LMFDB - Embedded newform 1148.4.a.c.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.a (trivial)

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:	yes
Analytic conductor:	$67.7341926866$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 270 x^{13} + 1158 x^{12} + 28413 x^{11} - 102669 x^{10} - 1445580 x^{9} + \cdots + 1052740152 $ x^15 - 5x^14 - 270x^13 + 1158x^12 + 28413x^11 - 102669x^10 - 1445580x^9 + 4412670x^8 + 35401830x^7 - 96527282x^6 - 348573059x^5 + 982943895x^4 + 672972471x^3 - 2124774315x^2 - 569503386x + 1052740152
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.92676$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.926761 q^{3} -7.25280 q^{5} +7.00000 q^{7} -26.1411 q^{9} +O(q^{10})$	$q-0.926761 q^{3} -7.25280 q^{5} +7.00000 q^{7} -26.1411 q^{9} +38.0756 q^{11} -50.9339 q^{13} +6.72161 q^{15} +138.053 q^{17} -98.3600 q^{19} -6.48733 q^{21} -2.99887 q^{23} -72.3969 q^{25} +49.2491 q^{27} -112.502 q^{29} -64.2742 q^{31} -35.2870 q^{33} -50.7696 q^{35} +160.922 q^{37} +47.2035 q^{39} -41.0000 q^{41} -236.422 q^{43} +189.596 q^{45} -124.665 q^{47} +49.0000 q^{49} -127.942 q^{51} +414.807 q^{53} -276.154 q^{55} +91.1562 q^{57} +557.553 q^{59} +178.867 q^{61} -182.988 q^{63} +369.413 q^{65} -35.9010 q^{67} +2.77924 q^{69} -23.5842 q^{71} -51.4409 q^{73} +67.0947 q^{75} +266.529 q^{77} +420.861 q^{79} +660.168 q^{81} +225.815 q^{83} -1001.27 q^{85} +104.262 q^{87} -81.7173 q^{89} -356.537 q^{91} +59.5669 q^{93} +713.385 q^{95} +42.4321 q^{97} -995.338 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) $ 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9	$ 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9} + 62 q^{11} + 148 q^{13} + 152 q^{15} + 132 q^{17} + 260 q^{19} + 70 q^{21} - 26 q^{23} + 453 q^{25} + 454 q^{27} + 12 q^{29} + 144 q^{31} + 336 q^{33} + 42 q^{35} + 274 q^{37} + 218 q^{39} - 615 q^{41} + 986 q^{43} + 648 q^{45} + 44 q^{47} + 735 q^{49} + 202 q^{51} - 366 q^{53} + 928 q^{55} - 294 q^{57} - 8 q^{59} + 1396 q^{61} + 1155 q^{63} + 156 q^{65} - 24 q^{67} + 892 q^{69} + 1464 q^{71} + 1174 q^{73} + 318 q^{75} + 434 q^{77} + 1590 q^{79} + 1959 q^{81} + 1970 q^{83} + 2348 q^{85} + 2544 q^{87} + 1972 q^{89} + 1036 q^{91} + 4034 q^{93} + 1070 q^{95} + 954 q^{97} + 3398 q^{99}+O(q^{100}) $ 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9 + 62 * q^11 + 148 * q^13 + 152 * q^15 + 132 * q^17 + 260 * q^19 + 70 * q^21 - 26 * q^23 + 453 * q^25 + 454 * q^27 + 12 * q^29 + 144 * q^31 + 336 * q^33 + 42 * q^35 + 274 * q^37 + 218 * q^39 - 615 * q^41 + 986 * q^43 + 648 * q^45 + 44 * q^47 + 735 * q^49 + 202 * q^51 - 366 * q^53 + 928 * q^55 - 294 * q^57 - 8 * q^59 + 1396 * q^61 + 1155 * q^63 + 156 * q^65 - 24 * q^67 + 892 * q^69 + 1464 * q^71 + 1174 * q^73 + 318 * q^75 + 434 * q^77 + 1590 * q^79 + 1959 * q^81 + 1970 * q^83 + 2348 * q^85 + 2544 * q^87 + 1972 * q^89 + 1036 * q^91 + 4034 * q^93 + 1070 * q^95 + 954 * q^97 + 3398 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−0.926761	−0.178355	−0.0891777	−	0.996016i	$-0.528424\pi$
$3$	−0.926761	−0.178355	−0.0891777	+	0.996016i	$0.528424\pi$
$4$	0	0
$5$	−7.25280	−0.648710	−0.324355	−	0.945935i	$-0.605147\pi$
$5$	−7.25280	−0.648710	−0.324355	+	0.945935i	$0.605147\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−26.1411	−0.968189
$10$	0	0
$11$	38.0756	1.04366	0.521828	−	0.853051i	$-0.325250\pi$
$11$	38.0756	1.04366	0.521828	+	0.853051i	$0.325250\pi$
$12$	0	0
$13$	−50.9339	−1.08665	−0.543327	−	0.839521i	$-0.682836\pi$
$13$	−50.9339	−1.08665	−0.543327	+	0.839521i	$0.682836\pi$
$14$	0	0
$15$	6.72161	0.115701
$16$	0	0
$17$	138.053	1.96957	0.984787	−	0.173767i	$-0.0555939\pi$
$17$	138.053	1.96957	0.984787	+	0.173767i	$0.0555939\pi$
$18$	0	0
$19$	−98.3600	−1.18765	−0.593824	−	0.804595i	$-0.702383\pi$
$19$	−98.3600	−1.18765	−0.593824	+	0.804595i	$0.702383\pi$
$20$	0	0
$21$	−6.48733	−0.0674120
$22$	0	0
$23$	−2.99887	−0.0271873	−0.0135936	−	0.999908i	$-0.504327\pi$
$23$	−2.99887	−0.0271873	−0.0135936	+	0.999908i	$0.504327\pi$
$24$	0	0
$25$	−72.3969	−0.579175
$26$	0	0
$27$	49.2491	0.351037
$28$	0	0
$29$	−112.502	−0.720382	−0.360191	−	0.932879i	$-0.617288\pi$
$29$	−112.502	−0.720382	−0.360191	+	0.932879i	$0.617288\pi$
$30$	0	0
$31$	−64.2742	−0.372387	−0.186193	−	0.982513i	$-0.559615\pi$
$31$	−64.2742	−0.372387	−0.186193	+	0.982513i	$0.559615\pi$
$32$	0	0
$33$	−35.2870	−0.186142
$34$	0	0
$35$	−50.7696	−0.245189
$36$	0	0
$37$	160.922	0.715012	0.357506	−	0.933911i	$-0.383627\pi$
$37$	160.922	0.715012	0.357506	+	0.933911i	$0.383627\pi$
$38$	0	0
$39$	47.2035	0.193811
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	−236.422	−0.838465	−0.419233	−	0.907879i	$-0.637701\pi$
$43$	−236.422	−0.838465	−0.419233	+	0.907879i	$0.637701\pi$
$44$	0	0
$45$	189.596	0.628074
$46$	0	0
$47$	−124.665	−0.386900	−0.193450	−	0.981110i	$-0.561968\pi$
$47$	−124.665	−0.386900	−0.193450	+	0.981110i	$0.561968\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−127.942	−0.351284
$52$	0	0
$53$	414.807	1.07506	0.537529	−	0.843245i	$-0.319358\pi$
$53$	414.807	1.07506	0.537529	+	0.843245i	$0.319358\pi$
$54$	0	0
$55$	−276.154	−0.677030
$56$	0	0
$57$	91.1562	0.211823
$58$	0	0
$59$	557.553	1.23029	0.615146	−	0.788413i	$-0.289097\pi$
$59$	557.553	1.23029	0.615146	+	0.788413i	$0.289097\pi$
$60$	0	0
$61$	178.867	0.375436	0.187718	−	0.982223i	$-0.439891\pi$
$61$	178.867	0.375436	0.187718	+	0.982223i	$0.439891\pi$
$62$	0	0
$63$	−182.988	−0.365941
$64$	0	0
$65$	369.413	0.704924
$66$	0	0
$67$	−35.9010	−0.0654628	−0.0327314	−	0.999464i	$-0.510421\pi$
$67$	−35.9010	−0.0654628	−0.0327314	+	0.999464i	$0.510421\pi$
$68$	0	0
$69$	2.77924	0.00484900
$70$	0	0
$71$	−23.5842	−0.0394216	−0.0197108	−	0.999806i	$-0.506275\pi$
$71$	−23.5842	−0.0394216	−0.0197108	+	0.999806i	$0.506275\pi$
$72$	0	0
$73$	−51.4409	−0.0824754	−0.0412377	−	0.999149i	$-0.513130\pi$
$73$	−51.4409	−0.0824754	−0.0412377	+	0.999149i	$0.513130\pi$
$74$	0	0
$75$	67.0947	0.103299
$76$	0	0
$77$	266.529	0.394465
$78$	0	0
$79$	420.861	0.599374	0.299687	−	0.954038i	$-0.403118\pi$
$79$	420.861	0.599374	0.299687	+	0.954038i	$0.403118\pi$
$80$	0	0
$81$	660.168	0.905580
$82$	0	0
$83$	225.815	0.298632	0.149316	−	0.988790i	$-0.452293\pi$
$83$	225.815	0.298632	0.149316	+	0.988790i	$0.452293\pi$
$84$	0	0
$85$	−1001.27	−1.27768
$86$	0	0
$87$	104.262	0.128484
$88$	0	0
$89$	−81.7173	−0.0973260	−0.0486630	−	0.998815i	$-0.515496\pi$
$89$	−81.7173	−0.0973260	−0.0486630	+	0.998815i	$0.515496\pi$
$90$	0	0
$91$	−356.537	−0.410717
$92$	0	0
$93$	59.5669	0.0664172
$94$	0	0
$95$	713.385	0.770440
$96$	0	0
$97$	42.4321	0.0444157	0.0222078	−	0.999753i	$-0.492930\pi$
$97$	42.4321	0.0444157	0.0222078	+	0.999753i	$0.492930\pi$
$98$	0	0
$99$	−995.338	−1.01046
$100$	0	0
$101$	1717.84	1.69239	0.846196	−	0.532871i	$-0.178887\pi$
$101$	1717.84	1.69239	0.846196	+	0.532871i	$0.178887\pi$
$102$	0	0
$103$	−351.834	−0.336575	−0.168288	−	0.985738i	$-0.553824\pi$
$103$	−351.834	−0.336575	−0.168288	+	0.985738i	$0.553824\pi$
$104$	0	0
$105$	47.0513	0.0437308
$106$	0	0
$107$	−739.677	−0.668292	−0.334146	−	0.942521i	$-0.608448\pi$
$107$	−739.677	−0.668292	−0.334146	+	0.942521i	$0.608448\pi$
$108$	0	0
$109$	994.456	0.873868	0.436934	−	0.899494i	$-0.356064\pi$
$109$	994.456	0.873868	0.436934	+	0.899494i	$0.356064\pi$
$110$	0	0
$111$	−149.136	−0.127526
$112$	0	0
$113$	285.527	0.237700	0.118850	−	0.992912i	$-0.462079\pi$
$113$	285.527	0.237700	0.118850	+	0.992912i	$0.462079\pi$
$114$	0	0
$115$	21.7502	0.0176367
$116$	0	0
$117$	1331.47	1.05209
$118$	0	0
$119$	966.370	0.744429
$120$	0	0
$121$	118.748	0.0892173
$122$	0	0
$123$	37.9972	0.0278544
$124$	0	0
$125$	1431.68	1.02443
$126$	0	0
$127$	−1196.87	−0.836258	−0.418129	−	0.908388i	$-0.637314\pi$
$127$	−1196.87	−0.836258	−0.418129	+	0.908388i	$0.637314\pi$
$128$	0	0
$129$	219.107	0.149545
$130$	0	0
$131$	2018.68	1.34636	0.673178	−	0.739481i	$-0.264929\pi$
$131$	2018.68	1.34636	0.673178	+	0.739481i	$0.264929\pi$
$132$	0	0
$133$	−688.520	−0.448889
$134$	0	0
$135$	−357.194	−0.227721
$136$	0	0
$137$	2139.06	1.33396	0.666978	−	0.745078i	$-0.267588\pi$
$137$	2139.06	1.33396	0.666978	+	0.745078i	$0.267588\pi$
$138$	0	0
$139$	1293.41	0.789252	0.394626	−	0.918842i	$-0.370874\pi$
$139$	1293.41	0.789252	0.394626	+	0.918842i	$0.370874\pi$
$140$	0	0
$141$	115.535	0.0690056
$142$	0	0
$143$	−1939.34	−1.13409
$144$	0	0
$145$	815.953	0.467319
$146$	0	0
$147$	−45.4113	−0.0254793
$148$	0	0
$149$	1076.14	0.591686	0.295843	−	0.955237i	$-0.404400\pi$
$149$	1076.14	0.591686	0.295843	+	0.955237i	$0.404400\pi$
$150$	0	0
$151$	1700.94	0.916693	0.458347	−	0.888774i	$-0.348442\pi$
$151$	1700.94	0.916693	0.458347	+	0.888774i	$0.348442\pi$
$152$	0	0
$153$	−3608.86	−1.90692
$154$	0	0
$155$	466.168	0.241571
$156$	0	0
$157$	2400.70	1.22036	0.610181	−	0.792262i	$-0.291097\pi$
$157$	2400.70	1.22036	0.610181	+	0.792262i	$0.291097\pi$
$158$	0	0
$159$	−384.427	−0.191742
$160$	0	0
$161$	−20.9921	−0.0102758
$162$	0	0
$163$	−591.992	−0.284469	−0.142234	−	0.989833i	$-0.545429\pi$
$163$	−591.992	−0.284469	−0.142234	+	0.989833i	$0.545429\pi$
$164$	0	0
$165$	255.929	0.120752
$166$	0	0
$167$	3285.29	1.52230	0.761148	−	0.648579i	$-0.224636\pi$
$167$	3285.29	1.52230	0.761148	+	0.648579i	$0.224636\pi$
$168$	0	0
$169$	397.259	0.180819
$170$	0	0
$171$	2571.24	1.14987
$172$	0	0
$173$	−536.205	−0.235647	−0.117824	−	0.993035i	$-0.537592\pi$
$173$	−536.205	−0.235647	−0.117824	+	0.993035i	$0.537592\pi$
$174$	0	0
$175$	−506.779	−0.218908
$176$	0	0
$177$	−516.719	−0.219429
$178$	0	0
$179$	1786.79	0.746096	0.373048	−	0.927812i	$-0.378313\pi$
$179$	1786.79	0.746096	0.373048	+	0.927812i	$0.378313\pi$
$180$	0	0
$181$	−1179.03	−0.484179	−0.242090	−	0.970254i	$-0.577833\pi$
$181$	−1179.03	−0.484179	−0.242090	+	0.970254i	$0.577833\pi$
$182$	0	0
$183$	−165.767	−0.0669610
$184$	0	0
$185$	−1167.14	−0.463835
$186$	0	0
$187$	5256.44	2.05556
$188$	0	0
$189$	344.744	0.132680
$190$	0	0
$191$	−1056.45	−0.400219	−0.200110	−	0.979773i	$-0.564130\pi$
$191$	−1056.45	−0.400219	−0.200110	+	0.979773i	$0.564130\pi$
$192$	0	0
$193$	−4479.29	−1.67060	−0.835302	−	0.549792i	$-0.814707\pi$
$193$	−4479.29	−1.67060	−0.835302	+	0.549792i	$0.814707\pi$
$194$	0	0
$195$	−342.358	−0.125727
$196$	0	0
$197$	2720.36	0.983847	0.491924	−	0.870638i	$-0.336294\pi$
$197$	2720.36	0.983847	0.491924	+	0.870638i	$0.336294\pi$
$198$	0	0
$199$	535.496	0.190755	0.0953776	−	0.995441i	$-0.469594\pi$
$199$	535.496	0.190755	0.0953776	+	0.995441i	$0.469594\pi$
$200$	0	0
$201$	33.2717	0.0116756
$202$	0	0
$203$	−787.513	−0.272279
$204$	0	0
$205$	297.365	0.101311
$206$	0	0
$207$	78.3938	0.0263224
$208$	0	0
$209$	−3745.11	−1.23950
$210$	0	0
$211$	1217.60	0.397266	0.198633	−	0.980074i	$-0.436350\pi$
$211$	1217.60	0.397266	0.198633	+	0.980074i	$0.436350\pi$
$212$	0	0
$213$	21.8570	0.00703105
$214$	0	0
$215$	1714.72	0.543921
$216$	0	0
$217$	−449.920	−0.140749
$218$	0	0
$219$	47.6735	0.0147099
$220$	0	0
$221$	−7031.57	−2.14025
$222$	0	0
$223$	5509.67	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$223$	5509.67	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$224$	0	0
$225$	1892.54	0.560752
$226$	0	0
$227$	−3569.18	−1.04359	−0.521795	−	0.853071i	$-0.674737\pi$
$227$	−3569.18	−1.04359	−0.521795	+	0.853071i	$0.674737\pi$
$228$	0	0
$229$	6321.42	1.82415	0.912076	−	0.410020i	$-0.134478\pi$
$229$	6321.42	1.82415	0.912076	+	0.410020i	$0.134478\pi$
$230$	0	0
$231$	−247.009	−0.0703549
$232$	0	0
$233$	−1257.94	−0.353694	−0.176847	−	0.984238i	$-0.556590\pi$
$233$	−1257.94	−0.353694	−0.176847	+	0.984238i	$0.556590\pi$
$234$	0	0
$235$	904.171	0.250986
$236$	0	0
$237$	−390.038	−0.106902
$238$	0	0
$239$	5806.34	1.57147	0.785733	−	0.618565i	$-0.212286\pi$
$239$	5806.34	1.57147	0.785733	+	0.618565i	$0.212286\pi$
$240$	0	0
$241$	−4443.21	−1.18760	−0.593801	−	0.804612i	$-0.702374\pi$
$241$	−4443.21	−1.18760	−0.593801	+	0.804612i	$0.702374\pi$
$242$	0	0
$243$	−1941.54	−0.512552
$244$	0	0
$245$	−355.387	−0.0926728
$246$	0	0
$247$	5009.85	1.29056
$248$	0	0
$249$	−209.277	−0.0532626
$250$	0	0
$251$	4309.57	1.08374	0.541868	−	0.840463i	$-0.317717\pi$
$251$	4309.57	1.08374	0.541868	+	0.840463i	$0.317717\pi$
$252$	0	0
$253$	−114.184	−0.0283742
$254$	0	0
$255$	927.938	0.227881
$256$	0	0
$257$	−3233.45	−0.784813	−0.392407	−	0.919792i	$-0.628357\pi$
$257$	−3233.45	−0.784813	−0.392407	+	0.919792i	$0.628357\pi$
$258$	0	0
$259$	1126.46	0.270249
$260$	0	0
$261$	2940.92	0.697466
$262$	0	0
$263$	−3590.51	−0.841827	−0.420914	−	0.907101i	$-0.638290\pi$
$263$	−3590.51	−0.841827	−0.420914	+	0.907101i	$0.638290\pi$
$264$	0	0
$265$	−3008.51	−0.697401
$266$	0	0
$267$	75.7324	0.0173586
$268$	0	0
$269$	−2912.41	−0.660123	−0.330061	−	0.943959i	$-0.607069\pi$
$269$	−2912.41	−0.660123	−0.330061	+	0.943959i	$0.607069\pi$
$270$	0	0
$271$	7819.09	1.75268	0.876339	−	0.481695i	$-0.159979\pi$
$271$	7819.09	1.75268	0.876339	+	0.481695i	$0.159979\pi$
$272$	0	0
$273$	330.425	0.0732535
$274$	0	0
$275$	−2756.55	−0.604460
$276$	0	0
$277$	2409.51	0.522648	0.261324	−	0.965251i	$-0.415841\pi$
$277$	2409.51	0.522648	0.261324	+	0.965251i	$0.415841\pi$
$278$	0	0
$279$	1680.20	0.360541
$280$	0	0
$281$	−3484.36	−0.739714	−0.369857	−	0.929089i	$-0.620593\pi$
$281$	−3484.36	−0.739714	−0.369857	+	0.929089i	$0.620593\pi$
$282$	0	0
$283$	4733.36	0.994238	0.497119	−	0.867682i	$-0.334391\pi$
$283$	4733.36	0.994238	0.497119	+	0.867682i	$0.334391\pi$
$284$	0	0
$285$	−661.138	−0.137412
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	14145.6	2.87922
$290$	0	0
$291$	−39.3244	−0.00792178
$292$	0	0
$293$	−5118.57	−1.02058	−0.510290	−	0.860002i	$-0.670462\pi$
$293$	−5118.57	−1.02058	−0.510290	+	0.860002i	$0.670462\pi$
$294$	0	0
$295$	−4043.82	−0.798103
$296$	0	0
$297$	1875.19	0.366362
$298$	0	0
$299$	152.744	0.0295432
$300$	0	0
$301$	−1654.95	−0.316910
$302$	0	0
$303$	−1592.03	−0.301847
$304$	0	0
$305$	−1297.29	−0.243549
$306$	0	0
$307$	−60.9871	−0.0113378	−0.00566892	−	0.999984i	$-0.501804\pi$
$307$	−60.9871	−0.0113378	−0.00566892	+	0.999984i	$0.501804\pi$
$308$	0	0
$309$	326.066	0.0600300
$310$	0	0
$311$	−2532.13	−0.461685	−0.230843	−	0.972991i	$-0.574148\pi$
$311$	−2532.13	−0.461685	−0.230843	+	0.972991i	$0.574148\pi$
$312$	0	0
$313$	−6793.68	−1.22684	−0.613421	−	0.789756i	$-0.710207\pi$
$313$	−6793.68	−1.22684	−0.613421	+	0.789756i	$0.710207\pi$
$314$	0	0
$315$	1327.17	0.237390
$316$	0	0
$317$	1846.24	0.327114	0.163557	−	0.986534i	$-0.447703\pi$
$317$	1846.24	0.327114	0.163557	+	0.986534i	$0.447703\pi$
$318$	0	0
$319$	−4283.57	−0.751831
$320$	0	0
$321$	685.504	0.119193
$322$	0	0
$323$	−13578.9	−2.33916
$324$	0	0
$325$	3687.46	0.629364
$326$	0	0
$327$	−921.624	−0.155859
$328$	0	0
$329$	−872.656	−0.146234
$330$	0	0
$331$	−3749.99	−0.622713	−0.311357	−	0.950293i	$-0.600783\pi$
$331$	−3749.99	−0.622713	−0.311357	+	0.950293i	$0.600783\pi$
$332$	0	0
$333$	−4206.69	−0.692267
$334$	0	0
$335$	260.383	0.0424664
$336$	0	0
$337$	6943.56	1.12237	0.561187	−	0.827689i	$-0.310345\pi$
$337$	6943.56	1.12237	0.561187	+	0.827689i	$0.310345\pi$
$338$	0	0
$339$	−264.615	−0.0423951
$340$	0	0
$341$	−2447.28	−0.388644
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−20.1572	−0.00314559
$346$	0	0
$347$	2229.36	0.344894	0.172447	−	0.985019i	$-0.444833\pi$
$347$	2229.36	0.344894	0.172447	+	0.985019i	$0.444833\pi$
$348$	0	0
$349$	−3263.12	−0.500489	−0.250245	−	0.968183i	$-0.580511\pi$
$349$	−3263.12	−0.500489	−0.250245	+	0.968183i	$0.580511\pi$
$350$	0	0
$351$	−2508.45	−0.381456
$352$	0	0
$353$	3649.53	0.550269	0.275135	−	0.961406i	$-0.411278\pi$
$353$	3649.53	0.550269	0.275135	+	0.961406i	$0.411278\pi$
$354$	0	0
$355$	171.052	0.0255732
$356$	0	0
$357$	−895.595	−0.132773
$358$	0	0
$359$	3045.91	0.447792	0.223896	−	0.974613i	$-0.428122\pi$
$359$	3045.91	0.447792	0.223896	+	0.974613i	$0.428122\pi$
$360$	0	0
$361$	2815.69	0.410510
$362$	0	0
$363$	−110.051	−0.0159124
$364$	0	0
$365$	373.091	0.0535026
$366$	0	0
$367$	1770.69	0.251851	0.125926	−	0.992040i	$-0.459810\pi$
$367$	1770.69	0.251851	0.125926	+	0.992040i	$0.459810\pi$
$368$	0	0
$369$	1071.79	0.151206
$370$	0	0
$371$	2903.65	0.406334
$372$	0	0
$373$	1283.96	0.178233	0.0891164	−	0.996021i	$-0.471596\pi$
$373$	1283.96	0.178233	0.0891164	+	0.996021i	$0.471596\pi$
$374$	0	0
$375$	−1326.83	−0.182712
$376$	0	0
$377$	5730.16	0.782807
$378$	0	0
$379$	989.889	0.134161	0.0670807	−	0.997748i	$-0.478631\pi$
$379$	989.889	0.134161	0.0670807	+	0.997748i	$0.478631\pi$
$380$	0	0
$381$	1109.21	0.149151
$382$	0	0
$383$	−3646.86	−0.486543	−0.243271	−	0.969958i	$-0.578221\pi$
$383$	−3646.86	−0.486543	−0.243271	+	0.969958i	$0.578221\pi$
$384$	0	0
$385$	−1933.08	−0.255893
$386$	0	0
$387$	6180.33	0.811793
$388$	0	0
$389$	−12481.7	−1.62686	−0.813429	−	0.581665i	$-0.802402\pi$
$389$	−12481.7	−1.62686	−0.813429	+	0.581665i	$0.802402\pi$
$390$	0	0
$391$	−414.003	−0.0535474
$392$	0	0
$393$	−1870.83	−0.240130
$394$	0	0
$395$	−3052.42	−0.388820
$396$	0	0
$397$	10469.6	1.32356	0.661782	−	0.749696i	$-0.269800\pi$
$397$	10469.6	1.32356	0.661782	+	0.749696i	$0.269800\pi$
$398$	0	0
$399$	638.094	0.0800617
$400$	0	0
$401$	7100.74	0.884275	0.442137	−	0.896947i	$-0.354220\pi$
$401$	7100.74	0.884275	0.442137	+	0.896947i	$0.354220\pi$
$402$	0	0
$403$	3273.74	0.404656
$404$	0	0
$405$	−4788.06	−0.587459
$406$	0	0
$407$	6127.20	0.746226
$408$	0	0
$409$	4028.15	0.486991	0.243496	−	0.969902i	$-0.421706\pi$
$409$	4028.15	0.486991	0.243496	+	0.969902i	$0.421706\pi$
$410$	0	0
$411$	−1982.39	−0.237918
$412$	0	0
$413$	3902.87	0.465007
$414$	0	0
$415$	−1637.79	−0.193725
$416$	0	0
$417$	−1198.69	−0.140767
$418$	0	0
$419$	2979.52	0.347396	0.173698	−	0.984799i	$-0.444428\pi$
$419$	2979.52	0.347396	0.173698	+	0.984799i	$0.444428\pi$
$420$	0	0
$421$	−10851.2	−1.25619	−0.628093	−	0.778138i	$-0.716164\pi$
$421$	−10851.2	−1.25619	−0.628093	+	0.778138i	$0.716164\pi$
$422$	0	0
$423$	3258.89	0.374592
$424$	0	0
$425$	−9994.61	−1.14073
$426$	0	0
$427$	1252.07	0.141901
$428$	0	0
$429$	1797.30	0.202272
$430$	0	0
$431$	−7784.72	−0.870016	−0.435008	−	0.900427i	$-0.643254\pi$
$431$	−7784.72	−0.870016	−0.435008	+	0.900427i	$0.643254\pi$
$432$	0	0
$433$	15297.3	1.69779	0.848893	−	0.528565i	$-0.177270\pi$
$433$	15297.3	1.69779	0.848893	+	0.528565i	$0.177270\pi$
$434$	0	0
$435$	−756.194	−0.0833488
$436$	0	0
$437$	294.969	0.0322890
$438$	0	0
$439$	−6829.20	−0.742460	−0.371230	−	0.928541i	$-0.621064\pi$
$439$	−6829.20	−0.742460	−0.371230	+	0.928541i	$0.621064\pi$
$440$	0	0
$441$	−1280.91	−0.138313
$442$	0	0
$443$	4274.77	0.458466	0.229233	−	0.973372i	$-0.426378\pi$
$443$	4274.77	0.458466	0.229233	+	0.973372i	$0.426378\pi$
$444$	0	0
$445$	592.679	0.0631363
$446$	0	0
$447$	−997.329	−0.105530
$448$	0	0
$449$	−8607.16	−0.904670	−0.452335	−	0.891848i	$-0.649409\pi$
$449$	−8607.16	−0.904670	−0.452335	+	0.891848i	$0.649409\pi$
$450$	0	0
$451$	−1561.10	−0.162992
$452$	0	0
$453$	−1576.37	−0.163497
$454$	0	0
$455$	2585.89	0.266436
$456$	0	0
$457$	−11524.5	−1.17963	−0.589816	−	0.807538i	$-0.700800\pi$
$457$	−11524.5	−1.17963	−0.589816	+	0.807538i	$0.700800\pi$
$458$	0	0
$459$	6798.99	0.691393
$460$	0	0
$461$	−1976.92	−0.199727	−0.0998637	−	0.995001i	$-0.531841\pi$
$461$	−1976.92	−0.199727	−0.0998637	+	0.995001i	$0.531841\pi$
$462$	0	0
$463$	−17431.5	−1.74970	−0.874852	−	0.484391i	$-0.839041\pi$
$463$	−17431.5	−1.74970	−0.874852	+	0.484391i	$0.839041\pi$
$464$	0	0
$465$	−432.027	−0.0430855
$466$	0	0
$467$	−6908.55	−0.684560	−0.342280	−	0.939598i	$-0.611199\pi$
$467$	−6908.55	−0.684560	−0.342280	+	0.939598i	$0.611199\pi$
$468$	0	0
$469$	−251.307	−0.0247426
$470$	0	0
$471$	−2224.88	−0.217658
$472$	0	0
$473$	−9001.90	−0.875069
$474$	0	0
$475$	7120.96	0.687857
$476$	0	0
$477$	−10843.5	−1.04086
$478$	0	0
$479$	3277.55	0.312641	0.156321	−	0.987706i	$-0.450037\pi$
$479$	3277.55	0.312641	0.156321	+	0.987706i	$0.450037\pi$
$480$	0	0
$481$	−8196.39	−0.776971
$482$	0	0
$483$	19.4547	0.00183275
$484$	0	0
$485$	−307.751	−0.0288129
$486$	0	0
$487$	−7573.64	−0.704711	−0.352356	−	0.935866i	$-0.614619\pi$
$487$	−7573.64	−0.704711	−0.352356	+	0.935866i	$0.614619\pi$
$488$	0	0
$489$	548.635	0.0507365
$490$	0	0
$491$	2227.09	0.204699	0.102350	−	0.994748i	$-0.467364\pi$
$491$	2227.09	0.204699	0.102350	+	0.994748i	$0.467364\pi$
$492$	0	0
$493$	−15531.2	−1.41885
$494$	0	0
$495$	7218.98	0.655493
$496$	0	0
$497$	−165.090	−0.0149000
$498$	0	0
$499$	−19312.1	−1.73252	−0.866262	−	0.499589i	$-0.833484\pi$
$499$	−19312.1	−1.73252	−0.866262	+	0.499589i	$0.833484\pi$
$500$	0	0
$501$	−3044.68	−0.271509
$502$	0	0
$503$	7377.87	0.654001	0.327001	−	0.945024i	$-0.393962\pi$
$503$	7377.87	0.654001	0.327001	+	0.945024i	$0.393962\pi$
$504$	0	0
$505$	−12459.2	−1.09787
$506$	0	0
$507$	−368.164	−0.0322500
$508$	0	0
$509$	−15498.4	−1.34962	−0.674809	−	0.737992i	$-0.735774\pi$
$509$	−15498.4	−1.34962	−0.674809	+	0.737992i	$0.735774\pi$
$510$	0	0
$511$	−360.087	−0.0311728
$512$	0	0
$513$	−4844.14	−0.416909
$514$	0	0
$515$	2551.78	0.218340
$516$	0	0
$517$	−4746.70	−0.403790
$518$	0	0
$519$	496.934	0.0420289
$520$	0	0
$521$	−19337.4	−1.62608	−0.813040	−	0.582208i	$-0.802189\pi$
$521$	−19337.4	−1.62608	−0.813040	+	0.582208i	$0.802189\pi$
$522$	0	0
$523$	8280.60	0.692324	0.346162	−	0.938175i	$-0.387485\pi$
$523$	8280.60	0.692324	0.346162	+	0.938175i	$0.387485\pi$
$524$	0	0
$525$	469.663	0.0390434
$526$	0	0
$527$	−8873.25	−0.733444
$528$	0	0
$529$	−12158.0	−0.999261
$530$	0	0
$531$	−14575.1	−1.19116
$532$	0	0
$533$	2088.29	0.169707
$534$	0	0
$535$	5364.73	0.433528
$536$	0	0
$537$	−1655.93	−0.133070
$538$	0	0
$539$	1865.70	0.149094
$540$	0	0
$541$	−11597.0	−0.921617	−0.460809	−	0.887500i	$-0.652441\pi$
$541$	−11597.0	−0.921617	−0.460809	+	0.887500i	$0.652441\pi$
$542$	0	0
$543$	1092.68	0.0863559
$544$	0	0
$545$	−7212.59	−0.566887
$546$	0	0
$547$	625.886	0.0489231	0.0244616	−	0.999701i	$-0.492213\pi$
$547$	625.886	0.0489231	0.0244616	+	0.999701i	$0.492213\pi$
$548$	0	0
$549$	−4675.79	−0.363493
$550$	0	0
$551$	11065.7	0.855561
$552$	0	0
$553$	2946.03	0.226542
$554$	0	0
$555$	1081.66	0.0827275
$556$	0	0
$557$	−1840.91	−0.140040	−0.0700198	−	0.997546i	$-0.522306\pi$
$557$	−1840.91	−0.140040	−0.0700198	+	0.997546i	$0.522306\pi$
$558$	0	0
$559$	12041.9	0.911123
$560$	0	0
$561$	−4871.47	−0.366619
$562$	0	0
$563$	−18246.1	−1.36587	−0.682933	−	0.730481i	$-0.739296\pi$
$563$	−18246.1	−1.36587	−0.682933	+	0.730481i	$0.739296\pi$
$564$	0	0
$565$	−2070.87	−0.154198
$566$	0	0
$567$	4621.18	0.342277
$568$	0	0
$569$	−21131.1	−1.55687	−0.778437	−	0.627722i	$-0.783987\pi$
$569$	−21131.1	−1.55687	−0.778437	+	0.627722i	$0.783987\pi$
$570$	0	0
$571$	19803.5	1.45140	0.725702	−	0.688009i	$-0.241515\pi$
$571$	19803.5	1.45140	0.725702	+	0.688009i	$0.241515\pi$
$572$	0	0
$573$	979.075	0.0713813
$574$	0	0
$575$	217.109	0.0157462
$576$	0	0
$577$	419.154	0.0302419	0.0151210	−	0.999886i	$-0.495187\pi$
$577$	419.154	0.0302419	0.0151210	+	0.999886i	$0.495187\pi$
$578$	0	0
$579$	4151.23	0.297961
$580$	0	0
$581$	1580.71	0.112872
$582$	0	0
$583$	15794.0	1.12199
$584$	0	0
$585$	−9656.87	−0.682500
$586$	0	0
$587$	−5218.53	−0.366937	−0.183468	−	0.983026i	$-0.558733\pi$
$587$	−5218.53	−0.366937	−0.183468	+	0.983026i	$0.558733\pi$
$588$	0	0
$589$	6322.01	0.442265
$590$	0	0
$591$	−2521.13	−0.175474
$592$	0	0
$593$	11135.9	0.771157	0.385578	−	0.922675i	$-0.374002\pi$
$593$	11135.9	0.771157	0.385578	+	0.922675i	$0.374002\pi$
$594$	0	0
$595$	−7008.89	−0.482918
$596$	0	0
$597$	−496.277	−0.0340222
$598$	0	0
$599$	27726.0	1.89124	0.945622	−	0.325268i	$-0.105454\pi$
$599$	27726.0	1.89124	0.945622	+	0.325268i	$0.105454\pi$
$600$	0	0
$601$	22008.9	1.49378	0.746890	−	0.664948i	$-0.231546\pi$
$601$	22008.9	1.49378	0.746890	+	0.664948i	$0.231546\pi$
$602$	0	0
$603$	938.493	0.0633804
$604$	0	0
$605$	−861.257	−0.0578761
$606$	0	0
$607$	24262.0	1.62235	0.811173	−	0.584806i	$-0.198829\pi$
$607$	24262.0	1.62235	0.811173	+	0.584806i	$0.198829\pi$
$608$	0	0
$609$	729.837	0.0485624
$610$	0	0
$611$	6349.68	0.420426
$612$	0	0
$613$	327.226	0.0215604	0.0107802	−	0.999942i	$-0.496568\pi$
$613$	327.226	0.0215604	0.0107802	+	0.999942i	$0.496568\pi$
$614$	0	0
$615$	−275.586	−0.0180694
$616$	0	0
$617$	21314.7	1.39076	0.695378	−	0.718644i	$-0.255237\pi$
$617$	21314.7	1.39076	0.695378	+	0.718644i	$0.255237\pi$
$618$	0	0
$619$	9233.54	0.599560	0.299780	−	0.954008i	$-0.403087\pi$
$619$	9233.54	0.599560	0.299780	+	0.954008i	$0.403087\pi$
$620$	0	0
$621$	−147.692	−0.00954375
$622$	0	0
$623$	−572.021	−0.0367858
$624$	0	0
$625$	−1334.07	−0.0853804
$626$	0	0
$627$	3470.82	0.221071
$628$	0	0
$629$	22215.8	1.40827
$630$	0	0
$631$	−29890.7	−1.88578	−0.942892	−	0.333099i	$-0.891906\pi$
$631$	−29890.7	−1.88578	−0.942892	+	0.333099i	$0.891906\pi$
$632$	0	0
$633$	−1128.43	−0.0708546
$634$	0	0
$635$	8680.64	0.542489
$636$	0	0
$637$	−2495.76	−0.155236
$638$	0	0
$639$	616.518	0.0381676
$640$	0	0
$641$	6751.25	0.416004	0.208002	−	0.978128i	$-0.433304\pi$
$641$	6751.25	0.416004	0.208002	+	0.978128i	$0.433304\pi$
$642$	0	0
$643$	6861.83	0.420846	0.210423	−	0.977610i	$-0.432516\pi$
$643$	6861.83	0.420846	0.210423	+	0.977610i	$0.432516\pi$
$644$	0	0
$645$	−1589.14	−0.0970112
$646$	0	0
$647$	−1604.65	−0.0975044	−0.0487522	−	0.998811i	$-0.515524\pi$
$647$	−1604.65	−0.0975044	−0.0487522	+	0.998811i	$0.515524\pi$
$648$	0	0
$649$	21229.1	1.28400
$650$	0	0
$651$	416.968	0.0251033
$652$	0	0
$653$	−20278.0	−1.21522	−0.607610	−	0.794236i	$-0.707872\pi$
$653$	−20278.0	−1.21522	−0.607610	+	0.794236i	$0.707872\pi$
$654$	0	0
$655$	−14641.1	−0.873394
$656$	0	0
$657$	1344.72	0.0798518
$658$	0	0
$659$	3044.53	0.179967	0.0899835	−	0.995943i	$-0.471319\pi$
$659$	3044.53	0.179967	0.0899835	+	0.995943i	$0.471319\pi$
$660$	0	0
$661$	−9506.93	−0.559420	−0.279710	−	0.960085i	$-0.590238\pi$
$661$	−9506.93	−0.559420	−0.279710	+	0.960085i	$0.590238\pi$
$662$	0	0
$663$	6516.59	0.381724
$664$	0	0
$665$	4993.70	0.291199
$666$	0	0
$667$	337.379	0.0195852
$668$	0	0
$669$	−5106.15	−0.295090
$670$	0	0
$671$	6810.47	0.391826
$672$	0	0
$673$	−54.0937	−0.00309830	−0.00154915	−	0.999999i	$-0.500493\pi$
$673$	−54.0937	−0.00309830	−0.00154915	+	0.999999i	$0.500493\pi$
$674$	0	0
$675$	−3565.49	−0.203312
$676$	0	0
$677$	31449.2	1.78536	0.892681	−	0.450688i	$-0.148821\pi$
$677$	31449.2	1.78536	0.892681	+	0.450688i	$0.148821\pi$
$678$	0	0
$679$	297.024	0.0167876
$680$	0	0
$681$	3307.78	0.186130
$682$	0	0
$683$	21763.7	1.21928	0.609638	−	0.792680i	$-0.291315\pi$
$683$	21763.7	1.21928	0.609638	+	0.792680i	$0.291315\pi$
$684$	0	0
$685$	−15514.1	−0.865350
$686$	0	0
$687$	−5858.44	−0.325347
$688$	0	0
$689$	−21127.7	−1.16822
$690$	0	0
$691$	−8549.18	−0.470660	−0.235330	−	0.971916i	$-0.575617\pi$
$691$	−8549.18	−0.470660	−0.235330	+	0.971916i	$0.575617\pi$
$692$	0	0
$693$	−6967.36	−0.381917
$694$	0	0
$695$	−9380.88	−0.511996
$696$	0	0
$697$	−5660.17	−0.307596
$698$	0	0
$699$	1165.81	0.0630831
$700$	0	0
$701$	6782.97	0.365462	0.182731	−	0.983163i	$-0.441506\pi$
$701$	6782.97	0.365462	0.182731	+	0.983163i	$0.441506\pi$
$702$	0	0
$703$	−15828.3	−0.849183
$704$	0	0
$705$	−837.951	−0.0447646
$706$	0	0
$707$	12024.9	0.639664
$708$	0	0
$709$	−3999.34	−0.211845	−0.105923	−	0.994374i	$-0.533780\pi$
$709$	−3999.34	−0.211845	−0.105923	+	0.994374i	$0.533780\pi$
$710$	0	0
$711$	−11001.8	−0.580308
$712$	0	0
$713$	192.750	0.0101242
$714$	0	0
$715$	14065.6	0.735698
$716$	0	0
$717$	−5381.09	−0.280279
$718$	0	0
$719$	−14538.9	−0.754115	−0.377058	−	0.926190i	$-0.623064\pi$
$719$	−14538.9	−0.754115	−0.377058	+	0.926190i	$0.623064\pi$
$720$	0	0
$721$	−2462.84	−0.127214
$722$	0	0
$723$	4117.80	0.211815
$724$	0	0
$725$	8144.79	0.417228
$726$	0	0
$727$	−22445.8	−1.14507	−0.572536	−	0.819880i	$-0.694040\pi$
$727$	−22445.8	−1.14507	−0.572536	+	0.819880i	$0.694040\pi$
$728$	0	0
$729$	−16025.2	−0.814164
$730$	0	0
$731$	−32638.7	−1.65142
$732$	0	0
$733$	5504.36	0.277365	0.138682	−	0.990337i	$-0.455713\pi$
$733$	5504.36	0.277365	0.138682	+	0.990337i	$0.455713\pi$
$734$	0	0
$735$	329.359	0.0165287
$736$	0	0
$737$	−1366.95	−0.0683206
$738$	0	0
$739$	−511.159	−0.0254442	−0.0127221	−	0.999919i	$-0.504050\pi$
$739$	−511.159	−0.0254442	−0.0127221	+	0.999919i	$0.504050\pi$
$740$	0	0
$741$	−4642.94	−0.230179
$742$	0	0
$743$	−9448.92	−0.466551	−0.233275	−	0.972411i	$-0.574944\pi$
$743$	−9448.92	−0.466551	−0.233275	+	0.972411i	$0.574944\pi$
$744$	0	0
$745$	−7805.06	−0.383832
$746$	0	0
$747$	−5903.06	−0.289132
$748$	0	0
$749$	−5177.74	−0.252591
$750$	0	0
$751$	12528.9	0.608771	0.304385	−	0.952549i	$-0.401549\pi$
$751$	12528.9	0.608771	0.304385	+	0.952549i	$0.401549\pi$
$752$	0	0
$753$	−3993.95	−0.193290
$754$	0	0
$755$	−12336.6	−0.594668
$756$	0	0
$757$	−1212.93	−0.0582360	−0.0291180	−	0.999576i	$-0.509270\pi$
$757$	−1212.93	−0.0582360	−0.0291180	+	0.999576i	$0.509270\pi$
$758$	0	0
$759$	105.821	0.00506068
$760$	0	0
$761$	−11174.2	−0.532281	−0.266140	−	0.963934i	$-0.585749\pi$
$761$	−11174.2	−0.532281	−0.266140	+	0.963934i	$0.585749\pi$
$762$	0	0
$763$	6961.19	0.330291
$764$	0	0
$765$	26174.3	1.23704
$766$	0	0
$767$	−28398.3	−1.33690
$768$	0	0
$769$	−3525.40	−0.165317	−0.0826587	−	0.996578i	$-0.526341\pi$
$769$	−3525.40	−0.165317	−0.0826587	+	0.996578i	$0.526341\pi$
$770$	0	0
$771$	2996.64	0.139976
$772$	0	0
$773$	8719.68	0.405725	0.202862	−	0.979207i	$-0.434976\pi$
$773$	8719.68	0.405725	0.202862	+	0.979207i	$0.434976\pi$
$774$	0	0
$775$	4653.26	0.215677
$776$	0	0
$777$	−1043.96	−0.0482004
$778$	0	0
$779$	4032.76	0.185480
$780$	0	0
$781$	−897.983	−0.0411426
$782$	0	0
$783$	−5540.62	−0.252881
$784$	0	0
$785$	−17411.8	−0.791661
$786$	0	0
$787$	5883.67	0.266493	0.133247	−	0.991083i	$-0.457460\pi$
$787$	5883.67	0.266493	0.133247	+	0.991083i	$0.457460\pi$
$788$	0	0
$789$	3327.55	0.150144
$790$	0	0
$791$	1998.69	0.0898422
$792$	0	0
$793$	−9110.40	−0.407969
$794$	0	0
$795$	2788.17	0.124385
$796$	0	0
$797$	16433.2	0.730356	0.365178	−	0.930938i	$-0.381008\pi$
$797$	16433.2	0.730356	0.365178	+	0.930938i	$0.381008\pi$
$798$	0	0
$799$	−17210.4	−0.762028
$800$	0	0
$801$	2136.18	0.0942300
$802$	0	0
$803$	−1958.64	−0.0860760
$804$	0	0
$805$	152.251	0.00666603
$806$	0	0
$807$	2699.11	0.117736
$808$	0	0
$809$	−19422.9	−0.844094	−0.422047	−	0.906574i	$-0.638688\pi$
$809$	−19422.9	−0.844094	−0.422047	+	0.906574i	$0.638688\pi$
$810$	0	0
$811$	17774.3	0.769595	0.384798	−	0.923001i	$-0.374271\pi$
$811$	17774.3	0.769595	0.384798	+	0.923001i	$0.374271\pi$
$812$	0	0
$813$	−7246.43	−0.312599
$814$	0	0
$815$	4293.60	0.184538
$816$	0	0
$817$	23254.5	0.995803
$818$	0	0
$819$	9320.28	0.397652
$820$	0	0
$821$	26424.4	1.12328	0.561642	−	0.827380i	$-0.310170\pi$
$821$	26424.4	1.12328	0.561642	+	0.827380i	$0.310170\pi$
$822$	0	0
$823$	39178.5	1.65939	0.829694	−	0.558218i	$-0.188515\pi$
$823$	39178.5	1.65939	0.829694	+	0.558218i	$0.188515\pi$
$824$	0	0
$825$	2554.67	0.107809
$826$	0	0
$827$	−17485.1	−0.735206	−0.367603	−	0.929983i	$-0.619821\pi$
$827$	−17485.1	−0.735206	−0.367603	+	0.929983i	$0.619821\pi$
$828$	0	0
$829$	−3296.73	−0.138119	−0.0690593	−	0.997613i	$-0.522000\pi$
$829$	−3296.73	−0.138119	−0.0690593	+	0.997613i	$0.522000\pi$
$830$	0	0
$831$	−2233.04	−0.0932171
$832$	0	0
$833$	6764.59	0.281368
$834$	0	0
$835$	−23827.5	−0.987528
$836$	0	0
$837$	−3165.45	−0.130722
$838$	0	0
$839$	38868.3	1.59938	0.799692	−	0.600411i	$-0.204996\pi$
$839$	38868.3	1.59938	0.799692	+	0.600411i	$0.204996\pi$
$840$	0	0
$841$	−11732.3	−0.481050
$842$	0	0
$843$	3229.17	0.131932
$844$	0	0
$845$	−2881.24	−0.117299
$846$	0	0
$847$	831.237	0.0337210
$848$	0	0
$849$	−4386.70	−0.177328
$850$	0	0
$851$	−482.585	−0.0194392
$852$	0	0
$853$	19221.9	0.771565	0.385783	−	0.922590i	$-0.373931\pi$
$853$	19221.9	0.771565	0.385783	+	0.922590i	$0.373931\pi$
$854$	0	0
$855$	−18648.7	−0.745931
$856$	0	0
$857$	−45199.1	−1.80160	−0.900800	−	0.434234i	$-0.857019\pi$
$857$	−45199.1	−1.80160	−0.900800	+	0.434234i	$0.857019\pi$
$858$	0	0
$859$	39550.3	1.57094	0.785471	−	0.618898i	$-0.212421\pi$
$859$	39550.3	1.57094	0.785471	+	0.618898i	$0.212421\pi$
$860$	0	0
$861$	265.981	0.0105280
$862$	0	0
$863$	−10106.6	−0.398649	−0.199324	−	0.979934i	$-0.563875\pi$
$863$	−10106.6	−0.398649	−0.199324	+	0.979934i	$0.563875\pi$
$864$	0	0
$865$	3888.99	0.152867
$866$	0	0
$867$	−13109.6	−0.513524
$868$	0	0
$869$	16024.5	0.625540
$870$	0	0
$871$	1828.58	0.0711355
$872$	0	0
$873$	−1109.22	−0.0430028
$874$	0	0
$875$	10021.8	0.387197
$876$	0	0
$877$	12319.4	0.474340	0.237170	−	0.971468i	$-0.423780\pi$
$877$	12319.4	0.474340	0.237170	+	0.971468i	$0.423780\pi$
$878$	0	0
$879$	4743.69	0.182026
$880$	0	0
$881$	13680.4	0.523159	0.261579	−	0.965182i	$-0.415757\pi$
$881$	13680.4	0.523159	0.261579	+	0.965182i	$0.415757\pi$
$882$	0	0
$883$	13692.3	0.521839	0.260920	−	0.965361i	$-0.415974\pi$
$883$	13692.3	0.521839	0.260920	+	0.965361i	$0.415974\pi$
$884$	0	0
$885$	3747.66	0.142346
$886$	0	0
$887$	6996.24	0.264837	0.132419	−	0.991194i	$-0.457726\pi$
$887$	6996.24	0.264837	0.132419	+	0.991194i	$0.457726\pi$
$888$	0	0
$889$	−8378.07	−0.316076
$890$	0	0
$891$	25136.3	0.945114
$892$	0	0
$893$	12262.1	0.459501
$894$	0	0
$895$	−12959.2	−0.484000
$896$	0	0
$897$	−141.557	−0.00526919
$898$	0	0
$899$	7230.97	0.268261
$900$	0	0
$901$	57265.3	2.11741
$902$	0	0
$903$	1533.75	0.0565226
$904$	0	0
$905$	8551.25	0.314092
$906$	0	0
$907$	30350.0	1.11109	0.555544	−	0.831487i	$-0.312510\pi$
$907$	30350.0	1.11109	0.555544	+	0.831487i	$0.312510\pi$
$908$	0	0
$909$	−44906.3	−1.63856
$910$	0	0
$911$	−34820.9	−1.26638	−0.633188	−	0.773998i	$-0.718254\pi$
$911$	−34820.9	−1.26638	−0.633188	+	0.773998i	$0.718254\pi$
$912$	0	0
$913$	8598.04	0.311669
$914$	0	0
$915$	1202.28	0.0434383
$916$	0	0
$917$	14130.7	0.508875
$918$	0	0
$919$	45487.9	1.63276	0.816380	−	0.577515i	$-0.195977\pi$
$919$	45487.9	1.63276	0.816380	+	0.577515i	$0.195977\pi$
$920$	0	0
$921$	56.5204	0.00202216
$922$	0	0
$923$	1201.24	0.0428377
$924$	0	0
$925$	−11650.3	−0.414117
$926$	0	0
$927$	9197.34	0.325869
$928$	0	0
$929$	−13418.7	−0.473899	−0.236950	−	0.971522i	$-0.576148\pi$
$929$	−13418.7	−0.473899	−0.236950	+	0.971522i	$0.576148\pi$
$930$	0	0
$931$	−4819.64	−0.169664
$932$	0	0
$933$	2346.68	0.0823440
$934$	0	0
$935$	−38123.9	−1.33346
$936$	0	0
$937$	−5360.01	−0.186877	−0.0934386	−	0.995625i	$-0.529786\pi$
$937$	−5360.01	−0.186877	−0.0934386	+	0.995625i	$0.529786\pi$
$938$	0	0
$939$	6296.12	0.218814
$940$	0	0
$941$	23732.7	0.822171	0.411086	−	0.911597i	$-0.365150\pi$
$941$	23732.7	0.822171	0.411086	+	0.911597i	$0.365150\pi$
$942$	0	0
$943$	122.954	0.00424594
$944$	0	0
$945$	−2500.36	−0.0860705
$946$	0	0
$947$	52536.0	1.80274	0.901369	−	0.433053i	$-0.142564\pi$
$947$	52536.0	1.80274	0.901369	+	0.433053i	$0.142564\pi$
$948$	0	0
$949$	2620.09	0.0896223
$950$	0	0
$951$	−1711.02	−0.0583424
$952$	0	0
$953$	−47763.6	−1.62352	−0.811761	−	0.583990i	$-0.801491\pi$
$953$	−47763.6	−1.62352	−0.811761	+	0.583990i	$0.801491\pi$
$954$	0	0
$955$	7662.20	0.259626
$956$	0	0
$957$	3969.85	0.134093
$958$	0	0
$959$	14973.4	0.504188
$960$	0	0
$961$	−25659.8	−0.861328
$962$	0	0
$963$	19336.0	0.647033
$964$	0	0
$965$	32487.4	1.08374
$966$	0	0
$967$	−10379.9	−0.345185	−0.172592	−	0.984993i	$-0.555214\pi$
$967$	−10379.9	−0.345185	−0.172592	+	0.984993i	$0.555214\pi$
$968$	0	0
$969$	12584.4	0.417202
$970$	0	0
$971$	−20850.4	−0.689104	−0.344552	−	0.938767i	$-0.611969\pi$
$971$	−20850.4	−0.689104	−0.344552	+	0.938767i	$0.611969\pi$
$972$	0	0
$973$	9053.90	0.298309
$974$	0	0
$975$	−3417.39	−0.112250
$976$	0	0
$977$	19413.8	0.635723	0.317862	−	0.948137i	$-0.397035\pi$
$977$	19413.8	0.635723	0.317862	+	0.948137i	$0.397035\pi$
$978$	0	0
$979$	−3111.43	−0.101575
$980$	0	0
$981$	−25996.2	−0.846070
$982$	0	0
$983$	38878.7	1.26148	0.630741	−	0.775994i	$-0.282751\pi$
$983$	38878.7	1.26148	0.630741	+	0.775994i	$0.282751\pi$
$984$	0	0
$985$	−19730.2	−0.638231
$986$	0	0
$987$	808.744	0.0260817
$988$	0	0
$989$	708.999	0.0227956
$990$	0	0
$991$	−22136.6	−0.709580	−0.354790	−	0.934946i	$-0.615448\pi$
$991$	−22136.6	−0.709580	−0.354790	+	0.934946i	$0.615448\pi$
$992$	0	0
$993$	3475.35	0.111064
$994$	0	0
$995$	−3883.84	−0.123745
$996$	0	0
$997$	−49224.5	−1.56365	−0.781823	−	0.623500i	$-0.785710\pi$
$997$	−49224.5	−1.56365	−0.781823	+	0.623500i	$0.785710\pi$
$998$	0	0
$999$	7925.28	0.250996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.c.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.c.1.7	✓	15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-0.926761 q^{3} -7.25280 q^{5} +7.00000 q^{7} -26.1411 q^{9} +O(q^{10})\)	\(q-0.926761 q^{3} -7.25280 q^{5} +7.00000 q^{7} -26.1411 q^{9} +38.0756 q^{11} -50.9339 q^{13} +6.72161 q^{15} +138.053 q^{17} -98.3600 q^{19} -6.48733 q^{21} -2.99887 q^{23} -72.3969 q^{25} +49.2491 q^{27} -112.502 q^{29} -64.2742 q^{31} -35.2870 q^{33} -50.7696 q^{35} +160.922 q^{37} +47.2035 q^{39} -41.0000 q^{41} -236.422 q^{43} +189.596 q^{45} -124.665 q^{47} +49.0000 q^{49} -127.942 q^{51} +414.807 q^{53} -276.154 q^{55} +91.1562 q^{57} +557.553 q^{59} +178.867 q^{61} -182.988 q^{63} +369.413 q^{65} -35.9010 q^{67} +2.77924 q^{69} -23.5842 q^{71} -51.4409 q^{73} +67.0947 q^{75} +266.529 q^{77} +420.861 q^{79} +660.168 q^{81} +225.815 q^{83} -1001.27 q^{85} +104.262 q^{87} -81.7173 q^{89} -356.537 q^{91} +59.5669 q^{93} +713.385 q^{95} +42.4321 q^{97} -995.338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) \) 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9	\( 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9} + 62 q^{11} + 148 q^{13} + 152 q^{15} + 132 q^{17} + 260 q^{19} + 70 q^{21} - 26 q^{23} + 453 q^{25} + 454 q^{27} + 12 q^{29} + 144 q^{31} + 336 q^{33} + 42 q^{35} + 274 q^{37} + 218 q^{39} - 615 q^{41} + 986 q^{43} + 648 q^{45} + 44 q^{47} + 735 q^{49} + 202 q^{51} - 366 q^{53} + 928 q^{55} - 294 q^{57} - 8 q^{59} + 1396 q^{61} + 1155 q^{63} + 156 q^{65} - 24 q^{67} + 892 q^{69} + 1464 q^{71} + 1174 q^{73} + 318 q^{75} + 434 q^{77} + 1590 q^{79} + 1959 q^{81} + 1970 q^{83} + 2348 q^{85} + 2544 q^{87} + 1972 q^{89} + 1036 q^{91} + 4034 q^{93} + 1070 q^{95} + 954 q^{97} + 3398 q^{99}+O(q^{100}) \) 15 * q + 10 * q^3 + 6 * q^5 + 105 * q^7 + 165 * q^9 + 62 * q^11 + 148 * q^13 + 152 * q^15 + 132 * q^17 + 260 * q^19 + 70 * q^21 - 26 * q^23 + 453 * q^25 + 454 * q^27 + 12 * q^29 + 144 * q^31 + 336 * q^33 + 42 * q^35 + 274 * q^37 + 218 * q^39 - 615 * q^41 + 986 * q^43 + 648 * q^45 + 44 * q^47 + 735 * q^49 + 202 * q^51 - 366 * q^53 + 928 * q^55 - 294 * q^57 - 8 * q^59 + 1396 * q^61 + 1155 * q^63 + 156 * q^65 - 24 * q^67 + 892 * q^69 + 1464 * q^71 + 1174 * q^73 + 318 * q^75 + 434 * q^77 + 1590 * q^79 + 1959 * q^81 + 1970 * q^83 + 2348 * q^85 + 2544 * q^87 + 1972 * q^89 + 1036 * q^91 + 4034 * q^93 + 1070 * q^95 + 954 * q^97 + 3398 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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