LMFDB - Embedded newform 10000.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$10000 = 2^{4} \cdot 5^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1$ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.08529$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19849 q^{3} +0.992398 q^{7} +1.83337 q^{9} -2.00000 q^{11} +3.37406 q^{13} -2.89451 q^{17} +2.58448 q^{19} -2.18178 q^{21} -4.54963 q^{23} +2.56484 q^{27} -5.38430 q^{29} -0.136538 q^{31} +4.39698 q^{33} +2.14910 q^{37} -7.41785 q^{39} +8.63318 q^{41} +4.64398 q^{43} +9.92630 q^{47} -6.01515 q^{49} +6.36356 q^{51} -7.56521 q^{53} -5.68196 q^{57} -4.91775 q^{59} -2.76972 q^{61} +1.81943 q^{63} +2.18577 q^{67} +10.0023 q^{69} -9.64254 q^{71} -0.775929 q^{73} -1.98480 q^{77} -15.8508 q^{79} -11.1389 q^{81} -1.77110 q^{83} +11.8373 q^{87} +14.5080 q^{89} +3.34841 q^{91} +0.300177 q^{93} +17.0291 q^{97} -3.66673 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89}+ \cdots - 8 q^{99}+O(q^{100})$ 8 * q + 4 * q^9 - 16 * q^11 - 10 * q^19 + 6 * q^21 + 20 * q^29 - 16 * q^31 - 18 * q^39 + 26 * q^41 - 14 * q^49 + 4 * q^51 - 30 * q^59 + 6 * q^61 + 8 * q^69 - 46 * q^71 - 10 * q^79 - 32 * q^81 + 30 * q^89 + 14 * q^91 - 8 * 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
$3$	−2.19849	−1.26930	−0.634650	−	0.772800i	$-0.718856\pi$
$3$	−2.19849	−1.26930	−0.634650	+	0.772800i	$0.718856\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.992398	0.375091	0.187546	−	0.982256i	$-0.439947\pi$
$7$	0.992398	0.375091	0.187546	+	0.982256i	$0.439947\pi$
$8$	0	0
$9$	1.83337	0.611122
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	3.37406	0.935796	0.467898	−	0.883782i	$-0.345011\pi$
$13$	3.37406	0.935796	0.467898	+	0.883782i	$0.345011\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.89451	−0.702022	−0.351011	−	0.936371i	$-0.614162\pi$
$17$	−2.89451	−0.702022	−0.351011	+	0.936371i	$0.614162\pi$
$18$	0	0
$19$	2.58448	0.592921	0.296460	−	0.955045i	$-0.404194\pi$
$19$	2.58448	0.592921	0.296460	+	0.955045i	$0.404194\pi$
$20$	0	0
$21$	−2.18178	−0.476103
$22$	0	0
$23$	−4.54963	−0.948664	−0.474332	−	0.880346i	$-0.657310\pi$
$23$	−4.54963	−0.948664	−0.474332	+	0.880346i	$0.657310\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.56484	0.493603
$28$	0	0
$29$	−5.38430	−0.999839	−0.499919	−	0.866072i	$-0.666637\pi$
$29$	−5.38430	−0.999839	−0.499919	+	0.866072i	$0.666637\pi$
$30$	0	0
$31$	−0.136538	−0.0245229	−0.0122614	−	0.999925i	$-0.503903\pi$
$31$	−0.136538	−0.0245229	−0.0122614	+	0.999925i	$0.503903\pi$
$32$	0	0
$33$	4.39698	0.765417
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.14910	0.353311	0.176655	−	0.984273i	$-0.443472\pi$
$37$	2.14910	0.353311	0.176655	+	0.984273i	$0.443472\pi$
$38$	0	0
$39$	−7.41785	−1.18781
$40$	0	0
$41$	8.63318	1.34828	0.674138	−	0.738605i	$-0.264515\pi$
$41$	8.63318	1.34828	0.674138	+	0.738605i	$0.264515\pi$
$42$	0	0
$43$	4.64398	0.708200	0.354100	−	0.935208i	$-0.384787\pi$
$43$	4.64398	0.708200	0.354100	+	0.935208i	$0.384787\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.92630	1.44790	0.723950	−	0.689853i	$-0.242325\pi$
$47$	9.92630	1.44790	0.723950	+	0.689853i	$0.242325\pi$
$48$	0	0
$49$	−6.01515	−0.859306
$50$	0	0
$51$	6.36356	0.891077
$52$	0	0
$53$	−7.56521	−1.03916	−0.519581	−	0.854421i	$-0.673912\pi$
$53$	−7.56521	−1.03916	−0.519581	+	0.854421i	$0.673912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.68196	−0.752594
$58$	0	0
$59$	−4.91775	−0.640237	−0.320118	−	0.947378i	$-0.603723\pi$
$59$	−4.91775	−0.640237	−0.320118	+	0.947378i	$0.603723\pi$
$60$	0	0
$61$	−2.76972	−0.354626	−0.177313	−	0.984155i	$-0.556740\pi$
$61$	−2.76972	−0.354626	−0.177313	+	0.984155i	$0.556740\pi$
$62$	0	0
$63$	1.81943	0.229227
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.18577	0.267035	0.133517	−	0.991046i	$-0.457373\pi$
$67$	2.18577	0.267035	0.133517	+	0.991046i	$0.457373\pi$
$68$	0	0
$69$	10.0023	1.20414
$70$	0	0
$71$	−9.64254	−1.14436	−0.572179	−	0.820128i	$-0.693902\pi$
$71$	−9.64254	−1.14436	−0.572179	+	0.820128i	$0.693902\pi$
$72$	0	0
$73$	−0.775929	−0.0908157	−0.0454078	−	0.998969i	$-0.514459\pi$
$73$	−0.775929	−0.0908157	−0.0454078	+	0.998969i	$0.514459\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.98480	−0.226189
$78$	0	0
$79$	−15.8508	−1.78336	−0.891679	−	0.452667i	$-0.850473\pi$
$79$	−15.8508	−1.78336	−0.891679	+	0.452667i	$0.850473\pi$
$80$	0	0
$81$	−11.1389	−1.23765
$82$	0	0
$83$	−1.77110	−0.194404	−0.0972019	−	0.995265i	$-0.530989\pi$
$83$	−1.77110	−0.194404	−0.0972019	+	0.995265i	$0.530989\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.8373	1.26909
$88$	0	0
$89$	14.5080	1.53785	0.768923	−	0.639341i	$-0.220793\pi$
$89$	14.5080	1.53785	0.768923	+	0.639341i	$0.220793\pi$
$90$	0	0
$91$	3.34841	0.351009
$92$	0	0
$93$	0.300177	0.0311269
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.0291	1.72904	0.864522	−	0.502595i	$-0.167621\pi$
$97$	17.0291	1.72904	0.864522	+	0.502595i	$0.167621\pi$
$98$	0	0
$99$	−3.66673	−0.368520
$100$	0	0
$101$	−2.54716	−0.253452	−0.126726	−	0.991938i	$-0.540447\pi$
$101$	−2.54716	−0.253452	−0.126726	+	0.991938i	$0.540447\pi$
$102$	0	0
$103$	10.1654	1.00163	0.500815	−	0.865555i	$-0.333034\pi$
$103$	10.1654	1.00163	0.500815	+	0.865555i	$0.333034\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.81720	−0.465697	−0.232848	−	0.972513i	$-0.574805\pi$
$107$	−4.81720	−0.465697	−0.232848	+	0.972513i	$0.574805\pi$
$108$	0	0
$109$	16.2743	1.55879	0.779397	−	0.626531i	$-0.215526\pi$
$109$	16.2743	1.55879	0.779397	+	0.626531i	$0.215526\pi$
$110$	0	0
$111$	−4.72479	−0.448457
$112$	0	0
$113$	−6.75704	−0.635649	−0.317825	−	0.948150i	$-0.602952\pi$
$113$	−6.75704	−0.635649	−0.317825	+	0.948150i	$0.602952\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.18589	0.571886
$118$	0	0
$119$	−2.87251	−0.263322
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−18.9800	−1.71137
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.49081	−0.132288	−0.0661441	−	0.997810i	$-0.521070\pi$
$127$	−1.49081	−0.132288	−0.0661441	+	0.997810i	$0.521070\pi$
$128$	0	0
$129$	−10.2097	−0.898918
$130$	0	0
$131$	−14.1147	−1.23320	−0.616602	−	0.787275i	$-0.711491\pi$
$131$	−14.1147	−1.23320	−0.616602	+	0.787275i	$0.711491\pi$
$132$	0	0
$133$	2.56484	0.222399
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.689447	0.0589035	0.0294517	−	0.999566i	$-0.490624\pi$
$137$	0.689447	0.0589035	0.0294517	+	0.999566i	$0.490624\pi$
$138$	0	0
$139$	16.5719	1.40561	0.702803	−	0.711384i	$-0.251931\pi$
$139$	16.5719	1.40561	0.702803	+	0.711384i	$0.251931\pi$
$140$	0	0
$141$	−21.8229	−1.83782
$142$	0	0
$143$	−6.74812	−0.564307
$144$	0	0
$145$	0	0
$146$	0	0
$147$	13.2242	1.09072
$148$	0	0
$149$	3.21156	0.263101	0.131551	−	0.991309i	$-0.458004\pi$
$149$	3.21156	0.263101	0.131551	+	0.991309i	$0.458004\pi$
$150$	0	0
$151$	−17.6863	−1.43929	−0.719647	−	0.694340i	$-0.755696\pi$
$151$	−17.6863	−1.43929	−0.719647	+	0.694340i	$0.755696\pi$
$152$	0	0
$153$	−5.30670	−0.429021
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.65512	−0.132093	−0.0660465	−	0.997817i	$-0.521039\pi$
$157$	−1.65512	−0.132093	−0.0660465	+	0.997817i	$0.521039\pi$
$158$	0	0
$159$	16.6321	1.31901
$160$	0	0
$161$	−4.51505	−0.355836
$162$	0	0
$163$	0.892934	0.0699400	0.0349700	−	0.999388i	$-0.488866\pi$
$163$	0.892934	0.0699400	0.0349700	+	0.999388i	$0.488866\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.19558	0.402046	0.201023	−	0.979587i	$-0.435573\pi$
$167$	5.19558	0.402046	0.201023	+	0.979587i	$0.435573\pi$
$168$	0	0
$169$	−1.61570	−0.124285
$170$	0	0
$171$	4.73830	0.362347
$172$	0	0
$173$	5.76465	0.438278	0.219139	−	0.975694i	$-0.429675\pi$
$173$	5.76465	0.438278	0.219139	+	0.975694i	$0.429675\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.8116	0.812652
$178$	0	0
$179$	−8.66887	−0.647942	−0.323971	−	0.946067i	$-0.605018\pi$
$179$	−8.66887	−0.647942	−0.323971	+	0.946067i	$0.605018\pi$
$180$	0	0
$181$	14.2909	1.06223	0.531116	−	0.847299i	$-0.321773\pi$
$181$	14.2909	1.06223	0.531116	+	0.847299i	$0.321773\pi$
$182$	0	0
$183$	6.08920	0.450127
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.78902	0.423335
$188$	0	0
$189$	2.54534	0.185146
$190$	0	0
$191$	1.26636	0.0916305	0.0458153	−	0.998950i	$-0.485411\pi$
$191$	1.26636	0.0916305	0.0458153	+	0.998950i	$0.485411\pi$
$192$	0	0
$193$	−21.1730	−1.52406	−0.762031	−	0.647540i	$-0.775798\pi$
$193$	−21.1730	−1.52406	−0.762031	+	0.647540i	$0.775798\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.2013	0.869308	0.434654	−	0.900597i	$-0.356871\pi$
$197$	12.2013	0.869308	0.434654	+	0.900597i	$0.356871\pi$
$198$	0	0
$199$	−10.4065	−0.737695	−0.368848	−	0.929490i	$-0.620248\pi$
$199$	−10.4065	−0.737695	−0.368848	+	0.929490i	$0.620248\pi$
$200$	0	0
$201$	−4.80540	−0.338947
$202$	0	0
$203$	−5.34337	−0.375031
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.34114	−0.579749
$208$	0	0
$209$	−5.16896	−0.357545
$210$	0	0
$211$	8.65769	0.596020	0.298010	−	0.954563i	$-0.403677\pi$
$211$	8.65769	0.596020	0.298010	+	0.954563i	$0.403677\pi$
$212$	0	0
$213$	21.1990	1.45253
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.135500	−0.00919832
$218$	0	0
$219$	1.70587	0.115272
$220$	0	0
$221$	−9.76626	−0.656950
$222$	0	0
$223$	28.3434	1.89801	0.949007	−	0.315256i	$-0.102091\pi$
$223$	28.3434	1.89801	0.949007	+	0.315256i	$0.102091\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.2415	1.47622	0.738109	−	0.674682i	$-0.235719\pi$
$227$	22.2415	1.47622	0.738109	+	0.674682i	$0.235719\pi$
$228$	0	0
$229$	2.47559	0.163592	0.0817958	−	0.996649i	$-0.473934\pi$
$229$	2.47559	0.163592	0.0817958	+	0.996649i	$0.473934\pi$
$230$	0	0
$231$	4.36356	0.287101
$232$	0	0
$233$	−5.95605	−0.390194	−0.195097	−	0.980784i	$-0.562502\pi$
$233$	−5.95605	−0.390194	−0.195097	+	0.980784i	$0.562502\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	34.8479	2.26362
$238$	0	0
$239$	7.03243	0.454890	0.227445	−	0.973791i	$-0.426963\pi$
$239$	7.03243	0.454890	0.227445	+	0.973791i	$0.426963\pi$
$240$	0	0
$241$	1.17976	0.0759953	0.0379976	−	0.999278i	$-0.487902\pi$
$241$	1.17976	0.0759953	0.0379976	+	0.999278i	$0.487902\pi$
$242$	0	0
$243$	16.7942	1.07735
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.72020	0.554853
$248$	0	0
$249$	3.89375	0.246757
$250$	0	0
$251$	−4.60867	−0.290897	−0.145448	−	0.989366i	$-0.546462\pi$
$251$	−4.60867	−0.290897	−0.145448	+	0.989366i	$0.546462\pi$
$252$	0	0
$253$	9.09927	0.572066
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.75542	0.608526	0.304263	−	0.952588i	$-0.401590\pi$
$257$	9.75542	0.608526	0.304263	+	0.952588i	$0.401590\pi$
$258$	0	0
$259$	2.13277	0.132524
$260$	0	0
$261$	−9.87138	−0.611023
$262$	0	0
$263$	0.995828	0.0614054	0.0307027	−	0.999529i	$-0.490225\pi$
$263$	0.995828	0.0614054	0.0307027	+	0.999529i	$0.490225\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−31.8958	−1.95199
$268$	0	0
$269$	−3.28853	−0.200506	−0.100253	−	0.994962i	$-0.531965\pi$
$269$	−3.28853	−0.200506	−0.100253	+	0.994962i	$0.531965\pi$
$270$	0	0
$271$	−12.1500	−0.738063	−0.369031	−	0.929417i	$-0.620311\pi$
$271$	−12.1500	−0.738063	−0.369031	+	0.929417i	$0.620311\pi$
$272$	0	0
$273$	−7.36146	−0.445536
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.8666	0.712993	0.356496	−	0.934297i	$-0.383971\pi$
$277$	11.8666	0.712993	0.356496	+	0.934297i	$0.383971\pi$
$278$	0	0
$279$	−0.250324	−0.0149865
$280$	0	0
$281$	−24.6416	−1.47000	−0.734998	−	0.678070i	$-0.762817\pi$
$281$	−24.6416	−1.47000	−0.734998	+	0.678070i	$0.762817\pi$
$282$	0	0
$283$	−3.36343	−0.199935	−0.0999675	−	0.994991i	$-0.531874\pi$
$283$	−3.36343	−0.199935	−0.0999675	+	0.994991i	$0.531874\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.56755	0.505727
$288$	0	0
$289$	−8.62180	−0.507165
$290$	0	0
$291$	−37.4383	−2.19467
$292$	0	0
$293$	8.96340	0.523647	0.261824	−	0.965116i	$-0.415676\pi$
$293$	8.96340	0.523647	0.261824	+	0.965116i	$0.415676\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.12967	−0.297654
$298$	0	0
$299$	−15.3507	−0.887756
$300$	0	0
$301$	4.60867	0.265640
$302$	0	0
$303$	5.59991	0.321706
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.48133	0.541128	0.270564	−	0.962702i	$-0.412790\pi$
$307$	9.48133	0.541128	0.270564	+	0.962702i	$0.412790\pi$
$308$	0	0
$309$	−22.3486	−1.27137
$310$	0	0
$311$	−29.3320	−1.66327	−0.831633	−	0.555325i	$-0.812594\pi$
$311$	−29.3320	−1.66327	−0.831633	+	0.555325i	$0.812594\pi$
$312$	0	0
$313$	−18.8901	−1.06773	−0.533865	−	0.845570i	$-0.679261\pi$
$313$	−18.8901	−1.06773	−0.533865	+	0.845570i	$0.679261\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.7893	−1.27998	−0.639988	−	0.768385i	$-0.721061\pi$
$317$	−22.7893	−1.27998	−0.639988	+	0.768385i	$0.721061\pi$
$318$	0	0
$319$	10.7686	0.602925
$320$	0	0
$321$	10.5906	0.591109
$322$	0	0
$323$	−7.48081	−0.416244
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−35.7789	−1.97858
$328$	0	0
$329$	9.85084	0.543094
$330$	0	0
$331$	−2.96299	−0.162861	−0.0814304	−	0.996679i	$-0.525949\pi$
$331$	−2.96299	−0.162861	−0.0814304	+	0.996679i	$0.525949\pi$
$332$	0	0
$333$	3.94010	0.215916
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.8123	1.02477	0.512385	−	0.858756i	$-0.328762\pi$
$337$	18.8123	1.02477	0.512385	+	0.858756i	$0.328762\pi$
$338$	0	0
$339$	14.8553	0.806829
$340$	0	0
$341$	0.273075	0.0147879
$342$	0	0
$343$	−12.9162	−0.697410
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.7382	−1.22065	−0.610325	−	0.792151i	$-0.708961\pi$
$347$	−22.7382	−1.22065	−0.610325	+	0.792151i	$0.708961\pi$
$348$	0	0
$349$	1.93849	0.103765	0.0518824	−	0.998653i	$-0.483478\pi$
$349$	1.93849	0.103765	0.0518824	+	0.998653i	$0.483478\pi$
$350$	0	0
$351$	8.65392	0.461912
$352$	0	0
$353$	−5.24945	−0.279400	−0.139700	−	0.990194i	$-0.544614\pi$
$353$	−5.24945	−0.279400	−0.139700	+	0.990194i	$0.544614\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.31519	0.334235
$358$	0	0
$359$	−22.5937	−1.19245	−0.596226	−	0.802817i	$-0.703334\pi$
$359$	−22.5937	−1.19245	−0.596226	+	0.802817i	$0.703334\pi$
$360$	0	0
$361$	−12.3205	−0.648445
$362$	0	0
$363$	15.3894	0.807736
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.29872	−0.380990	−0.190495	−	0.981688i	$-0.561009\pi$
$367$	−7.29872	−0.380990	−0.190495	+	0.981688i	$0.561009\pi$
$368$	0	0
$369$	15.8278	0.823961
$370$	0	0
$371$	−7.50770	−0.389781
$372$	0	0
$373$	22.3074	1.15503	0.577516	−	0.816380i	$-0.304022\pi$
$373$	22.3074	1.15503	0.577516	+	0.816380i	$0.304022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.1669	−0.935645
$378$	0	0
$379$	32.9466	1.69235	0.846177	−	0.532903i	$-0.178899\pi$
$379$	32.9466	1.69235	0.846177	+	0.532903i	$0.178899\pi$
$380$	0	0
$381$	3.27754	0.167913
$382$	0	0
$383$	−20.7002	−1.05773	−0.528865	−	0.848706i	$-0.677382\pi$
$383$	−20.7002	−1.05773	−0.528865	+	0.848706i	$0.677382\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.51410	0.432796
$388$	0	0
$389$	−1.14446	−0.0580263	−0.0290132	−	0.999579i	$-0.509236\pi$
$389$	−1.14446	−0.0580263	−0.0290132	+	0.999579i	$0.509236\pi$
$390$	0	0
$391$	13.1690	0.665983
$392$	0	0
$393$	31.0310	1.56531
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.68513	0.235140	0.117570	−	0.993065i	$-0.462490\pi$
$397$	4.68513	0.235140	0.117570	+	0.993065i	$0.462490\pi$
$398$	0	0
$399$	−5.63877	−0.282292
$400$	0	0
$401$	−24.0851	−1.20275	−0.601376	−	0.798966i	$-0.705381\pi$
$401$	−24.0851	−1.20275	−0.601376	+	0.798966i	$0.705381\pi$
$402$	0	0
$403$	−0.460687	−0.0229484
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.29821	−0.213054
$408$	0	0
$409$	−1.89934	−0.0939165	−0.0469583	−	0.998897i	$-0.514953\pi$
$409$	−1.89934	−0.0939165	−0.0469583	+	0.998897i	$0.514953\pi$
$410$	0	0
$411$	−1.51574	−0.0747661
$412$	0	0
$413$	−4.88037	−0.240147
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−36.4331	−1.78414
$418$	0	0
$419$	2.32806	0.113733	0.0568666	−	0.998382i	$-0.481889\pi$
$419$	2.32806	0.113733	0.0568666	+	0.998382i	$0.481889\pi$
$420$	0	0
$421$	−23.9501	−1.16725	−0.583627	−	0.812022i	$-0.698367\pi$
$421$	−23.9501	−1.16725	−0.583627	+	0.812022i	$0.698367\pi$
$422$	0	0
$423$	18.1985	0.884843
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.74866	−0.133017
$428$	0	0
$429$	14.8357	0.716274
$430$	0	0
$431$	−1.19227	−0.0574294	−0.0287147	−	0.999588i	$-0.509141\pi$
$431$	−1.19227	−0.0574294	−0.0287147	+	0.999588i	$0.509141\pi$
$432$	0	0
$433$	−25.6138	−1.23092	−0.615461	−	0.788167i	$-0.711030\pi$
$433$	−25.6138	−1.23092	−0.615461	+	0.788167i	$0.711030\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.7584	−0.562483
$438$	0	0
$439$	−19.3741	−0.924676	−0.462338	−	0.886704i	$-0.652989\pi$
$439$	−19.3741	−0.924676	−0.462338	+	0.886704i	$0.652989\pi$
$440$	0	0
$441$	−11.0280	−0.525141
$442$	0	0
$443$	−2.46263	−0.117003	−0.0585016	−	0.998287i	$-0.518632\pi$
$443$	−2.46263	−0.117003	−0.0585016	+	0.998287i	$0.518632\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.06059	−0.333955
$448$	0	0
$449$	−14.3585	−0.677618	−0.338809	−	0.940855i	$-0.610024\pi$
$449$	−14.3585	−0.677618	−0.338809	+	0.940855i	$0.610024\pi$
$450$	0	0
$451$	−17.2664	−0.813041
$452$	0	0
$453$	38.8833	1.82690
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.1964	−1.17864	−0.589319	−	0.807901i	$-0.700604\pi$
$457$	−25.1964	−1.17864	−0.589319	+	0.807901i	$0.700604\pi$
$458$	0	0
$459$	−7.42395	−0.346520
$460$	0	0
$461$	28.8255	1.34254	0.671269	−	0.741214i	$-0.265749\pi$
$461$	28.8255	1.34254	0.671269	+	0.741214i	$0.265749\pi$
$462$	0	0
$463$	−31.8796	−1.48157	−0.740786	−	0.671742i	$-0.765547\pi$
$463$	−31.8796	−1.48157	−0.740786	+	0.671742i	$0.765547\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−43.0996	−1.99441	−0.997206	−	0.0747039i	$-0.976199\pi$
$467$	−43.0996	−1.99441	−0.997206	+	0.0747039i	$0.976199\pi$
$468$	0	0
$469$	2.16916	0.100162
$470$	0	0
$471$	3.63877	0.167666
$472$	0	0
$473$	−9.28795	−0.427060
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.8698	−0.635054
$478$	0	0
$479$	−21.0263	−0.960717	−0.480359	−	0.877072i	$-0.659494\pi$
$479$	−21.0263	−0.960717	−0.480359	+	0.877072i	$0.659494\pi$
$480$	0	0
$481$	7.25121	0.330627
$482$	0	0
$483$	9.92630	0.451662
$484$	0	0
$485$	0	0
$486$	0	0
$487$	27.9190	1.26513	0.632565	−	0.774507i	$-0.282002\pi$
$487$	27.9190	1.26513	0.632565	+	0.774507i	$0.282002\pi$
$488$	0	0
$489$	−1.96311	−0.0887748
$490$	0	0
$491$	−14.9611	−0.675183	−0.337591	−	0.941293i	$-0.609612\pi$
$491$	−14.9611	−0.675183	−0.337591	+	0.941293i	$0.609612\pi$
$492$	0	0
$493$	15.5849	0.701909
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.56924	−0.429239
$498$	0	0
$499$	44.3253	1.98427	0.992137	−	0.125160i	$-0.0399443\pi$
$499$	44.3253	1.98427	0.992137	+	0.125160i	$0.0399443\pi$
$500$	0	0
$501$	−11.4224	−0.510316
$502$	0	0
$503$	−23.6212	−1.05322	−0.526609	−	0.850108i	$-0.676537\pi$
$503$	−23.6212	−1.05322	−0.526609	+	0.850108i	$0.676537\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.55211	0.157755
$508$	0	0
$509$	−26.7154	−1.18414	−0.592070	−	0.805886i	$-0.701689\pi$
$509$	−26.7154	−1.18414	−0.592070	+	0.805886i	$0.701689\pi$
$510$	0	0
$511$	−0.770031	−0.0340642
$512$	0	0
$513$	6.62877	0.292667
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.8526	−0.873116
$518$	0	0
$519$	−12.6735	−0.556306
$520$	0	0
$521$	32.7073	1.43293	0.716466	−	0.697622i	$-0.245759\pi$
$521$	32.7073	1.43293	0.716466	+	0.697622i	$0.245759\pi$
$522$	0	0
$523$	0.235966	0.0103181	0.00515904	−	0.999987i	$-0.498358\pi$
$523$	0.235966	0.0103181	0.00515904	+	0.999987i	$0.498358\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.395210	0.0172156
$528$	0	0
$529$	−2.30084	−0.100037
$530$	0	0
$531$	−9.01604	−0.391263
$532$	0	0
$533$	29.1289	1.26171
$534$	0	0
$535$	0	0
$536$	0	0
$537$	19.0584	0.822432
$538$	0	0
$539$	12.0303	0.518181
$540$	0	0
$541$	−33.5572	−1.44274	−0.721369	−	0.692551i	$-0.756487\pi$
$541$	−33.5572	−1.44274	−0.721369	+	0.692551i	$0.756487\pi$
$542$	0	0
$543$	−31.4183	−1.34829
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.5125	1.64668	0.823338	−	0.567552i	$-0.192109\pi$
$547$	38.5125	1.64668	0.823338	+	0.567552i	$0.192109\pi$
$548$	0	0
$549$	−5.07790	−0.216720
$550$	0	0
$551$	−13.9156	−0.592825
$552$	0	0
$553$	−15.7303	−0.668922
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.33445	−0.183657	−0.0918283	−	0.995775i	$-0.529271\pi$
$557$	−4.33445	−0.183657	−0.0918283	+	0.995775i	$0.529271\pi$
$558$	0	0
$559$	15.6691	0.662731
$560$	0	0
$561$	−12.7271	−0.537339
$562$	0	0
$563$	−34.9018	−1.47094	−0.735468	−	0.677559i	$-0.763038\pi$
$563$	−34.9018	−1.47094	−0.735468	+	0.677559i	$0.763038\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−11.0542	−0.464233
$568$	0	0
$569$	−41.9646	−1.75925	−0.879623	−	0.475671i	$-0.842205\pi$
$569$	−41.9646	−1.75925	−0.879623	+	0.475671i	$0.842205\pi$
$570$	0	0
$571$	13.8332	0.578900	0.289450	−	0.957193i	$-0.406528\pi$
$571$	13.8332	0.578900	0.289450	+	0.957193i	$0.406528\pi$
$572$	0	0
$573$	−2.78408	−0.116307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−12.7793	−0.532008	−0.266004	−	0.963972i	$-0.585704\pi$
$577$	−12.7793	−0.532008	−0.266004	+	0.963972i	$0.585704\pi$
$578$	0	0
$579$	46.5486	1.93449
$580$	0	0
$581$	−1.75764	−0.0729191
$582$	0	0
$583$	15.1304	0.626638
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.1870	0.503009	0.251505	−	0.967856i	$-0.419075\pi$
$587$	12.1870	0.503009	0.251505	+	0.967856i	$0.419075\pi$
$588$	0	0
$589$	−0.352879	−0.0145401
$590$	0	0
$591$	−26.8245	−1.10341
$592$	0	0
$593$	−31.2580	−1.28361	−0.641807	−	0.766866i	$-0.721815\pi$
$593$	−31.2580	−1.28361	−0.641807	+	0.766866i	$0.721815\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.8785	0.936356
$598$	0	0
$599$	33.3707	1.36349	0.681746	−	0.731589i	$-0.261221\pi$
$599$	33.3707	1.36349	0.681746	+	0.731589i	$0.261221\pi$
$600$	0	0
$601$	−46.8052	−1.90922	−0.954611	−	0.297854i	$-0.903729\pi$
$601$	−46.8052	−1.90922	−0.954611	+	0.297854i	$0.903729\pi$
$602$	0	0
$603$	4.00732	0.163191
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−30.7401	−1.24770	−0.623851	−	0.781543i	$-0.714433\pi$
$607$	−30.7401	−1.24770	−0.623851	+	0.781543i	$0.714433\pi$
$608$	0	0
$609$	11.7473	0.476027
$610$	0	0
$611$	33.4919	1.35494
$612$	0	0
$613$	−38.2895	−1.54650	−0.773248	−	0.634103i	$-0.781369\pi$
$613$	−38.2895	−1.54650	−0.773248	+	0.634103i	$0.781369\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.425306	0.0171222	0.00856109	−	0.999963i	$-0.497275\pi$
$617$	0.425306	0.0171222	0.00856109	+	0.999963i	$0.497275\pi$
$618$	0	0
$619$	−7.51147	−0.301912	−0.150956	−	0.988541i	$-0.548235\pi$
$619$	−7.51147	−0.301912	−0.150956	+	0.988541i	$0.548235\pi$
$620$	0	0
$621$	−11.6691	−0.468263
$622$	0	0
$623$	14.3977	0.576833
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.3639	0.453831
$628$	0	0
$629$	−6.22061	−0.248032
$630$	0	0
$631$	11.2716	0.448714	0.224357	−	0.974507i	$-0.427972\pi$
$631$	11.2716	0.448714	0.224357	+	0.974507i	$0.427972\pi$
$632$	0	0
$633$	−19.0338	−0.756528
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−20.2955	−0.804136
$638$	0	0
$639$	−17.6783	−0.699343
$640$	0	0
$641$	26.0825	1.03020	0.515099	−	0.857131i	$-0.327755\pi$
$641$	26.0825	1.03020	0.515099	+	0.857131i	$0.327755\pi$
$642$	0	0
$643$	31.9492	1.25995	0.629977	−	0.776614i	$-0.283064\pi$
$643$	31.9492	1.25995	0.629977	+	0.776614i	$0.283064\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.39433	0.290701	0.145351	−	0.989380i	$-0.453569\pi$
$647$	7.39433	0.290701	0.145351	+	0.989380i	$0.453569\pi$
$648$	0	0
$649$	9.83550	0.386077
$650$	0	0
$651$	0.297895	0.0116754
$652$	0	0
$653$	18.6853	0.731212	0.365606	−	0.930770i	$-0.380862\pi$
$653$	18.6853	0.731212	0.365606	+	0.930770i	$0.380862\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.42256	−0.0554994
$658$	0	0
$659$	−9.80157	−0.381815	−0.190907	−	0.981608i	$-0.561143\pi$
$659$	−9.80157	−0.381815	−0.190907	+	0.981608i	$0.561143\pi$
$660$	0	0
$661$	−28.1585	−1.09524	−0.547619	−	0.836728i	$-0.684466\pi$
$661$	−28.1585	−1.09524	−0.547619	+	0.836728i	$0.684466\pi$
$662$	0	0
$663$	21.4710	0.833866
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.4966	0.948511
$668$	0	0
$669$	−62.3127	−2.40915
$670$	0	0
$671$	5.53943	0.213847
$672$	0	0
$673$	−39.0253	−1.50432	−0.752158	−	0.658983i	$-0.770987\pi$
$673$	−39.0253	−1.50432	−0.752158	+	0.658983i	$0.770987\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.03533	0.193523	0.0967617	−	0.995308i	$-0.469152\pi$
$677$	5.03533	0.193523	0.0967617	+	0.995308i	$0.469152\pi$
$678$	0	0
$679$	16.8997	0.648549
$680$	0	0
$681$	−48.8977	−1.87376
$682$	0	0
$683$	30.3312	1.16059	0.580295	−	0.814406i	$-0.302937\pi$
$683$	30.3312	1.16059	0.580295	+	0.814406i	$0.302937\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−5.44257	−0.207647
$688$	0	0
$689$	−25.5255	−0.972444
$690$	0	0
$691$	21.2329	0.807739	0.403869	−	0.914817i	$-0.367665\pi$
$691$	21.2329	0.807739	0.403869	+	0.914817i	$0.367665\pi$
$692$	0	0
$693$	−3.63886	−0.138229
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−24.9888	−0.946520
$698$	0	0
$699$	13.0943	0.495273
$700$	0	0
$701$	32.7698	1.23770	0.618849	−	0.785510i	$-0.287599\pi$
$701$	32.7698	1.23770	0.618849	+	0.785510i	$0.287599\pi$
$702$	0	0
$703$	5.55432	0.209485
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.52780	−0.0950676
$708$	0	0
$709$	19.8459	0.745330	0.372665	−	0.927966i	$-0.378444\pi$
$709$	19.8459	0.745330	0.372665	+	0.927966i	$0.378444\pi$
$710$	0	0
$711$	−29.0604	−1.08985
$712$	0	0
$713$	0.621196	0.0232640
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.4607	−0.577392
$718$	0	0
$719$	−24.2201	−0.903258	−0.451629	−	0.892206i	$-0.649157\pi$
$719$	−24.2201	−0.903258	−0.451629	+	0.892206i	$0.649157\pi$
$720$	0	0
$721$	10.0882	0.375703
$722$	0	0
$723$	−2.59370	−0.0964608
$724$	0	0
$725$	0	0
$726$	0	0
$727$	5.86510	0.217524	0.108762	−	0.994068i	$-0.465311\pi$
$727$	5.86510	0.217524	0.108762	+	0.994068i	$0.465311\pi$
$728$	0	0
$729$	−3.50531	−0.129826
$730$	0	0
$731$	−13.4420	−0.497172
$732$	0	0
$733$	−34.5015	−1.27434	−0.637171	−	0.770723i	$-0.719895\pi$
$733$	−34.5015	−1.27434	−0.637171	+	0.770723i	$0.719895\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.37155	−0.161028
$738$	0	0
$739$	−39.5712	−1.45565	−0.727826	−	0.685762i	$-0.759469\pi$
$739$	−39.5712	−1.45565	−0.727826	+	0.685762i	$0.759469\pi$
$740$	0	0
$741$	−19.1713	−0.704275
$742$	0	0
$743$	−29.7058	−1.08980	−0.544900	−	0.838501i	$-0.683433\pi$
$743$	−29.7058	−1.08980	−0.544900	+	0.838501i	$0.683433\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3.24708	−0.118804
$748$	0	0
$749$	−4.78058	−0.174679
$750$	0	0
$751$	−26.8870	−0.981122	−0.490561	−	0.871407i	$-0.663208\pi$
$751$	−26.8870	−0.981122	−0.490561	+	0.871407i	$0.663208\pi$
$752$	0	0
$753$	10.1321	0.369235
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−44.6792	−1.62389	−0.811947	−	0.583731i	$-0.801592\pi$
$757$	−44.6792	−1.62389	−0.811947	+	0.583731i	$0.801592\pi$
$758$	0	0
$759$	−20.0047	−0.726123
$760$	0	0
$761$	20.3080	0.736163	0.368081	−	0.929794i	$-0.380015\pi$
$761$	20.3080	0.736163	0.368081	+	0.929794i	$0.380015\pi$
$762$	0	0
$763$	16.1506	0.584690
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.5928	−0.599131
$768$	0	0
$769$	26.0577	0.939665	0.469832	−	0.882756i	$-0.344314\pi$
$769$	26.0577	0.939665	0.469832	+	0.882756i	$0.344314\pi$
$770$	0	0
$771$	−21.4472	−0.772402
$772$	0	0
$773$	−13.7305	−0.493851	−0.246926	−	0.969034i	$-0.579420\pi$
$773$	−13.7305	−0.493851	−0.246926	+	0.969034i	$0.579420\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.68887	−0.168212
$778$	0	0
$779$	22.3123	0.799421
$780$	0	0
$781$	19.2851	0.690074
$782$	0	0
$783$	−13.8098	−0.493523
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−19.6660	−0.701018	−0.350509	−	0.936559i	$-0.613991\pi$
$787$	−19.6660	−0.701018	−0.350509	+	0.936559i	$0.613991\pi$
$788$	0	0
$789$	−2.18932	−0.0779418
$790$	0	0
$791$	−6.70568	−0.238427
$792$	0	0
$793$	−9.34520	−0.331858
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.9441	−0.387660	−0.193830	−	0.981035i	$-0.562091\pi$
$797$	−10.9441	−0.387660	−0.193830	+	0.981035i	$0.562091\pi$
$798$	0	0
$799$	−28.7318	−1.01646
$800$	0	0
$801$	26.5985	0.939812
$802$	0	0
$803$	1.55186	0.0547639
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.22982	0.254502
$808$	0	0
$809$	16.4427	0.578096	0.289048	−	0.957315i	$-0.406661\pi$
$809$	16.4427	0.578096	0.289048	+	0.957315i	$0.406661\pi$
$810$	0	0
$811$	−22.6473	−0.795253	−0.397627	−	0.917547i	$-0.630166\pi$
$811$	−22.6473	−0.795253	−0.397627	+	0.917547i	$0.630166\pi$
$812$	0	0
$813$	26.7118	0.936823
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.0023	0.419906
$818$	0	0
$819$	6.13887	0.214509
$820$	0	0
$821$	−39.7792	−1.38830	−0.694151	−	0.719829i	$-0.744220\pi$
$821$	−39.7792	−1.38830	−0.694151	+	0.719829i	$0.744220\pi$
$822$	0	0
$823$	16.5602	0.577252	0.288626	−	0.957442i	$-0.406802\pi$
$823$	16.5602	0.577252	0.288626	+	0.957442i	$0.406802\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.0361	−0.905365	−0.452683	−	0.891672i	$-0.649533\pi$
$827$	−26.0361	−0.905365	−0.452683	+	0.891672i	$0.649533\pi$
$828$	0	0
$829$	5.14357	0.178644	0.0893218	−	0.996003i	$-0.471530\pi$
$829$	5.14357	0.178644	0.0893218	+	0.996003i	$0.471530\pi$
$830$	0	0
$831$	−26.0885	−0.905002
$832$	0	0
$833$	17.4109	0.603252
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−0.350197	−0.0121046
$838$	0	0
$839$	−38.5664	−1.33146	−0.665730	−	0.746193i	$-0.731880\pi$
$839$	−38.5664	−1.33146	−0.665730	+	0.746193i	$0.731880\pi$
$840$	0	0
$841$	−0.00936035	−0.000322771 0
$842$	0	0
$843$	54.1744	1.86586
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.94679	−0.238694
$848$	0	0
$849$	7.39447	0.253777
$850$	0	0
$851$	−9.77764	−0.335173
$852$	0	0
$853$	−9.14763	−0.313209	−0.156604	−	0.987661i	$-0.550055\pi$
$853$	−9.14763	−0.313209	−0.156604	+	0.987661i	$0.550055\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.6712	0.466998	0.233499	−	0.972357i	$-0.424982\pi$
$857$	13.6712	0.466998	0.233499	+	0.972357i	$0.424982\pi$
$858$	0	0
$859$	35.6556	1.21655	0.608277	−	0.793725i	$-0.291861\pi$
$859$	35.6556	1.21655	0.608277	+	0.793725i	$0.291861\pi$
$860$	0	0
$861$	−18.8357	−0.641919
$862$	0	0
$863$	33.9333	1.15510	0.577552	−	0.816354i	$-0.304008\pi$
$863$	33.9333	1.15510	0.577552	+	0.816354i	$0.304008\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18.9550	0.643744
$868$	0	0
$869$	31.7017	1.07541
$870$	0	0
$871$	7.37494	0.249890
$872$	0	0
$873$	31.2206	1.05666
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.6991	−0.969099	−0.484550	−	0.874764i	$-0.661017\pi$
$877$	−28.6991	−0.969099	−0.484550	+	0.874764i	$0.661017\pi$
$878$	0	0
$879$	−19.7060	−0.664665
$880$	0	0
$881$	8.33039	0.280658	0.140329	−	0.990105i	$-0.455184\pi$
$881$	8.33039	0.280658	0.140329	+	0.990105i	$0.455184\pi$
$882$	0	0
$883$	−50.3165	−1.69329	−0.846643	−	0.532161i	$-0.821380\pi$
$883$	−50.3165	−1.69329	−0.846643	+	0.532161i	$0.821380\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.1186	0.406903	0.203452	−	0.979085i	$-0.434784\pi$
$887$	12.1186	0.406903	0.203452	+	0.979085i	$0.434784\pi$
$888$	0	0
$889$	−1.47948	−0.0496202
$890$	0	0
$891$	22.2777	0.746332
$892$	0	0
$893$	25.6543	0.858490
$894$	0	0
$895$	0	0
$896$	0	0
$897$	33.7485	1.12683
$898$	0	0
$899$	0.735159	0.0245189
$900$	0	0
$901$	21.8976	0.729514
$902$	0	0
$903$	−10.1321	−0.337176
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−31.9105	−1.05957	−0.529786	−	0.848132i	$-0.677728\pi$
$907$	−31.9105	−1.05957	−0.529786	+	0.848132i	$0.677728\pi$
$908$	0	0
$909$	−4.66988	−0.154890
$910$	0	0
$911$	24.6880	0.817949	0.408975	−	0.912546i	$-0.365886\pi$
$911$	24.6880	0.817949	0.408975	+	0.912546i	$0.365886\pi$
$912$	0	0
$913$	3.54220	0.117230
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14.0074	−0.462565
$918$	0	0
$919$	19.4850	0.642752	0.321376	−	0.946952i	$-0.395855\pi$
$919$	19.4850	0.642752	0.321376	+	0.946952i	$0.395855\pi$
$920$	0	0
$921$	−20.8446	−0.686854
$922$	0	0
$923$	−32.5345	−1.07089
$924$	0	0
$925$	0	0
$926$	0	0
$927$	18.6369	0.612118
$928$	0	0
$929$	11.7642	0.385972	0.192986	−	0.981201i	$-0.438183\pi$
$929$	11.7642	0.385972	0.192986	+	0.981201i	$0.438183\pi$
$930$	0	0
$931$	−15.5460	−0.509501
$932$	0	0
$933$	64.4862	2.11118
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.0683	0.688271	0.344136	−	0.938920i	$-0.388172\pi$
$937$	21.0683	0.688271	0.344136	+	0.938920i	$0.388172\pi$
$938$	0	0
$939$	41.5296	1.35527
$940$	0	0
$941$	−2.24706	−0.0732521	−0.0366261	−	0.999329i	$-0.511661\pi$
$941$	−2.24706	−0.0732521	−0.0366261	+	0.999329i	$0.511661\pi$
$942$	0	0
$943$	−39.2778	−1.27906
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.75625	−0.219549	−0.109774	−	0.993957i	$-0.535013\pi$
$947$	−6.75625	−0.219549	−0.109774	+	0.993957i	$0.535013\pi$
$948$	0	0
$949$	−2.61803	−0.0849850
$950$	0	0
$951$	50.1021	1.62467
$952$	0	0
$953$	−59.9534	−1.94208	−0.971040	−	0.238918i	$-0.923207\pi$
$953$	−59.9534	−1.94208	−0.971040	+	0.238918i	$0.923207\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−23.6747	−0.765293
$958$	0	0
$959$	0.684206	0.0220942
$960$	0	0
$961$	−30.9814	−0.999399
$962$	0	0
$963$	−8.83169	−0.284597
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−9.05599	−0.291221	−0.145610	−	0.989342i	$-0.546515\pi$
$967$	−9.05599	−0.291221	−0.145610	+	0.989342i	$0.546515\pi$
$968$	0	0
$969$	16.4465	0.528338
$970$	0	0
$971$	−47.3508	−1.51956	−0.759780	−	0.650180i	$-0.774693\pi$
$971$	−47.3508	−1.51956	−0.759780	+	0.650180i	$0.774693\pi$
$972$	0	0
$973$	16.4459	0.527231
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.74467	0.151795	0.0758977	−	0.997116i	$-0.475818\pi$
$977$	4.74467	0.151795	0.0758977	+	0.997116i	$0.475818\pi$
$978$	0	0
$979$	−29.0160	−0.927357
$980$	0	0
$981$	29.8367	0.952613
$982$	0	0
$983$	18.5656	0.592150	0.296075	−	0.955165i	$-0.404322\pi$
$983$	18.5656	0.592150	0.296075	+	0.955165i	$0.404322\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−21.6570	−0.689350
$988$	0	0
$989$	−21.1284	−0.671843
$990$	0	0
$991$	39.7199	1.26174	0.630871	−	0.775887i	$-0.282698\pi$
$991$	39.7199	1.26174	0.630871	+	0.775887i	$0.282698\pi$
$992$	0	0
$993$	6.51411	0.206719
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−30.2914	−0.959338	−0.479669	−	0.877449i	$-0.659243\pi$
$997$	−30.2914	−0.959338	−0.479669	+	0.877449i	$0.659243\pi$
$998$	0	0
$999$	5.51210	0.174395

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bj.1.2		8
4.3	odd	2		625.2.a.f.1.7		8
5.4	even	2	inner	10000.2.a.bj.1.7		8
12.11	even	2		5625.2.a.x.1.2		8
20.3	even	4		625.2.b.c.624.2		8
20.7	even	4		625.2.b.c.624.7		8
20.19	odd	2		625.2.a.f.1.2		8
25.12	odd	20		400.2.y.c.369.1		8
25.23	odd	20		400.2.y.c.129.1		8
60.59	even	2		5625.2.a.x.1.7		8
100.3	even	20		625.2.e.a.249.1		8
100.11	odd	10		125.2.d.b.101.4		16
100.19	odd	10		625.2.d.o.251.4		16
100.23	even	20		25.2.e.a.4.1	✓	8
100.27	even	20		125.2.e.b.24.2		8
100.31	odd	10		625.2.d.o.251.1		16
100.39	odd	10		125.2.d.b.101.1		16
100.47	even	20		625.2.e.i.249.2		8
100.59	odd	10		125.2.d.b.26.1		16
100.63	even	20		125.2.e.b.99.2		8
100.67	even	20		625.2.e.a.374.1		8
100.71	odd	10		625.2.d.o.376.1		16
100.79	odd	10		625.2.d.o.376.4		16
100.83	even	20		625.2.e.i.374.2		8
100.87	even	20		25.2.e.a.19.1	yes	8
100.91	odd	10		125.2.d.b.26.4		16
300.23	odd	20		225.2.m.a.154.2		8
300.287	odd	20		225.2.m.a.19.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.4.1	✓	8	100.23	even	20
25.2.e.a.19.1	yes	8	100.87	even	20
125.2.d.b.26.1		16	100.59	odd	10
125.2.d.b.26.4		16	100.91	odd	10
125.2.d.b.101.1		16	100.39	odd	10
125.2.d.b.101.4		16	100.11	odd	10
125.2.e.b.24.2		8	100.27	even	20
125.2.e.b.99.2		8	100.63	even	20
225.2.m.a.19.2		8	300.287	odd	20
225.2.m.a.154.2		8	300.23	odd	20
400.2.y.c.129.1		8	25.23	odd	20
400.2.y.c.369.1		8	25.12	odd	20
625.2.a.f.1.2		8	20.19	odd	2
625.2.a.f.1.7		8	4.3	odd	2
625.2.b.c.624.2		8	20.3	even	4
625.2.b.c.624.7		8	20.7	even	4
625.2.d.o.251.1		16	100.31	odd	10
625.2.d.o.251.4		16	100.19	odd	10
625.2.d.o.376.1		16	100.71	odd	10
625.2.d.o.376.4		16	100.79	odd	10
625.2.e.a.249.1		8	100.3	even	20
625.2.e.a.374.1		8	100.67	even	20
625.2.e.i.249.2		8	100.47	even	20
625.2.e.i.374.2		8	100.83	even	20
5625.2.a.x.1.2		8	12.11	even	2
5625.2.a.x.1.7		8	60.59	even	2
10000.2.a.bj.1.2		8	1.1	even	1	trivial
10000.2.a.bj.1.7		8	5.4	even	2	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	10000.2.a.bj.1.2

Level	$10000$
Weight	$2$
Character	10000.1
Self dual	yes
Analytic conductor	$79.850$
Analytic rank	$1$
Dimension	$8$
Inner twists	$2$