LMFDB - Embedded newform 1575.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-3,0,1,0,0,-7,21,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1575.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} -7.00000 q^{7} +21.0000 q^{8} +36.0000 q^{11} +34.0000 q^{13} +21.0000 q^{14} -71.0000 q^{16} +42.0000 q^{17} -124.000 q^{19} -108.000 q^{22} -102.000 q^{26} -7.00000 q^{28} -102.000 q^{29} -160.000 q^{31} +45.0000 q^{32} -126.000 q^{34} -398.000 q^{37} +372.000 q^{38} +318.000 q^{41} +268.000 q^{43} +36.0000 q^{44} +240.000 q^{47} +49.0000 q^{49} +34.0000 q^{52} -498.000 q^{53} -147.000 q^{56} +306.000 q^{58} +132.000 q^{59} +398.000 q^{61} +480.000 q^{62} +433.000 q^{64} -92.0000 q^{67} +42.0000 q^{68} +720.000 q^{71} +502.000 q^{73} +1194.00 q^{74} -124.000 q^{76} -252.000 q^{77} -1024.00 q^{79} -954.000 q^{82} -204.000 q^{83} -804.000 q^{86} +756.000 q^{88} -354.000 q^{89} -238.000 q^{91} -720.000 q^{94} +286.000 q^{97} -147.000 q^{98} +O(q^{100})$

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	21.0000	0.928078
$9$	0	0
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	34.0000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	34.0000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	21.0000	0.400892
$15$	0	0
$16$	−71.0000	−1.10938
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	0	0
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	0	0
$21$	0	0
$22$	−108.000	−1.04662
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	−102.000	−0.769379
$27$	0	0
$28$	−7.00000	−0.0472456
$29$	−102.000	−0.653135	−0.326568	−	0.945174i	$-0.605892\pi$
$29$	−102.000	−0.653135	−0.326568	+	0.945174i	$0.605892\pi$
$30$	0	0
$31$	−160.000	−0.926995	−0.463498	−	0.886098i	$-0.653406\pi$
$31$	−160.000	−0.926995	−0.463498	+	0.886098i	$0.653406\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	−126.000	−0.635554
$35$	0	0
$36$	0	0
$37$	−398.000	−1.76840	−0.884200	−	0.467109i	$-0.845296\pi$
$37$	−398.000	−1.76840	−0.884200	+	0.467109i	$0.845296\pi$
$38$	372.000	1.58806
$39$	0	0
$40$	0	0
$41$	318.000	1.21130	0.605649	−	0.795732i	$-0.292913\pi$
$41$	318.000	1.21130	0.605649	+	0.795732i	$0.292913\pi$
$42$	0	0
$43$	268.000	0.950456	0.475228	−	0.879863i	$-0.342366\pi$
$43$	268.000	0.950456	0.475228	+	0.879863i	$0.342366\pi$
$44$	36.0000	0.123346
$45$	0	0
$46$	0	0
$47$	240.000	0.744843	0.372421	−	0.928064i	$-0.378528\pi$
$47$	240.000	0.744843	0.372421	+	0.928064i	$0.378528\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	34.0000	0.0906721
$53$	−498.000	−1.29067	−0.645335	−	0.763899i	$-0.723282\pi$
$53$	−498.000	−1.29067	−0.645335	+	0.763899i	$0.723282\pi$
$54$	0	0
$55$	0	0
$56$	−147.000	−0.350780
$57$	0	0
$58$	306.000	0.692755
$59$	132.000	0.291270	0.145635	−	0.989338i	$-0.453477\pi$
$59$	132.000	0.291270	0.145635	+	0.989338i	$0.453477\pi$
$60$	0	0
$61$	398.000	0.835388	0.417694	−	0.908588i	$-0.362838\pi$
$61$	398.000	0.835388	0.417694	+	0.908588i	$0.362838\pi$
$62$	480.000	0.983227
$63$	0	0
$64$	433.000	0.845703
$65$	0	0
$66$	0	0
$67$	−92.0000	−0.167755	−0.0838775	−	0.996476i	$-0.526730\pi$
$67$	−92.0000	−0.167755	−0.0838775	+	0.996476i	$0.526730\pi$
$68$	42.0000	0.0749007
$69$	0	0
$70$	0	0
$71$	720.000	1.20350	0.601748	−	0.798686i	$-0.294471\pi$
$71$	720.000	1.20350	0.601748	+	0.798686i	$0.294471\pi$
$72$	0	0
$73$	502.000	0.804858	0.402429	−	0.915451i	$-0.368166\pi$
$73$	502.000	0.804858	0.402429	+	0.915451i	$0.368166\pi$
$74$	1194.00	1.87567
$75$	0	0
$76$	−124.000	−0.187155
$77$	−252.000	−0.372962
$78$	0	0
$79$	−1024.00	−1.45834	−0.729171	−	0.684332i	$-0.760094\pi$
$79$	−1024.00	−1.45834	−0.729171	+	0.684332i	$0.760094\pi$
$80$	0	0
$81$	0	0
$82$	−954.000	−1.28478
$83$	−204.000	−0.269782	−0.134891	−	0.990860i	$-0.543068\pi$
$83$	−204.000	−0.269782	−0.134891	+	0.990860i	$0.543068\pi$
$84$	0	0
$85$	0	0
$86$	−804.000	−1.00811
$87$	0	0
$88$	756.000	0.915794
$89$	−354.000	−0.421617	−0.210809	−	0.977527i	$-0.567610\pi$
$89$	−354.000	−0.421617	−0.210809	+	0.977527i	$0.567610\pi$
$90$	0	0
$91$	−238.000	−0.274167
$92$	0	0
$93$	0	0
$94$	−720.000	−0.790025
$95$	0	0
$96$	0	0
$97$	286.000	0.299370	0.149685	−	0.988734i	$-0.452174\pi$
$97$	286.000	0.299370	0.149685	+	0.988734i	$0.452174\pi$
$98$	−147.000	−0.151523
$99$	0	0
$100$	0	0
$101$	−414.000	−0.407867	−0.203933	−	0.978985i	$-0.565373\pi$
$101$	−414.000	−0.407867	−0.203933	+	0.978985i	$0.565373\pi$
$102$	0	0
$103$	−56.0000	−0.0535713	−0.0267857	−	0.999641i	$-0.508527\pi$
$103$	−56.0000	−0.0535713	−0.0267857	+	0.999641i	$0.508527\pi$
$104$	714.000	0.673206
$105$	0	0
$106$	1494.00	1.36896
$107$	12.0000	0.0108419	0.00542095	−	0.999985i	$-0.498274\pi$
$107$	12.0000	0.0108419	0.00542095	+	0.999985i	$0.498274\pi$
$108$	0	0
$109$	1478.00	1.29878	0.649389	−	0.760457i	$-0.275025\pi$
$109$	1478.00	1.29878	0.649389	+	0.760457i	$0.275025\pi$
$110$	0	0
$111$	0	0
$112$	497.000	0.419304
$113$	402.000	0.334664	0.167332	−	0.985901i	$-0.446485\pi$
$113$	402.000	0.334664	0.167332	+	0.985901i	$0.446485\pi$
$114$	0	0
$115$	0	0
$116$	−102.000	−0.0816419
$117$	0	0
$118$	−396.000	−0.308939
$119$	−294.000	−0.226478
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	−1194.00	−0.886063
$123$	0	0
$124$	−160.000	−0.115874
$125$	0	0
$126$	0	0
$127$	−1280.00	−0.894344	−0.447172	−	0.894448i	$-0.647569\pi$
$127$	−1280.00	−0.894344	−0.447172	+	0.894448i	$0.647569\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	0	0
$131$	−1764.00	−1.17650	−0.588250	−	0.808679i	$-0.700183\pi$
$131$	−1764.00	−1.17650	−0.588250	+	0.808679i	$0.700183\pi$
$132$	0	0
$133$	868.000	0.565903
$134$	276.000	0.177931
$135$	0	0
$136$	882.000	0.556109
$137$	−2358.00	−1.47049	−0.735246	−	0.677800i	$-0.762934\pi$
$137$	−2358.00	−1.47049	−0.735246	+	0.677800i	$0.762934\pi$
$138$	0	0
$139$	−52.0000	−0.0317308	−0.0158654	−	0.999874i	$-0.505050\pi$
$139$	−52.0000	−0.0317308	−0.0158654	+	0.999874i	$0.505050\pi$
$140$	0	0
$141$	0	0
$142$	−2160.00	−1.27650
$143$	1224.00	0.715776
$144$	0	0
$145$	0	0
$146$	−1506.00	−0.853681
$147$	0	0
$148$	−398.000	−0.221050
$149$	1746.00	0.959986	0.479993	−	0.877272i	$-0.340639\pi$
$149$	1746.00	0.959986	0.479993	+	0.877272i	$0.340639\pi$
$150$	0	0
$151$	−232.000	−0.125032	−0.0625162	−	0.998044i	$-0.519913\pi$
$151$	−232.000	−0.125032	−0.0625162	+	0.998044i	$0.519913\pi$
$152$	−2604.00	−1.38955
$153$	0	0
$154$	756.000	0.395586
$155$	0	0
$156$	0	0
$157$	−1694.00	−0.861120	−0.430560	−	0.902562i	$-0.641684\pi$
$157$	−1694.00	−0.861120	−0.430560	+	0.902562i	$0.641684\pi$
$158$	3072.00	1.54681
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2932.00	1.40891	0.704454	−	0.709750i	$-0.251192\pi$
$163$	2932.00	1.40891	0.704454	+	0.709750i	$0.251192\pi$
$164$	318.000	0.151412
$165$	0	0
$166$	612.000	0.286147
$167$	1176.00	0.544920	0.272460	−	0.962167i	$-0.412163\pi$
$167$	1176.00	0.544920	0.272460	+	0.962167i	$0.412163\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	0	0
$172$	268.000	0.118807
$173$	870.000	0.382340	0.191170	−	0.981557i	$-0.438772\pi$
$173$	870.000	0.382340	0.191170	+	0.981557i	$0.438772\pi$
$174$	0	0
$175$	0	0
$176$	−2556.00	−1.09469
$177$	0	0
$178$	1062.00	0.447193
$179$	2316.00	0.967072	0.483536	−	0.875324i	$-0.339352\pi$
$179$	2316.00	0.967072	0.483536	+	0.875324i	$0.339352\pi$
$180$	0	0
$181$	−106.000	−0.0435299	−0.0217650	−	0.999763i	$-0.506929\pi$
$181$	−106.000	−0.0435299	−0.0217650	+	0.999763i	$0.506929\pi$
$182$	714.000	0.290798
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1512.00	0.591275
$188$	240.000	0.0931053
$189$	0	0
$190$	0	0
$191$	1128.00	0.427326	0.213663	−	0.976907i	$-0.431461\pi$
$191$	1128.00	0.427326	0.213663	+	0.976907i	$0.431461\pi$
$192$	0	0
$193$	−4034.00	−1.50453	−0.752263	−	0.658862i	$-0.771038\pi$
$193$	−4034.00	−1.50453	−0.752263	+	0.658862i	$0.771038\pi$
$194$	−858.000	−0.317530
$195$	0	0
$196$	49.0000	0.0178571
$197$	−1314.00	−0.475221	−0.237611	−	0.971360i	$-0.576364\pi$
$197$	−1314.00	−0.475221	−0.237611	+	0.971360i	$0.576364\pi$
$198$	0	0
$199$	5096.00	1.81531	0.907653	−	0.419722i	$-0.137872\pi$
$199$	5096.00	1.81531	0.907653	+	0.419722i	$0.137872\pi$
$200$	0	0
$201$	0	0
$202$	1242.00	0.432608
$203$	714.000	0.246862
$204$	0	0
$205$	0	0
$206$	168.000	0.0568209
$207$	0	0
$208$	−2414.00	−0.804715
$209$	−4464.00	−1.47742
$210$	0	0
$211$	−3076.00	−1.00360	−0.501802	−	0.864982i	$-0.667330\pi$
$211$	−3076.00	−1.00360	−0.501802	+	0.864982i	$0.667330\pi$
$212$	−498.000	−0.161334
$213$	0	0
$214$	−36.0000	−0.0114996
$215$	0	0
$216$	0	0
$217$	1120.00	0.350371
$218$	−4434.00	−1.37756
$219$	0	0
$220$	0	0
$221$	1428.00	0.434650
$222$	0	0
$223$	1888.00	0.566950	0.283475	−	0.958980i	$-0.408513\pi$
$223$	1888.00	0.566950	0.283475	+	0.958980i	$0.408513\pi$
$224$	−315.000	−0.0939590
$225$	0	0
$226$	−1206.00	−0.354964
$227$	−4716.00	−1.37891	−0.689454	−	0.724330i	$-0.742149\pi$
$227$	−4716.00	−1.37891	−0.689454	+	0.724330i	$0.742149\pi$
$228$	0	0
$229$	−1690.00	−0.487678	−0.243839	−	0.969816i	$-0.578407\pi$
$229$	−1690.00	−0.487678	−0.243839	+	0.969816i	$0.578407\pi$
$230$	0	0
$231$	0	0
$232$	−2142.00	−0.606160
$233$	138.000	0.0388012	0.0194006	−	0.999812i	$-0.493824\pi$
$233$	138.000	0.0388012	0.0194006	+	0.999812i	$0.493824\pi$
$234$	0	0
$235$	0	0
$236$	132.000	0.0364088
$237$	0	0
$238$	882.000	0.240217
$239$	−1896.00	−0.513147	−0.256573	−	0.966525i	$-0.582594\pi$
$239$	−1896.00	−0.513147	−0.256573	+	0.966525i	$0.582594\pi$
$240$	0	0
$241$	−3598.00	−0.961691	−0.480846	−	0.876805i	$-0.659670\pi$
$241$	−3598.00	−0.961691	−0.480846	+	0.876805i	$0.659670\pi$
$242$	105.000	0.0278911
$243$	0	0
$244$	398.000	0.104424
$245$	0	0
$246$	0	0
$247$	−4216.00	−1.08606
$248$	−3360.00	−0.860323
$249$	0	0
$250$	0	0
$251$	3060.00	0.769504	0.384752	−	0.923020i	$-0.374287\pi$
$251$	3060.00	0.769504	0.384752	+	0.923020i	$0.374287\pi$
$252$	0	0
$253$	0	0
$254$	3840.00	0.948595
$255$	0	0
$256$	1513.00	0.369385
$257$	−6822.00	−1.65582	−0.827908	−	0.560864i	$-0.810469\pi$
$257$	−6822.00	−1.65582	−0.827908	+	0.560864i	$0.810469\pi$
$258$	0	0
$259$	2786.00	0.668392
$260$	0	0
$261$	0	0
$262$	5292.00	1.24787
$263$	2592.00	0.607717	0.303858	−	0.952717i	$-0.401725\pi$
$263$	2592.00	0.607717	0.303858	+	0.952717i	$0.401725\pi$
$264$	0	0
$265$	0	0
$266$	−2604.00	−0.600231
$267$	0	0
$268$	−92.0000	−0.0209694
$269$	−8214.00	−1.86177	−0.930886	−	0.365311i	$-0.880963\pi$
$269$	−8214.00	−1.86177	−0.930886	+	0.365311i	$0.880963\pi$
$270$	0	0
$271$	−5344.00	−1.19788	−0.598939	−	0.800795i	$-0.704411\pi$
$271$	−5344.00	−1.19788	−0.598939	+	0.800795i	$0.704411\pi$
$272$	−2982.00	−0.664744
$273$	0	0
$274$	7074.00	1.55969
$275$	0	0
$276$	0	0
$277$	6514.00	1.41295	0.706477	−	0.707736i	$-0.250283\pi$
$277$	6514.00	1.41295	0.706477	+	0.707736i	$0.250283\pi$
$278$	156.000	0.0336556
$279$	0	0
$280$	0	0
$281$	−6618.00	−1.40497	−0.702485	−	0.711698i	$-0.747926\pi$
$281$	−6618.00	−1.40497	−0.702485	+	0.711698i	$0.747926\pi$
$282$	0	0
$283$	−3260.00	−0.684759	−0.342380	−	0.939562i	$-0.611233\pi$
$283$	−3260.00	−0.684759	−0.342380	+	0.939562i	$0.611233\pi$
$284$	720.000	0.150437
$285$	0	0
$286$	−3672.00	−0.759195
$287$	−2226.00	−0.457828
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	0	0
$292$	502.000	0.100607
$293$	5118.00	1.02047	0.510233	−	0.860036i	$-0.329559\pi$
$293$	5118.00	1.02047	0.510233	+	0.860036i	$0.329559\pi$
$294$	0	0
$295$	0	0
$296$	−8358.00	−1.64121
$297$	0	0
$298$	−5238.00	−1.01822
$299$	0	0
$300$	0	0
$301$	−1876.00	−0.359239
$302$	696.000	0.132617
$303$	0	0
$304$	8804.00	1.66100
$305$	0	0
$306$	0	0
$307$	−452.000	−0.0840293	−0.0420147	−	0.999117i	$-0.513378\pi$
$307$	−452.000	−0.0840293	−0.0420147	+	0.999117i	$0.513378\pi$
$308$	−252.000	−0.0466202
$309$	0	0
$310$	0	0
$311$	−5016.00	−0.914570	−0.457285	−	0.889320i	$-0.651178\pi$
$311$	−5016.00	−0.914570	−0.457285	+	0.889320i	$0.651178\pi$
$312$	0	0
$313$	−5402.00	−0.975524	−0.487762	−	0.872977i	$-0.662187\pi$
$313$	−5402.00	−0.975524	−0.487762	+	0.872977i	$0.662187\pi$
$314$	5082.00	0.913356
$315$	0	0
$316$	−1024.00	−0.182293
$317$	10086.0	1.78702	0.893511	−	0.449041i	$-0.148234\pi$
$317$	10086.0	1.78702	0.893511	+	0.449041i	$0.148234\pi$
$318$	0	0
$319$	−3672.00	−0.644491
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5208.00	−0.897154
$324$	0	0
$325$	0	0
$326$	−8796.00	−1.49437
$327$	0	0
$328$	6678.00	1.12418
$329$	−1680.00	−0.281524
$330$	0	0
$331$	−8044.00	−1.33577	−0.667883	−	0.744267i	$-0.732799\pi$
$331$	−8044.00	−1.33577	−0.667883	+	0.744267i	$0.732799\pi$
$332$	−204.000	−0.0337228
$333$	0	0
$334$	−3528.00	−0.577975
$335$	0	0
$336$	0	0
$337$	−4178.00	−0.675342	−0.337671	−	0.941264i	$-0.609639\pi$
$337$	−4178.00	−0.675342	−0.337671	+	0.941264i	$0.609639\pi$
$338$	3123.00	0.502570
$339$	0	0
$340$	0	0
$341$	−5760.00	−0.914726
$342$	0	0
$343$	−343.000	−0.0539949
$344$	5628.00	0.882097
$345$	0	0
$346$	−2610.00	−0.405533
$347$	156.000	0.0241341	0.0120670	−	0.999927i	$-0.496159\pi$
$347$	156.000	0.0241341	0.0120670	+	0.999927i	$0.496159\pi$
$348$	0	0
$349$	−12418.0	−1.90464	−0.952321	−	0.305097i	$-0.901311\pi$
$349$	−12418.0	−1.90464	−0.952321	+	0.305097i	$0.901311\pi$
$350$	0	0
$351$	0	0
$352$	1620.00	0.245302
$353$	−7830.00	−1.18059	−0.590296	−	0.807187i	$-0.700989\pi$
$353$	−7830.00	−1.18059	−0.590296	+	0.807187i	$0.700989\pi$
$354$	0	0
$355$	0	0
$356$	−354.000	−0.0527021
$357$	0	0
$358$	−6948.00	−1.02574
$359$	9312.00	1.36899	0.684497	−	0.729016i	$-0.260022\pi$
$359$	9312.00	1.36899	0.684497	+	0.729016i	$0.260022\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	318.000	0.0461705
$363$	0	0
$364$	−238.000	−0.0342709
$365$	0	0
$366$	0	0
$367$	3760.00	0.534797	0.267398	−	0.963586i	$-0.413836\pi$
$367$	3760.00	0.534797	0.267398	+	0.963586i	$0.413836\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3486.00	0.487828
$372$	0	0
$373$	−5870.00	−0.814845	−0.407422	−	0.913240i	$-0.633572\pi$
$373$	−5870.00	−0.814845	−0.407422	+	0.913240i	$0.633572\pi$
$374$	−4536.00	−0.627142
$375$	0	0
$376$	5040.00	0.691272
$377$	−3468.00	−0.473769
$378$	0	0
$379$	−1852.00	−0.251005	−0.125502	−	0.992093i	$-0.540054\pi$
$379$	−1852.00	−0.251005	−0.125502	+	0.992093i	$0.540054\pi$
$380$	0	0
$381$	0	0
$382$	−3384.00	−0.453247
$383$	2160.00	0.288175	0.144087	−	0.989565i	$-0.453975\pi$
$383$	2160.00	0.288175	0.144087	+	0.989565i	$0.453975\pi$
$384$	0	0
$385$	0	0
$386$	12102.0	1.59579
$387$	0	0
$388$	286.000	0.0374213
$389$	6786.00	0.884483	0.442241	−	0.896896i	$-0.354183\pi$
$389$	6786.00	0.884483	0.442241	+	0.896896i	$0.354183\pi$
$390$	0	0
$391$	0	0
$392$	1029.00	0.132583
$393$	0	0
$394$	3942.00	0.504048
$395$	0	0
$396$	0	0
$397$	6514.00	0.823497	0.411748	−	0.911298i	$-0.364918\pi$
$397$	6514.00	0.823497	0.411748	+	0.911298i	$0.364918\pi$
$398$	−15288.0	−1.92542
$399$	0	0
$400$	0	0
$401$	−3330.00	−0.414694	−0.207347	−	0.978267i	$-0.566483\pi$
$401$	−3330.00	−0.414694	−0.207347	+	0.978267i	$0.566483\pi$
$402$	0	0
$403$	−5440.00	−0.672421
$404$	−414.000	−0.0509833
$405$	0	0
$406$	−2142.00	−0.261837
$407$	−14328.0	−1.74499
$408$	0	0
$409$	−5398.00	−0.652601	−0.326301	−	0.945266i	$-0.605802\pi$
$409$	−5398.00	−0.652601	−0.326301	+	0.945266i	$0.605802\pi$
$410$	0	0
$411$	0	0
$412$	−56.0000	−0.00669641
$413$	−924.000	−0.110090
$414$	0	0
$415$	0	0
$416$	1530.00	0.180323
$417$	0	0
$418$	13392.0	1.56704
$419$	−13092.0	−1.52646	−0.763229	−	0.646128i	$-0.776387\pi$
$419$	−13092.0	−1.52646	−0.763229	+	0.646128i	$0.776387\pi$
$420$	0	0
$421$	−322.000	−0.0372763	−0.0186381	−	0.999826i	$-0.505933\pi$
$421$	−322.000	−0.0372763	−0.0186381	+	0.999826i	$0.505933\pi$
$422$	9228.00	1.06448
$423$	0	0
$424$	−10458.0	−1.19784
$425$	0	0
$426$	0	0
$427$	−2786.00	−0.315747
$428$	12.0000	0.00135524
$429$	0	0
$430$	0	0
$431$	−2616.00	−0.292363	−0.146181	−	0.989258i	$-0.546698\pi$
$431$	−2616.00	−0.292363	−0.146181	+	0.989258i	$0.546698\pi$
$432$	0	0
$433$	−4322.00	−0.479681	−0.239841	−	0.970812i	$-0.577095\pi$
$433$	−4322.00	−0.479681	−0.239841	+	0.970812i	$0.577095\pi$
$434$	−3360.00	−0.371625
$435$	0	0
$436$	1478.00	0.162347
$437$	0	0
$438$	0	0
$439$	−9016.00	−0.980205	−0.490103	−	0.871665i	$-0.663041\pi$
$439$	−9016.00	−0.980205	−0.490103	+	0.871665i	$0.663041\pi$
$440$	0	0
$441$	0	0
$442$	−4284.00	−0.461016
$443$	−5268.00	−0.564989	−0.282495	−	0.959269i	$-0.591162\pi$
$443$	−5268.00	−0.564989	−0.282495	+	0.959269i	$0.591162\pi$
$444$	0	0
$445$	0	0
$446$	−5664.00	−0.601341
$447$	0	0
$448$	−3031.00	−0.319646
$449$	5310.00	0.558117	0.279058	−	0.960274i	$-0.409978\pi$
$449$	5310.00	0.558117	0.279058	+	0.960274i	$0.409978\pi$
$450$	0	0
$451$	11448.0	1.19527
$452$	402.000	0.0418329
$453$	0	0
$454$	14148.0	1.46255
$455$	0	0
$456$	0	0
$457$	−15770.0	−1.61420	−0.807100	−	0.590415i	$-0.798964\pi$
$457$	−15770.0	−1.61420	−0.807100	+	0.590415i	$0.798964\pi$
$458$	5070.00	0.517261
$459$	0	0
$460$	0	0
$461$	5370.00	0.542529	0.271264	−	0.962505i	$-0.412558\pi$
$461$	5370.00	0.542529	0.271264	+	0.962505i	$0.412558\pi$
$462$	0	0
$463$	3328.00	0.334050	0.167025	−	0.985953i	$-0.446584\pi$
$463$	3328.00	0.334050	0.167025	+	0.985953i	$0.446584\pi$
$464$	7242.00	0.724572
$465$	0	0
$466$	−414.000	−0.0411549
$467$	4548.00	0.450656	0.225328	−	0.974283i	$-0.427655\pi$
$467$	4548.00	0.450656	0.225328	+	0.974283i	$0.427655\pi$
$468$	0	0
$469$	644.000	0.0634055
$470$	0	0
$471$	0	0
$472$	2772.00	0.270321
$473$	9648.00	0.937876
$474$	0	0
$475$	0	0
$476$	−294.000	−0.0283098
$477$	0	0
$478$	5688.00	0.544274
$479$	8064.00	0.769214	0.384607	−	0.923080i	$-0.374337\pi$
$479$	8064.00	0.769214	0.384607	+	0.923080i	$0.374337\pi$
$480$	0	0
$481$	−13532.0	−1.28276
$482$	10794.0	1.02003
$483$	0	0
$484$	−35.0000	−0.00328700
$485$	0	0
$486$	0	0
$487$	−16616.0	−1.54608	−0.773042	−	0.634355i	$-0.781266\pi$
$487$	−16616.0	−1.54608	−0.773042	+	0.634355i	$0.781266\pi$
$488$	8358.00	0.775305
$489$	0	0
$490$	0	0
$491$	7140.00	0.656260	0.328130	−	0.944633i	$-0.393582\pi$
$491$	7140.00	0.656260	0.328130	+	0.944633i	$0.393582\pi$
$492$	0	0
$493$	−4284.00	−0.391362
$494$	12648.0	1.15194
$495$	0	0
$496$	11360.0	1.02839
$497$	−5040.00	−0.454879
$498$	0	0
$499$	−9124.00	−0.818530	−0.409265	−	0.912416i	$-0.634215\pi$
$499$	−9124.00	−0.818530	−0.409265	+	0.912416i	$0.634215\pi$
$500$	0	0
$501$	0	0
$502$	−9180.00	−0.816182
$503$	−6552.00	−0.580794	−0.290397	−	0.956906i	$-0.593787\pi$
$503$	−6552.00	−0.580794	−0.290397	+	0.956906i	$0.593787\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−1280.00	−0.111793
$509$	−2790.00	−0.242956	−0.121478	−	0.992594i	$-0.538763\pi$
$509$	−2790.00	−0.242956	−0.121478	+	0.992594i	$0.538763\pi$
$510$	0	0
$511$	−3514.00	−0.304208
$512$	8733.00	0.753804
$513$	0	0
$514$	20466.0	1.75626
$515$	0	0
$516$	0	0
$517$	8640.00	0.734984
$518$	−8358.00	−0.708937
$519$	0	0
$520$	0	0
$521$	14862.0	1.24974	0.624871	−	0.780728i	$-0.285151\pi$
$521$	14862.0	1.24974	0.624871	+	0.780728i	$0.285151\pi$
$522$	0	0
$523$	−17660.0	−1.47652	−0.738258	−	0.674518i	$-0.764351\pi$
$523$	−17660.0	−1.47652	−0.738258	+	0.674518i	$0.764351\pi$
$524$	−1764.00	−0.147062
$525$	0	0
$526$	−7776.00	−0.644581
$527$	−6720.00	−0.555461
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	868.000	0.0707379
$533$	10812.0	0.878649
$534$	0	0
$535$	0	0
$536$	−1932.00	−0.155690
$537$	0	0
$538$	24642.0	1.97471
$539$	1764.00	0.140966
$540$	0	0
$541$	−19834.0	−1.57621	−0.788106	−	0.615540i	$-0.788938\pi$
$541$	−19834.0	−1.57621	−0.788106	+	0.615540i	$0.788938\pi$
$542$	16032.0	1.27054
$543$	0	0
$544$	1890.00	0.148958
$545$	0	0
$546$	0	0
$547$	−20972.0	−1.63930	−0.819651	−	0.572863i	$-0.805833\pi$
$547$	−20972.0	−1.63930	−0.819651	+	0.572863i	$0.805833\pi$
$548$	−2358.00	−0.183812
$549$	0	0
$550$	0	0
$551$	12648.0	0.977900
$552$	0	0
$553$	7168.00	0.551201
$554$	−19542.0	−1.49866
$555$	0	0
$556$	−52.0000	−0.00396635
$557$	21174.0	1.61072	0.805360	−	0.592786i	$-0.201972\pi$
$557$	21174.0	1.61072	0.805360	+	0.592786i	$0.201972\pi$
$558$	0	0
$559$	9112.00	0.689439
$560$	0	0
$561$	0	0
$562$	19854.0	1.49020
$563$	−17772.0	−1.33037	−0.665187	−	0.746677i	$-0.731648\pi$
$563$	−17772.0	−1.33037	−0.665187	+	0.746677i	$0.731648\pi$
$564$	0	0
$565$	0	0
$566$	9780.00	0.726297
$567$	0	0
$568$	15120.0	1.11694
$569$	−8250.00	−0.607835	−0.303917	−	0.952698i	$-0.598295\pi$
$569$	−8250.00	−0.607835	−0.303917	+	0.952698i	$0.598295\pi$
$570$	0	0
$571$	20756.0	1.52121	0.760606	−	0.649214i	$-0.224902\pi$
$571$	20756.0	1.52121	0.760606	+	0.649214i	$0.224902\pi$
$572$	1224.00	0.0894720
$573$	0	0
$574$	6678.00	0.485600
$575$	0	0
$576$	0	0
$577$	−2.00000	−0.000144300 0	−7.21500e−5	−	1.00000i	$-0.500023\pi$
$577$	−2.00000	−0.000144300 0	−7.21500e−5		1.00000i	$0.500023\pi$
$578$	9447.00	0.679833
$579$	0	0
$580$	0	0
$581$	1428.00	0.101968
$582$	0	0
$583$	−17928.0	−1.27359
$584$	10542.0	0.746971
$585$	0	0
$586$	−15354.0	−1.08237
$587$	26364.0	1.85376	0.926881	−	0.375354i	$-0.122479\pi$
$587$	26364.0	1.85376	0.926881	+	0.375354i	$0.122479\pi$
$588$	0	0
$589$	19840.0	1.38793
$590$	0	0
$591$	0	0
$592$	28258.0	1.96182
$593$	2298.00	0.159136	0.0795679	−	0.996829i	$-0.474646\pi$
$593$	2298.00	0.159136	0.0795679	+	0.996829i	$0.474646\pi$
$594$	0	0
$595$	0	0
$596$	1746.00	0.119998
$597$	0	0
$598$	0	0
$599$	−3072.00	−0.209547	−0.104773	−	0.994496i	$-0.533412\pi$
$599$	−3072.00	−0.209547	−0.104773	+	0.994496i	$0.533412\pi$
$600$	0	0
$601$	24554.0	1.66652	0.833260	−	0.552881i	$-0.186472\pi$
$601$	24554.0	1.66652	0.833260	+	0.552881i	$0.186472\pi$
$602$	5628.00	0.381030
$603$	0	0
$604$	−232.000	−0.0156290
$605$	0	0
$606$	0	0
$607$	−16832.0	−1.12552	−0.562759	−	0.826621i	$-0.690260\pi$
$607$	−16832.0	−1.12552	−0.562759	+	0.826621i	$0.690260\pi$
$608$	−5580.00	−0.372202
$609$	0	0
$610$	0	0
$611$	8160.00	0.540292
$612$	0	0
$613$	2482.00	0.163535	0.0817676	−	0.996651i	$-0.473943\pi$
$613$	2482.00	0.163535	0.0817676	+	0.996651i	$0.473943\pi$
$614$	1356.00	0.0891266
$615$	0	0
$616$	−5292.00	−0.346138
$617$	−15798.0	−1.03080	−0.515400	−	0.856950i	$-0.672357\pi$
$617$	−15798.0	−1.03080	−0.515400	+	0.856950i	$0.672357\pi$
$618$	0	0
$619$	−15460.0	−1.00386	−0.501930	−	0.864908i	$-0.667377\pi$
$619$	−15460.0	−1.00386	−0.501930	+	0.864908i	$0.667377\pi$
$620$	0	0
$621$	0	0
$622$	15048.0	0.970048
$623$	2478.00	0.159356
$624$	0	0
$625$	0	0
$626$	16206.0	1.03470
$627$	0	0
$628$	−1694.00	−0.107640
$629$	−16716.0	−1.05964
$630$	0	0
$631$	−7720.00	−0.487050	−0.243525	−	0.969895i	$-0.578304\pi$
$631$	−7720.00	−0.487050	−0.243525	+	0.969895i	$0.578304\pi$
$632$	−21504.0	−1.35345
$633$	0	0
$634$	−30258.0	−1.89542
$635$	0	0
$636$	0	0
$637$	1666.00	0.103625
$638$	11016.0	0.683586
$639$	0	0
$640$	0	0
$641$	17262.0	1.06366	0.531832	−	0.846850i	$-0.321504\pi$
$641$	17262.0	1.06366	0.531832	+	0.846850i	$0.321504\pi$
$642$	0	0
$643$	12220.0	0.749471	0.374735	−	0.927132i	$-0.377734\pi$
$643$	12220.0	0.749471	0.374735	+	0.927132i	$0.377734\pi$
$644$	0	0
$645$	0	0
$646$	15624.0	0.951576
$647$	13560.0	0.823955	0.411977	−	0.911194i	$-0.364838\pi$
$647$	13560.0	0.823955	0.411977	+	0.911194i	$0.364838\pi$
$648$	0	0
$649$	4752.00	0.287415
$650$	0	0
$651$	0	0
$652$	2932.00	0.176113
$653$	23094.0	1.38398	0.691989	−	0.721908i	$-0.256735\pi$
$653$	23094.0	1.38398	0.691989	+	0.721908i	$0.256735\pi$
$654$	0	0
$655$	0	0
$656$	−22578.0	−1.34378
$657$	0	0
$658$	5040.00	0.298601
$659$	−22548.0	−1.33285	−0.666423	−	0.745574i	$-0.732175\pi$
$659$	−22548.0	−1.33285	−0.666423	+	0.745574i	$0.732175\pi$
$660$	0	0
$661$	17462.0	1.02752	0.513762	−	0.857933i	$-0.328252\pi$
$661$	17462.0	1.02752	0.513762	+	0.857933i	$0.328252\pi$
$662$	24132.0	1.41679
$663$	0	0
$664$	−4284.00	−0.250379
$665$	0	0
$666$	0	0
$667$	0	0
$668$	1176.00	0.0681150
$669$	0	0
$670$	0	0
$671$	14328.0	0.824331
$672$	0	0
$673$	22462.0	1.28655	0.643274	−	0.765636i	$-0.277576\pi$
$673$	22462.0	1.28655	0.643274	+	0.765636i	$0.277576\pi$
$674$	12534.0	0.716308
$675$	0	0
$676$	−1041.00	−0.0592285
$677$	−25554.0	−1.45069	−0.725347	−	0.688383i	$-0.758321\pi$
$677$	−25554.0	−1.45069	−0.725347	+	0.688383i	$0.758321\pi$
$678$	0	0
$679$	−2002.00	−0.113151
$680$	0	0
$681$	0	0
$682$	17280.0	0.970213
$683$	9276.00	0.519672	0.259836	−	0.965653i	$-0.416331\pi$
$683$	9276.00	0.519672	0.259836	+	0.965653i	$0.416331\pi$
$684$	0	0
$685$	0	0
$686$	1029.00	0.0572703
$687$	0	0
$688$	−19028.0	−1.05441
$689$	−16932.0	−0.936223
$690$	0	0
$691$	27380.0	1.50736	0.753679	−	0.657243i	$-0.228277\pi$
$691$	27380.0	1.50736	0.753679	+	0.657243i	$0.228277\pi$
$692$	870.000	0.0477925
$693$	0	0
$694$	−468.000	−0.0255980
$695$	0	0
$696$	0	0
$697$	13356.0	0.725817
$698$	37254.0	2.02018
$699$	0	0
$700$	0	0
$701$	−25830.0	−1.39171	−0.695853	−	0.718184i	$-0.744973\pi$
$701$	−25830.0	−1.39171	−0.695853	+	0.718184i	$0.744973\pi$
$702$	0	0
$703$	49352.0	2.64772
$704$	15588.0	0.834510
$705$	0	0
$706$	23490.0	1.25221
$707$	2898.00	0.154159
$708$	0	0
$709$	−6226.00	−0.329792	−0.164896	−	0.986311i	$-0.552729\pi$
$709$	−6226.00	−0.329792	−0.164896	+	0.986311i	$0.552729\pi$
$710$	0	0
$711$	0	0
$712$	−7434.00	−0.391293
$713$	0	0
$714$	0	0
$715$	0	0
$716$	2316.00	0.120884
$717$	0	0
$718$	−27936.0	−1.45204
$719$	15072.0	0.781767	0.390884	−	0.920440i	$-0.372169\pi$
$719$	15072.0	0.781767	0.390884	+	0.920440i	$0.372169\pi$
$720$	0	0
$721$	392.000	0.0202480
$722$	−25551.0	−1.31705
$723$	0	0
$724$	−106.000	−0.00544124
$725$	0	0
$726$	0	0
$727$	32920.0	1.67942	0.839708	−	0.543038i	$-0.182726\pi$
$727$	32920.0	1.67942	0.839708	+	0.543038i	$0.182726\pi$
$728$	−4998.00	−0.254448
$729$	0	0
$730$	0	0
$731$	11256.0	0.569519
$732$	0	0
$733$	6946.00	0.350009	0.175004	−	0.984568i	$-0.444006\pi$
$733$	6946.00	0.350009	0.175004	+	0.984568i	$0.444006\pi$
$734$	−11280.0	−0.567238
$735$	0	0
$736$	0	0
$737$	−3312.00	−0.165535
$738$	0	0
$739$	−2356.00	−0.117276	−0.0586379	−	0.998279i	$-0.518676\pi$
$739$	−2356.00	−0.117276	−0.0586379	+	0.998279i	$0.518676\pi$
$740$	0	0
$741$	0	0
$742$	−10458.0	−0.517419
$743$	−23520.0	−1.16133	−0.580663	−	0.814144i	$-0.697207\pi$
$743$	−23520.0	−1.16133	−0.580663	+	0.814144i	$0.697207\pi$
$744$	0	0
$745$	0	0
$746$	17610.0	0.864273
$747$	0	0
$748$	1512.00	0.0739094
$749$	−84.0000	−0.00409785
$750$	0	0
$751$	3008.00	0.146156	0.0730782	−	0.997326i	$-0.476718\pi$
$751$	3008.00	0.146156	0.0730782	+	0.997326i	$0.476718\pi$
$752$	−17040.0	−0.826310
$753$	0	0
$754$	10404.0	0.502508
$755$	0	0
$756$	0	0
$757$	20770.0	0.997224	0.498612	−	0.866825i	$-0.333843\pi$
$757$	20770.0	0.997224	0.498612	+	0.866825i	$0.333843\pi$
$758$	5556.00	0.266231
$759$	0	0
$760$	0	0
$761$	−11538.0	−0.549609	−0.274804	−	0.961500i	$-0.588613\pi$
$761$	−11538.0	−0.549609	−0.274804	+	0.961500i	$0.588613\pi$
$762$	0	0
$763$	−10346.0	−0.490892
$764$	1128.00	0.0534157
$765$	0	0
$766$	−6480.00	−0.305655
$767$	4488.00	0.211281
$768$	0	0
$769$	8498.00	0.398499	0.199249	−	0.979949i	$-0.436150\pi$
$769$	8498.00	0.398499	0.199249	+	0.979949i	$0.436150\pi$
$770$	0	0
$771$	0	0
$772$	−4034.00	−0.188066
$773$	−32322.0	−1.50393	−0.751967	−	0.659200i	$-0.770895\pi$
$773$	−32322.0	−1.50393	−0.751967	+	0.659200i	$0.770895\pi$
$774$	0	0
$775$	0	0
$776$	6006.00	0.277839
$777$	0	0
$778$	−20358.0	−0.938136
$779$	−39432.0	−1.81360
$780$	0	0
$781$	25920.0	1.18757
$782$	0	0
$783$	0	0
$784$	−3479.00	−0.158482
$785$	0	0
$786$	0	0
$787$	−26228.0	−1.18796	−0.593982	−	0.804479i	$-0.702445\pi$
$787$	−26228.0	−1.18796	−0.593982	+	0.804479i	$0.702445\pi$
$788$	−1314.00	−0.0594027
$789$	0	0
$790$	0	0
$791$	−2814.00	−0.126491
$792$	0	0
$793$	13532.0	0.605972
$794$	−19542.0	−0.873450
$795$	0	0
$796$	5096.00	0.226913
$797$	−43338.0	−1.92611	−0.963056	−	0.269302i	$-0.913207\pi$
$797$	−43338.0	−1.92611	−0.963056	+	0.269302i	$0.913207\pi$
$798$	0	0
$799$	10080.0	0.446314
$800$	0	0
$801$	0	0
$802$	9990.00	0.439849
$803$	18072.0	0.794206
$804$	0	0
$805$	0	0
$806$	16320.0	0.713210
$807$	0	0
$808$	−8694.00	−0.378532
$809$	28902.0	1.25604	0.628022	−	0.778195i	$-0.283865\pi$
$809$	28902.0	1.25604	0.628022	+	0.778195i	$0.283865\pi$
$810$	0	0
$811$	27164.0	1.17615	0.588075	−	0.808807i	$-0.299886\pi$
$811$	27164.0	1.17615	0.588075	+	0.808807i	$0.299886\pi$
$812$	714.000	0.0308577
$813$	0	0
$814$	42984.0	1.85085
$815$	0	0
$816$	0	0
$817$	−33232.0	−1.42306
$818$	16194.0	0.692188
$819$	0	0
$820$	0	0
$821$	17202.0	0.731247	0.365624	−	0.930763i	$-0.380856\pi$
$821$	17202.0	0.731247	0.365624	+	0.930763i	$0.380856\pi$
$822$	0	0
$823$	5992.00	0.253789	0.126894	−	0.991916i	$-0.459499\pi$
$823$	5992.00	0.253789	0.126894	+	0.991916i	$0.459499\pi$
$824$	−1176.00	−0.0497183
$825$	0	0
$826$	2772.00	0.116768
$827$	25884.0	1.08836	0.544181	−	0.838968i	$-0.316841\pi$
$827$	25884.0	1.08836	0.544181	+	0.838968i	$0.316841\pi$
$828$	0	0
$829$	−1474.00	−0.0617541	−0.0308770	−	0.999523i	$-0.509830\pi$
$829$	−1474.00	−0.0617541	−0.0308770	+	0.999523i	$0.509830\pi$
$830$	0	0
$831$	0	0
$832$	14722.0	0.613454
$833$	2058.00	0.0856008
$834$	0	0
$835$	0	0
$836$	−4464.00	−0.184678
$837$	0	0
$838$	39276.0	1.61905
$839$	−33528.0	−1.37964	−0.689818	−	0.723983i	$-0.742310\pi$
$839$	−33528.0	−1.37964	−0.689818	+	0.723983i	$0.742310\pi$
$840$	0	0
$841$	−13985.0	−0.573414
$842$	966.000	0.0395375
$843$	0	0
$844$	−3076.00	−0.125451
$845$	0	0
$846$	0	0
$847$	245.000	0.00993896
$848$	35358.0	1.43184
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1190.00	−0.0477665	−0.0238832	−	0.999715i	$-0.507603\pi$
$853$	−1190.00	−0.0477665	−0.0238832	+	0.999715i	$0.507603\pi$
$854$	8358.00	0.334900
$855$	0	0
$856$	252.000	0.0100621
$857$	34578.0	1.37825	0.689126	−	0.724642i	$-0.257995\pi$
$857$	34578.0	1.37825	0.689126	+	0.724642i	$0.257995\pi$
$858$	0	0
$859$	−44404.0	−1.76373	−0.881865	−	0.471501i	$-0.843712\pi$
$859$	−44404.0	−1.76373	−0.881865	+	0.471501i	$0.843712\pi$
$860$	0	0
$861$	0	0
$862$	7848.00	0.310097
$863$	−38328.0	−1.51182	−0.755910	−	0.654676i	$-0.772805\pi$
$863$	−38328.0	−1.51182	−0.755910	+	0.654676i	$0.772805\pi$
$864$	0	0
$865$	0	0
$866$	12966.0	0.508779
$867$	0	0
$868$	1120.00	0.0437964
$869$	−36864.0	−1.43904
$870$	0	0
$871$	−3128.00	−0.121686
$872$	31038.0	1.20537
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38842.0	1.49555	0.747777	−	0.663950i	$-0.231121\pi$
$877$	38842.0	1.49555	0.747777	+	0.663950i	$0.231121\pi$
$878$	27048.0	1.03966
$879$	0	0
$880$	0	0
$881$	35046.0	1.34022	0.670108	−	0.742264i	$-0.266248\pi$
$881$	35046.0	1.34022	0.670108	+	0.742264i	$0.266248\pi$
$882$	0	0
$883$	−14204.0	−0.541339	−0.270670	−	0.962672i	$-0.587245\pi$
$883$	−14204.0	−0.541339	−0.270670	+	0.962672i	$0.587245\pi$
$884$	1428.00	0.0543313
$885$	0	0
$886$	15804.0	0.599262
$887$	−26136.0	−0.989359	−0.494679	−	0.869076i	$-0.664714\pi$
$887$	−26136.0	−0.989359	−0.494679	+	0.869076i	$0.664714\pi$
$888$	0	0
$889$	8960.00	0.338030
$890$	0	0
$891$	0	0
$892$	1888.00	0.0708687
$893$	−29760.0	−1.11521
$894$	0	0
$895$	0	0
$896$	11613.0	0.432995
$897$	0	0
$898$	−15930.0	−0.591972
$899$	16320.0	0.605453
$900$	0	0
$901$	−20916.0	−0.773377
$902$	−34344.0	−1.26777
$903$	0	0
$904$	8442.00	0.310594
$905$	0	0
$906$	0	0
$907$	9052.00	0.331386	0.165693	−	0.986177i	$-0.447014\pi$
$907$	9052.00	0.331386	0.165693	+	0.986177i	$0.447014\pi$
$908$	−4716.00	−0.172363
$909$	0	0
$910$	0	0
$911$	−5016.00	−0.182423	−0.0912116	−	0.995832i	$-0.529074\pi$
$911$	−5016.00	−0.182423	−0.0912116	+	0.995832i	$0.529074\pi$
$912$	0	0
$913$	−7344.00	−0.266211
$914$	47310.0	1.71212
$915$	0	0
$916$	−1690.00	−0.0609598
$917$	12348.0	0.444675
$918$	0	0
$919$	44552.0	1.59917	0.799584	−	0.600555i	$-0.205054\pi$
$919$	44552.0	1.59917	0.799584	+	0.600555i	$0.205054\pi$
$920$	0	0
$921$	0	0
$922$	−16110.0	−0.575439
$923$	24480.0	0.872989
$924$	0	0
$925$	0	0
$926$	−9984.00	−0.354314
$927$	0	0
$928$	−4590.00	−0.162364
$929$	−24234.0	−0.855858	−0.427929	−	0.903812i	$-0.640757\pi$
$929$	−24234.0	−0.855858	−0.427929	+	0.903812i	$0.640757\pi$
$930$	0	0
$931$	−6076.00	−0.213891
$932$	138.000	0.00485015
$933$	0	0
$934$	−13644.0	−0.477993
$935$	0	0
$936$	0	0
$937$	13894.0	0.484415	0.242208	−	0.970224i	$-0.422128\pi$
$937$	13894.0	0.484415	0.242208	+	0.970224i	$0.422128\pi$
$938$	−1932.00	−0.0672516
$939$	0	0
$940$	0	0
$941$	−46758.0	−1.61984	−0.809919	−	0.586542i	$-0.800489\pi$
$941$	−46758.0	−1.61984	−0.809919	+	0.586542i	$0.800489\pi$
$942$	0	0
$943$	0	0
$944$	−9372.00	−0.323128
$945$	0	0
$946$	−28944.0	−0.994768
$947$	13812.0	0.473949	0.236974	−	0.971516i	$-0.423844\pi$
$947$	13812.0	0.473949	0.236974	+	0.971516i	$0.423844\pi$
$948$	0	0
$949$	17068.0	0.583826
$950$	0	0
$951$	0	0
$952$	−6174.00	−0.210190
$953$	−58518.0	−1.98907	−0.994535	−	0.104402i	$-0.966707\pi$
$953$	−58518.0	−1.98907	−0.994535	+	0.104402i	$0.966707\pi$
$954$	0	0
$955$	0	0
$956$	−1896.00	−0.0641433
$957$	0	0
$958$	−24192.0	−0.815875
$959$	16506.0	0.555794
$960$	0	0
$961$	−4191.00	−0.140680
$962$	40596.0	1.36057
$963$	0	0
$964$	−3598.00	−0.120211
$965$	0	0
$966$	0	0
$967$	−19640.0	−0.653133	−0.326567	−	0.945174i	$-0.605892\pi$
$967$	−19640.0	−0.653133	−0.326567	+	0.945174i	$0.605892\pi$
$968$	−735.000	−0.0244047
$969$	0	0
$970$	0	0
$971$	58308.0	1.92708	0.963539	−	0.267568i	$-0.0862200\pi$
$971$	58308.0	1.92708	0.963539	+	0.267568i	$0.0862200\pi$
$972$	0	0
$973$	364.000	0.0119931
$974$	49848.0	1.63987
$975$	0	0
$976$	−28258.0	−0.926759
$977$	−23550.0	−0.771168	−0.385584	−	0.922673i	$-0.626000\pi$
$977$	−23550.0	−0.771168	−0.385584	+	0.922673i	$0.626000\pi$
$978$	0	0
$979$	−12744.0	−0.416037
$980$	0	0
$981$	0	0
$982$	−21420.0	−0.696069
$983$	15768.0	0.511619	0.255809	−	0.966727i	$-0.417658\pi$
$983$	15768.0	0.511619	0.255809	+	0.966727i	$0.417658\pi$
$984$	0	0
$985$	0	0
$986$	12852.0	0.415102
$987$	0	0
$988$	−4216.00	−0.135758
$989$	0	0
$990$	0	0
$991$	35264.0	1.13037	0.565186	−	0.824964i	$-0.308805\pi$
$991$	35264.0	1.13037	0.565186	+	0.824964i	$0.308805\pi$
$992$	−7200.00	−0.230444
$993$	0	0
$994$	15120.0	0.482472
$995$	0	0
$996$	0	0
$997$	29338.0	0.931940	0.465970	−	0.884801i	$-0.345706\pi$
$997$	29338.0	0.931940	0.465970	+	0.884801i	$0.345706\pi$
$998$	27372.0	0.868182
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.b.1.1		1
3.2	odd	2		525.4.a.g.1.1		1
5.4	even	2		63.4.a.c.1.1		1
15.2	even	4		525.4.d.c.274.2		2
15.8	even	4		525.4.d.c.274.1		2
15.14	odd	2		21.4.a.a.1.1	✓	1
20.19	odd	2		1008.4.a.v.1.1		1
35.4	even	6		441.4.e.b.226.1		2
35.9	even	6		441.4.e.b.361.1		2
35.19	odd	6		441.4.e.d.361.1		2
35.24	odd	6		441.4.e.d.226.1		2
35.34	odd	2		441.4.a.j.1.1		1
60.59	even	2		336.4.a.f.1.1		1
105.44	odd	6		147.4.e.i.67.1		2
105.59	even	6		147.4.e.g.79.1		2
105.74	odd	6		147.4.e.i.79.1		2
105.89	even	6		147.4.e.g.67.1		2
105.104	even	2		147.4.a.c.1.1		1
120.29	odd	2		1344.4.a.ba.1.1		1
120.59	even	2		1344.4.a.n.1.1		1
420.419	odd	2		2352.4.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.a.1.1	✓	1	15.14	odd	2
63.4.a.c.1.1		1	5.4	even	2
147.4.a.c.1.1		1	105.104	even	2
147.4.e.g.67.1		2	105.89	even	6
147.4.e.g.79.1		2	105.59	even	6
147.4.e.i.67.1		2	105.44	odd	6
147.4.e.i.79.1		2	105.74	odd	6
336.4.a.f.1.1		1	60.59	even	2
441.4.a.j.1.1		1	35.34	odd	2
441.4.e.b.226.1		2	35.4	even	6
441.4.e.b.361.1		2	35.9	even	6
441.4.e.d.226.1		2	35.24	odd	6
441.4.e.d.361.1		2	35.19	odd	6
525.4.a.g.1.1		1	3.2	odd	2
525.4.d.c.274.1		2	15.8	even	4
525.4.d.c.274.2		2	15.2	even	4
1008.4.a.v.1.1		1	20.19	odd	2
1344.4.a.n.1.1		1	120.59	even	2
1344.4.a.ba.1.1		1	120.29	odd	2
1575.4.a.b.1.1		1	1.1	even	1	trivial
2352.4.a.r.1.1		1	420.419	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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