LMFDB - Embedded newform 16.7.c.b.15.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,7,Mod(15,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("16.15");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	16.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.68086533792$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	16.15
Dual form			16.7.c.b.15.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.7128i q^{3} -150.000 q^{5} -609.682i q^{7} -39.0000 q^{9} +O(q^{10})$	$q-27.7128i q^{3} -150.000 q^{5} -609.682i q^{7} -39.0000 q^{9} +914.523i q^{11} +154.000 q^{13} +4156.92i q^{15} +7458.00 q^{17} -2133.89i q^{19} -16896.0 q^{21} -8480.12i q^{23} +6875.00 q^{25} -19121.8i q^{27} -10758.0 q^{29} +1995.32i q^{31} +25344.0 q^{33} +91452.3i q^{35} -11350.0 q^{37} -4267.77i q^{39} +67122.0 q^{41} -79563.5i q^{43} +5850.00 q^{45} +69503.7i q^{47} -254063. q^{49} -206682. i q^{51} +109962. q^{53} -137178. i q^{55} -59136.0 q^{57} +305866. i q^{59} +306746. q^{61} +23777.6i q^{63} -23100.0 q^{65} +220345. i q^{67} -235008. q^{69} +372294. i q^{71} +165682. q^{73} -190526. i q^{75} +557568. q^{77} -762102. i q^{79} -558351. q^{81} -494757. i q^{83} -1.11870e6 q^{85} +298134. i q^{87} +471954. q^{89} -93891.0i q^{91} +55296.0 q^{93} +320083. i q^{95} +910594. q^{97} -35666.4i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 300 q^{5} - 78 q^{9}+O(q^{10}) $ 2 * q - 300 * q^5 - 78 * q^9	$ 2 q - 300 q^{5} - 78 q^{9} + 308 q^{13} + 14916 q^{17} - 33792 q^{21} + 13750 q^{25} - 21516 q^{29} + 50688 q^{33} - 22700 q^{37} + 134244 q^{41} + 11700 q^{45} - 508126 q^{49} + 219924 q^{53} - 118272 q^{57} + 613492 q^{61} - 46200 q^{65} - 470016 q^{69} + 331364 q^{73} + 1115136 q^{77} - 1116702 q^{81} - 2237400 q^{85} + 943908 q^{89} + 110592 q^{93} + 1821188 q^{97}+O(q^{100}) $ 2 * q - 300 * q^5 - 78 * q^9 + 308 * q^13 + 14916 * q^17 - 33792 * q^21 + 13750 * q^25 - 21516 * q^29 + 50688 * q^33 - 22700 * q^37 + 134244 * q^41 + 11700 * q^45 - 508126 * q^49 + 219924 * q^53 - 118272 * q^57 + 613492 * q^61 - 46200 * q^65 - 470016 * q^69 + 331364 * q^73 + 1115136 * q^77 - 1116702 * q^81 - 2237400 * q^85 + 943908 * q^89 + 110592 * q^93 + 1821188 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$.

$n$	$5$	$15$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 27.7128i	− 1.02640i	−0.858269	−	0.513200i	$-0.828460\pi$
$3$	− 27.7128i	− 1.02640i	0.858269	−	0.513200i	$-0.171540\pi$
$4$	0	0
$5$	−150.000	−1.20000	−0.600000	−	0.800000i	$-0.704833\pi$
$5$	−150.000	−1.20000	−0.600000	+	0.800000i	$0.704833\pi$
$6$	0	0
$7$	− 609.682i	− 1.77750i	−0.458394	−	0.888749i	$-0.651575\pi$
$7$	− 609.682i	− 1.77750i	0.458394	−	0.888749i	$-0.348425\pi$
$8$	0	0
$9$	−39.0000	−0.0534979
$10$	0	0
$11$	914.523i	0.687095i	0.939135	+	0.343547i	$0.111629\pi$
$11$	914.523i	0.687095i	−0.939135	+	0.343547i	$0.888371\pi$
$12$	0	0
$13$	154.000	0.0700956	0.0350478	−	0.999386i	$-0.488842\pi$
$13$	154.000	0.0700956	0.0350478	+	0.999386i	$0.488842\pi$
$14$	0	0
$15$	4156.92i	1.23168i
$16$	0	0
$17$	7458.00	1.51801	0.759007	−	0.651083i	$-0.225685\pi$
$17$	7458.00	1.51801	0.759007	+	0.651083i	$0.225685\pi$
$18$	0	0
$19$	− 2133.89i	− 0.311108i	−0.987827	−	0.155554i	$-0.950284\pi$
$19$	− 2133.89i	− 0.311108i	0.987827	−	0.155554i	$-0.0497162\pi$
$20$	0	0
$21$	−16896.0	−1.82443
$22$	0	0
$23$	− 8480.12i	− 0.696977i	−0.937313	−	0.348489i	$-0.886695\pi$
$23$	− 8480.12i	− 0.696977i	0.937313	−	0.348489i	$-0.113305\pi$
$24$	0	0
$25$	6875.00	0.440000
$26$	0	0
$27$	− 19121.8i	− 0.971490i
$28$	0	0
$29$	−10758.0	−0.441100	−0.220550	−	0.975376i	$-0.570785\pi$
$29$	−10758.0	−0.441100	−0.220550	+	0.975376i	$0.570785\pi$
$30$	0	0
$31$	1995.32i	0.0669774i	0.999439	+	0.0334887i	$0.0106618\pi$
$31$	1995.32i	0.0669774i	−0.999439	+	0.0334887i	$0.989338\pi$
$32$	0	0
$33$	25344.0	0.705234
$34$	0	0
$35$	91452.3i	2.13300i
$36$	0	0
$37$	−11350.0	−0.224074	−0.112037	−	0.993704i	$-0.535737\pi$
$37$	−11350.0	−0.224074	−0.112037	+	0.993704i	$0.535737\pi$
$38$	0	0
$39$	− 4267.77i	− 0.0719461i
$40$	0	0
$41$	67122.0	0.973898	0.486949	−	0.873431i	$-0.338110\pi$
$41$	67122.0	0.973898	0.486949	+	0.873431i	$0.338110\pi$
$42$	0	0
$43$	− 79563.5i	− 1.00071i	−0.865820	−	0.500355i	$-0.833203\pi$
$43$	− 79563.5i	− 1.00071i	0.865820	−	0.500355i	$-0.166797\pi$
$44$	0	0
$45$	5850.00	0.0641975
$46$	0	0
$47$	69503.7i	0.669444i	0.942317	+	0.334722i	$0.108642\pi$
$47$	69503.7i	0.669444i	−0.942317	+	0.334722i	$0.891358\pi$
$48$	0	0
$49$	−254063.	−2.15950
$50$	0	0
$51$	− 206682.i	− 1.55809i
$52$	0	0
$53$	109962.	0.738610	0.369305	−	0.929308i	$-0.379596\pi$
$53$	109962.	0.738610	0.369305	+	0.929308i	$0.379596\pi$
$54$	0	0
$55$	− 137178.i	− 0.824513i
$56$	0	0
$57$	−59136.0	−0.319321
$58$	0	0
$59$	305866.i	1.48928i	0.667468	+	0.744639i	$0.267378\pi$
$59$	305866.i	1.48928i	−0.667468	+	0.744639i	$0.732622\pi$
$60$	0	0
$61$	306746.	1.35142	0.675709	−	0.737169i	$-0.263838\pi$
$61$	306746.	1.35142	0.675709	+	0.737169i	$0.263838\pi$
$62$	0	0
$63$	23777.6i	0.0950925i
$64$	0	0
$65$	−23100.0	−0.0841147
$66$	0	0
$67$	220345.i	0.732619i	0.930493	+	0.366309i	$0.119379\pi$
$67$	220345.i	0.732619i	−0.930493	+	0.366309i	$0.880621\pi$
$68$	0	0
$69$	−235008.	−0.715378
$70$	0	0
$71$	372294.i	1.04019i	0.854110	+	0.520093i	$0.174103\pi$
$71$	372294.i	1.04019i	−0.854110	+	0.520093i	$0.825897\pi$
$72$	0	0
$73$	165682.	0.425899	0.212950	−	0.977063i	$-0.431693\pi$
$73$	165682.	0.425899	0.212950	+	0.977063i	$0.431693\pi$
$74$	0	0
$75$	− 190526.i	− 0.451616i
$76$	0	0
$77$	557568.	1.22131
$78$	0	0
$79$	− 762102.i	− 1.54572i	−0.634574	−	0.772862i	$-0.718824\pi$
$79$	− 762102.i	− 1.54572i	0.634574	−	0.772862i	$-0.281176\pi$
$80$	0	0
$81$	−558351.	−1.05064
$82$	0	0
$83$	− 494757.i	− 0.865282i	−0.901566	−	0.432641i	$-0.857582\pi$
$83$	− 494757.i	− 0.865282i	0.901566	−	0.432641i	$-0.142418\pi$
$84$	0	0
$85$	−1.11870e6	−1.82162
$86$	0	0
$87$	298134.i	0.452746i
$88$	0	0
$89$	471954.	0.669468	0.334734	−	0.942313i	$-0.391354\pi$
$89$	471954.	0.669468	0.334734	+	0.942313i	$0.391354\pi$
$90$	0	0
$91$	− 93891.0i	− 0.124595i
$92$	0	0
$93$	55296.0	0.0687456
$94$	0	0
$95$	320083.i	0.373329i
$96$	0	0
$97$	910594.	0.997722	0.498861	−	0.866682i	$-0.333752\pi$
$97$	910594.	0.997722	0.498861	+	0.866682i	$0.333752\pi$
$98$	0	0
$99$	− 35666.4i	− 0.0367581i
$100$	0	0
$101$	−955350.	−0.927253	−0.463627	−	0.886031i	$-0.653452\pi$
$101$	−955350.	−0.927253	−0.463627	+	0.886031i	$0.653452\pi$
$102$	0	0
$103$	723692.i	0.662281i	0.943581	+	0.331141i	$0.107433\pi$
$103$	723692.i	0.662281i	−0.943581	+	0.331141i	$0.892567\pi$
$104$	0	0
$105$	2.53440e6	2.18931
$106$	0	0
$107$	1.33978e6i	1.09366i	0.837245	+	0.546828i	$0.184165\pi$
$107$	1.33978e6i	1.09366i	−0.837245	+	0.546828i	$0.815835\pi$
$108$	0	0
$109$	−61094.0	−0.0471758	−0.0235879	−	0.999722i	$-0.507509\pi$
$109$	−61094.0	−0.0471758	−0.0235879	+	0.999722i	$0.507509\pi$
$110$	0	0
$111$	314540.i	0.229989i
$112$	0	0
$113$	−2.81609e6	−1.95169	−0.975847	−	0.218454i	$-0.929899\pi$
$113$	−2.81609e6	−1.95169	−0.975847	+	0.218454i	$0.929899\pi$
$114$	0	0
$115$	1.27202e6i	0.836373i
$116$	0	0
$117$	−6006.00	−0.00374997
$118$	0	0
$119$	− 4.54701e6i	− 2.69827i
$120$	0	0
$121$	935209.	0.527901
$122$	0	0
$123$	− 1.86014e6i	− 0.999609i
$124$	0	0
$125$	1.31250e6	0.672000
$126$	0	0
$127$	2.01927e6i	0.985786i	0.870090	+	0.492893i	$0.164061\pi$
$127$	2.01927e6i	0.985786i	−0.870090	+	0.492893i	$0.835939\pi$
$128$	0	0
$129$	−2.20493e6	−1.02713
$130$	0	0
$131$	− 2.81033e6i	− 1.25010i	−0.780586	−	0.625048i	$-0.785079\pi$
$131$	− 2.81033e6i	− 1.25010i	0.780586	−	0.625048i	$-0.214921\pi$
$132$	0	0
$133$	−1.30099e6	−0.552993
$134$	0	0
$135$	2.86828e6i	1.16579i
$136$	0	0
$137$	−1.38536e6	−0.538766	−0.269383	−	0.963033i	$-0.586820\pi$
$137$	−1.38536e6	−0.538766	−0.269383	+	0.963033i	$0.586820\pi$
$138$	0	0
$139$	− 1.58914e6i	− 0.591721i	−0.955231	−	0.295860i	$-0.904394\pi$
$139$	− 1.58914e6i	− 0.591721i	0.955231	−	0.295860i	$-0.0956063\pi$
$140$	0	0
$141$	1.92614e6	0.687118
$142$	0	0
$143$	140837.i	0.0481623i
$144$	0	0
$145$	1.61370e6	0.529321
$146$	0	0
$147$	7.04080e6i	2.21651i
$148$	0	0
$149$	3.11764e6	0.942470	0.471235	−	0.882008i	$-0.343808\pi$
$149$	3.11764e6	0.942470	0.471235	+	0.882008i	$0.343808\pi$
$150$	0	0
$151$	900500.i	0.261549i	0.991412	+	0.130774i	$0.0417464\pi$
$151$	900500.i	0.261549i	−0.991412	+	0.130774i	$0.958254\pi$
$152$	0	0
$153$	−290862.	−0.0812106
$154$	0	0
$155$	− 299298.i	− 0.0803728i
$156$	0	0
$157$	5.33772e6	1.37929	0.689647	−	0.724145i	$-0.257766\pi$
$157$	5.33772e6	1.37929	0.689647	+	0.724145i	$0.257766\pi$
$158$	0	0
$159$	− 3.04736e6i	− 0.758109i
$160$	0	0
$161$	−5.17018e6	−1.23888
$162$	0	0
$163$	− 881295.i	− 0.203497i	−0.994810	−	0.101749i	$-0.967556\pi$
$163$	− 881295.i	− 0.203497i	0.994810	−	0.101749i	$-0.0324437\pi$
$164$	0	0
$165$	−3.80160e6	−0.846281
$166$	0	0
$167$	1.71382e6i	0.367972i	0.982929	+	0.183986i	$0.0589001\pi$
$167$	1.71382e6i	0.367972i	−0.982929	+	0.183986i	$0.941100\pi$
$168$	0	0
$169$	−4.80309e6	−0.995087
$170$	0	0
$171$	83221.6i	0.0166436i
$172$	0	0
$173$	−1.02135e6	−0.197259	−0.0986294	−	0.995124i	$-0.531446\pi$
$173$	−1.02135e6	−0.197259	−0.0986294	+	0.995124i	$0.531446\pi$
$174$	0	0
$175$	− 4.19156e6i	− 0.782099i
$176$	0	0
$177$	8.47642e6	1.52860
$178$	0	0
$179$	− 8.58978e6i	− 1.49769i	−0.662743	−	0.748847i	$-0.730608\pi$
$179$	− 8.58978e6i	− 1.49769i	0.662743	−	0.748847i	$-0.269392\pi$
$180$	0	0
$181$	−2.69797e6	−0.454990	−0.227495	−	0.973779i	$-0.573054\pi$
$181$	−2.69797e6	−0.454990	−0.227495	+	0.973779i	$0.573054\pi$
$182$	0	0
$183$	− 8.50079e6i	− 1.38710i
$184$	0	0
$185$	1.70250e6	0.268888
$186$	0	0
$187$	6.82051e6i	1.04302i
$188$	0	0
$189$	−1.16582e7	−1.72682
$190$	0	0
$191$	9.60016e6i	1.37778i	0.724868	+	0.688888i	$0.241901\pi$
$191$	9.60016e6i	1.37778i	−0.724868	+	0.688888i	$0.758099\pi$
$192$	0	0
$193$	5.35891e6	0.745427	0.372713	−	0.927947i	$-0.378428\pi$
$193$	5.35891e6	0.745427	0.372713	+	0.927947i	$0.378428\pi$
$194$	0	0
$195$	640166.i	0.0863354i
$196$	0	0
$197$	8.00587e6	1.04715	0.523576	−	0.851979i	$-0.324598\pi$
$197$	8.00587e6	1.04715	0.523576	+	0.851979i	$0.324598\pi$
$198$	0	0
$199$	9.17388e6i	1.16411i	0.813149	+	0.582055i	$0.197751\pi$
$199$	9.17388e6i	1.16411i	−0.813149	+	0.582055i	$0.802249\pi$
$200$	0	0
$201$	6.10637e6	0.751960
$202$	0	0
$203$	6.55896e6i	0.784055i
$204$	0	0
$205$	−1.00683e7	−1.16868
$206$	0	0
$207$	330725.i	0.0372868i
$208$	0	0
$209$	1.95149e6	0.213760
$210$	0	0
$211$	− 1.23275e7i	− 1.31228i	−0.754639	−	0.656140i	$-0.772188\pi$
$211$	− 1.23275e7i	− 1.31228i	0.754639	−	0.656140i	$-0.227812\pi$
$212$	0	0
$213$	1.03173e7	1.06765
$214$	0	0
$215$	1.19345e7i	1.20085i
$216$	0	0
$217$	1.21651e6	0.119052
$218$	0	0
$219$	− 4.59151e6i	− 0.437143i
$220$	0	0
$221$	1.14853e6	0.106406
$222$	0	0
$223$	− 1.76726e7i	− 1.59362i	−0.604229	−	0.796811i	$-0.706519\pi$
$223$	− 1.76726e7i	− 1.59362i	0.604229	−	0.796811i	$-0.293481\pi$
$224$	0	0
$225$	−268125.	−0.0235391
$226$	0	0
$227$	1.93650e7i	1.65554i	0.561066	+	0.827771i	$0.310392\pi$
$227$	1.93650e7i	1.65554i	−0.561066	+	0.827771i	$0.689608\pi$
$228$	0	0
$229$	−1.60536e7	−1.33680	−0.668399	−	0.743803i	$-0.733020\pi$
$229$	−1.60536e7	−1.33680	−0.668399	+	0.743803i	$0.733020\pi$
$230$	0	0
$231$	− 1.54518e7i	− 1.25355i
$232$	0	0
$233$	−1.68901e7	−1.33526	−0.667629	−	0.744494i	$-0.732691\pi$
$233$	−1.68901e7	−1.33526	−0.667629	+	0.744494i	$0.732691\pi$
$234$	0	0
$235$	− 1.04256e7i	− 0.803333i
$236$	0	0
$237$	−2.11200e7	−1.58653
$238$	0	0
$239$	− 1.11243e7i	− 0.814849i	−0.913239	−	0.407425i	$-0.866427\pi$
$239$	− 1.11243e7i	− 0.814849i	0.913239	−	0.407425i	$-0.133573\pi$
$240$	0	0
$241$	1.69703e7	1.21238	0.606191	−	0.795319i	$-0.292697\pi$
$241$	1.69703e7	1.21238	0.606191	+	0.795319i	$0.292697\pi$
$242$	0	0
$243$	1.53365e6i	0.106883i
$244$	0	0
$245$	3.81094e7	2.59140
$246$	0	0
$247$	− 328619.i	− 0.0218073i
$248$	0	0
$249$	−1.37111e7	−0.888126
$250$	0	0
$251$	− 655879.i	− 0.0414766i	−0.999785	−	0.0207383i	$-0.993398\pi$
$251$	− 655879.i	− 0.0414766i	0.999785	−	0.0207383i	$-0.00660167\pi$
$252$	0	0
$253$	7.75526e6	0.478889
$254$	0	0
$255$	3.10023e7i	1.86971i
$256$	0	0
$257$	−3.34592e6	−0.197113	−0.0985566	−	0.995131i	$-0.531423\pi$
$257$	−3.34592e6	−0.197113	−0.0985566	+	0.995131i	$0.531423\pi$
$258$	0	0
$259$	6.91989e6i	0.398290i
$260$	0	0
$261$	419562.	0.0235980
$262$	0	0
$263$	1.96988e6i	0.108286i	0.998533	+	0.0541431i	$0.0172427\pi$
$263$	1.96988e6i	0.108286i	−0.998533	+	0.0541431i	$0.982757\pi$
$264$	0	0
$265$	−1.64943e7	−0.886332
$266$	0	0
$267$	− 1.30792e7i	− 0.687142i
$268$	0	0
$269$	2.72658e7	1.40075	0.700377	−	0.713773i	$-0.253015\pi$
$269$	2.72658e7	1.40075	0.700377	+	0.713773i	$0.253015\pi$
$270$	0	0
$271$	3.22924e7i	1.62253i	0.584679	+	0.811265i	$0.301220\pi$
$271$	3.22924e7i	1.62253i	−0.584679	+	0.811265i	$0.698780\pi$
$272$	0	0
$273$	−2.60198e6	−0.127884
$274$	0	0
$275$	6.28734e6i	0.302322i
$276$	0	0
$277$	1.07054e7	0.503688	0.251844	−	0.967768i	$-0.418963\pi$
$277$	1.07054e7	0.503688	0.251844	+	0.967768i	$0.418963\pi$
$278$	0	0
$279$	− 77817.6i	− 0.00358315i
$280$	0	0
$281$	−2.81925e7	−1.27062	−0.635308	−	0.772259i	$-0.719127\pi$
$281$	−2.81925e7	−1.27062	−0.635308	+	0.772259i	$0.719127\pi$
$282$	0	0
$283$	1.30127e7i	0.574129i	0.957911	+	0.287065i	$0.0926794\pi$
$283$	1.30127e7i	0.574129i	−0.957911	+	0.287065i	$0.907321\pi$
$284$	0	0
$285$	8.87040e6	0.383185
$286$	0	0
$287$	− 4.09231e7i	− 1.73110i
$288$	0	0
$289$	3.14842e7	1.30436
$290$	0	0
$291$	− 2.52351e7i	− 1.02406i
$292$	0	0
$293$	−2.95772e7	−1.17586	−0.587928	−	0.808914i	$-0.700056\pi$
$293$	−2.95772e7	−1.17586	−0.587928	+	0.808914i	$0.700056\pi$
$294$	0	0
$295$	− 4.58799e7i	− 1.78713i
$296$	0	0
$297$	1.74874e7	0.667506
$298$	0	0
$299$	− 1.30594e6i	− 0.0488550i
$300$	0	0
$301$	−4.85084e7	−1.77876
$302$	0	0
$303$	2.64754e7i	0.951733i
$304$	0	0
$305$	−4.60119e7	−1.62170
$306$	0	0
$307$	4.85337e6i	0.167737i	0.996477	+	0.0838684i	$0.0267275\pi$
$307$	4.85337e6i	0.167737i	−0.996477	+	0.0838684i	$0.973272\pi$
$308$	0	0
$309$	2.00556e7	0.679766
$310$	0	0
$311$	1.91593e7i	0.636938i	0.947933	+	0.318469i	$0.103169\pi$
$311$	1.91593e7i	0.636938i	−0.947933	+	0.318469i	$0.896831\pi$
$312$	0	0
$313$	1.76042e7	0.574096	0.287048	−	0.957916i	$-0.407326\pi$
$313$	1.76042e7	0.574096	0.287048	+	0.957916i	$0.407326\pi$
$314$	0	0
$315$	− 3.56664e6i	− 0.114111i
$316$	0	0
$317$	−4.72378e6	−0.148290	−0.0741450	−	0.997247i	$-0.523623\pi$
$317$	−4.72378e6	−0.148290	−0.0741450	+	0.997247i	$0.523623\pi$
$318$	0	0
$319$	− 9.83844e6i	− 0.303078i
$320$	0	0
$321$	3.71290e7	1.12253
$322$	0	0
$323$	− 1.59145e7i	− 0.472265i
$324$	0	0
$325$	1.05875e6	0.0308421
$326$	0	0
$327$	1.69309e6i	0.0484212i
$328$	0	0
$329$	4.23752e7	1.18994
$330$	0	0
$331$	5.68459e7i	1.56753i	0.621059	+	0.783763i	$0.286703\pi$
$331$	5.68459e7i	1.56753i	−0.621059	+	0.783763i	$0.713297\pi$
$332$	0	0
$333$	442650.	0.0119875
$334$	0	0
$335$	− 3.30517e7i	− 0.879142i
$336$	0	0
$337$	−2.60511e7	−0.680670	−0.340335	−	0.940304i	$-0.610540\pi$
$337$	−2.60511e7	−0.680670	−0.340335	+	0.940304i	$0.610540\pi$
$338$	0	0
$339$	7.80419e7i	2.00322i
$340$	0	0
$341$	−1.82477e6	−0.0460198
$342$	0	0
$343$	8.31691e7i	2.06101i
$344$	0	0
$345$	3.52512e7	0.858453
$346$	0	0
$347$	− 6.90309e7i	− 1.65217i	−0.563544	−	0.826086i	$-0.690562\pi$
$347$	− 6.90309e7i	− 1.65217i	0.563544	−	0.826086i	$-0.309438\pi$
$348$	0	0
$349$	2.06010e7	0.484631	0.242315	−	0.970198i	$-0.422093\pi$
$349$	2.06010e7	0.484631	0.242315	+	0.970198i	$0.422093\pi$
$350$	0	0
$351$	− 2.94476e6i	− 0.0680972i
$352$	0	0
$353$	−9.92173e6	−0.225561	−0.112780	−	0.993620i	$-0.535976\pi$
$353$	−9.92173e6	−0.225561	−0.112780	+	0.993620i	$0.535976\pi$
$354$	0	0
$355$	− 5.58441e7i	− 1.24822i
$356$	0	0
$357$	−1.26010e8	−2.76950
$358$	0	0
$359$	2.24771e7i	0.485800i	0.970051	+	0.242900i	$0.0780987\pi$
$359$	2.24771e7i	0.485800i	−0.970051	+	0.242900i	$0.921901\pi$
$360$	0	0
$361$	4.24924e7	0.903212
$362$	0	0
$363$	− 2.59173e7i	− 0.541838i
$364$	0	0
$365$	−2.48523e7	−0.511079
$366$	0	0
$367$	− 3.45032e7i	− 0.698010i	−0.937121	−	0.349005i	$-0.886520\pi$
$367$	− 3.45032e7i	− 0.698010i	0.937121	−	0.349005i	$-0.113480\pi$
$368$	0	0
$369$	−2.61776e6	−0.0521015
$370$	0	0
$371$	− 6.70418e7i	− 1.31288i
$372$	0	0
$373$	7.04574e7	1.35769	0.678844	−	0.734283i	$-0.262481\pi$
$373$	7.04574e7	1.35769	0.678844	+	0.734283i	$0.262481\pi$
$374$	0	0
$375$	− 3.63731e7i	− 0.689741i
$376$	0	0
$377$	−1.65673e6	−0.0309192
$378$	0	0
$379$	4.83111e7i	0.887420i	0.896170	+	0.443710i	$0.146338\pi$
$379$	4.83111e7i	0.887420i	−0.896170	+	0.443710i	$0.853662\pi$
$380$	0	0
$381$	5.59596e7	1.01181
$382$	0	0
$383$	− 8.40430e6i	− 0.149591i	−0.997199	−	0.0747954i	$-0.976170\pi$
$383$	− 8.40430e6i	− 0.149591i	0.997199	−	0.0747954i	$-0.0238304\pi$
$384$	0	0
$385$	−8.36352e7	−1.46557
$386$	0	0
$387$	3.10298e6i	0.0535360i
$388$	0	0
$389$	−8.57915e7	−1.45746	−0.728728	−	0.684803i	$-0.759888\pi$
$389$	−8.57915e7	−1.45746	−0.728728	+	0.684803i	$0.759888\pi$
$390$	0	0
$391$	− 6.32447e7i	− 1.05802i
$392$	0	0
$393$	−7.78821e7	−1.28310
$394$	0	0
$395$	1.14315e8i	1.85487i
$396$	0	0
$397$	5.94256e7	0.949734	0.474867	−	0.880058i	$-0.342496\pi$
$397$	5.94256e7	0.949734	0.474867	+	0.880058i	$0.342496\pi$
$398$	0	0
$399$	3.60541e7i	0.567592i
$400$	0	0
$401$	3.24424e7	0.503130	0.251565	−	0.967840i	$-0.419055\pi$
$401$	3.24424e7	0.503130	0.251565	+	0.967840i	$0.419055\pi$
$402$	0	0
$403$	307280.i	0.00469482i
$404$	0	0
$405$	8.37526e7	1.26076
$406$	0	0
$407$	− 1.03798e7i	− 0.153960i
$408$	0	0
$409$	1.01480e7	0.148324	0.0741622	−	0.997246i	$-0.476372\pi$
$409$	1.01480e7	0.148324	0.0741622	+	0.997246i	$0.476372\pi$
$410$	0	0
$411$	3.83922e7i	0.552990i
$412$	0	0
$413$	1.86481e8	2.64719
$414$	0	0
$415$	7.42135e7i	1.03834i
$416$	0	0
$417$	−4.40394e7	−0.607342
$418$	0	0
$419$	8.85907e7i	1.20433i	0.798371	+	0.602166i	$0.205695\pi$
$419$	8.85907e7i	1.20433i	−0.798371	+	0.602166i	$0.794305\pi$
$420$	0	0
$421$	−1.09866e8	−1.47237	−0.736185	−	0.676781i	$-0.763375\pi$
$421$	−1.09866e8	−1.47237	−0.736185	+	0.676781i	$0.763375\pi$
$422$	0	0
$423$	− 2.71065e6i	− 0.0358139i
$424$	0	0
$425$	5.12738e7	0.667926
$426$	0	0
$427$	− 1.87017e8i	− 2.40214i
$428$	0	0
$429$	3.90298e6	0.0494338
$430$	0	0
$431$	7.26168e7i	0.906995i	0.891257	+	0.453498i	$0.149824\pi$
$431$	7.26168e7i	0.906995i	−0.891257	+	0.453498i	$0.850176\pi$
$432$	0	0
$433$	−6.49224e7	−0.799707	−0.399853	−	0.916579i	$-0.630939\pi$
$433$	−6.49224e7	−0.799707	−0.399853	+	0.916579i	$0.630939\pi$
$434$	0	0
$435$	− 4.47202e7i	− 0.543295i
$436$	0	0
$437$	−1.80956e7	−0.216835
$438$	0	0
$439$	1.91885e7i	0.226803i	0.993549	+	0.113401i	$0.0361746\pi$
$439$	1.91885e7i	0.226803i	−0.993549	+	0.113401i	$0.963825\pi$
$440$	0	0
$441$	9.90846e6	0.115529
$442$	0	0
$443$	− 8.64648e6i	− 0.0994554i	−0.998763	−	0.0497277i	$-0.984165\pi$
$443$	− 8.64648e6i	− 0.0994554i	0.998763	−	0.0497277i	$-0.0158353\pi$
$444$	0	0
$445$	−7.07931e7	−0.803361
$446$	0	0
$447$	− 8.63986e7i	− 0.967351i
$448$	0	0
$449$	−5.59372e7	−0.617962	−0.308981	−	0.951068i	$-0.599988\pi$
$449$	−5.59372e7	−0.617962	−0.308981	+	0.951068i	$0.599988\pi$
$450$	0	0
$451$	6.13846e7i	0.669160i
$452$	0	0
$453$	2.49554e7	0.268454
$454$	0	0
$455$	1.40837e7i	0.149514i
$456$	0	0
$457$	−3.24097e6	−0.0339568	−0.0169784	−	0.999856i	$-0.505405\pi$
$457$	−3.24097e6	−0.0339568	−0.0169784	+	0.999856i	$0.505405\pi$
$458$	0	0
$459$	− 1.42611e8i	− 1.47474i
$460$	0	0
$461$	−6.91355e7	−0.705664	−0.352832	−	0.935687i	$-0.614781\pi$
$461$	−6.91355e7	−0.705664	−0.352832	+	0.935687i	$0.614781\pi$
$462$	0	0
$463$	− 1.10064e8i	− 1.10893i	−0.832208	−	0.554463i	$-0.812924\pi$
$463$	− 1.10064e8i	− 1.10893i	0.832208	−	0.554463i	$-0.187076\pi$
$464$	0	0
$465$	−8.29440e6	−0.0824947
$466$	0	0
$467$	1.10914e7i	0.108902i	0.998516	+	0.0544511i	$0.0173409\pi$
$467$	1.10914e7i	0.108902i	−0.998516	+	0.0544511i	$0.982659\pi$
$468$	0	0
$469$	1.34340e8	1.30223
$470$	0	0
$471$	− 1.47923e8i	− 1.41571i
$472$	0	0
$473$	7.27626e7	0.687583
$474$	0	0
$475$	− 1.46705e7i	− 0.136887i
$476$	0	0
$477$	−4.28852e6	−0.0395141
$478$	0	0
$479$	1.44736e8i	1.31695i	0.752601	+	0.658476i	$0.228799\pi$
$479$	1.44736e8i	1.31695i	−0.752601	+	0.658476i	$0.771201\pi$
$480$	0	0
$481$	−1.74790e6	−0.0157066
$482$	0	0
$483$	1.43280e8i	1.27158i
$484$	0	0
$485$	−1.36589e8	−1.19727
$486$	0	0
$487$	2.11294e8i	1.82937i	0.404173	+	0.914683i	$0.367559\pi$
$487$	2.11294e8i	1.82937i	−0.404173	+	0.914683i	$0.632441\pi$
$488$	0	0
$489$	−2.44232e7	−0.208870
$490$	0	0
$491$	1.35050e8i	1.14091i	0.821329	+	0.570455i	$0.193233\pi$
$491$	1.35050e8i	1.14091i	−0.821329	+	0.570455i	$0.806767\pi$
$492$	0	0
$493$	−8.02332e7	−0.669596
$494$	0	0
$495$	5.34996e6i	0.0441098i
$496$	0	0
$497$	2.26981e8	1.84893
$498$	0	0
$499$	− 8.35993e7i	− 0.672823i	−0.941715	−	0.336412i	$-0.890787\pi$
$499$	− 8.35993e7i	− 0.672823i	0.941715	−	0.336412i	$-0.109213\pi$
$500$	0	0
$501$	4.74947e7	0.377687
$502$	0	0
$503$	− 6.44940e7i	− 0.506775i	−0.967365	−	0.253388i	$-0.918455\pi$
$503$	− 6.44940e7i	− 0.506775i	0.967365	−	0.253388i	$-0.0815448\pi$
$504$	0	0
$505$	1.43302e8	1.11270
$506$	0	0
$507$	1.33107e8i	1.02136i
$508$	0	0
$509$	−4.99175e7	−0.378529	−0.189265	−	0.981926i	$-0.560610\pi$
$509$	−4.99175e7	−0.378529	−0.189265	+	0.981926i	$0.560610\pi$
$510$	0	0
$511$	− 1.01013e8i	− 0.757035i
$512$	0	0
$513$	−4.08038e7	−0.302238
$514$	0	0
$515$	− 1.08554e8i	− 0.794737i
$516$	0	0
$517$	−6.35628e7	−0.459972
$518$	0	0
$519$	2.83045e7i	0.202466i
$520$	0	0
$521$	1.45291e8	1.02737	0.513684	−	0.857980i	$-0.328280\pi$
$521$	1.45291e8	1.02737	0.513684	+	0.857980i	$0.328280\pi$
$522$	0	0
$523$	1.28688e8i	0.899564i	0.893138	+	0.449782i	$0.148498\pi$
$523$	1.28688e8i	0.899564i	−0.893138	+	0.449782i	$0.851502\pi$
$524$	0	0
$525$	−1.16160e8	−0.802747
$526$	0	0
$527$	1.48811e7i	0.101673i
$528$	0	0
$529$	7.61234e7	0.514223
$530$	0	0
$531$	− 1.19288e7i	− 0.0796733i
$532$	0	0
$533$	1.03368e7	0.0682659
$534$	0	0
$535$	− 2.00966e8i	− 1.31239i
$536$	0	0
$537$	−2.38047e8	−1.53723
$538$	0	0
$539$	− 2.32346e8i	− 1.48378i
$540$	0	0
$541$	−4.07372e7	−0.257276	−0.128638	−	0.991692i	$-0.541060\pi$
$541$	−4.07372e7	−0.257276	−0.128638	+	0.991692i	$0.541060\pi$
$542$	0	0
$543$	7.47684e7i	0.467002i
$544$	0	0
$545$	9.16410e6	0.0566109
$546$	0	0
$547$	− 1.10887e8i	− 0.677517i	−0.940873	−	0.338759i	$-0.889993\pi$
$547$	− 1.10887e8i	− 0.677517i	0.940873	−	0.338759i	$-0.110007\pi$
$548$	0	0
$549$	−1.19631e7	−0.0722980
$550$	0	0
$551$	2.29564e7i	0.137230i
$552$	0	0
$553$	−4.64640e8	−2.74752
$554$	0	0
$555$	− 4.71811e7i	− 0.275987i
$556$	0	0
$557$	−5.78426e7	−0.334720	−0.167360	−	0.985896i	$-0.553524\pi$
$557$	−5.78426e7	−0.334720	−0.167360	+	0.985896i	$0.553524\pi$
$558$	0	0
$559$	− 1.22528e7i	− 0.0701454i
$560$	0	0
$561$	1.89016e8	1.07055
$562$	0	0
$563$	− 2.25694e8i	− 1.26472i	−0.774674	−	0.632361i	$-0.782086\pi$
$563$	− 2.25694e8i	− 1.26472i	0.774674	−	0.632361i	$-0.217914\pi$
$564$	0	0
$565$	4.22414e8	2.34203
$566$	0	0
$567$	3.40416e8i	1.86750i
$568$	0	0
$569$	−6.40162e7	−0.347499	−0.173749	−	0.984790i	$-0.555588\pi$
$569$	−6.40162e7	−0.347499	−0.173749	+	0.984790i	$0.555588\pi$
$570$	0	0
$571$	1.46301e8i	0.785851i	0.919570	+	0.392925i	$0.128537\pi$
$571$	1.46301e8i	0.785851i	−0.919570	+	0.392925i	$0.871463\pi$
$572$	0	0
$573$	2.66047e8	1.41415
$574$	0	0
$575$	− 5.83008e7i	− 0.306670i
$576$	0	0
$577$	2.68202e8	1.39616	0.698079	−	0.716020i	$-0.254038\pi$
$577$	2.68202e8	1.39616	0.698079	+	0.716020i	$0.254038\pi$
$578$	0	0
$579$	− 1.48511e8i	− 0.765106i
$580$	0	0
$581$	−3.01644e8	−1.53804
$582$	0	0
$583$	1.00563e8i	0.507495i
$584$	0	0
$585$	900900.	0.00449996
$586$	0	0
$587$	3.26903e8i	1.61623i	0.589022	+	0.808117i	$0.299513\pi$
$587$	3.26903e8i	1.61623i	−0.589022	+	0.808117i	$0.700487\pi$
$588$	0	0
$589$	4.25779e6	0.0208372
$590$	0	0
$591$	− 2.21865e8i	− 1.07480i
$592$	0	0
$593$	−3.17792e8	−1.52398	−0.761988	−	0.647591i	$-0.775777\pi$
$593$	−3.17792e8	−1.52398	−0.761988	+	0.647591i	$0.775777\pi$
$594$	0	0
$595$	6.82051e8i	3.23792i
$596$	0	0
$597$	2.54234e8	1.19484
$598$	0	0
$599$	− 2.85128e8i	− 1.32666i	−0.748327	−	0.663330i	$-0.769143\pi$
$599$	− 2.85128e8i	− 1.32666i	0.748327	−	0.663330i	$-0.230857\pi$
$600$	0	0
$601$	−1.83806e8	−0.846713	−0.423357	−	0.905963i	$-0.639148\pi$
$601$	−1.83806e8	−0.846713	−0.423357	+	0.905963i	$0.639148\pi$
$602$	0	0
$603$	− 8.59344e6i	− 0.0391936i
$604$	0	0
$605$	−1.40281e8	−0.633481
$606$	0	0
$607$	1.24826e8i	0.558136i	0.960271	+	0.279068i	$0.0900255\pi$
$607$	1.24826e8i	0.558136i	−0.960271	+	0.279068i	$0.909975\pi$
$608$	0	0
$609$	1.81767e8	0.804755
$610$	0	0
$611$	1.07036e7i	0.0469251i
$612$	0	0
$613$	4.13232e8	1.79396	0.896979	−	0.442073i	$-0.145757\pi$
$613$	4.13232e8	1.79396	0.896979	+	0.442073i	$0.145757\pi$
$614$	0	0
$615$	2.79021e8i	1.19953i
$616$	0	0
$617$	2.80810e8	1.19552	0.597760	−	0.801675i	$-0.296058\pi$
$617$	2.80810e8	1.19552	0.597760	+	0.801675i	$0.296058\pi$
$618$	0	0
$619$	− 3.94504e8i	− 1.66333i	−0.555275	−	0.831667i	$-0.687387\pi$
$619$	− 3.94504e8i	− 1.66333i	0.555275	−	0.831667i	$-0.312613\pi$
$620$	0	0
$621$	−1.62156e8	−0.677106
$622$	0	0
$623$	− 2.87742e8i	− 1.18998i
$624$	0	0
$625$	−3.04297e8	−1.24640
$626$	0	0
$627$	− 5.40812e7i	− 0.219404i
$628$	0	0
$629$	−8.46483e7	−0.340147
$630$	0	0
$631$	1.62705e7i	0.0647608i	0.999476	+	0.0323804i	$0.0103088\pi$
$631$	1.62705e7i	0.0647608i	−0.999476	+	0.0323804i	$0.989691\pi$
$632$	0	0
$633$	−3.41629e8	−1.34692
$634$	0	0
$635$	− 3.02890e8i	− 1.18294i
$636$	0	0
$637$	−3.91257e7	−0.151371
$638$	0	0
$639$	− 1.45195e7i	− 0.0556478i
$640$	0	0
$641$	5.98983e7	0.227426	0.113713	−	0.993514i	$-0.463726\pi$
$641$	5.98983e7	0.227426	0.113713	+	0.993514i	$0.463726\pi$
$642$	0	0
$643$	− 2.84603e8i	− 1.07055i	−0.844678	−	0.535275i	$-0.820208\pi$
$643$	− 2.84603e8i	− 1.07055i	0.844678	−	0.535275i	$-0.179792\pi$
$644$	0	0
$645$	3.30739e8	1.23256
$646$	0	0
$647$	− 3.15576e8i	− 1.16518i	−0.812768	−	0.582588i	$-0.802040\pi$
$647$	− 3.15576e8i	− 1.16518i	0.812768	−	0.582588i	$-0.197960\pi$
$648$	0	0
$649$	−2.79722e8	−1.02327
$650$	0	0
$651$	− 3.37130e7i	− 0.122195i
$652$	0	0
$653$	−5.41537e7	−0.194486	−0.0972431	−	0.995261i	$-0.531002\pi$
$653$	−5.41537e7	−0.194486	−0.0972431	+	0.995261i	$0.531002\pi$
$654$	0	0
$655$	4.21549e8i	1.50011i
$656$	0	0
$657$	−6.46160e6	−0.0227847
$658$	0	0
$659$	3.29333e8i	1.15075i	0.817891	+	0.575373i	$0.195143\pi$
$659$	3.29333e8i	1.15075i	−0.817891	+	0.575373i	$0.804857\pi$
$660$	0	0
$661$	−3.27505e8	−1.13400	−0.567001	−	0.823717i	$-0.691896\pi$
$661$	−3.27505e8	−1.13400	−0.567001	+	0.823717i	$0.691896\pi$
$662$	0	0
$663$	− 3.18291e7i	− 0.109215i
$664$	0	0
$665$	1.95149e8	0.663592
$666$	0	0
$667$	9.12291e7i	0.307437i
$668$	0	0
$669$	−4.89757e8	−1.63569
$670$	0	0
$671$	2.80526e8i	0.928551i
$672$	0	0
$673$	3.90277e8	1.28035	0.640174	−	0.768230i	$-0.278862\pi$
$673$	3.90277e8	1.28035	0.640174	+	0.768230i	$0.278862\pi$
$674$	0	0
$675$	− 1.31463e8i	− 0.427456i
$676$	0	0
$677$	5.32174e7	0.171509	0.0857547	−	0.996316i	$-0.472670\pi$
$677$	5.32174e7	0.171509	0.0857547	+	0.996316i	$0.472670\pi$
$678$	0	0
$679$	− 5.55173e8i	− 1.77345i
$680$	0	0
$681$	5.36659e8	1.69925
$682$	0	0
$683$	1.05420e8i	0.330872i	0.986221	+	0.165436i	$0.0529031\pi$
$683$	1.05420e8i	0.330872i	−0.986221	+	0.165436i	$0.947097\pi$
$684$	0	0
$685$	2.07804e8	0.646519
$686$	0	0
$687$	4.44890e8i	1.37209i
$688$	0	0
$689$	1.69341e7	0.0517733
$690$	0	0
$691$	− 2.74005e8i	− 0.830471i	−0.909714	−	0.415235i	$-0.863699\pi$
$691$	− 2.74005e8i	− 0.830471i	0.909714	−	0.415235i	$-0.136301\pi$
$692$	0	0
$693$	−2.17452e7	−0.0653375
$694$	0	0
$695$	2.38370e8i	0.710065i
$696$	0	0
$697$	5.00596e8	1.47839
$698$	0	0
$699$	4.68073e8i	1.37051i
$700$	0	0
$701$	1.51827e8	0.440753	0.220377	−	0.975415i	$-0.429271\pi$
$701$	1.51827e8	0.440753	0.220377	+	0.975415i	$0.429271\pi$
$702$	0	0
$703$	2.42196e7i	0.0697110i
$704$	0	0
$705$	−2.88922e8	−0.824542
$706$	0	0
$707$	5.82460e8i	1.64819i
$708$	0	0
$709$	−6.05603e8	−1.69922	−0.849610	−	0.527412i	$-0.823162\pi$
$709$	−6.05603e8	−1.69922	−0.849610	+	0.527412i	$0.823162\pi$
$710$	0	0
$711$	2.97220e7i	0.0826931i
$712$	0	0
$713$	1.69206e7	0.0466817
$714$	0	0
$715$	− 2.11255e7i	− 0.0577948i
$716$	0	0
$717$	−3.08284e8	−0.836362
$718$	0	0
$719$	3.81343e7i	0.102596i	0.998683	+	0.0512978i	$0.0163358\pi$
$719$	3.81343e7i	0.102596i	−0.998683	+	0.0512978i	$0.983664\pi$
$720$	0	0
$721$	4.41222e8	1.17720
$722$	0	0
$723$	− 4.70296e8i	− 1.24439i
$724$	0	0
$725$	−7.39612e7	−0.194084
$726$	0	0
$727$	− 3.15705e7i	− 0.0821633i	−0.999156	−	0.0410817i	$-0.986920\pi$
$727$	− 3.15705e7i	− 0.0821633i	0.999156	−	0.0410817i	$-0.0130804\pi$
$728$	0	0
$729$	−3.64536e8	−0.940931
$730$	0	0
$731$	− 5.93384e8i	− 1.51909i
$732$	0	0
$733$	2.41122e8	0.612244	0.306122	−	0.951992i	$-0.400968\pi$
$733$	2.41122e8	0.612244	0.306122	+	0.951992i	$0.400968\pi$
$734$	0	0
$735$	− 1.05612e9i	− 2.65981i
$736$	0	0
$737$	−2.01510e8	−0.503378
$738$	0	0
$739$	4.74607e8i	1.17598i	0.808867	+	0.587991i	$0.200081\pi$
$739$	4.74607e8i	1.17598i	−0.808867	+	0.587991i	$0.799919\pi$
$740$	0	0
$741$	−9.10694e6	−0.0223830
$742$	0	0
$743$	3.18139e8i	0.775622i	0.921739	+	0.387811i	$0.126769\pi$
$743$	3.18139e8i	0.775622i	−0.921739	+	0.387811i	$0.873231\pi$
$744$	0	0
$745$	−4.67646e8	−1.13096
$746$	0	0
$747$	1.92955e7i	0.0462908i
$748$	0	0
$749$	8.16837e8	1.94397
$750$	0	0
$751$	− 3.81587e8i	− 0.900895i	−0.892803	−	0.450447i	$-0.851264\pi$
$751$	− 3.81587e8i	− 0.900895i	0.892803	−	0.450447i	$-0.148736\pi$
$752$	0	0
$753$	−1.81763e7	−0.0425716
$754$	0	0
$755$	− 1.35075e8i	− 0.313859i
$756$	0	0
$757$	−7.36714e8	−1.69829	−0.849144	−	0.528161i	$-0.822882\pi$
$757$	−7.36714e8	−1.69829	−0.849144	+	0.528161i	$0.822882\pi$
$758$	0	0
$759$	− 2.14920e8i	− 0.491532i
$760$	0	0
$761$	3.30915e8	0.750866	0.375433	−	0.926849i	$-0.377494\pi$
$761$	3.30915e8	0.750866	0.375433	+	0.926849i	$0.377494\pi$
$762$	0	0
$763$	3.72479e7i	0.0838549i
$764$	0	0
$765$	4.36293e7	0.0974527
$766$	0	0
$767$	4.71034e7i	0.104392i
$768$	0	0
$769$	6.68947e6	0.0147100	0.00735500	−	0.999973i	$-0.497659\pi$
$769$	6.68947e6	0.0147100	0.00735500	+	0.999973i	$0.497659\pi$
$770$	0	0
$771$	9.27248e7i	0.202317i
$772$	0	0
$773$	4.18447e7	0.0905946	0.0452973	−	0.998974i	$-0.485576\pi$
$773$	4.18447e7	0.0905946	0.0452973	+	0.998974i	$0.485576\pi$
$774$	0	0
$775$	1.37178e7i	0.0294700i
$776$	0	0
$777$	1.91770e8	0.408805
$778$	0	0
$779$	− 1.43231e8i	− 0.302987i
$780$	0	0
$781$	−3.40471e8	−0.714706
$782$	0	0
$783$	2.05713e8i	0.428525i
$784$	0	0
$785$	−8.00658e8	−1.65515
$786$	0	0
$787$	− 6.21936e8i	− 1.27591i	−0.770072	−	0.637957i	$-0.779780\pi$
$787$	− 6.21936e8i	− 1.27591i	0.770072	−	0.637957i	$-0.220220\pi$
$788$	0	0
$789$	5.45910e7	0.111145
$790$	0	0
$791$	1.71692e9i	3.46913i
$792$	0	0
$793$	4.72389e7	0.0947284
$794$	0	0
$795$	4.57103e8i	0.909731i
$796$	0	0
$797$	6.72512e8	1.32839	0.664194	−	0.747560i	$-0.268775\pi$
$797$	6.72512e8	1.32839	0.664194	+	0.747560i	$0.268775\pi$
$798$	0	0
$799$	5.18359e8i	1.01623i
$800$	0	0
$801$	−1.84062e7	−0.0358151
$802$	0	0
$803$	1.51520e8i	0.292633i
$804$	0	0
$805$	7.75526e8	1.48665
$806$	0	0
$807$	− 7.55613e8i	− 1.43773i
$808$	0	0
$809$	6.87717e8	1.29887	0.649433	−	0.760419i	$-0.275006\pi$
$809$	6.87717e8	1.29887	0.649433	+	0.760419i	$0.275006\pi$
$810$	0	0
$811$	− 1.91471e8i	− 0.358955i	−0.983762	−	0.179478i	$-0.942559\pi$
$811$	− 1.91471e8i	− 0.358955i	0.983762	−	0.179478i	$-0.0574407\pi$
$812$	0	0
$813$	8.94914e8	1.66537
$814$	0	0
$815$	1.32194e8i	0.244197i
$816$	0	0
$817$	−1.69779e8	−0.311329
$818$	0	0
$819$	3.66175e6i	0.00666556i
$820$	0	0
$821$	−5.47707e8	−0.989734	−0.494867	−	0.868969i	$-0.664783\pi$
$821$	−5.47707e8	−0.989734	−0.494867	+	0.868969i	$0.664783\pi$
$822$	0	0
$823$	− 3.65856e7i	− 0.0656313i	−0.999461	−	0.0328156i	$-0.989553\pi$
$823$	− 3.65856e7i	− 0.0656313i	0.999461	−	0.0328156i	$-0.0104474\pi$
$824$	0	0
$825$	1.74240e8	0.310303
$826$	0	0
$827$	4.24921e8i	0.751263i	0.926769	+	0.375631i	$0.122574\pi$
$827$	4.24921e8i	0.751263i	−0.926769	+	0.375631i	$0.877426\pi$
$828$	0	0
$829$	−1.32771e8	−0.233045	−0.116522	−	0.993188i	$-0.537175\pi$
$829$	−1.32771e8	−0.233045	−0.116522	+	0.993188i	$0.537175\pi$
$830$	0	0
$831$	− 2.96675e8i	− 0.516986i
$832$	0	0
$833$	−1.89480e9	−3.27815
$834$	0	0
$835$	− 2.57072e8i	− 0.441566i
$836$	0	0
$837$	3.81542e7	0.0650678
$838$	0	0
$839$	− 6.80810e8i	− 1.15276i	−0.817181	−	0.576381i	$-0.804464\pi$
$839$	− 6.80810e8i	− 1.15276i	0.817181	−	0.576381i	$-0.195536\pi$
$840$	0	0
$841$	−4.79089e8	−0.805430
$842$	0	0
$843$	7.81293e8i	1.30416i
$844$	0	0
$845$	7.20464e8	1.19410
$846$	0	0
$847$	− 5.70180e8i	− 0.938343i
$848$	0	0
$849$	3.60620e8	0.589286
$850$	0	0
$851$	9.62494e7i	0.156174i
$852$	0	0
$853$	−2.95564e7	−0.0476217	−0.0238108	−	0.999716i	$-0.507580\pi$
$853$	−2.95564e7	−0.0476217	−0.0238108	+	0.999716i	$0.507580\pi$
$854$	0	0
$855$	− 1.24832e7i	− 0.0199723i
$856$	0	0
$857$	−5.13041e7	−0.0815098	−0.0407549	−	0.999169i	$-0.512976\pi$
$857$	−5.13041e7	−0.0815098	−0.0407549	+	0.999169i	$0.512976\pi$
$858$	0	0
$859$	1.95973e8i	0.309183i	0.987978	+	0.154592i	$0.0494062\pi$
$859$	1.95973e8i	0.309183i	−0.987978	+	0.154592i	$0.950594\pi$
$860$	0	0
$861$	−1.13409e9	−1.77680
$862$	0	0
$863$	− 7.64022e8i	− 1.18870i	−0.804205	−	0.594351i	$-0.797409\pi$
$863$	− 7.64022e8i	− 1.18870i	0.804205	−	0.594351i	$-0.202591\pi$
$864$	0	0
$865$	1.53202e8	0.236711
$866$	0	0
$867$	− 8.72516e8i	− 1.33880i
$868$	0	0
$869$	6.96960e8	1.06206
$870$	0	0
$871$	3.39331e7i	0.0513533i
$872$	0	0
$873$	−3.55132e7	−0.0533761
$874$	0	0
$875$	− 8.00207e8i	− 1.19448i
$876$	0	0
$877$	−4.83881e8	−0.717364	−0.358682	−	0.933460i	$-0.616774\pi$
$877$	−4.83881e8	−0.717364	−0.358682	+	0.933460i	$0.616774\pi$
$878$	0	0
$879$	8.19667e8i	1.20690i
$880$	0	0
$881$	3.11966e8	0.456226	0.228113	−	0.973635i	$-0.426744\pi$
$881$	3.11966e8	0.456226	0.228113	+	0.973635i	$0.426744\pi$
$882$	0	0
$883$	3.13104e8i	0.454785i	0.973803	+	0.227392i	$0.0730200\pi$
$883$	3.13104e8i	0.454785i	−0.973803	+	0.227392i	$0.926980\pi$
$884$	0	0
$885$	−1.27146e9	−1.83431
$886$	0	0
$887$	− 1.35592e9i	− 1.94295i	−0.237136	−	0.971477i	$-0.576209\pi$
$887$	− 1.35592e9i	− 1.94295i	0.237136	−	0.971477i	$-0.423791\pi$
$888$	0	0
$889$	1.23111e9	1.75223
$890$	0	0
$891$	− 5.10625e8i	− 0.721886i
$892$	0	0
$893$	1.48313e8	0.208269
$894$	0	0
$895$	1.28847e9i	1.79723i
$896$	0	0
$897$	−3.61912e7	−0.0501448
$898$	0	0
$899$	− 2.14657e7i	− 0.0295437i
$900$	0	0
$901$	8.20097e8	1.12122
$902$	0	0
$903$	1.34430e9i	1.82572i
$904$	0	0
$905$	4.04696e8	0.545988
$906$	0	0
$907$	2.15341e8i	0.288605i	0.989534	+	0.144303i	$0.0460938\pi$
$907$	2.15341e8i	0.288605i	−0.989534	+	0.144303i	$0.953906\pi$
$908$	0	0
$909$	3.72586e7	0.0496061
$910$	0	0
$911$	− 1.07950e7i	− 0.0142780i	−0.999975	−	0.00713902i	$-0.997728\pi$
$911$	− 1.07950e7i	− 0.0142780i	0.999975	−	0.00713902i	$-0.00227244\pi$
$912$	0	0
$913$	4.52466e8	0.594530
$914$	0	0
$915$	1.27512e9i	1.66451i
$916$	0	0
$917$	−1.71341e9	−2.22204
$918$	0	0
$919$	8.14011e8i	1.04878i	0.851479	+	0.524389i	$0.175706\pi$
$919$	8.14011e8i	1.04878i	−0.851479	+	0.524389i	$0.824294\pi$
$920$	0	0
$921$	1.34501e8	0.172165
$922$	0	0
$923$	5.73333e7i	0.0729124i
$924$	0	0
$925$	−7.80312e7	−0.0985924
$926$	0	0
$927$	− 2.82240e7i	− 0.0354307i
$928$	0	0
$929$	−2.58385e8	−0.322271	−0.161135	−	0.986932i	$-0.551516\pi$
$929$	−2.58385e8	−0.322271	−0.161135	+	0.986932i	$0.551516\pi$
$930$	0	0
$931$	5.42142e8i	0.671837i
$932$	0	0
$933$	5.30957e8	0.653754
$934$	0	0
$935$	− 1.02308e9i	− 1.25162i
$936$	0	0
$937$	−4.19101e8	−0.509448	−0.254724	−	0.967014i	$-0.581985\pi$
$937$	−4.19101e8	−0.509448	−0.254724	+	0.967014i	$0.581985\pi$
$938$	0	0
$939$	− 4.87863e8i	− 0.589252i
$940$	0	0
$941$	9.59808e8	1.15190	0.575951	−	0.817484i	$-0.304632\pi$
$941$	9.59808e8	1.15190	0.575951	+	0.817484i	$0.304632\pi$
$942$	0	0
$943$	− 5.69203e8i	− 0.678784i
$944$	0	0
$945$	1.74874e9	2.07219
$946$	0	0
$947$	3.10896e8i	0.366071i	0.983106	+	0.183035i	$0.0585923\pi$
$947$	3.10896e8i	0.366071i	−0.983106	+	0.183035i	$0.941408\pi$
$948$	0	0
$949$	2.55150e7	0.0298536
$950$	0	0
$951$	1.30909e8i	0.152205i
$952$	0	0
$953$	−6.13111e8	−0.708370	−0.354185	−	0.935175i	$-0.615242\pi$
$953$	−6.13111e8	−0.708370	−0.354185	+	0.935175i	$0.615242\pi$
$954$	0	0
$955$	− 1.44002e9i	− 1.65333i
$956$	0	0
$957$	−2.72651e8	−0.311079
$958$	0	0
$959$	8.44628e8i	0.957656i
$960$	0	0
$961$	8.83522e8	0.995514
$962$	0	0
$963$	− 5.22513e7i	− 0.0585084i
$964$	0	0
$965$	−8.03837e8	−0.894512
$966$	0	0
$967$	1.10566e8i	0.122277i	0.998129	+	0.0611384i	$0.0194731\pi$
$967$	1.10566e8i	0.122277i	−0.998129	+	0.0611384i	$0.980527\pi$
$968$	0	0
$969$	−4.41036e8	−0.484733
$970$	0	0
$971$	− 1.56745e8i	− 0.171213i	−0.996329	−	0.0856063i	$-0.972717\pi$
$971$	− 1.56745e8i	− 0.171213i	0.996329	−	0.0856063i	$-0.0272827\pi$
$972$	0	0
$973$	−9.68867e8	−1.05178
$974$	0	0
$975$	− 2.93409e7i	− 0.0316563i
$976$	0	0
$977$	−1.21722e9	−1.30523	−0.652615	−	0.757690i	$-0.726328\pi$
$977$	−1.21722e9	−1.30523	−0.652615	+	0.757690i	$0.726328\pi$
$978$	0	0
$979$	4.31613e8i	0.459988i
$980$	0	0
$981$	2.38267e6	0.00252381
$982$	0	0
$983$	− 6.76677e8i	− 0.712395i	−0.934411	−	0.356197i	$-0.884073\pi$
$983$	− 6.76677e8i	− 0.712395i	0.934411	−	0.356197i	$-0.115927\pi$
$984$	0	0
$985$	−1.20088e9	−1.25658
$986$	0	0
$987$	− 1.17434e9i	− 1.22135i
$988$	0	0
$989$	−6.74708e8	−0.697472
$990$	0	0
$991$	− 1.69699e9i	− 1.74365i	−0.489820	−	0.871824i	$-0.662938\pi$
$991$	− 1.69699e9i	− 1.74365i	0.489820	−	0.871824i	$-0.337062\pi$
$992$	0	0
$993$	1.57536e9	1.60891
$994$	0	0
$995$	− 1.37608e9i	− 1.39693i
$996$	0	0
$997$	6.20677e8	0.626296	0.313148	−	0.949704i	$-0.398616\pi$
$997$	6.20677e8	0.626296	0.313148	+	0.949704i	$0.398616\pi$
$998$	0	0
$999$	2.17033e8i	0.217685i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.7.c.b.15.1	✓	2
3.2	odd	2		144.7.g.f.127.1		2
4.3	odd	2	inner	16.7.c.b.15.2	yes	2
5.2	odd	4		400.7.h.b.399.2		4
5.3	odd	4		400.7.h.b.399.4		4
5.4	even	2		400.7.b.c.351.2		2
8.3	odd	2		64.7.c.d.63.1		2
8.5	even	2		64.7.c.d.63.2		2
12.11	even	2		144.7.g.f.127.2		2
16.3	odd	4		256.7.d.e.127.2		4
16.5	even	4		256.7.d.e.127.1		4
16.11	odd	4		256.7.d.e.127.3		4
16.13	even	4		256.7.d.e.127.4		4
20.3	even	4		400.7.h.b.399.1		4
20.7	even	4		400.7.h.b.399.3		4
20.19	odd	2		400.7.b.c.351.1		2
24.5	odd	2		576.7.g.d.127.1		2
24.11	even	2		576.7.g.d.127.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.7.c.b.15.1	✓	2	1.1	even	1	trivial
16.7.c.b.15.2	yes	2	4.3	odd	2	inner
64.7.c.d.63.1		2	8.3	odd	2
64.7.c.d.63.2		2	8.5	even	2
144.7.g.f.127.1		2	3.2	odd	2
144.7.g.f.127.2		2	12.11	even	2
256.7.d.e.127.1		4	16.5	even	4
256.7.d.e.127.2		4	16.3	odd	4
256.7.d.e.127.3		4	16.11	odd	4
256.7.d.e.127.4		4	16.13	even	4
400.7.b.c.351.1		2	20.19	odd	2
400.7.b.c.351.2		2	5.4	even	2
400.7.h.b.399.1		4	20.3	even	4
400.7.h.b.399.2		4	5.2	odd	4
400.7.h.b.399.3		4	20.7	even	4
400.7.h.b.399.4		4	5.3	odd	4
576.7.g.d.127.1		2	24.5	odd	2
576.7.g.d.127.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-27.7128i q^{3} -150.000 q^{5} -609.682i q^{7} -39.0000 q^{9} +O(q^{10})\)	\(q-27.7128i q^{3} -150.000 q^{5} -609.682i q^{7} -39.0000 q^{9} +914.523i q^{11} +154.000 q^{13} +4156.92i q^{15} +7458.00 q^{17} -2133.89i q^{19} -16896.0 q^{21} -8480.12i q^{23} +6875.00 q^{25} -19121.8i q^{27} -10758.0 q^{29} +1995.32i q^{31} +25344.0 q^{33} +91452.3i q^{35} -11350.0 q^{37} -4267.77i q^{39} +67122.0 q^{41} -79563.5i q^{43} +5850.00 q^{45} +69503.7i q^{47} -254063. q^{49} -206682. i q^{51} +109962. q^{53} -137178. i q^{55} -59136.0 q^{57} +305866. i q^{59} +306746. q^{61} +23777.6i q^{63} -23100.0 q^{65} +220345. i q^{67} -235008. q^{69} +372294. i q^{71} +165682. q^{73} -190526. i q^{75} +557568. q^{77} -762102. i q^{79} -558351. q^{81} -494757. i q^{83} -1.11870e6 q^{85} +298134. i q^{87} +471954. q^{89} -93891.0i q^{91} +55296.0 q^{93} +320083. i q^{95} +910594. q^{97} -35666.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 300 q^{5} - 78 q^{9}+O(q^{10}) \) 2 * q - 300 * q^5 - 78 * q^9	\( 2 q - 300 q^{5} - 78 q^{9} + 308 q^{13} + 14916 q^{17} - 33792 q^{21} + 13750 q^{25} - 21516 q^{29} + 50688 q^{33} - 22700 q^{37} + 134244 q^{41} + 11700 q^{45} - 508126 q^{49} + 219924 q^{53} - 118272 q^{57} + 613492 q^{61} - 46200 q^{65} - 470016 q^{69} + 331364 q^{73} + 1115136 q^{77} - 1116702 q^{81} - 2237400 q^{85} + 943908 q^{89} + 110592 q^{93} + 1821188 q^{97}+O(q^{100}) \) 2 * q - 300 * q^5 - 78 * q^9 + 308 * q^13 + 14916 * q^17 - 33792 * q^21 + 13750 * q^25 - 21516 * q^29 + 50688 * q^33 - 22700 * q^37 + 134244 * q^41 + 11700 * q^45 - 508126 * q^49 + 219924 * q^53 - 118272 * q^57 + 613492 * q^61 - 46200 * q^65 - 470016 * q^69 + 331364 * q^73 + 1115136 * q^77 - 1116702 * q^81 - 2237400 * q^85 + 943908 * q^89 + 110592 * q^93 + 1821188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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