LMFDB - Embedded newform 225.10.b.g.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$115.883063137$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{4729})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2365x^{2} + 1397124 $ x^4 + 2365*x^2 + 1397124
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$-33.8839i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.10.b.g.199.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.8839i q^{2} -107.207 q^{4} +4010.50i q^{7} -10072.8i q^{8} +O(q^{10})$	$q-24.8839i q^{2} -107.207 q^{4} +4010.50i q^{7} -10072.8i q^{8} -84861.3 q^{11} +119425. i q^{13} +99796.8 q^{14} -305541. q^{16} +116934. i q^{17} +234932. q^{19} +2.11168e6i q^{22} -2.34570e6i q^{23} +2.97176e6 q^{26} -429953. i q^{28} -464196. q^{29} -5.11766e6 q^{31} +2.44574e6i q^{32} +2.90976e6 q^{34} -8.69354e6i q^{37} -5.84601e6i q^{38} +9.05805e6 q^{41} +8.63491e6i q^{43} +9.09769e6 q^{44} -5.83701e7 q^{46} +3.31511e7i q^{47} +2.42695e7 q^{49} -1.28032e7i q^{52} +6.41254e7i q^{53} +4.03971e7 q^{56} +1.15510e7i q^{58} +1.49407e8 q^{59} +1.54634e8 q^{61} +1.27347e8i q^{62} -9.55772e7 q^{64} -2.72755e8i q^{67} -1.25360e7i q^{68} +3.56924e8 q^{71} +2.06253e8i q^{73} -2.16329e8 q^{74} -2.51862e7 q^{76} -3.40337e8i q^{77} +4.04380e8 q^{79} -2.25399e8i q^{82} +5.17034e6i q^{83} +2.14870e8 q^{86} +8.54793e8i q^{88} +4.32242e8 q^{89} -4.78955e8 q^{91} +2.51475e8i q^{92} +8.24928e8 q^{94} +1.32066e9i q^{97} -6.03918e8i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3042 q^{4}+O(q^{10}) $ 4 * q - 3042 * q^4	$ 4 q - 3042 q^{4} - 70976 q^{11} - 490392 q^{14} + 1414530 q^{16} + 806592 q^{19} + 3815092 q^{26} - 149144 q^{29} - 10054256 q^{31} - 17721764 q^{34} - 28422664 q^{41} - 121411904 q^{44} - 302983296 q^{46} + 5639900 q^{49} + 703018680 q^{56} + 375726272 q^{59} + 308160120 q^{61} - 1276602178 q^{64} + 456541952 q^{71} - 724038452 q^{74} - 526437448 q^{76} + 1864813520 q^{79} - 1247853416 q^{86} + 449036328 q^{89} - 1339205056 q^{91} - 3863301632 q^{94}+O(q^{100}) $ 4 * q - 3042 * q^4 - 70976 * q^11 - 490392 * q^14 + 1414530 * q^16 + 806592 * q^19 + 3815092 * q^26 - 149144 * q^29 - 10054256 * q^31 - 17721764 * q^34 - 28422664 * q^41 - 121411904 * q^44 - 302983296 * q^46 + 5639900 * q^49 + 703018680 * q^56 + 375726272 * q^59 + 308160120 * q^61 - 1276602178 * q^64 + 456541952 * q^71 - 724038452 * q^74 - 526437448 * q^76 + 1864813520 * q^79 - 1247853416 * q^86 + 449036328 * q^89 - 1339205056 * q^91 - 3863301632 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 24.8839i	− 1.09972i	−0.835256	−	0.549861i	$-0.814681\pi$
$2$	− 24.8839i	− 1.09972i	0.835256	−	0.549861i	$-0.185319\pi$
$3$	0	0
$3$	0	0
$4$	−107.207	−0.209388
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4010.50i	0.631332i	0.948870	+	0.315666i	$0.102228\pi$
$7$	4010.50i	0.631332i	−0.948870	+	0.315666i	$0.897772\pi$
$8$	− 10072.8i	− 0.869453i
$9$	0	0
$10$	0	0
$11$	−84861.3	−1.74760	−0.873801	−	0.486283i	$-0.838352\pi$
$11$	−84861.3	−1.74760	−0.873801	+	0.486283i	$0.838352\pi$
$12$	0	0
$13$	119425.i	1.15971i	0.814718	+	0.579857i	$0.196892\pi$
$13$	119425.i	1.15971i	−0.814718	+	0.579857i	$0.803108\pi$
$14$	99796.8	0.694289
$15$	0	0
$16$	−305541.	−1.16554
$17$	116934.i	0.339562i	0.985482	+	0.169781i	$0.0543060\pi$
$17$	116934.i	0.339562i	−0.985482	+	0.169781i	$0.945694\pi$
$18$	0	0
$19$	234932.	0.413571	0.206786	−	0.978386i	$-0.433700\pi$
$19$	234932.	0.413571	0.206786	+	0.978386i	$0.433700\pi$
$20$	0	0
$21$	0	0
$22$	2.11168e6i	1.92188i
$23$	− 2.34570e6i	− 1.74782i	−0.486084	−	0.873912i	$-0.661575\pi$
$23$	− 2.34570e6i	− 1.74782i	0.486084	−	0.873912i	$-0.338425\pi$
$24$	0	0
$25$	0	0
$26$	2.97176e6	1.27536
$27$	0	0
$28$	− 429953.i	− 0.132193i
$29$	−464196.	−0.121874	−0.0609369	−	0.998142i	$-0.519409\pi$
$29$	−464196.	−0.121874	−0.0609369	+	0.998142i	$0.519409\pi$
$30$	0	0
$31$	−5.11766e6	−0.995277	−0.497638	−	0.867385i	$-0.665799\pi$
$31$	−5.11766e6	−0.995277	−0.497638	+	0.867385i	$0.665799\pi$
$32$	2.44574e6i	0.412321i
$33$	0	0
$34$	2.90976e6	0.373423
$35$	0	0
$36$	0	0
$37$	− 8.69354e6i	− 0.762586i	−0.924454	−	0.381293i	$-0.875479\pi$
$37$	− 8.69354e6i	− 0.762586i	0.924454	−	0.381293i	$-0.124521\pi$
$38$	− 5.84601e6i	− 0.454813i
$39$	0	0
$40$	0	0
$41$	9.05805e6	0.500619	0.250310	−	0.968166i	$-0.419468\pi$
$41$	9.05805e6	0.500619	0.250310	+	0.968166i	$0.419468\pi$
$42$	0	0
$43$	8.63491e6i	0.385168i	0.981281	+	0.192584i	$0.0616867\pi$
$43$	8.63491e6i	0.385168i	−0.981281	+	0.192584i	$0.938313\pi$
$44$	9.09769e6	0.365927
$45$	0	0
$46$	−5.83701e7	−1.92212
$47$	3.31511e7i	0.990964i	0.868618	+	0.495482i	$0.165009\pi$
$47$	3.31511e7i	0.990964i	−0.868618	+	0.495482i	$0.834991\pi$
$48$	0	0
$49$	2.42695e7	0.601420
$50$	0	0
$51$	0	0
$52$	− 1.28032e7i	− 0.242830i
$53$	6.41254e7i	1.11632i	0.829733	+	0.558160i	$0.188492\pi$
$53$	6.41254e7i	1.11632i	−0.829733	+	0.558160i	$0.811508\pi$
$54$	0	0
$55$	0	0
$56$	4.03971e7	0.548914
$57$	0	0
$58$	1.15510e7i	0.134027i
$59$	1.49407e8	1.60523	0.802613	−	0.596500i	$-0.203442\pi$
$59$	1.49407e8	1.60523	0.802613	+	0.596500i	$0.203442\pi$
$60$	0	0
$61$	1.54634e8	1.42995	0.714973	−	0.699152i	$-0.246439\pi$
$61$	1.54634e8	1.42995	0.714973	+	0.699152i	$0.246439\pi$
$62$	1.27347e8i	1.09453i
$63$	0	0
$64$	−9.55772e7	−0.712106
$65$	0	0
$66$	0	0
$67$	− 2.72755e8i	− 1.65362i	−0.562479	−	0.826811i	$-0.690152\pi$
$67$	− 2.72755e8i	− 1.65362i	0.562479	−	0.826811i	$-0.309848\pi$
$68$	− 1.25360e7i	− 0.0711001i
$69$	0	0
$70$	0	0
$71$	3.56924e8	1.66691	0.833457	−	0.552584i	$-0.186358\pi$
$71$	3.56924e8	1.66691	0.833457	+	0.552584i	$0.186358\pi$
$72$	0	0
$73$	2.06253e8i	0.850057i	0.905180	+	0.425029i	$0.139736\pi$
$73$	2.06253e8i	0.850057i	−0.905180	+	0.425029i	$0.860264\pi$
$74$	−2.16329e8	−0.838632
$75$	0	0
$76$	−2.51862e7	−0.0865968
$77$	− 3.40337e8i	− 1.10332i
$78$	0	0
$79$	4.04380e8	1.16807	0.584033	−	0.811730i	$-0.301474\pi$
$79$	4.04380e8	1.16807	0.584033	+	0.811730i	$0.301474\pi$
$80$	0	0
$81$	0	0
$82$	− 2.25399e8i	− 0.550542i
$83$	5.17034e6i	0.0119582i	0.999982	+	0.00597912i	$0.00190323\pi$
$83$	5.17034e6i	0.0119582i	−0.999982	+	0.00597912i	$0.998097\pi$
$84$	0	0
$85$	0	0
$86$	2.14870e8	0.423577
$87$	0	0
$88$	8.54793e8i	1.51946i
$89$	4.32242e8	0.730250	0.365125	−	0.930958i	$-0.381026\pi$
$89$	4.32242e8	0.730250	0.365125	+	0.930958i	$0.381026\pi$
$90$	0	0
$91$	−4.78955e8	−0.732165
$92$	2.51475e8i	0.365973i
$93$	0	0
$94$	8.24928e8	1.08978
$95$	0	0
$96$	0	0
$97$	1.32066e9i	1.51468i	0.653023	+	0.757338i	$0.273501\pi$
$97$	1.32066e9i	1.51468i	−0.653023	+	0.757338i	$0.726499\pi$
$98$	− 6.03918e8i	− 0.661395i
$99$	0	0
$100$	0	0
$101$	−5.30458e8	−0.507230	−0.253615	−	0.967305i	$-0.581620\pi$
$101$	−5.30458e8	−0.507230	−0.253615	+	0.967305i	$0.581620\pi$
$102$	0	0
$103$	− 6.07207e7i	− 0.0531580i	−0.999647	−	0.0265790i	$-0.991539\pi$
$103$	− 6.07207e7i	− 0.0531580i	0.999647	−	0.0265790i	$-0.00846136\pi$
$104$	1.20295e9	1.00832
$105$	0	0
$106$	1.59569e9	1.22764
$107$	− 1.00828e9i	− 0.743625i	−0.928308	−	0.371812i	$-0.878736\pi$
$107$	− 1.00828e9i	− 0.743625i	0.928308	−	0.371812i	$-0.121264\pi$
$108$	0	0
$109$	1.77424e9	1.20391	0.601954	−	0.798531i	$-0.294389\pi$
$109$	1.77424e9	1.20391	0.601954	+	0.798531i	$0.294389\pi$
$110$	0	0
$111$	0	0
$112$	− 1.22537e9i	− 0.735846i
$113$	− 9.45495e8i	− 0.545514i	−0.962083	−	0.272757i	$-0.912064\pi$
$113$	− 9.45495e8i	− 0.545514i	0.962083	−	0.272757i	$-0.0879356\pi$
$114$	0	0
$115$	0	0
$116$	4.97649e7	0.0255189
$117$	0	0
$118$	− 3.71782e9i	− 1.76530i
$119$	−4.68962e8	−0.214376
$120$	0	0
$121$	4.84349e9	2.05411
$122$	− 3.84788e9i	− 1.57254i
$123$	0	0
$124$	5.48647e8	0.208399
$125$	0	0
$126$	0	0
$127$	5.19758e9i	1.77290i	0.462825	+	0.886450i	$0.346836\pi$
$127$	5.19758e9i	1.77290i	−0.462825	+	0.886450i	$0.653164\pi$
$128$	3.63055e9i	1.19544i
$129$	0	0
$130$	0	0
$131$	−5.28408e8	−0.156765	−0.0783824	−	0.996923i	$-0.524976\pi$
$131$	−5.28408e8	−0.156765	−0.0783824	+	0.996923i	$0.524976\pi$
$132$	0	0
$133$	9.42194e8i	0.261101i
$134$	−6.78720e9	−1.81852
$135$	0	0
$136$	1.17785e9	0.295233
$137$	5.01761e9i	1.21690i	0.793593	+	0.608449i	$0.208208\pi$
$137$	5.01761e9i	1.21690i	−0.793593	+	0.608449i	$0.791792\pi$
$138$	0	0
$139$	−3.51872e9	−0.799499	−0.399750	−	0.916624i	$-0.630903\pi$
$139$	−3.51872e9	−0.799499	−0.399750	+	0.916624i	$0.630903\pi$
$140$	0	0
$141$	0	0
$142$	− 8.88165e9i	− 1.83314i
$143$	− 1.01346e10i	− 2.02672i
$144$	0	0
$145$	0	0
$146$	5.13238e9	0.934827
$147$	0	0
$148$	9.32005e8i	0.159676i
$149$	4.32815e9	0.719390	0.359695	−	0.933070i	$-0.382881\pi$
$149$	4.32815e9	0.719390	0.359695	+	0.933070i	$0.382881\pi$
$150$	0	0
$151$	−5.61832e9	−0.879448	−0.439724	−	0.898133i	$-0.644924\pi$
$151$	−5.61832e9	−0.879448	−0.439724	+	0.898133i	$0.644924\pi$
$152$	− 2.36642e9i	− 0.359581i
$153$	0	0
$154$	−8.46889e9	−1.21334
$155$	0	0
$156$	0	0
$157$	− 1.55603e9i	− 0.204394i	−0.994764	−	0.102197i	$-0.967413\pi$
$157$	− 1.55603e9i	− 0.204394i	0.994764	−	0.102197i	$-0.0325873\pi$
$158$	− 1.00625e10i	− 1.28455i
$159$	0	0
$160$	0	0
$161$	9.40745e9	1.10346
$162$	0	0
$163$	− 1.15580e10i	− 1.28245i	−0.767354	−	0.641224i	$-0.778427\pi$
$163$	− 1.15580e10i	− 1.28245i	0.767354	−	0.641224i	$-0.221573\pi$
$164$	−9.71083e8	−0.104824
$165$	0	0
$166$	1.28658e8	0.0131507
$167$	− 1.50486e10i	− 1.49717i	−0.663039	−	0.748585i	$-0.730734\pi$
$167$	− 1.50486e10i	− 1.49717i	0.663039	−	0.748585i	$-0.269266\pi$
$168$	0	0
$169$	−3.65789e9	−0.344938
$170$	0	0
$171$	0	0
$172$	− 9.25719e8i	− 0.0806494i
$173$	− 2.23157e10i	− 1.89410i	−0.321081	−	0.947052i	$-0.604046\pi$
$173$	− 2.23157e10i	− 1.89410i	0.321081	−	0.947052i	$-0.395954\pi$
$174$	0	0
$175$	0	0
$176$	2.59286e10	2.03691
$177$	0	0
$178$	− 1.07558e10i	− 0.803072i
$179$	−1.73543e10	−1.26348	−0.631742	−	0.775179i	$-0.717660\pi$
$179$	−1.73543e10	−1.26348	−0.631742	+	0.775179i	$0.717660\pi$
$180$	0	0
$181$	−1.34040e10	−0.928282	−0.464141	−	0.885761i	$-0.653637\pi$
$181$	−1.34040e10	−0.928282	−0.464141	+	0.885761i	$0.653637\pi$
$182$	1.19183e10i	0.805178i
$183$	0	0
$184$	−2.36278e10	−1.51965
$185$	0	0
$186$	0	0
$187$	− 9.92313e9i	− 0.593419i
$188$	− 3.55402e9i	− 0.207496i
$189$	0	0
$190$	0	0
$191$	6.01312e9	0.326926	0.163463	−	0.986549i	$-0.447734\pi$
$191$	6.01312e9	0.326926	0.163463	+	0.986549i	$0.447734\pi$
$192$	0	0
$193$	− 6.91844e9i	− 0.358922i	−0.983765	−	0.179461i	$-0.942565\pi$
$193$	− 6.91844e9i	− 0.358922i	0.983765	−	0.179461i	$-0.0574354\pi$
$194$	3.28632e10	1.66572
$195$	0	0
$196$	−2.60185e9	−0.125930
$197$	1.66139e10i	0.785909i	0.919558	+	0.392955i	$0.128547\pi$
$197$	1.66139e10i	0.785909i	−0.919558	+	0.392955i	$0.871453\pi$
$198$	0	0
$199$	3.06711e10	1.38641	0.693204	−	0.720742i	$-0.256199\pi$
$199$	3.06711e10	1.38641	0.693204	+	0.720742i	$0.256199\pi$
$200$	0	0
$201$	0	0
$202$	1.31998e10i	0.557812i
$203$	− 1.86166e9i	− 0.0769428i
$204$	0	0
$205$	0	0
$206$	−1.51096e9	−0.0584591
$207$	0	0
$208$	− 3.64893e10i	− 1.35170i
$209$	−1.99366e10	−0.722758
$210$	0	0
$211$	3.85598e10	1.33926	0.669628	−	0.742697i	$-0.266454\pi$
$211$	3.85598e10	1.33926	0.669628	+	0.742697i	$0.266454\pi$
$212$	− 6.87467e9i	− 0.233744i
$213$	0	0
$214$	−2.50899e10	−0.817780
$215$	0	0
$216$	0	0
$217$	− 2.05244e10i	− 0.628350i
$218$	− 4.41500e10i	− 1.32396i
$219$	0	0
$220$	0	0
$221$	−1.39648e10	−0.393795
$222$	0	0
$223$	− 3.11357e10i	− 0.843115i	−0.906802	−	0.421558i	$-0.861484\pi$
$223$	− 3.11357e10i	− 0.843115i	0.906802	−	0.421558i	$-0.138516\pi$
$224$	−9.80866e9	−0.260312
$225$	0	0
$226$	−2.35276e10	−0.599914
$227$	− 1.14589e10i	− 0.286436i	−0.989691	−	0.143218i	$-0.954255\pi$
$227$	− 1.14589e10i	− 0.286436i	0.989691	−	0.143218i	$-0.0457451\pi$
$228$	0	0
$229$	3.04556e10	0.731825	0.365913	−	0.930649i	$-0.380757\pi$
$229$	3.04556e10	0.731825	0.365913	+	0.930649i	$0.380757\pi$
$230$	0	0
$231$	0	0
$232$	4.67576e9i	0.105964i
$233$	− 2.83630e9i	− 0.0630451i	−0.999503	−	0.0315225i	$-0.989964\pi$
$233$	− 2.83630e9i	− 0.0630451i	0.999503	−	0.0315225i	$-0.0100356\pi$
$234$	0	0
$235$	0	0
$236$	−1.60174e10	−0.336115
$237$	0	0
$238$	1.16696e10i	0.235754i
$239$	−6.25862e10	−1.24076	−0.620381	−	0.784301i	$-0.713022\pi$
$239$	−6.25862e10	−1.24076	−0.620381	+	0.784301i	$0.713022\pi$
$240$	0	0
$241$	7.24015e10	1.38252	0.691259	−	0.722607i	$-0.257056\pi$
$241$	7.24015e10	1.38252	0.691259	+	0.722607i	$0.257056\pi$
$242$	− 1.20525e11i	− 2.25895i
$243$	0	0
$244$	−1.65778e10	−0.299414
$245$	0	0
$246$	0	0
$247$	2.80568e10i	0.479624i
$248$	5.15493e10i	0.865347i
$249$	0	0
$250$	0	0
$251$	5.65927e10	0.899971	0.449986	−	0.893036i	$-0.351429\pi$
$251$	5.65927e10	0.899971	0.449986	+	0.893036i	$0.351429\pi$
$252$	0	0
$253$	1.99059e11i	3.05450i
$254$	1.29336e11	1.94970
$255$	0	0
$256$	4.14066e10	0.602545
$257$	4.95688e10i	0.708777i	0.935098	+	0.354388i	$0.115311\pi$
$257$	4.95688e10i	0.708777i	−0.935098	+	0.354388i	$0.884689\pi$
$258$	0	0
$259$	3.48655e10	0.481445
$260$	0	0
$261$	0	0
$262$	1.31488e10i	0.172398i
$263$	− 3.59498e10i	− 0.463336i	−0.972795	−	0.231668i	$-0.925582\pi$
$263$	− 3.59498e10i	− 0.463336i	0.972795	−	0.231668i	$-0.0744183\pi$
$264$	0	0
$265$	0	0
$266$	2.34454e10	0.287138
$267$	0	0
$268$	2.92412e10i	0.346249i
$269$	−7.09650e10	−0.826340	−0.413170	−	0.910654i	$-0.635579\pi$
$269$	−7.09650e10	−0.826340	−0.413170	+	0.910654i	$0.635579\pi$
$270$	0	0
$271$	1.47201e11	1.65786	0.828930	−	0.559352i	$-0.188950\pi$
$271$	1.47201e11	1.65786	0.828930	+	0.559352i	$0.188950\pi$
$272$	− 3.57279e10i	− 0.395774i
$273$	0	0
$274$	1.24857e11	1.33825
$275$	0	0
$276$	0	0
$277$	2.87336e10i	0.293245i	0.989193	+	0.146622i	$0.0468402\pi$
$277$	2.87336e10i	0.293245i	−0.989193	+	0.146622i	$0.953160\pi$
$278$	8.75593e10i	0.879227i
$279$	0	0
$280$	0	0
$281$	−5.17071e10	−0.494734	−0.247367	−	0.968922i	$-0.579565\pi$
$281$	−5.17071e10	−0.494734	−0.247367	+	0.968922i	$0.579565\pi$
$282$	0	0
$283$	− 3.22434e9i	− 0.0298814i	−0.999888	−	0.0149407i	$-0.995244\pi$
$283$	− 3.22434e9i	− 0.0298814i	0.999888	−	0.0149407i	$-0.00475595\pi$
$284$	−3.82646e10	−0.349032
$285$	0	0
$286$	−2.52188e11	−2.22883
$287$	3.63273e10i	0.316057i
$288$	0	0
$289$	1.04914e11	0.884698
$290$	0	0
$291$	0	0
$292$	− 2.21117e10i	− 0.177992i
$293$	1.13591e11i	0.900409i	0.892926	+	0.450204i	$0.148649\pi$
$293$	1.13591e11i	0.900409i	−0.892926	+	0.450204i	$0.851351\pi$
$294$	0	0
$295$	0	0
$296$	−8.75685e10	−0.663033
$297$	0	0
$298$	− 1.07701e11i	− 0.791129i
$299$	2.80136e11	2.02698
$300$	0	0
$301$	−3.46303e10	−0.243169
$302$	1.39805e11i	0.967148i
$303$	0	0
$304$	−7.17811e10	−0.482036
$305$	0	0
$306$	0	0
$307$	2.30543e10i	0.148125i	0.997254	+	0.0740627i	$0.0235965\pi$
$307$	2.30543e10i	0.148125i	−0.997254	+	0.0740627i	$0.976403\pi$
$308$	3.64863e10i	0.231021i
$309$	0	0
$310$	0	0
$311$	−7.71709e10	−0.467769	−0.233885	−	0.972264i	$-0.575144\pi$
$311$	−7.71709e10	−0.467769	−0.233885	+	0.972264i	$0.575144\pi$
$312$	0	0
$313$	− 4.68832e10i	− 0.276101i	−0.990425	−	0.138050i	$-0.955916\pi$
$313$	− 4.68832e10i	− 0.276101i	0.990425	−	0.138050i	$-0.0440836\pi$
$314$	−3.87200e10	−0.224777
$315$	0	0
$316$	−4.33522e10	−0.244579
$317$	2.19420e11i	1.22042i	0.792240	+	0.610210i	$0.208915\pi$
$317$	2.19420e11i	1.22042i	−0.792240	+	0.610210i	$0.791085\pi$
$318$	0	0
$319$	3.93923e10	0.212987
$320$	0	0
$321$	0	0
$322$	− 2.34094e11i	− 1.21350i
$323$	2.74714e10i	0.140433i
$324$	0	0
$325$	0	0
$326$	−2.87609e11	−1.41034
$327$	0	0
$328$	− 9.12401e10i	− 0.435265i
$329$	−1.32953e11	−0.625627
$330$	0	0
$331$	3.89075e9	0.0178159	0.00890794	−	0.999960i	$-0.497164\pi$
$331$	3.89075e9	0.0178159	0.00890794	+	0.999960i	$0.497164\pi$
$332$	− 5.54295e8i	− 0.00250391i
$333$	0	0
$334$	−3.74466e11	−1.64647
$335$	0	0
$336$	0	0
$337$	1.48259e11i	0.626163i	0.949726	+	0.313082i	$0.101361\pi$
$337$	1.48259e11i	0.626163i	−0.949726	+	0.313082i	$0.898639\pi$
$338$	9.10225e10i	0.379336i
$339$	0	0
$340$	0	0
$341$	4.34291e11	1.73935
$342$	0	0
$343$	2.59171e11i	1.01103i
$344$	8.69779e10	0.334885
$345$	0	0
$346$	−5.55302e11	−2.08299
$347$	− 1.63695e11i	− 0.606114i	−0.952972	−	0.303057i	$-0.901993\pi$
$347$	− 1.63695e11i	− 0.606114i	0.952972	−	0.303057i	$-0.0980072\pi$
$348$	0	0
$349$	−7.02057e10	−0.253313	−0.126657	−	0.991947i	$-0.540425\pi$
$349$	−7.02057e10	−0.253313	−0.126657	+	0.991947i	$0.540425\pi$
$350$	0	0
$351$	0	0
$352$	− 2.07549e11i	− 0.720574i
$353$	1.55727e10i	0.0533798i	0.999644	+	0.0266899i	$0.00849667\pi$
$353$	1.55727e10i	0.0533798i	−0.999644	+	0.0266899i	$0.991503\pi$
$354$	0	0
$355$	0	0
$356$	−4.63392e10	−0.152906
$357$	0	0
$358$	4.31843e11i	1.38948i
$359$	3.28525e10	0.104386	0.0521931	−	0.998637i	$-0.483379\pi$
$359$	3.28525e10	0.104386	0.0521931	+	0.998637i	$0.483379\pi$
$360$	0	0
$361$	−2.67495e11	−0.828959
$362$	3.33543e11i	1.02085i
$363$	0	0
$364$	5.13472e10	0.153306
$365$	0	0
$366$	0	0
$367$	4.76030e11i	1.36974i	0.728667	+	0.684868i	$0.240140\pi$
$367$	4.76030e11i	1.36974i	−0.728667	+	0.684868i	$0.759860\pi$
$368$	7.16707e11i	2.03717i
$369$	0	0
$370$	0	0
$371$	−2.57175e11	−0.704768
$372$	0	0
$373$	− 9.72745e10i	− 0.260201i	−0.991501	−	0.130101i	$-0.958470\pi$
$373$	− 9.72745e10i	− 0.260201i	0.991501	−	0.130101i	$-0.0415300\pi$
$374$	−2.46926e11	−0.652596
$375$	0	0
$376$	3.33925e11	0.861597
$377$	− 5.54367e10i	− 0.141339i
$378$	0	0
$379$	−4.45171e11	−1.10828	−0.554141	−	0.832423i	$-0.686953\pi$
$379$	−4.45171e11	−1.10828	−0.554141	+	0.832423i	$0.686953\pi$
$380$	0	0
$381$	0	0
$382$	− 1.49630e11i	− 0.359528i
$383$	− 2.19414e11i	− 0.521039i	−0.965469	−	0.260519i	$-0.916106\pi$
$383$	− 2.19414e11i	− 0.521039i	0.965469	−	0.260519i	$-0.0838938\pi$
$384$	0	0
$385$	0	0
$386$	−1.72157e11	−0.394714
$387$	0	0
$388$	− 1.41584e11i	− 0.317155i
$389$	−3.71956e11	−0.823603	−0.411801	−	0.911274i	$-0.635100\pi$
$389$	−3.71956e11	−0.823603	−0.411801	+	0.911274i	$0.635100\pi$
$390$	0	0
$391$	2.74291e11	0.593494
$392$	− 2.44462e11i	− 0.522907i
$393$	0	0
$394$	4.13417e11	0.864281
$395$	0	0
$396$	0	0
$397$	1.38786e11i	0.280406i	0.990123	+	0.140203i	$0.0447756\pi$
$397$	1.38786e11i	0.280406i	−0.990123	+	0.140203i	$0.955224\pi$
$398$	− 7.63216e11i	− 1.52466i
$399$	0	0
$400$	0	0
$401$	−6.95127e11	−1.34250	−0.671251	−	0.741231i	$-0.734243\pi$
$401$	−6.95127e11	−1.34250	−0.671251	+	0.741231i	$0.734243\pi$
$402$	0	0
$403$	− 6.11178e11i	− 1.15424i
$404$	5.68686e10	0.106208
$405$	0	0
$406$	−4.63253e10	−0.0846157
$407$	7.37745e11i	1.33270i
$408$	0	0
$409$	−9.10800e11	−1.60942	−0.804708	−	0.593671i	$-0.797678\pi$
$409$	−9.10800e11	−1.60942	−0.804708	+	0.593671i	$0.797678\pi$
$410$	0	0
$411$	0	0
$412$	6.50966e9i	0.0111307i
$413$	5.99196e11i	1.01343i
$414$	0	0
$415$	0	0
$416$	−2.92084e11	−0.478175
$417$	0	0
$418$	4.96100e11i	0.794832i
$419$	3.11152e11	0.493184	0.246592	−	0.969119i	$-0.420689\pi$
$419$	3.11152e11	0.493184	0.246592	+	0.969119i	$0.420689\pi$
$420$	0	0
$421$	7.63941e11	1.18520	0.592598	−	0.805498i	$-0.298102\pi$
$421$	7.63941e11	1.18520	0.592598	+	0.805498i	$0.298102\pi$
$422$	− 9.59516e11i	− 1.47281i
$423$	0	0
$424$	6.45924e11	0.970588
$425$	0	0
$426$	0	0
$427$	6.20159e11i	0.902771i
$428$	1.08094e11i	0.155706i
$429$	0	0
$430$	0	0
$431$	−6.54629e10	−0.0913792	−0.0456896	−	0.998956i	$-0.514549\pi$
$431$	−6.54629e10	−0.0913792	−0.0456896	+	0.998956i	$0.514549\pi$
$432$	0	0
$433$	− 7.87434e11i	− 1.07651i	−0.842781	−	0.538256i	$-0.819083\pi$
$433$	− 7.87434e11i	− 1.07651i	0.842781	−	0.538256i	$-0.180917\pi$
$434$	−5.10726e11	−0.691010
$435$	0	0
$436$	−1.90210e11	−0.252084
$437$	− 5.51080e11i	− 0.722850i
$438$	0	0
$439$	8.39211e11	1.07840	0.539201	−	0.842177i	$-0.318726\pi$
$439$	8.39211e11	1.07840	0.539201	+	0.842177i	$0.318726\pi$
$440$	0	0
$441$	0	0
$442$	3.47499e11i	0.433065i
$443$	1.28252e12i	1.58215i	0.611720	+	0.791074i	$0.290478\pi$
$443$	1.28252e12i	1.58215i	−0.611720	+	0.791074i	$0.709522\pi$
$444$	0	0
$445$	0	0
$446$	−7.74777e11	−0.927192
$447$	0	0
$448$	− 3.83313e11i	− 0.449575i
$449$	8.02030e10	0.0931284	0.0465642	−	0.998915i	$-0.485173\pi$
$449$	8.02030e10	0.0931284	0.0465642	+	0.998915i	$0.485173\pi$
$450$	0	0
$451$	−7.68678e11	−0.874883
$452$	1.01363e11i	0.114224i
$453$	0	0
$454$	−2.85143e11	−0.315000
$455$	0	0
$456$	0	0
$457$	− 1.27085e11i	− 0.136292i	−0.997675	−	0.0681461i	$-0.978292\pi$
$457$	− 1.27085e11i	− 0.136292i	0.997675	−	0.0681461i	$-0.0217084\pi$
$458$	− 7.57853e11i	− 0.804804i
$459$	0	0
$460$	0	0
$461$	7.39880e11	0.762970	0.381485	−	0.924375i	$-0.375413\pi$
$461$	7.39880e11	0.762970	0.381485	+	0.924375i	$0.375413\pi$
$462$	0	0
$463$	5.71025e11i	0.577485i	0.957407	+	0.288742i	$0.0932371\pi$
$463$	5.71025e11i	0.577485i	−0.957407	+	0.288742i	$0.906763\pi$
$464$	1.41831e11	0.142049
$465$	0	0
$466$	−7.05782e10	−0.0693320
$467$	1.19075e12i	1.15850i	0.815150	+	0.579249i	$0.196654\pi$
$467$	1.19075e12i	1.15850i	−0.815150	+	0.579249i	$0.803346\pi$
$468$	0	0
$469$	1.09389e12	1.04398
$470$	0	0
$471$	0	0
$472$	− 1.50495e12i	− 1.39567i
$473$	− 7.32770e11i	− 0.673120i
$474$	0	0
$475$	0	0
$476$	5.02759e10	0.0448878
$477$	0	0
$478$	1.55739e12i	1.36449i
$479$	1.46074e12	1.26784	0.633919	−	0.773400i	$-0.281445\pi$
$479$	1.46074e12	1.26784	0.633919	+	0.773400i	$0.281445\pi$
$480$	0	0
$481$	1.03823e12	0.884382
$482$	− 1.80163e12i	− 1.52038i
$483$	0	0
$484$	−5.19254e11	−0.430107
$485$	0	0
$486$	0	0
$487$	1.46931e12i	1.18368i	0.806056	+	0.591839i	$0.201598\pi$
$487$	1.46931e12i	1.18368i	−0.806056	+	0.591839i	$0.798402\pi$
$488$	− 1.55760e12i	− 1.24327i
$489$	0	0
$490$	0	0
$491$	1.76138e12	1.36769	0.683843	−	0.729629i	$-0.260307\pi$
$491$	1.76138e12	1.36769	0.683843	+	0.729629i	$0.260307\pi$
$492$	0	0
$493$	− 5.42801e10i	− 0.0413837i
$494$	6.98161e11	0.527453
$495$	0	0
$496$	1.56365e12	1.16004
$497$	1.43145e12i	1.05238i
$498$	0	0
$499$	1.63016e12	1.17700	0.588502	−	0.808496i	$-0.299718\pi$
$499$	1.63016e12	1.17700	0.588502	+	0.808496i	$0.299718\pi$
$500$	0	0
$501$	0	0
$502$	− 1.40825e12i	− 0.989718i
$503$	− 9.47625e11i	− 0.660056i	−0.943971	−	0.330028i	$-0.892942\pi$
$503$	− 9.47625e11i	− 0.660056i	0.943971	−	0.330028i	$-0.107058\pi$
$504$	0	0
$505$	0	0
$506$	4.95337e12	3.35910
$507$	0	0
$508$	− 5.57214e11i	− 0.371224i
$509$	−2.90709e11	−0.191968	−0.0959838	−	0.995383i	$-0.530600\pi$
$509$	−2.90709e11	−0.191968	−0.0959838	+	0.995383i	$0.530600\pi$
$510$	0	0
$511$	−8.27180e11	−0.536668
$512$	8.28486e11i	0.532808i
$513$	0	0
$514$	1.23346e12	0.779457
$515$	0	0
$516$	0	0
$517$	− 2.81325e12i	− 1.73181i
$518$	− 8.67587e11i	− 0.529455i
$519$	0	0
$520$	0	0
$521$	2.04811e12	1.21782	0.608910	−	0.793240i	$-0.291607\pi$
$521$	2.04811e12	1.21782	0.608910	+	0.793240i	$0.291607\pi$
$522$	0	0
$523$	8.49299e10i	0.0496367i	0.999692	+	0.0248184i	$0.00790074\pi$
$523$	8.49299e10i	0.0496367i	−0.999692	+	0.0248184i	$0.992099\pi$
$524$	5.66488e10	0.0328247
$525$	0	0
$526$	−8.94570e11	−0.509540
$527$	− 5.98426e11i	− 0.337958i
$528$	0	0
$529$	−3.70117e12	−2.05489
$530$	0	0
$531$	0	0
$532$	− 1.01009e11i	− 0.0546713i
$533$	1.08176e12i	0.580575i
$534$	0	0
$535$	0	0
$536$	−2.74741e12	−1.43775
$537$	0	0
$538$	1.76588e12i	0.908745i
$539$	−2.05954e12	−1.05104
$540$	0	0
$541$	−1.29101e12	−0.647949	−0.323974	−	0.946066i	$-0.605019\pi$
$541$	−1.29101e12	−0.647949	−0.323974	+	0.946066i	$0.605019\pi$
$542$	− 3.66292e12i	− 1.82318i
$543$	0	0
$544$	−2.85989e11	−0.140009
$545$	0	0
$546$	0	0
$547$	5.30222e11i	0.253230i	0.991952	+	0.126615i	$0.0404112\pi$
$547$	5.30222e11i	0.253230i	−0.991952	+	0.126615i	$0.959589\pi$
$548$	− 5.37921e11i	− 0.254804i
$549$	0	0
$550$	0	0
$551$	−1.09054e11	−0.0504035
$552$	0	0
$553$	1.62177e12i	0.737438i
$554$	7.15002e11	0.322488
$555$	0	0
$556$	3.77230e11	0.167405
$557$	8.17386e11i	0.359815i	0.983684	+	0.179907i	$0.0575798\pi$
$557$	8.17386e11i	0.359815i	−0.983684	+	0.179907i	$0.942420\pi$
$558$	0	0
$559$	−1.03123e12	−0.446684
$560$	0	0
$561$	0	0
$562$	1.28667e12i	0.544069i
$563$	1.42197e12i	0.596488i	0.954490	+	0.298244i	$0.0964009\pi$
$563$	1.42197e12i	0.596488i	−0.954490	+	0.298244i	$0.903599\pi$
$564$	0	0
$565$	0	0
$566$	−8.02339e10	−0.0328613
$567$	0	0
$568$	− 3.59523e12i	− 1.44930i
$569$	−1.04614e12	−0.418394	−0.209197	−	0.977874i	$-0.567085\pi$
$569$	−1.04614e12	−0.418394	−0.209197	+	0.977874i	$0.567085\pi$
$570$	0	0
$571$	2.09508e12	0.824780	0.412390	−	0.911007i	$-0.364694\pi$
$571$	2.09508e12	0.824780	0.412390	+	0.911007i	$0.364694\pi$
$572$	1.08649e12i	0.424371i
$573$	0	0
$574$	9.03965e11	0.347575
$575$	0	0
$576$	0	0
$577$	− 2.68760e12i	− 1.00942i	−0.863288	−	0.504711i	$-0.831599\pi$
$577$	− 2.68760e12i	− 1.00942i	0.863288	−	0.504711i	$-0.168401\pi$
$578$	− 2.61068e12i	− 0.972921i
$579$	0	0
$580$	0	0
$581$	−2.07357e10	−0.00754962
$582$	0	0
$583$	− 5.44176e12i	− 1.95088i
$584$	2.07755e12	0.739085
$585$	0	0
$586$	2.82658e12	0.990199
$587$	1.61925e12i	0.562913i	0.959574	+	0.281457i	$0.0908176\pi$
$587$	1.61925e12i	0.562913i	−0.959574	+	0.281457i	$0.909182\pi$
$588$	0	0
$589$	−1.20230e12	−0.411618
$590$	0	0
$591$	0	0
$592$	2.65623e12i	0.888828i
$593$	− 5.82677e12i	− 1.93500i	−0.252863	−	0.967502i	$-0.581372\pi$
$593$	− 5.82677e12i	− 1.93500i	0.252863	−	0.967502i	$-0.418628\pi$
$594$	0	0
$595$	0	0
$596$	−4.64007e11	−0.150632
$597$	0	0
$598$	− 6.97087e12i	− 2.22911i
$599$	−3.89086e12	−1.23488	−0.617441	−	0.786617i	$-0.711831\pi$
$599$	−3.89086e12	−1.23488	−0.617441	+	0.786617i	$0.711831\pi$
$600$	0	0
$601$	2.13051e12	0.666115	0.333058	−	0.942907i	$-0.391920\pi$
$601$	2.13051e12	0.666115	0.333058	+	0.942907i	$0.391920\pi$
$602$	8.61737e11i	0.267418i
$603$	0	0
$604$	6.02321e11	0.184146
$605$	0	0
$606$	0	0
$607$	2.52733e12i	0.755637i	0.925880	+	0.377818i	$0.123326\pi$
$607$	2.52733e12i	0.755637i	−0.925880	+	0.377818i	$0.876674\pi$
$608$	5.74582e11i	0.170524i
$609$	0	0
$610$	0	0
$611$	−3.95908e12	−1.14924
$612$	0	0
$613$	1.89788e12i	0.542870i	0.962457	+	0.271435i	$0.0874983\pi$
$613$	1.89788e12i	0.542870i	−0.962457	+	0.271435i	$0.912502\pi$
$614$	5.73681e11	0.162897
$615$	0	0
$616$	−3.42815e12	−0.959283
$617$	− 1.40480e12i	− 0.390240i	−0.980779	−	0.195120i	$-0.937490\pi$
$617$	− 1.40480e12i	− 0.390240i	0.980779	−	0.195120i	$-0.0625096\pi$
$618$	0	0
$619$	2.81250e12	0.769989	0.384995	−	0.922919i	$-0.374203\pi$
$619$	2.81250e12	0.769989	0.384995	+	0.922919i	$0.374203\pi$
$620$	0	0
$621$	0	0
$622$	1.92031e12i	0.514416i
$623$	1.73351e12i	0.461030i
$624$	0	0
$625$	0	0
$626$	−1.16664e12	−0.303634
$627$	0	0
$628$	1.66817e11i	0.0427977i
$629$	1.01657e12	0.258945
$630$	0	0
$631$	−1.92653e12	−0.483775	−0.241888	−	0.970304i	$-0.577767\pi$
$631$	−1.92653e12	−0.483775	−0.241888	+	0.970304i	$0.577767\pi$
$632$	− 4.07325e12i	− 1.01558i
$633$	0	0
$634$	5.46001e12	1.34212
$635$	0	0
$636$	0	0
$637$	2.89839e12i	0.697476i
$638$	− 9.80232e11i	− 0.234226i
$639$	0	0
$640$	0	0
$641$	7.10318e12	1.66185	0.830924	−	0.556385i	$-0.187812\pi$
$641$	7.10318e12	1.66185	0.830924	+	0.556385i	$0.187812\pi$
$642$	0	0
$643$	3.40733e12i	0.786077i	0.919522	+	0.393038i	$0.128576\pi$
$643$	3.40733e12i	0.786077i	−0.919522	+	0.393038i	$0.871424\pi$
$644$	−1.00854e12	−0.231051
$645$	0	0
$646$	6.83594e11	0.154437
$647$	− 4.31347e12i	− 0.967737i	−0.875141	−	0.483868i	$-0.839231\pi$
$647$	− 4.31347e12i	− 0.967737i	0.875141	−	0.483868i	$-0.160769\pi$
$648$	0	0
$649$	−1.26789e13	−2.80530
$650$	0	0
$651$	0	0
$652$	1.23910e12i	0.268529i
$653$	2.53169e12i	0.544880i	0.962173	+	0.272440i	$0.0878306\pi$
$653$	2.53169e12i	0.544880i	−0.962173	+	0.272440i	$0.912169\pi$
$654$	0	0
$655$	0	0
$656$	−2.76760e12	−0.583494
$657$	0	0
$658$	3.30838e12i	0.688016i
$659$	−4.43263e12	−0.915540	−0.457770	−	0.889071i	$-0.651352\pi$
$659$	−4.43263e12	−0.915540	−0.457770	+	0.889071i	$0.651352\pi$
$660$	0	0
$661$	9.79827e11	0.199638	0.0998189	−	0.995006i	$-0.468174\pi$
$661$	9.79827e11	0.199638	0.0998189	+	0.995006i	$0.468174\pi$
$662$	− 9.68168e10i	− 0.0195925i
$663$	0	0
$664$	5.20799e10	0.0103971
$665$	0	0
$666$	0	0
$667$	1.08887e12i	0.213014i
$668$	1.61331e12i	0.313489i
$669$	0	0
$670$	0	0
$671$	−1.31224e13	−2.49898
$672$	0	0
$673$	7.69434e12i	1.44578i	0.690961	+	0.722892i	$0.257188\pi$
$673$	7.69434e12i	1.44578i	−0.690961	+	0.722892i	$0.742812\pi$
$674$	3.68927e12	0.688605
$675$	0	0
$676$	3.92151e11	0.0722258
$677$	2.45111e12i	0.448451i	0.974537	+	0.224225i	$0.0719852\pi$
$677$	2.45111e12i	0.448451i	−0.974537	+	0.224225i	$0.928015\pi$
$678$	0	0
$679$	−5.29653e12	−0.956264
$680$	0	0
$681$	0	0
$682$	− 1.08068e13i	− 1.91280i
$683$	− 3.13070e12i	− 0.550488i	−0.961374	−	0.275244i	$-0.911241\pi$
$683$	− 3.13070e12i	− 0.550488i	0.961374	−	0.275244i	$-0.0887587\pi$
$684$	0	0
$685$	0	0
$686$	6.44918e12	1.11185
$687$	0	0
$688$	− 2.63831e12i	− 0.448930i
$689$	−7.65819e12	−1.29461
$690$	0	0
$691$	−8.92815e12	−1.48974	−0.744870	−	0.667210i	$-0.767488\pi$
$691$	−8.92815e12	−1.48974	−0.744870	+	0.667210i	$0.767488\pi$
$692$	2.39239e12i	0.396602i
$693$	0	0
$694$	−4.07338e12	−0.666556
$695$	0	0
$696$	0	0
$697$	1.05919e12i	0.169991i
$698$	1.74699e12i	0.278574i
$699$	0	0
$700$	0	0
$701$	1.44295e12	0.225695	0.112847	−	0.993612i	$-0.464003\pi$
$701$	1.44295e12	0.225695	0.112847	+	0.993612i	$0.464003\pi$
$702$	0	0
$703$	− 2.04239e12i	− 0.315383i
$704$	8.11081e12	1.24448
$705$	0	0
$706$	3.87509e11	0.0587030
$707$	− 2.12740e12i	− 0.320230i
$708$	0	0
$709$	7.40550e12	1.10064	0.550321	−	0.834953i	$-0.314505\pi$
$709$	7.40550e12	1.10064	0.550321	+	0.834953i	$0.314505\pi$
$710$	0	0
$711$	0	0
$712$	− 4.35390e12i	− 0.634919i
$713$	1.20045e13i	1.73957i
$714$	0	0
$715$	0	0
$716$	1.86050e12	0.264558
$717$	0	0
$718$	− 8.17496e11i	− 0.114796i
$719$	−3.90289e12	−0.544636	−0.272318	−	0.962207i	$-0.587790\pi$
$719$	−3.90289e12	−0.544636	−0.272318	+	0.962207i	$0.587790\pi$
$720$	0	0
$721$	2.43520e11	0.0335604
$722$	6.65631e12i	0.911624i
$723$	0	0
$724$	1.43699e12	0.194371
$725$	0	0
$726$	0	0
$727$	− 1.33674e12i	− 0.177477i	−0.996055	−	0.0887384i	$-0.971716\pi$
$727$	− 1.33674e12i	− 0.177477i	0.996055	−	0.0887384i	$-0.0282835\pi$
$728$	4.82443e12i	0.636583i
$729$	0	0
$730$	0	0
$731$	−1.00971e12	−0.130788
$732$	0	0
$733$	− 5.98863e11i	− 0.0766231i	−0.999266	−	0.0383116i	$-0.987802\pi$
$733$	− 5.98863e11i	− 0.0766231i	0.999266	−	0.0383116i	$-0.0121979\pi$
$734$	1.18455e13	1.50633
$735$	0	0
$736$	5.73699e12	0.720665
$737$	2.31464e13i	2.88987i
$738$	0	0
$739$	1.40697e13	1.73534	0.867671	−	0.497138i	$-0.165616\pi$
$739$	1.40697e13	1.73534	0.867671	+	0.497138i	$0.165616\pi$
$740$	0	0
$741$	0	0
$742$	6.39951e12i	0.775049i
$743$	2.52604e12i	0.304082i	0.988374	+	0.152041i	$0.0485846\pi$
$743$	2.52604e12i	0.304082i	−0.988374	+	0.152041i	$0.951415\pi$
$744$	0	0
$745$	0	0
$746$	−2.42056e12	−0.286149
$747$	0	0
$748$	1.06383e12i	0.124255i
$749$	4.04371e12	0.469474
$750$	0	0
$751$	3.78704e12	0.434430	0.217215	−	0.976124i	$-0.430303\pi$
$751$	3.78704e12	0.434430	0.217215	+	0.976124i	$0.430303\pi$
$752$	− 1.01290e13i	− 1.15501i
$753$	0	0
$754$	−1.37948e12	−0.155433
$755$	0	0
$756$	0	0
$757$	− 3.55646e12i	− 0.393629i	−0.980441	−	0.196814i	$-0.936940\pi$
$757$	− 3.55646e12i	− 0.393629i	0.980441	−	0.196814i	$-0.0630596\pi$
$758$	1.10776e13i	1.21880i
$759$	0	0
$760$	0	0
$761$	−1.67100e13	−1.80612	−0.903058	−	0.429519i	$-0.858683\pi$
$761$	−1.67100e13	−1.80612	−0.903058	+	0.429519i	$0.858683\pi$
$762$	0	0
$763$	7.11560e12i	0.760066i
$764$	−6.44647e11	−0.0684544
$765$	0	0
$766$	−5.45987e12	−0.572997
$767$	1.78429e13i	1.86160i
$768$	0	0
$769$	7.51512e12	0.774939	0.387469	−	0.921883i	$-0.373349\pi$
$769$	7.51512e12	0.774939	0.387469	+	0.921883i	$0.373349\pi$
$770$	0	0
$771$	0	0
$772$	7.41702e11i	0.0751540i
$773$	8.52580e12i	0.858870i	0.903098	+	0.429435i	$0.141287\pi$
$773$	8.52580e12i	0.858870i	−0.903098	+	0.429435i	$0.858713\pi$
$774$	0	0
$775$	0	0
$776$	1.33028e13	1.31694
$777$	0	0
$778$	9.25569e12i	0.905734i
$779$	2.12802e12	0.207042
$780$	0	0
$781$	−3.02890e13	−2.91310
$782$	− 6.82543e12i	− 0.652679i
$783$	0	0
$784$	−7.41531e12	−0.700982
$785$	0	0
$786$	0	0
$787$	− 3.20742e12i	− 0.298037i	−0.988834	−	0.149018i	$-0.952389\pi$
$787$	− 3.20742e12i	− 0.298037i	0.988834	−	0.149018i	$-0.0476114\pi$
$788$	− 1.78112e12i	− 0.164560i
$789$	0	0
$790$	0	0
$791$	3.79191e12	0.344401
$792$	0	0
$793$	1.84672e13i	1.65833i
$794$	3.45353e12	0.308369
$795$	0	0
$796$	−3.28815e12	−0.290297
$797$	− 1.83036e13i	− 1.60684i	−0.595412	−	0.803421i	$-0.703011\pi$
$797$	− 1.83036e13i	− 1.60684i	0.595412	−	0.803421i	$-0.296989\pi$
$798$	0	0
$799$	−3.87648e12	−0.336494
$800$	0	0
$801$	0	0
$802$	1.72975e13i	1.47638i
$803$	− 1.75029e13i	− 1.48556i
$804$	0	0
$805$	0	0
$806$	−1.52085e13	−1.26934
$807$	0	0
$808$	5.34321e12i	0.441013i
$809$	9.38919e12	0.770655	0.385327	−	0.922780i	$-0.374089\pi$
$809$	9.38919e12	0.770655	0.385327	+	0.922780i	$0.374089\pi$
$810$	0	0
$811$	2.03949e13	1.65549	0.827746	−	0.561103i	$-0.189623\pi$
$811$	2.03949e13	1.65549	0.827746	+	0.561103i	$0.189623\pi$
$812$	1.99582e11i	0.0161109i
$813$	0	0
$814$	1.83579e13	1.46560
$815$	0	0
$816$	0	0
$817$	2.02861e12i	0.159294i
$818$	2.26642e13i	1.76991i
$819$	0	0
$820$	0	0
$821$	1.52259e13	1.16961	0.584803	−	0.811176i	$-0.301172\pi$
$821$	1.52259e13	1.16961	0.584803	+	0.811176i	$0.301172\pi$
$822$	0	0
$823$	− 2.70705e12i	− 0.205682i	−0.994698	−	0.102841i	$-0.967207\pi$
$823$	− 2.70705e12i	− 0.205682i	0.994698	−	0.102841i	$-0.0327933\pi$
$824$	−6.11629e11	−0.0462184
$825$	0	0
$826$	1.49103e13	1.11449
$827$	5.67203e10i	0.00421661i	0.999998	+	0.00210831i	$0.000671095\pi$
$827$	5.67203e10i	0.00421661i	−0.999998	+	0.00210831i	$0.999329\pi$
$828$	0	0
$829$	1.43206e13	1.05309	0.526547	−	0.850146i	$-0.323487\pi$
$829$	1.43206e13	1.05309	0.526547	+	0.850146i	$0.323487\pi$
$830$	0	0
$831$	0	0
$832$	− 1.14143e13i	− 0.825839i
$833$	2.83791e12i	0.204219i
$834$	0	0
$835$	0	0
$836$	2.13734e12	0.151337
$837$	0	0
$838$	− 7.74265e12i	− 0.542365i
$839$	−2.13693e13	−1.48889	−0.744443	−	0.667686i	$-0.767285\pi$
$839$	−2.13693e13	−1.48889	−0.744443	+	0.667686i	$0.767285\pi$
$840$	0	0
$841$	−1.42917e13	−0.985147
$842$	− 1.90098e13i	− 1.30339i
$843$	0	0
$844$	−4.13386e12	−0.280424
$845$	0	0
$846$	0	0
$847$	1.94248e13i	1.29683i
$848$	− 1.95929e13i	− 1.30112i
$849$	0	0
$850$	0	0
$851$	−2.03925e13	−1.33287
$852$	0	0
$853$	− 1.18123e13i	− 0.763948i	−0.924173	−	0.381974i	$-0.875244\pi$
$853$	− 1.18123e13i	− 0.763948i	0.924173	−	0.381974i	$-0.124756\pi$
$854$	1.54319e13	0.992797
$855$	0	0
$856$	−1.01562e13	−0.646547
$857$	2.69459e13i	1.70640i	0.521587	+	0.853198i	$0.325340\pi$
$857$	2.69459e13i	1.70640i	−0.521587	+	0.853198i	$0.674660\pi$
$858$	0	0
$859$	−9.66669e12	−0.605771	−0.302885	−	0.953027i	$-0.597950\pi$
$859$	−9.66669e12	−0.605771	−0.302885	+	0.953027i	$0.597950\pi$
$860$	0	0
$861$	0	0
$862$	1.62897e12i	0.100492i
$863$	− 2.80982e12i	− 0.172437i	−0.996276	−	0.0862185i	$-0.972522\pi$
$863$	− 2.80982e12i	− 0.172437i	0.996276	−	0.0862185i	$-0.0274783\pi$
$864$	0	0
$865$	0	0
$866$	−1.95944e13	−1.18386
$867$	0	0
$868$	2.20035e12i	0.131569i
$869$	−3.43162e13	−2.04132
$870$	0	0
$871$	3.25739e13	1.91773
$872$	− 1.78716e13i	− 1.04674i
$873$	0	0
$874$	−1.37130e13	−0.794933
$875$	0	0
$876$	0	0
$877$	− 3.37045e13i	− 1.92393i	−0.273169	−	0.961966i	$-0.588072\pi$
$877$	− 3.37045e13i	− 1.92393i	0.273169	−	0.961966i	$-0.411928\pi$
$878$	− 2.08828e13i	− 1.18594i
$879$	0	0
$880$	0	0
$881$	1.20571e13	0.674297	0.337148	−	0.941452i	$-0.390538\pi$
$881$	1.20571e13	0.674297	0.337148	+	0.941452i	$0.390538\pi$
$882$	0	0
$883$	− 1.91595e13i	− 1.06062i	−0.847803	−	0.530311i	$-0.822075\pi$
$883$	− 1.91595e13i	− 1.06062i	0.847803	−	0.530311i	$-0.177925\pi$
$884$	1.49712e12	0.0824559
$885$	0	0
$886$	3.19140e13	1.73992
$887$	− 2.24991e13i	− 1.22042i	−0.792241	−	0.610209i	$-0.791086\pi$
$887$	− 2.24991e13i	− 1.22042i	0.792241	−	0.610209i	$-0.208914\pi$
$888$	0	0
$889$	−2.08449e13	−1.11929
$890$	0	0
$891$	0	0
$892$	3.33795e12i	0.176538i
$893$	7.78825e12i	0.409834i
$894$	0	0
$895$	0	0
$896$	−1.45603e13	−0.754719
$897$	0	0
$898$	− 1.99576e12i	− 0.102415i
$899$	2.37560e12	0.121298
$900$	0	0
$901$	−7.49841e12	−0.379060
$902$	1.91277e13i	0.962128i
$903$	0	0
$904$	−9.52380e12	−0.474299
$905$	0	0
$906$	0	0
$907$	2.39724e13i	1.17619i	0.808791	+	0.588096i	$0.200122\pi$
$907$	2.39724e13i	1.17619i	−0.808791	+	0.588096i	$0.799878\pi$
$908$	1.22847e12i	0.0599763i
$909$	0	0
$910$	0	0
$911$	6.54470e12	0.314816	0.157408	−	0.987534i	$-0.449686\pi$
$911$	6.54470e12	0.314816	0.157408	+	0.987534i	$0.449686\pi$
$912$	0	0
$913$	− 4.38762e11i	− 0.0208983i
$914$	−3.16236e12	−0.149884
$915$	0	0
$916$	−3.26504e12	−0.153235
$917$	− 2.11918e12i	− 0.0989706i
$918$	0	0
$919$	−2.63348e12	−0.121789	−0.0608947	−	0.998144i	$-0.519395\pi$
$919$	−2.63348e12	−0.121789	−0.0608947	+	0.998144i	$0.519395\pi$
$920$	0	0
$921$	0	0
$922$	− 1.84111e13i	− 0.839055i
$923$	4.26258e13i	1.93314i
$924$	0	0
$925$	0	0
$926$	1.42093e13	0.635073
$927$	0	0
$928$	− 1.13530e12i	− 0.0502512i
$929$	2.94271e13	1.29621	0.648106	−	0.761550i	$-0.275561\pi$
$929$	2.94271e13	1.29621	0.648106	+	0.761550i	$0.275561\pi$
$930$	0	0
$931$	5.70166e12	0.248730
$932$	3.04070e11i	0.0132009i
$933$	0	0
$934$	2.96305e13	1.27403
$935$	0	0
$936$	0	0
$937$	1.71769e13i	0.727974i	0.931404	+	0.363987i	$0.118585\pi$
$937$	1.71769e13i	0.727974i	−0.931404	+	0.363987i	$0.881415\pi$
$938$	− 2.72201e13i	− 1.14809i
$939$	0	0
$940$	0	0
$941$	2.46944e13	1.02670	0.513352	−	0.858178i	$-0.328404\pi$
$941$	2.46944e13	1.02670	0.513352	+	0.858178i	$0.328404\pi$
$942$	0	0
$943$	− 2.12475e13i	− 0.874994i
$944$	−4.56498e13	−1.87096
$945$	0	0
$946$	−1.82341e13	−0.740244
$947$	− 1.19428e12i	− 0.0482537i	−0.999709	−	0.0241268i	$-0.992319\pi$
$947$	− 1.19428e12i	− 0.0482537i	0.999709	−	0.0241268i	$-0.00768056\pi$
$948$	0	0
$949$	−2.46319e13	−0.985824
$950$	0	0
$951$	0	0
$952$	4.72377e12i	0.186390i
$953$	− 1.47694e13i	− 0.580023i	−0.957023	−	0.290011i	$-0.906341\pi$
$953$	− 1.47694e13i	− 0.580023i	0.957023	−	0.290011i	$-0.0936591\pi$
$954$	0	0
$955$	0	0
$956$	6.70966e12	0.259800
$957$	0	0
$958$	− 3.63489e13i	− 1.39427i
$959$	−2.01231e13	−0.768267
$960$	0	0
$961$	−2.49176e11	−0.00942435
$962$	− 2.58351e13i	− 0.972574i
$963$	0	0
$964$	−7.76192e12	−0.289483
$965$	0	0
$966$	0	0
$967$	8.95584e12i	0.329372i	0.986346	+	0.164686i	$0.0526611\pi$
$967$	8.95584e12i	0.329372i	−0.986346	+	0.164686i	$0.947339\pi$
$968$	− 4.87876e13i	− 1.78596i
$969$	0	0
$970$	0	0
$971$	2.20587e13	0.796330	0.398165	−	0.917314i	$-0.369647\pi$
$971$	2.20587e13	0.796330	0.398165	+	0.917314i	$0.369647\pi$
$972$	0	0
$973$	− 1.41118e13i	− 0.504749i
$974$	3.65621e13	1.30172
$975$	0	0
$976$	−4.72469e13	−1.66667
$977$	3.56972e13i	1.25345i	0.779239	+	0.626727i	$0.215606\pi$
$977$	3.56972e13i	1.25345i	−0.779239	+	0.626727i	$0.784394\pi$
$978$	0	0
$979$	−3.66806e13	−1.27619
$980$	0	0
$981$	0	0
$982$	− 4.38300e13i	− 1.50408i
$983$	− 3.19525e13i	− 1.09147i	−0.837956	−	0.545737i	$-0.816250\pi$
$983$	− 3.19525e13i	− 1.09147i	0.837956	−	0.545737i	$-0.183750\pi$
$984$	0	0
$985$	0	0
$986$	−1.35070e12	−0.0455105
$987$	0	0
$988$	− 3.00787e12i	− 0.100428i
$989$	2.02549e13	0.673205
$990$	0	0
$991$	1.56883e13	0.516706	0.258353	−	0.966051i	$-0.416820\pi$
$991$	1.56883e13	0.516706	0.258353	+	0.966051i	$0.416820\pi$
$992$	− 1.25165e13i	− 0.410374i
$993$	0	0
$994$	3.56199e13	1.15732
$995$	0	0
$996$	0	0
$997$	− 6.52145e12i	− 0.209033i	−0.994523	−	0.104517i	$-0.966670\pi$
$997$	− 6.52145e12i	− 0.209033i	0.994523	−	0.104517i	$-0.0333296\pi$
$998$	− 4.05647e13i	− 1.29438i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.b.g.199.2		4
3.2	odd	2		75.10.b.e.49.3		4
5.2	odd	4		45.10.a.e.1.2		2
5.3	odd	4		225.10.a.j.1.1		2
5.4	even	2	inner	225.10.b.g.199.3		4
15.2	even	4		15.10.a.c.1.1	✓	2
15.8	even	4		75.10.a.g.1.2		2
15.14	odd	2		75.10.b.e.49.2		4
60.47	odd	4		240.10.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.c.1.1	✓	2	15.2	even	4
45.10.a.e.1.2		2	5.2	odd	4
75.10.a.g.1.2		2	15.8	even	4
75.10.b.e.49.2		4	15.14	odd	2
75.10.b.e.49.3		4	3.2	odd	2
225.10.a.j.1.1		2	5.3	odd	4
225.10.b.g.199.2		4	1.1	even	1	trivial
225.10.b.g.199.3		4	5.4	even	2	inner
240.10.a.m.1.1		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-24.8839i q^{2} -107.207 q^{4} +4010.50i q^{7} -10072.8i q^{8} +O(q^{10})\)	\(q-24.8839i q^{2} -107.207 q^{4} +4010.50i q^{7} -10072.8i q^{8} -84861.3 q^{11} +119425. i q^{13} +99796.8 q^{14} -305541. q^{16} +116934. i q^{17} +234932. q^{19} +2.11168e6i q^{22} -2.34570e6i q^{23} +2.97176e6 q^{26} -429953. i q^{28} -464196. q^{29} -5.11766e6 q^{31} +2.44574e6i q^{32} +2.90976e6 q^{34} -8.69354e6i q^{37} -5.84601e6i q^{38} +9.05805e6 q^{41} +8.63491e6i q^{43} +9.09769e6 q^{44} -5.83701e7 q^{46} +3.31511e7i q^{47} +2.42695e7 q^{49} -1.28032e7i q^{52} +6.41254e7i q^{53} +4.03971e7 q^{56} +1.15510e7i q^{58} +1.49407e8 q^{59} +1.54634e8 q^{61} +1.27347e8i q^{62} -9.55772e7 q^{64} -2.72755e8i q^{67} -1.25360e7i q^{68} +3.56924e8 q^{71} +2.06253e8i q^{73} -2.16329e8 q^{74} -2.51862e7 q^{76} -3.40337e8i q^{77} +4.04380e8 q^{79} -2.25399e8i q^{82} +5.17034e6i q^{83} +2.14870e8 q^{86} +8.54793e8i q^{88} +4.32242e8 q^{89} -4.78955e8 q^{91} +2.51475e8i q^{92} +8.24928e8 q^{94} +1.32066e9i q^{97} -6.03918e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3042 q^{4}+O(q^{10}) \) 4 * q - 3042 * q^4	\( 4 q - 3042 q^{4} - 70976 q^{11} - 490392 q^{14} + 1414530 q^{16} + 806592 q^{19} + 3815092 q^{26} - 149144 q^{29} - 10054256 q^{31} - 17721764 q^{34} - 28422664 q^{41} - 121411904 q^{44} - 302983296 q^{46} + 5639900 q^{49} + 703018680 q^{56} + 375726272 q^{59} + 308160120 q^{61} - 1276602178 q^{64} + 456541952 q^{71} - 724038452 q^{74} - 526437448 q^{76} + 1864813520 q^{79} - 1247853416 q^{86} + 449036328 q^{89} - 1339205056 q^{91} - 3863301632 q^{94}+O(q^{100}) \) 4 * q - 3042 * q^4 - 70976 * q^11 - 490392 * q^14 + 1414530 * q^16 + 806592 * q^19 + 3815092 * q^26 - 149144 * q^29 - 10054256 * q^31 - 17721764 * q^34 - 28422664 * q^41 - 121411904 * q^44 - 302983296 * q^46 + 5639900 * q^49 + 703018680 * q^56 + 375726272 * q^59 + 308160120 * q^61 - 1276602178 * q^64 + 456541952 * q^71 - 724038452 * q^74 - 526437448 * q^76 + 1864813520 * q^79 - 1247853416 * q^86 + 449036328 * q^89 - 1339205056 * q^91 - 3863301632 * q^94

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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