LMFDB - Embedded newform 816.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,4,Mod(1,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, 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("816.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	816.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:	$48.1455585647$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	816.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -20.0000 q^{5} +2.00000 q^{7} +9.00000 q^{9} +48.0000 q^{11} -14.0000 q^{13} +60.0000 q^{15} -17.0000 q^{17} -92.0000 q^{19} -6.00000 q^{21} +122.000 q^{23} +275.000 q^{25} -27.0000 q^{27} -36.0000 q^{29} +182.000 q^{31} -144.000 q^{33} -40.0000 q^{35} +76.0000 q^{37} +42.0000 q^{39} +294.000 q^{41} +428.000 q^{43} -180.000 q^{45} +12.0000 q^{47} -339.000 q^{49} +51.0000 q^{51} -234.000 q^{53} -960.000 q^{55} +276.000 q^{57} +540.000 q^{59} -820.000 q^{61} +18.0000 q^{63} +280.000 q^{65} -700.000 q^{67} -366.000 q^{69} -794.000 q^{71} -1038.00 q^{73} -825.000 q^{75} +96.0000 q^{77} -858.000 q^{79} +81.0000 q^{81} -1052.00 q^{83} +340.000 q^{85} +108.000 q^{87} +1102.00 q^{89} -28.0000 q^{91} -546.000 q^{93} +1840.00 q^{95} +710.000 q^{97} +432.000 q^{99} +O(q^{100})$

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
$5$	−20.0000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−20.0000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	2.00000	0.107990	0.0539949	−	0.998541i	$-0.482805\pi$
$7$	2.00000	0.107990	0.0539949	+	0.998541i	$0.482805\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	0	0
$13$	−14.0000	−0.298685	−0.149342	−	0.988786i	$-0.547716\pi$
$13$	−14.0000	−0.298685	−0.149342	+	0.988786i	$0.547716\pi$
$14$	0	0
$15$	60.0000	1.03280
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	−6.00000	−0.0623480
$22$	0	0
$23$	122.000	1.10603	0.553016	−	0.833170i	$-0.313477\pi$
$23$	122.000	1.10603	0.553016	+	0.833170i	$0.313477\pi$
$24$	0	0
$25$	275.000	2.20000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−36.0000	−0.230518	−0.115259	−	0.993335i	$-0.536770\pi$
$29$	−36.0000	−0.230518	−0.115259	+	0.993335i	$0.536770\pi$
$30$	0	0
$31$	182.000	1.05446	0.527228	−	0.849724i	$-0.323231\pi$
$31$	182.000	1.05446	0.527228	+	0.849724i	$0.323231\pi$
$32$	0	0
$33$	−144.000	−0.759612
$34$	0	0
$35$	−40.0000	−0.193178
$36$	0	0
$37$	76.0000	0.337684	0.168842	−	0.985643i	$-0.445997\pi$
$37$	76.0000	0.337684	0.168842	+	0.985643i	$0.445997\pi$
$38$	0	0
$39$	42.0000	0.172446
$40$	0	0
$41$	294.000	1.11988	0.559940	−	0.828533i	$-0.310824\pi$
$41$	294.000	1.11988	0.559940	+	0.828533i	$0.310824\pi$
$42$	0	0
$43$	428.000	1.51789	0.758946	−	0.651153i	$-0.225714\pi$
$43$	428.000	1.51789	0.758946	+	0.651153i	$0.225714\pi$
$44$	0	0
$45$	−180.000	−0.596285
$46$	0	0
$47$	12.0000	0.0372421	0.0186211	−	0.999827i	$-0.494072\pi$
$47$	12.0000	0.0372421	0.0186211	+	0.999827i	$0.494072\pi$
$48$	0	0
$49$	−339.000	−0.988338
$50$	0	0
$51$	51.0000	0.140028
$52$	0	0
$53$	−234.000	−0.606460	−0.303230	−	0.952917i	$-0.598065\pi$
$53$	−234.000	−0.606460	−0.303230	+	0.952917i	$0.598065\pi$
$54$	0	0
$55$	−960.000	−2.35357
$56$	0	0
$57$	276.000	0.641353
$58$	0	0
$59$	540.000	1.19156	0.595780	−	0.803148i	$-0.296843\pi$
$59$	540.000	1.19156	0.595780	+	0.803148i	$0.296843\pi$
$60$	0	0
$61$	−820.000	−1.72115	−0.860576	−	0.509322i	$-0.829896\pi$
$61$	−820.000	−1.72115	−0.860576	+	0.509322i	$0.829896\pi$
$62$	0	0
$63$	18.0000	0.0359966
$64$	0	0
$65$	280.000	0.534303
$66$	0	0
$67$	−700.000	−1.27640	−0.638199	−	0.769872i	$-0.720320\pi$
$67$	−700.000	−1.27640	−0.638199	+	0.769872i	$0.720320\pi$
$68$	0	0
$69$	−366.000	−0.638568
$70$	0	0
$71$	−794.000	−1.32719	−0.663595	−	0.748092i	$-0.730970\pi$
$71$	−794.000	−1.32719	−0.663595	+	0.748092i	$0.730970\pi$
$72$	0	0
$73$	−1038.00	−1.66423	−0.832114	−	0.554604i	$-0.812870\pi$
$73$	−1038.00	−1.66423	−0.832114	+	0.554604i	$0.812870\pi$
$74$	0	0
$75$	−825.000	−1.27017
$76$	0	0
$77$	96.0000	0.142081
$78$	0	0
$79$	−858.000	−1.22193	−0.610965	−	0.791657i	$-0.709219\pi$
$79$	−858.000	−1.22193	−0.610965	+	0.791657i	$0.709219\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1052.00	−1.39123	−0.695614	−	0.718415i	$-0.744868\pi$
$83$	−1052.00	−1.39123	−0.695614	+	0.718415i	$0.744868\pi$
$84$	0	0
$85$	340.000	0.433861
$86$	0	0
$87$	108.000	0.133090
$88$	0	0
$89$	1102.00	1.31249	0.656246	−	0.754547i	$-0.272143\pi$
$89$	1102.00	1.31249	0.656246	+	0.754547i	$0.272143\pi$
$90$	0	0
$91$	−28.0000	−0.0322549
$92$	0	0
$93$	−546.000	−0.608791
$94$	0	0
$95$	1840.00	1.98716
$96$	0	0
$97$	710.000	0.743192	0.371596	−	0.928395i	$-0.378811\pi$
$97$	710.000	0.743192	0.371596	+	0.928395i	$0.378811\pi$
$98$	0	0
$99$	432.000	0.438562
$100$	0	0
$101$	−1210.00	−1.19207	−0.596037	−	0.802957i	$-0.703259\pi$
$101$	−1210.00	−1.19207	−0.596037	+	0.802957i	$0.703259\pi$
$102$	0	0
$103$	−644.000	−0.616070	−0.308035	−	0.951375i	$-0.599671\pi$
$103$	−644.000	−0.616070	−0.308035	+	0.951375i	$0.599671\pi$
$104$	0	0
$105$	120.000	0.111531
$106$	0	0
$107$	256.000	0.231294	0.115647	−	0.993290i	$-0.463106\pi$
$107$	256.000	0.231294	0.115647	+	0.993290i	$0.463106\pi$
$108$	0	0
$109$	−1248.00	−1.09667	−0.548334	−	0.836260i	$-0.684738\pi$
$109$	−1248.00	−1.09667	−0.548334	+	0.836260i	$0.684738\pi$
$110$	0	0
$111$	−228.000	−0.194962
$112$	0	0
$113$	−350.000	−0.291374	−0.145687	−	0.989331i	$-0.546539\pi$
$113$	−350.000	−0.291374	−0.145687	+	0.989331i	$0.546539\pi$
$114$	0	0
$115$	−2440.00	−1.97853
$116$	0	0
$117$	−126.000	−0.0995616
$118$	0	0
$119$	−34.0000	−0.0261914
$120$	0	0
$121$	973.000	0.731029
$122$	0	0
$123$	−882.000	−0.646563
$124$	0	0
$125$	−3000.00	−2.14663
$126$	0	0
$127$	328.000	0.229176	0.114588	−	0.993413i	$-0.463445\pi$
$127$	328.000	0.229176	0.114588	+	0.993413i	$0.463445\pi$
$128$	0	0
$129$	−1284.00	−0.876356
$130$	0	0
$131$	1952.00	1.30189	0.650943	−	0.759127i	$-0.274374\pi$
$131$	1952.00	1.30189	0.650943	+	0.759127i	$0.274374\pi$
$132$	0	0
$133$	−184.000	−0.119961
$134$	0	0
$135$	540.000	0.344265
$136$	0	0
$137$	2634.00	1.64261	0.821306	−	0.570488i	$-0.193246\pi$
$137$	2634.00	1.64261	0.821306	+	0.570488i	$0.193246\pi$
$138$	0	0
$139$	−1864.00	−1.13743	−0.568714	−	0.822536i	$-0.692559\pi$
$139$	−1864.00	−1.13743	−0.568714	+	0.822536i	$0.692559\pi$
$140$	0	0
$141$	−36.0000	−0.0215018
$142$	0	0
$143$	−672.000	−0.392975
$144$	0	0
$145$	720.000	0.412364
$146$	0	0
$147$	1017.00	0.570617
$148$	0	0
$149$	−286.000	−0.157249	−0.0786243	−	0.996904i	$-0.525053\pi$
$149$	−286.000	−0.157249	−0.0786243	+	0.996904i	$0.525053\pi$
$150$	0	0
$151$	1624.00	0.875227	0.437613	−	0.899163i	$-0.355824\pi$
$151$	1624.00	0.875227	0.437613	+	0.899163i	$0.355824\pi$
$152$	0	0
$153$	−153.000	−0.0808452
$154$	0	0
$155$	−3640.00	−1.88627
$156$	0	0
$157$	2542.00	1.29219	0.646095	−	0.763257i	$-0.276401\pi$
$157$	2542.00	1.29219	0.646095	+	0.763257i	$0.276401\pi$
$158$	0	0
$159$	702.000	0.350140
$160$	0	0
$161$	244.000	0.119440
$162$	0	0
$163$	−684.000	−0.328681	−0.164341	−	0.986404i	$-0.552550\pi$
$163$	−684.000	−0.328681	−0.164341	+	0.986404i	$0.552550\pi$
$164$	0	0
$165$	2880.00	1.35883
$166$	0	0
$167$	−542.000	−0.251145	−0.125573	−	0.992084i	$-0.540077\pi$
$167$	−542.000	−0.251145	−0.125573	+	0.992084i	$0.540077\pi$
$168$	0	0
$169$	−2001.00	−0.910787
$170$	0	0
$171$	−828.000	−0.370285
$172$	0	0
$173$	−836.000	−0.367398	−0.183699	−	0.982983i	$-0.558807\pi$
$173$	−836.000	−0.367398	−0.183699	+	0.982983i	$0.558807\pi$
$174$	0	0
$175$	550.000	0.237578
$176$	0	0
$177$	−1620.00	−0.687947
$178$	0	0
$179$	3444.00	1.43808	0.719041	−	0.694968i	$-0.244581\pi$
$179$	3444.00	1.43808	0.719041	+	0.694968i	$0.244581\pi$
$180$	0	0
$181$	−3284.00	−1.34861	−0.674303	−	0.738454i	$-0.735556\pi$
$181$	−3284.00	−1.34861	−0.674303	+	0.738454i	$0.735556\pi$
$182$	0	0
$183$	2460.00	0.993707
$184$	0	0
$185$	−1520.00	−0.604068
$186$	0	0
$187$	−816.000	−0.319101
$188$	0	0
$189$	−54.0000	−0.0207827
$190$	0	0
$191$	340.000	0.128804	0.0644019	−	0.997924i	$-0.479486\pi$
$191$	340.000	0.128804	0.0644019	+	0.997924i	$0.479486\pi$
$192$	0	0
$193$	−1498.00	−0.558696	−0.279348	−	0.960190i	$-0.590118\pi$
$193$	−1498.00	−0.558696	−0.279348	+	0.960190i	$0.590118\pi$
$194$	0	0
$195$	−840.000	−0.308480
$196$	0	0
$197$	−1176.00	−0.425312	−0.212656	−	0.977127i	$-0.568211\pi$
$197$	−1176.00	−0.425312	−0.212656	+	0.977127i	$0.568211\pi$
$198$	0	0
$199$	2450.00	0.872743	0.436372	−	0.899767i	$-0.356263\pi$
$199$	2450.00	0.872743	0.436372	+	0.899767i	$0.356263\pi$
$200$	0	0
$201$	2100.00	0.736928
$202$	0	0
$203$	−72.0000	−0.0248936
$204$	0	0
$205$	−5880.00	−2.00330
$206$	0	0
$207$	1098.00	0.368678
$208$	0	0
$209$	−4416.00	−1.46154
$210$	0	0
$211$	−1064.00	−0.347151	−0.173575	−	0.984821i	$-0.555532\pi$
$211$	−1064.00	−0.347151	−0.173575	+	0.984821i	$0.555532\pi$
$212$	0	0
$213$	2382.00	0.766253
$214$	0	0
$215$	−8560.00	−2.71529
$216$	0	0
$217$	364.000	0.113871
$218$	0	0
$219$	3114.00	0.960843
$220$	0	0
$221$	238.000	0.0724417
$222$	0	0
$223$	2244.00	0.673854	0.336927	−	0.941531i	$-0.390613\pi$
$223$	2244.00	0.673854	0.336927	+	0.941531i	$0.390613\pi$
$224$	0	0
$225$	2475.00	0.733333
$226$	0	0
$227$	−516.000	−0.150873	−0.0754364	−	0.997151i	$-0.524035\pi$
$227$	−516.000	−0.150873	−0.0754364	+	0.997151i	$0.524035\pi$
$228$	0	0
$229$	−2922.00	−0.843193	−0.421596	−	0.906784i	$-0.638530\pi$
$229$	−2922.00	−0.843193	−0.421596	+	0.906784i	$0.638530\pi$
$230$	0	0
$231$	−288.000	−0.0820303
$232$	0	0
$233$	3114.00	0.875558	0.437779	−	0.899083i	$-0.355765\pi$
$233$	3114.00	0.875558	0.437779	+	0.899083i	$0.355765\pi$
$234$	0	0
$235$	−240.000	−0.0666207
$236$	0	0
$237$	2574.00	0.705482
$238$	0	0
$239$	−4124.00	−1.11615	−0.558074	−	0.829791i	$-0.688459\pi$
$239$	−4124.00	−1.11615	−0.558074	+	0.829791i	$0.688459\pi$
$240$	0	0
$241$	−2034.00	−0.543658	−0.271829	−	0.962346i	$-0.587628\pi$
$241$	−2034.00	−0.543658	−0.271829	+	0.962346i	$0.587628\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	6780.00	1.76799
$246$	0	0
$247$	1288.00	0.331795
$248$	0	0
$249$	3156.00	0.803226
$250$	0	0
$251$	−996.000	−0.250466	−0.125233	−	0.992127i	$-0.539968\pi$
$251$	−996.000	−0.250466	−0.125233	+	0.992127i	$0.539968\pi$
$252$	0	0
$253$	5856.00	1.45519
$254$	0	0
$255$	−1020.00	−0.250490
$256$	0	0
$257$	−3246.00	−0.787860	−0.393930	−	0.919141i	$-0.628885\pi$
$257$	−3246.00	−0.787860	−0.393930	+	0.919141i	$0.628885\pi$
$258$	0	0
$259$	152.000	0.0364665
$260$	0	0
$261$	−324.000	−0.0768395
$262$	0	0
$263$	−932.000	−0.218516	−0.109258	−	0.994013i	$-0.534847\pi$
$263$	−932.000	−0.218516	−0.109258	+	0.994013i	$0.534847\pi$
$264$	0	0
$265$	4680.00	1.08487
$266$	0	0
$267$	−3306.00	−0.757767
$268$	0	0
$269$	−3884.00	−0.880341	−0.440170	−	0.897914i	$-0.645082\pi$
$269$	−3884.00	−0.880341	−0.440170	+	0.897914i	$0.645082\pi$
$270$	0	0
$271$	1936.00	0.433962	0.216981	−	0.976176i	$-0.430379\pi$
$271$	1936.00	0.433962	0.216981	+	0.976176i	$0.430379\pi$
$272$	0	0
$273$	84.0000	0.0186224
$274$	0	0
$275$	13200.0	2.89451
$276$	0	0
$277$	872.000	0.189146	0.0945729	−	0.995518i	$-0.469851\pi$
$277$	872.000	0.189146	0.0945729	+	0.995518i	$0.469851\pi$
$278$	0	0
$279$	1638.00	0.351486
$280$	0	0
$281$	3198.00	0.678921	0.339460	−	0.940620i	$-0.389756\pi$
$281$	3198.00	0.678921	0.339460	+	0.940620i	$0.389756\pi$
$282$	0	0
$283$	1936.00	0.406655	0.203327	−	0.979111i	$-0.434824\pi$
$283$	1936.00	0.406655	0.203327	+	0.979111i	$0.434824\pi$
$284$	0	0
$285$	−5520.00	−1.14729
$286$	0	0
$287$	588.000	0.120936
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−2130.00	−0.429082
$292$	0	0
$293$	−5718.00	−1.14010	−0.570050	−	0.821610i	$-0.693076\pi$
$293$	−5718.00	−1.14010	−0.570050	+	0.821610i	$0.693076\pi$
$294$	0	0
$295$	−10800.0	−2.13153
$296$	0	0
$297$	−1296.00	−0.253204
$298$	0	0
$299$	−1708.00	−0.330355
$300$	0	0
$301$	856.000	0.163917
$302$	0	0
$303$	3630.00	0.688244
$304$	0	0
$305$	16400.0	3.07889
$306$	0	0
$307$	−684.000	−0.127159	−0.0635797	−	0.997977i	$-0.520252\pi$
$307$	−684.000	−0.127159	−0.0635797	+	0.997977i	$0.520252\pi$
$308$	0	0
$309$	1932.00	0.355688
$310$	0	0
$311$	290.000	0.0528759	0.0264379	−	0.999650i	$-0.491584\pi$
$311$	290.000	0.0528759	0.0264379	+	0.999650i	$0.491584\pi$
$312$	0	0
$313$	−11050.0	−1.99547	−0.997736	−	0.0672478i	$-0.978578\pi$
$313$	−11050.0	−1.99547	−0.997736	+	0.0672478i	$0.978578\pi$
$314$	0	0
$315$	−360.000	−0.0643927
$316$	0	0
$317$	992.000	0.175761	0.0878806	−	0.996131i	$-0.471991\pi$
$317$	992.000	0.175761	0.0878806	+	0.996131i	$0.471991\pi$
$318$	0	0
$319$	−1728.00	−0.303290
$320$	0	0
$321$	−768.000	−0.133538
$322$	0	0
$323$	1564.00	0.269422
$324$	0	0
$325$	−3850.00	−0.657106
$326$	0	0
$327$	3744.00	0.633161
$328$	0	0
$329$	24.0000	0.00402177
$330$	0	0
$331$	−2860.00	−0.474924	−0.237462	−	0.971397i	$-0.576316\pi$
$331$	−2860.00	−0.474924	−0.237462	+	0.971397i	$0.576316\pi$
$332$	0	0
$333$	684.000	0.112561
$334$	0	0
$335$	14000.0	2.28329
$336$	0	0
$337$	6298.00	1.01802	0.509012	−	0.860760i	$-0.330011\pi$
$337$	6298.00	1.01802	0.509012	+	0.860760i	$0.330011\pi$
$338$	0	0
$339$	1050.00	0.168225
$340$	0	0
$341$	8736.00	1.38733
$342$	0	0
$343$	−1364.00	−0.214720
$344$	0	0
$345$	7320.00	1.14231
$346$	0	0
$347$	3508.00	0.542707	0.271353	−	0.962480i	$-0.412529\pi$
$347$	3508.00	0.542707	0.271353	+	0.962480i	$0.412529\pi$
$348$	0	0
$349$	−2406.00	−0.369026	−0.184513	−	0.982830i	$-0.559071\pi$
$349$	−2406.00	−0.369026	−0.184513	+	0.982830i	$0.559071\pi$
$350$	0	0
$351$	378.000	0.0574819
$352$	0	0
$353$	−1842.00	−0.277733	−0.138867	−	0.990311i	$-0.544346\pi$
$353$	−1842.00	−0.277733	−0.138867	+	0.990311i	$0.544346\pi$
$354$	0	0
$355$	15880.0	2.37415
$356$	0	0
$357$	102.000	0.0151216
$358$	0	0
$359$	−3264.00	−0.479853	−0.239927	−	0.970791i	$-0.577123\pi$
$359$	−3264.00	−0.479853	−0.239927	+	0.970791i	$0.577123\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	−2919.00	−0.422060
$364$	0	0
$365$	20760.0	2.97706
$366$	0	0
$367$	−7354.00	−1.04598	−0.522991	−	0.852338i	$-0.675184\pi$
$367$	−7354.00	−1.04598	−0.522991	+	0.852338i	$0.675184\pi$
$368$	0	0
$369$	2646.00	0.373293
$370$	0	0
$371$	−468.000	−0.0654915
$372$	0	0
$373$	11322.0	1.57166	0.785832	−	0.618440i	$-0.212235\pi$
$373$	11322.0	1.57166	0.785832	+	0.618440i	$0.212235\pi$
$374$	0	0
$375$	9000.00	1.23935
$376$	0	0
$377$	504.000	0.0688523
$378$	0	0
$379$	−12796.0	−1.73426	−0.867132	−	0.498078i	$-0.834039\pi$
$379$	−12796.0	−1.73426	−0.867132	+	0.498078i	$0.834039\pi$
$380$	0	0
$381$	−984.000	−0.132315
$382$	0	0
$383$	420.000	0.0560339	0.0280170	−	0.999607i	$-0.491081\pi$
$383$	420.000	0.0560339	0.0280170	+	0.999607i	$0.491081\pi$
$384$	0	0
$385$	−1920.00	−0.254162
$386$	0	0
$387$	3852.00	0.505964
$388$	0	0
$389$	6426.00	0.837561	0.418780	−	0.908088i	$-0.362458\pi$
$389$	6426.00	0.837561	0.418780	+	0.908088i	$0.362458\pi$
$390$	0	0
$391$	−2074.00	−0.268252
$392$	0	0
$393$	−5856.00	−0.751644
$394$	0	0
$395$	17160.0	2.18586
$396$	0	0
$397$	−13212.0	−1.67026	−0.835128	−	0.550056i	$-0.814606\pi$
$397$	−13212.0	−1.67026	−0.835128	+	0.550056i	$0.814606\pi$
$398$	0	0
$399$	552.000	0.0692596
$400$	0	0
$401$	−6242.00	−0.777333	−0.388667	−	0.921378i	$-0.627064\pi$
$401$	−6242.00	−0.777333	−0.388667	+	0.921378i	$0.627064\pi$
$402$	0	0
$403$	−2548.00	−0.314950
$404$	0	0
$405$	−1620.00	−0.198762
$406$	0	0
$407$	3648.00	0.444287
$408$	0	0
$409$	−5194.00	−0.627938	−0.313969	−	0.949433i	$-0.601659\pi$
$409$	−5194.00	−0.627938	−0.313969	+	0.949433i	$0.601659\pi$
$410$	0	0
$411$	−7902.00	−0.948362
$412$	0	0
$413$	1080.00	0.128676
$414$	0	0
$415$	21040.0	2.48871
$416$	0	0
$417$	5592.00	0.656694
$418$	0	0
$419$	−15956.0	−1.86039	−0.930193	−	0.367071i	$-0.880361\pi$
$419$	−15956.0	−1.86039	−0.930193	+	0.367071i	$0.880361\pi$
$420$	0	0
$421$	−15254.0	−1.76588	−0.882939	−	0.469488i	$-0.844438\pi$
$421$	−15254.0	−1.76588	−0.882939	+	0.469488i	$0.844438\pi$
$422$	0	0
$423$	108.000	0.0124140
$424$	0	0
$425$	−4675.00	−0.533578
$426$	0	0
$427$	−1640.00	−0.185867
$428$	0	0
$429$	2016.00	0.226884
$430$	0	0
$431$	5538.00	0.618924	0.309462	−	0.950912i	$-0.399851\pi$
$431$	5538.00	0.618924	0.309462	+	0.950912i	$0.399851\pi$
$432$	0	0
$433$	11342.0	1.25880	0.629402	−	0.777080i	$-0.283300\pi$
$433$	11342.0	1.25880	0.629402	+	0.777080i	$0.283300\pi$
$434$	0	0
$435$	−2160.00	−0.238078
$436$	0	0
$437$	−11224.0	−1.22864
$438$	0	0
$439$	−982.000	−0.106762	−0.0533808	−	0.998574i	$-0.517000\pi$
$439$	−982.000	−0.106762	−0.0533808	+	0.998574i	$0.517000\pi$
$440$	0	0
$441$	−3051.00	−0.329446
$442$	0	0
$443$	2492.00	0.267265	0.133633	−	0.991031i	$-0.457336\pi$
$443$	2492.00	0.267265	0.133633	+	0.991031i	$0.457336\pi$
$444$	0	0
$445$	−22040.0	−2.34786
$446$	0	0
$447$	858.000	0.0907875
$448$	0	0
$449$	−5498.00	−0.577877	−0.288938	−	0.957348i	$-0.593302\pi$
$449$	−5498.00	−0.577877	−0.288938	+	0.957348i	$0.593302\pi$
$450$	0	0
$451$	14112.0	1.47341
$452$	0	0
$453$	−4872.00	−0.505312
$454$	0	0
$455$	560.000	0.0576994
$456$	0	0
$457$	4998.00	0.511590	0.255795	−	0.966731i	$-0.417663\pi$
$457$	4998.00	0.511590	0.255795	+	0.966731i	$0.417663\pi$
$458$	0	0
$459$	459.000	0.0466760
$460$	0	0
$461$	−7586.00	−0.766411	−0.383205	−	0.923663i	$-0.625180\pi$
$461$	−7586.00	−0.766411	−0.383205	+	0.923663i	$0.625180\pi$
$462$	0	0
$463$	−5900.00	−0.592217	−0.296108	−	0.955154i	$-0.595689\pi$
$463$	−5900.00	−0.592217	−0.296108	+	0.955154i	$0.595689\pi$
$464$	0	0
$465$	10920.0	1.08904
$466$	0	0
$467$	19404.0	1.92272	0.961360	−	0.275295i	$-0.0887755\pi$
$467$	19404.0	1.92272	0.961360	+	0.275295i	$0.0887755\pi$
$468$	0	0
$469$	−1400.00	−0.137838
$470$	0	0
$471$	−7626.00	−0.746046
$472$	0	0
$473$	20544.0	1.99707
$474$	0	0
$475$	−25300.0	−2.44388
$476$	0	0
$477$	−2106.00	−0.202153
$478$	0	0
$479$	−2586.00	−0.246675	−0.123338	−	0.992365i	$-0.539360\pi$
$479$	−2586.00	−0.246675	−0.123338	+	0.992365i	$0.539360\pi$
$480$	0	0
$481$	−1064.00	−0.100861
$482$	0	0
$483$	−732.000	−0.0689589
$484$	0	0
$485$	−14200.0	−1.32946
$486$	0	0
$487$	−10106.0	−0.940342	−0.470171	−	0.882575i	$-0.655808\pi$
$487$	−10106.0	−0.940342	−0.470171	+	0.882575i	$0.655808\pi$
$488$	0	0
$489$	2052.00	0.189764
$490$	0	0
$491$	76.0000	0.00698540	0.00349270	−	0.999994i	$-0.498888\pi$
$491$	76.0000	0.00698540	0.00349270	+	0.999994i	$0.498888\pi$
$492$	0	0
$493$	612.000	0.0559089
$494$	0	0
$495$	−8640.00	−0.784523
$496$	0	0
$497$	−1588.00	−0.143323
$498$	0	0
$499$	−8096.00	−0.726306	−0.363153	−	0.931730i	$-0.618300\pi$
$499$	−8096.00	−0.726306	−0.363153	+	0.931730i	$0.618300\pi$
$500$	0	0
$501$	1626.00	0.144999
$502$	0	0
$503$	−15942.0	−1.41316	−0.706579	−	0.707634i	$-0.749763\pi$
$503$	−15942.0	−1.41316	−0.706579	+	0.707634i	$0.749763\pi$
$504$	0	0
$505$	24200.0	2.13245
$506$	0	0
$507$	6003.00	0.525843
$508$	0	0
$509$	−13742.0	−1.19667	−0.598333	−	0.801247i	$-0.704170\pi$
$509$	−13742.0	−1.19667	−0.598333	+	0.801247i	$0.704170\pi$
$510$	0	0
$511$	−2076.00	−0.179720
$512$	0	0
$513$	2484.00	0.213784
$514$	0	0
$515$	12880.0	1.10206
$516$	0	0
$517$	576.000	0.0489989
$518$	0	0
$519$	2508.00	0.212117
$520$	0	0
$521$	11942.0	1.00420	0.502100	−	0.864809i	$-0.332561\pi$
$521$	11942.0	1.00420	0.502100	+	0.864809i	$0.332561\pi$
$522$	0	0
$523$	6012.00	0.502651	0.251325	−	0.967903i	$-0.419134\pi$
$523$	6012.00	0.502651	0.251325	+	0.967903i	$0.419134\pi$
$524$	0	0
$525$	−1650.00	−0.137166
$526$	0	0
$527$	−3094.00	−0.255743
$528$	0	0
$529$	2717.00	0.223309
$530$	0	0
$531$	4860.00	0.397187
$532$	0	0
$533$	−4116.00	−0.334491
$534$	0	0
$535$	−5120.00	−0.413751
$536$	0	0
$537$	−10332.0	−0.830277
$538$	0	0
$539$	−16272.0	−1.30034
$540$	0	0
$541$	6420.00	0.510198	0.255099	−	0.966915i	$-0.417892\pi$
$541$	6420.00	0.510198	0.255099	+	0.966915i	$0.417892\pi$
$542$	0	0
$543$	9852.00	0.778618
$544$	0	0
$545$	24960.0	1.96178
$546$	0	0
$547$	−1576.00	−0.123190	−0.0615950	−	0.998101i	$-0.519619\pi$
$547$	−1576.00	−0.123190	−0.0615950	+	0.998101i	$0.519619\pi$
$548$	0	0
$549$	−7380.00	−0.573717
$550$	0	0
$551$	3312.00	0.256072
$552$	0	0
$553$	−1716.00	−0.131956
$554$	0	0
$555$	4560.00	0.348759
$556$	0	0
$557$	15318.0	1.16525	0.582625	−	0.812741i	$-0.302026\pi$
$557$	15318.0	1.16525	0.582625	+	0.812741i	$0.302026\pi$
$558$	0	0
$559$	−5992.00	−0.453371
$560$	0	0
$561$	2448.00	0.184233
$562$	0	0
$563$	−13220.0	−0.989621	−0.494810	−	0.869001i	$-0.664763\pi$
$563$	−13220.0	−0.989621	−0.494810	+	0.869001i	$0.664763\pi$
$564$	0	0
$565$	7000.00	0.521225
$566$	0	0
$567$	162.000	0.0119989
$568$	0	0
$569$	2794.00	0.205853	0.102927	−	0.994689i	$-0.467179\pi$
$569$	2794.00	0.205853	0.102927	+	0.994689i	$0.467179\pi$
$570$	0	0
$571$	14756.0	1.08147	0.540735	−	0.841193i	$-0.318146\pi$
$571$	14756.0	1.08147	0.540735	+	0.841193i	$0.318146\pi$
$572$	0	0
$573$	−1020.00	−0.0743649
$574$	0	0
$575$	33550.0	2.43327
$576$	0	0
$577$	−2846.00	−0.205339	−0.102669	−	0.994716i	$-0.532738\pi$
$577$	−2846.00	−0.205339	−0.102669	+	0.994716i	$0.532738\pi$
$578$	0	0
$579$	4494.00	0.322563
$580$	0	0
$581$	−2104.00	−0.150239
$582$	0	0
$583$	−11232.0	−0.797911
$584$	0	0
$585$	2520.00	0.178101
$586$	0	0
$587$	16820.0	1.18268	0.591342	−	0.806421i	$-0.298598\pi$
$587$	16820.0	1.18268	0.591342	+	0.806421i	$0.298598\pi$
$588$	0	0
$589$	−16744.0	−1.17135
$590$	0	0
$591$	3528.00	0.245554
$592$	0	0
$593$	13314.0	0.921991	0.460995	−	0.887403i	$-0.347492\pi$
$593$	13314.0	0.921991	0.460995	+	0.887403i	$0.347492\pi$
$594$	0	0
$595$	680.000	0.0468526
$596$	0	0
$597$	−7350.00	−0.503879
$598$	0	0
$599$	−2880.00	−0.196450	−0.0982250	−	0.995164i	$-0.531317\pi$
$599$	−2880.00	−0.196450	−0.0982250	+	0.995164i	$0.531317\pi$
$600$	0	0
$601$	−24854.0	−1.68688	−0.843441	−	0.537222i	$-0.819474\pi$
$601$	−24854.0	−1.68688	−0.843441	+	0.537222i	$0.819474\pi$
$602$	0	0
$603$	−6300.00	−0.425466
$604$	0	0
$605$	−19460.0	−1.30770
$606$	0	0
$607$	−6122.00	−0.409365	−0.204682	−	0.978828i	$-0.565616\pi$
$607$	−6122.00	−0.409365	−0.204682	+	0.978828i	$0.565616\pi$
$608$	0	0
$609$	216.000	0.0143724
$610$	0	0
$611$	−168.000	−0.0111237
$612$	0	0
$613$	17398.0	1.14633	0.573164	−	0.819441i	$-0.305716\pi$
$613$	17398.0	1.14633	0.573164	+	0.819441i	$0.305716\pi$
$614$	0	0
$615$	17640.0	1.15661
$616$	0	0
$617$	2922.00	0.190657	0.0953284	−	0.995446i	$-0.469610\pi$
$617$	2922.00	0.190657	0.0953284	+	0.995446i	$0.469610\pi$
$618$	0	0
$619$	−9660.00	−0.627251	−0.313625	−	0.949547i	$-0.601544\pi$
$619$	−9660.00	−0.627251	−0.313625	+	0.949547i	$0.601544\pi$
$620$	0	0
$621$	−3294.00	−0.212856
$622$	0	0
$623$	2204.00	0.141736
$624$	0	0
$625$	25625.0	1.64000
$626$	0	0
$627$	13248.0	0.843818
$628$	0	0
$629$	−1292.00	−0.0819005
$630$	0	0
$631$	−2788.00	−0.175893	−0.0879465	−	0.996125i	$-0.528030\pi$
$631$	−2788.00	−0.175893	−0.0879465	+	0.996125i	$0.528030\pi$
$632$	0	0
$633$	3192.00	0.200428
$634$	0	0
$635$	−6560.00	−0.409962
$636$	0	0
$637$	4746.00	0.295202
$638$	0	0
$639$	−7146.00	−0.442397
$640$	0	0
$641$	−16290.0	−1.00377	−0.501885	−	0.864934i	$-0.667360\pi$
$641$	−16290.0	−1.00377	−0.501885	+	0.864934i	$0.667360\pi$
$642$	0	0
$643$	16588.0	1.01737	0.508683	−	0.860954i	$-0.330132\pi$
$643$	16588.0	1.01737	0.508683	+	0.860954i	$0.330132\pi$
$644$	0	0
$645$	25680.0	1.56767
$646$	0	0
$647$	−22364.0	−1.35892	−0.679459	−	0.733714i	$-0.737785\pi$
$647$	−22364.0	−1.35892	−0.679459	+	0.733714i	$0.737785\pi$
$648$	0	0
$649$	25920.0	1.56772
$650$	0	0
$651$	−1092.00	−0.0657432
$652$	0	0
$653$	−19356.0	−1.15997	−0.579984	−	0.814628i	$-0.696941\pi$
$653$	−19356.0	−1.15997	−0.579984	+	0.814628i	$0.696941\pi$
$654$	0	0
$655$	−39040.0	−2.32888
$656$	0	0
$657$	−9342.00	−0.554743
$658$	0	0
$659$	4220.00	0.249450	0.124725	−	0.992191i	$-0.460195\pi$
$659$	4220.00	0.249450	0.124725	+	0.992191i	$0.460195\pi$
$660$	0	0
$661$	−12070.0	−0.710240	−0.355120	−	0.934821i	$-0.615560\pi$
$661$	−12070.0	−0.710240	−0.355120	+	0.934821i	$0.615560\pi$
$662$	0	0
$663$	−714.000	−0.0418242
$664$	0	0
$665$	3680.00	0.214593
$666$	0	0
$667$	−4392.00	−0.254961
$668$	0	0
$669$	−6732.00	−0.389050
$670$	0	0
$671$	−39360.0	−2.26449
$672$	0	0
$673$	−5914.00	−0.338734	−0.169367	−	0.985553i	$-0.554172\pi$
$673$	−5914.00	−0.338734	−0.169367	+	0.985553i	$0.554172\pi$
$674$	0	0
$675$	−7425.00	−0.423390
$676$	0	0
$677$	−27624.0	−1.56821	−0.784104	−	0.620630i	$-0.786877\pi$
$677$	−27624.0	−1.56821	−0.784104	+	0.620630i	$0.786877\pi$
$678$	0	0
$679$	1420.00	0.0802571
$680$	0	0
$681$	1548.00	0.0871064
$682$	0	0
$683$	−29132.0	−1.63207	−0.816036	−	0.578001i	$-0.803833\pi$
$683$	−29132.0	−1.63207	−0.816036	+	0.578001i	$0.803833\pi$
$684$	0	0
$685$	−52680.0	−2.93839
$686$	0	0
$687$	8766.00	0.486818
$688$	0	0
$689$	3276.00	0.181140
$690$	0	0
$691$	20336.0	1.11956	0.559781	−	0.828640i	$-0.310885\pi$
$691$	20336.0	1.11956	0.559781	+	0.828640i	$0.310885\pi$
$692$	0	0
$693$	864.000	0.0473602
$694$	0	0
$695$	37280.0	2.03469
$696$	0	0
$697$	−4998.00	−0.271611
$698$	0	0
$699$	−9342.00	−0.505503
$700$	0	0
$701$	17142.0	0.923601	0.461801	−	0.886984i	$-0.347204\pi$
$701$	17142.0	0.923601	0.461801	+	0.886984i	$0.347204\pi$
$702$	0	0
$703$	−6992.00	−0.375118
$704$	0	0
$705$	720.000	0.0384635
$706$	0	0
$707$	−2420.00	−0.128732
$708$	0	0
$709$	22784.0	1.20687	0.603435	−	0.797412i	$-0.293798\pi$
$709$	22784.0	1.20687	0.603435	+	0.797412i	$0.293798\pi$
$710$	0	0
$711$	−7722.00	−0.407310
$712$	0	0
$713$	22204.0	1.16626
$714$	0	0
$715$	13440.0	0.702976
$716$	0	0
$717$	12372.0	0.644408
$718$	0	0
$719$	−58.0000	−0.00300839	−0.00150420	−	0.999999i	$-0.500479\pi$
$719$	−58.0000	−0.00300839	−0.00150420	+	0.999999i	$0.500479\pi$
$720$	0	0
$721$	−1288.00	−0.0665293
$722$	0	0
$723$	6102.00	0.313881
$724$	0	0
$725$	−9900.00	−0.507140
$726$	0	0
$727$	−712.000	−0.0363227	−0.0181614	−	0.999835i	$-0.505781\pi$
$727$	−712.000	−0.0363227	−0.0181614	+	0.999835i	$0.505781\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−7276.00	−0.368143
$732$	0	0
$733$	23050.0	1.16149	0.580744	−	0.814086i	$-0.302762\pi$
$733$	23050.0	1.16149	0.580744	+	0.814086i	$0.302762\pi$
$734$	0	0
$735$	−20340.0	−1.02075
$736$	0	0
$737$	−33600.0	−1.67934
$738$	0	0
$739$	38708.0	1.92679	0.963394	−	0.268088i	$-0.0863919\pi$
$739$	38708.0	1.92679	0.963394	+	0.268088i	$0.0863919\pi$
$740$	0	0
$741$	−3864.00	−0.191562
$742$	0	0
$743$	11034.0	0.544816	0.272408	−	0.962182i	$-0.412180\pi$
$743$	11034.0	0.544816	0.272408	+	0.962182i	$0.412180\pi$
$744$	0	0
$745$	5720.00	0.281295
$746$	0	0
$747$	−9468.00	−0.463743
$748$	0	0
$749$	512.000	0.0249774
$750$	0	0
$751$	−7502.00	−0.364516	−0.182258	−	0.983251i	$-0.558341\pi$
$751$	−7502.00	−0.364516	−0.182258	+	0.983251i	$0.558341\pi$
$752$	0	0
$753$	2988.00	0.144607
$754$	0	0
$755$	−32480.0	−1.56565
$756$	0	0
$757$	−20954.0	−1.00606	−0.503029	−	0.864269i	$-0.667781\pi$
$757$	−20954.0	−1.00606	−0.503029	+	0.864269i	$0.667781\pi$
$758$	0	0
$759$	−17568.0	−0.840155
$760$	0	0
$761$	8186.00	0.389937	0.194969	−	0.980809i	$-0.437539\pi$
$761$	8186.00	0.389937	0.194969	+	0.980809i	$0.437539\pi$
$762$	0	0
$763$	−2496.00	−0.118429
$764$	0	0
$765$	3060.00	0.144620
$766$	0	0
$767$	−7560.00	−0.355901
$768$	0	0
$769$	−5798.00	−0.271887	−0.135944	−	0.990717i	$-0.543407\pi$
$769$	−5798.00	−0.271887	−0.135944	+	0.990717i	$0.543407\pi$
$770$	0	0
$771$	9738.00	0.454871
$772$	0	0
$773$	−39950.0	−1.85886	−0.929432	−	0.368994i	$-0.879702\pi$
$773$	−39950.0	−1.85886	−0.929432	+	0.368994i	$0.879702\pi$
$774$	0	0
$775$	50050.0	2.31981
$776$	0	0
$777$	−456.000	−0.0210539
$778$	0	0
$779$	−27048.0	−1.24402
$780$	0	0
$781$	−38112.0	−1.74616
$782$	0	0
$783$	972.000	0.0443633
$784$	0	0
$785$	−50840.0	−2.31154
$786$	0	0
$787$	−20656.0	−0.935587	−0.467793	−	0.883838i	$-0.654951\pi$
$787$	−20656.0	−0.935587	−0.467793	+	0.883838i	$0.654951\pi$
$788$	0	0
$789$	2796.00	0.126160
$790$	0	0
$791$	−700.000	−0.0314654
$792$	0	0
$793$	11480.0	0.514082
$794$	0	0
$795$	−14040.0	−0.626349
$796$	0	0
$797$	−3722.00	−0.165420	−0.0827102	−	0.996574i	$-0.526358\pi$
$797$	−3722.00	−0.165420	−0.0827102	+	0.996574i	$0.526358\pi$
$798$	0	0
$799$	−204.000	−0.00903254
$800$	0	0
$801$	9918.00	0.437497
$802$	0	0
$803$	−49824.0	−2.18960
$804$	0	0
$805$	−4880.00	−0.213661
$806$	0	0
$807$	11652.0	0.508265
$808$	0	0
$809$	6518.00	0.283264	0.141632	−	0.989919i	$-0.454765\pi$
$809$	6518.00	0.283264	0.141632	+	0.989919i	$0.454765\pi$
$810$	0	0
$811$	33068.0	1.43178	0.715891	−	0.698212i	$-0.246021\pi$
$811$	33068.0	1.43178	0.715891	+	0.698212i	$0.246021\pi$
$812$	0	0
$813$	−5808.00	−0.250548
$814$	0	0
$815$	13680.0	0.587963
$816$	0	0
$817$	−39376.0	−1.68616
$818$	0	0
$819$	−252.000	−0.0107516
$820$	0	0
$821$	−5328.00	−0.226490	−0.113245	−	0.993567i	$-0.536125\pi$
$821$	−5328.00	−0.226490	−0.113245	+	0.993567i	$0.536125\pi$
$822$	0	0
$823$	−25874.0	−1.09588	−0.547941	−	0.836517i	$-0.684588\pi$
$823$	−25874.0	−1.09588	−0.547941	+	0.836517i	$0.684588\pi$
$824$	0	0
$825$	−39600.0	−1.67115
$826$	0	0
$827$	−29184.0	−1.22712	−0.613559	−	0.789649i	$-0.710263\pi$
$827$	−29184.0	−1.22712	−0.613559	+	0.789649i	$0.710263\pi$
$828$	0	0
$829$	−9394.00	−0.393567	−0.196784	−	0.980447i	$-0.563050\pi$
$829$	−9394.00	−0.393567	−0.196784	+	0.980447i	$0.563050\pi$
$830$	0	0
$831$	−2616.00	−0.109203
$832$	0	0
$833$	5763.00	0.239707
$834$	0	0
$835$	10840.0	0.449262
$836$	0	0
$837$	−4914.00	−0.202930
$838$	0	0
$839$	32062.0	1.31931	0.659656	−	0.751567i	$-0.270702\pi$
$839$	32062.0	1.31931	0.659656	+	0.751567i	$0.270702\pi$
$840$	0	0
$841$	−23093.0	−0.946861
$842$	0	0
$843$	−9594.00	−0.391975
$844$	0	0
$845$	40020.0	1.62927
$846$	0	0
$847$	1946.00	0.0789437
$848$	0	0
$849$	−5808.00	−0.234782
$850$	0	0
$851$	9272.00	0.373490
$852$	0	0
$853$	9152.00	0.367361	0.183680	−	0.982986i	$-0.441199\pi$
$853$	9152.00	0.367361	0.183680	+	0.982986i	$0.441199\pi$
$854$	0	0
$855$	16560.0	0.662386
$856$	0	0
$857$	−19706.0	−0.785466	−0.392733	−	0.919653i	$-0.628470\pi$
$857$	−19706.0	−0.785466	−0.392733	+	0.919653i	$0.628470\pi$
$858$	0	0
$859$	−28684.0	−1.13933	−0.569666	−	0.821877i	$-0.692927\pi$
$859$	−28684.0	−1.13933	−0.569666	+	0.821877i	$0.692927\pi$
$860$	0	0
$861$	−1764.00	−0.0698223
$862$	0	0
$863$	6320.00	0.249288	0.124644	−	0.992202i	$-0.460221\pi$
$863$	6320.00	0.249288	0.124644	+	0.992202i	$0.460221\pi$
$864$	0	0
$865$	16720.0	0.657222
$866$	0	0
$867$	−867.000	−0.0339618
$868$	0	0
$869$	−41184.0	−1.60768
$870$	0	0
$871$	9800.00	0.381240
$872$	0	0
$873$	6390.00	0.247731
$874$	0	0
$875$	−6000.00	−0.231814
$876$	0	0
$877$	49272.0	1.89715	0.948573	−	0.316558i	$-0.102527\pi$
$877$	49272.0	1.89715	0.948573	+	0.316558i	$0.102527\pi$
$878$	0	0
$879$	17154.0	0.658237
$880$	0	0
$881$	−16462.0	−0.629533	−0.314767	−	0.949169i	$-0.601926\pi$
$881$	−16462.0	−0.629533	−0.314767	+	0.949169i	$0.601926\pi$
$882$	0	0
$883$	−21300.0	−0.811780	−0.405890	−	0.913922i	$-0.633038\pi$
$883$	−21300.0	−0.811780	−0.405890	+	0.913922i	$0.633038\pi$
$884$	0	0
$885$	32400.0	1.23064
$886$	0	0
$887$	16590.0	0.628002	0.314001	−	0.949423i	$-0.398330\pi$
$887$	16590.0	0.628002	0.314001	+	0.949423i	$0.398330\pi$
$888$	0	0
$889$	656.000	0.0247486
$890$	0	0
$891$	3888.00	0.146187
$892$	0	0
$893$	−1104.00	−0.0413706
$894$	0	0
$895$	−68880.0	−2.57252
$896$	0	0
$897$	5124.00	0.190731
$898$	0	0
$899$	−6552.00	−0.243072
$900$	0	0
$901$	3978.00	0.147088
$902$	0	0
$903$	−2568.00	−0.0946375
$904$	0	0
$905$	65680.0	2.41246
$906$	0	0
$907$	36184.0	1.32466	0.662332	−	0.749211i	$-0.269567\pi$
$907$	36184.0	1.32466	0.662332	+	0.749211i	$0.269567\pi$
$908$	0	0
$909$	−10890.0	−0.397358
$910$	0	0
$911$	−15626.0	−0.568290	−0.284145	−	0.958781i	$-0.591710\pi$
$911$	−15626.0	−0.568290	−0.284145	+	0.958781i	$0.591710\pi$
$912$	0	0
$913$	−50496.0	−1.83042
$914$	0	0
$915$	−49200.0	−1.77760
$916$	0	0
$917$	3904.00	0.140590
$918$	0	0
$919$	−36672.0	−1.31632	−0.658160	−	0.752878i	$-0.728665\pi$
$919$	−36672.0	−1.31632	−0.658160	+	0.752878i	$0.728665\pi$
$920$	0	0
$921$	2052.00	0.0734155
$922$	0	0
$923$	11116.0	0.396411
$924$	0	0
$925$	20900.0	0.742906
$926$	0	0
$927$	−5796.00	−0.205357
$928$	0	0
$929$	−15810.0	−0.558352	−0.279176	−	0.960240i	$-0.590061\pi$
$929$	−15810.0	−0.558352	−0.279176	+	0.960240i	$0.590061\pi$
$930$	0	0
$931$	31188.0	1.09790
$932$	0	0
$933$	−870.000	−0.0305279
$934$	0	0
$935$	16320.0	0.570825
$936$	0	0
$937$	48646.0	1.69605	0.848023	−	0.529959i	$-0.177793\pi$
$937$	48646.0	1.69605	0.848023	+	0.529959i	$0.177793\pi$
$938$	0	0
$939$	33150.0	1.15209
$940$	0	0
$941$	43872.0	1.51986	0.759929	−	0.650006i	$-0.225234\pi$
$941$	43872.0	1.51986	0.759929	+	0.650006i	$0.225234\pi$
$942$	0	0
$943$	35868.0	1.23862
$944$	0	0
$945$	1080.00	0.0371771
$946$	0	0
$947$	−38552.0	−1.32288	−0.661442	−	0.749996i	$-0.730055\pi$
$947$	−38552.0	−1.32288	−0.661442	+	0.749996i	$0.730055\pi$
$948$	0	0
$949$	14532.0	0.497080
$950$	0	0
$951$	−2976.00	−0.101476
$952$	0	0
$953$	−52954.0	−1.79995	−0.899973	−	0.435946i	$-0.856414\pi$
$953$	−52954.0	−1.79995	−0.899973	+	0.435946i	$0.856414\pi$
$954$	0	0
$955$	−6800.00	−0.230411
$956$	0	0
$957$	5184.00	0.175104
$958$	0	0
$959$	5268.00	0.177385
$960$	0	0
$961$	3333.00	0.111879
$962$	0	0
$963$	2304.00	0.0770980
$964$	0	0
$965$	29960.0	0.999426
$966$	0	0
$967$	46428.0	1.54398	0.771988	−	0.635638i	$-0.219263\pi$
$967$	46428.0	1.54398	0.771988	+	0.635638i	$0.219263\pi$
$968$	0	0
$969$	−4692.00	−0.155551
$970$	0	0
$971$	40980.0	1.35439	0.677194	−	0.735804i	$-0.263196\pi$
$971$	40980.0	1.35439	0.677194	+	0.735804i	$0.263196\pi$
$972$	0	0
$973$	−3728.00	−0.122831
$974$	0	0
$975$	11550.0	0.379381
$976$	0	0
$977$	−10206.0	−0.334206	−0.167103	−	0.985939i	$-0.553441\pi$
$977$	−10206.0	−0.334206	−0.167103	+	0.985939i	$0.553441\pi$
$978$	0	0
$979$	52896.0	1.72683
$980$	0	0
$981$	−11232.0	−0.365556
$982$	0	0
$983$	44934.0	1.45796	0.728979	−	0.684536i	$-0.239995\pi$
$983$	44934.0	1.45796	0.728979	+	0.684536i	$0.239995\pi$
$984$	0	0
$985$	23520.0	0.760822
$986$	0	0
$987$	−72.0000	−0.00232197
$988$	0	0
$989$	52216.0	1.67884
$990$	0	0
$991$	−20526.0	−0.657951	−0.328976	−	0.944338i	$-0.606703\pi$
$991$	−20526.0	−0.657951	−0.328976	+	0.944338i	$0.606703\pi$
$992$	0	0
$993$	8580.00	0.274197
$994$	0	0
$995$	−49000.0	−1.56121
$996$	0	0
$997$	29260.0	0.929462	0.464731	−	0.885452i	$-0.346151\pi$
$997$	29260.0	0.929462	0.464731	+	0.885452i	$0.346151\pi$
$998$	0	0
$999$	−2052.00	−0.0649874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.4.a.a.1.1		1
3.2	odd	2		2448.4.a.r.1.1		1
4.3	odd	2		51.4.a.b.1.1	✓	1
12.11	even	2		153.4.a.c.1.1		1
20.19	odd	2		1275.4.a.e.1.1		1
28.27	even	2		2499.4.a.d.1.1		1
68.67	odd	2		867.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.b.1.1	✓	1	4.3	odd	2
153.4.a.c.1.1		1	12.11	even	2
816.4.a.a.1.1		1	1.1	even	1	trivial
867.4.a.c.1.1		1	68.67	odd	2
1275.4.a.e.1.1		1	20.19	odd	2
2448.4.a.r.1.1		1	3.2	odd	2
2499.4.a.d.1.1		1	28.27	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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	816.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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