LMFDB - Embedded newform 432.7.e.k.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,7,Mod(161,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("432.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	432.e (of order $2$, degree $1$, not 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:	$99.3833641238$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 75x^{4} - 90x^{3} + 1861x^{2} + 7864x + 10098 $ x^6 - 2x^5 - 75x^4 - 90x^3 + 1861x^2 + 7864*x + 10098
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{23}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$7.70376 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	432.161
Dual form			432.7.e.k.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+62.7833i q^{5} +496.882 q^{7} +O(q^{10})$	$q+62.7833i q^{5} +496.882 q^{7} +1437.06i q^{11} +3905.03 q^{13} -8385.53i q^{17} +3625.19 q^{19} -24070.3i q^{23} +11683.3 q^{25} +38898.5i q^{29} +10728.8 q^{31} +31195.9i q^{35} -45162.5 q^{37} -35120.1i q^{41} +21584.6 q^{43} -103509. i q^{47} +129242. q^{49} +70152.9i q^{53} -90223.6 q^{55} -185774. i q^{59} -301079. q^{61} +245171. i q^{65} +130275. q^{67} -464741. i q^{71} +274352. q^{73} +714051. i q^{77} -705830. q^{79} -233654. i q^{83} +526471. q^{85} +1.16112e6i q^{89} +1.94034e6 q^{91} +227602. i q^{95} +129972. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 114 q^{7}+O(q^{10}) $ 6 * q + 114 * q^7	$ 6 q + 114 q^{7} + 4026 q^{13} - 1854 q^{19} + 7686 q^{25} - 708 q^{31} - 66198 q^{37} + 116172 q^{43} + 237072 q^{49} + 158544 q^{55} + 170154 q^{61} + 84162 q^{67} + 748986 q^{73} + 560130 q^{79} + 1073040 q^{85} + 2795982 q^{91} + 1967418 q^{97}+O(q^{100}) $ 6 * q + 114 * q^7 + 4026 * q^13 - 1854 * q^19 + 7686 * q^25 - 708 * q^31 - 66198 * q^37 + 116172 * q^43 + 237072 * q^49 + 158544 * q^55 + 170154 * q^61 + 84162 * q^67 + 748986 * q^73 + 560130 * q^79 + 1073040 * q^85 + 2795982 * q^91 + 1967418 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	62.7833i	0.502266i	0.967953	+	0.251133i	$0.0808032\pi$
$5$	62.7833i	0.502266i	−0.967953	+	0.251133i	$0.919197\pi$
$6$	0	0
$7$	496.882	1.44863	0.724317	−	0.689467i	$-0.242155\pi$
$7$	496.882	1.44863	0.724317	+	0.689467i	$0.242155\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1437.06i	1.07969i	0.841765	+	0.539844i	$0.181517\pi$
$11$	1437.06i	1.07969i	−0.841765	+	0.539844i	$0.818483\pi$
$12$	0	0
$13$	3905.03	1.77744	0.888719	−	0.458452i	$-0.151596\pi$
$13$	3905.03	1.77744	0.888719	+	0.458452i	$0.151596\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 8385.53i	− 1.70681i	−0.521252	−	0.853403i	$-0.674535\pi$
$17$	− 8385.53i	− 1.70681i	0.521252	−	0.853403i	$-0.325465\pi$
$18$	0	0
$19$	3625.19	0.528531	0.264265	−	0.964450i	$-0.414871\pi$
$19$	3625.19	0.528531	0.264265	+	0.964450i	$0.414871\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 24070.3i	− 1.97833i	−0.146820	−	0.989163i	$-0.546904\pi$
$23$	− 24070.3i	− 1.97833i	0.146820	−	0.989163i	$-0.453096\pi$
$24$	0	0
$25$	11683.3	0.747729
$26$	0	0
$27$	0	0
$28$	0	0
$29$	38898.5i	1.59492i	0.603372	+	0.797460i	$0.293823\pi$
$29$	38898.5i	1.59492i	−0.603372	+	0.797460i	$0.706177\pi$
$30$	0	0
$31$	10728.8	0.360136	0.180068	−	0.983654i	$-0.442368\pi$
$31$	10728.8	0.360136	0.180068	+	0.983654i	$0.442368\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	31195.9i	0.727600i
$36$	0	0
$37$	−45162.5	−0.891606	−0.445803	−	0.895131i	$-0.647082\pi$
$37$	−45162.5	−0.891606	−0.445803	+	0.895131i	$0.647082\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 35120.1i	− 0.509571i	−0.966998	−	0.254785i	$-0.917995\pi$
$41$	− 35120.1i	− 0.509571i	0.966998	−	0.254785i	$-0.0820048\pi$
$42$	0	0
$43$	21584.6	0.271481	0.135740	−	0.990744i	$-0.456659\pi$
$43$	21584.6	0.271481	0.135740	+	0.990744i	$0.456659\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 103509.i	− 0.996974i	−0.866897	−	0.498487i	$-0.833889\pi$
$47$	− 103509.i	− 0.996974i	0.866897	−	0.498487i	$-0.166111\pi$
$48$	0	0
$49$	129242.	1.09854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	70152.9i	0.471214i	0.971848	+	0.235607i	$0.0757078\pi$
$53$	70152.9i	0.471214i	−0.971848	+	0.235607i	$0.924292\pi$
$54$	0	0
$55$	−90223.6	−0.542291
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 185774.i	− 0.904541i	−0.891881	−	0.452271i	$-0.850614\pi$
$59$	− 185774.i	− 0.904541i	0.891881	−	0.452271i	$-0.149386\pi$
$60$	0	0
$61$	−301079.	−1.32645	−0.663226	−	0.748419i	$-0.730813\pi$
$61$	−301079.	−1.32645	−0.663226	+	0.748419i	$0.730813\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	245171.i	0.892748i
$66$	0	0
$67$	130275.	0.433148	0.216574	−	0.976266i	$-0.430512\pi$
$67$	130275.	0.433148	0.216574	+	0.976266i	$0.430512\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 464741.i	− 1.29848i	−0.760583	−	0.649241i	$-0.775087\pi$
$71$	− 464741.i	− 1.29848i	0.760583	−	0.649241i	$-0.224913\pi$
$72$	0	0
$73$	274352.	0.705245	0.352623	−	0.935766i	$-0.385290\pi$
$73$	274352.	0.705245	0.352623	+	0.935766i	$0.385290\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	714051.i	1.56407i
$78$	0	0
$79$	−705830.	−1.43159	−0.715795	−	0.698311i	$-0.753935\pi$
$79$	−705830.	−1.43159	−0.715795	+	0.698311i	$0.753935\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 233654.i	− 0.408639i	−0.978904	−	0.204319i	$-0.934502\pi$
$83$	− 233654.i	− 0.408639i	0.978904	−	0.204319i	$-0.0654981\pi$
$84$	0	0
$85$	526471.	0.857271
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.16112e6i	1.64706i	0.567275	+	0.823529i	$0.307998\pi$
$89$	1.16112e6i	1.64706i	−0.567275	+	0.823529i	$0.692002\pi$
$90$	0	0
$91$	1.94034e6	2.57486
$92$	0	0
$93$	0	0
$94$	0	0
$95$	227602.i	0.265463i
$96$	0	0
$97$	129972.	0.142408	0.0712038	−	0.997462i	$-0.477316\pi$
$97$	129972.	0.142408	0.0712038	+	0.997462i	$0.477316\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	44362.7i	0.0430580i	0.999768	+	0.0215290i	$0.00685342\pi$
$101$	44362.7i	0.0430580i	−0.999768	+	0.0215290i	$0.993147\pi$
$102$	0	0
$103$	1.21733e6	1.11403	0.557013	−	0.830504i	$-0.311947\pi$
$103$	1.21733e6	1.11403	0.557013	+	0.830504i	$0.311947\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 194040.i	− 0.158394i	−0.996859	−	0.0791972i	$-0.974764\pi$
$107$	− 194040.i	− 0.158394i	0.996859	−	0.0791972i	$-0.0252357\pi$
$108$	0	0
$109$	2.22984e6	1.72184	0.860921	−	0.508738i	$-0.169888\pi$
$109$	2.22984e6	1.72184	0.860921	+	0.508738i	$0.169888\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.17695e6i	− 0.815684i	−0.913052	−	0.407842i	$-0.866281\pi$
$113$	− 1.17695e6i	− 0.815684i	0.913052	−	0.407842i	$-0.133719\pi$
$114$	0	0
$115$	1.51121e6	0.993647
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 4.16662e6i	− 2.47254i
$120$	0	0
$121$	−293593.	−0.165725
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.71450e6i	0.877825i
$126$	0	0
$127$	−2.57158e6	−1.25542	−0.627710	−	0.778448i	$-0.716007\pi$
$127$	−2.57158e6	−1.25542	−0.627710	+	0.778448i	$0.716007\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.08627e6i	− 0.928020i	−0.885830	−	0.464010i	$-0.846410\pi$
$131$	− 2.08627e6i	− 0.928020i	0.885830	−	0.464010i	$-0.153590\pi$
$132$	0	0
$133$	1.80129e6	0.765648
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.52222e6i	0.980892i	0.871471	+	0.490446i	$0.163166\pi$
$137$	2.52222e6i	0.980892i	−0.871471	+	0.490446i	$0.836834\pi$
$138$	0	0
$139$	−558638.	−0.208011	−0.104006	−	0.994577i	$-0.533166\pi$
$139$	−558638.	−0.208011	−0.104006	+	0.994577i	$0.533166\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.61178e6i	1.91908i
$144$	0	0
$145$	−2.44218e6	−0.801074
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 951789.i	− 0.287728i	−0.989597	−	0.143864i	$-0.954047\pi$
$149$	− 951789.i	− 0.287728i	0.989597	−	0.143864i	$-0.0459528\pi$
$150$	0	0
$151$	3.52962e6	1.02517	0.512586	−	0.858636i	$-0.328687\pi$
$151$	3.52962e6	1.02517	0.512586	+	0.858636i	$0.328687\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	673590.i	0.180884i
$156$	0	0
$157$	678623.	0.175360	0.0876798	−	0.996149i	$-0.472055\pi$
$157$	678623.	0.175360	0.0876798	+	0.996149i	$0.472055\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.19601e7i	− 2.86587i
$162$	0	0
$163$	2.74597e6	0.634064	0.317032	−	0.948415i	$-0.397314\pi$
$163$	2.74597e6	0.634064	0.317032	+	0.948415i	$0.397314\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.51214e6i	1.39822i	0.715016	+	0.699108i	$0.246420\pi$
$167$	6.51214e6i	1.39822i	−0.715016	+	0.699108i	$0.753580\pi$
$168$	0	0
$169$	1.04225e7	2.15929
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.02492e6i	0.584220i	0.956385	+	0.292110i	$0.0943573\pi$
$173$	3.02492e6i	0.584220i	−0.956385	+	0.292110i	$0.905643\pi$
$174$	0	0
$175$	5.80520e6	1.08319
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3.43892e6i	− 0.599602i	−0.954002	−	0.299801i	$-0.903080\pi$
$179$	− 3.43892e6i	− 0.599602i	0.954002	−	0.299801i	$-0.0969202\pi$
$180$	0	0
$181$	4.76769e6	0.804030	0.402015	−	0.915633i	$-0.368310\pi$
$181$	4.76769e6	0.804030	0.402015	+	0.915633i	$0.368310\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.83545e6i	− 0.447824i
$186$	0	0
$187$	1.20506e7	1.84282
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.43616e6i	0.206112i	0.994676	+	0.103056i	$0.0328621\pi$
$191$	1.43616e6i	0.206112i	−0.994676	+	0.103056i	$0.967138\pi$
$192$	0	0
$193$	6.88018e6	0.957036	0.478518	−	0.878078i	$-0.341174\pi$
$193$	6.88018e6	0.957036	0.478518	+	0.878078i	$0.341174\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.46415e6i	0.583902i	0.956433	+	0.291951i	$0.0943045\pi$
$197$	4.46415e6i	0.583902i	−0.956433	+	0.291951i	$0.905696\pi$
$198$	0	0
$199$	1.21686e7	1.54412	0.772058	−	0.635552i	$-0.219227\pi$
$199$	1.21686e7	1.54412	0.772058	+	0.635552i	$0.219227\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.93280e7i	2.31046i
$204$	0	0
$205$	2.20496e6	0.255940
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.20964e6i	0.570648i
$210$	0	0
$211$	−1.68577e7	−1.79453	−0.897265	−	0.441491i	$-0.854450\pi$
$211$	−1.68577e7	−1.79453	−0.897265	+	0.441491i	$0.854450\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.35515e6i	0.136356i
$216$	0	0
$217$	5.33095e6	0.521706
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 3.27458e7i	− 3.03374i
$222$	0	0
$223$	1.09013e6	0.0983019	0.0491510	−	0.998791i	$-0.484348\pi$
$223$	1.09013e6	0.0983019	0.0491510	+	0.998791i	$0.484348\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.12524e6i	− 0.523656i	−0.965115	−	0.261828i	$-0.915675\pi$
$227$	− 6.12524e6i	− 0.523656i	0.965115	−	0.261828i	$-0.0843253\pi$
$228$	0	0
$229$	−9.73155e6	−0.810356	−0.405178	−	0.914238i	$-0.632790\pi$
$229$	−9.73155e6	−0.810356	−0.405178	+	0.914238i	$0.632790\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5.98923e6i	− 0.473482i	−0.971573	−	0.236741i	$-0.923921\pi$
$233$	− 5.98923e6i	− 0.473482i	0.971573	−	0.236741i	$-0.0760793\pi$
$234$	0	0
$235$	6.49862e6	0.500746
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 7.14769e6i	− 0.523567i	−0.965127	−	0.261783i	$-0.915689\pi$
$239$	− 7.14769e6i	− 0.523567i	0.965127	−	0.261783i	$-0.0843106\pi$
$240$	0	0
$241$	−2.98986e6	−0.213599	−0.106800	−	0.994281i	$-0.534060\pi$
$241$	−2.98986e6	−0.213599	−0.106800	+	0.994281i	$0.534060\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.11427e6i	0.551761i
$246$	0	0
$247$	1.41565e7	0.939431
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.85894e7i	1.80794i	0.427598	+	0.903969i	$0.359360\pi$
$251$	2.85894e7i	1.80794i	−0.427598	+	0.903969i	$0.640640\pi$
$252$	0	0
$253$	3.45906e7	2.13597
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.45391e6i	0.556945i	0.960444	+	0.278472i	$0.0898281\pi$
$257$	9.45391e6i	0.556945i	−0.960444	+	0.278472i	$0.910172\pi$
$258$	0	0
$259$	−2.24404e7	−1.29161
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.26436e7i	− 0.695030i	−0.937674	−	0.347515i	$-0.887026\pi$
$263$	− 1.26436e7i	− 0.695030i	0.937674	−	0.347515i	$-0.112974\pi$
$264$	0	0
$265$	−4.40443e6	−0.236675
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 1.91952e7i	− 0.986133i	−0.869992	−	0.493067i	$-0.835876\pi$
$269$	− 1.91952e7i	− 0.986133i	0.869992	−	0.493067i	$-0.164124\pi$
$270$	0	0
$271$	3.10242e6	0.155881	0.0779404	−	0.996958i	$-0.475166\pi$
$271$	3.10242e6	0.155881	0.0779404	+	0.996958i	$0.475166\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.67896e7i	0.807313i
$276$	0	0
$277$	−1.97207e7	−0.927861	−0.463930	−	0.885872i	$-0.653561\pi$
$277$	−1.97207e7	−0.927861	−0.463930	+	0.885872i	$0.653561\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.26616e7i	1.47203i	0.676963	+	0.736017i	$0.263296\pi$
$281$	3.26616e7i	1.47203i	−0.676963	+	0.736017i	$0.736704\pi$
$282$	0	0
$283$	1.16099e7	0.512233	0.256117	−	0.966646i	$-0.417557\pi$
$283$	1.16099e7	0.512233	0.256117	+	0.966646i	$0.417557\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.74506e7i	− 0.738182i
$288$	0	0
$289$	−4.61796e7	−1.91318
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.03466e7i	1.20645i	0.797573	+	0.603223i	$0.206117\pi$
$293$	3.03466e7i	1.20645i	−0.797573	+	0.603223i	$0.793883\pi$
$294$	0	0
$295$	1.16635e7	0.454321
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.39953e7i	− 3.51635i
$300$	0	0
$301$	1.07250e7	0.393276
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.89027e7i	− 0.666232i
$306$	0	0
$307$	−4.99035e6	−0.172471	−0.0862355	−	0.996275i	$-0.527484\pi$
$307$	−4.99035e6	−0.172471	−0.0862355	+	0.996275i	$0.527484\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.75448e7i	0.583266i	0.956530	+	0.291633i	$0.0941985\pi$
$311$	1.75448e7i	0.583266i	−0.956530	+	0.291633i	$0.905801\pi$
$312$	0	0
$313$	1.42107e7	0.463429	0.231714	−	0.972784i	$-0.425567\pi$
$313$	1.42107e7	0.463429	0.231714	+	0.972784i	$0.425567\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.79914e7i	1.19264i	0.802749	+	0.596318i	$0.203370\pi$
$317$	3.79914e7i	1.19264i	−0.802749	+	0.596318i	$0.796630\pi$
$318$	0	0
$319$	−5.58996e7	−1.72202
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.03992e7i	− 0.902099i
$324$	0	0
$325$	4.56235e7	1.32904
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.14316e7i	− 1.44425i
$330$	0	0
$331$	−3.85294e6	−0.106245	−0.0531224	−	0.998588i	$-0.516917\pi$
$331$	−3.85294e6	−0.106245	−0.0531224	+	0.998588i	$0.516917\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.17909e6i	0.217556i
$336$	0	0
$337$	−4.67260e7	−1.22087	−0.610434	−	0.792067i	$-0.709005\pi$
$337$	−4.67260e7	−1.22087	−0.610434	+	0.792067i	$0.709005\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.54180e7i	0.388835i
$342$	0	0
$343$	5.76057e6	0.142752
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.68841e6i	0.207947i	0.994580	+	0.103973i	$0.0331557\pi$
$347$	8.68841e6i	0.207947i	−0.994580	+	0.103973i	$0.966844\pi$
$348$	0	0
$349$	3.56685e7	0.839089	0.419545	−	0.907735i	$-0.362190\pi$
$349$	3.56685e7	0.839089	0.419545	+	0.907735i	$0.362190\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.34071e7i	1.44150i	0.693196	+	0.720749i	$0.256202\pi$
$353$	6.34071e7i	1.44150i	−0.693196	+	0.720749i	$0.743798\pi$
$354$	0	0
$355$	2.91780e7	0.652184
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.47602e7i	0.967405i	0.875233	+	0.483702i	$0.160708\pi$
$359$	4.47602e7i	0.967405i	−0.875233	+	0.483702i	$0.839292\pi$
$360$	0	0
$361$	−3.39039e7	−0.720655
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.72247e7i	0.354221i
$366$	0	0
$367$	−1.51940e7	−0.307378	−0.153689	−	0.988119i	$-0.549115\pi$
$367$	−1.51940e7	−0.307378	−0.153689	+	0.988119i	$0.549115\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.48577e7i	0.682616i
$372$	0	0
$373$	−9.43540e7	−1.81817	−0.909084	−	0.416614i	$-0.863217\pi$
$373$	−9.43540e7	−1.81817	−0.909084	+	0.416614i	$0.863217\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.51900e8i	2.83487i
$378$	0	0
$379$	−8.08369e6	−0.148488	−0.0742441	−	0.997240i	$-0.523654\pi$
$379$	−8.08369e6	−0.148488	−0.0742441	+	0.997240i	$0.523654\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 9.63217e7i	− 1.71446i	−0.514932	−	0.857231i	$-0.672183\pi$
$383$	− 9.63217e7i	− 1.71446i	0.514932	−	0.857231i	$-0.327817\pi$
$384$	0	0
$385$	−4.48305e7	−0.785581
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.43211e7i	− 0.413176i	−0.978428	−	0.206588i	$-0.933764\pi$
$389$	− 2.43211e7i	− 0.413176i	0.978428	−	0.206588i	$-0.0662360\pi$
$390$	0	0
$391$	−2.01842e8	−3.37662
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4.43143e7i	− 0.719039i
$396$	0	0
$397$	4.70196e7	0.751463	0.375732	−	0.926729i	$-0.377391\pi$
$397$	4.70196e7	0.751463	0.375732	+	0.926729i	$0.377391\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 6.79234e6i	− 0.105338i	−0.998612	−	0.0526692i	$-0.983227\pi$
$401$	− 6.79234e6i	− 0.105338i	0.998612	−	0.0526692i	$-0.0167729\pi$
$402$	0	0
$403$	4.18964e7	0.640120
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.49015e7i	− 0.962656i
$408$	0	0
$409$	−8.43151e7	−1.23235	−0.616177	−	0.787608i	$-0.711319\pi$
$409$	−8.43151e7	−1.23235	−0.616177	+	0.787608i	$0.711319\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 9.23076e7i	− 1.31035i
$414$	0	0
$415$	1.46696e7	0.205245
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.72937e7i	0.235097i	0.993067	+	0.117548i	$0.0375035\pi$
$419$	1.72937e7i	0.235097i	−0.993067	+	0.117548i	$0.962496\pi$
$420$	0	0
$421$	−1.25301e8	−1.67922	−0.839612	−	0.543187i	$-0.817217\pi$
$421$	−1.25301e8	−1.67922	−0.839612	+	0.543187i	$0.817217\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 9.79704e7i	− 1.27623i
$426$	0	0
$427$	−1.49601e8	−1.92154
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.28715e7i	0.785275i	0.919693	+	0.392638i	$0.128437\pi$
$431$	6.28715e7i	0.785275i	−0.919693	+	0.392638i	$0.871563\pi$
$432$	0	0
$433$	−5.66021e7	−0.697219	−0.348609	−	0.937268i	$-0.613346\pi$
$433$	−5.66021e7	−0.697219	−0.348609	+	0.937268i	$0.613346\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.72595e7i	− 1.04561i
$438$	0	0
$439$	−3.82688e7	−0.452326	−0.226163	−	0.974089i	$-0.572618\pi$
$439$	−3.82688e7	−0.452326	−0.226163	+	0.974089i	$0.572618\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.79866e7i	1.01206i	0.862517	+	0.506029i	$0.168887\pi$
$443$	8.79866e7i	1.01206i	−0.862517	+	0.506029i	$0.831113\pi$
$444$	0	0
$445$	−7.28992e7	−0.827262
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 8.97838e7i	− 0.991879i	−0.868357	−	0.495940i	$-0.834824\pi$
$449$	− 8.97838e7i	− 0.991879i	0.868357	−	0.495940i	$-0.165176\pi$
$450$	0	0
$451$	5.04699e7	0.550177
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.21821e8i	1.29327i
$456$	0	0
$457$	1.76314e8	1.84731	0.923653	−	0.383229i	$-0.125188\pi$
$457$	1.76314e8	1.84731	0.923653	+	0.383229i	$0.125188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.28602e7i	0.539543i	0.962924	+	0.269771i	$0.0869481\pi$
$461$	5.28602e7i	0.539543i	−0.962924	+	0.269771i	$0.913052\pi$
$462$	0	0
$463$	6.74136e6	0.0679211	0.0339605	−	0.999423i	$-0.489188\pi$
$463$	6.74136e6	0.0679211	0.0339605	+	0.999423i	$0.489188\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.62620e7i	− 0.159670i	−0.996808	−	0.0798351i	$-0.974561\pi$
$467$	− 1.62620e7i	− 0.159670i	0.996808	−	0.0798351i	$-0.0254394\pi$
$468$	0	0
$469$	6.47312e7	0.627473
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.10185e7i	0.293114i
$474$	0	0
$475$	4.23541e7	0.395198
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.65819e6i	0.0605829i	0.999541	+	0.0302914i	$0.00964354\pi$
$479$	6.65819e6i	0.0605829i	−0.999541	+	0.0302914i	$0.990356\pi$
$480$	0	0
$481$	−1.76361e8	−1.58478
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.16004e6i	0.0715265i
$486$	0	0
$487$	8.41405e7	0.728481	0.364241	−	0.931305i	$-0.381329\pi$
$487$	8.41405e7	0.728481	0.364241	+	0.931305i	$0.381329\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.06143e7i	0.174150i	0.996202	+	0.0870751i	$0.0277520\pi$
$491$	2.06143e7i	0.174150i	−0.996202	+	0.0870751i	$0.972248\pi$
$492$	0	0
$493$	3.26185e8	2.72222
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.30921e8i	− 1.88103i
$498$	0	0
$499$	−6.76628e7	−0.544563	−0.272282	−	0.962218i	$-0.587778\pi$
$499$	−6.76628e7	−0.544563	−0.272282	+	0.962218i	$0.587778\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.12807e7i	0.638680i	0.947640	+	0.319340i	$0.103461\pi$
$503$	8.12807e7i	0.638680i	−0.947640	+	0.319340i	$0.896539\pi$
$504$	0	0
$505$	−2.78523e6	−0.0216266
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.46456e8i	1.11059i	0.831655	+	0.555293i	$0.187394\pi$
$509$	1.46456e8i	1.11059i	−0.831655	+	0.555293i	$0.812606\pi$
$510$	0	0
$511$	1.36321e8	1.02164
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.64278e7i	0.559538i
$516$	0	0
$517$	1.48749e8	1.07642
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.00866e8i	− 0.713232i	−0.934251	−	0.356616i	$-0.883931\pi$
$521$	− 1.00866e8i	− 0.713232i	0.934251	−	0.356616i	$-0.116069\pi$
$522$	0	0
$523$	−1.23966e8	−0.866555	−0.433277	−	0.901261i	$-0.642643\pi$
$523$	−1.23966e8	−0.866555	−0.433277	+	0.901261i	$0.642643\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.99669e7i	− 0.614682i
$528$	0	0
$529$	−4.31343e8	−2.91378
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.37145e8i	− 0.905731i
$534$	0	0
$535$	1.21825e7	0.0795562
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.85730e8i	1.18608i
$540$	0	0
$541$	1.57932e8	0.997419	0.498709	−	0.866769i	$-0.333807\pi$
$541$	1.57932e8	0.997419	0.498709	+	0.866769i	$0.333807\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.39996e8i	0.864823i
$546$	0	0
$547$	−1.50060e7	−0.0916862	−0.0458431	−	0.998949i	$-0.514597\pi$
$547$	−1.50060e7	−0.0916862	−0.0458431	+	0.998949i	$0.514597\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.41015e8i	0.842964i
$552$	0	0
$553$	−3.50714e8	−2.07385
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.89257e7i	0.572458i	0.958161	+	0.286229i	$0.0924018\pi$
$557$	9.89257e7i	0.572458i	−0.958161	+	0.286229i	$0.907598\pi$
$558$	0	0
$559$	8.42886e7	0.482540
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 8.24717e7i	− 0.462146i	−0.972936	−	0.231073i	$-0.925776\pi$
$563$	− 8.24717e7i	− 0.462146i	0.972936	−	0.231073i	$-0.0742237\pi$
$564$	0	0
$565$	7.38927e7	0.409691
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.93991e8i	− 1.05304i	−0.850162	−	0.526521i	$-0.823496\pi$
$569$	− 1.93991e8i	− 1.05304i	0.850162	−	0.526521i	$-0.176504\pi$
$570$	0	0
$571$	−1.87396e8	−1.00659	−0.503293	−	0.864116i	$-0.667878\pi$
$571$	−1.87396e8	−1.00659	−0.503293	+	0.864116i	$0.667878\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 2.81220e8i	− 1.47925i
$576$	0	0
$577$	1.29755e8	0.675457	0.337729	−	0.941244i	$-0.390341\pi$
$577$	1.29755e8	0.675457	0.337729	+	0.941244i	$0.390341\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.16099e8i	− 0.591968i
$582$	0	0
$583$	−1.00814e8	−0.508764
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.63761e8i	1.79846i	0.437473	+	0.899231i	$0.355873\pi$
$587$	3.63761e8i	1.79846i	−0.437473	+	0.899231i	$0.644127\pi$
$588$	0	0
$589$	3.88940e7	0.190343
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.88503e7i	0.330173i	0.986279	+	0.165086i	$0.0527903\pi$
$593$	6.88503e7i	0.330173i	−0.986279	+	0.165086i	$0.947210\pi$
$594$	0	0
$595$	2.61594e8	1.24187
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.17543e8i	− 0.546911i	−0.961885	−	0.273455i	$-0.911833\pi$
$599$	− 1.17543e8i	− 0.546911i	0.961885	−	0.273455i	$-0.0881666\pi$
$600$	0	0
$601$	−1.38103e7	−0.0636179	−0.0318090	−	0.999494i	$-0.510127\pi$
$601$	−1.38103e7	−0.0636179	−0.0318090	+	0.999494i	$0.510127\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1.84327e7i	− 0.0832383i
$606$	0	0
$607$	2.93369e8	1.31174	0.655871	−	0.754873i	$-0.272302\pi$
$607$	2.93369e8	1.31174	0.655871	+	0.754873i	$0.272302\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.04205e8i	− 1.77206i
$612$	0	0
$613$	2.86835e8	1.24523	0.622617	−	0.782526i	$-0.286069\pi$
$613$	2.86835e8	1.24523	0.622617	+	0.782526i	$0.286069\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 7.64357e7i	− 0.325417i	−0.986674	−	0.162709i	$-0.947977\pi$
$617$	− 7.64357e7i	− 0.325417i	0.986674	−	0.162709i	$-0.0520230\pi$
$618$	0	0
$619$	−3.68179e7	−0.155234	−0.0776170	−	0.996983i	$-0.524731\pi$
$619$	−3.68179e7	−0.155234	−0.0776170	+	0.996983i	$0.524731\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.76942e8i	2.38598i
$624$	0	0
$625$	7.49088e7	0.306826
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.78712e8i	1.52180i
$630$	0	0
$631$	1.91018e8	0.760301	0.380150	−	0.924925i	$-0.375872\pi$
$631$	1.91018e8	0.760301	0.380150	+	0.924925i	$0.375872\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.61452e8i	− 0.630555i
$636$	0	0
$637$	5.04696e8	1.95259
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.16010e8i	1.95922i	0.200899	+	0.979612i	$0.435614\pi$
$641$	5.16010e8i	1.95922i	−0.200899	+	0.979612i	$0.564386\pi$
$642$	0	0
$643$	−3.35849e8	−1.26332	−0.631658	−	0.775248i	$-0.717625\pi$
$643$	−3.35849e8	−1.26332	−0.631658	+	0.775248i	$0.717625\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.13350e8i	− 1.15695i	−0.815698	−	0.578477i	$-0.803647\pi$
$647$	− 3.13350e8i	− 1.15695i	0.815698	−	0.578477i	$-0.196353\pi$
$648$	0	0
$649$	2.66969e8	0.976622
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.59248e8i	1.29019i	0.764102	+	0.645096i	$0.223183\pi$
$653$	3.59248e8i	1.29019i	−0.764102	+	0.645096i	$0.776817\pi$
$654$	0	0
$655$	1.30983e8	0.466113
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 5.36947e8i	− 1.87618i	−0.346389	−	0.938091i	$-0.612592\pi$
$659$	− 5.36947e8i	− 1.87618i	0.346389	−	0.938091i	$-0.387408\pi$
$660$	0	0
$661$	2.98758e8	1.03446	0.517231	−	0.855846i	$-0.326963\pi$
$661$	2.98758e8	1.03446	0.517231	+	0.855846i	$0.326963\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.13091e8i	0.384559i
$666$	0	0
$667$	9.36298e8	3.15527
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 4.32670e8i	− 1.43215i
$672$	0	0
$673$	−3.23451e8	−1.06112	−0.530558	−	0.847649i	$-0.678018\pi$
$673$	−3.23451e8	−1.06112	−0.530558	+	0.847649i	$0.678018\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4.09203e8i	− 1.31878i	−0.751800	−	0.659391i	$-0.770814\pi$
$677$	− 4.09203e8i	− 1.31878i	0.751800	−	0.659391i	$-0.229186\pi$
$678$	0	0
$679$	6.45805e7	0.206297
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.86785e8i	− 0.586244i	−0.956075	−	0.293122i	$-0.905306\pi$
$683$	− 1.86785e8i	− 0.586244i	0.956075	−	0.293122i	$-0.0946943\pi$
$684$	0	0
$685$	−1.58353e8	−0.492669
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.73949e8i	0.837553i
$690$	0	0
$691$	3.75447e8	1.13793	0.568964	−	0.822362i	$-0.307344\pi$
$691$	3.75447e8	1.13793	0.568964	+	0.822362i	$0.307344\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 3.50732e7i	− 0.104477i
$696$	0	0
$697$	−2.94501e8	−0.869738
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 5.11037e8i	− 1.48354i	−0.670657	−	0.741768i	$-0.733988\pi$
$701$	− 5.11037e8i	− 1.48354i	0.670657	−	0.741768i	$-0.266012\pi$
$702$	0	0
$703$	−1.63723e8	−0.471242
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.20430e7i	0.0623753i
$708$	0	0
$709$	4.42507e8	1.24160	0.620800	−	0.783969i	$-0.286808\pi$
$709$	4.42507e8	1.24160	0.620800	+	0.783969i	$0.286808\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.58246e8i	− 0.712467i
$714$	0	0
$715$	−3.52326e8	−0.963889
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.20610e8i	1.93871i	0.245659	+	0.969356i	$0.420996\pi$
$719$	7.20610e8i	1.93871i	−0.245659	+	0.969356i	$0.579004\pi$
$720$	0	0
$721$	6.04867e8	1.61382
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.54461e8i	1.19257i
$726$	0	0
$727$	2.46407e8	0.641282	0.320641	−	0.947201i	$-0.396102\pi$
$727$	2.46407e8	0.641282	0.320641	+	0.947201i	$0.396102\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.80998e8i	− 0.463364i
$732$	0	0
$733$	−1.34987e8	−0.342751	−0.171375	−	0.985206i	$-0.554821\pi$
$733$	−1.34987e8	−0.342751	−0.171375	+	0.985206i	$0.554821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.87213e8i	0.467665i
$738$	0	0
$739$	−5.93121e8	−1.46964	−0.734819	−	0.678264i	$-0.762733\pi$
$739$	−5.93121e8	−1.46964	−0.734819	+	0.678264i	$0.762733\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.95231e6i	− 0.0120737i	−0.999982	−	0.00603686i	$-0.998078\pi$
$743$	− 4.95231e6i	− 0.0120737i	0.999982	−	0.00603686i	$-0.00192160\pi$
$744$	0	0
$745$	5.97564e7	0.144516
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 9.64149e7i	− 0.229456i
$750$	0	0
$751$	−8.35754e7	−0.197314	−0.0986572	−	0.995121i	$-0.531455\pi$
$751$	−8.35754e7	−0.197314	−0.0986572	+	0.995121i	$0.531455\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.21601e8i	0.514910i
$756$	0	0
$757$	8.99345e7	0.207319	0.103659	−	0.994613i	$-0.466945\pi$
$757$	8.99345e7	0.207319	0.103659	+	0.994613i	$0.466945\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.68572e8i	0.836312i	0.908375	+	0.418156i	$0.137323\pi$
$761$	3.68572e8i	0.836312i	−0.908375	+	0.418156i	$0.862677\pi$
$762$	0	0
$763$	1.10796e9	2.49432
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 7.25453e8i	− 1.60777i
$768$	0	0
$769$	−5.28138e8	−1.16136	−0.580682	−	0.814130i	$-0.697214\pi$
$769$	−5.28138e8	−1.16136	−0.580682	+	0.814130i	$0.697214\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.87278e8i	0.838465i	0.907879	+	0.419232i	$0.137701\pi$
$773$	3.87278e8i	0.838465i	−0.907879	+	0.419232i	$0.862299\pi$
$774$	0	0
$775$	1.25348e8	0.269284
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.27317e8i	− 0.269324i
$780$	0	0
$781$	6.67862e8	1.40195
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.26062e7i	0.0880772i
$786$	0	0
$787$	6.36033e8	1.30483	0.652417	−	0.757860i	$-0.273755\pi$
$787$	6.36033e8	1.30483	0.652417	+	0.757860i	$0.273755\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 5.84804e8i	− 1.18163i
$792$	0	0
$793$	−1.17572e9	−2.35769
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 4.61669e8i	− 0.911918i	−0.890001	−	0.455959i	$-0.849296\pi$
$797$	− 4.61669e8i	− 0.911918i	0.890001	−	0.455959i	$-0.150704\pi$
$798$	0	0
$799$	−8.67976e8	−1.70164
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.94262e8i	0.761444i
$804$	0	0
$805$	7.50894e8	1.43943
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 2.25161e8i	− 0.425252i	−0.977134	−	0.212626i	$-0.931798\pi$
$809$	− 2.25161e8i	− 0.425252i	0.977134	−	0.212626i	$-0.0682016\pi$
$810$	0	0
$811$	−3.06936e8	−0.575420	−0.287710	−	0.957718i	$-0.592894\pi$
$811$	−3.06936e8	−0.575420	−0.287710	+	0.957718i	$0.592894\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.72401e8i	0.318469i
$816$	0	0
$817$	7.82484e7	0.143486
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.73840e8i	0.494843i	0.968908	+	0.247422i	$0.0795832\pi$
$821$	2.73840e8i	0.494843i	−0.968908	+	0.247422i	$0.920417\pi$
$822$	0	0
$823$	−1.03168e9	−1.85073	−0.925366	−	0.379074i	$-0.876243\pi$
$823$	−1.03168e9	−1.85073	−0.925366	+	0.379074i	$0.876243\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.97417e8i	1.05624i	0.849171	+	0.528118i	$0.177102\pi$
$827$	5.97417e8i	1.05624i	−0.849171	+	0.528118i	$0.822898\pi$
$828$	0	0
$829$	2.40482e8	0.422103	0.211051	−	0.977475i	$-0.432311\pi$
$829$	2.40482e8	0.422103	0.211051	+	0.977475i	$0.432311\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.08377e9i	− 1.87500i
$834$	0	0
$835$	−4.08854e8	−0.702277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.75223e8i	1.14330i	0.820497	+	0.571651i	$0.193697\pi$
$839$	6.75223e8i	1.14330i	−0.820497	+	0.571651i	$0.806303\pi$
$840$	0	0
$841$	−9.18270e8	−1.54377
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.54357e8i	1.08454i
$846$	0	0
$847$	−1.45881e8	−0.240076
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.08708e9i	1.76389i
$852$	0	0
$853$	2.33689e8	0.376522	0.188261	−	0.982119i	$-0.439715\pi$
$853$	2.33689e8	0.376522	0.188261	+	0.982119i	$0.439715\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.09818e8i	0.651102i	0.945525	+	0.325551i	$0.105550\pi$
$857$	4.09818e8i	0.651102i	−0.945525	+	0.325551i	$0.894450\pi$
$858$	0	0
$859$	4.80708e8	0.758406	0.379203	−	0.925314i	$-0.376198\pi$
$859$	4.80708e8	0.758406	0.379203	+	0.925314i	$0.376198\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.08940e8i	− 0.169495i	−0.996402	−	0.0847475i	$-0.972992\pi$
$863$	− 1.08940e8i	− 0.169495i	0.996402	−	0.0847475i	$-0.0270083\pi$
$864$	0	0
$865$	−1.89915e8	−0.293434
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.01432e9i	− 1.54567i
$870$	0	0
$871$	5.08728e8	0.769894
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.51905e8i	1.27165i
$876$	0	0
$877$	1.29815e7	0.0192454	0.00962271	−	0.999954i	$-0.496937\pi$
$877$	1.29815e7	0.0192454	0.00962271	+	0.999954i	$0.496937\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 4.01837e8i	− 0.587655i	−0.955859	−	0.293827i	$-0.905071\pi$
$881$	− 4.01837e8i	− 0.587655i	0.955859	−	0.293827i	$-0.0949290\pi$
$882$	0	0
$883$	−3.93666e8	−0.571803	−0.285901	−	0.958259i	$-0.592293\pi$
$883$	−3.93666e8	−0.571803	−0.285901	+	0.958259i	$0.592293\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.78240e8i	1.25847i	0.777216	+	0.629234i	$0.216631\pi$
$887$	8.78240e8i	1.25847i	−0.777216	+	0.629234i	$0.783369\pi$
$888$	0	0
$889$	−1.27777e9	−1.81864
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 3.75239e8i	− 0.526931i
$894$	0	0
$895$	2.15907e8	0.301160
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.17335e8i	0.574388i
$900$	0	0
$901$	5.88269e8	0.804270
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.99331e8i	0.403837i
$906$	0	0
$907$	−5.02203e8	−0.673066	−0.336533	−	0.941672i	$-0.609254\pi$
$907$	−5.02203e8	−0.673066	−0.336533	+	0.941672i	$0.609254\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.26644e9i	− 1.67506i	−0.546393	−	0.837529i	$-0.683999\pi$
$911$	− 1.26644e9i	− 1.67506i	0.546393	−	0.837529i	$-0.316001\pi$
$912$	0	0
$913$	3.35776e8	0.441202
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.03663e9i	− 1.34436i
$918$	0	0
$919$	1.07637e9	1.38681	0.693405	−	0.720548i	$-0.256110\pi$
$919$	1.07637e9	1.38681	0.693405	+	0.720548i	$0.256110\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.81483e9i	− 2.30797i
$924$	0	0
$925$	−5.27646e8	−0.666680
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8.92619e8i	− 1.11332i	−0.830741	−	0.556658i	$-0.812083\pi$
$929$	− 8.92619e8i	− 1.11332i	0.830741	−	0.556658i	$-0.187917\pi$
$930$	0	0
$931$	4.68529e8	0.580614
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.56573e8i	0.925585i
$936$	0	0
$937$	1.04330e8	0.126821	0.0634105	−	0.997988i	$-0.479802\pi$
$937$	1.04330e8	0.126821	0.0634105	+	0.997988i	$0.479802\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.03692e9i	− 1.24445i	−0.782839	−	0.622224i	$-0.786229\pi$
$941$	− 1.03692e9i	− 1.24445i	0.782839	−	0.622224i	$-0.213771\pi$
$942$	0	0
$943$	−8.45352e8	−1.00810
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.27799e9i	− 1.50479i	−0.658710	−	0.752397i	$-0.728898\pi$
$947$	− 1.27799e9i	− 1.50479i	0.658710	−	0.752397i	$-0.271102\pi$
$948$	0	0
$949$	1.07135e9	1.25353
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 5.38519e8i	− 0.622189i	−0.950379	−	0.311095i	$-0.899304\pi$
$953$	− 5.38519e8i	− 0.622189i	0.950379	−	0.311095i	$-0.100696\pi$
$954$	0	0
$955$	−9.01670e7	−0.103523
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.25325e9i	1.42095i
$960$	0	0
$961$	−7.72396e8	−0.870302
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.31961e8i	0.480687i
$966$	0	0
$967$	2.17113e8	0.240108	0.120054	−	0.992767i	$-0.461693\pi$
$967$	2.17113e8	0.240108	0.120054	+	0.992767i	$0.461693\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.72634e8i	0.188568i	0.995545	+	0.0942841i	$0.0300562\pi$
$971$	1.72634e8i	0.188568i	−0.995545	+	0.0942841i	$0.969944\pi$
$972$	0	0
$973$	−2.77577e8	−0.301332
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.23627e8i	0.775946i	0.921671	+	0.387973i	$0.126825\pi$
$977$	7.23627e8i	0.775946i	−0.921671	+	0.387973i	$0.873175\pi$
$978$	0	0
$979$	−1.66861e9	−1.77831
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.66788e7i	0.0807263i	0.999185	+	0.0403631i	$0.0128515\pi$
$983$	7.66788e7i	0.0807263i	−0.999185	+	0.0403631i	$0.987149\pi$
$984$	0	0
$985$	−2.80274e8	−0.293274
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 5.19548e8i	− 0.537077i
$990$	0	0
$991$	−4.77700e8	−0.490834	−0.245417	−	0.969418i	$-0.578925\pi$
$991$	−4.77700e8	−0.490834	−0.245417	+	0.969418i	$0.578925\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.63983e8i	0.775558i
$996$	0	0
$997$	−1.42542e9	−1.43833	−0.719165	−	0.694839i	$-0.755475\pi$
$997$	−1.42542e9	−1.43833	−0.719165	+	0.694839i	$0.755475\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.7.e.k.161.4		6
3.2	odd	2	inner	432.7.e.k.161.3		6
4.3	odd	2		216.7.e.b.161.4	yes	6
12.11	even	2		216.7.e.b.161.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.7.e.b.161.3	✓	6	12.11	even	2
216.7.e.b.161.4	yes	6	4.3	odd	2
432.7.e.k.161.3		6	3.2	odd	2	inner
432.7.e.k.161.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	432.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+62.7833i q^{5} +496.882 q^{7} +O(q^{10})\)	\(q+62.7833i q^{5} +496.882 q^{7} +1437.06i q^{11} +3905.03 q^{13} -8385.53i q^{17} +3625.19 q^{19} -24070.3i q^{23} +11683.3 q^{25} +38898.5i q^{29} +10728.8 q^{31} +31195.9i q^{35} -45162.5 q^{37} -35120.1i q^{41} +21584.6 q^{43} -103509. i q^{47} +129242. q^{49} +70152.9i q^{53} -90223.6 q^{55} -185774. i q^{59} -301079. q^{61} +245171. i q^{65} +130275. q^{67} -464741. i q^{71} +274352. q^{73} +714051. i q^{77} -705830. q^{79} -233654. i q^{83} +526471. q^{85} +1.16112e6i q^{89} +1.94034e6 q^{91} +227602. i q^{95} +129972. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 114 q^{7}+O(q^{10}) \) 6 * q + 114 * q^7	\( 6 q + 114 q^{7} + 4026 q^{13} - 1854 q^{19} + 7686 q^{25} - 708 q^{31} - 66198 q^{37} + 116172 q^{43} + 237072 q^{49} + 158544 q^{55} + 170154 q^{61} + 84162 q^{67} + 748986 q^{73} + 560130 q^{79} + 1073040 q^{85} + 2795982 q^{91} + 1967418 q^{97}+O(q^{100}) \) 6 * q + 114 * q^7 + 4026 * q^13 - 1854 * q^19 + 7686 * q^25 - 708 * q^31 - 66198 * q^37 + 116172 * q^43 + 237072 * q^49 + 158544 * q^55 + 170154 * q^61 + 84162 * q^67 + 748986 * q^73 + 560130 * q^79 + 1073040 * q^85 + 2795982 * q^91 + 1967418 * q^97

Introduction

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