LMFDB - Embedded newform 1232.4.a.j.1.1

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:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.54138$ 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)

$f(q)$	$=$	$q-9.08276 q^{3} +6.91724 q^{5} -7.00000 q^{7} +55.4966 q^{9} +O(q^{10})$	$q-9.08276 q^{3} +6.91724 q^{5} -7.00000 q^{7} +55.4966 q^{9} -11.0000 q^{11} +7.08276 q^{13} -62.8276 q^{15} -68.3311 q^{17} +94.5725 q^{19} +63.5793 q^{21} -104.331 q^{23} -77.1518 q^{25} -258.828 q^{27} +48.4966 q^{29} -38.8413 q^{31} +99.9104 q^{33} -48.4207 q^{35} +144.317 q^{37} -64.3311 q^{39} +409.986 q^{41} +16.8413 q^{43} +383.883 q^{45} +535.449 q^{47} +49.0000 q^{49} +620.635 q^{51} -155.807 q^{53} -76.0896 q^{55} -858.979 q^{57} -449.235 q^{59} -529.096 q^{61} -388.476 q^{63} +48.9932 q^{65} +296.814 q^{67} +947.614 q^{69} +942.414 q^{71} -87.1587 q^{73} +700.752 q^{75} +77.0000 q^{77} +1328.97 q^{79} +852.462 q^{81} -611.001 q^{83} -472.662 q^{85} -440.483 q^{87} +1195.21 q^{89} -49.5793 q^{91} +352.787 q^{93} +654.180 q^{95} -1854.79 q^{97} -610.462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 26 q^{5} - 14 q^{7} + 38 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 26 * q^5 - 14 * q^7 + 38 * q^9	$ 2 q - 6 q^{3} + 26 q^{5} - 14 q^{7} + 38 q^{9} - 22 q^{11} + 2 q^{13} - 4 q^{15} - 88 q^{17} - 42 q^{19} + 42 q^{21} - 160 q^{23} + 162 q^{25} - 396 q^{27} + 24 q^{29} - 248 q^{31} + 66 q^{33} - 182 q^{35} - 52 q^{37} - 80 q^{39} + 528 q^{41} + 204 q^{43} + 50 q^{45} - 24 q^{47} + 98 q^{49} + 560 q^{51} + 248 q^{53} - 286 q^{55} - 1280 q^{57} - 570 q^{59} - 1338 q^{61} - 266 q^{63} - 48 q^{65} + 180 q^{67} + 776 q^{69} + 60 q^{71} - 4 q^{73} + 1438 q^{75} + 154 q^{77} + 1928 q^{79} + 902 q^{81} + 542 q^{83} - 848 q^{85} - 516 q^{87} - 140 q^{89} - 14 q^{91} - 292 q^{93} - 1952 q^{95} - 2712 q^{97} - 418 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 26 * q^5 - 14 * q^7 + 38 * q^9 - 22 * q^11 + 2 * q^13 - 4 * q^15 - 88 * q^17 - 42 * q^19 + 42 * q^21 - 160 * q^23 + 162 * q^25 - 396 * q^27 + 24 * q^29 - 248 * q^31 + 66 * q^33 - 182 * q^35 - 52 * q^37 - 80 * q^39 + 528 * q^41 + 204 * q^43 + 50 * q^45 - 24 * q^47 + 98 * q^49 + 560 * q^51 + 248 * q^53 - 286 * q^55 - 1280 * q^57 - 570 * q^59 - 1338 * q^61 - 266 * q^63 - 48 * q^65 + 180 * q^67 + 776 * q^69 + 60 * q^71 - 4 * q^73 + 1438 * q^75 + 154 * q^77 + 1928 * q^79 + 902 * q^81 + 542 * q^83 - 848 * q^85 - 516 * q^87 - 140 * q^89 - 14 * q^91 - 292 * q^93 - 1952 * q^95 - 2712 * q^97 - 418 * 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
$3$	−9.08276	−1.74798	−0.873989	−	0.485945i	$-0.838475\pi$
$3$	−9.08276	−1.74798	−0.873989	+	0.485945i	$0.838475\pi$
$4$	0	0
$5$	6.91724	0.618697	0.309348	−	0.950949i	$-0.399889\pi$
$5$	6.91724	0.618697	0.309348	+	0.950949i	$0.399889\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	55.4966	2.05543
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	7.08276	0.151108	0.0755540	−	0.997142i	$-0.475927\pi$
$13$	7.08276	0.151108	0.0755540	+	0.997142i	$0.475927\pi$
$14$	0	0
$15$	−62.8276	−1.08147
$16$	0	0
$17$	−68.3311	−0.974866	−0.487433	−	0.873161i	$-0.662067\pi$
$17$	−68.3311	−0.974866	−0.487433	+	0.873161i	$0.662067\pi$
$18$	0	0
$19$	94.5725	1.14192	0.570958	−	0.820979i	$-0.306572\pi$
$19$	94.5725	1.14192	0.570958	+	0.820979i	$0.306572\pi$
$20$	0	0
$21$	63.5793	0.660674
$22$	0	0
$23$	−104.331	−0.945849	−0.472925	−	0.881103i	$-0.656802\pi$
$23$	−104.331	−0.945849	−0.472925	+	0.881103i	$0.656802\pi$
$24$	0	0
$25$	−77.1518	−0.617215
$26$	0	0
$27$	−258.828	−1.84487
$28$	0	0
$29$	48.4966	0.310538	0.155269	−	0.987872i	$-0.450376\pi$
$29$	48.4966	0.310538	0.155269	+	0.987872i	$0.450376\pi$
$30$	0	0
$31$	−38.8413	−0.225036	−0.112518	−	0.993650i	$-0.535892\pi$
$31$	−38.8413	−0.225036	−0.112518	+	0.993650i	$0.535892\pi$
$32$	0	0
$33$	99.9104	0.527035
$34$	0	0
$35$	−48.4207	−0.233845
$36$	0	0
$37$	144.317	0.641233	0.320617	−	0.947209i	$-0.396110\pi$
$37$	144.317	0.641233	0.320617	+	0.947209i	$0.396110\pi$
$38$	0	0
$39$	−64.3311	−0.264134
$40$	0	0
$41$	409.986	1.56169	0.780843	−	0.624728i	$-0.214790\pi$
$41$	409.986	1.56169	0.780843	+	0.624728i	$0.214790\pi$
$42$	0	0
$43$	16.8413	0.0597274	0.0298637	−	0.999554i	$-0.490493\pi$
$43$	16.8413	0.0597274	0.0298637	+	0.999554i	$0.490493\pi$
$44$	0	0
$45$	383.883	1.27169
$46$	0	0
$47$	535.449	1.66177	0.830885	−	0.556444i	$-0.187835\pi$
$47$	535.449	1.66177	0.830885	+	0.556444i	$0.187835\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	620.635	1.70404
$52$	0	0
$53$	−155.807	−0.403807	−0.201903	−	0.979405i	$-0.564713\pi$
$53$	−155.807	−0.403807	−0.201903	+	0.979405i	$0.564713\pi$
$54$	0	0
$55$	−76.0896	−0.186544
$56$	0	0
$57$	−858.979	−1.99605
$58$	0	0
$59$	−449.235	−0.991277	−0.495639	−	0.868529i	$-0.665066\pi$
$59$	−449.235	−0.991277	−0.495639	+	0.868529i	$0.665066\pi$
$60$	0	0
$61$	−529.096	−1.11056	−0.555278	−	0.831665i	$-0.687388\pi$
$61$	−529.096	−1.11056	−0.555278	+	0.831665i	$0.687388\pi$
$62$	0	0
$63$	−388.476	−0.776879
$64$	0	0
$65$	48.9932	0.0934900
$66$	0	0
$67$	296.814	0.541218	0.270609	−	0.962689i	$-0.412775\pi$
$67$	296.814	0.541218	0.270609	+	0.962689i	$0.412775\pi$
$68$	0	0
$69$	947.614	1.65332
$70$	0	0
$71$	942.414	1.57527	0.787634	−	0.616144i	$-0.211306\pi$
$71$	942.414	1.57527	0.787634	+	0.616144i	$0.211306\pi$
$72$	0	0
$73$	−87.1587	−0.139742	−0.0698709	−	0.997556i	$-0.522259\pi$
$73$	−87.1587	−0.139742	−0.0698709	+	0.997556i	$0.522259\pi$
$74$	0	0
$75$	700.752	1.07888
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	1328.97	1.89266	0.946331	−	0.323198i	$-0.104758\pi$
$79$	1328.97	1.89266	0.946331	+	0.323198i	$0.104758\pi$
$80$	0	0
$81$	852.462	1.16936
$82$	0	0
$83$	−611.001	−0.808024	−0.404012	−	0.914754i	$-0.632385\pi$
$83$	−611.001	−0.808024	−0.404012	+	0.914754i	$0.632385\pi$
$84$	0	0
$85$	−472.662	−0.603146
$86$	0	0
$87$	−440.483	−0.542813
$88$	0	0
$89$	1195.21	1.42351	0.711756	−	0.702427i	$-0.247900\pi$
$89$	1195.21	1.42351	0.711756	+	0.702427i	$0.247900\pi$
$90$	0	0
$91$	−49.5793	−0.0571135
$92$	0	0
$93$	352.787	0.393358
$94$	0	0
$95$	654.180	0.706500
$96$	0	0
$97$	−1854.79	−1.94150	−0.970748	−	0.240102i	$-0.922819\pi$
$97$	−1854.79	−1.94150	−0.970748	+	0.240102i	$0.922819\pi$
$98$	0	0
$99$	−610.462	−0.619735
$100$	0	0
$101$	−1434.95	−1.41369	−0.706844	−	0.707370i	$-0.749882\pi$
$101$	−1434.95	−1.41369	−0.706844	+	0.707370i	$0.749882\pi$
$102$	0	0
$103$	−568.745	−0.544079	−0.272040	−	0.962286i	$-0.587698\pi$
$103$	−568.745	−0.544079	−0.272040	+	0.962286i	$0.587698\pi$
$104$	0	0
$105$	439.793	0.408757
$106$	0	0
$107$	−1783.71	−1.61157	−0.805784	−	0.592209i	$-0.798256\pi$
$107$	−1783.71	−1.61157	−0.805784	+	0.592209i	$0.798256\pi$
$108$	0	0
$109$	−529.587	−0.465369	−0.232684	−	0.972552i	$-0.574751\pi$
$109$	−529.587	−0.465369	−0.232684	+	0.972552i	$0.574751\pi$
$110$	0	0
$111$	−1310.80	−1.12086
$112$	0	0
$113$	1995.09	1.66091	0.830453	−	0.557089i	$-0.188082\pi$
$113$	1995.09	1.66091	0.830453	+	0.557089i	$0.188082\pi$
$114$	0	0
$115$	−721.683	−0.585194
$116$	0	0
$117$	393.069	0.310592
$118$	0	0
$119$	478.317	0.368465
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−3723.81	−2.72979
$124$	0	0
$125$	−1398.33	−1.00057
$126$	0	0
$127$	1150.65	0.803965	0.401982	−	0.915647i	$-0.368321\pi$
$127$	1150.65	0.803965	0.401982	+	0.915647i	$0.368321\pi$
$128$	0	0
$129$	−152.966	−0.104402
$130$	0	0
$131$	2230.85	1.48786	0.743932	−	0.668255i	$-0.232959\pi$
$131$	2230.85	1.48786	0.743932	+	0.668255i	$0.232959\pi$
$132$	0	0
$133$	−662.007	−0.431604
$134$	0	0
$135$	−1790.37	−1.14141
$136$	0	0
$137$	656.594	0.409464	0.204732	−	0.978818i	$-0.434368\pi$
$137$	656.594	0.409464	0.204732	+	0.978818i	$0.434368\pi$
$138$	0	0
$139$	−1012.17	−0.617636	−0.308818	−	0.951121i	$-0.599933\pi$
$139$	−1012.17	−0.617636	−0.308818	+	0.951121i	$0.599933\pi$
$140$	0	0
$141$	−4863.35	−2.90474
$142$	0	0
$143$	−77.9104	−0.0455608
$144$	0	0
$145$	335.462	0.192128
$146$	0	0
$147$	−445.055	−0.249711
$148$	0	0
$149$	−786.677	−0.432531	−0.216265	−	0.976335i	$-0.569388\pi$
$149$	−786.677	−0.432531	−0.216265	+	0.976335i	$0.569388\pi$
$150$	0	0
$151$	−205.573	−0.110790	−0.0553950	−	0.998465i	$-0.517642\pi$
$151$	−205.573	−0.110790	−0.0553950	+	0.998465i	$0.517642\pi$
$152$	0	0
$153$	−3792.14	−2.00377
$154$	0	0
$155$	−268.675	−0.139229
$156$	0	0
$157$	−3284.08	−1.66941	−0.834707	−	0.550695i	$-0.814363\pi$
$157$	−3284.08	−1.66941	−0.834707	+	0.550695i	$0.814363\pi$
$158$	0	0
$159$	1415.16	0.705845
$160$	0	0
$161$	730.317	0.357497
$162$	0	0
$163$	1403.88	0.674601	0.337301	−	0.941397i	$-0.390486\pi$
$163$	1403.88	0.674601	0.337301	+	0.941397i	$0.390486\pi$
$164$	0	0
$165$	691.104	0.326075
$166$	0	0
$167$	−4161.31	−1.92822	−0.964108	−	0.265512i	$-0.914459\pi$
$167$	−4161.31	−1.92822	−0.964108	+	0.265512i	$0.914459\pi$
$168$	0	0
$169$	−2146.83	−0.977166
$170$	0	0
$171$	5248.45	2.34713
$172$	0	0
$173$	−2693.66	−1.18379	−0.591894	−	0.806016i	$-0.701620\pi$
$173$	−2693.66	−1.18379	−0.591894	+	0.806016i	$0.701620\pi$
$174$	0	0
$175$	540.063	0.233285
$176$	0	0
$177$	4080.29	1.73273
$178$	0	0
$179$	−1877.74	−0.784071	−0.392035	−	0.919950i	$-0.628229\pi$
$179$	−1877.74	−0.784071	−0.392035	+	0.919950i	$0.628229\pi$
$180$	0	0
$181$	639.291	0.262531	0.131265	−	0.991347i	$-0.458096\pi$
$181$	639.291	0.262531	0.131265	+	0.991347i	$0.458096\pi$
$182$	0	0
$183$	4805.66	1.94123
$184$	0	0
$185$	998.277	0.396729
$186$	0	0
$187$	751.642	0.293933
$188$	0	0
$189$	1811.79	0.697294
$190$	0	0
$191$	−3705.56	−1.40379	−0.701897	−	0.712278i	$-0.747663\pi$
$191$	−3705.56	−1.40379	−0.701897	+	0.712278i	$0.747663\pi$
$192$	0	0
$193$	2147.38	0.800891	0.400445	−	0.916321i	$-0.368855\pi$
$193$	2147.38	0.800891	0.400445	+	0.916321i	$0.368855\pi$
$194$	0	0
$195$	−444.993	−0.163419
$196$	0	0
$197$	2356.98	0.852426	0.426213	−	0.904623i	$-0.359847\pi$
$197$	2356.98	0.852426	0.426213	+	0.904623i	$0.359847\pi$
$198$	0	0
$199$	−5191.26	−1.84924	−0.924619	−	0.380893i	$-0.875617\pi$
$199$	−5191.26	−1.84924	−0.924619	+	0.380893i	$0.875617\pi$
$200$	0	0
$201$	−2695.89	−0.946037
$202$	0	0
$203$	−339.476	−0.117372
$204$	0	0
$205$	2835.97	0.966209
$206$	0	0
$207$	−5790.02	−1.94413
$208$	0	0
$209$	−1040.30	−0.344301
$210$	0	0
$211$	−4020.35	−1.31172	−0.655858	−	0.754884i	$-0.727693\pi$
$211$	−4020.35	−1.31172	−0.655858	+	0.754884i	$0.727693\pi$
$212$	0	0
$213$	−8559.73	−2.75353
$214$	0	0
$215$	116.495	0.0369531
$216$	0	0
$217$	271.889	0.0850555
$218$	0	0
$219$	791.642	0.244266
$220$	0	0
$221$	−483.973	−0.147310
$222$	0	0
$223$	444.758	0.133557	0.0667785	−	0.997768i	$-0.478728\pi$
$223$	444.758	0.133557	0.0667785	+	0.997768i	$0.478728\pi$
$224$	0	0
$225$	−4281.66	−1.26864
$226$	0	0
$227$	−4757.40	−1.39101	−0.695506	−	0.718520i	$-0.744820\pi$
$227$	−4757.40	−1.39101	−0.695506	+	0.718520i	$0.744820\pi$
$228$	0	0
$229$	1032.94	0.298073	0.149037	−	0.988832i	$-0.452383\pi$
$229$	1032.94	0.298073	0.149037	+	0.988832i	$0.452383\pi$
$230$	0	0
$231$	−699.373	−0.199201
$232$	0	0
$233$	−2952.57	−0.830168	−0.415084	−	0.909783i	$-0.636248\pi$
$233$	−2952.57	−0.830168	−0.415084	+	0.909783i	$0.636248\pi$
$234$	0	0
$235$	3703.83	1.02813
$236$	0	0
$237$	−12070.7	−3.30833
$238$	0	0
$239$	−5382.65	−1.45680	−0.728399	−	0.685154i	$-0.759735\pi$
$239$	−5382.65	−1.45680	−0.728399	+	0.685154i	$0.759735\pi$
$240$	0	0
$241$	2758.35	0.737265	0.368632	−	0.929575i	$-0.379826\pi$
$241$	2758.35	0.737265	0.368632	+	0.929575i	$0.379826\pi$
$242$	0	0
$243$	−754.367	−0.199147
$244$	0	0
$245$	338.945	0.0883852
$246$	0	0
$247$	669.834	0.172553
$248$	0	0
$249$	5549.57	1.41241
$250$	0	0
$251$	3814.42	0.959220	0.479610	−	0.877482i	$-0.340778\pi$
$251$	3814.42	0.959220	0.479610	+	0.877482i	$0.340778\pi$
$252$	0	0
$253$	1147.64	0.285184
$254$	0	0
$255$	4293.08	1.05429
$256$	0	0
$257$	−6959.85	−1.68927	−0.844637	−	0.535339i	$-0.820184\pi$
$257$	−6959.85	−1.68927	−0.844637	+	0.535339i	$0.820184\pi$
$258$	0	0
$259$	−1010.22	−0.242363
$260$	0	0
$261$	2691.39	0.638288
$262$	0	0
$263$	−956.774	−0.224324	−0.112162	−	0.993690i	$-0.535778\pi$
$263$	−956.774	−0.224324	−0.112162	+	0.993690i	$0.535778\pi$
$264$	0	0
$265$	−1077.75	−0.249834
$266$	0	0
$267$	−10855.9	−2.48827
$268$	0	0
$269$	−974.888	−0.220966	−0.110483	−	0.993878i	$-0.535240\pi$
$269$	−974.888	−0.220966	−0.110483	+	0.993878i	$0.535240\pi$
$270$	0	0
$271$	−3520.03	−0.789027	−0.394514	−	0.918890i	$-0.629087\pi$
$271$	−3520.03	−0.789027	−0.394514	+	0.918890i	$0.629087\pi$
$272$	0	0
$273$	450.317	0.0998331
$274$	0	0
$275$	848.670	0.186097
$276$	0	0
$277$	2310.59	0.501192	0.250596	−	0.968092i	$-0.419374\pi$
$277$	2310.59	0.501192	0.250596	+	0.968092i	$0.419374\pi$
$278$	0	0
$279$	−2155.56	−0.462545
$280$	0	0
$281$	5616.65	1.19239	0.596194	−	0.802840i	$-0.296679\pi$
$281$	5616.65	1.19239	0.596194	+	0.802840i	$0.296679\pi$
$282$	0	0
$283$	1065.17	0.223737	0.111869	−	0.993723i	$-0.464316\pi$
$283$	1065.17	0.223737	0.111869	+	0.993723i	$0.464316\pi$
$284$	0	0
$285$	−5941.76	−1.23495
$286$	0	0
$287$	−2869.90	−0.590262
$288$	0	0
$289$	−243.868	−0.0496372
$290$	0	0
$291$	16846.6	3.39369
$292$	0	0
$293$	3453.14	0.688514	0.344257	−	0.938875i	$-0.388131\pi$
$293$	3453.14	0.688514	0.344257	+	0.938875i	$0.388131\pi$
$294$	0	0
$295$	−3107.46	−0.613300
$296$	0	0
$297$	2847.10	0.556248
$298$	0	0
$299$	−738.952	−0.142925
$300$	0	0
$301$	−117.889	−0.0225748
$302$	0	0
$303$	13033.3	2.47110
$304$	0	0
$305$	−3659.89	−0.687097
$306$	0	0
$307$	−4864.91	−0.904413	−0.452207	−	0.891913i	$-0.649363\pi$
$307$	−4864.91	−0.904413	−0.452207	+	0.891913i	$0.649363\pi$
$308$	0	0
$309$	5165.78	0.951039
$310$	0	0
$311$	−378.550	−0.0690213	−0.0345106	−	0.999404i	$-0.510987\pi$
$311$	−378.550	−0.0690213	−0.0345106	+	0.999404i	$0.510987\pi$
$312$	0	0
$313$	1912.52	0.345375	0.172687	−	0.984977i	$-0.444755\pi$
$313$	1912.52	0.345375	0.172687	+	0.984977i	$0.444755\pi$
$314$	0	0
$315$	−2687.18	−0.480652
$316$	0	0
$317$	−6640.27	−1.17651	−0.588256	−	0.808674i	$-0.700185\pi$
$317$	−6640.27	−1.17651	−0.588256	+	0.808674i	$0.700185\pi$
$318$	0	0
$319$	−533.462	−0.0936306
$320$	0	0
$321$	16201.0	2.81699
$322$	0	0
$323$	−6462.24	−1.11322
$324$	0	0
$325$	−546.448	−0.0932661
$326$	0	0
$327$	4810.11	0.813455
$328$	0	0
$329$	−3748.14	−0.628090
$330$	0	0
$331$	5342.84	0.887218	0.443609	−	0.896220i	$-0.353698\pi$
$331$	5342.84	0.887218	0.443609	+	0.896220i	$0.353698\pi$
$332$	0	0
$333$	8009.12	1.31801
$334$	0	0
$335$	2053.13	0.334850
$336$	0	0
$337$	4264.37	0.689303	0.344652	−	0.938731i	$-0.387997\pi$
$337$	4264.37	0.689303	0.344652	+	0.938731i	$0.387997\pi$
$338$	0	0
$339$	−18120.9	−2.90323
$340$	0	0
$341$	427.255	0.0678508
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	6554.87	1.02291
$346$	0	0
$347$	3767.11	0.582792	0.291396	−	0.956602i	$-0.405880\pi$
$347$	3767.11	0.582792	0.291396	+	0.956602i	$0.405880\pi$
$348$	0	0
$349$	5616.88	0.861503	0.430752	−	0.902471i	$-0.358249\pi$
$349$	5616.88	0.861503	0.430752	+	0.902471i	$0.358249\pi$
$350$	0	0
$351$	−1833.21	−0.278774
$352$	0	0
$353$	−1338.83	−0.201866	−0.100933	−	0.994893i	$-0.532183\pi$
$353$	−1338.83	−0.201866	−0.100933	+	0.994893i	$0.532183\pi$
$354$	0	0
$355$	6518.90	0.974613
$356$	0	0
$357$	−4344.44	−0.644068
$358$	0	0
$359$	3323.84	0.488650	0.244325	−	0.969693i	$-0.421434\pi$
$359$	3323.84	0.488650	0.244325	+	0.969693i	$0.421434\pi$
$360$	0	0
$361$	2084.96	0.303974
$362$	0	0
$363$	−1099.01	−0.158907
$364$	0	0
$365$	−602.897	−0.0864578
$366$	0	0
$367$	−612.596	−0.0871315	−0.0435657	−	0.999051i	$-0.513872\pi$
$367$	−612.596	−0.0871315	−0.0435657	+	0.999051i	$0.513872\pi$
$368$	0	0
$369$	22752.8	3.20993
$370$	0	0
$371$	1090.65	0.152625
$372$	0	0
$373$	−11858.9	−1.64619	−0.823094	−	0.567905i	$-0.807754\pi$
$373$	−11858.9	−1.64619	−0.823094	+	0.567905i	$0.807754\pi$
$374$	0	0
$375$	12700.7	1.74897
$376$	0	0
$377$	343.490	0.0469247
$378$	0	0
$379$	−6906.97	−0.936114	−0.468057	−	0.883698i	$-0.655046\pi$
$379$	−6906.97	−0.936114	−0.468057	+	0.883698i	$0.655046\pi$
$380$	0	0
$381$	−10451.1	−1.40531
$382$	0	0
$383$	5189.68	0.692377	0.346189	−	0.938165i	$-0.387476\pi$
$383$	5189.68	0.692377	0.346189	+	0.938165i	$0.387476\pi$
$384$	0	0
$385$	532.627	0.0705070
$386$	0	0
$387$	934.636	0.122765
$388$	0	0
$389$	4600.72	0.599654	0.299827	−	0.953994i	$-0.403071\pi$
$389$	4600.72	0.599654	0.299827	+	0.953994i	$0.403071\pi$
$390$	0	0
$391$	7129.05	0.922076
$392$	0	0
$393$	−20262.3	−2.60075
$394$	0	0
$395$	9192.77	1.17098
$396$	0	0
$397$	−5313.96	−0.671788	−0.335894	−	0.941900i	$-0.609038\pi$
$397$	−5313.96	−0.671788	−0.335894	+	0.941900i	$0.609038\pi$
$398$	0	0
$399$	6012.86	0.754434
$400$	0	0
$401$	15051.5	1.87440	0.937199	−	0.348794i	$-0.113409\pi$
$401$	15051.5	1.87440	0.937199	+	0.348794i	$0.113409\pi$
$402$	0	0
$403$	−275.104	−0.0340047
$404$	0	0
$405$	5896.68	0.723478
$406$	0	0
$407$	−1587.49	−0.193339
$408$	0	0
$409$	9547.20	1.15423	0.577113	−	0.816664i	$-0.304179\pi$
$409$	9547.20	1.15423	0.577113	+	0.816664i	$0.304179\pi$
$410$	0	0
$411$	−5963.68	−0.715734
$412$	0	0
$413$	3144.64	0.374668
$414$	0	0
$415$	−4226.44	−0.499922
$416$	0	0
$417$	9193.33	1.07961
$418$	0	0
$419$	643.721	0.0750545	0.0375272	−	0.999296i	$-0.488052\pi$
$419$	643.721	0.0750545	0.0375272	+	0.999296i	$0.488052\pi$
$420$	0	0
$421$	12480.4	1.44479	0.722396	−	0.691480i	$-0.243041\pi$
$421$	12480.4	1.44479	0.722396	+	0.691480i	$0.243041\pi$
$422$	0	0
$423$	29715.6	3.41565
$424$	0	0
$425$	5271.87	0.601701
$426$	0	0
$427$	3703.68	0.419750
$428$	0	0
$429$	707.642	0.0796393
$430$	0	0
$431$	5652.33	0.631701	0.315850	−	0.948809i	$-0.397710\pi$
$431$	5652.33	0.631701	0.315850	+	0.948809i	$0.397710\pi$
$432$	0	0
$433$	−2179.80	−0.241928	−0.120964	−	0.992657i	$-0.538599\pi$
$433$	−2179.80	−0.241928	−0.120964	+	0.992657i	$0.538599\pi$
$434$	0	0
$435$	−3046.92	−0.335836
$436$	0	0
$437$	−9866.85	−1.08008
$438$	0	0
$439$	3515.01	0.382146	0.191073	−	0.981576i	$-0.438803\pi$
$439$	3515.01	0.382146	0.191073	+	0.981576i	$0.438803\pi$
$440$	0	0
$441$	2719.33	0.293633
$442$	0	0
$443$	−6329.77	−0.678863	−0.339432	−	0.940631i	$-0.610235\pi$
$443$	−6329.77	−0.678863	−0.339432	+	0.940631i	$0.610235\pi$
$444$	0	0
$445$	8267.58	0.880721
$446$	0	0
$447$	7145.20	0.756054
$448$	0	0
$449$	−10624.6	−1.11672	−0.558361	−	0.829598i	$-0.688570\pi$
$449$	−10624.6	−1.11672	−0.558361	+	0.829598i	$0.688570\pi$
$450$	0	0
$451$	−4509.85	−0.470866
$452$	0	0
$453$	1867.17	0.193659
$454$	0	0
$455$	−342.952	−0.0353359
$456$	0	0
$457$	−11373.9	−1.16422	−0.582110	−	0.813110i	$-0.697773\pi$
$457$	−11373.9	−1.16422	−0.582110	+	0.813110i	$0.697773\pi$
$458$	0	0
$459$	17686.0	1.79850
$460$	0	0
$461$	−8473.86	−0.856111	−0.428055	−	0.903753i	$-0.640801\pi$
$461$	−8473.86	−0.856111	−0.428055	+	0.903753i	$0.640801\pi$
$462$	0	0
$463$	−9796.71	−0.983352	−0.491676	−	0.870778i	$-0.663615\pi$
$463$	−9796.71	−0.983352	−0.491676	+	0.870778i	$0.663615\pi$
$464$	0	0
$465$	2440.31	0.243369
$466$	0	0
$467$	3000.81	0.297347	0.148674	−	0.988886i	$-0.452500\pi$
$467$	3000.81	0.297347	0.148674	+	0.988886i	$0.452500\pi$
$468$	0	0
$469$	−2077.70	−0.204561
$470$	0	0
$471$	29828.5	2.91810
$472$	0	0
$473$	−185.255	−0.0180085
$474$	0	0
$475$	−7296.44	−0.704808
$476$	0	0
$477$	−8646.76	−0.829996
$478$	0	0
$479$	−14384.2	−1.37209	−0.686047	−	0.727557i	$-0.740655\pi$
$479$	−14384.2	−1.37209	−0.686047	+	0.727557i	$0.740655\pi$
$480$	0	0
$481$	1022.17	0.0968955
$482$	0	0
$483$	−6633.30	−0.624898
$484$	0	0
$485$	−12830.0	−1.20120
$486$	0	0
$487$	19562.5	1.82025	0.910126	−	0.414332i	$-0.135985\pi$
$487$	19562.5	1.82025	0.910126	+	0.414332i	$0.135985\pi$
$488$	0	0
$489$	−12751.1	−1.17919
$490$	0	0
$491$	−1724.88	−0.158539	−0.0792697	−	0.996853i	$-0.525259\pi$
$491$	−1724.88	−0.158539	−0.0792697	+	0.996853i	$0.525259\pi$
$492$	0	0
$493$	−3313.82	−0.302732
$494$	0	0
$495$	−4222.71	−0.383428
$496$	0	0
$497$	−6596.90	−0.595395
$498$	0	0
$499$	−20898.7	−1.87486	−0.937430	−	0.348173i	$-0.886802\pi$
$499$	−20898.7	−1.87486	−0.937430	+	0.348173i	$0.886802\pi$
$500$	0	0
$501$	37796.2	3.37048
$502$	0	0
$503$	6062.13	0.537369	0.268685	−	0.963228i	$-0.413411\pi$
$503$	6062.13	0.537369	0.268685	+	0.963228i	$0.413411\pi$
$504$	0	0
$505$	−9925.86	−0.874644
$506$	0	0
$507$	19499.2	1.70807
$508$	0	0
$509$	310.270	0.0270186	0.0135093	−	0.999909i	$-0.495700\pi$
$509$	310.270	0.0270186	0.0135093	+	0.999909i	$0.495700\pi$
$510$	0	0
$511$	610.111	0.0528174
$512$	0	0
$513$	−24478.0	−2.10668
$514$	0	0
$515$	−3934.15	−0.336620
$516$	0	0
$517$	−5889.93	−0.501043
$518$	0	0
$519$	24465.9	2.06924
$520$	0	0
$521$	−6406.10	−0.538688	−0.269344	−	0.963044i	$-0.586807\pi$
$521$	−6406.10	−0.538688	−0.269344	+	0.963044i	$0.586807\pi$
$522$	0	0
$523$	−21061.8	−1.76094	−0.880468	−	0.474106i	$-0.842771\pi$
$523$	−21061.8	−1.76094	−0.880468	+	0.474106i	$0.842771\pi$
$524$	0	0
$525$	−4905.26	−0.407777
$526$	0	0
$527$	2654.07	0.219380
$528$	0	0
$529$	−1282.03	−0.105370
$530$	0	0
$531$	−24931.0	−2.03750
$532$	0	0
$533$	2903.84	0.235983
$534$	0	0
$535$	−12338.4	−0.997072
$536$	0	0
$537$	17055.0	1.37054
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−6597.32	−0.524290	−0.262145	−	0.965029i	$-0.584430\pi$
$541$	−6597.32	−0.524290	−0.262145	+	0.965029i	$0.584430\pi$
$542$	0	0
$543$	−5806.52	−0.458898
$544$	0	0
$545$	−3663.28	−0.287922
$546$	0	0
$547$	−13719.3	−1.07238	−0.536192	−	0.844096i	$-0.680138\pi$
$547$	−13719.3	−1.07238	−0.536192	+	0.844096i	$0.680138\pi$
$548$	0	0
$549$	−29363.0	−2.28267
$550$	0	0
$551$	4586.44	0.354608
$552$	0	0
$553$	−9302.76	−0.715359
$554$	0	0
$555$	−9067.12	−0.693473
$556$	0	0
$557$	9528.54	0.724842	0.362421	−	0.932014i	$-0.381950\pi$
$557$	9528.54	0.724842	0.362421	+	0.932014i	$0.381950\pi$
$558$	0	0
$559$	119.283	0.00902529
$560$	0	0
$561$	−6826.98	−0.513789
$562$	0	0
$563$	12164.1	0.910576	0.455288	−	0.890344i	$-0.349536\pi$
$563$	12164.1	0.910576	0.455288	+	0.890344i	$0.349536\pi$
$564$	0	0
$565$	13800.5	1.02760
$566$	0	0
$567$	−5967.24	−0.441976
$568$	0	0
$569$	7553.23	0.556499	0.278249	−	0.960509i	$-0.410246\pi$
$569$	7553.23	0.556499	0.278249	+	0.960509i	$0.410246\pi$
$570$	0	0
$571$	−18632.0	−1.36555	−0.682773	−	0.730631i	$-0.739226\pi$
$571$	−18632.0	−1.36555	−0.682773	+	0.730631i	$0.739226\pi$
$572$	0	0
$573$	33656.7	2.45380
$574$	0	0
$575$	8049.33	0.583792
$576$	0	0
$577$	−8150.64	−0.588069	−0.294034	−	0.955795i	$-0.594998\pi$
$577$	−8150.64	−0.588069	−0.294034	+	0.955795i	$0.594998\pi$
$578$	0	0
$579$	−19504.2	−1.39994
$580$	0	0
$581$	4277.00	0.305405
$582$	0	0
$583$	1713.88	0.121752
$584$	0	0
$585$	2718.95	0.192162
$586$	0	0
$587$	−14337.5	−1.00813	−0.504065	−	0.863666i	$-0.668163\pi$
$587$	−14337.5	−1.00813	−0.504065	+	0.863666i	$0.668163\pi$
$588$	0	0
$589$	−3673.32	−0.256972
$590$	0	0
$591$	−21407.9	−1.49002
$592$	0	0
$593$	−27017.3	−1.87094	−0.935471	−	0.353402i	$-0.885025\pi$
$593$	−27017.3	−1.87094	−0.935471	+	0.353402i	$0.885025\pi$
$594$	0	0
$595$	3308.63	0.227968
$596$	0	0
$597$	47151.0	3.23243
$598$	0	0
$599$	−18429.3	−1.25710	−0.628549	−	0.777770i	$-0.716351\pi$
$599$	−18429.3	−1.25710	−0.628549	+	0.777770i	$0.716351\pi$
$600$	0	0
$601$	−4515.02	−0.306442	−0.153221	−	0.988192i	$-0.548965\pi$
$601$	−4515.02	−0.306442	−0.153221	+	0.988192i	$0.548965\pi$
$602$	0	0
$603$	16472.2	1.11243
$604$	0	0
$605$	836.986	0.0562451
$606$	0	0
$607$	−9714.87	−0.649612	−0.324806	−	0.945781i	$-0.605299\pi$
$607$	−9714.87	−0.649612	−0.324806	+	0.945781i	$0.605299\pi$
$608$	0	0
$609$	3083.38	0.205164
$610$	0	0
$611$	3792.46	0.251107
$612$	0	0
$613$	10724.5	0.706618	0.353309	−	0.935507i	$-0.385056\pi$
$613$	10724.5	0.706618	0.353309	+	0.935507i	$0.385056\pi$
$614$	0	0
$615$	−25758.5	−1.68891
$616$	0	0
$617$	28206.5	1.84044	0.920219	−	0.391404i	$-0.128010\pi$
$617$	28206.5	1.84044	0.920219	+	0.391404i	$0.128010\pi$
$618$	0	0
$619$	−156.324	−0.0101505	−0.00507526	−	0.999987i	$-0.501616\pi$
$619$	−156.324	−0.0101505	−0.00507526	+	0.999987i	$0.501616\pi$
$620$	0	0
$621$	27003.8	1.74497
$622$	0	0
$623$	−8366.50	−0.538037
$624$	0	0
$625$	−28.6176	−0.00183152
$626$	0	0
$627$	9448.77	0.601830
$628$	0	0
$629$	−9861.36	−0.625116
$630$	0	0
$631$	−16349.5	−1.03148	−0.515740	−	0.856745i	$-0.672483\pi$
$631$	−16349.5	−1.03148	−0.515740	+	0.856745i	$0.672483\pi$
$632$	0	0
$633$	36515.8	2.29285
$634$	0	0
$635$	7959.31	0.497410
$636$	0	0
$637$	347.055	0.0215869
$638$	0	0
$639$	52300.8	3.23785
$640$	0	0
$641$	−16933.8	−1.04344	−0.521721	−	0.853116i	$-0.674710\pi$
$641$	−16933.8	−1.04344	−0.521721	+	0.853116i	$0.674710\pi$
$642$	0	0
$643$	−14717.7	−0.902657	−0.451328	−	0.892358i	$-0.649050\pi$
$643$	−14717.7	−0.902657	−0.451328	+	0.892358i	$0.649050\pi$
$644$	0	0
$645$	−1058.10	−0.0645933
$646$	0	0
$647$	4572.55	0.277845	0.138922	−	0.990303i	$-0.455636\pi$
$647$	4572.55	0.277845	0.138922	+	0.990303i	$0.455636\pi$
$648$	0	0
$649$	4941.58	0.298881
$650$	0	0
$651$	−2469.51	−0.148675
$652$	0	0
$653$	400.476	0.0239997	0.0119999	−	0.999928i	$-0.496180\pi$
$653$	400.476	0.0239997	0.0119999	+	0.999928i	$0.496180\pi$
$654$	0	0
$655$	15431.3	0.920536
$656$	0	0
$657$	−4837.01	−0.287229
$658$	0	0
$659$	−1761.23	−0.104109	−0.0520544	−	0.998644i	$-0.516577\pi$
$659$	−1761.23	−0.104109	−0.0520544	+	0.998644i	$0.516577\pi$
$660$	0	0
$661$	−20410.8	−1.20104	−0.600520	−	0.799610i	$-0.705040\pi$
$661$	−20410.8	−1.20104	−0.600520	+	0.799610i	$0.705040\pi$
$662$	0	0
$663$	4395.81	0.257495
$664$	0	0
$665$	−4579.26	−0.267032
$666$	0	0
$667$	−5059.70	−0.293722
$668$	0	0
$669$	−4039.63	−0.233455
$670$	0	0
$671$	5820.06	0.334845
$672$	0	0
$673$	4578.55	0.262244	0.131122	−	0.991366i	$-0.458142\pi$
$673$	4578.55	0.262244	0.131122	+	0.991366i	$0.458142\pi$
$674$	0	0
$675$	19969.0	1.13868
$676$	0	0
$677$	−16330.1	−0.927057	−0.463528	−	0.886082i	$-0.653417\pi$
$677$	−16330.1	−0.927057	−0.463528	+	0.886082i	$0.653417\pi$
$678$	0	0
$679$	12983.5	0.733816
$680$	0	0
$681$	43210.4	2.43146
$682$	0	0
$683$	25379.8	1.42186	0.710929	−	0.703263i	$-0.248275\pi$
$683$	25379.8	1.42186	0.710929	+	0.703263i	$0.248275\pi$
$684$	0	0
$685$	4541.81	0.253334
$686$	0	0
$687$	−9381.98	−0.521026
$688$	0	0
$689$	−1103.54	−0.0610184
$690$	0	0
$691$	−18557.9	−1.02167	−0.510837	−	0.859678i	$-0.670664\pi$
$691$	−18557.9	−1.02167	−0.510837	+	0.859678i	$0.670664\pi$
$692$	0	0
$693$	4273.24	0.234238
$694$	0	0
$695$	−7001.44	−0.382129
$696$	0	0
$697$	−28014.8	−1.52243
$698$	0	0
$699$	26817.5	1.45112
$700$	0	0
$701$	25115.0	1.35318	0.676590	−	0.736360i	$-0.263457\pi$
$701$	25115.0	1.35318	0.676590	+	0.736360i	$0.263457\pi$
$702$	0	0
$703$	13648.5	0.732235
$704$	0	0
$705$	−33641.0	−1.79715
$706$	0	0
$707$	10044.6	0.534324
$708$	0	0
$709$	36408.7	1.92857	0.964286	−	0.264864i	$-0.0853272\pi$
$709$	36408.7	1.92857	0.964286	+	0.264864i	$0.0853272\pi$
$710$	0	0
$711$	73753.0	3.89023
$712$	0	0
$713$	4052.36	0.212850
$714$	0	0
$715$	−538.925	−0.0281883
$716$	0	0
$717$	48889.3	2.54645
$718$	0	0
$719$	−3503.59	−0.181727	−0.0908637	−	0.995863i	$-0.528963\pi$
$719$	−3503.59	−0.181727	−0.0908637	+	0.995863i	$0.528963\pi$
$720$	0	0
$721$	3981.22	0.205643
$722$	0	0
$723$	−25053.4	−1.28872
$724$	0	0
$725$	−3741.60	−0.191668
$726$	0	0
$727$	18869.2	0.962613	0.481306	−	0.876552i	$-0.340162\pi$
$727$	18869.2	0.962613	0.481306	+	0.876552i	$0.340162\pi$
$728$	0	0
$729$	−16164.7	−0.821254
$730$	0	0
$731$	−1150.79	−0.0582262
$732$	0	0
$733$	−26123.3	−1.31635	−0.658176	−	0.752864i	$-0.728672\pi$
$733$	−26123.3	−1.31635	−0.658176	+	0.752864i	$0.728672\pi$
$734$	0	0
$735$	−3078.55	−0.154495
$736$	0	0
$737$	−3264.95	−0.163183
$738$	0	0
$739$	1259.95	0.0627170	0.0313585	−	0.999508i	$-0.490017\pi$
$739$	1259.95	0.0627170	0.0313585	+	0.999508i	$0.490017\pi$
$740$	0	0
$741$	−6083.95	−0.301619
$742$	0	0
$743$	6987.68	0.345024	0.172512	−	0.985007i	$-0.444812\pi$
$743$	6987.68	0.345024	0.172512	+	0.985007i	$0.444812\pi$
$744$	0	0
$745$	−5441.63	−0.267605
$746$	0	0
$747$	−33908.4	−1.66084
$748$	0	0
$749$	12486.0	0.609116
$750$	0	0
$751$	−26501.3	−1.28768	−0.643838	−	0.765162i	$-0.722659\pi$
$751$	−26501.3	−1.28768	−0.643838	+	0.765162i	$0.722659\pi$
$752$	0	0
$753$	−34645.5	−1.67670
$754$	0	0
$755$	−1422.00	−0.0685454
$756$	0	0
$757$	−18638.9	−0.894905	−0.447453	−	0.894308i	$-0.647669\pi$
$757$	−18638.9	−0.894905	−0.447453	+	0.894308i	$0.647669\pi$
$758$	0	0
$759$	−10423.8	−0.498496
$760$	0	0
$761$	10502.8	0.500298	0.250149	−	0.968207i	$-0.419520\pi$
$761$	10502.8	0.500298	0.250149	+	0.968207i	$0.419520\pi$
$762$	0	0
$763$	3707.11	0.175893
$764$	0	0
$765$	−26231.1	−1.23972
$766$	0	0
$767$	−3181.82	−0.149790
$768$	0	0
$769$	−31451.8	−1.47488	−0.737440	−	0.675413i	$-0.763965\pi$
$769$	−31451.8	−1.47488	−0.737440	+	0.675413i	$0.763965\pi$
$770$	0	0
$771$	63214.7	2.95282
$772$	0	0
$773$	−12785.3	−0.594897	−0.297448	−	0.954738i	$-0.596136\pi$
$773$	−12785.3	−0.594897	−0.297448	+	0.954738i	$0.596136\pi$
$774$	0	0
$775$	2996.68	0.138895
$776$	0	0
$777$	9175.60	0.423646
$778$	0	0
$779$	38773.4	1.78331
$780$	0	0
$781$	−10366.6	−0.474961
$782$	0	0
$783$	−12552.3	−0.572900
$784$	0	0
$785$	−22716.7	−1.03286
$786$	0	0
$787$	26171.7	1.18542	0.592708	−	0.805418i	$-0.298059\pi$
$787$	26171.7	1.18542	0.592708	+	0.805418i	$0.298059\pi$
$788$	0	0
$789$	8690.15	0.392114
$790$	0	0
$791$	−13965.6	−0.627763
$792$	0	0
$793$	−3747.46	−0.167814
$794$	0	0
$795$	9788.99	0.436704
$796$	0	0
$797$	−9956.37	−0.442500	−0.221250	−	0.975217i	$-0.571014\pi$
$797$	−9956.37	−0.442500	−0.221250	+	0.975217i	$0.571014\pi$
$798$	0	0
$799$	−36587.8	−1.62000
$800$	0	0
$801$	66330.3	2.92593
$802$	0	0
$803$	958.745	0.0421337
$804$	0	0
$805$	5051.78	0.221182
$806$	0	0
$807$	8854.67	0.386244
$808$	0	0
$809$	27460.5	1.19340	0.596699	−	0.802465i	$-0.296478\pi$
$809$	27460.5	1.19340	0.596699	+	0.802465i	$0.296478\pi$
$810$	0	0
$811$	−13721.5	−0.594116	−0.297058	−	0.954859i	$-0.596006\pi$
$811$	−13721.5	−0.594116	−0.297058	+	0.954859i	$0.596006\pi$
$812$	0	0
$813$	31971.6	1.37920
$814$	0	0
$815$	9710.94	0.417374
$816$	0	0
$817$	1592.73	0.0682037
$818$	0	0
$819$	−2751.48	−0.117393
$820$	0	0
$821$	−7959.34	−0.338347	−0.169173	−	0.985586i	$-0.554110\pi$
$821$	−7959.34	−0.338347	−0.169173	+	0.985586i	$0.554110\pi$
$822$	0	0
$823$	−15031.6	−0.636659	−0.318329	−	0.947980i	$-0.603122\pi$
$823$	−15031.6	−0.636659	−0.318329	+	0.947980i	$0.603122\pi$
$824$	0	0
$825$	−7708.27	−0.325294
$826$	0	0
$827$	−34367.2	−1.44506	−0.722531	−	0.691339i	$-0.757021\pi$
$827$	−34367.2	−1.44506	−0.722531	+	0.691339i	$0.757021\pi$
$828$	0	0
$829$	34355.0	1.43932	0.719662	−	0.694324i	$-0.244297\pi$
$829$	34355.0	1.43932	0.719662	+	0.694324i	$0.244297\pi$
$830$	0	0
$831$	−20986.6	−0.876072
$832$	0	0
$833$	−3348.22	−0.139267
$834$	0	0
$835$	−28784.8	−1.19298
$836$	0	0
$837$	10053.2	0.415161
$838$	0	0
$839$	−25095.0	−1.03263	−0.516314	−	0.856399i	$-0.672696\pi$
$839$	−25095.0	−1.03263	−0.516314	+	0.856399i	$0.672696\pi$
$840$	0	0
$841$	−22037.1	−0.903566
$842$	0	0
$843$	−51014.7	−2.08427
$844$	0	0
$845$	−14850.2	−0.604569
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−9674.67	−0.391088
$850$	0	0
$851$	−15056.8	−0.606510
$852$	0	0
$853$	−8877.88	−0.356357	−0.178179	−	0.983998i	$-0.557020\pi$
$853$	−8877.88	−0.356357	−0.178179	+	0.983998i	$0.557020\pi$
$854$	0	0
$855$	36304.8	1.45216
$856$	0	0
$857$	−28281.8	−1.12729	−0.563646	−	0.826016i	$-0.690602\pi$
$857$	−28281.8	−1.12729	−0.563646	+	0.826016i	$0.690602\pi$
$858$	0	0
$859$	25282.1	1.00421	0.502104	−	0.864807i	$-0.332559\pi$
$859$	25282.1	1.00421	0.502104	+	0.864807i	$0.332559\pi$
$860$	0	0
$861$	26066.7	1.03176
$862$	0	0
$863$	38721.6	1.52735	0.763673	−	0.645603i	$-0.223394\pi$
$863$	38721.6	1.52735	0.763673	+	0.645603i	$0.223394\pi$
$864$	0	0
$865$	−18632.7	−0.732405
$866$	0	0
$867$	2214.99	0.0867648
$868$	0	0
$869$	−14618.6	−0.570659
$870$	0	0
$871$	2102.26	0.0817824
$872$	0	0
$873$	−102934.	−3.99061
$874$	0	0
$875$	9788.33	0.378178
$876$	0	0
$877$	47198.6	1.81731	0.908656	−	0.417546i	$-0.137110\pi$
$877$	47198.6	1.81731	0.908656	+	0.417546i	$0.137110\pi$
$878$	0	0
$879$	−31364.1	−1.20351
$880$	0	0
$881$	12794.1	0.489266	0.244633	−	0.969616i	$-0.421333\pi$
$881$	12794.1	0.489266	0.244633	+	0.969616i	$0.421333\pi$
$882$	0	0
$883$	22310.6	0.850295	0.425148	−	0.905124i	$-0.360222\pi$
$883$	22310.6	0.850295	0.425148	+	0.905124i	$0.360222\pi$
$884$	0	0
$885$	28224.3	1.07203
$886$	0	0
$887$	31281.7	1.18415	0.592073	−	0.805884i	$-0.298310\pi$
$887$	31281.7	1.18415	0.592073	+	0.805884i	$0.298310\pi$
$888$	0	0
$889$	−8054.54	−0.303870
$890$	0	0
$891$	−9377.09	−0.352575
$892$	0	0
$893$	50638.7	1.89760
$894$	0	0
$895$	−12988.8	−0.485102
$896$	0	0
$897$	6711.73	0.249831
$898$	0	0
$899$	−1883.67	−0.0698820
$900$	0	0
$901$	10646.5	0.393657
$902$	0	0
$903$	1070.76	0.0394603
$904$	0	0
$905$	4422.12	0.162427
$906$	0	0
$907$	31863.9	1.16651	0.583255	−	0.812289i	$-0.301779\pi$
$907$	31863.9	1.16651	0.583255	+	0.812289i	$0.301779\pi$
$908$	0	0
$909$	−79634.6	−2.90573
$910$	0	0
$911$	38643.5	1.40540	0.702698	−	0.711488i	$-0.251978\pi$
$911$	38643.5	1.40540	0.702698	+	0.711488i	$0.251978\pi$
$912$	0	0
$913$	6721.01	0.243629
$914$	0	0
$915$	33241.9	1.20103
$916$	0	0
$917$	−15615.9	−0.562360
$918$	0	0
$919$	22868.7	0.820857	0.410428	−	0.911893i	$-0.365379\pi$
$919$	22868.7	0.820857	0.410428	+	0.911893i	$0.365379\pi$
$920$	0	0
$921$	44186.8	1.58090
$922$	0	0
$923$	6674.90	0.238036
$924$	0	0
$925$	−11134.3	−0.395778
$926$	0	0
$927$	−31563.4	−1.11832
$928$	0	0
$929$	9352.39	0.330293	0.165146	−	0.986269i	$-0.447190\pi$
$929$	9352.39	0.330293	0.165146	+	0.986269i	$0.447190\pi$
$930$	0	0
$931$	4634.05	0.163131
$932$	0	0
$933$	3438.28	0.120648
$934$	0	0
$935$	5199.28	0.181855
$936$	0	0
$937$	6943.74	0.242094	0.121047	−	0.992647i	$-0.461375\pi$
$937$	6943.74	0.242094	0.121047	+	0.992647i	$0.461375\pi$
$938$	0	0
$939$	−17371.0	−0.603707
$940$	0	0
$941$	−17294.1	−0.599120	−0.299560	−	0.954077i	$-0.596840\pi$
$941$	−17294.1	−0.599120	−0.299560	+	0.954077i	$0.596840\pi$
$942$	0	0
$943$	−42774.3	−1.47712
$944$	0	0
$945$	12532.6	0.431413
$946$	0	0
$947$	−52139.8	−1.78914	−0.894570	−	0.446928i	$-0.852518\pi$
$947$	−52139.8	−1.78914	−0.894570	+	0.446928i	$0.852518\pi$
$948$	0	0
$949$	−617.324	−0.0211161
$950$	0	0
$951$	60312.0	2.05652
$952$	0	0
$953$	13921.4	0.473197	0.236599	−	0.971607i	$-0.423967\pi$
$953$	13921.4	0.473197	0.236599	+	0.971607i	$0.423967\pi$
$954$	0	0
$955$	−25632.2	−0.868523
$956$	0	0
$957$	4845.31	0.163664
$958$	0	0
$959$	−4596.16	−0.154763
$960$	0	0
$961$	−28282.4	−0.949359
$962$	0	0
$963$	−98989.9	−3.31246
$964$	0	0
$965$	14853.9	0.495508
$966$	0	0
$967$	18698.2	0.621812	0.310906	−	0.950441i	$-0.399368\pi$
$967$	18698.2	0.621812	0.310906	+	0.950441i	$0.399368\pi$
$968$	0	0
$969$	58695.0	1.94588
$970$	0	0
$971$	46060.5	1.52230	0.761149	−	0.648578i	$-0.224636\pi$
$971$	46060.5	1.52230	0.761149	+	0.648578i	$0.224636\pi$
$972$	0	0
$973$	7085.21	0.233444
$974$	0	0
$975$	4963.26	0.163027
$976$	0	0
$977$	−2625.78	−0.0859839	−0.0429919	−	0.999075i	$-0.513689\pi$
$977$	−2625.78	−0.0859839	−0.0429919	+	0.999075i	$0.513689\pi$
$978$	0	0
$979$	−13147.4	−0.429205
$980$	0	0
$981$	−29390.3	−0.956533
$982$	0	0
$983$	21740.7	0.705412	0.352706	−	0.935734i	$-0.385261\pi$
$983$	21740.7	0.705412	0.352706	+	0.935734i	$0.385261\pi$
$984$	0	0
$985$	16303.8	0.527393
$986$	0	0
$987$	34043.5	1.09789
$988$	0	0
$989$	−1757.07	−0.0564931
$990$	0	0
$991$	−8494.35	−0.272282	−0.136141	−	0.990689i	$-0.543470\pi$
$991$	−8494.35	−0.272282	−0.136141	+	0.990689i	$0.543470\pi$
$992$	0	0
$993$	−48527.8	−1.55084
$994$	0	0
$995$	−35909.2	−1.14412
$996$	0	0
$997$	−9802.62	−0.311386	−0.155693	−	0.987805i	$-0.549761\pi$
$997$	−9802.62	−0.311386	−0.155693	+	0.987805i	$0.549761\pi$
$998$	0	0
$999$	−37353.3	−1.18299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.j.1.1		2
4.3	odd	2		154.4.a.g.1.2	✓	2
12.11	even	2		1386.4.a.u.1.2		2
28.27	even	2		1078.4.a.i.1.1		2
44.43	even	2		1694.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.a.g.1.2	✓	2	4.3	odd	2
1078.4.a.i.1.1		2	28.27	even	2
1232.4.a.j.1.1		2	1.1	even	1	trivial
1386.4.a.u.1.2		2	12.11	even	2
1694.4.a.p.1.2		2	44.43	even	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)

\(f(q)\)	\(=\)	\(q-9.08276 q^{3} +6.91724 q^{5} -7.00000 q^{7} +55.4966 q^{9} +O(q^{10})\)	\(q-9.08276 q^{3} +6.91724 q^{5} -7.00000 q^{7} +55.4966 q^{9} -11.0000 q^{11} +7.08276 q^{13} -62.8276 q^{15} -68.3311 q^{17} +94.5725 q^{19} +63.5793 q^{21} -104.331 q^{23} -77.1518 q^{25} -258.828 q^{27} +48.4966 q^{29} -38.8413 q^{31} +99.9104 q^{33} -48.4207 q^{35} +144.317 q^{37} -64.3311 q^{39} +409.986 q^{41} +16.8413 q^{43} +383.883 q^{45} +535.449 q^{47} +49.0000 q^{49} +620.635 q^{51} -155.807 q^{53} -76.0896 q^{55} -858.979 q^{57} -449.235 q^{59} -529.096 q^{61} -388.476 q^{63} +48.9932 q^{65} +296.814 q^{67} +947.614 q^{69} +942.414 q^{71} -87.1587 q^{73} +700.752 q^{75} +77.0000 q^{77} +1328.97 q^{79} +852.462 q^{81} -611.001 q^{83} -472.662 q^{85} -440.483 q^{87} +1195.21 q^{89} -49.5793 q^{91} +352.787 q^{93} +654.180 q^{95} -1854.79 q^{97} -610.462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 26 q^{5} - 14 q^{7} + 38 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 26 * q^5 - 14 * q^7 + 38 * q^9	\( 2 q - 6 q^{3} + 26 q^{5} - 14 q^{7} + 38 q^{9} - 22 q^{11} + 2 q^{13} - 4 q^{15} - 88 q^{17} - 42 q^{19} + 42 q^{21} - 160 q^{23} + 162 q^{25} - 396 q^{27} + 24 q^{29} - 248 q^{31} + 66 q^{33} - 182 q^{35} - 52 q^{37} - 80 q^{39} + 528 q^{41} + 204 q^{43} + 50 q^{45} - 24 q^{47} + 98 q^{49} + 560 q^{51} + 248 q^{53} - 286 q^{55} - 1280 q^{57} - 570 q^{59} - 1338 q^{61} - 266 q^{63} - 48 q^{65} + 180 q^{67} + 776 q^{69} + 60 q^{71} - 4 q^{73} + 1438 q^{75} + 154 q^{77} + 1928 q^{79} + 902 q^{81} + 542 q^{83} - 848 q^{85} - 516 q^{87} - 140 q^{89} - 14 q^{91} - 292 q^{93} - 1952 q^{95} - 2712 q^{97} - 418 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 26 * q^5 - 14 * q^7 + 38 * q^9 - 22 * q^11 + 2 * q^13 - 4 * q^15 - 88 * q^17 - 42 * q^19 + 42 * q^21 - 160 * q^23 + 162 * q^25 - 396 * q^27 + 24 * q^29 - 248 * q^31 + 66 * q^33 - 182 * q^35 - 52 * q^37 - 80 * q^39 + 528 * q^41 + 204 * q^43 + 50 * q^45 - 24 * q^47 + 98 * q^49 + 560 * q^51 + 248 * q^53 - 286 * q^55 - 1280 * q^57 - 570 * q^59 - 1338 * q^61 - 266 * q^63 - 48 * q^65 + 180 * q^67 + 776 * q^69 + 60 * q^71 - 4 * q^73 + 1438 * q^75 + 154 * q^77 + 1928 * q^79 + 902 * q^81 + 542 * q^83 - 848 * q^85 - 516 * q^87 - 140 * q^89 - 14 * q^91 - 292 * q^93 - 1952 * q^95 - 2712 * q^97 - 418 * q^99

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