LMFDB - Embedded newform 1134.2.d.a.1133.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(1133,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1134.1133");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.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:	$9.05503558921$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 $ x^16 - 6x^14 + 9x^12 + 54x^10 - 288x^8 + 486x^6 + 729x^4 - 4374*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1133.12
Root			$-0.0967785 + 1.72934i$ of defining polynomial
Character	$\chi$	$=$	1134.1133
Dual form			1134.2.d.a.1133.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -0.366598 q^{5} +(-1.91449 - 1.82612i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} -0.366598 q^{5} +(-1.91449 - 1.82612i) q^{7} -1.00000i q^{8} -0.366598i q^{10} +0.669453i q^{11} -1.00156i q^{13} +(1.82612 - 1.91449i) q^{14} +1.00000 q^{16} +4.98906 q^{17} +6.35722i q^{19} +0.366598 q^{20} -0.669453 q^{22} +7.69459i q^{23} -4.86561 q^{25} +1.00156 q^{26} +(1.91449 + 1.82612i) q^{28} +1.82898i q^{29} +6.32588i q^{31} +1.00000i q^{32} +4.98906i q^{34} +(0.701849 + 0.669453i) q^{35} -5.16789 q^{37} -6.35722 q^{38} +0.366598i q^{40} -4.31856 q^{41} -4.49843 q^{43} -0.669453i q^{44} -7.69459 q^{46} +8.32901 q^{47} +(0.330547 + 6.99219i) q^{49} -4.86561i q^{50} +1.00156i q^{52} -0.245420i q^{55} +(-1.82612 + 1.91449i) q^{56} -1.82898 q^{58} +8.72695 q^{59} +4.95771i q^{61} -6.32588 q^{62} -1.00000 q^{64} +0.367172i q^{65} -10.8907 q^{67} -4.98906 q^{68} +(-0.669453 + 0.701849i) q^{70} +5.49843i q^{71} -4.07314i q^{73} -5.16789i q^{74} -6.35722i q^{76} +(1.22250 - 1.28166i) q^{77} +8.35568 q^{79} -0.366598 q^{80} -4.31856i q^{82} +17.0142 q^{83} -1.82898 q^{85} -4.49843i q^{86} +0.669453 q^{88} -10.7113 q^{89} +(-1.82898 + 1.91749i) q^{91} -7.69459i q^{92} +8.32901i q^{94} -2.33055i q^{95} +17.2157i q^{97} +(-6.99219 + 0.330547i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 4 q^{7}+O(q^{10}) $ 16 * q - 16 * q^4 - 4 * q^7	$ 16 q - 16 q^{4} - 4 q^{7} + 16 q^{16} + 16 q^{25} + 4 q^{28} - 8 q^{37} - 8 q^{43} + 24 q^{46} + 16 q^{49} + 24 q^{58} - 16 q^{64} + 56 q^{67} + 8 q^{79} + 24 q^{85} + 24 q^{91}+O(q^{100}) $ 16 * q - 16 * q^4 - 4 * q^7 + 16 * q^16 + 16 * q^25 + 4 * q^28 - 8 * q^37 - 8 * q^43 + 24 * q^46 + 16 * q^49 + 24 * q^58 - 16 * q^64 + 56 * q^67 + 8 * q^79 + 24 * q^85 + 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−0.366598			−0.163948			−0.0819738	−	0.996634i	$-0.526122\pi$
$5$	−0.366598			−0.163948			−0.0819738	+	0.996634i	$0.526122\pi$
$6$		0			0
$7$	−1.91449	−	1.82612i	−0.723609	−	0.690210i
$7$	−1.91449	−	1.82612i	−0.723609	−	0.690210i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	0.366598i		−	0.115929i
$11$			0.669453i			0.201848i	0.994894	+	0.100924i	$0.0321799\pi$
$11$			0.669453i			0.201848i	−0.994894	+	0.100924i	$0.967820\pi$
$12$		0			0
$13$		−	1.00156i		−	0.277784i	−0.990308	−	0.138892i	$-0.955646\pi$
$13$		−	1.00156i		−	0.277784i	0.990308	−	0.138892i	$-0.0443541\pi$
$14$	1.82612	−	1.91449i	0.488052	−	0.511669i
$15$		0			0
$16$	1.00000			0.250000
$17$	4.98906			1.21003			0.605013	−	0.796216i	$-0.293168\pi$
$17$	4.98906			1.21003			0.605013	+	0.796216i	$0.293168\pi$
$18$		0			0
$19$			6.35722i			1.45845i	0.684275	+	0.729224i	$0.260119\pi$
$19$			6.35722i			1.45845i	−0.684275	+	0.729224i	$0.739881\pi$
$20$	0.366598			0.0819738
$21$		0			0
$22$	−0.669453			−0.142728
$23$			7.69459i			1.60443i	0.597034	+	0.802216i	$0.296346\pi$
$23$			7.69459i			1.60443i	−0.597034	+	0.802216i	$0.703654\pi$
$24$		0			0
$25$	−4.86561			−0.973121
$26$	1.00156			0.196423
$27$		0			0
$28$	1.91449	+	1.82612i	0.361805	+	0.345105i
$29$			1.82898i			0.339633i	0.985476	+	0.169817i	$0.0543175\pi$
$29$			1.82898i			0.339633i	−0.985476	+	0.169817i	$0.945682\pi$
$30$		0			0
$31$			6.32588i			1.13616i	0.822973	+	0.568081i	$0.192314\pi$
$31$			6.32588i			1.13616i	−0.822973	+	0.568081i	$0.807686\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			4.98906i			0.855617i
$35$	0.701849	+	0.669453i	0.118634	+	0.113158i
$36$		0			0
$37$	−5.16789			−0.849595			−0.424798	−	0.905288i	$-0.639655\pi$
$37$	−5.16789			−0.849595			−0.424798	+	0.905288i	$0.639655\pi$
$38$	−6.35722			−1.03128
$39$		0			0
$40$			0.366598i			0.0579643i
$41$	−4.31856			−0.674446			−0.337223	−	0.941425i	$-0.609488\pi$
$41$	−4.31856			−0.674446			−0.337223	+	0.941425i	$0.609488\pi$
$42$		0			0
$43$	−4.49843			−0.686005			−0.343002	−	0.939335i	$-0.611444\pi$
$43$	−4.49843			−0.686005			−0.343002	+	0.939335i	$0.611444\pi$
$44$		−	0.669453i		−	0.100924i
$45$		0			0
$46$	−7.69459			−1.13450
$47$	8.32901			1.21491			0.607455	−	0.794354i	$-0.292190\pi$
$47$	8.32901			1.21491			0.607455	+	0.794354i	$0.292190\pi$
$48$		0			0
$49$	0.330547	+	6.99219i	0.0472209	+	0.998884i
$50$		−	4.86561i		−	0.688101i
$51$		0			0
$52$			1.00156i			0.138892i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		−	0.245420i		−	0.0330925i
$56$	−1.82612	+	1.91449i	−0.244026	+	0.255835i
$57$		0			0
$58$	−1.82898			−0.240157
$59$	8.72695			1.13615			0.568076	−	0.822976i	$-0.307688\pi$
$59$	8.72695			1.13615			0.568076	+	0.822976i	$0.307688\pi$
$60$		0			0
$61$			4.95771i			0.634770i	0.948297	+	0.317385i	$0.102805\pi$
$61$			4.95771i			0.634770i	−0.948297	+	0.317385i	$0.897195\pi$
$62$	−6.32588			−0.803387
$63$		0			0
$64$	−1.00000			−0.125000
$65$			0.367172i			0.0455420i
$66$		0			0
$67$	−10.8907			−1.33052			−0.665258	−	0.746614i	$-0.731678\pi$
$67$	−10.8907			−1.33052			−0.665258	+	0.746614i	$0.731678\pi$
$68$	−4.98906			−0.605013
$69$		0			0
$70$	−0.669453	+	0.701849i	−0.0800150	+	0.0838869i
$71$			5.49843i			0.652544i	0.945276	+	0.326272i	$0.105793\pi$
$71$			5.49843i			0.652544i	−0.945276	+	0.326272i	$0.894207\pi$
$72$		0			0
$73$		−	4.07314i		−	0.476725i	−0.971176	−	0.238363i	$-0.923389\pi$
$73$		−	4.07314i		−	0.476725i	0.971176	−	0.238363i	$-0.0766106\pi$
$74$		−	5.16789i		−	0.600755i
$75$		0			0
$76$		−	6.35722i		−	0.729224i
$77$	1.22250	−	1.28166i	0.139317	−	0.146059i
$78$		0			0
$79$	8.35568			0.940087			0.470044	−	0.882643i	$-0.344238\pi$
$79$	8.35568			0.940087			0.470044	+	0.882643i	$0.344238\pi$
$80$	−0.366598			−0.0409869
$81$		0			0
$82$		−	4.31856i		−	0.476905i
$83$	17.0142			1.86756			0.933778	−	0.357852i	$-0.116491\pi$
$83$	17.0142			1.86756			0.933778	+	0.357852i	$0.116491\pi$
$84$		0			0
$85$	−1.82898			−0.198381
$86$		−	4.49843i		−	0.485079i
$87$		0			0
$88$	0.669453			0.0713640
$89$	−10.7113			−1.13540			−0.567699	−	0.823236i	$-0.692166\pi$
$89$	−10.7113			−1.13540			−0.567699	+	0.823236i	$0.692166\pi$
$90$		0			0
$91$	−1.82898	+	1.91749i	−0.191729	+	0.201007i
$92$		−	7.69459i		−	0.802216i
$93$		0			0
$94$			8.32901i			0.859071i
$95$		−	2.33055i		−	0.239109i
$96$		0			0
$97$			17.2157i			1.74799i	0.485932	+	0.873997i	$0.338480\pi$
$97$			17.2157i			1.74799i	−0.485932	+	0.873997i	$0.661520\pi$
$98$	−6.99219	+	0.330547i	−0.706318	+	0.0333902i
$99$		0			0
$100$	4.86561			0.486561
$101$	−15.7317			−1.56537			−0.782683	−	0.622421i	$-0.786149\pi$
$101$	−15.7317			−1.56537			−0.782683	+	0.622421i	$0.786149\pi$
$102$		0			0
$103$		−	11.4445i		−	1.12766i	−0.825890	−	0.563831i	$-0.809327\pi$
$103$		−	11.4445i		−	1.12766i	0.825890	−	0.563831i	$-0.190673\pi$
$104$	−1.00156			−0.0982115
$105$		0			0
$106$		0			0
$107$			11.0618i			1.06938i	0.845048	+	0.534690i	$0.179572\pi$
$107$			11.0618i			1.06938i	−0.845048	+	0.534690i	$0.820428\pi$
$108$		0			0
$109$	−10.5633			−1.01178			−0.505891	−	0.862597i	$-0.668836\pi$
$109$	−10.5633			−1.01178			−0.505891	+	0.862597i	$0.668836\pi$
$110$	0.245420			0.0233999
$111$		0			0
$112$	−1.91449	−	1.82612i	−0.180902	−	0.172552i
$113$		−	4.15953i		−	0.391295i	−0.980674	−	0.195648i	$-0.937319\pi$
$113$		−	4.15953i		−	0.391295i	0.980674	−	0.195648i	$-0.0626809\pi$
$114$		0			0
$115$		−	2.82082i		−	0.263043i
$116$		−	1.82898i		−	0.169817i
$117$		0			0
$118$			8.72695i			0.803381i
$119$	−9.55151	−	9.11064i	−0.875586	−	0.835171i
$120$		0			0
$121$	10.5518			0.959257
$122$	−4.95771			−0.448850
$123$		0			0
$124$		−	6.32588i		−	0.568081i
$125$	3.61671			0.323489
$126$		0			0
$127$	−1.66945			−0.148140			−0.0740700	−	0.997253i	$-0.523599\pi$
$127$	−1.66945			−0.148140			−0.0740700	+	0.997253i	$0.523599\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−0.367172			−0.0322031
$131$	13.5321			1.18231			0.591154	−	0.806558i	$-0.298672\pi$
$131$	13.5321			1.18231			0.591154	+	0.806558i	$0.298672\pi$
$132$		0			0
$133$	11.6091	−	12.1708i	1.00663	−	1.05535i
$134$		−	10.8907i		−	0.940817i
$135$		0			0
$136$		−	4.98906i		−	0.427809i
$137$			8.98851i			0.767940i	0.923346	+	0.383970i	$0.125443\pi$
$137$			8.98851i			0.767940i	−0.923346	+	0.383970i	$0.874557\pi$
$138$		0			0
$139$		−	9.29922i		−	0.788750i	−0.918950	−	0.394375i	$-0.870961\pi$
$139$		−	9.29922i		−	0.788750i	0.918950	−	0.394375i	$-0.129039\pi$
$140$	−0.701849	−	0.669453i	−0.0593170	−	0.0565791i
$141$		0			0
$142$	−5.49843			−0.461418
$143$	0.670501			0.0560701
$144$		0			0
$145$		−	0.670501i		−	0.0556821i
$146$	4.07314			0.337096
$147$		0			0
$148$	5.16789			0.424798
$149$			2.83211i			0.232016i	0.993248	+	0.116008i	$0.0370098\pi$
$149$			2.83211i			0.232016i	−0.993248	+	0.116008i	$0.962990\pi$
$150$		0			0
$151$	−16.5518			−1.34697			−0.673484	−	0.739201i	$-0.735203\pi$
$151$	−16.5518			−1.34697			−0.673484	+	0.739201i	$0.735203\pi$
$152$	6.35722			0.515639
$153$		0			0
$154$	1.28166	+	1.22250i	0.103279	+	0.0985122i
$155$		−	2.31905i		−	0.186271i
$156$		0			0
$157$			2.83456i			0.226222i	0.993582	+	0.113111i	$0.0360816\pi$
$157$			2.83456i			0.226222i	−0.993582	+	0.113111i	$0.963918\pi$
$158$			8.35568i			0.664742i
$159$		0			0
$160$		−	0.366598i		−	0.0289821i
$161$	14.0513	−	14.7312i	1.10739	−	1.16098i
$162$		0			0
$163$	24.7281			1.93685			0.968426	−	0.249300i	$-0.0802005\pi$
$163$	24.7281			1.93685			0.968426	+	0.249300i	$0.0802005\pi$
$164$	4.31856			0.337223
$165$		0			0
$166$			17.0142i			1.32056i
$167$	−19.3484			−1.49723			−0.748614	−	0.663006i	$-0.769280\pi$
$167$	−19.3484			−1.49723			−0.748614	+	0.663006i	$0.769280\pi$
$168$		0			0
$169$	11.9969			0.922836
$170$		−	1.82898i		−	0.140276i
$171$		0			0
$172$	4.49843			0.343002
$173$	−4.83654			−0.367715			−0.183858	−	0.982953i	$-0.558859\pi$
$173$	−4.83654			−0.367715			−0.183858	+	0.982953i	$0.558859\pi$
$174$		0			0
$175$	9.31516	+	8.88520i	0.704160	+	0.671658i
$176$			0.669453i			0.0504619i
$177$		0			0
$178$		−	10.7113i		−	0.802847i
$179$			3.65796i			0.273409i	0.990612	+	0.136704i	$0.0436511\pi$
$179$			3.65796i			0.273409i	−0.990612	+	0.136704i	$0.956349\pi$
$180$		0			0
$181$		−	5.66796i		−	0.421296i	−0.977562	−	0.210648i	$-0.932443\pi$
$181$		−	5.66796i		−	0.421296i	0.977562	−	0.210648i	$-0.0675574\pi$
$182$	−1.91749	−	1.82898i	−0.142134	−	0.135573i
$183$		0			0
$184$	7.69459			0.567252
$185$	1.89454			0.139289
$186$		0			0
$187$			3.33994i			0.244241i
$188$	−8.32901			−0.607455
$189$		0			0
$190$	2.33055			0.169076
$191$			27.3777i			1.98098i	0.137587	+	0.990490i	$0.456065\pi$
$191$			27.3777i			1.98098i	−0.137587	+	0.990490i	$0.543935\pi$
$192$		0			0
$193$	−10.0283			−0.721850			−0.360925	−	0.932595i	$-0.617539\pi$
$193$	−10.0283			−0.721850			−0.360925	+	0.932595i	$0.617539\pi$
$194$	−17.2157			−1.23602
$195$		0			0
$196$	−0.330547	−	6.99219i	−0.0236105	−	0.499442i
$197$		−	18.8258i		−	1.34129i	−0.741780	−	0.670643i	$-0.766018\pi$
$197$		−	18.8258i		−	1.34129i	0.741780	−	0.670643i	$-0.233982\pi$
$198$		0			0
$199$			5.36406i			0.380248i	0.981760	+	0.190124i	$0.0608890\pi$
$199$			5.36406i			0.380248i	−0.981760	+	0.190124i	$0.939111\pi$
$200$			4.86561i			0.344050i
$201$		0			0
$202$		−	15.7317i		−	1.10688i
$203$	3.33994	−	3.50157i	0.234418	−	0.245762i
$204$		0			0
$205$	1.58318			0.110574
$206$	11.4445			0.797377
$207$		0			0
$208$		−	1.00156i		−	0.0694460i
$209$	−4.25587			−0.294384
$210$		0			0
$211$	1.65796			0.114139			0.0570694	−	0.998370i	$-0.481824\pi$
$211$	1.65796			0.114139			0.0570694	+	0.998370i	$0.481824\pi$
$212$		0			0
$213$		0			0
$214$	−11.0618			−0.756166
$215$	1.64912			0.112469
$216$		0			0
$217$	11.5518	−	12.1108i	0.784189	−	0.822137i
$218$		−	10.5633i		−	0.715439i
$219$		0			0
$220$			0.245420i			0.0165462i
$221$		−	4.99687i		−	0.336126i
$222$		0			0
$223$		−	17.0372i		−	1.14090i	−0.821334	−	0.570448i	$-0.806770\pi$
$223$		−	17.0372i		−	1.14090i	0.821334	−	0.570448i	$-0.193230\pi$
$224$	1.82612	−	1.91449i	0.122013	−	0.127917i
$225$		0			0
$226$	4.15953			0.276688
$227$	5.11024			0.339179			0.169589	−	0.985515i	$-0.445756\pi$
$227$	5.11024			0.339179			0.169589	+	0.985515i	$0.445756\pi$
$228$		0			0
$229$		−	15.2669i		−	1.00887i	−0.863451	−	0.504433i	$-0.831702\pi$
$229$		−	15.2669i		−	1.00887i	0.863451	−	0.504433i	$-0.168298\pi$
$230$	2.82082			0.185999
$231$		0			0
$232$	1.82898			0.120078
$233$			10.1930i			0.667767i	0.942614	+	0.333883i	$0.108359\pi$
$233$			10.1930i			0.667767i	−0.942614	+	0.333883i	$0.891641\pi$
$234$		0			0
$235$	−3.05340			−0.199182
$236$	−8.72695			−0.568076
$237$		0			0
$238$	9.11064	−	9.55151i	0.590555	−	0.619132i
$239$			19.1815i			1.24075i	0.784305	+	0.620375i	$0.213020\pi$
$239$			19.1815i			1.24075i	−0.784305	+	0.620375i	$0.786980\pi$
$240$		0			0
$241$		−	20.6853i		−	1.33245i	−0.745749	−	0.666227i	$-0.767908\pi$
$241$		−	20.6853i		−	1.33245i	0.745749	−	0.666227i	$-0.232092\pi$
$242$			10.5518i			0.678297i
$243$		0			0
$244$		−	4.95771i		−	0.317385i
$245$	−0.121178	−	2.56332i	−0.00774176	−	0.163765i
$246$		0			0
$247$	6.36717			0.405133
$248$	6.32588			0.401694
$249$		0			0
$250$			3.61671i			0.228741i
$251$	1.81200			0.114373			0.0571864	−	0.998364i	$-0.481787\pi$
$251$	1.81200			0.114373			0.0571864	+	0.998364i	$0.481787\pi$
$252$		0			0
$253$	−5.15117			−0.323851
$254$		−	1.66945i		−	0.104751i
$255$		0			0
$256$	1.00000			0.0625000
$257$	6.45545			0.402680			0.201340	−	0.979521i	$-0.435470\pi$
$257$	6.45545			0.402680			0.201340	+	0.979521i	$0.435470\pi$
$258$		0			0
$259$	9.89387	+	9.43720i	0.614775	+	0.586399i
$260$		−	0.367172i		−	0.0227710i
$261$		0			0
$262$			13.5321i			0.836018i
$263$		−	8.82062i		−	0.543903i	−0.962311	−	0.271951i	$-0.912331\pi$
$263$		−	8.82062i		−	0.543903i	0.962311	−	0.271951i	$-0.0876690\pi$
$264$		0			0
$265$		0			0
$266$	12.1708	+	11.6091i	0.746242	+	0.711798i
$267$		0			0
$268$	10.8907			0.665258
$269$	−14.2653			−0.869773			−0.434886	−	0.900485i	$-0.643212\pi$
$269$	−14.2653			−0.869773			−0.434886	+	0.900485i	$0.643212\pi$
$270$		0			0
$271$		−	3.05281i		−	0.185445i	−0.995692	−	0.0927226i	$-0.970443\pi$
$271$		−	3.05281i		−	0.185445i	0.995692	−	0.0927226i	$-0.0295570\pi$
$272$	4.98906			0.302506
$273$		0			0
$274$	−8.98851			−0.543016
$275$		−	3.25730i		−	0.196422i
$276$		0			0
$277$	1.26566			0.0760459			0.0380230	−	0.999277i	$-0.487894\pi$
$277$	1.26566			0.0760459			0.0380230	+	0.999277i	$0.487894\pi$
$278$	9.29922			0.557730
$279$		0			0
$280$	0.669453	−	0.701849i	0.0400075	−	0.0419435i
$281$			10.5267i			0.627970i	0.949428	+	0.313985i	$0.101664\pi$
$281$			10.5267i			0.627970i	−0.949428	+	0.313985i	$0.898336\pi$
$282$		0			0
$283$		−	19.8718i		−	1.18125i	−0.806945	−	0.590627i	$-0.798881\pi$
$283$		−	19.8718i		−	1.18125i	0.806945	−	0.590627i	$-0.201119\pi$
$284$		−	5.49843i		−	0.326272i
$285$		0			0
$286$			0.670501i			0.0396475i
$287$	8.26784	+	7.88623i	0.488035	+	0.465509i
$288$		0			0
$289$	7.89074			0.464161
$290$	0.670501			0.0393732
$291$		0			0
$292$			4.07314i			0.238363i
$293$	13.4121			0.783544			0.391772	−	0.920062i	$-0.371862\pi$
$293$	13.4121			0.783544			0.391772	+	0.920062i	$0.371862\pi$
$294$		0			0
$295$	−3.19928			−0.186270
$296$			5.16789i			0.300377i
$297$		0			0
$298$	−2.83211			−0.164060
$299$	7.70663			0.445686
$300$		0			0
$301$	8.61221	+	8.21470i	0.496399	+	0.473487i
$302$		−	16.5518i		−	0.952451i
$303$		0			0
$304$			6.35722i			0.364612i
$305$		−	1.81749i		−	0.104069i
$306$		0			0
$307$		−	0.653728i		−	0.0373102i	−0.999826	−	0.0186551i	$-0.994062\pi$
$307$		−	0.653728i		−	0.0373102i	0.999826	−	0.0186551i	$-0.00593845\pi$
$308$	−1.22250	+	1.28166i	−0.0696587	+	0.0730295i
$309$		0			0
$310$	2.31905			0.131713
$311$	9.24493			0.524232			0.262116	−	0.965036i	$-0.415580\pi$
$311$	9.24493			0.524232			0.262116	+	0.965036i	$0.415580\pi$
$312$		0			0
$313$			6.16414i			0.348418i	0.984709	+	0.174209i	$0.0557368\pi$
$313$			6.16414i			0.348418i	−0.984709	+	0.174209i	$0.944263\pi$
$314$	−2.83456			−0.159963
$315$		0			0
$316$	−8.35568			−0.470044
$317$		−	20.6548i		−	1.16009i	−0.814584	−	0.580045i	$-0.803035\pi$
$317$		−	20.6548i		−	1.16009i	0.814584	−	0.580045i	$-0.196965\pi$
$318$		0			0
$319$	−1.22442			−0.0685542
$320$	0.366598			0.0204935
$321$		0			0
$322$	14.7312	+	14.0513i	0.820938	+	0.783046i
$323$			31.7166i			1.76476i
$324$		0			0
$325$			4.87322i			0.270318i
$326$			24.7281i			1.36956i
$327$		0			0
$328$			4.31856i			0.238453i
$329$	−15.9458	−	15.2098i	−0.879121	−	0.838543i
$330$		0			0
$331$	10.7114			0.588750			0.294375	−	0.955690i	$-0.404889\pi$
$331$	10.7114			0.588750			0.294375	+	0.955690i	$0.404889\pi$
$332$	−17.0142			−0.933778
$333$		0			0
$334$		−	19.3484i		−	1.05870i
$335$	3.99252			0.218135
$336$		0			0
$337$	−7.55183			−0.411375			−0.205687	−	0.978618i	$-0.565943\pi$
$337$	−7.55183			−0.411375			−0.205687	+	0.978618i	$0.565943\pi$
$338$			11.9969i			0.652544i
$339$		0			0
$340$	1.82898			0.0991904
$341$	−4.23488			−0.229332
$342$		0			0
$343$	12.1358	−	13.9901i	0.655270	−	0.755394i
$344$			4.49843i			0.242539i
$345$		0			0
$346$		−	4.83654i		−	0.260014i
$347$			10.9320i			0.586859i	0.955981	+	0.293430i	$0.0947967\pi$
$347$			10.9320i			0.586859i	−0.955981	+	0.293430i	$0.905203\pi$
$348$		0			0
$349$		−	1.18429i		−	0.0633936i	−0.999498	−	0.0316968i	$-0.989909\pi$
$349$		−	1.18429i		−	0.0633936i	0.999498	−	0.0316968i	$-0.0100911\pi$
$350$	−8.88520	+	9.31516i	−0.474934	+	0.497916i
$351$		0			0
$352$	−0.669453			−0.0356820
$353$	33.5824			1.78741			0.893706	−	0.448653i	$-0.148096\pi$
$353$	33.5824			1.78741			0.893706	+	0.448653i	$0.148096\pi$
$354$		0			0
$355$		−	2.01572i		−	0.106983i
$356$	10.7113			0.567699
$357$		0			0
$358$	−3.65796			−0.193329
$359$		−	10.1281i		−	0.534542i	−0.963621	−	0.267271i	$-0.913878\pi$
$359$		−	10.1281i		−	0.534542i	0.963621	−	0.267271i	$-0.0861219\pi$
$360$		0			0
$361$	−21.4143			−1.12707
$362$	5.66796			0.297901
$363$		0			0
$364$	1.82898	−	1.91749i	0.0958646	−	0.100504i
$365$			1.49321i			0.0781580i
$366$		0			0
$367$			18.0021i			0.939701i	0.882746	+	0.469850i	$0.155692\pi$
$367$			18.0021i			0.939701i	−0.882746	+	0.469850i	$0.844308\pi$
$368$			7.69459i			0.401108i
$369$		0			0
$370$			1.89454i			0.0984923i
$371$		0			0
$372$		0			0
$373$	16.4090			0.849627			0.424814	−	0.905281i	$-0.360340\pi$
$373$	16.4090			0.849627			0.424814	+	0.905281i	$0.360340\pi$
$374$	−3.33994			−0.172704
$375$		0			0
$376$		−	8.32901i		−	0.429536i
$377$	1.83184			0.0943447
$378$		0			0
$379$	−2.91372			−0.149668			−0.0748339	−	0.997196i	$-0.523843\pi$
$379$	−2.91372			−0.149668			−0.0748339	+	0.997196i	$0.523843\pi$
$380$			2.33055i			0.119555i
$381$		0			0
$382$	−27.3777			−1.40076
$383$	−8.57443			−0.438133			−0.219066	−	0.975710i	$-0.570301\pi$
$383$	−8.57443			−0.438133			−0.219066	+	0.975710i	$0.570301\pi$
$384$		0			0
$385$	−0.448168	+	0.469855i	−0.0228407	+	0.0239460i
$386$		−	10.0283i		−	0.510425i
$387$		0			0
$388$		−	17.2157i		−	0.873997i
$389$		−	35.5539i		−	1.80266i	−0.433137	−	0.901328i	$-0.642594\pi$
$389$		−	35.5539i		−	1.80266i	0.433137	−	0.901328i	$-0.357406\pi$
$390$		0			0
$391$			38.3888i			1.94140i
$392$	6.99219	−	0.330547i	0.353159	−	0.0166951i
$393$		0			0
$394$	18.8258			0.948433
$395$	−3.06318			−0.154125
$396$		0			0
$397$			3.58034i			0.179692i	0.995956	+	0.0898460i	$0.0286375\pi$
$397$			3.58034i			0.179692i	−0.995956	+	0.0898460i	$0.971363\pi$
$398$	−5.36406			−0.268876
$399$		0			0
$400$	−4.86561			−0.243280
$401$		−	0.190871i		−	0.00953167i	−0.999989	−	0.00476583i	$-0.998483\pi$
$401$		−	0.190871i		−	0.00953167i	0.999989	−	0.00476583i	$-0.00151702\pi$
$402$		0			0
$403$	6.33577			0.315607
$404$	15.7317			0.782683
$405$		0			0
$406$	3.50157	+	3.33994i	0.173780	+	0.165759i
$407$		−	3.45966i		−	0.171489i
$408$		0			0
$409$		−	3.47371i		−	0.171764i	−0.996305	−	0.0858819i	$-0.972629\pi$
$409$		−	3.47371i		−	0.171764i	0.996305	−	0.0858819i	$-0.0273708\pi$
$410$			1.58318i			0.0781875i
$411$		0			0
$412$			11.4445i			0.563831i
$413$	−16.7077	−	15.9365i	−0.822130	−	0.784183i
$414$		0			0
$415$	−6.23739			−0.306182
$416$	1.00156			0.0491057
$417$		0			0
$418$		−	4.25587i		−	0.208161i
$419$	1.40791			0.0687809			0.0343905	−	0.999408i	$-0.489051\pi$
$419$	1.40791			0.0687809			0.0343905	+	0.999408i	$0.489051\pi$
$420$		0			0
$421$	−30.3860			−1.48093			−0.740463	−	0.672098i	$-0.765393\pi$
$421$	−30.3860			−1.48093			−0.740463	+	0.672098i	$0.765393\pi$
$422$			1.65796i			0.0807083i
$423$		0			0
$424$		0			0
$425$	−24.2748			−1.17750
$426$		0			0
$427$	9.05340	−	9.49150i	0.438125	−	0.459326i
$428$		−	11.0618i		−	0.534690i
$429$		0			0
$430$			1.64912i			0.0795275i
$431$		−	27.2747i		−	1.31378i	−0.753988	−	0.656888i	$-0.771873\pi$
$431$		−	27.2747i		−	1.31378i	0.753988	−	0.656888i	$-0.228127\pi$
$432$		0			0
$433$		−	8.15047i		−	0.391686i	−0.980635	−	0.195843i	$-0.937256\pi$
$433$		−	8.15047i		−	0.391686i	0.980635	−	0.195843i	$-0.0627444\pi$
$434$	12.1108	+	11.5518i	0.581338	+	0.554506i
$435$		0			0
$436$	10.5633			0.505891
$437$	−48.9162			−2.33998
$438$		0			0
$439$		−	12.2404i		−	0.584203i	−0.956387	−	0.292101i	$-0.905646\pi$
$439$		−	12.2404i		−	0.584203i	0.956387	−	0.292101i	$-0.0943545\pi$
$440$	−0.245420			−0.0117000
$441$		0			0
$442$	4.99687			0.237677
$443$			8.00836i			0.380489i	0.981737	+	0.190244i	$0.0609280\pi$
$443$			8.00836i			0.380489i	−0.981737	+	0.190244i	$0.939072\pi$
$444$		0			0
$445$	3.92675			0.186146
$446$	17.0372			0.806735
$447$		0			0
$448$	1.91449	+	1.82612i	0.0904512	+	0.0862762i
$449$			14.5183i			0.685163i	0.939488	+	0.342581i	$0.111301\pi$
$449$			14.5183i			0.685163i	−0.939488	+	0.342581i	$0.888699\pi$
$450$		0			0
$451$		−	2.89108i		−	0.136135i
$452$			4.15953i			0.195648i
$453$		0			0
$454$			5.11024i			0.239835i
$455$	0.670501	−	0.702947i	0.0314336	−	0.0329547i
$456$		0			0
$457$	9.95501			0.465676			0.232838	−	0.972516i	$-0.425199\pi$
$457$	9.95501			0.465676			0.232838	+	0.972516i	$0.425199\pi$
$458$	15.2669			0.713375
$459$		0			0
$460$			2.82082i			0.131521i
$461$	−32.3270			−1.50562			−0.752810	−	0.658237i	$-0.771302\pi$
$461$	−32.3270			−1.50562			−0.752810	+	0.658237i	$0.771302\pi$
$462$		0			0
$463$	9.45032			0.439193			0.219597	−	0.975591i	$-0.429526\pi$
$463$	9.45032			0.439193			0.219597	+	0.975591i	$0.429526\pi$
$464$			1.82898i			0.0849083i
$465$		0			0
$466$	−10.1930			−0.472183
$467$	−20.6623			−0.956138			−0.478069	−	0.878322i	$-0.658663\pi$
$467$	−20.6623			−0.956138			−0.478069	+	0.878322i	$0.658663\pi$
$468$		0			0
$469$	20.8502	+	19.8878i	0.962773	+	0.918335i
$470$		−	3.05340i		−	0.140843i
$471$		0			0
$472$		−	8.72695i		−	0.401691i
$473$		−	3.01149i		−	0.138469i
$474$		0			0
$475$		−	30.9317i		−	1.41925i
$476$	9.55151	+	9.11064i	0.437793	+	0.417586i
$477$		0			0
$478$	−19.1815			−0.877343
$479$	−10.1608			−0.464261			−0.232131	−	0.972685i	$-0.574570\pi$
$479$	−10.1608			−0.464261			−0.232131	+	0.972685i	$0.574570\pi$
$480$		0			0
$481$			5.17597i			0.236004i
$482$	20.6853			0.942188
$483$		0			0
$484$	−10.5518			−0.479629
$485$		−	6.31126i		−	0.286579i
$486$		0			0
$487$	−31.2296			−1.41515			−0.707575	−	0.706638i	$-0.750211\pi$
$487$	−31.2296			−1.41515			−0.707575	+	0.706638i	$0.750211\pi$
$488$	4.95771			0.224425
$489$		0			0
$490$	2.56332	−	0.121178i	0.115799	−	0.00547425i
$491$		−	20.5899i		−	0.929211i	−0.885518	−	0.464605i	$-0.846196\pi$
$491$		−	20.5899i		−	0.929211i	0.885518	−	0.464605i	$-0.153804\pi$
$492$		0			0
$493$			9.12490i			0.410965i
$494$			6.36717i			0.286473i
$495$		0			0
$496$			6.32588i			0.284040i
$497$	10.0408	−	10.5267i	0.450392	−	0.472187i
$498$		0			0
$499$	−25.1533			−1.12601			−0.563007	−	0.826452i	$-0.690356\pi$
$499$	−25.1533			−1.12601			−0.563007	+	0.826452i	$0.690356\pi$
$500$	−3.61671			−0.161744
$501$		0			0
$502$			1.81200i			0.0808737i
$503$	31.1553			1.38915			0.694574	−	0.719421i	$-0.255593\pi$
$503$	31.1553			1.38915			0.694574	+	0.719421i	$0.255593\pi$
$504$		0			0
$505$	5.76722			0.256638
$506$		−	5.15117i		−	0.228997i
$507$		0			0
$508$	1.66945			0.0740700
$509$	−4.83347			−0.214240			−0.107120	−	0.994246i	$-0.534163\pi$
$509$	−4.83347			−0.214240			−0.107120	+	0.994246i	$0.534163\pi$
$510$		0			0
$511$	−7.43806	+	7.79799i	−0.329040	+	0.344963i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			6.45545i			0.284738i
$515$			4.19554i			0.184877i
$516$		0			0
$517$			5.57588i			0.245227i
$518$	−9.43720	+	9.89387i	−0.414647	+	0.434712i
$519$		0			0
$520$	0.367172			0.0161015
$521$	−17.5322			−0.768101			−0.384050	−	0.923312i	$-0.625471\pi$
$521$	−17.5322			−0.768101			−0.384050	+	0.923312i	$0.625471\pi$
$522$		0			0
$523$			19.1019i			0.835267i	0.908616	+	0.417633i	$0.137140\pi$
$523$			19.1019i			0.835267i	−0.908616	+	0.417633i	$0.862860\pi$
$524$	−13.5321			−0.591154
$525$		0			0
$526$	8.82062			0.384597
$527$			31.5602i			1.37478i
$528$		0			0
$529$	−36.2067			−1.57420
$530$		0			0
$531$		0			0
$532$	−11.6091	+	12.1708i	−0.503317	+	0.527673i
$533$			4.32532i			0.187350i
$534$		0			0
$535$		−	4.05522i		−	0.175322i
$536$			10.8907i			0.470408i
$537$		0			0
$538$		−	14.2653i		−	0.615022i
$539$	−4.68095	+	0.221286i	−0.201623	+	0.00953144i
$540$		0			0
$541$	13.6642			0.587471			0.293735	−	0.955887i	$-0.405102\pi$
$541$	13.6642			0.587471			0.293735	+	0.955887i	$0.405102\pi$
$542$	3.05281			0.131130
$543$		0			0
$544$			4.98906i			0.213904i
$545$	3.87249			0.165879
$546$		0			0
$547$	−9.88761			−0.422764			−0.211382	−	0.977404i	$-0.567796\pi$
$547$	−9.88761			−0.422764			−0.211382	+	0.977404i	$0.567796\pi$
$548$		−	8.98851i		−	0.383970i
$549$		0			0
$550$	3.25730			0.138892
$551$	−11.6272			−0.495337
$552$		0			0
$553$	−15.9969	−	15.2585i	−0.680256	−	0.648858i
$554$			1.26566i			0.0537726i
$555$		0			0
$556$			9.29922i			0.394375i
$557$		−	12.5800i		−	0.533034i	−0.963830	−	0.266517i	$-0.914127\pi$
$557$		−	12.5800i		−	0.533034i	0.963830	−	0.266517i	$-0.0858728\pi$
$558$		0			0
$559$			4.50547i			0.190561i
$560$	0.701849	+	0.669453i	0.0296585	+	0.0282896i
$561$		0			0
$562$	−10.5267			−0.444042
$563$	24.3333			1.02553			0.512763	−	0.858530i	$-0.328622\pi$
$563$	24.3333			1.02553			0.512763	+	0.858530i	$0.328622\pi$
$564$		0			0
$565$			1.52487i			0.0641520i
$566$	19.8718			0.835272
$567$		0			0
$568$	5.49843			0.230709
$569$		−	9.45406i		−	0.396335i	−0.980168	−	0.198167i	$-0.936501\pi$
$569$		−	9.45406i		−	0.396335i	0.980168	−	0.198167i	$-0.0634990\pi$
$570$		0			0
$571$	−31.5686			−1.32110			−0.660551	−	0.750781i	$-0.729677\pi$
$571$	−31.5686			−1.32110			−0.660551	+	0.750781i	$0.729677\pi$
$572$	−0.670501			−0.0280351
$573$		0			0
$574$	−7.88623	+	8.26784i	−0.329165	+	0.345093i
$575$		−	37.4388i		−	1.56131i
$576$		0			0
$577$			33.5794i			1.39793i	0.715157	+	0.698964i	$0.246355\pi$
$577$			33.5794i			1.39793i	−0.715157	+	0.698964i	$0.753645\pi$
$578$			7.89074i			0.328211i
$579$		0			0
$580$			0.670501i			0.0278410i
$581$	−32.5736	−	31.0701i	−1.35138	−	1.28901i
$582$		0			0
$583$		0			0
$584$	−4.07314			−0.168548
$585$		0			0
$586$			13.4121i			0.554049i
$587$	−19.3171			−0.797302			−0.398651	−	0.917103i	$-0.630521\pi$
$587$	−19.3171			−0.797302			−0.398651	+	0.917103i	$0.630521\pi$
$588$		0			0
$589$	−40.2150			−1.65703
$590$		−	3.19928i		−	0.131712i
$591$		0			0
$592$	−5.16789			−0.212399
$593$	−0.733196			−0.0301088			−0.0150544	−	0.999887i	$-0.504792\pi$
$593$	−0.733196			−0.0301088			−0.0150544	+	0.999887i	$0.504792\pi$
$594$		0			0
$595$	3.50157	+	3.33994i	0.143550	+	0.136924i
$596$		−	2.83211i		−	0.116008i
$597$		0			0
$598$			7.70663i			0.315147i
$599$		−	30.7783i		−	1.25757i	−0.777580	−	0.628785i	$-0.783553\pi$
$599$		−	30.7783i		−	1.25757i	0.777580	−	0.628785i	$-0.216447\pi$
$600$		0			0
$601$			0.908670i			0.0370654i	0.999828	+	0.0185327i	$0.00589948\pi$
$601$			0.908670i			0.0370654i	−0.999828	+	0.0185327i	$0.994101\pi$
$602$	−8.21470	+	8.61221i	−0.334806	+	0.351007i
$603$		0			0
$604$	16.5518			0.673484
$605$	−3.86828			−0.157268
$606$		0			0
$607$			44.7773i			1.81746i	0.417389	+	0.908728i	$0.362945\pi$
$607$			44.7773i			1.81746i	−0.417389	+	0.908728i	$0.637055\pi$
$608$	−6.35722			−0.257820
$609$		0			0
$610$	1.81749			0.0735880
$611$		−	8.34204i		−	0.337483i
$612$		0			0
$613$	18.1480			0.732992			0.366496	−	0.930420i	$-0.380557\pi$
$613$	18.1480			0.732992			0.366496	+	0.930420i	$0.380557\pi$
$614$	0.653728			0.0263823
$615$		0			0
$616$	−1.28166	−	1.22250i	−0.0516396	−	0.0492561i
$617$			22.7930i			0.917610i	0.888537	+	0.458805i	$0.151722\pi$
$617$			22.7930i			0.917610i	−0.888537	+	0.458805i	$0.848278\pi$
$618$		0			0
$619$			44.3668i			1.78325i	0.452772	+	0.891626i	$0.350435\pi$
$619$			44.3668i			1.78325i	−0.452772	+	0.891626i	$0.649565\pi$
$620$			2.31905i			0.0931355i
$621$		0			0
$622$			9.24493i			0.370688i
$623$	20.5067	+	19.5602i	0.821584	+	0.783663i
$624$		0			0
$625$	23.0021			0.920086
$626$	−6.16414			−0.246368
$627$		0			0
$628$		−	2.83456i		−	0.113111i
$629$	−25.7829			−1.02803
$630$		0			0
$631$	−32.5707			−1.29662			−0.648310	−	0.761377i	$-0.724524\pi$
$631$	−32.5707			−1.29662			−0.648310	+	0.761377i	$0.724524\pi$
$632$		−	8.35568i		−	0.332371i
$633$		0			0
$634$	20.6548			0.820308
$635$	0.612018			0.0242872
$636$		0			0
$637$	7.00313	−	0.331064i	0.277474	−	0.0131172i
$638$		−	1.22442i		−	0.0484751i
$639$		0			0
$640$			0.366598i			0.0144911i
$641$		−	11.8091i		−	0.466433i	−0.972425	−	0.233216i	$-0.925075\pi$
$641$		−	11.8091i		−	0.466433i	0.972425	−	0.233216i	$-0.0749250\pi$
$642$		0			0
$643$			29.2964i			1.15534i	0.816272	+	0.577668i	$0.196037\pi$
$643$			29.2964i			1.15534i	−0.816272	+	0.577668i	$0.803963\pi$
$644$	−14.0513	+	14.7312i	−0.553697	+	0.580491i
$645$		0			0
$646$	−31.7166			−1.24787
$647$	28.1683			1.10741			0.553705	−	0.832713i	$-0.313214\pi$
$647$	28.1683			1.10741			0.553705	+	0.832713i	$0.313214\pi$
$648$		0			0
$649$			5.84229i			0.229330i
$650$	−4.87322			−0.191143
$651$		0			0
$652$	−24.7281			−0.968426
$653$		−	45.0974i		−	1.76480i	−0.470502	−	0.882399i	$-0.655927\pi$
$653$		−	45.0974i		−	1.76480i	0.470502	−	0.882399i	$-0.344073\pi$
$654$		0			0
$655$	−4.96086			−0.193837
$656$	−4.31856			−0.168611
$657$		0			0
$658$	15.2098	−	15.9458i	0.592939	−	0.621632i
$659$			31.8045i			1.23893i	0.785026	+	0.619463i	$0.212650\pi$
$659$			31.8045i			1.23893i	−0.785026	+	0.619463i	$0.787350\pi$
$660$		0			0
$661$			19.7724i			0.769056i	0.923113	+	0.384528i	$0.125636\pi$
$661$			19.7724i			0.769056i	−0.923113	+	0.384528i	$0.874364\pi$
$662$			10.7114i			0.416309i
$663$		0			0
$664$		−	17.0142i		−	0.660281i
$665$	−4.25587	+	4.46181i	−0.165035	+	0.173022i
$666$		0			0
$667$	−14.0733			−0.544918
$668$	19.3484			0.748614
$669$		0			0
$670$			3.99252i			0.154245i
$671$	−3.31896			−0.128127
$672$		0			0
$673$	1.89074			0.0728826			0.0364413	−	0.999336i	$-0.488398\pi$
$673$	1.89074			0.0728826			0.0364413	+	0.999336i	$0.488398\pi$
$674$		−	7.55183i		−	0.290886i
$675$		0			0
$676$	−11.9969			−0.461418
$677$	21.1322			0.812175			0.406088	−	0.913834i	$-0.366893\pi$
$677$	21.1322			0.812175			0.406088	+	0.913834i	$0.366893\pi$
$678$		0			0
$679$	31.4381	−	32.9594i	1.20648	−	1.26486i
$680$			1.82898i			0.0701382i
$681$		0			0
$682$		−	4.23488i		−	0.162162i
$683$		−	8.71972i		−	0.333651i	−0.985986	−	0.166825i	$-0.946648\pi$
$683$		−	8.71972i		−	0.333651i	0.985986	−	0.166825i	$-0.0533516\pi$
$684$		0			0
$685$		−	3.29517i		−	0.125902i
$686$	13.9901	+	12.1358i	0.534145	+	0.463346i
$687$		0			0
$688$	−4.49843			−0.171501
$689$		0			0
$690$		0			0
$691$			18.1370i			0.689964i	0.938609	+	0.344982i	$0.112115\pi$
$691$			18.1370i			0.689964i	−0.938609	+	0.344982i	$0.887885\pi$
$692$	4.83654			0.183858
$693$		0			0
$694$	−10.9320			−0.414972
$695$			3.40908i			0.129314i
$696$		0			0
$697$	−21.5456			−0.816097
$698$	1.18429			0.0448260
$699$		0			0
$700$	−9.31516	−	8.88520i	−0.352080	−	0.335829i
$701$			35.6167i			1.34523i	0.739995	+	0.672613i	$0.234828\pi$
$701$			35.6167i			1.34523i	−0.739995	+	0.672613i	$0.765172\pi$
$702$		0			0
$703$		−	32.8534i		−	1.23909i
$704$		−	0.669453i		−	0.0252310i
$705$		0			0
$706$			33.5824i			1.26389i
$707$	30.1182	+	28.7281i	1.13271	+	1.08043i
$708$		0			0
$709$	−3.60770			−0.135490			−0.0677449	−	0.997703i	$-0.521580\pi$
$709$	−3.60770			−0.135490			−0.0677449	+	0.997703i	$0.521580\pi$
$710$	2.01572			0.0756485
$711$		0			0
$712$			10.7113i			0.401424i
$713$	−48.6750			−1.82289
$714$		0			0
$715$	−0.245804			−0.00919256
$716$		−	3.65796i		−	0.136704i
$717$		0			0
$718$	10.1281			0.377978
$719$	25.7829			0.961540			0.480770	−	0.876847i	$-0.340357\pi$
$719$	25.7829			0.961540			0.480770	+	0.876847i	$0.340357\pi$
$720$		0			0
$721$	−20.8991	+	21.9104i	−0.778323	+	0.815987i
$722$		−	21.4143i		−	0.796958i
$723$		0			0
$724$			5.66796i			0.210648i
$725$		−	8.89910i		−	0.330504i
$726$		0			0
$727$		−	1.52909i		−	0.0567107i	−0.999598	−	0.0283554i	$-0.990973\pi$
$727$		−	1.52909i		−	0.0567107i	0.999598	−	0.0283554i	$-0.00902700\pi$
$728$	1.91749	+	1.82898i	0.0710668	+	0.0677865i
$729$		0			0
$730$	−1.49321			−0.0552660
$731$	−22.4430			−0.830083
$732$		0			0
$733$		−	20.7739i		−	0.767303i	−0.923478	−	0.383651i	$-0.874666\pi$
$733$		−	20.7739i		−	0.767303i	0.923478	−	0.383651i	$-0.125334\pi$
$734$	−18.0021			−0.664469
$735$		0			0
$736$	−7.69459			−0.283626
$737$		−	7.29084i		−	0.268562i
$738$		0			0
$739$	−11.8709			−0.436678			−0.218339	−	0.975873i	$-0.570064\pi$
$739$	−11.8709			−0.436678			−0.218339	+	0.975873i	$0.570064\pi$
$740$	−1.89454			−0.0696446
$741$		0			0
$742$		0			0
$743$			43.4059i			1.59241i	0.605028	+	0.796204i	$0.293162\pi$
$743$			43.4059i			1.59241i	−0.605028	+	0.796204i	$0.706838\pi$
$744$		0			0
$745$		−	1.03825i		−	0.0380384i
$746$			16.4090i			0.600777i
$747$		0			0
$748$		−	3.33994i		−	0.122120i
$749$	20.2001	−	21.1776i	0.738097	−	0.773814i
$750$		0			0
$751$	2.31383			0.0844327			0.0422164	−	0.999108i	$-0.486558\pi$
$751$	2.31383			0.0844327			0.0422164	+	0.999108i	$0.486558\pi$
$752$	8.32901			0.303728
$753$		0			0
$754$			1.83184i			0.0667118i
$755$	6.06787			0.220832
$756$		0			0
$757$	−15.0946			−0.548624			−0.274312	−	0.961641i	$-0.588450\pi$
$757$	−15.0946			−0.548624			−0.274312	+	0.961641i	$0.588450\pi$
$758$		−	2.91372i		−	0.105831i
$759$		0			0
$760$	−2.33055			−0.0845378
$761$	23.3379			0.845998			0.422999	−	0.906130i	$-0.360977\pi$
$761$	23.3379			0.845998			0.422999	+	0.906130i	$0.360977\pi$
$762$		0			0
$763$	20.2234	+	19.2899i	0.732136	+	0.698342i
$764$		−	27.3777i		−	0.990490i
$765$		0			0
$766$		−	8.57443i		−	0.309807i
$767$		−	8.74061i		−	0.315605i
$768$		0			0
$769$		−	18.2750i		−	0.659012i	−0.944153	−	0.329506i	$-0.893118\pi$
$769$		−	18.2750i		−	0.659012i	0.944153	−	0.329506i	$-0.106882\pi$
$770$	−0.469855	−	0.448168i	−0.0169324	−	0.0161508i
$771$		0			0
$772$	10.0283			0.360925
$773$	0.438507			0.0157720			0.00788600	−	0.999969i	$-0.497490\pi$
$773$	0.438507			0.0157720			0.00788600	+	0.999969i	$0.497490\pi$
$774$		0			0
$775$		−	30.7792i		−	1.10562i
$776$	17.2157			0.618009
$777$		0			0
$778$	35.5539			1.27467
$779$		−	27.4541i		−	0.983644i
$780$		0			0
$781$	−3.68095			−0.131715
$782$	−38.3888			−1.37278
$783$		0			0
$784$	0.330547	+	6.99219i	0.0118052	+	0.249721i
$785$		−	1.03914i		−	0.0370886i
$786$		0			0
$787$		−	38.2572i		−	1.36372i	−0.731481	−	0.681861i	$-0.761171\pi$
$787$		−	38.2572i		−	1.36372i	0.731481	−	0.681861i	$-0.238829\pi$
$788$			18.8258i			0.670643i
$789$		0			0
$790$		−	3.06318i		−	0.108983i
$791$	−7.59581	+	7.96337i	−0.270076	+	0.283145i
$792$		0			0
$793$	4.96547			0.176329
$794$	−3.58034			−0.127061
$795$		0			0
$796$		−	5.36406i		−	0.190124i
$797$	35.3225			1.25119			0.625594	−	0.780149i	$-0.284857\pi$
$797$	35.3225			1.25119			0.625594	+	0.780149i	$0.284857\pi$
$798$		0			0
$799$	41.5539			1.47007
$800$		−	4.86561i		−	0.172025i
$801$		0			0
$802$	0.190871			0.00673991
$803$	2.72678			0.0962259
$804$		0			0
$805$	−5.15117	+	5.40043i	−0.181555	+	0.190340i
$806$			6.33577i			0.223168i
$807$		0			0
$808$			15.7317i			0.553440i
$809$			21.7669i			0.765282i	0.923897	+	0.382641i	$0.124985\pi$
$809$			21.7669i			0.765282i	−0.923897	+	0.382641i	$0.875015\pi$
$810$		0			0
$811$			17.0184i			0.597598i	0.954316	+	0.298799i	$0.0965860\pi$
$811$			17.0184i			0.597598i	−0.954316	+	0.298799i	$0.903414\pi$
$812$	−3.33994	+	3.50157i	−0.117209	+	0.122881i
$813$		0			0
$814$	3.45966			0.121261
$815$	−9.06527			−0.317542
$816$		0			0
$817$		−	28.5976i		−	1.00050i
$818$	3.47371			0.121455
$819$		0			0
$820$	−1.58318			−0.0552869
$821$		−	24.8013i		−	0.865572i	−0.901497	−	0.432786i	$-0.857531\pi$
$821$		−	24.8013i		−	0.865572i	0.901497	−	0.432786i	$-0.142469\pi$
$822$		0			0
$823$	21.3144			0.742973			0.371486	−	0.928438i	$-0.378848\pi$
$823$	21.3144			0.742973			0.371486	+	0.928438i	$0.378848\pi$
$824$	−11.4445			−0.398689
$825$		0			0
$826$	15.9365	−	16.7077i	0.554501	−	0.581334i
$827$			49.7585i			1.73027i	0.501537	+	0.865136i	$0.332768\pi$
$827$			49.7585i			1.73027i	−0.501537	+	0.865136i	$0.667232\pi$
$828$		0			0
$829$		−	43.1190i		−	1.49759i	−0.662804	−	0.748793i	$-0.730634\pi$
$829$		−	43.1190i		−	1.49759i	0.662804	−	0.748793i	$-0.269366\pi$
$830$		−	6.23739i		−	0.216503i
$831$		0			0
$832$			1.00156i			0.0347230i
$833$	1.64912	+	34.8845i	0.0571385	+	1.20868i
$834$		0			0
$835$	7.09310			0.245467
$836$	4.25587			0.147192
$837$		0			0
$838$			1.40791i			0.0486354i
$839$	29.9971			1.03561			0.517807	−	0.855498i	$-0.326749\pi$
$839$	29.9971			1.03561			0.517807	+	0.855498i	$0.326749\pi$
$840$		0			0
$841$	25.6548			0.884649
$842$		−	30.3860i		−	1.04717i
$843$		0			0
$844$	−1.65796			−0.0570694
$845$	−4.39803			−0.151297
$846$		0			0
$847$	−20.2014	−	19.2689i	−0.694128	−	0.662089i
$848$		0			0
$849$		0			0
$850$		−	24.2748i		−	0.832619i
$851$		−	39.7648i		−	1.36312i
$852$		0			0
$853$			29.7559i			1.01882i	0.860524	+	0.509411i	$0.170137\pi$
$853$			29.7559i			1.01882i	−0.860524	+	0.509411i	$0.829863\pi$
$854$	9.49150	+	9.05340i	0.324792	+	0.309801i
$855$		0			0
$856$	11.0618			0.378083
$857$	45.8592			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$857$	45.8592			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$858$		0			0
$859$		−	3.74208i		−	0.127678i	−0.997960	−	0.0638390i	$-0.979666\pi$
$859$		−	3.74208i		−	0.127678i	0.997960	−	0.0638390i	$-0.0203344\pi$
$860$	−1.64912			−0.0562344
$861$		0			0
$862$	27.2747			0.928979
$863$			31.3944i			1.06868i	0.845270	+	0.534339i	$0.179439\pi$
$863$			31.3944i			1.06868i	−0.845270	+	0.534339i	$0.820561\pi$
$864$		0			0
$865$	1.77307			0.0602860
$866$	8.15047			0.276964
$867$		0			0
$868$	−11.5518	+	12.1108i	−0.392095	+	0.411068i
$869$			5.59374i			0.189755i
$870$		0			0
$871$			10.9078i			0.369596i
$872$			10.5633i			0.357719i
$873$		0			0
$874$		−	48.9162i		−	1.65462i
$875$	−6.92416	−	6.60456i	−0.234079	−	0.223275i
$876$		0			0
$877$	−20.3923			−0.688599			−0.344300	−	0.938860i	$-0.611884\pi$
$877$	−20.3923			−0.688599			−0.344300	+	0.938860i	$0.611884\pi$
$878$	12.2404			0.413094
$879$		0			0
$880$		−	0.245420i		−	0.00827312i
$881$	−21.2010			−0.714280			−0.357140	−	0.934051i	$-0.616248\pi$
$881$	−21.2010			−0.714280			−0.357140	+	0.934051i	$0.616248\pi$
$882$		0			0
$883$	38.6157			1.29952			0.649761	−	0.760139i	$-0.274869\pi$
$883$	38.6157			1.29952			0.649761	+	0.760139i	$0.274869\pi$
$884$			4.99687i			0.168063i
$885$		0			0
$886$	−8.00836			−0.269046
$887$	−6.19211			−0.207911			−0.103955	−	0.994582i	$-0.533150\pi$
$887$	−6.19211			−0.207911			−0.103955	+	0.994582i	$0.533150\pi$
$888$		0			0
$889$	3.19615	+	3.04863i	0.107196	+	0.102248i
$890$			3.92675i			0.131625i
$891$		0			0
$892$			17.0372i			0.570448i
$893$			52.9494i			1.77188i
$894$		0			0
$895$		−	1.34100i		−	0.0448247i
$896$	−1.82612	+	1.91449i	−0.0610065	+	0.0639586i
$897$		0			0
$898$	−14.5183			−0.484483
$899$	−11.5699			−0.385878
$900$		0			0
$901$		0			0
$902$	2.89108			0.0962623
$903$		0			0
$904$	−4.15953			−0.138344
$905$			2.07786i			0.0690705i
$906$		0			0
$907$	−0.129113			−0.00428714			−0.00214357	−	0.999998i	$-0.500682\pi$
$907$	−0.129113			−0.00428714			−0.00214357	+	0.999998i	$0.500682\pi$
$908$	−5.11024			−0.169589
$909$		0			0
$910$	0.702947	+	0.670501i	0.0233025	+	0.0222269i
$911$			34.2579i			1.13502i	0.823368	+	0.567508i	$0.192092\pi$
$911$			34.2579i			1.13502i	−0.823368	+	0.567508i	$0.807908\pi$
$912$		0			0
$913$			11.3902i			0.376962i
$914$			9.95501i			0.329283i
$915$		0			0
$916$			15.2669i			0.504433i
$917$	−25.9072	−	24.7114i	−0.855530	−	0.816041i
$918$		0			0
$919$	14.3054			0.471892			0.235946	−	0.971766i	$-0.424181\pi$
$919$	14.3054			0.471892			0.235946	+	0.971766i	$0.424181\pi$
$920$	−2.82082			−0.0929997
$921$		0			0
$922$		−	32.3270i		−	1.06463i
$923$	5.50704			0.181266
$924$		0			0
$925$	25.1449			0.826759
$926$			9.45032i			0.310557i
$927$		0			0
$928$	−1.82898			−0.0600392
$929$	11.7473			0.385415			0.192708	−	0.981256i	$-0.438273\pi$
$929$	11.7473			0.385415			0.192708	+	0.981256i	$0.438273\pi$
$930$		0			0
$931$	−44.4509	+	2.10136i	−1.45682	+	0.0688693i
$932$		−	10.1930i		−	0.333883i
$933$		0			0
$934$		−	20.6623i		−	0.676091i
$935$		−	1.22442i		−	0.0400427i
$936$		0			0
$937$		−	2.63611i		−	0.0861179i	−0.999073	−	0.0430589i	$-0.986290\pi$
$937$		−	2.63611i		−	0.0861179i	0.999073	−	0.0430589i	$-0.0137103\pi$
$938$	−19.8878	+	20.8502i	−0.649361	+	0.680784i
$939$		0			0
$940$	3.05340			0.0995909
$941$	−11.9311			−0.388944			−0.194472	−	0.980908i	$-0.562299\pi$
$941$	−11.9311			−0.388944			−0.194472	+	0.980908i	$0.562299\pi$
$942$		0			0
$943$		−	33.2295i		−	1.08210i
$944$	8.72695			0.284038
$945$		0			0
$946$	3.01149			0.0979121
$947$			7.73957i			0.251502i	0.992062	+	0.125751i	$0.0401341\pi$
$947$			7.73957i			0.251502i	−0.992062	+	0.125751i	$0.959866\pi$
$948$		0			0
$949$	−4.07951			−0.132427
$950$	30.9317			1.00356
$951$		0			0
$952$	−9.11064	+	9.55151i	−0.295278	+	0.309566i
$953$			3.76685i			0.122020i	0.998137	+	0.0610102i	$0.0194322\pi$
$953$			3.76685i			0.122020i	−0.998137	+	0.0610102i	$0.980568\pi$
$954$		0			0
$955$		−	10.0366i		−	0.324777i
$956$		−	19.1815i		−	0.620375i
$957$		0			0
$958$		−	10.1608i		−	0.328282i
$959$	16.4141	−	17.2084i	0.530040	−	0.555689i
$960$		0			0
$961$	−9.01672			−0.290862
$962$	−5.17597			−0.166880
$963$		0			0
$964$			20.6853i			0.666227i
$965$	3.67634			0.118346
$966$		0			0
$967$	4.57482			0.147116			0.0735581	−	0.997291i	$-0.476565\pi$
$967$	4.57482			0.147116			0.0735581	+	0.997291i	$0.476565\pi$
$968$		−	10.5518i		−	0.339149i
$969$		0			0
$970$	6.31126			0.202642
$971$	25.8445			0.829388			0.414694	−	0.909961i	$-0.363889\pi$
$971$	25.8445			0.829388			0.414694	+	0.909961i	$0.363889\pi$
$972$		0			0
$973$	−16.9815	+	17.8033i	−0.544403	+	0.570747i
$974$		−	31.2296i		−	1.00066i
$975$		0			0
$976$			4.95771i			0.158693i
$977$			30.1319i			0.964004i	0.876170	+	0.482002i	$0.160090\pi$
$977$			30.1319i			0.964004i	−0.876170	+	0.482002i	$0.839910\pi$
$978$		0			0
$979$		−	7.17073i		−	0.229178i
$980$	0.121178	+	2.56332i	0.00387088	+	0.0818824i
$981$		0			0
$982$	20.5899			0.657051
$983$	−12.6059			−0.402064			−0.201032	−	0.979585i	$-0.564430\pi$
$983$	−12.6059			−0.402064			−0.201032	+	0.979585i	$0.564430\pi$
$984$		0			0
$985$			6.90152i			0.219901i
$986$	−9.12490			−0.290596
$987$		0			0
$988$	−6.36717			−0.202567
$989$		−	34.6136i		−	1.10065i
$990$		0			0
$991$	51.6852			1.64184			0.820918	−	0.571046i	$-0.193462\pi$
$991$	51.6852			1.64184			0.820918	+	0.571046i	$0.193462\pi$
$992$	−6.32588			−0.200847
$993$		0			0
$994$	10.5267	+	10.0408i	0.333887	+	0.318475i
$995$		−	1.96645i		−	0.0623408i
$996$		0			0
$997$		−	40.5841i		−	1.28531i	−0.766155	−	0.642656i	$-0.777832\pi$
$997$		−	40.5841i		−	1.28531i	0.766155	−	0.642656i	$-0.222168\pi$
$998$		−	25.1533i		−	0.796213i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.2.d.a.1133.12		16
3.2	odd	2	inner	1134.2.d.a.1133.5		16
7.6	odd	2	inner	1134.2.d.a.1133.13		16
9.2	odd	6		378.2.m.a.125.2		16
9.4	even	3		378.2.m.a.251.3		16
9.5	odd	6		126.2.m.a.83.6	yes	16
9.7	even	3		126.2.m.a.41.7	yes	16
21.20	even	2	inner	1134.2.d.a.1133.4		16
36.7	odd	6		1008.2.cc.b.545.4		16
36.11	even	6		3024.2.cc.b.881.4		16
36.23	even	6		1008.2.cc.b.209.5		16
36.31	odd	6		3024.2.cc.b.2897.5		16
63.2	odd	6		2646.2.t.a.2285.7		16
63.4	even	3		2646.2.t.a.1979.6		16
63.5	even	6		882.2.l.a.227.1		16
63.11	odd	6		2646.2.l.b.1097.2		16
63.13	odd	6		378.2.m.a.251.2		16
63.16	even	3		882.2.t.b.815.4		16
63.20	even	6		378.2.m.a.125.3		16
63.23	odd	6		882.2.l.a.227.4		16
63.25	even	3		882.2.l.a.509.5		16
63.31	odd	6		2646.2.t.a.1979.7		16
63.32	odd	6		882.2.t.b.803.1		16
63.34	odd	6		126.2.m.a.41.6	✓	16
63.38	even	6		2646.2.l.b.1097.3		16
63.40	odd	6		2646.2.l.b.521.6		16
63.41	even	6		126.2.m.a.83.7	yes	16
63.47	even	6		2646.2.t.a.2285.6		16
63.52	odd	6		882.2.l.a.509.8		16
63.58	even	3		2646.2.l.b.521.7		16
63.59	even	6		882.2.t.b.803.4		16
63.61	odd	6		882.2.t.b.815.1		16
252.83	odd	6		3024.2.cc.b.881.5		16
252.139	even	6		3024.2.cc.b.2897.4		16
252.167	odd	6		1008.2.cc.b.209.4		16
252.223	even	6		1008.2.cc.b.545.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.m.a.41.6	✓	16	63.34	odd	6
126.2.m.a.41.7	yes	16	9.7	even	3
126.2.m.a.83.6	yes	16	9.5	odd	6
126.2.m.a.83.7	yes	16	63.41	even	6
378.2.m.a.125.2		16	9.2	odd	6
378.2.m.a.125.3		16	63.20	even	6
378.2.m.a.251.2		16	63.13	odd	6
378.2.m.a.251.3		16	9.4	even	3
882.2.l.a.227.1		16	63.5	even	6
882.2.l.a.227.4		16	63.23	odd	6
882.2.l.a.509.5		16	63.25	even	3
882.2.l.a.509.8		16	63.52	odd	6
882.2.t.b.803.1		16	63.32	odd	6
882.2.t.b.803.4		16	63.59	even	6
882.2.t.b.815.1		16	63.61	odd	6
882.2.t.b.815.4		16	63.16	even	3
1008.2.cc.b.209.4		16	252.167	odd	6
1008.2.cc.b.209.5		16	36.23	even	6
1008.2.cc.b.545.4		16	36.7	odd	6
1008.2.cc.b.545.5		16	252.223	even	6
1134.2.d.a.1133.4		16	21.20	even	2	inner
1134.2.d.a.1133.5		16	3.2	odd	2	inner
1134.2.d.a.1133.12		16	1.1	even	1	trivial
1134.2.d.a.1133.13		16	7.6	odd	2	inner
2646.2.l.b.521.6		16	63.40	odd	6
2646.2.l.b.521.7		16	63.58	even	3
2646.2.l.b.1097.2		16	63.11	odd	6
2646.2.l.b.1097.3		16	63.38	even	6
2646.2.t.a.1979.6		16	63.4	even	3
2646.2.t.a.1979.7		16	63.31	odd	6
2646.2.t.a.2285.6		16	63.47	even	6
2646.2.t.a.2285.7		16	63.2	odd	6
3024.2.cc.b.881.4		16	36.11	even	6
3024.2.cc.b.881.5		16	252.83	odd	6
3024.2.cc.b.2897.4		16	252.139	even	6
3024.2.cc.b.2897.5		16	36.31	odd	6

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:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.366598 q^{5} +(-1.91449 - 1.82612i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -0.366598 q^{5} +(-1.91449 - 1.82612i) q^{7} -1.00000i q^{8} -0.366598i q^{10} +0.669453i q^{11} -1.00156i q^{13} +(1.82612 - 1.91449i) q^{14} +1.00000 q^{16} +4.98906 q^{17} +6.35722i q^{19} +0.366598 q^{20} -0.669453 q^{22} +7.69459i q^{23} -4.86561 q^{25} +1.00156 q^{26} +(1.91449 + 1.82612i) q^{28} +1.82898i q^{29} +6.32588i q^{31} +1.00000i q^{32} +4.98906i q^{34} +(0.701849 + 0.669453i) q^{35} -5.16789 q^{37} -6.35722 q^{38} +0.366598i q^{40} -4.31856 q^{41} -4.49843 q^{43} -0.669453i q^{44} -7.69459 q^{46} +8.32901 q^{47} +(0.330547 + 6.99219i) q^{49} -4.86561i q^{50} +1.00156i q^{52} -0.245420i q^{55} +(-1.82612 + 1.91449i) q^{56} -1.82898 q^{58} +8.72695 q^{59} +4.95771i q^{61} -6.32588 q^{62} -1.00000 q^{64} +0.367172i q^{65} -10.8907 q^{67} -4.98906 q^{68} +(-0.669453 + 0.701849i) q^{70} +5.49843i q^{71} -4.07314i q^{73} -5.16789i q^{74} -6.35722i q^{76} +(1.22250 - 1.28166i) q^{77} +8.35568 q^{79} -0.366598 q^{80} -4.31856i q^{82} +17.0142 q^{83} -1.82898 q^{85} -4.49843i q^{86} +0.669453 q^{88} -10.7113 q^{89} +(-1.82898 + 1.91749i) q^{91} -7.69459i q^{92} +8.32901i q^{94} -2.33055i q^{95} +17.2157i q^{97} +(-6.99219 + 0.330547i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 4 q^{7}+O(q^{10}) \) 16 * q - 16 * q^4 - 4 * q^7	\( 16 q - 16 q^{4} - 4 q^{7} + 16 q^{16} + 16 q^{25} + 4 q^{28} - 8 q^{37} - 8 q^{43} + 24 q^{46} + 16 q^{49} + 24 q^{58} - 16 q^{64} + 56 q^{67} + 8 q^{79} + 24 q^{85} + 24 q^{91}+O(q^{100}) \) 16 * q - 16 * q^4 - 4 * q^7 + 16 * q^16 + 16 * q^25 + 4 * q^28 - 8 * q^37 - 8 * q^43 + 24 * q^46 + 16 * q^49 + 24 * q^58 - 16 * q^64 + 56 * q^67 + 8 * q^79 + 24 * q^85 + 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more