LMFDB - Embedded newform 5472.2.g.b.2737.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(2737,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("5472.2737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.g (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:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2737.13
Root			$0.340606 + 1.37258i$ of defining polynomial
Character	$\chi$	$=$	5472.2737
Dual form			5472.2.g.b.2737.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13486i q^{5} -3.29464 q^{7} +O(q^{10})$	$q+2.13486i q^{5} -3.29464 q^{7} +3.71210i q^{11} -2.32843i q^{13} +6.48822 q^{17} -1.00000i q^{19} +7.32651 q^{23} +0.442384 q^{25} -2.59857i q^{29} +1.34204 q^{31} -7.03360i q^{35} +3.72986i q^{37} -6.52385 q^{41} -1.97202i q^{43} +5.45991 q^{47} +3.85468 q^{49} +4.98640i q^{53} -7.92480 q^{55} -9.67136i q^{59} -8.15570i q^{61} +4.97088 q^{65} -0.524986i q^{67} +7.17489 q^{71} -6.33130 q^{73} -12.2300i q^{77} +8.75644 q^{79} +7.74008i q^{83} +13.8514i q^{85} +1.04368 q^{89} +7.67136i q^{91} +2.13486 q^{95} -0.117594 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{7}+O(q^{10}) $ 16 * q + 8 * q^7	$ 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) $ 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.13486i	0.954737i	0.878703	+	0.477369i	$0.158409\pi$
$5$	2.13486i	0.954737i	−0.878703	+	0.477369i	$0.841591\pi$
$6$	0	0
$7$	−3.29464	−1.24526	−0.622629	−	0.782517i	$-0.713936\pi$
$7$	−3.29464	−1.24526	−0.622629	+	0.782517i	$0.713936\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.71210i	1.11924i	0.828749	+	0.559620i	$0.189053\pi$
$11$	3.71210i	1.11924i	−0.828749	+	0.559620i	$0.810947\pi$
$12$	0	0
$13$	− 2.32843i	− 0.645792i	−0.946435	−	0.322896i	$-0.895344\pi$
$13$	− 2.32843i	− 0.645792i	0.946435	−	0.322896i	$-0.104656\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.48822	1.57362	0.786812	−	0.617192i	$-0.211730\pi$
$17$	6.48822	1.57362	0.786812	+	0.617192i	$0.211730\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.32651	1.52768	0.763842	−	0.645404i	$-0.223311\pi$
$23$	7.32651	1.52768	0.763842	+	0.645404i	$0.223311\pi$
$24$	0	0
$25$	0.442384	0.0884769
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.59857i	− 0.482542i	−0.970458	−	0.241271i	$-0.922436\pi$
$29$	− 2.59857i	− 0.482542i	0.970458	−	0.241271i	$-0.0775643\pi$
$30$	0	0
$31$	1.34204	0.241037	0.120518	−	0.992711i	$-0.461544\pi$
$31$	1.34204	0.241037	0.120518	+	0.992711i	$0.461544\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 7.03360i	− 1.18889i
$36$	0	0
$37$	3.72986i	0.613186i	0.951841	+	0.306593i	$0.0991890\pi$
$37$	3.72986i	0.613186i	−0.951841	+	0.306593i	$0.900811\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.52385	−1.01885	−0.509427	−	0.860514i	$-0.670143\pi$
$41$	−6.52385	−1.01885	−0.509427	+	0.860514i	$0.670143\pi$
$42$	0	0
$43$	− 1.97202i	− 0.300729i	−0.988631	−	0.150365i	$-0.951955\pi$
$43$	− 1.97202i	− 0.300729i	0.988631	−	0.150365i	$-0.0480448\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.45991	0.796410	0.398205	−	0.917297i	$-0.369633\pi$
$47$	5.45991	0.796410	0.398205	+	0.917297i	$0.369633\pi$
$48$	0	0
$49$	3.85468	0.550669
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.98640i	0.684935i	0.939530	+	0.342467i	$0.111263\pi$
$53$	4.98640i	0.684935i	−0.939530	+	0.342467i	$0.888737\pi$
$54$	0	0
$55$	−7.92480	−1.06858
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 9.67136i	− 1.25910i	−0.776958	−	0.629552i	$-0.783238\pi$
$59$	− 9.67136i	− 1.25910i	0.776958	−	0.629552i	$-0.216762\pi$
$60$	0	0
$61$	− 8.15570i	− 1.04423i	−0.852875	−	0.522115i	$-0.825143\pi$
$61$	− 8.15570i	− 1.04423i	0.852875	−	0.522115i	$-0.174857\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.97088	0.616561
$66$	0	0
$67$	− 0.524986i	− 0.0641372i	−0.999486	−	0.0320686i	$-0.989790\pi$
$67$	− 0.524986i	− 0.0641372i	0.999486	−	0.0320686i	$-0.0102095\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.17489	0.851503	0.425751	−	0.904840i	$-0.360010\pi$
$71$	7.17489	0.851503	0.425751	+	0.904840i	$0.360010\pi$
$72$	0	0
$73$	−6.33130	−0.741022	−0.370511	−	0.928828i	$-0.620817\pi$
$73$	−6.33130	−0.741022	−0.370511	+	0.928828i	$0.620817\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.2300i	− 1.39374i
$78$	0	0
$79$	8.75644	0.985176	0.492588	−	0.870263i	$-0.336051\pi$
$79$	8.75644	0.985176	0.492588	+	0.870263i	$0.336051\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.74008i	0.849585i	0.905291	+	0.424792i	$0.139653\pi$
$83$	7.74008i	0.849585i	−0.905291	+	0.424792i	$0.860347\pi$
$84$	0	0
$85$	13.8514i	1.50240i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.04368	0.110630	0.0553148	−	0.998469i	$-0.482384\pi$
$89$	1.04368	0.110630	0.0553148	+	0.998469i	$0.482384\pi$
$90$	0	0
$91$	7.67136i	0.804178i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.13486	0.219032
$96$	0	0
$97$	−0.117594	−0.0119398	−0.00596992	−	0.999982i	$-0.501900\pi$
$97$	−0.117594	−0.0119398	−0.00596992	+	0.999982i	$0.501900\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.6142i	− 1.05615i	−0.849197	−	0.528077i	$-0.822913\pi$
$101$	− 10.6142i	− 1.05615i	0.849197	−	0.528077i	$-0.177087\pi$
$102$	0	0
$103$	−12.4190	−1.22368	−0.611840	−	0.790982i	$-0.709570\pi$
$103$	−12.4190	−1.22368	−0.611840	+	0.790982i	$0.709570\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.5439i	1.59936i	0.600430	+	0.799678i	$0.294996\pi$
$107$	16.5439i	1.59936i	−0.600430	+	0.799678i	$0.705004\pi$
$108$	0	0
$109$	17.4437i	1.67080i	0.549641	+	0.835401i	$0.314765\pi$
$109$	17.4437i	1.67080i	−0.549641	+	0.835401i	$0.685235\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.3193	1.34705	0.673523	−	0.739166i	$-0.264780\pi$
$113$	14.3193	1.34705	0.673523	+	0.739166i	$0.264780\pi$
$114$	0	0
$115$	15.6411i	1.45854i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−21.3764	−1.95957
$120$	0	0
$121$	−2.77968	−0.252698
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.6187i	1.03921i
$126$	0	0
$127$	−2.63985	−0.234248	−0.117124	−	0.993117i	$-0.537368\pi$
$127$	−2.63985	−0.234248	−0.117124	+	0.993117i	$0.537368\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.3670i	1.34263i	0.741174	+	0.671313i	$0.234269\pi$
$131$	15.3670i	1.34263i	−0.741174	+	0.671313i	$0.765731\pi$
$132$	0	0
$133$	3.29464i	0.285682i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.91853	0.420218	0.210109	−	0.977678i	$-0.432618\pi$
$137$	4.91853	0.420218	0.210109	+	0.977678i	$0.432618\pi$
$138$	0	0
$139$	9.80377i	0.831545i	0.909469	+	0.415773i	$0.136489\pi$
$139$	9.80377i	0.831545i	−0.909469	+	0.415773i	$0.863511\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.64338	0.722796
$144$	0	0
$145$	5.54758	0.460701
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 0.724147i	− 0.0593244i	−0.999560	−	0.0296622i	$-0.990557\pi$
$149$	− 0.724147i	− 0.0593244i	0.999560	−	0.0296622i	$-0.00944316\pi$
$150$	0	0
$151$	15.7252	1.27970	0.639850	−	0.768500i	$-0.278997\pi$
$151$	15.7252	1.27970	0.639850	+	0.768500i	$0.278997\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.86505i	0.230127i
$156$	0	0
$157$	− 0.141127i	− 0.0112632i	−0.999984	−	0.00563160i	$-0.998207\pi$
$157$	− 0.141127i	− 0.0112632i	0.999984	−	0.00563160i	$-0.00179260\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−24.1383	−1.90236
$162$	0	0
$163$	8.41859i	0.659395i	0.944087	+	0.329697i	$0.106947\pi$
$163$	8.41859i	0.659395i	−0.944087	+	0.329697i	$0.893053\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.16538	0.399709	0.199854	−	0.979826i	$-0.435953\pi$
$167$	5.16538	0.399709	0.199854	+	0.979826i	$0.435953\pi$
$168$	0	0
$169$	7.57839	0.582953
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.13988i	− 0.238721i	−0.992851	−	0.119360i	$-0.961916\pi$
$173$	− 3.13988i	− 0.238721i	0.992851	−	0.119360i	$-0.0380844\pi$
$174$	0	0
$175$	−1.45750	−0.110177
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.1898i	1.35957i	0.733410	+	0.679786i	$0.237927\pi$
$179$	18.1898i	1.35957i	−0.733410	+	0.679786i	$0.762073\pi$
$180$	0	0
$181$	14.0798i	1.04654i	0.852166	+	0.523271i	$0.175288\pi$
$181$	14.0798i	1.04654i	−0.852166	+	0.523271i	$0.824712\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.96273	−0.585431
$186$	0	0
$187$	24.0849i	1.76126i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.87409	0.352677	0.176338	−	0.984330i	$-0.443575\pi$
$191$	4.87409	0.352677	0.176338	+	0.984330i	$0.443575\pi$
$192$	0	0
$193$	−9.99845	−0.719704	−0.359852	−	0.933009i	$-0.617173\pi$
$193$	−9.99845	−0.719704	−0.359852	+	0.933009i	$0.617173\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.2209i	1.36943i	0.728810	+	0.684716i	$0.240074\pi$
$197$	19.2209i	1.36943i	−0.728810	+	0.684716i	$0.759926\pi$
$198$	0	0
$199$	11.7323	0.831683	0.415842	−	0.909437i	$-0.363487\pi$
$199$	11.7323	0.831683	0.415842	+	0.909437i	$0.363487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.56137i	0.600890i
$204$	0	0
$205$	− 13.9275i	− 0.972737i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.71210	0.256771
$210$	0	0
$211$	0.399383i	0.0274947i	0.999906	+	0.0137473i	$0.00437605\pi$
$211$	0.399383i	0.0274947i	−0.999906	+	0.0137473i	$0.995624\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.20997	0.287118
$216$	0	0
$217$	−4.42153	−0.300153
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 15.1074i	− 1.01623i
$222$	0	0
$223$	19.2548	1.28939	0.644697	−	0.764438i	$-0.276983\pi$
$223$	19.2548	1.28939	0.644697	+	0.764438i	$0.276983\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 20.0414i	− 1.33019i	−0.746758	−	0.665096i	$-0.768391\pi$
$227$	− 20.0414i	− 1.33019i	0.746758	−	0.665096i	$-0.231609\pi$
$228$	0	0
$229$	− 18.2013i	− 1.20277i	−0.798958	−	0.601387i	$-0.794615\pi$
$229$	− 18.2013i	− 1.20277i	0.798958	−	0.601387i	$-0.205385\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.2376	−0.932739	−0.466369	−	0.884590i	$-0.654438\pi$
$233$	−14.2376	−0.932739	−0.466369	+	0.884590i	$0.654438\pi$
$234$	0	0
$235$	11.6561i	0.760362i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.31368	0.537767	0.268884	−	0.963173i	$-0.413345\pi$
$239$	8.31368	0.537767	0.268884	+	0.963173i	$0.413345\pi$
$240$	0	0
$241$	−26.1776	−1.68625	−0.843125	−	0.537718i	$-0.819287\pi$
$241$	−26.1776	−1.68625	−0.843125	+	0.537718i	$0.819287\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.22920i	0.525744i
$246$	0	0
$247$	−2.32843	−0.148155
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.4740i	1.86038i	0.367075	+	0.930191i	$0.380359\pi$
$251$	29.4740i	1.86038i	−0.367075	+	0.930191i	$0.619641\pi$
$252$	0	0
$253$	27.1967i	1.70984i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.28195	−0.0799657	−0.0399828	−	0.999200i	$-0.512730\pi$
$257$	−1.28195	−0.0799657	−0.0399828	+	0.999200i	$0.512730\pi$
$258$	0	0
$259$	− 12.2886i	− 0.763575i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−29.2736	−1.80509	−0.902544	−	0.430597i	$-0.858303\pi$
$263$	−29.2736	−1.80509	−0.902544	+	0.430597i	$0.858303\pi$
$264$	0	0
$265$	−10.6453	−0.653933
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 14.1850i	− 0.864875i	−0.901664	−	0.432437i	$-0.857654\pi$
$269$	− 14.1850i	− 0.864875i	0.901664	−	0.432437i	$-0.142346\pi$
$270$	0	0
$271$	−26.3185	−1.59874	−0.799369	−	0.600840i	$-0.794833\pi$
$271$	−26.3185	−1.59874	−0.799369	+	0.600840i	$0.794833\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.64217i	0.0990269i
$276$	0	0
$277$	− 3.58863i	− 0.215620i	−0.994172	−	0.107810i	$-0.965616\pi$
$277$	− 3.58863i	− 0.215620i	0.994172	−	0.107810i	$-0.0343838\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.42597	0.264031	0.132016	−	0.991248i	$-0.457855\pi$
$281$	4.42597	0.264031	0.132016	+	0.991248i	$0.457855\pi$
$282$	0	0
$283$	0.493151i	0.0293148i	0.999893	+	0.0146574i	$0.00466576\pi$
$283$	0.493151i	0.0293148i	−0.999893	+	0.0146574i	$0.995334\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	21.4938	1.26874
$288$	0	0
$289$	25.0970	1.47630
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.7228i	0.743275i	0.928378	+	0.371637i	$0.121204\pi$
$293$	12.7228i	0.743275i	−0.928378	+	0.371637i	$0.878796\pi$
$294$	0	0
$295$	20.6470	1.20211
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 17.0593i	− 0.986565i
$300$	0	0
$301$	6.49709i	0.374486i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	17.4113	0.996966
$306$	0	0
$307$	− 10.2166i	− 0.583092i	−0.956557	−	0.291546i	$-0.905830\pi$
$307$	− 10.2166i	− 0.583092i	0.956557	−	0.291546i	$-0.0941697\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0.184934	0.0104867	0.00524333	−	0.999986i	$-0.498331\pi$
$311$	0.184934	0.0104867	0.00524333	+	0.999986i	$0.498331\pi$
$312$	0	0
$313$	−19.0733	−1.07809	−0.539044	−	0.842278i	$-0.681214\pi$
$313$	−19.0733	−1.07809	−0.539044	+	0.842278i	$0.681214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.911635i	0.0512025i	0.999672	+	0.0256013i	$0.00815002\pi$
$317$	0.911635i	0.0512025i	−0.999672	+	0.0256013i	$0.991850\pi$
$318$	0	0
$319$	9.64615	0.540081
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.48822i	− 0.361014i
$324$	0	0
$325$	− 1.03006i	− 0.0571376i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17.9885	−0.991736
$330$	0	0
$331$	2.45230i	0.134791i	0.997726	+	0.0673953i	$0.0214689\pi$
$331$	2.45230i	0.134791i	−0.997726	+	0.0673953i	$0.978531\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.12077	0.0612342
$336$	0	0
$337$	35.1467	1.91456	0.957282	−	0.289156i	$-0.0933746\pi$
$337$	35.1467	1.91456	0.957282	+	0.289156i	$0.0933746\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.98177i	0.269778i
$342$	0	0
$343$	10.3627	0.559533
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5.30643i	− 0.284864i	−0.989805	−	0.142432i	$-0.954508\pi$
$347$	− 5.30643i	− 0.284864i	0.989805	−	0.142432i	$-0.0454922\pi$
$348$	0	0
$349$	26.8913i	1.43946i	0.694255	+	0.719729i	$0.255734\pi$
$349$	26.8913i	1.43946i	−0.694255	+	0.719729i	$0.744266\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.21383	0.171055	0.0855275	−	0.996336i	$-0.472742\pi$
$353$	3.21383	0.171055	0.0855275	+	0.996336i	$0.472742\pi$
$354$	0	0
$355$	15.3174i	0.812961i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.6827	−0.827699	−0.413850	−	0.910345i	$-0.635816\pi$
$359$	−15.6827	−0.827699	−0.413850	+	0.910345i	$0.635816\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 13.5164i	− 0.707481i
$366$	0	0
$367$	−14.3509	−0.749111	−0.374555	−	0.927205i	$-0.622205\pi$
$367$	−14.3509	−0.749111	−0.374555	+	0.927205i	$0.622205\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 16.4284i	− 0.852921i
$372$	0	0
$373$	9.08701i	0.470508i	0.971934	+	0.235254i	$0.0755921\pi$
$373$	9.08701i	0.470508i	−0.971934	+	0.235254i	$0.924408\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.05060	−0.311622
$378$	0	0
$379$	− 18.2006i	− 0.934900i	−0.884020	−	0.467450i	$-0.845173\pi$
$379$	− 18.2006i	− 0.934900i	0.884020	−	0.467450i	$-0.154827\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.4154	−1.14537	−0.572686	−	0.819775i	$-0.694099\pi$
$383$	−22.4154	−1.14537	−0.572686	+	0.819775i	$0.694099\pi$
$384$	0	0
$385$	26.1094	1.33066
$386$	0	0
$387$	0	0
$388$	0	0
$389$	34.9213i	1.77058i	0.465038	+	0.885291i	$0.346041\pi$
$389$	34.9213i	1.77058i	−0.465038	+	0.885291i	$0.653959\pi$
$390$	0	0
$391$	47.5360	2.40400
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.6937i	0.940584i
$396$	0	0
$397$	− 31.3443i	− 1.57313i	−0.617510	−	0.786563i	$-0.711858\pi$
$397$	− 31.3443i	− 1.57313i	0.617510	−	0.786563i	$-0.288142\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.64321	−0.0820579	−0.0410290	−	0.999158i	$-0.513064\pi$
$401$	−1.64321	−0.0820579	−0.0410290	+	0.999158i	$0.513064\pi$
$402$	0	0
$403$	− 3.12484i	− 0.155659i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.8456	−0.686302
$408$	0	0
$409$	−2.46645	−0.121958	−0.0609791	−	0.998139i	$-0.519422\pi$
$409$	−2.46645	−0.121958	−0.0609791	+	0.998139i	$0.519422\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	31.8637i	1.56791i
$414$	0	0
$415$	−16.5240	−0.811130
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 4.25063i	− 0.207657i	−0.994595	−	0.103828i	$-0.966891\pi$
$419$	− 4.25063i	− 0.207657i	0.994595	−	0.103828i	$-0.0331093\pi$
$420$	0	0
$421$	− 31.4900i	− 1.53473i	−0.641212	−	0.767364i	$-0.721568\pi$
$421$	− 31.4900i	− 1.53473i	0.641212	−	0.767364i	$-0.278432\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.87029	0.139229
$426$	0	0
$427$	26.8701i	1.30034i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.0564	−1.01425	−0.507125	−	0.861872i	$-0.669292\pi$
$431$	−21.0564	−1.01425	−0.507125	+	0.861872i	$0.669292\pi$
$432$	0	0
$433$	−7.53125	−0.361929	−0.180964	−	0.983490i	$-0.557922\pi$
$433$	−7.53125	−0.361929	−0.180964	+	0.983490i	$0.557922\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.32651i	− 0.350475i
$438$	0	0
$439$	−0.823995	−0.0393271	−0.0196636	−	0.999807i	$-0.506260\pi$
$439$	−0.823995	−0.0393271	−0.0196636	+	0.999807i	$0.506260\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.58526i	0.407898i	0.978982	+	0.203949i	$0.0653777\pi$
$443$	8.58526i	0.407898i	−0.978982	+	0.203949i	$0.934622\pi$
$444$	0	0
$445$	2.22810i	0.105622i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.49451	−0.117723	−0.0588615	−	0.998266i	$-0.518747\pi$
$449$	−2.49451	−0.117723	−0.0588615	+	0.998266i	$0.518747\pi$
$450$	0	0
$451$	− 24.2172i	− 1.14034i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16.3773	−0.767778
$456$	0	0
$457$	14.5004	0.678301	0.339150	−	0.940732i	$-0.389860\pi$
$457$	14.5004	0.678301	0.339150	+	0.940732i	$0.389860\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 23.1930i	− 1.08020i	−0.841600	−	0.540102i	$-0.818386\pi$
$461$	− 23.1930i	− 1.08020i	0.841600	−	0.540102i	$-0.181614\pi$
$462$	0	0
$463$	−26.0637	−1.21128	−0.605642	−	0.795737i	$-0.707084\pi$
$463$	−26.0637	−1.21128	−0.605642	+	0.795737i	$0.707084\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.1953i	− 0.564332i	−0.959366	−	0.282166i	$-0.908947\pi$
$467$	− 12.1953i	− 0.564332i	0.959366	−	0.282166i	$-0.0910529\pi$
$468$	0	0
$469$	1.72964i	0.0798674i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.32031	0.336588
$474$	0	0
$475$	− 0.442384i	− 0.0202980i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.1883	0.831045	0.415523	−	0.909583i	$-0.363599\pi$
$479$	18.1883	0.831045	0.415523	+	0.909583i	$0.363599\pi$
$480$	0	0
$481$	8.68475	0.395990
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 0.251046i	− 0.0113994i
$486$	0	0
$487$	2.13693	0.0968334	0.0484167	−	0.998827i	$-0.484582\pi$
$487$	2.13693	0.0968334	0.0484167	+	0.998827i	$0.484582\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.32773i	0.105049i	0.998620	+	0.0525244i	$0.0167267\pi$
$491$	2.32773i	0.105049i	−0.998620	+	0.0525244i	$0.983273\pi$
$492$	0	0
$493$	− 16.8601i	− 0.759341i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23.6387	−1.06034
$498$	0	0
$499$	16.9382i	0.758260i	0.925343	+	0.379130i	$0.123777\pi$
$499$	16.9382i	0.758260i	−0.925343	+	0.379130i	$0.876223\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.22413	0.366696	0.183348	−	0.983048i	$-0.441307\pi$
$503$	8.22413	0.366696	0.183348	+	0.983048i	$0.441307\pi$
$504$	0	0
$505$	22.6598	1.00835
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.8145i	0.878261i	0.898423	+	0.439131i	$0.144713\pi$
$509$	19.8145i	0.878261i	−0.898423	+	0.439131i	$0.855287\pi$
$510$	0	0
$511$	20.8594	0.922764
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 26.5128i	− 1.16829i
$516$	0	0
$517$	20.2677i	0.891374i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	36.0978	1.58147	0.790737	−	0.612156i	$-0.209698\pi$
$521$	36.0978	1.58147	0.790737	+	0.612156i	$0.209698\pi$
$522$	0	0
$523$	4.19475i	0.183424i	0.995786	+	0.0917119i	$0.0292339\pi$
$523$	4.19475i	0.183424i	−0.995786	+	0.0917119i	$0.970766\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.70743	0.379301
$528$	0	0
$529$	30.6778	1.33382
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.1903i	0.657967i
$534$	0	0
$535$	−35.3188	−1.52696
$536$	0	0
$537$	0	0
$538$	0	0
$539$	14.3090i	0.616331i
$540$	0	0
$541$	6.12211i	0.263210i	0.991302	+	0.131605i	$0.0420131\pi$
$541$	6.12211i	0.263210i	−0.991302	+	0.131605i	$0.957987\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−37.2398	−1.59518
$546$	0	0
$547$	− 4.88879i	− 0.209029i	−0.994523	−	0.104515i	$-0.966671\pi$
$547$	− 4.88879i	− 0.209029i	0.994523	−	0.104515i	$-0.0333289\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.59857	−0.110703
$552$	0	0
$553$	−28.8493	−1.22680
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.5987i	1.21176i	0.795554	+	0.605882i	$0.207180\pi$
$557$	28.5987i	1.21176i	−0.795554	+	0.605882i	$0.792820\pi$
$558$	0	0
$559$	−4.59171	−0.194209
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 9.83345i	− 0.414431i	−0.978295	−	0.207215i	$-0.933560\pi$
$563$	− 9.83345i	− 0.414431i	0.978295	−	0.207215i	$-0.0664401\pi$
$564$	0	0
$565$	30.5697i	1.28608i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.3543	0.434075	0.217037	−	0.976163i	$-0.430361\pi$
$569$	10.3543	0.434075	0.217037	+	0.976163i	$0.430361\pi$
$570$	0	0
$571$	− 36.7988i	− 1.53998i	−0.638055	−	0.769991i	$-0.720261\pi$
$571$	− 36.7988i	− 1.53998i	0.638055	−	0.769991i	$-0.279739\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.24113	0.135165
$576$	0	0
$577$	−8.22871	−0.342566	−0.171283	−	0.985222i	$-0.554791\pi$
$577$	−8.22871	−0.342566	−0.171283	+	0.985222i	$0.554791\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 25.5008i	− 1.05795i
$582$	0	0
$583$	−18.5100	−0.766606
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.18047i	0.296370i	0.988960	+	0.148185i	$0.0473431\pi$
$587$	7.18047i	0.296370i	−0.988960	+	0.148185i	$0.952657\pi$
$588$	0	0
$589$	− 1.34204i	− 0.0552976i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.7687	1.01713	0.508564	−	0.861024i	$-0.330177\pi$
$593$	24.7687	1.01713	0.508564	+	0.861024i	$0.330177\pi$
$594$	0	0
$595$	− 45.6355i	− 1.87087i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	41.9307	1.71324	0.856621	−	0.515946i	$-0.172560\pi$
$599$	41.9307	1.71324	0.856621	+	0.515946i	$0.172560\pi$
$600$	0	0
$601$	2.10027	0.0856718	0.0428359	−	0.999082i	$-0.486361\pi$
$601$	2.10027	0.0856718	0.0428359	+	0.999082i	$0.486361\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 5.93422i	− 0.241260i
$606$	0	0
$607$	11.2877	0.458152	0.229076	−	0.973409i	$-0.426430\pi$
$607$	11.2877	0.458152	0.229076	+	0.973409i	$0.426430\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 12.7130i	− 0.514315i
$612$	0	0
$613$	31.4983i	1.27220i	0.771605	+	0.636102i	$0.219454\pi$
$613$	31.4983i	1.27220i	−0.771605	+	0.636102i	$0.780546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.4563	0.461214	0.230607	−	0.973047i	$-0.425929\pi$
$617$	11.4563	0.461214	0.230607	+	0.973047i	$0.425929\pi$
$618$	0	0
$619$	37.6816i	1.51455i	0.653095	+	0.757276i	$0.273470\pi$
$619$	37.6816i	1.51455i	−0.653095	+	0.757276i	$0.726530\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.43855	−0.137763
$624$	0	0
$625$	−22.5924	−0.903695
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.2002i	0.964925i
$630$	0	0
$631$	−6.99670	−0.278534	−0.139267	−	0.990255i	$-0.544475\pi$
$631$	−6.99670	−0.278534	−0.139267	+	0.990255i	$0.544475\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 5.63569i	− 0.223646i
$636$	0	0
$637$	− 8.97538i	− 0.355617i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.20749	0.245181	0.122591	−	0.992457i	$-0.460880\pi$
$641$	6.20749	0.245181	0.122591	+	0.992457i	$0.460880\pi$
$642$	0	0
$643$	29.4008i	1.15945i	0.814811	+	0.579726i	$0.196841\pi$
$643$	29.4008i	1.15945i	−0.814811	+	0.579726i	$0.803159\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.62285	0.221057	0.110529	−	0.993873i	$-0.464746\pi$
$647$	5.62285	0.221057	0.110529	+	0.993873i	$0.464746\pi$
$648$	0	0
$649$	35.9011	1.40924
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 34.9447i	− 1.36749i	−0.729721	−	0.683745i	$-0.760350\pi$
$653$	− 34.9447i	− 1.36749i	0.729721	−	0.683745i	$-0.239650\pi$
$654$	0	0
$655$	−32.8064	−1.28185
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 37.8768i	− 1.47547i	−0.675090	−	0.737736i	$-0.735895\pi$
$659$	− 37.8768i	− 1.47547i	0.675090	−	0.737736i	$-0.264105\pi$
$660$	0	0
$661$	6.77264i	0.263425i	0.991288	+	0.131713i	$0.0420476\pi$
$661$	6.77264i	0.263425i	−0.991288	+	0.131713i	$0.957952\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.03360	−0.272751
$666$	0	0
$667$	− 19.0385i	− 0.737172i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	30.2748	1.16874
$672$	0	0
$673$	24.6355	0.949630	0.474815	−	0.880086i	$-0.342515\pi$
$673$	24.6355	0.949630	0.474815	+	0.880086i	$0.342515\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.2956i	− 1.08749i	−0.839251	−	0.543744i	$-0.817006\pi$
$677$	− 28.2956i	− 1.08749i	0.839251	−	0.543744i	$-0.182994\pi$
$678$	0	0
$679$	0.387430	0.0148682
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 34.4333i	− 1.31755i	−0.752339	−	0.658777i	$-0.771074\pi$
$683$	− 34.4333i	− 1.31755i	0.752339	−	0.658777i	$-0.228926\pi$
$684$	0	0
$685$	10.5004i	0.401198i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.6105	0.442325
$690$	0	0
$691$	2.19928i	0.0836646i	0.999125	+	0.0418323i	$0.0133195\pi$
$691$	2.19928i	0.0836646i	−0.999125	+	0.0418323i	$0.986680\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.9297	−0.793907
$696$	0	0
$697$	−42.3282	−1.60329
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.0608i	1.36200i	0.732285	+	0.680999i	$0.238454\pi$
$701$	36.0608i	1.36200i	−0.732285	+	0.680999i	$0.761546\pi$
$702$	0	0
$703$	3.72986	0.140675
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.9701i	1.31518i
$708$	0	0
$709$	10.7299i	0.402970i	0.979492	+	0.201485i	$0.0645768\pi$
$709$	10.7299i	0.402970i	−0.979492	+	0.201485i	$0.935423\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.83244	0.368228
$714$	0	0
$715$	18.4524i	0.690080i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29.1037	1.08539	0.542693	−	0.839931i	$-0.317405\pi$
$719$	29.1037	1.08539	0.542693	+	0.839931i	$0.317405\pi$
$720$	0	0
$721$	40.9162	1.52380
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.14957i	− 0.0426938i
$726$	0	0
$727$	40.5850	1.50521	0.752607	−	0.658470i	$-0.228796\pi$
$727$	40.5850	1.50521	0.752607	+	0.658470i	$0.228796\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 12.7949i	− 0.473235i
$732$	0	0
$733$	− 18.6206i	− 0.687766i	−0.939012	−	0.343883i	$-0.888258\pi$
$733$	− 18.6206i	− 0.687766i	0.939012	−	0.343883i	$-0.111742\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.94880	0.0717849
$738$	0	0
$739$	7.89939i	0.290584i	0.989389	+	0.145292i	$0.0464121\pi$
$739$	7.89939i	0.290584i	−0.989389	+	0.145292i	$0.953588\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.4064	−0.528521	−0.264261	−	0.964451i	$-0.585128\pi$
$743$	−14.4064	−0.528521	−0.264261	+	0.964451i	$0.585128\pi$
$744$	0	0
$745$	1.54595	0.0566392
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 54.5061i	− 1.99161i
$750$	0	0
$751$	−14.0914	−0.514204	−0.257102	−	0.966384i	$-0.582768\pi$
$751$	−14.0914	−0.514204	−0.257102	+	0.966384i	$0.582768\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	33.5711i	1.22178i
$756$	0	0
$757$	− 50.2311i	− 1.82568i	−0.408318	−	0.912840i	$-0.633884\pi$
$757$	− 50.2311i	− 1.82568i	0.408318	−	0.912840i	$-0.366116\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.7483	−0.860875	−0.430437	−	0.902620i	$-0.641641\pi$
$761$	−23.7483	−0.860875	−0.430437	+	0.902620i	$0.641641\pi$
$762$	0	0
$763$	− 57.4707i	− 2.08058i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.5191	−0.813119
$768$	0	0
$769$	−47.4035	−1.70942	−0.854708	−	0.519109i	$-0.826264\pi$
$769$	−47.4035	−1.70942	−0.854708	+	0.519109i	$0.826264\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	11.8508i	0.426242i	0.977026	+	0.213121i	$0.0683628\pi$
$773$	11.8508i	0.426242i	−0.977026	+	0.213121i	$0.931637\pi$
$774$	0	0
$775$	0.593696	0.0213262
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.52385i	0.233741i
$780$	0	0
$781$	26.6339i	0.953036i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.301287	0.0107534
$786$	0	0
$787$	26.3903i	0.940713i	0.882477	+	0.470356i	$0.155875\pi$
$787$	26.3903i	0.940713i	−0.882477	+	0.470356i	$0.844125\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−47.1770	−1.67742
$792$	0	0
$793$	−18.9900	−0.674356
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.1893i	0.502610i	0.967908	+	0.251305i	$0.0808597\pi$
$797$	14.1893i	0.502610i	−0.967908	+	0.251305i	$0.919140\pi$
$798$	0	0
$799$	35.4251	1.25325
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 23.5024i	− 0.829382i
$804$	0	0
$805$	− 51.5317i	− 1.81625i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.9998	1.05474	0.527368	−	0.849637i	$-0.323179\pi$
$809$	29.9998	1.05474	0.527368	+	0.849637i	$0.323179\pi$
$810$	0	0
$811$	− 39.8833i	− 1.40049i	−0.713901	−	0.700246i	$-0.753073\pi$
$811$	− 39.8833i	− 1.40049i	0.713901	−	0.700246i	$-0.246927\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−17.9725	−0.629549
$816$	0	0
$817$	−1.97202	−0.0689921
$818$	0	0
$819$	0	0
$820$	0	0
$821$	56.4395i	1.96975i	0.173256	+	0.984877i	$0.444571\pi$
$821$	56.4395i	1.96975i	−0.173256	+	0.984877i	$0.555429\pi$
$822$	0	0
$823$	41.7360	1.45482	0.727412	−	0.686201i	$-0.240723\pi$
$823$	41.7360	1.45482	0.727412	+	0.686201i	$0.240723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17.4248i	− 0.605921i	−0.953003	−	0.302960i	$-0.902025\pi$
$827$	− 17.4248i	− 0.605921i	0.953003	−	0.302960i	$-0.0979750\pi$
$828$	0	0
$829$	52.2723i	1.81549i	0.419519	+	0.907747i	$0.362199\pi$
$829$	52.2723i	1.81549i	−0.419519	+	0.907747i	$0.637801\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25.0100	0.866547
$834$	0	0
$835$	11.0273i	0.381617i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.8429	1.41005	0.705027	−	0.709180i	$-0.250935\pi$
$839$	40.8429	1.41005	0.705027	+	0.709180i	$0.250935\pi$
$840$	0	0
$841$	22.2474	0.767153
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.1788i	0.556567i
$846$	0	0
$847$	9.15805	0.314674
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27.3269i	0.936754i
$852$	0	0
$853$	42.8153i	1.46597i	0.680245	+	0.732984i	$0.261873\pi$
$853$	42.8153i	1.46597i	−0.680245	+	0.732984i	$0.738127\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.2947	−0.795732	−0.397866	−	0.917444i	$-0.630249\pi$
$857$	−23.2947	−0.795732	−0.397866	+	0.917444i	$0.630249\pi$
$858$	0	0
$859$	17.5879i	0.600091i	0.953925	+	0.300045i	$0.0970018\pi$
$859$	17.5879i	0.600091i	−0.953925	+	0.300045i	$0.902998\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.8886	−0.404692	−0.202346	−	0.979314i	$-0.564857\pi$
$863$	−11.8886	−0.404692	−0.202346	+	0.979314i	$0.564857\pi$
$864$	0	0
$865$	6.70319	0.227915
$866$	0	0
$867$	0	0
$868$	0	0
$869$	32.5048i	1.10265i
$870$	0	0
$871$	−1.22239	−0.0414193
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 38.2795i	− 1.29408i
$876$	0	0
$877$	− 28.6698i	− 0.968111i	−0.875037	−	0.484056i	$-0.839163\pi$
$877$	− 28.6698i	− 0.968111i	0.875037	−	0.484056i	$-0.160837\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44.2297	1.49014	0.745069	−	0.666988i	$-0.232417\pi$
$881$	44.2297	1.49014	0.745069	+	0.666988i	$0.232417\pi$
$882$	0	0
$883$	− 26.5869i	− 0.894719i	−0.894354	−	0.447359i	$-0.852364\pi$
$883$	− 26.5869i	− 0.894719i	0.894354	−	0.447359i	$-0.147636\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.4124	0.551074	0.275537	−	0.961290i	$-0.411144\pi$
$887$	16.4124	0.551074	0.275537	+	0.961290i	$0.411144\pi$
$888$	0	0
$889$	8.69735	0.291700
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.45991i	− 0.182709i
$894$	0	0
$895$	−38.8327	−1.29803
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.48737i	− 0.116310i
$900$	0	0
$901$	32.3529i	1.07783i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−30.0583	−0.999172
$906$	0	0
$907$	− 47.3380i	− 1.57183i	−0.618333	−	0.785916i	$-0.712192\pi$
$907$	− 47.3380i	− 1.57183i	0.618333	−	0.785916i	$-0.287808\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.30896	0.175894	0.0879469	−	0.996125i	$-0.471969\pi$
$911$	5.30896	0.175894	0.0879469	+	0.996125i	$0.471969\pi$
$912$	0	0
$913$	−28.7320	−0.950889
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 50.6290i	− 1.67192i
$918$	0	0
$919$	−49.2513	−1.62465	−0.812326	−	0.583204i	$-0.801799\pi$
$919$	−49.2513	−1.62465	−0.812326	+	0.583204i	$0.801799\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 16.7063i	− 0.549893i
$924$	0	0
$925$	1.65003i	0.0542528i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29.2990	0.961270	0.480635	−	0.876921i	$-0.340406\pi$
$929$	29.2990	0.961270	0.480635	+	0.876921i	$0.340406\pi$
$930$	0	0
$931$	− 3.85468i	− 0.126332i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−51.4179	−1.68154
$936$	0	0
$937$	−6.35345	−0.207558	−0.103779	−	0.994600i	$-0.533093\pi$
$937$	−6.35345	−0.207558	−0.103779	+	0.994600i	$0.533093\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.07523i	0.0676505i	0.999428	+	0.0338253i	$0.0107690\pi$
$941$	2.07523i	0.0676505i	−0.999428	+	0.0338253i	$0.989231\pi$
$942$	0	0
$943$	−47.7970	−1.55649
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13.3618i	− 0.434201i	−0.976149	−	0.217101i	$-0.930340\pi$
$947$	− 13.3618i	− 0.434201i	0.976149	−	0.217101i	$-0.0696600\pi$
$948$	0	0
$949$	14.7420i	0.478546i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52.6796	1.70646	0.853231	−	0.521534i	$-0.174640\pi$
$953$	52.6796	1.70646	0.853231	+	0.521534i	$0.174640\pi$
$954$	0	0
$955$	10.4055i	0.336714i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.2048	−0.523281
$960$	0	0
$961$	−29.1989	−0.941901
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 21.3453i	− 0.687128i
$966$	0	0
$967$	7.17519	0.230739	0.115369	−	0.993323i	$-0.463195\pi$
$967$	7.17519	0.230739	0.115369	+	0.993323i	$0.463195\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 14.6991i	− 0.471718i	−0.971787	−	0.235859i	$-0.924210\pi$
$971$	− 14.6991i	− 0.471718i	0.971787	−	0.235859i	$-0.0757903\pi$
$972$	0	0
$973$	− 32.3000i	− 1.03549i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.5470	−0.625365	−0.312682	−	0.949858i	$-0.601228\pi$
$977$	−19.5470	−0.625365	−0.312682	+	0.949858i	$0.601228\pi$
$978$	0	0
$979$	3.87424i	0.123821i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	24.0339	0.766562	0.383281	−	0.923632i	$-0.374794\pi$
$983$	24.0339	0.766562	0.383281	+	0.923632i	$0.374794\pi$
$984$	0	0
$985$	−41.0339	−1.30745
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 14.4480i	− 0.459419i
$990$	0	0
$991$	−50.5419	−1.60552	−0.802758	−	0.596305i	$-0.796635\pi$
$991$	−50.5419	−1.60552	−0.802758	+	0.596305i	$0.796635\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.0469i	0.794039i
$996$	0	0
$997$	21.9225i	0.694293i	0.937811	+	0.347147i	$0.112849\pi$
$997$	21.9225i	0.694293i	−0.937811	+	0.347147i	$0.887151\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.g.b.2737.13		16
3.2	odd	2		608.2.c.b.305.15		16
4.3	odd	2		1368.2.g.b.685.14		16
8.3	odd	2		1368.2.g.b.685.13		16
8.5	even	2	inner	5472.2.g.b.2737.4		16
12.11	even	2		152.2.c.b.77.3	✓	16
24.5	odd	2		608.2.c.b.305.2		16
24.11	even	2		152.2.c.b.77.4	yes	16
48.5	odd	4		4864.2.a.bp.1.8		8
48.11	even	4		4864.2.a.bq.1.1		8
48.29	odd	4		4864.2.a.bn.1.1		8
48.35	even	4		4864.2.a.bo.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.3	✓	16	12.11	even	2
152.2.c.b.77.4	yes	16	24.11	even	2
608.2.c.b.305.2		16	24.5	odd	2
608.2.c.b.305.15		16	3.2	odd	2
1368.2.g.b.685.13		16	8.3	odd	2
1368.2.g.b.685.14		16	4.3	odd	2
4864.2.a.bn.1.1		8	48.29	odd	4
4864.2.a.bo.1.8		8	48.35	even	4
4864.2.a.bp.1.8		8	48.5	odd	4
4864.2.a.bq.1.1		8	48.11	even	4
5472.2.g.b.2737.4		16	8.5	even	2	inner
5472.2.g.b.2737.13		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.13486i q^{5} -3.29464 q^{7} +O(q^{10})\)	\(q+2.13486i q^{5} -3.29464 q^{7} +3.71210i q^{11} -2.32843i q^{13} +6.48822 q^{17} -1.00000i q^{19} +7.32651 q^{23} +0.442384 q^{25} -2.59857i q^{29} +1.34204 q^{31} -7.03360i q^{35} +3.72986i q^{37} -6.52385 q^{41} -1.97202i q^{43} +5.45991 q^{47} +3.85468 q^{49} +4.98640i q^{53} -7.92480 q^{55} -9.67136i q^{59} -8.15570i q^{61} +4.97088 q^{65} -0.524986i q^{67} +7.17489 q^{71} -6.33130 q^{73} -12.2300i q^{77} +8.75644 q^{79} +7.74008i q^{83} +13.8514i q^{85} +1.04368 q^{89} +7.67136i q^{91} +2.13486 q^{95} -0.117594 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{7}+O(q^{10}) \) 16 * q + 8 * q^7	\( 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) \) 16 * q + 8 * q^7 + 8 * q^17 - 24 * q^25 - 16 * q^31 - 16 * q^41 + 24 * q^47 + 24 * q^49 - 16 * q^55 - 16 * q^65 + 48 * q^71 + 48 * q^79 + 16 * q^89 + 16 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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