LMFDB - Embedded newform 4560.2.d.k.2431.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.d (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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} + \cdots + 64 $ x^12 - 4x^11 + 8x^10 + 4x^9 + 36x^8 - 128x^7 + 232x^6 + 104x^5 + 324x^4 - 784x^3 + 800x^2 - 320*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.4
Root			$0.280289 - 0.280289i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.k.2431.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.20519i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -2.20519i q^{7} +1.00000 q^{9} +4.04806i q^{11} +1.23424i q^{13} -1.00000 q^{15} -4.40345 q^{17} +(3.61178 + 2.44030i) q^{19} -2.20519i q^{21} -9.21663i q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.22295i q^{29} -7.30698 q^{31} +4.04806i q^{33} +2.20519i q^{35} -3.55426i q^{37} +1.23424i q^{39} -4.31329i q^{41} +7.01837i q^{43} -1.00000 q^{45} -10.4035i q^{47} +2.13716 q^{49} -4.40345 q^{51} +7.55958i q^{53} -4.04806i q^{55} +(3.61178 + 2.44030i) q^{57} -0.470233 q^{59} +0.970304 q^{61} -2.20519i q^{63} -1.23424i q^{65} -13.8255 q^{67} -9.21663i q^{69} +15.6510 q^{71} +0.320027 q^{73} +1.00000 q^{75} +8.92673 q^{77} +11.4452 q^{79} +1.00000 q^{81} -9.32538i q^{83} +4.40345 q^{85} -8.22295i q^{87} -4.17986i q^{89} +2.72172 q^{91} -7.30698 q^{93} +(-3.61178 - 2.44030i) q^{95} -11.6504i q^{97} +4.04806i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 - 12 * q^5 + 12 * q^9	$ 12 q + 12 q^{3} - 12 q^{5} + 12 q^{9} - 12 q^{15} - 8 q^{17} + 8 q^{19} + 12 q^{25} + 12 q^{27} - 12 q^{45} - 20 q^{49} - 8 q^{51} + 8 q^{57} + 8 q^{59} - 8 q^{67} + 32 q^{71} - 24 q^{73} + 12 q^{75} + 56 q^{77} + 16 q^{79} + 12 q^{81} + 8 q^{85} - 16 q^{91} - 8 q^{95}+O(q^{100}) $ 12 * q + 12 * q^3 - 12 * q^5 + 12 * q^9 - 12 * q^15 - 8 * q^17 + 8 * q^19 + 12 * q^25 + 12 * q^27 - 12 * q^45 - 20 * q^49 - 8 * q^51 + 8 * q^57 + 8 * q^59 - 8 * q^67 + 32 * q^71 - 24 * q^73 + 12 * q^75 + 56 * q^77 + 16 * q^79 + 12 * q^81 + 8 * q^85 - 16 * q^91 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	2.20519i		−	0.833482i	−0.909025	−	0.416741i	$-0.863172\pi$
$7$		−	2.20519i		−	0.833482i	0.909025	−	0.416741i	$-0.136828\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			4.04806i			1.22054i	0.792195	+	0.610268i	$0.208938\pi$
$11$			4.04806i			1.22054i	−0.792195	+	0.610268i	$0.791062\pi$
$12$		0			0
$13$			1.23424i			0.342316i	0.985244	+	0.171158i	$0.0547508\pi$
$13$			1.23424i			0.342316i	−0.985244	+	0.171158i	$0.945249\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−4.40345			−1.06799			−0.533997	−	0.845486i	$-0.679311\pi$
$17$	−4.40345			−1.06799			−0.533997	+	0.845486i	$0.679311\pi$
$18$		0			0
$19$	3.61178	+	2.44030i	0.828598	+	0.559844i
$19$	3.61178	+	2.44030i	0.828598	+	0.559844i
$20$		0			0
$21$		−	2.20519i		−	0.481211i
$22$		0			0
$23$		−	9.21663i		−	1.92180i	−0.276894	−	0.960901i	$-0.589305\pi$
$23$		−	9.21663i		−	1.92180i	0.276894	−	0.960901i	$-0.410695\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	8.22295i		−	1.52696i	−0.645830	−	0.763481i	$-0.723488\pi$
$29$		−	8.22295i		−	1.52696i	0.645830	−	0.763481i	$-0.276512\pi$
$30$		0			0
$31$	−7.30698			−1.31237			−0.656186	−	0.754599i	$-0.727831\pi$
$31$	−7.30698			−1.31237			−0.656186	+	0.754599i	$0.727831\pi$
$32$		0			0
$33$			4.04806i			0.704677i
$34$		0			0
$35$			2.20519i			0.372744i
$36$		0			0
$37$		−	3.55426i		−	0.584318i	−0.956370	−	0.292159i	$-0.905626\pi$
$37$		−	3.55426i		−	0.584318i	0.956370	−	0.292159i	$-0.0943736\pi$
$38$		0			0
$39$			1.23424i			0.197636i
$40$		0			0
$41$		−	4.31329i		−	0.673623i	−0.941572	−	0.336811i	$-0.890652\pi$
$41$		−	4.31329i		−	0.673623i	0.941572	−	0.336811i	$-0.109348\pi$
$42$		0			0
$43$			7.01837i			1.07029i	0.844760	+	0.535145i	$0.179743\pi$
$43$			7.01837i			1.07029i	−0.844760	+	0.535145i	$0.820257\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$		−	10.4035i		−	1.51750i	−0.651382	−	0.758750i	$-0.725810\pi$
$47$		−	10.4035i		−	1.51750i	0.651382	−	0.758750i	$-0.274190\pi$
$48$		0			0
$49$	2.13716			0.305308
$50$		0			0
$51$	−4.40345			−0.616607
$52$		0			0
$53$			7.55958i			1.03839i	0.854656	+	0.519194i	$0.173768\pi$
$53$			7.55958i			1.03839i	−0.854656	+	0.519194i	$0.826232\pi$
$54$		0			0
$55$		−	4.04806i		−	0.545841i
$56$		0			0
$57$	3.61178	+	2.44030i	0.478391	+	0.323226i
$58$		0			0
$59$	−0.470233			−0.0612192			−0.0306096	−	0.999531i	$-0.509745\pi$
$59$	−0.470233			−0.0612192			−0.0306096	+	0.999531i	$0.509745\pi$
$60$		0			0
$61$	0.970304			0.124235			0.0621174	−	0.998069i	$-0.480215\pi$
$61$	0.970304			0.124235			0.0621174	+	0.998069i	$0.480215\pi$
$62$		0			0
$63$		−	2.20519i		−	0.277827i
$64$		0			0
$65$		−	1.23424i		−	0.153088i
$66$		0			0
$67$	−13.8255			−1.68905			−0.844524	−	0.535518i	$-0.820117\pi$
$67$	−13.8255			−1.68905			−0.844524	+	0.535518i	$0.820117\pi$
$68$		0			0
$69$		−	9.21663i		−	1.10955i
$70$		0			0
$71$	15.6510			1.85744			0.928718	−	0.370786i	$-0.120912\pi$
$71$	15.6510			1.85744			0.928718	+	0.370786i	$0.120912\pi$
$72$		0			0
$73$	0.320027			0.0374564			0.0187282	−	0.999825i	$-0.494038\pi$
$73$	0.320027			0.0374564			0.0187282	+	0.999825i	$0.494038\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	8.92673			1.01730
$78$		0			0
$79$	11.4452			1.28769			0.643843	−	0.765158i	$-0.277339\pi$
$79$	11.4452			1.28769			0.643843	+	0.765158i	$0.277339\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	9.32538i		−	1.02359i	−0.859106	−	0.511797i	$-0.828980\pi$
$83$		−	9.32538i		−	1.02359i	0.859106	−	0.511797i	$-0.171020\pi$
$84$		0			0
$85$	4.40345			0.477622
$86$		0			0
$87$		−	8.22295i		−	0.881592i
$88$		0			0
$89$		−	4.17986i		−	0.443065i	−0.975153	−	0.221532i	$-0.928894\pi$
$89$		−	4.17986i		−	0.443065i	0.975153	−	0.221532i	$-0.0711058\pi$
$90$		0			0
$91$	2.72172			0.285314
$92$		0			0
$93$	−7.30698			−0.757698
$94$		0			0
$95$	−3.61178	−	2.44030i	−0.370560	−	0.250370i
$96$		0			0
$97$		−	11.6504i		−	1.18292i	−0.806335	−	0.591459i	$-0.798552\pi$
$97$		−	11.6504i		−	1.18292i	0.806335	−	0.591459i	$-0.201448\pi$
$98$		0			0
$99$			4.04806i			0.406846i
$100$		0			0
$101$	14.9156			1.48416			0.742078	−	0.670314i	$-0.233840\pi$
$101$	14.9156			1.48416			0.742078	+	0.670314i	$0.233840\pi$
$102$		0			0
$103$	−14.4655			−1.42533			−0.712664	−	0.701505i	$-0.752512\pi$
$103$	−14.4655			−1.42533			−0.712664	+	0.701505i	$0.752512\pi$
$104$		0			0
$105$			2.20519i			0.215204i
$106$		0			0
$107$	14.4989			1.40166			0.700832	−	0.713326i	$-0.252812\pi$
$107$	14.4989			1.40166			0.700832	+	0.713326i	$0.252812\pi$
$108$		0			0
$109$			5.28922i			0.506615i	0.967386	+	0.253308i	$0.0815185\pi$
$109$			5.28922i			0.506615i	−0.967386	+	0.253308i	$0.918482\pi$
$110$		0			0
$111$		−	3.55426i		−	0.337356i
$112$		0			0
$113$		−	13.8980i		−	1.30742i	−0.756747	−	0.653708i	$-0.773213\pi$
$113$		−	13.8980i		−	1.30742i	0.756747	−	0.653708i	$-0.226787\pi$
$114$		0			0
$115$			9.21663i			0.859456i
$116$		0			0
$117$			1.23424i			0.114105i
$118$		0			0
$119$			9.71043i			0.890154i
$120$		0			0
$121$	−5.38681			−0.489710
$122$		0			0
$123$		−	4.31329i		−	0.388916i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−14.3384			−1.27233			−0.636165	−	0.771553i	$-0.719480\pi$
$127$	−14.3384			−1.27233			−0.636165	+	0.771553i	$0.719480\pi$
$128$		0			0
$129$			7.01837i			0.617933i
$130$		0			0
$131$		−	17.6624i		−	1.54317i	−0.636127	−	0.771584i	$-0.719465\pi$
$131$		−	17.6624i		−	1.54317i	0.636127	−	0.771584i	$-0.280535\pi$
$132$		0			0
$133$	5.38132	−	7.96464i	0.466620	−	0.690622i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	−14.6157			−1.24870			−0.624352	−	0.781143i	$-0.714637\pi$
$137$	−14.6157			−1.24870			−0.624352	+	0.781143i	$0.714637\pi$
$138$		0			0
$139$		−	7.86064i		−	0.666731i	−0.942798	−	0.333365i	$-0.891816\pi$
$139$		−	7.86064i		−	0.666731i	0.942798	−	0.333365i	$-0.108184\pi$
$140$		0			0
$141$		−	10.4035i		−	0.876129i
$142$		0			0
$143$	−4.99627			−0.417809
$144$		0			0
$145$			8.22295i			0.682879i
$146$		0			0
$147$	2.13716			0.176270
$148$		0			0
$149$	17.5175			1.43509			0.717544	−	0.696513i	$-0.245266\pi$
$149$	17.5175			1.43509			0.717544	+	0.696513i	$0.245266\pi$
$150$		0			0
$151$	−15.3374			−1.24814			−0.624071	−	0.781367i	$-0.714523\pi$
$151$	−15.3374			−1.24814			−0.624071	+	0.781367i	$0.714523\pi$
$152$		0			0
$153$	−4.40345			−0.355998
$154$		0			0
$155$	7.30698			0.586911
$156$		0			0
$157$	−10.9126			−0.870921			−0.435461	−	0.900208i	$-0.643414\pi$
$157$	−10.9126			−0.870921			−0.435461	+	0.900208i	$0.643414\pi$
$158$		0			0
$159$			7.55958i			0.599514i
$160$		0			0
$161$	−20.3244			−1.60179
$162$		0			0
$163$		−	15.9737i		−	1.25116i	−0.780161	−	0.625579i	$-0.784863\pi$
$163$		−	15.9737i		−	1.25116i	0.780161	−	0.625579i	$-0.215137\pi$
$164$		0			0
$165$		−	4.04806i		−	0.315141i
$166$		0			0
$167$	−0.356216			−0.0275648			−0.0137824	−	0.999905i	$-0.504387\pi$
$167$	−0.356216			−0.0275648			−0.0137824	+	0.999905i	$0.504387\pi$
$168$		0			0
$169$	11.4767			0.882820
$170$		0			0
$171$	3.61178	+	2.44030i	0.276199	+	0.186615i
$172$		0			0
$173$			4.81713i			0.366240i	0.983091	+	0.183120i	$0.0586196\pi$
$173$			4.81713i			0.366240i	−0.983091	+	0.183120i	$0.941380\pi$
$174$		0			0
$175$		−	2.20519i		−	0.166696i
$176$		0			0
$177$	−0.470233			−0.0353449
$178$		0			0
$179$	−22.3596			−1.67123			−0.835616	−	0.549313i	$-0.814889\pi$
$179$	−22.3596			−1.67123			−0.835616	+	0.549313i	$0.814889\pi$
$180$		0			0
$181$			9.96763i			0.740888i	0.928855	+	0.370444i	$0.120794\pi$
$181$			9.96763i			0.740888i	−0.928855	+	0.370444i	$0.879206\pi$
$182$		0			0
$183$	0.970304			0.0717270
$184$		0			0
$185$			3.55426i			0.261315i
$186$		0			0
$187$		−	17.8255i		−	1.30353i
$188$		0			0
$189$		−	2.20519i		−	0.160404i
$190$		0			0
$191$		−	22.1221i		−	1.60070i	−0.599534	−	0.800349i	$-0.704647\pi$
$191$		−	22.1221i		−	1.60070i	0.599534	−	0.800349i	$-0.295353\pi$
$192$		0			0
$193$			4.52803i			0.325934i	0.986631	+	0.162967i	$0.0521065\pi$
$193$			4.52803i			0.325934i	−0.986631	+	0.162967i	$0.947894\pi$
$194$		0			0
$195$		−	1.23424i		−	0.0883856i
$196$		0			0
$197$	−8.46460			−0.603078			−0.301539	−	0.953454i	$-0.597500\pi$
$197$	−8.46460			−0.603078			−0.301539	+	0.953454i	$0.597500\pi$
$198$		0			0
$199$		−	7.21423i		−	0.511403i	−0.966756	−	0.255702i	$-0.917694\pi$
$199$		−	7.21423i		−	0.511403i	0.966756	−	0.255702i	$-0.0823065\pi$
$200$		0			0
$201$	−13.8255			−0.975172
$202$		0			0
$203$	−18.1331			−1.27270
$204$		0			0
$205$			4.31329i			0.301253i
$206$		0			0
$207$		−	9.21663i		−	0.640600i
$208$		0			0
$209$	−9.87849	+	14.6207i	−0.683310	+	1.01133i
$210$		0			0
$211$	−14.9776			−1.03110			−0.515549	−	0.856860i	$-0.672412\pi$
$211$	−14.9776			−1.03110			−0.515549	+	0.856860i	$0.672412\pi$
$212$		0			0
$213$	15.6510			1.07239
$214$		0			0
$215$		−	7.01837i		−	0.478649i
$216$		0			0
$217$			16.1132i			1.09384i
$218$		0			0
$219$	0.320027			0.0216254
$220$		0			0
$221$		−	5.43491i		−	0.365591i
$222$		0			0
$223$	−9.24952			−0.619394			−0.309697	−	0.950835i	$-0.600228\pi$
$223$	−9.24952			−0.619394			−0.309697	+	0.950835i	$0.600228\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−23.8867			−1.58541			−0.792707	−	0.609603i	$-0.791329\pi$
$227$	−23.8867			−1.58541			−0.792707	+	0.609603i	$0.791329\pi$
$228$		0			0
$229$	12.4239			0.820994			0.410497	−	0.911862i	$-0.365355\pi$
$229$	12.4239			0.820994			0.410497	+	0.911862i	$0.365355\pi$
$230$		0			0
$231$	8.92673			0.587336
$232$		0			0
$233$	29.1146			1.90736			0.953682	−	0.300816i	$-0.0972592\pi$
$233$	29.1146			1.90736			0.953682	+	0.300816i	$0.0972592\pi$
$234$		0			0
$235$			10.4035i			0.678646i
$236$		0			0
$237$	11.4452			0.743446
$238$		0			0
$239$		−	3.27796i		−	0.212034i	−0.994364	−	0.106017i	$-0.966190\pi$
$239$		−	3.27796i		−	0.212034i	0.994364	−	0.106017i	$-0.0338098\pi$
$240$		0			0
$241$			1.28166i			0.0825587i	0.999148	+	0.0412794i	$0.0131434\pi$
$241$			1.28166i			0.0825587i	−0.999148	+	0.0412794i	$0.986857\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−2.13716			−0.136538
$246$		0			0
$247$	−3.01191	+	4.45779i	−0.191643	+	0.283642i
$248$		0			0
$249$		−	9.32538i		−	0.590972i
$250$		0			0
$251$			0.323718i			0.0204329i	0.999948	+	0.0102165i	$0.00325206\pi$
$251$			0.323718i			0.0204329i	−0.999948	+	0.0102165i	$0.996748\pi$
$252$		0			0
$253$	37.3095			2.34563
$254$		0			0
$255$	4.40345			0.275755
$256$		0			0
$257$		−	11.3497i		−	0.707976i	−0.935250	−	0.353988i	$-0.884825\pi$
$257$		−	11.3497i		−	0.707976i	0.935250	−	0.353988i	$-0.115175\pi$
$258$		0			0
$259$	−7.83781			−0.487018
$260$		0			0
$261$		−	8.22295i		−	0.508988i
$262$		0			0
$263$			11.2534i			0.693912i	0.937881	+	0.346956i	$0.112785\pi$
$263$			11.2534i			0.693912i	−0.937881	+	0.346956i	$0.887215\pi$
$264$		0			0
$265$		−	7.55958i		−	0.464381i
$266$		0			0
$267$		−	4.17986i		−	0.255803i
$268$		0			0
$269$		−	15.0809i		−	0.919500i	−0.888048	−	0.459750i	$-0.847939\pi$
$269$		−	15.0809i		−	0.919500i	0.888048	−	0.459750i	$-0.152061\pi$
$270$		0			0
$271$		−	30.9368i		−	1.87928i	−0.342169	−	0.939638i	$-0.611162\pi$
$271$		−	30.9368i		−	1.87928i	0.342169	−	0.939638i	$-0.388838\pi$
$272$		0			0
$273$	2.72172			0.164726
$274$		0			0
$275$			4.04806i			0.244107i
$276$		0			0
$277$	9.96077			0.598485			0.299242	−	0.954177i	$-0.403266\pi$
$277$	9.96077			0.598485			0.299242	+	0.954177i	$0.403266\pi$
$278$		0			0
$279$	−7.30698			−0.437457
$280$		0			0
$281$			3.42360i			0.204235i	0.994772	+	0.102117i	$0.0325617\pi$
$281$			3.42360i			0.204235i	−0.994772	+	0.102117i	$0.967438\pi$
$282$		0			0
$283$			6.15914i			0.366123i	0.983102	+	0.183061i	$0.0586007\pi$
$283$			6.15914i			0.366123i	−0.983102	+	0.183061i	$0.941399\pi$
$284$		0			0
$285$	−3.61178	−	2.44030i	−0.213943	−	0.144551i
$286$		0			0
$287$	−9.51161			−0.561452
$288$		0			0
$289$	2.39040			0.140612
$290$		0			0
$291$		−	11.6504i		−	0.682958i
$292$		0			0
$293$		−	24.7112i		−	1.44364i	−0.692079	−	0.721822i	$-0.743305\pi$
$293$		−	24.7112i		−	1.44364i	0.692079	−	0.721822i	$-0.256695\pi$
$294$		0			0
$295$	0.470233			0.0273780
$296$		0			0
$297$			4.04806i			0.234892i
$298$		0			0
$299$	11.3755			0.657863
$300$		0			0
$301$	15.4768			0.892068
$302$		0			0
$303$	14.9156			0.856878
$304$		0			0
$305$	−0.970304			−0.0555595
$306$		0			0
$307$	7.93681			0.452978			0.226489	−	0.974014i	$-0.427275\pi$
$307$	7.93681			0.452978			0.226489	+	0.974014i	$0.427275\pi$
$308$		0			0
$309$	−14.4655			−0.822914
$310$		0			0
$311$		−	0.578158i		−	0.0327843i	−0.999866	−	0.0163922i	$-0.994782\pi$
$311$		−	0.578158i		−	0.0327843i	0.999866	−	0.0163922i	$-0.00521802\pi$
$312$		0			0
$313$	7.18892			0.406342			0.203171	−	0.979143i	$-0.434875\pi$
$313$	7.18892			0.406342			0.203171	+	0.979143i	$0.434875\pi$
$314$		0			0
$315$			2.20519i			0.124248i
$316$		0			0
$317$		−	1.24733i		−	0.0700568i	−0.999386	−	0.0350284i	$-0.988848\pi$
$317$		−	1.24733i		−	0.0700568i	0.999386	−	0.0350284i	$-0.0111522\pi$
$318$		0			0
$319$	33.2870			1.86371
$320$		0			0
$321$	14.4989			0.809252
$322$		0			0
$323$	−15.9043	−	10.7458i	−0.884938	−	0.597910i
$324$		0			0
$325$			1.23424i			0.0684632i
$326$		0			0
$327$			5.28922i			0.292494i
$328$		0			0
$329$	−22.9415			−1.26481
$330$		0			0
$331$	−12.4754			−0.685713			−0.342856	−	0.939388i	$-0.611394\pi$
$331$	−12.4754			−0.685713			−0.342856	+	0.939388i	$0.611394\pi$
$332$		0			0
$333$		−	3.55426i		−	0.194773i
$334$		0			0
$335$	13.8255			0.755365
$336$		0			0
$337$			20.5603i			1.11999i	0.828496	+	0.559994i	$0.189197\pi$
$337$			20.5603i			1.11999i	−0.828496	+	0.559994i	$0.810803\pi$
$338$		0			0
$339$		−	13.8980i		−	0.754837i
$340$		0			0
$341$		−	29.5791i		−	1.60180i
$342$		0			0
$343$		−	20.1491i		−	1.08795i
$344$		0			0
$345$			9.21663i			0.496207i
$346$		0			0
$347$			3.00585i			0.161362i	0.996740	+	0.0806812i	$0.0257096\pi$
$347$			3.00585i			0.161362i	−0.996740	+	0.0806812i	$0.974290\pi$
$348$		0			0
$349$	15.9230			0.852339			0.426169	−	0.904643i	$-0.359863\pi$
$349$	15.9230			0.852339			0.426169	+	0.904643i	$0.359863\pi$
$350$		0			0
$351$			1.23424i			0.0658787i
$352$		0			0
$353$	26.4303			1.40674			0.703372	−	0.710822i	$-0.251677\pi$
$353$	26.4303			1.40674			0.703372	+	0.710822i	$0.251677\pi$
$354$		0			0
$355$	−15.6510			−0.830671
$356$		0			0
$357$			9.71043i			0.513931i
$358$		0			0
$359$			26.5186i			1.39960i	0.714339	+	0.699800i	$0.246728\pi$
$359$			26.5186i			1.39960i	−0.714339	+	0.699800i	$0.753272\pi$
$360$		0			0
$361$	7.08985	+	17.6276i	0.373150	+	0.927771i
$362$		0			0
$363$	−5.38681			−0.282734
$364$		0			0
$365$	−0.320027			−0.0167510
$366$		0			0
$367$			25.3776i			1.32470i	0.749195	+	0.662350i	$0.230441\pi$
$367$			25.3776i			1.32470i	−0.749195	+	0.662350i	$0.769559\pi$
$368$		0			0
$369$		−	4.31329i		−	0.224541i
$370$		0			0
$371$	16.6703			0.865478
$372$		0			0
$373$			17.9214i			0.927936i	0.885852	+	0.463968i	$0.153575\pi$
$373$			17.9214i			0.927936i	−0.885852	+	0.463968i	$0.846425\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	10.1491			0.522704
$378$		0			0
$379$	25.3455			1.30191			0.650957	−	0.759115i	$-0.274368\pi$
$379$	25.3455			1.30191			0.650957	+	0.759115i	$0.274368\pi$
$380$		0			0
$381$	−14.3384			−0.734580
$382$		0			0
$383$	−22.0924			−1.12887			−0.564436	−	0.825477i	$-0.690906\pi$
$383$	−22.0924			−1.12887			−0.564436	+	0.825477i	$0.690906\pi$
$384$		0			0
$385$	−8.92673			−0.454948
$386$		0			0
$387$			7.01837i			0.356764i
$388$		0			0
$389$	−24.2984			−1.23198			−0.615989	−	0.787754i	$-0.711244\pi$
$389$	−24.2984			−1.23198			−0.615989	+	0.787754i	$0.711244\pi$
$390$		0			0
$391$			40.5850i			2.05247i
$392$		0			0
$393$		−	17.6624i		−	0.890949i
$394$		0			0
$395$	−11.4452			−0.575871
$396$		0			0
$397$	−4.02883			−0.202201			−0.101101	−	0.994876i	$-0.532236\pi$
$397$	−4.02883			−0.202201			−0.101101	+	0.994876i	$0.532236\pi$
$398$		0			0
$399$	5.38132	−	7.96464i	0.269403	−	0.398731i
$400$		0			0
$401$		−	6.14264i		−	0.306749i	−0.988168	−	0.153374i	$-0.950986\pi$
$401$		−	6.14264i		−	0.306749i	0.988168	−	0.153374i	$-0.0490140\pi$
$402$		0			0
$403$		−	9.01855i		−	0.449246i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	14.3879			0.713181
$408$		0			0
$409$			31.1883i			1.54216i	0.636736	+	0.771082i	$0.280284\pi$
$409$			31.1883i			1.54216i	−0.636736	+	0.771082i	$0.719716\pi$
$410$		0			0
$411$	−14.6157			−0.720940
$412$		0			0
$413$			1.03695i			0.0510251i
$414$		0			0
$415$			9.32538i			0.457765i
$416$		0			0
$417$		−	7.86064i		−	0.384937i
$418$		0			0
$419$		−	0.977475i		−	0.0477528i	−0.999715	−	0.0238764i	$-0.992399\pi$
$419$		−	0.977475i		−	0.0477528i	0.999715	−	0.0238764i	$-0.00760081\pi$
$420$		0			0
$421$		−	37.8086i		−	1.84268i	−0.388761	−	0.921339i	$-0.627097\pi$
$421$		−	37.8086i		−	1.84268i	0.388761	−	0.921339i	$-0.372903\pi$
$422$		0			0
$423$		−	10.4035i		−	0.505833i
$424$		0			0
$425$	−4.40345			−0.213599
$426$		0			0
$427$		−	2.13970i		−	0.103547i
$428$		0			0
$429$	−4.99627			−0.241222
$430$		0			0
$431$	−22.8298			−1.09967			−0.549837	−	0.835272i	$-0.685310\pi$
$431$	−22.8298			−1.09967			−0.549837	+	0.835272i	$0.685310\pi$
$432$		0			0
$433$			36.0259i			1.73129i	0.500654	+	0.865647i	$0.333093\pi$
$433$			36.0259i			1.73129i	−0.500654	+	0.865647i	$0.666907\pi$
$434$		0			0
$435$			8.22295i			0.394260i
$436$		0			0
$437$	22.4914	−	33.2884i	1.07591	−	1.59240i
$438$		0			0
$439$	−23.8206			−1.13690			−0.568448	−	0.822719i	$-0.692456\pi$
$439$	−23.8206			−1.13690			−0.568448	+	0.822719i	$0.692456\pi$
$440$		0			0
$441$	2.13716			0.101769
$442$		0			0
$443$			17.7105i			0.841452i	0.907188	+	0.420726i	$0.138225\pi$
$443$			17.7105i			0.841452i	−0.907188	+	0.420726i	$0.861775\pi$
$444$		0			0
$445$			4.17986i			0.198145i
$446$		0			0
$447$	17.5175			0.828548
$448$		0			0
$449$			3.21702i			0.151821i	0.997115	+	0.0759104i	$0.0241863\pi$
$449$			3.21702i			0.151821i	−0.997115	+	0.0759104i	$0.975814\pi$
$450$		0			0
$451$	17.4605			0.822181
$452$		0			0
$453$	−15.3374			−0.720616
$454$		0			0
$455$	−2.72172			−0.127596
$456$		0			0
$457$	3.93491			0.184067			0.0920336	−	0.995756i	$-0.470663\pi$
$457$	3.93491			0.184067			0.0920336	+	0.995756i	$0.470663\pi$
$458$		0			0
$459$	−4.40345			−0.205536
$460$		0			0
$461$	29.0719			1.35401			0.677007	−	0.735977i	$-0.263277\pi$
$461$	29.0719			1.35401			0.677007	+	0.735977i	$0.263277\pi$
$462$		0			0
$463$			3.13917i			0.145890i	0.997336	+	0.0729448i	$0.0232397\pi$
$463$			3.13917i			0.145890i	−0.997336	+	0.0729448i	$0.976760\pi$
$464$		0			0
$465$	7.30698			0.338853
$466$		0			0
$467$		−	15.0274i		−	0.695384i	−0.937609	−	0.347692i	$-0.886966\pi$
$467$		−	15.0274i		−	0.695384i	0.937609	−	0.347692i	$-0.113034\pi$
$468$		0			0
$469$			30.4877i			1.40779i
$470$		0			0
$471$	−10.9126			−0.502827
$472$		0			0
$473$	−28.4108			−1.30633
$474$		0			0
$475$	3.61178	+	2.44030i	0.165720	+	0.111969i
$476$		0			0
$477$			7.55958i			0.346129i
$478$		0			0
$479$		−	0.457146i		−	0.0208875i	−0.999945	−	0.0104438i	$-0.996676\pi$
$479$		−	0.457146i		−	0.0208875i	0.999945	−	0.0104438i	$-0.00332441\pi$
$480$		0			0
$481$	4.38681			0.200021
$482$		0			0
$483$	−20.3244			−0.924792
$484$		0			0
$485$			11.6504i			0.529017i
$486$		0			0
$487$	41.0347			1.85946			0.929729	−	0.368243i	$-0.120041\pi$
$487$	41.0347			1.85946			0.929729	+	0.368243i	$0.120041\pi$
$488$		0			0
$489$		−	15.9737i		−	0.722357i
$490$		0			0
$491$			18.3616i			0.828648i	0.910130	+	0.414324i	$0.135982\pi$
$491$			18.3616i			0.828648i	−0.910130	+	0.414324i	$0.864018\pi$
$492$		0			0
$493$			36.2094i			1.63079i
$494$		0			0
$495$		−	4.04806i		−	0.181947i
$496$		0			0
$497$		−	34.5134i		−	1.54814i
$498$		0			0
$499$			34.6857i			1.55275i	0.630273	+	0.776374i	$0.282943\pi$
$499$			34.6857i			1.55275i	−0.630273	+	0.776374i	$0.717057\pi$
$500$		0			0
$501$	−0.356216			−0.0159145
$502$		0			0
$503$		−	24.4735i		−	1.09122i	−0.838040	−	0.545609i	$-0.816298\pi$
$503$		−	24.4735i		−	1.09122i	0.838040	−	0.545609i	$-0.183702\pi$
$504$		0			0
$505$	−14.9156			−0.663735
$506$		0			0
$507$	11.4767			0.509696
$508$		0			0
$509$			36.1023i			1.60020i	0.599864	+	0.800102i	$0.295221\pi$
$509$			36.1023i			1.60020i	−0.599864	+	0.800102i	$0.704779\pi$
$510$		0			0
$511$		−	0.705720i		−	0.0312192i
$512$		0			0
$513$	3.61178	+	2.44030i	0.159464	+	0.107742i
$514$		0			0
$515$	14.4655			0.637426
$516$		0			0
$517$	42.1138			1.85216
$518$		0			0
$519$			4.81713i			0.211449i
$520$		0			0
$521$			25.9388i			1.13640i	0.822891	+	0.568200i	$0.192360\pi$
$521$			25.9388i			1.13640i	−0.822891	+	0.568200i	$0.807640\pi$
$522$		0			0
$523$	1.59448			0.0697219			0.0348609	−	0.999392i	$-0.488901\pi$
$523$	1.59448			0.0697219			0.0348609	+	0.999392i	$0.488901\pi$
$524$		0			0
$525$		−	2.20519i		−	0.0962422i
$526$		0			0
$527$	32.1759			1.40161
$528$		0			0
$529$	−61.9464			−2.69332
$530$		0			0
$531$	−0.470233			−0.0204064
$532$		0			0
$533$	5.32363			0.230592
$534$		0			0
$535$	−14.4989			−0.626844
$536$		0			0
$537$	−22.3596			−0.964887
$538$		0			0
$539$			8.65135i			0.372640i
$540$		0			0
$541$	45.1720			1.94210			0.971049	−	0.238881i	$-0.0767806\pi$
$541$	45.1720			1.94210			0.971049	+	0.238881i	$0.0767806\pi$
$542$		0			0
$543$			9.96763i			0.427752i
$544$		0			0
$545$		−	5.28922i		−	0.226565i
$546$		0			0
$547$	−35.6483			−1.52421			−0.762106	−	0.647452i	$-0.775835\pi$
$547$	−35.6483			−1.52421			−0.762106	+	0.647452i	$0.775835\pi$
$548$		0			0
$549$	0.970304			0.0414116
$550$		0			0
$551$	20.0665	−	29.6994i	0.854861	−	1.26524i
$552$		0			0
$553$		−	25.2388i		−	1.07326i
$554$		0			0
$555$			3.55426i			0.150870i
$556$		0			0
$557$	−39.2984			−1.66513			−0.832563	−	0.553930i	$-0.813127\pi$
$557$	−39.2984			−1.66513			−0.832563	+	0.553930i	$0.813127\pi$
$558$		0			0
$559$	−8.66233			−0.366378
$560$		0			0
$561$		−	17.8255i		−	0.752591i
$562$		0			0
$563$	9.81444			0.413629			0.206815	−	0.978380i	$-0.433690\pi$
$563$	9.81444			0.413629			0.206815	+	0.978380i	$0.433690\pi$
$564$		0			0
$565$			13.8980i			0.584694i
$566$		0			0
$567$		−	2.20519i		−	0.0926091i
$568$		0			0
$569$			26.3042i			1.10273i	0.834265	+	0.551364i	$0.185893\pi$
$569$			26.3042i			1.10273i	−0.834265	+	0.551364i	$0.814107\pi$
$570$		0			0
$571$		−	24.3658i		−	1.01968i	−0.860270	−	0.509839i	$-0.829705\pi$
$571$		−	24.3658i		−	1.01968i	0.860270	−	0.509839i	$-0.170295\pi$
$572$		0			0
$573$		−	22.1221i		−	0.924164i
$574$		0			0
$575$		−	9.21663i		−	0.384360i
$576$		0			0
$577$	27.4885			1.14436			0.572182	−	0.820127i	$-0.306097\pi$
$577$	27.4885			1.14436			0.572182	+	0.820127i	$0.306097\pi$
$578$		0			0
$579$			4.52803i			0.188178i
$580$		0			0
$581$	−20.5642			−0.853147
$582$		0			0
$583$	−30.6017			−1.26739
$584$		0			0
$585$		−	1.23424i		−	0.0510294i
$586$		0			0
$587$		−	0.523540i		−	0.0216088i	−0.999942	−	0.0108044i	$-0.996561\pi$
$587$		−	0.523540i		−	0.0216088i	0.999942	−	0.0108044i	$-0.00343921\pi$
$588$		0			0
$589$	−26.3912	−	17.8312i	−1.08743	−	0.734723i
$590$		0			0
$591$	−8.46460			−0.348187
$592$		0			0
$593$	11.4009			0.468180			0.234090	−	0.972215i	$-0.424789\pi$
$593$	11.4009			0.468180			0.234090	+	0.972215i	$0.424789\pi$
$594$		0			0
$595$		−	9.71043i		−	0.398089i
$596$		0			0
$597$		−	7.21423i		−	0.295259i
$598$		0			0
$599$	17.1713			0.701601			0.350800	−	0.936450i	$-0.385910\pi$
$599$	17.1713			0.701601			0.350800	+	0.936450i	$0.385910\pi$
$600$		0			0
$601$		−	39.8582i		−	1.62585i	−0.582369	−	0.812924i	$-0.697874\pi$
$601$		−	39.8582i		−	1.62585i	0.582369	−	0.812924i	$-0.302126\pi$
$602$		0			0
$603$	−13.8255			−0.563016
$604$		0			0
$605$	5.38681			0.219005
$606$		0			0
$607$	3.19661			0.129746			0.0648732	−	0.997894i	$-0.479336\pi$
$607$	3.19661			0.129746			0.0648732	+	0.997894i	$0.479336\pi$
$608$		0			0
$609$	−18.1331			−0.734791
$610$		0			0
$611$	12.8403			0.519464
$612$		0			0
$613$	−14.3847			−0.580994			−0.290497	−	0.956876i	$-0.593821\pi$
$613$	−14.3847			−0.580994			−0.290497	+	0.956876i	$0.593821\pi$
$614$		0			0
$615$			4.31329i			0.173929i
$616$		0			0
$617$	−13.4439			−0.541232			−0.270616	−	0.962687i	$-0.587227\pi$
$617$	−13.4439			−0.541232			−0.270616	+	0.962687i	$0.587227\pi$
$618$		0			0
$619$			14.8410i			0.596511i	0.954486	+	0.298256i	$0.0964048\pi$
$619$			14.8410i			0.596511i	−0.954486	+	0.298256i	$0.903595\pi$
$620$		0			0
$621$		−	9.21663i		−	0.369851i
$622$		0			0
$623$	−9.21737			−0.369286
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−9.87849	+	14.6207i	−0.394509	+	0.583894i
$628$		0			0
$629$			15.6510i			0.624048i
$630$		0			0
$631$			9.16341i			0.364789i	0.983225	+	0.182395i	$0.0583849\pi$
$631$			9.16341i			0.364789i	−0.983225	+	0.182395i	$0.941615\pi$
$632$		0			0
$633$	−14.9776			−0.595305
$634$		0			0
$635$	14.3384			0.569003
$636$		0			0
$637$			2.63776i			0.104512i
$638$		0			0
$639$	15.6510			0.619146
$640$		0			0
$641$			22.7951i			0.900353i	0.892940	+	0.450176i	$0.148639\pi$
$641$			22.7951i			0.900353i	−0.892940	+	0.450176i	$0.851361\pi$
$642$		0			0
$643$		−	42.9599i		−	1.69417i	−0.531456	−	0.847086i	$-0.678355\pi$
$643$		−	42.9599i		−	1.69417i	0.531456	−	0.847086i	$-0.321645\pi$
$644$		0			0
$645$		−	7.01837i		−	0.276348i
$646$		0			0
$647$		−	6.17729i		−	0.242854i	−0.992600	−	0.121427i	$-0.961253\pi$
$647$		−	6.17729i		−	0.242854i	0.992600	−	0.121427i	$-0.0387471\pi$
$648$		0			0
$649$		−	1.90353i		−	0.0747203i
$650$		0			0
$651$			16.1132i			0.631528i
$652$		0			0
$653$	−15.7151			−0.614979			−0.307490	−	0.951551i	$-0.599489\pi$
$653$	−15.7151			−0.614979			−0.307490	+	0.951551i	$0.599489\pi$
$654$		0			0
$655$			17.6624i			0.690126i
$656$		0			0
$657$	0.320027			0.0124855
$658$		0			0
$659$	23.7943			0.926896			0.463448	−	0.886124i	$-0.346612\pi$
$659$	23.7943			0.926896			0.463448	+	0.886124i	$0.346612\pi$
$660$		0			0
$661$		−	3.30405i		−	0.128513i	−0.997933	−	0.0642564i	$-0.979532\pi$
$661$		−	3.30405i		−	0.128513i	0.997933	−	0.0642564i	$-0.0204676\pi$
$662$		0			0
$663$		−	5.43491i		−	0.211074i
$664$		0			0
$665$	−5.38132	+	7.96464i	−0.208679	+	0.308855i
$666$		0			0
$667$	−75.7879			−2.93452
$668$		0			0
$669$	−9.24952			−0.357607
$670$		0			0
$671$			3.92785i			0.151633i
$672$		0			0
$673$		−	34.9847i		−	1.34856i	−0.738475	−	0.674281i	$-0.764454\pi$
$673$		−	34.9847i		−	1.34856i	0.738475	−	0.674281i	$-0.235546\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$			33.4637i			1.28611i	0.765818	+	0.643057i	$0.222334\pi$
$677$			33.4637i			1.28611i	−0.765818	+	0.643057i	$0.777666\pi$
$678$		0			0
$679$	−25.6913			−0.985940
$680$		0			0
$681$	−23.8867			−0.915339
$682$		0			0
$683$	38.9199			1.48923			0.744615	−	0.667494i	$-0.232633\pi$
$683$	38.9199			1.48923			0.744615	+	0.667494i	$0.232633\pi$
$684$		0			0
$685$	14.6157			0.558438
$686$		0			0
$687$	12.4239			0.474001
$688$		0			0
$689$	−9.33032			−0.355457
$690$		0			0
$691$		−	4.35686i		−	0.165743i	−0.996560	−	0.0828714i	$-0.973591\pi$
$691$		−	4.35686i		−	0.165743i	0.996560	−	0.0828714i	$-0.0264091\pi$
$692$		0			0
$693$	8.92673			0.339098
$694$		0			0
$695$			7.86064i			0.298171i
$696$		0			0
$697$			18.9934i			0.719425i
$698$		0			0
$699$	29.1146			1.10122
$700$		0			0
$701$	−1.49337			−0.0564039			−0.0282020	−	0.999602i	$-0.508978\pi$
$701$	−1.49337			−0.0564039			−0.0282020	+	0.999602i	$0.508978\pi$
$702$		0			0
$703$	8.67348	−	12.8372i	0.327126	−	0.484164i
$704$		0			0
$705$			10.4035i			0.391817i
$706$		0			0
$707$		−	32.8916i		−	1.23702i
$708$		0			0
$709$	−44.1781			−1.65914			−0.829571	−	0.558401i	$-0.811415\pi$
$709$	−44.1781			−1.65914			−0.829571	+	0.558401i	$0.811415\pi$
$710$		0			0
$711$	11.4452			0.429229
$712$		0			0
$713$			67.3457i			2.52212i
$714$		0			0
$715$	4.99627			0.186850
$716$		0			0
$717$		−	3.27796i		−	0.122418i
$718$		0			0
$719$			48.9398i			1.82515i	0.408911	+	0.912574i	$0.365909\pi$
$719$			48.9398i			1.82515i	−0.408911	+	0.912574i	$0.634091\pi$
$720$		0			0
$721$			31.8991i			1.18799i
$722$		0			0
$723$			1.28166i			0.0476653i
$724$		0			0
$725$		−	8.22295i		−	0.305393i
$726$		0			0
$727$			30.2687i			1.12260i	0.827611	+	0.561302i	$0.189699\pi$
$727$			30.2687i			1.12260i	−0.827611	+	0.561302i	$0.810301\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	30.9051i		−	1.14306i
$732$		0			0
$733$	16.4834			0.608827			0.304413	−	0.952540i	$-0.401540\pi$
$733$	16.4834			0.608827			0.304413	+	0.952540i	$0.401540\pi$
$734$		0			0
$735$	−2.13716			−0.0788302
$736$		0			0
$737$		−	55.9663i		−	2.06155i
$738$		0			0
$739$			36.2349i			1.33292i	0.745540	+	0.666461i	$0.232192\pi$
$739$			36.2349i			1.33292i	−0.745540	+	0.666461i	$0.767808\pi$
$740$		0			0
$741$	−3.01191	+	4.45779i	−0.110645	+	0.163761i
$742$		0			0
$743$	−21.5620			−0.791032			−0.395516	−	0.918459i	$-0.629434\pi$
$743$	−21.5620			−0.791032			−0.395516	+	0.918459i	$0.629434\pi$
$744$		0			0
$745$	−17.5175			−0.641791
$746$		0			0
$747$		−	9.32538i		−	0.341198i
$748$		0			0
$749$		−	31.9728i		−	1.16826i
$750$		0			0
$751$	21.5768			0.787348			0.393674	−	0.919250i	$-0.371204\pi$
$751$	21.5768			0.787348			0.393674	+	0.919250i	$0.371204\pi$
$752$		0			0
$753$			0.323718i			0.0117970i
$754$		0			0
$755$	15.3374			0.558187
$756$		0			0
$757$	−23.1484			−0.841341			−0.420671	−	0.907213i	$-0.638205\pi$
$757$	−23.1484			−0.841341			−0.420671	+	0.907213i	$0.638205\pi$
$758$		0			0
$759$	37.3095			1.35425
$760$		0			0
$761$	5.89987			0.213870			0.106935	−	0.994266i	$-0.465896\pi$
$761$	5.89987			0.213870			0.106935	+	0.994266i	$0.465896\pi$
$762$		0			0
$763$	11.6637			0.422254
$764$		0			0
$765$	4.40345			0.159207
$766$		0			0
$767$		−	0.580380i		−	0.0209563i
$768$		0			0
$769$	13.8068			0.497885			0.248942	−	0.968518i	$-0.419917\pi$
$769$	13.8068			0.497885			0.248942	+	0.968518i	$0.419917\pi$
$770$		0			0
$771$		−	11.3497i		−	0.408750i
$772$		0			0
$773$		−	23.3296i		−	0.839109i	−0.907730	−	0.419554i	$-0.862186\pi$
$773$		−	23.3296i		−	0.839109i	0.907730	−	0.419554i	$-0.137814\pi$
$774$		0			0
$775$	−7.30698			−0.262474
$776$		0			0
$777$	−7.83781			−0.281180
$778$		0			0
$779$	10.5257	−	15.5786i	0.377123	−	0.558163i
$780$		0			0
$781$			63.3564i			2.26707i
$782$		0			0
$783$		−	8.22295i		−	0.293864i
$784$		0			0
$785$	10.9126			0.389488
$786$		0			0
$787$	32.6037			1.16220			0.581098	−	0.813834i	$-0.302623\pi$
$787$	32.6037			1.16220			0.581098	+	0.813834i	$0.302623\pi$
$788$		0			0
$789$			11.2534i			0.400630i
$790$		0			0
$791$	−30.6477			−1.08971
$792$		0			0
$793$			1.19759i			0.0425275i
$794$		0			0
$795$		−	7.55958i		−	0.268111i
$796$		0			0
$797$		−	18.9938i		−	0.672794i	−0.941720	−	0.336397i	$-0.890792\pi$
$797$		−	18.9938i		−	0.672794i	0.941720	−	0.336397i	$-0.109208\pi$
$798$		0			0
$799$			45.8111i			1.62068i
$800$		0			0
$801$		−	4.17986i		−	0.147688i
$802$		0			0
$803$			1.29549i			0.0457169i
$804$		0			0
$805$	20.3244			0.716341
$806$		0			0
$807$		−	15.0809i		−	0.530873i
$808$		0			0
$809$	−35.7741			−1.25775			−0.628875	−	0.777507i	$-0.716484\pi$
$809$	−35.7741			−1.25775			−0.628875	+	0.777507i	$0.716484\pi$
$810$		0			0
$811$	28.7302			1.00885			0.504427	−	0.863455i	$-0.331704\pi$
$811$	28.7302			1.00885			0.504427	+	0.863455i	$0.331704\pi$
$812$		0			0
$813$		−	30.9368i		−	1.08500i
$814$		0			0
$815$			15.9737i			0.559535i
$816$		0			0
$817$	−17.1269	+	25.3488i	−0.599196	+	0.886841i
$818$		0			0
$819$	2.72172			0.0951047
$820$		0			0
$821$	20.2681			0.707360			0.353680	−	0.935366i	$-0.384930\pi$
$821$	20.2681			0.707360			0.353680	+	0.935366i	$0.384930\pi$
$822$		0			0
$823$		−	15.4615i		−	0.538953i	−0.963007	−	0.269476i	$-0.913149\pi$
$823$		−	15.4615i		−	0.538953i	0.963007	−	0.269476i	$-0.0868506\pi$
$824$		0			0
$825$			4.04806i			0.140935i
$826$		0			0
$827$	−48.0865			−1.67213			−0.836065	−	0.548630i	$-0.815150\pi$
$827$	−48.0865			−1.67213			−0.836065	+	0.548630i	$0.815150\pi$
$828$		0			0
$829$		−	18.9194i		−	0.657096i	−0.944487	−	0.328548i	$-0.893441\pi$
$829$		−	18.9194i		−	0.657096i	0.944487	−	0.328548i	$-0.106559\pi$
$830$		0			0
$831$	9.96077			0.345535
$832$		0			0
$833$	−9.41087			−0.326067
$834$		0			0
$835$	0.356216			0.0123273
$836$		0			0
$837$	−7.30698			−0.252566
$838$		0			0
$839$	1.51947			0.0524580			0.0262290	−	0.999656i	$-0.491650\pi$
$839$	1.51947			0.0524580			0.0262290	+	0.999656i	$0.491650\pi$
$840$		0			0
$841$	−38.6169			−1.33162
$842$		0			0
$843$			3.42360i			0.117915i
$844$		0			0
$845$	−11.4767			−0.394809
$846$		0			0
$847$			11.8789i			0.408164i
$848$		0			0
$849$			6.15914i			0.211381i
$850$		0			0
$851$	−32.7584			−1.12294
$852$		0			0
$853$	26.3427			0.901957			0.450978	−	0.892535i	$-0.351075\pi$
$853$	26.3427			0.901957			0.450978	+	0.892535i	$0.351075\pi$
$854$		0			0
$855$	−3.61178	−	2.44030i	−0.123520	−	0.0834566i
$856$		0			0
$857$		−	5.88515i		−	0.201033i	−0.994935	−	0.100516i	$-0.967950\pi$
$857$		−	5.88515i		−	0.201033i	0.994935	−	0.100516i	$-0.0320495\pi$
$858$		0			0
$859$		−	18.3191i		−	0.625040i	−0.949911	−	0.312520i	$-0.898827\pi$
$859$		−	18.3191i		−	0.625040i	0.949911	−	0.312520i	$-0.101173\pi$
$860$		0			0
$861$	−9.51161			−0.324155
$862$		0			0
$863$	46.6356			1.58750			0.793748	−	0.608247i	$-0.208127\pi$
$863$	46.6356			1.58750			0.793748	+	0.608247i	$0.208127\pi$
$864$		0			0
$865$		−	4.81713i		−	0.163787i
$866$		0			0
$867$	2.39040			0.0811824
$868$		0			0
$869$			46.3309i			1.57167i
$870$		0			0
$871$		−	17.0639i		−	0.578188i
$872$		0			0
$873$		−	11.6504i		−	0.394306i
$874$		0			0
$875$			2.20519i			0.0745489i
$876$		0			0
$877$		−	29.5433i		−	0.997607i	−0.866715	−	0.498803i	$-0.833773\pi$
$877$		−	29.5433i		−	0.997607i	0.866715	−	0.498803i	$-0.166227\pi$
$878$		0			0
$879$		−	24.7112i		−	0.833488i
$880$		0			0
$881$	−25.7237			−0.866653			−0.433327	−	0.901237i	$-0.642660\pi$
$881$	−25.7237			−0.866653			−0.433327	+	0.901237i	$0.642660\pi$
$882$		0			0
$883$			45.5906i			1.53425i	0.641499	+	0.767124i	$0.278313\pi$
$883$			45.5906i			1.53425i	−0.641499	+	0.767124i	$0.721687\pi$
$884$		0			0
$885$	0.470233			0.0158067
$886$		0			0
$887$	37.1405			1.24706			0.623528	−	0.781801i	$-0.285699\pi$
$887$	37.1405			1.24706			0.623528	+	0.781801i	$0.285699\pi$
$888$		0			0
$889$			31.6189i			1.06046i
$890$		0			0
$891$			4.04806i			0.135615i
$892$		0			0
$893$	25.3876	−	37.5749i	0.849563	−	1.25740i
$894$		0			0
$895$	22.3596			0.747398
$896$		0			0
$897$	11.3755			0.379817
$898$		0			0
$899$			60.0849i			2.00394i
$900$		0			0
$901$		−	33.2883i		−	1.10899i
$902$		0			0
$903$	15.4768			0.515036
$904$		0			0
$905$		−	9.96763i		−	0.331335i
$906$		0			0
$907$	55.9876			1.85904			0.929519	−	0.368775i	$-0.120223\pi$
$907$	55.9876			1.85904			0.929519	+	0.368775i	$0.120223\pi$
$908$		0			0
$909$	14.9156			0.494719
$910$		0			0
$911$	28.5424			0.945651			0.472826	−	0.881156i	$-0.343234\pi$
$911$	28.5424			0.945651			0.472826	+	0.881156i	$0.343234\pi$
$912$		0			0
$913$	37.7497			1.24933
$914$		0			0
$915$	−0.970304			−0.0320773
$916$		0			0
$917$	−38.9488			−1.28620
$918$		0			0
$919$			30.4320i			1.00386i	0.864909	+	0.501929i	$0.167376\pi$
$919$			30.4320i			1.00386i	−0.864909	+	0.501929i	$0.832624\pi$
$920$		0			0
$921$	7.93681			0.261527
$922$		0			0
$923$			19.3171i			0.635830i
$924$		0			0
$925$		−	3.55426i		−	0.116864i
$926$		0			0
$927$	−14.4655			−0.475110
$928$		0			0
$929$	32.0254			1.05072			0.525359	−	0.850881i	$-0.323931\pi$
$929$	32.0254			1.05072			0.525359	+	0.850881i	$0.323931\pi$
$930$		0			0
$931$	7.71893	+	5.21531i	0.252978	+	0.170925i
$932$		0			0
$933$		−	0.578158i		−	0.0189280i
$934$		0			0
$935$			17.8255i			0.582955i
$936$		0			0
$937$	48.2194			1.57526			0.787629	−	0.616149i	$-0.211308\pi$
$937$	48.2194			1.57526			0.787629	+	0.616149i	$0.211308\pi$
$938$		0			0
$939$	7.18892			0.234601
$940$		0			0
$941$		−	59.2280i		−	1.93078i	−0.260815	−	0.965389i	$-0.583991\pi$
$941$		−	59.2280i		−	1.93078i	0.260815	−	0.965389i	$-0.416009\pi$
$942$		0			0
$943$	−39.7540			−1.29457
$944$		0			0
$945$			2.20519i			0.0717347i
$946$		0			0
$947$			44.1262i			1.43391i	0.697121	+	0.716954i	$0.254464\pi$
$947$			44.1262i			1.43391i	−0.697121	+	0.716954i	$0.745536\pi$
$948$		0			0
$949$			0.394990i			0.0128219i
$950$		0			0
$951$		−	1.24733i		−	0.0404473i
$952$		0			0
$953$			11.2648i			0.364903i	0.983215	+	0.182451i	$0.0584032\pi$
$953$			11.2648i			0.364903i	−0.983215	+	0.182451i	$0.941597\pi$
$954$		0			0
$955$			22.1221i			0.715854i
$956$		0			0
$957$	33.2870			1.07602
$958$		0			0
$959$			32.2304i			1.04077i
$960$		0			0
$961$	22.3919			0.722320
$962$		0			0
$963$	14.4989			0.467222
$964$		0			0
$965$		−	4.52803i		−	0.145762i
$966$		0			0
$967$			11.2657i			0.362282i	0.983457	+	0.181141i	$0.0579790\pi$
$967$			11.2657i			0.362282i	−0.983457	+	0.181141i	$0.942021\pi$
$968$		0			0
$969$	−15.9043	−	10.7458i	−0.510919	−	0.345203i
$970$		0			0
$971$	−33.0946			−1.06206			−0.531028	−	0.847354i	$-0.678194\pi$
$971$	−33.0946			−1.06206			−0.531028	+	0.847354i	$0.678194\pi$
$972$		0			0
$973$	−17.3342			−0.555708
$974$		0			0
$975$			1.23424i			0.0395272i
$976$		0			0
$977$		−	21.7074i		−	0.694482i	−0.937776	−	0.347241i	$-0.887119\pi$
$977$		−	21.7074i		−	0.694482i	0.937776	−	0.347241i	$-0.112881\pi$
$978$		0			0
$979$	16.9203			0.540777
$980$		0			0
$981$			5.28922i			0.168872i
$982$		0			0
$983$	−31.2350			−0.996242			−0.498121	−	0.867107i	$-0.665977\pi$
$983$	−31.2350			−0.996242			−0.498121	+	0.867107i	$0.665977\pi$
$984$		0			0
$985$	8.46460			0.269705
$986$		0			0
$987$	−22.9415			−0.730237
$988$		0			0
$989$	64.6857			2.05689
$990$		0			0
$991$	−48.6036			−1.54394			−0.771972	−	0.635657i	$-0.780729\pi$
$991$	−48.6036			−1.54394			−0.771972	+	0.635657i	$0.780729\pi$
$992$		0			0
$993$	−12.4754			−0.395896
$994$		0			0
$995$			7.21423i			0.228707i
$996$		0			0
$997$	20.5887			0.652050			0.326025	−	0.945361i	$-0.394291\pi$
$997$	20.5887			0.652050			0.326025	+	0.945361i	$0.394291\pi$
$998$		0			0
$999$		−	3.55426i		−	0.112452i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.k.2431.4	yes	12
4.3	odd	2		4560.2.d.i.2431.9	yes	12
19.18	odd	2		4560.2.d.i.2431.4	✓	12
76.75	even	2	inner	4560.2.d.k.2431.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.i.2431.4	✓	12	19.18	odd	2
4560.2.d.i.2431.9	yes	12	4.3	odd	2
4560.2.d.k.2431.4	yes	12	1.1	even	1	trivial
4560.2.d.k.2431.9	yes	12	76.75	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.20519i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.20519i q^{7} +1.00000 q^{9} +4.04806i q^{11} +1.23424i q^{13} -1.00000 q^{15} -4.40345 q^{17} +(3.61178 + 2.44030i) q^{19} -2.20519i q^{21} -9.21663i q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.22295i q^{29} -7.30698 q^{31} +4.04806i q^{33} +2.20519i q^{35} -3.55426i q^{37} +1.23424i q^{39} -4.31329i q^{41} +7.01837i q^{43} -1.00000 q^{45} -10.4035i q^{47} +2.13716 q^{49} -4.40345 q^{51} +7.55958i q^{53} -4.04806i q^{55} +(3.61178 + 2.44030i) q^{57} -0.470233 q^{59} +0.970304 q^{61} -2.20519i q^{63} -1.23424i q^{65} -13.8255 q^{67} -9.21663i q^{69} +15.6510 q^{71} +0.320027 q^{73} +1.00000 q^{75} +8.92673 q^{77} +11.4452 q^{79} +1.00000 q^{81} -9.32538i q^{83} +4.40345 q^{85} -8.22295i q^{87} -4.17986i q^{89} +2.72172 q^{91} -7.30698 q^{93} +(-3.61178 - 2.44030i) q^{95} -11.6504i q^{97} +4.04806i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 - 12 * q^5 + 12 * q^9	\( 12 q + 12 q^{3} - 12 q^{5} + 12 q^{9} - 12 q^{15} - 8 q^{17} + 8 q^{19} + 12 q^{25} + 12 q^{27} - 12 q^{45} - 20 q^{49} - 8 q^{51} + 8 q^{57} + 8 q^{59} - 8 q^{67} + 32 q^{71} - 24 q^{73} + 12 q^{75} + 56 q^{77} + 16 q^{79} + 12 q^{81} + 8 q^{85} - 16 q^{91} - 8 q^{95}+O(q^{100}) \) 12 * q + 12 * q^3 - 12 * q^5 + 12 * q^9 - 12 * q^15 - 8 * q^17 + 8 * q^19 + 12 * q^25 + 12 * q^27 - 12 * q^45 - 20 * q^49 - 8 * q^51 + 8 * q^57 + 8 * q^59 - 8 * q^67 + 32 * q^71 - 24 * q^73 + 12 * q^75 + 56 * q^77 + 16 * q^79 + 12 * q^81 + 8 * q^85 - 16 * q^91 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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