LMFDB - Embedded newform 1428.2.d.c.169.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1428,2,Mod(169,1428)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1428, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1428.169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1428.d (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:	$11.4026374086$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 43x^{10} + 647x^{8} + 4049x^{6} + 10288x^{4} + 9088x^{2} + 2304 $ x^12 + 43x^10 + 647x^8 + 4049x^6 + 10288x^4 + 9088*x^2 + 2304
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.4
Root			$0.981755i$ of defining polynomial
Character	$\chi$	$=$	1428.169
Dual form			1428.2.d.c.169.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +0.981755i q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +0.981755i q^{5} +1.00000i q^{7} -1.00000 q^{9} +0.478569i q^{11} -6.19199 q^{13} +0.981755 q^{15} +(2.75736 - 3.06544i) q^{17} +6.75943 q^{19} +1.00000 q^{21} +6.21975i q^{23} +4.03616 q^{25} +1.00000i q^{27} +1.43256i q^{29} +2.44411i q^{31} +0.478569 q^{33} -0.981755 q^{35} +10.1488i q^{37} +6.19199i q^{39} +7.68008i q^{41} -0.741517 q^{43} -0.981755i q^{45} +2.44411 q^{47} -1.00000 q^{49} +(-3.06544 - 2.75736i) q^{51} +13.2282 q^{53} -0.469838 q^{55} -6.75943i q^{57} -7.45048 q^{59} +0.680410i q^{61} -1.00000i q^{63} -6.07902i q^{65} -15.0823 q^{67} +6.21975 q^{69} -3.73705i q^{71} +7.17375i q^{73} -4.03616i q^{75} -0.478569 q^{77} +13.2646i q^{79} +1.00000 q^{81} +9.34113 q^{83} +(3.00951 + 2.70705i) q^{85} +1.43256 q^{87} +17.7765 q^{89} -6.19199i q^{91} +2.44411 q^{93} +6.63610i q^{95} -3.24673i q^{97} -0.478569i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 26 q^{25} + 10 q^{33} - 2 q^{35} - 18 q^{43} - 20 q^{47} - 12 q^{49} - 2 q^{51} + 16 q^{53} + 22 q^{55} - 12 q^{59} + 32 q^{67} - 6 q^{69} - 10 q^{77} + 12 q^{81} - 16 q^{83} + 14 q^{85} + 24 q^{87} + 4 q^{89} - 20 q^{93}+O(q^{100}) $ 12 * q - 12 * q^9 - 6 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 26 * q^25 + 10 * q^33 - 2 * q^35 - 18 * q^43 - 20 * q^47 - 12 * q^49 - 2 * q^51 + 16 * q^53 + 22 * q^55 - 12 * q^59 + 32 * q^67 - 6 * q^69 - 10 * q^77 + 12 * q^81 - 16 * q^83 + 14 * q^85 + 24 * q^87 + 4 * q^89 - 20 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times$.

$n$	$409$	$715$	$953$	$1261$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$			0.981755i			0.439054i	0.975606	+	0.219527i	$0.0704514\pi$
$5$			0.981755i			0.439054i	−0.975606	+	0.219527i	$0.929549\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			0.478569i			0.144294i	0.997394	+	0.0721470i	$0.0229851\pi$
$11$			0.478569i			0.144294i	−0.997394	+	0.0721470i	$0.977015\pi$
$12$		0			0
$13$	−6.19199			−1.71735			−0.858675	−	0.512520i	$-0.828712\pi$
$13$	−6.19199			−1.71735			−0.858675	+	0.512520i	$0.828712\pi$
$14$		0			0
$15$	0.981755			0.253488
$16$		0			0
$17$	2.75736	−	3.06544i	0.668759	−	0.743479i
$17$	2.75736	−	3.06544i	0.668759	−	0.743479i
$18$		0			0
$19$	6.75943			1.55072			0.775360	−	0.631520i	$-0.217569\pi$
$19$	6.75943			1.55072			0.775360	+	0.631520i	$0.217569\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$			6.21975i			1.29691i	0.761254	+	0.648454i	$0.224584\pi$
$23$			6.21975i			1.29691i	−0.761254	+	0.648454i	$0.775416\pi$
$24$		0			0
$25$	4.03616			0.807232
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$			1.43256i			0.266021i	0.991115	+	0.133010i	$0.0424643\pi$
$29$			1.43256i			0.266021i	−0.991115	+	0.133010i	$0.957536\pi$
$30$		0			0
$31$			2.44411i			0.438975i	0.975615	+	0.219487i	$0.0704384\pi$
$31$			2.44411i			0.438975i	−0.975615	+	0.219487i	$0.929562\pi$
$32$		0			0
$33$	0.478569			0.0833082
$34$		0			0
$35$	−0.981755			−0.165947
$36$		0			0
$37$			10.1488i			1.66845i	0.551422	+	0.834226i	$0.314085\pi$
$37$			10.1488i			1.66845i	−0.551422	+	0.834226i	$0.685915\pi$
$38$		0			0
$39$			6.19199i			0.991513i
$40$		0			0
$41$			7.68008i			1.19943i	0.800215	+	0.599713i	$0.204719\pi$
$41$			7.68008i			1.19943i	−0.800215	+	0.599713i	$0.795281\pi$
$42$		0			0
$43$	−0.741517			−0.113080			−0.0565401	−	0.998400i	$-0.518007\pi$
$43$	−0.741517			−0.113080			−0.0565401	+	0.998400i	$0.518007\pi$
$44$		0			0
$45$		−	0.981755i		−	0.146351i
$46$		0			0
$47$	2.44411			0.356510			0.178255	−	0.983984i	$-0.442955\pi$
$47$	2.44411			0.356510			0.178255	+	0.983984i	$0.442955\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.06544	−	2.75736i	−0.429248	−	0.386108i
$52$		0			0
$53$	13.2282			1.81703			0.908513	−	0.417856i	$-0.137218\pi$
$53$	13.2282			1.81703			0.908513	+	0.417856i	$0.137218\pi$
$54$		0			0
$55$	−0.469838			−0.0633529
$56$		0			0
$57$		−	6.75943i		−	0.895308i
$58$		0			0
$59$	−7.45048			−0.969970			−0.484985	−	0.874523i	$-0.661175\pi$
$59$	−7.45048			−0.969970			−0.484985	+	0.874523i	$0.661175\pi$
$60$		0			0
$61$			0.680410i			0.0871176i	0.999051	+	0.0435588i	$0.0138696\pi$
$61$			0.680410i			0.0871176i	−0.999051	+	0.0435588i	$0.986130\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$		−	6.07902i		−	0.754010i
$66$		0			0
$67$	−15.0823			−1.84260			−0.921299	−	0.388855i	$-0.872871\pi$
$67$	−15.0823			−1.84260			−0.921299	+	0.388855i	$0.872871\pi$
$68$		0			0
$69$	6.21975			0.748770
$70$		0			0
$71$		−	3.73705i		−	0.443507i	−0.975103	−	0.221753i	$-0.928822\pi$
$71$		−	3.73705i		−	0.443507i	0.975103	−	0.221753i	$-0.0711780\pi$
$72$		0			0
$73$			7.17375i			0.839624i	0.907611	+	0.419812i	$0.137904\pi$
$73$			7.17375i			0.839624i	−0.907611	+	0.419812i	$0.862096\pi$
$74$		0			0
$75$		−	4.03616i		−	0.466055i
$76$		0			0
$77$	−0.478569			−0.0545380
$78$		0			0
$79$			13.2646i			1.49239i	0.665728	+	0.746194i	$0.268121\pi$
$79$			13.2646i			1.49239i	−0.665728	+	0.746194i	$0.731879\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	9.34113			1.02532			0.512661	−	0.858591i	$-0.328660\pi$
$83$	9.34113			1.02532			0.512661	+	0.858591i	$0.328660\pi$
$84$		0			0
$85$	3.00951	+	2.70705i	0.326428	+	0.293621i
$86$		0			0
$87$	1.43256			0.153587
$88$		0			0
$89$	17.7765			1.88431			0.942153	−	0.335184i	$-0.108798\pi$
$89$	17.7765			1.88431			0.942153	+	0.335184i	$0.108798\pi$
$90$		0			0
$91$		−	6.19199i		−	0.649097i
$92$		0			0
$93$	2.44411			0.253442
$94$		0			0
$95$			6.63610i			0.680850i
$96$		0			0
$97$		−	3.24673i		−	0.329656i	−0.986322	−	0.164828i	$-0.947293\pi$
$97$		−	3.24673i		−	0.329656i	0.986322	−	0.164828i	$-0.0527068\pi$
$98$		0			0
$99$		−	0.478569i		−	0.0480980i
$100$		0			0
$101$	5.99127			0.596153			0.298077	−	0.954542i	$-0.403655\pi$
$101$	5.99127			0.596153			0.298077	+	0.954542i	$0.403655\pi$
$102$		0			0
$103$	−1.01825			−0.100331			−0.0501653	−	0.998741i	$-0.515975\pi$
$103$	−1.01825			−0.100331			−0.0501653	+	0.998741i	$0.515975\pi$
$104$		0			0
$105$			0.981755i			0.0958094i
$106$		0			0
$107$		−	16.6402i		−	1.60867i	−0.594175	−	0.804336i	$-0.702521\pi$
$107$		−	16.6402i		−	1.60867i	0.594175	−	0.804336i	$-0.297479\pi$
$108$		0			0
$109$			17.0694i			1.63496i	0.575959	+	0.817478i	$0.304629\pi$
$109$			17.0694i			1.63496i	−0.575959	+	0.817478i	$0.695371\pi$
$110$		0			0
$111$	10.1488			0.963281
$112$		0			0
$113$			1.21694i			0.114480i	0.998360	+	0.0572402i	$0.0182301\pi$
$113$			1.21694i			0.114480i	−0.998360	+	0.0572402i	$0.981770\pi$
$114$		0			0
$115$	−6.10627			−0.569413
$116$		0			0
$117$	6.19199			0.572450
$118$		0			0
$119$	3.06544	+	2.75736i	0.281009	+	0.252767i
$120$		0			0
$121$	10.7710			0.979179
$122$		0			0
$123$	7.68008			0.692489
$124$		0			0
$125$			8.87129i			0.793472i
$126$		0			0
$127$	−10.8584			−0.963530			−0.481765	−	0.876301i	$-0.660004\pi$
$127$	−10.8584			−0.963530			−0.481765	+	0.876301i	$0.660004\pi$
$128$		0			0
$129$			0.741517i			0.0652869i
$130$		0			0
$131$			8.11901i			0.709361i	0.934987	+	0.354681i	$0.115410\pi$
$131$			8.11901i			0.709361i	−0.934987	+	0.354681i	$0.884590\pi$
$132$		0			0
$133$			6.75943i			0.586117i
$134$		0			0
$135$	−0.981755			−0.0844960
$136$		0			0
$137$	−6.19870			−0.529590			−0.264795	−	0.964305i	$-0.585304\pi$
$137$	−6.19870			−0.529590			−0.264795	+	0.964305i	$0.585304\pi$
$138$		0			0
$139$		−	14.2983i		−	1.21276i	−0.795174	−	0.606382i	$-0.792620\pi$
$139$		−	14.2983i		−	1.21276i	0.795174	−	0.606382i	$-0.207380\pi$
$140$		0			0
$141$		−	2.44411i		−	0.205831i
$142$		0			0
$143$		−	2.96330i		−	0.247803i
$144$		0			0
$145$	−1.40643			−0.116797
$146$		0			0
$147$			1.00000i			0.0824786i
$148$		0			0
$149$	−6.38399			−0.522997			−0.261498	−	0.965204i	$-0.584217\pi$
$149$	−6.38399			−0.522997			−0.261498	+	0.965204i	$0.584217\pi$
$150$		0			0
$151$	−7.52696			−0.612535			−0.306268	−	0.951945i	$-0.599080\pi$
$151$	−7.52696			−0.612535			−0.306268	+	0.951945i	$0.599080\pi$
$152$		0			0
$153$	−2.75736	+	3.06544i	−0.222920	+	0.247826i
$154$		0			0
$155$	−2.39951			−0.192734
$156$		0			0
$157$	−7.81097			−0.623383			−0.311691	−	0.950183i	$-0.600896\pi$
$157$	−7.81097			−0.623383			−0.311691	+	0.950183i	$0.600896\pi$
$158$		0			0
$159$		−	13.2282i		−	1.04906i
$160$		0			0
$161$	−6.21975			−0.490185
$162$		0			0
$163$		−	0.275182i		−	0.0215539i	−0.999942	−	0.0107770i	$-0.996570\pi$
$163$		−	0.275182i		−	0.0215539i	0.999942	−	0.0107770i	$-0.00343048\pi$
$164$		0			0
$165$			0.469838i			0.0365768i
$166$		0			0
$167$		−	21.7913i		−	1.68626i	−0.537710	−	0.843130i	$-0.680711\pi$
$167$		−	21.7913i		−	1.68626i	0.537710	−	0.843130i	$-0.319289\pi$
$168$		0			0
$169$	25.3408			1.94929
$170$		0			0
$171$	−6.75943			−0.516906
$172$		0			0
$173$		−	15.1271i		−	1.15009i	−0.818122	−	0.575044i	$-0.804985\pi$
$173$		−	15.1271i		−	1.15009i	0.818122	−	0.575044i	$-0.195015\pi$
$174$		0			0
$175$			4.03616i			0.305105i
$176$		0			0
$177$			7.45048i			0.560012i
$178$		0			0
$179$	−9.93107			−0.742283			−0.371142	−	0.928576i	$-0.621034\pi$
$179$	−9.93107			−0.742283			−0.371142	+	0.928576i	$0.621034\pi$
$180$		0			0
$181$		−	15.5577i		−	1.15640i	−0.815896	−	0.578199i	$-0.803756\pi$
$181$		−	15.5577i		−	1.15640i	0.815896	−	0.578199i	$-0.196244\pi$
$182$		0			0
$183$	0.680410			0.0502974
$184$		0			0
$185$	−9.96363			−0.732541
$186$		0			0
$187$	1.46703	+	1.31959i	0.107280	+	0.0964980i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	20.5252			1.48515			0.742577	−	0.669761i	$-0.233603\pi$
$191$	20.5252			1.48515			0.742577	+	0.669761i	$0.233603\pi$
$192$		0			0
$193$		−	7.29708i		−	0.525255i	−0.964897	−	0.262628i	$-0.915411\pi$
$193$		−	7.29708i		−	0.525255i	0.964897	−	0.262628i	$-0.0845891\pi$
$194$		0			0
$195$	−6.07902			−0.435328
$196$		0			0
$197$		−	25.5835i		−	1.82275i	−0.411581	−	0.911373i	$-0.635023\pi$
$197$		−	25.5835i		−	1.82275i	0.411581	−	0.911373i	$-0.364977\pi$
$198$		0			0
$199$			15.0613i			1.06767i	0.845589	+	0.533835i	$0.179250\pi$
$199$			15.0613i			1.06767i	−0.845589	+	0.533835i	$0.820750\pi$
$200$		0			0
$201$			15.0823i			1.06382i
$202$		0			0
$203$	−1.43256			−0.100546
$204$		0			0
$205$	−7.53995			−0.526613
$206$		0			0
$207$		−	6.21975i		−	0.432303i
$208$		0			0
$209$			3.23486i			0.223760i
$210$		0			0
$211$			11.6728i			0.803589i	0.915730	+	0.401794i	$0.131613\pi$
$211$			11.6728i			0.803589i	−0.915730	+	0.401794i	$0.868387\pi$
$212$		0			0
$213$	−3.73705			−0.256059
$214$		0			0
$215$		−	0.727988i		−	0.0496484i
$216$		0			0
$217$	−2.44411			−0.165917
$218$		0			0
$219$	7.17375			0.484757
$220$		0			0
$221$	−17.0736	+	18.9812i	−1.14849	+	1.27681i
$222$		0			0
$223$	−18.3952			−1.23183			−0.615917	−	0.787811i	$-0.711214\pi$
$223$	−18.3952			−1.23183			−0.615917	+	0.787811i	$0.711214\pi$
$224$		0			0
$225$	−4.03616			−0.269077
$226$		0			0
$227$			13.1538i			0.873046i	0.899693	+	0.436523i	$0.143790\pi$
$227$			13.1538i			0.873046i	−0.899693	+	0.436523i	$0.856210\pi$
$228$		0			0
$229$	0.569823			0.0376550			0.0188275	−	0.999823i	$-0.494007\pi$
$229$	0.569823			0.0376550			0.0188275	+	0.999823i	$0.494007\pi$
$230$		0			0
$231$			0.478569i			0.0314876i
$232$		0			0
$233$			17.8788i			1.17128i	0.810572	+	0.585639i	$0.199156\pi$
$233$			17.8788i			1.17128i	−0.810572	+	0.585639i	$0.800844\pi$
$234$		0			0
$235$			2.39951i			0.156527i
$236$		0			0
$237$	13.2646			0.861631
$238$		0			0
$239$	−8.89968			−0.575673			−0.287836	−	0.957680i	$-0.592936\pi$
$239$	−8.89968			−0.575673			−0.287836	+	0.957680i	$0.592936\pi$
$240$		0			0
$241$		−	10.2721i		−	0.661686i	−0.943686	−	0.330843i	$-0.892667\pi$
$241$		−	10.2721i		−	0.661686i	0.943686	−	0.330843i	$-0.107333\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	0.981755i		−	0.0627220i
$246$		0			0
$247$	−41.8543			−2.66313
$248$		0			0
$249$		−	9.34113i		−	0.591970i
$250$		0			0
$251$	17.1953			1.08536			0.542678	−	0.839941i	$-0.317410\pi$
$251$	17.1953			1.08536			0.542678	+	0.839941i	$0.317410\pi$
$252$		0			0
$253$	−2.97658			−0.187136
$254$		0			0
$255$	2.70705	−	3.00951i	0.169522	−	0.188463i
$256$		0			0
$257$	8.48067			0.529010			0.264505	−	0.964384i	$-0.414791\pi$
$257$	8.48067			0.529010			0.264505	+	0.964384i	$0.414791\pi$
$258$		0			0
$259$	−10.1488			−0.630616
$260$		0			0
$261$		−	1.43256i		−	0.0886735i
$262$		0			0
$263$	5.85809			0.361226			0.180613	−	0.983554i	$-0.442192\pi$
$263$	5.85809			0.361226			0.180613	+	0.983554i	$0.442192\pi$
$264$		0			0
$265$			12.9868i			0.797773i
$266$		0			0
$267$		−	17.7765i		−	1.08790i
$268$		0			0
$269$		−	21.5471i		−	1.31375i	−0.754000	−	0.656875i	$-0.771878\pi$
$269$		−	21.5471i		−	1.31375i	0.754000	−	0.656875i	$-0.228122\pi$
$270$		0			0
$271$	0.161218			0.00979332			0.00489666	−	0.999988i	$-0.498441\pi$
$271$	0.161218			0.00979332			0.00489666	+	0.999988i	$0.498441\pi$
$272$		0			0
$273$	−6.19199			−0.374757
$274$		0			0
$275$			1.93158i			0.116479i
$276$		0			0
$277$			32.4753i			1.95125i	0.219436	+	0.975627i	$0.429578\pi$
$277$			32.4753i			1.95125i	−0.219436	+	0.975627i	$0.570422\pi$
$278$		0			0
$279$		−	2.44411i		−	0.146325i
$280$		0			0
$281$	3.34113			0.199315			0.0996575	−	0.995022i	$-0.468225\pi$
$281$	3.34113			0.199315			0.0996575	+	0.995022i	$0.468225\pi$
$282$		0			0
$283$			5.61547i			0.333805i	0.985973	+	0.166902i	$0.0533765\pi$
$283$			5.61547i			0.333805i	−0.985973	+	0.166902i	$0.946624\pi$
$284$		0			0
$285$	6.63610			0.393089
$286$		0			0
$287$	−7.68008			−0.453341
$288$		0			0
$289$	−1.79389	−	16.9051i	−0.105523	−	0.994417i
$290$		0			0
$291$	−3.24673			−0.190327
$292$		0			0
$293$	−20.3747			−1.19030			−0.595152	−	0.803613i	$-0.702908\pi$
$293$	−20.3747			−1.19030			−0.595152	+	0.803613i	$0.702908\pi$
$294$		0			0
$295$		−	7.31454i		−	0.425869i
$296$		0			0
$297$	−0.478569			−0.0277694
$298$		0			0
$299$		−	38.5127i		−	2.22725i
$300$		0			0
$301$		−	0.741517i		−	0.0427403i
$302$		0			0
$303$		−	5.99127i		−	0.344189i
$304$		0			0
$305$	−0.667996			−0.0382493
$306$		0			0
$307$	−17.7637			−1.01383			−0.506914	−	0.861997i	$-0.669214\pi$
$307$	−17.7637			−1.01383			−0.506914	+	0.861997i	$0.669214\pi$
$308$		0			0
$309$			1.01825i			0.0579260i
$310$		0			0
$311$		−	22.9822i		−	1.30320i	−0.758562	−	0.651600i	$-0.774098\pi$
$311$		−	22.9822i		−	1.30320i	0.758562	−	0.651600i	$-0.225902\pi$
$312$		0			0
$313$		−	17.6044i		−	0.995062i	−0.867446	−	0.497531i	$-0.834240\pi$
$313$		−	17.6044i		−	0.995062i	0.867446	−	0.497531i	$-0.165760\pi$
$314$		0			0
$315$	0.981755			0.0553156
$316$		0			0
$317$			28.6237i			1.60767i	0.594855	+	0.803833i	$0.297210\pi$
$317$			28.6237i			1.60767i	−0.594855	+	0.803833i	$0.702790\pi$
$318$		0			0
$319$	−0.685582			−0.0383852
$320$		0			0
$321$	−16.6402			−0.928767
$322$		0			0
$323$	18.6382	−	20.7207i	1.03706	−	1.15293i
$324$		0			0
$325$	−24.9919			−1.38630
$326$		0			0
$327$	17.0694			0.943943
$328$		0			0
$329$			2.44411i			0.134748i
$330$		0			0
$331$	−31.8104			−1.74846			−0.874229	−	0.485513i	$-0.838633\pi$
$331$	−31.8104			−1.74846			−0.874229	+	0.485513i	$0.838633\pi$
$332$		0			0
$333$		−	10.1488i		−	0.556151i
$334$		0			0
$335$		−	14.8071i		−	0.809000i
$336$		0			0
$337$			8.63820i			0.470553i	0.971928	+	0.235276i	$0.0755996\pi$
$337$			8.63820i			0.470553i	−0.971928	+	0.235276i	$0.924400\pi$
$338$		0			0
$339$	1.21694			0.0660953
$340$		0			0
$341$	−1.16967			−0.0633414
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			6.10627i			0.328751i
$346$		0			0
$347$		−	11.9988i		−	0.644131i	−0.946717	−	0.322065i	$-0.895623\pi$
$347$		−	11.9988i		−	0.644131i	0.946717	−	0.322065i	$-0.104377\pi$
$348$		0			0
$349$	−6.48730			−0.347257			−0.173629	−	0.984811i	$-0.555549\pi$
$349$	−6.48730			−0.347257			−0.173629	+	0.984811i	$0.555549\pi$
$350$		0			0
$351$		−	6.19199i		−	0.330504i
$352$		0			0
$353$	18.9741			1.00989			0.504946	−	0.863151i	$-0.331513\pi$
$353$	18.9741			1.00989			0.504946	+	0.863151i	$0.331513\pi$
$354$		0			0
$355$	3.66887			0.194723
$356$		0			0
$357$	2.75736	−	3.06544i	0.145935	−	0.162240i
$358$		0			0
$359$	−32.1374			−1.69615			−0.848073	−	0.529880i	$-0.822237\pi$
$359$	−32.1374			−1.69615			−0.848073	+	0.529880i	$0.822237\pi$
$360$		0			0
$361$	26.6899			1.40473
$362$		0			0
$363$		−	10.7710i		−	0.565329i
$364$		0			0
$365$	−7.04286			−0.368640
$366$		0			0
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		0			0
$369$		−	7.68008i		−	0.399809i
$370$		0			0
$371$			13.2282i			0.686771i
$372$		0			0
$373$	14.7800			0.765279			0.382639	−	0.923898i	$-0.375015\pi$
$373$	14.7800			0.765279			0.382639	+	0.923898i	$0.375015\pi$
$374$		0			0
$375$	8.87129			0.458111
$376$		0			0
$377$		−	8.87043i		−	0.456851i
$378$		0			0
$379$			9.33707i			0.479613i	0.970821	+	0.239807i	$0.0770840\pi$
$379$			9.33707i			0.479613i	−0.970821	+	0.239807i	$0.922916\pi$
$380$		0			0
$381$			10.8584i			0.556294i
$382$		0			0
$383$	26.7417			1.36644			0.683219	−	0.730214i	$-0.260580\pi$
$383$	26.7417			1.36644			0.683219	+	0.730214i	$0.260580\pi$
$384$		0			0
$385$		−	0.469838i		−	0.0239451i
$386$		0			0
$387$	0.741517			0.0376934
$388$		0			0
$389$	−25.8581			−1.31106			−0.655529	−	0.755170i	$-0.727554\pi$
$389$	−25.8581			−1.31106			−0.655529	+	0.755170i	$0.727554\pi$
$390$		0			0
$391$	19.0663	+	17.1501i	0.964224	+	0.867319i
$392$		0			0
$393$	8.11901			0.409550
$394$		0			0
$395$	−13.0226			−0.655239
$396$		0			0
$397$		−	6.01327i		−	0.301798i	−0.988549	−	0.150899i	$-0.951783\pi$
$397$		−	6.01327i		−	0.301798i	0.988549	−	0.150899i	$-0.0482168\pi$
$398$		0			0
$399$	6.75943			0.338395
$400$		0			0
$401$			6.26543i			0.312880i	0.987687	+	0.156440i	$0.0500018\pi$
$401$			6.26543i			0.312880i	−0.987687	+	0.156440i	$0.949998\pi$
$402$		0			0
$403$		−	15.1339i		−	0.753873i
$404$		0			0
$405$			0.981755i			0.0487838i
$406$		0			0
$407$	−4.85690			−0.240748
$408$		0			0
$409$	−10.3872			−0.513613			−0.256807	−	0.966463i	$-0.582670\pi$
$409$	−10.3872			−0.513613			−0.256807	+	0.966463i	$0.582670\pi$
$410$		0			0
$411$			6.19870i			0.305759i
$412$		0			0
$413$		−	7.45048i		−	0.366614i
$414$		0			0
$415$			9.17070i			0.450172i
$416$		0			0
$417$	−14.2983			−0.700189
$418$		0			0
$419$		−	31.3933i		−	1.53366i	−0.641848	−	0.766832i	$-0.721832\pi$
$419$		−	31.3933i		−	1.53366i	0.641848	−	0.766832i	$-0.278168\pi$
$420$		0			0
$421$	33.1560			1.61593			0.807963	−	0.589233i	$-0.200570\pi$
$421$	33.1560			1.61593			0.807963	+	0.589233i	$0.200570\pi$
$422$		0			0
$423$	−2.44411			−0.118837
$424$		0			0
$425$	11.1292	−	12.3726i	0.539843	−	0.600160i
$426$		0			0
$427$	−0.680410			−0.0329274
$428$		0			0
$429$	−2.96330			−0.143069
$430$		0			0
$431$			15.8257i			0.762299i	0.924513	+	0.381149i	$0.124472\pi$
$431$			15.8257i			0.762299i	−0.924513	+	0.381149i	$0.875528\pi$
$432$		0			0
$433$	7.87360			0.378381			0.189191	−	0.981940i	$-0.439414\pi$
$433$	7.87360			0.378381			0.189191	+	0.981940i	$0.439414\pi$
$434$		0			0
$435$			1.40643i			0.0674330i
$436$		0			0
$437$			42.0420i			2.01114i
$438$		0			0
$439$		−	40.3850i		−	1.92747i	−0.266859	−	0.963736i	$-0.585986\pi$
$439$		−	40.3850i		−	1.92747i	0.266859	−	0.963736i	$-0.414014\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$	−26.2583			−1.24757			−0.623784	−	0.781597i	$-0.714406\pi$
$443$	−26.2583			−1.24757			−0.623784	+	0.781597i	$0.714406\pi$
$444$		0			0
$445$			17.4522i			0.827312i
$446$		0			0
$447$			6.38399i			0.301952i
$448$		0			0
$449$			16.0822i			0.758968i	0.925198	+	0.379484i	$0.123898\pi$
$449$			16.0822i			0.758968i	−0.925198	+	0.379484i	$0.876102\pi$
$450$		0			0
$451$	−3.67545			−0.173070
$452$		0			0
$453$			7.52696i			0.353648i
$454$		0			0
$455$	6.07902			0.284989
$456$		0			0
$457$	−22.6042			−1.05738			−0.528691	−	0.848814i	$-0.677317\pi$
$457$	−22.6042			−1.05738			−0.528691	+	0.848814i	$0.677317\pi$
$458$		0			0
$459$	3.06544	+	2.75736i	0.143083	+	0.128703i
$460$		0			0
$461$	25.9582			1.20899			0.604496	−	0.796608i	$-0.293374\pi$
$461$	25.9582			1.20899			0.604496	+	0.796608i	$0.293374\pi$
$462$		0			0
$463$	9.36125			0.435054			0.217527	−	0.976054i	$-0.430201\pi$
$463$	9.36125			0.435054			0.217527	+	0.976054i	$0.430201\pi$
$464$		0			0
$465$			2.39951i			0.111275i
$466$		0			0
$467$	−10.7677			−0.498269			−0.249134	−	0.968469i	$-0.580146\pi$
$467$	−10.7677			−0.498269			−0.249134	+	0.968469i	$0.580146\pi$
$468$		0			0
$469$		−	15.0823i		−	0.696436i
$470$		0			0
$471$			7.81097i			0.359910i
$472$		0			0
$473$		−	0.354867i		−	0.0163168i
$474$		0			0
$475$	27.2821			1.25179
$476$		0			0
$477$	−13.2282			−0.605675
$478$		0			0
$479$		−	19.3883i		−	0.885872i	−0.896553	−	0.442936i	$-0.853937\pi$
$479$		−	19.3883i		−	0.885872i	0.896553	−	0.442936i	$-0.146063\pi$
$480$		0			0
$481$		−	62.8413i		−	2.86532i
$482$		0			0
$483$			6.21975i			0.283009i
$484$		0			0
$485$	3.18749			0.144737
$486$		0			0
$487$		−	16.8727i		−	0.764576i	−0.924043	−	0.382288i	$-0.875136\pi$
$487$		−	16.8727i		−	0.764576i	0.924043	−	0.382288i	$-0.124864\pi$
$488$		0			0
$489$	−0.275182			−0.0124442
$490$		0			0
$491$	−1.40112			−0.0632318			−0.0316159	−	0.999500i	$-0.510065\pi$
$491$	−1.40112			−0.0632318			−0.0316159	+	0.999500i	$0.510065\pi$
$492$		0			0
$493$	4.39145	+	3.95010i	0.197781	+	0.177904i
$494$		0			0
$495$	0.469838			0.0211176
$496$		0			0
$497$	3.73705			0.167630
$498$		0			0
$499$		−	43.8565i		−	1.96328i	−0.190732	−	0.981642i	$-0.561086\pi$
$499$		−	43.8565i		−	1.96328i	0.190732	−	0.981642i	$-0.438914\pi$
$500$		0			0
$501$	−21.7913			−0.973562
$502$		0			0
$503$			42.2970i			1.88593i	0.332889	+	0.942966i	$0.391977\pi$
$503$			42.2970i			1.88593i	−0.332889	+	0.942966i	$0.608023\pi$
$504$		0			0
$505$			5.88196i			0.261744i
$506$		0			0
$507$		−	25.3408i		−	1.12542i
$508$		0			0
$509$	43.2398			1.91657			0.958286	−	0.285810i	$-0.0922628\pi$
$509$	43.2398			1.91657			0.958286	+	0.285810i	$0.0922628\pi$
$510$		0			0
$511$	−7.17375			−0.317348
$512$		0			0
$513$			6.75943i			0.298436i
$514$		0			0
$515$		−	0.999667i		−	0.0440506i
$516$		0			0
$517$			1.16967i			0.0514422i
$518$		0			0
$519$	−15.1271			−0.664004
$520$		0			0
$521$			19.0895i			0.836325i	0.908372	+	0.418162i	$0.137326\pi$
$521$			19.0895i			0.836325i	−0.908372	+	0.418162i	$0.862674\pi$
$522$		0			0
$523$	0.455302			0.0199090			0.00995448	−	0.999950i	$-0.496831\pi$
$523$	0.455302			0.0199090			0.00995448	+	0.999950i	$0.496831\pi$
$524$		0			0
$525$	4.03616			0.176152
$526$		0			0
$527$	7.49227	+	6.73929i	0.326369	+	0.293568i
$528$		0			0
$529$	−15.6853			−0.681971
$530$		0			0
$531$	7.45048			0.323323
$532$		0			0
$533$		−	47.5550i		−	2.05984i
$534$		0			0
$535$	16.3366			0.706294
$536$		0			0
$537$			9.93107i			0.428558i
$538$		0			0
$539$		−	0.478569i		−	0.0206134i
$540$		0			0
$541$			14.2190i			0.611322i	0.952140	+	0.305661i	$0.0988775\pi$
$541$			14.2190i			0.611322i	−0.952140	+	0.305661i	$0.901122\pi$
$542$		0			0
$543$	−15.5577			−0.667646
$544$		0			0
$545$	−16.7580			−0.717834
$546$		0			0
$547$			18.5565i			0.793420i	0.917944	+	0.396710i	$0.129848\pi$
$547$			18.5565i			0.793420i	−0.917944	+	0.396710i	$0.870152\pi$
$548$		0			0
$549$		−	0.680410i		−	0.0290392i
$550$		0			0
$551$			9.68332i			0.412523i
$552$		0			0
$553$	−13.2646			−0.564070
$554$		0			0
$555$			9.96363i			0.422933i
$556$		0			0
$557$	−45.3071			−1.91972			−0.959861	−	0.280477i	$-0.909507\pi$
$557$	−45.3071			−1.91972			−0.959861	+	0.280477i	$0.909507\pi$
$558$		0			0
$559$	4.59147			0.194198
$560$		0			0
$561$	1.31959	−	1.46703i	0.0557131	−	0.0619379i
$562$		0			0
$563$	24.6705			1.03974			0.519869	−	0.854246i	$-0.325981\pi$
$563$	24.6705			1.03974			0.519869	+	0.854246i	$0.325981\pi$
$564$		0			0
$565$	−1.19474			−0.0502631
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	36.3661			1.52455			0.762273	−	0.647255i	$-0.224083\pi$
$569$	36.3661			1.52455			0.762273	+	0.647255i	$0.224083\pi$
$570$		0			0
$571$			25.8299i			1.08095i	0.841361	+	0.540473i	$0.181755\pi$
$571$			25.8299i			1.08095i	−0.841361	+	0.540473i	$0.818245\pi$
$572$		0			0
$573$		−	20.5252i		−	0.857454i
$574$		0			0
$575$			25.1039i			1.04691i
$576$		0			0
$577$	35.8676			1.49319			0.746595	−	0.665279i	$-0.231688\pi$
$577$	35.8676			1.49319			0.746595	+	0.665279i	$0.231688\pi$
$578$		0			0
$579$	−7.29708			−0.303256
$580$		0			0
$581$			9.34113i			0.387535i
$582$		0			0
$583$			6.33059i			0.262186i
$584$		0			0
$585$			6.07902i			0.251337i
$586$		0			0
$587$	−26.2130			−1.08193			−0.540963	−	0.841046i	$-0.681940\pi$
$587$	−26.2130			−1.08193			−0.540963	+	0.841046i	$0.681940\pi$
$588$		0			0
$589$			16.5208i			0.680726i
$590$		0			0
$591$	−25.5835			−1.05236
$592$		0			0
$593$	4.25752			0.174835			0.0874177	−	0.996172i	$-0.472139\pi$
$593$	4.25752			0.174835			0.0874177	+	0.996172i	$0.472139\pi$
$594$		0			0
$595$	−2.70705	+	3.00951i	−0.110978	+	0.123378i
$596$		0			0
$597$	15.0613			0.616420
$598$		0			0
$599$	17.5605			0.717502			0.358751	−	0.933433i	$-0.383203\pi$
$599$	17.5605			0.717502			0.358751	+	0.933433i	$0.383203\pi$
$600$		0			0
$601$			6.19927i			0.252873i	0.991975	+	0.126437i	$0.0403541\pi$
$601$			6.19927i			0.252873i	−0.991975	+	0.126437i	$0.959646\pi$
$602$		0			0
$603$	15.0823			0.614199
$604$		0			0
$605$			10.5745i			0.429913i
$606$		0			0
$607$		−	15.8262i		−	0.642366i	−0.947017	−	0.321183i	$-0.895920\pi$
$607$		−	15.8262i		−	0.642366i	0.947017	−	0.321183i	$-0.104080\pi$
$608$		0			0
$609$			1.43256i			0.0580505i
$610$		0			0
$611$	−15.1339			−0.612252
$612$		0			0
$613$	−29.6250			−1.19654			−0.598271	−	0.801294i	$-0.704146\pi$
$613$	−29.6250			−1.19654			−0.598271	+	0.801294i	$0.704146\pi$
$614$		0			0
$615$			7.53995i			0.304040i
$616$		0			0
$617$			22.4121i			0.902278i	0.892454	+	0.451139i	$0.148982\pi$
$617$			22.4121i			0.902278i	−0.892454	+	0.451139i	$0.851018\pi$
$618$		0			0
$619$			23.2873i			0.935995i	0.883730	+	0.467998i	$0.155024\pi$
$619$			23.2873i			0.935995i	−0.883730	+	0.467998i	$0.844976\pi$
$620$		0			0
$621$	−6.21975			−0.249590
$622$		0			0
$623$			17.7765i			0.712201i
$624$		0			0
$625$	11.4714			0.458854
$626$		0			0
$627$	3.23486			0.129188
$628$		0			0
$629$	31.1106	+	27.9839i	1.24046	+	1.11579i
$630$		0			0
$631$	41.2209			1.64098			0.820489	−	0.571662i	$-0.193701\pi$
$631$	41.2209			1.64098			0.820489	+	0.571662i	$0.193701\pi$
$632$		0			0
$633$	11.6728			0.463952
$634$		0			0
$635$		−	10.6603i		−	0.423042i
$636$		0			0
$637$	6.19199			0.245336
$638$		0			0
$639$			3.73705i			0.147836i
$640$		0			0
$641$			23.5909i			0.931783i	0.884842	+	0.465891i	$0.154266\pi$
$641$			23.5909i			0.931783i	−0.884842	+	0.465891i	$0.845734\pi$
$642$		0			0
$643$		−	23.7061i		−	0.934877i	−0.884026	−	0.467438i	$-0.845177\pi$
$643$		−	23.7061i		−	0.934877i	0.884026	−	0.467438i	$-0.154823\pi$
$644$		0			0
$645$	−0.727988			−0.0286645
$646$		0			0
$647$	15.8664			0.623771			0.311885	−	0.950120i	$-0.399039\pi$
$647$	15.8664			0.623771			0.311885	+	0.950120i	$0.399039\pi$
$648$		0			0
$649$		−	3.56557i		−	0.139961i
$650$		0			0
$651$			2.44411i			0.0957921i
$652$		0			0
$653$		−	28.4040i		−	1.11153i	−0.831338	−	0.555767i	$-0.812425\pi$
$653$		−	28.4040i		−	1.11153i	0.831338	−	0.555767i	$-0.187575\pi$
$654$		0			0
$655$	−7.97088			−0.311448
$656$		0			0
$657$		−	7.17375i		−	0.279875i
$658$		0			0
$659$	−5.37006			−0.209188			−0.104594	−	0.994515i	$-0.533354\pi$
$659$	−5.37006			−0.209188			−0.104594	+	0.994515i	$0.533354\pi$
$660$		0			0
$661$	11.0808			0.430993			0.215496	−	0.976505i	$-0.430863\pi$
$661$	11.0808			0.430993			0.215496	+	0.976505i	$0.430863\pi$
$662$		0			0
$663$	18.9812	+	17.0736i	0.737169	+	0.663083i
$664$		0			0
$665$	−6.63610			−0.257337
$666$		0			0
$667$	−8.91020			−0.345004
$668$		0			0
$669$			18.3952i			0.711199i
$670$		0			0
$671$	−0.325624			−0.0125706
$672$		0			0
$673$		−	30.0526i		−	1.15844i	−0.815170	−	0.579222i	$-0.803357\pi$
$673$		−	30.0526i		−	1.15844i	0.815170	−	0.579222i	$-0.196643\pi$
$674$		0			0
$675$			4.03616i			0.155352i
$676$		0			0
$677$		−	5.19473i		−	0.199650i	−0.995005	−	0.0998248i	$-0.968172\pi$
$677$		−	5.19473i		−	0.199650i	0.995005	−	0.0998248i	$-0.0318282\pi$
$678$		0			0
$679$	3.24673			0.124598
$680$		0			0
$681$	13.1538			0.504054
$682$		0			0
$683$		−	32.2719i		−	1.23485i	−0.786629	−	0.617426i	$-0.788176\pi$
$683$		−	32.2719i		−	1.23485i	0.786629	−	0.617426i	$-0.211824\pi$
$684$		0			0
$685$		−	6.08560i		−	0.232519i
$686$		0			0
$687$		−	0.569823i		−	0.0217401i
$688$		0			0
$689$	−81.9086			−3.12047
$690$		0			0
$691$			43.0141i			1.63633i	0.574981	+	0.818167i	$0.305009\pi$
$691$			43.0141i			1.63633i	−0.574981	+	0.818167i	$0.694991\pi$
$692$		0			0
$693$	0.478569			0.0181793
$694$		0			0
$695$	14.0374			0.532469
$696$		0			0
$697$	23.5428	+	21.1768i	0.891749	+	0.802127i
$698$		0			0
$699$	17.8788			0.676237
$700$		0			0
$701$	33.3405			1.25925			0.629626	−	0.776898i	$-0.283208\pi$
$701$	33.3405			1.25925			0.629626	+	0.776898i	$0.283208\pi$
$702$		0			0
$703$			68.6001i			2.58730i
$704$		0			0
$705$	2.39951			0.0903709
$706$		0			0
$707$			5.99127i			0.225325i
$708$		0			0
$709$		−	8.00000i		−	0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$		−	8.00000i		−	0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$		0			0
$711$		−	13.2646i		−	0.497463i
$712$		0			0
$713$	−15.2017			−0.569310
$714$		0			0
$715$	2.90923			0.108799
$716$		0			0
$717$			8.89968i			0.332365i
$718$		0			0
$719$			18.1701i			0.677629i	0.940853	+	0.338814i	$0.110026\pi$
$719$			18.1701i			0.677629i	−0.940853	+	0.338814i	$0.889974\pi$
$720$		0			0
$721$		−	1.01825i		−	0.0379214i
$722$		0			0
$723$	−10.2721			−0.382025
$724$		0			0
$725$			5.78206i			0.214740i
$726$		0			0
$727$	7.03407			0.260879			0.130440	−	0.991456i	$-0.458361\pi$
$727$	7.03407			0.260879			0.130440	+	0.991456i	$0.458361\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−2.04463	+	2.27308i	−0.0756234	+	0.0840729i
$732$		0			0
$733$	−1.27611			−0.0471342			−0.0235671	−	0.999722i	$-0.507502\pi$
$733$	−1.27611			−0.0471342			−0.0235671	+	0.999722i	$0.507502\pi$
$734$		0			0
$735$	−0.981755			−0.0362126
$736$		0			0
$737$		−	7.21793i		−	0.265876i
$738$		0			0
$739$	−31.1114			−1.14445			−0.572226	−	0.820096i	$-0.693920\pi$
$739$	−31.1114			−1.14445			−0.572226	+	0.820096i	$0.693920\pi$
$740$		0			0
$741$			41.8543i			1.53756i
$742$		0			0
$743$			14.8570i			0.545049i	0.962149	+	0.272524i	$0.0878585\pi$
$743$			14.8570i			0.545049i	−0.962149	+	0.272524i	$0.912141\pi$
$744$		0			0
$745$		−	6.26751i		−	0.229624i
$746$		0			0
$747$	−9.34113			−0.341774
$748$		0			0
$749$	16.6402			0.608021
$750$		0			0
$751$			20.0765i			0.732601i	0.930497	+	0.366301i	$0.119376\pi$
$751$			20.0765i			0.732601i	−0.930497	+	0.366301i	$0.880624\pi$
$752$		0			0
$753$		−	17.1953i		−	0.626631i
$754$		0			0
$755$		−	7.38963i		−	0.268936i
$756$		0			0
$757$	28.5851			1.03894			0.519472	−	0.854488i	$-0.326129\pi$
$757$	28.5851			1.03894			0.519472	+	0.854488i	$0.326129\pi$
$758$		0			0
$759$			2.97658i			0.108043i
$760$		0			0
$761$	−9.61218			−0.348441			−0.174221	−	0.984707i	$-0.555741\pi$
$761$	−9.61218			−0.348441			−0.174221	+	0.984707i	$0.555741\pi$
$762$		0			0
$763$	−17.0694			−0.617956
$764$		0			0
$765$	−3.00951	−	2.70705i	−0.108809	−	0.0978738i
$766$		0			0
$767$	46.1333			1.66578
$768$		0			0
$769$	16.7410			0.603695			0.301847	−	0.953356i	$-0.402397\pi$
$769$	16.7410			0.603695			0.301847	+	0.953356i	$0.402397\pi$
$770$		0			0
$771$		−	8.48067i		−	0.305424i
$772$		0			0
$773$	−33.1130			−1.19099			−0.595496	−	0.803358i	$-0.703045\pi$
$773$	−33.1130			−1.19099			−0.595496	+	0.803358i	$0.703045\pi$
$774$		0			0
$775$			9.86480i			0.354354i
$776$		0			0
$777$			10.1488i			0.364086i
$778$		0			0
$779$			51.9129i			1.85997i
$780$		0			0
$781$	1.78844			0.0639954
$782$		0			0
$783$	−1.43256			−0.0511957
$784$		0			0
$785$		−	7.66845i		−	0.273699i
$786$		0			0
$787$		−	9.30644i		−	0.331739i	−0.986148	−	0.165869i	$-0.946957\pi$
$787$		−	9.30644i		−	0.331739i	0.986148	−	0.165869i	$-0.0530430\pi$
$788$		0			0
$789$		−	5.85809i		−	0.208554i
$790$		0			0
$791$	−1.21694			−0.0432695
$792$		0			0
$793$		−	4.21310i		−	0.149611i
$794$		0			0
$795$	12.9868			0.460594
$796$		0			0
$797$	2.29134			0.0811633			0.0405816	−	0.999176i	$-0.487079\pi$
$797$	2.29134			0.0811633			0.0405816	+	0.999176i	$0.487079\pi$
$798$		0			0
$799$	6.73929	−	7.49227i	0.238419	−	0.265058i
$800$		0			0
$801$	−17.7765			−0.628102
$802$		0			0
$803$	−3.43314			−0.121153
$804$		0			0
$805$		−	6.10627i		−	0.215218i
$806$		0			0
$807$	−21.5471			−0.758494
$808$		0			0
$809$		−	23.0582i		−	0.810685i	−0.914165	−	0.405342i	$-0.867152\pi$
$809$		−	23.0582i		−	0.810685i	0.914165	−	0.405342i	$-0.132848\pi$
$810$		0			0
$811$		−	31.9136i		−	1.12064i	−0.828277	−	0.560319i	$-0.810678\pi$
$811$		−	31.9136i		−	1.12064i	0.828277	−	0.560319i	$-0.189322\pi$
$812$		0			0
$813$		−	0.161218i		−	0.00565418i
$814$		0			0
$815$	0.270162			0.00946335
$816$		0			0
$817$	−5.01223			−0.175356
$818$		0			0
$819$			6.19199i			0.216366i
$820$		0			0
$821$			8.99462i			0.313914i	0.987605	+	0.156957i	$0.0501685\pi$
$821$			8.99462i			0.313914i	−0.987605	+	0.156957i	$0.949832\pi$
$822$		0			0
$823$			23.9666i			0.835423i	0.908580	+	0.417711i	$0.137168\pi$
$823$			23.9666i			0.835423i	−0.908580	+	0.417711i	$0.862832\pi$
$824$		0			0
$825$	1.93158			0.0672490
$826$		0			0
$827$		−	34.3456i		−	1.19431i	−0.802125	−	0.597157i	$-0.796297\pi$
$827$		−	34.3456i		−	1.19431i	0.802125	−	0.597157i	$-0.203703\pi$
$828$		0			0
$829$	2.60897			0.0906134			0.0453067	−	0.998973i	$-0.485573\pi$
$829$	2.60897			0.0906134			0.0453067	+	0.998973i	$0.485573\pi$
$830$		0			0
$831$	32.4753			1.12656
$832$		0			0
$833$	−2.75736	+	3.06544i	−0.0955370	+	0.106211i
$834$		0			0
$835$	21.3937			0.740359
$836$		0			0
$837$	−2.44411			−0.0844807
$838$		0			0
$839$			7.08425i			0.244576i	0.992495	+	0.122288i	$0.0390231\pi$
$839$			7.08425i			0.244576i	−0.992495	+	0.122288i	$0.960977\pi$
$840$		0			0
$841$	26.9478			0.929233
$842$		0			0
$843$		−	3.34113i		−	0.115075i
$844$		0			0
$845$			24.8784i			0.855844i
$846$		0			0
$847$			10.7710i			0.370095i
$848$		0			0
$849$	5.61547			0.192722
$850$		0			0
$851$	−63.1230			−2.16383
$852$		0			0
$853$		−	32.2509i		−	1.10425i	−0.833762	−	0.552125i	$-0.813817\pi$
$853$		−	32.2509i		−	1.10425i	0.833762	−	0.552125i	$-0.186183\pi$
$854$		0			0
$855$		−	6.63610i		−	0.226950i
$856$		0			0
$857$		−	27.4904i		−	0.939053i	−0.882918	−	0.469527i	$-0.844425\pi$
$857$		−	27.4904i		−	0.939053i	0.882918	−	0.469527i	$-0.155575\pi$
$858$		0			0
$859$	−12.7376			−0.434602			−0.217301	−	0.976105i	$-0.569725\pi$
$859$	−12.7376			−0.434602			−0.217301	+	0.976105i	$0.569725\pi$
$860$		0			0
$861$			7.68008i			0.261736i
$862$		0			0
$863$	−14.3637			−0.488947			−0.244474	−	0.969656i	$-0.578615\pi$
$863$	−14.3637			−0.488947			−0.244474	+	0.969656i	$0.578615\pi$
$864$		0			0
$865$	14.8511			0.504951
$866$		0			0
$867$	−16.9051	+	1.79389i	−0.574127	+	0.0609238i
$868$		0			0
$869$	−6.34805			−0.215343
$870$		0			0
$871$	93.3896			3.16439
$872$		0			0
$873$			3.24673i			0.109885i
$874$		0			0
$875$	−8.87129			−0.299904
$876$		0			0
$877$		−	24.6010i		−	0.830718i	−0.909657	−	0.415359i	$-0.863656\pi$
$877$		−	24.6010i		−	0.830718i	0.909657	−	0.415359i	$-0.136344\pi$
$878$		0			0
$879$			20.3747i			0.687222i
$880$		0			0
$881$			53.9581i			1.81789i	0.416911	+	0.908947i	$0.363113\pi$
$881$			53.9581i			1.81789i	−0.416911	+	0.908947i	$0.636887\pi$
$882$		0			0
$883$	17.7546			0.597490			0.298745	−	0.954333i	$-0.403432\pi$
$883$	17.7546			0.597490			0.298745	+	0.954333i	$0.403432\pi$
$884$		0			0
$885$	−7.31454			−0.245876
$886$		0			0
$887$			26.2348i			0.880880i	0.897782	+	0.440440i	$0.145177\pi$
$887$			26.2348i			0.880880i	−0.897782	+	0.440440i	$0.854823\pi$
$888$		0			0
$889$		−	10.8584i		−	0.364180i
$890$		0			0
$891$			0.478569i			0.0160327i
$892$		0			0
$893$	16.5208			0.552846
$894$		0			0
$895$		−	9.74988i		−	0.325903i
$896$		0			0
$897$	−38.5127			−1.28590
$898$		0			0
$899$	−3.50134			−0.116776
$900$		0			0
$901$	36.4748	−	40.5502i	1.21515	−	1.35092i
$902$		0			0
$903$	−0.741517			−0.0246761
$904$		0			0
$905$	15.2739			0.507721
$906$		0			0
$907$		−	24.2658i		−	0.805734i	−0.915259	−	0.402867i	$-0.868014\pi$
$907$		−	24.2658i		−	0.805734i	0.915259	−	0.402867i	$-0.131986\pi$
$908$		0			0
$909$	−5.99127			−0.198718
$910$		0			0
$911$		−	16.6008i		−	0.550008i	−0.961443	−	0.275004i	$-0.911321\pi$
$911$		−	16.6008i		−	0.550008i	0.961443	−	0.275004i	$-0.0886793\pi$
$912$		0			0
$913$			4.47038i			0.147948i
$914$		0			0
$915$			0.667996i			0.0220833i
$916$		0			0
$917$	−8.11901			−0.268113
$918$		0			0
$919$	30.4120			1.00320			0.501600	−	0.865100i	$-0.332745\pi$
$919$	30.4120			1.00320			0.501600	+	0.865100i	$0.332745\pi$
$920$		0			0
$921$			17.7637i			0.585334i
$922$		0			0
$923$			23.1398i			0.761656i
$924$		0			0
$925$			40.9622i			1.34683i
$926$		0			0
$927$	1.01825			0.0334436
$928$		0			0
$929$		−	27.6790i		−	0.908118i	−0.890972	−	0.454059i	$-0.849975\pi$
$929$		−	27.6790i		−	0.908118i	0.890972	−	0.454059i	$-0.150025\pi$
$930$		0			0
$931$	−6.75943			−0.221531
$932$		0			0
$933$	−22.9822			−0.752403
$934$		0			0
$935$	−1.29551	+	1.44026i	−0.0423678	+	0.0471016i
$936$		0			0
$937$	7.78940			0.254469			0.127234	−	0.991873i	$-0.459390\pi$
$937$	7.78940			0.254469			0.127234	+	0.991873i	$0.459390\pi$
$938$		0			0
$939$	−17.6044			−0.574499
$940$		0			0
$941$		−	1.16746i		−	0.0380582i	−0.999819	−	0.0190291i	$-0.993942\pi$
$941$		−	1.16746i		−	0.0380582i	0.999819	−	0.0190291i	$-0.00605752\pi$
$942$		0			0
$943$	−47.7682			−1.55555
$944$		0			0
$945$		−	0.981755i		−	0.0319365i
$946$		0			0
$947$		−	11.8703i		−	0.385734i	−0.981225	−	0.192867i	$-0.938221\pi$
$947$		−	11.8703i		−	0.385734i	0.981225	−	0.192867i	$-0.0617786\pi$
$948$		0			0
$949$		−	44.4198i		−	1.44193i
$950$		0			0
$951$	28.6237			0.928187
$952$		0			0
$953$	−19.7585			−0.640041			−0.320021	−	0.947411i	$-0.603690\pi$
$953$	−19.7585			−0.640041			−0.320021	+	0.947411i	$0.603690\pi$
$954$		0			0
$955$			20.1507i			0.652063i
$956$		0			0
$957$			0.685582i			0.0221617i
$958$		0			0
$959$		−	6.19870i		−	0.200166i
$960$		0			0
$961$	25.0263			0.807301
$962$		0			0
$963$			16.6402i			0.536224i
$964$		0			0
$965$	7.16394			0.230615
$966$		0			0
$967$	12.6197			0.405823			0.202912	−	0.979197i	$-0.434960\pi$
$967$	12.6197			0.405823			0.202912	+	0.979197i	$0.434960\pi$
$968$		0			0
$969$	−20.7207	−	18.6382i	−0.665643	−	0.598745i
$970$		0			0
$971$	50.7951			1.63009			0.815046	−	0.579396i	$-0.196711\pi$
$971$	50.7951			1.63009			0.815046	+	0.579396i	$0.196711\pi$
$972$		0			0
$973$	14.2983			0.458381
$974$		0			0
$975$			24.9919i			0.800380i
$976$		0			0
$977$	−21.6003			−0.691054			−0.345527	−	0.938409i	$-0.612300\pi$
$977$	−21.6003			−0.691054			−0.345527	+	0.938409i	$0.612300\pi$
$978$		0			0
$979$			8.50729i			0.271894i
$980$		0			0
$981$		−	17.0694i		−	0.544986i
$982$		0			0
$983$			30.3369i			0.967596i	0.875180	+	0.483798i	$0.160743\pi$
$983$			30.3369i			0.967596i	−0.875180	+	0.483798i	$0.839257\pi$
$984$		0			0
$985$	25.1167			0.800284
$986$		0			0
$987$	2.44411			0.0777968
$988$		0			0
$989$		−	4.61205i		−	0.146655i
$990$		0			0
$991$			12.6099i			0.400568i	0.979738	+	0.200284i	$0.0641865\pi$
$991$			12.6099i			0.400568i	−0.979738	+	0.200284i	$0.935814\pi$
$992$		0			0
$993$			31.8104i			1.00947i
$994$		0			0
$995$	−14.7865			−0.468765
$996$		0			0
$997$		−	46.4095i		−	1.46980i	−0.678174	−	0.734901i	$-0.737229\pi$
$997$		−	46.4095i		−	1.46980i	0.678174	−	0.734901i	$-0.262771\pi$
$998$		0			0
$999$	−10.1488			−0.321094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1428.2.d.c.169.4	✓	12
3.2	odd	2		4284.2.d.f.3025.5		12
17.16	even	2	inner	1428.2.d.c.169.9	yes	12
51.50	odd	2		4284.2.d.f.3025.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1428.2.d.c.169.4	✓	12	1.1	even	1	trivial
1428.2.d.c.169.9	yes	12	17.16	even	2	inner
4284.2.d.f.3025.5		12	3.2	odd	2
4284.2.d.f.3025.8		12	51.50	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1428.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +0.981755i q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +0.981755i q^{5} +1.00000i q^{7} -1.00000 q^{9} +0.478569i q^{11} -6.19199 q^{13} +0.981755 q^{15} +(2.75736 - 3.06544i) q^{17} +6.75943 q^{19} +1.00000 q^{21} +6.21975i q^{23} +4.03616 q^{25} +1.00000i q^{27} +1.43256i q^{29} +2.44411i q^{31} +0.478569 q^{33} -0.981755 q^{35} +10.1488i q^{37} +6.19199i q^{39} +7.68008i q^{41} -0.741517 q^{43} -0.981755i q^{45} +2.44411 q^{47} -1.00000 q^{49} +(-3.06544 - 2.75736i) q^{51} +13.2282 q^{53} -0.469838 q^{55} -6.75943i q^{57} -7.45048 q^{59} +0.680410i q^{61} -1.00000i q^{63} -6.07902i q^{65} -15.0823 q^{67} +6.21975 q^{69} -3.73705i q^{71} +7.17375i q^{73} -4.03616i q^{75} -0.478569 q^{77} +13.2646i q^{79} +1.00000 q^{81} +9.34113 q^{83} +(3.00951 + 2.70705i) q^{85} +1.43256 q^{87} +17.7765 q^{89} -6.19199i q^{91} +2.44411 q^{93} +6.63610i q^{95} -3.24673i q^{97} -0.478569i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 26 q^{25} + 10 q^{33} - 2 q^{35} - 18 q^{43} - 20 q^{47} - 12 q^{49} - 2 q^{51} + 16 q^{53} + 22 q^{55} - 12 q^{59} + 32 q^{67} - 6 q^{69} - 10 q^{77} + 12 q^{81} - 16 q^{83} + 14 q^{85} + 24 q^{87} + 4 q^{89} - 20 q^{93}+O(q^{100}) \) 12 * q - 12 * q^9 - 6 * q^13 + 2 * q^15 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 26 * q^25 + 10 * q^33 - 2 * q^35 - 18 * q^43 - 20 * q^47 - 12 * q^49 - 2 * q^51 + 16 * q^53 + 22 * q^55 - 12 * q^59 + 32 * q^67 - 6 * q^69 - 10 * q^77 + 12 * q^81 - 16 * q^83 + 14 * q^85 + 24 * q^87 + 4 * q^89 - 20 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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