LMFDB - Embedded newform 4560.2.d.j.2431.11

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} + 35x^{10} + 202x^{8} + 362x^{6} + 245x^{4} + 63x^{2} + 4 $ x^12 + 35x^10 + 202x^8 + 362x^6 + 245x^4 + 63*x^2 + 4
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			2431.11
Root			$0.687758i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.j.2431.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +3.95229i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +3.95229i q^{7} +1.00000 q^{9} -2.16866i q^{11} +6.98743i q^{13} -1.00000 q^{15} -2.03991 q^{17} +(-2.20175 + 3.76196i) q^{19} -3.95229i q^{21} +8.87858i q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.63276i q^{29} -1.31415 q^{31} +2.16866i q^{33} +3.95229i q^{35} +1.77374i q^{37} -6.98743i q^{39} -4.92629i q^{41} +9.57516i q^{43} +1.00000 q^{45} -9.15609i q^{47} -8.62061 q^{49} +2.03991 q^{51} -5.41448i q^{53} -2.16866i q^{55} +(2.20175 - 3.76196i) q^{57} -12.9348 q^{59} +1.62061 q^{61} +3.95229i q^{63} +6.98743i q^{65} -6.56189 q^{67} -8.87858i q^{69} -5.35406 q^{71} +2.95056 q^{73} -1.00000 q^{75} +8.57117 q^{77} +11.1906 q^{79} +1.00000 q^{81} -4.37136i q^{83} -2.03991 q^{85} +5.63276i q^{87} -6.91831i q^{89} -27.6164 q^{91} +1.31415 q^{93} +(-2.20175 + 3.76196i) q^{95} +13.0393i q^{97} -2.16866i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{17} + 12 q^{25} - 12 q^{27} + 12 q^{31} + 12 q^{45} - 28 q^{49} - 4 q^{51} - 52 q^{59} - 56 q^{61} + 32 q^{67} - 8 q^{71} + 32 q^{73} - 12 q^{75} + 24 q^{77} + 28 q^{79} + 12 q^{81} + 4 q^{85} - 32 q^{91} - 12 q^{93}+O(q^{100}) $ 12 * q - 12 * q^3 + 12 * q^5 + 12 * q^9 - 12 * q^15 + 4 * q^17 + 12 * q^25 - 12 * q^27 + 12 * q^31 + 12 * q^45 - 28 * q^49 - 4 * q^51 - 52 * q^59 - 56 * q^61 + 32 * q^67 - 8 * q^71 + 32 * q^73 - 12 * q^75 + 24 * q^77 + 28 * q^79 + 12 * q^81 + 4 * q^85 - 32 * q^91 - 12 * q^93

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
$7$			3.95229i			1.49383i	0.664922	+	0.746913i	$0.268465\pi$
$7$			3.95229i			1.49383i	−0.664922	+	0.746913i	$0.731535\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	2.16866i		−	0.653875i	−0.945046	−	0.326937i	$-0.893983\pi$
$11$		−	2.16866i		−	0.653875i	0.945046	−	0.326937i	$-0.106017\pi$
$12$		0			0
$13$			6.98743i			1.93796i	0.247130	+	0.968982i	$0.420513\pi$
$13$			6.98743i			1.93796i	−0.247130	+	0.968982i	$0.579487\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−2.03991			−0.494750			−0.247375	−	0.968920i	$-0.579568\pi$
$17$	−2.03991			−0.494750			−0.247375	+	0.968920i	$0.579568\pi$
$18$		0			0
$19$	−2.20175	+	3.76196i	−0.505115	+	0.863052i
$19$	−2.20175	+	3.76196i	−0.505115	+	0.863052i
$20$		0			0
$21$		−	3.95229i		−	0.862461i
$22$		0			0
$23$			8.87858i			1.85131i	0.378366	+	0.925656i	$0.376486\pi$
$23$			8.87858i			1.85131i	−0.378366	+	0.925656i	$0.623514\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	5.63276i		−	1.04598i	−0.852340	−	0.522989i	$-0.824817\pi$
$29$		−	5.63276i		−	1.04598i	0.852340	−	0.522989i	$-0.175183\pi$
$30$		0			0
$31$	−1.31415			−0.236028			−0.118014	−	0.993012i	$-0.537653\pi$
$31$	−1.31415			−0.236028			−0.118014	+	0.993012i	$0.537653\pi$
$32$		0			0
$33$			2.16866i			0.377515i
$34$		0			0
$35$			3.95229i			0.668059i
$36$		0			0
$37$			1.77374i			0.291601i	0.989314	+	0.145801i	$0.0465758\pi$
$37$			1.77374i			0.291601i	−0.989314	+	0.145801i	$0.953424\pi$
$38$		0			0
$39$		−	6.98743i		−	1.11888i
$40$		0			0
$41$		−	4.92629i		−	0.769357i	−0.923051	−	0.384679i	$-0.874312\pi$
$41$		−	4.92629i		−	0.769357i	0.923051	−	0.384679i	$-0.125688\pi$
$42$		0			0
$43$			9.57516i			1.46020i	0.683341	+	0.730099i	$0.260526\pi$
$43$			9.57516i			1.46020i	−0.683341	+	0.730099i	$0.739474\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$		−	9.15609i		−	1.33555i	−0.744362	−	0.667776i	$-0.767246\pi$
$47$		−	9.15609i		−	1.33555i	0.744362	−	0.667776i	$-0.232754\pi$
$48$		0			0
$49$	−8.62061			−1.23152
$50$		0			0
$51$	2.03991			0.285644
$52$		0			0
$53$		−	5.41448i		−	0.743736i	−0.928286	−	0.371868i	$-0.878717\pi$
$53$		−	5.41448i		−	0.743736i	0.928286	−	0.371868i	$-0.121283\pi$
$54$		0			0
$55$		−	2.16866i		−	0.292422i
$56$		0			0
$57$	2.20175	−	3.76196i	0.291629	−	0.498283i
$58$		0			0
$59$	−12.9348			−1.68396			−0.841981	−	0.539507i	$-0.818611\pi$
$59$	−12.9348			−1.68396			−0.841981	+	0.539507i	$0.818611\pi$
$60$		0			0
$61$	1.62061			0.207497			0.103749	−	0.994604i	$-0.466916\pi$
$61$	1.62061			0.207497			0.103749	+	0.994604i	$0.466916\pi$
$62$		0			0
$63$			3.95229i			0.497942i
$64$		0			0
$65$			6.98743i			0.866684i
$66$		0			0
$67$	−6.56189			−0.801662			−0.400831	−	0.916152i	$-0.631279\pi$
$67$	−6.56189			−0.801662			−0.400831	+	0.916152i	$0.631279\pi$
$68$		0			0
$69$		−	8.87858i		−	1.06886i
$70$		0			0
$71$	−5.35406			−0.635410			−0.317705	−	0.948190i	$-0.602912\pi$
$71$	−5.35406			−0.635410			−0.317705	+	0.948190i	$0.602912\pi$
$72$		0			0
$73$	2.95056			0.345337			0.172669	−	0.984980i	$-0.444761\pi$
$73$	2.95056			0.345337			0.172669	+	0.984980i	$0.444761\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	8.57117			0.976775
$78$		0			0
$79$	11.1906			1.25904			0.629521	−	0.776984i	$-0.283251\pi$
$79$	11.1906			1.25904			0.629521	+	0.776984i	$0.283251\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.37136i		−	0.479819i	−0.970795	−	0.239910i	$-0.922882\pi$
$83$		−	4.37136i		−	0.479819i	0.970795	−	0.239910i	$-0.0771178\pi$
$84$		0			0
$85$	−2.03991			−0.221259
$86$		0			0
$87$			5.63276i			0.603895i
$88$		0			0
$89$		−	6.91831i		−	0.733340i	−0.930351	−	0.366670i	$-0.880498\pi$
$89$		−	6.91831i		−	0.733340i	0.930351	−	0.366670i	$-0.119502\pi$
$90$		0			0
$91$	−27.6164			−2.89498
$92$		0			0
$93$	1.31415			0.136271
$94$		0			0
$95$	−2.20175	+	3.76196i	−0.225894	+	0.385968i
$96$		0			0
$97$			13.0393i			1.32394i	0.749532	+	0.661968i	$0.230279\pi$
$97$			13.0393i			1.32394i	−0.749532	+	0.661968i	$0.769721\pi$
$98$		0			0
$99$		−	2.16866i		−	0.217958i
$100$		0			0
$101$	6.88415			0.684998			0.342499	−	0.939518i	$-0.388727\pi$
$101$	6.88415			0.684998			0.342499	+	0.939518i	$0.388727\pi$
$102$		0			0
$103$	6.56189			0.646562			0.323281	−	0.946303i	$-0.395214\pi$
$103$	6.56189			0.646562			0.323281	+	0.946303i	$0.395214\pi$
$104$		0			0
$105$		−	3.95229i		−	0.385704i
$106$		0			0
$107$	−9.20783			−0.890155			−0.445077	−	0.895492i	$-0.646824\pi$
$107$	−9.20783			−0.890155			−0.445077	+	0.895492i	$0.646824\pi$
$108$		0			0
$109$		−	17.5873i		−	1.68455i	−0.539045	−	0.842277i	$-0.681215\pi$
$109$		−	17.5873i		−	1.68455i	0.539045	−	0.842277i	$-0.318785\pi$
$110$		0			0
$111$		−	1.77374i		−	0.168356i
$112$		0			0
$113$			0.870836i			0.0819213i	0.999161	+	0.0409607i	$0.0130418\pi$
$113$			0.870836i			0.0819213i	−0.999161	+	0.0409607i	$0.986958\pi$
$114$		0			0
$115$			8.87858i			0.827932i
$116$		0			0
$117$			6.98743i			0.645988i
$118$		0			0
$119$		−	8.06230i		−	0.739070i
$120$		0			0
$121$	6.29692			0.572447
$122$		0			0
$123$			4.92629i			0.444189i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	5.61133			0.497925			0.248962	−	0.968513i	$-0.419910\pi$
$127$	5.61133			0.497925			0.248962	+	0.968513i	$0.419910\pi$
$128$		0			0
$129$		−	9.57516i		−	0.843046i
$130$		0			0
$131$		−	13.1751i		−	1.15112i	−0.817761	−	0.575558i	$-0.804785\pi$
$131$		−	13.1751i		−	1.15112i	0.817761	−	0.575558i	$-0.195215\pi$
$132$		0			0
$133$	−14.8683	−	8.70195i	−1.28925	−	0.754554i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	13.9976			1.19590			0.597948	−	0.801535i	$-0.295983\pi$
$137$	13.9976			1.19590			0.597948	+	0.801535i	$0.295983\pi$
$138$		0			0
$139$			21.0062i			1.78173i	0.454273	+	0.890863i	$0.349899\pi$
$139$			21.0062i			1.78173i	−0.454273	+	0.890863i	$0.650101\pi$
$140$		0			0
$141$			9.15609i			0.771082i
$142$		0			0
$143$	15.1533			1.26719
$144$		0			0
$145$		−	5.63276i		−	0.467775i
$146$		0			0
$147$	8.62061			0.711016
$148$		0			0
$149$	5.98236			0.490094			0.245047	−	0.969511i	$-0.421197\pi$
$149$	5.98236			0.490094			0.245047	+	0.969511i	$0.421197\pi$
$150$		0			0
$151$	12.2317			0.995400			0.497700	−	0.867349i	$-0.334178\pi$
$151$	12.2317			0.995400			0.497700	+	0.867349i	$0.334178\pi$
$152$		0			0
$153$	−2.03991			−0.164917
$154$		0			0
$155$	−1.31415			−0.105555
$156$		0			0
$157$	−6.13042			−0.489261			−0.244630	−	0.969616i	$-0.578667\pi$
$157$	−6.13042			−0.489261			−0.244630	+	0.969616i	$0.578667\pi$
$158$		0			0
$159$			5.41448i			0.429396i
$160$		0			0
$161$	−35.0907			−2.76554
$162$		0			0
$163$		−	9.47515i		−	0.742151i	−0.928603	−	0.371076i	$-0.878989\pi$
$163$		−	9.47515i		−	0.742151i	0.928603	−	0.371076i	$-0.121011\pi$
$164$		0			0
$165$			2.16866i			0.168830i
$166$		0			0
$167$	−21.7817			−1.68551			−0.842757	−	0.538294i	$-0.819069\pi$
$167$	−21.7817			−1.68551			−0.842757	+	0.538294i	$0.819069\pi$
$168$		0			0
$169$	−35.8242			−2.75571
$170$		0			0
$171$	−2.20175	+	3.76196i	−0.168372	+	0.287684i
$172$		0			0
$173$		−	22.4728i		−	1.70857i	−0.519802	−	0.854287i	$-0.673994\pi$
$173$		−	22.4728i		−	1.70857i	0.519802	−	0.854287i	$-0.326006\pi$
$174$		0			0
$175$			3.95229i			0.298765i
$176$		0			0
$177$	12.9348			0.972236
$178$		0			0
$179$	3.11079			0.232512			0.116256	−	0.993219i	$-0.462911\pi$
$179$	3.11079			0.232512			0.116256	+	0.993219i	$0.462911\pi$
$180$		0			0
$181$			2.28171i			0.169598i	0.996398	+	0.0847992i	$0.0270249\pi$
$181$			2.28171i			0.169598i	−0.996398	+	0.0847992i	$0.972975\pi$
$182$		0			0
$183$	−1.62061			−0.119799
$184$		0			0
$185$			1.77374i			0.130408i
$186$		0			0
$187$			4.42386i			0.323505i
$188$		0			0
$189$		−	3.95229i		−	0.287487i
$190$		0			0
$191$			25.4170i			1.83911i	0.392960	+	0.919555i	$0.371451\pi$
$191$			25.4170i			1.83911i	−0.392960	+	0.919555i	$0.628549\pi$
$192$		0			0
$193$		−	12.2011i		−	0.878256i	−0.898425	−	0.439128i	$-0.855287\pi$
$193$		−	12.2011i		−	0.878256i	0.898425	−	0.439128i	$-0.144713\pi$
$194$		0			0
$195$		−	6.98743i		−	0.500380i
$196$		0			0
$197$	−23.2941			−1.65964			−0.829818	−	0.558034i	$-0.811556\pi$
$197$	−23.2941			−1.65964			−0.829818	+	0.558034i	$0.811556\pi$
$198$		0			0
$199$		−	5.68961i		−	0.403326i	−0.979455	−	0.201663i	$-0.935365\pi$
$199$		−	5.68961i		−	0.403326i	0.979455	−	0.201663i	$-0.0646345\pi$
$200$		0			0
$201$	6.56189			0.462840
$202$		0			0
$203$	22.2623			1.56251
$204$		0			0
$205$		−	4.92629i		−	0.344067i
$206$		0			0
$207$			8.87858i			0.617104i
$208$		0			0
$209$	8.15839	+	4.77484i	0.564328	+	0.330282i
$210$		0			0
$211$	20.4764			1.40965			0.704827	−	0.709379i	$-0.251025\pi$
$211$	20.4764			1.40965			0.704827	+	0.709379i	$0.251025\pi$
$212$		0			0
$213$	5.35406			0.366854
$214$		0			0
$215$			9.57516i			0.653021i
$216$		0			0
$217$		−	5.19391i		−	0.352585i
$218$		0			0
$219$	−2.95056			−0.199381
$220$		0			0
$221$		−	14.2537i		−	0.958808i
$222$		0			0
$223$	−24.2530			−1.62410			−0.812051	−	0.583586i	$-0.801649\pi$
$223$	−24.2530			−1.62410			−0.812051	+	0.583586i	$0.801649\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−11.0735			−0.734976			−0.367488	−	0.930028i	$-0.619782\pi$
$227$	−11.0735			−0.734976			−0.367488	+	0.930028i	$0.619782\pi$
$228$		0			0
$229$	−20.6524			−1.36475			−0.682375	−	0.731003i	$-0.739053\pi$
$229$	−20.6524			−1.36475			−0.682375	+	0.731003i	$0.739053\pi$
$230$		0			0
$231$	−8.57117			−0.563941
$232$		0			0
$233$	−3.98277			−0.260920			−0.130460	−	0.991454i	$-0.541645\pi$
$233$	−3.98277			−0.260920			−0.130460	+	0.991454i	$0.541645\pi$
$234$		0			0
$235$		−	9.15609i		−	0.597277i
$236$		0			0
$237$	−11.1906			−0.726908
$238$		0			0
$239$			13.8555i			0.896241i	0.893973	+	0.448120i	$0.147906\pi$
$239$			13.8555i			0.896241i	−0.893973	+	0.448120i	$0.852094\pi$
$240$		0			0
$241$			11.1943i			0.721091i	0.932742	+	0.360545i	$0.117409\pi$
$241$			11.1943i			0.721091i	−0.932742	+	0.360545i	$0.882591\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−8.62061			−0.550750
$246$		0			0
$247$	−26.2864	−	15.3846i	−1.67256	−	0.978896i
$248$		0			0
$249$			4.37136i			0.277024i
$250$		0			0
$251$			17.3655i			1.09610i	0.836446	+	0.548050i	$0.184630\pi$
$251$			17.3655i			1.09610i	−0.836446	+	0.548050i	$0.815370\pi$
$252$		0			0
$253$	19.2546			1.21053
$254$		0			0
$255$	2.03991			0.127744
$256$		0			0
$257$			1.35467i			0.0845021i	0.999107	+	0.0422510i	$0.0134529\pi$
$257$			1.35467i			0.0845021i	−0.999107	+	0.0422510i	$0.986547\pi$
$258$		0			0
$259$	−7.01035			−0.435602
$260$		0			0
$261$		−	5.63276i		−	0.348659i
$262$		0			0
$263$			5.19286i			0.320205i	0.987100	+	0.160103i	$0.0511826\pi$
$263$			5.19286i			0.320205i	−0.987100	+	0.160103i	$0.948817\pi$
$264$		0			0
$265$		−	5.41448i		−	0.332609i
$266$		0			0
$267$			6.91831i			0.423394i
$268$		0			0
$269$		−	0.931395i		−	0.0567882i	−0.999597	−	0.0283941i	$-0.990961\pi$
$269$		−	0.931395i		−	0.0567882i	0.999597	−	0.0283941i	$-0.00903933\pi$
$270$		0			0
$271$		−	26.4790i		−	1.60848i	−0.594302	−	0.804242i	$-0.702572\pi$
$271$		−	26.4790i		−	1.60848i	0.594302	−	0.804242i	$-0.297428\pi$
$272$		0			0
$273$	27.6164			1.67142
$274$		0			0
$275$		−	2.16866i		−	0.130775i
$276$		0			0
$277$	6.62871			0.398281			0.199140	−	0.979971i	$-0.436185\pi$
$277$	6.62871			0.398281			0.199140	+	0.979971i	$0.436185\pi$
$278$		0			0
$279$	−1.31415			−0.0786762
$280$		0			0
$281$			8.67106i			0.517272i	0.965975	+	0.258636i	$0.0832730\pi$
$281$			8.67106i			0.517272i	−0.965975	+	0.258636i	$0.916727\pi$
$282$		0			0
$283$		−	26.2409i		−	1.55986i	−0.625866	−	0.779930i	$-0.715254\pi$
$283$		−	26.2409i		−	1.55986i	0.625866	−	0.779930i	$-0.284746\pi$
$284$		0			0
$285$	2.20175	−	3.76196i	0.130420	−	0.222839i
$286$		0			0
$287$	19.4701			1.14929
$288$		0			0
$289$	−12.8388			−0.755223
$290$		0			0
$291$		−	13.0393i		−	0.764375i
$292$		0			0
$293$			24.0239i			1.40349i	0.712426	+	0.701747i	$0.247596\pi$
$293$			24.0239i			1.40349i	−0.712426	+	0.701747i	$0.752404\pi$
$294$		0			0
$295$	−12.9348			−0.753091
$296$		0			0
$297$			2.16866i			0.125838i
$298$		0			0
$299$	−62.0385			−3.58778
$300$		0			0
$301$	−37.8438			−2.18128
$302$		0			0
$303$	−6.88415			−0.395484
$304$		0			0
$305$	1.62061			0.0927956
$306$		0			0
$307$	−15.6908			−0.895523			−0.447761	−	0.894153i	$-0.647779\pi$
$307$	−15.6908			−0.895523			−0.447761	+	0.894153i	$0.647779\pi$
$308$		0			0
$309$	−6.56189			−0.373293
$310$		0			0
$311$			5.71614i			0.324133i	0.986780	+	0.162066i	$0.0518159\pi$
$311$			5.71614i			0.324133i	−0.986780	+	0.162066i	$0.948184\pi$
$312$		0			0
$313$	2.66076			0.150395			0.0751976	−	0.997169i	$-0.476041\pi$
$313$	2.66076			0.150395			0.0751976	+	0.997169i	$0.476041\pi$
$314$		0			0
$315$			3.95229i			0.222686i
$316$		0			0
$317$			21.7787i			1.22321i	0.791163	+	0.611606i	$0.209476\pi$
$317$			21.7787i			1.22321i	−0.791163	+	0.611606i	$0.790524\pi$
$318$		0			0
$319$	−12.2155			−0.683938
$320$		0			0
$321$	9.20783			0.513931
$322$		0			0
$323$	4.49136	−	7.67403i	0.249906	−	0.426995i
$324$		0			0
$325$			6.98743i			0.387593i
$326$		0			0
$327$			17.5873i			0.972578i
$328$		0			0
$329$	36.1875			1.99508
$330$		0			0
$331$	−18.6834			−1.02693			−0.513467	−	0.858109i	$-0.671639\pi$
$331$	−18.6834			−1.02693			−0.513467	+	0.858109i	$0.671639\pi$
$332$		0			0
$333$			1.77374i			0.0972005i
$334$		0			0
$335$	−6.56189			−0.358514
$336$		0			0
$337$			21.2652i			1.15839i	0.815189	+	0.579195i	$0.196633\pi$
$337$			21.2652i			1.15839i	−0.815189	+	0.579195i	$0.803367\pi$
$338$		0			0
$339$		−	0.870836i		−	0.0472973i
$340$		0			0
$341$			2.84994i			0.154333i
$342$		0			0
$343$		−	6.40510i		−	0.345843i
$344$		0			0
$345$		−	8.87858i		−	0.478007i
$346$		0			0
$347$			18.6435i			1.00084i	0.865784	+	0.500418i	$0.166820\pi$
$347$			18.6435i			1.00084i	−0.865784	+	0.500418i	$0.833180\pi$
$348$		0			0
$349$	33.9270			1.81607			0.908036	−	0.418891i	$-0.137581\pi$
$349$	33.9270			1.81607			0.908036	+	0.418891i	$0.137581\pi$
$350$		0			0
$351$		−	6.98743i		−	0.372961i
$352$		0			0
$353$	−1.05835			−0.0563303			−0.0281652	−	0.999603i	$-0.508966\pi$
$353$	−1.05835			−0.0563303			−0.0281652	+	0.999603i	$0.508966\pi$
$354$		0			0
$355$	−5.35406			−0.284164
$356$		0			0
$357$			8.06230i			0.426702i
$358$		0			0
$359$			27.4331i			1.44786i	0.689871	+	0.723932i	$0.257667\pi$
$359$			27.4331i			1.44786i	−0.689871	+	0.723932i	$0.742333\pi$
$360$		0			0
$361$	−9.30462	−	16.5657i	−0.489717	−	0.871882i
$362$		0			0
$363$	−6.29692			−0.330503
$364$		0			0
$365$	2.95056			0.154439
$366$		0			0
$367$		−	2.09954i		−	0.109595i	−0.998497	−	0.0547975i	$-0.982549\pi$
$367$		−	2.09954i		−	0.109595i	0.998497	−	0.0547975i	$-0.0174513\pi$
$368$		0			0
$369$		−	4.92629i		−	0.256452i
$370$		0			0
$371$	21.3996			1.11101
$372$		0			0
$373$		−	7.00721i		−	0.362820i	−0.983408	−	0.181410i	$-0.941934\pi$
$373$		−	7.00721i		−	0.362820i	0.983408	−	0.181410i	$-0.0580661\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	39.3585			2.02707
$378$		0			0
$379$	−13.9907			−0.718655			−0.359328	−	0.933212i	$-0.616994\pi$
$379$	−13.9907			−0.718655			−0.359328	+	0.933212i	$0.616994\pi$
$380$		0			0
$381$	−5.61133			−0.287477
$382$		0			0
$383$	−7.72187			−0.394569			−0.197284	−	0.980346i	$-0.563212\pi$
$383$	−7.72187			−0.394569			−0.197284	+	0.980346i	$0.563212\pi$
$384$		0			0
$385$	8.57117			0.436827
$386$		0			0
$387$			9.57516i			0.486733i
$388$		0			0
$389$	−10.8868			−0.551983			−0.275991	−	0.961160i	$-0.589006\pi$
$389$	−10.8868			−0.551983			−0.275991	+	0.961160i	$0.589006\pi$
$390$		0			0
$391$		−	18.1115i		−	0.915936i
$392$		0			0
$393$			13.1751i			0.664597i
$394$		0			0
$395$	11.1906			0.563060
$396$		0			0
$397$	−23.7551			−1.19224			−0.596118	−	0.802897i	$-0.703291\pi$
$397$	−23.7551			−1.19224			−0.596118	+	0.802897i	$0.703291\pi$
$398$		0			0
$399$	14.8683	+	8.70195i	0.744348	+	0.435642i
$400$		0			0
$401$			26.6810i			1.33238i	0.745780	+	0.666192i	$0.232077\pi$
$401$			26.6810i			1.33238i	−0.745780	+	0.666192i	$0.767923\pi$
$402$		0			0
$403$		−	9.18254i		−	0.457415i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	3.84664			0.190671
$408$		0			0
$409$			24.1730i			1.19528i	0.801765	+	0.597640i	$0.203895\pi$
$409$			24.1730i			1.19528i	−0.801765	+	0.597640i	$0.796105\pi$
$410$		0			0
$411$	−13.9976			−0.690450
$412$		0			0
$413$		−	51.1219i		−	2.51555i
$414$		0			0
$415$		−	4.37136i		−	0.214582i
$416$		0			0
$417$		−	21.0062i		−	1.02868i
$418$		0			0
$419$		−	2.72367i		−	0.133060i	−0.997784	−	0.0665300i	$-0.978807\pi$
$419$		−	2.72367i		−	0.133060i	0.997784	−	0.0665300i	$-0.0211928\pi$
$420$		0			0
$421$			31.1453i			1.51793i	0.651131	+	0.758966i	$0.274295\pi$
$421$			31.1453i			1.51793i	−0.651131	+	0.758966i	$0.725705\pi$
$422$		0			0
$423$		−	9.15609i		−	0.445184i
$424$		0			0
$425$	−2.03991			−0.0989500
$426$		0			0
$427$			6.40510i			0.309965i
$428$		0			0
$429$	−15.1533			−0.731611
$430$		0			0
$431$	15.7900			0.760578			0.380289	−	0.924868i	$-0.375824\pi$
$431$	15.7900			0.760578			0.380289	+	0.924868i	$0.375824\pi$
$432$		0			0
$433$			37.1696i			1.78626i	0.449801	+	0.893129i	$0.351495\pi$
$433$			37.1696i			1.78626i	−0.449801	+	0.893129i	$0.648505\pi$
$434$		0			0
$435$			5.63276i			0.270070i
$436$		0			0
$437$	−33.4008	−	19.5484i	−1.59778	−	0.935127i
$438$		0			0
$439$	6.94673			0.331549			0.165775	−	0.986164i	$-0.446988\pi$
$439$	6.94673			0.331549			0.165775	+	0.986164i	$0.446988\pi$
$440$		0			0
$441$	−8.62061			−0.410505
$442$		0			0
$443$		−	17.3896i		−	0.826206i	−0.910684	−	0.413103i	$-0.864445\pi$
$443$		−	17.3896i		−	0.826206i	0.910684	−	0.413103i	$-0.135555\pi$
$444$		0			0
$445$		−	6.91831i		−	0.327959i
$446$		0			0
$447$	−5.98236			−0.282956
$448$		0			0
$449$		−	20.9370i		−	0.988079i	−0.869439	−	0.494040i	$-0.835520\pi$
$449$		−	20.9370i		−	0.988079i	0.869439	−	0.494040i	$-0.164480\pi$
$450$		0			0
$451$	−10.6834			−0.503064
$452$		0			0
$453$	−12.2317			−0.574694
$454$		0			0
$455$	−27.6164			−1.29468
$456$		0			0
$457$	2.09321			0.0979162			0.0489581	−	0.998801i	$-0.484410\pi$
$457$	2.09321			0.0979162			0.0489581	+	0.998801i	$0.484410\pi$
$458$		0			0
$459$	2.03991			0.0952146
$460$		0			0
$461$	−15.2212			−0.708923			−0.354462	−	0.935071i	$-0.615336\pi$
$461$	−15.2212			−0.708923			−0.354462	+	0.935071i	$0.615336\pi$
$462$		0			0
$463$			23.1211i			1.07453i	0.843415	+	0.537263i	$0.180542\pi$
$463$			23.1211i			1.07453i	−0.843415	+	0.537263i	$0.819458\pi$
$464$		0			0
$465$	1.31415			0.0609423
$466$		0			0
$467$		−	23.8388i		−	1.10313i	−0.834133	−	0.551563i	$-0.814032\pi$
$467$		−	23.8388i		−	1.10313i	0.834133	−	0.551563i	$-0.185968\pi$
$468$		0			0
$469$		−	25.9345i		−	1.19754i
$470$		0			0
$471$	6.13042			0.282475
$472$		0			0
$473$	20.7652			0.954787
$474$		0			0
$475$	−2.20175	+	3.76196i	−0.101023	+	0.172610i
$476$		0			0
$477$		−	5.41448i		−	0.247912i
$478$		0			0
$479$			3.04636i			0.139192i	0.997575	+	0.0695959i	$0.0221710\pi$
$479$			3.04636i			0.139192i	−0.997575	+	0.0695959i	$0.977829\pi$
$480$		0			0
$481$	−12.3939			−0.565113
$482$		0			0
$483$	35.0907			1.59668
$484$		0			0
$485$			13.0393i			0.592082i
$486$		0			0
$487$	31.7706			1.43967			0.719833	−	0.694148i	$-0.244219\pi$
$487$	31.7706			1.43967			0.719833	+	0.694148i	$0.244219\pi$
$488$		0			0
$489$			9.47515i			0.428481i
$490$		0			0
$491$			19.9060i			0.898347i	0.893445	+	0.449174i	$0.148282\pi$
$491$			19.9060i			0.898347i	−0.893445	+	0.449174i	$0.851718\pi$
$492$		0			0
$493$			11.4903i			0.517497i
$494$		0			0
$495$		−	2.16866i		−	0.0974739i
$496$		0			0
$497$		−	21.1608i		−	0.949191i
$498$		0			0
$499$		−	9.90847i		−	0.443564i	−0.975096	−	0.221782i	$-0.928813\pi$
$499$		−	9.90847i		−	0.443564i	0.975096	−	0.221782i	$-0.0711873\pi$
$500$		0			0
$501$	21.7817			0.973132
$502$		0			0
$503$			1.97119i			0.0878911i	0.999034	+	0.0439456i	$0.0139928\pi$
$503$			1.97119i			0.0878911i	−0.999034	+	0.0439456i	$0.986007\pi$
$504$		0			0
$505$	6.88415			0.306341
$506$		0			0
$507$	35.8242			1.59101
$508$		0			0
$509$			23.7299i			1.05181i	0.850544	+	0.525905i	$0.176273\pi$
$509$			23.7299i			1.05181i	−0.850544	+	0.525905i	$0.823727\pi$
$510$		0			0
$511$			11.6615i			0.515874i
$512$		0			0
$513$	2.20175	−	3.76196i	0.0972095	−	0.166094i
$514$		0			0
$515$	6.56189			0.289151
$516$		0			0
$517$	−19.8564			−0.873285
$518$		0			0
$519$			22.4728i			0.986445i
$520$		0			0
$521$		−	33.2733i		−	1.45773i	−0.684657	−	0.728865i	$-0.740048\pi$
$521$		−	33.2733i		−	1.45773i	0.684657	−	0.728865i	$-0.259952\pi$
$522$		0			0
$523$	40.3328			1.76363			0.881816	−	0.471594i	$-0.156321\pi$
$523$	40.3328			1.76363			0.881816	+	0.471594i	$0.156321\pi$
$524$		0			0
$525$		−	3.95229i		−	0.172492i
$526$		0			0
$527$	2.68074			0.116775
$528$		0			0
$529$	−55.8292			−2.42736
$530$		0			0
$531$	−12.9348			−0.561321
$532$		0			0
$533$	34.4221			1.49099
$534$		0			0
$535$	−9.20783			−0.398089
$536$		0			0
$537$	−3.11079			−0.134241
$538$		0			0
$539$			18.6951i			0.805257i
$540$		0			0
$541$	18.3966			0.790932			0.395466	−	0.918481i	$-0.370583\pi$
$541$	18.3966			0.790932			0.395466	+	0.918481i	$0.370583\pi$
$542$		0			0
$543$		−	2.28171i		−	0.0979177i
$544$		0			0
$545$		−	17.5873i		−	0.753355i
$546$		0			0
$547$	−8.64717			−0.369726			−0.184863	−	0.982764i	$-0.559184\pi$
$547$	−8.64717			−0.369726			−0.184863	+	0.982764i	$0.559184\pi$
$548$		0			0
$549$	1.62061			0.0691658
$550$		0			0
$551$	21.1902	+	12.4019i	0.902732	+	0.528339i
$552$		0			0
$553$			44.2285i			1.88079i
$554$		0			0
$555$		−	1.77374i		−	0.0752912i
$556$		0			0
$557$	−28.6277			−1.21300			−0.606498	−	0.795085i	$-0.707426\pi$
$557$	−28.6277			−1.21300			−0.606498	+	0.795085i	$0.707426\pi$
$558$		0			0
$559$	−66.9058			−2.82981
$560$		0			0
$561$		−	4.42386i		−	0.186775i
$562$		0			0
$563$	−9.69899			−0.408764			−0.204382	−	0.978891i	$-0.565518\pi$
$563$	−9.69899			−0.408764			−0.204382	+	0.978891i	$0.565518\pi$
$564$		0			0
$565$			0.870836i			0.0366363i
$566$		0			0
$567$			3.95229i			0.165981i
$568$		0			0
$569$		−	16.1831i		−	0.678431i	−0.940709	−	0.339215i	$-0.889838\pi$
$569$		−	16.1831i		−	0.678431i	0.940709	−	0.339215i	$-0.110162\pi$
$570$		0			0
$571$		−	5.53056i		−	0.231447i	−0.993281	−	0.115723i	$-0.963081\pi$
$571$		−	5.53056i		−	0.231447i	0.993281	−	0.115723i	$-0.0369186\pi$
$572$		0			0
$573$		−	25.4170i		−	1.06181i
$574$		0			0
$575$			8.87858i			0.370262i
$576$		0			0
$577$	−10.8363			−0.451121			−0.225560	−	0.974229i	$-0.572421\pi$
$577$	−10.8363			−0.451121			−0.225560	+	0.974229i	$0.572421\pi$
$578$		0			0
$579$			12.2011i			0.507061i
$580$		0			0
$581$	17.2769			0.716766
$582$		0			0
$583$	−11.7422			−0.486310
$584$		0			0
$585$			6.98743i			0.288895i
$586$		0			0
$587$		−	44.2720i		−	1.82730i	−0.406500	−	0.913651i	$-0.633251\pi$
$587$		−	44.2720i		−	1.82730i	0.406500	−	0.913651i	$-0.366749\pi$
$588$		0			0
$589$	2.89343	−	4.94378i	0.119222	−	0.203705i
$590$		0			0
$591$	23.2941			0.958191
$592$		0			0
$593$	13.1477			0.539912			0.269956	−	0.962873i	$-0.412991\pi$
$593$	13.1477			0.539912			0.269956	+	0.962873i	$0.412991\pi$
$594$		0			0
$595$		−	8.06230i		−	0.330522i
$596$		0			0
$597$			5.68961i			0.232860i
$598$		0			0
$599$	−16.0279			−0.654883			−0.327441	−	0.944872i	$-0.606186\pi$
$599$	−16.0279			−0.654883			−0.327441	+	0.944872i	$0.606186\pi$
$600$		0			0
$601$			7.74553i			0.315947i	0.987443	+	0.157973i	$0.0504960\pi$
$601$			7.74553i			0.315947i	−0.987443	+	0.157973i	$0.949504\pi$
$602$		0			0
$603$	−6.56189			−0.267221
$604$		0			0
$605$	6.29692			0.256006
$606$		0			0
$607$	4.88150			0.198134			0.0990670	−	0.995081i	$-0.468414\pi$
$607$	4.88150			0.198134			0.0990670	+	0.995081i	$0.468414\pi$
$608$		0			0
$609$	−22.2623			−0.902114
$610$		0			0
$611$	63.9775			2.58825
$612$		0			0
$613$	−19.2515			−0.777563			−0.388781	−	0.921330i	$-0.627104\pi$
$613$	−19.2515			−0.777563			−0.388781	+	0.921330i	$0.627104\pi$
$614$		0			0
$615$			4.92629i			0.198647i
$616$		0			0
$617$	−19.1822			−0.772248			−0.386124	−	0.922447i	$-0.626186\pi$
$617$	−19.1822			−0.772248			−0.386124	+	0.922447i	$0.626186\pi$
$618$		0			0
$619$		−	11.0144i		−	0.442705i	−0.975194	−	0.221353i	$-0.928953\pi$
$619$		−	11.0144i		−	0.442705i	0.975194	−	0.221353i	$-0.0710472\pi$
$620$		0			0
$621$		−	8.87858i		−	0.356285i
$622$		0			0
$623$	27.3432			1.09548
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−8.15839	−	4.77484i	−0.325815	−	0.190689i
$628$		0			0
$629$		−	3.61827i		−	0.144270i
$630$		0			0
$631$		−	14.9494i		−	0.595127i	−0.954702	−	0.297563i	$-0.903826\pi$
$631$		−	14.9494i		−	0.595127i	0.954702	−	0.297563i	$-0.0961740\pi$
$632$		0			0
$633$	−20.4764			−0.813864
$634$		0			0
$635$	5.61133			0.222679
$636$		0			0
$637$		−	60.2359i		−	2.38663i
$638$		0			0
$639$	−5.35406			−0.211803
$640$		0			0
$641$			42.1693i			1.66559i	0.553584	+	0.832794i	$0.313260\pi$
$641$			42.1693i			1.66559i	−0.553584	+	0.832794i	$0.686740\pi$
$642$		0			0
$643$		−	16.9247i		−	0.667446i	−0.942671	−	0.333723i	$-0.891695\pi$
$643$		−	16.9247i		−	0.667446i	0.942671	−	0.333723i	$-0.108305\pi$
$644$		0			0
$645$		−	9.57516i		−	0.377022i
$646$		0			0
$647$		−	5.87195i		−	0.230850i	−0.993316	−	0.115425i	$-0.963177\pi$
$647$		−	5.87195i		−	0.230850i	0.993316	−	0.115425i	$-0.0368230\pi$
$648$		0			0
$649$			28.0511i			1.10110i
$650$		0			0
$651$			5.19391i			0.203565i
$652$		0			0
$653$	34.4871			1.34959			0.674793	−	0.738007i	$-0.264233\pi$
$653$	34.4871			1.34959			0.674793	+	0.738007i	$0.264233\pi$
$654$		0			0
$655$		−	13.1751i		−	0.514795i
$656$		0			0
$657$	2.95056			0.115112
$658$		0			0
$659$	5.72034			0.222833			0.111416	−	0.993774i	$-0.464461\pi$
$659$	5.72034			0.222833			0.111416	+	0.993774i	$0.464461\pi$
$660$		0			0
$661$		−	23.5610i		−	0.916415i	−0.888845	−	0.458208i	$-0.848492\pi$
$661$		−	23.5610i		−	0.916415i	0.888845	−	0.458208i	$-0.151508\pi$
$662$		0			0
$663$			14.2537i			0.553568i
$664$		0			0
$665$	−14.8683	−	8.70195i	−0.576570	−	0.337447i
$666$		0			0
$667$	50.0109			1.93643
$668$		0			0
$669$	24.2530			0.937676
$670$		0			0
$671$		−	3.51454i		−	0.135677i
$672$		0			0
$673$			34.9200i			1.34607i	0.739611	+	0.673034i	$0.235009\pi$
$673$			34.9200i			1.34607i	−0.739611	+	0.673034i	$0.764991\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$		−	39.6191i		−	1.52269i	−0.648349	−	0.761343i	$-0.724540\pi$
$677$		−	39.6191i		−	1.52269i	0.648349	−	0.761343i	$-0.275460\pi$
$678$		0			0
$679$	−51.5350			−1.97773
$680$		0			0
$681$	11.0735			0.424339
$682$		0			0
$683$	11.9690			0.457980			0.228990	−	0.973429i	$-0.426458\pi$
$683$	11.9690			0.457980			0.228990	+	0.973429i	$0.426458\pi$
$684$		0			0
$685$	13.9976			0.534821
$686$		0			0
$687$	20.6524			0.787938
$688$		0			0
$689$	37.8333			1.44133
$690$		0			0
$691$		−	12.6282i		−	0.480399i	−0.970724	−	0.240200i	$-0.922787\pi$
$691$		−	12.6282i		−	0.480399i	0.970724	−	0.240200i	$-0.0772129\pi$
$692$		0			0
$693$	8.57117			0.325592
$694$		0			0
$695$			21.0062i			0.796812i
$696$		0			0
$697$			10.0492i			0.380639i
$698$		0			0
$699$	3.98277			0.150642
$700$		0			0
$701$	−8.33677			−0.314875			−0.157438	−	0.987529i	$-0.550323\pi$
$701$	−8.33677			−0.314875			−0.157438	+	0.987529i	$0.550323\pi$
$702$		0			0
$703$	−6.67274	−	3.90533i	−0.251667	−	0.147292i
$704$		0			0
$705$			9.15609i			0.344838i
$706$		0			0
$707$			27.2082i			1.02327i
$708$		0			0
$709$	−9.89450			−0.371596			−0.185798	−	0.982588i	$-0.559487\pi$
$709$	−9.89450			−0.371596			−0.185798	+	0.982588i	$0.559487\pi$
$710$		0			0
$711$	11.1906			0.419680
$712$		0			0
$713$		−	11.6678i		−	0.436962i
$714$		0			0
$715$	15.1533			0.566703
$716$		0			0
$717$		−	13.8555i		−	0.517445i
$718$		0			0
$719$			3.30971i			0.123431i	0.998094	+	0.0617157i	$0.0196572\pi$
$719$			3.30971i			0.123431i	−0.998094	+	0.0617157i	$0.980343\pi$
$720$		0			0
$721$			25.9345i			0.965851i
$722$		0			0
$723$		−	11.1943i		−	0.416322i
$724$		0			0
$725$		−	5.63276i		−	0.209195i
$726$		0			0
$727$		−	39.2801i		−	1.45682i	−0.685143	−	0.728409i	$-0.740260\pi$
$727$		−	39.2801i		−	1.45682i	0.685143	−	0.728409i	$-0.259740\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	19.5324i		−	0.722433i
$732$		0			0
$733$	−1.89079			−0.0698379			−0.0349189	−	0.999390i	$-0.511117\pi$
$733$	−1.89079			−0.0698379			−0.0349189	+	0.999390i	$0.511117\pi$
$734$		0			0
$735$	8.62061			0.317976
$736$		0			0
$737$			14.2305i			0.524187i
$738$		0			0
$739$		−	28.4982i		−	1.04832i	−0.851619	−	0.524161i	$-0.824379\pi$
$739$		−	28.4982i		−	1.04832i	0.851619	−	0.524161i	$-0.175621\pi$
$740$		0			0
$741$	26.2864	+	15.3846i	0.965655	+	0.565166i
$742$		0			0
$743$	−4.36639			−0.160187			−0.0800936	−	0.996787i	$-0.525522\pi$
$743$	−4.36639			−0.160187			−0.0800936	+	0.996787i	$0.525522\pi$
$744$		0			0
$745$	5.98236			0.219177
$746$		0			0
$747$		−	4.37136i		−	0.159940i
$748$		0			0
$749$		−	36.3920i		−	1.32974i
$750$		0			0
$751$	2.16726			0.0790845			0.0395422	−	0.999218i	$-0.487410\pi$
$751$	2.16726			0.0790845			0.0395422	+	0.999218i	$0.487410\pi$
$752$		0			0
$753$		−	17.3655i		−	0.632834i
$754$		0			0
$755$	12.2317			0.445156
$756$		0			0
$757$	42.5274			1.54568			0.772842	−	0.634599i	$-0.218835\pi$
$757$	42.5274			1.54568			0.772842	+	0.634599i	$0.218835\pi$
$758$		0			0
$759$	−19.2546			−0.698898
$760$		0			0
$761$	18.7893			0.681113			0.340557	−	0.940224i	$-0.389384\pi$
$761$	18.7893			0.681113			0.340557	+	0.940224i	$0.389384\pi$
$762$		0			0
$763$	69.5100			2.51643
$764$		0			0
$765$	−2.03991			−0.0737529
$766$		0			0
$767$		−	90.3807i		−	3.26346i
$768$		0			0
$769$	44.0559			1.58870			0.794349	−	0.607462i	$-0.207812\pi$
$769$	44.0559			1.58870			0.794349	+	0.607462i	$0.207812\pi$
$770$		0			0
$771$		−	1.35467i		−	0.0487873i
$772$		0			0
$773$			31.9855i			1.15044i	0.817999	+	0.575220i	$0.195084\pi$
$773$			31.9855i			1.15044i	−0.817999	+	0.575220i	$0.804916\pi$
$774$		0			0
$775$	−1.31415			−0.0472057
$776$		0			0
$777$	7.01035			0.251495
$778$		0			0
$779$	18.5325	+	10.8464i	0.663995	+	0.388614i
$780$		0			0
$781$			11.6111i			0.415478i
$782$		0			0
$783$			5.63276i			0.201298i
$784$		0			0
$785$	−6.13042			−0.218804
$786$		0			0
$787$	3.82963			0.136512			0.0682558	−	0.997668i	$-0.478257\pi$
$787$	3.82963			0.136512			0.0682558	+	0.997668i	$0.478257\pi$
$788$		0			0
$789$		−	5.19286i		−	0.184871i
$790$		0			0
$791$	−3.44180			−0.122376
$792$		0			0
$793$			11.3239i			0.402122i
$794$		0			0
$795$			5.41448i			0.192032i
$796$		0			0
$797$		−	24.2446i		−	0.858788i	−0.903117	−	0.429394i	$-0.858727\pi$
$797$		−	24.2446i		−	0.858788i	0.903117	−	0.429394i	$-0.141273\pi$
$798$		0			0
$799$			18.6776i			0.660765i
$800$		0			0
$801$		−	6.91831i		−	0.244447i
$802$		0			0
$803$		−	6.39876i		−	0.225807i
$804$		0			0
$805$	−35.0907			−1.23679
$806$		0			0
$807$			0.931395i			0.0327867i
$808$		0			0
$809$	19.6643			0.691361			0.345681	−	0.938352i	$-0.387648\pi$
$809$	19.6643			0.691361			0.345681	+	0.938352i	$0.387648\pi$
$810$		0			0
$811$	4.14796			0.145655			0.0728273	−	0.997345i	$-0.476798\pi$
$811$	4.14796			0.145655			0.0728273	+	0.997345i	$0.476798\pi$
$812$		0			0
$813$			26.4790i			0.928658i
$814$		0			0
$815$		−	9.47515i		−	0.331900i
$816$		0			0
$817$	−36.0213	−	21.0821i	−1.26023	−	0.737569i
$818$		0			0
$819$	−27.6164			−0.964994
$820$		0			0
$821$	6.77273			0.236370			0.118185	−	0.992992i	$-0.462292\pi$
$821$	6.77273			0.236370			0.118185	+	0.992992i	$0.462292\pi$
$822$		0			0
$823$			10.4578i			0.364536i	0.983249	+	0.182268i	$0.0583438\pi$
$823$			10.4578i			0.364536i	−0.983249	+	0.182268i	$0.941656\pi$
$824$		0			0
$825$			2.16866i			0.0755030i
$826$		0			0
$827$	−56.0931			−1.95055			−0.975274	−	0.220997i	$-0.929069\pi$
$827$	−56.0931			−1.95055			−0.975274	+	0.220997i	$0.929069\pi$
$828$		0			0
$829$		−	24.6745i		−	0.856981i	−0.903546	−	0.428491i	$-0.859045\pi$
$829$		−	24.6745i		−	0.856981i	0.903546	−	0.428491i	$-0.140955\pi$
$830$		0			0
$831$	−6.62871			−0.229948
$832$		0			0
$833$	17.5852			0.609292
$834$		0			0
$835$	−21.7817			−0.753785
$836$		0			0
$837$	1.31415			0.0454237
$838$		0			0
$839$	13.9966			0.483217			0.241608	−	0.970374i	$-0.422325\pi$
$839$	13.9966			0.483217			0.241608	+	0.970374i	$0.422325\pi$
$840$		0			0
$841$	−2.72798			−0.0940683
$842$		0			0
$843$		−	8.67106i		−	0.298647i
$844$		0			0
$845$	−35.8242			−1.23239
$846$		0			0
$847$			24.8873i			0.855137i
$848$		0			0
$849$			26.2409i			0.900586i
$850$		0			0
$851$	−15.7483			−0.539845
$852$		0			0
$853$	6.62871			0.226963			0.113481	−	0.993540i	$-0.463800\pi$
$853$	6.62871			0.226963			0.113481	+	0.993540i	$0.463800\pi$
$854$		0			0
$855$	−2.20175	+	3.76196i	−0.0752982	+	0.128656i
$856$		0			0
$857$			52.9338i			1.80818i	0.427338	+	0.904092i	$0.359452\pi$
$857$			52.9338i			1.80818i	−0.427338	+	0.904092i	$0.640548\pi$
$858$		0			0
$859$			24.9080i			0.849851i	0.905228	+	0.424925i	$0.139700\pi$
$859$			24.9080i			0.849851i	−0.905228	+	0.424925i	$0.860300\pi$
$860$		0			0
$861$	−19.4701			−0.663541
$862$		0			0
$863$	44.7187			1.52224			0.761121	−	0.648610i	$-0.224649\pi$
$863$	44.7187			1.52224			0.761121	+	0.648610i	$0.224649\pi$
$864$		0			0
$865$		−	22.4728i		−	0.764097i
$866$		0			0
$867$	12.8388			0.436028
$868$		0			0
$869$		−	24.2686i		−	0.823256i
$870$		0			0
$871$		−	45.8507i		−	1.55359i
$872$		0			0
$873$			13.0393i			0.441312i
$874$		0			0
$875$			3.95229i			0.133612i
$876$		0			0
$877$			20.8670i			0.704630i	0.935881	+	0.352315i	$0.114605\pi$
$877$			20.8670i			0.704630i	−0.935881	+	0.352315i	$0.885395\pi$
$878$		0			0
$879$		−	24.0239i		−	0.810308i
$880$		0			0
$881$	38.2852			1.28986			0.644930	−	0.764241i	$-0.276886\pi$
$881$	38.2852			1.28986			0.644930	+	0.764241i	$0.276886\pi$
$882$		0			0
$883$			46.0029i			1.54812i	0.633111	+	0.774061i	$0.281778\pi$
$883$			46.0029i			1.54812i	−0.633111	+	0.774061i	$0.718222\pi$
$884$		0			0
$885$	12.9348			0.434797
$886$		0			0
$887$	48.4722			1.62754			0.813768	−	0.581190i	$-0.197413\pi$
$887$	48.4722			1.62754			0.813768	+	0.581190i	$0.197413\pi$
$888$		0			0
$889$			22.1776i			0.743813i
$890$		0			0
$891$		−	2.16866i		−	0.0726528i
$892$		0			0
$893$	34.4448	+	20.1594i	1.15265	+	0.674608i
$894$		0			0
$895$	3.11079			0.103982
$896$		0			0
$897$	62.0385			2.07140
$898$		0			0
$899$			7.40230i			0.246880i
$900$		0			0
$901$			11.0450i			0.367963i
$902$		0			0
$903$	37.8438			1.25936
$904$		0			0
$905$			2.28171i			0.0758467i
$906$		0			0
$907$	32.5139			1.07961			0.539803	−	0.841792i	$-0.318499\pi$
$907$	32.5139			1.07961			0.539803	+	0.841792i	$0.318499\pi$
$908$		0			0
$909$	6.88415			0.228333
$910$		0			0
$911$	−13.4971			−0.447177			−0.223589	−	0.974684i	$-0.571777\pi$
$911$	−13.4971			−0.447177			−0.223589	+	0.974684i	$0.571777\pi$
$912$		0			0
$913$	−9.47999			−0.313742
$914$		0			0
$915$	−1.62061			−0.0535756
$916$		0			0
$917$	52.0719			1.71957
$918$		0			0
$919$			1.05976i			0.0349583i	0.999847	+	0.0174792i	$0.00556408\pi$
$919$			1.05976i			0.0349583i	−0.999847	+	0.0174792i	$0.994436\pi$
$920$		0			0
$921$	15.6908			0.517030
$922$		0			0
$923$		−	37.4111i		−	1.23140i
$924$		0			0
$925$			1.77374i			0.0583203i
$926$		0			0
$927$	6.56189			0.215521
$928$		0			0
$929$	39.3492			1.29101			0.645503	−	0.763757i	$-0.276648\pi$
$929$	39.3492			1.29101			0.645503	+	0.763757i	$0.276648\pi$
$930$		0			0
$931$	18.9804	−	32.4303i	0.622057	−	1.06286i
$932$		0			0
$933$		−	5.71614i		−	0.187138i
$934$		0			0
$935$			4.42386i			0.144676i
$936$		0			0
$937$	−12.6710			−0.413944			−0.206972	−	0.978347i	$-0.566361\pi$
$937$	−12.6710			−0.413944			−0.206972	+	0.978347i	$0.566361\pi$
$938$		0			0
$939$	−2.66076			−0.0868308
$940$		0			0
$941$			28.5980i			0.932268i	0.884714	+	0.466134i	$0.154354\pi$
$941$			28.5980i			0.932268i	−0.884714	+	0.466134i	$0.845646\pi$
$942$		0			0
$943$	43.7385			1.42432
$944$		0			0
$945$		−	3.95229i		−	0.128568i
$946$		0			0
$947$			31.7407i			1.03143i	0.856759	+	0.515717i	$0.172474\pi$
$947$			31.7407i			1.03143i	−0.856759	+	0.515717i	$0.827526\pi$
$948$		0			0
$949$			20.6169i			0.669251i
$950$		0			0
$951$		−	21.7787i		−	0.706221i
$952$		0			0
$953$			59.1874i			1.91727i	0.284642	+	0.958634i	$0.408125\pi$
$953$			59.1874i			1.91727i	−0.284642	+	0.958634i	$0.591875\pi$
$954$		0			0
$955$			25.4170i			0.822475i
$956$		0			0
$957$	12.2155			0.394872
$958$		0			0
$959$			55.3226i			1.78646i
$960$		0			0
$961$	−29.2730			−0.944291
$962$		0			0
$963$	−9.20783			−0.296718
$964$		0			0
$965$		−	12.2011i		−	0.392768i
$966$		0			0
$967$		−	23.4451i		−	0.753943i	−0.926225	−	0.376972i	$-0.876965\pi$
$967$		−	23.4451i		−	0.753943i	0.926225	−	0.376972i	$-0.123035\pi$
$968$		0			0
$969$	−4.49136	+	7.67403i	−0.144283	+	0.246526i
$970$		0			0
$971$	−23.0763			−0.740555			−0.370277	−	0.928921i	$-0.620737\pi$
$971$	−23.0763			−0.740555			−0.370277	+	0.928921i	$0.620737\pi$
$972$		0			0
$973$	−83.0227			−2.66159
$974$		0			0
$975$		−	6.98743i		−	0.223777i
$976$		0			0
$977$			52.1745i			1.66921i	0.550848	+	0.834606i	$0.314305\pi$
$977$			52.1745i			1.66921i	−0.550848	+	0.834606i	$0.685695\pi$
$978$		0			0
$979$	−15.0035			−0.479512
$980$		0			0
$981$		−	17.5873i		−	0.561518i
$982$		0			0
$983$	−13.6841			−0.436455			−0.218228	−	0.975898i	$-0.570028\pi$
$983$	−13.6841			−0.436455			−0.218228	+	0.975898i	$0.570028\pi$
$984$		0			0
$985$	−23.2941			−0.742212
$986$		0			0
$987$	−36.1875			−1.15186
$988$		0			0
$989$	−85.0139			−2.70328
$990$		0			0
$991$	48.0734			1.52710			0.763550	−	0.645748i	$-0.223455\pi$
$991$	48.0734			1.52710			0.763550	+	0.645748i	$0.223455\pi$
$992$		0			0
$993$	18.6834			0.592901
$994$		0			0
$995$		−	5.68961i		−	0.180373i
$996$		0			0
$997$	3.43801			0.108883			0.0544414	−	0.998517i	$-0.482662\pi$
$997$	3.43801			0.108883			0.0544414	+	0.998517i	$0.482662\pi$
$998$		0			0
$999$		−	1.77374i		−	0.0561187i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.j.2431.11	yes	12
4.3	odd	2		4560.2.d.l.2431.2	yes	12
19.18	odd	2		4560.2.d.l.2431.11	yes	12
76.75	even	2	inner	4560.2.d.j.2431.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.j.2431.2	✓	12	76.75	even	2	inner
4560.2.d.j.2431.11	yes	12	1.1	even	1	trivial
4560.2.d.l.2431.2	yes	12	4.3	odd	2
4560.2.d.l.2431.11	yes	12	19.18	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.95229i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +3.95229i q^{7} +1.00000 q^{9} -2.16866i q^{11} +6.98743i q^{13} -1.00000 q^{15} -2.03991 q^{17} +(-2.20175 + 3.76196i) q^{19} -3.95229i q^{21} +8.87858i q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.63276i q^{29} -1.31415 q^{31} +2.16866i q^{33} +3.95229i q^{35} +1.77374i q^{37} -6.98743i q^{39} -4.92629i q^{41} +9.57516i q^{43} +1.00000 q^{45} -9.15609i q^{47} -8.62061 q^{49} +2.03991 q^{51} -5.41448i q^{53} -2.16866i q^{55} +(2.20175 - 3.76196i) q^{57} -12.9348 q^{59} +1.62061 q^{61} +3.95229i q^{63} +6.98743i q^{65} -6.56189 q^{67} -8.87858i q^{69} -5.35406 q^{71} +2.95056 q^{73} -1.00000 q^{75} +8.57117 q^{77} +11.1906 q^{79} +1.00000 q^{81} -4.37136i q^{83} -2.03991 q^{85} +5.63276i q^{87} -6.91831i q^{89} -27.6164 q^{91} +1.31415 q^{93} +(-2.20175 + 3.76196i) q^{95} +13.0393i q^{97} -2.16866i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 + 12 * q^5 + 12 * q^9	\( 12 q - 12 q^{3} + 12 q^{5} + 12 q^{9} - 12 q^{15} + 4 q^{17} + 12 q^{25} - 12 q^{27} + 12 q^{31} + 12 q^{45} - 28 q^{49} - 4 q^{51} - 52 q^{59} - 56 q^{61} + 32 q^{67} - 8 q^{71} + 32 q^{73} - 12 q^{75} + 24 q^{77} + 28 q^{79} + 12 q^{81} + 4 q^{85} - 32 q^{91} - 12 q^{93}+O(q^{100}) \) 12 * q - 12 * q^3 + 12 * q^5 + 12 * q^9 - 12 * q^15 + 4 * q^17 + 12 * q^25 - 12 * q^27 + 12 * q^31 + 12 * q^45 - 28 * q^49 - 4 * q^51 - 52 * q^59 - 56 * q^61 + 32 * q^67 - 8 * q^71 + 32 * q^73 - 12 * q^75 + 24 * q^77 + 28 * q^79 + 12 * q^81 + 4 * q^85 - 32 * q^91 - 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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