LMFDB - Embedded newform 320.9.b.d.191.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,9,Mod(191,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.191"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	320.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-38800] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$130.361155220$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 $ x^16 - 5x^15 + 26x^14 - 834x^13 + 4390x^12 - 61783x^11 + 466168x^10 - 1105435x^9 + 46850799x^8 - 116275535x^7 + 626274432x^6 - 8558999923x^5 + 34408048994x^4 - 448299413930x^3 + 1712647133330x^2 + 15986651928135*x + 206161212459445
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{120}\cdot 5^{16} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.7
Root			$-4.16577 - 5.52698i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.9.b.d.191.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-27.2434i q^{3} -279.508 q^{5} -3325.58i q^{7} +5818.80 q^{9} +6361.21i q^{11} +30777.4 q^{13} +7614.76i q^{15} +122375. q^{17} -7554.43i q^{19} -90600.1 q^{21} +406311. i q^{23} +78125.0 q^{25} -337268. i q^{27} -828838. q^{29} +1.39549e6i q^{31} +173301. q^{33} +929527. i q^{35} +386059. q^{37} -838482. i q^{39} -972335. q^{41} +4.61450e6i q^{43} -1.62640e6 q^{45} -1.87219e6i q^{47} -5.29467e6 q^{49} -3.33392e6i q^{51} +8.01944e6 q^{53} -1.77801e6i q^{55} -205808. q^{57} -6.18263e6i q^{59} -9.79320e6 q^{61} -1.93509e7i q^{63} -8.60255e6 q^{65} -1.19411e6i q^{67} +1.10693e7 q^{69} +3.62332e7i q^{71} +3.35548e7 q^{73} -2.12839e6i q^{75} +2.11547e7 q^{77} -1.22874e6i q^{79} +2.89888e7 q^{81} +6.96923e7i q^{83} -3.42049e7 q^{85} +2.25804e7i q^{87} +5.58347e7 q^{89} -1.02353e8i q^{91} +3.80180e7 q^{93} +2.11153e6i q^{95} -5.97582e7 q^{97} +3.70146e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) $ 16 * q - 38800 * q^9 - 51392 * q^13 + 27552 * q^17 - 414496 * q^21 + 1250000 * q^25 - 2764896 * q^29 - 5521600 * q^33 - 9009472 * q^37 - 8576448 * q^41 - 1580000 * q^45 - 32803600 * q^49 - 2452032 * q^53 + 11957760 * q^57 - 8371712 * q^61 + 9060000 * q^65 - 7527264 * q^69 + 61907232 * q^73 + 156997440 * q^77 + 140586672 * q^81 + 106960000 * q^85 + 106647456 * q^89 - 105563840 * q^93 + 171851232 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 27.2434i	− 0.336338i	−0.985758	−	0.168169i	$-0.946215\pi$
$3$	− 27.2434i	− 0.336338i	0.985758	−	0.168169i	$-0.0537855\pi$
$4$	0	0
$5$	−279.508	−0.447214
$5$	−279.508	−0.447214
$6$	0	0
$7$	− 3325.58i	− 1.38508i	−0.721379	−	0.692540i	$-0.756492\pi$
$7$	− 3325.58i	− 1.38508i	0.721379	−	0.692540i	$-0.243508\pi$
$8$	0	0
$9$	5818.80	0.886877
$10$	0	0
$11$	6361.21i	0.434479i	0.976118	+	0.217240i	$0.0697053\pi$
$11$	6361.21i	0.434479i	−0.976118	+	0.217240i	$0.930295\pi$
$12$	0	0
$13$	30777.4	1.07760	0.538801	−	0.842433i	$-0.318877\pi$
$13$	30777.4	1.07760	0.538801	+	0.842433i	$0.318877\pi$
$14$	0	0
$15$	7614.76i	0.150415i
$16$	0	0
$17$	122375.	1.46520	0.732601	−	0.680658i	$-0.238306\pi$
$17$	122375.	1.46520	0.732601	+	0.680658i	$0.238306\pi$
$18$	0	0
$19$	− 7554.43i	− 0.0579679i	−0.999580	−	0.0289839i	$-0.990773\pi$
$19$	− 7554.43i	− 0.0579679i	0.999580	−	0.0289839i	$-0.00922717\pi$
$20$	0	0
$21$	−90600.1	−0.465856
$22$	0	0
$23$	406311.i	1.45194i	0.687729	+	0.725968i	$0.258608\pi$
$23$	406311.i	1.45194i	−0.687729	+	0.725968i	$0.741392\pi$
$24$	0	0
$25$	78125.0	0.200000
$26$	0	0
$27$	− 337268.i	− 0.634629i
$28$	0	0
$29$	−828838.	−1.17187	−0.585933	−	0.810360i	$-0.699272\pi$
$29$	−828838.	−1.17187	−0.585933	+	0.810360i	$0.699272\pi$
$30$	0	0
$31$	1.39549e6i	1.51106i	0.655116	+	0.755528i	$0.272620\pi$
$31$	1.39549e6i	1.51106i	−0.655116	+	0.755528i	$0.727380\pi$
$32$	0	0
$33$	173301.	0.146132
$34$	0	0
$35$	929527.i	0.619427i
$36$	0	0
$37$	386059.	0.205990	0.102995	−	0.994682i	$-0.467157\pi$
$37$	386059.	0.205990	0.102995	+	0.994682i	$0.467157\pi$
$38$	0	0
$39$	− 838482.i	− 0.362439i
$40$	0	0
$41$	−972335.	−0.344097	−0.172048	−	0.985089i	$-0.555039\pi$
$41$	−972335.	−0.344097	−0.172048	+	0.985089i	$0.555039\pi$
$42$	0	0
$43$	4.61450e6i	1.34974i	0.737935	+	0.674871i	$0.235801\pi$
$43$	4.61450e6i	1.34974i	−0.737935	+	0.674871i	$0.764199\pi$
$44$	0	0
$45$	−1.62640e6	−0.396623
$46$	0	0
$47$	− 1.87219e6i	− 0.383670i	−0.981427	−	0.191835i	$-0.938556\pi$
$47$	− 1.87219e6i	− 0.383670i	0.981427	−	0.191835i	$-0.0614438\pi$
$48$	0	0
$49$	−5.29467e6	−0.918447
$50$	0	0
$51$	− 3.33392e6i	− 0.492804i
$52$	0	0
$53$	8.01944e6	1.01634	0.508172	−	0.861256i	$-0.330322\pi$
$53$	8.01944e6	1.01634	0.508172	+	0.861256i	$0.330322\pi$
$54$	0	0
$55$	− 1.77801e6i	− 0.194305i
$56$	0	0
$57$	−205808.	−0.0194968
$58$	0	0
$59$	− 6.18263e6i	− 0.510229i	−0.966911	−	0.255115i	$-0.917887\pi$
$59$	− 6.18263e6i	− 0.510229i	0.966911	−	0.255115i	$-0.0821132\pi$
$60$	0	0
$61$	−9.79320e6	−0.707303	−0.353651	−	0.935377i	$-0.615060\pi$
$61$	−9.79320e6	−0.707303	−0.353651	+	0.935377i	$0.615060\pi$
$62$	0	0
$63$	− 1.93509e7i	− 1.22840i
$64$	0	0
$65$	−8.60255e6	−0.481919
$66$	0	0
$67$	− 1.19411e6i	− 0.0592579i	−0.999561	−	0.0296290i	$-0.990567\pi$
$67$	− 1.19411e6i	− 0.0592579i	0.999561	−	0.0296290i	$-0.00943257\pi$
$68$	0	0
$69$	1.10693e7	0.488342
$70$	0	0
$71$	3.62332e7i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$71$	3.62332e7i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$72$	0	0
$73$	3.35548e7	1.18158	0.590790	−	0.806826i	$-0.298816\pi$
$73$	3.35548e7	1.18158	0.590790	+	0.806826i	$0.298816\pi$
$74$	0	0
$75$	− 2.12839e6i	− 0.0672677i
$76$	0	0
$77$	2.11547e7	0.601789
$78$	0	0
$79$	− 1.22874e6i	− 0.0315466i	−0.999876	−	0.0157733i	$-0.994979\pi$
$79$	− 1.22874e6i	− 0.0315466i	0.999876	−	0.0157733i	$-0.00502101\pi$
$80$	0	0
$81$	2.89888e7	0.673426
$82$	0	0
$83$	6.96923e7i	1.46849i	0.678882	+	0.734247i	$0.262465\pi$
$83$	6.96923e7i	1.46849i	−0.678882	+	0.734247i	$0.737535\pi$
$84$	0	0
$85$	−3.42049e7	−0.655258
$86$	0	0
$87$	2.25804e7i	0.394143i
$88$	0	0
$89$	5.58347e7	0.889907	0.444953	−	0.895554i	$-0.353220\pi$
$89$	5.58347e7	0.889907	0.444953	+	0.895554i	$0.353220\pi$
$90$	0	0
$91$	− 1.02353e8i	− 1.49257i
$92$	0	0
$93$	3.80180e7	0.508226
$94$	0	0
$95$	2.11153e6i	0.0259240i
$96$	0	0
$97$	−5.97582e7	−0.675010	−0.337505	−	0.941324i	$-0.609583\pi$
$97$	−5.97582e7	−0.675010	−0.337505	+	0.941324i	$0.609583\pi$
$98$	0	0
$99$	3.70146e7i	0.385329i
$100$	0	0
$101$	−9.62751e7	−0.925185	−0.462593	−	0.886571i	$-0.653081\pi$
$101$	−9.62751e7	−0.925185	−0.462593	+	0.886571i	$0.653081\pi$
$102$	0	0
$103$	1.96312e8i	1.74421i	0.489320	+	0.872104i	$0.337245\pi$
$103$	1.96312e8i	1.74421i	−0.489320	+	0.872104i	$0.662755\pi$
$104$	0	0
$105$	2.53235e7	0.208337
$106$	0	0
$107$	1.64435e8i	1.25447i	0.778832	+	0.627233i	$0.215813\pi$
$107$	1.64435e8i	1.25447i	−0.778832	+	0.627233i	$0.784187\pi$
$108$	0	0
$109$	1.81680e8	1.28707	0.643534	−	0.765417i	$-0.277467\pi$
$109$	1.81680e8	1.28707	0.643534	+	0.765417i	$0.277467\pi$
$110$	0	0
$111$	− 1.05176e7i	− 0.0692824i
$112$	0	0
$113$	−1.30629e8	−0.801172	−0.400586	−	0.916259i	$-0.631193\pi$
$113$	−1.30629e8	−0.801172	−0.400586	+	0.916259i	$0.631193\pi$
$114$	0	0
$115$	− 1.13567e8i	− 0.649325i
$116$	0	0
$117$	1.79087e8	0.955700
$118$	0	0
$119$	− 4.06968e8i	− 2.02942i
$120$	0	0
$121$	1.73894e8	0.811228
$122$	0	0
$123$	2.64897e7i	0.115733i
$124$	0	0
$125$	−2.18366e7	−0.0894427
$126$	0	0
$127$	− 4.45982e8i	− 1.71436i	−0.515015	−	0.857181i	$-0.672214\pi$
$127$	− 4.45982e8i	− 1.71436i	0.515015	−	0.857181i	$-0.327786\pi$
$128$	0	0
$129$	1.25715e8	0.453970
$130$	0	0
$131$	1.31555e8i	0.446707i	0.974738	+	0.223353i	$0.0717004\pi$
$131$	1.31555e8i	0.446707i	−0.974738	+	0.223353i	$0.928300\pi$
$132$	0	0
$133$	−2.51228e7	−0.0802902
$134$	0	0
$135$	9.42692e7i	0.283815i
$136$	0	0
$137$	−3.26396e8	−0.926536	−0.463268	−	0.886218i	$-0.653323\pi$
$137$	−3.26396e8	−0.926536	−0.463268	+	0.886218i	$0.653323\pi$
$138$	0	0
$139$	− 6.48927e8i	− 1.73835i	−0.494506	−	0.869174i	$-0.664651\pi$
$139$	− 6.48927e8i	− 1.73835i	0.494506	−	0.869174i	$-0.335349\pi$
$140$	0	0
$141$	−5.10048e7	−0.129043
$142$	0	0
$143$	1.95782e8i	0.468196i
$144$	0	0
$145$	2.31667e8	0.524074
$146$	0	0
$147$	1.44245e8i	0.308909i
$148$	0	0
$149$	7.30660e8	1.48242	0.741208	−	0.671276i	$-0.234253\pi$
$149$	7.30660e8	1.48242	0.741208	+	0.671276i	$0.234253\pi$
$150$	0	0
$151$	7.74114e8i	1.48901i	0.667618	+	0.744504i	$0.267314\pi$
$151$	7.74114e8i	1.48901i	−0.667618	+	0.744504i	$0.732686\pi$
$152$	0	0
$153$	7.12076e8	1.29945
$154$	0	0
$155$	− 3.90052e8i	− 0.675765i
$156$	0	0
$157$	−3.17523e8	−0.522609	−0.261304	−	0.965256i	$-0.584153\pi$
$157$	−3.17523e8	−0.522609	−0.261304	+	0.965256i	$0.584153\pi$
$158$	0	0
$159$	− 2.18477e8i	− 0.341835i
$160$	0	0
$161$	1.35122e9	2.01105
$162$	0	0
$163$	− 2.02004e7i	− 0.0286161i	−0.999898	−	0.0143081i	$-0.995445\pi$
$163$	− 2.02004e7i	− 0.0286161i	0.999898	−	0.0143081i	$-0.00455455\pi$
$164$	0	0
$165$	−4.84391e7	−0.0653522
$166$	0	0
$167$	5.65743e7i	0.0727366i	0.999338	+	0.0363683i	$0.0115789\pi$
$167$	5.65743e7i	0.0727366i	−0.999338	+	0.0363683i	$0.988421\pi$
$168$	0	0
$169$	1.31518e8	0.161228
$170$	0	0
$171$	− 4.39577e7i	− 0.0514103i
$172$	0	0
$173$	1.09785e9	1.22563	0.612813	−	0.790228i	$-0.290038\pi$
$173$	1.09785e9	1.22563	0.612813	+	0.790228i	$0.290038\pi$
$174$	0	0
$175$	− 2.59811e8i	− 0.277016i
$176$	0	0
$177$	−1.68436e8	−0.171610
$178$	0	0
$179$	6.44793e8i	0.628070i	0.949411	+	0.314035i	$0.101681\pi$
$179$	6.44793e8i	0.628070i	−0.949411	+	0.314035i	$0.898319\pi$
$180$	0	0
$181$	6.79018e8	0.632655	0.316328	−	0.948650i	$-0.397550\pi$
$181$	6.79018e8	0.632655	0.316328	+	0.948650i	$0.397550\pi$
$182$	0	0
$183$	2.66800e8i	0.237893i
$184$	0	0
$185$	−1.07907e8	−0.0921217
$186$	0	0
$187$	7.78454e8i	0.636600i
$188$	0	0
$189$	−1.12161e9	−0.879012
$190$	0	0
$191$	− 1.53738e8i	− 0.115518i	−0.998331	−	0.0577588i	$-0.981605\pi$
$191$	− 1.53738e8i	− 0.115518i	0.998331	−	0.0577588i	$-0.0183954\pi$
$192$	0	0
$193$	9.67761e7	0.0697491	0.0348746	−	0.999392i	$-0.488897\pi$
$193$	9.67761e7	0.0697491	0.0348746	+	0.999392i	$0.488897\pi$
$194$	0	0
$195$	2.34363e8i	0.162088i
$196$	0	0
$197$	2.84766e9	1.89070	0.945352	−	0.326051i	$-0.105718\pi$
$197$	2.84766e9	1.89070	0.945352	+	0.326051i	$0.105718\pi$
$198$	0	0
$199$	− 8.71254e8i	− 0.555562i	−0.960644	−	0.277781i	$-0.910401\pi$
$199$	− 8.71254e8i	− 0.555562i	0.960644	−	0.277781i	$-0.0895990\pi$
$200$	0	0
$201$	−3.25317e7	−0.0199307
$202$	0	0
$203$	2.75637e9i	1.62313i
$204$	0	0
$205$	2.71776e8	0.153885
$206$	0	0
$207$	2.36424e9i	1.28769i
$208$	0	0
$209$	4.80553e7	0.0251858
$210$	0	0
$211$	− 2.22378e9i	− 1.12192i	−0.827843	−	0.560960i	$-0.810432\pi$
$211$	− 2.22378e9i	− 1.12192i	0.827843	−	0.560960i	$-0.189568\pi$
$212$	0	0
$213$	9.87114e8	0.479567
$214$	0	0
$215$	− 1.28979e9i	− 0.603623i
$216$	0	0
$217$	4.64082e9	2.09293
$218$	0	0
$219$	− 9.14146e8i	− 0.397410i
$220$	0	0
$221$	3.76639e9	1.57891
$222$	0	0
$223$	− 1.76089e9i	− 0.712054i	−0.934476	−	0.356027i	$-0.884131\pi$
$223$	− 1.76089e9i	− 0.712054i	0.934476	−	0.356027i	$-0.115869\pi$
$224$	0	0
$225$	4.54593e8	0.177375
$226$	0	0
$227$	− 1.85447e9i	− 0.698418i	−0.937045	−	0.349209i	$-0.886450\pi$
$227$	− 1.85447e9i	− 0.698418i	0.937045	−	0.349209i	$-0.113550\pi$
$228$	0	0
$229$	1.13087e9	0.411218	0.205609	−	0.978634i	$-0.434083\pi$
$229$	1.13087e9	0.411218	0.205609	+	0.978634i	$0.434083\pi$
$230$	0	0
$231$	− 5.76326e8i	− 0.202405i
$232$	0	0
$233$	2.24935e9	0.763191	0.381596	−	0.924329i	$-0.375375\pi$
$233$	2.24935e9	0.763191	0.381596	+	0.924329i	$0.375375\pi$
$234$	0	0
$235$	5.23292e8i	0.171582i
$236$	0	0
$237$	−3.34752e7	−0.0106103
$238$	0	0
$239$	2.77234e9i	0.849679i	0.905269	+	0.424840i	$0.139669\pi$
$239$	2.77234e9i	0.849679i	−0.905269	+	0.424840i	$0.860331\pi$
$240$	0	0
$241$	−1.60515e8	−0.0475826	−0.0237913	−	0.999717i	$-0.507574\pi$
$241$	−1.60515e8	−0.0475826	−0.0237913	+	0.999717i	$0.507574\pi$
$242$	0	0
$243$	− 3.00257e9i	− 0.861128i
$244$	0	0
$245$	1.47990e9	0.410742
$246$	0	0
$247$	− 2.32506e8i	− 0.0624663i
$248$	0	0
$249$	1.89866e9	0.493911
$250$	0	0
$251$	5.77680e9i	1.45543i	0.685877	+	0.727717i	$0.259419\pi$
$251$	5.77680e9i	1.45543i	−0.685877	+	0.727717i	$0.740581\pi$
$252$	0	0
$253$	−2.58463e9	−0.630836
$254$	0	0
$255$	9.31858e8i	0.220389i
$256$	0	0
$257$	4.57743e9	1.04928	0.524638	−	0.851326i	$-0.324201\pi$
$257$	4.57743e9	1.04928	0.524638	+	0.851326i	$0.324201\pi$
$258$	0	0
$259$	− 1.28387e9i	− 0.285313i
$260$	0	0
$261$	−4.82284e9	−1.03930
$262$	0	0
$263$	4.85941e8i	0.101569i	0.998710	+	0.0507844i	$0.0161721\pi$
$263$	4.85941e8i	0.101569i	−0.998710	+	0.0507844i	$0.983828\pi$
$264$	0	0
$265$	−2.24150e9	−0.454523
$266$	0	0
$267$	− 1.52113e9i	− 0.299310i
$268$	0	0
$269$	1.48731e9	0.284049	0.142024	−	0.989863i	$-0.454639\pi$
$269$	1.48731e9	0.284049	0.142024	+	0.989863i	$0.454639\pi$
$270$	0	0
$271$	− 6.79398e9i	− 1.25964i	−0.776740	−	0.629821i	$-0.783128\pi$
$271$	− 6.79398e9i	− 1.25964i	0.776740	−	0.629821i	$-0.216872\pi$
$272$	0	0
$273$	−2.78844e9	−0.502007
$274$	0	0
$275$	4.96970e8i	0.0868959i
$276$	0	0
$277$	−1.54730e9	−0.262818	−0.131409	−	0.991328i	$-0.541950\pi$
$277$	−1.54730e9	−0.262818	−0.131409	+	0.991328i	$0.541950\pi$
$278$	0	0
$279$	8.12009e9i	1.34012i
$280$	0	0
$281$	−9.33090e9	−1.49657	−0.748287	−	0.663375i	$-0.769123\pi$
$281$	−9.33090e9	−1.49657	−0.748287	+	0.663375i	$0.769123\pi$
$282$	0	0
$283$	− 7.43824e9i	− 1.15964i	−0.814743	−	0.579822i	$-0.803122\pi$
$283$	− 7.43824e9i	− 1.15964i	0.814743	−	0.579822i	$-0.196878\pi$
$284$	0	0
$285$	5.75252e7	0.00871924
$286$	0	0
$287$	3.23358e9i	0.476602i
$288$	0	0
$289$	7.99992e9	1.14682
$290$	0	0
$291$	1.62802e9i	0.227032i
$292$	0	0
$293$	1.59214e9	0.216028	0.108014	−	0.994149i	$-0.465551\pi$
$293$	1.59214e9	0.216028	0.108014	+	0.994149i	$0.465551\pi$
$294$	0	0
$295$	1.72810e9i	0.228181i
$296$	0	0
$297$	2.14543e9	0.275733
$298$	0	0
$299$	1.25052e10i	1.56461i
$300$	0	0
$301$	1.53459e10	1.86950
$302$	0	0
$303$	2.62286e9i	0.311175i
$304$	0	0
$305$	2.73728e9	0.316315
$306$	0	0
$307$	3.36362e9i	0.378663i	0.981913	+	0.189332i	$0.0606321\pi$
$307$	3.36362e9i	0.378663i	−0.981913	+	0.189332i	$0.939368\pi$
$308$	0	0
$309$	5.34821e9	0.586644
$310$	0	0
$311$	− 1.23254e10i	− 1.31752i	−0.752352	−	0.658761i	$-0.771081\pi$
$311$	− 1.23254e10i	− 1.31752i	0.752352	−	0.658761i	$-0.228919\pi$
$312$	0	0
$313$	2.77987e9	0.289633	0.144816	−	0.989459i	$-0.453741\pi$
$313$	2.77987e9	0.289633	0.144816	+	0.989459i	$0.453741\pi$
$314$	0	0
$315$	5.40873e9i	0.549355i
$316$	0	0
$317$	5.40177e8	0.0534933	0.0267466	−	0.999642i	$-0.491485\pi$
$317$	5.40177e8	0.0534933	0.0267466	+	0.999642i	$0.491485\pi$
$318$	0	0
$319$	− 5.27241e9i	− 0.509151i
$320$	0	0
$321$	4.47977e9	0.421925
$322$	0	0
$323$	− 9.24475e8i	− 0.0849347i
$324$	0	0
$325$	2.40449e9	0.215521
$326$	0	0
$327$	− 4.94959e9i	− 0.432891i
$328$	0	0
$329$	−6.22610e9	−0.531414
$330$	0	0
$331$	− 1.21115e10i	− 1.00899i	−0.863414	−	0.504495i	$-0.831679\pi$
$331$	− 1.21115e10i	− 1.00899i	0.863414	−	0.504495i	$-0.168321\pi$
$332$	0	0
$333$	2.24640e9	0.182688
$334$	0	0
$335$	3.33765e8i	0.0265009i
$336$	0	0
$337$	−9.48812e9	−0.735632	−0.367816	−	0.929899i	$-0.619894\pi$
$337$	−9.48812e9	−0.735632	−0.367816	+	0.929899i	$0.619894\pi$
$338$	0	0
$339$	3.55878e9i	0.269465i
$340$	0	0
$341$	−8.87702e9	−0.656523
$342$	0	0
$343$	− 1.56347e9i	− 0.112957i
$344$	0	0
$345$	−3.09396e9	−0.218393
$346$	0	0
$347$	− 9.01032e9i	− 0.621473i	−0.950496	−	0.310737i	$-0.899424\pi$
$347$	− 9.01032e9i	− 0.621473i	0.950496	−	0.310737i	$-0.100576\pi$
$348$	0	0
$349$	−2.17409e10	−1.46547	−0.732733	−	0.680517i	$-0.761755\pi$
$349$	−2.17409e10	−1.46547	−0.732733	+	0.680517i	$0.761755\pi$
$350$	0	0
$351$	− 1.03802e10i	− 0.683878i
$352$	0	0
$353$	−5.60018e9	−0.360664	−0.180332	−	0.983606i	$-0.557717\pi$
$353$	−5.60018e9	−0.360664	−0.180332	+	0.983606i	$0.557717\pi$
$354$	0	0
$355$	− 1.01275e10i	− 0.637658i
$356$	0	0
$357$	−1.10872e10	−0.682573
$358$	0	0
$359$	1.31213e10i	0.789952i	0.918692	+	0.394976i	$0.129247\pi$
$359$	1.31213e10i	0.789952i	−0.918692	+	0.394976i	$0.870753\pi$
$360$	0	0
$361$	1.69265e10	0.996640
$362$	0	0
$363$	− 4.73746e9i	− 0.272847i
$364$	0	0
$365$	−9.37884e9	−0.528418
$366$	0	0
$367$	9.89380e9i	0.545379i	0.962102	+	0.272690i	$0.0879132\pi$
$367$	9.89380e9i	0.545379i	−0.962102	+	0.272690i	$0.912087\pi$
$368$	0	0
$369$	−5.65782e9	−0.305171
$370$	0	0
$371$	− 2.66693e10i	− 1.40772i
$372$	0	0
$373$	−1.01496e10	−0.524339	−0.262170	−	0.965022i	$-0.584438\pi$
$373$	−1.01496e10	−0.524339	−0.262170	+	0.965022i	$0.584438\pi$
$374$	0	0
$375$	5.94903e8i	0.0300830i
$376$	0	0
$377$	−2.55095e10	−1.26281
$378$	0	0
$379$	− 1.39748e10i	− 0.677312i	−0.940910	−	0.338656i	$-0.890028\pi$
$379$	− 1.39748e10i	− 0.677312i	0.940910	−	0.338656i	$-0.109972\pi$
$380$	0	0
$381$	−1.21501e10	−0.576606
$382$	0	0
$383$	− 2.31617e10i	− 1.07640i	−0.842816	−	0.538201i	$-0.819104\pi$
$383$	− 2.31617e10i	− 1.07640i	0.842816	−	0.538201i	$-0.180896\pi$
$384$	0	0
$385$	−5.91292e9	−0.269128
$386$	0	0
$387$	2.68508e10i	1.19706i
$388$	0	0
$389$	1.13177e10	0.494267	0.247133	−	0.968981i	$-0.420511\pi$
$389$	1.13177e10	0.494267	0.247133	+	0.968981i	$0.420511\pi$
$390$	0	0
$391$	4.97224e10i	2.12738i
$392$	0	0
$393$	3.58401e9	0.150245
$394$	0	0
$395$	3.43444e8i	0.0141081i
$396$	0	0
$397$	−1.01291e10	−0.407765	−0.203882	−	0.978995i	$-0.565356\pi$
$397$	−1.01291e10	−0.407765	−0.203882	+	0.978995i	$0.565356\pi$
$398$	0	0
$399$	6.84432e8i	0.0270047i
$400$	0	0
$401$	1.52259e10	0.588850	0.294425	−	0.955675i	$-0.404872\pi$
$401$	1.52259e10	0.588850	0.294425	+	0.955675i	$0.404872\pi$
$402$	0	0
$403$	4.29496e10i	1.62832i
$404$	0	0
$405$	−8.10262e9	−0.301165
$406$	0	0
$407$	2.45580e9i	0.0894985i
$408$	0	0
$409$	−5.16482e10	−1.84570	−0.922851	−	0.385156i	$-0.874148\pi$
$409$	−5.16482e10	−1.84570	−0.922851	+	0.385156i	$0.874148\pi$
$410$	0	0
$411$	8.89213e9i	0.311629i
$412$	0	0
$413$	−2.05608e10	−0.706708
$414$	0	0
$415$	− 1.94796e10i	− 0.656731i
$416$	0	0
$417$	−1.76790e10	−0.584673
$418$	0	0
$419$	2.90780e10i	0.943427i	0.881752	+	0.471714i	$0.156364\pi$
$419$	2.90780e10i	0.943427i	−0.881752	+	0.471714i	$0.843636\pi$
$420$	0	0
$421$	2.02634e9	0.0645037	0.0322519	−	0.999480i	$-0.489732\pi$
$421$	2.02634e9	0.0645037	0.0322519	+	0.999480i	$0.489732\pi$
$422$	0	0
$423$	− 1.08939e10i	− 0.340268i
$424$	0	0
$425$	9.56056e9	0.293040
$426$	0	0
$427$	3.25681e10i	0.979671i
$428$	0	0
$429$	5.33376e9	0.157472
$430$	0	0
$431$	2.31188e10i	0.669972i	0.942223	+	0.334986i	$0.108732\pi$
$431$	2.31188e10i	0.669972i	−0.942223	+	0.334986i	$0.891268\pi$
$432$	0	0
$433$	−5.16117e10	−1.46824	−0.734119	−	0.679021i	$-0.762405\pi$
$433$	−5.16117e10	−1.46824	−0.734119	+	0.679021i	$0.762405\pi$
$434$	0	0
$435$	− 6.31141e9i	− 0.176266i
$436$	0	0
$437$	3.06945e9	0.0841656
$438$	0	0
$439$	− 4.07914e10i	− 1.09827i	−0.835733	−	0.549137i	$-0.814957\pi$
$439$	− 4.07914e10i	− 1.09827i	0.835733	−	0.549137i	$-0.185043\pi$
$440$	0	0
$441$	−3.08086e10	−0.814549
$442$	0	0
$443$	− 1.78192e9i	− 0.0462671i	−0.999732	−	0.0231336i	$-0.992636\pi$
$443$	− 1.78192e9i	− 0.0462671i	0.999732	−	0.0231336i	$-0.00736430\pi$
$444$	0	0
$445$	−1.56063e10	−0.397978
$446$	0	0
$447$	− 1.99057e10i	− 0.498593i
$448$	0	0
$449$	5.74811e10	1.41429	0.707147	−	0.707067i	$-0.249982\pi$
$449$	5.74811e10	1.41429	0.707147	+	0.707067i	$0.249982\pi$
$450$	0	0
$451$	− 6.18523e9i	− 0.149503i
$452$	0	0
$453$	2.10895e10	0.500811
$454$	0	0
$455$	2.86084e10i	0.667496i
$456$	0	0
$457$	−4.50957e10	−1.03388	−0.516941	−	0.856021i	$-0.672929\pi$
$457$	−4.50957e10	−1.03388	−0.516941	+	0.856021i	$0.672929\pi$
$458$	0	0
$459$	− 4.12732e10i	− 0.929860i
$460$	0	0
$461$	−7.73092e10	−1.71170	−0.855850	−	0.517225i	$-0.826965\pi$
$461$	−7.73092e10	−1.71170	−0.855850	+	0.517225i	$0.826965\pi$
$462$	0	0
$463$	8.52854e10i	1.85588i	0.372726	+	0.927941i	$0.378423\pi$
$463$	8.52854e10i	1.85588i	−0.372726	+	0.927941i	$0.621577\pi$
$464$	0	0
$465$	−1.06263e10	−0.227286
$466$	0	0
$467$	− 3.20726e10i	− 0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$467$	− 3.20726e10i	− 0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$468$	0	0
$469$	−3.97112e9	−0.0820770
$470$	0	0
$471$	8.65041e9i	0.175773i
$472$	0	0
$473$	−2.93538e10	−0.586435
$474$	0	0
$475$	− 5.90190e8i	− 0.0115936i
$476$	0	0
$477$	4.66635e10	0.901371
$478$	0	0
$479$	− 3.58893e10i	− 0.681746i	−0.940109	−	0.340873i	$-0.889277\pi$
$479$	− 3.58893e10i	− 0.681746i	0.940109	−	0.340873i	$-0.110723\pi$
$480$	0	0
$481$	1.18819e10	0.221976
$482$	0	0
$483$	− 3.68118e10i	− 0.676393i
$484$	0	0
$485$	1.67029e10	0.301874
$486$	0	0
$487$	− 5.98567e10i	− 1.06414i	−0.846702	−	0.532068i	$-0.821415\pi$
$487$	− 5.98567e10i	− 1.06414i	0.846702	−	0.532068i	$-0.178585\pi$
$488$	0	0
$489$	−5.50329e8	−0.00962470
$490$	0	0
$491$	3.00275e10i	0.516645i	0.966059	+	0.258323i	$0.0831698\pi$
$491$	3.00275e10i	0.516645i	−0.966059	+	0.258323i	$0.916830\pi$
$492$	0	0
$493$	−1.01429e11	−1.71702
$494$	0	0
$495$	− 1.03459e10i	− 0.172325i
$496$	0	0
$497$	1.20496e11	1.97491
$498$	0	0
$499$	− 2.32602e10i	− 0.375156i	−0.982250	−	0.187578i	$-0.939936\pi$
$499$	− 2.32602e10i	− 0.375156i	0.982250	−	0.187578i	$-0.0600637\pi$
$500$	0	0
$501$	1.54128e9	0.0244641
$502$	0	0
$503$	2.16574e10i	0.338326i	0.985588	+	0.169163i	$0.0541064\pi$
$503$	2.16574e10i	0.338326i	−0.985588	+	0.169163i	$0.945894\pi$
$504$	0	0
$505$	2.69097e10	0.413755
$506$	0	0
$507$	− 3.58300e9i	− 0.0542270i
$508$	0	0
$509$	−1.21565e10	−0.181108	−0.0905541	−	0.995892i	$-0.528864\pi$
$509$	−1.21565e10	−0.181108	−0.0905541	+	0.995892i	$0.528864\pi$
$510$	0	0
$511$	− 1.11589e11i	− 1.63658i
$512$	0	0
$513$	−2.54787e9	−0.0367881
$514$	0	0
$515$	− 5.48709e10i	− 0.780034i
$516$	0	0
$517$	1.19094e10	0.166697
$518$	0	0
$519$	− 2.99091e10i	− 0.412225i
$520$	0	0
$521$	−5.20805e10	−0.706845	−0.353422	−	0.935464i	$-0.614982\pi$
$521$	−5.20805e10	−0.706845	−0.353422	+	0.935464i	$0.614982\pi$
$522$	0	0
$523$	2.42007e10i	0.323461i	0.986835	+	0.161731i	$0.0517075\pi$
$523$	2.42007e10i	0.323461i	−0.986835	+	0.161731i	$0.948292\pi$
$524$	0	0
$525$	−7.07813e9	−0.0931711
$526$	0	0
$527$	1.70774e11i	2.21400i
$528$	0	0
$529$	−8.67778e10	−1.10812
$530$	0	0
$531$	− 3.59755e10i	− 0.452510i
$532$	0	0
$533$	−2.99260e10	−0.370800
$534$	0	0
$535$	− 4.59609e10i	− 0.561014i
$536$	0	0
$537$	1.75664e10	0.211244
$538$	0	0
$539$	− 3.36805e10i	− 0.399046i
$540$	0	0
$541$	7.39316e10	0.863060	0.431530	−	0.902099i	$-0.357974\pi$
$541$	7.39316e10	0.863060	0.431530	+	0.902099i	$0.357974\pi$
$542$	0	0
$543$	− 1.84988e10i	− 0.212786i
$544$	0	0
$545$	−5.07812e10	−0.575595
$546$	0	0
$547$	9.08736e10i	1.01505i	0.861636	+	0.507526i	$0.169440\pi$
$547$	9.08736e10i	1.01505i	−0.861636	+	0.507526i	$0.830560\pi$
$548$	0	0
$549$	−5.69847e10	−0.627290
$550$	0	0
$551$	6.26140e9i	0.0679305i
$552$	0	0
$553$	−4.08628e9	−0.0436946
$554$	0	0
$555$	2.93975e9i	0.0309840i
$556$	0	0
$557$	7.25820e9	0.0754064	0.0377032	−	0.999289i	$-0.487996\pi$
$557$	7.25820e9	0.0754064	0.0377032	+	0.999289i	$0.487996\pi$
$558$	0	0
$559$	1.42022e11i	1.45449i
$560$	0	0
$561$	2.12077e10	0.214113
$562$	0	0
$563$	4.02851e10i	0.400969i	0.979697	+	0.200485i	$0.0642517\pi$
$563$	4.02851e10i	0.400969i	−0.979697	+	0.200485i	$0.935748\pi$
$564$	0	0
$565$	3.65119e10	0.358295
$566$	0	0
$567$	− 9.64045e10i	− 0.932750i
$568$	0	0
$569$	4.42204e10	0.421866	0.210933	−	0.977501i	$-0.432350\pi$
$569$	4.42204e10	0.421866	0.210933	+	0.977501i	$0.432350\pi$
$570$	0	0
$571$	1.92416e11i	1.81008i	0.425327	+	0.905040i	$0.360159\pi$
$571$	1.92416e11i	1.81008i	−0.425327	+	0.905040i	$0.639841\pi$
$572$	0	0
$573$	−4.18835e9	−0.0388530
$574$	0	0
$575$	3.17431e10i	0.290387i
$576$	0	0
$577$	5.62180e10	0.507192	0.253596	−	0.967310i	$-0.418387\pi$
$577$	5.62180e10	0.507192	0.253596	+	0.967310i	$0.418387\pi$
$578$	0	0
$579$	− 2.63651e9i	− 0.0234593i
$580$	0	0
$581$	2.31767e11	2.03398
$582$	0	0
$583$	5.10133e10i	0.441580i
$584$	0	0
$585$	−5.00565e10	−0.427402
$586$	0	0
$587$	− 2.12589e11i	− 1.79056i	−0.445508	−	0.895278i	$-0.646977\pi$
$587$	− 2.12589e11i	− 1.79056i	0.445508	−	0.895278i	$-0.353023\pi$
$588$	0	0
$589$	1.05422e10	0.0875927
$590$	0	0
$591$	− 7.75800e10i	− 0.635916i
$592$	0	0
$593$	−3.64674e10	−0.294908	−0.147454	−	0.989069i	$-0.547108\pi$
$593$	−3.64674e10	−0.294908	−0.147454	+	0.989069i	$0.547108\pi$
$594$	0	0
$595$	1.13751e11i	0.907585i
$596$	0	0
$597$	−2.37359e10	−0.186857
$598$	0	0
$599$	− 1.54791e11i	− 1.20237i	−0.799111	−	0.601184i	$-0.794696\pi$
$599$	− 1.54791e11i	− 1.20237i	0.799111	−	0.601184i	$-0.205304\pi$
$600$	0	0
$601$	1.37850e11	1.05659	0.528297	−	0.849059i	$-0.322831\pi$
$601$	1.37850e11	1.05659	0.528297	+	0.849059i	$0.322831\pi$
$602$	0	0
$603$	− 6.94830e9i	− 0.0525545i
$604$	0	0
$605$	−4.86048e10	−0.362792
$606$	0	0
$607$	4.21916e10i	0.310793i	0.987852	+	0.155396i	$0.0496655\pi$
$607$	4.21916e10i	0.310793i	−0.987852	+	0.155396i	$0.950335\pi$
$608$	0	0
$609$	7.50928e10	0.545920
$610$	0	0
$611$	− 5.76211e10i	− 0.413444i
$612$	0	0
$613$	1.76436e11	1.24953	0.624763	−	0.780815i	$-0.285196\pi$
$613$	1.76436e11	1.24953	0.624763	+	0.780815i	$0.285196\pi$
$614$	0	0
$615$	− 7.40410e9i	− 0.0517573i
$616$	0	0
$617$	3.80722e10	0.262705	0.131352	−	0.991336i	$-0.458068\pi$
$617$	3.80722e10	0.262705	0.131352	+	0.991336i	$0.458068\pi$
$618$	0	0
$619$	4.67363e10i	0.318340i	0.987251	+	0.159170i	$0.0508818\pi$
$619$	4.67363e10i	0.318340i	−0.987251	+	0.159170i	$0.949118\pi$
$620$	0	0
$621$	1.37036e11	0.921441
$622$	0	0
$623$	− 1.85683e11i	− 1.23259i
$624$	0	0
$625$	6.10352e9	0.0400000
$626$	0	0
$627$	− 1.30919e9i	− 0.00847096i
$628$	0	0
$629$	4.72440e10	0.301817
$630$	0	0
$631$	1.82629e11i	1.15200i	0.817450	+	0.576000i	$0.195387\pi$
$631$	1.82629e11i	1.15200i	−0.817450	+	0.576000i	$0.804613\pi$
$632$	0	0
$633$	−6.05834e10	−0.377345
$634$	0	0
$635$	1.24656e11i	0.766686i
$636$	0	0
$637$	−1.62956e11	−0.989721
$638$	0	0
$639$	2.10833e11i	1.26455i
$640$	0	0
$641$	1.05531e11	0.625100	0.312550	−	0.949901i	$-0.398817\pi$
$641$	1.05531e11	0.625100	0.312550	+	0.949901i	$0.398817\pi$
$642$	0	0
$643$	− 1.09834e11i	− 0.642528i	−0.946990	−	0.321264i	$-0.895892\pi$
$643$	− 1.09834e11i	− 0.642528i	0.946990	−	0.321264i	$-0.104108\pi$
$644$	0	0
$645$	−3.51383e10	−0.203022
$646$	0	0
$647$	4.48807e9i	0.0256119i	0.999918	+	0.0128060i	$0.00407638\pi$
$647$	4.48807e9i	0.0256119i	−0.999918	+	0.0128060i	$0.995924\pi$
$648$	0	0
$649$	3.93290e10	0.221684
$650$	0	0
$651$	− 1.26432e11i	− 0.703934i
$652$	0	0
$653$	−2.96586e11	−1.63117	−0.815583	−	0.578640i	$-0.803584\pi$
$653$	−2.96586e11	−1.63117	−0.815583	+	0.578640i	$0.803584\pi$
$654$	0	0
$655$	− 3.67708e10i	− 0.199773i
$656$	0	0
$657$	1.95248e11	1.04791
$658$	0	0
$659$	8.22505e10i	0.436111i	0.975936	+	0.218056i	$0.0699714\pi$
$659$	8.22505e10i	0.436111i	−0.975936	+	0.218056i	$0.930029\pi$
$660$	0	0
$661$	1.54172e11	0.807608	0.403804	−	0.914845i	$-0.367688\pi$
$661$	1.54172e11	0.807608	0.403804	+	0.914845i	$0.367688\pi$
$662$	0	0
$663$	− 1.02609e11i	− 0.531047i
$664$	0	0
$665$	7.02205e9	0.0359068
$666$	0	0
$667$	− 3.36766e11i	− 1.70147i
$668$	0	0
$669$	−4.79726e10	−0.239491
$670$	0	0
$671$	− 6.22966e10i	− 0.307308i
$672$	0	0
$673$	2.82157e11	1.37541	0.687703	−	0.725992i	$-0.258619\pi$
$673$	2.82157e11	1.37541	0.687703	+	0.725992i	$0.258619\pi$
$674$	0	0
$675$	− 2.63491e10i	− 0.126926i
$676$	0	0
$677$	2.70699e11	1.28864	0.644320	−	0.764756i	$-0.277140\pi$
$677$	2.70699e11	1.28864	0.644320	+	0.764756i	$0.277140\pi$
$678$	0	0
$679$	1.98730e11i	0.934943i
$680$	0	0
$681$	−5.05220e10	−0.234905
$682$	0	0
$683$	3.96155e10i	0.182046i	0.995849	+	0.0910232i	$0.0290137\pi$
$683$	3.96155e10i	0.182046i	−0.995849	+	0.0910232i	$0.970986\pi$
$684$	0	0
$685$	9.12304e10	0.414359
$686$	0	0
$687$	− 3.08088e10i	− 0.138308i
$688$	0	0
$689$	2.46818e11	1.09521
$690$	0	0
$691$	− 1.26869e11i	− 0.556472i	−0.960513	−	0.278236i	$-0.910250\pi$
$691$	− 1.26869e11i	− 0.556472i	0.960513	−	0.278236i	$-0.0897497\pi$
$692$	0	0
$693$	1.23095e11	0.533712
$694$	0	0
$695$	1.81381e11i	0.777413i
$696$	0	0
$697$	−1.18990e11	−0.504171
$698$	0	0
$699$	− 6.12800e10i	− 0.256691i
$700$	0	0
$701$	6.26836e10	0.259587	0.129793	−	0.991541i	$-0.458569\pi$
$701$	6.26836e10	0.259587	0.129793	+	0.991541i	$0.458569\pi$
$702$	0	0
$703$	− 2.91646e9i	− 0.0119408i
$704$	0	0
$705$	1.42563e10	0.0577098
$706$	0	0
$707$	3.20170e11i	1.28146i
$708$	0	0
$709$	3.57762e11	1.41582	0.707912	−	0.706300i	$-0.249637\pi$
$709$	3.57762e11	1.41582	0.707912	+	0.706300i	$0.249637\pi$
$710$	0	0
$711$	− 7.14981e9i	− 0.0279780i
$712$	0	0
$713$	−5.67004e11	−2.19396
$714$	0	0
$715$	− 5.47226e10i	− 0.209384i
$716$	0	0
$717$	7.55280e10	0.285780
$718$	0	0
$719$	− 1.70342e11i	− 0.637392i	−0.947857	−	0.318696i	$-0.896755\pi$
$719$	− 1.70342e11i	− 0.637392i	0.947857	−	0.318696i	$-0.103245\pi$
$720$	0	0
$721$	6.52852e11	2.41587
$722$	0	0
$723$	4.37298e9i	0.0160038i
$724$	0	0
$725$	−6.47530e10	−0.234373
$726$	0	0
$727$	− 6.56094e10i	− 0.234870i	−0.993081	−	0.117435i	$-0.962533\pi$
$727$	− 6.56094e10i	− 0.234870i	0.993081	−	0.117435i	$-0.0374672\pi$
$728$	0	0
$729$	1.08395e11	0.383796
$730$	0	0
$731$	5.64700e11i	1.97765i
$732$	0	0
$733$	−2.60967e11	−0.904003	−0.452001	−	0.892017i	$-0.649290\pi$
$733$	−2.60967e11	−0.904003	−0.452001	+	0.892017i	$0.649290\pi$
$734$	0	0
$735$	− 4.03176e10i	− 0.138148i
$736$	0	0
$737$	7.59601e9	0.0257463
$738$	0	0
$739$	4.19763e11i	1.40743i	0.710483	+	0.703715i	$0.248477\pi$
$739$	4.19763e11i	1.40743i	−0.710483	+	0.703715i	$0.751523\pi$
$740$	0	0
$741$	−6.33425e9	−0.0210098
$742$	0	0
$743$	− 1.42604e11i	− 0.467925i	−0.972246	−	0.233962i	$-0.924831\pi$
$743$	− 1.42604e11i	− 0.467925i	0.972246	−	0.233962i	$-0.0751693\pi$
$744$	0	0
$745$	−2.04226e11	−0.662957
$746$	0	0
$747$	4.05525e11i	1.30237i
$748$	0	0
$749$	5.46841e11	1.73754
$750$	0	0
$751$	− 3.26311e10i	− 0.102582i	−0.998684	−	0.0512911i	$-0.983666\pi$
$751$	− 3.26311e10i	− 0.102582i	0.998684	−	0.0512911i	$-0.0163336\pi$
$752$	0	0
$753$	1.57380e11	0.489518
$754$	0	0
$755$	− 2.16371e11i	− 0.665905i
$756$	0	0
$757$	4.36612e11	1.32957	0.664787	−	0.747033i	$-0.268522\pi$
$757$	4.36612e11	1.32957	0.664787	+	0.747033i	$0.268522\pi$
$758$	0	0
$759$	7.04142e10i	0.212174i
$760$	0	0
$761$	−1.98059e11	−0.590548	−0.295274	−	0.955413i	$-0.595411\pi$
$761$	−1.98059e11	−0.590548	−0.295274	+	0.955413i	$0.595411\pi$
$762$	0	0
$763$	− 6.04192e11i	− 1.78269i
$764$	0	0
$765$	−1.99031e11	−0.581133
$766$	0	0
$767$	− 1.90285e11i	− 0.549824i
$768$	0	0
$769$	−1.42097e10	−0.0406329	−0.0203165	−	0.999794i	$-0.506467\pi$
$769$	−1.42097e10	−0.0406329	−0.0203165	+	0.999794i	$0.506467\pi$
$770$	0	0
$771$	− 1.24705e11i	− 0.352912i
$772$	0	0
$773$	−6.62483e10	−0.185548	−0.0927741	−	0.995687i	$-0.529573\pi$
$773$	−6.62483e10	−0.185548	−0.0927741	+	0.995687i	$0.529573\pi$
$774$	0	0
$775$	1.09023e11i	0.302211i
$776$	0	0
$777$	−3.49770e10	−0.0959617
$778$	0	0
$779$	7.34544e9i	0.0199466i
$780$	0	0
$781$	−2.30487e11	−0.619501
$782$	0	0
$783$	2.79540e11i	0.743700i
$784$	0	0
$785$	8.87504e10	0.233718
$786$	0	0
$787$	7.13437e10i	0.185976i	0.995667	+	0.0929880i	$0.0296418\pi$
$787$	7.13437e10i	0.185976i	−0.995667	+	0.0929880i	$0.970358\pi$
$788$	0	0
$789$	1.32387e10	0.0341615
$790$	0	0
$791$	4.34417e11i	1.10969i
$792$	0	0
$793$	−3.01409e11	−0.762191
$794$	0	0
$795$	6.10661e10i	0.152873i
$796$	0	0
$797$	−1.11980e11	−0.277527	−0.138764	−	0.990326i	$-0.544313\pi$
$797$	−1.11980e11	−0.277527	−0.138764	+	0.990326i	$0.544313\pi$
$798$	0	0
$799$	− 2.29109e11i	− 0.562154i
$800$	0	0
$801$	3.24891e11	0.789237
$802$	0	0
$803$	2.13449e11i	0.513372i
$804$	0	0
$805$	−3.77677e11	−0.899368
$806$	0	0
$807$	− 4.05195e10i	− 0.0955365i
$808$	0	0
$809$	5.57023e10	0.130041	0.0650203	−	0.997884i	$-0.479289\pi$
$809$	5.57023e10	0.130041	0.0650203	+	0.997884i	$0.479289\pi$
$810$	0	0
$811$	− 7.06448e11i	− 1.63304i	−0.577317	−	0.816520i	$-0.695900\pi$
$811$	− 7.06448e11i	− 1.63304i	0.577317	−	0.816520i	$-0.304100\pi$
$812$	0	0
$813$	−1.85091e11	−0.423666
$814$	0	0
$815$	5.64620e9i	0.0127975i
$816$	0	0
$817$	3.48599e10	0.0782417
$818$	0	0
$819$	− 5.95569e11i	− 1.32372i
$820$	0	0
$821$	8.39733e10	0.184828	0.0924142	−	0.995721i	$-0.470542\pi$
$821$	8.39733e10	0.184828	0.0924142	+	0.995721i	$0.470542\pi$
$822$	0	0
$823$	1.90143e11i	0.414459i	0.978292	+	0.207229i	$0.0664446\pi$
$823$	1.90143e11i	0.414459i	−0.978292	+	0.207229i	$0.933555\pi$
$824$	0	0
$825$	1.35391e10	0.0292264
$826$	0	0
$827$	− 3.98982e11i	− 0.852965i	−0.904496	−	0.426483i	$-0.859753\pi$
$827$	− 3.98982e11i	− 0.852965i	0.904496	−	0.426483i	$-0.140247\pi$
$828$	0	0
$829$	−9.18641e11	−1.94504	−0.972518	−	0.232827i	$-0.925202\pi$
$829$	−9.18641e11	−1.94504	−0.972518	+	0.232827i	$0.925202\pi$
$830$	0	0
$831$	4.21536e10i	0.0883957i
$832$	0	0
$833$	−6.47936e11	−1.34571
$834$	0	0
$835$	− 1.58130e10i	− 0.0325288i
$836$	0	0
$837$	4.70655e11	0.958960
$838$	0	0
$839$	− 4.31925e11i	− 0.871688i	−0.900022	−	0.435844i	$-0.856450\pi$
$839$	− 4.31925e11i	− 0.871688i	0.900022	−	0.435844i	$-0.143550\pi$
$840$	0	0
$841$	1.86726e11	0.373268
$842$	0	0
$843$	2.54205e11i	0.503355i
$844$	0	0
$845$	−3.67605e10	−0.0721031
$846$	0	0
$847$	− 5.78298e11i	− 1.12362i
$848$	0	0
$849$	−2.02643e11	−0.390033
$850$	0	0
$851$	1.56860e11i	0.299085i
$852$	0	0
$853$	−9.29871e11	−1.75641	−0.878206	−	0.478282i	$-0.841260\pi$
$853$	−9.29871e11	−1.75641	−0.878206	+	0.478282i	$0.841260\pi$
$854$	0	0
$855$	1.22866e10i	0.0229914i
$856$	0	0
$857$	1.19713e11	0.221931	0.110965	−	0.993824i	$-0.464606\pi$
$857$	1.19713e11	0.221931	0.110965	+	0.993824i	$0.464606\pi$
$858$	0	0
$859$	4.22514e11i	0.776012i	0.921657	+	0.388006i	$0.126836\pi$
$859$	4.22514e11i	0.776012i	−0.921657	+	0.388006i	$0.873164\pi$
$860$	0	0
$861$	8.80936e10	0.160299
$862$	0	0
$863$	2.84698e11i	0.513264i	0.966509	+	0.256632i	$0.0826129\pi$
$863$	2.84698e11i	0.513264i	−0.966509	+	0.256632i	$0.917387\pi$
$864$	0	0
$865$	−3.06858e11	−0.548117
$866$	0	0
$867$	− 2.17945e11i	− 0.385719i
$868$	0	0
$869$	7.81630e9	0.0137064
$870$	0	0
$871$	− 3.67517e10i	− 0.0638565i
$872$	0	0
$873$	−3.47721e11	−0.598651
$874$	0	0
$875$	7.26193e10i	0.123885i
$876$	0	0
$877$	−1.94476e11	−0.328752	−0.164376	−	0.986398i	$-0.552561\pi$
$877$	−1.94476e11	−0.328752	−0.164376	+	0.986398i	$0.552561\pi$
$878$	0	0
$879$	− 4.33753e10i	− 0.0726585i
$880$	0	0
$881$	9.06691e11	1.50507	0.752533	−	0.658555i	$-0.228832\pi$
$881$	9.06691e11	1.50507	0.752533	+	0.658555i	$0.228832\pi$
$882$	0	0
$883$	2.26542e11i	0.372654i	0.982488	+	0.186327i	$0.0596584\pi$
$883$	2.26542e11i	0.372654i	−0.982488	+	0.186327i	$0.940342\pi$
$884$	0	0
$885$	4.70793e10	0.0767462
$886$	0	0
$887$	− 1.70744e11i	− 0.275837i	−0.990444	−	0.137918i	$-0.955959\pi$
$887$	− 1.70744e11i	− 0.275837i	0.990444	−	0.137918i	$-0.0440412\pi$
$888$	0	0
$889$	−1.48315e12	−2.37453
$890$	0	0
$891$	1.84404e11i	0.292590i
$892$	0	0
$893$	−1.41433e10	−0.0222405
$894$	0	0
$895$	− 1.80225e11i	− 0.280882i
$896$	0	0
$897$	3.40684e11	0.526238
$898$	0	0
$899$	− 1.15664e12i	− 1.77075i
$900$	0	0
$901$	9.81380e11	1.48915
$902$	0	0
$903$	− 4.18074e11i	− 0.628785i
$904$	0	0
$905$	−1.89791e11	−0.282932
$906$	0	0
$907$	5.20431e11i	0.769013i	0.923122	+	0.384507i	$0.125628\pi$
$907$	5.20431e11i	0.769013i	−0.923122	+	0.384507i	$0.874372\pi$
$908$	0	0
$909$	−5.60205e11	−0.820525
$910$	0	0
$911$	5.42499e11i	0.787635i	0.919189	+	0.393818i	$0.128846\pi$
$911$	5.42499e11i	0.787635i	−0.919189	+	0.393818i	$0.871154\pi$
$912$	0	0
$913$	−4.43327e11	−0.638030
$914$	0	0
$915$	− 7.45729e10i	− 0.106389i
$916$	0	0
$917$	4.37497e11	0.618725
$918$	0	0
$919$	− 6.49225e11i	− 0.910192i	−0.890442	−	0.455096i	$-0.849605\pi$
$919$	− 6.49225e11i	− 0.910192i	0.890442	−	0.455096i	$-0.150395\pi$
$920$	0	0
$921$	9.16364e10	0.127359
$922$	0	0
$923$	1.11516e12i	1.53650i
$924$	0	0
$925$	3.01609e10	0.0411981
$926$	0	0
$927$	1.14230e12i	1.54690i
$928$	0	0
$929$	4.98636e11	0.669455	0.334727	−	0.942315i	$-0.391356\pi$
$929$	4.98636e11	0.669455	0.334727	+	0.942315i	$0.391356\pi$
$930$	0	0
$931$	3.99982e10i	0.0532404i
$932$	0	0
$933$	−3.35785e11	−0.443133
$934$	0	0
$935$	− 2.17585e11i	− 0.284696i
$936$	0	0
$937$	4.01408e11	0.520748	0.260374	−	0.965508i	$-0.416154\pi$
$937$	4.01408e11	0.520748	0.260374	+	0.965508i	$0.416154\pi$
$938$	0	0
$939$	− 7.57332e10i	− 0.0974146i
$940$	0	0
$941$	2.64682e10	0.0337572	0.0168786	−	0.999858i	$-0.494627\pi$
$941$	2.64682e10	0.0337572	0.0168786	+	0.999858i	$0.494627\pi$
$942$	0	0
$943$	− 3.95071e11i	− 0.499606i
$944$	0	0
$945$	3.13500e11	0.393106
$946$	0	0
$947$	− 5.71352e11i	− 0.710401i	−0.934790	−	0.355201i	$-0.884413\pi$
$947$	− 5.71352e11i	− 0.710401i	0.934790	−	0.355201i	$-0.115587\pi$
$948$	0	0
$949$	1.03273e12	1.27327
$950$	0	0
$951$	− 1.47163e10i	− 0.0179918i
$952$	0	0
$953$	−1.09917e12	−1.33258	−0.666288	−	0.745694i	$-0.732118\pi$
$953$	−1.09917e12	−1.33258	−0.666288	+	0.745694i	$0.732118\pi$
$954$	0	0
$955$	4.29711e10i	0.0516611i
$956$	0	0
$957$	−1.43639e11	−0.171247
$958$	0	0
$959$	1.08545e12i	1.28333i
$960$	0	0
$961$	−1.09451e12	−1.28329
$962$	0	0
$963$	9.56813e11i	1.11256i
$964$	0	0
$965$	−2.70497e10	−0.0311928
$966$	0	0
$967$	8.78566e11i	1.00477i	0.864643	+	0.502387i	$0.167545\pi$
$967$	8.78566e11i	1.00477i	−0.864643	+	0.502387i	$0.832455\pi$
$968$	0	0
$969$	−2.51858e10	−0.0285668
$970$	0	0
$971$	7.91945e11i	0.890878i	0.895312	+	0.445439i	$0.146952\pi$
$971$	7.91945e11i	0.890878i	−0.895312	+	0.445439i	$0.853048\pi$
$972$	0	0
$973$	−2.15806e12	−2.40775
$974$	0	0
$975$	− 6.55064e10i	− 0.0724878i
$976$	0	0
$977$	−5.46919e11	−0.600268	−0.300134	−	0.953897i	$-0.597031\pi$
$977$	−5.46919e11	−0.600268	−0.300134	+	0.953897i	$0.597031\pi$
$978$	0	0
$979$	3.55177e11i	0.386646i
$980$	0	0
$981$	1.05716e12	1.14147
$982$	0	0
$983$	7.38083e11i	0.790481i	0.918578	+	0.395240i	$0.129339\pi$
$983$	7.38083e11i	0.790481i	−0.918578	+	0.395240i	$0.870661\pi$
$984$	0	0
$985$	−7.95946e11	−0.845549
$986$	0	0
$987$	1.69620e11i	0.178735i
$988$	0	0
$989$	−1.87492e12	−1.95974
$990$	0	0
$991$	− 3.73877e11i	− 0.387645i	−0.981037	−	0.193823i	$-0.937911\pi$
$991$	− 3.73877e11i	− 0.387645i	0.981037	−	0.193823i	$-0.0620886\pi$
$992$	0	0
$993$	−3.29959e11	−0.339362
$994$	0	0
$995$	2.43523e11i	0.248455i
$996$	0	0
$997$	−1.14224e12	−1.15605	−0.578025	−	0.816019i	$-0.696177\pi$
$997$	−1.14224e12	−1.15605	−0.578025	+	0.816019i	$0.696177\pi$
$998$	0	0
$999$	− 1.30205e11i	− 0.130727i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.9.b.d.191.7		16
4.3	odd	2	inner	320.9.b.d.191.10		16
8.3	odd	2		20.9.b.a.11.14	yes	16
8.5	even	2		20.9.b.a.11.13	✓	16
24.5	odd	2		180.9.c.a.91.4		16
24.11	even	2		180.9.c.a.91.3		16
40.3	even	4		100.9.d.c.99.23		32
40.13	odd	4		100.9.d.c.99.9		32
40.19	odd	2		100.9.b.d.51.3		16
40.27	even	4		100.9.d.c.99.10		32
40.29	even	2		100.9.b.d.51.4		16
40.37	odd	4		100.9.d.c.99.24		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.9.b.a.11.13	✓	16	8.5	even	2
20.9.b.a.11.14	yes	16	8.3	odd	2
100.9.b.d.51.3		16	40.19	odd	2
100.9.b.d.51.4		16	40.29	even	2
100.9.d.c.99.9		32	40.13	odd	4
100.9.d.c.99.10		32	40.27	even	4
100.9.d.c.99.23		32	40.3	even	4
100.9.d.c.99.24		32	40.37	odd	4
180.9.c.a.91.3		16	24.11	even	2
180.9.c.a.91.4		16	24.5	odd	2
320.9.b.d.191.7		16	1.1	even	1	trivial
320.9.b.d.191.10		16	4.3	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-27.2434i q^{3} -279.508 q^{5} -3325.58i q^{7} +5818.80 q^{9} +6361.21i q^{11} +30777.4 q^{13} +7614.76i q^{15} +122375. q^{17} -7554.43i q^{19} -90600.1 q^{21} +406311. i q^{23} +78125.0 q^{25} -337268. i q^{27} -828838. q^{29} +1.39549e6i q^{31} +173301. q^{33} +929527. i q^{35} +386059. q^{37} -838482. i q^{39} -972335. q^{41} +4.61450e6i q^{43} -1.62640e6 q^{45} -1.87219e6i q^{47} -5.29467e6 q^{49} -3.33392e6i q^{51} +8.01944e6 q^{53} -1.77801e6i q^{55} -205808. q^{57} -6.18263e6i q^{59} -9.79320e6 q^{61} -1.93509e7i q^{63} -8.60255e6 q^{65} -1.19411e6i q^{67} +1.10693e7 q^{69} +3.62332e7i q^{71} +3.35548e7 q^{73} -2.12839e6i q^{75} +2.11547e7 q^{77} -1.22874e6i q^{79} +2.89888e7 q^{81} +6.96923e7i q^{83} -3.42049e7 q^{85} +2.25804e7i q^{87} +5.58347e7 q^{89} -1.02353e8i q^{91} +3.80180e7 q^{93} +2.11153e6i q^{95} -5.97582e7 q^{97} +3.70146e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53}+ \cdots + 171851232 q^{97}+O(q^{100}) \) 16 * q - 38800 * q^9 - 51392 * q^13 + 27552 * q^17 - 414496 * q^21 + 1250000 * q^25 - 2764896 * q^29 - 5521600 * q^33 - 9009472 * q^37 - 8576448 * q^41 - 1580000 * q^45 - 32803600 * q^49 - 2452032 * q^53 + 11957760 * q^57 - 8371712 * q^61 + 9060000 * q^65 - 7527264 * q^69 + 61907232 * q^73 + 156997440 * q^77 + 140586672 * q^81 + 106960000 * q^85 + 106647456 * q^89 - 105563840 * q^93 + 171851232 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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