LMFDB - Embedded newform 1520.2.d.h.609.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.16516096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 13x^{2} + 1 $ x^6 + 9x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.1
Root			$0.285442i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.h.609.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.21789i q^{3} +(-0.370556 - 2.20515i) q^{5} +2.59637i q^{7} -7.35482 q^{9} +O(q^{10})$	$q-3.21789i q^{3} +(-0.370556 - 2.20515i) q^{5} +2.59637i q^{7} -7.35482 q^{9} -0.741113 q^{11} -3.78878i q^{13} +(-7.09593 + 1.19241i) q^{15} +3.16725i q^{17} -1.00000 q^{19} +8.35482 q^{21} -0.570885i q^{23} +(-4.72538 + 1.63427i) q^{25} +14.0133i q^{27} -6.00000 q^{29} -5.83705 q^{31} +2.38482i q^{33} +(5.72538 - 0.962100i) q^{35} +1.40396i q^{37} -12.1919 q^{39} -3.83705 q^{41} +2.59637i q^{43} +(2.72538 + 16.2185i) q^{45} +5.08247i q^{47} +0.258887 q^{49} +10.1919 q^{51} -0.160905i q^{53} +(0.274624 + 1.63427i) q^{55} +3.21789i q^{57} +8.35482 q^{59} -8.57816 q^{61} -19.0958i q^{63} +(-8.35482 + 1.40396i) q^{65} -14.8464i q^{67} -1.83705 q^{69} -3.64518 q^{71} -10.8461i q^{73} +(5.25889 + 15.2057i) q^{75} -1.92420i q^{77} +1.83705 q^{79} +23.0289 q^{81} +4.19876i q^{83} +(6.98426 - 1.17365i) q^{85} +19.3073i q^{87} +16.9015 q^{89} +9.83705 q^{91} +18.7830i q^{93} +(0.370556 + 2.20515i) q^{95} +3.78878i q^{97} +5.45075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{5} - 14 q^{9}+O(q^{10}) $ 6 * q - q^5 - 14 * q^9	$ 6 q - q^{5} - 14 q^{9} - 2 q^{11} - 10 q^{15} - 6 q^{19} + 20 q^{21} + 3 q^{25} - 36 q^{29} + 3 q^{35} - 8 q^{39} + 12 q^{41} - 15 q^{45} + 4 q^{49} - 4 q^{51} + 33 q^{55} + 20 q^{59} - 14 q^{61} - 20 q^{65} + 24 q^{69} - 52 q^{71} + 34 q^{75} - 24 q^{79} + 38 q^{81} + 13 q^{85} - 24 q^{89} + 24 q^{91} + q^{95} - 30 q^{99}+O(q^{100}) $ 6 * q - q^5 - 14 * q^9 - 2 * q^11 - 10 * q^15 - 6 * q^19 + 20 * q^21 + 3 * q^25 - 36 * q^29 + 3 * q^35 - 8 * q^39 + 12 * q^41 - 15 * q^45 + 4 * q^49 - 4 * q^51 + 33 * q^55 + 20 * q^59 - 14 * q^61 - 20 * q^65 + 24 * q^69 - 52 * q^71 + 34 * q^75 - 24 * q^79 + 38 * q^81 + 13 * q^85 - 24 * q^89 + 24 * q^91 + q^95 - 30 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	3.21789i		−	1.85785i	−0.370268	−	0.928925i	$-0.620734\pi$
$3$		−	3.21789i		−	1.85785i	0.370268	−	0.928925i	$-0.379266\pi$
$4$		0			0
$5$	−0.370556	−	2.20515i	−0.165718	−	0.986173i
$5$	−0.370556	−	2.20515i	−0.165718	−	0.986173i
$6$		0			0
$7$			2.59637i			0.981334i	0.871347	+	0.490667i	$0.163247\pi$
$7$			2.59637i			0.981334i	−0.871347	+	0.490667i	$0.836753\pi$
$8$		0			0
$9$	−7.35482			−2.45161
$10$		0			0
$11$	−0.741113			−0.223454			−0.111727	−	0.993739i	$-0.535638\pi$
$11$	−0.741113			−0.223454			−0.111727	+	0.993739i	$0.535638\pi$
$12$		0			0
$13$		−	3.78878i		−	1.05082i	−0.850850	−	0.525409i	$-0.823912\pi$
$13$		−	3.78878i		−	1.05082i	0.850850	−	0.525409i	$-0.176088\pi$
$14$		0			0
$15$	−7.09593	+	1.19241i	−1.83216	+	0.307879i
$16$		0			0
$17$			3.16725i			0.768171i	0.923298	+	0.384086i	$0.125483\pi$
$17$			3.16725i			0.768171i	−0.923298	+	0.384086i	$0.874517\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	8.35482			1.82317
$22$		0			0
$23$		−	0.570885i		−	0.119038i	−0.998227	−	0.0595189i	$-0.981043\pi$
$23$		−	0.570885i		−	0.119038i	0.998227	−	0.0595189i	$-0.0189566\pi$
$24$		0			0
$25$	−4.72538	+	1.63427i	−0.945075	+	0.326853i
$26$		0			0
$27$			14.0133i			2.69687i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−5.83705			−1.04836			−0.524182	−	0.851606i	$-0.675629\pi$
$31$	−5.83705			−1.04836			−0.524182	+	0.851606i	$0.675629\pi$
$32$		0			0
$33$			2.38482i			0.415144i
$34$		0			0
$35$	5.72538	−	0.962100i	0.967765	−	0.162625i
$36$		0			0
$37$			1.40396i			0.230809i	0.993319	+	0.115404i	$0.0368164\pi$
$37$			1.40396i			0.230809i	−0.993319	+	0.115404i	$0.963184\pi$
$38$		0			0
$39$	−12.1919			−1.95226
$40$		0			0
$41$	−3.83705			−0.599246			−0.299623	−	0.954058i	$-0.596861\pi$
$41$	−3.83705			−0.599246			−0.299623	+	0.954058i	$0.596861\pi$
$42$		0			0
$43$			2.59637i			0.395942i	0.980208	+	0.197971i	$0.0634352\pi$
$43$			2.59637i			0.395942i	−0.980208	+	0.197971i	$0.936565\pi$
$44$		0			0
$45$	2.72538	+	16.2185i	0.406275	+	2.41771i
$46$		0			0
$47$			5.08247i			0.741354i	0.928762	+	0.370677i	$0.120874\pi$
$47$			5.08247i			0.741354i	−0.928762	+	0.370677i	$0.879126\pi$
$48$		0			0
$49$	0.258887			0.0369839
$50$		0			0
$51$	10.1919			1.42715
$52$		0			0
$53$		−	0.160905i		−	0.0221020i	−0.999939	−	0.0110510i	$-0.996482\pi$
$53$		−	0.160905i		−	0.0221020i	0.999939	−	0.0110510i	$-0.00351771\pi$
$54$		0			0
$55$	0.274624	+	1.63427i	0.0370303	+	0.220364i
$56$		0			0
$57$			3.21789i			0.426220i
$58$		0			0
$59$	8.35482			1.08770			0.543852	−	0.839181i	$-0.316965\pi$
$59$	8.35482			1.08770			0.543852	+	0.839181i	$0.316965\pi$
$60$		0			0
$61$	−8.57816			−1.09832			−0.549160	−	0.835717i	$-0.685052\pi$
$61$	−8.57816			−1.09832			−0.549160	+	0.835717i	$0.685052\pi$
$62$		0			0
$63$		−	19.0958i		−	2.40584i
$64$		0			0
$65$	−8.35482	+	1.40396i	−1.03629	+	0.174139i
$66$		0			0
$67$		−	14.8464i		−	1.81378i	−0.421371	−	0.906888i	$-0.638451\pi$
$67$		−	14.8464i		−	1.81378i	0.421371	−	0.906888i	$-0.361549\pi$
$68$		0			0
$69$	−1.83705			−0.221154
$70$		0			0
$71$	−3.64518			−0.432603			−0.216302	−	0.976327i	$-0.569399\pi$
$71$	−3.64518			−0.432603			−0.216302	+	0.976327i	$0.569399\pi$
$72$		0			0
$73$		−	10.8461i		−	1.26944i	−0.772743	−	0.634719i	$-0.781116\pi$
$73$		−	10.8461i		−	1.26944i	0.772743	−	0.634719i	$-0.218884\pi$
$74$		0			0
$75$	5.25889	+	15.2057i	0.607244	+	1.75581i
$76$		0			0
$77$		−	1.92420i		−	0.219283i
$78$		0			0
$79$	1.83705			0.206684			0.103342	−	0.994646i	$-0.467046\pi$
$79$	1.83705			0.206684			0.103342	+	0.994646i	$0.467046\pi$
$80$		0			0
$81$	23.0289			2.55877
$82$		0			0
$83$			4.19876i			0.460873i	0.973087	+	0.230437i	$0.0740154\pi$
$83$			4.19876i			0.460873i	−0.973087	+	0.230437i	$0.925985\pi$
$84$		0			0
$85$	6.98426	−	1.17365i	0.757550	−	0.127300i
$86$		0			0
$87$			19.3073i			2.06996i
$88$		0			0
$89$	16.9015			1.79156			0.895778	−	0.444502i	$-0.146619\pi$
$89$	16.9015			1.79156			0.895778	+	0.444502i	$0.146619\pi$
$90$		0			0
$91$	9.83705			1.03120
$92$		0			0
$93$			18.7830i			1.94770i
$94$		0			0
$95$	0.370556	+	2.20515i	0.0380183	+	0.226244i
$96$		0			0
$97$			3.78878i			0.384692i	0.981327	+	0.192346i	$0.0616096\pi$
$97$			3.78878i			0.384692i	−0.981327	+	0.192346i	$0.938390\pi$
$98$		0			0
$99$	5.45075			0.547821
$100$		0			0
$101$	8.35482			0.831336			0.415668	−	0.909517i	$-0.363548\pi$
$101$	8.35482			0.831336			0.415668	+	0.909517i	$0.363548\pi$
$102$		0			0
$103$		−	2.07612i		−	0.204566i	−0.994755	−	0.102283i	$-0.967385\pi$
$103$		−	2.07612i		−	0.204566i	0.994755	−	0.102283i	$-0.0326148\pi$
$104$		0			0
$105$	−3.09593	−	18.4236i	−0.302132	−	1.79796i
$106$		0			0
$107$		−	5.70399i		−	0.551426i	−0.961240	−	0.275713i	$-0.911086\pi$
$107$		−	5.70399i		−	0.551426i	0.961240	−	0.275713i	$-0.0889139\pi$
$108$		0			0
$109$	−1.64518			−0.157580			−0.0787899	−	0.996891i	$-0.525106\pi$
$109$	−1.64518			−0.157580			−0.0787899	+	0.996891i	$0.525106\pi$
$110$		0			0
$111$	4.51777			0.428808
$112$		0			0
$113$		−	3.89006i		−	0.365946i	−0.983118	−	0.182973i	$-0.941428\pi$
$113$		−	3.89006i		−	0.365946i	0.983118	−	0.182973i	$-0.0585720\pi$
$114$		0			0
$115$	−1.25889	+	0.211545i	−0.117392	+	0.0197267i
$116$		0			0
$117$			27.8658i			2.57619i
$118$		0			0
$119$	−8.22334			−0.753832
$120$		0			0
$121$	−10.4508			−0.950068
$122$		0			0
$123$			12.3472i			1.11331i
$124$		0			0
$125$	5.35482	+	9.81458i	0.478950	+	0.877842i
$126$		0			0
$127$		−	14.4233i		−	1.27986i	−0.768432	−	0.639931i	$-0.778963\pi$
$127$		−	14.4233i		−	1.27986i	0.768432	−	0.639931i	$-0.221037\pi$
$128$		0			0
$129$	8.35482			0.735601
$130$		0			0
$131$	−9.96853			−0.870954			−0.435477	−	0.900200i	$-0.643420\pi$
$131$	−9.96853			−0.870954			−0.435477	+	0.900200i	$0.643420\pi$
$132$		0			0
$133$		−	2.59637i		−	0.225133i
$134$		0			0
$135$	30.9015	−	5.19273i	2.65958	−	0.446919i
$136$		0			0
$137$			9.70431i			0.829095i	0.910028	+	0.414548i	$0.136060\pi$
$137$			9.70431i			0.829095i	−0.910028	+	0.414548i	$0.863940\pi$
$138$		0			0
$139$	−13.4508			−1.14088			−0.570439	−	0.821340i	$-0.693227\pi$
$139$	−13.4508			−1.14088			−0.570439	+	0.821340i	$0.693227\pi$
$140$		0			0
$141$	16.3548			1.37732
$142$		0			0
$143$			2.80791i			0.234809i
$144$		0			0
$145$	2.22334	+	13.2309i	0.184638	+	1.09877i
$146$		0			0
$147$		−	0.833070i		−	0.0687105i
$148$		0			0
$149$	−15.0959			−1.23671			−0.618353	−	0.785900i	$-0.712200\pi$
$149$	−15.0959			−1.23671			−0.618353	+	0.785900i	$0.712200\pi$
$150$		0			0
$151$	−14.1919			−1.15492			−0.577459	−	0.816420i	$-0.695956\pi$
$151$	−14.1919			−1.15492			−0.577459	+	0.816420i	$0.695956\pi$
$152$		0			0
$153$		−	23.2946i		−	1.88325i
$154$		0			0
$155$	2.16295	+	12.8716i	0.173733	+	1.03387i
$156$		0			0
$157$			7.57755i			0.604754i	0.953188	+	0.302377i	$0.0977802\pi$
$157$			7.57755i			0.604754i	−0.953188	+	0.302377i	$0.902220\pi$
$158$		0			0
$159$	−0.517774			−0.0410622
$160$		0			0
$161$	1.48223			0.116816
$162$		0			0
$163$		−	19.6757i		−	1.54112i	−0.637369	−	0.770559i	$-0.719977\pi$
$163$		−	19.6757i		−	1.54112i	0.637369	−	0.770559i	$-0.280023\pi$
$164$		0			0
$165$	5.25889	−	0.883711i	0.409404	−	0.0687968i
$166$		0			0
$167$			10.7954i			0.835376i	0.908590	+	0.417688i	$0.137160\pi$
$167$			10.7954i			0.835376i	−0.908590	+	0.417688i	$0.862840\pi$
$168$		0			0
$169$	−1.35482			−0.104217
$170$		0			0
$171$	7.35482			0.562437
$172$		0			0
$173$		−	20.3895i		−	1.55018i	−0.631848	−	0.775092i	$-0.717703\pi$
$173$		−	20.3895i		−	1.55018i	0.631848	−	0.775092i	$-0.282297\pi$
$174$		0			0
$175$	−4.24315	−	12.2688i	−0.320752	−	0.927434i
$176$		0			0
$177$		−	26.8849i		−	2.02079i
$178$		0			0
$179$	25.0645			1.87341			0.936703	−	0.350126i	$-0.113861\pi$
$179$	25.0645			1.87341			0.936703	+	0.350126i	$0.113861\pi$
$180$		0			0
$181$	−19.4193			−1.44342			−0.721712	−	0.692194i	$-0.756644\pi$
$181$	−19.4193			−1.44342			−0.721712	+	0.692194i	$0.756644\pi$
$182$		0			0
$183$			27.6036i			2.04051i
$184$		0			0
$185$	3.09593	−	0.520245i	0.227617	−	0.0382491i
$186$		0			0
$187$		−	2.34729i		−	0.171651i
$188$		0			0
$189$	−36.3837			−2.64653
$190$		0			0
$191$	−11.4508			−0.828547			−0.414274	−	0.910152i	$-0.635964\pi$
$191$	−11.4508			−0.828547			−0.414274	+	0.910152i	$0.635964\pi$
$192$		0			0
$193$		−	3.78878i		−	0.272722i	−0.990659	−	0.136361i	$-0.956459\pi$
$193$		−	3.78878i		−	0.272722i	0.990659	−	0.136361i	$-0.0435407\pi$
$194$		0			0
$195$	4.51777	+	26.8849i	0.323525	+	1.92527i
$196$		0			0
$197$			2.28354i			0.162695i	0.996686	+	0.0813477i	$0.0259224\pi$
$197$			2.28354i			0.162695i	−0.996686	+	0.0813477i	$0.974078\pi$
$198$		0			0
$199$	−19.4508			−1.37883			−0.689414	−	0.724368i	$-0.742132\pi$
$199$	−19.4508			−1.37883			−0.689414	+	0.724368i	$0.742132\pi$
$200$		0			0
$201$	−47.7741			−3.36972
$202$		0			0
$203$		−	15.5782i		−	1.09337i
$204$		0			0
$205$	1.42184	+	8.46126i	0.0993057	+	0.590960i
$206$		0			0
$207$			4.19876i			0.291834i
$208$		0			0
$209$	0.741113			0.0512639
$210$		0			0
$211$	−11.2274			−0.772927			−0.386463	−	0.922305i	$-0.626303\pi$
$211$	−11.2274			−0.772927			−0.386463	+	0.922305i	$0.626303\pi$
$212$		0			0
$213$			11.7298i			0.803712i
$214$		0			0
$215$	5.72538	−	0.962100i	0.390467	−	0.0656147i
$216$		0			0
$217$		−	15.1551i		−	1.02880i
$218$		0			0
$219$	−34.9015			−2.35843
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$		−	4.03785i		−	0.270394i	−0.990819	−	0.135197i	$-0.956833\pi$
$223$		−	4.03785i		−	0.270394i	0.990819	−	0.135197i	$-0.0431668\pi$
$224$		0			0
$225$	34.7543	−	12.0197i	2.31695	−	0.801315i
$226$		0			0
$227$			11.2185i			0.744600i	0.928112	+	0.372300i	$0.121431\pi$
$227$			11.2185i			0.744600i	−0.928112	+	0.372300i	$0.878569\pi$
$228$		0			0
$229$	−16.1315			−1.06600			−0.532999	−	0.846116i	$-0.678935\pi$
$229$	−16.1315			−1.06600			−0.532999	+	0.846116i	$0.678935\pi$
$230$		0			0
$231$	−6.19186			−0.407395
$232$		0			0
$233$			2.12676i			0.139329i	0.997570	+	0.0696644i	$0.0221928\pi$
$233$			2.12676i			0.139329i	−0.997570	+	0.0696644i	$0.977807\pi$
$234$		0			0
$235$	11.2076	−	1.88334i	0.731103	−	0.122856i
$236$		0			0
$237$		−	5.91141i		−	0.383987i
$238$		0			0
$239$	−14.4152			−0.932442			−0.466221	−	0.884668i	$-0.654385\pi$
$239$	−14.4152			−0.932442			−0.466221	+	0.884668i	$0.654385\pi$
$240$		0			0
$241$	−0.162955			−0.0104968			−0.00524842	−	0.999986i	$-0.501671\pi$
$241$	−0.162955			−0.0104968			−0.00524842	+	0.999986i	$0.501671\pi$
$242$		0			0
$243$		−	32.0645i		−	2.05694i
$244$		0			0
$245$	−0.0959323	−	0.570885i	−0.00612889	−	0.0364725i
$246$		0			0
$247$			3.78878i			0.241074i
$248$		0			0
$249$	13.5111			0.856233
$250$		0			0
$251$	−12.9330			−0.816322			−0.408161	−	0.912910i	$-0.633830\pi$
$251$	−12.9330			−0.816322			−0.408161	+	0.912910i	$0.633830\pi$
$252$		0			0
$253$			0.423090i			0.0265995i
$254$		0			0
$255$	−3.77666	−	22.4746i	−0.236504	−	1.40741i
$256$		0			0
$257$		−	11.0445i		−	0.688938i	−0.938798	−	0.344469i	$-0.888059\pi$
$257$		−	11.0445i		−	0.688938i	0.938798	−	0.344469i	$-0.111941\pi$
$258$		0			0
$259$	−3.64518			−0.226501
$260$		0			0
$261$	44.1289			2.73151
$262$		0			0
$263$			17.8527i			1.10085i	0.834885	+	0.550424i	$0.185534\pi$
$263$			17.8527i			1.10085i	−0.834885	+	0.550424i	$0.814466\pi$
$264$		0			0
$265$	−0.354819	+	0.0596243i	−0.0217964	+	0.00366269i
$266$		0			0
$267$		−	54.3872i		−	3.32844i
$268$		0			0
$269$	−24.9934			−1.52387			−0.761936	−	0.647652i	$-0.775751\pi$
$269$	−24.9934			−1.52387			−0.761936	+	0.647652i	$0.775751\pi$
$270$		0			0
$271$	23.8660			1.44975			0.724877	−	0.688879i	$-0.241897\pi$
$271$	23.8660			1.44975			0.724877	+	0.688879i	$0.241897\pi$
$272$		0			0
$273$		−	31.6545i		−	1.91582i
$274$		0			0
$275$	3.50204	−	1.21118i	0.211181	−	0.0730366i
$276$		0			0
$277$		−	21.2315i		−	1.27568i	−0.770169	−	0.637840i	$-0.779828\pi$
$277$		−	21.2315i		−	1.27568i	0.770169	−	0.637840i	$-0.220172\pi$
$278$		0			0
$279$	42.9304			2.57018
$280$		0			0
$281$	3.83705			0.228899			0.114449	−	0.993429i	$-0.463490\pi$
$281$	3.83705			0.228899			0.114449	+	0.993429i	$0.463490\pi$
$282$		0			0
$283$			0.211545i			0.0125751i	0.999980	+	0.00628753i	$0.00200139\pi$
$283$			0.211545i			0.0125751i	−0.999980	+	0.00628753i	$0.997999\pi$
$284$		0			0
$285$	7.09593	−	1.19241i	0.420327	−	0.0706323i
$286$		0			0
$287$		−	9.96237i		−	0.588060i
$288$		0			0
$289$	6.96853			0.409913
$290$		0			0
$291$	12.1919			0.714700
$292$		0			0
$293$			14.9942i			0.875970i	0.898982	+	0.437985i	$0.144308\pi$
$293$			14.9942i			0.875970i	−0.898982	+	0.437985i	$0.855692\pi$
$294$		0			0
$295$	−3.09593	−	18.4236i	−0.180252	−	1.07267i
$296$		0			0
$297$		−	10.3855i		−	0.602626i
$298$		0			0
$299$	−2.16295			−0.125087
$300$		0			0
$301$	−6.74111			−0.388551
$302$		0			0
$303$		−	26.8849i		−	1.54450i
$304$		0			0
$305$	3.17869	+	18.9161i	0.182011	+	1.08313i
$306$		0			0
$307$		−	1.65303i		−	0.0943434i	−0.998887	−	0.0471717i	$-0.984979\pi$
$307$		−	1.65303i		−	0.0943434i	0.998887	−	0.0471717i	$-0.0150208\pi$
$308$		0			0
$309$	−6.68073			−0.380053
$310$		0			0
$311$	0.741113			0.0420247			0.0210123	−	0.999779i	$-0.493311\pi$
$311$	0.741113			0.0420247			0.0210123	+	0.999779i	$0.493311\pi$
$312$		0			0
$313$			26.8849i			1.51962i	0.650143	+	0.759812i	$0.274709\pi$
$313$			26.8849i			1.51962i	−0.650143	+	0.759812i	$0.725291\pi$
$314$		0			0
$315$	−42.1091	+	7.07607i	−2.37258	+	0.398691i
$316$		0			0
$317$		−	8.16155i		−	0.458398i	−0.973380	−	0.229199i	$-0.926389\pi$
$317$		−	8.16155i		−	0.458398i	0.973380	−	0.229199i	$-0.0736107\pi$
$318$		0			0
$319$	4.44668			0.248966
$320$		0			0
$321$	−18.3548			−1.02447
$322$		0			0
$323$		−	3.16725i		−	0.176231i
$324$		0			0
$325$	6.19186	+	17.9034i	0.343463	+	0.993101i
$326$		0			0
$327$			5.29401i			0.292759i
$328$		0			0
$329$	−13.1959			−0.727516
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$		−	10.3258i		−	0.565852i
$334$		0			0
$335$	−32.7385	+	5.50143i	−1.78870	+	0.300575i
$336$		0			0
$337$		−	9.90275i		−	0.539437i	−0.962939	−	0.269718i	$-0.913069\pi$
$337$		−	9.90275i		−	0.539437i	0.962939	−	0.269718i	$-0.0869306\pi$
$338$		0			0
$339$	−12.5178			−0.679872
$340$		0			0
$341$	4.32591			0.234261
$342$		0			0
$343$			18.8467i			1.01763i
$344$		0			0
$345$	0.680729	+	4.05096i	0.0366492	+	0.218096i
$346$		0			0
$347$			21.2781i			1.14227i	0.820858	+	0.571133i	$0.193496\pi$
$347$			21.2781i			1.14227i	−0.820858	+	0.571133i	$0.806504\pi$
$348$		0			0
$349$	16.4152			0.878686			0.439343	−	0.898319i	$-0.355211\pi$
$349$	16.4152			0.878686			0.439343	+	0.898319i	$0.355211\pi$
$350$		0			0
$351$	53.0934			2.83391
$352$		0			0
$353$		−	23.8744i		−	1.27071i	−0.772221	−	0.635354i	$-0.780854\pi$
$353$		−	23.8744i		−	1.27071i	0.772221	−	0.635354i	$-0.219146\pi$
$354$		0			0
$355$	1.35075	+	8.03817i	0.0716901	+	0.426622i
$356$		0			0
$357$			26.4618i			1.40051i
$358$		0			0
$359$	2.22334			0.117343			0.0586717	−	0.998277i	$-0.481313\pi$
$359$	2.22334			0.117343			0.0586717	+	0.998277i	$0.481313\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			33.6294i			1.76508i
$364$		0			0
$365$	−23.9172	+	4.01909i	−1.25189	+	0.210369i
$366$		0			0
$367$			4.52057i			0.235972i	0.993015	+	0.117986i	$0.0376437\pi$
$367$			4.52057i			0.235972i	−0.993015	+	0.117986i	$0.962356\pi$
$368$		0			0
$369$	28.2208			1.46911
$370$		0			0
$371$	0.417768			0.0216894
$372$		0			0
$373$		−	15.5186i		−	0.803521i	−0.915745	−	0.401760i	$-0.868398\pi$
$373$		−	15.5186i		−	0.803521i	0.915745	−	0.401760i	$-0.131602\pi$
$374$		0			0
$375$	31.5822	−	17.2312i	1.63090	−	0.889817i
$376$		0			0
$377$			22.7327i			1.17079i
$378$		0			0
$379$	−18.9015			−0.970905			−0.485453	−	0.874263i	$-0.661345\pi$
$379$	−18.9015			−0.970905			−0.485453	+	0.874263i	$0.661345\pi$
$380$		0			0
$381$	−46.4126			−2.37779
$382$		0			0
$383$		−	13.7046i		−	0.700274i	−0.936699	−	0.350137i	$-0.886135\pi$
$383$		−	13.7046i		−	0.700274i	0.936699	−	0.350137i	$-0.113865\pi$
$384$		0			0
$385$	−4.24315	+	0.713025i	−0.216251	+	0.0363391i
$386$		0			0
$387$		−	19.0958i		−	0.970694i
$388$		0			0
$389$	−12.7411			−0.646000			−0.323000	−	0.946399i	$-0.604691\pi$
$389$	−12.7411			−0.646000			−0.323000	+	0.946399i	$0.604691\pi$
$390$		0			0
$391$	1.80814			0.0914413
$392$		0			0
$393$			32.0776i			1.61810i
$394$		0			0
$395$	−0.680729	−	4.05096i	−0.0342512	−	0.203826i
$396$		0			0
$397$		−	38.6522i		−	1.93990i	−0.243306	−	0.969950i	$-0.578232\pi$
$397$		−	38.6522i		−	1.93990i	0.243306	−	0.969950i	$-0.421768\pi$
$398$		0			0
$399$	−8.35482			−0.418264
$400$		0			0
$401$	31.8660			1.59131			0.795655	−	0.605750i	$-0.207127\pi$
$401$	31.8660			1.59131			0.795655	+	0.605750i	$0.207127\pi$
$402$		0			0
$403$			22.1153i			1.10164i
$404$		0			0
$405$	−8.53351	−	50.7822i	−0.424034	−	2.52339i
$406$		0			0
$407$		−	1.04049i		−	0.0515751i
$408$		0			0
$409$	−11.0645			−0.547102			−0.273551	−	0.961857i	$-0.588198\pi$
$409$	−11.0645			−0.547102			−0.273551	+	0.961857i	$0.588198\pi$
$410$		0			0
$411$	31.2274			1.54033
$412$		0			0
$413$			21.6922i			1.06740i
$414$		0			0
$415$	9.25889	−	1.55588i	0.454501	−	0.0763749i
$416$		0			0
$417$			43.2830i			2.11958i
$418$		0			0
$419$	−25.7452			−1.25773			−0.628867	−	0.777513i	$-0.716481\pi$
$419$	−25.7452			−1.25773			−0.628867	+	0.777513i	$0.716481\pi$
$420$		0			0
$421$	27.4482			1.33774			0.668871	−	0.743378i	$-0.266778\pi$
$421$	27.4482			1.33774			0.668871	+	0.743378i	$0.266778\pi$
$422$		0			0
$423$		−	37.3806i		−	1.81751i
$424$		0			0
$425$	−5.17613	−	14.9664i	−0.251079	−	0.725979i
$426$		0			0
$427$		−	22.2720i		−	1.07782i
$428$		0			0
$429$	9.03555			0.436240
$430$		0			0
$431$	−1.74519			−0.0840627			−0.0420314	−	0.999116i	$-0.513383\pi$
$431$	−1.74519			−0.0840627			−0.0420314	+	0.999116i	$0.513383\pi$
$432$		0			0
$433$			18.5208i			0.890052i	0.895518	+	0.445026i	$0.146806\pi$
$433$			18.5208i			0.890052i	−0.895518	+	0.445026i	$0.853194\pi$
$434$		0			0
$435$	42.5756	−	7.15446i	2.04134	−	0.343030i
$436$		0			0
$437$			0.570885i			0.0273091i
$438$		0			0
$439$	29.4482			1.40549			0.702743	−	0.711444i	$-0.251959\pi$
$439$	29.4482			1.40549			0.702743	+	0.711444i	$0.251959\pi$
$440$		0			0
$441$	−1.90407			−0.0906699
$442$		0			0
$443$			11.7388i			0.557726i	0.960331	+	0.278863i	$0.0899576\pi$
$443$			11.7388i			0.557726i	−0.960331	+	0.278863i	$0.910042\pi$
$444$		0			0
$445$	−6.26296	−	37.2704i	−0.296893	−	1.76678i
$446$		0			0
$447$			48.5771i			2.29762i
$448$		0			0
$449$	7.06446			0.333392			0.166696	−	0.986008i	$-0.446690\pi$
$449$	7.06446			0.333392			0.166696	+	0.986008i	$0.446690\pi$
$450$		0			0
$451$	2.84368			0.133904
$452$		0			0
$453$			45.6679i			2.14566i
$454$		0			0
$455$	−3.64518	−	21.6922i	−0.170889	−	1.01694i
$456$		0			0
$457$		−	34.5000i		−	1.61384i	−0.590660	−	0.806920i	$-0.701133\pi$
$457$		−	34.5000i		−	1.61384i	0.590660	−	0.806920i	$-0.298867\pi$
$458$		0			0
$459$	−44.3837			−2.07166
$460$		0			0
$461$	8.03147			0.374063			0.187032	−	0.982354i	$-0.440113\pi$
$461$	8.03147			0.374063			0.187032	+	0.982354i	$0.440113\pi$
$462$		0			0
$463$		−	25.3290i		−	1.17714i	−0.808447	−	0.588570i	$-0.799691\pi$
$463$		−	25.3290i		−	1.17714i	0.808447	−	0.588570i	$-0.200309\pi$
$464$		0			0
$465$	41.4193	−	6.96015i	1.92077	−	0.322769i
$466$		0			0
$467$		−	26.8759i		−	1.24367i	−0.783149	−	0.621834i	$-0.786388\pi$
$467$		−	26.8759i		−	1.24367i	0.783149	−	0.621834i	$-0.213612\pi$
$468$		0			0
$469$	38.5467			1.77992
$470$		0			0
$471$	24.3837			1.12354
$472$		0			0
$473$		−	1.92420i		−	0.0884748i
$474$		0			0
$475$	4.72538	−	1.63427i	0.216815	−	0.0749852i
$476$		0			0
$477$			1.18343i			0.0541854i
$478$		0			0
$479$	−28.9015			−1.32054			−0.660272	−	0.751027i	$-0.729559\pi$
$479$	−28.9015			−1.32054			−0.660272	+	0.751027i	$0.729559\pi$
$480$		0			0
$481$	5.31927			0.242538
$482$		0			0
$483$		−	4.76964i		−	0.217026i
$484$		0			0
$485$	8.35482	−	1.40396i	0.379373	−	0.0637503i
$486$		0			0
$487$			17.7294i			0.803395i	0.915773	+	0.401697i	$0.131580\pi$
$487$			17.7294i			0.803395i	−0.915773	+	0.401697i	$0.868420\pi$
$488$		0			0
$489$	−63.3141			−2.86316
$490$		0			0
$491$	35.1645			1.58695			0.793475	−	0.608603i	$-0.208270\pi$
$491$	35.1645			1.58695			0.793475	+	0.608603i	$0.208270\pi$
$492$		0			0
$493$		−	19.0035i		−	0.855875i
$494$		0			0
$495$	−2.01981	−	12.0197i	−0.0907838	−	0.540247i
$496$		0			0
$497$		−	9.46422i		−	0.424528i
$498$		0			0
$499$	21.4508			0.960268			0.480134	−	0.877195i	$-0.340588\pi$
$499$	21.4508			0.960268			0.480134	+	0.877195i	$0.340588\pi$
$500$		0			0
$501$	34.7385			1.55200
$502$		0			0
$503$		−	5.34053i		−	0.238122i	−0.992887	−	0.119061i	$-0.962012\pi$
$503$		−	5.34053i		−	0.238122i	0.992887	−	0.119061i	$-0.0379884\pi$
$504$		0			0
$505$	−3.09593	−	18.4236i	−0.137767	−	0.819841i
$506$		0			0
$507$			4.35966i			0.193619i
$508$		0			0
$509$	−36.1919			−1.60418			−0.802088	−	0.597206i	$-0.796278\pi$
$509$	−36.1919			−1.60418			−0.802088	+	0.597206i	$0.796278\pi$
$510$		0			0
$511$	28.1604			1.24574
$512$		0			0
$513$		−	14.0133i		−	0.618704i
$514$		0			0
$515$	−4.57816	+	0.769320i	−0.201738	+	0.0339003i
$516$		0			0
$517$		−	3.76668i		−	0.165658i
$518$		0			0
$519$	−65.6111			−2.88001
$520$		0			0
$521$	−2.77259			−0.121469			−0.0607346	−	0.998154i	$-0.519344\pi$
$521$	−2.77259			−0.121469			−0.0607346	+	0.998154i	$0.519344\pi$
$522$		0			0
$523$		−	20.5373i		−	0.898033i	−0.893524	−	0.449016i	$-0.851774\pi$
$523$		−	20.5373i		−	0.898033i	0.893524	−	0.449016i	$-0.148226\pi$
$524$		0			0
$525$	−39.4797	+	13.6540i	−1.72303	+	0.595909i
$526$		0			0
$527$		−	18.4874i		−	0.805323i
$528$		0			0
$529$	22.6741			0.985830
$530$		0			0
$531$	−61.4482			−2.66662
$532$		0			0
$533$			14.5377i			0.629698i
$534$		0			0
$535$	−12.5782	+	2.11365i	−0.543801	+	0.0913811i
$536$		0			0
$537$		−	80.6547i		−	3.48051i
$538$		0			0
$539$	−0.191865			−0.00826419
$540$		0			0
$541$	35.4797			1.52539			0.762695	−	0.646758i	$-0.223876\pi$
$541$	35.4797			1.52539			0.762695	+	0.646758i	$0.223876\pi$
$542$		0			0
$543$			62.4891i			2.68166i
$544$		0			0
$545$	0.609632	+	3.62787i	0.0261138	+	0.155401i
$546$		0			0
$547$		−	43.0756i		−	1.84178i	−0.389822	−	0.920890i	$-0.627463\pi$
$547$		−	43.0756i		−	1.84178i	0.389822	−	0.920890i	$-0.372537\pi$
$548$		0			0
$549$	63.0908			2.69265
$550$		0			0
$551$	6.00000			0.255609
$552$		0			0
$553$			4.76964i			0.202826i
$554$		0			0
$555$	−1.67409	−	9.96237i	−0.0710612	−	0.422879i
$556$		0			0
$557$			40.4376i			1.71340i	0.515818	+	0.856698i	$0.327488\pi$
$557$			40.4376i			1.71340i	−0.515818	+	0.856698i	$0.672512\pi$
$558$		0			0
$559$	9.83705			0.416063
$560$		0			0
$561$	−7.55332			−0.318902
$562$		0			0
$563$			19.7173i			0.830986i	0.909596	+	0.415493i	$0.136391\pi$
$563$			19.7173i			0.830986i	−0.909596	+	0.415493i	$0.863609\pi$
$564$		0			0
$565$	−8.57816	+	1.44149i	−0.360886	+	0.0606437i
$566$		0			0
$567$			59.7915i			2.51101i
$568$		0			0
$569$	−18.6807			−0.783137			−0.391568	−	0.920149i	$-0.628067\pi$
$569$	−18.6807			−0.783137			−0.391568	+	0.920149i	$0.628067\pi$
$570$		0			0
$571$	29.9371			1.25283			0.626413	−	0.779491i	$-0.284522\pi$
$571$	29.9371			1.25283			0.626413	+	0.779491i	$0.284522\pi$
$572$		0			0
$573$			36.8473i			1.53932i
$574$		0			0
$575$	0.932977	+	2.69765i	0.0389079	+	0.112500i
$576$		0			0
$577$		−	0.156779i		−	0.00652679i	−0.999995	−	0.00326339i	$-0.998961\pi$
$577$		−	0.156779i		−	0.00652679i	0.999995	−	0.00326339i	$-0.00103877\pi$
$578$		0			0
$579$	−12.1919			−0.506677
$580$		0			0
$581$	−10.9015			−0.452271
$582$		0			0
$583$			0.119249i			0.00493877i
$584$		0			0
$585$	61.4482	−	10.3258i	2.54057	−	0.426921i
$586$		0			0
$587$		−	31.1474i		−	1.28559i	−0.766038	−	0.642795i	$-0.777775\pi$
$587$		−	31.1474i		−	1.28559i	0.766038	−	0.642795i	$-0.222225\pi$
$588$		0			0
$589$	5.83705			0.240511
$590$		0			0
$591$	7.34818			0.302264
$592$		0			0
$593$		−	28.8728i		−	1.18567i	−0.805326	−	0.592833i	$-0.798009\pi$
$593$		−	28.8728i		−	1.18567i	0.805326	−	0.592833i	$-0.201991\pi$
$594$		0			0
$595$	3.04721	+	18.1337i	0.124923	+	0.743409i
$596$		0			0
$597$			62.5904i			2.56165i
$598$		0			0
$599$	25.3274			1.03485			0.517425	−	0.855728i	$-0.326891\pi$
$599$	25.3274			1.03485			0.517425	+	0.855728i	$0.326891\pi$
$600$		0			0
$601$	−19.8370			−0.809170			−0.404585	−	0.914500i	$-0.632584\pi$
$601$	−19.8370			−0.809170			−0.404585	+	0.914500i	$0.632584\pi$
$602$		0			0
$603$			109.193i			4.44667i
$604$		0			0
$605$	3.87259	+	23.0455i	0.157443	+	0.936932i
$606$		0			0
$607$			2.49921i			0.101440i	0.998713	+	0.0507199i	$0.0161516\pi$
$607$			2.49921i			0.101440i	−0.998713	+	0.0507199i	$0.983848\pi$
$608$		0			0
$609$	−50.1289			−2.03133
$610$		0			0
$611$	19.2563			0.779027
$612$		0			0
$613$			0.883711i			0.0356927i	0.999841	+	0.0178464i	$0.00568098\pi$
$613$			0.883711i			0.0356927i	−0.999841	+	0.0178464i	$0.994319\pi$
$614$		0			0
$615$	27.2274	−	4.57533i	1.09792	−	0.184495i
$616$		0			0
$617$		−	29.4085i		−	1.18394i	−0.805959	−	0.591971i	$-0.798350\pi$
$617$		−	29.4085i		−	1.18394i	0.805959	−	0.591971i	$-0.201650\pi$
$618$		0			0
$619$	30.3208			1.21870			0.609348	−	0.792903i	$-0.291431\pi$
$619$	30.3208			1.21870			0.609348	+	0.792903i	$0.291431\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$			43.8825i			1.75811i
$624$		0			0
$625$	19.6584	−	15.4450i	0.786334	−	0.617801i
$626$		0			0
$627$		−	2.38482i		−	0.0952405i
$628$		0			0
$629$	−4.44668			−0.177301
$630$		0			0
$631$	−17.7767			−0.707678			−0.353839	−	0.935306i	$-0.615124\pi$
$631$	−17.7767			−0.707678			−0.353839	+	0.935306i	$0.615124\pi$
$632$		0			0
$633$			36.1286i			1.43598i
$634$		0			0
$635$	−31.8056	+	5.34465i	−1.26217	+	0.212096i
$636$		0			0
$637$		−	0.980865i		−	0.0388633i
$638$		0			0
$639$	26.8096			1.06057
$640$		0			0
$641$	−32.6675			−1.29029			−0.645143	−	0.764062i	$-0.723202\pi$
$641$	−32.6675			−1.29029			−0.645143	+	0.764062i	$0.723202\pi$
$642$		0			0
$643$			31.8661i			1.25668i	0.777941	+	0.628338i	$0.216264\pi$
$643$			31.8661i			1.25668i	−0.777941	+	0.628338i	$0.783736\pi$
$644$		0			0
$645$	−3.09593	−	18.4236i	−0.121902	−	0.725430i
$646$		0			0
$647$		−	21.2601i		−	0.835820i	−0.908488	−	0.417910i	$-0.862763\pi$
$647$		−	21.2601i		−	0.835820i	0.908488	−	0.417910i	$-0.137237\pi$
$648$		0			0
$649$	−6.19186			−0.243052
$650$		0			0
$651$	−48.7675			−1.91135
$652$		0			0
$653$		−	12.8340i		−	0.502234i	−0.967957	−	0.251117i	$-0.919202\pi$
$653$		−	12.8340i		−	0.502234i	0.967957	−	0.251117i	$-0.0807980\pi$
$654$		0			0
$655$	3.69390	+	21.9821i	0.144333	+	0.858912i
$656$		0			0
$657$			79.7710i			3.11216i
$658$		0			0
$659$	20.3548			0.792911			0.396456	−	0.918054i	$-0.370240\pi$
$659$	20.3548			0.792911			0.396456	+	0.918054i	$0.370240\pi$
$660$		0			0
$661$	−30.7385			−1.19559			−0.597795	−	0.801649i	$-0.703957\pi$
$661$	−30.7385			−1.19559			−0.597795	+	0.801649i	$0.703957\pi$
$662$		0			0
$663$		−	38.6147i		−	1.49967i
$664$		0			0
$665$	−5.72538	+	0.962100i	−0.222021	+	0.0373086i
$666$		0			0
$667$			3.42531i			0.132629i
$668$		0			0
$669$	−12.9934			−0.502352
$670$		0			0
$671$	6.35738			0.245424
$672$		0			0
$673$		−	21.2094i		−	0.817564i	−0.912632	−	0.408782i	$-0.865954\pi$
$673$		−	21.2094i		−	0.817564i	0.912632	−	0.408782i	$-0.134046\pi$
$674$		0			0
$675$	−22.9015	−	66.2183i	−0.881479	−	2.54874i
$676$		0			0
$677$			11.2650i			0.432951i	0.976288	+	0.216475i	$0.0694561\pi$
$677$			11.2650i			0.432951i	−0.976288	+	0.216475i	$0.930544\pi$
$678$		0			0
$679$	−9.83705			−0.377511
$680$		0			0
$681$	36.1000			1.38336
$682$		0			0
$683$		−	12.3603i		−	0.472954i	−0.971637	−	0.236477i	$-0.924007\pi$
$683$		−	12.3603i		−	0.472954i	0.971637	−	0.236477i	$-0.0759928\pi$
$684$		0			0
$685$	21.3995	−	3.59600i	0.817632	−	0.137396i
$686$		0			0
$687$			51.9093i			1.98046i
$688$		0			0
$689$	−0.609632			−0.0232251
$690$		0			0
$691$	−22.7493			−0.865423			−0.432711	−	0.901533i	$-0.642443\pi$
$691$	−22.7493			−0.865423			−0.432711	+	0.901533i	$0.642443\pi$
$692$		0			0
$693$			14.1521i			0.537595i
$694$		0			0
$695$	4.98426	+	29.6609i	0.189064	+	1.12510i
$696$		0			0
$697$		−	12.1529i		−	0.460323i
$698$		0			0
$699$	6.84368			0.258852
$700$		0			0
$701$	−16.0289			−0.605404			−0.302702	−	0.953085i	$-0.597889\pi$
$701$	−16.0289			−0.605404			−0.302702	+	0.953085i	$0.597889\pi$
$702$		0			0
$703$		−	1.40396i		−	0.0529512i
$704$		0			0
$705$	−6.06038	−	36.0648i	−0.228247	−	1.35828i
$706$		0			0
$707$			21.6922i			0.815818i
$708$		0			0
$709$	−31.4193			−1.17998			−0.589988	−	0.807412i	$-0.700867\pi$
$709$	−31.4193			−1.17998			−0.589988	+	0.807412i	$0.700867\pi$
$710$		0			0
$711$	−13.5111			−0.506707
$712$		0			0
$713$			3.33228i			0.124795i
$714$		0			0
$715$	6.19186	−	1.04049i	0.231563	−	0.0389121i
$716$		0			0
$717$			46.3865i			1.73234i
$718$		0			0
$719$	11.2589			0.419886			0.209943	−	0.977714i	$-0.432672\pi$
$719$	11.2589			0.419886			0.209943	+	0.977714i	$0.432672\pi$
$720$		0			0
$721$	5.39037			0.200748
$722$		0			0
$723$			0.524371i			0.0195016i
$724$		0			0
$725$	28.3523	−	9.80559i	1.05298	−	0.364171i
$726$		0			0
$727$		−	48.9829i		−	1.81668i	−0.418237	−	0.908338i	$-0.637352\pi$
$727$		−	48.9829i		−	1.81668i	0.418237	−	0.908338i	$-0.362648\pi$
$728$		0			0
$729$	−34.0934			−1.26272
$730$		0			0
$731$	−8.22334			−0.304151
$732$		0			0
$733$			35.9260i			1.32696i	0.748195	+	0.663479i	$0.230921\pi$
$733$			35.9260i			1.32696i	−0.748195	+	0.663479i	$0.769079\pi$
$734$		0			0
$735$	−1.83705	+	0.308700i	−0.0677604	+	0.0113866i
$736$		0			0
$737$			11.0029i			0.405296i
$738$		0			0
$739$	14.3523			0.527956			0.263978	−	0.964529i	$-0.414965\pi$
$739$	14.3523			0.527956			0.263978	+	0.964529i	$0.414965\pi$
$740$		0			0
$741$	12.1919			0.447879
$742$		0			0
$743$			12.5629i			0.460887i	0.973086	+	0.230443i	$0.0740176\pi$
$743$			12.5629i			0.460887i	−0.973086	+	0.230443i	$0.925982\pi$
$744$		0			0
$745$	5.59390	+	33.2888i	0.204944	+	1.21961i
$746$		0			0
$747$		−	30.8811i		−	1.12988i
$748$		0			0
$749$	14.8096			0.541133
$750$		0			0
$751$	−26.4548			−0.965350			−0.482675	−	0.875799i	$-0.660335\pi$
$751$	−26.4548			−0.965350			−0.482675	+	0.875799i	$0.660335\pi$
$752$		0			0
$753$			41.6169i			1.51660i
$754$		0			0
$755$	5.25889	+	31.2952i	0.191390	+	1.13895i
$756$		0			0
$757$		−	15.7350i		−	0.571897i	−0.958245	−	0.285949i	$-0.907691\pi$
$757$		−	15.7350i		−	0.571897i	0.958245	−	0.285949i	$-0.0923087\pi$
$758$		0			0
$759$	1.36146			0.0494178
$760$		0			0
$761$	16.9619			0.614868			0.307434	−	0.951569i	$-0.400530\pi$
$761$	16.9619			0.614868			0.307434	+	0.951569i	$0.400530\pi$
$762$		0			0
$763$		−	4.27149i		−	0.154638i
$764$		0			0
$765$	−51.3680	+	8.63195i	−1.85721	+	0.312089i
$766$		0			0
$767$		−	31.6545i		−	1.14298i
$768$		0			0
$769$	−41.9974			−1.51447			−0.757233	−	0.653145i	$-0.773449\pi$
$769$	−41.9974			−1.51447			−0.757233	+	0.653145i	$0.773449\pi$
$770$		0			0
$771$	−35.5400			−1.27994
$772$		0			0
$773$			40.9579i			1.47315i	0.676355	+	0.736576i	$0.263559\pi$
$773$			40.9579i			1.47315i	−0.676355	+	0.736576i	$0.736441\pi$
$774$		0			0
$775$	27.5822	−	9.53928i	0.990783	−	0.342661i
$776$		0			0
$777$			11.7298i			0.420804i
$778$		0			0
$779$	3.83705			0.137476
$780$		0			0
$781$	2.70149			0.0966669
$782$		0			0
$783$		−	84.0800i		−	3.00477i
$784$		0			0
$785$	16.7096	−	2.80791i	0.596393	−	0.100219i
$786$		0			0
$787$			28.5379i			1.01727i	0.860983	+	0.508634i	$0.169849\pi$
$787$			28.5379i			1.01727i	−0.860983	+	0.508634i	$0.830151\pi$
$788$		0			0
$789$	57.4482			2.04521
$790$		0			0
$791$	10.1000			0.359115
$792$		0			0
$793$			32.5007i			1.15413i
$794$		0			0
$795$	0.191865	+	1.14177i	0.00680473	+	0.0404944i
$796$		0			0
$797$		−	33.2790i		−	1.17880i	−0.807840	−	0.589402i	$-0.799364\pi$
$797$		−	33.2790i		−	1.17880i	0.807840	−	0.589402i	$-0.200636\pi$
$798$		0			0
$799$	−16.0974			−0.569487
$800$		0			0
$801$	−124.308			−4.39219
$802$		0			0
$803$			8.03817i			0.283661i
$804$		0			0
$805$	−0.549248	−	3.26853i	−0.0193585	−	0.115201i
$806$		0			0
$807$			80.4259i			2.83113i
$808$		0			0
$809$	34.4234			1.21026			0.605130	−	0.796126i	$-0.293121\pi$
$809$	34.4234			1.21026			0.605130	+	0.796126i	$0.293121\pi$
$810$		0			0
$811$	11.2563			0.395263			0.197631	−	0.980276i	$-0.436675\pi$
$811$	11.2563			0.395263			0.197631	+	0.980276i	$0.436675\pi$
$812$		0			0
$813$		−	76.7980i		−	2.69342i
$814$		0			0
$815$	−43.3878	+	7.29095i	−1.51981	+	0.255391i
$816$		0			0
$817$		−	2.59637i		−	0.0908353i
$818$		0			0
$819$	−72.3497			−2.52810
$820$		0			0
$821$	−31.3167			−1.09296			−0.546480	−	0.837472i	$-0.684033\pi$
$821$	−31.3167			−1.09296			−0.546480	+	0.837472i	$0.684033\pi$
$822$		0			0
$823$			13.4800i			0.469882i	0.972010	+	0.234941i	$0.0754898\pi$
$823$			13.4800i			0.469882i	−0.972010	+	0.234941i	$0.924510\pi$
$824$		0			0
$825$	−3.89743	−	11.2692i	−0.135691	−	0.392342i
$826$		0			0
$827$			15.9882i			0.555963i	0.960586	+	0.277982i	$0.0896654\pi$
$827$			15.9882i			0.555963i	−0.960586	+	0.277982i	$0.910335\pi$
$828$		0			0
$829$	48.4837			1.68391			0.841955	−	0.539548i	$-0.181405\pi$
$829$	48.4837			1.68391			0.841955	+	0.539548i	$0.181405\pi$
$830$		0			0
$831$	−68.3208			−2.37002
$832$		0			0
$833$			0.819960i			0.0284099i
$834$		0			0
$835$	23.8056	−	4.00032i	0.823826	−	0.138437i
$836$		0			0
$837$		−	81.7965i		−	2.82730i
$838$		0			0
$839$	−18.9934			−0.655724			−0.327862	−	0.944726i	$-0.606328\pi$
$839$	−18.9934			−0.655724			−0.327862	+	0.944726i	$0.606328\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	12.3472i		−	0.425260i
$844$		0			0
$845$	0.502037	+	2.98758i	0.0172706	+	0.102776i
$846$		0			0
$847$		−	27.1340i		−	0.932334i
$848$		0			0
$849$	0.680729			0.0233626
$850$		0			0
$851$	0.801497			0.0274750
$852$		0			0
$853$		−	44.1210i		−	1.51067i	−0.655337	−	0.755337i	$-0.727473\pi$
$853$		−	44.1210i		−	1.51067i	0.655337	−	0.755337i	$-0.272527\pi$
$854$		0			0
$855$	−2.72538	−	16.2185i	−0.0932059	−	0.554660i
$856$		0			0
$857$			32.4149i			1.10727i	0.832759	+	0.553635i	$0.186760\pi$
$857$			32.4149i			1.10727i	−0.832759	+	0.553635i	$0.813240\pi$
$858$		0			0
$859$	6.29444			0.214763			0.107382	−	0.994218i	$-0.465753\pi$
$859$	6.29444			0.214763			0.107382	+	0.994218i	$0.465753\pi$
$860$		0			0
$861$	−32.0578			−1.09253
$862$		0			0
$863$		−	17.4338i		−	0.593453i	−0.954963	−	0.296726i	$-0.904105\pi$
$863$		−	17.4338i		−	0.593453i	0.954963	−	0.296726i	$-0.0958950\pi$
$864$		0			0
$865$	−44.9619	+	7.55546i	−1.52875	+	0.256893i
$866$		0			0
$867$		−	22.4240i		−	0.761557i
$868$		0			0
$869$	−1.36146			−0.0461843
$870$		0			0
$871$	−56.2497			−1.90595
$872$		0			0
$873$		−	27.8658i		−	0.943113i
$874$		0			0
$875$	−25.4822	+	13.9031i	−0.861456	+	0.470009i
$876$		0			0
$877$			23.2904i			0.786462i	0.919440	+	0.393231i	$0.128643\pi$
$877$			23.2904i			0.786462i	−0.919440	+	0.393231i	$0.871357\pi$
$878$		0			0
$879$	48.2497			1.62742
$880$		0			0
$881$	28.9619			0.975751			0.487875	−	0.872913i	$-0.337772\pi$
$881$	28.9619			0.975751			0.487875	+	0.872913i	$0.337772\pi$
$882$		0			0
$883$		−	37.3627i		−	1.25735i	−0.777667	−	0.628677i	$-0.783597\pi$
$883$		−	37.3627i		−	1.25735i	0.777667	−	0.628677i	$-0.216403\pi$
$884$		0			0
$885$	−59.2852	+	9.96237i	−1.99285	+	0.334881i
$886$		0			0
$887$			23.0234i			0.773050i	0.922279	+	0.386525i	$0.126325\pi$
$887$			23.0234i			0.773050i	−0.922279	+	0.386525i	$0.873675\pi$
$888$		0			0
$889$	37.4482			1.25597
$890$		0			0
$891$	−17.0670			−0.571767
$892$		0			0
$893$		−	5.08247i		−	0.170078i
$894$		0			0
$895$	−9.28780	−	55.2709i	−0.310457	−	1.84750i
$896$		0			0
$897$			6.96015i			0.232393i
$898$		0			0
$899$	35.0223			1.16806
$900$		0			0
$901$	0.509626			0.0169781
$902$		0			0
$903$			21.6922i			0.721870i
$904$		0			0
$905$	7.19594	+	42.8224i	0.239201	+	1.42347i
$906$		0			0
$907$			35.2693i			1.17110i	0.810637	+	0.585549i	$0.199121\pi$
$907$			35.2693i			1.17110i	−0.810637	+	0.585549i	$0.800879\pi$
$908$		0			0
$909$	−61.4482			−2.03811
$910$		0			0
$911$	4.97260			0.164750			0.0823748	−	0.996601i	$-0.473750\pi$
$911$	4.97260			0.164750			0.0823748	+	0.996601i	$0.473750\pi$
$912$		0			0
$913$		−	3.11175i		−	0.102984i
$914$		0			0
$915$	60.8700	−	10.2287i	2.01230	−	0.338150i
$916$		0			0
$917$		−	25.8819i		−	0.854697i
$918$		0			0
$919$	−48.7096			−1.60678			−0.803391	−	0.595451i	$-0.796973\pi$
$919$	−48.7096			−1.60678			−0.803391	+	0.595451i	$0.796973\pi$
$920$		0			0
$921$	−5.31927			−0.175276
$922$		0			0
$923$			13.8108i			0.454587i
$924$		0			0
$925$	−2.29444	−	6.63422i	−0.0754406	−	0.218132i
$926$		0			0
$927$			15.2695i			0.501516i
$928$		0			0
$929$	−32.5126			−1.06671			−0.533353	−	0.845893i	$-0.679068\pi$
$929$	−32.5126			−1.06671			−0.533353	+	0.845893i	$0.679068\pi$
$930$		0			0
$931$	−0.258887			−0.00848468
$932$		0			0
$933$		−	2.38482i		−	0.0780755i
$934$		0			0
$935$	−5.17613	+	0.869804i	−0.169277	+	0.0284456i
$936$		0			0
$937$		−	0.385560i		−	0.0125957i	−0.999980	−	0.00629785i	$-0.997995\pi$
$937$		−	0.385560i		−	0.0125957i	0.999980	−	0.00629785i	$-0.00200468\pi$
$938$		0			0
$939$	86.5126			2.82323
$940$		0			0
$941$	45.3482			1.47831			0.739154	−	0.673536i	$-0.235225\pi$
$941$	45.3482			1.47831			0.739154	+	0.673536i	$0.235225\pi$
$942$		0			0
$943$			2.19051i			0.0713329i
$944$		0			0
$945$	13.4822	+	80.2316i	0.438577	+	2.60993i
$946$		0			0
$947$		−	9.61202i		−	0.312349i	−0.987730	−	0.156174i	$-0.950084\pi$
$947$		−	9.61202i		−	0.312349i	0.987730	−	0.156174i	$-0.0499161\pi$
$948$		0			0
$949$	−41.0934			−1.33395
$950$		0			0
$951$	−26.2630			−0.851635
$952$		0			0
$953$			59.8421i			1.93848i	0.246126	+	0.969238i	$0.420842\pi$
$953$			59.8421i			1.93848i	−0.246126	+	0.969238i	$0.579158\pi$
$954$		0			0
$955$	4.24315	+	25.2506i	0.137305	+	0.817091i
$956$		0			0
$957$		−	14.3089i		−	0.462542i
$958$		0			0
$959$	−25.1959			−0.813619
$960$		0			0
$961$	3.07110			0.0990676
$962$		0			0
$963$			41.9518i			1.35188i
$964$		0			0
$965$	−8.35482	+	1.40396i	−0.268951	+	0.0451949i
$966$		0			0
$967$			0.368324i			0.0118445i	0.999982	+	0.00592225i	$0.00188512\pi$
$967$			0.368324i			0.0118445i	−0.999982	+	0.00592225i	$0.998115\pi$
$968$		0			0
$969$	−10.1919			−0.327410
$970$		0			0
$971$	45.8370			1.47098			0.735490	−	0.677535i	$-0.236952\pi$
$971$	45.8370			1.47098			0.735490	+	0.677535i	$0.236952\pi$
$972$		0			0
$973$		−	34.9231i		−	1.11958i
$974$		0			0
$975$	57.6111	−	19.9247i	1.84503	−	0.638102i
$976$		0			0
$977$			29.0337i			0.928872i	0.885607	+	0.464436i	$0.153743\pi$
$977$			29.0337i			0.928872i	−0.885607	+	0.464436i	$0.846257\pi$
$978$		0			0
$979$	−12.5259			−0.400330
$980$		0			0
$981$	12.1000			0.386323
$982$		0			0
$983$		−	11.0160i		−	0.351355i	−0.984448	−	0.175677i	$-0.943788\pi$
$983$		−	11.0160i		−	0.351355i	0.984448	−	0.175677i	$-0.0562116\pi$
$984$		0			0
$985$	5.03555	−	0.846180i	0.160446	−	0.0269615i
$986$		0			0
$987$			42.4631i			1.35161i
$988$		0			0
$989$	1.48223			0.0471320
$990$		0			0
$991$	25.6822			0.815823			0.407912	−	0.913021i	$-0.366257\pi$
$991$	25.6822			0.815823			0.407912	+	0.913021i	$0.366257\pi$
$992$		0			0
$993$			25.7431i			0.816933i
$994$		0			0
$995$	7.20760	+	42.8918i	0.228496	+	1.35976i
$996$		0			0
$997$			20.3854i			0.645611i	0.946465	+	0.322805i	$0.104626\pi$
$997$			20.3854i			0.645611i	−0.946465	+	0.322805i	$0.895374\pi$
$998$		0			0
$999$	−19.6741			−0.622461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.h.609.1		6
4.3	odd	2		95.2.b.b.39.4	yes	6
5.2	odd	4		7600.2.a.ck.1.1		6
5.3	odd	4		7600.2.a.ck.1.6		6
5.4	even	2	inner	1520.2.d.h.609.6		6
12.11	even	2		855.2.c.d.514.3		6
20.3	even	4		475.2.a.j.1.4		6
20.7	even	4		475.2.a.j.1.3		6
20.19	odd	2		95.2.b.b.39.3	✓	6
60.23	odd	4		4275.2.a.br.1.3		6
60.47	odd	4		4275.2.a.br.1.4		6
60.59	even	2		855.2.c.d.514.4		6
76.75	even	2		1805.2.b.e.1084.3		6
380.227	odd	4		9025.2.a.bx.1.4		6
380.303	odd	4		9025.2.a.bx.1.3		6
380.379	even	2		1805.2.b.e.1084.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.3	✓	6	20.19	odd	2
95.2.b.b.39.4	yes	6	4.3	odd	2
475.2.a.j.1.3		6	20.7	even	4
475.2.a.j.1.4		6	20.3	even	4
855.2.c.d.514.3		6	12.11	even	2
855.2.c.d.514.4		6	60.59	even	2
1520.2.d.h.609.1		6	1.1	even	1	trivial
1520.2.d.h.609.6		6	5.4	even	2	inner
1805.2.b.e.1084.3		6	76.75	even	2
1805.2.b.e.1084.4		6	380.379	even	2
4275.2.a.br.1.3		6	60.23	odd	4
4275.2.a.br.1.4		6	60.47	odd	4
7600.2.a.ck.1.1		6	5.2	odd	4
7600.2.a.ck.1.6		6	5.3	odd	4
9025.2.a.bx.1.3		6	380.303	odd	4
9025.2.a.bx.1.4		6	380.227	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.21789i q^{3} +(-0.370556 - 2.20515i) q^{5} +2.59637i q^{7} -7.35482 q^{9} +O(q^{10})\)	\(q-3.21789i q^{3} +(-0.370556 - 2.20515i) q^{5} +2.59637i q^{7} -7.35482 q^{9} -0.741113 q^{11} -3.78878i q^{13} +(-7.09593 + 1.19241i) q^{15} +3.16725i q^{17} -1.00000 q^{19} +8.35482 q^{21} -0.570885i q^{23} +(-4.72538 + 1.63427i) q^{25} +14.0133i q^{27} -6.00000 q^{29} -5.83705 q^{31} +2.38482i q^{33} +(5.72538 - 0.962100i) q^{35} +1.40396i q^{37} -12.1919 q^{39} -3.83705 q^{41} +2.59637i q^{43} +(2.72538 + 16.2185i) q^{45} +5.08247i q^{47} +0.258887 q^{49} +10.1919 q^{51} -0.160905i q^{53} +(0.274624 + 1.63427i) q^{55} +3.21789i q^{57} +8.35482 q^{59} -8.57816 q^{61} -19.0958i q^{63} +(-8.35482 + 1.40396i) q^{65} -14.8464i q^{67} -1.83705 q^{69} -3.64518 q^{71} -10.8461i q^{73} +(5.25889 + 15.2057i) q^{75} -1.92420i q^{77} +1.83705 q^{79} +23.0289 q^{81} +4.19876i q^{83} +(6.98426 - 1.17365i) q^{85} +19.3073i q^{87} +16.9015 q^{89} +9.83705 q^{91} +18.7830i q^{93} +(0.370556 + 2.20515i) q^{95} +3.78878i q^{97} +5.45075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{5} - 14 q^{9}+O(q^{10}) \) 6 * q - q^5 - 14 * q^9	\( 6 q - q^{5} - 14 q^{9} - 2 q^{11} - 10 q^{15} - 6 q^{19} + 20 q^{21} + 3 q^{25} - 36 q^{29} + 3 q^{35} - 8 q^{39} + 12 q^{41} - 15 q^{45} + 4 q^{49} - 4 q^{51} + 33 q^{55} + 20 q^{59} - 14 q^{61} - 20 q^{65} + 24 q^{69} - 52 q^{71} + 34 q^{75} - 24 q^{79} + 38 q^{81} + 13 q^{85} - 24 q^{89} + 24 q^{91} + q^{95} - 30 q^{99}+O(q^{100}) \) 6 * q - q^5 - 14 * q^9 - 2 * q^11 - 10 * q^15 - 6 * q^19 + 20 * q^21 + 3 * q^25 - 36 * q^29 + 3 * q^35 - 8 * q^39 + 12 * q^41 - 15 * q^45 + 4 * q^49 - 4 * q^51 + 33 * q^55 + 20 * q^59 - 14 * q^61 - 20 * q^65 + 24 * q^69 - 52 * q^71 + 34 * q^75 - 24 * q^79 + 38 * q^81 + 13 * q^85 - 24 * q^89 + 24 * q^91 + q^95 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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