LMFDB - Embedded newform 2112.2.h.b.1759.6

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)

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")

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);

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			1759.6
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	2112.1759
Dual form			2112.2.h.b.1759.4

$q$-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	$0.0891392\pi$
$5$			1.23607i			0.552786i	−0.961045	+	0.276393i	$0.910861\pi$
$6$		0			0
$7$	3.80423			1.43786			0.718931	−	0.695081i	$-0.244632\pi$
$7$	3.80423			1.43786			0.718931	+	0.695081i	$0.244632\pi$
$8$		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	$0.193148\pi$
$17$			4.70228i			1.14047i	−0.821481	+	0.570235i	$0.806852\pi$
$18$		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	$-0.934863\pi$
$37$		−	2.47214i		−	0.406417i	0.979136	−	0.203208i	$-0.0651369\pi$
$38$		0			0
$39$	−2.35114			−0.376484
$40$		0			0
$41$		−	7.60845i		−	1.18824i	−0.804376	−	0.594120i	$-0.797500\pi$
$41$		−	7.60845i		−	1.18824i	0.804376	−	0.594120i	$-0.202500\pi$
$42$		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	$0.0270550\pi$
$53$			1.23607i			0.169787i	−0.996390	+	0.0848935i	$0.972945\pi$
$54$		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	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		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	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$		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	$-0.853114\pi$
$73$		−	7.60845i		−	0.890502i	0.895406	−	0.445251i	$-0.146886\pi$
$74$		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	$-0.766623\pi$
$79$	−13.2088			−1.48610			−0.743052	+	0.669233i	$0.766623\pi$
$80$		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	$-0.300684\pi$
$127$	13.2088			1.17209			0.586045	+	0.810278i	$0.300684\pi$
$128$		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	$-0.473962\pi$
$151$	2.00811			0.163418			0.0817090	+	0.996656i	$0.473962\pi$
$152$		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	$-0.896574\pi$
$157$		−	8.00000i		−	0.638470i	0.947676	−	0.319235i	$-0.103426\pi$
$158$		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	$-0.827811\pi$
$163$	−21.8885			−1.71444			−0.857222	+	0.514948i	$0.827811\pi$
$164$		0			0
$165$	−0.898056	+	4.00000i	−0.0699136	+	0.311400i
$166$		0			0
$167$	22.8254			1.76628			0.883140	−	0.469110i	$-0.155425\pi$
$167$	22.8254			1.76628			0.883140	+	0.469110i	$0.155425\pi$
$168$		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	$-0.391513\pi$
$179$	8.94427			0.668526			0.334263	+	0.942480i	$0.391513\pi$
$180$		0			0
$181$		−	20.3607i		−	1.51340i	−0.653764	−	0.756699i	$-0.726811\pi$
$181$		−	20.3607i		−	1.51340i	0.653764	−	0.756699i	$-0.273189\pi$
$182$		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	$-0.790240\pi$
$193$		−	17.0130i		−	1.22462i	0.790616	−	0.612312i	$-0.209760\pi$
$194$		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	$-0.708527\pi$
$229$		−	24.0000i		−	1.58596i	0.609245	−	0.792982i	$-0.291473\pi$
$230$		0			0
$231$	12.3107	+	2.76393i	0.809987	+	0.181853i
$232$		0			0
$233$			13.4208i			0.879225i	0.898187	+	0.439613i	$0.144884\pi$
$233$			13.4208i			0.879225i	−0.898187	+	0.439613i	$0.855116\pi$
$234$		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	$-0.764336\pi$
$239$	−22.8254			−1.47645			−0.738225	+	0.674555i	$0.764336\pi$
$240$		0			0
$241$		−	1.79611i		−	0.115698i	−0.998325	−	0.0578489i	$-0.981576\pi$
$241$		−	1.79611i		−	0.115698i	0.998325	−	0.0578489i	$-0.0184242\pi$
$242$		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	$-0.465613\pi$
$251$	3.41641			0.215642			0.107821	+	0.994170i	$0.465613\pi$
$252$		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	$-0.675760\pi$
$263$	−17.0130			−1.04907			−0.524534	+	0.851390i	$0.675760\pi$
$264$		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	$0.236366\pi$
$269$			22.1803i			1.35236i	−0.736737	+	0.676180i	$0.763634\pi$
$270$		0			0
$271$	−26.6296			−1.61763			−0.808816	−	0.588062i	$-0.799891\pi$
$271$	−26.6296			−1.61763			−0.808816	+	0.588062i	$0.799891\pi$
$272$		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	$-0.955207\pi$
$281$		−	4.70228i		−	0.280515i	0.990115	−	0.140257i	$-0.0447930\pi$
$282$		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	$-0.869627\pi$
$317$		−	14.1803i		−	0.796447i	0.917288	−	0.398224i	$-0.130373\pi$
$318$		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	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		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	$0.0664447\pi$
$337$			7.60845i			0.414459i	−0.978292	+	0.207229i	$0.933555\pi$
$338$		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	$-0.450984\pi$
$359$	5.81234			0.306764			0.153382	+	0.988167i	$0.450984\pi$
$360$		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	$-0.383901\pi$
$379$	13.8885			0.713407			0.356703	+	0.934218i	$0.383901\pi$
$380$		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	$0.360308\pi$
$389$			35.7082i			1.81048i	−0.424903	+	0.905239i	$0.639692\pi$
$390$		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	$-0.710924\pi$
$397$		−	31.4164i		−	1.57674i	0.615199	−	0.788372i	$-0.289076\pi$
$398$		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	$-0.877226\pi$
$409$		−	15.2169i		−	0.752427i	0.926533	−	0.376214i	$-0.122774\pi$
$410$		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	$-0.790159\pi$
$419$	−32.3607			−1.58092			−0.790461	+	0.612512i	$0.790159\pi$
$420$		0			0
$421$		−	15.4164i		−	0.751350i	−0.926752	−	0.375675i	$-0.877411\pi$
$421$		−	15.4164i		−	0.751350i	0.926752	−	0.375675i	$-0.122589\pi$
$422$		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	$-0.280487\pi$
$431$	26.4176			1.27249			0.636245	+	0.771487i	$0.280487\pi$
$432$		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	$-0.484740\pi$
$439$	2.00811			0.0958421			0.0479210	+	0.998851i	$0.484740\pi$
$440$		0			0
$441$	7.47214			0.355816
$442$		0			0
$443$	−1.52786			−0.0725910			−0.0362955	−	0.999341i	$-0.511556\pi$
$443$	−1.52786			−0.0725910			−0.0362955	+	0.999341i	$0.511556\pi$
$444$		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	$0.0434070\pi$
$457$			5.81234i			0.271890i	−0.990716	+	0.135945i	$0.956593\pi$
$458$		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	$-0.325953\pi$
$467$	22.4721			1.03989			0.519943	+	0.854201i	$0.325953\pi$
$468$		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	$-0.486935\pi$
$479$	1.79611			0.0820664			0.0410332	+	0.999158i	$0.486935\pi$
$480$		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	$-0.390590\pi$
$499$	15.0557			0.673987			0.336993	+	0.941507i	$0.390590\pi$
$500$		0			0
$501$	22.8254			1.01976
$502$		0			0
$503$	−13.4208			−0.598404			−0.299202	−	0.954190i	$-0.596720\pi$
$503$	−13.4208			−0.598404			−0.299202	+	0.954190i	$0.596720\pi$
$504$		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	$-0.789530\pi$
$509$		−	27.7082i		−	1.22814i	0.789250	−	0.614072i	$-0.210470\pi$
$510$		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	$-0.671506\pi$
$569$		−	40.9484i		−	1.71665i	0.513108	−	0.858324i	$-0.328494\pi$
$570$		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	$-0.342156\pi$
$587$	23.0557			0.951612			0.475806	+	0.879550i	$0.342156\pi$
$588$		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	$-0.632559\pi$
$593$		−	44.5407i		−	1.82907i	0.404512	−	0.914533i	$-0.367441\pi$
$594$		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	$-0.887205\pi$
$601$		−	17.0130i		−	0.693975i	0.937870	−	0.346988i	$-0.112795\pi$
$602$		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	$-0.681739\pi$
$607$	−26.6296			−1.08086			−0.540431	+	0.841389i	$0.681739\pi$
$608$		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	$-0.617053\pi$
$619$	−17.8885			−0.719001			−0.359501	+	0.933145i	$0.617053\pi$
$620$		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	$-0.267058\pi$
$643$	33.8885			1.33643			0.668217	+	0.743967i	$0.267058\pi$
$644$		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	$-0.930154\pi$
$653$		−	11.1246i		−	0.435340i	0.976022	−	0.217670i	$-0.0698456\pi$
$654$		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	$-0.635472\pi$
$661$		−	46.8328i		−	1.82159i	0.412864	−	0.910793i	$-0.364528\pi$
$662$		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	$-0.881917\pi$
$673$		−	18.8091i		−	0.725039i	0.931976	−	0.362519i	$-0.118083\pi$
$674$		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	$-0.577393\pi$
$683$	−12.5836			−0.481498			−0.240749	+	0.970587i	$0.577393\pi$
$684$		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	$-0.738243\pi$
$691$	−35.7771			−1.36102			−0.680512	+	0.732737i	$0.738243\pi$
$692$		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	$-0.890956\pi$
$709$		−	17.8885i		−	0.671818i	0.941894	−	0.335909i	$-0.109044\pi$
$710$		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	$-0.444812\pi$
$743$	9.40456			0.345020			0.172510	+	0.985008i	$0.444812\pi$
$744$		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	$-0.761116\pi$
$757$		−	37.5279i		−	1.36397i	0.731365	−	0.681987i	$-0.238884\pi$
$758$		0			0
$759$	−2.35114	+	10.4721i	−0.0853410	+	0.380114i
$760$		0			0
$761$		−	10.5146i		−	0.381155i	−0.981672	−	0.190577i	$-0.938964\pi$
$761$		−	10.5146i		−	0.381155i	0.981672	−	0.190577i	$-0.0610360\pi$
$762$		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	$0.270278\pi$
$769$			41.6345i			1.50138i	−0.660656	+	0.750689i	$0.729722\pi$
$770$		0			0
$771$	−21.4164			−0.771293
$772$		0			0
$773$			30.1803i			1.08551i	0.839891	+	0.542756i	$0.182619\pi$
$773$			30.1803i			1.08551i	−0.839891	+	0.542756i	$0.817381\pi$
$774$		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	$0.0415399\pi$
$797$			7.34752i			0.260263i	−0.991497	+	0.130131i	$0.958460\pi$
$798$		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	$-0.924188\pi$
$809$		−	13.4208i		−	0.471850i	0.971771	−	0.235925i	$-0.0758120\pi$
$810$		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	$0.302323\pi$
$829$			46.8328i			1.62657i	−0.581865	+	0.813285i	$0.697677\pi$
$830$		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	$0.209567\pi$
$857$			35.8221i			1.22366i	−0.790989	+	0.611831i	$0.790433\pi$
$858$		0			0
$859$	−2.11146			−0.0720420			−0.0360210	−	0.999351i	$-0.511468\pi$
$859$	−2.11146			−0.0720420			−0.0360210	+	0.999351i	$0.511468\pi$
$860$		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	$-0.430110\pi$
$883$	12.9443			0.435609			0.217805	+	0.975992i	$0.430110\pi$
$884$		0			0
$885$			4.94427i			0.166200i
$886$		0			0
$887$	5.81234			0.195159			0.0975796	−	0.995228i	$-0.468890\pi$
$887$	5.81234			0.195159			0.0975796	+	0.995228i	$0.468890\pi$
$888$		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	$-0.386955\pi$
$907$	20.9443			0.695443			0.347722	+	0.937598i	$0.386955\pi$
$908$		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	$-0.320281\pi$
$919$	32.4419			1.07016			0.535080	+	0.844801i	$0.320281\pi$
$920$		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	$-0.981312\pi$
$937$		−	3.59222i		−	0.117353i	0.998277	−	0.0586764i	$-0.0186880\pi$
$938$		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	$-0.718823\pi$
$947$	−39.0557			−1.26914			−0.634570	+	0.772865i	$0.718823\pi$
$948$		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	$0.197024\pi$
$953$			35.8221i			1.16039i	−0.814477	+	0.580197i	$0.802976\pi$
$954$		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	$-0.577561\pi$
$967$	−15.0049			−0.482525			−0.241263	+	0.970460i	$0.577561\pi$
$968$		0			0
$969$	28.9443			0.929824
$970$		0			0
$971$	29.3050			0.940441			0.470220	−	0.882549i	$-0.344174\pi$
$971$	29.3050			0.940441			0.470220	+	0.882549i	$0.344174\pi$
$972$		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.b.1759.6	yes	8
4.3	odd	2		2112.2.h.a.1759.5	yes	8
8.3	odd	2	inner	2112.2.h.b.1759.3	yes	8
8.5	even	2		2112.2.h.a.1759.4	yes	8
11.10	odd	2	inner	2112.2.h.b.1759.5	yes	8
44.43	even	2		2112.2.h.a.1759.6	yes	8
88.21	odd	2		2112.2.h.a.1759.3	✓	8
88.43	even	2	inner	2112.2.h.b.1759.4	yes	8

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

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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