LMFDB - Embedded newform 4368.2.h.p.337.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.5
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.p.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.63090i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.63090i q^{5} -1.00000i q^{7} +1.00000 q^{9} -3.70928i q^{11} +(2.17009 + 2.87936i) q^{13} +2.63090i q^{15} +3.41855 q^{17} +1.26180i q^{19} -1.00000i q^{21} -3.07838 q^{23} -1.92162 q^{25} +1.00000 q^{27} +4.15676 q^{29} -4.68035i q^{31} -3.70928i q^{33} +2.63090 q^{35} +1.26180i q^{37} +(2.17009 + 2.87936i) q^{39} +1.36910i q^{41} +6.83710 q^{43} +2.63090i q^{45} +5.70928i q^{47} -1.00000 q^{49} +3.41855 q^{51} +6.68035 q^{53} +9.75872 q^{55} +1.26180i q^{57} -7.86603i q^{59} -3.65983 q^{61} -1.00000i q^{63} +(-7.57531 + 5.70928i) q^{65} +2.73820i q^{67} -3.07838 q^{69} +11.7093i q^{71} +0.496928i q^{73} -1.92162 q^{75} -3.70928 q^{77} +7.02052 q^{79} +1.00000 q^{81} -12.5464i q^{83} +8.99386i q^{85} +4.15676 q^{87} -6.94441i q^{89} +(2.87936 - 2.17009i) q^{91} -4.68035i q^{93} -3.31965 q^{95} +7.91548i q^{97} -3.70928i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 6 * q^9	$ 6 q + 6 q^{3} + 6 q^{9} + 2 q^{13} - 8 q^{17} - 12 q^{23} - 18 q^{25} + 6 q^{27} + 12 q^{29} + 8 q^{35} + 2 q^{39} - 16 q^{43} - 6 q^{49} - 8 q^{51} - 4 q^{53} + 8 q^{55} - 44 q^{61} - 4 q^{65} - 12 q^{69} - 18 q^{75} - 8 q^{77} - 24 q^{79} + 6 q^{81} + 12 q^{87} - 8 q^{91} - 64 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 + 6 * q^9 + 2 * q^13 - 8 * q^17 - 12 * q^23 - 18 * q^25 + 6 * q^27 + 12 * q^29 + 8 * q^35 + 2 * q^39 - 16 * q^43 - 6 * q^49 - 8 * q^51 - 4 * q^53 + 8 * q^55 - 44 * q^61 - 4 * q^65 - 12 * q^69 - 18 * q^75 - 8 * q^77 - 24 * q^79 + 6 * q^81 + 12 * q^87 - 8 * q^91 - 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			2.63090i			1.17657i	0.808653	+	0.588287i	$0.200197\pi$
$5$			2.63090i			1.17657i	−0.808653	+	0.588287i	$0.799803\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.70928i		−	1.11839i	−0.829037	−	0.559194i	$-0.811111\pi$
$11$		−	3.70928i		−	1.11839i	0.829037	−	0.559194i	$-0.188889\pi$
$12$		0			0
$13$	2.17009	+	2.87936i	0.601874	+	0.798591i
$13$	2.17009	+	2.87936i	0.601874	+	0.798591i
$14$		0			0
$15$			2.63090i			0.679295i
$16$		0			0
$17$	3.41855			0.829120			0.414560	−	0.910022i	$-0.363935\pi$
$17$	3.41855			0.829120			0.414560	+	0.910022i	$0.363935\pi$
$18$		0			0
$19$			1.26180i			0.289476i	0.989470	+	0.144738i	$0.0462339\pi$
$19$			1.26180i			0.289476i	−0.989470	+	0.144738i	$0.953766\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$	−3.07838			−0.641886			−0.320943	−	0.947098i	$-0.604000\pi$
$23$	−3.07838			−0.641886			−0.320943	+	0.947098i	$0.604000\pi$
$24$		0			0
$25$	−1.92162			−0.384324
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	4.15676			0.771890			0.385945	−	0.922522i	$-0.373875\pi$
$29$	4.15676			0.771890			0.385945	+	0.922522i	$0.373875\pi$
$30$		0			0
$31$		−	4.68035i		−	0.840615i	−0.907382	−	0.420307i	$-0.861922\pi$
$31$		−	4.68035i		−	0.840615i	0.907382	−	0.420307i	$-0.138078\pi$
$32$		0			0
$33$		−	3.70928i		−	0.645702i
$34$		0			0
$35$	2.63090			0.444703
$36$		0			0
$37$			1.26180i			0.207438i	0.994607	+	0.103719i	$0.0330742\pi$
$37$			1.26180i			0.207438i	−0.994607	+	0.103719i	$0.966926\pi$
$38$		0			0
$39$	2.17009	+	2.87936i	0.347492	+	0.461067i
$40$		0			0
$41$			1.36910i			0.213818i	0.994269	+	0.106909i	$0.0340953\pi$
$41$			1.36910i			0.213818i	−0.994269	+	0.106909i	$0.965905\pi$
$42$		0			0
$43$	6.83710			1.04265			0.521324	−	0.853359i	$-0.325438\pi$
$43$	6.83710			1.04265			0.521324	+	0.853359i	$0.325438\pi$
$44$		0			0
$45$			2.63090i			0.392191i
$46$		0			0
$47$			5.70928i			0.832783i	0.909185	+	0.416392i	$0.136705\pi$
$47$			5.70928i			0.832783i	−0.909185	+	0.416392i	$0.863295\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	3.41855			0.478693
$52$		0			0
$53$	6.68035			0.917616			0.458808	−	0.888535i	$-0.348277\pi$
$53$	6.68035			0.917616			0.458808	+	0.888535i	$0.348277\pi$
$54$		0			0
$55$	9.75872			1.31587
$56$		0			0
$57$			1.26180i			0.167129i
$58$		0			0
$59$		−	7.86603i		−	1.02407i	−0.858965	−	0.512035i	$-0.828892\pi$
$59$		−	7.86603i		−	1.02407i	0.858965	−	0.512035i	$-0.171108\pi$
$60$		0			0
$61$	−3.65983			−0.468593			−0.234296	−	0.972165i	$-0.575279\pi$
$61$	−3.65983			−0.468593			−0.234296	+	0.972165i	$0.575279\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	−7.57531	+	5.70928i	−0.939601	+	0.708148i
$66$		0			0
$67$			2.73820i			0.334525i	0.985912	+	0.167262i	$0.0534927\pi$
$67$			2.73820i			0.334525i	−0.985912	+	0.167262i	$0.946507\pi$
$68$		0			0
$69$	−3.07838			−0.370593
$70$		0			0
$71$			11.7093i			1.38964i	0.719186	+	0.694818i	$0.244515\pi$
$71$			11.7093i			1.38964i	−0.719186	+	0.694818i	$0.755485\pi$
$72$		0			0
$73$			0.496928i			0.0581611i	0.999577	+	0.0290805i	$0.00925793\pi$
$73$			0.496928i			0.0581611i	−0.999577	+	0.0290805i	$0.990742\pi$
$74$		0			0
$75$	−1.92162			−0.221890
$76$		0			0
$77$	−3.70928			−0.422711
$78$		0			0
$79$	7.02052			0.789870			0.394935	−	0.918709i	$-0.370767\pi$
$79$	7.02052			0.789870			0.394935	+	0.918709i	$0.370767\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	12.5464i		−	1.37714i	−0.725168	−	0.688572i	$-0.758238\pi$
$83$		−	12.5464i		−	1.37714i	0.725168	−	0.688572i	$-0.241762\pi$
$84$		0			0
$85$			8.99386i			0.975521i
$86$		0			0
$87$	4.15676			0.445651
$88$		0			0
$89$		−	6.94441i		−	0.736106i	−0.929805	−	0.368053i	$-0.880025\pi$
$89$		−	6.94441i		−	0.736106i	0.929805	−	0.368053i	$-0.119975\pi$
$90$		0			0
$91$	2.87936	−	2.17009i	0.301839	−	0.227487i
$92$		0			0
$93$		−	4.68035i		−	0.485329i
$94$		0			0
$95$	−3.31965			−0.340589
$96$		0			0
$97$			7.91548i			0.803695i	0.915707	+	0.401848i	$0.131632\pi$
$97$			7.91548i			0.803695i	−0.915707	+	0.401848i	$0.868368\pi$
$98$		0			0
$99$		−	3.70928i		−	0.372796i
$100$		0			0
$101$	−5.26180			−0.523568			−0.261784	−	0.965126i	$-0.584311\pi$
$101$	−5.26180			−0.523568			−0.261784	+	0.965126i	$0.584311\pi$
$102$		0			0
$103$	−5.07838			−0.500387			−0.250194	−	0.968196i	$-0.580494\pi$
$103$	−5.07838			−0.500387			−0.250194	+	0.968196i	$0.580494\pi$
$104$		0			0
$105$	2.63090			0.256749
$106$		0			0
$107$	11.0784			1.07099			0.535494	−	0.844539i	$-0.320126\pi$
$107$	11.0784			1.07099			0.535494	+	0.844539i	$0.320126\pi$
$108$		0			0
$109$			11.7854i			1.12884i	0.825489	+	0.564418i	$0.190899\pi$
$109$			11.7854i			1.12884i	−0.825489	+	0.564418i	$0.809101\pi$
$110$		0			0
$111$			1.26180i			0.119764i
$112$		0			0
$113$	8.15676			0.767323			0.383662	−	0.923474i	$-0.374663\pi$
$113$	8.15676			0.767323			0.383662	+	0.923474i	$0.374663\pi$
$114$		0			0
$115$		−	8.09890i		−	0.755226i
$116$		0			0
$117$	2.17009	+	2.87936i	0.200625	+	0.266197i
$118$		0			0
$119$		−	3.41855i		−	0.313378i
$120$		0			0
$121$	−2.75872			−0.250793
$122$		0			0
$123$			1.36910i			0.123448i
$124$		0			0
$125$			8.09890i			0.724387i
$126$		0			0
$127$	19.3340			1.71562			0.857809	−	0.513969i	$-0.171825\pi$
$127$	19.3340			1.71562			0.857809	+	0.513969i	$0.171825\pi$
$128$		0			0
$129$	6.83710			0.601973
$130$		0			0
$131$	−6.15676			−0.537918			−0.268959	−	0.963152i	$-0.586680\pi$
$131$	−6.15676			−0.537918			−0.268959	+	0.963152i	$0.586680\pi$
$132$		0			0
$133$	1.26180			0.109412
$134$		0			0
$135$			2.63090i			0.226432i
$136$		0			0
$137$		−	8.44748i		−	0.721717i	−0.932621	−	0.360858i	$-0.882484\pi$
$137$		−	8.44748i		−	0.721717i	0.932621	−	0.360858i	$-0.117516\pi$
$138$		0			0
$139$	4.76487			0.404151			0.202075	−	0.979370i	$-0.435231\pi$
$139$	4.76487			0.404151			0.202075	+	0.979370i	$0.435231\pi$
$140$		0			0
$141$			5.70928i			0.480808i
$142$		0			0
$143$	10.6803	−	8.04945i	0.893135	−	0.673129i
$144$		0			0
$145$			10.9360i			0.908185i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$		−	3.55252i		−	0.291034i	−0.989356	−	0.145517i	$-0.953515\pi$
$149$		−	3.55252i		−	0.291034i	0.989356	−	0.145517i	$-0.0464845\pi$
$150$		0			0
$151$			16.9939i			1.38294i	0.722405	+	0.691470i	$0.243037\pi$
$151$			16.9939i			1.38294i	−0.722405	+	0.691470i	$0.756963\pi$
$152$		0			0
$153$	3.41855			0.276373
$154$		0			0
$155$	12.3135			0.989045
$156$		0			0
$157$	−3.84324			−0.306724			−0.153362	−	0.988170i	$-0.549010\pi$
$157$	−3.84324			−0.306724			−0.153362	+	0.988170i	$0.549010\pi$
$158$		0			0
$159$	6.68035			0.529786
$160$		0			0
$161$			3.07838i			0.242610i
$162$		0			0
$163$			8.58145i			0.672151i	0.941835	+	0.336075i	$0.109100\pi$
$163$			8.58145i			0.672151i	−0.941835	+	0.336075i	$0.890900\pi$
$164$		0			0
$165$	9.75872			0.759716
$166$		0			0
$167$		−	4.13397i		−	0.319896i	−0.987125	−	0.159948i	$-0.948867\pi$
$167$		−	4.13397i		−	0.319896i	0.987125	−	0.159948i	$-0.0511327\pi$
$168$		0			0
$169$	−3.58145	+	12.4969i	−0.275496	+	0.961302i
$170$		0			0
$171$			1.26180i			0.0964919i
$172$		0			0
$173$	2.73820			0.208182			0.104091	−	0.994568i	$-0.466807\pi$
$173$	2.73820			0.208182			0.104091	+	0.994568i	$0.466807\pi$
$174$		0			0
$175$			1.92162i			0.145261i
$176$		0			0
$177$		−	7.86603i		−	0.591247i
$178$		0			0
$179$	0.241276			0.0180338			0.00901692	−	0.999959i	$-0.497130\pi$
$179$	0.241276			0.0180338			0.00901692	+	0.999959i	$0.497130\pi$
$180$		0			0
$181$	−5.31965			−0.395407			−0.197703	−	0.980262i	$-0.563348\pi$
$181$	−5.31965			−0.395407			−0.197703	+	0.980262i	$0.563348\pi$
$182$		0			0
$183$	−3.65983			−0.270542
$184$		0			0
$185$	−3.31965			−0.244066
$186$		0			0
$187$		−	12.6803i		−	0.927279i
$188$		0			0
$189$		−	1.00000i		−	0.0727393i
$190$		0			0
$191$	4.43907			0.321200			0.160600	−	0.987020i	$-0.448657\pi$
$191$	4.43907			0.321200			0.160600	+	0.987020i	$0.448657\pi$
$192$		0			0
$193$			20.7792i			1.49572i	0.663855	+	0.747861i	$0.268919\pi$
$193$			20.7792i			1.49572i	−0.663855	+	0.747861i	$0.731081\pi$
$194$		0			0
$195$	−7.57531	+	5.70928i	−0.542479	+	0.408850i
$196$		0			0
$197$		−	0.814315i		−	0.0580175i	−0.999579	−	0.0290088i	$-0.990765\pi$
$197$		−	0.814315i		−	0.0580175i	0.999579	−	0.0290088i	$-0.00923508\pi$
$198$		0			0
$199$	−13.8432			−0.981322			−0.490661	−	0.871351i	$-0.663245\pi$
$199$	−13.8432			−0.981322			−0.490661	+	0.871351i	$0.663245\pi$
$200$		0			0
$201$			2.73820i			0.193138i
$202$		0			0
$203$		−	4.15676i		−	0.291747i
$204$		0			0
$205$	−3.60197			−0.251572
$206$		0			0
$207$	−3.07838			−0.213962
$208$		0			0
$209$	4.68035			0.323746
$210$		0			0
$211$	19.7009			1.35626			0.678132	−	0.734940i	$-0.262790\pi$
$211$	19.7009			1.35626			0.678132	+	0.734940i	$0.262790\pi$
$212$		0			0
$213$			11.7093i			0.802306i
$214$		0			0
$215$			17.9877i			1.22675i
$216$		0			0
$217$	−4.68035			−0.317723
$218$		0			0
$219$			0.496928i			0.0335793i
$220$		0			0
$221$	7.41855	+	9.84324i	0.499026	+	0.662128i
$222$		0			0
$223$			4.58145i			0.306797i	0.988164	+	0.153398i	$0.0490217\pi$
$223$			4.58145i			0.306797i	−0.988164	+	0.153398i	$0.950978\pi$
$224$		0			0
$225$	−1.92162			−0.128108
$226$		0			0
$227$		−	2.60424i		−	0.172849i	−0.996258	−	0.0864246i	$-0.972456\pi$
$227$		−	2.60424i		−	0.172849i	0.996258	−	0.0864246i	$-0.0275442\pi$
$228$		0			0
$229$			7.33403i			0.484646i	0.970196	+	0.242323i	$0.0779094\pi$
$229$			7.33403i			0.484646i	−0.970196	+	0.242323i	$0.922091\pi$
$230$		0			0
$231$	−3.70928			−0.244052
$232$		0			0
$233$	−24.0410			−1.57498			−0.787490	−	0.616327i	$-0.788620\pi$
$233$	−24.0410			−1.57498			−0.787490	+	0.616327i	$0.788620\pi$
$234$		0			0
$235$	−15.0205			−0.979831
$236$		0			0
$237$	7.02052			0.456032
$238$		0			0
$239$			17.0700i			1.10416i	0.833790	+	0.552082i	$0.186167\pi$
$239$			17.0700i			1.10416i	−0.833790	+	0.552082i	$0.813833\pi$
$240$		0			0
$241$			15.3340i			0.987752i	0.869532	+	0.493876i	$0.164420\pi$
$241$			15.3340i			0.987752i	−0.869532	+	0.493876i	$0.835580\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	2.63090i		−	0.168082i
$246$		0			0
$247$	−3.63317	+	2.73820i	−0.231173	+	0.174228i
$248$		0			0
$249$		−	12.5464i		−	0.795094i
$250$		0			0
$251$	−27.5174			−1.73689			−0.868443	−	0.495789i	$-0.834879\pi$
$251$	−27.5174			−1.73689			−0.868443	+	0.495789i	$0.834879\pi$
$252$		0			0
$253$			11.4186i			0.717878i
$254$		0			0
$255$			8.99386i			0.563217i
$256$		0			0
$257$	−1.57531			−0.0982649			−0.0491325	−	0.998792i	$-0.515646\pi$
$257$	−1.57531			−0.0982649			−0.0491325	+	0.998792i	$0.515646\pi$
$258$		0			0
$259$	1.26180			0.0784041
$260$		0			0
$261$	4.15676			0.257297
$262$		0			0
$263$	−6.28231			−0.387384			−0.193692	−	0.981062i	$-0.562046\pi$
$263$	−6.28231			−0.387384			−0.193692	+	0.981062i	$0.562046\pi$
$264$		0			0
$265$			17.5753i			1.07964i
$266$		0			0
$267$		−	6.94441i		−	0.424991i
$268$		0			0
$269$	29.1461			1.77707			0.888534	−	0.458811i	$-0.151725\pi$
$269$	29.1461			1.77707			0.888534	+	0.458811i	$0.151725\pi$
$270$		0			0
$271$		−	6.73820i		−	0.409317i	−0.978833	−	0.204658i	$-0.934392\pi$
$271$		−	6.73820i		−	0.409317i	0.978833	−	0.204658i	$-0.0656083\pi$
$272$		0			0
$273$	2.87936	−	2.17009i	0.174267	−	0.131340i
$274$		0			0
$275$			7.12783i			0.429824i
$276$		0			0
$277$	27.9877			1.68162			0.840809	−	0.541331i	$-0.182080\pi$
$277$	27.9877			1.68162			0.840809	+	0.541331i	$0.182080\pi$
$278$		0			0
$279$		−	4.68035i		−	0.280205i
$280$		0			0
$281$		−	14.1217i		−	0.842429i	−0.906961	−	0.421214i	$-0.861604\pi$
$281$		−	14.1217i		−	0.842429i	0.906961	−	0.421214i	$-0.138396\pi$
$282$		0			0
$283$	−10.4703			−0.622393			−0.311196	−	0.950346i	$-0.600730\pi$
$283$	−10.4703			−0.622393			−0.311196	+	0.950346i	$0.600730\pi$
$284$		0			0
$285$	−3.31965			−0.196639
$286$		0			0
$287$	1.36910			0.0808156
$288$		0			0
$289$	−5.31351			−0.312559
$290$		0			0
$291$			7.91548i			0.464014i
$292$		0			0
$293$		−	11.5259i		−	0.673348i	−0.941621	−	0.336674i	$-0.890698\pi$
$293$		−	11.5259i		−	0.673348i	0.941621	−	0.336674i	$-0.109302\pi$
$294$		0			0
$295$	20.6947			1.20489
$296$		0			0
$297$		−	3.70928i		−	0.215234i
$298$		0			0
$299$	−6.68035	−	8.86376i	−0.386334	−	0.512605i
$300$		0			0
$301$		−	6.83710i		−	0.394084i
$302$		0			0
$303$	−5.26180			−0.302282
$304$		0			0
$305$		−	9.62863i		−	0.551334i
$306$		0			0
$307$		−	16.9939i		−	0.969891i	−0.874544	−	0.484945i	$-0.838839\pi$
$307$		−	16.9939i		−	0.969891i	0.874544	−	0.484945i	$-0.161161\pi$
$308$		0			0
$309$	−5.07838			−0.288899
$310$		0			0
$311$	−9.04718			−0.513019			−0.256509	−	0.966542i	$-0.582572\pi$
$311$	−9.04718			−0.513019			−0.256509	+	0.966542i	$0.582572\pi$
$312$		0			0
$313$	−22.3812			−1.26506			−0.632530	−	0.774536i	$-0.717984\pi$
$313$	−22.3812			−1.26506			−0.632530	+	0.774536i	$0.717984\pi$
$314$		0			0
$315$	2.63090			0.148234
$316$		0			0
$317$		−	34.1750i		−	1.91946i	−0.280925	−	0.959730i	$-0.590641\pi$
$317$		−	34.1750i		−	1.91946i	0.280925	−	0.959730i	$-0.409359\pi$
$318$		0			0
$319$		−	15.4186i		−	0.863273i
$320$		0			0
$321$	11.0784			0.618335
$322$		0			0
$323$			4.31351i			0.240010i
$324$		0			0
$325$	−4.17009	−	5.53305i	−0.231315	−	0.306918i
$326$		0			0
$327$			11.7854i			0.651733i
$328$		0			0
$329$	5.70928			0.314763
$330$		0			0
$331$			19.8843i			1.09294i	0.837479	+	0.546470i	$0.184029\pi$
$331$			19.8843i			1.09294i	−0.837479	+	0.546470i	$0.815971\pi$
$332$		0			0
$333$			1.26180i			0.0691460i
$334$		0			0
$335$	−7.20394			−0.393593
$336$		0			0
$337$	−5.28846			−0.288081			−0.144040	−	0.989572i	$-0.546009\pi$
$337$	−5.28846			−0.288081			−0.144040	+	0.989572i	$0.546009\pi$
$338$		0			0
$339$	8.15676			0.443014
$340$		0			0
$341$	−17.3607			−0.940134
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	8.09890i		−	0.436030i
$346$		0			0
$347$	7.75872			0.416510			0.208255	−	0.978075i	$-0.433222\pi$
$347$	7.75872			0.416510			0.208255	+	0.978075i	$0.433222\pi$
$348$		0			0
$349$		−	0.130094i		−	0.00696375i	−0.999994	−	0.00348187i	$-0.998892\pi$
$349$		−	0.130094i		−	0.00696375i	0.999994	−	0.00348187i	$-0.00110832\pi$
$350$		0			0
$351$	2.17009	+	2.87936i	0.115831	+	0.153689i
$352$		0			0
$353$			3.21235i			0.170976i	0.996339	+	0.0854880i	$0.0272449\pi$
$353$			3.21235i			0.170976i	−0.996339	+	0.0854880i	$0.972755\pi$
$354$		0			0
$355$	−30.8059			−1.63501
$356$		0			0
$357$		−	3.41855i		−	0.180929i
$358$		0			0
$359$		−	5.18568i		−	0.273690i	−0.990592	−	0.136845i	$-0.956304\pi$
$359$		−	5.18568i		−	0.273690i	0.990592	−	0.136845i	$-0.0436962\pi$
$360$		0			0
$361$	17.4079			0.916204
$362$		0			0
$363$	−2.75872			−0.144795
$364$		0			0
$365$	−1.30737			−0.0684308
$366$		0			0
$367$	31.6020			1.64961			0.824805	−	0.565418i	$-0.191285\pi$
$367$	31.6020			1.64961			0.824805	+	0.565418i	$0.191285\pi$
$368$		0			0
$369$			1.36910i			0.0712726i
$370$		0			0
$371$		−	6.68035i		−	0.346826i
$372$		0			0
$373$	26.9627			1.39607			0.698037	−	0.716062i	$-0.254057\pi$
$373$	26.9627			1.39607			0.698037	+	0.716062i	$0.254057\pi$
$374$		0			0
$375$			8.09890i			0.418225i
$376$		0			0
$377$	9.02052	+	11.9688i	0.464580	+	0.616425i
$378$		0			0
$379$			20.6803i			1.06228i	0.847285	+	0.531139i	$0.178236\pi$
$379$			20.6803i			1.06228i	−0.847285	+	0.531139i	$0.821764\pi$
$380$		0			0
$381$	19.3340			0.990512
$382$		0			0
$383$			34.5874i			1.76733i	0.468116	+	0.883667i	$0.344933\pi$
$383$			34.5874i			1.76733i	−0.468116	+	0.883667i	$0.655067\pi$
$384$		0			0
$385$		−	9.75872i		−	0.497351i
$386$		0			0
$387$	6.83710			0.347549
$388$		0			0
$389$	−21.2039			−1.07508			−0.537541	−	0.843238i	$-0.680647\pi$
$389$	−21.2039			−1.07508			−0.537541	+	0.843238i	$0.680647\pi$
$390$		0			0
$391$	−10.5236			−0.532201
$392$		0			0
$393$	−6.15676			−0.310567
$394$		0			0
$395$			18.4703i			0.929340i
$396$		0			0
$397$		−	23.9155i		−	1.20028i	−0.799894	−	0.600142i	$-0.795111\pi$
$397$		−	23.9155i		−	1.20028i	0.799894	−	0.600142i	$-0.204889\pi$
$398$		0			0
$399$	1.26180			0.0631688
$400$		0			0
$401$			0.930033i			0.0464436i	0.999730	+	0.0232218i	$0.00739240\pi$
$401$			0.930033i			0.0464436i	−0.999730	+	0.0232218i	$0.992608\pi$
$402$		0			0
$403$	13.4764	−	10.1568i	0.671308	−	0.505944i
$404$		0			0
$405$			2.63090i			0.130730i
$406$		0			0
$407$	4.68035			0.231996
$408$		0			0
$409$		−	18.0722i		−	0.893614i	−0.894630	−	0.446807i	$-0.852561\pi$
$409$		−	18.0722i		−	0.893614i	0.894630	−	0.446807i	$-0.147439\pi$
$410$		0			0
$411$		−	8.44748i		−	0.416683i
$412$		0			0
$413$	−7.86603			−0.387062
$414$		0			0
$415$	33.0082			1.62031
$416$		0			0
$417$	4.76487			0.233337
$418$		0			0
$419$	−25.1917			−1.23069			−0.615346	−	0.788257i	$-0.710984\pi$
$419$	−25.1917			−1.23069			−0.615346	+	0.788257i	$0.710984\pi$
$420$		0			0
$421$		−	9.78992i		−	0.477132i	−0.971126	−	0.238566i	$-0.923323\pi$
$421$		−	9.78992i		−	0.477132i	0.971126	−	0.238566i	$-0.0766773\pi$
$422$		0			0
$423$			5.70928i			0.277594i
$424$		0			0
$425$	−6.56916			−0.318651
$426$		0			0
$427$			3.65983i			0.177111i
$428$		0			0
$429$	10.6803	−	8.04945i	0.515652	−	0.388631i
$430$		0			0
$431$			1.33791i			0.0644446i	0.999481	+	0.0322223i	$0.0102585\pi$
$431$			1.33791i			0.0644446i	−0.999481	+	0.0322223i	$0.989742\pi$
$432$		0			0
$433$	3.65983			0.175880			0.0879400	−	0.996126i	$-0.471972\pi$
$433$	3.65983			0.175880			0.0879400	+	0.996126i	$0.471972\pi$
$434$		0			0
$435$			10.9360i			0.524341i
$436$		0			0
$437$		−	3.88428i		−	0.185810i
$438$		0			0
$439$	20.6491			0.985530			0.492765	−	0.870162i	$-0.335986\pi$
$439$	20.6491			0.985530			0.492765	+	0.870162i	$0.335986\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	23.2762			1.10588			0.552942	−	0.833220i	$-0.313505\pi$
$443$	23.2762			1.10588			0.552942	+	0.833220i	$0.313505\pi$
$444$		0			0
$445$	18.2700			0.866082
$446$		0			0
$447$		−	3.55252i		−	0.168028i
$448$		0			0
$449$		−	21.9071i		−	1.03386i	−0.856028	−	0.516929i	$-0.827075\pi$
$449$		−	21.9071i		−	1.03386i	0.856028	−	0.516929i	$-0.172925\pi$
$450$		0			0
$451$	5.07838			0.239131
$452$		0			0
$453$			16.9939i			0.798441i
$454$		0			0
$455$	5.70928	+	7.57531i	0.267655	+	0.355136i
$456$		0			0
$457$		−	20.7792i		−	0.972012i	−0.873955	−	0.486006i	$-0.838453\pi$
$457$		−	20.7792i		−	0.972012i	0.873955	−	0.486006i	$-0.161547\pi$
$458$		0			0
$459$	3.41855			0.159564
$460$		0			0
$461$		−	39.6079i		−	1.84473i	−0.386325	−	0.922363i	$-0.626256\pi$
$461$		−	39.6079i		−	1.84473i	0.386325	−	0.922363i	$-0.373744\pi$
$462$		0			0
$463$		−	10.7382i		−	0.499047i	−0.968369	−	0.249523i	$-0.919726\pi$
$463$		−	10.7382i		−	0.499047i	0.968369	−	0.249523i	$-0.0802739\pi$
$464$		0			0
$465$	12.3135			0.571025
$466$		0			0
$467$	−31.0349			−1.43612			−0.718062	−	0.695979i	$-0.754971\pi$
$467$	−31.0349			−1.43612			−0.718062	+	0.695979i	$0.754971\pi$
$468$		0			0
$469$	2.73820			0.126439
$470$		0			0
$471$	−3.84324			−0.177087
$472$		0			0
$473$		−	25.3607i		−	1.16609i
$474$		0			0
$475$		−	2.42469i		−	0.111253i
$476$		0			0
$477$	6.68035			0.305872
$478$		0			0
$479$		−	13.9071i		−	0.635430i	−0.948186	−	0.317715i	$-0.897084\pi$
$479$		−	13.9071i		−	0.635430i	0.948186	−	0.317715i	$-0.102916\pi$
$480$		0			0
$481$	−3.63317	+	2.73820i	−0.165658	+	0.124851i
$482$		0			0
$483$			3.07838i			0.140071i
$484$		0			0
$485$	−20.8248			−0.945606
$486$		0			0
$487$		−	18.3545i		−	0.831724i	−0.909428	−	0.415862i	$-0.863480\pi$
$487$		−	18.3545i		−	0.831724i	0.909428	−	0.415862i	$-0.136520\pi$
$488$		0			0
$489$			8.58145i			0.388067i
$490$		0			0
$491$	−42.5958			−1.92232			−0.961161	−	0.275987i	$-0.910995\pi$
$491$	−42.5958			−1.92232			−0.961161	+	0.275987i	$0.910995\pi$
$492$		0			0
$493$	14.2101			0.639990
$494$		0			0
$495$	9.75872			0.438622
$496$		0			0
$497$	11.7093			0.525233
$498$		0			0
$499$		−	22.8371i		−	1.02233i	−0.859483	−	0.511165i	$-0.829214\pi$
$499$		−	22.8371i		−	1.02233i	0.859483	−	0.511165i	$-0.170786\pi$
$500$		0			0
$501$		−	4.13397i		−	0.184692i
$502$		0			0
$503$	−17.9877			−0.802033			−0.401016	−	0.916071i	$-0.631343\pi$
$503$	−17.9877			−0.802033			−0.401016	+	0.916071i	$0.631343\pi$
$504$		0			0
$505$		−	13.8432i		−	0.616016i
$506$		0			0
$507$	−3.58145	+	12.4969i	−0.159058	+	0.555008i
$508$		0			0
$509$			5.15449i			0.228469i	0.993454	+	0.114234i	$0.0364415\pi$
$509$			5.15449i			0.228469i	−0.993454	+	0.114234i	$0.963559\pi$
$510$		0			0
$511$	0.496928			0.0219828
$512$		0			0
$513$			1.26180i			0.0557096i
$514$		0			0
$515$		−	13.3607i		−	0.588742i
$516$		0			0
$517$	21.1773			0.931375
$518$		0			0
$519$	2.73820			0.120194
$520$		0			0
$521$	20.0456			0.878212			0.439106	−	0.898435i	$-0.355295\pi$
$521$	20.0456			0.878212			0.439106	+	0.898435i	$0.355295\pi$
$522$		0			0
$523$	25.0784			1.09660			0.548300	−	0.836281i	$-0.315275\pi$
$523$	25.0784			1.09660			0.548300	+	0.836281i	$0.315275\pi$
$524$		0			0
$525$			1.92162i			0.0838665i
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	−13.5236			−0.587982
$530$		0			0
$531$		−	7.86603i		−	0.341357i
$532$		0			0
$533$	−3.94214	+	2.97107i	−0.170753	+	0.128691i
$534$		0			0
$535$			29.1461i			1.26009i
$536$		0			0
$537$	0.241276			0.0104118
$538$		0			0
$539$			3.70928i			0.159770i
$540$		0			0
$541$		−	8.62702i		−	0.370905i	−0.982653	−	0.185452i	$-0.940625\pi$
$541$		−	8.62702i		−	0.370905i	0.982653	−	0.185452i	$-0.0593750\pi$
$542$		0			0
$543$	−5.31965			−0.228288
$544$		0			0
$545$	−31.0061			−1.32816
$546$		0			0
$547$	−44.9315			−1.92113			−0.960565	−	0.278054i	$-0.910310\pi$
$547$	−44.9315			−1.92113			−0.960565	+	0.278054i	$0.910310\pi$
$548$		0			0
$549$	−3.65983			−0.156198
$550$		0			0
$551$			5.24497i			0.223443i
$552$		0			0
$553$		−	7.02052i		−	0.298543i
$554$		0			0
$555$	−3.31965			−0.140911
$556$		0			0
$557$			35.0166i			1.48370i	0.670564	+	0.741851i	$0.266052\pi$
$557$			35.0166i			1.48370i	−0.670564	+	0.741851i	$0.733948\pi$
$558$		0			0
$559$	14.8371	+	19.6865i	0.627543	+	0.832650i
$560$		0			0
$561$		−	12.6803i		−	0.535365i
$562$		0			0
$563$	−20.8781			−0.879909			−0.439954	−	0.898020i	$-0.645005\pi$
$563$	−20.8781			−0.879909			−0.439954	+	0.898020i	$0.645005\pi$
$564$		0			0
$565$			21.4596i			0.902812i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	44.8371			1.87967			0.939835	−	0.341630i	$-0.110979\pi$
$569$	44.8371			1.87967			0.939835	+	0.341630i	$0.110979\pi$
$570$		0			0
$571$	−18.8515			−0.788910			−0.394455	−	0.918915i	$-0.629067\pi$
$571$	−18.8515			−0.788910			−0.394455	+	0.918915i	$0.629067\pi$
$572$		0			0
$573$	4.43907			0.185445
$574$		0			0
$575$	5.91548			0.246693
$576$		0			0
$577$		−	7.91548i		−	0.329526i	−0.986333	−	0.164763i	$-0.947314\pi$
$577$		−	7.91548i		−	0.329526i	0.986333	−	0.164763i	$-0.0526859\pi$
$578$		0			0
$579$			20.7792i			0.863556i
$580$		0			0
$581$	−12.5464			−0.520511
$582$		0			0
$583$		−	24.7792i		−	1.02625i
$584$		0			0
$585$	−7.57531	+	5.70928i	−0.313200	+	0.236049i
$586$		0			0
$587$		−	38.7565i		−	1.59965i	−0.600233	−	0.799825i	$-0.704926\pi$
$587$		−	38.7565i		−	1.59965i	0.600233	−	0.799825i	$-0.295074\pi$
$588$		0			0
$589$	5.90564			0.243338
$590$		0			0
$591$		−	0.814315i		−	0.0334964i
$592$		0			0
$593$			0.886550i			0.0364062i	0.999834	+	0.0182031i	$0.00579455\pi$
$593$			0.886550i			0.0364062i	−0.999834	+	0.0182031i	$0.994205\pi$
$594$		0			0
$595$	8.99386			0.368712
$596$		0			0
$597$	−13.8432			−0.566566
$598$		0			0
$599$	−13.7998			−0.563843			−0.281921	−	0.959437i	$-0.590972\pi$
$599$	−13.7998			−0.563843			−0.281921	+	0.959437i	$0.590972\pi$
$600$		0			0
$601$	40.8515			1.66637			0.833183	−	0.552997i	$-0.186516\pi$
$601$	40.8515			1.66637			0.833183	+	0.552997i	$0.186516\pi$
$602$		0			0
$603$			2.73820i			0.111508i
$604$		0			0
$605$		−	7.25792i		−	0.295076i
$606$		0			0
$607$	−11.9155			−0.483634			−0.241817	−	0.970322i	$-0.577743\pi$
$607$	−11.9155			−0.483634			−0.241817	+	0.970322i	$0.577743\pi$
$608$		0			0
$609$		−	4.15676i		−	0.168440i
$610$		0			0
$611$	−16.4391	+	12.3896i	−0.665054	+	0.501230i
$612$		0			0
$613$			22.4703i			0.907566i	0.891112	+	0.453783i	$0.149926\pi$
$613$			22.4703i			0.907566i	−0.891112	+	0.453783i	$0.850074\pi$
$614$		0			0
$615$	−3.60197			−0.145245
$616$		0			0
$617$		−	0.859888i		−	0.0346178i	−0.999850	−	0.0173089i	$-0.994490\pi$
$617$		−	0.859888i		−	0.0346178i	0.999850	−	0.0173089i	$-0.00550987\pi$
$618$		0			0
$619$		−	38.1399i		−	1.53297i	−0.642260	−	0.766487i	$-0.722003\pi$
$619$		−	38.1399i		−	1.53297i	0.642260	−	0.766487i	$-0.277997\pi$
$620$		0			0
$621$	−3.07838			−0.123531
$622$		0			0
$623$	−6.94441			−0.278222
$624$		0			0
$625$	−30.9155			−1.23662
$626$		0			0
$627$	4.68035			0.186915
$628$		0			0
$629$			4.31351i			0.171991i
$630$		0			0
$631$			34.1399i			1.35909i	0.733634	+	0.679545i	$0.237823\pi$
$631$			34.1399i			1.35909i	−0.733634	+	0.679545i	$0.762177\pi$
$632$		0			0
$633$	19.7009			0.783039
$634$		0			0
$635$			50.8659i			2.01855i
$636$		0			0
$637$	−2.17009	−	2.87936i	−0.0859820	−	0.114084i
$638$		0			0
$639$			11.7093i			0.463212i
$640$		0			0
$641$	−1.31965			−0.0521232			−0.0260616	−	0.999660i	$-0.508297\pi$
$641$	−1.31965			−0.0521232			−0.0260616	+	0.999660i	$0.508297\pi$
$642$		0			0
$643$		−	36.2511i		−	1.42960i	−0.699327	−	0.714802i	$-0.746517\pi$
$643$		−	36.2511i		−	1.42960i	0.699327	−	0.714802i	$-0.253483\pi$
$644$		0			0
$645$			17.9877i			0.708266i
$646$		0			0
$647$	26.4079			1.03820			0.519100	−	0.854713i	$-0.326267\pi$
$647$	26.4079			1.03820			0.519100	+	0.854713i	$0.326267\pi$
$648$		0			0
$649$	−29.1773			−1.14531
$650$		0			0
$651$	−4.68035			−0.183437
$652$		0			0
$653$	−44.4079			−1.73781			−0.868907	−	0.494975i	$-0.835177\pi$
$653$	−44.4079			−1.73781			−0.868907	+	0.494975i	$0.835177\pi$
$654$		0			0
$655$		−	16.1978i		−	0.632900i
$656$		0			0
$657$			0.496928i			0.0193870i
$658$		0			0
$659$	−21.0661			−0.820618			−0.410309	−	0.911946i	$-0.634579\pi$
$659$	−21.0661			−0.820618			−0.410309	+	0.911946i	$0.634579\pi$
$660$		0			0
$661$		−	29.3751i		−	1.14256i	−0.820756	−	0.571279i	$-0.806447\pi$
$661$		−	29.3751i		−	1.14256i	0.820756	−	0.571279i	$-0.193553\pi$
$662$		0			0
$663$	7.41855	+	9.84324i	0.288113	+	0.382280i
$664$		0			0
$665$			3.31965i			0.128731i
$666$		0			0
$667$	−12.7961			−0.495466
$668$		0			0
$669$			4.58145i			0.177129i
$670$		0			0
$671$			13.5753i			0.524069i
$672$		0			0
$673$	27.3607			1.05468			0.527339	−	0.849655i	$-0.323190\pi$
$673$	27.3607			1.05468			0.527339	+	0.849655i	$0.323190\pi$
$674$		0			0
$675$	−1.92162			−0.0739633
$676$		0			0
$677$	−14.9893			−0.576086			−0.288043	−	0.957617i	$-0.593005\pi$
$677$	−14.9893			−0.576086			−0.288043	+	0.957617i	$0.593005\pi$
$678$		0			0
$679$	7.91548			0.303768
$680$		0			0
$681$		−	2.60424i		−	0.0997945i
$682$		0			0
$683$		−	46.7442i		−	1.78862i	−0.447452	−	0.894308i	$-0.647668\pi$
$683$		−	46.7442i		−	1.78862i	0.447452	−	0.894308i	$-0.352332\pi$
$684$		0			0
$685$	22.2245			0.849153
$686$		0			0
$687$			7.33403i			0.279811i
$688$		0			0
$689$	14.4969	+	19.2351i	0.552289	+	0.732800i
$690$		0			0
$691$			8.46573i			0.322052i	0.986950	+	0.161026i	$0.0514802\pi$
$691$			8.46573i			0.322052i	−0.986950	+	0.161026i	$0.948520\pi$
$692$		0			0
$693$	−3.70928			−0.140904
$694$		0			0
$695$			12.5359i			0.475513i
$696$		0			0
$697$			4.68035i			0.177281i
$698$		0			0
$699$	−24.0410			−0.909316
$700$		0			0
$701$	−31.3607			−1.18448			−0.592238	−	0.805763i	$-0.701756\pi$
$701$	−31.3607			−1.18448			−0.592238	+	0.805763i	$0.701756\pi$
$702$		0			0
$703$	−1.59213			−0.0600482
$704$		0			0
$705$	−15.0205			−0.565705
$706$		0			0
$707$			5.26180i			0.197890i
$708$		0			0
$709$		−	15.4719i		−	0.581058i	−0.956866	−	0.290529i	$-0.906169\pi$
$709$		−	15.4719i		−	0.581058i	0.956866	−	0.290529i	$-0.0938313\pi$
$710$		0			0
$711$	7.02052			0.263290
$712$		0			0
$713$			14.4079i			0.539579i
$714$		0			0
$715$	21.1773	+	28.0989i	0.791985	+	1.05084i
$716$		0			0
$717$			17.0700i			0.637490i
$718$		0			0
$719$	−37.8432			−1.41131			−0.705657	−	0.708553i	$-0.749348\pi$
$719$	−37.8432			−1.41131			−0.705657	+	0.708553i	$0.749348\pi$
$720$		0			0
$721$			5.07838i			0.189129i
$722$		0			0
$723$			15.3340i			0.570279i
$724$		0			0
$725$	−7.98771			−0.296656
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	23.3730			0.864481
$732$		0			0
$733$			33.8576i			1.25056i	0.780401	+	0.625280i	$0.215015\pi$
$733$			33.8576i			1.25056i	−0.780401	+	0.625280i	$0.784985\pi$
$734$		0			0
$735$		−	2.63090i		−	0.0970421i
$736$		0			0
$737$	10.1568			0.374129
$738$		0			0
$739$		−	10.9360i		−	0.402287i	−0.979562	−	0.201144i	$-0.935534\pi$
$739$		−	10.9360i		−	0.402287i	0.979562	−	0.201144i	$-0.0644658\pi$
$740$		0			0
$741$	−3.63317	+	2.73820i	−0.133468	+	0.100590i
$742$		0			0
$743$		−	17.8492i		−	0.654824i	−0.944882	−	0.327412i	$-0.893824\pi$
$743$		−	17.8492i		−	0.654824i	0.944882	−	0.327412i	$-0.106176\pi$
$744$		0			0
$745$	9.34632			0.342423
$746$		0			0
$747$		−	12.5464i		−	0.459048i
$748$		0			0
$749$		−	11.0784i		−	0.404795i
$750$		0			0
$751$	39.8843			1.45540			0.727699	−	0.685897i	$-0.240590\pi$
$751$	39.8843			1.45540			0.727699	+	0.685897i	$0.240590\pi$
$752$		0			0
$753$	−27.5174			−1.00279
$754$		0			0
$755$	−44.7091			−1.62713
$756$		0			0
$757$	−48.2700			−1.75440			−0.877202	−	0.480121i	$-0.840593\pi$
$757$	−48.2700			−1.75440			−0.877202	+	0.480121i	$0.840593\pi$
$758$		0			0
$759$			11.4186i			0.414467i
$760$		0			0
$761$		−	10.5320i		−	0.381785i	−0.981611	−	0.190892i	$-0.938862\pi$
$761$		−	10.5320i		−	0.381785i	0.981611	−	0.190892i	$-0.0611381\pi$
$762$		0			0
$763$	11.7854			0.426660
$764$		0			0
$765$			8.99386i			0.325174i
$766$		0			0
$767$	22.6491	−	17.0700i	0.817813	−	0.616361i
$768$		0			0
$769$			13.1773i			0.475185i	0.971365	+	0.237592i	$0.0763582\pi$
$769$			13.1773i			0.475185i	−0.971365	+	0.237592i	$0.923642\pi$
$770$		0			0
$771$	−1.57531			−0.0567333
$772$		0			0
$773$			9.20006i			0.330903i	0.986218	+	0.165452i	$0.0529082\pi$
$773$			9.20006i			0.330903i	−0.986218	+	0.165452i	$0.947092\pi$
$774$		0			0
$775$			8.99386i			0.323069i
$776$		0			0
$777$	1.26180			0.0452667
$778$		0			0
$779$	−1.72753			−0.0618951
$780$		0			0
$781$	43.4329			1.55415
$782$		0			0
$783$	4.15676			0.148550
$784$		0			0
$785$		−	10.1112i		−	0.360884i
$786$		0			0
$787$		−	24.8950i		−	0.887409i	−0.896173	−	0.443705i	$-0.853664\pi$
$787$		−	24.8950i		−	0.887409i	0.896173	−	0.443705i	$-0.146336\pi$
$788$		0			0
$789$	−6.28231			−0.223656
$790$		0			0
$791$		−	8.15676i		−	0.290021i
$792$		0			0
$793$	−7.94214	−	10.5380i	−0.282034	−	0.374214i
$794$		0			0
$795$			17.5753i			0.623332i
$796$		0			0
$797$	39.6163			1.40328			0.701641	−	0.712530i	$-0.252451\pi$
$797$	39.6163			1.40328			0.701641	+	0.712530i	$0.252451\pi$
$798$		0			0
$799$			19.5174i			0.690478i
$800$		0			0
$801$		−	6.94441i		−	0.245369i
$802$		0			0
$803$	1.84324			0.0650467
$804$		0			0
$805$	−8.09890			−0.285449
$806$		0			0
$807$	29.1461			1.02599
$808$		0			0
$809$	−27.9877			−0.983996			−0.491998	−	0.870596i	$-0.663733\pi$
$809$	−27.9877			−0.983996			−0.491998	+	0.870596i	$0.663733\pi$
$810$		0			0
$811$			38.3012i			1.34494i	0.740125	+	0.672469i	$0.234766\pi$
$811$			38.3012i			1.34494i	−0.740125	+	0.672469i	$0.765234\pi$
$812$		0			0
$813$		−	6.73820i		−	0.236319i
$814$		0			0
$815$	−22.5769			−0.790835
$816$		0			0
$817$			8.62702i			0.301821i
$818$		0			0
$819$	2.87936	−	2.17009i	0.100613	−	0.0758290i
$820$		0			0
$821$			33.4947i			1.16897i	0.811404	+	0.584486i	$0.198704\pi$
$821$			33.4947i			1.16897i	−0.811404	+	0.584486i	$0.801296\pi$
$822$		0			0
$823$	−14.2101			−0.495332			−0.247666	−	0.968845i	$-0.579664\pi$
$823$	−14.2101			−0.495332			−0.247666	+	0.968845i	$0.579664\pi$
$824$		0			0
$825$			7.12783i			0.248159i
$826$		0			0
$827$		−	8.87217i		−	0.308516i	−0.988031	−	0.154258i	$-0.950701\pi$
$827$		−	8.87217i		−	0.308516i	0.988031	−	0.154258i	$-0.0492986\pi$
$828$		0			0
$829$	−25.2807			−0.878035			−0.439018	−	0.898478i	$-0.644673\pi$
$829$	−25.2807			−0.878035			−0.439018	+	0.898478i	$0.644673\pi$
$830$		0			0
$831$	27.9877			0.970883
$832$		0			0
$833$	−3.41855			−0.118446
$834$		0			0
$835$	10.8760			0.376381
$836$		0			0
$837$		−	4.68035i		−	0.161776i
$838$		0			0
$839$		−	45.2267i		−	1.56140i	−0.624906	−	0.780700i	$-0.714863\pi$
$839$		−	45.2267i		−	1.56140i	0.624906	−	0.780700i	$-0.285137\pi$
$840$		0			0
$841$	−11.7214			−0.404186
$842$		0			0
$843$		−	14.1217i		−	0.486377i
$844$		0			0
$845$	−32.8781	−	9.42243i	−1.13104	−	0.324141i
$846$		0			0
$847$			2.75872i			0.0947909i
$848$		0			0
$849$	−10.4703			−0.359339
$850$		0			0
$851$		−	3.88428i		−	0.133151i
$852$		0			0
$853$		−	36.5958i		−	1.25302i	−0.779415	−	0.626509i	$-0.784483\pi$
$853$		−	36.5958i		−	1.25302i	0.779415	−	0.626509i	$-0.215517\pi$
$854$		0			0
$855$	−3.31965			−0.113530
$856$		0			0
$857$	25.7731			0.880392			0.440196	−	0.897902i	$-0.354909\pi$
$857$	25.7731			0.880392			0.440196	+	0.897902i	$0.354909\pi$
$858$		0			0
$859$	−19.2039			−0.655230			−0.327615	−	0.944811i	$-0.606245\pi$
$859$	−19.2039			−0.655230			−0.327615	+	0.944811i	$0.606245\pi$
$860$		0			0
$861$	1.36910			0.0466589
$862$		0			0
$863$		−	11.5936i		−	0.394649i	−0.980338	−	0.197325i	$-0.936775\pi$
$863$		−	11.5936i		−	0.394649i	0.980338	−	0.197325i	$-0.0632253\pi$
$864$		0			0
$865$			7.20394i			0.244941i
$866$		0			0
$867$	−5.31351			−0.180456
$868$		0			0
$869$		−	26.0410i		−	0.883382i
$870$		0			0
$871$	−7.88428	+	5.94214i	−0.267149	+	0.201342i
$872$		0			0
$873$			7.91548i			0.267898i
$874$		0			0
$875$	8.09890			0.273793
$876$		0			0
$877$			28.2511i			0.953972i	0.878911	+	0.476986i	$0.158271\pi$
$877$			28.2511i			0.953972i	−0.878911	+	0.476986i	$0.841729\pi$
$878$		0			0
$879$		−	11.5259i		−	0.388758i
$880$		0			0
$881$	48.2967			1.62716			0.813578	−	0.581455i	$-0.197516\pi$
$881$	48.2967			1.62716			0.813578	+	0.581455i	$0.197516\pi$
$882$		0			0
$883$	28.6947			0.965654			0.482827	−	0.875716i	$-0.339610\pi$
$883$	28.6947			0.965654			0.482827	+	0.875716i	$0.339610\pi$
$884$		0			0
$885$	20.6947			0.695645
$886$		0			0
$887$	−12.7337			−0.427555			−0.213777	−	0.976882i	$-0.568577\pi$
$887$	−12.7337			−0.427555			−0.213777	+	0.976882i	$0.568577\pi$
$888$		0			0
$889$		−	19.3340i		−	0.648443i
$890$		0			0
$891$		−	3.70928i		−	0.124265i
$892$		0			0
$893$	−7.20394			−0.241071
$894$		0			0
$895$			0.634773i			0.0212181i
$896$		0			0
$897$	−6.68035	−	8.86376i	−0.223050	−	0.295952i
$898$		0			0
$899$		−	19.4551i		−	0.648862i
$900$		0			0
$901$	22.8371			0.760814
$902$		0			0
$903$		−	6.83710i		−	0.227524i
$904$		0			0
$905$		−	13.9955i		−	0.465225i
$906$		0			0
$907$	−28.6947			−0.952793			−0.476396	−	0.879231i	$-0.658057\pi$
$907$	−28.6947			−0.952793			−0.476396	+	0.879231i	$0.658057\pi$
$908$		0			0
$909$	−5.26180			−0.174523
$910$		0			0
$911$	15.7587			0.522110			0.261055	−	0.965324i	$-0.415930\pi$
$911$	15.7587			0.522110			0.261055	+	0.965324i	$0.415930\pi$
$912$		0			0
$913$	−46.5380			−1.54018
$914$		0			0
$915$		−	9.62863i		−	0.318313i
$916$		0			0
$917$			6.15676i			0.203314i
$918$		0			0
$919$	−13.9109			−0.458880			−0.229440	−	0.973323i	$-0.573689\pi$
$919$	−13.9109			−0.458880			−0.229440	+	0.973323i	$0.573689\pi$
$920$		0			0
$921$		−	16.9939i		−	0.559967i
$922$		0			0
$923$	−33.7152	+	25.4101i	−1.10975	+	0.836385i
$924$		0			0
$925$		−	2.42469i		−	0.0797234i
$926$		0			0
$927$	−5.07838			−0.166796
$928$		0			0
$929$			27.0433i			0.887262i	0.896210	+	0.443631i	$0.146310\pi$
$929$			27.0433i			0.887262i	−0.896210	+	0.443631i	$0.853690\pi$
$930$		0			0
$931$		−	1.26180i		−	0.0413537i
$932$		0			0
$933$	−9.04718			−0.296191
$934$		0			0
$935$	33.3607			1.09101
$936$		0			0
$937$	−22.8638			−0.746927			−0.373463	−	0.927645i	$-0.621830\pi$
$937$	−22.8638			−0.746927			−0.373463	+	0.927645i	$0.621830\pi$
$938$		0			0
$939$	−22.3812			−0.730383
$940$		0			0
$941$		−	38.8287i		−	1.26578i	−0.774242	−	0.632890i	$-0.781869\pi$
$941$		−	38.8287i		−	1.26578i	0.774242	−	0.632890i	$-0.218131\pi$
$942$		0			0
$943$		−	4.21461i		−	0.137247i
$944$		0			0
$945$	2.63090			0.0855831
$946$		0			0
$947$		−	8.48852i		−	0.275840i	−0.990443	−	0.137920i	$-0.955958\pi$
$947$		−	8.48852i		−	0.275840i	0.990443	−	0.137920i	$-0.0440416\pi$
$948$		0			0
$949$	−1.43084	+	1.07838i	−0.0464469	+	0.0350056i
$950$		0			0
$951$		−	34.1750i		−	1.10820i
$952$		0			0
$953$	−23.1917			−0.751251			−0.375626	−	0.926771i	$-0.622572\pi$
$953$	−23.1917			−0.751251			−0.375626	+	0.926771i	$0.622572\pi$
$954$		0			0
$955$			11.6787i			0.377915i
$956$		0			0
$957$		−	15.4186i		−	0.498411i
$958$		0			0
$959$	−8.44748			−0.272783
$960$		0			0
$961$	9.09436			0.293367
$962$		0			0
$963$	11.0784			0.356996
$964$		0			0
$965$	−54.6681			−1.75983
$966$		0			0
$967$			11.3028i			0.363475i	0.983347	+	0.181737i	$0.0581721\pi$
$967$			11.3028i			0.363475i	−0.983347	+	0.181737i	$0.941828\pi$
$968$		0			0
$969$			4.31351i			0.138570i
$970$		0			0
$971$	−7.68649			−0.246671			−0.123336	−	0.992365i	$-0.539359\pi$
$971$	−7.68649			−0.246671			−0.123336	+	0.992365i	$0.539359\pi$
$972$		0			0
$973$		−	4.76487i		−	0.152755i
$974$		0			0
$975$	−4.17009	−	5.53305i	−0.133550	−	0.177199i
$976$		0			0
$977$			11.9194i			0.381334i	0.981655	+	0.190667i	$0.0610651\pi$
$977$			11.9194i			0.381334i	−0.981655	+	0.190667i	$0.938935\pi$
$978$		0			0
$979$	−25.7587			−0.823252
$980$		0			0
$981$			11.7854i			0.376278i
$982$		0			0
$983$		−	41.8537i		−	1.33493i	−0.744642	−	0.667464i	$-0.767380\pi$
$983$		−	41.8537i		−	1.33493i	0.744642	−	0.667464i	$-0.232620\pi$
$984$		0			0
$985$	2.14238			0.0682619
$986$		0			0
$987$	5.70928			0.181728
$988$		0			0
$989$	−21.0472			−0.669261
$990$		0			0
$991$	−2.21008			−0.0702055			−0.0351027	−	0.999384i	$-0.511176\pi$
$991$	−2.21008			−0.0702055			−0.0351027	+	0.999384i	$0.511176\pi$
$992$		0			0
$993$			19.8843i			0.631009i
$994$		0			0
$995$		−	36.4202i		−	1.15460i
$996$		0			0
$997$	−31.5052			−0.997778			−0.498889	−	0.866666i	$-0.666259\pi$
$997$	−31.5052			−0.997778			−0.498889	+	0.866666i	$0.666259\pi$
$998$		0			0
$999$			1.26180i			0.0399214i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.p.337.5		6
4.3	odd	2		2184.2.h.d.337.5	yes	6
13.12	even	2	inner	4368.2.h.p.337.2		6
52.51	odd	2		2184.2.h.d.337.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.d.337.2	✓	6	52.51	odd	2
2184.2.h.d.337.5	yes	6	4.3	odd	2
4368.2.h.p.337.2		6	13.12	even	2	inner
4368.2.h.p.337.5		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.63090i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.63090i q^{5} -1.00000i q^{7} +1.00000 q^{9} -3.70928i q^{11} +(2.17009 + 2.87936i) q^{13} +2.63090i q^{15} +3.41855 q^{17} +1.26180i q^{19} -1.00000i q^{21} -3.07838 q^{23} -1.92162 q^{25} +1.00000 q^{27} +4.15676 q^{29} -4.68035i q^{31} -3.70928i q^{33} +2.63090 q^{35} +1.26180i q^{37} +(2.17009 + 2.87936i) q^{39} +1.36910i q^{41} +6.83710 q^{43} +2.63090i q^{45} +5.70928i q^{47} -1.00000 q^{49} +3.41855 q^{51} +6.68035 q^{53} +9.75872 q^{55} +1.26180i q^{57} -7.86603i q^{59} -3.65983 q^{61} -1.00000i q^{63} +(-7.57531 + 5.70928i) q^{65} +2.73820i q^{67} -3.07838 q^{69} +11.7093i q^{71} +0.496928i q^{73} -1.92162 q^{75} -3.70928 q^{77} +7.02052 q^{79} +1.00000 q^{81} -12.5464i q^{83} +8.99386i q^{85} +4.15676 q^{87} -6.94441i q^{89} +(2.87936 - 2.17009i) q^{91} -4.68035i q^{93} -3.31965 q^{95} +7.91548i q^{97} -3.70928i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 6 * q^9	\( 6 q + 6 q^{3} + 6 q^{9} + 2 q^{13} - 8 q^{17} - 12 q^{23} - 18 q^{25} + 6 q^{27} + 12 q^{29} + 8 q^{35} + 2 q^{39} - 16 q^{43} - 6 q^{49} - 8 q^{51} - 4 q^{53} + 8 q^{55} - 44 q^{61} - 4 q^{65} - 12 q^{69} - 18 q^{75} - 8 q^{77} - 24 q^{79} + 6 q^{81} + 12 q^{87} - 8 q^{91} - 64 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 + 6 * q^9 + 2 * q^13 - 8 * q^17 - 12 * q^23 - 18 * q^25 + 6 * q^27 + 12 * q^29 + 8 * q^35 + 2 * q^39 - 16 * q^43 - 6 * q^49 - 8 * q^51 - 4 * q^53 + 8 * q^55 - 44 * q^61 - 4 * q^65 - 12 * q^69 - 18 * q^75 - 8 * q^77 - 24 * q^79 + 6 * q^81 + 12 * q^87 - 8 * q^91 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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