LMFDB - Embedded newform 1232.4.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 $ x^6 - x^5 - 47x^4 + 10x^3 + 612x^2 + 240x - 1440
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.61124$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.60307 q^{3} -19.7567 q^{5} -7.00000 q^{7} -14.0179 q^{9} +O(q^{10})$	$q-3.60307 q^{3} -19.7567 q^{5} -7.00000 q^{7} -14.0179 q^{9} +11.0000 q^{11} +0.0182143 q^{13} +71.1849 q^{15} +41.9558 q^{17} +20.9878 q^{19} +25.2215 q^{21} -47.5358 q^{23} +265.329 q^{25} +147.790 q^{27} +141.154 q^{29} +1.39728 q^{31} -39.6338 q^{33} +138.297 q^{35} -67.5533 q^{37} -0.0656275 q^{39} +211.907 q^{41} +101.721 q^{43} +276.948 q^{45} -548.347 q^{47} +49.0000 q^{49} -151.170 q^{51} +701.099 q^{53} -217.324 q^{55} -75.6205 q^{57} -66.9253 q^{59} +175.267 q^{61} +98.1252 q^{63} -0.359856 q^{65} -168.099 q^{67} +171.275 q^{69} -100.908 q^{71} -163.811 q^{73} -955.999 q^{75} -77.0000 q^{77} -119.758 q^{79} -154.016 q^{81} +86.6715 q^{83} -828.910 q^{85} -508.589 q^{87} +507.100 q^{89} -0.127500 q^{91} -5.03451 q^{93} -414.651 q^{95} -1225.15 q^{97} -154.197 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 42 q^{7} + 60 q^{9}+O(q^{10}) $ 6 * q + 2 * q^5 - 42 * q^7 + 60 * q^9	$ 6 q + 2 q^{5} - 42 q^{7} + 60 q^{9} + 66 q^{11} - 6 q^{13} - 126 q^{15} - 14 q^{17} - 80 q^{19} - 254 q^{23} + 220 q^{25} - 90 q^{27} + 132 q^{29} + 52 q^{31} - 14 q^{35} - 518 q^{37} - 332 q^{39} + 486 q^{41} - 428 q^{43} - 244 q^{45} - 790 q^{47} + 294 q^{49} - 640 q^{51} - 40 q^{53} + 22 q^{55} + 2276 q^{57} - 436 q^{59} + 1034 q^{61} - 420 q^{63} + 800 q^{65} - 562 q^{67} + 530 q^{69} - 2474 q^{71} - 902 q^{73} - 1986 q^{75} - 462 q^{77} - 1636 q^{79} + 570 q^{81} - 3016 q^{83} + 236 q^{85} - 4340 q^{87} + 1750 q^{89} + 42 q^{91} + 150 q^{93} - 3628 q^{95} - 1250 q^{97} + 660 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 - 42 * q^7 + 60 * q^9 + 66 * q^11 - 6 * q^13 - 126 * q^15 - 14 * q^17 - 80 * q^19 - 254 * q^23 + 220 * q^25 - 90 * q^27 + 132 * q^29 + 52 * q^31 - 14 * q^35 - 518 * q^37 - 332 * q^39 + 486 * q^41 - 428 * q^43 - 244 * q^45 - 790 * q^47 + 294 * q^49 - 640 * q^51 - 40 * q^53 + 22 * q^55 + 2276 * q^57 - 436 * q^59 + 1034 * q^61 - 420 * q^63 + 800 * q^65 - 562 * q^67 + 530 * q^69 - 2474 * q^71 - 902 * q^73 - 1986 * q^75 - 462 * q^77 - 1636 * q^79 + 570 * q^81 - 3016 * q^83 + 236 * q^85 - 4340 * q^87 + 1750 * q^89 + 42 * q^91 + 150 * q^93 - 3628 * q^95 - 1250 * q^97 + 660 * 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
$3$	−3.60307	−0.693411	−0.346706	−	0.937974i	$-0.612700\pi$
$3$	−3.60307	−0.693411	−0.346706	+	0.937974i	$0.612700\pi$
$4$	0	0
$5$	−19.7567	−1.76710	−0.883549	−	0.468340i	$-0.844852\pi$
$5$	−19.7567	−1.76710	−0.883549	+	0.468340i	$0.844852\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−14.0179	−0.519181
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	0.0182143	0.000388596 0	0.000194298	−	1.00000i	$-0.499938\pi$
$13$	0.0182143	0.000388596 0	0.000194298		1.00000i	$0.499938\pi$
$14$	0	0
$15$	71.1849	1.22532
$16$	0	0
$17$	41.9558	0.598575	0.299288	−	0.954163i	$-0.403251\pi$
$17$	41.9558	0.598575	0.299288	+	0.954163i	$0.403251\pi$
$18$	0	0
$19$	20.9878	0.253417	0.126709	−	0.991940i	$-0.459559\pi$
$19$	20.9878	0.253417	0.126709	+	0.991940i	$0.459559\pi$
$20$	0	0
$21$	25.2215	0.262085
$22$	0	0
$23$	−47.5358	−0.430952	−0.215476	−	0.976509i	$-0.569130\pi$
$23$	−47.5358	−0.430952	−0.215476	+	0.976509i	$0.569130\pi$
$24$	0	0
$25$	265.329	2.12263
$26$	0	0
$27$	147.790	1.05342
$28$	0	0
$29$	141.154	0.903852	0.451926	−	0.892056i	$-0.350737\pi$
$29$	141.154	0.903852	0.451926	+	0.892056i	$0.350737\pi$
$30$	0	0
$31$	1.39728	0.00809547	0.00404773	−	0.999992i	$-0.498712\pi$
$31$	1.39728	0.00809547	0.00404773	+	0.999992i	$0.498712\pi$
$32$	0	0
$33$	−39.6338	−0.209071
$34$	0	0
$35$	138.297	0.667900
$36$	0	0
$37$	−67.5533	−0.300154	−0.150077	−	0.988674i	$-0.547952\pi$
$37$	−67.5533	−0.300154	−0.150077	+	0.988674i	$0.547952\pi$
$38$	0	0
$39$	−0.0656275	−0.000269457 0
$40$	0	0
$41$	211.907	0.807179	0.403589	−	0.914940i	$-0.367762\pi$
$41$	211.907	0.807179	0.403589	+	0.914940i	$0.367762\pi$
$42$	0	0
$43$	101.721	0.360751	0.180375	−	0.983598i	$-0.442269\pi$
$43$	101.721	0.360751	0.180375	+	0.983598i	$0.442269\pi$
$44$	0	0
$45$	276.948	0.917443
$46$	0	0
$47$	−548.347	−1.70180	−0.850900	−	0.525328i	$-0.823943\pi$
$47$	−548.347	−1.70180	−0.850900	+	0.525328i	$0.823943\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−151.170	−0.415059
$52$	0	0
$53$	701.099	1.81704	0.908522	−	0.417836i	$-0.137211\pi$
$53$	701.099	1.81704	0.908522	+	0.417836i	$0.137211\pi$
$54$	0	0
$55$	−217.324	−0.532800
$56$	0	0
$57$	−75.6205	−0.175722
$58$	0	0
$59$	−66.9253	−0.147677	−0.0738384	−	0.997270i	$-0.523525\pi$
$59$	−66.9253	−0.147677	−0.0738384	+	0.997270i	$0.523525\pi$
$60$	0	0
$61$	175.267	0.367878	0.183939	−	0.982938i	$-0.441115\pi$
$61$	175.267	0.367878	0.183939	+	0.982938i	$0.441115\pi$
$62$	0	0
$63$	98.1252	0.196232
$64$	0	0
$65$	−0.359856	−0.000686687 0
$66$	0	0
$67$	−168.099	−0.306515	−0.153258	−	0.988186i	$-0.548976\pi$
$67$	−168.099	−0.306515	−0.153258	+	0.988186i	$0.548976\pi$
$68$	0	0
$69$	171.275	0.298827
$70$	0	0
$71$	−100.908	−0.168670	−0.0843349	−	0.996437i	$-0.526877\pi$
$71$	−100.908	−0.168670	−0.0843349	+	0.996437i	$0.526877\pi$
$72$	0	0
$73$	−163.811	−0.262639	−0.131320	−	0.991340i	$-0.541921\pi$
$73$	−163.811	−0.262639	−0.131320	+	0.991340i	$0.541921\pi$
$74$	0	0
$75$	−955.999	−1.47186
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−119.758	−0.170555	−0.0852777	−	0.996357i	$-0.527178\pi$
$79$	−119.758	−0.170555	−0.0852777	+	0.996357i	$0.527178\pi$
$80$	0	0
$81$	−154.016	−0.211270
$82$	0	0
$83$	86.6715	0.114620	0.0573098	−	0.998356i	$-0.481748\pi$
$83$	86.6715	0.114620	0.0573098	+	0.998356i	$0.481748\pi$
$84$	0	0
$85$	−828.910	−1.05774
$86$	0	0
$87$	−508.589	−0.626741
$88$	0	0
$89$	507.100	0.603961	0.301980	−	0.953314i	$-0.402352\pi$
$89$	507.100	0.603961	0.301980	+	0.953314i	$0.402352\pi$
$90$	0	0
$91$	−0.127500	−0.000146876 0
$92$	0	0
$93$	−5.03451	−0.00561349
$94$	0	0
$95$	−414.651	−0.447813
$96$	0	0
$97$	−1225.15	−1.28242	−0.641212	−	0.767364i	$-0.721568\pi$
$97$	−1225.15	−1.28242	−0.641212	+	0.767364i	$0.721568\pi$
$98$	0	0
$99$	−154.197	−0.156539
$100$	0	0
$101$	−588.082	−0.579369	−0.289685	−	0.957122i	$-0.593550\pi$
$101$	−588.082	−0.579369	−0.289685	+	0.957122i	$0.593550\pi$
$102$	0	0
$103$	−1153.18	−1.10316	−0.551582	−	0.834121i	$-0.685976\pi$
$103$	−1153.18	−1.10316	−0.551582	+	0.834121i	$0.685976\pi$
$104$	0	0
$105$	−498.295	−0.463129
$106$	0	0
$107$	−6.16932	−0.00557393	−0.00278697	−	0.999996i	$-0.500887\pi$
$107$	−6.16932	−0.00557393	−0.00278697	+	0.999996i	$0.500887\pi$
$108$	0	0
$109$	20.6168	0.0181168	0.00905840	−	0.999959i	$-0.497117\pi$
$109$	20.6168	0.0181168	0.00905840	+	0.999959i	$0.497117\pi$
$110$	0	0
$111$	243.399	0.208130
$112$	0	0
$113$	264.323	0.220048	0.110024	−	0.993929i	$-0.464907\pi$
$113$	264.323	0.220048	0.110024	+	0.993929i	$0.464907\pi$
$114$	0	0
$115$	939.153	0.761534
$116$	0	0
$117$	−0.255327	−0.000201752 0
$118$	0	0
$119$	−293.691	−0.226240
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−763.516	−0.559707
$124$	0	0
$125$	−2772.45	−1.98380
$126$	0	0
$127$	299.822	0.209488	0.104744	−	0.994499i	$-0.466598\pi$
$127$	299.822	0.209488	0.104744	+	0.994499i	$0.466598\pi$
$128$	0	0
$129$	−366.507	−0.250148
$130$	0	0
$131$	−1132.34	−0.755217	−0.377608	−	0.925965i	$-0.623253\pi$
$131$	−1132.34	−0.755217	−0.377608	+	0.925965i	$0.623253\pi$
$132$	0	0
$133$	−146.915	−0.0957828
$134$	0	0
$135$	−2919.86	−1.86149
$136$	0	0
$137$	2433.35	1.51748	0.758741	−	0.651393i	$-0.225815\pi$
$137$	2433.35	1.51748	0.758741	+	0.651393i	$0.225815\pi$
$138$	0	0
$139$	3148.13	1.92101	0.960506	−	0.278259i	$-0.0897573\pi$
$139$	3148.13	1.92101	0.960506	+	0.278259i	$0.0897573\pi$
$140$	0	0
$141$	1975.73	1.18005
$142$	0	0
$143$	0.200358	0.000117166 0
$144$	0	0
$145$	−2788.75	−1.59719
$146$	0	0
$147$	−176.550	−0.0990587
$148$	0	0
$149$	696.259	0.382817	0.191409	−	0.981510i	$-0.438694\pi$
$149$	696.259	0.382817	0.191409	+	0.981510i	$0.438694\pi$
$150$	0	0
$151$	−1793.14	−0.966383	−0.483191	−	0.875515i	$-0.660522\pi$
$151$	−1793.14	−0.966383	−0.483191	+	0.875515i	$0.660522\pi$
$152$	0	0
$153$	−588.132	−0.310769
$154$	0	0
$155$	−27.6058	−0.0143055
$156$	0	0
$157$	223.532	0.113629	0.0568146	−	0.998385i	$-0.481906\pi$
$157$	223.532	0.113629	0.0568146	+	0.998385i	$0.481906\pi$
$158$	0	0
$159$	−2526.11	−1.25996
$160$	0	0
$161$	332.751	0.162885
$162$	0	0
$163$	−223.081	−0.107197	−0.0535984	−	0.998563i	$-0.517069\pi$
$163$	−223.081	−0.107197	−0.0535984	+	0.998563i	$0.517069\pi$
$164$	0	0
$165$	783.034	0.369449
$166$	0	0
$167$	−2926.00	−1.35581	−0.677907	−	0.735147i	$-0.737113\pi$
$167$	−2926.00	−1.35581	−0.677907	+	0.735147i	$0.737113\pi$
$168$	0	0
$169$	−2197.00	−1.00000
$170$	0	0
$171$	−294.205	−0.131570
$172$	0	0
$173$	314.639	0.138275	0.0691374	−	0.997607i	$-0.477975\pi$
$173$	314.639	0.138275	0.0691374	+	0.997607i	$0.477975\pi$
$174$	0	0
$175$	−1857.30	−0.802280
$176$	0	0
$177$	241.136	0.102401
$178$	0	0
$179$	−4216.32	−1.76057	−0.880286	−	0.474443i	$-0.842650\pi$
$179$	−4216.32	−1.76057	−0.880286	+	0.474443i	$0.842650\pi$
$180$	0	0
$181$	−4070.18	−1.67146	−0.835729	−	0.549142i	$-0.814955\pi$
$181$	−4070.18	−1.67146	−0.835729	+	0.549142i	$0.814955\pi$
$182$	0	0
$183$	−631.498	−0.255091
$184$	0	0
$185$	1334.63	0.530401
$186$	0	0
$187$	461.514	0.180477
$188$	0	0
$189$	−1034.53	−0.398154
$190$	0	0
$191$	1277.35	0.483905	0.241953	−	0.970288i	$-0.422212\pi$
$191$	1277.35	0.483905	0.241953	+	0.970288i	$0.422212\pi$
$192$	0	0
$193$	2143.02	0.799264	0.399632	−	0.916676i	$-0.369138\pi$
$193$	2143.02	0.799264	0.399632	+	0.916676i	$0.369138\pi$
$194$	0	0
$195$	1.29659	0.000476156 0
$196$	0	0
$197$	4498.68	1.62699	0.813496	−	0.581571i	$-0.197562\pi$
$197$	4498.68	1.62699	0.813496	+	0.581571i	$0.197562\pi$
$198$	0	0
$199$	−522.831	−0.186244	−0.0931219	−	0.995655i	$-0.529685\pi$
$199$	−522.831	−0.186244	−0.0931219	+	0.995655i	$0.529685\pi$
$200$	0	0
$201$	605.671	0.212541
$202$	0	0
$203$	−988.080	−0.341624
$204$	0	0
$205$	−4186.59	−1.42636
$206$	0	0
$207$	666.352	0.223742
$208$	0	0
$209$	230.866	0.0764082
$210$	0	0
$211$	3623.82	1.18234	0.591171	−	0.806546i	$-0.298666\pi$
$211$	3623.82	1.18234	0.591171	+	0.806546i	$0.298666\pi$
$212$	0	0
$213$	363.578	0.116957
$214$	0	0
$215$	−2009.67	−0.637481
$216$	0	0
$217$	−9.78099	−0.00305980
$218$	0	0
$219$	590.224	0.182117
$220$	0	0
$221$	0.764197	0.000232604 0
$222$	0	0
$223$	−5398.40	−1.62109	−0.810547	−	0.585674i	$-0.800830\pi$
$223$	−5398.40	−1.62109	−0.810547	+	0.585674i	$0.800830\pi$
$224$	0	0
$225$	−3719.35	−1.10203
$226$	0	0
$227$	4566.64	1.33524	0.667619	−	0.744504i	$-0.267314\pi$
$227$	4566.64	1.33524	0.667619	+	0.744504i	$0.267314\pi$
$228$	0	0
$229$	5268.77	1.52039	0.760196	−	0.649694i	$-0.225103\pi$
$229$	5268.77	1.52039	0.760196	+	0.649694i	$0.225103\pi$
$230$	0	0
$231$	277.436	0.0790215
$232$	0	0
$233$	−4894.41	−1.37615	−0.688076	−	0.725639i	$-0.741544\pi$
$233$	−4894.41	−1.37615	−0.688076	+	0.725639i	$0.741544\pi$
$234$	0	0
$235$	10833.5	3.00725
$236$	0	0
$237$	431.498	0.118265
$238$	0	0
$239$	3332.53	0.901938	0.450969	−	0.892540i	$-0.351078\pi$
$239$	3332.53	0.901938	0.450969	+	0.892540i	$0.351078\pi$
$240$	0	0
$241$	836.643	0.223622	0.111811	−	0.993729i	$-0.464335\pi$
$241$	836.643	0.223622	0.111811	+	0.993729i	$0.464335\pi$
$242$	0	0
$243$	−3435.41	−0.906920
$244$	0	0
$245$	−968.081	−0.252442
$246$	0	0
$247$	0.382279	9.84770e−5 0
$248$	0	0
$249$	−312.283	−0.0794785
$250$	0	0
$251$	5592.18	1.40628	0.703139	−	0.711053i	$-0.251781\pi$
$251$	5592.18	1.40628	0.703139	+	0.711053i	$0.251781\pi$
$252$	0	0
$253$	−522.894	−0.129937
$254$	0	0
$255$	2986.62	0.733449
$256$	0	0
$257$	−5854.56	−1.42100	−0.710501	−	0.703696i	$-0.751532\pi$
$257$	−5854.56	−1.42100	−0.710501	+	0.703696i	$0.751532\pi$
$258$	0	0
$259$	472.873	0.113448
$260$	0	0
$261$	−1978.69	−0.469263
$262$	0	0
$263$	−722.763	−0.169458	−0.0847291	−	0.996404i	$-0.527002\pi$
$263$	−722.763	−0.169458	−0.0847291	+	0.996404i	$0.527002\pi$
$264$	0	0
$265$	−13851.4	−3.21089
$266$	0	0
$267$	−1827.12	−0.418793
$268$	0	0
$269$	−8082.40	−1.83194	−0.915972	−	0.401243i	$-0.868578\pi$
$269$	−8082.40	−1.83194	−0.915972	+	0.401243i	$0.868578\pi$
$270$	0	0
$271$	3529.76	0.791209	0.395604	−	0.918421i	$-0.370535\pi$
$271$	3529.76	0.791209	0.395604	+	0.918421i	$0.370535\pi$
$272$	0	0
$273$	0.459393	0.000101845 0
$274$	0	0
$275$	2918.62	0.639998
$276$	0	0
$277$	−653.323	−0.141713	−0.0708563	−	0.997487i	$-0.522573\pi$
$277$	−653.323	−0.141713	−0.0708563	+	0.997487i	$0.522573\pi$
$278$	0	0
$279$	−19.5870	−0.00420301
$280$	0	0
$281$	604.579	0.128349	0.0641746	−	0.997939i	$-0.479559\pi$
$281$	604.579	0.128349	0.0641746	+	0.997939i	$0.479559\pi$
$282$	0	0
$283$	−5267.13	−1.10636	−0.553178	−	0.833063i	$-0.686585\pi$
$283$	−5267.13	−1.10636	−0.553178	+	0.833063i	$0.686585\pi$
$284$	0	0
$285$	1494.01	0.310519
$286$	0	0
$287$	−1483.35	−0.305085
$288$	0	0
$289$	−3152.71	−0.641708
$290$	0	0
$291$	4414.30	0.889246
$292$	0	0
$293$	3670.60	0.731874	0.365937	−	0.930640i	$-0.380749\pi$
$293$	3670.60	0.731874	0.365937	+	0.930640i	$0.380749\pi$
$294$	0	0
$295$	1322.23	0.260959
$296$	0	0
$297$	1625.69	0.317617
$298$	0	0
$299$	−0.865833	−0.000167466 0
$300$	0	0
$301$	−712.045	−0.136351
$302$	0	0
$303$	2118.90	0.401741
$304$	0	0
$305$	−3462.70	−0.650077
$306$	0	0
$307$	−2302.42	−0.428033	−0.214016	−	0.976830i	$-0.568655\pi$
$307$	−2302.42	−0.428033	−0.214016	+	0.976830i	$0.568655\pi$
$308$	0	0
$309$	4154.97	0.764946
$310$	0	0
$311$	−4940.57	−0.900818	−0.450409	−	0.892822i	$-0.648722\pi$
$311$	−4940.57	−0.900818	−0.450409	+	0.892822i	$0.648722\pi$
$312$	0	0
$313$	−7477.37	−1.35031	−0.675154	−	0.737677i	$-0.735923\pi$
$313$	−7477.37	−1.35031	−0.675154	+	0.737677i	$0.735923\pi$
$314$	0	0
$315$	−1938.64	−0.346761
$316$	0	0
$317$	4784.21	0.847660	0.423830	−	0.905742i	$-0.360685\pi$
$317$	4784.21	0.847660	0.423830	+	0.905742i	$0.360685\pi$
$318$	0	0
$319$	1552.70	0.272521
$320$	0	0
$321$	22.2285	0.00386503
$322$	0	0
$323$	880.560	0.151689
$324$	0	0
$325$	4.83279	0.000824847 0
$326$	0	0
$327$	−74.2838	−0.0125624
$328$	0	0
$329$	3838.43	0.643220
$330$	0	0
$331$	−8015.32	−1.33100	−0.665501	−	0.746397i	$-0.731782\pi$
$331$	−8015.32	−1.33100	−0.665501	+	0.746397i	$0.731782\pi$
$332$	0	0
$333$	946.955	0.155834
$334$	0	0
$335$	3321.08	0.541642
$336$	0	0
$337$	10422.2	1.68467	0.842333	−	0.538957i	$-0.181182\pi$
$337$	10422.2	1.68467	0.842333	+	0.538957i	$0.181182\pi$
$338$	0	0
$339$	−952.374	−0.152584
$340$	0	0
$341$	15.3701	0.00244088
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3383.83	−0.528056
$346$	0	0
$347$	12183.0	1.88478	0.942389	−	0.334519i	$-0.108574\pi$
$347$	12183.0	1.88478	0.942389	+	0.334519i	$0.108574\pi$
$348$	0	0
$349$	−8441.05	−1.29467	−0.647334	−	0.762207i	$-0.724116\pi$
$349$	−8441.05	−1.29467	−0.647334	+	0.762207i	$0.724116\pi$
$350$	0	0
$351$	2.69190	0.000409354 0
$352$	0	0
$353$	2207.73	0.332878	0.166439	−	0.986052i	$-0.446773\pi$
$353$	2207.73	0.332878	0.166439	+	0.986052i	$0.446773\pi$
$354$	0	0
$355$	1993.61	0.298056
$356$	0	0
$357$	1058.19	0.156877
$358$	0	0
$359$	−1289.60	−0.189589	−0.0947943	−	0.995497i	$-0.530219\pi$
$359$	−1289.60	−0.189589	−0.0947943	+	0.995497i	$0.530219\pi$
$360$	0	0
$361$	−6418.51	−0.935780
$362$	0	0
$363$	−435.971	−0.0630374
$364$	0	0
$365$	3236.38	0.464109
$366$	0	0
$367$	−5634.79	−0.801454	−0.400727	−	0.916197i	$-0.631242\pi$
$367$	−5634.79	−0.801454	−0.400727	+	0.916197i	$0.631242\pi$
$368$	0	0
$369$	−2970.49	−0.419072
$370$	0	0
$371$	−4907.69	−0.686778
$372$	0	0
$373$	4597.03	0.638138	0.319069	−	0.947732i	$-0.396630\pi$
$373$	4597.03	0.638138	0.319069	+	0.947732i	$0.396630\pi$
$374$	0	0
$375$	9989.31	1.37559
$376$	0	0
$377$	2.57103	0.000351233 0
$378$	0	0
$379$	−9634.49	−1.30578	−0.652890	−	0.757453i	$-0.726443\pi$
$379$	−9634.49	−1.30578	−0.652890	+	0.757453i	$0.726443\pi$
$380$	0	0
$381$	−1080.28	−0.145261
$382$	0	0
$383$	−417.635	−0.0557184	−0.0278592	−	0.999612i	$-0.508869\pi$
$383$	−417.635	−0.0557184	−0.0278592	+	0.999612i	$0.508869\pi$
$384$	0	0
$385$	1521.27	0.201379
$386$	0	0
$387$	−1425.91	−0.187295
$388$	0	0
$389$	7502.68	0.977895	0.488947	−	0.872313i	$-0.337381\pi$
$389$	7502.68	0.977895	0.488947	+	0.872313i	$0.337381\pi$
$390$	0	0
$391$	−1994.40	−0.257957
$392$	0	0
$393$	4079.92	0.523676
$394$	0	0
$395$	2366.04	0.301388
$396$	0	0
$397$	−7994.97	−1.01072	−0.505360	−	0.862909i	$-0.668640\pi$
$397$	−7994.97	−1.01072	−0.505360	+	0.862909i	$0.668640\pi$
$398$	0	0
$399$	529.343	0.0664168
$400$	0	0
$401$	8619.78	1.07344	0.536722	−	0.843759i	$-0.319662\pi$
$401$	8619.78	1.07344	0.536722	+	0.843759i	$0.319662\pi$
$402$	0	0
$403$	0.0254506	3.14587e−6 0
$404$	0	0
$405$	3042.85	0.373334
$406$	0	0
$407$	−743.087	−0.0904998
$408$	0	0
$409$	15630.3	1.88965	0.944826	−	0.327572i	$-0.106230\pi$
$409$	15630.3	1.88965	0.944826	+	0.327572i	$0.106230\pi$
$410$	0	0
$411$	−8767.52	−1.05224
$412$	0	0
$413$	468.477	0.0558166
$414$	0	0
$415$	−1712.35	−0.202544
$416$	0	0
$417$	−11342.9	−1.33205
$418$	0	0
$419$	126.343	0.0147309	0.00736544	−	0.999973i	$-0.497655\pi$
$419$	126.343	0.0147309	0.00736544	+	0.999973i	$0.497655\pi$
$420$	0	0
$421$	445.948	0.0516251	0.0258125	−	0.999667i	$-0.491783\pi$
$421$	445.948	0.0516251	0.0258125	+	0.999667i	$0.491783\pi$
$422$	0	0
$423$	7686.67	0.883542
$424$	0	0
$425$	11132.1	1.27055
$426$	0	0
$427$	−1226.87	−0.139045
$428$	0	0
$429$	−0.721903	−8.12443e−5 0
$430$	0	0
$431$	−19.8480	−0.00221820	−0.00110910	−	0.999999i	$-0.500353\pi$
$431$	−19.8480	−0.00221820	−0.00110910	+	0.999999i	$0.500353\pi$
$432$	0	0
$433$	−7746.37	−0.859739	−0.429869	−	0.902891i	$-0.641440\pi$
$433$	−7746.37	−0.859739	−0.429869	+	0.902891i	$0.641440\pi$
$434$	0	0
$435$	10048.1	1.10751
$436$	0	0
$437$	−997.671	−0.109211
$438$	0	0
$439$	−4588.45	−0.498849	−0.249424	−	0.968394i	$-0.580241\pi$
$439$	−4588.45	−0.498849	−0.249424	+	0.968394i	$0.580241\pi$
$440$	0	0
$441$	−686.877	−0.0741687
$442$	0	0
$443$	3311.04	0.355107	0.177554	−	0.984111i	$-0.443182\pi$
$443$	3311.04	0.355107	0.177554	+	0.984111i	$0.443182\pi$
$444$	0	0
$445$	−10018.6	−1.06726
$446$	0	0
$447$	−2508.67	−0.265450
$448$	0	0
$449$	2422.34	0.254604	0.127302	−	0.991864i	$-0.459368\pi$
$449$	2422.34	0.254604	0.127302	+	0.991864i	$0.459368\pi$
$450$	0	0
$451$	2330.98	0.243374
$452$	0	0
$453$	6460.81	0.670101
$454$	0	0
$455$	2.51899	0.000259543 0
$456$	0	0
$457$	11076.2	1.13375	0.566873	−	0.823805i	$-0.308153\pi$
$457$	11076.2	1.13375	0.566873	+	0.823805i	$0.308153\pi$
$458$	0	0
$459$	6200.66	0.630549
$460$	0	0
$461$	−7809.47	−0.788987	−0.394494	−	0.918899i	$-0.629080\pi$
$461$	−7809.47	−0.788987	−0.394494	+	0.918899i	$0.629080\pi$
$462$	0	0
$463$	−7713.55	−0.774253	−0.387126	−	0.922027i	$-0.626532\pi$
$463$	−7713.55	−0.774253	−0.387126	+	0.922027i	$0.626532\pi$
$464$	0	0
$465$	99.4655	0.00991958
$466$	0	0
$467$	1228.60	0.121740	0.0608701	−	0.998146i	$-0.480612\pi$
$467$	1228.60	0.121740	0.0608701	+	0.998146i	$0.480612\pi$
$468$	0	0
$469$	1176.69	0.115852
$470$	0	0
$471$	−805.401	−0.0787918
$472$	0	0
$473$	1118.93	0.108770
$474$	0	0
$475$	5568.67	0.537912
$476$	0	0
$477$	−9827.93	−0.943375
$478$	0	0
$479$	14448.2	1.37819	0.689095	−	0.724671i	$-0.258008\pi$
$479$	14448.2	1.37819	0.689095	+	0.724671i	$0.258008\pi$
$480$	0	0
$481$	−1.23044	−0.000116639 0
$482$	0	0
$483$	−1198.92	−0.112946
$484$	0	0
$485$	24205.0	2.26617
$486$	0	0
$487$	975.281	0.0907479	0.0453739	−	0.998970i	$-0.485552\pi$
$487$	975.281	0.0907479	0.0453739	+	0.998970i	$0.485552\pi$
$488$	0	0
$489$	803.777	0.0743314
$490$	0	0
$491$	−15085.7	−1.38658	−0.693288	−	0.720661i	$-0.743839\pi$
$491$	−15085.7	−1.38658	−0.693288	+	0.720661i	$0.743839\pi$
$492$	0	0
$493$	5922.24	0.541023
$494$	0	0
$495$	3046.43	0.276620
$496$	0	0
$497$	706.355	0.0637512
$498$	0	0
$499$	14015.4	1.25735	0.628674	−	0.777669i	$-0.283598\pi$
$499$	14015.4	1.25735	0.628674	+	0.777669i	$0.283598\pi$
$500$	0	0
$501$	10542.6	0.940137
$502$	0	0
$503$	−4923.00	−0.436393	−0.218197	−	0.975905i	$-0.570017\pi$
$503$	−4923.00	−0.436393	−0.218197	+	0.975905i	$0.570017\pi$
$504$	0	0
$505$	11618.6	1.02380
$506$	0	0
$507$	7915.94	0.693411
$508$	0	0
$509$	−5551.73	−0.483450	−0.241725	−	0.970345i	$-0.577713\pi$
$509$	−5551.73	−0.483450	−0.241725	+	0.970345i	$0.577713\pi$
$510$	0	0
$511$	1146.68	0.0992683
$512$	0	0
$513$	3101.79	0.266954
$514$	0	0
$515$	22783.0	1.94940
$516$	0	0
$517$	−6031.81	−0.513112
$518$	0	0
$519$	−1133.66	−0.0958812
$520$	0	0
$521$	17576.4	1.47800	0.738999	−	0.673706i	$-0.235299\pi$
$521$	17576.4	1.47800	0.738999	+	0.673706i	$0.235299\pi$
$522$	0	0
$523$	−7511.42	−0.628014	−0.314007	−	0.949421i	$-0.601672\pi$
$523$	−7511.42	−0.628014	−0.314007	+	0.949421i	$0.601672\pi$
$524$	0	0
$525$	6691.99	0.556310
$526$	0	0
$527$	58.6242	0.00484575
$528$	0	0
$529$	−9907.35	−0.814280
$530$	0	0
$531$	938.151	0.0766710
$532$	0	0
$533$	3.85975	0.000313667 0
$534$	0	0
$535$	121.886	0.00984968
$536$	0	0
$537$	15191.7	1.22080
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	8054.90	0.640124	0.320062	−	0.947397i	$-0.396296\pi$
$541$	8054.90	0.640124	0.320062	+	0.947397i	$0.396296\pi$
$542$	0	0
$543$	14665.1	1.15901
$544$	0	0
$545$	−407.321	−0.0320142
$546$	0	0
$547$	−23349.5	−1.82514	−0.912572	−	0.408916i	$-0.865907\pi$
$547$	−23349.5	−1.82514	−0.912572	+	0.408916i	$0.865907\pi$
$548$	0	0
$549$	−2456.87	−0.190996
$550$	0	0
$551$	2962.52	0.229052
$552$	0	0
$553$	838.309	0.0644639
$554$	0	0
$555$	−4808.78	−0.367786
$556$	0	0
$557$	18939.2	1.44072	0.720358	−	0.693603i	$-0.243978\pi$
$557$	18939.2	1.44072	0.720358	+	0.693603i	$0.243978\pi$
$558$	0	0
$559$	1.85278	0.000140186 0
$560$	0	0
$561$	−1662.87	−0.125145
$562$	0	0
$563$	−16253.6	−1.21671	−0.608357	−	0.793664i	$-0.708171\pi$
$563$	−16253.6	−1.21671	−0.608357	+	0.793664i	$0.708171\pi$
$564$	0	0
$565$	−5222.16	−0.388846
$566$	0	0
$567$	1078.11	0.0798525
$568$	0	0
$569$	19058.2	1.40415	0.702076	−	0.712102i	$-0.252257\pi$
$569$	19058.2	1.40415	0.702076	+	0.712102i	$0.252257\pi$
$570$	0	0
$571$	−12738.5	−0.933606	−0.466803	−	0.884361i	$-0.654594\pi$
$571$	−12738.5	−0.933606	−0.466803	+	0.884361i	$0.654594\pi$
$572$	0	0
$573$	−4602.39	−0.335545
$574$	0	0
$575$	−12612.6	−0.914753
$576$	0	0
$577$	−20068.9	−1.44797	−0.723986	−	0.689814i	$-0.757692\pi$
$577$	−20068.9	−1.44797	−0.723986	+	0.689814i	$0.757692\pi$
$578$	0	0
$579$	−7721.45	−0.554219
$580$	0	0
$581$	−606.700	−0.0433222
$582$	0	0
$583$	7712.09	0.547860
$584$	0	0
$585$	5.04442	0.000356515 0
$586$	0	0
$587$	21358.6	1.50181	0.750906	−	0.660409i	$-0.229617\pi$
$587$	21358.6	1.50181	0.750906	+	0.660409i	$0.229617\pi$
$588$	0	0
$589$	29.3259	0.00205153
$590$	0	0
$591$	−16209.0	−1.12817
$592$	0	0
$593$	14535.8	1.00660	0.503299	−	0.864113i	$-0.332120\pi$
$593$	14535.8	1.00660	0.503299	+	0.864113i	$0.332120\pi$
$594$	0	0
$595$	5802.37	0.399788
$596$	0	0
$597$	1883.80	0.129144
$598$	0	0
$599$	−9745.25	−0.664742	−0.332371	−	0.943149i	$-0.607849\pi$
$599$	−9745.25	−0.664742	−0.332371	+	0.943149i	$0.607849\pi$
$600$	0	0
$601$	16431.8	1.11525	0.557626	−	0.830092i	$-0.311712\pi$
$601$	16431.8	1.11525	0.557626	+	0.830092i	$0.311712\pi$
$602$	0	0
$603$	2356.39	0.159137
$604$	0	0
$605$	−2390.57	−0.160645
$606$	0	0
$607$	4270.19	0.285538	0.142769	−	0.989756i	$-0.454399\pi$
$607$	4270.19	0.285538	0.142769	+	0.989756i	$0.454399\pi$
$608$	0	0
$609$	3560.12	0.236886
$610$	0	0
$611$	−9.98778	−0.000661313 0
$612$	0	0
$613$	28529.3	1.87975	0.939876	−	0.341516i	$-0.110940\pi$
$613$	28529.3	1.87975	0.939876	+	0.341516i	$0.110940\pi$
$614$	0	0
$615$	15084.6	0.989056
$616$	0	0
$617$	−14310.8	−0.933761	−0.466880	−	0.884320i	$-0.654622\pi$
$617$	−14310.8	−0.933761	−0.466880	+	0.884320i	$0.654622\pi$
$618$	0	0
$619$	15177.9	0.985542	0.492771	−	0.870159i	$-0.335984\pi$
$619$	15177.9	0.985542	0.492771	+	0.870159i	$0.335984\pi$
$620$	0	0
$621$	−7025.33	−0.453972
$622$	0	0
$623$	−3549.70	−0.228276
$624$	0	0
$625$	21608.4	1.38294
$626$	0	0
$627$	−831.825	−0.0529823
$628$	0	0
$629$	−2834.25	−0.179665
$630$	0	0
$631$	−14998.5	−0.946246	−0.473123	−	0.880996i	$-0.656873\pi$
$631$	−14998.5	−0.946246	−0.473123	+	0.880996i	$0.656873\pi$
$632$	0	0
$633$	−13056.9	−0.819849
$634$	0	0
$635$	−5923.51	−0.370185
$636$	0	0
$637$	0.892503	5.55137e−5 0
$638$	0	0
$639$	1414.51	0.0875702
$640$	0	0
$641$	−14138.0	−0.871169	−0.435584	−	0.900148i	$-0.643458\pi$
$641$	−14138.0	−0.871169	−0.435584	+	0.900148i	$0.643458\pi$
$642$	0	0
$643$	−15748.4	−0.965872	−0.482936	−	0.875656i	$-0.660430\pi$
$643$	−15748.4	−0.965872	−0.482936	+	0.875656i	$0.660430\pi$
$644$	0	0
$645$	7240.99	0.442036
$646$	0	0
$647$	−17544.5	−1.06606	−0.533032	−	0.846095i	$-0.678948\pi$
$647$	−17544.5	−1.06606	−0.533032	+	0.846095i	$0.678948\pi$
$648$	0	0
$649$	−736.178	−0.0445262
$650$	0	0
$651$	35.2416	0.00212170
$652$	0	0
$653$	5024.02	0.301080	0.150540	−	0.988604i	$-0.451899\pi$
$653$	5024.02	0.301080	0.150540	+	0.988604i	$0.451899\pi$
$654$	0	0
$655$	22371.4	1.33454
$656$	0	0
$657$	2296.29	0.136357
$658$	0	0
$659$	31316.9	1.85119	0.925593	−	0.378519i	$-0.123567\pi$
$659$	31316.9	1.85119	0.925593	+	0.378519i	$0.123567\pi$
$660$	0	0
$661$	−14852.3	−0.873959	−0.436979	−	0.899472i	$-0.643952\pi$
$661$	−14852.3	−0.873959	−0.436979	+	0.899472i	$0.643952\pi$
$662$	0	0
$663$	−2.75346	−0.000161290 0
$664$	0	0
$665$	2902.55	0.169257
$666$	0	0
$667$	−6709.88	−0.389517
$668$	0	0
$669$	19450.8	1.12408
$670$	0	0
$671$	1927.93	0.110920
$672$	0	0
$673$	16168.6	0.926081	0.463040	−	0.886337i	$-0.346758\pi$
$673$	16168.6	0.926081	0.463040	+	0.886337i	$0.346758\pi$
$674$	0	0
$675$	39213.1	2.23602
$676$	0	0
$677$	−21355.9	−1.21237	−0.606185	−	0.795324i	$-0.707301\pi$
$677$	−21355.9	−1.21237	−0.606185	+	0.795324i	$0.707301\pi$
$678$	0	0
$679$	8576.04	0.484710
$680$	0	0
$681$	−16453.9	−0.925868
$682$	0	0
$683$	−5894.79	−0.330246	−0.165123	−	0.986273i	$-0.552802\pi$
$683$	−5894.79	−0.330246	−0.165123	+	0.986273i	$0.552802\pi$
$684$	0	0
$685$	−48075.0	−2.68154
$686$	0	0
$687$	−18983.7	−1.05426
$688$	0	0
$689$	12.7701	0.000706097 0
$690$	0	0
$691$	15676.2	0.863026	0.431513	−	0.902107i	$-0.357980\pi$
$691$	15676.2	0.863026	0.431513	+	0.902107i	$0.357980\pi$
$692$	0	0
$693$	1079.38	0.0591662
$694$	0	0
$695$	−62196.8	−3.39462
$696$	0	0
$697$	8890.73	0.483157
$698$	0	0
$699$	17634.9	0.954239
$700$	0	0
$701$	−35134.8	−1.89304	−0.946521	−	0.322642i	$-0.895429\pi$
$701$	−35134.8	−1.89304	−0.946521	+	0.322642i	$0.895429\pi$
$702$	0	0
$703$	−1417.80	−0.0760642
$704$	0	0
$705$	−39034.0	−2.08526
$706$	0	0
$707$	4116.57	0.218981
$708$	0	0
$709$	14524.4	0.769357	0.384678	−	0.923051i	$-0.374312\pi$
$709$	14524.4	0.769357	0.384678	+	0.923051i	$0.374312\pi$
$710$	0	0
$711$	1678.76	0.0885491
$712$	0	0
$713$	−66.4210	−0.00348876
$714$	0	0
$715$	−3.95842	−0.000207044 0
$716$	0	0
$717$	−12007.3	−0.625414
$718$	0	0
$719$	−31740.2	−1.64633	−0.823163	−	0.567805i	$-0.807793\pi$
$719$	−31740.2	−1.64633	−0.823163	+	0.567805i	$0.807793\pi$
$720$	0	0
$721$	8072.23	0.416957
$722$	0	0
$723$	−3014.48	−0.155062
$724$	0	0
$725$	37452.3	1.91854
$726$	0	0
$727$	−13098.8	−0.668236	−0.334118	−	0.942531i	$-0.608438\pi$
$727$	−13098.8	−0.668236	−0.334118	+	0.942531i	$0.608438\pi$
$728$	0	0
$729$	16536.4	0.840138
$730$	0	0
$731$	4267.78	0.215936
$732$	0	0
$733$	15118.2	0.761808	0.380904	−	0.924615i	$-0.375613\pi$
$733$	15118.2	0.761808	0.380904	+	0.924615i	$0.375613\pi$
$734$	0	0
$735$	3488.06	0.175046
$736$	0	0
$737$	−1849.08	−0.0924178
$738$	0	0
$739$	−18486.4	−0.920209	−0.460105	−	0.887865i	$-0.652188\pi$
$739$	−18486.4	−0.920209	−0.460105	+	0.887865i	$0.652188\pi$
$740$	0	0
$741$	−1.37738	−6.82850e−5 0
$742$	0	0
$743$	−30178.4	−1.49009	−0.745045	−	0.667014i	$-0.767572\pi$
$743$	−30178.4	−1.49009	−0.745045	+	0.667014i	$0.767572\pi$
$744$	0	0
$745$	−13755.8	−0.676475
$746$	0	0
$747$	−1214.95	−0.0595084
$748$	0	0
$749$	43.1853	0.00210675
$750$	0	0
$751$	−17412.8	−0.846072	−0.423036	−	0.906113i	$-0.639036\pi$
$751$	−17412.8	−0.846072	−0.423036	+	0.906113i	$0.639036\pi$
$752$	0	0
$753$	−20149.0	−0.975128
$754$	0	0
$755$	35426.6	1.70769
$756$	0	0
$757$	−21680.2	−1.04093	−0.520463	−	0.853885i	$-0.674240\pi$
$757$	−21680.2	−1.04093	−0.520463	+	0.853885i	$0.674240\pi$
$758$	0	0
$759$	1884.02	0.0900997
$760$	0	0
$761$	38057.3	1.81285	0.906423	−	0.422372i	$-0.138802\pi$
$761$	38057.3	1.81285	0.906423	+	0.422372i	$0.138802\pi$
$762$	0	0
$763$	−144.318	−0.00684751
$764$	0	0
$765$	11619.6	0.549159
$766$	0	0
$767$	−1.21900	−5.73866e−5 0
$768$	0	0
$769$	−10764.9	−0.504799	−0.252400	−	0.967623i	$-0.581220\pi$
$769$	−10764.9	−0.504799	−0.252400	+	0.967623i	$0.581220\pi$
$770$	0	0
$771$	21094.4	0.985339
$772$	0	0
$773$	−38553.1	−1.79387	−0.896934	−	0.442164i	$-0.854211\pi$
$773$	−38553.1	−1.79387	−0.896934	+	0.442164i	$0.854211\pi$
$774$	0	0
$775$	370.740	0.0171837
$776$	0	0
$777$	−1703.80	−0.0786658
$778$	0	0
$779$	4447.46	0.204553
$780$	0	0
$781$	−1109.99	−0.0508559
$782$	0	0
$783$	20861.2	0.952133
$784$	0	0
$785$	−4416.26	−0.200794
$786$	0	0
$787$	25991.8	1.17727	0.588633	−	0.808400i	$-0.299666\pi$
$787$	25991.8	1.17727	0.588633	+	0.808400i	$0.299666\pi$
$788$	0	0
$789$	2604.17	0.117504
$790$	0	0
$791$	−1850.26	−0.0831703
$792$	0	0
$793$	3.19236	0.000142956 0
$794$	0	0
$795$	49907.7	2.22647
$796$	0	0
$797$	24168.6	1.07415	0.537075	−	0.843535i	$-0.319529\pi$
$797$	24168.6	1.07415	0.537075	+	0.843535i	$0.319529\pi$
$798$	0	0
$799$	−23006.3	−1.01866
$800$	0	0
$801$	−7108.47	−0.313565
$802$	0	0
$803$	−1801.92	−0.0791887
$804$	0	0
$805$	−6574.07	−0.287833
$806$	0	0
$807$	29121.5	1.27029
$808$	0	0
$809$	−31498.7	−1.36889	−0.684447	−	0.729063i	$-0.739956\pi$
$809$	−31498.7	−1.36889	−0.684447	+	0.729063i	$0.739956\pi$
$810$	0	0
$811$	36557.5	1.58287	0.791435	−	0.611253i	$-0.209334\pi$
$811$	36557.5	1.58287	0.791435	+	0.611253i	$0.209334\pi$
$812$	0	0
$813$	−12718.0	−0.548633
$814$	0	0
$815$	4407.36	0.189427
$816$	0	0
$817$	2134.89	0.0914204
$818$	0	0
$819$	1.78729	7.62550e−5 0
$820$	0	0
$821$	18550.4	0.788567	0.394284	−	0.918989i	$-0.370993\pi$
$821$	18550.4	0.788567	0.394284	+	0.918989i	$0.370993\pi$
$822$	0	0
$823$	−15610.7	−0.661186	−0.330593	−	0.943773i	$-0.607249\pi$
$823$	−15610.7	−0.661186	−0.330593	+	0.943773i	$0.607249\pi$
$824$	0	0
$825$	−10516.0	−0.443781
$826$	0	0
$827$	−41474.6	−1.74391	−0.871956	−	0.489585i	$-0.837148\pi$
$827$	−41474.6	−1.74391	−0.871956	+	0.489585i	$0.837148\pi$
$828$	0	0
$829$	−14751.7	−0.618031	−0.309015	−	0.951057i	$-0.599999\pi$
$829$	−14751.7	−0.618031	−0.309015	+	0.951057i	$0.599999\pi$
$830$	0	0
$831$	2353.97	0.0982650
$832$	0	0
$833$	2055.83	0.0855107
$834$	0	0
$835$	57808.3	2.39586
$836$	0	0
$837$	206.505	0.00852790
$838$	0	0
$839$	−8387.23	−0.345124	−0.172562	−	0.984999i	$-0.555205\pi$
$839$	−8387.23	−0.345124	−0.172562	+	0.984999i	$0.555205\pi$
$840$	0	0
$841$	−4464.47	−0.183052
$842$	0	0
$843$	−2178.34	−0.0889988
$844$	0	0
$845$	43405.6	1.76710
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	18977.8	0.767159
$850$	0	0
$851$	3211.20	0.129352
$852$	0	0
$853$	−34914.0	−1.40145	−0.700723	−	0.713434i	$-0.747139\pi$
$853$	−34914.0	−1.40145	−0.700723	+	0.713434i	$0.747139\pi$
$854$	0	0
$855$	5812.53	0.232496
$856$	0	0
$857$	20481.8	0.816389	0.408194	−	0.912895i	$-0.366159\pi$
$857$	20481.8	0.816389	0.408194	+	0.912895i	$0.366159\pi$
$858$	0	0
$859$	−12310.1	−0.488959	−0.244480	−	0.969654i	$-0.578617\pi$
$859$	−12310.1	−0.488959	−0.244480	+	0.969654i	$0.578617\pi$
$860$	0	0
$861$	5344.61	0.211549
$862$	0	0
$863$	−42003.0	−1.65678	−0.828389	−	0.560153i	$-0.810742\pi$
$863$	−42003.0	−1.65678	−0.828389	+	0.560153i	$0.810742\pi$
$864$	0	0
$865$	−6216.23	−0.244345
$866$	0	0
$867$	11359.4	0.444967
$868$	0	0
$869$	−1317.34	−0.0514244
$870$	0	0
$871$	−3.06181	−0.000119111 0
$872$	0	0
$873$	17174.0	0.665810
$874$	0	0
$875$	19407.1	0.749806
$876$	0	0
$877$	−45845.5	−1.76521	−0.882606	−	0.470113i	$-0.844213\pi$
$877$	−45845.5	−1.76521	−0.882606	+	0.470113i	$0.844213\pi$
$878$	0	0
$879$	−13225.4	−0.507489
$880$	0	0
$881$	26032.0	0.995506	0.497753	−	0.867319i	$-0.334159\pi$
$881$	26032.0	0.995506	0.497753	+	0.867319i	$0.334159\pi$
$882$	0	0
$883$	−36576.2	−1.39398	−0.696992	−	0.717079i	$-0.745479\pi$
$883$	−36576.2	−1.39398	−0.696992	+	0.717079i	$0.745479\pi$
$884$	0	0
$885$	−4764.07	−0.180952
$886$	0	0
$887$	−21330.3	−0.807443	−0.403721	−	0.914882i	$-0.632283\pi$
$887$	−21330.3	−0.807443	−0.403721	+	0.914882i	$0.632283\pi$
$888$	0	0
$889$	−2098.76	−0.0791788
$890$	0	0
$891$	−1694.17	−0.0637003
$892$	0	0
$893$	−11508.6	−0.431266
$894$	0	0
$895$	83300.8	3.11110
$896$	0	0
$897$	3.11966	0.000116123 0
$898$	0	0
$899$	197.233	0.00731710
$900$	0	0
$901$	29415.2	1.08764
$902$	0	0
$903$	2565.55	0.0945472
$904$	0	0
$905$	80413.5	2.95363
$906$	0	0
$907$	19146.0	0.700918	0.350459	−	0.936578i	$-0.386026\pi$
$907$	19146.0	0.700918	0.350459	+	0.936578i	$0.386026\pi$
$908$	0	0
$909$	8243.66	0.300798
$910$	0	0
$911$	−23738.7	−0.863334	−0.431667	−	0.902033i	$-0.642074\pi$
$911$	−23738.7	−0.863334	−0.431667	+	0.902033i	$0.642074\pi$
$912$	0	0
$913$	953.386	0.0345591
$914$	0	0
$915$	12476.3	0.450770
$916$	0	0
$917$	7926.41	0.285445
$918$	0	0
$919$	−20401.4	−0.732298	−0.366149	−	0.930556i	$-0.619324\pi$
$919$	−20401.4	−0.732298	−0.366149	+	0.930556i	$0.619324\pi$
$920$	0	0
$921$	8295.78	0.296803
$922$	0	0
$923$	−1.83797	−6.55444e−5 0
$924$	0	0
$925$	−17923.9	−0.637117
$926$	0	0
$927$	16165.1	0.572742
$928$	0	0
$929$	−9767.25	−0.344944	−0.172472	−	0.985014i	$-0.555175\pi$
$929$	−9767.25	−0.344944	−0.172472	+	0.985014i	$0.555175\pi$
$930$	0	0
$931$	1028.40	0.0362025
$932$	0	0
$933$	17801.2	0.624637
$934$	0	0
$935$	−9118.01	−0.318921
$936$	0	0
$937$	−22232.7	−0.775145	−0.387572	−	0.921839i	$-0.626686\pi$
$937$	−22232.7	−0.775145	−0.387572	+	0.921839i	$0.626686\pi$
$938$	0	0
$939$	26941.5	0.936318
$940$	0	0
$941$	−15489.2	−0.536591	−0.268296	−	0.963337i	$-0.586460\pi$
$941$	−15489.2	−0.536591	−0.268296	+	0.963337i	$0.586460\pi$
$942$	0	0
$943$	−10073.2	−0.347855
$944$	0	0
$945$	20439.0	0.703577
$946$	0	0
$947$	37543.3	1.28827	0.644136	−	0.764911i	$-0.277217\pi$
$947$	37543.3	1.28827	0.644136	+	0.764911i	$0.277217\pi$
$948$	0	0
$949$	−2.98372	−0.000102061 0
$950$	0	0
$951$	−17237.9	−0.587777
$952$	0	0
$953$	−35368.1	−1.20219	−0.601095	−	0.799178i	$-0.705269\pi$
$953$	−35368.1	−1.20219	−0.601095	+	0.799178i	$0.705269\pi$
$954$	0	0
$955$	−25236.3	−0.855108
$956$	0	0
$957$	−5594.48	−0.188969
$958$	0	0
$959$	−17033.4	−0.573554
$960$	0	0
$961$	−29789.0	−0.999934
$962$	0	0
$963$	86.4809	0.00289388
$964$	0	0
$965$	−42339.1	−1.41238
$966$	0	0
$967$	5195.61	0.172781	0.0863907	−	0.996261i	$-0.472467\pi$
$967$	5195.61	0.172781	0.0863907	+	0.996261i	$0.472467\pi$
$968$	0	0
$969$	−3172.72	−0.105183
$970$	0	0
$971$	14597.0	0.482431	0.241215	−	0.970472i	$-0.422454\pi$
$971$	14597.0	0.482431	0.241215	+	0.970472i	$0.422454\pi$
$972$	0	0
$973$	−22036.9	−0.726074
$974$	0	0
$975$	−17.4129	−0.000571958 0
$976$	0	0
$977$	−7870.81	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$977$	−7870.81	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$978$	0	0
$979$	5578.10	0.182101
$980$	0	0
$981$	−289.004	−0.00940590
$982$	0	0
$983$	20006.6	0.649146	0.324573	−	0.945861i	$-0.394779\pi$
$983$	20006.6	0.649146	0.324573	+	0.945861i	$0.394779\pi$
$984$	0	0
$985$	−88879.2	−2.87505
$986$	0	0
$987$	−13830.1	−0.446016
$988$	0	0
$989$	−4835.38	−0.155466
$990$	0	0
$991$	−30910.5	−0.990823	−0.495412	−	0.868658i	$-0.664983\pi$
$991$	−30910.5	−0.990823	−0.495412	+	0.868658i	$0.664983\pi$
$992$	0	0
$993$	28879.7	0.922932
$994$	0	0
$995$	10329.4	0.329111
$996$	0	0
$997$	31305.6	0.994441	0.497221	−	0.867624i	$-0.334354\pi$
$997$	31305.6	0.994441	0.497221	+	0.867624i	$0.334354\pi$
$998$	0	0
$999$	−9983.73	−0.316187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.bb.1.3		6
4.3	odd	2		616.4.a.i.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.i.1.4	✓	6	4.3	odd	2
1232.4.a.bb.1.3		6	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-3.60307 q^{3} -19.7567 q^{5} -7.00000 q^{7} -14.0179 q^{9} +O(q^{10})\)	\(q-3.60307 q^{3} -19.7567 q^{5} -7.00000 q^{7} -14.0179 q^{9} +11.0000 q^{11} +0.0182143 q^{13} +71.1849 q^{15} +41.9558 q^{17} +20.9878 q^{19} +25.2215 q^{21} -47.5358 q^{23} +265.329 q^{25} +147.790 q^{27} +141.154 q^{29} +1.39728 q^{31} -39.6338 q^{33} +138.297 q^{35} -67.5533 q^{37} -0.0656275 q^{39} +211.907 q^{41} +101.721 q^{43} +276.948 q^{45} -548.347 q^{47} +49.0000 q^{49} -151.170 q^{51} +701.099 q^{53} -217.324 q^{55} -75.6205 q^{57} -66.9253 q^{59} +175.267 q^{61} +98.1252 q^{63} -0.359856 q^{65} -168.099 q^{67} +171.275 q^{69} -100.908 q^{71} -163.811 q^{73} -955.999 q^{75} -77.0000 q^{77} -119.758 q^{79} -154.016 q^{81} +86.6715 q^{83} -828.910 q^{85} -508.589 q^{87} +507.100 q^{89} -0.127500 q^{91} -5.03451 q^{93} -414.651 q^{95} -1225.15 q^{97} -154.197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 42 q^{7} + 60 q^{9}+O(q^{10}) \) 6 * q + 2 * q^5 - 42 * q^7 + 60 * q^9	\( 6 q + 2 q^{5} - 42 q^{7} + 60 q^{9} + 66 q^{11} - 6 q^{13} - 126 q^{15} - 14 q^{17} - 80 q^{19} - 254 q^{23} + 220 q^{25} - 90 q^{27} + 132 q^{29} + 52 q^{31} - 14 q^{35} - 518 q^{37} - 332 q^{39} + 486 q^{41} - 428 q^{43} - 244 q^{45} - 790 q^{47} + 294 q^{49} - 640 q^{51} - 40 q^{53} + 22 q^{55} + 2276 q^{57} - 436 q^{59} + 1034 q^{61} - 420 q^{63} + 800 q^{65} - 562 q^{67} + 530 q^{69} - 2474 q^{71} - 902 q^{73} - 1986 q^{75} - 462 q^{77} - 1636 q^{79} + 570 q^{81} - 3016 q^{83} + 236 q^{85} - 4340 q^{87} + 1750 q^{89} + 42 q^{91} + 150 q^{93} - 3628 q^{95} - 1250 q^{97} + 660 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 - 42 * q^7 + 60 * q^9 + 66 * q^11 - 6 * q^13 - 126 * q^15 - 14 * q^17 - 80 * q^19 - 254 * q^23 + 220 * q^25 - 90 * q^27 + 132 * q^29 + 52 * q^31 - 14 * q^35 - 518 * q^37 - 332 * q^39 + 486 * q^41 - 428 * q^43 - 244 * q^45 - 790 * q^47 + 294 * q^49 - 640 * q^51 - 40 * q^53 + 22 * q^55 + 2276 * q^57 - 436 * q^59 + 1034 * q^61 - 420 * q^63 + 800 * q^65 - 562 * q^67 + 530 * q^69 - 2474 * q^71 - 902 * q^73 - 1986 * q^75 - 462 * q^77 - 1636 * q^79 + 570 * q^81 - 3016 * q^83 + 236 * q^85 - 4340 * q^87 + 1750 * q^89 + 42 * q^91 + 150 * q^93 - 3628 * q^95 - 1250 * q^97 + 660 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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