LMFDB - Embedded newform 7728.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} +4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} -4.00000 q^{15} +4.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} +11.0000 q^{25} -1.00000 q^{27} +10.0000 q^{29} +2.00000 q^{31} +3.00000 q^{33} -4.00000 q^{35} -10.0000 q^{37} +2.00000 q^{39} +7.00000 q^{41} +4.00000 q^{43} +4.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} -9.00000 q^{53} -12.0000 q^{55} -7.00000 q^{57} -9.00000 q^{59} -1.00000 q^{61} -1.00000 q^{63} -8.00000 q^{65} -1.00000 q^{69} -8.00000 q^{71} +4.00000 q^{73} -11.0000 q^{75} +3.00000 q^{77} -14.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} +16.0000 q^{85} -10.0000 q^{87} +6.00000 q^{89} +2.00000 q^{91} -2.00000 q^{93} +28.0000 q^{95} -2.00000 q^{97} -3.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	11.0000	2.20000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−8.00000	−0.992278
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	−11.0000	−1.27017
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	16.0000	1.73544
$86$	0	0
$87$	−10.0000	−1.07211
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	28.0000	2.87274
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−7.00000	−0.631169
$124$	0	0
$125$	24.0000	2.14663
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	40.0000	3.32182
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	−19.0000	−1.54620	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	−19.0000	−1.54620	−0.773099	+	0.634285i	$0.781294\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	0	0
$177$	9.00000	0.676481
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	0	0
$185$	−40.0000	−2.94086
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−5.00000	−0.361787	−0.180894	−	0.983503i	$-0.557899\pi$
$191$	−5.00000	−0.361787	−0.180894	+	0.983503i	$0.557899\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	8.00000	0.572892
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.0000	−0.701862
$204$	0	0
$205$	28.0000	1.95560
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	23.0000	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	0	0
$237$	14.0000	0.909398
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.00000	0.255551
$246$	0	0
$247$	−14.0000	−0.890799
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−16.0000	−1.00196
$256$	0	0
$257$	−5.00000	−0.311891	−0.155946	−	0.987766i	$-0.549842\pi$
$257$	−5.00000	−0.311891	−0.155946	+	0.987766i	$0.549842\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	29.0000	1.78822	0.894108	−	0.447851i	$-0.147810\pi$
$263$	29.0000	1.78822	0.894108	+	0.447851i	$0.147810\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−33.0000	−1.98997
$276$	0	0
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	−28.0000	−1.65858
$286$	0	0
$287$	−7.00000	−0.413197
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	−36.0000	−2.09600
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	−3.00000	−0.172345
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	0	0
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	−30.0000	−1.67968
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	28.0000	1.55796
$324$	0	0
$325$	−22.0000	−1.22034
$326$	0	0
$327$	8.00000	0.442401
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−23.0000	−1.26419	−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000	−1.26419	−0.632097	+	0.774889i	$0.717806\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	−32.0000	−1.69838
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	16.0000	0.837478
$366$	0	0
$367$	−13.0000	−0.678594	−0.339297	−	0.940679i	$-0.610189\pi$
$367$	−13.0000	−0.678594	−0.339297	+	0.940679i	$0.610189\pi$
$368$	0	0
$369$	7.00000	0.364405
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	0	0
$375$	−24.0000	−1.23935
$376$	0	0
$377$	−20.0000	−1.03005
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	0	0
$381$	−17.0000	−0.870936
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	−56.0000	−2.81767
$396$	0	0
$397$	34.0000	1.70641	0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000	1.70641	0.853206	+	0.521575i	$0.174655\pi$
$398$	0	0
$399$	7.00000	0.350438
$400$	0	0
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	4.00000	0.198762
$406$	0	0
$407$	30.0000	1.48704
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	0	0
$413$	9.00000	0.442861
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	44.0000	2.13431
$426$	0	0
$427$	1.00000	0.0483934
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	−7.00000	−0.337178	−0.168589	−	0.985686i	$-0.553921\pi$
$431$	−7.00000	−0.337178	−0.168589	+	0.985686i	$0.553921\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	−40.0000	−1.91785
$436$	0	0
$437$	7.00000	0.334855
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	11.0000	0.520282
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−21.0000	−0.988851
$452$	0	0
$453$	19.0000	0.892698
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	21.0000	0.975953	0.487976	−	0.872857i	$-0.337735\pi$
$463$	21.0000	0.975953	0.487976	+	0.872857i	$0.337735\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	13.0000	0.599008
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	77.0000	3.53300
$476$	0	0
$477$	−9.00000	−0.412082
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−8.00000	−0.363261
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	0	0
$489$	−17.0000	−0.768767
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	40.0000	1.80151
$494$	0	0
$495$	−12.0000	−0.539360
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−29.0000	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$509$	−29.0000	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	52.0000	2.29139
$516$	0	0
$517$	9.00000	0.395820
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−1.00000	−0.0437269	−0.0218635	−	0.999761i	$-0.506960\pi$
$523$	−1.00000	−0.0437269	−0.0218635	+	0.999761i	$0.506960\pi$
$524$	0	0
$525$	11.0000	0.480079
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.00000	−0.390567
$532$	0	0
$533$	−14.0000	−0.606407
$534$	0	0
$535$	80.0000	3.45870
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−9.00000	−0.386940	−0.193470	−	0.981106i	$-0.561974\pi$
$541$	−9.00000	−0.386940	−0.193470	+	0.981106i	$0.561974\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	−32.0000	−1.37073
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	70.0000	2.98210
$552$	0	0
$553$	14.0000	0.595341
$554$	0	0
$555$	40.0000	1.69791
$556$	0	0
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	40.0000	1.68281
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−35.0000	−1.46728	−0.733638	−	0.679540i	$-0.762179\pi$
$569$	−35.0000	−1.46728	−0.733638	+	0.679540i	$0.762179\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	5.00000	0.208878
$574$	0	0
$575$	11.0000	0.458732
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	19.0000	0.789613
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	27.0000	1.11823
$584$	0	0
$585$	−8.00000	−0.330759
$586$	0	0
$587$	9.00000	0.371470	0.185735	−	0.982600i	$-0.440533\pi$
$587$	9.00000	0.371470	0.185735	+	0.982600i	$0.440533\pi$
$588$	0	0
$589$	14.0000	0.576860
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	−11.0000	−0.450200
$598$	0	0
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.00000	−0.325246
$606$	0	0
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	0	0
$609$	10.0000	0.405220
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	−28.0000	−1.12907
$616$	0	0
$617$	−34.0000	−1.36879	−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000	−1.36879	−0.684394	+	0.729112i	$0.739933\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	41.0000	1.64000
$626$	0	0
$627$	21.0000	0.838659
$628$	0	0
$629$	−40.0000	−1.59490
$630$	0	0
$631$	46.0000	1.83123	0.915616	−	0.402055i	$-0.131704\pi$
$631$	46.0000	1.83123	0.915616	+	0.402055i	$0.131704\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	68.0000	2.69850
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	13.0000	0.513469	0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	0	0
$643$	−35.0000	−1.38027	−0.690133	−	0.723683i	$-0.742448\pi$
$643$	−35.0000	−1.38027	−0.690133	+	0.723683i	$0.742448\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	27.0000	1.05984
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−1.00000	−0.0388955	−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000	−0.0388955	−0.0194477	+	0.999811i	$0.506191\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	−28.0000	−1.08579
$666$	0	0
$667$	10.0000	0.387202
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	3.00000	0.115814
$672$	0	0
$673$	−15.0000	−0.578208	−0.289104	−	0.957298i	$-0.593357\pi$
$673$	−15.0000	−0.578208	−0.289104	+	0.957298i	$0.593357\pi$
$674$	0	0
$675$	−11.0000	−0.423390
$676$	0	0
$677$	−28.0000	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$677$	−28.0000	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	8.00000	0.306111	0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000	0.306111	0.153056	+	0.988218i	$0.451089\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	−23.0000	−0.877505
$688$	0	0
$689$	18.0000	0.685745
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	64.0000	2.42766
$696$	0	0
$697$	28.0000	1.06058
$698$	0	0
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	0	0
$703$	−70.0000	−2.64010
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	−3.00000	−0.112827
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	0	0
$713$	2.00000	0.0749006
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	−12.0000	−0.448148
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	0	0
$723$	−25.0000	−0.929760
$724$	0	0
$725$	110.000	4.08530
$726$	0	0
$727$	17.0000	0.630495	0.315248	−	0.949009i	$-0.397912\pi$
$727$	17.0000	0.630495	0.315248	+	0.949009i	$0.397912\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	14.0000	0.514303
$742$	0	0
$743$	45.0000	1.65089	0.825445	−	0.564483i	$-0.190924\pi$
$743$	45.0000	1.65089	0.825445	+	0.564483i	$0.190924\pi$
$744$	0	0
$745$	−44.0000	−1.61204
$746$	0	0
$747$	6.00000	0.219529
$748$	0	0
$749$	−20.0000	−0.730784
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	−76.0000	−2.76592
$756$	0	0
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	−15.0000	−0.543750	−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000	−0.543750	−0.271875	+	0.962333i	$0.587644\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	0	0
$765$	16.0000	0.578481
$766$	0	0
$767$	18.0000	0.649942
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	5.00000	0.180071
$772$	0	0
$773$	−44.0000	−1.58257	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−44.0000	−1.58257	−0.791285	+	0.611448i	$0.790588\pi$
$774$	0	0
$775$	22.0000	0.790263
$776$	0	0
$777$	−10.0000	−0.358748
$778$	0	0
$779$	49.0000	1.75561
$780$	0	0
$781$	24.0000	0.858788
$782$	0	0
$783$	−10.0000	−0.357371
$784$	0	0
$785$	−52.0000	−1.85596
$786$	0	0
$787$	13.0000	0.463400	0.231700	−	0.972787i	$-0.425571\pi$
$787$	13.0000	0.463400	0.231700	+	0.972787i	$0.425571\pi$
$788$	0	0
$789$	−29.0000	−1.03243
$790$	0	0
$791$	−10.0000	−0.355559
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	36.0000	1.27679
$796$	0	0
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−12.0000	−0.423471
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	68.0000	2.38194
$816$	0	0
$817$	28.0000	0.979596
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	0	0
$825$	33.0000	1.14891
$826$	0	0
$827$	−19.0000	−0.660695	−0.330347	−	0.943859i	$-0.607166\pi$
$827$	−19.0000	−0.660695	−0.330347	+	0.943859i	$0.607166\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	−3.00000	−0.104069
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	36.0000	1.24583
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	2.00000	0.0688837
$844$	0	0
$845$	−36.0000	−1.23844
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	−10.0000	−0.342796
$852$	0	0
$853$	−44.0000	−1.50653	−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000	−1.50653	−0.753266	+	0.657716i	$0.771523\pi$
$854$	0	0
$855$	28.0000	0.957580
$856$	0	0
$857$	−13.0000	−0.444072	−0.222036	−	0.975039i	$-0.571270\pi$
$857$	−13.0000	−0.444072	−0.222036	+	0.975039i	$0.571270\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	7.00000	0.238559
$862$	0	0
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	88.0000	2.99209
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	42.0000	1.42475
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−21.0000	−0.709120	−0.354560	−	0.935033i	$-0.615369\pi$
$877$	−21.0000	−0.709120	−0.354560	+	0.935033i	$0.615369\pi$
$878$	0	0
$879$	−16.0000	−0.539667
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	36.0000	1.21013
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	−17.0000	−0.570162
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−21.0000	−0.702738
$894$	0	0
$895$	−48.0000	−1.60446
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	20.0000	0.667037
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	24.0000	0.796907	0.398453	−	0.917189i	$-0.369547\pi$
$907$	24.0000	0.796907	0.398453	+	0.917189i	$0.369547\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	0	0
$915$	4.00000	0.132236
$916$	0	0
$917$	3.00000	0.0990687
$918$	0	0
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	−110.000	−3.61678
$926$	0	0
$927$	13.0000	0.426976
$928$	0	0
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	0	0
$933$	−11.0000	−0.360124
$934$	0	0
$935$	−48.0000	−1.56977
$936$	0	0
$937$	−59.0000	−1.92745	−0.963723	−	0.266904i	$-0.913999\pi$
$937$	−59.0000	−1.92745	−0.963723	+	0.266904i	$0.913999\pi$
$938$	0	0
$939$	17.0000	0.554774
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	7.00000	0.227951
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	35.0000	1.13376	0.566881	−	0.823800i	$-0.308150\pi$
$953$	35.0000	1.13376	0.566881	+	0.823800i	$0.308150\pi$
$954$	0	0
$955$	−20.0000	−0.647185
$956$	0	0
$957$	30.0000	0.969762
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	20.0000	0.644491
$964$	0	0
$965$	−76.0000	−2.44653
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	0	0
$969$	−28.0000	−0.899490
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	22.0000	0.704564
$976$	0	0
$977$	47.0000	1.50366	0.751832	−	0.659355i	$-0.229171\pi$
$977$	47.0000	1.50366	0.751832	+	0.659355i	$0.229171\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−8.00000	−0.255420
$982$	0	0
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	0	0
$985$	72.0000	2.29411
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	0	0
$993$	23.0000	0.729883
$994$	0	0
$995$	44.0000	1.39489
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.k.1.1		1
4.3	odd	2		3864.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.f.1.1	✓	1	4.3	odd	2
7728.2.a.k.1.1		1	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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