LMFDB - Embedded newform 2496.4.a.ca.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 20x^{3} - 33x^{2} + 17x + 40 $ x^5 - 20x^3 - 33x^2 + 17*x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.52754$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +4.16805 q^{5} -28.4346 q^{7} +9.00000 q^{9} -19.3149 q^{11} +13.0000 q^{13} -12.5041 q^{15} +57.1862 q^{17} -40.1726 q^{19} +85.3039 q^{21} +203.259 q^{23} -107.627 q^{25} -27.0000 q^{27} -163.622 q^{29} -143.802 q^{31} +57.9447 q^{33} -118.517 q^{35} +345.752 q^{37} -39.0000 q^{39} +334.327 q^{41} -179.620 q^{43} +37.5124 q^{45} -506.595 q^{47} +465.528 q^{49} -171.559 q^{51} +606.740 q^{53} -80.5054 q^{55} +120.518 q^{57} +785.475 q^{59} +458.960 q^{61} -255.912 q^{63} +54.1846 q^{65} -96.9382 q^{67} -609.777 q^{69} +403.353 q^{71} -893.467 q^{73} +322.882 q^{75} +549.212 q^{77} -139.564 q^{79} +81.0000 q^{81} +988.603 q^{83} +238.355 q^{85} +490.867 q^{87} +165.627 q^{89} -369.650 q^{91} +431.407 q^{93} -167.441 q^{95} -1506.76 q^{97} -173.834 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 15 q^{3} + 10 q^{5} - 14 q^{7} + 45 q^{9} + 22 q^{11} + 65 q^{13} - 30 q^{15} - 34 q^{17} - 90 q^{19} + 42 q^{21} - 96 q^{23} + 107 q^{25} - 135 q^{27} + 54 q^{29} - 378 q^{31} - 66 q^{33} + 84 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) $ 5 * q - 15 * q^3 + 10 * q^5 - 14 * q^7 + 45 * q^9 + 22 * q^11 + 65 * q^13 - 30 * q^15 - 34 * q^17 - 90 * q^19 + 42 * q^21 - 96 * q^23 + 107 * q^25 - 135 * q^27 + 54 * q^29 - 378 * q^31 - 66 * q^33 + 84 * q^35 + 318 * q^37 - 195 * q^39 - 458 * q^41 + 104 * q^43 + 90 * q^45 - 474 * q^47 + 25 * q^49 + 102 * q^51 + 450 * q^53 - 1116 * q^55 + 270 * q^57 + 822 * q^59 + 546 * q^61 - 126 * q^63 + 130 * q^65 - 130 * q^67 + 288 * q^69 - 1646 * q^71 - 166 * q^73 - 321 * q^75 + 900 * q^77 - 196 * q^79 + 405 * q^81 - 118 * q^83 + 3340 * q^85 - 162 * q^87 - 186 * q^89 - 182 * q^91 + 1134 * q^93 - 564 * q^95 - 2582 * q^97 + 198 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	4.16805	0.372802	0.186401	−	0.982474i	$-0.440318\pi$
$5$	4.16805	0.372802	0.186401	+	0.982474i	$0.440318\pi$
$6$	0	0
$7$	−28.4346	−1.53533	−0.767663	−	0.640854i	$-0.778580\pi$
$7$	−28.4346	−1.53533	−0.767663	+	0.640854i	$0.778580\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−19.3149	−0.529424	−0.264712	−	0.964328i	$-0.585277\pi$
$11$	−19.3149	−0.529424	−0.264712	+	0.964328i	$0.585277\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−12.5041	−0.215237
$16$	0	0
$17$	57.1862	0.815864	0.407932	−	0.913012i	$-0.366250\pi$
$17$	57.1862	0.815864	0.407932	+	0.913012i	$0.366250\pi$
$18$	0	0
$19$	−40.1726	−0.485064	−0.242532	−	0.970143i	$-0.577978\pi$
$19$	−40.1726	−0.485064	−0.242532	+	0.970143i	$0.577978\pi$
$20$	0	0
$21$	85.3039	0.886421
$22$	0	0
$23$	203.259	1.84271	0.921357	−	0.388717i	$-0.127082\pi$
$23$	203.259	1.84271	0.921357	+	0.388717i	$0.127082\pi$
$24$	0	0
$25$	−107.627	−0.861019
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−163.622	−1.04772	−0.523861	−	0.851804i	$-0.675509\pi$
$29$	−163.622	−1.04772	−0.523861	+	0.851804i	$0.675509\pi$
$30$	0	0
$31$	−143.802	−0.833150	−0.416575	−	0.909101i	$-0.636770\pi$
$31$	−143.802	−0.833150	−0.416575	+	0.909101i	$0.636770\pi$
$32$	0	0
$33$	57.9447	0.305663
$34$	0	0
$35$	−118.517	−0.572372
$36$	0	0
$37$	345.752	1.53625	0.768126	−	0.640299i	$-0.221190\pi$
$37$	345.752	1.53625	0.768126	+	0.640299i	$0.221190\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	334.327	1.27349	0.636746	−	0.771074i	$-0.280280\pi$
$41$	334.327	1.27349	0.636746	+	0.771074i	$0.280280\pi$
$42$	0	0
$43$	−179.620	−0.637019	−0.318509	−	0.947920i	$-0.603182\pi$
$43$	−179.620	−0.637019	−0.318509	+	0.947920i	$0.603182\pi$
$44$	0	0
$45$	37.5124	0.124267
$46$	0	0
$47$	−506.595	−1.57222	−0.786111	−	0.618085i	$-0.787909\pi$
$47$	−506.595	−1.57222	−0.786111	+	0.618085i	$0.787909\pi$
$48$	0	0
$49$	465.528	1.35722
$50$	0	0
$51$	−171.559	−0.471039
$52$	0	0
$53$	606.740	1.57249	0.786247	−	0.617913i	$-0.212022\pi$
$53$	606.740	1.57249	0.786247	+	0.617913i	$0.212022\pi$
$54$	0	0
$55$	−80.5054	−0.197370
$56$	0	0
$57$	120.518	0.280052
$58$	0	0
$59$	785.475	1.73322	0.866611	−	0.498984i	$-0.166294\pi$
$59$	785.475	1.73322	0.866611	+	0.498984i	$0.166294\pi$
$60$	0	0
$61$	458.960	0.963340	0.481670	−	0.876353i	$-0.340030\pi$
$61$	458.960	0.963340	0.481670	+	0.876353i	$0.340030\pi$
$62$	0	0
$63$	−255.912	−0.511775
$64$	0	0
$65$	54.1846	0.103397
$66$	0	0
$67$	−96.9382	−0.176759	−0.0883797	−	0.996087i	$-0.528169\pi$
$67$	−96.9382	−0.176759	−0.0883797	+	0.996087i	$0.528169\pi$
$68$	0	0
$69$	−609.777	−1.06389
$70$	0	0
$71$	403.353	0.674214	0.337107	−	0.941466i	$-0.390552\pi$
$71$	403.353	0.674214	0.337107	+	0.941466i	$0.390552\pi$
$72$	0	0
$73$	−893.467	−1.43250	−0.716249	−	0.697844i	$-0.754143\pi$
$73$	−893.467	−1.43250	−0.716249	+	0.697844i	$0.754143\pi$
$74$	0	0
$75$	322.882	0.497110
$76$	0	0
$77$	549.212	0.812838
$78$	0	0
$79$	−139.564	−0.198761	−0.0993805	−	0.995050i	$-0.531686\pi$
$79$	−139.564	−0.198761	−0.0993805	+	0.995050i	$0.531686\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	988.603	1.30739	0.653695	−	0.756759i	$-0.273218\pi$
$83$	988.603	1.30739	0.653695	+	0.756759i	$0.273218\pi$
$84$	0	0
$85$	238.355	0.304155
$86$	0	0
$87$	490.867	0.604902
$88$	0	0
$89$	165.627	0.197264	0.0986318	−	0.995124i	$-0.468553\pi$
$89$	165.627	0.197264	0.0986318	+	0.995124i	$0.468553\pi$
$90$	0	0
$91$	−369.650	−0.425823
$92$	0	0
$93$	431.407	0.481019
$94$	0	0
$95$	−167.441	−0.180833
$96$	0	0
$97$	−1506.76	−1.57720	−0.788599	−	0.614908i	$-0.789193\pi$
$97$	−1506.76	−1.57720	−0.788599	+	0.614908i	$0.789193\pi$
$98$	0	0
$99$	−173.834	−0.176475
$100$	0	0
$101$	−193.102	−0.190241	−0.0951207	−	0.995466i	$-0.530324\pi$
$101$	−193.102	−0.190241	−0.0951207	+	0.995466i	$0.530324\pi$
$102$	0	0
$103$	−1973.16	−1.88759	−0.943794	−	0.330534i	$-0.892771\pi$
$103$	−1973.16	−1.88759	−0.943794	+	0.330534i	$0.892771\pi$
$104$	0	0
$105$	355.551	0.330459
$106$	0	0
$107$	−732.944	−0.662209	−0.331104	−	0.943594i	$-0.607421\pi$
$107$	−732.944	−0.662209	−0.331104	+	0.943594i	$0.607421\pi$
$108$	0	0
$109$	837.449	0.735900	0.367950	−	0.929846i	$-0.380060\pi$
$109$	837.449	0.735900	0.367950	+	0.929846i	$0.380060\pi$
$110$	0	0
$111$	−1037.26	−0.886955
$112$	0	0
$113$	−574.885	−0.478589	−0.239295	−	0.970947i	$-0.576916\pi$
$113$	−574.885	−0.478589	−0.239295	+	0.970947i	$0.576916\pi$
$114$	0	0
$115$	847.193	0.686967
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−1626.07	−1.25262
$120$	0	0
$121$	−957.935	−0.719711
$122$	0	0
$123$	−1002.98	−0.735251
$124$	0	0
$125$	−969.602	−0.693791
$126$	0	0
$127$	2053.37	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$127$	2053.37	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$128$	0	0
$129$	538.860	0.367783
$130$	0	0
$131$	−217.920	−0.145341	−0.0726707	−	0.997356i	$-0.523152\pi$
$131$	−217.920	−0.145341	−0.0726707	+	0.997356i	$0.523152\pi$
$132$	0	0
$133$	1142.29	0.744732
$134$	0	0
$135$	−112.537	−0.0717457
$136$	0	0
$137$	−24.0095	−0.0149728	−0.00748638	−	0.999972i	$-0.502383\pi$
$137$	−24.0095	−0.0149728	−0.00748638	+	0.999972i	$0.502383\pi$
$138$	0	0
$139$	−2117.71	−1.29224	−0.646121	−	0.763235i	$-0.723610\pi$
$139$	−2117.71	−1.29224	−0.646121	+	0.763235i	$0.723610\pi$
$140$	0	0
$141$	1519.79	0.907723
$142$	0	0
$143$	−251.094	−0.146836
$144$	0	0
$145$	−681.986	−0.390592
$146$	0	0
$147$	−1396.58	−0.783594
$148$	0	0
$149$	1567.38	0.861776	0.430888	−	0.902405i	$-0.358200\pi$
$149$	1567.38	0.861776	0.430888	+	0.902405i	$0.358200\pi$
$150$	0	0
$151$	2563.06	1.38132	0.690659	−	0.723181i	$-0.257321\pi$
$151$	2563.06	1.38132	0.690659	+	0.723181i	$0.257321\pi$
$152$	0	0
$153$	514.676	0.271955
$154$	0	0
$155$	−599.374	−0.310599
$156$	0	0
$157$	1371.47	0.697166	0.348583	−	0.937278i	$-0.386663\pi$
$157$	1371.47	0.697166	0.348583	+	0.937278i	$0.386663\pi$
$158$	0	0
$159$	−1820.22	−0.907879
$160$	0	0
$161$	−5779.59	−2.82917
$162$	0	0
$163$	−2487.37	−1.19525	−0.597625	−	0.801776i	$-0.703889\pi$
$163$	−2487.37	−1.19525	−0.597625	+	0.801776i	$0.703889\pi$
$164$	0	0
$165$	241.516	0.113952
$166$	0	0
$167$	2645.80	1.22598	0.612988	−	0.790092i	$-0.289967\pi$
$167$	2645.80	1.22598	0.612988	+	0.790092i	$0.289967\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−361.553	−0.161688
$172$	0	0
$173$	3808.37	1.67367	0.836836	−	0.547454i	$-0.184403\pi$
$173$	3808.37	1.67367	0.836836	+	0.547454i	$0.184403\pi$
$174$	0	0
$175$	3060.34	1.32194
$176$	0	0
$177$	−2356.42	−1.00068
$178$	0	0
$179$	−1204.70	−0.503035	−0.251517	−	0.967853i	$-0.580930\pi$
$179$	−1204.70	−0.503035	−0.251517	+	0.967853i	$0.580930\pi$
$180$	0	0
$181$	−3524.03	−1.44718	−0.723589	−	0.690231i	$-0.757509\pi$
$181$	−3524.03	−1.44718	−0.723589	+	0.690231i	$0.757509\pi$
$182$	0	0
$183$	−1376.88	−0.556185
$184$	0	0
$185$	1441.11	0.572717
$186$	0	0
$187$	−1104.55	−0.431938
$188$	0	0
$189$	767.735	0.295474
$190$	0	0
$191$	1319.77	0.499975	0.249987	−	0.968249i	$-0.419574\pi$
$191$	1319.77	0.499975	0.249987	+	0.968249i	$0.419574\pi$
$192$	0	0
$193$	1624.93	0.606036	0.303018	−	0.952985i	$-0.402006\pi$
$193$	1624.93	0.606036	0.303018	+	0.952985i	$0.402006\pi$
$194$	0	0
$195$	−162.554	−0.0596960
$196$	0	0
$197$	−3366.18	−1.21741	−0.608707	−	0.793395i	$-0.708312\pi$
$197$	−3366.18	−1.21741	−0.608707	+	0.793395i	$0.708312\pi$
$198$	0	0
$199$	−3072.12	−1.09435	−0.547177	−	0.837017i	$-0.684298\pi$
$199$	−3072.12	−1.09435	−0.547177	+	0.837017i	$0.684298\pi$
$200$	0	0
$201$	290.815	0.102052
$202$	0	0
$203$	4652.54	1.60859
$204$	0	0
$205$	1393.49	0.474760
$206$	0	0
$207$	1829.33	0.614238
$208$	0	0
$209$	775.929	0.256805
$210$	0	0
$211$	2689.84	0.877614	0.438807	−	0.898581i	$-0.355401\pi$
$211$	2689.84	0.877614	0.438807	+	0.898581i	$0.355401\pi$
$212$	0	0
$213$	−1210.06	−0.389258
$214$	0	0
$215$	−748.665	−0.237482
$216$	0	0
$217$	4088.96	1.27916
$218$	0	0
$219$	2680.40	0.827054
$220$	0	0
$221$	743.420	0.226280
$222$	0	0
$223$	−4427.48	−1.32953	−0.664767	−	0.747051i	$-0.731469\pi$
$223$	−4427.48	−1.32953	−0.664767	+	0.747051i	$0.731469\pi$
$224$	0	0
$225$	−968.646	−0.287006
$226$	0	0
$227$	−4345.30	−1.27052	−0.635260	−	0.772299i	$-0.719107\pi$
$227$	−4345.30	−1.27052	−0.635260	+	0.772299i	$0.719107\pi$
$228$	0	0
$229$	966.407	0.278873	0.139437	−	0.990231i	$-0.455471\pi$
$229$	966.407	0.278873	0.139437	+	0.990231i	$0.455471\pi$
$230$	0	0
$231$	−1647.64	−0.469292
$232$	0	0
$233$	−5764.75	−1.62086	−0.810432	−	0.585833i	$-0.800767\pi$
$233$	−5764.75	−1.62086	−0.810432	+	0.585833i	$0.800767\pi$
$234$	0	0
$235$	−2111.51	−0.586127
$236$	0	0
$237$	418.691	0.114755
$238$	0	0
$239$	1248.65	0.337942	0.168971	−	0.985621i	$-0.445956\pi$
$239$	1248.65	0.337942	0.168971	+	0.985621i	$0.445956\pi$
$240$	0	0
$241$	−6121.62	−1.63622	−0.818108	−	0.575065i	$-0.804977\pi$
$241$	−6121.62	−1.63622	−0.818108	+	0.575065i	$0.804977\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1940.34	0.505975
$246$	0	0
$247$	−522.244	−0.134533
$248$	0	0
$249$	−2965.81	−0.754821
$250$	0	0
$251$	2295.82	0.577335	0.288667	−	0.957429i	$-0.406788\pi$
$251$	2295.82	0.577335	0.288667	+	0.957429i	$0.406788\pi$
$252$	0	0
$253$	−3925.93	−0.975577
$254$	0	0
$255$	−715.064	−0.175604
$256$	0	0
$257$	−447.329	−0.108574	−0.0542871	−	0.998525i	$-0.517289\pi$
$257$	−447.329	−0.108574	−0.0542871	+	0.998525i	$0.517289\pi$
$258$	0	0
$259$	−9831.33	−2.35865
$260$	0	0
$261$	−1472.60	−0.349240
$262$	0	0
$263$	−6253.75	−1.46625	−0.733124	−	0.680095i	$-0.761938\pi$
$263$	−6253.75	−1.46625	−0.733124	+	0.680095i	$0.761938\pi$
$264$	0	0
$265$	2528.92	0.586228
$266$	0	0
$267$	−496.882	−0.113890
$268$	0	0
$269$	8274.78	1.87555	0.937774	−	0.347247i	$-0.112884\pi$
$269$	8274.78	1.87555	0.937774	+	0.347247i	$0.112884\pi$
$270$	0	0
$271$	−8010.21	−1.79552	−0.897760	−	0.440485i	$-0.854806\pi$
$271$	−8010.21	−1.79552	−0.897760	+	0.440485i	$0.854806\pi$
$272$	0	0
$273$	1108.95	0.245849
$274$	0	0
$275$	2078.81	0.455844
$276$	0	0
$277$	5517.01	1.19670	0.598348	−	0.801236i	$-0.295824\pi$
$277$	5517.01	1.19670	0.598348	+	0.801236i	$0.295824\pi$
$278$	0	0
$279$	−1294.22	−0.277717
$280$	0	0
$281$	−1533.33	−0.325518	−0.162759	−	0.986666i	$-0.552039\pi$
$281$	−1533.33	−0.325518	−0.162759	+	0.986666i	$0.552039\pi$
$282$	0	0
$283$	−8486.53	−1.78259	−0.891293	−	0.453429i	$-0.850201\pi$
$283$	−8486.53	−1.78259	−0.891293	+	0.453429i	$0.850201\pi$
$284$	0	0
$285$	502.324	0.104404
$286$	0	0
$287$	−9506.47	−1.95522
$288$	0	0
$289$	−1642.74	−0.334366
$290$	0	0
$291$	4520.28	0.910596
$292$	0	0
$293$	−2388.22	−0.476182	−0.238091	−	0.971243i	$-0.576522\pi$
$293$	−2388.22	−0.476182	−0.238091	+	0.971243i	$0.576522\pi$
$294$	0	0
$295$	3273.90	0.646148
$296$	0	0
$297$	521.502	0.101888
$298$	0	0
$299$	2642.37	0.511077
$300$	0	0
$301$	5107.43	0.978031
$302$	0	0
$303$	579.306	0.109836
$304$	0	0
$305$	1912.97	0.359135
$306$	0	0
$307$	4487.43	0.834238	0.417119	−	0.908852i	$-0.363040\pi$
$307$	4487.43	0.834238	0.417119	+	0.908852i	$0.363040\pi$
$308$	0	0
$309$	5919.49	1.08980
$310$	0	0
$311$	9390.66	1.71220	0.856102	−	0.516806i	$-0.172879\pi$
$311$	9390.66	1.71220	0.856102	+	0.516806i	$0.172879\pi$
$312$	0	0
$313$	5881.72	1.06215	0.531077	−	0.847323i	$-0.321787\pi$
$313$	5881.72	1.06215	0.531077	+	0.847323i	$0.321787\pi$
$314$	0	0
$315$	−1066.65	−0.190791
$316$	0	0
$317$	−1628.91	−0.288607	−0.144304	−	0.989533i	$-0.546094\pi$
$317$	−1628.91	−0.288607	−0.144304	+	0.989533i	$0.546094\pi$
$318$	0	0
$319$	3160.35	0.554689
$320$	0	0
$321$	2198.83	0.382326
$322$	0	0
$323$	−2297.32	−0.395747
$324$	0	0
$325$	−1399.16	−0.238804
$326$	0	0
$327$	−2512.35	−0.424872
$328$	0	0
$329$	14404.8	2.41387
$330$	0	0
$331$	−6570.00	−1.09100	−0.545498	−	0.838112i	$-0.683660\pi$
$331$	−6570.00	−1.09100	−0.545498	+	0.838112i	$0.683660\pi$
$332$	0	0
$333$	3111.77	0.512084
$334$	0	0
$335$	−404.043	−0.0658962
$336$	0	0
$337$	−6499.74	−1.05063	−0.525317	−	0.850907i	$-0.676053\pi$
$337$	−6499.74	−1.05063	−0.525317	+	0.850907i	$0.676053\pi$
$338$	0	0
$339$	1724.65	0.276314
$340$	0	0
$341$	2777.52	0.441089
$342$	0	0
$343$	−3484.03	−0.548455
$344$	0	0
$345$	−2541.58	−0.396620
$346$	0	0
$347$	−337.205	−0.0521675	−0.0260838	−	0.999660i	$-0.508304\pi$
$347$	−337.205	−0.0521675	−0.0260838	+	0.999660i	$0.508304\pi$
$348$	0	0
$349$	−8949.55	−1.37266	−0.686330	−	0.727290i	$-0.740779\pi$
$349$	−8949.55	−1.37266	−0.686330	+	0.727290i	$0.740779\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−8207.17	−1.23746	−0.618730	−	0.785604i	$-0.712353\pi$
$353$	−8207.17	−1.23746	−0.618730	+	0.785604i	$0.712353\pi$
$354$	0	0
$355$	1681.19	0.251348
$356$	0	0
$357$	4878.20	0.723199
$358$	0	0
$359$	−4959.16	−0.729065	−0.364532	−	0.931191i	$-0.618771\pi$
$359$	−4959.16	−0.729065	−0.364532	+	0.931191i	$0.618771\pi$
$360$	0	0
$361$	−5245.16	−0.764713
$362$	0	0
$363$	2873.80	0.415525
$364$	0	0
$365$	−3724.01	−0.534038
$366$	0	0
$367$	115.538	0.0164334	0.00821669	−	0.999966i	$-0.497385\pi$
$367$	115.538	0.0164334	0.00821669	+	0.999966i	$0.497385\pi$
$368$	0	0
$369$	3008.95	0.424497
$370$	0	0
$371$	−17252.4	−2.41429
$372$	0	0
$373$	−1150.57	−0.159717	−0.0798583	−	0.996806i	$-0.525447\pi$
$373$	−1150.57	−0.159717	−0.0798583	+	0.996806i	$0.525447\pi$
$374$	0	0
$375$	2908.81	0.400560
$376$	0	0
$377$	−2127.09	−0.290586
$378$	0	0
$379$	−1151.03	−0.156001	−0.0780006	−	0.996953i	$-0.524854\pi$
$379$	−1151.03	−0.156001	−0.0780006	+	0.996953i	$0.524854\pi$
$380$	0	0
$381$	−6160.10	−0.828325
$382$	0	0
$383$	−8185.19	−1.09202	−0.546010	−	0.837779i	$-0.683854\pi$
$383$	−8185.19	−1.09202	−0.546010	+	0.837779i	$0.683854\pi$
$384$	0	0
$385$	2289.14	0.303027
$386$	0	0
$387$	−1616.58	−0.212340
$388$	0	0
$389$	−3063.00	−0.399230	−0.199615	−	0.979874i	$-0.563969\pi$
$389$	−3063.00	−0.399230	−0.199615	+	0.979874i	$0.563969\pi$
$390$	0	0
$391$	11623.6	1.50340
$392$	0	0
$393$	653.759	0.0839129
$394$	0	0
$395$	−581.707	−0.0740984
$396$	0	0
$397$	−6264.17	−0.791913	−0.395957	−	0.918269i	$-0.629587\pi$
$397$	−6264.17	−0.791913	−0.395957	+	0.918269i	$0.629587\pi$
$398$	0	0
$399$	−3426.88	−0.429971
$400$	0	0
$401$	−2035.40	−0.253473	−0.126737	−	0.991936i	$-0.540450\pi$
$401$	−2035.40	−0.253473	−0.126737	+	0.991936i	$0.540450\pi$
$402$	0	0
$403$	−1869.43	−0.231074
$404$	0	0
$405$	337.612	0.0414224
$406$	0	0
$407$	−6678.17	−0.813328
$408$	0	0
$409$	−5386.56	−0.651218	−0.325609	−	0.945505i	$-0.605569\pi$
$409$	−5386.56	−0.651218	−0.325609	+	0.945505i	$0.605569\pi$
$410$	0	0
$411$	72.0284	0.00864452
$412$	0	0
$413$	−22334.7	−2.66106
$414$	0	0
$415$	4120.55	0.487397
$416$	0	0
$417$	6353.12	0.746076
$418$	0	0
$419$	4899.94	0.571308	0.285654	−	0.958333i	$-0.407789\pi$
$419$	4899.94	0.571308	0.285654	+	0.958333i	$0.407789\pi$
$420$	0	0
$421$	10054.7	1.16398	0.581992	−	0.813194i	$-0.302273\pi$
$421$	10054.7	1.16398	0.581992	+	0.813194i	$0.302273\pi$
$422$	0	0
$423$	−4559.36	−0.524074
$424$	0	0
$425$	−6154.80	−0.702474
$426$	0	0
$427$	−13050.3	−1.47904
$428$	0	0
$429$	753.281	0.0847756
$430$	0	0
$431$	−12428.5	−1.38900	−0.694500	−	0.719493i	$-0.744374\pi$
$431$	−12428.5	−1.38900	−0.694500	+	0.719493i	$0.744374\pi$
$432$	0	0
$433$	−852.216	−0.0945841	−0.0472920	−	0.998881i	$-0.515059\pi$
$433$	−852.216	−0.0945841	−0.0472920	+	0.998881i	$0.515059\pi$
$434$	0	0
$435$	2045.96	0.225508
$436$	0	0
$437$	−8165.44	−0.893835
$438$	0	0
$439$	4624.92	0.502814	0.251407	−	0.967881i	$-0.419107\pi$
$439$	4624.92	0.502814	0.251407	+	0.967881i	$0.419107\pi$
$440$	0	0
$441$	4189.75	0.452408
$442$	0	0
$443$	700.492	0.0751273	0.0375636	−	0.999294i	$-0.488040\pi$
$443$	700.492	0.0751273	0.0375636	+	0.999294i	$0.488040\pi$
$444$	0	0
$445$	690.343	0.0735402
$446$	0	0
$447$	−4702.14	−0.497547
$448$	0	0
$449$	−225.522	−0.0237039	−0.0118519	−	0.999930i	$-0.503773\pi$
$449$	−225.522	−0.0237039	−0.0118519	+	0.999930i	$0.503773\pi$
$450$	0	0
$451$	−6457.50	−0.674217
$452$	0	0
$453$	−7689.18	−0.797504
$454$	0	0
$455$	−1540.72	−0.158747
$456$	0	0
$457$	2656.86	0.271953	0.135976	−	0.990712i	$-0.456583\pi$
$457$	2656.86	0.271953	0.135976	+	0.990712i	$0.456583\pi$
$458$	0	0
$459$	−1544.03	−0.157013
$460$	0	0
$461$	11577.8	1.16970	0.584851	−	0.811141i	$-0.301153\pi$
$461$	11577.8	1.16970	0.584851	+	0.811141i	$0.301153\pi$
$462$	0	0
$463$	−16475.9	−1.65378	−0.826889	−	0.562366i	$-0.809891\pi$
$463$	−16475.9	−1.65378	−0.826889	+	0.562366i	$0.809891\pi$
$464$	0	0
$465$	1798.12	0.179325
$466$	0	0
$467$	2939.46	0.291267	0.145634	−	0.989339i	$-0.453478\pi$
$467$	2939.46	0.291267	0.145634	+	0.989339i	$0.453478\pi$
$468$	0	0
$469$	2756.40	0.271383
$470$	0	0
$471$	−4114.40	−0.402509
$472$	0	0
$473$	3469.34	0.337253
$474$	0	0
$475$	4323.67	0.417650
$476$	0	0
$477$	5460.66	0.524164
$478$	0	0
$479$	−17274.2	−1.64776	−0.823882	−	0.566761i	$-0.808196\pi$
$479$	−17274.2	−1.64776	−0.823882	+	0.566761i	$0.808196\pi$
$480$	0	0
$481$	4494.78	0.426079
$482$	0	0
$483$	17338.8	1.63342
$484$	0	0
$485$	−6280.24	−0.587982
$486$	0	0
$487$	6256.38	0.582143	0.291072	−	0.956701i	$-0.405988\pi$
$487$	6256.38	0.582143	0.291072	+	0.956701i	$0.405988\pi$
$488$	0	0
$489$	7462.11	0.690078
$490$	0	0
$491$	−11644.8	−1.07031	−0.535156	−	0.844753i	$-0.679747\pi$
$491$	−11644.8	−1.07031	−0.535156	+	0.844753i	$0.679747\pi$
$492$	0	0
$493$	−9356.94	−0.854798
$494$	0	0
$495$	−724.549	−0.0657900
$496$	0	0
$497$	−11469.2	−1.03514
$498$	0	0
$499$	−6903.82	−0.619354	−0.309677	−	0.950842i	$-0.600221\pi$
$499$	−6903.82	−0.619354	−0.309677	+	0.950842i	$0.600221\pi$
$500$	0	0
$501$	−7937.40	−0.707818
$502$	0	0
$503$	4303.25	0.381456	0.190728	−	0.981643i	$-0.438915\pi$
$503$	4303.25	0.381456	0.190728	+	0.981643i	$0.438915\pi$
$504$	0	0
$505$	−804.859	−0.0709223
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−13723.9	−1.19509	−0.597547	−	0.801834i	$-0.703858\pi$
$509$	−13723.9	−1.19509	−0.597547	+	0.801834i	$0.703858\pi$
$510$	0	0
$511$	25405.4	2.19935
$512$	0	0
$513$	1084.66	0.0933507
$514$	0	0
$515$	−8224.24	−0.703696
$516$	0	0
$517$	9784.83	0.832372
$518$	0	0
$519$	−11425.1	−0.966295
$520$	0	0
$521$	−9220.39	−0.775341	−0.387671	−	0.921798i	$-0.626720\pi$
$521$	−9220.39	−0.775341	−0.387671	+	0.921798i	$0.626720\pi$
$522$	0	0
$523$	−2527.06	−0.211282	−0.105641	−	0.994404i	$-0.533689\pi$
$523$	−2527.06	−0.211282	−0.105641	+	0.994404i	$0.533689\pi$
$524$	0	0
$525$	−9181.03	−0.763225
$526$	0	0
$527$	−8223.50	−0.679737
$528$	0	0
$529$	29147.2	2.39560
$530$	0	0
$531$	7069.27	0.577741
$532$	0	0
$533$	4346.25	0.353203
$534$	0	0
$535$	−3054.95	−0.246873
$536$	0	0
$537$	3614.09	0.290427
$538$	0	0
$539$	−8991.62	−0.718547
$540$	0	0
$541$	−6684.92	−0.531251	−0.265626	−	0.964076i	$-0.585579\pi$
$541$	−6684.92	−0.531251	−0.265626	+	0.964076i	$0.585579\pi$
$542$	0	0
$543$	10572.1	0.835529
$544$	0	0
$545$	3490.53	0.274345
$546$	0	0
$547$	10787.0	0.843177	0.421589	−	0.906787i	$-0.361473\pi$
$547$	10787.0	0.843177	0.421589	+	0.906787i	$0.361473\pi$
$548$	0	0
$549$	4130.64	0.321113
$550$	0	0
$551$	6573.14	0.508212
$552$	0	0
$553$	3968.44	0.305163
$554$	0	0
$555$	−4323.33	−0.330658
$556$	0	0
$557$	−20131.1	−1.53139	−0.765694	−	0.643205i	$-0.777604\pi$
$557$	−20131.1	−1.53139	−0.765694	+	0.643205i	$0.777604\pi$
$558$	0	0
$559$	−2335.06	−0.176677
$560$	0	0
$561$	3313.64	0.249379
$562$	0	0
$563$	−2190.13	−0.163948	−0.0819741	−	0.996634i	$-0.526122\pi$
$563$	−2190.13	−0.163948	−0.0819741	+	0.996634i	$0.526122\pi$
$564$	0	0
$565$	−2396.15	−0.178419
$566$	0	0
$567$	−2303.20	−0.170592
$568$	0	0
$569$	12391.7	0.912982	0.456491	−	0.889728i	$-0.349106\pi$
$569$	12391.7	0.912982	0.456491	+	0.889728i	$0.349106\pi$
$570$	0	0
$571$	−2375.87	−0.174128	−0.0870638	−	0.996203i	$-0.527748\pi$
$571$	−2375.87	−0.174128	−0.0870638	+	0.996203i	$0.527748\pi$
$572$	0	0
$573$	−3959.31	−0.288661
$574$	0	0
$575$	−21876.2	−1.58661
$576$	0	0
$577$	−16750.3	−1.20853	−0.604267	−	0.796782i	$-0.706534\pi$
$577$	−16750.3	−1.20853	−0.604267	+	0.796782i	$0.706534\pi$
$578$	0	0
$579$	−4874.79	−0.349895
$580$	0	0
$581$	−28110.6	−2.00727
$582$	0	0
$583$	−11719.1	−0.832515
$584$	0	0
$585$	487.662	0.0344655
$586$	0	0
$587$	4709.53	0.331147	0.165573	−	0.986197i	$-0.447053\pi$
$587$	4709.53	0.331147	0.165573	+	0.986197i	$0.447053\pi$
$588$	0	0
$589$	5776.91	0.404131
$590$	0	0
$591$	10098.6	0.702875
$592$	0	0
$593$	1608.81	0.111410	0.0557048	−	0.998447i	$-0.482259\pi$
$593$	1608.81	0.111410	0.0557048	+	0.998447i	$0.482259\pi$
$594$	0	0
$595$	−6777.53	−0.466977
$596$	0	0
$597$	9216.35	0.631826
$598$	0	0
$599$	−7017.34	−0.478665	−0.239333	−	0.970938i	$-0.576929\pi$
$599$	−7017.34	−0.478665	−0.239333	+	0.970938i	$0.576929\pi$
$600$	0	0
$601$	19124.5	1.29801	0.649005	−	0.760784i	$-0.275186\pi$
$601$	19124.5	1.29801	0.649005	+	0.760784i	$0.275186\pi$
$602$	0	0
$603$	−872.444	−0.0589198
$604$	0	0
$605$	−3992.72	−0.268309
$606$	0	0
$607$	22042.4	1.47392	0.736962	−	0.675934i	$-0.236260\pi$
$607$	22042.4	1.47392	0.736962	+	0.675934i	$0.236260\pi$
$608$	0	0
$609$	−13957.6	−0.928722
$610$	0	0
$611$	−6585.74	−0.436056
$612$	0	0
$613$	5797.39	0.381981	0.190991	−	0.981592i	$-0.438830\pi$
$613$	5797.39	0.381981	0.190991	+	0.981592i	$0.438830\pi$
$614$	0	0
$615$	−4180.48	−0.274103
$616$	0	0
$617$	11262.6	0.734871	0.367435	−	0.930049i	$-0.380236\pi$
$617$	11262.6	0.734871	0.367435	+	0.930049i	$0.380236\pi$
$618$	0	0
$619$	−14224.3	−0.923621	−0.461810	−	0.886979i	$-0.652800\pi$
$619$	−14224.3	−0.923621	−0.461810	+	0.886979i	$0.652800\pi$
$620$	0	0
$621$	−5487.99	−0.354631
$622$	0	0
$623$	−4709.55	−0.302864
$624$	0	0
$625$	9412.07	0.602373
$626$	0	0
$627$	−2327.79	−0.148266
$628$	0	0
$629$	19772.2	1.25337
$630$	0	0
$631$	16541.9	1.04362	0.521809	−	0.853062i	$-0.325257\pi$
$631$	16541.9	1.04362	0.521809	+	0.853062i	$0.325257\pi$
$632$	0	0
$633$	−8069.53	−0.506691
$634$	0	0
$635$	8558.54	0.534859
$636$	0	0
$637$	6051.86	0.376426
$638$	0	0
$639$	3630.18	0.224738
$640$	0	0
$641$	4089.63	0.251998	0.125999	−	0.992030i	$-0.459786\pi$
$641$	4089.63	0.251998	0.125999	+	0.992030i	$0.459786\pi$
$642$	0	0
$643$	5156.76	0.316272	0.158136	−	0.987417i	$-0.449452\pi$
$643$	5156.76	0.316272	0.158136	+	0.987417i	$0.449452\pi$
$644$	0	0
$645$	2246.00	0.137110
$646$	0	0
$647$	10997.9	0.668271	0.334135	−	0.942525i	$-0.391556\pi$
$647$	10997.9	0.668271	0.334135	+	0.942525i	$0.391556\pi$
$648$	0	0
$649$	−15171.4	−0.917609
$650$	0	0
$651$	−12266.9	−0.738521
$652$	0	0
$653$	−10824.3	−0.648680	−0.324340	−	0.945940i	$-0.605142\pi$
$653$	−10824.3	−0.648680	−0.324340	+	0.945940i	$0.605142\pi$
$654$	0	0
$655$	−908.299	−0.0541835
$656$	0	0
$657$	−8041.20	−0.477500
$658$	0	0
$659$	12526.7	0.740470	0.370235	−	0.928938i	$-0.379277\pi$
$659$	12526.7	0.740470	0.370235	+	0.928938i	$0.379277\pi$
$660$	0	0
$661$	−13928.5	−0.819598	−0.409799	−	0.912176i	$-0.634401\pi$
$661$	−13928.5	−0.819598	−0.409799	+	0.912176i	$0.634401\pi$
$662$	0	0
$663$	−2230.26	−0.130643
$664$	0	0
$665$	4761.13	0.277637
$666$	0	0
$667$	−33257.7	−1.93065
$668$	0	0
$669$	13282.4	0.767607
$670$	0	0
$671$	−8864.76	−0.510015
$672$	0	0
$673$	−8979.99	−0.514344	−0.257172	−	0.966366i	$-0.582791\pi$
$673$	−8979.99	−0.514344	−0.257172	+	0.966366i	$0.582791\pi$
$674$	0	0
$675$	2905.94	0.165703
$676$	0	0
$677$	−5984.79	−0.339755	−0.169878	−	0.985465i	$-0.554337\pi$
$677$	−5984.79	−0.339755	−0.169878	+	0.985465i	$0.554337\pi$
$678$	0	0
$679$	42844.1	2.42151
$680$	0	0
$681$	13035.9	0.733535
$682$	0	0
$683$	21531.8	1.20628	0.603142	−	0.797634i	$-0.293915\pi$
$683$	21531.8	1.20628	0.603142	+	0.797634i	$0.293915\pi$
$684$	0	0
$685$	−100.073	−0.00558187
$686$	0	0
$687$	−2899.22	−0.161008
$688$	0	0
$689$	7887.62	0.436131
$690$	0	0
$691$	−12902.6	−0.710329	−0.355164	−	0.934804i	$-0.615575\pi$
$691$	−12902.6	−0.710329	−0.355164	+	0.934804i	$0.615575\pi$
$692$	0	0
$693$	4942.91	0.270946
$694$	0	0
$695$	−8826.71	−0.481750
$696$	0	0
$697$	19118.9	1.03900
$698$	0	0
$699$	17294.2	0.935806
$700$	0	0
$701$	−26032.1	−1.40259	−0.701297	−	0.712869i	$-0.747396\pi$
$701$	−26032.1	−1.40259	−0.701297	+	0.712869i	$0.747396\pi$
$702$	0	0
$703$	−13889.8	−0.745181
$704$	0	0
$705$	6334.54	0.338401
$706$	0	0
$707$	5490.79	0.292082
$708$	0	0
$709$	−20810.3	−1.10232	−0.551160	−	0.834399i	$-0.685815\pi$
$709$	−20810.3	−1.10232	−0.551160	+	0.834399i	$0.685815\pi$
$710$	0	0
$711$	−1256.07	−0.0662537
$712$	0	0
$713$	−29229.1	−1.53526
$714$	0	0
$715$	−1046.57	−0.0547406
$716$	0	0
$717$	−3745.94	−0.195111
$718$	0	0
$719$	15301.1	0.793649	0.396825	−	0.917894i	$-0.370112\pi$
$719$	15301.1	0.793649	0.396825	+	0.917894i	$0.370112\pi$
$720$	0	0
$721$	56106.2	2.89806
$722$	0	0
$723$	18364.9	0.944670
$724$	0	0
$725$	17610.3	0.902108
$726$	0	0
$727$	−1716.29	−0.0875565	−0.0437782	−	0.999041i	$-0.513940\pi$
$727$	−1716.29	−0.0875565	−0.0437782	+	0.999041i	$0.513940\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−10271.8	−0.519721
$732$	0	0
$733$	22161.9	1.11674	0.558369	−	0.829593i	$-0.311427\pi$
$733$	22161.9	1.11674	0.558369	+	0.829593i	$0.311427\pi$
$734$	0	0
$735$	−5821.03	−0.292125
$736$	0	0
$737$	1872.35	0.0935807
$738$	0	0
$739$	−9810.07	−0.488321	−0.244161	−	0.969735i	$-0.578512\pi$
$739$	−9810.07	−0.488321	−0.244161	+	0.969735i	$0.578512\pi$
$740$	0	0
$741$	1566.73	0.0776725
$742$	0	0
$743$	10376.9	0.512372	0.256186	−	0.966628i	$-0.417534\pi$
$743$	10376.9	0.512372	0.256186	+	0.966628i	$0.417534\pi$
$744$	0	0
$745$	6532.91	0.321272
$746$	0	0
$747$	8897.43	0.435796
$748$	0	0
$749$	20841.0	1.01671
$750$	0	0
$751$	−419.090	−0.0203633	−0.0101816	−	0.999948i	$-0.503241\pi$
$751$	−419.090	−0.0203633	−0.0101816	+	0.999948i	$0.503241\pi$
$752$	0	0
$753$	−6887.47	−0.333324
$754$	0	0
$755$	10683.0	0.514957
$756$	0	0
$757$	40437.4	1.94151	0.970755	−	0.240074i	$-0.0771716\pi$
$757$	40437.4	1.94151	0.970755	+	0.240074i	$0.0771716\pi$
$758$	0	0
$759$	11777.8	0.563249
$760$	0	0
$761$	−33507.9	−1.59614	−0.798069	−	0.602567i	$-0.794145\pi$
$761$	−33507.9	−1.59614	−0.798069	+	0.602567i	$0.794145\pi$
$762$	0	0
$763$	−23812.5	−1.12985
$764$	0	0
$765$	2145.19	0.101385
$766$	0	0
$767$	10211.2	0.480709
$768$	0	0
$769$	26117.8	1.22475	0.612375	−	0.790568i	$-0.290214\pi$
$769$	26117.8	1.22475	0.612375	+	0.790568i	$0.290214\pi$
$770$	0	0
$771$	1341.99	0.0626854
$772$	0	0
$773$	33310.4	1.54993	0.774963	−	0.632007i	$-0.217769\pi$
$773$	33310.4	1.54993	0.774963	+	0.632007i	$0.217769\pi$
$774$	0	0
$775$	15477.1	0.717358
$776$	0	0
$777$	29494.0	1.36176
$778$	0	0
$779$	−13430.8	−0.617725
$780$	0	0
$781$	−7790.72	−0.356945
$782$	0	0
$783$	4417.81	0.201634
$784$	0	0
$785$	5716.34	0.259904
$786$	0	0
$787$	18976.6	0.859521	0.429761	−	0.902943i	$-0.358598\pi$
$787$	18976.6	0.859521	0.429761	+	0.902943i	$0.358598\pi$
$788$	0	0
$789$	18761.3	0.846538
$790$	0	0
$791$	16346.6	0.734791
$792$	0	0
$793$	5966.47	0.267183
$794$	0	0
$795$	−7586.76	−0.338459
$796$	0	0
$797$	−13163.1	−0.585021	−0.292511	−	0.956262i	$-0.594491\pi$
$797$	−13163.1	−0.585021	−0.292511	+	0.956262i	$0.594491\pi$
$798$	0	0
$799$	−28970.2	−1.28272
$800$	0	0
$801$	1490.65	0.0657545
$802$	0	0
$803$	17257.2	0.758399
$804$	0	0
$805$	−24089.6	−1.05472
$806$	0	0
$807$	−24824.3	−1.08285
$808$	0	0
$809$	−8357.17	−0.363192	−0.181596	−	0.983373i	$-0.558126\pi$
$809$	−8357.17	−0.363192	−0.181596	+	0.983373i	$0.558126\pi$
$810$	0	0
$811$	−9527.89	−0.412540	−0.206270	−	0.978495i	$-0.566132\pi$
$811$	−9527.89	−0.412540	−0.206270	+	0.978495i	$0.566132\pi$
$812$	0	0
$813$	24030.6	1.03664
$814$	0	0
$815$	−10367.5	−0.445591
$816$	0	0
$817$	7215.80	0.308995
$818$	0	0
$819$	−3326.85	−0.141941
$820$	0	0
$821$	−20151.8	−0.856642	−0.428321	−	0.903627i	$-0.640895\pi$
$821$	−20151.8	−0.856642	−0.428321	+	0.903627i	$0.640895\pi$
$822$	0	0
$823$	−20811.6	−0.881467	−0.440734	−	0.897638i	$-0.645282\pi$
$823$	−20811.6	−0.881467	−0.440734	+	0.897638i	$0.645282\pi$
$824$	0	0
$825$	−6236.44	−0.263182
$826$	0	0
$827$	−4991.30	−0.209873	−0.104936	−	0.994479i	$-0.533464\pi$
$827$	−4991.30	−0.209873	−0.104936	+	0.994479i	$0.533464\pi$
$828$	0	0
$829$	−16837.9	−0.705433	−0.352717	−	0.935730i	$-0.614742\pi$
$829$	−16837.9	−0.705433	−0.352717	+	0.935730i	$0.614742\pi$
$830$	0	0
$831$	−16551.0	−0.690913
$832$	0	0
$833$	26621.8	1.10731
$834$	0	0
$835$	11027.8	0.457046
$836$	0	0
$837$	3882.66	0.160340
$838$	0	0
$839$	13025.9	0.536000	0.268000	−	0.963419i	$-0.413637\pi$
$839$	13025.9	0.536000	0.268000	+	0.963419i	$0.413637\pi$
$840$	0	0
$841$	2383.29	0.0977200
$842$	0	0
$843$	4599.98	0.187938
$844$	0	0
$845$	704.400	0.0286770
$846$	0	0
$847$	27238.5	1.10499
$848$	0	0
$849$	25459.6	1.02918
$850$	0	0
$851$	70277.2	2.83087
$852$	0	0
$853$	11278.0	0.452698	0.226349	−	0.974046i	$-0.427321\pi$
$853$	11278.0	0.452698	0.226349	+	0.974046i	$0.427321\pi$
$854$	0	0
$855$	−1506.97	−0.0602776
$856$	0	0
$857$	−7278.15	−0.290101	−0.145051	−	0.989424i	$-0.546335\pi$
$857$	−7278.15	−0.290101	−0.145051	+	0.989424i	$0.546335\pi$
$858$	0	0
$859$	43829.0	1.74089	0.870446	−	0.492263i	$-0.163830\pi$
$859$	43829.0	1.74089	0.870446	+	0.492263i	$0.163830\pi$
$860$	0	0
$861$	28519.4	1.12885
$862$	0	0
$863$	−1132.98	−0.0446896	−0.0223448	−	0.999750i	$-0.507113\pi$
$863$	−1132.98	−0.0446896	−0.0223448	+	0.999750i	$0.507113\pi$
$864$	0	0
$865$	15873.5	0.623948
$866$	0	0
$867$	4928.22	0.193046
$868$	0	0
$869$	2695.66	0.105229
$870$	0	0
$871$	−1260.20	−0.0490243
$872$	0	0
$873$	−13560.8	−0.525733
$874$	0	0
$875$	27570.3	1.06519
$876$	0	0
$877$	37968.2	1.46191	0.730955	−	0.682426i	$-0.239075\pi$
$877$	37968.2	1.46191	0.730955	+	0.682426i	$0.239075\pi$
$878$	0	0
$879$	7164.67	0.274924
$880$	0	0
$881$	−40587.2	−1.55212	−0.776060	−	0.630659i	$-0.782785\pi$
$881$	−40587.2	−1.55212	−0.776060	+	0.630659i	$0.782785\pi$
$882$	0	0
$883$	30722.7	1.17090	0.585449	−	0.810709i	$-0.300918\pi$
$883$	30722.7	1.17090	0.585449	+	0.810709i	$0.300918\pi$
$884$	0	0
$885$	−9821.69	−0.373054
$886$	0	0
$887$	8589.35	0.325143	0.162572	−	0.986697i	$-0.448021\pi$
$887$	8589.35	0.325143	0.162572	+	0.986697i	$0.448021\pi$
$888$	0	0
$889$	−58386.7	−2.20273
$890$	0	0
$891$	−1564.51	−0.0588249
$892$	0	0
$893$	20351.2	0.762629
$894$	0	0
$895$	−5021.23	−0.187532
$896$	0	0
$897$	−7927.10	−0.295070
$898$	0	0
$899$	23529.3	0.872909
$900$	0	0
$901$	34697.1	1.28294
$902$	0	0
$903$	−15322.3	−0.564666
$904$	0	0
$905$	−14688.3	−0.539510
$906$	0	0
$907$	−4784.14	−0.175143	−0.0875715	−	0.996158i	$-0.527911\pi$
$907$	−4784.14	−0.175143	−0.0875715	+	0.996158i	$0.527911\pi$
$908$	0	0
$909$	−1737.92	−0.0634138
$910$	0	0
$911$	14336.1	0.521381	0.260690	−	0.965422i	$-0.416050\pi$
$911$	14336.1	0.521381	0.260690	+	0.965422i	$0.416050\pi$
$912$	0	0
$913$	−19094.8	−0.692163
$914$	0	0
$915$	−5738.90	−0.207347
$916$	0	0
$917$	6196.46	0.223146
$918$	0	0
$919$	−42216.6	−1.51534	−0.757669	−	0.652639i	$-0.773662\pi$
$919$	−42216.6	−1.51534	−0.757669	+	0.652639i	$0.773662\pi$
$920$	0	0
$921$	−13462.3	−0.481647
$922$	0	0
$923$	5243.59	0.186993
$924$	0	0
$925$	−37212.4	−1.32274
$926$	0	0
$927$	−17758.5	−0.629196
$928$	0	0
$929$	−48513.1	−1.71331	−0.856655	−	0.515890i	$-0.827461\pi$
$929$	−48513.1	−1.71331	−0.856655	+	0.515890i	$0.827461\pi$
$930$	0	0
$931$	−18701.5	−0.658341
$932$	0	0
$933$	−28172.0	−0.988542
$934$	0	0
$935$	−4603.80	−0.161027
$936$	0	0
$937$	39496.4	1.37705	0.688523	−	0.725215i	$-0.258260\pi$
$937$	39496.4	1.37705	0.688523	+	0.725215i	$0.258260\pi$
$938$	0	0
$939$	−17645.2	−0.613235
$940$	0	0
$941$	−49619.6	−1.71897	−0.859485	−	0.511160i	$-0.829216\pi$
$941$	−49619.6	−1.71897	−0.859485	+	0.511160i	$0.829216\pi$
$942$	0	0
$943$	67955.0	2.34668
$944$	0	0
$945$	3199.96	0.110153
$946$	0	0
$947$	−33759.8	−1.15844	−0.579221	−	0.815170i	$-0.696643\pi$
$947$	−33759.8	−1.15844	−0.579221	+	0.815170i	$0.696643\pi$
$948$	0	0
$949$	−11615.1	−0.397304
$950$	0	0
$951$	4886.72	0.166627
$952$	0	0
$953$	27222.9	0.925328	0.462664	−	0.886534i	$-0.346894\pi$
$953$	27222.9	0.925328	0.462664	+	0.886534i	$0.346894\pi$
$954$	0	0
$955$	5500.86	0.186391
$956$	0	0
$957$	−9481.05	−0.320250
$958$	0	0
$959$	682.700	0.0229880
$960$	0	0
$961$	−9111.93	−0.305862
$962$	0	0
$963$	−6596.49	−0.220736
$964$	0	0
$965$	6772.79	0.225931
$966$	0	0
$967$	44305.1	1.47338	0.736689	−	0.676232i	$-0.236388\pi$
$967$	44305.1	1.47338	0.736689	+	0.676232i	$0.236388\pi$
$968$	0	0
$969$	6891.95	0.228484
$970$	0	0
$971$	6092.31	0.201351	0.100675	−	0.994919i	$-0.467900\pi$
$971$	6092.31	0.201351	0.100675	+	0.994919i	$0.467900\pi$
$972$	0	0
$973$	60216.2	1.98401
$974$	0	0
$975$	4197.47	0.137873
$976$	0	0
$977$	−4983.98	−0.163205	−0.0816027	−	0.996665i	$-0.526004\pi$
$977$	−4983.98	−0.163205	−0.0816027	+	0.996665i	$0.526004\pi$
$978$	0	0
$979$	−3199.08	−0.104436
$980$	0	0
$981$	7537.04	0.245300
$982$	0	0
$983$	8228.82	0.266998	0.133499	−	0.991049i	$-0.457379\pi$
$983$	8228.82	0.266998	0.133499	+	0.991049i	$0.457379\pi$
$984$	0	0
$985$	−14030.4	−0.453854
$986$	0	0
$987$	−43214.5	−1.39365
$988$	0	0
$989$	−36509.4	−1.17384
$990$	0	0
$991$	−46521.1	−1.49121	−0.745606	−	0.666388i	$-0.767840\pi$
$991$	−46521.1	−1.49121	−0.745606	+	0.666388i	$0.767840\pi$
$992$	0	0
$993$	19710.0	0.629887
$994$	0	0
$995$	−12804.7	−0.407977
$996$	0	0
$997$	−22290.5	−0.708073	−0.354036	−	0.935232i	$-0.615191\pi$
$997$	−22290.5	−0.708073	−0.354036	+	0.935232i	$0.615191\pi$
$998$	0	0
$999$	−9335.31	−0.295652

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.ca.1.4		5
4.3	odd	2		2496.4.a.cf.1.4		5
8.3	odd	2		1248.4.a.g.1.2	✓	5
8.5	even	2		1248.4.a.l.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.4.a.g.1.2	✓	5	8.3	odd	2
1248.4.a.l.1.2	yes	5	8.5	even	2
2496.4.a.ca.1.4		5	1.1	even	1	trivial
2496.4.a.cf.1.4		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +4.16805 q^{5} -28.4346 q^{7} +9.00000 q^{9} -19.3149 q^{11} +13.0000 q^{13} -12.5041 q^{15} +57.1862 q^{17} -40.1726 q^{19} +85.3039 q^{21} +203.259 q^{23} -107.627 q^{25} -27.0000 q^{27} -163.622 q^{29} -143.802 q^{31} +57.9447 q^{33} -118.517 q^{35} +345.752 q^{37} -39.0000 q^{39} +334.327 q^{41} -179.620 q^{43} +37.5124 q^{45} -506.595 q^{47} +465.528 q^{49} -171.559 q^{51} +606.740 q^{53} -80.5054 q^{55} +120.518 q^{57} +785.475 q^{59} +458.960 q^{61} -255.912 q^{63} +54.1846 q^{65} -96.9382 q^{67} -609.777 q^{69} +403.353 q^{71} -893.467 q^{73} +322.882 q^{75} +549.212 q^{77} -139.564 q^{79} +81.0000 q^{81} +988.603 q^{83} +238.355 q^{85} +490.867 q^{87} +165.627 q^{89} -369.650 q^{91} +431.407 q^{93} -167.441 q^{95} -1506.76 q^{97} -173.834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} + 10 q^{5} - 14 q^{7} + 45 q^{9} + 22 q^{11} + 65 q^{13} - 30 q^{15} - 34 q^{17} - 90 q^{19} + 42 q^{21} - 96 q^{23} + 107 q^{25} - 135 q^{27} + 54 q^{29} - 378 q^{31} - 66 q^{33} + 84 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 5 * q - 15 * q^3 + 10 * q^5 - 14 * q^7 + 45 * q^9 + 22 * q^11 + 65 * q^13 - 30 * q^15 - 34 * q^17 - 90 * q^19 + 42 * q^21 - 96 * q^23 + 107 * q^25 - 135 * q^27 + 54 * q^29 - 378 * q^31 - 66 * q^33 + 84 * q^35 + 318 * q^37 - 195 * q^39 - 458 * q^41 + 104 * q^43 + 90 * q^45 - 474 * q^47 + 25 * q^49 + 102 * q^51 + 450 * q^53 - 1116 * q^55 + 270 * q^57 + 822 * q^59 + 546 * q^61 - 126 * q^63 + 130 * q^65 - 130 * q^67 + 288 * q^69 - 1646 * q^71 - 166 * q^73 - 321 * q^75 + 900 * q^77 - 196 * q^79 + 405 * q^81 - 118 * q^83 + 3340 * q^85 - 162 * q^87 - 186 * q^89 - 182 * q^91 + 1134 * q^93 - 564 * q^95 - 2582 * q^97 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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