LMFDB - Embedded newform 3120.2.l.n.1249.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1249,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3120.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (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:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.57815240704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 $ x^8 - 2x^7 + 2x^6 + 89x^4 - 170x^3 + 162x^2 - 72x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.6
Root			$0.594137 + 0.594137i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.n.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.594137 + 2.15569i) q^{5} -4.92778i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-0.594137 + 2.15569i) q^{5} -4.92778i q^{7} -1.00000 q^{9} +1.38360 q^{11} +1.00000i q^{13} +(-2.15569 - 0.594137i) q^{15} -0.195329i q^{17} +4.92778 q^{21} +2.19533i q^{23} +(-4.29400 - 2.56155i) q^{25} -1.00000i q^{27} -7.49966 q^{29} +1.38360i q^{33} +(10.6228 + 2.92778i) q^{35} -6.05088i q^{37} -1.00000 q^{39} -2.11605 q^{41} -3.49966i q^{43} +(0.594137 - 2.15569i) q^{45} +2.31138i q^{47} -17.2830 q^{49} +0.195329 q^{51} -6.05088i q^{53} +(-0.822051 + 2.98262i) q^{55} +9.43449 q^{59} -3.69498 q^{61} +4.92778i q^{63} +(-2.15569 - 0.594137i) q^{65} -12.6228i q^{67} -2.19533 q^{69} -13.9716 q^{71} +4.73245i q^{73} +(2.56155 - 4.29400i) q^{75} -6.81809i q^{77} -8.07153 q^{79} +1.00000 q^{81} +13.3997i q^{83} +(0.421068 + 0.116052i) q^{85} -7.49966i q^{87} -3.02770 q^{89} +4.92778 q^{91} -6.01612i q^{97} -1.38360 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^5 - 8 * q^9	$ 8 q - 2 q^{5} - 8 q^{9} - 2 q^{11} + 2 q^{15} + 14 q^{21} - 16 q^{29} + 8 q^{35} - 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} - 6 q^{51} - 10 q^{55} + 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} - 30 q^{71} + 4 q^{75} - 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} + 14 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 2 * q^5 - 8 * q^9 - 2 * q^11 + 2 * q^15 + 14 * q^21 - 16 * q^29 + 8 * q^35 - 8 * q^39 + 14 * q^41 + 2 * q^45 - 18 * q^49 - 6 * q^51 - 10 * q^55 + 4 * q^59 + 22 * q^61 + 2 * q^65 - 10 * q^69 - 30 * q^71 + 4 * q^75 - 2 * q^79 + 8 * q^81 + 24 * q^85 - 18 * q^89 + 14 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−0.594137	+	2.15569i	−0.265706	+	0.964054i
$5$	−0.594137	+	2.15569i	−0.265706	+	0.964054i
$6$		0			0
$7$		−	4.92778i		−	1.86252i	−0.364349	−	0.931262i	$-0.618709\pi$
$7$		−	4.92778i		−	1.86252i	0.364349	−	0.931262i	$-0.381291\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.38360			0.417172			0.208586	−	0.978004i	$-0.433114\pi$
$11$	1.38360			0.417172			0.208586	+	0.978004i	$0.433114\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	−2.15569	−	0.594137i	−0.556597	−	0.153406i
$16$		0			0
$17$		−	0.195329i		−	0.0473741i	−0.999719	−	0.0236871i	$-0.992459\pi$
$17$		−	0.195329i		−	0.0473741i	0.999719	−	0.0236871i	$-0.00754053\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	4.92778			1.07533
$22$		0			0
$23$			2.19533i			0.457758i	0.973455	+	0.228879i	$0.0735060\pi$
$23$			2.19533i			0.457758i	−0.973455	+	0.228879i	$0.926494\pi$
$24$		0			0
$25$	−4.29400	−	2.56155i	−0.858800	−	0.512311i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−7.49966			−1.39265			−0.696326	−	0.717726i	$-0.745183\pi$
$29$	−7.49966			−1.39265			−0.696326	+	0.717726i	$0.745183\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$			1.38360i			0.240854i
$34$		0			0
$35$	10.6228	+	2.92778i	1.79557	+	0.494885i
$36$		0			0
$37$		−	6.05088i		−	0.994759i	−0.867533	−	0.497379i	$-0.834296\pi$
$37$		−	6.05088i		−	0.994759i	0.867533	−	0.497379i	$-0.165704\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	−2.11605			−0.330472			−0.165236	−	0.986254i	$-0.552839\pi$
$41$	−2.11605			−0.330472			−0.165236	+	0.986254i	$0.552839\pi$
$42$		0			0
$43$		−	3.49966i		−	0.533692i	−0.963739	−	0.266846i	$-0.914018\pi$
$43$		−	3.49966i		−	0.533692i	0.963739	−	0.266846i	$-0.0859816\pi$
$44$		0			0
$45$	0.594137	−	2.15569i	0.0885688	−	0.321351i
$46$		0			0
$47$			2.31138i			0.337150i	0.985689	+	0.168575i	$0.0539165\pi$
$47$			2.31138i			0.337150i	−0.985689	+	0.168575i	$0.946084\pi$
$48$		0			0
$49$	−17.2830			−2.46900
$50$		0			0
$51$	0.195329			0.0273515
$52$		0			0
$53$		−	6.05088i		−	0.831153i	−0.909558	−	0.415576i	$-0.863580\pi$
$53$		−	6.05088i		−	0.831153i	0.909558	−	0.415576i	$-0.136420\pi$
$54$		0			0
$55$	−0.822051	+	2.98262i	−0.110845	+	0.402176i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.43449			1.22827			0.614133	−	0.789203i	$-0.289506\pi$
$59$	9.43449			1.22827			0.614133	+	0.789203i	$0.289506\pi$
$60$		0			0
$61$	−3.69498			−0.473094			−0.236547	−	0.971620i	$-0.576016\pi$
$61$	−3.69498			−0.473094			−0.236547	+	0.971620i	$0.576016\pi$
$62$		0			0
$63$			4.92778i			0.620842i
$64$		0			0
$65$	−2.15569	−	0.594137i	−0.267380	−	0.0736937i
$66$		0			0
$67$		−	12.6228i		−	1.54212i	−0.636765	−	0.771058i	$-0.719728\pi$
$67$		−	12.6228i		−	1.54212i	0.636765	−	0.771058i	$-0.280272\pi$
$68$		0			0
$69$	−2.19533			−0.264287
$70$		0			0
$71$	−13.9716			−1.65812			−0.829062	−	0.559156i	$-0.811125\pi$
$71$	−13.9716			−1.65812			−0.829062	+	0.559156i	$0.811125\pi$
$72$		0			0
$73$			4.73245i			0.553891i	0.960886	+	0.276946i	$0.0893222\pi$
$73$			4.73245i			0.553891i	−0.960886	+	0.276946i	$0.910678\pi$
$74$		0			0
$75$	2.56155	−	4.29400i	0.295783	−	0.495829i
$76$		0			0
$77$		−	6.81809i		−	0.776993i
$78$		0			0
$79$	−8.07153			−0.908119			−0.454059	−	0.890971i	$-0.650025\pi$
$79$	−8.07153			−0.908119			−0.454059	+	0.890971i	$0.650025\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			13.3997i			1.47081i	0.677627	+	0.735406i	$0.263008\pi$
$83$			13.3997i			1.47081i	−0.677627	+	0.735406i	$0.736992\pi$
$84$		0			0
$85$	0.421068	+	0.116052i	0.0456712	+	0.0125876i
$86$		0			0
$87$		−	7.49966i		−	0.804047i
$88$		0			0
$89$	−3.02770			−0.320936			−0.160468	−	0.987041i	$-0.551300\pi$
$89$	−3.02770			−0.320936			−0.160468	+	0.987041i	$0.551300\pi$
$90$		0			0
$91$	4.92778			0.516571
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	6.01612i		−	0.610845i	−0.952217	−	0.305422i	$-0.901202\pi$
$97$		−	6.01612i		−	0.610845i	0.952217	−	0.305422i	$-0.0987977\pi$
$98$		0			0
$99$	−1.38360			−0.139057
$100$		0			0
$101$	−3.10900			−0.309357			−0.154678	−	0.987965i	$-0.549434\pi$
$101$	−3.10900			−0.309357			−0.154678	+	0.987965i	$0.549434\pi$
$102$		0			0
$103$		−	16.8690i		−	1.66215i	−0.556161	−	0.831075i	$-0.687726\pi$
$103$		−	16.8690i		−	1.66215i	0.556161	−	0.831075i	$-0.312274\pi$
$104$		0			0
$105$	−2.92778	+	10.6228i	−0.285722	+	1.03668i
$106$		0			0
$107$		−	17.1946i		−	1.66227i	−0.556072	−	0.831134i	$-0.687692\pi$
$107$		−	17.1946i		−	1.66227i	0.556072	−	0.831134i	$-0.312308\pi$
$108$		0			0
$109$	−8.76721			−0.839746			−0.419873	−	0.907583i	$-0.637925\pi$
$109$	−8.76721			−0.839746			−0.419873	+	0.907583i	$0.637925\pi$
$110$		0			0
$111$	6.05088			0.574324
$112$		0			0
$113$			7.46490i			0.702238i	0.936331	+	0.351119i	$0.114199\pi$
$113$			7.46490i			0.702238i	−0.936331	+	0.351119i	$0.885801\pi$
$114$		0			0
$115$	−4.73245	−	1.30433i	−0.441303	−	0.121629i
$116$		0			0
$117$		−	1.00000i		−	0.0924500i
$118$		0			0
$119$	−0.962536			−0.0882355
$120$		0			0
$121$	−9.08564			−0.825967
$122$		0			0
$123$		−	2.11605i		−	0.190798i
$124$		0			0
$125$	8.07314	−	7.73462i	0.722084	−	0.691806i
$126$		0			0
$127$		−	11.1231i		−	0.987016i	−0.869741	−	0.493508i	$-0.835714\pi$
$127$		−	11.1231i		−	0.987016i	0.869741	−	0.493508i	$-0.164286\pi$
$128$		0			0
$129$	3.49966			0.308127
$130$		0			0
$131$	1.82080			0.159084			0.0795418	−	0.996832i	$-0.474654\pi$
$131$	1.82080			0.159084			0.0795418	+	0.996832i	$0.474654\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.15569	+	0.594137i	0.185532	+	0.0511352i
$136$		0			0
$137$			8.70204i			0.743465i	0.928340	+	0.371733i	$0.121236\pi$
$137$			8.70204i			0.743465i	−0.928340	+	0.371733i	$0.878764\pi$
$138$		0			0
$139$	6.81809			0.578303			0.289151	−	0.957283i	$-0.406627\pi$
$139$	6.81809			0.578303			0.289151	+	0.957283i	$0.406627\pi$
$140$		0			0
$141$	−2.31138			−0.194653
$142$		0			0
$143$			1.38360i			0.115703i
$144$		0			0
$145$	4.45583	−	16.1669i	0.370036	−	1.34259i
$146$		0			0
$147$		−	17.2830i		−	1.42548i
$148$		0			0
$149$	7.48537			0.613225			0.306613	−	0.951834i	$-0.400804\pi$
$149$	7.48537			0.613225			0.306613	+	0.951834i	$0.400804\pi$
$150$		0			0
$151$	−12.8549			−1.04611			−0.523057	−	0.852298i	$-0.675209\pi$
$151$	−12.8549			−1.04611			−0.523057	+	0.852298i	$0.675209\pi$
$152$		0			0
$153$			0.195329i			0.0157914i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			17.7318i			1.41515i	0.706639	+	0.707574i	$0.250210\pi$
$157$			17.7318i			1.41515i	−0.706639	+	0.707574i	$0.749790\pi$
$158$		0			0
$159$	6.05088			0.479866
$160$		0			0
$161$	10.8181			0.852585
$162$		0			0
$163$		−	18.1385i		−	1.42072i	−0.703838	−	0.710360i	$-0.748532\pi$
$163$		−	18.1385i		−	1.42072i	0.703838	−	0.710360i	$-0.251468\pi$
$164$		0			0
$165$	−2.98262	−	0.822051i	−0.232197	−	0.0639966i
$166$		0			0
$167$		−	4.06517i		−	0.314572i	−0.987553	−	0.157286i	$-0.949726\pi$
$167$		−	4.06517i		−	0.314572i	0.987553	−	0.157286i	$-0.0502745\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	23.2455i		−	1.76732i	−0.468125	−	0.883662i	$-0.655070\pi$
$173$		−	23.2455i		−	1.76732i	0.468125	−	0.883662i	$-0.344930\pi$
$174$		0			0
$175$	−12.6228	+	21.1599i	−0.954191	+	1.59954i
$176$		0			0
$177$			9.43449i			0.709139i
$178$		0			0
$179$	14.1224			1.05556			0.527779	−	0.849381i	$-0.323025\pi$
$179$	14.1224			1.05556			0.527779	+	0.849381i	$0.323025\pi$
$180$		0			0
$181$	3.67433			0.273111			0.136556	−	0.990632i	$-0.456397\pi$
$181$	3.67433			0.273111			0.136556	+	0.990632i	$0.456397\pi$
$182$		0			0
$183$		−	3.69498i		−	0.273141i
$184$		0			0
$185$	13.0438	+	3.59506i	0.959001	+	0.264314i
$186$		0			0
$187$		−	0.270257i		−	0.0197632i
$188$		0			0
$189$	−4.92778			−0.358443
$190$		0			0
$191$	16.8549			1.21958			0.609788	−	0.792565i	$-0.291255\pi$
$191$	16.8549			1.21958			0.609788	+	0.792565i	$0.291255\pi$
$192$		0			0
$193$			6.53712i			0.470552i	0.971929	+	0.235276i	$0.0755994\pi$
$193$			6.53712i			0.470552i	−0.971929	+	0.235276i	$0.924401\pi$
$194$		0			0
$195$	0.594137	−	2.15569i	0.0425471	−	0.154372i
$196$		0			0
$197$		−	25.7890i		−	1.83739i	−0.394967	−	0.918695i	$-0.629244\pi$
$197$		−	25.7890i		−	1.83739i	0.394967	−	0.918695i	$-0.370756\pi$
$198$		0			0
$199$	−16.9646			−1.20259			−0.601293	−	0.799029i	$-0.705347\pi$
$199$	−16.9646			−1.20259			−0.601293	+	0.799029i	$0.705347\pi$
$200$		0			0
$201$	12.6228			0.890341
$202$		0			0
$203$			36.9566i			2.59385i
$204$		0			0
$205$	1.25723	−	4.56155i	0.0878085	−	0.318593i
$206$		0			0
$207$		−	2.19533i		−	0.152586i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	26.4577			1.82142			0.910710	−	0.413046i	$-0.135535\pi$
$211$	26.4577			1.82142			0.910710	+	0.413046i	$0.135535\pi$
$212$		0			0
$213$		−	13.9716i		−	0.957319i
$214$		0			0
$215$	7.54417	+	2.07928i	0.514508	+	0.141805i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−4.73245			−0.319789
$220$		0			0
$221$	0.195329			0.0131392
$222$		0			0
$223$			2.02065i			0.135313i	0.997709	+	0.0676564i	$0.0215522\pi$
$223$			2.02065i			0.135313i	−0.997709	+	0.0676564i	$0.978448\pi$
$224$		0			0
$225$	4.29400	+	2.56155i	0.286267	+	0.170770i
$226$		0			0
$227$		−	17.7904i		−	1.18079i	−0.807115	−	0.590395i	$-0.798972\pi$
$227$		−	17.7904i		−	1.18079i	0.807115	−	0.590395i	$-0.201028\pi$
$228$		0			0
$229$	18.3339			1.21154			0.605768	−	0.795641i	$-0.292866\pi$
$229$	18.3339			1.21154			0.605768	+	0.795641i	$0.292866\pi$
$230$		0			0
$231$	6.81809			0.448597
$232$		0			0
$233$		−	23.0361i		−	1.50914i	−0.656217	−	0.754572i	$-0.727844\pi$
$233$		−	23.0361i		−	1.50914i	0.656217	−	0.754572i	$-0.272156\pi$
$234$		0			0
$235$	−4.98262	−	1.37328i	−0.325030	−	0.0895828i
$236$		0			0
$237$		−	8.07153i		−	0.524302i
$238$		0			0
$239$	−0.986402			−0.0638051			−0.0319025	−	0.999491i	$-0.510157\pi$
$239$	−0.986402			−0.0638051			−0.0319025	+	0.999491i	$0.510157\pi$
$240$		0			0
$241$	1.52031			0.0979316			0.0489658	−	0.998800i	$-0.484407\pi$
$241$	1.52031			0.0979316			0.0489658	+	0.998800i	$0.484407\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	10.2685	−	37.2568i	0.656028	−	2.38025i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−13.3997			−0.849173
$250$		0			0
$251$	−11.6763			−0.737005			−0.368502	−	0.929627i	$-0.620129\pi$
$251$	−11.6763			−0.737005			−0.368502	+	0.929627i	$0.620129\pi$
$252$		0			0
$253$			3.03746i			0.190964i
$254$		0			0
$255$	−0.116052	+	0.421068i	−0.00726746	+	0.0263683i
$256$		0			0
$257$			3.00069i			0.187178i	0.995611	+	0.0935889i	$0.0298339\pi$
$257$			3.00069i			0.187178i	−0.995611	+	0.0935889i	$0.970166\pi$
$258$		0			0
$259$	−29.8174			−1.85276
$260$		0			0
$261$	7.49966			0.464217
$262$		0			0
$263$		−	25.8683i		−	1.59511i	−0.603248	−	0.797553i	$-0.706127\pi$
$263$		−	25.8683i		−	1.59511i	0.603248	−	0.797553i	$-0.293873\pi$
$264$		0			0
$265$	13.0438	+	3.59506i	0.801276	+	0.220843i
$266$		0			0
$267$		−	3.02770i		−	0.185293i
$268$		0			0
$269$	−11.1090			−0.677328			−0.338664	−	0.940907i	$-0.609975\pi$
$269$	−11.1090			−0.677328			−0.338664	+	0.940907i	$0.609975\pi$
$270$		0			0
$271$	13.9859			0.849582			0.424791	−	0.905291i	$-0.360347\pi$
$271$	13.9859			0.849582			0.424791	+	0.905291i	$0.360347\pi$
$272$		0			0
$273$			4.92778i			0.298243i
$274$		0			0
$275$	−5.94120	−	3.54417i	−0.358268	−	0.213722i
$276$		0			0
$277$		−	10.0529i		−	0.604020i	−0.953305	−	0.302010i	$-0.902342\pi$
$277$		−	10.0529i		−	0.604020i	0.953305	−	0.302010i	$-0.0976576\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−19.6318			−1.17114			−0.585568	−	0.810623i	$-0.699129\pi$
$281$	−19.6318			−1.17114			−0.585568	+	0.810623i	$0.699129\pi$
$282$		0			0
$283$			25.5667i			1.51978i	0.650051	+	0.759890i	$0.274747\pi$
$283$			25.5667i			1.51978i	−0.650051	+	0.759890i	$0.725253\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.4274i			0.615512i
$288$		0			0
$289$	16.9618			0.997756
$290$		0			0
$291$	6.01612			0.352671
$292$		0			0
$293$			6.71614i			0.392361i	0.980568	+	0.196181i	$0.0628539\pi$
$293$			6.71614i			0.392361i	−0.980568	+	0.196181i	$0.937146\pi$
$294$		0			0
$295$	−5.60538	+	20.3378i	−0.326358	+	1.18411i
$296$		0			0
$297$		−	1.38360i		−	0.0802848i
$298$		0			0
$299$	−2.19533			−0.126959
$300$		0			0
$301$	−17.2455			−0.994015
$302$		0			0
$303$		−	3.10900i		−	0.178607i
$304$		0			0
$305$	2.19533	−	7.96524i	0.125704	−	0.456088i
$306$		0			0
$307$			14.2689i			0.814368i	0.913346	+	0.407184i	$0.133489\pi$
$307$			14.2689i			0.814368i	−0.913346	+	0.407184i	$0.866511\pi$
$308$		0			0
$309$	16.8690			0.959642
$310$		0			0
$311$	7.15786			0.405885			0.202943	−	0.979191i	$-0.434950\pi$
$311$	7.15786			0.405885			0.202943	+	0.979191i	$0.434950\pi$
$312$		0			0
$313$			21.6568i			1.22412i	0.790813	+	0.612058i	$0.209658\pi$
$313$			21.6568i			1.22412i	−0.790813	+	0.612058i	$0.790342\pi$
$314$		0			0
$315$	−10.6228	−	2.92778i	−0.598525	−	0.164962i
$316$		0			0
$317$			29.0232i			1.63010i	0.579388	+	0.815052i	$0.303292\pi$
$317$			29.0232i			1.63010i	−0.579388	+	0.815052i	$0.696708\pi$
$318$		0			0
$319$	−10.3765			−0.580975
$320$		0			0
$321$	17.1946			0.959711
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.56155	−	4.29400i	0.142089	−	0.238188i
$326$		0			0
$327$		−	8.76721i		−	0.484828i
$328$		0			0
$329$	11.3900			0.627949
$330$		0			0
$331$	−21.8683			−1.20199			−0.600995	−	0.799253i	$-0.705229\pi$
$331$	−21.8683			−1.20199			−0.600995	+	0.799253i	$0.705229\pi$
$332$		0			0
$333$			6.05088i			0.331586i
$334$		0			0
$335$	27.2108	+	7.49966i	1.48668	+	0.409750i
$336$		0			0
$337$			22.2991i			1.21471i	0.794431	+	0.607355i	$0.207769\pi$
$337$			22.2991i			1.21471i	−0.794431	+	0.607355i	$0.792231\pi$
$338$		0			0
$339$	−7.46490			−0.405438
$340$		0			0
$341$		0			0
$342$		0			0
$343$			50.6723i			2.73605i
$344$		0			0
$345$	1.30433	−	4.73245i	0.0702226	−	0.254786i
$346$		0			0
$347$			22.9199i			1.23040i	0.788370	+	0.615201i	$0.210925\pi$
$347$			22.9199i			1.23040i	−0.788370	+	0.615201i	$0.789075\pi$
$348$		0			0
$349$	−6.75310			−0.361485			−0.180743	−	0.983530i	$-0.557850\pi$
$349$	−6.75310			−0.361485			−0.180743	+	0.983530i	$0.557850\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			3.07859i			0.163857i	0.996638	+	0.0819283i	$0.0261079\pi$
$353$			3.07859i			0.163857i	−0.996638	+	0.0819283i	$0.973892\pi$
$354$		0			0
$355$	8.30105	−	30.1185i	0.440574	−	1.59852i
$356$		0			0
$357$		−	0.962536i		−	0.0509428i
$358$		0			0
$359$	−13.0091			−0.686592			−0.343296	−	0.939227i	$-0.611543\pi$
$359$	−13.0091			−0.686592			−0.343296	+	0.939227i	$0.611543\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	9.08564i		−	0.476872i
$364$		0			0
$365$	−10.2017	−	2.81173i	−0.533981	−	0.147172i
$366$		0			0
$367$		−	27.4041i		−	1.43048i	−0.698878	−	0.715241i	$-0.746317\pi$
$367$		−	27.4041i		−	1.43048i	0.698878	−	0.715241i	$-0.253683\pi$
$368$		0			0
$369$	2.11605			0.110157
$370$		0			0
$371$	−29.8174			−1.54804
$372$		0			0
$373$			19.9573i			1.03335i	0.856181	+	0.516675i	$0.172831\pi$
$373$			19.9573i			1.03335i	−0.856181	+	0.516675i	$0.827169\pi$
$374$		0			0
$375$	7.73462	+	8.07314i	0.399414	+	0.416895i
$376$		0			0
$377$		−	7.49966i		−	0.386252i
$378$		0			0
$379$	−29.2042			−1.50012			−0.750060	−	0.661370i	$-0.769975\pi$
$379$	−29.2042			−1.50012			−0.750060	+	0.661370i	$0.769975\pi$
$380$		0			0
$381$	11.1231			0.569854
$382$		0			0
$383$		−	15.5764i		−	0.795918i	−0.917403	−	0.397959i	$-0.869719\pi$
$383$		−	15.5764i		−	0.795918i	0.917403	−	0.397959i	$-0.130281\pi$
$384$		0			0
$385$	14.6977	+	4.05088i	0.749064	+	0.206452i
$386$		0			0
$387$			3.49966i			0.177897i
$388$		0			0
$389$	−20.0529			−1.01672			−0.508361	−	0.861144i	$-0.669749\pi$
$389$	−20.0529			−1.01672			−0.508361	+	0.861144i	$0.669749\pi$
$390$		0			0
$391$	0.428810			0.0216859
$392$		0			0
$393$			1.82080i			0.0918470i
$394$		0			0
$395$	4.79560	−	17.3997i	0.241293	−	0.875475i
$396$		0			0
$397$			30.9057i			1.55112i	0.631277	+	0.775558i	$0.282531\pi$
$397$			30.9057i			1.55112i	−0.631277	+	0.775558i	$0.717469\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.9902			−0.598764			−0.299382	−	0.954133i	$-0.596780\pi$
$401$	−11.9902			−0.598764			−0.299382	+	0.954133i	$0.596780\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.594137	+	2.15569i	−0.0295229	+	0.107117i
$406$		0			0
$407$		−	8.37202i		−	0.414986i
$408$		0			0
$409$	−39.2169			−1.93915			−0.969577	−	0.244788i	$-0.921282\pi$
$409$	−39.2169			−1.93915			−0.969577	+	0.244788i	$0.921282\pi$
$410$		0			0
$411$	−8.70204			−0.429240
$412$		0			0
$413$		−	46.4910i		−	2.28767i
$414$		0			0
$415$	−28.8857	−	7.96128i	−1.41794	−	0.390804i
$416$		0			0
$417$			6.81809i			0.333883i
$418$		0			0
$419$	−17.8621			−0.872621			−0.436310	−	0.899796i	$-0.643715\pi$
$419$	−17.8621			−0.872621			−0.436310	+	0.899796i	$0.643715\pi$
$420$		0			0
$421$	28.8690			1.40699			0.703494	−	0.710701i	$-0.251622\pi$
$421$	28.8690			1.40699			0.703494	+	0.710701i	$0.251622\pi$
$422$		0			0
$423$		−	2.31138i		−	0.112383i
$424$		0			0
$425$	−0.500344	+	0.838741i	−0.0242703	+	0.0406849i
$426$		0			0
$427$			18.2081i			0.881150i
$428$		0			0
$429$	−1.38360			−0.0668010
$430$		0			0
$431$	−17.0373			−0.820657			−0.410329	−	0.911938i	$-0.634586\pi$
$431$	−17.0373			−0.820657			−0.410329	+	0.911938i	$0.634586\pi$
$432$		0			0
$433$		−	29.1554i		−	1.40112i	−0.713595	−	0.700558i	$-0.752934\pi$
$433$		−	29.1554i		−	1.40112i	0.713595	−	0.700558i	$-0.247066\pi$
$434$		0			0
$435$	16.1669	+	4.45583i	0.775145	+	0.213641i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−23.2087			−1.10769			−0.553847	−	0.832619i	$-0.686841\pi$
$439$	−23.2087			−1.10769			−0.553847	+	0.832619i	$0.686841\pi$
$440$		0			0
$441$	17.2830			0.822999
$442$		0			0
$443$			14.9484i			0.710221i	0.934824	+	0.355111i	$0.115557\pi$
$443$			14.9484i			0.710221i	−0.934824	+	0.355111i	$0.884443\pi$
$444$		0			0
$445$	1.79887	−	6.52679i	0.0852748	−	0.309400i
$446$		0			0
$447$			7.48537i			0.354046i
$448$		0			0
$449$	42.2023			1.99165			0.995826	−	0.0912763i	$-0.0290947\pi$
$449$	42.2023			1.99165			0.995826	+	0.0912763i	$0.0290947\pi$
$450$		0			0
$451$	−2.92778			−0.137864
$452$		0			0
$453$		−	12.8549i		−	0.603974i
$454$		0			0
$455$	−2.92778	+	10.6228i	−0.137256	+	0.498003i
$456$		0			0
$457$			15.1599i			0.709149i	0.935028	+	0.354575i	$0.115374\pi$
$457$			15.1599i			0.709149i	−0.935028	+	0.354575i	$0.884626\pi$
$458$		0			0
$459$	−0.195329			−0.00911716
$460$		0			0
$461$	−0.848501			−0.0395186			−0.0197593	−	0.999805i	$-0.506290\pi$
$461$	−0.848501			−0.0395186			−0.0197593	+	0.999805i	$0.506290\pi$
$462$		0			0
$463$		−	13.8109i		−	0.641845i	−0.947105	−	0.320922i	$-0.896007\pi$
$463$		−	13.8109i		−	0.641845i	0.947105	−	0.320922i	$-0.103993\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.5165i			0.486644i	0.969946	+	0.243322i	$0.0782372\pi$
$467$			10.5165i			0.486644i	−0.969946	+	0.243322i	$0.921763\pi$
$468$		0			0
$469$	−62.2022			−2.87223
$470$		0			0
$471$	−17.7318			−0.817036
$472$		0			0
$473$		−	4.84214i		−	0.222642i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			6.05088i			0.277051i
$478$		0			0
$479$	−6.41905			−0.293294			−0.146647	−	0.989189i	$-0.546848\pi$
$479$	−6.41905			−0.293294			−0.146647	+	0.989189i	$0.546848\pi$
$480$		0			0
$481$	6.05088			0.275897
$482$		0			0
$483$			10.8181i			0.492240i
$484$		0			0
$485$	12.9689	+	3.57440i	0.588887	+	0.162305i
$486$		0			0
$487$			23.5492i			1.06711i	0.845764	+	0.533557i	$0.179145\pi$
$487$			23.5492i			1.06711i	−0.845764	+	0.533557i	$0.820855\pi$
$488$		0			0
$489$	18.1385			0.820253
$490$		0			0
$491$	−3.99346			−0.180222			−0.0901111	−	0.995932i	$-0.528722\pi$
$491$	−3.99346			−0.180222			−0.0901111	+	0.995932i	$0.528722\pi$
$492$		0			0
$493$			1.46490i			0.0659756i
$494$		0			0
$495$	0.822051	−	2.98262i	0.0369484	−	0.134059i
$496$		0			0
$497$			68.8490i			3.08830i
$498$		0			0
$499$	−5.30231			−0.237364			−0.118682	−	0.992932i	$-0.537867\pi$
$499$	−5.30231			−0.237364			−0.118682	+	0.992932i	$0.537867\pi$
$500$		0			0
$501$	4.06517			0.181618
$502$		0			0
$503$		−	34.8408i		−	1.55347i	−0.629826	−	0.776736i	$-0.716874\pi$
$503$		−	34.8408i		−	1.55347i	0.629826	−	0.776736i	$-0.283126\pi$
$504$		0			0
$505$	1.84717	−	6.70204i	0.0821981	−	0.298237i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	−42.2526			−1.87281			−0.936406	−	0.350918i	$-0.885870\pi$
$509$	−42.2526			−1.87281			−0.936406	+	0.350918i	$0.885870\pi$
$510$		0			0
$511$	23.3205			1.03164
$512$		0			0
$513$		0			0
$514$		0			0
$515$	36.3643	+	10.0225i	1.60240	+	0.441644i
$516$		0			0
$517$			3.19803i			0.140649i
$518$		0			0
$519$	23.2455			1.02037
$520$		0			0
$521$	−2.65098			−0.116141			−0.0580707	−	0.998312i	$-0.518495\pi$
$521$	−2.65098			−0.116141			−0.0580707	+	0.998312i	$0.518495\pi$
$522$		0			0
$523$			31.6014i			1.38183i	0.722934	+	0.690917i	$0.242793\pi$
$523$			31.6014i			1.38183i	−0.722934	+	0.690917i	$0.757207\pi$
$524$		0			0
$525$	−21.1599	−	12.6228i	−0.923493	−	0.550902i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	18.1805			0.790458
$530$		0			0
$531$	−9.43449			−0.409422
$532$		0			0
$533$		−	2.11605i		−	0.0916564i
$534$		0			0
$535$	37.0663	+	10.2160i	1.60252	+	0.441675i
$536$		0			0
$537$			14.1224i			0.609427i
$538$		0			0
$539$	−23.9128			−1.03000
$540$		0			0
$541$	1.54852			0.0665761			0.0332881	−	0.999446i	$-0.489402\pi$
$541$	1.54852			0.0665761			0.0332881	+	0.999446i	$0.489402\pi$
$542$		0			0
$543$			3.67433i			0.157681i
$544$		0			0
$545$	5.20893	−	18.8994i	0.223126	−	0.809561i
$546$		0			0
$547$		−	33.2376i		−	1.42114i	−0.703628	−	0.710569i	$-0.748438\pi$
$547$		−	33.2376i		−	1.42114i	0.703628	−	0.710569i	$-0.251562\pi$
$548$		0			0
$549$	3.69498			0.157698
$550$		0			0
$551$		0			0
$552$		0			0
$553$			39.7747i			1.69139i
$554$		0			0
$555$	−3.59506	+	13.0438i	−0.152602	+	0.553680i
$556$		0			0
$557$		−	1.64594i		−	0.0697407i	−0.999392	−	0.0348703i	$-0.988898\pi$
$557$		−	1.64594i		−	0.0697407i	0.999392	−	0.0348703i	$-0.0111018\pi$
$558$		0			0
$559$	3.49966			0.148020
$560$		0			0
$561$	0.270257			0.0114103
$562$		0			0
$563$			18.7486i			0.790158i	0.918647	+	0.395079i	$0.129283\pi$
$563$			18.7486i			0.790158i	−0.918647	+	0.395079i	$0.870717\pi$
$564$		0			0
$565$	−16.0920	−	4.43518i	−0.676996	−	0.186589i
$566$		0			0
$567$		−	4.92778i		−	0.206947i
$568$		0			0
$569$	41.6543			1.74624			0.873120	−	0.487505i	$-0.162093\pi$
$569$	41.6543			1.74624			0.873120	+	0.487505i	$0.162093\pi$
$570$		0			0
$571$	27.3184			1.14324			0.571620	−	0.820518i	$-0.306315\pi$
$571$	27.3184			1.14324			0.571620	+	0.820518i	$0.306315\pi$
$572$		0			0
$573$			16.8549i			0.704122i
$574$		0			0
$575$	5.62345	−	9.42674i	0.234514	−	0.393122i
$576$		0			0
$577$			24.0715i			1.00211i	0.865415	+	0.501056i	$0.167055\pi$
$577$			24.0715i			1.00211i	−0.865415	+	0.501056i	$0.832945\pi$
$578$		0			0
$579$	−6.53712			−0.271673
$580$		0			0
$581$	66.0309			2.73942
$582$		0			0
$583$		−	8.37202i		−	0.346734i
$584$		0			0
$585$	2.15569	+	0.594137i	0.0891268	+	0.0245646i
$586$		0			0
$587$			23.6872i			0.977677i	0.872374	+	0.488839i	$0.162579\pi$
$587$			23.6872i			0.977677i	−0.872374	+	0.488839i	$0.837421\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	25.7890			1.06082
$592$		0			0
$593$			38.8632i			1.59592i	0.602709	+	0.797961i	$0.294088\pi$
$593$			38.8632i			1.59592i	−0.602709	+	0.797961i	$0.705912\pi$
$594$		0			0
$595$	0.571878	−	2.07493i	0.0234447	−	0.0850638i
$596$		0			0
$597$		−	16.9646i		−	0.694313i
$598$		0			0
$599$	18.4783			0.755003			0.377502	−	0.926009i	$-0.376783\pi$
$599$	18.4783			0.755003			0.377502	+	0.926009i	$0.376783\pi$
$600$		0			0
$601$	−2.08702			−0.0851313			−0.0425656	−	0.999094i	$-0.513553\pi$
$601$	−2.08702			−0.0851313			−0.0425656	+	0.999094i	$0.513553\pi$
$602$		0			0
$603$			12.6228i			0.514039i
$604$		0			0
$605$	5.39812	−	19.5858i	0.219465	−	0.796277i
$606$		0			0
$607$		−	3.59120i		−	0.145762i	−0.997341	−	0.0728812i	$-0.976781\pi$
$607$		−	3.59120i		−	0.145762i	0.997341	−	0.0728812i	$-0.0232194\pi$
$608$		0			0
$609$	−36.9566			−1.49756
$610$		0			0
$611$	−2.31138			−0.0935084
$612$		0			0
$613$		−	24.7895i		−	1.00124i	−0.865667	−	0.500620i	$-0.833106\pi$
$613$		−	24.7895i		−	1.00124i	0.865667	−	0.500620i	$-0.166894\pi$
$614$		0			0
$615$	4.56155	+	1.25723i	0.183940	+	0.0506962i
$616$		0			0
$617$			14.7574i			0.594112i	0.954860	+	0.297056i	$0.0960049\pi$
$617$			14.7574i			0.594112i	−0.954860	+	0.297056i	$0.903995\pi$
$618$		0			0
$619$	25.4081			1.02124			0.510619	−	0.859807i	$-0.329416\pi$
$619$	25.4081			1.02124			0.510619	+	0.859807i	$0.329416\pi$
$620$		0			0
$621$	2.19533			0.0880955
$622$		0			0
$623$			14.9199i			0.597751i
$624$		0			0
$625$	11.8769	+	21.9986i	0.475076	+	0.879945i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.18191			−0.0471258
$630$		0			0
$631$	−9.46490			−0.376792			−0.188396	−	0.982093i	$-0.560329\pi$
$631$	−9.46490			−0.376792			−0.188396	+	0.982093i	$0.560329\pi$
$632$		0			0
$633$			26.4577i			1.05160i
$634$		0			0
$635$	23.9780	+	6.60865i	0.951537	+	0.262256i
$636$		0			0
$637$		−	17.2830i		−	0.684777i
$638$		0			0
$639$	13.9716			0.552708
$640$		0			0
$641$	7.89686			0.311907			0.155954	−	0.987764i	$-0.450155\pi$
$641$	7.89686			0.311907			0.155954	+	0.987764i	$0.450155\pi$
$642$		0			0
$643$		−	16.1204i		−	0.635727i	−0.948137	−	0.317863i	$-0.897035\pi$
$643$		−	16.1204i		−	0.635727i	0.948137	−	0.317863i	$-0.102965\pi$
$644$		0			0
$645$	−2.07928	+	7.54417i	−0.0818714	+	0.297052i
$646$		0			0
$647$			2.99616i			0.117791i	0.998264	+	0.0588956i	$0.0187579\pi$
$647$			2.99616i			0.117791i	−0.998264	+	0.0588956i	$0.981242\pi$
$648$		0			0
$649$	13.0536			0.512398
$650$		0			0
$651$		0			0
$652$		0			0
$653$			21.5330i			0.842653i	0.906909	+	0.421326i	$0.138435\pi$
$653$			21.5330i			0.842653i	−0.906909	+	0.421326i	$0.861565\pi$
$654$		0			0
$655$	−1.08180	+	3.92507i	−0.0422695	+	0.153365i
$656$		0			0
$657$		−	4.73245i		−	0.184630i
$658$		0			0
$659$	17.4429			0.679478			0.339739	−	0.940520i	$-0.389661\pi$
$659$	17.4429			0.679478			0.339739	+	0.940520i	$0.389661\pi$
$660$		0			0
$661$	−22.4642			−0.873756			−0.436878	−	0.899521i	$-0.643916\pi$
$661$	−22.4642			−0.873756			−0.436878	+	0.899521i	$0.643916\pi$
$662$		0			0
$663$			0.195329i			0.00758593i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	16.4642i		−	0.637497i
$668$		0			0
$669$	−2.02065			−0.0781229
$670$		0			0
$671$	−5.11239			−0.197362
$672$		0			0
$673$		−	46.8122i		−	1.80448i	−0.431237	−	0.902239i	$-0.641923\pi$
$673$		−	46.8122i		−	1.80448i	0.431237	−	0.902239i	$-0.358077\pi$
$674$		0			0
$675$	−2.56155	+	4.29400i	−0.0985942	+	0.165276i
$676$		0			0
$677$			31.2682i			1.20173i	0.799349	+	0.600867i	$0.205178\pi$
$677$			31.2682i			1.20173i	−0.799349	+	0.600867i	$0.794822\pi$
$678$		0			0
$679$	−29.6461			−1.13771
$680$		0			0
$681$	17.7904			0.681729
$682$		0			0
$683$		−	44.1797i		−	1.69049i	−0.534381	−	0.845244i	$-0.679455\pi$
$683$		−	44.1797i		−	1.69049i	0.534381	−	0.845244i	$-0.320545\pi$
$684$		0			0
$685$	−18.7589	−	5.17021i	−0.716741	−	0.197543i
$686$		0			0
$687$			18.3339i			0.699481i
$688$		0			0
$689$	6.05088			0.230520
$690$		0			0
$691$	25.8360			0.982849			0.491425	−	0.870920i	$-0.336476\pi$
$691$	25.8360			0.982849			0.491425	+	0.870920i	$0.336476\pi$
$692$		0			0
$693$			6.81809i			0.258998i
$694$		0			0
$695$	−4.05088	+	14.6977i	−0.153659	+	0.557515i
$696$		0			0
$697$			0.413325i			0.0156558i
$698$		0			0
$699$	23.0361			0.871305
$700$		0			0
$701$	15.5887			0.588777			0.294388	−	0.955686i	$-0.404884\pi$
$701$	15.5887			0.588777			0.294388	+	0.955686i	$0.404884\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	1.37328	−	4.98262i	0.0517206	−	0.187656i
$706$		0			0
$707$			15.3205i			0.576185i
$708$		0			0
$709$	4.05541			0.152304			0.0761521	−	0.997096i	$-0.475737\pi$
$709$	4.05541			0.152304			0.0761521	+	0.997096i	$0.475737\pi$
$710$		0			0
$711$	8.07153			0.302706
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−2.98262	−	0.822051i	−0.111544	−	0.0307430i
$716$		0			0
$717$		−	0.986402i		−	0.0368379i
$718$		0			0
$719$	13.4081			0.500038			0.250019	−	0.968241i	$-0.419563\pi$
$719$	13.4081			0.500038			0.250019	+	0.968241i	$0.419563\pi$
$720$		0			0
$721$	−83.1265			−3.09579
$722$		0			0
$723$			1.52031i			0.0565408i
$724$		0			0
$725$	32.2035	+	19.2108i	1.19601	+	0.713470i
$726$		0			0
$727$		−	12.9787i		−	0.481352i	−0.970606	−	0.240676i	$-0.922631\pi$
$727$		−	12.9787i		−	0.481352i	0.970606	−	0.240676i	$-0.0773691\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−0.683583			−0.0252832
$732$		0			0
$733$		−	49.2410i		−	1.81876i	−0.415969	−	0.909379i	$-0.636557\pi$
$733$		−	49.2410i		−	1.81876i	0.415969	−	0.909379i	$-0.363443\pi$
$734$		0			0
$735$	37.2568	+	10.2685i	1.37424	+	0.378758i
$736$		0			0
$737$		−	17.4649i		−	0.643328i
$738$		0			0
$739$	−11.3359			−0.416999			−0.208500	−	0.978022i	$-0.566858\pi$
$739$	−11.3359			−0.416999			−0.208500	+	0.978022i	$0.566858\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			8.07790i			0.296349i	0.988961	+	0.148175i	$0.0473398\pi$
$743$			8.07790i			0.296349i	−0.988961	+	0.148175i	$0.952660\pi$
$744$		0			0
$745$	−4.44734	+	16.1361i	−0.162938	+	0.591182i
$746$		0			0
$747$		−	13.3997i		−	0.490270i
$748$		0			0
$749$	−84.7314			−3.09602
$750$		0			0
$751$	−18.2403			−0.665598			−0.332799	−	0.942998i	$-0.607993\pi$
$751$	−18.2403			−0.665598			−0.332799	+	0.942998i	$0.607993\pi$
$752$		0			0
$753$		−	11.6763i		−	0.425510i
$754$		0			0
$755$	7.63756	−	27.7111i	0.277959	−	1.00851i
$756$		0			0
$757$		−	5.71111i		−	0.207574i	−0.994600	−	0.103787i	$-0.966904\pi$
$757$		−	5.71111i		−	0.207574i	0.994600	−	0.103787i	$-0.0330960\pi$
$758$		0			0
$759$	−3.03746			−0.110253
$760$		0			0
$761$	12.6543			0.458718			0.229359	−	0.973342i	$-0.426337\pi$
$761$	12.6543			0.458718			0.229359	+	0.973342i	$0.426337\pi$
$762$		0			0
$763$			43.2028i			1.56405i
$764$		0			0
$765$	−0.421068	−	0.116052i	−0.0152237	−	0.00419587i
$766$		0			0
$767$			9.43449i			0.340660i
$768$		0			0
$769$	21.4222			0.772505			0.386252	−	0.922393i	$-0.373769\pi$
$769$	21.4222			0.772505			0.386252	+	0.922393i	$0.373769\pi$
$770$		0			0
$771$	−3.00069			−0.108067
$772$		0			0
$773$			36.0547i			1.29680i	0.761300	+	0.648399i	$0.224561\pi$
$773$			36.0547i			1.29680i	−0.761300	+	0.648399i	$0.775439\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	29.8174i		−	1.06969i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−19.3312			−0.691723
$782$		0			0
$783$			7.49966i			0.268016i
$784$		0			0
$785$	−38.2242	−	10.5351i	−1.36428	−	0.376014i
$786$		0			0
$787$		−	40.8394i		−	1.45577i	−0.685701	−	0.727883i	$-0.740504\pi$
$787$		−	40.8394i		−	1.45577i	0.685701	−	0.727883i	$-0.259496\pi$
$788$		0			0
$789$	25.8683			0.920935
$790$		0			0
$791$	36.7853			1.30794
$792$		0			0
$793$		−	3.69498i		−	0.131213i
$794$		0			0
$795$	−3.59506	+	13.0438i	−0.127504	+	0.462617i
$796$		0			0
$797$			25.4691i			0.902161i	0.892483	+	0.451080i	$0.148961\pi$
$797$			25.4691i			0.902161i	−0.892483	+	0.451080i	$0.851039\pi$
$798$		0			0
$799$	0.451479			0.0159722
$800$		0			0
$801$	3.02770			0.106979
$802$		0			0
$803$			6.54783i			0.231068i
$804$		0			0
$805$	−6.42743	+	23.3205i	−0.226537	+	0.821938i
$806$		0			0
$807$		−	11.1090i		−	0.391055i
$808$		0			0
$809$	−15.7252			−0.552869			−0.276435	−	0.961033i	$-0.589153\pi$
$809$	−15.7252			−0.552869			−0.276435	+	0.961033i	$0.589153\pi$
$810$		0			0
$811$	49.6815			1.74455			0.872277	−	0.489012i	$-0.162643\pi$
$811$	49.6815			1.74455			0.872277	+	0.489012i	$0.162643\pi$
$812$		0			0
$813$			13.9859i			0.490507i
$814$		0			0
$815$	39.1011	+	10.7768i	1.36965	+	0.377494i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−4.92778			−0.172190
$820$		0			0
$821$	23.9936			0.837384			0.418692	−	0.908128i	$-0.362489\pi$
$821$	23.9936			0.837384			0.418692	+	0.908128i	$0.362489\pi$
$822$		0			0
$823$			23.0457i			0.803321i	0.915789	+	0.401661i	$0.131567\pi$
$823$			23.0457i			0.803321i	−0.915789	+	0.401661i	$0.868433\pi$
$824$		0			0
$825$	3.54417	−	5.94120i	0.123392	−	0.206846i
$826$		0			0
$827$			49.1195i			1.70805i	0.520229	+	0.854027i	$0.325847\pi$
$827$			49.1195i			1.70805i	−0.520229	+	0.854027i	$0.674153\pi$
$828$		0			0
$829$	5.54739			0.192669			0.0963344	−	0.995349i	$-0.469288\pi$
$829$	5.54739			0.192669			0.0963344	+	0.995349i	$0.469288\pi$
$830$		0			0
$831$	10.0529			0.348731
$832$		0			0
$833$			3.37586i			0.116967i
$834$		0			0
$835$	8.76325	+	2.41527i	0.303265	+	0.0835838i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−28.7106			−0.991200			−0.495600	−	0.868551i	$-0.665052\pi$
$839$	−28.7106			−0.991200			−0.495600	+	0.868551i	$0.665052\pi$
$840$		0			0
$841$	27.2448			0.939477
$842$		0			0
$843$		−	19.6318i		−	0.676156i
$844$		0			0
$845$	0.594137	−	2.15569i	0.0204390	−	0.0741580i
$846$		0			0
$847$			44.7720i			1.53838i
$848$		0			0
$849$	−25.5667			−0.877446
$850$		0			0
$851$	13.2837			0.455359
$852$		0			0
$853$		−	27.7606i		−	0.950505i	−0.879849	−	0.475253i	$-0.842357\pi$
$853$		−	27.7606i		−	0.950505i	0.879849	−	0.475253i	$-0.157643\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			35.8869i			1.22587i	0.790132	+	0.612937i	$0.210012\pi$
$857$			35.8869i			1.22587i	−0.790132	+	0.612937i	$0.789988\pi$
$858$		0			0
$859$	−26.6107			−0.907944			−0.453972	−	0.891016i	$-0.649993\pi$
$859$	−26.6107			−0.907944			−0.453972	+	0.891016i	$0.649993\pi$
$860$		0			0
$861$	−10.4274			−0.355366
$862$		0			0
$863$			20.0366i			0.682054i	0.940054	+	0.341027i	$0.110775\pi$
$863$			20.0366i			0.682054i	−0.940054	+	0.341027i	$0.889225\pi$
$864$		0			0
$865$	50.1101	+	13.8110i	1.70380	+	0.469589i
$866$		0			0
$867$			16.9618i			0.576055i
$868$		0			0
$869$	−11.1678			−0.378842
$870$		0			0
$871$	12.6228			0.427706
$872$		0			0
$873$			6.01612i			0.203615i
$874$		0			0
$875$	−38.1145	−	39.7826i	−1.28851	−	1.34490i
$876$		0			0
$877$		−	2.15855i		−	0.0728892i	−0.999336	−	0.0364446i	$-0.988397\pi$
$877$		−	2.15855i		−	0.0728892i	0.999336	−	0.0364446i	$-0.0116032\pi$
$878$		0			0
$879$	−6.71614			−0.226530
$880$		0			0
$881$	−14.8421			−0.500044			−0.250022	−	0.968240i	$-0.580438\pi$
$881$	−14.8421			−0.500044			−0.250022	+	0.968240i	$0.580438\pi$
$882$		0			0
$883$		−	13.0742i		−	0.439983i	−0.975502	−	0.219992i	$-0.929397\pi$
$883$		−	13.0742i		−	0.439983i	0.975502	−	0.219992i	$-0.0706030\pi$
$884$		0			0
$885$	−20.3378	−	5.60538i	−0.683649	−	0.188423i
$886$		0			0
$887$		−	39.4254i		−	1.32377i	−0.749604	−	0.661887i	$-0.769756\pi$
$887$		−	39.4254i		−	1.32377i	0.749604	−	0.661887i	$-0.230244\pi$
$888$		0			0
$889$	−54.8122			−1.83834
$890$		0			0
$891$	1.38360			0.0463525
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−8.39066	+	30.4436i	−0.280469	+	1.01762i
$896$		0			0
$897$		−	2.19533i		−	0.0732999i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−1.18191			−0.0393751
$902$		0			0
$903$		−	17.2455i		−	0.573895i
$904$		0			0
$905$	−2.18306	+	7.92072i	−0.0725673	+	0.263294i
$906$		0			0
$907$			49.8397i			1.65490i	0.561539	+	0.827450i	$0.310209\pi$
$907$			49.8397i			1.65490i	−0.561539	+	0.827450i	$0.689791\pi$
$908$		0			0
$909$	3.10900			0.103119
$910$		0			0
$911$	−8.92438			−0.295678			−0.147839	−	0.989011i	$-0.547232\pi$
$911$	−8.92438			−0.295678			−0.147839	+	0.989011i	$0.547232\pi$
$912$		0			0
$913$			18.5399i			0.613581i
$914$		0			0
$915$	7.96524	+	2.19533i	0.263323	+	0.0725753i
$916$		0			0
$917$		−	8.97247i		−	0.296297i
$918$		0			0
$919$	−21.9876			−0.725302			−0.362651	−	0.931925i	$-0.618128\pi$
$919$	−21.9876			−0.725302			−0.362651	+	0.931925i	$0.618128\pi$
$920$		0			0
$921$	−14.2689			−0.470176
$922$		0			0
$923$		−	13.9716i		−	0.459881i
$924$		0			0
$925$	−15.4997	+	25.9825i	−0.509626	+	0.854299i
$926$		0			0
$927$			16.8690i			0.554050i
$928$		0			0
$929$	−4.14289			−0.135924			−0.0679619	−	0.997688i	$-0.521650\pi$
$929$	−4.14289			−0.135924			−0.0679619	+	0.997688i	$0.521650\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			7.15786i			0.234338i
$934$		0			0
$935$	0.582591	+	0.160570i	0.0190528	+	0.00525120i
$936$		0			0
$937$		−	59.2235i		−	1.93475i	−0.253354	−	0.967374i	$-0.581534\pi$
$937$		−	59.2235i		−	1.93475i	0.253354	−	0.967374i	$-0.418466\pi$
$938$		0			0
$939$	−21.6568			−0.706744
$940$		0			0
$941$	11.7615			0.383415			0.191707	−	0.981452i	$-0.438598\pi$
$941$	11.7615			0.383415			0.191707	+	0.981452i	$0.438598\pi$
$942$		0			0
$943$		−	4.64543i		−	0.151276i
$944$		0			0
$945$	2.92778	−	10.6228i	0.0952406	−	0.345558i
$946$		0			0
$947$		−	8.51629i		−	0.276742i	−0.990380	−	0.138371i	$-0.955813\pi$
$947$		−	8.51629i		−	0.276742i	0.990380	−	0.138371i	$-0.0441867\pi$
$948$		0			0
$949$	−4.73245			−0.153622
$950$		0			0
$951$	−29.0232			−0.941141
$952$		0			0
$953$			35.4872i			1.14954i	0.818314	+	0.574772i	$0.194909\pi$
$953$			35.4872i			1.14954i	−0.818314	+	0.574772i	$0.805091\pi$
$954$		0			0
$955$	−10.0141	+	36.3339i	−0.324049	+	1.17574i
$956$		0			0
$957$		−	10.3765i		−	0.335426i
$958$		0			0
$959$	42.8817			1.38472
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$			17.1946i			0.554090i
$964$		0			0
$965$	−14.0920	−	3.88395i	−0.453638	−	0.125029i
$966$		0			0
$967$		−	8.51445i		−	0.273806i	−0.990584	−	0.136903i	$-0.956285\pi$
$967$		−	8.51445i		−	0.273806i	0.990584	−	0.136903i	$-0.0437149\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	50.5013			1.62066			0.810331	−	0.585972i	$-0.199287\pi$
$971$	50.5013			1.62066			0.810331	+	0.585972i	$0.199287\pi$
$972$		0			0
$973$		−	33.5980i		−	1.07710i
$974$		0			0
$975$	4.29400	+	2.56155i	0.137518	+	0.0820353i
$976$		0			0
$977$			39.0540i			1.24945i	0.780845	+	0.624725i	$0.214789\pi$
$977$			39.0540i			1.24945i	−0.780845	+	0.624725i	$0.785211\pi$
$978$		0			0
$979$	−4.18914			−0.133886
$980$		0			0
$981$	8.76721			0.279915
$982$		0			0
$983$		−	54.9882i		−	1.75385i	−0.480627	−	0.876925i	$-0.659591\pi$
$983$		−	54.9882i		−	1.75385i	0.480627	−	0.876925i	$-0.340409\pi$
$984$		0			0
$985$	55.5931	+	15.3222i	1.77134	+	0.488206i
$986$		0			0
$987$			11.3900i			0.362547i
$988$		0			0
$989$	7.68289			0.244302
$990$		0			0
$991$	15.0052			0.476656			0.238328	−	0.971185i	$-0.423401\pi$
$991$	15.0052			0.476656			0.238328	+	0.971185i	$0.423401\pi$
$992$		0			0
$993$		−	21.8683i		−	0.693969i
$994$		0			0
$995$	10.0793	−	36.5703i	0.319535	−	1.15936i
$996$		0			0
$997$		−	22.4113i		−	0.709773i	−0.934909	−	0.354887i	$-0.884519\pi$
$997$		−	22.4113i		−	0.709773i	0.934909	−	0.354887i	$-0.115481\pi$
$998$		0			0
$999$	−6.05088			−0.191441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.n.1249.6		8
4.3	odd	2		1560.2.l.d.1249.2	✓	8
5.4	even	2	inner	3120.2.l.n.1249.2		8
12.11	even	2		4680.2.l.g.2809.5		8
20.3	even	4		7800.2.a.by.1.4		4
20.7	even	4		7800.2.a.bt.1.1		4
20.19	odd	2		1560.2.l.d.1249.6	yes	8
60.59	even	2		4680.2.l.g.2809.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.d.1249.2	✓	8	4.3	odd	2
1560.2.l.d.1249.6	yes	8	20.19	odd	2
3120.2.l.n.1249.2		8	5.4	even	2	inner
3120.2.l.n.1249.6		8	1.1	even	1	trivial
4680.2.l.g.2809.5		8	12.11	even	2
4680.2.l.g.2809.6		8	60.59	even	2
7800.2.a.bt.1.1		4	20.7	even	4
7800.2.a.by.1.4		4	20.3	even	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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.594137 + 2.15569i) q^{5} -4.92778i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-0.594137 + 2.15569i) q^{5} -4.92778i q^{7} -1.00000 q^{9} +1.38360 q^{11} +1.00000i q^{13} +(-2.15569 - 0.594137i) q^{15} -0.195329i q^{17} +4.92778 q^{21} +2.19533i q^{23} +(-4.29400 - 2.56155i) q^{25} -1.00000i q^{27} -7.49966 q^{29} +1.38360i q^{33} +(10.6228 + 2.92778i) q^{35} -6.05088i q^{37} -1.00000 q^{39} -2.11605 q^{41} -3.49966i q^{43} +(0.594137 - 2.15569i) q^{45} +2.31138i q^{47} -17.2830 q^{49} +0.195329 q^{51} -6.05088i q^{53} +(-0.822051 + 2.98262i) q^{55} +9.43449 q^{59} -3.69498 q^{61} +4.92778i q^{63} +(-2.15569 - 0.594137i) q^{65} -12.6228i q^{67} -2.19533 q^{69} -13.9716 q^{71} +4.73245i q^{73} +(2.56155 - 4.29400i) q^{75} -6.81809i q^{77} -8.07153 q^{79} +1.00000 q^{81} +13.3997i q^{83} +(0.421068 + 0.116052i) q^{85} -7.49966i q^{87} -3.02770 q^{89} +4.92778 q^{91} -6.01612i q^{97} -1.38360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^5 - 8 * q^9	\( 8 q - 2 q^{5} - 8 q^{9} - 2 q^{11} + 2 q^{15} + 14 q^{21} - 16 q^{29} + 8 q^{35} - 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} - 6 q^{51} - 10 q^{55} + 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} - 30 q^{71} + 4 q^{75} - 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} + 14 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 2 * q^5 - 8 * q^9 - 2 * q^11 + 2 * q^15 + 14 * q^21 - 16 * q^29 + 8 * q^35 - 8 * q^39 + 14 * q^41 + 2 * q^45 - 18 * q^49 - 6 * q^51 - 10 * q^55 + 4 * q^59 + 22 * q^61 + 2 * q^65 - 10 * q^69 - 30 * q^71 + 4 * q^75 - 2 * q^79 + 8 * q^81 + 24 * q^85 - 18 * q^89 + 14 * q^91 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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