LMFDB - Embedded newform 572.2.f.c.441.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(441,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("572.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.f (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:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 21x^{6} + 136x^{4} + 309x^{2} + 225 $ x^8 + 21x^6 + 136x^4 + 309*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.8
Root			$2.51091i$ of defining polynomial
Character	$\chi$	$=$	572.441
Dual form			572.2.f.c.441.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51091 q^{3} +3.81560i q^{5} +1.51091i q^{7} +3.30469 q^{9} +O(q^{10})$	$q+2.51091 q^{3} +3.81560i q^{5} +1.51091i q^{7} +3.30469 q^{9} -1.00000i q^{11} +(-3.24595 - 1.56965i) q^{13} +9.58065i q^{15} +5.13930 q^{17} -1.83462i q^{19} +3.79377i q^{21} -5.32652 q^{23} -9.55882 q^{25} +0.765046 q^{27} +9.51373 q^{29} +9.81560i q^{31} -2.51091i q^{33} -5.76505 q^{35} -3.20623i q^{37} +(-8.15030 - 3.94126i) q^{39} -6.74603i q^{41} +7.51373 q^{43} +12.6094i q^{45} -9.02183i q^{47} +4.71714 q^{49} +12.9044 q^{51} -1.06973 q^{53} +3.81560 q^{55} -4.60656i q^{57} -8.19003i q^{59} +5.51373 q^{61} +4.99310i q^{63} +(5.98917 - 12.3853i) q^{65} +4.46301i q^{67} -13.3744 q^{69} -3.58065i q^{71} -9.71714i q^{73} -24.0014 q^{75} +1.51091 q^{77} -4.25695 q^{79} -7.99310 q^{81} -4.98099i q^{83} +19.6095i q^{85} +23.8882 q^{87} +17.8593i q^{89} +(2.37161 - 4.90435i) q^{91} +24.6461i q^{93} +7.00017 q^{95} +4.18440i q^{97} -3.30469i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 + 18 * q^9	$ 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) $ 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	2.51091			1.44968			0.724838	−	0.688919i	$-0.241914\pi$
$3$	2.51091			1.44968			0.724838	+	0.688919i	$0.241914\pi$
$4$		0			0
$5$			3.81560i			1.70639i	0.521593	+	0.853195i	$0.325338\pi$
$5$			3.81560i			1.70639i	−0.521593	+	0.853195i	$0.674662\pi$
$6$		0			0
$7$			1.51091i			0.571072i	0.958368	+	0.285536i	$0.0921716\pi$
$7$			1.51091i			0.571072i	−0.958368	+	0.285536i	$0.907828\pi$
$8$		0			0
$9$	3.30469			1.10156
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−3.24595	−	1.56965i	−0.900265	−	0.435343i
$13$	−3.24595	−	1.56965i	−0.900265	−	0.435343i
$14$		0			0
$15$			9.58065i			2.47371i
$16$		0			0
$17$	5.13930			1.24646			0.623232	−	0.782037i	$-0.285819\pi$
$17$	5.13930			1.24646			0.623232	+	0.782037i	$0.285819\pi$
$18$		0			0
$19$		−	1.83462i		−	0.420890i	−0.977606	−	0.210445i	$-0.932509\pi$
$19$		−	1.83462i		−	0.420890i	0.977606	−	0.210445i	$-0.0674913\pi$
$20$		0			0
$21$			3.79377i			0.827869i
$22$		0			0
$23$	−5.32652			−1.11066			−0.555328	−	0.831632i	$-0.687407\pi$
$23$	−5.32652			−1.11066			−0.555328	+	0.831632i	$0.687407\pi$
$24$		0			0
$25$	−9.55882			−1.91176
$26$		0			0
$27$	0.765046			0.147233
$28$		0			0
$29$	9.51373			1.76665			0.883327	−	0.468756i	$-0.155298\pi$
$29$	9.51373			1.76665			0.883327	+	0.468756i	$0.155298\pi$
$30$		0			0
$31$			9.81560i			1.76293i	0.472245	+	0.881467i	$0.343444\pi$
$31$			9.81560i			1.76293i	−0.472245	+	0.881467i	$0.656556\pi$
$32$		0			0
$33$		−	2.51091i		−	0.437094i
$34$		0			0
$35$	−5.76505			−0.974471
$36$		0			0
$37$		−	3.20623i		−	0.527100i	−0.964646	−	0.263550i	$-0.915107\pi$
$37$		−	3.20623i		−	0.527100i	0.964646	−	0.263550i	$-0.0848934\pi$
$38$		0			0
$39$	−8.15030	−	3.94126i	−1.30509	−	0.631107i
$40$		0			0
$41$		−	6.74603i		−	1.05355i	−0.850004	−	0.526777i	$-0.823400\pi$
$41$		−	6.74603i		−	1.05355i	0.850004	−	0.526777i	$-0.176600\pi$
$42$		0			0
$43$	7.51373			1.14583			0.572916	−	0.819614i	$-0.305812\pi$
$43$	7.51373			1.14583			0.572916	+	0.819614i	$0.305812\pi$
$44$		0			0
$45$			12.6094i			1.87969i
$46$		0			0
$47$		−	9.02183i		−	1.31597i	−0.753032	−	0.657984i	$-0.771409\pi$
$47$		−	9.02183i		−	1.31597i	0.753032	−	0.657984i	$-0.228591\pi$
$48$		0			0
$49$	4.71714			0.673877
$50$		0			0
$51$	12.9044			1.80697
$52$		0			0
$53$	−1.06973			−0.146939			−0.0734697	−	0.997297i	$-0.523407\pi$
$53$	−1.06973			−0.146939			−0.0734697	+	0.997297i	$0.523407\pi$
$54$		0			0
$55$	3.81560			0.514496
$56$		0			0
$57$		−	4.60656i		−	0.610154i
$58$		0			0
$59$		−	8.19003i		−	1.06625i	−0.846036	−	0.533125i	$-0.821017\pi$
$59$		−	8.19003i		−	1.06625i	0.846036	−	0.533125i	$-0.178983\pi$
$60$		0			0
$61$	5.51373			0.705961			0.352980	−	0.935631i	$-0.385168\pi$
$61$	5.51373			0.705961			0.352980	+	0.935631i	$0.385168\pi$
$62$		0			0
$63$			4.99310i			0.629071i
$64$		0			0
$65$	5.98917	−	12.3853i	0.742865	−	1.53620i
$66$		0			0
$67$			4.46301i			0.545243i	0.962121	+	0.272622i	$0.0878907\pi$
$67$			4.46301i			0.545243i	−0.962121	+	0.272622i	$0.912109\pi$
$68$		0			0
$69$	−13.3744			−1.61009
$70$		0			0
$71$		−	3.58065i		−	0.424945i	−0.977167	−	0.212472i	$-0.931848\pi$
$71$		−	3.58065i		−	0.424945i	0.977167	−	0.212472i	$-0.0681516\pi$
$72$		0			0
$73$		−	9.71714i		−	1.13731i	−0.822578	−	0.568653i	$-0.807465\pi$
$73$		−	9.71714i		−	1.13731i	0.822578	−	0.568653i	$-0.192535\pi$
$74$		0			0
$75$	−24.0014			−2.77144
$76$		0			0
$77$	1.51091			0.172185
$78$		0			0
$79$	−4.25695			−0.478944			−0.239472	−	0.970903i	$-0.576974\pi$
$79$	−4.25695			−0.478944			−0.239472	+	0.970903i	$0.576974\pi$
$80$		0			0
$81$	−7.99310			−0.888122
$82$		0			0
$83$		−	4.98099i		−	0.546734i	−0.961910	−	0.273367i	$-0.911863\pi$
$83$		−	4.98099i		−	0.546734i	0.961910	−	0.273367i	$-0.0881374\pi$
$84$		0			0
$85$			19.6095i			2.12695i
$86$		0			0
$87$	23.8882			2.56108
$88$		0			0
$89$			17.8593i			1.89308i	0.322590	+	0.946539i	$0.395446\pi$
$89$			17.8593i			1.89308i	−0.322590	+	0.946539i	$0.604554\pi$
$90$		0			0
$91$	2.37161	−	4.90435i	0.248612	−	0.514116i
$92$		0			0
$93$			24.6461i			2.55568i
$94$		0			0
$95$	7.00017			0.718202
$96$		0			0
$97$			4.18440i			0.424861i	0.977176	+	0.212431i	$0.0681379\pi$
$97$			4.18440i			0.424861i	−0.977176	+	0.212431i	$0.931862\pi$
$98$		0			0
$99$		−	3.30469i		−	0.332134i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	5.60673			0.552447			0.276224	−	0.961093i	$-0.410917\pi$
$103$	5.60673			0.552447			0.276224	+	0.961093i	$0.410917\pi$
$104$		0			0
$105$	−14.4755			−1.41267
$106$		0			0
$107$	−5.88252			−0.568685			−0.284342	−	0.958723i	$-0.591775\pi$
$107$	−5.88252			−0.568685			−0.284342	+	0.958723i	$0.591775\pi$
$108$		0			0
$109$			3.18721i			0.305280i	0.988282	+	0.152640i	$0.0487774\pi$
$109$			3.18721i			0.305280i	−0.988282	+	0.152640i	$0.951223\pi$
$110$		0			0
$111$		−	8.05056i		−	0.764125i
$112$		0			0
$113$	7.90154			0.743314			0.371657	−	0.928370i	$-0.378790\pi$
$113$	7.90154			0.743314			0.371657	+	0.928370i	$0.378790\pi$
$114$		0			0
$115$		−	20.3239i		−	1.89521i
$116$		0			0
$117$	−10.7269	−	5.18721i	−0.991698	−	0.479558i
$118$		0			0
$119$			7.76505i			0.711821i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	16.9387i		−	1.52731i
$124$		0			0
$125$		−	17.3946i		−	1.55582i
$126$		0			0
$127$	−12.9838			−1.15213			−0.576063	−	0.817405i	$-0.695412\pi$
$127$	−12.9838			−1.15213			−0.576063	+	0.817405i	$0.695412\pi$
$128$		0			0
$129$	18.8663			1.66109
$130$		0			0
$131$	−10.4125			−0.909740			−0.454870	−	0.890558i	$-0.650314\pi$
$131$	−10.4125			−0.909740			−0.454870	+	0.890558i	$0.650314\pi$
$132$		0			0
$133$	2.77195			0.240358
$134$		0			0
$135$			2.91911i			0.251237i
$136$		0			0
$137$		−	16.6983i		−	1.42663i	−0.700843	−	0.713316i	$-0.747193\pi$
$137$		−	16.6983i		−	1.42663i	0.700843	−	0.713316i	$-0.252807\pi$
$138$		0			0
$139$	−13.7650			−1.16754			−0.583768	−	0.811921i	$-0.698422\pi$
$139$	−13.7650			−1.16754			−0.583768	+	0.811921i	$0.698422\pi$
$140$		0			0
$141$		−	22.6530i		−	1.90773i
$142$		0			0
$143$	−1.56965	+	3.24595i	−0.131261	+	0.271440i
$144$		0			0
$145$			36.3006i			3.01460i
$146$		0			0
$147$	11.8443			0.976904
$148$		0			0
$149$			10.1639i			0.832663i	0.909213	+	0.416331i	$0.136684\pi$
$149$			10.1639i			0.832663i	−0.909213	+	0.416331i	$0.863316\pi$
$150$		0			0
$151$		−	21.6532i		−	1.76211i	−0.473012	−	0.881056i	$-0.656833\pi$
$151$		−	21.6532i		−	1.76211i	0.473012	−	0.881056i	$-0.343167\pi$
$152$		0			0
$153$	16.9838			1.37306
$154$		0			0
$155$	−37.4524			−3.00825
$156$		0			0
$157$	−11.0916			−0.885203			−0.442602	−	0.896718i	$-0.645944\pi$
$157$	−11.0916			−0.885203			−0.442602	+	0.896718i	$0.645944\pi$
$158$		0			0
$159$	−2.68601			−0.213015
$160$		0			0
$161$		−	8.04791i		−	0.634264i
$162$		0			0
$163$			8.27861i			0.648431i	0.945983	+	0.324215i	$0.105100\pi$
$163$			8.27861i			0.648431i	−0.945983	+	0.324215i	$0.894900\pi$
$164$		0			0
$165$	9.58065			0.745852
$166$		0			0
$167$		−	22.4660i		−	1.73847i	−0.494399	−	0.869235i	$-0.664612\pi$
$167$		−	22.4660i		−	1.73847i	0.494399	−	0.869235i	$-0.335388\pi$
$168$		0			0
$169$	8.07238	+	10.1900i	0.620953	+	0.783848i
$170$		0			0
$171$		−	6.06283i		−	0.463637i
$172$		0			0
$173$	−7.18296			−0.546110			−0.273055	−	0.961998i	$-0.588034\pi$
$173$	−7.18296			−0.546110			−0.273055	+	0.961998i	$0.588034\pi$
$174$		0			0
$175$		−	14.4426i		−	1.09175i
$176$		0			0
$177$		−	20.5644i		−	1.54572i
$178$		0			0
$179$	11.9114			0.890301			0.445151	−	0.895456i	$-0.353150\pi$
$179$	11.9114			0.890301			0.445151	+	0.895456i	$0.353150\pi$
$180$		0			0
$181$	9.02448			0.670784			0.335392	−	0.942079i	$-0.391131\pi$
$181$	9.02448			0.670784			0.335392	+	0.942079i	$0.391131\pi$
$182$		0			0
$183$	13.8445			1.02341
$184$		0			0
$185$	12.2337			0.899438
$186$		0			0
$187$		−	5.13930i		−	0.375823i
$188$		0			0
$189$			1.15592i			0.0840808i
$190$		0			0
$191$	−11.7677			−0.851481			−0.425740	−	0.904845i	$-0.639986\pi$
$191$	−11.7677			−0.851481			−0.425740	+	0.904845i	$0.639986\pi$
$192$		0			0
$193$			18.5384i			1.33442i	0.744869	+	0.667211i	$0.232512\pi$
$193$			18.5384i			1.33442i	−0.744869	+	0.667211i	$0.767488\pi$
$194$		0			0
$195$	15.0383	−	31.0983i	1.07691	−	2.22700i
$196$		0			0
$197$			4.46036i			0.317787i	0.987296	+	0.158894i	$0.0507927\pi$
$197$			4.46036i			0.317787i	−0.987296	+	0.158894i	$0.949207\pi$
$198$		0			0
$199$	6.23070			0.441683			0.220841	−	0.975310i	$-0.429120\pi$
$199$	6.23070			0.441683			0.220841	+	0.975310i	$0.429120\pi$
$200$		0			0
$201$			11.2062i			0.790426i
$202$		0			0
$203$			14.3744i			1.00889i
$204$		0			0
$205$	25.7402			1.79777
$206$		0			0
$207$	−17.6025			−1.22346
$208$		0			0
$209$	−1.83462			−0.126903
$210$		0			0
$211$	−15.9098			−1.09528			−0.547638	−	0.836715i	$-0.684473\pi$
$211$	−15.9098			−1.09528			−0.547638	+	0.836715i	$0.684473\pi$
$212$		0			0
$213$		−	8.99070i		−	0.616033i
$214$		0			0
$215$			28.6694i			1.95524i
$216$		0			0
$217$	−14.8305			−1.00676
$218$		0			0
$219$		−	24.3989i		−	1.64872i
$220$		0			0
$221$	−16.6819	−	8.06692i	−1.12215	−	0.542640i
$222$		0			0
$223$			10.3292i			0.691692i	0.938291	+	0.345846i	$0.112408\pi$
$223$			10.3292i			0.691692i	−0.938291	+	0.345846i	$0.887592\pi$
$224$		0			0
$225$	−31.5889			−2.10593
$226$		0			0
$227$			6.53418i			0.433689i	0.976206	+	0.216844i	$0.0695764\pi$
$227$			6.53418i			0.433689i	−0.976206	+	0.216844i	$0.930424\pi$
$228$		0			0
$229$			22.1957i			1.46673i	0.679835	+	0.733365i	$0.262051\pi$
$229$			22.1957i			1.46673i	−0.679835	+	0.733365i	$0.737949\pi$
$230$		0			0
$231$	3.79377			0.249612
$232$		0			0
$233$	−5.30044			−0.347243			−0.173622	−	0.984812i	$-0.555547\pi$
$233$	−5.30044			−0.347243			−0.173622	+	0.984812i	$0.555547\pi$
$234$		0			0
$235$	34.4237			2.24556
$236$		0			0
$237$	−10.6888			−0.694314
$238$		0			0
$239$		−	4.59950i		−	0.297517i	−0.988874	−	0.148758i	$-0.952472\pi$
$239$		−	4.59950i		−	0.297517i	0.988874	−	0.148758i	$-0.0475277\pi$
$240$		0			0
$241$		−	4.27580i		−	0.275428i	−0.990472	−	0.137714i	$-0.956024\pi$
$241$		−	4.27580i		−	0.275428i	0.990472	−	0.137714i	$-0.0439755\pi$
$242$		0			0
$243$	−22.3651			−1.43472
$244$		0			0
$245$			17.9987i			1.14990i
$246$		0			0
$247$	−2.87971	+	5.95507i	−0.183232	+	0.378912i
$248$		0			0
$249$		−	12.5068i		−	0.792588i
$250$		0			0
$251$	−18.6791			−1.17902			−0.589508	−	0.807763i	$-0.700678\pi$
$251$	−18.6791			−1.17902			−0.589508	+	0.807763i	$0.700678\pi$
$252$		0			0
$253$			5.32652i			0.334875i
$254$		0			0
$255$			49.2379i			3.08340i
$256$		0			0
$257$	14.1259			0.881151			0.440575	−	0.897716i	$-0.354774\pi$
$257$	14.1259			0.881151			0.440575	+	0.897716i	$0.354774\pi$
$258$		0			0
$259$	4.84433			0.301012
$260$		0			0
$261$	31.4399			1.94608
$262$		0			0
$263$	19.8087			1.22146			0.610728	−	0.791840i	$-0.290877\pi$
$263$	19.8087			1.22146			0.610728	+	0.791840i	$0.290877\pi$
$264$		0			0
$265$		−	4.08168i		−	0.250736i
$266$		0			0
$267$			44.8431i			2.74435i
$268$		0			0
$269$	−25.8910			−1.57860			−0.789300	−	0.614008i	$-0.789556\pi$
$269$	−25.8910			−1.57860			−0.789300	+	0.614008i	$0.789556\pi$
$270$		0			0
$271$		−	8.46566i		−	0.514252i	−0.966378	−	0.257126i	$-0.917224\pi$
$271$		−	8.46566i		−	0.514252i	0.966378	−	0.257126i	$-0.0827755\pi$
$272$		0			0
$273$	5.95491	−	12.3144i	0.360407	−	0.745302i
$274$		0			0
$275$			9.55882i			0.576419i
$276$		0			0
$277$	27.1613			1.63196			0.815982	−	0.578077i	$-0.196197\pi$
$277$	27.1613			1.63196			0.815982	+	0.578077i	$0.196197\pi$
$278$		0			0
$279$			32.4375i			1.94198i
$280$		0			0
$281$		−	29.0373i		−	1.73222i	−0.499852	−	0.866111i	$-0.666612\pi$
$281$		−	29.0373i		−	1.73222i	0.499852	−	0.866111i	$-0.333388\pi$
$282$		0			0
$283$	1.92618			0.114499			0.0572497	−	0.998360i	$-0.481767\pi$
$283$	1.92618			0.114499			0.0572497	+	0.998360i	$0.481767\pi$
$284$		0			0
$285$	17.5768			1.04116
$286$		0			0
$287$	10.1927			0.601655
$288$		0			0
$289$	9.41245			0.553674
$290$		0			0
$291$			10.5067i			0.615911i
$292$		0			0
$293$			29.9318i			1.74863i	0.485355	+	0.874317i	$0.338691\pi$
$293$			29.9318i			1.74863i	−0.485355	+	0.874317i	$0.661309\pi$
$294$		0			0
$295$	31.2499			1.81944
$296$		0			0
$297$		−	0.765046i		−	0.0443925i
$298$		0			0
$299$	17.2896	+	8.36078i	0.999884	+	0.483516i
$300$		0			0
$301$			11.3526i			0.654353i
$302$		0			0
$303$	−15.0655			−0.865489
$304$		0			0
$305$			21.0382i			1.20464i
$306$		0			0
$307$			0.748847i			0.0427389i	0.999772	+	0.0213695i	$0.00680263\pi$
$307$			0.748847i			0.0427389i	−0.999772	+	0.0213695i	$0.993197\pi$
$308$		0			0
$309$	14.0780			0.800870
$310$		0			0
$311$	−15.5453			−0.881491			−0.440746	−	0.897632i	$-0.645286\pi$
$311$	−15.5453			−0.881491			−0.440746	+	0.897632i	$0.645286\pi$
$312$		0			0
$313$	−32.3159			−1.82661			−0.913303	−	0.407282i	$-0.866477\pi$
$313$	−32.3159			−1.82661			−0.913303	+	0.407282i	$0.866477\pi$
$314$		0			0
$315$	−19.0517			−1.07344
$316$		0			0
$317$		−	31.6243i		−	1.77620i	−0.459652	−	0.888099i	$-0.652026\pi$
$317$		−	31.6243i		−	1.77620i	0.459652	−	0.888099i	$-0.347974\pi$
$318$		0			0
$319$		−	9.51373i		−	0.532666i
$320$		0			0
$321$	−14.7705			−0.824409
$322$		0			0
$323$		−	9.42865i		−	0.524624i
$324$		0			0
$325$	31.0275	+	15.0040i	1.72109	+	0.832274i
$326$		0			0
$327$			8.00281i			0.442557i
$328$		0			0
$329$	13.6312			0.751513
$330$		0			0
$331$			25.4905i			1.40108i	0.713612	+	0.700541i	$0.247058\pi$
$331$			25.4905i			1.40108i	−0.713612	+	0.700541i	$0.752942\pi$
$332$		0			0
$333$		−	10.5956i		−	0.580634i
$334$		0			0
$335$	−17.0291			−0.930397
$336$		0			0
$337$	−15.8445			−0.863105			−0.431552	−	0.902088i	$-0.642034\pi$
$337$	−15.8445			−0.863105			−0.431552	+	0.902088i	$0.642034\pi$
$338$		0			0
$339$	19.8401			1.07757
$340$		0			0
$341$	9.81560			0.531545
$342$		0			0
$343$			17.7036i			0.955904i
$344$		0			0
$345$		−	51.0315i		−	2.74744i
$346$		0			0
$347$	2.48627			0.133470			0.0667350	−	0.997771i	$-0.478742\pi$
$347$	2.48627			0.133470			0.0667350	+	0.997771i	$0.478742\pi$
$348$		0			0
$349$			24.8908i			1.33237i	0.745785	+	0.666187i	$0.232075\pi$
$349$			24.8908i			1.33237i	−0.745785	+	0.666187i	$0.767925\pi$
$350$		0			0
$351$	−2.48330	−	1.20086i	−0.132549	−	0.0640970i
$352$		0			0
$353$			5.99310i			0.318981i	0.987200	+	0.159490i	$0.0509851\pi$
$353$			5.99310i			0.318981i	−0.987200	+	0.159490i	$0.949015\pi$
$354$		0			0
$355$	13.6623			0.725121
$356$		0			0
$357$			19.4974i			1.03191i
$358$		0			0
$359$			24.9345i			1.31599i	0.753022	+	0.657995i	$0.228595\pi$
$359$			24.9345i			1.31599i	−0.753022	+	0.657995i	$0.771405\pi$
$360$		0			0
$361$	15.6342			0.822852
$362$		0			0
$363$	−2.51091			−0.131789
$364$		0			0
$365$	37.0767			1.94069
$366$		0			0
$367$	10.3718			0.541402			0.270701	−	0.962663i	$-0.412744\pi$
$367$	10.3718			0.541402			0.270701	+	0.962663i	$0.412744\pi$
$368$		0			0
$369$		−	22.2935i		−	1.16056i
$370$		0			0
$371$		−	1.61628i		−	0.0839129i
$372$		0			0
$373$	8.93164			0.462463			0.231231	−	0.972899i	$-0.425725\pi$
$373$	8.93164			0.462463			0.231231	+	0.972899i	$0.425725\pi$
$374$		0			0
$375$		−	43.6765i		−	2.25544i
$376$		0			0
$377$	−30.8811	−	14.9332i	−1.59046	−	0.769101i
$378$		0			0
$379$			26.3675i			1.35441i	0.735795	+	0.677204i	$0.236809\pi$
$379$			26.3675i			1.35441i	−0.735795	+	0.677204i	$0.763191\pi$
$380$		0			0
$381$	−32.6012			−1.67021
$382$		0			0
$383$		−	21.8212i		−	1.11501i	−0.830173	−	0.557506i	$-0.811758\pi$
$383$		−	21.8212i		−	1.11501i	0.830173	−	0.557506i	$-0.188242\pi$
$384$		0			0
$385$			5.76505i			0.293814i
$386$		0			0
$387$	24.8305			1.26221
$388$		0			0
$389$	7.53274			0.381925			0.190963	−	0.981597i	$-0.438839\pi$
$389$	7.53274			0.381925			0.190963	+	0.981597i	$0.438839\pi$
$390$		0			0
$391$	−27.3746			−1.38439
$392$		0			0
$393$	−26.1448			−1.31883
$394$		0			0
$395$		−	16.2428i		−	0.817265i
$396$		0			0
$397$		−	0.748847i		−	0.0375835i	−0.999823	−	0.0187918i	$-0.994018\pi$
$397$		−	0.748847i		−	0.0375835i	0.999823	−	0.0187918i	$-0.00598196\pi$
$398$		0			0
$399$	6.96012			0.348442
$400$		0			0
$401$		−	6.47024i		−	0.323108i	−0.986864	−	0.161554i	$-0.948349\pi$
$401$		−	6.47024i		−	0.323108i	0.986864	−	0.161554i	$-0.0516506\pi$
$402$		0			0
$403$	15.4071	−	31.8610i	0.767481	−	1.58711i
$404$		0			0
$405$		−	30.4985i		−	1.51548i
$406$		0			0
$407$	−3.20623			−0.158927
$408$		0			0
$409$		−	2.98380i		−	0.147539i	−0.997275	−	0.0737697i	$-0.976497\pi$
$409$		−	2.98380i		−	0.147539i	0.997275	−	0.0737697i	$-0.0235030\pi$
$410$		0			0
$411$		−	41.9280i		−	2.06815i
$412$		0			0
$413$	12.3744			0.608906
$414$		0			0
$415$	19.0055			0.932941
$416$		0			0
$417$	−34.5628			−1.69255
$418$		0			0
$419$	10.6339			0.519498			0.259749	−	0.965676i	$-0.416360\pi$
$419$	10.6339			0.519498			0.259749	+	0.965676i	$0.416360\pi$
$420$		0			0
$421$		−	4.78125i		−	0.233024i	−0.993189	−	0.116512i	$-0.962829\pi$
$421$		−	4.78125i		−	0.233024i	0.993189	−	0.116512i	$-0.0371713\pi$
$422$		0			0
$423$		−	29.8143i		−	1.44962i
$424$		0			0
$425$	−49.1257			−2.38295
$426$		0			0
$427$			8.33077i			0.403154i
$428$		0			0
$429$	−3.94126	+	8.15030i	−0.190286	+	0.393500i
$430$		0			0
$431$		−	28.3251i		−	1.36437i	−0.731179	−	0.682186i	$-0.761030\pi$
$431$		−	28.3251i		−	1.36437i	0.731179	−	0.682186i	$-0.238970\pi$
$432$		0			0
$433$	4.74620			0.228088			0.114044	−	0.993476i	$-0.463620\pi$
$433$	4.74620			0.228088			0.114044	+	0.993476i	$0.463620\pi$
$434$		0			0
$435$			91.1477i			4.37020i
$436$		0			0
$437$			9.77211i			0.467464i
$438$		0			0
$439$	−18.3386			−0.875255			−0.437628	−	0.899156i	$-0.644181\pi$
$439$	−18.3386			−0.875255			−0.437628	+	0.899156i	$0.644181\pi$
$440$		0			0
$441$	15.5887			0.742318
$442$		0			0
$443$	1.98805			0.0944552			0.0472276	−	0.998884i	$-0.484961\pi$
$443$	1.98805			0.0944552			0.0472276	+	0.998884i	$0.484961\pi$
$444$		0			0
$445$	−68.1438			−3.23033
$446$		0			0
$447$			25.5208i			1.20709i
$448$		0			0
$449$			2.23368i			0.105414i	0.998610	+	0.0527070i	$0.0167849\pi$
$449$			2.23368i			0.105414i	−0.998610	+	0.0527070i	$0.983215\pi$
$450$		0			0
$451$	−6.74603			−0.317658
$452$		0			0
$453$		−	54.3693i		−	2.55449i
$454$		0			0
$455$	18.7131	+	9.04912i	0.877281	+	0.424229i
$456$		0			0
$457$		−	2.37017i		−	0.110872i	−0.998462	−	0.0554360i	$-0.982345\pi$
$457$		−	2.37017i		−	0.110872i	0.998462	−	0.0554360i	$-0.0176549\pi$
$458$		0			0
$459$	3.93181			0.183521
$460$		0			0
$461$			13.1816i			0.613928i	0.951721	+	0.306964i	$0.0993130\pi$
$461$			13.1816i			0.613928i	−0.951721	+	0.306964i	$0.900687\pi$
$462$		0			0
$463$			24.3675i			1.13245i	0.824249	+	0.566227i	$0.191598\pi$
$463$			24.3675i			1.13245i	−0.824249	+	0.566227i	$0.808402\pi$
$464$		0			0
$465$	−94.0398			−4.36099
$466$		0			0
$467$	18.3239			0.847927			0.423964	−	0.905679i	$-0.360638\pi$
$467$	18.3239			0.847927			0.423964	+	0.905679i	$0.360638\pi$
$468$		0			0
$469$	−6.74322			−0.311373
$470$		0			0
$471$	−27.8500			−1.28326
$472$		0			0
$473$		−	7.51373i		−	0.345482i
$474$		0			0
$475$			17.5368i			0.804642i
$476$		0			0
$477$	−3.53514			−0.161863
$478$		0			0
$479$		−	16.8249i		−	0.768749i	−0.923177	−	0.384375i	$-0.874417\pi$
$479$		−	16.8249i		−	0.768749i	0.923177	−	0.384375i	$-0.125583\pi$
$480$		0			0
$481$	−5.03266	+	10.4072i	−0.229469	+	0.474530i
$482$		0			0
$483$		−	20.2076i		−	0.919478i
$484$		0			0
$485$	−15.9660			−0.724979
$486$		0			0
$487$		−	14.8811i		−	0.674326i	−0.941446	−	0.337163i	$-0.890533\pi$
$487$		−	14.8811i		−	0.674326i	0.941446	−	0.337163i	$-0.109467\pi$
$488$		0			0
$489$			20.7869i			0.940015i
$490$		0			0
$491$	39.3582			1.77621			0.888106	−	0.459639i	$-0.152021\pi$
$491$	39.3582			1.77621			0.888106	+	0.459639i	$0.152021\pi$
$492$		0			0
$493$	48.8939			2.20207
$494$		0			0
$495$	12.6094			0.566749
$496$		0			0
$497$	5.41005			0.242674
$498$		0			0
$499$		−	14.3223i		−	0.641153i	−0.947223	−	0.320576i	$-0.896123\pi$
$499$		−	14.3223i		−	0.641153i	0.947223	−	0.320576i	$-0.103877\pi$
$500$		0			0
$501$		−	56.4102i		−	2.52022i
$502$		0			0
$503$	24.0578			1.07268			0.536342	−	0.844001i	$-0.319806\pi$
$503$	24.0578			1.07268			0.536342	+	0.844001i	$0.319806\pi$
$504$		0			0
$505$		−	22.8936i		−	1.01875i
$506$		0			0
$507$	20.2691	+	25.5863i	0.900181	+	1.13633i
$508$		0			0
$509$		−	0.755747i		−	0.0334979i	−0.999860	−	0.0167490i	$-0.994668\pi$
$509$		−	0.755747i		−	0.0334979i	0.999860	−	0.0167490i	$-0.00533161\pi$
$510$		0			0
$511$	14.6818			0.649483
$512$		0			0
$513$		−	1.40357i		−	0.0619690i
$514$		0			0
$515$			21.3930i			0.942690i
$516$		0			0
$517$	−9.02183			−0.396780
$518$		0			0
$519$	−18.0358			−0.791683
$520$		0			0
$521$	13.2307			0.579648			0.289824	−	0.957080i	$-0.406403\pi$
$521$	13.2307			0.579648			0.289824	+	0.957080i	$0.406403\pi$
$522$		0			0
$523$	−41.0167			−1.79354			−0.896768	−	0.442501i	$-0.854091\pi$
$523$	−41.0167			−1.79354			−0.896768	+	0.442501i	$0.854091\pi$
$524$		0			0
$525$		−	36.2640i		−	1.58269i
$526$		0			0
$527$			50.4454i			2.19743i
$528$		0			0
$529$	5.37177			0.233555
$530$		0			0
$531$		−	27.0655i		−	1.17454i
$532$		0			0
$533$	−10.5889	+	21.8973i	−0.458657	+	0.948477i
$534$		0			0
$535$		−	22.4454i		−	0.970398i
$536$		0			0
$537$	29.9085			1.29065
$538$		0			0
$539$		−	4.71714i		−	0.203182i
$540$		0			0
$541$			10.4783i			0.450499i	0.974301	+	0.225250i	$0.0723198\pi$
$541$			10.4783i			0.450499i	−0.974301	+	0.225250i	$0.927680\pi$
$542$		0			0
$543$	22.6597			0.972420
$544$		0			0
$545$	−12.1611			−0.520926
$546$		0			0
$547$	−10.6150			−0.453865			−0.226932	−	0.973911i	$-0.572870\pi$
$547$	−10.6150			−0.453865			−0.226932	+	0.973911i	$0.572870\pi$
$548$		0			0
$549$	18.2212			0.777660
$550$		0			0
$551$		−	17.4540i		−	0.743567i
$552$		0			0
$553$		−	6.43188i		−	0.273511i
$554$		0			0
$555$	30.7177			1.30389
$556$		0			0
$557$		−	14.6122i		−	0.619138i	−0.950877	−	0.309569i	$-0.899815\pi$
$557$		−	14.6122i		−	0.619138i	0.950877	−	0.309569i	$-0.100185\pi$
$558$		0			0
$559$	−24.3892	−	11.7939i	−1.03155	−	0.498831i
$560$		0			0
$561$		−	12.9044i		−	0.544822i
$562$		0			0
$563$	−41.5577			−1.75145			−0.875724	−	0.482811i	$-0.839616\pi$
$563$	−41.5577			−1.75145			−0.875724	+	0.482811i	$0.839616\pi$
$564$		0			0
$565$			30.1491i			1.26838i
$566$		0			0
$567$		−	12.0769i		−	0.507181i
$568$		0			0
$569$	32.6530			1.36889			0.684443	−	0.729066i	$-0.260045\pi$
$569$	32.6530			1.36889			0.684443	+	0.729066i	$0.260045\pi$
$570$		0			0
$571$	−2.69106			−0.112617			−0.0563087	−	0.998413i	$-0.517933\pi$
$571$	−2.69106			−0.112617			−0.0563087	+	0.998413i	$0.517933\pi$
$572$		0			0
$573$	−29.5477			−1.23437
$574$		0			0
$575$	50.9152			2.12331
$576$		0			0
$577$		−	28.7036i		−	1.19495i	−0.801889	−	0.597473i	$-0.796172\pi$
$577$		−	28.7036i		−	1.19495i	0.801889	−	0.597473i	$-0.203828\pi$
$578$		0			0
$579$			46.5482i			1.93448i
$580$		0			0
$581$	7.52584			0.312225
$582$		0			0
$583$			1.06973i			0.0443039i
$584$		0			0
$585$	19.7923	−	40.9294i	0.818312	−	1.69222i
$586$		0			0
$587$		−	4.28711i		−	0.176948i	−0.996078	−	0.0884740i	$-0.971801\pi$
$587$		−	4.28711i		−	0.176948i	0.996078	−	0.0884740i	$-0.0281990\pi$
$588$		0			0
$589$	18.0079			0.742001
$590$		0			0
$591$			11.1996i			0.460689i
$592$		0			0
$593$			2.02448i			0.0831353i	0.999136	+	0.0415676i	$0.0132352\pi$
$593$			2.02448i			0.0831353i	−0.999136	+	0.0415676i	$0.986765\pi$
$594$		0			0
$595$	−29.6283			−1.21464
$596$		0			0
$597$	15.6448			0.640297
$598$		0			0
$599$	−10.0916			−0.412330			−0.206165	−	0.978517i	$-0.566098\pi$
$599$	−10.0916			−0.412330			−0.206165	+	0.978517i	$0.566098\pi$
$600$		0			0
$601$	4.94014			0.201513			0.100756	−	0.994911i	$-0.467874\pi$
$601$	4.94014			0.201513			0.100756	+	0.994911i	$0.467874\pi$
$602$		0			0
$603$			14.7488i			0.600619i
$604$		0			0
$605$		−	3.81560i		−	0.155126i
$606$		0			0
$607$	−22.9697			−0.932310			−0.466155	−	0.884703i	$-0.654361\pi$
$607$	−22.9697			−0.932310			−0.466155	+	0.884703i	$0.654361\pi$
$608$		0			0
$609$			36.0929i			1.46256i
$610$		0			0
$611$	−14.1611	+	29.2844i	−0.572898	+	1.18472i
$612$		0			0
$613$		−	29.8459i		−	1.20546i	−0.797944	−	0.602732i	$-0.794079\pi$
$613$		−	29.8459i		−	1.20546i	0.797944	−	0.602732i	$-0.205921\pi$
$614$		0			0
$615$	64.6314			2.60619
$616$		0			0
$617$		−	12.5083i		−	0.503564i	−0.967784	−	0.251782i	$-0.918983\pi$
$617$		−	12.5083i		−	0.503564i	0.967784	−	0.251782i	$-0.0810166\pi$
$618$		0			0
$619$			15.1597i			0.609320i	0.952461	+	0.304660i	$0.0985428\pi$
$619$			15.1597i			0.609320i	−0.952461	+	0.304660i	$0.901457\pi$
$620$		0			0
$621$	−4.07503			−0.163525
$622$		0			0
$623$	−26.9838			−1.08108
$624$		0			0
$625$	18.5770			0.743078
$626$		0			0
$627$	−4.60656			−0.183968
$628$		0			0
$629$		−	16.4778i		−	0.657012i
$630$		0			0
$631$			5.94382i			0.236620i	0.992977	+	0.118310i	$0.0377476\pi$
$631$			5.94382i			0.236620i	−0.992977	+	0.118310i	$0.962252\pi$
$632$		0			0
$633$	−39.9482			−1.58780
$634$		0			0
$635$		−	49.5410i		−	1.96598i
$636$		0			0
$637$	−15.3116	−	7.40427i	−0.606668	−	0.293368i
$638$		0			0
$639$		−	11.8329i		−	0.468103i
$640$		0			0
$641$	41.0592			1.62174			0.810870	−	0.585227i	$-0.198994\pi$
$641$	41.0592			1.62174			0.810870	+	0.585227i	$0.198994\pi$
$642$		0			0
$643$		−	2.46863i		−	0.0973534i	−0.998815	−	0.0486767i	$-0.984500\pi$
$643$		−	2.46863i		−	0.0973534i	0.998815	−	0.0486767i	$-0.0155004\pi$
$644$		0			0
$645$			71.9864i			2.83446i
$646$		0			0
$647$	−2.97552			−0.116980			−0.0584900	−	0.998288i	$-0.518629\pi$
$647$	−2.97552			−0.116980			−0.0584900	+	0.998288i	$0.518629\pi$
$648$		0			0
$649$	−8.19003			−0.321487
$650$		0			0
$651$	−37.2382			−1.45948
$652$		0			0
$653$	30.2915			1.18540			0.592698	−	0.805425i	$-0.298063\pi$
$653$	30.2915			1.18540			0.592698	+	0.805425i	$0.298063\pi$
$654$		0			0
$655$		−	39.7298i		−	1.55237i
$656$		0			0
$657$		−	32.1121i		−	1.25281i
$658$		0			0
$659$	8.07095			0.314399			0.157200	−	0.987567i	$-0.449753\pi$
$659$	8.07095			0.314399			0.157200	+	0.987567i	$0.449753\pi$
$660$		0			0
$661$			34.8952i			1.35727i	0.734477	+	0.678633i	$0.237427\pi$
$661$			34.8952i			1.35727i	−0.734477	+	0.678633i	$0.762573\pi$
$662$		0			0
$663$	−41.8869	−	20.2553i	−1.62675	−	0.786652i
$664$		0			0
$665$			10.5766i			0.410145i
$666$		0			0
$667$	−50.6750			−1.96214
$668$		0			0
$669$			25.9356i			1.00273i
$670$		0			0
$671$		−	5.51373i		−	0.212855i
$672$		0			0
$673$	10.9424			0.421798			0.210899	−	0.977508i	$-0.432361\pi$
$673$	10.9424			0.421798			0.210899	+	0.977508i	$0.432361\pi$
$674$		0			0
$675$	−7.31294			−0.281475
$676$		0			0
$677$	8.79250			0.337923			0.168962	−	0.985623i	$-0.445959\pi$
$677$	8.79250			0.337923			0.168962	+	0.985623i	$0.445959\pi$
$678$		0			0
$679$	−6.32226			−0.242626
$680$		0			0
$681$			16.4068i			0.628708i
$682$		0			0
$683$			25.4917i			0.975414i	0.873007	+	0.487707i	$0.162167\pi$
$683$			25.4917i			0.975414i	−0.873007	+	0.487707i	$0.837833\pi$
$684$		0			0
$685$	63.7140			2.43439
$686$		0			0
$687$			55.7314i			2.12628i
$688$		0			0
$689$	3.47231	+	1.67911i	0.132284	+	0.0639691i
$690$		0			0
$691$			22.3675i			0.850901i	0.904982	+	0.425450i	$0.139884\pi$
$691$			22.3675i			0.850901i	−0.904982	+	0.425450i	$0.860116\pi$
$692$		0			0
$693$	4.99310			0.189672
$694$		0			0
$695$		−	52.5219i		−	1.99227i
$696$		0			0
$697$		−	34.6699i		−	1.31322i
$698$		0			0
$699$	−13.3089			−0.503390
$700$		0			0
$701$	−46.8362			−1.76898			−0.884489	−	0.466562i	$-0.845493\pi$
$701$	−46.8362			−1.76898			−0.884489	+	0.466562i	$0.845493\pi$
$702$		0			0
$703$	−5.88219			−0.221851
$704$		0			0
$705$	86.4350			3.25533
$706$		0			0
$707$		−	9.06548i		−	0.340943i
$708$		0			0
$709$			48.4168i			1.81833i	0.416435	+	0.909166i	$0.363279\pi$
$709$			48.4168i			1.81833i	−0.416435	+	0.909166i	$0.636721\pi$
$710$		0			0
$711$	−14.0679			−0.527587
$712$		0			0
$713$		−	52.2830i		−	1.95801i
$714$		0			0
$715$	−12.3853	−	5.98917i	−0.463182	−	0.223982i
$716$		0			0
$717$		−	11.5489i		−	0.431303i
$718$		0			0
$719$	−23.9114			−0.891745			−0.445873	−	0.895096i	$-0.647107\pi$
$719$	−23.9114			−0.891745			−0.445873	+	0.895096i	$0.647107\pi$
$720$		0			0
$721$			8.47128i			0.315487i
$722$		0			0
$723$		−	10.7362i		−	0.399282i
$724$		0			0
$725$	−90.9400			−3.37743
$726$		0			0
$727$	3.51781			0.130469			0.0652343	−	0.997870i	$-0.479221\pi$
$727$	3.51781			0.130469			0.0652343	+	0.997870i	$0.479221\pi$
$728$		0			0
$729$	−32.1776			−1.19176
$730$		0			0
$731$	38.6153			1.42824
$732$		0			0
$733$		−	27.5709i		−	1.01836i	−0.860661	−	0.509178i	$-0.829950\pi$
$733$		−	27.5709i		−	1.01836i	0.860661	−	0.509178i	$-0.170050\pi$
$734$		0			0
$735$			45.1933i			1.66698i
$736$		0			0
$737$	4.46301			0.164397
$738$		0			0
$739$			1.19428i			0.0439322i	0.999759	+	0.0219661i	$0.00699259\pi$
$739$			1.19428i			0.0439322i	−0.999759	+	0.0219661i	$0.993007\pi$
$740$		0			0
$741$	−7.23070	+	14.9527i	−0.265626	+	0.549300i
$742$		0			0
$743$		−	31.6391i		−	1.16072i	−0.814358	−	0.580362i	$-0.802911\pi$
$743$		−	31.6391i		−	1.16072i	0.814358	−	0.580362i	$-0.197089\pi$
$744$		0			0
$745$	−38.7816			−1.42085
$746$		0			0
$747$		−	16.4606i		−	0.602262i
$748$		0			0
$749$		−	8.88799i		−	0.324760i
$750$		0			0
$751$	−42.2294			−1.54097			−0.770487	−	0.637456i	$-0.779987\pi$
$751$	−42.2294			−1.54097			−0.770487	+	0.637456i	$0.779987\pi$
$752$		0			0
$753$	−46.9016			−1.70919
$754$		0			0
$755$	82.6200			3.00685
$756$		0			0
$757$	−2.39360			−0.0869970			−0.0434985	−	0.999053i	$-0.513850\pi$
$757$	−2.39360			−0.0869970			−0.0434985	+	0.999053i	$0.513850\pi$
$758$		0			0
$759$			13.3744i			0.485461i
$760$		0			0
$761$			20.5604i			0.745313i	0.927969	+	0.372656i	$0.121553\pi$
$761$			20.5604i			0.745313i	−0.927969	+	0.372656i	$0.878447\pi$
$762$		0			0
$763$	−4.81560			−0.174337
$764$		0			0
$765$			64.8034i			2.34297i
$766$		0			0
$767$	−12.8555	+	26.5844i	−0.464185	+	0.959908i
$768$		0			0
$769$			48.5003i			1.74897i	0.485055	+	0.874484i	$0.338800\pi$
$769$			48.5003i			1.74897i	−0.485055	+	0.874484i	$0.661200\pi$
$770$		0			0
$771$	35.4690			1.27738
$772$		0			0
$773$			2.13384i			0.0767490i	0.999263	+	0.0383745i	$0.0122180\pi$
$773$			2.13384i			0.0767490i	−0.999263	+	0.0383745i	$0.987782\pi$
$774$		0			0
$775$		−	93.8256i		−	3.37031i
$776$		0			0
$777$	12.1637			0.436370
$778$		0			0
$779$	−12.3764			−0.443430
$780$		0			0
$781$	−3.58065			−0.128126
$782$		0			0
$783$	7.27844			0.260110
$784$		0			0
$785$		−	42.3210i		−	1.51050i
$786$		0			0
$787$		−	34.3609i		−	1.22483i	−0.790535	−	0.612416i	$-0.790198\pi$
$787$		−	34.3609i		−	1.22483i	0.790535	−	0.612416i	$-0.209802\pi$
$788$		0			0
$789$	49.7379			1.77072
$790$		0			0
$791$			11.9385i			0.424486i
$792$		0			0
$793$	−17.8973	−	8.65464i	−0.635551	−	0.307335i
$794$		0			0
$795$		−	10.2488i		−	0.363486i
$796$		0			0
$797$	−16.4715			−0.583451			−0.291725	−	0.956502i	$-0.594229\pi$
$797$	−16.4715			−0.583451			−0.291725	+	0.956502i	$0.594229\pi$
$798$		0			0
$799$		−	46.3659i		−	1.64031i
$800$		0			0
$801$			59.0193i			2.08534i
$802$		0			0
$803$	−9.71714			−0.342910
$804$		0			0
$805$	30.7076			1.08230
$806$		0			0
$807$	−65.0100			−2.28846
$808$		0			0
$809$	12.7978			0.449947			0.224973	−	0.974365i	$-0.427770\pi$
$809$	12.7978			0.449947			0.224973	+	0.974365i	$0.427770\pi$
$810$		0			0
$811$			52.5928i			1.84678i	0.383861	+	0.923391i	$0.374594\pi$
$811$			52.5928i			1.84678i	−0.383861	+	0.923391i	$0.625406\pi$
$812$		0			0
$813$		−	21.2565i		−	0.745499i
$814$		0			0
$815$	−31.5879			−1.10648
$816$		0			0
$817$		−	13.7848i		−	0.482269i
$818$		0			0
$819$	7.83743	−	16.2074i	0.273862	−	0.566331i
$820$		0			0
$821$			44.5792i			1.55583i	0.628372	+	0.777913i	$0.283721\pi$
$821$			44.5792i			1.55583i	−0.628372	+	0.777913i	$0.716279\pi$
$822$		0			0
$823$	−26.3864			−0.919771			−0.459886	−	0.887978i	$-0.652110\pi$
$823$	−26.3864			−0.919771			−0.459886	+	0.887978i	$0.652110\pi$
$824$		0			0
$825$			24.0014i			0.835621i
$826$		0			0
$827$			34.9688i			1.21598i	0.793943	+	0.607992i	$0.208025\pi$
$827$			34.9688i			1.21598i	−0.793943	+	0.607992i	$0.791975\pi$
$828$		0			0
$829$	46.4300			1.61258			0.806291	−	0.591519i	$-0.201472\pi$
$829$	46.4300			1.61258			0.806291	+	0.591519i	$0.201472\pi$
$830$		0			0
$831$	68.1997			2.36582
$832$		0			0
$833$	24.2428			0.839964
$834$		0			0
$835$	85.7213			2.96651
$836$		0			0
$837$			7.50939i			0.259563i
$838$		0			0
$839$		−	45.4274i		−	1.56833i	−0.620554	−	0.784164i	$-0.713092\pi$
$839$		−	45.4274i		−	1.56833i	0.620554	−	0.784164i	$-0.286908\pi$
$840$		0			0
$841$	61.5110			2.12107
$842$		0			0
$843$		−	72.9102i		−	2.51116i
$844$		0			0
$845$	−38.8811	+	30.8010i	−1.33755	+	1.05959i
$846$		0			0
$847$		−	1.51091i		−	0.0519156i
$848$		0			0
$849$	4.83647			0.165987
$850$		0			0
$851$			17.0780i			0.585427i
$852$		0			0
$853$		−	31.7143i		−	1.08588i	−0.839773	−	0.542938i	$-0.817312\pi$
$853$		−	31.7143i		−	1.08588i	0.839773	−	0.542938i	$-0.182688\pi$
$854$		0			0
$855$	23.1334			0.791144
$856$		0			0
$857$	8.27331			0.282611			0.141305	−	0.989966i	$-0.454870\pi$
$857$	8.27331			0.282611			0.141305	+	0.989966i	$0.454870\pi$
$858$		0			0
$859$	27.2598			0.930091			0.465046	−	0.885287i	$-0.346038\pi$
$859$	27.2598			0.930091			0.465046	+	0.885287i	$0.346038\pi$
$860$		0			0
$861$	25.5929			0.872205
$862$		0			0
$863$			30.0929i			1.02438i	0.858873	+	0.512188i	$0.171165\pi$
$863$			30.0929i			1.02438i	−0.858873	+	0.512188i	$0.828835\pi$
$864$		0			0
$865$		−	27.4073i		−	0.931877i
$866$		0			0
$867$	23.6339			0.802648
$868$		0			0
$869$			4.25695i			0.144407i
$870$		0			0
$871$	7.00537	−	14.4867i	0.237368	−	0.490863i
$872$		0			0
$873$			13.8281i			0.468011i
$874$		0			0
$875$	26.2818			0.888488
$876$		0			0
$877$		−	26.4876i		−	0.894424i	−0.894428	−	0.447212i	$-0.852417\pi$
$877$		−	26.4876i		−	0.894424i	0.894428	−	0.447212i	$-0.147583\pi$
$878$		0			0
$879$			75.1562i			2.53496i
$880$		0			0
$881$	17.7155			0.596852			0.298426	−	0.954433i	$-0.403538\pi$
$881$	17.7155			0.596852			0.298426	+	0.954433i	$0.403538\pi$
$882$		0			0
$883$	−0.585366			−0.0196991			−0.00984956	−	0.999951i	$-0.503135\pi$
$883$	−0.585366			−0.0196991			−0.00984956	+	0.999951i	$0.503135\pi$
$884$		0			0
$885$	78.4658			2.63760
$886$		0			0
$887$	−38.2865			−1.28553			−0.642767	−	0.766062i	$-0.722214\pi$
$887$	−38.2865			−1.28553			−0.642767	+	0.766062i	$0.722214\pi$
$888$		0			0
$889$		−	19.6174i		−	0.657947i
$890$		0			0
$891$			7.99310i			0.267779i
$892$		0			0
$893$	−16.5516			−0.553878
$894$		0			0
$895$			45.4492i			1.51920i
$896$		0			0
$897$	43.4127	+	20.9932i	1.44951	+	0.700942i
$898$		0			0
$899$			93.3830i			3.11450i
$900$		0			0
$901$	−5.49769			−0.183155
$902$		0			0
$903$			28.5054i			0.948600i
$904$		0			0
$905$			34.4338i			1.14462i
$906$		0			0
$907$	9.98115			0.331419			0.165709	−	0.986175i	$-0.447009\pi$
$907$	9.98115			0.331419			0.165709	+	0.986175i	$0.447009\pi$
$908$		0			0
$909$	−19.8281			−0.657658
$910$		0			0
$911$	−10.1696			−0.336933			−0.168467	−	0.985707i	$-0.553882\pi$
$911$	−10.1696			−0.336933			−0.168467	+	0.985707i	$0.553882\pi$
$912$		0			0
$913$	−4.98099			−0.164847
$914$		0			0
$915$			52.8251i			1.74634i
$916$		0			0
$917$		−	15.7323i		−	0.519527i
$918$		0			0
$919$	1.37989			0.0455182			0.0227591	−	0.999741i	$-0.492755\pi$
$919$	1.37989			0.0455182			0.0227591	+	0.999741i	$0.492755\pi$
$920$		0			0
$921$			1.88029i			0.0619576i
$922$		0			0
$923$	−5.62037	+	11.6226i	−0.184997	+	0.382563i
$924$		0			0
$925$			30.6477i			1.00769i
$926$		0			0
$927$	18.5285			0.608555
$928$		0			0
$929$			43.0767i			1.41330i	0.707562	+	0.706651i	$0.249795\pi$
$929$			43.0767i			1.41330i	−0.707562	+	0.706651i	$0.750205\pi$
$930$		0			0
$931$		−	8.65414i		−	0.283628i
$932$		0			0
$933$	−39.0328			−1.27788
$934$		0			0
$935$	19.6095			0.641301
$936$		0			0
$937$	−52.2704			−1.70760			−0.853800	−	0.520601i	$-0.825708\pi$
$937$	−52.2704			−1.70760			−0.853800	+	0.520601i	$0.825708\pi$
$938$		0			0
$939$	−81.1426			−2.64799
$940$		0			0
$941$			11.7969i			0.384569i	0.981339	+	0.192284i	$0.0615896\pi$
$941$			11.7969i			0.384569i	−0.981339	+	0.192284i	$0.938410\pi$
$942$		0			0
$943$			35.9329i			1.17013i
$944$		0			0
$945$	−4.41053			−0.143474
$946$		0			0
$947$			40.2915i			1.30930i	0.755934	+	0.654648i	$0.227183\pi$
$947$			40.2915i			1.30930i	−0.755934	+	0.654648i	$0.772817\pi$
$948$		0			0
$949$	−15.2525	+	31.5413i	−0.495118	+	1.02388i
$950$		0			0
$951$		−	79.4059i		−	2.57491i
$952$		0			0
$953$	58.5685			1.89722			0.948609	−	0.316449i	$-0.102491\pi$
$953$	58.5685			1.89722			0.948609	+	0.316449i	$0.102491\pi$
$954$		0			0
$955$		−	44.9008i		−	1.45296i
$956$		0			0
$957$		−	23.8882i		−	0.772194i
$958$		0			0
$959$	25.2297			0.814709
$960$		0			0
$961$	−65.3460			−2.10794
$962$		0			0
$963$	−19.4399			−0.626442
$964$		0			0
$965$	−70.7350			−2.27704
$966$		0			0
$967$			2.77699i			0.0893021i	0.999003	+	0.0446511i	$0.0142176\pi$
$967$			2.77699i			0.0893021i	−0.999003	+	0.0446511i	$0.985782\pi$
$968$		0			0
$969$		−	23.6745i		−	0.760535i
$970$		0			0
$971$	48.6039			1.55977			0.779886	−	0.625922i	$-0.215277\pi$
$971$	48.6039			1.55977			0.779886	+	0.625922i	$0.215277\pi$
$972$		0			0
$973$		−	20.7978i		−	0.666747i
$974$		0			0
$975$	77.9073	+	37.6738i	2.49503	+	1.20653i
$976$		0			0
$977$		−	23.7201i		−	0.758874i	−0.925218	−	0.379437i	$-0.876118\pi$
$977$		−	23.7201i		−	0.758874i	0.925218	−	0.379437i	$-0.123882\pi$
$978$		0			0
$979$	17.8593			0.570784
$980$		0			0
$981$			10.5327i			0.336285i
$982$		0			0
$983$		−	22.5123i		−	0.718031i	−0.933332	−	0.359015i	$-0.883113\pi$
$983$		−	22.5123i		−	0.718031i	0.933332	−	0.359015i	$-0.116887\pi$
$984$		0			0
$985$	−17.0190			−0.542269
$986$		0			0
$987$	34.2268			1.08945
$988$		0			0
$989$	−40.0220			−1.27263
$990$		0			0
$991$	−54.7939			−1.74058			−0.870292	−	0.492535i	$-0.836070\pi$
$991$	−54.7939			−1.74058			−0.870292	+	0.492535i	$0.836070\pi$
$992$		0			0
$993$			64.0044i			2.03112i
$994$		0			0
$995$			23.7739i			0.753683i
$996$		0			0
$997$	0.229656			0.00727328			0.00363664	−	0.999993i	$-0.498842\pi$
$997$	0.229656			0.00727328			0.00363664	+	0.999993i	$0.498842\pi$
$998$		0			0
$999$		−	2.45291i		−	0.0776067i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.f.c.441.8	yes	8
3.2	odd	2		5148.2.e.c.1585.2		8
4.3	odd	2		2288.2.j.i.1585.2		8
13.5	odd	4		7436.2.a.p.1.4		4
13.8	odd	4		7436.2.a.o.1.4		4
13.12	even	2	inner	572.2.f.c.441.7	✓	8
39.38	odd	2		5148.2.e.c.1585.7		8
52.51	odd	2		2288.2.j.i.1585.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.f.c.441.7	✓	8	13.12	even	2	inner
572.2.f.c.441.8	yes	8	1.1	even	1	trivial
2288.2.j.i.1585.1		8	52.51	odd	2
2288.2.j.i.1585.2		8	4.3	odd	2
5148.2.e.c.1585.2		8	3.2	odd	2
5148.2.e.c.1585.7		8	39.38	odd	2
7436.2.a.o.1.4		4	13.8	odd	4
7436.2.a.p.1.4		4	13.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.51091 q^{3} +3.81560i q^{5} +1.51091i q^{7} +3.30469 q^{9} +O(q^{10})\)	\(q+2.51091 q^{3} +3.81560i q^{5} +1.51091i q^{7} +3.30469 q^{9} -1.00000i q^{11} +(-3.24595 - 1.56965i) q^{13} +9.58065i q^{15} +5.13930 q^{17} -1.83462i q^{19} +3.79377i q^{21} -5.32652 q^{23} -9.55882 q^{25} +0.765046 q^{27} +9.51373 q^{29} +9.81560i q^{31} -2.51091i q^{33} -5.76505 q^{35} -3.20623i q^{37} +(-8.15030 - 3.94126i) q^{39} -6.74603i q^{41} +7.51373 q^{43} +12.6094i q^{45} -9.02183i q^{47} +4.71714 q^{49} +12.9044 q^{51} -1.06973 q^{53} +3.81560 q^{55} -4.60656i q^{57} -8.19003i q^{59} +5.51373 q^{61} +4.99310i q^{63} +(5.98917 - 12.3853i) q^{65} +4.46301i q^{67} -13.3744 q^{69} -3.58065i q^{71} -9.71714i q^{73} -24.0014 q^{75} +1.51091 q^{77} -4.25695 q^{79} -7.99310 q^{81} -4.98099i q^{83} +19.6095i q^{85} +23.8882 q^{87} +17.8593i q^{89} +(2.37161 - 4.90435i) q^{91} +24.6461i q^{93} +7.00017 q^{95} +4.18440i q^{97} -3.30469i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 18 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 + 18 * q^9	\( 8 q - 2 q^{3} + 18 q^{9} - 8 q^{13} + 16 q^{17} + 10 q^{23} - 52 q^{25} - 32 q^{27} - 4 q^{29} - 8 q^{35} + 16 q^{39} - 20 q^{43} + 2 q^{49} + 40 q^{51} + 38 q^{53} - 36 q^{61} + 36 q^{65} - 52 q^{69} + 10 q^{75} - 10 q^{77} + 40 q^{79} + 32 q^{81} + 56 q^{87} + 22 q^{91} - 12 q^{95}+O(q^{100}) \) 8 * q - 2 * q^3 + 18 * q^9 - 8 * q^13 + 16 * q^17 + 10 * q^23 - 52 * q^25 - 32 * q^27 - 4 * q^29 - 8 * q^35 + 16 * q^39 - 20 * q^43 + 2 * q^49 + 40 * q^51 + 38 * q^53 - 36 * q^61 + 36 * q^65 - 52 * q^69 + 10 * q^75 - 10 * q^77 + 40 * q^79 + 32 * q^81 + 56 * q^87 + 22 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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