LMFDB - Embedded newform 2112.2.h.a.1759.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,2,Mod(1759,2112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2112.1759"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1759.3
Root			$-0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	2112.1759
Dual form			2112.2.h.a.1759.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.23607i q^{5} -3.80423 q^{7} +1.00000 q^{9} +(-3.23607 + 0.726543i) q^{11} -2.35114 q^{13} +1.23607i q^{15} -4.70228i q^{17} -6.15537i q^{19} +3.80423 q^{21} +3.23607i q^{23} +3.47214 q^{25} -1.00000 q^{27} +1.45309 q^{29} +4.47214i q^{31} +(3.23607 - 0.726543i) q^{33} +4.70228i q^{35} +2.47214i q^{37} +2.35114 q^{39} +7.60845i q^{41} +1.45309i q^{43} -1.23607i q^{45} +0.763932i q^{47} +7.47214 q^{49} +4.70228i q^{51} -1.23607i q^{53} +(0.898056 + 4.00000i) q^{55} +6.15537i q^{57} -4.00000 q^{59} -9.95959 q^{61} -3.80423 q^{63} +2.90617i q^{65} +8.00000 q^{67} -3.23607i q^{69} +13.7082i q^{71} +7.60845i q^{73} -3.47214 q^{75} +(12.3107 - 2.76393i) q^{77} +13.2088 q^{79} +1.00000 q^{81} -9.06154i q^{83} -5.81234 q^{85} -1.45309 q^{87} +9.41641 q^{89} +8.94427 q^{91} -4.47214i q^{93} -7.60845 q^{95} +3.52786 q^{97} +(-3.23607 + 0.726543i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{9} - 8 q^{11} - 8 q^{25} - 8 q^{27} + 8 q^{33} + 24 q^{49} - 32 q^{59} + 64 q^{67} + 8 q^{75} + 8 q^{81} - 32 q^{89} + 64 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^11 - 8 * q^25 - 8 * q^27 + 8 * q^33 + 24 * q^49 - 32 * q^59 + 64 * q^67 + 8 * q^75 + 8 * q^81 - 32 * q^89 + 64 * q^97 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	1.23607i		−	0.552786i	−0.961045	−	0.276393i	$-0.910861\pi$
$5$		−	1.23607i		−	0.552786i	0.961045	−	0.276393i	$-0.0891392\pi$
$6$		0			0
$7$	−3.80423			−1.43786			−0.718931	−	0.695081i	$-0.755368\pi$
$7$	−3.80423			−1.43786			−0.718931	+	0.695081i	$0.755368\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	−3.23607	+	0.726543i	−0.975711	+	0.219061i
$11$	−3.23607	+	0.726543i	−0.975711	+	0.219061i
$12$		0			0
$13$	−2.35114			−0.652089			−0.326045	−	0.945354i	$-0.605716\pi$
$13$	−2.35114			−0.652089			−0.326045	+	0.945354i	$0.605716\pi$
$14$		0			0
$15$			1.23607i			0.319151i
$16$		0			0
$17$		−	4.70228i		−	1.14047i	−0.821481	−	0.570235i	$-0.806852\pi$
$17$		−	4.70228i		−	1.14047i	0.821481	−	0.570235i	$-0.193148\pi$
$18$		0			0
$19$		−	6.15537i		−	1.41214i	−0.708143	−	0.706069i	$-0.750467\pi$
$19$		−	6.15537i		−	1.41214i	0.708143	−	0.706069i	$-0.249533\pi$
$20$		0			0
$21$	3.80423			0.830150
$22$		0			0
$23$			3.23607i			0.674767i	0.941367	+	0.337383i	$0.109542\pi$
$23$			3.23607i			0.674767i	−0.941367	+	0.337383i	$0.890458\pi$
$24$		0			0
$25$	3.47214			0.694427
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	1.45309			0.269831			0.134916	−	0.990857i	$-0.456924\pi$
$29$	1.45309			0.269831			0.134916	+	0.990857i	$0.456924\pi$
$30$		0			0
$31$			4.47214i			0.803219i	0.915811	+	0.401610i	$0.131549\pi$
$31$			4.47214i			0.803219i	−0.915811	+	0.401610i	$0.868451\pi$
$32$		0			0
$33$	3.23607	−	0.726543i	0.563327	−	0.126475i
$34$		0			0
$35$			4.70228i			0.794831i
$36$		0			0
$37$			2.47214i			0.406417i	0.979136	+	0.203208i	$0.0651369\pi$
$37$			2.47214i			0.406417i	−0.979136	+	0.203208i	$0.934863\pi$
$38$		0			0
$39$	2.35114			0.376484
$40$		0			0
$41$			7.60845i			1.18824i	0.804376	+	0.594120i	$0.202500\pi$
$41$			7.60845i			1.18824i	−0.804376	+	0.594120i	$0.797500\pi$
$42$		0			0
$43$			1.45309i			0.221593i	0.993843	+	0.110797i	$0.0353402\pi$
$43$			1.45309i			0.221593i	−0.993843	+	0.110797i	$0.964660\pi$
$44$		0			0
$45$		−	1.23607i		−	0.184262i
$46$		0			0
$47$			0.763932i			0.111431i	0.998447	+	0.0557155i	$0.0177440\pi$
$47$			0.763932i			0.111431i	−0.998447	+	0.0557155i	$0.982256\pi$
$48$		0			0
$49$	7.47214			1.06745
$50$		0			0
$51$			4.70228i			0.658451i
$52$		0			0
$53$		−	1.23607i		−	0.169787i	−0.996390	−	0.0848935i	$-0.972945\pi$
$53$		−	1.23607i		−	0.169787i	0.996390	−	0.0848935i	$-0.0270550\pi$
$54$		0			0
$55$	0.898056	+	4.00000i	0.121094	+	0.539360i
$56$		0			0
$57$			6.15537i			0.815298i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	−9.95959			−1.27520			−0.637598	−	0.770370i	$-0.720072\pi$
$61$	−9.95959			−1.27520			−0.637598	+	0.770370i	$0.720072\pi$
$62$		0			0
$63$	−3.80423			−0.479287
$64$		0			0
$65$			2.90617i			0.360466i
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$		0			0
$69$		−	3.23607i		−	0.389577i
$70$		0			0
$71$			13.7082i			1.62686i	0.581660	+	0.813432i	$0.302404\pi$
$71$			13.7082i			1.62686i	−0.581660	+	0.813432i	$0.697596\pi$
$72$		0			0
$73$			7.60845i			0.890502i	0.895406	+	0.445251i	$0.146886\pi$
$73$			7.60845i			0.890502i	−0.895406	+	0.445251i	$0.853114\pi$
$74$		0			0
$75$	−3.47214			−0.400928
$76$		0			0
$77$	12.3107	−	2.76393i	1.40294	−	0.314979i
$78$		0			0
$79$	13.2088			1.48610			0.743052	−	0.669233i	$-0.233377\pi$
$79$	13.2088			1.48610			0.743052	+	0.669233i	$0.233377\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	9.06154i		−	0.994633i	−0.867569	−	0.497316i	$-0.834319\pi$
$83$		−	9.06154i		−	0.994633i	0.867569	−	0.497316i	$-0.165681\pi$
$84$		0			0
$85$	−5.81234			−0.630437
$86$		0			0
$87$	−1.45309			−0.155787
$88$		0			0
$89$	9.41641			0.998137			0.499069	−	0.866562i	$-0.333676\pi$
$89$	9.41641			0.998137			0.499069	+	0.866562i	$0.333676\pi$
$90$		0			0
$91$	8.94427			0.937614
$92$		0			0
$93$		−	4.47214i		−	0.463739i
$94$		0			0
$95$	−7.60845			−0.780611
$96$		0			0
$97$	3.52786			0.358200			0.179100	−	0.983831i	$-0.442681\pi$
$97$	3.52786			0.358200			0.179100	+	0.983831i	$0.442681\pi$
$98$		0			0
$99$	−3.23607	+	0.726543i	−0.325237	+	0.0730203i
$100$		0			0
$101$	−6.15537			−0.612482			−0.306241	−	0.951954i	$-0.599071\pi$
$101$	−6.15537			−0.612482			−0.306241	+	0.951954i	$0.599071\pi$
$102$		0			0
$103$			9.41641i			0.927826i	0.885881	+	0.463913i	$0.153555\pi$
$103$			9.41641i			0.927826i	−0.885881	+	0.463913i	$0.846445\pi$
$104$		0			0
$105$		−	4.70228i		−	0.458896i
$106$		0			0
$107$		−	13.7638i		−	1.33060i	−0.746577	−	0.665299i	$-0.768304\pi$
$107$		−	13.7638i		−	1.33060i	0.746577	−	0.665299i	$-0.231696\pi$
$108$		0			0
$109$	14.6619			1.40435			0.702176	−	0.712003i	$-0.252212\pi$
$109$	14.6619			1.40435			0.702176	+	0.712003i	$0.252212\pi$
$110$		0			0
$111$		−	2.47214i		−	0.234645i
$112$		0			0
$113$	9.41641			0.885821			0.442911	−	0.896566i	$-0.353946\pi$
$113$	9.41641			0.885821			0.442911	+	0.896566i	$0.353946\pi$
$114$		0			0
$115$	4.00000			0.373002
$116$		0			0
$117$	−2.35114			−0.217363
$118$		0			0
$119$			17.8885i			1.63984i
$120$		0			0
$121$	9.94427	−	4.70228i	0.904025	−	0.427480i
$122$		0			0
$123$		−	7.60845i		−	0.686031i
$124$		0			0
$125$		−	10.4721i		−	0.936656i
$126$		0			0
$127$	−13.2088			−1.17209			−0.586045	−	0.810278i	$-0.699316\pi$
$127$	−13.2088			−1.17209			−0.586045	+	0.810278i	$0.699316\pi$
$128$		0			0
$129$		−	1.45309i		−	0.127937i
$130$		0			0
$131$			21.3723i			1.86730i	0.358181	+	0.933652i	$0.383397\pi$
$131$			21.3723i			1.86730i	−0.358181	+	0.933652i	$0.616603\pi$
$132$		0			0
$133$			23.4164i			2.03046i
$134$		0			0
$135$			1.23607i			0.106384i
$136$		0			0
$137$	7.52786			0.643149			0.321574	−	0.946884i	$-0.395788\pi$
$137$	7.52786			0.643149			0.321574	+	0.946884i	$0.395788\pi$
$138$		0			0
$139$			16.6700i			1.41393i	0.707249	+	0.706965i	$0.249936\pi$
$139$			16.6700i			1.41393i	−0.707249	+	0.706965i	$0.750064\pi$
$140$		0			0
$141$		−	0.763932i		−	0.0643347i
$142$		0			0
$143$	7.60845	−	1.70820i	0.636251	−	0.142847i
$144$		0			0
$145$		−	1.79611i		−	0.149159i
$146$		0			0
$147$	−7.47214			−0.616291
$148$		0			0
$149$	3.24920			0.266185			0.133092	−	0.991104i	$-0.457509\pi$
$149$	3.24920			0.266185			0.133092	+	0.991104i	$0.457509\pi$
$150$		0			0
$151$	−2.00811			−0.163418			−0.0817090	−	0.996656i	$-0.526038\pi$
$151$	−2.00811			−0.163418			−0.0817090	+	0.996656i	$0.526038\pi$
$152$		0			0
$153$		−	4.70228i		−	0.380157i
$154$		0			0
$155$	5.52786			0.444009
$156$		0			0
$157$			8.00000i			0.638470i	0.947676	+	0.319235i	$0.103426\pi$
$157$			8.00000i			0.638470i	−0.947676	+	0.319235i	$0.896574\pi$
$158$		0			0
$159$			1.23607i			0.0980266i
$160$		0			0
$161$		−	12.3107i		−	0.970222i
$162$		0			0
$163$	21.8885			1.71444			0.857222	−	0.514948i	$-0.172189\pi$
$163$	21.8885			1.71444			0.857222	+	0.514948i	$0.172189\pi$
$164$		0			0
$165$	−0.898056	−	4.00000i	−0.0699136	−	0.311400i
$166$		0			0
$167$	−22.8254			−1.76628			−0.883140	−	0.469110i	$-0.844575\pi$
$167$	−22.8254			−1.76628			−0.883140	+	0.469110i	$0.844575\pi$
$168$		0			0
$169$	−7.47214			−0.574780
$170$		0			0
$171$		−	6.15537i		−	0.470713i
$172$		0			0
$173$	−23.1684			−1.76146			−0.880730	−	0.473619i	$-0.842947\pi$
$173$	−23.1684			−1.76146			−0.880730	+	0.473619i	$0.842947\pi$
$174$		0			0
$175$	−13.2088			−0.998491
$176$		0			0
$177$	4.00000			0.300658
$178$		0			0
$179$	−8.94427			−0.668526			−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427			−0.668526			−0.334263	+	0.942480i	$0.608487\pi$
$180$		0			0
$181$			20.3607i			1.51340i	0.653764	+	0.756699i	$0.273189\pi$
$181$			20.3607i			1.51340i	−0.653764	+	0.756699i	$0.726811\pi$
$182$		0			0
$183$	9.95959			0.736234
$184$		0			0
$185$	3.05573			0.224662
$186$		0			0
$187$	3.41641	+	15.2169i	0.249832	+	1.11277i
$188$		0			0
$189$	3.80423			0.276717
$190$		0			0
$191$		−	12.7639i		−	0.923566i	−0.886993	−	0.461783i	$-0.847210\pi$
$191$		−	12.7639i		−	0.923566i	0.886993	−	0.461783i	$-0.152790\pi$
$192$		0			0
$193$			17.0130i			1.22462i	0.790616	+	0.612312i	$0.209760\pi$
$193$			17.0130i			1.22462i	−0.790616	+	0.612312i	$0.790240\pi$
$194$		0			0
$195$		−	2.90617i		−	0.208115i
$196$		0			0
$197$	−0.343027			−0.0244397			−0.0122198	−	0.999925i	$-0.503890\pi$
$197$	−0.343027			−0.0244397			−0.0122198	+	0.999925i	$0.503890\pi$
$198$		0			0
$199$			10.0000i			0.708881i	0.935079	+	0.354441i	$0.115329\pi$
$199$			10.0000i			0.708881i	−0.935079	+	0.354441i	$0.884671\pi$
$200$		0			0
$201$	−8.00000			−0.564276
$202$		0			0
$203$	−5.52786			−0.387980
$204$		0			0
$205$	9.40456			0.656843
$206$		0			0
$207$			3.23607i			0.224922i
$208$		0			0
$209$	4.47214	+	19.9192i	0.309344	+	1.37784i
$210$		0			0
$211$		−	0.343027i		−	0.0236149i	−0.999930	−	0.0118075i	$-0.996241\pi$
$211$		−	0.343027i		−	0.0236149i	0.999930	−	0.0118075i	$-0.00375852\pi$
$212$		0			0
$213$		−	13.7082i		−	0.939271i
$214$		0			0
$215$	1.79611			0.122494
$216$		0			0
$217$		−	17.0130i		−	1.15492i
$218$		0			0
$219$		−	7.60845i		−	0.514132i
$220$		0			0
$221$			11.0557i			0.743689i
$222$		0			0
$223$		−	23.8885i		−	1.59970i	−0.600203	−	0.799848i	$-0.704914\pi$
$223$		−	23.8885i		−	1.59970i	0.600203	−	0.799848i	$-0.295086\pi$
$224$		0			0
$225$	3.47214			0.231476
$226$		0			0
$227$		−	23.1684i		−	1.53774i	−0.639405	−	0.768870i	$-0.720819\pi$
$227$		−	23.1684i		−	1.53774i	0.639405	−	0.768870i	$-0.279181\pi$
$228$		0			0
$229$			24.0000i			1.58596i	0.609245	+	0.792982i	$0.291473\pi$
$229$			24.0000i			1.58596i	−0.609245	+	0.792982i	$0.708527\pi$
$230$		0			0
$231$	−12.3107	+	2.76393i	−0.809987	+	0.181853i
$232$		0			0
$233$		−	13.4208i		−	0.879225i	−0.898187	−	0.439613i	$-0.855116\pi$
$233$		−	13.4208i		−	0.879225i	0.898187	−	0.439613i	$-0.144884\pi$
$234$		0			0
$235$	0.944272			0.0615975
$236$		0			0
$237$	−13.2088			−0.858003
$238$		0			0
$239$	22.8254			1.47645			0.738225	−	0.674555i	$-0.235664\pi$
$239$	22.8254			1.47645			0.738225	+	0.674555i	$0.235664\pi$
$240$		0			0
$241$			1.79611i			0.115698i	0.998325	+	0.0578489i	$0.0184242\pi$
$241$			1.79611i			0.115698i	−0.998325	+	0.0578489i	$0.981576\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$		−	9.23607i		−	0.590071i
$246$		0			0
$247$			14.4721i			0.920840i
$248$		0			0
$249$			9.06154i			0.574252i
$250$		0			0
$251$	−3.41641			−0.215642			−0.107821	−	0.994170i	$-0.534387\pi$
$251$	−3.41641			−0.215642			−0.107821	+	0.994170i	$0.534387\pi$
$252$		0			0
$253$	−2.35114	−	10.4721i	−0.147815	−	0.658378i
$254$		0			0
$255$	5.81234			0.363983
$256$		0			0
$257$	−21.4164			−1.33592			−0.667959	−	0.744198i	$-0.732832\pi$
$257$	−21.4164			−1.33592			−0.667959	+	0.744198i	$0.732832\pi$
$258$		0			0
$259$		−	9.40456i		−	0.584371i
$260$		0			0
$261$	1.45309			0.0899437
$262$		0			0
$263$	17.0130			1.04907			0.524534	−	0.851390i	$-0.324240\pi$
$263$	17.0130			1.04907			0.524534	+	0.851390i	$0.324240\pi$
$264$		0			0
$265$	−1.52786			−0.0938559
$266$		0			0
$267$	−9.41641			−0.576275
$268$		0			0
$269$		−	22.1803i		−	1.35236i	−0.736737	−	0.676180i	$-0.763634\pi$
$269$		−	22.1803i		−	1.35236i	0.736737	−	0.676180i	$-0.236366\pi$
$270$		0			0
$271$	26.6296			1.61763			0.808816	−	0.588062i	$-0.200109\pi$
$271$	26.6296			1.61763			0.808816	+	0.588062i	$0.200109\pi$
$272$		0			0
$273$	−8.94427			−0.541332
$274$		0			0
$275$	−11.2361	+	2.52265i	−0.677560	+	0.152122i
$276$		0			0
$277$	8.84953			0.531717			0.265859	−	0.964012i	$-0.414345\pi$
$277$	8.84953			0.531717			0.265859	+	0.964012i	$0.414345\pi$
$278$		0			0
$279$			4.47214i			0.267740i
$280$		0			0
$281$			4.70228i			0.280515i	0.990115	+	0.140257i	$0.0447930\pi$
$281$			4.70228i			0.280515i	−0.990115	+	0.140257i	$0.955207\pi$
$282$		0			0
$283$			3.24920i			0.193145i	0.995326	+	0.0965724i	$0.0307879\pi$
$283$			3.24920i			0.193145i	−0.995326	+	0.0965724i	$0.969212\pi$
$284$		0			0
$285$	7.60845			0.450686
$286$		0			0
$287$		−	28.9443i		−	1.70853i
$288$		0			0
$289$	−5.11146			−0.300674
$290$		0			0
$291$	−3.52786			−0.206807
$292$		0			0
$293$	16.6700			0.973871			0.486936	−	0.873438i	$-0.338115\pi$
$293$	16.6700			0.973871			0.486936	+	0.873438i	$0.338115\pi$
$294$		0			0
$295$			4.94427i			0.287867i
$296$		0			0
$297$	3.23607	−	0.726543i	0.187776	−	0.0421583i
$298$		0			0
$299$		−	7.60845i		−	0.440008i
$300$		0			0
$301$		−	5.52786i		−	0.318621i
$302$		0			0
$303$	6.15537			0.353617
$304$		0			0
$305$			12.3107i			0.704911i
$306$		0			0
$307$			16.6700i			0.951407i	0.879606	+	0.475703i	$0.157806\pi$
$307$			16.6700i			0.951407i	−0.879606	+	0.475703i	$0.842194\pi$
$308$		0			0
$309$		−	9.41641i		−	0.535681i
$310$		0			0
$311$		−	4.18034i		−	0.237045i	−0.992951	−	0.118523i	$-0.962184\pi$
$311$		−	4.18034i		−	0.237045i	0.992951	−	0.118523i	$-0.0378158\pi$
$312$		0			0
$313$	−2.94427			−0.166420			−0.0832100	−	0.996532i	$-0.526517\pi$
$313$	−2.94427			−0.166420			−0.0832100	+	0.996532i	$0.526517\pi$
$314$		0			0
$315$			4.70228i			0.264944i
$316$		0			0
$317$			14.1803i			0.796447i	0.917288	+	0.398224i	$0.130373\pi$
$317$			14.1803i			0.796447i	−0.917288	+	0.398224i	$0.869627\pi$
$318$		0			0
$319$	−4.70228	+	1.05573i	−0.263277	+	0.0591094i
$320$		0			0
$321$			13.7638i			0.768221i
$322$		0			0
$323$	−28.9443			−1.61050
$324$		0			0
$325$	−8.16348			−0.452828
$326$		0			0
$327$	−14.6619			−0.810804
$328$		0			0
$329$		−	2.90617i		−	0.160222i
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$		0			0
$333$			2.47214i			0.135472i
$334$		0			0
$335$		−	9.88854i		−	0.540269i
$336$		0			0
$337$		−	7.60845i		−	0.414459i	−0.978292	−	0.207229i	$-0.933555\pi$
$337$		−	7.60845i		−	0.414459i	0.978292	−	0.207229i	$-0.0664447\pi$
$338$		0			0
$339$	−9.41641			−0.511429
$340$		0			0
$341$	−3.24920	−	14.4721i	−0.175954	−	0.783710i
$342$		0			0
$343$	−1.79611			−0.0969809
$344$		0			0
$345$	−4.00000			−0.215353
$346$		0			0
$347$			15.5599i			0.835301i	0.908608	+	0.417650i	$0.137146\pi$
$347$			15.5599i			0.835301i	−0.908608	+	0.417650i	$0.862854\pi$
$348$		0			0
$349$	−11.7557			−0.629268			−0.314634	−	0.949213i	$-0.601882\pi$
$349$	−11.7557			−0.629268			−0.314634	+	0.949213i	$0.601882\pi$
$350$		0			0
$351$	2.35114			0.125495
$352$		0			0
$353$	−31.8885			−1.69726			−0.848628	−	0.528990i	$-0.822571\pi$
$353$	−31.8885			−1.69726			−0.848628	+	0.528990i	$0.822571\pi$
$354$		0			0
$355$	16.9443			0.899309
$356$		0			0
$357$		−	17.8885i		−	0.946762i
$358$		0			0
$359$	−5.81234			−0.306764			−0.153382	−	0.988167i	$-0.549016\pi$
$359$	−5.81234			−0.306764			−0.153382	+	0.988167i	$0.549016\pi$
$360$		0			0
$361$	−18.8885			−0.994134
$362$		0			0
$363$	−9.94427	+	4.70228i	−0.521939	+	0.246806i
$364$		0			0
$365$	9.40456			0.492257
$366$		0			0
$367$		−	26.9443i		−	1.40648i	−0.710953	−	0.703240i	$-0.751736\pi$
$367$		−	26.9443i		−	1.40648i	0.710953	−	0.703240i	$-0.248264\pi$
$368$		0			0
$369$			7.60845i			0.396080i
$370$		0			0
$371$			4.70228i			0.244130i
$372$		0			0
$373$	−17.5680			−0.909639			−0.454819	−	0.890584i	$-0.650296\pi$
$373$	−17.5680			−0.909639			−0.454819	+	0.890584i	$0.650296\pi$
$374$		0			0
$375$			10.4721i			0.540779i
$376$		0			0
$377$	−3.41641			−0.175954
$378$		0			0
$379$	−13.8885			−0.713407			−0.356703	−	0.934218i	$-0.616099\pi$
$379$	−13.8885			−0.713407			−0.356703	+	0.934218i	$0.616099\pi$
$380$		0			0
$381$	13.2088			0.676707
$382$		0			0
$383$		−	12.1803i		−	0.622386i	−0.950347	−	0.311193i	$-0.899271\pi$
$383$		−	12.1803i		−	0.622386i	0.950347	−	0.311193i	$-0.100729\pi$
$384$		0			0
$385$	−3.41641	−	15.2169i	−0.174116	−	0.775525i
$386$		0			0
$387$			1.45309i			0.0738645i
$388$		0			0
$389$		−	35.7082i		−	1.81048i	−0.424903	−	0.905239i	$-0.639692\pi$
$389$		−	35.7082i		−	1.81048i	0.424903	−	0.905239i	$-0.360308\pi$
$390$		0			0
$391$	15.2169			0.769552
$392$		0			0
$393$		−	21.3723i		−	1.07809i
$394$		0			0
$395$		−	16.3270i		−	0.821499i
$396$		0			0
$397$			31.4164i			1.57674i	0.615199	+	0.788372i	$0.289076\pi$
$397$			31.4164i			1.57674i	−0.615199	+	0.788372i	$0.710924\pi$
$398$		0			0
$399$		−	23.4164i		−	1.17229i
$400$		0			0
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$		−	10.5146i		−	0.523771i
$404$		0			0
$405$		−	1.23607i		−	0.0614207i
$406$		0			0
$407$	−1.79611	−	8.00000i	−0.0890300	−	0.396545i
$408$		0			0
$409$			15.2169i			0.752427i	0.926533	+	0.376214i	$0.122774\pi$
$409$			15.2169i			0.752427i	−0.926533	+	0.376214i	$0.877226\pi$
$410$		0			0
$411$	−7.52786			−0.371322
$412$		0			0
$413$	15.2169			0.748775
$414$		0			0
$415$	−11.2007			−0.549820
$416$		0			0
$417$		−	16.6700i		−	0.816333i
$418$		0			0
$419$	32.3607			1.58092			0.790461	−	0.612512i	$-0.209841\pi$
$419$	32.3607			1.58092			0.790461	+	0.612512i	$0.209841\pi$
$420$		0			0
$421$			15.4164i			0.751350i	0.926752	+	0.375675i	$0.122589\pi$
$421$			15.4164i			0.751350i	−0.926752	+	0.375675i	$0.877411\pi$
$422$		0			0
$423$			0.763932i			0.0371436i
$424$		0			0
$425$		−	16.3270i		−	0.791974i
$426$		0			0
$427$	37.8885			1.83356
$428$		0			0
$429$	−7.60845	+	1.70820i	−0.367340	+	0.0824729i
$430$		0			0
$431$	−26.4176			−1.27249			−0.636245	−	0.771487i	$-0.719513\pi$
$431$	−26.4176			−1.27249			−0.636245	+	0.771487i	$0.719513\pi$
$432$		0			0
$433$	−23.8885			−1.14801			−0.574005	−	0.818852i	$-0.694611\pi$
$433$	−23.8885			−1.14801			−0.574005	+	0.818852i	$0.694611\pi$
$434$		0			0
$435$			1.79611i			0.0861170i
$436$		0			0
$437$	19.9192			0.952864
$438$		0			0
$439$	−2.00811			−0.0958421			−0.0479210	−	0.998851i	$-0.515260\pi$
$439$	−2.00811			−0.0958421			−0.0479210	+	0.998851i	$0.515260\pi$
$440$		0			0
$441$	7.47214			0.355816
$442$		0			0
$443$	1.52786			0.0725910			0.0362955	−	0.999341i	$-0.488444\pi$
$443$	1.52786			0.0725910			0.0362955	+	0.999341i	$0.488444\pi$
$444$		0			0
$445$		−	11.6393i		−	0.551757i
$446$		0			0
$447$	−3.24920			−0.153682
$448$		0			0
$449$	−0.472136			−0.0222815			−0.0111407	−	0.999938i	$-0.503546\pi$
$449$	−0.472136			−0.0222815			−0.0111407	+	0.999938i	$0.503546\pi$
$450$		0			0
$451$	−5.52786	−	24.6215i	−0.260297	−	1.15938i
$452$		0			0
$453$	2.00811			0.0943494
$454$		0			0
$455$		−	11.0557i		−	0.518301i
$456$		0			0
$457$		−	5.81234i		−	0.271890i	−0.990716	−	0.135945i	$-0.956593\pi$
$457$		−	5.81234i		−	0.271890i	0.990716	−	0.135945i	$-0.0434070\pi$
$458$		0			0
$459$			4.70228i			0.219484i
$460$		0			0
$461$	−4.35926			−0.203031			−0.101515	−	0.994834i	$-0.532369\pi$
$461$	−4.35926			−0.203031			−0.101515	+	0.994834i	$0.532369\pi$
$462$		0			0
$463$		−	26.3607i		−	1.22508i	−0.790438	−	0.612542i	$-0.790147\pi$
$463$		−	26.3607i		−	1.22508i	0.790438	−	0.612542i	$-0.209853\pi$
$464$		0			0
$465$	−5.52786			−0.256349
$466$		0			0
$467$	−22.4721			−1.03989			−0.519943	−	0.854201i	$-0.674047\pi$
$467$	−22.4721			−1.03989			−0.519943	+	0.854201i	$0.674047\pi$
$468$		0			0
$469$	−30.4338			−1.40530
$470$		0			0
$471$		−	8.00000i		−	0.368621i
$472$		0			0
$473$	−1.05573	−	4.70228i	−0.0485424	−	0.216211i
$474$		0			0
$475$		−	21.3723i		−	0.980627i
$476$		0			0
$477$		−	1.23607i		−	0.0565957i
$478$		0			0
$479$	−1.79611			−0.0820664			−0.0410332	−	0.999158i	$-0.513065\pi$
$479$	−1.79611			−0.0820664			−0.0410332	+	0.999158i	$0.513065\pi$
$480$		0			0
$481$		−	5.81234i		−	0.265020i
$482$		0			0
$483$			12.3107i			0.560158i
$484$		0			0
$485$		−	4.36068i		−	0.198008i
$486$		0			0
$487$		−	9.05573i		−	0.410354i	−0.978725	−	0.205177i	$-0.934223\pi$
$487$		−	9.05573i		−	0.410354i	0.978725	−	0.205177i	$-0.0657770\pi$
$488$		0			0
$489$	−21.8885			−0.989834
$490$		0			0
$491$			16.6700i			0.752306i	0.926558	+	0.376153i	$0.122753\pi$
$491$			16.6700i			0.752306i	−0.926558	+	0.376153i	$0.877247\pi$
$492$		0			0
$493$		−	6.83282i		−	0.307735i
$494$		0			0
$495$	0.898056	+	4.00000i	0.0403646	+	0.179787i
$496$		0			0
$497$		−	52.1491i		−	2.33921i
$498$		0			0
$499$	−15.0557			−0.673987			−0.336993	−	0.941507i	$-0.609410\pi$
$499$	−15.0557			−0.673987			−0.336993	+	0.941507i	$0.609410\pi$
$500$		0			0
$501$	22.8254			1.01976
$502$		0			0
$503$	13.4208			0.598404			0.299202	−	0.954190i	$-0.403280\pi$
$503$	13.4208			0.598404			0.299202	+	0.954190i	$0.403280\pi$
$504$		0			0
$505$			7.60845i			0.338572i
$506$		0			0
$507$	7.47214			0.331849
$508$		0			0
$509$			27.7082i			1.22814i	0.789250	+	0.614072i	$0.210470\pi$
$509$			27.7082i			1.22814i	−0.789250	+	0.614072i	$0.789530\pi$
$510$		0			0
$511$		−	28.9443i		−	1.28042i
$512$		0			0
$513$			6.15537i			0.271766i
$514$		0			0
$515$	11.6393			0.512890
$516$		0			0
$517$	−0.555029	−	2.47214i	−0.0244102	−	0.108724i
$518$		0			0
$519$	23.1684			1.01698
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$		0			0
$523$			7.26543i			0.317695i	0.987303	+	0.158847i	$0.0507778\pi$
$523$			7.26543i			0.317695i	−0.987303	+	0.158847i	$0.949222\pi$
$524$		0			0
$525$	13.2088			0.576479
$526$		0			0
$527$	21.0292			0.916048
$528$		0			0
$529$	12.5279			0.544690
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$		−	17.8885i		−	0.774839i
$534$		0			0
$535$	−17.0130			−0.735537
$536$		0			0
$537$	8.94427			0.385974
$538$		0			0
$539$	−24.1803	+	5.42882i	−1.04152	+	0.233836i
$540$		0			0
$541$	−19.3642			−0.832530			−0.416265	−	0.909243i	$-0.636661\pi$
$541$	−19.3642			−0.832530			−0.416265	+	0.909243i	$0.636661\pi$
$542$		0			0
$543$		−	20.3607i		−	0.873760i
$544$		0			0
$545$		−	18.1231i		−	0.776307i
$546$		0			0
$547$			30.0908i			1.28659i	0.765619	+	0.643294i	$0.222433\pi$
$547$			30.0908i			1.28659i	−0.765619	+	0.643294i	$0.777567\pi$
$548$		0			0
$549$	−9.95959			−0.425065
$550$		0			0
$551$		−	8.94427i		−	0.381039i
$552$		0			0
$553$	−50.2492			−2.13681
$554$		0			0
$555$	−3.05573			−0.129708
$556$		0			0
$557$	−38.3853			−1.62644			−0.813218	−	0.581959i	$-0.802287\pi$
$557$	−38.3853			−1.62644			−0.813218	+	0.581959i	$0.802287\pi$
$558$		0			0
$559$		−	3.41641i		−	0.144499i
$560$		0			0
$561$	−3.41641	−	15.2169i	−0.144241	−	0.642458i
$562$		0			0
$563$			26.0746i			1.09891i	0.835523	+	0.549456i	$0.185165\pi$
$563$			26.0746i			1.09891i	−0.835523	+	0.549456i	$0.814835\pi$
$564$		0			0
$565$		−	11.6393i		−	0.489670i
$566$		0			0
$567$	−3.80423			−0.159762
$568$		0			0
$569$			40.9484i			1.71665i	0.513108	+	0.858324i	$0.328494\pi$
$569$			40.9484i			1.71665i	−0.513108	+	0.858324i	$0.671506\pi$
$570$		0			0
$571$		−	38.3853i		−	1.60637i	−0.595727	−	0.803187i	$-0.703136\pi$
$571$		−	38.3853i		−	1.60637i	0.595727	−	0.803187i	$-0.296864\pi$
$572$		0			0
$573$			12.7639i			0.533221i
$574$		0			0
$575$			11.2361i			0.468576i
$576$		0			0
$577$	−12.4721			−0.519222			−0.259611	−	0.965713i	$-0.583594\pi$
$577$	−12.4721			−0.519222			−0.259611	+	0.965713i	$0.583594\pi$
$578$		0			0
$579$		−	17.0130i		−	0.707037i
$580$		0			0
$581$			34.4721i			1.43015i
$582$		0			0
$583$	0.898056	+	4.00000i	0.0371937	+	0.165663i
$584$		0			0
$585$			2.90617i			0.120155i
$586$		0			0
$587$	−23.0557			−0.951612			−0.475806	−	0.879550i	$-0.657844\pi$
$587$	−23.0557			−0.951612			−0.475806	+	0.879550i	$0.657844\pi$
$588$		0			0
$589$	27.5276			1.13426
$590$		0			0
$591$	0.343027			0.0141102
$592$		0			0
$593$			44.5407i			1.82907i	0.404512	+	0.914533i	$0.367441\pi$
$593$			44.5407i			1.82907i	−0.404512	+	0.914533i	$0.632559\pi$
$594$		0			0
$595$	22.1115			0.906481
$596$		0			0
$597$		−	10.0000i		−	0.409273i
$598$		0			0
$599$			40.1803i			1.64172i	0.571126	+	0.820862i	$0.306507\pi$
$599$			40.1803i			1.64172i	−0.571126	+	0.820862i	$0.693493\pi$
$600$		0			0
$601$			17.0130i			0.693975i	0.937870	+	0.346988i	$0.112795\pi$
$601$			17.0130i			0.693975i	−0.937870	+	0.346988i	$0.887205\pi$
$602$		0			0
$603$	8.00000			0.325785
$604$		0			0
$605$	−5.81234	−	12.2918i	−0.236305	−	0.499733i
$606$		0			0
$607$	26.6296			1.08086			0.540431	−	0.841389i	$-0.318261\pi$
$607$	26.6296			1.08086			0.540431	+	0.841389i	$0.318261\pi$
$608$		0			0
$609$	5.52786			0.224000
$610$		0			0
$611$		−	1.79611i		−	0.0726629i
$612$		0			0
$613$	29.8788			1.20679			0.603396	−	0.797441i	$-0.293814\pi$
$613$	29.8788			1.20679			0.603396	+	0.797441i	$0.293814\pi$
$614$		0			0
$615$	−9.40456			−0.379229
$616$		0			0
$617$	−32.4721			−1.30728			−0.653639	−	0.756806i	$-0.726759\pi$
$617$	−32.4721			−1.30728			−0.653639	+	0.756806i	$0.726759\pi$
$618$		0			0
$619$	17.8885			0.719001			0.359501	−	0.933145i	$-0.382947\pi$
$619$	17.8885			0.719001			0.359501	+	0.933145i	$0.382947\pi$
$620$		0			0
$621$		−	3.23607i		−	0.129859i
$622$		0			0
$623$	−35.8221			−1.43518
$624$		0			0
$625$	4.41641			0.176656
$626$		0			0
$627$	−4.47214	−	19.9192i	−0.178600	−	0.795496i
$628$		0			0
$629$	11.6247			0.463506
$630$		0			0
$631$		−	10.3607i		−	0.412452i	−0.978504	−	0.206226i	$-0.933882\pi$
$631$		−	10.3607i		−	0.412452i	0.978504	−	0.206226i	$-0.0661182\pi$
$632$		0			0
$633$			0.343027i			0.0136341i
$634$		0			0
$635$			16.3270i			0.647916i
$636$		0			0
$637$	−17.5680			−0.696071
$638$		0			0
$639$			13.7082i			0.542288i
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$		0			0
$643$	−33.8885			−1.33643			−0.668217	−	0.743967i	$-0.732942\pi$
$643$	−33.8885			−1.33643			−0.668217	+	0.743967i	$0.732942\pi$
$644$		0			0
$645$	−1.79611			−0.0707218
$646$		0			0
$647$		−	11.5967i		−	0.455915i	−0.973671	−	0.227958i	$-0.926795\pi$
$647$		−	11.5967i		−	0.455915i	0.973671	−	0.227958i	$-0.0732048\pi$
$648$		0			0
$649$	12.9443	−	2.90617i	0.508107	−	0.114077i
$650$		0			0
$651$			17.0130i			0.666793i
$652$		0			0
$653$			11.1246i			0.435340i	0.976022	+	0.217670i	$0.0698456\pi$
$653$			11.1246i			0.435340i	−0.976022	+	0.217670i	$0.930154\pi$
$654$		0			0
$655$	26.4176			1.03222
$656$		0			0
$657$			7.60845i			0.296834i
$658$		0			0
$659$		−	41.9775i		−	1.63521i	−0.575778	−	0.817606i	$-0.695301\pi$
$659$		−	41.9775i		−	1.63521i	0.575778	−	0.817606i	$-0.304699\pi$
$660$		0			0
$661$			46.8328i			1.82159i	0.412864	+	0.910793i	$0.364528\pi$
$661$			46.8328i			1.82159i	−0.412864	+	0.910793i	$0.635472\pi$
$662$		0			0
$663$		−	11.0557i		−	0.429369i
$664$		0			0
$665$	28.9443			1.12241
$666$		0			0
$667$			4.70228i			0.182073i
$668$		0			0
$669$			23.8885i			0.923584i
$670$		0			0
$671$	32.2299	−	7.23607i	1.24422	−	0.279345i
$672$		0			0
$673$			18.8091i			0.725039i	0.931976	+	0.362519i	$0.118083\pi$
$673$			18.8091i			0.725039i	−0.931976	+	0.362519i	$0.881917\pi$
$674$		0			0
$675$	−3.47214			−0.133643
$676$		0			0
$677$	−24.9645			−0.959463			−0.479732	−	0.877415i	$-0.659266\pi$
$677$	−24.9645			−0.959463			−0.479732	+	0.877415i	$0.659266\pi$
$678$		0			0
$679$	−13.4208			−0.515043
$680$		0			0
$681$			23.1684i			0.887814i
$682$		0			0
$683$	12.5836			0.481498			0.240749	−	0.970587i	$-0.422607\pi$
$683$	12.5836			0.481498			0.240749	+	0.970587i	$0.422607\pi$
$684$		0			0
$685$		−	9.30495i		−	0.355524i
$686$		0			0
$687$		−	24.0000i		−	0.915657i
$688$		0			0
$689$			2.90617i			0.110716i
$690$		0			0
$691$	35.7771			1.36102			0.680512	−	0.732737i	$-0.261757\pi$
$691$	35.7771			1.36102			0.680512	+	0.732737i	$0.261757\pi$
$692$		0			0
$693$	12.3107	−	2.76393i	0.467646	−	0.104993i
$694$		0			0
$695$	20.6052			0.781601
$696$		0			0
$697$	35.7771			1.35515
$698$		0			0
$699$			13.4208i			0.507621i
$700$		0			0
$701$	35.4791			1.34003			0.670014	−	0.742349i	$-0.266288\pi$
$701$	35.4791			1.34003			0.670014	+	0.742349i	$0.266288\pi$
$702$		0			0
$703$	15.2169			0.573916
$704$		0			0
$705$	−0.944272			−0.0355633
$706$		0			0
$707$	23.4164			0.880665
$708$		0			0
$709$			17.8885i			0.671818i	0.941894	+	0.335909i	$0.109044\pi$
$709$			17.8885i			0.671818i	−0.941894	+	0.335909i	$0.890956\pi$
$710$		0			0
$711$	13.2088			0.495368
$712$		0			0
$713$	−14.4721			−0.541986
$714$		0			0
$715$	−2.11146	−	9.40456i	−0.0789640	−	0.351711i
$716$		0			0
$717$	−22.8254			−0.852429
$718$		0			0
$719$			5.70820i			0.212880i	0.994319	+	0.106440i	$0.0339452\pi$
$719$			5.70820i			0.212880i	−0.994319	+	0.106440i	$0.966055\pi$
$720$		0			0
$721$		−	35.8221i		−	1.33409i
$722$		0			0
$723$		−	1.79611i		−	0.0667981i
$724$		0			0
$725$	5.04531			0.187378
$726$		0			0
$727$			24.8328i			0.920998i	0.887660	+	0.460499i	$0.152330\pi$
$727$			24.8328i			0.920998i	−0.887660	+	0.460499i	$0.847670\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	6.83282			0.252721
$732$		0			0
$733$	11.0697			0.408867			0.204433	−	0.978880i	$-0.434465\pi$
$733$	11.0697			0.408867			0.204433	+	0.978880i	$0.434465\pi$
$734$		0			0
$735$			9.23607i			0.340677i
$736$		0			0
$737$	−25.8885	+	5.81234i	−0.953617	+	0.214100i
$738$		0			0
$739$			52.4921i			1.93095i	0.260488	+	0.965477i	$0.416117\pi$
$739$			52.4921i			1.93095i	−0.260488	+	0.965477i	$0.583883\pi$
$740$		0			0
$741$		−	14.4721i		−	0.531647i
$742$		0			0
$743$	−9.40456			−0.345020			−0.172510	−	0.985008i	$-0.555188\pi$
$743$	−9.40456			−0.345020			−0.172510	+	0.985008i	$0.555188\pi$
$744$		0			0
$745$		−	4.01623i		−	0.147143i
$746$		0			0
$747$		−	9.06154i		−	0.331544i
$748$		0			0
$749$			52.3607i			1.91322i
$750$		0			0
$751$			1.41641i			0.0516855i	0.999666	+	0.0258427i	$0.00822691\pi$
$751$			1.41641i			0.0516855i	−0.999666	+	0.0258427i	$0.991773\pi$
$752$		0			0
$753$	3.41641			0.124501
$754$		0			0
$755$			2.48217i			0.0903353i
$756$		0			0
$757$			37.5279i			1.36397i	0.731365	+	0.681987i	$0.238884\pi$
$757$			37.5279i			1.36397i	−0.731365	+	0.681987i	$0.761116\pi$
$758$		0			0
$759$	2.35114	+	10.4721i	0.0853410	+	0.380114i
$760$		0			0
$761$			10.5146i			0.381155i	0.981672	+	0.190577i	$0.0610360\pi$
$761$			10.5146i			0.381155i	−0.981672	+	0.190577i	$0.938964\pi$
$762$		0			0
$763$	−55.7771			−2.01927
$764$		0			0
$765$	−5.81234			−0.210146
$766$		0			0
$767$	9.40456			0.339579
$768$		0			0
$769$		−	41.6345i		−	1.50138i	−0.660656	−	0.750689i	$-0.729722\pi$
$769$		−	41.6345i		−	1.50138i	0.660656	−	0.750689i	$-0.270278\pi$
$770$		0			0
$771$	21.4164			0.771293
$772$		0			0
$773$		−	30.1803i		−	1.08551i	−0.839891	−	0.542756i	$-0.817381\pi$
$773$		−	30.1803i		−	1.08551i	0.839891	−	0.542756i	$-0.182619\pi$
$774$		0			0
$775$			15.5279i			0.557777i
$776$		0			0
$777$			9.40456i			0.337387i
$778$		0			0
$779$	46.8328			1.67796
$780$		0			0
$781$	−9.95959	−	44.3607i	−0.356382	−	1.58735i
$782$		0			0
$783$	−1.45309			−0.0519290
$784$		0			0
$785$	9.88854			0.352937
$786$		0			0
$787$			29.6668i			1.05751i	0.848776	+	0.528753i	$0.177340\pi$
$787$			29.6668i			1.05751i	−0.848776	+	0.528753i	$0.822660\pi$
$788$		0			0
$789$	−17.0130			−0.605679
$790$		0			0
$791$	−35.8221			−1.27369
$792$		0			0
$793$	23.4164			0.831541
$794$		0			0
$795$	1.52786			0.0541878
$796$		0			0
$797$		−	7.34752i		−	0.260263i	−0.991497	−	0.130131i	$-0.958460\pi$
$797$		−	7.34752i		−	0.260263i	0.991497	−	0.130131i	$-0.0415399\pi$
$798$		0			0
$799$	3.59222			0.127084
$800$		0			0
$801$	9.41641			0.332712
$802$		0			0
$803$	−5.52786	−	24.6215i	−0.195074	−	0.868873i
$804$		0			0
$805$	−15.2169			−0.536325
$806$		0			0
$807$			22.1803i			0.780785i
$808$		0			0
$809$			13.4208i			0.471850i	0.971771	+	0.235925i	$0.0758120\pi$
$809$			13.4208i			0.471850i	−0.971771	+	0.235925i	$0.924188\pi$
$810$		0			0
$811$		−	7.95148i		−	0.279214i	−0.990207	−	0.139607i	$-0.955416\pi$
$811$		−	7.95148i		−	0.279214i	0.990207	−	0.139607i	$-0.0445840\pi$
$812$		0			0
$813$	−26.6296			−0.933940
$814$		0			0
$815$		−	27.0557i		−	0.947721i
$816$		0			0
$817$	8.94427			0.312920
$818$		0			0
$819$	8.94427			0.312538
$820$		0			0
$821$	−11.9677			−0.417676			−0.208838	−	0.977950i	$-0.566968\pi$
$821$	−11.9677			−0.417676			−0.208838	+	0.977950i	$0.566968\pi$
$822$		0			0
$823$		−	14.0000i		−	0.488009i	−0.969774	−	0.244005i	$-0.921539\pi$
$823$		−	14.0000i		−	0.488009i	0.969774	−	0.244005i	$-0.0784612\pi$
$824$		0			0
$825$	11.2361	−	2.52265i	0.391190	−	0.0878276i
$826$		0			0
$827$		−	43.0876i		−	1.49830i	−0.662399	−	0.749151i	$-0.730462\pi$
$827$		−	43.0876i		−	1.49830i	0.662399	−	0.749151i	$-0.269538\pi$
$828$		0			0
$829$		−	46.8328i		−	1.62657i	−0.581865	−	0.813285i	$-0.697677\pi$
$829$		−	46.8328i		−	1.62657i	0.581865	−	0.813285i	$-0.302323\pi$
$830$		0			0
$831$	−8.84953			−0.306987
$832$		0			0
$833$		−	35.1361i		−	1.21739i
$834$		0			0
$835$			28.2137i			0.976375i
$836$		0			0
$837$		−	4.47214i		−	0.154580i
$838$		0			0
$839$			45.1246i			1.55787i	0.627102	+	0.778937i	$0.284241\pi$
$839$			45.1246i			1.55787i	−0.627102	+	0.778937i	$0.715759\pi$
$840$		0			0
$841$	−26.8885			−0.927191
$842$		0			0
$843$		−	4.70228i		−	0.161955i
$844$		0			0
$845$			9.23607i			0.317730i
$846$		0			0
$847$	−37.8303	+	17.8885i	−1.29986	+	0.614658i
$848$		0			0
$849$		−	3.24920i		−	0.111512i
$850$		0			0
$851$	−8.00000			−0.274236
$852$		0			0
$853$	28.0827			0.961532			0.480766	−	0.876849i	$-0.340359\pi$
$853$	28.0827			0.961532			0.480766	+	0.876849i	$0.340359\pi$
$854$		0			0
$855$	−7.60845			−0.260204
$856$		0			0
$857$		−	35.8221i		−	1.22366i	−0.790989	−	0.611831i	$-0.790433\pi$
$857$		−	35.8221i		−	1.22366i	0.790989	−	0.611831i	$-0.209567\pi$
$858$		0			0
$859$	2.11146			0.0720420			0.0360210	−	0.999351i	$-0.488532\pi$
$859$	2.11146			0.0720420			0.0360210	+	0.999351i	$0.488532\pi$
$860$		0			0
$861$			28.9443i			0.986418i
$862$		0			0
$863$			8.76393i			0.298328i	0.988812	+	0.149164i	$0.0476582\pi$
$863$			8.76393i			0.298328i	−0.988812	+	0.149164i	$0.952342\pi$
$864$		0			0
$865$			28.6377i			0.973711i
$866$		0			0
$867$	5.11146			0.173594
$868$		0			0
$869$	−42.7445	+	9.59675i	−1.45001	+	0.325547i
$870$		0			0
$871$	−18.8091			−0.637323
$872$		0			0
$873$	3.52786			0.119400
$874$		0			0
$875$			39.8384i			1.34678i
$876$		0			0
$877$	54.5002			1.84034			0.920171	−	0.391516i	$-0.128049\pi$
$877$	54.5002			1.84034			0.920171	+	0.391516i	$0.128049\pi$
$878$		0			0
$879$	−16.6700			−0.562265
$880$		0			0
$881$	−28.8328			−0.971402			−0.485701	−	0.874125i	$-0.661436\pi$
$881$	−28.8328			−0.971402			−0.485701	+	0.874125i	$0.661436\pi$
$882$		0			0
$883$	−12.9443			−0.435609			−0.217805	−	0.975992i	$-0.569890\pi$
$883$	−12.9443			−0.435609			−0.217805	+	0.975992i	$0.569890\pi$
$884$		0			0
$885$		−	4.94427i		−	0.166200i
$886$		0			0
$887$	−5.81234			−0.195159			−0.0975796	−	0.995228i	$-0.531110\pi$
$887$	−5.81234			−0.195159			−0.0975796	+	0.995228i	$0.531110\pi$
$888$		0			0
$889$	50.2492			1.68530
$890$		0			0
$891$	−3.23607	+	0.726543i	−0.108412	+	0.0243401i
$892$		0			0
$893$	4.70228			0.157356
$894$		0			0
$895$			11.0557i			0.369552i
$896$		0			0
$897$			7.60845i			0.254039i
$898$		0			0
$899$			6.49839i			0.216734i
$900$		0			0
$901$	−5.81234			−0.193637
$902$		0			0
$903$			5.52786i			0.183956i
$904$		0			0
$905$	25.1672			0.836586
$906$		0			0
$907$	−20.9443			−0.695443			−0.347722	−	0.937598i	$-0.613045\pi$
$907$	−20.9443			−0.695443			−0.347722	+	0.937598i	$0.613045\pi$
$908$		0			0
$909$	−6.15537			−0.204161
$910$		0			0
$911$			9.34752i			0.309697i	0.987938	+	0.154849i	$0.0494890\pi$
$911$			9.34752i			0.309697i	−0.987938	+	0.154849i	$0.950511\pi$
$912$		0			0
$913$	6.58359	+	29.3238i	0.217885	+	0.970474i
$914$		0			0
$915$		−	12.3107i		−	0.406980i
$916$		0			0
$917$		−	81.3050i		−	2.68493i
$918$		0			0
$919$	−32.4419			−1.07016			−0.535080	−	0.844801i	$-0.679719\pi$
$919$	−32.4419			−1.07016			−0.535080	+	0.844801i	$0.679719\pi$
$920$		0			0
$921$		−	16.6700i		−	0.549295i
$922$		0			0
$923$		−	32.2299i		−	1.06086i
$924$		0			0
$925$			8.58359i			0.282227i
$926$		0			0
$927$			9.41641i			0.309275i
$928$		0			0
$929$	51.3050			1.68326			0.841630	−	0.540054i	$-0.181596\pi$
$929$	51.3050			1.68326			0.841630	+	0.540054i	$0.181596\pi$
$930$		0			0
$931$		−	45.9937i		−	1.50738i
$932$		0			0
$933$			4.18034i			0.136858i
$934$		0			0
$935$	18.8091	−	4.22291i	0.615124	−	0.138104i
$936$		0			0
$937$			3.59222i			0.117353i	0.998277	+	0.0586764i	$0.0186880\pi$
$937$			3.59222i			0.117353i	−0.998277	+	0.0586764i	$0.981312\pi$
$938$		0			0
$939$	2.94427			0.0960827
$940$		0			0
$941$	12.6538			0.412501			0.206250	−	0.978499i	$-0.433874\pi$
$941$	12.6538			0.412501			0.206250	+	0.978499i	$0.433874\pi$
$942$		0			0
$943$	−24.6215			−0.801785
$944$		0			0
$945$		−	4.70228i		−	0.152965i
$946$		0			0
$947$	39.0557			1.26914			0.634570	−	0.772865i	$-0.281177\pi$
$947$	39.0557			1.26914			0.634570	+	0.772865i	$0.281177\pi$
$948$		0			0
$949$		−	17.8885i		−	0.580687i
$950$		0			0
$951$		−	14.1803i		−	0.459829i
$952$		0			0
$953$		−	35.8221i		−	1.16039i	−0.814477	−	0.580197i	$-0.802976\pi$
$953$		−	35.8221i		−	1.16039i	0.814477	−	0.580197i	$-0.197024\pi$
$954$		0			0
$955$	−15.7771			−0.510535
$956$		0			0
$957$	4.70228	−	1.05573i	0.152003	−	0.0341268i
$958$		0			0
$959$	−28.6377			−0.924759
$960$		0			0
$961$	11.0000			0.354839
$962$		0			0
$963$		−	13.7638i		−	0.443533i
$964$		0			0
$965$	21.0292			0.676955
$966$		0			0
$967$	15.0049			0.482525			0.241263	−	0.970460i	$-0.422439\pi$
$967$	15.0049			0.482525			0.241263	+	0.970460i	$0.422439\pi$
$968$		0			0
$969$	28.9443			0.929824
$970$		0			0
$971$	−29.3050			−0.940441			−0.470220	−	0.882549i	$-0.655826\pi$
$971$	−29.3050			−0.940441			−0.470220	+	0.882549i	$0.655826\pi$
$972$		0			0
$973$		−	63.4164i		−	2.03304i
$974$		0			0
$975$	8.16348			0.261441
$976$		0			0
$977$	42.7214			1.36678			0.683389	−	0.730055i	$-0.260505\pi$
$977$	42.7214			1.36678			0.683389	+	0.730055i	$0.260505\pi$
$978$		0			0
$979$	−30.4721	+	6.84142i	−0.973894	+	0.218653i
$980$		0			0
$981$	14.6619			0.468118
$982$		0			0
$983$		−	30.0689i		−	0.959048i	−0.877529	−	0.479524i	$-0.840809\pi$
$983$		−	30.0689i		−	0.959048i	0.877529	−	0.479524i	$-0.159191\pi$
$984$		0			0
$985$			0.424005i			0.0135099i
$986$		0			0
$987$			2.90617i			0.0925044i
$988$		0			0
$989$	−4.70228			−0.149524
$990$		0			0
$991$			51.3050i			1.62976i	0.579633	+	0.814878i	$0.303196\pi$
$991$			51.3050i			1.62976i	−0.579633	+	0.814878i	$0.696804\pi$
$992$		0			0
$993$	−28.0000			−0.888553
$994$		0			0
$995$	12.3607			0.391860
$996$		0			0
$997$	31.6749			1.00315			0.501577	−	0.865113i	$-0.332753\pi$
$997$	31.6749			1.00315			0.501577	+	0.865113i	$0.332753\pi$
$998$		0			0
$999$		−	2.47214i		−	0.0782149i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.h.a.1759.3	✓	8
4.3	odd	2		2112.2.h.b.1759.4	yes	8
8.3	odd	2	inner	2112.2.h.a.1759.6	yes	8
8.5	even	2		2112.2.h.b.1759.5	yes	8
11.10	odd	2	inner	2112.2.h.a.1759.4	yes	8
44.43	even	2		2112.2.h.b.1759.3	yes	8
88.21	odd	2		2112.2.h.b.1759.6	yes	8
88.43	even	2	inner	2112.2.h.a.1759.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.h.a.1759.3	✓	8	1.1	even	1	trivial
2112.2.h.a.1759.4	yes	8	11.10	odd	2	inner
2112.2.h.a.1759.5	yes	8	88.43	even	2	inner
2112.2.h.a.1759.6	yes	8	8.3	odd	2	inner
2112.2.h.b.1759.3	yes	8	44.43	even	2
2112.2.h.b.1759.4	yes	8	4.3	odd	2
2112.2.h.b.1759.5	yes	8	8.5	even	2
2112.2.h.b.1759.6	yes	8	88.21	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.23607i q^{5} -3.80423 q^{7} +1.00000 q^{9} +(-3.23607 + 0.726543i) q^{11} -2.35114 q^{13} +1.23607i q^{15} -4.70228i q^{17} -6.15537i q^{19} +3.80423 q^{21} +3.23607i q^{23} +3.47214 q^{25} -1.00000 q^{27} +1.45309 q^{29} +4.47214i q^{31} +(3.23607 - 0.726543i) q^{33} +4.70228i q^{35} +2.47214i q^{37} +2.35114 q^{39} +7.60845i q^{41} +1.45309i q^{43} -1.23607i q^{45} +0.763932i q^{47} +7.47214 q^{49} +4.70228i q^{51} -1.23607i q^{53} +(0.898056 + 4.00000i) q^{55} +6.15537i q^{57} -4.00000 q^{59} -9.95959 q^{61} -3.80423 q^{63} +2.90617i q^{65} +8.00000 q^{67} -3.23607i q^{69} +13.7082i q^{71} +7.60845i q^{73} -3.47214 q^{75} +(12.3107 - 2.76393i) q^{77} +13.2088 q^{79} +1.00000 q^{81} -9.06154i q^{83} -5.81234 q^{85} -1.45309 q^{87} +9.41641 q^{89} +8.94427 q^{91} -4.47214i q^{93} -7.60845 q^{95} +3.52786 q^{97} +(-3.23607 + 0.726543i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{9} - 8 q^{11} - 8 q^{25} - 8 q^{27} + 8 q^{33} + 24 q^{49} - 32 q^{59} + 64 q^{67} + 8 q^{75} + 8 q^{81} - 32 q^{89} + 64 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^11 - 8 * q^25 - 8 * q^27 + 8 * q^33 + 24 * q^49 - 32 * q^59 + 64 * q^67 + 8 * q^75 + 8 * q^81 - 32 * q^89 + 64 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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