LMFDB - Embedded newform 784.4.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.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:	$46.2574974445$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+5.00000 q^{3} -9.00000 q^{5} -2.00000 q^{9} +O(q^{10})$

$q+5.00000 q^{3} -9.00000 q^{5} -2.00000 q^{9} +57.0000 q^{11} -70.0000 q^{13} -45.0000 q^{15} +51.0000 q^{17} -5.00000 q^{19} -69.0000 q^{23} -44.0000 q^{25} -145.000 q^{27} +114.000 q^{29} -23.0000 q^{31} +285.000 q^{33} -253.000 q^{37} -350.000 q^{39} -42.0000 q^{41} +124.000 q^{43} +18.0000 q^{45} -201.000 q^{47} +255.000 q^{51} -393.000 q^{53} -513.000 q^{55} -25.0000 q^{57} -219.000 q^{59} -709.000 q^{61} +630.000 q^{65} -419.000 q^{67} -345.000 q^{69} +96.0000 q^{71} -313.000 q^{73} -220.000 q^{75} -461.000 q^{79} -671.000 q^{81} +588.000 q^{83} -459.000 q^{85} +570.000 q^{87} -1017.00 q^{89} -115.000 q^{93} +45.0000 q^{95} -1834.00 q^{97} -114.000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	5.00000	0.962250	0.481125	−	0.876652i	$-0.340228\pi$
$3$	5.00000	0.962250	0.481125	+	0.876652i	$0.340228\pi$
$4$	0	0
$5$	−9.00000	−0.804984	−0.402492	−	0.915423i	$-0.631856\pi$
$5$	−9.00000	−0.804984	−0.402492	+	0.915423i	$0.631856\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	57.0000	1.56238	0.781188	−	0.624295i	$-0.214614\pi$
$11$	57.0000	1.56238	0.781188	+	0.624295i	$0.214614\pi$
$12$	0	0
$13$	−70.0000	−1.49342	−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−70.0000	−1.49342	−0.746712	+	0.665148i	$0.768369\pi$
$14$	0	0
$15$	−45.0000	−0.774597
$16$	0	0
$17$	51.0000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	51.0000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−5.00000	−0.0603726	−0.0301863	−	0.999544i	$-0.509610\pi$
$19$	−5.00000	−0.0603726	−0.0301863	+	0.999544i	$0.509610\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−69.0000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−69.0000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	0	0
$27$	−145.000	−1.03353
$28$	0	0
$29$	114.000	0.729975	0.364987	−	0.931012i	$-0.381073\pi$
$29$	114.000	0.729975	0.364987	+	0.931012i	$0.381073\pi$
$30$	0	0
$31$	−23.0000	−0.133256	−0.0666278	−	0.997778i	$-0.521224\pi$
$31$	−23.0000	−0.133256	−0.0666278	+	0.997778i	$0.521224\pi$
$32$	0	0
$33$	285.000	1.50340
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−253.000	−1.12413	−0.562067	−	0.827092i	$-0.689994\pi$
$37$	−253.000	−1.12413	−0.562067	+	0.827092i	$0.689994\pi$
$38$	0	0
$39$	−350.000	−1.43705
$40$	0	0
$41$	−42.0000	−0.159983	−0.0799914	−	0.996796i	$-0.525489\pi$
$41$	−42.0000	−0.159983	−0.0799914	+	0.996796i	$0.525489\pi$
$42$	0	0
$43$	124.000	0.439763	0.219882	−	0.975527i	$-0.429433\pi$
$43$	124.000	0.439763	0.219882	+	0.975527i	$0.429433\pi$
$44$	0	0
$45$	18.0000	0.0596285
$46$	0	0
$47$	−201.000	−0.623806	−0.311903	−	0.950114i	$-0.600966\pi$
$47$	−201.000	−0.623806	−0.311903	+	0.950114i	$0.600966\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	255.000	0.700140
$52$	0	0
$53$	−393.000	−1.01854	−0.509271	−	0.860606i	$-0.670085\pi$
$53$	−393.000	−1.01854	−0.509271	+	0.860606i	$0.670085\pi$
$54$	0	0
$55$	−513.000	−1.25769
$56$	0	0
$57$	−25.0000	−0.0580935
$58$	0	0
$59$	−219.000	−0.483244	−0.241622	−	0.970371i	$-0.577679\pi$
$59$	−219.000	−0.483244	−0.241622	+	0.970371i	$0.577679\pi$
$60$	0	0
$61$	−709.000	−1.48817	−0.744083	−	0.668087i	$-0.767113\pi$
$61$	−709.000	−1.48817	−0.744083	+	0.668087i	$0.767113\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	630.000	1.20218
$66$	0	0
$67$	−419.000	−0.764015	−0.382007	−	0.924159i	$-0.624767\pi$
$67$	−419.000	−0.764015	−0.382007	+	0.924159i	$0.624767\pi$
$68$	0	0
$69$	−345.000	−0.601929
$70$	0	0
$71$	96.0000	0.160466	0.0802331	−	0.996776i	$-0.474434\pi$
$71$	96.0000	0.160466	0.0802331	+	0.996776i	$0.474434\pi$
$72$	0	0
$73$	−313.000	−0.501834	−0.250917	−	0.968009i	$-0.580732\pi$
$73$	−313.000	−0.501834	−0.250917	+	0.968009i	$0.580732\pi$
$74$	0	0
$75$	−220.000	−0.338712
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−461.000	−0.656539	−0.328269	−	0.944584i	$-0.606465\pi$
$79$	−461.000	−0.656539	−0.328269	+	0.944584i	$0.606465\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	588.000	0.777607	0.388804	−	0.921321i	$-0.372888\pi$
$83$	588.000	0.777607	0.388804	+	0.921321i	$0.372888\pi$
$84$	0	0
$85$	−459.000	−0.585712
$86$	0	0
$87$	570.000	0.702419
$88$	0	0
$89$	−1017.00	−1.21126	−0.605628	−	0.795748i	$-0.707078\pi$
$89$	−1017.00	−1.21126	−0.605628	+	0.795748i	$0.707078\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−115.000	−0.128225
$94$	0	0
$95$	45.0000	0.0485990
$96$	0	0
$97$	−1834.00	−1.91974	−0.959868	−	0.280451i	$-0.909516\pi$
$97$	−1834.00	−1.91974	−0.959868	+	0.280451i	$0.909516\pi$
$98$	0	0
$99$	−114.000	−0.115732
$100$	0	0
$101$	−285.000	−0.280778	−0.140389	−	0.990096i	$-0.544835\pi$
$101$	−285.000	−0.280778	−0.140389	+	0.990096i	$0.544835\pi$
$102$	0	0
$103$	499.000	0.477359	0.238679	−	0.971098i	$-0.423286\pi$
$103$	499.000	0.477359	0.238679	+	0.971098i	$0.423286\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1107.00	1.00017	0.500083	−	0.865978i	$-0.333303\pi$
$107$	1107.00	1.00017	0.500083	+	0.865978i	$0.333303\pi$
$108$	0	0
$109$	923.000	0.811077	0.405538	−	0.914078i	$-0.367084\pi$
$109$	923.000	0.811077	0.405538	+	0.914078i	$0.367084\pi$
$110$	0	0
$111$	−1265.00	−1.08170
$112$	0	0
$113$	1542.00	1.28371	0.641855	−	0.766826i	$-0.278165\pi$
$113$	1542.00	1.28371	0.641855	+	0.766826i	$0.278165\pi$
$114$	0	0
$115$	621.000	0.503553
$116$	0	0
$117$	140.000	0.110624
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1918.00	1.44102
$122$	0	0
$123$	−210.000	−0.153944
$124$	0	0
$125$	1521.00	1.08834
$126$	0	0
$127$	2056.00	1.43654	0.718270	−	0.695765i	$-0.244934\pi$
$127$	2056.00	1.43654	0.718270	+	0.695765i	$0.244934\pi$
$128$	0	0
$129$	620.000	0.423162
$130$	0	0
$131$	−2049.00	−1.36658	−0.683290	−	0.730147i	$-0.739451\pi$
$131$	−2049.00	−1.36658	−0.683290	+	0.730147i	$0.739451\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1305.00	0.831974
$136$	0	0
$137$	−141.000	−0.0879302	−0.0439651	−	0.999033i	$-0.513999\pi$
$137$	−141.000	−0.0879302	−0.0439651	+	0.999033i	$0.513999\pi$
$138$	0	0
$139$	−1484.00	−0.905548	−0.452774	−	0.891625i	$-0.649566\pi$
$139$	−1484.00	−0.905548	−0.452774	+	0.891625i	$0.649566\pi$
$140$	0	0
$141$	−1005.00	−0.600257
$142$	0	0
$143$	−3990.00	−2.33329
$144$	0	0
$145$	−1026.00	−0.587618
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−57.0000	−0.0313397	−0.0156699	−	0.999877i	$-0.504988\pi$
$149$	−57.0000	−0.0313397	−0.0156699	+	0.999877i	$0.504988\pi$
$150$	0	0
$151$	−839.000	−0.452165	−0.226082	−	0.974108i	$-0.572592\pi$
$151$	−839.000	−0.452165	−0.226082	+	0.974108i	$0.572592\pi$
$152$	0	0
$153$	−102.000	−0.0538968
$154$	0	0
$155$	207.000	0.107269
$156$	0	0
$157$	−2833.00	−1.44011	−0.720057	−	0.693915i	$-0.755885\pi$
$157$	−2833.00	−1.44011	−0.720057	+	0.693915i	$0.755885\pi$
$158$	0	0
$159$	−1965.00	−0.980092
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2311.00	1.11050	0.555250	−	0.831684i	$-0.312623\pi$
$163$	2311.00	1.11050	0.555250	+	0.831684i	$0.312623\pi$
$164$	0	0
$165$	−2565.00	−1.21021
$166$	0	0
$167$	−1260.00	−0.583843	−0.291921	−	0.956442i	$-0.594295\pi$
$167$	−1260.00	−0.583843	−0.291921	+	0.956442i	$0.594295\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	0	0
$171$	10.0000	0.00447204
$172$	0	0
$173$	3267.00	1.43575	0.717877	−	0.696170i	$-0.245114\pi$
$173$	3267.00	1.43575	0.717877	+	0.696170i	$0.245114\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1095.00	−0.465001
$178$	0	0
$179$	−1287.00	−0.537402	−0.268701	−	0.963224i	$-0.586594\pi$
$179$	−1287.00	−0.537402	−0.268701	+	0.963224i	$0.586594\pi$
$180$	0	0
$181$	−2674.00	−1.09810	−0.549052	−	0.835788i	$-0.685011\pi$
$181$	−2674.00	−1.09810	−0.549052	+	0.835788i	$0.685011\pi$
$182$	0	0
$183$	−3545.00	−1.43199
$184$	0	0
$185$	2277.00	0.904910
$186$	0	0
$187$	2907.00	1.13680
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4185.00	−1.58542	−0.792712	−	0.609596i	$-0.791332\pi$
$191$	−4185.00	−1.58542	−0.792712	+	0.609596i	$0.791332\pi$
$192$	0	0
$193$	−85.0000	−0.0317017	−0.0158509	−	0.999874i	$-0.505046\pi$
$193$	−85.0000	−0.0317017	−0.0158509	+	0.999874i	$0.505046\pi$
$194$	0	0
$195$	3150.00	1.15680
$196$	0	0
$197$	−390.000	−0.141047	−0.0705237	−	0.997510i	$-0.522467\pi$
$197$	−390.000	−0.141047	−0.0705237	+	0.997510i	$0.522467\pi$
$198$	0	0
$199$	2833.00	1.00918	0.504588	−	0.863360i	$-0.331644\pi$
$199$	2833.00	1.00918	0.504588	+	0.863360i	$0.331644\pi$
$200$	0	0
$201$	−2095.00	−0.735174
$202$	0	0
$203$	0	0
$204$	0	0
$205$	378.000	0.128784
$206$	0	0
$207$	138.000	0.0463365
$208$	0	0
$209$	−285.000	−0.0943247
$210$	0	0
$211$	124.000	0.0404574	0.0202287	−	0.999795i	$-0.493561\pi$
$211$	124.000	0.0404574	0.0202287	+	0.999795i	$0.493561\pi$
$212$	0	0
$213$	480.000	0.154409
$214$	0	0
$215$	−1116.00	−0.354003
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1565.00	−0.482890
$220$	0	0
$221$	−3570.00	−1.08663
$222$	0	0
$223$	−56.0000	−0.0168163	−0.00840816	−	0.999965i	$-0.502676\pi$
$223$	−56.0000	−0.0168163	−0.00840816	+	0.999965i	$0.502676\pi$
$224$	0	0
$225$	88.0000	0.0260741
$226$	0	0
$227$	3057.00	0.893834	0.446917	−	0.894576i	$-0.352522\pi$
$227$	3057.00	0.893834	0.446917	+	0.894576i	$0.352522\pi$
$228$	0	0
$229$	−961.000	−0.277313	−0.138656	−	0.990341i	$-0.544278\pi$
$229$	−961.000	−0.277313	−0.138656	+	0.990341i	$0.544278\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2829.00	−0.795425	−0.397712	−	0.917510i	$-0.630196\pi$
$233$	−2829.00	−0.795425	−0.397712	+	0.917510i	$0.630196\pi$
$234$	0	0
$235$	1809.00	0.502154
$236$	0	0
$237$	−2305.00	−0.631755
$238$	0	0
$239$	3540.00	0.958090	0.479045	−	0.877790i	$-0.340983\pi$
$239$	3540.00	0.958090	0.479045	+	0.877790i	$0.340983\pi$
$240$	0	0
$241$	5231.00	1.39817	0.699084	−	0.715040i	$-0.253591\pi$
$241$	5231.00	1.39817	0.699084	+	0.715040i	$0.253591\pi$
$242$	0	0
$243$	560.000	0.147835
$244$	0	0
$245$	0	0
$246$	0	0
$247$	350.000	0.0901618
$248$	0	0
$249$	2940.00	0.748253
$250$	0	0
$251$	−5040.00	−1.26742	−0.633709	−	0.773571i	$-0.718468\pi$
$251$	−5040.00	−1.26742	−0.633709	+	0.773571i	$0.718468\pi$
$252$	0	0
$253$	−3933.00	−0.977334
$254$	0	0
$255$	−2295.00	−0.563602
$256$	0	0
$257$	−1437.00	−0.348784	−0.174392	−	0.984676i	$-0.555796\pi$
$257$	−1437.00	−0.348784	−0.174392	+	0.984676i	$0.555796\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−228.000	−0.0540722
$262$	0	0
$263$	2325.00	0.545117	0.272558	−	0.962139i	$-0.412130\pi$
$263$	2325.00	0.545117	0.272558	+	0.962139i	$0.412130\pi$
$264$	0	0
$265$	3537.00	0.819910
$266$	0	0
$267$	−5085.00	−1.16553
$268$	0	0
$269$	−2385.00	−0.540580	−0.270290	−	0.962779i	$-0.587120\pi$
$269$	−2385.00	−0.540580	−0.270290	+	0.962779i	$0.587120\pi$
$270$	0	0
$271$	331.000	0.0741949	0.0370975	−	0.999312i	$-0.488189\pi$
$271$	331.000	0.0741949	0.0370975	+	0.999312i	$0.488189\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2508.00	−0.549957
$276$	0	0
$277$	4871.00	1.05657	0.528285	−	0.849067i	$-0.322835\pi$
$277$	4871.00	1.05657	0.528285	+	0.849067i	$0.322835\pi$
$278$	0	0
$279$	46.0000	0.00987078
$280$	0	0
$281$	−7026.00	−1.49159	−0.745794	−	0.666177i	$-0.767930\pi$
$281$	−7026.00	−1.49159	−0.745794	+	0.666177i	$0.767930\pi$
$282$	0	0
$283$	5353.00	1.12439	0.562196	−	0.827004i	$-0.309957\pi$
$283$	5353.00	1.12439	0.562196	+	0.827004i	$0.309957\pi$
$284$	0	0
$285$	225.000	0.0467644
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2312.00	−0.470588
$290$	0	0
$291$	−9170.00	−1.84727
$292$	0	0
$293$	4158.00	0.829054	0.414527	−	0.910037i	$-0.363947\pi$
$293$	4158.00	0.829054	0.414527	+	0.910037i	$0.363947\pi$
$294$	0	0
$295$	1971.00	0.389004
$296$	0	0
$297$	−8265.00	−1.61476
$298$	0	0
$299$	4830.00	0.934201
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1425.00	−0.270179
$304$	0	0
$305$	6381.00	1.19795
$306$	0	0
$307$	9604.00	1.78544	0.892719	−	0.450615i	$-0.148795\pi$
$307$	9604.00	1.78544	0.892719	+	0.450615i	$0.148795\pi$
$308$	0	0
$309$	2495.00	0.459338
$310$	0	0
$311$	−10131.0	−1.84719	−0.923595	−	0.383369i	$-0.874764\pi$
$311$	−10131.0	−1.84719	−0.923595	+	0.383369i	$0.874764\pi$
$312$	0	0
$313$	10799.0	1.95015	0.975073	−	0.221885i	$-0.0712210\pi$
$313$	10799.0	1.95015	0.975073	+	0.221885i	$0.0712210\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	531.000	0.0940818	0.0470409	−	0.998893i	$-0.485021\pi$
$317$	531.000	0.0940818	0.0470409	+	0.998893i	$0.485021\pi$
$318$	0	0
$319$	6498.00	1.14050
$320$	0	0
$321$	5535.00	0.962410
$322$	0	0
$323$	−255.000	−0.0439275
$324$	0	0
$325$	3080.00	0.525685
$326$	0	0
$327$	4615.00	0.780459
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7015.00	1.16489	0.582446	−	0.812869i	$-0.302096\pi$
$331$	7015.00	1.16489	0.582446	+	0.812869i	$0.302096\pi$
$332$	0	0
$333$	506.000	0.0832692
$334$	0	0
$335$	3771.00	0.615020
$336$	0	0
$337$	8990.00	1.45316	0.726582	−	0.687079i	$-0.241108\pi$
$337$	8990.00	1.45316	0.726582	+	0.687079i	$0.241108\pi$
$338$	0	0
$339$	7710.00	1.23525
$340$	0	0
$341$	−1311.00	−0.208195
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3105.00	0.484544
$346$	0	0
$347$	8709.00	1.34733	0.673665	−	0.739037i	$-0.264719\pi$
$347$	8709.00	1.34733	0.673665	+	0.739037i	$0.264719\pi$
$348$	0	0
$349$	6482.00	0.994193	0.497097	−	0.867695i	$-0.334399\pi$
$349$	6482.00	0.994193	0.497097	+	0.867695i	$0.334399\pi$
$350$	0	0
$351$	10150.0	1.54350
$352$	0	0
$353$	−2133.00	−0.321609	−0.160805	−	0.986986i	$-0.551409\pi$
$353$	−2133.00	−0.321609	−0.160805	+	0.986986i	$0.551409\pi$
$354$	0	0
$355$	−864.000	−0.129173
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3849.00	−0.565856	−0.282928	−	0.959141i	$-0.591306\pi$
$359$	−3849.00	−0.565856	−0.282928	+	0.959141i	$0.591306\pi$
$360$	0	0
$361$	−6834.00	−0.996355
$362$	0	0
$363$	9590.00	1.38662
$364$	0	0
$365$	2817.00	0.403969
$366$	0	0
$367$	−6491.00	−0.923236	−0.461618	−	0.887079i	$-0.652731\pi$
$367$	−6491.00	−0.923236	−0.461618	+	0.887079i	$0.652731\pi$
$368$	0	0
$369$	84.0000	0.0118506
$370$	0	0
$371$	0	0
$372$	0	0
$373$	923.000	0.128126	0.0640632	−	0.997946i	$-0.479594\pi$
$373$	923.000	0.128126	0.0640632	+	0.997946i	$0.479594\pi$
$374$	0	0
$375$	7605.00	1.04725
$376$	0	0
$377$	−7980.00	−1.09016
$378$	0	0
$379$	−6344.00	−0.859814	−0.429907	−	0.902873i	$-0.641454\pi$
$379$	−6344.00	−0.859814	−0.429907	+	0.902873i	$0.641454\pi$
$380$	0	0
$381$	10280.0	1.38231
$382$	0	0
$383$	5007.00	0.668005	0.334002	−	0.942572i	$-0.391601\pi$
$383$	5007.00	0.668005	0.334002	+	0.942572i	$0.391601\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−248.000	−0.0325751
$388$	0	0
$389$	12291.0	1.60200	0.801001	−	0.598664i	$-0.204301\pi$
$389$	12291.0	1.60200	0.801001	+	0.598664i	$0.204301\pi$
$390$	0	0
$391$	−3519.00	−0.455150
$392$	0	0
$393$	−10245.0	−1.31499
$394$	0	0
$395$	4149.00	0.528503
$396$	0	0
$397$	887.000	0.112134	0.0560671	−	0.998427i	$-0.482144\pi$
$397$	887.000	0.112134	0.0560671	+	0.998427i	$0.482144\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11955.0	1.48879	0.744394	−	0.667740i	$-0.232738\pi$
$401$	11955.0	1.48879	0.744394	+	0.667740i	$0.232738\pi$
$402$	0	0
$403$	1610.00	0.199007
$404$	0	0
$405$	6039.00	0.740939
$406$	0	0
$407$	−14421.0	−1.75632
$408$	0	0
$409$	−3421.00	−0.413588	−0.206794	−	0.978384i	$-0.566303\pi$
$409$	−3421.00	−0.413588	−0.206794	+	0.978384i	$0.566303\pi$
$410$	0	0
$411$	−705.000	−0.0846109
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5292.00	−0.625962
$416$	0	0
$417$	−7420.00	−0.871364
$418$	0	0
$419$	5460.00	0.636607	0.318304	−	0.947989i	$-0.396887\pi$
$419$	5460.00	0.636607	0.318304	+	0.947989i	$0.396887\pi$
$420$	0	0
$421$	7730.00	0.894863	0.447431	−	0.894318i	$-0.352339\pi$
$421$	7730.00	0.894863	0.447431	+	0.894318i	$0.352339\pi$
$422$	0	0
$423$	402.000	0.0462078
$424$	0	0
$425$	−2244.00	−0.256118
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−19950.0	−2.24521
$430$	0	0
$431$	11313.0	1.26433	0.632167	−	0.774832i	$-0.282166\pi$
$431$	11313.0	1.26433	0.632167	+	0.774832i	$0.282166\pi$
$432$	0	0
$433$	4214.00	0.467695	0.233847	−	0.972273i	$-0.424868\pi$
$433$	4214.00	0.467695	0.233847	+	0.972273i	$0.424868\pi$
$434$	0	0
$435$	−5130.00	−0.565436
$436$	0	0
$437$	345.000	0.0377656
$438$	0	0
$439$	−16553.0	−1.79962	−0.899808	−	0.436286i	$-0.856294\pi$
$439$	−16553.0	−1.79962	−0.899808	+	0.436286i	$0.856294\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16395.0	1.75835	0.879176	−	0.476497i	$-0.158094\pi$
$443$	16395.0	1.75835	0.879176	+	0.476497i	$0.158094\pi$
$444$	0	0
$445$	9153.00	0.975042
$446$	0	0
$447$	−285.000	−0.0301567
$448$	0	0
$449$	−15090.0	−1.58606	−0.793030	−	0.609182i	$-0.791498\pi$
$449$	−15090.0	−1.58606	−0.793030	+	0.609182i	$0.791498\pi$
$450$	0	0
$451$	−2394.00	−0.249954
$452$	0	0
$453$	−4195.00	−0.435096
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14785.0	−1.51338	−0.756688	−	0.653776i	$-0.773184\pi$
$457$	−14785.0	−1.51338	−0.756688	+	0.653776i	$0.773184\pi$
$458$	0	0
$459$	−7395.00	−0.752002
$460$	0	0
$461$	2898.00	0.292784	0.146392	−	0.989227i	$-0.453234\pi$
$461$	2898.00	0.292784	0.146392	+	0.989227i	$0.453234\pi$
$462$	0	0
$463$	−464.000	−0.0465743	−0.0232872	−	0.999729i	$-0.507413\pi$
$463$	−464.000	−0.0465743	−0.0232872	+	0.999729i	$0.507413\pi$
$464$	0	0
$465$	1035.00	0.103219
$466$	0	0
$467$	−4233.00	−0.419443	−0.209721	−	0.977761i	$-0.567256\pi$
$467$	−4233.00	−0.419443	−0.209721	+	0.977761i	$0.567256\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−14165.0	−1.38575
$472$	0	0
$473$	7068.00	0.687076
$474$	0	0
$475$	220.000	0.0212511
$476$	0	0
$477$	786.000	0.0754475
$478$	0	0
$479$	−2739.00	−0.261270	−0.130635	−	0.991431i	$-0.541702\pi$
$479$	−2739.00	−0.261270	−0.130635	+	0.991431i	$0.541702\pi$
$480$	0	0
$481$	17710.0	1.67881
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16506.0	1.54536
$486$	0	0
$487$	−17051.0	−1.58656	−0.793280	−	0.608857i	$-0.791628\pi$
$487$	−17051.0	−1.58656	−0.793280	+	0.608857i	$0.791628\pi$
$488$	0	0
$489$	11555.0	1.06858
$490$	0	0
$491$	4296.00	0.394859	0.197429	−	0.980317i	$-0.436741\pi$
$491$	4296.00	0.394859	0.197429	+	0.980317i	$0.436741\pi$
$492$	0	0
$493$	5814.00	0.531135
$494$	0	0
$495$	1026.00	0.0931622
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3401.00	−0.305110	−0.152555	−	0.988295i	$-0.548750\pi$
$499$	−3401.00	−0.305110	−0.152555	+	0.988295i	$0.548750\pi$
$500$	0	0
$501$	−6300.00	−0.561803
$502$	0	0
$503$	−16800.0	−1.48921	−0.744607	−	0.667503i	$-0.767363\pi$
$503$	−16800.0	−1.48921	−0.744607	+	0.667503i	$0.767363\pi$
$504$	0	0
$505$	2565.00	0.226022
$506$	0	0
$507$	13515.0	1.18387
$508$	0	0
$509$	1839.00	0.160142	0.0800710	−	0.996789i	$-0.474485\pi$
$509$	1839.00	0.160142	0.0800710	+	0.996789i	$0.474485\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	725.000	0.0623967
$514$	0	0
$515$	−4491.00	−0.384266
$516$	0	0
$517$	−11457.0	−0.974620
$518$	0	0
$519$	16335.0	1.38155
$520$	0	0
$521$	303.000	0.0254792	0.0127396	−	0.999919i	$-0.495945\pi$
$521$	303.000	0.0254792	0.0127396	+	0.999919i	$0.495945\pi$
$522$	0	0
$523$	21667.0	1.81153	0.905767	−	0.423777i	$-0.139296\pi$
$523$	21667.0	1.81153	0.905767	+	0.423777i	$0.139296\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1173.00	−0.0969577
$528$	0	0
$529$	−7406.00	−0.608696
$530$	0	0
$531$	438.000	0.0357958
$532$	0	0
$533$	2940.00	0.238922
$534$	0	0
$535$	−9963.00	−0.805118
$536$	0	0
$537$	−6435.00	−0.517115
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5039.00	0.400450	0.200225	−	0.979750i	$-0.435833\pi$
$541$	5039.00	0.400450	0.200225	+	0.979750i	$0.435833\pi$
$542$	0	0
$543$	−13370.0	−1.05665
$544$	0	0
$545$	−8307.00	−0.652904
$546$	0	0
$547$	2392.00	0.186974	0.0934868	−	0.995621i	$-0.470199\pi$
$547$	2392.00	0.186974	0.0934868	+	0.995621i	$0.470199\pi$
$548$	0	0
$549$	1418.00	0.110235
$550$	0	0
$551$	−570.000	−0.0440704
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11385.0	0.870750
$556$	0	0
$557$	−22149.0	−1.68489	−0.842445	−	0.538783i	$-0.818884\pi$
$557$	−22149.0	−1.68489	−0.842445	+	0.538783i	$0.818884\pi$
$558$	0	0
$559$	−8680.00	−0.656753
$560$	0	0
$561$	14535.0	1.09388
$562$	0	0
$563$	8349.00	0.624988	0.312494	−	0.949920i	$-0.398836\pi$
$563$	8349.00	0.624988	0.312494	+	0.949920i	$0.398836\pi$
$564$	0	0
$565$	−13878.0	−1.03337
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15345.0	−1.13057	−0.565286	−	0.824895i	$-0.691234\pi$
$569$	−15345.0	−1.13057	−0.565286	+	0.824895i	$0.691234\pi$
$570$	0	0
$571$	11593.0	0.849653	0.424827	−	0.905275i	$-0.360335\pi$
$571$	11593.0	0.849653	0.424827	+	0.905275i	$0.360335\pi$
$572$	0	0
$573$	−20925.0	−1.52557
$574$	0	0
$575$	3036.00	0.220191
$576$	0	0
$577$	−14593.0	−1.05288	−0.526442	−	0.850211i	$-0.676474\pi$
$577$	−14593.0	−1.05288	−0.526442	+	0.850211i	$0.676474\pi$
$578$	0	0
$579$	−425.000	−0.0305050
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−22401.0	−1.59135
$584$	0	0
$585$	−1260.00	−0.0890506
$586$	0	0
$587$	15372.0	1.08087	0.540435	−	0.841386i	$-0.318260\pi$
$587$	15372.0	1.08087	0.540435	+	0.841386i	$0.318260\pi$
$588$	0	0
$589$	115.000	0.00804498
$590$	0	0
$591$	−1950.00	−0.135723
$592$	0	0
$593$	−14373.0	−0.995326	−0.497663	−	0.867370i	$-0.665808\pi$
$593$	−14373.0	−0.995326	−0.497663	+	0.867370i	$0.665808\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14165.0	0.971080
$598$	0	0
$599$	−2547.00	−0.173736	−0.0868678	−	0.996220i	$-0.527686\pi$
$599$	−2547.00	−0.173736	−0.0868678	+	0.996220i	$0.527686\pi$
$600$	0	0
$601$	−7042.00	−0.477952	−0.238976	−	0.971025i	$-0.576812\pi$
$601$	−7042.00	−0.477952	−0.238976	+	0.971025i	$0.576812\pi$
$602$	0	0
$603$	838.000	0.0565937
$604$	0	0
$605$	−17262.0	−1.16000
$606$	0	0
$607$	22591.0	1.51061	0.755305	−	0.655373i	$-0.227489\pi$
$607$	22591.0	1.51061	0.755305	+	0.655373i	$0.227489\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14070.0	0.931606
$612$	0	0
$613$	−8485.00	−0.559063	−0.279532	−	0.960136i	$-0.590179\pi$
$613$	−8485.00	−0.559063	−0.279532	+	0.960136i	$0.590179\pi$
$614$	0	0
$615$	1890.00	0.123922
$616$	0	0
$617$	−18282.0	−1.19288	−0.596439	−	0.802658i	$-0.703418\pi$
$617$	−18282.0	−1.19288	−0.596439	+	0.802658i	$0.703418\pi$
$618$	0	0
$619$	−2291.00	−0.148761	−0.0743805	−	0.997230i	$-0.523698\pi$
$619$	−2291.00	−0.148761	−0.0743805	+	0.997230i	$0.523698\pi$
$620$	0	0
$621$	10005.0	0.646517
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8189.00	−0.524096
$626$	0	0
$627$	−1425.00	−0.0907640
$628$	0	0
$629$	−12903.0	−0.817927
$630$	0	0
$631$	6928.00	0.437083	0.218541	−	0.975828i	$-0.429870\pi$
$631$	6928.00	0.437083	0.218541	+	0.975828i	$0.429870\pi$
$632$	0	0
$633$	620.000	0.0389302
$634$	0	0
$635$	−18504.0	−1.15639
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−192.000	−0.0118864
$640$	0	0
$641$	24975.0	1.53893	0.769464	−	0.638690i	$-0.220523\pi$
$641$	24975.0	1.53893	0.769464	+	0.638690i	$0.220523\pi$
$642$	0	0
$643$	−9548.00	−0.585593	−0.292797	−	0.956175i	$-0.594586\pi$
$643$	−9548.00	−0.585593	−0.292797	+	0.956175i	$0.594586\pi$
$644$	0	0
$645$	−5580.00	−0.340639
$646$	0	0
$647$	−10131.0	−0.615596	−0.307798	−	0.951452i	$-0.599592\pi$
$647$	−10131.0	−0.615596	−0.307798	+	0.951452i	$0.599592\pi$
$648$	0	0
$649$	−12483.0	−0.755009
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16659.0	0.998342	0.499171	−	0.866504i	$-0.333638\pi$
$653$	16659.0	0.998342	0.499171	+	0.866504i	$0.333638\pi$
$654$	0	0
$655$	18441.0	1.10008
$656$	0	0
$657$	626.000	0.0371729
$658$	0	0
$659$	−29556.0	−1.74710	−0.873550	−	0.486735i	$-0.838188\pi$
$659$	−29556.0	−1.74710	−0.873550	+	0.486735i	$0.838188\pi$
$660$	0	0
$661$	191.000	0.0112391	0.00561955	−	0.999984i	$-0.498211\pi$
$661$	191.000	0.0112391	0.00561955	+	0.999984i	$0.498211\pi$
$662$	0	0
$663$	−17850.0	−1.04561
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7866.00	−0.456631
$668$	0	0
$669$	−280.000	−0.0161815
$670$	0	0
$671$	−40413.0	−2.32508
$672$	0	0
$673$	2606.00	0.149263	0.0746314	−	0.997211i	$-0.476222\pi$
$673$	2606.00	0.149263	0.0746314	+	0.997211i	$0.476222\pi$
$674$	0	0
$675$	6380.00	0.363802
$676$	0	0
$677$	−4209.00	−0.238944	−0.119472	−	0.992838i	$-0.538120\pi$
$677$	−4209.00	−0.238944	−0.119472	+	0.992838i	$0.538120\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	15285.0	0.860092
$682$	0	0
$683$	−24303.0	−1.36154	−0.680768	−	0.732500i	$-0.738354\pi$
$683$	−24303.0	−1.36154	−0.680768	+	0.732500i	$0.738354\pi$
$684$	0	0
$685$	1269.00	0.0707825
$686$	0	0
$687$	−4805.00	−0.266845
$688$	0	0
$689$	27510.0	1.52111
$690$	0	0
$691$	−15041.0	−0.828056	−0.414028	−	0.910264i	$-0.635878\pi$
$691$	−15041.0	−0.828056	−0.414028	+	0.910264i	$0.635878\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13356.0	0.728952
$696$	0	0
$697$	−2142.00	−0.116405
$698$	0	0
$699$	−14145.0	−0.765398
$700$	0	0
$701$	24726.0	1.33222	0.666111	−	0.745852i	$-0.267958\pi$
$701$	24726.0	1.33222	0.666111	+	0.745852i	$0.267958\pi$
$702$	0	0
$703$	1265.00	0.0678668
$704$	0	0
$705$	9045.00	0.483198
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4957.00	−0.262573	−0.131286	−	0.991344i	$-0.541911\pi$
$709$	−4957.00	−0.262573	−0.131286	+	0.991344i	$0.541911\pi$
$710$	0	0
$711$	922.000	0.0486325
$712$	0	0
$713$	1587.00	0.0833571
$714$	0	0
$715$	35910.0	1.87826
$716$	0	0
$717$	17700.0	0.921923
$718$	0	0
$719$	−27669.0	−1.43516	−0.717580	−	0.696476i	$-0.754750\pi$
$719$	−27669.0	−1.43516	−0.717580	+	0.696476i	$0.754750\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	26155.0	1.34539
$724$	0	0
$725$	−5016.00	−0.256951
$726$	0	0
$727$	13888.0	0.708497	0.354249	−	0.935151i	$-0.384737\pi$
$727$	13888.0	0.708497	0.354249	+	0.935151i	$0.384737\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	6324.00	0.319975
$732$	0	0
$733$	14243.0	0.717704	0.358852	−	0.933394i	$-0.383168\pi$
$733$	14243.0	0.717704	0.358852	+	0.933394i	$0.383168\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23883.0	−1.19368
$738$	0	0
$739$	−36959.0	−1.83973	−0.919864	−	0.392238i	$-0.871701\pi$
$739$	−36959.0	−1.83973	−0.919864	+	0.392238i	$0.871701\pi$
$740$	0	0
$741$	1750.00	0.0867582
$742$	0	0
$743$	12528.0	0.618584	0.309292	−	0.950967i	$-0.399908\pi$
$743$	12528.0	0.618584	0.309292	+	0.950967i	$0.399908\pi$
$744$	0	0
$745$	513.000	0.0252280
$746$	0	0
$747$	−1176.00	−0.0576005
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17767.0	0.863285	0.431643	−	0.902045i	$-0.357934\pi$
$751$	17767.0	0.863285	0.431643	+	0.902045i	$0.357934\pi$
$752$	0	0
$753$	−25200.0	−1.21957
$754$	0	0
$755$	7551.00	0.363985
$756$	0	0
$757$	−28726.0	−1.37921	−0.689606	−	0.724184i	$-0.742216\pi$
$757$	−28726.0	−1.37921	−0.689606	+	0.724184i	$0.742216\pi$
$758$	0	0
$759$	−19665.0	−0.940440
$760$	0	0
$761$	−26469.0	−1.26084	−0.630421	−	0.776254i	$-0.717118\pi$
$761$	−26469.0	−1.26084	−0.630421	+	0.776254i	$0.717118\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	918.000	0.0433861
$766$	0	0
$767$	15330.0	0.721687
$768$	0	0
$769$	5054.00	0.236999	0.118499	−	0.992954i	$-0.462192\pi$
$769$	5054.00	0.236999	0.118499	+	0.992954i	$0.462192\pi$
$770$	0	0
$771$	−7185.00	−0.335618
$772$	0	0
$773$	−35565.0	−1.65483	−0.827415	−	0.561590i	$-0.810190\pi$
$773$	−35565.0	−1.65483	−0.827415	+	0.561590i	$0.810190\pi$
$774$	0	0
$775$	1012.00	0.0469060
$776$	0	0
$777$	0	0
$778$	0	0
$779$	210.000	0.00965858
$780$	0	0
$781$	5472.00	0.250709
$782$	0	0
$783$	−16530.0	−0.754450
$784$	0	0
$785$	25497.0	1.15927
$786$	0	0
$787$	8629.00	0.390839	0.195420	−	0.980720i	$-0.437393\pi$
$787$	8629.00	0.390839	0.195420	+	0.980720i	$0.437393\pi$
$788$	0	0
$789$	11625.0	0.524539
$790$	0	0
$791$	0	0
$792$	0	0
$793$	49630.0	2.22246
$794$	0	0
$795$	17685.0	0.788959
$796$	0	0
$797$	20706.0	0.920256	0.460128	−	0.887853i	$-0.347803\pi$
$797$	20706.0	0.920256	0.460128	+	0.887853i	$0.347803\pi$
$798$	0	0
$799$	−10251.0	−0.453885
$800$	0	0
$801$	2034.00	0.0897227
$802$	0	0
$803$	−17841.0	−0.784054
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11925.0	−0.520173
$808$	0	0
$809$	−16185.0	−0.703380	−0.351690	−	0.936117i	$-0.614393\pi$
$809$	−16185.0	−0.703380	−0.351690	+	0.936117i	$0.614393\pi$
$810$	0	0
$811$	11788.0	0.510398	0.255199	−	0.966889i	$-0.417859\pi$
$811$	11788.0	0.510398	0.255199	+	0.966889i	$0.417859\pi$
$812$	0	0
$813$	1655.00	0.0713941
$814$	0	0
$815$	−20799.0	−0.893935
$816$	0	0
$817$	−620.000	−0.0265496
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29793.0	−1.26648	−0.633242	−	0.773954i	$-0.718276\pi$
$821$	−29793.0	−1.26648	−0.633242	+	0.773954i	$0.718276\pi$
$822$	0	0
$823$	−30323.0	−1.28432	−0.642159	−	0.766572i	$-0.721961\pi$
$823$	−30323.0	−1.28432	−0.642159	+	0.766572i	$0.721961\pi$
$824$	0	0
$825$	−12540.0	−0.529196
$826$	0	0
$827$	−21156.0	−0.889560	−0.444780	−	0.895640i	$-0.646718\pi$
$827$	−21156.0	−0.889560	−0.444780	+	0.895640i	$0.646718\pi$
$828$	0	0
$829$	−5269.00	−0.220748	−0.110374	−	0.993890i	$-0.535205\pi$
$829$	−5269.00	−0.220748	−0.110374	+	0.993890i	$0.535205\pi$
$830$	0	0
$831$	24355.0	1.01669
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11340.0	0.469984
$836$	0	0
$837$	3335.00	0.137723
$838$	0	0
$839$	39816.0	1.63838	0.819190	−	0.573522i	$-0.194423\pi$
$839$	39816.0	1.63838	0.819190	+	0.573522i	$0.194423\pi$
$840$	0	0
$841$	−11393.0	−0.467137
$842$	0	0
$843$	−35130.0	−1.43528
$844$	0	0
$845$	−24327.0	−0.990384
$846$	0	0
$847$	0	0
$848$	0	0
$849$	26765.0	1.08195
$850$	0	0
$851$	17457.0	0.703194
$852$	0	0
$853$	14546.0	0.583875	0.291938	−	0.956437i	$-0.405700\pi$
$853$	14546.0	0.583875	0.291938	+	0.956437i	$0.405700\pi$
$854$	0	0
$855$	−90.0000	−0.00359992
$856$	0	0
$857$	−31449.0	−1.25353	−0.626766	−	0.779207i	$-0.715622\pi$
$857$	−31449.0	−1.25353	−0.626766	+	0.779207i	$0.715622\pi$
$858$	0	0
$859$	24523.0	0.974056	0.487028	−	0.873386i	$-0.338081\pi$
$859$	24523.0	0.974056	0.487028	+	0.873386i	$0.338081\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	8163.00	0.321983	0.160992	−	0.986956i	$-0.448531\pi$
$863$	8163.00	0.321983	0.160992	+	0.986956i	$0.448531\pi$
$864$	0	0
$865$	−29403.0	−1.15576
$866$	0	0
$867$	−11560.0	−0.452824
$868$	0	0
$869$	−26277.0	−1.02576
$870$	0	0
$871$	29330.0	1.14100
$872$	0	0
$873$	3668.00	0.142203
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4367.00	0.168145	0.0840725	−	0.996460i	$-0.473207\pi$
$877$	4367.00	0.168145	0.0840725	+	0.996460i	$0.473207\pi$
$878$	0	0
$879$	20790.0	0.797758
$880$	0	0
$881$	−50190.0	−1.91935	−0.959673	−	0.281118i	$-0.909295\pi$
$881$	−50190.0	−1.91935	−0.959673	+	0.281118i	$0.909295\pi$
$882$	0	0
$883$	−12308.0	−0.469079	−0.234540	−	0.972107i	$-0.575358\pi$
$883$	−12308.0	−0.469079	−0.234540	+	0.972107i	$0.575358\pi$
$884$	0	0
$885$	9855.00	0.374319
$886$	0	0
$887$	−31617.0	−1.19684	−0.598419	−	0.801183i	$-0.704204\pi$
$887$	−31617.0	−1.19684	−0.598419	+	0.801183i	$0.704204\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−38247.0	−1.43807
$892$	0	0
$893$	1005.00	0.0376607
$894$	0	0
$895$	11583.0	0.432600
$896$	0	0
$897$	24150.0	0.898935
$898$	0	0
$899$	−2622.00	−0.0972732
$900$	0	0
$901$	−20043.0	−0.741098
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24066.0	0.883957
$906$	0	0
$907$	13525.0	0.495138	0.247569	−	0.968870i	$-0.420368\pi$
$907$	13525.0	0.495138	0.247569	+	0.968870i	$0.420368\pi$
$908$	0	0
$909$	570.000	0.0207984
$910$	0	0
$911$	19248.0	0.700016	0.350008	−	0.936747i	$-0.386179\pi$
$911$	19248.0	0.700016	0.350008	+	0.936747i	$0.386179\pi$
$912$	0	0
$913$	33516.0	1.21492
$914$	0	0
$915$	31905.0	1.15273
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8695.00	0.312102	0.156051	−	0.987749i	$-0.450124\pi$
$919$	8695.00	0.312102	0.156051	+	0.987749i	$0.450124\pi$
$920$	0	0
$921$	48020.0	1.71804
$922$	0	0
$923$	−6720.00	−0.239644
$924$	0	0
$925$	11132.0	0.395695
$926$	0	0
$927$	−998.000	−0.0353599
$928$	0	0
$929$	19479.0	0.687928	0.343964	−	0.938983i	$-0.388230\pi$
$929$	19479.0	0.687928	0.343964	+	0.938983i	$0.388230\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−50655.0	−1.77746
$934$	0	0
$935$	−26163.0	−0.915103
$936$	0	0
$937$	−12502.0	−0.435883	−0.217942	−	0.975962i	$-0.569934\pi$
$937$	−12502.0	−0.435883	−0.217942	+	0.975962i	$0.569934\pi$
$938$	0	0
$939$	53995.0	1.87653
$940$	0	0
$941$	−15993.0	−0.554046	−0.277023	−	0.960863i	$-0.589348\pi$
$941$	−15993.0	−0.554046	−0.277023	+	0.960863i	$0.589348\pi$
$942$	0	0
$943$	2898.00	0.100076
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44001.0	−1.50986	−0.754932	−	0.655804i	$-0.772330\pi$
$947$	−44001.0	−1.50986	−0.754932	+	0.655804i	$0.772330\pi$
$948$	0	0
$949$	21910.0	0.749451
$950$	0	0
$951$	2655.00	0.0905303
$952$	0	0
$953$	−4002.00	−0.136031	−0.0680155	−	0.997684i	$-0.521667\pi$
$953$	−4002.00	−0.136031	−0.0680155	+	0.997684i	$0.521667\pi$
$954$	0	0
$955$	37665.0	1.27624
$956$	0	0
$957$	32490.0	1.09744
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29262.0	−0.982243
$962$	0	0
$963$	−2214.00	−0.0740863
$964$	0	0
$965$	765.000	0.0255194
$966$	0	0
$967$	−10544.0	−0.350643	−0.175322	−	0.984511i	$-0.556097\pi$
$967$	−10544.0	−0.350643	−0.175322	+	0.984511i	$0.556097\pi$
$968$	0	0
$969$	−1275.00	−0.0422692
$970$	0	0
$971$	6183.00	0.204348	0.102174	−	0.994767i	$-0.467420\pi$
$971$	6183.00	0.204348	0.102174	+	0.994767i	$0.467420\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	15400.0	0.505841
$976$	0	0
$977$	3723.00	0.121913	0.0609567	−	0.998140i	$-0.480585\pi$
$977$	3723.00	0.121913	0.0609567	+	0.998140i	$0.480585\pi$
$978$	0	0
$979$	−57969.0	−1.89244
$980$	0	0
$981$	−1846.00	−0.0600798
$982$	0	0
$983$	45897.0	1.48920	0.744602	−	0.667509i	$-0.232639\pi$
$983$	45897.0	1.48920	0.744602	+	0.667509i	$0.232639\pi$
$984$	0	0
$985$	3510.00	0.113541
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8556.00	−0.275091
$990$	0	0
$991$	−6467.00	−0.207297	−0.103648	−	0.994614i	$-0.533052\pi$
$991$	−6467.00	−0.207297	−0.103648	+	0.994614i	$0.533052\pi$
$992$	0	0
$993$	35075.0	1.12092
$994$	0	0
$995$	−25497.0	−0.812371
$996$	0	0
$997$	23039.0	0.731848	0.365924	−	0.930645i	$-0.380753\pi$
$997$	23039.0	0.731848	0.365924	+	0.930645i	$0.380753\pi$
$998$	0	0
$999$	36685.0	1.16182

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.p.1.1		1
4.3	odd	2		98.4.a.d.1.1		1
7.2	even	3		112.4.i.a.81.1		2
7.4	even	3		112.4.i.a.65.1		2
7.6	odd	2		784.4.a.c.1.1		1
12.11	even	2		882.4.a.f.1.1		1
20.19	odd	2		2450.4.a.q.1.1		1
28.3	even	6		98.4.c.a.79.1		2
28.11	odd	6		14.4.c.a.9.1	✓	2
28.19	even	6		98.4.c.a.67.1		2
28.23	odd	6		14.4.c.a.11.1	yes	2
28.27	even	2		98.4.a.f.1.1		1
56.11	odd	6		448.4.i.b.65.1		2
56.37	even	6		448.4.i.e.193.1		2
56.51	odd	6		448.4.i.b.193.1		2
56.53	even	6		448.4.i.e.65.1		2
84.11	even	6		126.4.g.d.37.1		2
84.23	even	6		126.4.g.d.109.1		2
84.47	odd	6		882.4.g.u.361.1		2
84.59	odd	6		882.4.g.u.667.1		2
84.83	odd	2		882.4.a.c.1.1		1
140.23	even	12		350.4.j.b.249.1		4
140.39	odd	6		350.4.e.e.51.1		2
140.67	even	12		350.4.j.b.149.1		4
140.79	odd	6		350.4.e.e.151.1		2
140.107	even	12		350.4.j.b.249.2		4
140.123	even	12		350.4.j.b.149.2		4
140.139	even	2		2450.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.c.a.9.1	✓	2	28.11	odd	6
14.4.c.a.11.1	yes	2	28.23	odd	6
98.4.a.d.1.1		1	4.3	odd	2
98.4.a.f.1.1		1	28.27	even	2
98.4.c.a.67.1		2	28.19	even	6
98.4.c.a.79.1		2	28.3	even	6
112.4.i.a.65.1		2	7.4	even	3
112.4.i.a.81.1		2	7.2	even	3
126.4.g.d.37.1		2	84.11	even	6
126.4.g.d.109.1		2	84.23	even	6
350.4.e.e.51.1		2	140.39	odd	6
350.4.e.e.151.1		2	140.79	odd	6
350.4.j.b.149.1		4	140.67	even	12
350.4.j.b.149.2		4	140.123	even	12
350.4.j.b.249.1		4	140.23	even	12
350.4.j.b.249.2		4	140.107	even	12
448.4.i.b.65.1		2	56.11	odd	6
448.4.i.b.193.1		2	56.51	odd	6
448.4.i.e.65.1		2	56.53	even	6
448.4.i.e.193.1		2	56.37	even	6
784.4.a.c.1.1		1	7.6	odd	2
784.4.a.p.1.1		1	1.1	even	1	trivial
882.4.a.c.1.1		1	84.83	odd	2
882.4.a.f.1.1		1	12.11	even	2
882.4.g.u.361.1		2	84.47	odd	6
882.4.g.u.667.1		2	84.59	odd	6
2450.4.a.d.1.1		1	140.139	even	2
2450.4.a.q.1.1		1	20.19	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Introduction

L-functions

Modular forms

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