LMFDB - Embedded newform 1520.2.d.h.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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([0, 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("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (of order $2$, degree $1$, not 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:	$6$
Coefficient field:	6.0.16516096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 13x^{2} + 1 $ x^6 + 9x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.2
Root			$-2.68667i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.h.609.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31446i q^{3} +(1.94827 + 1.09737i) q^{5} +1.45033i q^{7} -2.35673 q^{9} +3.89655 q^{11} +3.05888i q^{13} +(2.53982 - 4.50920i) q^{15} -3.92301i q^{17} -1.00000 q^{19} +3.35673 q^{21} +5.37334i q^{23} +(2.59155 + 4.27596i) q^{25} -1.48883i q^{27} -6.00000 q^{29} +8.43637 q^{31} -9.01841i q^{33} +(-1.59155 + 2.82564i) q^{35} +5.95953i q^{37} +7.07965 q^{39} +10.4364 q^{41} +1.45033i q^{43} +(-4.59155 - 2.58620i) q^{45} +4.90686i q^{47} +4.89655 q^{49} -9.07965 q^{51} -4.23127i q^{53} +(7.59155 + 4.27596i) q^{55} +2.31446i q^{57} +3.35673 q^{59} +10.3329 q^{61} -3.41802i q^{63} +(-3.35673 + 5.95953i) q^{65} -9.84404i q^{67} +12.4364 q^{69} -8.64327 q^{71} -2.43418i q^{73} +(9.89655 - 5.99804i) q^{75} +5.65127i q^{77} -12.4364 q^{79} -10.5160 q^{81} -12.6635i q^{83} +(4.30500 - 7.64310i) q^{85} +13.8868i q^{87} -12.3662 q^{89} -4.43637 q^{91} -19.5256i q^{93} +(-1.94827 - 1.09737i) q^{95} -3.05888i q^{97} -9.18310 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{5} - 14 q^{9} - 2 q^{11} - 10 q^{15} - 6 q^{19} + 20 q^{21} + 3 q^{25} - 36 q^{29} + 3 q^{35} - 8 q^{39} + 12 q^{41} - 15 q^{45} + 4 q^{49} - 4 q^{51} + 33 q^{55} + 20 q^{59} - 14 q^{61} - 20 q^{65}+ \cdots - 30 q^{99}+O(q^{100}) $ 6 * q - q^5 - 14 * q^9 - 2 * q^11 - 10 * q^15 - 6 * q^19 + 20 * q^21 + 3 * q^25 - 36 * q^29 + 3 * q^35 - 8 * q^39 + 12 * q^41 - 15 * q^45 + 4 * q^49 - 4 * q^51 + 33 * q^55 + 20 * q^59 - 14 * q^61 - 20 * q^65 + 24 * q^69 - 52 * q^71 + 34 * q^75 - 24 * q^79 + 38 * q^81 + 13 * q^85 - 24 * q^89 + 24 * q^91 + q^95 - 30 * q^99

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
$3$		−	2.31446i		−	1.33625i	−0.744047	−	0.668127i	$-0.767096\pi$
$3$		−	2.31446i		−	1.33625i	0.744047	−	0.668127i	$-0.232904\pi$
$4$		0			0
$5$	1.94827	+	1.09737i	0.871295	+	0.490760i
$5$	1.94827	+	1.09737i	0.871295	+	0.490760i
$6$		0			0
$7$			1.45033i			0.548172i	0.961705	+	0.274086i	$0.0883754\pi$
$7$			1.45033i			0.548172i	−0.961705	+	0.274086i	$0.911625\pi$
$8$		0			0
$9$	−2.35673			−0.785575
$10$		0			0
$11$	3.89655			1.17485			0.587427	−	0.809277i	$-0.300141\pi$
$11$	3.89655			1.17485			0.587427	+	0.809277i	$0.300141\pi$
$12$		0			0
$13$			3.05888i			0.848380i	0.905573	+	0.424190i	$0.139441\pi$
$13$			3.05888i			0.848380i	−0.905573	+	0.424190i	$0.860559\pi$
$14$		0			0
$15$	2.53982	−	4.50920i	0.655780	−	1.16427i
$16$		0			0
$17$		−	3.92301i		−	0.951469i	−0.879589	−	0.475735i	$-0.842182\pi$
$17$		−	3.92301i		−	0.951469i	0.879589	−	0.475735i	$-0.157818\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	3.35673			0.732498
$22$		0			0
$23$			5.37334i			1.12042i	0.828351	+	0.560209i	$0.189279\pi$
$23$			5.37334i			1.12042i	−0.828351	+	0.560209i	$0.810721\pi$
$24$		0			0
$25$	2.59155	+	4.27596i	0.518310	+	0.855193i
$26$		0			0
$27$		−	1.48883i		−	0.286526i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	8.43637			1.51522			0.757609	−	0.652709i	$-0.226368\pi$
$31$	8.43637			1.51522			0.757609	+	0.652709i	$0.226368\pi$
$32$		0			0
$33$		−	9.01841i		−	1.56990i
$34$		0			0
$35$	−1.59155	+	2.82564i	−0.269021	+	0.477620i
$36$		0			0
$37$			5.95953i			0.979741i	0.871795	+	0.489871i	$0.162956\pi$
$37$			5.95953i			0.979741i	−0.871795	+	0.489871i	$0.837044\pi$
$38$		0			0
$39$	7.07965			1.13365
$40$		0			0
$41$	10.4364			1.62989			0.814944	−	0.579540i	$-0.196768\pi$
$41$	10.4364			1.62989			0.814944	+	0.579540i	$0.196768\pi$
$42$		0			0
$43$			1.45033i			0.221173i	0.993867	+	0.110586i	$0.0352729\pi$
$43$			1.45033i			0.221173i	−0.993867	+	0.110586i	$0.964727\pi$
$44$		0			0
$45$	−4.59155	−	2.58620i	−0.684468	−	0.385529i
$46$		0			0
$47$			4.90686i			0.715739i	0.933772	+	0.357869i	$0.116497\pi$
$47$			4.90686i			0.715739i	−0.933772	+	0.357869i	$0.883503\pi$
$48$		0			0
$49$	4.89655			0.699507
$50$		0			0
$51$	−9.07965			−1.27140
$52$		0			0
$53$		−	4.23127i		−	0.581209i	−0.956843	−	0.290605i	$-0.906144\pi$
$53$		−	4.23127i		−	0.581209i	0.956843	−	0.290605i	$-0.0938564\pi$
$54$		0			0
$55$	7.59155	+	4.27596i	1.02364	+	0.576571i
$56$		0			0
$57$			2.31446i			0.306558i
$58$		0			0
$59$	3.35673			0.437008			0.218504	−	0.975836i	$-0.429882\pi$
$59$	3.35673			0.437008			0.218504	+	0.975836i	$0.429882\pi$
$60$		0			0
$61$	10.3329			1.32300			0.661498	−	0.749947i	$-0.269921\pi$
$61$	10.3329			1.32300			0.661498	+	0.749947i	$0.269921\pi$
$62$		0			0
$63$		−	3.41802i		−	0.430631i
$64$		0			0
$65$	−3.35673	+	5.95953i	−0.416351	+	0.739189i
$66$		0			0
$67$		−	9.84404i		−	1.20264i	−0.799008	−	0.601320i	$-0.794642\pi$
$67$		−	9.84404i		−	1.20264i	0.799008	−	0.601320i	$-0.205358\pi$
$68$		0			0
$69$	12.4364			1.49716
$70$		0			0
$71$	−8.64327			−1.02577			−0.512884	−	0.858458i	$-0.671423\pi$
$71$	−8.64327			−1.02577			−0.512884	+	0.858458i	$0.671423\pi$
$72$		0			0
$73$		−	2.43418i		−	0.284899i	−0.989802	−	0.142449i	$-0.954502\pi$
$73$		−	2.43418i		−	0.284899i	0.989802	−	0.142449i	$-0.0454978\pi$
$74$		0			0
$75$	9.89655	−	5.99804i	1.14276	−	0.692594i
$76$		0			0
$77$			5.65127i			0.644022i
$78$		0			0
$79$	−12.4364			−1.39920			−0.699601	−	0.714534i	$-0.746639\pi$
$79$	−12.4364			−1.39920			−0.699601	+	0.714534i	$0.746639\pi$
$80$		0			0
$81$	−10.5160			−1.16845
$82$		0			0
$83$		−	12.6635i		−	1.39000i	−0.719011	−	0.694999i	$-0.755405\pi$
$83$		−	12.6635i		−	1.39000i	0.719011	−	0.694999i	$-0.244595\pi$
$84$		0			0
$85$	4.30500	−	7.64310i	0.466943	−	0.829011i
$86$		0			0
$87$			13.8868i			1.48882i
$88$		0			0
$89$	−12.3662			−1.31081			−0.655407	−	0.755276i	$-0.727503\pi$
$89$	−12.3662			−1.31081			−0.655407	+	0.755276i	$0.727503\pi$
$90$		0			0
$91$	−4.43637			−0.465058
$92$		0			0
$93$		−	19.5256i		−	2.02472i
$94$		0			0
$95$	−1.94827	−	1.09737i	−0.199889	−	0.112588i
$96$		0			0
$97$		−	3.05888i		−	0.310582i	−0.987869	−	0.155291i	$-0.950369\pi$
$97$		−	3.05888i		−	0.310582i	0.987869	−	0.155291i	$-0.0496315\pi$
$98$		0			0
$99$	−9.18310			−0.922936
$100$		0			0
$101$	3.35673			0.334007			0.167003	−	0.985956i	$-0.446591\pi$
$101$	3.35673			0.334007			0.167003	+	0.985956i	$0.446591\pi$
$102$		0			0
$103$		−	13.0611i		−	1.28695i	−0.765466	−	0.643476i	$-0.777492\pi$
$103$		−	13.0611i		−	1.28695i	0.765466	−	0.643476i	$-0.222508\pi$
$104$		0			0
$105$	6.53982	+	3.68358i	0.638221	+	0.359480i
$106$		0			0
$107$		−	5.77099i		−	0.557903i	−0.960305	−	0.278951i	$-0.910013\pi$
$107$		−	5.77099i		−	0.557903i	0.960305	−	0.278951i	$-0.0899868\pi$
$108$		0			0
$109$	−6.64327			−0.636310			−0.318155	−	0.948039i	$-0.603063\pi$
$109$	−6.64327			−0.636310			−0.318155	+	0.948039i	$0.603063\pi$
$110$		0			0
$111$	13.7931			1.30918
$112$		0			0
$113$		−	9.41606i		−	0.885789i	−0.896574	−	0.442894i	$-0.853952\pi$
$113$		−	9.41606i		−	0.885789i	0.896574	−	0.442894i	$-0.146048\pi$
$114$		0			0
$115$	−5.89655	+	10.4687i	−0.549856	+	0.976215i
$116$		0			0
$117$		−	7.20893i		−	0.666466i
$118$		0			0
$119$	5.68965			0.521569
$120$		0			0
$121$	4.18310			0.380282
$122$		0			0
$123$		−	24.1546i		−	2.17794i
$124$		0			0
$125$	0.356726	+	11.1746i	0.0319065	+	0.999491i
$126$		0			0
$127$			11.0934i			0.984383i	0.870487	+	0.492192i	$0.163804\pi$
$127$			11.0934i			0.984383i	−0.870487	+	0.492192i	$0.836196\pi$
$128$		0			0
$129$	3.35673			0.295543
$130$		0			0
$131$	−4.61000			−0.402778			−0.201389	−	0.979511i	$-0.564545\pi$
$131$	−4.61000			−0.402778			−0.201389	+	0.979511i	$0.564545\pi$
$132$		0			0
$133$		−	1.45033i		−	0.125759i
$134$		0			0
$135$	1.63380	−	2.90066i	0.140615	−	0.249649i
$136$		0			0
$137$			13.1808i			1.12612i	0.826417	+	0.563058i	$0.190375\pi$
$137$			13.1808i			1.12612i	−0.826417	+	0.563058i	$0.809625\pi$
$138$		0			0
$139$	1.18310			0.100349			0.0501745	−	0.998740i	$-0.484022\pi$
$139$	1.18310			0.100349			0.0501745	+	0.998740i	$0.484022\pi$
$140$		0			0
$141$	11.3567			0.956409
$142$		0			0
$143$			11.9191i			0.996722i
$144$		0			0
$145$	−11.6896	−	6.58423i	−0.970772	−	0.546791i
$146$		0			0
$147$		−	11.3329i		−	0.934719i
$148$		0			0
$149$	−5.46018			−0.447315			−0.223658	−	0.974668i	$-0.571800\pi$
$149$	−5.46018			−0.447315			−0.223658	+	0.974668i	$0.571800\pi$
$150$		0			0
$151$	5.07965			0.413376			0.206688	−	0.978407i	$-0.433732\pi$
$151$	5.07965			0.413376			0.206688	+	0.978407i	$0.433732\pi$
$152$		0			0
$153$			9.24546i			0.747451i
$154$		0			0
$155$	16.4364	+	9.25784i	1.32020	+	0.743608i
$156$		0			0
$157$		−	6.11775i		−	0.488250i	−0.969744	−	0.244125i	$-0.921499\pi$
$157$		−	6.11775i		−	0.488250i	0.969744	−	0.244125i	$-0.0785007\pi$
$158$		0			0
$159$	−9.79310			−0.776643
$160$		0			0
$161$	−7.79310			−0.614182
$162$		0			0
$163$			16.4365i			1.28740i	0.765277	+	0.643701i	$0.222602\pi$
$163$			16.4365i			1.28740i	−0.765277	+	0.643701i	$0.777398\pi$
$164$		0			0
$165$	9.89655	−	17.5703i	0.770445	−	1.36785i
$166$		0			0
$167$		−	3.80329i		−	0.294308i	−0.989114	−	0.147154i	$-0.952989\pi$
$167$		−	3.80329i		−	0.294308i	0.989114	−	0.147154i	$-0.0470112\pi$
$168$		0			0
$169$	3.64327			0.280252
$170$		0			0
$171$	2.35673			0.180223
$172$		0			0
$173$		−	11.3838i		−	0.865491i	−0.901516	−	0.432746i	$-0.857545\pi$
$173$		−	11.3838i		−	0.865491i	0.901516	−	0.432746i	$-0.142455\pi$
$174$		0			0
$175$	−6.20155	+	3.75860i	−0.468793	+	0.284123i
$176$		0			0
$177$		−	7.76901i		−	0.583954i
$178$		0			0
$179$	10.0702			0.752680			0.376340	−	0.926482i	$-0.377182\pi$
$179$	10.0702			0.752680			0.376340	+	0.926482i	$0.377182\pi$
$180$		0			0
$181$	0.573097			0.0425980			0.0212990	−	0.999773i	$-0.493220\pi$
$181$	0.573097			0.0425980			0.0212990	+	0.999773i	$0.493220\pi$
$182$		0			0
$183$		−	23.9151i		−	1.76786i
$184$		0			0
$185$	−6.53982	+	11.6108i	−0.480817	+	0.853643i
$186$		0			0
$187$		−	15.2862i		−	1.11784i
$188$		0			0
$189$	2.15930			0.157066
$190$		0			0
$191$	3.18310			0.230321			0.115160	−	0.993347i	$-0.463262\pi$
$191$	3.18310			0.230321			0.115160	+	0.993347i	$0.463262\pi$
$192$		0			0
$193$			3.05888i			0.220183i	0.993921	+	0.110091i	$0.0351143\pi$
$193$			3.05888i			0.220183i	−0.993921	+	0.110091i	$0.964886\pi$
$194$		0			0
$195$	13.7931	+	7.76901i	0.987744	+	0.556350i
$196$		0			0
$197$		−	21.4933i		−	1.53134i	−0.643235	−	0.765669i	$-0.722408\pi$
$197$		−	21.4933i		−	1.53134i	0.643235	−	0.765669i	$-0.277592\pi$
$198$		0			0
$199$	−4.81690			−0.341461			−0.170731	−	0.985318i	$-0.554613\pi$
$199$	−4.81690			−0.341461			−0.170731	+	0.985318i	$0.554613\pi$
$200$		0			0
$201$	−22.7836			−1.60703
$202$		0			0
$203$		−	8.70197i		−	0.610758i
$204$		0			0
$205$	20.3329	+	11.4526i	1.42011	+	0.799883i
$206$		0			0
$207$		−	12.6635i		−	0.880173i
$208$		0			0
$209$	−3.89655			−0.269530
$210$		0			0
$211$	−10.5066			−0.723301			−0.361650	−	0.932314i	$-0.617787\pi$
$211$	−10.5066			−0.723301			−0.361650	+	0.932314i	$0.617787\pi$
$212$		0			0
$213$			20.0045i			1.37069i
$214$		0			0
$215$	−1.59155	+	2.82564i	−0.108543	+	0.192707i
$216$		0			0
$217$			12.2355i			0.830600i
$218$		0			0
$219$	−5.63380			−0.380697
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$			16.8947i			1.13136i	0.824626	+	0.565678i	$0.191385\pi$
$223$			16.8947i			1.13136i	−0.824626	+	0.565678i	$0.808615\pi$
$224$		0			0
$225$	−6.10757	−	10.0773i	−0.407171	−	0.671818i
$226$		0			0
$227$			17.1342i			1.13724i	0.822602	+	0.568618i	$0.192522\pi$
$227$			17.1342i			1.13724i	−0.822602	+	0.568618i	$0.807478\pi$
$228$		0			0
$229$	−25.0464			−1.65511			−0.827555	−	0.561384i	$-0.810269\pi$
$229$	−25.0464			−1.65511			−0.827555	+	0.561384i	$0.810269\pi$
$230$		0			0
$231$	13.0796			0.860578
$232$		0			0
$233$			19.2986i			1.26429i	0.774849	+	0.632147i	$0.217826\pi$
$233$			19.2986i			1.26429i	−0.774849	+	0.632147i	$0.782174\pi$
$234$		0			0
$235$	−5.38465	+	9.55991i	−0.351256	+	0.623620i
$236$		0			0
$237$			28.7835i			1.86969i
$238$		0			0
$239$	18.7693			1.21408			0.607042	−	0.794669i	$-0.292356\pi$
$239$	18.7693			1.21408			0.607042	+	0.794669i	$0.292356\pi$
$240$		0			0
$241$	−14.4364			−0.929929			−0.464964	−	0.885329i	$-0.653933\pi$
$241$	−14.4364			−0.929929			−0.464964	+	0.885329i	$0.653933\pi$
$242$		0			0
$243$			19.8724i			1.27482i
$244$		0			0
$245$	9.53982	+	5.37334i	0.609477	+	0.343290i
$246$		0			0
$247$		−	3.05888i		−	0.194632i
$248$		0			0
$249$	−29.3091			−1.85739
$250$		0			0
$251$	10.9762			0.692811			0.346406	−	0.938085i	$-0.387402\pi$
$251$	10.9762			0.692811			0.346406	+	0.938085i	$0.387402\pi$
$252$		0			0
$253$			20.9375i			1.31633i
$254$		0			0
$255$	−17.6896	−	9.96375i	−1.10777	−	0.623954i
$256$		0			0
$257$			17.6392i			1.10030i	0.835066	+	0.550150i	$0.185430\pi$
$257$			17.6392i			1.10030i	−0.835066	+	0.550150i	$0.814570\pi$
$258$		0			0
$259$	−8.64327			−0.537067
$260$		0			0
$261$	14.1404			0.875266
$262$		0			0
$263$			1.68976i			0.104195i	0.998642	+	0.0520975i	$0.0165907\pi$
$263$			1.68976i			0.104195i	−0.998642	+	0.0520975i	$0.983409\pi$
$264$		0			0
$265$	4.64327	−	8.24367i	0.285234	−	0.506405i
$266$		0			0
$267$			28.6211i			1.75158i
$268$		0			0
$269$	27.1022			1.65245			0.826226	−	0.563339i	$-0.190484\pi$
$269$	27.1022			1.65245			0.826226	+	0.563339i	$0.190484\pi$
$270$		0			0
$271$	−23.9524			−1.45500			−0.727502	−	0.686105i	$-0.759319\pi$
$271$	−23.9524			−1.45500			−0.727502	+	0.686105i	$0.759319\pi$
$272$		0			0
$273$			10.2678i			0.621436i
$274$		0			0
$275$	10.0981	+	16.6615i	0.608938	+	1.00473i
$276$		0			0
$277$		−	8.23549i		−	0.494822i	−0.968911	−	0.247411i	$-0.920420\pi$
$277$		−	8.23549i		−	0.494822i	0.968911	−	0.247411i	$-0.0795799\pi$
$278$		0			0
$279$	−19.8822			−1.19032
$280$		0			0
$281$	−10.4364			−0.622582			−0.311291	−	0.950315i	$-0.600761\pi$
$281$	−10.4364			−0.622582			−0.311291	+	0.950315i	$0.600761\pi$
$282$		0			0
$283$			10.4687i			0.622302i	0.950361	+	0.311151i	$0.100714\pi$
$283$			10.4687i			0.622302i	−0.950361	+	0.311151i	$0.899286\pi$
$284$		0			0
$285$	−2.53982	+	4.50920i	−0.150446	+	0.267102i
$286$		0			0
$287$			15.1362i			0.893459i
$288$		0			0
$289$	1.61000			0.0947059
$290$		0			0
$291$	−7.07965			−0.415016
$292$		0			0
$293$		−	16.4668i		−	0.961999i	−0.876721	−	0.481000i	$-0.840274\pi$
$293$		−	16.4668i		−	0.961999i	0.876721	−	0.481000i	$-0.159726\pi$
$294$		0			0
$295$	6.53982	+	3.68358i	0.380763	+	0.214466i
$296$		0			0
$297$		−	5.80131i		−	0.336626i
$298$		0			0
$299$	−16.4364			−0.950540
$300$		0			0
$301$	−2.10345			−0.121241
$302$		0			0
$303$		−	7.76901i		−	0.446318i
$304$		0			0
$305$	20.1314	+	11.3391i	1.15272	+	0.649273i
$306$		0			0
$307$			7.87634i			0.449526i	0.974413	+	0.224763i	$0.0721609\pi$
$307$			7.87634i			0.449526i	−0.974413	+	0.224763i	$0.927839\pi$
$308$		0			0
$309$	−30.2295			−1.71969
$310$		0			0
$311$	−3.89655			−0.220953			−0.110477	−	0.993879i	$-0.535238\pi$
$311$	−3.89655			−0.220953			−0.110477	+	0.993879i	$0.535238\pi$
$312$		0			0
$313$			7.76901i			0.439130i	0.975598	+	0.219565i	$0.0704638\pi$
$313$			7.76901i			0.439130i	−0.975598	+	0.219565i	$0.929536\pi$
$314$		0			0
$315$	3.75084	−	6.65925i	0.211336	−	0.375206i
$316$		0			0
$317$		−	19.0510i		−	1.07001i	−0.844849	−	0.535005i	$-0.820310\pi$
$317$		−	19.0510i		−	1.07001i	0.844849	−	0.535005i	$-0.179690\pi$
$318$		0			0
$319$	−23.3793			−1.30899
$320$		0			0
$321$	−13.3567			−0.745500
$322$		0			0
$323$			3.92301i			0.218282i
$324$		0			0
$325$	−13.0796	+	7.92723i	−0.725528	+	0.439724i
$326$		0			0
$327$			15.3756i			0.850272i
$328$		0			0
$329$	−7.11655			−0.392348
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$		−	14.0450i		−	0.769660i
$334$		0			0
$335$	10.8026	−	19.1789i	0.590207	−	1.04785i
$336$		0			0
$337$			6.89249i			0.375458i	0.982221	+	0.187729i	$0.0601127\pi$
$337$			6.89249i			0.375458i	−0.982221	+	0.187729i	$0.939887\pi$
$338$		0			0
$339$	−21.7931			−1.18364
$340$		0			0
$341$	32.8727			1.78016
$342$		0			0
$343$			17.2539i			0.931623i
$344$		0			0
$345$	24.2295	+	13.6473i	1.30447	+	0.734747i
$346$		0			0
$347$		−	30.5503i		−	1.64002i	−0.572346	−	0.820012i	$-0.693967\pi$
$347$		−	30.5503i		−	1.64002i	0.572346	−	0.820012i	$-0.306033\pi$
$348$		0			0
$349$	−16.7693			−0.897640			−0.448820	−	0.893622i	$-0.648156\pi$
$349$	−16.7693			−0.897640			−0.448820	+	0.893622i	$0.648156\pi$
$350$		0			0
$351$	4.55416			0.243083
$352$		0			0
$353$			29.0999i			1.54883i	0.632676	+	0.774417i	$0.281956\pi$
$353$			29.0999i			1.54883i	−0.632676	+	0.774417i	$0.718044\pi$
$354$		0			0
$355$	−16.8395	−	9.48489i	−0.893746	−	0.503406i
$356$		0			0
$357$		−	13.1685i		−	0.696949i
$358$		0			0
$359$	−11.6896			−0.616956			−0.308478	−	0.951231i	$-0.599820\pi$
$359$	−11.6896			−0.616956			−0.308478	+	0.951231i	$0.599820\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		−	9.68161i		−	0.508153i
$364$		0			0
$365$	2.67120	−	4.74244i	0.139817	−	0.248231i
$366$		0			0
$367$		−	4.20095i		−	0.219288i	−0.993971	−	0.109644i	$-0.965029\pi$
$367$		−	4.20095i		−	0.219288i	0.993971	−	0.109644i	$-0.0349710\pi$
$368$		0			0
$369$	−24.5957			−1.28040
$370$		0			0
$371$	6.13672			0.318603
$372$		0			0
$373$		−	16.9456i		−	0.877412i	−0.898631	−	0.438706i	$-0.855437\pi$
$373$		−	16.9456i		−	0.877412i	0.898631	−	0.438706i	$-0.144563\pi$
$374$		0			0
$375$	25.8633	−	0.825627i	1.33557	−	0.0426352i
$376$		0			0
$377$		−	18.3533i		−	0.945241i
$378$		0			0
$379$	10.3662			0.532476			0.266238	−	0.963907i	$-0.414219\pi$
$379$	10.3662			0.532476			0.266238	+	0.963907i	$0.414219\pi$
$380$		0			0
$381$	25.6753			1.31539
$382$		0			0
$383$		−	20.5907i		−	1.05214i	−0.850442	−	0.526068i	$-0.823666\pi$
$383$		−	20.5907i		−	1.05214i	0.850442	−	0.526068i	$-0.176334\pi$
$384$		0			0
$385$	−6.20155	+	11.0102i	−0.316060	+	0.561133i
$386$		0			0
$387$		−	3.41802i		−	0.173748i
$388$		0			0
$389$	−8.10345			−0.410861			−0.205431	−	0.978672i	$-0.565859\pi$
$389$	−8.10345			−0.410861			−0.205431	+	0.978672i	$0.565859\pi$
$390$		0			0
$391$	21.0796			1.06604
$392$		0			0
$393$			10.6697i			0.538213i
$394$		0			0
$395$	−24.2295	−	13.6473i	−1.21912	−	0.686672i
$396$		0			0
$397$		−	3.46891i		−	0.174100i	−0.996204	−	0.0870499i	$-0.972256\pi$
$397$		−	3.46891i		−	0.174100i	0.996204	−	0.0870499i	$-0.0277440\pi$
$398$		0			0
$399$	−3.35673			−0.168046
$400$		0			0
$401$	−15.9524			−0.796625			−0.398312	−	0.917250i	$-0.630404\pi$
$401$	−15.9524			−0.796625			−0.398312	+	0.917250i	$0.630404\pi$
$402$		0			0
$403$			25.8058i			1.28548i
$404$		0			0
$405$	−20.4881	−	11.5400i	−1.01806	−	0.573427i
$406$		0			0
$407$			23.2216i			1.15105i
$408$		0			0
$409$	3.92982			0.194317			0.0971586	−	0.995269i	$-0.469025\pi$
$409$	3.92982			0.194317			0.0971586	+	0.995269i	$0.469025\pi$
$410$		0			0
$411$	30.5066			1.50478
$412$		0			0
$413$			4.86835i			0.239556i
$414$		0			0
$415$	13.8965	−	24.6719i	0.682155	−	1.21110i
$416$		0			0
$417$		−	2.73823i		−	0.134092i
$418$		0			0
$419$	−34.2996			−1.67565			−0.837824	−	0.545941i	$-0.816172\pi$
$419$	−34.2996			−1.67565			−0.837824	+	0.545941i	$0.816172\pi$
$420$		0			0
$421$	−26.0891			−1.27151			−0.635753	−	0.771893i	$-0.719310\pi$
$421$	−26.0891			−1.27151			−0.635753	+	0.771893i	$0.719310\pi$
$422$		0			0
$423$		−	11.5641i		−	0.562267i
$424$		0			0
$425$	16.7746	−	10.1667i	0.813690	−	0.493156i
$426$		0			0
$427$			14.9861i			0.725229i
$428$		0			0
$429$	27.5862			1.33187
$430$		0			0
$431$	−10.2996			−0.496117			−0.248058	−	0.968745i	$-0.579792\pi$
$431$	−10.2996			−0.496117			−0.248058	+	0.968745i	$0.579792\pi$
$432$		0			0
$433$		−	36.2319i		−	1.74119i	−0.491999	−	0.870596i	$-0.663734\pi$
$433$		−	36.2319i		−	1.74119i	0.491999	−	0.870596i	$-0.336266\pi$
$434$		0			0
$435$	−15.2389	+	27.0552i	−0.730651	+	1.29720i
$436$		0			0
$437$		−	5.37334i		−	0.257042i
$438$		0			0
$439$	−24.0891			−1.14971			−0.574855	−	0.818255i	$-0.694942\pi$
$439$	−24.0891			−1.14971			−0.574855	+	0.818255i	$0.694942\pi$
$440$		0			0
$441$	−11.5398			−0.549515
$442$		0			0
$443$			5.52337i			0.262423i	0.991354	+	0.131212i	$0.0418868\pi$
$443$			5.52337i			0.262423i	−0.991354	+	0.131212i	$0.958113\pi$
$444$		0			0
$445$	−24.0927	−	13.5703i	−1.14211	−	0.643295i
$446$		0			0
$447$			12.6374i			0.597727i
$448$		0			0
$449$	−7.92982			−0.374231			−0.187116	−	0.982338i	$-0.559914\pi$
$449$	−7.92982			−0.374231			−0.187116	+	0.982338i	$0.559914\pi$
$450$		0			0
$451$	40.6658			1.91488
$452$		0			0
$453$		−	11.7566i		−	0.552375i
$454$		0			0
$455$	−8.64327	−	4.86835i	−0.405203	−	0.228232i
$456$		0			0
$457$			22.6534i			1.05968i	0.848098	+	0.529840i	$0.177748\pi$
$457$			22.6534i			1.05968i	−0.848098	+	0.529840i	$0.822252\pi$
$458$		0			0
$459$	−5.84070			−0.272621
$460$		0			0
$461$	13.3900			0.623634			0.311817	−	0.950142i	$-0.399062\pi$
$461$	13.3900			0.623634			0.311817	+	0.950142i	$0.399062\pi$
$462$		0			0
$463$			16.9029i			0.785546i	0.919635	+	0.392773i	$0.128484\pi$
$463$			16.9029i			0.785546i	−0.919635	+	0.392773i	$0.871516\pi$
$464$		0			0
$465$	21.4269	−	38.0413i	0.993649	−	1.76412i
$466$		0			0
$467$		−	22.2501i		−	1.02961i	−0.857306	−	0.514807i	$-0.827864\pi$
$467$		−	22.2501i		−	1.02961i	0.857306	−	0.514807i	$-0.172136\pi$
$468$		0			0
$469$	14.2771			0.659254
$470$		0			0
$471$	−14.1593			−0.652426
$472$		0			0
$473$			5.65127i			0.259846i
$474$		0			0
$475$	−2.59155	−	4.27596i	−0.118908	−	0.196195i
$476$		0			0
$477$			9.97194i			0.456584i
$478$		0			0
$479$	0.366196			0.0167319			0.00836597	−	0.999965i	$-0.497337\pi$
$479$	0.366196			0.0167319			0.00836597	+	0.999965i	$0.497337\pi$
$480$		0			0
$481$	−18.2295			−0.831192
$482$		0			0
$483$			18.0368i			0.820704i
$484$		0			0
$485$	3.35673	−	5.95953i	0.152421	−	0.270608i
$486$		0			0
$487$		−	26.8461i		−	1.21651i	−0.793740	−	0.608257i	$-0.791869\pi$
$487$		−	26.8461i		−	1.21651i	0.793740	−	0.608257i	$-0.208131\pi$
$488$		0			0
$489$	38.0415			1.72030
$490$		0			0
$491$	23.7266			1.07076			0.535382	−	0.844610i	$-0.320168\pi$
$491$	23.7266			1.07076			0.535382	+	0.844610i	$0.320168\pi$
$492$		0			0
$493$			23.5381i			1.06010i
$494$		0			0
$495$	−17.8912	−	10.0773i	−0.804150	−	0.452940i
$496$		0			0
$497$		−	12.5356i		−	0.562298i
$498$		0			0
$499$	6.81690			0.305166			0.152583	−	0.988291i	$-0.451241\pi$
$499$	6.81690			0.305166			0.152583	+	0.988291i	$0.451241\pi$
$500$		0			0
$501$	−8.80257			−0.393270
$502$		0			0
$503$			23.4102i			1.04381i	0.853004	+	0.521904i	$0.174778\pi$
$503$			23.4102i			1.04381i	−0.853004	+	0.521904i	$0.825222\pi$
$504$		0			0
$505$	6.53982	+	3.68358i	0.291018	+	0.163917i
$506$		0			0
$507$		−	8.43221i		−	0.374488i
$508$		0			0
$509$	−16.9204			−0.749981			−0.374991	−	0.927029i	$-0.622354\pi$
$509$	−16.9204			−0.749981			−0.374991	+	0.927029i	$0.622354\pi$
$510$		0			0
$511$	3.53035			0.156174
$512$		0			0
$513$			1.48883i			0.0657336i
$514$		0			0
$515$	14.3329	−	25.4467i	0.631584	−	1.12131i
$516$		0			0
$517$			19.1198i			0.840888i
$518$		0			0
$519$	−26.3473			−1.15652
$520$		0			0
$521$	−3.49345			−0.153051			−0.0765254	−	0.997068i	$-0.524383\pi$
$521$	−3.49345			−0.153051			−0.0765254	+	0.997068i	$0.524383\pi$
$522$		0			0
$523$			14.9271i			0.652714i	0.945247	+	0.326357i	$0.105821\pi$
$523$			14.9271i			0.652714i	−0.945247	+	0.326357i	$0.894179\pi$
$524$		0			0
$525$	8.69912	+	14.3532i	0.379661	+	0.626427i
$526$		0			0
$527$		−	33.0960i		−	1.44168i
$528$		0			0
$529$	−5.87275			−0.255337
$530$		0			0
$531$	−7.91088			−0.343303
$532$		0			0
$533$			31.9236i			1.38276i
$534$		0			0
$535$	6.33292	−	11.2435i	0.273796	−	0.486098i
$536$		0			0
$537$		−	23.3070i		−	1.00577i
$538$		0			0
$539$	19.0796			0.821819
$540$		0			0
$541$	−12.6991			−0.545978			−0.272989	−	0.962017i	$-0.588012\pi$
$541$	−12.6991			−0.545978			−0.272989	+	0.962017i	$0.588012\pi$
$542$		0			0
$543$		−	1.32641i		−	0.0569217i
$544$		0			0
$545$	−12.9429	−	7.29014i	−0.554414	−	0.312275i
$546$		0			0
$547$		−	31.8162i		−	1.36036i	−0.733043	−	0.680182i	$-0.761901\pi$
$547$		−	31.8162i		−	1.36036i	0.733043	−	0.680182i	$-0.238099\pi$
$548$		0			0
$549$	−24.3519			−1.03931
$550$		0			0
$551$	6.00000			0.255609
$552$		0			0
$553$		−	18.0368i		−	0.767003i
$554$		0			0
$555$	26.8727	+	15.1362i	1.14068	+	0.642494i
$556$		0			0
$557$			9.64731i			0.408770i	0.978891	+	0.204385i	$0.0655194\pi$
$557$			9.64731i			0.408770i	−0.978891	+	0.204385i	$0.934481\pi$
$558$		0			0
$559$	−4.43637			−0.187639
$560$		0			0
$561$	−35.3793			−1.49372
$562$		0			0
$563$			4.28216i			0.180471i	0.995920	+	0.0902357i	$0.0287620\pi$
$563$			4.28216i			0.180471i	−0.995920	+	0.0902357i	$0.971238\pi$
$564$		0			0
$565$	10.3329	−	18.3451i	0.434709	−	0.771783i
$566$		0			0
$567$		−	15.2517i		−	0.640510i
$568$		0			0
$569$	−42.2295			−1.77035			−0.885176	−	0.465257i	$-0.845962\pi$
$569$	−42.2295			−1.77035			−0.885176	+	0.465257i	$0.845962\pi$
$570$		0			0
$571$	19.2200			0.804332			0.402166	−	0.915567i	$-0.368257\pi$
$571$	19.2200			0.804332			0.402166	+	0.915567i	$0.368257\pi$
$572$		0			0
$573$		−	7.36715i		−	0.307767i
$574$		0			0
$575$	−22.9762	+	13.9253i	−0.958174	+	0.580724i
$576$		0			0
$577$			40.7919i			1.69819i	0.528239	+	0.849096i	$0.322852\pi$
$577$			40.7919i			1.69819i	−0.528239	+	0.849096i	$0.677148\pi$
$578$		0			0
$579$	7.07965			0.294220
$580$		0			0
$581$	18.3662			0.761958
$582$		0			0
$583$		−	16.4873i		−	0.682836i
$584$		0			0
$585$	7.91088	−	14.0450i	0.327075	−	0.580689i
$586$		0			0
$587$		−	31.8851i		−	1.31604i	−0.753001	−	0.658019i	$-0.771395\pi$
$587$		−	31.8851i		−	1.31604i	0.753001	−	0.658019i	$-0.228605\pi$
$588$		0			0
$589$	−8.43637			−0.347615
$590$		0			0
$591$	−49.7455			−2.04626
$592$		0			0
$593$		−	38.8973i		−	1.59732i	−0.601783	−	0.798660i	$-0.705543\pi$
$593$		−	38.8973i		−	1.59732i	0.601783	−	0.798660i	$-0.294457\pi$
$594$		0			0
$595$	11.0850	+	6.24366i	0.454441	+	0.255965i
$596$		0			0
$597$			11.1485i			0.456279i
$598$		0			0
$599$	28.1629			1.15071			0.575353	−	0.817905i	$-0.304865\pi$
$599$	28.1629			1.15071			0.575353	+	0.817905i	$0.304865\pi$
$600$		0			0
$601$	−5.56363			−0.226945			−0.113473	−	0.993541i	$-0.536197\pi$
$601$	−5.56363			−0.226945			−0.113473	+	0.993541i	$0.536197\pi$
$602$		0			0
$603$			23.1997i			0.944764i
$604$		0			0
$605$	8.14982	+	4.59042i	0.331337	+	0.186627i
$606$		0			0
$607$			33.9986i			1.37996i	0.723828	+	0.689980i	$0.242381\pi$
$607$			33.9986i			1.37996i	−0.723828	+	0.689980i	$0.757619\pi$
$608$		0			0
$609$	−20.1404			−0.816128
$610$		0			0
$611$	−15.0095			−0.607218
$612$		0			0
$613$			17.5703i			0.709659i	0.934931	+	0.354830i	$0.115461\pi$
$613$			17.5703i			0.709659i	−0.934931	+	0.354830i	$0.884539\pi$
$614$		0			0
$615$	26.5066	−	47.0597i	1.06885	−	1.89763i
$616$		0			0
$617$			13.0791i			0.526544i	0.964722	+	0.263272i	$0.0848016\pi$
$617$			13.0791i			0.526544i	−0.964722	+	0.263272i	$0.915198\pi$
$618$		0			0
$619$	−18.9393			−0.761234			−0.380617	−	0.924733i	$-0.624288\pi$
$619$	−18.9393			−0.761234			−0.380617	+	0.924733i	$0.624288\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$		−	17.9350i		−	0.718552i
$624$		0			0
$625$	−11.5677	+	22.1627i	−0.462710	+	0.886510i
$626$		0			0
$627$			9.01841i			0.360161i
$628$		0			0
$629$	23.3793			0.932194
$630$		0			0
$631$	−31.6896			−1.26154			−0.630772	−	0.775968i	$-0.717262\pi$
$631$	−31.6896			−1.26154			−0.630772	+	0.775968i	$0.717262\pi$
$632$		0			0
$633$			24.3170i			0.966514i
$634$		0			0
$635$	−12.1736	+	21.6131i	−0.483096	+	0.857688i
$636$		0			0
$637$			14.9779i			0.593448i
$638$		0			0
$639$	20.3698			0.805818
$640$		0			0
$641$	47.9750			1.89490			0.947449	−	0.319908i	$-0.103652\pi$
$641$	47.9750			1.89490			0.947449	+	0.319908i	$0.103652\pi$
$642$		0			0
$643$			0.200927i			0.00792378i	0.999992	+	0.00396189i	$0.00126111\pi$
$643$			0.200927i			0.00792378i	−0.999992	+	0.00396189i	$0.998739\pi$
$644$		0			0
$645$	6.53982	+	3.68358i	0.257505	+	0.145041i
$646$		0			0
$647$			1.58798i			0.0624299i	0.999513	+	0.0312150i	$0.00993765\pi$
$647$			1.58798i			0.0624299i	−0.999513	+	0.0312150i	$0.990062\pi$
$648$		0			0
$649$	13.0796			0.513421
$650$		0			0
$651$	28.3186			1.10989
$652$		0			0
$653$		−	33.5624i		−	1.31340i	−0.754152	−	0.656700i	$-0.771952\pi$
$653$		−	33.5624i		−	1.31340i	0.754152	−	0.656700i	$-0.228048\pi$
$654$		0			0
$655$	−8.98155	−	5.05889i	−0.350938	−	0.197667i
$656$		0			0
$657$			5.73669i			0.223809i
$658$		0			0
$659$	15.3567			0.598213			0.299107	−	0.954220i	$-0.403311\pi$
$659$	15.3567			0.598213			0.299107	+	0.954220i	$0.403311\pi$
$660$		0			0
$661$	12.8026			0.497962			0.248981	−	0.968508i	$-0.419904\pi$
$661$	12.8026			0.497962			0.248981	+	0.968508i	$0.419904\pi$
$662$		0			0
$663$		−	27.7735i		−	1.07863i
$664$		0			0
$665$	1.59155	−	2.82564i	0.0617176	−	0.109573i
$666$		0			0
$667$		−	32.2400i		−	1.24834i
$668$		0			0
$669$	39.1022			1.51178
$670$		0			0
$671$	40.2627			1.55433
$672$		0			0
$673$			7.82545i			0.301649i	0.988561	+	0.150824i	$0.0481928\pi$
$673$			7.82545i			0.301649i	−0.988561	+	0.150824i	$0.951807\pi$
$674$		0			0
$675$	6.36620	−	3.85838i	0.245035	−	0.148509i
$676$		0			0
$677$		−	21.6516i		−	0.832137i	−0.909333	−	0.416069i	$-0.863408\pi$
$677$		−	21.6516i		−	0.832137i	0.909333	−	0.416069i	$-0.136592\pi$
$678$		0			0
$679$	4.43637			0.170252
$680$		0			0
$681$	39.6564			1.51964
$682$		0			0
$683$		−	6.38751i		−	0.244411i	−0.992505	−	0.122206i	$-0.961003\pi$
$683$		−	6.38751i		−	0.244411i	0.992505	−	0.122206i	$-0.0389967\pi$
$684$		0			0
$685$	−14.4643	+	25.6799i	−0.552652	+	0.981179i
$686$		0			0
$687$			57.9688i			2.21165i
$688$		0			0
$689$	12.9429			0.493086
$690$		0			0
$691$	−44.4958			−1.69270			−0.846351	−	0.532626i	$-0.821205\pi$
$691$	−44.4958			−1.69270			−0.846351	+	0.532626i	$0.821205\pi$
$692$		0			0
$693$		−	13.3185i		−	0.505928i
$694$		0			0
$695$	2.30500	+	1.29830i	0.0874336	+	0.0492473i
$696$		0			0
$697$		−	40.9420i		−	1.55079i
$698$		0			0
$699$	44.6658			1.68942
$700$		0			0
$701$	17.5160			0.661571			0.330785	−	0.943706i	$-0.392686\pi$
$701$	17.5160			0.661571			0.330785	+	0.943706i	$0.392686\pi$
$702$		0			0
$703$		−	5.95953i		−	0.224768i
$704$		0			0
$705$	22.1260	+	12.4626i	0.833314	+	0.469367i
$706$		0			0
$707$			4.86835i			0.183093i
$708$		0			0
$709$	−11.4269			−0.429146			−0.214573	−	0.976708i	$-0.568836\pi$
$709$	−11.4269			−0.429146			−0.214573	+	0.976708i	$0.568836\pi$
$710$		0			0
$711$	29.3091			1.09918
$712$		0			0
$713$			45.3315i			1.69768i
$714$		0			0
$715$	−13.0796	+	23.2216i	−0.489151	+	0.868439i
$716$		0			0
$717$		−	43.4408i		−	1.62233i
$718$		0			0
$719$	15.8965			0.592841			0.296421	−	0.955057i	$-0.404207\pi$
$719$	15.8965			0.592841			0.296421	+	0.955057i	$0.404207\pi$
$720$		0			0
$721$	18.9429			0.705471
$722$		0			0
$723$			33.4124i			1.24262i
$724$		0			0
$725$	−15.5493	−	25.6558i	−0.577486	−	0.952832i
$726$		0			0
$727$			41.9905i			1.55734i	0.627434	+	0.778670i	$0.284105\pi$
$727$			41.9905i			1.55734i	−0.627434	+	0.778670i	$0.715895\pi$
$728$		0			0
$729$	14.4458			0.535031
$730$		0			0
$731$	5.68965			0.210439
$732$		0			0
$733$		−	0.632884i		−	0.0233761i	−0.999932	−	0.0116881i	$-0.996279\pi$
$733$		−	0.632884i		−	0.0233761i	0.999932	−	0.0116881i	$-0.00372051\pi$
$734$		0			0
$735$	12.4364	−	22.0795i	0.458723	−	0.814416i
$736$		0			0
$737$		−	38.3578i		−	1.41293i
$738$		0			0
$739$	−29.5493			−1.08699			−0.543494	−	0.839413i	$-0.682899\pi$
$739$	−29.5493			−1.08699			−0.543494	+	0.839413i	$0.682899\pi$
$740$		0			0
$741$	−7.07965			−0.260077
$742$		0			0
$743$			31.3374i			1.14966i	0.818274	+	0.574829i	$0.194931\pi$
$743$			31.3374i			1.14966i	−0.818274	+	0.574829i	$0.805069\pi$
$744$		0			0
$745$	−10.6379	−	5.99184i	−0.389743	−	0.219524i
$746$		0			0
$747$			29.8443i			1.09195i
$748$		0			0
$749$	8.36983			0.305827
$750$		0			0
$751$	−25.0131			−0.912741			−0.456371	−	0.889790i	$-0.650851\pi$
$751$	−25.0131			−0.912741			−0.456371	+	0.889790i	$0.650851\pi$
$752$		0			0
$753$		−	25.4040i		−	0.925772i
$754$		0			0
$755$	9.89655	+	5.57426i	0.360172	+	0.202868i
$756$		0			0
$757$			32.0900i			1.16633i	0.812354	+	0.583165i	$0.198186\pi$
$757$			32.0900i			1.16633i	−0.812354	+	0.583165i	$0.801814\pi$
$758$		0			0
$759$	48.4589			1.75895
$760$		0			0
$761$	−40.4922			−1.46784			−0.733921	−	0.679235i	$-0.762312\pi$
$761$	−40.4922			−1.46784			−0.733921	+	0.679235i	$0.762312\pi$
$762$		0			0
$763$		−	9.63492i		−	0.348808i
$764$		0			0
$765$	−10.1457	+	18.0127i	−0.366819	+	0.651250i
$766$		0			0
$767$			10.2678i			0.370749i
$768$		0			0
$769$	−3.09398			−0.111572			−0.0557859	−	0.998443i	$-0.517766\pi$
$769$	−3.09398			−0.111572			−0.0557859	+	0.998443i	$0.517766\pi$
$770$		0			0
$771$	40.8251			1.47028
$772$		0			0
$773$		−	1.96350i		−	0.0706220i	−0.999376	−	0.0353110i	$-0.988758\pi$
$773$		−	1.96350i		−	0.0706220i	0.999376	−	0.0353110i	$-0.0112422\pi$
$774$		0			0
$775$	21.8633	+	36.0736i	0.785352	+	1.29580i
$776$		0			0
$777$			20.0045i			0.717658i
$778$		0			0
$779$	−10.4364			−0.373922
$780$		0			0
$781$	−33.6789			−1.20513
$782$		0			0
$783$			8.93300i			0.319239i
$784$		0			0
$785$	6.71345	−	11.9191i	0.239613	−	0.425410i
$786$		0			0
$787$		−	0.107331i		−	0.00382595i	−0.999998	−	0.00191297i	$-0.999391\pi$
$787$		−	0.107331i		−	0.00382595i	0.999998	−	0.00191297i	$-0.000608919\pi$
$788$		0			0
$789$	3.91088			0.139231
$790$		0			0
$791$	13.6564			0.485565
$792$		0			0
$793$			31.6071i			1.12240i
$794$		0			0
$795$	−19.0796	−	10.7467i	−0.676685	−	0.381145i
$796$		0			0
$797$			8.32068i			0.294734i	0.989082	+	0.147367i	$0.0470798\pi$
$797$			8.32068i			0.294734i	−0.989082	+	0.147367i	$0.952920\pi$
$798$		0			0
$799$	19.2496			0.681003
$800$		0			0
$801$	29.1437			1.02974
$802$		0			0
$803$		−	9.48489i		−	0.334714i
$804$		0			0
$805$	−15.1831	−	8.55193i	−0.535134	−	0.301416i
$806$		0			0
$807$		−	62.7270i		−	2.20810i
$808$		0			0
$809$	27.6231			0.971177			0.485588	−	0.874188i	$-0.338605\pi$
$809$	27.6231			0.971177			0.485588	+	0.874188i	$0.338605\pi$
$810$		0			0
$811$	−23.0095			−0.807972			−0.403986	−	0.914765i	$-0.632376\pi$
$811$	−23.0095			−0.807972			−0.403986	+	0.914765i	$0.632376\pi$
$812$		0			0
$813$			55.4369i			1.94426i
$814$		0			0
$815$	−18.0369	+	32.0227i	−0.631805	+	1.12171i
$816$		0			0
$817$		−	1.45033i		−	0.0507405i
$818$		0			0
$819$	10.4553			0.365338
$820$		0			0
$821$	31.1355			1.08664			0.543318	−	0.839527i	$-0.317168\pi$
$821$	31.1355			1.08664			0.543318	+	0.839527i	$0.317168\pi$
$822$		0			0
$823$		−	20.4201i		−	0.711800i	−0.934524	−	0.355900i	$-0.884174\pi$
$823$		−	20.4201i		−	0.711800i	0.934524	−	0.355900i	$-0.115826\pi$
$824$		0			0
$825$	38.5624	−	23.3716i	1.34257	−	0.813696i
$826$		0			0
$827$		−	0.902638i		−	0.0313878i	−0.999877	−	0.0156939i	$-0.995004\pi$
$827$		−	0.902638i		−	0.0313878i	0.999877	−	0.0156939i	$-0.00499573\pi$
$828$		0			0
$829$	13.4971			0.468773			0.234386	−	0.972143i	$-0.424692\pi$
$829$	13.4971			0.468773			0.234386	+	0.972143i	$0.424692\pi$
$830$		0			0
$831$	−19.0607			−0.661209
$832$		0			0
$833$		−	19.2092i		−	0.665560i
$834$		0			0
$835$	4.17363	−	7.40986i	0.144434	−	0.256429i
$836$		0			0
$837$		−	12.5603i		−	0.434149i
$838$		0			0
$839$	33.1022			1.14282			0.571408	−	0.820666i	$-0.306397\pi$
$839$	33.1022			1.14282			0.571408	+	0.820666i	$0.306397\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			24.1546i			0.831928i
$844$		0			0
$845$	7.09810	+	3.99803i	0.244182	+	0.137536i
$846$		0			0
$847$			6.06686i			0.208460i
$848$		0			0
$849$	24.2295			0.831553
$850$		0			0
$851$	−32.0226			−1.09772
$852$		0			0
$853$			50.9097i			1.74312i	0.490293	+	0.871558i	$0.336890\pi$
$853$			50.9097i			1.74312i	−0.490293	+	0.871558i	$0.663110\pi$
$854$		0			0
$855$	4.59155	+	2.58620i	0.157028	+	0.0884463i
$856$		0			0
$857$		−	21.2333i		−	0.725317i	−0.931922	−	0.362659i	$-0.881869\pi$
$857$		−	21.2333i		−	0.725317i	0.931922	−	0.362659i	$-0.118131\pi$
$858$		0			0
$859$	29.4827			1.00594			0.502969	−	0.864304i	$-0.332241\pi$
$859$	29.4827			1.00594			0.502969	+	0.864304i	$0.332241\pi$
$860$		0			0
$861$	35.0320			1.19389
$862$		0			0
$863$		−	25.7755i		−	0.877408i	−0.898632	−	0.438704i	$-0.855438\pi$
$863$		−	25.7755i		−	0.877408i	0.898632	−	0.438704i	$-0.144562\pi$
$864$		0			0
$865$	12.4922	−	22.1787i	0.424748	−	0.754098i
$866$		0			0
$867$		−	3.72628i		−	0.126551i
$868$		0			0
$869$	−48.4589			−1.64386
$870$		0			0
$871$	30.1117			1.02030
$872$		0			0
$873$			7.20893i			0.243985i
$874$		0			0
$875$	−16.2069	+	0.517369i	−0.547893	+	0.0174903i
$876$		0			0
$877$		−	54.2687i		−	1.83252i	−0.400581	−	0.916261i	$-0.631192\pi$
$877$		−	54.2687i		−	1.83252i	0.400581	−	0.916261i	$-0.368808\pi$
$878$		0			0
$879$	−38.1117			−1.28548
$880$		0			0
$881$	−28.4922			−0.959927			−0.479964	−	0.877288i	$-0.659350\pi$
$881$	−28.4922			−0.959927			−0.479964	+	0.877288i	$0.659350\pi$
$882$		0			0
$883$		−	40.5264i		−	1.36382i	−0.731435	−	0.681911i	$-0.761149\pi$
$883$		−	40.5264i		−	1.36382i	0.731435	−	0.681911i	$-0.238851\pi$
$884$		0			0
$885$	8.52549	−	15.1362i	0.286581	−	0.508797i
$886$		0			0
$887$		−	11.4705i		−	0.385142i	−0.981283	−	0.192571i	$-0.938317\pi$
$887$		−	11.4705i		−	0.385142i	0.981283	−	0.192571i	$-0.0616826\pi$
$888$		0			0
$889$	−16.0891			−0.539612
$890$		0			0
$891$	−40.9762			−1.37275
$892$		0			0
$893$		−	4.90686i		−	0.164202i
$894$		0			0
$895$	19.6195	+	11.0507i	0.655807	+	0.369385i
$896$		0			0
$897$			38.0413i			1.27016i
$898$		0			0
$899$	−50.6182			−1.68821
$900$		0			0
$901$	−16.5993			−0.553003
$902$		0			0
$903$			4.86835i			0.162009i
$904$		0			0
$905$	1.11655	+	0.628901i	0.0371154	+	0.0209054i
$906$		0			0
$907$		−	48.1000i		−	1.59714i	−0.601905	−	0.798568i	$-0.705591\pi$
$907$		−	48.1000i		−	1.59714i	0.601905	−	0.798568i	$-0.294409\pi$
$908$		0			0
$909$	−7.91088			−0.262387
$910$		0			0
$911$	12.8062			0.424288			0.212144	−	0.977238i	$-0.431955\pi$
$911$	12.8062			0.424288			0.212144	+	0.977238i	$0.431955\pi$
$912$		0			0
$913$		−	49.3439i		−	1.63304i
$914$		0			0
$915$	26.2438	−	46.5933i	0.867593	−	1.54033i
$916$		0			0
$917$		−	6.68601i		−	0.220792i
$918$		0			0
$919$	−38.7135			−1.27704			−0.638519	−	0.769606i	$-0.720453\pi$
$919$	−38.7135			−1.27704			−0.638519	+	0.769606i	$0.720453\pi$
$920$		0			0
$921$	18.2295			0.600682
$922$		0			0
$923$		−	26.4387i		−	0.870241i
$924$		0			0
$925$	−25.4827	+	15.4444i	−0.837868	+	0.507809i
$926$		0			0
$927$			30.7815i			1.01100i
$928$		0			0
$929$	36.0189			1.18174			0.590872	−	0.806766i	$-0.298784\pi$
$929$	36.0189			1.18174			0.590872	+	0.806766i	$0.298784\pi$
$930$		0			0
$931$	−4.89655			−0.160478
$932$		0			0
$933$			9.01841i			0.295249i
$934$		0			0
$935$	16.7746	−	29.7817i	0.548590	−	0.973966i
$936$		0			0
$937$		−	45.2421i		−	1.47799i	−0.673709	−	0.738997i	$-0.735300\pi$
$937$		−	45.2421i		−	1.47799i	0.673709	−	0.738997i	$-0.264700\pi$
$938$		0			0
$939$	17.9811			0.586790
$940$		0			0
$941$	−11.7455			−0.382892			−0.191446	−	0.981503i	$-0.561318\pi$
$941$	−11.7455			−0.382892			−0.191446	+	0.981503i	$0.561318\pi$
$942$		0			0
$943$			56.0781i			1.82616i
$944$		0			0
$945$	4.20690	+	2.36955i	0.136850	+	0.0770815i
$946$		0			0
$947$			13.7752i			0.447635i	0.974631	+	0.223817i	$0.0718519\pi$
$947$			13.7752i			0.447635i	−0.974631	+	0.223817i	$0.928148\pi$
$948$		0			0
$949$	7.44584			0.241702
$950$		0			0
$951$	−44.0927			−1.42981
$952$		0			0
$953$		−	9.01421i		−	0.291999i	−0.989285	−	0.145999i	$-0.953360\pi$
$953$		−	9.01421i		−	0.291999i	0.989285	−	0.145999i	$-0.0466398\pi$
$954$		0			0
$955$	6.20155	+	3.49304i	0.200677	+	0.113032i
$956$		0			0
$957$			54.1105i			1.74914i
$958$		0			0
$959$	−19.1166			−0.617306
$960$		0			0
$961$	40.1724			1.29588
$962$		0			0
$963$			13.6006i			0.438274i
$964$		0			0
$965$	−3.35673	+	5.95953i	−0.108057	+	0.191844i
$966$		0			0
$967$		−	30.3232i		−	0.975129i	−0.873087	−	0.487564i	$-0.837885\pi$
$967$		−	30.3232i		−	0.975129i	0.873087	−	0.487564i	$-0.162115\pi$
$968$		0			0
$969$	9.07965			0.291680
$970$		0			0
$971$	31.5636			1.01292			0.506462	−	0.862262i	$-0.330953\pi$
$971$	31.5636			1.01292			0.506462	+	0.862262i	$0.330953\pi$
$972$		0			0
$973$			1.71588i			0.0550086i
$974$		0			0
$975$	18.3473	+	30.2723i	0.587582	+	0.969490i
$976$		0			0
$977$			43.1285i			1.37980i	0.723903	+	0.689902i	$0.242346\pi$
$977$			43.1285i			1.37980i	−0.723903	+	0.689902i	$0.757654\pi$
$978$		0			0
$979$	−48.1855			−1.54002
$980$		0			0
$981$	15.6564			0.499870
$982$		0			0
$983$			7.81570i			0.249282i	0.992202	+	0.124641i	$0.0397779\pi$
$983$			7.81570i			0.249282i	−0.992202	+	0.124641i	$0.960222\pi$
$984$		0			0
$985$	23.5862	−	41.8749i	0.751519	−	1.33425i
$986$		0			0
$987$			16.4710i			0.524277i
$988$		0			0
$989$	−7.79310			−0.247806
$990$		0			0
$991$	23.5197			0.747126			0.373563	−	0.927605i	$-0.378136\pi$
$991$	23.5197			0.747126			0.373563	+	0.927605i	$0.378136\pi$
$992$		0			0
$993$			18.5157i			0.587577i
$994$		0			0
$995$	−9.38465	−	5.28593i	−0.297513	−	0.167575i
$996$		0			0
$997$		−	33.6395i		−	1.06537i	−0.846313	−	0.532686i	$-0.821183\pi$
$997$		−	33.6395i		−	1.06537i	0.846313	−	0.532686i	$-0.178817\pi$
$998$		0			0
$999$	8.87275			0.280721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.h.609.2		6
4.3	odd	2		95.2.b.b.39.2	✓	6
5.2	odd	4		7600.2.a.ck.1.2		6
5.3	odd	4		7600.2.a.ck.1.5		6
5.4	even	2	inner	1520.2.d.h.609.5		6
12.11	even	2		855.2.c.d.514.5		6
20.3	even	4		475.2.a.j.1.2		6
20.7	even	4		475.2.a.j.1.5		6
20.19	odd	2		95.2.b.b.39.5	yes	6
60.23	odd	4		4275.2.a.br.1.5		6
60.47	odd	4		4275.2.a.br.1.2		6
60.59	even	2		855.2.c.d.514.2		6
76.75	even	2		1805.2.b.e.1084.5		6
380.227	odd	4		9025.2.a.bx.1.2		6
380.303	odd	4		9025.2.a.bx.1.5		6
380.379	even	2		1805.2.b.e.1084.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.b.39.2	✓	6	4.3	odd	2
95.2.b.b.39.5	yes	6	20.19	odd	2
475.2.a.j.1.2		6	20.3	even	4
475.2.a.j.1.5		6	20.7	even	4
855.2.c.d.514.2		6	60.59	even	2
855.2.c.d.514.5		6	12.11	even	2
1520.2.d.h.609.2		6	1.1	even	1	trivial
1520.2.d.h.609.5		6	5.4	even	2	inner
1805.2.b.e.1084.2		6	380.379	even	2
1805.2.b.e.1084.5		6	76.75	even	2
4275.2.a.br.1.2		6	60.47	odd	4
4275.2.a.br.1.5		6	60.23	odd	4
7600.2.a.ck.1.2		6	5.2	odd	4
7600.2.a.ck.1.5		6	5.3	odd	4
9025.2.a.bx.1.2		6	380.227	odd	4
9025.2.a.bx.1.5		6	380.303	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.31446i q^{3} +(1.94827 + 1.09737i) q^{5} +1.45033i q^{7} -2.35673 q^{9} +3.89655 q^{11} +3.05888i q^{13} +(2.53982 - 4.50920i) q^{15} -3.92301i q^{17} -1.00000 q^{19} +3.35673 q^{21} +5.37334i q^{23} +(2.59155 + 4.27596i) q^{25} -1.48883i q^{27} -6.00000 q^{29} +8.43637 q^{31} -9.01841i q^{33} +(-1.59155 + 2.82564i) q^{35} +5.95953i q^{37} +7.07965 q^{39} +10.4364 q^{41} +1.45033i q^{43} +(-4.59155 - 2.58620i) q^{45} +4.90686i q^{47} +4.89655 q^{49} -9.07965 q^{51} -4.23127i q^{53} +(7.59155 + 4.27596i) q^{55} +2.31446i q^{57} +3.35673 q^{59} +10.3329 q^{61} -3.41802i q^{63} +(-3.35673 + 5.95953i) q^{65} -9.84404i q^{67} +12.4364 q^{69} -8.64327 q^{71} -2.43418i q^{73} +(9.89655 - 5.99804i) q^{75} +5.65127i q^{77} -12.4364 q^{79} -10.5160 q^{81} -12.6635i q^{83} +(4.30500 - 7.64310i) q^{85} +13.8868i q^{87} -12.3662 q^{89} -4.43637 q^{91} -19.5256i q^{93} +(-1.94827 - 1.09737i) q^{95} -3.05888i q^{97} -9.18310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{5} - 14 q^{9} - 2 q^{11} - 10 q^{15} - 6 q^{19} + 20 q^{21} + 3 q^{25} - 36 q^{29} + 3 q^{35} - 8 q^{39} + 12 q^{41} - 15 q^{45} + 4 q^{49} - 4 q^{51} + 33 q^{55} + 20 q^{59} - 14 q^{61} - 20 q^{65}+ \cdots - 30 q^{99}+O(q^{100}) \) 6 * q - q^5 - 14 * q^9 - 2 * q^11 - 10 * q^15 - 6 * q^19 + 20 * q^21 + 3 * q^25 - 36 * q^29 + 3 * q^35 - 8 * q^39 + 12 * q^41 - 15 * q^45 + 4 * q^49 - 4 * q^51 + 33 * q^55 + 20 * q^59 - 14 * q^61 - 20 * q^65 + 24 * q^69 - 52 * q^71 + 34 * q^75 - 24 * q^79 + 38 * q^81 + 13 * q^85 - 24 * q^89 + 24 * q^91 + q^95 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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