LMFDB - Embedded newform 160.6.d.a.81.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(81,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.d (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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{90}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.9
Root			$0.593959 + 3.95566i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.93089i q^{3} -25.0000i q^{5} -47.1406 q^{7} +194.963 q^{9} +O(q^{10})$	$q-6.93089i q^{3} -25.0000i q^{5} -47.1406 q^{7} +194.963 q^{9} +253.791i q^{11} -1032.09i q^{13} -173.272 q^{15} +756.546 q^{17} -344.235i q^{19} +326.727i q^{21} -4976.22 q^{23} -625.000 q^{25} -3035.47i q^{27} +372.003i q^{29} +134.803 q^{31} +1759.00 q^{33} +1178.52i q^{35} -6653.34i q^{37} -7153.30 q^{39} -15933.8 q^{41} -4771.42i q^{43} -4874.07i q^{45} -14043.7 q^{47} -14584.8 q^{49} -5243.54i q^{51} -5893.80i q^{53} +6344.77 q^{55} -2385.86 q^{57} -30117.5i q^{59} -23143.8i q^{61} -9190.67 q^{63} -25802.3 q^{65} +22646.3i q^{67} +34489.6i q^{69} +53900.1 q^{71} -51287.1 q^{73} +4331.81i q^{75} -11963.9i q^{77} -40838.6 q^{79} +26337.4 q^{81} -108060. i q^{83} -18913.7i q^{85} +2578.32 q^{87} +81947.9 q^{89} +48653.4i q^{91} -934.304i q^{93} -8605.88 q^{95} +52534.7 q^{97} +49479.7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) $ 20 * q + 196 * q^7 - 1620 * q^9	$ 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) $ 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 6.93089i	− 0.444617i	−0.974976	−	0.222308i	$-0.928641\pi$
$3$	− 6.93089i	− 0.444617i	0.974976	−	0.222308i	$-0.0713592\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	−47.1406	−0.363622	−0.181811	−	0.983333i	$-0.558196\pi$
$7$	−47.1406	−0.363622	−0.181811	+	0.983333i	$0.558196\pi$
$8$	0	0
$9$	194.963	0.802316
$10$	0	0
$11$	253.791i	0.632403i	0.948692	+	0.316202i	$0.102408\pi$
$11$	253.791i	0.632403i	−0.948692	+	0.316202i	$0.897592\pi$
$12$	0	0
$13$	− 1032.09i	− 1.69379i	−0.531761	−	0.846894i	$-0.678470\pi$
$13$	− 1032.09i	− 1.69379i	0.531761	−	0.846894i	$-0.321530\pi$
$14$	0	0
$15$	−173.272	−0.198839
$16$	0	0
$17$	756.546	0.634912	0.317456	−	0.948273i	$-0.397171\pi$
$17$	756.546	0.634912	0.317456	+	0.948273i	$0.397171\pi$
$18$	0	0
$19$	− 344.235i	− 0.218762i	−0.994000	−	0.109381i	$-0.965113\pi$
$19$	− 344.235i	− 0.218762i	0.994000	−	0.109381i	$-0.0348868\pi$
$20$	0	0
$21$	326.727i	0.161673i
$22$	0	0
$23$	−4976.22	−1.96146	−0.980731	−	0.195363i	$-0.937411\pi$
$23$	−4976.22	−1.96146	−0.980731	+	0.195363i	$0.937411\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	− 3035.47i	− 0.801340i
$28$	0	0
$29$	372.003i	0.0821395i	0.999156	+	0.0410697i	$0.0130766\pi$
$29$	372.003i	0.0821395i	−0.999156	+	0.0410697i	$0.986923\pi$
$30$	0	0
$31$	134.803	0.0251939	0.0125969	−	0.999921i	$-0.495990\pi$
$31$	134.803	0.0251939	0.0125969	+	0.999921i	$0.495990\pi$
$32$	0	0
$33$	1759.00	0.281177
$34$	0	0
$35$	1178.52i	0.162617i
$36$	0	0
$37$	− 6653.34i	− 0.798980i	−0.916738	−	0.399490i	$-0.869187\pi$
$37$	− 6653.34i	− 0.798980i	0.916738	−	0.399490i	$-0.130813\pi$
$38$	0	0
$39$	−7153.30	−0.753087
$40$	0	0
$41$	−15933.8	−1.48034	−0.740168	−	0.672421i	$-0.765254\pi$
$41$	−15933.8	−1.48034	−0.740168	+	0.672421i	$0.765254\pi$
$42$	0	0
$43$	− 4771.42i	− 0.393529i	−0.980451	−	0.196764i	$-0.936957\pi$
$43$	− 4771.42i	− 0.393529i	0.980451	−	0.196764i	$-0.0630434\pi$
$44$	0	0
$45$	− 4874.07i	− 0.358807i
$46$	0	0
$47$	−14043.7	−0.927337	−0.463668	−	0.886009i	$-0.653467\pi$
$47$	−14043.7	−0.927337	−0.463668	+	0.886009i	$0.653467\pi$
$48$	0	0
$49$	−14584.8	−0.867779
$50$	0	0
$51$	− 5243.54i	− 0.282292i
$52$	0	0
$53$	− 5893.80i	− 0.288208i	−0.989563	−	0.144104i	$-0.953970\pi$
$53$	− 5893.80i	− 0.288208i	0.989563	−	0.144104i	$-0.0460300\pi$
$54$	0	0
$55$	6344.77	0.282819
$56$	0	0
$57$	−2385.86	−0.0972651
$58$	0	0
$59$	− 30117.5i	− 1.12639i	−0.826324	−	0.563195i	$-0.809572\pi$
$59$	− 30117.5i	− 1.12639i	0.826324	−	0.563195i	$-0.190428\pi$
$60$	0	0
$61$	− 23143.8i	− 0.796362i	−0.917307	−	0.398181i	$-0.869642\pi$
$61$	− 23143.8i	− 0.796362i	0.917307	−	0.398181i	$-0.130358\pi$
$62$	0	0
$63$	−9190.67	−0.291740
$64$	0	0
$65$	−25802.3	−0.757485
$66$	0	0
$67$	22646.3i	0.616326i	0.951334	+	0.308163i	$0.0997142\pi$
$67$	22646.3i	0.616326i	−0.951334	+	0.308163i	$0.900286\pi$
$68$	0	0
$69$	34489.6i	0.872099i
$70$	0	0
$71$	53900.1	1.26895	0.634473	−	0.772945i	$-0.281217\pi$
$71$	53900.1	1.26895	0.634473	+	0.772945i	$0.281217\pi$
$72$	0	0
$73$	−51287.1	−1.12642	−0.563211	−	0.826313i	$-0.690434\pi$
$73$	−51287.1	−1.12642	−0.563211	+	0.826313i	$0.690434\pi$
$74$	0	0
$75$	4331.81i	0.0889234i
$76$	0	0
$77$	− 11963.9i	− 0.229956i
$78$	0	0
$79$	−40838.6	−0.736212	−0.368106	−	0.929784i	$-0.619994\pi$
$79$	−40838.6	−0.736212	−0.368106	+	0.929784i	$0.619994\pi$
$80$	0	0
$81$	26337.4	0.446027
$82$	0	0
$83$	− 108060.i	− 1.72175i	−0.508814	−	0.860877i	$-0.669916\pi$
$83$	− 108060.i	− 1.72175i	0.508814	−	0.860877i	$-0.330084\pi$
$84$	0	0
$85$	− 18913.7i	− 0.283941i
$86$	0	0
$87$	2578.32	0.0365206
$88$	0	0
$89$	81947.9	1.09664	0.548318	−	0.836270i	$-0.315268\pi$
$89$	81947.9	1.09664	0.548318	+	0.836270i	$0.315268\pi$
$90$	0	0
$91$	48653.4i	0.615899i
$92$	0	0
$93$	− 934.304i	− 0.0112016i
$94$	0	0
$95$	−8605.88	−0.0978332
$96$	0	0
$97$	52534.7	0.566914	0.283457	−	0.958985i	$-0.408519\pi$
$97$	52534.7	0.566914	0.283457	+	0.958985i	$0.408519\pi$
$98$	0	0
$99$	49479.7i	0.507387i
$100$	0	0
$101$	66010.1i	0.643883i	0.946760	+	0.321942i	$0.104335\pi$
$101$	66010.1i	0.643883i	−0.946760	+	0.321942i	$0.895665\pi$
$102$	0	0
$103$	4624.62	0.0429520	0.0214760	−	0.999769i	$-0.493163\pi$
$103$	4624.62	0.0429520	0.0214760	+	0.999769i	$0.493163\pi$
$104$	0	0
$105$	8168.16	0.0723022
$106$	0	0
$107$	177733.i	1.50075i	0.661013	+	0.750375i	$0.270127\pi$
$107$	177733.i	1.50075i	−0.661013	+	0.750375i	$0.729873\pi$
$108$	0	0
$109$	157581.i	1.27039i	0.772352	+	0.635194i	$0.219080\pi$
$109$	157581.i	1.27039i	−0.772352	+	0.635194i	$0.780920\pi$
$110$	0	0
$111$	−46113.6	−0.355240
$112$	0	0
$113$	203402.	1.49851	0.749255	−	0.662282i	$-0.230412\pi$
$113$	203402.	1.49851	0.749255	+	0.662282i	$0.230412\pi$
$114$	0	0
$115$	124405.i	0.877192i
$116$	0	0
$117$	− 201219.i	− 1.35895i
$118$	0	0
$119$	−35664.1	−0.230868
$120$	0	0
$121$	96641.3	0.600066
$122$	0	0
$123$	110436.i	0.658183i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	275318.	1.51469	0.757347	−	0.653012i	$-0.226495\pi$
$127$	275318.	1.51469	0.757347	+	0.653012i	$0.226495\pi$
$128$	0	0
$129$	−33070.2	−0.174969
$130$	0	0
$131$	− 172997.i	− 0.880765i	−0.897810	−	0.440383i	$-0.854843\pi$
$131$	− 172997.i	− 0.880765i	0.897810	−	0.440383i	$-0.145157\pi$
$132$	0	0
$133$	16227.5i	0.0795466i
$134$	0	0
$135$	−75886.8	−0.358370
$136$	0	0
$137$	−177934.	−0.809947	−0.404973	−	0.914328i	$-0.632719\pi$
$137$	−177934.	−0.809947	−0.404973	+	0.914328i	$0.632719\pi$
$138$	0	0
$139$	− 133948.i	− 0.588028i	−0.955801	−	0.294014i	$-0.905009\pi$
$139$	− 133948.i	− 0.588028i	0.955801	−	0.294014i	$-0.0949912\pi$
$140$	0	0
$141$	97335.5i	0.412309i
$142$	0	0
$143$	261935.	1.07116
$144$	0	0
$145$	9300.09	0.0367339
$146$	0	0
$147$	101085.i	0.385829i
$148$	0	0
$149$	494438.i	1.82451i	0.409623	+	0.912255i	$0.365660\pi$
$149$	494438.i	1.82451i	−0.409623	+	0.912255i	$0.634340\pi$
$150$	0	0
$151$	162378.	0.579543	0.289772	−	0.957096i	$-0.406421\pi$
$151$	162378.	0.579543	0.289772	+	0.957096i	$0.406421\pi$
$152$	0	0
$153$	147498.	0.509400
$154$	0	0
$155$	− 3370.07i	− 0.0112670i
$156$	0	0
$157$	− 152314.i	− 0.493163i	−0.969122	−	0.246581i	$-0.920693\pi$
$157$	− 152314.i	− 0.493163i	0.969122	−	0.246581i	$-0.0793073\pi$
$158$	0	0
$159$	−40849.3	−0.128142
$160$	0	0
$161$	234582.	0.713231
$162$	0	0
$163$	− 620995.i	− 1.83071i	−0.402652	−	0.915353i	$-0.631911\pi$
$163$	− 620995.i	− 1.83071i	0.402652	−	0.915353i	$-0.368089\pi$
$164$	0	0
$165$	− 43974.9i	− 0.125746i
$166$	0	0
$167$	252049.	0.699348	0.349674	−	0.936871i	$-0.386292\pi$
$167$	252049.	0.699348	0.349674	+	0.936871i	$0.386292\pi$
$168$	0	0
$169$	−693917.	−1.86892
$170$	0	0
$171$	− 67113.0i	− 0.175516i
$172$	0	0
$173$	− 258619.i	− 0.656971i	−0.944509	−	0.328485i	$-0.893462\pi$
$173$	− 258619.i	− 0.656971i	0.944509	−	0.328485i	$-0.106538\pi$
$174$	0	0
$175$	29462.9	0.0727244
$176$	0	0
$177$	−208741.	−0.500812
$178$	0	0
$179$	574713.i	1.34066i	0.742063	+	0.670330i	$0.233847\pi$
$179$	574713.i	1.34066i	−0.742063	+	0.670330i	$0.766153\pi$
$180$	0	0
$181$	− 428043.i	− 0.971160i	−0.874192	−	0.485580i	$-0.838609\pi$
$181$	− 428043.i	− 0.971160i	0.874192	−	0.485580i	$-0.161391\pi$
$182$	0	0
$183$	−160407.	−0.354076
$184$	0	0
$185$	−166334.	−0.357315
$186$	0	0
$187$	192004.i	0.401520i
$188$	0	0
$189$	143094.i	0.291385i
$190$	0	0
$191$	−590225.	−1.17067	−0.585335	−	0.810792i	$-0.699037\pi$
$191$	−590225.	−1.17067	−0.585335	+	0.810792i	$0.699037\pi$
$192$	0	0
$193$	−186892.	−0.361158	−0.180579	−	0.983560i	$-0.557797\pi$
$193$	−186892.	−0.361158	−0.180579	+	0.983560i	$0.557797\pi$
$194$	0	0
$195$	178833.i	0.336791i
$196$	0	0
$197$	− 292002.i	− 0.536069i	−0.963409	−	0.268034i	$-0.913626\pi$
$197$	− 292002.i	− 0.536069i	0.963409	−	0.268034i	$-0.0863741\pi$
$198$	0	0
$199$	−666267.	−1.19266	−0.596329	−	0.802740i	$-0.703375\pi$
$199$	−666267.	−1.19266	−0.596329	+	0.802740i	$0.703375\pi$
$200$	0	0
$201$	156959.	0.274029
$202$	0	0
$203$	− 17536.5i	− 0.0298677i
$204$	0	0
$205$	398346.i	0.662027i
$206$	0	0
$207$	−970177.	−1.57371
$208$	0	0
$209$	87363.7	0.138346
$210$	0	0
$211$	− 881071.i	− 1.36240i	−0.732097	−	0.681200i	$-0.761458\pi$
$211$	− 881071.i	− 1.36240i	0.732097	−	0.681200i	$-0.238542\pi$
$212$	0	0
$213$	− 373576.i	− 0.564195i
$214$	0	0
$215$	−119285.	−0.175991
$216$	0	0
$217$	−6354.69	−0.00916105
$218$	0	0
$219$	355466.i	0.500826i
$220$	0	0
$221$	− 780824.i	− 1.07541i
$222$	0	0
$223$	1.01162e6	1.36225	0.681124	−	0.732168i	$-0.261491\pi$
$223$	1.01162e6	1.36225	0.681124	+	0.732168i	$0.261491\pi$
$224$	0	0
$225$	−121852.	−0.160463
$226$	0	0
$227$	− 1.03227e6i	− 1.32962i	−0.747012	−	0.664811i	$-0.768512\pi$
$227$	− 1.03227e6i	− 1.32962i	0.747012	−	0.664811i	$-0.231488\pi$
$228$	0	0
$229$	269099.i	0.339097i	0.985522	+	0.169549i	$0.0542309\pi$
$229$	269099.i	0.339097i	−0.985522	+	0.169549i	$0.945769\pi$
$230$	0	0
$231$	−82920.2	−0.102242
$232$	0	0
$233$	1.25412e6	1.51339	0.756693	−	0.653770i	$-0.226814\pi$
$233$	1.25412e6	1.51339	0.756693	+	0.653770i	$0.226814\pi$
$234$	0	0
$235$	351093.i	0.414718i
$236$	0	0
$237$	283048.i	0.327332i
$238$	0	0
$239$	−995406.	−1.12721	−0.563606	−	0.826044i	$-0.690586\pi$
$239$	−995406.	−1.12721	−0.563606	+	0.826044i	$0.690586\pi$
$240$	0	0
$241$	−1.38569e6	−1.53682	−0.768409	−	0.639959i	$-0.778951\pi$
$241$	−1.38569e6	−1.53682	−0.768409	+	0.639959i	$0.778951\pi$
$242$	0	0
$243$	− 920162.i	− 0.999651i
$244$	0	0
$245$	364619.i	0.388083i
$246$	0	0
$247$	−355282.	−0.370536
$248$	0	0
$249$	−748954.	−0.765521
$250$	0	0
$251$	856001.i	0.857610i	0.903397	+	0.428805i	$0.141065\pi$
$251$	856001.i	0.857610i	−0.903397	+	0.428805i	$0.858935\pi$
$252$	0	0
$253$	− 1.26292e6i	− 1.24043i
$254$	0	0
$255$	−131089.	−0.126245
$256$	0	0
$257$	1.14881e6	1.08497	0.542484	−	0.840066i	$-0.317484\pi$
$257$	1.14881e6	1.08497	0.542484	+	0.840066i	$0.317484\pi$
$258$	0	0
$259$	313643.i	0.290527i
$260$	0	0
$261$	72526.8i	0.0659018i
$262$	0	0
$263$	133562.	0.119067	0.0595337	−	0.998226i	$-0.481039\pi$
$263$	133562.	0.119067	0.0595337	+	0.998226i	$0.481039\pi$
$264$	0	0
$265$	−147345.	−0.128891
$266$	0	0
$267$	− 567972.i	− 0.487583i
$268$	0	0
$269$	476429.i	0.401437i	0.979649	+	0.200718i	$0.0643276\pi$
$269$	476429.i	0.401437i	−0.979649	+	0.200718i	$0.935672\pi$
$270$	0	0
$271$	2.28597e6	1.89081	0.945404	−	0.325902i	$-0.105668\pi$
$271$	2.28597e6	1.89081	0.945404	+	0.325902i	$0.105668\pi$
$272$	0	0
$273$	337211.	0.273839
$274$	0	0
$275$	− 158619.i	− 0.126481i
$276$	0	0
$277$	1.02985e6i	0.806448i	0.915101	+	0.403224i	$0.132110\pi$
$277$	1.02985e6i	0.806448i	−0.915101	+	0.403224i	$0.867890\pi$
$278$	0	0
$279$	26281.5	0.0202134
$280$	0	0
$281$	494955.	0.373938	0.186969	−	0.982366i	$-0.440134\pi$
$281$	494955.	0.373938	0.186969	+	0.982366i	$0.440134\pi$
$282$	0	0
$283$	299542.i	0.222327i	0.993802	+	0.111163i	$0.0354577\pi$
$283$	299542.i	0.222327i	−0.993802	+	0.111163i	$0.964542\pi$
$284$	0	0
$285$	59646.4i	0.0434983i
$286$	0	0
$287$	751131.	0.538283
$288$	0	0
$289$	−847494.	−0.596887
$290$	0	0
$291$	− 364112.i	− 0.252059i
$292$	0	0
$293$	− 525785.i	− 0.357799i	−0.983867	−	0.178900i	$-0.942746\pi$
$293$	− 525785.i	− 0.357799i	0.983867	−	0.178900i	$-0.0572537\pi$
$294$	0	0
$295$	−752937.	−0.503737
$296$	0	0
$297$	770375.	0.506770
$298$	0	0
$299$	5.13591e6i	3.32230i
$300$	0	0
$301$	224928.i	0.143096i
$302$	0	0
$303$	457509.	0.286281
$304$	0	0
$305$	−578596.	−0.356144
$306$	0	0
$307$	− 1.90796e6i	− 1.15538i	−0.816257	−	0.577689i	$-0.803955\pi$
$307$	− 1.90796e6i	− 1.15538i	0.816257	−	0.577689i	$-0.196045\pi$
$308$	0	0
$309$	− 32052.8i	− 0.0190972i
$310$	0	0
$311$	2.53102e6	1.48387	0.741933	−	0.670474i	$-0.233909\pi$
$311$	2.53102e6	1.48387	0.741933	+	0.670474i	$0.233909\pi$
$312$	0	0
$313$	−1.14121e6	−0.658423	−0.329212	−	0.944256i	$-0.606783\pi$
$313$	−1.14121e6	−0.658423	−0.329212	+	0.944256i	$0.606783\pi$
$314$	0	0
$315$	229767.i	0.130470i
$316$	0	0
$317$	2.71214e6i	1.51588i	0.652326	+	0.757938i	$0.273793\pi$
$317$	2.71214e6i	1.51588i	−0.652326	+	0.757938i	$0.726207\pi$
$318$	0	0
$319$	−94411.0	−0.0519453
$320$	0	0
$321$	1.23185e6	0.667258
$322$	0	0
$323$	− 260430.i	− 0.138894i
$324$	0	0
$325$	645056.i	0.338758i
$326$	0	0
$327$	1.09217e6	0.564836
$328$	0	0
$329$	662030.	0.337200
$330$	0	0
$331$	− 238802.i	− 0.119803i	−0.998204	−	0.0599015i	$-0.980921\pi$
$331$	− 238802.i	− 0.119803i	0.998204	−	0.0599015i	$-0.0190787\pi$
$332$	0	0
$333$	− 1.29715e6i	− 0.641034i
$334$	0	0
$335$	566158.	0.275629
$336$	0	0
$337$	−21973.1	−0.0105394	−0.00526971	−	0.999986i	$-0.501677\pi$
$337$	−21973.1	−0.0105394	−0.00526971	+	0.999986i	$0.501677\pi$
$338$	0	0
$339$	− 1.40976e6i	− 0.666262i
$340$	0	0
$341$	34211.7i	0.0159327i
$342$	0	0
$343$	1.47983e6	0.679166
$344$	0	0
$345$	862241.	0.390015
$346$	0	0
$347$	1.79508e6i	0.800312i	0.916447	+	0.400156i	$0.131044\pi$
$347$	1.79508e6i	0.800312i	−0.916447	+	0.400156i	$0.868956\pi$
$348$	0	0
$349$	2.42117e6i	1.06405i	0.846729	+	0.532025i	$0.178569\pi$
$349$	2.42117e6i	1.06405i	−0.846729	+	0.532025i	$0.821431\pi$
$350$	0	0
$351$	−3.13288e6	−1.35730
$352$	0	0
$353$	2.66389e6	1.13784	0.568918	−	0.822395i	$-0.307362\pi$
$353$	2.66389e6	1.13784	0.568918	+	0.822395i	$0.307362\pi$
$354$	0	0
$355$	− 1.34750e6i	− 0.567490i
$356$	0	0
$357$	247184.i	0.102648i
$358$	0	0
$359$	1.36013e6	0.556986	0.278493	−	0.960438i	$-0.410165\pi$
$359$	1.36013e6	0.556986	0.278493	+	0.960438i	$0.410165\pi$
$360$	0	0
$361$	2.35760e6	0.952143
$362$	0	0
$363$	− 669810.i	− 0.266800i
$364$	0	0
$365$	1.28218e6i	0.503751i
$366$	0	0
$367$	−2.21077e6	−0.856796	−0.428398	−	0.903590i	$-0.640922\pi$
$367$	−2.21077e6	−0.856796	−0.428398	+	0.903590i	$0.640922\pi$
$368$	0	0
$369$	−3.10650e6	−1.18770
$370$	0	0
$371$	277838.i	0.104799i
$372$	0	0
$373$	− 573375.i	− 0.213386i	−0.994292	−	0.106693i	$-0.965974\pi$
$373$	− 573375.i	− 0.213386i	0.994292	−	0.106693i	$-0.0340263\pi$
$374$	0	0
$375$	108295.	0.0397677
$376$	0	0
$377$	383941.	0.139127
$378$	0	0
$379$	3.23007e6i	1.15509i	0.816360	+	0.577543i	$0.195988\pi$
$379$	3.23007e6i	1.15509i	−0.816360	+	0.577543i	$0.804012\pi$
$380$	0	0
$381$	− 1.90820e6i	− 0.673459i
$382$	0	0
$383$	−452284.	−0.157549	−0.0787743	−	0.996892i	$-0.525101\pi$
$383$	−452284.	−0.157549	−0.0787743	+	0.996892i	$0.525101\pi$
$384$	0	0
$385$	−299096.	−0.102839
$386$	0	0
$387$	− 930249.i	− 0.315734i
$388$	0	0
$389$	− 2.60438e6i	− 0.872632i	−0.899794	−	0.436316i	$-0.856283\pi$
$389$	− 2.60438e6i	− 0.872632i	0.899794	−	0.436316i	$-0.143717\pi$
$390$	0	0
$391$	−3.76474e6	−1.24536
$392$	0	0
$393$	−1.19902e6	−0.391603
$394$	0	0
$395$	1.02097e6i	0.329244i
$396$	0	0
$397$	− 5.23930e6i	− 1.66839i	−0.551472	−	0.834193i	$-0.685934\pi$
$397$	− 5.23930e6i	− 1.66839i	0.551472	−	0.834193i	$-0.314066\pi$
$398$	0	0
$399$	112471.	0.0353677
$400$	0	0
$401$	2.42996e6	0.754639	0.377319	−	0.926083i	$-0.376846\pi$
$401$	2.42996e6	0.754639	0.377319	+	0.926083i	$0.376846\pi$
$402$	0	0
$403$	− 139129.i	− 0.0426731i
$404$	0	0
$405$	− 658436.i	− 0.199469i
$406$	0	0
$407$	1.68856e6	0.505277
$408$	0	0
$409$	2.21703e6	0.655336	0.327668	−	0.944793i	$-0.393737\pi$
$409$	2.21703e6	0.655336	0.327668	+	0.944793i	$0.393737\pi$
$410$	0	0
$411$	1.23324e6i	0.360116i
$412$	0	0
$413$	1.41976e6i	0.409580i
$414$	0	0
$415$	−2.70151e6	−0.769991
$416$	0	0
$417$	−928376.	−0.261447
$418$	0	0
$419$	− 5.47841e6i	− 1.52447i	−0.647299	−	0.762236i	$-0.724101\pi$
$419$	− 5.47841e6i	− 1.52447i	0.647299	−	0.762236i	$-0.275899\pi$
$420$	0	0
$421$	1.25486e6i	0.345055i	0.985005	+	0.172528i	$0.0551934\pi$
$421$	1.25486e6i	0.345055i	−0.985005	+	0.172528i	$0.944807\pi$
$422$	0	0
$423$	−2.73800e6	−0.744017
$424$	0	0
$425$	−472842.	−0.126982
$426$	0	0
$427$	1.09101e6i	0.289575i
$428$	0	0
$429$	− 1.81544e6i	− 0.476255i
$430$	0	0
$431$	1.51168e6	0.391984	0.195992	−	0.980606i	$-0.437207\pi$
$431$	1.51168e6	0.391984	0.195992	+	0.980606i	$0.437207\pi$
$432$	0	0
$433$	−4.14633e6	−1.06278	−0.531390	−	0.847127i	$-0.678330\pi$
$433$	−4.14633e6	−1.06278	−0.531390	+	0.847127i	$0.678330\pi$
$434$	0	0
$435$	− 64457.9i	− 0.0163325i
$436$	0	0
$437$	1.71299e6i	0.429093i
$438$	0	0
$439$	−4.17351e6	−1.03357	−0.516786	−	0.856115i	$-0.672872\pi$
$439$	−4.17351e6	−1.03357	−0.516786	+	0.856115i	$0.672872\pi$
$440$	0	0
$441$	−2.84349e6	−0.696233
$442$	0	0
$443$	− 1.12969e6i	− 0.273496i	−0.990606	−	0.136748i	$-0.956335\pi$
$443$	− 1.12969e6i	− 0.273496i	0.990606	−	0.136748i	$-0.0436651\pi$
$444$	0	0
$445$	− 2.04870e6i	− 0.490431i
$446$	0	0
$447$	3.42690e6	0.811208
$448$	0	0
$449$	−584617.	−0.136853	−0.0684266	−	0.997656i	$-0.521798\pi$
$449$	−584617.	−0.136853	−0.0684266	+	0.997656i	$0.521798\pi$
$450$	0	0
$451$	− 4.04386e6i	− 0.936170i
$452$	0	0
$453$	− 1.12543e6i	− 0.257675i
$454$	0	0
$455$	1.21633e6	0.275438
$456$	0	0
$457$	−5.96781e6	−1.33667	−0.668335	−	0.743860i	$-0.732993\pi$
$457$	−5.96781e6	−1.33667	−0.668335	+	0.743860i	$0.732993\pi$
$458$	0	0
$459$	− 2.29648e6i	− 0.508780i
$460$	0	0
$461$	5.60132e6i	1.22755i	0.789482	+	0.613774i	$0.210349\pi$
$461$	5.60132e6i	1.22755i	−0.789482	+	0.613774i	$0.789651\pi$
$462$	0	0
$463$	−2.41865e6	−0.524348	−0.262174	−	0.965021i	$-0.584439\pi$
$463$	−2.41865e6	−0.524348	−0.262174	+	0.965021i	$0.584439\pi$
$464$	0	0
$465$	−23357.6	−0.00500952
$466$	0	0
$467$	598924.i	0.127081i	0.997979	+	0.0635403i	$0.0202392\pi$
$467$	598924.i	0.127081i	−0.997979	+	0.0635403i	$0.979761\pi$
$468$	0	0
$469$	− 1.06756e6i	− 0.224110i
$470$	0	0
$471$	−1.05567e6	−0.219269
$472$	0	0
$473$	1.21094e6	0.248869
$474$	0	0
$475$	215147.i	0.0437523i
$476$	0	0
$477$	− 1.14907e6i	− 0.231234i
$478$	0	0
$479$	6.30180e6	1.25495	0.627474	−	0.778637i	$-0.284089\pi$
$479$	6.30180e6	1.25495	0.627474	+	0.778637i	$0.284089\pi$
$480$	0	0
$481$	−6.86685e6	−1.35330
$482$	0	0
$483$	− 1.62586e6i	− 0.317115i
$484$	0	0
$485$	− 1.31337e6i	− 0.253532i
$486$	0	0
$487$	−4.59697e6	−0.878314	−0.439157	−	0.898410i	$-0.644723\pi$
$487$	−4.59697e6	−0.878314	−0.439157	+	0.898410i	$0.644723\pi$
$488$	0	0
$489$	−4.30405e6	−0.813963
$490$	0	0
$491$	2.80880e6i	0.525795i	0.964824	+	0.262898i	$0.0846782\pi$
$491$	2.80880e6i	0.525795i	−0.964824	+	0.262898i	$0.915322\pi$
$492$	0	0
$493$	281438.i	0.0521513i
$494$	0	0
$495$	1.23699e6	0.226910
$496$	0	0
$497$	−2.54088e6	−0.461417
$498$	0	0
$499$	− 8.10859e6i	− 1.45779i	−0.684628	−	0.728893i	$-0.740035\pi$
$499$	− 8.10859e6i	− 1.45779i	0.684628	−	0.728893i	$-0.259965\pi$
$500$	0	0
$501$	− 1.74692e6i	− 0.310942i
$502$	0	0
$503$	2.07254e6	0.365244	0.182622	−	0.983183i	$-0.441542\pi$
$503$	2.07254e6	0.365244	0.182622	+	0.983183i	$0.441542\pi$
$504$	0	0
$505$	1.65025e6	0.287953
$506$	0	0
$507$	4.80946e6i	0.830953i
$508$	0	0
$509$	5.79862e6i	0.992043i	0.868310	+	0.496021i	$0.165206\pi$
$509$	5.79862e6i	0.992043i	−0.868310	+	0.496021i	$0.834794\pi$
$510$	0	0
$511$	2.41771e6	0.409592
$512$	0	0
$513$	−1.04492e6	−0.175302
$514$	0	0
$515$	− 115616.i	− 0.0192087i
$516$	0	0
$517$	− 3.56417e6i	− 0.586451i
$518$	0	0
$519$	−1.79246e6	−0.292100
$520$	0	0
$521$	−3.54910e6	−0.572828	−0.286414	−	0.958106i	$-0.592463\pi$
$521$	−3.54910e6	−0.572828	−0.286414	+	0.958106i	$0.592463\pi$
$522$	0	0
$523$	1.24052e7i	1.98312i	0.129651	+	0.991560i	$0.458614\pi$
$523$	1.24052e7i	1.98312i	−0.129651	+	0.991560i	$0.541386\pi$
$524$	0	0
$525$	− 204204.i	− 0.0323345i
$526$	0	0
$527$	101985.	0.0159959
$528$	0	0
$529$	1.83264e7	2.84733
$530$	0	0
$531$	− 5.87179e6i	− 0.903721i
$532$	0	0
$533$	1.64451e7i	2.50738i
$534$	0	0
$535$	4.44332e6	0.671155
$536$	0	0
$537$	3.98327e6	0.596080
$538$	0	0
$539$	− 3.70148e6i	− 0.548786i
$540$	0	0
$541$	− 9.81000e6i	− 1.44104i	−0.693434	−	0.720520i	$-0.743903\pi$
$541$	− 9.81000e6i	− 1.44104i	0.693434	−	0.720520i	$-0.256097\pi$
$542$	0	0
$543$	−2.96672e6	−0.431794
$544$	0	0
$545$	3.93952e6	0.568135
$546$	0	0
$547$	493400.i	0.0705067i	0.999378	+	0.0352534i	$0.0112238\pi$
$547$	493400.i	0.0705067i	−0.999378	+	0.0352534i	$0.988776\pi$
$548$	0	0
$549$	− 4.51218e6i	− 0.638934i
$550$	0	0
$551$	128057.	0.0179690
$552$	0	0
$553$	1.92516e6	0.267703
$554$	0	0
$555$	1.15284e6i	0.158868i
$556$	0	0
$557$	− 8.46020e6i	− 1.15543i	−0.816239	−	0.577714i	$-0.803945\pi$
$557$	− 8.46020e6i	− 1.15543i	0.816239	−	0.577714i	$-0.196055\pi$
$558$	0	0
$559$	−4.92453e6	−0.666554
$560$	0	0
$561$	1.33076e6	0.178523
$562$	0	0
$563$	− 2.63523e6i	− 0.350387i	−0.984534	−	0.175194i	$-0.943945\pi$
$563$	− 2.63523e6i	− 0.350387i	0.984534	−	0.175194i	$-0.0560551\pi$
$564$	0	0
$565$	− 5.08505e6i	− 0.670154i
$566$	0	0
$567$	−1.24156e6	−0.162185
$568$	0	0
$569$	3.68731e6	0.477451	0.238725	−	0.971087i	$-0.423270\pi$
$569$	3.68731e6	0.477451	0.238725	+	0.971087i	$0.423270\pi$
$570$	0	0
$571$	7.92295e6i	1.01694i	0.861079	+	0.508471i	$0.169789\pi$
$571$	7.92295e6i	1.01694i	−0.861079	+	0.508471i	$0.830211\pi$
$572$	0	0
$573$	4.09079e6i	0.520499i
$574$	0	0
$575$	3.11014e6	0.392292
$576$	0	0
$577$	7.78576e6	0.973558	0.486779	−	0.873525i	$-0.338172\pi$
$577$	7.78576e6	0.973558	0.486779	+	0.873525i	$0.338172\pi$
$578$	0	0
$579$	1.29533e6i	0.160577i
$580$	0	0
$581$	5.09403e6i	0.626068i
$582$	0	0
$583$	1.49579e6	0.182264
$584$	0	0
$585$	−5.03048e6	−0.607742
$586$	0	0
$587$	8.90940e6i	1.06722i	0.845731	+	0.533609i	$0.179165\pi$
$587$	8.90940e6i	1.06722i	−0.845731	+	0.533609i	$0.820835\pi$
$588$	0	0
$589$	− 46403.9i	− 0.00551145i
$590$	0	0
$591$	−2.02384e6	−0.238345
$592$	0	0
$593$	2.33553e6	0.272740	0.136370	−	0.990658i	$-0.456456\pi$
$593$	2.33553e6	0.272740	0.136370	+	0.990658i	$0.456456\pi$
$594$	0	0
$595$	891602.i	0.103247i
$596$	0	0
$597$	4.61783e6i	0.530276i
$598$	0	0
$599$	1.19930e7	1.36572	0.682859	−	0.730550i	$-0.260736\pi$
$599$	1.19930e7	1.36572	0.682859	+	0.730550i	$0.260736\pi$
$600$	0	0
$601$	2.37038e6	0.267689	0.133845	−	0.991002i	$-0.457268\pi$
$601$	2.37038e6	0.267689	0.133845	+	0.991002i	$0.457268\pi$
$602$	0	0
$603$	4.41519e6i	0.494488i
$604$	0	0
$605$	− 2.41603e6i	− 0.268358i
$606$	0	0
$607$	−3.22254e6	−0.354998	−0.177499	−	0.984121i	$-0.556801\pi$
$607$	−3.22254e6	−0.354998	−0.177499	+	0.984121i	$0.556801\pi$
$608$	0	0
$609$	−121543.	−0.0132797
$610$	0	0
$611$	1.44944e7i	1.57071i
$612$	0	0
$613$	1.82202e6i	0.195840i	0.995194	+	0.0979199i	$0.0312189\pi$
$613$	1.82202e6i	0.195840i	−0.995194	+	0.0979199i	$0.968781\pi$
$614$	0	0
$615$	2.76089e6	0.294348
$616$	0	0
$617$	8.01310e6	0.847399	0.423699	−	0.905803i	$-0.360731\pi$
$617$	8.01310e6	0.847399	0.423699	+	0.905803i	$0.360731\pi$
$618$	0	0
$619$	− 1.75655e7i	− 1.84261i	−0.388839	−	0.921306i	$-0.627124\pi$
$619$	− 1.75655e7i	− 1.84261i	0.388839	−	0.921306i	$-0.372876\pi$
$620$	0	0
$621$	1.51052e7i	1.57180i
$622$	0	0
$623$	−3.86308e6	−0.398761
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 605508.i	− 0.0615108i
$628$	0	0
$629$	− 5.03356e6i	− 0.507281i
$630$	0	0
$631$	−1.66158e6	−0.166130	−0.0830650	−	0.996544i	$-0.526471\pi$
$631$	−1.66158e6	−0.166130	−0.0830650	+	0.996544i	$0.526471\pi$
$632$	0	0
$633$	−6.10661e6	−0.605746
$634$	0	0
$635$	− 6.88295e6i	− 0.677392i
$636$	0	0
$637$	1.50528e7i	1.46983i
$638$	0	0
$639$	1.05085e7	1.01810
$640$	0	0
$641$	4.45104e6	0.427875	0.213937	−	0.976847i	$-0.431371\pi$
$641$	4.45104e6	0.427875	0.213937	+	0.976847i	$0.431371\pi$
$642$	0	0
$643$	508859.i	0.0485367i	0.999705	+	0.0242684i	$0.00772561\pi$
$643$	508859.i	0.0485367i	−0.999705	+	0.0242684i	$0.992274\pi$
$644$	0	0
$645$	826754.i	0.0782487i
$646$	0	0
$647$	5.45061e6	0.511899	0.255949	−	0.966690i	$-0.417612\pi$
$647$	5.45061e6	0.511899	0.255949	+	0.966690i	$0.417612\pi$
$648$	0	0
$649$	7.64354e6	0.712333
$650$	0	0
$651$	44043.7i	0.00407316i
$652$	0	0
$653$	5.95121e6i	0.546163i	0.961991	+	0.273082i	$0.0880429\pi$
$653$	5.95121e6i	0.546163i	−0.961991	+	0.273082i	$0.911957\pi$
$654$	0	0
$655$	−4.32492e6	−0.393890
$656$	0	0
$657$	−9.99908e6	−0.903747
$658$	0	0
$659$	3.61561e6i	0.324316i	0.986765	+	0.162158i	$0.0518454\pi$
$659$	3.61561e6i	0.324316i	−0.986765	+	0.162158i	$0.948155\pi$
$660$	0	0
$661$	8.17394e6i	0.727659i	0.931466	+	0.363829i	$0.118531\pi$
$661$	8.17394e6i	0.727659i	−0.931466	+	0.363829i	$0.881469\pi$
$662$	0	0
$663$	−5.41181e6	−0.478144
$664$	0	0
$665$	405686.	0.0355743
$666$	0	0
$667$	− 1.85117e6i	− 0.161113i
$668$	0	0
$669$	− 7.01145e6i	− 0.605679i
$670$	0	0
$671$	5.87369e6	0.503622
$672$	0	0
$673$	−1.07614e7	−0.915867	−0.457933	−	0.888987i	$-0.651410\pi$
$673$	−1.07614e7	−0.915867	−0.457933	+	0.888987i	$0.651410\pi$
$674$	0	0
$675$	1.89717e6i	0.160268i
$676$	0	0
$677$	− 1.44154e7i	− 1.20880i	−0.796680	−	0.604401i	$-0.793412\pi$
$677$	− 1.44154e7i	− 1.20880i	0.796680	−	0.604401i	$-0.206588\pi$
$678$	0	0
$679$	−2.47652e6	−0.206142
$680$	0	0
$681$	−7.15454e6	−0.591172
$682$	0	0
$683$	9.22051e6i	0.756316i	0.925741	+	0.378158i	$0.123442\pi$
$683$	9.22051e6i	0.756316i	−0.925741	+	0.378158i	$0.876558\pi$
$684$	0	0
$685$	4.44834e6i	0.362219i
$686$	0	0
$687$	1.86510e6	0.150768
$688$	0	0
$689$	−6.08294e6	−0.488163
$690$	0	0
$691$	− 7.56905e6i	− 0.603040i	−0.953460	−	0.301520i	$-0.902506\pi$
$691$	− 7.56905e6i	− 0.603040i	0.953460	−	0.301520i	$-0.0974940\pi$
$692$	0	0
$693$	− 2.33251e6i	− 0.184497i
$694$	0	0
$695$	−3.34869e6	−0.262974
$696$	0	0
$697$	−1.20547e7	−0.939883
$698$	0	0
$699$	− 8.69218e6i	− 0.672877i
$700$	0	0
$701$	− 1.80188e7i	− 1.38494i	−0.721448	−	0.692469i	$-0.756523\pi$
$701$	− 1.80188e7i	− 1.38494i	0.721448	−	0.692469i	$-0.243477\pi$
$702$	0	0
$703$	−2.29031e6	−0.174786
$704$	0	0
$705$	2.43339e6	0.184390
$706$	0	0
$707$	− 3.11176e6i	− 0.234130i
$708$	0	0
$709$	− 1.63602e7i	− 1.22229i	−0.791520	−	0.611143i	$-0.790710\pi$
$709$	− 1.63602e7i	− 1.22229i	0.791520	−	0.611143i	$-0.209290\pi$
$710$	0	0
$711$	−7.96201e6	−0.590675
$712$	0	0
$713$	−670808.	−0.0494168
$714$	0	0
$715$	− 6.54837e6i	− 0.479036i
$716$	0	0
$717$	6.89905e6i	0.501177i
$718$	0	0
$719$	1.78536e7	1.28796	0.643981	−	0.765042i	$-0.277282\pi$
$719$	1.78536e7	1.28796	0.643981	+	0.765042i	$0.277282\pi$
$720$	0	0
$721$	−218008.	−0.0156183
$722$	0	0
$723$	9.60404e6i	0.683295i
$724$	0	0
$725$	− 232502.i	− 0.0164279i
$726$	0	0
$727$	1.64661e7	1.15546	0.577731	−	0.816227i	$-0.303938\pi$
$727$	1.64661e7	1.15546	0.577731	+	0.816227i	$0.303938\pi$
$728$	0	0
$729$	22455.1	0.00156494
$730$	0	0
$731$	− 3.60980e6i	− 0.249856i
$732$	0	0
$733$	− 1.89907e7i	− 1.30551i	−0.757568	−	0.652756i	$-0.773613\pi$
$733$	− 1.89907e7i	− 1.30551i	0.757568	−	0.652756i	$-0.226387\pi$
$734$	0	0
$735$	2.52713e6	0.172548
$736$	0	0
$737$	−5.74742e6	−0.389766
$738$	0	0
$739$	1.31894e7i	0.888414i	0.895924	+	0.444207i	$0.146514\pi$
$739$	1.31894e7i	0.888414i	−0.895924	+	0.444207i	$0.853486\pi$
$740$	0	0
$741$	2.46242e6i	0.164747i
$742$	0	0
$743$	−1.07718e7	−0.715839	−0.357919	−	0.933752i	$-0.616514\pi$
$743$	−1.07718e7	−0.715839	−0.357919	+	0.933752i	$0.616514\pi$
$744$	0	0
$745$	1.23610e7	0.815946
$746$	0	0
$747$	− 2.10677e7i	− 1.38139i
$748$	0	0
$749$	− 8.37844e6i	− 0.545706i
$750$	0	0
$751$	−9.68507e6	−0.626618	−0.313309	−	0.949651i	$-0.601438\pi$
$751$	−9.68507e6	−0.626618	−0.313309	+	0.949651i	$0.601438\pi$
$752$	0	0
$753$	5.93285e6	0.381308
$754$	0	0
$755$	− 4.05946e6i	− 0.259180i
$756$	0	0
$757$	1.71192e7i	1.08578i	0.839803	+	0.542891i	$0.182670\pi$
$757$	1.71192e7i	1.08578i	−0.839803	+	0.542891i	$0.817330\pi$
$758$	0	0
$759$	−8.75315e6	−0.551518
$760$	0	0
$761$	−6.88918e6	−0.431227	−0.215614	−	0.976479i	$-0.569175\pi$
$761$	−6.88918e6	−0.431227	−0.215614	+	0.976479i	$0.569175\pi$
$762$	0	0
$763$	− 7.42845e6i	− 0.461941i
$764$	0	0
$765$	− 3.68746e6i	− 0.227810i
$766$	0	0
$767$	−3.10840e7	−1.90787
$768$	0	0
$769$	−2.01582e7	−1.22924	−0.614618	−	0.788825i	$-0.710690\pi$
$769$	−2.01582e7	−1.22924	−0.614618	+	0.788825i	$0.710690\pi$
$770$	0	0
$771$	− 7.96230e6i	− 0.482395i
$772$	0	0
$773$	9.58833e6i	0.577158i	0.957456	+	0.288579i	$0.0931827\pi$
$773$	9.58833e6i	0.577158i	−0.957456	+	0.288579i	$0.906817\pi$
$774$	0	0
$775$	−84251.8	−0.00503877
$776$	0	0
$777$	2.17382e6	0.129173
$778$	0	0
$779$	5.48498e6i	0.323841i
$780$	0	0
$781$	1.36793e7i	0.802486i
$782$	0	0
$783$	1.12921e6	0.0658217
$784$	0	0
$785$	−3.80785e6	−0.220549
$786$	0	0
$787$	3.07200e6i	0.176801i	0.996085	+	0.0884003i	$0.0281755\pi$
$787$	3.07200e6i	0.176801i	−0.996085	+	0.0884003i	$0.971825\pi$
$788$	0	0
$789$	− 925703.i	− 0.0529394i
$790$	0	0
$791$	−9.58851e6	−0.544891
$792$	0	0
$793$	−2.38865e7	−1.34887
$794$	0	0
$795$	1.02123e6i	0.0573069i
$796$	0	0
$797$	2.69531e6i	0.150301i	0.997172	+	0.0751506i	$0.0239438\pi$
$797$	2.69531e6i	0.150301i	−0.997172	+	0.0751506i	$0.976056\pi$
$798$	0	0
$799$	−1.06247e7	−0.588777
$800$	0	0
$801$	1.59768e7	0.879849
$802$	0	0
$803$	− 1.30162e7i	− 0.712353i
$804$	0	0
$805$	− 5.86455e6i	− 0.318967i
$806$	0	0
$807$	3.30207e6	0.178486
$808$	0	0
$809$	1.37752e7	0.739991	0.369996	−	0.929034i	$-0.379359\pi$
$809$	1.37752e7	0.739991	0.369996	+	0.929034i	$0.379359\pi$
$810$	0	0
$811$	− 1.00830e7i	− 0.538316i	−0.963096	−	0.269158i	$-0.913255\pi$
$811$	− 1.00830e7i	− 0.538316i	0.963096	−	0.269158i	$-0.0867454\pi$
$812$	0	0
$813$	− 1.58438e7i	− 0.840685i
$814$	0	0
$815$	−1.55249e7	−0.818717
$816$	0	0
$817$	−1.64249e6	−0.0860890
$818$	0	0
$819$	9.48560e6i	0.494146i
$820$	0	0
$821$	− 1.21926e7i	− 0.631303i	−0.948875	−	0.315651i	$-0.897777\pi$
$821$	− 1.21926e7i	− 0.631303i	0.948875	−	0.315651i	$-0.102223\pi$
$822$	0	0
$823$	383000.	0.0197105	0.00985527	−	0.999951i	$-0.496863\pi$
$823$	383000.	0.0197105	0.00985527	+	0.999951i	$0.496863\pi$
$824$	0	0
$825$	−1.09937e6	−0.0562354
$826$	0	0
$827$	− 9.81451e6i	− 0.499005i	−0.968374	−	0.249502i	$-0.919733\pi$
$827$	− 9.81451e6i	− 0.499005i	0.968374	−	0.249502i	$-0.0802671\pi$
$828$	0	0
$829$	− 1.94664e6i	− 0.0983782i	−0.998789	−	0.0491891i	$-0.984336\pi$
$829$	− 1.94664e6i	− 0.0983782i	0.998789	−	0.0491891i	$-0.0156637\pi$
$830$	0	0
$831$	7.13781e6	0.358560
$832$	0	0
$833$	−1.10340e7	−0.550963
$834$	0	0
$835$	− 6.30122e6i	− 0.312758i
$836$	0	0
$837$	− 409190.i	− 0.0201889i
$838$	0	0
$839$	−1.04571e7	−0.512868	−0.256434	−	0.966562i	$-0.582548\pi$
$839$	−1.04571e7	−0.512868	−0.256434	+	0.966562i	$0.582548\pi$
$840$	0	0
$841$	2.03728e7	0.993253
$842$	0	0
$843$	− 3.43048e6i	− 0.166259i
$844$	0	0
$845$	1.73479e7i	0.835806i
$846$	0	0
$847$	−4.55573e6	−0.218197
$848$	0	0
$849$	2.07609e6	0.0988502
$850$	0	0
$851$	3.31085e7i	1.56717i
$852$	0	0
$853$	− 762753.i	− 0.0358931i	−0.999839	−	0.0179466i	$-0.994287\pi$
$853$	− 762753.i	− 0.0358931i	0.999839	−	0.0179466i	$-0.00571288\pi$
$854$	0	0
$855$	−1.67783e6	−0.0784931
$856$	0	0
$857$	1.88500e7	0.876717	0.438358	−	0.898800i	$-0.355560\pi$
$857$	1.88500e7	0.876717	0.438358	+	0.898800i	$0.355560\pi$
$858$	0	0
$859$	2.00534e7i	0.927269i	0.886027	+	0.463634i	$0.153455\pi$
$859$	2.00534e7i	0.927269i	−0.886027	+	0.463634i	$0.846545\pi$
$860$	0	0
$861$	− 5.20601e6i	− 0.239330i
$862$	0	0
$863$	3.43348e6	0.156931	0.0784653	−	0.996917i	$-0.474998\pi$
$863$	3.43348e6	0.156931	0.0784653	+	0.996917i	$0.474998\pi$
$864$	0	0
$865$	−6.46549e6	−0.293806
$866$	0	0
$867$	5.87389e6i	0.265386i
$868$	0	0
$869$	− 1.03645e7i	− 0.465583i
$870$	0	0
$871$	2.33730e7	1.04393
$872$	0	0
$873$	1.02423e7	0.454844
$874$	0	0
$875$	− 736572.i	− 0.0325234i
$876$	0	0
$877$	522042.i	0.0229196i	0.999934	+	0.0114598i	$0.00364784\pi$
$877$	522042.i	0.0229196i	−0.999934	+	0.0114598i	$0.996352\pi$
$878$	0	0
$879$	−3.64416e6	−0.159084
$880$	0	0
$881$	−7.98479e6	−0.346596	−0.173298	−	0.984869i	$-0.555442\pi$
$881$	−7.98479e6	−0.346596	−0.173298	+	0.984869i	$0.555442\pi$
$882$	0	0
$883$	3.50061e7i	1.51092i	0.655195	+	0.755460i	$0.272587\pi$
$883$	3.50061e7i	1.51092i	−0.655195	+	0.755460i	$0.727413\pi$
$884$	0	0
$885$	5.21853e6i	0.223970i
$886$	0	0
$887$	1.12935e7	0.481967	0.240984	−	0.970529i	$-0.422530\pi$
$887$	1.12935e7	0.481967	0.240984	+	0.970529i	$0.422530\pi$
$888$	0	0
$889$	−1.29787e7	−0.550776
$890$	0	0
$891$	6.68419e6i	0.282069i
$892$	0	0
$893$	4.83434e6i	0.202866i
$894$	0	0
$895$	1.43678e7	0.599561
$896$	0	0
$897$	3.55964e7	1.47715
$898$	0	0
$899$	50147.1i	0.00206941i
$900$	0	0
$901$	− 4.45894e6i	− 0.182987i
$902$	0	0
$903$	1.55895e6	0.0636228
$904$	0	0
$905$	−1.07011e7	−0.434316
$906$	0	0
$907$	− 2.01754e7i	− 0.814338i	−0.913353	−	0.407169i	$-0.866516\pi$
$907$	− 2.01754e7i	− 0.814338i	0.913353	−	0.407169i	$-0.133484\pi$
$908$	0	0
$909$	1.28695e7i	0.516598i
$910$	0	0
$911$	−1.61032e7	−0.642859	−0.321430	−	0.946933i	$-0.604163\pi$
$911$	−1.61032e7	−0.642859	−0.321430	+	0.946933i	$0.604163\pi$
$912$	0	0
$913$	2.74247e7	1.08884
$914$	0	0
$915$	4.01018e6i	0.158348i
$916$	0	0
$917$	8.15518e6i	0.320266i
$918$	0	0
$919$	7.96458e6	0.311082	0.155541	−	0.987829i	$-0.450288\pi$
$919$	7.96458e6	0.311082	0.155541	+	0.987829i	$0.450288\pi$
$920$	0	0
$921$	−1.32239e7	−0.513700
$922$	0	0
$923$	− 5.56297e7i	− 2.14933i
$924$	0	0
$925$	4.15834e6i	0.159796i
$926$	0	0
$927$	901629.	0.0344611
$928$	0	0
$929$	−1.21673e7	−0.462547	−0.231273	−	0.972889i	$-0.574289\pi$
$929$	−1.21673e7	−0.462547	−0.231273	+	0.972889i	$0.574289\pi$
$930$	0	0
$931$	5.02059e6i	0.189837i
$932$	0	0
$933$	− 1.75422e7i	− 0.659752i
$934$	0	0
$935$	4.80011e6	0.179565
$936$	0	0
$937$	−9.87499e6	−0.367441	−0.183720	−	0.982979i	$-0.558814\pi$
$937$	−9.87499e6	−0.367441	−0.183720	+	0.982979i	$0.558814\pi$
$938$	0	0
$939$	7.90961e6i	0.292746i
$940$	0	0
$941$	− 1.60023e7i	− 0.589126i	−0.955632	−	0.294563i	$-0.904826\pi$
$941$	− 1.60023e7i	− 0.589126i	0.955632	−	0.294563i	$-0.0951741\pi$
$942$	0	0
$943$	7.92902e7	2.90362
$944$	0	0
$945$	3.57735e6	0.130311
$946$	0	0
$947$	− 1.12341e6i	− 0.0407066i	−0.999793	−	0.0203533i	$-0.993521\pi$
$947$	− 1.12341e6i	− 0.0407066i	0.999793	−	0.0203533i	$-0.00647910\pi$
$948$	0	0
$949$	5.29329e7i	1.90792i
$950$	0	0
$951$	1.87975e7	0.673984
$952$	0	0
$953$	4.80409e7	1.71348	0.856740	−	0.515748i	$-0.172486\pi$
$953$	4.80409e7	1.71348	0.856740	+	0.515748i	$0.172486\pi$
$954$	0	0
$955$	1.47556e7i	0.523539i
$956$	0	0
$957$	654353.i	0.0230957i
$958$	0	0
$959$	8.38790e6	0.294515
$960$	0	0
$961$	−2.86110e7	−0.999365
$962$	0	0
$963$	3.46513e7i	1.20407i
$964$	0	0
$965$	4.67230e6i	0.161515i
$966$	0	0
$967$	−6.17173e6	−0.212247	−0.106123	−	0.994353i	$-0.533844\pi$
$967$	−6.17173e6	−0.212247	−0.106123	+	0.994353i	$0.533844\pi$
$968$	0	0
$969$	−1.80501e6	−0.0617547
$970$	0	0
$971$	− 4.28206e7i	− 1.45749i	−0.684788	−	0.728743i	$-0.740105\pi$
$971$	− 4.28206e7i	− 1.45749i	0.684788	−	0.728743i	$-0.259895\pi$
$972$	0	0
$973$	6.31437e6i	0.213820i
$974$	0	0
$975$	4.47081e6	0.150617
$976$	0	0
$977$	−3.34844e7	−1.12229	−0.561147	−	0.827716i	$-0.689640\pi$
$977$	−3.34844e7	−1.12229	−0.561147	+	0.827716i	$0.689640\pi$
$978$	0	0
$979$	2.07976e7i	0.693517i
$980$	0	0
$981$	3.07223e7i	1.01925i
$982$	0	0
$983$	−1.31219e7	−0.433124	−0.216562	−	0.976269i	$-0.569484\pi$
$983$	−1.31219e7	−0.433124	−0.216562	+	0.976269i	$0.569484\pi$
$984$	0	0
$985$	−7.30005e6	−0.239737
$986$	0	0
$987$	− 4.58846e6i	− 0.149925i
$988$	0	0
$989$	2.37436e7i	0.771891i
$990$	0	0
$991$	3.19619e7	1.03383	0.516914	−	0.856037i	$-0.327081\pi$
$991$	3.19619e7	1.03383	0.516914	+	0.856037i	$0.327081\pi$
$992$	0	0
$993$	−1.65511e6	−0.0532664
$994$	0	0
$995$	1.66567e7i	0.533373i
$996$	0	0
$997$	1.01694e6i	0.0324009i	0.999869	+	0.0162005i	$0.00515699\pi$
$997$	1.01694e6i	0.0324009i	−0.999869	+	0.0162005i	$0.994843\pi$
$998$	0	0
$999$	−2.01960e7	−0.640254

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.9		20
4.3	odd	2		40.6.d.a.21.5	✓	20
5.2	odd	4		800.6.f.b.49.9		20
5.3	odd	4		800.6.f.c.49.12		20
5.4	even	2		800.6.d.c.401.12		20
8.3	odd	2		40.6.d.a.21.6	yes	20
8.5	even	2	inner	160.6.d.a.81.12		20
12.11	even	2		360.6.k.b.181.16		20
20.3	even	4		200.6.f.b.149.6		20
20.7	even	4		200.6.f.c.149.15		20
20.19	odd	2		200.6.d.b.101.16		20
24.11	even	2		360.6.k.b.181.15		20
40.3	even	4		200.6.f.c.149.16		20
40.13	odd	4		800.6.f.b.49.10		20
40.19	odd	2		200.6.d.b.101.15		20
40.27	even	4		200.6.f.b.149.5		20
40.29	even	2		800.6.d.c.401.9		20
40.37	odd	4		800.6.f.c.49.11		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.5	✓	20	4.3	odd	2
40.6.d.a.21.6	yes	20	8.3	odd	2
160.6.d.a.81.9		20	1.1	even	1	trivial
160.6.d.a.81.12		20	8.5	even	2	inner
200.6.d.b.101.15		20	40.19	odd	2
200.6.d.b.101.16		20	20.19	odd	2
200.6.f.b.149.5		20	40.27	even	4
200.6.f.b.149.6		20	20.3	even	4
200.6.f.c.149.15		20	20.7	even	4
200.6.f.c.149.16		20	40.3	even	4
360.6.k.b.181.15		20	24.11	even	2
360.6.k.b.181.16		20	12.11	even	2
800.6.d.c.401.9		20	40.29	even	2
800.6.d.c.401.12		20	5.4	even	2
800.6.f.b.49.9		20	5.2	odd	4
800.6.f.b.49.10		20	40.13	odd	4
800.6.f.c.49.11		20	40.37	odd	4
800.6.f.c.49.12		20	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.93089i q^{3} -25.0000i q^{5} -47.1406 q^{7} +194.963 q^{9} +O(q^{10})\)	\(q-6.93089i q^{3} -25.0000i q^{5} -47.1406 q^{7} +194.963 q^{9} +253.791i q^{11} -1032.09i q^{13} -173.272 q^{15} +756.546 q^{17} -344.235i q^{19} +326.727i q^{21} -4976.22 q^{23} -625.000 q^{25} -3035.47i q^{27} +372.003i q^{29} +134.803 q^{31} +1759.00 q^{33} +1178.52i q^{35} -6653.34i q^{37} -7153.30 q^{39} -15933.8 q^{41} -4771.42i q^{43} -4874.07i q^{45} -14043.7 q^{47} -14584.8 q^{49} -5243.54i q^{51} -5893.80i q^{53} +6344.77 q^{55} -2385.86 q^{57} -30117.5i q^{59} -23143.8i q^{61} -9190.67 q^{63} -25802.3 q^{65} +22646.3i q^{67} +34489.6i q^{69} +53900.1 q^{71} -51287.1 q^{73} +4331.81i q^{75} -11963.9i q^{77} -40838.6 q^{79} +26337.4 q^{81} -108060. i q^{83} -18913.7i q^{85} +2578.32 q^{87} +81947.9 q^{89} +48653.4i q^{91} -934.304i q^{93} -8605.88 q^{95} +52534.7 q^{97} +49479.7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) \) 20 * q + 196 * q^7 - 1620 * q^9	\( 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) \) 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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