LMFDB - Embedded newform 1520.2.j.e.911.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(911,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1520.911");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.j (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.433300527120384.59
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4x^{6} - 10x^{4} + 676x^{2} + 28561 $ x^8 + 4x^6 - 10x^4 + 676*x^2 + 28561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			911.5
Root			$-1.14437 - 3.41912i$ of defining polynomial
Character	$\chi$	$=$	1520.911
Dual form			1520.2.j.e.911.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.14437 q^{3} +1.00000 q^{5} -0.786873i q^{7} -1.69042 q^{9} -0.786873i q^{11} -3.91275i q^{13} +1.14437 q^{15} +(-1.14437 - 4.20600i) q^{19} -0.900474i q^{21} -6.05138i q^{23} +1.00000 q^{25} -5.36757 q^{27} +7.65632 q^{31} -0.900474i q^{33} -0.786873i q^{35} +4.81322i q^{37} -4.47763i q^{39} -8.72597i q^{41} -7.62512i q^{43} -1.69042 q^{45} -0.786873i q^{47} +6.38083 q^{49} +12.6387i q^{53} -0.786873i q^{55} +(-1.30958 - 4.81322i) q^{57} +9.94506 q^{59} -4.69042 q^{61} +1.33014i q^{63} -3.91275i q^{65} +11.0894 q^{67} -6.92502i q^{69} -9.94506 q^{71} -7.38083 q^{73} +1.14437 q^{75} -0.619168 q^{77} -7.65632 q^{79} -1.07125 q^{81} -12.8896i q^{83} +17.4519i q^{89} -3.07883 q^{91} +8.76166 q^{93} +(-1.14437 - 4.20600i) q^{95} +12.6387i q^{97} +1.33014i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} + 24 q^{9} + 8 q^{25} + 24 q^{45} - 24 q^{49} - 48 q^{57} + 16 q^{73} - 80 q^{77} + 104 q^{81} - 80 q^{93}+O(q^{100}) $ 8 * q + 8 * q^5 + 24 * q^9 + 8 * q^25 + 24 * q^45 - 24 * q^49 - 48 * q^57 + 16 * q^73 - 80 * q^77 + 104 * q^81 - 80 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	1.14437			0.660703			0.330351	−	0.943858i	$-0.392833\pi$
$3$	1.14437			0.660703			0.330351	+	0.943858i	$0.392833\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	0.786873i		−	0.297410i	−0.988882	−	0.148705i	$-0.952490\pi$
$7$		−	0.786873i		−	0.297410i	0.988882	−	0.148705i	$-0.0475105\pi$
$8$		0			0
$9$	−1.69042			−0.563472
$10$		0			0
$11$		−	0.786873i		−	0.237251i	−0.992939	−	0.118626i	$-0.962151\pi$
$11$		−	0.786873i		−	0.237251i	0.992939	−	0.118626i	$-0.0378488\pi$
$12$		0			0
$13$		−	3.91275i		−	1.08520i	−0.839991	−	0.542600i	$-0.817440\pi$
$13$		−	3.91275i		−	1.08520i	0.839991	−	0.542600i	$-0.182560\pi$
$14$		0			0
$15$	1.14437			0.295475
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−1.14437	−	4.20600i	−0.262537	−	0.964922i
$19$	−1.14437	−	4.20600i	−0.262537	−	0.964922i
$20$		0			0
$21$		−	0.900474i		−	0.196500i
$22$		0			0
$23$		−	6.05138i		−	1.26180i	−0.775864	−	0.630900i	$-0.782686\pi$
$23$		−	6.05138i		−	1.26180i	0.775864	−	0.630900i	$-0.217314\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−5.36757			−1.03299
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	7.65632			1.37511			0.687557	−	0.726130i	$-0.258683\pi$
$31$	7.65632			1.37511			0.687557	+	0.726130i	$0.258683\pi$
$32$		0			0
$33$		−	0.900474i		−	0.156752i
$34$		0			0
$35$		−	0.786873i		−	0.133006i
$36$		0			0
$37$			4.81322i			0.791289i	0.918404	+	0.395644i	$0.129479\pi$
$37$			4.81322i			0.791289i	−0.918404	+	0.395644i	$0.870521\pi$
$38$		0			0
$39$		−	4.47763i		−	0.716995i
$40$		0			0
$41$		−	8.72597i		−	1.36277i	−0.731927	−	0.681384i	$-0.761379\pi$
$41$		−	8.72597i		−	1.36277i	0.731927	−	0.681384i	$-0.238621\pi$
$42$		0			0
$43$		−	7.62512i		−	1.16282i	−0.813611	−	0.581410i	$-0.802501\pi$
$43$		−	7.62512i		−	1.16282i	0.813611	−	0.581410i	$-0.197499\pi$
$44$		0			0
$45$	−1.69042			−0.251992
$46$		0			0
$47$		−	0.786873i		−	0.114777i	−0.998352	−	0.0573886i	$-0.981723\pi$
$47$		−	0.786873i		−	0.114777i	0.998352	−	0.0573886i	$-0.0182774\pi$
$48$		0			0
$49$	6.38083			0.911547
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.6387i			1.73606i	0.496511	+	0.868030i	$0.334614\pi$
$53$			12.6387i			1.73606i	−0.496511	+	0.868030i	$0.665386\pi$
$54$		0			0
$55$		−	0.786873i		−	0.106102i
$56$		0			0
$57$	−1.30958	−	4.81322i	−0.173459	−	0.637527i
$58$		0			0
$59$	9.94506			1.29474			0.647368	−	0.762178i	$-0.275870\pi$
$59$	9.94506			1.29474			0.647368	+	0.762178i	$0.275870\pi$
$60$		0			0
$61$	−4.69042			−0.600546			−0.300273	−	0.953853i	$-0.597078\pi$
$61$	−4.69042			−0.600546			−0.300273	+	0.953853i	$0.597078\pi$
$62$		0			0
$63$			1.33014i			0.167582i
$64$		0			0
$65$		−	3.91275i		−	0.485316i
$66$		0			0
$67$	11.0894			1.35479			0.677395	−	0.735620i	$-0.263109\pi$
$67$	11.0894			1.35479			0.677395	+	0.735620i	$0.263109\pi$
$68$		0			0
$69$		−	6.92502i		−	0.833674i
$70$		0			0
$71$	−9.94506			−1.18026			−0.590131	−	0.807308i	$-0.700924\pi$
$71$	−9.94506			−1.18026			−0.590131	+	0.807308i	$0.700924\pi$
$72$		0			0
$73$	−7.38083			−0.863861			−0.431930	−	0.901907i	$-0.642167\pi$
$73$	−7.38083			−0.863861			−0.431930	+	0.901907i	$0.642167\pi$
$74$		0			0
$75$	1.14437			0.132141
$76$		0			0
$77$	−0.619168			−0.0705608
$78$		0			0
$79$	−7.65632			−0.861403			−0.430701	−	0.902494i	$-0.641734\pi$
$79$	−7.65632			−0.861403			−0.430701	+	0.902494i	$0.641734\pi$
$80$		0			0
$81$	−1.07125			−0.119027
$82$		0			0
$83$		−	12.8896i		−	1.41482i	−0.706803	−	0.707410i	$-0.749863\pi$
$83$		−	12.8896i		−	1.41482i	0.706803	−	0.707410i	$-0.250137\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			17.4519i			1.84990i	0.380087	+	0.924951i	$0.375894\pi$
$89$			17.4519i			1.84990i	−0.380087	+	0.924951i	$0.624106\pi$
$90$		0			0
$91$	−3.07883			−0.322749
$92$		0			0
$93$	8.76166			0.908542
$94$		0			0
$95$	−1.14437	−	4.20600i	−0.117410	−	0.431526i
$96$		0			0
$97$			12.6387i			1.28327i	0.767011	+	0.641633i	$0.221743\pi$
$97$			12.6387i			1.28327i	−0.767011	+	0.641633i	$0.778257\pi$
$98$		0			0
$99$			1.33014i			0.133684i
$100$		0			0
$101$	2.69042			0.267706			0.133853	−	0.991001i	$-0.457265\pi$
$101$	2.69042			0.267706			0.133853	+	0.991001i	$0.457265\pi$
$102$		0			0
$103$	15.6669			1.54371			0.771853	−	0.635801i	$-0.219330\pi$
$103$	15.6669			1.54371			0.771853	+	0.635801i	$0.219330\pi$
$104$		0			0
$105$		−	0.900474i		−	0.0878773i
$106$		0			0
$107$	−6.51194			−0.629533			−0.314767	−	0.949169i	$-0.601926\pi$
$107$	−6.51194			−0.629533			−0.314767	+	0.949169i	$0.601926\pi$
$108$		0			0
$109$			8.72597i			0.835796i	0.908494	+	0.417898i	$0.137233\pi$
$109$			8.72597i			0.835796i	−0.908494	+	0.417898i	$0.862767\pi$
$110$		0			0
$111$			5.50811i			0.522806i
$112$		0			0
$113$		−	11.7382i		−	1.10424i	−0.833764	−	0.552120i	$-0.813819\pi$
$113$		−	11.7382i		−	1.10424i	0.833764	−	0.552120i	$-0.186181\pi$
$114$		0			0
$115$		−	6.05138i		−	0.564294i
$116$		0			0
$117$			6.61417i			0.611480i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.3808			0.943712
$122$		0			0
$123$		−	9.98574i		−	0.900384i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−5.72185			−0.507732			−0.253866	−	0.967239i	$-0.581702\pi$
$127$	−5.72185			−0.507732			−0.253866	+	0.967239i	$0.581702\pi$
$128$		0			0
$129$		−	8.72597i		−	0.768279i
$130$		0			0
$131$			1.57375i			0.137499i	0.997634	+	0.0687494i	$0.0219009\pi$
$131$			1.57375i			0.137499i	−0.997634	+	0.0687494i	$0.978099\pi$
$132$		0			0
$133$	−3.30958	+	0.900474i	−0.286977	+	0.0780810i
$134$		0			0
$135$	−5.36757			−0.461967
$136$		0			0
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$		−	12.8896i		−	1.09328i	−0.837366	−	0.546642i	$-0.815906\pi$
$139$		−	12.8896i		−	1.09328i	0.837366	−	0.546642i	$-0.184094\pi$
$140$		0			0
$141$		−	0.900474i		−	0.0758336i
$142$		0			0
$143$	−3.07883			−0.257465
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.30204			0.602262
$148$		0			0
$149$	−8.69042			−0.711947			−0.355973	−	0.934496i	$-0.615851\pi$
$149$	−8.69042			−0.711947			−0.355973	+	0.934496i	$0.615851\pi$
$150$		0			0
$151$	−4.57748			−0.372510			−0.186255	−	0.982501i	$-0.559635\pi$
$151$	−4.57748			−0.372510			−0.186255	+	0.982501i	$0.559635\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.65632			0.614970
$156$		0			0
$157$	−1.38083			−0.110202			−0.0551012	−	0.998481i	$-0.517548\pi$
$157$	−1.38083			−0.110202			−0.0551012	+	0.998481i	$0.517548\pi$
$158$		0			0
$159$			14.4634i			1.14702i
$160$		0			0
$161$	−4.76166			−0.375272
$162$		0			0
$163$		−	22.8754i		−	1.79174i	−0.444319	−	0.895869i	$-0.646554\pi$
$163$		−	22.8754i		−	1.79174i	0.444319	−	0.895869i	$-0.353446\pi$
$164$		0			0
$165$		−	0.900474i		−	0.0701018i
$166$		0			0
$167$	16.4570			1.27348			0.636741	−	0.771078i	$-0.280282\pi$
$167$	16.4570			1.27348			0.636741	+	0.771078i	$0.280282\pi$
$168$		0			0
$169$	−2.30958			−0.177660
$170$		0			0
$171$	1.93446	+	7.10988i	0.147932	+	0.543706i
$172$		0			0
$173$			3.91275i			0.297481i	0.988876	+	0.148740i	$0.0475218\pi$
$173$			3.91275i			0.297481i	−0.988876	+	0.148740i	$0.952478\pi$
$174$		0			0
$175$		−	0.786873i		−	0.0594820i
$176$		0			0
$177$	11.3808			0.855436
$178$		0			0
$179$	9.94506			0.743328			0.371664	−	0.928367i	$-0.378787\pi$
$179$	9.94506			0.743328			0.371664	+	0.928367i	$0.378787\pi$
$180$		0			0
$181$		−	7.82549i		−	0.581664i	−0.956774	−	0.290832i	$-0.906068\pi$
$181$		−	7.82549i		−	0.581664i	0.956774	−	0.290832i	$-0.0939321\pi$
$182$		0			0
$183$	−5.36757			−0.396783
$184$		0			0
$185$			4.81322i			0.353875i
$186$		0			0
$187$		0			0
$188$		0			0
$189$			4.22360i			0.307221i
$190$		0			0
$191$			20.5147i			1.48440i	0.670181	+	0.742198i	$0.266216\pi$
$191$			20.5147i			1.48440i	−0.670181	+	0.742198i	$0.733784\pi$
$192$		0			0
$193$			3.01227i			0.216828i	0.994106	+	0.108414i	$0.0345772\pi$
$193$			3.01227i			0.216828i	−0.994106	+	0.108414i	$0.965423\pi$
$194$		0			0
$195$		−	4.47763i		−	0.320650i
$196$		0			0
$197$	5.38083			0.383368			0.191684	−	0.981457i	$-0.438605\pi$
$197$	5.38083			0.383368			0.191684	+	0.981457i	$0.438605\pi$
$198$		0			0
$199$			6.83825i			0.484751i	0.970183	+	0.242375i	$0.0779265\pi$
$199$			6.83825i			0.484751i	−0.970183	+	0.242375i	$0.922073\pi$
$200$		0			0
$201$	12.6904			0.895113
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	8.72597i		−	0.609448i
$206$		0			0
$207$			10.2293i			0.710989i
$208$		0			0
$209$	−3.30958	+	0.900474i	−0.228929	+	0.0622871i
$210$		0			0
$211$	−15.3126			−1.05417			−0.527083	−	0.849814i	$-0.676714\pi$
$211$	−15.3126			−1.05417			−0.527083	+	0.849814i	$0.676714\pi$
$212$		0			0
$213$	−11.3808			−0.779802
$214$		0			0
$215$		−	7.62512i		−	0.520029i
$216$		0			0
$217$		−	6.02454i		−	0.408973i
$218$		0			0
$219$	−8.44641			−0.570755
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.14437			−0.0766327			−0.0383164	−	0.999266i	$-0.512199\pi$
$223$	−1.14437			−0.0766327			−0.0383164	+	0.999266i	$0.512199\pi$
$224$		0			0
$225$	−1.69042			−0.112694
$226$		0			0
$227$	−13.3782			−0.887940			−0.443970	−	0.896042i	$-0.646430\pi$
$227$	−13.3782			−0.887940			−0.443970	+	0.896042i	$0.646430\pi$
$228$		0			0
$229$	−1.30958			−0.0865398			−0.0432699	−	0.999063i	$-0.513778\pi$
$229$	−1.30958			−0.0865398			−0.0432699	+	0.999063i	$0.513778\pi$
$230$		0			0
$231$	−0.708558			−0.0466197
$232$		0			0
$233$	17.3808			1.13866			0.569328	−	0.822110i	$-0.307203\pi$
$233$	17.3808			1.13866			0.569328	+	0.822110i	$0.307203\pi$
$234$		0			0
$235$		−	0.786873i		−	0.0513299i
$236$		0			0
$237$	−8.76166			−0.569131
$238$		0			0
$239$			28.9267i			1.87112i	0.353174	+	0.935558i	$0.385102\pi$
$239$			28.9267i			1.87112i	−0.353174	+	0.935558i	$0.614898\pi$
$240$		0			0
$241$		−	18.3524i		−	1.18218i	−0.806605	−	0.591091i	$-0.798697\pi$
$241$		−	18.3524i		−	1.18218i	0.806605	−	0.591091i	$-0.201303\pi$
$242$		0			0
$243$	14.8768			0.954348
$244$		0			0
$245$	6.38083			0.407656
$246$		0			0
$247$	−16.4570	+	4.47763i	−1.04713	+	0.284905i
$248$		0			0
$249$		−	14.7505i		−	0.934776i
$250$		0			0
$251$			11.5595i			0.729628i	0.931080	+	0.364814i	$0.118867\pi$
$251$			11.5595i			0.729628i	−0.931080	+	0.364814i	$0.881133\pi$
$252$		0			0
$253$	−4.76166			−0.299363
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3.01227i			0.187900i	0.995577	+	0.0939502i	$0.0299494\pi$
$257$			3.01227i			0.187900i	−0.995577	+	0.0939502i	$0.970051\pi$
$258$		0			0
$259$	3.78739			0.235337
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.47763i			0.276103i	0.990425	+	0.138051i	$0.0440839\pi$
$263$			4.47763i			0.276103i	−0.990425	+	0.138051i	$0.955916\pi$
$264$		0			0
$265$			12.6387i			0.776390i
$266$		0			0
$267$			19.9715i			1.22223i
$268$		0			0
$269$		−	18.3524i		−	1.11897i	−0.828842	−	0.559483i	$-0.811000\pi$
$269$		−	18.3524i		−	1.11897i	0.828842	−	0.559483i	$-0.189000\pi$
$270$		0			0
$271$			28.1399i			1.70938i	0.519142	+	0.854688i	$0.326251\pi$
$271$			28.1399i			1.70938i	−0.519142	+	0.854688i	$0.673749\pi$
$272$		0			0
$273$	−3.52333			−0.213241
$274$		0			0
$275$		−	0.786873i		−	0.0474502i
$276$		0			0
$277$	13.3808			0.803976			0.401988	−	0.915645i	$-0.368319\pi$
$277$	13.3808			0.803976			0.401988	+	0.915645i	$0.368319\pi$
$278$		0			0
$279$	−12.9424			−0.774839
$280$		0			0
$281$			16.5515i			0.987377i	0.869639	+	0.493689i	$0.164352\pi$
$281$			16.5515i			0.987377i	−0.869639	+	0.493689i	$0.835648\pi$
$282$		0			0
$283$		−	17.6109i		−	1.04686i	−0.852070	−	0.523429i	$-0.824653\pi$
$283$		−	17.6109i		−	1.04686i	0.852070	−	0.523429i	$-0.175347\pi$
$284$		0			0
$285$	−1.30958	−	4.81322i	−0.0775731	−	0.285111i
$286$		0			0
$287$	−6.86622			−0.405300
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$			14.4634i			0.847858i
$292$		0			0
$293$			3.01227i			0.175979i	0.996121	+	0.0879894i	$0.0280442\pi$
$293$			3.01227i			0.175979i	−0.996121	+	0.0879894i	$0.971956\pi$
$294$		0			0
$295$	9.94506			0.579024
$296$		0			0
$297$			4.22360i			0.245078i
$298$		0			0
$299$	−23.6775			−1.36931
$300$		0			0
$301$	−6.00000			−0.345834
$302$		0			0
$303$	3.07883			0.176874
$304$		0			0
$305$	−4.69042			−0.268572
$306$		0			0
$307$	−11.0894			−0.632907			−0.316454	−	0.948608i	$-0.602492\pi$
$307$	−11.0894			−0.632907			−0.316454	+	0.948608i	$0.602492\pi$
$308$		0			0
$309$	17.9288			1.01993
$310$		0			0
$311$			19.1846i			1.08786i	0.839131	+	0.543930i	$0.183064\pi$
$311$			19.1846i			1.08786i	−0.839131	+	0.543930i	$0.816936\pi$
$312$		0			0
$313$	2.76166			0.156098			0.0780492	−	0.996950i	$-0.475131\pi$
$313$	2.76166			0.156098			0.0780492	+	0.996950i	$0.475131\pi$
$314$		0			0
$315$			1.33014i			0.0749450i
$316$		0			0
$317$		−	5.71369i		−	0.320913i	−0.987043	−	0.160457i	$-0.948703\pi$
$317$		−	5.71369i		−	0.320913i	0.987043	−	0.160457i	$-0.0512966\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−7.45208			−0.415934
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	3.91275i		−	0.217040i
$326$		0			0
$327$			9.98574i			0.552213i
$328$		0			0
$329$	−0.619168			−0.0341359
$330$		0			0
$331$	22.1789			1.21906			0.609530	−	0.792763i	$-0.291358\pi$
$331$	22.1789			1.21906			0.609530	+	0.792763i	$0.291358\pi$
$332$		0			0
$333$		−	8.13634i		−	0.445869i
$334$		0			0
$335$	11.0894			0.605880
$336$		0			0
$337$		−	11.7382i		−	0.639423i	−0.947515	−	0.319711i	$-0.896414\pi$
$337$		−	11.7382i		−	0.639423i	0.947515	−	0.319711i	$-0.103586\pi$
$338$		0			0
$339$		−	13.4329i		−	0.729575i
$340$		0			0
$341$		−	6.02454i		−	0.326247i
$342$		0			0
$343$		−	10.5290i		−	0.568513i
$344$		0			0
$345$		−	6.92502i		−	0.372830i
$346$		0			0
$347$			18.1541i			0.974565i	0.873244	+	0.487282i	$0.162012\pi$
$347$			18.1541i			0.974565i	−0.873244	+	0.487282i	$0.837988\pi$
$348$		0			0
$349$	25.3808			1.35860			0.679302	−	0.733858i	$-0.262282\pi$
$349$	25.3808			1.35860			0.679302	+	0.733858i	$0.262282\pi$
$350$		0			0
$351$			21.0020i			1.12100i
$352$		0			0
$353$	22.7617			1.21148			0.605741	−	0.795662i	$-0.292877\pi$
$353$	22.7617			1.21148			0.605741	+	0.795662i	$0.292877\pi$
$354$		0			0
$355$	−9.94506			−0.527829
$356$		0			0
$357$		0			0
$358$		0			0
$359$			16.0371i			0.846407i	0.906035	+	0.423203i	$0.139094\pi$
$359$			16.0371i			0.846407i	−0.906035	+	0.423203i	$0.860906\pi$
$360$		0			0
$361$	−16.3808	+	9.62644i	−0.862149	+	0.506655i
$362$		0			0
$363$	11.8795			0.623513
$364$		0			0
$365$	−7.38083			−0.386330
$366$		0			0
$367$			9.19887i			0.480177i	0.970751	+	0.240088i	$0.0771765\pi$
$367$			9.19887i			0.480177i	−0.970751	+	0.240088i	$0.922824\pi$
$368$		0			0
$369$			14.7505i			0.767881i
$370$		0			0
$371$	9.94506			0.516322
$372$		0			0
$373$			3.01227i			0.155970i	0.996955	+	0.0779848i	$0.0248486\pi$
$373$			3.01227i			0.155970i	−0.996955	+	0.0779848i	$0.975151\pi$
$374$		0			0
$375$	1.14437			0.0590950
$376$		0			0
$377$		0			0
$378$		0			0
$379$	22.1789			1.13925			0.569626	−	0.821904i	$-0.307088\pi$
$379$	22.1789			1.13925			0.569626	+	0.821904i	$0.307088\pi$
$380$		0			0
$381$	−6.54792			−0.335460
$382$		0			0
$383$	−20.2444			−1.03444			−0.517220	−	0.855852i	$-0.673033\pi$
$383$	−20.2444			−1.03444			−0.517220	+	0.855852i	$0.673033\pi$
$384$		0			0
$385$	−0.619168			−0.0315557
$386$		0			0
$387$			12.8896i			0.655217i
$388$		0			0
$389$	−6.61917			−0.335605			−0.167803	−	0.985821i	$-0.553667\pi$
$389$	−6.61917			−0.335605			−0.167803	+	0.985821i	$0.553667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			1.80095i			0.0908458i
$394$		0			0
$395$	−7.65632			−0.385231
$396$		0			0
$397$	14.0000			0.702640			0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000			0.702640			0.351320	+	0.936255i	$0.385733\pi$
$398$		0			0
$399$	−3.78739	+	1.03048i	−0.189607	+	0.0515883i
$400$		0			0
$401$			1.80095i			0.0899350i	0.998988	+	0.0449675i	$0.0143184\pi$
$401$			1.80095i			0.0899350i	−0.998988	+	0.0449675i	$0.985682\pi$
$402$		0			0
$403$		−	29.9572i		−	1.49228i
$404$		0			0
$405$	−1.07125			−0.0532307
$406$		0			0
$407$	3.78739			0.187734
$408$		0			0
$409$			34.0034i			1.68136i	0.541533	+	0.840680i	$0.317844\pi$
$409$			34.0034i			1.68136i	−0.541533	+	0.840680i	$0.682156\pi$
$410$		0			0
$411$	−13.7324			−0.677372
$412$		0			0
$413$		−	7.82549i		−	0.385067i
$414$		0			0
$415$		−	12.8896i		−	0.632727i
$416$		0			0
$417$		−	14.7505i		−	0.722336i
$418$		0			0
$419$			10.5290i			0.514376i	0.966361	+	0.257188i	$0.0827959\pi$
$419$			10.5290i			0.514376i	−0.966361	+	0.257188i	$0.917204\pi$
$420$		0			0
$421$			40.0279i			1.95084i	0.220349	+	0.975421i	$0.429280\pi$
$421$			40.0279i			1.95084i	−0.220349	+	0.975421i	$0.570720\pi$
$422$		0			0
$423$			1.33014i			0.0646737i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			3.69076i			0.178608i
$428$		0			0
$429$	−3.52333			−0.170108
$430$		0			0
$431$	−33.6226			−1.61954			−0.809771	−	0.586746i	$-0.800409\pi$
$431$	−33.6226			−1.61954			−0.809771	+	0.586746i	$0.800409\pi$
$432$		0			0
$433$		−	13.5392i		−	0.650652i	−0.945602	−	0.325326i	$-0.894526\pi$
$433$		−	13.5392i		−	0.650652i	0.945602	−	0.325326i	$-0.105474\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−25.4521	+	6.92502i	−1.21754	+	0.331269i
$438$		0			0
$439$	12.2338			0.583887			0.291944	−	0.956436i	$-0.405698\pi$
$439$	12.2338			0.583887			0.291944	+	0.956436i	$0.405698\pi$
$440$		0			0
$441$	−10.7863			−0.513631
$442$		0			0
$443$		−	4.47763i		−	0.212739i	−0.994327	−	0.106369i	$-0.966077\pi$
$443$		−	4.47763i		−	0.212739i	0.994327	−	0.106369i	$-0.0339226\pi$
$444$		0			0
$445$			17.4519i			0.827301i
$446$		0			0
$447$	−9.94506			−0.470385
$448$		0			0
$449$			10.5269i			0.496796i	0.968658	+	0.248398i	$0.0799040\pi$
$449$			10.5269i			0.496796i	−0.968658	+	0.248398i	$0.920096\pi$
$450$		0			0
$451$	−6.86622			−0.323318
$452$		0			0
$453$	−5.23834			−0.246119
$454$		0			0
$455$	−3.07883			−0.144338
$456$		0			0
$457$	−30.1425			−1.41001			−0.705003	−	0.709204i	$-0.749054\pi$
$457$	−30.1425			−1.41001			−0.705003	+	0.709204i	$0.749054\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	23.3808			1.08895			0.544477	−	0.838776i	$-0.316728\pi$
$461$	23.3808			1.08895			0.544477	+	0.838776i	$0.316728\pi$
$462$		0			0
$463$		−	7.62512i		−	0.354369i	−0.984178	−	0.177185i	$-0.943301\pi$
$463$		−	7.62512i		−	0.354369i	0.984178	−	0.177185i	$-0.0566990\pi$
$464$		0			0
$465$	8.76166			0.406312
$466$		0			0
$467$			9.19887i			0.425673i	0.977088	+	0.212836i	$0.0682701\pi$
$467$			9.19887i			0.425673i	−0.977088	+	0.212836i	$0.931730\pi$
$468$		0			0
$469$		−	8.72597i		−	0.402928i
$470$		0			0
$471$	−1.58018			−0.0728110
$472$		0			0
$473$	−6.00000			−0.275880
$474$		0			0
$475$	−1.14437	−	4.20600i	−0.0525073	−	0.192984i
$476$		0			0
$477$		−	21.3647i		−	0.978221i
$478$		0			0
$479$			17.6109i			0.804661i	0.915495	+	0.402330i	$0.131800\pi$
$479$			17.6109i			0.804661i	−0.915495	+	0.402330i	$0.868200\pi$
$480$		0			0
$481$	18.8329			0.858707
$482$		0			0
$483$	−5.44911			−0.247943
$484$		0			0
$485$			12.6387i			0.573894i
$486$		0			0
$487$	17.9557			0.813648			0.406824	−	0.913507i	$-0.366636\pi$
$487$	17.9557			0.813648			0.406824	+	0.913507i	$0.366636\pi$
$488$		0			0
$489$		−	26.1779i		−	1.18381i
$490$		0			0
$491$			35.7650i			1.61405i	0.590516	+	0.807026i	$0.298924\pi$
$491$			35.7650i			1.61405i	−0.590516	+	0.807026i	$0.701076\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$			1.33014i			0.0597854i
$496$		0			0
$497$			7.82549i			0.351021i
$498$		0			0
$499$		−	32.8611i		−	1.47107i	−0.677489	−	0.735533i	$-0.736932\pi$
$499$		−	32.8611i		−	1.47107i	0.677489	−	0.735533i	$-0.263068\pi$
$500$		0			0
$501$	18.8329			0.841392
$502$		0			0
$503$		−	15.0066i		−	0.669113i	−0.942376	−	0.334557i	$-0.891413\pi$
$503$		−	15.0066i		−	0.669113i	0.942376	−	0.334557i	$-0.108587\pi$
$504$		0			0
$505$	2.69042			0.119722
$506$		0			0
$507$	−2.64302			−0.117381
$508$		0			0
$509$		−	23.4765i		−	1.04058i	−0.853991	−	0.520288i	$-0.825824\pi$
$509$		−	23.4765i		−	1.04058i	0.853991	−	0.520288i	$-0.174176\pi$
$510$		0			0
$511$			5.80777i			0.256921i
$512$		0			0
$513$	6.14249	+	22.5760i	0.271198	+	0.996755i
$514$		0			0
$515$	15.6669			0.690367
$516$		0			0
$517$	−0.619168			−0.0272310
$518$		0			0
$519$			4.47763i			0.196546i
$520$		0			0
$521$		−	6.02454i		−	0.263940i	−0.991254	−	0.131970i	$-0.957870\pi$
$521$		−	6.02454i		−	0.263940i	0.991254	−	0.131970i	$-0.0421303\pi$
$522$		0			0
$523$	27.9007			1.22001			0.610006	−	0.792396i	$-0.291167\pi$
$523$	27.9007			1.22001			0.610006	+	0.792396i	$0.291167\pi$
$524$		0			0
$525$		−	0.900474i		−	0.0392999i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−13.6192			−0.592138
$530$		0			0
$531$	−16.8113			−0.729547
$532$		0			0
$533$	−34.1425			−1.47888
$534$		0			0
$535$	−6.51194			−0.281536
$536$		0			0
$537$	11.3808			0.491119
$538$		0			0
$539$		−	5.02090i		−	0.216266i
$540$		0			0
$541$	−28.0712			−1.20688			−0.603439	−	0.797409i	$-0.706203\pi$
$541$	−28.0712			−1.20688			−0.603439	+	0.797409i	$0.706203\pi$
$542$		0			0
$543$		−	8.95526i		−	0.384307i
$544$		0			0
$545$			8.72597i			0.373779i
$546$		0			0
$547$	5.72185			0.244649			0.122324	−	0.992490i	$-0.460965\pi$
$547$	5.72185			0.244649			0.122324	+	0.992490i	$0.460965\pi$
$548$		0			0
$549$	7.92875			0.338391
$550$		0			0
$551$		0			0
$552$		0			0
$553$			6.02454i			0.256190i
$554$		0			0
$555$			5.50811i			0.233806i
$556$		0			0
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$		0			0
$559$	−29.8352			−1.26189
$560$		0			0
$561$		0			0
$562$		0			0
$563$	29.4809			1.24247			0.621236	−	0.783624i	$-0.286631\pi$
$563$	29.4809			1.24247			0.621236	+	0.783624i	$0.286631\pi$
$564$		0			0
$565$		−	11.7382i		−	0.493831i
$566$		0			0
$567$			0.842935i			0.0353999i
$568$		0			0
$569$			26.1779i			1.09743i	0.836008	+	0.548717i	$0.184884\pi$
$569$			26.1779i			1.09743i	−0.836008	+	0.548717i	$0.815116\pi$
$570$		0			0
$571$		−	28.1399i		−	1.17762i	−0.808273	−	0.588809i	$-0.799597\pi$
$571$		−	28.1399i		−	1.17762i	0.808273	−	0.588809i	$-0.200403\pi$
$572$		0			0
$573$			23.4765i			0.980744i
$574$		0			0
$575$		−	6.05138i		−	0.252360i
$576$		0			0
$577$	21.3808			0.890096			0.445048	−	0.895507i	$-0.353187\pi$
$577$	21.3808			0.890096			0.445048	+	0.895507i	$0.353187\pi$
$578$		0			0
$579$			3.44716i			0.143259i
$580$		0			0
$581$	−10.1425			−0.420782
$582$		0			0
$583$	9.94506			0.411882
$584$		0			0
$585$			6.61417i			0.273462i
$586$		0			0
$587$			17.6109i			0.726878i	0.931618	+	0.363439i	$0.118398\pi$
$587$			17.6109i			0.726878i	−0.931618	+	0.363439i	$0.881602\pi$
$588$		0			0
$589$	−8.76166	−	32.2024i	−0.361018	−	1.32688i
$590$		0			0
$591$	6.15767			0.253293
$592$		0			0
$593$	−39.5233			−1.62303			−0.811514	−	0.584333i	$-0.801356\pi$
$593$	−39.5233			−1.62303			−0.811514	+	0.584333i	$0.801356\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			7.82549i			0.320276i
$598$		0			0
$599$	22.9689			0.938486			0.469243	−	0.883069i	$-0.344527\pi$
$599$	22.9689			0.938486			0.469243	+	0.883069i	$0.344527\pi$
$600$		0			0
$601$			16.5515i			0.675148i	0.941299	+	0.337574i	$0.109606\pi$
$601$			16.5515i			0.675148i	−0.941299	+	0.337574i	$0.890394\pi$
$602$		0			0
$603$	−18.7457			−0.763386
$604$		0			0
$605$	10.3808			0.422041
$606$		0			0
$607$	11.0894			0.450106			0.225053	−	0.974347i	$-0.427744\pi$
$607$	11.0894			0.450106			0.225053	+	0.974347i	$0.427744\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.07883			−0.124556
$612$		0			0
$613$	38.1425			1.54056			0.770280	−	0.637705i	$-0.220116\pi$
$613$	38.1425			1.54056			0.770280	+	0.637705i	$0.220116\pi$
$614$		0			0
$615$		−	9.98574i		−	0.402664i
$616$		0			0
$617$	16.7617			0.674799			0.337400	−	0.941362i	$-0.390453\pi$
$617$	16.7617			0.674799			0.337400	+	0.941362i	$0.390453\pi$
$618$		0			0
$619$		−	17.6109i		−	0.707840i	−0.935276	−	0.353920i	$-0.884848\pi$
$619$		−	17.6109i		−	0.707840i	0.935276	−	0.353920i	$-0.115152\pi$
$620$		0			0
$621$			32.4812i			1.30343i
$622$		0			0
$623$	13.7324			0.550179
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−3.78739	+	1.03048i	−0.151254	+	0.0411532i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	2.36062i		−	0.0939747i	−0.998895	−	0.0469874i	$-0.985038\pi$
$631$		−	2.36062i		−	0.0939747i	0.998895	−	0.0469874i	$-0.0149620\pi$
$632$		0			0
$633$	−17.5233			−0.696490
$634$		0			0
$635$	−5.72185			−0.227065
$636$		0			0
$637$		−	24.9666i		−	0.989212i
$638$		0			0
$639$	16.8113			0.665044
$640$		0			0
$641$			16.5515i			0.653743i	0.945069	+	0.326872i	$0.105994\pi$
$641$			16.5515i			0.653743i	−0.945069	+	0.326872i	$0.894006\pi$
$642$		0			0
$643$		−	27.5966i		−	1.08830i	−0.838987	−	0.544152i	$-0.816852\pi$
$643$		−	27.5966i		−	1.08830i	0.838987	−	0.544152i	$-0.183148\pi$
$644$		0			0
$645$		−	8.72597i		−	0.343585i
$646$		0			0
$647$		−	31.2874i		−	1.23003i	−0.788514	−	0.615017i	$-0.789149\pi$
$647$		−	31.2874i		−	1.23003i	0.788514	−	0.615017i	$-0.210851\pi$
$648$		0			0
$649$		−	7.82549i		−	0.307177i
$650$		0			0
$651$		−	6.89431i		−	0.270209i
$652$		0			0
$653$	24.6192			0.963423			0.481711	−	0.876330i	$-0.340015\pi$
$653$	24.6192			0.963423			0.481711	+	0.876330i	$0.340015\pi$
$654$		0			0
$655$			1.57375i			0.0614913i
$656$		0			0
$657$	12.4767			0.486761
$658$		0			0
$659$	−16.8113			−0.654875			−0.327437	−	0.944873i	$-0.606185\pi$
$659$	−16.8113			−0.654875			−0.327437	+	0.944873i	$0.606185\pi$
$660$		0			0
$661$		−	24.3770i		−	0.948154i	−0.880484	−	0.474077i	$-0.842782\pi$
$661$		−	24.3770i		−	0.948154i	0.880484	−	0.474077i	$-0.157218\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.30958	+	0.900474i	−0.128340	+	0.0349189i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.30958			−0.0506315
$670$		0			0
$671$			3.69076i			0.142480i
$672$		0			0
$673$		−	38.8166i		−	1.49627i	−0.663546	−	0.748135i	$-0.730949\pi$
$673$		−	38.8166i		−	1.49627i	0.663546	−	0.748135i	$-0.269051\pi$
$674$		0			0
$675$	−5.36757			−0.206598
$676$		0			0
$677$		−	18.6633i		−	0.717287i	−0.933475	−	0.358644i	$-0.883239\pi$
$677$		−	18.6633i		−	0.717287i	0.933475	−	0.358644i	$-0.116761\pi$
$678$		0			0
$679$	9.94506			0.381656
$680$		0			0
$681$	−15.3096			−0.586665
$682$		0			0
$683$	−30.1894			−1.15517			−0.577584	−	0.816332i	$-0.696004\pi$
$683$	−30.1894			−1.15517			−0.577584	+	0.816332i	$0.696004\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$	−1.49865			−0.0571771
$688$		0			0
$689$	49.4521			1.88397
$690$		0			0
$691$		−	16.0371i		−	0.610081i	−0.952339	−	0.305040i	$-0.901330\pi$
$691$		−	16.0371i		−	0.610081i	0.952339	−	0.305040i	$-0.0986700\pi$
$692$		0			0
$693$	1.04665			0.0397590
$694$		0			0
$695$		−	12.8896i		−	0.488931i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	19.8901			0.752313
$700$		0			0
$701$	9.92875			0.375004			0.187502	−	0.982264i	$-0.439961\pi$
$701$	9.92875			0.375004			0.187502	+	0.982264i	$0.439961\pi$
$702$		0			0
$703$	20.2444	−	5.50811i	0.763532	−	0.207742i
$704$		0			0
$705$		−	0.900474i		−	0.0339138i
$706$		0			0
$707$		−	2.11701i		−	0.0796185i
$708$		0			0
$709$	−18.7617			−0.704609			−0.352305	−	0.935885i	$-0.614602\pi$
$709$	−18.7617			−0.704609			−0.352305	+	0.935885i	$0.614602\pi$
$710$		0			0
$711$	12.9424			0.485376
$712$		0			0
$713$		−	46.3313i		−	1.73512i
$714$		0			0
$715$	−3.07883			−0.115142
$716$		0			0
$717$			33.1029i			1.23625i
$718$		0			0
$719$			24.9924i			0.932059i	0.884769	+	0.466029i	$0.154316\pi$
$719$			24.9924i			0.932059i	−0.884769	+	0.466029i	$0.845684\pi$
$720$		0			0
$721$		−	12.3279i		−	0.459114i
$722$		0			0
$723$		−	21.0020i		−	0.781071i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	43.9334i		−	1.62940i	−0.579883	−	0.814700i	$-0.696902\pi$
$727$		−	43.9334i		−	1.62940i	0.579883	−	0.814700i	$-0.303098\pi$
$728$		0			0
$729$	20.2383			0.749568
$730$		0			0
$731$		0			0
$732$		0			0
$733$	5.23834			0.193482			0.0967412	−	0.995310i	$-0.469158\pi$
$733$	5.23834			0.193482			0.0967412	+	0.995310i	$0.469158\pi$
$734$		0			0
$735$	7.30204			0.269340
$736$		0			0
$737$		−	8.72597i		−	0.321425i
$738$		0			0
$739$		−	38.9125i		−	1.43142i	−0.698398	−	0.715709i	$-0.746103\pi$
$739$		−	38.9125i		−	1.43142i	0.698398	−	0.715709i	$-0.253897\pi$
$740$		0			0
$741$	−18.8329	+	5.12407i	−0.691844	+	0.188237i
$742$		0			0
$743$	−50.0796			−1.83724			−0.918621	−	0.395141i	$-0.870696\pi$
$743$	−50.0796			−1.83724			−0.918621	+	0.395141i	$0.870696\pi$
$744$		0			0
$745$	−8.69042			−0.318392
$746$		0			0
$747$			21.7888i			0.797212i
$748$		0			0
$749$			5.12407i			0.187229i
$750$		0			0
$751$	−10.7351			−0.391731			−0.195866	−	0.980631i	$-0.562752\pi$
$751$	−10.7351			−0.391731			−0.195866	+	0.980631i	$0.562752\pi$
$752$		0			0
$753$			13.2283i			0.482067i
$754$		0			0
$755$	−4.57748			−0.166592
$756$		0			0
$757$	20.0000			0.726912			0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000			0.726912			0.363456	+	0.931611i	$0.381597\pi$
$758$		0			0
$759$	−5.44911			−0.197790
$760$		0			0
$761$	−32.0712			−1.16258			−0.581291	−	0.813696i	$-0.697452\pi$
$761$	−32.0712			−1.16258			−0.581291	+	0.813696i	$0.697452\pi$
$762$		0			0
$763$	6.86622			0.248574
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	38.9125i		−	1.40505i
$768$		0			0
$769$	−33.4521			−1.20631			−0.603156	−	0.797623i	$-0.706091\pi$
$769$	−33.4521			−1.20631			−0.603156	+	0.797623i	$0.706091\pi$
$770$		0			0
$771$			3.44716i			0.124146i
$772$		0			0
$773$		−	30.9911i		−	1.11467i	−0.830287	−	0.557337i	$-0.811823\pi$
$773$		−	30.9911i		−	1.11467i	0.830287	−	0.557337i	$-0.188177\pi$
$774$		0			0
$775$	7.65632			0.275023
$776$		0			0
$777$	4.33418			0.155488
$778$		0			0
$779$	−36.7014	+	9.98574i	−1.31496	+	0.357776i
$780$		0			0
$781$			7.82549i			0.280018i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.38083			−0.0492840
$786$		0			0
$787$	40.9246			1.45880			0.729402	−	0.684085i	$-0.239798\pi$
$787$	40.9246			1.45880			0.729402	+	0.684085i	$0.239798\pi$
$788$		0			0
$789$			5.12407i			0.182422i
$790$		0			0
$791$	−9.23650			−0.328412
$792$		0			0
$793$			18.3524i			0.651713i
$794$		0			0
$795$			14.4634i			0.512963i
$796$		0			0
$797$			3.01227i			0.106700i	0.998576	+	0.0533501i	$0.0169899\pi$
$797$			3.01227i			0.106700i	−0.998576	+	0.0533501i	$0.983010\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	29.5010i		−	1.04237i
$802$		0			0
$803$			5.80777i			0.204952i
$804$		0			0
$805$	−4.76166			−0.167827
$806$		0			0
$807$		−	21.0020i		−	0.739304i
$808$		0			0
$809$	33.5233			1.17862			0.589309	−	0.807908i	$-0.299400\pi$
$809$	33.5233			1.17862			0.589309	+	0.807908i	$0.299400\pi$
$810$		0			0
$811$	38.9901			1.36913			0.684564	−	0.728952i	$-0.259992\pi$
$811$	38.9901			1.36913			0.684564	+	0.728952i	$0.259992\pi$
$812$		0			0
$813$			32.2024i			1.12939i
$814$		0			0
$815$		−	22.8754i		−	0.801289i
$816$		0			0
$817$	−32.0712	+	8.72597i	−1.12203	+	0.305283i
$818$		0			0
$819$	5.20451			0.181860
$820$		0			0
$821$	16.1425			0.563377			0.281688	−	0.959506i	$-0.409106\pi$
$821$	16.1425			0.563377			0.281688	+	0.959506i	$0.409106\pi$
$822$		0			0
$823$		−	18.1541i		−	0.632813i	−0.948624	−	0.316407i	$-0.897524\pi$
$823$		−	18.1541i		−	0.632813i	0.948624	−	0.316407i	$-0.102476\pi$
$824$		0			0
$825$		−	0.900474i		−	0.0313505i
$826$		0			0
$827$	−2.72455			−0.0947420			−0.0473710	−	0.998877i	$-0.515084\pi$
$827$	−2.72455			−0.0947420			−0.0473710	+	0.998877i	$0.515084\pi$
$828$		0			0
$829$			6.92502i			0.240516i	0.992743	+	0.120258i	$0.0383722\pi$
$829$			6.92502i			0.240516i	−0.992743	+	0.120258i	$0.961628\pi$
$830$		0			0
$831$	15.3126			0.531189
$832$		0			0
$833$		0			0
$834$		0			0
$835$	16.4570			0.569518
$836$		0			0
$837$	−41.0958			−1.42048
$838$		0			0
$839$	9.94506			0.343341			0.171671	−	0.985154i	$-0.445083\pi$
$839$	9.94506			0.343341			0.171671	+	0.985154i	$0.445083\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$			18.9410i			0.652363i
$844$		0			0
$845$	−2.30958			−0.0794521
$846$		0			0
$847$		−	8.16839i		−	0.280669i
$848$		0			0
$849$		−	20.1534i		−	0.691661i
$850$		0			0
$851$	29.1266			0.998447
$852$		0			0
$853$	24.7617			0.847823			0.423912	−	0.905704i	$-0.360657\pi$
$853$	24.7617			0.847823			0.423912	+	0.905704i	$0.360657\pi$
$854$		0			0
$855$	1.93446	+	7.10988i	0.0661572	+	0.243153i
$856$		0			0
$857$		−	31.8916i		−	1.08940i	−0.838632	−	0.544698i	$-0.816644\pi$
$857$		−	31.8916i		−	1.08940i	0.838632	−	0.544698i	$-0.183356\pi$
$858$		0			0
$859$		−	24.6927i		−	0.842505i	−0.906943	−	0.421252i	$-0.861591\pi$
$859$		−	24.6927i		−	0.842505i	0.906943	−	0.421252i	$-0.138409\pi$
$860$		0			0
$861$	−7.85751			−0.267783
$862$		0			0
$863$	56.2372			1.91434			0.957169	−	0.289531i	$-0.0934992\pi$
$863$	56.2372			1.91434			0.957169	+	0.289531i	$0.0934992\pi$
$864$		0			0
$865$			3.91275i			0.133037i
$866$		0			0
$867$	−19.4543			−0.660703
$868$		0			0
$869$			6.02454i			0.204369i
$870$		0			0
$871$		−	43.3901i		−	1.47022i
$872$		0			0
$873$		−	21.3647i		−	0.723085i
$874$		0			0
$875$		−	0.786873i		−	0.0266011i
$876$		0			0
$877$		−	39.7171i		−	1.34115i	−0.741841	−	0.670575i	$-0.766047\pi$
$877$		−	39.7171i		−	1.34115i	0.741841	−	0.670575i	$-0.233953\pi$
$878$		0			0
$879$			3.44716i			0.116270i
$880$		0			0
$881$	38.0712			1.28265			0.641326	−	0.767268i	$-0.278384\pi$
$881$	38.0712			1.28265			0.641326	+	0.767268i	$0.278384\pi$
$882$		0			0
$883$		−	14.4634i		−	0.486731i	−0.969935	−	0.243366i	$-0.921749\pi$
$883$		−	14.4634i		−	0.486731i	0.969935	−	0.243366i	$-0.0782515\pi$
$884$		0			0
$885$	11.3808			0.382562
$886$		0			0
$887$	0.354279			0.0118955			0.00594776	−	0.999982i	$-0.498107\pi$
$887$	0.354279			0.0118955			0.00594776	+	0.999982i	$0.498107\pi$
$888$		0			0
$889$			4.50237i			0.151005i
$890$		0			0
$891$			0.842935i			0.0282394i
$892$		0			0
$893$	−3.30958	+	0.900474i	−0.110751	+	0.0301332i
$894$		0			0
$895$	9.94506			0.332427
$896$		0			0
$897$	−27.0958			−0.904704
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−6.86622			−0.228494
$904$		0			0
$905$		−	7.82549i		−	0.260128i
$906$		0			0
$907$	−55.4471			−1.84109			−0.920546	−	0.390634i	$-0.872256\pi$
$907$	−55.4471			−1.84109			−0.920546	+	0.390634i	$0.872256\pi$
$908$		0			0
$909$	−4.54792			−0.150845
$910$		0			0
$911$	26.0478			0.863001			0.431501	−	0.902113i	$-0.357984\pi$
$911$	26.0478			0.863001			0.431501	+	0.902113i	$0.357984\pi$
$912$		0			0
$913$	−10.1425			−0.335668
$914$		0			0
$915$	−5.36757			−0.177447
$916$		0			0
$917$	1.23834			0.0408935
$918$		0			0
$919$		−	21.5452i		−	0.710711i	−0.934731	−	0.355356i	$-0.884360\pi$
$919$		−	21.5452i		−	0.710711i	0.934731	−	0.355356i	$-0.115640\pi$
$920$		0			0
$921$	−12.6904			−0.418163
$922$		0			0
$923$			38.9125i			1.28082i
$924$		0			0
$925$			4.81322i			0.158258i
$926$		0			0
$927$	−26.4836			−0.869835
$928$		0			0
$929$	−28.7617			−0.943640			−0.471820	−	0.881695i	$-0.656403\pi$
$929$	−28.7617			−0.943640			−0.471820	+	0.881695i	$0.656403\pi$
$930$		0			0
$931$	−7.30204	−	26.8378i	−0.239315	−	0.879572i
$932$		0			0
$933$			21.9543i			0.718752i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	48.7617			1.59297			0.796487	−	0.604656i	$-0.206689\pi$
$937$	48.7617			1.59297			0.796487	+	0.604656i	$0.206689\pi$
$938$		0			0
$939$	3.16037			0.103135
$940$		0			0
$941$			17.4519i			0.568917i	0.958688	+	0.284458i	$0.0918138\pi$
$941$			17.4519i			0.568917i	−0.958688	+	0.284458i	$0.908186\pi$
$942$		0			0
$943$	−52.8041			−1.71954
$944$		0			0
$945$			4.22360i			0.137394i
$946$		0			0
$947$		−	33.4044i		−	1.08550i	−0.839895	−	0.542748i	$-0.817384\pi$
$947$		−	33.4044i		−	1.08550i	0.839895	−	0.542748i	$-0.182616\pi$
$948$		0			0
$949$			28.8793i			0.937462i
$950$		0			0
$951$		−	6.53858i		−	0.212028i
$952$		0			0
$953$			22.2652i			0.721239i	0.932713	+	0.360620i	$0.117435\pi$
$953$			22.2652i			0.721239i	−0.932713	+	0.360620i	$0.882565\pi$
$954$		0			0
$955$			20.5147i			0.663842i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			9.44247i			0.304913i
$960$		0			0
$961$	27.6192			0.890941
$962$		0			0
$963$	11.0079			0.354724
$964$		0			0
$965$			3.01227i			0.0969685i
$966$		0			0
$967$		−	14.4634i		−	0.465111i	−0.972583	−	0.232555i	$-0.925291\pi$
$967$		−	14.4634i		−	0.465111i	0.972583	−	0.232555i	$-0.0747087\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.78739			−0.121543			−0.0607716	−	0.998152i	$-0.519356\pi$
$971$	−3.78739			−0.121543			−0.0607716	+	0.998152i	$0.519356\pi$
$972$		0			0
$973$	−10.1425			−0.325153
$974$		0			0
$975$		−	4.47763i		−	0.143399i
$976$		0			0
$977$			37.0157i			1.18424i	0.805851	+	0.592118i	$0.201708\pi$
$977$			37.0157i			1.18424i	−0.805851	+	0.592118i	$0.798292\pi$
$978$		0			0
$979$	13.7324			0.438891
$980$		0			0
$981$		−	14.7505i		−	0.470948i
$982$		0			0
$983$	46.2922			1.47649			0.738246	−	0.674532i	$-0.235655\pi$
$983$	46.2922			1.47649			0.738246	+	0.674532i	$0.235655\pi$
$984$		0			0
$985$	5.38083			0.171448
$986$		0			0
$987$	−0.708558			−0.0225537
$988$		0			0
$989$	−46.1425			−1.46725
$990$		0			0
$991$	−15.3126			−0.486422			−0.243211	−	0.969973i	$-0.578201\pi$
$991$	−15.3126			−0.486422			−0.243211	+	0.969973i	$0.578201\pi$
$992$		0			0
$993$	25.3808			0.805436
$994$		0			0
$995$			6.83825i			0.216787i
$996$		0			0
$997$	−48.1425			−1.52469			−0.762344	−	0.647172i	$-0.775952\pi$
$997$	−48.1425			−1.52469			−0.762344	+	0.647172i	$0.775952\pi$
$998$		0			0
$999$		−	25.8353i		−	0.817393i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.j.e.911.5	yes	8
4.3	odd	2	inner	1520.2.j.e.911.4	yes	8
19.18	odd	2	inner	1520.2.j.e.911.3	✓	8
76.75	even	2	inner	1520.2.j.e.911.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.j.e.911.3	✓	8	19.18	odd	2	inner
1520.2.j.e.911.4	yes	8	4.3	odd	2	inner
1520.2.j.e.911.5	yes	8	1.1	even	1	trivial
1520.2.j.e.911.6	yes	8	76.75	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.14437 q^{3} +1.00000 q^{5} -0.786873i q^{7} -1.69042 q^{9} -0.786873i q^{11} -3.91275i q^{13} +1.14437 q^{15} +(-1.14437 - 4.20600i) q^{19} -0.900474i q^{21} -6.05138i q^{23} +1.00000 q^{25} -5.36757 q^{27} +7.65632 q^{31} -0.900474i q^{33} -0.786873i q^{35} +4.81322i q^{37} -4.47763i q^{39} -8.72597i q^{41} -7.62512i q^{43} -1.69042 q^{45} -0.786873i q^{47} +6.38083 q^{49} +12.6387i q^{53} -0.786873i q^{55} +(-1.30958 - 4.81322i) q^{57} +9.94506 q^{59} -4.69042 q^{61} +1.33014i q^{63} -3.91275i q^{65} +11.0894 q^{67} -6.92502i q^{69} -9.94506 q^{71} -7.38083 q^{73} +1.14437 q^{75} -0.619168 q^{77} -7.65632 q^{79} -1.07125 q^{81} -12.8896i q^{83} +17.4519i q^{89} -3.07883 q^{91} +8.76166 q^{93} +(-1.14437 - 4.20600i) q^{95} +12.6387i q^{97} +1.33014i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} + 24 q^{9} + 8 q^{25} + 24 q^{45} - 24 q^{49} - 48 q^{57} + 16 q^{73} - 80 q^{77} + 104 q^{81} - 80 q^{93}+O(q^{100}) \) 8 * q + 8 * q^5 + 24 * q^9 + 8 * q^25 + 24 * q^45 - 24 * q^49 - 48 * q^57 + 16 * q^73 - 80 * q^77 + 104 * q^81 - 80 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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