LMFDB - Embedded newform 126.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,4,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("126.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	126.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:	$7.43424066072$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	126.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} -18.0000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} -18.0000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +36.0000 q^{10} +72.0000 q^{11} -34.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -6.00000 q^{17} +92.0000 q^{19} -72.0000 q^{20} -144.000 q^{22} +180.000 q^{23} +199.000 q^{25} +68.0000 q^{26} +28.0000 q^{28} +114.000 q^{29} +56.0000 q^{31} -32.0000 q^{32} +12.0000 q^{34} -126.000 q^{35} -34.0000 q^{37} -184.000 q^{38} +144.000 q^{40} -6.00000 q^{41} +164.000 q^{43} +288.000 q^{44} -360.000 q^{46} -168.000 q^{47} +49.0000 q^{49} -398.000 q^{50} -136.000 q^{52} -654.000 q^{53} -1296.00 q^{55} -56.0000 q^{56} -228.000 q^{58} +492.000 q^{59} -250.000 q^{61} -112.000 q^{62} +64.0000 q^{64} +612.000 q^{65} -124.000 q^{67} -24.0000 q^{68} +252.000 q^{70} -36.0000 q^{71} +1010.00 q^{73} +68.0000 q^{74} +368.000 q^{76} +504.000 q^{77} +56.0000 q^{79} -288.000 q^{80} +12.0000 q^{82} -228.000 q^{83} +108.000 q^{85} -328.000 q^{86} -576.000 q^{88} -390.000 q^{89} -238.000 q^{91} +720.000 q^{92} +336.000 q^{94} -1656.00 q^{95} -70.0000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−18.0000	−1.60997	−0.804984	−	0.593296i	$-0.797826\pi$
$5$	−18.0000	−1.60997	−0.804984	+	0.593296i	$0.797826\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	36.0000	1.13842
$11$	72.0000	1.97353	0.986764	−	0.162160i	$-0.0518462\pi$
$11$	72.0000	1.97353	0.986764	+	0.162160i	$0.0518462\pi$
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−6.00000	−0.0856008	−0.0428004	−	0.999084i	$-0.513628\pi$
$17$	−6.00000	−0.0856008	−0.0428004	+	0.999084i	$0.513628\pi$
$18$	0	0
$19$	92.0000	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$19$	92.0000	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$20$	−72.0000	−0.804984
$21$	0	0
$22$	−144.000	−1.39550
$23$	180.000	1.63185	0.815926	−	0.578156i	$-0.196228\pi$
$23$	180.000	1.63185	0.815926	+	0.578156i	$0.196228\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	68.0000	0.512919
$27$	0	0
$28$	28.0000	0.188982
$29$	114.000	0.729975	0.364987	−	0.931012i	$-0.381073\pi$
$29$	114.000	0.729975	0.364987	+	0.931012i	$0.381073\pi$
$30$	0	0
$31$	56.0000	0.324448	0.162224	−	0.986754i	$-0.448133\pi$
$31$	56.0000	0.324448	0.162224	+	0.986754i	$0.448133\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	12.0000	0.0605289
$35$	−126.000	−0.608511
$36$	0	0
$37$	−34.0000	−0.151069	−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000	−0.151069	−0.0755347	+	0.997143i	$0.524066\pi$
$38$	−184.000	−0.785493
$39$	0	0
$40$	144.000	0.569210
$41$	−6.00000	−0.0228547	−0.0114273	−	0.999935i	$-0.503638\pi$
$41$	−6.00000	−0.0228547	−0.0114273	+	0.999935i	$0.503638\pi$
$42$	0	0
$43$	164.000	0.581622	0.290811	−	0.956780i	$-0.406075\pi$
$43$	164.000	0.581622	0.290811	+	0.956780i	$0.406075\pi$
$44$	288.000	0.986764
$45$	0	0
$46$	−360.000	−1.15389
$47$	−168.000	−0.521390	−0.260695	−	0.965421i	$-0.583952\pi$
$47$	−168.000	−0.521390	−0.260695	+	0.965421i	$0.583952\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−398.000	−1.12571
$51$	0	0
$52$	−136.000	−0.362689
$53$	−654.000	−1.69498	−0.847489	−	0.530813i	$-0.821887\pi$
$53$	−654.000	−1.69498	−0.847489	+	0.530813i	$0.821887\pi$
$54$	0	0
$55$	−1296.00	−3.17732
$56$	−56.0000	−0.133631
$57$	0	0
$58$	−228.000	−0.516170
$59$	492.000	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$59$	492.000	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$60$	0	0
$61$	−250.000	−0.524741	−0.262371	−	0.964967i	$-0.584504\pi$
$61$	−250.000	−0.524741	−0.262371	+	0.964967i	$0.584504\pi$
$62$	−112.000	−0.229420
$63$	0	0
$64$	64.0000	0.125000
$65$	612.000	1.16783
$66$	0	0
$67$	−124.000	−0.226105	−0.113052	−	0.993589i	$-0.536063\pi$
$67$	−124.000	−0.226105	−0.113052	+	0.993589i	$0.536063\pi$
$68$	−24.0000	−0.0428004
$69$	0	0
$70$	252.000	0.430282
$71$	−36.0000	−0.0601748	−0.0300874	−	0.999547i	$-0.509579\pi$
$71$	−36.0000	−0.0601748	−0.0300874	+	0.999547i	$0.509579\pi$
$72$	0	0
$73$	1010.00	1.61934	0.809668	−	0.586888i	$-0.199647\pi$
$73$	1010.00	1.61934	0.809668	+	0.586888i	$0.199647\pi$
$74$	68.0000	0.106822
$75$	0	0
$76$	368.000	0.555428
$77$	504.000	0.745924
$78$	0	0
$79$	56.0000	0.0797531	0.0398765	−	0.999205i	$-0.487304\pi$
$79$	56.0000	0.0797531	0.0398765	+	0.999205i	$0.487304\pi$
$80$	−288.000	−0.402492
$81$	0	0
$82$	12.0000	0.0161607
$83$	−228.000	−0.301521	−0.150761	−	0.988570i	$-0.548172\pi$
$83$	−228.000	−0.301521	−0.150761	+	0.988570i	$0.548172\pi$
$84$	0	0
$85$	108.000	0.137815
$86$	−328.000	−0.411269
$87$	0	0
$88$	−576.000	−0.697748
$89$	−390.000	−0.464493	−0.232247	−	0.972657i	$-0.574608\pi$
$89$	−390.000	−0.464493	−0.232247	+	0.972657i	$0.574608\pi$
$90$	0	0
$91$	−238.000	−0.274167
$92$	720.000	0.815926
$93$	0	0
$94$	336.000	0.368678
$95$	−1656.00	−1.78844
$96$	0	0
$97$	−70.0000	−0.0732724	−0.0366362	−	0.999329i	$-0.511664\pi$
$97$	−70.0000	−0.0732724	−0.0366362	+	0.999329i	$0.511664\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	796.000	0.796000
$101$	1350.00	1.33000	0.665000	−	0.746843i	$-0.268431\pi$
$101$	1350.00	1.33000	0.665000	+	0.746843i	$0.268431\pi$
$102$	0	0
$103$	2000.00	1.91326	0.956630	−	0.291305i	$-0.0940893\pi$
$103$	2000.00	1.91326	0.956630	+	0.291305i	$0.0940893\pi$
$104$	272.000	0.256460
$105$	0	0
$106$	1308.00	1.19853
$107$	−696.000	−0.628830	−0.314415	−	0.949286i	$-0.601808\pi$
$107$	−696.000	−0.628830	−0.314415	+	0.949286i	$0.601808\pi$
$108$	0	0
$109$	−1114.00	−0.978916	−0.489458	−	0.872027i	$-0.662805\pi$
$109$	−1114.00	−0.978916	−0.489458	+	0.872027i	$0.662805\pi$
$110$	2592.00	2.24670
$111$	0	0
$112$	112.000	0.0944911
$113$	462.000	0.384613	0.192307	−	0.981335i	$-0.438403\pi$
$113$	462.000	0.384613	0.192307	+	0.981335i	$0.438403\pi$
$114$	0	0
$115$	−3240.00	−2.62723
$116$	456.000	0.364987
$117$	0	0
$118$	−984.000	−0.767666
$119$	−42.0000	−0.0323541
$120$	0	0
$121$	3853.00	2.89482
$122$	500.000	0.371048
$123$	0	0
$124$	224.000	0.162224
$125$	−1332.00	−0.953102
$126$	0	0
$127$	1064.00	0.743423	0.371712	−	0.928348i	$-0.378771\pi$
$127$	1064.00	0.743423	0.371712	+	0.928348i	$0.378771\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−1224.00	−0.825784
$131$	−180.000	−0.120051	−0.0600255	−	0.998197i	$-0.519118\pi$
$131$	−180.000	−0.120051	−0.0600255	+	0.998197i	$0.519118\pi$
$132$	0	0
$133$	644.000	0.419864
$134$	248.000	0.159880
$135$	0	0
$136$	48.0000	0.0302645
$137$	2718.00	1.69500	0.847498	−	0.530799i	$-0.178108\pi$
$137$	2718.00	1.69500	0.847498	+	0.530799i	$0.178108\pi$
$138$	0	0
$139$	−1348.00	−0.822560	−0.411280	−	0.911509i	$-0.634918\pi$
$139$	−1348.00	−0.822560	−0.411280	+	0.911509i	$0.634918\pi$
$140$	−504.000	−0.304256
$141$	0	0
$142$	72.0000	0.0425500
$143$	−2448.00	−1.43155
$144$	0	0
$145$	−2052.00	−1.17524
$146$	−2020.00	−1.14504
$147$	0	0
$148$	−136.000	−0.0755347
$149$	−558.000	−0.306800	−0.153400	−	0.988164i	$-0.549022\pi$
$149$	−558.000	−0.306800	−0.153400	+	0.988164i	$0.549022\pi$
$150$	0	0
$151$	1928.00	1.03906	0.519531	−	0.854451i	$-0.326107\pi$
$151$	1928.00	1.03906	0.519531	+	0.854451i	$0.326107\pi$
$152$	−736.000	−0.392747
$153$	0	0
$154$	−1008.00	−0.527448
$155$	−1008.00	−0.522352
$156$	0	0
$157$	−2410.00	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$157$	−2410.00	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$158$	−112.000	−0.0563939
$159$	0	0
$160$	576.000	0.284605
$161$	1260.00	0.616782
$162$	0	0
$163$	740.000	0.355591	0.177795	−	0.984067i	$-0.443104\pi$
$163$	740.000	0.355591	0.177795	+	0.984067i	$0.443104\pi$
$164$	−24.0000	−0.0114273
$165$	0	0
$166$	456.000	0.213208
$167$	−3984.00	−1.84605	−0.923027	−	0.384734i	$-0.874293\pi$
$167$	−3984.00	−1.84605	−0.923027	+	0.384734i	$0.874293\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	−216.000	−0.0974497
$171$	0	0
$172$	656.000	0.290811
$173$	1038.00	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	1038.00	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	1393.00	0.601719
$176$	1152.00	0.493382
$177$	0	0
$178$	780.000	0.328446
$179$	2568.00	1.07230	0.536149	−	0.844123i	$-0.319879\pi$
$179$	2568.00	1.07230	0.536149	+	0.844123i	$0.319879\pi$
$180$	0	0
$181$	−2698.00	−1.10796	−0.553980	−	0.832530i	$-0.686892\pi$
$181$	−2698.00	−1.10796	−0.553980	+	0.832530i	$0.686892\pi$
$182$	476.000	0.193865
$183$	0	0
$184$	−1440.00	−0.576947
$185$	612.000	0.243217
$186$	0	0
$187$	−432.000	−0.168936
$188$	−672.000	−0.260695
$189$	0	0
$190$	3312.00	1.26462
$191$	4116.00	1.55928	0.779642	−	0.626225i	$-0.215401\pi$
$191$	4116.00	1.55928	0.779642	+	0.626225i	$0.215401\pi$
$192$	0	0
$193$	−3310.00	−1.23450	−0.617251	−	0.786766i	$-0.711754\pi$
$193$	−3310.00	−1.23450	−0.617251	+	0.786766i	$0.711754\pi$
$194$	140.000	0.0518114
$195$	0	0
$196$	196.000	0.0714286
$197$	−1278.00	−0.462202	−0.231101	−	0.972930i	$-0.574233\pi$
$197$	−1278.00	−0.462202	−0.231101	+	0.972930i	$0.574233\pi$
$198$	0	0
$199$	2936.00	1.04587	0.522933	−	0.852374i	$-0.324838\pi$
$199$	2936.00	1.04587	0.522933	+	0.852374i	$0.324838\pi$
$200$	−1592.00	−0.562857
$201$	0	0
$202$	−2700.00	−0.940452
$203$	798.000	0.275905
$204$	0	0
$205$	108.000	0.0367954
$206$	−4000.00	−1.35288
$207$	0	0
$208$	−544.000	−0.181344
$209$	6624.00	2.19230
$210$	0	0
$211$	−3508.00	−1.14455	−0.572276	−	0.820061i	$-0.693940\pi$
$211$	−3508.00	−1.14455	−0.572276	+	0.820061i	$0.693940\pi$
$212$	−2616.00	−0.847489
$213$	0	0
$214$	1392.00	0.444650
$215$	−2952.00	−0.936394
$216$	0	0
$217$	392.000	0.122630
$218$	2228.00	0.692198
$219$	0	0
$220$	−5184.00	−1.58866
$221$	204.000	0.0620929
$222$	0	0
$223$	−1888.00	−0.566950	−0.283475	−	0.958980i	$-0.591487\pi$
$223$	−1888.00	−0.566950	−0.283475	+	0.958980i	$0.591487\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−924.000	−0.271963
$227$	−3564.00	−1.04207	−0.521037	−	0.853534i	$-0.674455\pi$
$227$	−3564.00	−1.04207	−0.521037	+	0.853534i	$0.674455\pi$
$228$	0	0
$229$	1334.00	0.384948	0.192474	−	0.981302i	$-0.438349\pi$
$229$	1334.00	0.384948	0.192474	+	0.981302i	$0.438349\pi$
$230$	6480.00	1.85773
$231$	0	0
$232$	−912.000	−0.258085
$233$	−2658.00	−0.747345	−0.373672	−	0.927561i	$-0.621902\pi$
$233$	−2658.00	−0.747345	−0.373672	+	0.927561i	$0.621902\pi$
$234$	0	0
$235$	3024.00	0.839421
$236$	1968.00	0.542822
$237$	0	0
$238$	84.0000	0.0228778
$239$	588.000	0.159140	0.0795702	−	0.996829i	$-0.474645\pi$
$239$	588.000	0.159140	0.0795702	+	0.996829i	$0.474645\pi$
$240$	0	0
$241$	5690.00	1.52085	0.760426	−	0.649425i	$-0.224990\pi$
$241$	5690.00	1.52085	0.760426	+	0.649425i	$0.224990\pi$
$242$	−7706.00	−2.04694
$243$	0	0
$244$	−1000.00	−0.262371
$245$	−882.000	−0.229996
$246$	0	0
$247$	−3128.00	−0.805789
$248$	−448.000	−0.114710
$249$	0	0
$250$	2664.00	0.673945
$251$	−180.000	−0.0452649	−0.0226325	−	0.999744i	$-0.507205\pi$
$251$	−180.000	−0.0452649	−0.0226325	+	0.999744i	$0.507205\pi$
$252$	0	0
$253$	12960.0	3.22051
$254$	−2128.00	−0.525680
$255$	0	0
$256$	256.000	0.0625000
$257$	−5310.00	−1.28883	−0.644414	−	0.764677i	$-0.722899\pi$
$257$	−5310.00	−1.28883	−0.644414	+	0.764677i	$0.722899\pi$
$258$	0	0
$259$	−238.000	−0.0570988
$260$	2448.00	0.583917
$261$	0	0
$262$	360.000	0.0848888
$263$	−828.000	−0.194132	−0.0970659	−	0.995278i	$-0.530946\pi$
$263$	−828.000	−0.194132	−0.0970659	+	0.995278i	$0.530946\pi$
$264$	0	0
$265$	11772.0	2.72886
$266$	−1288.00	−0.296889
$267$	0	0
$268$	−496.000	−0.113052
$269$	4134.00	0.937005	0.468503	−	0.883462i	$-0.344794\pi$
$269$	4134.00	0.937005	0.468503	+	0.883462i	$0.344794\pi$
$270$	0	0
$271$	−2968.00	−0.665288	−0.332644	−	0.943052i	$-0.607941\pi$
$271$	−2968.00	−0.665288	−0.332644	+	0.943052i	$0.607941\pi$
$272$	−96.0000	−0.0214002
$273$	0	0
$274$	−5436.00	−1.19854
$275$	14328.0	3.14186
$276$	0	0
$277$	−4786.00	−1.03813	−0.519067	−	0.854734i	$-0.673720\pi$
$277$	−4786.00	−1.03813	−0.519067	+	0.854734i	$0.673720\pi$
$278$	2696.00	0.581638
$279$	0	0
$280$	1008.00	0.215141
$281$	4398.00	0.933675	0.466838	−	0.884343i	$-0.345393\pi$
$281$	4398.00	0.933675	0.466838	+	0.884343i	$0.345393\pi$
$282$	0	0
$283$	4772.00	1.00235	0.501177	−	0.865345i	$-0.332901\pi$
$283$	4772.00	1.00235	0.501177	+	0.865345i	$0.332901\pi$
$284$	−144.000	−0.0300874
$285$	0	0
$286$	4896.00	1.01226
$287$	−42.0000	−0.00863826
$288$	0	0
$289$	−4877.00	−0.992673
$290$	4104.00	0.831018
$291$	0	0
$292$	4040.00	0.809668
$293$	−6522.00	−1.30041	−0.650204	−	0.759760i	$-0.725316\pi$
$293$	−6522.00	−1.30041	−0.650204	+	0.759760i	$0.725316\pi$
$294$	0	0
$295$	−8856.00	−1.74785
$296$	272.000	0.0534111
$297$	0	0
$298$	1116.00	0.216940
$299$	−6120.00	−1.18371
$300$	0	0
$301$	1148.00	0.219833
$302$	−3856.00	−0.734728
$303$	0	0
$304$	1472.00	0.277714
$305$	4500.00	0.844817
$306$	0	0
$307$	−6244.00	−1.16079	−0.580397	−	0.814333i	$-0.697103\pi$
$307$	−6244.00	−1.16079	−0.580397	+	0.814333i	$0.697103\pi$
$308$	2016.00	0.372962
$309$	0	0
$310$	2016.00	0.369358
$311$	528.000	0.0962705	0.0481353	−	0.998841i	$-0.484672\pi$
$311$	528.000	0.0962705	0.0481353	+	0.998841i	$0.484672\pi$
$312$	0	0
$313$	−5830.00	−1.05281	−0.526407	−	0.850232i	$-0.676461\pi$
$313$	−5830.00	−1.05281	−0.526407	+	0.850232i	$0.676461\pi$
$314$	4820.00	0.866269
$315$	0	0
$316$	224.000	0.0398765
$317$	−5046.00	−0.894043	−0.447021	−	0.894523i	$-0.647515\pi$
$317$	−5046.00	−0.894043	−0.447021	+	0.894523i	$0.647515\pi$
$318$	0	0
$319$	8208.00	1.44063
$320$	−1152.00	−0.201246
$321$	0	0
$322$	−2520.00	−0.436131
$323$	−552.000	−0.0950901
$324$	0	0
$325$	−6766.00	−1.15480
$326$	−1480.00	−0.251441
$327$	0	0
$328$	48.0000	0.00808036
$329$	−1176.00	−0.197067
$330$	0	0
$331$	−5020.00	−0.833608	−0.416804	−	0.908996i	$-0.636850\pi$
$331$	−5020.00	−0.833608	−0.416804	+	0.908996i	$0.636850\pi$
$332$	−912.000	−0.150761
$333$	0	0
$334$	7968.00	1.30536
$335$	2232.00	0.364021
$336$	0	0
$337$	−7486.00	−1.21005	−0.605027	−	0.796205i	$-0.706838\pi$
$337$	−7486.00	−1.21005	−0.605027	+	0.796205i	$0.706838\pi$
$338$	2082.00	0.335047
$339$	0	0
$340$	432.000	0.0689073
$341$	4032.00	0.640308
$342$	0	0
$343$	343.000	0.0539949
$344$	−1312.00	−0.205635
$345$	0	0
$346$	−2076.00	−0.322562
$347$	10032.0	1.55201	0.776003	−	0.630729i	$-0.217244\pi$
$347$	10032.0	1.55201	0.776003	+	0.630729i	$0.217244\pi$
$348$	0	0
$349$	5942.00	0.911370	0.455685	−	0.890141i	$-0.349394\pi$
$349$	5942.00	0.911370	0.455685	+	0.890141i	$0.349394\pi$
$350$	−2786.00	−0.425480
$351$	0	0
$352$	−2304.00	−0.348874
$353$	90.0000	0.0135700	0.00678501	−	0.999977i	$-0.497840\pi$
$353$	90.0000	0.0135700	0.00678501	+	0.999977i	$0.497840\pi$
$354$	0	0
$355$	648.000	0.0968796
$356$	−1560.00	−0.232247
$357$	0	0
$358$	−5136.00	−0.758229
$359$	−10596.0	−1.55776	−0.778880	−	0.627174i	$-0.784212\pi$
$359$	−10596.0	−1.55776	−0.778880	+	0.627174i	$0.784212\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	5396.00	0.783446
$363$	0	0
$364$	−952.000	−0.137083
$365$	−18180.0	−2.60708
$366$	0	0
$367$	4016.00	0.571208	0.285604	−	0.958348i	$-0.407806\pi$
$367$	4016.00	0.571208	0.285604	+	0.958348i	$0.407806\pi$
$368$	2880.00	0.407963
$369$	0	0
$370$	−1224.00	−0.171980
$371$	−4578.00	−0.640641
$372$	0	0
$373$	3278.00	0.455036	0.227518	−	0.973774i	$-0.426939\pi$
$373$	3278.00	0.455036	0.227518	+	0.973774i	$0.426939\pi$
$374$	864.000	0.119456
$375$	0	0
$376$	1344.00	0.184339
$377$	−3876.00	−0.529507
$378$	0	0
$379$	4628.00	0.627241	0.313621	−	0.949548i	$-0.398458\pi$
$379$	4628.00	0.627241	0.313621	+	0.949548i	$0.398458\pi$
$380$	−6624.00	−0.894221
$381$	0	0
$382$	−8232.00	−1.10258
$383$	2880.00	0.384233	0.192116	−	0.981372i	$-0.438465\pi$
$383$	2880.00	0.384233	0.192116	+	0.981372i	$0.438465\pi$
$384$	0	0
$385$	−9072.00	−1.20091
$386$	6620.00	0.872925
$387$	0	0
$388$	−280.000	−0.0366362
$389$	−7974.00	−1.03933	−0.519663	−	0.854371i	$-0.673943\pi$
$389$	−7974.00	−1.03933	−0.519663	+	0.854371i	$0.673943\pi$
$390$	0	0
$391$	−1080.00	−0.139688
$392$	−392.000	−0.0505076
$393$	0	0
$394$	2556.00	0.326826
$395$	−1008.00	−0.128400
$396$	0	0
$397$	−12346.0	−1.56078	−0.780388	−	0.625296i	$-0.784978\pi$
$397$	−12346.0	−1.56078	−0.780388	+	0.625296i	$0.784978\pi$
$398$	−5872.00	−0.739540
$399$	0	0
$400$	3184.00	0.398000
$401$	−9738.00	−1.21270	−0.606350	−	0.795198i	$-0.707367\pi$
$401$	−9738.00	−1.21270	−0.606350	+	0.795198i	$0.707367\pi$
$402$	0	0
$403$	−1904.00	−0.235347
$404$	5400.00	0.665000
$405$	0	0
$406$	−1596.00	−0.195094
$407$	−2448.00	−0.298140
$408$	0	0
$409$	−430.000	−0.0519857	−0.0259928	−	0.999662i	$-0.508275\pi$
$409$	−430.000	−0.0519857	−0.0259928	+	0.999662i	$0.508275\pi$
$410$	−216.000	−0.0260182
$411$	0	0
$412$	8000.00	0.956630
$413$	3444.00	0.410335
$414$	0	0
$415$	4104.00	0.485440
$416$	1088.00	0.128230
$417$	0	0
$418$	−13248.0	−1.55019
$419$	1812.00	0.211270	0.105635	−	0.994405i	$-0.466313\pi$
$419$	1812.00	0.211270	0.105635	+	0.994405i	$0.466313\pi$
$420$	0	0
$421$	−10690.0	−1.23753	−0.618763	−	0.785577i	$-0.712366\pi$
$421$	−10690.0	−1.23753	−0.618763	+	0.785577i	$0.712366\pi$
$422$	7016.00	0.809321
$423$	0	0
$424$	5232.00	0.599265
$425$	−1194.00	−0.136276
$426$	0	0
$427$	−1750.00	−0.198334
$428$	−2784.00	−0.314415
$429$	0	0
$430$	5904.00	0.662131
$431$	4116.00	0.460002	0.230001	−	0.973190i	$-0.426127\pi$
$431$	4116.00	0.460002	0.230001	+	0.973190i	$0.426127\pi$
$432$	0	0
$433$	9938.00	1.10298	0.551489	−	0.834182i	$-0.314060\pi$
$433$	9938.00	1.10298	0.551489	+	0.834182i	$0.314060\pi$
$434$	−784.000	−0.0867125
$435$	0	0
$436$	−4456.00	−0.489458
$437$	16560.0	1.81275
$438$	0	0
$439$	1784.00	0.193954	0.0969769	−	0.995287i	$-0.469083\pi$
$439$	1784.00	0.193954	0.0969769	+	0.995287i	$0.469083\pi$
$440$	10368.0	1.12335
$441$	0	0
$442$	−408.000	−0.0439063
$443$	11712.0	1.25610	0.628052	−	0.778172i	$-0.283853\pi$
$443$	11712.0	1.25610	0.628052	+	0.778172i	$0.283853\pi$
$444$	0	0
$445$	7020.00	0.747820
$446$	3776.00	0.400894
$447$	0	0
$448$	448.000	0.0472456
$449$	−7650.00	−0.804066	−0.402033	−	0.915625i	$-0.631696\pi$
$449$	−7650.00	−0.804066	−0.402033	+	0.915625i	$0.631696\pi$
$450$	0	0
$451$	−432.000	−0.0451044
$452$	1848.00	0.192307
$453$	0	0
$454$	7128.00	0.736858
$455$	4284.00	0.441400
$456$	0	0
$457$	3674.00	0.376067	0.188033	−	0.982163i	$-0.439789\pi$
$457$	3674.00	0.376067	0.188033	+	0.982163i	$0.439789\pi$
$458$	−2668.00	−0.272200
$459$	0	0
$460$	−12960.0	−1.31362
$461$	3102.00	0.313394	0.156697	−	0.987647i	$-0.449915\pi$
$461$	3102.00	0.313394	0.156697	+	0.987647i	$0.449915\pi$
$462$	0	0
$463$	8984.00	0.901775	0.450888	−	0.892581i	$-0.351108\pi$
$463$	8984.00	0.901775	0.450888	+	0.892581i	$0.351108\pi$
$464$	1824.00	0.182494
$465$	0	0
$466$	5316.00	0.528453
$467$	−3612.00	−0.357909	−0.178954	−	0.983857i	$-0.557271\pi$
$467$	−3612.00	−0.357909	−0.178954	+	0.983857i	$0.557271\pi$
$468$	0	0
$469$	−868.000	−0.0854595
$470$	−6048.00	−0.593561
$471$	0	0
$472$	−3936.00	−0.383833
$473$	11808.0	1.14785
$474$	0	0
$475$	18308.0	1.76848
$476$	−168.000	−0.0161770
$477$	0	0
$478$	−1176.00	−0.112529
$479$	9288.00	0.885970	0.442985	−	0.896529i	$-0.353920\pi$
$479$	9288.00	0.885970	0.442985	+	0.896529i	$0.353920\pi$
$480$	0	0
$481$	1156.00	0.109582
$482$	−11380.0	−1.07540
$483$	0	0
$484$	15412.0	1.44741
$485$	1260.00	0.117966
$486$	0	0
$487$	−5848.00	−0.544144	−0.272072	−	0.962277i	$-0.587709\pi$
$487$	−5848.00	−0.544144	−0.272072	+	0.962277i	$0.587709\pi$
$488$	2000.00	0.185524
$489$	0	0
$490$	1764.00	0.162631
$491$	5952.00	0.547067	0.273534	−	0.961862i	$-0.411808\pi$
$491$	5952.00	0.547067	0.273534	+	0.961862i	$0.411808\pi$
$492$	0	0
$493$	−684.000	−0.0624864
$494$	6256.00	0.569779
$495$	0	0
$496$	896.000	0.0811121
$497$	−252.000	−0.0227440
$498$	0	0
$499$	10748.0	0.964222	0.482111	−	0.876110i	$-0.339870\pi$
$499$	10748.0	0.964222	0.482111	+	0.876110i	$0.339870\pi$
$500$	−5328.00	−0.476551
$501$	0	0
$502$	360.000	0.0320071
$503$	16488.0	1.46156	0.730779	−	0.682614i	$-0.239157\pi$
$503$	16488.0	1.46156	0.730779	+	0.682614i	$0.239157\pi$
$504$	0	0
$505$	−24300.0	−2.14126
$506$	−25920.0	−2.27724
$507$	0	0
$508$	4256.00	0.371712
$509$	−14058.0	−1.22418	−0.612092	−	0.790786i	$-0.709672\pi$
$509$	−14058.0	−1.22418	−0.612092	+	0.790786i	$0.709672\pi$
$510$	0	0
$511$	7070.00	0.612052
$512$	−512.000	−0.0441942
$513$	0	0
$514$	10620.0	0.911339
$515$	−36000.0	−3.08029
$516$	0	0
$517$	−12096.0	−1.02898
$518$	476.000	0.0403750
$519$	0	0
$520$	−4896.00	−0.412892
$521$	14466.0	1.21644	0.608222	−	0.793767i	$-0.291883\pi$
$521$	14466.0	1.21644	0.608222	+	0.793767i	$0.291883\pi$
$522$	0	0
$523$	18524.0	1.54875	0.774377	−	0.632725i	$-0.218064\pi$
$523$	18524.0	1.54875	0.774377	+	0.632725i	$0.218064\pi$
$524$	−720.000	−0.0600255
$525$	0	0
$526$	1656.00	0.137272
$527$	−336.000	−0.0277730
$528$	0	0
$529$	20233.0	1.66294
$530$	−23544.0	−1.92960
$531$	0	0
$532$	2576.00	0.209932
$533$	204.000	0.0165783
$534$	0	0
$535$	12528.0	1.01240
$536$	992.000	0.0799401
$537$	0	0
$538$	−8268.00	−0.662563
$539$	3528.00	0.281933
$540$	0	0
$541$	4358.00	0.346331	0.173165	−	0.984893i	$-0.444600\pi$
$541$	4358.00	0.346331	0.173165	+	0.984893i	$0.444600\pi$
$542$	5936.00	0.470430
$543$	0	0
$544$	192.000	0.0151322
$545$	20052.0	1.57602
$546$	0	0
$547$	−2140.00	−0.167276	−0.0836378	−	0.996496i	$-0.526654\pi$
$547$	−2140.00	−0.167276	−0.0836378	+	0.996496i	$0.526654\pi$
$548$	10872.0	0.847498
$549$	0	0
$550$	−28656.0	−2.22163
$551$	10488.0	0.810896
$552$	0	0
$553$	392.000	0.0301438
$554$	9572.00	0.734071
$555$	0	0
$556$	−5392.00	−0.411280
$557$	−2022.00	−0.153815	−0.0769074	−	0.997038i	$-0.524505\pi$
$557$	−2022.00	−0.153815	−0.0769074	+	0.997038i	$0.524505\pi$
$558$	0	0
$559$	−5576.00	−0.421896
$560$	−2016.00	−0.152128
$561$	0	0
$562$	−8796.00	−0.660208
$563$	−7356.00	−0.550654	−0.275327	−	0.961351i	$-0.588786\pi$
$563$	−7356.00	−0.550654	−0.275327	+	0.961351i	$0.588786\pi$
$564$	0	0
$565$	−8316.00	−0.619215
$566$	−9544.00	−0.708771
$567$	0	0
$568$	288.000	0.0212750
$569$	−11202.0	−0.825329	−0.412665	−	0.910883i	$-0.635402\pi$
$569$	−11202.0	−0.825329	−0.412665	+	0.910883i	$0.635402\pi$
$570$	0	0
$571$	−10564.0	−0.774238	−0.387119	−	0.922030i	$-0.626530\pi$
$571$	−10564.0	−0.774238	−0.387119	+	0.922030i	$0.626530\pi$
$572$	−9792.00	−0.715776
$573$	0	0
$574$	84.0000	0.00610817
$575$	35820.0	2.59791
$576$	0	0
$577$	−18574.0	−1.34011	−0.670057	−	0.742310i	$-0.733730\pi$
$577$	−18574.0	−1.34011	−0.670057	+	0.742310i	$0.733730\pi$
$578$	9754.00	0.701925
$579$	0	0
$580$	−8208.00	−0.587618
$581$	−1596.00	−0.113964
$582$	0	0
$583$	−47088.0	−3.34509
$584$	−8080.00	−0.572522
$585$	0	0
$586$	13044.0	0.919527
$587$	−13188.0	−0.927303	−0.463652	−	0.886018i	$-0.653461\pi$
$587$	−13188.0	−0.927303	−0.463652	+	0.886018i	$0.653461\pi$
$588$	0	0
$589$	5152.00	0.360415
$590$	17712.0	1.23592
$591$	0	0
$592$	−544.000	−0.0377673
$593$	22506.0	1.55853	0.779267	−	0.626692i	$-0.215592\pi$
$593$	22506.0	1.55853	0.779267	+	0.626692i	$0.215592\pi$
$594$	0	0
$595$	756.000	0.0520890
$596$	−2232.00	−0.153400
$597$	0	0
$598$	12240.0	0.837008
$599$	−10596.0	−0.722773	−0.361386	−	0.932416i	$-0.617696\pi$
$599$	−10596.0	−0.722773	−0.361386	+	0.932416i	$0.617696\pi$
$600$	0	0
$601$	14618.0	0.992148	0.496074	−	0.868280i	$-0.334775\pi$
$601$	14618.0	0.992148	0.496074	+	0.868280i	$0.334775\pi$
$602$	−2296.00	−0.155445
$603$	0	0
$604$	7712.00	0.519531
$605$	−69354.0	−4.66056
$606$	0	0
$607$	5168.00	0.345573	0.172786	−	0.984959i	$-0.444723\pi$
$607$	5168.00	0.345573	0.172786	+	0.984959i	$0.444723\pi$
$608$	−2944.00	−0.196373
$609$	0	0
$610$	−9000.00	−0.597376
$611$	5712.00	0.378204
$612$	0	0
$613$	5726.00	0.377277	0.188639	−	0.982047i	$-0.439593\pi$
$613$	5726.00	0.377277	0.188639	+	0.982047i	$0.439593\pi$
$614$	12488.0	0.820806
$615$	0	0
$616$	−4032.00	−0.263724
$617$	7806.00	0.509332	0.254666	−	0.967029i	$-0.418035\pi$
$617$	7806.00	0.509332	0.254666	+	0.967029i	$0.418035\pi$
$618$	0	0
$619$	−18052.0	−1.17217	−0.586083	−	0.810251i	$-0.699331\pi$
$619$	−18052.0	−1.17217	−0.586083	+	0.810251i	$0.699331\pi$
$620$	−4032.00	−0.261176
$621$	0	0
$622$	−1056.00	−0.0680735
$623$	−2730.00	−0.175562
$624$	0	0
$625$	−899.000	−0.0575360
$626$	11660.0	0.744453
$627$	0	0
$628$	−9640.00	−0.612544
$629$	204.000	0.0129317
$630$	0	0
$631$	−6208.00	−0.391659	−0.195829	−	0.980638i	$-0.562740\pi$
$631$	−6208.00	−0.391659	−0.195829	+	0.980638i	$0.562740\pi$
$632$	−448.000	−0.0281970
$633$	0	0
$634$	10092.0	0.632184
$635$	−19152.0	−1.19689
$636$	0	0
$637$	−1666.00	−0.103625
$638$	−16416.0	−1.01868
$639$	0	0
$640$	2304.00	0.142302
$641$	21510.0	1.32542	0.662710	−	0.748876i	$-0.269406\pi$
$641$	21510.0	1.32542	0.662710	+	0.748876i	$0.269406\pi$
$642$	0	0
$643$	−11140.0	−0.683233	−0.341616	−	0.939839i	$-0.610974\pi$
$643$	−11140.0	−0.683233	−0.341616	+	0.939839i	$0.610974\pi$
$644$	5040.00	0.308391
$645$	0	0
$646$	1104.00	0.0672389
$647$	−9312.00	−0.565831	−0.282915	−	0.959145i	$-0.591302\pi$
$647$	−9312.00	−0.565831	−0.282915	+	0.959145i	$0.591302\pi$
$648$	0	0
$649$	35424.0	2.14255
$650$	13532.0	0.816567
$651$	0	0
$652$	2960.00	0.177795
$653$	−4878.00	−0.292329	−0.146165	−	0.989260i	$-0.546693\pi$
$653$	−4878.00	−0.292329	−0.146165	+	0.989260i	$0.546693\pi$
$654$	0	0
$655$	3240.00	0.193278
$656$	−96.0000	−0.00571367
$657$	0	0
$658$	2352.00	0.139347
$659$	9744.00	0.575982	0.287991	−	0.957633i	$-0.407013\pi$
$659$	9744.00	0.575982	0.287991	+	0.957633i	$0.407013\pi$
$660$	0	0
$661$	2990.00	0.175942	0.0879709	−	0.996123i	$-0.471962\pi$
$661$	2990.00	0.175942	0.0879709	+	0.996123i	$0.471962\pi$
$662$	10040.0	0.589450
$663$	0	0
$664$	1824.00	0.106604
$665$	−11592.0	−0.675968
$666$	0	0
$667$	20520.0	1.19121
$668$	−15936.0	−0.923027
$669$	0	0
$670$	−4464.00	−0.257402
$671$	−18000.0	−1.03559
$672$	0	0
$673$	33266.0	1.90536	0.952682	−	0.303969i	$-0.0983118\pi$
$673$	33266.0	1.90536	0.952682	+	0.303969i	$0.0983118\pi$
$674$	14972.0	0.855638
$675$	0	0
$676$	−4164.00	−0.236914
$677$	−5370.00	−0.304854	−0.152427	−	0.988315i	$-0.548709\pi$
$677$	−5370.00	−0.304854	−0.152427	+	0.988315i	$0.548709\pi$
$678$	0	0
$679$	−490.000	−0.0276944
$680$	−864.000	−0.0487248
$681$	0	0
$682$	−8064.00	−0.452766
$683$	−384.000	−0.0215130	−0.0107565	−	0.999942i	$-0.503424\pi$
$683$	−384.000	−0.0215130	−0.0107565	+	0.999942i	$0.503424\pi$
$684$	0	0
$685$	−48924.0	−2.72889
$686$	−686.000	−0.0381802
$687$	0	0
$688$	2624.00	0.145406
$689$	22236.0	1.22950
$690$	0	0
$691$	−14524.0	−0.799593	−0.399797	−	0.916604i	$-0.630919\pi$
$691$	−14524.0	−0.799593	−0.399797	+	0.916604i	$0.630919\pi$
$692$	4152.00	0.228086
$693$	0	0
$694$	−20064.0	−1.09743
$695$	24264.0	1.32430
$696$	0	0
$697$	36.0000	0.00195638
$698$	−11884.0	−0.644436
$699$	0	0
$700$	5572.00	0.300860
$701$	−24750.0	−1.33352	−0.666758	−	0.745274i	$-0.732318\pi$
$701$	−24750.0	−1.33352	−0.666758	+	0.745274i	$0.732318\pi$
$702$	0	0
$703$	−3128.00	−0.167816
$704$	4608.00	0.246691
$705$	0	0
$706$	−180.000	−0.00959545
$707$	9450.00	0.502693
$708$	0	0
$709$	−1042.00	−0.0551948	−0.0275974	−	0.999619i	$-0.508786\pi$
$709$	−1042.00	−0.0551948	−0.0275974	+	0.999619i	$0.508786\pi$
$710$	−1296.00	−0.0685042
$711$	0	0
$712$	3120.00	0.164223
$713$	10080.0	0.529452
$714$	0	0
$715$	44064.0	2.30476
$716$	10272.0	0.536149
$717$	0	0
$718$	21192.0	1.10150
$719$	36960.0	1.91707	0.958536	−	0.284970i	$-0.0919836\pi$
$719$	36960.0	1.91707	0.958536	+	0.284970i	$0.0919836\pi$
$720$	0	0
$721$	14000.0	0.723145
$722$	−3210.00	−0.165462
$723$	0	0
$724$	−10792.0	−0.553980
$725$	22686.0	1.16212
$726$	0	0
$727$	−16288.0	−0.830933	−0.415467	−	0.909608i	$-0.636382\pi$
$727$	−16288.0	−0.830933	−0.415467	+	0.909608i	$0.636382\pi$
$728$	1904.00	0.0969326
$729$	0	0
$730$	36360.0	1.84348
$731$	−984.000	−0.0497874
$732$	0	0
$733$	−7810.00	−0.393546	−0.196773	−	0.980449i	$-0.563046\pi$
$733$	−7810.00	−0.393546	−0.196773	+	0.980449i	$0.563046\pi$
$734$	−8032.00	−0.403905
$735$	0	0
$736$	−5760.00	−0.288473
$737$	−8928.00	−0.446224
$738$	0	0
$739$	−36700.0	−1.82684	−0.913418	−	0.407024i	$-0.866567\pi$
$739$	−36700.0	−1.82684	−0.913418	+	0.407024i	$0.866567\pi$
$740$	2448.00	0.121608
$741$	0	0
$742$	9156.00	0.453002
$743$	−29508.0	−1.45699	−0.728495	−	0.685051i	$-0.759780\pi$
$743$	−29508.0	−1.45699	−0.728495	+	0.685051i	$0.759780\pi$
$744$	0	0
$745$	10044.0	0.493938
$746$	−6556.00	−0.321759
$747$	0	0
$748$	−1728.00	−0.0844678
$749$	−4872.00	−0.237676
$750$	0	0
$751$	−15136.0	−0.735447	−0.367723	−	0.929935i	$-0.619863\pi$
$751$	−15136.0	−0.735447	−0.367723	+	0.929935i	$0.619863\pi$
$752$	−2688.00	−0.130347
$753$	0	0
$754$	7752.00	0.374418
$755$	−34704.0	−1.67286
$756$	0	0
$757$	3422.00	0.164299	0.0821497	−	0.996620i	$-0.473821\pi$
$757$	3422.00	0.164299	0.0821497	+	0.996620i	$0.473821\pi$
$758$	−9256.00	−0.443526
$759$	0	0
$760$	13248.0	0.632310
$761$	−31446.0	−1.49792	−0.748960	−	0.662616i	$-0.769446\pi$
$761$	−31446.0	−1.49792	−0.748960	+	0.662616i	$0.769446\pi$
$762$	0	0
$763$	−7798.00	−0.369995
$764$	16464.0	0.779642
$765$	0	0
$766$	−5760.00	−0.271694
$767$	−16728.0	−0.787501
$768$	0	0
$769$	−18718.0	−0.877748	−0.438874	−	0.898549i	$-0.644623\pi$
$769$	−18718.0	−0.877748	−0.438874	+	0.898549i	$0.644623\pi$
$770$	18144.0	0.849175
$771$	0	0
$772$	−13240.0	−0.617251
$773$	1686.00	0.0784492	0.0392246	−	0.999230i	$-0.487511\pi$
$773$	1686.00	0.0784492	0.0392246	+	0.999230i	$0.487511\pi$
$774$	0	0
$775$	11144.0	0.516522
$776$	560.000	0.0259057
$777$	0	0
$778$	15948.0	0.734915
$779$	−552.000	−0.0253883
$780$	0	0
$781$	−2592.00	−0.118757
$782$	2160.00	0.0987742
$783$	0	0
$784$	784.000	0.0357143
$785$	43380.0	1.97235
$786$	0	0
$787$	5492.00	0.248753	0.124377	−	0.992235i	$-0.460307\pi$
$787$	5492.00	0.248753	0.124377	+	0.992235i	$0.460307\pi$
$788$	−5112.00	−0.231101
$789$	0	0
$790$	2016.00	0.0907925
$791$	3234.00	0.145370
$792$	0	0
$793$	8500.00	0.380635
$794$	24692.0	1.10364
$795$	0	0
$796$	11744.0	0.522933
$797$	17310.0	0.769325	0.384662	−	0.923057i	$-0.374318\pi$
$797$	17310.0	0.769325	0.384662	+	0.923057i	$0.374318\pi$
$798$	0	0
$799$	1008.00	0.0446314
$800$	−6368.00	−0.281428
$801$	0	0
$802$	19476.0	0.857508
$803$	72720.0	3.19581
$804$	0	0
$805$	−22680.0	−0.993000
$806$	3808.00	0.166416
$807$	0	0
$808$	−10800.0	−0.470226
$809$	−35754.0	−1.55382	−0.776912	−	0.629609i	$-0.783215\pi$
$809$	−35754.0	−1.55382	−0.776912	+	0.629609i	$0.783215\pi$
$810$	0	0
$811$	33644.0	1.45672	0.728360	−	0.685194i	$-0.240283\pi$
$811$	33644.0	1.45672	0.728360	+	0.685194i	$0.240283\pi$
$812$	3192.00	0.137952
$813$	0	0
$814$	4896.00	0.210817
$815$	−13320.0	−0.572490
$816$	0	0
$817$	15088.0	0.646098
$818$	860.000	0.0367594
$819$	0	0
$820$	432.000	0.0183977
$821$	−28734.0	−1.22147	−0.610733	−	0.791837i	$-0.709125\pi$
$821$	−28734.0	−1.22147	−0.610733	+	0.791837i	$0.709125\pi$
$822$	0	0
$823$	−28672.0	−1.21439	−0.607195	−	0.794553i	$-0.707705\pi$
$823$	−28672.0	−1.21439	−0.607195	+	0.794553i	$0.707705\pi$
$824$	−16000.0	−0.676440
$825$	0	0
$826$	−6888.00	−0.290150
$827$	15912.0	0.669062	0.334531	−	0.942385i	$-0.391422\pi$
$827$	15912.0	0.669062	0.334531	+	0.942385i	$0.391422\pi$
$828$	0	0
$829$	17534.0	0.734597	0.367299	−	0.930103i	$-0.380283\pi$
$829$	17534.0	0.734597	0.367299	+	0.930103i	$0.380283\pi$
$830$	−8208.00	−0.343258
$831$	0	0
$832$	−2176.00	−0.0906721
$833$	−294.000	−0.0122287
$834$	0	0
$835$	71712.0	2.97209
$836$	26496.0	1.09615
$837$	0	0
$838$	−3624.00	−0.149390
$839$	−40656.0	−1.67295	−0.836473	−	0.548009i	$-0.815386\pi$
$839$	−40656.0	−1.67295	−0.836473	+	0.548009i	$0.815386\pi$
$840$	0	0
$841$	−11393.0	−0.467137
$842$	21380.0	0.875063
$843$	0	0
$844$	−14032.0	−0.572276
$845$	18738.0	0.762848
$846$	0	0
$847$	26971.0	1.09414
$848$	−10464.0	−0.423744
$849$	0	0
$850$	2388.00	0.0963620
$851$	−6120.00	−0.246523
$852$	0	0
$853$	23870.0	0.958140	0.479070	−	0.877777i	$-0.340974\pi$
$853$	23870.0	0.958140	0.479070	+	0.877777i	$0.340974\pi$
$854$	3500.00	0.140243
$855$	0	0
$856$	5568.00	0.222325
$857$	29610.0	1.18023	0.590116	−	0.807319i	$-0.299082\pi$
$857$	29610.0	1.18023	0.590116	+	0.807319i	$0.299082\pi$
$858$	0	0
$859$	−45484.0	−1.80663	−0.903314	−	0.428979i	$-0.858873\pi$
$859$	−45484.0	−1.80663	−0.903314	+	0.428979i	$0.858873\pi$
$860$	−11808.0	−0.468197
$861$	0	0
$862$	−8232.00	−0.325270
$863$	−46164.0	−1.82090	−0.910452	−	0.413614i	$-0.864266\pi$
$863$	−46164.0	−1.82090	−0.910452	+	0.413614i	$0.864266\pi$
$864$	0	0
$865$	−18684.0	−0.734422
$866$	−19876.0	−0.779924
$867$	0	0
$868$	1568.00	0.0613150
$869$	4032.00	0.157395
$870$	0	0
$871$	4216.00	0.164011
$872$	8912.00	0.346099
$873$	0	0
$874$	−33120.0	−1.28181
$875$	−9324.00	−0.360239
$876$	0	0
$877$	−2986.00	−0.114972	−0.0574858	−	0.998346i	$-0.518308\pi$
$877$	−2986.00	−0.114972	−0.0574858	+	0.998346i	$0.518308\pi$
$878$	−3568.00	−0.137146
$879$	0	0
$880$	−20736.0	−0.794330
$881$	−6534.00	−0.249871	−0.124935	−	0.992165i	$-0.539872\pi$
$881$	−6534.00	−0.249871	−0.124935	+	0.992165i	$0.539872\pi$
$882$	0	0
$883$	29756.0	1.13405	0.567027	−	0.823699i	$-0.308094\pi$
$883$	29756.0	1.13405	0.567027	+	0.823699i	$0.308094\pi$
$884$	816.000	0.0310464
$885$	0	0
$886$	−23424.0	−0.888199
$887$	−29952.0	−1.13381	−0.566905	−	0.823783i	$-0.691859\pi$
$887$	−29952.0	−1.13381	−0.566905	+	0.823783i	$0.691859\pi$
$888$	0	0
$889$	7448.00	0.280988
$890$	−14040.0	−0.528789
$891$	0	0
$892$	−7552.00	−0.283475
$893$	−15456.0	−0.579188
$894$	0	0
$895$	−46224.0	−1.72637
$896$	−896.000	−0.0334077
$897$	0	0
$898$	15300.0	0.568561
$899$	6384.00	0.236839
$900$	0	0
$901$	3924.00	0.145091
$902$	864.000	0.0318936
$903$	0	0
$904$	−3696.00	−0.135981
$905$	48564.0	1.78378
$906$	0	0
$907$	−36268.0	−1.32774	−0.663869	−	0.747848i	$-0.731087\pi$
$907$	−36268.0	−1.32774	−0.663869	+	0.747848i	$0.731087\pi$
$908$	−14256.0	−0.521037
$909$	0	0
$910$	−8568.00	−0.312117
$911$	23604.0	0.858436	0.429218	−	0.903201i	$-0.358789\pi$
$911$	23604.0	0.858436	0.429218	+	0.903201i	$0.358789\pi$
$912$	0	0
$913$	−16416.0	−0.595061
$914$	−7348.00	−0.265919
$915$	0	0
$916$	5336.00	0.192474
$917$	−1260.00	−0.0453750
$918$	0	0
$919$	34184.0	1.22701	0.613507	−	0.789689i	$-0.289758\pi$
$919$	34184.0	1.22701	0.613507	+	0.789689i	$0.289758\pi$
$920$	25920.0	0.928866
$921$	0	0
$922$	−6204.00	−0.221603
$923$	1224.00	0.0436495
$924$	0	0
$925$	−6766.00	−0.240502
$926$	−17968.0	−0.637651
$927$	0	0
$928$	−3648.00	−0.129043
$929$	53922.0	1.90433	0.952165	−	0.305583i	$-0.0988513\pi$
$929$	53922.0	1.90433	0.952165	+	0.305583i	$0.0988513\pi$
$930$	0	0
$931$	4508.00	0.158694
$932$	−10632.0	−0.373672
$933$	0	0
$934$	7224.00	0.253080
$935$	7776.00	0.271981
$936$	0	0
$937$	40538.0	1.41336	0.706680	−	0.707533i	$-0.250192\pi$
$937$	40538.0	1.41336	0.706680	+	0.707533i	$0.250192\pi$
$938$	1736.00	0.0604290
$939$	0	0
$940$	12096.0	0.419711
$941$	3606.00	0.124923	0.0624613	−	0.998047i	$-0.480105\pi$
$941$	3606.00	0.124923	0.0624613	+	0.998047i	$0.480105\pi$
$942$	0	0
$943$	−1080.00	−0.0372955
$944$	7872.00	0.271411
$945$	0	0
$946$	−23616.0	−0.811652
$947$	−14064.0	−0.482596	−0.241298	−	0.970451i	$-0.577573\pi$
$947$	−14064.0	−0.482596	−0.241298	+	0.970451i	$0.577573\pi$
$948$	0	0
$949$	−34340.0	−1.17463
$950$	−36616.0	−1.25051
$951$	0	0
$952$	336.000	0.0114389
$953$	−33066.0	−1.12394	−0.561969	−	0.827158i	$-0.689956\pi$
$953$	−33066.0	−1.12394	−0.561969	+	0.827158i	$0.689956\pi$
$954$	0	0
$955$	−74088.0	−2.51040
$956$	2352.00	0.0795702
$957$	0	0
$958$	−18576.0	−0.626475
$959$	19026.0	0.640648
$960$	0	0
$961$	−26655.0	−0.894733
$962$	−2312.00	−0.0774864
$963$	0	0
$964$	22760.0	0.760426
$965$	59580.0	1.98751
$966$	0	0
$967$	−26368.0	−0.876875	−0.438437	−	0.898762i	$-0.644468\pi$
$967$	−26368.0	−0.876875	−0.438437	+	0.898762i	$0.644468\pi$
$968$	−30824.0	−1.02347
$969$	0	0
$970$	−2520.00	−0.0834148
$971$	−55884.0	−1.84696	−0.923482	−	0.383641i	$-0.874670\pi$
$971$	−55884.0	−1.84696	−0.923482	+	0.383641i	$0.874670\pi$
$972$	0	0
$973$	−9436.00	−0.310899
$974$	11696.0	0.384768
$975$	0	0
$976$	−4000.00	−0.131185
$977$	51126.0	1.67417	0.837086	−	0.547072i	$-0.184257\pi$
$977$	51126.0	1.67417	0.837086	+	0.547072i	$0.184257\pi$
$978$	0	0
$979$	−28080.0	−0.916691
$980$	−3528.00	−0.114998
$981$	0	0
$982$	−11904.0	−0.386835
$983$	14184.0	0.460223	0.230112	−	0.973164i	$-0.426091\pi$
$983$	14184.0	0.460223	0.230112	+	0.973164i	$0.426091\pi$
$984$	0	0
$985$	23004.0	0.744130
$986$	1368.00	0.0441846
$987$	0	0
$988$	−12512.0	−0.402894
$989$	29520.0	0.949122
$990$	0	0
$991$	51680.0	1.65658	0.828289	−	0.560301i	$-0.189314\pi$
$991$	51680.0	1.65658	0.828289	+	0.560301i	$0.189314\pi$
$992$	−1792.00	−0.0573549
$993$	0	0
$994$	504.000	0.0160824
$995$	−52848.0	−1.68381
$996$	0	0
$997$	52094.0	1.65480	0.827399	−	0.561615i	$-0.189820\pi$
$997$	52094.0	1.65480	0.827399	+	0.561615i	$0.189820\pi$
$998$	−21496.0	−0.681808
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.4.a.a.1.1		1
3.2	odd	2		42.4.a.a.1.1	✓	1
4.3	odd	2		1008.4.a.b.1.1		1
7.2	even	3		882.4.g.w.361.1		2
7.3	odd	6		882.4.g.o.667.1		2
7.4	even	3		882.4.g.w.667.1		2
7.5	odd	6		882.4.g.o.361.1		2
7.6	odd	2		882.4.a.g.1.1		1
12.11	even	2		336.4.a.l.1.1		1
15.2	even	4		1050.4.g.a.799.2		2
15.8	even	4		1050.4.g.a.799.1		2
15.14	odd	2		1050.4.a.g.1.1		1
21.2	odd	6		294.4.e.c.67.1		2
21.5	even	6		294.4.e.b.67.1		2
21.11	odd	6		294.4.e.c.79.1		2
21.17	even	6		294.4.e.b.79.1		2
21.20	even	2		294.4.a.i.1.1		1
24.5	odd	2		1344.4.a.o.1.1		1
24.11	even	2		1344.4.a.a.1.1		1
84.83	odd	2		2352.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.a.a.1.1	✓	1	3.2	odd	2
126.4.a.a.1.1		1	1.1	even	1	trivial
294.4.a.i.1.1		1	21.20	even	2
294.4.e.b.67.1		2	21.5	even	6
294.4.e.b.79.1		2	21.17	even	6
294.4.e.c.67.1		2	21.2	odd	6
294.4.e.c.79.1		2	21.11	odd	6
336.4.a.l.1.1		1	12.11	even	2
882.4.a.g.1.1		1	7.6	odd	2
882.4.g.o.361.1		2	7.5	odd	6
882.4.g.o.667.1		2	7.3	odd	6
882.4.g.w.361.1		2	7.2	even	3
882.4.g.w.667.1		2	7.4	even	3
1008.4.a.b.1.1		1	4.3	odd	2
1050.4.a.g.1.1		1	15.14	odd	2
1050.4.g.a.799.1		2	15.8	even	4
1050.4.g.a.799.2		2	15.2	even	4
1344.4.a.a.1.1		1	24.11	even	2
1344.4.a.o.1.1		1	24.5	odd	2
2352.4.a.a.1.1		1	84.83	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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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