LMFDB - Embedded newform 3456.2.d.p.1729.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1729,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3456.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.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:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.959512576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 81 $ x^8 + 7*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.1
Root			$-1.52616 - 0.819051i$ of defining polynomial
Character	$\chi$	$=$	3456.1729
Dual form			3456.2.d.p.1729.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} -3.31662 q^{7} +O(q^{10})$	$q-1.41421i q^{5} -3.31662 q^{7} +4.69042i q^{11} -3.31662i q^{13} +4.69042 q^{17} -1.00000i q^{19} +1.41421 q^{23} +3.00000 q^{25} -5.65685i q^{29} +4.69042i q^{35} +9.94987i q^{37} -9.38083 q^{41} +6.00000i q^{43} -7.07107 q^{47} +4.00000 q^{49} -11.3137i q^{53} +6.63325 q^{55} -4.69042i q^{59} -3.31662i q^{61} -4.69042 q^{65} -13.0000i q^{67} +11.3137 q^{71} +1.00000 q^{73} -15.5563i q^{77} +9.94987 q^{79} -9.38083i q^{83} -6.63325i q^{85} +14.0712 q^{89} +11.0000i q^{91} -1.41421 q^{95} -7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 24 q^{25} + 32 q^{49} + 8 q^{73} - 56 q^{97}+O(q^{100}) $ 8 * q + 24 * q^25 + 32 * q^49 + 8 * q^73 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	−3.31662	−1.25357	−0.626783	−	0.779194i	$-0.715629\pi$
$7$	−3.31662	−1.25357	−0.626783	+	0.779194i	$0.715629\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.69042i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$11$	4.69042i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$12$	0	0
$13$	− 3.31662i	− 0.919866i	−0.887954	−	0.459933i	$-0.847873\pi$
$13$	− 3.31662i	− 0.919866i	0.887954	−	0.459933i	$-0.152127\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.69042	1.13759	0.568796	−	0.822478i	$-0.307409\pi$
$17$	4.69042	1.13759	0.568796	+	0.822478i	$0.307409\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$	− 1.00000i	− 0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.65685i	− 1.05045i	−0.850963	−	0.525226i	$-0.823981\pi$
$29$	− 5.65685i	− 1.05045i	0.850963	−	0.525226i	$-0.176019\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.69042i	0.792825i
$36$	0	0
$37$	9.94987i	1.63575i	0.575396	+	0.817875i	$0.304848\pi$
$37$	9.94987i	1.63575i	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.38083	−1.46504	−0.732520	−	0.680746i	$-0.761656\pi$
$41$	−9.38083	−1.46504	−0.732520	+	0.680746i	$0.761656\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.3137i	− 1.55406i	−0.629465	−	0.777029i	$-0.716726\pi$
$53$	− 11.3137i	− 1.55406i	0.629465	−	0.777029i	$-0.283274\pi$
$54$	0	0
$55$	6.63325	0.894427
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.69042i	− 0.610640i	−0.952250	−	0.305320i	$-0.901237\pi$
$59$	− 4.69042i	− 0.610640i	0.952250	−	0.305320i	$-0.0987634\pi$
$60$	0	0
$61$	− 3.31662i	− 0.424650i	−0.977199	−	0.212325i	$-0.931896\pi$
$61$	− 3.31662i	− 0.424650i	0.977199	−	0.212325i	$-0.0681036\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.69042	−0.581774
$66$	0	0
$67$	− 13.0000i	− 1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$	− 13.0000i	− 1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 15.5563i	− 1.77281i
$78$	0	0
$79$	9.94987	1.11945	0.559724	−	0.828679i	$-0.310907\pi$
$79$	9.94987	1.11945	0.559724	+	0.828679i	$0.310907\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.38083i	− 1.02968i	−0.857286	−	0.514840i	$-0.827851\pi$
$83$	− 9.38083i	− 1.02968i	0.857286	−	0.514840i	$-0.172149\pi$
$84$	0	0
$85$	− 6.63325i	− 0.719477i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.0712	1.49155	0.745775	−	0.666198i	$-0.232080\pi$
$89$	14.0712	1.49155	0.745775	+	0.666198i	$0.232080\pi$
$90$	0	0
$91$	11.0000i	1.15311i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.41421	−0.145095
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.1421i	− 1.40720i	−0.710599	−	0.703598i	$-0.751576\pi$
$101$	− 14.1421i	− 1.40720i	0.710599	−	0.703598i	$-0.248424\pi$
$102$	0	0
$103$	3.31662	0.326797	0.163398	−	0.986560i	$-0.447754\pi$
$103$	3.31662	0.326797	0.163398	+	0.986560i	$0.447754\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.0712i	− 1.36032i	−0.733064	−	0.680159i	$-0.761911\pi$
$107$	− 14.0712i	− 1.36032i	0.733064	−	0.680159i	$-0.238089\pi$
$108$	0	0
$109$	− 6.63325i	− 0.635350i	−0.948200	−	0.317675i	$-0.897098\pi$
$109$	− 6.63325i	− 0.635350i	0.948200	−	0.317675i	$-0.102902\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.69042	−0.441237	−0.220619	−	0.975360i	$-0.570808\pi$
$113$	−4.69042	−0.441237	−0.220619	+	0.975360i	$0.570808\pi$
$114$	0	0
$115$	− 2.00000i	− 0.186501i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−15.5563	−1.42605
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	−19.8997	−1.76582	−0.882909	−	0.469545i	$-0.844418\pi$
$127$	−19.8997	−1.76582	−0.882909	+	0.469545i	$0.844418\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	3.31662i	0.287588i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.69042	0.400729	0.200365	−	0.979721i	$-0.435787\pi$
$137$	4.69042	0.400729	0.200365	+	0.979721i	$0.435787\pi$
$138$	0	0
$139$	5.00000i	0.424094i	0.977259	+	0.212047i	$0.0680131\pi$
$139$	5.00000i	0.424094i	−0.977259	+	0.212047i	$0.931987\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.5563	1.30089
$144$	0	0
$145$	−8.00000	−0.664364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 11.3137i	− 0.926855i	−0.886135	−	0.463428i	$-0.846619\pi$
$149$	− 11.3137i	− 0.926855i	0.886135	−	0.463428i	$-0.153381\pi$
$150$	0	0
$151$	−9.94987	−0.809709	−0.404855	−	0.914381i	$-0.632678\pi$
$151$	−9.94987	−0.809709	−0.404855	+	0.914381i	$0.632678\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19.8997i	1.58817i	0.607805	+	0.794086i	$0.292050\pi$
$157$	19.8997i	1.58817i	−0.607805	+	0.794086i	$0.707950\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.69042	−0.369657
$162$	0	0
$163$	− 19.0000i	− 1.48819i	−0.668071	−	0.744097i	$-0.732880\pi$
$163$	− 19.0000i	− 1.48819i	0.668071	−	0.744097i	$-0.267120\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.07107	0.547176	0.273588	−	0.961847i	$-0.411790\pi$
$167$	7.07107	0.547176	0.273588	+	0.961847i	$0.411790\pi$
$168$	0	0
$169$	2.00000	0.153846
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 14.1421i	− 1.07521i	−0.843198	−	0.537603i	$-0.819330\pi$
$173$	− 14.1421i	− 1.07521i	0.843198	−	0.537603i	$-0.180670\pi$
$174$	0	0
$175$	−9.94987	−0.752140
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 23.2164i	− 1.72566i	−0.505495	−	0.862830i	$-0.668690\pi$
$181$	− 23.2164i	− 1.72566i	0.505495	−	0.862830i	$-0.331310\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.0712	1.03454
$186$	0	0
$187$	22.0000i	1.60880i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.0416	−1.73959	−0.869796	−	0.493412i	$-0.835749\pi$
$191$	−24.0416	−1.73959	−0.869796	+	0.493412i	$0.835749\pi$
$192$	0	0
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.2132i	− 1.51138i	−0.654931	−	0.755689i	$-0.727302\pi$
$197$	− 21.2132i	− 1.51138i	0.654931	−	0.755689i	$-0.272698\pi$
$198$	0	0
$199$	3.31662	0.235109	0.117555	−	0.993066i	$-0.462494\pi$
$199$	3.31662	0.235109	0.117555	+	0.993066i	$0.462494\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.7617i	1.31681i
$204$	0	0
$205$	13.2665i	0.926572i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.69042	0.324443
$210$	0	0
$211$	− 9.00000i	− 0.619586i	−0.950804	−	0.309793i	$-0.899740\pi$
$211$	− 9.00000i	− 0.619586i	0.950804	−	0.309793i	$-0.100260\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.48528	0.578691
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 15.5563i	− 1.04643i
$222$	0	0
$223$	13.2665	0.888390	0.444195	−	0.895930i	$-0.353490\pi$
$223$	13.2665	0.888390	0.444195	+	0.895930i	$0.353490\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	13.2665i	0.876675i	0.898810	+	0.438337i	$0.144433\pi$
$229$	13.2665i	0.876675i	−0.898810	+	0.438337i	$0.855567\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.1425	−1.84368	−0.921838	−	0.387575i	$-0.873313\pi$
$233$	−28.1425	−1.84368	−0.921838	+	0.387575i	$0.873313\pi$
$234$	0	0
$235$	10.0000i	0.652328i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.1421	0.914779	0.457389	−	0.889267i	$-0.348785\pi$
$239$	14.1421	0.914779	0.457389	+	0.889267i	$0.348785\pi$
$240$	0	0
$241$	−9.00000	−0.579741	−0.289870	−	0.957066i	$-0.593612\pi$
$241$	−9.00000	−0.579741	−0.289870	+	0.957066i	$0.593612\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.65685i	− 0.361403i
$246$	0	0
$247$	−3.31662	−0.211032
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.1425i	1.77634i	0.459516	+	0.888169i	$0.348023\pi$
$251$	28.1425i	1.77634i	−0.459516	+	0.888169i	$0.651977\pi$
$252$	0	0
$253$	6.63325i	0.417029i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	− 33.0000i	− 2.05052i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.4558	−1.56967	−0.784837	−	0.619702i	$-0.787254\pi$
$263$	−25.4558	−1.56967	−0.784837	+	0.619702i	$0.787254\pi$
$264$	0	0
$265$	−16.0000	−0.982872
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.24264i	0.258678i	0.991600	+	0.129339i	$0.0412856\pi$
$269$	4.24264i	0.258678i	−0.991600	+	0.129339i	$0.958714\pi$
$270$	0	0
$271$	23.2164	1.41029	0.705147	−	0.709061i	$-0.250881\pi$
$271$	23.2164	1.41029	0.705147	+	0.709061i	$0.250881\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.0712i	0.848528i
$276$	0	0
$277$	− 19.8997i	− 1.19566i	−0.801623	−	0.597830i	$-0.796030\pi$
$277$	− 19.8997i	− 1.19566i	0.801623	−	0.597830i	$-0.203970\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.7617	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$281$	−18.7617	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$282$	0	0
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.1127	1.83652
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.07107i	0.413096i	0.978436	+	0.206548i	$0.0662230\pi$
$293$	7.07107i	0.413096i	−0.978436	+	0.206548i	$0.933777\pi$
$294$	0	0
$295$	−6.63325	−0.386203
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.69042i	− 0.271254i
$300$	0	0
$301$	− 19.8997i	− 1.14700i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.69042	−0.268572
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.89949	−0.561349	−0.280674	−	0.959803i	$-0.590558\pi$
$311$	−9.89949	−0.561349	−0.280674	+	0.959803i	$0.590558\pi$
$312$	0	0
$313$	31.0000	1.75222	0.876112	−	0.482108i	$-0.160129\pi$
$313$	31.0000	1.75222	0.876112	+	0.482108i	$0.160129\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.48528i	− 0.476581i	−0.971194	−	0.238290i	$-0.923413\pi$
$317$	− 8.48528i	− 0.476581i	0.971194	−	0.238290i	$-0.0765870\pi$
$318$	0	0
$319$	26.5330	1.48556
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 4.69042i	− 0.260982i
$324$	0	0
$325$	− 9.94987i	− 0.551920i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	23.4521	1.29295
$330$	0	0
$331$	1.00000i	0.0549650i	0.999622	+	0.0274825i	$0.00874905\pi$
$331$	1.00000i	0.0549650i	−0.999622	+	0.0274825i	$0.991251\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−18.3848	−1.00447
$336$	0	0
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	9.94987	0.537243
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.1425i	− 1.51077i	−0.655282	−	0.755384i	$-0.727450\pi$
$347$	− 28.1425i	− 1.51077i	0.655282	−	0.755384i	$-0.272550\pi$
$348$	0	0
$349$	− 3.31662i	− 0.177535i	−0.996052	−	0.0887674i	$-0.971707\pi$
$349$	− 3.31662i	− 0.177535i	0.996052	−	0.0887674i	$-0.0282928\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.1425	1.49787	0.748937	−	0.662641i	$-0.230565\pi$
$353$	28.1425	1.49787	0.748937	+	0.662641i	$0.230565\pi$
$354$	0	0
$355$	− 16.0000i	− 0.849192i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.8701	1.41815	0.709074	−	0.705134i	$-0.249113\pi$
$359$	26.8701	1.41815	0.709074	+	0.705134i	$0.249113\pi$
$360$	0	0
$361$	18.0000	0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.41421i	− 0.0740233i
$366$	0	0
$367$	−29.8496	−1.55814	−0.779069	−	0.626938i	$-0.784308\pi$
$367$	−29.8496	−1.55814	−0.779069	+	0.626938i	$0.784308\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	37.5233i	1.94811i
$372$	0	0
$373$	9.94987i	0.515185i	0.966254	+	0.257592i	$0.0829292\pi$
$373$	9.94987i	0.515185i	−0.966254	+	0.257592i	$0.917071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.7617	−0.966275
$378$	0	0
$379$	19.0000i	0.975964i	0.872854	+	0.487982i	$0.162267\pi$
$379$	19.0000i	0.975964i	−0.872854	+	0.487982i	$0.837733\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.82843	0.144526	0.0722629	−	0.997386i	$-0.476978\pi$
$383$	2.82843	0.144526	0.0722629	+	0.997386i	$0.476978\pi$
$384$	0	0
$385$	−22.0000	−1.12122
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.07107i	0.358517i	0.983802	+	0.179259i	$0.0573699\pi$
$389$	7.07107i	0.358517i	−0.983802	+	0.179259i	$0.942630\pi$
$390$	0	0
$391$	6.63325	0.335458
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 14.0712i	− 0.708001i
$396$	0	0
$397$	19.8997i	0.998740i	0.866389	+	0.499370i	$0.166435\pi$
$397$	19.8997i	0.998740i	−0.866389	+	0.499370i	$0.833565\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.1425	1.40537	0.702685	−	0.711502i	$-0.251985\pi$
$401$	28.1425	1.40537	0.702685	+	0.711502i	$0.251985\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−46.6690	−2.31330
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15.5563i	0.765478i
$414$	0	0
$415$	−13.2665	−0.651227
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 32.8329i	− 1.60399i	−0.597329	−	0.801996i	$-0.703771\pi$
$419$	− 32.8329i	− 1.60399i	0.597329	−	0.801996i	$-0.296229\pi$
$420$	0	0
$421$	29.8496i	1.45478i	0.686223	+	0.727391i	$0.259267\pi$
$421$	29.8496i	1.45478i	−0.686223	+	0.727391i	$0.740733\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	14.0712	0.682556
$426$	0	0
$427$	11.0000i	0.532327i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.3848	0.885564	0.442782	−	0.896629i	$-0.353992\pi$
$431$	18.3848	0.885564	0.442782	+	0.896629i	$0.353992\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.41421i	− 0.0676510i
$438$	0	0
$439$	13.2665	0.633175	0.316588	−	0.948563i	$-0.397463\pi$
$439$	13.2665	0.633175	0.316588	+	0.948563i	$0.397463\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.7617i	− 0.891394i	−0.895184	−	0.445697i	$-0.852956\pi$
$443$	− 18.7617i	− 0.891394i	0.895184	−	0.445697i	$-0.147044\pi$
$444$	0	0
$445$	− 19.8997i	− 0.943339i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.0712	−0.664063	−0.332032	−	0.943268i	$-0.607734\pi$
$449$	−14.0712	−0.664063	−0.332032	+	0.943268i	$0.607734\pi$
$450$	0	0
$451$	− 44.0000i	− 2.07188i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15.5563	0.729293
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.2132i	0.987997i	0.869463	+	0.493999i	$0.164465\pi$
$461$	21.2132i	0.987997i	−0.869463	+	0.493999i	$0.835535\pi$
$462$	0	0
$463$	−36.4829	−1.69550	−0.847751	−	0.530394i	$-0.822044\pi$
$463$	−36.4829	−1.69550	−0.847751	+	0.530394i	$0.822044\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 32.8329i	− 1.51933i	−0.650317	−	0.759663i	$-0.725364\pi$
$467$	− 32.8329i	− 1.51933i	0.650317	−	0.759663i	$-0.274636\pi$
$468$	0	0
$469$	43.1161i	1.99092i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−28.1425	−1.29399
$474$	0	0
$475$	− 3.00000i	− 0.137649i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.6274	−1.03387	−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274	−1.03387	−0.516937	+	0.856024i	$0.672928\pi$
$480$	0	0
$481$	33.0000	1.50467
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.89949i	0.449513i
$486$	0	0
$487$	−36.4829	−1.65320	−0.826598	−	0.562792i	$-0.809727\pi$
$487$	−36.4829	−1.65320	−0.826598	+	0.562792i	$0.809727\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.69042i	0.211676i	0.994383	+	0.105838i	$0.0337524\pi$
$491$	4.69042i	0.211676i	−0.994383	+	0.105838i	$0.966248\pi$
$492$	0	0
$493$	− 26.5330i	− 1.19499i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−37.5233	−1.68315
$498$	0	0
$499$	32.0000i	1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$	32.0000i	1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	32.5269	1.45030	0.725152	−	0.688589i	$-0.241770\pi$
$503$	32.5269	1.45030	0.725152	+	0.688589i	$0.241770\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.5269i	1.44173i	0.693075	+	0.720865i	$0.256255\pi$
$509$	32.5269i	1.44173i	−0.693075	+	0.720865i	$0.743745\pi$
$510$	0	0
$511$	−3.31662	−0.146719
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.69042i	− 0.206684i
$516$	0	0
$517$	− 33.1662i	− 1.45865i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	42.2137	1.84942	0.924709	−	0.380675i	$-0.124308\pi$
$521$	42.2137	1.84942	0.924709	+	0.380675i	$0.124308\pi$
$522$	0	0
$523$	1.00000i	0.0437269i	0.999761	+	0.0218635i	$0.00695991\pi$
$523$	1.00000i	0.0437269i	−0.999761	+	0.0218635i	$0.993040\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	31.1127i	1.34764i
$534$	0	0
$535$	−19.8997	−0.860341
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.7617i	0.808122i
$540$	0	0
$541$	− 9.94987i	− 0.427779i	−0.976858	−	0.213889i	$-0.931387\pi$
$541$	− 9.94987i	− 0.427779i	0.976858	−	0.213889i	$-0.0686132\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.38083	−0.401831
$546$	0	0
$547$	− 29.0000i	− 1.23995i	−0.784621	−	0.619975i	$-0.787143\pi$
$547$	− 29.0000i	− 1.23995i	0.784621	−	0.619975i	$-0.212857\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.65685	−0.240990
$552$	0	0
$553$	−33.0000	−1.40330
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.24264i	− 0.179766i	−0.995952	−	0.0898832i	$-0.971351\pi$
$557$	− 4.24264i	− 0.179766i	0.995952	−	0.0898832i	$-0.0286494\pi$
$558$	0	0
$559$	19.8997	0.841670
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 9.38083i	− 0.395355i	−0.980267	−	0.197677i	$-0.936660\pi$
$563$	− 9.38083i	− 0.395355i	0.980267	−	0.197677i	$-0.0633399\pi$
$564$	0	0
$565$	6.63325i	0.279063i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.0712	0.589897	0.294949	−	0.955513i	$-0.404697\pi$
$569$	14.0712	0.589897	0.294949	+	0.955513i	$0.404697\pi$
$570$	0	0
$571$	35.0000i	1.46470i	0.680926	+	0.732352i	$0.261578\pi$
$571$	35.0000i	1.46470i	−0.680926	+	0.732352i	$0.738422\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.24264	0.176930
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	31.1127i	1.29077i
$582$	0	0
$583$	53.0660	2.19777
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 14.0712i	− 0.580783i	−0.956908	−	0.290391i	$-0.906215\pi$
$587$	− 14.0712i	− 0.580783i	0.956908	−	0.290391i	$-0.0937855\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.38083	−0.385225	−0.192612	−	0.981275i	$-0.561696\pi$
$593$	−9.38083	−0.385225	−0.192612	+	0.981275i	$0.561696\pi$
$594$	0	0
$595$	22.0000i	0.901912i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.5563i	0.632456i
$606$	0	0
$607$	23.2164	0.942324	0.471162	−	0.882047i	$-0.343835\pi$
$607$	23.2164	0.942324	0.471162	+	0.882047i	$0.343835\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23.4521i	0.948770i
$612$	0	0
$613$	3.31662i	0.133957i	0.997754	+	0.0669786i	$0.0213359\pi$
$613$	3.31662i	0.133957i	−0.997754	+	0.0669786i	$0.978664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.69042	−0.188829	−0.0944145	−	0.995533i	$-0.530098\pi$
$617$	−4.69042	−0.188829	−0.0944145	+	0.995533i	$0.530098\pi$
$618$	0	0
$619$	− 3.00000i	− 0.120580i	−0.998181	−	0.0602901i	$-0.980797\pi$
$619$	− 3.00000i	− 0.120580i	0.998181	−	0.0602901i	$-0.0192026\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−46.6690	−1.86976
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	46.6690i	1.86082i
$630$	0	0
$631$	−9.94987	−0.396098	−0.198049	−	0.980192i	$-0.563461\pi$
$631$	−9.94987	−0.396098	−0.198049	+	0.980192i	$0.563461\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	28.1425i	1.11680i
$636$	0	0
$637$	− 13.2665i	− 0.525638i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.7617	−0.741041	−0.370521	−	0.928824i	$-0.620821\pi$
$641$	−18.7617	−0.741041	−0.370521	+	0.928824i	$0.620821\pi$
$642$	0	0
$643$	− 46.0000i	− 1.81406i	−0.421063	−	0.907031i	$-0.638343\pi$
$643$	− 46.0000i	− 1.81406i	0.421063	−	0.907031i	$-0.361657\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.82843	0.111197	0.0555985	−	0.998453i	$-0.482293\pi$
$647$	2.82843	0.111197	0.0555985	+	0.998453i	$0.482293\pi$
$648$	0	0
$649$	22.0000	0.863576
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.4558i	0.996164i	0.867130	+	0.498082i	$0.165962\pi$
$653$	25.4558i	0.996164i	−0.867130	+	0.498082i	$0.834038\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 18.7617i	− 0.730851i	−0.930841	−	0.365425i	$-0.880924\pi$
$659$	− 18.7617i	− 0.730851i	0.930841	−	0.365425i	$-0.119076\pi$
$660$	0	0
$661$	16.5831i	0.645009i	0.946568	+	0.322504i	$0.104525\pi$
$661$	16.5831i	0.645009i	−0.946568	+	0.322504i	$0.895475\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.69042	0.181887
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15.5563	0.600546
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.07107i	− 0.271763i	−0.990725	−	0.135882i	$-0.956613\pi$
$677$	− 7.07107i	− 0.271763i	0.990725	−	0.135882i	$-0.0433867\pi$
$678$	0	0
$679$	23.2164	0.890963
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.38083i	0.358948i	0.983763	+	0.179474i	$0.0574395\pi$
$683$	9.38083i	0.358948i	−0.983763	+	0.179474i	$0.942561\pi$
$684$	0	0
$685$	− 6.63325i	− 0.253443i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−37.5233	−1.42952
$690$	0	0
$691$	22.0000i	0.836919i	0.908235	+	0.418460i	$0.137430\pi$
$691$	22.0000i	0.836919i	−0.908235	+	0.418460i	$0.862570\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.07107	0.268221
$696$	0	0
$697$	−44.0000	−1.66662
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.3848i	0.694383i	0.937794	+	0.347192i	$0.112865\pi$
$701$	18.3848i	0.694383i	−0.937794	+	0.347192i	$0.887135\pi$
$702$	0	0
$703$	9.94987	0.375267
$704$	0	0
$705$	0	0
$706$	0	0
$707$	46.9042i	1.76401i
$708$	0	0
$709$	− 29.8496i	− 1.12103i	−0.828146	−	0.560513i	$-0.810604\pi$
$709$	− 29.8496i	− 1.12103i	0.828146	−	0.560513i	$-0.189396\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 22.0000i	− 0.822753i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.4558	0.949343	0.474671	−	0.880163i	$-0.342567\pi$
$719$	25.4558	0.949343	0.474671	+	0.880163i	$0.342567\pi$
$720$	0	0
$721$	−11.0000	−0.409661
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 16.9706i	− 0.630271i
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.1425i	1.04089i
$732$	0	0
$733$	19.8997i	0.735014i	0.930021	+	0.367507i	$0.119789\pi$
$733$	19.8997i	0.735014i	−0.930021	+	0.367507i	$0.880211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.9754	2.24606
$738$	0	0
$739$	6.00000i	0.220714i	0.993892	+	0.110357i	$0.0351994\pi$
$739$	6.00000i	0.220714i	−0.993892	+	0.110357i	$0.964801\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.41421	−0.0518825	−0.0259412	−	0.999663i	$-0.508258\pi$
$743$	−1.41421	−0.0518825	−0.0259412	+	0.999663i	$0.508258\pi$
$744$	0	0
$745$	−16.0000	−0.586195
$746$	0	0
$747$	0	0
$748$	0	0
$749$	46.6690i	1.70525i
$750$	0	0
$751$	16.5831	0.605127	0.302563	−	0.953129i	$-0.402158\pi$
$751$	16.5831	0.605127	0.302563	+	0.953129i	$0.402158\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.0712i	0.512105i
$756$	0	0
$757$	− 3.31662i	− 0.120545i	−0.998182	−	0.0602724i	$-0.980803\pi$
$757$	− 3.31662i	− 0.120545i	0.998182	−	0.0602724i	$-0.0191969\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−14.0712	−0.510082	−0.255041	−	0.966930i	$-0.582089\pi$
$761$	−14.0712	−0.510082	−0.255041	+	0.966930i	$0.582089\pi$
$762$	0	0
$763$	22.0000i	0.796453i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15.5563	−0.561707
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	48.0833i	1.72943i	0.502259	+	0.864717i	$0.332502\pi$
$773$	48.0833i	1.72943i	−0.502259	+	0.864717i	$0.667498\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.38083i	0.336103i
$780$	0	0
$781$	53.0660i	1.89885i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	28.1425	1.00445
$786$	0	0
$787$	31.0000i	1.10503i	0.833503	+	0.552515i	$0.186332\pi$
$787$	31.0000i	1.10503i	−0.833503	+	0.552515i	$0.813668\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.5563	0.553120
$792$	0	0
$793$	−11.0000	−0.390621
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.7990i	0.701316i	0.936504	+	0.350658i	$0.114042\pi$
$797$	19.7990i	0.701316i	−0.936504	+	0.350658i	$0.885958\pi$
$798$	0	0
$799$	−33.1662	−1.17334
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.69042i	0.165521i
$804$	0	0
$805$	6.63325i	0.233791i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.7617	0.659625	0.329812	−	0.944046i	$-0.393014\pi$
$809$	18.7617	0.659625	0.329812	+	0.944046i	$0.393014\pi$
$810$	0	0
$811$	− 14.0000i	− 0.491606i	−0.969320	−	0.245803i	$-0.920948\pi$
$811$	− 14.0000i	− 0.491606i	0.969320	−	0.245803i	$-0.0790517\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−26.8701	−0.941217
$816$	0	0
$817$	6.00000	0.209913
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26.8701i	0.937771i	0.883259	+	0.468886i	$0.155344\pi$
$821$	26.8701i	0.937771i	−0.883259	+	0.468886i	$0.844656\pi$
$822$	0	0
$823$	−16.5831	−0.578051	−0.289026	−	0.957321i	$-0.593331\pi$
$823$	−16.5831	−0.578051	−0.289026	+	0.957321i	$0.593331\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 42.2137i	− 1.46792i	−0.679195	−	0.733958i	$-0.737671\pi$
$827$	− 42.2137i	− 1.46792i	0.679195	−	0.733958i	$-0.262329\pi$
$828$	0	0
$829$	− 29.8496i	− 1.03672i	−0.855162	−	0.518360i	$-0.826543\pi$
$829$	− 29.8496i	− 1.03672i	0.855162	−	0.518360i	$-0.173457\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.7617	0.650053
$834$	0	0
$835$	− 10.0000i	− 0.346064i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.2548	−1.56237	−0.781185	−	0.624299i	$-0.785385\pi$
$839$	−45.2548	−1.56237	−0.781185	+	0.624299i	$0.785385\pi$
$840$	0	0
$841$	−3.00000	−0.103448
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.82843i	− 0.0973009i
$846$	0	0
$847$	36.4829	1.25357
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.0712i	0.482356i
$852$	0	0
$853$	16.5831i	0.567795i	0.958855	+	0.283898i	$0.0916276\pi$
$853$	16.5831i	0.567795i	−0.958855	+	0.283898i	$0.908372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	− 43.0000i	− 1.46714i	−0.679613	−	0.733571i	$-0.737852\pi$
$859$	− 43.0000i	− 1.46714i	0.679613	−	0.733571i	$-0.262148\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.41421	−0.0481404	−0.0240702	−	0.999710i	$-0.507663\pi$
$863$	−1.41421	−0.0481404	−0.0240702	+	0.999710i	$0.507663\pi$
$864$	0	0
$865$	−20.0000	−0.680020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.6690i	1.58314i
$870$	0	0
$871$	−43.1161	−1.46093
$872$	0	0
$873$	0	0
$874$	0	0
$875$	37.5233i	1.26852i
$876$	0	0
$877$	43.1161i	1.45593i	0.685615	+	0.727964i	$0.259533\pi$
$877$	43.1161i	1.45593i	−0.685615	+	0.727964i	$0.740467\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.4521	0.790121	0.395060	−	0.918655i	$-0.370724\pi$
$881$	23.4521	0.790121	0.395060	+	0.918655i	$0.370724\pi$
$882$	0	0
$883$	27.0000i	0.908622i	0.890843	+	0.454311i	$0.150115\pi$
$883$	27.0000i	0.908622i	−0.890843	+	0.454311i	$0.849885\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	66.0000	2.21357
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.07107i	0.236624i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 53.0660i	− 1.76788i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−32.8329	−1.09140
$906$	0	0
$907$	− 31.0000i	− 1.02934i	−0.857389	−	0.514669i	$-0.827915\pi$
$907$	− 31.0000i	− 1.02934i	0.857389	−	0.514669i	$-0.172085\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.7990	0.655970	0.327985	−	0.944683i	$-0.393630\pi$
$911$	19.7990	0.655970	0.327985	+	0.944683i	$0.393630\pi$
$912$	0	0
$913$	44.0000	1.45619
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−13.2665	−0.437621	−0.218811	−	0.975767i	$-0.570218\pi$
$919$	−13.2665	−0.437621	−0.218811	+	0.975767i	$0.570218\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 37.5233i	− 1.23510i
$924$	0	0
$925$	29.8496i	0.981450i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.1425	−0.923325	−0.461663	−	0.887056i	$-0.652747\pi$
$929$	−28.1425	−0.923325	−0.461663	+	0.887056i	$0.652747\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	31.1127	1.01749
$936$	0	0
$937$	39.0000	1.27407	0.637037	−	0.770833i	$-0.280160\pi$
$937$	39.0000	1.27407	0.637037	+	0.770833i	$0.280160\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.5269i	1.06035i	0.847889	+	0.530174i	$0.177873\pi$
$941$	32.5269i	1.06035i	−0.847889	+	0.530174i	$0.822127\pi$
$942$	0	0
$943$	−13.2665	−0.432017
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 42.2137i	− 1.37176i	−0.727714	−	0.685881i	$-0.759417\pi$
$947$	− 42.2137i	− 1.37176i	0.727714	−	0.685881i	$-0.240583\pi$
$948$	0	0
$949$	− 3.31662i	− 0.107662i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−14.0712	−0.455812	−0.227906	−	0.973683i	$-0.573188\pi$
$953$	−14.0712	−0.455812	−0.227906	+	0.973683i	$0.573188\pi$
$954$	0	0
$955$	34.0000i	1.10021i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.5563	−0.502341
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 12.7279i	− 0.409726i
$966$	0	0
$967$	−43.1161	−1.38652	−0.693261	−	0.720687i	$-0.743826\pi$
$967$	−43.1161	−1.38652	−0.693261	+	0.720687i	$0.743826\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 14.0712i	− 0.451568i	−0.974177	−	0.225784i	$-0.927506\pi$
$971$	− 14.0712i	− 0.451568i	0.974177	−	0.225784i	$-0.0724943\pi$
$972$	0	0
$973$	− 16.5831i	− 0.531631i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.38083	−0.300119	−0.150060	−	0.988677i	$-0.547947\pi$
$977$	−9.38083	−0.300119	−0.150060	+	0.988677i	$0.547947\pi$
$978$	0	0
$979$	66.0000i	2.10937i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.2132	0.676596	0.338298	−	0.941039i	$-0.390149\pi$
$983$	21.2132	0.676596	0.338298	+	0.941039i	$0.390149\pi$
$984$	0	0
$985$	−30.0000	−0.955879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.48528i	0.269816i
$990$	0	0
$991$	−29.8496	−0.948205	−0.474102	−	0.880470i	$-0.657227\pi$
$991$	−29.8496	−0.948205	−0.474102	+	0.880470i	$0.657227\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 4.69042i	− 0.148696i
$996$	0	0
$997$	− 59.6992i	− 1.89069i	−0.326066	−	0.945347i	$-0.605723\pi$
$997$	− 59.6992i	− 1.89069i	0.326066	−	0.945347i	$-0.394277\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.d.p.1729.1	✓	8
3.2	odd	2	inner	3456.2.d.p.1729.5	yes	8
4.3	odd	2	inner	3456.2.d.p.1729.3	yes	8
8.3	odd	2	inner	3456.2.d.p.1729.8	yes	8
8.5	even	2	inner	3456.2.d.p.1729.6	yes	8
12.11	even	2	inner	3456.2.d.p.1729.7	yes	8
16.3	odd	4		6912.2.a.cf.1.1		4
16.5	even	4		6912.2.a.cf.1.4		4
16.11	odd	4		6912.2.a.cg.1.3		4
16.13	even	4		6912.2.a.cg.1.2		4
24.5	odd	2	inner	3456.2.d.p.1729.2	yes	8
24.11	even	2	inner	3456.2.d.p.1729.4	yes	8
48.5	odd	4		6912.2.a.cf.1.2		4
48.11	even	4		6912.2.a.cg.1.1		4
48.29	odd	4		6912.2.a.cg.1.4		4
48.35	even	4		6912.2.a.cf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3456.2.d.p.1729.1	✓	8	1.1	even	1	trivial
3456.2.d.p.1729.2	yes	8	24.5	odd	2	inner
3456.2.d.p.1729.3	yes	8	4.3	odd	2	inner
3456.2.d.p.1729.4	yes	8	24.11	even	2	inner
3456.2.d.p.1729.5	yes	8	3.2	odd	2	inner
3456.2.d.p.1729.6	yes	8	8.5	even	2	inner
3456.2.d.p.1729.7	yes	8	12.11	even	2	inner
3456.2.d.p.1729.8	yes	8	8.3	odd	2	inner
6912.2.a.cf.1.1		4	16.3	odd	4
6912.2.a.cf.1.2		4	48.5	odd	4
6912.2.a.cf.1.3		4	48.35	even	4
6912.2.a.cf.1.4		4	16.5	even	4
6912.2.a.cg.1.1		4	48.11	even	4
6912.2.a.cg.1.2		4	16.13	even	4
6912.2.a.cg.1.3		4	16.11	odd	4
6912.2.a.cg.1.4		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} -3.31662 q^{7} +O(q^{10})\)	\(q-1.41421i q^{5} -3.31662 q^{7} +4.69042i q^{11} -3.31662i q^{13} +4.69042 q^{17} -1.00000i q^{19} +1.41421 q^{23} +3.00000 q^{25} -5.65685i q^{29} +4.69042i q^{35} +9.94987i q^{37} -9.38083 q^{41} +6.00000i q^{43} -7.07107 q^{47} +4.00000 q^{49} -11.3137i q^{53} +6.63325 q^{55} -4.69042i q^{59} -3.31662i q^{61} -4.69042 q^{65} -13.0000i q^{67} +11.3137 q^{71} +1.00000 q^{73} -15.5563i q^{77} +9.94987 q^{79} -9.38083i q^{83} -6.63325i q^{85} +14.0712 q^{89} +11.0000i q^{91} -1.41421 q^{95} -7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 24 q^{25} + 32 q^{49} + 8 q^{73} - 56 q^{97}+O(q^{100}) \) 8 * q + 24 * q^25 + 32 * q^49 + 8 * q^73 - 56 * q^97

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