LMFDB - Embedded newform 2925.2.c.u.2224.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.c (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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.u.2224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{2} -3.82843 q^{4} -2.82843i q^{7} -4.41421i q^{8} +O(q^{10})$	$q+2.41421i q^{2} -3.82843 q^{4} -2.82843i q^{7} -4.41421i q^{8} +2.00000 q^{11} +1.00000i q^{13} +6.82843 q^{14} +3.00000 q^{16} +3.65685i q^{17} -2.82843 q^{19} +4.82843i q^{22} -4.00000i q^{23} -2.41421 q^{26} +10.8284i q^{28} +2.00000 q^{29} -6.82843 q^{31} -1.58579i q^{32} -8.82843 q^{34} +3.65685i q^{37} -6.82843i q^{38} -10.8284 q^{41} -9.65685i q^{43} -7.65685 q^{44} +9.65685 q^{46} +0.343146i q^{47} -1.00000 q^{49} -3.82843i q^{52} -2.00000i q^{53} -12.4853 q^{56} +4.82843i q^{58} -3.65685 q^{59} -9.31371 q^{61} -16.4853i q^{62} +9.82843 q^{64} +1.17157i q^{67} -14.0000i q^{68} -2.00000 q^{71} -11.6569i q^{73} -8.82843 q^{74} +10.8284 q^{76} -5.65685i q^{77} -11.3137 q^{79} -26.1421i q^{82} -7.65685i q^{83} +23.3137 q^{86} -8.82843i q^{88} +9.17157 q^{89} +2.82843 q^{91} +15.3137i q^{92} -0.828427 q^{94} -7.65685i q^{97} -2.41421i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 8 q^{11} + 16 q^{14} + 12 q^{16} - 4 q^{26} + 8 q^{29} - 16 q^{31} - 24 q^{34} - 32 q^{41} - 8 q^{44} + 16 q^{46} - 4 q^{49} - 16 q^{56} + 8 q^{59} + 8 q^{61} + 28 q^{64} - 8 q^{71} - 24 q^{74} + 32 q^{76} + 48 q^{86} + 48 q^{89} + 8 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^11 + 16 * q^14 + 12 * q^16 - 4 * q^26 + 8 * q^29 - 16 * q^31 - 24 * q^34 - 32 * q^41 - 8 * q^44 + 16 * q^46 - 4 * q^49 - 16 * q^56 + 8 * q^59 + 8 * q^61 + 28 * q^64 - 8 * q^71 - 24 * q^74 + 32 * q^76 + 48 * q^86 + 48 * q^89 + 8 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	0	0
$3$	0	0
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	− 4.41421i	− 1.56066i
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	6.82843	1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	3.65685i	0.886917i	0.896295	+	0.443459i	$0.146249\pi$
$17$	3.65685i	0.886917i	−0.896295	+	0.443459i	$0.853751\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	4.82843i	1.02942i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	−2.41421	−0.473466
$27$	0	0
$28$	10.8284i	2.04638i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	− 1.58579i	− 0.280330i
$33$	0	0
$34$	−8.82843	−1.51406
$35$	0	0
$36$	0	0
$37$	3.65685i	0.601183i	0.953753	+	0.300592i	$0.0971841\pi$
$37$	3.65685i	0.601183i	−0.953753	+	0.300592i	$0.902816\pi$
$38$	− 6.82843i	− 1.10772i
$39$	0	0
$40$	0	0
$41$	−10.8284	−1.69112	−0.845558	−	0.533883i	$-0.820732\pi$
$41$	−10.8284	−1.69112	−0.845558	+	0.533883i	$0.820732\pi$
$42$	0	0
$43$	− 9.65685i	− 1.47266i	−0.676625	−	0.736328i	$-0.736558\pi$
$43$	− 9.65685i	− 1.47266i	0.676625	−	0.736328i	$-0.263442\pi$
$44$	−7.65685	−1.15431
$45$	0	0
$46$	9.65685	1.42383
$47$	0.343146i	0.0500530i	0.999687	+	0.0250265i	$0.00796701\pi$
$47$	0.343146i	0.0500530i	−0.999687	+	0.0250265i	$0.992033\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 3.82843i	− 0.530907i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	−12.4853	−1.66842
$57$	0	0
$58$	4.82843i	0.634004i
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	− 16.4853i	− 2.09363i
$63$	0	0
$64$	9.82843	1.22855
$65$	0	0
$66$	0	0
$67$	1.17157i	0.143130i	0.997436	+	0.0715652i	$0.0227994\pi$
$67$	1.17157i	0.143130i	−0.997436	+	0.0715652i	$0.977201\pi$
$68$	− 14.0000i	− 1.69775i
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	− 11.6569i	− 1.36433i	−0.731198	−	0.682166i	$-0.761038\pi$
$73$	− 11.6569i	− 1.36433i	0.731198	−	0.682166i	$-0.238962\pi$
$74$	−8.82843	−1.02628
$75$	0	0
$76$	10.8284	1.24211
$77$	− 5.65685i	− 0.644658i
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	0	0
$82$	− 26.1421i	− 2.88692i
$83$	− 7.65685i	− 0.840449i	−0.907420	−	0.420224i	$-0.861951\pi$
$83$	− 7.65685i	− 0.840449i	0.907420	−	0.420224i	$-0.138049\pi$
$84$	0	0
$85$	0	0
$86$	23.3137	2.51398
$87$	0	0
$88$	− 8.82843i	− 0.941113i
$89$	9.17157	0.972185	0.486092	−	0.873907i	$-0.338422\pi$
$89$	9.17157	0.972185	0.486092	+	0.873907i	$0.338422\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	15.3137i	1.59656i
$93$	0	0
$94$	−0.828427	−0.0854457
$95$	0	0
$96$	0	0
$97$	− 7.65685i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$97$	− 7.65685i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$98$	− 2.41421i	− 0.243872i
$99$	0	0
$100$	0	0
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	0	0
$103$	− 13.6569i	− 1.34565i	−0.739802	−	0.672825i	$-0.765081\pi$
$103$	− 13.6569i	− 1.34565i	0.739802	−	0.672825i	$-0.234919\pi$
$104$	4.41421	0.432849
$105$	0	0
$106$	4.82843	0.468978
$107$	− 11.3137i	− 1.09374i	−0.837218	−	0.546869i	$-0.815820\pi$
$107$	− 11.3137i	− 1.09374i	0.837218	−	0.546869i	$-0.184180\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	0	0
$112$	− 8.48528i	− 0.801784i
$113$	17.3137i	1.62874i	0.580348	+	0.814368i	$0.302916\pi$
$113$	17.3137i	1.62874i	−0.580348	+	0.814368i	$0.697084\pi$
$114$	0	0
$115$	0	0
$116$	−7.65685	−0.710921
$117$	0	0
$118$	− 8.82843i	− 0.812723i
$119$	10.3431	0.948155
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 22.4853i	− 2.03572i
$123$	0	0
$124$	26.1421	2.34763
$125$	0	0
$126$	0	0
$127$	− 5.65685i	− 0.501965i	−0.967992	−	0.250982i	$-0.919246\pi$
$127$	− 5.65685i	− 0.501965i	0.967992	−	0.250982i	$-0.0807536\pi$
$128$	20.5563i	1.81694i
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	−2.82843	−0.244339
$135$	0	0
$136$	16.1421	1.38418
$137$	5.17157i	0.441837i	0.975292	+	0.220919i	$0.0709055\pi$
$137$	5.17157i	0.441837i	−0.975292	+	0.220919i	$0.929094\pi$
$138$	0	0
$139$	−15.3137	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$139$	−15.3137	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$140$	0	0
$141$	0	0
$142$	− 4.82843i	− 0.405193i
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	28.1421	2.32906
$147$	0	0
$148$	− 14.0000i	− 1.15079i
$149$	−14.8284	−1.21479	−0.607396	−	0.794399i	$-0.707786\pi$
$149$	−14.8284	−1.21479	−0.607396	+	0.794399i	$0.707786\pi$
$150$	0	0
$151$	−20.4853	−1.66707	−0.833534	−	0.552468i	$-0.813686\pi$
$151$	−20.4853	−1.66707	−0.833534	+	0.552468i	$0.813686\pi$
$152$	12.4853i	1.01269i
$153$	0	0
$154$	13.6569	1.10050
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	− 27.3137i	− 2.17296i
$159$	0	0
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	− 13.1716i	− 1.03168i	−0.856686	−	0.515839i	$-0.827480\pi$
$163$	− 13.1716i	− 1.03168i	0.856686	−	0.515839i	$-0.172520\pi$
$164$	41.4558	3.23716
$165$	0	0
$166$	18.4853	1.43474
$167$	− 7.65685i	− 0.592505i	−0.955110	−	0.296253i	$-0.904263\pi$
$167$	− 7.65685i	− 0.592505i	0.955110	−	0.296253i	$-0.0957370\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	36.9706i	2.81898i
$173$	− 0.343146i	− 0.0260889i	−0.999915	−	0.0130444i	$-0.995848\pi$
$173$	− 0.343146i	− 0.0260889i	0.999915	−	0.0130444i	$-0.00415229\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	22.1421i	1.65962i
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	6.82843i	0.506157i
$183$	0	0
$184$	−17.6569	−1.30168
$185$	0	0
$186$	0	0
$187$	7.31371i	0.534831i
$188$	− 1.31371i	− 0.0958120i
$189$	0	0
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	17.3137i	1.24627i	0.782115	+	0.623134i	$0.214141\pi$
$193$	17.3137i	1.24627i	−0.782115	+	0.623134i	$0.785859\pi$
$194$	18.4853	1.32717
$195$	0	0
$196$	3.82843	0.273459
$197$	16.4853i	1.17453i	0.809396	+	0.587264i	$0.199795\pi$
$197$	16.4853i	1.17453i	−0.809396	+	0.587264i	$0.800205\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	0	0
$202$	8.82843i	0.621166i
$203$	− 5.65685i	− 0.397033i
$204$	0	0
$205$	0	0
$206$	32.9706	2.29717
$207$	0	0
$208$	3.00000i	0.208013i
$209$	−5.65685	−0.391293
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	7.65685i	0.525875i
$213$	0	0
$214$	27.3137	1.86713
$215$	0	0
$216$	0	0
$217$	19.3137i	1.31110i
$218$	41.7990i	2.83098i
$219$	0	0
$220$	0	0
$221$	−3.65685	−0.245987
$222$	0	0
$223$	− 4.48528i	− 0.300357i	−0.988659	−	0.150178i	$-0.952015\pi$
$223$	− 4.48528i	− 0.300357i	0.988659	−	0.150178i	$-0.0479848\pi$
$224$	−4.48528	−0.299685
$225$	0	0
$226$	−41.7990	−2.78043
$227$	− 5.31371i	− 0.352683i	−0.984329	−	0.176342i	$-0.943574\pi$
$227$	− 5.31371i	− 0.352683i	0.984329	−	0.176342i	$-0.0564263\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	0	0
$232$	− 8.82843i	− 0.579615i
$233$	− 26.9706i	− 1.76690i	−0.468525	−	0.883450i	$-0.655214\pi$
$233$	− 26.9706i	− 1.76690i	0.468525	−	0.883450i	$-0.344786\pi$
$234$	0	0
$235$	0	0
$236$	14.0000	0.911322
$237$	0	0
$238$	24.9706i	1.61860i
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	11.6569	0.750884	0.375442	−	0.926846i	$-0.377491\pi$
$241$	11.6569	0.750884	0.375442	+	0.926846i	$0.377491\pi$
$242$	− 16.8995i	− 1.08634i
$243$	0	0
$244$	35.6569	2.28270
$245$	0	0
$246$	0	0
$247$	− 2.82843i	− 0.179969i
$248$	30.1421i	1.91403i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	13.6569	0.856907
$255$	0	0
$256$	−29.9706	−1.87316
$257$	15.6569i	0.976648i	0.872662	+	0.488324i	$0.162392\pi$
$257$	15.6569i	0.976648i	−0.872662	+	0.488324i	$0.837608\pi$
$258$	0	0
$259$	10.3431	0.642692
$260$	0	0
$261$	0	0
$262$	19.3137i	1.19320i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	−19.3137	−1.18420
$267$	0	0
$268$	− 4.48528i	− 0.273982i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	11.7990	0.716738	0.358369	−	0.933580i	$-0.383333\pi$
$271$	11.7990	0.716738	0.358369	+	0.933580i	$0.383333\pi$
$272$	10.9706i	0.665188i
$273$	0	0
$274$	−12.4853	−0.754263
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 36.9706i	− 2.21735i
$279$	0	0
$280$	0	0
$281$	−26.8284	−1.60045	−0.800225	−	0.599700i	$-0.795287\pi$
$281$	−26.8284	−1.60045	−0.800225	+	0.599700i	$0.795287\pi$
$282$	0	0
$283$	4.97056i	0.295469i	0.989027	+	0.147735i	$0.0471982\pi$
$283$	4.97056i	0.295469i	−0.989027	+	0.147735i	$0.952802\pi$
$284$	7.65685	0.454351
$285$	0	0
$286$	−4.82843	−0.285511
$287$	30.6274i	1.80788i
$288$	0	0
$289$	3.62742	0.213377
$290$	0	0
$291$	0	0
$292$	44.6274i	2.61162i
$293$	− 26.1421i	− 1.52724i	−0.645666	−	0.763620i	$-0.723420\pi$
$293$	− 26.1421i	− 1.52724i	0.645666	−	0.763620i	$-0.276580\pi$
$294$	0	0
$295$	0	0
$296$	16.1421	0.938243
$297$	0	0
$298$	− 35.7990i	− 2.07378i
$299$	4.00000	0.231326
$300$	0	0
$301$	−27.3137	−1.57434
$302$	− 49.4558i	− 2.84586i
$303$	0	0
$304$	−8.48528	−0.486664
$305$	0	0
$306$	0	0
$307$	− 17.1716i	− 0.980033i	−0.871713	−	0.490017i	$-0.836991\pi$
$307$	− 17.1716i	− 0.980033i	0.871713	−	0.490017i	$-0.163009\pi$
$308$	21.6569i	1.23401i
$309$	0	0
$310$	0	0
$311$	−34.6274	−1.96354	−0.981770	−	0.190071i	$-0.939128\pi$
$311$	−34.6274	−1.96354	−0.981770	+	0.190071i	$0.939128\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	24.1421	1.36242
$315$	0	0
$316$	43.3137	2.43659
$317$	8.48528i	0.476581i	0.971194	+	0.238290i	$0.0765870\pi$
$317$	8.48528i	0.476581i	−0.971194	+	0.238290i	$0.923413\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	− 27.3137i	− 1.52213i
$323$	− 10.3431i	− 0.575508i
$324$	0	0
$325$	0	0
$326$	31.7990	1.76118
$327$	0	0
$328$	47.7990i	2.63926i
$329$	0.970563	0.0535089
$330$	0	0
$331$	−2.14214	−0.117742	−0.0588712	−	0.998266i	$-0.518750\pi$
$331$	−2.14214	−0.117742	−0.0588712	+	0.998266i	$0.518750\pi$
$332$	29.3137i	1.60880i
$333$	0	0
$334$	18.4853	1.01147
$335$	0	0
$336$	0	0
$337$	− 13.3137i	− 0.725244i	−0.931936	−	0.362622i	$-0.881882\pi$
$337$	− 13.3137i	− 0.725244i	0.931936	−	0.362622i	$-0.118118\pi$
$338$	− 2.41421i	− 0.131316i
$339$	0	0
$340$	0	0
$341$	−13.6569	−0.739560
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	−42.6274	−2.29832
$345$	0	0
$346$	0.828427	0.0445365
$347$	31.3137i	1.68101i	0.541805	+	0.840504i	$0.317741\pi$
$347$	31.3137i	1.68101i	−0.541805	+	0.840504i	$0.682259\pi$
$348$	0	0
$349$	7.65685	0.409862	0.204931	−	0.978776i	$-0.434303\pi$
$349$	7.65685	0.409862	0.204931	+	0.978776i	$0.434303\pi$
$350$	0	0
$351$	0	0
$352$	− 3.17157i	− 0.169045i
$353$	17.4558i	0.929081i	0.885552	+	0.464540i	$0.153780\pi$
$353$	17.4558i	0.929081i	−0.885552	+	0.464540i	$0.846220\pi$
$354$	0	0
$355$	0	0
$356$	−35.1127	−1.86097
$357$	0	0
$358$	− 1.65685i	− 0.0875675i
$359$	1.02944	0.0543316	0.0271658	−	0.999631i	$-0.491352\pi$
$359$	1.02944	0.0543316	0.0271658	+	0.999631i	$0.491352\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	33.7990i	1.77644i
$363$	0	0
$364$	−10.8284	−0.567564
$365$	0	0
$366$	0	0
$367$	− 24.0000i	− 1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$	− 24.0000i	− 1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$	− 12.0000i	− 0.625543i
$369$	0	0
$370$	0	0
$371$	−5.65685	−0.293689
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	−17.6569	−0.913014
$375$	0	0
$376$	1.51472	0.0781156
$377$	2.00000i	0.103005i
$378$	0	0
$379$	16.4853	0.846792	0.423396	−	0.905945i	$-0.360838\pi$
$379$	16.4853	0.846792	0.423396	+	0.905945i	$0.360838\pi$
$380$	0	0
$381$	0	0
$382$	46.6274i	2.38567i
$383$	− 2.97056i	− 0.151789i	−0.997116	−	0.0758943i	$-0.975819\pi$
$383$	− 2.97056i	− 0.151789i	0.997116	−	0.0758943i	$-0.0241812\pi$
$384$	0	0
$385$	0	0
$386$	−41.7990	−2.12751
$387$	0	0
$388$	29.3137i	1.48818i
$389$	6.97056	0.353422	0.176711	−	0.984263i	$-0.443454\pi$
$389$	6.97056	0.353422	0.176711	+	0.984263i	$0.443454\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	4.41421i	0.222951i
$393$	0	0
$394$	−39.7990	−2.00504
$395$	0	0
$396$	0	0
$397$	− 2.97056i	− 0.149088i	−0.997218	−	0.0745441i	$-0.976250\pi$
$397$	− 2.97056i	− 0.149088i	0.997218	−	0.0745441i	$-0.0237502\pi$
$398$	− 24.9706i	− 1.25166i
$399$	0	0
$400$	0	0
$401$	2.14214	0.106973	0.0534866	−	0.998569i	$-0.482967\pi$
$401$	2.14214	0.106973	0.0534866	+	0.998569i	$0.482967\pi$
$402$	0	0
$403$	− 6.82843i	− 0.340148i
$404$	−14.0000	−0.696526
$405$	0	0
$406$	13.6569	0.677778
$407$	7.31371i	0.362527i
$408$	0	0
$409$	1.02944	0.0509024	0.0254512	−	0.999676i	$-0.491898\pi$
$409$	1.02944	0.0509024	0.0254512	+	0.999676i	$0.491898\pi$
$410$	0	0
$411$	0	0
$412$	52.2843i	2.57586i
$413$	10.3431i	0.508953i
$414$	0	0
$415$	0	0
$416$	1.58579	0.0777496
$417$	0	0
$418$	− 13.6569i	− 0.667979i
$419$	−30.6274	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$419$	−30.6274	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$420$	0	0
$421$	14.6863	0.715766	0.357883	−	0.933766i	$-0.383499\pi$
$421$	14.6863	0.715766	0.357883	+	0.933766i	$0.383499\pi$
$422$	− 28.9706i	− 1.41026i
$423$	0	0
$424$	−8.82843	−0.428746
$425$	0	0
$426$	0	0
$427$	26.3431i	1.27483i
$428$	43.3137i	2.09365i
$429$	0	0
$430$	0	0
$431$	−19.6569	−0.946837	−0.473419	−	0.880838i	$-0.656980\pi$
$431$	−19.6569	−0.946837	−0.473419	+	0.880838i	$0.656980\pi$
$432$	0	0
$433$	− 1.31371i	− 0.0631328i	−0.999502	−	0.0315664i	$-0.989950\pi$
$433$	− 1.31371i	− 0.0631328i	0.999502	−	0.0315664i	$-0.0100496\pi$
$434$	−46.6274	−2.23819
$435$	0	0
$436$	−66.2843	−3.17444
$437$	11.3137i	0.541208i
$438$	0	0
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	0	0
$442$	− 8.82843i	− 0.419925i
$443$	41.9411i	1.99268i	0.0854611	+	0.996342i	$0.472764\pi$
$443$	41.9411i	1.99268i	−0.0854611	+	0.996342i	$0.527236\pi$
$444$	0	0
$445$	0	0
$446$	10.8284	0.512741
$447$	0	0
$448$	− 27.7990i	− 1.31338i
$449$	7.79899	0.368057	0.184029	−	0.982921i	$-0.441086\pi$
$449$	7.79899	0.368057	0.184029	+	0.982921i	$0.441086\pi$
$450$	0	0
$451$	−21.6569	−1.01978
$452$	− 66.2843i	− 3.11775i
$453$	0	0
$454$	12.8284	0.602068
$455$	0	0
$456$	0	0
$457$	3.65685i	0.171060i	0.996336	+	0.0855302i	$0.0272584\pi$
$457$	3.65685i	0.171060i	−0.996336	+	0.0855302i	$0.972742\pi$
$458$	− 51.4558i	− 2.40437i
$459$	0	0
$460$	0	0
$461$	−10.8284	−0.504330	−0.252165	−	0.967684i	$-0.581143\pi$
$461$	−10.8284	−0.504330	−0.252165	+	0.967684i	$0.581143\pi$
$462$	0	0
$463$	7.51472i	0.349239i	0.984636	+	0.174619i	$0.0558695\pi$
$463$	7.51472i	0.349239i	−0.984636	+	0.174619i	$0.944131\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	65.1127	3.01629
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	0	0
$469$	3.31371	0.153013
$470$	0	0
$471$	0	0
$472$	16.1421i	0.743002i
$473$	− 19.3137i	− 0.888045i
$474$	0	0
$475$	0	0
$476$	−39.5980	−1.81497
$477$	0	0
$478$	4.82843i	0.220847i
$479$	2.68629	0.122740	0.0613699	−	0.998115i	$-0.480453\pi$
$479$	2.68629	0.122740	0.0613699	+	0.998115i	$0.480453\pi$
$480$	0	0
$481$	−3.65685	−0.166738
$482$	28.1421i	1.28184i
$483$	0	0
$484$	26.7990	1.21814
$485$	0	0
$486$	0	0
$487$	31.7990i	1.44095i	0.693481	+	0.720475i	$0.256076\pi$
$487$	31.7990i	1.44095i	−0.693481	+	0.720475i	$0.743924\pi$
$488$	41.1127i	1.86108i
$489$	0	0
$490$	0	0
$491$	14.6274	0.660126	0.330063	−	0.943959i	$-0.392930\pi$
$491$	14.6274	0.660126	0.330063	+	0.943959i	$0.392930\pi$
$492$	0	0
$493$	7.31371i	0.329393i
$494$	6.82843	0.307225
$495$	0	0
$496$	−20.4853	−0.919816
$497$	5.65685i	0.253745i
$498$	0	0
$499$	2.14214	0.0958952	0.0479476	−	0.998850i	$-0.484732\pi$
$499$	2.14214	0.0958952	0.0479476	+	0.998850i	$0.484732\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.3137i	0.682805i	0.939917	+	0.341402i	$0.110902\pi$
$503$	15.3137i	0.682805i	−0.939917	+	0.341402i	$0.889098\pi$
$504$	0	0
$505$	0	0
$506$	19.3137	0.858599
$507$	0	0
$508$	21.6569i	0.960868i
$509$	−27.7990	−1.23217	−0.616084	−	0.787680i	$-0.711282\pi$
$509$	−27.7990	−1.23217	−0.616084	+	0.787680i	$0.711282\pi$
$510$	0	0
$511$	−32.9706	−1.45853
$512$	− 31.2426i	− 1.38074i
$513$	0	0
$514$	−37.7990	−1.66724
$515$	0	0
$516$	0	0
$517$	0.686292i	0.0301831i
$518$	24.9706i	1.09714i
$519$	0	0
$520$	0	0
$521$	−2.68629	−0.117689	−0.0588443	−	0.998267i	$-0.518742\pi$
$521$	−2.68629	−0.117689	−0.0588443	+	0.998267i	$0.518742\pi$
$522$	0	0
$523$	− 7.31371i	− 0.319806i	−0.987133	−	0.159903i	$-0.948882\pi$
$523$	− 7.31371i	− 0.319806i	0.987133	−	0.159903i	$-0.0511182\pi$
$524$	−30.6274	−1.33796
$525$	0	0
$526$	−28.9706	−1.26318
$527$	− 24.9706i	− 1.08773i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	− 30.6274i	− 1.32787i
$533$	− 10.8284i	− 0.469031i
$534$	0	0
$535$	0	0
$536$	5.17157	0.223378
$537$	0	0
$538$	43.4558i	1.87351i
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	28.4853i	1.22355i
$543$	0	0
$544$	5.79899	0.248630
$545$	0	0
$546$	0	0
$547$	0.686292i	0.0293437i	0.999892	+	0.0146719i	$0.00467036\pi$
$547$	0.686292i	0.0293437i	−0.999892	+	0.0146719i	$0.995330\pi$
$548$	− 19.7990i	− 0.845771i
$549$	0	0
$550$	0	0
$551$	−5.65685	−0.240990
$552$	0	0
$553$	32.0000i	1.36078i
$554$	4.82843	0.205140
$555$	0	0
$556$	58.6274	2.48636
$557$	− 31.7990i	− 1.34737i	−0.739020	−	0.673683i	$-0.764711\pi$
$557$	− 31.7990i	− 1.34737i	0.739020	−	0.673683i	$-0.235289\pi$
$558$	0	0
$559$	9.65685	0.408441
$560$	0	0
$561$	0	0
$562$	− 64.7696i	− 2.73214i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	8.82843i	0.370433i
$569$	−9.02944	−0.378534	−0.189267	−	0.981926i	$-0.560611\pi$
$569$	−9.02944	−0.378534	−0.189267	+	0.981926i	$0.560611\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	− 7.65685i	− 0.320149i
$573$	0	0
$574$	−73.9411	−3.08624
$575$	0	0
$576$	0	0
$577$	35.9411i	1.49625i	0.663559	+	0.748124i	$0.269045\pi$
$577$	35.9411i	1.49625i	−0.663559	+	0.748124i	$0.730955\pi$
$578$	8.75736i	0.364258i
$579$	0	0
$580$	0	0
$581$	−21.6569	−0.898478
$582$	0	0
$583$	− 4.00000i	− 0.165663i
$584$	−51.4558	−2.12926
$585$	0	0
$586$	63.1127	2.60716
$587$	− 22.9706i	− 0.948097i	−0.880499	−	0.474048i	$-0.842792\pi$
$587$	− 22.9706i	− 0.948097i	0.880499	−	0.474048i	$-0.157208\pi$
$588$	0	0
$589$	19.3137	0.795807
$590$	0	0
$591$	0	0
$592$	10.9706i	0.450887i
$593$	− 3.51472i	− 0.144332i	−0.997393	−	0.0721661i	$-0.977009\pi$
$593$	− 3.51472i	− 0.144332i	0.997393	−	0.0721661i	$-0.0229912\pi$
$594$	0	0
$595$	0	0
$596$	56.7696	2.32537
$597$	0	0
$598$	9.65685i	0.394898i
$599$	−0.686292	−0.0280411	−0.0140206	−	0.999902i	$-0.504463\pi$
$599$	−0.686292	−0.0280411	−0.0140206	+	0.999902i	$0.504463\pi$
$600$	0	0
$601$	44.6274	1.82039	0.910195	−	0.414180i	$-0.135931\pi$
$601$	44.6274	1.82039	0.910195	+	0.414180i	$0.135931\pi$
$602$	− 65.9411i	− 2.68756i
$603$	0	0
$604$	78.4264	3.19113
$605$	0	0
$606$	0	0
$607$	− 25.9411i	− 1.05292i	−0.850201	−	0.526459i	$-0.823519\pi$
$607$	− 25.9411i	− 1.05292i	0.850201	−	0.526459i	$-0.176481\pi$
$608$	4.48528i	0.181902i
$609$	0	0
$610$	0	0
$611$	−0.343146	−0.0138822
$612$	0	0
$613$	36.3431i	1.46789i	0.679211	+	0.733943i	$0.262322\pi$
$613$	36.3431i	1.46789i	−0.679211	+	0.733943i	$0.737678\pi$
$614$	41.4558	1.67302
$615$	0	0
$616$	−24.9706	−1.00609
$617$	29.1716i	1.17440i	0.809441	+	0.587202i	$0.199770\pi$
$617$	29.1716i	1.17440i	−0.809441	+	0.587202i	$0.800230\pi$
$618$	0	0
$619$	15.7990	0.635015	0.317508	−	0.948256i	$-0.397154\pi$
$619$	15.7990	0.635015	0.317508	+	0.948256i	$0.397154\pi$
$620$	0	0
$621$	0	0
$622$	− 83.5980i	− 3.35197i
$623$	− 25.9411i	− 1.03931i
$624$	0	0
$625$	0	0
$626$	14.4853	0.578948
$627$	0	0
$628$	38.2843i	1.52771i
$629$	−13.3726	−0.533200
$630$	0	0
$631$	−19.1127	−0.760865	−0.380432	−	0.924809i	$-0.624225\pi$
$631$	−19.1127	−0.760865	−0.380432	+	0.924809i	$0.624225\pi$
$632$	49.9411i	1.98655i
$633$	0	0
$634$	−20.4853	−0.813574
$635$	0	0
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	9.65685i	0.382319i
$639$	0	0
$640$	0	0
$641$	26.2843	1.03817	0.519083	−	0.854724i	$-0.326273\pi$
$641$	26.2843	1.03817	0.519083	+	0.854724i	$0.326273\pi$
$642$	0	0
$643$	− 17.1716i	− 0.677181i	−0.940934	−	0.338590i	$-0.890050\pi$
$643$	− 17.1716i	− 0.677181i	0.940934	−	0.338590i	$-0.109950\pi$
$644$	43.3137	1.70680
$645$	0	0
$646$	24.9706	0.982454
$647$	11.3137i	0.444788i	0.974957	+	0.222394i	$0.0713871\pi$
$647$	11.3137i	0.444788i	−0.974957	+	0.222394i	$0.928613\pi$
$648$	0	0
$649$	−7.31371	−0.287088
$650$	0	0
$651$	0	0
$652$	50.4264i	1.97485i
$653$	− 2.68629i	− 0.105123i	−0.998618	−	0.0525614i	$-0.983261\pi$
$653$	− 2.68629i	− 0.105123i	0.998618	−	0.0525614i	$-0.0167385\pi$
$654$	0	0
$655$	0	0
$656$	−32.4853	−1.26834
$657$	0	0
$658$	2.34315i	0.0913453i
$659$	−24.6863	−0.961641	−0.480821	−	0.876819i	$-0.659661\pi$
$659$	−24.6863	−0.961641	−0.480821	+	0.876819i	$0.659661\pi$
$660$	0	0
$661$	−1.02944	−0.0400405	−0.0200202	−	0.999800i	$-0.506373\pi$
$661$	−1.02944	−0.0400405	−0.0200202	+	0.999800i	$0.506373\pi$
$662$	− 5.17157i	− 0.200999i
$663$	0	0
$664$	−33.7990	−1.31166
$665$	0	0
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	29.3137i	1.13418i
$669$	0	0
$670$	0	0
$671$	−18.6274	−0.719103
$672$	0	0
$673$	28.6274i	1.10351i	0.834008	+	0.551753i	$0.186041\pi$
$673$	28.6274i	1.10351i	−0.834008	+	0.551753i	$0.813959\pi$
$674$	32.1421	1.23807
$675$	0	0
$676$	3.82843	0.147247
$677$	− 49.3137i	− 1.89528i	−0.319341	−	0.947640i	$-0.603462\pi$
$677$	− 49.3137i	− 1.89528i	0.319341	−	0.947640i	$-0.396538\pi$
$678$	0	0
$679$	−21.6569	−0.831114
$680$	0	0
$681$	0	0
$682$	− 32.9706i	− 1.26251i
$683$	− 19.9411i	− 0.763026i	−0.924363	−	0.381513i	$-0.875403\pi$
$683$	− 19.9411i	− 0.763026i	0.924363	−	0.381513i	$-0.124597\pi$
$684$	0	0
$685$	0	0
$686$	40.9706	1.56426
$687$	0	0
$688$	− 28.9706i	− 1.10449i
$689$	2.00000	0.0761939
$690$	0	0
$691$	−34.1421	−1.29883	−0.649414	−	0.760435i	$-0.724986\pi$
$691$	−34.1421	−1.29883	−0.649414	+	0.760435i	$0.724986\pi$
$692$	1.31371i	0.0499397i
$693$	0	0
$694$	−75.5980	−2.86966
$695$	0	0
$696$	0	0
$697$	− 39.5980i	− 1.49988i
$698$	18.4853i	0.699678i
$699$	0	0
$700$	0	0
$701$	−38.9706	−1.47190	−0.735949	−	0.677037i	$-0.763264\pi$
$701$	−38.9706	−1.47190	−0.735949	+	0.677037i	$0.763264\pi$
$702$	0	0
$703$	− 10.3431i	− 0.390099i
$704$	19.6569	0.740846
$705$	0	0
$706$	−42.1421	−1.58604
$707$	− 10.3431i	− 0.388994i
$708$	0	0
$709$	−40.6274	−1.52579	−0.762897	−	0.646520i	$-0.776224\pi$
$709$	−40.6274	−1.52579	−0.762897	+	0.646520i	$0.776224\pi$
$710$	0	0
$711$	0	0
$712$	− 40.4853i	− 1.51725i
$713$	27.3137i	1.02291i
$714$	0	0
$715$	0	0
$716$	2.62742	0.0981912
$717$	0	0
$718$	2.48528i	0.0927499i
$719$	37.9411	1.41497	0.707483	−	0.706731i	$-0.249831\pi$
$719$	37.9411	1.41497	0.707483	+	0.706731i	$0.249831\pi$
$720$	0	0
$721$	−38.6274	−1.43856
$722$	− 26.5563i	− 0.988325i
$723$	0	0
$724$	−53.5980	−1.99195
$725$	0	0
$726$	0	0
$727$	− 21.6569i	− 0.803208i	−0.915813	−	0.401604i	$-0.868453\pi$
$727$	− 21.6569i	− 0.803208i	0.915813	−	0.401604i	$-0.131547\pi$
$728$	− 12.4853i	− 0.462735i
$729$	0	0
$730$	0	0
$731$	35.3137	1.30612
$732$	0	0
$733$	− 8.62742i	− 0.318661i	−0.987225	−	0.159330i	$-0.949066\pi$
$733$	− 8.62742i	− 0.318661i	0.987225	−	0.159330i	$-0.0509335\pi$
$734$	57.9411	2.13865
$735$	0	0
$736$	−6.34315	−0.233811
$737$	2.34315i	0.0863109i
$738$	0	0
$739$	−10.1421	−0.373084	−0.186542	−	0.982447i	$-0.559728\pi$
$739$	−10.1421	−0.373084	−0.186542	+	0.982447i	$0.559728\pi$
$740$	0	0
$741$	0	0
$742$	− 13.6569i	− 0.501359i
$743$	2.00000i	0.0733729i	0.999327	+	0.0366864i	$0.0116803\pi$
$743$	2.00000i	0.0733729i	−0.999327	+	0.0366864i	$0.988320\pi$
$744$	0	0
$745$	0	0
$746$	24.1421	0.883906
$747$	0	0
$748$	− 28.0000i	− 1.02378i
$749$	−32.0000	−1.16925
$750$	0	0
$751$	32.9706	1.20311	0.601556	−	0.798830i	$-0.294547\pi$
$751$	32.9706	1.20311	0.601556	+	0.798830i	$0.294547\pi$
$752$	1.02944i	0.0375397i
$753$	0	0
$754$	−4.82843	−0.175841
$755$	0	0
$756$	0	0
$757$	− 15.9411i	− 0.579390i	−0.957119	−	0.289695i	$-0.906446\pi$
$757$	− 15.9411i	− 0.579390i	0.957119	−	0.289695i	$-0.0935539\pi$
$758$	39.7990i	1.44556i
$759$	0	0
$760$	0	0
$761$	−15.5147	−0.562408	−0.281204	−	0.959648i	$-0.590734\pi$
$761$	−15.5147	−0.562408	−0.281204	+	0.959648i	$0.590734\pi$
$762$	0	0
$763$	− 48.9706i	− 1.77285i
$764$	−73.9411	−2.67510
$765$	0	0
$766$	7.17157	0.259119
$767$	− 3.65685i	− 0.132041i
$768$	0	0
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	0	0
$772$	− 66.2843i	− 2.38562i
$773$	5.85786i	0.210693i	0.994436	+	0.105346i	$0.0335951\pi$
$773$	5.85786i	0.210693i	−0.994436	+	0.105346i	$0.966405\pi$
$774$	0	0
$775$	0	0
$776$	−33.7990	−1.21331
$777$	0	0
$778$	16.8284i	0.603328i
$779$	30.6274	1.09734
$780$	0	0
$781$	−4.00000	−0.143131
$782$	35.3137i	1.26282i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	32.7696i	1.16811i	0.811715	+	0.584054i	$0.198534\pi$
$787$	32.7696i	1.16811i	−0.811715	+	0.584054i	$0.801466\pi$
$788$	− 63.1127i	− 2.24830i
$789$	0	0
$790$	0	0
$791$	48.9706	1.74119
$792$	0	0
$793$	− 9.31371i	− 0.330739i
$794$	7.17157	0.254510
$795$	0	0
$796$	39.5980	1.40351
$797$	− 35.6569i	− 1.26303i	−0.775363	−	0.631515i	$-0.782433\pi$
$797$	− 35.6569i	− 1.26303i	0.775363	−	0.631515i	$-0.217567\pi$
$798$	0	0
$799$	−1.25483	−0.0443928
$800$	0	0
$801$	0	0
$802$	5.17157i	0.182615i
$803$	− 23.3137i	− 0.822723i
$804$	0	0
$805$	0	0
$806$	16.4853	0.580669
$807$	0	0
$808$	− 16.1421i	− 0.567878i
$809$	41.3137	1.45251	0.726256	−	0.687424i	$-0.241259\pi$
$809$	41.3137	1.45251	0.726256	+	0.687424i	$0.241259\pi$
$810$	0	0
$811$	−1.85786	−0.0652384	−0.0326192	−	0.999468i	$-0.510385\pi$
$811$	−1.85786	−0.0652384	−0.0326192	+	0.999468i	$0.510385\pi$
$812$	21.6569i	0.760007i
$813$	0	0
$814$	−17.6569	−0.618872
$815$	0	0
$816$	0	0
$817$	27.3137i	0.955586i
$818$	2.48528i	0.0868958i
$819$	0	0
$820$	0	0
$821$	−15.7990	−0.551389	−0.275694	−	0.961245i	$-0.588908\pi$
$821$	−15.7990	−0.551389	−0.275694	+	0.961245i	$0.588908\pi$
$822$	0	0
$823$	− 48.9706i	− 1.70701i	−0.521088	−	0.853503i	$-0.674473\pi$
$823$	− 48.9706i	− 1.70701i	0.521088	−	0.853503i	$-0.325527\pi$
$824$	−60.2843	−2.10010
$825$	0	0
$826$	−24.9706	−0.868837
$827$	26.0000i	0.904109i	0.891990	+	0.452054i	$0.149309\pi$
$827$	26.0000i	0.904109i	−0.891990	+	0.452054i	$0.850691\pi$
$828$	0	0
$829$	−5.31371	−0.184553	−0.0922764	−	0.995733i	$-0.529414\pi$
$829$	−5.31371	−0.184553	−0.0922764	+	0.995733i	$0.529414\pi$
$830$	0	0
$831$	0	0
$832$	9.82843i	0.340739i
$833$	− 3.65685i	− 0.126702i
$834$	0	0
$835$	0	0
$836$	21.6569	0.749018
$837$	0	0
$838$	− 73.9411i	− 2.55425i
$839$	−47.2548	−1.63142	−0.815709	−	0.578462i	$-0.803653\pi$
$839$	−47.2548	−1.63142	−0.815709	+	0.578462i	$0.803653\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	35.4558i	1.22189i
$843$	0	0
$844$	45.9411	1.58136
$845$	0	0
$846$	0	0
$847$	19.7990i	0.680301i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	14.6274	0.501421
$852$	0	0
$853$	7.65685i	0.262166i	0.991371	+	0.131083i	$0.0418454\pi$
$853$	7.65685i	0.262166i	−0.991371	+	0.131083i	$0.958155\pi$
$854$	−63.5980	−2.17628
$855$	0	0
$856$	−49.9411	−1.70695
$857$	− 29.5980i	− 1.01105i	−0.862813	−	0.505524i	$-0.831299\pi$
$857$	− 29.5980i	− 1.01105i	0.862813	−	0.505524i	$-0.168701\pi$
$858$	0	0
$859$	23.3137	0.795453	0.397727	−	0.917504i	$-0.369799\pi$
$859$	23.3137	0.795453	0.397727	+	0.917504i	$0.369799\pi$
$860$	0	0
$861$	0	0
$862$	− 47.4558i	− 1.61635i
$863$	− 39.6569i	− 1.34994i	−0.737847	−	0.674968i	$-0.764158\pi$
$863$	− 39.6569i	− 1.34994i	0.737847	−	0.674968i	$-0.235842\pi$
$864$	0	0
$865$	0	0
$866$	3.17157	0.107774
$867$	0	0
$868$	− 73.9411i	− 2.50973i
$869$	−22.6274	−0.767583
$870$	0	0
$871$	−1.17157	−0.0396972
$872$	− 76.4264i	− 2.58812i
$873$	0	0
$874$	−27.3137	−0.923900
$875$	0	0
$876$	0	0
$877$	− 14.2843i	− 0.482346i	−0.970482	−	0.241173i	$-0.922468\pi$
$877$	− 14.2843i	− 0.482346i	0.970482	−	0.241173i	$-0.0775321\pi$
$878$	40.9706i	1.38269i
$879$	0	0
$880$	0	0
$881$	−53.5980	−1.80576	−0.902881	−	0.429891i	$-0.858552\pi$
$881$	−53.5980	−1.80576	−0.902881	+	0.429891i	$0.858552\pi$
$882$	0	0
$883$	51.5980i	1.73641i	0.496205	+	0.868205i	$0.334726\pi$
$883$	51.5980i	1.73641i	−0.496205	+	0.868205i	$0.665274\pi$
$884$	14.0000	0.470871
$885$	0	0
$886$	−101.255	−3.40172
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	17.1716i	0.574947i
$893$	− 0.970563i	− 0.0324786i
$894$	0	0
$895$	0	0
$896$	58.1421	1.94239
$897$	0	0
$898$	18.8284i	0.628313i
$899$	−13.6569	−0.455482
$900$	0	0
$901$	7.31371	0.243655
$902$	− 52.2843i	− 1.74088i
$903$	0	0
$904$	76.4264	2.54190
$905$	0	0
$906$	0	0
$907$	20.9706i	0.696316i	0.937436	+	0.348158i	$0.113193\pi$
$907$	20.9706i	0.696316i	−0.937436	+	0.348158i	$0.886807\pi$
$908$	20.3431i	0.675111i
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	− 15.3137i	− 0.506810i
$914$	−8.82843	−0.292018
$915$	0	0
$916$	81.5980	2.69607
$917$	− 22.6274i	− 0.747223i
$918$	0	0
$919$	19.3137	0.637100	0.318550	−	0.947906i	$-0.396804\pi$
$919$	19.3137	0.637100	0.318550	+	0.947906i	$0.396804\pi$
$920$	0	0
$921$	0	0
$922$	− 26.1421i	− 0.860945i
$923$	− 2.00000i	− 0.0658308i
$924$	0	0
$925$	0	0
$926$	−18.1421	−0.596188
$927$	0	0
$928$	− 3.17157i	− 0.104112i
$929$	−27.7990	−0.912055	−0.456028	−	0.889966i	$-0.650728\pi$
$929$	−27.7990	−0.912055	−0.456028	+	0.889966i	$0.650728\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	103.255i	3.38222i
$933$	0	0
$934$	−19.3137	−0.631964
$935$	0	0
$936$	0	0
$937$	1.31371i	0.0429170i	0.999770	+	0.0214585i	$0.00683097\pi$
$937$	1.31371i	0.0429170i	−0.999770	+	0.0214585i	$0.993169\pi$
$938$	8.00000i	0.261209i
$939$	0	0
$940$	0	0
$941$	−5.85786	−0.190961	−0.0954805	−	0.995431i	$-0.530439\pi$
$941$	−5.85786	−0.190961	−0.0954805	+	0.995431i	$0.530439\pi$
$942$	0	0
$943$	43.3137i	1.41049i
$944$	−10.9706	−0.357061
$945$	0	0
$946$	46.6274	1.51599
$947$	54.9706i	1.78630i	0.449756	+	0.893152i	$0.351511\pi$
$947$	54.9706i	1.78630i	−0.449756	+	0.893152i	$0.648489\pi$
$948$	0	0
$949$	11.6569	0.378398
$950$	0	0
$951$	0	0
$952$	− 45.6569i	− 1.47975i
$953$	51.6569i	1.67333i	0.547715	+	0.836665i	$0.315498\pi$
$953$	51.6569i	1.67333i	−0.547715	+	0.836665i	$0.684502\pi$
$954$	0	0
$955$	0	0
$956$	−7.65685	−0.247640
$957$	0	0
$958$	6.48528i	0.209530i
$959$	14.6274	0.472344
$960$	0	0
$961$	15.6274	0.504110
$962$	− 8.82843i	− 0.284640i
$963$	0	0
$964$	−44.6274	−1.43735
$965$	0	0
$966$	0	0
$967$	− 10.1421i	− 0.326149i	−0.986614	−	0.163075i	$-0.947859\pi$
$967$	− 10.1421i	− 0.326149i	0.986614	−	0.163075i	$-0.0521411\pi$
$968$	30.8995i	0.993147i
$969$	0	0
$970$	0	0
$971$	−7.31371	−0.234708	−0.117354	−	0.993090i	$-0.537441\pi$
$971$	−7.31371	−0.234708	−0.117354	+	0.993090i	$0.537441\pi$
$972$	0	0
$973$	43.3137i	1.38857i
$974$	−76.7696	−2.45986
$975$	0	0
$976$	−27.9411	−0.894374
$977$	− 13.8579i	− 0.443352i	−0.975120	−	0.221676i	$-0.928847\pi$
$977$	− 13.8579i	− 0.443352i	0.975120	−	0.221676i	$-0.0711528\pi$
$978$	0	0
$979$	18.3431	0.586249
$980$	0	0
$981$	0	0
$982$	35.3137i	1.12691i
$983$	2.68629i	0.0856794i	0.999082	+	0.0428397i	$0.0136405\pi$
$983$	2.68629i	0.0856794i	−0.999082	+	0.0428397i	$0.986360\pi$
$984$	0	0
$985$	0	0
$986$	−17.6569	−0.562309
$987$	0	0
$988$	10.8284i	0.344498i
$989$	−38.6274	−1.22828
$990$	0	0
$991$	27.3137	0.867649	0.433824	−	0.900998i	$-0.357164\pi$
$991$	27.3137	0.867649	0.433824	+	0.900998i	$0.357164\pi$
$992$	10.8284i	0.343803i
$993$	0	0
$994$	−13.6569	−0.433169
$995$	0	0
$996$	0	0
$997$	51.2548i	1.62326i	0.584174	+	0.811628i	$0.301419\pi$
$997$	51.2548i	1.62326i	−0.584174	+	0.811628i	$0.698581\pi$
$998$	5.17157i	0.163703i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.u.2224.4		4
3.2	odd	2		975.2.c.h.274.1		4
5.2	odd	4		2925.2.a.v.1.1		2
5.3	odd	4		117.2.a.c.1.2		2
5.4	even	2	inner	2925.2.c.u.2224.1		4
15.2	even	4		975.2.a.l.1.2		2
15.8	even	4		39.2.a.b.1.1	✓	2
15.14	odd	2		975.2.c.h.274.4		4
20.3	even	4		1872.2.a.w.1.1		2
35.13	even	4		5733.2.a.u.1.2		2
40.3	even	4		7488.2.a.co.1.2		2
40.13	odd	4		7488.2.a.cl.1.2		2
45.13	odd	12		1053.2.e.e.703.1		4
45.23	even	12		1053.2.e.m.703.2		4
45.38	even	12		1053.2.e.m.352.2		4
45.43	odd	12		1053.2.e.e.352.1		4
60.23	odd	4		624.2.a.k.1.2		2
65.8	even	4		1521.2.b.j.1351.4		4
65.18	even	4		1521.2.b.j.1351.1		4
65.38	odd	4		1521.2.a.f.1.1		2
105.83	odd	4		1911.2.a.h.1.1		2
120.53	even	4		2496.2.a.bf.1.1		2
120.83	odd	4		2496.2.a.bi.1.1		2
165.98	odd	4		4719.2.a.p.1.2		2
195.8	odd	4		507.2.b.e.337.1		4
195.23	even	12		507.2.e.d.22.1		4
195.38	even	4		507.2.a.h.1.2		2
195.68	even	12		507.2.e.h.22.2		4
195.83	odd	4		507.2.b.e.337.4		4
195.98	odd	12		507.2.j.f.361.1		8
195.113	even	12		507.2.e.h.484.2		4
195.128	odd	12		507.2.j.f.316.4		8
195.158	odd	12		507.2.j.f.316.1		8
195.173	even	12		507.2.e.d.484.1		4
195.188	odd	12		507.2.j.f.361.4		8
780.623	odd	4		8112.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.1	✓	2	15.8	even	4
117.2.a.c.1.2		2	5.3	odd	4
507.2.a.h.1.2		2	195.38	even	4
507.2.b.e.337.1		4	195.8	odd	4
507.2.b.e.337.4		4	195.83	odd	4
507.2.e.d.22.1		4	195.23	even	12
507.2.e.d.484.1		4	195.173	even	12
507.2.e.h.22.2		4	195.68	even	12
507.2.e.h.484.2		4	195.113	even	12
507.2.j.f.316.1		8	195.158	odd	12
507.2.j.f.316.4		8	195.128	odd	12
507.2.j.f.361.1		8	195.98	odd	12
507.2.j.f.361.4		8	195.188	odd	12
624.2.a.k.1.2		2	60.23	odd	4
975.2.a.l.1.2		2	15.2	even	4
975.2.c.h.274.1		4	3.2	odd	2
975.2.c.h.274.4		4	15.14	odd	2
1053.2.e.e.352.1		4	45.43	odd	12
1053.2.e.e.703.1		4	45.13	odd	12
1053.2.e.m.352.2		4	45.38	even	12
1053.2.e.m.703.2		4	45.23	even	12
1521.2.a.f.1.1		2	65.38	odd	4
1521.2.b.j.1351.1		4	65.18	even	4
1521.2.b.j.1351.4		4	65.8	even	4
1872.2.a.w.1.1		2	20.3	even	4
1911.2.a.h.1.1		2	105.83	odd	4
2496.2.a.bf.1.1		2	120.53	even	4
2496.2.a.bi.1.1		2	120.83	odd	4
2925.2.a.v.1.1		2	5.2	odd	4
2925.2.c.u.2224.1		4	5.4	even	2	inner
2925.2.c.u.2224.4		4	1.1	even	1	trivial
4719.2.a.p.1.2		2	165.98	odd	4
5733.2.a.u.1.2		2	35.13	even	4
7488.2.a.cl.1.2		2	40.13	odd	4
7488.2.a.co.1.2		2	40.3	even	4
8112.2.a.bm.1.1		2	780.623	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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{2} -3.82843 q^{4} -2.82843i q^{7} -4.41421i q^{8} +O(q^{10})\)	\(q+2.41421i q^{2} -3.82843 q^{4} -2.82843i q^{7} -4.41421i q^{8} +2.00000 q^{11} +1.00000i q^{13} +6.82843 q^{14} +3.00000 q^{16} +3.65685i q^{17} -2.82843 q^{19} +4.82843i q^{22} -4.00000i q^{23} -2.41421 q^{26} +10.8284i q^{28} +2.00000 q^{29} -6.82843 q^{31} -1.58579i q^{32} -8.82843 q^{34} +3.65685i q^{37} -6.82843i q^{38} -10.8284 q^{41} -9.65685i q^{43} -7.65685 q^{44} +9.65685 q^{46} +0.343146i q^{47} -1.00000 q^{49} -3.82843i q^{52} -2.00000i q^{53} -12.4853 q^{56} +4.82843i q^{58} -3.65685 q^{59} -9.31371 q^{61} -16.4853i q^{62} +9.82843 q^{64} +1.17157i q^{67} -14.0000i q^{68} -2.00000 q^{71} -11.6569i q^{73} -8.82843 q^{74} +10.8284 q^{76} -5.65685i q^{77} -11.3137 q^{79} -26.1421i q^{82} -7.65685i q^{83} +23.3137 q^{86} -8.82843i q^{88} +9.17157 q^{89} +2.82843 q^{91} +15.3137i q^{92} -0.828427 q^{94} -7.65685i q^{97} -2.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 8 q^{11} + 16 q^{14} + 12 q^{16} - 4 q^{26} + 8 q^{29} - 16 q^{31} - 24 q^{34} - 32 q^{41} - 8 q^{44} + 16 q^{46} - 4 q^{49} - 16 q^{56} + 8 q^{59} + 8 q^{61} + 28 q^{64} - 8 q^{71} - 24 q^{74} + 32 q^{76} + 48 q^{86} + 48 q^{89} + 8 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^11 + 16 * q^14 + 12 * q^16 - 4 * q^26 + 8 * q^29 - 16 * q^31 - 24 * q^34 - 32 * q^41 - 8 * q^44 + 16 * q^46 - 4 * q^49 - 16 * q^56 + 8 * q^59 + 8 * q^61 + 28 * q^64 - 8 * q^71 - 24 * q^74 + 32 * q^76 + 48 * q^86 + 48 * q^89 + 8 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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