LMFDB - Embedded newform 378.4.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("378.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ of defining polynomial
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -3.94987 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -3.94987 q^{5} +7.00000 q^{7} +8.00000 q^{8} -7.89975 q^{10} +21.9499 q^{11} -39.6992 q^{13} +14.0000 q^{14} +16.0000 q^{16} +91.5990 q^{17} +130.398 q^{19} -15.7995 q^{20} +43.8997 q^{22} +91.7494 q^{23} -109.398 q^{25} -79.3985 q^{26} +28.0000 q^{28} -69.1980 q^{29} +124.699 q^{31} +32.0000 q^{32} +183.198 q^{34} -27.6491 q^{35} +70.6992 q^{37} +260.797 q^{38} -31.5990 q^{40} +333.048 q^{41} -82.3008 q^{43} +87.7995 q^{44} +183.499 q^{46} +592.496 q^{47} +49.0000 q^{49} -218.797 q^{50} -158.797 q^{52} -39.1980 q^{53} -86.6992 q^{55} +56.0000 q^{56} -138.396 q^{58} -399.293 q^{59} -125.805 q^{61} +249.398 q^{62} +64.0000 q^{64} +156.807 q^{65} -505.895 q^{67} +366.396 q^{68} -55.2982 q^{70} +919.639 q^{71} -199.895 q^{73} +141.398 q^{74} +521.594 q^{76} +153.649 q^{77} -1315.89 q^{79} -63.1980 q^{80} +666.095 q^{82} +686.802 q^{83} -361.805 q^{85} -164.602 q^{86} +175.599 q^{88} -393.539 q^{89} -277.895 q^{91} +366.997 q^{92} +1184.99 q^{94} -515.058 q^{95} -953.805 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 14 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 14 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 14 q^{7} + 16 q^{8} + 24 q^{10} + 24 q^{11} + 40 q^{13} + 28 q^{14} + 32 q^{16} + 24 q^{17} + 22 q^{19} + 48 q^{20} + 48 q^{22} + 84 q^{23} + 20 q^{25} + 80 q^{26} + 56 q^{28} + 180 q^{29} + 130 q^{31} + 64 q^{32} + 48 q^{34} + 84 q^{35} + 22 q^{37} + 44 q^{38} + 96 q^{40} + 288 q^{41} - 284 q^{43} + 96 q^{44} + 168 q^{46} + 588 q^{47} + 98 q^{49} + 40 q^{50} + 160 q^{52} + 240 q^{53} - 54 q^{55} + 112 q^{56} + 360 q^{58} + 276 q^{59} - 968 q^{61} + 260 q^{62} + 128 q^{64} + 1428 q^{65} - 176 q^{67} + 96 q^{68} + 168 q^{70} + 108 q^{71} + 436 q^{73} + 44 q^{74} + 88 q^{76} + 168 q^{77} - 1796 q^{79} + 192 q^{80} + 576 q^{82} + 1692 q^{83} - 1440 q^{85} - 568 q^{86} + 192 q^{88} + 984 q^{89} + 280 q^{91} + 336 q^{92} + 1176 q^{94} - 2244 q^{95} - 2624 q^{97} + 196 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 14 * q^7 + 16 * q^8 + 24 * q^10 + 24 * q^11 + 40 * q^13 + 28 * q^14 + 32 * q^16 + 24 * q^17 + 22 * q^19 + 48 * q^20 + 48 * q^22 + 84 * q^23 + 20 * q^25 + 80 * q^26 + 56 * q^28 + 180 * q^29 + 130 * q^31 + 64 * q^32 + 48 * q^34 + 84 * q^35 + 22 * q^37 + 44 * q^38 + 96 * q^40 + 288 * q^41 - 284 * q^43 + 96 * q^44 + 168 * q^46 + 588 * q^47 + 98 * q^49 + 40 * q^50 + 160 * q^52 + 240 * q^53 - 54 * q^55 + 112 * q^56 + 360 * q^58 + 276 * q^59 - 968 * q^61 + 260 * q^62 + 128 * q^64 + 1428 * q^65 - 176 * q^67 + 96 * q^68 + 168 * q^70 + 108 * q^71 + 436 * q^73 + 44 * q^74 + 88 * q^76 + 168 * q^77 - 1796 * q^79 + 192 * q^80 + 576 * q^82 + 1692 * q^83 - 1440 * q^85 - 568 * q^86 + 192 * q^88 + 984 * q^89 + 280 * q^91 + 336 * q^92 + 1176 * q^94 - 2244 * q^95 - 2624 * q^97 + 196 * q^98

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
$5$	−3.94987	−0.353288	−0.176644	−	0.984275i	$-0.556524\pi$
$5$	−3.94987	−0.353288	−0.176644	+	0.984275i	$0.556524\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−7.89975	−0.249812
$11$	21.9499	0.601649	0.300824	−	0.953680i	$-0.402738\pi$
$11$	21.9499	0.601649	0.300824	+	0.953680i	$0.402738\pi$
$12$	0	0
$13$	−39.6992	−0.846968	−0.423484	−	0.905903i	$-0.639193\pi$
$13$	−39.6992	−0.846968	−0.423484	+	0.905903i	$0.639193\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	91.5990	1.30682	0.653412	−	0.757002i	$-0.273337\pi$
$17$	91.5990	1.30682	0.653412	+	0.757002i	$0.273337\pi$
$18$	0	0
$19$	130.398	1.57450	0.787249	−	0.616635i	$-0.211505\pi$
$19$	130.398	1.57450	0.787249	+	0.616635i	$0.211505\pi$
$20$	−15.7995	−0.176644
$21$	0	0
$22$	43.8997	0.425430
$23$	91.7494	0.831786	0.415893	−	0.909414i	$-0.363469\pi$
$23$	91.7494	0.831786	0.415893	+	0.909414i	$0.363469\pi$
$24$	0	0
$25$	−109.398	−0.875188
$26$	−79.3985	−0.598897
$27$	0	0
$28$	28.0000	0.188982
$29$	−69.1980	−0.443095	−0.221547	−	0.975150i	$-0.571111\pi$
$29$	−69.1980	−0.443095	−0.221547	+	0.975150i	$0.571111\pi$
$30$	0	0
$31$	124.699	0.722472	0.361236	−	0.932474i	$-0.382355\pi$
$31$	124.699	0.722472	0.361236	+	0.932474i	$0.382355\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	183.198	0.924065
$35$	−27.6491	−0.133530
$36$	0	0
$37$	70.6992	0.314132	0.157066	−	0.987588i	$-0.449796\pi$
$37$	70.6992	0.314132	0.157066	+	0.987588i	$0.449796\pi$
$38$	260.797	1.11334
$39$	0	0
$40$	−31.5990	−0.124906
$41$	333.048	1.26862	0.634309	−	0.773080i	$-0.281285\pi$
$41$	333.048	1.26862	0.634309	+	0.773080i	$0.281285\pi$
$42$	0	0
$43$	−82.3008	−0.291878	−0.145939	−	0.989294i	$-0.546620\pi$
$43$	−82.3008	−0.291878	−0.145939	+	0.989294i	$0.546620\pi$
$44$	87.7995	0.300824
$45$	0	0
$46$	183.499	0.588161
$47$	592.496	1.83882	0.919409	−	0.393303i	$-0.128667\pi$
$47$	592.496	1.83882	0.919409	+	0.393303i	$0.128667\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−218.797	−0.618851
$51$	0	0
$52$	−158.797	−0.423484
$53$	−39.1980	−0.101590	−0.0507949	−	0.998709i	$-0.516175\pi$
$53$	−39.1980	−0.101590	−0.0507949	+	0.998709i	$0.516175\pi$
$54$	0	0
$55$	−86.6992	−0.212555
$56$	56.0000	0.133631
$57$	0	0
$58$	−138.396	−0.313315
$59$	−399.293	−0.881077	−0.440539	−	0.897734i	$-0.645212\pi$
$59$	−399.293	−0.881077	−0.440539	+	0.897734i	$0.645212\pi$
$60$	0	0
$61$	−125.805	−0.264059	−0.132030	−	0.991246i	$-0.542149\pi$
$61$	−125.805	−0.264059	−0.132030	+	0.991246i	$0.542149\pi$
$62$	249.398	0.510865
$63$	0	0
$64$	64.0000	0.125000
$65$	156.807	0.299223
$66$	0	0
$67$	−505.895	−0.922461	−0.461230	−	0.887280i	$-0.652592\pi$
$67$	−505.895	−0.922461	−0.461230	+	0.887280i	$0.652592\pi$
$68$	366.396	0.653412
$69$	0	0
$70$	−55.2982	−0.0944201
$71$	919.639	1.53720	0.768599	−	0.639731i	$-0.220954\pi$
$71$	919.639	1.53720	0.768599	+	0.639731i	$0.220954\pi$
$72$	0	0
$73$	−199.895	−0.320492	−0.160246	−	0.987077i	$-0.551229\pi$
$73$	−199.895	−0.320492	−0.160246	+	0.987077i	$0.551229\pi$
$74$	141.398	0.222125
$75$	0	0
$76$	521.594	0.787249
$77$	153.649	0.227402
$78$	0	0
$79$	−1315.89	−1.87405	−0.937024	−	0.349266i	$-0.886431\pi$
$79$	−1315.89	−1.87405	−0.937024	+	0.349266i	$0.886431\pi$
$80$	−63.1980	−0.0883219
$81$	0	0
$82$	666.095	0.897048
$83$	686.802	0.908269	0.454134	−	0.890933i	$-0.349949\pi$
$83$	686.802	0.908269	0.454134	+	0.890933i	$0.349949\pi$
$84$	0	0
$85$	−361.805	−0.461685
$86$	−164.602	−0.206389
$87$	0	0
$88$	175.599	0.212715
$89$	−393.539	−0.468708	−0.234354	−	0.972151i	$-0.575298\pi$
$89$	−393.539	−0.468708	−0.234354	+	0.972151i	$0.575298\pi$
$90$	0	0
$91$	−277.895	−0.320124
$92$	366.997	0.415893
$93$	0	0
$94$	1184.99	1.30024
$95$	−515.058	−0.556251
$96$	0	0
$97$	−953.805	−0.998394	−0.499197	−	0.866489i	$-0.666372\pi$
$97$	−953.805	−0.998394	−0.499197	+	0.866489i	$0.666372\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−437.594	−0.437594
$101$	665.604	0.655743	0.327872	−	0.944722i	$-0.393669\pi$
$101$	665.604	0.655743	0.327872	+	0.944722i	$0.393669\pi$
$102$	0	0
$103$	−11.5113	−0.0110121	−0.00550603	−	0.999985i	$-0.501753\pi$
$103$	−11.5113	−0.0110121	−0.00550603	+	0.999985i	$0.501753\pi$
$104$	−317.594	−0.299449
$105$	0	0
$106$	−78.3960	−0.0718348
$107$	529.093	0.478031	0.239016	−	0.971016i	$-0.423175\pi$
$107$	529.093	0.478031	0.239016	+	0.971016i	$0.423175\pi$
$108$	0	0
$109$	658.098	0.578297	0.289148	−	0.957284i	$-0.406628\pi$
$109$	658.098	0.578297	0.289148	+	0.957284i	$0.406628\pi$
$110$	−173.398	−0.150299
$111$	0	0
$112$	112.000	0.0944911
$113$	−1531.90	−1.27530	−0.637650	−	0.770326i	$-0.720094\pi$
$113$	−1531.90	−1.27530	−0.637650	+	0.770326i	$0.720094\pi$
$114$	0	0
$115$	−362.398	−0.293859
$116$	−276.792	−0.221547
$117$	0	0
$118$	−798.586	−0.623016
$119$	641.193	0.493933
$120$	0	0
$121$	−849.203	−0.638019
$122$	−251.609	−0.186718
$123$	0	0
$124$	498.797	0.361236
$125$	925.845	0.662480
$126$	0	0
$127$	1188.48	0.830399	0.415199	−	0.909730i	$-0.363712\pi$
$127$	1188.48	0.830399	0.415199	+	0.909730i	$0.363712\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	313.614	0.211583
$131$	−2420.30	−1.61422	−0.807108	−	0.590404i	$-0.798968\pi$
$131$	−2420.30	−1.61422	−0.807108	+	0.590404i	$0.798968\pi$
$132$	0	0
$133$	912.789	0.595104
$134$	−1011.79	−0.652278
$135$	0	0
$136$	732.792	0.462032
$137$	−725.810	−0.452628	−0.226314	−	0.974054i	$-0.572668\pi$
$137$	−725.810	−0.452628	−0.226314	+	0.974054i	$0.572668\pi$
$138$	0	0
$139$	−1692.99	−1.03308	−0.516539	−	0.856264i	$-0.672780\pi$
$139$	−1692.99	−1.03308	−0.516539	+	0.856264i	$0.672780\pi$
$140$	−110.596	−0.0667651
$141$	0	0
$142$	1839.28	1.08696
$143$	−871.393	−0.509577
$144$	0	0
$145$	273.323	0.156540
$146$	−399.789	−0.226622
$147$	0	0
$148$	282.797	0.157066
$149$	−1962.51	−1.07903	−0.539513	−	0.841977i	$-0.681392\pi$
$149$	−1962.51	−1.07903	−0.539513	+	0.841977i	$0.681392\pi$
$150$	0	0
$151$	−221.789	−0.119530	−0.0597648	−	0.998212i	$-0.519035\pi$
$151$	−221.789	−0.119530	−0.0597648	+	0.998212i	$0.519035\pi$
$152$	1043.19	0.556669
$153$	0	0
$154$	307.298	0.160797
$155$	−492.546	−0.255240
$156$	0	0
$157$	170.316	0.0865776	0.0432888	−	0.999063i	$-0.486216\pi$
$157$	170.316	0.0865776	0.0432888	+	0.999063i	$0.486216\pi$
$158$	−2631.79	−1.32515
$159$	0	0
$160$	−126.396	−0.0624530
$161$	642.246	0.314385
$162$	0	0
$163$	−2232.09	−1.07258	−0.536291	−	0.844033i	$-0.680175\pi$
$163$	−2232.09	−1.07258	−0.536291	+	0.844033i	$0.680175\pi$
$164$	1332.19	0.634309
$165$	0	0
$166$	1373.60	0.642243
$167$	−2343.09	−1.08571	−0.542855	−	0.839826i	$-0.682657\pi$
$167$	−2343.09	−1.08571	−0.542855	+	0.839826i	$0.682657\pi$
$168$	0	0
$169$	−620.970	−0.282644
$170$	−723.609	−0.326460
$171$	0	0
$172$	−329.203	−0.145939
$173$	416.842	0.183190	0.0915951	−	0.995796i	$-0.470803\pi$
$173$	416.842	0.183190	0.0915951	+	0.995796i	$0.470803\pi$
$174$	0	0
$175$	−765.789	−0.330790
$176$	351.198	0.150412
$177$	0	0
$178$	−787.078	−0.331427
$179$	−3526.69	−1.47261	−0.736304	−	0.676651i	$-0.763431\pi$
$179$	−3526.69	−1.47261	−0.736304	+	0.676651i	$0.763431\pi$
$180$	0	0
$181$	1256.59	0.516029	0.258015	−	0.966141i	$-0.416932\pi$
$181$	1256.59	0.516029	0.258015	+	0.966141i	$0.416932\pi$
$182$	−555.789	−0.226362
$183$	0	0
$184$	733.995	0.294081
$185$	−279.253	−0.110979
$186$	0	0
$187$	2010.59	0.786249
$188$	2369.98	0.919409
$189$	0	0
$190$	−1030.12	−0.393329
$191$	242.256	0.0917749	0.0458874	−	0.998947i	$-0.485388\pi$
$191$	242.256	0.0917749	0.0458874	+	0.998947i	$0.485388\pi$
$192$	0	0
$193$	4815.74	1.79609	0.898044	−	0.439906i	$-0.144988\pi$
$193$	4815.74	1.79609	0.898044	+	0.439906i	$0.144988\pi$
$194$	−1907.61	−0.705971
$195$	0	0
$196$	196.000	0.0714286
$197$	785.113	0.283944	0.141972	−	0.989871i	$-0.454656\pi$
$197$	785.113	0.283944	0.141972	+	0.989871i	$0.454656\pi$
$198$	0	0
$199$	3605.26	1.28427	0.642135	−	0.766591i	$-0.278049\pi$
$199$	3605.26	1.28427	0.642135	+	0.766591i	$0.278049\pi$
$200$	−875.188	−0.309426
$201$	0	0
$202$	1331.21	0.463681
$203$	−484.386	−0.167474
$204$	0	0
$205$	−1315.50	−0.448187
$206$	−23.0226	−0.00778671
$207$	0	0
$208$	−635.188	−0.211742
$209$	2862.23	0.947295
$210$	0	0
$211$	−4699.56	−1.53332	−0.766662	−	0.642051i	$-0.778084\pi$
$211$	−4699.56	−1.53332	−0.766662	+	0.642051i	$0.778084\pi$
$212$	−156.792	−0.0507949
$213$	0	0
$214$	1058.19	0.338019
$215$	325.078	0.103117
$216$	0	0
$217$	872.895	0.273069
$218$	1316.20	0.408917
$219$	0	0
$220$	−346.797	−0.106277
$221$	−3636.41	−1.10684
$222$	0	0
$223$	−920.534	−0.276428	−0.138214	−	0.990402i	$-0.544136\pi$
$223$	−920.534	−0.276428	−0.138214	+	0.990402i	$0.544136\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−3063.80	−0.901774
$227$	5249.90	1.53502	0.767508	−	0.641040i	$-0.221497\pi$
$227$	5249.90	1.53502	0.767508	+	0.641040i	$0.221497\pi$
$228$	0	0
$229$	−2470.56	−0.712921	−0.356461	−	0.934310i	$-0.616017\pi$
$229$	−2470.56	−0.712921	−0.356461	+	0.934310i	$0.616017\pi$
$230$	−724.797	−0.207790
$231$	0	0
$232$	−553.584	−0.156658
$233$	−5531.45	−1.55527	−0.777634	−	0.628718i	$-0.783580\pi$
$233$	−5531.45	−1.55527	−0.777634	+	0.628718i	$0.783580\pi$
$234$	0	0
$235$	−2340.29	−0.649632
$236$	−1597.17	−0.440539
$237$	0	0
$238$	1282.39	0.349264
$239$	3614.87	0.978353	0.489176	−	0.872185i	$-0.337297\pi$
$239$	3614.87	0.978353	0.489176	+	0.872185i	$0.337297\pi$
$240$	0	0
$241$	4198.09	1.12209	0.561043	−	0.827787i	$-0.310400\pi$
$241$	4198.09	1.12209	0.561043	+	0.827787i	$0.310400\pi$
$242$	−1698.41	−0.451147
$243$	0	0
$244$	−503.218	−0.132030
$245$	−193.544	−0.0504696
$246$	0	0
$247$	−5176.72	−1.33355
$248$	997.594	0.255433
$249$	0	0
$250$	1851.69	0.468444
$251$	1540.21	0.387320	0.193660	−	0.981069i	$-0.437964\pi$
$251$	1540.21	0.387320	0.193660	+	0.981069i	$0.437964\pi$
$252$	0	0
$253$	2013.89	0.500443
$254$	2376.96	0.587181
$255$	0	0
$256$	256.000	0.0625000
$257$	−5552.42	−1.34767	−0.673833	−	0.738884i	$-0.735353\pi$
$257$	−5552.42	−1.34767	−0.673833	+	0.738884i	$0.735353\pi$
$258$	0	0
$259$	494.895	0.118731
$260$	627.228	0.149612
$261$	0	0
$262$	−4840.59	−1.14142
$263$	−3474.85	−0.814708	−0.407354	−	0.913270i	$-0.633549\pi$
$263$	−3474.85	−0.814708	−0.407354	+	0.913270i	$0.633549\pi$
$264$	0	0
$265$	154.827	0.0358904
$266$	1825.58	0.420802
$267$	0	0
$268$	−2023.58	−0.461230
$269$	1021.16	0.231455	0.115728	−	0.993281i	$-0.463080\pi$
$269$	1021.16	0.231455	0.115728	+	0.993281i	$0.463080\pi$
$270$	0	0
$271$	−530.165	−0.118839	−0.0594193	−	0.998233i	$-0.518925\pi$
$271$	−530.165	−0.118839	−0.0594193	+	0.998233i	$0.518925\pi$
$272$	1465.58	0.326706
$273$	0	0
$274$	−1451.62	−0.320057
$275$	−2401.28	−0.526556
$276$	0	0
$277$	352.714	0.0765074	0.0382537	−	0.999268i	$-0.487820\pi$
$277$	352.714	0.0765074	0.0382537	+	0.999268i	$0.487820\pi$
$278$	−3385.98	−0.730496
$279$	0	0
$280$	−221.193	−0.0472100
$281$	3503.49	0.743775	0.371888	−	0.928278i	$-0.378711\pi$
$281$	3503.49	0.743775	0.371888	+	0.928278i	$0.378711\pi$
$282$	0	0
$283$	1290.17	0.270998	0.135499	−	0.990777i	$-0.456736\pi$
$283$	1290.17	0.270998	0.135499	+	0.990777i	$0.456736\pi$
$284$	3678.56	0.768599
$285$	0	0
$286$	−1742.79	−0.360326
$287$	2331.33	0.479492
$288$	0	0
$289$	3477.38	0.707791
$290$	546.647	0.110690
$291$	0	0
$292$	−799.579	−0.160246
$293$	8929.33	1.78040	0.890200	−	0.455570i	$-0.150564\pi$
$293$	8929.33	1.78040	0.890200	+	0.455570i	$0.150564\pi$
$294$	0	0
$295$	1577.16	0.311274
$296$	565.594	0.111062
$297$	0	0
$298$	−3925.01	−0.762986
$299$	−3642.38	−0.704496
$300$	0	0
$301$	−576.105	−0.110319
$302$	−443.579	−0.0845202
$303$	0	0
$304$	2086.38	0.393625
$305$	496.912	0.0932889
$306$	0	0
$307$	−8240.70	−1.53199	−0.765996	−	0.642845i	$-0.777754\pi$
$307$	−8240.70	−1.53199	−0.765996	+	0.642845i	$0.777754\pi$
$308$	614.596	0.113701
$309$	0	0
$310$	−985.093	−0.180482
$311$	3492.65	0.636816	0.318408	−	0.947954i	$-0.396852\pi$
$311$	3492.65	0.636816	0.318408	+	0.947954i	$0.396852\pi$
$312$	0	0
$313$	778.707	0.140623	0.0703117	−	0.997525i	$-0.477601\pi$
$313$	778.707	0.140623	0.0703117	+	0.997525i	$0.477601\pi$
$314$	340.632	0.0612196
$315$	0	0
$316$	−5263.58	−0.937024
$317$	5027.05	0.890686	0.445343	−	0.895360i	$-0.353082\pi$
$317$	5027.05	0.890686	0.445343	+	0.895360i	$0.353082\pi$
$318$	0	0
$319$	−1518.89	−0.266587
$320$	−252.792	−0.0441609
$321$	0	0
$322$	1284.49	0.222304
$323$	11944.4	2.05759
$324$	0	0
$325$	4343.04	0.741257
$326$	−4464.18	−0.758430
$327$	0	0
$328$	2664.38	0.448524
$329$	4147.47	0.695008
$330$	0	0
$331$	8074.89	1.34089	0.670447	−	0.741957i	$-0.266102\pi$
$331$	8074.89	1.34089	0.670447	+	0.741957i	$0.266102\pi$
$332$	2747.21	0.454134
$333$	0	0
$334$	−4686.18	−0.767713
$335$	1998.22	0.325894
$336$	0	0
$337$	11525.5	1.86301	0.931504	−	0.363731i	$-0.118497\pi$
$337$	11525.5	1.86301	0.931504	+	0.363731i	$0.118497\pi$
$338$	−1241.94	−0.199860
$339$	0	0
$340$	−1447.22	−0.230842
$341$	2737.13	0.434675
$342$	0	0
$343$	343.000	0.0539949
$344$	−658.406	−0.103194
$345$	0	0
$346$	833.684	0.129535
$347$	−5326.24	−0.823999	−0.411999	−	0.911184i	$-0.635169\pi$
$347$	−5326.24	−0.823999	−0.411999	+	0.911184i	$0.635169\pi$
$348$	0	0
$349$	−939.414	−0.144085	−0.0720425	−	0.997402i	$-0.522952\pi$
$349$	−939.414	−0.144085	−0.0720425	+	0.997402i	$0.522952\pi$
$350$	−1531.58	−0.233904
$351$	0	0
$352$	702.396	0.106357
$353$	8340.33	1.25754	0.628769	−	0.777592i	$-0.283559\pi$
$353$	8340.33	1.25754	0.628769	+	0.777592i	$0.283559\pi$
$354$	0	0
$355$	−3632.46	−0.543073
$356$	−1574.16	−0.234354
$357$	0	0
$358$	−7053.37	−1.04129
$359$	−3504.71	−0.515241	−0.257621	−	0.966246i	$-0.582938\pi$
$359$	−3504.71	−0.515241	−0.257621	+	0.966246i	$0.582938\pi$
$360$	0	0
$361$	10144.8	1.47904
$362$	2513.17	0.364888
$363$	0	0
$364$	−1111.58	−0.160062
$365$	789.559	0.113226
$366$	0	0
$367$	−5977.21	−0.850158	−0.425079	−	0.905156i	$-0.639754\pi$
$367$	−5977.21	−0.850158	−0.425079	+	0.905156i	$0.639754\pi$
$368$	1467.99	0.207946
$369$	0	0
$370$	−558.506	−0.0784739
$371$	−274.386	−0.0383973
$372$	0	0
$373$	−7523.41	−1.04436	−0.522181	−	0.852835i	$-0.674882\pi$
$373$	−7523.41	−1.04436	−0.522181	+	0.852835i	$0.674882\pi$
$374$	4021.17	0.555962
$375$	0	0
$376$	4739.97	0.650120
$377$	2747.11	0.375287
$378$	0	0
$379$	10603.6	1.43713	0.718564	−	0.695461i	$-0.244800\pi$
$379$	10603.6	1.43713	0.718564	+	0.695461i	$0.244800\pi$
$380$	−2060.23	−0.278125
$381$	0	0
$382$	484.511	0.0648946
$383$	1050.00	0.140085	0.0700427	−	0.997544i	$-0.477686\pi$
$383$	1050.00	0.140085	0.0700427	+	0.997544i	$0.477686\pi$
$384$	0	0
$385$	−606.895	−0.0803382
$386$	9631.49	1.27003
$387$	0	0
$388$	−3815.22	−0.499197
$389$	−12986.8	−1.69269	−0.846343	−	0.532639i	$-0.821201\pi$
$389$	−12986.8	−1.69269	−0.846343	+	0.532639i	$0.821201\pi$
$390$	0	0
$391$	8404.15	1.08700
$392$	392.000	0.0505076
$393$	0	0
$394$	1570.23	0.200779
$395$	5197.62	0.662077
$396$	0	0
$397$	11194.3	1.41518	0.707589	−	0.706624i	$-0.249783\pi$
$397$	11194.3	1.41518	0.707589	+	0.706624i	$0.249783\pi$
$398$	7210.51	0.908116
$399$	0	0
$400$	−1750.38	−0.218797
$401$	10903.5	1.35784	0.678919	−	0.734213i	$-0.262449\pi$
$401$	10903.5	1.35784	0.678919	+	0.734213i	$0.262449\pi$
$402$	0	0
$403$	−4950.47	−0.611911
$404$	2662.42	0.327872
$405$	0	0
$406$	−968.772	−0.118422
$407$	1551.84	0.188997
$408$	0	0
$409$	754.151	0.0911744	0.0455872	−	0.998960i	$-0.485484\pi$
$409$	754.151	0.0911744	0.0455872	+	0.998960i	$0.485484\pi$
$410$	−2630.99	−0.316916
$411$	0	0
$412$	−46.0452	−0.00550603
$413$	−2795.05	−0.333016
$414$	0	0
$415$	−2712.78	−0.320880
$416$	−1270.38	−0.149724
$417$	0	0
$418$	5724.46	0.669839
$419$	7798.09	0.909216	0.454608	−	0.890692i	$-0.349779\pi$
$419$	7798.09	0.909216	0.454608	+	0.890692i	$0.349779\pi$
$420$	0	0
$421$	14024.8	1.62358	0.811792	−	0.583947i	$-0.198492\pi$
$421$	14024.8	1.62358	0.811792	+	0.583947i	$0.198492\pi$
$422$	−9399.13	−1.08422
$423$	0	0
$424$	−313.584	−0.0359174
$425$	−10020.8	−1.14372
$426$	0	0
$427$	−880.632	−0.0998050
$428$	2116.37	0.239016
$429$	0	0
$430$	650.155	0.0729146
$431$	−17663.1	−1.97402	−0.987008	−	0.160674i	$-0.948633\pi$
$431$	−17663.1	−1.97402	−0.987008	+	0.160674i	$0.948633\pi$
$432$	0	0
$433$	3670.68	0.407394	0.203697	−	0.979034i	$-0.434704\pi$
$433$	3670.68	0.407394	0.203697	+	0.979034i	$0.434704\pi$
$434$	1745.79	0.193089
$435$	0	0
$436$	2632.39	0.289148
$437$	11964.0	1.30964
$438$	0	0
$439$	−8504.78	−0.924627	−0.462313	−	0.886717i	$-0.652980\pi$
$439$	−8504.78	−0.924627	−0.462313	+	0.886717i	$0.652980\pi$
$440$	−693.594	−0.0751495
$441$	0	0
$442$	−7272.82	−0.782654
$443$	−15151.2	−1.62495	−0.812476	−	0.582994i	$-0.801881\pi$
$443$	−15151.2	−1.62495	−0.812476	+	0.582994i	$0.801881\pi$
$444$	0	0
$445$	1554.43	0.165589
$446$	−1841.07	−0.195464
$447$	0	0
$448$	448.000	0.0472456
$449$	−2288.00	−0.240484	−0.120242	−	0.992745i	$-0.538367\pi$
$449$	−2288.00	−0.240484	−0.120242	+	0.992745i	$0.538367\pi$
$450$	0	0
$451$	7310.35	0.763262
$452$	−6127.60	−0.637650
$453$	0	0
$454$	10499.8	1.08542
$455$	1097.65	0.113096
$456$	0	0
$457$	2427.89	0.248516	0.124258	−	0.992250i	$-0.460345\pi$
$457$	2427.89	0.248516	0.124258	+	0.992250i	$0.460345\pi$
$458$	−4941.11	−0.504111
$459$	0	0
$460$	−1449.59	−0.146930
$461$	−10377.8	−1.04847	−0.524233	−	0.851575i	$-0.675648\pi$
$461$	−10377.8	−1.04847	−0.524233	+	0.851575i	$0.675648\pi$
$462$	0	0
$463$	−17448.9	−1.75144	−0.875722	−	0.482816i	$-0.839614\pi$
$463$	−17448.9	−1.75144	−0.875722	+	0.482816i	$0.839614\pi$
$464$	−1107.17	−0.110774
$465$	0	0
$466$	−11062.9	−1.09974
$467$	−2126.50	−0.210713	−0.105356	−	0.994435i	$-0.533598\pi$
$467$	−2126.50	−0.210713	−0.105356	+	0.994435i	$0.533598\pi$
$468$	0	0
$469$	−3541.26	−0.348657
$470$	−4680.57	−0.459359
$471$	0	0
$472$	−3194.35	−0.311508
$473$	−1806.49	−0.175608
$474$	0	0
$475$	−14265.4	−1.37798
$476$	2564.77	0.246967
$477$	0	0
$478$	7229.73	0.691800
$479$	141.092	0.0134586	0.00672931	−	0.999977i	$-0.497858\pi$
$479$	141.092	0.0134586	0.00672931	+	0.999977i	$0.497858\pi$
$480$	0	0
$481$	−2806.71	−0.266060
$482$	8396.18	0.793435
$483$	0	0
$484$	−3396.81	−0.319009
$485$	3767.41	0.352720
$486$	0	0
$487$	11451.4	1.06553	0.532764	−	0.846264i	$-0.321154\pi$
$487$	11451.4	1.06553	0.532764	+	0.846264i	$0.321154\pi$
$488$	−1006.44	−0.0933591
$489$	0	0
$490$	−387.088	−0.0356874
$491$	−6569.93	−0.603863	−0.301932	−	0.953330i	$-0.597631\pi$
$491$	−6569.93	−0.603863	−0.301932	+	0.953330i	$0.597631\pi$
$492$	0	0
$493$	−6338.47	−0.579047
$494$	−10353.4	−0.942962
$495$	0	0
$496$	1995.19	0.180618
$497$	6437.47	0.581006
$498$	0	0
$499$	−5324.53	−0.477672	−0.238836	−	0.971060i	$-0.576766\pi$
$499$	−5324.53	−0.477672	−0.238836	+	0.971060i	$0.576766\pi$
$500$	3703.38	0.331240
$501$	0	0
$502$	3080.42	0.273876
$503$	18499.6	1.63987	0.819936	−	0.572455i	$-0.194009\pi$
$503$	18499.6	1.63987	0.819936	+	0.572455i	$0.194009\pi$
$504$	0	0
$505$	−2629.05	−0.231666
$506$	4027.77	0.353866
$507$	0	0
$508$	4753.92	0.415199
$509$	−19841.9	−1.72786	−0.863928	−	0.503616i	$-0.832003\pi$
$509$	−19841.9	−1.72786	−0.863928	+	0.503616i	$0.832003\pi$
$510$	0	0
$511$	−1399.26	−0.121135
$512$	512.000	0.0441942
$513$	0	0
$514$	−11104.8	−0.952944
$515$	45.4682	0.00389043
$516$	0	0
$517$	13005.2	1.10632
$518$	989.789	0.0839553
$519$	0	0
$520$	1254.46	0.105791
$521$	−17985.9	−1.51243	−0.756215	−	0.654323i	$-0.772954\pi$
$521$	−17985.9	−1.51243	−0.756215	+	0.654323i	$0.772954\pi$
$522$	0	0
$523$	−16949.2	−1.41709	−0.708545	−	0.705666i	$-0.750648\pi$
$523$	−16949.2	−1.41709	−0.708545	+	0.705666i	$0.750648\pi$
$524$	−9681.18	−0.807108
$525$	0	0
$526$	−6949.69	−0.576086
$527$	11422.3	0.944145
$528$	0	0
$529$	−3749.05	−0.308133
$530$	309.654	0.0253783
$531$	0	0
$532$	3651.16	0.297552
$533$	−13221.7	−1.07448
$534$	0	0
$535$	−2089.85	−0.168882
$536$	−4047.16	−0.326139
$537$	0	0
$538$	2042.33	0.163663
$539$	1075.54	0.0859498
$540$	0	0
$541$	18599.2	1.47808	0.739041	−	0.673660i	$-0.235279\pi$
$541$	18599.2	1.47808	0.739041	+	0.673660i	$0.235279\pi$
$542$	−1060.33	−0.0840316
$543$	0	0
$544$	2931.17	0.231016
$545$	−2599.40	−0.204305
$546$	0	0
$547$	9774.11	0.764005	0.382002	−	0.924161i	$-0.375235\pi$
$547$	9774.11	0.764005	0.382002	+	0.924161i	$0.375235\pi$
$548$	−2903.24	−0.226314
$549$	0	0
$550$	−4802.57	−0.372331
$551$	−9023.31	−0.697652
$552$	0	0
$553$	−9211.26	−0.708323
$554$	705.429	0.0540989
$555$	0	0
$556$	−6771.97	−0.516539
$557$	−21311.3	−1.62116	−0.810581	−	0.585627i	$-0.800848\pi$
$557$	−21311.3	−1.62116	−0.810581	+	0.585627i	$0.800848\pi$
$558$	0	0
$559$	3267.28	0.247211
$560$	−442.386	−0.0333825
$561$	0	0
$562$	7006.99	0.525929
$563$	6186.09	0.463077	0.231539	−	0.972826i	$-0.425624\pi$
$563$	6186.09	0.463077	0.231539	+	0.972826i	$0.425624\pi$
$564$	0	0
$565$	6050.81	0.450548
$566$	2580.33	0.191624
$567$	0	0
$568$	7357.11	0.543482
$569$	−12081.2	−0.890104	−0.445052	−	0.895505i	$-0.646815\pi$
$569$	−12081.2	−0.890104	−0.445052	+	0.895505i	$0.646815\pi$
$570$	0	0
$571$	−18005.1	−1.31960	−0.659800	−	0.751442i	$-0.729359\pi$
$571$	−18005.1	−1.31960	−0.659800	+	0.751442i	$0.729359\pi$
$572$	−3485.57	−0.254789
$573$	0	0
$574$	4662.67	0.339052
$575$	−10037.2	−0.727969
$576$	0	0
$577$	−6957.31	−0.501970	−0.250985	−	0.967991i	$-0.580754\pi$
$577$	−6957.31	−0.501970	−0.250985	+	0.967991i	$0.580754\pi$
$578$	6954.75	0.500484
$579$	0	0
$580$	1093.29	0.0782699
$581$	4807.61	0.343293
$582$	0	0
$583$	−860.391	−0.0611214
$584$	−1599.16	−0.113311
$585$	0	0
$586$	17858.7	1.25893
$587$	1638.67	0.115221	0.0576107	−	0.998339i	$-0.481652\pi$
$587$	1638.67	0.115221	0.0576107	+	0.998339i	$0.481652\pi$
$588$	0	0
$589$	16260.6	1.13753
$590$	3154.32	0.220104
$591$	0	0
$592$	1131.19	0.0785330
$593$	18419.5	1.27554	0.637771	−	0.770226i	$-0.279857\pi$
$593$	18419.5	1.27554	0.637771	+	0.770226i	$0.279857\pi$
$594$	0	0
$595$	−2532.63	−0.174500
$596$	−7850.03	−0.539513
$597$	0	0
$598$	−7284.76	−0.498154
$599$	15906.9	1.08504	0.542519	−	0.840044i	$-0.317471\pi$
$599$	15906.9	1.08504	0.542519	+	0.840044i	$0.317471\pi$
$600$	0	0
$601$	−26144.3	−1.77445	−0.887227	−	0.461333i	$-0.847371\pi$
$601$	−26144.3	−1.77445	−0.887227	+	0.461333i	$0.847371\pi$
$602$	−1152.21	−0.0780076
$603$	0	0
$604$	−887.158	−0.0597648
$605$	3354.25	0.225404
$606$	0	0
$607$	−22984.5	−1.53692	−0.768461	−	0.639897i	$-0.778977\pi$
$607$	−22984.5	−1.53692	−0.768461	+	0.639897i	$0.778977\pi$
$608$	4172.75	0.278335
$609$	0	0
$610$	993.824	0.0659652
$611$	−23521.7	−1.55742
$612$	0	0
$613$	−28133.8	−1.85369	−0.926846	−	0.375441i	$-0.877491\pi$
$613$	−28133.8	−1.85369	−0.926846	+	0.375441i	$0.877491\pi$
$614$	−16481.4	−1.08328
$615$	0	0
$616$	1229.19	0.0803987
$617$	25254.8	1.64785	0.823923	−	0.566702i	$-0.191781\pi$
$617$	25254.8	1.64785	0.823923	+	0.566702i	$0.191781\pi$
$618$	0	0
$619$	−16262.7	−1.05598	−0.527992	−	0.849250i	$-0.677055\pi$
$619$	−16262.7	−1.05598	−0.527992	+	0.849250i	$0.677055\pi$
$620$	−1970.19	−0.127620
$621$	0	0
$622$	6985.29	0.450297
$623$	−2754.77	−0.177155
$624$	0	0
$625$	10017.8	0.641142
$626$	1557.41	0.0994357
$627$	0	0
$628$	681.263	0.0432888
$629$	6475.98	0.410515
$630$	0	0
$631$	7489.19	0.472488	0.236244	−	0.971694i	$-0.424084\pi$
$631$	7489.19	0.472488	0.236244	+	0.971694i	$0.424084\pi$
$632$	−10527.2	−0.662576
$633$	0	0
$634$	10054.1	0.629810
$635$	−4694.35	−0.293370
$636$	0	0
$637$	−1945.26	−0.120995
$638$	−3037.77	−0.188506
$639$	0	0
$640$	−505.584	−0.0312265
$641$	−19555.8	−1.20500	−0.602502	−	0.798117i	$-0.705830\pi$
$641$	−19555.8	−1.20500	−0.602502	+	0.798117i	$0.705830\pi$
$642$	0	0
$643$	−9939.92	−0.609630	−0.304815	−	0.952412i	$-0.598595\pi$
$643$	−9939.92	−0.609630	−0.304815	+	0.952412i	$0.598595\pi$
$644$	2568.98	0.157193
$645$	0	0
$646$	23888.7	1.45494
$647$	−2343.39	−0.142393	−0.0711965	−	0.997462i	$-0.522682\pi$
$647$	−2343.39	−0.142393	−0.0711965	+	0.997462i	$0.522682\pi$
$648$	0	0
$649$	−8764.44	−0.530099
$650$	8686.08	0.524148
$651$	0	0
$652$	−8928.36	−0.536291
$653$	−3117.98	−0.186854	−0.0934272	−	0.995626i	$-0.529782\pi$
$653$	−3117.98	−0.186854	−0.0934272	+	0.995626i	$0.529782\pi$
$654$	0	0
$655$	9559.86	0.570282
$656$	5328.76	0.317154
$657$	0	0
$658$	8294.95	0.491445
$659$	4568.28	0.270038	0.135019	−	0.990843i	$-0.456891\pi$
$659$	4568.28	0.270038	0.135019	+	0.990843i	$0.456891\pi$
$660$	0	0
$661$	28322.1	1.66657	0.833285	−	0.552843i	$-0.186457\pi$
$661$	28322.1	1.66657	0.833285	+	0.552843i	$0.186457\pi$
$662$	16149.8	0.948155
$663$	0	0
$664$	5494.42	0.321122
$665$	−3605.40	−0.210243
$666$	0	0
$667$	−6348.87	−0.368560
$668$	−9372.35	−0.542855
$669$	0	0
$670$	3996.44	0.230442
$671$	−2761.39	−0.158871
$672$	0	0
$673$	8604.29	0.492824	0.246412	−	0.969165i	$-0.420748\pi$
$673$	8604.29	0.492824	0.246412	+	0.969165i	$0.420748\pi$
$674$	23051.0	1.31735
$675$	0	0
$676$	−2483.88	−0.141322
$677$	18174.0	1.03173	0.515867	−	0.856669i	$-0.327470\pi$
$677$	18174.0	1.03173	0.515867	+	0.856669i	$0.327470\pi$
$678$	0	0
$679$	−6676.63	−0.377357
$680$	−2894.44	−0.163230
$681$	0	0
$682$	5474.27	0.307361
$683$	−26536.9	−1.48669	−0.743343	−	0.668911i	$-0.766761\pi$
$683$	−26536.9	−1.48669	−0.743343	+	0.668911i	$0.766761\pi$
$684$	0	0
$685$	2866.86	0.159908
$686$	686.000	0.0381802
$687$	0	0
$688$	−1316.81	−0.0729695
$689$	1556.13	0.0860433
$690$	0	0
$691$	−21463.3	−1.18162	−0.590812	−	0.806810i	$-0.701192\pi$
$691$	−21463.3	−1.18162	−0.590812	+	0.806810i	$0.701192\pi$
$692$	1667.37	0.0915951
$693$	0	0
$694$	−10652.5	−0.582655
$695$	6687.11	0.364973
$696$	0	0
$697$	30506.8	1.65786
$698$	−1878.83	−0.101883
$699$	0	0
$700$	−3063.16	−0.165395
$701$	11648.9	0.627639	0.313819	−	0.949483i	$-0.398391\pi$
$701$	11648.9	0.627639	0.313819	+	0.949483i	$0.398391\pi$
$702$	0	0
$703$	9219.08	0.494600
$704$	1404.79	0.0752061
$705$	0	0
$706$	16680.7	0.889213
$707$	4659.23	0.247848
$708$	0	0
$709$	−11918.3	−0.631311	−0.315656	−	0.948874i	$-0.602224\pi$
$709$	−11918.3	−0.631311	−0.315656	+	0.948874i	$0.602224\pi$
$710$	−7264.92	−0.384011
$711$	0	0
$712$	−3148.31	−0.165713
$713$	11441.1	0.600942
$714$	0	0
$715$	3441.89	0.180027
$716$	−14106.7	−0.736304
$717$	0	0
$718$	−7009.42	−0.364331
$719$	36229.6	1.87919	0.939595	−	0.342288i	$-0.111202\pi$
$719$	36229.6	1.87919	0.939595	+	0.342288i	$0.111202\pi$
$720$	0	0
$721$	−80.5791	−0.00416217
$722$	20289.5	1.04584
$723$	0	0
$724$	5026.35	0.258015
$725$	7570.16	0.387791
$726$	0	0
$727$	22972.9	1.17196	0.585982	−	0.810324i	$-0.300709\pi$
$727$	22972.9	1.17196	0.585982	+	0.810324i	$0.300709\pi$
$728$	−2223.16	−0.113181
$729$	0	0
$730$	1579.12	0.0800627
$731$	−7538.67	−0.381433
$732$	0	0
$733$	−10209.7	−0.514466	−0.257233	−	0.966349i	$-0.582811\pi$
$733$	−10209.7	−0.514466	−0.257233	+	0.966349i	$0.582811\pi$
$734$	−11954.4	−0.601152
$735$	0	0
$736$	2935.98	0.147040
$737$	−11104.3	−0.554997
$738$	0	0
$739$	−17101.7	−0.851280	−0.425640	−	0.904893i	$-0.639951\pi$
$739$	−17101.7	−0.851280	−0.425640	+	0.904893i	$0.639951\pi$
$740$	−1117.01	−0.0554895
$741$	0	0
$742$	−548.772	−0.0271510
$743$	−32574.9	−1.60842	−0.804211	−	0.594344i	$-0.797412\pi$
$743$	−32574.9	−1.60842	−0.804211	+	0.594344i	$0.797412\pi$
$744$	0	0
$745$	7751.65	0.381206
$746$	−15046.8	−0.738476
$747$	0	0
$748$	8042.35	0.393125
$749$	3703.65	0.180679
$750$	0	0
$751$	−14264.4	−0.693097	−0.346548	−	0.938032i	$-0.612646\pi$
$751$	−14264.4	−0.693097	−0.346548	+	0.938032i	$0.612646\pi$
$752$	9479.94	0.459705
$753$	0	0
$754$	5494.22	0.265368
$755$	876.040	0.0422283
$756$	0	0
$757$	720.376	0.0345872	0.0172936	−	0.999850i	$-0.494495\pi$
$757$	720.376	0.0345872	0.0172936	+	0.999850i	$0.494495\pi$
$758$	21207.2	1.01620
$759$	0	0
$760$	−4120.46	−0.196664
$761$	−20313.2	−0.967614	−0.483807	−	0.875175i	$-0.660746\pi$
$761$	−20313.2	−0.967614	−0.483807	+	0.875175i	$0.660746\pi$
$762$	0	0
$763$	4606.68	0.218576
$764$	969.023	0.0458874
$765$	0	0
$766$	2100.01	0.0990554
$767$	15851.6	0.746245
$768$	0	0
$769$	11694.0	0.548371	0.274185	−	0.961677i	$-0.411592\pi$
$769$	11694.0	0.548371	0.274185	+	0.961677i	$0.411592\pi$
$770$	−1213.79	−0.0568077
$771$	0	0
$772$	19263.0	0.898044
$773$	30720.8	1.42943	0.714716	−	0.699415i	$-0.246556\pi$
$773$	30720.8	1.42943	0.714716	+	0.699415i	$0.246556\pi$
$774$	0	0
$775$	−13641.9	−0.632299
$776$	−7630.44	−0.352985
$777$	0	0
$778$	−25973.5	−1.19691
$779$	43428.9	1.99744
$780$	0	0
$781$	20186.0	0.924853
$782$	16808.3	0.768624
$783$	0	0
$784$	784.000	0.0357143
$785$	−672.726	−0.0305868
$786$	0	0
$787$	−18800.5	−0.851545	−0.425772	−	0.904830i	$-0.639998\pi$
$787$	−18800.5	−0.851545	−0.425772	+	0.904830i	$0.639998\pi$
$788$	3140.45	0.141972
$789$	0	0
$790$	10395.2	0.468159
$791$	−10723.3	−0.482018
$792$	0	0
$793$	4994.34	0.223650
$794$	22388.6	1.00068
$795$	0	0
$796$	14421.0	0.642135
$797$	−13169.0	−0.585280	−0.292640	−	0.956223i	$-0.594534\pi$
$797$	−13169.0	−0.585280	−0.292640	+	0.956223i	$0.594534\pi$
$798$	0	0
$799$	54272.1	2.40301
$800$	−3500.75	−0.154713
$801$	0	0
$802$	21806.9	0.960136
$803$	−4387.66	−0.192824
$804$	0	0
$805$	−2536.79	−0.111068
$806$	−9900.93	−0.432687
$807$	0	0
$808$	5324.83	0.231840
$809$	43648.1	1.89689	0.948445	−	0.316941i	$-0.102656\pi$
$809$	43648.1	1.89689	0.948445	+	0.316941i	$0.102656\pi$
$810$	0	0
$811$	−33794.5	−1.46324	−0.731619	−	0.681713i	$-0.761235\pi$
$811$	−33794.5	−1.46324	−0.731619	+	0.681713i	$0.761235\pi$
$812$	−1937.54	−0.0837370
$813$	0	0
$814$	3103.68	0.133641
$815$	8816.48	0.378930
$816$	0	0
$817$	−10731.9	−0.459561
$818$	1508.30	0.0644701
$819$	0	0
$820$	−5261.98	−0.224093
$821$	23524.8	1.00003	0.500013	−	0.866018i	$-0.333329\pi$
$821$	23524.8	1.00003	0.500013	+	0.866018i	$0.333329\pi$
$822$	0	0
$823$	9113.52	0.385999	0.193000	−	0.981199i	$-0.438178\pi$
$823$	9113.52	0.385999	0.193000	+	0.981199i	$0.438178\pi$
$824$	−92.0905	−0.00389335
$825$	0	0
$826$	−5590.11	−0.235478
$827$	39909.5	1.67810	0.839051	−	0.544052i	$-0.183111\pi$
$827$	39909.5	1.67810	0.839051	+	0.544052i	$0.183111\pi$
$828$	0	0
$829$	−18413.9	−0.771461	−0.385731	−	0.922611i	$-0.626051\pi$
$829$	−18413.9	−0.771461	−0.385731	+	0.922611i	$0.626051\pi$
$830$	−5425.56	−0.226896
$831$	0	0
$832$	−2540.75	−0.105871
$833$	4488.35	0.186689
$834$	0	0
$835$	9254.90	0.383568
$836$	11448.9	0.473647
$837$	0	0
$838$	15596.2	0.642913
$839$	13666.1	0.562343	0.281171	−	0.959658i	$-0.409277\pi$
$839$	13666.1	0.562343	0.281171	+	0.959658i	$0.409277\pi$
$840$	0	0
$841$	−19600.6	−0.803667
$842$	28049.7	1.14805
$843$	0	0
$844$	−18798.3	−0.766662
$845$	2452.75	0.0998548
$846$	0	0
$847$	−5944.42	−0.241148
$848$	−627.168	−0.0253974
$849$	0	0
$850$	−20041.6	−0.808730
$851$	6486.61	0.261290
$852$	0	0
$853$	−37360.4	−1.49964	−0.749821	−	0.661641i	$-0.769860\pi$
$853$	−37360.4	−1.49964	−0.749821	+	0.661641i	$0.769860\pi$
$854$	−1761.26	−0.0705728
$855$	0	0
$856$	4232.74	0.169010
$857$	−13673.7	−0.545023	−0.272511	−	0.962153i	$-0.587854\pi$
$857$	−13673.7	−0.545023	−0.272511	+	0.962153i	$0.587854\pi$
$858$	0	0
$859$	−893.452	−0.0354880	−0.0177440	−	0.999843i	$-0.505648\pi$
$859$	−893.452	−0.0354880	−0.0177440	+	0.999843i	$0.505648\pi$
$860$	1300.31	0.0515584
$861$	0	0
$862$	−35326.2	−1.39584
$863$	−17449.6	−0.688285	−0.344142	−	0.938917i	$-0.611830\pi$
$863$	−17449.6	−0.688285	−0.344142	+	0.938917i	$0.611830\pi$
$864$	0	0
$865$	−1646.47	−0.0647188
$866$	7341.35	0.288071
$867$	0	0
$868$	3491.58	0.136534
$869$	−28883.7	−1.12752
$870$	0	0
$871$	20083.6	0.781295
$872$	5264.78	0.204459
$873$	0	0
$874$	23928.0	0.926059
$875$	6480.91	0.250394
$876$	0	0
$877$	6454.45	0.248519	0.124260	−	0.992250i	$-0.460344\pi$
$877$	6454.45	0.248519	0.124260	+	0.992250i	$0.460344\pi$
$878$	−17009.6	−0.653810
$879$	0	0
$880$	−1387.19	−0.0531387
$881$	−21689.1	−0.829427	−0.414714	−	0.909952i	$-0.636118\pi$
$881$	−21689.1	−0.829427	−0.414714	+	0.909952i	$0.636118\pi$
$882$	0	0
$883$	7981.14	0.304175	0.152088	−	0.988367i	$-0.451400\pi$
$883$	7981.14	0.304175	0.152088	+	0.988367i	$0.451400\pi$
$884$	−14545.6	−0.553420
$885$	0	0
$886$	−30302.3	−1.14901
$887$	−20714.5	−0.784133	−0.392066	−	0.919937i	$-0.628240\pi$
$887$	−20714.5	−0.784133	−0.392066	+	0.919937i	$0.628240\pi$
$888$	0	0
$889$	8319.37	0.313861
$890$	3108.86	0.117089
$891$	0	0
$892$	−3682.14	−0.138214
$893$	77260.6	2.89522
$894$	0	0
$895$	13930.0	0.520254
$896$	896.000	0.0334077
$897$	0	0
$898$	−4576.00	−0.170048
$899$	−8628.94	−0.320124
$900$	0	0
$901$	−3590.50	−0.132760
$902$	14620.7	0.539708
$903$	0	0
$904$	−12255.2	−0.450887
$905$	−4963.36	−0.182307
$906$	0	0
$907$	−12412.2	−0.454400	−0.227200	−	0.973848i	$-0.572957\pi$
$907$	−12412.2	−0.454400	−0.227200	+	0.973848i	$0.572957\pi$
$908$	20999.6	0.767508
$909$	0	0
$910$	2195.30	0.0799708
$911$	15009.3	0.545862	0.272931	−	0.962034i	$-0.412007\pi$
$911$	15009.3	0.545862	0.272931	+	0.962034i	$0.412007\pi$
$912$	0	0
$913$	15075.2	0.546459
$914$	4855.77	0.175727
$915$	0	0
$916$	−9882.23	−0.356461
$917$	−16942.1	−0.610116
$918$	0	0
$919$	−8189.62	−0.293962	−0.146981	−	0.989139i	$-0.546956\pi$
$919$	−8189.62	−0.293962	−0.146981	+	0.989139i	$0.546956\pi$
$920$	−2899.19	−0.103895
$921$	0	0
$922$	−20755.6	−0.741378
$923$	−36509.0	−1.30196
$924$	0	0
$925$	−7734.39	−0.274925
$926$	−34897.8	−1.23846
$927$	0	0
$928$	−2214.34	−0.0783288
$929$	−16223.0	−0.572939	−0.286470	−	0.958089i	$-0.592482\pi$
$929$	−16223.0	−0.572939	−0.286470	+	0.958089i	$0.592482\pi$
$930$	0	0
$931$	6389.53	0.224928
$932$	−22125.8	−0.777634
$933$	0	0
$934$	−4253.00	−0.148996
$935$	−7941.56	−0.277772
$936$	0	0
$937$	−38312.1	−1.33575	−0.667877	−	0.744271i	$-0.732797\pi$
$937$	−38312.1	−1.33575	−0.667877	+	0.744271i	$0.732797\pi$
$938$	−7082.53	−0.246538
$939$	0	0
$940$	−9361.14	−0.324816
$941$	−7363.94	−0.255109	−0.127555	−	0.991832i	$-0.540713\pi$
$941$	−7363.94	−0.255109	−0.127555	+	0.991832i	$0.540713\pi$
$942$	0	0
$943$	30556.9	1.05522
$944$	−6388.69	−0.220269
$945$	0	0
$946$	−3612.98	−0.124174
$947$	23385.9	0.802470	0.401235	−	0.915975i	$-0.368581\pi$
$947$	23385.9	0.802470	0.401235	+	0.915975i	$0.368581\pi$
$948$	0	0
$949$	7935.67	0.271447
$950$	−28530.8	−0.974380
$951$	0	0
$952$	5129.54	0.174632
$953$	30666.7	1.04238	0.521191	−	0.853440i	$-0.325488\pi$
$953$	30666.7	1.04238	0.521191	+	0.853440i	$0.325488\pi$
$954$	0	0
$955$	−956.879	−0.0324229
$956$	14459.5	0.489176
$957$	0	0
$958$	282.185	0.00951668
$959$	−5080.67	−0.171077
$960$	0	0
$961$	−14241.1	−0.478034
$962$	−5613.41	−0.188133
$963$	0	0
$964$	16792.4	0.561043
$965$	−19021.6	−0.634535
$966$	0	0
$967$	35356.2	1.17578	0.587890	−	0.808941i	$-0.299959\pi$
$967$	35356.2	1.17578	0.587890	+	0.808941i	$0.299959\pi$
$968$	−6793.62	−0.225574
$969$	0	0
$970$	7534.82	0.249411
$971$	46018.1	1.52090	0.760449	−	0.649398i	$-0.224979\pi$
$971$	46018.1	1.52090	0.760449	+	0.649398i	$0.224979\pi$
$972$	0	0
$973$	−11850.9	−0.390467
$974$	22902.8	0.753441
$975$	0	0
$976$	−2012.87	−0.0660148
$977$	23244.5	0.761166	0.380583	−	0.924747i	$-0.375723\pi$
$977$	23244.5	0.761166	0.380583	+	0.924747i	$0.375723\pi$
$978$	0	0
$979$	−8638.13	−0.281998
$980$	−774.175	−0.0252348
$981$	0	0
$982$	−13139.9	−0.426996
$983$	−47392.4	−1.53772	−0.768862	−	0.639414i	$-0.779177\pi$
$983$	−47392.4	−1.53772	−0.768862	+	0.639414i	$0.779177\pi$
$984$	0	0
$985$	−3101.10	−0.100314
$986$	−12676.9	−0.409448
$987$	0	0
$988$	−20706.9	−0.666775
$989$	−7551.04	−0.242780
$990$	0	0
$991$	2427.01	0.0777966	0.0388983	−	0.999243i	$-0.487615\pi$
$991$	2427.01	0.0777966	0.0388983	+	0.999243i	$0.487615\pi$
$992$	3990.38	0.127716
$993$	0	0
$994$	12874.9	0.410834
$995$	−14240.3	−0.453717
$996$	0	0
$997$	−17491.5	−0.555629	−0.277815	−	0.960635i	$-0.589610\pi$
$997$	−17491.5	−0.555629	−0.277815	+	0.960635i	$0.589610\pi$
$998$	−10649.1	−0.337765
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.a.r.1.1	yes	2
3.2	odd	2		378.4.a.m.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.a.m.1.2	✓	2	3.2	odd	2
378.4.a.r.1.1	yes	2	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -3.94987 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -3.94987 q^{5} +7.00000 q^{7} +8.00000 q^{8} -7.89975 q^{10} +21.9499 q^{11} -39.6992 q^{13} +14.0000 q^{14} +16.0000 q^{16} +91.5990 q^{17} +130.398 q^{19} -15.7995 q^{20} +43.8997 q^{22} +91.7494 q^{23} -109.398 q^{25} -79.3985 q^{26} +28.0000 q^{28} -69.1980 q^{29} +124.699 q^{31} +32.0000 q^{32} +183.198 q^{34} -27.6491 q^{35} +70.6992 q^{37} +260.797 q^{38} -31.5990 q^{40} +333.048 q^{41} -82.3008 q^{43} +87.7995 q^{44} +183.499 q^{46} +592.496 q^{47} +49.0000 q^{49} -218.797 q^{50} -158.797 q^{52} -39.1980 q^{53} -86.6992 q^{55} +56.0000 q^{56} -138.396 q^{58} -399.293 q^{59} -125.805 q^{61} +249.398 q^{62} +64.0000 q^{64} +156.807 q^{65} -505.895 q^{67} +366.396 q^{68} -55.2982 q^{70} +919.639 q^{71} -199.895 q^{73} +141.398 q^{74} +521.594 q^{76} +153.649 q^{77} -1315.89 q^{79} -63.1980 q^{80} +666.095 q^{82} +686.802 q^{83} -361.805 q^{85} -164.602 q^{86} +175.599 q^{88} -393.539 q^{89} -277.895 q^{91} +366.997 q^{92} +1184.99 q^{94} -515.058 q^{95} -953.805 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 14 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 14 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 12 q^{5} + 14 q^{7} + 16 q^{8} + 24 q^{10} + 24 q^{11} + 40 q^{13} + 28 q^{14} + 32 q^{16} + 24 q^{17} + 22 q^{19} + 48 q^{20} + 48 q^{22} + 84 q^{23} + 20 q^{25} + 80 q^{26} + 56 q^{28} + 180 q^{29} + 130 q^{31} + 64 q^{32} + 48 q^{34} + 84 q^{35} + 22 q^{37} + 44 q^{38} + 96 q^{40} + 288 q^{41} - 284 q^{43} + 96 q^{44} + 168 q^{46} + 588 q^{47} + 98 q^{49} + 40 q^{50} + 160 q^{52} + 240 q^{53} - 54 q^{55} + 112 q^{56} + 360 q^{58} + 276 q^{59} - 968 q^{61} + 260 q^{62} + 128 q^{64} + 1428 q^{65} - 176 q^{67} + 96 q^{68} + 168 q^{70} + 108 q^{71} + 436 q^{73} + 44 q^{74} + 88 q^{76} + 168 q^{77} - 1796 q^{79} + 192 q^{80} + 576 q^{82} + 1692 q^{83} - 1440 q^{85} - 568 q^{86} + 192 q^{88} + 984 q^{89} + 280 q^{91} + 336 q^{92} + 1176 q^{94} - 2244 q^{95} - 2624 q^{97} + 196 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 12 * q^5 + 14 * q^7 + 16 * q^8 + 24 * q^10 + 24 * q^11 + 40 * q^13 + 28 * q^14 + 32 * q^16 + 24 * q^17 + 22 * q^19 + 48 * q^20 + 48 * q^22 + 84 * q^23 + 20 * q^25 + 80 * q^26 + 56 * q^28 + 180 * q^29 + 130 * q^31 + 64 * q^32 + 48 * q^34 + 84 * q^35 + 22 * q^37 + 44 * q^38 + 96 * q^40 + 288 * q^41 - 284 * q^43 + 96 * q^44 + 168 * q^46 + 588 * q^47 + 98 * q^49 + 40 * q^50 + 160 * q^52 + 240 * q^53 - 54 * q^55 + 112 * q^56 + 360 * q^58 + 276 * q^59 - 968 * q^61 + 260 * q^62 + 128 * q^64 + 1428 * q^65 - 176 * q^67 + 96 * q^68 + 168 * q^70 + 108 * q^71 + 436 * q^73 + 44 * q^74 + 88 * q^76 + 168 * q^77 - 1796 * q^79 + 192 * q^80 + 576 * q^82 + 1692 * q^83 - 1440 * q^85 - 568 * q^86 + 192 * q^88 + 984 * q^89 + 280 * q^91 + 336 * q^92 + 1176 * q^94 - 2244 * q^95 - 2624 * q^97 + 196 * q^98

Introduction

L-functions

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