LMFDB - Embedded newform 1232.4.a.s.1.2

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:	$4$
Coefficient field:	4.4.522072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 5x + 1 $ x^4 - x^3 - 12x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.20317$ 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)

$f(q)$	$=$	$q-6.57251 q^{3} +15.5514 q^{5} +7.00000 q^{7} +16.1978 q^{9} +O(q^{10})$	$q-6.57251 q^{3} +15.5514 q^{5} +7.00000 q^{7} +16.1978 q^{9} +11.0000 q^{11} +74.3459 q^{13} -102.211 q^{15} -94.0836 q^{17} -135.682 q^{19} -46.0075 q^{21} -81.1793 q^{23} +116.845 q^{25} +70.9972 q^{27} -53.4259 q^{29} +9.50536 q^{31} -72.2976 q^{33} +108.860 q^{35} -9.14224 q^{37} -488.639 q^{39} -339.461 q^{41} -433.078 q^{43} +251.899 q^{45} +54.4784 q^{47} +49.0000 q^{49} +618.365 q^{51} +123.830 q^{53} +171.065 q^{55} +891.773 q^{57} +534.396 q^{59} -358.624 q^{61} +113.385 q^{63} +1156.18 q^{65} +694.318 q^{67} +533.551 q^{69} +278.330 q^{71} -886.688 q^{73} -767.963 q^{75} +77.0000 q^{77} +185.631 q^{79} -903.972 q^{81} +122.624 q^{83} -1463.13 q^{85} +351.142 q^{87} -847.086 q^{89} +520.422 q^{91} -62.4740 q^{93} -2110.04 q^{95} +1002.49 q^{97} +178.176 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10}) $ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9	$ 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100}) $ 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 + 44 * q^11 + 58 * q^13 - 284 * q^15 + 4 * q^17 - 258 * q^19 - 98 * q^21 - 8 * q^23 + 80 * q^25 - 428 * q^27 - 396 * q^29 + 56 * q^31 - 154 * q^33 + 70 * q^35 + 84 * q^37 + 412 * q^39 + 52 * q^41 - 408 * q^43 + 826 * q^45 - 8 * q^47 + 196 * q^49 + 388 * q^51 + 624 * q^53 + 110 * q^55 + 48 * q^57 + 238 * q^59 - 162 * q^61 + 532 * q^63 - 32 * q^65 - 1340 * q^67 + 2416 * q^69 - 1788 * q^71 + 1456 * q^73 + 806 * q^75 + 308 * q^77 + 1324 * q^79 + 1444 * q^81 - 450 * q^83 - 1736 * q^85 - 588 * q^87 - 3072 * q^89 + 406 * q^91 - 1264 * q^93 - 24 * q^95 - 652 * q^97 + 836 * 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
$3$	−6.57251	−1.26488	−0.632440	−	0.774610i	$-0.717946\pi$
$3$	−6.57251	−1.26488	−0.632440	+	0.774610i	$0.717946\pi$
$4$	0	0
$5$	15.5514	1.39096	0.695478	−	0.718547i	$-0.255193\pi$
$5$	15.5514	1.39096	0.695478	+	0.718547i	$0.255193\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	16.1978	0.599920
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	74.3459	1.58614	0.793071	−	0.609129i	$-0.208481\pi$
$13$	74.3459	1.58614	0.793071	+	0.609129i	$0.208481\pi$
$14$	0	0
$15$	−102.211	−1.75939
$16$	0	0
$17$	−94.0836	−1.34227	−0.671136	−	0.741334i	$-0.734193\pi$
$17$	−94.0836	−1.34227	−0.671136	+	0.741334i	$0.734193\pi$
$18$	0	0
$19$	−135.682	−1.63830	−0.819149	−	0.573581i	$-0.805554\pi$
$19$	−135.682	−1.63830	−0.819149	+	0.573581i	$0.805554\pi$
$20$	0	0
$21$	−46.0075	−0.478080
$22$	0	0
$23$	−81.1793	−0.735959	−0.367979	−	0.929834i	$-0.619950\pi$
$23$	−81.1793	−0.735959	−0.367979	+	0.929834i	$0.619950\pi$
$24$	0	0
$25$	116.845	0.934758
$26$	0	0
$27$	70.9972	0.506053
$28$	0	0
$29$	−53.4259	−0.342102	−0.171051	−	0.985262i	$-0.554716\pi$
$29$	−53.4259	−0.342102	−0.171051	+	0.985262i	$0.554716\pi$
$30$	0	0
$31$	9.50536	0.0550714	0.0275357	−	0.999621i	$-0.491234\pi$
$31$	9.50536	0.0550714	0.0275357	+	0.999621i	$0.491234\pi$
$32$	0	0
$33$	−72.2976	−0.381376
$34$	0	0
$35$	108.860	0.525732
$36$	0	0
$37$	−9.14224	−0.0406209	−0.0203105	−	0.999794i	$-0.506465\pi$
$37$	−9.14224	−0.0406209	−0.0203105	+	0.999794i	$0.506465\pi$
$38$	0	0
$39$	−488.639	−2.00628
$40$	0	0
$41$	−339.461	−1.29305	−0.646523	−	0.762894i	$-0.723778\pi$
$41$	−339.461	−1.29305	−0.646523	+	0.762894i	$0.723778\pi$
$42$	0	0
$43$	−433.078	−1.53590	−0.767950	−	0.640509i	$-0.778723\pi$
$43$	−433.078	−1.53590	−0.767950	+	0.640509i	$0.778723\pi$
$44$	0	0
$45$	251.899	0.834463
$46$	0	0
$47$	54.4784	0.169074	0.0845371	−	0.996420i	$-0.473059\pi$
$47$	54.4784	0.169074	0.0845371	+	0.996420i	$0.473059\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	618.365	1.69781
$52$	0	0
$53$	123.830	0.320932	0.160466	−	0.987041i	$-0.448700\pi$
$53$	123.830	0.320932	0.160466	+	0.987041i	$0.448700\pi$
$54$	0	0
$55$	171.065	0.419389
$56$	0	0
$57$	891.773	2.07225
$58$	0	0
$59$	534.396	1.17919	0.589596	−	0.807698i	$-0.299287\pi$
$59$	534.396	1.17919	0.589596	+	0.807698i	$0.299287\pi$
$60$	0	0
$61$	−358.624	−0.752740	−0.376370	−	0.926469i	$-0.622828\pi$
$61$	−358.624	−0.752740	−0.376370	+	0.926469i	$0.622828\pi$
$62$	0	0
$63$	113.385	0.226749
$64$	0	0
$65$	1156.18	2.20625
$66$	0	0
$67$	694.318	1.26604	0.633019	−	0.774137i	$-0.281816\pi$
$67$	694.318	1.26604	0.633019	+	0.774137i	$0.281816\pi$
$68$	0	0
$69$	533.551	0.930899
$70$	0	0
$71$	278.330	0.465236	0.232618	−	0.972568i	$-0.425271\pi$
$71$	278.330	0.465236	0.232618	+	0.972568i	$0.425271\pi$
$72$	0	0
$73$	−886.688	−1.42163	−0.710815	−	0.703379i	$-0.751674\pi$
$73$	−886.688	−1.42163	−0.710815	+	0.703379i	$0.751674\pi$
$74$	0	0
$75$	−767.963	−1.18236
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	185.631	0.264369	0.132184	−	0.991225i	$-0.457801\pi$
$79$	185.631	0.264369	0.132184	+	0.991225i	$0.457801\pi$
$80$	0	0
$81$	−903.972	−1.24002
$82$	0	0
$83$	122.624	0.162166	0.0810830	−	0.996707i	$-0.474162\pi$
$83$	122.624	0.162166	0.0810830	+	0.996707i	$0.474162\pi$
$84$	0	0
$85$	−1463.13	−1.86704
$86$	0	0
$87$	351.142	0.432717
$88$	0	0
$89$	−847.086	−1.00889	−0.504443	−	0.863445i	$-0.668302\pi$
$89$	−847.086	−1.00889	−0.504443	+	0.863445i	$0.668302\pi$
$90$	0	0
$91$	520.422	0.599506
$92$	0	0
$93$	−62.4740	−0.0696586
$94$	0	0
$95$	−2110.04	−2.27880
$96$	0	0
$97$	1002.49	1.04935	0.524676	−	0.851302i	$-0.324186\pi$
$97$	1002.49	1.04935	0.524676	+	0.851302i	$0.324186\pi$
$98$	0	0
$99$	178.176	0.180883
$100$	0	0
$101$	1124.79	1.10812	0.554062	−	0.832476i	$-0.313077\pi$
$101$	1124.79	1.10812	0.554062	+	0.832476i	$0.313077\pi$
$102$	0	0
$103$	−966.118	−0.924218	−0.462109	−	0.886823i	$-0.652907\pi$
$103$	−966.118	−0.924218	−0.462109	+	0.886823i	$0.652907\pi$
$104$	0	0
$105$	−715.480	−0.664988
$106$	0	0
$107$	144.202	0.130286	0.0651428	−	0.997876i	$-0.479250\pi$
$107$	144.202	0.130286	0.0651428	+	0.997876i	$0.479250\pi$
$108$	0	0
$109$	−1875.32	−1.64792	−0.823961	−	0.566647i	$-0.808240\pi$
$109$	−1875.32	−1.64792	−0.823961	+	0.566647i	$0.808240\pi$
$110$	0	0
$111$	60.0874	0.0513806
$112$	0	0
$113$	1207.46	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$113$	1207.46	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$114$	0	0
$115$	−1262.45	−1.02369
$116$	0	0
$117$	1204.24	0.951559
$118$	0	0
$119$	−658.585	−0.507331
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	2231.11	1.63555
$124$	0	0
$125$	−126.825	−0.0907483
$126$	0	0
$127$	−1143.56	−0.799009	−0.399504	−	0.916731i	$-0.630818\pi$
$127$	−1143.56	−0.799009	−0.399504	+	0.916731i	$0.630818\pi$
$128$	0	0
$129$	2846.41	1.94273
$130$	0	0
$131$	−2478.40	−1.65297	−0.826484	−	0.562961i	$-0.809662\pi$
$131$	−2478.40	−1.65297	−0.826484	+	0.562961i	$0.809662\pi$
$132$	0	0
$133$	−949.776	−0.619218
$134$	0	0
$135$	1104.10	0.703897
$136$	0	0
$137$	−835.661	−0.521134	−0.260567	−	0.965456i	$-0.583909\pi$
$137$	−835.661	−0.521134	−0.260567	+	0.965456i	$0.583909\pi$
$138$	0	0
$139$	1726.99	1.05382	0.526912	−	0.849920i	$-0.323350\pi$
$139$	1726.99	1.05382	0.526912	+	0.849920i	$0.323350\pi$
$140$	0	0
$141$	−358.060	−0.213859
$142$	0	0
$143$	817.805	0.478240
$144$	0	0
$145$	−830.846	−0.475848
$146$	0	0
$147$	−322.053	−0.180697
$148$	0	0
$149$	−3454.95	−1.89960	−0.949799	−	0.312859i	$-0.898713\pi$
$149$	−3454.95	−1.89960	−0.949799	+	0.312859i	$0.898713\pi$
$150$	0	0
$151$	2468.17	1.33018	0.665089	−	0.746764i	$-0.268394\pi$
$151$	2468.17	1.33018	0.665089	+	0.746764i	$0.268394\pi$
$152$	0	0
$153$	−1523.95	−0.805256
$154$	0	0
$155$	147.821	0.0766018
$156$	0	0
$157$	−1561.27	−0.793649	−0.396824	−	0.917895i	$-0.629888\pi$
$157$	−1561.27	−0.793649	−0.396824	+	0.917895i	$0.629888\pi$
$158$	0	0
$159$	−813.875	−0.405940
$160$	0	0
$161$	−568.255	−0.278166
$162$	0	0
$163$	−3458.59	−1.66195	−0.830973	−	0.556312i	$-0.812216\pi$
$163$	−3458.59	−1.66195	−0.830973	+	0.556312i	$0.812216\pi$
$164$	0	0
$165$	−1124.33	−0.530477
$166$	0	0
$167$	972.527	0.450637	0.225319	−	0.974285i	$-0.427658\pi$
$167$	972.527	0.450637	0.225319	+	0.974285i	$0.427658\pi$
$168$	0	0
$169$	3330.32	1.51585
$170$	0	0
$171$	−2197.76	−0.982848
$172$	0	0
$173$	−1154.18	−0.507230	−0.253615	−	0.967305i	$-0.581620\pi$
$173$	−1154.18	−0.507230	−0.253615	+	0.967305i	$0.581620\pi$
$174$	0	0
$175$	817.914	0.353305
$176$	0	0
$177$	−3512.32	−1.49154
$178$	0	0
$179$	−259.234	−0.108246	−0.0541231	−	0.998534i	$-0.517236\pi$
$179$	−259.234	−0.108246	−0.0541231	+	0.998534i	$0.517236\pi$
$180$	0	0
$181$	−2121.49	−0.871209	−0.435604	−	0.900138i	$-0.643465\pi$
$181$	−2121.49	−0.871209	−0.435604	+	0.900138i	$0.643465\pi$
$182$	0	0
$183$	2357.06	0.952126
$184$	0	0
$185$	−142.174	−0.0565019
$186$	0	0
$187$	−1034.92	−0.404710
$188$	0	0
$189$	496.981	0.191270
$190$	0	0
$191$	−2918.27	−1.10554	−0.552772	−	0.833332i	$-0.686430\pi$
$191$	−2918.27	−1.10554	−0.552772	+	0.833332i	$0.686430\pi$
$192$	0	0
$193$	3757.48	1.40139	0.700697	−	0.713459i	$-0.252872\pi$
$193$	3757.48	1.40139	0.700697	+	0.713459i	$0.252872\pi$
$194$	0	0
$195$	−7599.00	−2.79065
$196$	0	0
$197$	1608.39	0.581689	0.290845	−	0.956770i	$-0.406064\pi$
$197$	1608.39	0.581689	0.290845	+	0.956770i	$0.406064\pi$
$198$	0	0
$199$	−2865.53	−1.02076	−0.510382	−	0.859948i	$-0.670496\pi$
$199$	−2865.53	−1.02076	−0.510382	+	0.859948i	$0.670496\pi$
$200$	0	0
$201$	−4563.41	−1.60138
$202$	0	0
$203$	−373.981	−0.129302
$204$	0	0
$205$	−5279.08	−1.79857
$206$	0	0
$207$	−1314.93	−0.441517
$208$	0	0
$209$	−1492.50	−0.493965
$210$	0	0
$211$	−821.996	−0.268192	−0.134096	−	0.990968i	$-0.542813\pi$
$211$	−821.996	−0.268192	−0.134096	+	0.990968i	$0.542813\pi$
$212$	0	0
$213$	−1829.33	−0.588467
$214$	0	0
$215$	−6734.95	−2.13637
$216$	0	0
$217$	66.5375	0.0208150
$218$	0	0
$219$	5827.76	1.79819
$220$	0	0
$221$	−6994.73	−2.12903
$222$	0	0
$223$	−109.532	−0.0328916	−0.0164458	−	0.999865i	$-0.505235\pi$
$223$	−109.532	−0.0328916	−0.0164458	+	0.999865i	$0.505235\pi$
$224$	0	0
$225$	1892.63	0.560780
$226$	0	0
$227$	−3023.03	−0.883900	−0.441950	−	0.897040i	$-0.645713\pi$
$227$	−3023.03	−0.883900	−0.441950	+	0.897040i	$0.645713\pi$
$228$	0	0
$229$	−2278.75	−0.657571	−0.328786	−	0.944405i	$-0.606639\pi$
$229$	−2278.75	−0.657571	−0.328786	+	0.944405i	$0.606639\pi$
$230$	0	0
$231$	−506.083	−0.144146
$232$	0	0
$233$	−1864.08	−0.524121	−0.262061	−	0.965051i	$-0.584402\pi$
$233$	−1864.08	−0.524121	−0.262061	+	0.965051i	$0.584402\pi$
$234$	0	0
$235$	847.213	0.235175
$236$	0	0
$237$	−1220.06	−0.334395
$238$	0	0
$239$	−1404.69	−0.380174	−0.190087	−	0.981767i	$-0.560877\pi$
$239$	−1404.69	−0.380174	−0.190087	+	0.981767i	$0.560877\pi$
$240$	0	0
$241$	3879.32	1.03688	0.518442	−	0.855113i	$-0.326512\pi$
$241$	3879.32	1.03688	0.518442	+	0.855113i	$0.326512\pi$
$242$	0	0
$243$	4024.43	1.06242
$244$	0	0
$245$	762.017	0.198708
$246$	0	0
$247$	−10087.4	−2.59857
$248$	0	0
$249$	−805.950	−0.205121
$250$	0	0
$251$	2143.54	0.539040	0.269520	−	0.962995i	$-0.413135\pi$
$251$	2143.54	0.539040	0.269520	+	0.962995i	$0.413135\pi$
$252$	0	0
$253$	−892.972	−0.221900
$254$	0	0
$255$	9616.42	2.36158
$256$	0	0
$257$	7288.62	1.76907	0.884537	−	0.466471i	$-0.154475\pi$
$257$	7288.62	1.76907	0.884537	+	0.466471i	$0.154475\pi$
$258$	0	0
$259$	−63.9957	−0.0153533
$260$	0	0
$261$	−865.385	−0.205234
$262$	0	0
$263$	−2670.52	−0.626127	−0.313064	−	0.949732i	$-0.601355\pi$
$263$	−2670.52	−0.626127	−0.313064	+	0.949732i	$0.601355\pi$
$264$	0	0
$265$	1925.73	0.446402
$266$	0	0
$267$	5567.48	1.27612
$268$	0	0
$269$	1073.80	0.243387	0.121693	−	0.992568i	$-0.461168\pi$
$269$	1073.80	0.243387	0.121693	+	0.992568i	$0.461168\pi$
$270$	0	0
$271$	−3624.88	−0.812530	−0.406265	−	0.913755i	$-0.633169\pi$
$271$	−3624.88	−0.812530	−0.406265	+	0.913755i	$0.633169\pi$
$272$	0	0
$273$	−3420.47	−0.758302
$274$	0	0
$275$	1285.29	0.281840
$276$	0	0
$277$	−4905.35	−1.06402	−0.532010	−	0.846738i	$-0.678563\pi$
$277$	−4905.35	−1.06402	−0.532010	+	0.846738i	$0.678563\pi$
$278$	0	0
$279$	153.966	0.0330384
$280$	0	0
$281$	−2661.90	−0.565109	−0.282555	−	0.959251i	$-0.591182\pi$
$281$	−2661.90	−0.565109	−0.282555	+	0.959251i	$0.591182\pi$
$282$	0	0
$283$	4367.36	0.917359	0.458680	−	0.888602i	$-0.348323\pi$
$283$	4367.36	0.917359	0.458680	+	0.888602i	$0.348323\pi$
$284$	0	0
$285$	13868.3	2.88241
$286$	0	0
$287$	−2376.23	−0.488726
$288$	0	0
$289$	3938.72	0.801693
$290$	0	0
$291$	−6588.86	−1.32730
$292$	0	0
$293$	1992.30	0.397240	0.198620	−	0.980077i	$-0.436354\pi$
$293$	1992.30	0.397240	0.198620	+	0.980077i	$0.436354\pi$
$294$	0	0
$295$	8310.58	1.64021
$296$	0	0
$297$	780.969	0.152581
$298$	0	0
$299$	−6035.35	−1.16734
$300$	0	0
$301$	−3031.54	−0.580516
$302$	0	0
$303$	−7392.67	−1.40164
$304$	0	0
$305$	−5577.10	−1.04703
$306$	0	0
$307$	−7633.53	−1.41912	−0.709558	−	0.704647i	$-0.751105\pi$
$307$	−7633.53	−1.41912	−0.709558	+	0.704647i	$0.751105\pi$
$308$	0	0
$309$	6349.82	1.16902
$310$	0	0
$311$	−1453.52	−0.265020	−0.132510	−	0.991182i	$-0.542304\pi$
$311$	−1453.52	−0.265020	−0.132510	+	0.991182i	$0.542304\pi$
$312$	0	0
$313$	1936.27	0.349663	0.174831	−	0.984598i	$-0.444062\pi$
$313$	1936.27	0.349663	0.174831	+	0.984598i	$0.444062\pi$
$314$	0	0
$315$	1763.29	0.315397
$316$	0	0
$317$	−1534.36	−0.271856	−0.135928	−	0.990719i	$-0.543402\pi$
$317$	−1534.36	−0.271856	−0.135928	+	0.990719i	$0.543402\pi$
$318$	0	0
$319$	−587.685	−0.103148
$320$	0	0
$321$	−947.770	−0.164795
$322$	0	0
$323$	12765.5	2.19904
$324$	0	0
$325$	8686.94	1.48266
$326$	0	0
$327$	12325.6	2.08442
$328$	0	0
$329$	381.349	0.0639041
$330$	0	0
$331$	−3807.42	−0.632251	−0.316125	−	0.948717i	$-0.602382\pi$
$331$	−3807.42	−0.632251	−0.316125	+	0.948717i	$0.602382\pi$
$332$	0	0
$333$	−148.085	−0.0243693
$334$	0	0
$335$	10797.6	1.76100
$336$	0	0
$337$	−4724.09	−0.763614	−0.381807	−	0.924242i	$-0.624698\pi$
$337$	−4724.09	−0.763614	−0.381807	+	0.924242i	$0.624698\pi$
$338$	0	0
$339$	−7936.05	−1.27147
$340$	0	0
$341$	104.559	0.0166046
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	8297.45	1.29484
$346$	0	0
$347$	−6122.41	−0.947171	−0.473586	−	0.880748i	$-0.657041\pi$
$347$	−6122.41	−0.947171	−0.473586	+	0.880748i	$0.657041\pi$
$348$	0	0
$349$	6778.93	1.03974	0.519868	−	0.854247i	$-0.325981\pi$
$349$	6778.93	1.03974	0.519868	+	0.854247i	$0.325981\pi$
$350$	0	0
$351$	5278.35	0.802672
$352$	0	0
$353$	−3486.98	−0.525760	−0.262880	−	0.964829i	$-0.584672\pi$
$353$	−3486.98	−0.525760	−0.262880	+	0.964829i	$0.584672\pi$
$354$	0	0
$355$	4328.42	0.647123
$356$	0	0
$357$	4328.55	0.641713
$358$	0	0
$359$	−3230.16	−0.474878	−0.237439	−	0.971402i	$-0.576308\pi$
$359$	−3230.16	−0.474878	−0.237439	+	0.971402i	$0.576308\pi$
$360$	0	0
$361$	11550.7	1.68402
$362$	0	0
$363$	−795.273	−0.114989
$364$	0	0
$365$	−13789.2	−1.97742
$366$	0	0
$367$	8190.16	1.16491	0.582456	−	0.812862i	$-0.302092\pi$
$367$	8190.16	1.16491	0.582456	+	0.812862i	$0.302092\pi$
$368$	0	0
$369$	−5498.54	−0.775725
$370$	0	0
$371$	866.812	0.121301
$372$	0	0
$373$	5008.53	0.695260	0.347630	−	0.937632i	$-0.386987\pi$
$373$	5008.53	0.695260	0.347630	+	0.937632i	$0.386987\pi$
$374$	0	0
$375$	833.556	0.114786
$376$	0	0
$377$	−3972.00	−0.542622
$378$	0	0
$379$	−1522.26	−0.206314	−0.103157	−	0.994665i	$-0.532894\pi$
$379$	−1522.26	−0.206314	−0.103157	+	0.994665i	$0.532894\pi$
$380$	0	0
$381$	7516.02	1.01065
$382$	0	0
$383$	−8520.08	−1.13670	−0.568349	−	0.822787i	$-0.692418\pi$
$383$	−8520.08	−1.13670	−0.568349	+	0.822787i	$0.692418\pi$
$384$	0	0
$385$	1197.45	0.158514
$386$	0	0
$387$	−7014.93	−0.921418
$388$	0	0
$389$	1588.83	0.207088	0.103544	−	0.994625i	$-0.466982\pi$
$389$	1588.83	0.207088	0.103544	+	0.994625i	$0.466982\pi$
$390$	0	0
$391$	7637.64	0.987856
$392$	0	0
$393$	16289.3	2.09080
$394$	0	0
$395$	2886.82	0.367725
$396$	0	0
$397$	−1483.46	−0.187539	−0.0937694	−	0.995594i	$-0.529892\pi$
$397$	−1483.46	−0.187539	−0.0937694	+	0.995594i	$0.529892\pi$
$398$	0	0
$399$	6242.41	0.783236
$400$	0	0
$401$	10189.5	1.26892	0.634460	−	0.772956i	$-0.281222\pi$
$401$	10189.5	1.26892	0.634460	+	0.772956i	$0.281222\pi$
$402$	0	0
$403$	706.685	0.0873510
$404$	0	0
$405$	−14058.0	−1.72481
$406$	0	0
$407$	−100.565	−0.0122477
$408$	0	0
$409$	3465.96	0.419024	0.209512	−	0.977806i	$-0.432812\pi$
$409$	3465.96	0.419024	0.209512	+	0.977806i	$0.432812\pi$
$410$	0	0
$411$	5492.39	0.659171
$412$	0	0
$413$	3740.77	0.445693
$414$	0	0
$415$	1906.98	0.225566
$416$	0	0
$417$	−11350.7	−1.33296
$418$	0	0
$419$	−2754.83	−0.321199	−0.160599	−	0.987020i	$-0.551343\pi$
$419$	−2754.83	−0.321199	−0.160599	+	0.987020i	$0.551343\pi$
$420$	0	0
$421$	−3140.19	−0.363524	−0.181762	−	0.983343i	$-0.558180\pi$
$421$	−3140.19	−0.363524	−0.181762	+	0.983343i	$0.558180\pi$
$422$	0	0
$423$	882.433	0.101431
$424$	0	0
$425$	−10993.2	−1.25470
$426$	0	0
$427$	−2510.37	−0.284509
$428$	0	0
$429$	−5375.03	−0.604916
$430$	0	0
$431$	4476.22	0.500260	0.250130	−	0.968212i	$-0.419527\pi$
$431$	4476.22	0.500260	0.250130	+	0.968212i	$0.419527\pi$
$432$	0	0
$433$	1646.00	0.182683	0.0913416	−	0.995820i	$-0.470884\pi$
$433$	1646.00	0.182683	0.0913416	+	0.995820i	$0.470884\pi$
$434$	0	0
$435$	5460.74	0.601891
$436$	0	0
$437$	11014.6	1.20572
$438$	0	0
$439$	7241.05	0.787236	0.393618	−	0.919274i	$-0.371223\pi$
$439$	7241.05	0.787236	0.393618	+	0.919274i	$0.371223\pi$
$440$	0	0
$441$	793.695	0.0857029
$442$	0	0
$443$	−5887.27	−0.631405	−0.315703	−	0.948858i	$-0.602240\pi$
$443$	−5887.27	−0.631405	−0.315703	+	0.948858i	$0.602240\pi$
$444$	0	0
$445$	−13173.3	−1.40332
$446$	0	0
$447$	22707.7	2.40276
$448$	0	0
$449$	6378.42	0.670415	0.335207	−	0.942144i	$-0.391194\pi$
$449$	6378.42	0.670415	0.335207	+	0.942144i	$0.391194\pi$
$450$	0	0
$451$	−3734.07	−0.389868
$452$	0	0
$453$	−16222.1	−1.68252
$454$	0	0
$455$	8093.26	0.833886
$456$	0	0
$457$	−18368.0	−1.88013	−0.940064	−	0.340998i	$-0.889235\pi$
$457$	−18368.0	−1.88013	−0.940064	+	0.340998i	$0.889235\pi$
$458$	0	0
$459$	−6679.67	−0.679260
$460$	0	0
$461$	17510.6	1.76909	0.884546	−	0.466453i	$-0.154468\pi$
$461$	17510.6	1.76909	0.884546	+	0.466453i	$0.154468\pi$
$462$	0	0
$463$	12732.5	1.27803	0.639016	−	0.769193i	$-0.279342\pi$
$463$	12732.5	1.27803	0.639016	+	0.769193i	$0.279342\pi$
$464$	0	0
$465$	−971.556	−0.0968921
$466$	0	0
$467$	4997.23	0.495170	0.247585	−	0.968866i	$-0.420363\pi$
$467$	4997.23	0.495170	0.247585	+	0.968866i	$0.420363\pi$
$468$	0	0
$469$	4860.23	0.478517
$470$	0	0
$471$	10261.5	1.00387
$472$	0	0
$473$	−4763.85	−0.463091
$474$	0	0
$475$	−15853.8	−1.53141
$476$	0	0
$477$	2005.78	0.192534
$478$	0	0
$479$	9075.53	0.865703	0.432851	−	0.901465i	$-0.357507\pi$
$479$	9075.53	0.865703	0.432851	+	0.901465i	$0.357507\pi$
$480$	0	0
$481$	−679.688	−0.0644306
$482$	0	0
$483$	3734.86	0.351847
$484$	0	0
$485$	15590.0	1.45960
$486$	0	0
$487$	3867.63	0.359875	0.179937	−	0.983678i	$-0.442410\pi$
$487$	3867.63	0.359875	0.179937	+	0.983678i	$0.442410\pi$
$488$	0	0
$489$	22731.6	2.10216
$490$	0	0
$491$	−7334.24	−0.674113	−0.337057	−	0.941484i	$-0.609431\pi$
$491$	−7334.24	−0.674113	−0.337057	+	0.941484i	$0.609431\pi$
$492$	0	0
$493$	5026.50	0.459193
$494$	0	0
$495$	2770.88	0.251600
$496$	0	0
$497$	1948.31	0.175843
$498$	0	0
$499$	4365.93	0.391675	0.195838	−	0.980636i	$-0.437257\pi$
$499$	4365.93	0.391675	0.195838	+	0.980636i	$0.437257\pi$
$500$	0	0
$501$	−6391.94	−0.570002
$502$	0	0
$503$	16352.6	1.44956	0.724779	−	0.688981i	$-0.241942\pi$
$503$	16352.6	1.44956	0.724779	+	0.688981i	$0.241942\pi$
$504$	0	0
$505$	17492.0	1.54135
$506$	0	0
$507$	−21888.5	−1.91737
$508$	0	0
$509$	−12108.2	−1.05440	−0.527199	−	0.849742i	$-0.676758\pi$
$509$	−12108.2	−1.05440	−0.527199	+	0.849742i	$0.676758\pi$
$510$	0	0
$511$	−6206.82	−0.537326
$512$	0	0
$513$	−9633.06	−0.829065
$514$	0	0
$515$	−15024.4	−1.28555
$516$	0	0
$517$	599.262	0.0509778
$518$	0	0
$519$	7585.86	0.641584
$520$	0	0
$521$	−5625.67	−0.473062	−0.236531	−	0.971624i	$-0.576010\pi$
$521$	−5625.67	−0.473062	−0.236531	+	0.971624i	$0.576010\pi$
$522$	0	0
$523$	3280.32	0.274261	0.137130	−	0.990553i	$-0.456212\pi$
$523$	3280.32	0.274261	0.137130	+	0.990553i	$0.456212\pi$
$524$	0	0
$525$	−5375.74	−0.446889
$526$	0	0
$527$	−894.298	−0.0739207
$528$	0	0
$529$	−5576.93	−0.458365
$530$	0	0
$531$	8656.06	0.707422
$532$	0	0
$533$	−25237.6	−2.05096
$534$	0	0
$535$	2242.54	0.181221
$536$	0	0
$537$	1703.82	0.136918
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−7772.10	−0.617650	−0.308825	−	0.951119i	$-0.599936\pi$
$541$	−7772.10	−0.617650	−0.308825	+	0.951119i	$0.599936\pi$
$542$	0	0
$543$	13943.5	1.10197
$544$	0	0
$545$	−29163.8	−2.29219
$546$	0	0
$547$	−10022.2	−0.783398	−0.391699	−	0.920094i	$-0.628112\pi$
$547$	−10022.2	−0.783398	−0.391699	+	0.920094i	$0.628112\pi$
$548$	0	0
$549$	−5808.94	−0.451584
$550$	0	0
$551$	7248.95	0.560464
$552$	0	0
$553$	1299.42	0.0999220
$554$	0	0
$555$	934.441	0.0714681
$556$	0	0
$557$	−10849.3	−0.825310	−0.412655	−	0.910887i	$-0.635399\pi$
$557$	−10849.3	−0.825310	−0.412655	+	0.910887i	$0.635399\pi$
$558$	0	0
$559$	−32197.6	−2.43616
$560$	0	0
$561$	6802.01	0.511910
$562$	0	0
$563$	−22019.9	−1.64836	−0.824180	−	0.566328i	$-0.808363\pi$
$563$	−22019.9	−1.64836	−0.824180	+	0.566328i	$0.808363\pi$
$564$	0	0
$565$	18777.7	1.39820
$566$	0	0
$567$	−6327.80	−0.468682
$568$	0	0
$569$	13742.9	1.01253	0.506267	−	0.862377i	$-0.331025\pi$
$569$	13742.9	1.01253	0.506267	+	0.862377i	$0.331025\pi$
$570$	0	0
$571$	−19382.4	−1.42054	−0.710269	−	0.703931i	$-0.751427\pi$
$571$	−19382.4	−1.42054	−0.710269	+	0.703931i	$0.751427\pi$
$572$	0	0
$573$	19180.4	1.39838
$574$	0	0
$575$	−9485.37	−0.687943
$576$	0	0
$577$	−565.348	−0.0407899	−0.0203949	−	0.999792i	$-0.506492\pi$
$577$	−565.348	−0.0407899	−0.0203949	+	0.999792i	$0.506492\pi$
$578$	0	0
$579$	−24696.0	−1.77260
$580$	0	0
$581$	858.371	0.0612930
$582$	0	0
$583$	1362.13	0.0967646
$584$	0	0
$585$	18727.6	1.32358
$586$	0	0
$587$	20727.4	1.45743	0.728714	−	0.684818i	$-0.240118\pi$
$587$	20727.4	1.45743	0.728714	+	0.684818i	$0.240118\pi$
$588$	0	0
$589$	−1289.71	−0.0902233
$590$	0	0
$591$	−10571.1	−0.735767
$592$	0	0
$593$	−2787.10	−0.193006	−0.0965030	−	0.995333i	$-0.530766\pi$
$593$	−2787.10	−0.193006	−0.0965030	+	0.995333i	$0.530766\pi$
$594$	0	0
$595$	−10241.9	−0.705675
$596$	0	0
$597$	18833.7	1.29114
$598$	0	0
$599$	−18935.5	−1.29162	−0.645812	−	0.763497i	$-0.723481\pi$
$599$	−18935.5	−1.29162	−0.645812	+	0.763497i	$0.723481\pi$
$600$	0	0
$601$	15821.3	1.07382	0.536908	−	0.843641i	$-0.319592\pi$
$601$	15821.3	1.07382	0.536908	+	0.843641i	$0.319592\pi$
$602$	0	0
$603$	11246.5	0.759521
$604$	0	0
$605$	1881.71	0.126451
$606$	0	0
$607$	2084.63	0.139394	0.0696972	−	0.997568i	$-0.477797\pi$
$607$	2084.63	0.139394	0.0696972	+	0.997568i	$0.477797\pi$
$608$	0	0
$609$	2458.00	0.163552
$610$	0	0
$611$	4050.25	0.268176
$612$	0	0
$613$	−4395.15	−0.289590	−0.144795	−	0.989462i	$-0.546252\pi$
$613$	−4395.15	−0.289590	−0.144795	+	0.989462i	$0.546252\pi$
$614$	0	0
$615$	34696.8	2.27498
$616$	0	0
$617$	−98.5856	−0.00643259	−0.00321629	−	0.999995i	$-0.501024\pi$
$617$	−98.5856	−0.00643259	−0.00321629	+	0.999995i	$0.501024\pi$
$618$	0	0
$619$	16533.9	1.07359	0.536796	−	0.843712i	$-0.319634\pi$
$619$	16533.9	1.07359	0.536796	+	0.843712i	$0.319634\pi$
$620$	0	0
$621$	−5763.50	−0.372434
$622$	0	0
$623$	−5929.60	−0.381323
$624$	0	0
$625$	−16577.9	−1.06099
$626$	0	0
$627$	9809.50	0.624806
$628$	0	0
$629$	860.135	0.0545243
$630$	0	0
$631$	−22032.9	−1.39004	−0.695022	−	0.718989i	$-0.744605\pi$
$631$	−22032.9	−1.39004	−0.695022	+	0.718989i	$0.744605\pi$
$632$	0	0
$633$	5402.57	0.339231
$634$	0	0
$635$	−17783.8	−1.11139
$636$	0	0
$637$	3642.95	0.226592
$638$	0	0
$639$	4508.35	0.279104
$640$	0	0
$641$	434.281	0.0267598	0.0133799	−	0.999910i	$-0.495741\pi$
$641$	434.281	0.0267598	0.0133799	+	0.999910i	$0.495741\pi$
$642$	0	0
$643$	−10963.1	−0.672381	−0.336190	−	0.941794i	$-0.609139\pi$
$643$	−10963.1	−0.672381	−0.336190	+	0.941794i	$0.609139\pi$
$644$	0	0
$645$	44265.5	2.70225
$646$	0	0
$647$	452.083	0.0274702	0.0137351	−	0.999906i	$-0.495628\pi$
$647$	452.083	0.0274702	0.0137351	+	0.999906i	$0.495628\pi$
$648$	0	0
$649$	5878.35	0.355540
$650$	0	0
$651$	−437.318	−0.0263285
$652$	0	0
$653$	−10438.7	−0.625572	−0.312786	−	0.949824i	$-0.601262\pi$
$653$	−10438.7	−0.625572	−0.312786	+	0.949824i	$0.601262\pi$
$654$	0	0
$655$	−38542.5	−2.29920
$656$	0	0
$657$	−14362.4	−0.852865
$658$	0	0
$659$	8738.79	0.516563	0.258281	−	0.966070i	$-0.416844\pi$
$659$	8738.79	0.516563	0.258281	+	0.966070i	$0.416844\pi$
$660$	0	0
$661$	12849.4	0.756102	0.378051	−	0.925785i	$-0.376594\pi$
$661$	12849.4	0.756102	0.378051	+	0.925785i	$0.376594\pi$
$662$	0	0
$663$	45972.9	2.69297
$664$	0	0
$665$	−14770.3	−0.861305
$666$	0	0
$667$	4337.08	0.251773
$668$	0	0
$669$	719.903	0.0416040
$670$	0	0
$671$	−3944.87	−0.226960
$672$	0	0
$673$	−22926.5	−1.31315	−0.656577	−	0.754259i	$-0.727996\pi$
$673$	−22926.5	−1.31315	−0.656577	+	0.754259i	$0.727996\pi$
$674$	0	0
$675$	8295.66	0.473037
$676$	0	0
$677$	13633.4	0.773965	0.386983	−	0.922087i	$-0.373517\pi$
$677$	13633.4	0.773965	0.386983	+	0.922087i	$0.373517\pi$
$678$	0	0
$679$	7017.41	0.396618
$680$	0	0
$681$	19868.9	1.11803
$682$	0	0
$683$	−33307.5	−1.86600	−0.932998	−	0.359882i	$-0.882817\pi$
$683$	−33307.5	−1.86600	−0.932998	+	0.359882i	$0.882817\pi$
$684$	0	0
$685$	−12995.7	−0.724874
$686$	0	0
$687$	14977.1	0.831748
$688$	0	0
$689$	9206.28	0.509044
$690$	0	0
$691$	10077.3	0.554786	0.277393	−	0.960757i	$-0.410530\pi$
$691$	10077.3	0.554786	0.277393	+	0.960757i	$0.410530\pi$
$692$	0	0
$693$	1247.23	0.0683673
$694$	0	0
$695$	26857.1	1.46582
$696$	0	0
$697$	31937.7	1.73562
$698$	0	0
$699$	12251.7	0.662950
$700$	0	0
$701$	−4621.74	−0.249017	−0.124508	−	0.992219i	$-0.539735\pi$
$701$	−4621.74	−0.249017	−0.124508	+	0.992219i	$0.539735\pi$
$702$	0	0
$703$	1240.44	0.0665492
$704$	0	0
$705$	−5568.31	−0.297468
$706$	0	0
$707$	7873.51	0.418831
$708$	0	0
$709$	17746.6	0.940038	0.470019	−	0.882656i	$-0.344247\pi$
$709$	17746.6	0.940038	0.470019	+	0.882656i	$0.344247\pi$
$710$	0	0
$711$	3006.82	0.158600
$712$	0	0
$713$	−771.638	−0.0405302
$714$	0	0
$715$	12718.0	0.665211
$716$	0	0
$717$	9232.31	0.480875
$718$	0	0
$719$	31652.1	1.64176	0.820879	−	0.571103i	$-0.193484\pi$
$719$	31652.1	1.64176	0.820879	+	0.571103i	$0.193484\pi$
$720$	0	0
$721$	−6762.83	−0.349322
$722$	0	0
$723$	−25496.9	−1.31153
$724$	0	0
$725$	−6242.54	−0.319782
$726$	0	0
$727$	−21610.8	−1.10248	−0.551239	−	0.834348i	$-0.685845\pi$
$727$	−21610.8	−1.10248	−0.551239	+	0.834348i	$0.685845\pi$
$728$	0	0
$729$	−2043.39	−0.103815
$730$	0	0
$731$	40745.5	2.06160
$732$	0	0
$733$	−7665.43	−0.386261	−0.193130	−	0.981173i	$-0.561864\pi$
$733$	−7665.43	−0.386261	−0.193130	+	0.981173i	$0.561864\pi$
$734$	0	0
$735$	−5008.36	−0.251342
$736$	0	0
$737$	7637.50	0.381725
$738$	0	0
$739$	13841.5	0.688996	0.344498	−	0.938787i	$-0.388049\pi$
$739$	13841.5	0.688996	0.344498	+	0.938787i	$0.388049\pi$
$740$	0	0
$741$	66299.7	3.28688
$742$	0	0
$743$	12136.3	0.599243	0.299621	−	0.954058i	$-0.403140\pi$
$743$	12136.3	0.599243	0.299621	+	0.954058i	$0.403140\pi$
$744$	0	0
$745$	−53729.1	−2.64226
$746$	0	0
$747$	1986.25	0.0972867
$748$	0	0
$749$	1009.42	0.0492433
$750$	0	0
$751$	−28866.7	−1.40261	−0.701305	−	0.712861i	$-0.747399\pi$
$751$	−28866.7	−1.40261	−0.701305	+	0.712861i	$0.747399\pi$
$752$	0	0
$753$	−14088.4	−0.681820
$754$	0	0
$755$	38383.4	1.85022
$756$	0	0
$757$	8370.48	0.401889	0.200945	−	0.979603i	$-0.435599\pi$
$757$	8370.48	0.401889	0.200945	+	0.979603i	$0.435599\pi$
$758$	0	0
$759$	5869.06	0.280677
$760$	0	0
$761$	30458.1	1.45086	0.725430	−	0.688296i	$-0.241641\pi$
$761$	30458.1	1.45086	0.725430	+	0.688296i	$0.241641\pi$
$762$	0	0
$763$	−13127.3	−0.622856
$764$	0	0
$765$	−23699.5	−1.12008
$766$	0	0
$767$	39730.1	1.87037
$768$	0	0
$769$	−32239.5	−1.51182	−0.755908	−	0.654678i	$-0.772804\pi$
$769$	−32239.5	−1.51182	−0.755908	+	0.654678i	$0.772804\pi$
$770$	0	0
$771$	−47904.5	−2.23766
$772$	0	0
$773$	−7524.47	−0.350112	−0.175056	−	0.984559i	$-0.556011\pi$
$773$	−7524.47	−0.350112	−0.175056	+	0.984559i	$0.556011\pi$
$774$	0	0
$775$	1110.65	0.0514784
$776$	0	0
$777$	420.612	0.0194200
$778$	0	0
$779$	46058.8	2.11839
$780$	0	0
$781$	3061.63	0.140274
$782$	0	0
$783$	−3793.09	−0.173121
$784$	0	0
$785$	−24279.9	−1.10393
$786$	0	0
$787$	−33196.6	−1.50360	−0.751798	−	0.659394i	$-0.770813\pi$
$787$	−33196.6	−1.50360	−0.751798	+	0.659394i	$0.770813\pi$
$788$	0	0
$789$	17552.0	0.791976
$790$	0	0
$791$	8452.23	0.379932
$792$	0	0
$793$	−26662.3	−1.19395
$794$	0	0
$795$	−12656.9	−0.564645
$796$	0	0
$797$	8817.06	0.391865	0.195932	−	0.980617i	$-0.437227\pi$
$797$	8817.06	0.391865	0.195932	+	0.980617i	$0.437227\pi$
$798$	0	0
$799$	−5125.52	−0.226944
$800$	0	0
$801$	−13721.0	−0.605252
$802$	0	0
$803$	−9753.57	−0.428638
$804$	0	0
$805$	−8837.14	−0.386917
$806$	0	0
$807$	−7057.59	−0.307855
$808$	0	0
$809$	−28930.3	−1.25727	−0.628637	−	0.777699i	$-0.716387\pi$
$809$	−28930.3	−1.25727	−0.628637	+	0.777699i	$0.716387\pi$
$810$	0	0
$811$	45923.3	1.98839	0.994195	−	0.107594i	$-0.0343146\pi$
$811$	45923.3	1.98839	0.994195	+	0.107594i	$0.0343146\pi$
$812$	0	0
$813$	23824.5	1.02775
$814$	0	0
$815$	−53785.7	−2.31169
$816$	0	0
$817$	58761.0	2.51626
$818$	0	0
$819$	8429.71	0.359656
$820$	0	0
$821$	16382.8	0.696423	0.348211	−	0.937416i	$-0.386789\pi$
$821$	16382.8	0.696423	0.348211	+	0.937416i	$0.386789\pi$
$822$	0	0
$823$	32395.3	1.37209	0.686045	−	0.727559i	$-0.259345\pi$
$823$	32395.3	1.37209	0.686045	+	0.727559i	$0.259345\pi$
$824$	0	0
$825$	−8447.60	−0.356494
$826$	0	0
$827$	14329.2	0.602508	0.301254	−	0.953544i	$-0.402595\pi$
$827$	14329.2	0.602508	0.301254	+	0.953544i	$0.402595\pi$
$828$	0	0
$829$	16445.8	0.689005	0.344503	−	0.938785i	$-0.388048\pi$
$829$	16445.8	0.689005	0.344503	+	0.938785i	$0.388048\pi$
$830$	0	0
$831$	32240.4	1.34586
$832$	0	0
$833$	−4610.10	−0.191753
$834$	0	0
$835$	15124.1	0.626816
$836$	0	0
$837$	674.854	0.0278690
$838$	0	0
$839$	−40514.7	−1.66713	−0.833565	−	0.552422i	$-0.813704\pi$
$839$	−40514.7	−1.66713	−0.833565	+	0.552422i	$0.813704\pi$
$840$	0	0
$841$	−21534.7	−0.882967
$842$	0	0
$843$	17495.4	0.714795
$844$	0	0
$845$	51791.0	2.10848
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−28704.5	−1.16035
$850$	0	0
$851$	742.160	0.0298953
$852$	0	0
$853$	−27962.7	−1.12242	−0.561211	−	0.827673i	$-0.689664\pi$
$853$	−27962.7	−1.12242	−0.561211	+	0.827673i	$0.689664\pi$
$854$	0	0
$855$	−34178.2	−1.36710
$856$	0	0
$857$	−4639.52	−0.184928	−0.0924639	−	0.995716i	$-0.529474\pi$
$857$	−4639.52	−0.184928	−0.0924639	+	0.995716i	$0.529474\pi$
$858$	0	0
$859$	−18521.2	−0.735663	−0.367831	−	0.929892i	$-0.619900\pi$
$859$	−18521.2	−0.735663	−0.367831	+	0.929892i	$0.619900\pi$
$860$	0	0
$861$	15617.8	0.618179
$862$	0	0
$863$	−40491.4	−1.59715	−0.798577	−	0.601893i	$-0.794413\pi$
$863$	−40491.4	−1.59715	−0.798577	+	0.601893i	$0.794413\pi$
$864$	0	0
$865$	−17949.1	−0.705534
$866$	0	0
$867$	−25887.3	−1.01405
$868$	0	0
$869$	2041.94	0.0797102
$870$	0	0
$871$	51619.8	2.00812
$872$	0	0
$873$	16238.1	0.629528
$874$	0	0
$875$	−887.772	−0.0342996
$876$	0	0
$877$	−48530.3	−1.86859	−0.934295	−	0.356502i	$-0.883969\pi$
$877$	−48530.3	−1.86859	−0.934295	+	0.356502i	$0.883969\pi$
$878$	0	0
$879$	−13094.4	−0.502461
$880$	0	0
$881$	−11590.0	−0.443219	−0.221609	−	0.975135i	$-0.571131\pi$
$881$	−11590.0	−0.443219	−0.221609	+	0.975135i	$0.571131\pi$
$882$	0	0
$883$	−41900.7	−1.59691	−0.798455	−	0.602054i	$-0.794349\pi$
$883$	−41900.7	−1.59691	−0.798455	+	0.602054i	$0.794349\pi$
$884$	0	0
$885$	−54621.3	−2.07466
$886$	0	0
$887$	17136.8	0.648700	0.324350	−	0.945937i	$-0.394854\pi$
$887$	17136.8	0.648700	0.324350	+	0.945937i	$0.394854\pi$
$888$	0	0
$889$	−8004.89	−0.301997
$890$	0	0
$891$	−9943.69	−0.373879
$892$	0	0
$893$	−7391.75	−0.276994
$894$	0	0
$895$	−4031.44	−0.150566
$896$	0	0
$897$	39667.4	1.47654
$898$	0	0
$899$	−507.832	−0.0188400
$900$	0	0
$901$	−11650.4	−0.430778
$902$	0	0
$903$	19924.8	0.734283
$904$	0	0
$905$	−32992.0	−1.21181
$906$	0	0
$907$	−40350.7	−1.47720	−0.738601	−	0.674143i	$-0.764513\pi$
$907$	−40350.7	−1.47720	−0.738601	+	0.674143i	$0.764513\pi$
$908$	0	0
$909$	18219.1	0.664786
$910$	0	0
$911$	32322.7	1.17552	0.587760	−	0.809035i	$-0.300010\pi$
$911$	32322.7	1.17552	0.587760	+	0.809035i	$0.300010\pi$
$912$	0	0
$913$	1348.87	0.0488949
$914$	0	0
$915$	36655.5	1.32437
$916$	0	0
$917$	−17348.8	−0.624763
$918$	0	0
$919$	−2431.28	−0.0872694	−0.0436347	−	0.999048i	$-0.513894\pi$
$919$	−2431.28	−0.0872694	−0.0436347	+	0.999048i	$0.513894\pi$
$920$	0	0
$921$	50171.4	1.79501
$922$	0	0
$923$	20692.7	0.737930
$924$	0	0
$925$	−1068.22	−0.0379708
$926$	0	0
$927$	−15649.0	−0.554457
$928$	0	0
$929$	40221.0	1.42046	0.710230	−	0.703970i	$-0.248591\pi$
$929$	40221.0	1.42046	0.710230	+	0.703970i	$0.248591\pi$
$930$	0	0
$931$	−6648.43	−0.234042
$932$	0	0
$933$	9553.25	0.335219
$934$	0	0
$935$	−16094.4	−0.562934
$936$	0	0
$937$	41503.4	1.44702	0.723510	−	0.690314i	$-0.242528\pi$
$937$	41503.4	1.44702	0.723510	+	0.690314i	$0.242528\pi$
$938$	0	0
$939$	−12726.2	−0.442282
$940$	0	0
$941$	−9032.78	−0.312923	−0.156461	−	0.987684i	$-0.550009\pi$
$941$	−9032.78	−0.312923	−0.156461	+	0.987684i	$0.550009\pi$
$942$	0	0
$943$	27557.2	0.951629
$944$	0	0
$945$	7728.72	0.266048
$946$	0	0
$947$	−5565.96	−0.190992	−0.0954960	−	0.995430i	$-0.530444\pi$
$947$	−5565.96	−0.190992	−0.0954960	+	0.995430i	$0.530444\pi$
$948$	0	0
$949$	−65921.7	−2.25491
$950$	0	0
$951$	10084.6	0.343865
$952$	0	0
$953$	32657.4	1.11005	0.555024	−	0.831834i	$-0.312709\pi$
$953$	32657.4	1.11005	0.555024	+	0.831834i	$0.312709\pi$
$954$	0	0
$955$	−45383.1	−1.53776
$956$	0	0
$957$	3862.56	0.130469
$958$	0	0
$959$	−5849.63	−0.196970
$960$	0	0
$961$	−29700.6	−0.996967
$962$	0	0
$963$	2335.76	0.0781609
$964$	0	0
$965$	58433.9	1.94928
$966$	0	0
$967$	610.079	0.0202883	0.0101442	−	0.999949i	$-0.496771\pi$
$967$	610.079	0.0202883	0.0101442	+	0.999949i	$0.496771\pi$
$968$	0	0
$969$	−83901.2	−2.78152
$970$	0	0
$971$	45371.4	1.49952	0.749762	−	0.661708i	$-0.230168\pi$
$971$	45371.4	1.49952	0.749762	+	0.661708i	$0.230168\pi$
$972$	0	0
$973$	12088.9	0.398308
$974$	0	0
$975$	−57094.9	−1.87539
$976$	0	0
$977$	27229.0	0.891639	0.445820	−	0.895123i	$-0.352912\pi$
$977$	27229.0	0.891639	0.445820	+	0.895123i	$0.352912\pi$
$978$	0	0
$979$	−9317.94	−0.304191
$980$	0	0
$981$	−30376.2	−0.988622
$982$	0	0
$983$	−7458.49	−0.242003	−0.121002	−	0.992652i	$-0.538611\pi$
$983$	−7458.49	−0.242003	−0.121002	+	0.992652i	$0.538611\pi$
$984$	0	0
$985$	25012.6	0.809104
$986$	0	0
$987$	−2506.42	−0.0808310
$988$	0	0
$989$	35156.9	1.13036
$990$	0	0
$991$	24357.8	0.780778	0.390389	−	0.920650i	$-0.372340\pi$
$991$	24357.8	0.780778	0.390389	+	0.920650i	$0.372340\pi$
$992$	0	0
$993$	25024.3	0.799721
$994$	0	0
$995$	−44562.9	−1.41984
$996$	0	0
$997$	19954.3	0.633862	0.316931	−	0.948449i	$-0.397348\pi$
$997$	19954.3	0.633862	0.316931	+	0.948449i	$0.397348\pi$
$998$	0	0
$999$	−649.074	−0.0205563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.s.1.2		4
4.3	odd	2		77.4.a.d.1.1	✓	4
12.11	even	2		693.4.a.l.1.4		4
20.19	odd	2		1925.4.a.p.1.4		4
28.27	even	2		539.4.a.g.1.1		4
44.43	even	2		847.4.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.d.1.1	✓	4	4.3	odd	2
539.4.a.g.1.1		4	28.27	even	2
693.4.a.l.1.4		4	12.11	even	2
847.4.a.d.1.4		4	44.43	even	2
1232.4.a.s.1.2		4	1.1	even	1	trivial
1925.4.a.p.1.4		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-6.57251 q^{3} +15.5514 q^{5} +7.00000 q^{7} +16.1978 q^{9} +O(q^{10})\)	\(q-6.57251 q^{3} +15.5514 q^{5} +7.00000 q^{7} +16.1978 q^{9} +11.0000 q^{11} +74.3459 q^{13} -102.211 q^{15} -94.0836 q^{17} -135.682 q^{19} -46.0075 q^{21} -81.1793 q^{23} +116.845 q^{25} +70.9972 q^{27} -53.4259 q^{29} +9.50536 q^{31} -72.2976 q^{33} +108.860 q^{35} -9.14224 q^{37} -488.639 q^{39} -339.461 q^{41} -433.078 q^{43} +251.899 q^{45} +54.4784 q^{47} +49.0000 q^{49} +618.365 q^{51} +123.830 q^{53} +171.065 q^{55} +891.773 q^{57} +534.396 q^{59} -358.624 q^{61} +113.385 q^{63} +1156.18 q^{65} +694.318 q^{67} +533.551 q^{69} +278.330 q^{71} -886.688 q^{73} -767.963 q^{75} +77.0000 q^{77} +185.631 q^{79} -903.972 q^{81} +122.624 q^{83} -1463.13 q^{85} +351.142 q^{87} -847.086 q^{89} +520.422 q^{91} -62.4740 q^{93} -2110.04 q^{95} +1002.49 q^{97} +178.176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10}) \) 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9	\( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100}) \) 4 * q - 14 * q^3 + 10 * q^5 + 28 * q^7 + 76 * q^9 + 44 * q^11 + 58 * q^13 - 284 * q^15 + 4 * q^17 - 258 * q^19 - 98 * q^21 - 8 * q^23 + 80 * q^25 - 428 * q^27 - 396 * q^29 + 56 * q^31 - 154 * q^33 + 70 * q^35 + 84 * q^37 + 412 * q^39 + 52 * q^41 - 408 * q^43 + 826 * q^45 - 8 * q^47 + 196 * q^49 + 388 * q^51 + 624 * q^53 + 110 * q^55 + 48 * q^57 + 238 * q^59 - 162 * q^61 + 532 * q^63 - 32 * q^65 - 1340 * q^67 + 2416 * q^69 - 1788 * q^71 + 1456 * q^73 + 806 * q^75 + 308 * q^77 + 1324 * q^79 + 1444 * q^81 - 450 * q^83 - 1736 * q^85 - 588 * q^87 - 3072 * q^89 + 406 * q^91 - 1264 * q^93 - 24 * q^95 - 652 * q^97 + 836 * q^99

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