LMFDB - Embedded newform 464.3.h.c.463.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,3,Mod(463,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("464.463");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	464.h (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:	$12.6430842663$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 58 x^{18} + 2256 x^{16} + 48540 x^{14} + 757617 x^{12} + 7081908 x^{10} + 46200312 x^{8} + \cdots + 529984 $ x^20 + 58x^18 + 2256x^16 + 48540x^14 + 757617x^12 + 7081908x^10 + 46200312x^8 + 108855714x^6 + 191707065x^4 + 10229128*x^2 + 529984
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{39}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			463.10
Root			$-0.115536 - 0.200115i$ of defining polynomial
Character	$\chi$	$=$	464.463
Dual form			464.3.h.c.463.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.231073 q^{3} +4.30394 q^{5} +6.80119i q^{7} -8.94661 q^{9} +O(q^{10})$	$q-0.231073 q^{3} +4.30394 q^{5} +6.80119i q^{7} -8.94661 q^{9} -19.6581 q^{11} -7.70811 q^{13} -0.994523 q^{15} -21.6001i q^{17} -7.18543 q^{19} -1.57157i q^{21} +9.72519i q^{23} -6.47606 q^{25} +4.14697 q^{27} +(14.1842 - 25.2945i) q^{29} +3.38352 q^{31} +4.54244 q^{33} +29.2719i q^{35} +41.4808i q^{37} +1.78113 q^{39} -24.1263i q^{41} -59.7926 q^{43} -38.5057 q^{45} -51.6582 q^{47} +2.74381 q^{49} +4.99119i q^{51} -29.5243 q^{53} -84.6072 q^{55} +1.66036 q^{57} +36.7723i q^{59} -91.5419i q^{61} -60.8476i q^{63} -33.1753 q^{65} +80.0439i q^{67} -2.24722i q^{69} +112.333i q^{71} +88.8678i q^{73} +1.49644 q^{75} -133.698i q^{77} +57.8571 q^{79} +79.5612 q^{81} -132.894i q^{83} -92.9656i q^{85} +(-3.27757 + 5.84485i) q^{87} +24.3633i q^{89} -52.4243i q^{91} -0.781838 q^{93} -30.9257 q^{95} +116.255i q^{97} +175.873 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{5} + 52 q^{9}+O(q^{10}) $ 20 * q - 8 * q^5 + 52 * q^9	$ 20 q - 8 q^{5} + 52 q^{9} - 16 q^{13} + 12 q^{25} + 4 q^{29} - 96 q^{33} - 112 q^{45} - 76 q^{49} - 112 q^{53} + 120 q^{57} + 136 q^{65} - 52 q^{81}+O(q^{100}) $ 20 * q - 8 * q^5 + 52 * q^9 - 16 * q^13 + 12 * q^25 + 4 * q^29 - 96 * q^33 - 112 * q^45 - 76 * q^49 - 112 * q^53 + 120 * q^57 + 136 * q^65 - 52 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$.

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.231073			−0.0770242			−0.0385121	−	0.999258i	$-0.512262\pi$
$3$	−0.231073			−0.0770242			−0.0385121	+	0.999258i	$0.512262\pi$
$4$		0			0
$5$	4.30394			0.860789			0.430394	−	0.902641i	$-0.358374\pi$
$5$	4.30394			0.860789			0.430394	+	0.902641i	$0.358374\pi$
$6$		0			0
$7$			6.80119i			0.971599i	0.874070	+	0.485799i	$0.161471\pi$
$7$			6.80119i			0.971599i	−0.874070	+	0.485799i	$0.838529\pi$
$8$		0			0
$9$	−8.94661			−0.994067
$10$		0			0
$11$	−19.6581			−1.78710			−0.893549	−	0.448966i	$-0.851792\pi$
$11$	−19.6581			−1.78710			−0.893549	+	0.448966i	$0.851792\pi$
$12$		0			0
$13$	−7.70811			−0.592932			−0.296466	−	0.955043i	$-0.595808\pi$
$13$	−7.70811			−0.592932			−0.296466	+	0.955043i	$0.595808\pi$
$14$		0			0
$15$	−0.994523			−0.0663015
$16$		0			0
$17$		−	21.6001i		−	1.27059i	−0.772268	−	0.635297i	$-0.780878\pi$
$17$		−	21.6001i		−	1.27059i	0.772268	−	0.635297i	$-0.219122\pi$
$18$		0			0
$19$	−7.18543			−0.378181			−0.189090	−	0.981960i	$-0.560554\pi$
$19$	−7.18543			−0.378181			−0.189090	+	0.981960i	$0.560554\pi$
$20$		0			0
$21$		−	1.57157i		−	0.0748366i
$22$		0			0
$23$			9.72519i			0.422834i	0.977396	+	0.211417i	$0.0678079\pi$
$23$			9.72519i			0.422834i	−0.977396	+	0.211417i	$0.932192\pi$
$24$		0			0
$25$	−6.47606			−0.259043
$26$		0			0
$27$	4.14697			0.153591
$28$		0			0
$29$	14.1842	−	25.2945i	0.489109	−	0.872222i
$29$	14.1842	−	25.2945i	0.489109	−	0.872222i
$30$		0			0
$31$	3.38352			0.109146			0.0545728	−	0.998510i	$-0.482620\pi$
$31$	3.38352			0.109146			0.0545728	+	0.998510i	$0.482620\pi$
$32$		0			0
$33$	4.54244			0.137650
$34$		0			0
$35$			29.2719i			0.836341i
$36$		0			0
$37$			41.4808i			1.12110i	0.828120	+	0.560551i	$0.189411\pi$
$37$			41.4808i			1.12110i	−0.828120	+	0.560551i	$0.810589\pi$
$38$		0			0
$39$	1.78113			0.0456701
$40$		0			0
$41$		−	24.1263i		−	0.588446i	−0.955737	−	0.294223i	$-0.904939\pi$
$41$		−	24.1263i		−	0.588446i	0.955737	−	0.294223i	$-0.0950608\pi$
$42$		0			0
$43$	−59.7926			−1.39052			−0.695262	−	0.718756i	$-0.744712\pi$
$43$	−59.7926			−1.39052			−0.695262	+	0.718756i	$0.744712\pi$
$44$		0			0
$45$	−38.5057			−0.855682
$46$		0			0
$47$	−51.6582			−1.09911			−0.549555	−	0.835457i	$-0.685203\pi$
$47$	−51.6582			−1.09911			−0.549555	+	0.835457i	$0.685203\pi$
$48$		0			0
$49$	2.74381			0.0559961
$50$		0			0
$51$			4.99119i			0.0978664i
$52$		0			0
$53$	−29.5243			−0.557063			−0.278531	−	0.960427i	$-0.589848\pi$
$53$	−29.5243			−0.557063			−0.278531	+	0.960427i	$0.589848\pi$
$54$		0			0
$55$	−84.6072			−1.53831
$56$		0			0
$57$	1.66036			0.0291290
$58$		0			0
$59$			36.7723i			0.623259i	0.950204	+	0.311630i	$0.100875\pi$
$59$			36.7723i			0.623259i	−0.950204	+	0.311630i	$0.899125\pi$
$60$		0			0
$61$		−	91.5419i		−	1.50069i	−0.661049	−	0.750343i	$-0.729888\pi$
$61$		−	91.5419i		−	1.50069i	0.661049	−	0.750343i	$-0.270112\pi$
$62$		0			0
$63$		−	60.8476i		−	0.965834i
$64$		0			0
$65$	−33.1753			−0.510389
$66$		0			0
$67$			80.0439i			1.19469i	0.801986	+	0.597343i	$0.203777\pi$
$67$			80.0439i			1.19469i	−0.801986	+	0.597343i	$0.796223\pi$
$68$		0			0
$69$		−	2.24722i		−	0.0325685i
$70$		0			0
$71$			112.333i			1.58216i	0.611715	+	0.791078i	$0.290480\pi$
$71$			112.333i			1.58216i	−0.611715	+	0.791078i	$0.709520\pi$
$72$		0			0
$73$			88.8678i			1.21737i	0.793413	+	0.608684i	$0.208302\pi$
$73$			88.8678i			1.21737i	−0.793413	+	0.608684i	$0.791698\pi$
$74$		0			0
$75$	1.49644			0.0199525
$76$		0			0
$77$		−	133.698i		−	1.73634i
$78$		0			0
$79$	57.8571			0.732369			0.366184	−	0.930542i	$-0.380664\pi$
$79$	57.8571			0.732369			0.366184	+	0.930542i	$0.380664\pi$
$80$		0			0
$81$	79.5612			0.982237
$82$		0			0
$83$		−	132.894i		−	1.60114i	−0.599242	−	0.800568i	$-0.704531\pi$
$83$		−	132.894i		−	1.60114i	0.599242	−	0.800568i	$-0.295469\pi$
$84$		0			0
$85$		−	92.9656i		−	1.09371i
$86$		0			0
$87$	−3.27757	+	5.84485i	−0.0376732	+	0.0671822i
$88$		0			0
$89$			24.3633i			0.273744i	0.990589	+	0.136872i	$0.0437050\pi$
$89$			24.3633i			0.273744i	−0.990589	+	0.136872i	$0.956295\pi$
$90$		0			0
$91$		−	52.4243i		−	0.576091i
$92$		0			0
$93$	−0.781838			−0.00840686
$94$		0			0
$95$	−30.9257			−0.325534
$96$		0			0
$97$			116.255i			1.19850i	0.800561	+	0.599251i	$0.204535\pi$
$97$			116.255i			1.19850i	−0.800561	+	0.599251i	$0.795465\pi$
$98$		0			0
$99$	175.873			1.77649
$100$		0			0
$101$			130.349i			1.29058i	0.763938	+	0.645290i	$0.223263\pi$
$101$			130.349i			1.29058i	−0.763938	+	0.645290i	$0.776737\pi$
$102$		0			0
$103$		−	23.7727i		−	0.230803i	−0.993319	−	0.115401i	$-0.963185\pi$
$103$		−	23.7727i		−	0.230803i	0.993319	−	0.115401i	$-0.0368155\pi$
$104$		0			0
$105$		−	6.76394i		−	0.0644185i
$106$		0			0
$107$		−	98.7482i		−	0.922881i	−0.887171	−	0.461440i	$-0.847333\pi$
$107$		−	98.7482i		−	0.922881i	0.887171	−	0.461440i	$-0.152667\pi$
$108$		0			0
$109$	−128.857			−1.18217			−0.591085	−	0.806609i	$-0.701300\pi$
$109$	−128.857			−1.18217			−0.591085	+	0.806609i	$0.701300\pi$
$110$		0			0
$111$		−	9.58506i		−	0.0863519i
$112$		0			0
$113$			187.341i			1.65789i	0.559333	+	0.828943i	$0.311057\pi$
$113$			187.341i			1.65789i	−0.559333	+	0.828943i	$0.688943\pi$
$114$		0			0
$115$			41.8567i			0.363971i
$116$		0			0
$117$	68.9614			0.589414
$118$		0			0
$119$	146.906			1.23451
$120$		0			0
$121$	265.440			2.19372
$122$		0			0
$123$			5.57492i			0.0453245i
$124$		0			0
$125$	−135.471			−1.08377
$126$		0			0
$127$	24.1239			0.189952			0.0949761	−	0.995480i	$-0.469723\pi$
$127$	24.1239			0.189952			0.0949761	+	0.995480i	$0.469723\pi$
$128$		0			0
$129$	13.8164			0.107104
$130$		0			0
$131$	−59.1156			−0.451264			−0.225632	−	0.974213i	$-0.572445\pi$
$131$	−59.1156			−0.451264			−0.225632	+	0.974213i	$0.572445\pi$
$132$		0			0
$133$		−	48.8695i		−	0.367440i
$134$		0			0
$135$	17.8483			0.132210
$136$		0			0
$137$			15.4531i			0.112796i	0.998408	+	0.0563982i	$0.0179616\pi$
$137$			15.4531i			0.112796i	−0.998408	+	0.0563982i	$0.982038\pi$
$138$		0			0
$139$		−	46.2451i		−	0.332698i	−0.986067	−	0.166349i	$-0.946802\pi$
$139$		−	46.2451i		−	0.332698i	0.986067	−	0.166349i	$-0.0531979\pi$
$140$		0			0
$141$	11.9368			0.0846581
$142$		0			0
$143$	151.527			1.05963
$144$		0			0
$145$	61.0479	−	108.866i	0.421020	−	0.750799i
$146$		0			0
$147$	−0.634019			−0.00431306
$148$		0			0
$149$	160.675			1.07835			0.539177	−	0.842193i	$-0.318735\pi$
$149$	160.675			1.07835			0.539177	+	0.842193i	$0.318735\pi$
$150$		0			0
$151$			25.8590i			0.171252i	0.996327	+	0.0856260i	$0.0272890\pi$
$151$			25.8590i			0.171252i	−0.996327	+	0.0856260i	$0.972711\pi$
$152$		0			0
$153$			193.247i			1.26306i
$154$		0			0
$155$	14.5625			0.0939514
$156$		0			0
$157$		−	147.491i		−	0.939431i	−0.882818	−	0.469716i	$-0.844356\pi$
$157$		−	147.491i		−	0.939431i	0.882818	−	0.469716i	$-0.155644\pi$
$158$		0			0
$159$	6.82226			0.0429073
$160$		0			0
$161$	−66.1429			−0.410825
$162$		0			0
$163$	97.0774			0.595567			0.297783	−	0.954633i	$-0.403753\pi$
$163$	97.0774			0.595567			0.297783	+	0.954633i	$0.403753\pi$
$164$		0			0
$165$	19.5504			0.118487
$166$		0			0
$167$		−	190.831i		−	1.14270i	−0.820707	−	0.571349i	$-0.806420\pi$
$167$		−	190.831i		−	1.14270i	0.820707	−	0.571349i	$-0.193580\pi$
$168$		0			0
$169$	−109.585			−0.648432
$170$		0			0
$171$	64.2852			0.375937
$172$		0			0
$173$	−18.7517			−0.108391			−0.0541956	−	0.998530i	$-0.517259\pi$
$173$	−18.7517			−0.108391			−0.0541956	+	0.998530i	$0.517259\pi$
$174$		0			0
$175$		−	44.0449i		−	0.251685i
$176$		0			0
$177$		−	8.49707i		−	0.0480060i
$178$		0			0
$179$		−	309.856i		−	1.73104i	−0.500876	−	0.865519i	$-0.666989\pi$
$179$		−	309.856i		−	1.73104i	0.500876	−	0.865519i	$-0.333011\pi$
$180$		0			0
$181$	−288.536			−1.59412			−0.797062	−	0.603897i	$-0.793614\pi$
$181$	−288.536			−1.59412			−0.797062	+	0.603897i	$0.793614\pi$
$182$		0			0
$183$			21.1528i			0.115589i
$184$		0			0
$185$			178.531i			0.965032i
$186$		0			0
$187$			424.616i			2.27067i
$188$		0			0
$189$			28.2043i			0.149229i
$190$		0			0
$191$	188.884			0.988921			0.494460	−	0.869200i	$-0.335366\pi$
$191$	188.884			0.988921			0.494460	+	0.869200i	$0.335366\pi$
$192$		0			0
$193$			188.056i			0.974383i	0.873295	+	0.487191i	$0.161979\pi$
$193$			188.056i			0.974383i	−0.873295	+	0.487191i	$0.838021\pi$
$194$		0			0
$195$	7.66589			0.0393123
$196$		0			0
$197$	177.823			0.902655			0.451327	−	0.892358i	$-0.350951\pi$
$197$	177.823			0.902655			0.451327	+	0.892358i	$0.350951\pi$
$198$		0			0
$199$		−	303.810i		−	1.52669i	−0.645994	−	0.763343i	$-0.723557\pi$
$199$		−	303.810i		−	1.52669i	0.645994	−	0.763343i	$-0.276443\pi$
$200$		0			0
$201$		−	18.4960i		−	0.0920197i
$202$		0			0
$203$	172.032	+	96.4693i	0.847450	+	0.475218i
$204$		0			0
$205$		−	103.838i		−	0.506527i
$206$		0			0
$207$		−	87.0075i		−	0.420326i
$208$		0			0
$209$	141.252			0.675846
$210$		0			0
$211$	−98.5614			−0.467116			−0.233558	−	0.972343i	$-0.575037\pi$
$211$	−98.5614			−0.467116			−0.233558	+	0.972343i	$0.575037\pi$
$212$		0			0
$213$		−	25.9571i		−	0.121864i
$214$		0			0
$215$	−257.344			−1.19695
$216$		0			0
$217$			23.0119i			0.106046i
$218$		0			0
$219$		−	20.5349i		−	0.0937667i
$220$		0			0
$221$			166.496i			0.753375i
$222$		0			0
$223$			72.7331i			0.326157i	0.986613	+	0.163079i	$0.0521424\pi$
$223$			72.7331i			0.326157i	−0.986613	+	0.163079i	$0.947858\pi$
$224$		0			0
$225$	57.9388			0.257506
$226$		0			0
$227$			65.0082i			0.286380i	0.989695	+	0.143190i	$0.0457359\pi$
$227$			65.0082i			0.286380i	−0.989695	+	0.143190i	$0.954264\pi$
$228$		0			0
$229$		−	218.440i		−	0.953887i	−0.878934	−	0.476943i	$-0.841745\pi$
$229$		−	218.440i		−	0.953887i	0.878934	−	0.476943i	$-0.158255\pi$
$230$		0			0
$231$			30.8940i			0.133740i
$232$		0			0
$233$	276.750			1.18777			0.593884	−	0.804551i	$-0.297594\pi$
$233$	276.750			1.18777			0.593884	+	0.804551i	$0.297594\pi$
$234$		0			0
$235$	−222.334			−0.946102
$236$		0			0
$237$	−13.3692			−0.0564101
$238$		0			0
$239$			272.798i			1.14141i	0.821154	+	0.570706i	$0.193331\pi$
$239$			272.798i			1.14141i	−0.821154	+	0.570706i	$0.806669\pi$
$240$		0			0
$241$	−0.924675			−0.00383683			−0.00191841	−	0.999998i	$-0.500611\pi$
$241$	−0.924675			−0.00383683			−0.00191841	+	0.999998i	$0.500611\pi$
$242$		0			0
$243$	−55.7071			−0.229247
$244$		0			0
$245$	11.8092			0.0482008
$246$		0			0
$247$	55.3861			0.224235
$248$		0			0
$249$			30.7082i			0.123326i
$250$		0			0
$251$	309.338			1.23242			0.616212	−	0.787580i	$-0.288666\pi$
$251$	309.338			1.23242			0.616212	+	0.787580i	$0.288666\pi$
$252$		0			0
$253$		−	191.179i		−	0.755646i
$254$		0			0
$255$			21.4818i			0.0842423i
$256$		0			0
$257$	−94.6179			−0.368163			−0.184082	−	0.982911i	$-0.558931\pi$
$257$	−94.6179			−0.368163			−0.184082	+	0.982911i	$0.558931\pi$
$258$		0			0
$259$	−282.119			−1.08926
$260$		0			0
$261$	−126.900	+	226.299i	−0.486208	+	0.867048i
$262$		0			0
$263$	155.836			0.592531			0.296265	−	0.955106i	$-0.404259\pi$
$263$	155.836			0.592531			0.296265	+	0.955106i	$0.404259\pi$
$264$		0			0
$265$	−127.071			−0.479513
$266$		0			0
$267$		−	5.62968i		−	0.0210849i
$268$		0			0
$269$		−	270.950i		−	1.00725i	−0.863923	−	0.503625i	$-0.831999\pi$
$269$		−	270.950i		−	1.00725i	0.863923	−	0.503625i	$-0.168001\pi$
$270$		0			0
$271$	−390.591			−1.44129			−0.720647	−	0.693302i	$-0.756155\pi$
$271$	−390.591			−1.44129			−0.720647	+	0.693302i	$0.756155\pi$
$272$		0			0
$273$			12.1138i			0.0443730i
$274$		0			0
$275$	127.307			0.462934
$276$		0			0
$277$	−320.161			−1.15582			−0.577909	−	0.816102i	$-0.696131\pi$
$277$	−320.161			−1.15582			−0.577909	+	0.816102i	$0.696131\pi$
$278$		0			0
$279$	−30.2710			−0.108498
$280$		0			0
$281$	40.6284			0.144585			0.0722925	−	0.997383i	$-0.476969\pi$
$281$	40.6284			0.144585			0.0722925	+	0.997383i	$0.476969\pi$
$282$		0			0
$283$		−	106.865i		−	0.377614i	−0.982014	−	0.188807i	$-0.939538\pi$
$283$		−	106.865i		−	0.377614i	0.982014	−	0.188807i	$-0.0604621\pi$
$284$		0			0
$285$	7.14608			0.0250740
$286$		0			0
$287$	164.087			0.571733
$288$		0			0
$289$	−177.564			−0.614408
$290$		0			0
$291$		−	26.8632i		−	0.0923136i
$292$		0			0
$293$			79.8459i			0.272512i	0.990674	+	0.136256i	$0.0435069\pi$
$293$			79.8459i			0.272512i	−0.990674	+	0.136256i	$0.956493\pi$
$294$		0			0
$295$			158.266i			0.536495i
$296$		0			0
$297$	−81.5214			−0.274483
$298$		0			0
$299$		−	74.9629i		−	0.250712i
$300$		0			0
$301$		−	406.661i		−	1.35103i
$302$		0			0
$303$		−	30.1200i		−	0.0994059i
$304$		0			0
$305$		−	393.991i		−	1.29177i
$306$		0			0
$307$	−548.532			−1.78675			−0.893375	−	0.449312i	$-0.851669\pi$
$307$	−548.532			−1.78675			−0.893375	+	0.449312i	$0.851669\pi$
$308$		0			0
$309$			5.49322i			0.0177774i
$310$		0			0
$311$	−559.013			−1.79747			−0.898735	−	0.438491i	$-0.855513\pi$
$311$	−559.013			−1.79747			−0.898735	+	0.438491i	$0.855513\pi$
$312$		0			0
$313$	31.9388			0.102041			0.0510204	−	0.998698i	$-0.483753\pi$
$313$	31.9388			0.102041			0.0510204	+	0.998698i	$0.483753\pi$
$314$		0			0
$315$		−	261.885i		−	0.831380i
$316$		0			0
$317$			308.715i			0.973866i	0.873439	+	0.486933i	$0.161884\pi$
$317$			308.715i			0.973866i	−0.873439	+	0.486933i	$0.838116\pi$
$318$		0			0
$319$	−278.833	+	497.240i	−0.874086	+	1.55875i
$320$		0			0
$321$			22.8180i			0.0710841i
$322$		0			0
$323$			155.206i			0.480514i
$324$		0			0
$325$	49.9182			0.153594
$326$		0			0
$327$	29.7752			0.0910557
$328$		0			0
$329$		−	351.337i		−	1.06789i
$330$		0			0
$331$	−214.596			−0.648327			−0.324164	−	0.946001i	$-0.605083\pi$
$331$	−214.596			−0.648327			−0.324164	+	0.946001i	$0.605083\pi$
$332$		0			0
$333$		−	371.112i		−	1.11445i
$334$		0			0
$335$			344.505i			1.02837i
$336$		0			0
$337$		−	243.746i		−	0.723282i	−0.932317	−	0.361641i	$-0.882217\pi$
$337$		−	243.746i		−	0.723282i	0.932317	−	0.361641i	$-0.117783\pi$
$338$		0			0
$339$		−	43.2894i		−	0.127697i
$340$		0			0
$341$	−66.5134			−0.195054
$342$		0			0
$343$			351.920i			1.02600i
$344$		0			0
$345$		−	9.67193i		−	0.0280346i
$346$		0			0
$347$			171.176i			0.493304i	0.969104	+	0.246652i	$0.0793304\pi$
$347$			171.176i			0.493304i	−0.969104	+	0.246652i	$0.920670\pi$
$348$		0			0
$349$	20.5336			0.0588356			0.0294178	−	0.999567i	$-0.490635\pi$
$349$	20.5336			0.0588356			0.0294178	+	0.999567i	$0.490635\pi$
$350$		0			0
$351$	−31.9653			−0.0910692
$352$		0			0
$353$	198.062			0.561083			0.280542	−	0.959842i	$-0.409486\pi$
$353$	198.062			0.561083			0.280542	+	0.959842i	$0.409486\pi$
$354$		0			0
$355$			483.475i			1.36190i
$356$		0			0
$357$	−33.9460			−0.0950869
$358$		0			0
$359$	237.929			0.662754			0.331377	−	0.943498i	$-0.392487\pi$
$359$	237.929			0.662754			0.331377	+	0.943498i	$0.392487\pi$
$360$		0			0
$361$	−309.370			−0.856979
$362$		0			0
$363$	−61.3358			−0.168969
$364$		0			0
$365$			382.482i			1.04790i
$366$		0			0
$367$	−616.145			−1.67887			−0.839435	−	0.543460i	$-0.817114\pi$
$367$	−616.145			−1.67887			−0.839435	+	0.543460i	$0.817114\pi$
$368$		0			0
$369$			215.848i			0.584954i
$370$		0			0
$371$		−	200.800i		−	0.541241i
$372$		0			0
$373$	549.310			1.47268			0.736341	−	0.676611i	$-0.236552\pi$
$373$	549.310			1.47268			0.736341	+	0.676611i	$0.236552\pi$
$374$		0			0
$375$	31.3037			0.0834765
$376$		0			0
$377$	−109.333	+	194.972i	−0.290008	+	0.517168i
$378$		0			0
$379$	−651.622			−1.71932			−0.859659	−	0.510868i	$-0.829324\pi$
$379$	−651.622			−1.71932			−0.859659	+	0.510868i	$0.829324\pi$
$380$		0			0
$381$	−5.57438			−0.0146309
$382$		0			0
$383$		−	449.975i		−	1.17487i	−0.809272	−	0.587434i	$-0.800138\pi$
$383$		−	449.975i		−	1.17487i	0.809272	−	0.587434i	$-0.199862\pi$
$384$		0			0
$385$		−	575.430i		−	1.49462i
$386$		0			0
$387$	534.940			1.38227
$388$		0			0
$389$			405.267i			1.04182i	0.853612	+	0.520909i	$0.174407\pi$
$389$			405.267i			1.04182i	−0.853612	+	0.520909i	$0.825593\pi$
$390$		0			0
$391$	210.065			0.537251
$392$		0			0
$393$	13.6600			0.0347583
$394$		0			0
$395$	249.014			0.630415
$396$		0			0
$397$	−88.5209			−0.222975			−0.111487	−	0.993766i	$-0.535561\pi$
$397$	−88.5209			−0.222975			−0.111487	+	0.993766i	$0.535561\pi$
$398$		0			0
$399$			11.2924i			0.0283017i
$400$		0			0
$401$	−592.146			−1.47667			−0.738337	−	0.674432i	$-0.764388\pi$
$401$	−592.146			−1.47667			−0.738337	+	0.674432i	$0.764388\pi$
$402$		0			0
$403$	−26.0805			−0.0647159
$404$		0			0
$405$	342.427			0.845499
$406$		0			0
$407$		−	815.432i		−	2.00352i
$408$		0			0
$409$			771.560i			1.88645i	0.332148	+	0.943227i	$0.392227\pi$
$409$			771.560i			1.88645i	−0.332148	+	0.943227i	$0.607773\pi$
$410$		0			0
$411$		−	3.57079i		−	0.00868805i
$412$		0			0
$413$	−250.095			−0.605558
$414$		0			0
$415$		−	571.970i		−	1.37824i
$416$		0			0
$417$			10.6860i			0.0256258i
$418$		0			0
$419$		−	28.5209i		−	0.0680689i	−0.999421	−	0.0340344i	$-0.989164\pi$
$419$		−	28.5209i		−	0.0680689i	0.999421	−	0.0340344i	$-0.0108356\pi$
$420$		0			0
$421$		−	169.786i		−	0.403292i	−0.979459	−	0.201646i	$-0.935371\pi$
$421$		−	169.786i		−	0.403292i	0.979459	−	0.201646i	$-0.0646290\pi$
$422$		0			0
$423$	462.165			1.09259
$424$		0			0
$425$			139.884i			0.329138i
$426$		0			0
$427$	622.594			1.45806
$428$		0			0
$429$	−35.0136			−0.0816168
$430$		0			0
$431$		−	107.397i		−	0.249181i	−0.992208	−	0.124591i	$-0.960238\pi$
$431$		−	107.397i		−	0.249181i	0.992208	−	0.124591i	$-0.0397617\pi$
$432$		0			0
$433$		−	45.2546i		−	0.104514i	−0.998634	−	0.0522570i	$-0.983358\pi$
$433$		−	45.2546i		−	0.104514i	0.998634	−	0.0522570i	$-0.0166415\pi$
$434$		0			0
$435$	−14.1065	+	25.1559i	−0.0324287	+	0.0578297i
$436$		0			0
$437$		−	69.8797i		−	0.159908i
$438$		0			0
$439$			769.460i			1.75276i	0.481624	+	0.876378i	$0.340047\pi$
$439$			769.460i			1.75276i	−0.481624	+	0.876378i	$0.659953\pi$
$440$		0			0
$441$	−24.5478			−0.0556639
$442$		0			0
$443$	389.900			0.880136			0.440068	−	0.897965i	$-0.354954\pi$
$443$	389.900			0.880136			0.440068	+	0.897965i	$0.354954\pi$
$444$		0			0
$445$			104.858i			0.235636i
$446$		0			0
$447$	−37.1275			−0.0830593
$448$		0			0
$449$		−	657.473i		−	1.46431i	−0.681140	−	0.732153i	$-0.738516\pi$
$449$		−	657.473i		−	1.46431i	0.681140	−	0.732153i	$-0.261484\pi$
$450$		0			0
$451$			474.276i			1.05161i
$452$		0			0
$453$		−	5.97531i		−	0.0131905i
$454$		0			0
$455$		−	225.631i		−	0.495893i
$456$		0			0
$457$	519.676			1.13715			0.568573	−	0.822633i	$-0.307496\pi$
$457$	519.676			1.13715			0.568573	+	0.822633i	$0.307496\pi$
$458$		0			0
$459$		−	89.5749i		−	0.195152i
$460$		0			0
$461$		−	190.635i		−	0.413524i	−0.978391	−	0.206762i	$-0.933707\pi$
$461$		−	190.635i		−	0.413524i	0.978391	−	0.206762i	$-0.0662926\pi$
$462$		0			0
$463$		−	246.775i		−	0.532990i	−0.963836	−	0.266495i	$-0.914134\pi$
$463$		−	246.775i		−	0.532990i	0.963836	−	0.266495i	$-0.0858657\pi$
$464$		0			0
$465$	−3.36499			−0.00723653
$466$		0			0
$467$	−463.418			−0.992330			−0.496165	−	0.868228i	$-0.665259\pi$
$467$	−463.418			−0.992330			−0.496165	+	0.868228i	$0.665259\pi$
$468$		0			0
$469$	−544.394			−1.16075
$470$		0			0
$471$			34.0810i			0.0723589i
$472$		0			0
$473$	1175.41			2.48500
$474$		0			0
$475$	46.5333			0.0979649
$476$		0			0
$477$	264.142			0.553758
$478$		0			0
$479$	335.918			0.701291			0.350646	−	0.936508i	$-0.385962\pi$
$479$	335.918			0.701291			0.350646	+	0.936508i	$0.385962\pi$
$480$		0			0
$481$		−	319.738i		−	0.664737i
$482$		0			0
$483$	15.2838			0.0316435
$484$		0			0
$485$			500.353i			1.03166i
$486$		0			0
$487$			235.047i			0.482642i	0.970445	+	0.241321i	$0.0775806\pi$
$487$			235.047i			0.482642i	−0.970445	+	0.241321i	$0.922419\pi$
$488$		0			0
$489$	−22.4319			−0.0458730
$490$		0			0
$491$	−548.826			−1.11777			−0.558886	−	0.829245i	$-0.688771\pi$
$491$	−548.826			−1.11777			−0.558886	+	0.829245i	$0.688771\pi$
$492$		0			0
$493$	−546.362	−	306.379i	−1.10824	−	0.621459i
$494$		0			0
$495$	756.948			1.52919
$496$		0			0
$497$	−763.999			−1.53722
$498$		0			0
$499$			479.046i			0.960011i	0.877265	+	0.480006i	$0.159365\pi$
$499$			479.046i			0.960011i	−0.877265	+	0.480006i	$0.840635\pi$
$500$		0			0
$501$			44.0957i			0.0880154i
$502$		0			0
$503$	371.364			0.738297			0.369149	−	0.929370i	$-0.379649\pi$
$503$	371.364			0.738297			0.369149	+	0.929370i	$0.379649\pi$
$504$		0			0
$505$			561.013i			1.11092i
$506$		0			0
$507$	25.3221			0.0499449
$508$		0			0
$509$	212.182			0.416860			0.208430	−	0.978037i	$-0.433165\pi$
$509$	212.182			0.416860			0.208430	+	0.978037i	$0.433165\pi$
$510$		0			0
$511$	−604.407			−1.18279
$512$		0			0
$513$	−29.7978			−0.0580853
$514$		0			0
$515$		−	102.316i		−	0.198673i
$516$		0			0
$517$	1015.50			1.96422
$518$		0			0
$519$	4.33300			0.00834874
$520$		0			0
$521$	−171.625			−0.329415			−0.164707	−	0.986342i	$-0.552668\pi$
$521$	−171.625			−0.329415			−0.164707	+	0.986342i	$0.552668\pi$
$522$		0			0
$523$		−	889.718i		−	1.70118i	−0.525829	−	0.850591i	$-0.676245\pi$
$523$		−	889.718i		−	1.70118i	0.525829	−	0.850591i	$-0.323755\pi$
$524$		0			0
$525$			10.1776i			0.0193859i
$526$		0			0
$527$		−	73.0843i		−	0.138680i
$528$		0			0
$529$	434.421			0.821211
$530$		0			0
$531$		−	328.987i		−	0.619562i
$532$		0			0
$533$			185.968i			0.348908i
$534$		0			0
$535$		−	425.007i		−	0.794405i
$536$		0			0
$537$			71.5992i			0.133332i
$538$		0			0
$539$	−53.9380			−0.100071
$540$		0			0
$541$			336.328i			0.621679i	0.950462	+	0.310840i	$0.100610\pi$
$541$			336.328i			0.621679i	−0.950462	+	0.310840i	$0.899390\pi$
$542$		0			0
$543$	66.6729			0.122786
$544$		0			0
$545$	−554.592			−1.01760
$546$		0			0
$547$		−	360.971i		−	0.659911i	−0.943997	−	0.329955i	$-0.892966\pi$
$547$		−	360.971i		−	0.659911i	0.943997	−	0.329955i	$-0.107034\pi$
$548$		0			0
$549$			818.989i			1.49178i
$550$		0			0
$551$	−101.919	+	181.752i	−0.184972	+	0.329858i
$552$		0			0
$553$			393.497i			0.711568i
$554$		0			0
$555$		−	41.2536i		−	0.0743308i
$556$		0			0
$557$	−440.285			−0.790458			−0.395229	−	0.918583i	$-0.629335\pi$
$557$	−440.285			−0.790458			−0.395229	+	0.918583i	$0.629335\pi$
$558$		0			0
$559$	460.888			0.824486
$560$		0			0
$561$		−	98.1171i		−	0.174897i
$562$		0			0
$563$	1029.19			1.82804			0.914019	−	0.405671i	$-0.132962\pi$
$563$	1029.19			1.82804			0.914019	+	0.405671i	$0.132962\pi$
$564$		0			0
$565$			806.306i			1.42709i
$566$		0			0
$567$			541.111i			0.954340i
$568$		0			0
$569$			176.077i			0.309450i	0.987958	+	0.154725i	$0.0494492\pi$
$569$			176.077i			0.309450i	−0.987958	+	0.154725i	$0.950551\pi$
$570$		0			0
$571$		−	367.028i		−	0.642781i	−0.946947	−	0.321391i	$-0.895850\pi$
$571$		−	367.028i		−	0.642781i	0.946947	−	0.321391i	$-0.104150\pi$
$572$		0			0
$573$	−43.6459			−0.0761708
$574$		0			0
$575$		−	62.9810i		−	0.109532i
$576$		0			0
$577$		−	777.390i		−	1.34730i	−0.739053	−	0.673648i	$-0.764726\pi$
$577$		−	777.390i		−	1.34730i	0.739053	−	0.673648i	$-0.235274\pi$
$578$		0			0
$579$		−	43.4545i		−	0.0750510i
$580$		0			0
$581$	903.840			1.55566
$582$		0			0
$583$	580.391			0.995525
$584$		0			0
$585$	296.806			0.507361
$586$		0			0
$587$			558.784i			0.951932i	0.879464	+	0.475966i	$0.157901\pi$
$587$			558.784i			0.951932i	−0.879464	+	0.475966i	$0.842099\pi$
$588$		0			0
$589$	−24.3120			−0.0412768
$590$		0			0
$591$	−41.0900			−0.0695262
$592$		0			0
$593$	−678.701			−1.14452			−0.572261	−	0.820072i	$-0.693933\pi$
$593$	−678.701			−1.14452			−0.572261	+	0.820072i	$0.693933\pi$
$594$		0			0
$595$	632.277			1.06265
$596$		0			0
$597$			70.2022i			0.117592i
$598$		0			0
$599$	−142.960			−0.238664			−0.119332	−	0.992854i	$-0.538075\pi$
$599$	−142.960			−0.238664			−0.119332	+	0.992854i	$0.538075\pi$
$600$		0			0
$601$		−	423.145i		−	0.704068i	−0.935987	−	0.352034i	$-0.885490\pi$
$601$		−	423.145i		−	0.704068i	0.935987	−	0.352034i	$-0.114510\pi$
$602$		0			0
$603$		−	716.121i		−	1.18760i
$604$		0			0
$605$	1142.44			1.88833
$606$		0			0
$607$	908.045			1.49596			0.747978	−	0.663724i	$-0.231025\pi$
$607$	908.045			1.49596			0.747978	+	0.663724i	$0.231025\pi$
$608$		0			0
$609$	−39.7520	−	22.2914i	−0.0652741	−	0.0366033i
$610$		0			0
$611$	398.187			0.651697
$612$		0			0
$613$	−372.883			−0.608292			−0.304146	−	0.952625i	$-0.598371\pi$
$613$	−372.883			−0.608292			−0.304146	+	0.952625i	$0.598371\pi$
$614$		0			0
$615$			23.9941i			0.0390148i
$616$		0			0
$617$			682.584i			1.10630i	0.833083	+	0.553148i	$0.186573\pi$
$617$			682.584i			1.10630i	−0.833083	+	0.553148i	$0.813427\pi$
$618$		0			0
$619$	−368.152			−0.594752			−0.297376	−	0.954760i	$-0.596112\pi$
$619$	−368.152			−0.594752			−0.297376	+	0.954760i	$0.596112\pi$
$620$		0			0
$621$			40.3301i			0.0649437i
$622$		0			0
$623$	−165.699			−0.265970
$624$		0			0
$625$	−421.159			−0.673854
$626$		0			0
$627$	−32.6394			−0.0520564
$628$		0			0
$629$	895.988			1.42446
$630$		0			0
$631$			625.046i			0.990564i	0.868732	+	0.495282i	$0.164935\pi$
$631$			625.046i			0.990564i	−0.868732	+	0.495282i	$0.835065\pi$
$632$		0			0
$633$	22.7748			0.0359792
$634$		0			0
$635$	103.828			0.163509
$636$		0			0
$637$	−21.1496			−0.0332019
$638$		0			0
$639$		−	1005.00i		−	1.57277i
$640$		0			0
$641$		−	989.562i		−	1.54378i	−0.635757	−	0.771889i	$-0.719312\pi$
$641$		−	989.562i		−	1.54378i	0.635757	−	0.771889i	$-0.280688\pi$
$642$		0			0
$643$		−	405.454i		−	0.630565i	−0.948998	−	0.315283i	$-0.897901\pi$
$643$		−	405.454i		−	0.630565i	0.948998	−	0.315283i	$-0.102099\pi$
$644$		0			0
$645$	59.4651			0.0921939
$646$		0			0
$647$		−	1045.04i		−	1.61521i	−0.589722	−	0.807606i	$-0.700763\pi$
$647$		−	1045.04i		−	1.61521i	0.589722	−	0.807606i	$-0.299237\pi$
$648$		0			0
$649$		−	722.872i		−	1.11382i
$650$		0			0
$651$		−	5.31743i		−	0.00816809i
$652$		0			0
$653$			539.784i			0.826622i	0.910590	+	0.413311i	$0.135628\pi$
$653$			539.784i			0.826622i	−0.910590	+	0.413311i	$0.864372\pi$
$654$		0			0
$655$	−254.430			−0.388443
$656$		0			0
$657$		−	795.065i		−	1.21015i
$658$		0			0
$659$	238.329			0.361652			0.180826	−	0.983515i	$-0.442123\pi$
$659$	238.329			0.361652			0.180826	+	0.983515i	$0.442123\pi$
$660$		0			0
$661$	69.9519			0.105827			0.0529137	−	0.998599i	$-0.483149\pi$
$661$	69.9519			0.105827			0.0529137	+	0.998599i	$0.483149\pi$
$662$		0			0
$663$		−	38.4726i		−	0.0580281i
$664$		0			0
$665$		−	210.332i		−	0.316288i
$666$		0			0
$667$	245.993	+	137.944i	0.368806	+	0.206812i
$668$		0			0
$669$		−	16.8066i		−	0.0251220i
$670$		0			0
$671$			1799.54i			2.68187i
$672$		0			0
$673$	−829.767			−1.23294			−0.616469	−	0.787379i	$-0.711437\pi$
$673$	−829.767			−1.23294			−0.616469	+	0.787379i	$0.711437\pi$
$674$		0			0
$675$	−26.8560			−0.0397867
$676$		0			0
$677$			622.228i			0.919095i	0.888153	+	0.459548i	$0.151988\pi$
$677$			622.228i			0.919095i	−0.888153	+	0.459548i	$0.848012\pi$
$678$		0			0
$679$	−790.670			−1.16446
$680$		0			0
$681$		−	15.0216i		−	0.0220581i
$682$		0			0
$683$			900.254i			1.31809i	0.752105	+	0.659044i	$0.229039\pi$
$683$			900.254i			1.31809i	−0.752105	+	0.659044i	$0.770961\pi$
$684$		0			0
$685$			66.5093i			0.0970939i
$686$		0			0
$687$			50.4755i			0.0734723i
$688$		0			0
$689$	227.577			0.330300
$690$		0			0
$691$		−	574.448i		−	0.831328i	−0.909518	−	0.415664i	$-0.863549\pi$
$691$		−	574.448i		−	0.831328i	0.909518	−	0.415664i	$-0.136451\pi$
$692$		0			0
$693$			1196.15i			1.72604i
$694$		0			0
$695$		−	199.036i		−	0.286383i
$696$		0			0
$697$	−521.129			−0.747675
$698$		0			0
$699$	−63.9493			−0.0914868
$700$		0			0
$701$	−795.987			−1.13550			−0.567751	−	0.823200i	$-0.692186\pi$
$701$	−795.987			−1.13550			−0.567751	+	0.823200i	$0.692186\pi$
$702$		0			0
$703$		−	298.057i		−	0.423979i
$704$		0			0
$705$	51.3753			0.0728727
$706$		0			0
$707$	−886.526			−1.25393
$708$		0			0
$709$	−360.327			−0.508219			−0.254109	−	0.967176i	$-0.581782\pi$
$709$	−360.327			−0.508219			−0.254109	+	0.967176i	$0.581782\pi$
$710$		0			0
$711$	−517.625			−0.728024
$712$		0			0
$713$			32.9054i			0.0461506i
$714$		0			0
$715$	652.162			0.912115
$716$		0			0
$717$		−	63.0360i		−	0.0879164i
$718$		0			0
$719$		−	554.656i		−	0.771427i	−0.922619	−	0.385714i	$-0.873955\pi$
$719$		−	554.656i		−	0.771427i	0.922619	−	0.385714i	$-0.126045\pi$
$720$		0			0
$721$	161.683			0.224248
$722$		0			0
$723$	0.213667			0.000295528
$724$		0			0
$725$	−91.8576	+	163.808i	−0.126700	+	0.225943i
$726$		0			0
$727$	−888.134			−1.22164			−0.610821	−	0.791769i	$-0.709161\pi$
$727$	−888.134			−1.22164			−0.610821	+	0.791769i	$0.709161\pi$
$728$		0			0
$729$	−703.178			−0.964579
$730$		0			0
$731$			1291.52i			1.76679i
$732$		0			0
$733$			320.351i			0.437041i	0.975832	+	0.218520i	$0.0701230\pi$
$733$			320.351i			0.437041i	−0.975832	+	0.218520i	$0.929877\pi$
$734$		0			0
$735$	−2.72878			−0.00371263
$736$		0			0
$737$		−	1573.51i		−	2.13502i
$738$		0			0
$739$	−889.912			−1.20421			−0.602106	−	0.798416i	$-0.705671\pi$
$739$	−889.912			−1.20421			−0.602106	+	0.798416i	$0.705671\pi$
$740$		0			0
$741$	−12.7982			−0.0172715
$742$		0			0
$743$	81.5000			0.109690			0.0548452	−	0.998495i	$-0.482533\pi$
$743$	81.5000			0.109690			0.0548452	+	0.998495i	$0.482533\pi$
$744$		0			0
$745$	691.535			0.928235
$746$		0			0
$747$			1188.95i			1.59164i
$748$		0			0
$749$	671.606			0.896670
$750$		0			0
$751$	−1059.96			−1.41140			−0.705699	−	0.708512i	$-0.749367\pi$
$751$	−1059.96			−1.41140			−0.705699	+	0.708512i	$0.749367\pi$
$752$		0			0
$753$	−71.4796			−0.0949264
$754$		0			0
$755$			111.296i			0.147412i
$756$		0			0
$757$			299.069i			0.395071i	0.980296	+	0.197536i	$0.0632938\pi$
$757$			299.069i			0.395071i	−0.980296	+	0.197536i	$0.936706\pi$
$758$		0			0
$759$			44.1761i			0.0582030i
$760$		0			0
$761$	67.2911			0.0884245			0.0442123	−	0.999022i	$-0.485922\pi$
$761$	67.2911			0.0884245			0.0442123	+	0.999022i	$0.485922\pi$
$762$		0			0
$763$		−	876.378i		−	1.14860i
$764$		0			0
$765$			831.726i			1.08722i
$766$		0			0
$767$		−	283.445i		−	0.369550i
$768$		0			0
$769$			28.8673i			0.0375387i	0.999824	+	0.0187694i	$0.00597483\pi$
$769$			28.8673i			0.0375387i	−0.999824	+	0.0187694i	$0.994025\pi$
$770$		0			0
$771$	21.8636			0.0283575
$772$		0			0
$773$		−	365.876i		−	0.473320i	−0.971593	−	0.236660i	$-0.923947\pi$
$773$		−	365.876i		−	0.473320i	0.971593	−	0.236660i	$-0.0760527\pi$
$774$		0			0
$775$	−21.9119			−0.0282734
$776$		0			0
$777$	65.1898			0.0838994
$778$		0			0
$779$			173.358i			0.222539i
$780$		0			0
$781$		−	2208.25i		−	2.82747i
$782$		0			0
$783$	58.8213	−	104.895i	0.0751230	−	0.133966i
$784$		0			0
$785$		−	634.792i		−	0.808652i
$786$		0			0
$787$			1111.05i			1.41175i	0.708336	+	0.705875i	$0.249446\pi$
$787$			1111.05i			1.41175i	−0.708336	+	0.705875i	$0.750554\pi$
$788$		0			0
$789$	−36.0093			−0.0456392
$790$		0			0
$791$	−1274.14			−1.61080
$792$		0			0
$793$			705.615i			0.889804i
$794$		0			0
$795$	29.3626			0.0369341
$796$		0			0
$797$		−	530.802i		−	0.665999i	−0.942927	−	0.333000i	$-0.891939\pi$
$797$		−	530.802i		−	0.665999i	0.942927	−	0.333000i	$-0.108061\pi$
$798$		0			0
$799$			1115.82i			1.39652i
$800$		0			0
$801$		−	217.968i		−	0.272120i
$802$		0			0
$803$		−	1746.97i		−	2.17555i
$804$		0			0
$805$	−284.675			−0.353634
$806$		0			0
$807$			62.6091i			0.0775826i
$808$		0			0
$809$			1027.63i			1.27025i	0.772411	+	0.635123i	$0.219050\pi$
$809$			1027.63i			1.27025i	−0.772411	+	0.635123i	$0.780950\pi$
$810$		0			0
$811$			1073.28i			1.32340i	0.749768	+	0.661701i	$0.230165\pi$
$811$			1073.28i			1.32340i	−0.749768	+	0.661701i	$0.769835\pi$
$812$		0			0
$813$	90.2548			0.111015
$814$		0			0
$815$	417.816			0.512657
$816$		0			0
$817$	429.635			0.525869
$818$		0			0
$819$			469.020i			0.572674i
$820$		0			0
$821$	681.241			0.829769			0.414885	−	0.909874i	$-0.363822\pi$
$821$	681.241			0.829769			0.414885	+	0.909874i	$0.363822\pi$
$822$		0			0
$823$	839.775			1.02038			0.510192	−	0.860061i	$-0.329575\pi$
$823$	839.775			1.02038			0.510192	+	0.860061i	$0.329575\pi$
$824$		0			0
$825$	−29.4171			−0.0356571
$826$		0			0
$827$	−1539.73			−1.86183			−0.930915	−	0.365237i	$-0.880988\pi$
$827$	−1539.73			−1.86183			−0.930915	+	0.365237i	$0.880988\pi$
$828$		0			0
$829$		−	591.109i		−	0.713039i	−0.934288	−	0.356519i	$-0.883963\pi$
$829$		−	591.109i		−	0.713039i	0.934288	−	0.356519i	$-0.116037\pi$
$830$		0			0
$831$	73.9805			0.0890258
$832$		0			0
$833$		−	59.2665i		−	0.0711483i
$834$		0			0
$835$		−	821.324i		−	0.983622i
$836$		0			0
$837$	14.0313			0.0167638
$838$		0			0
$839$	−249.139			−0.296948			−0.148474	−	0.988916i	$-0.547436\pi$
$839$	−249.139			−0.296948			−0.148474	+	0.988916i	$0.547436\pi$
$840$		0			0
$841$	−438.618	−	717.562i	−0.521544	−	0.853224i
$842$		0			0
$843$	−9.38810			−0.0111365
$844$		0			0
$845$	−471.648			−0.558163
$846$		0			0
$847$			1805.31i			2.13141i
$848$		0			0
$849$			24.6935i			0.0290854i
$850$		0			0
$851$	−403.408			−0.474041
$852$		0			0
$853$		−	832.091i		−	0.975488i	−0.872987	−	0.487744i	$-0.837820\pi$
$853$		−	832.091i		−	0.975488i	0.872987	−	0.487744i	$-0.162180\pi$
$854$		0			0
$855$	276.680			0.323602
$856$		0			0
$857$	−548.077			−0.639530			−0.319765	−	0.947497i	$-0.603604\pi$
$857$	−548.077			−0.639530			−0.319765	+	0.947497i	$0.603604\pi$
$858$		0			0
$859$	706.412			0.822365			0.411183	−	0.911553i	$-0.365116\pi$
$859$	706.412			0.822365			0.411183	+	0.911553i	$0.365116\pi$
$860$		0			0
$861$	−37.9161			−0.0440372
$862$		0			0
$863$			54.8305i			0.0635348i	0.999495	+	0.0317674i	$0.0101136\pi$
$863$			54.8305i			0.0635348i	−0.999495	+	0.0317674i	$0.989886\pi$
$864$		0			0
$865$	−80.7062			−0.0933020
$866$		0			0
$867$	41.0301			0.0473242
$868$		0			0
$869$	−1137.36			−1.30881
$870$		0			0
$871$		−	616.987i		−	0.708367i
$872$		0			0
$873$		−	1040.08i		−	1.19139i
$874$		0			0
$875$		−	921.366i		−	1.05299i
$876$		0			0
$877$	−802.388			−0.914924			−0.457462	−	0.889229i	$-0.651241\pi$
$877$	−802.388			−0.914924			−0.457462	+	0.889229i	$0.651241\pi$
$878$		0			0
$879$		−	18.4502i		−	0.0209900i
$880$		0			0
$881$			513.331i			0.582669i	0.956621	+	0.291334i	$0.0940992\pi$
$881$			513.331i			0.582669i	−0.956621	+	0.291334i	$0.905901\pi$
$882$		0			0
$883$			1548.18i			1.75332i	0.481109	+	0.876661i	$0.340234\pi$
$883$			1548.18i			1.75332i	−0.481109	+	0.876661i	$0.659766\pi$
$884$		0			0
$885$		−	36.5709i		−	0.0413231i
$886$		0			0
$887$	166.748			0.187991			0.0939954	−	0.995573i	$-0.470036\pi$
$887$	166.748			0.187991			0.0939954	+	0.995573i	$0.470036\pi$
$888$		0			0
$889$			164.071i			0.184557i
$890$		0			0
$891$	−1564.02			−1.75535
$892$		0			0
$893$	371.186			0.415662
$894$		0			0
$895$		−	1333.60i		−	1.49006i
$896$		0			0
$897$			17.3219i			0.0193109i
$898$		0			0
$899$	47.9924	−	85.5842i	0.0533842	−	0.0951993i
$900$		0			0
$901$			637.728i			0.707800i
$902$		0			0
$903$			93.9681i			0.104062i
$904$		0			0
$905$	−1241.85			−1.37220
$906$		0			0
$907$	−1350.48			−1.48895			−0.744476	−	0.667649i	$-0.767301\pi$
$907$	−1350.48			−1.48895			−0.744476	+	0.667649i	$0.767301\pi$
$908$		0			0
$909$		−	1166.18i		−	1.28292i
$910$		0			0
$911$	885.235			0.971717			0.485859	−	0.874037i	$-0.338507\pi$
$911$	885.235			0.971717			0.485859	+	0.874037i	$0.338507\pi$
$912$		0			0
$913$			2612.45i			2.86139i
$914$		0			0
$915$			91.0405i			0.0994978i
$916$		0			0
$917$		−	402.057i		−	0.438448i
$918$		0			0
$919$		−	1362.45i		−	1.48253i	−0.671212	−	0.741266i	$-0.734226\pi$
$919$		−	1362.45i		−	1.48253i	0.671212	−	0.741266i	$-0.265774\pi$
$920$		0			0
$921$	126.751			0.137623
$922$		0			0
$923$		−	865.876i		−	0.938110i
$924$		0			0
$925$		−	268.632i		−	0.290413i
$926$		0			0
$927$			212.685i			0.229434i
$928$		0			0
$929$	473.689			0.509891			0.254946	−	0.966955i	$-0.417942\pi$
$929$	473.689			0.509891			0.254946	+	0.966955i	$0.417942\pi$
$930$		0			0
$931$	−19.7155			−0.0211767
$932$		0			0
$933$	129.173			0.138449
$934$		0			0
$935$			1827.52i			1.95457i
$936$		0			0
$937$	1762.54			1.88104			0.940521	−	0.339736i	$-0.110338\pi$
$937$	1762.54			1.88104			0.940521	+	0.339736i	$0.110338\pi$
$938$		0			0
$939$	−7.38017			−0.00785961
$940$		0			0
$941$	−197.827			−0.210230			−0.105115	−	0.994460i	$-0.533521\pi$
$941$	−197.827			−0.210230			−0.105115	+	0.994460i	$0.533521\pi$
$942$		0			0
$943$	234.633			0.248815
$944$		0			0
$945$			121.390i			0.128455i
$946$		0			0
$947$	−397.069			−0.419291			−0.209646	−	0.977777i	$-0.567231\pi$
$947$	−397.069			−0.419291			−0.209646	+	0.977777i	$0.567231\pi$
$948$		0			0
$949$		−	685.003i		−	0.721816i
$950$		0			0
$951$		−	71.3356i		−	0.0750112i
$952$		0			0
$953$	46.1185			0.0483929			0.0241965	−	0.999707i	$-0.492297\pi$
$953$	46.1185			0.0483929			0.0241965	+	0.999707i	$0.492297\pi$
$954$		0			0
$955$	812.945			0.851252
$956$		0			0
$957$	64.4307	−	114.899i	0.0673258	−	0.120061i
$958$		0			0
$959$	−105.100			−0.109593
$960$		0			0
$961$	−949.552			−0.988087
$962$		0			0
$963$			883.462i			0.917406i
$964$		0			0
$965$			809.382i			0.838738i
$966$		0			0
$967$	−1575.16			−1.62891			−0.814457	−	0.580225i	$-0.802965\pi$
$967$	−1575.16			−1.62891			−0.814457	+	0.580225i	$0.802965\pi$
$968$		0			0
$969$		−	35.8638i		−	0.0370112i
$970$		0			0
$971$	725.761			0.747437			0.373718	−	0.927542i	$-0.378083\pi$
$971$	725.761			0.747437			0.373718	+	0.927542i	$0.378083\pi$
$972$		0			0
$973$	314.522			0.323249
$974$		0			0
$975$	−11.5347			−0.0118305
$976$		0			0
$977$	512.873			0.524946			0.262473	−	0.964939i	$-0.415462\pi$
$977$	512.873			0.524946			0.262473	+	0.964939i	$0.415462\pi$
$978$		0			0
$979$		−	478.935i		−	0.489208i
$980$		0			0
$981$	1152.83			1.17516
$982$		0			0
$983$	766.524			0.779780			0.389890	−	0.920861i	$-0.372513\pi$
$983$	766.524			0.779780			0.389890	+	0.920861i	$0.372513\pi$
$984$		0			0
$985$	765.340			0.776995
$986$		0			0
$987$			81.1844i			0.0822537i
$988$		0			0
$989$		−	581.494i		−	0.587962i
$990$		0			0
$991$			835.945i			0.843537i	0.906704	+	0.421768i	$0.138590\pi$
$991$			835.945i			0.843537i	−0.906704	+	0.421768i	$0.861410\pi$
$992$		0			0
$993$	49.5873			0.0499369
$994$		0			0
$995$		−	1307.58i		−	1.31415i
$996$		0			0
$997$			319.504i			0.320465i	0.987079	+	0.160233i	$0.0512244\pi$
$997$			319.504i			0.320465i	−0.987079	+	0.160233i	$0.948776\pi$
$998$		0			0
$999$			172.019i			0.172192i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.3.h.c.463.10	yes	20
4.3	odd	2	inner	464.3.h.c.463.11	yes	20
29.28	even	2	inner	464.3.h.c.463.12	yes	20
116.115	odd	2	inner	464.3.h.c.463.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
464.3.h.c.463.9	✓	20	116.115	odd	2	inner
464.3.h.c.463.10	yes	20	1.1	even	1	trivial
464.3.h.c.463.11	yes	20	4.3	odd	2	inner
464.3.h.c.463.12	yes	20	29.28	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	464.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.231073 q^{3} +4.30394 q^{5} +6.80119i q^{7} -8.94661 q^{9} +O(q^{10})\)	\(q-0.231073 q^{3} +4.30394 q^{5} +6.80119i q^{7} -8.94661 q^{9} -19.6581 q^{11} -7.70811 q^{13} -0.994523 q^{15} -21.6001i q^{17} -7.18543 q^{19} -1.57157i q^{21} +9.72519i q^{23} -6.47606 q^{25} +4.14697 q^{27} +(14.1842 - 25.2945i) q^{29} +3.38352 q^{31} +4.54244 q^{33} +29.2719i q^{35} +41.4808i q^{37} +1.78113 q^{39} -24.1263i q^{41} -59.7926 q^{43} -38.5057 q^{45} -51.6582 q^{47} +2.74381 q^{49} +4.99119i q^{51} -29.5243 q^{53} -84.6072 q^{55} +1.66036 q^{57} +36.7723i q^{59} -91.5419i q^{61} -60.8476i q^{63} -33.1753 q^{65} +80.0439i q^{67} -2.24722i q^{69} +112.333i q^{71} +88.8678i q^{73} +1.49644 q^{75} -133.698i q^{77} +57.8571 q^{79} +79.5612 q^{81} -132.894i q^{83} -92.9656i q^{85} +(-3.27757 + 5.84485i) q^{87} +24.3633i q^{89} -52.4243i q^{91} -0.781838 q^{93} -30.9257 q^{95} +116.255i q^{97} +175.873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{5} + 52 q^{9}+O(q^{10}) \) 20 * q - 8 * q^5 + 52 * q^9	\( 20 q - 8 q^{5} + 52 q^{9} - 16 q^{13} + 12 q^{25} + 4 q^{29} - 96 q^{33} - 112 q^{45} - 76 q^{49} - 112 q^{53} + 120 q^{57} + 136 q^{65} - 52 q^{81}+O(q^{100}) \) 20 * q - 8 * q^5 + 52 * q^9 - 16 * q^13 + 12 * q^25 + 4 * q^29 - 96 * q^33 - 112 * q^45 - 76 * q^49 - 112 * q^53 + 120 * q^57 + 136 * q^65 - 52 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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