LMFDB - Embedded newform 116.2.c.a.57.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,2,Mod(57,116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("116.57");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 116 = 2^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	116.c (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:	$0.926264663447$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			57.1
Root			$0.500000 + 1.32288i$ of defining polynomial
Character	$\chi$	$=$	116.57
Dual form			116.2.c.a.57.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575i q^{3} +1.00000 q^{5} -2.00000 q^{7} -4.00000 q^{9} +O(q^{10})$	$q-2.64575i q^{3} +1.00000 q^{5} -2.00000 q^{7} -4.00000 q^{9} -2.64575i q^{11} +5.00000 q^{13} -2.64575i q^{15} +5.29150i q^{17} +5.29150i q^{19} +5.29150i q^{21} +6.00000 q^{23} -4.00000 q^{25} +2.64575i q^{27} +(-1.00000 - 5.29150i) q^{29} +2.64575i q^{31} -7.00000 q^{33} -2.00000 q^{35} +10.5830i q^{37} -13.2288i q^{39} -5.29150i q^{41} +7.93725i q^{43} -4.00000 q^{45} -7.93725i q^{47} -3.00000 q^{49} +14.0000 q^{51} +5.00000 q^{53} -2.64575i q^{55} +14.0000 q^{57} -14.0000 q^{59} -5.29150i q^{61} +8.00000 q^{63} +5.00000 q^{65} -4.00000 q^{67} -15.8745i q^{69} -8.00000 q^{71} -10.5830i q^{73} +10.5830i q^{75} +5.29150i q^{77} +2.64575i q^{79} -5.00000 q^{81} +2.00000 q^{83} +5.29150i q^{85} +(-14.0000 + 2.64575i) q^{87} -5.29150i q^{89} -10.0000 q^{91} +7.00000 q^{93} +5.29150i q^{95} +5.29150i q^{97} +10.5830i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7} - 8 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 4 * q^7 - 8 * q^9	$ 2 q + 2 q^{5} - 4 q^{7} - 8 q^{9} + 10 q^{13} + 12 q^{23} - 8 q^{25} - 2 q^{29} - 14 q^{33} - 4 q^{35} - 8 q^{45} - 6 q^{49} + 28 q^{51} + 10 q^{53} + 28 q^{57} - 28 q^{59} + 16 q^{63} + 10 q^{65} - 8 q^{67} - 16 q^{71} - 10 q^{81} + 4 q^{83} - 28 q^{87} - 20 q^{91} + 14 q^{93}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 - 8 * q^9 + 10 * q^13 + 12 * q^23 - 8 * q^25 - 2 * q^29 - 14 * q^33 - 4 * q^35 - 8 * q^45 - 6 * q^49 + 28 * q^51 + 10 * q^53 + 28 * q^57 - 28 * q^59 + 16 * q^63 + 10 * q^65 - 8 * q^67 - 16 * q^71 - 10 * q^81 + 4 * q^83 - 28 * q^87 - 20 * q^91 + 14 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/116\mathbb{Z}\right)^\times$.

$n$	$59$	$89$
$\chi(n)$	$1$	$-1$

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
$3$		−	2.64575i		−	1.52753i	−0.645497	−	0.763763i	$-0.723350\pi$
$3$		−	2.64575i		−	1.52753i	0.645497	−	0.763763i	$-0.276650\pi$
$4$		0			0
$5$	1.00000			0.447214			0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000			0.447214			0.223607	+	0.974679i	$0.428217\pi$
$6$		0			0
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$		0			0
$9$	−4.00000			−1.33333
$10$		0			0
$11$		−	2.64575i		−	0.797724i	−0.917011	−	0.398862i	$-0.869405\pi$
$11$		−	2.64575i		−	0.797724i	0.917011	−	0.398862i	$-0.130595\pi$
$12$		0			0
$13$	5.00000			1.38675			0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000			1.38675			0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		−	2.64575i		−	0.683130i
$16$		0			0
$17$			5.29150i			1.28338i	0.766965	+	0.641689i	$0.221766\pi$
$17$			5.29150i			1.28338i	−0.766965	+	0.641689i	$0.778234\pi$
$18$		0			0
$19$			5.29150i			1.21395i	0.794719	+	0.606977i	$0.207618\pi$
$19$			5.29150i			1.21395i	−0.794719	+	0.606977i	$0.792382\pi$
$20$		0			0
$21$			5.29150i			1.15470i
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$			2.64575i			0.509175i
$28$		0			0
$29$	−1.00000	−	5.29150i	−0.185695	−	0.982607i
$29$	−1.00000	−	5.29150i	−0.185695	−	0.982607i
$30$		0			0
$31$			2.64575i			0.475191i	0.971364	+	0.237595i	$0.0763593\pi$
$31$			2.64575i			0.475191i	−0.971364	+	0.237595i	$0.923641\pi$
$32$		0			0
$33$	−7.00000			−1.21854
$34$		0			0
$35$	−2.00000			−0.338062
$36$		0			0
$37$			10.5830i			1.73984i	0.493197	+	0.869918i	$0.335828\pi$
$37$			10.5830i			1.73984i	−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		−	13.2288i		−	2.11830i
$40$		0			0
$41$		−	5.29150i		−	0.826394i	−0.910642	−	0.413197i	$-0.864412\pi$
$41$		−	5.29150i		−	0.826394i	0.910642	−	0.413197i	$-0.135588\pi$
$42$		0			0
$43$			7.93725i			1.21042i	0.796066	+	0.605210i	$0.206911\pi$
$43$			7.93725i			1.21042i	−0.796066	+	0.605210i	$0.793089\pi$
$44$		0			0
$45$	−4.00000			−0.596285
$46$		0			0
$47$		−	7.93725i		−	1.15777i	−0.815410	−	0.578884i	$-0.803489\pi$
$47$		−	7.93725i		−	1.15777i	0.815410	−	0.578884i	$-0.196511\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$	14.0000			1.96039
$52$		0			0
$53$	5.00000			0.686803			0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000			0.686803			0.343401	+	0.939189i	$0.388421\pi$
$54$		0			0
$55$		−	2.64575i		−	0.356753i
$56$		0			0
$57$	14.0000			1.85435
$58$		0			0
$59$	−14.0000			−1.82264			−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000			−1.82264			−0.911322	+	0.411693i	$0.864937\pi$
$60$		0			0
$61$		−	5.29150i		−	0.677507i	−0.940875	−	0.338754i	$-0.889995\pi$
$61$		−	5.29150i		−	0.677507i	0.940875	−	0.338754i	$-0.110005\pi$
$62$		0			0
$63$	8.00000			1.00791
$64$		0			0
$65$	5.00000			0.620174
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$		−	15.8745i		−	1.91107i
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$		−	10.5830i		−	1.23865i	−0.785136	−	0.619324i	$-0.787407\pi$
$73$		−	10.5830i		−	1.23865i	0.785136	−	0.619324i	$-0.212593\pi$
$74$		0			0
$75$			10.5830i			1.22202i
$76$		0			0
$77$			5.29150i			0.603023i
$78$		0			0
$79$			2.64575i			0.297670i	0.988862	+	0.148835i	$0.0475524\pi$
$79$			2.64575i			0.297670i	−0.988862	+	0.148835i	$0.952448\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	2.00000			0.219529			0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000			0.219529			0.109764	+	0.993958i	$0.464990\pi$
$84$		0			0
$85$			5.29150i			0.573944i
$86$		0			0
$87$	−14.0000	+	2.64575i	−1.50096	+	0.283654i
$88$		0			0
$89$		−	5.29150i		−	0.560898i	−0.959869	−	0.280449i	$-0.909517\pi$
$89$		−	5.29150i		−	0.560898i	0.959869	−	0.280449i	$-0.0904834\pi$
$90$		0			0
$91$	−10.0000			−1.04828
$92$		0			0
$93$	7.00000			0.725866
$94$		0			0
$95$			5.29150i			0.542897i
$96$		0			0
$97$			5.29150i			0.537271i	0.963242	+	0.268635i	$0.0865727\pi$
$97$			5.29150i			0.537271i	−0.963242	+	0.268635i	$0.913427\pi$
$98$		0			0
$99$			10.5830i			1.06363i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−8.00000			−0.788263			−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$			5.29150i			0.516398i
$106$		0			0
$107$	10.0000			0.966736			0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000			0.966736			0.483368	+	0.875417i	$0.339413\pi$
$108$		0			0
$109$	9.00000			0.862044			0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000			0.862044			0.431022	+	0.902342i	$0.358153\pi$
$110$		0			0
$111$	28.0000			2.65764
$112$		0			0
$113$			10.5830i			0.995565i	0.867302	+	0.497783i	$0.165852\pi$
$113$			10.5830i			0.995565i	−0.867302	+	0.497783i	$0.834148\pi$
$114$		0			0
$115$	6.00000			0.559503
$116$		0			0
$117$	−20.0000			−1.84900
$118$		0			0
$119$		−	10.5830i		−	0.970143i
$120$		0			0
$121$	4.00000			0.363636
$122$		0			0
$123$	−14.0000			−1.26234
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$			15.8745i			1.40863i	0.709885	+	0.704317i	$0.248747\pi$
$127$			15.8745i			1.40863i	−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$	21.0000			1.84895
$130$		0			0
$131$			15.8745i			1.38696i	0.720475	+	0.693481i	$0.243924\pi$
$131$			15.8745i			1.38696i	−0.720475	+	0.693481i	$0.756076\pi$
$132$		0			0
$133$		−	10.5830i		−	0.917663i
$134$		0			0
$135$			2.64575i			0.227710i
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	−2.00000			−0.169638			−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000			−0.169638			−0.0848189	+	0.996396i	$0.527031\pi$
$140$		0			0
$141$	−21.0000			−1.76852
$142$		0			0
$143$		−	13.2288i		−	1.10624i
$144$		0			0
$145$	−1.00000	−	5.29150i	−0.0830455	−	0.439435i
$146$		0			0
$147$			7.93725i			0.654654i
$148$		0			0
$149$	−3.00000			−0.245770			−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000			−0.245770			−0.122885	+	0.992421i	$0.539215\pi$
$150$		0			0
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$		0			0
$153$		−	21.1660i		−	1.71117i
$154$		0			0
$155$			2.64575i			0.212512i
$156$		0			0
$157$		−	15.8745i		−	1.26692i	−0.773774	−	0.633462i	$-0.781633\pi$
$157$		−	15.8745i		−	1.26692i	0.773774	−	0.633462i	$-0.218367\pi$
$158$		0			0
$159$		−	13.2288i		−	1.04911i
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$		−	2.64575i		−	0.207231i	−0.994617	−	0.103616i	$-0.966959\pi$
$163$		−	2.64575i		−	0.207231i	0.994617	−	0.103616i	$-0.0330412\pi$
$164$		0			0
$165$	−7.00000			−0.544949
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		−	21.1660i		−	1.61861i
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	8.00000			0.604743
$176$		0			0
$177$			37.0405i			2.78414i
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	−14.0000			−1.03491
$184$		0			0
$185$			10.5830i			0.778078i
$186$		0			0
$187$	14.0000			1.02378
$188$		0			0
$189$		−	5.29150i		−	0.384900i
$190$		0			0
$191$		−	15.8745i		−	1.14864i	−0.818631	−	0.574320i	$-0.805267\pi$
$191$		−	15.8745i		−	1.14864i	0.818631	−	0.574320i	$-0.194733\pi$
$192$		0			0
$193$		−	26.4575i		−	1.90445i	−0.305392	−	0.952227i	$-0.598787\pi$
$193$		−	26.4575i		−	1.90445i	0.305392	−	0.952227i	$-0.401213\pi$
$194$		0			0
$195$		−	13.2288i		−	0.947331i
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$			10.5830i			0.746468i
$202$		0			0
$203$	2.00000	+	10.5830i	0.140372	+	0.742781i
$204$		0			0
$205$		−	5.29150i		−	0.369575i
$206$		0			0
$207$	−24.0000			−1.66812
$208$		0			0
$209$	14.0000			0.968400
$210$		0			0
$211$		−	13.2288i		−	0.910705i	−0.890311	−	0.455353i	$-0.849513\pi$
$211$		−	13.2288i		−	0.910705i	0.890311	−	0.455353i	$-0.150487\pi$
$212$		0			0
$213$			21.1660i			1.45027i
$214$		0			0
$215$			7.93725i			0.541316i
$216$		0			0
$217$		−	5.29150i		−	0.359211i
$218$		0			0
$219$	−28.0000			−1.89206
$220$		0			0
$221$			26.4575i			1.77972i
$222$		0			0
$223$	−12.0000			−0.803579			−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000			−0.803579			−0.401790	+	0.915732i	$0.631612\pi$
$224$		0			0
$225$	16.0000			1.06667
$226$		0			0
$227$	−24.0000			−1.59294			−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000			−1.59294			−0.796468	+	0.604681i	$0.793301\pi$
$228$		0			0
$229$			15.8745i			1.04902i	0.851405	+	0.524509i	$0.175751\pi$
$229$			15.8745i			1.04902i	−0.851405	+	0.524509i	$0.824249\pi$
$230$		0			0
$231$	14.0000			0.921132
$232$		0			0
$233$	−25.0000			−1.63780			−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000			−1.63780			−0.818902	+	0.573933i	$0.805417\pi$
$234$		0			0
$235$		−	7.93725i		−	0.517769i
$236$		0			0
$237$	7.00000			0.454699
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	−17.0000			−1.09507			−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000			−1.09507			−0.547533	+	0.836784i	$0.684433\pi$
$242$		0			0
$243$			21.1660i			1.35780i
$244$		0			0
$245$	−3.00000			−0.191663
$246$		0			0
$247$			26.4575i			1.68345i
$248$		0			0
$249$		−	5.29150i		−	0.335335i
$250$		0			0
$251$		−	23.8118i		−	1.50299i	−0.659742	−	0.751493i	$-0.729334\pi$
$251$		−	23.8118i		−	1.50299i	0.659742	−	0.751493i	$-0.270666\pi$
$252$		0			0
$253$		−	15.8745i		−	0.998022i
$254$		0			0
$255$	14.0000			0.876714
$256$		0			0
$257$	−13.0000			−0.810918			−0.405459	−	0.914113i	$-0.632888\pi$
$257$	−13.0000			−0.810918			−0.405459	+	0.914113i	$0.632888\pi$
$258$		0			0
$259$		−	21.1660i		−	1.31519i
$260$		0			0
$261$	4.00000	+	21.1660i	0.247594	+	1.31014i
$262$		0			0
$263$			2.64575i			0.163144i	0.996667	+	0.0815720i	$0.0259941\pi$
$263$			2.64575i			0.163144i	−0.996667	+	0.0815720i	$0.974006\pi$
$264$		0			0
$265$	5.00000			0.307148
$266$		0			0
$267$	−14.0000			−0.856786
$268$		0			0
$269$			5.29150i			0.322629i	0.986903	+	0.161314i	$0.0515733\pi$
$269$			5.29150i			0.322629i	−0.986903	+	0.161314i	$0.948427\pi$
$270$		0			0
$271$			13.2288i			0.803590i	0.915730	+	0.401795i	$0.131614\pi$
$271$			13.2288i			0.803590i	−0.915730	+	0.401795i	$0.868386\pi$
$272$		0			0
$273$			26.4575i			1.60128i
$274$		0			0
$275$			10.5830i			0.638179i
$276$		0			0
$277$	30.0000			1.80253			0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000			1.80253			0.901263	+	0.433273i	$0.142641\pi$
$278$		0			0
$279$		−	10.5830i		−	0.633588i
$280$		0			0
$281$	−13.0000			−0.775515			−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000			−0.775515			−0.387757	+	0.921761i	$0.626750\pi$
$282$		0			0
$283$	18.0000			1.06999			0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000			1.06999			0.534994	+	0.844856i	$0.320314\pi$
$284$		0			0
$285$	14.0000			0.829288
$286$		0			0
$287$			10.5830i			0.624695i
$288$		0			0
$289$	−11.0000			−0.647059
$290$		0			0
$291$	14.0000			0.820695
$292$		0			0
$293$			10.5830i			0.618266i	0.951019	+	0.309133i	$0.100039\pi$
$293$			10.5830i			0.618266i	−0.951019	+	0.309133i	$0.899961\pi$
$294$		0			0
$295$	−14.0000			−0.815112
$296$		0			0
$297$	7.00000			0.406181
$298$		0			0
$299$	30.0000			1.73494
$300$		0			0
$301$		−	15.8745i		−	0.914991i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5.29150i		−	0.302991i
$306$		0			0
$307$		−	23.8118i		−	1.35901i	−0.733671	−	0.679505i	$-0.762195\pi$
$307$		−	23.8118i		−	1.35901i	0.733671	−	0.679505i	$-0.237805\pi$
$308$		0			0
$309$			21.1660i			1.20409i
$310$		0			0
$311$		−	15.8745i		−	0.900161i	−0.892988	−	0.450080i	$-0.851395\pi$
$311$		−	15.8745i		−	0.900161i	0.892988	−	0.450080i	$-0.148605\pi$
$312$		0			0
$313$	−13.0000			−0.734803			−0.367402	−	0.930062i	$-0.619753\pi$
$313$	−13.0000			−0.734803			−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$	8.00000			0.450749
$316$		0			0
$317$		−	5.29150i		−	0.297200i	−0.988897	−	0.148600i	$-0.952523\pi$
$317$		−	5.29150i		−	0.297200i	0.988897	−	0.148600i	$-0.0474767\pi$
$318$		0			0
$319$	−14.0000	+	2.64575i	−0.783850	+	0.148134i
$320$		0			0
$321$		−	26.4575i		−	1.47671i
$322$		0			0
$323$	−28.0000			−1.55796
$324$		0			0
$325$	−20.0000			−1.10940
$326$		0			0
$327$		−	23.8118i		−	1.31679i
$328$		0			0
$329$			15.8745i			0.875190i
$330$		0			0
$331$		−	2.64575i		−	0.145424i	−0.997353	−	0.0727118i	$-0.976835\pi$
$331$		−	2.64575i		−	0.145424i	0.997353	−	0.0727118i	$-0.0231653\pi$
$332$		0			0
$333$		−	42.3320i		−	2.31978i
$334$		0			0
$335$	−4.00000			−0.218543
$336$		0			0
$337$		−	5.29150i		−	0.288247i	−0.989560	−	0.144123i	$-0.953964\pi$
$337$		−	5.29150i		−	0.288247i	0.989560	−	0.144123i	$-0.0460362\pi$
$338$		0			0
$339$	28.0000			1.52075
$340$		0			0
$341$	7.00000			0.379071
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		−	15.8745i		−	0.854655i
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$	−19.0000			−1.01705			−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000			−1.01705			−0.508523	+	0.861048i	$0.669808\pi$
$350$		0			0
$351$			13.2288i			0.706099i
$352$		0			0
$353$	−18.0000			−0.958043			−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000			−0.958043			−0.479022	+	0.877803i	$0.659008\pi$
$354$		0			0
$355$	−8.00000			−0.424596
$356$		0			0
$357$	−28.0000			−1.48192
$358$		0			0
$359$			13.2288i			0.698187i	0.937088	+	0.349094i	$0.113510\pi$
$359$			13.2288i			0.698187i	−0.937088	+	0.349094i	$0.886490\pi$
$360$		0			0
$361$	−9.00000			−0.473684
$362$		0			0
$363$		−	10.5830i		−	0.555464i
$364$		0			0
$365$		−	10.5830i		−	0.553940i
$366$		0			0
$367$			5.29150i			0.276214i	0.990417	+	0.138107i	$0.0441018\pi$
$367$			5.29150i			0.276214i	−0.990417	+	0.138107i	$0.955898\pi$
$368$		0			0
$369$			21.1660i			1.10186i
$370$		0			0
$371$	−10.0000			−0.519174
$372$		0			0
$373$	13.0000			0.673114			0.336557	−	0.941663i	$-0.390737\pi$
$373$	13.0000			0.673114			0.336557	+	0.941663i	$0.390737\pi$
$374$		0			0
$375$			23.8118i			1.22963i
$376$		0			0
$377$	−5.00000	−	26.4575i	−0.257513	−	1.36263i
$378$		0			0
$379$			5.29150i			0.271806i	0.990722	+	0.135903i	$0.0433936\pi$
$379$			5.29150i			0.271806i	−0.990722	+	0.135903i	$0.956606\pi$
$380$		0			0
$381$	42.0000			2.15173
$382$		0			0
$383$	6.00000			0.306586			0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000			0.306586			0.153293	+	0.988181i	$0.451012\pi$
$384$		0			0
$385$			5.29150i			0.269680i
$386$		0			0
$387$		−	31.7490i		−	1.61389i
$388$		0			0
$389$		−	31.7490i		−	1.60974i	−0.593452	−	0.804869i	$-0.702235\pi$
$389$		−	31.7490i		−	1.60974i	0.593452	−	0.804869i	$-0.297765\pi$
$390$		0			0
$391$			31.7490i			1.60562i
$392$		0			0
$393$	42.0000			2.11862
$394$		0			0
$395$			2.64575i			0.133122i
$396$		0			0
$397$	21.0000			1.05396			0.526980	−	0.849878i	$-0.323324\pi$
$397$	21.0000			1.05396			0.526980	+	0.849878i	$0.323324\pi$
$398$		0			0
$399$	−28.0000			−1.40175
$400$		0			0
$401$	−17.0000			−0.848939			−0.424470	−	0.905442i	$-0.639539\pi$
$401$	−17.0000			−0.848939			−0.424470	+	0.905442i	$0.639539\pi$
$402$		0			0
$403$			13.2288i			0.658971i
$404$		0			0
$405$	−5.00000			−0.248452
$406$		0			0
$407$	28.0000			1.38791
$408$		0			0
$409$		−	10.5830i		−	0.523296i	−0.965163	−	0.261648i	$-0.915734\pi$
$409$		−	10.5830i		−	0.523296i	0.965163	−	0.261648i	$-0.0842659\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	28.0000			1.37779
$414$		0			0
$415$	2.00000			0.0981761
$416$		0			0
$417$			5.29150i			0.259126i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$			26.4575i			1.28946i	0.764410	+	0.644730i	$0.223030\pi$
$421$			26.4575i			1.28946i	−0.764410	+	0.644730i	$0.776970\pi$
$422$		0			0
$423$			31.7490i			1.54369i
$424$		0			0
$425$		−	21.1660i		−	1.02670i
$426$		0			0
$427$			10.5830i			0.512148i
$428$		0			0
$429$	−35.0000			−1.68982
$430$		0			0
$431$	−2.00000			−0.0963366			−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000			−0.0963366			−0.0481683	+	0.998839i	$0.515338\pi$
$432$		0			0
$433$		−	10.5830i		−	0.508587i	−0.967127	−	0.254293i	$-0.918157\pi$
$433$		−	10.5830i		−	0.508587i	0.967127	−	0.254293i	$-0.0818429\pi$
$434$		0			0
$435$	−14.0000	+	2.64575i	−0.671249	+	0.126854i
$436$		0			0
$437$			31.7490i			1.51876i
$438$		0			0
$439$	6.00000			0.286364			0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000			0.286364			0.143182	+	0.989696i	$0.454267\pi$
$440$		0			0
$441$	12.0000			0.571429
$442$		0			0
$443$			26.4575i			1.25703i	0.777796	+	0.628517i	$0.216338\pi$
$443$			26.4575i			1.25703i	−0.777796	+	0.628517i	$0.783662\pi$
$444$		0			0
$445$		−	5.29150i		−	0.250841i
$446$		0			0
$447$			7.93725i			0.375419i
$448$		0			0
$449$			31.7490i			1.49833i	0.662384	+	0.749164i	$0.269545\pi$
$449$			31.7490i			1.49833i	−0.662384	+	0.749164i	$0.730455\pi$
$450$		0			0
$451$	−14.0000			−0.659234
$452$		0			0
$453$		−	5.29150i		−	0.248616i
$454$		0			0
$455$	−10.0000			−0.468807
$456$		0			0
$457$	−18.0000			−0.842004			−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000			−0.842004			−0.421002	+	0.907060i	$0.638322\pi$
$458$		0			0
$459$	−14.0000			−0.653464
$460$		0			0
$461$		−	21.1660i		−	0.985799i	−0.870086	−	0.492900i	$-0.835937\pi$
$461$		−	21.1660i		−	0.985799i	0.870086	−	0.492900i	$-0.164063\pi$
$462$		0			0
$463$	−22.0000			−1.02243			−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000			−1.02243			−0.511213	+	0.859454i	$0.670804\pi$
$464$		0			0
$465$	7.00000			0.324617
$466$		0			0
$467$		−	23.8118i		−	1.10188i	−0.834546	−	0.550938i	$-0.814270\pi$
$467$		−	23.8118i		−	1.10188i	0.834546	−	0.550938i	$-0.185730\pi$
$468$		0			0
$469$	8.00000			0.369406
$470$		0			0
$471$	−42.0000			−1.93526
$472$		0			0
$473$	21.0000			0.965581
$474$		0			0
$475$		−	21.1660i		−	0.971163i
$476$		0			0
$477$	−20.0000			−0.915737
$478$		0			0
$479$		−	18.5203i		−	0.846212i	−0.906080	−	0.423106i	$-0.860940\pi$
$479$		−	18.5203i		−	0.846212i	0.906080	−	0.423106i	$-0.139060\pi$
$480$		0			0
$481$			52.9150i			2.41272i
$482$		0			0
$483$			31.7490i			1.44463i
$484$		0			0
$485$			5.29150i			0.240275i
$486$		0			0
$487$	26.0000			1.17817			0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000			1.17817			0.589086	+	0.808070i	$0.299488\pi$
$488$		0			0
$489$	−7.00000			−0.316551
$490$		0			0
$491$			7.93725i			0.358203i	0.983831	+	0.179102i	$0.0573191\pi$
$491$			7.93725i			0.358203i	−0.983831	+	0.179102i	$0.942681\pi$
$492$		0			0
$493$	28.0000	−	5.29150i	1.26106	−	0.238317i
$494$		0			0
$495$			10.5830i			0.475671i
$496$		0			0
$497$	16.0000			0.717698
$498$		0			0
$499$	−6.00000			−0.268597			−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000			−0.268597			−0.134298	+	0.990941i	$0.542878\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7.93725i		−	0.353905i	−0.984219	−	0.176952i	$-0.943376\pi$
$503$		−	7.93725i		−	0.353905i	0.984219	−	0.176952i	$-0.0566238\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	31.7490i		−	1.41002i
$508$		0			0
$509$	−27.0000			−1.19675			−0.598377	−	0.801215i	$-0.704187\pi$
$509$	−27.0000			−1.19675			−0.598377	+	0.801215i	$0.704187\pi$
$510$		0			0
$511$			21.1660i			0.936329i
$512$		0			0
$513$	−14.0000			−0.618115
$514$		0			0
$515$	−8.00000			−0.352522
$516$		0			0
$517$	−21.0000			−0.923579
$518$		0			0
$519$		−	15.8745i		−	0.696814i
$520$		0			0
$521$	35.0000			1.53338			0.766689	−	0.642019i	$-0.221903\pi$
$521$	35.0000			1.53338			0.766689	+	0.642019i	$0.221903\pi$
$522$		0			0
$523$	28.0000			1.22435			0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000			1.22435			0.612177	+	0.790721i	$0.290294\pi$
$524$		0			0
$525$		−	21.1660i		−	0.923760i
$526$		0			0
$527$	−14.0000			−0.609850
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	56.0000			2.43019
$532$		0			0
$533$		−	26.4575i		−	1.14600i
$534$		0			0
$535$	10.0000			0.432338
$536$		0			0
$537$		−	63.4980i		−	2.74014i
$538$		0			0
$539$			7.93725i			0.341882i
$540$		0			0
$541$			10.5830i			0.454999i	0.973778	+	0.227499i	$0.0730550\pi$
$541$			10.5830i			0.454999i	−0.973778	+	0.227499i	$0.926945\pi$
$542$		0			0
$543$			18.5203i			0.794780i
$544$		0			0
$545$	9.00000			0.385518
$546$		0			0
$547$	−22.0000			−0.940652			−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000			−0.940652			−0.470326	+	0.882493i	$0.655864\pi$
$548$		0			0
$549$			21.1660i			0.903343i
$550$		0			0
$551$	28.0000	−	5.29150i	1.19284	−	0.225426i
$552$		0			0
$553$		−	5.29150i		−	0.225018i
$554$		0			0
$555$	28.0000			1.18853
$556$		0			0
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		0			0
$559$			39.6863i			1.67855i
$560$		0			0
$561$		−	37.0405i		−	1.56385i
$562$		0			0
$563$		−	13.2288i		−	0.557526i	−0.960360	−	0.278763i	$-0.910076\pi$
$563$		−	13.2288i		−	0.557526i	0.960360	−	0.278763i	$-0.0899243\pi$
$564$		0			0
$565$			10.5830i			0.445230i
$566$		0			0
$567$	10.0000			0.419961
$568$		0			0
$569$		−	31.7490i		−	1.33099i	−0.746403	−	0.665494i	$-0.768221\pi$
$569$		−	31.7490i		−	1.33099i	0.746403	−	0.665494i	$-0.231779\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$	−42.0000			−1.75458
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$			37.0405i			1.54202i	0.636825	+	0.771008i	$0.280247\pi$
$577$			37.0405i			1.54202i	−0.636825	+	0.771008i	$0.719753\pi$
$578$		0			0
$579$	−70.0000			−2.90910
$580$		0			0
$581$	−4.00000			−0.165948
$582$		0			0
$583$		−	13.2288i		−	0.547879i
$584$		0			0
$585$	−20.0000			−0.826898
$586$		0			0
$587$	30.0000			1.23823			0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000			1.23823			0.619116	+	0.785299i	$0.287491\pi$
$588$		0			0
$589$	−14.0000			−0.576860
$590$		0			0
$591$		−	58.2065i		−	2.39430i
$592$		0			0
$593$	−1.00000			−0.0410651			−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000			−0.0410651			−0.0205325	+	0.999789i	$0.506536\pi$
$594$		0			0
$595$		−	10.5830i		−	0.433861i
$596$		0			0
$597$		−	26.4575i		−	1.08283i
$598$		0			0
$599$			23.8118i			0.972922i	0.873702	+	0.486461i	$0.161712\pi$
$599$			23.8118i			0.972922i	−0.873702	+	0.486461i	$0.838288\pi$
$600$		0			0
$601$			15.8745i			0.647535i	0.946137	+	0.323767i	$0.104950\pi$
$601$			15.8745i			0.647535i	−0.946137	+	0.323767i	$0.895050\pi$
$602$		0			0
$603$	16.0000			0.651570
$604$		0			0
$605$	4.00000			0.162623
$606$		0			0
$607$		−	18.5203i		−	0.751714i	−0.926678	−	0.375857i	$-0.877348\pi$
$607$		−	18.5203i		−	0.751714i	0.926678	−	0.375857i	$-0.122652\pi$
$608$		0			0
$609$	28.0000	−	5.29150i	1.13462	−	0.214423i
$610$		0			0
$611$		−	39.6863i		−	1.60553i
$612$		0			0
$613$	5.00000			0.201948			0.100974	−	0.994889i	$-0.467804\pi$
$613$	5.00000			0.201948			0.100974	+	0.994889i	$0.467804\pi$
$614$		0			0
$615$	−14.0000			−0.564534
$616$		0			0
$617$			21.1660i			0.852111i	0.904697	+	0.426056i	$0.140097\pi$
$617$			21.1660i			0.852111i	−0.904697	+	0.426056i	$0.859903\pi$
$618$		0			0
$619$		−	2.64575i		−	0.106342i	−0.998585	−	0.0531709i	$-0.983067\pi$
$619$		−	2.64575i		−	0.106342i	0.998585	−	0.0531709i	$-0.0169328\pi$
$620$		0			0
$621$			15.8745i			0.637022i
$622$		0			0
$623$			10.5830i			0.423999i
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		−	37.0405i		−	1.47926i
$628$		0			0
$629$	−56.0000			−2.23287
$630$		0			0
$631$	−44.0000			−1.75161			−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000			−1.75161			−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−35.0000			−1.39113
$634$		0			0
$635$			15.8745i			0.629961i
$636$		0			0
$637$	−15.0000			−0.594322
$638$		0			0
$639$	32.0000			1.26590
$640$		0			0
$641$		−	47.6235i		−	1.88102i	−0.339771	−	0.940508i	$-0.610350\pi$
$641$		−	47.6235i		−	1.88102i	0.339771	−	0.940508i	$-0.389650\pi$
$642$		0			0
$643$	14.0000			0.552106			0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000			0.552106			0.276053	+	0.961142i	$0.410973\pi$
$644$		0			0
$645$	21.0000			0.826874
$646$		0			0
$647$	38.0000			1.49393			0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000			1.49393			0.746967	+	0.664861i	$0.231509\pi$
$648$		0			0
$649$			37.0405i			1.45397i
$650$		0			0
$651$	−14.0000			−0.548703
$652$		0			0
$653$			47.6235i			1.86365i	0.362905	+	0.931826i	$0.381785\pi$
$653$			47.6235i			1.86365i	−0.362905	+	0.931826i	$0.618215\pi$
$654$		0			0
$655$			15.8745i			0.620268i
$656$		0			0
$657$			42.3320i			1.65153i
$658$		0			0
$659$			18.5203i			0.721447i	0.932673	+	0.360723i	$0.117470\pi$
$659$			18.5203i			0.721447i	−0.932673	+	0.360723i	$0.882530\pi$
$660$		0			0
$661$	−38.0000			−1.47803			−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000			−1.47803			−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$	70.0000			2.71857
$664$		0			0
$665$		−	10.5830i		−	0.410391i
$666$		0			0
$667$	−6.00000	−	31.7490i	−0.232321	−	1.22933i
$668$		0			0
$669$			31.7490i			1.22749i
$670$		0			0
$671$	−14.0000			−0.540464
$672$		0			0
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$		0			0
$675$		−	10.5830i		−	0.407340i
$676$		0			0
$677$		−	21.1660i		−	0.813476i	−0.913545	−	0.406738i	$-0.866666\pi$
$677$		−	21.1660i		−	0.813476i	0.913545	−	0.406738i	$-0.133334\pi$
$678$		0			0
$679$		−	10.5830i		−	0.406138i
$680$		0			0
$681$			63.4980i			2.43325i
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	42.0000			1.60240
$688$		0			0
$689$	25.0000			0.952424
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$		−	21.1660i		−	0.804030i
$694$		0			0
$695$	−2.00000			−0.0758643
$696$		0			0
$697$	28.0000			1.06058
$698$		0			0
$699$			66.1438i			2.50179i
$700$		0			0
$701$	−15.0000			−0.566542			−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000			−0.566542			−0.283271	+	0.959040i	$0.591420\pi$
$702$		0			0
$703$	−56.0000			−2.11208
$704$		0			0
$705$	−21.0000			−0.790906
$706$		0			0
$707$		0			0
$708$		0			0
$709$	25.0000			0.938895			0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000			0.938895			0.469447	+	0.882960i	$0.344453\pi$
$710$		0			0
$711$		−	10.5830i		−	0.396894i
$712$		0			0
$713$			15.8745i			0.594505i
$714$		0			0
$715$		−	13.2288i		−	0.494727i
$716$		0			0
$717$		−	15.8745i		−	0.592844i
$718$		0			0
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$			44.9778i			1.67274i
$724$		0			0
$725$	4.00000	+	21.1660i	0.148556	+	0.786086i
$726$		0			0
$727$		−	15.8745i		−	0.588753i	−0.955690	−	0.294376i	$-0.904888\pi$
$727$		−	15.8745i		−	0.588753i	0.955690	−	0.294376i	$-0.0951119\pi$
$728$		0			0
$729$	41.0000			1.51852
$730$		0			0
$731$	−42.0000			−1.55343
$732$		0			0
$733$			31.7490i			1.17268i	0.810066	+	0.586338i	$0.199431\pi$
$733$			31.7490i			1.17268i	−0.810066	+	0.586338i	$0.800569\pi$
$734$		0			0
$735$			7.93725i			0.292770i
$736$		0			0
$737$			10.5830i			0.389830i
$738$		0			0
$739$		−	23.8118i		−	0.875930i	−0.898992	−	0.437965i	$-0.855699\pi$
$739$		−	23.8118i		−	0.875930i	0.898992	−	0.437965i	$-0.144301\pi$
$740$		0			0
$741$	70.0000			2.57151
$742$		0			0
$743$			5.29150i			0.194126i	0.995278	+	0.0970632i	$0.0309449\pi$
$743$			5.29150i			0.194126i	−0.995278	+	0.0970632i	$0.969055\pi$
$744$		0			0
$745$	−3.00000			−0.109911
$746$		0			0
$747$	−8.00000			−0.292705
$748$		0			0
$749$	−20.0000			−0.730784
$750$		0			0
$751$		−	37.0405i		−	1.35163i	−0.737072	−	0.675814i	$-0.763792\pi$
$751$		−	37.0405i		−	1.35163i	0.737072	−	0.675814i	$-0.236208\pi$
$752$		0			0
$753$	−63.0000			−2.29585
$754$		0			0
$755$	2.00000			0.0727875
$756$		0			0
$757$			5.29150i			0.192323i	0.995366	+	0.0961615i	$0.0306565\pi$
$757$			5.29150i			0.192323i	−0.995366	+	0.0961615i	$0.969343\pi$
$758$		0			0
$759$	−42.0000			−1.52450
$760$		0			0
$761$	14.0000			0.507500			0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000			0.507500			0.253750	+	0.967270i	$0.418336\pi$
$762$		0			0
$763$	−18.0000			−0.651644
$764$		0			0
$765$		−	21.1660i		−	0.765259i
$766$		0			0
$767$	−70.0000			−2.52755
$768$		0			0
$769$			5.29150i			0.190816i	0.995438	+	0.0954082i	$0.0304156\pi$
$769$			5.29150i			0.190816i	−0.995438	+	0.0954082i	$0.969584\pi$
$770$		0			0
$771$			34.3948i			1.23870i
$772$		0			0
$773$			5.29150i			0.190322i	0.995462	+	0.0951611i	$0.0303366\pi$
$773$			5.29150i			0.190322i	−0.995462	+	0.0951611i	$0.969663\pi$
$774$		0			0
$775$		−	10.5830i		−	0.380153i
$776$		0			0
$777$	−56.0000			−2.00899
$778$		0			0
$779$	28.0000			1.00320
$780$		0			0
$781$			21.1660i			0.757379i
$782$		0			0
$783$	14.0000	−	2.64575i	0.500319	−	0.0945514i
$784$		0			0
$785$		−	15.8745i		−	0.566585i
$786$		0			0
$787$	28.0000			0.998092			0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000			0.998092			0.499046	+	0.866575i	$0.333684\pi$
$788$		0			0
$789$	7.00000			0.249207
$790$		0			0
$791$		−	21.1660i		−	0.752577i
$792$		0			0
$793$		−	26.4575i		−	0.939534i
$794$		0			0
$795$		−	13.2288i		−	0.469176i
$796$		0			0
$797$		−	26.4575i		−	0.937173i	−0.883418	−	0.468587i	$-0.844763\pi$
$797$		−	26.4575i		−	0.937173i	0.883418	−	0.468587i	$-0.155237\pi$
$798$		0			0
$799$	42.0000			1.48585
$800$		0			0
$801$			21.1660i			0.747864i
$802$		0			0
$803$	−28.0000			−0.988099
$804$		0			0
$805$	−12.0000			−0.422944
$806$		0			0
$807$	14.0000			0.492823
$808$		0			0
$809$			21.1660i			0.744157i	0.928201	+	0.372079i	$0.121355\pi$
$809$			21.1660i			0.744157i	−0.928201	+	0.372079i	$0.878645\pi$
$810$		0			0
$811$	−30.0000			−1.05344			−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000			−1.05344			−0.526721	+	0.850038i	$0.676579\pi$
$812$		0			0
$813$	35.0000			1.22750
$814$		0			0
$815$		−	2.64575i		−	0.0926766i
$816$		0			0
$817$	−42.0000			−1.46939
$818$		0			0
$819$	40.0000			1.39771
$820$		0			0
$821$	45.0000			1.57051			0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000			1.57051			0.785255	+	0.619172i	$0.212532\pi$
$822$		0			0
$823$		−	5.29150i		−	0.184450i	−0.995738	−	0.0922251i	$-0.970602\pi$
$823$		−	5.29150i		−	0.184450i	0.995738	−	0.0922251i	$-0.0293979\pi$
$824$		0			0
$825$	28.0000			0.974835
$826$		0			0
$827$		−	23.8118i		−	0.828016i	−0.910273	−	0.414008i	$-0.864129\pi$
$827$		−	23.8118i		−	0.828016i	0.910273	−	0.414008i	$-0.135871\pi$
$828$		0			0
$829$		−	26.4575i		−	0.918907i	−0.888202	−	0.459454i	$-0.848045\pi$
$829$		−	26.4575i		−	0.918907i	0.888202	−	0.459454i	$-0.151955\pi$
$830$		0			0
$831$		−	79.3725i		−	2.75340i
$832$		0			0
$833$		−	15.8745i		−	0.550019i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−7.00000			−0.241955
$838$		0			0
$839$		−	7.93725i		−	0.274024i	−0.990569	−	0.137012i	$-0.956250\pi$
$839$		−	7.93725i		−	0.274024i	0.990569	−	0.137012i	$-0.0437500\pi$
$840$		0			0
$841$	−27.0000	+	10.5830i	−0.931034	+	0.364931i
$842$		0			0
$843$			34.3948i			1.18462i
$844$		0			0
$845$	12.0000			0.412813
$846$		0			0
$847$	−8.00000			−0.274883
$848$		0			0
$849$		−	47.6235i		−	1.63444i
$850$		0			0
$851$			63.4980i			2.17668i
$852$		0			0
$853$		−	21.1660i		−	0.724710i	−0.932040	−	0.362355i	$-0.881973\pi$
$853$		−	21.1660i		−	0.724710i	0.932040	−	0.362355i	$-0.118027\pi$
$854$		0			0
$855$		−	21.1660i		−	0.723862i
$856$		0			0
$857$	7.00000			0.239115			0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000			0.239115			0.119558	+	0.992827i	$0.461852\pi$
$858$		0			0
$859$			29.1033i			0.992991i	0.868039	+	0.496495i	$0.165380\pi$
$859$			29.1033i			0.992991i	−0.868039	+	0.496495i	$0.834620\pi$
$860$		0			0
$861$	28.0000			0.954237
$862$		0			0
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$	6.00000			0.204006
$866$		0			0
$867$			29.1033i			0.988399i
$868$		0			0
$869$	7.00000			0.237459
$870$		0			0
$871$	−20.0000			−0.677674
$872$		0			0
$873$		−	21.1660i		−	0.716361i
$874$		0			0
$875$	18.0000			0.608511
$876$		0			0
$877$	1.00000			0.0337676			0.0168838	−	0.999857i	$-0.494625\pi$
$877$	1.00000			0.0337676			0.0168838	+	0.999857i	$0.494625\pi$
$878$		0			0
$879$	28.0000			0.944417
$880$		0			0
$881$			42.3320i			1.42620i	0.701061	+	0.713101i	$0.252710\pi$
$881$			42.3320i			1.42620i	−0.701061	+	0.713101i	$0.747290\pi$
$882$		0			0
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$		0			0
$885$			37.0405i			1.24510i
$886$		0			0
$887$		−	39.6863i		−	1.33253i	−0.745713	−	0.666267i	$-0.767891\pi$
$887$		−	39.6863i		−	1.33253i	0.745713	−	0.666267i	$-0.232109\pi$
$888$		0			0
$889$		−	31.7490i		−	1.06483i
$890$		0			0
$891$			13.2288i			0.443180i
$892$		0			0
$893$	42.0000			1.40548
$894$		0			0
$895$	24.0000			0.802232
$896$		0			0
$897$		−	79.3725i		−	2.65017i
$898$		0			0
$899$	14.0000	−	2.64575i	0.466926	−	0.0882407i
$900$		0			0
$901$			26.4575i			0.881428i
$902$		0			0
$903$	−42.0000			−1.39767
$904$		0			0
$905$	−7.00000			−0.232688
$906$		0			0
$907$			5.29150i			0.175701i	0.996134	+	0.0878507i	$0.0279999\pi$
$907$			5.29150i			0.175701i	−0.996134	+	0.0878507i	$0.972000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	7.93725i		−	0.262973i	−0.991318	−	0.131486i	$-0.958025\pi$
$911$		−	7.93725i		−	0.262973i	0.991318	−	0.131486i	$-0.0419750\pi$
$912$		0			0
$913$		−	5.29150i		−	0.175123i
$914$		0			0
$915$	−14.0000			−0.462826
$916$		0			0
$917$		−	31.7490i		−	1.04844i
$918$		0			0
$919$	52.0000			1.71532			0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000			1.71532			0.857661	+	0.514216i	$0.171917\pi$
$920$		0			0
$921$	−63.0000			−2.07592
$922$		0			0
$923$	−40.0000			−1.31662
$924$		0			0
$925$		−	42.3320i		−	1.39187i
$926$		0			0
$927$	32.0000			1.05102
$928$		0			0
$929$	−14.0000			−0.459325			−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000			−0.459325			−0.229663	+	0.973270i	$0.573762\pi$
$930$		0			0
$931$		−	15.8745i		−	0.520266i
$932$		0			0
$933$	−42.0000			−1.37502
$934$		0			0
$935$	14.0000			0.457849
$936$		0			0
$937$	14.0000			0.457360			0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000			0.457360			0.228680	+	0.973502i	$0.426559\pi$
$938$		0			0
$939$			34.3948i			1.12243i
$940$		0			0
$941$	−39.0000			−1.27136			−0.635682	−	0.771951i	$-0.719281\pi$
$941$	−39.0000			−1.27136			−0.635682	+	0.771951i	$0.719281\pi$
$942$		0			0
$943$		−	31.7490i		−	1.03389i
$944$		0			0
$945$		−	5.29150i		−	0.172133i
$946$		0			0
$947$			39.6863i			1.28963i	0.764338	+	0.644815i	$0.223066\pi$
$947$			39.6863i			1.28963i	−0.764338	+	0.644815i	$0.776934\pi$
$948$		0			0
$949$		−	52.9150i		−	1.71769i
$950$		0			0
$951$	−14.0000			−0.453981
$952$		0			0
$953$	−41.0000			−1.32812			−0.664060	−	0.747679i	$-0.731168\pi$
$953$	−41.0000			−1.32812			−0.664060	+	0.747679i	$0.731168\pi$
$954$		0			0
$955$		−	15.8745i		−	0.513687i
$956$		0			0
$957$	7.00000	+	37.0405i	0.226278	+	1.19735i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	24.0000			0.774194
$962$		0			0
$963$	−40.0000			−1.28898
$964$		0			0
$965$		−	26.4575i		−	0.851697i
$966$		0			0
$967$			44.9778i			1.44639i	0.690645	+	0.723194i	$0.257327\pi$
$967$			44.9778i			1.44639i	−0.690645	+	0.723194i	$0.742673\pi$
$968$		0			0
$969$			74.0810i			2.37983i
$970$		0			0
$971$			5.29150i			0.169812i	0.996389	+	0.0849062i	$0.0270591\pi$
$971$			5.29150i			0.169812i	−0.996389	+	0.0849062i	$0.972941\pi$
$972$		0			0
$973$	4.00000			0.128234
$974$		0			0
$975$			52.9150i			1.69464i
$976$		0			0
$977$	−33.0000			−1.05576			−0.527882	−	0.849318i	$-0.677014\pi$
$977$	−33.0000			−1.05576			−0.527882	+	0.849318i	$0.677014\pi$
$978$		0			0
$979$	−14.0000			−0.447442
$980$		0			0
$981$	−36.0000			−1.14939
$982$		0			0
$983$			44.9778i			1.43457i	0.696781	+	0.717284i	$0.254615\pi$
$983$			44.9778i			1.43457i	−0.696781	+	0.717284i	$0.745385\pi$
$984$		0			0
$985$	22.0000			0.700978
$986$		0			0
$987$	42.0000			1.33687
$988$		0			0
$989$			47.6235i			1.51434i
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$	−7.00000			−0.222138
$994$		0			0
$995$	10.0000			0.317021
$996$		0			0
$997$			52.9150i			1.67584i	0.545796	+	0.837918i	$0.316227\pi$
$997$			52.9150i			1.67584i	−0.545796	+	0.837918i	$0.683773\pi$
$998$		0			0
$999$	−28.0000			−0.885881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.c.a.57.1	✓	2
3.2	odd	2		1044.2.h.c.289.2		2
4.3	odd	2		464.2.e.b.289.2		2
5.2	odd	4		2900.2.f.b.1449.1		4
5.3	odd	4		2900.2.f.b.1449.4		4
5.4	even	2		2900.2.d.a.1101.2		2
8.3	odd	2		1856.2.e.c.1217.1		2
8.5	even	2		1856.2.e.a.1217.2		2
12.11	even	2		4176.2.o.e.289.1		2
29.12	odd	4		3364.2.a.f.1.2		2
29.17	odd	4		3364.2.a.f.1.1		2
29.28	even	2	inner	116.2.c.a.57.2	yes	2
87.86	odd	2		1044.2.h.c.289.1		2
116.115	odd	2		464.2.e.b.289.1		2
145.28	odd	4		2900.2.f.b.1449.2		4
145.57	odd	4		2900.2.f.b.1449.3		4
145.144	even	2		2900.2.d.a.1101.1		2
232.115	odd	2		1856.2.e.c.1217.2		2
232.173	even	2		1856.2.e.a.1217.1		2
348.347	even	2		4176.2.o.e.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.c.a.57.1	✓	2	1.1	even	1	trivial
116.2.c.a.57.2	yes	2	29.28	even	2	inner
464.2.e.b.289.1		2	116.115	odd	2
464.2.e.b.289.2		2	4.3	odd	2
1044.2.h.c.289.1		2	87.86	odd	2
1044.2.h.c.289.2		2	3.2	odd	2
1856.2.e.a.1217.1		2	232.173	even	2
1856.2.e.a.1217.2		2	8.5	even	2
1856.2.e.c.1217.1		2	8.3	odd	2
1856.2.e.c.1217.2		2	232.115	odd	2
2900.2.d.a.1101.1		2	145.144	even	2
2900.2.d.a.1101.2		2	5.4	even	2
2900.2.f.b.1449.1		4	5.2	odd	4
2900.2.f.b.1449.2		4	145.28	odd	4
2900.2.f.b.1449.3		4	145.57	odd	4
2900.2.f.b.1449.4		4	5.3	odd	4
3364.2.a.f.1.1		2	29.17	odd	4
3364.2.a.f.1.2		2	29.12	odd	4
4176.2.o.e.289.1		2	12.11	even	2
4176.2.o.e.289.2		2	348.347	even	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:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64575i q^{3} +1.00000 q^{5} -2.00000 q^{7} -4.00000 q^{9} +O(q^{10})\)	\(q-2.64575i q^{3} +1.00000 q^{5} -2.00000 q^{7} -4.00000 q^{9} -2.64575i q^{11} +5.00000 q^{13} -2.64575i q^{15} +5.29150i q^{17} +5.29150i q^{19} +5.29150i q^{21} +6.00000 q^{23} -4.00000 q^{25} +2.64575i q^{27} +(-1.00000 - 5.29150i) q^{29} +2.64575i q^{31} -7.00000 q^{33} -2.00000 q^{35} +10.5830i q^{37} -13.2288i q^{39} -5.29150i q^{41} +7.93725i q^{43} -4.00000 q^{45} -7.93725i q^{47} -3.00000 q^{49} +14.0000 q^{51} +5.00000 q^{53} -2.64575i q^{55} +14.0000 q^{57} -14.0000 q^{59} -5.29150i q^{61} +8.00000 q^{63} +5.00000 q^{65} -4.00000 q^{67} -15.8745i q^{69} -8.00000 q^{71} -10.5830i q^{73} +10.5830i q^{75} +5.29150i q^{77} +2.64575i q^{79} -5.00000 q^{81} +2.00000 q^{83} +5.29150i q^{85} +(-14.0000 + 2.64575i) q^{87} -5.29150i q^{89} -10.0000 q^{91} +7.00000 q^{93} +5.29150i q^{95} +5.29150i q^{97} +10.5830i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 4 * q^7 - 8 * q^9	\( 2 q + 2 q^{5} - 4 q^{7} - 8 q^{9} + 10 q^{13} + 12 q^{23} - 8 q^{25} - 2 q^{29} - 14 q^{33} - 4 q^{35} - 8 q^{45} - 6 q^{49} + 28 q^{51} + 10 q^{53} + 28 q^{57} - 28 q^{59} + 16 q^{63} + 10 q^{65} - 8 q^{67} - 16 q^{71} - 10 q^{81} + 4 q^{83} - 28 q^{87} - 20 q^{91} + 14 q^{93}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 - 8 * q^9 + 10 * q^13 + 12 * q^23 - 8 * q^25 - 2 * q^29 - 14 * q^33 - 4 * q^35 - 8 * q^45 - 6 * q^49 + 28 * q^51 + 10 * q^53 + 28 * q^57 - 28 * q^59 + 16 * q^63 + 10 * q^65 - 8 * q^67 - 16 * q^71 - 10 * q^81 + 4 * q^83 - 28 * q^87 - 20 * q^91 + 14 * q^93

Introduction

L-functions

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Database

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