0	0
$28$	−28.0000	−0.188982
$29$	146.000	0.934880	0.467440	−	0.884025i	$-0.345176\pi$
$29$	146.000	0.934880	0.467440	+	0.884025i	$0.345176\pi$
$30$	0	0
$31$	−128.000	−0.741596	−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000	−0.741596	−0.370798	+	0.928714i	$0.620916\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	92.0000	0.464055
$35$	14.0000	0.0676123
$36$	0	0
$37$	−26.0000	−0.115524	−0.0577618	−	0.998330i	$-0.518396\pi$
$37$	−26.0000	−0.115524	−0.0577618	+	0.998330i	$0.518396\pi$
$38$	−96.0000	−0.409823
$39$	0	0
$40$	−16.0000	−0.0632456
$41$	−10.0000	−0.0380912	−0.0190456	−	0.999819i	$-0.506063\pi$
$41$	−10.0000	−0.0380912	−0.0190456	+	0.999819i	$0.506063\pi$
$42$	0	0
$43$	52.0000	0.184417	0.0922084	−	0.995740i	$-0.470607\pi$
$43$	52.0000	0.184417	0.0922084	+	0.995740i	$0.470607\pi$
$44$	−44.0000	−0.150756
$45$	0	0
$46$	256.000	0.820547
$47$	544.000	1.68831	0.844155	−	0.536099i	$-0.180103\pi$
$47$	544.000	1.68831	0.844155	+	0.536099i	$0.180103\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−242.000	−0.684479
$51$	0	0
$52$	104.000	0.277350
$53$	−318.000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−318.000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	22.0000	0.0539360
$56$	−56.0000	−0.133631
$57$	0	0
$58$	292.000	0.661060
$59$	48.0000	0.105916	0.0529582	−	0.998597i	$-0.483135\pi$
$59$	48.0000	0.105916	0.0529582	+	0.998597i	$0.483135\pi$
$60$	0	0
$61$	466.000	0.978118	0.489059	−	0.872251i	$-0.337340\pi$
$61$	466.000	0.978118	0.489059	+	0.872251i	$0.337340\pi$
$62$	−256.000	−0.524388
$63$	0	0
$64$	64.0000	0.125000
$65$	−52.0000	−0.0992278
$66$	0	0
$67$	516.000	0.940887	0.470444	−	0.882430i	$-0.344094\pi$
$67$	516.000	0.940887	0.470444	+	0.882430i	$0.344094\pi$
$68$	184.000	0.328136
$69$	0	0
$70$	28.0000	0.0478091
$71$	392.000	0.655237	0.327619	−	0.944810i	$-0.393754\pi$
$71$	392.000	0.655237	0.327619	+	0.944810i	$0.393754\pi$
$72$	0	0
$73$	754.000	1.20889	0.604445	−	0.796647i	$-0.293395\pi$
$73$	754.000	1.20889	0.604445	+	0.796647i	$0.293395\pi$
$74$	−52.0000	−0.0816875
$75$	0	0
$76$	−192.000	−0.289788
$77$	77.0000	0.113961
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−32.0000	−0.0447214
$81$	0	0
$82$	−20.0000	−0.0269345
$83$	−624.000	−0.825216	−0.412608	−	0.910909i	$-0.635382\pi$
$83$	−624.000	−0.825216	−0.412608	+	0.910909i	$0.635382\pi$
$84$	0	0
$85$	−92.0000	−0.117398
$86$	104.000	0.130402
$87$	0	0
$88$	−88.0000	−0.106600
$89$	1590.00	1.89370	0.946852	−	0.321669i	$-0.104244\pi$
$89$	1590.00	1.89370	0.946852	+	0.321669i	$0.104244\pi$
$90$	0	0
$91$	−182.000	−0.209657
$92$	512.000	0.580214
$93$	0	0
$94$	1088.00	1.19382
$95$	96.0000	0.103678
$96$	0	0
$97$	1018.00	1.06559	0.532795	−	0.846244i	$-0.321142\pi$
$97$	1018.00	1.06559	0.532795	+	0.846244i	$0.321142\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−484.000	−0.484000
$101$	−474.000	−0.466978	−0.233489	−	0.972359i	$-0.575014\pi$
$101$	−474.000	−0.466978	−0.233489	+	0.972359i	$0.575014\pi$
$102$	0	0
$103$	−984.000	−0.941324	−0.470662	−	0.882314i	$-0.655985\pi$
$103$	−984.000	−0.941324	−0.470662	+	0.882314i	$0.655985\pi$
$104$	208.000	0.196116
$105$	0	0
$106$	−636.000	−0.582772
$107$	−92.0000	−0.0831213	−0.0415606	−	0.999136i	$-0.513233\pi$
$107$	−92.0000	−0.0831213	−0.0415606	+	0.999136i	$0.513233\pi$
$108$	0	0
$109$	1246.00	1.09491	0.547455	−	0.836835i	$-0.315597\pi$
$109$	1246.00	1.09491	0.547455	+	0.836835i	$0.315597\pi$
$110$	44.0000	0.0381385
$111$	0	0
$112$	−112.000	−0.0944911
$113$	1630.00	1.35697	0.678485	−	0.734615i	$-0.262637\pi$
$113$	1630.00	1.35697	0.678485	+	0.734615i	$0.262637\pi$
$114$	0	0
$115$	−256.000	−0.207584
$116$	584.000	0.467440
$117$	0	0
$118$	96.0000	0.0748942
$119$	−322.000	−0.248048
$120$	0	0
$121$	121.000	0.0909091
$122$	932.000	0.691634
$123$	0	0
$124$	−512.000	−0.370798
$125$	492.000	0.352047
$126$	0	0
$127$	1016.00	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	1016.00	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−104.000	−0.0701646
$131$	1920.00	1.28054	0.640272	−	0.768149i	$-0.278822\pi$
$131$	1920.00	1.28054	0.640272	+	0.768149i	$0.278822\pi$
$132$	0	0
$133$	336.000	0.219059
$134$	1032.00	0.665308
$135$	0	0
$136$	368.000	0.232027
$137$	−1482.00	−0.924203	−0.462101	−	0.886827i	$-0.652904\pi$
$137$	−1482.00	−0.924203	−0.462101	+	0.886827i	$0.652904\pi$
$138$	0	0
$139$	−2608.00	−1.59142	−0.795711	−	0.605676i	$-0.792903\pi$
$139$	−2608.00	−1.59142	−0.795711	+	0.605676i	$0.792903\pi$
$140$	56.0000	0.0338062
$141$	0	0
$142$	784.000	0.463323
$143$	−286.000	−0.167248
$144$	0	0
$145$	−292.000	−0.167236
$146$	1508.00	0.854815
$147$	0	0
$148$	−104.000	−0.0577618
$149$	−1310.00	−0.720264	−0.360132	−	0.932901i	$-0.617268\pi$
$149$	−1310.00	−0.720264	−0.360132	+	0.932901i	$0.617268\pi$
$150$	0	0
$151$	−192.000	−0.103475	−0.0517375	−	0.998661i	$-0.516476\pi$
$151$	−192.000	−0.103475	−0.0517375	+	0.998661i	$0.516476\pi$
$152$	−384.000	−0.204911
$153$	0	0
$154$	154.000	0.0805823
$155$	256.000	0.132661
$156$	0	0
$157$	658.000	0.334485	0.167242	−	0.985916i	$-0.446514\pi$
$157$	658.000	0.334485	0.167242	+	0.985916i	$0.446514\pi$
$158$	0	0
$159$	0	0
$160$	−64.0000	−0.0316228
$161$	−896.000	−0.438601
$162$	0	0
$163$	2236.00	1.07446	0.537230	−	0.843436i	$-0.319471\pi$
$163$	2236.00	1.07446	0.537230	+	0.843436i	$0.319471\pi$
$164$	−40.0000	−0.0190456
$165$	0	0
$166$	−1248.00	−0.583516
$167$	1664.00	0.771043	0.385522	−	0.922699i	$-0.374022\pi$
$167$	1664.00	0.771043	0.385522	+	0.922699i	$0.374022\pi$
$168$	0	0
$169$	−1521.00	−0.692308
$170$	−184.000	−0.0830127
$171$	0	0
$172$	208.000	0.0922084
$173$	662.000	0.290930	0.145465	−	0.989363i	$-0.453532\pi$
$173$	662.000	0.290930	0.145465	+	0.989363i	$0.453532\pi$
$174$	0	0
$175$	847.000	0.365870
$176$	−176.000	−0.0753778
$177$	0	0
$178$	3180.00	1.33905
$179$	2540.00	1.06061	0.530303	−	0.847808i	$-0.322078\pi$
$179$	2540.00	1.06061	0.530303	+	0.847808i	$0.322078\pi$
$180$	0	0
$181$	2762.00	1.13424	0.567121	−	0.823634i	$-0.308057\pi$
$181$	2762.00	1.13424	0.567121	+	0.823634i	$0.308057\pi$
$182$	−364.000	−0.148250
$183$	0	0
$184$	1024.00	0.410273
$185$	52.0000	0.0206655
$186$	0	0
$187$	−506.000	−0.197874
$188$	2176.00	0.844155
$189$	0	0
$190$	192.000	0.0733113
$191$	16.0000	0.00606136	0.00303068	−	0.999995i	$-0.499035\pi$
$191$	16.0000	0.00606136	0.00303068	+	0.999995i	$0.499035\pi$
$192$	0	0
$193$	5138.00	1.91628	0.958138	−	0.286306i	$-0.0924275\pi$
$193$	5138.00	1.91628	0.958138	+	0.286306i	$0.0924275\pi$
$194$	2036.00	0.753486
$195$	0	0
$196$	196.000	0.0714286
$197$	−4350.00	−1.57322	−0.786611	−	0.617449i	$-0.788166\pi$
$197$	−4350.00	−1.57322	−0.786611	+	0.617449i	$0.788166\pi$
$198$	0	0
$199$	−4040.00	−1.43914	−0.719568	−	0.694422i	$-0.755660\pi$
$199$	−4040.00	−1.43914	−0.719568	+	0.694422i	$0.755660\pi$
$200$	−968.000	−0.342240
$201$	0	0
$202$	−948.000	−0.330203
$203$	−1022.00	−0.353351
$204$	0	0
$205$	20.0000	0.00681395
$206$	−1968.00	−0.665617
$207$	0	0
$208$	416.000	0.138675
$209$	528.000	0.174749
$210$	0	0
$211$	−1820.00	−0.593810	−0.296905	−	0.954907i	$-0.595955\pi$
$211$	−1820.00	−0.593810	−0.296905	+	0.954907i	$0.595955\pi$
$212$	−1272.00	−0.412082
$213$	0	0
$214$	−184.000	−0.0587756
$215$	−104.000	−0.0329895
$216$	0	0
$217$	896.000	0.280297
$218$	2492.00	0.774218
$219$	0	0
$220$	88.0000	0.0269680
$221$	1196.00	0.364035
$222$	0	0
$223$	−2360.00	−0.708687	−0.354344	−	0.935115i	$-0.615296\pi$
$223$	−2360.00	−0.708687	−0.354344	+	0.935115i	$0.615296\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	3260.00	0.959522
$227$	6416.00	1.87597	0.937984	−	0.346678i	$-0.112690\pi$
$227$	6416.00	1.87597	0.937984	+	0.346678i	$0.112690\pi$
$228$	0	0
$229$	−1558.00	−0.449588	−0.224794	−	0.974406i	$-0.572171\pi$
$229$	−1558.00	−0.449588	−0.224794	+	0.974406i	$0.572171\pi$
$230$	−512.000	−0.146784
$231$	0	0
$232$	1168.00	0.330530
$233$	−522.000	−0.146770	−0.0733849	−	0.997304i	$-0.523380\pi$
$233$	−522.000	−0.146770	−0.0733849	+	0.997304i	$0.523380\pi$
$234$	0	0
$235$	−1088.00	−0.302014
$236$	192.000	0.0529582
$237$	0	0
$238$	−644.000	−0.175396
$239$	−2152.00	−0.582432	−0.291216	−	0.956657i	$-0.594060\pi$
$239$	−2152.00	−0.582432	−0.291216	+	0.956657i	$0.594060\pi$
$240$	0	0
$241$	−606.000	−0.161975	−0.0809873	−	0.996715i	$-0.525807\pi$
$241$	−606.000	−0.161975	−0.0809873	+	0.996715i	$0.525807\pi$
$242$	242.000	0.0642824
$243$	0	0
$244$	1864.00	0.489059
$245$	−98.0000	−0.0255551
$246$	0	0
$247$	−1248.00	−0.321491
$248$	−1024.00	−0.262194
$249$	0	0
$250$	984.000	0.248934
$251$	1608.00	0.404367	0.202183	−	0.979348i	$-0.435196\pi$
$251$	1608.00	0.404367	0.202183	+	0.979348i	$0.435196\pi$
$252$	0	0
$253$	−1408.00	−0.349882
$254$	2032.00	0.501965
$255$	0	0
$256$	256.000	0.0625000
$257$	4446.00	1.07912	0.539560	−	0.841947i	$-0.318591\pi$
$257$	4446.00	1.07912	0.539560	+	0.841947i	$0.318591\pi$
$258$	0	0
$259$	182.000	0.0436638
$260$	−208.000	−0.0496139
$261$	0	0
$262$	3840.00	0.905481
$263$	−6600.00	−1.54743	−0.773714	−	0.633535i	$-0.781603\pi$
$263$	−6600.00	−1.54743	−0.773714	+	0.633535i	$0.781603\pi$
$264$	0	0
$265$	636.000	0.147431
$266$	672.000	0.154898
$267$	0	0
$268$	2064.00	0.470444
$269$	1854.00	0.420224	0.210112	−	0.977677i	$-0.432617\pi$
$269$	1854.00	0.420224	0.210112	+	0.977677i	$0.432617\pi$
$270$	0	0
$271$	−272.000	−0.0609698	−0.0304849	−	0.999535i	$-0.509705\pi$
$271$	−272.000	−0.0609698	−0.0304849	+	0.999535i	$0.509705\pi$
$272$	736.000	0.164068
$273$	0	0
$274$	−2964.00	−0.653510
$275$	1331.00	0.291863
$276$	0	0
$277$	−5010.00	−1.08672	−0.543361	−	0.839499i	$-0.682848\pi$
$277$	−5010.00	−1.08672	−0.543361	+	0.839499i	$0.682848\pi$
$278$	−5216.00	−1.12531
$279$	0	0
$280$	112.000	0.0239046
$281$	−314.000	−0.0666607	−0.0333304	−	0.999444i	$-0.510611\pi$
$281$	−314.000	−0.0666607	−0.0333304	+	0.999444i	$0.510611\pi$
$282$	0	0
$283$	3480.00	0.730970	0.365485	−	0.930817i	$-0.380903\pi$
$283$	3480.00	0.730970	0.365485	+	0.930817i	$0.380903\pi$
$284$	1568.00	0.327619
$285$	0	0
$286$	−572.000	−0.118262
$287$	70.0000	0.0143971
$288$	0	0
$289$	−2797.00	−0.569306
$290$	−584.000	−0.118254
$291$	0	0
$292$	3016.00	0.604445
$293$	4230.00	0.843410	0.421705	−	0.906733i	$-0.361432\pi$
$293$	4230.00	0.843410	0.421705	+	0.906733i	$0.361432\pi$
$294$	0	0
$295$	−96.0000	−0.0189469
$296$	−208.000	−0.0408438
$297$	0	0
$298$	−2620.00	−0.509304
$299$	3328.00	0.643690
$300$	0	0
$301$	−364.000	−0.0697030
$302$	−384.000	−0.0731679
$303$	0	0
$304$	−768.000	−0.144894
$305$	−932.000	−0.174971
$306$	0	0
$307$	−1552.00	−0.288525	−0.144263	−	0.989539i	$-0.546081\pi$
$307$	−1552.00	−0.288525	−0.144263	+	0.989539i	$0.546081\pi$
$308$	308.000	0.0569803
$309$	0	0
$310$	512.000	0.0938053
$311$	4864.00	0.886856	0.443428	−	0.896310i	$-0.353762\pi$
$311$	4864.00	0.886856	0.443428	+	0.896310i	$0.353762\pi$
$312$	0	0
$313$	4786.00	0.864283	0.432142	−	0.901806i	$-0.357758\pi$
$313$	4786.00	0.864283	0.432142	+	0.901806i	$0.357758\pi$
$314$	1316.00	0.236516
$315$	0	0
$316$	0	0
$317$	1530.00	0.271083	0.135542	−	0.990772i	$-0.456723\pi$
$317$	1530.00	0.271083	0.135542	+	0.990772i	$0.456723\pi$
$318$	0	0
$319$	−1606.00	−0.281877
$320$	−128.000	−0.0223607
$321$	0	0
$322$	−1792.00	−0.310137
$323$	−2208.00	−0.380360
$324$	0	0
$325$	−3146.00	−0.536950
$326$	4472.00	0.759758
$327$	0	0
$328$	−80.0000	−0.0134673
$329$	−3808.00	−0.638121
$330$	0	0
$331$	−10844.0	−1.80073	−0.900363	−	0.435140i	$-0.856699\pi$
$331$	−10844.0	−1.80073	−0.900363	+	0.435140i	$0.856699\pi$
$332$	−2496.00	−0.412608
$333$	0	0
$334$	3328.00	0.545210
$335$	−1032.00	−0.168311
$336$	0	0
$337$	402.000	0.0649802	0.0324901	−	0.999472i	$-0.489656\pi$
$337$	402.000	0.0649802	0.0324901	+	0.999472i	$0.489656\pi$
$338$	−3042.00	−0.489535
$339$	0	0
$340$	−368.000	−0.0586988
$341$	1408.00	0.223600
$342$	0	0
$343$	−343.000	−0.0539949
$344$	416.000	0.0652012
$345$	0	0
$346$	1324.00	0.205719
$347$	5980.00	0.925139	0.462569	−	0.886583i	$-0.346928\pi$
$347$	5980.00	0.925139	0.462569	+	0.886583i	$0.346928\pi$
$348$	0	0
$349$	−3094.00	−0.474550	−0.237275	−	0.971442i	$-0.576254\pi$
$349$	−3094.00	−0.474550	−0.237275	+	0.971442i	$0.576254\pi$
$350$	1694.00	0.258709
$351$	0	0
$352$	−352.000	−0.0533002
$353$	4494.00	0.677596	0.338798	−	0.940859i	$-0.389980\pi$
$353$	4494.00	0.677596	0.338798	+	0.940859i	$0.389980\pi$
$354$	0	0
$355$	−784.000	−0.117212
$356$	6360.00	0.946852
$357$	0	0
$358$	5080.00	0.749962
$359$	2752.00	0.404582	0.202291	−	0.979325i	$-0.435161\pi$
$359$	2752.00	0.404582	0.202291	+	0.979325i	$0.435161\pi$
$360$	0	0
$361$	−4555.00	−0.664091
$362$	5524.00	0.802030
$363$	0	0
$364$	−728.000	−0.104828
$365$	−1508.00	−0.216253
$366$	0	0
$367$	−2024.00	−0.287880	−0.143940	−	0.989586i	$-0.545977\pi$
$367$	−2024.00	−0.287880	−0.143940	+	0.989586i	$0.545977\pi$
$368$	2048.00	0.290107
$369$	0	0
$370$	104.000	0.0146127
$371$	2226.00	0.311504
$372$	0	0
$373$	5246.00	0.728224	0.364112	−	0.931355i	$-0.381373\pi$
$373$	5246.00	0.728224	0.364112	+	0.931355i	$0.381373\pi$
$374$	−1012.00	−0.139918
$375$	0	0
$376$	4352.00	0.596908
$377$	3796.00	0.518578
$378$	0	0
$379$	3892.00	0.527490	0.263745	−	0.964592i	$-0.415042\pi$
$379$	3892.00	0.527490	0.263745	+	0.964592i	$0.415042\pi$
$380$	384.000	0.0518389
$381$	0	0
$382$	32.0000	0.00428603
$383$	−6752.00	−0.900812	−0.450406	−	0.892824i	$-0.648721\pi$
$383$	−6752.00	−0.900812	−0.450406	+	0.892824i	$0.648721\pi$
$384$	0	0
$385$	−154.000	−0.0203859
$386$	10276.0	1.35501
$387$	0	0
$388$	4072.00	0.532795
$389$	−12486.0	−1.62742	−0.813709	−	0.581273i	$-0.802555\pi$
$389$	−12486.0	−1.62742	−0.813709	+	0.581273i	$0.802555\pi$
$390$	0	0
$391$	5888.00	0.761557
$392$	392.000	0.0505076
$393$	0	0
$394$	−8700.00	−1.11244
$395$	0	0
$396$	0	0
$397$	1938.00	0.245001	0.122501	−	0.992468i	$-0.460909\pi$
$397$	1938.00	0.245001	0.122501	+	0.992468i	$0.460909\pi$
$398$	−8080.00	−1.01762
$399$	0	0
$400$	−1936.00	−0.242000
$401$	−4530.00	−0.564133	−0.282067	−	0.959395i	$-0.591020\pi$
$401$	−4530.00	−0.564133	−0.282067	+	0.959395i	$0.591020\pi$
$402$	0	0
$403$	−3328.00	−0.411363
$404$	−1896.00	−0.233489
$405$	0	0
$406$	−2044.00	−0.249857
$407$	286.000	0.0348317
$408$	0	0
$409$	−13718.0	−1.65846	−0.829232	−	0.558905i	$-0.811222\pi$
$409$	−13718.0	−1.65846	−0.829232	+	0.558905i	$0.811222\pi$
$410$	40.0000	0.00481819
$411$	0	0
$412$	−3936.00	−0.470662
$413$	−336.000	−0.0400326
$414$	0	0
$415$	1248.00	0.147619
$416$	832.000	0.0980581
$417$	0	0
$418$	1056.00	0.123566
$419$	−15280.0	−1.78157	−0.890784	−	0.454427i	$-0.849844\pi$
$419$	−15280.0	−1.78157	−0.890784	+	0.454427i	$0.849844\pi$
$420$	0	0
$421$	478.000	0.0553356	0.0276678	−	0.999617i	$-0.491192\pi$
$421$	478.000	0.0553356	0.0276678	+	0.999617i	$0.491192\pi$
$422$	−3640.00	−0.419887
$423$	0	0
$424$	−2544.00	−0.291386
$425$	−5566.00	−0.635272
$426$	0	0
$427$	−3262.00	−0.369694
$428$	−368.000	−0.0415606
$429$	0	0
$430$	−208.000	−0.0233271
$431$	−6280.00	−0.701849	−0.350925	−	0.936404i	$-0.614133\pi$
$431$	−6280.00	−0.701849	−0.350925	+	0.936404i	$0.614133\pi$
$432$	0	0
$433$	13802.0	1.53183	0.765914	−	0.642943i	$-0.222287\pi$
$433$	13802.0	1.53183	0.765914	+	0.642943i	$0.222287\pi$
$434$	1792.00	0.198200
$435$	0	0
$436$	4984.00	0.547455
$437$	−6144.00	−0.672557
$438$	0	0
$439$	8728.00	0.948895	0.474447	−	0.880284i	$-0.342648\pi$
$439$	8728.00	0.948895	0.474447	+	0.880284i	$0.342648\pi$
$440$	176.000	0.0190693
$441$	0	0
$442$	2392.00	0.257411
$443$	3540.00	0.379662	0.189831	−	0.981817i	$-0.439206\pi$
$443$	3540.00	0.379662	0.189831	+	0.981817i	$0.439206\pi$
$444$	0	0
$445$	−3180.00	−0.338756
$446$	−4720.00	−0.501118
$447$	0	0
$448$	−448.000	−0.0472456
$449$	−4194.00	−0.440818	−0.220409	−	0.975408i	$-0.570739\pi$
$449$	−4194.00	−0.440818	−0.220409	+	0.975408i	$0.570739\pi$
$450$	0	0
$451$	110.000	0.0114849
$452$	6520.00	0.678485
$453$	0	0
$454$	12832.0	1.32651
$455$	364.000	0.0375046
$456$	0	0
$457$	−14134.0	−1.44674	−0.723370	−	0.690460i	$-0.757408\pi$
$457$	−14134.0	−1.44674	−0.723370	+	0.690460i	$0.757408\pi$
$458$	−3116.00	−0.317906
$459$	0	0
$460$	−1024.00	−0.103792
$461$	−234.000	−0.0236409	−0.0118205	−	0.999930i	$-0.503763\pi$
$461$	−234.000	−0.0236409	−0.0118205	+	0.999930i	$0.503763\pi$
$462$	0	0
$463$	13696.0	1.37475	0.687373	−	0.726305i	$-0.258764\pi$
$463$	13696.0	1.37475	0.687373	+	0.726305i	$0.258764\pi$
$464$	2336.00	0.233720
$465$	0	0
$466$	−1044.00	−0.103782
$467$	−16104.0	−1.59573	−0.797863	−	0.602839i	$-0.794036\pi$
$467$	−16104.0	−1.59573	−0.797863	+	0.602839i	$0.794036\pi$
$468$	0	0
$469$	−3612.00	−0.355622
$470$	−2176.00	−0.213556
$471$	0	0
$472$	384.000	0.0374471
$473$	−572.000	−0.0556038
$474$	0	0
$475$	5808.00	0.561030
$476$	−1288.00	−0.124024
$477$	0	0
$478$	−4304.00	−0.411842
$479$	11272.0	1.07522	0.537610	−	0.843193i	$-0.319327\pi$
$479$	11272.0	1.07522	0.537610	+	0.843193i	$0.319327\pi$
$480$	0	0
$481$	−676.000	−0.0640810
$482$	−1212.00	−0.114533
$483$	0	0
$484$	484.000	0.0454545
$485$	−2036.00	−0.190619
$486$	0	0
$487$	−304.000	−0.0282866	−0.0141433	−	0.999900i	$-0.504502\pi$
$487$	−304.000	−0.0282866	−0.0141433	+	0.999900i	$0.504502\pi$
$488$	3728.00	0.345817
$489$	0	0
$490$	−196.000	−0.0180702
$491$	−10572.0	−0.971706	−0.485853	−	0.874041i	$-0.661491\pi$
$491$	−10572.0	−0.971706	−0.485853	+	0.874041i	$0.661491\pi$
$492$	0	0
$493$	6716.00	0.613536
$494$	−2496.00	−0.227329
$495$	0	0
$496$	−2048.00	−0.185399
$497$	−2744.00	−0.247656
$498$	0	0
$499$	−15004.0	−1.34603	−0.673017	−	0.739627i	$-0.735002\pi$
$499$	−15004.0	−1.34603	−0.673017	+	0.739627i	$0.735002\pi$
$500$	1968.00	0.176023
$501$	0	0
$502$	3216.00	0.285930
$503$	−16872.0	−1.49560	−0.747799	−	0.663926i	$-0.768889\pi$
$503$	−16872.0	−1.49560	−0.747799	+	0.663926i	$0.768889\pi$
$504$	0	0
$505$	948.000	0.0835355
$506$	−2816.00	−0.247404
$507$	0	0
$508$	4064.00	0.354943
$509$	−818.000	−0.0712322	−0.0356161	−	0.999366i	$-0.511339\pi$
$509$	−818.000	−0.0712322	−0.0356161	+	0.999366i	$0.511339\pi$
$510$	0	0
$511$	−5278.00	−0.456918
$512$	512.000	0.0441942
$513$	0	0
$514$	8892.00	0.763053
$515$	1968.00	0.168389
$516$	0	0
$517$	−5984.00	−0.509045
$518$	364.000	0.0308750
$519$	0	0
$520$	−416.000	−0.0350823
$521$	2270.00	0.190884	0.0954419	−	0.995435i	$-0.469574\pi$
$521$	2270.00	0.190884	0.0954419	+	0.995435i	$0.469574\pi$
$522$	0	0
$523$	12776.0	1.06817	0.534087	−	0.845429i	$-0.320655\pi$
$523$	12776.0	1.06817	0.534087	+	0.845429i	$0.320655\pi$
$524$	7680.00	0.640272
$525$	0	0
$526$	−13200.0	−1.09420
$527$	−5888.00	−0.486689
$528$	0	0
$529$	4217.00	0.346593
$530$	1272.00	0.104249
$531$	0	0
$532$	1344.00	0.109530
$533$	−260.000	−0.0211292
$534$	0	0
$535$	184.000	0.0148692
$536$	4128.00	0.332654
$537$	0	0
$538$	3708.00	0.297144
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−23050.0	−1.83179	−0.915894	−	0.401421i	$-0.868516\pi$
$541$	−23050.0	−1.83179	−0.915894	+	0.401421i	$0.868516\pi$
$542$	−544.000	−0.0431122
$543$	0	0
$544$	1472.00	0.116014
$545$	−2492.00	−0.195863
$546$	0	0
$547$	−6564.00	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−6564.00	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−5928.00	−0.462101
$549$	0	0
$550$	2662.00	0.206378
$551$	−7008.00	−0.541835
$552$	0	0
$553$	0	0
$554$	−10020.0	−0.768428
$555$	0	0
$556$	−10432.0	−0.795711
$557$	4522.00	0.343992	0.171996	−	0.985098i	$-0.444978\pi$
$557$	4522.00	0.343992	0.171996	+	0.985098i	$0.444978\pi$
$558$	0	0
$559$	1352.00	0.102296
$560$	224.000	0.0169031
$561$	0	0
$562$	−628.000	−0.0471363
$563$	20440.0	1.53009	0.765047	−	0.643974i	$-0.222716\pi$
$563$	20440.0	1.53009	0.765047	+	0.643974i	$0.222716\pi$
$564$	0	0
$565$	−3260.00	−0.242742
$566$	6960.00	0.516874
$567$	0	0
$568$	3136.00	0.231661
$569$	16518.0	1.21700	0.608498	−	0.793556i	$-0.291772\pi$
$569$	16518.0	1.21700	0.608498	+	0.793556i	$0.291772\pi$
$570$	0	0
$571$	−8828.00	−0.647006	−0.323503	−	0.946227i	$-0.604861\pi$
$571$	−8828.00	−0.647006	−0.323503	+	0.946227i	$0.604861\pi$
$572$	−1144.00	−0.0836242
$573$	0	0
$574$	140.000	0.0101803
$575$	−15488.0	−1.12329
$576$	0	0
$577$	−15550.0	−1.12193	−0.560966	−	0.827839i	$-0.689570\pi$
$577$	−15550.0	−1.12193	−0.560966	+	0.827839i	$0.689570\pi$
$578$	−5594.00	−0.402560
$579$	0	0
$580$	−1168.00	−0.0836182
$581$	4368.00	0.311902
$582$	0	0
$583$	3498.00	0.248495
$584$	6032.00	0.427407
$585$	0	0
$586$	8460.00	0.596381
$587$	4536.00	0.318945	0.159473	−	0.987202i	$-0.449021\pi$
$587$	4536.00	0.318945	0.159473	+	0.987202i	$0.449021\pi$
$588$	0	0
$589$	6144.00	0.429812
$590$	−192.000	−0.0133975
$591$	0	0
$592$	−416.000	−0.0288809
$593$	11142.0	0.771580	0.385790	−	0.922587i	$-0.373929\pi$
$593$	11142.0	0.771580	0.385790	+	0.922587i	$0.373929\pi$
$594$	0	0
$595$	644.000	0.0443721
$596$	−5240.00	−0.360132
$597$	0	0
$598$	6656.00	0.455157
$599$	16248.0	1.10831	0.554153	−	0.832415i	$-0.313042\pi$
$599$	16248.0	1.10831	0.554153	+	0.832415i	$0.313042\pi$
$600$	0	0
$601$	−9646.00	−0.654690	−0.327345	−	0.944905i	$-0.606154\pi$
$601$	−9646.00	−0.654690	−0.327345	+	0.944905i	$0.606154\pi$
$602$	−728.000	−0.0492875
$603$	0	0
$604$	−768.000	−0.0517375
$605$	−242.000	−0.0162623
$606$	0	0
$607$	4064.00	0.271751	0.135875	−	0.990726i	$-0.456615\pi$
$607$	4064.00	0.271751	0.135875	+	0.990726i	$0.456615\pi$
$608$	−1536.00	−0.102456
$609$	0	0
$610$	−1864.00	−0.123723
$611$	14144.0	0.936506
$612$	0	0
$613$	−15098.0	−0.994784	−0.497392	−	0.867526i	$-0.665709\pi$
$613$	−15098.0	−0.994784	−0.497392	+	0.867526i	$0.665709\pi$
$614$	−3104.00	−0.204018
$615$	0	0
$616$	616.000	0.0402911
$617$	4470.00	0.291662	0.145831	−	0.989310i	$-0.453414\pi$
$617$	4470.00	0.291662	0.145831	+	0.989310i	$0.453414\pi$
$618$	0	0
$619$	−21184.0	−1.37554	−0.687768	−	0.725931i	$-0.741409\pi$
$619$	−21184.0	−1.37554	−0.687768	+	0.725931i	$0.741409\pi$
$620$	1024.00	0.0663304
$621$	0	0
$622$	9728.00	0.627102
$623$	−11130.0	−0.715753
$624$	0	0
$625$	14141.0	0.905024
$626$	9572.00	0.611141
$627$	0	0
$628$	2632.00	0.167242
$629$	−1196.00	−0.0758150
$630$	0	0
$631$	−8760.00	−0.552663	−0.276331	−	0.961062i	$-0.589119\pi$
$631$	−8760.00	−0.552663	−0.276331	+	0.961062i	$0.589119\pi$
$632$	0	0
$633$	0	0
$634$	3060.00	0.191685
$635$	−2032.00	−0.126988
$636$	0	0
$637$	1274.00	0.0792429
$638$	−3212.00	−0.199317
$639$	0	0
$640$	−256.000	−0.0158114
$641$	3582.00	0.220718	0.110359	−	0.993892i	$-0.464800\pi$
$641$	3582.00	0.220718	0.110359	+	0.993892i	$0.464800\pi$
$642$	0	0
$643$	−23168.0	−1.42093	−0.710464	−	0.703734i	$-0.751515\pi$
$643$	−23168.0	−1.42093	−0.710464	+	0.703734i	$0.751515\pi$
$644$	−3584.00	−0.219300
$645$	0	0
$646$	−4416.00	−0.268955
$647$	−30216.0	−1.83603	−0.918017	−	0.396542i	$-0.870210\pi$
$647$	−30216.0	−1.83603	−0.918017	+	0.396542i	$0.870210\pi$
$648$	0	0
$649$	−528.000	−0.0319350
$650$	−6292.00	−0.379681
$651$	0	0
$652$	8944.00	0.537230
$653$	−8158.00	−0.488893	−0.244447	−	0.969663i	$-0.578606\pi$
$653$	−8158.00	−0.488893	−0.244447	+	0.969663i	$0.578606\pi$
$654$	0	0
$655$	−3840.00	−0.229071
$656$	−160.000	−0.00952279
$657$	0	0
$658$	−7616.00	−0.451220
$659$	−11932.0	−0.705318	−0.352659	−	0.935752i	$-0.614722\pi$
$659$	−11932.0	−0.705318	−0.352659	+	0.935752i	$0.614722\pi$
$660$	0	0
$661$	26882.0	1.58183	0.790914	−	0.611927i	$-0.209605\pi$
$661$	26882.0	1.58183	0.790914	+	0.611927i	$0.209605\pi$
$662$	−21688.0	−1.27331
$663$	0	0
$664$	−4992.00	−0.291758
$665$	−672.000	−0.0391865
$666$	0	0
$667$	18688.0	1.08486
$668$	6656.00	0.385522
$669$	0	0
$670$	−2064.00	−0.119014
$671$	−5126.00	−0.294914
$672$	0	0
$673$	13090.0	0.749751	0.374875	−	0.927075i	$-0.377685\pi$
$673$	13090.0	0.749751	0.374875	+	0.927075i	$0.377685\pi$
$674$	804.000	0.0459480
$675$	0	0
$676$	−6084.00	−0.346154
$677$	33790.0	1.91825	0.959125	−	0.282983i	$-0.0913240\pi$
$677$	33790.0	1.91825	0.959125	+	0.282983i	$0.0913240\pi$
$678$	0	0
$679$	−7126.00	−0.402755
$680$	−736.000	−0.0415063
$681$	0	0
$682$	2816.00	0.158109
$683$	−24588.0	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−24588.0	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	2964.00	0.165326
$686$	−686.000	−0.0381802
$687$	0	0
$688$	832.000	0.0461042
$689$	−8268.00	−0.457164
$690$	0	0
$691$	−1128.00	−0.0621001	−0.0310500	−	0.999518i	$-0.509885\pi$
$691$	−1128.00	−0.0621001	−0.0310500	+	0.999518i	$0.509885\pi$
$692$	2648.00	0.145465
$693$	0	0
$694$	11960.0	0.654172
$695$	5216.00	0.284682
$696$	0	0
$697$	−460.000	−0.0249982
$698$	−6188.00	−0.335558
$699$	0	0
$700$	3388.00	0.182935
$701$	18786.0	1.01218	0.506089	−	0.862481i	$-0.331091\pi$
$701$	18786.0	1.01218	0.506089	+	0.862481i	$0.331091\pi$
$702$	0	0
$703$	1248.00	0.0669548
$704$	−704.000	−0.0376889
$705$	0	0
$706$	8988.00	0.479133
$707$	3318.00	0.176501
$708$	0	0
$709$	12102.0	0.641044	0.320522	−	0.947241i	$-0.396142\pi$
$709$	12102.0	0.641044	0.320522	+	0.947241i	$0.396142\pi$
$710$	−1568.00	−0.0828817
$711$	0	0
$712$	12720.0	0.669525
$713$	−16384.0	−0.860569
$714$	0	0
$715$	572.000	0.0299183
$716$	10160.0	0.530303
$717$	0	0
$718$	5504.00	0.286083
$719$	−18112.0	−0.939449	−0.469724	−	0.882813i	$-0.655647\pi$
$719$	−18112.0	−0.939449	−0.469724	+	0.882813i	$0.655647\pi$
$720$	0	0
$721$	6888.00	0.355787
$722$	−9110.00	−0.469583
$723$	0	0
$724$	11048.0	0.567121
$725$	−17666.0	−0.904964
$726$	0	0
$727$	12728.0	0.649320	0.324660	−	0.945831i	$-0.394750\pi$
$727$	12728.0	0.649320	0.324660	+	0.945831i	$0.394750\pi$
$728$	−1456.00	−0.0741249
$729$	0	0
$730$	−3016.00	−0.152914
$731$	2392.00	0.121028
$732$	0	0
$733$	17138.0	0.863583	0.431792	−	0.901973i	$-0.357882\pi$
$733$	17138.0	0.863583	0.431792	+	0.901973i	$0.357882\pi$
$734$	−4048.00	−0.203562
$735$	0	0
$736$	4096.00	0.205137
$737$	−5676.00	−0.283688
$738$	0	0
$739$	−8340.00	−0.415145	−0.207572	−	0.978220i	$-0.566556\pi$
$739$	−8340.00	−0.415145	−0.207572	+	0.978220i	$0.566556\pi$
$740$	208.000	0.0103327
$741$	0	0
$742$	4452.00	0.220267
$743$	−8304.00	−0.410019	−0.205010	−	0.978760i	$-0.565723\pi$
$743$	−8304.00	−0.410019	−0.205010	+	0.978760i	$0.565723\pi$
$744$	0	0
$745$	2620.00	0.128845
$746$	10492.0	0.514932
$747$	0	0
$748$	−2024.00	−0.0989369
$749$	644.000	0.0314169
$750$	0	0
$751$	2152.00	0.104564	0.0522820	−	0.998632i	$-0.483351\pi$
$751$	2152.00	0.104564	0.0522820	+	0.998632i	$0.483351\pi$
$752$	8704.00	0.422077
$753$	0	0
$754$	7592.00	0.366690
$755$	384.000	0.0185102
$756$	0	0
$757$	−25594.0	−1.22884	−0.614419	−	0.788980i	$-0.710609\pi$
$757$	−25594.0	−1.22884	−0.614419	+	0.788980i	$0.710609\pi$
$758$	7784.00	0.372992
$759$	0	0
$760$	768.000	0.0366556
$761$	17190.0	0.818840	0.409420	−	0.912346i	$-0.365731\pi$
$761$	17190.0	0.818840	0.409420	+	0.912346i	$0.365731\pi$
$762$	0	0
$763$	−8722.00	−0.413837
$764$	64.0000	0.00303068
$765$	0	0
$766$	−13504.0	−0.636970
$767$	1248.00	0.0587518
$768$	0	0
$769$	−15086.0	−0.707432	−0.353716	−	0.935353i	$-0.615082\pi$
$769$	−15086.0	−0.707432	−0.353716	+	0.935353i	$0.615082\pi$
$770$	−308.000	−0.0144150
$771$	0	0
$772$	20552.0	0.958138
$773$	−14178.0	−0.659699	−0.329849	−	0.944034i	$-0.606998\pi$
$773$	−14178.0	−0.659699	−0.329849	+	0.944034i	$0.606998\pi$
$774$	0	0
$775$	15488.0	0.717865
$776$	8144.00	0.376743
$777$	0	0
$778$	−24972.0	−1.15076
$779$	480.000	0.0220767
$780$	0	0
$781$	−4312.00	−0.197561
$782$	11776.0	0.538502
$783$	0	0
$784$	784.000	0.0357143
$785$	−1316.00	−0.0598345
$786$	0	0
$787$	−18304.0	−0.829056	−0.414528	−	0.910037i	$-0.636053\pi$
$787$	−18304.0	−0.829056	−0.414528	+	0.910037i	$0.636053\pi$
$788$	−17400.0	−0.786611
$789$	0	0
$790$	0	0
$791$	−11410.0	−0.512886
$792$	0	0
$793$	12116.0	0.542562
$794$	3876.00	0.173242
$795$	0	0
$796$	−16160.0	−0.719568
$797$	38206.0	1.69803	0.849013	−	0.528373i	$-0.177198\pi$
$797$	38206.0	1.69803	0.849013	+	0.528373i	$0.177198\pi$
$798$	0	0
$799$	25024.0	1.10799
$800$	−3872.00	−0.171120
$801$	0	0
$802$	−9060.00	−0.398902
$803$	−8294.00	−0.364494
$804$	0	0
$805$	1792.00	0.0784593
$806$	−6656.00	−0.290878
$807$	0	0
$808$	−3792.00	−0.165102
$809$	−7146.00	−0.310556	−0.155278	−	0.987871i	$-0.549627\pi$
$809$	−7146.00	−0.310556	−0.155278	+	0.987871i	$0.549627\pi$
$810$	0	0
$811$	−21256.0	−0.920344	−0.460172	−	0.887830i	$-0.652212\pi$
$811$	−21256.0	−0.920344	−0.460172	+	0.887830i	$0.652212\pi$
$812$	−4088.00	−0.176676
$813$	0	0
$814$	572.000	0.0246297
$815$	−4472.00	−0.192205
$816$	0	0
$817$	−2496.00	−0.106884
$818$	−27436.0	−1.17271
$819$	0	0
$820$	80.0000	0.00340698
$821$	−38670.0	−1.64384	−0.821920	−	0.569603i	$-0.807097\pi$
$821$	−38670.0	−1.64384	−0.821920	+	0.569603i	$0.807097\pi$
$822$	0	0
$823$	−21112.0	−0.894190	−0.447095	−	0.894487i	$-0.647541\pi$
$823$	−21112.0	−0.894190	−0.447095	+	0.894487i	$0.647541\pi$
$824$	−7872.00	−0.332808
$825$	0	0
$826$	−672.000	−0.0283073
$827$	3172.00	0.133375	0.0666876	−	0.997774i	$-0.478757\pi$
$827$	3172.00	0.133375	0.0666876	+	0.997774i	$0.478757\pi$
$828$	0	0
$829$	30346.0	1.27136	0.635682	−	0.771951i	$-0.280719\pi$
$829$	30346.0	1.27136	0.635682	+	0.771951i	$0.280719\pi$
$830$	2496.00	0.104382
$831$	0	0
$832$	1664.00	0.0693375
$833$	2254.00	0.0937533
$834$	0	0
$835$	−3328.00	−0.137928
$836$	2112.00	0.0873745
$837$	0	0
$838$	−30560.0	−1.25976
$839$	9480.00	0.390091	0.195045	−	0.980794i	$-0.437515\pi$
$839$	9480.00	0.390091	0.195045	+	0.980794i	$0.437515\pi$
$840$	0	0
$841$	−3073.00	−0.125999
$842$	956.000	0.0391282
$843$	0	0
$844$	−7280.00	−0.296905
$845$	3042.00	0.123844
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−5088.00	−0.206041
$849$	0	0
$850$	−11132.0	−0.449205
$851$	−3328.00	−0.134057
$852$	0	0
$853$	−24958.0	−1.00181	−0.500906	−	0.865502i	$-0.667000\pi$
$853$	−24958.0	−1.00181	−0.500906	+	0.865502i	$0.667000\pi$
$854$	−6524.00	−0.261413
$855$	0	0
$856$	−736.000	−0.0293878
$857$	26806.0	1.06847	0.534233	−	0.845337i	$-0.320600\pi$
$857$	26806.0	1.06847	0.534233	+	0.845337i	$0.320600\pi$
$858$	0	0
$859$	23128.0	0.918646	0.459323	−	0.888269i	$-0.348092\pi$
$859$	23128.0	0.918646	0.459323	+	0.888269i	$0.348092\pi$
$860$	−416.000	−0.0164947
$861$	0	0
$862$	−12560.0	−0.496282
$863$	−12496.0	−0.492895	−0.246448	−	0.969156i	$-0.579263\pi$
$863$	−12496.0	−0.492895	−0.246448	+	0.969156i	$0.579263\pi$
$864$	0	0
$865$	−1324.00	−0.0520432
$866$	27604.0	1.08317
$867$	0	0
$868$	3584.00	0.140148
$869$	0	0
$870$	0	0
$871$	13416.0	0.521910
$872$	9968.00	0.387109
$873$	0	0
$874$	−12288.0	−0.475570
$875$	−3444.00	−0.133061
$876$	0	0
$877$	30478.0	1.17351	0.586755	−	0.809764i	$-0.300405\pi$
$877$	30478.0	1.17351	0.586755	+	0.809764i	$0.300405\pi$
$878$	17456.0	0.670970
$879$	0	0
$880$	352.000	0.0134840
$881$	−25506.0	−0.975390	−0.487695	−	0.873014i	$-0.662162\pi$
$881$	−25506.0	−0.975390	−0.487695	+	0.873014i	$0.662162\pi$
$882$	0	0
$883$	−13244.0	−0.504752	−0.252376	−	0.967629i	$-0.581212\pi$
$883$	−13244.0	−0.504752	−0.252376	+	0.967629i	$0.581212\pi$
$884$	4784.00	0.182017
$885$	0	0
$886$	7080.00	0.268462
$887$	−25456.0	−0.963618	−0.481809	−	0.876276i	$-0.660020\pi$
$887$	−25456.0	−0.963618	−0.481809	+	0.876276i	$0.660020\pi$
$888$	0	0
$889$	−7112.00	−0.268311
$890$	−6360.00	−0.239537
$891$	0	0
$892$	−9440.00	−0.354344
$893$	−26112.0	−0.978505
$894$	0	0
$895$	−5080.00	−0.189727
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−8388.00	−0.311705
$899$	−18688.0	−0.693303
$900$	0	0
$901$	−14628.0	−0.540876
$902$	220.000	0.00812106
$903$	0	0
$904$	13040.0	0.479761
$905$	−5524.00	−0.202899
$906$	0	0
$907$	−51652.0	−1.89093	−0.945467	−	0.325719i	$-0.894394\pi$
$907$	−51652.0	−1.89093	−0.945467	+	0.325719i	$0.894394\pi$
$908$	25664.0	0.937984
$909$	0	0
$910$	728.000	0.0265197
$911$	46392.0	1.68720	0.843598	−	0.536975i	$-0.180433\pi$
$911$	46392.0	1.68720	0.843598	+	0.536975i	$0.180433\pi$
$912$	0	0
$913$	6864.00	0.248812
$914$	−28268.0	−1.02300
$915$	0	0
$916$	−6232.00	−0.224794
$917$	−13440.0	−0.484000
$918$	0	0
$919$	17832.0	0.640069	0.320034	−	0.947406i	$-0.396305\pi$
$919$	17832.0	0.640069	0.320034	+	0.947406i	$0.396305\pi$
$920$	−2048.00	−0.0733919
$921$	0	0
$922$	−468.000	−0.0167167
$923$	10192.0	0.363460
$924$	0	0
$925$	3146.00	0.111827
$926$	27392.0	0.972092
$927$	0	0
$928$	4672.00	0.165265
$929$	41334.0	1.45977	0.729884	−	0.683571i	$-0.239574\pi$
$929$	41334.0	1.45977	0.729884	+	0.683571i	$0.239574\pi$
$930$	0	0
$931$	−2352.00	−0.0827967
$932$	−2088.00	−0.0733849
$933$	0	0
$934$	−32208.0	−1.12835
$935$	1012.00	0.0353967
$936$	0	0
$937$	23058.0	0.803919	0.401959	−	0.915657i	$-0.368329\pi$
$937$	23058.0	0.803919	0.401959	+	0.915657i	$0.368329\pi$
$938$	−7224.00	−0.251463
$939$	0	0
$940$	−4352.00	−0.151007
$941$	11678.0	0.404561	0.202281	−	0.979328i	$-0.435165\pi$
$941$	11678.0	0.404561	0.202281	+	0.979328i	$0.435165\pi$
$942$	0	0
$943$	−1280.00	−0.0442021
$944$	768.000	0.0264791
$945$	0	0
$946$	−1144.00	−0.0393178
$947$	36436.0	1.25028	0.625138	−	0.780514i	$-0.285043\pi$
$947$	36436.0	1.25028	0.625138	+	0.780514i	$0.285043\pi$
$948$	0	0
$949$	19604.0	0.670572
$950$	11616.0	0.396708
$951$	0	0
$952$	−2576.00	−0.0876982
$953$	−21098.0	−0.717137	−0.358568	−	0.933503i	$-0.616735\pi$
$953$	−21098.0	−0.717137	−0.358568	+	0.933503i	$0.616735\pi$
$954$	0	0
$955$	−32.0000	−0.00108429
$956$	−8608.00	−0.291216
$957$	0	0
$958$	22544.0	0.760296
$959$	10374.0	0.349316
$960$	0	0
$961$	−13407.0	−0.450035
$962$	−1352.00	−0.0453121
$963$	0	0
$964$	−2424.00	−0.0809873
$965$	−10276.0	−0.342794
$966$	0	0
$967$	7184.00	0.238906	0.119453	−	0.992840i	$-0.461886\pi$
$967$	7184.00	0.238906	0.119453	+	0.992840i	$0.461886\pi$
$968$	968.000	0.0321412
$969$	0	0
$970$	−4072.00	−0.134788
$971$	40048.0	1.32359	0.661793	−	0.749687i	$-0.269796\pi$
$971$	40048.0	1.32359	0.661793	+	0.749687i	$0.269796\pi$
$972$	0	0
$973$	18256.0	0.601501
$974$	−608.000	−0.0200016
$975$	0	0
$976$	7456.00	0.244529
$977$	−51938.0	−1.70076	−0.850381	−	0.526168i	$-0.823628\pi$
$977$	−51938.0	−1.70076	−0.850381	+	0.526168i	$0.823628\pi$
$978$	0	0
$979$	−17490.0	−0.570973
$980$	−392.000	−0.0127775
$981$	0	0
$982$	−21144.0	−0.687100
$983$	28968.0	0.939914	0.469957	−	0.882689i	$-0.344269\pi$
$983$	28968.0	0.939914	0.469957	+	0.882689i	$0.344269\pi$
$984$	0	0
$985$	8700.00	0.281426
$986$	13432.0	0.433836
$987$	0	0
$988$	−4992.00	−0.160746
$989$	6656.00	0.214003
$990$	0	0
$991$	37504.0	1.20217	0.601087	−	0.799184i	$-0.294735\pi$
$991$	37504.0	1.20217	0.601087	+	0.799184i	$0.294735\pi$
$992$	−4096.00	−0.131097
$993$	0	0
$994$	−5488.00	−0.175120
$995$	8080.00	0.257440
$996$	0	0
$997$	20298.0	0.644778	0.322389	−	0.946607i	$-0.395514\pi$
$997$	20298.0	0.644778	0.322389	+	0.946607i	$0.395514\pi$
$998$	−30008.0	−0.951790
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.4.a.j.1.1		1
3.2	odd	2		154.4.a.b.1.1	✓	1
12.11	even	2		1232.4.a.e.1.1		1
21.20	even	2		1078.4.a.b.1.1		1
33.32	even	2		1694.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.a.b.1.1	✓	1	3.2	odd	2
1078.4.a.b.1.1		1	21.20	even	2
1232.4.a.e.1.1		1	12.11	even	2
1386.4.a.j.1.1		1	1.1	even	1	trivial
1694.4.a.f.1.1		1	33.32	even	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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