LMFDB - Embedded newform 1960.2.g.f.1569.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1569,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1960.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.g (of order $2$, degree $1$, 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:	$15.6506787962$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 25x^{10} + 230x^{8} + 950x^{6} + 1657x^{4} + 785x^{2} + 64 $ x^12 + 25x^10 + 230x^8 + 950x^6 + 1657x^4 + 785*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.3
Root			$-2.10821i$ of defining polynomial
Character	$\chi$	$=$	1960.1569
Dual form			1960.2.g.f.1569.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10821i q^{3} +(-2.19509 - 0.426103i) q^{5} -1.44457 q^{9} +O(q^{10})$	$q-2.10821i q^{3} +(-2.19509 - 0.426103i) q^{5} -1.44457 q^{9} +1.77465 q^{11} -1.44457i q^{13} +(-0.898317 + 4.62773i) q^{15} +5.86806i q^{17} -7.16205 q^{19} -0.663645i q^{23} +(4.63687 + 1.87067i) q^{25} -3.27918i q^{27} -6.45763 q^{29} -10.0732 q^{31} -3.74135i q^{33} +5.01705i q^{37} -3.04546 q^{39} -1.92363 q^{41} +5.81995i q^{43} +(3.17097 + 0.615536i) q^{45} +3.70322i q^{47} +12.3711 q^{51} -0.592363i q^{53} +(-3.89553 - 0.756185i) q^{55} +15.0991i q^{57} +7.59076 q^{59} +8.72111 q^{61} +(-0.615536 + 3.17097i) q^{65} +2.16832i q^{67} -1.39911 q^{69} +6.49826 q^{71} +15.5320i q^{73} +(3.94378 - 9.77552i) q^{75} +2.70720 q^{79} -11.2469 q^{81} -6.35340i q^{83} +(2.50040 - 12.8809i) q^{85} +13.6141i q^{87} -5.86157 q^{89} +21.2364i q^{93} +(15.7214 + 3.05177i) q^{95} +14.7748i q^{97} -2.56361 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{9}+O(q^{10}) $ 12 * q - 14 * q^9	$ 12 q - 14 q^{9} + 2 q^{11} + 6 q^{15} + 10 q^{19} + 2 q^{25} + 6 q^{29} - 4 q^{31} - 20 q^{39} + 12 q^{41} + 8 q^{45} - 6 q^{55} + 48 q^{59} + 18 q^{61} + 26 q^{65} - 30 q^{69} + 8 q^{71} + 14 q^{75} + 44 q^{79} - 12 q^{81} - 22 q^{85} - 30 q^{89} + 26 q^{95} - 22 q^{99}+O(q^{100}) $ 12 * q - 14 * q^9 + 2 * q^11 + 6 * q^15 + 10 * q^19 + 2 * q^25 + 6 * q^29 - 4 * q^31 - 20 * q^39 + 12 * q^41 + 8 * q^45 - 6 * q^55 + 48 * q^59 + 18 * q^61 + 26 * q^65 - 30 * q^69 + 8 * q^71 + 14 * q^75 + 44 * q^79 - 12 * q^81 - 22 * q^85 - 30 * q^89 + 26 * q^95 - 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$.

$n$	$981$	$1081$	$1177$	$1471$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	2.10821i		−	1.21718i	−0.793485	−	0.608589i	$-0.791736\pi$
$3$		−	2.10821i		−	1.21718i	0.793485	−	0.608589i	$-0.208264\pi$
$4$		0			0
$5$	−2.19509	−	0.426103i	−0.981676	−	0.190559i
$5$	−2.19509	−	0.426103i	−0.981676	−	0.190559i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.44457			−0.481523
$10$		0			0
$11$	1.77465			0.535078			0.267539	−	0.963547i	$-0.413790\pi$
$11$	1.77465			0.535078			0.267539	+	0.963547i	$0.413790\pi$
$12$		0			0
$13$		−	1.44457i		−	0.400652i	−0.979729	−	0.200326i	$-0.935800\pi$
$13$		−	1.44457i		−	0.400652i	0.979729	−	0.200326i	$-0.0642001\pi$
$14$		0			0
$15$	−0.898317	+	4.62773i	−0.231945	+	1.19487i
$16$		0			0
$17$			5.86806i			1.42321i	0.702578	+	0.711607i	$0.252032\pi$
$17$			5.86806i			1.42321i	−0.702578	+	0.711607i	$0.747968\pi$
$18$		0			0
$19$	−7.16205			−1.64309			−0.821543	−	0.570146i	$-0.806886\pi$
$19$	−7.16205			−1.64309			−0.821543	+	0.570146i	$0.806886\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.663645i		−	0.138380i	−0.997604	−	0.0691898i	$-0.977959\pi$
$23$		−	0.663645i		−	0.138380i	0.997604	−	0.0691898i	$-0.0220414\pi$
$24$		0			0
$25$	4.63687	+	1.87067i	0.927374	+	0.374135i
$26$		0			0
$27$		−	3.27918i		−	0.631079i
$28$		0			0
$29$	−6.45763			−1.19915			−0.599576	−	0.800318i	$-0.704664\pi$
$29$	−6.45763			−1.19915			−0.599576	+	0.800318i	$0.704664\pi$
$30$		0			0
$31$	−10.0732			−1.80919			−0.904597	−	0.426267i	$-0.859828\pi$
$31$	−10.0732			−1.80919			−0.904597	+	0.426267i	$0.859828\pi$
$32$		0			0
$33$		−	3.74135i		−	0.651285i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			5.01705i			0.824797i	0.911004	+	0.412399i	$0.135309\pi$
$37$			5.01705i			0.824797i	−0.911004	+	0.412399i	$0.864691\pi$
$38$		0			0
$39$	−3.04546			−0.487664
$40$		0			0
$41$	−1.92363			−0.300421			−0.150211	−	0.988654i	$-0.547995\pi$
$41$	−1.92363			−0.300421			−0.150211	+	0.988654i	$0.547995\pi$
$42$		0			0
$43$			5.81995i			0.887535i	0.896142	+	0.443767i	$0.146358\pi$
$43$			5.81995i			0.887535i	−0.896142	+	0.443767i	$0.853642\pi$
$44$		0			0
$45$	3.17097	+	0.615536i	0.472700	+	0.0917587i
$46$		0			0
$47$			3.70322i			0.540171i	0.962836	+	0.270085i	$0.0870519\pi$
$47$			3.70322i			0.540171i	−0.962836	+	0.270085i	$0.912948\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	12.3711			1.73231
$52$		0			0
$53$		−	0.592363i		−	0.0813674i	−0.999172	−	0.0406837i	$-0.987046\pi$
$53$		−	0.592363i		−	0.0813674i	0.999172	−	0.0406837i	$-0.0129536\pi$
$54$		0			0
$55$	−3.89553	−	0.756185i	−0.525273	−	0.101964i
$56$		0			0
$57$			15.0991i			1.99993i
$58$		0			0
$59$	7.59076			0.988233			0.494116	−	0.869396i	$-0.335492\pi$
$59$	7.59076			0.988233			0.494116	+	0.869396i	$0.335492\pi$
$60$		0			0
$61$	8.72111			1.11662			0.558312	−	0.829631i	$-0.311449\pi$
$61$	8.72111			1.11662			0.558312	+	0.829631i	$0.311449\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.615536	+	3.17097i	−0.0763478	+	0.393310i
$66$		0			0
$67$			2.16832i			0.264903i	0.991190	+	0.132451i	$0.0422848\pi$
$67$			2.16832i			0.264903i	−0.991190	+	0.132451i	$0.957715\pi$
$68$		0			0
$69$	−1.39911			−0.168433
$70$		0			0
$71$	6.49826			0.771201			0.385601	−	0.922666i	$-0.373994\pi$
$71$	6.49826			0.771201			0.385601	+	0.922666i	$0.373994\pi$
$72$		0			0
$73$			15.5320i			1.81788i	0.416924	+	0.908941i	$0.363108\pi$
$73$			15.5320i			1.81788i	−0.416924	+	0.908941i	$0.636892\pi$
$74$		0			0
$75$	3.94378	−	9.77552i	0.455389	−	1.12878i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	2.70720			0.304584			0.152292	−	0.988336i	$-0.451335\pi$
$79$	2.70720			0.304584			0.152292	+	0.988336i	$0.451335\pi$
$80$		0			0
$81$	−11.2469			−1.24966
$82$		0			0
$83$		−	6.35340i		−	0.697376i	−0.937239	−	0.348688i	$-0.886627\pi$
$83$		−	6.35340i		−	0.697376i	0.937239	−	0.348688i	$-0.113373\pi$
$84$		0			0
$85$	2.50040	−	12.8809i	0.271207	−	1.39713i
$86$		0			0
$87$			13.6141i			1.45958i
$88$		0			0
$89$	−5.86157			−0.621326			−0.310663	−	0.950520i	$-0.600551\pi$
$89$	−5.86157			−0.621326			−0.310663	+	0.950520i	$0.600551\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			21.2364i			2.20211i
$94$		0			0
$95$	15.7214	+	3.05177i	1.61298	+	0.313105i
$96$		0			0
$97$			14.7748i			1.50015i	0.661351	+	0.750076i	$0.269983\pi$
$97$			14.7748i			1.50015i	−0.661351	+	0.750076i	$0.730017\pi$
$98$		0			0
$99$	−2.56361			−0.257652
$100$		0			0
$101$	3.48186			0.346458			0.173229	−	0.984882i	$-0.444580\pi$
$101$	3.48186			0.346458			0.173229	+	0.984882i	$0.444580\pi$
$102$		0			0
$103$		−	8.07861i		−	0.796009i	−0.917384	−	0.398004i	$-0.869703\pi$
$103$		−	8.07861i		−	0.796009i	0.917384	−	0.398004i	$-0.130297\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.129645i		−	0.0125333i	−0.999980	−	0.00626664i	$-0.998005\pi$
$107$		−	0.129645i		−	0.0125333i	0.999980	−	0.00626664i	$-0.00199475\pi$
$108$		0			0
$109$	−12.2648			−1.17476			−0.587379	−	0.809312i	$-0.699840\pi$
$109$	−12.2648			−1.17476			−0.587379	+	0.809312i	$0.699840\pi$
$110$		0			0
$111$	10.5770			1.00393
$112$		0			0
$113$			11.7467i			1.10503i	0.833501	+	0.552517i	$0.186333\pi$
$113$			11.7467i			1.10503i	−0.833501	+	0.552517i	$0.813667\pi$
$114$		0			0
$115$	−0.282781	+	1.45676i	−0.0263695	+	0.135844i
$116$		0			0
$117$			2.08678i			0.192923i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.85061			−0.713692
$122$		0			0
$123$			4.05543i			0.365666i
$124$		0			0
$125$	−9.38127	−	6.08209i	−0.839086	−	0.543999i
$126$		0			0
$127$			12.6983i			1.12679i	0.826187	+	0.563395i	$0.190505\pi$
$127$			12.6983i			1.12679i	−0.826187	+	0.563395i	$0.809495\pi$
$128$		0			0
$129$	12.2697			1.08029
$130$		0			0
$131$	−0.230932			−0.0201766			−0.0100883	−	0.999949i	$-0.503211\pi$
$131$	−0.230932			−0.0201766			−0.0100883	+	0.999949i	$0.503211\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.39727	+	7.19811i	−0.120258	+	0.619515i
$136$		0			0
$137$		−	16.8117i		−	1.43632i	−0.695877	−	0.718161i	$-0.744984\pi$
$137$		−	16.8117i		−	1.43632i	0.695877	−	0.718161i	$-0.255016\pi$
$138$		0			0
$139$	20.5531			1.74329			0.871644	−	0.490139i	$-0.163054\pi$
$139$	20.5531			1.74329			0.871644	+	0.490139i	$0.163054\pi$
$140$		0			0
$141$	7.80719			0.657484
$142$		0			0
$143$		−	2.56361i		−	0.214380i
$144$		0			0
$145$	14.1751	+	2.75162i	1.17718	+	0.228510i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−21.2725			−1.74271			−0.871356	−	0.490652i	$-0.836759\pi$
$149$	−21.2725			−1.74271			−0.871356	+	0.490652i	$0.836759\pi$
$150$		0			0
$151$	6.47990			0.527327			0.263663	−	0.964615i	$-0.415069\pi$
$151$	6.47990			0.527327			0.263663	+	0.964615i	$0.415069\pi$
$152$		0			0
$153$		−	8.47683i		−	0.685311i
$154$		0			0
$155$	22.1116	+	4.29221i	1.77604	+	0.344759i
$156$		0			0
$157$		−	5.44990i		−	0.434950i	−0.976066	−	0.217475i	$-0.930218\pi$
$157$		−	5.44990i		−	0.434950i	0.976066	−	0.217475i	$-0.0697820\pi$
$158$		0			0
$159$	−1.24883			−0.0990386
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	6.26921i		−	0.491042i	−0.969391	−	0.245521i	$-0.921041\pi$
$163$		−	6.26921i		−	0.491042i	0.969391	−	0.245521i	$-0.0789591\pi$
$164$		0			0
$165$	−1.59420	+	8.21261i	−0.124108	+	0.639350i
$166$		0			0
$167$		−	18.5728i		−	1.43720i	−0.695422	−	0.718602i	$-0.744782\pi$
$167$		−	18.5728i		−	1.43720i	0.695422	−	0.718602i	$-0.255218\pi$
$168$		0			0
$169$	10.9132			0.839478
$170$		0			0
$171$	10.3461			0.791184
$172$		0			0
$173$		−	13.9672i		−	1.06191i	−0.847401	−	0.530954i	$-0.821834\pi$
$173$		−	13.9672i		−	1.06191i	0.847401	−	0.530954i	$-0.178166\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	16.0030i		−	1.20286i
$178$		0			0
$179$	−7.92539			−0.592372			−0.296186	−	0.955130i	$-0.595715\pi$
$179$	−7.92539			−0.592372			−0.296186	+	0.955130i	$0.595715\pi$
$180$		0			0
$181$	−11.6887			−0.868815			−0.434407	−	0.900717i	$-0.643042\pi$
$181$	−11.6887			−0.868815			−0.434407	+	0.900717i	$0.643042\pi$
$182$		0			0
$183$		−	18.3860i		−	1.35913i
$184$		0			0
$185$	2.13778	−	11.0129i	0.157173	−	0.809683i
$186$		0			0
$187$			10.4138i			0.761530i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.9782			−0.939066			−0.469533	−	0.882915i	$-0.655578\pi$
$191$	−12.9782			−0.939066			−0.469533	+	0.882915i	$0.655578\pi$
$192$		0			0
$193$			21.4053i			1.54079i	0.637570	+	0.770393i	$0.279940\pi$
$193$			21.4053i			1.54079i	−0.637570	+	0.770393i	$0.720060\pi$
$194$		0			0
$195$	6.68508	+	1.29768i	0.478728	+	0.0929289i
$196$		0			0
$197$			8.42871i			0.600521i	0.953857	+	0.300261i	$0.0970736\pi$
$197$			8.42871i			0.600521i	−0.953857	+	0.300261i	$0.902926\pi$
$198$		0			0
$199$	−4.21481			−0.298780			−0.149390	−	0.988778i	$-0.547731\pi$
$199$	−4.21481			−0.298780			−0.149390	+	0.988778i	$0.547731\pi$
$200$		0			0
$201$	4.57129			0.322434
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.22256	+	0.819667i	0.294916	+	0.0572480i
$206$		0			0
$207$			0.958682i			0.0666330i
$208$		0			0
$209$	−12.7101			−0.879179
$210$		0			0
$211$	−0.629004			−0.0433025			−0.0216512	−	0.999766i	$-0.506892\pi$
$211$	−0.629004			−0.0433025			−0.0216512	+	0.999766i	$0.506892\pi$
$212$		0			0
$213$		−	13.6997i		−	0.938689i
$214$		0			0
$215$	2.47990	−	12.7753i	0.169128	−	0.871271i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	32.7448			2.21269
$220$		0			0
$221$	8.47683			0.570213
$222$		0			0
$223$			21.4138i			1.43398i	0.697086	+	0.716988i	$0.254480\pi$
$223$			21.4138i			1.43398i	−0.697086	+	0.716988i	$0.745520\pi$
$224$		0			0
$225$	−6.69829	−	2.70232i	−0.446552	−	0.180155i
$226$		0			0
$227$			22.0493i			1.46346i	0.681593	+	0.731732i	$0.261288\pi$
$227$			22.0493i			1.46346i	−0.681593	+	0.731732i	$0.738712\pi$
$228$		0			0
$229$	−11.6359			−0.768925			−0.384462	−	0.923141i	$-0.625613\pi$
$229$	−11.6359			−0.768925			−0.384462	+	0.923141i	$0.625613\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	11.2193i		−	0.735004i	−0.930023	−	0.367502i	$-0.880213\pi$
$233$		−	11.2193i		−	0.735004i	0.930023	−	0.367502i	$-0.119787\pi$
$234$		0			0
$235$	1.57796	−	8.12892i	0.102934	−	0.530272i
$236$		0			0
$237$		−	5.70737i		−	0.370733i
$238$		0			0
$239$	−22.8279			−1.47662			−0.738308	−	0.674464i	$-0.764375\pi$
$239$	−22.8279			−1.47662			−0.738308	+	0.674464i	$0.764375\pi$
$240$		0			0
$241$	4.20270			0.270720			0.135360	−	0.990796i	$-0.456781\pi$
$241$	4.20270			0.270720			0.135360	+	0.990796i	$0.456781\pi$
$242$		0			0
$243$			13.8734i			0.889979i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			10.3461i			0.658305i
$248$		0			0
$249$	−13.3943			−0.848831
$250$		0			0
$251$	−16.9587			−1.07042			−0.535211	−	0.844718i	$-0.679768\pi$
$251$	−16.9587			−1.07042			−0.535211	+	0.844718i	$0.679768\pi$
$252$		0			0
$253$		−	1.17774i		−	0.0740438i
$254$		0			0
$255$	−27.1558	−	5.27138i	−1.70056	−	0.330107i
$256$		0			0
$257$			26.9303i			1.67986i	0.542692	+	0.839932i	$0.317405\pi$
$257$			26.9303i			1.67986i	−0.542692	+	0.839932i	$0.682595\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	9.32850			0.577420
$262$		0			0
$263$		−	21.9520i		−	1.35362i	−0.736158	−	0.676809i	$-0.763362\pi$
$263$		−	21.9520i		−	1.35362i	0.736158	−	0.676809i	$-0.236638\pi$
$264$		0			0
$265$	−0.252408	+	1.30029i	−0.0155053	+	0.0798764i
$266$		0			0
$267$			12.3575i			0.756264i
$268$		0			0
$269$	−24.0854			−1.46851			−0.734256	−	0.678872i	$-0.762469\pi$
$269$	−24.0854			−1.46851			−0.734256	+	0.678872i	$0.762469\pi$
$270$		0			0
$271$	2.06731			0.125580			0.0627900	−	0.998027i	$-0.480000\pi$
$271$	2.06731			0.125580			0.0627900	+	0.998027i	$0.480000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.22883	+	3.31979i	0.496217	+	0.200191i
$276$		0			0
$277$			17.0757i			1.02598i	0.858395	+	0.512990i	$0.171462\pi$
$277$			17.0757i			1.02598i	−0.858395	+	0.512990i	$0.828538\pi$
$278$		0			0
$279$	14.5514			0.871169
$280$		0			0
$281$	−13.2035			−0.787655			−0.393828	−	0.919184i	$-0.628849\pi$
$281$	−13.2035			−0.787655			−0.393828	+	0.919184i	$0.628849\pi$
$282$		0			0
$283$		−	26.5989i		−	1.58114i	−0.612372	−	0.790570i	$-0.709784\pi$
$283$		−	26.5989i		−	1.58114i	0.612372	−	0.790570i	$-0.290216\pi$
$284$		0			0
$285$	6.43379	−	33.1440i	0.381105	−	1.96328i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.4342			−1.02554
$290$		0			0
$291$	31.1484			1.82595
$292$		0			0
$293$			0.472681i			0.0276143i	0.999905	+	0.0138071i	$0.00439509\pi$
$293$			0.472681i			0.0276143i	−0.999905	+	0.0138071i	$0.995605\pi$
$294$		0			0
$295$	−16.6624	−	3.23445i	−0.970124	−	0.188317i
$296$		0			0
$297$		−	5.81940i		−	0.337676i
$298$		0			0
$299$	−0.958682			−0.0554420
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	7.34050i		−	0.421701i
$304$		0			0
$305$	−19.1436	−	3.71609i	−1.09616	−	0.212783i
$306$		0			0
$307$			15.2073i			0.867929i	0.900930	+	0.433965i	$0.142886\pi$
$307$			15.2073i			0.867929i	−0.900930	+	0.433965i	$0.857114\pi$
$308$		0			0
$309$	−17.0314			−0.968885
$310$		0			0
$311$	−16.5806			−0.940200			−0.470100	−	0.882613i	$-0.655782\pi$
$311$	−16.5806			−0.940200			−0.470100	+	0.882613i	$0.655782\pi$
$312$		0			0
$313$			1.17950i			0.0666695i	0.999444	+	0.0333348i	$0.0106128\pi$
$313$			1.17950i			0.0666695i	−0.999444	+	0.0333348i	$0.989387\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	27.4288i		−	1.54056i	−0.637709	−	0.770278i	$-0.720118\pi$
$317$		−	27.4288i		−	1.54056i	0.637709	−	0.770278i	$-0.279882\pi$
$318$		0			0
$319$	−11.4600			−0.641640
$320$		0			0
$321$	−0.273320			−0.0152552
$322$		0			0
$323$		−	42.0273i		−	2.33846i
$324$		0			0
$325$	2.70232	−	6.69829i	0.149898	−	0.371554i
$326$		0			0
$327$			25.8569i			1.42989i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5.21160			0.286455			0.143228	−	0.989690i	$-0.454252\pi$
$331$	5.21160			0.286455			0.143228	+	0.989690i	$0.454252\pi$
$332$		0			0
$333$		−	7.24747i		−	0.397159i
$334$		0			0
$335$	0.923929	−	4.75967i	0.0504796	−	0.260048i
$336$		0			0
$337$		−	9.52496i		−	0.518858i	−0.965762	−	0.259429i	$-0.916466\pi$
$337$		−	9.52496i		−	0.518858i	0.965762	−	0.259429i	$-0.0835343\pi$
$338$		0			0
$339$	24.7645			1.34502
$340$		0			0
$341$	−17.8764			−0.968059
$342$		0			0
$343$		0			0
$344$		0			0
$345$	3.07117	+	0.596164i	0.165346	+	0.0320964i
$346$		0			0
$347$		−	27.2781i		−	1.46436i	−0.681109	−	0.732182i	$-0.738502\pi$
$347$		−	27.2781i		−	1.46436i	0.681109	−	0.732182i	$-0.261498\pi$
$348$		0			0
$349$	−6.78145			−0.363003			−0.181501	−	0.983391i	$-0.558096\pi$
$349$	−6.78145			−0.363003			−0.181501	+	0.983391i	$0.558096\pi$
$350$		0			0
$351$	−4.73701			−0.252843
$352$		0			0
$353$			15.6791i			0.834517i	0.908788	+	0.417258i	$0.137009\pi$
$353$			15.6791i			0.834517i	−0.908788	+	0.417258i	$0.862991\pi$
$354$		0			0
$355$	−14.2643	−	2.76893i	−0.757069	−	0.146959i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.33467			0.334331			0.167165	−	0.985929i	$-0.446539\pi$
$359$	6.33467			0.334331			0.167165	+	0.985929i	$0.446539\pi$
$360$		0			0
$361$	32.2949			1.69973
$362$		0			0
$363$			16.5508i			0.868691i
$364$		0			0
$365$	6.61823	−	34.0942i	0.346414	−	1.78457i
$366$		0			0
$367$			12.7599i			0.666059i	0.942916	+	0.333030i	$0.108071\pi$
$367$			12.7599i			0.666059i	−0.942916	+	0.333030i	$0.891929\pi$
$368$		0			0
$369$	2.77882			0.144660
$370$		0			0
$371$		0			0
$372$		0			0
$373$			1.66915i			0.0864255i	0.999066	+	0.0432127i	$0.0137593\pi$
$373$			1.66915i			0.0864255i	−0.999066	+	0.0432127i	$0.986241\pi$
$374$		0			0
$375$	−12.8224	+	19.7777i	−0.662143	+	1.02132i
$376$		0			0
$377$			9.32850i			0.480442i
$378$		0			0
$379$	21.5557			1.10724			0.553621	−	0.832768i	$-0.313245\pi$
$379$	21.5557			1.10724			0.553621	+	0.832768i	$0.313245\pi$
$380$		0			0
$381$	26.7707			1.37151
$382$		0			0
$383$			5.04425i			0.257749i	0.991661	+	0.128875i	$0.0411365\pi$
$383$			5.04425i			0.257749i	−0.991661	+	0.128875i	$0.958864\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	8.40733i		−	0.427369i
$388$		0			0
$389$	12.5195			0.634764			0.317382	−	0.948298i	$-0.397196\pi$
$389$	12.5195			0.634764			0.317382	+	0.948298i	$0.397196\pi$
$390$		0			0
$391$	3.89431			0.196944
$392$		0			0
$393$			0.486853i			0.0245585i
$394$		0			0
$395$	−5.94257	−	1.15355i	−0.299003	−	0.0580413i
$396$		0			0
$397$			22.3935i			1.12390i	0.827171	+	0.561950i	$0.189949\pi$
$397$			22.3935i			1.12390i	−0.827171	+	0.561950i	$0.810051\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.3931			−0.768694			−0.384347	−	0.923189i	$-0.625573\pi$
$401$	−15.3931			−0.768694			−0.384347	+	0.923189i	$0.625573\pi$
$402$		0			0
$403$			14.5514i			0.724857i
$404$		0			0
$405$	24.6881	+	4.79235i	1.22676	+	0.238134i
$406$		0			0
$407$			8.90351i			0.441330i
$408$		0			0
$409$	−30.9701			−1.53137			−0.765687	−	0.643213i	$-0.777601\pi$
$409$	−30.9701			−1.53137			−0.765687	+	0.643213i	$0.777601\pi$
$410$		0			0
$411$	−35.4427			−1.74826
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.70720	+	13.9463i	−0.132891	+	0.684597i
$416$		0			0
$417$		−	43.3303i		−	2.12189i
$418$		0			0
$419$	−0.434578			−0.0212305			−0.0106153	−	0.999944i	$-0.503379\pi$
$419$	−0.434578			−0.0212305			−0.0106153	+	0.999944i	$0.503379\pi$
$420$		0			0
$421$	−27.3152			−1.33126			−0.665630	−	0.746282i	$-0.731837\pi$
$421$	−27.3152			−1.33126			−0.665630	+	0.746282i	$0.731837\pi$
$422$		0			0
$423$		−	5.34957i		−	0.260105i
$424$		0			0
$425$	−10.9772	+	27.2095i	−0.532474	+	1.31985i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−5.40464			−0.260938
$430$		0			0
$431$	−20.5275			−0.988773			−0.494386	−	0.869242i	$-0.664607\pi$
$431$	−20.5275			−0.988773			−0.494386	+	0.869242i	$0.664607\pi$
$432$		0			0
$433$		−	16.0082i		−	0.769304i	−0.923062	−	0.384652i	$-0.874321\pi$
$433$		−	16.0082i		−	0.769304i	0.923062	−	0.384652i	$-0.125679\pi$
$434$		0			0
$435$	5.80100	−	29.8842i	0.278137	−	1.43284i
$436$		0			0
$437$			4.75306i			0.227370i
$438$		0			0
$439$	−8.06958			−0.385140			−0.192570	−	0.981283i	$-0.561682\pi$
$439$	−8.06958			−0.385140			−0.192570	+	0.981283i	$0.561682\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	7.19214i		−	0.341709i	−0.985296	−	0.170854i	$-0.945347\pi$
$443$		−	7.19214i		−	0.341709i	0.985296	−	0.170854i	$-0.0546528\pi$
$444$		0			0
$445$	12.8667	+	2.49764i	0.609940	+	0.118399i
$446$		0			0
$447$			44.8470i			2.12119i
$448$		0			0
$449$	4.16729			0.196667			0.0983334	−	0.995154i	$-0.468649\pi$
$449$	4.16729			0.196667			0.0983334	+	0.995154i	$0.468649\pi$
$450$		0			0
$451$	−3.41378			−0.160749
$452$		0			0
$453$		−	13.6610i		−	0.641851i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			9.17950i			0.429399i	0.976680	+	0.214700i	$0.0688772\pi$
$457$			9.17950i			0.429399i	−0.976680	+	0.214700i	$0.931123\pi$
$458$		0			0
$459$	19.2424			0.898160
$460$		0			0
$461$	27.3537			1.27399			0.636995	−	0.770868i	$-0.280177\pi$
$461$	27.3537			1.27399			0.636995	+	0.770868i	$0.280177\pi$
$462$		0			0
$463$		−	12.8660i		−	0.597932i	−0.954264	−	0.298966i	$-0.903358\pi$
$463$		−	12.8660i		−	0.597932i	0.954264	−	0.298966i	$-0.0966418\pi$
$464$		0			0
$465$	9.04890	−	46.6159i	0.419633	−	2.16176i
$466$		0			0
$467$			13.9010i			0.643260i	0.946865	+	0.321630i	$0.104231\pi$
$467$			13.9010i			0.643260i	−0.946865	+	0.321630i	$0.895769\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−11.4896			−0.529411
$472$		0			0
$473$			10.3284i			0.474900i
$474$		0			0
$475$	−33.2095	−	13.3978i	−1.52376	−	0.614735i
$476$		0			0
$477$			0.855710i			0.0391803i
$478$		0			0
$479$	0.371944			0.0169946			0.00849729	−	0.999964i	$-0.497295\pi$
$479$	0.371944			0.0169946			0.00849729	+	0.999964i	$0.497295\pi$
$480$		0			0
$481$	7.24747			0.330456
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.29559	−	32.4321i	0.285868	−	1.47266i
$486$		0			0
$487$		−	19.2340i		−	0.871577i	−0.900049	−	0.435788i	$-0.856470\pi$
$487$		−	19.2340i		−	0.871577i	0.900049	−	0.435788i	$-0.143530\pi$
$488$		0			0
$489$	−13.2168			−0.597686
$490$		0			0
$491$	15.2267			0.687170			0.343585	−	0.939122i	$-0.388359\pi$
$491$	15.2267			0.687170			0.343585	+	0.939122i	$0.388359\pi$
$492$		0			0
$493$		−	37.8938i		−	1.70665i
$494$		0			0
$495$	5.62736	+	1.09236i	0.252931	+	0.0490980i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.9430			−0.624173			−0.312087	−	0.950054i	$-0.601028\pi$
$499$	−13.9430			−0.624173			−0.312087	+	0.950054i	$0.601028\pi$
$500$		0			0
$501$	−39.1554			−1.74933
$502$		0			0
$503$		−	41.6394i		−	1.85661i	−0.371821	−	0.928305i	$-0.621266\pi$
$503$		−	41.6394i		−	1.85661i	0.371821	−	0.928305i	$-0.378734\pi$
$504$		0			0
$505$	−7.64300	−	1.48363i	−0.340109	−	0.0660207i
$506$		0			0
$507$		−	23.0074i		−	1.02179i
$508$		0			0
$509$	31.1663			1.38142			0.690712	−	0.723130i	$-0.257297\pi$
$509$	31.1663			1.38142			0.690712	+	0.723130i	$0.257297\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			23.4856i			1.03692i
$514$		0			0
$515$	−3.44232	+	17.7333i	−0.151687	+	0.781422i
$516$		0			0
$517$			6.57193i			0.289033i
$518$		0			0
$519$	−29.4459			−1.29253
$520$		0			0
$521$	−9.91918			−0.434567			−0.217284	−	0.976109i	$-0.569720\pi$
$521$	−9.91918			−0.434567			−0.217284	+	0.976109i	$0.569720\pi$
$522$		0			0
$523$		−	19.2340i		−	0.841045i	−0.907282	−	0.420523i	$-0.861847\pi$
$523$		−	19.2340i		−	0.841045i	0.907282	−	0.420523i	$-0.138153\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	59.1100i		−	2.57487i
$528$		0			0
$529$	22.5596			0.980851
$530$		0			0
$531$	−10.9654			−0.475857
$532$		0			0
$533$			2.77882i			0.120364i
$534$		0			0
$535$	−0.0552422	+	0.284583i	−0.00238833	+	0.0123036i
$536$		0			0
$537$			16.7084i			0.721022i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.60558			−0.155016			−0.0775081	−	0.996992i	$-0.524696\pi$
$541$	−3.60558			−0.155016			−0.0775081	+	0.996992i	$0.524696\pi$
$542$		0			0
$543$			24.6423i			1.05750i
$544$		0			0
$545$	26.9224	+	5.22608i	1.15323	+	0.223861i
$546$		0			0
$547$			11.1011i			0.474647i	0.971431	+	0.237324i	$0.0762702\pi$
$547$			11.1011i			0.474647i	−0.971431	+	0.237324i	$0.923730\pi$
$548$		0			0
$549$	−12.5982			−0.537680
$550$		0			0
$551$	46.2499			1.97031
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−23.2175	−	4.50690i	−0.985529	−	0.191307i
$556$		0			0
$557$			24.8346i			1.05228i	0.850399	+	0.526138i	$0.176360\pi$
$557$			24.8346i			1.05228i	−0.850399	+	0.526138i	$0.823640\pi$
$558$		0			0
$559$	8.40733			0.355592
$560$		0			0
$561$	21.9545			0.926918
$562$		0			0
$563$		−	2.82819i		−	0.119194i	−0.998223	−	0.0595970i	$-0.981018\pi$
$563$		−	2.82819i		−	0.119194i	0.998223	−	0.0595970i	$-0.0189816\pi$
$564$		0			0
$565$	5.00530	−	25.7851i	0.210575	−	1.08479i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	25.1318			1.05358			0.526791	−	0.849995i	$-0.323395\pi$
$569$	25.1318			1.05358			0.526791	+	0.849995i	$0.323395\pi$
$570$		0			0
$571$	−35.2093			−1.47347			−0.736733	−	0.676184i	$-0.763632\pi$
$571$	−35.2093			−1.47347			−0.736733	+	0.676184i	$0.763632\pi$
$572$		0			0
$573$			27.3607i			1.14301i
$574$		0			0
$575$	1.24146	−	3.07724i	0.0517726	−	0.128330i
$576$		0			0
$577$			27.8365i			1.15885i	0.815027	+	0.579424i	$0.196722\pi$
$577$			27.8365i			1.15885i	−0.815027	+	0.579424i	$0.803278\pi$
$578$		0			0
$579$	45.1269			1.87541
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	1.05124i		−	0.0435378i
$584$		0			0
$585$	0.889185	−	4.58068i	0.0367633	−	0.189388i
$586$		0			0
$587$		−	4.74640i		−	0.195905i	−0.995191	−	0.0979524i	$-0.968771\pi$
$587$		−	4.74640i		−	0.195905i	0.995191	−	0.0979524i	$-0.0312293\pi$
$588$		0			0
$589$	72.1445			2.97266
$590$		0			0
$591$	17.7695			0.730941
$592$		0			0
$593$		−	7.48084i		−	0.307201i	−0.988133	−	0.153601i	$-0.950913\pi$
$593$		−	7.48084i		−	0.307201i	0.988133	−	0.153601i	$-0.0490869\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			8.88573i			0.363669i
$598$		0			0
$599$	4.56165			0.186384			0.0931920	−	0.995648i	$-0.470293\pi$
$599$	4.56165			0.186384			0.0931920	+	0.995648i	$0.470293\pi$
$600$		0			0
$601$	−4.96538			−0.202542			−0.101271	−	0.994859i	$-0.532291\pi$
$601$	−4.96538			−0.202542			−0.101271	+	0.994859i	$0.532291\pi$
$602$		0			0
$603$		−	3.13229i		−	0.127557i
$604$		0			0
$605$	17.2328	+	3.34517i	0.700614	+	0.136001i
$606$		0			0
$607$			4.40325i			0.178722i	0.995999	+	0.0893612i	$0.0284825\pi$
$607$			4.40325i			0.178722i	−0.995999	+	0.0893612i	$0.971517\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.34957			0.216420
$612$		0			0
$613$		−	42.6950i		−	1.72444i	−0.506538	−	0.862218i	$-0.669075\pi$
$613$		−	42.6950i		−	1.72444i	0.506538	−	0.862218i	$-0.330925\pi$
$614$		0			0
$615$	1.72803	−	8.90206i	0.0696810	−	0.358966i
$616$		0			0
$617$		−	23.2135i		−	0.934541i	−0.884114	−	0.467270i	$-0.845237\pi$
$617$		−	23.2135i		−	0.934541i	0.884114	−	0.467270i	$-0.154763\pi$
$618$		0			0
$619$	28.0134			1.12595			0.562977	−	0.826473i	$-0.309656\pi$
$619$	28.0134			1.12595			0.562977	+	0.826473i	$0.309656\pi$
$620$		0			0
$621$	−2.17621			−0.0873284
$622$		0			0
$623$		0			0
$624$		0			0
$625$	18.0012	+	17.3481i	0.720047	+	0.693926i
$626$		0			0
$627$			26.7957i			1.07012i
$628$		0			0
$629$	−29.4403			−1.17386
$630$		0			0
$631$	−28.8446			−1.14829			−0.574144	−	0.818755i	$-0.694665\pi$
$631$	−28.8446			−1.14829			−0.574144	+	0.818755i	$0.694665\pi$
$632$		0			0
$633$			1.32608i			0.0527068i
$634$		0			0
$635$	5.41078	−	27.8739i	0.214720	−	1.10614i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−9.38718			−0.371351
$640$		0			0
$641$	−41.1854			−1.62673			−0.813363	−	0.581757i	$-0.802366\pi$
$641$	−41.1854			−1.62673			−0.813363	+	0.581757i	$0.802366\pi$
$642$		0			0
$643$		−	7.64360i		−	0.301434i	−0.988577	−	0.150717i	$-0.951842\pi$
$643$		−	7.64360i		−	0.301434i	0.988577	−	0.150717i	$-0.0481583\pi$
$644$		0			0
$645$	−26.9332	−	5.22816i	−1.06049	−	0.205859i
$646$		0			0
$647$		−	22.1376i		−	0.870318i	−0.900354	−	0.435159i	$-0.856692\pi$
$647$		−	22.1376i		−	0.870318i	0.900354	−	0.435159i	$-0.143308\pi$
$648$		0			0
$649$	13.4710			0.528781
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	18.3026i		−	0.716238i	−0.933676	−	0.358119i	$-0.883418\pi$
$653$		−	18.3026i		−	0.716238i	0.933676	−	0.358119i	$-0.116582\pi$
$654$		0			0
$655$	0.506917	+	0.0984007i	0.0198069	+	0.00384483i
$656$		0			0
$657$		−	22.4370i		−	0.875353i
$658$		0			0
$659$	28.1377			1.09609			0.548044	−	0.836449i	$-0.315373\pi$
$659$	28.1377			1.09609			0.548044	+	0.836449i	$0.315373\pi$
$660$		0			0
$661$	−23.2742			−0.905261			−0.452630	−	0.891698i	$-0.649514\pi$
$661$	−23.2742			−0.905261			−0.452630	+	0.891698i	$0.649514\pi$
$662$		0			0
$663$		−	17.8710i		−	0.694051i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			4.28558i			0.165938i
$668$		0			0
$669$	45.1449			1.74540
$670$		0			0
$671$	15.4769			0.597480
$672$		0			0
$673$		−	19.0282i		−	0.733484i	−0.930323	−	0.366742i	$-0.880473\pi$
$673$		−	19.0282i		−	0.733484i	0.930323	−	0.366742i	$-0.119527\pi$
$674$		0			0
$675$	6.13428	−	15.2051i	0.236108	−	0.585246i
$676$		0			0
$677$			40.7526i			1.56625i	0.621866	+	0.783124i	$0.286375\pi$
$677$			40.7526i			1.56625i	−0.621866	+	0.783124i	$0.713625\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	46.4847			1.78130
$682$		0			0
$683$		−	31.9224i		−	1.22148i	−0.791833	−	0.610738i	$-0.790873\pi$
$683$		−	31.9224i		−	1.22148i	0.791833	−	0.610738i	$-0.209127\pi$
$684$		0			0
$685$	−7.16353	+	36.9033i	−0.273704	+	1.41000i
$686$		0			0
$687$			24.5311i			0.935918i
$688$		0			0
$689$	−0.855710			−0.0326000
$690$		0			0
$691$	−12.1124			−0.460779			−0.230389	−	0.973099i	$-0.574000\pi$
$691$	−12.1124			−0.460779			−0.230389	+	0.973099i	$0.574000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−45.1159	−	8.75773i	−1.71134	−	0.332200i
$696$		0			0
$697$		−	11.2880i		−	0.427564i
$698$		0			0
$699$	−23.6528			−0.894631
$700$		0			0
$701$	−9.83067			−0.371299			−0.185650	−	0.982616i	$-0.559439\pi$
$701$	−9.83067			−0.371299			−0.185650	+	0.982616i	$0.559439\pi$
$702$		0			0
$703$		−	35.9323i		−	1.35521i
$704$		0			0
$705$	−17.1375	−	3.32667i	−0.645436	−	0.125290i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9.22203			0.346341			0.173170	−	0.984892i	$-0.444599\pi$
$709$	9.22203			0.346341			0.173170	+	0.984892i	$0.444599\pi$
$710$		0			0
$711$	−3.91075			−0.146664
$712$		0			0
$713$			6.68501i			0.250356i
$714$		0			0
$715$	−1.09236	+	5.62736i	−0.0408520	+	0.210451i
$716$		0			0
$717$			48.1262i			1.79731i
$718$		0			0
$719$	32.5773			1.21493			0.607464	−	0.794347i	$-0.292187\pi$
$719$	32.5773			1.21493			0.607464	+	0.794347i	$0.292187\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	8.86020i		−	0.329515i
$724$		0			0
$725$	−29.9432	−	12.0801i	−1.11206	−	0.448644i
$726$		0			0
$727$		−	28.4462i		−	1.05501i	−0.849552	−	0.527505i	$-0.823128\pi$
$727$		−	28.4462i		−	1.05501i	0.849552	−	0.527505i	$-0.176872\pi$
$728$		0			0
$729$	−4.49268			−0.166396
$730$		0			0
$731$	−34.1518			−1.26315
$732$		0			0
$733$			9.61408i			0.355104i	0.984111	+	0.177552i	$0.0568178\pi$
$733$			9.61408i			0.355104i	−0.984111	+	0.177552i	$0.943182\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			3.84801i			0.141743i
$738$		0			0
$739$	−3.88409			−0.142879			−0.0714393	−	0.997445i	$-0.522759\pi$
$739$	−3.88409			−0.142879			−0.0714393	+	0.997445i	$0.522759\pi$
$740$		0			0
$741$	21.8118			0.801275
$742$		0			0
$743$			16.8174i			0.616971i	0.951229	+	0.308486i	$0.0998222\pi$
$743$			16.8174i			0.616971i	−0.951229	+	0.308486i	$0.900178\pi$
$744$		0			0
$745$	46.6951	+	9.06428i	1.71078	+	0.332090i
$746$		0			0
$747$			9.17793i			0.335803i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−33.3654			−1.21752			−0.608760	−	0.793354i	$-0.708333\pi$
$751$	−33.3654			−1.21752			−0.608760	+	0.793354i	$0.708333\pi$
$752$		0			0
$753$			35.7525i			1.30290i
$754$		0			0
$755$	−14.2240	−	2.76111i	−0.517664	−	0.100487i
$756$		0			0
$757$			31.2781i			1.13682i	0.822744	+	0.568412i	$0.192442\pi$
$757$			31.2781i			1.13682i	−0.822744	+	0.568412i	$0.807558\pi$
$758$		0			0
$759$	−2.48293			−0.0901245
$760$		0			0
$761$	2.82448			0.102387			0.0511937	−	0.998689i	$-0.483697\pi$
$761$	2.82448			0.102387			0.0511937	+	0.998689i	$0.483697\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−3.61200	+	18.6074i	−0.130592	+	0.672753i
$766$		0			0
$767$		−	10.9654i		−	0.395937i
$768$		0			0
$769$	29.1637			1.05167			0.525836	−	0.850586i	$-0.323753\pi$
$769$	29.1637			1.05167			0.525836	+	0.850586i	$0.323753\pi$
$770$		0			0
$771$	56.7748			2.04469
$772$		0			0
$773$			40.7784i			1.46670i	0.679853	+	0.733348i	$0.262043\pi$
$773$			40.7784i			1.46670i	−0.679853	+	0.733348i	$0.737957\pi$
$774$		0			0
$775$	−46.7080	−	18.8436i	−1.67780	−	0.676882i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.7772			0.493618
$780$		0			0
$781$	11.5321			0.412652
$782$		0			0
$783$			21.1758i			0.756760i
$784$		0			0
$785$	−2.32222	+	11.9631i	−0.0828837	+	0.426980i
$786$		0			0
$787$			8.25206i			0.294154i	0.989125	+	0.147077i	$0.0469866\pi$
$787$			8.25206i			0.294154i	−0.989125	+	0.147077i	$0.953013\pi$
$788$		0			0
$789$	−46.2795			−1.64760
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	12.5982i		−	0.447377i
$794$		0			0
$795$	2.74130	+	0.532130i	0.0972238	+	0.0188727i
$796$		0			0
$797$		−	39.6687i		−	1.40514i	−0.711616	−	0.702569i	$-0.752036\pi$
$797$		−	39.6687i		−	1.40514i	0.711616	−	0.702569i	$-0.247964\pi$
$798$		0			0
$799$	−21.7307			−0.768779
$800$		0			0
$801$	8.46745			0.299183
$802$		0			0
$803$			27.5639i			0.972708i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			50.7772i			1.78744i
$808$		0			0
$809$	32.3156			1.13616			0.568078	−	0.822975i	$-0.307687\pi$
$809$	32.3156			1.13616			0.568078	+	0.822975i	$0.307687\pi$
$810$		0			0
$811$	7.67248			0.269417			0.134709	−	0.990885i	$-0.456990\pi$
$811$	7.67248			0.269417			0.134709	+	0.990885i	$0.456990\pi$
$812$		0			0
$813$		−	4.35833i		−	0.152853i
$814$		0			0
$815$	−2.67133	+	13.7615i	−0.0935726	+	0.482044i
$816$		0			0
$817$		−	41.6828i		−	1.45830i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17.0562			0.595266			0.297633	−	0.954680i	$-0.403803\pi$
$821$	17.0562			0.595266			0.297633	+	0.954680i	$0.403803\pi$
$822$		0			0
$823$			33.1024i			1.15388i	0.816787	+	0.576939i	$0.195753\pi$
$823$			33.1024i			1.15388i	−0.816787	+	0.576939i	$0.804247\pi$
$824$		0			0
$825$	6.99884	−	17.3481i	0.243668	−	0.603985i
$826$		0			0
$827$			20.1051i			0.699123i	0.936913	+	0.349561i	$0.113669\pi$
$827$			20.1051i			0.699123i	−0.936913	+	0.349561i	$0.886331\pi$
$828$		0			0
$829$	−18.9667			−0.658740			−0.329370	−	0.944201i	$-0.606836\pi$
$829$	−18.9667			−0.658740			−0.329370	+	0.944201i	$0.606836\pi$
$830$		0			0
$831$	35.9993			1.24880
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−7.91391	+	40.7690i	−0.273872	+	1.41087i
$836$		0			0
$837$			33.0317i			1.14174i
$838$		0			0
$839$	−4.15025			−0.143283			−0.0716413	−	0.997430i	$-0.522824\pi$
$839$	−4.15025			−0.143283			−0.0716413	+	0.997430i	$0.522824\pi$
$840$		0			0
$841$	12.7010			0.437967
$842$		0			0
$843$			27.8358i			0.958717i
$844$		0			0
$845$	−23.9555	−	4.65016i	−0.824095	−	0.159970i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−56.0762			−1.92453
$850$		0			0
$851$	3.32954			0.114135
$852$		0			0
$853$		−	1.53140i		−	0.0524340i	−0.999656	−	0.0262170i	$-0.991654\pi$
$853$		−	1.53140i		−	0.0524340i	0.999656	−	0.0262170i	$-0.00834609\pi$
$854$		0			0
$855$	−22.7106	−	4.40850i	−0.776686	−	0.150767i
$856$		0			0
$857$			23.8235i			0.813794i	0.913474	+	0.406897i	$0.133389\pi$
$857$			23.8235i			0.813794i	−0.913474	+	0.406897i	$0.866611\pi$
$858$		0			0
$859$	−14.6833			−0.500987			−0.250494	−	0.968118i	$-0.580593\pi$
$859$	−14.6833			−0.500987			−0.250494	+	0.968118i	$0.580593\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			42.3325i			1.44102i	0.693446	+	0.720508i	$0.256091\pi$
$863$			42.3325i			1.44102i	−0.693446	+	0.720508i	$0.743909\pi$
$864$		0			0
$865$	−5.95147	+	30.6593i	−0.202356	+	1.04245i
$866$		0			0
$867$			36.7549i			1.24826i
$868$		0			0
$869$	4.80434			0.162976
$870$		0			0
$871$	3.13229			0.106134
$872$		0			0
$873$		−	21.3432i		−	0.722359i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			10.5940i			0.357735i	0.983873	+	0.178868i	$0.0572434\pi$
$877$			10.5940i			0.357735i	−0.983873	+	0.178868i	$0.942757\pi$
$878$		0			0
$879$	0.996512			0.0336115
$880$		0			0
$881$	27.1174			0.913609			0.456805	−	0.889567i	$-0.348994\pi$
$881$	27.1174			0.913609			0.456805	+	0.889567i	$0.348994\pi$
$882$		0			0
$883$			21.7240i			0.731072i	0.930797	+	0.365536i	$0.119114\pi$
$883$			21.7240i			0.731072i	−0.930797	+	0.365536i	$0.880886\pi$
$884$		0			0
$885$	−6.81891	+	35.1280i	−0.229215	+	1.18081i
$886$		0			0
$887$		−	21.1605i		−	0.710501i	−0.934771	−	0.355251i	$-0.884396\pi$
$887$		−	21.1605i		−	0.710501i	0.934771	−	0.355251i	$-0.115604\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−19.9594			−0.668664
$892$		0			0
$893$		−	26.5227i		−	0.887547i
$894$		0			0
$895$	17.3970	+	3.37704i	0.581517	+	0.112882i
$896$		0			0
$897$			2.02111i			0.0674828i
$898$		0			0
$899$	65.0488			2.16950
$900$		0			0
$901$	3.47602			0.115803
$902$		0			0
$903$		0			0
$904$		0			0
$905$	25.6578	+	4.98060i	0.852894	+	0.165561i
$906$		0			0
$907$		−	37.8642i		−	1.25726i	−0.777705	−	0.628630i	$-0.783616\pi$
$907$		−	37.8642i		−	1.25726i	0.777705	−	0.628630i	$-0.216384\pi$
$908$		0			0
$909$	−5.02978			−0.166827
$910$		0			0
$911$	−30.1221			−0.997990			−0.498995	−	0.866605i	$-0.666297\pi$
$911$	−30.1221			−0.997990			−0.498995	+	0.866605i	$0.666297\pi$
$912$		0			0
$913$		−	11.2751i		−	0.373150i
$914$		0			0
$915$	−7.83432	+	40.3589i	−0.258995	+	1.33422i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	46.1132			1.52113			0.760566	−	0.649260i	$-0.224921\pi$
$919$	46.1132			1.52113			0.760566	+	0.649260i	$0.224921\pi$
$920$		0			0
$921$	32.0604			1.05642
$922$		0			0
$923$		−	9.38718i		−	0.308983i
$924$		0			0
$925$	−9.38525	+	23.2634i	−0.308585	+	0.764896i
$926$		0			0
$927$			11.6701i			0.383297i
$928$		0			0
$929$	−27.5174			−0.902816			−0.451408	−	0.892318i	$-0.649078\pi$
$929$	−27.5174			−0.902816			−0.451408	+	0.892318i	$0.649078\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			34.9555i			1.14439i
$934$		0			0
$935$	4.43734	−	22.8592i	0.145117	−	0.747575i
$936$		0			0
$937$			50.4704i			1.64880i	0.566010	+	0.824399i	$0.308486\pi$
$937$			50.4704i			1.64880i	−0.566010	+	0.824399i	$0.691514\pi$
$938$		0			0
$939$	2.48665			0.0811487
$940$		0			0
$941$	1.85663			0.0605243			0.0302622	−	0.999542i	$-0.490366\pi$
$941$	1.85663			0.0605243			0.0302622	+	0.999542i	$0.490366\pi$
$942$		0			0
$943$			1.27661i			0.0415721i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	36.1689i		−	1.17533i	−0.809104	−	0.587666i	$-0.800047\pi$
$947$		−	36.1689i		−	1.17533i	0.809104	−	0.587666i	$-0.199953\pi$
$948$		0			0
$949$	22.4370			0.728337
$950$		0			0
$951$	−57.8258			−1.87513
$952$		0			0
$953$		−	17.3367i		−	0.561590i	−0.959768	−	0.280795i	$-0.909402\pi$
$953$		−	17.3367i		−	0.561590i	0.959768	−	0.280795i	$-0.0905981\pi$
$954$		0			0
$955$	28.4883	+	5.53004i	0.921859	+	0.178948i
$956$		0			0
$957$			24.1602i			0.780990i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	70.4688			2.27319
$962$		0			0
$963$			0.187282i			0.00603506i
$964$		0			0
$965$	9.12086	−	46.9866i	0.293611	−	1.51255i
$966$		0			0
$967$			23.4618i			0.754480i	0.926116	+	0.377240i	$0.123127\pi$
$967$			23.4618i			0.754480i	−0.926116	+	0.377240i	$0.876873\pi$
$968$		0			0
$969$	−88.6027			−2.84633
$970$		0			0
$971$	−56.5553			−1.81495			−0.907473	−	0.420110i	$-0.861991\pi$
$971$	−56.5553			−1.81495			−0.907473	+	0.420110i	$0.861991\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−14.1214	−	5.69707i	−0.452248	−	0.182452i
$976$		0			0
$977$		−	28.9783i		−	0.927099i	−0.886071	−	0.463549i	$-0.846576\pi$
$977$		−	28.9783i		−	0.927099i	0.886071	−	0.463549i	$-0.153424\pi$
$978$		0			0
$979$	−10.4022			−0.332457
$980$		0			0
$981$	17.7174			0.565673
$982$		0			0
$983$			41.3280i			1.31816i	0.752074	+	0.659079i	$0.229054\pi$
$983$			41.3280i			1.31816i	−0.752074	+	0.659079i	$0.770946\pi$
$984$		0			0
$985$	3.59150	−	18.5018i	0.114435	−	0.589517i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.86238			0.122817
$990$		0			0
$991$	−29.0897			−0.924065			−0.462032	−	0.886863i	$-0.652880\pi$
$991$	−29.0897			−0.924065			−0.462032	+	0.886863i	$0.652880\pi$
$992$		0			0
$993$		−	10.9872i		−	0.348667i
$994$		0			0
$995$	9.25191	+	1.79595i	0.293305	+	0.0569353i
$996$		0			0
$997$		−	21.4734i		−	0.680069i	−0.940413	−	0.340035i	$-0.889561\pi$
$997$		−	21.4734i		−	0.680069i	0.940413	−	0.340035i	$-0.110439\pi$
$998$		0			0
$999$	16.4518			0.520512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.g.f.1569.3		12
5.2	odd	4		9800.2.a.cv.1.2		6
5.3	odd	4		9800.2.a.cx.1.5		6
5.4	even	2	inner	1960.2.g.f.1569.10		12
7.2	even	3		280.2.bg.a.249.10	yes	24
7.4	even	3		280.2.bg.a.9.3	✓	24
7.6	odd	2		1960.2.g.e.1569.10		12
28.11	odd	6		560.2.bw.f.289.10		24
28.23	odd	6		560.2.bw.f.529.3		24
35.2	odd	12		1400.2.q.o.1201.5		12
35.4	even	6		280.2.bg.a.9.10	yes	24
35.9	even	6		280.2.bg.a.249.3	yes	24
35.13	even	4		9800.2.a.cw.1.2		6
35.18	odd	12		1400.2.q.n.401.2		12
35.23	odd	12		1400.2.q.n.1201.2		12
35.27	even	4		9800.2.a.cy.1.5		6
35.32	odd	12		1400.2.q.o.401.5		12
35.34	odd	2		1960.2.g.e.1569.3		12
140.39	odd	6		560.2.bw.f.289.3		24
140.79	odd	6		560.2.bw.f.529.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.bg.a.9.3	✓	24	7.4	even	3
280.2.bg.a.9.10	yes	24	35.4	even	6
280.2.bg.a.249.3	yes	24	35.9	even	6
280.2.bg.a.249.10	yes	24	7.2	even	3
560.2.bw.f.289.3		24	140.39	odd	6
560.2.bw.f.289.10		24	28.11	odd	6
560.2.bw.f.529.3		24	28.23	odd	6
560.2.bw.f.529.10		24	140.79	odd	6
1400.2.q.n.401.2		12	35.18	odd	12
1400.2.q.n.1201.2		12	35.23	odd	12
1400.2.q.o.401.5		12	35.32	odd	12
1400.2.q.o.1201.5		12	35.2	odd	12
1960.2.g.e.1569.3		12	35.34	odd	2
1960.2.g.e.1569.10		12	7.6	odd	2
1960.2.g.f.1569.3		12	1.1	even	1	trivial
1960.2.g.f.1569.10		12	5.4	even	2	inner
9800.2.a.cv.1.2		6	5.2	odd	4
9800.2.a.cw.1.2		6	35.13	even	4
9800.2.a.cx.1.5		6	5.3	odd	4
9800.2.a.cy.1.5		6	35.27	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:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.10821i q^{3} +(-2.19509 - 0.426103i) q^{5} -1.44457 q^{9} +O(q^{10})\)	\(q-2.10821i q^{3} +(-2.19509 - 0.426103i) q^{5} -1.44457 q^{9} +1.77465 q^{11} -1.44457i q^{13} +(-0.898317 + 4.62773i) q^{15} +5.86806i q^{17} -7.16205 q^{19} -0.663645i q^{23} +(4.63687 + 1.87067i) q^{25} -3.27918i q^{27} -6.45763 q^{29} -10.0732 q^{31} -3.74135i q^{33} +5.01705i q^{37} -3.04546 q^{39} -1.92363 q^{41} +5.81995i q^{43} +(3.17097 + 0.615536i) q^{45} +3.70322i q^{47} +12.3711 q^{51} -0.592363i q^{53} +(-3.89553 - 0.756185i) q^{55} +15.0991i q^{57} +7.59076 q^{59} +8.72111 q^{61} +(-0.615536 + 3.17097i) q^{65} +2.16832i q^{67} -1.39911 q^{69} +6.49826 q^{71} +15.5320i q^{73} +(3.94378 - 9.77552i) q^{75} +2.70720 q^{79} -11.2469 q^{81} -6.35340i q^{83} +(2.50040 - 12.8809i) q^{85} +13.6141i q^{87} -5.86157 q^{89} +21.2364i q^{93} +(15.7214 + 3.05177i) q^{95} +14.7748i q^{97} -2.56361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{9}+O(q^{10}) \) 12 * q - 14 * q^9	\( 12 q - 14 q^{9} + 2 q^{11} + 6 q^{15} + 10 q^{19} + 2 q^{25} + 6 q^{29} - 4 q^{31} - 20 q^{39} + 12 q^{41} + 8 q^{45} - 6 q^{55} + 48 q^{59} + 18 q^{61} + 26 q^{65} - 30 q^{69} + 8 q^{71} + 14 q^{75} + 44 q^{79} - 12 q^{81} - 22 q^{85} - 30 q^{89} + 26 q^{95} - 22 q^{99}+O(q^{100}) \) 12 * q - 14 * q^9 + 2 * q^11 + 6 * q^15 + 10 * q^19 + 2 * q^25 + 6 * q^29 - 4 * q^31 - 20 * q^39 + 12 * q^41 + 8 * q^45 - 6 * q^55 + 48 * q^59 + 18 * q^61 + 26 * q^65 - 30 * q^69 + 8 * q^71 + 14 * q^75 + 44 * q^79 - 12 * q^81 - 22 * q^85 - 30 * q^89 + 26 * q^95 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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