LMFDB - Embedded newform 1148.4.a.c.1.10

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			1.10
Root			$-1.16028$ 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)

$f(q)$	$=$	$q+2.16028 q^{3} -14.3655 q^{5} +7.00000 q^{7} -22.3332 q^{9} +O(q^{10})$	$q+2.16028 q^{3} -14.3655 q^{5} +7.00000 q^{7} -22.3332 q^{9} -58.4700 q^{11} -16.4312 q^{13} -31.0335 q^{15} -7.42853 q^{17} +1.95888 q^{19} +15.1220 q^{21} +4.12992 q^{23} +81.3679 q^{25} -106.574 q^{27} -213.528 q^{29} +144.598 q^{31} -126.312 q^{33} -100.559 q^{35} +267.155 q^{37} -35.4959 q^{39} -41.0000 q^{41} +482.282 q^{43} +320.828 q^{45} -292.866 q^{47} +49.0000 q^{49} -16.0477 q^{51} +174.859 q^{53} +839.951 q^{55} +4.23173 q^{57} -48.2199 q^{59} -35.3366 q^{61} -156.332 q^{63} +236.042 q^{65} +165.683 q^{67} +8.92179 q^{69} -190.733 q^{71} +215.024 q^{73} +175.778 q^{75} -409.290 q^{77} -522.220 q^{79} +372.767 q^{81} -435.012 q^{83} +106.715 q^{85} -461.280 q^{87} +1547.40 q^{89} -115.018 q^{91} +312.372 q^{93} -28.1403 q^{95} -123.909 q^{97} +1305.82 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.16028	0.415746	0.207873	−	0.978156i	$-0.433346\pi$
$3$	2.16028	0.415746	0.207873	+	0.978156i	$0.433346\pi$
$4$	0	0
$5$	−14.3655	−1.28489	−0.642445	−	0.766332i	$-0.722080\pi$
$5$	−14.3655	−1.28489	−0.642445	+	0.766332i	$0.722080\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−22.3332	−0.827155
$10$	0	0
$11$	−58.4700	−1.60267	−0.801335	−	0.598216i	$-0.795876\pi$
$11$	−58.4700	−1.60267	−0.801335	+	0.598216i	$0.795876\pi$
$12$	0	0
$13$	−16.4312	−0.350553	−0.175276	−	0.984519i	$-0.556082\pi$
$13$	−16.4312	−0.350553	−0.175276	+	0.984519i	$0.556082\pi$
$14$	0	0
$15$	−31.0335	−0.534188
$16$	0	0
$17$	−7.42853	−0.105981	−0.0529907	−	0.998595i	$-0.516875\pi$
$17$	−7.42853	−0.105981	−0.0529907	+	0.998595i	$0.516875\pi$
$18$	0	0
$19$	1.95888	0.0236525	0.0118263	−	0.999930i	$-0.496235\pi$
$19$	1.95888	0.0236525	0.0118263	+	0.999930i	$0.496235\pi$
$20$	0	0
$21$	15.1220	0.157137
$22$	0	0
$23$	4.12992	0.0374412	0.0187206	−	0.999825i	$-0.494041\pi$
$23$	4.12992	0.0374412	0.0187206	+	0.999825i	$0.494041\pi$
$24$	0	0
$25$	81.3679	0.650943
$26$	0	0
$27$	−106.574	−0.759633
$28$	0	0
$29$	−213.528	−1.36728	−0.683639	−	0.729820i	$-0.739604\pi$
$29$	−213.528	−1.36728	−0.683639	+	0.729820i	$0.739604\pi$
$30$	0	0
$31$	144.598	0.837760	0.418880	−	0.908042i	$-0.362423\pi$
$31$	144.598	0.837760	0.418880	+	0.908042i	$0.362423\pi$
$32$	0	0
$33$	−126.312	−0.666304
$34$	0	0
$35$	−100.559	−0.485643
$36$	0	0
$37$	267.155	1.18703	0.593514	−	0.804823i	$-0.297740\pi$
$37$	267.155	1.18703	0.593514	+	0.804823i	$0.297740\pi$
$38$	0	0
$39$	−35.4959	−0.145741
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	482.282	1.71040	0.855202	−	0.518295i	$-0.173433\pi$
$43$	482.282	1.71040	0.855202	+	0.518295i	$0.173433\pi$
$44$	0	0
$45$	320.828	1.06280
$46$	0	0
$47$	−292.866	−0.908913	−0.454456	−	0.890769i	$-0.650166\pi$
$47$	−292.866	−0.908913	−0.454456	+	0.890769i	$0.650166\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−16.0477	−0.0440613
$52$	0	0
$53$	174.859	0.453184	0.226592	−	0.973990i	$-0.427242\pi$
$53$	174.859	0.453184	0.226592	+	0.973990i	$0.427242\pi$
$54$	0	0
$55$	839.951	2.05925
$56$	0	0
$57$	4.23173	0.00983344
$58$	0	0
$59$	−48.2199	−0.106402	−0.0532008	−	0.998584i	$-0.516942\pi$
$59$	−48.2199	−0.106402	−0.0532008	+	0.998584i	$0.516942\pi$
$60$	0	0
$61$	−35.3366	−0.0741702	−0.0370851	−	0.999312i	$-0.511807\pi$
$61$	−35.3366	−0.0741702	−0.0370851	+	0.999312i	$0.511807\pi$
$62$	0	0
$63$	−156.332	−0.312635
$64$	0	0
$65$	236.042	0.450422
$66$	0	0
$67$	165.683	0.302110	0.151055	−	0.988525i	$-0.451733\pi$
$67$	165.683	0.302110	0.151055	+	0.988525i	$0.451733\pi$
$68$	0	0
$69$	8.92179	0.0155660
$70$	0	0
$71$	−190.733	−0.318814	−0.159407	−	0.987213i	$-0.550958\pi$
$71$	−190.733	−0.318814	−0.159407	+	0.987213i	$0.550958\pi$
$72$	0	0
$73$	215.024	0.344749	0.172375	−	0.985031i	$-0.444856\pi$
$73$	215.024	0.344749	0.172375	+	0.985031i	$0.444856\pi$
$74$	0	0
$75$	175.778	0.270627
$76$	0	0
$77$	−409.290	−0.605752
$78$	0	0
$79$	−522.220	−0.743726	−0.371863	−	0.928288i	$-0.621281\pi$
$79$	−522.220	−0.743726	−0.371863	+	0.928288i	$0.621281\pi$
$80$	0	0
$81$	372.767	0.511340
$82$	0	0
$83$	−435.012	−0.575287	−0.287643	−	0.957738i	$-0.592872\pi$
$83$	−435.012	−0.575287	−0.287643	+	0.957738i	$0.592872\pi$
$84$	0	0
$85$	106.715	0.136174
$86$	0	0
$87$	−461.280	−0.568441
$88$	0	0
$89$	1547.40	1.84297	0.921485	−	0.388414i	$-0.126977\pi$
$89$	1547.40	1.84297	0.921485	+	0.388414i	$0.126977\pi$
$90$	0	0
$91$	−115.018	−0.132496
$92$	0	0
$93$	312.372	0.348296
$94$	0	0
$95$	−28.1403	−0.0303909
$96$	0	0
$97$	−123.909	−0.129701	−0.0648507	−	0.997895i	$-0.520657\pi$
$97$	−123.909	−0.129701	−0.0648507	+	0.997895i	$0.520657\pi$
$98$	0	0
$99$	1305.82	1.32566
$100$	0	0
$101$	−1005.36	−0.990465	−0.495233	−	0.868760i	$-0.664917\pi$
$101$	−1005.36	−0.990465	−0.495233	+	0.868760i	$0.664917\pi$
$102$	0	0
$103$	−1513.78	−1.44812	−0.724062	−	0.689735i	$-0.757727\pi$
$103$	−1513.78	−1.44812	−0.724062	+	0.689735i	$0.757727\pi$
$104$	0	0
$105$	−217.235	−0.201904
$106$	0	0
$107$	2185.34	1.97444	0.987218	−	0.159373i	$-0.0509473\pi$
$107$	2185.34	1.97444	0.987218	+	0.159373i	$0.0509473\pi$
$108$	0	0
$109$	2178.77	1.91457	0.957285	−	0.289145i	$-0.0933709\pi$
$109$	2178.77	1.91457	0.957285	+	0.289145i	$0.0933709\pi$
$110$	0	0
$111$	577.131	0.493503
$112$	0	0
$113$	227.714	0.189571	0.0947855	−	0.995498i	$-0.469783\pi$
$113$	227.714	0.189571	0.0947855	+	0.995498i	$0.469783\pi$
$114$	0	0
$115$	−59.3284	−0.0481078
$116$	0	0
$117$	366.960	0.289961
$118$	0	0
$119$	−51.9997	−0.0400572
$120$	0	0
$121$	2087.74	1.56855
$122$	0	0
$123$	−88.5715	−0.0649287
$124$	0	0
$125$	626.798	0.448500
$126$	0	0
$127$	−515.646	−0.360285	−0.180143	−	0.983641i	$-0.557656\pi$
$127$	−515.646	−0.360285	−0.180143	+	0.983641i	$0.557656\pi$
$128$	0	0
$129$	1041.87	0.711094
$130$	0	0
$131$	106.863	0.0712721	0.0356360	−	0.999365i	$-0.488654\pi$
$131$	106.863	0.0712721	0.0356360	+	0.999365i	$0.488654\pi$
$132$	0	0
$133$	13.7122	0.00893981
$134$	0	0
$135$	1530.98	0.976045
$136$	0	0
$137$	−2734.07	−1.70501	−0.852507	−	0.522715i	$-0.824919\pi$
$137$	−2734.07	−1.70501	−0.852507	+	0.522715i	$0.824919\pi$
$138$	0	0
$139$	−1065.27	−0.650036	−0.325018	−	0.945708i	$-0.605370\pi$
$139$	−1065.27	−0.650036	−0.325018	+	0.945708i	$0.605370\pi$
$140$	0	0
$141$	−632.673	−0.377877
$142$	0	0
$143$	960.729	0.561820
$144$	0	0
$145$	3067.43	1.75680
$146$	0	0
$147$	105.854	0.0593923
$148$	0	0
$149$	2858.57	1.57170	0.785850	−	0.618418i	$-0.212226\pi$
$149$	2858.57	1.57170	0.785850	+	0.618418i	$0.212226\pi$
$150$	0	0
$151$	−2165.81	−1.16723	−0.583614	−	0.812031i	$-0.698362\pi$
$151$	−2165.81	−1.16723	−0.583614	+	0.812031i	$0.698362\pi$
$152$	0	0
$153$	165.903	0.0876630
$154$	0	0
$155$	−2077.22	−1.07643
$156$	0	0
$157$	2633.94	1.33893	0.669463	−	0.742846i	$-0.266524\pi$
$157$	2633.94	1.33893	0.669463	+	0.742846i	$0.266524\pi$
$158$	0	0
$159$	377.745	0.188410
$160$	0	0
$161$	28.9094	0.0141514
$162$	0	0
$163$	637.326	0.306253	0.153126	−	0.988207i	$-0.451066\pi$
$163$	637.326	0.306253	0.153126	+	0.988207i	$0.451066\pi$
$164$	0	0
$165$	1814.53	0.856127
$166$	0	0
$167$	−731.389	−0.338902	−0.169451	−	0.985539i	$-0.554199\pi$
$167$	−731.389	−0.338902	−0.169451	+	0.985539i	$0.554199\pi$
$168$	0	0
$169$	−1927.02	−0.877113
$170$	0	0
$171$	−43.7480	−0.0195643
$172$	0	0
$173$	472.476	0.207640	0.103820	−	0.994596i	$-0.466893\pi$
$173$	472.476	0.207640	0.103820	+	0.994596i	$0.466893\pi$
$174$	0	0
$175$	569.575	0.246033
$176$	0	0
$177$	−104.168	−0.0442361
$178$	0	0
$179$	477.175	0.199250	0.0996250	−	0.995025i	$-0.468236\pi$
$179$	477.175	0.199250	0.0996250	+	0.995025i	$0.468236\pi$
$180$	0	0
$181$	3432.40	1.40955	0.704774	−	0.709432i	$-0.251048\pi$
$181$	3432.40	1.40955	0.704774	+	0.709432i	$0.251048\pi$
$182$	0	0
$183$	−76.3369	−0.0308360
$184$	0	0
$185$	−3837.82	−1.52520
$186$	0	0
$187$	434.346	0.169853
$188$	0	0
$189$	−746.015	−0.287114
$190$	0	0
$191$	1015.05	0.384536	0.192268	−	0.981342i	$-0.438416\pi$
$191$	1015.05	0.384536	0.192268	+	0.981342i	$0.438416\pi$
$192$	0	0
$193$	408.093	0.152203	0.0761015	−	0.997100i	$-0.475753\pi$
$193$	408.093	0.152203	0.0761015	+	0.997100i	$0.475753\pi$
$194$	0	0
$195$	509.917	0.187261
$196$	0	0
$197$	−1338.88	−0.484220	−0.242110	−	0.970249i	$-0.577839\pi$
$197$	−1338.88	−0.484220	−0.242110	+	0.970249i	$0.577839\pi$
$198$	0	0
$199$	3629.46	1.29289	0.646447	−	0.762959i	$-0.276254\pi$
$199$	3629.46	1.29289	0.646447	+	0.762959i	$0.276254\pi$
$200$	0	0
$201$	357.921	0.125601
$202$	0	0
$203$	−1494.69	−0.516783
$204$	0	0
$205$	588.986	0.200666
$206$	0	0
$207$	−92.2342	−0.0309697
$208$	0	0
$209$	−114.536	−0.0379071
$210$	0	0
$211$	−1584.91	−0.517109	−0.258555	−	0.965997i	$-0.583246\pi$
$211$	−1584.91	−0.517109	−0.258555	+	0.965997i	$0.583246\pi$
$212$	0	0
$213$	−412.037	−0.132546
$214$	0	0
$215$	−6928.23	−2.19768
$216$	0	0
$217$	1012.19	0.316644
$218$	0	0
$219$	464.513	0.143328
$220$	0	0
$221$	122.059	0.0371520
$222$	0	0
$223$	−1453.89	−0.436590	−0.218295	−	0.975883i	$-0.570049\pi$
$223$	−1453.89	−0.436590	−0.218295	+	0.975883i	$0.570049\pi$
$224$	0	0
$225$	−1817.20	−0.538431
$226$	0	0
$227$	4808.71	1.40601	0.703007	−	0.711183i	$-0.251840\pi$
$227$	4808.71	1.40601	0.703007	+	0.711183i	$0.251840\pi$
$228$	0	0
$229$	−5592.66	−1.61386	−0.806929	−	0.590648i	$-0.798872\pi$
$229$	−5592.66	−1.61386	−0.806929	+	0.590648i	$0.798872\pi$
$230$	0	0
$231$	−884.181	−0.251839
$232$	0	0
$233$	−3409.16	−0.958548	−0.479274	−	0.877665i	$-0.659100\pi$
$233$	−3409.16	−0.958548	−0.479274	+	0.877665i	$0.659100\pi$
$234$	0	0
$235$	4207.17	1.16785
$236$	0	0
$237$	−1128.14	−0.309201
$238$	0	0
$239$	175.297	0.0474437	0.0237218	−	0.999719i	$-0.492448\pi$
$239$	175.297	0.0474437	0.0237218	+	0.999719i	$0.492448\pi$
$240$	0	0
$241$	2844.19	0.760209	0.380104	−	0.924944i	$-0.375888\pi$
$241$	2844.19	0.760209	0.380104	+	0.924944i	$0.375888\pi$
$242$	0	0
$243$	3682.77	0.972221
$244$	0	0
$245$	−703.910	−0.183556
$246$	0	0
$247$	−32.1866	−0.00829145
$248$	0	0
$249$	−939.749	−0.239173
$250$	0	0
$251$	−3209.86	−0.807190	−0.403595	−	0.914938i	$-0.632240\pi$
$251$	−3209.86	−0.807190	−0.403595	+	0.914938i	$0.632240\pi$
$252$	0	0
$253$	−241.476	−0.0600059
$254$	0	0
$255$	230.533	0.0566140
$256$	0	0
$257$	4037.57	0.979988	0.489994	−	0.871726i	$-0.336999\pi$
$257$	4037.57	0.979988	0.489994	+	0.871726i	$0.336999\pi$
$258$	0	0
$259$	1870.09	0.448655
$260$	0	0
$261$	4768.75	1.13095
$262$	0	0
$263$	−4192.52	−0.982973	−0.491486	−	0.870885i	$-0.663546\pi$
$263$	−4192.52	−0.982973	−0.491486	+	0.870885i	$0.663546\pi$
$264$	0	0
$265$	−2511.94	−0.582292
$266$	0	0
$267$	3342.82	0.766208
$268$	0	0
$269$	836.173	0.189526	0.0947628	−	0.995500i	$-0.469791\pi$
$269$	836.173	0.189526	0.0947628	+	0.995500i	$0.469791\pi$
$270$	0	0
$271$	6398.98	1.43435	0.717177	−	0.696891i	$-0.245434\pi$
$271$	6398.98	1.43435	0.717177	+	0.696891i	$0.245434\pi$
$272$	0	0
$273$	−248.471	−0.0550849
$274$	0	0
$275$	−4757.58	−1.04325
$276$	0	0
$277$	−2162.97	−0.469170	−0.234585	−	0.972096i	$-0.575373\pi$
$277$	−2162.97	−0.469170	−0.234585	+	0.972096i	$0.575373\pi$
$278$	0	0
$279$	−3229.33	−0.692957
$280$	0	0
$281$	−2300.57	−0.488400	−0.244200	−	0.969725i	$-0.578525\pi$
$281$	−2300.57	−0.488400	−0.244200	+	0.969725i	$0.578525\pi$
$282$	0	0
$283$	7740.88	1.62596	0.812981	−	0.582290i	$-0.197843\pi$
$283$	7740.88	1.62596	0.812981	+	0.582290i	$0.197843\pi$
$284$	0	0
$285$	−60.7910	−0.0126349
$286$	0	0
$287$	−287.000	−0.0590281
$288$	0	0
$289$	−4857.82	−0.988768
$290$	0	0
$291$	−267.678	−0.0539229
$292$	0	0
$293$	3182.00	0.634452	0.317226	−	0.948350i	$-0.397249\pi$
$293$	3182.00	0.634452	0.317226	+	0.948350i	$0.397249\pi$
$294$	0	0
$295$	692.703	0.136714
$296$	0	0
$297$	6231.35	1.21744
$298$	0	0
$299$	−67.8593	−0.0131251
$300$	0	0
$301$	3375.98	0.646472
$302$	0	0
$303$	−2171.86	−0.411782
$304$	0	0
$305$	507.628	0.0953006
$306$	0	0
$307$	−6194.26	−1.15155	−0.575774	−	0.817609i	$-0.695299\pi$
$307$	−6194.26	−1.15155	−0.575774	+	0.817609i	$0.695299\pi$
$308$	0	0
$309$	−3270.18	−0.602052
$310$	0	0
$311$	7406.31	1.35040	0.675199	−	0.737636i	$-0.264058\pi$
$311$	7406.31	1.35040	0.675199	+	0.737636i	$0.264058\pi$
$312$	0	0
$313$	10294.4	1.85902	0.929509	−	0.368799i	$-0.120231\pi$
$313$	10294.4	1.85902	0.929509	+	0.368799i	$0.120231\pi$
$314$	0	0
$315$	2245.79	0.401702
$316$	0	0
$317$	5232.73	0.927127	0.463563	−	0.886064i	$-0.346571\pi$
$317$	5232.73	0.927127	0.463563	+	0.886064i	$0.346571\pi$
$318$	0	0
$319$	12485.0	2.19130
$320$	0	0
$321$	4720.95	0.820865
$322$	0	0
$323$	−14.5516	−0.00250672
$324$	0	0
$325$	−1336.97	−0.228190
$326$	0	0
$327$	4706.75	0.795976
$328$	0	0
$329$	−2050.06	−0.343537
$330$	0	0
$331$	10308.4	1.71179	0.855894	−	0.517151i	$-0.173007\pi$
$331$	10308.4	1.71179	0.855894	+	0.517151i	$0.173007\pi$
$332$	0	0
$333$	−5966.43	−0.981857
$334$	0	0
$335$	−2380.12	−0.388178
$336$	0	0
$337$	−5862.56	−0.947637	−0.473819	−	0.880622i	$-0.657125\pi$
$337$	−5862.56	−0.947637	−0.473819	+	0.880622i	$0.657125\pi$
$338$	0	0
$339$	491.926	0.0788134
$340$	0	0
$341$	−8454.64	−1.34265
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−128.166	−0.0200007
$346$	0	0
$347$	2531.06	0.391569	0.195784	−	0.980647i	$-0.437275\pi$
$347$	2531.06	0.391569	0.195784	+	0.980647i	$0.437275\pi$
$348$	0	0
$349$	5178.95	0.794334	0.397167	−	0.917746i	$-0.369993\pi$
$349$	5178.95	0.794334	0.397167	+	0.917746i	$0.369993\pi$
$350$	0	0
$351$	1751.13	0.266291
$352$	0	0
$353$	−8868.65	−1.33720	−0.668599	−	0.743623i	$-0.733106\pi$
$353$	−8868.65	−1.33720	−0.668599	+	0.743623i	$0.733106\pi$
$354$	0	0
$355$	2739.97	0.409641
$356$	0	0
$357$	−112.334	−0.0166536
$358$	0	0
$359$	3595.62	0.528606	0.264303	−	0.964440i	$-0.414858\pi$
$359$	3595.62	0.528606	0.264303	+	0.964440i	$0.414858\pi$
$360$	0	0
$361$	−6855.16	−0.999441
$362$	0	0
$363$	4510.10	0.652118
$364$	0	0
$365$	−3088.94	−0.442965
$366$	0	0
$367$	−7881.15	−1.12096	−0.560480	−	0.828168i	$-0.689383\pi$
$367$	−7881.15	−1.12096	−0.560480	+	0.828168i	$0.689383\pi$
$368$	0	0
$369$	915.661	0.129180
$370$	0	0
$371$	1224.01	0.171288
$372$	0	0
$373$	−6307.42	−0.875566	−0.437783	−	0.899081i	$-0.644236\pi$
$373$	−6307.42	−0.875566	−0.437783	+	0.899081i	$0.644236\pi$
$374$	0	0
$375$	1354.06	0.186462
$376$	0	0
$377$	3508.51	0.479303
$378$	0	0
$379$	−3452.77	−0.467960	−0.233980	−	0.972241i	$-0.575175\pi$
$379$	−3452.77	−0.467960	−0.233980	+	0.972241i	$0.575175\pi$
$380$	0	0
$381$	−1113.94	−0.149787
$382$	0	0
$383$	6820.43	0.909942	0.454971	−	0.890506i	$-0.349650\pi$
$383$	6820.43	0.909942	0.454971	+	0.890506i	$0.349650\pi$
$384$	0	0
$385$	5879.66	0.778325
$386$	0	0
$387$	−10770.9	−1.41477
$388$	0	0
$389$	−551.416	−0.0718712	−0.0359356	−	0.999354i	$-0.511441\pi$
$389$	−551.416	−0.0718712	−0.0359356	+	0.999354i	$0.511441\pi$
$390$	0	0
$391$	−30.6792	−0.00396807
$392$	0	0
$393$	230.854	0.0296311
$394$	0	0
$395$	7501.95	0.955606
$396$	0	0
$397$	−3312.81	−0.418804	−0.209402	−	0.977830i	$-0.567152\pi$
$397$	−3312.81	−0.418804	−0.209402	+	0.977830i	$0.567152\pi$
$398$	0	0
$399$	29.6221	0.00371669
$400$	0	0
$401$	1512.07	0.188303	0.0941513	−	0.995558i	$-0.469986\pi$
$401$	1512.07	0.188303	0.0941513	+	0.995558i	$0.469986\pi$
$402$	0	0
$403$	−2375.91	−0.293679
$404$	0	0
$405$	−5354.99	−0.657016
$406$	0	0
$407$	−15620.6	−1.90241
$408$	0	0
$409$	2613.76	0.315995	0.157998	−	0.987439i	$-0.449496\pi$
$409$	2613.76	0.315995	0.157998	+	0.987439i	$0.449496\pi$
$410$	0	0
$411$	−5906.35	−0.708854
$412$	0	0
$413$	−337.539	−0.0402160
$414$	0	0
$415$	6249.17	0.739180
$416$	0	0
$417$	−2301.28	−0.270250
$418$	0	0
$419$	650.467	0.0758411	0.0379205	−	0.999281i	$-0.487927\pi$
$419$	650.467	0.0758411	0.0379205	+	0.999281i	$0.487927\pi$
$420$	0	0
$421$	−7035.78	−0.814496	−0.407248	−	0.913318i	$-0.633512\pi$
$421$	−7035.78	−0.814496	−0.407248	+	0.913318i	$0.633512\pi$
$422$	0	0
$423$	6540.63	0.751812
$424$	0	0
$425$	−604.443	−0.0689878
$426$	0	0
$427$	−247.356	−0.0280337
$428$	0	0
$429$	2075.45	0.233575
$430$	0	0
$431$	10531.0	1.17694	0.588470	−	0.808519i	$-0.299731\pi$
$431$	10531.0	1.17694	0.588470	+	0.808519i	$0.299731\pi$
$432$	0	0
$433$	−8743.38	−0.970393	−0.485196	−	0.874405i	$-0.661252\pi$
$433$	−8743.38	−0.970393	−0.485196	+	0.874405i	$0.661252\pi$
$434$	0	0
$435$	6626.52	0.730385
$436$	0	0
$437$	8.09001	0.000885578 0
$438$	0	0
$439$	3124.18	0.339656	0.169828	−	0.985474i	$-0.445679\pi$
$439$	3124.18	0.339656	0.169828	+	0.985474i	$0.445679\pi$
$440$	0	0
$441$	−1094.33	−0.118165
$442$	0	0
$443$	−4524.40	−0.485239	−0.242619	−	0.970122i	$-0.578007\pi$
$443$	−4524.40	−0.485239	−0.242619	+	0.970122i	$0.578007\pi$
$444$	0	0
$445$	−22229.2	−2.36801
$446$	0	0
$447$	6175.32	0.653428
$448$	0	0
$449$	−15024.1	−1.57913	−0.789567	−	0.613664i	$-0.789695\pi$
$449$	−15024.1	−1.57913	−0.789567	+	0.613664i	$0.789695\pi$
$450$	0	0
$451$	2397.27	0.250295
$452$	0	0
$453$	−4678.77	−0.485271
$454$	0	0
$455$	1652.29	0.170243
$456$	0	0
$457$	−12949.4	−1.32548	−0.662742	−	0.748848i	$-0.730607\pi$
$457$	−12949.4	−1.32548	−0.662742	+	0.748848i	$0.730607\pi$
$458$	0	0
$459$	791.684	0.0805069
$460$	0	0
$461$	4898.82	0.494926	0.247463	−	0.968897i	$-0.420403\pi$
$461$	4898.82	0.494926	0.247463	+	0.968897i	$0.420403\pi$
$462$	0	0
$463$	−3251.66	−0.326388	−0.163194	−	0.986594i	$-0.552180\pi$
$463$	−3251.66	−0.326388	−0.163194	+	0.986594i	$0.552180\pi$
$464$	0	0
$465$	−4487.39	−0.447522
$466$	0	0
$467$	−17400.1	−1.72416	−0.862079	−	0.506773i	$-0.830838\pi$
$467$	−17400.1	−1.72416	−0.862079	+	0.506773i	$0.830838\pi$
$468$	0	0
$469$	1159.78	0.114187
$470$	0	0
$471$	5690.05	0.556653
$472$	0	0
$473$	−28199.0	−2.74121
$474$	0	0
$475$	159.390	0.0153964
$476$	0	0
$477$	−3905.16	−0.374854
$478$	0	0
$479$	−15859.0	−1.51277	−0.756386	−	0.654126i	$-0.773037\pi$
$479$	−15859.0	−1.51277	−0.756386	+	0.654126i	$0.773037\pi$
$480$	0	0
$481$	−4389.67	−0.416116
$482$	0	0
$483$	62.4525	0.00588341
$484$	0	0
$485$	1780.01	0.166652
$486$	0	0
$487$	−10360.1	−0.963982	−0.481991	−	0.876176i	$-0.660086\pi$
$487$	−10360.1	−0.963982	−0.481991	+	0.876176i	$0.660086\pi$
$488$	0	0
$489$	1376.80	0.127324
$490$	0	0
$491$	3872.05	0.355892	0.177946	−	0.984040i	$-0.443055\pi$
$491$	3872.05	0.355892	0.177946	+	0.984040i	$0.443055\pi$
$492$	0	0
$493$	1586.20	0.144906
$494$	0	0
$495$	−18758.8	−1.70332
$496$	0	0
$497$	−1335.13	−0.120500
$498$	0	0
$499$	−5117.79	−0.459126	−0.229563	−	0.973294i	$-0.573730\pi$
$499$	−5117.79	−0.459126	−0.229563	+	0.973294i	$0.573730\pi$
$500$	0	0
$501$	−1580.01	−0.140897
$502$	0	0
$503$	510.164	0.0452228	0.0226114	−	0.999744i	$-0.492802\pi$
$503$	510.164	0.0452228	0.0226114	+	0.999744i	$0.492802\pi$
$504$	0	0
$505$	14442.5	1.27264
$506$	0	0
$507$	−4162.90	−0.364657
$508$	0	0
$509$	4652.17	0.405115	0.202558	−	0.979270i	$-0.435075\pi$
$509$	4652.17	0.405115	0.202558	+	0.979270i	$0.435075\pi$
$510$	0	0
$511$	1505.17	0.130303
$512$	0	0
$513$	−208.765	−0.0179672
$514$	0	0
$515$	21746.2	1.86068
$516$	0	0
$517$	17123.9	1.45669
$518$	0	0
$519$	1020.68	0.0863254
$520$	0	0
$521$	−11889.3	−0.999767	−0.499884	−	0.866093i	$-0.666624\pi$
$521$	−11889.3	−0.999767	−0.499884	+	0.866093i	$0.666624\pi$
$522$	0	0
$523$	4248.18	0.355182	0.177591	−	0.984104i	$-0.443170\pi$
$523$	4248.18	0.355182	0.177591	+	0.984104i	$0.443170\pi$
$524$	0	0
$525$	1230.44	0.102287
$526$	0	0
$527$	−1074.15	−0.0887869
$528$	0	0
$529$	−12149.9	−0.998598
$530$	0	0
$531$	1076.90	0.0880106
$532$	0	0
$533$	673.677	0.0547471
$534$	0	0
$535$	−31393.5	−2.53693
$536$	0	0
$537$	1030.83	0.0828375
$538$	0	0
$539$	−2865.03	−0.228953
$540$	0	0
$541$	11545.2	0.917497	0.458749	−	0.888566i	$-0.348298\pi$
$541$	11545.2	0.917497	0.458749	+	0.888566i	$0.348298\pi$
$542$	0	0
$543$	7414.95	0.586015
$544$	0	0
$545$	−31299.1	−2.46001
$546$	0	0
$547$	−3574.35	−0.279393	−0.139697	−	0.990194i	$-0.544613\pi$
$547$	−3574.35	−0.279393	−0.139697	+	0.990194i	$0.544613\pi$
$548$	0	0
$549$	789.178	0.0613503
$550$	0	0
$551$	−418.275	−0.0323396
$552$	0	0
$553$	−3655.54	−0.281102
$554$	0	0
$555$	−8290.78	−0.634097
$556$	0	0
$557$	16758.9	1.27486	0.637429	−	0.770509i	$-0.279998\pi$
$557$	16758.9	1.27486	0.637429	+	0.770509i	$0.279998\pi$
$558$	0	0
$559$	−7924.46	−0.599586
$560$	0	0
$561$	938.309	0.0706158
$562$	0	0
$563$	13916.8	1.04178	0.520890	−	0.853624i	$-0.325600\pi$
$563$	13916.8	1.04178	0.520890	+	0.853624i	$0.325600\pi$
$564$	0	0
$565$	−3271.23	−0.243578
$566$	0	0
$567$	2609.37	0.193268
$568$	0	0
$569$	14108.5	1.03947	0.519736	−	0.854327i	$-0.326030\pi$
$569$	14108.5	1.03947	0.519736	+	0.854327i	$0.326030\pi$
$570$	0	0
$571$	−9539.21	−0.699130	−0.349565	−	0.936912i	$-0.613671\pi$
$571$	−9539.21	−0.699130	−0.349565	+	0.936912i	$0.613671\pi$
$572$	0	0
$573$	2192.79	0.159869
$574$	0	0
$575$	336.043	0.0243721
$576$	0	0
$577$	6249.14	0.450875	0.225438	−	0.974258i	$-0.427619\pi$
$577$	6249.14	0.450875	0.225438	+	0.974258i	$0.427619\pi$
$578$	0	0
$579$	881.596	0.0632779
$580$	0	0
$581$	−3045.09	−0.217438
$582$	0	0
$583$	−10224.0	−0.726304
$584$	0	0
$585$	−5271.57	−0.372568
$586$	0	0
$587$	−5435.92	−0.382222	−0.191111	−	0.981568i	$-0.561209\pi$
$587$	−5435.92	−0.382222	−0.191111	+	0.981568i	$0.561209\pi$
$588$	0	0
$589$	283.250	0.0198151
$590$	0	0
$591$	−2892.36	−0.201313
$592$	0	0
$593$	−22648.6	−1.56841	−0.784204	−	0.620503i	$-0.786929\pi$
$593$	−22648.6	−1.56841	−0.784204	+	0.620503i	$0.786929\pi$
$594$	0	0
$595$	747.002	0.0514691
$596$	0	0
$597$	7840.66	0.537516
$598$	0	0
$599$	16796.2	1.14570	0.572851	−	0.819660i	$-0.305838\pi$
$599$	16796.2	1.14570	0.572851	+	0.819660i	$0.305838\pi$
$600$	0	0
$601$	20835.4	1.41413	0.707067	−	0.707146i	$-0.250018\pi$
$601$	20835.4	1.41413	0.707067	+	0.707146i	$0.250018\pi$
$602$	0	0
$603$	−3700.22	−0.249892
$604$	0	0
$605$	−29991.4	−2.01541
$606$	0	0
$607$	−4557.85	−0.304773	−0.152387	−	0.988321i	$-0.548696\pi$
$607$	−4557.85	−0.304773	−0.152387	+	0.988321i	$0.548696\pi$
$608$	0	0
$609$	−3228.96	−0.214851
$610$	0	0
$611$	4812.13	0.318622
$612$	0	0
$613$	8975.72	0.591396	0.295698	−	0.955281i	$-0.404448\pi$
$613$	8975.72	0.591396	0.295698	+	0.955281i	$0.404448\pi$
$614$	0	0
$615$	1272.38	0.0834262
$616$	0	0
$617$	−14842.3	−0.968439	−0.484219	−	0.874947i	$-0.660896\pi$
$617$	−14842.3	−0.968439	−0.484219	+	0.874947i	$0.660896\pi$
$618$	0	0
$619$	15890.1	1.03179	0.515896	−	0.856651i	$-0.327459\pi$
$619$	15890.1	1.03179	0.515896	+	0.856651i	$0.327459\pi$
$620$	0	0
$621$	−440.140	−0.0284416
$622$	0	0
$623$	10831.8	0.696577
$624$	0	0
$625$	−19175.3	−1.22722
$626$	0	0
$627$	−247.429	−0.0157598
$628$	0	0
$629$	−1984.57	−0.125803
$630$	0	0
$631$	24875.0	1.56935	0.784674	−	0.619909i	$-0.212831\pi$
$631$	24875.0	1.56935	0.784674	+	0.619909i	$0.212831\pi$
$632$	0	0
$633$	−3423.86	−0.214986
$634$	0	0
$635$	7407.52	0.462927
$636$	0	0
$637$	−805.127	−0.0500789
$638$	0	0
$639$	4259.67	0.263709
$640$	0	0
$641$	25423.1	1.56654	0.783271	−	0.621680i	$-0.213550\pi$
$641$	25423.1	1.56654	0.783271	+	0.621680i	$0.213550\pi$
$642$	0	0
$643$	10255.4	0.628980	0.314490	−	0.949261i	$-0.398166\pi$
$643$	10255.4	0.628980	0.314490	+	0.949261i	$0.398166\pi$
$644$	0	0
$645$	−14966.9	−0.913678
$646$	0	0
$647$	−17194.4	−1.04479	−0.522396	−	0.852703i	$-0.674962\pi$
$647$	−17194.4	−1.04479	−0.522396	+	0.852703i	$0.674962\pi$
$648$	0	0
$649$	2819.41	0.170526
$650$	0	0
$651$	2186.61	0.131643
$652$	0	0
$653$	10100.8	0.605322	0.302661	−	0.953098i	$-0.402125\pi$
$653$	10100.8	0.605322	0.302661	+	0.953098i	$0.402125\pi$
$654$	0	0
$655$	−1535.14	−0.0915768
$656$	0	0
$657$	−4802.18	−0.285161
$658$	0	0
$659$	−2177.99	−0.128744	−0.0643721	−	0.997926i	$-0.520504\pi$
$659$	−2177.99	−0.128744	−0.0643721	+	0.997926i	$0.520504\pi$
$660$	0	0
$661$	−16231.3	−0.955105	−0.477553	−	0.878603i	$-0.658476\pi$
$661$	−16231.3	−0.955105	−0.477553	+	0.878603i	$0.658476\pi$
$662$	0	0
$663$	263.682	0.0154458
$664$	0	0
$665$	−196.982	−0.0114867
$666$	0	0
$667$	−881.852	−0.0511926
$668$	0	0
$669$	−3140.81	−0.181511
$670$	0	0
$671$	2066.13	0.118870
$672$	0	0
$673$	−31814.6	−1.82223	−0.911117	−	0.412147i	$-0.864779\pi$
$673$	−31814.6	−1.82223	−0.911117	+	0.412147i	$0.864779\pi$
$674$	0	0
$675$	−8671.66	−0.494478
$676$	0	0
$677$	24450.9	1.38807	0.694036	−	0.719940i	$-0.255831\pi$
$677$	24450.9	1.38807	0.694036	+	0.719940i	$0.255831\pi$
$678$	0	0
$679$	−867.362	−0.0490225
$680$	0	0
$681$	10388.2	0.584545
$682$	0	0
$683$	−11603.0	−0.650041	−0.325021	−	0.945707i	$-0.605371\pi$
$683$	−11603.0	−0.650041	−0.325021	+	0.945707i	$0.605371\pi$
$684$	0	0
$685$	39276.2	2.19076
$686$	0	0
$687$	−12081.7	−0.670956
$688$	0	0
$689$	−2873.14	−0.158865
$690$	0	0
$691$	−21733.1	−1.19648	−0.598238	−	0.801319i	$-0.704132\pi$
$691$	−21733.1	−1.19648	−0.598238	+	0.801319i	$0.704132\pi$
$692$	0	0
$693$	9140.74	0.501051
$694$	0	0
$695$	15303.1	0.835225
$696$	0	0
$697$	304.570	0.0165515
$698$	0	0
$699$	−7364.75	−0.398513
$700$	0	0
$701$	−14069.6	−0.758062	−0.379031	−	0.925384i	$-0.623743\pi$
$701$	−14069.6	−0.758062	−0.379031	+	0.925384i	$0.623743\pi$
$702$	0	0
$703$	523.325	0.0280762
$704$	0	0
$705$	9088.67	0.485531
$706$	0	0
$707$	−7037.52	−0.374361
$708$	0	0
$709$	10459.1	0.554022	0.277011	−	0.960867i	$-0.410656\pi$
$709$	10459.1	0.554022	0.277011	+	0.960867i	$0.410656\pi$
$710$	0	0
$711$	11662.8	0.615176
$712$	0	0
$713$	597.178	0.0313667
$714$	0	0
$715$	−13801.4	−0.721877
$716$	0	0
$717$	378.691	0.0197245
$718$	0	0
$719$	14102.2	0.731466	0.365733	−	0.930720i	$-0.380818\pi$
$719$	14102.2	0.731466	0.365733	+	0.930720i	$0.380818\pi$
$720$	0	0
$721$	−10596.4	−0.547339
$722$	0	0
$723$	6144.25	0.316054
$724$	0	0
$725$	−17374.3	−0.890021
$726$	0	0
$727$	37554.5	1.91584	0.957922	−	0.287030i	$-0.0926679\pi$
$727$	37554.5	1.91584	0.957922	+	0.287030i	$0.0926679\pi$
$728$	0	0
$729$	−2108.89	−0.107143
$730$	0	0
$731$	−3582.65	−0.181271
$732$	0	0
$733$	12554.1	0.632602	0.316301	−	0.948659i	$-0.397559\pi$
$733$	12554.1	0.632602	0.316301	+	0.948659i	$0.397559\pi$
$734$	0	0
$735$	−1520.64	−0.0763126
$736$	0	0
$737$	−9687.46	−0.484182
$738$	0	0
$739$	24498.0	1.21945	0.609724	−	0.792614i	$-0.291280\pi$
$739$	24498.0	1.21945	0.609724	+	0.792614i	$0.291280\pi$
$740$	0	0
$741$	−69.5322	−0.00344714
$742$	0	0
$743$	1463.79	0.0722764	0.0361382	−	0.999347i	$-0.488494\pi$
$743$	1463.79	0.0722764	0.0361382	+	0.999347i	$0.488494\pi$
$744$	0	0
$745$	−41064.8	−2.01946
$746$	0	0
$747$	9715.21	0.475851
$748$	0	0
$749$	15297.4	0.746267
$750$	0	0
$751$	25856.4	1.25634	0.628172	−	0.778075i	$-0.283803\pi$
$751$	25856.4	1.25634	0.628172	+	0.778075i	$0.283803\pi$
$752$	0	0
$753$	−6934.21	−0.335586
$754$	0	0
$755$	31113.0	1.49976
$756$	0	0
$757$	24176.5	1.16078	0.580389	−	0.814339i	$-0.302900\pi$
$757$	24176.5	1.16078	0.580389	+	0.814339i	$0.302900\pi$
$758$	0	0
$759$	−521.657	−0.0249472
$760$	0	0
$761$	20105.5	0.957720	0.478860	−	0.877891i	$-0.341050\pi$
$761$	20105.5	0.957720	0.478860	+	0.877891i	$0.341050\pi$
$762$	0	0
$763$	15251.4	0.723640
$764$	0	0
$765$	−2383.28	−0.112637
$766$	0	0
$767$	792.308	0.0372993
$768$	0	0
$769$	10724.4	0.502904	0.251452	−	0.967870i	$-0.419092\pi$
$769$	10724.4	0.502904	0.251452	+	0.967870i	$0.419092\pi$
$770$	0	0
$771$	8722.30	0.407427
$772$	0	0
$773$	−28774.3	−1.33886	−0.669430	−	0.742875i	$-0.733462\pi$
$773$	−28774.3	−1.33886	−0.669430	+	0.742875i	$0.733462\pi$
$774$	0	0
$775$	11765.6	0.545334
$776$	0	0
$777$	4039.92	0.186527
$778$	0	0
$779$	−80.3140	−0.00369390
$780$	0	0
$781$	11152.1	0.510954
$782$	0	0
$783$	22756.4	1.03863
$784$	0	0
$785$	−37837.9	−1.72037
$786$	0	0
$787$	5866.93	0.265735	0.132867	−	0.991134i	$-0.457582\pi$
$787$	5866.93	0.265735	0.132867	+	0.991134i	$0.457582\pi$
$788$	0	0
$789$	−9057.02	−0.408667
$790$	0	0
$791$	1594.00	0.0716511
$792$	0	0
$793$	580.621	0.0260006
$794$	0	0
$795$	−5426.50	−0.242086
$796$	0	0
$797$	8760.86	0.389367	0.194684	−	0.980866i	$-0.437632\pi$
$797$	8760.86	0.389367	0.194684	+	0.980866i	$0.437632\pi$
$798$	0	0
$799$	2175.56	0.0963277
$800$	0	0
$801$	−34558.4	−1.52442
$802$	0	0
$803$	−12572.5	−0.552519
$804$	0	0
$805$	−415.299	−0.0181831
$806$	0	0
$807$	1806.37	0.0787945
$808$	0	0
$809$	18318.4	0.796096	0.398048	−	0.917365i	$-0.369688\pi$
$809$	18318.4	0.796096	0.398048	+	0.917365i	$0.369688\pi$
$810$	0	0
$811$	−28402.2	−1.22976	−0.614880	−	0.788621i	$-0.710796\pi$
$811$	−28402.2	−1.22976	−0.614880	+	0.788621i	$0.710796\pi$
$812$	0	0
$813$	13823.6	0.596328
$814$	0	0
$815$	−9155.51	−0.393501
$816$	0	0
$817$	944.733	0.0404553
$818$	0	0
$819$	2568.72	0.109595
$820$	0	0
$821$	−46061.8	−1.95806	−0.979030	−	0.203716i	$-0.934698\pi$
$821$	−46061.8	−1.95806	−0.979030	+	0.203716i	$0.934698\pi$
$822$	0	0
$823$	−28822.5	−1.22077	−0.610383	−	0.792107i	$-0.708984\pi$
$823$	−28822.5	−1.22077	−0.610383	+	0.792107i	$0.708984\pi$
$824$	0	0
$825$	−10277.7	−0.433726
$826$	0	0
$827$	−25691.0	−1.08025	−0.540123	−	0.841586i	$-0.681622\pi$
$827$	−25691.0	−1.08025	−0.540123	+	0.841586i	$0.681622\pi$
$828$	0	0
$829$	−12170.1	−0.509874	−0.254937	−	0.966958i	$-0.582055\pi$
$829$	−12170.1	−0.509874	−0.254937	+	0.966958i	$0.582055\pi$
$830$	0	0
$831$	−4672.62	−0.195056
$832$	0	0
$833$	−363.998	−0.0151402
$834$	0	0
$835$	10506.8	0.435452
$836$	0	0
$837$	−15410.3	−0.636390
$838$	0	0
$839$	7617.28	0.313442	0.156721	−	0.987643i	$-0.449908\pi$
$839$	7617.28	0.313442	0.156721	+	0.987643i	$0.449908\pi$
$840$	0	0
$841$	21205.1	0.869452
$842$	0	0
$843$	−4969.88	−0.203051
$844$	0	0
$845$	27682.6	1.12699
$846$	0	0
$847$	14614.2	0.592856
$848$	0	0
$849$	16722.5	0.675988
$850$	0	0
$851$	1103.33	0.0444438
$852$	0	0
$853$	−9969.56	−0.400177	−0.200089	−	0.979778i	$-0.564123\pi$
$853$	−9969.56	−0.400177	−0.200089	+	0.979778i	$0.564123\pi$
$854$	0	0
$855$	628.462	0.0251380
$856$	0	0
$857$	−20967.1	−0.835733	−0.417867	−	0.908508i	$-0.637222\pi$
$857$	−20967.1	−0.835733	−0.417867	+	0.908508i	$0.637222\pi$
$858$	0	0
$859$	27826.4	1.10527	0.552634	−	0.833424i	$-0.313623\pi$
$859$	27826.4	1.10527	0.552634	+	0.833424i	$0.313623\pi$
$860$	0	0
$861$	−620.001	−0.0245407
$862$	0	0
$863$	34651.5	1.36680	0.683402	−	0.730042i	$-0.260500\pi$
$863$	34651.5	1.36680	0.683402	+	0.730042i	$0.260500\pi$
$864$	0	0
$865$	−6787.35	−0.266794
$866$	0	0
$867$	−10494.3	−0.411077
$868$	0	0
$869$	30534.2	1.19195
$870$	0	0
$871$	−2722.36	−0.105905
$872$	0	0
$873$	2767.28	0.107283
$874$	0	0
$875$	4387.58	0.169517
$876$	0	0
$877$	−13758.6	−0.529753	−0.264877	−	0.964282i	$-0.585331\pi$
$877$	−13758.6	−0.529753	−0.264877	+	0.964282i	$0.585331\pi$
$878$	0	0
$879$	6874.02	0.263771
$880$	0	0
$881$	−22584.0	−0.863649	−0.431824	−	0.901958i	$-0.642130\pi$
$881$	−22584.0	−0.863649	−0.431824	+	0.901958i	$0.642130\pi$
$882$	0	0
$883$	−25051.8	−0.954768	−0.477384	−	0.878695i	$-0.658415\pi$
$883$	−25051.8	−0.954768	−0.477384	+	0.878695i	$0.658415\pi$
$884$	0	0
$885$	1496.43	0.0568385
$886$	0	0
$887$	45947.2	1.73929	0.869647	−	0.493673i	$-0.164346\pi$
$887$	45947.2	1.73929	0.869647	+	0.493673i	$0.164346\pi$
$888$	0	0
$889$	−3609.53	−0.136175
$890$	0	0
$891$	−21795.7	−0.819509
$892$	0	0
$893$	−573.689	−0.0214981
$894$	0	0
$895$	−6854.86	−0.256014
$896$	0	0
$897$	−146.595	−0.00545672
$898$	0	0
$899$	−30875.7	−1.14545
$900$	0	0
$901$	−1298.95	−0.0480291
$902$	0	0
$903$	7293.06	0.268768
$904$	0	0
$905$	−49308.2	−1.81111
$906$	0	0
$907$	15162.8	0.555098	0.277549	−	0.960711i	$-0.410478\pi$
$907$	15162.8	0.555098	0.277549	+	0.960711i	$0.410478\pi$
$908$	0	0
$909$	22452.9	0.819268
$910$	0	0
$911$	−46314.8	−1.68439	−0.842194	−	0.539174i	$-0.818737\pi$
$911$	−46314.8	−1.68439	−0.842194	+	0.539174i	$0.818737\pi$
$912$	0	0
$913$	25435.2	0.921994
$914$	0	0
$915$	1096.62	0.0396209
$916$	0	0
$917$	748.039	0.0269383
$918$	0	0
$919$	50374.6	1.80817	0.904083	−	0.427357i	$-0.140555\pi$
$919$	50374.6	1.80817	0.904083	+	0.427357i	$0.140555\pi$
$920$	0	0
$921$	−13381.3	−0.478752
$922$	0	0
$923$	3133.96	0.111761
$924$	0	0
$925$	21737.9	0.772688
$926$	0	0
$927$	33807.4	1.19782
$928$	0	0
$929$	32777.3	1.15758	0.578789	−	0.815477i	$-0.303525\pi$
$929$	32777.3	1.15758	0.578789	+	0.815477i	$0.303525\pi$
$930$	0	0
$931$	95.9851	0.00337893
$932$	0	0
$933$	15999.7	0.561423
$934$	0	0
$935$	−6239.60	−0.218242
$936$	0	0
$937$	19088.0	0.665504	0.332752	−	0.943014i	$-0.392023\pi$
$937$	19088.0	0.665504	0.332752	+	0.943014i	$0.392023\pi$
$938$	0	0
$939$	22238.8	0.772880
$940$	0	0
$941$	41211.5	1.42769	0.713846	−	0.700303i	$-0.246952\pi$
$941$	41211.5	1.42769	0.713846	+	0.700303i	$0.246952\pi$
$942$	0	0
$943$	−169.327	−0.00584733
$944$	0	0
$945$	10716.9	0.368910
$946$	0	0
$947$	31669.7	1.08672	0.543362	−	0.839498i	$-0.317151\pi$
$947$	31669.7	1.08672	0.543362	+	0.839498i	$0.317151\pi$
$948$	0	0
$949$	−3533.10	−0.120853
$950$	0	0
$951$	11304.2	0.385450
$952$	0	0
$953$	3397.10	0.115470	0.0577350	−	0.998332i	$-0.481612\pi$
$953$	3397.10	0.115470	0.0577350	+	0.998332i	$0.481612\pi$
$954$	0	0
$955$	−14581.7	−0.494087
$956$	0	0
$957$	26971.0	0.911023
$958$	0	0
$959$	−19138.5	−0.644435
$960$	0	0
$961$	−8882.42	−0.298158
$962$	0	0
$963$	−48805.6	−1.63317
$964$	0	0
$965$	−5862.47	−0.195564
$966$	0	0
$967$	16425.3	0.546226	0.273113	−	0.961982i	$-0.411947\pi$
$967$	16425.3	0.546226	0.273113	+	0.961982i	$0.411947\pi$
$968$	0	0
$969$	−31.4355	−0.00104216
$970$	0	0
$971$	−13263.2	−0.438349	−0.219175	−	0.975686i	$-0.570336\pi$
$971$	−13263.2	−0.438349	−0.219175	+	0.975686i	$0.570336\pi$
$972$	0	0
$973$	−7456.89	−0.245690
$974$	0	0
$975$	−2888.23	−0.0948690
$976$	0	0
$977$	58733.4	1.92328	0.961642	−	0.274307i	$-0.0884483\pi$
$977$	58733.4	1.92328	0.961642	+	0.274307i	$0.0884483\pi$
$978$	0	0
$979$	−90476.6	−2.95367
$980$	0	0
$981$	−48658.8	−1.58365
$982$	0	0
$983$	42028.9	1.36370	0.681849	−	0.731493i	$-0.261176\pi$
$983$	42028.9	1.36370	0.681849	+	0.731493i	$0.261176\pi$
$984$	0	0
$985$	19233.7	0.622169
$986$	0	0
$987$	−4428.71	−0.142824
$988$	0	0
$989$	1991.79	0.0640396
$990$	0	0
$991$	40315.8	1.29230	0.646152	−	0.763208i	$-0.276377\pi$
$991$	40315.8	1.29230	0.646152	+	0.763208i	$0.276377\pi$
$992$	0	0
$993$	22269.1	0.711670
$994$	0	0
$995$	−52139.1	−1.66123
$996$	0	0
$997$	−15803.5	−0.502008	−0.251004	−	0.967986i	$-0.580761\pi$
$997$	−15803.5	−0.502008	−0.251004	+	0.967986i	$0.580761\pi$
$998$	0	0
$999$	−28471.7	−0.901706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.c.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.c.1.10	✓	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)

\(f(q)\)	\(=\)	\(q+2.16028 q^{3} -14.3655 q^{5} +7.00000 q^{7} -22.3332 q^{9} +O(q^{10})\)	\(q+2.16028 q^{3} -14.3655 q^{5} +7.00000 q^{7} -22.3332 q^{9} -58.4700 q^{11} -16.4312 q^{13} -31.0335 q^{15} -7.42853 q^{17} +1.95888 q^{19} +15.1220 q^{21} +4.12992 q^{23} +81.3679 q^{25} -106.574 q^{27} -213.528 q^{29} +144.598 q^{31} -126.312 q^{33} -100.559 q^{35} +267.155 q^{37} -35.4959 q^{39} -41.0000 q^{41} +482.282 q^{43} +320.828 q^{45} -292.866 q^{47} +49.0000 q^{49} -16.0477 q^{51} +174.859 q^{53} +839.951 q^{55} +4.23173 q^{57} -48.2199 q^{59} -35.3366 q^{61} -156.332 q^{63} +236.042 q^{65} +165.683 q^{67} +8.92179 q^{69} -190.733 q^{71} +215.024 q^{73} +175.778 q^{75} -409.290 q^{77} -522.220 q^{79} +372.767 q^{81} -435.012 q^{83} +106.715 q^{85} -461.280 q^{87} +1547.40 q^{89} -115.018 q^{91} +312.372 q^{93} -28.1403 q^{95} -123.909 q^{97} +1305.82 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more