LMFDB - Embedded newform 1148.4.a.b.1.15

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:	$1$
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} - 247 x^{13} - 6 x^{12} + 23870 x^{11} + 940 x^{10} - 1147074 x^{9} - 8966 x^{8} + \cdots + 1720288256 $ x^15 - 247x^13 - 6x^12 + 23870x^11 + 940x^10 - 1147074x^9 - 8966x^8 + 28798759x^7 - 3139244x^6 - 363586916x^5 + 114207044x^4 + 2060613491x^3 - 892545712x^2 - 4098298668*x + 1720288256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$8.45579$ 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)

$f(q)$	$=$	$q+8.45579 q^{3} +6.41990 q^{5} -7.00000 q^{7} +44.5003 q^{9} +O(q^{10})$	$q+8.45579 q^{3} +6.41990 q^{5} -7.00000 q^{7} +44.5003 q^{9} -62.1224 q^{11} -68.2648 q^{13} +54.2853 q^{15} +0.454400 q^{17} -77.2213 q^{19} -59.1905 q^{21} +134.195 q^{23} -83.7849 q^{25} +147.979 q^{27} -232.809 q^{29} +72.9818 q^{31} -525.294 q^{33} -44.9393 q^{35} +153.552 q^{37} -577.233 q^{39} -41.0000 q^{41} -124.774 q^{43} +285.687 q^{45} -584.803 q^{47} +49.0000 q^{49} +3.84231 q^{51} -335.147 q^{53} -398.820 q^{55} -652.966 q^{57} +364.232 q^{59} +485.627 q^{61} -311.502 q^{63} -438.253 q^{65} -784.960 q^{67} +1134.72 q^{69} +1118.81 q^{71} -259.120 q^{73} -708.467 q^{75} +434.857 q^{77} -632.126 q^{79} +49.7687 q^{81} +295.878 q^{83} +2.91720 q^{85} -1968.58 q^{87} +848.357 q^{89} +477.854 q^{91} +617.119 q^{93} -495.753 q^{95} +1281.58 q^{97} -2764.47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * 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
$3$	8.45579	1.62732	0.813658	−	0.581343i	$-0.197473\pi$
$3$	8.45579	1.62732	0.813658	+	0.581343i	$0.197473\pi$
$4$	0	0
$5$	6.41990	0.574213	0.287107	−	0.957899i	$-0.407307\pi$
$5$	6.41990	0.574213	0.287107	+	0.957899i	$0.407307\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	44.5003	1.64816
$10$	0	0
$11$	−62.1224	−1.70278	−0.851391	−	0.524531i	$-0.824241\pi$
$11$	−62.1224	−1.70278	−0.851391	+	0.524531i	$0.824241\pi$
$12$	0	0
$13$	−68.2648	−1.45640	−0.728202	−	0.685363i	$-0.759644\pi$
$13$	−68.2648	−1.45640	−0.728202	+	0.685363i	$0.759644\pi$
$14$	0	0
$15$	54.2853	0.934427
$16$	0	0
$17$	0.454400	0.00648284	0.00324142	−	0.999995i	$-0.498968\pi$
$17$	0.454400	0.00648284	0.00324142	+	0.999995i	$0.498968\pi$
$18$	0	0
$19$	−77.2213	−0.932409	−0.466204	−	0.884677i	$-0.654379\pi$
$19$	−77.2213	−0.932409	−0.466204	+	0.884677i	$0.654379\pi$
$20$	0	0
$21$	−59.1905	−0.615068
$22$	0	0
$23$	134.195	1.21659	0.608294	−	0.793711i	$-0.291854\pi$
$23$	134.195	1.21659	0.608294	+	0.793711i	$0.291854\pi$
$24$	0	0
$25$	−83.7849	−0.670279
$26$	0	0
$27$	147.979	1.05476
$28$	0	0
$29$	−232.809	−1.49074	−0.745370	−	0.666651i	$-0.767727\pi$
$29$	−232.809	−1.49074	−0.745370	+	0.666651i	$0.767727\pi$
$30$	0	0
$31$	72.9818	0.422836	0.211418	−	0.977396i	$-0.432192\pi$
$31$	72.9818	0.422836	0.211418	+	0.977396i	$0.432192\pi$
$32$	0	0
$33$	−525.294	−2.77097
$34$	0	0
$35$	−44.9393	−0.217032
$36$	0	0
$37$	153.552	0.682266	0.341133	−	0.940015i	$-0.389189\pi$
$37$	153.552	0.682266	0.341133	+	0.940015i	$0.389189\pi$
$38$	0	0
$39$	−577.233	−2.37003
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	−124.774	−0.442509	−0.221254	−	0.975216i	$-0.571015\pi$
$43$	−124.774	−0.442509	−0.221254	+	0.975216i	$0.571015\pi$
$44$	0	0
$45$	285.687	0.946395
$46$	0	0
$47$	−584.803	−1.81494	−0.907471	−	0.420114i	$-0.861990\pi$
$47$	−584.803	−1.81494	−0.907471	+	0.420114i	$0.861990\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	3.84231	0.0105496
$52$	0	0
$53$	−335.147	−0.868603	−0.434301	−	0.900768i	$-0.643005\pi$
$53$	−335.147	−0.868603	−0.434301	+	0.900768i	$0.643005\pi$
$54$	0	0
$55$	−398.820	−0.977760
$56$	0	0
$57$	−652.966	−1.51732
$58$	0	0
$59$	364.232	0.803710	0.401855	−	0.915703i	$-0.368366\pi$
$59$	364.232	0.803710	0.401855	+	0.915703i	$0.368366\pi$
$60$	0	0
$61$	485.627	1.01931	0.509657	−	0.860378i	$-0.329772\pi$
$61$	485.627	1.01931	0.509657	+	0.860378i	$0.329772\pi$
$62$	0	0
$63$	−311.502	−0.622946
$64$	0	0
$65$	−438.253	−0.836286
$66$	0	0
$67$	−784.960	−1.43131	−0.715657	−	0.698452i	$-0.753873\pi$
$67$	−784.960	−1.43131	−0.715657	+	0.698452i	$0.753873\pi$
$68$	0	0
$69$	1134.72	1.97978
$70$	0	0
$71$	1118.81	1.87012	0.935058	−	0.354496i	$-0.115347\pi$
$71$	1118.81	1.87012	0.935058	+	0.354496i	$0.115347\pi$
$72$	0	0
$73$	−259.120	−0.415449	−0.207724	−	0.978187i	$-0.566606\pi$
$73$	−259.120	−0.415449	−0.207724	+	0.978187i	$0.566606\pi$
$74$	0	0
$75$	−708.467	−1.09076
$76$	0	0
$77$	434.857	0.643591
$78$	0	0
$79$	−632.126	−0.900250	−0.450125	−	0.892966i	$-0.648621\pi$
$79$	−632.126	−0.900250	−0.450125	+	0.892966i	$0.648621\pi$
$80$	0	0
$81$	49.7687	0.0682698
$82$	0	0
$83$	295.878	0.391288	0.195644	−	0.980675i	$-0.437320\pi$
$83$	295.878	0.391288	0.195644	+	0.980675i	$0.437320\pi$
$84$	0	0
$85$	2.91720	0.00372253
$86$	0	0
$87$	−1968.58	−2.42591
$88$	0	0
$89$	848.357	1.01040	0.505200	−	0.863002i	$-0.331419\pi$
$89$	848.357	1.01040	0.505200	+	0.863002i	$0.331419\pi$
$90$	0	0
$91$	477.854	0.550469
$92$	0	0
$93$	617.119	0.688088
$94$	0	0
$95$	−495.753	−0.535402
$96$	0	0
$97$	1281.58	1.34149	0.670744	−	0.741689i	$-0.265975\pi$
$97$	1281.58	1.34149	0.670744	+	0.741689i	$0.265975\pi$
$98$	0	0
$99$	−2764.47	−2.80646
$100$	0	0
$101$	−1922.36	−1.89388	−0.946941	−	0.321409i	$-0.895844\pi$
$101$	−1922.36	−1.89388	−0.946941	+	0.321409i	$0.895844\pi$
$102$	0	0
$103$	106.579	0.101957	0.0509784	−	0.998700i	$-0.483766\pi$
$103$	106.579	0.101957	0.0509784	+	0.998700i	$0.483766\pi$
$104$	0	0
$105$	−379.997	−0.353180
$106$	0	0
$107$	−1619.27	−1.46299	−0.731497	−	0.681845i	$-0.761178\pi$
$107$	−1619.27	−1.46299	−0.731497	+	0.681845i	$0.761178\pi$
$108$	0	0
$109$	1137.55	0.999613	0.499806	−	0.866137i	$-0.333405\pi$
$109$	1137.55	0.999613	0.499806	+	0.866137i	$0.333405\pi$
$110$	0	0
$111$	1298.41	1.11026
$112$	0	0
$113$	896.973	0.746727	0.373363	−	0.927685i	$-0.378204\pi$
$113$	896.973	0.746727	0.373363	+	0.927685i	$0.378204\pi$
$114$	0	0
$115$	861.517	0.698581
$116$	0	0
$117$	−3037.80	−2.40039
$118$	0	0
$119$	−3.18080	−0.00245028
$120$	0	0
$121$	2528.19	1.89947
$122$	0	0
$123$	−346.687	−0.254144
$124$	0	0
$125$	−1340.38	−0.959096
$126$	0	0
$127$	1063.49	0.743068	0.371534	−	0.928419i	$-0.378832\pi$
$127$	1063.49	0.743068	0.371534	+	0.928419i	$0.378832\pi$
$128$	0	0
$129$	−1055.06	−0.720102
$130$	0	0
$131$	−2861.76	−1.90865	−0.954326	−	0.298769i	$-0.903424\pi$
$131$	−2861.76	−1.90865	−0.954326	+	0.298769i	$0.903424\pi$
$132$	0	0
$133$	540.549	0.352417
$134$	0	0
$135$	950.009	0.605657
$136$	0	0
$137$	1442.02	0.899270	0.449635	−	0.893212i	$-0.351554\pi$
$137$	1442.02	0.899270	0.449635	+	0.893212i	$0.351554\pi$
$138$	0	0
$139$	3233.12	1.97287	0.986437	−	0.164142i	$-0.0524857\pi$
$139$	3233.12	1.97287	0.986437	+	0.164142i	$0.0524857\pi$
$140$	0	0
$141$	−4944.97	−2.95349
$142$	0	0
$143$	4240.77	2.47994
$144$	0	0
$145$	−1494.61	−0.856003
$146$	0	0
$147$	414.333	0.232474
$148$	0	0
$149$	−3149.56	−1.73169	−0.865846	−	0.500310i	$-0.833219\pi$
$149$	−3149.56	−1.73169	−0.865846	+	0.500310i	$0.833219\pi$
$150$	0	0
$151$	−63.9496	−0.0344645	−0.0172323	−	0.999852i	$-0.505485\pi$
$151$	−63.9496	−0.0344645	−0.0172323	+	0.999852i	$0.505485\pi$
$152$	0	0
$153$	20.2210	0.0106848
$154$	0	0
$155$	468.536	0.242798
$156$	0	0
$157$	−771.724	−0.392295	−0.196147	−	0.980574i	$-0.562843\pi$
$157$	−771.724	−0.392295	−0.196147	+	0.980574i	$0.562843\pi$
$158$	0	0
$159$	−2833.93	−1.41349
$160$	0	0
$161$	−939.363	−0.459827
$162$	0	0
$163$	−1496.93	−0.719319	−0.359659	−	0.933084i	$-0.617107\pi$
$163$	−1496.93	−0.719319	−0.359659	+	0.933084i	$0.617107\pi$
$164$	0	0
$165$	−3372.33	−1.59113
$166$	0	0
$167$	3370.22	1.56165	0.780824	−	0.624750i	$-0.214799\pi$
$167$	3370.22	1.56165	0.780824	+	0.624750i	$0.214799\pi$
$168$	0	0
$169$	2463.08	1.12111
$170$	0	0
$171$	−3436.37	−1.53676
$172$	0	0
$173$	4242.73	1.86456	0.932281	−	0.361736i	$-0.117816\pi$
$173$	4242.73	1.86456	0.932281	+	0.361736i	$0.117816\pi$
$174$	0	0
$175$	586.494	0.253342
$176$	0	0
$177$	3079.86	1.30789
$178$	0	0
$179$	1705.82	0.712285	0.356143	−	0.934432i	$-0.384092\pi$
$179$	1705.82	0.712285	0.356143	+	0.934432i	$0.384092\pi$
$180$	0	0
$181$	−2273.85	−0.933780	−0.466890	−	0.884315i	$-0.654626\pi$
$181$	−2273.85	−0.933780	−0.466890	+	0.884315i	$0.654626\pi$
$182$	0	0
$183$	4106.36	1.65875
$184$	0	0
$185$	985.791	0.391766
$186$	0	0
$187$	−28.2284	−0.0110389
$188$	0	0
$189$	−1035.85	−0.398662
$190$	0	0
$191$	4502.27	1.70562	0.852809	−	0.522223i	$-0.174897\pi$
$191$	4502.27	1.70562	0.852809	+	0.522223i	$0.174897\pi$
$192$	0	0
$193$	−2687.47	−1.00232	−0.501161	−	0.865354i	$-0.667093\pi$
$193$	−2687.47	−1.00232	−0.501161	+	0.865354i	$0.667093\pi$
$194$	0	0
$195$	−3705.77	−1.36090
$196$	0	0
$197$	−1385.62	−0.501125	−0.250562	−	0.968100i	$-0.580616\pi$
$197$	−1385.62	−0.501125	−0.250562	+	0.968100i	$0.580616\pi$
$198$	0	0
$199$	−3673.09	−1.30844	−0.654218	−	0.756306i	$-0.727002\pi$
$199$	−3673.09	−1.30844	−0.654218	+	0.756306i	$0.727002\pi$
$200$	0	0
$201$	−6637.45	−2.32920
$202$	0	0
$203$	1629.66	0.563447
$204$	0	0
$205$	−263.216	−0.0896770
$206$	0	0
$207$	5971.71	2.00513
$208$	0	0
$209$	4797.17	1.58769
$210$	0	0
$211$	5370.87	1.75235	0.876175	−	0.481992i	$-0.160087\pi$
$211$	5370.87	1.75235	0.876175	+	0.481992i	$0.160087\pi$
$212$	0	0
$213$	9460.41	3.04327
$214$	0	0
$215$	−801.037	−0.254094
$216$	0	0
$217$	−510.873	−0.159817
$218$	0	0
$219$	−2191.07	−0.676066
$220$	0	0
$221$	−31.0196	−0.00944163
$222$	0	0
$223$	−4911.25	−1.47480	−0.737402	−	0.675454i	$-0.763948\pi$
$223$	−4911.25	−1.47480	−0.737402	+	0.675454i	$0.763948\pi$
$224$	0	0
$225$	−3728.45	−1.10473
$226$	0	0
$227$	−1742.66	−0.509534	−0.254767	−	0.967002i	$-0.581999\pi$
$227$	−1742.66	−0.509534	−0.254767	+	0.967002i	$0.581999\pi$
$228$	0	0
$229$	5162.79	1.48981	0.744906	−	0.667169i	$-0.232494\pi$
$229$	5162.79	1.48981	0.744906	+	0.667169i	$0.232494\pi$
$230$	0	0
$231$	3677.06	1.04733
$232$	0	0
$233$	4676.86	1.31498	0.657492	−	0.753462i	$-0.271617\pi$
$233$	4676.86	1.31498	0.657492	+	0.753462i	$0.271617\pi$
$234$	0	0
$235$	−3754.38	−1.04216
$236$	0	0
$237$	−5345.12	−1.46499
$238$	0	0
$239$	−3947.76	−1.06845	−0.534225	−	0.845342i	$-0.679396\pi$
$239$	−3947.76	−1.06845	−0.534225	+	0.845342i	$0.679396\pi$
$240$	0	0
$241$	1312.55	0.350825	0.175413	−	0.984495i	$-0.443874\pi$
$241$	1312.55	0.350825	0.175413	+	0.984495i	$0.443874\pi$
$242$	0	0
$243$	−3574.59	−0.943664
$244$	0	0
$245$	314.575	0.0820305
$246$	0	0
$247$	5271.49	1.35796
$248$	0	0
$249$	2501.88	0.636749
$250$	0	0
$251$	−6553.82	−1.64810	−0.824051	−	0.566515i	$-0.808291\pi$
$251$	−6553.82	−1.64810	−0.824051	+	0.566515i	$0.808291\pi$
$252$	0	0
$253$	−8336.50	−2.07159
$254$	0	0
$255$	24.6673	0.00605774
$256$	0	0
$257$	−1395.50	−0.338712	−0.169356	−	0.985555i	$-0.554169\pi$
$257$	−1395.50	−0.338712	−0.169356	+	0.985555i	$0.554169\pi$
$258$	0	0
$259$	−1074.87	−0.257872
$260$	0	0
$261$	−10360.1	−2.45698
$262$	0	0
$263$	−5105.11	−1.19694	−0.598469	−	0.801146i	$-0.704224\pi$
$263$	−5105.11	−1.19694	−0.598469	+	0.801146i	$0.704224\pi$
$264$	0	0
$265$	−2151.61	−0.498763
$266$	0	0
$267$	7173.53	1.64424
$268$	0	0
$269$	−5948.12	−1.34819	−0.674096	−	0.738644i	$-0.735467\pi$
$269$	−5948.12	−1.34819	−0.674096	+	0.738644i	$0.735467\pi$
$270$	0	0
$271$	−5018.10	−1.12483	−0.562413	−	0.826857i	$-0.690127\pi$
$271$	−5018.10	−1.12483	−0.562413	+	0.826857i	$0.690127\pi$
$272$	0	0
$273$	4040.63	0.895787
$274$	0	0
$275$	5204.92	1.14134
$276$	0	0
$277$	−6199.71	−1.34478	−0.672390	−	0.740197i	$-0.734732\pi$
$277$	−6199.71	−1.34478	−0.672390	+	0.740197i	$0.734732\pi$
$278$	0	0
$279$	3247.71	0.696901
$280$	0	0
$281$	−2381.83	−0.505651	−0.252826	−	0.967512i	$-0.581360\pi$
$281$	−2381.83	−0.505651	−0.252826	+	0.967512i	$0.581360\pi$
$282$	0	0
$283$	−1883.36	−0.395598	−0.197799	−	0.980243i	$-0.563379\pi$
$283$	−1883.36	−0.395598	−0.197799	+	0.980243i	$0.563379\pi$
$284$	0	0
$285$	−4191.98	−0.871268
$286$	0	0
$287$	287.000	0.0590281
$288$	0	0
$289$	−4912.79	−0.999958
$290$	0	0
$291$	10836.7	2.18303
$292$	0	0
$293$	−5898.11	−1.17601	−0.588005	−	0.808857i	$-0.700087\pi$
$293$	−5898.11	−1.17601	−0.588005	+	0.808857i	$0.700087\pi$
$294$	0	0
$295$	2338.33	0.461501
$296$	0	0
$297$	−9192.80	−1.79603
$298$	0	0
$299$	−9160.78	−1.77184
$300$	0	0
$301$	873.419	0.167253
$302$	0	0
$303$	−16255.1	−3.08194
$304$	0	0
$305$	3117.67	0.585303
$306$	0	0
$307$	−3973.77	−0.738746	−0.369373	−	0.929281i	$-0.620427\pi$
$307$	−3973.77	−0.738746	−0.369373	+	0.929281i	$0.620427\pi$
$308$	0	0
$309$	901.209	0.165916
$310$	0	0
$311$	1066.43	0.194443	0.0972213	−	0.995263i	$-0.469005\pi$
$311$	1066.43	0.194443	0.0972213	+	0.995263i	$0.469005\pi$
$312$	0	0
$313$	−1817.17	−0.328155	−0.164078	−	0.986447i	$-0.552465\pi$
$313$	−1817.17	−0.328155	−0.164078	+	0.986447i	$0.552465\pi$
$314$	0	0
$315$	−1999.81	−0.357704
$316$	0	0
$317$	1365.32	0.241906	0.120953	−	0.992658i	$-0.461405\pi$
$317$	1365.32	0.241906	0.120953	+	0.992658i	$0.461405\pi$
$318$	0	0
$319$	14462.6	2.53841
$320$	0	0
$321$	−13692.2	−2.38075
$322$	0	0
$323$	−35.0894	−0.00604466
$324$	0	0
$325$	5719.56	0.976197
$326$	0	0
$327$	9618.90	1.62669
$328$	0	0
$329$	4093.62	0.685984
$330$	0	0
$331$	4320.26	0.717412	0.358706	−	0.933451i	$-0.383218\pi$
$331$	4320.26	0.717412	0.358706	+	0.933451i	$0.383218\pi$
$332$	0	0
$333$	6833.13	1.12448
$334$	0	0
$335$	−5039.36	−0.821880
$336$	0	0
$337$	6986.84	1.12937	0.564684	−	0.825307i	$-0.308998\pi$
$337$	6986.84	1.12937	0.564684	+	0.825307i	$0.308998\pi$
$338$	0	0
$339$	7584.61	1.21516
$340$	0	0
$341$	−4533.80	−0.719998
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	7284.80	1.13681
$346$	0	0
$347$	870.538	0.134677	0.0673385	−	0.997730i	$-0.478549\pi$
$347$	870.538	0.134677	0.0673385	+	0.997730i	$0.478549\pi$
$348$	0	0
$349$	−5079.34	−0.779057	−0.389529	−	0.921014i	$-0.627362\pi$
$349$	−5079.34	−0.779057	−0.389529	+	0.921014i	$0.627362\pi$
$350$	0	0
$351$	−10101.7	−1.53616
$352$	0	0
$353$	1528.95	0.230533	0.115266	−	0.993335i	$-0.463228\pi$
$353$	1528.95	0.230533	0.115266	+	0.993335i	$0.463228\pi$
$354$	0	0
$355$	7182.64	1.07384
$356$	0	0
$357$	−26.8962	−0.00398739
$358$	0	0
$359$	10994.4	1.61634	0.808168	−	0.588953i	$-0.200460\pi$
$359$	10994.4	1.61634	0.808168	+	0.588953i	$0.200460\pi$
$360$	0	0
$361$	−895.878	−0.130613
$362$	0	0
$363$	21377.8	3.09104
$364$	0	0
$365$	−1663.53	−0.238556
$366$	0	0
$367$	304.815	0.0433548	0.0216774	−	0.999765i	$-0.493099\pi$
$367$	304.815	0.0433548	0.0216774	+	0.999765i	$0.493099\pi$
$368$	0	0
$369$	−1824.51	−0.257399
$370$	0	0
$371$	2346.03	0.328301
$372$	0	0
$373$	−10245.1	−1.42218	−0.711091	−	0.703100i	$-0.751798\pi$
$373$	−10245.1	−1.42218	−0.711091	+	0.703100i	$0.751798\pi$
$374$	0	0
$375$	−11333.9	−1.56075
$376$	0	0
$377$	15892.6	2.17112
$378$	0	0
$379$	−2068.33	−0.280324	−0.140162	−	0.990129i	$-0.544762\pi$
$379$	−2068.33	−0.280324	−0.140162	+	0.990129i	$0.544762\pi$
$380$	0	0
$381$	8992.65	1.20921
$382$	0	0
$383$	−6138.31	−0.818937	−0.409469	−	0.912324i	$-0.634286\pi$
$383$	−6138.31	−0.818937	−0.409469	+	0.912324i	$0.634286\pi$
$384$	0	0
$385$	2791.74	0.369559
$386$	0	0
$387$	−5552.48	−0.729325
$388$	0	0
$389$	13541.6	1.76500	0.882499	−	0.470314i	$-0.155859\pi$
$389$	13541.6	1.76500	0.882499	+	0.470314i	$0.155859\pi$
$390$	0	0
$391$	60.9782	0.00788695
$392$	0	0
$393$	−24198.4	−3.10598
$394$	0	0
$395$	−4058.18	−0.516935
$396$	0	0
$397$	8754.31	1.10672	0.553358	−	0.832944i	$-0.313346\pi$
$397$	8754.31	1.10672	0.553358	+	0.832944i	$0.313346\pi$
$398$	0	0
$399$	4570.76	0.573495
$400$	0	0
$401$	7602.40	0.946748	0.473374	−	0.880862i	$-0.343036\pi$
$401$	7602.40	0.946748	0.473374	+	0.880862i	$0.343036\pi$
$402$	0	0
$403$	−4982.09	−0.615820
$404$	0	0
$405$	319.510	0.0392014
$406$	0	0
$407$	−9539.04	−1.16175
$408$	0	0
$409$	13641.4	1.64921	0.824604	−	0.565710i	$-0.191398\pi$
$409$	13641.4	1.64921	0.824604	+	0.565710i	$0.191398\pi$
$410$	0	0
$411$	12193.4	1.46340
$412$	0	0
$413$	−2549.62	−0.303774
$414$	0	0
$415$	1899.51	0.224683
$416$	0	0
$417$	27338.5	3.21049
$418$	0	0
$419$	160.662	0.0187323	0.00936617	−	0.999956i	$-0.497019\pi$
$419$	160.662	0.0187323	0.00936617	+	0.999956i	$0.497019\pi$
$420$	0	0
$421$	−14396.0	−1.66655	−0.833273	−	0.552861i	$-0.813536\pi$
$421$	−14396.0	−1.66655	−0.833273	+	0.552861i	$0.813536\pi$
$422$	0	0
$423$	−26023.9	−2.99131
$424$	0	0
$425$	−38.0719	−0.00434531
$426$	0	0
$427$	−3399.39	−0.385264
$428$	0	0
$429$	35859.1	4.03565
$430$	0	0
$431$	−15611.1	−1.74469	−0.872346	−	0.488888i	$-0.837403\pi$
$431$	−15611.1	−1.74469	−0.872346	+	0.488888i	$0.837403\pi$
$432$	0	0
$433$	8705.45	0.966183	0.483091	−	0.875570i	$-0.339514\pi$
$433$	8705.45	0.966183	0.483091	+	0.875570i	$0.339514\pi$
$434$	0	0
$435$	−12638.1	−1.39299
$436$	0	0
$437$	−10362.7	−1.13436
$438$	0	0
$439$	1255.79	0.136527	0.0682637	−	0.997667i	$-0.478254\pi$
$439$	1255.79	0.136527	0.0682637	+	0.997667i	$0.478254\pi$
$440$	0	0
$441$	2180.51	0.235451
$442$	0	0
$443$	3690.79	0.395835	0.197918	−	0.980219i	$-0.436582\pi$
$443$	3690.79	0.395835	0.197918	+	0.980219i	$0.436582\pi$
$444$	0	0
$445$	5446.37	0.580186
$446$	0	0
$447$	−26632.0	−2.81801
$448$	0	0
$449$	−4534.61	−0.476618	−0.238309	−	0.971189i	$-0.576593\pi$
$449$	−4534.61	−0.476618	−0.238309	+	0.971189i	$0.576593\pi$
$450$	0	0
$451$	2547.02	0.265930
$452$	0	0
$453$	−540.744	−0.0560847
$454$	0	0
$455$	3067.77	0.316087
$456$	0	0
$457$	8221.40	0.841534	0.420767	−	0.907169i	$-0.361761\pi$
$457$	8221.40	0.841534	0.420767	+	0.907169i	$0.361761\pi$
$458$	0	0
$459$	67.2416	0.00683784
$460$	0	0
$461$	−2989.24	−0.302002	−0.151001	−	0.988534i	$-0.548250\pi$
$461$	−2989.24	−0.302002	−0.151001	+	0.988534i	$0.548250\pi$
$462$	0	0
$463$	−1028.22	−0.103209	−0.0516043	−	0.998668i	$-0.516433\pi$
$463$	−1028.22	−0.103209	−0.0516043	+	0.998668i	$0.516433\pi$
$464$	0	0
$465$	3961.84	0.395109
$466$	0	0
$467$	3420.72	0.338955	0.169478	−	0.985534i	$-0.445792\pi$
$467$	3420.72	0.338955	0.169478	+	0.985534i	$0.445792\pi$
$468$	0	0
$469$	5494.72	0.540986
$470$	0	0
$471$	−6525.53	−0.638388
$472$	0	0
$473$	7751.26	0.753496
$474$	0	0
$475$	6469.97	0.624974
$476$	0	0
$477$	−14914.1	−1.43160
$478$	0	0
$479$	−2479.52	−0.236518	−0.118259	−	0.992983i	$-0.537731\pi$
$479$	−2479.52	−0.236518	−0.118259	+	0.992983i	$0.537731\pi$
$480$	0	0
$481$	−10482.2	−0.993655
$482$	0	0
$483$	−7943.06	−0.748285
$484$	0	0
$485$	8227.59	0.770300
$486$	0	0
$487$	9552.49	0.888839	0.444420	−	0.895819i	$-0.353410\pi$
$487$	9552.49	0.888839	0.444420	+	0.895819i	$0.353410\pi$
$488$	0	0
$489$	−12657.8	−1.17056
$490$	0	0
$491$	7071.61	0.649974	0.324987	−	0.945718i	$-0.394640\pi$
$491$	7071.61	0.649974	0.324987	+	0.945718i	$0.394640\pi$
$492$	0	0
$493$	−105.788	−0.00966423
$494$	0	0
$495$	−17747.6	−1.61150
$496$	0	0
$497$	−7831.66	−0.706837
$498$	0	0
$499$	−6084.67	−0.545867	−0.272933	−	0.962033i	$-0.587994\pi$
$499$	−6084.67	−0.545867	−0.272933	+	0.962033i	$0.587994\pi$
$500$	0	0
$501$	28497.8	2.54130
$502$	0	0
$503$	−12849.1	−1.13899	−0.569494	−	0.821995i	$-0.692861\pi$
$503$	−12849.1	−1.13899	−0.569494	+	0.821995i	$0.692861\pi$
$504$	0	0
$505$	−12341.4	−1.08749
$506$	0	0
$507$	20827.3	1.82440
$508$	0	0
$509$	−3079.34	−0.268152	−0.134076	−	0.990971i	$-0.542807\pi$
$509$	−3079.34	−0.268152	−0.134076	+	0.990971i	$0.542807\pi$
$510$	0	0
$511$	1813.84	0.157025
$512$	0	0
$513$	−11427.1	−0.983468
$514$	0	0
$515$	684.227	0.0585449
$516$	0	0
$517$	36329.4	3.09045
$518$	0	0
$519$	35875.6	3.03423
$520$	0	0
$521$	−5450.75	−0.458353	−0.229176	−	0.973385i	$-0.573603\pi$
$521$	−5450.75	−0.458353	−0.229176	+	0.973385i	$0.573603\pi$
$522$	0	0
$523$	−16969.5	−1.41878	−0.709391	−	0.704815i	$-0.751030\pi$
$523$	−16969.5	−1.41878	−0.709391	+	0.704815i	$0.751030\pi$
$524$	0	0
$525$	4959.27	0.412267
$526$	0	0
$527$	33.1630	0.00274118
$528$	0	0
$529$	5841.24	0.480088
$530$	0	0
$531$	16208.4	1.32464
$532$	0	0
$533$	2798.86	0.227452
$534$	0	0
$535$	−10395.5	−0.840070
$536$	0	0
$537$	14424.1	1.15911
$538$	0	0
$539$	−3044.00	−0.243255
$540$	0	0
$541$	6518.89	0.518057	0.259028	−	0.965870i	$-0.416598\pi$
$541$	6518.89	0.518057	0.259028	+	0.965870i	$0.416598\pi$
$542$	0	0
$543$	−19227.2	−1.51956
$544$	0	0
$545$	7302.97	0.573991
$546$	0	0
$547$	6453.13	0.504416	0.252208	−	0.967673i	$-0.418843\pi$
$547$	6453.13	0.504416	0.252208	+	0.967673i	$0.418843\pi$
$548$	0	0
$549$	21610.5	1.67999
$550$	0	0
$551$	17977.8	1.38998
$552$	0	0
$553$	4424.88	0.340262
$554$	0	0
$555$	8335.63	0.637528
$556$	0	0
$557$	8706.27	0.662291	0.331146	−	0.943580i	$-0.392565\pi$
$557$	8706.27	0.662291	0.331146	+	0.943580i	$0.392565\pi$
$558$	0	0
$559$	8517.68	0.644471
$560$	0	0
$561$	−238.694	−0.0179637
$562$	0	0
$563$	4534.17	0.339418	0.169709	−	0.985494i	$-0.445717\pi$
$563$	4534.17	0.339418	0.169709	+	0.985494i	$0.445717\pi$
$564$	0	0
$565$	5758.48	0.428780
$566$	0	0
$567$	−348.381	−0.0258036
$568$	0	0
$569$	7788.45	0.573829	0.286914	−	0.957956i	$-0.407370\pi$
$569$	7788.45	0.573829	0.286914	+	0.957956i	$0.407370\pi$
$570$	0	0
$571$	18011.6	1.32008	0.660038	−	0.751232i	$-0.270540\pi$
$571$	18011.6	1.32008	0.660038	+	0.751232i	$0.270540\pi$
$572$	0	0
$573$	38070.3	2.77558
$574$	0	0
$575$	−11243.5	−0.815454
$576$	0	0
$577$	7126.29	0.514162	0.257081	−	0.966390i	$-0.417239\pi$
$577$	7126.29	0.514162	0.257081	+	0.966390i	$0.417239\pi$
$578$	0	0
$579$	−22724.6	−1.63109
$580$	0	0
$581$	−2071.15	−0.147893
$582$	0	0
$583$	20820.1	1.47904
$584$	0	0
$585$	−19502.4	−1.37833
$586$	0	0
$587$	410.109	0.0288365	0.0144183	−	0.999896i	$-0.495410\pi$
$587$	410.109	0.0288365	0.0144183	+	0.999896i	$0.495410\pi$
$588$	0	0
$589$	−5635.75	−0.394256
$590$	0	0
$591$	−11716.5	−0.815489
$592$	0	0
$593$	18591.3	1.28744	0.643721	−	0.765260i	$-0.277390\pi$
$593$	18591.3	1.28744	0.643721	+	0.765260i	$0.277390\pi$
$594$	0	0
$595$	−20.4204	−0.00140699
$596$	0	0
$597$	−31058.9	−2.12924
$598$	0	0
$599$	8192.51	0.558826	0.279413	−	0.960171i	$-0.409860\pi$
$599$	8192.51	0.558826	0.279413	+	0.960171i	$0.409860\pi$
$600$	0	0
$601$	−9329.66	−0.633219	−0.316609	−	0.948556i	$-0.602544\pi$
$601$	−9329.66	−0.633219	−0.316609	+	0.948556i	$0.602544\pi$
$602$	0	0
$603$	−34930.9	−2.35903
$604$	0	0
$605$	16230.7	1.09070
$606$	0	0
$607$	−335.931	−0.0224630	−0.0112315	−	0.999937i	$-0.503575\pi$
$607$	−335.931	−0.0224630	−0.0112315	+	0.999937i	$0.503575\pi$
$608$	0	0
$609$	13780.1	0.916907
$610$	0	0
$611$	39921.5	2.64329
$612$	0	0
$613$	451.466	0.0297464	0.0148732	−	0.999889i	$-0.495266\pi$
$613$	451.466	0.0297464	0.0148732	+	0.999889i	$0.495266\pi$
$614$	0	0
$615$	−2225.70	−0.145933
$616$	0	0
$617$	9312.61	0.607636	0.303818	−	0.952730i	$-0.401738\pi$
$617$	9312.61	0.607636	0.303818	+	0.952730i	$0.401738\pi$
$618$	0	0
$619$	2683.44	0.174243	0.0871217	−	0.996198i	$-0.472233\pi$
$619$	2683.44	0.174243	0.0871217	+	0.996198i	$0.472233\pi$
$620$	0	0
$621$	19858.0	1.28321
$622$	0	0
$623$	−5938.50	−0.381896
$624$	0	0
$625$	1868.02	0.119553
$626$	0	0
$627$	40563.8	2.58367
$628$	0	0
$629$	69.7742	0.00442302
$630$	0	0
$631$	−6903.47	−0.435535	−0.217768	−	0.976001i	$-0.569877\pi$
$631$	−6903.47	−0.435535	−0.217768	+	0.976001i	$0.569877\pi$
$632$	0	0
$633$	45414.9	2.85163
$634$	0	0
$635$	6827.51	0.426679
$636$	0	0
$637$	−3344.98	−0.208058
$638$	0	0
$639$	49787.3	3.08225
$640$	0	0
$641$	4294.58	0.264627	0.132313	−	0.991208i	$-0.457759\pi$
$641$	4294.58	0.264627	0.132313	+	0.991208i	$0.457759\pi$
$642$	0	0
$643$	5543.93	0.340018	0.170009	−	0.985443i	$-0.445620\pi$
$643$	5543.93	0.340018	0.170009	+	0.985443i	$0.445620\pi$
$644$	0	0
$645$	−6773.40	−0.413492
$646$	0	0
$647$	−10940.1	−0.664763	−0.332381	−	0.943145i	$-0.607852\pi$
$647$	−10940.1	−0.664763	−0.332381	+	0.943145i	$0.607852\pi$
$648$	0	0
$649$	−22626.9	−1.36854
$650$	0	0
$651$	−4319.83	−0.260073
$652$	0	0
$653$	12430.8	0.744955	0.372477	−	0.928041i	$-0.378508\pi$
$653$	12430.8	0.744955	0.372477	+	0.928041i	$0.378508\pi$
$654$	0	0
$655$	−18372.2	−1.09597
$656$	0	0
$657$	−11530.9	−0.684725
$658$	0	0
$659$	−25700.5	−1.51919	−0.759597	−	0.650394i	$-0.774604\pi$
$659$	−25700.5	−1.51919	−0.759597	+	0.650394i	$0.774604\pi$
$660$	0	0
$661$	−21722.3	−1.27822	−0.639108	−	0.769117i	$-0.720696\pi$
$661$	−21722.3	−1.27822	−0.639108	+	0.769117i	$0.720696\pi$
$662$	0	0
$663$	−262.295	−0.0153645
$664$	0	0
$665$	3470.27	0.202363
$666$	0	0
$667$	−31241.7	−1.81362
$668$	0	0
$669$	−41528.4	−2.39997
$670$	0	0
$671$	−30168.3	−1.73567
$672$	0	0
$673$	19641.9	1.12502	0.562512	−	0.826789i	$-0.309835\pi$
$673$	19641.9	1.12502	0.562512	+	0.826789i	$0.309835\pi$
$674$	0	0
$675$	−12398.4	−0.706984
$676$	0	0
$677$	13481.6	0.765349	0.382674	−	0.923883i	$-0.375003\pi$
$677$	13481.6	0.765349	0.382674	+	0.923883i	$0.375003\pi$
$678$	0	0
$679$	−8971.03	−0.507035
$680$	0	0
$681$	−14735.5	−0.829173
$682$	0	0
$683$	−8919.22	−0.499684	−0.249842	−	0.968287i	$-0.580379\pi$
$683$	−8919.22	−0.499684	−0.249842	+	0.968287i	$0.580379\pi$
$684$	0	0
$685$	9257.62	0.516373
$686$	0	0
$687$	43655.5	2.42440
$688$	0	0
$689$	22878.7	1.26504
$690$	0	0
$691$	−17828.5	−0.981514	−0.490757	−	0.871296i	$-0.663280\pi$
$691$	−17828.5	−0.981514	−0.490757	+	0.871296i	$0.663280\pi$
$692$	0	0
$693$	19351.3	1.06074
$694$	0	0
$695$	20756.3	1.13285
$696$	0	0
$697$	−18.6304	−0.00101245
$698$	0	0
$699$	39546.5	2.13989
$700$	0	0
$701$	14526.4	0.782673	0.391336	−	0.920248i	$-0.372013\pi$
$701$	14526.4	0.782673	0.391336	+	0.920248i	$0.372013\pi$
$702$	0	0
$703$	−11857.5	−0.636151
$704$	0	0
$705$	−31746.2	−1.69593
$706$	0	0
$707$	13456.5	0.715820
$708$	0	0
$709$	−7955.44	−0.421400	−0.210700	−	0.977551i	$-0.567574\pi$
$709$	−7955.44	−0.421400	−0.210700	+	0.977551i	$0.567574\pi$
$710$	0	0
$711$	−28129.8	−1.48375
$712$	0	0
$713$	9793.78	0.514418
$714$	0	0
$715$	27225.3	1.42401
$716$	0	0
$717$	−33381.4	−1.73871
$718$	0	0
$719$	−28116.1	−1.45835	−0.729176	−	0.684326i	$-0.760096\pi$
$719$	−28116.1	−1.45835	−0.729176	+	0.684326i	$0.760096\pi$
$720$	0	0
$721$	−746.053	−0.0385360
$722$	0	0
$723$	11098.7	0.570904
$724$	0	0
$725$	19505.8	0.999212
$726$	0	0
$727$	−13706.1	−0.699219	−0.349610	−	0.936895i	$-0.613686\pi$
$727$	−13706.1	−0.699219	−0.349610	+	0.936895i	$0.613686\pi$
$728$	0	0
$729$	−31569.8	−1.60391
$730$	0	0
$731$	−56.6974	−0.00286871
$732$	0	0
$733$	36910.3	1.85991	0.929955	−	0.367672i	$-0.119845\pi$
$733$	36910.3	1.85991	0.929955	+	0.367672i	$0.119845\pi$
$734$	0	0
$735$	2659.98	0.133490
$736$	0	0
$737$	48763.6	2.43722
$738$	0	0
$739$	436.182	0.0217121	0.0108560	−	0.999941i	$-0.496544\pi$
$739$	436.182	0.0217121	0.0108560	+	0.999941i	$0.496544\pi$
$740$	0	0
$741$	44574.6	2.20984
$742$	0	0
$743$	6713.72	0.331497	0.165749	−	0.986168i	$-0.446996\pi$
$743$	6713.72	0.331497	0.165749	+	0.986168i	$0.446996\pi$
$744$	0	0
$745$	−20219.9	−0.994361
$746$	0	0
$747$	13166.7	0.644905
$748$	0	0
$749$	11334.9	0.552960
$750$	0	0
$751$	−17188.6	−0.835183	−0.417591	−	0.908635i	$-0.637126\pi$
$751$	−17188.6	−0.835183	−0.417591	+	0.908635i	$0.637126\pi$
$752$	0	0
$753$	−55417.7	−2.68198
$754$	0	0
$755$	−410.550	−0.0197900
$756$	0	0
$757$	−11005.5	−0.528406	−0.264203	−	0.964467i	$-0.585109\pi$
$757$	−11005.5	−0.528406	−0.264203	+	0.964467i	$0.585109\pi$
$758$	0	0
$759$	−70491.7	−3.37113
$760$	0	0
$761$	−4143.71	−0.197384	−0.0986921	−	0.995118i	$-0.531466\pi$
$761$	−4143.71	−0.197384	−0.0986921	+	0.995118i	$0.531466\pi$
$762$	0	0
$763$	−7962.87	−0.377818
$764$	0	0
$765$	129.816	0.00613533
$766$	0	0
$767$	−24864.2	−1.17053
$768$	0	0
$769$	4888.87	0.229255	0.114628	−	0.993409i	$-0.463433\pi$
$769$	4888.87	0.229255	0.114628	+	0.993409i	$0.463433\pi$
$770$	0	0
$771$	−11800.1	−0.551192
$772$	0	0
$773$	−19442.8	−0.904670	−0.452335	−	0.891848i	$-0.649409\pi$
$773$	−19442.8	−0.904670	−0.452335	+	0.891848i	$0.649409\pi$
$774$	0	0
$775$	−6114.77	−0.283418
$776$	0	0
$777$	−9088.84	−0.419640
$778$	0	0
$779$	3166.07	0.145618
$780$	0	0
$781$	−69503.1	−3.18440
$782$	0	0
$783$	−34450.7	−1.57237
$784$	0	0
$785$	−4954.39	−0.225261
$786$	0	0
$787$	19938.9	0.903107	0.451554	−	0.892244i	$-0.350870\pi$
$787$	19938.9	0.903107	0.451554	+	0.892244i	$0.350870\pi$
$788$	0	0
$789$	−43167.7	−1.94780
$790$	0	0
$791$	−6278.81	−0.282236
$792$	0	0
$793$	−33151.2	−1.48453
$794$	0	0
$795$	−18193.5	−0.811645
$796$	0	0
$797$	−7001.95	−0.311194	−0.155597	−	0.987821i	$-0.549730\pi$
$797$	−7001.95	−0.311194	−0.155597	+	0.987821i	$0.549730\pi$
$798$	0	0
$799$	−265.735	−0.0117660
$800$	0	0
$801$	37752.1	1.66530
$802$	0	0
$803$	16097.2	0.707419
$804$	0	0
$805$	−6030.62	−0.264039
$806$	0	0
$807$	−50296.1	−2.19393
$808$	0	0
$809$	−12322.9	−0.535538	−0.267769	−	0.963483i	$-0.586286\pi$
$809$	−12322.9	−0.535538	−0.267769	+	0.963483i	$0.586286\pi$
$810$	0	0
$811$	5840.68	0.252890	0.126445	−	0.991974i	$-0.459643\pi$
$811$	5840.68	0.252890	0.126445	+	0.991974i	$0.459643\pi$
$812$	0	0
$813$	−42432.0	−1.83045
$814$	0	0
$815$	−9610.17	−0.413042
$816$	0	0
$817$	9635.21	0.412599
$818$	0	0
$819$	21264.6	0.907260
$820$	0	0
$821$	−3193.47	−0.135753	−0.0678763	−	0.997694i	$-0.521622\pi$
$821$	−3193.47	−0.135753	−0.0678763	+	0.997694i	$0.521622\pi$
$822$	0	0
$823$	−32361.0	−1.37064	−0.685318	−	0.728244i	$-0.740337\pi$
$823$	−32361.0	−1.37064	−0.685318	+	0.728244i	$0.740337\pi$
$824$	0	0
$825$	44011.7	1.85732
$826$	0	0
$827$	−32247.0	−1.35591	−0.677955	−	0.735103i	$-0.737134\pi$
$827$	−32247.0	−1.35591	−0.677955	+	0.735103i	$0.737134\pi$
$828$	0	0
$829$	−3829.12	−0.160423	−0.0802116	−	0.996778i	$-0.525560\pi$
$829$	−3829.12	−0.160423	−0.0802116	+	0.996778i	$0.525560\pi$
$830$	0	0
$831$	−52423.4	−2.18838
$832$	0	0
$833$	22.2656	0.000926120 0
$834$	0	0
$835$	21636.5	0.896720
$836$	0	0
$837$	10799.8	0.445991
$838$	0	0
$839$	31906.3	1.31291	0.656453	−	0.754367i	$-0.272056\pi$
$839$	31906.3	1.31291	0.656453	+	0.754367i	$0.272056\pi$
$840$	0	0
$841$	29810.8	1.22231
$842$	0	0
$843$	−20140.2	−0.822855
$844$	0	0
$845$	15812.7	0.643757
$846$	0	0
$847$	−17697.3	−0.717931
$848$	0	0
$849$	−15925.3	−0.643763
$850$	0	0
$851$	20605.9	0.830037
$852$	0	0
$853$	23001.0	0.923259	0.461629	−	0.887073i	$-0.347265\pi$
$853$	23001.0	0.923259	0.461629	+	0.887073i	$0.347265\pi$
$854$	0	0
$855$	−22061.1	−0.882427
$856$	0	0
$857$	5740.70	0.228820	0.114410	−	0.993434i	$-0.463502\pi$
$857$	5740.70	0.228820	0.114410	+	0.993434i	$0.463502\pi$
$858$	0	0
$859$	−1670.70	−0.0663604	−0.0331802	−	0.999449i	$-0.510564\pi$
$859$	−1670.70	−0.0663604	−0.0331802	+	0.999449i	$0.510564\pi$
$860$	0	0
$861$	2426.81	0.0960575
$862$	0	0
$863$	13328.4	0.525729	0.262865	−	0.964833i	$-0.415333\pi$
$863$	13328.4	0.525729	0.262865	+	0.964833i	$0.415333\pi$
$864$	0	0
$865$	27237.9	1.07066
$866$	0	0
$867$	−41541.5	−1.62725
$868$	0	0
$869$	39269.2	1.53293
$870$	0	0
$871$	53585.1	2.08457
$872$	0	0
$873$	57030.5	2.21099
$874$	0	0
$875$	9382.65	0.362504
$876$	0	0
$877$	−17788.5	−0.684920	−0.342460	−	0.939532i	$-0.611260\pi$
$877$	−17788.5	−0.684920	−0.342460	+	0.939532i	$0.611260\pi$
$878$	0	0
$879$	−49873.1	−1.91374
$880$	0	0
$881$	15881.0	0.607315	0.303657	−	0.952781i	$-0.401792\pi$
$881$	15881.0	0.607315	0.303657	+	0.952781i	$0.401792\pi$
$882$	0	0
$883$	−26802.8	−1.02150	−0.510751	−	0.859729i	$-0.670633\pi$
$883$	−26802.8	−1.02150	−0.510751	+	0.859729i	$0.670633\pi$
$884$	0	0
$885$	19772.4	0.751008
$886$	0	0
$887$	−10210.7	−0.386517	−0.193259	−	0.981148i	$-0.561906\pi$
$887$	−10210.7	−0.386517	−0.193259	+	0.981148i	$0.561906\pi$
$888$	0	0
$889$	−7444.44	−0.280853
$890$	0	0
$891$	−3091.75	−0.116249
$892$	0	0
$893$	45159.2	1.69227
$894$	0	0
$895$	10951.2	0.409004
$896$	0	0
$897$	−77461.6	−2.88335
$898$	0	0
$899$	−16990.8	−0.630339
$900$	0	0
$901$	−152.291	−0.00563101
$902$	0	0
$903$	7385.44	0.272173
$904$	0	0
$905$	−14597.9	−0.536189
$906$	0	0
$907$	−22862.8	−0.836985	−0.418493	−	0.908220i	$-0.637441\pi$
$907$	−22862.8	−0.836985	−0.418493	+	0.908220i	$0.637441\pi$
$908$	0	0
$909$	−85545.6	−3.12142
$910$	0	0
$911$	5529.15	0.201086	0.100543	−	0.994933i	$-0.467942\pi$
$911$	5529.15	0.201086	0.100543	+	0.994933i	$0.467942\pi$
$912$	0	0
$913$	−18380.7	−0.666278
$914$	0	0
$915$	26362.4	0.952474
$916$	0	0
$917$	20032.3	0.721402
$918$	0	0
$919$	−19007.1	−0.682248	−0.341124	−	0.940018i	$-0.610808\pi$
$919$	−19007.1	−0.682248	−0.341124	+	0.940018i	$0.610808\pi$
$920$	0	0
$921$	−33601.3	−1.20217
$922$	0	0
$923$	−76375.3	−2.72364
$924$	0	0
$925$	−12865.4	−0.457309
$926$	0	0
$927$	4742.80	0.168041
$928$	0	0
$929$	37121.9	1.31101	0.655505	−	0.755191i	$-0.272456\pi$
$929$	37121.9	1.31101	0.655505	+	0.755191i	$0.272456\pi$
$930$	0	0
$931$	−3783.84	−0.133201
$932$	0	0
$933$	9017.49	0.316420
$934$	0	0
$935$	−181.224	−0.00633866
$936$	0	0
$937$	14459.0	0.504115	0.252057	−	0.967712i	$-0.418893\pi$
$937$	14459.0	0.504115	0.252057	+	0.967712i	$0.418893\pi$
$938$	0	0
$939$	−15365.6	−0.534013
$940$	0	0
$941$	3781.22	0.130993	0.0654964	−	0.997853i	$-0.479137\pi$
$941$	3781.22	0.130993	0.0654964	+	0.997853i	$0.479137\pi$
$942$	0	0
$943$	−5501.99	−0.189999
$944$	0	0
$945$	−6650.06	−0.228917
$946$	0	0
$947$	−11543.5	−0.396106	−0.198053	−	0.980191i	$-0.563462\pi$
$947$	−11543.5	−0.396106	−0.198053	+	0.980191i	$0.563462\pi$
$948$	0	0
$949$	17688.8	0.605061
$950$	0	0
$951$	11544.9	0.393657
$952$	0	0
$953$	−14755.8	−0.501562	−0.250781	−	0.968044i	$-0.580687\pi$
$953$	−14755.8	−0.501562	−0.250781	+	0.968044i	$0.580687\pi$
$954$	0	0
$955$	28904.1	0.979389
$956$	0	0
$957$	122293.	4.13079
$958$	0	0
$959$	−10094.1	−0.339892
$960$	0	0
$961$	−24464.7	−0.821210
$962$	0	0
$963$	−72057.8	−2.41125
$964$	0	0
$965$	−17253.3	−0.575546
$966$	0	0
$967$	−22331.1	−0.742627	−0.371314	−	0.928507i	$-0.621093\pi$
$967$	−22331.1	−0.742627	−0.371314	+	0.928507i	$0.621093\pi$
$968$	0	0
$969$	−296.708	−0.00983657
$970$	0	0
$971$	53571.4	1.77053	0.885267	−	0.465083i	$-0.153976\pi$
$971$	53571.4	1.77053	0.885267	+	0.465083i	$0.153976\pi$
$972$	0	0
$973$	−22631.8	−0.745676
$974$	0	0
$975$	48363.4	1.58858
$976$	0	0
$977$	46492.5	1.52244	0.761221	−	0.648492i	$-0.224600\pi$
$977$	46492.5	1.52244	0.761221	+	0.648492i	$0.224600\pi$
$978$	0	0
$979$	−52702.0	−1.72049
$980$	0	0
$981$	50621.4	1.64752
$982$	0	0
$983$	28347.9	0.919794	0.459897	−	0.887972i	$-0.347886\pi$
$983$	28347.9	0.919794	0.459897	+	0.887972i	$0.347886\pi$
$984$	0	0
$985$	−8895.56	−0.287753
$986$	0	0
$987$	34614.8	1.11631
$988$	0	0
$989$	−16744.0	−0.538351
$990$	0	0
$991$	−34217.6	−1.09683	−0.548414	−	0.836207i	$-0.684768\pi$
$991$	−34217.6	−1.09683	−0.548414	+	0.836207i	$0.684768\pi$
$992$	0	0
$993$	36531.2	1.16746
$994$	0	0
$995$	−23580.9	−0.751321
$996$	0	0
$997$	23455.0	0.745063	0.372532	−	0.928020i	$-0.378490\pi$
$997$	23455.0	0.745063	0.372532	+	0.928020i	$0.378490\pi$
$998$	0	0
$999$	22722.5	0.719627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.b.1.15	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.b.1.15	✓	15	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+8.45579 q^{3} +6.41990 q^{5} -7.00000 q^{7} +44.5003 q^{9} +O(q^{10})\)	\(q+8.45579 q^{3} +6.41990 q^{5} -7.00000 q^{7} +44.5003 q^{9} -62.1224 q^{11} -68.2648 q^{13} +54.2853 q^{15} +0.454400 q^{17} -77.2213 q^{19} -59.1905 q^{21} +134.195 q^{23} -83.7849 q^{25} +147.979 q^{27} -232.809 q^{29} +72.9818 q^{31} -525.294 q^{33} -44.9393 q^{35} +153.552 q^{37} -577.233 q^{39} -41.0000 q^{41} -124.774 q^{43} +285.687 q^{45} -584.803 q^{47} +49.0000 q^{49} +3.84231 q^{51} -335.147 q^{53} -398.820 q^{55} -652.966 q^{57} +364.232 q^{59} +485.627 q^{61} -311.502 q^{63} -438.253 q^{65} -784.960 q^{67} +1134.72 q^{69} +1118.81 q^{71} -259.120 q^{73} -708.467 q^{75} +434.857 q^{77} -632.126 q^{79} +49.7687 q^{81} +295.878 q^{83} +2.91720 q^{85} -1968.58 q^{87} +848.357 q^{89} +477.854 q^{91} +617.119 q^{93} -495.753 q^{95} +1281.58 q^{97} -2764.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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