LMFDB - Embedded newform 2475.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53919 q^{2} +0.369102 q^{4} -0.290725 q^{7} -2.51026 q^{8} +1.00000 q^{11} -6.97107 q^{13} -0.447480 q^{14} -4.60197 q^{16} +4.78765 q^{17} +7.75872 q^{19} +1.53919 q^{22} +4.00000 q^{23} -10.7298 q^{26} -0.107307 q^{28} +7.41855 q^{29} +6.34017 q^{31} -2.06278 q^{32} +7.36910 q^{34} +3.41855 q^{37} +11.9421 q^{38} +7.41855 q^{41} -0.290725 q^{43} +0.369102 q^{44} +6.15676 q^{46} +5.26180 q^{47} -6.91548 q^{49} -2.57304 q^{52} +5.75872 q^{53} +0.729794 q^{56} +11.4186 q^{58} -3.60197 q^{59} -6.68035 q^{61} +9.75872 q^{62} +6.02893 q^{64} -6.15676 q^{67} +1.76713 q^{68} +5.07838 q^{71} +1.12783 q^{73} +5.26180 q^{74} +2.86376 q^{76} -0.290725 q^{77} -0.921622 q^{79} +11.4186 q^{82} +1.70928 q^{83} -0.447480 q^{86} -2.51026 q^{88} -4.34017 q^{89} +2.02666 q^{91} +1.47641 q^{92} +8.09890 q^{94} -4.68035 q^{97} -10.6442 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} + 3 q^{11} - 6 q^{13} - 2 q^{14} + 5 q^{16} + 4 q^{17} - 2 q^{19} + 3 q^{22} + 12 q^{23} + 8 q^{26} - 12 q^{28} + 8 q^{29} + 8 q^{31} + 11 q^{32} + 26 q^{34}+ \cdots - 29 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 + 3 * q^11 - 6 * q^13 - 2 * q^14 + 5 * q^16 + 4 * q^17 - 2 * q^19 + 3 * q^22 + 12 * q^23 + 8 * q^26 - 12 * q^28 + 8 * q^29 + 8 * q^31 + 11 * q^32 + 26 * q^34 - 4 * q^37 + 6 * q^38 + 8 * q^41 - 8 * q^43 + 5 * q^44 + 12 * q^46 + 8 * q^47 + 11 * q^49 + 26 * q^52 - 8 * q^53 - 38 * q^56 + 20 * q^58 + 8 * q^59 + 2 * q^61 + 4 * q^62 + 33 * q^64 - 12 * q^67 + 28 * q^68 + 12 * q^71 - 18 * q^73 + 8 * q^74 - 18 * q^76 - 8 * q^77 - 6 * q^79 + 20 * q^82 - 2 * q^83 - 2 * q^86 + 9 * q^88 - 2 * q^89 + 8 * q^91 + 20 * q^92 - 12 * q^94 + 8 * q^97 - 29 * q^98

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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	0	0
$3$	0	0
$4$	0.369102	0.184551
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.290725	−0.109884	−0.0549418	−	0.998490i	$-0.517497\pi$
$7$	−0.290725	−0.109884	−0.0549418	+	0.998490i	$0.517497\pi$
$8$	−2.51026	−0.887511
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.97107	−1.93343	−0.966714	−	0.255861i	$-0.917641\pi$
$13$	−6.97107	−1.93343	−0.966714	+	0.255861i	$0.917641\pi$
$14$	−0.447480	−0.119594
$15$	0	0
$16$	−4.60197	−1.15049
$17$	4.78765	1.16118	0.580588	−	0.814197i	$-0.302823\pi$
$17$	4.78765	1.16118	0.580588	+	0.814197i	$0.302823\pi$
$18$	0	0
$19$	7.75872	1.77997	0.889987	−	0.455987i	$-0.150714\pi$
$19$	7.75872	1.77997	0.889987	+	0.455987i	$0.150714\pi$
$20$	0	0
$21$	0	0
$22$	1.53919	0.328156
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	−10.7298	−2.10429
$27$	0	0
$28$	−0.107307	−0.0202791
$29$	7.41855	1.37759	0.688795	−	0.724956i	$-0.258140\pi$
$29$	7.41855	1.37759	0.688795	+	0.724956i	$0.258140\pi$
$30$	0	0
$31$	6.34017	1.13873	0.569364	−	0.822085i	$-0.307189\pi$
$31$	6.34017	1.13873	0.569364	+	0.822085i	$0.307189\pi$
$32$	−2.06278	−0.364651
$33$	0	0
$34$	7.36910	1.26379
$35$	0	0
$36$	0	0
$37$	3.41855	0.562006	0.281003	−	0.959707i	$-0.409333\pi$
$37$	3.41855	0.562006	0.281003	+	0.959707i	$0.409333\pi$
$38$	11.9421	1.93727
$39$	0	0
$40$	0	0
$41$	7.41855	1.15858	0.579291	−	0.815120i	$-0.303329\pi$
$41$	7.41855	1.15858	0.579291	+	0.815120i	$0.303329\pi$
$42$	0	0
$43$	−0.290725	−0.0443351	−0.0221675	−	0.999754i	$-0.507057\pi$
$43$	−0.290725	−0.0443351	−0.0221675	+	0.999754i	$0.507057\pi$
$44$	0.369102	0.0556443
$45$	0	0
$46$	6.15676	0.907764
$47$	5.26180	0.767512	0.383756	−	0.923435i	$-0.374630\pi$
$47$	5.26180	0.767512	0.383756	+	0.923435i	$0.374630\pi$
$48$	0	0
$49$	−6.91548	−0.987926
$50$	0	0
$51$	0	0
$52$	−2.57304	−0.356816
$53$	5.75872	0.791022	0.395511	−	0.918461i	$-0.370568\pi$
$53$	5.75872	0.791022	0.395511	+	0.918461i	$0.370568\pi$
$54$	0	0
$55$	0	0
$56$	0.729794	0.0975229
$57$	0	0
$58$	11.4186	1.49933
$59$	−3.60197	−0.468936	−0.234468	−	0.972124i	$-0.575335\pi$
$59$	−3.60197	−0.468936	−0.234468	+	0.972124i	$0.575335\pi$
$60$	0	0
$61$	−6.68035	−0.855331	−0.427665	−	0.903937i	$-0.640664\pi$
$61$	−6.68035	−0.855331	−0.427665	+	0.903937i	$0.640664\pi$
$62$	9.75872	1.23936
$63$	0	0
$64$	6.02893	0.753616
$65$	0	0
$66$	0	0
$67$	−6.15676	−0.752167	−0.376084	−	0.926586i	$-0.622729\pi$
$67$	−6.15676	−0.752167	−0.376084	+	0.926586i	$0.622729\pi$
$68$	1.76713	0.214296
$69$	0	0
$70$	0	0
$71$	5.07838	0.602693	0.301346	−	0.953515i	$-0.402564\pi$
$71$	5.07838	0.602693	0.301346	+	0.953515i	$0.402564\pi$
$72$	0	0
$73$	1.12783	0.132002	0.0660010	−	0.997820i	$-0.478976\pi$
$73$	1.12783	0.132002	0.0660010	+	0.997820i	$0.478976\pi$
$74$	5.26180	0.611671
$75$	0	0
$76$	2.86376	0.328496
$77$	−0.290725	−0.0331311
$78$	0	0
$79$	−0.921622	−0.103691	−0.0518453	−	0.998655i	$-0.516510\pi$
$79$	−0.921622	−0.103691	−0.0518453	+	0.998655i	$0.516510\pi$
$80$	0	0
$81$	0	0
$82$	11.4186	1.26097
$83$	1.70928	0.187617	0.0938087	−	0.995590i	$-0.470096\pi$
$83$	1.70928	0.187617	0.0938087	+	0.995590i	$0.470096\pi$
$84$	0	0
$85$	0	0
$86$	−0.447480	−0.0482530
$87$	0	0
$88$	−2.51026	−0.267595
$89$	−4.34017	−0.460057	−0.230029	−	0.973184i	$-0.573882\pi$
$89$	−4.34017	−0.460057	−0.230029	+	0.973184i	$0.573882\pi$
$90$	0	0
$91$	2.02666	0.212452
$92$	1.47641	0.153926
$93$	0	0
$94$	8.09890	0.835337
$95$	0	0
$96$	0	0
$97$	−4.68035	−0.475217	−0.237609	−	0.971361i	$-0.576364\pi$
$97$	−4.68035	−0.475217	−0.237609	+	0.971361i	$0.576364\pi$
$98$	−10.6442	−1.07523
$99$	0	0
$100$	0	0
$101$	8.58145	0.853886	0.426943	−	0.904279i	$-0.359590\pi$
$101$	8.58145	0.853886	0.426943	+	0.904279i	$0.359590\pi$
$102$	0	0
$103$	−6.73820	−0.663935	−0.331968	−	0.943291i	$-0.607712\pi$
$103$	−6.73820	−0.663935	−0.331968	+	0.943291i	$0.607712\pi$
$104$	17.4992	1.71594
$105$	0	0
$106$	8.86376	0.860925
$107$	12.2329	1.18260	0.591298	−	0.806453i	$-0.298616\pi$
$107$	12.2329	1.18260	0.591298	+	0.806453i	$0.298616\pi$
$108$	0	0
$109$	−6.31351	−0.604725	−0.302362	−	0.953193i	$-0.597775\pi$
$109$	−6.31351	−0.604725	−0.302362	+	0.953193i	$0.597775\pi$
$110$	0	0
$111$	0	0
$112$	1.33791	0.126420
$113$	−16.4969	−1.55190	−0.775950	−	0.630794i	$-0.782729\pi$
$113$	−16.4969	−1.55190	−0.775950	+	0.630794i	$0.782729\pi$
$114$	0	0
$115$	0	0
$116$	2.73820	0.254236
$117$	0	0
$118$	−5.54411	−0.510377
$119$	−1.39189	−0.127594
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.2823	−0.930917
$123$	0	0
$124$	2.34017	0.210154
$125$	0	0
$126$	0	0
$127$	−4.87217	−0.432336	−0.216168	−	0.976356i	$-0.569356\pi$
$127$	−4.87217	−0.432336	−0.216168	+	0.976356i	$0.569356\pi$
$128$	13.4052	1.18487
$129$	0	0
$130$	0	0
$131$	8.68035	0.758405	0.379203	−	0.925314i	$-0.376198\pi$
$131$	8.68035	0.758405	0.379203	+	0.925314i	$0.376198\pi$
$132$	0	0
$133$	−2.25565	−0.195590
$134$	−9.47641	−0.818637
$135$	0	0
$136$	−12.0183	−1.03056
$137$	−12.5958	−1.07613	−0.538067	−	0.842902i	$-0.680845\pi$
$137$	−12.5958	−1.07613	−0.538067	+	0.842902i	$0.680845\pi$
$138$	0	0
$139$	−9.91548	−0.841020	−0.420510	−	0.907288i	$-0.638149\pi$
$139$	−9.91548	−0.841020	−0.420510	+	0.907288i	$0.638149\pi$
$140$	0	0
$141$	0	0
$142$	7.81658	0.655953
$143$	−6.97107	−0.582950
$144$	0	0
$145$	0	0
$146$	1.73594	0.143667
$147$	0	0
$148$	1.26180	0.103719
$149$	1.26180	0.103370	0.0516851	−	0.998663i	$-0.483541\pi$
$149$	1.26180	0.103370	0.0516851	+	0.998663i	$0.483541\pi$
$150$	0	0
$151$	1.60197	0.130366	0.0651832	−	0.997873i	$-0.479237\pi$
$151$	1.60197	0.130366	0.0651832	+	0.997873i	$0.479237\pi$
$152$	−19.4764	−1.57975
$153$	0	0
$154$	−0.447480	−0.0360590
$155$	0	0
$156$	0	0
$157$	7.10504	0.567044	0.283522	−	0.958966i	$-0.408497\pi$
$157$	7.10504	0.567044	0.283522	+	0.958966i	$0.408497\pi$
$158$	−1.41855	−0.112854
$159$	0	0
$160$	0	0
$161$	−1.16290	−0.0916492
$162$	0	0
$163$	22.9360	1.79649	0.898243	−	0.439499i	$-0.144844\pi$
$163$	22.9360	1.79649	0.898243	+	0.439499i	$0.144844\pi$
$164$	2.73820	0.213818
$165$	0	0
$166$	2.63090	0.204197
$167$	4.81432	0.372543	0.186271	−	0.982498i	$-0.440360\pi$
$167$	4.81432	0.372543	0.186271	+	0.982498i	$0.440360\pi$
$168$	0	0
$169$	35.5958	2.73814
$170$	0	0
$171$	0	0
$172$	−0.107307	−0.00818209
$173$	−12.8865	−0.979746	−0.489873	−	0.871794i	$-0.662957\pi$
$173$	−12.8865	−0.979746	−0.489873	+	0.871794i	$0.662957\pi$
$174$	0	0
$175$	0	0
$176$	−4.60197	−0.346886
$177$	0	0
$178$	−6.68035	−0.500713
$179$	−1.84324	−0.137771	−0.0688853	−	0.997625i	$-0.521944\pi$
$179$	−1.84324	−0.137771	−0.0688853	+	0.997625i	$0.521944\pi$
$180$	0	0
$181$	10.2823	0.764278	0.382139	−	0.924105i	$-0.375187\pi$
$181$	10.2823	0.764278	0.382139	+	0.924105i	$0.375187\pi$
$182$	3.11942	0.231226
$183$	0	0
$184$	−10.0410	−0.740235
$185$	0	0
$186$	0	0
$187$	4.78765	0.350108
$188$	1.94214	0.141645
$189$	0	0
$190$	0	0
$191$	11.5174	0.833373	0.416687	−	0.909050i	$-0.363191\pi$
$191$	11.5174	0.833373	0.416687	+	0.909050i	$0.363191\pi$
$192$	0	0
$193$	−3.86603	−0.278283	−0.139141	−	0.990273i	$-0.544434\pi$
$193$	−3.86603	−0.278283	−0.139141	+	0.990273i	$0.544434\pi$
$194$	−7.20394	−0.517212
$195$	0	0
$196$	−2.55252	−0.182323
$197$	−8.57304	−0.610804	−0.305402	−	0.952224i	$-0.598791\pi$
$197$	−8.57304	−0.610804	−0.305402	+	0.952224i	$0.598791\pi$
$198$	0	0
$199$	8.31351	0.589329	0.294665	−	0.955601i	$-0.404792\pi$
$199$	8.31351	0.589329	0.294665	+	0.955601i	$0.404792\pi$
$200$	0	0
$201$	0	0
$202$	13.2085	0.929345
$203$	−2.15676	−0.151375
$204$	0	0
$205$	0	0
$206$	−10.3714	−0.722608
$207$	0	0
$208$	32.0806	2.22439
$209$	7.75872	0.536682
$210$	0	0
$211$	25.9155	1.78410	0.892048	−	0.451941i	$-0.149268\pi$
$211$	25.9155	1.78410	0.892048	+	0.451941i	$0.149268\pi$
$212$	2.12556	0.145984
$213$	0	0
$214$	18.8287	1.28710
$215$	0	0
$216$	0	0
$217$	−1.84324	−0.125128
$218$	−9.71769	−0.658165
$219$	0	0
$220$	0	0
$221$	−33.3751	−2.24505
$222$	0	0
$223$	9.62863	0.644781	0.322390	−	0.946607i	$-0.395514\pi$
$223$	9.62863	0.644781	0.322390	+	0.946607i	$0.395514\pi$
$224$	0.599701	0.0400692
$225$	0	0
$226$	−25.3919	−1.68904
$227$	−18.3896	−1.22056	−0.610281	−	0.792185i	$-0.708943\pi$
$227$	−18.3896	−1.22056	−0.610281	+	0.792185i	$0.708943\pi$
$228$	0	0
$229$	17.1506	1.13334	0.566672	−	0.823943i	$-0.308231\pi$
$229$	17.1506	1.13334	0.566672	+	0.823943i	$0.308231\pi$
$230$	0	0
$231$	0	0
$232$	−18.6225	−1.22263
$233$	−4.10731	−0.269079	−0.134539	−	0.990908i	$-0.542955\pi$
$233$	−4.10731	−0.269079	−0.134539	+	0.990908i	$0.542955\pi$
$234$	0	0
$235$	0	0
$236$	−1.32950	−0.0865428
$237$	0	0
$238$	−2.14238	−0.138870
$239$	4.36683	0.282467	0.141234	−	0.989976i	$-0.454893\pi$
$239$	4.36683	0.282467	0.141234	+	0.989976i	$0.454893\pi$
$240$	0	0
$241$	−5.20394	−0.335215	−0.167608	−	0.985854i	$-0.553604\pi$
$241$	−5.20394	−0.335215	−0.167608	+	0.985854i	$0.553604\pi$
$242$	1.53919	0.0989428
$243$	0	0
$244$	−2.46573	−0.157852
$245$	0	0
$246$	0	0
$247$	−54.0866	−3.44145
$248$	−15.9155	−1.01063
$249$	0	0
$250$	0	0
$251$	8.28231	0.522775	0.261388	−	0.965234i	$-0.415820\pi$
$251$	8.28231	0.522775	0.261388	+	0.965234i	$0.415820\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	−7.49920	−0.470541
$255$	0	0
$256$	8.57531	0.535957
$257$	11.0205	0.687441	0.343721	−	0.939072i	$-0.388313\pi$
$257$	11.0205	0.687441	0.343721	+	0.939072i	$0.388313\pi$
$258$	0	0
$259$	−0.993857	−0.0617553
$260$	0	0
$261$	0	0
$262$	13.3607	0.825426
$263$	−2.97107	−0.183204	−0.0916020	−	0.995796i	$-0.529199\pi$
$263$	−2.97107	−0.183204	−0.0916020	+	0.995796i	$0.529199\pi$
$264$	0	0
$265$	0	0
$266$	−3.47187	−0.212874
$267$	0	0
$268$	−2.27247	−0.138813
$269$	15.8432	0.965980	0.482990	−	0.875626i	$-0.339551\pi$
$269$	15.8432	0.965980	0.482990	+	0.875626i	$0.339551\pi$
$270$	0	0
$271$	−6.28231	−0.381623	−0.190812	−	0.981627i	$-0.561112\pi$
$271$	−6.28231	−0.381623	−0.190812	+	0.981627i	$0.561112\pi$
$272$	−22.0326	−1.33592
$273$	0	0
$274$	−19.3874	−1.17123
$275$	0	0
$276$	0	0
$277$	−5.12783	−0.308101	−0.154051	−	0.988063i	$-0.549232\pi$
$277$	−5.12783	−0.308101	−0.154051	+	0.988063i	$0.549232\pi$
$278$	−15.2618	−0.915342
$279$	0	0
$280$	0	0
$281$	−21.8888	−1.30578	−0.652889	−	0.757454i	$-0.726443\pi$
$281$	−21.8888	−1.30578	−0.652889	+	0.757454i	$0.726443\pi$
$282$	0	0
$283$	−25.9649	−1.54345	−0.771727	−	0.635954i	$-0.780607\pi$
$283$	−25.9649	−1.54345	−0.771727	+	0.635954i	$0.780607\pi$
$284$	1.87444	0.111228
$285$	0	0
$286$	−10.7298	−0.634466
$287$	−2.15676	−0.127309
$288$	0	0
$289$	5.92162	0.348331
$290$	0	0
$291$	0	0
$292$	0.416283	0.0243611
$293$	10.4163	0.608526	0.304263	−	0.952588i	$-0.401590\pi$
$293$	10.4163	0.608526	0.304263	+	0.952588i	$0.401590\pi$
$294$	0	0
$295$	0	0
$296$	−8.58145	−0.498787
$297$	0	0
$298$	1.94214	0.112505
$299$	−27.8843	−1.61259
$300$	0	0
$301$	0.0845208	0.00487170
$302$	2.46573	0.141887
$303$	0	0
$304$	−35.7054	−2.04785
$305$	0	0
$306$	0	0
$307$	29.0700	1.65911	0.829555	−	0.558425i	$-0.188594\pi$
$307$	29.0700	1.65911	0.829555	+	0.558425i	$0.188594\pi$
$308$	−0.107307	−0.00611439
$309$	0	0
$310$	0	0
$311$	−5.44521	−0.308770	−0.154385	−	0.988011i	$-0.549340\pi$
$311$	−5.44521	−0.308770	−0.154385	+	0.988011i	$0.549340\pi$
$312$	0	0
$313$	−25.0928	−1.41833	−0.709163	−	0.705044i	$-0.750927\pi$
$313$	−25.0928	−1.41833	−0.709163	+	0.705044i	$0.750927\pi$
$314$	10.9360	0.617154
$315$	0	0
$316$	−0.340173	−0.0191362
$317$	21.1773	1.18943	0.594717	−	0.803935i	$-0.297264\pi$
$317$	21.1773	1.18943	0.594717	+	0.803935i	$0.297264\pi$
$318$	0	0
$319$	7.41855	0.415359
$320$	0	0
$321$	0	0
$322$	−1.78992	−0.0997484
$323$	37.1461	2.06686
$324$	0	0
$325$	0	0
$326$	35.3028	1.95524
$327$	0	0
$328$	−18.6225	−1.02825
$329$	−1.52973	−0.0843369
$330$	0	0
$331$	−6.70701	−0.368650	−0.184325	−	0.982865i	$-0.559010\pi$
$331$	−6.70701	−0.368650	−0.184325	+	0.982865i	$0.559010\pi$
$332$	0.630898	0.0346250
$333$	0	0
$334$	7.41014	0.405465
$335$	0	0
$336$	0	0
$337$	−23.7503	−1.29376	−0.646881	−	0.762591i	$-0.723927\pi$
$337$	−23.7503	−1.29376	−0.646881	+	0.762591i	$0.723927\pi$
$338$	54.7887	2.98011
$339$	0	0
$340$	0	0
$341$	6.34017	0.343340
$342$	0	0
$343$	4.04557	0.218440
$344$	0.729794	0.0393479
$345$	0	0
$346$	−19.8348	−1.06633
$347$	−31.1689	−1.67323	−0.836616	−	0.547790i	$-0.815469\pi$
$347$	−31.1689	−1.67323	−0.836616	+	0.547790i	$0.815469\pi$
$348$	0	0
$349$	11.0472	0.591342	0.295671	−	0.955290i	$-0.404457\pi$
$349$	11.0472	0.591342	0.295671	+	0.955290i	$0.404457\pi$
$350$	0	0
$351$	0	0
$352$	−2.06278	−0.109947
$353$	−27.4329	−1.46011	−0.730054	−	0.683390i	$-0.760505\pi$
$353$	−27.4329	−1.46011	−0.730054	+	0.683390i	$0.760505\pi$
$354$	0	0
$355$	0	0
$356$	−1.60197	−0.0849041
$357$	0	0
$358$	−2.83710	−0.149945
$359$	19.1506	1.01073	0.505365	−	0.862905i	$-0.331358\pi$
$359$	19.1506	1.01073	0.505365	+	0.862905i	$0.331358\pi$
$360$	0	0
$361$	41.1978	2.16830
$362$	15.8264	0.831818
$363$	0	0
$364$	0.748046	0.0392083
$365$	0	0
$366$	0	0
$367$	14.5692	0.760504	0.380252	−	0.924883i	$-0.375837\pi$
$367$	14.5692	0.760504	0.380252	+	0.924883i	$0.375837\pi$
$368$	−18.4079	−0.959577
$369$	0	0
$370$	0	0
$371$	−1.67420	−0.0869203
$372$	0	0
$373$	8.81432	0.456388	0.228194	−	0.973616i	$-0.426718\pi$
$373$	8.81432	0.456388	0.228194	+	0.973616i	$0.426718\pi$
$374$	7.36910	0.381047
$375$	0	0
$376$	−13.2085	−0.681175
$377$	−51.7152	−2.66347
$378$	0	0
$379$	−23.5174	−1.20801	−0.604005	−	0.796980i	$-0.706429\pi$
$379$	−23.5174	−1.20801	−0.604005	+	0.796980i	$0.706429\pi$
$380$	0	0
$381$	0	0
$382$	17.7275	0.907019
$383$	−19.3028	−0.986329	−0.493164	−	0.869936i	$-0.664160\pi$
$383$	−19.3028	−0.986329	−0.493164	+	0.869936i	$0.664160\pi$
$384$	0	0
$385$	0	0
$386$	−5.95055	−0.302875
$387$	0	0
$388$	−1.72753	−0.0877019
$389$	20.0410	1.01612	0.508060	−	0.861321i	$-0.330363\pi$
$389$	20.0410	1.01612	0.508060	+	0.861321i	$0.330363\pi$
$390$	0	0
$391$	19.1506	0.968488
$392$	17.3596	0.876795
$393$	0	0
$394$	−13.1955	−0.664781
$395$	0	0
$396$	0	0
$397$	33.3607	1.67433	0.837163	−	0.546954i	$-0.184213\pi$
$397$	33.3607	1.67433	0.837163	+	0.546954i	$0.184213\pi$
$398$	12.7961	0.641409
$399$	0	0
$400$	0	0
$401$	37.6475	1.88003	0.940014	−	0.341135i	$-0.110811\pi$
$401$	37.6475	1.88003	0.940014	+	0.341135i	$0.110811\pi$
$402$	0	0
$403$	−44.1978	−2.20165
$404$	3.16743	0.157586
$405$	0	0
$406$	−3.31965	−0.164752
$407$	3.41855	0.169451
$408$	0	0
$409$	−25.2039	−1.24625	−0.623127	−	0.782120i	$-0.714138\pi$
$409$	−25.2039	−1.24625	−0.623127	+	0.782120i	$0.714138\pi$
$410$	0	0
$411$	0	0
$412$	−2.48709	−0.122530
$413$	1.04718	0.0515284
$414$	0	0
$415$	0	0
$416$	14.3798	0.705027
$417$	0	0
$418$	11.9421	0.584109
$419$	−17.8432	−0.871700	−0.435850	−	0.900019i	$-0.643552\pi$
$419$	−17.8432	−0.871700	−0.435850	+	0.900019i	$0.643552\pi$
$420$	0	0
$421$	−11.8120	−0.575684	−0.287842	−	0.957678i	$-0.592938\pi$
$421$	−11.8120	−0.575684	−0.287842	+	0.957678i	$0.592938\pi$
$422$	39.8888	1.94176
$423$	0	0
$424$	−14.4559	−0.702040
$425$	0	0
$426$	0	0
$427$	1.94214	0.0939868
$428$	4.51518	0.218249
$429$	0	0
$430$	0	0
$431$	−16.6803	−0.803464	−0.401732	−	0.915757i	$-0.631592\pi$
$431$	−16.6803	−0.803464	−0.401732	+	0.915757i	$0.631592\pi$
$432$	0	0
$433$	28.3135	1.36066	0.680330	−	0.732906i	$-0.261836\pi$
$433$	28.3135	1.36066	0.680330	+	0.732906i	$0.261836\pi$
$434$	−2.83710	−0.136185
$435$	0	0
$436$	−2.33033	−0.111603
$437$	31.0349	1.48460
$438$	0	0
$439$	−27.4452	−1.30989	−0.654944	−	0.755677i	$-0.727308\pi$
$439$	−27.4452	−1.30989	−0.654944	+	0.755677i	$0.727308\pi$
$440$	0	0
$441$	0	0
$442$	−51.3705	−2.44345
$443$	0.412408	0.0195941	0.00979704	−	0.999952i	$-0.496881\pi$
$443$	0.412408	0.0195941	0.00979704	+	0.999952i	$0.496881\pi$
$444$	0	0
$445$	0	0
$446$	14.8203	0.701761
$447$	0	0
$448$	−1.75276	−0.0828100
$449$	−1.33403	−0.0629568	−0.0314784	−	0.999504i	$-0.510022\pi$
$449$	−1.33403	−0.0629568	−0.0314784	+	0.999504i	$0.510022\pi$
$450$	0	0
$451$	7.41855	0.349326
$452$	−6.08906	−0.286405
$453$	0	0
$454$	−28.3051	−1.32842
$455$	0	0
$456$	0	0
$457$	−11.6514	−0.545030	−0.272515	−	0.962151i	$-0.587855\pi$
$457$	−11.6514	−0.545030	−0.272515	+	0.962151i	$0.587855\pi$
$458$	26.3980	1.23350
$459$	0	0
$460$	0	0
$461$	2.05786	0.0958440	0.0479220	−	0.998851i	$-0.484740\pi$
$461$	2.05786	0.0958440	0.0479220	+	0.998851i	$0.484740\pi$
$462$	0	0
$463$	−28.7792	−1.33748	−0.668742	−	0.743494i	$-0.733167\pi$
$463$	−28.7792	−1.33748	−0.668742	+	0.743494i	$0.733167\pi$
$464$	−34.1399	−1.58491
$465$	0	0
$466$	−6.32192	−0.292857
$467$	1.84324	0.0852952	0.0426476	−	0.999090i	$-0.486421\pi$
$467$	1.84324	0.0852952	0.0426476	+	0.999090i	$0.486421\pi$
$468$	0	0
$469$	1.78992	0.0826509
$470$	0	0
$471$	0	0
$472$	9.04187	0.416186
$473$	−0.290725	−0.0133675
$474$	0	0
$475$	0	0
$476$	−0.513749	−0.0235477
$477$	0	0
$478$	6.72138	0.307429
$479$	−26.8371	−1.22622	−0.613109	−	0.789998i	$-0.710081\pi$
$479$	−26.8371	−1.22622	−0.613109	+	0.789998i	$0.710081\pi$
$480$	0	0
$481$	−23.8310	−1.08660
$482$	−8.00984	−0.364838
$483$	0	0
$484$	0.369102	0.0167774
$485$	0	0
$486$	0	0
$487$	−28.5646	−1.29439	−0.647193	−	0.762326i	$-0.724057\pi$
$487$	−28.5646	−1.29439	−0.647193	+	0.762326i	$0.724057\pi$
$488$	16.7694	0.759115
$489$	0	0
$490$	0	0
$491$	−25.9877	−1.17281	−0.586405	−	0.810018i	$-0.699457\pi$
$491$	−25.9877	−1.17281	−0.586405	+	0.810018i	$0.699457\pi$
$492$	0	0
$493$	35.5174	1.59963
$494$	−83.2495	−3.74557
$495$	0	0
$496$	−29.1773	−1.31010
$497$	−1.47641	−0.0662260
$498$	0	0
$499$	−27.5174	−1.23185	−0.615925	−	0.787805i	$-0.711218\pi$
$499$	−27.5174	−1.23185	−0.615925	+	0.787805i	$0.711218\pi$
$500$	0	0
$501$	0	0
$502$	12.7480	0.568973
$503$	22.6576	1.01025	0.505125	−	0.863046i	$-0.331446\pi$
$503$	22.6576	1.01025	0.505125	+	0.863046i	$0.331446\pi$
$504$	0	0
$505$	0	0
$506$	6.15676	0.273701
$507$	0	0
$508$	−1.79833	−0.0797880
$509$	27.8432	1.23413	0.617065	−	0.786912i	$-0.288322\pi$
$509$	27.8432	1.23413	0.617065	+	0.786912i	$0.288322\pi$
$510$	0	0
$511$	−0.327887	−0.0145049
$512$	−13.6114	−0.601546
$513$	0	0
$514$	16.9627	0.748191
$515$	0	0
$516$	0	0
$517$	5.26180	0.231413
$518$	−1.52973	−0.0672126
$519$	0	0
$520$	0	0
$521$	−29.7009	−1.30122	−0.650609	−	0.759413i	$-0.725486\pi$
$521$	−29.7009	−1.30122	−0.650609	+	0.759413i	$0.725486\pi$
$522$	0	0
$523$	−24.7565	−1.08252	−0.541262	−	0.840854i	$-0.682053\pi$
$523$	−24.7565	−1.08252	−0.541262	+	0.840854i	$0.682053\pi$
$524$	3.20394	0.139965
$525$	0	0
$526$	−4.57304	−0.199394
$527$	30.3545	1.32226
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−0.832567	−0.0360963
$533$	−51.7152	−2.24004
$534$	0	0
$535$	0	0
$536$	15.4551	0.667557
$537$	0	0
$538$	24.3857	1.05134
$539$	−6.91548	−0.297871
$540$	0	0
$541$	−12.5236	−0.538431	−0.269216	−	0.963080i	$-0.586764\pi$
$541$	−12.5236	−0.538431	−0.269216	+	0.963080i	$0.586764\pi$
$542$	−9.66967	−0.415348
$543$	0	0
$544$	−9.87587	−0.423425
$545$	0	0
$546$	0	0
$547$	14.2784	0.610502	0.305251	−	0.952272i	$-0.401260\pi$
$547$	14.2784	0.610502	0.305251	+	0.952272i	$0.401260\pi$
$548$	−4.64915	−0.198602
$549$	0	0
$550$	0	0
$551$	57.5585	2.45207
$552$	0	0
$553$	0.267938	0.0113939
$554$	−7.89269	−0.335328
$555$	0	0
$556$	−3.65983	−0.155211
$557$	−3.57918	−0.151655	−0.0758274	−	0.997121i	$-0.524160\pi$
$557$	−3.57918	−0.151655	−0.0758274	+	0.997121i	$0.524160\pi$
$558$	0	0
$559$	2.02666	0.0857187
$560$	0	0
$561$	0	0
$562$	−33.6910	−1.42117
$563$	28.9588	1.22047	0.610234	−	0.792222i	$-0.291076\pi$
$563$	28.9588	1.22047	0.610234	+	0.792222i	$0.291076\pi$
$564$	0	0
$565$	0	0
$566$	−39.9649	−1.67985
$567$	0	0
$568$	−12.7480	−0.534896
$569$	25.8264	1.08270	0.541350	−	0.840797i	$-0.317913\pi$
$569$	25.8264	1.08270	0.541350	+	0.840797i	$0.317913\pi$
$570$	0	0
$571$	17.1194	0.716425	0.358213	−	0.933640i	$-0.383386\pi$
$571$	17.1194	0.716425	0.358213	+	0.933640i	$0.383386\pi$
$572$	−2.57304	−0.107584
$573$	0	0
$574$	−3.31965	−0.138560
$575$	0	0
$576$	0	0
$577$	−22.5692	−0.939567	−0.469783	−	0.882782i	$-0.655668\pi$
$577$	−22.5692	−0.939567	−0.469783	+	0.882782i	$0.655668\pi$
$578$	9.11450	0.379113
$579$	0	0
$580$	0	0
$581$	−0.496928	−0.0206161
$582$	0	0
$583$	5.75872	0.238502
$584$	−2.83114	−0.117153
$585$	0	0
$586$	16.0326	0.662302
$587$	3.63317	0.149957	0.0749784	−	0.997185i	$-0.476111\pi$
$587$	3.63317	0.149957	0.0749784	+	0.997185i	$0.476111\pi$
$588$	0	0
$589$	49.1917	2.02691
$590$	0	0
$591$	0	0
$592$	−15.7321	−0.646584
$593$	12.3051	0.505310	0.252655	−	0.967556i	$-0.418696\pi$
$593$	12.3051	0.505310	0.252655	+	0.967556i	$0.418696\pi$
$594$	0	0
$595$	0	0
$596$	0.465732	0.0190771
$597$	0	0
$598$	−42.9192	−1.75510
$599$	−19.6865	−0.804368	−0.402184	−	0.915559i	$-0.631749\pi$
$599$	−19.6865	−0.804368	−0.402184	+	0.915559i	$0.631749\pi$
$600$	0	0
$601$	25.8843	1.05584	0.527921	−	0.849294i	$-0.322972\pi$
$601$	25.8843	1.05584	0.527921	+	0.849294i	$0.322972\pi$
$602$	0.130094	0.00530222
$603$	0	0
$604$	0.591290	0.0240593
$605$	0	0
$606$	0	0
$607$	41.5357	1.68588	0.842941	−	0.538006i	$-0.180822\pi$
$607$	41.5357	1.68588	0.842941	+	0.538006i	$0.180822\pi$
$608$	−16.0045	−0.649070
$609$	0	0
$610$	0	0
$611$	−36.6803	−1.48393
$612$	0	0
$613$	20.6453	0.833855	0.416927	−	0.908940i	$-0.363107\pi$
$613$	20.6453	0.833855	0.416927	+	0.908940i	$0.363107\pi$
$614$	44.7442	1.80573
$615$	0	0
$616$	0.729794	0.0294042
$617$	8.69472	0.350036	0.175018	−	0.984565i	$-0.444002\pi$
$617$	8.69472	0.350036	0.175018	+	0.984565i	$0.444002\pi$
$618$	0	0
$619$	2.65368	0.106661	0.0533303	−	0.998577i	$-0.483016\pi$
$619$	2.65368	0.106661	0.0533303	+	0.998577i	$0.483016\pi$
$620$	0	0
$621$	0	0
$622$	−8.38121	−0.336056
$623$	1.26180	0.0505528
$624$	0	0
$625$	0	0
$626$	−38.6225	−1.54367
$627$	0	0
$628$	2.62249	0.104649
$629$	16.3668	0.652588
$630$	0	0
$631$	−10.6393	−0.423544	−0.211772	−	0.977319i	$-0.567923\pi$
$631$	−10.6393	−0.423544	−0.211772	+	0.977319i	$0.567923\pi$
$632$	2.31351	0.0920265
$633$	0	0
$634$	32.5958	1.29455
$635$	0	0
$636$	0	0
$637$	48.2083	1.91008
$638$	11.4186	0.452065
$639$	0	0
$640$	0	0
$641$	35.8576	1.41629	0.708145	−	0.706067i	$-0.249532\pi$
$641$	35.8576	1.41629	0.708145	+	0.706067i	$0.249532\pi$
$642$	0	0
$643$	−12.2146	−0.481697	−0.240849	−	0.970563i	$-0.577426\pi$
$643$	−12.2146	−0.481697	−0.240849	+	0.970563i	$0.577426\pi$
$644$	−0.429229	−0.0169140
$645$	0	0
$646$	57.1748	2.24951
$647$	25.9421	1.01989	0.509945	−	0.860207i	$-0.329666\pi$
$647$	25.9421	1.01989	0.509945	+	0.860207i	$0.329666\pi$
$648$	0	0
$649$	−3.60197	−0.141390
$650$	0	0
$651$	0	0
$652$	8.46573	0.331544
$653$	−14.3402	−0.561174	−0.280587	−	0.959829i	$-0.590529\pi$
$653$	−14.3402	−0.561174	−0.280587	+	0.959829i	$0.590529\pi$
$654$	0	0
$655$	0	0
$656$	−34.1399	−1.33294
$657$	0	0
$658$	−2.35455	−0.0917899
$659$	−6.52359	−0.254123	−0.127062	−	0.991895i	$-0.540555\pi$
$659$	−6.52359	−0.254123	−0.127062	+	0.991895i	$0.540555\pi$
$660$	0	0
$661$	3.16290	0.123022	0.0615112	−	0.998106i	$-0.480408\pi$
$661$	3.16290	0.123022	0.0615112	+	0.998106i	$0.480408\pi$
$662$	−10.3234	−0.401228
$663$	0	0
$664$	−4.29072	−0.166512
$665$	0	0
$666$	0	0
$667$	29.6742	1.14899
$668$	1.77698	0.0687532
$669$	0	0
$670$	0	0
$671$	−6.68035	−0.257892
$672$	0	0
$673$	18.4885	0.712680	0.356340	−	0.934356i	$-0.384024\pi$
$673$	18.4885	0.712680	0.356340	+	0.934356i	$0.384024\pi$
$674$	−36.5562	−1.40809
$675$	0	0
$676$	13.1385	0.505327
$677$	15.0966	0.580211	0.290105	−	0.956995i	$-0.406310\pi$
$677$	15.0966	0.580211	0.290105	+	0.956995i	$0.406310\pi$
$678$	0	0
$679$	1.36069	0.0522186
$680$	0	0
$681$	0	0
$682$	9.75872	0.373681
$683$	−12.4657	−0.476988	−0.238494	−	0.971144i	$-0.576654\pi$
$683$	−12.4657	−0.476988	−0.238494	+	0.971144i	$0.576654\pi$
$684$	0	0
$685$	0	0
$686$	6.22690	0.237744
$687$	0	0
$688$	1.33791	0.0510072
$689$	−40.1445	−1.52938
$690$	0	0
$691$	−30.7214	−1.16870	−0.584348	−	0.811503i	$-0.698650\pi$
$691$	−30.7214	−1.16870	−0.584348	+	0.811503i	$0.698650\pi$
$692$	−4.75646	−0.180813
$693$	0	0
$694$	−47.9748	−1.82110
$695$	0	0
$696$	0	0
$697$	35.5174	1.34532
$698$	17.0037	0.643599
$699$	0	0
$700$	0	0
$701$	−17.9955	−0.679679	−0.339840	−	0.940483i	$-0.610373\pi$
$701$	−17.9955	−0.679679	−0.339840	+	0.940483i	$0.610373\pi$
$702$	0	0
$703$	26.5236	1.00036
$704$	6.02893	0.227224
$705$	0	0
$706$	−42.2245	−1.58914
$707$	−2.49484	−0.0938281
$708$	0	0
$709$	23.5897	0.885929	0.442965	−	0.896539i	$-0.353927\pi$
$709$	23.5897	0.885929	0.442965	+	0.896539i	$0.353927\pi$
$710$	0	0
$711$	0	0
$712$	10.8950	0.408306
$713$	25.3607	0.949765
$714$	0	0
$715$	0	0
$716$	−0.680346	−0.0254257
$717$	0	0
$718$	29.4764	1.10005
$719$	−24.5646	−0.916106	−0.458053	−	0.888925i	$-0.651453\pi$
$719$	−24.5646	−0.916106	−0.458053	+	0.888925i	$0.651453\pi$
$720$	0	0
$721$	1.95896	0.0729556
$722$	63.4112	2.35992
$723$	0	0
$724$	3.79523	0.141048
$725$	0	0
$726$	0	0
$727$	−8.51130	−0.315667	−0.157833	−	0.987466i	$-0.550451\pi$
$727$	−8.51130	−0.315667	−0.157833	+	0.987466i	$0.550451\pi$
$728$	−5.08745	−0.188553
$729$	0	0
$730$	0	0
$731$	−1.39189	−0.0514809
$732$	0	0
$733$	29.8615	1.10296	0.551480	−	0.834188i	$-0.314063\pi$
$733$	29.8615	1.10296	0.551480	+	0.834188i	$0.314063\pi$
$734$	22.4247	0.827711
$735$	0	0
$736$	−8.25112	−0.304140
$737$	−6.15676	−0.226787
$738$	0	0
$739$	−36.7526	−1.35197	−0.675983	−	0.736917i	$-0.736281\pi$
$739$	−36.7526	−1.35197	−0.675983	+	0.736917i	$0.736281\pi$
$740$	0	0
$741$	0	0
$742$	−2.57691	−0.0946015
$743$	−2.17501	−0.0797933	−0.0398966	−	0.999204i	$-0.512703\pi$
$743$	−2.17501	−0.0797933	−0.0398966	+	0.999204i	$0.512703\pi$
$744$	0	0
$745$	0	0
$746$	13.5669	0.496719
$747$	0	0
$748$	1.76713	0.0646128
$749$	−3.55640	−0.129948
$750$	0	0
$751$	−44.4580	−1.62229	−0.811147	−	0.584842i	$-0.801157\pi$
$751$	−44.4580	−1.62229	−0.811147	+	0.584842i	$0.801157\pi$
$752$	−24.2146	−0.883016
$753$	0	0
$754$	−79.5995	−2.89884
$755$	0	0
$756$	0	0
$757$	26.4247	0.960422	0.480211	−	0.877153i	$-0.340560\pi$
$757$	26.4247	0.960422	0.480211	+	0.877153i	$0.340560\pi$
$758$	−36.1978	−1.31476
$759$	0	0
$760$	0	0
$761$	27.6163	1.00109	0.500546	−	0.865710i	$-0.333133\pi$
$761$	27.6163	1.00109	0.500546	+	0.865710i	$0.333133\pi$
$762$	0	0
$763$	1.83549	0.0664493
$764$	4.25112	0.153800
$765$	0	0
$766$	−29.7107	−1.07349
$767$	25.1096	0.906654
$768$	0	0
$769$	27.8432	1.00405	0.502027	−	0.864852i	$-0.332588\pi$
$769$	27.8432	1.00405	0.502027	+	0.864852i	$0.332588\pi$
$770$	0	0
$771$	0	0
$772$	−1.42696	−0.0513575
$773$	−18.0267	−0.648374	−0.324187	−	0.945993i	$-0.605091\pi$
$773$	−18.0267	−0.648374	−0.324187	+	0.945993i	$0.605091\pi$
$774$	0	0
$775$	0	0
$776$	11.7489	0.421760
$777$	0	0
$778$	30.8469	1.10592
$779$	57.5585	2.06225
$780$	0	0
$781$	5.07838	0.181719
$782$	29.4764	1.05407
$783$	0	0
$784$	31.8248	1.13660
$785$	0	0
$786$	0	0
$787$	−35.6391	−1.27040	−0.635199	−	0.772349i	$-0.719082\pi$
$787$	−35.6391	−1.27040	−0.635199	+	0.772349i	$0.719082\pi$
$788$	−3.16433	−0.112725
$789$	0	0
$790$	0	0
$791$	4.79606	0.170528
$792$	0	0
$793$	46.5692	1.65372
$794$	51.3484	1.82229
$795$	0	0
$796$	3.06854	0.108761
$797$	−4.97948	−0.176382	−0.0881911	−	0.996104i	$-0.528109\pi$
$797$	−4.97948	−0.176382	−0.0881911	+	0.996104i	$0.528109\pi$
$798$	0	0
$799$	25.1917	0.891217
$800$	0	0
$801$	0	0
$802$	57.9467	2.04617
$803$	1.12783	0.0398001
$804$	0	0
$805$	0	0
$806$	−68.0288	−2.39621
$807$	0	0
$808$	−21.5417	−0.757833
$809$	−31.6697	−1.11345	−0.556723	−	0.830698i	$-0.687942\pi$
$809$	−31.6697	−1.11345	−0.556723	+	0.830698i	$0.687942\pi$
$810$	0	0
$811$	−21.7998	−0.765493	−0.382747	−	0.923853i	$-0.625022\pi$
$811$	−21.7998	−0.765493	−0.382747	+	0.923853i	$0.625022\pi$
$812$	−0.796064	−0.0279364
$813$	0	0
$814$	5.26180	0.184426
$815$	0	0
$816$	0	0
$817$	−2.25565	−0.0789153
$818$	−38.7936	−1.35639
$819$	0	0
$820$	0	0
$821$	10.9360	0.381669	0.190834	−	0.981622i	$-0.438881\pi$
$821$	10.9360	0.381669	0.190834	+	0.981622i	$0.438881\pi$
$822$	0	0
$823$	−45.9709	−1.60244	−0.801222	−	0.598367i	$-0.795817\pi$
$823$	−45.9709	−1.60244	−0.801222	+	0.598367i	$0.795817\pi$
$824$	16.9146	0.589249
$825$	0	0
$826$	1.61181	0.0560820
$827$	−22.9711	−0.798782	−0.399391	−	0.916781i	$-0.630778\pi$
$827$	−22.9711	−0.798782	−0.399391	+	0.916781i	$0.630778\pi$
$828$	0	0
$829$	−48.1543	−1.67247	−0.836234	−	0.548373i	$-0.815248\pi$
$829$	−48.1543	−1.67247	−0.836234	+	0.548373i	$0.815248\pi$
$830$	0	0
$831$	0	0
$832$	−42.0281	−1.45706
$833$	−33.1089	−1.14716
$834$	0	0
$835$	0	0
$836$	2.86376	0.0990453
$837$	0	0
$838$	−27.4641	−0.948732
$839$	17.9278	0.618935	0.309468	−	0.950910i	$-0.399849\pi$
$839$	17.9278	0.618935	0.309468	+	0.950910i	$0.399849\pi$
$840$	0	0
$841$	26.0349	0.897755
$842$	−18.1810	−0.626558
$843$	0	0
$844$	9.56547	0.329257
$845$	0	0
$846$	0	0
$847$	−0.290725	−0.00998942
$848$	−26.5015	−0.910064
$849$	0	0
$850$	0	0
$851$	13.6742	0.468746
$852$	0	0
$853$	9.54799	0.326917	0.163458	−	0.986550i	$-0.447735\pi$
$853$	9.54799	0.326917	0.163458	+	0.986550i	$0.447735\pi$
$854$	2.98932	0.102292
$855$	0	0
$856$	−30.7077	−1.04957
$857$	10.5776	0.361323	0.180662	−	0.983545i	$-0.442176\pi$
$857$	10.5776	0.361323	0.180662	+	0.983545i	$0.442176\pi$
$858$	0	0
$859$	−31.3340	−1.06910	−0.534552	−	0.845136i	$-0.679520\pi$
$859$	−31.3340	−1.06910	−0.534552	+	0.845136i	$0.679520\pi$
$860$	0	0
$861$	0	0
$862$	−25.6742	−0.874467
$863$	56.7792	1.93279	0.966394	−	0.257066i	$-0.0827557\pi$
$863$	56.7792	1.93279	0.966394	+	0.257066i	$0.0827557\pi$
$864$	0	0
$865$	0	0
$866$	43.5798	1.48090
$867$	0	0
$868$	−0.680346	−0.0230924
$869$	−0.921622	−0.0312639
$870$	0	0
$871$	42.9192	1.45426
$872$	15.8486	0.536700
$873$	0	0
$874$	47.7686	1.61580
$875$	0	0
$876$	0	0
$877$	−5.87832	−0.198497	−0.0992483	−	0.995063i	$-0.531644\pi$
$877$	−5.87832	−0.198497	−0.0992483	+	0.995063i	$0.531644\pi$
$878$	−42.2434	−1.42564
$879$	0	0
$880$	0	0
$881$	−8.21008	−0.276605	−0.138302	−	0.990390i	$-0.544165\pi$
$881$	−8.21008	−0.276605	−0.138302	+	0.990390i	$0.544165\pi$
$882$	0	0
$883$	27.0349	0.909797	0.454898	−	0.890543i	$-0.349676\pi$
$883$	27.0349	0.909797	0.454898	+	0.890543i	$0.349676\pi$
$884$	−12.3188	−0.414327
$885$	0	0
$886$	0.634773	0.0213256
$887$	−48.7442	−1.63667	−0.818335	−	0.574742i	$-0.805102\pi$
$887$	−48.7442	−1.63667	−0.818335	+	0.574742i	$0.805102\pi$
$888$	0	0
$889$	1.41646	0.0475066
$890$	0	0
$891$	0	0
$892$	3.55395	0.118995
$893$	40.8248	1.36615
$894$	0	0
$895$	0	0
$896$	−3.89723	−0.130197
$897$	0	0
$898$	−2.05332	−0.0685203
$899$	47.0349	1.56870
$900$	0	0
$901$	27.5708	0.918516
$902$	11.4186	0.380196
$903$	0	0
$904$	41.4116	1.37733
$905$	0	0
$906$	0	0
$907$	32.4657	1.07801	0.539003	−	0.842304i	$-0.318801\pi$
$907$	32.4657	1.07801	0.539003	+	0.842304i	$0.318801\pi$
$908$	−6.78765	−0.225256
$909$	0	0
$910$	0	0
$911$	4.84939	0.160667	0.0803337	−	0.996768i	$-0.474401\pi$
$911$	4.84939	0.160667	0.0803337	+	0.996768i	$0.474401\pi$
$912$	0	0
$913$	1.70928	0.0565688
$914$	−17.9337	−0.593195
$915$	0	0
$916$	6.33033	0.209160
$917$	−2.52359	−0.0833363
$918$	0	0
$919$	−14.3980	−0.474947	−0.237474	−	0.971394i	$-0.576319\pi$
$919$	−14.3980	−0.474947	−0.237474	+	0.971394i	$0.576319\pi$
$920$	0	0
$921$	0	0
$922$	3.16743	0.104314
$923$	−35.4017	−1.16526
$924$	0	0
$925$	0	0
$926$	−44.2967	−1.45568
$927$	0	0
$928$	−15.3028	−0.502340
$929$	11.1629	0.366243	0.183121	−	0.983090i	$-0.441380\pi$
$929$	11.1629	0.366243	0.183121	+	0.983090i	$0.441380\pi$
$930$	0	0
$931$	−53.6553	−1.75848
$932$	−1.51602	−0.0496588
$933$	0	0
$934$	2.83710	0.0928328
$935$	0	0
$936$	0	0
$937$	−48.6453	−1.58917	−0.794586	−	0.607152i	$-0.792312\pi$
$937$	−48.6453	−1.58917	−0.794586	+	0.607152i	$0.792312\pi$
$938$	2.75503	0.0899548
$939$	0	0
$940$	0	0
$941$	−16.4124	−0.535029	−0.267515	−	0.963554i	$-0.586202\pi$
$941$	−16.4124	−0.535029	−0.267515	+	0.963554i	$0.586202\pi$
$942$	0	0
$943$	29.6742	0.966325
$944$	16.5761	0.539508
$945$	0	0
$946$	−0.447480	−0.0145488
$947$	−45.7198	−1.48569	−0.742847	−	0.669462i	$-0.766525\pi$
$947$	−45.7198	−1.48569	−0.742847	+	0.669462i	$0.766525\pi$
$948$	0	0
$949$	−7.86216	−0.255216
$950$	0	0
$951$	0	0
$952$	3.49400	0.113241
$953$	−26.4619	−0.857184	−0.428592	−	0.903498i	$-0.640990\pi$
$953$	−26.4619	−0.857184	−0.428592	+	0.903498i	$0.640990\pi$
$954$	0	0
$955$	0	0
$956$	1.61181	0.0521296
$957$	0	0
$958$	−41.3074	−1.33458
$959$	3.66192	0.118249
$960$	0	0
$961$	9.19779	0.296703
$962$	−36.6803	−1.18262
$963$	0	0
$964$	−1.92079	−0.0618643
$965$	0	0
$966$	0	0
$967$	−50.5464	−1.62546	−0.812731	−	0.582639i	$-0.802020\pi$
$967$	−50.5464	−1.62546	−0.812731	+	0.582639i	$0.802020\pi$
$968$	−2.51026	−0.0806828
$969$	0	0
$970$	0	0
$971$	57.3074	1.83908	0.919540	−	0.392995i	$-0.128561\pi$
$971$	57.3074	1.83908	0.919540	+	0.392995i	$0.128561\pi$
$972$	0	0
$973$	2.88267	0.0924143
$974$	−43.9664	−1.40877
$975$	0	0
$976$	30.7427	0.984051
$977$	−45.7875	−1.46487	−0.732436	−	0.680836i	$-0.761616\pi$
$977$	−45.7875	−1.46487	−0.732436	+	0.680836i	$0.761616\pi$
$978$	0	0
$979$	−4.34017	−0.138713
$980$	0	0
$981$	0	0
$982$	−40.0000	−1.27645
$983$	55.0805	1.75679	0.878397	−	0.477932i	$-0.158613\pi$
$983$	55.0805	1.75679	0.878397	+	0.477932i	$0.158613\pi$
$984$	0	0
$985$	0	0
$986$	54.6681	1.74099
$987$	0	0
$988$	−19.9635	−0.635123
$989$	−1.16290	−0.0369780
$990$	0	0
$991$	40.6947	1.29271	0.646355	−	0.763037i	$-0.276292\pi$
$991$	40.6947	1.29271	0.646355	+	0.763037i	$0.276292\pi$
$992$	−13.0784	−0.415239
$993$	0	0
$994$	−2.27247	−0.0720785
$995$	0	0
$996$	0	0
$997$	32.7610	1.03755	0.518775	−	0.854911i	$-0.326388\pi$
$997$	32.7610	1.03755	0.518775	+	0.854911i	$0.326388\pi$
$998$	−42.3545	−1.34071
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.be.1.2		3
3.2	odd	2		825.2.a.h.1.2		3
5.2	odd	4		495.2.c.d.199.5		6
5.3	odd	4		495.2.c.d.199.2		6
5.4	even	2		2475.2.a.y.1.2		3
15.2	even	4		165.2.c.a.34.2	✓	6
15.8	even	4		165.2.c.a.34.5	yes	6
15.14	odd	2		825.2.a.n.1.2		3
33.32	even	2		9075.2.a.ck.1.2		3
60.23	odd	4		2640.2.d.i.529.3		6
60.47	odd	4		2640.2.d.i.529.6		6
165.32	odd	4		1815.2.c.d.364.5		6
165.98	odd	4		1815.2.c.d.364.2		6
165.164	even	2		9075.2.a.cc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.c.a.34.2	✓	6	15.2	even	4
165.2.c.a.34.5	yes	6	15.8	even	4
495.2.c.d.199.2		6	5.3	odd	4
495.2.c.d.199.5		6	5.2	odd	4
825.2.a.h.1.2		3	3.2	odd	2
825.2.a.n.1.2		3	15.14	odd	2
1815.2.c.d.364.2		6	165.98	odd	4
1815.2.c.d.364.5		6	165.32	odd	4
2475.2.a.y.1.2		3	5.4	even	2
2475.2.a.be.1.2		3	1.1	even	1	trivial
2640.2.d.i.529.3		6	60.23	odd	4
2640.2.d.i.529.6		6	60.47	odd	4
9075.2.a.cc.1.2		3	165.164	even	2
9075.2.a.ck.1.2		3	33.32	even	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} +0.369102 q^{4} -0.290725 q^{7} -2.51026 q^{8} +1.00000 q^{11} -6.97107 q^{13} -0.447480 q^{14} -4.60197 q^{16} +4.78765 q^{17} +7.75872 q^{19} +1.53919 q^{22} +4.00000 q^{23} -10.7298 q^{26} -0.107307 q^{28} +7.41855 q^{29} +6.34017 q^{31} -2.06278 q^{32} +7.36910 q^{34} +3.41855 q^{37} +11.9421 q^{38} +7.41855 q^{41} -0.290725 q^{43} +0.369102 q^{44} +6.15676 q^{46} +5.26180 q^{47} -6.91548 q^{49} -2.57304 q^{52} +5.75872 q^{53} +0.729794 q^{56} +11.4186 q^{58} -3.60197 q^{59} -6.68035 q^{61} +9.75872 q^{62} +6.02893 q^{64} -6.15676 q^{67} +1.76713 q^{68} +5.07838 q^{71} +1.12783 q^{73} +5.26180 q^{74} +2.86376 q^{76} -0.290725 q^{77} -0.921622 q^{79} +11.4186 q^{82} +1.70928 q^{83} -0.447480 q^{86} -2.51026 q^{88} -4.34017 q^{89} +2.02666 q^{91} +1.47641 q^{92} +8.09890 q^{94} -4.68035 q^{97} -10.6442 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 5 q^{4} - 8 q^{7} + 9 q^{8} + 3 q^{11} - 6 q^{13} - 2 q^{14} + 5 q^{16} + 4 q^{17} - 2 q^{19} + 3 q^{22} + 12 q^{23} + 8 q^{26} - 12 q^{28} + 8 q^{29} + 8 q^{31} + 11 q^{32} + 26 q^{34}+ \cdots - 29 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 5 * q^4 - 8 * q^7 + 9 * q^8 + 3 * q^11 - 6 * q^13 - 2 * q^14 + 5 * q^16 + 4 * q^17 - 2 * q^19 + 3 * q^22 + 12 * q^23 + 8 * q^26 - 12 * q^28 + 8 * q^29 + 8 * q^31 + 11 * q^32 + 26 * q^34 - 4 * q^37 + 6 * q^38 + 8 * q^41 - 8 * q^43 + 5 * q^44 + 12 * q^46 + 8 * q^47 + 11 * q^49 + 26 * q^52 - 8 * q^53 - 38 * q^56 + 20 * q^58 + 8 * q^59 + 2 * q^61 + 4 * q^62 + 33 * q^64 - 12 * q^67 + 28 * q^68 + 12 * q^71 - 18 * q^73 + 8 * q^74 - 18 * q^76 - 8 * q^77 - 6 * q^79 + 20 * q^82 - 2 * q^83 - 2 * q^86 + 9 * q^88 - 2 * q^89 + 8 * q^91 + 20 * q^92 - 12 * q^94 + 8 * q^97 - 29 * q^98

Introduction

L-functions

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