LMFDB - Embedded newform 3120.2.l.h.1249.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.h.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} +4.00000i q^{17} +2.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +2.00000i q^{33} +(-4.00000 - 8.00000i) q^{35} -6.00000i q^{37} -1.00000 q^{39} -6.00000 q^{41} +8.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -8.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -10.0000i q^{53} +(-4.00000 + 2.00000i) q^{55} -2.00000i q^{57} -14.0000 q^{59} +10.0000 q^{61} +4.00000i q^{63} +(-1.00000 - 2.00000i) q^{65} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} -10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +8.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +(4.00000 + 8.00000i) q^{85} -2.00000i q^{87} +18.0000 q^{89} -4.00000 q^{91} -4.00000i q^{93} +(4.00000 - 2.00000i) q^{95} +6.00000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^5 - 2 * q^9	$ 2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 4 q^{29} + 8 q^{31} - 8 q^{35} - 2 q^{39} - 12 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} - 8 q^{55} - 28 q^{59} + 20 q^{61} - 2 q^{65} - 12 q^{69} - 16 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} + 8 q^{85} + 36 q^{89} - 8 q^{91} + 8 q^{95} + 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 + 4 * q^19 - 8 * q^21 + 6 * q^25 + 4 * q^29 + 8 * q^31 - 8 * q^35 - 2 * q^39 - 12 * q^41 - 4 * q^45 - 18 * q^49 + 8 * q^51 - 8 * q^55 - 28 * q^59 + 20 * q^61 - 2 * q^65 - 12 * q^69 - 16 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 + 8 * q^85 + 36 * q^89 - 8 * q^91 + 8 * q^95 + 4 * 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
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	−1.00000	−	2.00000i	−0.258199	−	0.516398i
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	−4.00000	−	8.00000i	−0.676123	−	1.35225i
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$	−2.00000	+	1.00000i	−0.298142	+	0.149071i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$		0			0
$55$	−4.00000	+	2.00000i	−0.539360	+	0.269680i
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$	−14.0000			−1.82264			−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000			−1.82264			−0.911322	+	0.411693i	$0.864937\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$			4.00000i			0.503953i
$64$		0			0
$65$	−1.00000	−	2.00000i	−0.124035	−	0.248069i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$		0			0
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$			8.00000i			0.911685i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$	4.00000	+	8.00000i	0.433861	+	0.867722i
$86$		0			0
$87$		−	2.00000i		−	0.214423i
$88$		0			0
$89$	18.0000			1.90800			0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000			1.90800			0.953998	+	0.299813i	$0.0969242\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		−	4.00000i		−	0.414781i
$94$		0			0
$95$	4.00000	−	2.00000i	0.410391	−	0.205196i
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−14.0000			−1.39305			−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000			−1.39305			−0.696526	+	0.717532i	$0.745272\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$	−8.00000	+	4.00000i	−0.780720	+	0.390360i
$106$		0			0
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$		0			0
$109$	−12.0000			−1.14939			−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000			−1.14939			−0.574696	+	0.818367i	$0.694880\pi$
$110$		0			0
$111$	−6.00000			−0.569495
$112$		0			0
$113$			12.0000i			1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$			12.0000i			1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$		0			0
$115$	−6.00000	−	12.0000i	−0.559503	−	1.11901i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	16.0000			1.46672
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$			6.00000i			0.541002i
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$		0			0
$135$	1.00000	+	2.00000i	0.0860663	+	0.172133i
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$	16.0000			1.35710			0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000			1.35710			0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$			2.00000i			0.167248i
$144$		0			0
$145$	4.00000	−	2.00000i	0.332182	−	0.166091i
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$	8.00000	−	4.00000i	0.642575	−	0.321288i
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$		0			0
$159$	−10.0000			−0.793052
$160$		0			0
$161$	−24.0000			−1.89146
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$	2.00000	+	4.00000i	0.155700	+	0.311400i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−2.00000			−0.152944
$172$		0			0
$173$		−	10.0000i		−	0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$		−	10.0000i		−	0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$		0			0
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$		0			0
$177$			14.0000i			1.05230i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$	−6.00000	−	12.0000i	−0.441129	−	0.882258i
$186$		0			0
$187$		−	8.00000i		−	0.585018i
$188$		0			0
$189$	4.00000			0.290957
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	6.00000i		−	0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$		−	6.00000i		−	0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$		0			0
$195$	−2.00000	+	1.00000i	−0.143223	+	0.0716115i
$196$		0			0
$197$			26.0000i			1.85242i	0.377004	+	0.926212i	$0.376954\pi$
$197$			26.0000i			1.85242i	−0.377004	+	0.926212i	$0.623046\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$		−	8.00000i		−	0.561490i
$204$		0			0
$205$	−12.0000	+	6.00000i	−0.838116	+	0.419058i
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	28.0000			1.92760			0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000			1.92760			0.963800	+	0.266627i	$0.0859092\pi$
$212$		0			0
$213$			8.00000i			0.548151i
$214$		0			0
$215$	8.00000	+	16.0000i	0.545595	+	1.09119i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$	−10.0000			−0.675737
$220$		0			0
$221$	4.00000			0.269069
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	16.0000			1.05731			0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000			1.05731			0.528655	+	0.848837i	$0.322697\pi$
$230$		0			0
$231$	8.00000			0.526361
$232$		0			0
$233$			4.00000i			0.262049i	0.991379	+	0.131024i	$0.0418266\pi$
$233$			4.00000i			0.262049i	−0.991379	+	0.131024i	$0.958173\pi$
$234$		0			0
$235$	−8.00000	−	16.0000i	−0.521862	−	1.04372i
$236$		0			0
$237$			8.00000i			0.519656i
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−30.0000			−1.93247			−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000			−1.93247			−0.966235	+	0.257663i	$0.917048\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−18.0000	+	9.00000i	−1.14998	+	0.574989i
$246$		0			0
$247$		−	2.00000i		−	0.127257i
$248$		0			0
$249$	12.0000			0.760469
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$		0			0
$255$	8.00000	−	4.00000i	0.500979	−	0.250490i
$256$		0			0
$257$		−	12.0000i		−	0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$		−	12.0000i		−	0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$		0			0
$259$	−24.0000			−1.49129
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$		−	10.0000i		−	0.616626i	−0.951285	−	0.308313i	$-0.900236\pi$
$263$		−	10.0000i		−	0.616626i	0.951285	−	0.308313i	$-0.0997645\pi$
$264$		0			0
$265$	−10.0000	−	20.0000i	−0.614295	−	1.22859i
$266$		0			0
$267$		−	18.0000i		−	1.10158i
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$			4.00000i			0.242091i
$274$		0			0
$275$	−6.00000	+	8.00000i	−0.361814	+	0.482418i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	−4.00000			−0.239474
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		0			0
$285$	−2.00000	−	4.00000i	−0.118470	−	0.236940i
$286$		0			0
$287$			24.0000i			1.41668i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	6.00000			0.351726
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	−28.0000	+	14.0000i	−1.63022	+	0.815112i
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	32.0000			1.84445
$302$		0			0
$303$			14.0000i			0.804279i
$304$		0			0
$305$	20.0000	−	10.0000i	1.14520	−	0.572598i
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$		0			0
$309$	6.00000			0.341328
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$		0			0
$315$	4.00000	+	8.00000i	0.225374	+	0.450749i
$316$		0			0
$317$		−	30.0000i		−	1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$		−	30.0000i		−	1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$		0			0
$319$	−4.00000			−0.223957
$320$		0			0
$321$	8.00000			0.446516
$322$		0			0
$323$			8.00000i			0.445132i
$324$		0			0
$325$	−4.00000	−	3.00000i	−0.221880	−	0.166410i
$326$		0			0
$327$			12.0000i			0.663602i
$328$		0			0
$329$	−32.0000			−1.76422
$330$		0			0
$331$	−18.0000			−0.989369			−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000			−0.989369			−0.494685	+	0.869072i	$0.664716\pi$
$332$		0			0
$333$			6.00000i			0.328798i
$334$		0			0
$335$	−4.00000	−	8.00000i	−0.218543	−	0.437087i
$336$		0			0
$337$		−	32.0000i		−	1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$		−	32.0000i		−	1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$		0			0
$339$	12.0000			0.651751
$340$		0			0
$341$	−8.00000			−0.433224
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$	−12.0000	+	6.00000i	−0.646058	+	0.323029i
$346$		0			0
$347$			24.0000i			1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$			24.0000i			1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$		0			0
$349$	28.0000			1.49881			0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000			1.49881			0.749403	+	0.662114i	$0.230341\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			26.0000i			1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$		0			0
$355$	−16.0000	+	8.00000i	−0.849192	+	0.424596i
$356$		0			0
$357$		−	16.0000i		−	0.846810i
$358$		0			0
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$			7.00000i			0.367405i
$364$		0			0
$365$	−10.0000	−	20.0000i	−0.523424	−	1.04685i
$366$		0			0
$367$		−	22.0000i		−	1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$		−	22.0000i		−	1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	−40.0000			−2.07670
$372$		0			0
$373$			2.00000i			0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$			2.00000i			0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$		0			0
$375$	−11.0000	−	2.00000i	−0.568038	−	0.103280i
$376$		0			0
$377$		−	2.00000i		−	0.103005i
$378$		0			0
$379$	26.0000			1.33553			0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000			1.33553			0.667765	+	0.744372i	$0.267251\pi$
$380$		0			0
$381$	18.0000			0.922168
$382$		0			0
$383$		−	12.0000i		−	0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$		−	12.0000i		−	0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$		0			0
$385$	8.00000	+	16.0000i	0.407718	+	0.815436i
$386$		0			0
$387$		−	8.00000i		−	0.406663i
$388$		0			0
$389$	26.0000			1.31825			0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000			1.31825			0.659126	+	0.752032i	$0.270926\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.0000	+	8.00000i	−0.805047	+	0.402524i
$396$		0			0
$397$			6.00000i			0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$			6.00000i			0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$		0			0
$399$	−8.00000			−0.400501
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$		0			0
$405$	2.00000	−	1.00000i	0.0993808	−	0.0496904i
$406$		0			0
$407$			12.0000i			0.594818i
$408$		0			0
$409$	2.00000			0.0988936			0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000			0.0988936			0.0494468	+	0.998777i	$0.484254\pi$
$410$		0			0
$411$	2.00000			0.0986527
$412$		0			0
$413$			56.0000i			2.75558i
$414$		0			0
$415$	12.0000	+	24.0000i	0.589057	+	1.17811i
$416$		0			0
$417$		−	16.0000i		−	0.783523i
$418$		0			0
$419$	16.0000			0.781651			0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000			0.781651			0.390826	+	0.920465i	$0.372190\pi$
$420$		0			0
$421$	28.0000			1.36464			0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000			1.36464			0.682318	+	0.731055i	$0.260972\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	16.0000	+	12.0000i	0.776114	+	0.582086i
$426$		0			0
$427$		−	40.0000i		−	1.93574i
$428$		0			0
$429$	2.00000			0.0965609
$430$		0			0
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−2.00000	−	4.00000i	−0.0958927	−	0.191785i
$436$		0			0
$437$		−	12.0000i		−	0.574038i
$438$		0			0
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$			8.00000i			0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$			8.00000i			0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$		0			0
$445$	36.0000	−	18.0000i	1.70656	−	0.853282i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$	−8.00000	+	4.00000i	−0.375046	+	0.187523i
$456$		0			0
$457$		−	26.0000i		−	1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$		−	26.0000i		−	1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$		0			0
$459$	−4.00000			−0.186704
$460$		0			0
$461$	−4.00000			−0.186299			−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000			−0.186299			−0.0931493	+	0.995652i	$0.529693\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$		0			0
$465$	−4.00000	−	8.00000i	−0.185496	−	0.370991i
$466$		0			0
$467$			16.0000i			0.740392i	0.928954	+	0.370196i	$0.120709\pi$
$467$			16.0000i			0.740392i	−0.928954	+	0.370196i	$0.879291\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	−10.0000			−0.460776
$472$		0			0
$473$		−	16.0000i		−	0.735681i
$474$		0			0
$475$	6.00000	−	8.00000i	0.275299	−	0.367065i
$476$		0			0
$477$			10.0000i			0.457869i
$478$		0			0
$479$	−12.0000			−0.548294			−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000			−0.548294			−0.274147	+	0.961688i	$0.588395\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$		0			0
$483$			24.0000i			1.09204i
$484$		0			0
$485$	6.00000	+	12.0000i	0.272446	+	0.544892i
$486$		0			0
$487$			40.0000i			1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$			40.0000i			1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$			8.00000i			0.360302i
$494$		0			0
$495$	4.00000	−	2.00000i	0.179787	−	0.0898933i
$496$		0			0
$497$			32.0000i			1.43540i
$498$		0			0
$499$	30.0000			1.34298			0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000			1.34298			0.671492	+	0.741012i	$0.265654\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			18.0000i			0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$			18.0000i			0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$		0			0
$505$	−28.0000	+	14.0000i	−1.24598	+	0.622992i
$506$		0			0
$507$			1.00000i			0.0444116i
$508$		0			0
$509$	16.0000			0.709188			0.354594	−	0.935020i	$-0.384619\pi$
$509$	16.0000			0.709188			0.354594	+	0.935020i	$0.384619\pi$
$510$		0			0
$511$	−40.0000			−1.76950
$512$		0			0
$513$			2.00000i			0.0883022i
$514$		0			0
$515$	6.00000	+	12.0000i	0.264392	+	0.528783i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$	−10.0000			−0.438951
$520$		0			0
$521$	38.0000			1.66481			0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000			1.66481			0.832405	+	0.554168i	$0.186963\pi$
$522$		0			0
$523$			8.00000i			0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$			8.00000i			0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$		0			0
$525$	−12.0000	+	16.0000i	−0.523723	+	0.698297i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	14.0000			0.607548
$532$		0			0
$533$			6.00000i			0.259889i
$534$		0			0
$535$	8.00000	+	16.0000i	0.345870	+	0.691740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.0000			0.775315
$540$		0			0
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$		0			0
$543$		−	6.00000i		−	0.257485i
$544$		0			0
$545$	−24.0000	+	12.0000i	−1.02805	+	0.514024i
$546$		0			0
$547$		−	44.0000i		−	1.88130i	−0.339372	−	0.940652i	$-0.610215\pi$
$547$		−	44.0000i		−	1.88130i	0.339372	−	0.940652i	$-0.389785\pi$
$548$		0			0
$549$	−10.0000			−0.426790
$550$		0			0
$551$	4.00000			0.170406
$552$		0			0
$553$			32.0000i			1.36078i
$554$		0			0
$555$	−12.0000	+	6.00000i	−0.509372	+	0.254686i
$556$		0			0
$557$			14.0000i			0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$			14.0000i			0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	−8.00000			−0.337760
$562$		0			0
$563$		−	16.0000i		−	0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$563$		−	16.0000i		−	0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$564$		0			0
$565$	12.0000	+	24.0000i	0.504844	+	1.00969i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$	36.0000			1.50655			0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000			1.50655			0.753277	+	0.657704i	$0.228472\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000	−	18.0000i	−1.00087	−	0.750652i
$576$		0			0
$577$			26.0000i			1.08239i	0.840896	+	0.541197i	$0.182029\pi$
$577$			26.0000i			1.08239i	−0.840896	+	0.541197i	$0.817971\pi$
$578$		0			0
$579$	−6.00000			−0.249351
$580$		0			0
$581$	48.0000			1.99138
$582$		0			0
$583$			20.0000i			0.828315i
$584$		0			0
$585$	1.00000	+	2.00000i	0.0413449	+	0.0826898i
$586$		0			0
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	8.00000			0.329634
$590$		0			0
$591$	26.0000			1.06950
$592$		0			0
$593$		−	14.0000i		−	0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$		−	14.0000i		−	0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$		0			0
$595$	32.0000	−	16.0000i	1.31187	−	0.655936i
$596$		0			0
$597$		−	24.0000i		−	0.982255i
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$		0			0
$605$	−14.0000	+	7.00000i	−0.569181	+	0.284590i
$606$		0			0
$607$		−	18.0000i		−	0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$		−	18.0000i		−	0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$		0			0
$609$	−8.00000			−0.324176
$610$		0			0
$611$	−8.00000			−0.323645
$612$		0			0
$613$		−	26.0000i		−	1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$		−	26.0000i		−	1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$		0			0
$615$	6.00000	+	12.0000i	0.241943	+	0.483887i
$616$		0			0
$617$		−	34.0000i		−	1.36879i	−0.729112	−	0.684394i	$-0.760067\pi$
$617$		−	34.0000i		−	1.36879i	0.729112	−	0.684394i	$-0.239933\pi$
$618$		0			0
$619$	42.0000			1.68812			0.844061	−	0.536247i	$-0.180158\pi$
$619$	42.0000			1.68812			0.844061	+	0.536247i	$0.180158\pi$
$620$		0			0
$621$	6.00000			0.240772
$622$		0			0
$623$		−	72.0000i		−	2.88462i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			4.00000i			0.159745i
$628$		0			0
$629$	24.0000			0.956943
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$		−	28.0000i		−	1.11290i
$634$		0			0
$635$	18.0000	+	36.0000i	0.714308	+	1.42862i
$636$		0			0
$637$			9.00000i			0.356593i
$638$		0			0
$639$	8.00000			0.316475
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		−	20.0000i		−	0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$		−	20.0000i		−	0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$		0			0
$645$	16.0000	−	8.00000i	0.629999	−	0.315000i
$646$		0			0
$647$			6.00000i			0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$			6.00000i			0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$		0			0
$649$	28.0000			1.09910
$650$		0			0
$651$	−16.0000			−0.627089
$652$		0			0
$653$			26.0000i			1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$			26.0000i			1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			10.0000i			0.390137i
$658$		0			0
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$	24.0000			0.933492			0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000			0.933492			0.466746	+	0.884391i	$0.345426\pi$
$662$		0			0
$663$		−	4.00000i		−	0.155347i
$664$		0			0
$665$	−8.00000	−	16.0000i	−0.310227	−	0.620453i
$666$		0			0
$667$		−	12.0000i		−	0.464642i
$668$		0			0
$669$	−16.0000			−0.618596
$670$		0			0
$671$	−20.0000			−0.772091
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$		0			0
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$		0			0
$677$			22.0000i			0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$			22.0000i			0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	2.00000	+	4.00000i	0.0764161	+	0.152832i
$686$		0			0
$687$		−	16.0000i		−	0.610438i
$688$		0			0
$689$	−10.0000			−0.380970
$690$		0			0
$691$	−2.00000			−0.0760836			−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000			−0.0760836			−0.0380418	+	0.999276i	$0.512112\pi$
$692$		0			0
$693$		−	8.00000i		−	0.303895i
$694$		0			0
$695$	32.0000	−	16.0000i	1.21383	−	0.606915i
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		0			0
$699$	4.00000			0.151294
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$		0			0
$705$	−16.0000	+	8.00000i	−0.602595	+	0.301297i
$706$		0			0
$707$			56.0000i			2.10610i
$708$		0			0
$709$	32.0000			1.20179			0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000			1.20179			0.600893	+	0.799330i	$0.294812\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$		0			0
$713$		−	24.0000i		−	0.898807i
$714$		0			0
$715$	2.00000	+	4.00000i	0.0747958	+	0.149592i
$716$		0			0
$717$			12.0000i			0.448148i
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$		0			0
$723$			30.0000i			1.11571i
$724$		0			0
$725$	6.00000	−	8.00000i	0.222834	−	0.297113i
$726$		0			0
$727$		−	14.0000i		−	0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$727$		−	14.0000i		−	0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−32.0000			−1.18356
$732$		0			0
$733$			2.00000i			0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$			2.00000i			0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$		0			0
$735$	9.00000	+	18.0000i	0.331970	+	0.663940i
$736$		0			0
$737$			8.00000i			0.294684i
$738$		0			0
$739$	−2.00000			−0.0735712			−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000			−0.0735712			−0.0367856	+	0.999323i	$0.511712\pi$
$740$		0			0
$741$	−2.00000			−0.0734718
$742$		0			0
$743$		−	4.00000i		−	0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$		−	4.00000i		−	0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	12.0000i		−	0.439057i
$748$		0			0
$749$	32.0000			1.16925
$750$		0			0
$751$	−48.0000			−1.75154			−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000			−1.75154			−0.875772	+	0.482724i	$0.839647\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$	−16.0000	+	8.00000i	−0.582300	+	0.291150i
$756$		0			0
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$		0			0
$759$	12.0000			0.435572
$760$		0			0
$761$	14.0000			0.507500			0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000			0.507500			0.253750	+	0.967270i	$0.418336\pi$
$762$		0			0
$763$			48.0000i			1.73772i
$764$		0			0
$765$	−4.00000	−	8.00000i	−0.144620	−	0.289241i
$766$		0			0
$767$			14.0000i			0.505511i
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$		−	14.0000i		−	0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$		−	14.0000i		−	0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$		0			0
$775$	12.0000	−	16.0000i	0.431053	−	0.574737i
$776$		0			0
$777$			24.0000i			0.860995i
$778$		0			0
$779$	−12.0000			−0.429945
$780$		0			0
$781$	16.0000			0.572525
$782$		0			0
$783$			2.00000i			0.0714742i
$784$		0			0
$785$	−10.0000	−	20.0000i	−0.356915	−	0.713831i
$786$		0			0
$787$			28.0000i			0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$			28.0000i			0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$		0			0
$789$	−10.0000			−0.356009
$790$		0			0
$791$	48.0000			1.70668
$792$		0			0
$793$		−	10.0000i		−	0.355110i
$794$		0			0
$795$	−20.0000	+	10.0000i	−0.709327	+	0.354663i
$796$		0			0
$797$			34.0000i			1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$			34.0000i			1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$		0			0
$799$	32.0000			1.13208
$800$		0			0
$801$	−18.0000			−0.635999
$802$		0			0
$803$			20.0000i			0.705785i
$804$		0			0
$805$	−48.0000	+	24.0000i	−1.69178	+	0.845889i
$806$		0			0
$807$		−	14.0000i		−	0.492823i
$808$		0			0
$809$	−14.0000			−0.492214			−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000			−0.492214			−0.246107	+	0.969243i	$0.579151\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$		0			0
$813$			16.0000i			0.561144i
$814$		0			0
$815$	−4.00000	−	8.00000i	−0.140114	−	0.280228i
$816$		0			0
$817$			16.0000i			0.559769i
$818$		0			0
$819$	4.00000			0.139771
$820$		0			0
$821$	−12.0000			−0.418803			−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000			−0.418803			−0.209401	+	0.977830i	$0.567152\pi$
$822$		0			0
$823$			26.0000i			0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$			26.0000i			0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$		0			0
$825$	8.00000	+	6.00000i	0.278524	+	0.208893i
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	−42.0000			−1.45872			−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000			−1.45872			−0.729360	+	0.684130i	$0.760182\pi$
$830$		0			0
$831$	2.00000			0.0693792
$832$		0			0
$833$		−	36.0000i		−	1.24733i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			4.00000i			0.138260i
$838$		0			0
$839$	16.0000			0.552381			0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000			0.552381			0.276191	+	0.961103i	$0.410928\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			18.0000i			0.619953i
$844$		0			0
$845$	−2.00000	+	1.00000i	−0.0688021	+	0.0344010i
$846$		0			0
$847$			28.0000i			0.962091i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$	−36.0000			−1.23406
$852$		0			0
$853$			42.0000i			1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$			42.0000i			1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$		0			0
$855$	−4.00000	+	2.00000i	−0.136797	+	0.0683986i
$856$		0			0
$857$		−	40.0000i		−	1.36637i	−0.730243	−	0.683187i	$-0.760593\pi$
$857$		−	40.0000i		−	1.36637i	0.730243	−	0.683187i	$-0.239407\pi$
$858$		0			0
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		0			0
$861$	24.0000			0.817918
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	−10.0000	−	20.0000i	−0.340010	−	0.680020i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	16.0000			0.542763
$870$		0			0
$871$	−4.00000			−0.135535
$872$		0			0
$873$		−	6.00000i		−	0.203069i
$874$		0			0
$875$	−44.0000	−	8.00000i	−1.48747	−	0.270449i
$876$		0			0
$877$		−	30.0000i		−	1.01303i	−0.862232	−	0.506514i	$-0.830934\pi$
$877$		−	30.0000i		−	1.01303i	0.862232	−	0.506514i	$-0.169066\pi$
$878$		0			0
$879$	14.0000			0.472208
$880$		0			0
$881$	−46.0000			−1.54978			−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000			−1.54978			−0.774890	+	0.632096i	$0.782195\pi$
$882$		0			0
$883$			48.0000i			1.61533i	0.589643	+	0.807664i	$0.299269\pi$
$883$			48.0000i			1.61533i	−0.589643	+	0.807664i	$0.700731\pi$
$884$		0			0
$885$	14.0000	+	28.0000i	0.470605	+	0.941210i
$886$		0			0
$887$			6.00000i			0.201460i	0.994914	+	0.100730i	$0.0321179\pi$
$887$			6.00000i			0.201460i	−0.994914	+	0.100730i	$0.967882\pi$
$888$		0			0
$889$	72.0000			2.41480
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$		−	16.0000i		−	0.535420i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			6.00000i			0.200334i
$898$		0			0
$899$	8.00000			0.266815
$900$		0			0
$901$	40.0000			1.33259
$902$		0			0
$903$		−	32.0000i		−	1.06489i
$904$		0			0
$905$	12.0000	−	6.00000i	0.398893	−	0.199447i
$906$		0			0
$907$			32.0000i			1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$			32.0000i			1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$		0			0
$909$	14.0000			0.464351
$910$		0			0
$911$	32.0000			1.06021			0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000			1.06021			0.530104	+	0.847933i	$0.322153\pi$
$912$		0			0
$913$		−	24.0000i		−	0.794284i
$914$		0			0
$915$	−10.0000	−	20.0000i	−0.330590	−	0.661180i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$	−28.0000			−0.922631
$922$		0			0
$923$			8.00000i			0.263323i
$924$		0			0
$925$	−24.0000	−	18.0000i	−0.789115	−	0.591836i
$926$		0			0
$927$		−	6.00000i		−	0.197066i
$928$		0			0
$929$	−46.0000			−1.50921			−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000			−1.50921			−0.754606	+	0.656179i	$0.772172\pi$
$930$		0			0
$931$	−18.0000			−0.589926
$932$		0			0
$933$			8.00000i			0.261908i
$934$		0			0
$935$	−8.00000	−	16.0000i	−0.261628	−	0.523256i
$936$		0			0
$937$			16.0000i			0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$			16.0000i			0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$		0			0
$939$	−8.00000			−0.261070
$940$		0			0
$941$	40.0000			1.30396			0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000			1.30396			0.651981	+	0.758235i	$0.273938\pi$
$942$		0			0
$943$			36.0000i			1.17232i
$944$		0			0
$945$	8.00000	−	4.00000i	0.260240	−	0.130120i
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	−10.0000			−0.324614
$950$		0			0
$951$	−30.0000			−0.972817
$952$		0			0
$953$			48.0000i			1.55487i	0.628962	+	0.777436i	$0.283480\pi$
$953$			48.0000i			1.55487i	−0.628962	+	0.777436i	$0.716520\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			4.00000i			0.129302i
$958$		0			0
$959$	8.00000			0.258333
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		−	8.00000i		−	0.257796i
$964$		0			0
$965$	−6.00000	−	12.0000i	−0.193147	−	0.386294i
$966$		0			0
$967$			28.0000i			0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$			28.0000i			0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$		0			0
$969$	8.00000			0.256997
$970$		0			0
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$		−	64.0000i		−	2.05175i
$974$		0			0
$975$	−3.00000	+	4.00000i	−0.0960769	+	0.128103i
$976$		0			0
$977$			42.0000i			1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$			42.0000i			1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$		0			0
$979$	−36.0000			−1.15056
$980$		0			0
$981$	12.0000			0.383131
$982$		0			0
$983$			4.00000i			0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$			4.00000i			0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$		0			0
$985$	26.0000	+	52.0000i	0.828429	+	1.65686i
$986$		0			0
$987$			32.0000i			1.01857i
$988$		0			0
$989$	48.0000			1.52631
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$			18.0000i			0.571213i
$994$		0			0
$995$	48.0000	−	24.0000i	1.52170	−	0.760851i
$996$		0			0
$997$			38.0000i			1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$			38.0000i			1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$		0			0
$999$	6.00000			0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.h.1249.1		2
4.3	odd	2		390.2.e.b.79.2	yes	2
5.4	even	2	inner	3120.2.l.h.1249.2		2
12.11	even	2		1170.2.e.b.469.1		2
20.3	even	4		1950.2.a.u.1.1		1
20.7	even	4		1950.2.a.g.1.1		1
20.19	odd	2		390.2.e.b.79.1	✓	2
60.23	odd	4		5850.2.a.x.1.1		1
60.47	odd	4		5850.2.a.bd.1.1		1
60.59	even	2		1170.2.e.b.469.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.b.79.1	✓	2	20.19	odd	2
390.2.e.b.79.2	yes	2	4.3	odd	2
1170.2.e.b.469.1		2	12.11	even	2
1170.2.e.b.469.2		2	60.59	even	2
1950.2.a.g.1.1		1	20.7	even	4
1950.2.a.u.1.1		1	20.3	even	4
3120.2.l.h.1249.1		2	1.1	even	1	trivial
3120.2.l.h.1249.2		2	5.4	even	2	inner
5850.2.a.x.1.1		1	60.23	odd	4
5850.2.a.bd.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} -1.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} +4.00000i q^{17} +2.00000 q^{19} -4.00000 q^{21} -6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +2.00000i q^{33} +(-4.00000 - 8.00000i) q^{35} -6.00000i q^{37} -1.00000 q^{39} -6.00000 q^{41} +8.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -8.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -10.0000i q^{53} +(-4.00000 + 2.00000i) q^{55} -2.00000i q^{57} -14.0000 q^{59} +10.0000 q^{61} +4.00000i q^{63} +(-1.00000 - 2.00000i) q^{65} -4.00000i q^{67} -6.00000 q^{69} -8.00000 q^{71} -10.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +8.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +(4.00000 + 8.00000i) q^{85} -2.00000i q^{87} +18.0000 q^{89} -4.00000 q^{91} -4.00000i q^{93} +(4.00000 - 2.00000i) q^{95} +6.00000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^5 - 2 * q^9	\( 2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 4 q^{29} + 8 q^{31} - 8 q^{35} - 2 q^{39} - 12 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} - 8 q^{55} - 28 q^{59} + 20 q^{61} - 2 q^{65} - 12 q^{69} - 16 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} + 8 q^{85} + 36 q^{89} - 8 q^{91} + 8 q^{95} + 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 + 4 * q^19 - 8 * q^21 + 6 * q^25 + 4 * q^29 + 8 * q^31 - 8 * q^35 - 2 * q^39 - 12 * q^41 - 4 * q^45 - 18 * q^49 + 8 * q^51 - 8 * q^55 - 28 * q^59 + 20 * q^61 - 2 * q^65 - 12 * q^69 - 16 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 + 8 * q^85 + 36 * q^89 - 8 * q^91 + 8 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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