LMFDB - Embedded newform 414.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,4,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, 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("414.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	414.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:	$24.4267907424$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -15.3137 q^{5} -5.31371 q^{7} -8.00000 q^{8} +30.6274 q^{10} +21.2548 q^{11} +87.2548 q^{13} +10.6274 q^{14} +16.0000 q^{16} -18.0589 q^{17} +47.9411 q^{19} -61.2548 q^{20} -42.5097 q^{22} -23.0000 q^{23} +109.510 q^{25} -174.510 q^{26} -21.2548 q^{28} -179.137 q^{29} -117.137 q^{31} -32.0000 q^{32} +36.1177 q^{34} +81.3726 q^{35} -410.902 q^{37} -95.8823 q^{38} +122.510 q^{40} +205.765 q^{41} -149.549 q^{43} +85.0193 q^{44} +46.0000 q^{46} +299.882 q^{47} -314.765 q^{49} -219.019 q^{50} +349.019 q^{52} +295.862 q^{53} -325.490 q^{55} +42.5097 q^{56} +358.274 q^{58} -737.450 q^{59} +550.431 q^{61} +234.274 q^{62} +64.0000 q^{64} -1336.20 q^{65} -527.470 q^{67} -72.2355 q^{68} -162.745 q^{70} +297.568 q^{71} -714.313 q^{73} +821.803 q^{74} +191.765 q^{76} -112.942 q^{77} -834.686 q^{79} -245.019 q^{80} -411.529 q^{82} -795.606 q^{83} +276.548 q^{85} +299.098 q^{86} -170.039 q^{88} -46.9605 q^{89} -463.647 q^{91} -92.0000 q^{92} -599.765 q^{94} -734.156 q^{95} +1037.72 q^{97} +629.529 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 8 q^{5} + 12 q^{7} - 16 q^{8} + 16 q^{10} - 48 q^{11} + 84 q^{13} - 24 q^{14} + 32 q^{16} - 104 q^{17} + 28 q^{19} - 32 q^{20} + 96 q^{22} - 46 q^{23} + 38 q^{25} - 168 q^{26}+ \cdots + 716 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 8 * q^5 + 12 * q^7 - 16 * q^8 + 16 * q^10 - 48 * q^11 + 84 * q^13 - 24 * q^14 + 32 * q^16 - 104 * q^17 + 28 * q^19 - 32 * q^20 + 96 * q^22 - 46 * q^23 + 38 * q^25 - 168 * q^26 + 48 * q^28 - 132 * q^29 - 8 * q^31 - 64 * q^32 + 208 * q^34 + 208 * q^35 - 324 * q^37 - 56 * q^38 + 64 * q^40 + 140 * q^41 - 548 * q^43 - 192 * q^44 + 92 * q^46 + 464 * q^47 - 358 * q^49 - 76 * q^50 + 336 * q^52 - 336 * q^53 - 832 * q^55 - 96 * q^56 + 264 * q^58 - 72 * q^59 + 60 * q^61 + 16 * q^62 + 128 * q^64 - 1360 * q^65 - 444 * q^67 - 416 * q^68 - 416 * q^70 - 672 * q^71 - 252 * q^73 + 648 * q^74 + 112 * q^76 - 1312 * q^77 - 1692 * q^79 - 128 * q^80 - 280 * q^82 + 400 * q^83 - 352 * q^85 + 1096 * q^86 + 384 * q^88 + 336 * q^89 - 520 * q^91 - 184 * q^92 - 928 * q^94 - 880 * q^95 + 220 * q^97 + 716 * q^98

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
$5$	−15.3137	−1.36970	−0.684850	−	0.728684i	$-0.740132\pi$
$5$	−15.3137	−1.36970	−0.684850	+	0.728684i	$0.740132\pi$
$6$	0	0
$7$	−5.31371	−0.286913	−0.143457	−	0.989657i	$-0.545822\pi$
$7$	−5.31371	−0.286913	−0.143457	+	0.989657i	$0.545822\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	30.6274	0.968524
$11$	21.2548	0.582598	0.291299	−	0.956632i	$-0.405913\pi$
$11$	21.2548	0.582598	0.291299	+	0.956632i	$0.405913\pi$
$12$	0	0
$13$	87.2548	1.86155	0.930774	−	0.365594i	$-0.119134\pi$
$13$	87.2548	1.86155	0.930774	+	0.365594i	$0.119134\pi$
$14$	10.6274	0.202878
$15$	0	0
$16$	16.0000	0.250000
$17$	−18.0589	−0.257642	−0.128821	−	0.991668i	$-0.541119\pi$
$17$	−18.0589	−0.257642	−0.128821	+	0.991668i	$0.541119\pi$
$18$	0	0
$19$	47.9411	0.578866	0.289433	−	0.957198i	$-0.406533\pi$
$19$	47.9411	0.578866	0.289433	+	0.957198i	$0.406533\pi$
$20$	−61.2548	−0.684850
$21$	0	0
$22$	−42.5097	−0.411959
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	109.510	0.876077
$26$	−174.510	−1.31631
$27$	0	0
$28$	−21.2548	−0.143457
$29$	−179.137	−1.14707	−0.573533	−	0.819182i	$-0.694428\pi$
$29$	−179.137	−1.14707	−0.573533	+	0.819182i	$0.694428\pi$
$30$	0	0
$31$	−117.137	−0.678659	−0.339330	−	0.940668i	$-0.610200\pi$
$31$	−117.137	−0.678659	−0.339330	+	0.940668i	$0.610200\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	36.1177	0.182181
$35$	81.3726	0.392985
$36$	0	0
$37$	−410.902	−1.82572	−0.912862	−	0.408268i	$-0.866133\pi$
$37$	−410.902	−1.82572	−0.912862	+	0.408268i	$0.866133\pi$
$38$	−95.8823	−0.409320
$39$	0	0
$40$	122.510	0.484262
$41$	205.765	0.783781	0.391890	−	0.920012i	$-0.371821\pi$
$41$	205.765	0.783781	0.391890	+	0.920012i	$0.371821\pi$
$42$	0	0
$43$	−149.549	−0.530373	−0.265187	−	0.964197i	$-0.585434\pi$
$43$	−149.549	−0.530373	−0.265187	+	0.964197i	$0.585434\pi$
$44$	85.0193	0.291299
$45$	0	0
$46$	46.0000	0.147442
$47$	299.882	0.930688	0.465344	−	0.885130i	$-0.345931\pi$
$47$	299.882	0.930688	0.465344	+	0.885130i	$0.345931\pi$
$48$	0	0
$49$	−314.765	−0.917681
$50$	−219.019	−0.619480
$51$	0	0
$52$	349.019	0.930774
$53$	295.862	0.766788	0.383394	−	0.923585i	$-0.374755\pi$
$53$	295.862	0.766788	0.383394	+	0.923585i	$0.374755\pi$
$54$	0	0
$55$	−325.490	−0.797984
$56$	42.5097	0.101439
$57$	0	0
$58$	358.274	0.811098
$59$	−737.450	−1.62725	−0.813625	−	0.581389i	$-0.802509\pi$
$59$	−737.450	−1.62725	−0.813625	+	0.581389i	$0.802509\pi$
$60$	0	0
$61$	550.431	1.15533	0.577667	−	0.816272i	$-0.303963\pi$
$61$	550.431	1.15533	0.577667	+	0.816272i	$0.303963\pi$
$62$	234.274	0.479885
$63$	0	0
$64$	64.0000	0.125000
$65$	−1336.20	−2.54976
$66$	0	0
$67$	−527.470	−0.961802	−0.480901	−	0.876775i	$-0.659690\pi$
$67$	−527.470	−0.961802	−0.480901	+	0.876775i	$0.659690\pi$
$68$	−72.2355	−0.128821
$69$	0	0
$70$	−162.745	−0.277882
$71$	297.568	0.497391	0.248696	−	0.968582i	$-0.419998\pi$
$71$	297.568	0.497391	0.248696	+	0.968582i	$0.419998\pi$
$72$	0	0
$73$	−714.313	−1.14526	−0.572630	−	0.819814i	$-0.694077\pi$
$73$	−714.313	−1.14526	−0.572630	+	0.819814i	$0.694077\pi$
$74$	821.803	1.29098
$75$	0	0
$76$	191.765	0.289433
$77$	−112.942	−0.167155
$78$	0	0
$79$	−834.686	−1.18873	−0.594364	−	0.804196i	$-0.702596\pi$
$79$	−834.686	−1.18873	−0.594364	+	0.804196i	$0.702596\pi$
$80$	−245.019	−0.342425
$81$	0	0
$82$	−411.529	−0.554217
$83$	−795.606	−1.05216	−0.526079	−	0.850436i	$-0.676338\pi$
$83$	−795.606	−1.05216	−0.526079	+	0.850436i	$0.676338\pi$
$84$	0	0
$85$	276.548	0.352893
$86$	299.098	0.375030
$87$	0	0
$88$	−170.039	−0.205979
$89$	−46.9605	−0.0559303	−0.0279652	−	0.999609i	$-0.508903\pi$
$89$	−46.9605	−0.0559303	−0.0279652	+	0.999609i	$0.508903\pi$
$90$	0	0
$91$	−463.647	−0.534103
$92$	−92.0000	−0.104257
$93$	0	0
$94$	−599.765	−0.658096
$95$	−734.156	−0.792872
$96$	0	0
$97$	1037.72	1.08624	0.543118	−	0.839656i	$-0.317244\pi$
$97$	1037.72	1.08624	0.543118	+	0.839656i	$0.317244\pi$
$98$	629.529	0.648898
$99$	0	0
$100$	438.039	0.438039
$101$	374.548	0.369000	0.184500	−	0.982833i	$-0.440934\pi$
$101$	374.548	0.369000	0.184500	+	0.982833i	$0.440934\pi$
$102$	0	0
$103$	−408.883	−0.391150	−0.195575	−	0.980689i	$-0.562657\pi$
$103$	−408.883	−0.391150	−0.195575	+	0.980689i	$0.562657\pi$
$104$	−698.039	−0.658157
$105$	0	0
$106$	−591.724	−0.542201
$107$	−1266.51	−1.14428	−0.572141	−	0.820155i	$-0.693887\pi$
$107$	−1266.51	−1.14428	−0.572141	+	0.820155i	$0.693887\pi$
$108$	0	0
$109$	−1367.57	−1.20174	−0.600868	−	0.799348i	$-0.705178\pi$
$109$	−1367.57	−1.20174	−0.600868	+	0.799348i	$0.705178\pi$
$110$	650.981	0.564260
$111$	0	0
$112$	−85.0193	−0.0717283
$113$	−94.5703	−0.0787294	−0.0393647	−	0.999225i	$-0.512533\pi$
$113$	−94.5703	−0.0787294	−0.0393647	+	0.999225i	$0.512533\pi$
$114$	0	0
$115$	352.215	0.285602
$116$	−716.548	−0.573533
$117$	0	0
$118$	1474.90	1.15064
$119$	95.9596	0.0739210
$120$	0	0
$121$	−879.232	−0.660580
$122$	−1100.86	−0.816945
$123$	0	0
$124$	−468.548	−0.339330
$125$	237.214	0.169737
$126$	0	0
$127$	2078.47	1.45224	0.726119	−	0.687569i	$-0.241322\pi$
$127$	2078.47	1.45224	0.726119	+	0.687569i	$0.241322\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	2672.39	1.80295
$131$	−749.568	−0.499924	−0.249962	−	0.968256i	$-0.580418\pi$
$131$	−749.568	−0.499924	−0.249962	+	0.968256i	$0.580418\pi$
$132$	0	0
$133$	−254.745	−0.166084
$134$	1054.94	0.680097
$135$	0	0
$136$	144.471	0.0910903
$137$	−1443.04	−0.899905	−0.449953	−	0.893052i	$-0.648559\pi$
$137$	−1443.04	−0.899905	−0.449953	+	0.893052i	$0.648559\pi$
$138$	0	0
$139$	−2228.63	−1.35992	−0.679962	−	0.733247i	$-0.738004\pi$
$139$	−2228.63	−1.35992	−0.679962	+	0.733247i	$0.738004\pi$
$140$	325.490	0.196493
$141$	0	0
$142$	−595.135	−0.351709
$143$	1854.59	1.08453
$144$	0	0
$145$	2743.25	1.57114
$146$	1428.63	0.809821
$147$	0	0
$148$	−1643.61	−0.912862
$149$	−1486.29	−0.817193	−0.408596	−	0.912715i	$-0.633982\pi$
$149$	−1486.29	−0.817193	−0.408596	+	0.912715i	$0.633982\pi$
$150$	0	0
$151$	348.865	0.188015	0.0940073	−	0.995572i	$-0.470032\pi$
$151$	348.865	0.188015	0.0940073	+	0.995572i	$0.470032\pi$
$152$	−383.529	−0.204660
$153$	0	0
$154$	225.884	0.118196
$155$	1793.80	0.929560
$156$	0	0
$157$	1751.14	0.890165	0.445083	−	0.895489i	$-0.353174\pi$
$157$	1751.14	0.890165	0.445083	+	0.895489i	$0.353174\pi$
$158$	1669.37	0.840558
$159$	0	0
$160$	490.039	0.242131
$161$	122.215	0.0598256
$162$	0	0
$163$	420.743	0.202179	0.101089	−	0.994877i	$-0.467767\pi$
$163$	420.743	0.202179	0.101089	+	0.994877i	$0.467767\pi$
$164$	823.058	0.391890
$165$	0	0
$166$	1591.21	0.743988
$167$	−651.606	−0.301933	−0.150967	−	0.988539i	$-0.548239\pi$
$167$	−651.606	−0.301933	−0.150967	+	0.988539i	$0.548239\pi$
$168$	0	0
$169$	5416.41	2.46536
$170$	−553.097	−0.249533
$171$	0	0
$172$	−598.197	−0.265187
$173$	1346.35	0.591684	0.295842	−	0.955237i	$-0.404400\pi$
$173$	1346.35	0.591684	0.295842	+	0.955237i	$0.404400\pi$
$174$	0	0
$175$	−581.902	−0.251358
$176$	340.077	0.145649
$177$	0	0
$178$	93.9209	0.0395487
$179$	3424.82	1.43007	0.715037	−	0.699086i	$-0.246410\pi$
$179$	3424.82	1.43007	0.715037	+	0.699086i	$0.246410\pi$
$180$	0	0
$181$	−3256.98	−1.33751	−0.668755	−	0.743482i	$-0.733173\pi$
$181$	−3256.98	−1.33751	−0.668755	+	0.743482i	$0.733173\pi$
$182$	927.294	0.377668
$183$	0	0
$184$	184.000	0.0737210
$185$	6292.43	2.50069
$186$	0	0
$187$	−383.838	−0.150102
$188$	1199.53	0.465344
$189$	0	0
$190$	1468.31	0.560645
$191$	4033.37	1.52798	0.763990	−	0.645228i	$-0.223237\pi$
$191$	4033.37	1.52798	0.763990	+	0.645228i	$0.223237\pi$
$192$	0	0
$193$	298.784	0.111435	0.0557174	−	0.998447i	$-0.482255\pi$
$193$	298.784	0.111435	0.0557174	+	0.998447i	$0.482255\pi$
$194$	−2075.45	−0.768085
$195$	0	0
$196$	−1259.06	−0.458840
$197$	−4254.19	−1.53857	−0.769286	−	0.638905i	$-0.779388\pi$
$197$	−4254.19	−1.53857	−0.769286	+	0.638905i	$0.779388\pi$
$198$	0	0
$199$	−4731.23	−1.68537	−0.842684	−	0.538409i	$-0.819026\pi$
$199$	−4731.23	−1.68537	−0.842684	+	0.538409i	$0.819026\pi$
$200$	−876.077	−0.309740
$201$	0	0
$202$	−749.097	−0.260922
$203$	951.882	0.329109
$204$	0	0
$205$	−3151.02	−1.07354
$206$	817.766	0.276585
$207$	0	0
$208$	1396.08	0.465387
$209$	1018.98	0.337246
$210$	0	0
$211$	5131.40	1.67422	0.837110	−	0.547035i	$-0.184243\pi$
$211$	5131.40	1.67422	0.837110	+	0.547035i	$0.184243\pi$
$212$	1183.45	0.383394
$213$	0	0
$214$	2533.02	0.809129
$215$	2290.15	0.726452
$216$	0	0
$217$	622.432	0.194716
$218$	2735.14	0.849756
$219$	0	0
$220$	−1301.96	−0.398992
$221$	−1575.72	−0.479614
$222$	0	0
$223$	2189.41	0.657461	0.328730	−	0.944424i	$-0.393379\pi$
$223$	2189.41	0.657461	0.328730	+	0.944424i	$0.393379\pi$
$224$	170.039	0.0507196
$225$	0	0
$226$	189.141	0.0556701
$227$	5853.37	1.71146	0.855731	−	0.517421i	$-0.173108\pi$
$227$	5853.37	1.71146	0.855731	+	0.517421i	$0.173108\pi$
$228$	0	0
$229$	−3427.17	−0.988970	−0.494485	−	0.869186i	$-0.664643\pi$
$229$	−3427.17	−0.988970	−0.494485	+	0.869186i	$0.664643\pi$
$230$	−704.431	−0.201951
$231$	0	0
$232$	1433.10	0.405549
$233$	−5723.33	−1.60922	−0.804609	−	0.593805i	$-0.797625\pi$
$233$	−5723.33	−1.60922	−0.804609	+	0.593805i	$0.797625\pi$
$234$	0	0
$235$	−4592.31	−1.27476
$236$	−2949.80	−0.813625
$237$	0	0
$238$	−191.919	−0.0522701
$239$	−3298.04	−0.892604	−0.446302	−	0.894882i	$-0.647259\pi$
$239$	−3298.04	−0.892604	−0.446302	+	0.894882i	$0.647259\pi$
$240$	0	0
$241$	−5391.60	−1.44109	−0.720547	−	0.693406i	$-0.756109\pi$
$241$	−5391.60	−1.44109	−0.720547	+	0.693406i	$0.756109\pi$
$242$	1758.46	0.467101
$243$	0	0
$244$	2201.72	0.577667
$245$	4820.21	1.25695
$246$	0	0
$247$	4183.09	1.07759
$248$	937.097	0.239942
$249$	0	0
$250$	−474.429	−0.120022
$251$	−3592.11	−0.903315	−0.451657	−	0.892191i	$-0.649167\pi$
$251$	−3592.11	−0.903315	−0.451657	+	0.892191i	$0.649167\pi$
$252$	0	0
$253$	−488.861	−0.121480
$254$	−4156.94	−1.02689
$255$	0	0
$256$	256.000	0.0625000
$257$	−5645.61	−1.37029	−0.685143	−	0.728409i	$-0.740260\pi$
$257$	−5645.61	−1.37029	−0.685143	+	0.728409i	$0.740260\pi$
$258$	0	0
$259$	2183.41	0.523825
$260$	−5344.78	−1.27488
$261$	0	0
$262$	1499.14	0.353500
$263$	−304.427	−0.0713756	−0.0356878	−	0.999363i	$-0.511362\pi$
$263$	−304.427	−0.0713756	−0.0356878	+	0.999363i	$0.511362\pi$
$264$	0	0
$265$	−4530.75	−1.05027
$266$	509.490	0.117439
$267$	0	0
$268$	−2109.88	−0.480901
$269$	−2748.79	−0.623035	−0.311517	−	0.950240i	$-0.600837\pi$
$269$	−2748.79	−0.623035	−0.311517	+	0.950240i	$0.600837\pi$
$270$	0	0
$271$	−1324.28	−0.296841	−0.148421	−	0.988924i	$-0.547419\pi$
$271$	−1324.28	−0.296841	−0.148421	+	0.988924i	$0.547419\pi$
$272$	−288.942	−0.0644106
$273$	0	0
$274$	2886.08	0.636329
$275$	2327.61	0.510401
$276$	0	0
$277$	5660.74	1.22787	0.613937	−	0.789355i	$-0.289585\pi$
$277$	5660.74	1.22787	0.613937	+	0.789355i	$0.289585\pi$
$278$	4457.25	0.961612
$279$	0	0
$280$	−650.981	−0.138941
$281$	−4981.86	−1.05763	−0.528813	−	0.848739i	$-0.677363\pi$
$281$	−4981.86	−1.05763	−0.528813	+	0.848739i	$0.677363\pi$
$282$	0	0
$283$	−5093.51	−1.06989	−0.534943	−	0.844888i	$-0.679667\pi$
$283$	−5093.51	−1.06989	−0.534943	+	0.844888i	$0.679667\pi$
$284$	1190.27	0.248696
$285$	0	0
$286$	−3709.17	−0.766881
$287$	−1093.37	−0.224877
$288$	0	0
$289$	−4586.88	−0.933620
$290$	−5486.51	−1.11096
$291$	0	0
$292$	−2857.25	−0.572630
$293$	7943.46	1.58383	0.791914	−	0.610632i	$-0.209085\pi$
$293$	7943.46	1.58383	0.791914	+	0.610632i	$0.209085\pi$
$294$	0	0
$295$	11293.1	2.22885
$296$	3287.21	0.645491
$297$	0	0
$298$	2972.58	0.577842
$299$	−2006.86	−0.388160
$300$	0	0
$301$	794.661	0.152171
$302$	−697.729	−0.132946
$303$	0	0
$304$	767.058	0.144716
$305$	−8429.13	−1.58246
$306$	0	0
$307$	7444.59	1.38399	0.691995	−	0.721902i	$-0.256732\pi$
$307$	7444.59	1.38399	0.691995	+	0.721902i	$0.256732\pi$
$308$	−451.768	−0.0835775
$309$	0	0
$310$	−3587.61	−0.657298
$311$	6584.66	1.20058	0.600292	−	0.799781i	$-0.295051\pi$
$311$	6584.66	1.20058	0.600292	+	0.799781i	$0.295051\pi$
$312$	0	0
$313$	2790.86	0.503990	0.251995	−	0.967728i	$-0.418913\pi$
$313$	2790.86	0.503990	0.251995	+	0.967728i	$0.418913\pi$
$314$	−3502.27	−0.629442
$315$	0	0
$316$	−3338.75	−0.594364
$317$	2083.69	0.369184	0.184592	−	0.982815i	$-0.440904\pi$
$317$	2083.69	0.369184	0.184592	+	0.982815i	$0.440904\pi$
$318$	0	0
$319$	−3807.53	−0.668278
$320$	−980.077	−0.171212
$321$	0	0
$322$	−244.431	−0.0423031
$323$	−865.763	−0.149140
$324$	0	0
$325$	9555.25	1.63086
$326$	−841.487	−0.142962
$327$	0	0
$328$	−1646.12	−0.277108
$329$	−1593.49	−0.267027
$330$	0	0
$331$	4687.57	0.778405	0.389203	−	0.921152i	$-0.372751\pi$
$331$	4687.57	0.778405	0.389203	+	0.921152i	$0.372751\pi$
$332$	−3182.43	−0.526079
$333$	0	0
$334$	1303.21	0.213499
$335$	8077.52	1.31738
$336$	0	0
$337$	−5987.61	−0.967851	−0.483926	−	0.875109i	$-0.660789\pi$
$337$	−5987.61	−0.967851	−0.483926	+	0.875109i	$0.660789\pi$
$338$	−10832.8	−1.74328
$339$	0	0
$340$	1106.19	0.176446
$341$	−2489.73	−0.395385
$342$	0	0
$343$	3495.17	0.550208
$344$	1196.39	0.187515
$345$	0	0
$346$	−2692.71	−0.418384
$347$	3378.30	0.522642	0.261321	−	0.965252i	$-0.415842\pi$
$347$	3378.30	0.522642	0.261321	+	0.965252i	$0.415842\pi$
$348$	0	0
$349$	−844.271	−0.129492	−0.0647461	−	0.997902i	$-0.520624\pi$
$349$	−844.271	−0.129492	−0.0647461	+	0.997902i	$0.520624\pi$
$350$	1163.80	0.177737
$351$	0	0
$352$	−680.155	−0.102990
$353$	−3167.13	−0.477534	−0.238767	−	0.971077i	$-0.576743\pi$
$353$	−3167.13	−0.477534	−0.238767	+	0.971077i	$0.576743\pi$
$354$	0	0
$355$	−4556.86	−0.681277
$356$	−187.842	−0.0279652
$357$	0	0
$358$	−6849.65	−1.01122
$359$	−10484.0	−1.54130	−0.770650	−	0.637259i	$-0.780068\pi$
$359$	−10484.0	−1.54130	−0.770650	+	0.637259i	$0.780068\pi$
$360$	0	0
$361$	−4560.65	−0.664914
$362$	6513.96	0.945763
$363$	0	0
$364$	−1854.59	−0.267052
$365$	10938.8	1.56866
$366$	0	0
$367$	2349.00	0.334106	0.167053	−	0.985948i	$-0.446575\pi$
$367$	2349.00	0.334106	0.167053	+	0.985948i	$0.446575\pi$
$368$	−368.000	−0.0521286
$369$	0	0
$370$	−12584.9	−1.76826
$371$	−1572.12	−0.220002
$372$	0	0
$373$	−12076.3	−1.67637	−0.838187	−	0.545384i	$-0.816384\pi$
$373$	−12076.3	−1.67637	−0.838187	+	0.545384i	$0.816384\pi$
$374$	767.677	0.106138
$375$	0	0
$376$	−2399.06	−0.329048
$377$	−15630.6	−2.13532
$378$	0	0
$379$	−1987.35	−0.269349	−0.134674	−	0.990890i	$-0.542999\pi$
$379$	−1987.35	−0.269349	−0.134674	+	0.990890i	$0.542999\pi$
$380$	−2936.63	−0.396436
$381$	0	0
$382$	−8066.74	−1.08045
$383$	−10474.6	−1.39746	−0.698731	−	0.715385i	$-0.746251\pi$
$383$	−10474.6	−1.39746	−0.698731	+	0.715385i	$0.746251\pi$
$384$	0	0
$385$	1729.56	0.228952
$386$	−597.568	−0.0787964
$387$	0	0
$388$	4150.90	0.543118
$389$	3978.53	0.518559	0.259279	−	0.965802i	$-0.416515\pi$
$389$	3978.53	0.518559	0.259279	+	0.965802i	$0.416515\pi$
$390$	0	0
$391$	415.354	0.0537221
$392$	2518.12	0.324449
$393$	0	0
$394$	8508.38	1.08793
$395$	12782.1	1.62820
$396$	0	0
$397$	5539.81	0.700340	0.350170	−	0.936686i	$-0.386124\pi$
$397$	5539.81	0.700340	0.350170	+	0.936686i	$0.386124\pi$
$398$	9462.46	1.19173
$399$	0	0
$400$	1752.15	0.219019
$401$	−2345.20	−0.292054	−0.146027	−	0.989281i	$-0.546649\pi$
$401$	−2345.20	−0.292054	−0.146027	+	0.989281i	$0.546649\pi$
$402$	0	0
$403$	−10220.8	−1.26336
$404$	1498.19	0.184500
$405$	0	0
$406$	−1903.76	−0.232715
$407$	−8733.65	−1.06366
$408$	0	0
$409$	−7263.96	−0.878190	−0.439095	−	0.898441i	$-0.644701\pi$
$409$	−7263.96	−0.878190	−0.439095	+	0.898441i	$0.644701\pi$
$410$	6302.04	0.759111
$411$	0	0
$412$	−1635.53	−0.195575
$413$	3918.59	0.466880
$414$	0	0
$415$	12183.7	1.44114
$416$	−2792.15	−0.329078
$417$	0	0
$418$	−2037.96	−0.238469
$419$	−4094.89	−0.477443	−0.238721	−	0.971088i	$-0.576728\pi$
$419$	−4094.89	−0.477443	−0.238721	+	0.971088i	$0.576728\pi$
$420$	0	0
$421$	4677.67	0.541510	0.270755	−	0.962648i	$-0.412727\pi$
$421$	4677.67	0.541510	0.270755	+	0.962648i	$0.412727\pi$
$422$	−10262.8	−1.18385
$423$	0	0
$424$	−2366.90	−0.271101
$425$	−1977.62	−0.225715
$426$	0	0
$427$	−2924.83	−0.331481
$428$	−5066.04	−0.572141
$429$	0	0
$430$	−4580.31	−0.513679
$431$	−13219.0	−1.47735	−0.738676	−	0.674060i	$-0.764549\pi$
$431$	−13219.0	−1.47735	−0.738676	+	0.674060i	$0.764549\pi$
$432$	0	0
$433$	3109.77	0.345141	0.172571	−	0.984997i	$-0.444793\pi$
$433$	3109.77	0.345141	0.172571	+	0.984997i	$0.444793\pi$
$434$	−1244.86	−0.137685
$435$	0	0
$436$	−5470.27	−0.600868
$437$	−1102.65	−0.120702
$438$	0	0
$439$	16425.0	1.78570	0.892851	−	0.450352i	$-0.148702\pi$
$439$	16425.0	1.78570	0.892851	+	0.450352i	$0.148702\pi$
$440$	2603.92	0.282130
$441$	0	0
$442$	3151.45	0.339138
$443$	4736.14	0.507948	0.253974	−	0.967211i	$-0.418262\pi$
$443$	4736.14	0.507948	0.253974	+	0.967211i	$0.418262\pi$
$444$	0	0
$445$	719.139	0.0766078
$446$	−4378.82	−0.464895
$447$	0	0
$448$	−340.077	−0.0358642
$449$	12341.8	1.29721	0.648605	−	0.761125i	$-0.275353\pi$
$449$	12341.8	1.29721	0.648605	+	0.761125i	$0.275353\pi$
$450$	0	0
$451$	4373.49	0.456629
$452$	−378.281	−0.0393647
$453$	0	0
$454$	−11706.7	−1.21019
$455$	7100.15	0.731561
$456$	0	0
$457$	5.80491	0.000594184 0	0.000297092	−	1.00000i	$-0.499905\pi$
$457$	5.80491	0.000594184 0	0.000297092		1.00000i	$0.499905\pi$
$458$	6854.35	0.699307
$459$	0	0
$460$	1408.86	0.142801
$461$	11502.8	1.16213	0.581063	−	0.813859i	$-0.302637\pi$
$461$	11502.8	1.16213	0.581063	+	0.813859i	$0.302637\pi$
$462$	0	0
$463$	11739.5	1.17836	0.589182	−	0.808001i	$-0.299450\pi$
$463$	11739.5	1.17836	0.589182	+	0.808001i	$0.299450\pi$
$464$	−2866.19	−0.286767
$465$	0	0
$466$	11446.7	1.13789
$467$	12169.8	1.20590	0.602948	−	0.797781i	$-0.293993\pi$
$467$	12169.8	1.20590	0.602948	+	0.797781i	$0.293993\pi$
$468$	0	0
$469$	2802.82	0.275954
$470$	9184.62	0.901393
$471$	0	0
$472$	5899.60	0.575320
$473$	−3178.64	−0.308994
$474$	0	0
$475$	5250.02	0.507131
$476$	383.838	0.0369605
$477$	0	0
$478$	6596.08	0.631166
$479$	−9096.85	−0.867737	−0.433868	−	0.900976i	$-0.642852\pi$
$479$	−9096.85	−0.867737	−0.433868	+	0.900976i	$0.642852\pi$
$480$	0	0
$481$	−35853.1	−3.39868
$482$	10783.2	1.01901
$483$	0	0
$484$	−3516.93	−0.330290
$485$	−15891.4	−1.48782
$486$	0	0
$487$	18741.0	1.74381	0.871906	−	0.489674i	$-0.162884\pi$
$487$	18741.0	1.74381	0.871906	+	0.489674i	$0.162884\pi$
$488$	−4403.44	−0.408472
$489$	0	0
$490$	−9640.42	−0.888796
$491$	16903.7	1.55367	0.776837	−	0.629702i	$-0.216823\pi$
$491$	16903.7	1.55367	0.776837	+	0.629702i	$0.216823\pi$
$492$	0	0
$493$	3235.01	0.295533
$494$	−8366.19	−0.761969
$495$	0	0
$496$	−1874.19	−0.169665
$497$	−1581.19	−0.142708
$498$	0	0
$499$	−2690.01	−0.241326	−0.120663	−	0.992694i	$-0.538502\pi$
$499$	−2690.01	−0.241326	−0.120663	+	0.992694i	$0.538502\pi$
$500$	948.858	0.0848684
$501$	0	0
$502$	7184.22	0.638740
$503$	−2009.56	−0.178135	−0.0890675	−	0.996026i	$-0.528389\pi$
$503$	−2009.56	−0.178135	−0.0890675	+	0.996026i	$0.528389\pi$
$504$	0	0
$505$	−5735.72	−0.505419
$506$	977.722	0.0858993
$507$	0	0
$508$	8313.88	0.726119
$509$	−16797.6	−1.46275	−0.731377	−	0.681973i	$-0.761122\pi$
$509$	−16797.6	−1.46275	−0.731377	+	0.681973i	$0.761122\pi$
$510$	0	0
$511$	3795.65	0.328590
$512$	−512.000	−0.0441942
$513$	0	0
$514$	11291.2	0.968938
$515$	6261.52	0.535758
$516$	0	0
$517$	6373.95	0.542216
$518$	−4366.82	−0.370400
$519$	0	0
$520$	10689.6	0.901477
$521$	−3387.35	−0.284841	−0.142421	−	0.989806i	$-0.545489\pi$
$521$	−3387.35	−0.284841	−0.142421	+	0.989806i	$0.545489\pi$
$522$	0	0
$523$	−4402.98	−0.368124	−0.184062	−	0.982915i	$-0.558925\pi$
$523$	−4402.98	−0.368124	−0.184062	+	0.982915i	$0.558925\pi$
$524$	−2998.27	−0.249962
$525$	0	0
$526$	608.854	0.0504702
$527$	2115.36	0.174851
$528$	0	0
$529$	529.000	0.0434783
$530$	9061.49	0.742653
$531$	0	0
$532$	−1018.98	−0.0830421
$533$	17953.9	1.45905
$534$	0	0
$535$	19395.0	1.56732
$536$	4219.76	0.340048
$537$	0	0
$538$	5497.57	0.440552
$539$	−6690.27	−0.534639
$540$	0	0
$541$	−7658.32	−0.608608	−0.304304	−	0.952575i	$-0.598424\pi$
$541$	−7658.32	−0.608608	−0.304304	+	0.952575i	$0.598424\pi$
$542$	2648.55	0.209899
$543$	0	0
$544$	577.884	0.0455452
$545$	20942.5	1.64602
$546$	0	0
$547$	3176.92	0.248327	0.124164	−	0.992262i	$-0.460375\pi$
$547$	3176.92	0.248327	0.124164	+	0.992262i	$0.460375\pi$
$548$	−5772.15	−0.449953
$549$	0	0
$550$	−4655.22	−0.360908
$551$	−8588.03	−0.663997
$552$	0	0
$553$	4435.28	0.341062
$554$	−11321.5	−0.868238
$555$	0	0
$556$	−8914.50	−0.679962
$557$	18450.7	1.40356	0.701780	−	0.712393i	$-0.252389\pi$
$557$	18450.7	1.40356	0.701780	+	0.712393i	$0.252389\pi$
$558$	0	0
$559$	−13048.9	−0.987315
$560$	1301.96	0.0982463
$561$	0	0
$562$	9963.71	0.747854
$563$	5515.81	0.412902	0.206451	−	0.978457i	$-0.433809\pi$
$563$	5515.81	0.412902	0.206451	+	0.978457i	$0.433809\pi$
$564$	0	0
$565$	1448.22	0.107836
$566$	10187.0	0.756523
$567$	0	0
$568$	−2380.54	−0.175854
$569$	−23452.9	−1.72794	−0.863968	−	0.503546i	$-0.832028\pi$
$569$	−23452.9	−1.72794	−0.863968	+	0.503546i	$0.832028\pi$
$570$	0	0
$571$	17731.8	1.29957	0.649784	−	0.760119i	$-0.274859\pi$
$571$	17731.8	1.29957	0.649784	+	0.760119i	$0.274859\pi$
$572$	7418.35	0.542267
$573$	0	0
$574$	2186.75	0.159012
$575$	−2518.72	−0.182675
$576$	0	0
$577$	−15350.1	−1.10751	−0.553756	−	0.832679i	$-0.686806\pi$
$577$	−15350.1	−1.10751	−0.553756	+	0.832679i	$0.686806\pi$
$578$	9173.75	0.660169
$579$	0	0
$580$	10973.0	0.785568
$581$	4227.62	0.301878
$582$	0	0
$583$	6288.50	0.446729
$584$	5714.50	0.404911
$585$	0	0
$586$	−15886.9	−1.11994
$587$	−23291.6	−1.63773	−0.818866	−	0.573984i	$-0.805397\pi$
$587$	−23291.6	−1.63773	−0.818866	+	0.573984i	$0.805397\pi$
$588$	0	0
$589$	−5615.68	−0.392853
$590$	−22586.2	−1.57603
$591$	0	0
$592$	−6574.43	−0.456431
$593$	20205.9	1.39925	0.699626	−	0.714509i	$-0.253350\pi$
$593$	20205.9	1.39925	0.699626	+	0.714509i	$0.253350\pi$
$594$	0	0
$595$	−1469.50	−0.101250
$596$	−5945.16	−0.408596
$597$	0	0
$598$	4013.72	0.274470
$599$	−11173.3	−0.762153	−0.381077	−	0.924543i	$-0.624447\pi$
$599$	−11173.3	−0.762153	−0.381077	+	0.924543i	$0.624447\pi$
$600$	0	0
$601$	−16631.6	−1.12881	−0.564405	−	0.825498i	$-0.690894\pi$
$601$	−16631.6	−1.12881	−0.564405	+	0.825498i	$0.690894\pi$
$602$	−1589.32	−0.107601
$603$	0	0
$604$	1395.46	0.0940073
$605$	13464.3	0.904796
$606$	0	0
$607$	−23958.5	−1.60205	−0.801025	−	0.598631i	$-0.795712\pi$
$607$	−23958.5	−1.60205	−0.801025	+	0.598631i	$0.795712\pi$
$608$	−1534.12	−0.102330
$609$	0	0
$610$	16858.3	1.11897
$611$	26166.2	1.73252
$612$	0	0
$613$	25615.0	1.68774	0.843868	−	0.536551i	$-0.180273\pi$
$613$	25615.0	1.68774	0.843868	+	0.536551i	$0.180273\pi$
$614$	−14889.2	−0.978629
$615$	0	0
$616$	903.536	0.0590982
$617$	11722.2	0.764859	0.382429	−	0.923985i	$-0.375088\pi$
$617$	11722.2	0.764859	0.382429	+	0.923985i	$0.375088\pi$
$618$	0	0
$619$	−18949.8	−1.23047	−0.615233	−	0.788346i	$-0.710938\pi$
$619$	−18949.8	−1.23047	−0.615233	+	0.788346i	$0.710938\pi$
$620$	7175.21	0.464780
$621$	0	0
$622$	−13169.3	−0.848941
$623$	249.534	0.0160472
$624$	0	0
$625$	−17321.3	−1.10857
$626$	−5581.73	−0.356375
$627$	0	0
$628$	7004.55	0.445083
$629$	7420.42	0.470384
$630$	0	0
$631$	−1559.47	−0.0983859	−0.0491930	−	0.998789i	$-0.515665\pi$
$631$	−1559.47	−0.0983859	−0.0491930	+	0.998789i	$0.515665\pi$
$632$	6677.49	0.420279
$633$	0	0
$634$	−4167.37	−0.261053
$635$	−31829.1	−1.98913
$636$	0	0
$637$	−27464.7	−1.70831
$638$	7615.06	0.472544
$639$	0	0
$640$	1960.15	0.121065
$641$	5937.72	0.365875	0.182938	−	0.983125i	$-0.441439\pi$
$641$	5937.72	0.365875	0.182938	+	0.983125i	$0.441439\pi$
$642$	0	0
$643$	10389.6	0.637209	0.318604	−	0.947888i	$-0.396786\pi$
$643$	10389.6	0.637209	0.318604	+	0.947888i	$0.396786\pi$
$644$	488.861	0.0299128
$645$	0	0
$646$	1731.53	0.105458
$647$	−23853.1	−1.44940	−0.724701	−	0.689064i	$-0.758022\pi$
$647$	−23853.1	−1.44940	−0.724701	+	0.689064i	$0.758022\pi$
$648$	0	0
$649$	−15674.4	−0.948032
$650$	−19110.5	−1.15319
$651$	0	0
$652$	1682.97	0.101089
$653$	−13748.3	−0.823912	−0.411956	−	0.911204i	$-0.635154\pi$
$653$	−13748.3	−0.823912	−0.411956	+	0.911204i	$0.635154\pi$
$654$	0	0
$655$	11478.7	0.684746
$656$	3292.23	0.195945
$657$	0	0
$658$	3186.97	0.188816
$659$	22821.1	1.34899	0.674495	−	0.738279i	$-0.264361\pi$
$659$	22821.1	1.34899	0.674495	+	0.738279i	$0.264361\pi$
$660$	0	0
$661$	−14131.9	−0.831567	−0.415783	−	0.909464i	$-0.636493\pi$
$661$	−14131.9	−0.831567	−0.415783	+	0.909464i	$0.636493\pi$
$662$	−9375.14	−0.550416
$663$	0	0
$664$	6364.85	0.371994
$665$	3901.09	0.227486
$666$	0	0
$667$	4120.15	0.239180
$668$	−2606.43	−0.150967
$669$	0	0
$670$	−16155.0	−0.931528
$671$	11699.3	0.673095
$672$	0	0
$673$	7960.88	0.455972	0.227986	−	0.973664i	$-0.426786\pi$
$673$	7960.88	0.455972	0.227986	+	0.973664i	$0.426786\pi$
$674$	11975.2	0.684374
$675$	0	0
$676$	21665.6	1.23268
$677$	−10547.0	−0.598751	−0.299376	−	0.954135i	$-0.596778\pi$
$677$	−10547.0	−0.598751	−0.299376	+	0.954135i	$0.596778\pi$
$678$	0	0
$679$	−5514.16	−0.311656
$680$	−2212.39	−0.124766
$681$	0	0
$682$	4979.46	0.279580
$683$	−870.274	−0.0487557	−0.0243778	−	0.999703i	$-0.507760\pi$
$683$	−870.274	−0.0487557	−0.0243778	+	0.999703i	$0.507760\pi$
$684$	0	0
$685$	22098.3	1.23260
$686$	−6990.34	−0.389056
$687$	0	0
$688$	−2392.79	−0.132593
$689$	25815.4	1.42741
$690$	0	0
$691$	19966.0	1.09919	0.549596	−	0.835431i	$-0.314782\pi$
$691$	19966.0	1.09919	0.549596	+	0.835431i	$0.314782\pi$
$692$	5385.41	0.295842
$693$	0	0
$694$	−6756.61	−0.369564
$695$	34128.5	1.86269
$696$	0	0
$697$	−3715.88	−0.201935
$698$	1688.54	0.0915648
$699$	0	0
$700$	−2327.61	−0.125679
$701$	28944.0	1.55949	0.779744	−	0.626099i	$-0.215349\pi$
$701$	28944.0	1.55949	0.779744	+	0.626099i	$0.215349\pi$
$702$	0	0
$703$	−19699.1	−1.05685
$704$	1360.31	0.0728247
$705$	0	0
$706$	6334.26	0.337667
$707$	−1990.24	−0.105871
$708$	0	0
$709$	8061.61	0.427024	0.213512	−	0.976940i	$-0.431510\pi$
$709$	8061.61	0.427024	0.213512	+	0.976940i	$0.431510\pi$
$710$	9113.73	0.481735
$711$	0	0
$712$	375.684	0.0197744
$713$	2694.15	0.141510
$714$	0	0
$715$	−28400.6	−1.48549
$716$	13699.3	0.715037
$717$	0	0
$718$	20968.1	1.08986
$719$	21709.2	1.12603	0.563015	−	0.826447i	$-0.309641\pi$
$719$	21709.2	1.12603	0.563015	+	0.826447i	$0.309641\pi$
$720$	0	0
$721$	2172.69	0.112226
$722$	9121.30	0.470166
$723$	0	0
$724$	−13027.9	−0.668755
$725$	−19617.2	−1.00492
$726$	0	0
$727$	11612.3	0.592403	0.296202	−	0.955125i	$-0.404280\pi$
$727$	11612.3	0.592403	0.296202	+	0.955125i	$0.404280\pi$
$728$	3709.17	0.188834
$729$	0	0
$730$	−21877.6	−1.10921
$731$	2700.69	0.136647
$732$	0	0
$733$	−25130.2	−1.26631	−0.633155	−	0.774025i	$-0.718240\pi$
$733$	−25130.2	−1.26631	−0.633155	+	0.774025i	$0.718240\pi$
$734$	−4698.01	−0.236249
$735$	0	0
$736$	736.000	0.0368605
$737$	−11211.3	−0.560344
$738$	0	0
$739$	−14011.9	−0.697478	−0.348739	−	0.937220i	$-0.613390\pi$
$739$	−14011.9	−0.697478	−0.348739	+	0.937220i	$0.613390\pi$
$740$	25169.7	1.25035
$741$	0	0
$742$	3144.25	0.155565
$743$	6207.90	0.306522	0.153261	−	0.988186i	$-0.451022\pi$
$743$	6207.90	0.306522	0.153261	+	0.988186i	$0.451022\pi$
$744$	0	0
$745$	22760.6	1.11931
$746$	24152.6	1.18537
$747$	0	0
$748$	−1535.35	−0.0750509
$749$	6729.86	0.328310
$750$	0	0
$751$	−8787.36	−0.426971	−0.213485	−	0.976946i	$-0.568482\pi$
$751$	−8787.36	−0.426971	−0.213485	+	0.976946i	$0.568482\pi$
$752$	4798.12	0.232672
$753$	0	0
$754$	31261.2	1.50990
$755$	−5342.41	−0.257524
$756$	0	0
$757$	35327.3	1.69616	0.848079	−	0.529870i	$-0.177759\pi$
$757$	35327.3	1.69616	0.848079	+	0.529870i	$0.177759\pi$
$758$	3974.70	0.190458
$759$	0	0
$760$	5873.25	0.280323
$761$	−20753.0	−0.988562	−0.494281	−	0.869302i	$-0.664569\pi$
$761$	−20753.0	−0.988562	−0.494281	+	0.869302i	$0.664569\pi$
$762$	0	0
$763$	7266.86	0.344794
$764$	16133.5	0.763990
$765$	0	0
$766$	20949.2	0.988154
$767$	−64346.1	−3.02921
$768$	0	0
$769$	−3409.43	−0.159879	−0.0799397	−	0.996800i	$-0.525473\pi$
$769$	−3409.43	−0.159879	−0.0799397	+	0.996800i	$0.525473\pi$
$770$	−3459.12	−0.161894
$771$	0	0
$772$	1195.14	0.0557174
$773$	504.444	0.0234716	0.0117358	−	0.999931i	$-0.496264\pi$
$773$	504.444	0.0234716	0.0117358	+	0.999931i	$0.496264\pi$
$774$	0	0
$775$	−12827.6	−0.594558
$776$	−8301.79	−0.384043
$777$	0	0
$778$	−7957.05	−0.366676
$779$	9864.58	0.453704
$780$	0	0
$781$	6324.75	0.289779
$782$	−830.708	−0.0379873
$783$	0	0
$784$	−5036.23	−0.229420
$785$	−26816.4	−1.21926
$786$	0	0
$787$	−8403.22	−0.380613	−0.190306	−	0.981725i	$-0.560948\pi$
$787$	−8403.22	−0.380613	−0.190306	+	0.981725i	$0.560948\pi$
$788$	−17016.8	−0.769286
$789$	0	0
$790$	−25564.3	−1.15131
$791$	502.519	0.0225885
$792$	0	0
$793$	48027.7	2.15071
$794$	−11079.6	−0.495215
$795$	0	0
$796$	−18924.9	−0.842684
$797$	−19884.7	−0.883754	−0.441877	−	0.897076i	$-0.645687\pi$
$797$	−19884.7	−0.883754	−0.441877	+	0.897076i	$0.645687\pi$
$798$	0	0
$799$	−5415.54	−0.239785
$800$	−3504.31	−0.154870
$801$	0	0
$802$	4690.40	0.206513
$803$	−15182.6	−0.667226
$804$	0	0
$805$	−1871.57	−0.0819430
$806$	20441.6	0.893329
$807$	0	0
$808$	−2996.39	−0.130461
$809$	−33037.5	−1.43577	−0.717883	−	0.696164i	$-0.754889\pi$
$809$	−33037.5	−1.43577	−0.717883	+	0.696164i	$0.754889\pi$
$810$	0	0
$811$	−11788.0	−0.510399	−0.255199	−	0.966888i	$-0.582141\pi$
$811$	−11788.0	−0.510399	−0.255199	+	0.966888i	$0.582141\pi$
$812$	3807.53	0.164554
$813$	0	0
$814$	17467.3	0.752123
$815$	−6443.14	−0.276924
$816$	0	0
$817$	−7169.56	−0.307015
$818$	14527.9	0.620974
$819$	0	0
$820$	−12604.1	−0.536772
$821$	−22695.5	−0.964772	−0.482386	−	0.875959i	$-0.660230\pi$
$821$	−22695.5	−0.964772	−0.482386	+	0.875959i	$0.660230\pi$
$822$	0	0
$823$	33804.9	1.43179	0.715896	−	0.698207i	$-0.246018\pi$
$823$	33804.9	1.43179	0.715896	+	0.698207i	$0.246018\pi$
$824$	3271.06	0.138292
$825$	0	0
$826$	−7837.19	−0.330134
$827$	−25825.6	−1.08591	−0.542953	−	0.839763i	$-0.682694\pi$
$827$	−25825.6	−1.08591	−0.542953	+	0.839763i	$0.682694\pi$
$828$	0	0
$829$	−11682.4	−0.489442	−0.244721	−	0.969594i	$-0.578696\pi$
$829$	−11682.4	−0.489442	−0.244721	+	0.969594i	$0.578696\pi$
$830$	−24367.4	−1.01904
$831$	0	0
$832$	5584.31	0.232694
$833$	5684.29	0.236433
$834$	0	0
$835$	9978.51	0.413558
$836$	4075.92	0.168623
$837$	0	0
$838$	8189.78	0.337603
$839$	20526.4	0.844635	0.422317	−	0.906448i	$-0.361217\pi$
$839$	20526.4	0.844635	0.422317	+	0.906448i	$0.361217\pi$
$840$	0	0
$841$	7701.10	0.315761
$842$	−9355.35	−0.382906
$843$	0	0
$844$	20525.6	0.837110
$845$	−82945.3	−3.37681
$846$	0	0
$847$	4671.98	0.189529
$848$	4733.79	0.191697
$849$	0	0
$850$	3955.24	0.159604
$851$	9450.74	0.380690
$852$	0	0
$853$	15186.7	0.609593	0.304797	−	0.952417i	$-0.401412\pi$
$853$	15186.7	0.609593	0.304797	+	0.952417i	$0.401412\pi$
$854$	5849.66	0.234392
$855$	0	0
$856$	10132.1	0.404565
$857$	−37394.9	−1.49053	−0.745266	−	0.666767i	$-0.767678\pi$
$857$	−37394.9	−1.49053	−0.745266	+	0.666767i	$0.767678\pi$
$858$	0	0
$859$	19572.9	0.777437	0.388719	−	0.921356i	$-0.372918\pi$
$859$	19572.9	0.777437	0.388719	+	0.921356i	$0.372918\pi$
$860$	9160.61	0.363226
$861$	0	0
$862$	26438.1	1.04465
$863$	−16086.0	−0.634500	−0.317250	−	0.948342i	$-0.602759\pi$
$863$	−16086.0	−0.634500	−0.317250	+	0.948342i	$0.602759\pi$
$864$	0	0
$865$	−20617.7	−0.810429
$866$	−6219.55	−0.244052
$867$	0	0
$868$	2489.73	0.0973582
$869$	−17741.1	−0.692550
$870$	0	0
$871$	−46024.3	−1.79044
$872$	10940.5	0.424878
$873$	0	0
$874$	2205.29	0.0853491
$875$	−1260.49	−0.0486998
$876$	0	0
$877$	1016.11	0.0391239	0.0195620	−	0.999809i	$-0.493773\pi$
$877$	1016.11	0.0391239	0.0195620	+	0.999809i	$0.493773\pi$
$878$	−32850.0	−1.26268
$879$	0	0
$880$	−5207.85	−0.199496
$881$	−7851.35	−0.300248	−0.150124	−	0.988667i	$-0.547967\pi$
$881$	−7851.35	−0.300248	−0.150124	+	0.988667i	$0.547967\pi$
$882$	0	0
$883$	17536.2	0.668336	0.334168	−	0.942513i	$-0.391545\pi$
$883$	17536.2	0.668336	0.334168	+	0.942513i	$0.391545\pi$
$884$	−6302.90	−0.239807
$885$	0	0
$886$	−9472.29	−0.359174
$887$	32681.8	1.23714	0.618572	−	0.785728i	$-0.287711\pi$
$887$	32681.8	1.23714	0.618572	+	0.785728i	$0.287711\pi$
$888$	0	0
$889$	−11044.4	−0.416667
$890$	−1438.28	−0.0541699
$891$	0	0
$892$	8757.64	0.328730
$893$	14376.7	0.538743
$894$	0	0
$895$	−52446.8	−1.95877
$896$	680.155	0.0253598
$897$	0	0
$898$	−24683.7	−0.917266
$899$	20983.6	0.778467
$900$	0	0
$901$	−5342.94	−0.197557
$902$	−8746.98	−0.322885
$903$	0	0
$904$	756.562	0.0278350
$905$	49876.5	1.83199
$906$	0	0
$907$	14992.8	0.548874	0.274437	−	0.961605i	$-0.411509\pi$
$907$	14992.8	0.548874	0.274437	+	0.961605i	$0.411509\pi$
$908$	23413.5	0.855731
$909$	0	0
$910$	−14200.3	−0.517292
$911$	28483.3	1.03589	0.517943	−	0.855415i	$-0.326698\pi$
$911$	28483.3	1.03589	0.517943	+	0.855415i	$0.326698\pi$
$912$	0	0
$913$	−16910.5	−0.612985
$914$	−11.6098	−0.000420152 0
$915$	0	0
$916$	−13708.7	−0.494485
$917$	3982.98	0.143435
$918$	0	0
$919$	48476.1	1.74002	0.870011	−	0.493032i	$-0.164111\pi$
$919$	48476.1	1.74002	0.870011	+	0.493032i	$0.164111\pi$
$920$	−2817.72	−0.100976
$921$	0	0
$922$	−23005.6	−0.821747
$923$	25964.2	0.925918
$924$	0	0
$925$	−44997.7	−1.59948
$926$	−23479.1	−0.833229
$927$	0	0
$928$	5732.39	0.202775
$929$	−44980.1	−1.58853	−0.794267	−	0.607568i	$-0.792145\pi$
$929$	−44980.1	−1.58853	−0.794267	+	0.607568i	$0.792145\pi$
$930$	0	0
$931$	−15090.2	−0.531214
$932$	−22893.3	−0.804609
$933$	0	0
$934$	−24339.7	−0.852697
$935$	5877.99	0.205594
$936$	0	0
$937$	1296.80	0.0452131	0.0226065	−	0.999744i	$-0.492804\pi$
$937$	1296.80	0.0452131	0.0226065	+	0.999744i	$0.492804\pi$
$938$	−5605.65	−0.195129
$939$	0	0
$940$	−18369.2	−0.637381
$941$	−42721.5	−1.48000	−0.740000	−	0.672607i	$-0.765175\pi$
$941$	−42721.5	−1.48000	−0.740000	+	0.672607i	$0.765175\pi$
$942$	0	0
$943$	−4732.58	−0.163430
$944$	−11799.2	−0.406813
$945$	0	0
$946$	6357.29	0.218492
$947$	13367.3	0.458688	0.229344	−	0.973345i	$-0.426342\pi$
$947$	13367.3	0.458688	0.229344	+	0.973345i	$0.426342\pi$
$948$	0	0
$949$	−62327.2	−2.13196
$950$	−10500.0	−0.358596
$951$	0	0
$952$	−767.677	−0.0261350
$953$	−7309.26	−0.248447	−0.124224	−	0.992254i	$-0.539644\pi$
$953$	−7309.26	−0.248447	−0.124224	+	0.992254i	$0.539644\pi$
$954$	0	0
$955$	−61765.8	−2.09287
$956$	−13192.2	−0.446302
$957$	0	0
$958$	18193.7	0.613583
$959$	7667.88	0.258195
$960$	0	0
$961$	−16069.9	−0.539421
$962$	71706.3	2.40323
$963$	0	0
$964$	−21566.4	−0.720547
$965$	−4575.49	−0.152632
$966$	0	0
$967$	41664.5	1.38556	0.692782	−	0.721147i	$-0.256385\pi$
$967$	41664.5	1.38556	0.692782	+	0.721147i	$0.256385\pi$
$968$	7033.86	0.233550
$969$	0	0
$970$	31782.8	1.05205
$971$	−54110.9	−1.78836	−0.894182	−	0.447705i	$-0.852242\pi$
$971$	−54110.9	−1.78836	−0.894182	+	0.447705i	$0.852242\pi$
$972$	0	0
$973$	11842.3	0.390181
$974$	−37482.0	−1.23306
$975$	0	0
$976$	8806.89	0.288834
$977$	−5877.98	−0.192480	−0.0962401	−	0.995358i	$-0.530682\pi$
$977$	−5877.98	−0.192480	−0.0962401	+	0.995358i	$0.530682\pi$
$978$	0	0
$979$	−998.137	−0.0325849
$980$	19280.8	0.628474
$981$	0	0
$982$	−33807.4	−1.09861
$983$	14598.5	0.473674	0.236837	−	0.971549i	$-0.423889\pi$
$983$	14598.5	0.473674	0.236837	+	0.971549i	$0.423889\pi$
$984$	0	0
$985$	65147.5	2.10738
$986$	−6470.03	−0.208973
$987$	0	0
$988$	16732.4	0.538793
$989$	3439.63	0.110590
$990$	0	0
$991$	36254.6	1.16212	0.581062	−	0.813859i	$-0.302637\pi$
$991$	36254.6	1.16212	0.581062	+	0.813859i	$0.302637\pi$
$992$	3748.39	0.119971
$993$	0	0
$994$	3162.38	0.100910
$995$	72452.7	2.30845
$996$	0	0
$997$	8640.87	0.274483	0.137241	−	0.990538i	$-0.456176\pi$
$997$	8640.87	0.274483	0.137241	+	0.990538i	$0.456176\pi$
$998$	5380.02	0.170643
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.4.a.g.1.1		2
3.2	odd	2		138.4.a.e.1.2	✓	2
12.11	even	2		1104.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.a.e.1.2	✓	2	3.2	odd	2
414.4.a.g.1.1		2	1.1	even	1	trivial
1104.4.a.p.1.2		2	12.11	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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -15.3137 q^{5} -5.31371 q^{7} -8.00000 q^{8} +30.6274 q^{10} +21.2548 q^{11} +87.2548 q^{13} +10.6274 q^{14} +16.0000 q^{16} -18.0589 q^{17} +47.9411 q^{19} -61.2548 q^{20} -42.5097 q^{22} -23.0000 q^{23} +109.510 q^{25} -174.510 q^{26} -21.2548 q^{28} -179.137 q^{29} -117.137 q^{31} -32.0000 q^{32} +36.1177 q^{34} +81.3726 q^{35} -410.902 q^{37} -95.8823 q^{38} +122.510 q^{40} +205.765 q^{41} -149.549 q^{43} +85.0193 q^{44} +46.0000 q^{46} +299.882 q^{47} -314.765 q^{49} -219.019 q^{50} +349.019 q^{52} +295.862 q^{53} -325.490 q^{55} +42.5097 q^{56} +358.274 q^{58} -737.450 q^{59} +550.431 q^{61} +234.274 q^{62} +64.0000 q^{64} -1336.20 q^{65} -527.470 q^{67} -72.2355 q^{68} -162.745 q^{70} +297.568 q^{71} -714.313 q^{73} +821.803 q^{74} +191.765 q^{76} -112.942 q^{77} -834.686 q^{79} -245.019 q^{80} -411.529 q^{82} -795.606 q^{83} +276.548 q^{85} +299.098 q^{86} -170.039 q^{88} -46.9605 q^{89} -463.647 q^{91} -92.0000 q^{92} -599.765 q^{94} -734.156 q^{95} +1037.72 q^{97} +629.529 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 8 q^{5} + 12 q^{7} - 16 q^{8} + 16 q^{10} - 48 q^{11} + 84 q^{13} - 24 q^{14} + 32 q^{16} - 104 q^{17} + 28 q^{19} - 32 q^{20} + 96 q^{22} - 46 q^{23} + 38 q^{25} - 168 q^{26}+ \cdots + 716 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 8 * q^5 + 12 * q^7 - 16 * q^8 + 16 * q^10 - 48 * q^11 + 84 * q^13 - 24 * q^14 + 32 * q^16 - 104 * q^17 + 28 * q^19 - 32 * q^20 + 96 * q^22 - 46 * q^23 + 38 * q^25 - 168 * q^26 + 48 * q^28 - 132 * q^29 - 8 * q^31 - 64 * q^32 + 208 * q^34 + 208 * q^35 - 324 * q^37 - 56 * q^38 + 64 * q^40 + 140 * q^41 - 548 * q^43 - 192 * q^44 + 92 * q^46 + 464 * q^47 - 358 * q^49 - 76 * q^50 + 336 * q^52 - 336 * q^53 - 832 * q^55 - 96 * q^56 + 264 * q^58 - 72 * q^59 + 60 * q^61 + 16 * q^62 + 128 * q^64 - 1360 * q^65 - 444 * q^67 - 416 * q^68 - 416 * q^70 - 672 * q^71 - 252 * q^73 + 648 * q^74 + 112 * q^76 - 1312 * q^77 - 1692 * q^79 - 128 * q^80 - 280 * q^82 + 400 * q^83 - 352 * q^85 + 1096 * q^86 + 384 * q^88 + 336 * q^89 - 520 * q^91 - 184 * q^92 - 928 * q^94 - 880 * q^95 + 220 * q^97 + 716 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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