LMFDB - Embedded newform 1170.2.e.f.469.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(469,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1170.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1170.e (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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3534400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40$ x^6 - 2x^5 - 3x^4 + 16x^3 + x^2 - 12x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.5
Root			$-1.81837 - 0.301352i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.f.469.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(0.301352 + 2.21567i) q^{5} -3.63675i q^{7} -1.00000i q^{8} +(-2.21567 + 0.301352i) q^{10} -2.00000 q^{11} +1.00000i q^{13} +3.63675 q^{14} +1.00000 q^{16} -6.67079i q^{17} -8.06808 q^{19} +(-0.301352 - 2.21567i) q^{20} -2.00000i q^{22} -0.794590i q^{23} +(-4.81837 + 1.33539i) q^{25} -1.00000 q^{26} +3.63675i q^{28} -8.06808 q^{29} +5.20541 q^{31} +1.00000i q^{32} +6.67079 q^{34} +(8.05783 - 1.09594i) q^{35} -0.431337i q^{37} -8.06808i q^{38} +(2.21567 - 0.301352i) q^{40} -6.86267 q^{41} -5.80811i q^{43} +2.00000 q^{44} +0.794590 q^{46} -7.63675i q^{47} -6.22593 q^{49} +(-1.33539 - 4.81837i) q^{50} -1.00000i q^{52} -0.794590i q^{53} +(-0.602705 - 4.43134i) q^{55} -3.63675 q^{56} -8.06808i q^{58} +8.06808 q^{59} -2.86267 q^{61} +5.20541i q^{62} -1.00000 q^{64} +(-2.21567 + 0.301352i) q^{65} +5.20541i q^{67} +6.67079i q^{68} +(1.09594 + 8.05783i) q^{70} +9.46538 q^{71} -6.00000i q^{73} +0.431337 q^{74} +8.06808 q^{76} +7.27349i q^{77} -1.93192 q^{79} +(0.301352 + 2.21567i) q^{80} -6.86267i q^{82} -2.06808i q^{83} +(14.7803 - 2.01026i) q^{85} +5.80811 q^{86} +2.00000i q^{88} -8.06808 q^{89} +3.63675 q^{91} +0.794590i q^{92} +7.63675 q^{94} +(-2.43134 - 17.8762i) q^{95} +13.6573i q^{97} -6.22593i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{4} - 4 q^{10} - 12 q^{11} - 4 q^{14} + 6 q^{16} - 4 q^{19} - 16 q^{25} - 6 q^{26} - 4 q^{29} + 24 q^{31} - 8 q^{34} + 6 q^{35} + 4 q^{40} - 4 q^{41} + 12 q^{44} + 12 q^{46} - 26 q^{49} + 16 q^{50}+ \cdots + 4 q^{95}+O(q^{100})$ 6 * q - 6 * q^4 - 4 * q^10 - 12 * q^11 - 4 * q^14 + 6 * q^16 - 4 * q^19 - 16 * q^25 - 6 * q^26 - 4 * q^29 + 24 * q^31 - 8 * q^34 + 6 * q^35 + 4 * q^40 - 4 * q^41 + 12 * q^44 + 12 * q^46 - 26 * q^49 + 16 * q^50 + 4 * q^56 + 4 * q^59 + 20 * q^61 - 6 * q^64 - 4 * q^65 + 12 * q^70 + 16 * q^71 - 16 * q^74 + 4 * q^76 - 56 * q^79 + 28 * q^85 + 24 * q^86 - 4 * q^89 - 4 * q^91 + 20 * q^94 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$ .

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	0.301352	+	2.21567i	0.134769	+	0.990877i
$5$	0.301352	+	2.21567i	0.134769	+	0.990877i
$6$		0			0
$7$		−	3.63675i		−	1.37456i	−0.726392	−	0.687281i	$-0.758804\pi$
$7$		−	3.63675i		−	1.37456i	0.726392	−	0.687281i	$-0.241196\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−2.21567	+	0.301352i	−0.700656	+	0.0952960i
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$	3.63675			0.971961
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.67079i		−	1.61790i	−0.587875	−	0.808952i	$-0.700035\pi$
$17$		−	6.67079i		−	1.61790i	0.587875	−	0.808952i	$-0.299965\pi$
$18$		0			0
$19$	−8.06808			−1.85095			−0.925473	−	0.378814i	$-0.876332\pi$
$19$	−8.06808			−1.85095			−0.925473	+	0.378814i	$0.876332\pi$
$20$	−0.301352	−	2.21567i	−0.0673845	−	0.495439i
$21$		0			0
$22$		−	2.00000i		−	0.426401i
$23$		−	0.794590i		−	0.165683i	−0.996563	−	0.0828417i	$-0.973600\pi$
$23$		−	0.794590i		−	0.165683i	0.996563	−	0.0828417i	$-0.0263996\pi$
$24$		0			0
$25$	−4.81837	+	1.33539i	−0.963675	+	0.267079i
$26$	−1.00000			−0.196116
$27$		0			0
$28$			3.63675i			0.687281i
$29$	−8.06808			−1.49821			−0.749103	−	0.662454i	$-0.769515\pi$
$29$	−8.06808			−1.49821			−0.749103	+	0.662454i	$0.769515\pi$
$30$		0			0
$31$	5.20541			0.934919			0.467460	−	0.884014i	$-0.345169\pi$
$31$	5.20541			0.934919			0.467460	+	0.884014i	$0.345169\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	6.67079			1.14403
$35$	8.05783	−	1.09594i	1.36202	−	0.185248i
$36$		0			0
$37$		−	0.431337i		−	0.0709114i	−0.999371	−	0.0354557i	$-0.988712\pi$
$37$		−	0.431337i		−	0.0709114i	0.999371	−	0.0354557i	$-0.0112883\pi$
$38$		−	8.06808i		−	1.30882i
$39$		0			0
$40$	2.21567	−	0.301352i	0.350328	−	0.0476480i
$41$	−6.86267			−1.07177			−0.535885	−	0.844291i	$-0.680022\pi$
$41$	−6.86267			−1.07177			−0.535885	+	0.844291i	$0.680022\pi$
$42$		0			0
$43$		−	5.80811i		−	0.885729i	−0.896589	−	0.442865i	$-0.853962\pi$
$43$		−	5.80811i		−	0.885729i	0.896589	−	0.442865i	$-0.146038\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	0.794590			0.117156
$47$		−	7.63675i		−	1.11393i	−0.830535	−	0.556967i	$-0.811965\pi$
$47$		−	7.63675i		−	1.11393i	0.830535	−	0.556967i	$-0.188035\pi$
$48$		0			0
$49$	−6.22593			−0.889418
$50$	−1.33539	−	4.81837i	−0.188853	−	0.681421i
$51$		0			0
$52$		−	1.00000i		−	0.138675i
$53$		−	0.794590i		−	0.109145i	−0.998510	−	0.0545727i	$-0.982620\pi$
$53$		−	0.794590i		−	0.109145i	0.998510	−	0.0545727i	$-0.0173797\pi$
$54$		0			0
$55$	−0.602705	−	4.43134i	−0.0812687	−	0.597521i
$56$	−3.63675			−0.485981
$57$		0			0
$58$		−	8.06808i		−	1.05939i
$59$	8.06808			1.05038			0.525188	−	0.850987i	$-0.323995\pi$
$59$	8.06808			1.05038			0.525188	+	0.850987i	$0.323995\pi$
$60$		0			0
$61$	−2.86267			−0.366528			−0.183264	−	0.983064i	$-0.558666\pi$
$61$	−2.86267			−0.366528			−0.183264	+	0.983064i	$0.558666\pi$
$62$			5.20541i			0.661088i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−2.21567	+	0.301352i	−0.274820	+	0.0373782i
$66$		0			0
$67$			5.20541i			0.635942i	0.948100	+	0.317971i	$0.103001\pi$
$67$			5.20541i			0.635942i	−0.948100	+	0.317971i	$0.896999\pi$
$68$			6.67079i			0.808952i
$69$		0			0
$70$	1.09594	+	8.05783i	0.130990	+	0.963094i
$71$	9.46538			1.12333			0.561667	−	0.827363i	$-0.310160\pi$
$71$	9.46538			1.12333			0.561667	+	0.827363i	$0.310160\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0.431337			0.0501419
$75$		0			0
$76$	8.06808			0.925473
$77$			7.27349i			0.828892i
$78$		0			0
$79$	−1.93192			−0.217358			−0.108679	−	0.994077i	$-0.534662\pi$
$79$	−1.93192			−0.217358			−0.108679	+	0.994077i	$0.534662\pi$
$80$	0.301352	+	2.21567i	0.0336922	+	0.247719i
$81$		0			0
$82$		−	6.86267i		−	0.757856i
$83$		−	2.06808i		−	0.227002i	−0.993538	−	0.113501i	$-0.963794\pi$
$83$		−	2.06808i		−	0.227002i	0.993538	−	0.113501i	$-0.0362065\pi$
$84$		0			0
$85$	14.7803	−	2.01026i	1.60314	−	0.218043i
$86$	5.80811			0.626305
$87$		0			0
$88$			2.00000i			0.213201i
$89$	−8.06808			−0.855215			−0.427608	−	0.903964i	$-0.640643\pi$
$89$	−8.06808			−0.855215			−0.427608	+	0.903964i	$0.640643\pi$
$90$		0			0
$91$	3.63675			0.381235
$92$			0.794590i			0.0828417i
$93$		0			0
$94$	7.63675			0.787670
$95$	−2.43134	−	17.8762i	−0.249450	−	1.83406i
$96$		0			0
$97$			13.6573i			1.38669i	0.720608	+	0.693343i	$0.243863\pi$
$97$			13.6573i			1.38669i	−0.720608	+	0.693343i	$0.756137\pi$
$98$		−	6.22593i		−	0.628914i
$99$		0			0
$100$	4.81837	−	1.33539i	0.481837	−	0.133539i
$101$	15.6843			1.56065			0.780324	−	0.625376i	$-0.215054\pi$
$101$	15.6843			1.56065			0.780324	+	0.625376i	$0.215054\pi$
$102$		0			0
$103$		−	15.3416i		−	1.51165i	−0.654773	−	0.755825i	$-0.727236\pi$
$103$		−	15.3416i		−	1.51165i	0.654773	−	0.755825i	$-0.272764\pi$
$104$	1.00000			0.0980581
$105$		0			0
$106$	0.794590			0.0771774
$107$			1.20541i			0.116531i	0.998301	+	0.0582657i	$0.0185571\pi$
$107$			1.20541i			0.116531i	−0.998301	+	0.0582657i	$0.981443\pi$
$108$		0			0
$109$	−9.80811			−0.939447			−0.469724	−	0.882814i	$-0.655646\pi$
$109$	−9.80811			−0.939447			−0.469724	+	0.882814i	$0.655646\pi$
$110$	4.43134	−	0.602705i	0.422511	−	0.0574657i
$111$		0			0
$112$		−	3.63675i		−	0.343640i
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$	1.76055	−	0.239452i	0.164172	−	0.0223290i
$116$	8.06808			0.749103
$117$		0			0
$118$			8.06808i			0.742727i
$119$	−24.2600			−2.22391
$120$		0			0
$121$	−7.00000			−0.636364
$122$		−	2.86267i		−	0.259174i
$123$		0			0
$124$	−5.20541			−0.467460
$125$	−4.41082	−	10.2735i	−0.394516	−	0.918889i
$126$		0			0
$127$		−	4.79459i		−	0.425451i	−0.977112	−	0.212726i	$-0.931766\pi$
$127$		−	4.79459i		−	0.425451i	0.977112	−	0.212726i	$-0.0682340\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−0.301352	−	2.21567i	−0.0264304	−	0.194327i
$131$	4.02052			0.351274			0.175637	−	0.984455i	$-0.443801\pi$
$131$	4.02052			0.351274			0.175637	+	0.984455i	$0.443801\pi$
$132$		0			0
$133$			29.3416i			2.54424i
$134$	−5.20541			−0.449679
$135$		0			0
$136$	−6.67079			−0.572015
$137$			11.2054i			0.957343i	0.877994	+	0.478671i	$0.158881\pi$
$137$			11.2054i			0.957343i	−0.877994	+	0.478671i	$0.841119\pi$
$138$		0			0
$139$	−8.84216			−0.749982			−0.374991	−	0.927028i	$-0.622354\pi$
$139$	−8.84216			−0.749982			−0.374991	+	0.927028i	$0.622354\pi$
$140$	−8.05783	+	1.09594i	−0.681011	+	0.0926241i
$141$		0			0
$142$			9.46538i			0.794317i
$143$		−	2.00000i		−	0.167248i
$144$		0			0
$145$	−2.43134	−	17.8762i	−0.201912	−	1.48454i
$146$	6.00000			0.496564
$147$		0			0
$148$			0.431337i			0.0354557i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	7.05456			0.574092			0.287046	−	0.957917i	$-0.407327\pi$
$151$	7.05456			0.574092			0.287046	+	0.957917i	$0.407327\pi$
$152$			8.06808i			0.654408i
$153$		0			0
$154$	−7.27349			−0.586115
$155$	1.56866	+	11.5335i	0.125998	+	0.926390i
$156$		0			0
$157$			0.0680837i			0.00543367i	0.999996	+	0.00271683i	$0.000864796\pi$
$157$			0.0680837i			0.00543367i	−0.999996	+	0.00271683i	$0.999135\pi$
$158$		−	1.93192i		−	0.153695i
$159$		0			0
$160$	−2.21567	+	0.301352i	−0.175164	+	0.0238240i
$161$	−2.88972			−0.227742
$162$		0			0
$163$		−	7.65726i		−	0.599763i	−0.953976	−	0.299882i	$-0.903053\pi$
$163$		−	7.65726i		−	0.599763i	0.953976	−	0.299882i	$-0.0969472\pi$
$164$	6.86267			0.535885
$165$		0			0
$166$	2.06808			0.160514
$167$		−	4.82164i		−	0.373110i	−0.982445	−	0.186555i	$-0.940268\pi$
$167$		−	4.82164i		−	0.373110i	0.982445	−	0.186555i	$-0.0597322\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$	2.01026	+	14.7803i	0.154180	+	1.13359i
$171$		0			0
$172$			5.80811i			0.442865i
$173$			12.7946i			0.972755i	0.873749	+	0.486377i	$0.161682\pi$
$173$			12.7946i			0.972755i	−0.873749	+	0.486377i	$0.838318\pi$
$174$		0			0
$175$	4.85649	+	17.5232i	0.367116	+	1.32463i
$176$	−2.00000			−0.150756
$177$		0			0
$178$		−	8.06808i		−	0.604728i
$179$	−8.84216			−0.660894			−0.330447	−	0.943825i	$-0.607199\pi$
$179$	−8.84216			−0.660894			−0.330447	+	0.943825i	$0.607199\pi$
$180$		0			0
$181$	1.61623			0.120133			0.0600667	−	0.998194i	$-0.480869\pi$
$181$	1.61623			0.120133			0.0600667	+	0.998194i	$0.480869\pi$
$182$			3.63675i			0.269574i
$183$		0			0
$184$	−0.794590			−0.0585780
$185$	0.955700	−	0.129984i	0.0702644	−	0.00955665i
$186$		0			0
$187$			13.3416i			0.975633i
$188$			7.63675i			0.556967i
$189$		0			0
$190$	17.8762	−	2.43134i	1.29688	−	0.176388i
$191$	−21.3416			−1.54422			−0.772111	−	0.635487i	$-0.780799\pi$
$191$	−21.3416			−1.54422			−0.772111	+	0.635487i	$0.780799\pi$
$192$		0			0
$193$		−	5.61623i		−	0.404265i	−0.979358	−	0.202133i	$-0.935213\pi$
$193$		−	5.61623i		−	0.404265i	0.979358	−	0.202133i	$-0.0647872\pi$
$194$	−13.6573			−0.980534
$195$		0			0
$196$	6.22593			0.444709
$197$			14.0205i			0.998920i	0.866337	+	0.499460i	$0.166468\pi$
$197$			14.0205i			0.998920i	−0.866337	+	0.499460i	$0.833532\pi$
$198$		0			0
$199$	15.7524			1.11666			0.558329	−	0.829620i	$-0.311443\pi$
$199$	15.7524			1.11666			0.558329	+	0.829620i	$0.311443\pi$
$200$	1.33539	+	4.81837i	0.0944266	+	0.340710i
$201$		0			0
$202$			15.6843i			1.10354i
$203$			29.3416i			2.05937i
$204$		0			0
$205$	−2.06808	−	15.2054i	−0.144441	−	1.06199i
$206$	15.3416			1.06890
$207$		0			0
$208$			1.00000i			0.0693375i
$209$	16.1362			1.11616
$210$		0			0
$211$	−1.70483			−0.117365			−0.0586827	−	0.998277i	$-0.518690\pi$
$211$	−1.70483			−0.117365			−0.0586827	+	0.998277i	$0.518690\pi$
$212$			0.794590i			0.0545727i
$213$		0			0
$214$	−1.20541			−0.0824001
$215$	12.8689	−	1.75029i	0.877649	−	0.119369i
$216$		0			0
$217$		−	18.9308i		−	1.28510i
$218$		−	9.80811i		−	0.664289i
$219$		0			0
$220$	0.602705	+	4.43134i	0.0406344	+	0.298761i
$221$	6.67079			0.448726
$222$		0			0
$223$			22.9513i			1.53693i	0.639891	+	0.768466i	$0.278979\pi$
$223$			22.9513i			1.53693i	−0.639891	+	0.768466i	$0.721021\pi$
$224$	3.63675			0.242990
$225$		0			0
$226$	4.00000			0.266076
$227$			10.5470i			0.700028i	0.936744	+	0.350014i	$0.113823\pi$
$227$			10.5470i			0.700028i	−0.936744	+	0.350014i	$0.886177\pi$
$228$		0			0
$229$	−25.9443			−1.71445			−0.857223	−	0.514945i	$-0.827812\pi$
$229$	−25.9443			−1.71445			−0.857223	+	0.514945i	$0.827812\pi$
$230$	0.239452	+	1.76055i	0.0157890	+	0.116087i
$231$		0			0
$232$			8.06808i			0.529696i
$233$		−	11.4924i		−	0.752894i	−0.926438	−	0.376447i	$-0.877146\pi$
$233$		−	11.4924i		−	0.752894i	0.926438	−	0.376447i	$-0.122854\pi$
$234$		0			0
$235$	16.9205	−	2.30135i	1.10377	−	0.150124i
$236$	−8.06808			−0.525188
$237$		0			0
$238$		−	24.2600i		−	1.57254i
$239$	14.6708			0.948974			0.474487	−	0.880262i	$-0.342634\pi$
$239$	14.6708			0.948974			0.474487	+	0.880262i	$0.342634\pi$
$240$		0			0
$241$	−18.6151			−1.19910			−0.599551	−	0.800337i	$-0.704654\pi$
$241$	−18.6151			−1.19910			−0.599551	+	0.800337i	$0.704654\pi$
$242$		−	7.00000i		−	0.449977i
$243$		0			0
$244$	2.86267			0.183264
$245$	−1.87620	−	13.7946i	−0.119866	−	0.881304i
$246$		0			0
$247$		−	8.06808i		−	0.513360i
$248$		−	5.20541i		−	0.330544i
$249$		0			0
$250$	10.2735	−	4.41082i	0.649753	−	0.278965i
$251$	14.4108			0.909603			0.454801	−	0.890593i	$-0.349710\pi$
$251$	14.4108			0.909603			0.454801	+	0.890593i	$0.349710\pi$
$252$		0			0
$253$			1.58918i			0.0999109i
$254$	4.79459			0.300839
$255$		0			0
$256$	1.00000			0.0625000
$257$			0.123802i			0.00772253i	0.999993	+	0.00386126i	$0.00122908\pi$
$257$			0.123802i			0.00772253i	−0.999993	+	0.00386126i	$0.998771\pi$
$258$		0			0
$259$	−1.56866			−0.0974720
$260$	2.21567	−	0.301352i	0.137410	−	0.0186891i
$261$		0			0
$262$			4.02052i			0.248388i
$263$		−	5.27349i		−	0.325178i	−0.986694	−	0.162589i	$-0.948016\pi$
$263$		−	5.27349i		−	0.325178i	0.986694	−	0.162589i	$-0.0519844\pi$
$264$		0			0
$265$	1.76055	−	0.239452i	0.108150	−	0.0147094i
$266$	−29.3416			−1.79905
$267$		0			0
$268$		−	5.20541i		−	0.317971i
$269$	1.27349			0.0776463			0.0388231	−	0.999246i	$-0.487639\pi$
$269$	1.27349			0.0776463			0.0388231	+	0.999246i	$0.487639\pi$
$270$		0			0
$271$	6.28702			0.381909			0.190955	−	0.981599i	$-0.438842\pi$
$271$	6.28702			0.381909			0.190955	+	0.981599i	$0.438842\pi$
$272$		−	6.67079i		−	0.404476i
$273$		0			0
$274$	−11.2054			−0.676944
$275$	9.63675	−	2.67079i	0.581118	−	0.161055i
$276$		0			0
$277$			22.9988i			1.38187i	0.722919	+	0.690933i	$0.242800\pi$
$277$			22.9988i			1.38187i	−0.722919	+	0.690933i	$0.757200\pi$
$278$		−	8.84216i		−	0.530317i
$279$		0			0
$280$	−1.09594	−	8.05783i	−0.0654951	−	0.481547i
$281$	−10.3427			−0.616996			−0.308498	−	0.951225i	$-0.599826\pi$
$281$	−10.3427			−0.616996			−0.308498	+	0.951225i	$0.599826\pi$
$282$		0			0
$283$			10.0270i			0.596046i	0.954559	+	0.298023i	$0.0963272\pi$
$283$			10.0270i			0.596046i	−0.954559	+	0.298023i	$0.903673\pi$
$284$	−9.46538			−0.561667
$285$		0			0
$286$	2.00000			0.118262
$287$			24.9578i			1.47321i
$288$		0			0
$289$	−27.4994			−1.61761
$290$	17.8762	−	2.43134i	1.04973	−	0.142773i
$291$		0			0
$292$			6.00000i			0.351123i
$293$			6.11565i			0.357280i	0.983914	+	0.178640i	$0.0571698\pi$
$293$			6.11565i			0.357280i	−0.983914	+	0.178640i	$0.942830\pi$
$294$		0			0
$295$	2.43134	+	17.8762i	0.141558	+	1.04079i
$296$	−0.431337			−0.0250709
$297$		0			0
$298$		0			0
$299$	0.794590			0.0459523
$300$		0			0
$301$	−21.1226			−1.21749
$302$			7.05456i			0.405944i
$303$		0			0
$304$	−8.06808			−0.462736
$305$	−0.862674	−	6.34274i	−0.0493966	−	0.363184i
$306$		0			0
$307$			16.8627i			0.962404i	0.876610	+	0.481202i	$0.159800\pi$
$307$			16.8627i			0.962404i	−0.876610	+	0.481202i	$0.840200\pi$
$308$		−	7.27349i		−	0.414446i
$309$		0			0
$310$	−11.5335	+	1.56866i	−0.655057	+	0.0890941i
$311$	−32.9988			−1.87119			−0.935596	−	0.353072i	$-0.885137\pi$
$311$	−32.9988			−1.87119			−0.935596	+	0.353072i	$0.885137\pi$
$312$		0			0
$313$			25.4654i			1.43939i	0.694291	+	0.719694i	$0.255718\pi$
$313$			25.4654i			1.43939i	−0.694291	+	0.719694i	$0.744282\pi$
$314$	−0.0680837			−0.00384218
$315$		0			0
$316$	1.93192			0.108679
$317$			14.1362i			0.793966i	0.917826	+	0.396983i	$0.129943\pi$
$317$			14.1362i			0.793966i	−0.917826	+	0.396983i	$0.870057\pi$
$318$		0			0
$319$	16.1362			0.903452
$320$	−0.301352	−	2.21567i	−0.0168461	−	0.123860i
$321$		0			0
$322$		−	2.88972i		−	0.161038i
$323$			53.8205i			2.99465i
$324$		0			0
$325$	−1.33539	−	4.81837i	−0.0740743	−	0.267275i
$326$	7.65726			0.424097
$327$		0			0
$328$			6.86267i			0.378928i
$329$	−27.7729			−1.53117
$330$		0			0
$331$	18.5199			1.01795			0.508974	−	0.860782i	$-0.330025\pi$
$331$	18.5199			1.01795			0.508974	+	0.860782i	$0.330025\pi$
$332$			2.06808i			0.113501i
$333$		0			0
$334$	4.82164			0.263828
$335$	−11.5335	+	1.56866i	−0.630140	+	0.0857052i
$336$		0			0
$337$		−	22.2870i		−	1.21405i	−0.794682	−	0.607026i	$-0.792363\pi$
$337$		−	22.2870i		−	1.21405i	0.794682	−	0.607026i	$-0.207637\pi$
$338$		−	1.00000i		−	0.0543928i
$339$		0			0
$340$	−14.7803	+	2.01026i	−0.801572	+	0.109022i
$341$	−10.4108			−0.563777
$342$		0			0
$343$		−	2.81511i		−	0.152002i
$344$	−5.80811			−0.313153
$345$		0			0
$346$	−12.7946			−0.687841
$347$		−	31.5335i		−	1.69280i	−0.532544	−	0.846402i	$-0.678764\pi$
$347$		−	31.5335i		−	1.69280i	0.532544	−	0.846402i	$-0.321236\pi$
$348$		0			0
$349$	−32.3551			−1.73193			−0.865964	−	0.500106i	$-0.833295\pi$
$349$	−32.3551			−1.73193			−0.865964	+	0.500106i	$0.833295\pi$
$350$	−17.5232	+	4.85649i	−0.936655	+	0.259590i
$351$		0			0
$352$		−	2.00000i		−	0.106600i
$353$		−	20.0270i		−	1.06593i	−0.846137	−	0.532966i	$-0.821077\pi$
$353$		−	20.0270i		−	1.06593i	0.846137	−	0.532966i	$-0.178923\pi$
$354$		0			0
$355$	2.85242	+	20.9721i	0.151390	+	1.11309i
$356$	8.06808			0.427608
$357$		0			0
$358$		−	8.84216i		−	0.467322i
$359$	22.9308			1.21024			0.605120	−	0.796135i	$-0.293125\pi$
$359$	22.9308			1.21024			0.605120	+	0.796135i	$0.293125\pi$
$360$		0			0
$361$	46.0940			2.42600
$362$			1.61623i			0.0849471i
$363$		0			0
$364$	−3.63675			−0.190617
$365$	13.2940	−	1.80811i	0.695840	−	0.0946411i
$366$		0			0
$367$		−	34.1362i		−	1.78189i	−0.454108	−	0.890947i	$-0.650042\pi$
$367$		−	34.1362i		−	1.78189i	0.454108	−	0.890947i	$-0.349958\pi$
$368$		−	0.794590i		−	0.0414209i
$369$		0			0
$370$	0.129984	+	0.955700i	0.00675757	+	0.0496845i
$371$	−2.88972			−0.150027
$372$		0			0
$373$		−	26.9988i		−	1.39795i	−0.715148	−	0.698974i	$-0.753641\pi$
$373$		−	26.9988i		−	1.39795i	0.715148	−	0.698974i	$-0.246359\pi$
$374$	−13.3416			−0.689877
$375$		0			0
$376$	−7.63675			−0.393835
$377$		−	8.06808i		−	0.415527i
$378$		0			0
$379$	−12.9308			−0.664208			−0.332104	−	0.943243i	$-0.607759\pi$
$379$	−12.9308			−0.664208			−0.332104	+	0.943243i	$0.607759\pi$
$380$	2.43134	+	17.8762i	0.124725	+	0.917030i
$381$		0			0
$382$		−	21.3416i		−	1.09193i
$383$			21.3621i			1.09155i	0.837931	+	0.545776i	$0.183765\pi$
$383$			21.3621i			1.09155i	−0.837931	+	0.545776i	$0.816235\pi$
$384$		0			0
$385$	−16.1157	+	2.19189i	−0.821330	+	0.111709i
$386$	5.61623			0.285859
$387$		0			0
$388$		−	13.6573i		−	0.693343i
$389$	−29.4507			−1.49321			−0.746605	−	0.665268i	$-0.768317\pi$
$389$	−29.4507			−1.49321			−0.746605	+	0.665268i	$0.768317\pi$
$390$		0			0
$391$	−5.30054			−0.268060
$392$			6.22593i			0.314457i
$393$		0			0
$394$	−14.0205			−0.706343
$395$	−0.582188	−	4.28049i	−0.0292930	−	0.215375i
$396$		0			0
$397$		−	16.6832i		−	0.837304i	−0.908147	−	0.418652i	$-0.862503\pi$
$397$		−	16.6832i		−	0.837304i	0.908147	−	0.418652i	$-0.137497\pi$
$398$			15.7524i			0.789596i
$399$		0			0
$400$	−4.81837	+	1.33539i	−0.240919	+	0.0667697i
$401$	7.34158			0.366621			0.183310	−	0.983055i	$-0.441319\pi$
$401$	7.34158			0.366621			0.183310	+	0.983055i	$0.441319\pi$
$402$		0			0
$403$			5.20541i			0.259300i
$404$	−15.6843			−0.780324
$405$		0			0
$406$	−29.3416			−1.45620
$407$			0.862674i			0.0427612i
$408$		0			0
$409$	35.4777			1.75426			0.877131	−	0.480252i	$-0.159455\pi$
$409$	35.4777			1.75426			0.877131	+	0.480252i	$0.159455\pi$
$410$	15.2054	−	2.06808i	0.750942	−	0.102135i
$411$		0			0
$412$			15.3416i			0.755825i
$413$		−	29.3416i		−	1.44380i
$414$		0			0
$415$	4.58219	−	0.623222i	0.224931	−	0.0305928i
$416$	−1.00000			−0.0490290
$417$		0			0
$418$			16.1362i			0.789246i
$419$	0.883191			0.0431467			0.0215734	−	0.999767i	$-0.493132\pi$
$419$	0.883191			0.0431467			0.0215734	+	0.999767i	$0.493132\pi$
$420$		0			0
$421$	27.2859			1.32983			0.664916	−	0.746918i	$-0.268467\pi$
$421$	27.2859			1.32983			0.664916	+	0.746918i	$0.268467\pi$
$422$		−	1.70483i		−	0.0829898i
$423$		0			0
$424$	−0.794590			−0.0385887
$425$	8.90813	+	32.1424i	0.432108	+	1.55913i
$426$		0			0
$427$			10.4108i			0.503815i
$428$		−	1.20541i		−	0.0582657i
$429$		0			0
$430$	1.75029	+	12.8689i	0.0844065	+	0.620591i
$431$	−0.259969			−0.0125223			−0.00626113	−	0.999980i	$-0.501993\pi$
$431$	−0.259969			−0.0125223			−0.00626113	+	0.999980i	$0.501993\pi$
$432$		0			0
$433$		−	17.4654i		−	0.839333i	−0.907678	−	0.419666i	$-0.862147\pi$
$433$		−	17.4654i		−	0.839333i	0.907678	−	0.419666i	$-0.137853\pi$
$434$	18.9308			0.908705
$435$		0			0
$436$	9.80811			0.469724
$437$			6.41082i			0.306671i
$438$		0			0
$439$	−17.6843			−0.844026			−0.422013	−	0.906590i	$-0.638676\pi$
$439$	−17.6843			−0.844026			−0.422013	+	0.906590i	$0.638676\pi$
$440$	−4.43134	+	0.602705i	−0.211256	+	0.0287328i
$441$		0			0
$442$			6.67079i			0.317297i
$443$		−	32.4913i		−	1.54371i	−0.635801	−	0.771853i	$-0.719330\pi$
$443$		−	32.4913i		−	1.54371i	0.635801	−	0.771853i	$-0.280670\pi$
$444$		0			0
$445$	−2.43134	−	17.8762i	−0.115256	−	0.847413i
$446$	−22.9513			−1.08677
$447$		0			0
$448$			3.63675i			0.171820i
$449$	35.0940			1.65619			0.828094	−	0.560590i	$-0.189426\pi$
$449$	35.0940			1.65619			0.828094	+	0.560590i	$0.189426\pi$
$450$		0			0
$451$	13.7253			0.646301
$452$			4.00000i			0.188144i
$453$		0			0
$454$	−10.5470			−0.494995
$455$	1.09594	+	8.05783i	0.0513786	+	0.377757i
$456$		0			0
$457$		−	20.5470i		−	0.961148i	−0.876954	−	0.480574i	$-0.840428\pi$
$457$		−	20.5470i		−	0.961148i	0.876954	−	0.480574i	$-0.159572\pi$
$458$		−	25.9443i		−	1.21230i
$459$		0			0
$460$	−1.76055	+	0.239452i	−0.0820860	+	0.0111645i
$461$	7.14969			0.332994			0.166497	−	0.986042i	$-0.446754\pi$
$461$	7.14969			0.332994			0.166497	+	0.986042i	$0.446754\pi$
$462$		0			0
$463$			22.6832i			1.05418i	0.849811	+	0.527088i	$0.176716\pi$
$463$			22.6832i			1.05418i	−0.849811	+	0.527088i	$0.823284\pi$
$464$	−8.06808			−0.374551
$465$		0			0
$466$	11.4924			0.532376
$467$		−	25.2054i		−	1.16637i	−0.812340	−	0.583184i	$-0.801807\pi$
$467$		−	25.2054i		−	1.16637i	0.812340	−	0.583184i	$-0.198193\pi$
$468$		0			0
$469$	18.9308			0.874141
$470$	2.30135	+	16.9205i	0.106153	+	0.780484i
$471$		0			0
$472$		−	8.06808i		−	0.371364i
$473$			11.6162i			0.534115i
$474$		0			0
$475$	38.8750	−	10.7741i	1.78371	−	0.494348i
$476$	24.2600			1.11195
$477$		0			0
$478$			14.6708i			0.671026i
$479$	4.01237			0.183330			0.0916648	−	0.995790i	$-0.470781\pi$
$479$	4.01237			0.183330			0.0916648	+	0.995790i	$0.470781\pi$
$480$		0			0
$481$	0.431337			0.0196673
$482$		−	18.6151i		−	0.847893i
$483$		0			0
$484$	7.00000			0.318182
$485$	−30.2600	+	4.11565i	−1.37403	+	0.186882i
$486$		0			0
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$			2.86267i			0.129587i
$489$		0			0
$490$	13.7946	−	1.87620i	0.623176	−	0.0847580i
$491$	8.88319			0.400893			0.200446	−	0.979705i	$-0.435761\pi$
$491$	8.88319			0.400893			0.200446	+	0.979705i	$0.435761\pi$
$492$		0			0
$493$			53.8205i			2.42395i
$494$	8.06808			0.363000
$495$		0			0
$496$	5.20541			0.233730
$497$		−	34.4232i		−	1.54409i
$498$		0			0
$499$	10.9988			0.492376			0.246188	−	0.969222i	$-0.420822\pi$
$499$	10.9988			0.492376			0.246188	+	0.969222i	$0.420822\pi$
$500$	4.41082	+	10.2735i	0.197258	+	0.459445i
$501$		0			0
$502$			14.4108i			0.643186i
$503$		−	5.65726i		−	0.252245i	−0.992015	−	0.126122i	$-0.959747\pi$
$503$		−	5.65726i		−	0.252245i	0.992015	−	0.126122i	$-0.0402532\pi$
$504$		0			0
$505$	4.72651	+	34.7512i	0.210327	+	1.54641i
$506$	−1.58918			−0.0706477
$507$		0			0
$508$			4.79459i			0.212726i
$509$	−9.72535			−0.431068			−0.215534	−	0.976496i	$-0.569149\pi$
$509$	−9.72535			−0.431068			−0.215534	+	0.976496i	$0.569149\pi$
$510$		0			0
$511$	−21.8205			−0.965281
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−0.123802			−0.00546065
$515$	33.9918	−	4.62322i	1.49786	−	0.203724i
$516$		0			0
$517$			15.2735i			0.671727i
$518$		−	1.56866i		−	0.0689231i
$519$		0			0
$520$	0.301352	+	2.21567i	0.0132152	+	0.0971635i
$521$	28.7741			1.26062			0.630308	−	0.776346i	$-0.282929\pi$
$521$	28.7741			1.26062			0.630308	+	0.776346i	$0.282929\pi$
$522$		0			0
$523$		−	37.6139i		−	1.64474i	−0.568952	−	0.822371i	$-0.692651\pi$
$523$		−	37.6139i		−	1.64474i	0.568952	−	0.822371i	$-0.307349\pi$
$524$	−4.02052			−0.175637
$525$		0			0
$526$	5.27349			0.229935
$527$		−	34.7242i		−	1.51261i
$528$		0			0
$529$	22.3686			0.972549
$530$	0.239452	+	1.76055i	0.0104011	+	0.0764733i
$531$		0			0
$532$		−	29.3416i		−	1.27212i
$533$		−	6.86267i		−	0.297255i
$534$		0			0
$535$	−2.67079	+	0.363253i	−0.115468	+	0.0157048i
$536$	5.20541			0.224839
$537$		0			0
$538$			1.27349i			0.0549042i
$539$	12.4519			0.536339
$540$		0			0
$541$	21.8081			0.937604			0.468802	−	0.883303i	$-0.344686\pi$
$541$	21.8081			0.937604			0.468802	+	0.883303i	$0.344686\pi$
$542$			6.28702i			0.270051i
$543$		0			0
$544$	6.67079			0.286008
$545$	−2.95570	−	21.7315i	−0.126608	−	0.930876i
$546$		0			0
$547$			4.73887i			0.202620i	0.994855	+	0.101310i	$0.0323033\pi$
$547$			4.73887i			0.202620i	−0.994855	+	0.101310i	$0.967697\pi$
$548$		−	11.2054i		−	0.478671i
$549$		0			0
$550$	2.67079	+	9.63675i	0.113883	+	0.410912i
$551$	65.0940			2.77310
$552$		0			0
$553$			7.02589i			0.298771i
$554$	−22.9988			−0.977127
$555$		0			0
$556$	8.84216			0.374991
$557$		−	6.74702i		−	0.285881i	−0.989731	−	0.142940i	$-0.954344\pi$
$557$		−	6.74702i		−	0.285881i	0.989731	−	0.142940i	$-0.0456557\pi$
$558$		0			0
$559$	5.80811			0.245657
$560$	8.05783	−	1.09594i	0.340505	−	0.0463120i
$561$		0			0
$562$		−	10.3427i		−	0.436282i
$563$			41.9443i			1.76774i	0.467732	+	0.883870i	$0.345071\pi$
$563$			41.9443i			1.76774i	−0.467732	+	0.883870i	$0.654929\pi$
$564$		0			0
$565$	8.86267	−	1.20541i	0.372855	−	0.0507120i
$566$	−10.0270			−0.421468
$567$		0			0
$568$		−	9.46538i		−	0.397158i
$569$	−7.95243			−0.333383			−0.166692	−	0.986009i	$-0.553308\pi$
$569$	−7.95243			−0.333383			−0.166692	+	0.986009i	$0.553308\pi$
$570$		0			0
$571$	−14.5265			−0.607914			−0.303957	−	0.952686i	$-0.598308\pi$
$571$	−14.5265			−0.607914			−0.303957	+	0.952686i	$0.598308\pi$
$572$			2.00000i			0.0836242i
$573$		0			0
$574$	−24.9578			−1.04172
$575$	1.06109	+	3.82863i	0.0442506	+	0.159665i
$576$		0			0
$577$		−	5.17836i		−	0.215578i	−0.994174	−	0.107789i	$-0.965623\pi$
$577$		−	5.17836i		−	0.215578i	0.994174	−	0.107789i	$-0.0343771\pi$
$578$		−	27.4994i		−	1.14383i
$579$		0			0
$580$	2.43134	+	17.8762i	0.100956	+	0.742269i
$581$	−7.52110			−0.312028
$582$		0			0
$583$			1.58918i			0.0658171i
$584$	−6.00000			−0.248282
$585$		0			0
$586$	−6.11565			−0.252635
$587$		−	22.8897i		−	0.944760i	−0.881395	−	0.472380i	$-0.843395\pi$
$587$		−	22.8897i		−	0.944760i	0.881395	−	0.472380i	$-0.156605\pi$
$588$		0			0
$589$	−41.9977			−1.73048
$590$	−17.8762	+	2.43134i	−0.735951	+	0.100097i
$591$		0			0
$592$		−	0.431337i		−	0.0177278i
$593$			14.3427i			0.588986i	0.955654	+	0.294493i	$0.0951507\pi$
$593$			14.3427i			0.588986i	−0.955654	+	0.294493i	$0.904849\pi$
$594$		0			0
$595$	−7.31080	−	53.7520i	−0.299714	−	2.20362i
$596$		0			0
$597$		0			0
$598$			0.794590i			0.0324932i
$599$	5.63021			0.230044			0.115022	−	0.993363i	$-0.463306\pi$
$599$	5.63021			0.230044			0.115022	+	0.993363i	$0.463306\pi$
$600$		0			0
$601$	−11.3211			−0.461796			−0.230898	−	0.972978i	$-0.574166\pi$
$601$	−11.3211			−0.461796			−0.230898	+	0.972978i	$0.574166\pi$
$602$		−	21.1226i		−	0.860895i
$603$		0			0
$604$	−7.05456			−0.287046
$605$	−2.10947	−	15.5097i	−0.0857620	−	0.630558i
$606$		0			0
$607$		−	25.0259i		−	1.01577i	−0.861425	−	0.507885i	$-0.830428\pi$
$607$		−	25.0259i		−	1.01577i	0.861425	−	0.507885i	$-0.169572\pi$
$608$		−	8.06808i		−	0.327204i
$609$		0			0
$610$	6.34274	−	0.862674i	0.256810	−	0.0349286i
$611$	7.63675			0.308950
$612$		0			0
$613$		−	25.4507i		−	1.02794i	−0.857807	−	0.513972i	$-0.828174\pi$
$613$		−	25.4507i		−	1.02794i	0.857807	−	0.513972i	$-0.171826\pi$
$614$	−16.8627			−0.680522
$615$		0			0
$616$	7.27349			0.293057
$617$			15.6843i			0.631427i	0.948855	+	0.315713i	$0.102244\pi$
$617$			15.6843i			0.631427i	−0.948855	+	0.315713i	$0.897756\pi$
$618$		0			0
$619$	44.5880			1.79214			0.896072	−	0.443909i	$-0.146409\pi$
$619$	44.5880			1.79214			0.896072	+	0.443909i	$0.146409\pi$
$620$	−1.56866	−	11.5335i	−0.0629990	−	0.463195i
$621$		0			0
$622$		−	32.9988i		−	1.32313i
$623$			29.3416i			1.17555i
$624$		0			0
$625$	21.4334	−	12.8689i	0.857338	−	0.514754i
$626$	−25.4654			−1.01780
$627$		0			0
$628$		−	0.0680837i		−	0.00271683i
$629$	−2.87736			−0.114728
$630$		0			0
$631$	−38.8070			−1.54488			−0.772440	−	0.635087i	$-0.780964\pi$
$631$	−38.8070			−1.54488			−0.772440	+	0.635087i	$0.780964\pi$
$632$			1.93192i			0.0768475i
$633$		0			0
$634$	−14.1362			−0.561419
$635$	10.6232	−	1.44486i	0.421570	−	0.0573376i
$636$		0			0
$637$		−	6.22593i		−	0.246680i
$638$			16.1362i			0.638837i
$639$		0			0
$640$	2.21567	−	0.301352i	0.0875820	−	0.0119120i
$641$	8.27465			0.326829			0.163415	−	0.986557i	$-0.447749\pi$
$641$	8.27465			0.326829			0.163415	+	0.986557i	$0.447749\pi$
$642$		0			0
$643$			6.16322i			0.243054i	0.992588	+	0.121527i	$0.0387790\pi$
$643$			6.16322i			0.243054i	−0.992588	+	0.121527i	$0.961221\pi$
$644$	2.88972			0.113871
$645$		0			0
$646$	−53.8205			−2.11754
$647$		−	7.24644i		−	0.284887i	−0.989803	−	0.142444i	$-0.954504\pi$
$647$		−	7.24644i		−	0.284887i	0.989803	−	0.142444i	$-0.0454959\pi$
$648$		0			0
$649$	−16.1362			−0.633400
$650$	4.81837	−	1.33539i	0.188992	−	0.0523785i
$651$		0			0
$652$			7.65726i			0.299882i
$653$		−	5.27349i		−	0.206368i	−0.994662	−	0.103184i	$-0.967097\pi$
$653$		−	5.27349i		−	0.206368i	0.994662	−	0.103184i	$-0.0329030\pi$
$654$		0			0
$655$	1.21159	+	8.90813i	0.0473408	+	0.348070i
$656$	−6.86267			−0.267942
$657$		0			0
$658$		−	27.7729i		−	1.08270i
$659$	−18.6832			−0.727792			−0.363896	−	0.931440i	$-0.618554\pi$
$659$	−18.6832			−0.727792			−0.363896	+	0.931440i	$0.618554\pi$
$660$		0			0
$661$	15.3145			0.595666			0.297833	−	0.954618i	$-0.403736\pi$
$661$	15.3145			0.595666			0.297833	+	0.954618i	$0.403736\pi$
$662$			18.5199i			0.719798i
$663$		0			0
$664$	−2.06808			−0.0802572
$665$	−65.0112	+	8.84216i	−2.52103	+	0.342884i
$666$		0			0
$667$			6.41082i			0.248228i
$668$			4.82164i			0.186555i
$669$		0			0
$670$	−1.56866	−	11.5335i	−0.0606027	−	0.445577i
$671$	5.72535			0.221025
$672$		0			0
$673$		−	10.2870i		−	0.396535i	−0.980148	−	0.198268i	$-0.936468\pi$
$673$		−	10.2870i		−	0.396535i	0.980148	−	0.198268i	$-0.0635315\pi$
$674$	22.2870			0.858464
$675$		0			0
$676$	1.00000			0.0384615
$677$			9.88856i			0.380048i	0.981779	+	0.190024i	$0.0608566\pi$
$677$			9.88856i			0.380048i	−0.981779	+	0.190024i	$0.939143\pi$
$678$		0			0
$679$	49.6680			1.90608
$680$	−2.01026	−	14.7803i	−0.0770899	−	0.566797i
$681$		0			0
$682$		−	10.4108i		−	0.398651i
$683$		−	2.06808i		−	0.0791330i	−0.999217	−	0.0395665i	$-0.987402\pi$
$683$		−	2.06808i		−	0.0791330i	0.999217	−	0.0395665i	$-0.0125977\pi$
$684$		0			0
$685$	−24.8275	+	3.37678i	−0.948609	+	0.129020i
$686$	2.81511			0.107481
$687$		0			0
$688$		−	5.80811i		−	0.221432i
$689$	0.794590			0.0302715
$690$		0			0
$691$	15.8205			0.601839			0.300920	−	0.953649i	$-0.402706\pi$
$691$	15.8205			0.601839			0.300920	+	0.953649i	$0.402706\pi$
$692$		−	12.7946i		−	0.486377i
$693$		0			0
$694$	31.5335			1.19699
$695$	−2.66461	−	19.5913i	−0.101074	−	0.743140i
$696$		0			0
$697$			45.7794i			1.73402i
$698$		−	32.3551i		−	1.22466i
$699$		0			0
$700$	−4.85649	−	17.5232i	−0.183558	−	0.662315i
$701$	−14.6561			−0.553553			−0.276777	−	0.960934i	$-0.589266\pi$
$701$	−14.6561			−0.553553			−0.276777	+	0.960934i	$0.589266\pi$
$702$		0			0
$703$			3.48006i			0.131253i
$704$	2.00000			0.0753778
$705$		0			0
$706$	20.0270			0.753728
$707$		−	57.0399i		−	2.14521i
$708$		0			0
$709$	23.3145			0.875595			0.437798	−	0.899073i	$-0.355759\pi$
$709$	23.3145			0.875595			0.437798	+	0.899073i	$0.355759\pi$
$710$	−20.9721	+	2.85242i	−0.787070	+	0.107049i
$711$		0			0
$712$			8.06808i			0.302364i
$713$		−	4.13617i		−	0.154901i
$714$		0			0
$715$	4.43134	−	0.602705i	0.165723	−	0.0225399i
$716$	8.84216			0.330447
$717$		0			0
$718$			22.9308i			0.855768i
$719$	−29.3416			−1.09426			−0.547128	−	0.837049i	$-0.684279\pi$
$719$	−29.3416			−1.09426			−0.547128	+	0.837049i	$0.684279\pi$
$720$		0			0
$721$	−55.7934			−2.07786
$722$			46.0940i			1.71544i
$723$		0			0
$724$	−1.61623			−0.0600667
$725$	38.8750	−	10.7741i	1.44378	−	0.400139i
$726$		0			0
$727$		−	15.4530i		−	0.573121i	−0.958062	−	0.286560i	$-0.907488\pi$
$727$		−	15.4530i		−	0.573121i	0.958062	−	0.286560i	$-0.0925119\pi$
$728$		−	3.63675i		−	0.134787i
$729$		0			0
$730$	1.80811	+	13.2940i	0.0669213	+	0.492033i
$731$	−38.7447			−1.43302
$732$		0			0
$733$		−	30.2108i		−	1.11586i	−0.829888	−	0.557930i	$-0.811596\pi$
$733$		−	30.2108i		−	1.11586i	0.829888	−	0.557930i	$-0.188404\pi$
$734$	34.1362			1.25999
$735$		0			0
$736$	0.794590			0.0292890
$737$		−	10.4108i		−	0.383487i
$738$		0			0
$739$	24.5880			0.904485			0.452242	−	0.891895i	$-0.350624\pi$
$739$	24.5880			0.904485			0.452242	+	0.891895i	$0.350624\pi$
$740$	−0.955700	+	0.129984i	−0.0351322	+	0.00477832i
$741$		0			0
$742$		−	2.88972i		−	0.106085i
$743$		−	48.0452i		−	1.76261i	−0.472549	−	0.881305i	$-0.656666\pi$
$743$		−	48.0452i		−	1.76261i	0.472549	−	0.881305i	$-0.343334\pi$
$744$		0			0
$745$		0			0
$746$	26.9988			0.988498
$747$		0			0
$748$		−	13.3416i		−	0.487816i
$749$	4.38377			0.160179
$750$		0			0
$751$	−4.90371			−0.178939			−0.0894694	−	0.995990i	$-0.528517\pi$
$751$	−4.90371			−0.178939			−0.0894694	+	0.995990i	$0.528517\pi$
$752$		−	7.63675i		−	0.278484i
$753$		0			0
$754$	8.06808			0.293822
$755$	2.12591	+	15.6306i	0.0773697	+	0.568854i
$756$		0			0
$757$			11.3416i			0.412217i	0.978529	+	0.206108i	$0.0660799\pi$
$757$			11.3416i			0.412217i	−0.978529	+	0.206108i	$0.933920\pi$
$758$		−	12.9308i		−	0.469666i
$759$		0			0
$760$	−17.8762	+	2.43134i	−0.648438	+	0.0881939i
$761$	−10.3427			−0.374924			−0.187462	−	0.982272i	$-0.560026\pi$
$761$	−10.3427			−0.374924			−0.187462	+	0.982272i	$0.560026\pi$
$762$		0			0
$763$			35.6696i			1.29133i
$764$	21.3416			0.772111
$765$		0			0
$766$	−21.3621			−0.771844
$767$			8.06808i			0.291322i
$768$		0			0
$769$	−27.6843			−0.998322			−0.499161	−	0.866509i	$-0.666358\pi$
$769$	−27.6843			−0.998322			−0.499161	+	0.866509i	$0.666358\pi$
$770$	−2.19189	−	16.1157i	−0.0789901	−	0.580768i
$771$		0			0
$772$			5.61623i			0.202133i
$773$			12.2952i			0.442227i	0.975248	+	0.221113i	$0.0709690\pi$
$773$			12.2952i			0.442227i	−0.975248	+	0.221113i	$0.929031\pi$
$774$		0			0
$775$	−25.0816	+	6.95127i	−0.900958	+	0.249697i
$776$	13.6573			0.490267
$777$		0			0
$778$		−	29.4507i		−	1.05586i
$779$	55.3686			1.98379
$780$		0			0
$781$	−18.9308			−0.677396
$782$		−	5.30054i		−	0.189547i
$783$		0			0
$784$	−6.22593			−0.222355
$785$	−0.150851	+	0.0205172i	−0.00538410	+	0.000732290i
$786$		0			0
$787$			43.5048i			1.55078i	0.631484	+	0.775389i	$0.282446\pi$
$787$			43.5048i			1.55078i	−0.631484	+	0.775389i	$0.717554\pi$
$788$		−	14.0205i		−	0.499460i
$789$		0			0
$790$	4.28049	−	0.582188i	0.152293	−	0.0207133i
$791$	−14.5470			−0.517231
$792$		0			0
$793$		−	2.86267i		−	0.101657i
$794$	16.6832			0.592063
$795$		0			0
$796$	−15.7524			−0.558329
$797$			12.9718i			0.459484i	0.973252	+	0.229742i	$0.0737883\pi$
$797$			12.9718i			0.459484i	−0.973252	+	0.229742i	$0.926212\pi$
$798$		0			0
$799$	−50.9431			−1.80224
$800$	−1.33539	−	4.81837i	−0.0472133	−	0.170355i
$801$		0			0
$802$			7.34158i			0.259240i
$803$			12.0000i			0.423471i
$804$		0			0
$805$	−0.870825	−	6.40267i	−0.0306926	−	0.225664i
$806$	−5.20541			−0.183353
$807$		0			0
$808$		−	15.6843i		−	0.551772i
$809$	8.73304			0.307037			0.153519	−	0.988146i	$-0.450939\pi$
$809$	8.73304			0.307037			0.153519	+	0.988146i	$0.450939\pi$
$810$		0			0
$811$	46.3134			1.62628			0.813141	−	0.582067i	$-0.197756\pi$
$811$	46.3134			1.62628			0.813141	+	0.582067i	$0.197756\pi$
$812$		−	29.3416i		−	1.02969i
$813$		0			0
$814$	−0.862674			−0.0302367
$815$	16.9660	−	2.30754i	0.594292	−	0.0808294i
$816$		0			0
$817$			46.8604i			1.63944i
$818$			35.4777i			1.24045i
$819$		0			0
$820$	2.06808	+	15.2054i	0.0722206	+	0.530996i
$821$	−31.9172			−1.11392			−0.556960	−	0.830540i	$-0.688032\pi$
$821$	−31.9172			−1.11392			−0.556960	+	0.830540i	$0.688032\pi$
$822$		0			0
$823$			6.42480i			0.223955i	0.993711	+	0.111977i	$0.0357184\pi$
$823$			6.42480i			0.223955i	−0.993711	+	0.111977i	$0.964282\pi$
$824$	−15.3416			−0.534449
$825$		0			0
$826$	29.3416			1.02092
$827$			38.7242i			1.34657i	0.739382	+	0.673286i	$0.235118\pi$
$827$			38.7242i			1.34657i	−0.739382	+	0.673286i	$0.764882\pi$
$828$		0			0
$829$	18.3427			0.637070			0.318535	−	0.947911i	$-0.396809\pi$
$829$	18.3427			0.637070			0.318535	+	0.947911i	$0.396809\pi$
$830$	0.623222	+	4.58219i	0.0216324	+	0.159050i
$831$		0			0
$832$		−	1.00000i		−	0.0346688i
$833$			41.5318i			1.43899i
$834$		0			0
$835$	10.6832	−	1.45301i	0.369706	−	0.0502836i
$836$	−16.1362			−0.558081
$837$		0			0
$838$			0.883191i			0.0305093i
$839$	−12.6561			−0.436937			−0.218469	−	0.975844i	$-0.570106\pi$
$839$	−12.6561			−0.436937			−0.218469	+	0.975844i	$0.570106\pi$
$840$		0			0
$841$	36.0940			1.24462
$842$			27.2859i			0.940333i
$843$		0			0
$844$	1.70483			0.0586827
$845$	−0.301352	−	2.21567i	−0.0103668	−	0.0762213i
$846$		0			0
$847$			25.4572i			0.874721i
$848$		−	0.794590i		−	0.0272863i
$849$		0			0
$850$	−32.1424	+	8.90813i	−1.10247	+	0.305546i
$851$	−0.342736			−0.0117488
$852$		0			0
$853$		−	44.7988i		−	1.53388i	−0.641718	−	0.766941i	$-0.721778\pi$
$853$		−	44.7988i		−	1.53388i	0.641718	−	0.766941i	$-0.278222\pi$
$854$	−10.4108			−0.356251
$855$		0			0
$856$	1.20541			0.0412001
$857$		−	46.8193i		−	1.59932i	−0.600455	−	0.799659i	$-0.705014\pi$
$857$		−	46.8193i		−	1.59932i	0.600455	−	0.799659i	$-0.294986\pi$
$858$		0			0
$859$	23.8638			0.814223			0.407112	−	0.913378i	$-0.366536\pi$
$859$	23.8638			0.814223			0.407112	+	0.913378i	$0.366536\pi$
$860$	−12.8689	+	1.75029i	−0.438824	+	0.0596844i
$861$		0			0
$862$		−	0.259969i		−	0.00885458i
$863$		−	12.2271i		−	0.416215i	−0.978106	−	0.208107i	$-0.933270\pi$
$863$		−	12.2271i		−	0.416215i	0.978106	−	0.208107i	$-0.0667304\pi$
$864$		0			0
$865$	−28.3486	+	3.85568i	−0.963880	+	0.131097i
$866$	17.4654			0.593498
$867$		0			0
$868$			18.9308i			0.642552i
$869$	3.86383			0.131072
$870$		0			0
$871$	−5.20541			−0.176379
$872$			9.80811i			0.332145i
$873$		0			0
$874$	−6.41082			−0.216849
$875$	−37.3621	+	16.0410i	−1.26307	+	0.542286i
$876$		0			0
$877$			0.336204i			0.0113528i	0.999984	+	0.00567640i	$0.00180686\pi$
$877$			0.336204i			0.0113528i	−0.999984	+	0.00567640i	$0.998193\pi$
$878$		−	17.6843i		−	0.596817i
$879$		0			0
$880$	−0.602705	−	4.43134i	−0.0203172	−	0.149380i
$881$	41.0464			1.38289			0.691444	−	0.722430i	$-0.256975\pi$
$881$	41.0464			1.38289			0.691444	+	0.722430i	$0.256975\pi$
$882$		0			0
$883$			40.0534i			1.34790i	0.738775	+	0.673952i	$0.235405\pi$
$883$			40.0534i			1.34790i	−0.738775	+	0.673952i	$0.764595\pi$
$884$	−6.67079			−0.224363
$885$		0			0
$886$	32.4913			1.09157
$887$			28.7242i			0.964464i	0.876044	+	0.482232i	$0.160174\pi$
$887$			28.7242i			0.964464i	−0.876044	+	0.482232i	$0.839826\pi$
$888$		0			0
$889$	−17.4367			−0.584808
$890$	17.8762	−	2.43134i	0.599212	−	0.0814986i
$891$		0			0
$892$		−	22.9513i		−	0.768466i
$893$			61.6139i			2.06183i
$894$		0			0
$895$	−2.66461	−	19.5913i	−0.0890679	−	0.654865i
$896$	−3.63675			−0.121495
$897$		0			0
$898$			35.0940i			1.17110i
$899$	−41.9977			−1.40070
$900$		0			0
$901$	−5.30054			−0.176587
$902$			13.7253i			0.457004i
$903$		0			0
$904$	−4.00000			−0.133038
$905$	0.487055	+	3.58103i	0.0161902	+	0.119037i
$906$		0			0
$907$		−	48.7636i		−	1.61917i	−0.587003	−	0.809584i	$-0.699692\pi$
$907$		−	48.7636i		−	1.61917i	0.587003	−	0.809584i	$-0.300308\pi$
$908$		−	10.5470i		−	0.350014i
$909$		0			0
$910$	−8.05783	+	1.09594i	−0.267114	+	0.0363301i
$911$	−15.4801			−0.512877			−0.256439	−	0.966561i	$-0.582549\pi$
$911$	−15.4801			−0.512877			−0.256439	+	0.966561i	$0.582549\pi$
$912$		0			0
$913$			4.13617i			0.136887i
$914$	20.5470			0.679634
$915$		0			0
$916$	25.9443			0.857223
$917$		−	14.6216i		−	0.482848i
$918$		0			0
$919$	−45.8615			−1.51283			−0.756416	−	0.654091i	$-0.773051\pi$
$919$	−45.8615			−1.51283			−0.756416	+	0.654091i	$0.773051\pi$
$920$	−0.239452	−	1.76055i	−0.00789449	−	0.0580436i
$921$		0			0
$922$			7.14969i			0.235463i
$923$			9.46538i			0.311557i
$924$		0			0
$925$	0.576005	+	2.07834i	0.0189389	+	0.0683355i
$926$	−22.6832			−0.745415
$927$		0			0
$928$		−	8.06808i		−	0.264848i
$929$	−26.9168			−0.883111			−0.441555	−	0.897234i	$-0.645573\pi$
$929$	−26.9168			−0.883111			−0.441555	+	0.897234i	$0.645573\pi$
$930$		0			0
$931$	50.2313			1.64626
$932$			11.4924i			0.376447i
$933$		0			0
$934$	25.2054			0.824746
$935$	−29.5605	+	4.02052i	−0.966732	+	0.131485i
$936$		0			0
$937$		−	4.90371i		−	0.160197i	−0.996787	−	0.0800986i	$-0.974476\pi$
$937$		−	4.90371i		−	0.160197i	0.996787	−	0.0800986i	$-0.0255235\pi$
$938$			18.9308i			0.618111i
$939$		0			0
$940$	−16.9205	+	2.30135i	−0.551886	+	0.0750618i
$941$	46.6004			1.51913			0.759565	−	0.650432i	$-0.225412\pi$
$941$	46.6004			1.51913			0.759565	+	0.650432i	$0.225412\pi$
$942$		0			0
$943$			5.45301i			0.177575i
$944$	8.06808			0.262594
$945$		0			0
$946$	−11.6162			−0.377676
$947$		−	27.7524i		−	0.901832i	−0.892566	−	0.450916i	$-0.851097\pi$
$947$		−	27.7524i		−	0.901832i	0.892566	−	0.450916i	$-0.148903\pi$
$948$		0			0
$949$	6.00000			0.194768
$950$	10.7741	+	38.8750i	0.349557	+	1.26127i
$951$		0			0
$952$			24.2600i			0.786270i
$953$		−	14.9725i		−	0.485007i	−0.970151	−	0.242503i	$-0.922031\pi$
$953$		−	14.9725i		−	0.485007i	0.970151	−	0.242503i	$-0.0779685\pi$
$954$		0			0
$955$	−6.43134	−	47.2859i	−0.208113	−	1.53013i
$956$	−14.6708			−0.474487
$957$		0			0
$958$			4.01237i			0.129634i
$959$	40.7512			1.31593
$960$		0			0
$961$	−3.90371			−0.125926
$962$			0.431337i			0.0139069i
$963$		0			0
$964$	18.6151			0.599551
$965$	12.4437	−	1.69246i	0.400577	−	0.0544824i
$966$		0			0
$967$		−	52.5946i		−	1.69133i	−0.533717	−	0.845663i	$-0.679205\pi$
$967$		−	52.5946i		−	1.69133i	0.533717	−	0.845663i	$-0.320795\pi$
$968$			7.00000i			0.224989i
$969$		0			0
$970$	−4.11565	−	30.2600i	−0.132146	−	0.971589i
$971$	15.5253			0.498231			0.249115	−	0.968474i	$-0.419860\pi$
$971$	15.5253			0.498231			0.249115	+	0.968474i	$0.419860\pi$
$972$		0			0
$973$			32.1567i			1.03090i
$974$	8.00000			0.256337
$975$		0			0
$976$	−2.86267			−0.0916320
$977$			47.5729i			1.52199i	0.648757	+	0.760996i	$0.275289\pi$
$977$			47.5729i			1.52199i	−0.648757	+	0.760996i	$0.724711\pi$
$978$		0			0
$979$	16.1362			0.515714
$980$	1.87620	+	13.7946i	0.0599330	+	0.440652i
$981$		0			0
$982$			8.88319i			0.283474i
$983$		−	37.3211i		−	1.19036i	−0.803594	−	0.595178i	$-0.797081\pi$
$983$		−	37.3211i		−	1.19036i	0.803594	−	0.595178i	$-0.202919\pi$
$984$		0			0
$985$	−31.0648	+	4.22512i	−0.989807	+	0.134623i
$986$	−53.8205			−1.71399
$987$		0			0
$988$			8.06808i			0.256680i
$989$	−4.61507			−0.146751
$990$		0			0
$991$	22.5060			0.714925			0.357463	−	0.933927i	$-0.383642\pi$
$991$	22.5060			0.714925			0.357463	+	0.933927i	$0.383642\pi$
$992$			5.20541i			0.165272i
$993$		0			0
$994$	34.4232			1.09184
$995$	4.74702	+	34.9021i	0.150491	+	1.10647i
$996$		0			0
$997$		−	24.4108i		−	0.773098i	−0.922269	−	0.386549i	$-0.873667\pi$
$997$		−	24.4108i		−	0.773098i	0.922269	−	0.386549i	$-0.126333\pi$
$998$			10.9988i			0.348162i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.e.f.469.5		6
3.2	odd	2		130.2.b.a.79.2	✓	6
5.2	odd	4		5850.2.a.cp.1.3		3
5.3	odd	4		5850.2.a.cs.1.1		3
5.4	even	2	inner	1170.2.e.f.469.2		6
12.11	even	2		1040.2.d.b.209.3		6
15.2	even	4		650.2.a.o.1.2		3
15.8	even	4		650.2.a.n.1.2		3
15.14	odd	2		130.2.b.a.79.5	yes	6
39.5	even	4		1690.2.c.a.1689.4		6
39.8	even	4		1690.2.c.d.1689.4		6
39.38	odd	2		1690.2.b.a.339.5		6
60.23	odd	4		5200.2.a.ce.1.2		3
60.47	odd	4		5200.2.a.cf.1.2		3
60.59	even	2		1040.2.d.b.209.4		6
195.38	even	4		8450.2.a.cc.1.2		3
195.44	even	4		1690.2.c.d.1689.3		6
195.77	even	4		8450.2.a.bs.1.2		3
195.164	even	4		1690.2.c.a.1689.3		6
195.194	odd	2		1690.2.b.a.339.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.b.a.79.2	✓	6	3.2	odd	2
130.2.b.a.79.5	yes	6	15.14	odd	2
650.2.a.n.1.2		3	15.8	even	4
650.2.a.o.1.2		3	15.2	even	4
1040.2.d.b.209.3		6	12.11	even	2
1040.2.d.b.209.4		6	60.59	even	2
1170.2.e.f.469.2		6	5.4	even	2	inner
1170.2.e.f.469.5		6	1.1	even	1	trivial
1690.2.b.a.339.2		6	195.194	odd	2
1690.2.b.a.339.5		6	39.38	odd	2
1690.2.c.a.1689.3		6	195.164	even	4
1690.2.c.a.1689.4		6	39.5	even	4
1690.2.c.d.1689.3		6	195.44	even	4
1690.2.c.d.1689.4		6	39.8	even	4
5200.2.a.ce.1.2		3	60.23	odd	4
5200.2.a.cf.1.2		3	60.47	odd	4
5850.2.a.cp.1.3		3	5.2	odd	4
5850.2.a.cs.1.1		3	5.3	odd	4
8450.2.a.bs.1.2		3	195.77	even	4
8450.2.a.cc.1.2		3	195.38	even	4

Overview	Random
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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

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Newspace parameters

Newform invariants

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$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1170.2.e.f.469.5

Level	$1170$
Weight	$2$
Character	1170.469
Analytic conductor	$9.342$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$