LMFDB - Embedded newform 416.2.u.b.47.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(47,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

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

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

chi := DirichletCharacter("416.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.u (of order $4$, degree $2$, 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:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 $ x^20 - 2x^19 + 2x^18 - 7x^16 + 14x^15 - 14x^14 + 8x^13 + 16x^12 - 40x^11 + 48x^10 - 80x^9 + 64x^8 + 64x^7 - 224x^6 + 448x^5 - 448x^4 + 512x^2 - 1024*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.3
Root			$1.15650 - 0.813947i$ of defining polynomial
Character	$\chi$	$=$	416.47
Dual form			416.2.u.b.239.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.38809 q^{3} +(-2.40872 + 2.40872i) q^{5} +(0.127019 + 0.127019i) q^{7} -1.07320 q^{9} +(2.63237 - 2.63237i) q^{11} +(1.93837 - 3.04019i) q^{13} +(3.34352 - 3.34352i) q^{15} -4.33908i q^{17} +(-4.97146 - 4.97146i) q^{19} +(-0.176315 - 0.176315i) q^{21} -3.98711 q^{23} -6.60383i q^{25} +5.65398 q^{27} +4.59378i q^{29} +(-1.07702 + 1.07702i) q^{31} +(-3.65398 + 3.65398i) q^{33} -0.611907 q^{35} +(-2.45494 - 2.45494i) q^{37} +(-2.69063 + 4.22006i) q^{39} +(0.388093 + 0.388093i) q^{41} -5.02419i q^{43} +(2.58503 - 2.58503i) q^{45} +(-1.00356 - 1.00356i) q^{47} -6.96773i q^{49} +6.02305i q^{51} +1.83922i q^{53} +12.6813i q^{55} +(6.90084 + 6.90084i) q^{57} +(-6.28635 + 6.28635i) q^{59} -10.5817i q^{61} +(-0.136317 - 0.136317i) q^{63} +(2.65398 + 11.9919i) q^{65} +(4.21315 + 4.21315i) q^{67} +5.53449 q^{69} +(2.59715 - 2.59715i) q^{71} +(-0.388093 + 0.388093i) q^{73} +9.16673i q^{75} +0.668725 q^{77} -15.3812i q^{79} -4.62865 q^{81} +(2.46502 + 2.46502i) q^{83} +(10.4516 + 10.4516i) q^{85} -6.37660i q^{87} +(-11.3583 + 11.3583i) q^{89} +(0.632373 - 0.139953i) q^{91} +(1.49500 - 1.49500i) q^{93} +23.9497 q^{95} +(3.31375 + 3.31375i) q^{97} +(-2.82506 + 2.82506i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 16 q^{9} + 8 q^{11} + 12 q^{19} + 52 q^{27} - 12 q^{33} - 44 q^{35} - 24 q^{41} + 8 q^{57} - 20 q^{59} - 8 q^{65} + 16 q^{67} + 24 q^{73} - 44 q^{81} - 16 q^{83} - 16 q^{89} - 32 q^{91}+ \cdots - 20 q^{99}+O(q^{100}) $ 20 * q + 4 * q^3 + 16 * q^9 + 8 * q^11 + 12 * q^19 + 52 * q^27 - 12 * q^33 - 44 * q^35 - 24 * q^41 + 8 * q^57 - 20 * q^59 - 8 * q^65 + 16 * q^67 + 24 * q^73 - 44 * q^81 - 16 * q^83 - 16 * q^89 - 32 * q^91 + 12 * q^97 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$	−1.38809			−0.801416			−0.400708	−	0.916206i	$-0.631236\pi$
$3$	−1.38809			−0.801416			−0.400708	+	0.916206i	$0.631236\pi$
$4$		0			0
$5$	−2.40872	+	2.40872i	−1.07721	+	1.07721i	−0.0804522	+	0.996758i	$0.525636\pi$
$5$	−2.40872	+	2.40872i	−1.07721	+	1.07721i	−0.996758	+	0.0804522i	$0.974364\pi$
$6$		0			0
$7$	0.127019	+	0.127019i	0.0480088	+	0.0480088i	0.730704	−	0.682695i	$-0.239192\pi$
$7$	0.127019	+	0.127019i	0.0480088	+	0.0480088i	−0.682695	+	0.730704i	$0.739192\pi$
$8$		0			0
$9$	−1.07320			−0.357733
$10$		0			0
$11$	2.63237	−	2.63237i	0.793690	−	0.793690i	−0.188402	−	0.982092i	$-0.560331\pi$
$11$	2.63237	−	2.63237i	0.793690	−	0.793690i	0.982092	+	0.188402i	$0.0603307\pi$
$12$		0			0
$13$	1.93837	−	3.04019i	0.537606	−	0.843196i
$13$	1.93837	−	3.04019i	0.537606	−	0.843196i
$14$		0			0
$15$	3.34352	−	3.34352i	0.863294	−	0.863294i
$16$		0			0
$17$		−	4.33908i		−	1.05238i	−0.850366	−	0.526191i	$-0.823620\pi$
$17$		−	4.33908i		−	1.05238i	0.850366	−	0.526191i	$-0.176380\pi$
$18$		0			0
$19$	−4.97146	−	4.97146i	−1.14053	−	1.14053i	−0.988353	−	0.152177i	$-0.951372\pi$
$19$	−4.97146	−	4.97146i	−1.14053	−	1.14053i	−0.152177	−	0.988353i	$-0.548628\pi$
$20$		0			0
$21$	−0.176315	−	0.176315i	−0.0384750	−	0.0384750i
$22$		0			0
$23$	−3.98711			−0.831371			−0.415685	−	0.909508i	$-0.636458\pi$
$23$	−3.98711			−0.831371			−0.415685	+	0.909508i	$0.636458\pi$
$24$		0			0
$25$		−	6.60383i		−	1.32077i
$26$		0			0
$27$	5.65398			1.08811
$28$		0			0
$29$			4.59378i			0.853044i	0.904477	+	0.426522i	$0.140261\pi$
$29$			4.59378i			0.853044i	−0.904477	+	0.426522i	$0.859739\pi$
$30$		0			0
$31$	−1.07702	+	1.07702i	−0.193438	+	0.193438i	−0.797180	−	0.603742i	$-0.793676\pi$
$31$	−1.07702	+	1.07702i	−0.193438	+	0.193438i	0.603742	+	0.797180i	$0.293676\pi$
$32$		0			0
$33$	−3.65398	+	3.65398i	−0.636076	+	0.636076i
$34$		0			0
$35$	−0.611907			−0.103431
$36$		0			0
$37$	−2.45494	−	2.45494i	−0.403590	−	0.403590i	0.475906	−	0.879496i	$-0.342120\pi$
$37$	−2.45494	−	2.45494i	−0.403590	−	0.403590i	−0.879496	+	0.475906i	$0.842120\pi$
$38$		0			0
$39$	−2.69063	+	4.22006i	−0.430846	+	0.675751i
$40$		0			0
$41$	0.388093	+	0.388093i	0.0606099	+	0.0606099i	0.736762	−	0.676152i	$-0.236354\pi$
$41$	0.388093	+	0.388093i	0.0606099	+	0.0606099i	−0.676152	+	0.736762i	$0.736354\pi$
$42$		0			0
$43$		−	5.02419i		−	0.766182i	−0.923711	−	0.383091i	$-0.874860\pi$
$43$		−	5.02419i		−	0.766182i	0.923711	−	0.383091i	$-0.125140\pi$
$44$		0			0
$45$	2.58503	−	2.58503i	0.385354	−	0.385354i
$46$		0			0
$47$	−1.00356	−	1.00356i	−0.146384	−	0.146384i	0.630116	−	0.776501i	$-0.283007\pi$
$47$	−1.00356	−	1.00356i	−0.146384	−	0.146384i	−0.776501	+	0.630116i	$0.783007\pi$
$48$		0			0
$49$		−	6.96773i		−	0.995390i
$50$		0			0
$51$			6.02305i			0.843396i
$52$		0			0
$53$			1.83922i			0.252636i	0.991990	+	0.126318i	$0.0403161\pi$
$53$			1.83922i			0.252636i	−0.991990	+	0.126318i	$0.959684\pi$
$54$		0			0
$55$			12.6813i			1.70994i
$56$		0			0
$57$	6.90084	+	6.90084i	0.914039	+	0.914039i
$58$		0			0
$59$	−6.28635	+	6.28635i	−0.818413	+	0.818413i	−0.985878	−	0.167465i	$-0.946442\pi$
$59$	−6.28635	+	6.28635i	−0.818413	+	0.818413i	0.167465	+	0.985878i	$0.446442\pi$
$60$		0			0
$61$		−	10.5817i		−	1.35485i	−0.735592	−	0.677425i	$-0.763096\pi$
$61$		−	10.5817i		−	1.35485i	0.735592	−	0.677425i	$-0.236904\pi$
$62$		0			0
$63$	−0.136317	−	0.136317i	−0.0171743	−	0.0171743i
$64$		0			0
$65$	2.65398	+	11.9919i	0.329185	+	1.48741i
$66$		0			0
$67$	4.21315	+	4.21315i	0.514718	+	0.514718i	0.915969	−	0.401250i	$-0.131424\pi$
$67$	4.21315	+	4.21315i	0.514718	+	0.514718i	−0.401250	+	0.915969i	$0.631424\pi$
$68$		0			0
$69$	5.53449			0.666274
$70$		0			0
$71$	2.59715	−	2.59715i	0.308225	−	0.308225i	−0.535995	−	0.844221i	$-0.680064\pi$
$71$	2.59715	−	2.59715i	0.308225	−	0.308225i	0.844221	+	0.535995i	$0.180064\pi$
$72$		0			0
$73$	−0.388093	+	0.388093i	−0.0454228	+	0.0454228i	−0.729453	−	0.684031i	$-0.760225\pi$
$73$	−0.388093	+	0.388093i	−0.0454228	+	0.0454228i	0.684031	+	0.729453i	$0.260225\pi$
$74$		0			0
$75$			9.16673i			1.05848i
$76$		0			0
$77$	0.668725			0.0762082
$78$		0			0
$79$		−	15.3812i		−	1.73052i	−0.501323	−	0.865260i	$-0.667153\pi$
$79$		−	15.3812i		−	1.73052i	0.501323	−	0.865260i	$-0.332847\pi$
$80$		0			0
$81$	−4.62865			−0.514294
$82$		0			0
$83$	2.46502	+	2.46502i	0.270571	+	0.270571i	0.829330	−	0.558759i	$-0.188722\pi$
$83$	2.46502	+	2.46502i	0.270571	+	0.270571i	−0.558759	+	0.829330i	$0.688722\pi$
$84$		0			0
$85$	10.4516	+	10.4516i	1.13364	+	1.13364i
$86$		0			0
$87$		−	6.37660i		−	0.683643i
$88$		0			0
$89$	−11.3583	+	11.3583i	−1.20397	+	1.20397i	−0.231027	+	0.972947i	$0.574209\pi$
$89$	−11.3583	+	11.3583i	−1.20397	+	1.20397i	−0.972947	+	0.231027i	$0.925791\pi$
$90$		0			0
$91$	0.632373	−	0.139953i	0.0662907	−	0.0146710i
$92$		0			0
$93$	1.49500	−	1.49500i	0.155025	−	0.155025i
$94$		0			0
$95$	23.9497			2.45718
$96$		0			0
$97$	3.31375	+	3.31375i	0.336461	+	0.336461i	0.855034	−	0.518573i	$-0.173536\pi$
$97$	3.31375	+	3.31375i	0.336461	+	0.336461i	−0.518573	+	0.855034i	$0.673536\pi$
$98$		0			0
$99$	−2.82506	+	2.82506i	−0.283929	+	0.283929i
$100$		0			0
$101$	−19.0522			−1.89577			−0.947884	−	0.318617i	$-0.896782\pi$
$101$	−19.0522			−1.89577			−0.947884	+	0.318617i	$0.896782\pi$
$102$		0			0
$103$	−13.3555			−1.31596			−0.657978	−	0.753037i	$-0.728588\pi$
$103$	−13.3555			−1.31596			−0.657978	+	0.753037i	$0.728588\pi$
$104$		0			0
$105$	0.849384			0.0828914
$106$		0			0
$107$	1.78249			0.172320			0.0861601	−	0.996281i	$-0.472540\pi$
$107$	1.78249			0.172320			0.0861601	+	0.996281i	$0.472540\pi$
$108$		0			0
$109$	−10.3367	+	10.3367i	−0.990079	+	0.990079i	−0.999951	−	0.00987275i	$-0.996857\pi$
$109$	−10.3367	+	10.3367i	−0.990079	+	0.990079i	0.00987275	+	0.999951i	$0.496857\pi$
$110$		0			0
$111$	3.40768	+	3.40768i	0.323443	+	0.323443i
$112$		0			0
$113$	11.9787			1.12686			0.563429	−	0.826164i	$-0.309482\pi$
$113$	11.9787			1.12686			0.563429	+	0.826164i	$0.309482\pi$
$114$		0			0
$115$	9.60383	−	9.60383i	0.895562	−	0.895562i
$116$		0			0
$117$	−2.08025	+	3.26273i	−0.192319	+	0.301639i
$118$		0			0
$119$	0.551148	−	0.551148i	0.0505236	−	0.0505236i
$120$		0			0
$121$		−	2.85877i		−	0.259888i
$122$		0			0
$123$	−0.538709	−	0.538709i	−0.0485737	−	0.0485737i
$124$		0			0
$125$	3.86317	+	3.86317i	0.345532	+	0.345532i
$126$		0			0
$127$	−1.46123			−0.129663			−0.0648316	−	0.997896i	$-0.520651\pi$
$127$	−1.46123			−0.129663			−0.0648316	+	0.997896i	$0.520651\pi$
$128$		0			0
$129$			6.97404i			0.614030i
$130$		0			0
$131$	−3.75944			−0.328464			−0.164232	−	0.986422i	$-0.552515\pi$
$131$	−3.75944			−0.328464			−0.164232	+	0.986422i	$0.552515\pi$
$132$		0			0
$133$		−	1.26294i		−	0.109511i
$134$		0			0
$135$	−13.6188	+	13.6188i	−1.17212	+	1.17212i
$136$		0			0
$137$	6.09739	−	6.09739i	0.520935	−	0.520935i	−0.396919	−	0.917854i	$-0.629921\pi$
$137$	6.09739	−	6.09739i	0.520935	−	0.520935i	0.917854	+	0.396919i	$0.129921\pi$
$138$		0			0
$139$	−19.3362			−1.64007			−0.820036	−	0.572311i	$-0.806047\pi$
$139$	−19.3362			−1.64007			−0.820036	+	0.572311i	$0.806047\pi$
$140$		0			0
$141$	1.39303	+	1.39303i	0.117315	+	0.117315i
$142$		0			0
$143$	−2.90041	−	13.1054i	−0.242544	−	1.09593i
$144$		0			0
$145$	−11.0651	−	11.0651i	−0.918908	−	0.918908i
$146$		0			0
$147$			9.67186i			0.797721i
$148$		0			0
$149$	5.83926	−	5.83926i	0.478371	−	0.478371i	−0.426240	−	0.904610i	$-0.640162\pi$
$149$	5.83926	−	5.83926i	0.478371	−	0.478371i	0.904610	+	0.426240i	$0.140162\pi$
$150$		0			0
$151$	13.5924	+	13.5924i	1.10613	+	1.10613i	0.993654	+	0.112479i	$0.0358791\pi$
$151$	13.5924	+	13.5924i	1.10613	+	1.10613i	0.112479	+	0.993654i	$0.464121\pi$
$152$		0			0
$153$			4.65670i			0.376472i
$154$		0			0
$155$		−	5.18847i		−	0.416748i
$156$		0			0
$157$		−	3.22500i		−	0.257383i	−0.991685	−	0.128691i	$-0.958922\pi$
$157$		−	3.22500i		−	0.257383i	0.991685	−	0.128691i	$-0.0410777\pi$
$158$		0			0
$159$		−	2.55301i		−	0.202467i
$160$		0			0
$161$	−0.506441	−	0.506441i	−0.0399131	−	0.0399131i
$162$		0			0
$163$	12.2685	−	12.2685i	0.960941	−	0.960941i	−0.0383247	−	0.999265i	$-0.512202\pi$
$163$	12.2685	−	12.2685i	0.960941	−	0.960941i	0.999265	+	0.0383247i	$0.0122021\pi$
$164$		0			0
$165$		−	17.6028i		−	1.37038i
$166$		0			0
$167$	−7.73367	−	7.73367i	−0.598450	−	0.598450i	0.341450	−	0.939900i	$-0.389082\pi$
$167$	−7.73367	−	7.73367i	−0.598450	−	0.598450i	−0.939900	+	0.341450i	$0.889082\pi$
$168$		0			0
$169$	−5.48548	−	11.7860i	−0.421960	−	0.906614i
$170$		0			0
$171$	5.33536	+	5.33536i	0.408005	+	0.408005i
$172$		0			0
$173$	13.2073			1.00413			0.502066	−	0.864829i	$-0.332573\pi$
$173$	13.2073			1.00413			0.502066	+	0.864829i	$0.332573\pi$
$174$		0			0
$175$	0.838814	−	0.838814i	0.0634084	−	0.0634084i
$176$		0			0
$177$	8.72604	−	8.72604i	0.655889	−	0.655889i
$178$		0			0
$179$			5.24686i			0.392169i	0.980587	+	0.196085i	$0.0628227\pi$
$179$			5.24686i			0.392169i	−0.980587	+	0.196085i	$0.937177\pi$
$180$		0			0
$181$	−3.49698			−0.259928			−0.129964	−	0.991519i	$-0.541486\pi$
$181$	−3.49698			−0.259928			−0.129964	+	0.991519i	$0.541486\pi$
$182$		0			0
$183$			14.6884i			1.08580i
$184$		0			0
$185$	11.8265			0.869502
$186$		0			0
$187$	−11.4221	−	11.4221i	−0.835266	−	0.835266i
$188$		0			0
$189$	0.718165	+	0.718165i	0.0522388	+	0.0522388i
$190$		0			0
$191$		−	9.17512i		−	0.663888i	−0.943299	−	0.331944i	$-0.892295\pi$
$191$		−	9.17512i		−	0.663888i	0.943299	−	0.331944i	$-0.107705\pi$
$192$		0			0
$193$	0.0825862	−	0.0825862i	0.00594469	−	0.00594469i	−0.704128	−	0.710073i	$-0.748662\pi$
$193$	0.0825862	−	0.0825862i	0.00594469	−	0.00594469i	0.710073	+	0.704128i	$0.248662\pi$
$194$		0			0
$195$	−3.68397	−	16.6459i	−0.263814	−	1.19204i
$196$		0			0
$197$	8.65184	−	8.65184i	0.616418	−	0.616418i	−0.328193	−	0.944611i	$-0.606439\pi$
$197$	8.65184	−	8.65184i	0.616418	−	0.616418i	0.944611	+	0.328193i	$0.106439\pi$
$198$		0			0
$199$	−16.3536			−1.15927			−0.579637	−	0.814875i	$-0.696806\pi$
$199$	−16.3536			−1.15927			−0.579637	+	0.814875i	$0.696806\pi$
$200$		0			0
$201$	−5.84824	−	5.84824i	−0.412503	−	0.412503i
$202$		0			0
$203$	−0.583499	+	0.583499i	−0.0409536	+	0.0409536i
$204$		0			0
$205$	−1.86961			−0.130579
$206$		0			0
$207$	4.27897			0.297409
$208$		0			0
$209$	−26.1734			−1.81046
$210$		0			0
$211$	12.7723			0.879283			0.439642	−	0.898173i	$-0.355105\pi$
$211$	12.7723			0.879283			0.439642	+	0.898173i	$0.355105\pi$
$212$		0			0
$213$	−3.60509	+	3.60509i	−0.247017	+	0.247017i
$214$		0			0
$215$	12.1018	+	12.1018i	0.825339	+	0.825339i
$216$		0			0
$217$	−0.273605			−0.0185735
$218$		0			0
$219$	0.538709	−	0.538709i	0.0364026	−	0.0364026i
$220$		0			0
$221$	−13.1916	−	8.41073i	−0.887365	−	0.565767i
$222$		0			0
$223$	12.8482	−	12.8482i	0.860380	−	0.860380i	−0.131002	−	0.991382i	$-0.541819\pi$
$223$	12.8482	−	12.8482i	0.860380	−	0.860380i	0.991382	+	0.131002i	$0.0418194\pi$
$224$		0			0
$225$			7.08722i			0.472481i
$226$		0			0
$227$	10.2863	+	10.2863i	0.682729	+	0.682729i	0.960614	−	0.277885i	$-0.0896335\pi$
$227$	10.2863	+	10.2863i	0.682729	+	0.682729i	−0.277885	+	0.960614i	$0.589634\pi$
$228$		0			0
$229$	−2.44653	−	2.44653i	−0.161671	−	0.161671i	0.621635	−	0.783307i	$-0.286469\pi$
$229$	−2.44653	−	2.44653i	−0.161671	−	0.161671i	−0.783307	+	0.621635i	$0.786469\pi$
$230$		0			0
$231$	−0.928252			−0.0610745
$232$		0			0
$233$			6.36599i			0.417050i	0.978017	+	0.208525i	$0.0668663\pi$
$233$			6.36599i			0.417050i	−0.978017	+	0.208525i	$0.933134\pi$
$234$		0			0
$235$	4.83458			0.315373
$236$		0			0
$237$			21.3505i			1.38687i
$238$		0			0
$239$	−12.2196	+	12.2196i	−0.790418	+	0.790418i	−0.981562	−	0.191144i	$-0.938780\pi$
$239$	−12.2196	+	12.2196i	−0.790418	+	0.790418i	0.191144	+	0.981562i	$0.438780\pi$
$240$		0			0
$241$	10.9752	−	10.9752i	0.706973	−	0.706973i	−0.258924	−	0.965898i	$-0.583368\pi$
$241$	10.9752	−	10.9752i	0.706973	−	0.706973i	0.965898	+	0.258924i	$0.0833680\pi$
$242$		0			0
$243$	−10.5369			−0.675945
$244$		0			0
$245$	16.7833	+	16.7833i	1.07225	+	1.07225i
$246$		0			0
$247$	−24.7507	+	5.47766i	−1.57485	+	0.348535i
$248$		0			0
$249$	−3.42167	−	3.42167i	−0.216839	−	0.216839i
$250$		0			0
$251$			0.692187i			0.0436905i	0.999761	+	0.0218452i	$0.00695411\pi$
$251$			0.692187i			0.0436905i	−0.999761	+	0.0218452i	$0.993046\pi$
$252$		0			0
$253$	−10.4956	+	10.4956i	−0.659851	+	0.659851i
$254$		0			0
$255$	−14.5078	−	14.5078i	−0.908515	−	0.908515i
$256$		0			0
$257$		−	16.5622i		−	1.03312i	−0.856251	−	0.516560i	$-0.827212\pi$
$257$		−	16.5622i		−	1.03312i	0.856251	−	0.516560i	$-0.172788\pi$
$258$		0			0
$259$		−	0.623650i		−	0.0387517i
$260$		0			0
$261$		−	4.93004i		−	0.305162i
$262$		0			0
$263$			3.54818i			0.218790i	0.993998	+	0.109395i	$0.0348914\pi$
$263$			3.54818i			0.218790i	−0.993998	+	0.109395i	$0.965109\pi$
$264$		0			0
$265$	−4.43016	−	4.43016i	−0.272143	−	0.272143i
$266$		0			0
$267$	15.7663	−	15.7663i	0.964884	−	0.964884i
$268$		0			0
$269$		−	12.6310i		−	0.770127i	−0.922890	−	0.385063i	$-0.874180\pi$
$269$		−	12.6310i		−	0.770127i	0.922890	−	0.385063i	$-0.125820\pi$
$270$		0			0
$271$	15.0265	+	15.0265i	0.912795	+	0.912795i	0.996491	−	0.0836966i	$-0.0266727\pi$
$271$	15.0265	+	15.0265i	0.912795	+	0.912795i	−0.0836966	+	0.996491i	$0.526673\pi$
$272$		0			0
$273$	−0.877792	+	0.194267i	−0.0531264	+	0.0117576i
$274$		0			0
$275$	−17.3837	−	17.3837i	−1.04828	−	1.04828i
$276$		0			0
$277$	11.4437			0.687585			0.343793	−	0.939046i	$-0.388288\pi$
$277$	11.4437			0.687585			0.343793	+	0.939046i	$0.388288\pi$
$278$		0			0
$279$	1.15586	−	1.15586i	0.0691993	−	0.0691993i
$280$		0			0
$281$	0.709295	−	0.709295i	0.0423130	−	0.0423130i	−0.685634	−	0.727947i	$-0.740475\pi$
$281$	0.709295	−	0.709295i	0.0423130	−	0.0423130i	0.727947	+	0.685634i	$0.240475\pi$
$282$		0			0
$283$		−	4.12466i		−	0.245185i	−0.992457	−	0.122593i	$-0.960879\pi$
$283$		−	4.12466i		−	0.245185i	0.992457	−	0.122593i	$-0.0391209\pi$
$284$		0			0
$285$	−33.2443			−1.96922
$286$		0			0
$287$			0.0985906i			0.00581962i
$288$		0			0
$289$	−1.82764			−0.107508
$290$		0			0
$291$	−4.59980	−	4.59980i	−0.269645	−	0.269645i
$292$		0			0
$293$	−3.00083	−	3.00083i	−0.175310	−	0.175310i	0.613998	−	0.789308i	$-0.289560\pi$
$293$	−3.00083	−	3.00083i	−0.175310	−	0.175310i	−0.789308	+	0.613998i	$0.789560\pi$
$294$		0			0
$295$		−	30.2841i		−	1.76321i
$296$		0			0
$297$	14.8834	−	14.8834i	0.863621	−	0.863621i
$298$		0			0
$299$	−7.72848	+	12.1216i	−0.446950	+	0.701009i
$300$		0			0
$301$	0.638169	−	0.638169i	0.0367835	−	0.0367835i
$302$		0			0
$303$	26.4463			1.51930
$304$		0			0
$305$	25.4883	+	25.4883i	1.45946	+	1.45946i
$306$		0			0
$307$	2.76089	−	2.76089i	0.157572	−	0.157572i	−0.623918	−	0.781490i	$-0.714460\pi$
$307$	2.76089	−	2.76089i	0.157572	−	0.157572i	0.781490	+	0.623918i	$0.214460\pi$
$308$		0			0
$309$	18.5387			1.05463
$310$		0			0
$311$	4.27897			0.242638			0.121319	−	0.992614i	$-0.461288\pi$
$311$	4.27897			0.242638			0.121319	+	0.992614i	$0.461288\pi$
$312$		0			0
$313$	−3.68528			−0.208304			−0.104152	−	0.994561i	$-0.533213\pi$
$313$	−3.68528			−0.208304			−0.104152	+	0.994561i	$0.533213\pi$
$314$		0			0
$315$	0.656698			0.0370007
$316$		0			0
$317$	2.94735	−	2.94735i	0.165540	−	0.165540i	−0.619476	−	0.785016i	$-0.712655\pi$
$317$	2.94735	−	2.94735i	0.165540	−	0.165540i	0.785016	+	0.619476i	$0.212655\pi$
$318$		0			0
$319$	12.0925	+	12.0925i	0.677053	+	0.677053i
$320$		0			0
$321$	−2.47427			−0.138100
$322$		0			0
$323$	−21.5716	+	21.5716i	−1.20027	+	1.20027i
$324$		0			0
$325$	−20.0769	−	12.8006i	−1.11366	−	0.710051i
$326$		0			0
$327$	14.3483	−	14.3483i	0.793464	−	0.793464i
$328$		0			0
$329$		−	0.254943i		−	0.0140555i
$330$		0			0
$331$	−8.72345	−	8.72345i	−0.479484	−	0.479484i	0.425482	−	0.904967i	$-0.360104\pi$
$331$	−8.72345	−	8.72345i	−0.479484	−	0.479484i	−0.904967	+	0.425482i	$0.860104\pi$
$332$		0			0
$333$	2.63464	+	2.63464i	0.144377	+	0.144377i
$334$		0			0
$335$	−20.2966			−1.10892
$336$		0			0
$337$			7.00939i			0.381826i	0.981607	+	0.190913i	$0.0611448\pi$
$337$			7.00939i			0.381826i	−0.981607	+	0.190913i	$0.938855\pi$
$338$		0			0
$339$	−16.6275			−0.903082
$340$		0			0
$341$			5.67023i			0.307060i
$342$		0			0
$343$	1.77417	−	1.77417i	0.0957963	−	0.0957963i
$344$		0			0
$345$	−13.3310	+	13.3310i	−0.717717	+	0.717717i
$346$		0			0
$347$	33.5948			1.80346			0.901730	−	0.432299i	$-0.142297\pi$
$347$	33.5948			1.80346			0.901730	+	0.432299i	$0.142297\pi$
$348$		0			0
$349$	14.5137	+	14.5137i	0.776901	+	0.776901i	0.979303	−	0.202401i	$-0.0648745\pi$
$349$	14.5137	+	14.5137i	0.776901	+	0.776901i	−0.202401	+	0.979303i	$0.564875\pi$
$350$		0			0
$351$	10.9595	−	17.1892i	0.584973	−	0.917489i
$352$		0			0
$353$	−5.02174	−	5.02174i	−0.267280	−	0.267280i	0.560723	−	0.828003i	$-0.310523\pi$
$353$	−5.02174	−	5.02174i	−0.267280	−	0.267280i	−0.828003	+	0.560723i	$0.810523\pi$
$354$		0			0
$355$			12.5116i			0.664047i
$356$		0			0
$357$	−0.765044	+	0.765044i	−0.0404904	+	0.0404904i
$358$		0			0
$359$	15.4781	+	15.4781i	0.816903	+	0.816903i	0.985658	−	0.168755i	$-0.0539746\pi$
$359$	15.4781	+	15.4781i	0.816903	+	0.816903i	−0.168755	+	0.985658i	$0.553975\pi$
$360$		0			0
$361$			30.4307i			1.60162i
$362$		0			0
$363$			3.96824i			0.208279i
$364$		0			0
$365$		−	1.86961i		−	0.0978599i
$366$		0			0
$367$		−	29.4166i		−	1.53553i	−0.640729	−	0.767767i	$-0.721368\pi$
$367$		−	29.4166i		−	1.53553i	0.640729	−	0.767767i	$-0.278632\pi$
$368$		0			0
$369$	−0.416501	−	0.416501i	−0.0216822	−	0.0216822i
$370$		0			0
$371$	−0.233617	+	0.233617i	−0.0121288	+	0.0121288i
$372$		0			0
$373$			2.12493i			0.110025i	0.998486	+	0.0550123i	$0.0175198\pi$
$373$			2.12493i			0.110025i	−0.998486	+	0.0550123i	$0.982480\pi$
$374$		0			0
$375$	−5.36244	−	5.36244i	−0.276915	−	0.276915i
$376$		0			0
$377$	13.9660	+	8.90443i	0.719284	+	0.458601i
$378$		0			0
$379$	9.31054	+	9.31054i	0.478250	+	0.478250i	0.904572	−	0.426321i	$-0.140191\pi$
$379$	9.31054	+	9.31054i	0.478250	+	0.478250i	−0.426321	+	0.904572i	$0.640191\pi$
$380$		0			0
$381$	2.02832			0.103914
$382$		0			0
$383$	−24.0746	+	24.0746i	−1.23015	+	1.23015i	−0.266249	+	0.963904i	$0.585784\pi$
$383$	−24.0746	+	24.0746i	−1.23015	+	1.23015i	−0.963904	+	0.266249i	$0.914216\pi$
$384$		0			0
$385$	−1.61077	+	1.61077i	−0.0820923	+	0.0820923i
$386$		0			0
$387$			5.39195i			0.274088i
$388$		0			0
$389$	−20.8431			−1.05679			−0.528394	−	0.848999i	$-0.677206\pi$
$389$	−20.8431			−1.05679			−0.528394	+	0.848999i	$0.677206\pi$
$390$		0			0
$391$			17.3004i			0.874920i
$392$		0			0
$393$	5.21846			0.263236
$394$		0			0
$395$	37.0490	+	37.0490i	1.86414	+	1.86414i
$396$		0			0
$397$	16.7224	+	16.7224i	0.839273	+	0.839273i	0.988763	−	0.149490i	$-0.0477633\pi$
$397$	16.7224	+	16.7224i	0.839273	+	0.839273i	−0.149490	+	0.988763i	$0.547763\pi$
$398$		0			0
$399$			1.75308i			0.0877638i
$400$		0			0
$401$	−19.8319	+	19.8319i	−0.990359	+	0.990359i	−0.999954	−	0.00959526i	$-0.996946\pi$
$401$	−19.8319	+	19.8319i	−0.990359	+	0.990359i	0.00959526	+	0.999954i	$0.496946\pi$
$402$		0			0
$403$	1.18668	+	5.36200i	0.0591129	+	0.267100i
$404$		0			0
$405$	11.1491	−	11.1491i	0.554003	−	0.554003i
$406$		0			0
$407$	−12.9246			−0.640650
$408$		0			0
$409$	−10.6165	−	10.6165i	−0.524954	−	0.524954i	0.394109	−	0.919064i	$-0.371053\pi$
$409$	−10.6165	−	10.6165i	−0.524954	−	0.524954i	−0.919064	+	0.394109i	$0.871053\pi$
$410$		0			0
$411$	−8.46374	+	8.46374i	−0.417485	+	0.417485i
$412$		0			0
$413$	−1.59698			−0.0785821
$414$		0			0
$415$	−11.8750			−0.582923
$416$		0			0
$417$	26.8404			1.31438
$418$		0			0
$419$	−19.6570			−0.960308			−0.480154	−	0.877184i	$-0.659419\pi$
$419$	−19.6570			−0.960308			−0.480154	+	0.877184i	$0.659419\pi$
$420$		0			0
$421$	−2.77602	+	2.77602i	−0.135295	+	0.135295i	−0.771511	−	0.636216i	$-0.780499\pi$
$421$	−2.77602	+	2.77602i	−0.135295	+	0.135295i	0.636216	+	0.771511i	$0.280499\pi$
$422$		0			0
$423$	1.07702	+	1.07702i	0.0523665	+	0.0523665i
$424$		0			0
$425$	−28.6546			−1.38995
$426$		0			0
$427$	1.34408	−	1.34408i	0.0650447	−	0.0650447i
$428$		0			0
$429$	4.02603	+	18.1915i	0.194379	+	0.878295i
$430$		0			0
$431$	−24.6412	+	24.6412i	−1.18692	+	1.18692i	−0.209009	+	0.977914i	$0.567024\pi$
$431$	−24.6412	+	24.6412i	−1.18692	+	1.18692i	−0.977914	+	0.209009i	$0.932976\pi$
$432$		0			0
$433$			1.73076i			0.0831752i	0.999135	+	0.0415876i	$0.0132416\pi$
$433$			1.73076i			0.0831752i	−0.999135	+	0.0415876i	$0.986758\pi$
$434$		0			0
$435$	15.3594	+	15.3594i	0.736427	+	0.736427i
$436$		0			0
$437$	19.8218	+	19.8218i	0.948204	+	0.948204i
$438$		0			0
$439$	25.0521			1.19567			0.597836	−	0.801618i	$-0.296027\pi$
$439$	25.0521			1.19567			0.597836	+	0.801618i	$0.296027\pi$
$440$		0			0
$441$			7.47776i			0.356084i
$442$		0			0
$443$	11.4098			0.542098			0.271049	−	0.962566i	$-0.412630\pi$
$443$	11.4098			0.542098			0.271049	+	0.962566i	$0.412630\pi$
$444$		0			0
$445$		−	54.7177i		−	2.59387i
$446$		0			0
$447$	−8.10543	+	8.10543i	−0.383374	+	0.383374i
$448$		0			0
$449$	−1.05015	+	1.05015i	−0.0495596	+	0.0495596i	−0.731452	−	0.681893i	$-0.761157\pi$
$449$	−1.05015	+	1.05015i	−0.0495596	+	0.0495596i	0.681893	+	0.731452i	$0.261157\pi$
$450$		0			0
$451$	2.04321			0.0962110
$452$		0			0
$453$	−18.8675	−	18.8675i	−0.886473	−	0.886473i
$454$		0			0
$455$	−1.18610	+	1.86031i	−0.0556052	+	0.0872128i
$456$		0			0
$457$	26.4630	+	26.4630i	1.23789	+	1.23789i	0.960863	+	0.277025i	$0.0893484\pi$
$457$	26.4630	+	26.4630i	1.23789	+	1.23789i	0.277025	+	0.960863i	$0.410652\pi$
$458$		0			0
$459$		−	24.5331i		−	1.14511i
$460$		0			0
$461$	15.0134	−	15.0134i	0.699243	−	0.699243i	−0.265004	−	0.964247i	$-0.585373\pi$
$461$	15.0134	−	15.0134i	0.699243	−	0.699243i	0.964247	+	0.265004i	$0.0853734\pi$
$462$		0			0
$463$	4.90869	+	4.90869i	0.228126	+	0.228126i	0.811909	−	0.583783i	$-0.198428\pi$
$463$	4.90869	+	4.90869i	0.228126	+	0.228126i	−0.583783	+	0.811909i	$0.698428\pi$
$464$		0			0
$465$			7.20207i			0.333988i
$466$		0			0
$467$			15.5284i			0.718566i	0.933229	+	0.359283i	$0.116979\pi$
$467$			15.5284i			0.718566i	−0.933229	+	0.359283i	$0.883021\pi$
$468$		0			0
$469$			1.07030i			0.0494220i
$470$		0			0
$471$			4.47660i			0.206271i
$472$		0			0
$473$	−13.2255	−	13.2255i	−0.608111	−	0.608111i
$474$		0			0
$475$	−32.8306	+	32.8306i	−1.50637	+	1.50637i
$476$		0			0
$477$		−	1.97385i		−	0.0903764i
$478$		0			0
$479$	13.9271	+	13.9271i	0.636345	+	0.636345i	0.949652	−	0.313307i	$-0.101437\pi$
$479$	13.9271	+	13.9271i	0.636345	+	0.636345i	−0.313307	+	0.949652i	$0.601437\pi$
$480$		0			0
$481$	−12.2220	+	2.70491i	−0.557277	+	0.123333i
$482$		0			0
$483$	0.702987	+	0.702987i	0.0319870	+	0.0319870i
$484$		0			0
$485$	−15.9638			−0.724878
$486$		0			0
$487$	10.2416	−	10.2416i	0.464092	−	0.464092i	−0.435902	−	0.899994i	$-0.643571\pi$
$487$	10.2416	−	10.2416i	0.464092	−	0.464092i	0.899994	+	0.435902i	$0.143571\pi$
$488$		0			0
$489$	−17.0298	+	17.0298i	−0.770113	+	0.770113i
$490$		0			0
$491$		−	38.4981i		−	1.73740i	−0.495341	−	0.868698i	$-0.664957\pi$
$491$		−	38.4981i		−	1.73740i	0.495341	−	0.868698i	$-0.335043\pi$
$492$		0			0
$493$	19.9328			0.897728
$494$		0			0
$495$		−	13.6095i		−	0.611703i
$496$		0			0
$497$	0.659778			0.0295951
$498$		0			0
$499$	−7.83831	−	7.83831i	−0.350891	−	0.350891i	0.509550	−	0.860441i	$-0.329812\pi$
$499$	−7.83831	−	7.83831i	−0.350891	−	0.350891i	−0.860441	+	0.509550i	$0.829812\pi$
$500$		0			0
$501$	10.7351	+	10.7351i	0.479607	+	0.479607i
$502$		0			0
$503$		−	30.2373i		−	1.34822i	−0.738633	−	0.674108i	$-0.764528\pi$
$503$		−	30.2373i		−	1.34822i	0.738633	−	0.674108i	$-0.235472\pi$
$504$		0			0
$505$	45.8914	−	45.8914i	2.04214	−	2.04214i
$506$		0			0
$507$	7.61436	+	16.3600i	0.338165	+	0.726575i
$508$		0			0
$509$	8.49635	−	8.49635i	0.376594	−	0.376594i	−0.493278	−	0.869872i	$-0.664201\pi$
$509$	8.49635	−	8.49635i	0.376594	−	0.376594i	0.869872	+	0.493278i	$0.164201\pi$
$510$		0			0
$511$	−0.0985906			−0.00436139
$512$		0			0
$513$	−28.1085	−	28.1085i	−1.24102	−	1.24102i
$514$		0			0
$515$	32.1696	−	32.1696i	1.41756	−	1.41756i
$516$		0			0
$517$	−5.28349			−0.232368
$518$		0			0
$519$	−18.3329			−0.804727
$520$		0			0
$521$	22.8258			1.00002			0.500008	−	0.866021i	$-0.333330\pi$
$521$	22.8258			1.00002			0.500008	+	0.866021i	$0.333330\pi$
$522$		0			0
$523$	−29.2368			−1.27844			−0.639219	−	0.769025i	$-0.720742\pi$
$523$	−29.2368			−1.27844			−0.639219	+	0.769025i	$0.720742\pi$
$524$		0			0
$525$	−1.16435	+	1.16435i	−0.0508165	+	0.0508165i
$526$		0			0
$527$	4.67328	+	4.67328i	0.203571	+	0.203571i
$528$		0			0
$529$	−7.10292			−0.308822
$530$		0			0
$531$	6.74650	−	6.74650i	0.292773	−	0.292773i
$532$		0			0
$533$	1.93214	−	0.427609i	0.0836903	−	0.0185218i
$534$		0			0
$535$	−4.29352	+	4.29352i	−0.185625	+	0.185625i
$536$		0			0
$537$		−	7.28313i		−	0.314290i
$538$		0			0
$539$	−18.3417	−	18.3417i	−0.790032	−	0.790032i
$540$		0			0
$541$	−4.53575	−	4.53575i	−0.195007	−	0.195007i	0.602849	−	0.797856i	$-0.294032\pi$
$541$	−4.53575	−	4.53575i	−0.195007	−	0.195007i	−0.797856	+	0.602849i	$0.794032\pi$
$542$		0			0
$543$	4.85413			0.208311
$544$		0			0
$545$		−	49.7965i		−	2.13305i
$546$		0			0
$547$	11.5183			0.492488			0.246244	−	0.969208i	$-0.420803\pi$
$547$	11.5183			0.492488			0.246244	+	0.969208i	$0.420803\pi$
$548$		0			0
$549$			11.3563i			0.484674i
$550$		0			0
$551$	22.8378	−	22.8378i	0.972923	−	0.972923i
$552$		0			0
$553$	1.95371	−	1.95371i	0.0830803	−	0.0830803i
$554$		0			0
$555$	−16.4163			−0.696832
$556$		0			0
$557$	−21.7184	−	21.7184i	−0.920237	−	0.920237i	0.0768092	−	0.997046i	$-0.475527\pi$
$557$	−21.7184	−	21.7184i	−0.920237	−	0.920237i	−0.997046	+	0.0768092i	$0.975527\pi$
$558$		0			0
$559$	−15.2745	−	9.73871i	−0.646042	−	0.411904i
$560$		0			0
$561$	15.8549	+	15.8549i	0.669395	+	0.669395i
$562$		0			0
$563$		−	22.2920i		−	0.939494i	−0.882801	−	0.469747i	$-0.844345\pi$
$563$		−	22.2920i		−	0.939494i	0.882801	−	0.469747i	$-0.155655\pi$
$564$		0			0
$565$	−28.8532	+	28.8532i	−1.21386	+	1.21386i
$566$		0			0
$567$	−0.587928	−	0.587928i	−0.0246907	−	0.0246907i
$568$		0			0
$569$		−	9.56035i		−	0.400791i	−0.979715	−	0.200395i	$-0.935777\pi$
$569$		−	9.56035i		−	0.400791i	0.979715	−	0.200395i	$-0.0642226\pi$
$570$		0			0
$571$		−	13.1869i		−	0.551854i	−0.961179	−	0.275927i	$-0.911015\pi$
$571$		−	13.1869i		−	0.551854i	0.961179	−	0.275927i	$-0.0889848\pi$
$572$		0			0
$573$			12.7359i			0.532050i
$574$		0			0
$575$			26.3302i			1.09805i
$576$		0			0
$577$	21.0017	+	21.0017i	0.874313	+	0.874313i	0.992939	−	0.118626i	$-0.0378489\pi$
$577$	21.0017	+	21.0017i	0.874313	+	0.874313i	−0.118626	+	0.992939i	$0.537849\pi$
$578$		0			0
$579$	−0.114637	+	0.114637i	−0.00476416	+	0.00476416i
$580$		0			0
$581$			0.626209i			0.0259795i
$582$		0			0
$583$	4.84152	+	4.84152i	0.200515	+	0.200515i
$584$		0			0
$585$	−2.84824	−	12.8697i	−0.117760	−	0.532097i
$586$		0			0
$587$	−17.6365	−	17.6365i	−0.727938	−	0.727938i	0.242271	−	0.970209i	$-0.422108\pi$
$587$	−17.6365	−	17.6365i	−0.727938	−	0.727938i	−0.970209	+	0.242271i	$0.922108\pi$
$588$		0			0
$589$	10.7087			0.441245
$590$		0			0
$591$	−12.0096	+	12.0096i	−0.494007	+	0.494007i
$592$		0			0
$593$	−8.59997	+	8.59997i	−0.353158	+	0.353158i	−0.861283	−	0.508125i	$-0.830339\pi$
$593$	−8.59997	+	8.59997i	−0.353158	+	0.353158i	0.508125	+	0.861283i	$0.330339\pi$
$594$		0			0
$595$			2.65512i			0.108849i
$596$		0			0
$597$	22.7003			0.929061
$598$		0			0
$599$		−	30.9851i		−	1.26602i	−0.774145	−	0.633009i	$-0.781820\pi$
$599$		−	30.9851i		−	1.26602i	0.774145	−	0.633009i	$-0.218180\pi$
$600$		0			0
$601$	−27.5087			−1.12210			−0.561052	−	0.827781i	$-0.689603\pi$
$601$	−27.5087			−1.12210			−0.561052	+	0.827781i	$0.689603\pi$
$602$		0			0
$603$	−4.52155	−	4.52155i	−0.184132	−	0.184132i
$604$		0			0
$605$	6.88597	+	6.88597i	0.279954	+	0.279954i
$606$		0			0
$607$			24.3086i			0.986655i	0.869844	+	0.493328i	$0.164220\pi$
$607$			24.3086i			0.986655i	−0.869844	+	0.493328i	$0.835780\pi$
$608$		0			0
$609$	0.809951	−	0.809951i	0.0328209	−	0.0328209i
$610$		0			0
$611$	−4.99628	+	1.10574i	−0.202128	+	0.0447336i
$612$		0			0
$613$	−18.8428	+	18.8428i	−0.761053	+	0.761053i	−0.976513	−	0.215460i	$-0.930875\pi$
$613$	−18.8428	+	18.8428i	−0.761053	+	0.761053i	0.215460	+	0.976513i	$0.430875\pi$
$614$		0			0
$615$	2.59519			0.104648
$616$		0			0
$617$	−17.2326	−	17.2326i	−0.693759	−	0.693759i	0.269298	−	0.963057i	$-0.413208\pi$
$617$	−17.2326	−	17.2326i	−0.693759	−	0.693759i	−0.963057	+	0.269298i	$0.913208\pi$
$618$		0			0
$619$	−3.52674	+	3.52674i	−0.141752	+	0.141752i	−0.774422	−	0.632670i	$-0.781959\pi$
$619$	−3.52674	+	3.52674i	−0.141752	+	0.141752i	0.632670	+	0.774422i	$0.281959\pi$
$620$		0			0
$621$	−22.5431			−0.904622
$622$		0			0
$623$	−2.88544			−0.115603
$624$		0			0
$625$	14.4086			0.576344
$626$		0			0
$627$	36.3312			1.45093
$628$		0			0
$629$	−10.6522	+	10.6522i	−0.424730	+	0.424730i
$630$		0			0
$631$	−6.65131	−	6.65131i	−0.264784	−	0.264784i	0.562210	−	0.826994i	$-0.309951\pi$
$631$	−6.65131	−	6.65131i	−0.264784	−	0.264784i	−0.826994	+	0.562210i	$0.809951\pi$
$632$		0			0
$633$	−17.7292			−0.704671
$634$		0			0
$635$	3.51969	−	3.51969i	0.139675	−	0.139675i
$636$		0			0
$637$	−21.1832	−	13.5060i	−0.839309	−	0.535128i
$638$		0			0
$639$	−2.78726	+	2.78726i	−0.110262	+	0.110262i
$640$		0			0
$641$		−	27.4814i		−	1.08545i	−0.839910	−	0.542725i	$-0.817393\pi$
$641$		−	27.4814i		−	1.08545i	0.839910	−	0.542725i	$-0.182607\pi$
$642$		0			0
$643$	8.78571	+	8.78571i	0.346475	+	0.346475i	0.858795	−	0.512320i	$-0.171214\pi$
$643$	8.78571	+	8.78571i	0.346475	+	0.346475i	−0.512320	+	0.858795i	$0.671214\pi$
$644$		0			0
$645$	−16.7985	−	16.7985i	−0.661440	−	0.661440i
$646$		0			0
$647$	−5.55484			−0.218383			−0.109192	−	0.994021i	$-0.534826\pi$
$647$	−5.55484			−0.218383			−0.109192	+	0.994021i	$0.534826\pi$
$648$		0			0
$649$			33.0960i			1.29913i
$650$		0			0
$651$	0.379789			0.0148851
$652$		0			0
$653$			20.7670i			0.812675i	0.913723	+	0.406338i	$0.133194\pi$
$653$			20.7670i			0.812675i	−0.913723	+	0.406338i	$0.866806\pi$
$654$		0			0
$655$	9.05543	−	9.05543i	0.353825	−	0.353825i
$656$		0			0
$657$	0.416501	−	0.416501i	0.0162492	−	0.0162492i
$658$		0			0
$659$	42.9802			1.67427			0.837135	−	0.546997i	$-0.184229\pi$
$659$	42.9802			1.67427			0.837135	+	0.546997i	$0.184229\pi$
$660$		0			0
$661$	24.6988	+	24.6988i	0.960674	+	0.960674i	0.999255	−	0.0385817i	$-0.0122840\pi$
$661$	24.6988	+	24.6988i	0.960674	+	0.960674i	−0.0385817	+	0.999255i	$0.512284\pi$
$662$		0			0
$663$	18.3112	+	11.6749i	0.711148	+	0.453414i
$664$		0			0
$665$	3.04207	+	3.04207i	0.117966	+	0.117966i
$666$		0			0
$667$		−	18.3159i		−	0.709196i
$668$		0			0
$669$	−17.8345	+	17.8345i	−0.689522	+	0.689522i
$670$		0			0
$671$	−27.8550	−	27.8550i	−1.07533	−	1.07533i
$672$		0			0
$673$			0.630317i			0.0242969i	0.999926	+	0.0121485i	$0.00386707\pi$
$673$			0.630317i			0.0242969i	−0.999926	+	0.0121485i	$0.996133\pi$
$674$		0			0
$675$		−	37.3379i		−	1.43714i
$676$		0			0
$677$		−	19.4398i		−	0.747132i	−0.927604	−	0.373566i	$-0.878135\pi$
$677$		−	19.4398i		−	0.747132i	0.927604	−	0.373566i	$-0.121865\pi$
$678$		0			0
$679$			0.841822i			0.0323062i
$680$		0			0
$681$	−14.2784	−	14.2784i	−0.547150	−	0.547150i
$682$		0			0
$683$	−23.5885	+	23.5885i	−0.902590	+	0.902590i	−0.995660	−	0.0930695i	$-0.970332\pi$
$683$	−23.5885	+	23.5885i	−0.902590	+	0.902590i	0.0930695	+	0.995660i	$0.470332\pi$
$684$		0			0
$685$			29.3738i			1.12231i
$686$		0			0
$687$	3.39601	+	3.39601i	0.129566	+	0.129566i
$688$		0			0
$689$	5.59158	+	3.56508i	0.213022	+	0.135819i
$690$		0			0
$691$	−10.6764	−	10.6764i	−0.406149	−	0.406149i	0.474244	−	0.880393i	$-0.342721\pi$
$691$	−10.6764	−	10.6764i	−0.406149	−	0.406149i	−0.880393	+	0.474244i	$0.842721\pi$
$692$		0			0
$693$	−0.717674			−0.0272622
$694$		0			0
$695$	46.5754	−	46.5754i	1.76670	−	1.76670i
$696$		0			0
$697$	1.68397	−	1.68397i	0.0637848	−	0.0637848i
$698$		0			0
$699$		−	8.83659i		−	0.334230i
$700$		0			0
$701$	28.6855			1.08344			0.541718	−	0.840560i	$-0.317774\pi$
$701$	28.6855			1.08344			0.541718	+	0.840560i	$0.317774\pi$
$702$		0			0
$703$			24.4092i			0.920612i
$704$		0			0
$705$	−6.71085			−0.252745
$706$		0			0
$707$	−2.42000	−	2.42000i	−0.0910135	−	0.0910135i
$708$		0			0
$709$	−32.5813	−	32.5813i	−1.22362	−	1.22362i	−0.966337	−	0.257279i	$-0.917174\pi$
$709$	−32.5813	−	32.5813i	−1.22362	−	1.22362i	−0.257279	−	0.966337i	$-0.582826\pi$
$710$		0			0
$711$			16.5071i			0.619064i
$712$		0			0
$713$	4.29420	−	4.29420i	0.160819	−	0.160819i
$714$		0			0
$715$	38.5535	+	24.5809i	1.44182	+	0.919275i
$716$		0			0
$717$	16.9619	−	16.9619i	0.633454	−	0.633454i
$718$		0			0
$719$	−13.6202			−0.507948			−0.253974	−	0.967211i	$-0.581738\pi$
$719$	−13.6202			−0.507948			−0.253974	+	0.967211i	$0.581738\pi$
$720$		0			0
$721$	−1.69641	−	1.69641i	−0.0631775	−	0.0631775i
$722$		0			0
$723$	−15.2346	+	15.2346i	−0.566580	+	0.566580i
$724$		0			0
$725$	30.3366			1.12667
$726$		0			0
$727$	6.72700			0.249491			0.124745	−	0.992189i	$-0.460189\pi$
$727$	6.72700			0.249491			0.124745	+	0.992189i	$0.460189\pi$
$728$		0			0
$729$	28.5122			1.05601
$730$		0			0
$731$	−21.8004			−0.806316
$732$		0			0
$733$	14.0245	−	14.0245i	0.518008	−	0.518008i	−0.398961	−	0.916968i	$-0.630629\pi$
$733$	14.0245	−	14.0245i	0.518008	−	0.518008i	0.916968	+	0.398961i	$0.130629\pi$
$734$		0			0
$735$	−23.2968	−	23.2968i	−0.859314	−	0.859314i
$736$		0			0
$737$	22.1812			0.817054
$738$		0			0
$739$	−23.8579	+	23.8579i	−0.877627	+	0.877627i	−0.993289	−	0.115661i	$-0.963101\pi$
$739$	−23.8579	+	23.8579i	−0.877627	+	0.877627i	0.115661	+	0.993289i	$0.463101\pi$
$740$		0			0
$741$	34.3562	−	7.60350i	1.26211	−	0.279322i
$742$		0			0
$743$	17.6669	−	17.6669i	0.648137	−	0.648137i	−0.304406	−	0.952542i	$-0.598458\pi$
$743$	17.6669	−	17.6669i	0.648137	−	0.648137i	0.952542	+	0.304406i	$0.0984579\pi$
$744$		0			0
$745$			28.1302i			1.03061i
$746$		0			0
$747$	−2.64545	−	2.64545i	−0.0967920	−	0.0967920i
$748$		0			0
$749$	0.226411	+	0.226411i	0.00827289	+	0.00827289i
$750$		0			0
$751$	−21.8413			−0.797002			−0.398501	−	0.917168i	$-0.630469\pi$
$751$	−21.8413			−0.797002			−0.398501	+	0.917168i	$0.630469\pi$
$752$		0			0
$753$		−	0.960820i		−	0.0350142i
$754$		0			0
$755$	−65.4804			−2.38308
$756$		0			0
$757$		−	4.18475i		−	0.152097i	−0.997104	−	0.0760486i	$-0.975770\pi$
$757$		−	4.18475i		−	0.152097i	0.997104	−	0.0760486i	$-0.0242304\pi$
$758$		0			0
$759$	14.5688	−	14.5688i	0.528815	−	0.528815i
$760$		0			0
$761$	7.52046	−	7.52046i	0.272617	−	0.272617i	−0.557536	−	0.830153i	$-0.688253\pi$
$761$	7.52046	−	7.52046i	0.272617	−	0.272617i	0.830153	+	0.557536i	$0.188253\pi$
$762$		0			0
$763$	−2.62593			−0.0950650
$764$		0			0
$765$	−11.2167	−	11.2167i	−0.405539	−	0.405539i
$766$		0			0
$767$	6.92644	+	31.2969i	0.250099	+	1.13007i
$768$		0			0
$769$	−9.42965	−	9.42965i	−0.340042	−	0.340042i	0.516341	−	0.856383i	$-0.327294\pi$
$769$	−9.42965	−	9.42965i	−0.340042	−	0.340042i	−0.856383	+	0.516341i	$0.827294\pi$
$770$		0			0
$771$			22.9898i			0.827959i
$772$		0			0
$773$	22.5741	−	22.5741i	0.811933	−	0.811933i	−0.172990	−	0.984924i	$-0.555343\pi$
$773$	22.5741	−	22.5741i	0.811933	−	0.811933i	0.984924	+	0.172990i	$0.0553429\pi$
$774$		0			0
$775$	7.11245	+	7.11245i	0.255487	+	0.255487i
$776$		0			0
$777$			0.865684i			0.0310562i
$778$		0			0
$779$		−	3.85877i		−	0.138255i
$780$		0			0
$781$		−	13.6733i		−	0.489271i
$782$		0			0
$783$			25.9731i			0.928204i
$784$		0			0
$785$	7.76811	+	7.76811i	0.277256	+	0.277256i
$786$		0			0
$787$	28.3997	−	28.3997i	1.01234	−	1.01234i	0.0124163	−	0.999923i	$-0.496048\pi$
$787$	28.3997	−	28.3997i	1.01234	−	1.01234i	0.999923	−	0.0124163i	$-0.00395233\pi$
$788$		0			0
$789$		−	4.92521i		−	0.175342i
$790$		0			0
$791$	1.52152	+	1.52152i	0.0540992	+	0.0540992i
$792$		0			0
$793$	−32.1704	−	20.5112i	−1.14240	−	0.728375i
$794$		0			0
$795$	6.14948	+	6.14948i	0.218099	+	0.218099i
$796$		0			0
$797$	−49.1622			−1.74142			−0.870708	−	0.491800i	$-0.836339\pi$
$797$	−49.1622			−1.74142			−0.870708	+	0.491800i	$0.836339\pi$
$798$		0			0
$799$	−4.35453	+	4.35453i	−0.154052	+	0.154052i
$800$		0			0
$801$	12.1897	−	12.1897i	0.430701	−	0.430701i
$802$		0			0
$803$			2.04321i			0.0721033i
$804$		0			0
$805$	2.43974			0.0859897
$806$		0			0
$807$			17.5330i			0.617192i
$808$		0			0
$809$	11.8224			0.415652			0.207826	−	0.978166i	$-0.433361\pi$
$809$	11.8224			0.415652			0.207826	+	0.978166i	$0.433361\pi$
$810$		0			0
$811$	24.6297	+	24.6297i	0.864866	+	0.864866i	0.991899	−	0.127033i	$-0.0405453\pi$
$811$	24.6297	+	24.6297i	0.864866	+	0.864866i	−0.127033	+	0.991899i	$0.540545\pi$
$812$		0			0
$813$	−20.8582	−	20.8582i	−0.731528	−	0.731528i
$814$		0			0
$815$			59.1025i			2.07027i
$816$		0			0
$817$	−24.9775	+	24.9775i	−0.873853	+	0.873853i
$818$		0			0
$819$	−0.678661	+	0.150197i	−0.0237144	+	0.00524831i
$820$		0			0
$821$	−25.0256	+	25.0256i	−0.873400	+	0.873400i	−0.992841	−	0.119441i	$-0.961890\pi$
$821$	−25.0256	+	25.0256i	−0.873400	+	0.873400i	0.119441	+	0.992841i	$0.461890\pi$
$822$		0			0
$823$	−32.0560			−1.11740			−0.558702	−	0.829369i	$-0.688700\pi$
$823$	−32.0560			−1.11740			−0.558702	+	0.829369i	$0.688700\pi$
$824$		0			0
$825$	24.1302	+	24.1302i	0.840107	+	0.840107i
$826$		0			0
$827$	12.2989	−	12.2989i	0.427674	−	0.427674i	−0.460162	−	0.887835i	$-0.652208\pi$
$827$	12.2989	−	12.2989i	0.427674	−	0.427674i	0.887835	+	0.460162i	$0.152208\pi$
$828$		0			0
$829$	44.2580			1.53714			0.768572	−	0.639763i	$-0.220967\pi$
$829$	44.2580			1.53714			0.768572	+	0.639763i	$0.220967\pi$
$830$		0			0
$831$	−15.8849			−0.551042
$832$		0			0
$833$	−30.2336			−1.04753
$834$		0			0
$835$	37.2564			1.28931
$836$		0			0
$837$	−6.08944	+	6.08944i	−0.210482	+	0.210482i
$838$		0			0
$839$	−11.5850	−	11.5850i	−0.399960	−	0.399960i	0.478259	−	0.878219i	$-0.341268\pi$
$839$	−11.5850	−	11.5850i	−0.399960	−	0.399960i	−0.878219	+	0.478259i	$0.841268\pi$
$840$		0			0
$841$	7.89716			0.272316
$842$		0			0
$843$	−0.984567	+	0.984567i	−0.0339103	+	0.0339103i
$844$		0			0
$845$	41.6021	+	15.1761i	1.43115	+	0.522075i
$846$		0			0
$847$	0.363119	−	0.363119i	0.0124769	−	0.0124769i
$848$		0			0
$849$			5.72541i			0.196495i
$850$		0			0
$851$	9.78812	+	9.78812i	0.335533	+	0.335533i
$852$		0			0
$853$	15.7316	+	15.7316i	0.538640	+	0.538640i	0.923129	−	0.384490i	$-0.125623\pi$
$853$	15.7316	+	15.7316i	0.538640	+	0.538640i	−0.384490	+	0.923129i	$0.625623\pi$
$854$		0			0
$855$	−25.7027			−0.879015
$856$		0			0
$857$		−	5.28481i		−	0.180526i	−0.995918	−	0.0902628i	$-0.971229\pi$
$857$		−	5.28481i		−	0.180526i	0.995918	−	0.0902628i	$-0.0287707\pi$
$858$		0			0
$859$	25.4764			0.869242			0.434621	−	0.900613i	$-0.356882\pi$
$859$	25.4764			0.869242			0.434621	+	0.900613i	$0.356882\pi$
$860$		0			0
$861$		−	0.136853i		−	0.00466393i
$862$		0			0
$863$	−12.9386	+	12.9386i	−0.440434	+	0.440434i	−0.892158	−	0.451724i	$-0.850809\pi$
$863$	−12.9386	+	12.9386i	−0.440434	+	0.440434i	0.451724	+	0.892158i	$0.350809\pi$
$864$		0			0
$865$	−31.8126	+	31.8126i	−1.08166	+	1.08166i
$866$		0			0
$867$	2.53694			0.0861589
$868$		0			0
$869$	−40.4891	−	40.4891i	−1.37350	−	1.37350i
$870$		0			0
$871$	20.9754	−	4.64214i	0.710724	−	0.157293i
$872$		0			0
$873$	−3.55632	−	3.55632i	−0.120363	−	0.120363i
$874$		0			0
$875$			0.981394i			0.0331772i
$876$		0			0
$877$	16.5941	−	16.5941i	0.560342	−	0.560342i	−0.369063	−	0.929404i	$-0.620321\pi$
$877$	16.5941	−	16.5941i	0.560342	−	0.560342i	0.929404	+	0.369063i	$0.120321\pi$
$878$		0			0
$879$	4.16543	+	4.16543i	0.140496	+	0.140496i
$880$		0			0
$881$			2.69158i			0.0906818i	0.998972	+	0.0453409i	$0.0144374\pi$
$881$			2.69158i			0.0906818i	−0.998972	+	0.0453409i	$0.985563\pi$
$882$		0			0
$883$			48.9980i			1.64892i	0.565924	+	0.824458i	$0.308520\pi$
$883$			48.9980i			1.64892i	−0.565924	+	0.824458i	$0.691480\pi$
$884$		0			0
$885$			42.0371i			1.41306i
$886$		0			0
$887$		−	18.5664i		−	0.623399i	−0.950181	−	0.311699i	$-0.899102\pi$
$887$		−	18.5664i		−	0.623399i	0.950181	−	0.311699i	$-0.100898\pi$
$888$		0			0
$889$	−0.185605	−	0.185605i	−0.00622498	−	0.00622498i
$890$		0			0
$891$	−12.1843	+	12.1843i	−0.408190	+	0.408190i
$892$		0			0
$893$			9.97831i			0.333911i
$894$		0			0
$895$	−12.6382	−	12.6382i	−0.422449	−	0.422449i
$896$		0			0
$897$	10.7279	−	16.8259i	0.358193	−	0.561800i
$898$		0			0
$899$	−4.94759	−	4.94759i	−0.165011	−	0.165011i
$900$		0			0
$901$	7.98054			0.265870
$902$		0			0
$903$	−0.885838	+	0.885838i	−0.0294789	+	0.0294789i
$904$		0			0
$905$	8.42322	−	8.42322i	0.279997	−	0.279997i
$906$		0			0
$907$			4.12839i			0.137081i	0.997648	+	0.0685404i	$0.0218342\pi$
$907$			4.12839i			0.137081i	−0.997648	+	0.0685404i	$0.978166\pi$
$908$		0			0
$909$	20.4468			0.678178
$910$		0			0
$911$		−	14.8565i		−	0.492218i	−0.969242	−	0.246109i	$-0.920848\pi$
$911$		−	14.8565i		−	0.492218i	0.969242	−	0.246109i	$-0.0791522\pi$
$912$		0			0
$913$	12.9777			0.429498
$914$		0			0
$915$	−35.3802	−	35.3802i	−1.16963	−	1.16963i
$916$		0			0
$917$	−0.477522	−	0.477522i	−0.0157692	−	0.0157692i
$918$		0			0
$919$			3.47888i			0.114758i	0.998352	+	0.0573788i	$0.0182743\pi$
$919$			3.47888i			0.114758i	−0.998352	+	0.0573788i	$0.981726\pi$
$920$		0			0
$921$	−3.83237	+	3.83237i	−0.126281	+	0.126281i
$922$		0			0
$923$	−2.86160	−	12.9301i	−0.0941907	−	0.425598i
$924$		0			0
$925$	−16.2120	+	16.2120i	−0.533047	+	0.533047i
$926$		0			0
$927$	14.3331			0.470761
$928$		0			0
$929$	8.41692	+	8.41692i	0.276150	+	0.276150i	0.831570	−	0.555420i	$-0.187442\pi$
$929$	8.41692	+	8.41692i	0.276150	+	0.276150i	−0.555420	+	0.831570i	$0.687442\pi$
$930$		0			0
$931$	−34.6398	+	34.6398i	−1.13527	+	1.13527i
$932$		0			0
$933$	−5.93960			−0.194454
$934$		0			0
$935$	55.0251			1.79951
$936$		0			0
$937$	−22.6059			−0.738503			−0.369251	−	0.929330i	$-0.620386\pi$
$937$	−22.6059			−0.738503			−0.369251	+	0.929330i	$0.620386\pi$
$938$		0			0
$939$	5.11550			0.166938
$940$		0			0
$941$	−24.2984	+	24.2984i	−0.792105	+	0.792105i	−0.981836	−	0.189731i	$-0.939238\pi$
$941$	−24.2984	+	24.2984i	−0.792105	+	0.792105i	0.189731	+	0.981836i	$0.439238\pi$
$942$		0			0
$943$	−1.54737	−	1.54737i	−0.0503893	−	0.0503893i
$944$		0			0
$945$	−3.45971			−0.112544
$946$		0			0
$947$	−11.1576	+	11.1576i	−0.362572	+	0.362572i	−0.864759	−	0.502187i	$-0.832529\pi$
$947$	−11.1576	+	11.1576i	−0.362572	+	0.362572i	0.502187	+	0.864759i	$0.332529\pi$
$948$		0			0
$949$	0.427609	+	1.93214i	0.0138808	+	0.0627199i
$950$		0			0
$951$	−4.09119	+	4.09119i	−0.132666	+	0.132666i
$952$		0			0
$953$			59.6199i			1.93128i	0.259888	+	0.965639i	$0.416314\pi$
$953$			59.6199i			1.93128i	−0.259888	+	0.965639i	$0.583686\pi$
$954$		0			0
$955$	22.1003	+	22.1003i	0.715147	+	0.715147i
$956$		0			0
$957$	−16.7856	−	16.7856i	−0.542601	−	0.542601i
$958$		0			0
$959$	1.54897			0.0500189
$960$		0			0
$961$			28.6801i			0.925163i
$962$		0			0
$963$	−1.91297			−0.0616446
$964$		0			0
$965$			0.397854i			0.0128074i
$966$		0			0
$967$	15.0331	−	15.0331i	0.483433	−	0.483433i	−0.422793	−	0.906226i	$-0.638950\pi$
$967$	15.0331	−	15.0331i	0.483433	−	0.483433i	0.906226	+	0.422793i	$0.138950\pi$
$968$		0			0
$969$	29.9433	−	29.9433i	0.961918	−	0.961918i
$970$		0			0
$971$	11.8204			0.379336			0.189668	−	0.981848i	$-0.439259\pi$
$971$	11.8204			0.379336			0.189668	+	0.981848i	$0.439259\pi$
$972$		0			0
$973$	−2.45607	−	2.45607i	−0.0787379	−	0.0787379i
$974$		0			0
$975$	27.8686	+	17.7685i	0.892508	+	0.569046i
$976$		0			0
$977$	3.25802	+	3.25802i	0.104233	+	0.104233i	0.757300	−	0.653067i	$-0.226518\pi$
$977$	3.25802	+	3.25802i	0.104233	+	0.104233i	−0.653067	+	0.757300i	$0.726518\pi$
$978$		0			0
$979$			59.7984i			1.91117i
$980$		0			0
$981$	11.0934	−	11.0934i	0.354184	−	0.354184i
$982$		0			0
$983$	−26.9767	−	26.9767i	−0.860422	−	0.860422i	0.130965	−	0.991387i	$-0.458192\pi$
$983$	−26.9767	−	26.9767i	−0.860422	−	0.860422i	−0.991387	+	0.130965i	$0.958192\pi$
$984$		0			0
$985$			41.6797i			1.32802i
$986$		0			0
$987$			0.353885i			0.0112643i
$988$		0			0
$989$			20.0320i			0.636981i
$990$		0			0
$991$		−	51.6574i		−	1.64095i	−0.571681	−	0.820476i	$-0.693709\pi$
$991$		−	51.6574i		−	1.64095i	0.571681	−	0.820476i	$-0.306291\pi$
$992$		0			0
$993$	12.1090	+	12.1090i	0.384266	+	0.384266i
$994$		0			0
$995$	39.3911	−	39.3911i	1.24878	−	1.24878i
$996$		0			0
$997$			25.6944i			0.813749i	0.913484	+	0.406874i	$0.133381\pi$
$997$			25.6944i			0.813749i	−0.913484	+	0.406874i	$0.866619\pi$
$998$		0			0
$999$	−13.8802	−	13.8802i	−0.439149	−	0.439149i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.u.b.47.3		20
4.3	odd	2		104.2.m.b.99.8	yes	20
8.3	odd	2	inner	416.2.u.b.47.4		20
8.5	even	2		104.2.m.b.99.3	yes	20
12.11	even	2		936.2.w.h.307.3		20
13.5	odd	4	inner	416.2.u.b.239.4		20
24.5	odd	2		936.2.w.h.307.8		20
52.31	even	4		104.2.m.b.83.3	✓	20
104.5	odd	4		104.2.m.b.83.8	yes	20
104.83	even	4	inner	416.2.u.b.239.3		20
156.83	odd	4		936.2.w.h.811.8		20
312.5	even	4		936.2.w.h.811.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.m.b.83.3	✓	20	52.31	even	4
104.2.m.b.83.8	yes	20	104.5	odd	4
104.2.m.b.99.3	yes	20	8.5	even	2
104.2.m.b.99.8	yes	20	4.3	odd	2
416.2.u.b.47.3		20	1.1	even	1	trivial
416.2.u.b.47.4		20	8.3	odd	2	inner
416.2.u.b.239.3		20	104.83	even	4	inner
416.2.u.b.239.4		20	13.5	odd	4	inner
936.2.w.h.307.3		20	12.11	even	2
936.2.w.h.307.8		20	24.5	odd	2
936.2.w.h.811.3		20	312.5	even	4
936.2.w.h.811.8		20	156.83	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.u (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.38809 q^{3} +(-2.40872 + 2.40872i) q^{5} +(0.127019 + 0.127019i) q^{7} -1.07320 q^{9} +(2.63237 - 2.63237i) q^{11} +(1.93837 - 3.04019i) q^{13} +(3.34352 - 3.34352i) q^{15} -4.33908i q^{17} +(-4.97146 - 4.97146i) q^{19} +(-0.176315 - 0.176315i) q^{21} -3.98711 q^{23} -6.60383i q^{25} +5.65398 q^{27} +4.59378i q^{29} +(-1.07702 + 1.07702i) q^{31} +(-3.65398 + 3.65398i) q^{33} -0.611907 q^{35} +(-2.45494 - 2.45494i) q^{37} +(-2.69063 + 4.22006i) q^{39} +(0.388093 + 0.388093i) q^{41} -5.02419i q^{43} +(2.58503 - 2.58503i) q^{45} +(-1.00356 - 1.00356i) q^{47} -6.96773i q^{49} +6.02305i q^{51} +1.83922i q^{53} +12.6813i q^{55} +(6.90084 + 6.90084i) q^{57} +(-6.28635 + 6.28635i) q^{59} -10.5817i q^{61} +(-0.136317 - 0.136317i) q^{63} +(2.65398 + 11.9919i) q^{65} +(4.21315 + 4.21315i) q^{67} +5.53449 q^{69} +(2.59715 - 2.59715i) q^{71} +(-0.388093 + 0.388093i) q^{73} +9.16673i q^{75} +0.668725 q^{77} -15.3812i q^{79} -4.62865 q^{81} +(2.46502 + 2.46502i) q^{83} +(10.4516 + 10.4516i) q^{85} -6.37660i q^{87} +(-11.3583 + 11.3583i) q^{89} +(0.632373 - 0.139953i) q^{91} +(1.49500 - 1.49500i) q^{93} +23.9497 q^{95} +(3.31375 + 3.31375i) q^{97} +(-2.82506 + 2.82506i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 16 q^{9} + 8 q^{11} + 12 q^{19} + 52 q^{27} - 12 q^{33} - 44 q^{35} - 24 q^{41} + 8 q^{57} - 20 q^{59} - 8 q^{65} + 16 q^{67} + 24 q^{73} - 44 q^{81} - 16 q^{83} - 16 q^{89} - 32 q^{91}+ \cdots - 20 q^{99}+O(q^{100}) \) 20 * q + 4 * q^3 + 16 * q^9 + 8 * q^11 + 12 * q^19 + 52 * q^27 - 12 * q^33 - 44 * q^35 - 24 * q^41 + 8 * q^57 - 20 * q^59 - 8 * q^65 + 16 * q^67 + 24 * q^73 - 44 * q^81 - 16 * q^83 - 16 * q^89 - 32 * q^91 + 12 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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