LMFDB - Embedded newform 1232.2.j.a.111.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(111,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1232.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.j (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:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2x^{14} - 11x^{12} + 48x^{10} + 768x^{6} - 2816x^{4} - 8192x^{2} + 65536 $ x^16 - 2x^14 - 11x^12 + 48x^10 + 768x^6 - 2816x^4 - 8192x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.13
Root			$1.82710 + 0.813447i$ of defining polynomial
Character	$\chi$	$=$	1232.111
Dual form			1232.2.j.a.111.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.26497 q^{3} -1.62689i q^{5} +(-2.64055 - 0.165823i) q^{7} +2.13007 q^{9} +O(q^{10})$	$q+2.26497 q^{3} -1.62689i q^{5} +(-2.64055 - 0.165823i) q^{7} +2.13007 q^{9} -1.00000i q^{11} -5.91917i q^{13} -3.68486i q^{15} +0.981534i q^{17} -2.02731 q^{19} +(-5.98076 - 0.375583i) q^{21} -5.94501i q^{23} +2.35321 q^{25} -1.97036 q^{27} -0.331646 q^{29} +4.63574 q^{31} -2.26497i q^{33} +(-0.269776 + 4.29590i) q^{35} -3.79336 q^{37} -13.4067i q^{39} -8.42179i q^{41} +4.81493i q^{43} -3.46540i q^{45} +2.27225 q^{47} +(6.94501 + 0.875728i) q^{49} +2.22314i q^{51} +4.48329 q^{53} -1.62689 q^{55} -4.59179 q^{57} +1.39653 q^{59} -13.5710i q^{61} +(-5.62456 - 0.353215i) q^{63} -9.62987 q^{65} +8.86844i q^{67} -13.4652i q^{69} +5.94501i q^{71} -1.73270i q^{73} +5.32995 q^{75} +(-0.165823 + 2.64055i) q^{77} +11.1466i q^{79} -10.8530 q^{81} -0.539553 q^{83} +1.59685 q^{85} -0.751167 q^{87} +3.58996i q^{89} +(-0.981534 + 15.6299i) q^{91} +10.4998 q^{93} +3.29822i q^{95} -6.63214i q^{97} -2.13007i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{9}+O(q^{10}) $ 16 * q + 24 * q^9	$ 16 q + 24 q^{9} + 12 q^{21} - 40 q^{25} + 8 q^{29} - 24 q^{37} - 16 q^{53} - 40 q^{57} + 48 q^{65} + 4 q^{77} + 96 q^{81} + 32 q^{85} - 24 q^{93}+O(q^{100}) $ 16 * q + 24 * q^9 + 12 * q^21 - 40 * q^25 + 8 * q^29 - 24 * q^37 - 16 * q^53 - 40 * q^57 + 48 * q^65 + 4 * q^77 + 96 * q^81 + 32 * q^85 - 24 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$.

$n$	$309$	$353$	$463$	$673$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	2.26497			1.30768			0.653839	−	0.756633i	$-0.273157\pi$
$3$	2.26497			1.30768			0.653839	+	0.756633i	$0.273157\pi$
$4$		0			0
$5$		−	1.62689i		−	0.727569i	−0.931483	−	0.363785i	$-0.881484\pi$
$5$		−	1.62689i		−	0.727569i	0.931483	−	0.363785i	$-0.118516\pi$
$6$		0			0
$7$	−2.64055	−	0.165823i	−0.998034	−	0.0626752i
$7$	−2.64055	−	0.165823i	−0.998034	−	0.0626752i
$8$		0			0
$9$	2.13007			0.710024
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$		−	5.91917i		−	1.64168i	−0.571156	−	0.820841i	$-0.693505\pi$
$13$		−	5.91917i		−	1.64168i	0.571156	−	0.820841i	$-0.306495\pi$
$14$		0			0
$15$		−	3.68486i		−	0.951427i
$16$		0			0
$17$			0.981534i			0.238057i	0.992891	+	0.119029i	$0.0379780\pi$
$17$			0.981534i			0.238057i	−0.992891	+	0.119029i	$0.962022\pi$
$18$		0			0
$19$	−2.02731			−0.465097			−0.232548	−	0.972585i	$-0.574706\pi$
$19$	−2.02731			−0.465097			−0.232548	+	0.972585i	$0.574706\pi$
$20$		0			0
$21$	−5.98076	−	0.375583i	−1.30511	−	0.0819590i
$22$		0			0
$23$		−	5.94501i		−	1.23962i	−0.784752	−	0.619810i	$-0.787210\pi$
$23$		−	5.94501i		−	1.23962i	0.784752	−	0.619810i	$-0.212790\pi$
$24$		0			0
$25$	2.35321			0.470643
$26$		0			0
$27$	−1.97036			−0.379195
$28$		0			0
$29$	−0.331646			−0.0615851			−0.0307925	−	0.999526i	$-0.509803\pi$
$29$	−0.331646			−0.0615851			−0.0307925	+	0.999526i	$0.509803\pi$
$30$		0			0
$31$	4.63574			0.832603			0.416302	−	0.909227i	$-0.363326\pi$
$31$	4.63574			0.832603			0.416302	+	0.909227i	$0.363326\pi$
$32$		0			0
$33$		−	2.26497i		−	0.394280i
$34$		0			0
$35$	−0.269776	+	4.29590i	−0.0456005	+	0.726139i
$36$		0			0
$37$	−3.79336			−0.623625			−0.311813	−	0.950144i	$-0.600936\pi$
$37$	−3.79336			−0.623625			−0.311813	+	0.950144i	$0.600936\pi$
$38$		0			0
$39$		−	13.4067i		−	2.14679i
$40$		0			0
$41$		−	8.42179i		−	1.31526i	−0.753340	−	0.657632i	$-0.771558\pi$
$41$		−	8.42179i		−	1.31526i	0.753340	−	0.657632i	$-0.228442\pi$
$42$		0			0
$43$			4.81493i			0.734270i	0.930168	+	0.367135i	$0.119661\pi$
$43$			4.81493i			0.734270i	−0.930168	+	0.367135i	$0.880339\pi$
$44$		0			0
$45$		−	3.46540i		−	0.516592i
$46$		0			0
$47$	2.27225			0.331442			0.165721	−	0.986173i	$-0.447005\pi$
$47$	2.27225			0.331442			0.165721	+	0.986173i	$0.447005\pi$
$48$		0			0
$49$	6.94501	+	0.875728i	0.992144	+	0.125104i
$50$		0			0
$51$			2.22314i			0.311302i
$52$		0			0
$53$	4.48329			0.615827			0.307913	−	0.951414i	$-0.400369\pi$
$53$	4.48329			0.615827			0.307913	+	0.951414i	$0.400369\pi$
$54$		0			0
$55$	−1.62689			−0.219370
$56$		0			0
$57$	−4.59179			−0.608198
$58$		0			0
$59$	1.39653			0.181812			0.0909061	−	0.995859i	$-0.471024\pi$
$59$	1.39653			0.181812			0.0909061	+	0.995859i	$0.471024\pi$
$60$		0			0
$61$		−	13.5710i		−	1.73759i	−0.495168	−	0.868797i	$-0.664894\pi$
$61$		−	13.5710i		−	1.73759i	0.495168	−	0.868797i	$-0.335106\pi$
$62$		0			0
$63$	−5.62456	−	0.353215i	−0.708628	−	0.0445009i
$64$		0			0
$65$	−9.62987			−1.19444
$66$		0			0
$67$			8.86844i			1.08345i	0.840555	+	0.541726i	$0.182229\pi$
$67$			8.86844i			1.08345i	−0.840555	+	0.541726i	$0.817771\pi$
$68$		0			0
$69$		−	13.4652i		−	1.62102i
$70$		0			0
$71$			5.94501i			0.705542i	0.935710	+	0.352771i	$0.114761\pi$
$71$			5.94501i			0.705542i	−0.935710	+	0.352771i	$0.885239\pi$
$72$		0			0
$73$		−	1.73270i		−	0.202797i	−0.994846	−	0.101399i	$-0.967668\pi$
$73$		−	1.73270i		−	0.202797i	0.994846	−	0.101399i	$-0.0323318\pi$
$74$		0			0
$75$	5.32995			0.615450
$76$		0			0
$77$	−0.165823	+	2.64055i	−0.0188973	+	0.300919i
$78$		0			0
$79$			11.1466i			1.25409i	0.778984	+	0.627044i	$0.215735\pi$
$79$			11.1466i			1.25409i	−0.778984	+	0.627044i	$0.784265\pi$
$80$		0			0
$81$	−10.8530			−1.20589
$82$		0			0
$83$	−0.539553			−0.0592236			−0.0296118	−	0.999561i	$-0.509427\pi$
$83$	−0.539553			−0.0592236			−0.0296118	+	0.999561i	$0.509427\pi$
$84$		0			0
$85$	1.59685			0.173203
$86$		0			0
$87$	−0.751167			−0.0805335
$88$		0			0
$89$			3.58996i			0.380535i	0.981732	+	0.190268i	$0.0609356\pi$
$89$			3.58996i			0.380535i	−0.981732	+	0.190268i	$0.939064\pi$
$90$		0			0
$91$	−0.981534	+	15.6299i	−0.102893	+	1.63846i
$92$		0			0
$93$	10.4998			1.08878
$94$		0			0
$95$			3.29822i			0.338390i
$96$		0			0
$97$		−	6.63214i		−	0.673392i	−0.941613	−	0.336696i	$-0.890691\pi$
$97$		−	6.63214i		−	0.673392i	0.941613	−	0.336696i	$-0.109309\pi$
$98$		0			0
$99$		−	2.13007i		−	0.214080i
$100$		0			0
$101$			7.13107i			0.709568i	0.934948	+	0.354784i	$0.115446\pi$
$101$			7.13107i			0.709568i	−0.934948	+	0.354784i	$0.884554\pi$
$102$		0			0
$103$	13.7029			1.35019			0.675093	−	0.737733i	$-0.264103\pi$
$103$	13.7029			1.35019			0.675093	+	0.737733i	$0.264103\pi$
$104$		0			0
$105$	−0.611035	+	9.73006i	−0.0596309	+	0.949556i
$106$		0			0
$107$			16.2932i			1.57512i	0.616239	+	0.787560i	$0.288656\pi$
$107$			16.2932i			1.57512i	−0.616239	+	0.787560i	$0.711344\pi$
$108$		0			0
$109$	−16.8849			−1.61728			−0.808642	−	0.588301i	$-0.799797\pi$
$109$	−16.8849			−1.61728			−0.808642	+	0.588301i	$0.799797\pi$
$110$		0			0
$111$	−8.59184			−0.815502
$112$		0			0
$113$	15.2001			1.42990			0.714952	−	0.699174i	$-0.246449\pi$
$113$	15.2001			1.42990			0.714952	+	0.699174i	$0.246449\pi$
$114$		0			0
$115$	−9.67190			−0.901909
$116$		0			0
$117$		−	12.6083i		−	1.16563i
$118$		0			0
$119$	0.162761	−	2.59179i	0.0149203	−	0.237589i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	19.0751i		−	1.71994i
$124$		0			0
$125$		−	11.9629i		−	1.06999i
$126$		0			0
$127$			3.29822i			0.292670i	0.989235	+	0.146335i	$0.0467477\pi$
$127$			3.29822i			0.292670i	−0.989235	+	0.146335i	$0.953252\pi$
$128$		0			0
$129$			10.9057i			0.960190i
$130$		0			0
$131$	−5.21686			−0.455799			−0.227899	−	0.973685i	$-0.573186\pi$
$131$	−5.21686			−0.455799			−0.227899	+	0.973685i	$0.573186\pi$
$132$		0			0
$133$	5.35321	+	0.336175i	0.464183	+	0.0291500i
$134$		0			0
$135$			3.20556i			0.275891i
$136$		0			0
$137$	19.6834			1.68166			0.840832	−	0.541296i	$-0.182066\pi$
$137$	19.6834			1.68166			0.840832	+	0.541296i	$0.182066\pi$
$138$		0			0
$139$	11.3630			0.963800			0.481900	−	0.876226i	$-0.339947\pi$
$139$	11.3630			0.963800			0.481900	+	0.876226i	$0.339947\pi$
$140$		0			0
$141$	5.14658			0.433420
$142$		0			0
$143$	−5.91917			−0.494986
$144$		0			0
$145$			0.539553i			0.0448074i
$146$		0			0
$147$	15.7302	+	1.98349i	1.29741	+	0.163596i
$148$		0			0
$149$	15.6299			1.28045			0.640224	−	0.768188i	$-0.278841\pi$
$149$	15.6299			1.28045			0.640224	+	0.768188i	$0.278841\pi$
$150$		0			0
$151$		−	9.82000i		−	0.799140i	−0.916703	−	0.399570i	$-0.869159\pi$
$151$		−	9.82000i		−	0.799140i	0.916703	−	0.399570i	$-0.130841\pi$
$152$		0			0
$153$			2.09074i			0.169026i
$154$		0			0
$155$		−	7.54186i		−	0.605777i
$156$		0			0
$157$			13.9260i			1.11141i	0.831378	+	0.555707i	$0.187552\pi$
$157$			13.9260i			1.11141i	−0.831378	+	0.555707i	$0.812448\pi$
$158$		0			0
$159$	10.1545			0.805304
$160$		0			0
$161$	−0.985818	+	15.6981i	−0.0776934	+	1.23718i
$162$		0			0
$163$		−	10.2932i		−	0.806222i	−0.915151	−	0.403111i	$-0.867929\pi$
$163$		−	10.2932i		−	0.806222i	0.915151	−	0.403111i	$-0.132071\pi$
$164$		0			0
$165$	−3.68486			−0.286866
$166$		0			0
$167$	−6.08193			−0.470634			−0.235317	−	0.971919i	$-0.575613\pi$
$167$	−6.08193			−0.470634			−0.235317	+	0.971919i	$0.575613\pi$
$168$		0			0
$169$	−22.0366			−1.69512
$170$		0			0
$171$	−4.31832			−0.330230
$172$		0			0
$173$			8.35419i			0.635157i	0.948232	+	0.317579i	$0.102870\pi$
$173$			8.35419i			0.635157i	−0.948232	+	0.317579i	$0.897130\pi$
$174$		0			0
$175$	−6.21378	−	0.390217i	−0.469718	−	0.0294976i
$176$		0			0
$177$	3.16309			0.237752
$178$		0			0
$179$		−	16.3482i		−	1.22192i	−0.791662	−	0.610959i	$-0.790784\pi$
$179$		−	16.3482i		−	1.22192i	0.791662	−	0.610959i	$-0.209216\pi$
$180$		0			0
$181$		−	16.7190i		−	1.24272i	−0.783527	−	0.621358i	$-0.786581\pi$
$181$		−	16.7190i		−	1.24272i	0.783527	−	0.621358i	$-0.213419\pi$
$182$		0			0
$183$		−	30.7380i		−	2.27222i
$184$		0			0
$185$			6.17140i			0.453731i
$186$		0			0
$187$	0.981534			0.0717769
$188$		0			0
$189$	5.20283	+	0.326730i	0.378450	+	0.0237661i
$190$		0			0
$191$			24.6514i			1.78372i	0.452316	+	0.891858i	$0.350598\pi$
$191$			24.6514i			1.78372i	−0.452316	+	0.891858i	$0.649402\pi$
$192$		0			0
$193$	4.99494			0.359543			0.179772	−	0.983708i	$-0.442464\pi$
$193$	4.99494			0.359543			0.179772	+	0.983708i	$0.442464\pi$
$194$		0			0
$195$	−21.8113			−1.56194
$196$		0			0
$197$	−25.2597			−1.79968			−0.899841	−	0.436219i	$-0.856317\pi$
$197$	−25.2597			−1.79968			−0.899841	+	0.436219i	$0.856317\pi$
$198$		0			0
$199$	−16.9567			−1.20203			−0.601014	−	0.799239i	$-0.705236\pi$
$199$	−16.9567			−1.20203			−0.601014	+	0.799239i	$0.705236\pi$
$200$		0			0
$201$			20.0867i			1.41681i
$202$		0			0
$203$	0.875728	+	0.0549945i	0.0614640	+	0.00385986i
$204$		0			0
$205$	−13.7014			−0.956945
$206$		0			0
$207$		−	12.6633i		−	0.880160i
$208$		0			0
$209$			2.02731i			0.140232i
$210$		0			0
$211$		−	15.2982i		−	1.05317i	−0.850122	−	0.526586i	$-0.823472\pi$
$211$		−	15.2982i		−	1.05317i	0.850122	−	0.526586i	$-0.176528\pi$
$212$		0			0
$213$			13.4652i			0.922623i
$214$		0			0
$215$	7.83339			0.534233
$216$		0			0
$217$	−12.2409	−	0.768712i	−0.830966	−	0.0521836i
$218$		0			0
$219$		−	3.92451i		−	0.265194i
$220$		0			0
$221$	5.80987			0.390814
$222$		0			0
$223$	15.3940			1.03086			0.515430	−	0.856932i	$-0.327632\pi$
$223$	15.3940			1.03086			0.515430	+	0.856932i	$0.327632\pi$
$224$		0			0
$225$	5.01252			0.334168
$226$		0			0
$227$	22.8116			1.51406			0.757030	−	0.653380i	$-0.226650\pi$
$227$	22.8116			1.51406			0.757030	+	0.653380i	$0.226650\pi$
$228$		0			0
$229$		−	23.3405i		−	1.54238i	−0.636602	−	0.771192i	$-0.719661\pi$
$229$		−	23.3405i		−	1.54238i	0.636602	−	0.771192i	$-0.280339\pi$
$230$		0			0
$231$	−0.375583	+	5.98076i	−0.0247116	+	0.393505i
$232$		0			0
$233$	−8.63493			−0.565693			−0.282846	−	0.959165i	$-0.591279\pi$
$233$	−8.63493			−0.565693			−0.282846	+	0.959165i	$0.591279\pi$
$234$		0			0
$235$		−	3.69672i		−	0.241147i
$236$		0			0
$237$			25.2466i			1.63994i
$238$		0			0
$239$			6.17852i			0.399655i	0.979831	+	0.199828i	$0.0640382\pi$
$239$			6.17852i			0.399655i	−0.979831	+	0.199828i	$0.935962\pi$
$240$		0			0
$241$		−	1.04914i		−	0.0675811i	−0.999429	−	0.0337906i	$-0.989242\pi$
$241$		−	1.04914i		−	0.0675811i	0.999429	−	0.0337906i	$-0.0107579\pi$
$242$		0			0
$243$	−18.6706			−1.19772
$244$		0			0
$245$	1.42472	−	11.2988i	0.0910218	−	0.721853i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		0			0
$249$	−1.22207			−0.0774455
$250$		0			0
$251$	3.14798			0.198699			0.0993494	−	0.995053i	$-0.468324\pi$
$251$	3.14798			0.198699			0.0993494	+	0.995053i	$0.468324\pi$
$252$		0			0
$253$	−5.94501			−0.373759
$254$		0			0
$255$	3.61682			0.226494
$256$		0			0
$257$			28.4328i			1.77359i	0.462163	+	0.886795i	$0.347073\pi$
$257$			28.4328i			1.77359i	−0.462163	+	0.886795i	$0.652927\pi$
$258$		0			0
$259$	10.0166	+	0.629027i	0.622399	+	0.0390858i
$260$		0			0
$261$	−0.706430			−0.0437269
$262$		0			0
$263$		−	18.7064i		−	1.15349i	−0.816925	−	0.576744i	$-0.804323\pi$
$263$		−	18.7064i		−	1.15349i	0.816925	−	0.576744i	$-0.195677\pi$
$264$		0			0
$265$		−	7.29383i		−	0.448057i
$266$		0			0
$267$			8.13115i			0.497618i
$268$		0			0
$269$			14.8805i			0.907281i	0.891185	+	0.453641i	$0.149875\pi$
$269$			14.8805i			0.907281i	−0.891185	+	0.453641i	$0.850125\pi$
$270$		0			0
$271$	4.28081			0.260041			0.130020	−	0.991511i	$-0.458496\pi$
$271$	4.28081			0.260041			0.130020	+	0.991511i	$0.458496\pi$
$272$		0			0
$273$	−2.22314	+	35.4011i	−0.134551	+	2.14257i
$274$		0			0
$275$		−	2.35321i		−	0.141904i
$276$		0			0
$277$	25.1882			1.51341			0.756707	−	0.653754i	$-0.226807\pi$
$277$	25.1882			1.51341			0.756707	+	0.653754i	$0.226807\pi$
$278$		0			0
$279$	9.87446			0.591168
$280$		0			0
$281$	17.4052			1.03831			0.519155	−	0.854680i	$-0.326247\pi$
$281$	17.4052			1.03831			0.519155	+	0.854680i	$0.326247\pi$
$282$		0			0
$283$	−19.1613			−1.13902			−0.569511	−	0.821983i	$-0.692868\pi$
$283$	−19.1613			−1.13902			−0.569511	+	0.821983i	$0.692868\pi$
$284$		0			0
$285$			7.47036i			0.442506i
$286$		0			0
$287$	−1.39653	+	22.2382i	−0.0824344	+	1.31268i
$288$		0			0
$289$	16.0366			0.943329
$290$		0			0
$291$		−	15.0216i		−	0.880580i
$292$		0			0
$293$			11.6080i			0.678145i	0.940760	+	0.339073i	$0.110113\pi$
$293$			11.6080i			0.678145i	−0.940760	+	0.339073i	$0.889887\pi$
$294$		0			0
$295$		−	2.27200i		−	0.132281i
$296$		0			0
$297$			1.97036i			0.114332i
$298$		0			0
$299$	−35.1895			−2.03506
$300$		0			0
$301$	0.798426	−	12.7141i	0.0460205	−	0.732827i
$302$		0			0
$303$			16.1516i			0.927888i
$304$		0			0
$305$	−22.0787			−1.26422
$306$		0			0
$307$	12.5253			0.714855			0.357427	−	0.933941i	$-0.383654\pi$
$307$	12.5253			0.714855			0.357427	+	0.933941i	$0.383654\pi$
$308$		0			0
$309$	31.0366			1.76561
$310$		0			0
$311$	25.7674			1.46114			0.730568	−	0.682840i	$-0.239255\pi$
$311$	25.7674			1.46114			0.730568	+	0.682840i	$0.239255\pi$
$312$		0			0
$313$			25.7643i			1.45629i	0.685426	+	0.728143i	$0.259616\pi$
$313$			25.7643i			1.45629i	−0.685426	+	0.728143i	$0.740384\pi$
$314$		0			0
$315$	−0.574643	+	9.15057i	−0.0323775	+	0.515576i
$316$		0			0
$317$	−7.87350			−0.442220			−0.221110	−	0.975249i	$-0.570968\pi$
$317$	−7.87350			−0.442220			−0.221110	+	0.975249i	$0.570968\pi$
$318$		0			0
$319$			0.331646i			0.0185686i
$320$		0			0
$321$			36.9035i			2.05975i
$322$		0			0
$323$		−	1.98988i		−	0.110720i
$324$		0			0
$325$		−	13.9291i		−	0.772646i
$326$		0			0
$327$	−38.2438			−2.11489
$328$		0			0
$329$	−6.00000	−	0.376792i	−0.330791	−	0.0207732i
$330$		0			0
$331$			22.1950i			1.21995i	0.792421	+	0.609974i	$0.208820\pi$
$331$			22.1950i			1.21995i	−0.792421	+	0.609974i	$0.791180\pi$
$332$		0			0
$333$	−8.08014			−0.442789
$334$		0			0
$335$	14.4280			0.788287
$336$		0			0
$337$	19.1836			1.04500			0.522498	−	0.852640i	$-0.325000\pi$
$337$	19.1836			1.04500			0.522498	+	0.852640i	$0.325000\pi$
$338$		0			0
$339$	34.4277			1.86986
$340$		0			0
$341$		−	4.63574i		−	0.251039i
$342$		0			0
$343$	−18.1934	−	3.46404i	−0.982352	−	0.187041i
$344$		0			0
$345$	−21.9065			−1.17941
$346$		0			0
$347$		−	4.26163i		−	0.228776i	−0.993436	−	0.114388i	$-0.963509\pi$
$347$		−	4.26163i		−	0.228776i	0.993436	−	0.114388i	$-0.0364908\pi$
$348$		0			0
$349$		−	23.8720i		−	1.27784i	−0.769274	−	0.638918i	$-0.779382\pi$
$349$		−	23.8720i		−	1.27784i	0.769274	−	0.638918i	$-0.220618\pi$
$350$		0			0
$351$			11.6629i			0.622518i
$352$		0			0
$353$			6.70854i			0.357059i	0.983935	+	0.178530i	$0.0571341\pi$
$353$			6.70854i			0.357059i	−0.983935	+	0.178530i	$0.942866\pi$
$354$		0			0
$355$	9.67190			0.513331
$356$		0			0
$357$	0.368648	−	5.87032i	0.0195109	−	0.310690i
$358$		0			0
$359$			29.8530i			1.57558i	0.615943	+	0.787791i	$0.288775\pi$
$359$			29.8530i			1.57558i	−0.615943	+	0.787791i	$0.711225\pi$
$360$		0			0
$361$	−14.8900			−0.783685
$362$		0			0
$363$	−2.26497			−0.118880
$364$		0			0
$365$	−2.81892			−0.147549
$366$		0			0
$367$	−14.4346			−0.753481			−0.376740	−	0.926319i	$-0.622955\pi$
$367$	−14.4346			−0.753481			−0.376740	+	0.926319i	$0.622955\pi$
$368$		0			0
$369$		−	17.9390i		−	0.933869i
$370$		0			0
$371$	−11.8383	−	0.743432i	−0.614616	−	0.0385971i
$372$		0			0
$373$	0.298632			0.0154626			0.00773130	−	0.999970i	$-0.497539\pi$
$373$	0.298632			0.0154626			0.00773130	+	0.999970i	$0.497539\pi$
$374$		0			0
$375$		−	27.0956i		−	1.39921i
$376$		0			0
$377$			1.96307i			0.101103i
$378$		0			0
$379$		−	0.761424i		−	0.0391117i	−0.999809	−	0.0195559i	$-0.993775\pi$
$379$		−	0.761424i		−	0.0391117i	0.999809	−	0.0195559i	$-0.00622522\pi$
$380$		0			0
$381$			7.47036i			0.382718i
$382$		0			0
$383$	−12.0118			−0.613772			−0.306886	−	0.951746i	$-0.599287\pi$
$383$	−12.0118			−0.613772			−0.306886	+	0.951746i	$0.599287\pi$
$384$		0			0
$385$	4.29590	+	0.269776i	0.218939	+	0.0137491i
$386$		0			0
$387$			10.2562i			0.521350i
$388$		0			0
$389$	7.87350			0.399203			0.199601	−	0.979877i	$-0.436035\pi$
$389$	7.87350			0.399203			0.199601	+	0.979877i	$0.436035\pi$
$390$		0			0
$391$	5.83523			0.295100
$392$		0			0
$393$	−11.8160			−0.596039
$394$		0			0
$395$	18.1343			0.912436
$396$		0			0
$397$		−	3.46540i		−	0.173924i	−0.996212	−	0.0869618i	$-0.972284\pi$
$397$		−	3.46540i		−	0.173924i	0.996212	−	0.0869618i	$-0.0277158\pi$
$398$		0			0
$399$	12.1249	+	0.761424i	0.607002	+	0.0381189i
$400$		0			0
$401$	13.3266			0.665498			0.332749	−	0.943015i	$-0.392024\pi$
$401$	13.3266			0.665498			0.332749	+	0.943015i	$0.392024\pi$
$402$		0			0
$403$		−	27.4397i		−	1.36687i
$404$		0			0
$405$			17.6567i			0.877368i
$406$		0			0
$407$			3.79336i			0.188030i
$408$		0			0
$409$		−	32.2449i		−	1.59441i	−0.603710	−	0.797204i	$-0.706312\pi$
$409$		−	32.2449i		−	1.59441i	0.603710	−	0.797204i	$-0.293688\pi$
$410$		0			0
$411$	44.5822			2.19908
$412$		0			0
$413$	−3.68760	−	0.231576i	−0.181455	−	0.0113951i
$414$		0			0
$415$			0.877795i			0.0430893i
$416$		0			0
$417$	25.7369			1.26034
$418$		0			0
$419$	−33.0613			−1.61515			−0.807574	−	0.589766i	$-0.799220\pi$
$419$	−33.0613			−1.61515			−0.807574	+	0.589766i	$0.799220\pi$
$420$		0			0
$421$	7.15670			0.348796			0.174398	−	0.984675i	$-0.444202\pi$
$421$	7.15670			0.348796			0.174398	+	0.984675i	$0.444202\pi$
$422$		0			0
$423$	4.84007			0.235332
$424$		0			0
$425$			2.30976i			0.112040i
$426$		0			0
$427$	−2.25039	+	35.8350i	−0.108904	+	1.73418i
$428$		0			0
$429$	−13.4067			−0.647283
$430$		0			0
$431$			29.9215i			1.44127i	0.693315	+	0.720635i	$0.256150\pi$
$431$			29.9215i			1.44127i	−0.693315	+	0.720635i	$0.743850\pi$
$432$		0			0
$433$			6.63214i			0.318720i	0.987221	+	0.159360i	$0.0509431\pi$
$433$			6.63214i			0.318720i	−0.987221	+	0.159360i	$0.949057\pi$
$434$		0			0
$435$			1.22207i			0.0585937i
$436$		0			0
$437$			12.0524i			0.576543i
$438$		0			0
$439$	−21.4523			−1.02386			−0.511932	−	0.859026i	$-0.671070\pi$
$439$	−21.4523			−1.02386			−0.511932	+	0.859026i	$0.671070\pi$
$440$		0			0
$441$	14.7934	+	1.86536i	0.704446	+	0.0888268i
$442$		0			0
$443$		−	31.8249i		−	1.51205i	−0.654545	−	0.756023i	$-0.727140\pi$
$443$		−	31.8249i		−	1.51205i	0.654545	−	0.756023i	$-0.272860\pi$
$444$		0			0
$445$	5.84049			0.276866
$446$		0			0
$447$	35.4011			1.67442
$448$		0			0
$449$	−29.3132			−1.38338			−0.691689	−	0.722196i	$-0.743133\pi$
$449$	−29.3132			−1.38338			−0.691689	+	0.722196i	$0.743133\pi$
$450$		0			0
$451$	−8.42179			−0.396567
$452$		0			0
$453$		−	22.2420i		−	1.04502i
$454$		0			0
$455$	25.4281	+	1.59685i	1.19209	+	0.0748616i
$456$		0			0
$457$	20.7064			0.968606			0.484303	−	0.874900i	$-0.339073\pi$
$457$	20.7064			0.968606			0.484303	+	0.874900i	$0.339073\pi$
$458$		0			0
$459$		−	1.93397i		−	0.0902701i
$460$		0			0
$461$			19.4739i			0.906989i	0.891259	+	0.453494i	$0.149823\pi$
$461$			19.4739i			0.906989i	−0.891259	+	0.453494i	$0.850177\pi$
$462$		0			0
$463$		−	21.7280i		−	1.00979i	−0.863182	−	0.504893i	$-0.831532\pi$
$463$		−	21.7280i		−	1.00979i	0.863182	−	0.504893i	$-0.168468\pi$
$464$		0			0
$465$		−	17.0821i		−	0.792161i
$466$		0			0
$467$	−16.5563			−0.766133			−0.383066	−	0.923721i	$-0.625132\pi$
$467$	−16.5563			−0.766133			−0.383066	+	0.923721i	$0.625132\pi$
$468$		0			0
$469$	1.47059	−	23.4176i	0.0679056	−	1.08132i
$470$		0			0
$471$			31.5419i			1.45337i
$472$		0			0
$473$	4.81493			0.221391
$474$		0			0
$475$	−4.77070			−0.218895
$476$		0			0
$477$	9.54973			0.437252
$478$		0			0
$479$	−37.9388			−1.73347			−0.866735	−	0.498769i	$-0.833785\pi$
$479$	−37.9388			−1.73347			−0.866735	+	0.498769i	$0.833785\pi$
$480$		0			0
$481$			22.4536i			1.02379i
$482$		0			0
$483$	−2.23285	+	35.5556i	−0.101598	+	1.61784i
$484$		0			0
$485$	−10.7898			−0.489939
$486$		0			0
$487$			7.54186i			0.341754i	0.985292	+	0.170877i	$0.0546601\pi$
$487$			7.54186i			0.341754i	−0.985292	+	0.170877i	$0.945340\pi$
$488$		0			0
$489$		−	23.3137i		−	1.05428i
$490$		0			0
$491$			28.0417i			1.26550i	0.774355	+	0.632751i	$0.218074\pi$
$491$			28.0417i			1.26550i	−0.774355	+	0.632751i	$0.781926\pi$
$492$		0			0
$493$		−	0.325522i		−	0.0146608i
$494$		0			0
$495$	−3.46540			−0.155758
$496$		0			0
$497$	0.985818	−	15.6981i	0.0442200	−	0.704155i
$498$		0			0
$499$		−	31.0635i		−	1.39059i	−0.718724	−	0.695296i	$-0.755273\pi$
$499$		−	31.0635i		−	1.39059i	0.718724	−	0.695296i	$-0.244727\pi$
$500$		0			0
$501$	−13.7754			−0.615438
$502$		0			0
$503$	−17.2479			−0.769047			−0.384524	−	0.923115i	$-0.625634\pi$
$503$	−17.2479			−0.769047			−0.384524	+	0.923115i	$0.625634\pi$
$504$		0			0
$505$	11.6015			0.516260
$506$		0			0
$507$	−49.9121			−2.21668
$508$		0			0
$509$			15.2167i			0.674468i	0.941421	+	0.337234i	$0.109491\pi$
$509$			15.2167i			0.674468i	−0.941421	+	0.337234i	$0.890509\pi$
$510$		0			0
$511$	−0.287322	+	4.57528i	−0.0127104	+	0.202399i
$512$		0			0
$513$	3.99453			0.176363
$514$		0			0
$515$		−	22.2932i		−	0.982354i
$516$		0			0
$517$		−	2.27225i		−	0.0999336i
$518$		0			0
$519$			18.9220i			0.830581i
$520$		0			0
$521$		−	32.0603i		−	1.40459i	−0.711888	−	0.702293i	$-0.752160\pi$
$521$		−	32.0603i		−	1.40459i	0.711888	−	0.702293i	$-0.247840\pi$
$522$		0			0
$523$	27.6695			1.20990			0.604951	−	0.796263i	$-0.293193\pi$
$523$	27.6695			1.20990			0.604951	+	0.796263i	$0.293193\pi$
$524$		0			0
$525$	−14.0740	−	0.883828i	−0.614240	−	0.0385734i
$526$		0			0
$527$			4.55014i			0.198207i
$528$		0			0
$529$	−12.3431			−0.536656
$530$		0			0
$531$	2.97470			0.129091
$532$		0			0
$533$	−49.8500			−2.15924
$534$		0			0
$535$	26.5072			1.14601
$536$		0			0
$537$		−	37.0280i		−	1.59788i
$538$		0			0
$539$	0.875728	−	6.94501i	0.0377203	−	0.299143i
$540$		0			0
$541$	23.3029			1.00187			0.500934	−	0.865485i	$-0.332990\pi$
$541$	23.3029			1.00187			0.500934	+	0.865485i	$0.332990\pi$
$542$		0			0
$543$		−	37.8680i		−	1.62507i
$544$		0			0
$545$			27.4700i			1.17669i
$546$		0			0
$547$		−	29.9331i		−	1.27985i	−0.768438	−	0.639925i	$-0.778966\pi$
$547$		−	29.9331i		−	1.27985i	0.768438	−	0.639925i	$-0.221034\pi$
$548$		0			0
$549$		−	28.9073i		−	1.23373i
$550$		0			0
$551$	0.672349			0.0286430
$552$		0			0
$553$	1.84836	−	29.4331i	0.0786002	−	1.25162i
$554$		0			0
$555$			13.9780i			0.593334i
$556$		0			0
$557$	−34.8896			−1.47832			−0.739160	−	0.673530i	$-0.764777\pi$
$557$	−34.8896			−1.47832			−0.739160	+	0.673530i	$0.764777\pi$
$558$		0			0
$559$	28.5004			1.20544
$560$		0			0
$561$	2.22314			0.0938612
$562$		0			0
$563$	37.6133			1.58521			0.792606	−	0.609734i	$-0.208724\pi$
$563$	37.6133			1.58521			0.792606	+	0.609734i	$0.208724\pi$
$564$		0			0
$565$		−	24.7289i		−	1.04035i
$566$		0			0
$567$	28.6579	+	1.79968i	1.20352	+	0.0755794i
$568$		0			0
$569$	19.3083			0.809448			0.404724	−	0.914439i	$-0.367368\pi$
$569$	19.3083			0.809448			0.404724	+	0.914439i	$0.367368\pi$
$570$		0			0
$571$		−	26.1070i		−	1.09254i	−0.837608	−	0.546272i	$-0.816046\pi$
$571$		−	26.1070i		−	1.09254i	0.837608	−	0.546272i	$-0.183954\pi$
$572$		0			0
$573$			55.8347i			2.33253i
$574$		0			0
$575$		−	13.9899i		−	0.583418i
$576$		0			0
$577$		−	12.1745i		−	0.506832i	−0.967357	−	0.253416i	$-0.918446\pi$
$577$		−	12.1745i		−	0.506832i	0.967357	−	0.253416i	$-0.0815541\pi$
$578$		0			0
$579$	11.3134			0.470167
$580$		0			0
$581$	1.42472	+	0.0894702i	0.0591072	+	0.00371185i
$582$		0			0
$583$		−	4.48329i		−	0.185679i
$584$		0			0
$585$	−20.5123			−0.848080
$586$		0			0
$587$	30.6375			1.26454			0.632272	−	0.774747i	$-0.282123\pi$
$587$	30.6375			1.26454			0.632272	+	0.774747i	$0.282123\pi$
$588$		0			0
$589$	−9.39808			−0.387241
$590$		0			0
$591$	−57.2124			−2.35341
$592$		0			0
$593$			18.7578i			0.770291i	0.922856	+	0.385145i	$0.125849\pi$
$593$			18.7578i			0.770291i	−0.922856	+	0.385145i	$0.874151\pi$
$594$		0			0
$595$	−4.21657	−	0.264795i	−0.172862	−	0.0108555i
$596$		0			0
$597$	−38.4063			−1.57187
$598$		0			0
$599$		−	32.7703i		−	1.33896i	−0.742831	−	0.669479i	$-0.766517\pi$
$599$		−	32.7703i		−	1.33896i	0.742831	−	0.669479i	$-0.233483\pi$
$600$		0			0
$601$		−	9.25178i		−	0.377388i	−0.982036	−	0.188694i	$-0.939575\pi$
$601$		−	9.25178i		−	0.377388i	0.982036	−	0.188694i	$-0.0604254\pi$
$602$		0			0
$603$			18.8904i			0.769277i
$604$		0			0
$605$			1.62689i			0.0661427i
$606$		0			0
$607$	−23.6621			−0.960415			−0.480208	−	0.877155i	$-0.659439\pi$
$607$	−23.6621			−0.960415			−0.480208	+	0.877155i	$0.659439\pi$
$608$		0			0
$609$	1.98349	+	0.124561i	0.0803752	+	0.00504745i
$610$		0			0
$611$		−	13.4499i		−	0.544123i
$612$		0			0
$613$	−1.58673			−0.0640874			−0.0320437	−	0.999486i	$-0.510202\pi$
$613$	−1.58673			−0.0640874			−0.0320437	+	0.999486i	$0.510202\pi$
$614$		0			0
$615$	−31.0331			−1.25138
$616$		0			0
$617$	−2.77645			−0.111775			−0.0558877	−	0.998437i	$-0.517799\pi$
$617$	−2.77645			−0.111775			−0.0558877	+	0.998437i	$0.517799\pi$
$618$		0			0
$619$	−7.02109			−0.282201			−0.141101	−	0.989995i	$-0.545064\pi$
$619$	−7.02109			−0.282201			−0.141101	+	0.989995i	$0.545064\pi$
$620$		0			0
$621$			11.7138i			0.470058i
$622$		0			0
$623$	0.595298	−	9.47948i	0.0238501	−	0.379787i
$624$		0			0
$625$	−7.69631			−0.307852
$626$		0			0
$627$			4.59179i			0.183378i
$628$		0			0
$629$		−	3.72332i		−	0.148458i
$630$		0			0
$631$		−	33.5317i		−	1.33488i	−0.744665	−	0.667439i	$-0.767391\pi$
$631$		−	33.5317i		−	1.33488i	0.744665	−	0.667439i	$-0.232609\pi$
$632$		0			0
$633$		−	34.6500i		−	1.37721i
$634$		0			0
$635$	5.36586			0.212937
$636$		0			0
$637$	5.18358	−	41.1087i	0.205381	−	1.62879i
$638$		0			0
$639$			12.6633i			0.500952i
$640$		0			0
$641$	0.239649			0.00946558			0.00473279	−	0.999989i	$-0.498494\pi$
$641$	0.239649			0.00946558			0.00473279	+	0.999989i	$0.498494\pi$
$642$		0			0
$643$	−19.4787			−0.768167			−0.384083	−	0.923298i	$-0.625483\pi$
$643$	−19.4787			−0.768167			−0.384083	+	0.923298i	$0.625483\pi$
$644$		0			0
$645$	17.7424			0.698605
$646$		0			0
$647$	28.2506			1.11065			0.555323	−	0.831635i	$-0.312595\pi$
$647$	28.2506			1.11065			0.555323	+	0.831635i	$0.312595\pi$
$648$		0			0
$649$		−	1.39653i		−	0.0548185i
$650$		0			0
$651$	−27.7252	−	1.74111i	−1.08664	−	0.0682394i
$652$		0			0
$653$	35.7864			1.40043			0.700215	−	0.713932i	$-0.253087\pi$
$653$	35.7864			1.40043			0.700215	+	0.713932i	$0.253087\pi$
$654$		0			0
$655$			8.48728i			0.331625i
$656$		0			0
$657$		−	3.69078i		−	0.143991i
$658$		0			0
$659$		−	9.43974i		−	0.367720i	−0.982952	−	0.183860i	$-0.941141\pi$
$659$		−	9.43974i		−	0.367720i	0.982952	−	0.183860i	$-0.0588593\pi$
$660$		0			0
$661$		−	46.6917i		−	1.81610i	−0.418866	−	0.908048i	$-0.637572\pi$
$661$		−	46.6917i		−	1.81610i	0.418866	−	0.908048i	$-0.362428\pi$
$662$		0			0
$663$	13.1592			0.511059
$664$		0			0
$665$	0.546921	−	8.70911i	0.0212087	−	0.337725i
$666$		0			0
$667$			1.97164i			0.0763421i
$668$		0			0
$669$	34.8670			1.34803
$670$		0			0
$671$	−13.5710			−0.523904
$672$		0			0
$673$	−6.89507			−0.265786			−0.132893	−	0.991130i	$-0.542427\pi$
$673$	−6.89507			−0.265786			−0.132893	+	0.991130i	$0.542427\pi$
$674$		0			0
$675$	−4.63667			−0.178466
$676$		0			0
$677$		−	14.5037i		−	0.557423i	−0.960375	−	0.278712i	$-0.910093\pi$
$677$		−	14.5037i		−	0.557423i	0.960375	−	0.278712i	$-0.0899074\pi$
$678$		0			0
$679$	−1.09976	+	17.5125i	−0.0422049	+	0.672068i
$680$		0			0
$681$	51.6675			1.97990
$682$		0			0
$683$		−	33.7161i		−	1.29011i	−0.764135	−	0.645056i	$-0.776834\pi$
$683$		−	33.7161i		−	1.29011i	0.764135	−	0.645056i	$-0.223166\pi$
$684$		0			0
$685$		−	32.0228i		−	1.22353i
$686$		0			0
$687$		−	52.8655i		−	2.01694i
$688$		0			0
$689$		−	26.5373i		−	1.01099i
$690$		0			0
$691$	3.89578			0.148203			0.0741013	−	0.997251i	$-0.476391\pi$
$691$	3.89578			0.148203			0.0741013	+	0.997251i	$0.476391\pi$
$692$		0			0
$693$	−0.353215	+	5.62456i	−0.0134175	+	0.213659i
$694$		0			0
$695$		−	18.4865i		−	0.701231i
$696$		0			0
$697$	8.26628			0.313108
$698$		0			0
$699$	−19.5578			−0.739745
$700$		0			0
$701$	37.3283			1.40987			0.704935	−	0.709272i	$-0.250976\pi$
$701$	37.3283			1.40987			0.704935	+	0.709272i	$0.250976\pi$
$702$		0			0
$703$	7.69033			0.290046
$704$		0			0
$705$		−	8.37294i		−	0.315343i
$706$		0			0
$707$	1.18250	−	18.8300i	0.0444723	−	0.708173i
$708$		0			0
$709$	−32.3067			−1.21330			−0.606652	−	0.794968i	$-0.707488\pi$
$709$	−32.3067			−1.21330			−0.606652	+	0.794968i	$0.707488\pi$
$710$		0			0
$711$			23.7430i			0.890433i
$712$		0			0
$713$		−	27.5595i		−	1.03211i
$714$		0			0
$715$			9.62987i			0.360137i
$716$		0			0
$717$			13.9941i			0.522621i
$718$		0			0
$719$	48.9527			1.82563			0.912814	−	0.408375i	$-0.133904\pi$
$719$	48.9527			1.82563			0.912814	+	0.408375i	$0.133904\pi$
$720$		0			0
$721$	−36.1832	−	2.27225i	−1.34753	−	0.0846232i
$722$		0			0
$723$		−	2.37627i		−	0.0883744i
$724$		0			0
$725$	−0.780434			−0.0289846
$726$		0			0
$727$	−10.3425			−0.383581			−0.191791	−	0.981436i	$-0.561429\pi$
$727$	−10.3425			−0.383581			−0.191791	+	0.981436i	$0.561429\pi$
$728$		0			0
$729$	−9.72932			−0.360345
$730$		0			0
$731$	−4.72602			−0.174798
$732$		0			0
$733$			13.0315i			0.481329i	0.970608	+	0.240665i	$0.0773654\pi$
$733$			13.0315i			0.481329i	−0.970608	+	0.240665i	$0.922635\pi$
$734$		0			0
$735$	3.22693	−	25.5914i	0.119027	−	0.943952i
$736$		0			0
$737$	8.86844			0.326673
$738$		0			0
$739$		−	2.30328i		−	0.0847276i	−0.999102	−	0.0423638i	$-0.986511\pi$
$739$		−	2.30328i		−	0.0847276i	0.999102	−	0.0423638i	$-0.0134889\pi$
$740$		0			0
$741$			27.1796i			0.998467i
$742$		0			0
$743$		−	0.780434i		−	0.0286313i	−0.999898	−	0.0143157i	$-0.995443\pi$
$743$		−	0.780434i		−	0.0286313i	0.999898	−	0.0143157i	$-0.00455698\pi$
$744$		0			0
$745$		−	25.4281i		−	0.931615i
$746$		0			0
$747$	−1.14929			−0.0420502
$748$		0			0
$749$	2.70178	−	43.0229i	0.0987209	−	1.57202i
$750$		0			0
$751$			21.7280i			0.792866i	0.918064	+	0.396433i	$0.129752\pi$
$751$			21.7280i			0.792866i	−0.918064	+	0.396433i	$0.870248\pi$
$752$		0			0
$753$	7.13007			0.259834
$754$		0			0
$755$	−15.9761			−0.581430
$756$		0			0
$757$	14.5964			0.530517			0.265258	−	0.964177i	$-0.414543\pi$
$757$	14.5964			0.530517			0.265258	+	0.964177i	$0.414543\pi$
$758$		0			0
$759$	−13.4652			−0.488757
$760$		0			0
$761$			33.3780i			1.20995i	0.796244	+	0.604976i	$0.206817\pi$
$761$			33.3780i			1.20995i	−0.796244	+	0.604976i	$0.793183\pi$
$762$		0			0
$763$	44.5855	+	2.79991i	1.61411	+	0.101364i
$764$		0			0
$765$	3.40141			0.122978
$766$		0			0
$767$		−	8.26628i		−	0.298478i
$768$		0			0
$769$			12.5620i			0.452996i	0.974012	+	0.226498i	$0.0727277\pi$
$769$			12.5620i			0.452996i	−0.974012	+	0.226498i	$0.927272\pi$
$770$		0			0
$771$			64.3994i			2.31929i
$772$		0			0
$773$			27.0444i			0.972719i	0.873759	+	0.486359i	$0.161675\pi$
$773$			27.0444i			0.972719i	−0.873759	+	0.486359i	$0.838325\pi$
$774$		0			0
$775$	10.9089			0.391859
$776$		0			0
$777$	22.6872	+	1.42472i	0.813898	+	0.0511117i
$778$		0			0
$779$			17.0736i			0.611725i
$780$		0			0
$781$	5.94501			0.212729
$782$		0			0
$783$	0.653461			0.0233528
$784$		0			0
$785$	22.6561			0.808630
$786$		0			0
$787$	−35.5860			−1.26850			−0.634252	−	0.773126i	$-0.718692\pi$
$787$	−35.5860			−1.26850			−0.634252	+	0.773126i	$0.718692\pi$
$788$		0			0
$789$		−	42.3694i		−	1.50839i
$790$		0			0
$791$	−40.1366	−	2.52052i	−1.42709	−	0.0896195i
$792$		0			0
$793$	−80.3293			−2.85258
$794$		0			0
$795$		−	16.5203i		−	0.585914i
$796$		0			0
$797$			30.5968i			1.08380i	0.840444	+	0.541898i	$0.182294\pi$
$797$			30.5968i			1.08380i	−0.840444	+	0.541898i	$0.817706\pi$
$798$		0			0
$799$			2.23030i			0.0789022i
$800$		0			0
$801$			7.64688i			0.270189i
$802$		0			0
$803$	−1.73270			−0.0611457
$804$		0			0
$805$	25.5391	+	1.60382i	0.900136	+	0.0565273i
$806$		0			0
$807$			33.7039i			1.18643i
$808$		0			0
$809$	29.6015			1.04073			0.520367	−	0.853943i	$-0.325795\pi$
$809$	29.6015			1.04073			0.520367	+	0.853943i	$0.325795\pi$
$810$		0			0
$811$	−39.0325			−1.37062			−0.685308	−	0.728253i	$-0.740333\pi$
$811$	−39.0325			−1.37062			−0.685308	+	0.728253i	$0.740333\pi$
$812$		0			0
$813$	9.69589			0.340050
$814$		0			0
$815$	−16.7459			−0.586583
$816$		0			0
$817$		−	9.76137i		−	0.341507i
$818$		0			0
$819$	−2.09074	+	33.2927i	−0.0730563	+	1.16334i
$820$		0			0
$821$	−40.2932			−1.40624			−0.703120	−	0.711071i	$-0.748211\pi$
$821$	−40.2932			−1.40624			−0.703120	+	0.711071i	$0.748211\pi$
$822$		0			0
$823$			19.5419i			0.681186i	0.940211	+	0.340593i	$0.110628\pi$
$823$			19.5419i			0.681186i	−0.940211	+	0.340593i	$0.889372\pi$
$824$		0			0
$825$		−	5.32995i		−	0.185565i
$826$		0			0
$827$			4.36614i			0.151826i	0.997114	+	0.0759129i	$0.0241871\pi$
$827$			4.36614i			0.151826i	−0.997114	+	0.0759129i	$0.975813\pi$
$828$		0			0
$829$			17.4289i			0.605330i	0.953097	+	0.302665i	$0.0978764\pi$
$829$			17.4289i			0.605330i	−0.953097	+	0.302665i	$0.902124\pi$
$830$		0			0
$831$	57.0505			1.97906
$832$		0			0
$833$	−0.859557	+	6.81676i	−0.0297819	+	0.236187i
$834$		0			0
$835$			9.89466i			0.342419i
$836$		0			0
$837$	−9.13406			−0.315719
$838$		0			0
$839$	37.0931			1.28059			0.640297	−	0.768127i	$-0.278811\pi$
$839$	37.0931			1.28059			0.640297	+	0.768127i	$0.278811\pi$
$840$		0			0
$841$	−28.8900			−0.996207
$842$		0			0
$843$	39.4223			1.35778
$844$		0			0
$845$			35.8512i			1.23332i
$846$		0			0
$847$	2.64055	+	0.165823i	0.0907304	+	0.00569774i
$848$		0			0
$849$	−43.3998			−1.48948
$850$		0			0
$851$			22.5516i			0.773058i
$852$		0			0
$853$			3.51667i			0.120409i	0.998186	+	0.0602043i	$0.0191752\pi$
$853$			3.51667i			0.120409i	−0.998186	+	0.0602043i	$0.980825\pi$
$854$		0			0
$855$			7.02545i			0.240265i
$856$		0			0
$857$		−	23.5251i		−	0.803603i	−0.915727	−	0.401802i	$-0.868384\pi$
$857$		−	23.5251i		−	0.803603i	0.915727	−	0.401802i	$-0.131616\pi$
$858$		0			0
$859$	2.00705			0.0684797			0.0342399	−	0.999414i	$-0.489099\pi$
$859$	2.00705			0.0684797			0.0342399	+	0.999414i	$0.489099\pi$
$860$		0			0
$861$	−3.16309	+	50.3687i	−0.107798	+	1.71656i
$862$		0			0
$863$		−	1.98988i		−	0.0677361i	−0.999426	−	0.0338681i	$-0.989217\pi$
$863$		−	1.98988i		−	0.0677361i	0.999426	−	0.0338681i	$-0.0107826\pi$
$864$		0			0
$865$	13.5914			0.462121
$866$		0			0
$867$	36.3223			1.23357
$868$		0			0
$869$	11.1466			0.378122
$870$		0			0
$871$	52.4938			1.77869
$872$		0			0
$873$		−	14.1269i		−	0.478124i
$874$		0			0
$875$	−1.98372	+	31.5886i	−0.0670621	+	1.06789i
$876$		0			0
$877$	31.2881			1.05652			0.528262	−	0.849081i	$-0.322844\pi$
$877$	31.2881			1.05652			0.528262	+	0.849081i	$0.322844\pi$
$878$		0			0
$879$			26.2917i			0.886796i
$880$		0			0
$881$		−	0.910818i		−	0.0306862i	−0.999882	−	0.0153431i	$-0.995116\pi$
$881$		−	0.910818i		−	0.0306862i	0.999882	−	0.0153431i	$-0.00488406\pi$
$882$		0			0
$883$			7.44373i			0.250501i	0.992125	+	0.125251i	$0.0399735\pi$
$883$			7.44373i			0.250501i	−0.992125	+	0.125251i	$0.960026\pi$
$884$		0			0
$885$		−	5.14601i		−	0.172981i
$886$		0			0
$887$	−37.2665			−1.25129			−0.625643	−	0.780109i	$-0.715163\pi$
$887$	−37.2665			−1.25129			−0.625643	+	0.780109i	$0.715163\pi$
$888$		0			0
$889$	0.546921	−	8.70911i	0.0183431	−	0.292094i
$890$		0			0
$891$			10.8530i			0.363589i
$892$		0			0
$893$	−4.60657			−0.154153
$894$		0			0
$895$	−26.5967			−0.889030
$896$		0			0
$897$	−79.7030			−2.66121
$898$		0			0
$899$	−1.53742			−0.0512760
$900$		0			0
$901$			4.40050i			0.146602i
$902$		0			0
$903$	1.80841	−	28.7969i	0.0601801	−	0.958302i
$904$		0			0
$905$	−27.2001			−0.904162
$906$		0			0
$907$			42.2162i			1.40177i	0.713277	+	0.700883i	$0.247210\pi$
$907$			42.2162i			1.40177i	−0.713277	+	0.700883i	$0.752790\pi$
$908$		0			0
$909$			15.1897i			0.503811i
$910$		0			0
$911$		−	30.5964i		−	1.01371i	−0.862033	−	0.506853i	$-0.830809\pi$
$911$		−	30.5964i		−	1.01371i	0.862033	−	0.506853i	$-0.169191\pi$
$912$		0			0
$913$			0.539553i			0.0178566i
$914$		0			0
$915$	−50.0074			−1.65319
$916$		0			0
$917$	13.7754	+	0.865075i	0.454903	+	0.0285673i
$918$		0			0
$919$			43.7085i			1.44181i	0.693033	+	0.720906i	$0.256274\pi$
$919$			43.7085i			1.44181i	−0.693033	+	0.720906i	$0.743726\pi$
$920$		0			0
$921$	28.3693			0.934801
$922$		0			0
$923$	35.1895			1.15828
$924$		0			0
$925$	−8.92660			−0.293505
$926$		0			0
$927$	29.1882			0.958665
$928$		0			0
$929$		−	39.4412i		−	1.29402i	−0.762480	−	0.647011i	$-0.776019\pi$
$929$		−	39.4412i		−	1.29402i	0.762480	−	0.647011i	$-0.223981\pi$
$930$		0			0
$931$	−14.0797	−	1.77537i	−0.461443	−	0.0581855i
$932$		0			0
$933$	58.3623			1.91070
$934$		0			0
$935$		−	1.59685i		−	0.0522227i
$936$		0			0
$937$			0.0299645i			0.000978896i	1.00000		0.000489448i	$0.000155796\pi$
$937$			0.0299645i			0.000978896i	−1.00000		0.000489448i	$0.999844\pi$
$938$		0			0
$939$			58.3553i			1.90435i
$940$		0			0
$941$			34.4960i			1.12454i	0.826955	+	0.562269i	$0.190071\pi$
$941$			34.4960i			1.12454i	−0.826955	+	0.562269i	$0.809929\pi$
$942$		0			0
$943$	−50.0676			−1.63043
$944$		0			0
$945$	0.531556	−	8.46445i	0.0172915	−	0.275348i
$946$		0			0
$947$		−	9.24870i		−	0.300542i	−0.988645	−	0.150271i	$-0.951985\pi$
$947$		−	9.24870i		−	0.300542i	0.988645	−	0.150271i	$-0.0480147\pi$
$948$		0			0
$949$	−10.2562			−0.332929
$950$		0			0
$951$	−17.8332			−0.578282
$952$		0			0
$953$	45.7775			1.48288			0.741440	−	0.671020i	$-0.234143\pi$
$953$	45.7775			1.48288			0.741440	+	0.671020i	$0.234143\pi$
$954$		0			0
$955$	40.1053			1.29778
$956$		0			0
$957$			0.751167i			0.0242818i
$958$		0			0
$959$	−51.9749	−	3.26396i	−1.67836	−	0.105399i
$960$		0			0
$961$	−9.50992			−0.306772
$962$		0			0
$963$			34.7056i			1.11837i
$964$		0			0
$965$		−	8.12624i		−	0.261593i
$966$		0			0
$967$		−	53.5285i		−	1.72136i	−0.509145	−	0.860681i	$-0.670038\pi$
$967$		−	53.5285i		−	1.72136i	0.509145	−	0.860681i	$-0.329962\pi$
$968$		0			0
$969$		−	4.50700i		−	0.144786i
$970$		0			0
$971$	26.6432			0.855019			0.427510	−	0.904011i	$-0.359391\pi$
$971$	26.6432			0.855019			0.427510	+	0.904011i	$0.359391\pi$
$972$		0			0
$973$	−30.0047	−	1.88425i	−0.961905	−	0.0604063i
$974$		0			0
$975$		−	31.5489i		−	1.01037i
$976$		0			0
$977$	−45.7965			−1.46516			−0.732580	−	0.680681i	$-0.761684\pi$
$977$	−45.7965			−1.46516			−0.732580	+	0.680681i	$0.761684\pi$
$978$		0			0
$979$	3.58996			0.114736
$980$		0			0
$981$	−35.9662			−1.14831
$982$		0			0
$983$	−38.2938			−1.22138			−0.610691	−	0.791869i	$-0.709108\pi$
$983$	−38.2938			−1.22138			−0.610691	+	0.791869i	$0.709108\pi$
$984$		0			0
$985$			41.0949i			1.30939i
$986$		0			0
$987$	−13.5898	−	0.853421i	−0.432568	−	0.0271647i
$988$		0			0
$989$	28.6248			0.910216
$990$		0			0
$991$			37.3266i			1.18572i	0.805306	+	0.592859i	$0.202001\pi$
$991$			37.3266i			1.18572i	−0.805306	+	0.592859i	$0.797999\pi$
$992$		0			0
$993$			50.2710i			1.59530i
$994$		0			0
$995$			27.5867i			0.874558i
$996$		0			0
$997$			20.9461i			0.663370i	0.943390	+	0.331685i	$0.107617\pi$
$997$			20.9461i			0.663370i	−0.943390	+	0.331685i	$0.892383\pi$
$998$		0			0
$999$	7.47428			0.236476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.j.a.111.13	yes	16
4.3	odd	2	inner	1232.2.j.a.111.3	✓	16
7.6	odd	2	inner	1232.2.j.a.111.4	yes	16
28.27	even	2	inner	1232.2.j.a.111.14	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1232.2.j.a.111.3	✓	16	4.3	odd	2	inner
1232.2.j.a.111.4	yes	16	7.6	odd	2	inner
1232.2.j.a.111.13	yes	16	1.1	even	1	trivial
1232.2.j.a.111.14	yes	16	28.27	even	2	inner

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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.26497 q^{3} -1.62689i q^{5} +(-2.64055 - 0.165823i) q^{7} +2.13007 q^{9} +O(q^{10})\)	\(q+2.26497 q^{3} -1.62689i q^{5} +(-2.64055 - 0.165823i) q^{7} +2.13007 q^{9} -1.00000i q^{11} -5.91917i q^{13} -3.68486i q^{15} +0.981534i q^{17} -2.02731 q^{19} +(-5.98076 - 0.375583i) q^{21} -5.94501i q^{23} +2.35321 q^{25} -1.97036 q^{27} -0.331646 q^{29} +4.63574 q^{31} -2.26497i q^{33} +(-0.269776 + 4.29590i) q^{35} -3.79336 q^{37} -13.4067i q^{39} -8.42179i q^{41} +4.81493i q^{43} -3.46540i q^{45} +2.27225 q^{47} +(6.94501 + 0.875728i) q^{49} +2.22314i q^{51} +4.48329 q^{53} -1.62689 q^{55} -4.59179 q^{57} +1.39653 q^{59} -13.5710i q^{61} +(-5.62456 - 0.353215i) q^{63} -9.62987 q^{65} +8.86844i q^{67} -13.4652i q^{69} +5.94501i q^{71} -1.73270i q^{73} +5.32995 q^{75} +(-0.165823 + 2.64055i) q^{77} +11.1466i q^{79} -10.8530 q^{81} -0.539553 q^{83} +1.59685 q^{85} -0.751167 q^{87} +3.58996i q^{89} +(-0.981534 + 15.6299i) q^{91} +10.4998 q^{93} +3.29822i q^{95} -6.63214i q^{97} -2.13007i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{9}+O(q^{10}) \) 16 * q + 24 * q^9	\( 16 q + 24 q^{9} + 12 q^{21} - 40 q^{25} + 8 q^{29} - 24 q^{37} - 16 q^{53} - 40 q^{57} + 48 q^{65} + 4 q^{77} + 96 q^{81} + 32 q^{85} - 24 q^{93}+O(q^{100}) \) 16 * q + 24 * q^9 + 12 * q^21 - 40 * q^25 + 8 * q^29 - 24 * q^37 - 16 * q^53 - 40 * q^57 + 48 * q^65 + 4 * q^77 + 96 * q^81 + 32 * q^85 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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