LMFDB - Embedded newform 4275.2.a.bm.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4275,2,Mod(1,4275)] 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("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,2,0,4,0,0,-4,3,0,0,-1,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.65109$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65109 q^{2} +0.726109 q^{4} +0.377203 q^{7} +2.10331 q^{8} +1.37720 q^{11} -2.82167 q^{13} -0.622797 q^{14} -4.92498 q^{16} +6.37720 q^{17} +1.00000 q^{19} -2.27389 q^{22} +6.19887 q^{23} +4.65884 q^{26} +0.273891 q^{28} +3.37720 q^{29} +2.48052 q^{31} +3.92498 q^{32} -10.5294 q^{34} -5.58383 q^{37} -1.65109 q^{38} -8.50106 q^{41} +12.1522 q^{43} +1.00000 q^{44} -10.2349 q^{46} -6.87826 q^{47} -6.85772 q^{49} -2.04884 q^{52} +11.5478 q^{53} +0.793375 q^{56} -5.57608 q^{58} -6.05659 q^{59} +5.02830 q^{61} -4.09556 q^{62} +3.36945 q^{64} -3.22717 q^{67} +4.63055 q^{68} +2.30219 q^{71} +3.19887 q^{73} +9.21942 q^{74} +0.726109 q^{76} +0.519485 q^{77} -6.71836 q^{79} +14.0360 q^{82} +18.2165 q^{83} -20.0643 q^{86} +2.89669 q^{88} -1.50106 q^{89} -1.06434 q^{91} +4.50106 q^{92} +11.3567 q^{94} +11.7827 q^{97} +11.3227 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{4} - 4 q^{7} + 3 q^{8} - q^{11} - 3 q^{13} - 7 q^{14} - 6 q^{16} + 14 q^{17} + 3 q^{19} - 5 q^{22} + 8 q^{23} + 11 q^{26} - q^{28} + 5 q^{29} - q^{31} + 3 q^{32} + 5 q^{34} - 5 q^{37}+ \cdots + 4 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^4 - 4 * q^7 + 3 * q^8 - q^11 - 3 * q^13 - 7 * q^14 - 6 * q^16 + 14 * q^17 + 3 * q^19 - 5 * q^22 + 8 * q^23 + 11 * q^26 - q^28 + 5 * q^29 - q^31 + 3 * q^32 + 5 * q^34 - 5 * q^37 + 2 * q^38 - q^41 + 5 * q^43 + 3 * q^44 - 12 * q^46 + 9 * q^47 - 7 * q^49 + 22 * q^52 + 31 * q^53 + 9 * q^56 - q^58 + 6 * q^59 + 3 * q^61 - 5 * q^62 + q^64 + 13 * q^67 + 23 * q^68 - 7 * q^71 - q^73 + q^74 + 4 * q^76 + 10 * q^77 - 18 * q^79 + 34 * q^82 + 3 * q^83 - 40 * q^86 + 12 * q^88 + 20 * q^89 + 17 * q^91 - 11 * q^92 + 45 * q^94 + 13 * q^97 + 4 * 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 $

$2$	−1.65109	−1.16750	−0.583750	−	0.811934i	$-0.698415\pi$
$2$	−1.65109	−1.16750	−0.583750	+	0.811934i	$0.698415\pi$
$3$	0	0
$3$	0	0
$4$	0.726109	0.363055
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.377203	0.142569	0.0712846	−	0.997456i	$-0.477290\pi$
$7$	0.377203	0.142569	0.0712846	+	0.997456i	$0.477290\pi$
$8$	2.10331	0.743633
$9$	0	0
$10$	0	0
$11$	1.37720	0.415242	0.207621	−	0.978209i	$-0.433428\pi$
$11$	1.37720	0.415242	0.207621	+	0.978209i	$0.433428\pi$
$12$	0	0
$13$	−2.82167	−0.782591	−0.391295	−	0.920265i	$-0.627973\pi$
$13$	−2.82167	−0.782591	−0.391295	+	0.920265i	$0.627973\pi$
$14$	−0.622797	−0.166450
$15$	0	0
$16$	−4.92498	−1.23125
$17$	6.37720	1.54670	0.773349	−	0.633980i	$-0.218580\pi$
$17$	6.37720	1.54670	0.773349	+	0.633980i	$0.218580\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−2.27389	−0.484795
$23$	6.19887	1.29255	0.646277	−	0.763103i	$-0.276325\pi$
$23$	6.19887	1.29255	0.646277	+	0.763103i	$0.276325\pi$
$24$	0	0
$25$	0	0
$26$	4.65884	0.913674
$27$	0	0
$28$	0.273891	0.0517604
$29$	3.37720	0.627131	0.313565	−	0.949567i	$-0.398476\pi$
$29$	3.37720	0.627131	0.313565	+	0.949567i	$0.398476\pi$
$30$	0	0
$31$	2.48052	0.445514	0.222757	−	0.974874i	$-0.428494\pi$
$31$	2.48052	0.445514	0.222757	+	0.974874i	$0.428494\pi$
$32$	3.92498	0.693846
$33$	0	0
$34$	−10.5294	−1.80577
$35$	0	0
$36$	0	0
$37$	−5.58383	−0.917976	−0.458988	−	0.888443i	$-0.651788\pi$
$37$	−5.58383	−0.917976	−0.458988	+	0.888443i	$0.651788\pi$
$38$	−1.65109	−0.267843
$39$	0	0
$40$	0	0
$41$	−8.50106	−1.32764	−0.663821	−	0.747891i	$-0.731066\pi$
$41$	−8.50106	−1.32764	−0.663821	+	0.747891i	$0.731066\pi$
$42$	0	0
$43$	12.1522	1.85319	0.926593	−	0.376065i	$-0.122723\pi$
$43$	12.1522	1.85319	0.926593	+	0.376065i	$0.122723\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−10.2349	−1.50906
$47$	−6.87826	−1.00330	−0.501649	−	0.865071i	$-0.667273\pi$
$47$	−6.87826	−1.00330	−0.501649	+	0.865071i	$0.667273\pi$
$48$	0	0
$49$	−6.85772	−0.979674
$50$	0	0
$51$	0	0
$52$	−2.04884	−0.284123
$53$	11.5478	1.58621	0.793105	−	0.609085i	$-0.208463\pi$
$53$	11.5478	1.58621	0.793105	+	0.609085i	$0.208463\pi$
$54$	0	0
$55$	0	0
$56$	0.793375	0.106019
$57$	0	0
$58$	−5.57608	−0.732175
$59$	−6.05659	−0.788501	−0.394251	−	0.919003i	$-0.628996\pi$
$59$	−6.05659	−0.788501	−0.394251	+	0.919003i	$0.628996\pi$
$60$	0	0
$61$	5.02830	0.643807	0.321904	−	0.946772i	$-0.395677\pi$
$61$	5.02830	0.643807	0.321904	+	0.946772i	$0.395677\pi$
$62$	−4.09556	−0.520137
$63$	0	0
$64$	3.36945	0.421182
$65$	0	0
$66$	0	0
$67$	−3.22717	−0.394262	−0.197131	−	0.980377i	$-0.563162\pi$
$67$	−3.22717	−0.394262	−0.197131	+	0.980377i	$0.563162\pi$
$68$	4.63055	0.561536
$69$	0	0
$70$	0	0
$71$	2.30219	0.273219	0.136610	−	0.990625i	$-0.456379\pi$
$71$	2.30219	0.273219	0.136610	+	0.990625i	$0.456379\pi$
$72$	0	0
$73$	3.19887	0.374400	0.187200	−	0.982322i	$-0.440059\pi$
$73$	3.19887	0.374400	0.187200	+	0.982322i	$0.440059\pi$
$74$	9.21942	1.07174
$75$	0	0
$76$	0.726109	0.0832905
$77$	0.519485	0.0592008
$78$	0	0
$79$	−6.71836	−0.755874	−0.377937	−	0.925831i	$-0.623366\pi$
$79$	−6.71836	−0.755874	−0.377937	+	0.925831i	$0.623366\pi$
$80$	0	0
$81$	0	0
$82$	14.0360	1.55002
$83$	18.2165	1.99952	0.999760	−	0.0218996i	$-0.00697141\pi$
$83$	18.2165	1.99952	0.999760	+	0.0218996i	$0.00697141\pi$
$84$	0	0
$85$	0	0
$86$	−20.0643	−2.16359
$87$	0	0
$88$	2.89669	0.308788
$89$	−1.50106	−0.159112	−0.0795561	−	0.996830i	$-0.525350\pi$
$89$	−1.50106	−0.159112	−0.0795561	+	0.996830i	$0.525350\pi$
$90$	0	0
$91$	−1.06434	−0.111573
$92$	4.50106	0.469268
$93$	0	0
$94$	11.3567	1.17135
$95$	0	0
$96$	0	0
$97$	11.7827	1.19635	0.598176	−	0.801365i	$-0.295892\pi$
$97$	11.7827	1.19635	0.598176	+	0.801365i	$0.295892\pi$
$98$	11.3227	1.14377
$99$	0	0
$100$	0	0
$101$	9.18820	0.914260	0.457130	−	0.889400i	$-0.348877\pi$
$101$	9.18820	0.914260	0.457130	+	0.889400i	$0.348877\pi$
$102$	0	0
$103$	9.47277	0.933379	0.466690	−	0.884421i	$-0.345447\pi$
$103$	9.47277	0.933379	0.466690	+	0.884421i	$0.345447\pi$
$104$	−5.93486	−0.581961
$105$	0	0
$106$	−19.0665	−1.85190
$107$	−15.3510	−1.48404	−0.742020	−	0.670378i	$-0.766132\pi$
$107$	−15.3510	−1.48404	−0.742020	+	0.670378i	$0.766132\pi$
$108$	0	0
$109$	−16.7643	−1.60573	−0.802863	−	0.596163i	$-0.796691\pi$
$109$	−16.7643	−1.60573	−0.802863	+	0.596163i	$0.796691\pi$
$110$	0	0
$111$	0	0
$112$	−1.85772	−0.175538
$113$	−13.3305	−1.25403	−0.627013	−	0.779009i	$-0.715723\pi$
$113$	−13.3305	−1.25403	−0.627013	+	0.779009i	$0.715723\pi$
$114$	0	0
$115$	0	0
$116$	2.45222	0.227683
$117$	0	0
$118$	10.0000	0.920575
$119$	2.40550	0.220512
$120$	0	0
$121$	−9.10331	−0.827574
$122$	−8.30219	−0.751645
$123$	0	0
$124$	1.80113	0.161746
$125$	0	0
$126$	0	0
$127$	3.04672	0.270353	0.135176	−	0.990822i	$-0.456840\pi$
$127$	3.04672	0.270353	0.135176	+	0.990822i	$0.456840\pi$
$128$	−13.4132	−1.18557
$129$	0	0
$130$	0	0
$131$	8.70769	0.760794	0.380397	−	0.924823i	$-0.375787\pi$
$131$	8.70769	0.760794	0.380397	+	0.924823i	$0.375787\pi$
$132$	0	0
$133$	0.377203	0.0327076
$134$	5.32836	0.460300
$135$	0	0
$136$	13.4132	1.15018
$137$	6.59450	0.563406	0.281703	−	0.959502i	$-0.409101\pi$
$137$	6.59450	0.563406	0.281703	+	0.959502i	$0.409101\pi$
$138$	0	0
$139$	−10.1677	−0.862409	−0.431205	−	0.902254i	$-0.641911\pi$
$139$	−10.1677	−0.862409	−0.431205	+	0.902254i	$0.641911\pi$
$140$	0	0
$141$	0	0
$142$	−3.80113	−0.318983
$143$	−3.88601	−0.324965
$144$	0	0
$145$	0	0
$146$	−5.28164	−0.437112
$147$	0	0
$148$	−4.05447	−0.333275
$149$	5.54003	0.453857	0.226929	−	0.973911i	$-0.427132\pi$
$149$	5.54003	0.453857	0.226929	+	0.973911i	$0.427132\pi$
$150$	0	0
$151$	5.12386	0.416974	0.208487	−	0.978025i	$-0.433146\pi$
$151$	5.12386	0.416974	0.208487	+	0.978025i	$0.433146\pi$
$152$	2.10331	0.170601
$153$	0	0
$154$	−0.857718	−0.0691169
$155$	0	0
$156$	0	0
$157$	−19.7643	−1.57736	−0.788681	−	0.614803i	$-0.789235\pi$
$157$	−19.7643	−1.57736	−0.788681	+	0.614803i	$0.789235\pi$
$158$	11.0926	0.882483
$159$	0	0
$160$	0	0
$161$	2.33823	0.184279
$162$	0	0
$163$	−13.5195	−1.05893	−0.529464	−	0.848332i	$-0.677607\pi$
$163$	−13.5195	−1.05893	−0.529464	+	0.848332i	$0.677607\pi$
$164$	−6.17270	−0.482007
$165$	0	0
$166$	−30.0771	−2.33444
$167$	13.1054	1.01413	0.507065	−	0.861908i	$-0.330731\pi$
$167$	13.1054	1.01413	0.507065	+	0.861908i	$0.330731\pi$
$168$	0	0
$169$	−5.03817	−0.387551
$170$	0	0
$171$	0	0
$172$	8.82379	0.672808
$173$	1.48827	0.113151	0.0565754	−	0.998398i	$-0.481982\pi$
$173$	1.48827	0.113151	0.0565754	+	0.998398i	$0.481982\pi$
$174$	0	0
$175$	0	0
$176$	−6.78270	−0.511265
$177$	0	0
$178$	2.47839	0.185763
$179$	17.1132	1.27910	0.639550	−	0.768750i	$-0.279121\pi$
$179$	17.1132	1.27910	0.639550	+	0.768750i	$0.279121\pi$
$180$	0	0
$181$	−5.73891	−0.426569	−0.213285	−	0.976990i	$-0.568416\pi$
$181$	−5.73891	−0.426569	−0.213285	+	0.976990i	$0.568416\pi$
$182$	1.75733	0.130262
$183$	0	0
$184$	13.0382	0.961187
$185$	0	0
$186$	0	0
$187$	8.78270	0.642255
$188$	−4.99437	−0.364252
$189$	0	0
$190$	0	0
$191$	−22.3948	−1.62043	−0.810216	−	0.586131i	$-0.800650\pi$
$191$	−22.3948	−1.62043	−0.810216	+	0.586131i	$0.800650\pi$
$192$	0	0
$193$	21.3687	1.53815	0.769075	−	0.639159i	$-0.220717\pi$
$193$	21.3687	1.53815	0.769075	+	0.639159i	$0.220717\pi$
$194$	−19.4543	−1.39674
$195$	0	0
$196$	−4.97945	−0.355675
$197$	1.11399	0.0793682	0.0396841	−	0.999212i	$-0.487365\pi$
$197$	1.11399	0.0793682	0.0396841	+	0.999212i	$0.487365\pi$
$198$	0	0
$199$	−2.22505	−0.157729	−0.0788647	−	0.996885i	$-0.525130\pi$
$199$	−2.22505	−0.157729	−0.0788647	+	0.996885i	$0.525130\pi$
$200$	0	0
$201$	0	0
$202$	−15.1706	−1.06740
$203$	1.27389	0.0894096
$204$	0	0
$205$	0	0
$206$	−15.6404	−1.08972
$207$	0	0
$208$	13.8967	0.963562
$209$	1.37720	0.0952631
$210$	0	0
$211$	−9.75441	−0.671521	−0.335760	−	0.941947i	$-0.608993\pi$
$211$	−9.75441	−0.671521	−0.335760	+	0.941947i	$0.608993\pi$
$212$	8.38495	0.575881
$213$	0	0
$214$	25.3460	1.73262
$215$	0	0
$216$	0	0
$217$	0.935657	0.0635166
$218$	27.6794	1.87468
$219$	0	0
$220$	0	0
$221$	−17.9944	−1.21043
$222$	0	0
$223$	18.6433	1.24845	0.624225	−	0.781244i	$-0.285415\pi$
$223$	18.6433	1.24845	0.624225	+	0.781244i	$0.285415\pi$
$224$	1.48052	0.0989211
$225$	0	0
$226$	22.0099	1.46407
$227$	−8.04672	−0.534080	−0.267040	−	0.963686i	$-0.586046\pi$
$227$	−8.04672	−0.534080	−0.267040	+	0.963686i	$0.586046\pi$
$228$	0	0
$229$	20.1415	1.33099	0.665493	−	0.746404i	$-0.268221\pi$
$229$	20.1415	1.33099	0.665493	+	0.746404i	$0.268221\pi$
$230$	0	0
$231$	0	0
$232$	7.10331	0.466355
$233$	−1.73386	−0.113589	−0.0567945	−	0.998386i	$-0.518088\pi$
$233$	−1.73386	−0.113589	−0.0567945	+	0.998386i	$0.518088\pi$
$234$	0	0
$235$	0	0
$236$	−4.39775	−0.286269
$237$	0	0
$238$	−3.97170	−0.257447
$239$	18.3150	1.18470	0.592349	−	0.805682i	$-0.298201\pi$
$239$	18.3150	1.18470	0.592349	+	0.805682i	$0.298201\pi$
$240$	0	0
$241$	−2.19675	−0.141505	−0.0707526	−	0.997494i	$-0.522540\pi$
$241$	−2.19675	−0.141505	−0.0707526	+	0.997494i	$0.522540\pi$
$242$	15.0304	0.966192
$243$	0	0
$244$	3.65109	0.233737
$245$	0	0
$246$	0	0
$247$	−2.82167	−0.179539
$248$	5.21730	0.331299
$249$	0	0
$250$	0	0
$251$	13.6716	0.862946	0.431473	−	0.902126i	$-0.357994\pi$
$251$	13.6716	0.862946	0.431473	+	0.902126i	$0.357994\pi$
$252$	0	0
$253$	8.53711	0.536723
$254$	−5.03042	−0.315637
$255$	0	0
$256$	15.4076	0.962976
$257$	28.4904	1.77718	0.888591	−	0.458701i	$-0.151685\pi$
$257$	28.4904	1.77718	0.888591	+	0.458701i	$0.151685\pi$
$258$	0	0
$259$	−2.10624	−0.130875
$260$	0	0
$261$	0	0
$262$	−14.3772	−0.888227
$263$	5.36945	0.331095	0.165547	−	0.986202i	$-0.447061\pi$
$263$	5.36945	0.331095	0.165547	+	0.986202i	$0.447061\pi$
$264$	0	0
$265$	0	0
$266$	−0.622797	−0.0381861
$267$	0	0
$268$	−2.34328	−0.143139
$269$	−7.06727	−0.430899	−0.215449	−	0.976515i	$-0.569122\pi$
$269$	−7.06727	−0.430899	−0.215449	+	0.976515i	$0.569122\pi$
$270$	0	0
$271$	21.6970	1.31800	0.659000	−	0.752143i	$-0.270980\pi$
$271$	21.6970	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$272$	−31.4076	−1.90437
$273$	0	0
$274$	−10.8881	−0.657776
$275$	0	0
$276$	0	0
$277$	−25.1132	−1.50891	−0.754453	−	0.656355i	$-0.772098\pi$
$277$	−25.1132	−1.50891	−0.754453	+	0.656355i	$0.772098\pi$
$278$	16.7877	1.00686
$279$	0	0
$280$	0	0
$281$	10.0411	0.599001	0.299501	−	0.954096i	$-0.403180\pi$
$281$	10.0411	0.599001	0.299501	+	0.954096i	$0.403180\pi$
$282$	0	0
$283$	20.9143	1.24323	0.621613	−	0.783324i	$-0.286478\pi$
$283$	20.9143	1.24323	0.621613	+	0.783324i	$0.286478\pi$
$284$	1.67164	0.0991936
$285$	0	0
$286$	6.41617	0.379396
$287$	−3.20662	−0.189281
$288$	0	0
$289$	23.6687	1.39228
$290$	0	0
$291$	0	0
$292$	2.32273	0.135928
$293$	−0.818748	−0.0478318	−0.0239159	−	0.999714i	$-0.507613\pi$
$293$	−0.818748	−0.0478318	−0.0239159	+	0.999714i	$0.507613\pi$
$294$	0	0
$295$	0	0
$296$	−11.7445	−0.682637
$297$	0	0
$298$	−9.14711	−0.529878
$299$	−17.4912	−1.01154
$300$	0	0
$301$	4.58383	0.264207
$302$	−8.45997	−0.486817
$303$	0	0
$304$	−4.92498	−0.282467
$305$	0	0
$306$	0	0
$307$	7.44447	0.424878	0.212439	−	0.977174i	$-0.431859\pi$
$307$	7.44447	0.424878	0.212439	+	0.977174i	$0.431859\pi$
$308$	0.377203	0.0214931
$309$	0	0
$310$	0	0
$311$	0.956204	0.0542213	0.0271107	−	0.999632i	$-0.491369\pi$
$311$	0.956204	0.0542213	0.0271107	+	0.999632i	$0.491369\pi$
$312$	0	0
$313$	−5.98158	−0.338099	−0.169049	−	0.985608i	$-0.554070\pi$
$313$	−5.98158	−0.338099	−0.169049	+	0.985608i	$0.554070\pi$
$314$	32.6327	1.84157
$315$	0	0
$316$	−4.87826	−0.274424
$317$	22.6511	1.27221	0.636106	−	0.771602i	$-0.280544\pi$
$317$	22.6511	1.27221	0.636106	+	0.771602i	$0.280544\pi$
$318$	0	0
$319$	4.65109	0.260411
$320$	0	0
$321$	0	0
$322$	−3.86064	−0.215145
$323$	6.37720	0.354837
$324$	0	0
$325$	0	0
$326$	22.3219	1.23630
$327$	0	0
$328$	−17.8804	−0.987279
$329$	−2.59450	−0.143039
$330$	0	0
$331$	4.16283	0.228810	0.114405	−	0.993434i	$-0.463504\pi$
$331$	4.16283	0.228810	0.114405	+	0.993434i	$0.463504\pi$
$332$	13.2272	0.725935
$333$	0	0
$334$	−21.6383	−1.18399
$335$	0	0
$336$	0	0
$337$	18.0304	0.982180	0.491090	−	0.871109i	$-0.336599\pi$
$337$	18.0304	0.982180	0.491090	+	0.871109i	$0.336599\pi$
$338$	8.31849	0.452466
$339$	0	0
$340$	0	0
$341$	3.41617	0.184996
$342$	0	0
$343$	−5.22717	−0.282241
$344$	25.5598	1.37809
$345$	0	0
$346$	−2.45726	−0.132103
$347$	7.43380	0.399067	0.199534	−	0.979891i	$-0.436057\pi$
$347$	7.43380	0.399067	0.199534	+	0.979891i	$0.436057\pi$
$348$	0	0
$349$	21.9194	1.17332	0.586658	−	0.809835i	$-0.300443\pi$
$349$	21.9194	1.17332	0.586658	+	0.809835i	$0.300443\pi$
$350$	0	0
$351$	0	0
$352$	5.40550	0.288114
$353$	10.3695	0.551910	0.275955	−	0.961171i	$-0.411006\pi$
$353$	10.3695	0.551910	0.275955	+	0.961171i	$0.411006\pi$
$354$	0	0
$355$	0	0
$356$	−1.08993	−0.0577664
$357$	0	0
$358$	−28.2555	−1.49335
$359$	23.9893	1.26611	0.633054	−	0.774108i	$-0.281801\pi$
$359$	23.9893	1.26611	0.633054	+	0.774108i	$0.281801\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	9.47547	0.498020
$363$	0	0
$364$	−0.772829	−0.0405073
$365$	0	0
$366$	0	0
$367$	−35.0275	−1.82842	−0.914210	−	0.405240i	$-0.867188\pi$
$367$	−35.0275	−1.82842	−0.914210	+	0.405240i	$0.867188\pi$
$368$	−30.5294	−1.59145
$369$	0	0
$370$	0	0
$371$	4.35586	0.226145
$372$	0	0
$373$	−30.8003	−1.59478	−0.797390	−	0.603464i	$-0.793787\pi$
$373$	−30.8003	−1.59478	−0.797390	+	0.603464i	$0.793787\pi$
$374$	−14.5011	−0.749832
$375$	0	0
$376$	−14.4671	−0.746086
$377$	−9.52936	−0.490787
$378$	0	0
$379$	0.671640	0.0344998	0.0172499	−	0.999851i	$-0.494509\pi$
$379$	0.671640	0.0344998	0.0172499	+	0.999851i	$0.494509\pi$
$380$	0	0
$381$	0	0
$382$	36.9759	1.89185
$383$	−30.1805	−1.54215	−0.771075	−	0.636745i	$-0.780280\pi$
$383$	−30.1805	−1.54215	−0.771075	+	0.636745i	$0.780280\pi$
$384$	0	0
$385$	0	0
$386$	−35.2816	−1.79579
$387$	0	0
$388$	8.55553	0.434341
$389$	31.5109	1.59767	0.798834	−	0.601552i	$-0.205451\pi$
$389$	31.5109	1.59767	0.798834	+	0.601552i	$0.205451\pi$
$390$	0	0
$391$	39.5315	1.99919
$392$	−14.4239	−0.728518
$393$	0	0
$394$	−1.83929	−0.0926623
$395$	0	0
$396$	0	0
$397$	12.5761	0.631175	0.315588	−	0.948896i	$-0.397798\pi$
$397$	12.5761	0.631175	0.315588	+	0.948896i	$0.397798\pi$
$398$	3.67376	0.184149
$399$	0	0
$400$	0	0
$401$	−24.6036	−1.22864	−0.614322	−	0.789056i	$-0.710570\pi$
$401$	−24.6036	−1.22864	−0.614322	+	0.789056i	$0.710570\pi$
$402$	0	0
$403$	−6.99920	−0.348655
$404$	6.67164	0.331926
$405$	0	0
$406$	−2.10331	−0.104386
$407$	−7.69006	−0.381182
$408$	0	0
$409$	13.6666	0.675770	0.337885	−	0.941187i	$-0.390289\pi$
$409$	13.6666	0.675770	0.337885	+	0.941187i	$0.390289\pi$
$410$	0	0
$411$	0	0
$412$	6.87826	0.338868
$413$	−2.28456	−0.112416
$414$	0	0
$415$	0	0
$416$	−11.0750	−0.542997
$417$	0	0
$418$	−2.27389	−0.111220
$419$	−12.6893	−0.619911	−0.309956	−	0.950751i	$-0.600314\pi$
$419$	−12.6893	−0.619911	−0.309956	+	0.950751i	$0.600314\pi$
$420$	0	0
$421$	29.1826	1.42227	0.711136	−	0.703055i	$-0.248181\pi$
$421$	29.1826	1.42227	0.711136	+	0.703055i	$0.248181\pi$
$422$	16.1054	0.784000
$423$	0	0
$424$	24.2886	1.17956
$425$	0	0
$426$	0	0
$427$	1.89669	0.0917872
$428$	−11.1465	−0.538788
$429$	0	0
$430$	0	0
$431$	29.9816	1.44416	0.722081	−	0.691809i	$-0.243186\pi$
$431$	29.9816	1.44416	0.722081	+	0.691809i	$0.243186\pi$
$432$	0	0
$433$	4.76216	0.228855	0.114427	−	0.993432i	$-0.463497\pi$
$433$	4.76216	0.228855	0.114427	+	0.993432i	$0.463497\pi$
$434$	−1.54486	−0.0741555
$435$	0	0
$436$	−12.1727	−0.582967
$437$	6.19887	0.296532
$438$	0	0
$439$	6.88601	0.328652	0.164326	−	0.986406i	$-0.447455\pi$
$439$	6.88601	0.328652	0.164326	+	0.986406i	$0.447455\pi$
$440$	0	0
$441$	0	0
$442$	29.7104	1.41318
$443$	8.65109	0.411026	0.205513	−	0.978654i	$-0.434114\pi$
$443$	8.65109	0.411026	0.205513	+	0.978654i	$0.434114\pi$
$444$	0	0
$445$	0	0
$446$	−30.7819	−1.45757
$447$	0	0
$448$	1.27097	0.0600476
$449$	−8.63055	−0.407301	−0.203650	−	0.979044i	$-0.565281\pi$
$449$	−8.63055	−0.407301	−0.203650	+	0.979044i	$0.565281\pi$
$450$	0	0
$451$	−11.7077	−0.551293
$452$	−9.67939	−0.455280
$453$	0	0
$454$	13.2859	0.623538
$455$	0	0
$456$	0	0
$457$	28.6092	1.33828	0.669141	−	0.743135i	$-0.266662\pi$
$457$	28.6092	1.33828	0.669141	+	0.743135i	$0.266662\pi$
$458$	−33.2555	−1.55393
$459$	0	0
$460$	0	0
$461$	39.1025	1.82119	0.910593	−	0.413305i	$-0.135626\pi$
$461$	39.1025	1.82119	0.910593	+	0.413305i	$0.135626\pi$
$462$	0	0
$463$	8.04380	0.373827	0.186913	−	0.982376i	$-0.440152\pi$
$463$	8.04380	0.373827	0.186913	+	0.982376i	$0.440152\pi$
$464$	−16.6327	−0.772152
$465$	0	0
$466$	2.86276	0.132615
$467$	17.0488	0.788926	0.394463	−	0.918912i	$-0.370931\pi$
$467$	17.0488	0.788926	0.394463	+	0.918912i	$0.370931\pi$
$468$	0	0
$469$	−1.21730	−0.0562096
$470$	0	0
$471$	0	0
$472$	−12.7389	−0.586356
$473$	16.7360	0.769521
$474$	0	0
$475$	0	0
$476$	1.74666	0.0800578
$477$	0	0
$478$	−30.2397	−1.38313
$479$	−22.5478	−1.03023	−0.515117	−	0.857120i	$-0.672252\pi$
$479$	−22.5478	−1.03023	−0.515117	+	0.857120i	$0.672252\pi$
$480$	0	0
$481$	15.7557	0.718399
$482$	3.62704	0.165207
$483$	0	0
$484$	−6.61000	−0.300455
$485$	0	0
$486$	0	0
$487$	24.5577	1.11281	0.556407	−	0.830910i	$-0.312180\pi$
$487$	24.5577	1.11281	0.556407	+	0.830910i	$0.312180\pi$
$488$	10.5761	0.478757
$489$	0	0
$490$	0	0
$491$	16.9298	0.764032	0.382016	−	0.924156i	$-0.375230\pi$
$491$	16.9298	0.764032	0.382016	+	0.924156i	$0.375230\pi$
$492$	0	0
$493$	21.5371	0.969983
$494$	4.65884	0.209611
$495$	0	0
$496$	−12.2165	−0.548537
$497$	0.868391	0.0389527
$498$	0	0
$499$	0.418295	0.0187255	0.00936273	−	0.999956i	$-0.497020\pi$
$499$	0.418295	0.0187255	0.00936273	+	0.999956i	$0.497020\pi$
$500$	0	0
$501$	0	0
$502$	−22.5732	−1.00749
$503$	−12.5993	−0.561776	−0.280888	−	0.959741i	$-0.590629\pi$
$503$	−12.5993	−0.561776	−0.280888	+	0.959741i	$0.590629\pi$
$504$	0	0
$505$	0	0
$506$	−14.0956	−0.626624
$507$	0	0
$508$	2.21225	0.0981528
$509$	−12.8209	−0.568275	−0.284138	−	0.958784i	$-0.591707\pi$
$509$	−12.8209	−0.568275	−0.284138	+	0.958784i	$0.591707\pi$
$510$	0	0
$511$	1.20662	0.0533779
$512$	1.38708	0.0613007
$513$	0	0
$514$	−47.0403	−2.07486
$515$	0	0
$516$	0	0
$517$	−9.47277	−0.416612
$518$	3.47759	0.152797
$519$	0	0
$520$	0	0
$521$	−11.3743	−0.498316	−0.249158	−	0.968463i	$-0.580154\pi$
$521$	−11.3743	−0.498316	−0.249158	+	0.968463i	$0.580154\pi$
$522$	0	0
$523$	−33.0948	−1.44713	−0.723566	−	0.690255i	$-0.757498\pi$
$523$	−33.0948	−1.44713	−0.723566	+	0.690255i	$0.757498\pi$
$524$	6.32273	0.276210
$525$	0	0
$526$	−8.86547	−0.386553
$527$	15.8187	0.689075
$528$	0	0
$529$	15.4260	0.670698
$530$	0	0
$531$	0	0
$532$	0.273891	0.0118747
$533$	23.9872	1.03900
$534$	0	0
$535$	0	0
$536$	−6.78775	−0.293186
$537$	0	0
$538$	11.6687	0.503074
$539$	−9.44447	−0.406802
$540$	0	0
$541$	31.4338	1.35144	0.675722	−	0.737156i	$-0.263832\pi$
$541$	31.4338	1.35144	0.675722	+	0.737156i	$0.263832\pi$
$542$	−35.8238	−1.53876
$543$	0	0
$544$	25.0304	1.07317
$545$	0	0
$546$	0	0
$547$	23.3460	0.998202	0.499101	−	0.866544i	$-0.333664\pi$
$547$	23.3460	0.998202	0.499101	+	0.866544i	$0.333664\pi$
$548$	4.78833	0.204547
$549$	0	0
$550$	0	0
$551$	3.37720	0.143874
$552$	0	0
$553$	−2.53418	−0.107764
$554$	41.4642	1.76165
$555$	0	0
$556$	−7.38283	−0.313102
$557$	15.9229	0.674673	0.337337	−	0.941384i	$-0.390474\pi$
$557$	15.9229	0.674673	0.337337	+	0.941384i	$0.390474\pi$
$558$	0	0
$559$	−34.2894	−1.45029
$560$	0	0
$561$	0	0
$562$	−16.5788	−0.699334
$563$	−29.6404	−1.24919	−0.624597	−	0.780947i	$-0.714737\pi$
$563$	−29.6404	−1.24919	−0.624597	+	0.780947i	$0.714737\pi$
$564$	0	0
$565$	0	0
$566$	−34.5315	−1.45147
$567$	0	0
$568$	4.84222	0.203175
$569$	7.64334	0.320426	0.160213	−	0.987082i	$-0.448782\pi$
$569$	7.64334	0.320426	0.160213	+	0.987082i	$0.448782\pi$
$570$	0	0
$571$	−10.5577	−0.441824	−0.220912	−	0.975294i	$-0.570903\pi$
$571$	−10.5577	−0.441824	−0.220912	+	0.975294i	$0.570903\pi$
$572$	−2.82167	−0.117980
$573$	0	0
$574$	5.29444	0.220986
$575$	0	0
$576$	0	0
$577$	−38.9447	−1.62129	−0.810645	−	0.585538i	$-0.800883\pi$
$577$	−38.9447	−1.62129	−0.810645	+	0.585538i	$0.800883\pi$
$578$	−39.0793	−1.62548
$579$	0	0
$580$	0	0
$581$	6.87131	0.285070
$582$	0	0
$583$	15.9036	0.658661
$584$	6.72823	0.278416
$585$	0	0
$586$	1.35183	0.0558436
$587$	−10.9992	−0.453986	−0.226993	−	0.973896i	$-0.572889\pi$
$587$	−10.9992	−0.453986	−0.226993	+	0.973896i	$0.572889\pi$
$588$	0	0
$589$	2.48052	0.102208
$590$	0	0
$591$	0	0
$592$	27.5003	1.13025
$593$	26.7848	1.09992	0.549960	−	0.835191i	$-0.314643\pi$
$593$	26.7848	1.09992	0.549960	+	0.835191i	$0.314643\pi$
$594$	0	0
$595$	0	0
$596$	4.02267	0.164775
$597$	0	0
$598$	28.8796	1.18097
$599$	3.44930	0.140934	0.0704672	−	0.997514i	$-0.477551\pi$
$599$	3.44930	0.140934	0.0704672	+	0.997514i	$0.477551\pi$
$600$	0	0
$601$	−37.0686	−1.51206	−0.756030	−	0.654537i	$-0.772863\pi$
$601$	−37.0686	−1.51206	−0.756030	+	0.654537i	$0.772863\pi$
$602$	−7.56833	−0.308462
$603$	0	0
$604$	3.72048	0.151384
$605$	0	0
$606$	0	0
$607$	14.8732	0.603685	0.301843	−	0.953358i	$-0.402398\pi$
$607$	14.8732	0.603685	0.301843	+	0.953358i	$0.402398\pi$
$608$	3.92498	0.159179
$609$	0	0
$610$	0	0
$611$	19.4082	0.785172
$612$	0	0
$613$	40.4671	1.63445	0.817226	−	0.576317i	$-0.195511\pi$
$613$	40.4671	1.63445	0.817226	+	0.576317i	$0.195511\pi$
$614$	−12.2915	−0.496045
$615$	0	0
$616$	1.09264	0.0440237
$617$	8.76991	0.353063	0.176532	−	0.984295i	$-0.443512\pi$
$617$	8.76991	0.353063	0.176532	+	0.984295i	$0.443512\pi$
$618$	0	0
$619$	32.0510	1.28824	0.644119	−	0.764926i	$-0.277224\pi$
$619$	32.0510	1.28824	0.644119	+	0.764926i	$0.277224\pi$
$620$	0	0
$621$	0	0
$622$	−1.57878	−0.0633034
$623$	−0.566205	−0.0226845
$624$	0	0
$625$	0	0
$626$	9.87614	0.394730
$627$	0	0
$628$	−14.3510	−0.572668
$629$	−35.6092	−1.41983
$630$	0	0
$631$	−18.9709	−0.755220	−0.377610	−	0.925965i	$-0.623254\pi$
$631$	−18.9709	−0.755220	−0.377610	+	0.925965i	$0.623254\pi$
$632$	−14.1308	−0.562093
$633$	0	0
$634$	−37.3991	−1.48531
$635$	0	0
$636$	0	0
$637$	19.3502	0.766684
$638$	−7.67939	−0.304030
$639$	0	0
$640$	0	0
$641$	44.3433	1.75145	0.875727	−	0.482806i	$-0.160383\pi$
$641$	44.3433	1.75145	0.875727	+	0.482806i	$0.160383\pi$
$642$	0	0
$643$	−42.9397	−1.69338	−0.846688	−	0.532090i	$-0.821407\pi$
$643$	−42.9397	−1.69338	−0.846688	+	0.532090i	$0.821407\pi$
$644$	1.69781	0.0669032
$645$	0	0
$646$	−10.5294	−0.414272
$647$	17.9864	0.707118	0.353559	−	0.935412i	$-0.384971\pi$
$647$	17.9864	0.707118	0.353559	+	0.935412i	$0.384971\pi$
$648$	0	0
$649$	−8.34116	−0.327419
$650$	0	0
$651$	0	0
$652$	−9.81663	−0.384449
$653$	−21.7274	−0.850260	−0.425130	−	0.905132i	$-0.639772\pi$
$653$	−21.7274	−0.850260	−0.425130	+	0.905132i	$0.639772\pi$
$654$	0	0
$655$	0	0
$656$	41.8676	1.63465
$657$	0	0
$658$	4.28376	0.166998
$659$	40.8590	1.59164	0.795821	−	0.605532i	$-0.207040\pi$
$659$	40.8590	1.59164	0.795821	+	0.605532i	$0.207040\pi$
$660$	0	0
$661$	33.7282	1.31188	0.655938	−	0.754815i	$-0.272273\pi$
$661$	33.7282	1.31188	0.655938	+	0.754815i	$0.272273\pi$
$662$	−6.87322	−0.267135
$663$	0	0
$664$	38.3150	1.48691
$665$	0	0
$666$	0	0
$667$	20.9349	0.810601
$668$	9.51598	0.368184
$669$	0	0
$670$	0	0
$671$	6.92498	0.267336
$672$	0	0
$673$	11.0622	0.426417	0.213209	−	0.977007i	$-0.431609\pi$
$673$	11.0622	0.426417	0.213209	+	0.977007i	$0.431609\pi$
$674$	−29.7699	−1.14669
$675$	0	0
$676$	−3.65826	−0.140702
$677$	−6.14711	−0.236253	−0.118126	−	0.992999i	$-0.537689\pi$
$677$	−6.14711	−0.236253	−0.118126	+	0.992999i	$0.537689\pi$
$678$	0	0
$679$	4.44447	0.170563
$680$	0	0
$681$	0	0
$682$	−5.64042	−0.215983
$683$	−13.6073	−0.520669	−0.260334	−	0.965519i	$-0.583833\pi$
$683$	−13.6073	−0.520669	−0.260334	+	0.965519i	$0.583833\pi$
$684$	0	0
$685$	0	0
$686$	8.63055	0.329516
$687$	0	0
$688$	−59.8492	−2.28173
$689$	−32.5840	−1.24135
$690$	0	0
$691$	−40.3043	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$691$	−40.3043	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$692$	1.08064	0.0410799
$693$	0	0
$694$	−12.2739	−0.465911
$695$	0	0
$696$	0	0
$697$	−54.2130	−2.05346
$698$	−36.1909	−1.36985
$699$	0	0
$700$	0	0
$701$	3.23704	0.122261	0.0611307	−	0.998130i	$-0.480529\pi$
$701$	3.23704	0.122261	0.0611307	+	0.998130i	$0.480529\pi$
$702$	0	0
$703$	−5.58383	−0.210598
$704$	4.64042	0.174892
$705$	0	0
$706$	−17.1209	−0.644355
$707$	3.46582	0.130345
$708$	0	0
$709$	20.2370	0.760018	0.380009	−	0.924983i	$-0.375921\pi$
$709$	20.2370	0.760018	0.380009	+	0.924983i	$0.375921\pi$
$710$	0	0
$711$	0	0
$712$	−3.15720	−0.118321
$713$	15.3764	0.575851
$714$	0	0
$715$	0	0
$716$	12.4260	0.464383
$717$	0	0
$718$	−39.6086	−1.47818
$719$	−47.8804	−1.78564	−0.892819	−	0.450416i	$-0.851276\pi$
$719$	−47.8804	−1.78564	−0.892819	+	0.450416i	$0.851276\pi$
$720$	0	0
$721$	3.57315	0.133071
$722$	−1.65109	−0.0614473
$723$	0	0
$724$	−4.16707	−0.154868
$725$	0	0
$726$	0	0
$727$	−17.7926	−0.659890	−0.329945	−	0.944000i	$-0.607030\pi$
$727$	−17.7926	−0.659890	−0.329945	+	0.944000i	$0.607030\pi$
$728$	−2.23864	−0.0829697
$729$	0	0
$730$	0	0
$731$	77.4968	2.86632
$732$	0	0
$733$	5.66097	0.209093	0.104546	−	0.994520i	$-0.466661\pi$
$733$	5.66097	0.209093	0.104546	+	0.994520i	$0.466661\pi$
$734$	57.8337	2.13468
$735$	0	0
$736$	24.3305	0.896834
$737$	−4.44447	−0.163714
$738$	0	0
$739$	−5.59955	−0.205983	−0.102991	−	0.994682i	$-0.532841\pi$
$739$	−5.59955	−0.205983	−0.102991	+	0.994682i	$0.532841\pi$
$740$	0	0
$741$	0	0
$742$	−7.19193	−0.264024
$743$	5.81663	0.213391	0.106696	−	0.994292i	$-0.465973\pi$
$743$	5.81663	0.213391	0.106696	+	0.994292i	$0.465973\pi$
$744$	0	0
$745$	0	0
$746$	50.8542	1.86191
$747$	0	0
$748$	6.37720	0.233174
$749$	−5.79045	−0.211579
$750$	0	0
$751$	−41.1797	−1.50267	−0.751333	−	0.659923i	$-0.770589\pi$
$751$	−41.1797	−1.50267	−0.751333	+	0.659923i	$0.770589\pi$
$752$	33.8753	1.23531
$753$	0	0
$754$	15.7339	0.572993
$755$	0	0
$756$	0	0
$757$	33.5208	1.21833	0.609167	−	0.793042i	$-0.291504\pi$
$757$	33.5208	1.21833	0.609167	+	0.793042i	$0.291504\pi$
$758$	−1.10894	−0.0402785
$759$	0	0
$760$	0	0
$761$	−31.5032	−1.14199	−0.570995	−	0.820954i	$-0.693442\pi$
$761$	−31.5032	−1.14199	−0.570995	+	0.820954i	$0.693442\pi$
$762$	0	0
$763$	−6.32353	−0.228927
$764$	−16.2611	−0.588306
$765$	0	0
$766$	49.8307	1.80046
$767$	17.0897	0.617074
$768$	0	0
$769$	−1.22425	−0.0441475	−0.0220737	−	0.999756i	$-0.507027\pi$
$769$	−1.22425	−0.0441475	−0.0220737	+	0.999756i	$0.507027\pi$
$770$	0	0
$771$	0	0
$772$	15.5160	0.558432
$773$	40.6687	1.46275	0.731376	−	0.681974i	$-0.238878\pi$
$773$	40.6687	1.46275	0.731376	+	0.681974i	$0.238878\pi$
$774$	0	0
$775$	0	0
$776$	24.7827	0.889647
$777$	0	0
$778$	−52.0275	−1.86528
$779$	−8.50106	−0.304582
$780$	0	0
$781$	3.17058	0.113452
$782$	−65.2702	−2.33406
$783$	0	0
$784$	33.7742	1.20622
$785$	0	0
$786$	0	0
$787$	39.9525	1.42415	0.712076	−	0.702102i	$-0.247755\pi$
$787$	39.9525	1.42415	0.712076	+	0.702102i	$0.247755\pi$
$788$	0.808876	0.0288150
$789$	0	0
$790$	0	0
$791$	−5.02830	−0.178786
$792$	0	0
$793$	−14.1882	−0.503838
$794$	−20.7643	−0.736897
$795$	0	0
$796$	−1.61563	−0.0572644
$797$	−4.53791	−0.160741	−0.0803705	−	0.996765i	$-0.525610\pi$
$797$	−4.53791	−0.160741	−0.0803705	+	0.996765i	$0.525610\pi$
$798$	0	0
$799$	−43.8641	−1.55180
$800$	0	0
$801$	0	0
$802$	40.6228	1.43444
$803$	4.40550	0.155467
$804$	0	0
$805$	0	0
$806$	11.5563	0.407054
$807$	0	0
$808$	19.3257	0.679874
$809$	−55.5958	−1.95465	−0.977323	−	0.211756i	$-0.932082\pi$
$809$	−55.5958	−1.95465	−0.977323	+	0.211756i	$0.932082\pi$
$810$	0	0
$811$	42.3326	1.48650	0.743249	−	0.669014i	$-0.233284\pi$
$811$	42.3326	1.48650	0.743249	+	0.669014i	$0.233284\pi$
$812$	0.924984	0.0324606
$813$	0	0
$814$	12.6970	0.445030
$815$	0	0
$816$	0	0
$817$	12.1522	0.425150
$818$	−22.5648	−0.788961
$819$	0	0
$820$	0	0
$821$	3.10543	0.108380	0.0541902	−	0.998531i	$-0.482742\pi$
$821$	3.10543	0.108380	0.0541902	+	0.998531i	$0.482742\pi$
$822$	0	0
$823$	45.2002	1.57558	0.787790	−	0.615944i	$-0.211225\pi$
$823$	45.2002	1.57558	0.787790	+	0.615944i	$0.211225\pi$
$824$	19.9242	0.694092
$825$	0	0
$826$	3.77203	0.131246
$827$	−35.6503	−1.23968	−0.619841	−	0.784727i	$-0.712803\pi$
$827$	−35.6503	−1.23968	−0.619841	+	0.784727i	$0.712803\pi$
$828$	0	0
$829$	−18.4330	−0.640204	−0.320102	−	0.947383i	$-0.603717\pi$
$829$	−18.4330	−0.640204	−0.320102	+	0.947383i	$0.603717\pi$
$830$	0	0
$831$	0	0
$832$	−9.50749	−0.329613
$833$	−43.7331	−1.51526
$834$	0	0
$835$	0	0
$836$	1.00000	0.0345857
$837$	0	0
$838$	20.9512	0.723746
$839$	14.1805	0.489564	0.244782	−	0.969578i	$-0.421284\pi$
$839$	14.1805	0.489564	0.244782	+	0.969578i	$0.421284\pi$
$840$	0	0
$841$	−17.5945	−0.606707
$842$	−48.1832	−1.66050
$843$	0	0
$844$	−7.08277	−0.243799
$845$	0	0
$846$	0	0
$847$	−3.43380	−0.117987
$848$	−56.8726	−1.95301
$849$	0	0
$850$	0	0
$851$	−34.6134	−1.18653
$852$	0	0
$853$	−42.6639	−1.46078	−0.730392	−	0.683028i	$-0.760663\pi$
$853$	−42.6639	−1.46078	−0.730392	+	0.683028i	$0.760663\pi$
$854$	−3.13161	−0.107161
$855$	0	0
$856$	−32.2880	−1.10358
$857$	−1.42392	−0.0486403	−0.0243201	−	0.999704i	$-0.507742\pi$
$857$	−1.42392	−0.0486403	−0.0243201	+	0.999704i	$0.507742\pi$
$858$	0	0
$859$	−8.27894	−0.282474	−0.141237	−	0.989976i	$-0.545108\pi$
$859$	−8.27894	−0.282474	−0.141237	+	0.989976i	$0.545108\pi$
$860$	0	0
$861$	0	0
$862$	−49.5024	−1.68606
$863$	30.4154	1.03535	0.517676	−	0.855577i	$-0.326797\pi$
$863$	30.4154	1.03535	0.517676	+	0.855577i	$0.326797\pi$
$864$	0	0
$865$	0	0
$866$	−7.86276	−0.267188
$867$	0	0
$868$	0.679390	0.0230600
$869$	−9.25254	−0.313871
$870$	0	0
$871$	9.10602	0.308546
$872$	−35.2605	−1.19407
$873$	0	0
$874$	−10.2349	−0.346201
$875$	0	0
$876$	0	0
$877$	−57.8484	−1.95340	−0.976700	−	0.214608i	$-0.931153\pi$
$877$	−57.8484	−1.95340	−0.976700	+	0.214608i	$0.931153\pi$
$878$	−11.3695	−0.383700
$879$	0	0
$880$	0	0
$881$	−18.4055	−0.620097	−0.310049	−	0.950721i	$-0.600345\pi$
$881$	−18.4055	−0.620097	−0.310049	+	0.950721i	$0.600345\pi$
$882$	0	0
$883$	18.7947	0.632492	0.316246	−	0.948677i	$-0.397578\pi$
$883$	18.7947	0.632492	0.316246	+	0.948677i	$0.397578\pi$
$884$	−13.0659	−0.439453
$885$	0	0
$886$	−14.2838	−0.479872
$887$	−11.1471	−0.374283	−0.187142	−	0.982333i	$-0.559922\pi$
$887$	−11.1471	−0.374283	−0.187142	+	0.982333i	$0.559922\pi$
$888$	0	0
$889$	1.14923	0.0385440
$890$	0	0
$891$	0	0
$892$	13.5371	0.453256
$893$	−6.87826	−0.230172
$894$	0	0
$895$	0	0
$896$	−5.05952	−0.169027
$897$	0	0
$898$	14.2498	0.475523
$899$	8.37720	0.279395
$900$	0	0
$901$	73.6425	2.45339
$902$	19.3305	0.643635
$903$	0	0
$904$	−28.0382	−0.932536
$905$	0	0
$906$	0	0
$907$	−38.5598	−1.28036	−0.640178	−	0.768226i	$-0.721139\pi$
$907$	−38.5598	−1.28036	−0.640178	+	0.768226i	$0.721139\pi$
$908$	−5.84280	−0.193900
$909$	0	0
$910$	0	0
$911$	1.79820	0.0595771	0.0297885	−	0.999556i	$-0.490517\pi$
$911$	1.79820	0.0595771	0.0297885	+	0.999556i	$0.490517\pi$
$912$	0	0
$913$	25.0878	0.830285
$914$	−47.2365	−1.56244
$915$	0	0
$916$	14.6249	0.483221
$917$	3.28456	0.108466
$918$	0	0
$919$	−32.4458	−1.07029	−0.535144	−	0.844761i	$-0.679743\pi$
$919$	−32.4458	−1.07029	−0.535144	+	0.844761i	$0.679743\pi$
$920$	0	0
$921$	0	0
$922$	−64.5619	−2.12623
$923$	−6.49602	−0.213819
$924$	0	0
$925$	0	0
$926$	−13.2811	−0.436443
$927$	0	0
$928$	13.2555	0.435132
$929$	18.4231	0.604443	0.302222	−	0.953238i	$-0.402272\pi$
$929$	18.4231	0.604443	0.302222	+	0.953238i	$0.402272\pi$
$930$	0	0
$931$	−6.85772	−0.224753
$932$	−1.25897	−0.0412390
$933$	0	0
$934$	−28.1492	−0.921071
$935$	0	0
$936$	0	0
$937$	−8.29926	−0.271125	−0.135563	−	0.990769i	$-0.543284\pi$
$937$	−8.29926	−0.271125	−0.135563	+	0.990769i	$0.543284\pi$
$938$	2.00987	0.0656247
$939$	0	0
$940$	0	0
$941$	−33.6687	−1.09757	−0.548784	−	0.835964i	$-0.684909\pi$
$941$	−33.6687	−1.09757	−0.548784	+	0.835964i	$0.684909\pi$
$942$	0	0
$943$	−52.6970	−1.71605
$944$	29.8286	0.970839
$945$	0	0
$946$	−27.6327	−0.898416
$947$	1.35103	0.0439026	0.0219513	−	0.999759i	$-0.493012\pi$
$947$	1.35103	0.0439026	0.0219513	+	0.999759i	$0.493012\pi$
$948$	0	0
$949$	−9.02617	−0.293002
$950$	0	0
$951$	0	0
$952$	5.05952	0.163980
$953$	−6.70769	−0.217283	−0.108642	−	0.994081i	$-0.534650\pi$
$953$	−6.70769	−0.217283	−0.108642	+	0.994081i	$0.534650\pi$
$954$	0	0
$955$	0	0
$956$	13.2987	0.430110
$957$	0	0
$958$	37.2285	1.20280
$959$	2.48746	0.0803244
$960$	0	0
$961$	−24.8470	−0.801518
$962$	−26.0142	−0.838731
$963$	0	0
$964$	−1.59508	−0.0513741
$965$	0	0
$966$	0	0
$967$	−19.5174	−0.627636	−0.313818	−	0.949483i	$-0.601608\pi$
$967$	−19.5174	−0.627636	−0.313818	+	0.949483i	$0.601608\pi$
$968$	−19.1471	−0.615411
$969$	0	0
$970$	0	0
$971$	8.50669	0.272993	0.136496	−	0.990641i	$-0.456416\pi$
$971$	8.50669	0.272993	0.136496	+	0.990641i	$0.456416\pi$
$972$	0	0
$973$	−3.83527	−0.122953
$974$	−40.5470	−1.29921
$975$	0	0
$976$	−24.7643	−0.792685
$977$	19.7048	0.630411	0.315206	−	0.949023i	$-0.397927\pi$
$977$	19.7048	0.630411	0.315206	+	0.949023i	$0.397927\pi$
$978$	0	0
$979$	−2.06727	−0.0660701
$980$	0	0
$981$	0	0
$982$	−27.9527	−0.892006
$983$	−12.4677	−0.397658	−0.198829	−	0.980034i	$-0.563714\pi$
$983$	−12.4677	−0.397658	−0.198829	+	0.980034i	$0.563714\pi$
$984$	0	0
$985$	0	0
$986$	−35.5598	−1.13245
$987$	0	0
$988$	−2.04884	−0.0651824
$989$	75.3297	2.39534
$990$	0	0
$991$	25.0841	0.796822	0.398411	−	0.917207i	$-0.369562\pi$
$991$	25.0841	0.796822	0.398411	+	0.917207i	$0.369562\pi$
$992$	9.73598	0.309118
$993$	0	0
$994$	−1.43380	−0.0454772
$995$	0	0
$996$	0	0
$997$	−11.8775	−0.376163	−0.188082	−	0.982153i	$-0.560227\pi$
$997$	−11.8775	−0.376163	−0.188082	+	0.982153i	$0.560227\pi$
$998$	−0.690644	−0.0218620
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bm.1.1		3
3.2	odd	2		475.2.a.e.1.3	✓	3
5.4	even	2		4275.2.a.ba.1.3		3
12.11	even	2		7600.2.a.cc.1.3		3
15.2	even	4		475.2.b.b.324.5		6
15.8	even	4		475.2.b.b.324.2		6
15.14	odd	2		475.2.a.g.1.1	yes	3
57.56	even	2		9025.2.a.bc.1.1		3
60.59	even	2		7600.2.a.bh.1.1		3
285.284	even	2		9025.2.a.y.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.3	✓	3	3.2	odd	2
475.2.a.g.1.1	yes	3	15.14	odd	2
475.2.b.b.324.2		6	15.8	even	4
475.2.b.b.324.5		6	15.2	even	4
4275.2.a.ba.1.3		3	5.4	even	2
4275.2.a.bm.1.1		3	1.1	even	1	trivial
7600.2.a.bh.1.1		3	60.59	even	2
7600.2.a.cc.1.3		3	12.11	even	2
9025.2.a.y.1.3		3	285.284	even	2
9025.2.a.bc.1.1		3	57.56	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q-1.65109 q^{2} +0.726109 q^{4} +0.377203 q^{7} +2.10331 q^{8} +1.37720 q^{11} -2.82167 q^{13} -0.622797 q^{14} -4.92498 q^{16} +6.37720 q^{17} +1.00000 q^{19} -2.27389 q^{22} +6.19887 q^{23} +4.65884 q^{26} +0.273891 q^{28} +3.37720 q^{29} +2.48052 q^{31} +3.92498 q^{32} -10.5294 q^{34} -5.58383 q^{37} -1.65109 q^{38} -8.50106 q^{41} +12.1522 q^{43} +1.00000 q^{44} -10.2349 q^{46} -6.87826 q^{47} -6.85772 q^{49} -2.04884 q^{52} +11.5478 q^{53} +0.793375 q^{56} -5.57608 q^{58} -6.05659 q^{59} +5.02830 q^{61} -4.09556 q^{62} +3.36945 q^{64} -3.22717 q^{67} +4.63055 q^{68} +2.30219 q^{71} +3.19887 q^{73} +9.21942 q^{74} +0.726109 q^{76} +0.519485 q^{77} -6.71836 q^{79} +14.0360 q^{82} +18.2165 q^{83} -20.0643 q^{86} +2.89669 q^{88} -1.50106 q^{89} -1.06434 q^{91} +4.50106 q^{92} +11.3567 q^{94} +11.7827 q^{97} +11.3227 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{4} - 4 q^{7} + 3 q^{8} - q^{11} - 3 q^{13} - 7 q^{14} - 6 q^{16} + 14 q^{17} + 3 q^{19} - 5 q^{22} + 8 q^{23} + 11 q^{26} - q^{28} + 5 q^{29} - q^{31} + 3 q^{32} + 5 q^{34} - 5 q^{37}+ \cdots + 4 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^4 - 4 * q^7 + 3 * q^8 - q^11 - 3 * q^13 - 7 * q^14 - 6 * q^16 + 14 * q^17 + 3 * q^19 - 5 * q^22 + 8 * q^23 + 11 * q^26 - q^28 + 5 * q^29 - q^31 + 3 * q^32 + 5 * q^34 - 5 * q^37 + 2 * q^38 - q^41 + 5 * q^43 + 3 * q^44 - 12 * q^46 + 9 * q^47 - 7 * q^49 + 22 * q^52 + 31 * q^53 + 9 * q^56 - q^58 + 6 * q^59 + 3 * q^61 - 5 * q^62 + q^64 + 13 * q^67 + 23 * q^68 - 7 * q^71 - q^73 + q^74 + 4 * q^76 + 10 * q^77 - 18 * q^79 + 34 * q^82 + 3 * q^83 - 40 * q^86 + 12 * q^88 + 20 * q^89 + 17 * q^91 - 11 * q^92 + 45 * q^94 + 13 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more