LMFDB - Embedded newform 1216.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+5.00000 q^{3} +12.0000 q^{5} +11.0000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q+5.00000 q^{3} +12.0000 q^{5} +11.0000 q^{7} -2.00000 q^{9} +54.0000 q^{11} -11.0000 q^{13} +60.0000 q^{15} -93.0000 q^{17} -19.0000 q^{19} +55.0000 q^{21} +183.000 q^{23} +19.0000 q^{25} -145.000 q^{27} +249.000 q^{29} +56.0000 q^{31} +270.000 q^{33} +132.000 q^{35} +250.000 q^{37} -55.0000 q^{39} +240.000 q^{41} +196.000 q^{43} -24.0000 q^{45} -168.000 q^{47} -222.000 q^{49} -465.000 q^{51} -435.000 q^{53} +648.000 q^{55} -95.0000 q^{57} -195.000 q^{59} +358.000 q^{61} -22.0000 q^{63} -132.000 q^{65} +961.000 q^{67} +915.000 q^{69} -246.000 q^{71} +353.000 q^{73} +95.0000 q^{75} +594.000 q^{77} -34.0000 q^{79} -671.000 q^{81} -234.000 q^{83} -1116.00 q^{85} +1245.00 q^{87} -168.000 q^{89} -121.000 q^{91} +280.000 q^{93} -228.000 q^{95} +758.000 q^{97} -108.000 q^{99} +O(q^{100})$

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
$3$	5.00000	0.962250	0.481125	−	0.876652i	$-0.340228\pi$
$3$	5.00000	0.962250	0.481125	+	0.876652i	$0.340228\pi$
$4$	0	0
$5$	12.0000	1.07331	0.536656	−	0.843801i	$-0.319687\pi$
$5$	12.0000	1.07331	0.536656	+	0.843801i	$0.319687\pi$
$6$	0	0
$7$	11.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	11.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	0	0
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	54.0000	1.48015	0.740073	−	0.672526i	$-0.234791\pi$
$11$	54.0000	1.48015	0.740073	+	0.672526i	$0.234791\pi$
$12$	0	0
$13$	−11.0000	−0.234681	−0.117340	−	0.993092i	$-0.537437\pi$
$13$	−11.0000	−0.234681	−0.117340	+	0.993092i	$0.537437\pi$
$14$	0	0
$15$	60.0000	1.03280
$16$	0	0
$17$	−93.0000	−1.32681	−0.663406	−	0.748259i	$-0.730890\pi$
$17$	−93.0000	−1.32681	−0.663406	+	0.748259i	$0.730890\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	55.0000	0.571523
$22$	0	0
$23$	183.000	1.65905	0.829525	−	0.558470i	$-0.188611\pi$
$23$	183.000	1.65905	0.829525	+	0.558470i	$0.188611\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	−145.000	−1.03353
$28$	0	0
$29$	249.000	1.59442	0.797209	−	0.603703i	$-0.206309\pi$
$29$	249.000	1.59442	0.797209	+	0.603703i	$0.206309\pi$
$30$	0	0
$31$	56.0000	0.324448	0.162224	−	0.986754i	$-0.448133\pi$
$31$	56.0000	0.324448	0.162224	+	0.986754i	$0.448133\pi$
$32$	0	0
$33$	270.000	1.42427
$34$	0	0
$35$	132.000	0.637488
$36$	0	0
$37$	250.000	1.11080	0.555402	−	0.831582i	$-0.312564\pi$
$37$	250.000	1.11080	0.555402	+	0.831582i	$0.312564\pi$
$38$	0	0
$39$	−55.0000	−0.225822
$40$	0	0
$41$	240.000	0.914188	0.457094	−	0.889418i	$-0.348890\pi$
$41$	240.000	0.914188	0.457094	+	0.889418i	$0.348890\pi$
$42$	0	0
$43$	196.000	0.695110	0.347555	−	0.937660i	$-0.387012\pi$
$43$	196.000	0.695110	0.347555	+	0.937660i	$0.387012\pi$
$44$	0	0
$45$	−24.0000	−0.0795046
$46$	0	0
$47$	−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
$49$	−222.000	−0.647230
$50$	0	0
$51$	−465.000	−1.27673
$52$	0	0
$53$	−435.000	−1.12739	−0.563697	−	0.825982i	$-0.690621\pi$
$53$	−435.000	−1.12739	−0.563697	+	0.825982i	$0.690621\pi$
$54$	0	0
$55$	648.000	1.58866
$56$	0	0
$57$	−95.0000	−0.220755
$58$	0	0
$59$	−195.000	−0.430285	−0.215143	−	0.976583i	$-0.569022\pi$
$59$	−195.000	−0.430285	−0.215143	+	0.976583i	$0.569022\pi$
$60$	0	0
$61$	358.000	0.751430	0.375715	−	0.926735i	$-0.377397\pi$
$61$	358.000	0.751430	0.375715	+	0.926735i	$0.377397\pi$
$62$	0	0
$63$	−22.0000	−0.0439959
$64$	0	0
$65$	−132.000	−0.251886
$66$	0	0
$67$	961.000	1.75231	0.876155	−	0.482029i	$-0.160100\pi$
$67$	961.000	1.75231	0.876155	+	0.482029i	$0.160100\pi$
$68$	0	0
$69$	915.000	1.59642
$70$	0	0
$71$	−246.000	−0.411195	−0.205597	−	0.978637i	$-0.565914\pi$
$71$	−246.000	−0.411195	−0.205597	+	0.978637i	$0.565914\pi$
$72$	0	0
$73$	353.000	0.565966	0.282983	−	0.959125i	$-0.408676\pi$
$73$	353.000	0.565966	0.282983	+	0.959125i	$0.408676\pi$
$74$	0	0
$75$	95.0000	0.146262
$76$	0	0
$77$	594.000	0.879124
$78$	0	0
$79$	−34.0000	−0.0484215	−0.0242108	−	0.999707i	$-0.507707\pi$
$79$	−34.0000	−0.0484215	−0.0242108	+	0.999707i	$0.507707\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	−234.000	−0.309456	−0.154728	−	0.987957i	$-0.549450\pi$
$83$	−234.000	−0.309456	−0.154728	+	0.987957i	$0.549450\pi$
$84$	0	0
$85$	−1116.00	−1.42408
$86$	0	0
$87$	1245.00	1.53423
$88$	0	0
$89$	−168.000	−0.200089	−0.100045	−	0.994983i	$-0.531899\pi$
$89$	−168.000	−0.200089	−0.100045	+	0.994983i	$0.531899\pi$
$90$	0	0
$91$	−121.000	−0.139387
$92$	0	0
$93$	280.000	0.312201
$94$	0	0
$95$	−228.000	−0.246235
$96$	0	0
$97$	758.000	0.793435	0.396718	−	0.917941i	$-0.370149\pi$
$97$	758.000	0.793435	0.396718	+	0.917941i	$0.370149\pi$
$98$	0	0
$99$	−108.000	−0.109640
$100$	0	0
$101$	726.000	0.715245	0.357622	−	0.933866i	$-0.383588\pi$
$101$	726.000	0.715245	0.357622	+	0.933866i	$0.383588\pi$
$102$	0	0
$103$	2.00000	0.00191326	0.000956630	−	1.00000i	$-0.499695\pi$
$103$	2.00000	0.00191326	0.000956630		1.00000i	$0.499695\pi$
$104$	0	0
$105$	660.000	0.613423
$106$	0	0
$107$	−1413.00	−1.27663	−0.638317	−	0.769773i	$-0.720369\pi$
$107$	−1413.00	−1.27663	−0.638317	+	0.769773i	$0.720369\pi$
$108$	0	0
$109$	−389.000	−0.341830	−0.170915	−	0.985286i	$-0.554672\pi$
$109$	−389.000	−0.341830	−0.170915	+	0.985286i	$0.554672\pi$
$110$	0	0
$111$	1250.00	1.06887
$112$	0	0
$113$	342.000	0.284714	0.142357	−	0.989815i	$-0.454532\pi$
$113$	342.000	0.284714	0.142357	+	0.989815i	$0.454532\pi$
$114$	0	0
$115$	2196.00	1.78068
$116$	0	0
$117$	22.0000	0.0173838
$118$	0	0
$119$	−1023.00	−0.788053
$120$	0	0
$121$	1585.00	1.19083
$122$	0	0
$123$	1200.00	0.879678
$124$	0	0
$125$	−1272.00	−0.910169
$126$	0	0
$127$	−1150.00	−0.803512	−0.401756	−	0.915747i	$-0.631600\pi$
$127$	−1150.00	−0.803512	−0.401756	+	0.915747i	$0.631600\pi$
$128$	0	0
$129$	980.000	0.668870
$130$	0	0
$131$	1452.00	0.968411	0.484205	−	0.874954i	$-0.339109\pi$
$131$	1452.00	0.968411	0.484205	+	0.874954i	$0.339109\pi$
$132$	0	0
$133$	−209.000	−0.136260
$134$	0	0
$135$	−1740.00	−1.10930
$136$	0	0
$137$	−1689.00	−1.05329	−0.526646	−	0.850085i	$-0.676551\pi$
$137$	−1689.00	−1.05329	−0.526646	+	0.850085i	$0.676551\pi$
$138$	0	0
$139$	−2144.00	−1.30829	−0.654143	−	0.756371i	$-0.726970\pi$
$139$	−2144.00	−1.30829	−0.654143	+	0.756371i	$0.726970\pi$
$140$	0	0
$141$	−840.000	−0.501708
$142$	0	0
$143$	−594.000	−0.347362
$144$	0	0
$145$	2988.00	1.71131
$146$	0	0
$147$	−1110.00	−0.622798
$148$	0	0
$149$	3000.00	1.64946	0.824730	−	0.565527i	$-0.191327\pi$
$149$	3000.00	1.64946	0.824730	+	0.565527i	$0.191327\pi$
$150$	0	0
$151$	−1006.00	−0.542166	−0.271083	−	0.962556i	$-0.587382\pi$
$151$	−1006.00	−0.542166	−0.271083	+	0.962556i	$0.587382\pi$
$152$	0	0
$153$	186.000	0.0982824
$154$	0	0
$155$	672.000	0.348234
$156$	0	0
$157$	−2846.00	−1.44672	−0.723362	−	0.690469i	$-0.757404\pi$
$157$	−2846.00	−1.44672	−0.723362	+	0.690469i	$0.757404\pi$
$158$	0	0
$159$	−2175.00	−1.08483
$160$	0	0
$161$	2013.00	0.985383
$162$	0	0
$163$	1600.00	0.768845	0.384422	−	0.923157i	$-0.374401\pi$
$163$	1600.00	0.768845	0.384422	+	0.923157i	$0.374401\pi$
$164$	0	0
$165$	3240.00	1.52869
$166$	0	0
$167$	−2004.00	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−2004.00	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−2076.00	−0.944925
$170$	0	0
$171$	38.0000	0.0169938
$172$	0	0
$173$	462.000	0.203036	0.101518	−	0.994834i	$-0.467630\pi$
$173$	462.000	0.203036	0.101518	+	0.994834i	$0.467630\pi$
$174$	0	0
$175$	209.000	0.0902795
$176$	0	0
$177$	−975.000	−0.414042
$178$	0	0
$179$	−720.000	−0.300644	−0.150322	−	0.988637i	$-0.548031\pi$
$179$	−720.000	−0.300644	−0.150322	+	0.988637i	$0.548031\pi$
$180$	0	0
$181$	2338.00	0.960122	0.480061	−	0.877235i	$-0.340614\pi$
$181$	2338.00	0.960122	0.480061	+	0.877235i	$0.340614\pi$
$182$	0	0
$183$	1790.00	0.723063
$184$	0	0
$185$	3000.00	1.19224
$186$	0	0
$187$	−5022.00	−1.96388
$188$	0	0
$189$	−1595.00	−0.613858
$190$	0	0
$191$	2871.00	1.08763	0.543817	−	0.839204i	$-0.316978\pi$
$191$	2871.00	1.08763	0.543817	+	0.839204i	$0.316978\pi$
$192$	0	0
$193$	1658.00	0.618370	0.309185	−	0.951002i	$-0.399944\pi$
$193$	1658.00	0.618370	0.309185	+	0.951002i	$0.399944\pi$
$194$	0	0
$195$	−660.000	−0.242377
$196$	0	0
$197$	4176.00	1.51029	0.755146	−	0.655556i	$-0.227566\pi$
$197$	4176.00	1.51029	0.755146	+	0.655556i	$0.227566\pi$
$198$	0	0
$199$	−241.000	−0.0858494	−0.0429247	−	0.999078i	$-0.513668\pi$
$199$	−241.000	−0.0858494	−0.0429247	+	0.999078i	$0.513668\pi$
$200$	0	0
$201$	4805.00	1.68616
$202$	0	0
$203$	2739.00	0.946996
$204$	0	0
$205$	2880.00	0.981209
$206$	0	0
$207$	−366.000	−0.122893
$208$	0	0
$209$	−1026.00	−0.339569
$210$	0	0
$211$	745.000	0.243071	0.121535	−	0.992587i	$-0.461218\pi$
$211$	745.000	0.243071	0.121535	+	0.992587i	$0.461218\pi$
$212$	0	0
$213$	−1230.00	−0.395672
$214$	0	0
$215$	2352.00	0.746070
$216$	0	0
$217$	616.000	0.192704
$218$	0	0
$219$	1765.00	0.544601
$220$	0	0
$221$	1023.00	0.311377
$222$	0	0
$223$	−1978.00	−0.593976	−0.296988	−	0.954881i	$-0.595982\pi$
$223$	−1978.00	−0.593976	−0.296988	+	0.954881i	$0.595982\pi$
$224$	0	0
$225$	−38.0000	−0.0112593
$226$	0	0
$227$	−5355.00	−1.56574	−0.782872	−	0.622183i	$-0.786246\pi$
$227$	−5355.00	−1.56574	−0.782872	+	0.622183i	$0.786246\pi$
$228$	0	0
$229$	6370.00	1.83817	0.919086	−	0.394057i	$-0.128929\pi$
$229$	6370.00	1.83817	0.919086	+	0.394057i	$0.128929\pi$
$230$	0	0
$231$	2970.00	0.845938
$232$	0	0
$233$	−2838.00	−0.797955	−0.398978	−	0.916961i	$-0.630635\pi$
$233$	−2838.00	−0.797955	−0.398978	+	0.916961i	$0.630635\pi$
$234$	0	0
$235$	−2016.00	−0.559614
$236$	0	0
$237$	−170.000	−0.0465936
$238$	0	0
$239$	−369.000	−0.0998687	−0.0499344	−	0.998753i	$-0.515901\pi$
$239$	−369.000	−0.0998687	−0.0499344	+	0.998753i	$0.515901\pi$
$240$	0	0
$241$	6608.00	1.76622	0.883109	−	0.469167i	$-0.155446\pi$
$241$	6608.00	1.76622	0.883109	+	0.469167i	$0.155446\pi$
$242$	0	0
$243$	560.000	0.147835
$244$	0	0
$245$	−2664.00	−0.694680
$246$	0	0
$247$	209.000	0.0538395
$248$	0	0
$249$	−1170.00	−0.297774
$250$	0	0
$251$	−4674.00	−1.17538	−0.587690	−	0.809086i	$-0.699962\pi$
$251$	−4674.00	−1.17538	−0.587690	+	0.809086i	$0.699962\pi$
$252$	0	0
$253$	9882.00	2.45564
$254$	0	0
$255$	−5580.00	−1.37033
$256$	0	0
$257$	4512.00	1.09514	0.547570	−	0.836760i	$-0.315553\pi$
$257$	4512.00	1.09514	0.547570	+	0.836760i	$0.315553\pi$
$258$	0	0
$259$	2750.00	0.659756
$260$	0	0
$261$	−498.000	−0.118105
$262$	0	0
$263$	3768.00	0.883440	0.441720	−	0.897153i	$-0.354368\pi$
$263$	3768.00	0.883440	0.441720	+	0.897153i	$0.354368\pi$
$264$	0	0
$265$	−5220.00	−1.21005
$266$	0	0
$267$	−840.000	−0.192536
$268$	0	0
$269$	−4758.00	−1.07844	−0.539220	−	0.842165i	$-0.681281\pi$
$269$	−4758.00	−1.07844	−0.539220	+	0.842165i	$0.681281\pi$
$270$	0	0
$271$	−2041.00	−0.457498	−0.228749	−	0.973485i	$-0.573463\pi$
$271$	−2041.00	−0.457498	−0.228749	+	0.973485i	$0.573463\pi$
$272$	0	0
$273$	−605.000	−0.134126
$274$	0	0
$275$	1026.00	0.224982
$276$	0	0
$277$	−1964.00	−0.426012	−0.213006	−	0.977051i	$-0.568325\pi$
$277$	−1964.00	−0.426012	−0.213006	+	0.977051i	$0.568325\pi$
$278$	0	0
$279$	−112.000	−0.0240332
$280$	0	0
$281$	−5496.00	−1.16678	−0.583388	−	0.812194i	$-0.698273\pi$
$281$	−5496.00	−1.16678	−0.583388	+	0.812194i	$0.698273\pi$
$282$	0	0
$283$	−3098.00	−0.650731	−0.325366	−	0.945588i	$-0.605487\pi$
$283$	−3098.00	−0.650731	−0.325366	+	0.945588i	$0.605487\pi$
$284$	0	0
$285$	−1140.00	−0.236940
$286$	0	0
$287$	2640.00	0.542977
$288$	0	0
$289$	3736.00	0.760432
$290$	0	0
$291$	3790.00	0.763484
$292$	0	0
$293$	−117.000	−0.0233284	−0.0116642	−	0.999932i	$-0.503713\pi$
$293$	−117.000	−0.0233284	−0.0116642	+	0.999932i	$0.503713\pi$
$294$	0	0
$295$	−2340.00	−0.461831
$296$	0	0
$297$	−7830.00	−1.52977
$298$	0	0
$299$	−2013.00	−0.389347
$300$	0	0
$301$	2156.00	0.412856
$302$	0	0
$303$	3630.00	0.688244
$304$	0	0
$305$	4296.00	0.806519
$306$	0	0
$307$	1420.00	0.263986	0.131993	−	0.991251i	$-0.457862\pi$
$307$	1420.00	0.263986	0.131993	+	0.991251i	$0.457862\pi$
$308$	0	0
$309$	10.0000	0.00184104
$310$	0	0
$311$	−6561.00	−1.19627	−0.598135	−	0.801395i	$-0.704091\pi$
$311$	−6561.00	−1.19627	−0.598135	+	0.801395i	$0.704091\pi$
$312$	0	0
$313$	−1483.00	−0.267809	−0.133904	−	0.990994i	$-0.542751\pi$
$313$	−1483.00	−0.267809	−0.133904	+	0.990994i	$0.542751\pi$
$314$	0	0
$315$	−264.000	−0.0472213
$316$	0	0
$317$	1239.00	0.219524	0.109762	−	0.993958i	$-0.464991\pi$
$317$	1239.00	0.219524	0.109762	+	0.993958i	$0.464991\pi$
$318$	0	0
$319$	13446.0	2.35997
$320$	0	0
$321$	−7065.00	−1.22844
$322$	0	0
$323$	1767.00	0.304392
$324$	0	0
$325$	−209.000	−0.0356715
$326$	0	0
$327$	−1945.00	−0.328926
$328$	0	0
$329$	−1848.00	−0.309676
$330$	0	0
$331$	8899.00	1.47774	0.738872	−	0.673846i	$-0.235359\pi$
$331$	8899.00	1.47774	0.738872	+	0.673846i	$0.235359\pi$
$332$	0	0
$333$	−500.000	−0.0822818
$334$	0	0
$335$	11532.0	1.88078
$336$	0	0
$337$	5816.00	0.940112	0.470056	−	0.882637i	$-0.344234\pi$
$337$	5816.00	0.940112	0.470056	+	0.882637i	$0.344234\pi$
$338$	0	0
$339$	1710.00	0.273966
$340$	0	0
$341$	3024.00	0.480231
$342$	0	0
$343$	−6215.00	−0.978363
$344$	0	0
$345$	10980.0	1.71346
$346$	0	0
$347$	1578.00	0.244125	0.122063	−	0.992522i	$-0.461049\pi$
$347$	1578.00	0.244125	0.122063	+	0.992522i	$0.461049\pi$
$348$	0	0
$349$	−1658.00	−0.254300	−0.127150	−	0.991883i	$-0.540583\pi$
$349$	−1658.00	−0.254300	−0.127150	+	0.991883i	$0.540583\pi$
$350$	0	0
$351$	1595.00	0.242549
$352$	0	0
$353$	−11367.0	−1.71389	−0.856947	−	0.515405i	$-0.827641\pi$
$353$	−11367.0	−1.71389	−0.856947	+	0.515405i	$0.827641\pi$
$354$	0	0
$355$	−2952.00	−0.441341
$356$	0	0
$357$	−5115.00	−0.758304
$358$	0	0
$359$	2553.00	0.375326	0.187663	−	0.982233i	$-0.439909\pi$
$359$	2553.00	0.375326	0.187663	+	0.982233i	$0.439909\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	7925.00	1.14588
$364$	0	0
$365$	4236.00	0.607459
$366$	0	0
$367$	−196.000	−0.0278777	−0.0139389	−	0.999903i	$-0.504437\pi$
$367$	−196.000	−0.0278777	−0.0139389	+	0.999903i	$0.504437\pi$
$368$	0	0
$369$	−480.000	−0.0677176
$370$	0	0
$371$	−4785.00	−0.669609
$372$	0	0
$373$	−9353.00	−1.29834	−0.649169	−	0.760644i	$-0.724883\pi$
$373$	−9353.00	−1.29834	−0.649169	+	0.760644i	$0.724883\pi$
$374$	0	0
$375$	−6360.00	−0.875811
$376$	0	0
$377$	−2739.00	−0.374180
$378$	0	0
$379$	−3827.00	−0.518680	−0.259340	−	0.965786i	$-0.583505\pi$
$379$	−3827.00	−0.518680	−0.259340	+	0.965786i	$0.583505\pi$
$380$	0	0
$381$	−5750.00	−0.773180
$382$	0	0
$383$	5694.00	0.759660	0.379830	−	0.925056i	$-0.375982\pi$
$383$	5694.00	0.759660	0.379830	+	0.925056i	$0.375982\pi$
$384$	0	0
$385$	7128.00	0.943575
$386$	0	0
$387$	−392.000	−0.0514896
$388$	0	0
$389$	−1290.00	−0.168138	−0.0840689	−	0.996460i	$-0.526792\pi$
$389$	−1290.00	−0.168138	−0.0840689	+	0.996460i	$0.526792\pi$
$390$	0	0
$391$	−17019.0	−2.20125
$392$	0	0
$393$	7260.00	0.931854
$394$	0	0
$395$	−408.000	−0.0519714
$396$	0	0
$397$	−6536.00	−0.826278	−0.413139	−	0.910668i	$-0.635568\pi$
$397$	−6536.00	−0.826278	−0.413139	+	0.910668i	$0.635568\pi$
$398$	0	0
$399$	−1045.00	−0.131116
$400$	0	0
$401$	2328.00	0.289912	0.144956	−	0.989438i	$-0.453696\pi$
$401$	2328.00	0.289912	0.144956	+	0.989438i	$0.453696\pi$
$402$	0	0
$403$	−616.000	−0.0761418
$404$	0	0
$405$	−8052.00	−0.987919
$406$	0	0
$407$	13500.0	1.64415
$408$	0	0
$409$	−6676.00	−0.807107	−0.403554	−	0.914956i	$-0.632225\pi$
$409$	−6676.00	−0.807107	−0.403554	+	0.914956i	$0.632225\pi$
$410$	0	0
$411$	−8445.00	−1.01353
$412$	0	0
$413$	−2145.00	−0.255565
$414$	0	0
$415$	−2808.00	−0.332143
$416$	0	0
$417$	−10720.0	−1.25890
$418$	0	0
$419$	8136.00	0.948615	0.474307	−	0.880359i	$-0.342699\pi$
$419$	8136.00	0.948615	0.474307	+	0.880359i	$0.342699\pi$
$420$	0	0
$421$	8665.00	1.00310	0.501551	−	0.865128i	$-0.332763\pi$
$421$	8665.00	1.00310	0.501551	+	0.865128i	$0.332763\pi$
$422$	0	0
$423$	336.000	0.0386215
$424$	0	0
$425$	−1767.00	−0.201676
$426$	0	0
$427$	3938.00	0.446307
$428$	0	0
$429$	−2970.00	−0.334249
$430$	0	0
$431$	750.000	0.0838196	0.0419098	−	0.999121i	$-0.486656\pi$
$431$	750.000	0.0838196	0.0419098	+	0.999121i	$0.486656\pi$
$432$	0	0
$433$	−4858.00	−0.539170	−0.269585	−	0.962977i	$-0.586887\pi$
$433$	−4858.00	−0.539170	−0.269585	+	0.962977i	$0.586887\pi$
$434$	0	0
$435$	14940.0	1.64671
$436$	0	0
$437$	−3477.00	−0.380612
$438$	0	0
$439$	6500.00	0.706670	0.353335	−	0.935497i	$-0.385048\pi$
$439$	6500.00	0.706670	0.353335	+	0.935497i	$0.385048\pi$
$440$	0	0
$441$	444.000	0.0479430
$442$	0	0
$443$	−3486.00	−0.373871	−0.186936	−	0.982372i	$-0.559856\pi$
$443$	−3486.00	−0.373871	−0.186936	+	0.982372i	$0.559856\pi$
$444$	0	0
$445$	−2016.00	−0.214759
$446$	0	0
$447$	15000.0	1.58719
$448$	0	0
$449$	−15030.0	−1.57975	−0.789877	−	0.613265i	$-0.789856\pi$
$449$	−15030.0	−1.57975	−0.789877	+	0.613265i	$0.789856\pi$
$450$	0	0
$451$	12960.0	1.35313
$452$	0	0
$453$	−5030.00	−0.521700
$454$	0	0
$455$	−1452.00	−0.149606
$456$	0	0
$457$	−2959.00	−0.302880	−0.151440	−	0.988466i	$-0.548391\pi$
$457$	−2959.00	−0.302880	−0.151440	+	0.988466i	$0.548391\pi$
$458$	0	0
$459$	13485.0	1.37130
$460$	0	0
$461$	156.000	0.0157606	0.00788031	−	0.999969i	$-0.497492\pi$
$461$	156.000	0.0157606	0.00788031	+	0.999969i	$0.497492\pi$
$462$	0	0
$463$	4484.00	0.450085	0.225042	−	0.974349i	$-0.427748\pi$
$463$	4484.00	0.450085	0.225042	+	0.974349i	$0.427748\pi$
$464$	0	0
$465$	3360.00	0.335089
$466$	0	0
$467$	−8766.00	−0.868613	−0.434306	−	0.900765i	$-0.643006\pi$
$467$	−8766.00	−0.868613	−0.434306	+	0.900765i	$0.643006\pi$
$468$	0	0
$469$	10571.0	1.04077
$470$	0	0
$471$	−14230.0	−1.39211
$472$	0	0
$473$	10584.0	1.02886
$474$	0	0
$475$	−361.000	−0.0348712
$476$	0	0
$477$	870.000	0.0835106
$478$	0	0
$479$	−18996.0	−1.81200	−0.906001	−	0.423275i	$-0.860881\pi$
$479$	−18996.0	−1.81200	−0.906001	+	0.423275i	$0.860881\pi$
$480$	0	0
$481$	−2750.00	−0.260684
$482$	0	0
$483$	10065.0	0.948185
$484$	0	0
$485$	9096.00	0.851604
$486$	0	0
$487$	−7450.00	−0.693207	−0.346603	−	0.938012i	$-0.612665\pi$
$487$	−7450.00	−0.693207	−0.346603	+	0.938012i	$0.612665\pi$
$488$	0	0
$489$	8000.00	0.739821
$490$	0	0
$491$	−6180.00	−0.568023	−0.284012	−	0.958821i	$-0.591665\pi$
$491$	−6180.00	−0.568023	−0.284012	+	0.958821i	$0.591665\pi$
$492$	0	0
$493$	−23157.0	−2.11549
$494$	0	0
$495$	−1296.00	−0.117679
$496$	0	0
$497$	−2706.00	−0.244227
$498$	0	0
$499$	−2576.00	−0.231097	−0.115549	−	0.993302i	$-0.536863\pi$
$499$	−2576.00	−0.231097	−0.115549	+	0.993302i	$0.536863\pi$
$500$	0	0
$501$	−10020.0	−0.893534
$502$	0	0
$503$	−10545.0	−0.934748	−0.467374	−	0.884060i	$-0.654800\pi$
$503$	−10545.0	−0.934748	−0.467374	+	0.884060i	$0.654800\pi$
$504$	0	0
$505$	8712.00	0.767681
$506$	0	0
$507$	−10380.0	−0.909254
$508$	0	0
$509$	14694.0	1.27957	0.639784	−	0.768555i	$-0.279024\pi$
$509$	14694.0	1.27957	0.639784	+	0.768555i	$0.279024\pi$
$510$	0	0
$511$	3883.00	0.336152
$512$	0	0
$513$	2755.00	0.237108
$514$	0	0
$515$	24.0000	0.00205353
$516$	0	0
$517$	−9072.00	−0.771733
$518$	0	0
$519$	2310.00	0.195371
$520$	0	0
$521$	10332.0	0.868816	0.434408	−	0.900716i	$-0.356958\pi$
$521$	10332.0	0.868816	0.434408	+	0.900716i	$0.356958\pi$
$522$	0	0
$523$	−10937.0	−0.914420	−0.457210	−	0.889359i	$-0.651151\pi$
$523$	−10937.0	−0.914420	−0.457210	+	0.889359i	$0.651151\pi$
$524$	0	0
$525$	1045.00	0.0868715
$526$	0	0
$527$	−5208.00	−0.430482
$528$	0	0
$529$	21322.0	1.75245
$530$	0	0
$531$	390.000	0.0318730
$532$	0	0
$533$	−2640.00	−0.214542
$534$	0	0
$535$	−16956.0	−1.37023
$536$	0	0
$537$	−3600.00	−0.289295
$538$	0	0
$539$	−11988.0	−0.957996
$540$	0	0
$541$	−18578.0	−1.47640	−0.738198	−	0.674584i	$-0.764323\pi$
$541$	−18578.0	−1.47640	−0.738198	+	0.674584i	$0.764323\pi$
$542$	0	0
$543$	11690.0	0.923878
$544$	0	0
$545$	−4668.00	−0.366890
$546$	0	0
$547$	−21404.0	−1.67307	−0.836535	−	0.547914i	$-0.815422\pi$
$547$	−21404.0	−1.67307	−0.836535	+	0.547914i	$0.815422\pi$
$548$	0	0
$549$	−716.000	−0.0556614
$550$	0	0
$551$	−4731.00	−0.365785
$552$	0	0
$553$	−374.000	−0.0287597
$554$	0	0
$555$	15000.0	1.14723
$556$	0	0
$557$	3948.00	0.300327	0.150163	−	0.988661i	$-0.452020\pi$
$557$	3948.00	0.300327	0.150163	+	0.988661i	$0.452020\pi$
$558$	0	0
$559$	−2156.00	−0.163129
$560$	0	0
$561$	−25110.0	−1.88974
$562$	0	0
$563$	−5724.00	−0.428486	−0.214243	−	0.976780i	$-0.568729\pi$
$563$	−5724.00	−0.428486	−0.214243	+	0.976780i	$0.568729\pi$
$564$	0	0
$565$	4104.00	0.305587
$566$	0	0
$567$	−7381.00	−0.546689
$568$	0	0
$569$	−20592.0	−1.51716	−0.758578	−	0.651582i	$-0.774105\pi$
$569$	−20592.0	−1.51716	−0.758578	+	0.651582i	$0.774105\pi$
$570$	0	0
$571$	−20684.0	−1.51593	−0.757967	−	0.652293i	$-0.773807\pi$
$571$	−20684.0	−1.51593	−0.757967	+	0.652293i	$0.773807\pi$
$572$	0	0
$573$	14355.0	1.04658
$574$	0	0
$575$	3477.00	0.252176
$576$	0	0
$577$	−19573.0	−1.41219	−0.706096	−	0.708116i	$-0.749545\pi$
$577$	−19573.0	−1.41219	−0.706096	+	0.708116i	$0.749545\pi$
$578$	0	0
$579$	8290.00	0.595027
$580$	0	0
$581$	−2574.00	−0.183800
$582$	0	0
$583$	−23490.0	−1.66871
$584$	0	0
$585$	264.000	0.0186582
$586$	0	0
$587$	−13524.0	−0.950929	−0.475464	−	0.879735i	$-0.657720\pi$
$587$	−13524.0	−0.950929	−0.475464	+	0.879735i	$0.657720\pi$
$588$	0	0
$589$	−1064.00	−0.0744335
$590$	0	0
$591$	20880.0	1.45328
$592$	0	0
$593$	8994.00	0.622832	0.311416	−	0.950274i	$-0.399197\pi$
$593$	8994.00	0.622832	0.311416	+	0.950274i	$0.399197\pi$
$594$	0	0
$595$	−12276.0	−0.845827
$596$	0	0
$597$	−1205.00	−0.0826087
$598$	0	0
$599$	10128.0	0.690850	0.345425	−	0.938446i	$-0.387735\pi$
$599$	10128.0	0.690850	0.345425	+	0.938446i	$0.387735\pi$
$600$	0	0
$601$	−22696.0	−1.54041	−0.770207	−	0.637794i	$-0.779847\pi$
$601$	−22696.0	−1.54041	−0.770207	+	0.637794i	$0.779847\pi$
$602$	0	0
$603$	−1922.00	−0.129801
$604$	0	0
$605$	19020.0	1.27814
$606$	0	0
$607$	−5182.00	−0.346509	−0.173254	−	0.984877i	$-0.555428\pi$
$607$	−5182.00	−0.346509	−0.173254	+	0.984877i	$0.555428\pi$
$608$	0	0
$609$	13695.0	0.911247
$610$	0	0
$611$	1848.00	0.122360
$612$	0	0
$613$	−10082.0	−0.664287	−0.332144	−	0.943229i	$-0.607772\pi$
$613$	−10082.0	−0.664287	−0.332144	+	0.943229i	$0.607772\pi$
$614$	0	0
$615$	14400.0	0.944169
$616$	0	0
$617$	−12174.0	−0.794338	−0.397169	−	0.917745i	$-0.630007\pi$
$617$	−12174.0	−0.794338	−0.397169	+	0.917745i	$0.630007\pi$
$618$	0	0
$619$	−7490.00	−0.486347	−0.243173	−	0.969983i	$-0.578188\pi$
$619$	−7490.00	−0.486347	−0.243173	+	0.969983i	$0.578188\pi$
$620$	0	0
$621$	−26535.0	−1.71467
$622$	0	0
$623$	−1848.00	−0.118842
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	−5130.00	−0.326750
$628$	0	0
$629$	−23250.0	−1.47383
$630$	0	0
$631$	11072.0	0.698525	0.349263	−	0.937025i	$-0.386432\pi$
$631$	11072.0	0.698525	0.349263	+	0.937025i	$0.386432\pi$
$632$	0	0
$633$	3725.00	0.233895
$634$	0	0
$635$	−13800.0	−0.862419
$636$	0	0
$637$	2442.00	0.151893
$638$	0	0
$639$	492.000	0.0304589
$640$	0	0
$641$	−18894.0	−1.16422	−0.582112	−	0.813108i	$-0.697774\pi$
$641$	−18894.0	−1.16422	−0.582112	+	0.813108i	$0.697774\pi$
$642$	0	0
$643$	19834.0	1.21645	0.608224	−	0.793765i	$-0.291882\pi$
$643$	19834.0	1.21645	0.608224	+	0.793765i	$0.291882\pi$
$644$	0	0
$645$	11760.0	0.717906
$646$	0	0
$647$	3375.00	0.205077	0.102539	−	0.994729i	$-0.467303\pi$
$647$	3375.00	0.205077	0.102539	+	0.994729i	$0.467303\pi$
$648$	0	0
$649$	−10530.0	−0.636885
$650$	0	0
$651$	3080.00	0.185430
$652$	0	0
$653$	24948.0	1.49509	0.747543	−	0.664214i	$-0.231234\pi$
$653$	24948.0	1.49509	0.747543	+	0.664214i	$0.231234\pi$
$654$	0	0
$655$	17424.0	1.03941
$656$	0	0
$657$	−706.000	−0.0419234
$658$	0	0
$659$	9879.00	0.583962	0.291981	−	0.956424i	$-0.405686\pi$
$659$	9879.00	0.583962	0.291981	+	0.956424i	$0.405686\pi$
$660$	0	0
$661$	14155.0	0.832928	0.416464	−	0.909152i	$-0.363269\pi$
$661$	14155.0	0.832928	0.416464	+	0.909152i	$0.363269\pi$
$662$	0	0
$663$	5115.00	0.299623
$664$	0	0
$665$	−2508.00	−0.146250
$666$	0	0
$667$	45567.0	2.64522
$668$	0	0
$669$	−9890.00	−0.571554
$670$	0	0
$671$	19332.0	1.11223
$672$	0	0
$673$	8948.00	0.512511	0.256256	−	0.966609i	$-0.417511\pi$
$673$	8948.00	0.512511	0.256256	+	0.966609i	$0.417511\pi$
$674$	0	0
$675$	−2755.00	−0.157096
$676$	0	0
$677$	11511.0	0.653477	0.326738	−	0.945115i	$-0.394050\pi$
$677$	11511.0	0.653477	0.326738	+	0.945115i	$0.394050\pi$
$678$	0	0
$679$	8338.00	0.471256
$680$	0	0
$681$	−26775.0	−1.50664
$682$	0	0
$683$	10476.0	0.586900	0.293450	−	0.955974i	$-0.405197\pi$
$683$	10476.0	0.586900	0.293450	+	0.955974i	$0.405197\pi$
$684$	0	0
$685$	−20268.0	−1.13051
$686$	0	0
$687$	31850.0	1.76878
$688$	0	0
$689$	4785.00	0.264578
$690$	0	0
$691$	−30098.0	−1.65699	−0.828496	−	0.559995i	$-0.810803\pi$
$691$	−30098.0	−1.65699	−0.828496	+	0.559995i	$0.810803\pi$
$692$	0	0
$693$	−1188.00	−0.0651203
$694$	0	0
$695$	−25728.0	−1.40420
$696$	0	0
$697$	−22320.0	−1.21296
$698$	0	0
$699$	−14190.0	−0.767833
$700$	0	0
$701$	14700.0	0.792028	0.396014	−	0.918245i	$-0.370393\pi$
$701$	14700.0	0.792028	0.396014	+	0.918245i	$0.370393\pi$
$702$	0	0
$703$	−4750.00	−0.254836
$704$	0	0
$705$	−10080.0	−0.538489
$706$	0	0
$707$	7986.00	0.424815
$708$	0	0
$709$	−31178.0	−1.65150	−0.825751	−	0.564035i	$-0.809248\pi$
$709$	−31178.0	−1.65150	−0.825751	+	0.564035i	$0.809248\pi$
$710$	0	0
$711$	68.0000	0.00358678
$712$	0	0
$713$	10248.0	0.538276
$714$	0	0
$715$	−7128.00	−0.372828
$716$	0	0
$717$	−1845.00	−0.0960987
$718$	0	0
$719$	−33285.0	−1.72645	−0.863227	−	0.504815i	$-0.831561\pi$
$719$	−33285.0	−1.72645	−0.863227	+	0.504815i	$0.831561\pi$
$720$	0	0
$721$	22.0000	0.00113637
$722$	0	0
$723$	33040.0	1.69954
$724$	0	0
$725$	4731.00	0.242352
$726$	0	0
$727$	−34729.0	−1.77170	−0.885851	−	0.463970i	$-0.846425\pi$
$727$	−34729.0	−1.77170	−0.885851	+	0.463970i	$0.846425\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	−18228.0	−0.922280
$732$	0	0
$733$	−4196.00	−0.211436	−0.105718	−	0.994396i	$-0.533714\pi$
$733$	−4196.00	−0.211436	−0.105718	+	0.994396i	$0.533714\pi$
$734$	0	0
$735$	−13320.0	−0.668457
$736$	0	0
$737$	51894.0	2.59368
$738$	0	0
$739$	10744.0	0.534810	0.267405	−	0.963584i	$-0.413834\pi$
$739$	10744.0	0.534810	0.267405	+	0.963584i	$0.413834\pi$
$740$	0	0
$741$	1045.00	0.0518071
$742$	0	0
$743$	−2208.00	−0.109022	−0.0545112	−	0.998513i	$-0.517360\pi$
$743$	−2208.00	−0.109022	−0.0545112	+	0.998513i	$0.517360\pi$
$744$	0	0
$745$	36000.0	1.77039
$746$	0	0
$747$	468.000	0.0229227
$748$	0	0
$749$	−15543.0	−0.758249
$750$	0	0
$751$	13160.0	0.639434	0.319717	−	0.947513i	$-0.396412\pi$
$751$	13160.0	0.639434	0.319717	+	0.947513i	$0.396412\pi$
$752$	0	0
$753$	−23370.0	−1.13101
$754$	0	0
$755$	−12072.0	−0.581914
$756$	0	0
$757$	−758.000	−0.0363936	−0.0181968	−	0.999834i	$-0.505793\pi$
$757$	−758.000	−0.0363936	−0.0181968	+	0.999834i	$0.505793\pi$
$758$	0	0
$759$	49410.0	2.36294
$760$	0	0
$761$	4851.00	0.231076	0.115538	−	0.993303i	$-0.463141\pi$
$761$	4851.00	0.231076	0.115538	+	0.993303i	$0.463141\pi$
$762$	0	0
$763$	−4279.00	−0.203028
$764$	0	0
$765$	2232.00	0.105488
$766$	0	0
$767$	2145.00	0.100980
$768$	0	0
$769$	−33091.0	−1.55175	−0.775873	−	0.630890i	$-0.782690\pi$
$769$	−33091.0	−1.55175	−0.775873	+	0.630890i	$0.782690\pi$
$770$	0	0
$771$	22560.0	1.05380
$772$	0	0
$773$	−42357.0	−1.97086	−0.985430	−	0.170079i	$-0.945598\pi$
$773$	−42357.0	−1.97086	−0.985430	+	0.170079i	$0.945598\pi$
$774$	0	0
$775$	1064.00	0.0493161
$776$	0	0
$777$	13750.0	0.634850
$778$	0	0
$779$	−4560.00	−0.209729
$780$	0	0
$781$	−13284.0	−0.608629
$782$	0	0
$783$	−36105.0	−1.64788
$784$	0	0
$785$	−34152.0	−1.55279
$786$	0	0
$787$	39877.0	1.80618	0.903089	−	0.429454i	$-0.141294\pi$
$787$	39877.0	1.80618	0.903089	+	0.429454i	$0.141294\pi$
$788$	0	0
$789$	18840.0	0.850091
$790$	0	0
$791$	3762.00	0.169104
$792$	0	0
$793$	−3938.00	−0.176346
$794$	0	0
$795$	−26100.0	−1.16437
$796$	0	0
$797$	30033.0	1.33478	0.667392	−	0.744706i	$-0.267410\pi$
$797$	30033.0	1.33478	0.667392	+	0.744706i	$0.267410\pi$
$798$	0	0
$799$	15624.0	0.691786
$800$	0	0
$801$	336.000	0.0148214
$802$	0	0
$803$	19062.0	0.837713
$804$	0	0
$805$	24156.0	1.05762
$806$	0	0
$807$	−23790.0	−1.03773
$808$	0	0
$809$	585.000	0.0254234	0.0127117	−	0.999919i	$-0.495954\pi$
$809$	585.000	0.0254234	0.0127117	+	0.999919i	$0.495954\pi$
$810$	0	0
$811$	−28361.0	−1.22798	−0.613989	−	0.789315i	$-0.710436\pi$
$811$	−28361.0	−1.22798	−0.613989	+	0.789315i	$0.710436\pi$
$812$	0	0
$813$	−10205.0	−0.440228
$814$	0	0
$815$	19200.0	0.825211
$816$	0	0
$817$	−3724.00	−0.159469
$818$	0	0
$819$	242.000	0.0103250
$820$	0	0
$821$	−25068.0	−1.06563	−0.532813	−	0.846233i	$-0.678865\pi$
$821$	−25068.0	−1.06563	−0.532813	+	0.846233i	$0.678865\pi$
$822$	0	0
$823$	10901.0	0.461707	0.230854	−	0.972989i	$-0.425848\pi$
$823$	10901.0	0.461707	0.230854	+	0.972989i	$0.425848\pi$
$824$	0	0
$825$	5130.00	0.216489
$826$	0	0
$827$	−12027.0	−0.505707	−0.252854	−	0.967505i	$-0.581369\pi$
$827$	−12027.0	−0.505707	−0.252854	+	0.967505i	$0.581369\pi$
$828$	0	0
$829$	19339.0	0.810219	0.405109	−	0.914268i	$-0.367233\pi$
$829$	19339.0	0.810219	0.405109	+	0.914268i	$0.367233\pi$
$830$	0	0
$831$	−9820.00	−0.409930
$832$	0	0
$833$	20646.0	0.858753
$834$	0	0
$835$	−24048.0	−0.996665
$836$	0	0
$837$	−8120.00	−0.335326
$838$	0	0
$839$	−13188.0	−0.542670	−0.271335	−	0.962485i	$-0.587465\pi$
$839$	−13188.0	−0.542670	−0.271335	+	0.962485i	$0.587465\pi$
$840$	0	0
$841$	37612.0	1.54217
$842$	0	0
$843$	−27480.0	−1.12273
$844$	0	0
$845$	−24912.0	−1.01420
$846$	0	0
$847$	17435.0	0.707289
$848$	0	0
$849$	−15490.0	−0.626167
$850$	0	0
$851$	45750.0	1.84288
$852$	0	0
$853$	4678.00	0.187775	0.0938873	−	0.995583i	$-0.470071\pi$
$853$	4678.00	0.187775	0.0938873	+	0.995583i	$0.470071\pi$
$854$	0	0
$855$	456.000	0.0182396
$856$	0	0
$857$	15252.0	0.607933	0.303966	−	0.952683i	$-0.401689\pi$
$857$	15252.0	0.607933	0.303966	+	0.952683i	$0.401689\pi$
$858$	0	0
$859$	610.000	0.0242293	0.0121146	−	0.999927i	$-0.496144\pi$
$859$	610.000	0.0242293	0.0121146	+	0.999927i	$0.496144\pi$
$860$	0	0
$861$	13200.0	0.522479
$862$	0	0
$863$	774.000	0.0305299	0.0152649	−	0.999883i	$-0.495141\pi$
$863$	774.000	0.0305299	0.0152649	+	0.999883i	$0.495141\pi$
$864$	0	0
$865$	5544.00	0.217921
$866$	0	0
$867$	18680.0	0.731726
$868$	0	0
$869$	−1836.00	−0.0716709
$870$	0	0
$871$	−10571.0	−0.411234
$872$	0	0
$873$	−1516.00	−0.0587730
$874$	0	0
$875$	−13992.0	−0.540590
$876$	0	0
$877$	31039.0	1.19511	0.597556	−	0.801827i	$-0.296139\pi$
$877$	31039.0	1.19511	0.597556	+	0.801827i	$0.296139\pi$
$878$	0	0
$879$	−585.000	−0.0224477
$880$	0	0
$881$	33678.0	1.28790	0.643950	−	0.765067i	$-0.277294\pi$
$881$	33678.0	1.28790	0.643950	+	0.765067i	$0.277294\pi$
$882$	0	0
$883$	42982.0	1.63812	0.819060	−	0.573708i	$-0.194496\pi$
$883$	42982.0	1.63812	0.819060	+	0.573708i	$0.194496\pi$
$884$	0	0
$885$	−11700.0	−0.444397
$886$	0	0
$887$	4494.00	0.170117	0.0850585	−	0.996376i	$-0.472892\pi$
$887$	4494.00	0.170117	0.0850585	+	0.996376i	$0.472892\pi$
$888$	0	0
$889$	−12650.0	−0.477241
$890$	0	0
$891$	−36234.0	−1.36238
$892$	0	0
$893$	3192.00	0.119615
$894$	0	0
$895$	−8640.00	−0.322685
$896$	0	0
$897$	−10065.0	−0.374649
$898$	0	0
$899$	13944.0	0.517306
$900$	0	0
$901$	40455.0	1.49584
$902$	0	0
$903$	10780.0	0.397271
$904$	0	0
$905$	28056.0	1.03051
$906$	0	0
$907$	23839.0	0.872724	0.436362	−	0.899771i	$-0.356267\pi$
$907$	23839.0	0.872724	0.436362	+	0.899771i	$0.356267\pi$
$908$	0	0
$909$	−1452.00	−0.0529811
$910$	0	0
$911$	−10332.0	−0.375757	−0.187878	−	0.982192i	$-0.560161\pi$
$911$	−10332.0	−0.375757	−0.187878	+	0.982192i	$0.560161\pi$
$912$	0	0
$913$	−12636.0	−0.458040
$914$	0	0
$915$	21480.0	0.776073
$916$	0	0
$917$	15972.0	0.575182
$918$	0	0
$919$	−14371.0	−0.515838	−0.257919	−	0.966166i	$-0.583037\pi$
$919$	−14371.0	−0.515838	−0.257919	+	0.966166i	$0.583037\pi$
$920$	0	0
$921$	7100.00	0.254021
$922$	0	0
$923$	2706.00	0.0964995
$924$	0	0
$925$	4750.00	0.168842
$926$	0	0
$927$	−4.00000	−0.000141723 0
$928$	0	0
$929$	26889.0	0.949623	0.474811	−	0.880088i	$-0.342516\pi$
$929$	26889.0	0.949623	0.474811	+	0.880088i	$0.342516\pi$
$930$	0	0
$931$	4218.00	0.148485
$932$	0	0
$933$	−32805.0	−1.15111
$934$	0	0
$935$	−60264.0	−2.10785
$936$	0	0
$937$	785.000	0.0273691	0.0136845	−	0.999906i	$-0.495644\pi$
$937$	785.000	0.0273691	0.0136845	+	0.999906i	$0.495644\pi$
$938$	0	0
$939$	−7415.00	−0.257699
$940$	0	0
$941$	18141.0	0.628459	0.314229	−	0.949347i	$-0.398254\pi$
$941$	18141.0	0.628459	0.314229	+	0.949347i	$0.398254\pi$
$942$	0	0
$943$	43920.0	1.51668
$944$	0	0
$945$	−19140.0	−0.658862
$946$	0	0
$947$	−23100.0	−0.792660	−0.396330	−	0.918108i	$-0.629716\pi$
$947$	−23100.0	−0.792660	−0.396330	+	0.918108i	$0.629716\pi$
$948$	0	0
$949$	−3883.00	−0.132821
$950$	0	0
$951$	6195.00	0.211237
$952$	0	0
$953$	45690.0	1.55304	0.776519	−	0.630094i	$-0.216984\pi$
$953$	45690.0	1.55304	0.776519	+	0.630094i	$0.216984\pi$
$954$	0	0
$955$	34452.0	1.16737
$956$	0	0
$957$	67230.0	2.27089
$958$	0	0
$959$	−18579.0	−0.625597
$960$	0	0
$961$	−26655.0	−0.894733
$962$	0	0
$963$	2826.00	0.0945655
$964$	0	0
$965$	19896.0	0.663705
$966$	0	0
$967$	21584.0	0.717781	0.358891	−	0.933380i	$-0.383155\pi$
$967$	21584.0	0.717781	0.358891	+	0.933380i	$0.383155\pi$
$968$	0	0
$969$	8835.00	0.292901
$970$	0	0
$971$	50556.0	1.67087	0.835437	−	0.549586i	$-0.185214\pi$
$971$	50556.0	1.67087	0.835437	+	0.549586i	$0.185214\pi$
$972$	0	0
$973$	−23584.0	−0.777049
$974$	0	0
$975$	−1045.00	−0.0343249
$976$	0	0
$977$	8568.00	0.280568	0.140284	−	0.990111i	$-0.455198\pi$
$977$	8568.00	0.280568	0.140284	+	0.990111i	$0.455198\pi$
$978$	0	0
$979$	−9072.00	−0.296162
$980$	0	0
$981$	778.000	0.0253207
$982$	0	0
$983$	29706.0	0.963860	0.481930	−	0.876210i	$-0.339936\pi$
$983$	29706.0	0.963860	0.481930	+	0.876210i	$0.339936\pi$
$984$	0	0
$985$	50112.0	1.62102
$986$	0	0
$987$	−9240.00	−0.297986
$988$	0	0
$989$	35868.0	1.15322
$990$	0	0
$991$	30512.0	0.978048	0.489024	−	0.872270i	$-0.337353\pi$
$991$	30512.0	0.978048	0.489024	+	0.872270i	$0.337353\pi$
$992$	0	0
$993$	44495.0	1.42196
$994$	0	0
$995$	−2892.00	−0.0921433
$996$	0	0
$997$	−47756.0	−1.51700	−0.758499	−	0.651674i	$-0.774067\pi$
$997$	−47756.0	−1.51700	−0.758499	+	0.651674i	$0.774067\pi$
$998$	0	0
$999$	−36250.0	−1.14805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.f.1.1		1
4.3	odd	2		1216.4.a.a.1.1		1
8.3	odd	2		304.4.a.b.1.1		1
8.5	even	2		19.4.a.a.1.1	✓	1
24.5	odd	2		171.4.a.d.1.1		1
40.13	odd	4		475.4.b.c.324.2		2
40.29	even	2		475.4.a.e.1.1		1
40.37	odd	4		475.4.b.c.324.1		2
56.13	odd	2		931.4.a.a.1.1		1
88.21	odd	2		2299.4.a.b.1.1		1
152.37	odd	2		361.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.a.1.1	✓	1	8.5	even	2
171.4.a.d.1.1		1	24.5	odd	2
304.4.a.b.1.1		1	8.3	odd	2
361.4.a.b.1.1		1	152.37	odd	2
475.4.a.e.1.1		1	40.29	even	2
475.4.b.c.324.1		2	40.37	odd	4
475.4.b.c.324.2		2	40.13	odd	4
931.4.a.a.1.1		1	56.13	odd	2
1216.4.a.a.1.1		1	4.3	odd	2
1216.4.a.f.1.1		1	1.1	even	1	trivial
2299.4.a.b.1.1		1	88.21	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

Introduction

L-functions

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Fields

Representations

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Database

Properties

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