LMFDB - Embedded newform 920.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,4,Mod(1,920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("920.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.2817572053$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 $ x^10 - x^9 - 204x^8 + 42x^7 + 12958x^6 + 5872x^5 - 259871x^4 - 149461x^3 + 1222472x^2 + 627136x - 43712
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.27510$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.27510 q^{3} -5.00000 q^{5} +33.0838 q^{7} +59.0275 q^{9} -54.6419 q^{11} +86.7693 q^{13} +46.3755 q^{15} +77.8072 q^{17} +138.051 q^{19} -306.856 q^{21} +23.0000 q^{23} +25.0000 q^{25} -297.058 q^{27} -212.229 q^{29} -147.641 q^{31} +506.810 q^{33} -165.419 q^{35} -185.803 q^{37} -804.794 q^{39} +322.883 q^{41} +3.59490 q^{43} -295.137 q^{45} -18.6351 q^{47} +751.539 q^{49} -721.670 q^{51} -6.84351 q^{53} +273.210 q^{55} -1280.44 q^{57} +414.113 q^{59} +258.968 q^{61} +1952.85 q^{63} -433.847 q^{65} -490.096 q^{67} -213.327 q^{69} -821.332 q^{71} +11.4667 q^{73} -231.877 q^{75} -1807.76 q^{77} +214.872 q^{79} +1161.50 q^{81} -793.747 q^{83} -389.036 q^{85} +1968.44 q^{87} -192.788 q^{89} +2870.66 q^{91} +1369.39 q^{93} -690.257 q^{95} +1135.34 q^{97} -3225.38 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{3} - 50 q^{5} + 28 q^{7} + 139 q^{9} - 14 q^{11} + 11 q^{13} + 5 q^{15} + 68 q^{17} + 114 q^{19} - 232 q^{21} + 230 q^{23} + 250 q^{25} - 433 q^{27} - 273 q^{29} - 129 q^{31} + 98 q^{33} - 140 q^{35}+ \cdots + 4356 q^{99}+O(q^{100}) $ 10 * q - q^3 - 50 * q^5 + 28 * q^7 + 139 * q^9 - 14 * q^11 + 11 * q^13 + 5 * q^15 + 68 * q^17 + 114 * q^19 - 232 * q^21 + 230 * q^23 + 250 * q^25 - 433 * q^27 - 273 * q^29 - 129 * q^31 + 98 * q^33 - 140 * q^35 + 62 * q^37 + 283 * q^39 + 767 * q^41 + 332 * q^43 - 695 * q^45 - 323 * q^47 + 1162 * q^49 + 176 * q^51 + 558 * q^53 + 70 * q^55 + 46 * q^57 + 822 * q^59 + 318 * q^61 + 2698 * q^63 - 55 * q^65 + 1152 * q^67 - 23 * q^69 + 1247 * q^71 + 1941 * q^73 - 25 * q^75 + 528 * q^77 + 3134 * q^79 + 6210 * q^81 + 482 * q^83 - 340 * q^85 + 1797 * q^87 + 4734 * q^89 + 4992 * q^91 + 4647 * q^93 - 570 * q^95 + 2326 * q^97 + 4356 * q^99

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
$3$	−9.27510	−1.78499	−0.892497	−	0.451054i	$-0.851048\pi$
$3$	−9.27510	−1.78499	−0.892497	+	0.451054i	$0.851048\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	33.0838	1.78636	0.893179	−	0.449701i	$-0.148470\pi$
$7$	33.0838	1.78636	0.893179	+	0.449701i	$0.148470\pi$
$8$	0	0
$9$	59.0275	2.18620
$10$	0	0
$11$	−54.6419	−1.49774	−0.748871	−	0.662716i	$-0.769404\pi$
$11$	−54.6419	−1.49774	−0.748871	+	0.662716i	$0.769404\pi$
$12$	0	0
$13$	86.7693	1.85119	0.925595	−	0.378514i	$-0.123565\pi$
$13$	86.7693	1.85119	0.925595	+	0.378514i	$0.123565\pi$
$14$	0	0
$15$	46.3755	0.798273
$16$	0	0
$17$	77.8072	1.11006	0.555030	−	0.831830i	$-0.312707\pi$
$17$	77.8072	1.11006	0.555030	+	0.831830i	$0.312707\pi$
$18$	0	0
$19$	138.051	1.66690	0.833452	−	0.552592i	$-0.186361\pi$
$19$	138.051	1.66690	0.833452	+	0.552592i	$0.186361\pi$
$20$	0	0
$21$	−306.856	−3.18864
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−297.058	−2.11736
$28$	0	0
$29$	−212.229	−1.35896	−0.679481	−	0.733693i	$-0.737795\pi$
$29$	−212.229	−1.35896	−0.679481	+	0.733693i	$0.737795\pi$
$30$	0	0
$31$	−147.641	−0.855393	−0.427696	−	0.903922i	$-0.640675\pi$
$31$	−147.641	−0.855393	−0.427696	+	0.903922i	$0.640675\pi$
$32$	0	0
$33$	506.810	2.67346
$34$	0	0
$35$	−165.419	−0.798884
$36$	0	0
$37$	−185.803	−0.825562	−0.412781	−	0.910830i	$-0.635443\pi$
$37$	−185.803	−0.825562	−0.412781	+	0.910830i	$0.635443\pi$
$38$	0	0
$39$	−804.794	−3.30436
$40$	0	0
$41$	322.883	1.22990	0.614950	−	0.788566i	$-0.289176\pi$
$41$	322.883	1.22990	0.614950	+	0.788566i	$0.289176\pi$
$42$	0	0
$43$	3.59490	0.0127492	0.00637461	−	0.999980i	$-0.497971\pi$
$43$	3.59490	0.0127492	0.00637461	+	0.999980i	$0.497971\pi$
$44$	0	0
$45$	−295.137	−0.977700
$46$	0	0
$47$	−18.6351	−0.0578341	−0.0289170	−	0.999582i	$-0.509206\pi$
$47$	−18.6351	−0.0578341	−0.0289170	+	0.999582i	$0.509206\pi$
$48$	0	0
$49$	751.539	2.19108
$50$	0	0
$51$	−721.670	−1.98145
$52$	0	0
$53$	−6.84351	−0.0177364	−0.00886820	−	0.999961i	$-0.502823\pi$
$53$	−6.84351	−0.0177364	−0.00886820	+	0.999961i	$0.502823\pi$
$54$	0	0
$55$	273.210	0.669811
$56$	0	0
$57$	−1280.44	−2.97541
$58$	0	0
$59$	414.113	0.913779	0.456889	−	0.889524i	$-0.348964\pi$
$59$	414.113	0.913779	0.456889	+	0.889524i	$0.348964\pi$
$60$	0	0
$61$	258.968	0.543564	0.271782	−	0.962359i	$-0.412387\pi$
$61$	258.968	0.543564	0.271782	+	0.962359i	$0.412387\pi$
$62$	0	0
$63$	1952.85	3.90534
$64$	0	0
$65$	−433.847	−0.827878
$66$	0	0
$67$	−490.096	−0.893653	−0.446826	−	0.894621i	$-0.647446\pi$
$67$	−490.096	−0.893653	−0.446826	+	0.894621i	$0.647446\pi$
$68$	0	0
$69$	−213.327	−0.372197
$70$	0	0
$71$	−821.332	−1.37288	−0.686438	−	0.727189i	$-0.740827\pi$
$71$	−821.332	−1.37288	−0.686438	+	0.727189i	$0.740827\pi$
$72$	0	0
$73$	11.4667	0.0183845	0.00919227	−	0.999958i	$-0.497074\pi$
$73$	11.4667	0.0183845	0.00919227	+	0.999958i	$0.497074\pi$
$74$	0	0
$75$	−231.877	−0.356999
$76$	0	0
$77$	−1807.76	−2.67550
$78$	0	0
$79$	214.872	0.306012	0.153006	−	0.988225i	$-0.451105\pi$
$79$	214.872	0.306012	0.153006	+	0.988225i	$0.451105\pi$
$80$	0	0
$81$	1161.50	1.59328
$82$	0	0
$83$	−793.747	−1.04970	−0.524850	−	0.851195i	$-0.675878\pi$
$83$	−793.747	−1.04970	−0.524850	+	0.851195i	$0.675878\pi$
$84$	0	0
$85$	−389.036	−0.496434
$86$	0	0
$87$	1968.44	2.42574
$88$	0	0
$89$	−192.788	−0.229612	−0.114806	−	0.993388i	$-0.536625\pi$
$89$	−192.788	−0.229612	−0.114806	+	0.993388i	$0.536625\pi$
$90$	0	0
$91$	2870.66	3.30689
$92$	0	0
$93$	1369.39	1.52687
$94$	0	0
$95$	−690.257	−0.745462
$96$	0	0
$97$	1135.34	1.18842	0.594208	−	0.804312i	$-0.297466\pi$
$97$	1135.34	1.18842	0.594208	+	0.804312i	$0.297466\pi$
$98$	0	0
$99$	−3225.38	−3.27437
$100$	0	0
$101$	1974.69	1.94543	0.972717	−	0.231995i	$-0.0745253\pi$
$101$	1974.69	1.94543	0.972717	+	0.231995i	$0.0745253\pi$
$102$	0	0
$103$	263.452	0.252026	0.126013	−	0.992029i	$-0.459782\pi$
$103$	263.452	0.252026	0.126013	+	0.992029i	$0.459782\pi$
$104$	0	0
$105$	1534.28	1.42600
$106$	0	0
$107$	−544.724	−0.492154	−0.246077	−	0.969250i	$-0.579142\pi$
$107$	−544.724	−0.492154	−0.246077	+	0.969250i	$0.579142\pi$
$108$	0	0
$109$	1850.62	1.62621	0.813105	−	0.582117i	$-0.197775\pi$
$109$	1850.62	1.62621	0.813105	+	0.582117i	$0.197775\pi$
$110$	0	0
$111$	1723.34	1.47362
$112$	0	0
$113$	−70.8472	−0.0589800	−0.0294900	−	0.999565i	$-0.509388\pi$
$113$	−70.8472	−0.0589800	−0.0294900	+	0.999565i	$0.509388\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	5121.77	4.04708
$118$	0	0
$119$	2574.16	1.98296
$120$	0	0
$121$	1654.74	1.24323
$122$	0	0
$123$	−2994.77	−2.19536
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1399.71	0.977982	0.488991	−	0.872289i	$-0.337365\pi$
$127$	1399.71	0.977982	0.488991	+	0.872289i	$0.337365\pi$
$128$	0	0
$129$	−33.3430	−0.0227573
$130$	0	0
$131$	1676.24	1.11797	0.558984	−	0.829178i	$-0.311191\pi$
$131$	1676.24	1.11797	0.558984	+	0.829178i	$0.311191\pi$
$132$	0	0
$133$	4567.27	2.97769
$134$	0	0
$135$	1485.29	0.946914
$136$	0	0
$137$	136.256	0.0849718	0.0424859	−	0.999097i	$-0.486472\pi$
$137$	136.256	0.0849718	0.0424859	+	0.999097i	$0.486472\pi$
$138$	0	0
$139$	−1640.19	−1.00086	−0.500430	−	0.865777i	$-0.666825\pi$
$139$	−1640.19	−1.00086	−0.500430	+	0.865777i	$0.666825\pi$
$140$	0	0
$141$	172.842	0.103234
$142$	0	0
$143$	−4741.25	−2.77261
$144$	0	0
$145$	1061.14	0.607746
$146$	0	0
$147$	−6970.60	−3.91106
$148$	0	0
$149$	−30.6106	−0.0168303	−0.00841516	−	0.999965i	$-0.502679\pi$
$149$	−30.6106	−0.0168303	−0.00841516	+	0.999965i	$0.502679\pi$
$150$	0	0
$151$	−2601.86	−1.40223	−0.701113	−	0.713050i	$-0.747313\pi$
$151$	−2601.86	−1.40223	−0.701113	+	0.713050i	$0.747313\pi$
$152$	0	0
$153$	4592.76	2.42682
$154$	0	0
$155$	738.207	0.382543
$156$	0	0
$157$	134.723	0.0684845	0.0342422	−	0.999414i	$-0.489098\pi$
$157$	134.723	0.0684845	0.0342422	+	0.999414i	$0.489098\pi$
$158$	0	0
$159$	63.4743	0.0316593
$160$	0	0
$161$	760.928	0.372481
$162$	0	0
$163$	−926.826	−0.445366	−0.222683	−	0.974891i	$-0.571481\pi$
$163$	−926.826	−0.445366	−0.222683	+	0.974891i	$0.571481\pi$
$164$	0	0
$165$	−2534.05	−1.19561
$166$	0	0
$167$	1934.24	0.896266	0.448133	−	0.893967i	$-0.352089\pi$
$167$	1934.24	0.896266	0.448133	+	0.893967i	$0.352089\pi$
$168$	0	0
$169$	5331.92	2.42691
$170$	0	0
$171$	8148.83	3.64419
$172$	0	0
$173$	−1627.32	−0.715162	−0.357581	−	0.933882i	$-0.616398\pi$
$173$	−1627.32	−0.715162	−0.357581	+	0.933882i	$0.616398\pi$
$174$	0	0
$175$	827.095	0.357272
$176$	0	0
$177$	−3840.94	−1.63109
$178$	0	0
$179$	−220.784	−0.0921910	−0.0460955	−	0.998937i	$-0.514678\pi$
$179$	−220.784	−0.0921910	−0.0460955	+	0.998937i	$0.514678\pi$
$180$	0	0
$181$	−2748.79	−1.12882	−0.564408	−	0.825496i	$-0.690895\pi$
$181$	−2748.79	−1.12882	−0.564408	+	0.825496i	$0.690895\pi$
$182$	0	0
$183$	−2401.95	−0.970259
$184$	0	0
$185$	929.014	0.369203
$186$	0	0
$187$	−4251.54	−1.66258
$188$	0	0
$189$	−9827.81	−3.78237
$190$	0	0
$191$	1759.57	0.666586	0.333293	−	0.942823i	$-0.391840\pi$
$191$	1759.57	0.666586	0.333293	+	0.942823i	$0.391840\pi$
$192$	0	0
$193$	3951.63	1.47381	0.736904	−	0.675998i	$-0.236287\pi$
$193$	3951.63	1.47381	0.736904	+	0.675998i	$0.236287\pi$
$194$	0	0
$195$	4023.97	1.47776
$196$	0	0
$197$	−2655.95	−0.960551	−0.480276	−	0.877118i	$-0.659463\pi$
$197$	−2655.95	−0.960551	−0.480276	+	0.877118i	$0.659463\pi$
$198$	0	0
$199$	4229.06	1.50648	0.753241	−	0.657745i	$-0.228490\pi$
$199$	4229.06	1.50648	0.753241	+	0.657745i	$0.228490\pi$
$200$	0	0
$201$	4545.69	1.59516
$202$	0	0
$203$	−7021.34	−2.42759
$204$	0	0
$205$	−1614.42	−0.550028
$206$	0	0
$207$	1357.63	0.455855
$208$	0	0
$209$	−7543.40	−2.49659
$210$	0	0
$211$	−88.7870	−0.0289685	−0.0144842	−	0.999895i	$-0.504611\pi$
$211$	−88.7870	−0.0289685	−0.0144842	+	0.999895i	$0.504611\pi$
$212$	0	0
$213$	7617.93	2.45057
$214$	0	0
$215$	−17.9745	−0.00570163
$216$	0	0
$217$	−4884.54	−1.52804
$218$	0	0
$219$	−106.354	−0.0328163
$220$	0	0
$221$	6751.28	2.05493
$222$	0	0
$223$	−4580.03	−1.37534	−0.687672	−	0.726022i	$-0.741367\pi$
$223$	−4580.03	−1.37534	−0.687672	+	0.726022i	$0.741367\pi$
$224$	0	0
$225$	1475.69	0.437241
$226$	0	0
$227$	−230.581	−0.0674195	−0.0337097	−	0.999432i	$-0.510732\pi$
$227$	−230.581	−0.0674195	−0.0337097	+	0.999432i	$0.510732\pi$
$228$	0	0
$229$	1673.24	0.482841	0.241421	−	0.970421i	$-0.422387\pi$
$229$	1673.24	0.482841	0.241421	+	0.970421i	$0.422387\pi$
$230$	0	0
$231$	16767.2	4.77576
$232$	0	0
$233$	4138.22	1.16354	0.581768	−	0.813355i	$-0.302361\pi$
$233$	4138.22	1.16354	0.581768	+	0.813355i	$0.302361\pi$
$234$	0	0
$235$	93.1753	0.0258642
$236$	0	0
$237$	−1992.96	−0.546230
$238$	0	0
$239$	−4784.02	−1.29478	−0.647390	−	0.762159i	$-0.724140\pi$
$239$	−4784.02	−1.29478	−0.647390	+	0.762159i	$0.724140\pi$
$240$	0	0
$241$	−67.4103	−0.0180178	−0.00900888	−	0.999959i	$-0.502868\pi$
$241$	−67.4103	−0.0180178	−0.00900888	+	0.999959i	$0.502868\pi$
$242$	0	0
$243$	−2752.47	−0.726630
$244$	0	0
$245$	−3757.69	−0.979879
$246$	0	0
$247$	11978.6	3.08576
$248$	0	0
$249$	7362.08	1.87371
$250$	0	0
$251$	216.815	0.0545230	0.0272615	−	0.999628i	$-0.491321\pi$
$251$	216.815	0.0545230	0.0272615	+	0.999628i	$0.491321\pi$
$252$	0	0
$253$	−1256.76	−0.312301
$254$	0	0
$255$	3608.35	0.886132
$256$	0	0
$257$	1285.87	0.312102	0.156051	−	0.987749i	$-0.450124\pi$
$257$	1285.87	0.312102	0.156051	+	0.987749i	$0.450124\pi$
$258$	0	0
$259$	−6147.07	−1.47475
$260$	0	0
$261$	−12527.3	−2.97097
$262$	0	0
$263$	1238.80	0.290447	0.145224	−	0.989399i	$-0.453610\pi$
$263$	1238.80	0.290447	0.145224	+	0.989399i	$0.453610\pi$
$264$	0	0
$265$	34.2176	0.00793196
$266$	0	0
$267$	1788.13	0.409857
$268$	0	0
$269$	2162.86	0.490230	0.245115	−	0.969494i	$-0.421174\pi$
$269$	2162.86	0.490230	0.245115	+	0.969494i	$0.421174\pi$
$270$	0	0
$271$	−4060.78	−0.910239	−0.455120	−	0.890430i	$-0.650403\pi$
$271$	−4060.78	−0.910239	−0.455120	+	0.890430i	$0.650403\pi$
$272$	0	0
$273$	−26625.7	−5.90278
$274$	0	0
$275$	−1366.05	−0.299548
$276$	0	0
$277$	373.897	0.0811022	0.0405511	−	0.999177i	$-0.487089\pi$
$277$	373.897	0.0811022	0.0405511	+	0.999177i	$0.487089\pi$
$278$	0	0
$279$	−8714.90	−1.87006
$280$	0	0
$281$	−3776.83	−0.801803	−0.400901	−	0.916121i	$-0.631303\pi$
$281$	−3776.83	−0.801803	−0.400901	+	0.916121i	$0.631303\pi$
$282$	0	0
$283$	3114.71	0.654241	0.327121	−	0.944983i	$-0.393922\pi$
$283$	3114.71	0.654241	0.327121	+	0.944983i	$0.393922\pi$
$284$	0	0
$285$	6402.20	1.33065
$286$	0	0
$287$	10682.2	2.19704
$288$	0	0
$289$	1140.96	0.232233
$290$	0	0
$291$	−10530.4	−2.12131
$292$	0	0
$293$	350.402	0.0698659	0.0349330	−	0.999390i	$-0.488878\pi$
$293$	350.402	0.0698659	0.0349330	+	0.999390i	$0.488878\pi$
$294$	0	0
$295$	−2070.57	−0.408654
$296$	0	0
$297$	16231.8	3.17127
$298$	0	0
$299$	1995.69	0.386000
$300$	0	0
$301$	118.933	0.0227747
$302$	0	0
$303$	−18315.4	−3.47259
$304$	0	0
$305$	−1294.84	−0.243089
$306$	0	0
$307$	−5292.26	−0.983861	−0.491931	−	0.870634i	$-0.663709\pi$
$307$	−5292.26	−0.983861	−0.491931	+	0.870634i	$0.663709\pi$
$308$	0	0
$309$	−2443.54	−0.449865
$310$	0	0
$311$	3921.72	0.715050	0.357525	−	0.933904i	$-0.383621\pi$
$311$	3921.72	0.715050	0.357525	+	0.933904i	$0.383621\pi$
$312$	0	0
$313$	−7396.40	−1.33569	−0.667843	−	0.744303i	$-0.732782\pi$
$313$	−7396.40	−1.33569	−0.667843	+	0.744303i	$0.732782\pi$
$314$	0	0
$315$	−9764.27	−1.74652
$316$	0	0
$317$	−8542.26	−1.51351	−0.756753	−	0.653701i	$-0.773215\pi$
$317$	−8542.26	−1.51351	−0.756753	+	0.653701i	$0.773215\pi$
$318$	0	0
$319$	11596.6	2.03537
$320$	0	0
$321$	5052.37	0.878492
$322$	0	0
$323$	10741.4	1.85036
$324$	0	0
$325$	2169.23	0.370238
$326$	0	0
$327$	−17164.7	−2.90277
$328$	0	0
$329$	−616.519	−0.103312
$330$	0	0
$331$	8197.69	1.36129	0.680643	−	0.732615i	$-0.261701\pi$
$331$	8197.69	1.36129	0.680643	+	0.732615i	$0.261701\pi$
$332$	0	0
$333$	−10967.5	−1.80485
$334$	0	0
$335$	2450.48	0.399654
$336$	0	0
$337$	−79.7920	−0.0128978	−0.00644888	−	0.999979i	$-0.502053\pi$
$337$	−79.7920	−0.0128978	−0.00644888	+	0.999979i	$0.502053\pi$
$338$	0	0
$339$	657.114	0.105279
$340$	0	0
$341$	8067.41	1.28116
$342$	0	0
$343$	13516.0	2.12769
$344$	0	0
$345$	1066.64	0.166452
$346$	0	0
$347$	5432.91	0.840501	0.420250	−	0.907408i	$-0.361942\pi$
$347$	5432.91	0.840501	0.420250	+	0.907408i	$0.361942\pi$
$348$	0	0
$349$	−5795.96	−0.888970	−0.444485	−	0.895786i	$-0.646613\pi$
$349$	−5795.96	−0.888970	−0.444485	+	0.895786i	$0.646613\pi$
$350$	0	0
$351$	−25775.5	−3.91965
$352$	0	0
$353$	309.167	0.0466156	0.0233078	−	0.999728i	$-0.492580\pi$
$353$	309.167	0.0466156	0.0233078	+	0.999728i	$0.492580\pi$
$354$	0	0
$355$	4106.66	0.613969
$356$	0	0
$357$	−23875.6	−3.53958
$358$	0	0
$359$	6416.01	0.943243	0.471622	−	0.881801i	$-0.343669\pi$
$359$	6416.01	0.943243	0.471622	+	0.881801i	$0.343669\pi$
$360$	0	0
$361$	12199.2	1.77857
$362$	0	0
$363$	−15347.9	−2.21916
$364$	0	0
$365$	−57.3333	−0.00822182
$366$	0	0
$367$	−7471.41	−1.06268	−0.531341	−	0.847158i	$-0.678312\pi$
$367$	−7471.41	−1.06268	−0.531341	+	0.847158i	$0.678312\pi$
$368$	0	0
$369$	19059.0	2.68881
$370$	0	0
$371$	−226.410	−0.0316835
$372$	0	0
$373$	5260.07	0.730177	0.365089	−	0.930973i	$-0.381039\pi$
$373$	5260.07	0.730177	0.365089	+	0.930973i	$0.381039\pi$
$374$	0	0
$375$	1159.39	0.159655
$376$	0	0
$377$	−18414.9	−2.51570
$378$	0	0
$379$	2919.86	0.395733	0.197867	−	0.980229i	$-0.436599\pi$
$379$	2919.86	0.395733	0.197867	+	0.980229i	$0.436599\pi$
$380$	0	0
$381$	−12982.4	−1.74569
$382$	0	0
$383$	−1320.08	−0.176117	−0.0880583	−	0.996115i	$-0.528066\pi$
$383$	−1320.08	−0.176117	−0.0880583	+	0.996115i	$0.528066\pi$
$384$	0	0
$385$	9038.82	1.19652
$386$	0	0
$387$	212.198	0.0278724
$388$	0	0
$389$	11445.1	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$389$	11445.1	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$390$	0	0
$391$	1789.57	0.231464
$392$	0	0
$393$	−15547.3	−1.99557
$394$	0	0
$395$	−1074.36	−0.136853
$396$	0	0
$397$	8985.05	1.13589	0.567943	−	0.823068i	$-0.307739\pi$
$397$	8985.05	1.13589	0.567943	+	0.823068i	$0.307739\pi$
$398$	0	0
$399$	−42361.9	−5.31515
$400$	0	0
$401$	−5793.76	−0.721513	−0.360756	−	0.932660i	$-0.617481\pi$
$401$	−5793.76	−0.721513	−0.360756	+	0.932660i	$0.617481\pi$
$402$	0	0
$403$	−12810.7	−1.58350
$404$	0	0
$405$	−5807.51	−0.712536
$406$	0	0
$407$	10152.6	1.23648
$408$	0	0
$409$	−12361.7	−1.49449	−0.747244	−	0.664550i	$-0.768623\pi$
$409$	−12361.7	−1.49449	−0.747244	+	0.664550i	$0.768623\pi$
$410$	0	0
$411$	−1263.79	−0.151674
$412$	0	0
$413$	13700.4	1.63234
$414$	0	0
$415$	3968.74	0.469440
$416$	0	0
$417$	15213.0	1.78653
$418$	0	0
$419$	14854.1	1.73191	0.865955	−	0.500122i	$-0.166711\pi$
$419$	14854.1	1.73191	0.865955	+	0.500122i	$0.166711\pi$
$420$	0	0
$421$	11243.5	1.30160	0.650799	−	0.759250i	$-0.274434\pi$
$421$	11243.5	1.30160	0.650799	+	0.759250i	$0.274434\pi$
$422$	0	0
$423$	−1099.98	−0.126437
$424$	0	0
$425$	1945.18	0.222012
$426$	0	0
$427$	8567.64	0.971001
$428$	0	0
$429$	43975.5	4.94909
$430$	0	0
$431$	11909.7	1.33102	0.665509	−	0.746389i	$-0.268214\pi$
$431$	11909.7	1.33102	0.665509	+	0.746389i	$0.268214\pi$
$432$	0	0
$433$	4466.68	0.495739	0.247869	−	0.968793i	$-0.420270\pi$
$433$	4466.68	0.495739	0.247869	+	0.968793i	$0.420270\pi$
$434$	0	0
$435$	−9842.22	−1.08482
$436$	0	0
$437$	3175.18	0.347573
$438$	0	0
$439$	2376.49	0.258369	0.129184	−	0.991621i	$-0.458764\pi$
$439$	2376.49	0.258369	0.129184	+	0.991621i	$0.458764\pi$
$440$	0	0
$441$	44361.4	4.79014
$442$	0	0
$443$	13887.4	1.48941	0.744705	−	0.667394i	$-0.232590\pi$
$443$	13887.4	1.48941	0.744705	+	0.667394i	$0.232590\pi$
$444$	0	0
$445$	963.941	0.102686
$446$	0	0
$447$	283.916	0.0300420
$448$	0	0
$449$	−2773.98	−0.291563	−0.145782	−	0.989317i	$-0.546570\pi$
$449$	−2773.98	−0.291563	−0.145782	+	0.989317i	$0.546570\pi$
$450$	0	0
$451$	−17643.0	−1.84207
$452$	0	0
$453$	24132.5	2.50296
$454$	0	0
$455$	−14353.3	−1.47889
$456$	0	0
$457$	−14117.6	−1.44506	−0.722531	−	0.691339i	$-0.757021\pi$
$457$	−14117.6	−1.44506	−0.722531	+	0.691339i	$0.757021\pi$
$458$	0	0
$459$	−23113.3	−2.35040
$460$	0	0
$461$	8221.44	0.830609	0.415305	−	0.909682i	$-0.363675\pi$
$461$	8221.44	0.830609	0.415305	+	0.909682i	$0.363675\pi$
$462$	0	0
$463$	9887.11	0.992426	0.496213	−	0.868201i	$-0.334724\pi$
$463$	9887.11	0.992426	0.496213	+	0.868201i	$0.334724\pi$
$464$	0	0
$465$	−6846.94	−0.682837
$466$	0	0
$467$	3685.63	0.365205	0.182603	−	0.983187i	$-0.441548\pi$
$467$	3685.63	0.365205	0.182603	+	0.983187i	$0.441548\pi$
$468$	0	0
$469$	−16214.2	−1.59638
$470$	0	0
$471$	−1249.57	−0.122244
$472$	0	0
$473$	−196.432	−0.0190951
$474$	0	0
$475$	3451.29	0.333381
$476$	0	0
$477$	−403.955	−0.0387753
$478$	0	0
$479$	−522.568	−0.0498470	−0.0249235	−	0.999689i	$-0.507934\pi$
$479$	−522.568	−0.0498470	−0.0249235	+	0.999689i	$0.507934\pi$
$480$	0	0
$481$	−16122.0	−1.52827
$482$	0	0
$483$	−7057.68	−0.664877
$484$	0	0
$485$	−5676.70	−0.531475
$486$	0	0
$487$	10610.0	0.987240	0.493620	−	0.869678i	$-0.335673\pi$
$487$	10610.0	0.987240	0.493620	+	0.869678i	$0.335673\pi$
$488$	0	0
$489$	8596.41	0.794975
$490$	0	0
$491$	10872.5	0.999328	0.499664	−	0.866219i	$-0.333457\pi$
$491$	10872.5	0.999328	0.499664	+	0.866219i	$0.333457\pi$
$492$	0	0
$493$	−16512.9	−1.50853
$494$	0	0
$495$	16126.9	1.46434
$496$	0	0
$497$	−27172.8	−2.45245
$498$	0	0
$499$	9152.41	0.821079	0.410539	−	0.911843i	$-0.365340\pi$
$499$	9152.41	0.821079	0.410539	+	0.911843i	$0.365340\pi$
$500$	0	0
$501$	−17940.3	−1.59983
$502$	0	0
$503$	−9969.46	−0.883730	−0.441865	−	0.897082i	$-0.645683\pi$
$503$	−9969.46	−0.883730	−0.441865	+	0.897082i	$0.645683\pi$
$504$	0	0
$505$	−9873.44	−0.870024
$506$	0	0
$507$	−49454.1	−4.33202
$508$	0	0
$509$	7978.16	0.694746	0.347373	−	0.937727i	$-0.387074\pi$
$509$	7978.16	0.694746	0.347373	+	0.937727i	$0.387074\pi$
$510$	0	0
$511$	379.361	0.0328414
$512$	0	0
$513$	−41009.3	−3.52944
$514$	0	0
$515$	−1317.26	−0.112709
$516$	0	0
$517$	1018.26	0.0866206
$518$	0	0
$519$	15093.6	1.27656
$520$	0	0
$521$	−7429.85	−0.624775	−0.312388	−	0.949955i	$-0.601129\pi$
$521$	−7429.85	−0.624775	−0.312388	+	0.949955i	$0.601129\pi$
$522$	0	0
$523$	9875.34	0.825656	0.412828	−	0.910809i	$-0.364541\pi$
$523$	9875.34	0.825656	0.412828	+	0.910809i	$0.364541\pi$
$524$	0	0
$525$	−7671.39	−0.637728
$526$	0	0
$527$	−11487.6	−0.949537
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	24444.1	1.99771
$532$	0	0
$533$	28016.4	2.27678
$534$	0	0
$535$	2723.62	0.220098
$536$	0	0
$537$	2047.80	0.164560
$538$	0	0
$539$	−41065.5	−3.28167
$540$	0	0
$541$	−11137.6	−0.885107	−0.442554	−	0.896742i	$-0.645927\pi$
$541$	−11137.6	−0.885107	−0.442554	+	0.896742i	$0.645927\pi$
$542$	0	0
$543$	25495.3	2.01493
$544$	0	0
$545$	−9253.08	−0.727263
$546$	0	0
$547$	−4180.81	−0.326798	−0.163399	−	0.986560i	$-0.552246\pi$
$547$	−4180.81	−0.326798	−0.163399	+	0.986560i	$0.552246\pi$
$548$	0	0
$549$	15286.2	1.18834
$550$	0	0
$551$	−29298.5	−2.26526
$552$	0	0
$553$	7108.78	0.546647
$554$	0	0
$555$	−8616.70	−0.659024
$556$	0	0
$557$	25536.3	1.94256	0.971281	−	0.237934i	$-0.0764703\pi$
$557$	25536.3	1.94256	0.971281	+	0.237934i	$0.0764703\pi$
$558$	0	0
$559$	311.927	0.0236013
$560$	0	0
$561$	39433.4	2.96770
$562$	0	0
$563$	12434.8	0.930843	0.465421	−	0.885089i	$-0.345903\pi$
$563$	12434.8	0.930843	0.465421	+	0.885089i	$0.345903\pi$
$564$	0	0
$565$	354.236	0.0263767
$566$	0	0
$567$	38426.9	2.84617
$568$	0	0
$569$	−2063.43	−0.152027	−0.0760136	−	0.997107i	$-0.524219\pi$
$569$	−2063.43	−0.152027	−0.0760136	+	0.997107i	$0.524219\pi$
$570$	0	0
$571$	11092.0	0.812931	0.406466	−	0.913666i	$-0.366761\pi$
$571$	11092.0	0.812931	0.406466	+	0.913666i	$0.366761\pi$
$572$	0	0
$573$	−16320.2	−1.18985
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	16235.7	1.17141	0.585704	−	0.810525i	$-0.300818\pi$
$577$	16235.7	1.17141	0.585704	+	0.810525i	$0.300818\pi$
$578$	0	0
$579$	−36651.8	−2.63074
$580$	0	0
$581$	−26260.2	−1.87514
$582$	0	0
$583$	373.943	0.0265645
$584$	0	0
$585$	−25608.9	−1.80991
$586$	0	0
$587$	17018.6	1.19665	0.598326	−	0.801253i	$-0.295833\pi$
$587$	17018.6	1.19665	0.598326	+	0.801253i	$0.295833\pi$
$588$	0	0
$589$	−20382.1	−1.42586
$590$	0	0
$591$	24634.2	1.71458
$592$	0	0
$593$	24478.2	1.69511	0.847553	−	0.530710i	$-0.178075\pi$
$593$	24478.2	1.69511	0.847553	+	0.530710i	$0.178075\pi$
$594$	0	0
$595$	−12870.8	−0.886809
$596$	0	0
$597$	−39224.9	−2.68906
$598$	0	0
$599$	17530.9	1.19581	0.597906	−	0.801566i	$-0.296000\pi$
$599$	17530.9	1.19581	0.597906	+	0.801566i	$0.296000\pi$
$600$	0	0
$601$	27897.8	1.89347	0.946735	−	0.322014i	$-0.104360\pi$
$601$	27897.8	1.89347	0.946735	+	0.322014i	$0.104360\pi$
$602$	0	0
$603$	−28929.1	−1.95371
$604$	0	0
$605$	−8273.71	−0.555990
$606$	0	0
$607$	17540.7	1.17291	0.586455	−	0.809982i	$-0.300523\pi$
$607$	17540.7	1.17291	0.586455	+	0.809982i	$0.300523\pi$
$608$	0	0
$609$	65123.6	4.33324
$610$	0	0
$611$	−1616.95	−0.107062
$612$	0	0
$613$	−18165.6	−1.19690	−0.598452	−	0.801159i	$-0.704217\pi$
$613$	−18165.6	−1.19690	−0.598452	+	0.801159i	$0.704217\pi$
$614$	0	0
$615$	14973.9	0.981796
$616$	0	0
$617$	−14018.2	−0.914670	−0.457335	−	0.889294i	$-0.651196\pi$
$617$	−14018.2	−0.914670	−0.457335	+	0.889294i	$0.651196\pi$
$618$	0	0
$619$	−9284.22	−0.602851	−0.301425	−	0.953490i	$-0.597462\pi$
$619$	−9284.22	−0.602851	−0.301425	+	0.953490i	$0.597462\pi$
$620$	0	0
$621$	−6832.33	−0.441501
$622$	0	0
$623$	−6378.17	−0.410170
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	69965.8	4.45640
$628$	0	0
$629$	−14456.8	−0.916423
$630$	0	0
$631$	16783.2	1.05884	0.529421	−	0.848359i	$-0.322409\pi$
$631$	16783.2	1.05884	0.529421	+	0.848359i	$0.322409\pi$
$632$	0	0
$633$	823.508	0.0517085
$634$	0	0
$635$	−6998.53	−0.437367
$636$	0	0
$637$	65210.5	4.05610
$638$	0	0
$639$	−48481.1	−3.00138
$640$	0	0
$641$	−15727.9	−0.969133	−0.484566	−	0.874755i	$-0.661023\pi$
$641$	−15727.9	−0.969133	−0.484566	+	0.874755i	$0.661023\pi$
$642$	0	0
$643$	−14597.8	−0.895304	−0.447652	−	0.894208i	$-0.647740\pi$
$643$	−14597.8	−0.895304	−0.447652	+	0.894208i	$0.647740\pi$
$644$	0	0
$645$	166.715	0.0101774
$646$	0	0
$647$	−22525.0	−1.36870	−0.684349	−	0.729155i	$-0.739914\pi$
$647$	−22525.0	−1.36870	−0.684349	+	0.729155i	$0.739914\pi$
$648$	0	0
$649$	−22627.9	−1.36860
$650$	0	0
$651$	45304.6	2.72754
$652$	0	0
$653$	−37.7999	−0.00226527	−0.00113264	−	0.999999i	$-0.500361\pi$
$653$	−37.7999	−0.00226527	−0.00113264	+	0.999999i	$0.500361\pi$
$654$	0	0
$655$	−8381.21	−0.499971
$656$	0	0
$657$	676.849	0.0401924
$658$	0	0
$659$	−28100.6	−1.66107	−0.830535	−	0.556966i	$-0.811965\pi$
$659$	−28100.6	−1.66107	−0.830535	+	0.556966i	$0.811965\pi$
$660$	0	0
$661$	−9234.91	−0.543414	−0.271707	−	0.962380i	$-0.587588\pi$
$661$	−9234.91	−0.543414	−0.271707	+	0.962380i	$0.587588\pi$
$662$	0	0
$663$	−62618.8	−3.66804
$664$	0	0
$665$	−22836.3	−1.33166
$666$	0	0
$667$	−4881.26	−0.283363
$668$	0	0
$669$	42480.2	2.45498
$670$	0	0
$671$	−14150.5	−0.814119
$672$	0	0
$673$	−20976.3	−1.20145	−0.600725	−	0.799456i	$-0.705121\pi$
$673$	−20976.3	−1.20145	−0.600725	+	0.799456i	$0.705121\pi$
$674$	0	0
$675$	−7426.45	−0.423473
$676$	0	0
$677$	21323.0	1.21050	0.605251	−	0.796034i	$-0.293073\pi$
$677$	21323.0	1.21050	0.605251	+	0.796034i	$0.293073\pi$
$678$	0	0
$679$	37561.4	2.12294
$680$	0	0
$681$	2138.66	0.120343
$682$	0	0
$683$	−15058.2	−0.843608	−0.421804	−	0.906687i	$-0.638603\pi$
$683$	−15058.2	−0.843608	−0.421804	+	0.906687i	$0.638603\pi$
$684$	0	0
$685$	−681.280	−0.0380005
$686$	0	0
$687$	−15519.4	−0.861868
$688$	0	0
$689$	−593.807	−0.0328334
$690$	0	0
$691$	−15327.4	−0.843821	−0.421911	−	0.906637i	$-0.638640\pi$
$691$	−15327.4	−0.843821	−0.421911	+	0.906637i	$0.638640\pi$
$692$	0	0
$693$	−106708.	−5.84920
$694$	0	0
$695$	8200.97	0.447598
$696$	0	0
$697$	25122.6	1.36526
$698$	0	0
$699$	−38382.4	−2.07691
$700$	0	0
$701$	−19724.0	−1.06272	−0.531358	−	0.847147i	$-0.678318\pi$
$701$	−19724.0	−1.06272	−0.531358	+	0.847147i	$0.678318\pi$
$702$	0	0
$703$	−25650.3	−1.37613
$704$	0	0
$705$	−864.210	−0.0461674
$706$	0	0
$707$	65330.2	3.47524
$708$	0	0
$709$	6024.59	0.319123	0.159562	−	0.987188i	$-0.448992\pi$
$709$	6024.59	0.319123	0.159562	+	0.987188i	$0.448992\pi$
$710$	0	0
$711$	12683.3	0.669004
$712$	0	0
$713$	−3395.75	−0.178362
$714$	0	0
$715$	23706.2	1.23995
$716$	0	0
$717$	44372.3	2.31118
$718$	0	0
$719$	−7618.85	−0.395181	−0.197591	−	0.980285i	$-0.563312\pi$
$719$	−7618.85	−0.395181	−0.197591	+	0.980285i	$0.563312\pi$
$720$	0	0
$721$	8715.99	0.450209
$722$	0	0
$723$	625.237	0.0321616
$724$	0	0
$725$	−5305.72	−0.271792
$726$	0	0
$727$	4030.71	0.205627	0.102813	−	0.994701i	$-0.467216\pi$
$727$	4030.71	0.205627	0.102813	+	0.994701i	$0.467216\pi$
$728$	0	0
$729$	−5831.08	−0.296250
$730$	0	0
$731$	279.709	0.0141524
$732$	0	0
$733$	−4186.96	−0.210981	−0.105490	−	0.994420i	$-0.533641\pi$
$733$	−4186.96	−0.210981	−0.105490	+	0.994420i	$0.533641\pi$
$734$	0	0
$735$	34853.0	1.74908
$736$	0	0
$737$	26779.8	1.33846
$738$	0	0
$739$	−15299.2	−0.761556	−0.380778	−	0.924666i	$-0.624344\pi$
$739$	−15299.2	−0.761556	−0.380778	+	0.924666i	$0.624344\pi$
$740$	0	0
$741$	−111103.	−5.50806
$742$	0	0
$743$	−32208.7	−1.59034	−0.795169	−	0.606388i	$-0.792618\pi$
$743$	−32208.7	−1.59034	−0.795169	+	0.606388i	$0.792618\pi$
$744$	0	0
$745$	153.053	0.00752674
$746$	0	0
$747$	−46852.9	−2.29486
$748$	0	0
$749$	−18021.6	−0.879163
$750$	0	0
$751$	−27095.1	−1.31653	−0.658265	−	0.752787i	$-0.728709\pi$
$751$	−27095.1	−1.31653	−0.658265	+	0.752787i	$0.728709\pi$
$752$	0	0
$753$	−2010.98	−0.0973232
$754$	0	0
$755$	13009.3	0.627094
$756$	0	0
$757$	−31829.6	−1.52822	−0.764112	−	0.645083i	$-0.776823\pi$
$757$	−31829.6	−1.52822	−0.764112	+	0.645083i	$0.776823\pi$
$758$	0	0
$759$	11656.6	0.557455
$760$	0	0
$761$	7041.33	0.335411	0.167706	−	0.985837i	$-0.446364\pi$
$761$	7041.33	0.335411	0.167706	+	0.985837i	$0.446364\pi$
$762$	0	0
$763$	61225.4	2.90499
$764$	0	0
$765$	−22963.8	−1.08531
$766$	0	0
$767$	35932.3	1.69158
$768$	0	0
$769$	16296.5	0.764198	0.382099	−	0.924121i	$-0.375201\pi$
$769$	16296.5	0.764198	0.382099	+	0.924121i	$0.375201\pi$
$770$	0	0
$771$	−11926.6	−0.557101
$772$	0	0
$773$	25481.8	1.18566	0.592830	−	0.805327i	$-0.298010\pi$
$773$	25481.8	1.18566	0.592830	+	0.805327i	$0.298010\pi$
$774$	0	0
$775$	−3691.03	−0.171079
$776$	0	0
$777$	57014.7	2.63242
$778$	0	0
$779$	44574.5	2.05012
$780$	0	0
$781$	44879.2	2.05621
$782$	0	0
$783$	63044.3	2.87742
$784$	0	0
$785$	−673.615	−0.0306272
$786$	0	0
$787$	22862.8	1.03554	0.517770	−	0.855520i	$-0.326762\pi$
$787$	22862.8	1.03554	0.517770	+	0.855520i	$0.326762\pi$
$788$	0	0
$789$	−11490.0	−0.518447
$790$	0	0
$791$	−2343.89	−0.105359
$792$	0	0
$793$	22470.5	1.00624
$794$	0	0
$795$	−317.371	−0.0141585
$796$	0	0
$797$	−9293.68	−0.413048	−0.206524	−	0.978442i	$-0.566215\pi$
$797$	−9293.68	−0.413048	−0.206524	+	0.978442i	$0.566215\pi$
$798$	0	0
$799$	−1449.94	−0.0641993
$800$	0	0
$801$	−11379.8	−0.501979
$802$	0	0
$803$	−626.561	−0.0275353
$804$	0	0
$805$	−3804.64	−0.166579
$806$	0	0
$807$	−20060.8	−0.875058
$808$	0	0
$809$	−27526.0	−1.19625	−0.598123	−	0.801405i	$-0.704086\pi$
$809$	−27526.0	−1.19625	−0.598123	+	0.801405i	$0.704086\pi$
$810$	0	0
$811$	20552.6	0.889888	0.444944	−	0.895558i	$-0.353224\pi$
$811$	20552.6	0.889888	0.444944	+	0.895558i	$0.353224\pi$
$812$	0	0
$813$	37664.1	1.62477
$814$	0	0
$815$	4634.13	0.199174
$816$	0	0
$817$	496.281	0.0212517
$818$	0	0
$819$	169448.	7.22953
$820$	0	0
$821$	22182.6	0.942971	0.471486	−	0.881874i	$-0.343718\pi$
$821$	22182.6	0.942971	0.471486	+	0.881874i	$0.343718\pi$
$822$	0	0
$823$	24099.2	1.02071	0.510355	−	0.859964i	$-0.329514\pi$
$823$	24099.2	1.02071	0.510355	+	0.859964i	$0.329514\pi$
$824$	0	0
$825$	12670.2	0.534692
$826$	0	0
$827$	−32947.6	−1.38537	−0.692685	−	0.721240i	$-0.743572\pi$
$827$	−32947.6	−1.38537	−0.692685	+	0.721240i	$0.743572\pi$
$828$	0	0
$829$	−47467.0	−1.98866	−0.994329	−	0.106352i	$-0.966083\pi$
$829$	−47467.0	−1.98866	−0.994329	+	0.106352i	$0.966083\pi$
$830$	0	0
$831$	−3467.94	−0.144767
$832$	0	0
$833$	58475.1	2.43223
$834$	0	0
$835$	−9671.22	−0.400822
$836$	0	0
$837$	43858.1	1.81118
$838$	0	0
$839$	35808.6	1.47348	0.736741	−	0.676175i	$-0.236364\pi$
$839$	35808.6	1.47348	0.736741	+	0.676175i	$0.236364\pi$
$840$	0	0
$841$	20652.1	0.846778
$842$	0	0
$843$	35030.5	1.43121
$844$	0	0
$845$	−26659.6	−1.08535
$846$	0	0
$847$	54745.2	2.22086
$848$	0	0
$849$	−28889.2	−1.16782
$850$	0	0
$851$	−4273.46	−0.172142
$852$	0	0
$853$	−29776.0	−1.19521	−0.597604	−	0.801791i	$-0.703880\pi$
$853$	−29776.0	−1.19521	−0.597604	+	0.801791i	$0.703880\pi$
$854$	0	0
$855$	−40744.1	−1.62973
$856$	0	0
$857$	21287.4	0.848500	0.424250	−	0.905545i	$-0.360538\pi$
$857$	21287.4	0.848500	0.424250	+	0.905545i	$0.360538\pi$
$858$	0	0
$859$	36073.0	1.43282	0.716412	−	0.697677i	$-0.245783\pi$
$859$	36073.0	1.43282	0.716412	+	0.697677i	$0.245783\pi$
$860$	0	0
$861$	−99078.5	−3.92170
$862$	0	0
$863$	8228.21	0.324556	0.162278	−	0.986745i	$-0.448116\pi$
$863$	8228.21	0.324556	0.162278	+	0.986745i	$0.448116\pi$
$864$	0	0
$865$	8136.61	0.319830
$866$	0	0
$867$	−10582.5	−0.414535
$868$	0	0
$869$	−11741.0	−0.458327
$870$	0	0
$871$	−42525.3	−1.65432
$872$	0	0
$873$	67016.2	2.59812
$874$	0	0
$875$	−4135.48	−0.159777
$876$	0	0
$877$	−35683.8	−1.37395	−0.686977	−	0.726679i	$-0.741063\pi$
$877$	−35683.8	−1.37395	−0.686977	+	0.726679i	$0.741063\pi$
$878$	0	0
$879$	−3250.02	−0.124710
$880$	0	0
$881$	14849.8	0.567881	0.283940	−	0.958842i	$-0.408358\pi$
$881$	14849.8	0.567881	0.283940	+	0.958842i	$0.408358\pi$
$882$	0	0
$883$	−11821.6	−0.450542	−0.225271	−	0.974296i	$-0.572327\pi$
$883$	−11821.6	−0.450542	−0.225271	+	0.974296i	$0.572327\pi$
$884$	0	0
$885$	19204.7	0.729445
$886$	0	0
$887$	11012.0	0.416851	0.208426	−	0.978038i	$-0.433166\pi$
$887$	11012.0	0.416851	0.208426	+	0.978038i	$0.433166\pi$
$888$	0	0
$889$	46307.6	1.74703
$890$	0	0
$891$	−63466.7	−2.38632
$892$	0	0
$893$	−2572.60	−0.0964039
$894$	0	0
$895$	1103.92	0.0412291
$896$	0	0
$897$	−18510.3	−0.689008
$898$	0	0
$899$	31333.8	1.16245
$900$	0	0
$901$	−532.475	−0.0196885
$902$	0	0
$903$	−1103.12	−0.0406527
$904$	0	0
$905$	13743.9	0.504822
$906$	0	0
$907$	5387.11	0.197217	0.0986086	−	0.995126i	$-0.468561\pi$
$907$	5387.11	0.197217	0.0986086	+	0.995126i	$0.468561\pi$
$908$	0	0
$909$	116561.	4.25311
$910$	0	0
$911$	20928.2	0.761122	0.380561	−	0.924756i	$-0.375731\pi$
$911$	20928.2	0.761122	0.380561	+	0.924756i	$0.375731\pi$
$912$	0	0
$913$	43371.9	1.57218
$914$	0	0
$915$	12009.8	0.433913
$916$	0	0
$917$	55456.5	1.99709
$918$	0	0
$919$	−44252.4	−1.58842	−0.794208	−	0.607647i	$-0.792114\pi$
$919$	−44252.4	−1.58842	−0.794208	+	0.607647i	$0.792114\pi$
$920$	0	0
$921$	49086.3	1.75619
$922$	0	0
$923$	−71266.4	−2.54145
$924$	0	0
$925$	−4645.07	−0.165112
$926$	0	0
$927$	15550.9	0.550980
$928$	0	0
$929$	354.349	0.0125143	0.00625717	−	0.999980i	$-0.498008\pi$
$929$	354.349	0.0125143	0.00625717	+	0.999980i	$0.498008\pi$
$930$	0	0
$931$	103751.	3.65231
$932$	0	0
$933$	−36374.4	−1.27636
$934$	0	0
$935$	21257.7	0.743530
$936$	0	0
$937$	26221.9	0.914228	0.457114	−	0.889408i	$-0.348883\pi$
$937$	26221.9	0.914228	0.457114	+	0.889408i	$0.348883\pi$
$938$	0	0
$939$	68602.4	2.38419
$940$	0	0
$941$	−17076.4	−0.591578	−0.295789	−	0.955253i	$-0.595583\pi$
$941$	−17076.4	−0.591578	−0.295789	+	0.955253i	$0.595583\pi$
$942$	0	0
$943$	7426.31	0.256452
$944$	0	0
$945$	49139.1	1.69153
$946$	0	0
$947$	−43933.6	−1.50755	−0.753775	−	0.657132i	$-0.771769\pi$
$947$	−43933.6	−1.50755	−0.753775	+	0.657132i	$0.771769\pi$
$948$	0	0
$949$	994.955	0.0340333
$950$	0	0
$951$	79230.3	2.70160
$952$	0	0
$953$	−15779.8	−0.536366	−0.268183	−	0.963368i	$-0.586423\pi$
$953$	−15779.8	−0.536366	−0.268183	+	0.963368i	$0.586423\pi$
$954$	0	0
$955$	−8797.85	−0.298106
$956$	0	0
$957$	−107560.	−3.63313
$958$	0	0
$959$	4507.87	0.151790
$960$	0	0
$961$	−7993.02	−0.268303
$962$	0	0
$963$	−32153.7	−1.07595
$964$	0	0
$965$	−19758.2	−0.659107
$966$	0	0
$967$	−22495.0	−0.748076	−0.374038	−	0.927413i	$-0.622027\pi$
$967$	−22495.0	−0.748076	−0.374038	+	0.927413i	$0.622027\pi$
$968$	0	0
$969$	−99627.5	−3.30289
$970$	0	0
$971$	30483.9	1.00749	0.503745	−	0.863852i	$-0.331955\pi$
$971$	30483.9	1.00749	0.503745	+	0.863852i	$0.331955\pi$
$972$	0	0
$973$	−54263.9	−1.78789
$974$	0	0
$975$	−20119.9	−0.660873
$976$	0	0
$977$	−20890.7	−0.684087	−0.342043	−	0.939684i	$-0.611119\pi$
$977$	−20890.7	−0.684087	−0.342043	+	0.939684i	$0.611119\pi$
$978$	0	0
$979$	10534.3	0.343900
$980$	0	0
$981$	109237.	3.55522
$982$	0	0
$983$	11588.6	0.376011	0.188006	−	0.982168i	$-0.439798\pi$
$983$	11588.6	0.376011	0.188006	+	0.982168i	$0.439798\pi$
$984$	0	0
$985$	13279.8	0.429572
$986$	0	0
$987$	5718.27	0.184412
$988$	0	0
$989$	82.6827	0.00265840
$990$	0	0
$991$	46035.4	1.47564	0.737822	−	0.674996i	$-0.235854\pi$
$991$	46035.4	1.47564	0.737822	+	0.674996i	$0.235854\pi$
$992$	0	0
$993$	−76034.4	−2.42989
$994$	0	0
$995$	−21145.3	−0.673719
$996$	0	0
$997$	−44523.8	−1.41433	−0.707163	−	0.707050i	$-0.750025\pi$
$997$	−44523.8	−1.41433	−0.707163	+	0.707050i	$0.750025\pi$
$998$	0	0
$999$	55194.2	1.74802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.4.a.g.1.2	✓	10
4.3	odd	2		1840.4.a.bb.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.g.1.2	✓	10	1.1	even	1	trivial
1840.4.a.bb.1.9		10	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-9.27510 q^{3} -5.00000 q^{5} +33.0838 q^{7} +59.0275 q^{9} -54.6419 q^{11} +86.7693 q^{13} +46.3755 q^{15} +77.8072 q^{17} +138.051 q^{19} -306.856 q^{21} +23.0000 q^{23} +25.0000 q^{25} -297.058 q^{27} -212.229 q^{29} -147.641 q^{31} +506.810 q^{33} -165.419 q^{35} -185.803 q^{37} -804.794 q^{39} +322.883 q^{41} +3.59490 q^{43} -295.137 q^{45} -18.6351 q^{47} +751.539 q^{49} -721.670 q^{51} -6.84351 q^{53} +273.210 q^{55} -1280.44 q^{57} +414.113 q^{59} +258.968 q^{61} +1952.85 q^{63} -433.847 q^{65} -490.096 q^{67} -213.327 q^{69} -821.332 q^{71} +11.4667 q^{73} -231.877 q^{75} -1807.76 q^{77} +214.872 q^{79} +1161.50 q^{81} -793.747 q^{83} -389.036 q^{85} +1968.44 q^{87} -192.788 q^{89} +2870.66 q^{91} +1369.39 q^{93} -690.257 q^{95} +1135.34 q^{97} -3225.38 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{3} - 50 q^{5} + 28 q^{7} + 139 q^{9} - 14 q^{11} + 11 q^{13} + 5 q^{15} + 68 q^{17} + 114 q^{19} - 232 q^{21} + 230 q^{23} + 250 q^{25} - 433 q^{27} - 273 q^{29} - 129 q^{31} + 98 q^{33} - 140 q^{35}+ \cdots + 4356 q^{99}+O(q^{100}) \) 10 * q - q^3 - 50 * q^5 + 28 * q^7 + 139 * q^9 - 14 * q^11 + 11 * q^13 + 5 * q^15 + 68 * q^17 + 114 * q^19 - 232 * q^21 + 230 * q^23 + 250 * q^25 - 433 * q^27 - 273 * q^29 - 129 * q^31 + 98 * q^33 - 140 * q^35 + 62 * q^37 + 283 * q^39 + 767 * q^41 + 332 * q^43 - 695 * q^45 - 323 * q^47 + 1162 * q^49 + 176 * q^51 + 558 * q^53 + 70 * q^55 + 46 * q^57 + 822 * q^59 + 318 * q^61 + 2698 * q^63 - 55 * q^65 + 1152 * q^67 - 23 * q^69 + 1247 * q^71 + 1941 * q^73 - 25 * q^75 + 528 * q^77 + 3134 * q^79 + 6210 * q^81 + 482 * q^83 - 340 * q^85 + 1797 * q^87 + 4734 * q^89 + 4992 * q^91 + 4647 * q^93 - 570 * q^95 + 2326 * q^97 + 4356 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more