LMFDB - Embedded newform 378.4.a.k.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -1.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} -1.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} -2.00000 q^{10} -44.0000 q^{11} -66.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +7.00000 q^{17} -4.00000 q^{19} -4.00000 q^{20} -88.0000 q^{22} -86.0000 q^{23} -124.000 q^{25} -132.000 q^{26} -28.0000 q^{28} +176.000 q^{29} +162.000 q^{31} +32.0000 q^{32} +14.0000 q^{34} +7.00000 q^{35} -199.000 q^{37} -8.00000 q^{38} -8.00000 q^{40} -363.000 q^{41} -451.000 q^{43} -176.000 q^{44} -172.000 q^{46} -9.00000 q^{47} +49.0000 q^{49} -248.000 q^{50} -264.000 q^{52} +174.000 q^{53} +44.0000 q^{55} -56.0000 q^{56} +352.000 q^{58} +587.000 q^{59} -156.000 q^{61} +324.000 q^{62} +64.0000 q^{64} +66.0000 q^{65} -560.000 q^{67} +28.0000 q^{68} +14.0000 q^{70} +532.000 q^{71} -854.000 q^{73} -398.000 q^{74} -16.0000 q^{76} +308.000 q^{77} -747.000 q^{79} -16.0000 q^{80} -726.000 q^{82} +613.000 q^{83} -7.00000 q^{85} -902.000 q^{86} -352.000 q^{88} +1266.00 q^{89} +462.000 q^{91} -344.000 q^{92} -18.0000 q^{94} +4.00000 q^{95} +64.0000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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
$5$	−1.00000	−0.0894427	−0.0447214	−	0.998999i	$-0.514240\pi$
$5$	−1.00000	−0.0894427	−0.0447214	+	0.998999i	$0.514240\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	−2.00000	−0.0632456
$11$	−44.0000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−44.0000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−66.0000	−1.40809	−0.704043	−	0.710158i	$-0.748624\pi$
$13$	−66.0000	−1.40809	−0.704043	+	0.710158i	$0.748624\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	7.00000	0.0998676	0.0499338	−	0.998753i	$-0.484099\pi$
$17$	7.00000	0.0998676	0.0499338	+	0.998753i	$0.484099\pi$
$18$	0	0
$19$	−4.00000	−0.0482980	−0.0241490	−	0.999708i	$-0.507688\pi$
$19$	−4.00000	−0.0482980	−0.0241490	+	0.999708i	$0.507688\pi$
$20$	−4.00000	−0.0447214
$21$	0	0
$22$	−88.0000	−0.852803
$23$	−86.0000	−0.779663	−0.389831	−	0.920886i	$-0.627467\pi$
$23$	−86.0000	−0.779663	−0.389831	+	0.920886i	$0.627467\pi$
$24$	0	0
$25$	−124.000	−0.992000
$26$	−132.000	−0.995667
$27$	0	0
$28$	−28.0000	−0.188982
$29$	176.000	1.12698	0.563489	−	0.826123i	$-0.309459\pi$
$29$	176.000	1.12698	0.563489	+	0.826123i	$0.309459\pi$
$30$	0	0
$31$	162.000	0.938583	0.469291	−	0.883043i	$-0.344509\pi$
$31$	162.000	0.938583	0.469291	+	0.883043i	$0.344509\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	14.0000	0.0706171
$35$	7.00000	0.0338062
$36$	0	0
$37$	−199.000	−0.884200	−0.442100	−	0.896966i	$-0.645766\pi$
$37$	−199.000	−0.884200	−0.442100	+	0.896966i	$0.645766\pi$
$38$	−8.00000	−0.0341519
$39$	0	0
$40$	−8.00000	−0.0316228
$41$	−363.000	−1.38271	−0.691355	−	0.722516i	$-0.742986\pi$
$41$	−363.000	−1.38271	−0.691355	+	0.722516i	$0.742986\pi$
$42$	0	0
$43$	−451.000	−1.59946	−0.799731	−	0.600359i	$-0.795025\pi$
$43$	−451.000	−1.59946	−0.799731	+	0.600359i	$0.795025\pi$
$44$	−176.000	−0.603023
$45$	0	0
$46$	−172.000	−0.551305
$47$	−9.00000	−0.0279316	−0.0139658	−	0.999902i	$-0.504446\pi$
$47$	−9.00000	−0.0279316	−0.0139658	+	0.999902i	$0.504446\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−248.000	−0.701450
$51$	0	0
$52$	−264.000	−0.704043
$53$	174.000	0.450957	0.225479	−	0.974248i	$-0.427605\pi$
$53$	174.000	0.450957	0.225479	+	0.974248i	$0.427605\pi$
$54$	0	0
$55$	44.0000	0.107872
$56$	−56.0000	−0.133631
$57$	0	0
$58$	352.000	0.796894
$59$	587.000	1.29527	0.647635	−	0.761951i	$-0.275758\pi$
$59$	587.000	1.29527	0.647635	+	0.761951i	$0.275758\pi$
$60$	0	0
$61$	−156.000	−0.327439	−0.163719	−	0.986507i	$-0.552349\pi$
$61$	−156.000	−0.327439	−0.163719	+	0.986507i	$0.552349\pi$
$62$	324.000	0.663678
$63$	0	0
$64$	64.0000	0.125000
$65$	66.0000	0.125943
$66$	0	0
$67$	−560.000	−1.02112	−0.510559	−	0.859843i	$-0.670561\pi$
$67$	−560.000	−1.02112	−0.510559	+	0.859843i	$0.670561\pi$
$68$	28.0000	0.0499338
$69$	0	0
$70$	14.0000	0.0239046
$71$	532.000	0.889250	0.444625	−	0.895717i	$-0.353337\pi$
$71$	532.000	0.889250	0.444625	+	0.895717i	$0.353337\pi$
$72$	0	0
$73$	−854.000	−1.36922	−0.684611	−	0.728909i	$-0.740028\pi$
$73$	−854.000	−1.36922	−0.684611	+	0.728909i	$0.740028\pi$
$74$	−398.000	−0.625224
$75$	0	0
$76$	−16.0000	−0.0241490
$77$	308.000	0.455842
$78$	0	0
$79$	−747.000	−1.06385	−0.531924	−	0.846792i	$-0.678531\pi$
$79$	−747.000	−1.06385	−0.531924	+	0.846792i	$0.678531\pi$
$80$	−16.0000	−0.0223607
$81$	0	0
$82$	−726.000	−0.977723
$83$	613.000	0.810669	0.405334	−	0.914169i	$-0.367155\pi$
$83$	613.000	0.810669	0.405334	+	0.914169i	$0.367155\pi$
$84$	0	0
$85$	−7.00000	−0.00893243
$86$	−902.000	−1.13099
$87$	0	0
$88$	−352.000	−0.426401
$89$	1266.00	1.50782	0.753909	−	0.656979i	$-0.228166\pi$
$89$	1266.00	1.50782	0.753909	+	0.656979i	$0.228166\pi$
$90$	0	0
$91$	462.000	0.532206
$92$	−344.000	−0.389831
$93$	0	0
$94$	−18.0000	−0.0197506
$95$	4.00000	0.00431991
$96$	0	0
$97$	64.0000	0.0669919	0.0334960	−	0.999439i	$-0.489336\pi$
$97$	64.0000	0.0669919	0.0334960	+	0.999439i	$0.489336\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−496.000	−0.496000
$101$	−670.000	−0.660074	−0.330037	−	0.943968i	$-0.607061\pi$
$101$	−670.000	−0.660074	−0.330037	+	0.943968i	$0.607061\pi$
$102$	0	0
$103$	554.000	0.529973	0.264987	−	0.964252i	$-0.414632\pi$
$103$	554.000	0.529973	0.264987	+	0.964252i	$0.414632\pi$
$104$	−528.000	−0.497833
$105$	0	0
$106$	348.000	0.318875
$107$	906.000	0.818564	0.409282	−	0.912408i	$-0.365779\pi$
$107$	906.000	0.818564	0.409282	+	0.912408i	$0.365779\pi$
$108$	0	0
$109$	1051.00	0.923555	0.461778	−	0.886996i	$-0.347212\pi$
$109$	1051.00	0.923555	0.461778	+	0.886996i	$0.347212\pi$
$110$	88.0000	0.0762770
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−1114.00	−0.927401	−0.463700	−	0.885992i	$-0.653479\pi$
$113$	−1114.00	−0.927401	−0.463700	+	0.885992i	$0.653479\pi$
$114$	0	0
$115$	86.0000	0.0697351
$116$	704.000	0.563489
$117$	0	0
$118$	1174.00	0.915894
$119$	−49.0000	−0.0377464
$120$	0	0
$121$	605.000	0.454545
$122$	−312.000	−0.231534
$123$	0	0
$124$	648.000	0.469291
$125$	249.000	0.178170
$126$	0	0
$127$	−189.000	−0.132055	−0.0660277	−	0.997818i	$-0.521033\pi$
$127$	−189.000	−0.132055	−0.0660277	+	0.997818i	$0.521033\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	132.000	0.0890551
$131$	908.000	0.605590	0.302795	−	0.953056i	$-0.402080\pi$
$131$	908.000	0.605590	0.302795	+	0.953056i	$0.402080\pi$
$132$	0	0
$133$	28.0000	0.0182549
$134$	−1120.00	−0.722039
$135$	0	0
$136$	56.0000	0.0353085
$137$	1334.00	0.831907	0.415954	−	0.909386i	$-0.363448\pi$
$137$	1334.00	0.831907	0.415954	+	0.909386i	$0.363448\pi$
$138$	0	0
$139$	1600.00	0.976333	0.488166	−	0.872751i	$-0.337666\pi$
$139$	1600.00	0.976333	0.488166	+	0.872751i	$0.337666\pi$
$140$	28.0000	0.0169031
$141$	0	0
$142$	1064.00	0.628795
$143$	2904.00	1.69821
$144$	0	0
$145$	−176.000	−0.100800
$146$	−1708.00	−0.968186
$147$	0	0
$148$	−796.000	−0.442100
$149$	−3156.00	−1.73523	−0.867616	−	0.497235i	$-0.834349\pi$
$149$	−3156.00	−1.73523	−0.867616	+	0.497235i	$0.834349\pi$
$150$	0	0
$151$	−2585.00	−1.39314	−0.696571	−	0.717488i	$-0.745292\pi$
$151$	−2585.00	−1.39314	−0.696571	+	0.717488i	$0.745292\pi$
$152$	−32.0000	−0.0170759
$153$	0	0
$154$	616.000	0.322329
$155$	−162.000	−0.0839494
$156$	0	0
$157$	3236.00	1.64497	0.822487	−	0.568784i	$-0.192586\pi$
$157$	3236.00	1.64497	0.822487	+	0.568784i	$0.192586\pi$
$158$	−1494.00	−0.752255
$159$	0	0
$160$	−32.0000	−0.0158114
$161$	602.000	0.294685
$162$	0	0
$163$	1307.00	0.628050	0.314025	−	0.949415i	$-0.398322\pi$
$163$	1307.00	0.628050	0.314025	+	0.949415i	$0.398322\pi$
$164$	−1452.00	−0.691355
$165$	0	0
$166$	1226.00	0.573229
$167$	−2281.00	−1.05694	−0.528470	−	0.848952i	$-0.677234\pi$
$167$	−2281.00	−1.05694	−0.528470	+	0.848952i	$0.677234\pi$
$168$	0	0
$169$	2159.00	0.982704
$170$	−14.0000	−0.00631618
$171$	0	0
$172$	−1804.00	−0.799731
$173$	−586.000	−0.257530	−0.128765	−	0.991675i	$-0.541101\pi$
$173$	−586.000	−0.257530	−0.128765	+	0.991675i	$0.541101\pi$
$174$	0	0
$175$	868.000	0.374941
$176$	−704.000	−0.301511
$177$	0	0
$178$	2532.00	1.06619
$179$	2766.00	1.15498	0.577488	−	0.816399i	$-0.304033\pi$
$179$	2766.00	1.15498	0.577488	+	0.816399i	$0.304033\pi$
$180$	0	0
$181$	2844.00	1.16792	0.583958	−	0.811784i	$-0.301503\pi$
$181$	2844.00	1.16792	0.583958	+	0.811784i	$0.301503\pi$
$182$	924.000	0.376327
$183$	0	0
$184$	−688.000	−0.275652
$185$	199.000	0.0790852
$186$	0	0
$187$	−308.000	−0.120445
$188$	−36.0000	−0.0139658
$189$	0	0
$190$	8.00000	0.00305464
$191$	−5058.00	−1.91615	−0.958073	−	0.286523i	$-0.907501\pi$
$191$	−5058.00	−1.91615	−0.958073	+	0.286523i	$0.907501\pi$
$192$	0	0
$193$	−4033.00	−1.50415	−0.752077	−	0.659075i	$-0.770948\pi$
$193$	−4033.00	−1.50415	−0.752077	+	0.659075i	$0.770948\pi$
$194$	128.000	0.0473704
$195$	0	0
$196$	196.000	0.0714286
$197$	2728.00	0.986609	0.493304	−	0.869857i	$-0.335789\pi$
$197$	2728.00	0.986609	0.493304	+	0.869857i	$0.335789\pi$
$198$	0	0
$199$	3248.00	1.15701	0.578504	−	0.815680i	$-0.303637\pi$
$199$	3248.00	1.15701	0.578504	+	0.815680i	$0.303637\pi$
$200$	−992.000	−0.350725
$201$	0	0
$202$	−1340.00	−0.466743
$203$	−1232.00	−0.425958
$204$	0	0
$205$	363.000	0.123673
$206$	1108.00	0.374748
$207$	0	0
$208$	−1056.00	−0.352021
$209$	176.000	0.0582496
$210$	0	0
$211$	1700.00	0.554658	0.277329	−	0.960775i	$-0.410551\pi$
$211$	1700.00	0.554658	0.277329	+	0.960775i	$0.410551\pi$
$212$	696.000	0.225479
$213$	0	0
$214$	1812.00	0.578812
$215$	451.000	0.143060
$216$	0	0
$217$	−1134.00	−0.354751
$218$	2102.00	0.653052
$219$	0	0
$220$	176.000	0.0539360
$221$	−462.000	−0.140622
$222$	0	0
$223$	2498.00	0.750128	0.375064	−	0.926999i	$-0.377621\pi$
$223$	2498.00	0.750128	0.375064	+	0.926999i	$0.377621\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−2228.00	−0.655771
$227$	4488.00	1.31224	0.656121	−	0.754656i	$-0.272196\pi$
$227$	4488.00	1.31224	0.656121	+	0.754656i	$0.272196\pi$
$228$	0	0
$229$	−4760.00	−1.37358	−0.686790	−	0.726856i	$-0.740981\pi$
$229$	−4760.00	−1.37358	−0.686790	+	0.726856i	$0.740981\pi$
$230$	172.000	0.0493102
$231$	0	0
$232$	1408.00	0.398447
$233$	−1562.00	−0.439185	−0.219592	−	0.975592i	$-0.570473\pi$
$233$	−1562.00	−0.439185	−0.219592	+	0.975592i	$0.570473\pi$
$234$	0	0
$235$	9.00000	0.00249828
$236$	2348.00	0.647635
$237$	0	0
$238$	−98.0000	−0.0266907
$239$	−954.000	−0.258197	−0.129099	−	0.991632i	$-0.541208\pi$
$239$	−954.000	−0.258197	−0.129099	+	0.991632i	$0.541208\pi$
$240$	0	0
$241$	446.000	0.119209	0.0596045	−	0.998222i	$-0.481016\pi$
$241$	446.000	0.119209	0.0596045	+	0.998222i	$0.481016\pi$
$242$	1210.00	0.321412
$243$	0	0
$244$	−624.000	−0.163719
$245$	−49.0000	−0.0127775
$246$	0	0
$247$	264.000	0.0680078
$248$	1296.00	0.331839
$249$	0	0
$250$	498.000	0.125985
$251$	−1995.00	−0.501686	−0.250843	−	0.968028i	$-0.580708\pi$
$251$	−1995.00	−0.501686	−0.250843	+	0.968028i	$0.580708\pi$
$252$	0	0
$253$	3784.00	0.940308
$254$	−378.000	−0.0933773
$255$	0	0
$256$	256.000	0.0625000
$257$	−4974.00	−1.20727	−0.603637	−	0.797259i	$-0.706282\pi$
$257$	−4974.00	−1.20727	−0.603637	+	0.797259i	$0.706282\pi$
$258$	0	0
$259$	1393.00	0.334196
$260$	264.000	0.0629715
$261$	0	0
$262$	1816.00	0.428217
$263$	1338.00	0.313706	0.156853	−	0.987622i	$-0.449865\pi$
$263$	1338.00	0.313706	0.156853	+	0.987622i	$0.449865\pi$
$264$	0	0
$265$	−174.000	−0.0403348
$266$	56.0000	0.0129082
$267$	0	0
$268$	−2240.00	−0.510559
$269$	−1885.00	−0.427251	−0.213625	−	0.976916i	$-0.568527\pi$
$269$	−1885.00	−0.427251	−0.213625	+	0.976916i	$0.568527\pi$
$270$	0	0
$271$	−1854.00	−0.415581	−0.207791	−	0.978173i	$-0.566627\pi$
$271$	−1854.00	−0.415581	−0.207791	+	0.978173i	$0.566627\pi$
$272$	112.000	0.0249669
$273$	0	0
$274$	2668.00	0.588247
$275$	5456.00	1.19640
$276$	0	0
$277$	−6415.00	−1.39148	−0.695740	−	0.718294i	$-0.744923\pi$
$277$	−6415.00	−1.39148	−0.695740	+	0.718294i	$0.744923\pi$
$278$	3200.00	0.690371
$279$	0	0
$280$	56.0000	0.0119523
$281$	718.000	0.152428	0.0762140	−	0.997091i	$-0.475717\pi$
$281$	718.000	0.152428	0.0762140	+	0.997091i	$0.475717\pi$
$282$	0	0
$283$	−7868.00	−1.65266	−0.826332	−	0.563183i	$-0.809577\pi$
$283$	−7868.00	−1.65266	−0.826332	+	0.563183i	$0.809577\pi$
$284$	2128.00	0.444625
$285$	0	0
$286$	5808.00	1.20082
$287$	2541.00	0.522615
$288$	0	0
$289$	−4864.00	−0.990026
$290$	−352.000	−0.0712764
$291$	0	0
$292$	−3416.00	−0.684611
$293$	−2127.00	−0.424098	−0.212049	−	0.977259i	$-0.568014\pi$
$293$	−2127.00	−0.424098	−0.212049	+	0.977259i	$0.568014\pi$
$294$	0	0
$295$	−587.000	−0.115852
$296$	−1592.00	−0.312612
$297$	0	0
$298$	−6312.00	−1.22699
$299$	5676.00	1.09783
$300$	0	0
$301$	3157.00	0.604540
$302$	−5170.00	−0.985100
$303$	0	0
$304$	−64.0000	−0.0120745
$305$	156.000	0.0292870
$306$	0	0
$307$	6022.00	1.11952	0.559762	−	0.828654i	$-0.310893\pi$
$307$	6022.00	1.11952	0.559762	+	0.828654i	$0.310893\pi$
$308$	1232.00	0.227921
$309$	0	0
$310$	−324.000	−0.0593612
$311$	4883.00	0.890320	0.445160	−	0.895451i	$-0.353147\pi$
$311$	4883.00	0.890320	0.445160	+	0.895451i	$0.353147\pi$
$312$	0	0
$313$	−7508.00	−1.35584	−0.677919	−	0.735137i	$-0.737118\pi$
$313$	−7508.00	−1.35584	−0.677919	+	0.735137i	$0.737118\pi$
$314$	6472.00	1.16317
$315$	0	0
$316$	−2988.00	−0.531924
$317$	−9486.00	−1.68072	−0.840358	−	0.542032i	$-0.817655\pi$
$317$	−9486.00	−1.68072	−0.840358	+	0.542032i	$0.817655\pi$
$318$	0	0
$319$	−7744.00	−1.35919
$320$	−64.0000	−0.0111803
$321$	0	0
$322$	1204.00	0.208374
$323$	−28.0000	−0.00482341
$324$	0	0
$325$	8184.00	1.39682
$326$	2614.00	0.444098
$327$	0	0
$328$	−2904.00	−0.488862
$329$	63.0000	0.0105572
$330$	0	0
$331$	−6103.00	−1.01345	−0.506724	−	0.862108i	$-0.669144\pi$
$331$	−6103.00	−1.01345	−0.506724	+	0.862108i	$0.669144\pi$
$332$	2452.00	0.405334
$333$	0	0
$334$	−4562.00	−0.747370
$335$	560.000	0.0913315
$336$	0	0
$337$	8847.00	1.43005	0.715025	−	0.699099i	$-0.246415\pi$
$337$	8847.00	1.43005	0.715025	+	0.699099i	$0.246415\pi$
$338$	4318.00	0.694876
$339$	0	0
$340$	−28.0000	−0.00446622
$341$	−7128.00	−1.13197
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−3608.00	−0.565495
$345$	0	0
$346$	−1172.00	−0.182101
$347$	−9462.00	−1.46382	−0.731912	−	0.681399i	$-0.761372\pi$
$347$	−9462.00	−1.46382	−0.731912	+	0.681399i	$0.761372\pi$
$348$	0	0
$349$	−6164.00	−0.945419	−0.472710	−	0.881218i	$-0.656724\pi$
$349$	−6164.00	−0.945419	−0.472710	+	0.881218i	$0.656724\pi$
$350$	1736.00	0.265123
$351$	0	0
$352$	−1408.00	−0.213201
$353$	−10077.0	−1.51939	−0.759695	−	0.650280i	$-0.774652\pi$
$353$	−10077.0	−1.51939	−0.759695	+	0.650280i	$0.774652\pi$
$354$	0	0
$355$	−532.000	−0.0795370
$356$	5064.00	0.753909
$357$	0	0
$358$	5532.00	0.816691
$359$	2078.00	0.305495	0.152747	−	0.988265i	$-0.451188\pi$
$359$	2078.00	0.305495	0.152747	+	0.988265i	$0.451188\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	5688.00	0.825842
$363$	0	0
$364$	1848.00	0.266103
$365$	854.000	0.122467
$366$	0	0
$367$	4958.00	0.705192	0.352596	−	0.935776i	$-0.385299\pi$
$367$	4958.00	0.705192	0.352596	+	0.935776i	$0.385299\pi$
$368$	−1376.00	−0.194916
$369$	0	0
$370$	398.000	0.0559217
$371$	−1218.00	−0.170446
$372$	0	0
$373$	−9281.00	−1.28834	−0.644172	−	0.764881i	$-0.722798\pi$
$373$	−9281.00	−1.28834	−0.644172	+	0.764881i	$0.722798\pi$
$374$	−616.000	−0.0851674
$375$	0	0
$376$	−72.0000	−0.00987531
$377$	−11616.0	−1.58688
$378$	0	0
$379$	−7085.00	−0.960243	−0.480121	−	0.877202i	$-0.659407\pi$
$379$	−7085.00	−0.960243	−0.480121	+	0.877202i	$0.659407\pi$
$380$	16.0000	0.00215995
$381$	0	0
$382$	−10116.0	−1.35492
$383$	3907.00	0.521249	0.260625	−	0.965440i	$-0.416072\pi$
$383$	3907.00	0.521249	0.260625	+	0.965440i	$0.416072\pi$
$384$	0	0
$385$	−308.000	−0.0407718
$386$	−8066.00	−1.06360
$387$	0	0
$388$	256.000	0.0334960
$389$	−5992.00	−0.780993	−0.390497	−	0.920604i	$-0.627697\pi$
$389$	−5992.00	−0.780993	−0.390497	+	0.920604i	$0.627697\pi$
$390$	0	0
$391$	−602.000	−0.0778630
$392$	392.000	0.0505076
$393$	0	0
$394$	5456.00	0.697638
$395$	747.000	0.0951535
$396$	0	0
$397$	−7446.00	−0.941320	−0.470660	−	0.882315i	$-0.655984\pi$
$397$	−7446.00	−0.941320	−0.470660	+	0.882315i	$0.655984\pi$
$398$	6496.00	0.818128
$399$	0	0
$400$	−1984.00	−0.248000
$401$	−3132.00	−0.390036	−0.195018	−	0.980800i	$-0.562477\pi$
$401$	−3132.00	−0.390036	−0.195018	+	0.980800i	$0.562477\pi$
$402$	0	0
$403$	−10692.0	−1.32160
$404$	−2680.00	−0.330037
$405$	0	0
$406$	−2464.00	−0.301198
$407$	8756.00	1.06639
$408$	0	0
$409$	10352.0	1.25152	0.625762	−	0.780014i	$-0.284788\pi$
$409$	10352.0	1.25152	0.625762	+	0.780014i	$0.284788\pi$
$410$	726.000	0.0874502
$411$	0	0
$412$	2216.00	0.264987
$413$	−4109.00	−0.489566
$414$	0	0
$415$	−613.000	−0.0725084
$416$	−2112.00	−0.248917
$417$	0	0
$418$	352.000	0.0411887
$419$	−891.000	−0.103886	−0.0519430	−	0.998650i	$-0.516541\pi$
$419$	−891.000	−0.103886	−0.0519430	+	0.998650i	$0.516541\pi$
$420$	0	0
$421$	1086.00	0.125721	0.0628603	−	0.998022i	$-0.479978\pi$
$421$	1086.00	0.125721	0.0628603	+	0.998022i	$0.479978\pi$
$422$	3400.00	0.392202
$423$	0	0
$424$	1392.00	0.159437
$425$	−868.000	−0.0990687
$426$	0	0
$427$	1092.00	0.123760
$428$	3624.00	0.409282
$429$	0	0
$430$	902.000	0.101159
$431$	4464.00	0.498894	0.249447	−	0.968388i	$-0.419751\pi$
$431$	4464.00	0.498894	0.249447	+	0.968388i	$0.419751\pi$
$432$	0	0
$433$	14762.0	1.63838	0.819188	−	0.573526i	$-0.194425\pi$
$433$	14762.0	1.63838	0.819188	+	0.573526i	$0.194425\pi$
$434$	−2268.00	−0.250847
$435$	0	0
$436$	4204.00	0.461778
$437$	344.000	0.0376562
$438$	0	0
$439$	−7116.00	−0.773640	−0.386820	−	0.922155i	$-0.626427\pi$
$439$	−7116.00	−0.773640	−0.386820	+	0.922155i	$0.626427\pi$
$440$	352.000	0.0381385
$441$	0	0
$442$	−924.000	−0.0994348
$443$	5428.00	0.582149	0.291075	−	0.956700i	$-0.405987\pi$
$443$	5428.00	0.582149	0.291075	+	0.956700i	$0.405987\pi$
$444$	0	0
$445$	−1266.00	−0.134863
$446$	4996.00	0.530420
$447$	0	0
$448$	−448.000	−0.0472456
$449$	−8998.00	−0.945750	−0.472875	−	0.881129i	$-0.656784\pi$
$449$	−8998.00	−0.945750	−0.472875	+	0.881129i	$0.656784\pi$
$450$	0	0
$451$	15972.0	1.66761
$452$	−4456.00	−0.463700
$453$	0	0
$454$	8976.00	0.927895
$455$	−462.000	−0.0476020
$456$	0	0
$457$	2266.00	0.231945	0.115973	−	0.993252i	$-0.463002\pi$
$457$	2266.00	0.231945	0.115973	+	0.993252i	$0.463002\pi$
$458$	−9520.00	−0.971267
$459$	0	0
$460$	344.000	0.0348676
$461$	−6895.00	−0.696599	−0.348300	−	0.937383i	$-0.613241\pi$
$461$	−6895.00	−0.696599	−0.348300	+	0.937383i	$0.613241\pi$
$462$	0	0
$463$	8107.00	0.813746	0.406873	−	0.913485i	$-0.366619\pi$
$463$	8107.00	0.813746	0.406873	+	0.913485i	$0.366619\pi$
$464$	2816.00	0.281745
$465$	0	0
$466$	−3124.00	−0.310550
$467$	−19108.0	−1.89339	−0.946695	−	0.322132i	$-0.895600\pi$
$467$	−19108.0	−1.89339	−0.946695	+	0.322132i	$0.895600\pi$
$468$	0	0
$469$	3920.00	0.385946
$470$	18.0000	0.00176655
$471$	0	0
$472$	4696.00	0.457947
$473$	19844.0	1.92902
$474$	0	0
$475$	496.000	0.0479117
$476$	−196.000	−0.0188732
$477$	0	0
$478$	−1908.00	−0.182573
$479$	−4631.00	−0.441745	−0.220872	−	0.975303i	$-0.570890\pi$
$479$	−4631.00	−0.441745	−0.220872	+	0.975303i	$0.570890\pi$
$480$	0	0
$481$	13134.0	1.24503
$482$	892.000	0.0842935
$483$	0	0
$484$	2420.00	0.227273
$485$	−64.0000	−0.00599194
$486$	0	0
$487$	−11656.0	−1.08457	−0.542283	−	0.840196i	$-0.682440\pi$
$487$	−11656.0	−1.08457	−0.542283	+	0.840196i	$0.682440\pi$
$488$	−1248.00	−0.115767
$489$	0	0
$490$	−98.0000	−0.00903508
$491$	15522.0	1.42668	0.713338	−	0.700820i	$-0.247182\pi$
$491$	15522.0	1.42668	0.713338	+	0.700820i	$0.247182\pi$
$492$	0	0
$493$	1232.00	0.112549
$494$	528.000	0.0480888
$495$	0	0
$496$	2592.00	0.234646
$497$	−3724.00	−0.336105
$498$	0	0
$499$	17059.0	1.53039	0.765196	−	0.643797i	$-0.222642\pi$
$499$	17059.0	1.53039	0.765196	+	0.643797i	$0.222642\pi$
$500$	996.000	0.0890849
$501$	0	0
$502$	−3990.00	−0.354746
$503$	7059.00	0.625736	0.312868	−	0.949797i	$-0.398710\pi$
$503$	7059.00	0.625736	0.312868	+	0.949797i	$0.398710\pi$
$504$	0	0
$505$	670.000	0.0590388
$506$	7568.00	0.664898
$507$	0	0
$508$	−756.000	−0.0660277
$509$	−3201.00	−0.278746	−0.139373	−	0.990240i	$-0.544509\pi$
$509$	−3201.00	−0.278746	−0.139373	+	0.990240i	$0.544509\pi$
$510$	0	0
$511$	5978.00	0.517517
$512$	512.000	0.0441942
$513$	0	0
$514$	−9948.00	−0.853672
$515$	−554.000	−0.0474022
$516$	0	0
$517$	396.000	0.0336868
$518$	2786.00	0.236312
$519$	0	0
$520$	528.000	0.0445276
$521$	8187.00	0.688443	0.344222	−	0.938888i	$-0.388143\pi$
$521$	8187.00	0.688443	0.344222	+	0.938888i	$0.388143\pi$
$522$	0	0
$523$	11918.0	0.996439	0.498220	−	0.867051i	$-0.333987\pi$
$523$	11918.0	0.996439	0.498220	+	0.867051i	$0.333987\pi$
$524$	3632.00	0.302795
$525$	0	0
$526$	2676.00	0.221823
$527$	1134.00	0.0937340
$528$	0	0
$529$	−4771.00	−0.392126
$530$	−348.000	−0.0285210
$531$	0	0
$532$	112.000	0.00912747
$533$	23958.0	1.94697
$534$	0	0
$535$	−906.000	−0.0732146
$536$	−4480.00	−0.361020
$537$	0	0
$538$	−3770.00	−0.302112
$539$	−2156.00	−0.172292
$540$	0	0
$541$	−10887.0	−0.865192	−0.432596	−	0.901588i	$-0.642402\pi$
$541$	−10887.0	−0.865192	−0.432596	+	0.901588i	$0.642402\pi$
$542$	−3708.00	−0.293860
$543$	0	0
$544$	224.000	0.0176543
$545$	−1051.00	−0.0826053
$546$	0	0
$547$	−4395.00	−0.343540	−0.171770	−	0.985137i	$-0.554949\pi$
$547$	−4395.00	−0.343540	−0.171770	+	0.985137i	$0.554949\pi$
$548$	5336.00	0.415954
$549$	0	0
$550$	10912.0	0.845980
$551$	−704.000	−0.0544309
$552$	0	0
$553$	5229.00	0.402097
$554$	−12830.0	−0.983925
$555$	0	0
$556$	6400.00	0.488166
$557$	−15676.0	−1.19248	−0.596242	−	0.802805i	$-0.703340\pi$
$557$	−15676.0	−1.19248	−0.596242	+	0.802805i	$0.703340\pi$
$558$	0	0
$559$	29766.0	2.25218
$560$	112.000	0.00845154
$561$	0	0
$562$	1436.00	0.107783
$563$	22468.0	1.68191	0.840953	−	0.541108i	$-0.181995\pi$
$563$	22468.0	1.68191	0.840953	+	0.541108i	$0.181995\pi$
$564$	0	0
$565$	1114.00	0.0829493
$566$	−15736.0	−1.16861
$567$	0	0
$568$	4256.00	0.314398
$569$	9222.00	0.679449	0.339724	−	0.940525i	$-0.389666\pi$
$569$	9222.00	0.679449	0.339724	+	0.940525i	$0.389666\pi$
$570$	0	0
$571$	−3075.00	−0.225367	−0.112684	−	0.993631i	$-0.535945\pi$
$571$	−3075.00	−0.225367	−0.112684	+	0.993631i	$0.535945\pi$
$572$	11616.0	0.849107
$573$	0	0
$574$	5082.00	0.369545
$575$	10664.0	0.773425
$576$	0	0
$577$	−12584.0	−0.907935	−0.453968	−	0.891018i	$-0.649992\pi$
$577$	−12584.0	−0.907935	−0.453968	+	0.891018i	$0.649992\pi$
$578$	−9728.00	−0.700054
$579$	0	0
$580$	−704.000	−0.0504000
$581$	−4291.00	−0.306404
$582$	0	0
$583$	−7656.00	−0.543875
$584$	−6832.00	−0.484093
$585$	0	0
$586$	−4254.00	−0.299882
$587$	24028.0	1.68951	0.844754	−	0.535154i	$-0.179746\pi$
$587$	24028.0	1.68951	0.844754	+	0.535154i	$0.179746\pi$
$588$	0	0
$589$	−648.000	−0.0453317
$590$	−1174.00	−0.0819200
$591$	0	0
$592$	−3184.00	−0.221050
$593$	−20565.0	−1.42412	−0.712060	−	0.702118i	$-0.752238\pi$
$593$	−20565.0	−1.42412	−0.712060	+	0.702118i	$0.752238\pi$
$594$	0	0
$595$	49.0000	0.00337614
$596$	−12624.0	−0.867616
$597$	0	0
$598$	11352.0	0.776284
$599$	15708.0	1.07147	0.535736	−	0.844386i	$-0.320034\pi$
$599$	15708.0	1.07147	0.535736	+	0.844386i	$0.320034\pi$
$600$	0	0
$601$	18326.0	1.24382	0.621908	−	0.783091i	$-0.286358\pi$
$601$	18326.0	1.24382	0.621908	+	0.783091i	$0.286358\pi$
$602$	6314.00	0.427474
$603$	0	0
$604$	−10340.0	−0.696571
$605$	−605.000	−0.0406558
$606$	0	0
$607$	−17466.0	−1.16791	−0.583956	−	0.811785i	$-0.698496\pi$
$607$	−17466.0	−1.16791	−0.583956	+	0.811785i	$0.698496\pi$
$608$	−128.000	−0.00853797
$609$	0	0
$610$	312.000	0.0207090
$611$	594.000	0.0393301
$612$	0	0
$613$	2118.00	0.139552	0.0697759	−	0.997563i	$-0.477772\pi$
$613$	2118.00	0.139552	0.0697759	+	0.997563i	$0.477772\pi$
$614$	12044.0	0.791623
$615$	0	0
$616$	2464.00	0.161165
$617$	2086.00	0.136109	0.0680545	−	0.997682i	$-0.478321\pi$
$617$	2086.00	0.136109	0.0680545	+	0.997682i	$0.478321\pi$
$618$	0	0
$619$	1960.00	0.127268	0.0636341	−	0.997973i	$-0.479731\pi$
$619$	1960.00	0.127268	0.0636341	+	0.997973i	$0.479731\pi$
$620$	−648.000	−0.0419747
$621$	0	0
$622$	9766.00	0.629551
$623$	−8862.00	−0.569901
$624$	0	0
$625$	15251.0	0.976064
$626$	−15016.0	−0.958722
$627$	0	0
$628$	12944.0	0.822487
$629$	−1393.00	−0.0883029
$630$	0	0
$631$	25343.0	1.59887	0.799437	−	0.600751i	$-0.205132\pi$
$631$	25343.0	1.59887	0.799437	+	0.600751i	$0.205132\pi$
$632$	−5976.00	−0.376127
$633$	0	0
$634$	−18972.0	−1.18845
$635$	189.000	0.0118114
$636$	0	0
$637$	−3234.00	−0.201155
$638$	−15488.0	−0.961091
$639$	0	0
$640$	−128.000	−0.00790569
$641$	−12236.0	−0.753967	−0.376984	−	0.926220i	$-0.623039\pi$
$641$	−12236.0	−0.753967	−0.376984	+	0.926220i	$0.623039\pi$
$642$	0	0
$643$	−6730.00	−0.412761	−0.206380	−	0.978472i	$-0.566168\pi$
$643$	−6730.00	−0.412761	−0.206380	+	0.978472i	$0.566168\pi$
$644$	2408.00	0.147342
$645$	0	0
$646$	−56.0000	−0.00341067
$647$	−20328.0	−1.23520	−0.617601	−	0.786491i	$-0.711895\pi$
$647$	−20328.0	−1.23520	−0.617601	+	0.786491i	$0.711895\pi$
$648$	0	0
$649$	−25828.0	−1.56215
$650$	16368.0	0.987701
$651$	0	0
$652$	5228.00	0.314025
$653$	10674.0	0.639672	0.319836	−	0.947473i	$-0.396372\pi$
$653$	10674.0	0.639672	0.319836	+	0.947473i	$0.396372\pi$
$654$	0	0
$655$	−908.000	−0.0541656
$656$	−5808.00	−0.345677
$657$	0	0
$658$	126.000	0.00746503
$659$	20104.0	1.18838	0.594189	−	0.804326i	$-0.297473\pi$
$659$	20104.0	1.18838	0.594189	+	0.804326i	$0.297473\pi$
$660$	0	0
$661$	910.000	0.0535475	0.0267738	−	0.999642i	$-0.491477\pi$
$661$	910.000	0.0535475	0.0267738	+	0.999642i	$0.491477\pi$
$662$	−12206.0	−0.716616
$663$	0	0
$664$	4904.00	0.286615
$665$	−28.0000	−0.00163277
$666$	0	0
$667$	−15136.0	−0.878663
$668$	−9124.00	−0.528470
$669$	0	0
$670$	1120.00	0.0645812
$671$	6864.00	0.394906
$672$	0	0
$673$	−7474.00	−0.428085	−0.214043	−	0.976824i	$-0.568663\pi$
$673$	−7474.00	−0.428085	−0.214043	+	0.976824i	$0.568663\pi$
$674$	17694.0	1.01120
$675$	0	0
$676$	8636.00	0.491352
$677$	25830.0	1.46636	0.733181	−	0.680033i	$-0.238035\pi$
$677$	25830.0	1.46636	0.733181	+	0.680033i	$0.238035\pi$
$678$	0	0
$679$	−448.000	−0.0253206
$680$	−56.0000	−0.00315809
$681$	0	0
$682$	−14256.0	−0.800426
$683$	30678.0	1.71868	0.859342	−	0.511401i	$-0.170873\pi$
$683$	30678.0	1.71868	0.859342	+	0.511401i	$0.170873\pi$
$684$	0	0
$685$	−1334.00	−0.0744081
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−7216.00	−0.399865
$689$	−11484.0	−0.634986
$690$	0	0
$691$	14570.0	0.802126	0.401063	−	0.916051i	$-0.368641\pi$
$691$	14570.0	0.802126	0.401063	+	0.916051i	$0.368641\pi$
$692$	−2344.00	−0.128765
$693$	0	0
$694$	−18924.0	−1.03508
$695$	−1600.00	−0.0873258
$696$	0	0
$697$	−2541.00	−0.138088
$698$	−12328.0	−0.668512
$699$	0	0
$700$	3472.00	0.187470
$701$	−16522.0	−0.890196	−0.445098	−	0.895482i	$-0.646831\pi$
$701$	−16522.0	−0.890196	−0.445098	+	0.895482i	$0.646831\pi$
$702$	0	0
$703$	796.000	0.0427051
$704$	−2816.00	−0.150756
$705$	0	0
$706$	−20154.0	−1.07437
$707$	4690.00	0.249485
$708$	0	0
$709$	27731.0	1.46891	0.734457	−	0.678656i	$-0.237437\pi$
$709$	27731.0	1.46891	0.734457	+	0.678656i	$0.237437\pi$
$710$	−1064.00	−0.0562411
$711$	0	0
$712$	10128.0	0.533094
$713$	−13932.0	−0.731778
$714$	0	0
$715$	−2904.00	−0.151893
$716$	11064.0	0.577488
$717$	0	0
$718$	4156.00	0.216017
$719$	−29601.0	−1.53537	−0.767685	−	0.640827i	$-0.778591\pi$
$719$	−29601.0	−1.53537	−0.767685	+	0.640827i	$0.778591\pi$
$720$	0	0
$721$	−3878.00	−0.200311
$722$	−13686.0	−0.705457
$723$	0	0
$724$	11376.0	0.583958
$725$	−21824.0	−1.11796
$726$	0	0
$727$	−20432.0	−1.04234	−0.521170	−	0.853453i	$-0.674504\pi$
$727$	−20432.0	−1.04234	−0.521170	+	0.853453i	$0.674504\pi$
$728$	3696.00	0.188163
$729$	0	0
$730$	1708.00	0.0865971
$731$	−3157.00	−0.159734
$732$	0	0
$733$	30228.0	1.52319	0.761594	−	0.648055i	$-0.224417\pi$
$733$	30228.0	1.52319	0.761594	+	0.648055i	$0.224417\pi$
$734$	9916.00	0.498646
$735$	0	0
$736$	−2752.00	−0.137826
$737$	24640.0	1.23151
$738$	0	0
$739$	16260.0	0.809383	0.404691	−	0.914453i	$-0.367379\pi$
$739$	16260.0	0.809383	0.404691	+	0.914453i	$0.367379\pi$
$740$	796.000	0.0395426
$741$	0	0
$742$	−2436.00	−0.120523
$743$	4440.00	0.219230	0.109615	−	0.993974i	$-0.465038\pi$
$743$	4440.00	0.219230	0.109615	+	0.993974i	$0.465038\pi$
$744$	0	0
$745$	3156.00	0.155204
$746$	−18562.0	−0.910996
$747$	0	0
$748$	−1232.00	−0.0602224
$749$	−6342.00	−0.309388
$750$	0	0
$751$	−11328.0	−0.550419	−0.275209	−	0.961384i	$-0.588747\pi$
$751$	−11328.0	−0.550419	−0.275209	+	0.961384i	$0.588747\pi$
$752$	−144.000	−0.00698290
$753$	0	0
$754$	−23232.0	−1.12209
$755$	2585.00	0.124606
$756$	0	0
$757$	−38253.0	−1.83663	−0.918315	−	0.395850i	$-0.870450\pi$
$757$	−38253.0	−1.83663	−0.918315	+	0.395850i	$0.870450\pi$
$758$	−14170.0	−0.678994
$759$	0	0
$760$	32.0000	0.00152732
$761$	−33455.0	−1.59362	−0.796809	−	0.604232i	$-0.793480\pi$
$761$	−33455.0	−1.59362	−0.796809	+	0.604232i	$0.793480\pi$
$762$	0	0
$763$	−7357.00	−0.349071
$764$	−20232.0	−0.958073
$765$	0	0
$766$	7814.00	0.368579
$767$	−38742.0	−1.82385
$768$	0	0
$769$	−37496.0	−1.75831	−0.879155	−	0.476536i	$-0.841892\pi$
$769$	−37496.0	−1.75831	−0.879155	+	0.476536i	$0.841892\pi$
$770$	−616.000	−0.0288300
$771$	0	0
$772$	−16132.0	−0.752077
$773$	37721.0	1.75515	0.877574	−	0.479440i	$-0.159160\pi$
$773$	37721.0	1.75515	0.877574	+	0.479440i	$0.159160\pi$
$774$	0	0
$775$	−20088.0	−0.931074
$776$	512.000	0.0236852
$777$	0	0
$778$	−11984.0	−0.552246
$779$	1452.00	0.0667822
$780$	0	0
$781$	−23408.0	−1.07248
$782$	−1204.00	−0.0550575
$783$	0	0
$784$	784.000	0.0357143
$785$	−3236.00	−0.147131
$786$	0	0
$787$	−34218.0	−1.54986	−0.774930	−	0.632047i	$-0.782215\pi$
$787$	−34218.0	−1.54986	−0.774930	+	0.632047i	$0.782215\pi$
$788$	10912.0	0.493304
$789$	0	0
$790$	1494.00	0.0672837
$791$	7798.00	0.350525
$792$	0	0
$793$	10296.0	0.461061
$794$	−14892.0	−0.665614
$795$	0	0
$796$	12992.0	0.578504
$797$	12954.0	0.575727	0.287863	−	0.957671i	$-0.407055\pi$
$797$	12954.0	0.575727	0.287863	+	0.957671i	$0.407055\pi$
$798$	0	0
$799$	−63.0000	−0.00278946
$800$	−3968.00	−0.175362
$801$	0	0
$802$	−6264.00	−0.275797
$803$	37576.0	1.65134
$804$	0	0
$805$	−602.000	−0.0263574
$806$	−21384.0	−0.934515
$807$	0	0
$808$	−5360.00	−0.233371
$809$	39590.0	1.72053	0.860266	−	0.509846i	$-0.170297\pi$
$809$	39590.0	1.72053	0.860266	+	0.509846i	$0.170297\pi$
$810$	0	0
$811$	1090.00	0.0471949	0.0235975	−	0.999722i	$-0.492488\pi$
$811$	1090.00	0.0471949	0.0235975	+	0.999722i	$0.492488\pi$
$812$	−4928.00	−0.212979
$813$	0	0
$814$	17512.0	0.754048
$815$	−1307.00	−0.0561745
$816$	0	0
$817$	1804.00	0.0772509
$818$	20704.0	0.884961
$819$	0	0
$820$	1452.00	0.0618366
$821$	−30560.0	−1.29909	−0.649544	−	0.760324i	$-0.725040\pi$
$821$	−30560.0	−1.29909	−0.649544	+	0.760324i	$0.725040\pi$
$822$	0	0
$823$	−30833.0	−1.30592	−0.652959	−	0.757393i	$-0.726473\pi$
$823$	−30833.0	−1.30592	−0.652959	+	0.757393i	$0.726473\pi$
$824$	4432.00	0.187374
$825$	0	0
$826$	−8218.00	−0.346175
$827$	37926.0	1.59470	0.797350	−	0.603518i	$-0.206235\pi$
$827$	37926.0	1.59470	0.797350	+	0.603518i	$0.206235\pi$
$828$	0	0
$829$	−5300.00	−0.222047	−0.111023	−	0.993818i	$-0.535413\pi$
$829$	−5300.00	−0.222047	−0.111023	+	0.993818i	$0.535413\pi$
$830$	−1226.00	−0.0512712
$831$	0	0
$832$	−4224.00	−0.176011
$833$	343.000	0.0142668
$834$	0	0
$835$	2281.00	0.0945356
$836$	704.000	0.0291248
$837$	0	0
$838$	−1782.00	−0.0734584
$839$	−42809.0	−1.76154	−0.880769	−	0.473546i	$-0.842974\pi$
$839$	−42809.0	−1.76154	−0.880769	+	0.473546i	$0.842974\pi$
$840$	0	0
$841$	6587.00	0.270081
$842$	2172.00	0.0888979
$843$	0	0
$844$	6800.00	0.277329
$845$	−2159.00	−0.0878957
$846$	0	0
$847$	−4235.00	−0.171802
$848$	2784.00	0.112739
$849$	0	0
$850$	−1736.00	−0.0700521
$851$	17114.0	0.689378
$852$	0	0
$853$	14948.0	0.600011	0.300006	−	0.953937i	$-0.403011\pi$
$853$	14948.0	0.600011	0.300006	+	0.953937i	$0.403011\pi$
$854$	2184.00	0.0875116
$855$	0	0
$856$	7248.00	0.289406
$857$	23349.0	0.930673	0.465336	−	0.885134i	$-0.345933\pi$
$857$	23349.0	0.930673	0.465336	+	0.885134i	$0.345933\pi$
$858$	0	0
$859$	−35578.0	−1.41316	−0.706581	−	0.707632i	$-0.749763\pi$
$859$	−35578.0	−1.41316	−0.706581	+	0.707632i	$0.749763\pi$
$860$	1804.00	0.0715301
$861$	0	0
$862$	8928.00	0.352771
$863$	−5710.00	−0.225227	−0.112613	−	0.993639i	$-0.535922\pi$
$863$	−5710.00	−0.225227	−0.112613	+	0.993639i	$0.535922\pi$
$864$	0	0
$865$	586.000	0.0230342
$866$	29524.0	1.15851
$867$	0	0
$868$	−4536.00	−0.177375
$869$	32868.0	1.28305
$870$	0	0
$871$	36960.0	1.43782
$872$	8408.00	0.326526
$873$	0	0
$874$	688.000	0.0266269
$875$	−1743.00	−0.0673419
$876$	0	0
$877$	−21023.0	−0.809460	−0.404730	−	0.914436i	$-0.632634\pi$
$877$	−21023.0	−0.809460	−0.404730	+	0.914436i	$0.632634\pi$
$878$	−14232.0	−0.547046
$879$	0	0
$880$	704.000	0.0269680
$881$	−31410.0	−1.20117	−0.600584	−	0.799561i	$-0.705065\pi$
$881$	−31410.0	−1.20117	−0.600584	+	0.799561i	$0.705065\pi$
$882$	0	0
$883$	−15937.0	−0.607387	−0.303694	−	0.952770i	$-0.598220\pi$
$883$	−15937.0	−0.607387	−0.303694	+	0.952770i	$0.598220\pi$
$884$	−1848.00	−0.0703110
$885$	0	0
$886$	10856.0	0.411642
$887$	−50565.0	−1.91410	−0.957050	−	0.289923i	$-0.906370\pi$
$887$	−50565.0	−1.91410	−0.957050	+	0.289923i	$0.906370\pi$
$888$	0	0
$889$	1323.00	0.0499123
$890$	−2532.00	−0.0953627
$891$	0	0
$892$	9992.00	0.375064
$893$	36.0000	0.00134904
$894$	0	0
$895$	−2766.00	−0.103304
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−17996.0	−0.668746
$899$	28512.0	1.05776
$900$	0	0
$901$	1218.00	0.0450360
$902$	31944.0	1.17918
$903$	0	0
$904$	−8912.00	−0.327886
$905$	−2844.00	−0.104462
$906$	0	0
$907$	43959.0	1.60930	0.804650	−	0.593750i	$-0.202353\pi$
$907$	43959.0	1.60930	0.804650	+	0.593750i	$0.202353\pi$
$908$	17952.0	0.656121
$909$	0	0
$910$	−924.000	−0.0336597
$911$	48420.0	1.76095	0.880475	−	0.474092i	$-0.157224\pi$
$911$	48420.0	1.76095	0.880475	+	0.474092i	$0.157224\pi$
$912$	0	0
$913$	−26972.0	−0.977703
$914$	4532.00	0.164010
$915$	0	0
$916$	−19040.0	−0.686790
$917$	−6356.00	−0.228892
$918$	0	0
$919$	−21425.0	−0.769038	−0.384519	−	0.923117i	$-0.625633\pi$
$919$	−21425.0	−0.769038	−0.384519	+	0.923117i	$0.625633\pi$
$920$	688.000	0.0246551
$921$	0	0
$922$	−13790.0	−0.492570
$923$	−35112.0	−1.25214
$924$	0	0
$925$	24676.0	0.877126
$926$	16214.0	0.575405
$927$	0	0
$928$	5632.00	0.199224
$929$	27801.0	0.981831	0.490916	−	0.871207i	$-0.336662\pi$
$929$	27801.0	0.981831	0.490916	+	0.871207i	$0.336662\pi$
$930$	0	0
$931$	−196.000	−0.00689972
$932$	−6248.00	−0.219592
$933$	0	0
$934$	−38216.0	−1.33883
$935$	308.000	0.0107729
$936$	0	0
$937$	−17896.0	−0.623945	−0.311973	−	0.950091i	$-0.600990\pi$
$937$	−17896.0	−0.623945	−0.311973	+	0.950091i	$0.600990\pi$
$938$	7840.00	0.272905
$939$	0	0
$940$	36.0000	0.00124914
$941$	−51673.0	−1.79011	−0.895054	−	0.445958i	$-0.852863\pi$
$941$	−51673.0	−1.79011	−0.895054	+	0.445958i	$0.852863\pi$
$942$	0	0
$943$	31218.0	1.07805
$944$	9392.00	0.323817
$945$	0	0
$946$	39688.0	1.36403
$947$	−46036.0	−1.57969	−0.789846	−	0.613305i	$-0.789840\pi$
$947$	−46036.0	−1.57969	−0.789846	+	0.613305i	$0.789840\pi$
$948$	0	0
$949$	56364.0	1.92798
$950$	992.000	0.0338787
$951$	0	0
$952$	−392.000	−0.0133454
$953$	−9134.00	−0.310471	−0.155236	−	0.987877i	$-0.549614\pi$
$953$	−9134.00	−0.310471	−0.155236	+	0.987877i	$0.549614\pi$
$954$	0	0
$955$	5058.00	0.171385
$956$	−3816.00	−0.129099
$957$	0	0
$958$	−9262.00	−0.312361
$959$	−9338.00	−0.314431
$960$	0	0
$961$	−3547.00	−0.119063
$962$	26268.0	0.880368
$963$	0	0
$964$	1784.00	0.0596045
$965$	4033.00	0.134536
$966$	0	0
$967$	30704.0	1.02107	0.510535	−	0.859857i	$-0.329447\pi$
$967$	30704.0	1.02107	0.510535	+	0.859857i	$0.329447\pi$
$968$	4840.00	0.160706
$969$	0	0
$970$	−128.000	−0.00423694
$971$	−22335.0	−0.738171	−0.369086	−	0.929395i	$-0.620329\pi$
$971$	−22335.0	−0.738171	−0.369086	+	0.929395i	$0.620329\pi$
$972$	0	0
$973$	−11200.0	−0.369019
$974$	−23312.0	−0.766904
$975$	0	0
$976$	−2496.00	−0.0818596
$977$	21168.0	0.693167	0.346584	−	0.938019i	$-0.387342\pi$
$977$	21168.0	0.693167	0.346584	+	0.938019i	$0.387342\pi$
$978$	0	0
$979$	−55704.0	−1.81850
$980$	−196.000	−0.00638877
$981$	0	0
$982$	31044.0	1.00881
$983$	10923.0	0.354415	0.177207	−	0.984174i	$-0.443294\pi$
$983$	10923.0	0.354415	0.177207	+	0.984174i	$0.443294\pi$
$984$	0	0
$985$	−2728.00	−0.0882450
$986$	2464.00	0.0795839
$987$	0	0
$988$	1056.00	0.0340039
$989$	38786.0	1.24704
$990$	0	0
$991$	52121.0	1.67071	0.835357	−	0.549707i	$-0.185261\pi$
$991$	52121.0	1.67071	0.835357	+	0.549707i	$0.185261\pi$
$992$	5184.00	0.165920
$993$	0	0
$994$	−7448.00	−0.237662
$995$	−3248.00	−0.103486
$996$	0	0
$997$	−26236.0	−0.833403	−0.416701	−	0.909043i	$-0.636814\pi$
$997$	−26236.0	−0.833403	−0.416701	+	0.909043i	$0.636814\pi$
$998$	34118.0	1.08215
$999$	0	0

Twists

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

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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