LMFDB - Embedded newform 3360.2.z.d.1231.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1231,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1231"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.z (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [28,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1231.25
Character	$\chi$	$=$	3360.1231
Dual form			3360.2.z.d.1231.26

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000 q^{5} +(2.43116 + 1.04376i) q^{7} -1.00000 q^{9} -5.54165 q^{11} -1.67913 q^{13} -1.00000i q^{15} +5.50143i q^{17} -1.73478i q^{19} +(1.04376 - 2.43116i) q^{21} -3.54807i q^{23} +1.00000 q^{25} +1.00000i q^{27} -8.46101i q^{29} -7.31172 q^{31} +5.54165i q^{33} +(2.43116 + 1.04376i) q^{35} +11.2477i q^{37} +1.67913i q^{39} -0.0892775i q^{41} -8.68405 q^{43} -1.00000 q^{45} -12.7949 q^{47} +(4.82112 + 5.07512i) q^{49} +5.50143 q^{51} -1.26667i q^{53} -5.54165 q^{55} -1.73478 q^{57} -13.0188i q^{59} -2.49724 q^{61} +(-2.43116 - 1.04376i) q^{63} -1.67913 q^{65} -7.26352 q^{67} -3.54807 q^{69} +6.75026i q^{71} -7.34731i q^{73} -1.00000i q^{75} +(-13.4727 - 5.78417i) q^{77} +1.15000i q^{79} +1.00000 q^{81} -9.14169i q^{83} +5.50143i q^{85} -8.46101 q^{87} -4.16614i q^{89} +(-4.08224 - 1.75261i) q^{91} +7.31172i q^{93} -1.73478i q^{95} +4.20427i q^{97} +5.54165 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 28 q^{5} - 4 q^{7} - 28 q^{9} - 8 q^{13} + 28 q^{25} - 24 q^{31} - 4 q^{35} - 24 q^{43} - 28 q^{45} + 24 q^{47} + 28 q^{49} + 8 q^{57} - 16 q^{61} + 4 q^{63} - 8 q^{65} + 8 q^{67} - 24 q^{69} + 16 q^{77}+ \cdots - 48 q^{91}+O(q^{100}) $ 28 * q + 28 * q^5 - 4 * q^7 - 28 * q^9 - 8 * q^13 + 28 * q^25 - 24 * q^31 - 4 * q^35 - 24 * q^43 - 28 * q^45 + 24 * q^47 + 28 * q^49 + 8 * q^57 - 16 * q^61 + 4 * q^63 - 8 * q^65 + 8 * q^67 - 24 * q^69 + 16 * q^77 + 28 * q^81 - 16 * q^87 - 48 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$-1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	2.43116	+	1.04376i	0.918894	+	0.394505i
$7$	2.43116	+	1.04376i	0.918894	+	0.394505i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.54165			−1.67087			−0.835435	−	0.549589i	$-0.814784\pi$
$11$	−5.54165			−1.67087			−0.835435	+	0.549589i	$0.814784\pi$
$12$		0			0
$13$	−1.67913			−0.465707			−0.232853	−	0.972512i	$-0.574806\pi$
$13$	−1.67913			−0.465707			−0.232853	+	0.972512i	$0.574806\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$			5.50143i			1.33429i	0.744926	+	0.667147i	$0.232485\pi$
$17$			5.50143i			1.33429i	−0.744926	+	0.667147i	$0.767515\pi$
$18$		0			0
$19$		−	1.73478i		−	0.397985i	−0.980001	−	0.198992i	$-0.936233\pi$
$19$		−	1.73478i		−	0.397985i	0.980001	−	0.198992i	$-0.0637669\pi$
$20$		0			0
$21$	1.04376	−	2.43116i	0.227768	−	0.530523i
$22$		0			0
$23$		−	3.54807i		−	0.739823i	−0.929067	−	0.369912i	$-0.879388\pi$
$23$		−	3.54807i		−	0.739823i	0.929067	−	0.369912i	$-0.120612\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		−	8.46101i		−	1.57117i	−0.618754	−	0.785585i	$-0.712362\pi$
$29$		−	8.46101i		−	1.57117i	0.618754	−	0.785585i	$-0.287638\pi$
$30$		0			0
$31$	−7.31172			−1.31322			−0.656612	−	0.754229i	$-0.728011\pi$
$31$	−7.31172			−1.31322			−0.656612	+	0.754229i	$0.728011\pi$
$32$		0			0
$33$			5.54165i			0.964677i
$34$		0			0
$35$	2.43116	+	1.04376i	0.410942	+	0.176428i
$36$		0			0
$37$			11.2477i			1.84912i	0.381039	+	0.924559i	$0.375567\pi$
$37$			11.2477i			1.84912i	−0.381039	+	0.924559i	$0.624433\pi$
$38$		0			0
$39$			1.67913i			0.268876i
$40$		0			0
$41$		−	0.0892775i		−	0.0139428i	−0.999976	−	0.00697140i	$-0.997781\pi$
$41$		−	0.0892775i		−	0.0139428i	0.999976	−	0.00697140i	$-0.00221908\pi$
$42$		0			0
$43$	−8.68405			−1.32430			−0.662152	−	0.749369i	$-0.730357\pi$
$43$	−8.68405			−1.32430			−0.662152	+	0.749369i	$0.730357\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$	−12.7949			−1.86633			−0.933164	−	0.359450i	$-0.882964\pi$
$47$	−12.7949			−1.86633			−0.933164	+	0.359450i	$0.882964\pi$
$48$		0			0
$49$	4.82112	+	5.07512i	0.688731	+	0.725017i
$50$		0			0
$51$	5.50143			0.770355
$52$		0			0
$53$		−	1.26667i		−	0.173991i	−0.996209	−	0.0869955i	$-0.972273\pi$
$53$		−	1.26667i		−	0.173991i	0.996209	−	0.0869955i	$-0.0277266\pi$
$54$		0			0
$55$	−5.54165			−0.747236
$56$		0			0
$57$	−1.73478			−0.229777
$58$		0			0
$59$		−	13.0188i		−	1.69490i	−0.530874	−	0.847451i	$-0.678136\pi$
$59$		−	13.0188i		−	1.69490i	0.530874	−	0.847451i	$-0.321864\pi$
$60$		0			0
$61$	−2.49724			−0.319739			−0.159869	−	0.987138i	$-0.551107\pi$
$61$	−2.49724			−0.319739			−0.159869	+	0.987138i	$0.551107\pi$
$62$		0			0
$63$	−2.43116	−	1.04376i	−0.306298	−	0.131502i
$64$		0			0
$65$	−1.67913			−0.208270
$66$		0			0
$67$	−7.26352			−0.887381			−0.443690	−	0.896180i	$-0.646331\pi$
$67$	−7.26352			−0.887381			−0.443690	+	0.896180i	$0.646331\pi$
$68$		0			0
$69$	−3.54807			−0.427137
$70$		0			0
$71$			6.75026i			0.801108i	0.916273	+	0.400554i	$0.131182\pi$
$71$			6.75026i			0.801108i	−0.916273	+	0.400554i	$0.868818\pi$
$72$		0			0
$73$		−	7.34731i		−	0.859938i	−0.902844	−	0.429969i	$-0.858525\pi$
$73$		−	7.34731i		−	0.859938i	0.902844	−	0.429969i	$-0.141475\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	−13.4727	−	5.78417i	−1.53535	−	0.659167i
$78$		0			0
$79$			1.15000i			0.129386i	0.997905	+	0.0646928i	$0.0206067\pi$
$79$			1.15000i			0.129386i	−0.997905	+	0.0646928i	$0.979393\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	9.14169i		−	1.00343i	−0.865033	−	0.501715i	$-0.832703\pi$
$83$		−	9.14169i		−	1.00343i	0.865033	−	0.501715i	$-0.167297\pi$
$84$		0			0
$85$			5.50143i			0.596714i
$86$		0			0
$87$	−8.46101			−0.907115
$88$		0			0
$89$		−	4.16614i		−	0.441610i	−0.975318	−	0.220805i	$-0.929132\pi$
$89$		−	4.16614i		−	0.441610i	0.975318	−	0.220805i	$-0.0708685\pi$
$90$		0			0
$91$	−4.08224	−	1.75261i	−0.427935	−	0.183724i
$92$		0			0
$93$			7.31172i			0.758190i
$94$		0			0
$95$		−	1.73478i		−	0.177984i
$96$		0			0
$97$			4.20427i			0.426878i	0.976956	+	0.213439i	$0.0684665\pi$
$97$			4.20427i			0.426878i	−0.976956	+	0.213439i	$0.931533\pi$
$98$		0			0
$99$	5.54165			0.556957
$100$		0			0
$101$	−15.0071			−1.49326			−0.746629	−	0.665240i	$-0.768329\pi$
$101$	−15.0071			−1.49326			−0.746629	+	0.665240i	$0.768329\pi$
$102$		0			0
$103$	1.31225			0.129300			0.0646501	−	0.997908i	$-0.479407\pi$
$103$	1.31225			0.129300			0.0646501	+	0.997908i	$0.479407\pi$
$104$		0			0
$105$	1.04376	−	2.43116i	0.101861	−	0.237257i
$106$		0			0
$107$	16.7713			1.62134			0.810672	−	0.585501i	$-0.199102\pi$
$107$	16.7713			1.62134			0.810672	+	0.585501i	$0.199102\pi$
$108$		0			0
$109$			2.48432i			0.237955i	0.992897	+	0.118978i	$0.0379616\pi$
$109$			2.48432i			0.237955i	−0.992897	+	0.118978i	$0.962038\pi$
$110$		0			0
$111$	11.2477			1.06759
$112$		0			0
$113$	−10.7975			−1.01575			−0.507873	−	0.861432i	$-0.669568\pi$
$113$	−10.7975			−1.01575			−0.507873	+	0.861432i	$0.669568\pi$
$114$		0			0
$115$		−	3.54807i		−	0.330859i
$116$		0			0
$117$	1.67913			0.155236
$118$		0			0
$119$	−5.74219	+	13.3749i	−0.526386	+	1.22607i
$120$		0			0
$121$	19.7099			1.79181
$122$		0			0
$123$	−0.0892775			−0.00804988
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	0.926602i		−	0.0822226i	−0.999155	−	0.0411113i	$-0.986910\pi$
$127$		−	0.926602i		−	0.0822226i	0.999155	−	0.0411113i	$-0.0130898\pi$
$128$		0			0
$129$			8.68405i			0.764588i
$130$		0			0
$131$			12.4963i			1.09181i	0.837847	+	0.545905i	$0.183814\pi$
$131$			12.4963i			1.09181i	−0.837847	+	0.545905i	$0.816186\pi$
$132$		0			0
$133$	1.81070	−	4.21753i	0.157007	−	0.365706i
$134$		0			0
$135$			1.00000i			0.0860663i
$136$		0			0
$137$	3.53259			0.301810			0.150905	−	0.988548i	$-0.451781\pi$
$137$	3.53259			0.301810			0.150905	+	0.988548i	$0.451781\pi$
$138$		0			0
$139$			12.1022i			1.02650i	0.858240	+	0.513249i	$0.171558\pi$
$139$			12.1022i			1.02650i	−0.858240	+	0.513249i	$0.828442\pi$
$140$		0			0
$141$			12.7949i			1.07753i
$142$		0			0
$143$	9.30514			0.778135
$144$		0			0
$145$		−	8.46101i		−	0.702649i
$146$		0			0
$147$	5.07512	−	4.82112i	0.418589	−	0.397639i
$148$		0			0
$149$		−	11.1107i		−	0.910221i	−0.890435	−	0.455110i	$-0.849600\pi$
$149$		−	11.1107i		−	0.910221i	0.890435	−	0.455110i	$-0.150400\pi$
$150$		0			0
$151$		−	3.03195i		−	0.246737i	−0.992361	−	0.123368i	$-0.960630\pi$
$151$		−	3.03195i		−	0.246737i	0.992361	−	0.123368i	$-0.0393696\pi$
$152$		0			0
$153$		−	5.50143i		−	0.444764i
$154$		0			0
$155$	−7.31172			−0.587292
$156$		0			0
$157$	−5.57251			−0.444735			−0.222367	−	0.974963i	$-0.571378\pi$
$157$	−5.57251			−0.444735			−0.222367	+	0.974963i	$0.571378\pi$
$158$		0			0
$159$	−1.26667			−0.100454
$160$		0			0
$161$	3.70334	−	8.62593i	0.291864	−	0.679819i
$162$		0			0
$163$	20.9634			1.64198			0.820989	−	0.570944i	$-0.193423\pi$
$163$	20.9634			1.64198			0.820989	+	0.570944i	$0.193423\pi$
$164$		0			0
$165$			5.54165i			0.431417i
$166$		0			0
$167$	5.73739			0.443973			0.221986	−	0.975050i	$-0.428746\pi$
$167$	5.73739			0.443973			0.221986	+	0.975050i	$0.428746\pi$
$168$		0			0
$169$	−10.1805			−0.783117
$170$		0			0
$171$			1.73478i			0.132662i
$172$		0			0
$173$	−17.1231			−1.30185			−0.650923	−	0.759144i	$-0.725618\pi$
$173$	−17.1231			−1.30185			−0.650923	+	0.759144i	$0.725618\pi$
$174$		0			0
$175$	2.43116	+	1.04376i	0.183779	+	0.0789011i
$176$		0			0
$177$	−13.0188			−0.978552
$178$		0			0
$179$	−14.4927			−1.08324			−0.541618	−	0.840625i	$-0.682188\pi$
$179$	−14.4927			−1.08324			−0.541618	+	0.840625i	$0.682188\pi$
$180$		0			0
$181$	−1.28496			−0.0955104			−0.0477552	−	0.998859i	$-0.515207\pi$
$181$	−1.28496			−0.0955104			−0.0477552	+	0.998859i	$0.515207\pi$
$182$		0			0
$183$			2.49724i			0.184601i
$184$		0			0
$185$			11.2477i			0.826951i
$186$		0			0
$187$		−	30.4870i		−	2.22943i
$188$		0			0
$189$	−1.04376	+	2.43116i	−0.0759226	+	0.176841i
$190$		0			0
$191$		−	9.06371i		−	0.655827i	−0.944708	−	0.327914i	$-0.893655\pi$
$191$		−	9.06371i		−	0.655827i	0.944708	−	0.327914i	$-0.106345\pi$
$192$		0			0
$193$	17.5065			1.26014			0.630072	−	0.776536i	$-0.283025\pi$
$193$	17.5065			1.26014			0.630072	+	0.776536i	$0.283025\pi$
$194$		0			0
$195$			1.67913i			0.120245i
$196$		0			0
$197$			13.1628i			0.937808i	0.883249	+	0.468904i	$0.155351\pi$
$197$			13.1628i			0.937808i	−0.883249	+	0.468904i	$0.844649\pi$
$198$		0			0
$199$	0.276416			0.0195946			0.00979732	−	0.999952i	$-0.496881\pi$
$199$	0.276416			0.0195946			0.00979732	+	0.999952i	$0.496881\pi$
$200$		0			0
$201$			7.26352i			0.512329i
$202$		0			0
$203$	8.83129	−	20.5701i	0.619835	−	1.44374i
$204$		0			0
$205$		−	0.0892775i		−	0.00623541i
$206$		0			0
$207$			3.54807i			0.246608i
$208$		0			0
$209$			9.61352i			0.664981i
$210$		0			0
$211$	−12.0851			−0.831973			−0.415986	−	0.909371i	$-0.636564\pi$
$211$	−12.0851			−0.831973			−0.415986	+	0.909371i	$0.636564\pi$
$212$		0			0
$213$	6.75026			0.462520
$214$		0			0
$215$	−8.68405			−0.592247
$216$		0			0
$217$	−17.7760	−	7.63171i	−1.20671	−	0.518074i
$218$		0			0
$219$	−7.34731			−0.496485
$220$		0			0
$221$		−	9.23762i		−	0.621389i
$222$		0			0
$223$	−17.8291			−1.19393			−0.596963	−	0.802268i	$-0.703626\pi$
$223$	−17.8291			−1.19393			−0.596963	+	0.802268i	$0.703626\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$			27.5789i			1.83048i	0.402911	+	0.915239i	$0.367999\pi$
$227$			27.5789i			1.83048i	−0.402911	+	0.915239i	$0.632001\pi$
$228$		0			0
$229$	−3.43380			−0.226912			−0.113456	−	0.993543i	$-0.536192\pi$
$229$	−3.43380			−0.226912			−0.113456	+	0.993543i	$0.536192\pi$
$230$		0			0
$231$	−5.78417	+	13.4727i	−0.380570	+	0.886436i
$232$		0			0
$233$	8.12381			0.532209			0.266104	−	0.963944i	$-0.414263\pi$
$233$	8.12381			0.532209			0.266104	+	0.963944i	$0.414263\pi$
$234$		0			0
$235$	−12.7949			−0.834648
$236$		0			0
$237$	1.15000			0.0747008
$238$		0			0
$239$		−	23.7708i		−	1.53761i	−0.639484	−	0.768804i	$-0.720852\pi$
$239$		−	23.7708i		−	1.53761i	0.639484	−	0.768804i	$-0.279148\pi$
$240$		0			0
$241$			13.8075i			0.889417i	0.895675	+	0.444708i	$0.146693\pi$
$241$			13.8075i			0.889417i	−0.895675	+	0.444708i	$0.853307\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	4.82112	+	5.07512i	0.308010	+	0.324238i
$246$		0			0
$247$			2.91291i			0.185344i
$248$		0			0
$249$	−9.14169			−0.579331
$250$		0			0
$251$			2.78833i			0.175998i	0.996121	+	0.0879990i	$0.0280472\pi$
$251$			2.78833i			0.175998i	−0.996121	+	0.0879990i	$0.971953\pi$
$252$		0			0
$253$			19.6621i			1.23615i
$254$		0			0
$255$	5.50143			0.344513
$256$		0			0
$257$		−	3.21699i		−	0.200670i	−0.994954	−	0.100335i	$-0.968009\pi$
$257$		−	3.21699i		−	0.200670i	0.994954	−	0.100335i	$-0.0319915\pi$
$258$		0			0
$259$	−11.7400	+	27.3451i	−0.729487	+	1.69914i
$260$		0			0
$261$			8.46101i			0.523723i
$262$		0			0
$263$			3.61223i			0.222740i	0.993779	+	0.111370i	$0.0355238\pi$
$263$			3.61223i			0.222740i	−0.993779	+	0.111370i	$0.964476\pi$
$264$		0			0
$265$		−	1.26667i		−	0.0778111i
$266$		0			0
$267$	−4.16614			−0.254964
$268$		0			0
$269$	−8.64776			−0.527263			−0.263632	−	0.964623i	$-0.584920\pi$
$269$	−8.64776			−0.527263			−0.263632	+	0.964623i	$0.584920\pi$
$270$		0			0
$271$	−3.09276			−0.187872			−0.0939358	−	0.995578i	$-0.529945\pi$
$271$	−3.09276			−0.187872			−0.0939358	+	0.995578i	$0.529945\pi$
$272$		0			0
$273$	−1.75261	+	4.08224i	−0.106073	+	0.247068i
$274$		0			0
$275$	−5.54165			−0.334174
$276$		0			0
$277$			21.5406i			1.29425i	0.762384	+	0.647125i	$0.224029\pi$
$277$			21.5406i			1.29425i	−0.762384	+	0.647125i	$0.775971\pi$
$278$		0			0
$279$	7.31172			0.437741
$280$		0			0
$281$	17.0871			1.01933			0.509664	−	0.860373i	$-0.329770\pi$
$281$	17.0871			1.01933			0.509664	+	0.860373i	$0.329770\pi$
$282$		0			0
$283$			25.9118i			1.54030i	0.637865	+	0.770148i	$0.279818\pi$
$283$			25.9118i			1.54030i	−0.637865	+	0.770148i	$0.720182\pi$
$284$		0			0
$285$	−1.73478			−0.102759
$286$		0			0
$287$	0.0931846	−	0.217048i	0.00550051	−	0.0128120i
$288$		0			0
$289$	−13.2658			−0.780339
$290$		0			0
$291$	4.20427			0.246458
$292$		0			0
$293$	−27.3161			−1.59583			−0.797913	−	0.602773i	$-0.794063\pi$
$293$	−27.3161			−1.59583			−0.797913	+	0.602773i	$0.794063\pi$
$294$		0			0
$295$		−	13.0188i		−	0.757983i
$296$		0			0
$297$		−	5.54165i		−	0.321559i
$298$		0			0
$299$			5.95767i			0.344541i
$300$		0			0
$301$	−21.1123	−	9.06409i	−1.21690	−	0.522445i
$302$		0			0
$303$			15.0071i			0.862133i
$304$		0			0
$305$	−2.49724			−0.142991
$306$		0			0
$307$			2.39283i			0.136566i	0.997666	+	0.0682831i	$0.0217521\pi$
$307$			2.39283i			0.136566i	−0.997666	+	0.0682831i	$0.978248\pi$
$308$		0			0
$309$		−	1.31225i		−	0.0746515i
$310$		0			0
$311$	−24.3201			−1.37907			−0.689533	−	0.724255i	$-0.742184\pi$
$311$	−24.3201			−1.37907			−0.689533	+	0.724255i	$0.742184\pi$
$312$		0			0
$313$		−	30.3185i		−	1.71370i	−0.515562	−	0.856852i	$-0.672417\pi$
$313$		−	30.3185i		−	1.71370i	0.515562	−	0.856852i	$-0.327583\pi$
$314$		0			0
$315$	−2.43116	−	1.04376i	−0.136981	−	0.0588094i
$316$		0			0
$317$			7.93499i			0.445673i	0.974856	+	0.222837i	$0.0715317\pi$
$317$			7.93499i			0.445673i	−0.974856	+	0.222837i	$0.928468\pi$
$318$		0			0
$319$			46.8879i			2.62522i
$320$		0			0
$321$		−	16.7713i		−	0.936083i
$322$		0			0
$323$	9.54375			0.531029
$324$		0			0
$325$	−1.67913			−0.0931413
$326$		0			0
$327$	2.48432			0.137383
$328$		0			0
$329$	−31.1065	−	13.3549i	−1.71496	−	0.736277i
$330$		0			0
$331$	14.2122			0.781172			0.390586	−	0.920566i	$-0.372272\pi$
$331$	14.2122			0.781172			0.390586	+	0.920566i	$0.372272\pi$
$332$		0			0
$333$		−	11.2477i		−	0.616373i
$334$		0			0
$335$	−7.26352			−0.396849
$336$		0			0
$337$	2.94854			0.160617			0.0803085	−	0.996770i	$-0.474409\pi$
$337$	2.94854			0.160617			0.0803085	+	0.996770i	$0.474409\pi$
$338$		0			0
$339$			10.7975i			0.586442i
$340$		0			0
$341$	40.5190			2.19423
$342$		0			0
$343$	6.42370	+	17.3706i	0.346847	+	0.937922i
$344$		0			0
$345$	−3.54807			−0.191022
$346$		0			0
$347$	30.9749			1.66282			0.831410	−	0.555659i	$-0.187534\pi$
$347$	30.9749			1.66282			0.831410	+	0.555659i	$0.187534\pi$
$348$		0			0
$349$	4.43452			0.237374			0.118687	−	0.992932i	$-0.462131\pi$
$349$	4.43452			0.237374			0.118687	+	0.992932i	$0.462131\pi$
$350$		0			0
$351$		−	1.67913i		−	0.0896253i
$352$		0			0
$353$		−	25.5750i		−	1.36122i	−0.732647	−	0.680609i	$-0.761715\pi$
$353$		−	25.5750i		−	1.36122i	0.732647	−	0.680609i	$-0.238285\pi$
$354$		0			0
$355$			6.75026i			0.358266i
$356$		0			0
$357$	13.3749	+	5.74219i	0.707874	+	0.303909i
$358$		0			0
$359$			15.8598i			0.837049i	0.908206	+	0.418525i	$0.137453\pi$
$359$			15.8598i			0.837049i	−0.908206	+	0.418525i	$0.862547\pi$
$360$		0			0
$361$	15.9906			0.841608
$362$		0			0
$363$		−	19.7099i		−	1.03450i
$364$		0			0
$365$		−	7.34731i		−	0.384576i
$366$		0			0
$367$	−20.7390			−1.08257			−0.541284	−	0.840840i	$-0.682062\pi$
$367$	−20.7390			−1.08257			−0.541284	+	0.840840i	$0.682062\pi$
$368$		0			0
$369$			0.0892775i			0.00464760i
$370$		0			0
$371$	1.32211	−	3.07949i	0.0686404	−	0.159879i
$372$		0			0
$373$		−	14.5992i		−	0.755920i	−0.925822	−	0.377960i	$-0.876626\pi$
$373$		−	14.5992i		−	0.755920i	0.925822	−	0.377960i	$-0.123374\pi$
$374$		0			0
$375$		−	1.00000i		−	0.0516398i
$376$		0			0
$377$			14.2071i			0.731704i
$378$		0			0
$379$	−35.0992			−1.80292			−0.901462	−	0.432859i	$-0.857505\pi$
$379$	−35.0992			−1.80292			−0.901462	+	0.432859i	$0.857505\pi$
$380$		0			0
$381$	−0.926602			−0.0474713
$382$		0			0
$383$	2.93845			0.150148			0.0750739	−	0.997178i	$-0.476081\pi$
$383$	2.93845			0.150148			0.0750739	+	0.997178i	$0.476081\pi$
$384$		0			0
$385$	−13.4727	−	5.78417i	−0.686630	−	0.294789i
$386$		0			0
$387$	8.68405			0.441435
$388$		0			0
$389$		−	29.8030i		−	1.51107i	−0.655106	−	0.755537i	$-0.727376\pi$
$389$		−	29.8030i		−	1.51107i	0.655106	−	0.755537i	$-0.272624\pi$
$390$		0			0
$391$	19.5195			0.987141
$392$		0			0
$393$	12.4963			0.630357
$394$		0			0
$395$			1.15000i			0.0578630i
$396$		0			0
$397$	20.3840			1.02304			0.511521	−	0.859271i	$-0.329082\pi$
$397$	20.3840			1.02304			0.511521	+	0.859271i	$0.329082\pi$
$398$		0			0
$399$	−4.21753	−	1.81070i	−0.211140	−	0.0906482i
$400$		0			0
$401$	−4.24845			−0.212157			−0.106079	−	0.994358i	$-0.533830\pi$
$401$	−4.24845			−0.212157			−0.106079	+	0.994358i	$0.533830\pi$
$402$		0			0
$403$	12.2773			0.611577
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$		−	62.3311i		−	3.08964i
$408$		0			0
$409$			16.3835i			0.810113i	0.914292	+	0.405057i	$0.132748\pi$
$409$			16.3835i			0.810113i	−0.914292	+	0.405057i	$0.867252\pi$
$410$		0			0
$411$		−	3.53259i		−	0.174250i
$412$		0			0
$413$	13.5885	−	31.6508i	0.668648	−	1.55743i
$414$		0			0
$415$		−	9.14169i		−	0.448748i
$416$		0			0
$417$	12.1022			0.592649
$418$		0			0
$419$			19.2720i			0.941497i	0.882268	+	0.470748i	$0.156016\pi$
$419$			19.2720i			0.941497i	−0.882268	+	0.470748i	$0.843984\pi$
$420$		0			0
$421$			14.8800i			0.725206i	0.931944	+	0.362603i	$0.118112\pi$
$421$			14.8800i			0.725206i	−0.931944	+	0.362603i	$0.881888\pi$
$422$		0			0
$423$	12.7949			0.622110
$424$		0			0
$425$			5.50143i			0.266859i
$426$		0			0
$427$	−6.07120	−	2.60653i	−0.293806	−	0.126139i
$428$		0			0
$429$		−	9.30514i		−	0.449257i
$430$		0			0
$431$		−	0.281619i		−	0.0135651i	−0.999977	−	0.00678255i	$-0.997841\pi$
$431$		−	0.281619i		−	0.0135651i	0.999977	−	0.00678255i	$-0.00215897\pi$
$432$		0			0
$433$		−	4.06965i		−	0.195575i	−0.995207	−	0.0977874i	$-0.968823\pi$
$433$		−	4.06965i		−	0.195575i	0.995207	−	0.0977874i	$-0.0311765\pi$
$434$		0			0
$435$	−8.46101			−0.405674
$436$		0			0
$437$	−6.15510			−0.294439
$438$		0			0
$439$	18.3292			0.874804			0.437402	−	0.899266i	$-0.355899\pi$
$439$	18.3292			0.874804			0.437402	+	0.899266i	$0.355899\pi$
$440$		0			0
$441$	−4.82112	−	5.07512i	−0.229577	−	0.241672i
$442$		0			0
$443$	16.2913			0.774023			0.387012	−	0.922075i	$-0.373507\pi$
$443$	16.2913			0.774023			0.387012	+	0.922075i	$0.373507\pi$
$444$		0			0
$445$		−	4.16614i		−	0.197494i
$446$		0			0
$447$	−11.1107			−0.525516
$448$		0			0
$449$	−36.2809			−1.71220			−0.856101	−	0.516809i	$-0.827120\pi$
$449$	−36.2809			−1.71220			−0.856101	+	0.516809i	$0.827120\pi$
$450$		0			0
$451$			0.494744i			0.0232966i
$452$		0			0
$453$	−3.03195			−0.142453
$454$		0			0
$455$	−4.08224	−	1.75261i	−0.191378	−	0.0821638i
$456$		0			0
$457$	21.7531			1.01757			0.508783	−	0.860895i	$-0.330096\pi$
$457$	21.7531			1.01757			0.508783	+	0.860895i	$0.330096\pi$
$458$		0			0
$459$	−5.50143			−0.256785
$460$		0			0
$461$	17.9992			0.838305			0.419153	−	0.907916i	$-0.362327\pi$
$461$	17.9992			0.838305			0.419153	+	0.907916i	$0.362327\pi$
$462$		0			0
$463$			22.6300i			1.05170i	0.850576	+	0.525852i	$0.176254\pi$
$463$			22.6300i			1.05170i	−0.850576	+	0.525852i	$0.823746\pi$
$464$		0			0
$465$			7.31172i			0.339073i
$466$		0			0
$467$			11.6927i			0.541072i	0.962710	+	0.270536i	$0.0872009\pi$
$467$			11.6927i			0.541072i	−0.962710	+	0.270536i	$0.912799\pi$
$468$		0			0
$469$	−17.6588	−	7.58140i	−0.815408	−	0.350077i
$470$		0			0
$471$			5.57251i			0.256768i
$472$		0			0
$473$	48.1239			2.21274
$474$		0			0
$475$		−	1.73478i		−	0.0795970i
$476$		0			0
$477$			1.26667i			0.0579970i
$478$		0			0
$479$	−26.6487			−1.21761			−0.608805	−	0.793320i	$-0.708351\pi$
$479$	−26.6487			−1.21761			−0.608805	+	0.793320i	$0.708351\pi$
$480$		0			0
$481$		−	18.8864i		−	0.861147i
$482$		0			0
$483$	−8.62593	−	3.70334i	−0.392494	−	0.168508i
$484$		0			0
$485$			4.20427i			0.190906i
$486$		0			0
$487$		−	9.67576i		−	0.438450i	−0.975674	−	0.219225i	$-0.929647\pi$
$487$		−	9.67576i		−	0.438450i	0.975674	−	0.219225i	$-0.0703530\pi$
$488$		0			0
$489$		−	20.9634i		−	0.947996i
$490$		0			0
$491$	21.3834			0.965021			0.482511	−	0.875890i	$-0.339725\pi$
$491$	21.3834			0.965021			0.482511	+	0.875890i	$0.339725\pi$
$492$		0			0
$493$	46.5477			2.09640
$494$		0			0
$495$	5.54165			0.249079
$496$		0			0
$497$	−7.04567	+	16.4110i	−0.316041	+	0.736133i
$498$		0			0
$499$	12.1520			0.543997			0.271998	−	0.962298i	$-0.412315\pi$
$499$	12.1520			0.543997			0.271998	+	0.962298i	$0.412315\pi$
$500$		0			0
$501$		−	5.73739i		−	0.256328i
$502$		0			0
$503$	−5.68562			−0.253509			−0.126755	−	0.991934i	$-0.540456\pi$
$503$	−5.68562			−0.253509			−0.126755	+	0.991934i	$0.540456\pi$
$504$		0			0
$505$	−15.0071			−0.667806
$506$		0			0
$507$			10.1805i			0.452133i
$508$		0			0
$509$	2.01211			0.0891852			0.0445926	−	0.999005i	$-0.485801\pi$
$509$	2.01211			0.0891852			0.0445926	+	0.999005i	$0.485801\pi$
$510$		0			0
$511$	7.66886	−	17.8625i	0.339250	−	0.790191i
$512$		0			0
$513$	1.73478			0.0765922
$514$		0			0
$515$	1.31225			0.0578248
$516$		0			0
$517$	70.9049			3.11839
$518$		0			0
$519$			17.1231i			0.751621i
$520$		0			0
$521$			4.06826i			0.178234i	0.996021	+	0.0891168i	$0.0284045\pi$
$521$			4.06826i			0.178234i	−0.996021	+	0.0891168i	$0.971596\pi$
$522$		0			0
$523$		−	23.4857i		−	1.02696i	−0.858102	−	0.513479i	$-0.828356\pi$
$523$		−	23.4857i		−	1.02696i	0.858102	−	0.513479i	$-0.171644\pi$
$524$		0			0
$525$	1.04376	−	2.43116i	0.0455536	−	0.106105i
$526$		0			0
$527$		−	40.2249i		−	1.75223i
$528$		0			0
$529$	10.4112			0.452661
$530$		0			0
$531$			13.0188i			0.564967i
$532$		0			0
$533$			0.149908i			0.00649326i
$534$		0			0
$535$	16.7713			0.725087
$536$		0			0
$537$			14.4927i			0.625407i
$538$		0			0
$539$	−26.7169	−	28.1245i	−1.15078	−	1.21141i
$540$		0			0
$541$			28.2718i			1.21550i	0.794129	+	0.607749i	$0.207928\pi$
$541$			28.2718i			1.21550i	−0.794129	+	0.607749i	$0.792072\pi$
$542$		0			0
$543$			1.28496i			0.0551430i
$544$		0			0
$545$			2.48432i			0.106417i
$546$		0			0
$547$	29.7498			1.27201			0.636004	−	0.771685i	$-0.280586\pi$
$547$	29.7498			1.27201			0.636004	+	0.771685i	$0.280586\pi$
$548$		0			0
$549$	2.49724			0.106580
$550$		0			0
$551$	−14.6780			−0.625302
$552$		0			0
$553$	−1.20033	+	2.79585i	−0.0510433	+	0.118892i
$554$		0			0
$555$	11.2477			0.477440
$556$		0			0
$557$		−	16.5286i		−	0.700340i	−0.936686	−	0.350170i	$-0.886124\pi$
$557$		−	16.5286i		−	0.700340i	0.936686	−	0.350170i	$-0.113876\pi$
$558$		0			0
$559$	14.5816			0.616738
$560$		0			0
$561$	−30.4870			−1.28716
$562$		0			0
$563$		−	15.9112i		−	0.670578i	−0.942115	−	0.335289i	$-0.891166\pi$
$563$		−	15.9112i		−	0.670578i	0.942115	−	0.335289i	$-0.108834\pi$
$564$		0			0
$565$	−10.7975			−0.454256
$566$		0			0
$567$	2.43116	+	1.04376i	0.102099	+	0.0438339i
$568$		0			0
$569$	13.2684			0.556239			0.278120	−	0.960546i	$-0.410289\pi$
$569$	13.2684			0.556239			0.278120	+	0.960546i	$0.410289\pi$
$570$		0			0
$571$	8.34007			0.349021			0.174511	−	0.984655i	$-0.444166\pi$
$571$	8.34007			0.349021			0.174511	+	0.984655i	$0.444166\pi$
$572$		0			0
$573$	−9.06371			−0.378642
$574$		0			0
$575$		−	3.54807i		−	0.147965i
$576$		0			0
$577$		−	23.6242i		−	0.983487i	−0.870740	−	0.491743i	$-0.836360\pi$
$577$		−	23.6242i		−	0.983487i	0.870740	−	0.491743i	$-0.163640\pi$
$578$		0			0
$579$		−	17.5065i		−	0.727545i
$580$		0			0
$581$	9.54176	−	22.2249i	0.395859	−	0.922046i
$582$		0			0
$583$			7.01946i			0.290716i
$584$		0			0
$585$	1.67913			0.0694235
$586$		0			0
$587$		−	6.67124i		−	0.275351i	−0.990477	−	0.137676i	$-0.956037\pi$
$587$		−	6.67124i		−	0.275351i	0.990477	−	0.137676i	$-0.0439632\pi$
$588$		0			0
$589$			12.6842i			0.522643i
$590$		0			0
$591$	13.1628			0.541444
$592$		0			0
$593$		−	25.3635i		−	1.04155i	−0.853693	−	0.520776i	$-0.825642\pi$
$593$		−	25.3635i		−	1.04155i	0.853693	−	0.520776i	$-0.174358\pi$
$594$		0			0
$595$	−5.74219	+	13.3749i	−0.235407	+	0.548317i
$596$		0			0
$597$		−	0.276416i		−	0.0113130i
$598$		0			0
$599$		−	32.0409i		−	1.30915i	−0.755995	−	0.654577i	$-0.772847\pi$
$599$		−	32.0409i		−	1.30915i	0.755995	−	0.654577i	$-0.227153\pi$
$600$		0			0
$601$			27.2798i			1.11277i	0.830925	+	0.556384i	$0.187812\pi$
$601$			27.2798i			1.11277i	−0.830925	+	0.556384i	$0.812188\pi$
$602$		0			0
$603$	7.26352			0.295794
$604$		0			0
$605$	19.7099			0.801320
$606$		0			0
$607$	29.6611			1.20391			0.601953	−	0.798532i	$-0.294390\pi$
$607$	29.6611			1.20391			0.601953	+	0.798532i	$0.294390\pi$
$608$		0			0
$609$	−20.5701	−	8.83129i	−0.833542	−	0.357862i
$610$		0			0
$611$	21.4843			0.869162
$612$		0			0
$613$		−	17.5337i		−	0.708180i	−0.935211	−	0.354090i	$-0.884791\pi$
$613$		−	17.5337i		−	0.708180i	0.935211	−	0.354090i	$-0.115209\pi$
$614$		0			0
$615$	−0.0892775			−0.00360002
$616$		0			0
$617$	−19.9979			−0.805085			−0.402543	−	0.915401i	$-0.631873\pi$
$617$	−19.9979			−0.805085			−0.402543	+	0.915401i	$0.631873\pi$
$618$		0			0
$619$			10.3616i			0.416466i	0.978079	+	0.208233i	$0.0667713\pi$
$619$			10.3616i			0.416466i	−0.978079	+	0.208233i	$0.933229\pi$
$620$		0			0
$621$	3.54807			0.142379
$622$		0			0
$623$	4.34847	−	10.1286i	0.174218	−	0.405793i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	9.61352			0.383927
$628$		0			0
$629$	−61.8787			−2.46727
$630$		0			0
$631$		−	1.55276i		−	0.0618145i	−0.999522	−	0.0309072i	$-0.990160\pi$
$631$		−	1.55276i		−	0.0618145i	0.999522	−	0.0309072i	$-0.00983964\pi$
$632$		0			0
$633$			12.0851i			0.480340i
$634$		0			0
$635$		−	0.926602i		−	0.0367711i
$636$		0			0
$637$	−8.09528	−	8.52178i	−0.320747	−	0.337645i
$638$		0			0
$639$		−	6.75026i		−	0.267036i
$640$		0			0
$641$	−13.3311			−0.526546			−0.263273	−	0.964721i	$-0.584802\pi$
$641$	−13.3311			−0.526546			−0.263273	+	0.964721i	$0.584802\pi$
$642$		0			0
$643$			20.6034i			0.812520i	0.913758	+	0.406260i	$0.133167\pi$
$643$			20.6034i			0.812520i	−0.913758	+	0.406260i	$0.866833\pi$
$644$		0			0
$645$			8.68405i			0.341934i
$646$		0			0
$647$	35.8391			1.40898			0.704490	−	0.709714i	$-0.251176\pi$
$647$	35.8391			1.40898			0.704490	+	0.709714i	$0.251176\pi$
$648$		0			0
$649$			72.1455i			2.83196i
$650$		0			0
$651$	−7.63171	+	17.7760i	−0.299110	+	0.696696i
$652$		0			0
$653$			2.28298i			0.0893398i	0.999002	+	0.0446699i	$0.0142236\pi$
$653$			2.28298i			0.0893398i	−0.999002	+	0.0446699i	$0.985776\pi$
$654$		0			0
$655$			12.4963i			0.488272i
$656$		0			0
$657$			7.34731i			0.286646i
$658$		0			0
$659$	−18.0400			−0.702737			−0.351369	−	0.936237i	$-0.614284\pi$
$659$	−18.0400			−0.702737			−0.351369	+	0.936237i	$0.614284\pi$
$660$		0			0
$661$	−29.8230			−1.15998			−0.579991	−	0.814623i	$-0.696944\pi$
$661$	−29.8230			−1.15998			−0.579991	+	0.814623i	$0.696944\pi$
$662$		0			0
$663$	−9.23762			−0.358759
$664$		0			0
$665$	1.81070	−	4.21753i	0.0702158	−	0.163549i
$666$		0			0
$667$	−30.0202			−1.16239
$668$		0			0
$669$			17.8291i			0.689314i
$670$		0			0
$671$	13.8388			0.534242
$672$		0			0
$673$	29.2670			1.12816			0.564080	−	0.825720i	$-0.309231\pi$
$673$	29.2670			1.12816			0.564080	+	0.825720i	$0.309231\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$	−25.9210			−0.996226			−0.498113	−	0.867112i	$-0.665974\pi$
$677$	−25.9210			−0.996226			−0.498113	+	0.867112i	$0.665974\pi$
$678$		0			0
$679$	−4.38826	+	10.2213i	−0.168406	+	0.392256i
$680$		0			0
$681$	27.5789			1.05683
$682$		0			0
$683$	−22.5472			−0.862745			−0.431373	−	0.902174i	$-0.641971\pi$
$683$	−22.5472			−0.862745			−0.431373	+	0.902174i	$0.641971\pi$
$684$		0			0
$685$	3.53259			0.134973
$686$		0			0
$687$			3.43380i			0.131008i
$688$		0			0
$689$			2.12691i			0.0810288i
$690$		0			0
$691$			8.74160i			0.332546i	0.986080	+	0.166273i	$0.0531733\pi$
$691$			8.74160i			0.332546i	−0.986080	+	0.166273i	$0.946827\pi$
$692$		0			0
$693$	13.4727	+	5.78417i	0.511784	+	0.219722i
$694$		0			0
$695$			12.1022i			0.459064i
$696$		0			0
$697$	0.491154			0.0186038
$698$		0			0
$699$		−	8.12381i		−	0.307271i
$700$		0			0
$701$			14.5041i			0.547811i	0.961757	+	0.273906i	$0.0883156\pi$
$701$			14.5041i			0.547811i	−0.961757	+	0.273906i	$0.911684\pi$
$702$		0			0
$703$	19.5123			0.735921
$704$		0			0
$705$			12.7949i			0.481884i
$706$		0			0
$707$	−36.4846	−	15.6638i	−1.37215	−	0.589099i
$708$		0			0
$709$		−	27.9863i		−	1.05105i	−0.850779	−	0.525523i	$-0.823870\pi$
$709$		−	27.9863i		−	1.05105i	0.850779	−	0.525523i	$-0.176130\pi$
$710$		0			0
$711$		−	1.15000i		−	0.0431285i
$712$		0			0
$713$			25.9425i			0.971554i
$714$		0			0
$715$	9.30514			0.347993
$716$		0			0
$717$	−23.7708			−0.887739
$718$		0			0
$719$	31.9104			1.19006			0.595029	−	0.803704i	$-0.297141\pi$
$719$	31.9104			1.19006			0.595029	+	0.803704i	$0.297141\pi$
$720$		0			0
$721$	3.19030	+	1.36968i	0.118813	+	0.0510096i
$722$		0			0
$723$	13.8075			0.513505
$724$		0			0
$725$		−	8.46101i		−	0.314234i
$726$		0			0
$727$	−37.3223			−1.38421			−0.692104	−	0.721798i	$-0.743316\pi$
$727$	−37.3223			−1.38421			−0.692104	+	0.721798i	$0.743316\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	47.7747i		−	1.76701i
$732$		0			0
$733$	−31.5170			−1.16411			−0.582053	−	0.813151i	$-0.697750\pi$
$733$	−31.5170			−1.16411			−0.582053	+	0.813151i	$0.697750\pi$
$734$		0			0
$735$	5.07512	−	4.82112i	0.187199	−	0.177830i
$736$		0			0
$737$	40.2519			1.48270
$738$		0			0
$739$	−4.86962			−0.179132			−0.0895659	−	0.995981i	$-0.528548\pi$
$739$	−4.86962			−0.179132			−0.0895659	+	0.995981i	$0.528548\pi$
$740$		0			0
$741$	2.91291			0.107009
$742$		0			0
$743$		−	24.0338i		−	0.881714i	−0.897577	−	0.440857i	$-0.854675\pi$
$743$		−	24.0338i		−	0.881714i	0.897577	−	0.440857i	$-0.145325\pi$
$744$		0			0
$745$		−	11.1107i		−	0.407063i
$746$		0			0
$747$			9.14169i			0.334477i
$748$		0			0
$749$	40.7738	+	17.5053i	1.48984	+	0.639629i
$750$		0			0
$751$			1.33007i			0.0485348i	0.999706	+	0.0242674i	$0.00772531\pi$
$751$			1.33007i			0.0485348i	−0.999706	+	0.0242674i	$0.992275\pi$
$752$		0			0
$753$	2.78833			0.101612
$754$		0			0
$755$		−	3.03195i		−	0.110344i
$756$		0			0
$757$			12.9373i			0.470213i	0.971970	+	0.235106i	$0.0755439\pi$
$757$			12.9373i			0.470213i	−0.971970	+	0.235106i	$0.924456\pi$
$758$		0			0
$759$	19.6621			0.713691
$760$		0			0
$761$		−	35.0635i		−	1.27105i	−0.772079	−	0.635526i	$-0.780783\pi$
$761$		−	35.0635i		−	1.27105i	0.772079	−	0.635526i	$-0.219217\pi$
$762$		0			0
$763$	−2.59305	+	6.03980i	−0.0938746	+	0.218655i
$764$		0			0
$765$		−	5.50143i		−	0.198905i
$766$		0			0
$767$			21.8602i			0.789327i
$768$		0			0
$769$		−	18.6210i		−	0.671491i	−0.941953	−	0.335746i	$-0.891012\pi$
$769$		−	18.6210i		−	0.671491i	0.941953	−	0.335746i	$-0.108988\pi$
$770$		0			0
$771$	−3.21699			−0.115857
$772$		0			0
$773$	26.4130			0.950010			0.475005	−	0.879983i	$-0.342446\pi$
$773$	26.4130			0.950010			0.475005	+	0.879983i	$0.342446\pi$
$774$		0			0
$775$	−7.31172			−0.262645
$776$		0			0
$777$	27.3451	+	11.7400i	0.981001	+	0.421170i
$778$		0			0
$779$	−0.154876			−0.00554903
$780$		0			0
$781$		−	37.4075i		−	1.33855i
$782$		0			0
$783$	8.46101			0.302372
$784$		0			0
$785$	−5.57251			−0.198892
$786$		0			0
$787$		−	18.3513i		−	0.654153i	−0.944998	−	0.327077i	$-0.893936\pi$
$787$		−	18.3513i		−	0.654153i	0.944998	−	0.327077i	$-0.106064\pi$
$788$		0			0
$789$	3.61223			0.128599
$790$		0			0
$791$	−26.2506	−	11.2701i	−0.933363	−	0.400718i
$792$		0			0
$793$	4.19319			0.148904
$794$		0			0
$795$	−1.26667			−0.0449243
$796$		0			0
$797$	16.6258			0.588917			0.294458	−	0.955664i	$-0.404861\pi$
$797$	16.6258			0.588917			0.294458	+	0.955664i	$0.404861\pi$
$798$		0			0
$799$		−	70.3903i		−	2.49023i
$800$		0			0
$801$			4.16614i			0.147203i
$802$		0			0
$803$			40.7162i			1.43684i
$804$		0			0
$805$	3.70334	−	8.62593i	0.130526	−	0.304024i
$806$		0			0
$807$			8.64776i			0.304416i
$808$		0			0
$809$	6.95098			0.244383			0.122192	−	0.992507i	$-0.461008\pi$
$809$	6.95098			0.244383			0.122192	+	0.992507i	$0.461008\pi$
$810$		0			0
$811$		−	49.6027i		−	1.74179i	−0.491471	−	0.870894i	$-0.663541\pi$
$811$		−	49.6027i		−	1.74179i	0.491471	−	0.870894i	$-0.336459\pi$
$812$		0			0
$813$			3.09276i			0.108468i
$814$		0			0
$815$	20.9634			0.734315
$816$		0			0
$817$			15.0649i			0.527053i
$818$		0			0
$819$	4.08224	+	1.75261i	0.142645	+	0.0612413i
$820$		0			0
$821$		−	46.2978i		−	1.61580i	−0.589317	−	0.807902i	$-0.700603\pi$
$821$		−	46.2978i		−	1.61580i	0.589317	−	0.807902i	$-0.299397\pi$
$822$		0			0
$823$		−	5.51492i		−	0.192238i	−0.995370	−	0.0961191i	$-0.969357\pi$
$823$		−	5.51492i		−	0.192238i	0.995370	−	0.0961191i	$-0.0306430\pi$
$824$		0			0
$825$			5.54165i			0.192935i
$826$		0			0
$827$	−56.0706			−1.94976			−0.974882	−	0.222720i	$-0.928506\pi$
$827$	−56.0706			−1.94976			−0.974882	+	0.222720i	$0.928506\pi$
$828$		0			0
$829$	44.7027			1.55259			0.776295	−	0.630370i	$-0.217097\pi$
$829$	44.7027			1.55259			0.776295	+	0.630370i	$0.217097\pi$
$830$		0			0
$831$	21.5406			0.747236
$832$		0			0
$833$	−27.9204	+	26.5230i	−0.967386	+	0.918969i
$834$		0			0
$835$	5.73739			0.198551
$836$		0			0
$837$		−	7.31172i		−	0.252730i
$838$		0			0
$839$	44.8497			1.54838			0.774191	−	0.632952i	$-0.218157\pi$
$839$	44.8497			1.54838			0.774191	+	0.632952i	$0.218157\pi$
$840$		0			0
$841$	−42.5887			−1.46857
$842$		0			0
$843$		−	17.0871i		−	0.588510i
$844$		0			0
$845$	−10.1805			−0.350221
$846$		0			0
$847$	47.9179	+	20.5724i	1.64648	+	0.706877i
$848$		0			0
$849$	25.9118			0.889290
$850$		0			0
$851$	39.9078			1.36802
$852$		0			0
$853$	−0.0388026			−0.00132858			−0.000664288	−	1.00000i	$-0.500211\pi$
$853$	−0.0388026			−0.00132858			−0.000664288		1.00000i	$0.500211\pi$
$854$		0			0
$855$			1.73478i			0.0593281i
$856$		0			0
$857$		−	57.0186i		−	1.94772i	−0.227154	−	0.973859i	$-0.572942\pi$
$857$		−	57.0186i		−	1.94772i	0.227154	−	0.973859i	$-0.427058\pi$
$858$		0			0
$859$		−	9.67639i		−	0.330154i	−0.986281	−	0.165077i	$-0.947213\pi$
$859$		−	9.67639i		−	0.330154i	0.986281	−	0.165077i	$-0.0527873\pi$
$860$		0			0
$861$	−0.217048	−	0.0931846i	−0.00739698	−	0.00317572i
$862$		0			0
$863$		−	35.9183i		−	1.22267i	−0.791371	−	0.611337i	$-0.790632\pi$
$863$		−	35.9183i		−	1.22267i	0.791371	−	0.611337i	$-0.209368\pi$
$864$		0			0
$865$	−17.1231			−0.582203
$866$		0			0
$867$			13.2658i			0.450529i
$868$		0			0
$869$		−	6.37291i		−	0.216186i
$870$		0			0
$871$	12.1964			0.413259
$872$		0			0
$873$		−	4.20427i		−	0.142293i
$874$		0			0
$875$	2.43116	+	1.04376i	0.0821883	+	0.0352856i
$876$		0			0
$877$		−	20.0801i		−	0.678057i	−0.940776	−	0.339028i	$-0.889902\pi$
$877$		−	20.0801i		−	0.678057i	0.940776	−	0.339028i	$-0.110098\pi$
$878$		0			0
$879$			27.3161i			0.921350i
$880$		0			0
$881$			37.0514i			1.24829i	0.781307	+	0.624147i	$0.214553\pi$
$881$			37.0514i			1.24829i	−0.781307	+	0.624147i	$0.785447\pi$
$882$		0			0
$883$	13.3489			0.449225			0.224613	−	0.974448i	$-0.427888\pi$
$883$	13.3489			0.449225			0.224613	+	0.974448i	$0.427888\pi$
$884$		0			0
$885$	−13.0188			−0.437622
$886$		0			0
$887$	−4.46449			−0.149903			−0.0749515	−	0.997187i	$-0.523880\pi$
$887$	−4.46449			−0.149903			−0.0749515	+	0.997187i	$0.523880\pi$
$888$		0			0
$889$	0.967153	−	2.25272i	0.0324373	−	0.0755539i
$890$		0			0
$891$	−5.54165			−0.185652
$892$		0			0
$893$			22.1963i			0.742771i
$894$		0			0
$895$	−14.4927			−0.484438
$896$		0			0
$897$	5.95767			0.198921
$898$		0			0
$899$			61.8645i			2.06330i
$900$		0			0
$901$	6.96852			0.232155
$902$		0			0
$903$	−9.06409	+	21.1123i	−0.301634	+	0.702575i
$904$		0			0
$905$	−1.28496			−0.0427135
$906$		0			0
$907$	−17.0639			−0.566598			−0.283299	−	0.959032i	$-0.591429\pi$
$907$	−17.0639			−0.566598			−0.283299	+	0.959032i	$0.591429\pi$
$908$		0			0
$909$	15.0071			0.497753
$910$		0			0
$911$		−	0.534170i		−	0.0176978i	−0.999961	−	0.00884891i	$-0.997183\pi$
$911$		−	0.534170i		−	0.0176978i	0.999961	−	0.00884891i	$-0.00281673\pi$
$912$		0			0
$913$			50.6600i			1.67660i
$914$		0			0
$915$			2.49724i			0.0825562i
$916$		0			0
$917$	−13.0432	+	30.3806i	−0.430725	+	1.00326i
$918$		0			0
$919$			7.87145i			0.259655i	0.991537	+	0.129828i	$0.0414424\pi$
$919$			7.87145i			0.259655i	−0.991537	+	0.129828i	$0.958558\pi$
$920$		0			0
$921$	2.39283			0.0788465
$922$		0			0
$923$		−	11.3346i		−	0.373081i
$924$		0			0
$925$			11.2477i			0.369824i
$926$		0			0
$927$	−1.31225			−0.0431001
$928$		0			0
$929$			29.4442i			0.966034i	0.875611	+	0.483017i	$0.160459\pi$
$929$			29.4442i			0.966034i	−0.875611	+	0.483017i	$0.839541\pi$
$930$		0			0
$931$	8.80420	−	8.36356i	0.288546	−	0.274105i
$932$		0			0
$933$			24.3201i			0.796204i
$934$		0			0
$935$		−	30.4870i		−	0.997032i
$936$		0			0
$937$		−	47.0734i		−	1.53782i	−0.639357	−	0.768910i	$-0.720799\pi$
$937$		−	47.0734i		−	1.53782i	0.639357	−	0.768910i	$-0.279201\pi$
$938$		0			0
$939$	−30.3185			−0.989407
$940$		0			0
$941$	7.55709			0.246354			0.123177	−	0.992385i	$-0.460692\pi$
$941$	7.55709			0.246354			0.123177	+	0.992385i	$0.460692\pi$
$942$		0			0
$943$	−0.316763			−0.0103152
$944$		0			0
$945$	−1.04376	+	2.43116i	−0.0339536	+	0.0790858i
$946$		0			0
$947$	35.7888			1.16298			0.581490	−	0.813554i	$-0.302470\pi$
$947$	35.7888			1.16298			0.581490	+	0.813554i	$0.302470\pi$
$948$		0			0
$949$			12.3371i			0.400479i
$950$		0			0
$951$	7.93499			0.257310
$952$		0			0
$953$	−0.983354			−0.0318540			−0.0159270	−	0.999873i	$-0.505070\pi$
$953$	−0.983354			−0.0318540			−0.0159270	+	0.999873i	$0.505070\pi$
$954$		0			0
$955$		−	9.06371i		−	0.293295i
$956$		0			0
$957$	46.8879			1.51567
$958$		0			0
$959$	8.58831	+	3.68719i	0.277331	+	0.119066i
$960$		0			0
$961$	22.4613			0.724557
$962$		0			0
$963$	−16.7713			−0.540448
$964$		0			0
$965$	17.5065			0.563554
$966$		0			0
$967$		−	18.3924i		−	0.591459i	−0.955272	−	0.295730i	$-0.904437\pi$
$967$		−	18.3924i		−	0.591459i	0.955272	−	0.295730i	$-0.0955627\pi$
$968$		0			0
$969$		−	9.54375i		−	0.306590i
$970$		0			0
$971$			1.49104i			0.0478497i	0.999714	+	0.0239248i	$0.00761624\pi$
$971$			1.49104i			0.0478497i	−0.999714	+	0.0239248i	$0.992384\pi$
$972$		0			0
$973$	−12.6319	+	29.4225i	−0.404959	+	0.943243i
$974$		0			0
$975$			1.67913i			0.0537752i
$976$		0			0
$977$	−24.8612			−0.795382			−0.397691	−	0.917519i	$-0.630188\pi$
$977$	−24.8612			−0.795382			−0.397691	+	0.917519i	$0.630188\pi$
$978$		0			0
$979$			23.0873i			0.737873i
$980$		0			0
$981$		−	2.48432i		−	0.0793183i
$982$		0			0
$983$	33.5242			1.06925			0.534627	−	0.845088i	$-0.320452\pi$
$983$	33.5242			1.06925			0.534627	+	0.845088i	$0.320452\pi$
$984$		0			0
$985$			13.1628i			0.419400i
$986$		0			0
$987$	−13.3549	+	31.1065i	−0.425090	+	0.990131i
$988$		0			0
$989$			30.8116i			0.979751i
$990$		0			0
$991$			32.3140i			1.02649i	0.858243	+	0.513244i	$0.171557\pi$
$991$			32.3140i			1.02649i	−0.858243	+	0.513244i	$0.828443\pi$
$992$		0			0
$993$		−	14.2122i		−	0.451010i
$994$		0			0
$995$	0.276416			0.00876299
$996$		0			0
$997$	−2.00398			−0.0634665			−0.0317333	−	0.999496i	$-0.510103\pi$
$997$	−2.00398			−0.0634665			−0.0317333	+	0.999496i	$0.510103\pi$
$998$		0			0
$999$	−11.2477			−0.355863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.z.d.1231.25		28
4.3	odd	2		840.2.z.d.811.11	yes	28
7.6	odd	2		3360.2.z.c.1231.4		28
8.3	odd	2		3360.2.z.c.1231.3		28
8.5	even	2		840.2.z.c.811.12	yes	28
28.27	even	2		840.2.z.c.811.11	✓	28
56.13	odd	2		840.2.z.d.811.12	yes	28
56.27	even	2	inner	3360.2.z.d.1231.26		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.z.c.811.11	✓	28	28.27	even	2
840.2.z.c.811.12	yes	28	8.5	even	2
840.2.z.d.811.11	yes	28	4.3	odd	2
840.2.z.d.811.12	yes	28	56.13	odd	2
3360.2.z.c.1231.3		28	8.3	odd	2
3360.2.z.c.1231.4		28	7.6	odd	2
3360.2.z.d.1231.25		28	1.1	even	1	trivial
3360.2.z.d.1231.26		28	56.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.z (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000 q^{5} +(2.43116 + 1.04376i) q^{7} -1.00000 q^{9} -5.54165 q^{11} -1.67913 q^{13} -1.00000i q^{15} +5.50143i q^{17} -1.73478i q^{19} +(1.04376 - 2.43116i) q^{21} -3.54807i q^{23} +1.00000 q^{25} +1.00000i q^{27} -8.46101i q^{29} -7.31172 q^{31} +5.54165i q^{33} +(2.43116 + 1.04376i) q^{35} +11.2477i q^{37} +1.67913i q^{39} -0.0892775i q^{41} -8.68405 q^{43} -1.00000 q^{45} -12.7949 q^{47} +(4.82112 + 5.07512i) q^{49} +5.50143 q^{51} -1.26667i q^{53} -5.54165 q^{55} -1.73478 q^{57} -13.0188i q^{59} -2.49724 q^{61} +(-2.43116 - 1.04376i) q^{63} -1.67913 q^{65} -7.26352 q^{67} -3.54807 q^{69} +6.75026i q^{71} -7.34731i q^{73} -1.00000i q^{75} +(-13.4727 - 5.78417i) q^{77} +1.15000i q^{79} +1.00000 q^{81} -9.14169i q^{83} +5.50143i q^{85} -8.46101 q^{87} -4.16614i q^{89} +(-4.08224 - 1.75261i) q^{91} +7.31172i q^{93} -1.73478i q^{95} +4.20427i q^{97} +5.54165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 28 q^{5} - 4 q^{7} - 28 q^{9} - 8 q^{13} + 28 q^{25} - 24 q^{31} - 4 q^{35} - 24 q^{43} - 28 q^{45} + 24 q^{47} + 28 q^{49} + 8 q^{57} - 16 q^{61} + 4 q^{63} - 8 q^{65} + 8 q^{67} - 24 q^{69} + 16 q^{77}+ \cdots - 48 q^{91}+O(q^{100}) \) 28 * q + 28 * q^5 - 4 * q^7 - 28 * q^9 - 8 * q^13 + 28 * q^25 - 24 * q^31 - 4 * q^35 - 24 * q^43 - 28 * q^45 + 24 * q^47 + 28 * q^49 + 8 * q^57 - 16 * q^61 + 4 * q^63 - 8 * q^65 + 8 * q^67 - 24 * q^69 + 16 * q^77 + 28 * q^81 - 16 * q^87 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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