LMFDB - Embedded newform 5625.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
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	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08529 q^{2} +2.34841 q^{4} -0.992398 q^{7} -0.726543 q^{8} +O(q^{10})$	$q-2.08529 q^{2} +2.34841 q^{4} -0.992398 q^{7} -0.726543 q^{8} -2.00000 q^{11} +3.37406 q^{13} +2.06943 q^{14} -3.18178 q^{16} +2.89451 q^{17} -2.58448 q^{19} +4.17057 q^{22} -4.54963 q^{23} -7.03588 q^{26} -2.33056 q^{28} +5.38430 q^{29} +0.136538 q^{31} +8.08800 q^{32} -6.03588 q^{34} +2.14910 q^{37} +5.38938 q^{38} -8.63318 q^{41} -4.64398 q^{43} -4.69683 q^{44} +9.48728 q^{46} +9.92630 q^{47} -6.01515 q^{49} +7.92369 q^{52} +7.56521 q^{53} +0.721020 q^{56} -11.2278 q^{58} -4.91775 q^{59} -2.76972 q^{61} -0.284720 q^{62} -10.5022 q^{64} -2.18577 q^{67} +6.79751 q^{68} -9.64254 q^{71} -0.775929 q^{73} -4.48150 q^{74} -6.06943 q^{76} +1.98480 q^{77} +15.8508 q^{79} +18.0026 q^{82} -1.77110 q^{83} +9.68401 q^{86} +1.45309 q^{88} -14.5080 q^{89} -3.34841 q^{91} -10.6844 q^{92} -20.6992 q^{94} +17.0291 q^{97} +12.5433 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4}+O(q^{10}) $ 8 * q + 6 * q^4	$ 8 q + 6 q^{4} - 16 q^{11} - 12 q^{14} - 2 q^{16} + 10 q^{19} - 6 q^{26} - 20 q^{29} + 16 q^{31} + 2 q^{34} - 26 q^{41} - 12 q^{44} + 6 q^{46} - 14 q^{49} - 10 q^{56} - 30 q^{59} + 6 q^{61} - 44 q^{64} - 46 q^{71} - 12 q^{74} - 20 q^{76} + 10 q^{79} + 14 q^{86} - 30 q^{89} - 14 q^{91} - 68 q^{94}+O(q^{100}) $ 8 * q + 6 * q^4 - 16 * q^11 - 12 * q^14 - 2 * q^16 + 10 * q^19 - 6 * q^26 - 20 * q^29 + 16 * q^31 + 2 * q^34 - 26 * q^41 - 12 * q^44 + 6 * q^46 - 14 * q^49 - 10 * q^56 - 30 * q^59 + 6 * q^61 - 44 * q^64 - 46 * q^71 - 12 * q^74 - 20 * q^76 + 10 * q^79 + 14 * q^86 - 30 * q^89 - 14 * q^91 - 68 * q^94

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 $

$2$	−2.08529	−1.47452	−0.737260	−	0.675610i	$-0.763881\pi$
$2$	−2.08529	−1.47452	−0.737260	+	0.675610i	$0.763881\pi$
$3$	0	0
$3$	0	0
$4$	2.34841	1.17421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.992398	−0.375091	−0.187546	−	0.982256i	$-0.560053\pi$
$7$	−0.992398	−0.375091	−0.187546	+	0.982256i	$0.560053\pi$
$8$	−0.726543	−0.256872
$9$	0	0
$10$	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	$0.345011\pi$
$14$	2.06943	0.553079
$15$	0	0
$16$	−3.18178	−0.795445
$17$	2.89451	0.702022	0.351011	−	0.936371i	$-0.385838\pi$
$17$	2.89451	0.702022	0.351011	+	0.936371i	$0.385838\pi$
$18$	0	0
$19$	−2.58448	−0.592921	−0.296460	−	0.955045i	$-0.595806\pi$
$19$	−2.58448	−0.592921	−0.296460	+	0.955045i	$0.595806\pi$
$20$	0	0
$21$	0	0
$22$	4.17057	0.889169
$23$	−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
$26$	−7.03588	−1.37985
$27$	0	0
$28$	−2.33056	−0.440435
$29$	5.38430	0.999839	0.499919	−	0.866072i	$-0.333363\pi$
$29$	5.38430	0.999839	0.499919	+	0.866072i	$0.333363\pi$
$30$	0	0
$31$	0.136538	0.0245229	0.0122614	−	0.999925i	$-0.496097\pi$
$31$	0.136538	0.0245229	0.0122614	+	0.999925i	$0.496097\pi$
$32$	8.08800	1.42977
$33$	0	0
$34$	−6.03588	−1.03515
$35$	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	$0.443472\pi$
$38$	5.38938	0.874273
$39$	0	0
$40$	0	0
$41$	−8.63318	−1.34828	−0.674138	−	0.738605i	$-0.735485\pi$
$41$	−8.63318	−1.34828	−0.674138	+	0.738605i	$0.735485\pi$
$42$	0	0
$43$	−4.64398	−0.708200	−0.354100	−	0.935208i	$-0.615213\pi$
$43$	−4.64398	−0.708200	−0.354100	+	0.935208i	$0.615213\pi$
$44$	−4.69683	−0.708073
$45$	0	0
$46$	9.48728	1.39882
$47$	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
$51$	0	0
$52$	7.92369	1.09882
$53$	7.56521	1.03916	0.519581	−	0.854421i	$-0.326088\pi$
$53$	7.56521	1.03916	0.519581	+	0.854421i	$0.326088\pi$
$54$	0	0
$55$	0	0
$56$	0.721020	0.0963503
$57$	0	0
$58$	−11.2278	−1.47428
$59$	−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	$0.556740\pi$
$62$	−0.284720	−0.0361595
$63$	0	0
$64$	−10.5022	−1.31278
$65$	0	0
$66$	0	0
$67$	−2.18577	−0.267035	−0.133517	−	0.991046i	$-0.542627\pi$
$67$	−2.18577	−0.267035	−0.133517	+	0.991046i	$0.542627\pi$
$68$	6.79751	0.824319
$69$	0	0
$70$	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	$0.514459\pi$
$74$	−4.48150	−0.520963
$75$	0	0
$76$	−6.06943	−0.696212
$77$	1.98480	0.226189
$78$	0	0
$79$	15.8508	1.78336	0.891679	−	0.452667i	$-0.149527\pi$
$79$	15.8508	1.78336	0.891679	+	0.452667i	$0.149527\pi$
$80$	0	0
$81$	0	0
$82$	18.0026	1.98806
$83$	−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
$86$	9.68401	1.04425
$87$	0	0
$88$	1.45309	0.154899
$89$	−14.5080	−1.53785	−0.768923	−	0.639341i	$-0.779207\pi$
$89$	−14.5080	−1.53785	−0.768923	+	0.639341i	$0.779207\pi$
$90$	0	0
$91$	−3.34841	−0.351009
$92$	−10.6844	−1.11393
$93$	0	0
$94$	−20.6992	−2.13496
$95$	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	$0.167621\pi$
$98$	12.5433	1.26706
$99$	0	0
$100$	0	0
$101$	2.54716	0.253452	0.126726	−	0.991938i	$-0.459553\pi$
$101$	2.54716	0.253452	0.126726	+	0.991938i	$0.459553\pi$
$102$	0	0
$103$	−10.1654	−1.00163	−0.500815	−	0.865555i	$-0.666966\pi$
$103$	−10.1654	−1.00163	−0.500815	+	0.865555i	$0.666966\pi$
$104$	−2.45140	−0.240380
$105$	0	0
$106$	−15.7756	−1.53226
$107$	−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
$111$	0	0
$112$	3.15759	0.298365
$113$	6.75704	0.635649	0.317825	−	0.948150i	$-0.397048\pi$
$113$	6.75704	0.635649	0.317825	+	0.948150i	$0.397048\pi$
$114$	0	0
$115$	0	0
$116$	12.6446	1.17402
$117$	0	0
$118$	10.2549	0.944041
$119$	−2.87251	−0.263322
$120$	0	0
$121$	−7.00000	−0.636364
$122$	5.77565	0.522903
$123$	0	0
$124$	0.320647	0.0287950
$125$	0	0
$126$	0	0
$127$	1.49081	0.132288	0.0661441	−	0.997810i	$-0.478930\pi$
$127$	1.49081	0.132288	0.0661441	+	0.997810i	$0.478930\pi$
$128$	5.72414	0.505948
$129$	0	0
$130$	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
$134$	4.55796	0.393748
$135$	0	0
$136$	−2.10299	−0.180330
$137$	−0.689447	−0.0589035	−0.0294517	−	0.999566i	$-0.509376\pi$
$137$	−0.689447	−0.0589035	−0.0294517	+	0.999566i	$0.509376\pi$
$138$	0	0
$139$	−16.5719	−1.40561	−0.702803	−	0.711384i	$-0.748069\pi$
$139$	−16.5719	−1.40561	−0.702803	+	0.711384i	$0.748069\pi$
$140$	0	0
$141$	0	0
$142$	20.1074	1.68738
$143$	−6.74812	−0.564307
$144$	0	0
$145$	0	0
$146$	1.61803	0.133909
$147$	0	0
$148$	5.04699	0.414860
$149$	−3.21156	−0.263101	−0.131551	−	0.991309i	$-0.541996\pi$
$149$	−3.21156	−0.263101	−0.131551	+	0.991309i	$0.541996\pi$
$150$	0	0
$151$	17.6863	1.43929	0.719647	−	0.694340i	$-0.244304\pi$
$151$	17.6863	1.43929	0.719647	+	0.694340i	$0.244304\pi$
$152$	1.87774	0.152305
$153$	0	0
$154$	−4.13887	−0.333519
$155$	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	$0.521039\pi$
$158$	−33.0535	−2.62960
$159$	0	0
$160$	0	0
$161$	4.51505	0.355836
$162$	0	0
$163$	−0.892934	−0.0699400	−0.0349700	−	0.999388i	$-0.511134\pi$
$163$	−0.892934	−0.0699400	−0.0349700	+	0.999388i	$0.511134\pi$
$164$	−20.2743	−1.58316
$165$	0	0
$166$	3.69325	0.286652
$167$	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
$171$	0	0
$172$	−10.9060	−0.831573
$173$	−5.76465	−0.438278	−0.219139	−	0.975694i	$-0.570325\pi$
$173$	−5.76465	−0.438278	−0.219139	+	0.975694i	$0.570325\pi$
$174$	0	0
$175$	0	0
$176$	6.36356	0.479671
$177$	0	0
$178$	30.2534	2.26758
$179$	−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	$0.321773\pi$
$182$	6.98240	0.517570
$183$	0	0
$184$	3.30550	0.243685
$185$	0	0
$186$	0	0
$187$	−5.78902	−0.423335
$188$	23.3111	1.70013
$189$	0	0
$190$	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	$0.775798\pi$
$194$	−35.5105	−2.54951
$195$	0	0
$196$	−14.1261	−1.00900
$197$	−12.2013	−0.869308	−0.434654	−	0.900597i	$-0.643129\pi$
$197$	−12.2013	−0.869308	−0.434654	+	0.900597i	$0.643129\pi$
$198$	0	0
$199$	10.4065	0.737695	0.368848	−	0.929490i	$-0.379752\pi$
$199$	10.4065	0.737695	0.368848	+	0.929490i	$0.379752\pi$
$200$	0	0
$201$	0	0
$202$	−5.31156	−0.373720
$203$	−5.34337	−0.375031
$204$	0	0
$205$	0	0
$206$	21.1978	1.47692
$207$	0	0
$208$	−10.7355	−0.744375
$209$	5.16896	0.357545
$210$	0	0
$211$	−8.65769	−0.596020	−0.298010	−	0.954563i	$-0.596323\pi$
$211$	−8.65769	−0.596020	−0.298010	+	0.954563i	$0.596323\pi$
$212$	17.7662	1.22019
$213$	0	0
$214$	10.0452	0.686679
$215$	0	0
$216$	0	0
$217$	−0.135500	−0.00919832
$218$	−33.9365	−2.29847
$219$	0	0
$220$	0	0
$221$	9.76626	0.656950
$222$	0	0
$223$	−28.3434	−1.89801	−0.949007	−	0.315256i	$-0.897909\pi$
$223$	−28.3434	−1.89801	−0.949007	+	0.315256i	$0.897909\pi$
$224$	−8.02652	−0.536295
$225$	0	0
$226$	−14.0904	−0.937277
$227$	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
$231$	0	0
$232$	−3.91192	−0.256830
$233$	5.95605	0.390194	0.195097	−	0.980784i	$-0.437498\pi$
$233$	5.95605	0.390194	0.195097	+	0.980784i	$0.437498\pi$
$234$	0	0
$235$	0	0
$236$	−11.5489	−0.751770
$237$	0	0
$238$	5.99000	0.388274
$239$	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	$0.487902\pi$
$242$	14.5970	0.938330
$243$	0	0
$244$	−6.50444	−0.416404
$245$	0	0
$246$	0	0
$247$	−8.72020	−0.554853
$248$	−0.0992004	−0.00629923
$249$	0	0
$250$	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
$254$	−3.10877	−0.195062
$255$	0	0
$256$	9.06799	0.566750
$257$	−9.75542	−0.608526	−0.304263	−	0.952588i	$-0.598410\pi$
$257$	−9.75542	−0.608526	−0.304263	+	0.952588i	$0.598410\pi$
$258$	0	0
$259$	−2.13277	−0.132524
$260$	0	0
$261$	0	0
$262$	29.4331	1.81838
$263$	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
$266$	−5.34841	−0.327932
$267$	0	0
$268$	−5.13310	−0.313554
$269$	3.28853	0.200506	0.100253	−	0.994962i	$-0.468035\pi$
$269$	3.28853	0.200506	0.100253	+	0.994962i	$0.468035\pi$
$270$	0	0
$271$	12.1500	0.738063	0.369031	−	0.929417i	$-0.379689\pi$
$271$	12.1500	0.738063	0.369031	+	0.929417i	$0.379689\pi$
$272$	−9.20970	−0.558420
$273$	0	0
$274$	1.43769	0.0868543
$275$	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	$0.383971\pi$
$278$	34.5571	2.07259
$279$	0	0
$280$	0	0
$281$	24.6416	1.47000	0.734998	−	0.678070i	$-0.237183\pi$
$281$	24.6416	1.47000	0.734998	+	0.678070i	$0.237183\pi$
$282$	0	0
$283$	3.36343	0.199935	0.0999675	−	0.994991i	$-0.468126\pi$
$283$	3.36343	0.199935	0.0999675	+	0.994991i	$0.468126\pi$
$284$	−22.6447	−1.34371
$285$	0	0
$286$	14.0718	0.832081
$287$	8.56755	0.505727
$288$	0	0
$289$	−8.62180	−0.507165
$290$	0	0
$291$	0	0
$292$	−1.82220	−0.106636
$293$	−8.96340	−0.523647	−0.261824	−	0.965116i	$-0.584324\pi$
$293$	−8.96340	−0.523647	−0.261824	+	0.965116i	$0.584324\pi$
$294$	0	0
$295$	0	0
$296$	−1.56142	−0.0907555
$297$	0	0
$298$	6.69702	0.387948
$299$	−15.3507	−0.887756
$300$	0	0
$301$	4.60867	0.265640
$302$	−36.8811	−2.12227
$303$	0	0
$304$	8.22325	0.471636
$305$	0	0
$306$	0	0
$307$	−9.48133	−0.541128	−0.270564	−	0.962702i	$-0.587210\pi$
$307$	−9.48133	−0.541128	−0.270564	+	0.962702i	$0.587210\pi$
$308$	4.66112	0.265592
$309$	0	0
$310$	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	$0.679261\pi$
$314$	3.45140	0.194774
$315$	0	0
$316$	37.2243	2.09403
$317$	22.7893	1.27998	0.639988	−	0.768385i	$-0.278939\pi$
$317$	22.7893	1.27998	0.639988	+	0.768385i	$0.278939\pi$
$318$	0	0
$319$	−10.7686	−0.602925
$320$	0	0
$321$	0	0
$322$	−9.41516	−0.524687
$323$	−7.48081	−0.416244
$324$	0	0
$325$	0	0
$326$	1.86202	0.103128
$327$	0	0
$328$	6.27237	0.346334
$329$	−9.85084	−0.543094
$330$	0	0
$331$	2.96299	0.162861	0.0814304	−	0.996679i	$-0.474051\pi$
$331$	2.96299	0.162861	0.0814304	+	0.996679i	$0.474051\pi$
$332$	−4.15928	−0.228270
$333$	0	0
$334$	−10.8343	−0.592824
$335$	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	$0.328762\pi$
$338$	3.36920	0.183261
$339$	0	0
$340$	0	0
$341$	−0.273075	−0.0147879
$342$	0	0
$343$	12.9162	0.697410
$344$	3.37405	0.181916
$345$	0	0
$346$	12.0209	0.646249
$347$	−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
$351$	0	0
$352$	−16.1760	−0.862184
$353$	5.24945	0.279400	0.139700	−	0.990194i	$-0.455386\pi$
$353$	5.24945	0.279400	0.139700	+	0.990194i	$0.455386\pi$
$354$	0	0
$355$	0	0
$356$	−34.0708	−1.80575
$357$	0	0
$358$	18.0771	0.955402
$359$	−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
$362$	−29.8005	−1.56628
$363$	0	0
$364$	−7.86346	−0.412157
$365$	0	0
$366$	0	0
$367$	7.29872	0.380990	0.190495	−	0.981688i	$-0.438991\pi$
$367$	7.29872	0.380990	0.190495	+	0.981688i	$0.438991\pi$
$368$	14.4759	0.754610
$369$	0	0
$370$	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	$0.304022\pi$
$374$	12.0718	0.624216
$375$	0	0
$376$	−7.21188	−0.371924
$377$	18.1669	0.935645
$378$	0	0
$379$	−32.9466	−1.69235	−0.846177	−	0.532903i	$-0.821101\pi$
$379$	−32.9466	−1.69235	−0.846177	+	0.532903i	$0.821101\pi$
$380$	0	0
$381$	0	0
$382$	−2.64072	−0.135111
$383$	−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
$386$	44.1516	2.24726
$387$	0	0
$388$	39.9914	2.03026
$389$	1.14446	0.0580263	0.0290132	−	0.999579i	$-0.490764\pi$
$389$	1.14446	0.0580263	0.0290132	+	0.999579i	$0.490764\pi$
$390$	0	0
$391$	−13.1690	−0.665983
$392$	4.37026	0.220731
$393$	0	0
$394$	25.4432	1.28181
$395$	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	$0.462490\pi$
$398$	−21.7005	−1.08775
$399$	0	0
$400$	0	0
$401$	24.0851	1.20275	0.601376	−	0.798966i	$-0.294619\pi$
$401$	24.0851	1.20275	0.601376	+	0.798966i	$0.294619\pi$
$402$	0	0
$403$	0.460687	0.0229484
$404$	5.98179	0.297605
$405$	0	0
$406$	11.1424	0.552990
$407$	−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
$411$	0	0
$412$	−23.8726	−1.17612
$413$	4.88037	0.240147
$414$	0	0
$415$	0	0
$416$	27.2894	1.33797
$417$	0	0
$418$	−10.7788	−0.527207
$419$	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	$0.698367\pi$
$422$	18.0537	0.878842
$423$	0	0
$424$	−5.49645	−0.266931
$425$	0	0
$426$	0	0
$427$	2.74866	0.133017
$428$	−11.3128	−0.546824
$429$	0	0
$430$	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	$0.711030\pi$
$434$	0.282556	0.0135631
$435$	0	0
$436$	38.2187	1.83035
$437$	11.7584	0.562483
$438$	0	0
$439$	19.3741	0.924676	0.462338	−	0.886704i	$-0.347011\pi$
$439$	19.3741	0.924676	0.462338	+	0.886704i	$0.347011\pi$
$440$	0	0
$441$	0	0
$442$	−20.3654	−0.968685
$443$	−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
$446$	59.1040	2.79866
$447$	0	0
$448$	10.4224	0.492412
$449$	14.3585	0.677618	0.338809	−	0.940855i	$-0.389976\pi$
$449$	14.3585	0.677618	0.338809	+	0.940855i	$0.389976\pi$
$450$	0	0
$451$	17.2664	0.813041
$452$	15.8683	0.746384
$453$	0	0
$454$	−46.3798	−2.17671
$455$	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	$0.700604\pi$
$458$	−5.16231	−0.241219
$459$	0	0
$460$	0	0
$461$	−28.8255	−1.34254	−0.671269	−	0.741214i	$-0.734251\pi$
$461$	−28.8255	−1.34254	−0.671269	+	0.741214i	$0.734251\pi$
$462$	0	0
$463$	31.8796	1.48157	0.740786	−	0.671742i	$-0.234453\pi$
$463$	31.8796	1.48157	0.740786	+	0.671742i	$0.234453\pi$
$464$	−17.1316	−0.795317
$465$	0	0
$466$	−12.4201	−0.575348
$467$	−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
$471$	0	0
$472$	3.57295	0.164459
$473$	9.28795	0.427060
$474$	0	0
$475$	0	0
$476$	−6.74584	−0.309195
$477$	0	0
$478$	−14.6646	−0.670744
$479$	−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
$482$	−2.46014	−0.112057
$483$	0	0
$484$	−16.4389	−0.747223
$485$	0	0
$486$	0	0
$487$	−27.9190	−1.26513	−0.632565	−	0.774507i	$-0.717998\pi$
$487$	−27.9190	−1.26513	−0.632565	+	0.774507i	$0.717998\pi$
$488$	2.01232	0.0910933
$489$	0	0
$490$	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
$494$	18.1841	0.818142
$495$	0	0
$496$	−0.434433	−0.0195066
$497$	9.56924	0.429239
$498$	0	0
$499$	−44.3253	−1.98427	−0.992137	−	0.125160i	$-0.960056\pi$
$499$	−44.3253	−1.98427	−0.992137	+	0.125160i	$0.960056\pi$
$500$	0	0
$501$	0	0
$502$	9.61040	0.428933
$503$	−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
$506$	−18.9746	−0.843522
$507$	0	0
$508$	3.50105	0.155334
$509$	26.7154	1.18414	0.592070	−	0.805886i	$-0.298311\pi$
$509$	26.7154	1.18414	0.592070	+	0.805886i	$0.298311\pi$
$510$	0	0
$511$	0.770031	0.0340642
$512$	−30.3576	−1.34163
$513$	0	0
$514$	20.3428	0.897283
$515$	0	0
$516$	0	0
$517$	−19.8526	−0.873116
$518$	4.44743	0.195409
$519$	0	0
$520$	0	0
$521$	−32.7073	−1.43293	−0.716466	−	0.697622i	$-0.754241\pi$
$521$	−32.7073	−1.43293	−0.716466	+	0.697622i	$0.754241\pi$
$522$	0	0
$523$	−0.235966	−0.0103181	−0.00515904	−	0.999987i	$-0.501642\pi$
$523$	−0.235966	−0.0103181	−0.00515904	+	0.999987i	$0.501642\pi$
$524$	−33.1471	−1.44804
$525$	0	0
$526$	−2.07658	−0.0905434
$527$	0.395210	0.0172156
$528$	0	0
$529$	−2.30084	−0.100037
$530$	0	0
$531$	0	0
$532$	6.02330	0.261143
$533$	−29.1289	−1.26171
$534$	0	0
$535$	0	0
$536$	1.58806	0.0685936
$537$	0	0
$538$	−6.85753	−0.295649
$539$	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	$0.756487\pi$
$542$	−25.3363	−1.08829
$543$	0	0
$544$	23.4108	1.00373
$545$	0	0
$546$	0	0
$547$	−38.5125	−1.64668	−0.823338	−	0.567552i	$-0.807891\pi$
$547$	−38.5125	−1.64668	−0.823338	+	0.567552i	$0.807891\pi$
$548$	−1.61911	−0.0691648
$549$	0	0
$550$	0	0
$551$	−13.9156	−0.592825
$552$	0	0
$553$	−15.7303	−0.668922
$554$	−24.7452	−1.05132
$555$	0	0
$556$	−38.9176	−1.65047
$557$	4.33445	0.183657	0.0918283	−	0.995775i	$-0.470729\pi$
$557$	4.33445	0.183657	0.0918283	+	0.995775i	$0.470729\pi$
$558$	0	0
$559$	−15.6691	−0.662731
$560$	0	0
$561$	0	0
$562$	−51.3848	−2.16754
$563$	−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
$566$	−7.01371	−0.294808
$567$	0	0
$568$	7.00572	0.293953
$569$	41.9646	1.75925	0.879623	−	0.475671i	$-0.157795\pi$
$569$	41.9646	1.75925	0.879623	+	0.475671i	$0.157795\pi$
$570$	0	0
$571$	−13.8332	−0.578900	−0.289450	−	0.957193i	$-0.593472\pi$
$571$	−13.8332	−0.578900	−0.289450	+	0.957193i	$0.593472\pi$
$572$	−15.8474	−0.662613
$573$	0	0
$574$	−17.8658	−0.745704
$575$	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	$0.585704\pi$
$578$	17.9789	0.747824
$579$	0	0
$580$	0	0
$581$	1.75764	0.0729191
$582$	0	0
$583$	−15.1304	−0.626638
$584$	0.563746	0.0233280
$585$	0	0
$586$	18.6912	0.772128
$587$	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
$591$	0	0
$592$	−6.83798	−0.281039
$593$	31.2580	1.28361	0.641807	−	0.766866i	$-0.278185\pi$
$593$	31.2580	1.28361	0.641807	+	0.766866i	$0.278185\pi$
$594$	0	0
$595$	0	0
$596$	−7.54208	−0.308936
$597$	0	0
$598$	32.0107	1.30901
$599$	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	$0.903729\pi$
$602$	−9.61040	−0.391691
$603$	0	0
$604$	41.5349	1.69003
$605$	0	0
$606$	0	0
$607$	30.7401	1.24770	0.623851	−	0.781543i	$-0.285567\pi$
$607$	30.7401	1.24770	0.623851	+	0.781543i	$0.285567\pi$
$608$	−20.9033	−0.847741
$609$	0	0
$610$	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	$0.781369\pi$
$614$	19.7713	0.797904
$615$	0	0
$616$	−1.44204	−0.0581014
$617$	−0.425306	−0.0171222	−0.00856109	−	0.999963i	$-0.502725\pi$
$617$	−0.425306	−0.0171222	−0.00856109	+	0.999963i	$0.502725\pi$
$618$	0	0
$619$	7.51147	0.301912	0.150956	−	0.988541i	$-0.451765\pi$
$619$	7.51147	0.301912	0.150956	+	0.988541i	$0.451765\pi$
$620$	0	0
$621$	0	0
$622$	61.1656	2.45252
$623$	14.3977	0.576833
$624$	0	0
$625$	0	0
$626$	39.3912	1.57439
$627$	0	0
$628$	−3.88691	−0.155105
$629$	6.22061	0.248032
$630$	0	0
$631$	−11.2716	−0.448714	−0.224357	−	0.974507i	$-0.572028\pi$
$631$	−11.2716	−0.448714	−0.224357	+	0.974507i	$0.572028\pi$
$632$	−11.5163	−0.458094
$633$	0	0
$634$	−47.5222	−1.88735
$635$	0	0
$636$	0	0
$637$	−20.2955	−0.804136
$638$	22.4556	0.889025
$639$	0	0
$640$	0	0
$641$	−26.0825	−1.03020	−0.515099	−	0.857131i	$-0.672245\pi$
$641$	−26.0825	−1.03020	−0.515099	+	0.857131i	$0.672245\pi$
$642$	0	0
$643$	−31.9492	−1.25995	−0.629977	−	0.776614i	$-0.716936\pi$
$643$	−31.9492	−1.25995	−0.629977	+	0.776614i	$0.716936\pi$
$644$	10.6032	0.417825
$645$	0	0
$646$	15.5996	0.613759
$647$	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	0
$652$	−2.09698	−0.0821240
$653$	−18.6853	−0.731212	−0.365606	−	0.930770i	$-0.619138\pi$
$653$	−18.6853	−0.731212	−0.365606	+	0.930770i	$0.619138\pi$
$654$	0	0
$655$	0	0
$656$	27.4689	1.07248
$657$	0	0
$658$	20.5418	0.800803
$659$	−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	$0.684466\pi$
$662$	−6.17868	−0.240141
$663$	0	0
$664$	1.28678	0.0499368
$665$	0	0
$666$	0	0
$667$	−24.4966	−0.948511
$668$	12.2014	0.472085
$669$	0	0
$670$	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	$0.770987\pi$
$674$	−39.2290	−1.51104
$675$	0	0
$676$	−3.79434	−0.145936
$677$	−5.03533	−0.193523	−0.0967617	−	0.995308i	$-0.530848\pi$
$677$	−5.03533	−0.193523	−0.0967617	+	0.995308i	$0.530848\pi$
$678$	0	0
$679$	−16.8997	−0.648549
$680$	0	0
$681$	0	0
$682$	0.569440	0.0218050
$683$	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
$686$	−26.9340	−1.02834
$687$	0	0
$688$	14.7761	0.563334
$689$	25.5255	0.972444
$690$	0	0
$691$	−21.2329	−0.807739	−0.403869	−	0.914817i	$-0.632335\pi$
$691$	−21.2329	−0.807739	−0.403869	+	0.914817i	$0.632335\pi$
$692$	−13.5378	−0.514629
$693$	0	0
$694$	47.4156	1.79987
$695$	0	0
$696$	0	0
$697$	−24.9888	−0.946520
$698$	−4.04230	−0.153003
$699$	0	0
$700$	0	0
$701$	−32.7698	−1.23770	−0.618849	−	0.785510i	$-0.712401\pi$
$701$	−32.7698	−1.23770	−0.618849	+	0.785510i	$0.712401\pi$
$702$	0	0
$703$	−5.55432	−0.209485
$704$	21.0045	0.791636
$705$	0	0
$706$	−10.9466	−0.411981
$707$	−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
$711$	0	0
$712$	10.5407	0.395029
$713$	−0.621196	−0.0232640
$714$	0	0
$715$	0	0
$716$	−20.3581	−0.760817
$717$	0	0
$718$	47.1144	1.75829
$719$	−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
$722$	25.6917	0.956144
$723$	0	0
$724$	33.5609	1.24728
$725$	0	0
$726$	0	0
$727$	−5.86510	−0.217524	−0.108762	−	0.994068i	$-0.534689\pi$
$727$	−5.86510	−0.217524	−0.108762	+	0.994068i	$0.534689\pi$
$728$	2.43277	0.0901643
$729$	0	0
$730$	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	$0.719895\pi$
$734$	−15.2199	−0.561777
$735$	0	0
$736$	−36.7974	−1.35637
$737$	4.37155	0.161028
$738$	0	0
$739$	39.5712	1.45565	0.727826	−	0.685762i	$-0.240531\pi$
$739$	39.5712	1.45565	0.727826	+	0.685762i	$0.240531\pi$
$740$	0	0
$741$	0	0
$742$	15.6557	0.574739
$743$	−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
$746$	−46.5172	−1.70312
$747$	0	0
$748$	−13.5950	−0.497083
$749$	4.78058	0.174679
$750$	0	0
$751$	26.8870	0.981122	0.490561	−	0.871407i	$-0.336792\pi$
$751$	26.8870	0.981122	0.490561	+	0.871407i	$0.336792\pi$
$752$	−31.5833	−1.15172
$753$	0	0
$754$	−37.8833	−1.37963
$755$	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	$0.801592\pi$
$758$	68.7031	2.49541
$759$	0	0
$760$	0	0
$761$	−20.3080	−0.736163	−0.368081	−	0.929794i	$-0.619985\pi$
$761$	−20.3080	−0.736163	−0.368081	+	0.929794i	$0.619985\pi$
$762$	0	0
$763$	−16.1506	−0.584690
$764$	2.97393	0.107593
$765$	0	0
$766$	43.1658	1.55964
$767$	−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
$771$	0	0
$772$	−49.7229	−1.78956
$773$	13.7305	0.493851	0.246926	−	0.969034i	$-0.420580\pi$
$773$	13.7305	0.493851	0.246926	+	0.969034i	$0.420580\pi$
$774$	0	0
$775$	0	0
$776$	−12.3724	−0.444142
$777$	0	0
$778$	−2.38652	−0.0855609
$779$	22.3123	0.799421
$780$	0	0
$781$	19.2851	0.690074
$782$	27.4610	0.982005
$783$	0	0
$784$	19.1389	0.683531
$785$	0	0
$786$	0	0
$787$	19.6660	0.701018	0.350509	−	0.936559i	$-0.386009\pi$
$787$	19.6660	0.701018	0.350509	+	0.936559i	$0.386009\pi$
$788$	−28.6538	−1.02075
$789$	0	0
$790$	0	0
$791$	−6.70568	−0.238427
$792$	0	0
$793$	−9.34520	−0.331858
$794$	−9.76984	−0.346719
$795$	0	0
$796$	24.4387	0.866207
$797$	10.9441	0.387660	0.193830	−	0.981035i	$-0.437909\pi$
$797$	10.9441	0.387660	0.193830	+	0.981035i	$0.437909\pi$
$798$	0	0
$799$	28.7318	1.01646
$800$	0	0
$801$	0	0
$802$	−50.2243	−1.77348
$803$	1.55186	0.0547639
$804$	0	0
$805$	0	0
$806$	−0.960663	−0.0338379
$807$	0	0
$808$	−1.85062	−0.0651046
$809$	−16.4427	−0.578096	−0.289048	−	0.957315i	$-0.593339\pi$
$809$	−16.4427	−0.578096	−0.289048	+	0.957315i	$0.593339\pi$
$810$	0	0
$811$	22.6473	0.795253	0.397627	−	0.917547i	$-0.369834\pi$
$811$	22.6473	0.795253	0.397627	+	0.917547i	$0.369834\pi$
$812$	−12.5484	−0.440364
$813$	0	0
$814$	8.96299	0.314153
$815$	0	0
$816$	0	0
$817$	12.0023	0.419906
$818$	3.96067	0.138482
$819$	0	0
$820$	0	0
$821$	39.7792	1.38830	0.694151	−	0.719829i	$-0.255780\pi$
$821$	39.7792	1.38830	0.694151	+	0.719829i	$0.255780\pi$
$822$	0	0
$823$	−16.5602	−0.577252	−0.288626	−	0.957442i	$-0.593198\pi$
$823$	−16.5602	−0.577252	−0.288626	+	0.957442i	$0.593198\pi$
$824$	7.38562	0.257290
$825$	0	0
$826$	−10.1770	−0.354102
$827$	−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
$831$	0	0
$832$	−35.4352	−1.22849
$833$	−17.4109	−0.603252
$834$	0	0
$835$	0	0
$836$	12.1389	0.419832
$837$	0	0
$838$	−4.85467	−0.167702
$839$	−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
$842$	49.9427	1.72114
$843$	0	0
$844$	−20.3318	−0.699850
$845$	0	0
$846$	0	0
$847$	6.94679	0.238694
$848$	−24.0708	−0.826596
$849$	0	0
$850$	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	$0.550055\pi$
$854$	−5.73175	−0.196136
$855$	0	0
$856$	3.49990	0.119624
$857$	−13.6712	−0.466998	−0.233499	−	0.972357i	$-0.575018\pi$
$857$	−13.6712	−0.466998	−0.233499	+	0.972357i	$0.575018\pi$
$858$	0	0
$859$	−35.6556	−1.21655	−0.608277	−	0.793725i	$-0.708139\pi$
$859$	−35.6556	−1.21655	−0.608277	+	0.793725i	$0.708139\pi$
$860$	0	0
$861$	0	0
$862$	2.48621	0.0846808
$863$	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
$866$	53.4121	1.81502
$867$	0	0
$868$	−0.318210	−0.0108007
$869$	−31.7017	−1.07541
$870$	0	0
$871$	−7.37494	−0.249890
$872$	−11.8240	−0.400410
$873$	0	0
$874$	−24.5197	−0.829391
$875$	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	$0.661017\pi$
$878$	−40.4006	−1.36345
$879$	0	0
$880$	0	0
$881$	−8.33039	−0.280658	−0.140329	−	0.990105i	$-0.544816\pi$
$881$	−8.33039	−0.280658	−0.140329	+	0.990105i	$0.544816\pi$
$882$	0	0
$883$	50.3165	1.69329	0.846643	−	0.532161i	$-0.178620\pi$
$883$	50.3165	1.69329	0.846643	+	0.532161i	$0.178620\pi$
$884$	22.9352	0.771395
$885$	0	0
$886$	5.13529	0.172524
$887$	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
$891$	0	0
$892$	−66.5620	−2.22866
$893$	−25.6543	−0.858490
$894$	0	0
$895$	0	0
$896$	−5.68063	−0.189777
$897$	0	0
$898$	−29.9415	−0.999161
$899$	0.735159	0.0245189
$900$	0	0
$901$	21.8976	0.729514
$902$	−36.0053	−1.19884
$903$	0	0
$904$	−4.90928	−0.163280
$905$	0	0
$906$	0	0
$907$	31.9105	1.05957	0.529786	−	0.848132i	$-0.322272\pi$
$907$	31.9105	1.05957	0.529786	+	0.848132i	$0.322272\pi$
$908$	52.2322	1.73339
$909$	0	0
$910$	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
$914$	52.5416	1.73792
$915$	0	0
$916$	5.81371	0.192090
$917$	14.0074	0.462565
$918$	0	0
$919$	−19.4850	−0.642752	−0.321376	−	0.946952i	$-0.604145\pi$
$919$	−19.4850	−0.642752	−0.321376	+	0.946952i	$0.604145\pi$
$920$	0	0
$921$	0	0
$922$	60.1094	1.97960
$923$	−32.5345	−1.07089
$924$	0	0
$925$	0	0
$926$	−66.4781	−2.18461
$927$	0	0
$928$	43.5482	1.42954
$929$	−11.7642	−0.385972	−0.192986	−	0.981201i	$-0.561817\pi$
$929$	−11.7642	−0.385972	−0.192986	+	0.981201i	$0.561817\pi$
$930$	0	0
$931$	15.5460	0.509501
$932$	13.9873	0.458168
$933$	0	0
$934$	89.8749	2.94080
$935$	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	$0.388172\pi$
$938$	−4.52331	−0.147691
$939$	0	0
$940$	0	0
$941$	2.24706	0.0732521	0.0366261	−	0.999329i	$-0.488339\pi$
$941$	2.24706	0.0732521	0.0366261	+	0.999329i	$0.488339\pi$
$942$	0	0
$943$	39.2778	1.27906
$944$	15.6472	0.509273
$945$	0	0
$946$	−19.3680	−0.629709
$947$	−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
$951$	0	0
$952$	2.08700	0.0676400
$953$	59.9534	1.94208	0.971040	−	0.238918i	$-0.0767926\pi$
$953$	59.9534	1.94208	0.971040	+	0.238918i	$0.0767926\pi$
$954$	0	0
$955$	0	0
$956$	16.5150	0.534135
$957$	0	0
$958$	43.8459	1.41660
$959$	0.684206	0.0220942
$960$	0	0
$961$	−30.9814	−0.999399
$962$	−15.1208	−0.487516
$963$	0	0
$964$	2.77057	0.0892342
$965$	0	0
$966$	0	0
$967$	9.05599	0.291221	0.145610	−	0.989342i	$-0.453485\pi$
$967$	9.05599	0.291221	0.145610	+	0.989342i	$0.453485\pi$
$968$	5.08580	0.163464
$969$	0	0
$970$	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
$974$	58.2191	1.86546
$975$	0	0
$976$	8.81263	0.282085
$977$	−4.74467	−0.151795	−0.0758977	−	0.997116i	$-0.524182\pi$
$977$	−4.74467	−0.151795	−0.0758977	+	0.997116i	$0.524182\pi$
$978$	0	0
$979$	29.0160	0.927357
$980$	0	0
$981$	0	0
$982$	31.1981	0.995570
$983$	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
$986$	−32.4990	−1.03498
$987$	0	0
$988$	−20.4786	−0.651513
$989$	21.1284	0.671843
$990$	0	0
$991$	−39.7199	−1.26174	−0.630871	−	0.775887i	$-0.717302\pi$
$991$	−39.7199	−1.26174	−0.630871	+	0.775887i	$0.717302\pi$
$992$	1.10432	0.0350621
$993$	0	0
$994$	−19.9546	−0.632921
$995$	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	$0.659243\pi$
$998$	92.4309	2.92585
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.x.1.2		8
3.2	odd	2		625.2.a.f.1.7		8
5.4	even	2	inner	5625.2.a.x.1.7		8
12.11	even	2		10000.2.a.bj.1.2		8
15.2	even	4		625.2.b.c.624.7		8
15.8	even	4		625.2.b.c.624.2		8
15.14	odd	2		625.2.a.f.1.2		8
25.12	odd	20		225.2.m.a.19.2		8
25.23	odd	20		225.2.m.a.154.2		8
60.59	even	2		10000.2.a.bj.1.7		8
75.2	even	20		125.2.e.b.24.2		8
75.8	even	20		625.2.e.i.374.2		8
75.11	odd	10		125.2.d.b.101.4		16
75.14	odd	10		125.2.d.b.101.1		16
75.17	even	20		625.2.e.a.374.1		8
75.23	even	20		25.2.e.a.4.1	✓	8
75.29	odd	10		625.2.d.o.376.4		16
75.38	even	20		125.2.e.b.99.2		8
75.41	odd	10		125.2.d.b.26.4		16
75.44	odd	10		625.2.d.o.251.4		16
75.47	even	20		625.2.e.i.249.2		8
75.53	even	20		625.2.e.a.249.1		8
75.56	odd	10		625.2.d.o.251.1		16
75.59	odd	10		125.2.d.b.26.1		16
75.62	even	20		25.2.e.a.19.1	yes	8
75.71	odd	10		625.2.d.o.376.1		16
300.23	odd	20		400.2.y.c.129.1		8
300.287	odd	20		400.2.y.c.369.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.4.1	✓	8	75.23	even	20
25.2.e.a.19.1	yes	8	75.62	even	20
125.2.d.b.26.1		16	75.59	odd	10
125.2.d.b.26.4		16	75.41	odd	10
125.2.d.b.101.1		16	75.14	odd	10
125.2.d.b.101.4		16	75.11	odd	10
125.2.e.b.24.2		8	75.2	even	20
125.2.e.b.99.2		8	75.38	even	20
225.2.m.a.19.2		8	25.12	odd	20
225.2.m.a.154.2		8	25.23	odd	20
400.2.y.c.129.1		8	300.23	odd	20
400.2.y.c.369.1		8	300.287	odd	20
625.2.a.f.1.2		8	15.14	odd	2
625.2.a.f.1.7		8	3.2	odd	2
625.2.b.c.624.2		8	15.8	even	4
625.2.b.c.624.7		8	15.2	even	4
625.2.d.o.251.1		16	75.56	odd	10
625.2.d.o.251.4		16	75.44	odd	10
625.2.d.o.376.1		16	75.71	odd	10
625.2.d.o.376.4		16	75.29	odd	10
625.2.e.a.249.1		8	75.53	even	20
625.2.e.a.374.1		8	75.17	even	20
625.2.e.i.249.2		8	75.47	even	20
625.2.e.i.374.2		8	75.8	even	20
5625.2.a.x.1.2		8	1.1	even	1	trivial
5625.2.a.x.1.7		8	5.4	even	2	inner
10000.2.a.bj.1.2		8	12.11	even	2
10000.2.a.bj.1.7		8	60.59	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08529 q^{2} +2.34841 q^{4} -0.992398 q^{7} -0.726543 q^{8} +O(q^{10})\)	\(q-2.08529 q^{2} +2.34841 q^{4} -0.992398 q^{7} -0.726543 q^{8} -2.00000 q^{11} +3.37406 q^{13} +2.06943 q^{14} -3.18178 q^{16} +2.89451 q^{17} -2.58448 q^{19} +4.17057 q^{22} -4.54963 q^{23} -7.03588 q^{26} -2.33056 q^{28} +5.38430 q^{29} +0.136538 q^{31} +8.08800 q^{32} -6.03588 q^{34} +2.14910 q^{37} +5.38938 q^{38} -8.63318 q^{41} -4.64398 q^{43} -4.69683 q^{44} +9.48728 q^{46} +9.92630 q^{47} -6.01515 q^{49} +7.92369 q^{52} +7.56521 q^{53} +0.721020 q^{56} -11.2278 q^{58} -4.91775 q^{59} -2.76972 q^{61} -0.284720 q^{62} -10.5022 q^{64} -2.18577 q^{67} +6.79751 q^{68} -9.64254 q^{71} -0.775929 q^{73} -4.48150 q^{74} -6.06943 q^{76} +1.98480 q^{77} +15.8508 q^{79} +18.0026 q^{82} -1.77110 q^{83} +9.68401 q^{86} +1.45309 q^{88} -14.5080 q^{89} -3.34841 q^{91} -10.6844 q^{92} -20.6992 q^{94} +17.0291 q^{97} +12.5433 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4}+O(q^{10}) \) 8 * q + 6 * q^4	\( 8 q + 6 q^{4} - 16 q^{11} - 12 q^{14} - 2 q^{16} + 10 q^{19} - 6 q^{26} - 20 q^{29} + 16 q^{31} + 2 q^{34} - 26 q^{41} - 12 q^{44} + 6 q^{46} - 14 q^{49} - 10 q^{56} - 30 q^{59} + 6 q^{61} - 44 q^{64} - 46 q^{71} - 12 q^{74} - 20 q^{76} + 10 q^{79} + 14 q^{86} - 30 q^{89} - 14 q^{91} - 68 q^{94}+O(q^{100}) \) 8 * q + 6 * q^4 - 16 * q^11 - 12 * q^14 - 2 * q^16 + 10 * q^19 - 6 * q^26 - 20 * q^29 + 16 * q^31 + 2 * q^34 - 26 * q^41 - 12 * q^44 + 6 * q^46 - 14 * q^49 - 10 * q^56 - 30 * q^59 + 6 * q^61 - 44 * q^64 - 46 * q^71 - 12 * q^74 - 20 * q^76 + 10 * q^79 + 14 * q^86 - 30 * q^89 - 14 * q^91 - 68 * q^94

Introduction

L-functions

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