LMFDB - Embedded newform 4014.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{8} +3.00000 q^{10} +4.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +3.00000 q^{20} +6.00000 q^{23} +4.00000 q^{25} +4.00000 q^{26} -8.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -9.00000 q^{37} +3.00000 q^{40} -4.00000 q^{41} -8.00000 q^{43} +6.00000 q^{46} +7.00000 q^{47} -7.00000 q^{49} +4.00000 q^{50} +4.00000 q^{52} +6.00000 q^{53} -8.00000 q^{58} +4.00000 q^{59} -4.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +7.00000 q^{67} +6.00000 q^{68} -4.00000 q^{71} -15.0000 q^{73} -9.00000 q^{74} -5.00000 q^{79} +3.00000 q^{80} -4.00000 q^{82} +11.0000 q^{83} +18.0000 q^{85} -8.00000 q^{86} -4.00000 q^{89} +6.00000 q^{92} +7.00000 q^{94} +12.0000 q^{97} -7.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.00000	0.948683
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	4.00000	0.784465
$27$	0	0
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	0	0
$40$	3.00000	0.474342
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	4.00000	0.565685
$51$	0	0
$52$	4.00000	0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	−9.00000	−1.04623
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−4.00000	−0.441726
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	−8.00000	−0.862662
$87$	0	0
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	7.00000	0.721995
$95$	0	0
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	−7.00000	−0.707107
$99$	0	0
$100$	4.00000	0.400000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	−8.00000	−0.742781
$117$	0	0
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−4.00000	−0.362143
$123$	0	0
$124$	2.00000	0.179605
$125$	−3.00000	−0.268328
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	12.0000	1.05247
$131$	11.0000	0.961074	0.480537	−	0.876974i	$-0.340442\pi$
$131$	11.0000	0.961074	0.480537	+	0.876974i	$0.340442\pi$
$132$	0	0
$133$	0	0
$134$	7.00000	0.604708
$135$	0	0
$136$	6.00000	0.514496
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	0	0
$142$	−4.00000	−0.335673
$143$	0	0
$144$	0	0
$145$	−24.0000	−1.99309
$146$	−15.0000	−1.24141
$147$	0	0
$148$	−9.00000	−0.739795
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	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
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−5.00000	−0.397779
$159$	0	0
$160$	3.00000	0.237171
$161$	0	0
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	11.0000	0.853766
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	18.0000	1.38054
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−4.00000	−0.299813
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	9.00000	0.668965	0.334482	−	0.942402i	$-0.391439\pi$
$181$	9.00000	0.668965	0.334482	+	0.942402i	$0.391439\pi$
$182$	0	0
$183$	0	0
$184$	6.00000	0.442326
$185$	−27.0000	−1.98508
$186$	0	0
$187$	0	0
$188$	7.00000	0.510527
$189$	0	0
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	10.0000	0.703598
$203$	0	0
$204$	0	0
$205$	−12.0000	−0.838116
$206$	9.00000	0.627060
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	0	0
$215$	−24.0000	−1.63679
$216$	0	0
$217$	0	0
$218$	−11.0000	−0.745014
$219$	0	0
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	14.0000	0.931266
$227$	25.0000	1.65931	0.829654	−	0.558278i	$-0.188538\pi$
$227$	25.0000	1.65931	0.829654	+	0.558278i	$0.188538\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	21.0000	1.36989
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−4.00000	−0.256074
$245$	−21.0000	−1.34164
$246$	0	0
$247$	0	0
$248$	2.00000	0.127000
$249$	0	0
$250$	−3.00000	−0.189737
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	0	0
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	0	0
$260$	12.0000	0.744208
$261$	0	0
$262$	11.0000	0.679582
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	0	0
$268$	7.00000	0.427593
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	−9.00000	−0.543710
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	−24.0000	−1.40933
$291$	0	0
$292$	−15.0000	−0.877809
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	−9.00000	−0.523114
$297$	0	0
$298$	−10.0000	−0.579284
$299$	24.0000	1.38796
$300$	0	0
$301$	0	0
$302$	−19.0000	−1.09333
$303$	0	0
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	13.0000	0.741949	0.370975	−	0.928643i	$-0.379024\pi$
$307$	13.0000	0.741949	0.370975	+	0.928643i	$0.379024\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−5.00000	−0.281272
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	0	0
$320$	3.00000	0.167705
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	16.0000	0.887520
$326$	17.0000	0.941543
$327$	0	0
$328$	−4.00000	−0.220863
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	11.0000	0.603703
$333$	0	0
$334$	6.00000	0.328305
$335$	21.0000	1.14735
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	18.0000	0.976187
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−5.00000	−0.268802
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	21.0000	1.12410	0.562052	−	0.827102i	$-0.310012\pi$
$349$	21.0000	1.12410	0.562052	+	0.827102i	$0.310012\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	−4.00000	−0.212000
$357$	0	0
$358$	3.00000	0.158555
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	9.00000	0.473029
$363$	0	0
$364$	0	0
$365$	−45.0000	−2.35541
$366$	0	0
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−27.0000	−1.40366
$371$	0	0
$372$	0	0
$373$	24.0000	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$374$	0	0
$375$	0	0
$376$	7.00000	0.360997
$377$	−32.0000	−1.64808
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	0	0
$381$	0	0
$382$	10.0000	0.511645
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	0	0
$385$	0	0
$386$	−24.0000	−1.22157
$387$	0	0
$388$	12.0000	0.609208
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	−7.00000	−0.353553
$393$	0	0
$394$	−22.0000	−1.10834
$395$	−15.0000	−0.754732
$396$	0	0
$397$	36.0000	1.80679	0.903394	−	0.428811i	$-0.141067\pi$
$397$	36.0000	1.80679	0.903394	+	0.428811i	$0.141067\pi$
$398$	6.00000	0.300753
$399$	0	0
$400$	4.00000	0.200000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	10.0000	0.497519
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	36.0000	1.78009	0.890043	−	0.455877i	$-0.150674\pi$
$409$	36.0000	1.78009	0.890043	+	0.455877i	$0.150674\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	9.00000	0.443398
$413$	0	0
$414$	0	0
$415$	33.0000	1.61991
$416$	4.00000	0.196116
$417$	0	0
$418$	0	0
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	−24.0000	−1.16969	−0.584844	−	0.811146i	$-0.698844\pi$
$421$	−24.0000	−1.16969	−0.584844	+	0.811146i	$0.698844\pi$
$422$	10.0000	0.486792
$423$	0	0
$424$	6.00000	0.291386
$425$	24.0000	1.16417
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−24.0000	−1.15738
$431$	−34.0000	−1.63772	−0.818861	−	0.573992i	$-0.805394\pi$
$431$	−34.0000	−1.63772	−0.818861	+	0.573992i	$0.805394\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	0	0
$436$	−11.0000	−0.526804
$437$	0	0
$438$	0	0
$439$	−21.0000	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$439$	−21.0000	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	5.00000	0.237557	0.118779	−	0.992921i	$-0.462102\pi$
$443$	5.00000	0.237557	0.118779	+	0.992921i	$0.462102\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	1.00000	0.0473514
$447$	0	0
$448$	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
$451$	0	0
$452$	14.0000	0.658505
$453$	0	0
$454$	25.0000	1.17331
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	8.00000	0.373815
$459$	0	0
$460$	18.0000	0.839254
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	−14.0000	−0.648537
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	21.0000	0.968658
$471$	0	0
$472$	4.00000	0.184115
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−15.0000	−0.686084
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	−13.0000	−0.592134
$483$	0	0
$484$	−11.0000	−0.500000
$485$	36.0000	1.63468
$486$	0	0
$487$	−14.0000	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$488$	−4.00000	−0.181071
$489$	0	0
$490$	−21.0000	−0.948683
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	−48.0000	−2.16181
$494$	0	0
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−15.0000	−0.669483
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	30.0000	1.33498
$506$	0	0
$507$	0	0
$508$	14.0000	0.621150
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	27.0000	1.18976
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	12.0000	0.526235
$521$	−9.00000	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$521$	−9.00000	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$522$	0	0
$523$	−40.0000	−1.74908	−0.874539	−	0.484955i	$-0.838836\pi$
$523$	−40.0000	−1.74908	−0.874539	+	0.484955i	$0.838836\pi$
$524$	11.0000	0.480537
$525$	0	0
$526$	−4.00000	−0.174408
$527$	12.0000	0.522728
$528$	0	0
$529$	13.0000	0.565217
$530$	18.0000	0.781870
$531$	0	0
$532$	0	0
$533$	−16.0000	−0.693037
$534$	0	0
$535$	0	0
$536$	7.00000	0.302354
$537$	0	0
$538$	−15.0000	−0.646696
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	6.00000	0.257248
$545$	−33.0000	−1.41356
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	−9.00000	−0.384461
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−6.00000	−0.254457
$557$	23.0000	0.974541	0.487271	−	0.873251i	$-0.337993\pi$
$557$	23.0000	0.974541	0.487271	+	0.873251i	$0.337993\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	0	0
$562$	−32.0000	−1.34984
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	42.0000	1.76695
$566$	−22.0000	−0.924729
$567$	0	0
$568$	−4.00000	−0.167836
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	1.00000	0.0416305	0.0208153	−	0.999783i	$-0.493374\pi$
$577$	1.00000	0.0416305	0.0208153	+	0.999783i	$0.493374\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−24.0000	−0.996546
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−15.0000	−0.620704
$585$	0	0
$586$	10.0000	0.413096
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	0	0
$589$	0	0
$590$	12.0000	0.494032
$591$	0	0
$592$	−9.00000	−0.369898
$593$	39.0000	1.60154	0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000	1.60154	0.800769	+	0.598973i	$0.204424\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	24.0000	0.981433
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	0	0
$604$	−19.0000	−0.773099
$605$	−33.0000	−1.34164
$606$	0	0
$607$	−21.0000	−0.852364	−0.426182	−	0.904638i	$-0.640142\pi$
$607$	−21.0000	−0.852364	−0.426182	+	0.904638i	$0.640142\pi$
$608$	0	0
$609$	0	0
$610$	−12.0000	−0.485866
$611$	28.0000	1.13276
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	13.0000	0.524637
$615$	0	0
$616$	0	0
$617$	−44.0000	−1.77137	−0.885687	−	0.464283i	$-0.846312\pi$
$617$	−44.0000	−1.77137	−0.885687	+	0.464283i	$0.846312\pi$
$618$	0	0
$619$	−49.0000	−1.96948	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−49.0000	−1.96948	−0.984738	+	0.174042i	$0.944317\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	14.0000	0.559553
$627$	0	0
$628$	4.00000	0.159617
$629$	−54.0000	−2.15312
$630$	0	0
$631$	19.0000	0.756378	0.378189	−	0.925728i	$-0.376547\pi$
$631$	19.0000	0.756378	0.378189	+	0.925728i	$0.376547\pi$
$632$	−5.00000	−0.198889
$633$	0	0
$634$	14.0000	0.556011
$635$	42.0000	1.66672
$636$	0	0
$637$	−28.0000	−1.10940
$638$	0	0
$639$	0	0
$640$	3.00000	0.118585
$641$	−19.0000	−0.750455	−0.375227	−	0.926933i	$-0.622435\pi$
$641$	−19.0000	−0.750455	−0.375227	+	0.926933i	$0.622435\pi$
$642$	0	0
$643$	−38.0000	−1.49857	−0.749287	−	0.662246i	$-0.769604\pi$
$643$	−38.0000	−1.49857	−0.749287	+	0.662246i	$0.769604\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	16.0000	0.627572
$651$	0	0
$652$	17.0000	0.665771
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	33.0000	1.28942
$656$	−4.00000	−0.156174
$657$	0	0
$658$	0	0
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	11.0000	0.426883
$665$	0	0
$666$	0	0
$667$	−48.0000	−1.85857
$668$	6.00000	0.232147
$669$	0	0
$670$	21.0000	0.811301
$671$	0	0
$672$	0	0
$673$	30.0000	1.15642	0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000	1.15642	0.578208	+	0.815890i	$0.303752\pi$
$674$	8.00000	0.308148
$675$	0	0
$676$	3.00000	0.115385
$677$	−4.00000	−0.153732	−0.0768662	−	0.997041i	$-0.524491\pi$
$677$	−4.00000	−0.153732	−0.0768662	+	0.997041i	$0.524491\pi$
$678$	0	0
$679$	0	0
$680$	18.0000	0.690268
$681$	0	0
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	−27.0000	−1.03162
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	24.0000	0.914327
$690$	0	0
$691$	−35.0000	−1.33146	−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000	−1.33146	−0.665731	+	0.746191i	$0.731880\pi$
$692$	−5.00000	−0.190071
$693$	0	0
$694$	−20.0000	−0.759190
$695$	−18.0000	−0.682779
$696$	0	0
$697$	−24.0000	−0.909065
$698$	21.0000	0.794862
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	−4.00000	−0.149906
$713$	12.0000	0.449404
$714$	0	0
$715$	0	0
$716$	3.00000	0.112115
$717$	0	0
$718$	−20.0000	−0.746393
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	0	0
$722$	−19.0000	−0.707107
$723$	0	0
$724$	9.00000	0.334482
$725$	−32.0000	−1.18845
$726$	0	0
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	0	0
$729$	0	0
$730$	−45.0000	−1.66552
$731$	−48.0000	−1.77534
$732$	0	0
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	−27.0000	−0.992540
$741$	0	0
$742$	0	0
$743$	−41.0000	−1.50414	−0.752072	−	0.659081i	$-0.770945\pi$
$743$	−41.0000	−1.50414	−0.752072	+	0.659081i	$0.770945\pi$
$744$	0	0
$745$	−30.0000	−1.09911
$746$	24.0000	0.878702
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	7.00000	0.255264
$753$	0	0
$754$	−32.0000	−1.16537
$755$	−57.0000	−2.07444
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	−34.0000	−1.23494
$759$	0	0
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	0	0
$764$	10.0000	0.361787
$765$	0	0
$766$	−2.00000	−0.0722629
$767$	16.0000	0.577727
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	0	0
$772$	−24.0000	−0.863779
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	12.0000	0.430775
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	36.0000	1.28736
$783$	0	0
$784$	−7.00000	−0.250000
$785$	12.0000	0.428298
$786$	0	0
$787$	23.0000	0.819861	0.409931	−	0.912117i	$-0.365553\pi$
$787$	23.0000	0.819861	0.409931	+	0.912117i	$0.365553\pi$
$788$	−22.0000	−0.783718
$789$	0	0
$790$	−15.0000	−0.533676
$791$	0	0
$792$	0	0
$793$	−16.0000	−0.568177
$794$	36.0000	1.27759
$795$	0	0
$796$	6.00000	0.212664
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	42.0000	1.48585
$800$	4.00000	0.141421
$801$	0	0
$802$	−8.00000	−0.282490
$803$	0	0
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	10.0000	0.351799
$809$	−19.0000	−0.668004	−0.334002	−	0.942572i	$-0.608399\pi$
$809$	−19.0000	−0.668004	−0.334002	+	0.942572i	$0.608399\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	51.0000	1.78645
$816$	0	0
$817$	0	0
$818$	36.0000	1.25871
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	21.0000	0.732014	0.366007	−	0.930612i	$-0.380725\pi$
$823$	21.0000	0.732014	0.366007	+	0.930612i	$0.380725\pi$
$824$	9.00000	0.313530
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	33.0000	1.14545
$831$	0	0
$832$	4.00000	0.138675
$833$	−42.0000	−1.45521
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	0	0
$838$	32.0000	1.10542
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−24.0000	−0.827095
$843$	0	0
$844$	10.0000	0.344214
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	24.0000	0.823193
$851$	−54.0000	−1.85110
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	0	0
$859$	−33.0000	−1.12595	−0.562973	−	0.826475i	$-0.690342\pi$
$859$	−33.0000	−1.12595	−0.562973	+	0.826475i	$0.690342\pi$
$860$	−24.0000	−0.818393
$861$	0	0
$862$	−34.0000	−1.15804
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	0	0
$865$	−15.0000	−0.510015
$866$	−5.00000	−0.169907
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	28.0000	0.948744
$872$	−11.0000	−0.372507
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	−21.0000	−0.708716
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	−13.0000	−0.437485	−0.218742	−	0.975783i	$-0.570195\pi$
$883$	−13.0000	−0.437485	−0.218742	+	0.975783i	$0.570195\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	5.00000	0.167978
$887$	−21.0000	−0.705111	−0.352555	−	0.935791i	$-0.614687\pi$
$887$	−21.0000	−0.705111	−0.352555	+	0.935791i	$0.614687\pi$
$888$	0	0
$889$	0	0
$890$	−12.0000	−0.402241
$891$	0	0
$892$	1.00000	0.0334825
$893$	0	0
$894$	0	0
$895$	9.00000	0.300837
$896$	0	0
$897$	0	0
$898$	6.00000	0.200223
$899$	−16.0000	−0.533630
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	14.0000	0.465633
$905$	27.0000	0.897510
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	25.0000	0.829654
$909$	0	0
$910$	0	0
$911$	53.0000	1.75597	0.877984	−	0.478690i	$-0.158888\pi$
$911$	53.0000	1.75597	0.877984	+	0.478690i	$0.158888\pi$
$912$	0	0
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	8.00000	0.264327
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	18.0000	0.593442
$921$	0	0
$922$	−30.0000	−0.987997
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−36.0000	−1.18367
$926$	6.00000	0.197172
$927$	0	0
$928$	−8.00000	−0.262613
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	0	0
$932$	−14.0000	−0.458585
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	0	0
$940$	21.0000	0.684944
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31.0000	1.00419	0.502094	−	0.864813i	$-0.332563\pi$
$953$	31.0000	1.00419	0.502094	+	0.864813i	$0.332563\pi$
$954$	0	0
$955$	30.0000	0.970777
$956$	−15.0000	−0.485135
$957$	0	0
$958$	−40.0000	−1.29234
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−36.0000	−1.16069
$963$	0	0
$964$	−13.0000	−0.418702
$965$	−72.0000	−2.31776
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	36.0000	1.15589
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	0	0
$974$	−14.0000	−0.448589
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−33.0000	−1.05576	−0.527882	−	0.849318i	$-0.677014\pi$
$977$	−33.0000	−1.05576	−0.527882	+	0.849318i	$0.677014\pi$
$978$	0	0
$979$	0	0
$980$	−21.0000	−0.670820
$981$	0	0
$982$	22.0000	0.702048
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	−66.0000	−2.10293
$986$	−48.0000	−1.52863
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	0	0
$995$	18.0000	0.570638
$996$	0	0
$997$	−17.0000	−0.538395	−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000	−0.538395	−0.269198	+	0.963085i	$0.586759\pi$
$998$	−14.0000	−0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.h.1.1		1
3.2	odd	2		1338.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.a.1.1	✓	1	3.2	odd	2
4014.2.a.h.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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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