LMFDB - Embedded newform 1232.4.a.bd.1.7

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 145x^{5} - 10x^{4} + 4790x^{3} - 2452x^{2} - 1496x + 320 $ x^7 - 145x^5 - 10x^4 + 4790x^3 - 2452x^2 - 1496*x + 320
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$10.0610$ 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)

$f(q)$	$=$	$q+10.0610 q^{3} +14.3419 q^{5} +7.00000 q^{7} +74.2235 q^{9} +11.0000 q^{11} +67.8481 q^{13} +144.294 q^{15} -17.1658 q^{17} +13.1002 q^{19} +70.4269 q^{21} -109.440 q^{23} +80.6901 q^{25} +475.115 q^{27} +8.40033 q^{29} +60.6206 q^{31} +110.671 q^{33} +100.393 q^{35} -409.456 q^{37} +682.619 q^{39} -438.921 q^{41} -141.177 q^{43} +1064.51 q^{45} -446.940 q^{47} +49.0000 q^{49} -172.705 q^{51} -138.795 q^{53} +157.761 q^{55} +131.801 q^{57} -681.593 q^{59} +687.672 q^{61} +519.564 q^{63} +973.071 q^{65} -584.435 q^{67} -1101.07 q^{69} -471.809 q^{71} +564.462 q^{73} +811.822 q^{75} +77.0000 q^{77} -175.668 q^{79} +2776.09 q^{81} +309.242 q^{83} -246.190 q^{85} +84.5156 q^{87} -878.091 q^{89} +474.937 q^{91} +609.903 q^{93} +187.882 q^{95} +103.599 q^{97} +816.458 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 6 q^{5} + 49 q^{7} + 101 q^{9} + 77 q^{11} + 88 q^{13} + 106 q^{15} + 134 q^{17} - 14 q^{19} - 42 q^{23} + 343 q^{25} + 30 q^{27} + 482 q^{29} - 50 q^{31} + 42 q^{35} + 152 q^{37} - 72 q^{39} + 234 q^{41}+ \cdots + 1111 q^{99}+O(q^{100}) $ 7 * q + 6 * q^5 + 49 * q^7 + 101 * q^9 + 77 * q^11 + 88 * q^13 + 106 * q^15 + 134 * q^17 - 14 * q^19 - 42 * q^23 + 343 * q^25 + 30 * q^27 + 482 * q^29 - 50 * q^31 + 42 * q^35 + 152 * q^37 - 72 * q^39 + 234 * q^41 - 472 * q^43 + 1384 * q^45 - 728 * q^47 + 343 * q^49 - 1572 * q^51 - 102 * q^53 + 66 * q^55 + 452 * q^57 - 1704 * q^59 + 656 * q^61 + 707 * q^63 + 2840 * q^65 - 1126 * q^67 - 350 * q^69 + 918 * q^71 + 1094 * q^73 - 274 * q^75 + 539 * q^77 - 672 * q^79 + 4503 * q^81 - 782 * q^83 + 1716 * q^85 - 3196 * q^87 - 464 * q^89 + 616 * q^91 + 1826 * q^93 + 1796 * q^95 + 3000 * q^97 + 1111 * q^99

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
$3$	10.0610	1.93624	0.968119	−	0.250491i	$-0.0805919\pi$
$3$	10.0610	1.93624	0.968119	+	0.250491i	$0.0805919\pi$
$4$	0	0
$5$	14.3419	1.28278	0.641389	−	0.767216i	$-0.278358\pi$
$5$	14.3419	1.28278	0.641389	+	0.767216i	$0.278358\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	74.2235	2.74902
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	67.8481	1.44751	0.723757	−	0.690055i	$-0.242414\pi$
$13$	67.8481	1.44751	0.723757	+	0.690055i	$0.242414\pi$
$14$	0	0
$15$	144.294	2.48376
$16$	0	0
$17$	−17.1658	−0.244901	−0.122451	−	0.992475i	$-0.539075\pi$
$17$	−17.1658	−0.244901	−0.122451	+	0.992475i	$0.539075\pi$
$18$	0	0
$19$	13.1002	0.158179	0.0790894	−	0.996868i	$-0.474799\pi$
$19$	13.1002	0.158179	0.0790894	+	0.996868i	$0.474799\pi$
$20$	0	0
$21$	70.4269	0.731829
$22$	0	0
$23$	−109.440	−0.992163	−0.496081	−	0.868276i	$-0.665228\pi$
$23$	−109.440	−0.992163	−0.496081	+	0.868276i	$0.665228\pi$
$24$	0	0
$25$	80.6901	0.645520
$26$	0	0
$27$	475.115	3.38652
$28$	0	0
$29$	8.40033	0.0537897	0.0268949	−	0.999638i	$-0.491438\pi$
$29$	8.40033	0.0537897	0.0268949	+	0.999638i	$0.491438\pi$
$30$	0	0
$31$	60.6206	0.351219	0.175609	−	0.984460i	$-0.443810\pi$
$31$	60.6206	0.351219	0.175609	+	0.984460i	$0.443810\pi$
$32$	0	0
$33$	110.671	0.583798
$34$	0	0
$35$	100.393	0.484845
$36$	0	0
$37$	−409.456	−1.81930	−0.909650	−	0.415375i	$-0.863650\pi$
$37$	−409.456	−1.81930	−0.909650	+	0.415375i	$0.863650\pi$
$38$	0	0
$39$	682.619	2.80273
$40$	0	0
$41$	−438.921	−1.67190	−0.835950	−	0.548806i	$-0.815082\pi$
$41$	−438.921	−1.67190	−0.835950	+	0.548806i	$0.815082\pi$
$42$	0	0
$43$	−141.177	−0.500680	−0.250340	−	0.968158i	$-0.580542\pi$
$43$	−141.177	−0.500680	−0.250340	+	0.968158i	$0.580542\pi$
$44$	0	0
$45$	1064.51	3.52638
$46$	0	0
$47$	−446.940	−1.38708	−0.693541	−	0.720417i	$-0.743950\pi$
$47$	−446.940	−1.38708	−0.693541	+	0.720417i	$0.743950\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−172.705	−0.474187
$52$	0	0
$53$	−138.795	−0.359716	−0.179858	−	0.983693i	$-0.557564\pi$
$53$	−138.795	−0.359716	−0.179858	+	0.983693i	$0.557564\pi$
$54$	0	0
$55$	157.761	0.386772
$56$	0	0
$57$	131.801	0.306272
$58$	0	0
$59$	−681.593	−1.50400	−0.751998	−	0.659165i	$-0.770910\pi$
$59$	−681.593	−1.50400	−0.751998	+	0.659165i	$0.770910\pi$
$60$	0	0
$61$	687.672	1.44340	0.721700	−	0.692206i	$-0.243361\pi$
$61$	687.672	1.44340	0.721700	+	0.692206i	$0.243361\pi$
$62$	0	0
$63$	519.564	1.03903
$64$	0	0
$65$	973.071	1.85684
$66$	0	0
$67$	−584.435	−1.06567	−0.532837	−	0.846218i	$-0.678874\pi$
$67$	−584.435	−1.06567	−0.532837	+	0.846218i	$0.678874\pi$
$68$	0	0
$69$	−1101.07	−1.92106
$70$	0	0
$71$	−471.809	−0.788640	−0.394320	−	0.918973i	$-0.629020\pi$
$71$	−471.809	−0.788640	−0.394320	+	0.918973i	$0.629020\pi$
$72$	0	0
$73$	564.462	0.905004	0.452502	−	0.891763i	$-0.350531\pi$
$73$	564.462	0.905004	0.452502	+	0.891763i	$0.350531\pi$
$74$	0	0
$75$	811.822	1.24988
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	−175.668	−0.250179	−0.125089	−	0.992145i	$-0.539922\pi$
$79$	−175.668	−0.250179	−0.125089	+	0.992145i	$0.539922\pi$
$80$	0	0
$81$	2776.09	3.80808
$82$	0	0
$83$	309.242	0.408961	0.204480	−	0.978871i	$-0.434450\pi$
$83$	309.242	0.408961	0.204480	+	0.978871i	$0.434450\pi$
$84$	0	0
$85$	−246.190	−0.314154
$86$	0	0
$87$	84.5156	0.104150
$88$	0	0
$89$	−878.091	−1.04581	−0.522907	−	0.852390i	$-0.675153\pi$
$89$	−878.091	−1.04581	−0.522907	+	0.852390i	$0.675153\pi$
$90$	0	0
$91$	474.937	0.547109
$92$	0	0
$93$	609.903	0.680043
$94$	0	0
$95$	187.882	0.202908
$96$	0	0
$97$	103.599	0.108443	0.0542213	−	0.998529i	$-0.482732\pi$
$97$	103.599	0.108443	0.0542213	+	0.998529i	$0.482732\pi$
$98$	0	0
$99$	816.458	0.828860
$100$	0	0
$101$	1776.97	1.75065	0.875324	−	0.483537i	$-0.160648\pi$
$101$	1776.97	1.75065	0.875324	+	0.483537i	$0.160648\pi$
$102$	0	0
$103$	−653.008	−0.624687	−0.312344	−	0.949969i	$-0.601114\pi$
$103$	−653.008	−0.624687	−0.312344	+	0.949969i	$0.601114\pi$
$104$	0	0
$105$	1010.06	0.938775
$106$	0	0
$107$	866.625	0.782989	0.391494	−	0.920180i	$-0.371958\pi$
$107$	866.625	0.782989	0.391494	+	0.920180i	$0.371958\pi$
$108$	0	0
$109$	461.141	0.405223	0.202611	−	0.979259i	$-0.435057\pi$
$109$	461.141	0.405223	0.202611	+	0.979259i	$0.435057\pi$
$110$	0	0
$111$	−4119.53	−3.52260
$112$	0	0
$113$	−285.685	−0.237831	−0.118916	−	0.992904i	$-0.537942\pi$
$113$	−285.685	−0.237831	−0.118916	+	0.992904i	$0.537942\pi$
$114$	0	0
$115$	−1569.57	−1.27272
$116$	0	0
$117$	5035.93	3.97924
$118$	0	0
$119$	−120.161	−0.0925640
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−4415.97	−3.23719
$124$	0	0
$125$	−635.489	−0.454719
$126$	0	0
$127$	2290.71	1.60053	0.800266	−	0.599645i	$-0.204691\pi$
$127$	2290.71	1.60053	0.800266	+	0.599645i	$0.204691\pi$
$128$	0	0
$129$	−1420.38	−0.969435
$130$	0	0
$131$	−2548.85	−1.69995	−0.849976	−	0.526821i	$-0.823384\pi$
$131$	−2548.85	−1.69995	−0.849976	+	0.526821i	$0.823384\pi$
$132$	0	0
$133$	91.7016	0.0597860
$134$	0	0
$135$	6814.05	4.34415
$136$	0	0
$137$	−2818.87	−1.75790	−0.878949	−	0.476916i	$-0.841755\pi$
$137$	−2818.87	−1.75790	−0.878949	+	0.476916i	$0.841755\pi$
$138$	0	0
$139$	−180.336	−0.110043	−0.0550213	−	0.998485i	$-0.517523\pi$
$139$	−180.336	−0.110043	−0.0550213	+	0.998485i	$0.517523\pi$
$140$	0	0
$141$	−4496.66	−2.68572
$142$	0	0
$143$	746.330	0.436442
$144$	0	0
$145$	120.477	0.0690003
$146$	0	0
$147$	492.988	0.276605
$148$	0	0
$149$	3222.44	1.77176	0.885882	−	0.463910i	$-0.153554\pi$
$149$	3222.44	1.77176	0.885882	+	0.463910i	$0.153554\pi$
$150$	0	0
$151$	−3385.37	−1.82449	−0.912243	−	0.409650i	$-0.865651\pi$
$151$	−3385.37	−1.82449	−0.912243	+	0.409650i	$0.865651\pi$
$152$	0	0
$153$	−1274.11	−0.673238
$154$	0	0
$155$	869.414	0.450536
$156$	0	0
$157$	2772.58	1.40940	0.704701	−	0.709505i	$-0.251081\pi$
$157$	2772.58	1.40940	0.704701	+	0.709505i	$0.251081\pi$
$158$	0	0
$159$	−1396.41	−0.696496
$160$	0	0
$161$	−766.077	−0.375002
$162$	0	0
$163$	1567.26	0.753113	0.376557	−	0.926394i	$-0.377108\pi$
$163$	1567.26	0.753113	0.376557	+	0.926394i	$0.377108\pi$
$164$	0	0
$165$	1587.23	0.748883
$166$	0	0
$167$	295.988	0.137151	0.0685755	−	0.997646i	$-0.478155\pi$
$167$	295.988	0.137151	0.0685755	+	0.997646i	$0.478155\pi$
$168$	0	0
$169$	2406.37	1.09530
$170$	0	0
$171$	972.345	0.434837
$172$	0	0
$173$	−120.518	−0.0529643	−0.0264822	−	0.999649i	$-0.508431\pi$
$173$	−120.518	−0.0529643	−0.0264822	+	0.999649i	$0.508431\pi$
$174$	0	0
$175$	564.830	0.243984
$176$	0	0
$177$	−6857.50	−2.91210
$178$	0	0
$179$	2892.59	1.20783	0.603917	−	0.797047i	$-0.293606\pi$
$179$	2892.59	1.20783	0.603917	+	0.797047i	$0.293606\pi$
$180$	0	0
$181$	3707.38	1.52247	0.761237	−	0.648474i	$-0.224592\pi$
$181$	3707.38	1.52247	0.761237	+	0.648474i	$0.224592\pi$
$182$	0	0
$183$	6918.66	2.79477
$184$	0	0
$185$	−5872.37	−2.33376
$186$	0	0
$187$	−188.824	−0.0738405
$188$	0	0
$189$	3325.80	1.27998
$190$	0	0
$191$	−421.426	−0.159651	−0.0798253	−	0.996809i	$-0.525436\pi$
$191$	−421.426	−0.159651	−0.0798253	+	0.996809i	$0.525436\pi$
$192$	0	0
$193$	−4215.98	−1.57240	−0.786198	−	0.617974i	$-0.787954\pi$
$193$	−4215.98	−1.57240	−0.786198	+	0.617974i	$0.787954\pi$
$194$	0	0
$195$	9790.06	3.59529
$196$	0	0
$197$	1718.73	0.621597	0.310799	−	0.950476i	$-0.399404\pi$
$197$	1718.73	0.621597	0.310799	+	0.950476i	$0.399404\pi$
$198$	0	0
$199$	126.081	0.0449126	0.0224563	−	0.999748i	$-0.492851\pi$
$199$	126.081	0.0449126	0.0224563	+	0.999748i	$0.492851\pi$
$200$	0	0
$201$	−5879.99	−2.06340
$202$	0	0
$203$	58.8023	0.0203306
$204$	0	0
$205$	−6294.95	−2.14468
$206$	0	0
$207$	−8122.99	−2.72747
$208$	0	0
$209$	144.103	0.0476927
$210$	0	0
$211$	−3816.42	−1.24518	−0.622591	−	0.782548i	$-0.713920\pi$
$211$	−3816.42	−1.24518	−0.622591	+	0.782548i	$0.713920\pi$
$212$	0	0
$213$	−4746.87	−1.52700
$214$	0	0
$215$	−2024.74	−0.642261
$216$	0	0
$217$	424.344	0.132748
$218$	0	0
$219$	5679.05	1.75230
$220$	0	0
$221$	−1164.67	−0.354498
$222$	0	0
$223$	−2772.63	−0.832598	−0.416299	−	0.909228i	$-0.636673\pi$
$223$	−2772.63	−0.832598	−0.416299	+	0.909228i	$0.636673\pi$
$224$	0	0
$225$	5989.10	1.77455
$226$	0	0
$227$	226.228	0.0661467	0.0330733	−	0.999453i	$-0.489471\pi$
$227$	226.228	0.0661467	0.0330733	+	0.999453i	$0.489471\pi$
$228$	0	0
$229$	3682.55	1.06266	0.531331	−	0.847165i	$-0.321692\pi$
$229$	3682.55	1.06266	0.531331	+	0.847165i	$0.321692\pi$
$230$	0	0
$231$	774.696	0.220655
$232$	0	0
$233$	−1924.70	−0.541166	−0.270583	−	0.962697i	$-0.587216\pi$
$233$	−1924.70	−0.541166	−0.270583	+	0.962697i	$0.587216\pi$
$234$	0	0
$235$	−6409.97	−1.77932
$236$	0	0
$237$	−1767.39	−0.484406
$238$	0	0
$239$	67.3679	0.0182329	0.00911646	−	0.999958i	$-0.497098\pi$
$239$	67.3679	0.0182329	0.00911646	+	0.999958i	$0.497098\pi$
$240$	0	0
$241$	1036.47	0.277032	0.138516	−	0.990360i	$-0.455767\pi$
$241$	1036.47	0.277032	0.138516	+	0.990360i	$0.455767\pi$
$242$	0	0
$243$	15102.1	3.98684
$244$	0	0
$245$	702.753	0.183254
$246$	0	0
$247$	888.826	0.228966
$248$	0	0
$249$	3111.28	0.791845
$250$	0	0
$251$	2832.00	0.712169	0.356085	−	0.934454i	$-0.384111\pi$
$251$	2832.00	0.712169	0.356085	+	0.934454i	$0.384111\pi$
$252$	0	0
$253$	−1203.84	−0.299148
$254$	0	0
$255$	−2476.92	−0.608277
$256$	0	0
$257$	6866.93	1.66672	0.833361	−	0.552730i	$-0.186414\pi$
$257$	6866.93	1.66672	0.833361	+	0.552730i	$0.186414\pi$
$258$	0	0
$259$	−2866.19	−0.687631
$260$	0	0
$261$	623.502	0.147869
$262$	0	0
$263$	−3900.01	−0.914391	−0.457196	−	0.889366i	$-0.651146\pi$
$263$	−3900.01	−0.914391	−0.457196	+	0.889366i	$0.651146\pi$
$264$	0	0
$265$	−1990.58	−0.461436
$266$	0	0
$267$	−8834.46	−2.02495
$268$	0	0
$269$	7754.91	1.75772	0.878858	−	0.477084i	$-0.158306\pi$
$269$	7754.91	1.75772	0.878858	+	0.477084i	$0.158306\pi$
$270$	0	0
$271$	638.600	0.143145	0.0715723	−	0.997435i	$-0.477198\pi$
$271$	638.600	0.143145	0.0715723	+	0.997435i	$0.477198\pi$
$272$	0	0
$273$	4778.34	1.05933
$274$	0	0
$275$	887.591	0.194632
$276$	0	0
$277$	2628.38	0.570123	0.285062	−	0.958509i	$-0.407986\pi$
$277$	2628.38	0.570123	0.285062	+	0.958509i	$0.407986\pi$
$278$	0	0
$279$	4499.47	0.965506
$280$	0	0
$281$	−1860.68	−0.395013	−0.197507	−	0.980302i	$-0.563284\pi$
$281$	−1860.68	−0.395013	−0.197507	+	0.980302i	$0.563284\pi$
$282$	0	0
$283$	−607.236	−0.127549	−0.0637746	−	0.997964i	$-0.520314\pi$
$283$	−607.236	−0.127549	−0.0637746	+	0.997964i	$0.520314\pi$
$284$	0	0
$285$	1890.28	0.392879
$286$	0	0
$287$	−3072.44	−0.631918
$288$	0	0
$289$	−4618.33	−0.940023
$290$	0	0
$291$	1042.31	0.209970
$292$	0	0
$293$	7800.95	1.55541	0.777707	−	0.628627i	$-0.216383\pi$
$293$	7800.95	1.55541	0.777707	+	0.628627i	$0.216383\pi$
$294$	0	0
$295$	−9775.33	−1.92929
$296$	0	0
$297$	5226.26	1.02107
$298$	0	0
$299$	−7425.28	−1.43617
$300$	0	0
$301$	−988.236	−0.189239
$302$	0	0
$303$	17878.1	3.38967
$304$	0	0
$305$	9862.52	1.85156
$306$	0	0
$307$	9064.65	1.68517	0.842584	−	0.538565i	$-0.181033\pi$
$307$	9064.65	1.68517	0.842584	+	0.538565i	$0.181033\pi$
$308$	0	0
$309$	−6569.91	−1.20954
$310$	0	0
$311$	5297.32	0.965863	0.482932	−	0.875658i	$-0.339572\pi$
$311$	5297.32	0.965863	0.482932	+	0.875658i	$0.339572\pi$
$312$	0	0
$313$	6918.42	1.24937	0.624684	−	0.780878i	$-0.285228\pi$
$313$	6918.42	1.24937	0.624684	+	0.780878i	$0.285228\pi$
$314$	0	0
$315$	7451.54	1.33285
$316$	0	0
$317$	3381.23	0.599082	0.299541	−	0.954083i	$-0.403166\pi$
$317$	3381.23	0.599082	0.299541	+	0.954083i	$0.403166\pi$
$318$	0	0
$319$	92.4036	0.0162182
$320$	0	0
$321$	8719.11	1.51605
$322$	0	0
$323$	−224.876	−0.0387382
$324$	0	0
$325$	5474.67	0.934400
$326$	0	0
$327$	4639.53	0.784608
$328$	0	0
$329$	−3128.58	−0.524268
$330$	0	0
$331$	−3209.23	−0.532917	−0.266458	−	0.963846i	$-0.585854\pi$
$331$	−3209.23	−0.532917	−0.266458	+	0.963846i	$0.585854\pi$
$332$	0	0
$333$	−30391.2	−5.00129
$334$	0	0
$335$	−8381.91	−1.36702
$336$	0	0
$337$	−4995.77	−0.807528	−0.403764	−	0.914863i	$-0.632298\pi$
$337$	−4995.77	−0.807528	−0.403764	+	0.914863i	$0.632298\pi$
$338$	0	0
$339$	−2874.27	−0.460498
$340$	0	0
$341$	666.826	0.105896
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−15791.4	−2.46430
$346$	0	0
$347$	10101.5	1.56276	0.781380	−	0.624055i	$-0.214516\pi$
$347$	10101.5	1.56276	0.781380	+	0.624055i	$0.214516\pi$
$348$	0	0
$349$	2958.46	0.453761	0.226880	−	0.973923i	$-0.427147\pi$
$349$	2958.46	0.453761	0.226880	+	0.973923i	$0.427147\pi$
$350$	0	0
$351$	32235.7	4.90203
$352$	0	0
$353$	4387.14	0.661485	0.330742	−	0.943721i	$-0.392701\pi$
$353$	4387.14	0.661485	0.330742	+	0.943721i	$0.392701\pi$
$354$	0	0
$355$	−6766.64	−1.01165
$356$	0	0
$357$	−1208.94	−0.179226
$358$	0	0
$359$	−9395.90	−1.38133	−0.690664	−	0.723176i	$-0.742682\pi$
$359$	−9395.90	−1.38133	−0.690664	+	0.723176i	$0.742682\pi$
$360$	0	0
$361$	−6687.38	−0.974979
$362$	0	0
$363$	1217.38	0.176022
$364$	0	0
$365$	8095.46	1.16092
$366$	0	0
$367$	13000.7	1.84913	0.924567	−	0.381019i	$-0.124427\pi$
$367$	13000.7	1.84913	0.924567	+	0.381019i	$0.124427\pi$
$368$	0	0
$369$	−32578.2	−4.59608
$370$	0	0
$371$	−971.565	−0.135960
$372$	0	0
$373$	85.0675	0.0118087	0.00590433	−	0.999983i	$-0.498121\pi$
$373$	85.0675	0.0118087	0.00590433	+	0.999983i	$0.498121\pi$
$374$	0	0
$375$	−6393.64	−0.880444
$376$	0	0
$377$	569.947	0.0778614
$378$	0	0
$379$	3927.28	0.532271	0.266136	−	0.963936i	$-0.414253\pi$
$379$	3927.28	0.532271	0.266136	+	0.963936i	$0.414253\pi$
$380$	0	0
$381$	23046.8	3.09901
$382$	0	0
$383$	10432.6	1.39186	0.695928	−	0.718112i	$-0.254993\pi$
$383$	10432.6	1.39186	0.695928	+	0.718112i	$0.254993\pi$
$384$	0	0
$385$	1104.33	0.146186
$386$	0	0
$387$	−10478.6	−1.37638
$388$	0	0
$389$	8843.08	1.15260	0.576301	−	0.817238i	$-0.304496\pi$
$389$	8843.08	1.15260	0.576301	+	0.817238i	$0.304496\pi$
$390$	0	0
$391$	1878.62	0.242982
$392$	0	0
$393$	−25643.9	−3.29151
$394$	0	0
$395$	−2519.41	−0.320924
$396$	0	0
$397$	−2586.92	−0.327038	−0.163519	−	0.986540i	$-0.552284\pi$
$397$	−2586.92	−0.327038	−0.163519	+	0.986540i	$0.552284\pi$
$398$	0	0
$399$	922.609	0.115760
$400$	0	0
$401$	−900.901	−0.112192	−0.0560958	−	0.998425i	$-0.517865\pi$
$401$	−900.901	−0.112192	−0.0560958	+	0.998425i	$0.517865\pi$
$402$	0	0
$403$	4112.99	0.508394
$404$	0	0
$405$	39814.4	4.88492
$406$	0	0
$407$	−4504.01	−0.548540
$408$	0	0
$409$	−3007.28	−0.363571	−0.181785	−	0.983338i	$-0.558188\pi$
$409$	−3007.28	−0.363571	−0.181785	+	0.983338i	$0.558188\pi$
$410$	0	0
$411$	−28360.6	−3.40371
$412$	0	0
$413$	−4771.15	−0.568457
$414$	0	0
$415$	4435.12	0.524606
$416$	0	0
$417$	−1814.36	−0.213069
$418$	0	0
$419$	−10957.7	−1.27761	−0.638806	−	0.769368i	$-0.720571\pi$
$419$	−10957.7	−1.27761	−0.638806	+	0.769368i	$0.720571\pi$
$420$	0	0
$421$	−9334.16	−1.08057	−0.540284	−	0.841483i	$-0.681683\pi$
$421$	−9334.16	−1.08057	−0.540284	+	0.841483i	$0.681683\pi$
$422$	0	0
$423$	−33173.4	−3.81311
$424$	0	0
$425$	−1385.11	−0.158089
$426$	0	0
$427$	4813.70	0.545554
$428$	0	0
$429$	7508.81	0.845056
$430$	0	0
$431$	−5103.12	−0.570322	−0.285161	−	0.958480i	$-0.592047\pi$
$431$	−5103.12	−0.570322	−0.285161	+	0.958480i	$0.592047\pi$
$432$	0	0
$433$	13928.9	1.54592	0.772959	−	0.634457i	$-0.218776\pi$
$433$	13928.9	1.54592	0.772959	+	0.634457i	$0.218776\pi$
$434$	0	0
$435$	1212.11	0.133601
$436$	0	0
$437$	−1433.68	−0.156939
$438$	0	0
$439$	−568.101	−0.0617631	−0.0308816	−	0.999523i	$-0.509831\pi$
$439$	−568.101	−0.0617631	−0.0308816	+	0.999523i	$0.509831\pi$
$440$	0	0
$441$	3636.95	0.392717
$442$	0	0
$443$	−5257.15	−0.563826	−0.281913	−	0.959440i	$-0.590969\pi$
$443$	−5257.15	−0.563826	−0.281913	+	0.959440i	$0.590969\pi$
$444$	0	0
$445$	−12593.5	−1.34155
$446$	0	0
$447$	32421.0	3.43056
$448$	0	0
$449$	7352.06	0.772751	0.386376	−	0.922342i	$-0.373727\pi$
$449$	7352.06	0.772751	0.386376	+	0.922342i	$0.373727\pi$
$450$	0	0
$451$	−4828.13	−0.504097
$452$	0	0
$453$	−34060.1	−3.53264
$454$	0	0
$455$	6811.50	0.701820
$456$	0	0
$457$	−3224.06	−0.330011	−0.165006	−	0.986293i	$-0.552764\pi$
$457$	−3224.06	−0.330011	−0.165006	+	0.986293i	$0.552764\pi$
$458$	0	0
$459$	−8155.73	−0.829362
$460$	0	0
$461$	6271.52	0.633609	0.316804	−	0.948491i	$-0.397390\pi$
$461$	6271.52	0.633609	0.316804	+	0.948491i	$0.397390\pi$
$462$	0	0
$463$	−3790.89	−0.380513	−0.190257	−	0.981734i	$-0.560932\pi$
$463$	−3790.89	−0.380513	−0.190257	+	0.981734i	$0.560932\pi$
$464$	0	0
$465$	8747.17	0.872344
$466$	0	0
$467$	−17870.2	−1.77074	−0.885370	−	0.464887i	$-0.846095\pi$
$467$	−17870.2	−1.77074	−0.885370	+	0.464887i	$0.846095\pi$
$468$	0	0
$469$	−4091.04	−0.402787
$470$	0	0
$471$	27894.9	2.72894
$472$	0	0
$473$	−1552.94	−0.150961
$474$	0	0
$475$	1057.06	0.102108
$476$	0	0
$477$	−10301.8	−0.988866
$478$	0	0
$479$	−5173.31	−0.493475	−0.246737	−	0.969082i	$-0.579358\pi$
$479$	−5173.31	−0.493475	−0.246737	+	0.969082i	$0.579358\pi$
$480$	0	0
$481$	−27780.8	−2.63346
$482$	0	0
$483$	−7707.50	−0.726094
$484$	0	0
$485$	1485.81	0.139108
$486$	0	0
$487$	−4340.73	−0.403895	−0.201948	−	0.979396i	$-0.564727\pi$
$487$	−4340.73	−0.403895	−0.201948	+	0.979396i	$0.564727\pi$
$488$	0	0
$489$	15768.2	1.45821
$490$	0	0
$491$	−14069.5	−1.29318	−0.646588	−	0.762839i	$-0.723805\pi$
$491$	−14069.5	−1.29318	−0.646588	+	0.762839i	$0.723805\pi$
$492$	0	0
$493$	−144.198	−0.0131732
$494$	0	0
$495$	11709.6	1.06324
$496$	0	0
$497$	−3302.66	−0.298078
$498$	0	0
$499$	10778.6	0.966969	0.483484	−	0.875353i	$-0.339371\pi$
$499$	10778.6	0.966969	0.483484	+	0.875353i	$0.339371\pi$
$500$	0	0
$501$	2977.93	0.265557
$502$	0	0
$503$	5264.94	0.466704	0.233352	−	0.972392i	$-0.425031\pi$
$503$	5264.94	0.466704	0.233352	+	0.972392i	$0.425031\pi$
$504$	0	0
$505$	25485.2	2.24569
$506$	0	0
$507$	24210.5	2.12076
$508$	0	0
$509$	2261.00	0.196890	0.0984448	−	0.995143i	$-0.468613\pi$
$509$	2261.00	0.196890	0.0984448	+	0.995143i	$0.468613\pi$
$510$	0	0
$511$	3951.24	0.342059
$512$	0	0
$513$	6224.11	0.535675
$514$	0	0
$515$	−9365.37	−0.801335
$516$	0	0
$517$	−4916.34	−0.418221
$518$	0	0
$519$	−1212.53	−0.102552
$520$	0	0
$521$	−14775.1	−1.24244	−0.621219	−	0.783637i	$-0.713362\pi$
$521$	−14775.1	−1.24244	−0.621219	+	0.783637i	$0.713362\pi$
$522$	0	0
$523$	−1812.18	−0.151512	−0.0757562	−	0.997126i	$-0.524137\pi$
$523$	−1812.18	−0.151512	−0.0757562	+	0.997126i	$0.524137\pi$
$524$	0	0
$525$	5682.75	0.472411
$526$	0	0
$527$	−1040.60	−0.0860139
$528$	0	0
$529$	−189.967	−0.0156133
$530$	0	0
$531$	−50590.2	−4.13451
$532$	0	0
$533$	−29779.9	−2.42010
$534$	0	0
$535$	12429.1	1.00440
$536$	0	0
$537$	29102.3	2.33866
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	8240.98	0.654912	0.327456	−	0.944866i	$-0.393809\pi$
$541$	8240.98	0.654912	0.327456	+	0.944866i	$0.393809\pi$
$542$	0	0
$543$	37300.0	2.94787
$544$	0	0
$545$	6613.64	0.519811
$546$	0	0
$547$	10443.2	0.816303	0.408151	−	0.912914i	$-0.366174\pi$
$547$	10443.2	0.816303	0.408151	+	0.912914i	$0.366174\pi$
$548$	0	0
$549$	51041.4	3.96793
$550$	0	0
$551$	110.046	0.00850840
$552$	0	0
$553$	−1229.67	−0.0945588
$554$	0	0
$555$	−59081.9	−4.51871
$556$	0	0
$557$	−17113.0	−1.30180	−0.650898	−	0.759166i	$-0.725607\pi$
$557$	−17113.0	−1.30180	−0.650898	+	0.759166i	$0.725607\pi$
$558$	0	0
$559$	−9578.57	−0.724741
$560$	0	0
$561$	−1899.76	−0.142973
$562$	0	0
$563$	7326.10	0.548416	0.274208	−	0.961670i	$-0.411584\pi$
$563$	7326.10	0.548416	0.274208	+	0.961670i	$0.411584\pi$
$564$	0	0
$565$	−4097.26	−0.305085
$566$	0	0
$567$	19432.6	1.43932
$568$	0	0
$569$	16201.1	1.19365	0.596823	−	0.802373i	$-0.296429\pi$
$569$	16201.1	1.19365	0.596823	+	0.802373i	$0.296429\pi$
$570$	0	0
$571$	−6085.63	−0.446017	−0.223008	−	0.974817i	$-0.571588\pi$
$571$	−6085.63	−0.446017	−0.223008	+	0.974817i	$0.571588\pi$
$572$	0	0
$573$	−4239.96	−0.309122
$574$	0	0
$575$	−8830.69	−0.640461
$576$	0	0
$577$	−24688.1	−1.78125	−0.890623	−	0.454742i	$-0.849731\pi$
$577$	−24688.1	−1.78125	−0.890623	+	0.454742i	$0.849731\pi$
$578$	0	0
$579$	−42416.9	−3.04453
$580$	0	0
$581$	2164.69	0.154573
$582$	0	0
$583$	−1526.74	−0.108458
$584$	0	0
$585$	72224.7	5.10449
$586$	0	0
$587$	−5267.95	−0.370412	−0.185206	−	0.982700i	$-0.559295\pi$
$587$	−5267.95	−0.370412	−0.185206	+	0.982700i	$0.559295\pi$
$588$	0	0
$589$	794.143	0.0555554
$590$	0	0
$591$	17292.2	1.20356
$592$	0	0
$593$	4931.50	0.341505	0.170752	−	0.985314i	$-0.445380\pi$
$593$	4931.50	0.341505	0.170752	+	0.985314i	$0.445380\pi$
$594$	0	0
$595$	−1723.33	−0.118739
$596$	0	0
$597$	1268.50	0.0869616
$598$	0	0
$599$	−3514.78	−0.239750	−0.119875	−	0.992789i	$-0.538249\pi$
$599$	−3514.78	−0.239750	−0.119875	+	0.992789i	$0.538249\pi$
$600$	0	0
$601$	−28812.3	−1.95554	−0.977770	−	0.209682i	$-0.932757\pi$
$601$	−28812.3	−1.95554	−0.977770	+	0.209682i	$0.932757\pi$
$602$	0	0
$603$	−43378.8	−2.92955
$604$	0	0
$605$	1735.37	0.116616
$606$	0	0
$607$	−19080.8	−1.27589	−0.637944	−	0.770083i	$-0.720215\pi$
$607$	−19080.8	−1.27589	−0.637944	+	0.770083i	$0.720215\pi$
$608$	0	0
$609$	591.609	0.0393649
$610$	0	0
$611$	−30324.0	−2.00782
$612$	0	0
$613$	−10521.7	−0.693260	−0.346630	−	0.938002i	$-0.612674\pi$
$613$	−10521.7	−0.693260	−0.346630	+	0.938002i	$0.612674\pi$
$614$	0	0
$615$	−63333.5	−4.15260
$616$	0	0
$617$	3227.78	0.210609	0.105304	−	0.994440i	$-0.466418\pi$
$617$	3227.78	0.210609	0.105304	+	0.994440i	$0.466418\pi$
$618$	0	0
$619$	11736.5	0.762081	0.381040	−	0.924558i	$-0.375566\pi$
$619$	11736.5	0.762081	0.381040	+	0.924558i	$0.375566\pi$
$620$	0	0
$621$	−51996.4	−3.35997
$622$	0	0
$623$	−6146.64	−0.395281
$624$	0	0
$625$	−19200.4	−1.22882
$626$	0	0
$627$	1449.81	0.0923445
$628$	0	0
$629$	7028.64	0.445549
$630$	0	0
$631$	−18613.0	−1.17428	−0.587141	−	0.809485i	$-0.699746\pi$
$631$	−18613.0	−1.17428	−0.587141	+	0.809485i	$0.699746\pi$
$632$	0	0
$633$	−38397.0	−2.41097
$634$	0	0
$635$	32853.1	2.05313
$636$	0	0
$637$	3324.56	0.206788
$638$	0	0
$639$	−35019.3	−2.16799
$640$	0	0
$641$	−15637.2	−0.963547	−0.481774	−	0.876296i	$-0.660007\pi$
$641$	−15637.2	−0.963547	−0.481774	+	0.876296i	$0.660007\pi$
$642$	0	0
$643$	−4556.56	−0.279461	−0.139730	−	0.990190i	$-0.544624\pi$
$643$	−4556.56	−0.279461	−0.139730	+	0.990190i	$0.544624\pi$
$644$	0	0
$645$	−20370.9	−1.24357
$646$	0	0
$647$	19522.0	1.18623	0.593113	−	0.805119i	$-0.297899\pi$
$647$	19522.0	1.18623	0.593113	+	0.805119i	$0.297899\pi$
$648$	0	0
$649$	−7497.52	−0.453472
$650$	0	0
$651$	4269.32	0.257032
$652$	0	0
$653$	−33019.5	−1.97879	−0.989396	−	0.145240i	$-0.953604\pi$
$653$	−33019.5	−1.97879	−0.989396	+	0.145240i	$0.953604\pi$
$654$	0	0
$655$	−36555.3	−2.18066
$656$	0	0
$657$	41896.4	2.48787
$658$	0	0
$659$	8821.98	0.521481	0.260740	−	0.965409i	$-0.416033\pi$
$659$	8821.98	0.521481	0.260740	+	0.965409i	$0.416033\pi$
$660$	0	0
$661$	−5136.74	−0.302264	−0.151132	−	0.988514i	$-0.548292\pi$
$661$	−5136.74	−0.302264	−0.151132	+	0.988514i	$0.548292\pi$
$662$	0	0
$663$	−11717.7	−0.686393
$664$	0	0
$665$	1315.18	0.0766922
$666$	0	0
$667$	−919.329	−0.0533682
$668$	0	0
$669$	−27895.4	−1.61211
$670$	0	0
$671$	7564.39	0.435201
$672$	0	0
$673$	30486.9	1.74618	0.873092	−	0.487556i	$-0.162111\pi$
$673$	30486.9	1.74618	0.873092	+	0.487556i	$0.162111\pi$
$674$	0	0
$675$	38337.1	2.18606
$676$	0	0
$677$	25819.3	1.46575	0.732876	−	0.680362i	$-0.238177\pi$
$677$	25819.3	1.46575	0.732876	+	0.680362i	$0.238177\pi$
$678$	0	0
$679$	725.196	0.0409874
$680$	0	0
$681$	2276.08	0.128076
$682$	0	0
$683$	−27914.2	−1.56385	−0.781923	−	0.623375i	$-0.785761\pi$
$683$	−27914.2	−1.56385	−0.781923	+	0.623375i	$0.785761\pi$
$684$	0	0
$685$	−40427.9	−2.25499
$686$	0	0
$687$	37050.0	2.05757
$688$	0	0
$689$	−9416.98	−0.520694
$690$	0	0
$691$	25028.4	1.37789	0.688946	−	0.724812i	$-0.258074\pi$
$691$	25028.4	1.37789	0.688946	+	0.724812i	$0.258074\pi$
$692$	0	0
$693$	5715.21	0.313280
$694$	0	0
$695$	−2586.36	−0.141160
$696$	0	0
$697$	7534.43	0.409450
$698$	0	0
$699$	−19364.4	−1.04783
$700$	0	0
$701$	−15094.8	−0.813300	−0.406650	−	0.913584i	$-0.633303\pi$
$701$	−15094.8	−0.813300	−0.406650	+	0.913584i	$0.633303\pi$
$702$	0	0
$703$	−5363.96	−0.287775
$704$	0	0
$705$	−64490.6	−3.44519
$706$	0	0
$707$	12438.8	0.661683
$708$	0	0
$709$	−8490.65	−0.449751	−0.224875	−	0.974388i	$-0.572197\pi$
$709$	−8490.65	−0.449751	−0.224875	+	0.974388i	$0.572197\pi$
$710$	0	0
$711$	−13038.7	−0.687746
$712$	0	0
$713$	−6634.29	−0.348466
$714$	0	0
$715$	10703.8	0.559858
$716$	0	0
$717$	677.788	0.0353033
$718$	0	0
$719$	−4495.09	−0.233155	−0.116578	−	0.993182i	$-0.537192\pi$
$719$	−4495.09	−0.233155	−0.116578	+	0.993182i	$0.537192\pi$
$720$	0	0
$721$	−4571.06	−0.236110
$722$	0	0
$723$	10427.9	0.536400
$724$	0	0
$725$	677.823	0.0347224
$726$	0	0
$727$	21560.6	1.09991	0.549957	−	0.835193i	$-0.314644\pi$
$727$	21560.6	1.09991	0.549957	+	0.835193i	$0.314644\pi$
$728$	0	0
$729$	76987.8	3.91138
$730$	0	0
$731$	2423.41	0.122617
$732$	0	0
$733$	−6787.76	−0.342035	−0.171017	−	0.985268i	$-0.554705\pi$
$733$	−6787.76	−0.342035	−0.171017	+	0.985268i	$0.554705\pi$
$734$	0	0
$735$	7070.39	0.354823
$736$	0	0
$737$	−6428.78	−0.321313
$738$	0	0
$739$	−4086.69	−0.203425	−0.101713	−	0.994814i	$-0.532432\pi$
$739$	−4086.69	−0.203425	−0.101713	+	0.994814i	$0.532432\pi$
$740$	0	0
$741$	8942.47	0.443333
$742$	0	0
$743$	−21237.7	−1.04863	−0.524316	−	0.851524i	$-0.675679\pi$
$743$	−21237.7	−1.04863	−0.524316	+	0.851524i	$0.675679\pi$
$744$	0	0
$745$	46216.0	2.27278
$746$	0	0
$747$	22953.0	1.12424
$748$	0	0
$749$	6066.38	0.295942
$750$	0	0
$751$	31705.5	1.54055	0.770274	−	0.637713i	$-0.220119\pi$
$751$	31705.5	1.54055	0.770274	+	0.637713i	$0.220119\pi$
$752$	0	0
$753$	28492.8	1.37893
$754$	0	0
$755$	−48552.6	−2.34041
$756$	0	0
$757$	4979.15	0.239063	0.119531	−	0.992830i	$-0.461861\pi$
$757$	4979.15	0.239063	0.119531	+	0.992830i	$0.461861\pi$
$758$	0	0
$759$	−12111.8	−0.579222
$760$	0	0
$761$	−1026.14	−0.0488796	−0.0244398	−	0.999701i	$-0.507780\pi$
$761$	−1026.14	−0.0488796	−0.0244398	+	0.999701i	$0.507780\pi$
$762$	0	0
$763$	3227.99	0.153160
$764$	0	0
$765$	−18273.1	−0.863615
$766$	0	0
$767$	−46244.8	−2.17706
$768$	0	0
$769$	6183.41	0.289960	0.144980	−	0.989435i	$-0.453688\pi$
$769$	6183.41	0.289960	0.144980	+	0.989435i	$0.453688\pi$
$770$	0	0
$771$	69088.1	3.22717
$772$	0	0
$773$	−6211.97	−0.289041	−0.144521	−	0.989502i	$-0.546164\pi$
$773$	−6211.97	−0.289041	−0.144521	+	0.989502i	$0.546164\pi$
$774$	0	0
$775$	4891.48	0.226719
$776$	0	0
$777$	−28836.7	−1.33142
$778$	0	0
$779$	−5749.96	−0.264459
$780$	0	0
$781$	−5189.90	−0.237784
$782$	0	0
$783$	3991.12	0.182160
$784$	0	0
$785$	39764.1	1.80795
$786$	0	0
$787$	33921.4	1.53642	0.768212	−	0.640195i	$-0.221146\pi$
$787$	33921.4	1.53642	0.768212	+	0.640195i	$0.221146\pi$
$788$	0	0
$789$	−39237.9	−1.77048
$790$	0	0
$791$	−1999.79	−0.0898919
$792$	0	0
$793$	46657.3	2.08934
$794$	0	0
$795$	−20027.2	−0.893450
$796$	0	0
$797$	−26779.1	−1.19017	−0.595083	−	0.803664i	$-0.702881\pi$
$797$	−26779.1	−1.19017	−0.595083	+	0.803664i	$0.702881\pi$
$798$	0	0
$799$	7672.09	0.339698
$800$	0	0
$801$	−65175.0	−2.87496
$802$	0	0
$803$	6209.09	0.272869
$804$	0	0
$805$	−10987.0	−0.481045
$806$	0	0
$807$	78022.1	3.40336
$808$	0	0
$809$	−7143.09	−0.310430	−0.155215	−	0.987881i	$-0.549607\pi$
$809$	−7143.09	−0.310430	−0.155215	+	0.987881i	$0.549607\pi$
$810$	0	0
$811$	36059.7	1.56132	0.780659	−	0.624957i	$-0.214884\pi$
$811$	36059.7	1.56132	0.780659	+	0.624957i	$0.214884\pi$
$812$	0	0
$813$	6424.95	0.277162
$814$	0	0
$815$	22477.5	0.966077
$816$	0	0
$817$	−1849.45	−0.0791969
$818$	0	0
$819$	35251.5	1.50401
$820$	0	0
$821$	2779.33	0.118148	0.0590738	−	0.998254i	$-0.481185\pi$
$821$	2779.33	0.118148	0.0590738	+	0.998254i	$0.481185\pi$
$822$	0	0
$823$	17594.9	0.745225	0.372613	−	0.927987i	$-0.378462\pi$
$823$	17594.9	0.745225	0.372613	+	0.927987i	$0.378462\pi$
$824$	0	0
$825$	8930.04	0.376853
$826$	0	0
$827$	24482.5	1.02943	0.514715	−	0.857361i	$-0.327898\pi$
$827$	24482.5	1.02943	0.514715	+	0.857361i	$0.327898\pi$
$828$	0	0
$829$	27032.5	1.13254	0.566270	−	0.824220i	$-0.308386\pi$
$829$	27032.5	1.13254	0.566270	+	0.824220i	$0.308386\pi$
$830$	0	0
$831$	26444.1	1.10389
$832$	0	0
$833$	−841.125	−0.0349859
$834$	0	0
$835$	4245.03	0.175934
$836$	0	0
$837$	28801.7	1.18941
$838$	0	0
$839$	−37534.3	−1.54449	−0.772246	−	0.635324i	$-0.780867\pi$
$839$	−37534.3	−1.54449	−0.772246	+	0.635324i	$0.780867\pi$
$840$	0	0
$841$	−24318.4	−0.997107
$842$	0	0
$843$	−18720.3	−0.764840
$844$	0	0
$845$	34511.9	1.40503
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−6109.39	−0.246966
$850$	0	0
$851$	44810.7	1.80504
$852$	0	0
$853$	−8716.67	−0.349886	−0.174943	−	0.984579i	$-0.555974\pi$
$853$	−8716.67	−0.349886	−0.174943	+	0.984579i	$0.555974\pi$
$854$	0	0
$855$	13945.3	0.557799
$856$	0	0
$857$	6145.31	0.244947	0.122474	−	0.992472i	$-0.460917\pi$
$857$	6145.31	0.244947	0.122474	+	0.992472i	$0.460917\pi$
$858$	0	0
$859$	−17466.3	−0.693764	−0.346882	−	0.937909i	$-0.612760\pi$
$859$	−17466.3	−0.693764	−0.346882	+	0.937909i	$0.612760\pi$
$860$	0	0
$861$	−30911.8	−1.22354
$862$	0	0
$863$	−11676.2	−0.460560	−0.230280	−	0.973124i	$-0.573964\pi$
$863$	−11676.2	−0.460560	−0.230280	+	0.973124i	$0.573964\pi$
$864$	0	0
$865$	−1728.46	−0.0679415
$866$	0	0
$867$	−46465.0	−1.82011
$868$	0	0
$869$	−1932.34	−0.0754318
$870$	0	0
$871$	−39652.8	−1.54258
$872$	0	0
$873$	7689.51	0.298110
$874$	0	0
$875$	−4448.42	−0.171868
$876$	0	0
$877$	35663.4	1.37317	0.686583	−	0.727052i	$-0.259110\pi$
$877$	35663.4	1.37317	0.686583	+	0.727052i	$0.259110\pi$
$878$	0	0
$879$	78485.3	3.01165
$880$	0	0
$881$	−10415.5	−0.398305	−0.199152	−	0.979969i	$-0.563819\pi$
$881$	−10415.5	−0.398305	−0.199152	+	0.979969i	$0.563819\pi$
$882$	0	0
$883$	−7393.46	−0.281778	−0.140889	−	0.990025i	$-0.544996\pi$
$883$	−7393.46	−0.281778	−0.140889	+	0.990025i	$0.544996\pi$
$884$	0	0
$885$	−98349.5	−3.73557
$886$	0	0
$887$	−46274.3	−1.75168	−0.875840	−	0.482601i	$-0.839692\pi$
$887$	−46274.3	−1.75168	−0.875840	+	0.482601i	$0.839692\pi$
$888$	0	0
$889$	16035.0	0.604945
$890$	0	0
$891$	30537.0	1.14818
$892$	0	0
$893$	−5855.01	−0.219407
$894$	0	0
$895$	41485.3	1.54938
$896$	0	0
$897$	−74705.6	−2.78077
$898$	0	0
$899$	509.233	0.0188920
$900$	0	0
$901$	2382.53	0.0880949
$902$	0	0
$903$	−9942.63	−0.366412
$904$	0	0
$905$	53170.9	1.95300
$906$	0	0
$907$	−20313.9	−0.743674	−0.371837	−	0.928298i	$-0.621272\pi$
$907$	−20313.9	−0.743674	−0.371837	+	0.928298i	$0.621272\pi$
$908$	0	0
$909$	131893.	4.81256
$910$	0	0
$911$	46553.4	1.69307	0.846534	−	0.532335i	$-0.178685\pi$
$911$	46553.4	1.69307	0.846534	+	0.532335i	$0.178685\pi$
$912$	0	0
$913$	3401.66	0.123306
$914$	0	0
$915$	99226.7	3.58507
$916$	0	0
$917$	−17841.9	−0.642521
$918$	0	0
$919$	19428.9	0.697389	0.348695	−	0.937236i	$-0.386625\pi$
$919$	19428.9	0.697389	0.348695	+	0.937236i	$0.386625\pi$
$920$	0	0
$921$	91199.3	3.26289
$922$	0	0
$923$	−32011.4	−1.14157
$924$	0	0
$925$	−33039.0	−1.17440
$926$	0	0
$927$	−48468.5	−1.71728
$928$	0	0
$929$	−19403.3	−0.685255	−0.342628	−	0.939471i	$-0.611317\pi$
$929$	−19403.3	−0.685255	−0.342628	+	0.939471i	$0.611317\pi$
$930$	0	0
$931$	641.911	0.0225970
$932$	0	0
$933$	53296.3	1.87014
$934$	0	0
$935$	−2708.09	−0.0947210
$936$	0	0
$937$	3532.47	0.123160	0.0615799	−	0.998102i	$-0.480386\pi$
$937$	3532.47	0.123160	0.0615799	+	0.998102i	$0.480386\pi$
$938$	0	0
$939$	69606.1	2.41907
$940$	0	0
$941$	52955.6	1.83454	0.917271	−	0.398263i	$-0.130387\pi$
$941$	52955.6	1.83454	0.917271	+	0.398263i	$0.130387\pi$
$942$	0	0
$943$	48035.3	1.65880
$944$	0	0
$945$	47698.3	1.64193
$946$	0	0
$947$	35064.1	1.20320	0.601600	−	0.798797i	$-0.294530\pi$
$947$	35064.1	1.20320	0.601600	+	0.798797i	$0.294530\pi$
$948$	0	0
$949$	38297.7	1.31001
$950$	0	0
$951$	34018.6	1.15997
$952$	0	0
$953$	−46657.0	−1.58591	−0.792953	−	0.609282i	$-0.791458\pi$
$953$	−46657.0	−1.58591	−0.792953	+	0.609282i	$0.791458\pi$
$954$	0	0
$955$	−6044.04	−0.204796
$956$	0	0
$957$	929.672	0.0314023
$958$	0	0
$959$	−19732.1	−0.664423
$960$	0	0
$961$	−26116.1	−0.876645
$962$	0	0
$963$	64323.9	2.15245
$964$	0	0
$965$	−60465.1	−2.01704
$966$	0	0
$967$	39518.1	1.31418	0.657092	−	0.753811i	$-0.271786\pi$
$967$	39518.1	1.31418	0.657092	+	0.753811i	$0.271786\pi$
$968$	0	0
$969$	−2262.48	−0.0750064
$970$	0	0
$971$	13173.8	0.435394	0.217697	−	0.976016i	$-0.430145\pi$
$971$	13173.8	0.435394	0.217697	+	0.976016i	$0.430145\pi$
$972$	0	0
$973$	−1262.35	−0.0415922
$974$	0	0
$975$	55080.6	1.80922
$976$	0	0
$977$	516.124	0.0169010	0.00845049	−	0.999964i	$-0.497310\pi$
$977$	516.124	0.0169010	0.00845049	+	0.999964i	$0.497310\pi$
$978$	0	0
$979$	−9659.00	−0.315325
$980$	0	0
$981$	34227.5	1.11397
$982$	0	0
$983$	−28044.2	−0.909941	−0.454971	−	0.890506i	$-0.650350\pi$
$983$	−28044.2	−0.909941	−0.454971	+	0.890506i	$0.650350\pi$
$984$	0	0
$985$	24649.9	0.797372
$986$	0	0
$987$	−31476.6	−1.01511
$988$	0	0
$989$	15450.3	0.496756
$990$	0	0
$991$	19654.1	0.630004	0.315002	−	0.949091i	$-0.397995\pi$
$991$	19654.1	0.630004	0.315002	+	0.949091i	$0.397995\pi$
$992$	0	0
$993$	−32288.1	−1.03185
$994$	0	0
$995$	1808.24	0.0576130
$996$	0	0
$997$	6672.91	0.211969	0.105985	−	0.994368i	$-0.466201\pi$
$997$	6672.91	0.211969	0.105985	+	0.994368i	$0.466201\pi$
$998$	0	0
$999$	−194539.	−6.16109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.bd.1.7		7
4.3	odd	2		616.4.a.j.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.j.1.1	✓	7	4.3	odd	2
1232.4.a.bd.1.7		7	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)

\(f(q)\)	\(=\)	\(q+10.0610 q^{3} +14.3419 q^{5} +7.00000 q^{7} +74.2235 q^{9} +11.0000 q^{11} +67.8481 q^{13} +144.294 q^{15} -17.1658 q^{17} +13.1002 q^{19} +70.4269 q^{21} -109.440 q^{23} +80.6901 q^{25} +475.115 q^{27} +8.40033 q^{29} +60.6206 q^{31} +110.671 q^{33} +100.393 q^{35} -409.456 q^{37} +682.619 q^{39} -438.921 q^{41} -141.177 q^{43} +1064.51 q^{45} -446.940 q^{47} +49.0000 q^{49} -172.705 q^{51} -138.795 q^{53} +157.761 q^{55} +131.801 q^{57} -681.593 q^{59} +687.672 q^{61} +519.564 q^{63} +973.071 q^{65} -584.435 q^{67} -1101.07 q^{69} -471.809 q^{71} +564.462 q^{73} +811.822 q^{75} +77.0000 q^{77} -175.668 q^{79} +2776.09 q^{81} +309.242 q^{83} -246.190 q^{85} +84.5156 q^{87} -878.091 q^{89} +474.937 q^{91} +609.903 q^{93} +187.882 q^{95} +103.599 q^{97} +816.458 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 6 q^{5} + 49 q^{7} + 101 q^{9} + 77 q^{11} + 88 q^{13} + 106 q^{15} + 134 q^{17} - 14 q^{19} - 42 q^{23} + 343 q^{25} + 30 q^{27} + 482 q^{29} - 50 q^{31} + 42 q^{35} + 152 q^{37} - 72 q^{39} + 234 q^{41}+ \cdots + 1111 q^{99}+O(q^{100}) \) 7 * q + 6 * q^5 + 49 * q^7 + 101 * q^9 + 77 * q^11 + 88 * q^13 + 106 * q^15 + 134 * q^17 - 14 * q^19 - 42 * q^23 + 343 * q^25 + 30 * q^27 + 482 * q^29 - 50 * q^31 + 42 * q^35 + 152 * q^37 - 72 * q^39 + 234 * q^41 - 472 * q^43 + 1384 * q^45 - 728 * q^47 + 343 * q^49 - 1572 * q^51 - 102 * q^53 + 66 * q^55 + 452 * q^57 - 1704 * q^59 + 656 * q^61 + 707 * q^63 + 2840 * q^65 - 1126 * q^67 - 350 * q^69 + 918 * q^71 + 1094 * q^73 - 274 * q^75 + 539 * q^77 - 672 * q^79 + 4503 * q^81 - 782 * q^83 + 1716 * q^85 - 3196 * q^87 - 464 * q^89 + 616 * q^91 + 1826 * q^93 + 1796 * q^95 + 3000 * q^97 + 1111 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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