LMFDB - Embedded newform 1960.2.g.g.1569.4

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 122x^{16} + 2481x^{12} + 15576x^{8} + 27792x^{4} + 16 $ x^20 + 122x^16 + 2481x^12 + 15576x^8 + 27792x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.4
Root			$2.22682 + 2.22682i$ of defining polynomial
Character	$\chi$	$=$	1960.1569
Dual form			1960.2.g.g.1569.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69859i q^{3} +(0.437748 + 2.19280i) q^{5} -4.28239 q^{9} +O(q^{10})$	$q-2.69859i q^{3} +(0.437748 + 2.19280i) q^{5} -4.28239 q^{9} +0.919787 q^{11} +5.24470i q^{13} +(5.91747 - 1.18130i) q^{15} -4.25577i q^{17} +1.83949 q^{19} +8.78042i q^{23} +(-4.61675 + 1.91979i) q^{25} +3.46065i q^{27} -6.71298 q^{29} +4.64200 q^{31} -2.48213i q^{33} -1.42175i q^{37} +14.1533 q^{39} +7.35699 q^{41} +9.80292i q^{43} +(-1.87461 - 9.39043i) q^{45} +6.80187i q^{47} -11.4846 q^{51} +11.2567i q^{53} +(0.402635 + 2.01691i) q^{55} -4.96404i q^{57} +11.3331 q^{59} +9.64670 q^{61} +(-11.5006 + 2.29585i) q^{65} -2.78505i q^{67} +23.6948 q^{69} +11.8349 q^{71} +5.11486i q^{73} +(5.18072 + 12.4587i) q^{75} -0.727336 q^{79} -3.50830 q^{81} -4.37545i q^{83} +(9.33205 - 1.86295i) q^{85} +18.1156i q^{87} -0.963998 q^{89} -12.5269i q^{93} +(0.805234 + 4.03364i) q^{95} -13.5588i q^{97} -3.93889 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 28 q^{9}+O(q^{10}) $ 20 * q - 28 * q^9	$ 20 q - 28 q^{9} - 24 q^{11} - 8 q^{15} + 8 q^{25} - 24 q^{29} + 40 q^{39} - 72 q^{51} - 20 q^{65} - 16 q^{71} + 8 q^{79} + 100 q^{81} + 60 q^{85} - 48 q^{95} + 144 q^{99}+O(q^{100}) $ 20 * q - 28 * q^9 - 24 * q^11 - 8 * q^15 + 8 * q^25 - 24 * q^29 + 40 * q^39 - 72 * q^51 - 20 * q^65 - 16 * q^71 + 8 * q^79 + 100 * q^81 + 60 * q^85 - 48 * q^95 + 144 * 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
$3$		−	2.69859i		−	1.55803i	−0.627004	−	0.779016i	$-0.715719\pi$
$3$		−	2.69859i		−	1.55803i	0.627004	−	0.779016i	$-0.284281\pi$
$4$		0			0
$5$	0.437748	+	2.19280i	0.195767	+	0.980650i
$5$	0.437748	+	2.19280i	0.195767	+	0.980650i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−4.28239			−1.42746
$10$		0			0
$11$	0.919787			0.277326			0.138663	−	0.990340i	$-0.455719\pi$
$11$	0.919787			0.277326			0.138663	+	0.990340i	$0.455719\pi$
$12$		0			0
$13$			5.24470i			1.45462i	0.686310	+	0.727309i	$0.259229\pi$
$13$			5.24470i			1.45462i	−0.686310	+	0.727309i	$0.740771\pi$
$14$		0			0
$15$	5.91747	−	1.18130i	1.52788	−	0.305011i
$16$		0			0
$17$		−	4.25577i		−	1.03217i	−0.856536	−	0.516087i	$-0.827388\pi$
$17$		−	4.25577i		−	1.03217i	0.856536	−	0.516087i	$-0.172612\pi$
$18$		0			0
$19$	1.83949			0.422009			0.211004	−	0.977485i	$-0.432327\pi$
$19$	1.83949			0.422009			0.211004	+	0.977485i	$0.432327\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			8.78042i			1.83085i	0.402494	+	0.915423i	$0.368143\pi$
$23$			8.78042i			1.83085i	−0.402494	+	0.915423i	$0.631857\pi$
$24$		0			0
$25$	−4.61675	+	1.91979i	−0.923351	+	0.383957i
$26$		0			0
$27$			3.46065i			0.666002i
$28$		0			0
$29$	−6.71298			−1.24657			−0.623284	−	0.781995i	$-0.714202\pi$
$29$	−6.71298			−1.24657			−0.623284	+	0.781995i	$0.714202\pi$
$30$		0			0
$31$	4.64200			0.833728			0.416864	−	0.908969i	$-0.363129\pi$
$31$	4.64200			0.833728			0.416864	+	0.908969i	$0.363129\pi$
$32$		0			0
$33$		−	2.48213i		−	0.432083i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	1.42175i		−	0.233735i	−0.993148	−	0.116867i	$-0.962715\pi$
$37$		−	1.42175i		−	0.233735i	0.993148	−	0.116867i	$-0.0372853\pi$
$38$		0			0
$39$	14.1533			2.26634
$40$		0			0
$41$	7.35699			1.14897			0.574484	−	0.818515i	$-0.305203\pi$
$41$	7.35699			1.14897			0.574484	+	0.818515i	$0.305203\pi$
$42$		0			0
$43$			9.80292i			1.49493i	0.664301	+	0.747466i	$0.268730\pi$
$43$			9.80292i			1.49493i	−0.664301	+	0.747466i	$0.731270\pi$
$44$		0			0
$45$	−1.87461	−	9.39043i	−0.279450	−	1.39984i
$46$		0			0
$47$			6.80187i			0.992155i	0.868278	+	0.496078i	$0.165227\pi$
$47$			6.80187i			0.992155i	−0.868278	+	0.496078i	$0.834773\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−11.4846			−1.60816
$52$		0			0
$53$			11.2567i			1.54623i	0.634268	+	0.773113i	$0.281302\pi$
$53$			11.2567i			1.54623i	−0.634268	+	0.773113i	$0.718698\pi$
$54$		0			0
$55$	0.402635	+	2.01691i	0.0542913	+	0.271960i
$56$		0			0
$57$		−	4.96404i		−	0.657503i
$58$		0			0
$59$	11.3331			1.47544			0.737721	−	0.675106i	$-0.235902\pi$
$59$	11.3331			1.47544			0.737721	+	0.675106i	$0.235902\pi$
$60$		0			0
$61$	9.64670			1.23513			0.617567	−	0.786519i	$-0.288119\pi$
$61$	9.64670			1.23513			0.617567	+	0.786519i	$0.288119\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−11.5006	+	2.29585i	−1.42647	+	0.284766i
$66$		0			0
$67$		−	2.78505i		−	0.340248i	−0.985423	−	0.170124i	$-0.945583\pi$
$67$		−	2.78505i		−	0.340248i	0.985423	−	0.170124i	$-0.0544169\pi$
$68$		0			0
$69$	23.6948			2.85252
$70$		0			0
$71$	11.8349			1.40455			0.702275	−	0.711906i	$-0.252168\pi$
$71$	11.8349			1.40455			0.702275	+	0.711906i	$0.252168\pi$
$72$		0			0
$73$			5.11486i			0.598649i	0.954151	+	0.299325i	$0.0967614\pi$
$73$			5.11486i			0.598649i	−0.954151	+	0.299325i	$0.903239\pi$
$74$		0			0
$75$	5.18072	+	12.4587i	0.598218	+	1.43861i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.727336			−0.0818317			−0.0409159	−	0.999163i	$-0.513028\pi$
$79$	−0.727336			−0.0818317			−0.0409159	+	0.999163i	$0.513028\pi$
$80$		0			0
$81$	−3.50830			−0.389811
$82$		0			0
$83$		−	4.37545i		−	0.480268i	−0.970740	−	0.240134i	$-0.922809\pi$
$83$		−	4.37545i		−	0.480268i	0.970740	−	0.240134i	$-0.0771914\pi$
$84$		0			0
$85$	9.33205	−	1.86295i	1.01220	−	0.202065i
$86$		0			0
$87$			18.1156i			1.94219i
$88$		0			0
$89$	−0.963998			−0.102184			−0.0510918	−	0.998694i	$-0.516270\pi$
$89$	−0.963998			−0.102184			−0.0510918	+	0.998694i	$0.516270\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	12.5269i		−	1.29898i
$94$		0			0
$95$	0.805234	+	4.03364i	0.0826153	+	0.413843i
$96$		0			0
$97$		−	13.5588i		−	1.37669i	−0.725384	−	0.688344i	$-0.758338\pi$
$97$		−	13.5588i		−	1.37669i	0.725384	−	0.688344i	$-0.241662\pi$
$98$		0			0
$99$	−3.93889			−0.395873
$100$		0			0
$101$	−19.2029			−1.91076			−0.955379	−	0.295384i	$-0.904552\pi$
$101$	−19.2029			−1.91076			−0.955379	+	0.295384i	$0.904552\pi$
$102$		0			0
$103$		−	1.67686i		−	0.165226i	−0.996582	−	0.0826130i	$-0.973673\pi$
$103$		−	1.67686i		−	0.165226i	0.996582	−	0.0826130i	$-0.0263265\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	18.1090i		−	1.75067i	−0.483520	−	0.875334i	$-0.660642\pi$
$107$		−	18.1090i		−	1.75067i	0.483520	−	0.875334i	$-0.339358\pi$
$108$		0			0
$109$	2.07628			0.198871			0.0994357	−	0.995044i	$-0.468296\pi$
$109$	2.07628			0.198871			0.0994357	+	0.995044i	$0.468296\pi$
$110$		0			0
$111$	−3.83673			−0.364167
$112$		0			0
$113$			1.45378i			0.136760i	0.997659	+	0.0683800i	$0.0217830\pi$
$113$			1.45378i			0.136760i	−0.997659	+	0.0683800i	$0.978217\pi$
$114$		0			0
$115$	−19.2537	+	3.84361i	−1.79542	+	0.358419i
$116$		0			0
$117$		−	22.4599i		−	2.07641i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.1540			−0.923090
$122$		0			0
$123$		−	19.8535i		−	1.79013i
$124$		0			0
$125$	−6.23069	−	9.28324i	−0.557289	−	0.830318i
$126$		0			0
$127$			7.25670i			0.643928i	0.946752	+	0.321964i	$0.104343\pi$
$127$			7.25670i			0.643928i	−0.946752	+	0.321964i	$0.895657\pi$
$128$		0			0
$129$	26.4541			2.32915
$130$		0			0
$131$	−4.04070			−0.353038			−0.176519	−	0.984297i	$-0.556484\pi$
$131$	−4.04070			−0.353038			−0.176519	+	0.984297i	$0.556484\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−7.58851	+	1.51489i	−0.653115	+	0.130381i
$136$		0			0
$137$		−	4.35288i		−	0.371891i	−0.982560	−	0.185946i	$-0.940465\pi$
$137$		−	4.35288i		−	0.371891i	0.982560	−	0.185946i	$-0.0595348\pi$
$138$		0			0
$139$	14.9602			1.26891			0.634455	−	0.772960i	$-0.281225\pi$
$139$	14.9602			1.26891			0.634455	+	0.772960i	$0.281225\pi$
$140$		0			0
$141$	18.3555			1.54581
$142$		0			0
$143$			4.82401i			0.403404i
$144$		0			0
$145$	−2.93859	−	14.7202i	−0.244037	−	1.22245i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.90223			0.811223			0.405611	−	0.914046i	$-0.367059\pi$
$149$	9.90223			0.811223			0.405611	+	0.914046i	$0.367059\pi$
$150$		0			0
$151$	0.794625			0.0646657			0.0323328	−	0.999477i	$-0.489706\pi$
$151$	0.794625			0.0646657			0.0323328	+	0.999477i	$0.489706\pi$
$152$		0			0
$153$			18.2249i			1.47339i
$154$		0			0
$155$	2.03203	+	10.1790i	0.163216	+	0.817596i
$156$		0			0
$157$			23.1524i			1.84777i	0.382675	+	0.923883i	$0.375003\pi$
$157$			23.1524i			1.84777i	−0.382675	+	0.923883i	$0.624997\pi$
$158$		0			0
$159$	30.3772			2.40907
$160$		0			0
$161$		0			0
$162$		0			0
$163$			7.04479i			0.551791i	0.961188	+	0.275895i	$0.0889743\pi$
$163$			7.04479i			0.551791i	−0.961188	+	0.275895i	$0.911026\pi$
$164$		0			0
$165$	5.44282	−	1.08655i	0.423723	−	0.0845875i
$166$		0			0
$167$		−	7.39931i		−	0.572576i	−0.958144	−	0.286288i	$-0.907579\pi$
$167$		−	7.39931i		−	0.572576i	0.958144	−	0.286288i	$-0.0924214\pi$
$168$		0			0
$169$	−14.5069			−1.11591
$170$		0			0
$171$	−7.87743			−0.602402
$172$		0			0
$173$			9.11776i			0.693211i	0.938011	+	0.346605i	$0.112666\pi$
$173$			9.11776i			0.693211i	−0.938011	+	0.346605i	$0.887334\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	30.5833i		−	2.29878i
$178$		0			0
$179$	13.8396			1.03442			0.517209	−	0.855859i	$-0.326971\pi$
$179$	13.8396			1.03442			0.517209	+	0.855859i	$0.326971\pi$
$180$		0			0
$181$	0.0875186			0.00650521			0.00325260	−	0.999995i	$-0.498965\pi$
$181$	0.0875186			0.00650521			0.00325260	+	0.999995i	$0.498965\pi$
$182$		0			0
$183$		−	26.0325i		−	1.92438i
$184$		0			0
$185$	3.11762	−	0.622370i	0.229212	−	0.0457575i
$186$		0			0
$187$		−	3.91440i		−	0.286249i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.6296			−0.913844			−0.456922	−	0.889507i	$-0.651048\pi$
$191$	−12.6296			−0.913844			−0.456922	+	0.889507i	$0.651048\pi$
$192$		0			0
$193$			20.5968i			1.48259i	0.671179	+	0.741295i	$0.265788\pi$
$193$			20.5968i			1.48259i	−0.671179	+	0.741295i	$0.734212\pi$
$194$		0			0
$195$	6.19557	+	31.0354i	0.443674	+	2.22249i
$196$		0			0
$197$		−	10.4851i		−	0.747033i	−0.927624	−	0.373517i	$-0.878152\pi$
$197$		−	10.4851i		−	0.747033i	0.927624	−	0.373517i	$-0.121848\pi$
$198$		0			0
$199$	11.8092			0.837133			0.418567	−	0.908186i	$-0.362533\pi$
$199$	11.8092			0.837133			0.418567	+	0.908186i	$0.362533\pi$
$200$		0			0
$201$	−7.51572			−0.530118
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.22051	+	16.1324i	0.224930	+	1.12674i
$206$		0			0
$207$		−	37.6012i		−	2.61346i
$208$		0			0
$209$	1.69194			0.117034
$210$		0			0
$211$	−11.3594			−0.782014			−0.391007	−	0.920388i	$-0.627873\pi$
$211$	−11.3594			−0.782014			−0.391007	+	0.920388i	$0.627873\pi$
$212$		0			0
$213$		−	31.9377i		−	2.18833i
$214$		0			0
$215$	−21.4959	+	4.29121i	−1.46601	+	0.292658i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	13.8029			0.932715
$220$		0			0
$221$	22.3202			1.50142
$222$		0			0
$223$		−	26.9321i		−	1.80351i	−0.432252	−	0.901753i	$-0.642281\pi$
$223$		−	26.9321i		−	1.80351i	0.432252	−	0.901753i	$-0.357719\pi$
$224$		0			0
$225$	19.7707	−	8.22128i	1.31805	−	0.548085i
$226$		0			0
$227$			17.9510i			1.19145i	0.803190	+	0.595724i	$0.203135\pi$
$227$			17.9510i			1.19145i	−0.803190	+	0.595724i	$0.796865\pi$
$228$		0			0
$229$	16.0850			1.06293			0.531463	−	0.847082i	$-0.321643\pi$
$229$	16.0850			1.06293			0.531463	+	0.847082i	$0.321643\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	19.5915i		−	1.28348i	−0.766922	−	0.641740i	$-0.778213\pi$
$233$		−	19.5915i		−	1.28348i	0.766922	−	0.641740i	$-0.221787\pi$
$234$		0			0
$235$	−14.9152	+	2.97750i	−0.972958	+	0.194231i
$236$		0			0
$237$			1.96278i			0.127496i
$238$		0			0
$239$	−26.0767			−1.68676			−0.843382	−	0.537314i	$-0.819439\pi$
$239$	−26.0767			−1.68676			−0.843382	+	0.537314i	$0.819439\pi$
$240$		0			0
$241$	23.2436			1.49725			0.748625	−	0.662994i	$-0.230714\pi$
$241$	23.2436			1.49725			0.748625	+	0.662994i	$0.230714\pi$
$242$		0			0
$243$			19.8494i			1.27334i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			9.64759i			0.613862i
$248$		0			0
$249$	−11.8075			−0.748273
$250$		0			0
$251$	6.75366			0.426287			0.213144	−	0.977021i	$-0.431630\pi$
$251$	6.75366			0.426287			0.213144	+	0.977021i	$0.431630\pi$
$252$		0			0
$253$			8.07612i			0.507742i
$254$		0			0
$255$	−5.02734	−	25.1834i	−0.314824	−	1.57704i
$256$		0			0
$257$		−	18.4392i		−	1.15021i	−0.818080	−	0.575104i	$-0.804962\pi$
$257$		−	18.4392i		−	1.15021i	0.818080	−	0.575104i	$-0.195038\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	28.7476			1.77943
$262$		0			0
$263$		−	9.92808i		−	0.612192i	−0.952001	−	0.306096i	$-0.900977\pi$
$263$		−	9.92808i		−	0.612192i	0.952001	−	0.306096i	$-0.0990228\pi$
$264$		0			0
$265$	−24.6837	+	4.92759i	−1.51631	+	0.302700i
$266$		0			0
$267$			2.60144i			0.159205i
$268$		0			0
$269$	−14.8533			−0.905624			−0.452812	−	0.891606i	$-0.649579\pi$
$269$	−14.8533			−0.905624			−0.452812	+	0.891606i	$0.649579\pi$
$270$		0			0
$271$	−3.34123			−0.202965			−0.101483	−	0.994837i	$-0.532359\pi$
$271$	−3.34123			−0.202965			−0.101483	+	0.994837i	$0.532359\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.24643	+	1.76580i	−0.256069	+	0.106482i
$276$		0			0
$277$			17.5428i			1.05404i	0.849852	+	0.527021i	$0.176691\pi$
$277$			17.5428i			1.05404i	−0.849852	+	0.527021i	$0.823309\pi$
$278$		0			0
$279$	−19.8789			−1.19012
$280$		0			0
$281$	1.54676			0.0922720			0.0461360	−	0.998935i	$-0.485309\pi$
$281$	1.54676			0.0922720			0.0461360	+	0.998935i	$0.485309\pi$
$282$		0			0
$283$		−	6.37758i		−	0.379108i	−0.981870	−	0.189554i	$-0.939296\pi$
$283$		−	6.37758i		−	0.379108i	0.981870	−	0.189554i	$-0.0607042\pi$
$284$		0			0
$285$	10.8852	−	2.17300i	0.644781	−	0.128717i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.11154			−0.0653848
$290$		0			0
$291$	−36.5897			−2.14492
$292$		0			0
$293$		−	2.65690i		−	0.155218i	−0.996984	−	0.0776090i	$-0.975271\pi$
$293$		−	2.65690i		−	0.155218i	0.996984	−	0.0776090i	$-0.0247286\pi$
$294$		0			0
$295$	4.96103	+	24.8512i	0.288842	+	1.44689i
$296$		0			0
$297$			3.18306i			0.184700i
$298$		0			0
$299$	−46.0507			−2.66318
$300$		0			0
$301$		0			0
$302$		0			0
$303$			51.8207i			2.97702i
$304$		0			0
$305$	4.22282	+	21.1533i	0.241798	+	1.21123i
$306$		0			0
$307$			11.0822i			0.632493i	0.948677	+	0.316246i	$0.102423\pi$
$307$			11.0822i			0.632493i	−0.948677	+	0.316246i	$0.897577\pi$
$308$		0			0
$309$	−4.52516			−0.257427
$310$		0			0
$311$	−22.9133			−1.29930			−0.649648	−	0.760235i	$-0.725084\pi$
$311$	−22.9133			−1.29930			−0.649648	+	0.760235i	$0.725084\pi$
$312$		0			0
$313$			20.7070i			1.17043i	0.810879	+	0.585214i	$0.198989\pi$
$313$			20.7070i			1.17043i	−0.810879	+	0.585214i	$0.801011\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			16.4624i			0.924620i	0.886718	+	0.462310i	$0.152979\pi$
$317$			16.4624i			0.924620i	−0.886718	+	0.462310i	$0.847021\pi$
$318$		0			0
$319$	−6.17451			−0.345706
$320$		0			0
$321$	−48.8689			−2.72760
$322$		0			0
$323$		−	7.82846i		−	0.435587i
$324$		0			0
$325$	−10.0687	−	24.2135i	−0.558511	−	1.34312i
$326$		0			0
$327$		−	5.60303i		−	0.309848i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.5231			1.01812			0.509062	−	0.860730i	$-0.329993\pi$
$331$	18.5231			1.01812			0.509062	+	0.860730i	$0.329993\pi$
$332$		0			0
$333$			6.08851i			0.333648i
$334$		0			0
$335$	6.10707	−	1.21915i	0.333665	−	0.0666093i
$336$		0			0
$337$			9.84378i			0.536225i	0.963388	+	0.268112i	$0.0863999\pi$
$337$			9.84378i			0.536225i	−0.963388	+	0.268112i	$0.913600\pi$
$338$		0			0
$339$	3.92316			0.213077
$340$		0			0
$341$	4.26965			0.231215
$342$		0			0
$343$		0			0
$344$		0			0
$345$	10.3723	+	51.9579i	0.558428	+	2.79732i
$346$		0			0
$347$			26.6878i			1.43267i	0.697755	+	0.716337i	$0.254183\pi$
$347$			26.6878i			1.43267i	−0.697755	+	0.716337i	$0.745817\pi$
$348$		0			0
$349$	−8.56272			−0.458352			−0.229176	−	0.973385i	$-0.573603\pi$
$349$	−8.56272			−0.458352			−0.229176	+	0.973385i	$0.573603\pi$
$350$		0			0
$351$	−18.1501			−0.968778
$352$		0			0
$353$		−	13.6743i		−	0.727808i	−0.931436	−	0.363904i	$-0.881444\pi$
$353$		−	13.6743i		−	0.727808i	0.931436	−	0.363904i	$-0.118556\pi$
$354$		0			0
$355$	5.18072	+	25.9517i	0.274964	+	1.37737i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5.99861			−0.316594			−0.158297	−	0.987392i	$-0.550600\pi$
$359$	−5.99861			−0.316594			−0.158297	+	0.987392i	$0.550600\pi$
$360$		0			0
$361$	−15.6163			−0.821909
$362$		0			0
$363$			27.4015i			1.43820i
$364$		0			0
$365$	−11.2159	+	2.23902i	−0.587066	+	0.117196i
$366$		0			0
$367$			3.14847i			0.164349i	0.996618	+	0.0821745i	$0.0261865\pi$
$367$			3.14847i			0.164349i	−0.996618	+	0.0821745i	$0.973814\pi$
$368$		0			0
$369$	−31.5055			−1.64011
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	3.95038i		−	0.204543i	−0.994757	−	0.102271i	$-0.967389\pi$
$373$		−	3.95038i		−	0.204543i	0.994757	−	0.102271i	$-0.0326110\pi$
$374$		0			0
$375$	−25.0517	+	16.8141i	−1.29366	+	0.868275i
$376$		0			0
$377$		−	35.2076i		−	1.81328i
$378$		0			0
$379$	29.9555			1.53871			0.769355	−	0.638822i	$-0.220578\pi$
$379$	29.9555			1.53871			0.769355	+	0.638822i	$0.220578\pi$
$380$		0			0
$381$	19.5829			1.00326
$382$		0			0
$383$		−	14.0242i		−	0.716602i	−0.933606	−	0.358301i	$-0.883356\pi$
$383$		−	14.0242i		−	0.716602i	0.933606	−	0.358301i	$-0.116644\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	41.9799i		−	2.13396i
$388$		0			0
$389$	10.3188			0.523185			0.261592	−	0.965178i	$-0.415752\pi$
$389$	10.3188			0.523185			0.261592	+	0.965178i	$0.415752\pi$
$390$		0			0
$391$	37.3674			1.88975
$392$		0			0
$393$			10.9042i			0.550044i
$394$		0			0
$395$	−0.318390	−	1.59490i	−0.0160199	−	0.0802483i
$396$		0			0
$397$			9.68961i			0.486308i	0.969988	+	0.243154i	$0.0781820\pi$
$397$			9.68961i			0.486308i	−0.969988	+	0.243154i	$0.921818\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−19.2731			−0.962454			−0.481227	−	0.876596i	$-0.659809\pi$
$401$	−19.2731			−0.962454			−0.481227	+	0.876596i	$0.659809\pi$
$402$		0			0
$403$			24.3459i			1.21276i
$404$		0			0
$405$	−1.53575	−	7.69301i	−0.0763121	−	0.382269i
$406$		0			0
$407$		−	1.30771i		−	0.0648208i
$408$		0			0
$409$	−22.8951			−1.13209			−0.566044	−	0.824375i	$-0.691527\pi$
$409$	−22.8951			−1.13209			−0.566044	+	0.824375i	$0.691527\pi$
$410$		0			0
$411$	−11.7466			−0.579419
$412$		0			0
$413$		0			0
$414$		0			0
$415$	9.59449	−	1.91534i	0.470975	−	0.0940205i
$416$		0			0
$417$		−	40.3715i		−	1.97700i
$418$		0			0
$419$	−30.7282			−1.50117			−0.750585	−	0.660774i	$-0.770228\pi$
$419$	−30.7282			−1.50117			−0.750585	+	0.660774i	$0.770228\pi$
$420$		0			0
$421$	−19.2411			−0.937754			−0.468877	−	0.883263i	$-0.655341\pi$
$421$	−19.2411			−0.937754			−0.468877	+	0.883263i	$0.655341\pi$
$422$		0			0
$423$		−	29.1283i		−	1.41627i
$424$		0			0
$425$	8.17017	+	19.6478i	0.396311	+	0.953059i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	13.0180			0.628516
$430$		0			0
$431$	14.0860			0.678499			0.339250	−	0.940696i	$-0.389827\pi$
$431$	14.0860			0.678499			0.339250	+	0.940696i	$0.389827\pi$
$432$		0			0
$433$			1.91785i			0.0921658i	0.998938	+	0.0460829i	$0.0146738\pi$
$433$			1.91785i			0.0921658i	−0.998938	+	0.0460829i	$0.985326\pi$
$434$		0			0
$435$	−39.7239	+	7.93005i	−1.90461	+	0.380217i
$436$		0			0
$437$			16.1515i			0.772633i
$438$		0			0
$439$	−12.6532			−0.603906			−0.301953	−	0.953323i	$-0.597639\pi$
$439$	−12.6532			−0.603906			−0.301953	+	0.953323i	$0.597639\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			21.6954i			1.03078i	0.856956	+	0.515390i	$0.172353\pi$
$443$			21.6954i			1.03078i	−0.856956	+	0.515390i	$0.827647\pi$
$444$		0			0
$445$	−0.421988	−	2.11386i	−0.0200042	−	0.100206i
$446$		0			0
$447$		−	26.7221i		−	1.26391i
$448$		0			0
$449$	−23.9706			−1.13124			−0.565621	−	0.824666i	$-0.691363\pi$
$449$	−23.9706			−1.13124			−0.565621	+	0.824666i	$0.691363\pi$
$450$		0			0
$451$	6.76687			0.318639
$452$		0			0
$453$		−	2.14437i		−	0.100751i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	7.29836i		−	0.341403i	−0.985323	−	0.170701i	$-0.945397\pi$
$457$		−	7.29836i		−	0.341403i	0.985323	−	0.170701i	$-0.0546033\pi$
$458$		0			0
$459$	14.7277			0.687431
$460$		0			0
$461$	−19.2232			−0.895312			−0.447656	−	0.894206i	$-0.647741\pi$
$461$	−19.2232			−0.895312			−0.447656	+	0.894206i	$0.647741\pi$
$462$		0			0
$463$			1.89096i			0.0878803i	0.999034	+	0.0439401i	$0.0139911\pi$
$463$			1.89096i			0.0878803i	−0.999034	+	0.0439401i	$0.986009\pi$
$464$		0			0
$465$	27.4689	−	5.48361i	1.27384	−	0.254296i
$466$		0			0
$467$		−	7.15284i		−	0.330994i	−0.986210	−	0.165497i	$-0.947077\pi$
$467$		−	7.15284i		−	0.330994i	0.986210	−	0.165497i	$-0.0529228\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	62.4790			2.87888
$472$		0			0
$473$			9.01660i			0.414584i
$474$		0			0
$475$	−8.49249	+	3.53144i	−0.389662	+	0.162033i
$476$		0			0
$477$		−	48.2056i		−	2.20718i
$478$		0			0
$479$	−7.46388			−0.341033			−0.170517	−	0.985355i	$-0.554544\pi$
$479$	−7.46388			−0.341033			−0.170517	+	0.985355i	$0.554544\pi$
$480$		0			0
$481$	7.45667			0.339995
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.7318	−	5.93534i	1.35005	−	0.269510i
$486$		0			0
$487$		−	38.1373i		−	1.72817i	−0.503348	−	0.864084i	$-0.667899\pi$
$487$		−	38.1373i		−	1.72817i	0.503348	−	0.864084i	$-0.332101\pi$
$488$		0			0
$489$	19.0110			0.859708
$490$		0			0
$491$	−4.25569			−0.192057			−0.0960283	−	0.995379i	$-0.530614\pi$
$491$	−4.25569			−0.192057			−0.0960283	+	0.995379i	$0.530614\pi$
$492$		0			0
$493$			28.5689i			1.28668i
$494$		0			0
$495$	−1.72424	−	8.63720i	−0.0774988	−	0.388213i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	36.0796			1.61515			0.807573	−	0.589768i	$-0.200781\pi$
$499$	36.0796			1.61515			0.807573	+	0.589768i	$0.200781\pi$
$500$		0			0
$501$	−19.9677			−0.892091
$502$		0			0
$503$		−	10.0442i		−	0.447848i	−0.974607	−	0.223924i	$-0.928113\pi$
$503$		−	10.0442i		−	0.447848i	0.974607	−	0.223924i	$-0.0718867\pi$
$504$		0			0
$505$	−8.40601	−	42.1081i	−0.374063	−	1.87378i
$506$		0			0
$507$			39.1481i			1.73863i
$508$		0			0
$509$	−22.2612			−0.986710			−0.493355	−	0.869828i	$-0.664230\pi$
$509$	−22.2612			−0.986710			−0.493355	+	0.869828i	$0.664230\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			6.36584i			0.281059i
$514$		0			0
$515$	3.67702	−	0.734042i	0.162029	−	0.0323457i
$516$		0			0
$517$			6.25628i			0.275151i
$518$		0			0
$519$	24.6051			1.08004
$520$		0			0
$521$	−4.18727			−0.183447			−0.0917237	−	0.995784i	$-0.529238\pi$
$521$	−4.18727			−0.183447			−0.0917237	+	0.995784i	$0.529238\pi$
$522$		0			0
$523$		−	6.49224i		−	0.283886i	−0.989875	−	0.141943i	$-0.954665\pi$
$523$		−	6.49224i		−	0.283886i	0.989875	−	0.141943i	$-0.0453350\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	19.7553i		−	0.860553i
$528$		0			0
$529$	−54.0958			−2.35199
$530$		0			0
$531$	−48.5327			−2.10614
$532$		0			0
$533$			38.5852i			1.67131i
$534$		0			0
$535$	39.7095	−	7.92719i	1.71679	−	0.342722i
$536$		0			0
$537$		−	37.3473i		−	1.61166i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−6.42739			−0.276335			−0.138168	−	0.990409i	$-0.544121\pi$
$541$	−6.42739			−0.276335			−0.138168	+	0.990409i	$0.544121\pi$
$542$		0			0
$543$		−	0.236177i		−	0.0101353i
$544$		0			0
$545$	0.908886	+	4.55287i	0.0389324	+	0.195023i
$546$		0			0
$547$		−	0.626973i		−	0.0268074i	−0.999910	−	0.0134037i	$-0.995733\pi$
$547$		−	0.626973i		−	0.0268074i	0.999910	−	0.0134037i	$-0.00426666\pi$
$548$		0			0
$549$	−41.3109			−1.76311
$550$		0			0
$551$	−12.3485			−0.526063
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.67952	−	8.41319i	−0.0712917	−	0.357120i
$556$		0			0
$557$		−	24.6950i		−	1.04636i	−0.852221	−	0.523181i	$-0.824745\pi$
$557$		−	24.6950i		−	1.04636i	0.852221	−	0.523181i	$-0.175255\pi$
$558$		0			0
$559$	−51.4134			−2.17455
$560$		0			0
$561$	−10.5634			−0.445985
$562$		0			0
$563$			8.88596i			0.374498i	0.982312	+	0.187249i	$0.0599572\pi$
$563$			8.88596i			0.374498i	−0.982312	+	0.187249i	$0.940043\pi$
$564$		0			0
$565$	−3.18785	+	0.636389i	−0.134114	+	0.0267731i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.1464			−0.467281			−0.233641	−	0.972323i	$-0.575064\pi$
$569$	−11.1464			−0.467281			−0.233641	+	0.972323i	$0.575064\pi$
$570$		0			0
$571$	21.2210			0.888072			0.444036	−	0.896009i	$-0.353546\pi$
$571$	21.2210			0.888072			0.444036	+	0.896009i	$0.353546\pi$
$572$		0			0
$573$			34.0820i			1.42380i
$574$		0			0
$575$	−16.8565	−	40.5371i	−0.702967	−	1.69051i
$576$		0			0
$577$			10.2379i			0.426209i	0.977029	+	0.213104i	$0.0683575\pi$
$577$			10.2379i			0.426209i	−0.977029	+	0.213104i	$0.931643\pi$
$578$		0			0
$579$	55.5824			2.30992
$580$		0			0
$581$		0			0
$582$		0			0
$583$			10.3538i			0.428809i
$584$		0			0
$585$	49.2500	−	9.83175i	2.03624	−	0.406493i
$586$		0			0
$587$			13.6732i			0.564354i	0.959362	+	0.282177i	$0.0910565\pi$
$587$			13.6732i			0.564354i	−0.959362	+	0.282177i	$0.908943\pi$
$588$		0			0
$589$	8.53893			0.351841
$590$		0			0
$591$	−28.2950			−1.16390
$592$		0			0
$593$		−	18.0203i		−	0.740004i	−0.929031	−	0.370002i	$-0.879357\pi$
$593$		−	18.0203i		−	0.740004i	0.929031	−	0.370002i	$-0.120643\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	31.8682i		−	1.30428i
$598$		0			0
$599$	2.02798			0.0828609			0.0414304	−	0.999141i	$-0.486809\pi$
$599$	2.02798			0.0828609			0.0414304	+	0.999141i	$0.486809\pi$
$600$		0			0
$601$	31.5443			1.28672			0.643359	−	0.765565i	$-0.277540\pi$
$601$	31.5443			1.28672			0.643359	+	0.765565i	$0.277540\pi$
$602$		0			0
$603$			11.9267i			0.485692i
$604$		0			0
$605$	−4.44489	−	22.2657i	−0.180710	−	0.905229i
$606$		0			0
$607$		−	36.5276i		−	1.48261i	−0.671169	−	0.741305i	$-0.734207\pi$
$607$		−	36.5276i		−	1.48261i	0.671169	−	0.741305i	$-0.265793\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−35.6738			−1.44321
$612$		0			0
$613$		−	27.1153i		−	1.09518i	−0.836748	−	0.547588i	$-0.815546\pi$
$613$		−	27.1153i		−	1.09518i	0.836748	−	0.547588i	$-0.184454\pi$
$614$		0			0
$615$	43.5348	−	8.69083i	1.75549	−	0.350448i
$616$		0			0
$617$			33.9078i			1.36508i	0.730849	+	0.682539i	$0.239124\pi$
$617$			33.9078i			1.36508i	−0.730849	+	0.682539i	$0.760876\pi$
$618$		0			0
$619$	1.51850			0.0610337			0.0305168	−	0.999534i	$-0.490285\pi$
$619$	1.51850			0.0610337			0.0305168	+	0.999534i	$0.490285\pi$
$620$		0			0
$621$	−30.3860			−1.21935
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.6288	−	17.7264i	0.705153	−	0.709055i
$626$		0			0
$627$		−	4.56586i		−	0.182343i
$628$		0			0
$629$	−6.05065			−0.241255
$630$		0			0
$631$	27.7605			1.10513			0.552565	−	0.833470i	$-0.313649\pi$
$631$	27.7605			1.10513			0.552565	+	0.833470i	$0.313649\pi$
$632$		0			0
$633$			30.6544i			1.21840i
$634$		0			0
$635$	−15.9125	+	3.17660i	−0.631468	+	0.126060i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−50.6819			−2.00494
$640$		0			0
$641$	21.4853			0.848619			0.424309	−	0.905517i	$-0.360517\pi$
$641$	21.4853			0.848619			0.424309	+	0.905517i	$0.360517\pi$
$642$		0			0
$643$		−	29.7709i		−	1.17405i	−0.809569	−	0.587025i	$-0.800299\pi$
$643$		−	29.7709i		−	1.17405i	0.809569	−	0.587025i	$-0.199701\pi$
$644$		0			0
$645$	11.5802	+	58.0085i	0.455970	+	2.28408i
$646$		0			0
$647$			18.2683i			0.718202i	0.933299	+	0.359101i	$0.116917\pi$
$647$			18.2683i			0.718202i	−0.933299	+	0.359101i	$0.883083\pi$
$648$		0			0
$649$	10.4240			0.409179
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	4.69936i		−	0.183900i	−0.995764	−	0.0919501i	$-0.970690\pi$
$653$		−	4.69936i		−	0.183900i	0.995764	−	0.0919501i	$-0.0293100\pi$
$654$		0			0
$655$	−1.76881	−	8.86045i	−0.0691130	−	0.346206i
$656$		0			0
$657$		−	21.9038i		−	0.854550i
$658$		0			0
$659$	46.9732			1.82982			0.914908	−	0.403663i	$-0.132263\pi$
$659$	46.9732			1.82982			0.914908	+	0.403663i	$0.132263\pi$
$660$		0			0
$661$	8.23347			0.320245			0.160122	−	0.987097i	$-0.448811\pi$
$661$	8.23347			0.320245			0.160122	+	0.987097i	$0.448811\pi$
$662$		0			0
$663$		−	60.2331i		−	2.33926i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	58.9428i		−	2.28227i
$668$		0			0
$669$	−72.6787			−2.80992
$670$		0			0
$671$	8.87291			0.342535
$672$		0			0
$673$		−	30.8570i		−	1.18945i	−0.803929	−	0.594725i	$-0.797261\pi$
$673$		−	30.8570i		−	1.18945i	0.803929	−	0.594725i	$-0.202739\pi$
$674$		0			0
$675$	−6.64371	−	15.9770i	−0.255716	−	0.614954i
$676$		0			0
$677$			17.8391i			0.685612i	0.939406	+	0.342806i	$0.111377\pi$
$677$			17.8391i			0.685612i	−0.939406	+	0.342806i	$0.888623\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	48.4423			1.85631
$682$		0			0
$683$			2.86296i			0.109548i	0.998499	+	0.0547741i	$0.0174439\pi$
$683$			2.86296i			0.109548i	−0.998499	+	0.0547741i	$0.982556\pi$
$684$		0			0
$685$	9.54499	−	1.90546i	0.364695	−	0.0728039i
$686$		0			0
$687$		−	43.4068i		−	1.65607i
$688$		0			0
$689$	−59.0380			−2.24917
$690$		0			0
$691$	39.3220			1.49588			0.747940	−	0.663767i	$-0.231043\pi$
$691$	39.3220			1.49588			0.747940	+	0.663767i	$0.231043\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.54881	+	32.8048i	0.248410	+	1.24436i
$696$		0			0
$697$		−	31.3096i		−	1.18594i
$698$		0			0
$699$	−52.8694			−1.99970
$700$		0			0
$701$	−8.95691			−0.338298			−0.169149	−	0.985591i	$-0.554102\pi$
$701$	−8.95691			−0.338298			−0.169149	+	0.985591i	$0.554102\pi$
$702$		0			0
$703$		−	2.61531i		−	0.0986382i
$704$		0			0
$705$	8.03507	+	40.2499i	0.302618	+	1.51590i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−13.1893			−0.495333			−0.247666	−	0.968845i	$-0.579664\pi$
$709$	−13.1893			−0.495333			−0.247666	+	0.968845i	$0.579664\pi$
$710$		0			0
$711$	3.11474			0.116812
$712$		0			0
$713$			40.7587i			1.52643i
$714$		0			0
$715$	−10.5781	+	2.11170i	−0.395598	+	0.0789730i
$716$		0			0
$717$			70.3705i			2.62803i
$718$		0			0
$719$	20.9300			0.780556			0.390278	−	0.920697i	$-0.372379\pi$
$719$	20.9300			0.780556			0.390278	+	0.920697i	$0.372379\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	62.7249i		−	2.33276i
$724$		0			0
$725$	30.9922	−	12.8875i	1.15102	−	0.478629i
$726$		0			0
$727$		−	34.8251i		−	1.29159i	−0.763510	−	0.645797i	$-0.776525\pi$
$727$		−	34.8251i		−	1.29159i	0.763510	−	0.645797i	$-0.223475\pi$
$728$		0			0
$729$	43.0405			1.59409
$730$		0			0
$731$	41.7189			1.54303
$732$		0			0
$733$			22.6429i			0.836335i	0.908370	+	0.418167i	$0.137327\pi$
$733$			22.6429i			0.836335i	−0.908370	+	0.418167i	$0.862673\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	2.56166i		−	0.0943599i
$738$		0			0
$739$	5.40246			0.198733			0.0993663	−	0.995051i	$-0.468318\pi$
$739$	5.40246			0.198733			0.0993663	+	0.995051i	$0.468318\pi$
$740$		0			0
$741$	26.0349			0.956416
$742$		0			0
$743$		−	13.6717i		−	0.501566i	−0.968043	−	0.250783i	$-0.919312\pi$
$743$		−	13.6717i		−	0.501566i	0.968043	−	0.250783i	$-0.0806881\pi$
$744$		0			0
$745$	4.33468	+	21.7136i	0.158810	+	0.795526i
$746$		0			0
$747$			18.7374i			0.685565i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.94579			−0.107493			−0.0537467	−	0.998555i	$-0.517116\pi$
$751$	−2.94579			−0.107493			−0.0537467	+	0.998555i	$0.517116\pi$
$752$		0			0
$753$		−	18.2254i		−	0.664169i
$754$		0			0
$755$	0.347845	+	1.74245i	0.0126594	+	0.0634144i
$756$		0			0
$757$		−	41.7761i		−	1.51838i	−0.650871	−	0.759188i	$-0.725596\pi$
$757$		−	41.7761i		−	1.51838i	0.650871	−	0.759188i	$-0.274404\pi$
$758$		0			0
$759$	21.7941			0.791077
$760$		0			0
$761$	−31.6836			−1.14853			−0.574264	−	0.818670i	$-0.694712\pi$
$761$	−31.6836			−1.14853			−0.574264	+	0.818670i	$0.694712\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−39.9635	+	7.97789i	−1.44488	+	0.288441i
$766$		0			0
$767$			59.4386i			2.14620i
$768$		0			0
$769$	33.7916			1.21856			0.609278	−	0.792957i	$-0.291459\pi$
$769$	33.7916			1.21856			0.609278	+	0.792957i	$0.291459\pi$
$770$		0			0
$771$	−49.7599			−1.79206
$772$		0			0
$773$		−	29.3136i		−	1.05434i	−0.849761	−	0.527168i	$-0.823254\pi$
$773$		−	29.3136i		−	1.05434i	0.849761	−	0.527168i	$-0.176746\pi$
$774$		0			0
$775$	−21.4310	+	8.91166i	−0.769824	+	0.320116i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.5331			0.484875
$780$		0			0
$781$	10.8856			0.389519
$782$		0			0
$783$		−	23.2313i		−	0.830217i
$784$		0			0
$785$	−50.7687	+	10.1349i	−1.81201	+	0.361731i
$786$		0			0
$787$		−	39.6786i		−	1.41439i	−0.707019	−	0.707195i	$-0.749960\pi$
$787$		−	39.6786i		−	1.41439i	0.707019	−	0.707195i	$-0.250040\pi$
$788$		0			0
$789$	−26.7918			−0.953814
$790$		0			0
$791$		0			0
$792$		0			0
$793$			50.5940i			1.79665i
$794$		0			0
$795$	13.2976	+	66.6112i	0.471616	+	2.36246i
$796$		0			0
$797$		−	6.82244i		−	0.241663i	−0.992673	−	0.120832i	$-0.961444\pi$
$797$		−	6.82244i		−	0.241663i	0.992673	−	0.120832i	$-0.0385561\pi$
$798$		0			0
$799$	28.9472			1.02408
$800$		0			0
$801$	4.12822			0.145863
$802$		0			0
$803$			4.70459i			0.166021i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			40.0831i			1.41099i
$808$		0			0
$809$	9.70620			0.341252			0.170626	−	0.985336i	$-0.445421\pi$
$809$	9.70620			0.341252			0.170626	+	0.985336i	$0.445421\pi$
$810$		0			0
$811$	23.8942			0.839037			0.419519	−	0.907747i	$-0.362199\pi$
$811$	23.8942			0.839037			0.419519	+	0.907747i	$0.362199\pi$
$812$		0			0
$813$			9.01660i			0.316226i
$814$		0			0
$815$	−15.4478	+	3.08384i	−0.541114	+	0.108022i
$816$		0			0
$817$			18.0324i			0.630874i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−48.5915			−1.69586			−0.847928	−	0.530111i	$-0.822150\pi$
$821$	−48.5915			−1.69586			−0.847928	+	0.530111i	$0.822150\pi$
$822$		0			0
$823$			0.0389997i			0.00135944i	1.00000		0.000679722i	$0.000216362\pi$
$823$			0.0389997i			0.00135944i	−1.00000		0.000679722i	$0.999784\pi$
$824$		0			0
$825$	4.76516	+	11.4594i	0.165902	+	0.398964i
$826$		0			0
$827$			26.7489i			0.930149i	0.885271	+	0.465075i	$0.153973\pi$
$827$			26.7489i			0.930149i	−0.885271	+	0.465075i	$0.846027\pi$
$828$		0			0
$829$	10.9119			0.378986			0.189493	−	0.981882i	$-0.439316\pi$
$829$	10.9119			0.378986			0.189493	+	0.981882i	$0.439316\pi$
$830$		0			0
$831$	47.3407			1.64223
$832$		0			0
$833$		0			0
$834$		0			0
$835$	16.2252	−	3.23903i	0.561497	−	0.112091i
$836$		0			0
$837$			16.0643i			0.555265i
$838$		0			0
$839$	3.86425			0.133409			0.0667043	−	0.997773i	$-0.478752\pi$
$839$	3.86425			0.133409			0.0667043	+	0.997773i	$0.478752\pi$
$840$		0			0
$841$	16.0641			0.553934
$842$		0			0
$843$		−	4.17407i		−	0.143763i
$844$		0			0
$845$	−6.35035	−	31.8107i	−0.218459	−	1.09432i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−17.2105			−0.590662
$850$		0			0
$851$	12.4836			0.427932
$852$		0			0
$853$		−	39.3320i		−	1.34670i	−0.739323	−	0.673351i	$-0.764854\pi$
$853$		−	39.3320i		−	1.34670i	0.739323	−	0.673351i	$-0.235146\pi$
$854$		0			0
$855$	−3.44833	−	17.2736i	−0.117930	−	0.590746i
$856$		0			0
$857$		−	31.1984i		−	1.06572i	−0.846205	−	0.532858i	$-0.821118\pi$
$857$		−	31.1984i		−	1.06572i	0.846205	−	0.532858i	$-0.178882\pi$
$858$		0			0
$859$	7.83869			0.267453			0.133726	−	0.991018i	$-0.457306\pi$
$859$	7.83869			0.267453			0.133726	+	0.991018i	$0.457306\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	6.16750i		−	0.209944i	−0.994475	−	0.104972i	$-0.966525\pi$
$863$		−	6.16750i		−	0.209944i	0.994475	−	0.104972i	$-0.0334753\pi$
$864$		0			0
$865$	−19.9934	+	3.99128i	−0.679798	+	0.135708i
$866$		0			0
$867$			2.99960i			0.101872i
$868$		0			0
$869$	−0.668995			−0.0226941
$870$		0			0
$871$	14.6068			0.494931
$872$		0			0
$873$			58.0641i			1.96517i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			25.0332i			0.845311i	0.906290	+	0.422655i	$0.138902\pi$
$877$			25.0332i			0.845311i	−0.906290	+	0.422655i	$0.861098\pi$
$878$		0			0
$879$	−7.16990			−0.241835
$880$		0			0
$881$	30.9200			1.04172			0.520861	−	0.853641i	$-0.325611\pi$
$881$	30.9200			1.04172			0.520861	+	0.853641i	$0.325611\pi$
$882$		0			0
$883$			40.5853i			1.36580i	0.730510	+	0.682902i	$0.239283\pi$
$883$			40.5853i			1.36580i	−0.730510	+	0.682902i	$0.760717\pi$
$884$		0			0
$885$	67.0632	−	13.3878i	2.25430	−	0.450026i
$886$		0			0
$887$		−	30.8084i		−	1.03444i	−0.855851	−	0.517222i	$-0.826966\pi$
$887$		−	30.8084i		−	1.03444i	0.855851	−	0.517222i	$-0.173034\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−3.22689			−0.108105
$892$		0			0
$893$			12.5120i			0.418698i
$894$		0			0
$895$	6.05824	+	30.3474i	0.202505	+	1.01440i
$896$		0			0
$897$			124.272i			4.14932i
$898$		0			0
$899$	−31.1617			−1.03930
$900$		0			0
$901$	47.9059			1.59598
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.0383111	+	0.191911i	0.00127350	+	0.00637934i
$906$		0			0
$907$			11.2998i			0.375205i	0.982245	+	0.187603i	$0.0600717\pi$
$907$			11.2998i			0.375205i	−0.982245	+	0.187603i	$0.939928\pi$
$908$		0			0
$909$	82.2342			2.72754
$910$		0			0
$911$	−31.4425			−1.04174			−0.520869	−	0.853637i	$-0.674392\pi$
$911$	−31.4425			−1.04174			−0.520869	+	0.853637i	$0.674392\pi$
$912$		0			0
$913$		−	4.02448i		−	0.133191i
$914$		0			0
$915$	57.0841	−	11.3957i	1.88714	−	0.376729i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−6.88776			−0.227206			−0.113603	−	0.993526i	$-0.536239\pi$
$919$	−6.88776			−0.227206			−0.113603	+	0.993526i	$0.536239\pi$
$920$		0			0
$921$	29.9062			0.985444
$922$		0			0
$923$			62.0707i			2.04308i
$924$		0			0
$925$	2.72947	+	6.56389i	0.0897443	+	0.215819i
$926$		0			0
$927$			7.18097i			0.235854i
$928$		0			0
$929$	−8.77677			−0.287956			−0.143978	−	0.989581i	$-0.545990\pi$
$929$	−8.77677			−0.287956			−0.143978	+	0.989581i	$0.545990\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			61.8337i			2.02435i
$934$		0			0
$935$	8.58350	−	1.71352i	0.280710	−	0.0560381i
$936$		0			0
$937$			49.7847i			1.62639i	0.581988	+	0.813197i	$0.302275\pi$
$937$			49.7847i			1.62639i	−0.581988	+	0.813197i	$0.697725\pi$
$938$		0			0
$939$	55.8797			1.82356
$940$		0			0
$941$	37.6116			1.22610			0.613052	−	0.790042i	$-0.289941\pi$
$941$	37.6116			1.22610			0.613052	+	0.790042i	$0.289941\pi$
$942$		0			0
$943$			64.5975i			2.10358i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			7.33224i			0.238266i	0.992878	+	0.119133i	$0.0380115\pi$
$947$			7.33224i			0.238266i	−0.992878	+	0.119133i	$0.961989\pi$
$948$		0			0
$949$	−26.8259			−0.870806
$950$		0			0
$951$	44.4252			1.44059
$952$		0			0
$953$		−	50.4457i		−	1.63410i	−0.576570	−	0.817048i	$-0.695609\pi$
$953$		−	50.4457i		−	1.63410i	0.576570	−	0.817048i	$-0.304391\pi$
$954$		0			0
$955$	−5.52857	−	27.6941i	−0.178900	−	0.896161i
$956$		0			0
$957$			16.6625i			0.538622i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−9.45182			−0.304897
$962$		0			0
$963$			77.5500i			2.49901i
$964$		0			0
$965$	−45.1647	+	9.01621i	−1.45390	+	0.290242i
$966$		0			0
$967$		−	20.1685i		−	0.648574i	−0.945959	−	0.324287i	$-0.894876\pi$
$967$		−	20.1685i		−	0.648574i	0.945959	−	0.324287i	$-0.105124\pi$
$968$		0			0
$969$	−21.1258			−0.678658
$970$		0			0
$971$	−61.7398			−1.98133			−0.990663	−	0.136334i	$-0.956468\pi$
$971$	−61.7398			−1.98133			−0.990663	+	0.136334i	$0.956468\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−65.3423	+	27.1713i	−2.09263	+	0.870179i
$976$		0			0
$977$		−	34.0613i		−	1.08972i	−0.838528	−	0.544858i	$-0.816583\pi$
$977$		−	34.0613i		−	1.08972i	0.838528	−	0.544858i	$-0.183417\pi$
$978$		0			0
$979$	−0.886674			−0.0283382
$980$		0			0
$981$	−8.89144			−0.283882
$982$		0			0
$983$		−	52.8543i		−	1.68579i	−0.538078	−	0.842895i	$-0.680849\pi$
$983$		−	52.8543i		−	1.68579i	0.538078	−	0.842895i	$-0.319151\pi$
$984$		0			0
$985$	22.9918	−	4.58983i	0.732578	−	0.146244i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−86.0738			−2.73699
$990$		0			0
$991$	−30.0746			−0.955352			−0.477676	−	0.878536i	$-0.658521\pi$
$991$	−30.0746			−0.955352			−0.477676	+	0.878536i	$0.658521\pi$
$992$		0			0
$993$		−	49.9863i		−	1.58627i
$994$		0			0
$995$	5.16946	+	25.8953i	0.163883	+	0.820935i
$996$		0			0
$997$			19.0097i			0.602044i	0.953617	+	0.301022i	$0.0973278\pi$
$997$			19.0097i			0.602044i	−0.953617	+	0.301022i	$0.902672\pi$
$998$		0			0
$999$	4.92019			0.155668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.g.g.1569.4	yes	20
5.2	odd	4		9800.2.a.dc.1.2		10
5.3	odd	4		9800.2.a.db.1.9		10
5.4	even	2	inner	1960.2.g.g.1569.18	yes	20
7.6	odd	2	inner	1960.2.g.g.1569.17	yes	20
35.13	even	4		9800.2.a.db.1.2		10
35.27	even	4		9800.2.a.dc.1.9		10
35.34	odd	2	inner	1960.2.g.g.1569.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.g.g.1569.3	✓	20	35.34	odd	2	inner
1960.2.g.g.1569.4	yes	20	1.1	even	1	trivial
1960.2.g.g.1569.17	yes	20	7.6	odd	2	inner
1960.2.g.g.1569.18	yes	20	5.4	even	2	inner
9800.2.a.db.1.2		10	35.13	even	4
9800.2.a.db.1.9		10	5.3	odd	4
9800.2.a.dc.1.2		10	5.2	odd	4
9800.2.a.dc.1.9		10	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)

\(f(q)\)	\(=\)	\(q-2.69859i q^{3} +(0.437748 + 2.19280i) q^{5} -4.28239 q^{9} +O(q^{10})\)	\(q-2.69859i q^{3} +(0.437748 + 2.19280i) q^{5} -4.28239 q^{9} +0.919787 q^{11} +5.24470i q^{13} +(5.91747 - 1.18130i) q^{15} -4.25577i q^{17} +1.83949 q^{19} +8.78042i q^{23} +(-4.61675 + 1.91979i) q^{25} +3.46065i q^{27} -6.71298 q^{29} +4.64200 q^{31} -2.48213i q^{33} -1.42175i q^{37} +14.1533 q^{39} +7.35699 q^{41} +9.80292i q^{43} +(-1.87461 - 9.39043i) q^{45} +6.80187i q^{47} -11.4846 q^{51} +11.2567i q^{53} +(0.402635 + 2.01691i) q^{55} -4.96404i q^{57} +11.3331 q^{59} +9.64670 q^{61} +(-11.5006 + 2.29585i) q^{65} -2.78505i q^{67} +23.6948 q^{69} +11.8349 q^{71} +5.11486i q^{73} +(5.18072 + 12.4587i) q^{75} -0.727336 q^{79} -3.50830 q^{81} -4.37545i q^{83} +(9.33205 - 1.86295i) q^{85} +18.1156i q^{87} -0.963998 q^{89} -12.5269i q^{93} +(0.805234 + 4.03364i) q^{95} -13.5588i q^{97} -3.93889 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 28 q^{9}+O(q^{10}) \) 20 * q - 28 * q^9	\( 20 q - 28 q^{9} - 24 q^{11} - 8 q^{15} + 8 q^{25} - 24 q^{29} + 40 q^{39} - 72 q^{51} - 20 q^{65} - 16 q^{71} + 8 q^{79} + 100 q^{81} + 60 q^{85} - 48 q^{95} + 144 q^{99}+O(q^{100}) \) 20 * q - 28 * q^9 - 24 * q^11 - 8 * q^15 + 8 * q^25 - 24 * q^29 + 40 * q^39 - 72 * q^51 - 20 * q^65 - 16 * q^71 + 8 * q^79 + 100 * q^81 + 60 * q^85 - 48 * q^95 + 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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