LMFDB - Embedded newform 48.7.g.a.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,7,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	48.g (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:	$11.0425960138$
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\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.7.g.a.31.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885i q^{3} -90.0000 q^{5} +187.061i q^{7} -243.000 q^{9} +O(q^{10})$	$q-15.5885i q^{3} -90.0000 q^{5} +187.061i q^{7} -243.000 q^{9} +1683.55i q^{11} +1762.00 q^{13} +1402.96i q^{15} -1638.00 q^{17} +12533.1i q^{19} +2916.00 q^{21} +13468.4i q^{23} -7525.00 q^{25} +3788.00i q^{27} -16002.0 q^{29} -15900.2i q^{31} +26244.0 q^{33} -16835.5i q^{35} +61130.0 q^{37} -27466.9i q^{39} -98550.0 q^{41} -49197.2i q^{43} +21870.0 q^{45} -178457. i q^{47} +82657.0 q^{49} +25533.9i q^{51} -275346. q^{53} -151520. i q^{55} +195372. q^{57} +254217. i q^{59} +106634. q^{61} -45455.9i q^{63} -158580. q^{65} +551644. i q^{67} +209952. q^{69} +74076.3i q^{71} +100978. q^{73} +117303. i q^{75} -314928. q^{77} -77256.4i q^{79} +59049.0 q^{81} +644801. i q^{83} +147420. q^{85} +249446. i q^{87} -819054. q^{89} +329602. i q^{91} -247860. q^{93} -1.12798e6i q^{95} +557026. q^{97} -409103. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 180 q^{5} - 486 q^{9}+O(q^{10}) $ 2 * q - 180 * q^5 - 486 * q^9	$ 2 q - 180 q^{5} - 486 q^{9} + 3524 q^{13} - 3276 q^{17} + 5832 q^{21} - 15050 q^{25} - 32004 q^{29} + 52488 q^{33} + 122260 q^{37} - 197100 q^{41} + 43740 q^{45} + 165314 q^{49} - 550692 q^{53} + 390744 q^{57} + 213268 q^{61} - 317160 q^{65} + 419904 q^{69} + 201956 q^{73} - 629856 q^{77} + 118098 q^{81} + 294840 q^{85} - 1638108 q^{89} - 495720 q^{93} + 1114052 q^{97}+O(q^{100}) $ 2 * q - 180 * q^5 - 486 * q^9 + 3524 * q^13 - 3276 * q^17 + 5832 * q^21 - 15050 * q^25 - 32004 * q^29 + 52488 * q^33 + 122260 * q^37 - 197100 * q^41 + 43740 * q^45 + 165314 * q^49 - 550692 * q^53 + 390744 * q^57 + 213268 * q^61 - 317160 * q^65 + 419904 * q^69 + 201956 * q^73 - 629856 * q^77 + 118098 * q^81 + 294840 * q^85 - 1638108 * q^89 - 495720 * q^93 + 1114052 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 15.5885i	− 0.577350i
$3$	− 15.5885i	− 0.577350i
$4$	0	0
$5$	−90.0000	−0.720000	−0.360000	−	0.932952i	$-0.617223\pi$
$5$	−90.0000	−0.720000	−0.360000	+	0.932952i	$0.617223\pi$
$6$	0	0
$7$	187.061i	0.545369i	0.962104	+	0.272684i	$0.0879115\pi$
$7$	187.061i	0.545369i	−0.962104	+	0.272684i	$0.912089\pi$
$8$	0	0
$9$	−243.000	−0.333333
$10$	0	0
$11$	1683.55i	1.26488i	0.774610	+	0.632439i	$0.217946\pi$
$11$	1683.55i	1.26488i	−0.774610	+	0.632439i	$0.782054\pi$
$12$	0	0
$13$	1762.00	0.802003	0.401001	−	0.916077i	$-0.368662\pi$
$13$	1762.00	0.802003	0.401001	+	0.916077i	$0.368662\pi$
$14$	0	0
$15$	1402.96i	0.415692i
$16$	0	0
$17$	−1638.00	−0.333401	−0.166701	−	0.986008i	$-0.553311\pi$
$17$	−1638.00	−0.333401	−0.166701	+	0.986008i	$0.553311\pi$
$18$	0	0
$19$	12533.1i	1.82725i	0.406556	+	0.913626i	$0.366730\pi$
$19$	12533.1i	1.82725i	−0.406556	+	0.913626i	$0.633270\pi$
$20$	0	0
$21$	2916.00	0.314869
$22$	0	0
$23$	13468.4i	1.10696i	0.832861	+	0.553482i	$0.186701\pi$
$23$	13468.4i	1.10696i	−0.832861	+	0.553482i	$0.813299\pi$
$24$	0	0
$25$	−7525.00	−0.481600
$26$	0	0
$27$	3788.00i	0.192450i
$28$	0	0
$29$	−16002.0	−0.656115	−0.328058	−	0.944658i	$-0.606394\pi$
$29$	−16002.0	−0.656115	−0.328058	+	0.944658i	$0.606394\pi$
$30$	0	0
$31$	− 15900.2i	− 0.533726i	−0.963734	−	0.266863i	$-0.914013\pi$
$31$	− 15900.2i	− 0.533726i	0.963734	−	0.266863i	$-0.0859871\pi$
$32$	0	0
$33$	26244.0	0.730278
$34$	0	0
$35$	− 16835.5i	− 0.392666i
$36$	0	0
$37$	61130.0	1.20684	0.603419	−	0.797424i	$-0.293805\pi$
$37$	61130.0	1.20684	0.603419	+	0.797424i	$0.293805\pi$
$38$	0	0
$39$	− 27466.9i	− 0.463036i
$40$	0	0
$41$	−98550.0	−1.42990	−0.714949	−	0.699177i	$-0.753550\pi$
$41$	−98550.0	−1.42990	−0.714949	+	0.699177i	$0.753550\pi$
$42$	0	0
$43$	− 49197.2i	− 0.618778i	−0.950936	−	0.309389i	$-0.899876\pi$
$43$	− 49197.2i	− 0.618778i	0.950936	−	0.309389i	$-0.100124\pi$
$44$	0	0
$45$	21870.0	0.240000
$46$	0	0
$47$	− 178457.i	− 1.71885i	−0.511258	−	0.859427i	$-0.670820\pi$
$47$	− 178457.i	− 1.71885i	0.511258	−	0.859427i	$-0.329180\pi$
$48$	0	0
$49$	82657.0	0.702573
$50$	0	0
$51$	25533.9i	0.192489i
$52$	0	0
$53$	−275346.	−1.84949	−0.924743	−	0.380592i	$-0.875720\pi$
$53$	−275346.	−1.84949	−0.924743	+	0.380592i	$0.875720\pi$
$54$	0	0
$55$	− 151520.i	− 0.910713i
$56$	0	0
$57$	195372.	1.05496
$58$	0	0
$59$	254217.i	1.23779i	0.785473	+	0.618896i	$0.212420\pi$
$59$	254217.i	1.23779i	−0.785473	+	0.618896i	$0.787580\pi$
$60$	0	0
$61$	106634.	0.469793	0.234896	−	0.972020i	$-0.424525\pi$
$61$	106634.	0.469793	0.234896	+	0.972020i	$0.424525\pi$
$62$	0	0
$63$	− 45455.9i	− 0.181790i
$64$	0	0
$65$	−158580.	−0.577442
$66$	0	0
$67$	551644.i	1.83415i	0.398715	+	0.917075i	$0.369456\pi$
$67$	551644.i	1.83415i	−0.398715	+	0.917075i	$0.630544\pi$
$68$	0	0
$69$	209952.	0.639106
$70$	0	0
$71$	74076.3i	0.206969i	0.994631	+	0.103484i	$0.0329992\pi$
$71$	74076.3i	0.206969i	−0.994631	+	0.103484i	$0.967001\pi$
$72$	0	0
$73$	100978.	0.259572	0.129786	−	0.991542i	$-0.458571\pi$
$73$	100978.	0.259572	0.129786	+	0.991542i	$0.458571\pi$
$74$	0	0
$75$	117303.i	0.278052i
$76$	0	0
$77$	−314928.	−0.689825
$78$	0	0
$79$	− 77256.4i	− 0.156694i	−0.996926	−	0.0783471i	$-0.975036\pi$
$79$	− 77256.4i	− 0.156694i	0.996926	−	0.0783471i	$-0.0249643\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	644801.i	1.12769i	0.825879	+	0.563847i	$0.190679\pi$
$83$	644801.i	1.12769i	−0.825879	+	0.563847i	$0.809321\pi$
$84$	0	0
$85$	147420.	0.240049
$86$	0	0
$87$	249446.i	0.378808i
$88$	0	0
$89$	−819054.	−1.16183	−0.580915	−	0.813964i	$-0.697305\pi$
$89$	−819054.	−1.16183	−0.580915	+	0.813964i	$0.697305\pi$
$90$	0	0
$91$	329602.i	0.437387i
$92$	0	0
$93$	−247860.	−0.308147
$94$	0	0
$95$	− 1.12798e6i	− 1.31562i
$96$	0	0
$97$	557026.	0.610324	0.305162	−	0.952300i	$-0.401289\pi$
$97$	557026.	0.610324	0.305162	+	0.952300i	$0.401289\pi$
$98$	0	0
$99$	− 409103.i	− 0.421626i
$100$	0	0
$101$	1.83800e6	1.78394	0.891971	−	0.452092i	$-0.149322\pi$
$101$	1.83800e6	1.78394	0.891971	+	0.452092i	$0.149322\pi$
$102$	0	0
$103$	− 1.25649e6i	− 1.14987i	−0.818200	−	0.574934i	$-0.805028\pi$
$103$	− 1.25649e6i	− 1.14987i	0.818200	−	0.574934i	$-0.194972\pi$
$104$	0	0
$105$	−262440.	−0.226706
$106$	0	0
$107$	− 459610.i	− 0.375179i	−0.982248	−	0.187589i	$-0.939933\pi$
$107$	− 459610.i	− 0.375179i	0.982248	−	0.187589i	$-0.0600674\pi$
$108$	0	0
$109$	−1.46027e6	−1.12760	−0.563798	−	0.825913i	$-0.690660\pi$
$109$	−1.46027e6	−1.12760	−0.563798	+	0.825913i	$0.690660\pi$
$110$	0	0
$111$	− 952922.i	− 0.696769i
$112$	0	0
$113$	563634.	0.390627	0.195313	−	0.980741i	$-0.437428\pi$
$113$	563634.	0.390627	0.195313	+	0.980741i	$0.437428\pi$
$114$	0	0
$115$	− 1.21216e6i	− 0.797014i
$116$	0	0
$117$	−428166.	−0.267334
$118$	0	0
$119$	− 306407.i	− 0.181827i
$120$	0	0
$121$	−1.06279e6	−0.599918
$122$	0	0
$123$	1.53624e6i	0.825552i
$124$	0	0
$125$	2.08350e6	1.06675
$126$	0	0
$127$	312954.i	0.152781i	0.997078	+	0.0763905i	$0.0243396\pi$
$127$	312954.i	0.152781i	−0.997078	+	0.0763905i	$0.975660\pi$
$128$	0	0
$129$	−766908.	−0.357252
$130$	0	0
$131$	− 705409.i	− 0.313781i	−0.987616	−	0.156891i	$-0.949853\pi$
$131$	− 705409.i	− 0.313781i	0.987616	−	0.156891i	$-0.0501470\pi$
$132$	0	0
$133$	−2.34446e6	−0.996526
$134$	0	0
$135$	− 340920.i	− 0.138564i
$136$	0	0
$137$	2.42937e6	0.944783	0.472391	−	0.881389i	$-0.343391\pi$
$137$	2.42937e6	0.944783	0.472391	+	0.881389i	$0.343391\pi$
$138$	0	0
$139$	1.93403e6i	0.720143i	0.932925	+	0.360071i	$0.117248\pi$
$139$	1.93403e6i	0.720143i	−0.932925	+	0.360071i	$0.882752\pi$
$140$	0	0
$141$	−2.78186e6	−0.992381
$142$	0	0
$143$	2.96642e6i	1.01444i
$144$	0	0
$145$	1.44018e6	0.472403
$146$	0	0
$147$	− 1.28850e6i	− 0.405631i
$148$	0	0
$149$	3.46397e6	1.04717	0.523583	−	0.851974i	$-0.324595\pi$
$149$	3.46397e6	1.04717	0.523583	+	0.851974i	$0.324595\pi$
$150$	0	0
$151$	5.05010e6i	1.46679i	0.679801	+	0.733397i	$0.262066\pi$
$151$	5.05010e6i	1.46679i	−0.679801	+	0.733397i	$0.737934\pi$
$152$	0	0
$153$	398034.	0.111134
$154$	0	0
$155$	1.43102e6i	0.384283i
$156$	0	0
$157$	1.34662e6	0.347973	0.173986	−	0.984748i	$-0.444335\pi$
$157$	1.34662e6	0.347973	0.173986	+	0.984748i	$0.444335\pi$
$158$	0	0
$159$	4.29222e6i	1.06780i
$160$	0	0
$161$	−2.51942e6	−0.603703
$162$	0	0
$163$	− 2.99093e6i	− 0.690626i	−0.938488	−	0.345313i	$-0.887773\pi$
$163$	− 2.99093e6i	− 0.690626i	0.938488	−	0.345313i	$-0.112227\pi$
$164$	0	0
$165$	−2.36196e6	−0.525800
$166$	0	0
$167$	6.39414e6i	1.37288i	0.727187	+	0.686440i	$0.240827\pi$
$167$	6.39414e6i	1.37288i	−0.727187	+	0.686440i	$0.759173\pi$
$168$	0	0
$169$	−1.72217e6	−0.356792
$170$	0	0
$171$	− 3.04555e6i	− 0.609084i
$172$	0	0
$173$	1.85391e6	0.358055	0.179028	−	0.983844i	$-0.442705\pi$
$173$	1.85391e6	0.358055	0.179028	+	0.983844i	$0.442705\pi$
$174$	0	0
$175$	− 1.40764e6i	− 0.262650i
$176$	0	0
$177$	3.96284e6	0.714640
$178$	0	0
$179$	− 7.89081e6i	− 1.37582i	−0.725794	−	0.687912i	$-0.758528\pi$
$179$	− 7.89081e6i	− 1.37582i	0.725794	−	0.687912i	$-0.241472\pi$
$180$	0	0
$181$	−5.68048e6	−0.957964	−0.478982	−	0.877825i	$-0.658994\pi$
$181$	−5.68048e6	−0.957964	−0.478982	+	0.877825i	$0.658994\pi$
$182$	0	0
$183$	− 1.66226e6i	− 0.271235i
$184$	0	0
$185$	−5.50170e6	−0.868924
$186$	0	0
$187$	− 2.75766e6i	− 0.421712i
$188$	0	0
$189$	−708588.	−0.104956
$190$	0	0
$191$	− 8.96997e6i	− 1.28733i	−0.765306	−	0.643667i	$-0.777412\pi$
$191$	− 8.96997e6i	− 1.28733i	0.765306	−	0.643667i	$-0.222588\pi$
$192$	0	0
$193$	5.39165e6	0.749980	0.374990	−	0.927029i	$-0.377646\pi$
$193$	5.39165e6	0.749980	0.374990	+	0.927029i	$0.377646\pi$
$194$	0	0
$195$	2.47202e6i	0.333386i
$196$	0	0
$197$	4.24870e6	0.555722	0.277861	−	0.960621i	$-0.410375\pi$
$197$	4.24870e6	0.555722	0.277861	+	0.960621i	$0.410375\pi$
$198$	0	0
$199$	2.99878e6i	0.380527i	0.981733	+	0.190264i	$0.0609343\pi$
$199$	2.99878e6i	0.380527i	−0.981733	+	0.190264i	$0.939066\pi$
$200$	0	0
$201$	8.59928e6	1.05895
$202$	0	0
$203$	− 2.99336e6i	− 0.357825i
$204$	0	0
$205$	8.86950e6	1.02953
$206$	0	0
$207$	− 3.27283e6i	− 0.368988i
$208$	0	0
$209$	−2.11002e7	−2.31125
$210$	0	0
$211$	− 1.54047e7i	− 1.63986i	−0.572466	−	0.819928i	$-0.694013\pi$
$211$	− 1.54047e7i	− 1.63986i	0.572466	−	0.819928i	$-0.305987\pi$
$212$	0	0
$213$	1.15474e6	0.119493
$214$	0	0
$215$	4.42775e6i	0.445520i
$216$	0	0
$217$	2.97432e6	0.291077
$218$	0	0
$219$	− 1.57409e6i	− 0.149864i
$220$	0	0
$221$	−2.88616e6	−0.267389
$222$	0	0
$223$	− 3.42902e6i	− 0.309212i	−0.987976	−	0.154606i	$-0.950589\pi$
$223$	− 3.42902e6i	− 0.309212i	0.987976	−	0.154606i	$-0.0494108\pi$
$224$	0	0
$225$	1.82858e6	0.160533
$226$	0	0
$227$	4.45637e6i	0.380981i	0.981689	+	0.190490i	$0.0610078\pi$
$227$	4.45637e6i	0.380981i	−0.981689	+	0.190490i	$0.938992\pi$
$228$	0	0
$229$	3.52299e6	0.293363	0.146682	−	0.989184i	$-0.453141\pi$
$229$	3.52299e6	0.293363	0.146682	+	0.989184i	$0.453141\pi$
$230$	0	0
$231$	4.90924e6i	0.398271i
$232$	0	0
$233$	−1.32900e7	−1.05065	−0.525325	−	0.850902i	$-0.676056\pi$
$233$	−1.32900e7	−1.05065	−0.525325	+	0.850902i	$0.676056\pi$
$234$	0	0
$235$	1.60611e7i	1.23758i
$236$	0	0
$237$	−1.20431e6	−0.0904675
$238$	0	0
$239$	646484.i	0.0473548i	0.999720	+	0.0236774i	$0.00753746\pi$
$239$	646484.i	0.0473548i	−0.999720	+	0.0236774i	$0.992463\pi$
$240$	0	0
$241$	1.56559e7	1.11847	0.559237	−	0.829008i	$-0.311094\pi$
$241$	1.56559e7	1.11847	0.559237	+	0.829008i	$0.311094\pi$
$242$	0	0
$243$	− 920483.i	− 0.0641500i
$244$	0	0
$245$	−7.43913e6	−0.505852
$246$	0	0
$247$	2.20834e7i	1.46546i
$248$	0	0
$249$	1.00515e7	0.651075
$250$	0	0
$251$	1.44466e7i	0.913574i	0.889576	+	0.456787i	$0.151000\pi$
$251$	1.44466e7i	0.913574i	−0.889576	+	0.456787i	$0.849000\pi$
$252$	0	0
$253$	−2.26748e7	−1.40017
$254$	0	0
$255$	− 2.29805e6i	− 0.138592i
$256$	0	0
$257$	−9.47072e6	−0.557935	−0.278967	−	0.960301i	$-0.589992\pi$
$257$	−9.47072e6	−0.557935	−0.278967	+	0.960301i	$0.589992\pi$
$258$	0	0
$259$	1.14351e7i	0.658172i
$260$	0	0
$261$	3.88849e6	0.218705
$262$	0	0
$263$	1.27916e7i	0.703168i	0.936156	+	0.351584i	$0.114357\pi$
$263$	1.27916e7i	0.703168i	−0.936156	+	0.351584i	$0.885643\pi$
$264$	0	0
$265$	2.47811e7	1.33163
$266$	0	0
$267$	1.27678e7i	0.670783i
$268$	0	0
$269$	3.18981e7	1.63873	0.819367	−	0.573269i	$-0.194325\pi$
$269$	3.18981e7	1.63873	0.819367	+	0.573269i	$0.194325\pi$
$270$	0	0
$271$	− 3.53215e7i	− 1.77473i	−0.461071	−	0.887363i	$-0.652535\pi$
$271$	− 3.53215e7i	− 1.77473i	0.461071	−	0.887363i	$-0.347465\pi$
$272$	0	0
$273$	5.13799e6	0.252526
$274$	0	0
$275$	− 1.26687e7i	− 0.609166i
$276$	0	0
$277$	−2.17022e7	−1.02109	−0.510546	−	0.859851i	$-0.670557\pi$
$277$	−2.17022e7	−1.02109	−0.510546	+	0.859851i	$0.670557\pi$
$278$	0	0
$279$	3.86376e6i	0.177909i
$280$	0	0
$281$	−5.81830e6	−0.262227	−0.131113	−	0.991367i	$-0.541855\pi$
$281$	−5.81830e6	−0.262227	−0.131113	+	0.991367i	$0.541855\pi$
$282$	0	0
$283$	2.40739e7i	1.06215i	0.847324	+	0.531076i	$0.178212\pi$
$283$	2.40739e7i	1.06215i	−0.847324	+	0.531076i	$0.821788\pi$
$284$	0	0
$285$	−1.75835e7	−0.759574
$286$	0	0
$287$	− 1.84349e7i	− 0.779822i
$288$	0	0
$289$	−2.14545e7	−0.888844
$290$	0	0
$291$	− 8.68318e6i	− 0.352371i
$292$	0	0
$293$	2.11884e7	0.842356	0.421178	−	0.906978i	$-0.361617\pi$
$293$	2.11884e7	0.842356	0.421178	+	0.906978i	$0.361617\pi$
$294$	0	0
$295$	− 2.28795e7i	− 0.891211i
$296$	0	0
$297$	−6.37729e6	−0.243426
$298$	0	0
$299$	2.37314e7i	0.887788i
$300$	0	0
$301$	9.20290e6	0.337462
$302$	0	0
$303$	− 2.86516e7i	− 1.02996i
$304$	0	0
$305$	−9.59706e6	−0.338251
$306$	0	0
$307$	− 6.60682e6i	− 0.228338i	−0.993461	−	0.114169i	$-0.963580\pi$
$307$	− 6.60682e6i	− 0.228338i	0.993461	−	0.114169i	$-0.0364205\pi$
$308$	0	0
$309$	−1.95868e7	−0.663877
$310$	0	0
$311$	− 3.68260e7i	− 1.22426i	−0.790757	−	0.612130i	$-0.790313\pi$
$311$	− 3.68260e7i	− 1.22426i	0.790757	−	0.612130i	$-0.209687\pi$
$312$	0	0
$313$	−211966.	−0.00691247	−0.00345623	−	0.999994i	$-0.501100\pi$
$313$	−211966.	−0.00691247	−0.00345623	+	0.999994i	$0.501100\pi$
$314$	0	0
$315$	4.09103e6i	0.130889i
$316$	0	0
$317$	3.59214e7	1.12765	0.563826	−	0.825894i	$-0.309329\pi$
$317$	3.59214e7	1.12765	0.563826	+	0.825894i	$0.309329\pi$
$318$	0	0
$319$	− 2.69402e7i	− 0.829906i
$320$	0	0
$321$	−7.16461e6	−0.216610
$322$	0	0
$323$	− 2.05292e7i	− 0.609208i
$324$	0	0
$325$	−1.32590e7	−0.386245
$326$	0	0
$327$	2.27634e7i	0.651018i
$328$	0	0
$329$	3.33824e7	0.937410
$330$	0	0
$331$	− 2.52849e7i	− 0.697232i	−0.937266	−	0.348616i	$-0.886652\pi$
$331$	− 2.52849e7i	− 0.697232i	0.937266	−	0.348616i	$-0.113348\pi$
$332$	0	0
$333$	−1.48546e7	−0.402280
$334$	0	0
$335$	− 4.96480e7i	− 1.32059i
$336$	0	0
$337$	1.09024e7	0.284861	0.142431	−	0.989805i	$-0.454508\pi$
$337$	1.09024e7	0.284861	0.142431	+	0.989805i	$0.454508\pi$
$338$	0	0
$339$	− 8.78618e6i	− 0.225528i
$340$	0	0
$341$	2.67689e7	0.675098
$342$	0	0
$343$	3.74695e7i	0.928530i
$344$	0	0
$345$	−1.88957e7	−0.460156
$346$	0	0
$347$	7.92432e7i	1.89659i	0.317390	+	0.948295i	$0.397193\pi$
$347$	7.92432e7i	1.89659i	−0.317390	+	0.948295i	$0.602807\pi$
$348$	0	0
$349$	−1.27972e7	−0.301050	−0.150525	−	0.988606i	$-0.548096\pi$
$349$	−1.27972e7	−0.301050	−0.150525	+	0.988606i	$0.548096\pi$
$350$	0	0
$351$	6.67445e6i	0.154345i
$352$	0	0
$353$	4.98344e7	1.13294	0.566468	−	0.824084i	$-0.308310\pi$
$353$	4.98344e7	1.13294	0.566468	+	0.824084i	$0.308310\pi$
$354$	0	0
$355$	− 6.66687e6i	− 0.149017i
$356$	0	0
$357$	−4.77641e6	−0.104978
$358$	0	0
$359$	5.39343e7i	1.16569i	0.812585	+	0.582843i	$0.198060\pi$
$359$	5.39343e7i	1.16569i	−0.812585	+	0.582843i	$0.801940\pi$
$360$	0	0
$361$	−1.10033e8	−2.33885
$362$	0	0
$363$	1.65673e7i	0.346363i
$364$	0	0
$365$	−9.08802e6	−0.186892
$366$	0	0
$367$	6.76278e7i	1.36813i	0.729422	+	0.684064i	$0.239789\pi$
$367$	6.76278e7i	1.36813i	−0.729422	+	0.684064i	$0.760211\pi$
$368$	0	0
$369$	2.39476e7	0.476633
$370$	0	0
$371$	− 5.15066e7i	− 1.00865i
$372$	0	0
$373$	5.55946e7	1.07129	0.535644	−	0.844444i	$-0.320069\pi$
$373$	5.55946e7	1.07129	0.535644	+	0.844444i	$0.320069\pi$
$374$	0	0
$375$	− 3.24786e7i	− 0.615890i
$376$	0	0
$377$	−2.81955e7	−0.526206
$378$	0	0
$379$	5.63506e7i	1.03510i	0.855654	+	0.517548i	$0.173155\pi$
$379$	5.63506e7i	1.03510i	−0.855654	+	0.517548i	$0.826845\pi$
$380$	0	0
$381$	4.87847e6	0.0882081
$382$	0	0
$383$	5.34023e7i	0.950525i	0.879844	+	0.475263i	$0.157647\pi$
$383$	5.34023e7i	0.950525i	−0.879844	+	0.475263i	$0.842353\pi$
$384$	0	0
$385$	2.83435e7	0.496674
$386$	0	0
$387$	1.19549e7i	0.206259i
$388$	0	0
$389$	−7.97251e6	−0.135440	−0.0677199	−	0.997704i	$-0.521572\pi$
$389$	−7.97251e6	−0.135440	−0.0677199	+	0.997704i	$0.521572\pi$
$390$	0	0
$391$	− 2.20613e7i	− 0.369063i
$392$	0	0
$393$	−1.09962e7	−0.181162
$394$	0	0
$395$	6.95308e6i	0.112820i
$396$	0	0
$397$	−8.32993e7	−1.33128	−0.665641	−	0.746272i	$-0.731842\pi$
$397$	−8.32993e7	−1.33128	−0.665641	+	0.746272i	$0.731842\pi$
$398$	0	0
$399$	3.65466e7i	0.575345i
$400$	0	0
$401$	−4.70511e6	−0.0729687	−0.0364844	−	0.999334i	$-0.511616\pi$
$401$	−4.70511e6	−0.0729687	−0.0364844	+	0.999334i	$0.511616\pi$
$402$	0	0
$403$	− 2.80162e7i	− 0.428050i
$404$	0	0
$405$	−5.31441e6	−0.0800000
$406$	0	0
$407$	1.02916e8i	1.52650i
$408$	0	0
$409$	1.31798e8	1.92637	0.963184	−	0.268845i	$-0.0866418\pi$
$409$	1.31798e8	1.92637	0.963184	+	0.268845i	$0.0866418\pi$
$410$	0	0
$411$	− 3.78701e7i	− 0.545471i
$412$	0	0
$413$	−4.75541e7	−0.675053
$414$	0	0
$415$	− 5.80321e7i	− 0.811940i
$416$	0	0
$417$	3.01485e7	0.415775
$418$	0	0
$419$	− 7.30006e7i	− 0.992394i	−0.868210	−	0.496197i	$-0.834729\pi$
$419$	− 7.30006e7i	− 0.992394i	0.868210	−	0.496197i	$-0.165271\pi$
$420$	0	0
$421$	1.06635e8	1.42906	0.714532	−	0.699602i	$-0.246639\pi$
$421$	1.06635e8	1.42906	0.714532	+	0.699602i	$0.246639\pi$
$422$	0	0
$423$	4.33650e7i	0.572952i
$424$	0	0
$425$	1.23260e7	0.160566
$426$	0	0
$427$	1.99471e7i	0.256210i
$428$	0	0
$429$	4.62419e7	0.585685
$430$	0	0
$431$	− 1.22021e8i	− 1.52406i	−0.647543	−	0.762029i	$-0.724204\pi$
$431$	− 1.22021e8i	− 1.52406i	0.647543	−	0.762029i	$-0.275796\pi$
$432$	0	0
$433$	−8.06280e7	−0.993167	−0.496584	−	0.867989i	$-0.665412\pi$
$433$	−8.06280e7	−0.993167	−0.496584	+	0.867989i	$0.665412\pi$
$434$	0	0
$435$	− 2.24502e7i	− 0.272742i
$436$	0	0
$437$	−1.68801e8	−2.02270
$438$	0	0
$439$	7.31274e7i	0.864344i	0.901791	+	0.432172i	$0.142253\pi$
$439$	7.31274e7i	0.864344i	−0.901791	+	0.432172i	$0.857747\pi$
$440$	0	0
$441$	−2.00857e7	−0.234191
$442$	0	0
$443$	− 1.16278e8i	− 1.33748i	−0.743498	−	0.668738i	$-0.766835\pi$
$443$	− 1.16278e8i	− 1.33748i	0.743498	−	0.668738i	$-0.233165\pi$
$444$	0	0
$445$	7.37149e7	0.836517
$446$	0	0
$447$	− 5.39980e7i	− 0.604582i
$448$	0	0
$449$	−1.03224e8	−1.14036	−0.570181	−	0.821519i	$-0.693127\pi$
$449$	−1.03224e8	−1.14036	−0.570181	+	0.821519i	$0.693127\pi$
$450$	0	0
$451$	− 1.65914e8i	− 1.80865i
$452$	0	0
$453$	7.87233e7	0.846854
$454$	0	0
$455$	− 2.96642e7i	− 0.314919i
$456$	0	0
$457$	1.17044e7	0.122631	0.0613153	−	0.998118i	$-0.480470\pi$
$457$	1.17044e7	0.122631	0.0613153	+	0.998118i	$0.480470\pi$
$458$	0	0
$459$	− 6.20474e6i	− 0.0641631i
$460$	0	0
$461$	−5.37243e7	−0.548363	−0.274182	−	0.961678i	$-0.588407\pi$
$461$	−5.37243e7	−0.548363	−0.274182	+	0.961678i	$0.588407\pi$
$462$	0	0
$463$	− 4.17310e7i	− 0.420451i	−0.977653	−	0.210226i	$-0.932580\pi$
$463$	− 4.17310e7i	− 0.420451i	0.977653	−	0.210226i	$-0.0674199\pi$
$464$	0	0
$465$	2.23074e7	0.221866
$466$	0	0
$467$	7.53205e7i	0.739541i	0.929123	+	0.369771i	$0.120564\pi$
$467$	7.53205e7i	0.739541i	−0.929123	+	0.369771i	$0.879436\pi$
$468$	0	0
$469$	−1.03191e8	−1.00029
$470$	0	0
$471$	− 2.09917e7i	− 0.200902i
$472$	0	0
$473$	8.28261e7	0.782679
$474$	0	0
$475$	− 9.43117e7i	− 0.880004i
$476$	0	0
$477$	6.69091e7	0.616495
$478$	0	0
$479$	1.54318e8i	1.40414i	0.712109	+	0.702069i	$0.247740\pi$
$479$	1.54318e8i	1.40414i	−0.712109	+	0.702069i	$0.752260\pi$
$480$	0	0
$481$	1.07711e8	0.967888
$482$	0	0
$483$	3.92739e7i	0.348548i
$484$	0	0
$485$	−5.01323e7	−0.439433
$486$	0	0
$487$	1.50446e8i	1.30254i	0.758844	+	0.651272i	$0.225764\pi$
$487$	1.50446e8i	1.30254i	−0.758844	+	0.651272i	$0.774236\pi$
$488$	0	0
$489$	−4.66239e7	−0.398733
$490$	0	0
$491$	− 1.25837e8i	− 1.06308i	−0.847034	−	0.531538i	$-0.821614\pi$
$491$	− 1.25837e8i	− 1.06308i	0.847034	−	0.531538i	$-0.178386\pi$
$492$	0	0
$493$	2.62113e7	0.218750
$494$	0	0
$495$	3.68193e7i	0.303571i
$496$	0	0
$497$	−1.38568e7	−0.112874
$498$	0	0
$499$	2.43975e7i	0.196356i	0.995169	+	0.0981779i	$0.0313014\pi$
$499$	2.43975e7i	0.196356i	−0.995169	+	0.0981779i	$0.968699\pi$
$500$	0	0
$501$	9.96747e7	0.792632
$502$	0	0
$503$	3.22030e7i	0.253042i	0.991964	+	0.126521i	$0.0403811\pi$
$503$	3.22030e7i	0.253042i	−0.991964	+	0.126521i	$0.959619\pi$
$504$	0	0
$505$	−1.65420e8	−1.28444
$506$	0	0
$507$	2.68459e7i	0.205994i
$508$	0	0
$509$	−1.70335e7	−0.129167	−0.0645833	−	0.997912i	$-0.520572\pi$
$509$	−1.70335e7	−0.129167	−0.0645833	+	0.997912i	$0.520572\pi$
$510$	0	0
$511$	1.88891e7i	0.141563i
$512$	0	0
$513$	−4.74754e7	−0.351655
$514$	0	0
$515$	1.13084e8i	0.827905i
$516$	0	0
$517$	3.00441e8	2.17414
$518$	0	0
$519$	− 2.88996e7i	− 0.206723i
$520$	0	0
$521$	6.08566e7	0.430323	0.215162	−	0.976578i	$-0.430972\pi$
$521$	6.08566e7	0.430323	0.215162	+	0.976578i	$0.430972\pi$
$522$	0	0
$523$	1.66240e8i	1.16207i	0.813880	+	0.581033i	$0.197351\pi$
$523$	1.66240e8i	1.16207i	−0.813880	+	0.581033i	$0.802649\pi$
$524$	0	0
$525$	−2.19429e7	−0.151641
$526$	0	0
$527$	2.60446e7i	0.177945i
$528$	0	0
$529$	−3.33626e7	−0.225369
$530$	0	0
$531$	− 6.17746e7i	− 0.412597i
$532$	0	0
$533$	−1.73645e8	−1.14678
$534$	0	0
$535$	4.13649e7i	0.270129i
$536$	0	0
$537$	−1.23006e8	−0.794332
$538$	0	0
$539$	1.39157e8i	0.888669i
$540$	0	0
$541$	−1.84782e8	−1.16699	−0.583496	−	0.812116i	$-0.698316\pi$
$541$	−1.84782e8	−1.16699	−0.583496	+	0.812116i	$0.698316\pi$
$542$	0	0
$543$	8.85499e7i	0.553081i
$544$	0	0
$545$	1.31424e8	0.811869
$546$	0	0
$547$	2.06644e8i	1.26259i	0.775544	+	0.631294i	$0.217476\pi$
$547$	2.06644e8i	1.26259i	−0.775544	+	0.631294i	$0.782524\pi$
$548$	0	0
$549$	−2.59121e7	−0.156598
$550$	0	0
$551$	− 2.00555e8i	− 1.19889i
$552$	0	0
$553$	1.44517e7	0.0854562
$554$	0	0
$555$	8.57630e7i	0.501673i
$556$	0	0
$557$	2.14641e8	1.24207	0.621037	−	0.783781i	$-0.286712\pi$
$557$	2.14641e8	1.24207	0.621037	+	0.783781i	$0.286712\pi$
$558$	0	0
$559$	− 8.66854e7i	− 0.496262i
$560$	0	0
$561$	−4.29877e7	−0.243476
$562$	0	0
$563$	2.29554e8i	1.28635i	0.765718	+	0.643176i	$0.222384\pi$
$563$	2.29554e8i	1.28635i	−0.765718	+	0.643176i	$0.777616\pi$
$564$	0	0
$565$	−5.07271e7	−0.281251
$566$	0	0
$567$	1.10458e7i	0.0605965i
$568$	0	0
$569$	2.21576e8	1.20278	0.601389	−	0.798956i	$-0.294614\pi$
$569$	2.21576e8	1.20278	0.601389	+	0.798956i	$0.294614\pi$
$570$	0	0
$571$	− 1.30875e8i	− 0.702989i	−0.936190	−	0.351495i	$-0.885674\pi$
$571$	− 1.30875e8i	− 0.702989i	0.936190	−	0.351495i	$-0.114326\pi$
$572$	0	0
$573$	−1.39828e8	−0.743242
$574$	0	0
$575$	− 1.01350e8i	− 0.533114i
$576$	0	0
$577$	−2.44188e8	−1.27115	−0.635574	−	0.772040i	$-0.719236\pi$
$577$	−2.44188e8	−1.27115	−0.635574	+	0.772040i	$0.719236\pi$
$578$	0	0
$579$	− 8.40475e7i	− 0.433001i
$580$	0	0
$581$	−1.20617e8	−0.615009
$582$	0	0
$583$	− 4.63560e8i	− 2.33938i
$584$	0	0
$585$	3.85349e7	0.192481
$586$	0	0
$587$	− 4.13902e7i	− 0.204636i	−0.994752	−	0.102318i	$-0.967374\pi$
$587$	− 4.13902e7i	− 0.204636i	0.994752	−	0.102318i	$-0.0326260\pi$
$588$	0	0
$589$	1.99279e8	0.975251
$590$	0	0
$591$	− 6.62307e7i	− 0.320846i
$592$	0	0
$593$	3.28345e8	1.57459	0.787293	−	0.616579i	$-0.211482\pi$
$593$	3.28345e8	1.57459	0.787293	+	0.616579i	$0.211482\pi$
$594$	0	0
$595$	2.75766e7i	0.130915i
$596$	0	0
$597$	4.67464e7	0.219698
$598$	0	0
$599$	− 7.55646e7i	− 0.351591i	−0.984427	−	0.175796i	$-0.943750\pi$
$599$	− 7.55646e7i	− 0.351591i	0.984427	−	0.175796i	$-0.0562498\pi$
$600$	0	0
$601$	−2.62430e7	−0.120890	−0.0604449	−	0.998172i	$-0.519252\pi$
$601$	−2.62430e7	−0.120890	−0.0604449	+	0.998172i	$0.519252\pi$
$602$	0	0
$603$	− 1.34050e8i	− 0.611383i
$604$	0	0
$605$	9.56512e7	0.431941
$606$	0	0
$607$	2.35756e8i	1.05414i	0.849823	+	0.527068i	$0.176709\pi$
$607$	2.35756e8i	1.05414i	−0.849823	+	0.527068i	$0.823291\pi$
$608$	0	0
$609$	−4.66618e7	−0.206590
$610$	0	0
$611$	− 3.14441e8i	− 1.37853i
$612$	0	0
$613$	−4.02537e8	−1.74753	−0.873763	−	0.486351i	$-0.838328\pi$
$613$	−4.02537e8	−1.74753	−0.873763	+	0.486351i	$0.838328\pi$
$614$	0	0
$615$	− 1.38262e8i	− 0.594397i
$616$	0	0
$617$	3.96288e8	1.68716	0.843578	−	0.537007i	$-0.180445\pi$
$617$	3.96288e8	1.68716	0.843578	+	0.537007i	$0.180445\pi$
$618$	0	0
$619$	4.07605e7i	0.171857i	0.996301	+	0.0859286i	$0.0273857\pi$
$619$	4.07605e7i	0.171857i	−0.996301	+	0.0859286i	$0.972614\pi$
$620$	0	0
$621$	−5.10183e7	−0.213035
$622$	0	0
$623$	− 1.53213e8i	− 0.633626i
$624$	0	0
$625$	−6.99369e7	−0.286461
$626$	0	0
$627$	3.28919e8i	1.33440i
$628$	0	0
$629$	−1.00131e8	−0.402361
$630$	0	0
$631$	− 3.93731e8i	− 1.56715i	−0.621296	−	0.783576i	$-0.713394\pi$
$631$	− 3.93731e8i	− 1.56715i	0.621296	−	0.783576i	$-0.286606\pi$
$632$	0	0
$633$	−2.40136e8	−0.946772
$634$	0	0
$635$	− 2.81658e7i	− 0.110002i
$636$	0	0
$637$	1.45642e8	0.563465
$638$	0	0
$639$	− 1.80006e7i	− 0.0689895i
$640$	0	0
$641$	−2.73944e8	−1.04013	−0.520066	−	0.854126i	$-0.674093\pi$
$641$	−2.73944e8	−1.04013	−0.520066	+	0.854126i	$0.674093\pi$
$642$	0	0
$643$	1.42472e8i	0.535915i	0.963431	+	0.267958i	$0.0863487\pi$
$643$	1.42472e8i	0.535915i	−0.963431	+	0.267958i	$0.913651\pi$
$644$	0	0
$645$	6.90217e7	0.257221
$646$	0	0
$647$	7.94839e7i	0.293472i	0.989176	+	0.146736i	$0.0468768\pi$
$647$	7.94839e7i	0.293472i	−0.989176	+	0.146736i	$0.953123\pi$
$648$	0	0
$649$	−4.27987e8	−1.56566
$650$	0	0
$651$	− 4.63651e7i	− 0.168054i
$652$	0	0
$653$	1.16064e8	0.416830	0.208415	−	0.978041i	$-0.433170\pi$
$653$	1.16064e8	0.416830	0.208415	+	0.978041i	$0.433170\pi$
$654$	0	0
$655$	6.34868e7i	0.225923i
$656$	0	0
$657$	−2.45377e7	−0.0865241
$658$	0	0
$659$	1.96149e8i	0.685378i	0.939449	+	0.342689i	$0.111338\pi$
$659$	1.96149e8i	0.685378i	−0.939449	+	0.342689i	$0.888662\pi$
$660$	0	0
$661$	4.94257e8	1.71139	0.855694	−	0.517481i	$-0.173130\pi$
$661$	4.94257e8	1.71139	0.855694	+	0.517481i	$0.173130\pi$
$662$	0	0
$663$	4.49907e7i	0.154377i
$664$	0	0
$665$	2.11002e8	0.717499
$666$	0	0
$667$	− 2.15522e8i	− 0.726296i
$668$	0	0
$669$	−5.34532e7	−0.178523
$670$	0	0
$671$	1.79524e8i	0.594231i
$672$	0	0
$673$	4.02582e8	1.32071	0.660357	−	0.750952i	$-0.270405\pi$
$673$	4.02582e8	1.32071	0.660357	+	0.750952i	$0.270405\pi$
$674$	0	0
$675$	− 2.85047e7i	− 0.0926840i
$676$	0	0
$677$	−1.98483e8	−0.639673	−0.319836	−	0.947473i	$-0.603628\pi$
$677$	−1.98483e8	−0.639673	−0.319836	+	0.947473i	$0.603628\pi$
$678$	0	0
$679$	1.04198e8i	0.332852i
$680$	0	0
$681$	6.94679e7	0.219959
$682$	0	0
$683$	1.79987e8i	0.564910i	0.959281	+	0.282455i	$0.0911487\pi$
$683$	1.79987e8i	0.564910i	−0.959281	+	0.282455i	$0.908851\pi$
$684$	0	0
$685$	−2.18643e8	−0.680244
$686$	0	0
$687$	− 5.49180e7i	− 0.169373i
$688$	0	0
$689$	−4.85160e8	−1.48329
$690$	0	0
$691$	4.07837e8i	1.23610i	0.786140	+	0.618049i	$0.212077\pi$
$691$	4.07837e8i	1.23610i	−0.786140	+	0.618049i	$0.787923\pi$
$692$	0	0
$693$	7.65275e7	0.229942
$694$	0	0
$695$	− 1.74063e8i	− 0.518503i
$696$	0	0
$697$	1.61425e8	0.476730
$698$	0	0
$699$	2.07171e8i	0.606593i
$700$	0	0
$701$	−8.89171e7	−0.258126	−0.129063	−	0.991636i	$-0.541197\pi$
$701$	−8.89171e7	−0.258126	−0.129063	+	0.991636i	$0.541197\pi$
$702$	0	0
$703$	7.66150e8i	2.20520i
$704$	0	0
$705$	2.50368e8	0.714515
$706$	0	0
$707$	3.43819e8i	0.972907i
$708$	0	0
$709$	−2.16180e7	−0.0606563	−0.0303282	−	0.999540i	$-0.509655\pi$
$709$	−2.16180e7	−0.0606563	−0.0303282	+	0.999540i	$0.509655\pi$
$710$	0	0
$711$	1.87733e7i	0.0522314i
$712$	0	0
$713$	2.14151e8	0.590815
$714$	0	0
$715$	− 2.66978e8i	− 0.730394i
$716$	0	0
$717$	1.00777e7	0.0273403
$718$	0	0
$719$	4.31754e8i	1.16158i	0.814053	+	0.580791i	$0.197257\pi$
$719$	4.31754e8i	1.16158i	−0.814053	+	0.580791i	$0.802743\pi$
$720$	0	0
$721$	2.35041e8	0.627102
$722$	0	0
$723$	− 2.44051e8i	− 0.645751i
$724$	0	0
$725$	1.20415e8	0.315985
$726$	0	0
$727$	− 7.47458e7i	− 0.194529i	−0.995259	−	0.0972644i	$-0.968991\pi$
$727$	− 7.47458e7i	− 0.194529i	0.995259	−	0.0972644i	$-0.0310092\pi$
$728$	0	0
$729$	−1.43489e7	−0.0370370
$730$	0	0
$731$	8.05850e7i	0.206301i
$732$	0	0
$733$	−1.25888e8	−0.319649	−0.159824	−	0.987145i	$-0.551093\pi$
$733$	−1.25888e8	−0.319649	−0.159824	+	0.987145i	$0.551093\pi$
$734$	0	0
$735$	1.15965e8i	0.292054i
$736$	0	0
$737$	−9.28723e8	−2.31998
$738$	0	0
$739$	1.16941e8i	0.289756i	0.989450	+	0.144878i	$0.0462789\pi$
$739$	1.16941e8i	0.289756i	−0.989450	+	0.144878i	$0.953721\pi$
$740$	0	0
$741$	3.44245e8	0.846084
$742$	0	0
$743$	− 4.12373e8i	− 1.00536i	−0.864471	−	0.502682i	$-0.832347\pi$
$743$	− 4.12373e8i	− 1.00536i	0.864471	−	0.502682i	$-0.167653\pi$
$744$	0	0
$745$	−3.11758e8	−0.753960
$746$	0	0
$747$	− 1.56687e8i	− 0.375898i
$748$	0	0
$749$	8.59753e7	0.204611
$750$	0	0
$751$	− 6.34823e8i	− 1.49876i	−0.662138	−	0.749382i	$-0.730351\pi$
$751$	− 6.34823e8i	− 1.49876i	0.662138	−	0.749382i	$-0.269649\pi$
$752$	0	0
$753$	2.25200e8	0.527452
$754$	0	0
$755$	− 4.54509e8i	− 1.05609i
$756$	0	0
$757$	−2.22740e7	−0.0513465	−0.0256733	−	0.999670i	$-0.508173\pi$
$757$	−2.22740e7	−0.0513465	−0.0256733	+	0.999670i	$0.508173\pi$
$758$	0	0
$759$	3.53465e8i	0.808391i
$760$	0	0
$761$	−5.94900e8	−1.34987	−0.674933	−	0.737879i	$-0.735827\pi$
$761$	−5.94900e8	−1.34987	−0.674933	+	0.737879i	$0.735827\pi$
$762$	0	0
$763$	− 2.73160e8i	− 0.614956i
$764$	0	0
$765$	−3.58231e7	−0.0800163
$766$	0	0
$767$	4.47930e8i	0.992713i
$768$	0	0
$769$	7.13646e8	1.56929	0.784646	−	0.619944i	$-0.212845\pi$
$769$	7.13646e8	1.56929	0.784646	+	0.619944i	$0.212845\pi$
$770$	0	0
$771$	1.47634e8i	0.322124i
$772$	0	0
$773$	−2.56623e8	−0.555594	−0.277797	−	0.960640i	$-0.589604\pi$
$773$	−2.56623e8	−0.555594	−0.277797	+	0.960640i	$0.589604\pi$
$774$	0	0
$775$	1.19649e8i	0.257042i
$776$	0	0
$777$	1.78255e8	0.379996
$778$	0	0
$779$	− 1.23514e9i	− 2.61278i
$780$	0	0
$781$	−1.24711e8	−0.261790
$782$	0	0
$783$	− 6.06155e7i	− 0.126269i
$784$	0	0
$785$	−1.21196e8	−0.250541
$786$	0	0
$787$	− 1.41932e8i	− 0.291176i	−0.989345	−	0.145588i	$-0.953493\pi$
$787$	− 1.41932e8i	− 0.291176i	0.989345	−	0.145588i	$-0.0465074\pi$
$788$	0	0
$789$	1.99402e8	0.405974
$790$	0	0
$791$	1.05434e8i	0.213036i
$792$	0	0
$793$	1.87889e8	0.376775
$794$	0	0
$795$	− 3.86300e8i	− 0.768817i
$796$	0	0
$797$	6.30998e7	0.124639	0.0623194	−	0.998056i	$-0.480150\pi$
$797$	6.30998e7	0.124639	0.0623194	+	0.998056i	$0.480150\pi$
$798$	0	0
$799$	2.92312e8i	0.573068i
$800$	0	0
$801$	1.99030e8	0.387277
$802$	0	0
$803$	1.70002e8i	0.328327i
$804$	0	0
$805$	2.26748e8	0.434666
$806$	0	0
$807$	− 4.97243e8i	− 0.946124i
$808$	0	0
$809$	1.46427e8	0.276551	0.138276	−	0.990394i	$-0.455844\pi$
$809$	1.46427e8	0.276551	0.138276	+	0.990394i	$0.455844\pi$
$810$	0	0
$811$	− 6.30861e8i	− 1.18269i	−0.806419	−	0.591345i	$-0.798597\pi$
$811$	− 6.30861e8i	− 1.18269i	0.806419	−	0.591345i	$-0.201403\pi$
$812$	0	0
$813$	−5.50608e8	−1.02464
$814$	0	0
$815$	2.69183e8i	0.497251i
$816$	0	0
$817$	6.16594e8	1.13066
$818$	0	0
$819$	− 8.00934e7i	− 0.145796i
$820$	0	0
$821$	−1.33109e8	−0.240534	−0.120267	−	0.992742i	$-0.538375\pi$
$821$	−1.33109e8	−0.240534	−0.120267	+	0.992742i	$0.538375\pi$
$822$	0	0
$823$	2.11396e8i	0.379225i	0.981859	+	0.189613i	$0.0607233\pi$
$823$	2.11396e8i	0.379225i	−0.981859	+	0.189613i	$0.939277\pi$
$824$	0	0
$825$	−1.97486e8	−0.351702
$826$	0	0
$827$	− 4.28614e8i	− 0.757792i	−0.925439	−	0.378896i	$-0.876304\pi$
$827$	− 4.28614e8i	− 0.757792i	0.925439	−	0.378896i	$-0.123696\pi$
$828$	0	0
$829$	−5.28303e7	−0.0927298	−0.0463649	−	0.998925i	$-0.514764\pi$
$829$	−5.28303e7	−0.0927298	−0.0463649	+	0.998925i	$0.514764\pi$
$830$	0	0
$831$	3.38304e8i	0.589527i
$832$	0	0
$833$	−1.35392e8	−0.234239
$834$	0	0
$835$	− 5.75472e8i	− 0.988473i
$836$	0	0
$837$	6.02300e7	0.102716
$838$	0	0
$839$	− 4.70210e8i	− 0.796170i	−0.917349	−	0.398085i	$-0.869675\pi$
$839$	− 4.70210e8i	− 0.796170i	0.917349	−	0.398085i	$-0.130325\pi$
$840$	0	0
$841$	−3.38759e8	−0.569513
$842$	0	0
$843$	9.06984e7i	0.151397i
$844$	0	0
$845$	1.54995e8	0.256890
$846$	0	0
$847$	− 1.98807e8i	− 0.327176i
$848$	0	0
$849$	3.75275e8	0.613234
$850$	0	0
$851$	8.23325e8i	1.33593i
$852$	0	0
$853$	−3.92907e8	−0.633057	−0.316528	−	0.948583i	$-0.602517\pi$
$853$	−3.92907e8	−0.633057	−0.316528	+	0.948583i	$0.602517\pi$
$854$	0	0
$855$	2.74099e8i	0.438540i
$856$	0	0
$857$	−1.14312e8	−0.181614	−0.0908071	−	0.995868i	$-0.528945\pi$
$857$	−1.14312e8	−0.181614	−0.0908071	+	0.995868i	$0.528945\pi$
$858$	0	0
$859$	2.34899e8i	0.370596i	0.982682	+	0.185298i	$0.0593251\pi$
$859$	2.34899e8i	0.370596i	−0.982682	+	0.185298i	$0.940675\pi$
$860$	0	0
$861$	−2.87372e8	−0.450230
$862$	0	0
$863$	1.18482e8i	0.184340i	0.995743	+	0.0921699i	$0.0293803\pi$
$863$	1.18482e8i	0.184340i	−0.995743	+	0.0921699i	$0.970620\pi$
$864$	0	0
$865$	−1.66852e8	−0.257800
$866$	0	0
$867$	3.34443e8i	0.513174i
$868$	0	0
$869$	1.30065e8	0.198199
$870$	0	0
$871$	9.71997e8i	1.47099i
$872$	0	0
$873$	−1.35357e8	−0.203441
$874$	0	0
$875$	3.89743e8i	0.581773i
$876$	0	0
$877$	−6.23833e8	−0.924846	−0.462423	−	0.886659i	$-0.653020\pi$
$877$	−6.23833e8	−0.924846	−0.462423	+	0.886659i	$0.653020\pi$
$878$	0	0
$879$	− 3.30295e8i	− 0.486335i
$880$	0	0
$881$	−2.79303e8	−0.408458	−0.204229	−	0.978923i	$-0.565469\pi$
$881$	−2.79303e8	−0.408458	−0.204229	+	0.978923i	$0.565469\pi$
$882$	0	0
$883$	2.09838e8i	0.304791i	0.988320	+	0.152395i	$0.0486987\pi$
$883$	2.09838e8i	0.304791i	−0.988320	+	0.152395i	$0.951301\pi$
$884$	0	0
$885$	−3.56656e8	−0.514541
$886$	0	0
$887$	5.19130e8i	0.743885i	0.928256	+	0.371942i	$0.121308\pi$
$887$	5.19130e8i	0.743885i	−0.928256	+	0.371942i	$0.878692\pi$
$888$	0	0
$889$	−5.85416e7	−0.0833219
$890$	0	0
$891$	9.94121e7i	0.140542i
$892$	0	0
$893$	2.23662e9	3.14078
$894$	0	0
$895$	7.10173e8i	0.990593i
$896$	0	0
$897$	3.69935e8	0.512565
$898$	0	0
$899$	2.54435e8i	0.350186i
$900$	0	0
$901$	4.51017e8	0.616621
$902$	0	0
$903$	− 1.43459e8i	− 0.194834i
$904$	0	0
$905$	5.11243e8	0.689734
$906$	0	0
$907$	6.15114e8i	0.824393i	0.911095	+	0.412196i	$0.135238\pi$
$907$	6.15114e8i	0.824393i	−0.911095	+	0.412196i	$0.864762\pi$
$908$	0	0
$909$	−4.46634e8	−0.594648
$910$	0	0
$911$	− 2.64035e8i	− 0.349226i	−0.984637	−	0.174613i	$-0.944133\pi$
$911$	− 2.64035e8i	− 0.349226i	0.984637	−	0.174613i	$-0.0558674\pi$
$912$	0	0
$913$	−1.08556e9	−1.42640
$914$	0	0
$915$	1.49603e8i	0.195289i
$916$	0	0
$917$	1.31955e8	0.171127
$918$	0	0
$919$	− 5.92856e8i	− 0.763841i	−0.924195	−	0.381920i	$-0.875263\pi$
$919$	− 5.92856e8i	− 0.763841i	0.924195	−	0.381920i	$-0.124737\pi$
$920$	0	0
$921$	−1.02990e8	−0.131831
$922$	0	0
$923$	1.30523e8i	0.165989i
$924$	0	0
$925$	−4.60003e8	−0.581214
$926$	0	0
$927$	3.05328e8i	0.383289i
$928$	0	0
$929$	−2.83619e8	−0.353743	−0.176871	−	0.984234i	$-0.556598\pi$
$929$	−2.83619e8	−0.353743	−0.176871	+	0.984234i	$0.556598\pi$
$930$	0	0
$931$	1.03595e9i	1.28378i
$932$	0	0
$933$	−5.74061e8	−0.706827
$934$	0	0
$935$	2.48189e8i	0.303633i
$936$	0	0
$937$	5.18936e7	0.0630804	0.0315402	−	0.999502i	$-0.489959\pi$
$937$	5.18936e7	0.0630804	0.0315402	+	0.999502i	$0.489959\pi$
$938$	0	0
$939$	3.30422e6i	0.00399092i
$940$	0	0
$941$	1.24148e9	1.48995	0.744976	−	0.667091i	$-0.232461\pi$
$941$	1.24148e9	1.48995	0.744976	+	0.667091i	$0.232461\pi$
$942$	0	0
$943$	− 1.32731e9i	− 1.58285i
$944$	0	0
$945$	6.37729e7	0.0755685
$946$	0	0
$947$	− 1.42668e9i	− 1.67988i	−0.542682	−	0.839938i	$-0.682591\pi$
$947$	− 1.42668e9i	− 1.67988i	0.542682	−	0.839938i	$-0.317409\pi$
$948$	0	0
$949$	1.77923e8	0.208178
$950$	0	0
$951$	− 5.59959e8i	− 0.651050i
$952$	0	0
$953$	1.00870e9	1.16542	0.582710	−	0.812680i	$-0.301992\pi$
$953$	1.00870e9	1.16542	0.582710	+	0.812680i	$0.301992\pi$
$954$	0	0
$955$	8.07298e8i	0.926880i
$956$	0	0
$957$	−4.19956e8	−0.479147
$958$	0	0
$959$	4.54442e8i	0.515255i
$960$	0	0
$961$	6.34686e8	0.715137
$962$	0	0
$963$	1.11685e8i	0.125060i
$964$	0	0
$965$	−4.85248e8	−0.539986
$966$	0	0
$967$	− 1.63537e9i	− 1.80858i	−0.426922	−	0.904289i	$-0.640402\pi$
$967$	− 1.63537e9i	− 1.80858i	0.426922	−	0.904289i	$-0.359598\pi$
$968$	0	0
$969$	−3.20019e8	−0.351726
$970$	0	0
$971$	1.36973e9i	1.49616i	0.663608	+	0.748081i	$0.269024\pi$
$971$	1.36973e9i	1.49616i	−0.663608	+	0.748081i	$0.730976\pi$
$972$	0	0
$973$	−3.61782e8	−0.392743
$974$	0	0
$975$	2.06688e8i	0.222998i
$976$	0	0
$977$	−1.26732e9	−1.35895	−0.679473	−	0.733701i	$-0.737791\pi$
$977$	−1.26732e9	−1.35895	−0.679473	+	0.733701i	$0.737791\pi$
$978$	0	0
$979$	− 1.37892e9i	− 1.46957i
$980$	0	0
$981$	3.54846e8	0.375865
$982$	0	0
$983$	2.57681e8i	0.271283i	0.990758	+	0.135641i	$0.0433095\pi$
$983$	2.57681e8i	0.271283i	−0.990758	+	0.135641i	$0.956690\pi$
$984$	0	0
$985$	−3.82383e8	−0.400120
$986$	0	0
$987$	− 5.20380e8i	− 0.541214i
$988$	0	0
$989$	6.62609e8	0.684965
$990$	0	0
$991$	7.74814e8i	0.796117i	0.917360	+	0.398058i	$0.130316\pi$
$991$	7.74814e8i	0.796117i	−0.917360	+	0.398058i	$0.869684\pi$
$992$	0	0
$993$	−3.94153e8	−0.402547
$994$	0	0
$995$	− 2.69890e8i	− 0.273980i
$996$	0	0
$997$	−4.94304e8	−0.498780	−0.249390	−	0.968403i	$-0.580230\pi$
$997$	−4.94304e8	−0.498780	−0.249390	+	0.968403i	$0.580230\pi$
$998$	0	0
$999$	2.31560e8i	0.232256i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.7.g.a.31.1	✓	2
3.2	odd	2		144.7.g.e.127.2		2
4.3	odd	2	inner	48.7.g.a.31.2	yes	2
8.3	odd	2		192.7.g.c.127.1		2
8.5	even	2		192.7.g.c.127.2		2
12.11	even	2		144.7.g.e.127.1		2
16.3	odd	4		768.7.b.c.127.2		4
16.5	even	4		768.7.b.c.127.1		4
16.11	odd	4		768.7.b.c.127.3		4
16.13	even	4		768.7.b.c.127.4		4
24.5	odd	2		576.7.g.e.127.2		2
24.11	even	2		576.7.g.e.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.7.g.a.31.1	✓	2	1.1	even	1	trivial
48.7.g.a.31.2	yes	2	4.3	odd	2	inner
144.7.g.e.127.1		2	12.11	even	2
144.7.g.e.127.2		2	3.2	odd	2
192.7.g.c.127.1		2	8.3	odd	2
192.7.g.c.127.2		2	8.5	even	2
576.7.g.e.127.1		2	24.11	even	2
576.7.g.e.127.2		2	24.5	odd	2
768.7.b.c.127.1		4	16.5	even	4
768.7.b.c.127.2		4	16.3	odd	4
768.7.b.c.127.3		4	16.11	odd	4
768.7.b.c.127.4		4	16.13	even	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.5885i q^{3} -90.0000 q^{5} +187.061i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q-15.5885i q^{3} -90.0000 q^{5} +187.061i q^{7} -243.000 q^{9} +1683.55i q^{11} +1762.00 q^{13} +1402.96i q^{15} -1638.00 q^{17} +12533.1i q^{19} +2916.00 q^{21} +13468.4i q^{23} -7525.00 q^{25} +3788.00i q^{27} -16002.0 q^{29} -15900.2i q^{31} +26244.0 q^{33} -16835.5i q^{35} +61130.0 q^{37} -27466.9i q^{39} -98550.0 q^{41} -49197.2i q^{43} +21870.0 q^{45} -178457. i q^{47} +82657.0 q^{49} +25533.9i q^{51} -275346. q^{53} -151520. i q^{55} +195372. q^{57} +254217. i q^{59} +106634. q^{61} -45455.9i q^{63} -158580. q^{65} +551644. i q^{67} +209952. q^{69} +74076.3i q^{71} +100978. q^{73} +117303. i q^{75} -314928. q^{77} -77256.4i q^{79} +59049.0 q^{81} +644801. i q^{83} +147420. q^{85} +249446. i q^{87} -819054. q^{89} +329602. i q^{91} -247860. q^{93} -1.12798e6i q^{95} +557026. q^{97} -409103. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 180 q^{5} - 486 q^{9}+O(q^{10}) \) 2 * q - 180 * q^5 - 486 * q^9	\( 2 q - 180 q^{5} - 486 q^{9} + 3524 q^{13} - 3276 q^{17} + 5832 q^{21} - 15050 q^{25} - 32004 q^{29} + 52488 q^{33} + 122260 q^{37} - 197100 q^{41} + 43740 q^{45} + 165314 q^{49} - 550692 q^{53} + 390744 q^{57} + 213268 q^{61} - 317160 q^{65} + 419904 q^{69} + 201956 q^{73} - 629856 q^{77} + 118098 q^{81} + 294840 q^{85} - 1638108 q^{89} - 495720 q^{93} + 1114052 q^{97}+O(q^{100}) \) 2 * q - 180 * q^5 - 486 * q^9 + 3524 * q^13 - 3276 * q^17 + 5832 * q^21 - 15050 * q^25 - 32004 * q^29 + 52488 * q^33 + 122260 * q^37 - 197100 * q^41 + 43740 * q^45 + 165314 * q^49 - 550692 * q^53 + 390744 * q^57 + 213268 * q^61 - 317160 * q^65 + 419904 * q^69 + 201956 * q^73 - 629856 * q^77 + 118098 * q^81 + 294840 * q^85 - 1638108 * q^89 - 495720 * q^93 + 1114052 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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