LMFDB - Embedded newform 4275.2.a.bd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$1$
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 285)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ 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)

$f(q)$	$=$	$q+1.48119 q^{2} +0.193937 q^{4} +4.48119 q^{7} -2.67513 q^{8} -3.67513 q^{11} +2.86907 q^{13} +6.63752 q^{14} -4.35026 q^{16} -6.15633 q^{17} -1.00000 q^{19} -5.44358 q^{22} -8.15633 q^{23} +4.24965 q^{26} +0.869067 q^{28} -4.63752 q^{29} -2.80606 q^{31} -1.09332 q^{32} -9.11871 q^{34} -3.44358 q^{37} -1.48119 q^{38} +5.59991 q^{41} -10.1441 q^{43} -0.712742 q^{44} -12.0811 q^{46} -3.84367 q^{47} +13.0811 q^{49} +0.556417 q^{52} +3.89446 q^{53} -11.9878 q^{56} -6.86907 q^{58} +9.53690 q^{59} -9.89446 q^{61} -4.15633 q^{62} +7.08110 q^{64} +8.70052 q^{67} -1.19394 q^{68} +2.00000 q^{71} -3.08840 q^{73} -5.10062 q^{74} -0.193937 q^{76} -16.4690 q^{77} +4.96239 q^{79} +8.29455 q^{82} -6.73084 q^{83} -15.0254 q^{86} +9.83146 q^{88} -11.4739 q^{89} +12.8568 q^{91} -1.58181 q^{92} -5.69323 q^{94} +9.63023 q^{97} +19.3757 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + 8 q^{7} - 3 q^{8} - 6 q^{11} + 4 q^{13} + 4 q^{14} - 3 q^{16} - 8 q^{17} - 3 q^{19} - 14 q^{23} - 4 q^{26} - 2 q^{28} + 2 q^{29} - 8 q^{31} + 3 q^{32} - 6 q^{34} + 6 q^{37} + q^{38}+ \cdots + 33 q^{98}+O(q^{100}) $ 3 * q - q^2 + q^4 + 8 * q^7 - 3 * q^8 - 6 * q^11 + 4 * q^13 + 4 * q^14 - 3 * q^16 - 8 * q^17 - 3 * q^19 - 14 * q^23 - 4 * q^26 - 2 * q^28 + 2 * q^29 - 8 * q^31 + 3 * q^32 - 6 * q^34 + 6 * q^37 + q^38 - 10 * q^41 + 6 * q^43 - 8 * q^44 - 4 * q^46 - 22 * q^47 + 7 * q^49 + 18 * q^52 - 8 * q^53 - 10 * q^56 - 16 * q^58 + 6 * q^59 - 10 * q^61 - 2 * q^62 - 11 * q^64 + 6 * q^67 - 4 * q^68 + 6 * q^71 + 10 * q^73 - 22 * q^74 - q^76 - 18 * q^77 + 4 * q^79 + 32 * q^82 + 2 * q^83 - 30 * q^86 + 14 * q^88 - 16 * q^89 + 8 * q^91 - 6 * q^92 + 16 * q^94 + 33 * 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 $

$2$	1.48119	1.04736	0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	0	0
$3$	0	0
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.48119	1.69373	0.846866	−	0.531806i	$-0.178486\pi$
$7$	4.48119	1.69373	0.846866	+	0.531806i	$0.178486\pi$
$8$	−2.67513	−0.945802
$9$	0	0
$10$	0	0
$11$	−3.67513	−1.10809	−0.554047	−	0.832486i	$-0.686917\pi$
$11$	−3.67513	−1.10809	−0.554047	+	0.832486i	$0.686917\pi$
$12$	0	0
$13$	2.86907	0.795736	0.397868	−	0.917443i	$-0.369750\pi$
$13$	2.86907	0.795736	0.397868	+	0.917443i	$0.369750\pi$
$14$	6.63752	1.77395
$15$	0	0
$16$	−4.35026	−1.08757
$17$	−6.15633	−1.49313	−0.746564	−	0.665314i	$-0.768298\pi$
$17$	−6.15633	−1.49313	−0.746564	+	0.665314i	$0.768298\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−5.44358	−1.16058
$23$	−8.15633	−1.70071	−0.850356	−	0.526208i	$-0.823613\pi$
$23$	−8.15633	−1.70071	−0.850356	+	0.526208i	$0.823613\pi$
$24$	0	0
$25$	0	0
$26$	4.24965	0.833424
$27$	0	0
$28$	0.869067	0.164238
$29$	−4.63752	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$29$	−4.63752	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$30$	0	0
$31$	−2.80606	−0.503984	−0.251992	−	0.967729i	$-0.581086\pi$
$31$	−2.80606	−0.503984	−0.251992	+	0.967729i	$0.581086\pi$
$32$	−1.09332	−0.193274
$33$	0	0
$34$	−9.11871	−1.56385
$35$	0	0
$36$	0	0
$37$	−3.44358	−0.566122	−0.283061	−	0.959102i	$-0.591350\pi$
$37$	−3.44358	−0.566122	−0.283061	+	0.959102i	$0.591350\pi$
$38$	−1.48119	−0.240281
$39$	0	0
$40$	0	0
$41$	5.59991	0.874559	0.437279	−	0.899326i	$-0.355942\pi$
$41$	5.59991	0.874559	0.437279	+	0.899326i	$0.355942\pi$
$42$	0	0
$43$	−10.1441	−1.54696	−0.773481	−	0.633820i	$-0.781486\pi$
$43$	−10.1441	−1.54696	−0.773481	+	0.633820i	$0.781486\pi$
$44$	−0.712742	−0.107450
$45$	0	0
$46$	−12.0811	−1.78126
$47$	−3.84367	−0.560658	−0.280329	−	0.959904i	$-0.590443\pi$
$47$	−3.84367	−0.560658	−0.280329	+	0.959904i	$0.590443\pi$
$48$	0	0
$49$	13.0811	1.86873
$50$	0	0
$51$	0	0
$52$	0.556417	0.0771612
$53$	3.89446	0.534945	0.267473	−	0.963565i	$-0.413812\pi$
$53$	3.89446	0.534945	0.267473	+	0.963565i	$0.413812\pi$
$54$	0	0
$55$	0	0
$56$	−11.9878	−1.60193
$57$	0	0
$58$	−6.86907	−0.901953
$59$	9.53690	1.24160	0.620800	−	0.783969i	$-0.286808\pi$
$59$	9.53690	1.24160	0.620800	+	0.783969i	$0.286808\pi$
$60$	0	0
$61$	−9.89446	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$61$	−9.89446	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$62$	−4.15633	−0.527854
$63$	0	0
$64$	7.08110	0.885138
$65$	0	0
$66$	0	0
$67$	8.70052	1.06294	0.531469	−	0.847078i	$-0.321640\pi$
$67$	8.70052	1.06294	0.531469	+	0.847078i	$0.321640\pi$
$68$	−1.19394	−0.144786
$69$	0	0
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	−3.08840	−0.361469	−0.180735	−	0.983532i	$-0.557848\pi$
$73$	−3.08840	−0.361469	−0.180735	+	0.983532i	$0.557848\pi$
$74$	−5.10062	−0.592934
$75$	0	0
$76$	−0.193937	−0.0222460
$77$	−16.4690	−1.87681
$78$	0	0
$79$	4.96239	0.558312	0.279156	−	0.960246i	$-0.409945\pi$
$79$	4.96239	0.558312	0.279156	+	0.960246i	$0.409945\pi$
$80$	0	0
$81$	0	0
$82$	8.29455	0.915980
$83$	−6.73084	−0.738806	−0.369403	−	0.929269i	$-0.620438\pi$
$83$	−6.73084	−0.738806	−0.369403	+	0.929269i	$0.620438\pi$
$84$	0	0
$85$	0	0
$86$	−15.0254	−1.62023
$87$	0	0
$88$	9.83146	1.04804
$89$	−11.4739	−1.21623	−0.608115	−	0.793849i	$-0.708074\pi$
$89$	−11.4739	−1.21623	−0.608115	+	0.793849i	$0.708074\pi$
$90$	0	0
$91$	12.8568	1.34776
$92$	−1.58181	−0.164915
$93$	0	0
$94$	−5.69323	−0.587212
$95$	0	0
$96$	0	0
$97$	9.63023	0.977801	0.488901	−	0.872339i	$-0.337398\pi$
$97$	9.63023	0.977801	0.488901	+	0.872339i	$0.337398\pi$
$98$	19.3757	1.95724
$99$	0	0
$100$	0	0
$101$	−5.61213	−0.558427	−0.279214	−	0.960229i	$-0.590074\pi$
$101$	−5.61213	−0.558427	−0.279214	+	0.960229i	$0.590074\pi$
$102$	0	0
$103$	−3.53690	−0.348502	−0.174251	−	0.984701i	$-0.555750\pi$
$103$	−3.53690	−0.348502	−0.174251	+	0.984701i	$0.555750\pi$
$104$	−7.67513	−0.752609
$105$	0	0
$106$	5.76845	0.560282
$107$	16.4387	1.58919	0.794593	−	0.607143i	$-0.207685\pi$
$107$	16.4387	1.58919	0.794593	+	0.607143i	$0.207685\pi$
$108$	0	0
$109$	−12.3879	−1.18654	−0.593272	−	0.805002i	$-0.702164\pi$
$109$	−12.3879	−1.18654	−0.593272	+	0.805002i	$0.702164\pi$
$110$	0	0
$111$	0	0
$112$	−19.4944	−1.84204
$113$	−11.2447	−1.05781	−0.528907	−	0.848680i	$-0.677398\pi$
$113$	−11.2447	−1.05781	−0.528907	+	0.848680i	$0.677398\pi$
$114$	0	0
$115$	0	0
$116$	−0.899385	−0.0835058
$117$	0	0
$118$	14.1260	1.30040
$119$	−27.5877	−2.52896
$120$	0	0
$121$	2.50659	0.227872
$122$	−14.6556	−1.32686
$123$	0	0
$124$	−0.544198	−0.0488705
$125$	0	0
$126$	0	0
$127$	−11.8641	−1.05277	−0.526386	−	0.850246i	$-0.676453\pi$
$127$	−11.8641	−1.05277	−0.526386	+	0.850246i	$0.676453\pi$
$128$	12.6751	1.12033
$129$	0	0
$130$	0	0
$131$	−1.54912	−0.135347	−0.0676737	−	0.997708i	$-0.521558\pi$
$131$	−1.54912	−0.135347	−0.0676737	+	0.997708i	$0.521558\pi$
$132$	0	0
$133$	−4.48119	−0.388569
$134$	12.8872	1.11328
$135$	0	0
$136$	16.4690	1.41220
$137$	17.0738	1.45871	0.729357	−	0.684133i	$-0.239819\pi$
$137$	17.0738	1.45871	0.729357	+	0.684133i	$0.239819\pi$
$138$	0	0
$139$	5.61213	0.476014	0.238007	−	0.971263i	$-0.423506\pi$
$139$	5.61213	0.476014	0.238007	+	0.971263i	$0.423506\pi$
$140$	0	0
$141$	0	0
$142$	2.96239	0.248598
$143$	−10.5442	−0.881750
$144$	0	0
$145$	0	0
$146$	−4.57452	−0.378590
$147$	0	0
$148$	−0.667837	−0.0548958
$149$	−5.53690	−0.453601	−0.226800	−	0.973941i	$-0.572827\pi$
$149$	−5.53690	−0.453601	−0.226800	+	0.973941i	$0.572827\pi$
$150$	0	0
$151$	0.0811024	0.00660002	0.00330001	−	0.999995i	$-0.498950\pi$
$151$	0.0811024	0.00660002	0.00330001	+	0.999995i	$0.498950\pi$
$152$	2.67513	0.216982
$153$	0	0
$154$	−24.3938	−1.96570
$155$	0	0
$156$	0	0
$157$	−2.64974	−0.211472	−0.105736	−	0.994394i	$-0.533720\pi$
$157$	−2.64974	−0.211472	−0.105736	+	0.994394i	$0.533720\pi$
$158$	7.35026	0.584755
$159$	0	0
$160$	0	0
$161$	−36.5501	−2.88055
$162$	0	0
$163$	−0.667837	−0.0523090	−0.0261545	−	0.999658i	$-0.508326\pi$
$163$	−0.667837	−0.0523090	−0.0261545	+	0.999658i	$0.508326\pi$
$164$	1.08603	0.0848045
$165$	0	0
$166$	−9.96968	−0.773797
$167$	−10.4182	−0.806184	−0.403092	−	0.915160i	$-0.632064\pi$
$167$	−10.4182	−0.806184	−0.403092	+	0.915160i	$0.632064\pi$
$168$	0	0
$169$	−4.76845	−0.366804
$170$	0	0
$171$	0	0
$172$	−1.96731	−0.150006
$173$	−4.54420	−0.345489	−0.172744	−	0.984967i	$-0.555263\pi$
$173$	−4.54420	−0.345489	−0.172744	+	0.984967i	$0.555263\pi$
$174$	0	0
$175$	0	0
$176$	15.9878	1.20512
$177$	0	0
$178$	−16.9951	−1.27383
$179$	15.0884	1.12776	0.563880	−	0.825857i	$-0.309308\pi$
$179$	15.0884	1.12776	0.563880	+	0.825857i	$0.309308\pi$
$180$	0	0
$181$	−8.44851	−0.627973	−0.313986	−	0.949428i	$-0.601665\pi$
$181$	−8.44851	−0.627973	−0.313986	+	0.949428i	$0.601665\pi$
$182$	19.0435	1.41160
$183$	0	0
$184$	21.8192	1.60854
$185$	0	0
$186$	0	0
$187$	22.6253	1.65453
$188$	−0.745429	−0.0543660
$189$	0	0
$190$	0	0
$191$	13.5393	0.979667	0.489834	−	0.871816i	$-0.337057\pi$
$191$	13.5393	0.979667	0.489834	+	0.871816i	$0.337057\pi$
$192$	0	0
$193$	−11.1817	−0.804878	−0.402439	−	0.915447i	$-0.631837\pi$
$193$	−11.1817	−0.804878	−0.402439	+	0.915447i	$0.631837\pi$
$194$	14.2642	1.02411
$195$	0	0
$196$	2.53690	0.181207
$197$	−7.24472	−0.516165	−0.258083	−	0.966123i	$-0.583091\pi$
$197$	−7.24472	−0.516165	−0.258083	+	0.966123i	$0.583091\pi$
$198$	0	0
$199$	−26.8627	−1.90425	−0.952124	−	0.305712i	$-0.901106\pi$
$199$	−26.8627	−1.90425	−0.952124	+	0.305712i	$0.901106\pi$
$200$	0	0
$201$	0	0
$202$	−8.31265	−0.584876
$203$	−20.7816	−1.45858
$204$	0	0
$205$	0	0
$206$	−5.23884	−0.365007
$207$	0	0
$208$	−12.4812	−0.865415
$209$	3.67513	0.254214
$210$	0	0
$211$	4.43866	0.305570	0.152785	−	0.988259i	$-0.451176\pi$
$211$	4.43866	0.305570	0.152785	+	0.988259i	$0.451176\pi$
$212$	0.755278	0.0518727
$213$	0	0
$214$	24.3488	1.66445
$215$	0	0
$216$	0	0
$217$	−12.5745	−0.853614
$218$	−18.3488	−1.24274
$219$	0	0
$220$	0	0
$221$	−17.6629	−1.18814
$222$	0	0
$223$	9.08840	0.608604	0.304302	−	0.952576i	$-0.401577\pi$
$223$	9.08840	0.608604	0.304302	+	0.952576i	$0.401577\pi$
$224$	−4.89938	−0.327354
$225$	0	0
$226$	−16.6556	−1.10792
$227$	−11.1939	−0.742968	−0.371484	−	0.928439i	$-0.621151\pi$
$227$	−11.1939	−0.742968	−0.371484	+	0.928439i	$0.621151\pi$
$228$	0	0
$229$	−23.6932	−1.56569	−0.782846	−	0.622215i	$-0.786233\pi$
$229$	−23.6932	−1.56569	−0.782846	+	0.622215i	$0.786233\pi$
$230$	0	0
$231$	0	0
$232$	12.4060	0.814492
$233$	12.0508	0.789473	0.394737	−	0.918794i	$-0.370836\pi$
$233$	12.0508	0.789473	0.394737	+	0.918794i	$0.370836\pi$
$234$	0	0
$235$	0	0
$236$	1.84955	0.120396
$237$	0	0
$238$	−40.8627	−2.64874
$239$	−13.9878	−0.904794	−0.452397	−	0.891817i	$-0.649431\pi$
$239$	−13.9878	−0.904794	−0.452397	+	0.891817i	$0.649431\pi$
$240$	0	0
$241$	23.7743	1.53144	0.765720	−	0.643174i	$-0.222383\pi$
$241$	23.7743	1.53144	0.765720	+	0.643174i	$0.222383\pi$
$242$	3.71274	0.238664
$243$	0	0
$244$	−1.91890	−0.122845
$245$	0	0
$246$	0	0
$247$	−2.86907	−0.182554
$248$	7.50659	0.476669
$249$	0	0
$250$	0	0
$251$	−6.76353	−0.426910	−0.213455	−	0.976953i	$-0.568472\pi$
$251$	−6.76353	−0.426910	−0.213455	+	0.976953i	$0.568472\pi$
$252$	0	0
$253$	29.9756	1.88455
$254$	−17.5731	−1.10263
$255$	0	0
$256$	4.61213	0.288258
$257$	19.9551	1.24476	0.622382	−	0.782713i	$-0.286165\pi$
$257$	19.9551	1.24476	0.622382	+	0.782713i	$0.286165\pi$
$258$	0	0
$259$	−15.4314	−0.958858
$260$	0	0
$261$	0	0
$262$	−2.29455	−0.141758
$263$	−5.14315	−0.317140	−0.158570	−	0.987348i	$-0.550688\pi$
$263$	−5.14315	−0.317140	−0.158570	+	0.987348i	$0.550688\pi$
$264$	0	0
$265$	0	0
$266$	−6.63752	−0.406972
$267$	0	0
$268$	1.68735	0.103071
$269$	11.1368	0.679023	0.339512	−	0.940602i	$-0.389738\pi$
$269$	11.1368	0.679023	0.339512	+	0.940602i	$0.389738\pi$
$270$	0	0
$271$	15.4763	0.940116	0.470058	−	0.882635i	$-0.344233\pi$
$271$	15.4763	0.940116	0.470058	+	0.882635i	$0.344233\pi$
$272$	26.7816	1.62387
$273$	0	0
$274$	25.2896	1.52780
$275$	0	0
$276$	0	0
$277$	12.6253	0.758581	0.379290	−	0.925278i	$-0.376168\pi$
$277$	12.6253	0.758581	0.379290	+	0.925278i	$0.376168\pi$
$278$	8.31265	0.498560
$279$	0	0
$280$	0	0
$281$	−2.66196	−0.158799	−0.0793995	−	0.996843i	$-0.525300\pi$
$281$	−2.66196	−0.158799	−0.0793995	+	0.996843i	$0.525300\pi$
$282$	0	0
$283$	1.63023	0.0969068	0.0484534	−	0.998825i	$-0.484571\pi$
$283$	1.63023	0.0969068	0.0484534	+	0.998825i	$0.484571\pi$
$284$	0.387873	0.0230160
$285$	0	0
$286$	−15.6180	−0.923512
$287$	25.0943	1.48127
$288$	0	0
$289$	20.9003	1.22943
$290$	0	0
$291$	0	0
$292$	−0.598953	−0.0350511
$293$	15.3707	0.897968	0.448984	−	0.893540i	$-0.351786\pi$
$293$	15.3707	0.897968	0.448984	+	0.893540i	$0.351786\pi$
$294$	0	0
$295$	0	0
$296$	9.21203	0.535439
$297$	0	0
$298$	−8.20123	−0.475085
$299$	−23.4010	−1.35332
$300$	0	0
$301$	−45.4577	−2.62014
$302$	0.120128	0.00691261
$303$	0	0
$304$	4.35026	0.249505
$305$	0	0
$306$	0	0
$307$	18.3634	1.04806	0.524028	−	0.851701i	$-0.324429\pi$
$307$	18.3634	1.04806	0.524028	+	0.851701i	$0.324429\pi$
$308$	−3.19394	−0.181991
$309$	0	0
$310$	0	0
$311$	−11.4739	−0.650625	−0.325313	−	0.945607i	$-0.605470\pi$
$311$	−11.4739	−0.650625	−0.325313	+	0.945607i	$0.605470\pi$
$312$	0	0
$313$	−4.57452	−0.258567	−0.129283	−	0.991608i	$-0.541268\pi$
$313$	−4.57452	−0.258567	−0.129283	+	0.991608i	$0.541268\pi$
$314$	−3.92478	−0.221488
$315$	0	0
$316$	0.962389	0.0541386
$317$	22.4083	1.25858	0.629289	−	0.777171i	$-0.283346\pi$
$317$	22.4083	1.25858	0.629289	+	0.777171i	$0.283346\pi$
$318$	0	0
$319$	17.0435	0.954252
$320$	0	0
$321$	0	0
$322$	−54.1378	−3.01698
$323$	6.15633	0.342547
$324$	0	0
$325$	0	0
$326$	−0.989196	−0.0547865
$327$	0	0
$328$	−14.9805	−0.827159
$329$	−17.2243	−0.949604
$330$	0	0
$331$	−31.8700	−1.75173	−0.875867	−	0.482552i	$-0.839710\pi$
$331$	−31.8700	−1.75173	−0.875867	+	0.482552i	$0.839710\pi$
$332$	−1.30536	−0.0716407
$333$	0	0
$334$	−15.4314	−0.844367
$335$	0	0
$336$	0	0
$337$	12.3815	0.674465	0.337233	−	0.941421i	$-0.390509\pi$
$337$	12.3815	0.674465	0.337233	+	0.941421i	$0.390509\pi$
$338$	−7.06300	−0.384177
$339$	0	0
$340$	0	0
$341$	10.3127	0.558461
$342$	0	0
$343$	27.2506	1.47139
$344$	27.1368	1.46312
$345$	0	0
$346$	−6.73084	−0.361852
$347$	−29.7440	−1.59674	−0.798371	−	0.602166i	$-0.794305\pi$
$347$	−29.7440	−1.59674	−0.798371	+	0.602166i	$0.794305\pi$
$348$	0	0
$349$	−3.42548	−0.183362	−0.0916810	−	0.995788i	$-0.529224\pi$
$349$	−3.42548	−0.183362	−0.0916810	+	0.995788i	$0.529224\pi$
$350$	0	0
$351$	0	0
$352$	4.01810	0.214165
$353$	7.48612	0.398446	0.199223	−	0.979954i	$-0.436158\pi$
$353$	7.48612	0.398446	0.199223	+	0.979954i	$0.436158\pi$
$354$	0	0
$355$	0	0
$356$	−2.22521	−0.117936
$357$	0	0
$358$	22.3488	1.18117
$359$	35.1632	1.85584	0.927920	−	0.372779i	$-0.121595\pi$
$359$	35.1632	1.85584	0.927920	+	0.372779i	$0.121595\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−12.5139	−0.657715
$363$	0	0
$364$	2.49341	0.130690
$365$	0	0
$366$	0	0
$367$	−2.14411	−0.111921	−0.0559607	−	0.998433i	$-0.517822\pi$
$367$	−2.14411	−0.111921	−0.0559607	+	0.998433i	$0.517822\pi$
$368$	35.4821	1.84963
$369$	0	0
$370$	0	0
$371$	17.4518	0.906054
$372$	0	0
$373$	36.4930	1.88953	0.944767	−	0.327743i	$-0.106288\pi$
$373$	36.4930	1.88953	0.944767	+	0.327743i	$0.106288\pi$
$374$	33.5125	1.73289
$375$	0	0
$376$	10.2823	0.530271
$377$	−13.3054	−0.685261
$378$	0	0
$379$	2.00588	0.103035	0.0515176	−	0.998672i	$-0.483594\pi$
$379$	2.00588	0.103035	0.0515176	+	0.998672i	$0.483594\pi$
$380$	0	0
$381$	0	0
$382$	20.0543	1.02607
$383$	−18.2374	−0.931889	−0.465945	−	0.884814i	$-0.654285\pi$
$383$	−18.2374	−0.931889	−0.465945	+	0.884814i	$0.654285\pi$
$384$	0	0
$385$	0	0
$386$	−16.5623	−0.842999
$387$	0	0
$388$	1.86765	0.0948157
$389$	−13.2144	−0.669997	−0.334998	−	0.942219i	$-0.608736\pi$
$389$	−13.2144	−0.669997	−0.334998	+	0.942219i	$0.608736\pi$
$390$	0	0
$391$	50.2130	2.53938
$392$	−34.9937	−1.76745
$393$	0	0
$394$	−10.7308	−0.540612
$395$	0	0
$396$	0	0
$397$	−9.46168	−0.474868	−0.237434	−	0.971404i	$-0.576306\pi$
$397$	−9.46168	−0.474868	−0.237434	+	0.971404i	$0.576306\pi$
$398$	−39.7889	−1.99444
$399$	0	0
$400$	0	0
$401$	28.0240	1.39945	0.699725	−	0.714412i	$-0.253306\pi$
$401$	28.0240	1.39945	0.699725	+	0.714412i	$0.253306\pi$
$402$	0	0
$403$	−8.05079	−0.401038
$404$	−1.08840	−0.0541498
$405$	0	0
$406$	−30.7816	−1.52767
$407$	12.6556	0.627316
$408$	0	0
$409$	−17.9756	−0.888834	−0.444417	−	0.895820i	$-0.646589\pi$
$409$	−17.9756	−0.888834	−0.444417	+	0.895820i	$0.646589\pi$
$410$	0	0
$411$	0	0
$412$	−0.685935	−0.0337936
$413$	42.7367	2.10294
$414$	0	0
$415$	0	0
$416$	−3.13681	−0.153795
$417$	0	0
$418$	5.44358	0.266254
$419$	−13.4377	−0.656475	−0.328237	−	0.944595i	$-0.606455\pi$
$419$	−13.4377	−0.656475	−0.328237	+	0.944595i	$0.606455\pi$
$420$	0	0
$421$	−24.3634	−1.18740	−0.593701	−	0.804686i	$-0.702334\pi$
$421$	−24.3634	−1.18740	−0.593701	+	0.804686i	$0.702334\pi$
$422$	6.57452	0.320042
$423$	0	0
$424$	−10.4182	−0.505952
$425$	0	0
$426$	0	0
$427$	−44.3390	−2.14571
$428$	3.18806	0.154101
$429$	0	0
$430$	0	0
$431$	1.42548	0.0686632	0.0343316	−	0.999410i	$-0.489070\pi$
$431$	1.42548	0.0686632	0.0343316	+	0.999410i	$0.489070\pi$
$432$	0	0
$433$	1.65466	0.0795180	0.0397590	−	0.999209i	$-0.487341\pi$
$433$	1.65466	0.0795180	0.0397590	+	0.999209i	$0.487341\pi$
$434$	−18.6253	−0.894043
$435$	0	0
$436$	−2.40246	−0.115057
$437$	8.15633	0.390170
$438$	0	0
$439$	−26.3996	−1.25999	−0.629993	−	0.776601i	$-0.716942\pi$
$439$	−26.3996	−1.25999	−0.629993	+	0.776601i	$0.716942\pi$
$440$	0	0
$441$	0	0
$442$	−26.1622	−1.24441
$443$	3.35614	0.159455	0.0797275	−	0.996817i	$-0.474595\pi$
$443$	3.35614	0.159455	0.0797275	+	0.996817i	$0.474595\pi$
$444$	0	0
$445$	0	0
$446$	13.4617	0.637429
$447$	0	0
$448$	31.7318	1.49919
$449$	−12.0122	−0.566892	−0.283446	−	0.958988i	$-0.591478\pi$
$449$	−12.0122	−0.566892	−0.283446	+	0.958988i	$0.591478\pi$
$450$	0	0
$451$	−20.5804	−0.969093
$452$	−2.18076	−0.102574
$453$	0	0
$454$	−16.5804	−0.778156
$455$	0	0
$456$	0	0
$457$	−31.5877	−1.47761	−0.738805	−	0.673919i	$-0.764609\pi$
$457$	−31.5877	−1.47761	−0.738805	+	0.673919i	$0.764609\pi$
$458$	−35.0943	−1.63985
$459$	0	0
$460$	0	0
$461$	−5.07381	−0.236311	−0.118155	−	0.992995i	$-0.537698\pi$
$461$	−5.07381	−0.236311	−0.118155	+	0.992995i	$0.537698\pi$
$462$	0	0
$463$	14.2701	0.663188	0.331594	−	0.943422i	$-0.392414\pi$
$463$	14.2701	0.663188	0.331594	+	0.943422i	$0.392414\pi$
$464$	20.1744	0.936574
$465$	0	0
$466$	17.8496	0.826865
$467$	−32.3839	−1.49855	−0.749274	−	0.662260i	$-0.769597\pi$
$467$	−32.3839	−1.49855	−0.749274	+	0.662260i	$0.769597\pi$
$468$	0	0
$469$	38.9887	1.80033
$470$	0	0
$471$	0	0
$472$	−25.5125	−1.17431
$473$	37.2809	1.71418
$474$	0	0
$475$	0	0
$476$	−5.35026	−0.245229
$477$	0	0
$478$	−20.7186	−0.947648
$479$	7.28726	0.332963	0.166482	−	0.986045i	$-0.446759\pi$
$479$	7.28726	0.332963	0.166482	+	0.986045i	$0.446759\pi$
$480$	0	0
$481$	−9.87987	−0.450483
$482$	35.2144	1.60397
$483$	0	0
$484$	0.486119	0.0220963
$485$	0	0
$486$	0	0
$487$	37.5731	1.70260	0.851300	−	0.524679i	$-0.175815\pi$
$487$	37.5731	1.70260	0.851300	+	0.524679i	$0.175815\pi$
$488$	26.4690	1.19819
$489$	0	0
$490$	0	0
$491$	−23.5901	−1.06460	−0.532302	−	0.846554i	$-0.678673\pi$
$491$	−23.5901	−1.06460	−0.532302	+	0.846554i	$0.678673\pi$
$492$	0	0
$493$	28.5501	1.28583
$494$	−4.24965	−0.191201
$495$	0	0
$496$	12.2071	0.548115
$497$	8.96239	0.402018
$498$	0	0
$499$	−25.7137	−1.15110	−0.575552	−	0.817765i	$-0.695213\pi$
$499$	−25.7137	−1.15110	−0.575552	+	0.817765i	$0.695213\pi$
$500$	0	0
$501$	0	0
$502$	−10.0181	−0.447130
$503$	−0.0956908	−0.00426664	−0.00213332	−	0.999998i	$-0.500679\pi$
$503$	−0.0956908	−0.00426664	−0.00213332	+	0.999998i	$0.500679\pi$
$504$	0	0
$505$	0	0
$506$	44.3996	1.97380
$507$	0	0
$508$	−2.30089	−0.102086
$509$	−23.0010	−1.01950	−0.509750	−	0.860323i	$-0.670262\pi$
$509$	−23.0010	−1.01950	−0.509750	+	0.860323i	$0.670262\pi$
$510$	0	0
$511$	−13.8397	−0.612233
$512$	−18.5188	−0.818423
$513$	0	0
$514$	29.5574	1.30372
$515$	0	0
$516$	0	0
$517$	14.1260	0.621261
$518$	−22.8568	−1.00427
$519$	0	0
$520$	0	0
$521$	−14.2398	−0.623857	−0.311928	−	0.950106i	$-0.600975\pi$
$521$	−14.2398	−0.623857	−0.311928	+	0.950106i	$0.600975\pi$
$522$	0	0
$523$	−17.9756	−0.786016	−0.393008	−	0.919535i	$-0.628566\pi$
$523$	−17.9756	−0.786016	−0.393008	+	0.919535i	$0.628566\pi$
$524$	−0.300432	−0.0131244
$525$	0	0
$526$	−7.61801	−0.332161
$527$	17.2750	0.752513
$528$	0	0
$529$	43.5256	1.89242
$530$	0	0
$531$	0	0
$532$	−0.869067	−0.0376789
$533$	16.0665	0.695918
$534$	0	0
$535$	0	0
$536$	−23.2750	−1.00533
$537$	0	0
$538$	16.4958	0.711184
$539$	−48.0748	−2.07073
$540$	0	0
$541$	−27.1490	−1.16723	−0.583614	−	0.812031i	$-0.698362\pi$
$541$	−27.1490	−1.16723	−0.583614	+	0.812031i	$0.698362\pi$
$542$	22.9234	0.984643
$543$	0	0
$544$	6.73084	0.288582
$545$	0	0
$546$	0	0
$547$	34.5256	1.47621	0.738105	−	0.674686i	$-0.235721\pi$
$547$	34.5256	1.47621	0.738105	+	0.674686i	$0.235721\pi$
$548$	3.31124	0.141449
$549$	0	0
$550$	0	0
$551$	4.63752	0.197565
$552$	0	0
$553$	22.2374	0.945632
$554$	18.7005	0.794509
$555$	0	0
$556$	1.08840	0.0461583
$557$	21.6873	0.918922	0.459461	−	0.888198i	$-0.348043\pi$
$557$	21.6873	0.918922	0.459461	+	0.888198i	$0.348043\pi$
$558$	0	0
$559$	−29.1041	−1.23097
$560$	0	0
$561$	0	0
$562$	−3.94288	−0.166320
$563$	8.80606	0.371131	0.185566	−	0.982632i	$-0.440588\pi$
$563$	8.80606	0.371131	0.185566	+	0.982632i	$0.440588\pi$
$564$	0	0
$565$	0	0
$566$	2.41468	0.101497
$567$	0	0
$568$	−5.35026	−0.224492
$569$	26.5379	1.11252	0.556262	−	0.831007i	$-0.312235\pi$
$569$	26.5379	1.11252	0.556262	+	0.831007i	$0.312235\pi$
$570$	0	0
$571$	31.0376	1.29888	0.649442	−	0.760411i	$-0.275003\pi$
$571$	31.0376	1.29888	0.649442	+	0.760411i	$0.275003\pi$
$572$	−2.04491	−0.0855018
$573$	0	0
$574$	37.1695	1.55142
$575$	0	0
$576$	0	0
$577$	23.3357	0.971477	0.485738	−	0.874104i	$-0.338551\pi$
$577$	23.3357	0.971477	0.485738	+	0.874104i	$0.338551\pi$
$578$	30.9575	1.28766
$579$	0	0
$580$	0	0
$581$	−30.1622	−1.25134
$582$	0	0
$583$	−14.3127	−0.592769
$584$	8.26187	0.341878
$585$	0	0
$586$	22.7670	0.940498
$587$	19.4168	0.801416	0.400708	−	0.916206i	$-0.368764\pi$
$587$	19.4168	0.801416	0.400708	+	0.916206i	$0.368764\pi$
$588$	0	0
$589$	2.80606	0.115622
$590$	0	0
$591$	0	0
$592$	14.9805	0.615694
$593$	−6.21108	−0.255058	−0.127529	−	0.991835i	$-0.540705\pi$
$593$	−6.21108	−0.255058	−0.127529	+	0.991835i	$0.540705\pi$
$594$	0	0
$595$	0	0
$596$	−1.07381	−0.0439849
$597$	0	0
$598$	−34.6615	−1.41741
$599$	−30.4993	−1.24617	−0.623084	−	0.782155i	$-0.714120\pi$
$599$	−30.4993	−1.24617	−0.623084	+	0.782155i	$0.714120\pi$
$600$	0	0
$601$	16.8218	0.686175	0.343088	−	0.939303i	$-0.388527\pi$
$601$	16.8218	0.686175	0.343088	+	0.939303i	$0.388527\pi$
$602$	−67.3317	−2.74424
$603$	0	0
$604$	0.0157287	0.000639993 0
$605$	0	0
$606$	0	0
$607$	37.7645	1.53281	0.766407	−	0.642356i	$-0.222043\pi$
$607$	37.7645	1.53281	0.766407	+	0.642356i	$0.222043\pi$
$608$	1.09332	0.0443400
$609$	0	0
$610$	0	0
$611$	−11.0278	−0.446136
$612$	0	0
$613$	25.7235	1.03896	0.519482	−	0.854481i	$-0.326125\pi$
$613$	25.7235	1.03896	0.519482	+	0.854481i	$0.326125\pi$
$614$	27.1998	1.09770
$615$	0	0
$616$	44.0567	1.77509
$617$	9.31994	0.375207	0.187603	−	0.982245i	$-0.439928\pi$
$617$	9.31994	0.375207	0.187603	+	0.982245i	$0.439928\pi$
$618$	0	0
$619$	33.9610	1.36501	0.682503	−	0.730882i	$-0.260891\pi$
$619$	33.9610	1.36501	0.682503	+	0.730882i	$0.260891\pi$
$620$	0	0
$621$	0	0
$622$	−16.9951	−0.681440
$623$	−51.4168	−2.05997
$624$	0	0
$625$	0	0
$626$	−6.77575	−0.270813
$627$	0	0
$628$	−0.513881	−0.0205061
$629$	21.1998	0.845292
$630$	0	0
$631$	31.3620	1.24850	0.624251	−	0.781224i	$-0.285404\pi$
$631$	31.3620	1.24850	0.624251	+	0.781224i	$0.285404\pi$
$632$	−13.2750	−0.528053
$633$	0	0
$634$	33.1911	1.31819
$635$	0	0
$636$	0	0
$637$	37.5306	1.48702
$638$	25.2447	0.999448
$639$	0	0
$640$	0	0
$641$	13.5853	0.536588	0.268294	−	0.963337i	$-0.413540\pi$
$641$	13.5853	0.536588	0.268294	+	0.963337i	$0.413540\pi$
$642$	0	0
$643$	41.0557	1.61908	0.809540	−	0.587065i	$-0.199717\pi$
$643$	41.0557	1.61908	0.809540	+	0.587065i	$0.199717\pi$
$644$	−7.08840	−0.279322
$645$	0	0
$646$	9.11871	0.358771
$647$	−16.2170	−0.637554	−0.318777	−	0.947830i	$-0.603272\pi$
$647$	−16.2170	−0.637554	−0.318777	+	0.947830i	$0.603272\pi$
$648$	0	0
$649$	−35.0494	−1.37581
$650$	0	0
$651$	0	0
$652$	−0.129518	−0.00507231
$653$	8.44263	0.330386	0.165193	−	0.986261i	$-0.447175\pi$
$653$	8.44263	0.330386	0.165193	+	0.986261i	$0.447175\pi$
$654$	0	0
$655$	0	0
$656$	−24.3611	−0.951140
$657$	0	0
$658$	−25.5125	−0.994579
$659$	5.71179	0.222500	0.111250	−	0.993792i	$-0.464515\pi$
$659$	5.71179	0.222500	0.111250	+	0.993792i	$0.464515\pi$
$660$	0	0
$661$	−12.5139	−0.486734	−0.243367	−	0.969934i	$-0.578252\pi$
$661$	−12.5139	−0.486734	−0.243367	+	0.969934i	$0.578252\pi$
$662$	−47.2057	−1.83470
$663$	0	0
$664$	18.0059	0.698764
$665$	0	0
$666$	0	0
$667$	37.8251	1.46459
$668$	−2.02047	−0.0781743
$669$	0	0
$670$	0	0
$671$	36.3634	1.40379
$672$	0	0
$673$	−27.7807	−1.07087	−0.535433	−	0.844578i	$-0.679852\pi$
$673$	−27.7807	−1.07087	−0.535433	+	0.844578i	$0.679852\pi$
$674$	18.3395	0.706410
$675$	0	0
$676$	−0.924777	−0.0355684
$677$	41.3923	1.59084	0.795418	−	0.606061i	$-0.207251\pi$
$677$	41.3923	1.59084	0.795418	+	0.606061i	$0.207251\pi$
$678$	0	0
$679$	43.1549	1.65613
$680$	0	0
$681$	0	0
$682$	15.2750	0.584911
$683$	13.9902	0.535318	0.267659	−	0.963514i	$-0.413750\pi$
$683$	13.9902	0.535318	0.267659	+	0.963514i	$0.413750\pi$
$684$	0	0
$685$	0	0
$686$	40.3634	1.54108
$687$	0	0
$688$	44.1295	1.68242
$689$	11.1735	0.425675
$690$	0	0
$691$	25.0640	0.953478	0.476739	−	0.879045i	$-0.341819\pi$
$691$	25.0640	0.953478	0.476739	+	0.879045i	$0.341819\pi$
$692$	−0.881286	−0.0335015
$693$	0	0
$694$	−44.0567	−1.67237
$695$	0	0
$696$	0	0
$697$	−34.4749	−1.30583
$698$	−5.07381	−0.192046
$699$	0	0
$700$	0	0
$701$	−5.62672	−0.212518	−0.106259	−	0.994338i	$-0.533887\pi$
$701$	−5.62672	−0.212518	−0.106259	+	0.994338i	$0.533887\pi$
$702$	0	0
$703$	3.44358	0.129877
$704$	−26.0240	−0.980816
$705$	0	0
$706$	11.0884	0.417317
$707$	−25.1490	−0.945827
$708$	0	0
$709$	−37.8653	−1.42206	−0.711030	−	0.703161i	$-0.751771\pi$
$709$	−37.8653	−1.42206	−0.711030	+	0.703161i	$0.751771\pi$
$710$	0	0
$711$	0	0
$712$	30.6942	1.15031
$713$	22.8872	0.857131
$714$	0	0
$715$	0	0
$716$	2.92619	0.109357
$717$	0	0
$718$	52.0835	1.94374
$719$	20.8749	0.778504	0.389252	−	0.921131i	$-0.372734\pi$
$719$	20.8749	0.778504	0.389252	+	0.921131i	$0.372734\pi$
$720$	0	0
$721$	−15.8496	−0.590268
$722$	1.48119	0.0551243
$723$	0	0
$724$	−1.63847	−0.0608934
$725$	0	0
$726$	0	0
$727$	−3.34392	−0.124019	−0.0620096	−	0.998076i	$-0.519751\pi$
$727$	−3.34392	−0.124019	−0.0620096	+	0.998076i	$0.519751\pi$
$728$	−34.3938	−1.27472
$729$	0	0
$730$	0	0
$731$	62.4504	2.30981
$732$	0	0
$733$	−31.0884	−1.14828	−0.574138	−	0.818759i	$-0.694663\pi$
$733$	−31.0884	−1.14828	−0.574138	+	0.818759i	$0.694663\pi$
$734$	−3.17584	−0.117222
$735$	0	0
$736$	8.91748	0.328703
$737$	−31.9756	−1.17783
$738$	0	0
$739$	3.13918	0.115477	0.0577383	−	0.998332i	$-0.481611\pi$
$739$	3.13918	0.115477	0.0577383	+	0.998332i	$0.481611\pi$
$740$	0	0
$741$	0	0
$742$	25.8496	0.948967
$743$	6.48944	0.238075	0.119037	−	0.992890i	$-0.462019\pi$
$743$	6.48944	0.238075	0.119037	+	0.992890i	$0.462019\pi$
$744$	0	0
$745$	0	0
$746$	54.0532	1.97903
$747$	0	0
$748$	4.38787	0.160437
$749$	73.6648	2.69165
$750$	0	0
$751$	24.6458	0.899337	0.449668	−	0.893196i	$-0.351542\pi$
$751$	24.6458	0.899337	0.449668	+	0.893196i	$0.351542\pi$
$752$	16.7210	0.609752
$753$	0	0
$754$	−19.7078	−0.717716
$755$	0	0
$756$	0	0
$757$	−37.3766	−1.35848	−0.679238	−	0.733918i	$-0.737690\pi$
$757$	−37.3766	−1.35848	−0.679238	+	0.733918i	$0.737690\pi$
$758$	2.97110	0.107915
$759$	0	0
$760$	0	0
$761$	−5.64832	−0.204752	−0.102376	−	0.994746i	$-0.532644\pi$
$761$	−5.64832	−0.204752	−0.102376	+	0.994746i	$0.532644\pi$
$762$	0	0
$763$	−55.5125	−2.00969
$764$	2.62576	0.0949967
$765$	0	0
$766$	−27.0132	−0.976026
$767$	27.3620	0.987985
$768$	0	0
$769$	−10.2520	−0.369697	−0.184849	−	0.982767i	$-0.559179\pi$
$769$	−10.2520	−0.369697	−0.184849	+	0.982767i	$0.559179\pi$
$770$	0	0
$771$	0	0
$772$	−2.16854	−0.0780476
$773$	−13.1939	−0.474553	−0.237276	−	0.971442i	$-0.576255\pi$
$773$	−13.1939	−0.474553	−0.237276	+	0.971442i	$0.576255\pi$
$774$	0	0
$775$	0	0
$776$	−25.7621	−0.924806
$777$	0	0
$778$	−19.5731	−0.701730
$779$	−5.59991	−0.200638
$780$	0	0
$781$	−7.35026	−0.263013
$782$	74.3752	2.65965
$783$	0	0
$784$	−56.9062	−2.03236
$785$	0	0
$786$	0	0
$787$	14.6497	0.522207	0.261103	−	0.965311i	$-0.415914\pi$
$787$	14.6497	0.522207	0.261103	+	0.965311i	$0.415914\pi$
$788$	−1.40502	−0.0500516
$789$	0	0
$790$	0	0
$791$	−50.3898	−1.79165
$792$	0	0
$793$	−28.3879	−1.00808
$794$	−14.0146	−0.497359
$795$	0	0
$796$	−5.20967	−0.184652
$797$	−36.7426	−1.30149	−0.650745	−	0.759296i	$-0.725543\pi$
$797$	−36.7426	−1.30149	−0.650745	+	0.759296i	$0.725543\pi$
$798$	0	0
$799$	23.6629	0.837134
$800$	0	0
$801$	0	0
$802$	41.5090	1.46573
$803$	11.3503	0.400542
$804$	0	0
$805$	0	0
$806$	−11.9248	−0.420032
$807$	0	0
$808$	15.0132	0.528162
$809$	−14.3127	−0.503206	−0.251603	−	0.967831i	$-0.580958\pi$
$809$	−14.3127	−0.503206	−0.251603	+	0.967831i	$0.580958\pi$
$810$	0	0
$811$	0.625301	0.0219573	0.0109786	−	0.999940i	$-0.496505\pi$
$811$	0.625301	0.0219573	0.0109786	+	0.999940i	$0.496505\pi$
$812$	−4.03032	−0.141436
$813$	0	0
$814$	18.7454	0.657027
$815$	0	0
$816$	0	0
$817$	10.1441	0.354897
$818$	−26.6253	−0.930932
$819$	0	0
$820$	0	0
$821$	14.8510	0.518302	0.259151	−	0.965837i	$-0.416557\pi$
$821$	14.8510	0.518302	0.259151	+	0.965837i	$0.416557\pi$
$822$	0	0
$823$	54.0181	1.88295	0.941476	−	0.337079i	$-0.109439\pi$
$823$	54.0181	1.88295	0.941476	+	0.337079i	$0.109439\pi$
$824$	9.46168	0.329613
$825$	0	0
$826$	63.3014	2.20254
$827$	−5.32979	−0.185335	−0.0926675	−	0.995697i	$-0.529539\pi$
$827$	−5.32979	−0.185335	−0.0926675	+	0.995697i	$0.529539\pi$
$828$	0	0
$829$	−26.5599	−0.922464	−0.461232	−	0.887279i	$-0.652592\pi$
$829$	−26.5599	−0.922464	−0.461232	+	0.887279i	$0.652592\pi$
$830$	0	0
$831$	0	0
$832$	20.3162	0.704336
$833$	−80.5315	−2.79025
$834$	0	0
$835$	0	0
$836$	0.712742	0.0246507
$837$	0	0
$838$	−19.9038	−0.687567
$839$	9.07381	0.313263	0.156631	−	0.987657i	$-0.449937\pi$
$839$	9.07381	0.313263	0.156631	+	0.987657i	$0.449937\pi$
$840$	0	0
$841$	−7.49341	−0.258394
$842$	−36.0870	−1.24364
$843$	0	0
$844$	0.860818	0.0296306
$845$	0	0
$846$	0	0
$847$	11.2325	0.385953
$848$	−16.9419	−0.581788
$849$	0	0
$850$	0	0
$851$	28.0870	0.962809
$852$	0	0
$853$	7.19982	0.246517	0.123259	−	0.992375i	$-0.460666\pi$
$853$	7.19982	0.246517	0.123259	+	0.992375i	$0.460666\pi$
$854$	−65.6747	−2.24734
$855$	0	0
$856$	−43.9756	−1.50305
$857$	25.5066	0.871288	0.435644	−	0.900119i	$-0.356521\pi$
$857$	25.5066	0.871288	0.435644	+	0.900119i	$0.356521\pi$
$858$	0	0
$859$	2.47295	0.0843758	0.0421879	−	0.999110i	$-0.486567\pi$
$859$	2.47295	0.0843758	0.0421879	+	0.999110i	$0.486567\pi$
$860$	0	0
$861$	0	0
$862$	2.11142	0.0719152
$863$	−47.0903	−1.60297	−0.801486	−	0.598013i	$-0.795957\pi$
$863$	−47.0903	−1.60297	−0.801486	+	0.598013i	$0.795957\pi$
$864$	0	0
$865$	0	0
$866$	2.45088	0.0832842
$867$	0	0
$868$	−2.43866	−0.0827735
$869$	−18.2374	−0.618662
$870$	0	0
$871$	24.9624	0.845818
$872$	33.1392	1.12223
$873$	0	0
$874$	12.0811	0.408649
$875$	0	0
$876$	0	0
$877$	−52.2067	−1.76289	−0.881447	−	0.472284i	$-0.843430\pi$
$877$	−52.2067	−1.76289	−0.881447	+	0.472284i	$0.843430\pi$
$878$	−39.1030	−1.31966
$879$	0	0
$880$	0	0
$881$	2.06537	0.0695842	0.0347921	−	0.999395i	$-0.488923\pi$
$881$	2.06537	0.0695842	0.0347921	+	0.999395i	$0.488923\pi$
$882$	0	0
$883$	−42.9053	−1.44388	−0.721939	−	0.691957i	$-0.756749\pi$
$883$	−42.9053	−1.44388	−0.721939	+	0.691957i	$0.756749\pi$
$884$	−3.42548	−0.115212
$885$	0	0
$886$	4.97110	0.167007
$887$	−22.1768	−0.744624	−0.372312	−	0.928108i	$-0.621435\pi$
$887$	−22.1768	−0.744624	−0.372312	+	0.928108i	$0.621435\pi$
$888$	0	0
$889$	−53.1655	−1.78311
$890$	0	0
$891$	0	0
$892$	1.76257	0.0590153
$893$	3.84367	0.128624
$894$	0	0
$895$	0	0
$896$	56.7997	1.89755
$897$	0	0
$898$	−17.7924	−0.593741
$899$	13.0132	0.434014
$900$	0	0
$901$	−23.9756	−0.798742
$902$	−30.4836	−1.01499
$903$	0	0
$904$	30.0811	1.00048
$905$	0	0
$906$	0	0
$907$	−25.6629	−0.852123	−0.426062	−	0.904694i	$-0.640099\pi$
$907$	−25.6629	−0.852123	−0.426062	+	0.904694i	$0.640099\pi$
$908$	−2.17091	−0.0720443
$909$	0	0
$910$	0	0
$911$	2.65959	0.0881161	0.0440580	−	0.999029i	$-0.485971\pi$
$911$	2.65959	0.0881161	0.0440580	+	0.999029i	$0.485971\pi$
$912$	0	0
$913$	24.7367	0.818666
$914$	−46.7875	−1.54759
$915$	0	0
$916$	−4.59498	−0.151823
$917$	−6.94192	−0.229242
$918$	0	0
$919$	−6.94780	−0.229187	−0.114593	−	0.993412i	$-0.536557\pi$
$919$	−6.94780	−0.229187	−0.114593	+	0.993412i	$0.536557\pi$
$920$	0	0
$921$	0	0
$922$	−7.51530	−0.247503
$923$	5.73813	0.188873
$924$	0	0
$925$	0	0
$926$	21.1368	0.694599
$927$	0	0
$928$	5.07030	0.166441
$929$	−53.3112	−1.74908	−0.874542	−	0.484949i	$-0.838838\pi$
$929$	−53.3112	−1.74908	−0.874542	+	0.484949i	$0.838838\pi$
$930$	0	0
$931$	−13.0811	−0.428716
$932$	2.33709	0.0765539
$933$	0	0
$934$	−47.9669	−1.56952
$935$	0	0
$936$	0	0
$937$	−40.9135	−1.33659	−0.668293	−	0.743898i	$-0.732975\pi$
$937$	−40.9135	−1.33659	−0.668293	+	0.743898i	$0.732975\pi$
$938$	57.7499	1.88560
$939$	0	0
$940$	0	0
$941$	27.0762	0.882658	0.441329	−	0.897345i	$-0.354507\pi$
$941$	27.0762	0.882658	0.441329	+	0.897345i	$0.354507\pi$
$942$	0	0
$943$	−45.6747	−1.48737
$944$	−41.4880	−1.35032
$945$	0	0
$946$	55.2203	1.79537
$947$	−13.7929	−0.448209	−0.224104	−	0.974565i	$-0.571946\pi$
$947$	−13.7929	−0.448209	−0.224104	+	0.974565i	$0.571946\pi$
$948$	0	0
$949$	−8.86082	−0.287634
$950$	0	0
$951$	0	0
$952$	73.8007	2.39189
$953$	−13.7948	−0.446857	−0.223429	−	0.974720i	$-0.571725\pi$
$953$	−13.7948	−0.446857	−0.223429	+	0.974720i	$0.571725\pi$
$954$	0	0
$955$	0	0
$956$	−2.71274	−0.0877364
$957$	0	0
$958$	10.7938	0.348733
$959$	76.5111	2.47067
$960$	0	0
$961$	−23.1260	−0.746000
$962$	−14.6340	−0.471819
$963$	0	0
$964$	4.61071	0.148501
$965$	0	0
$966$	0	0
$967$	20.6072	0.662683	0.331341	−	0.943511i	$-0.392499\pi$
$967$	20.6072	0.662683	0.331341	+	0.943511i	$0.392499\pi$
$968$	−6.70545	−0.215521
$969$	0	0
$970$	0	0
$971$	−1.93463	−0.0620851	−0.0310426	−	0.999518i	$-0.509883\pi$
$971$	−1.93463	−0.0620851	−0.0310426	+	0.999518i	$0.509883\pi$
$972$	0	0
$973$	25.1490	0.806241
$974$	55.6531	1.78324
$975$	0	0
$976$	43.0435	1.37779
$977$	−30.6312	−0.979978	−0.489989	−	0.871729i	$-0.662999\pi$
$977$	−30.6312	−0.979978	−0.489989	+	0.871729i	$0.662999\pi$
$978$	0	0
$979$	42.1681	1.34770
$980$	0	0
$981$	0	0
$982$	−34.9415	−1.11503
$983$	51.2663	1.63514	0.817571	−	0.575828i	$-0.195320\pi$
$983$	51.2663	1.63514	0.817571	+	0.575828i	$0.195320\pi$
$984$	0	0
$985$	0	0
$986$	42.2882	1.34673
$987$	0	0
$988$	−0.556417	−0.0177020
$989$	82.7386	2.63094
$990$	0	0
$991$	−31.5778	−1.00310	−0.501552	−	0.865128i	$-0.667237\pi$
$991$	−31.5778	−1.00310	−0.501552	+	0.865128i	$0.667237\pi$
$992$	3.06793	0.0974068
$993$	0	0
$994$	13.2750	0.421059
$995$	0	0
$996$	0	0
$997$	−35.7090	−1.13091	−0.565457	−	0.824778i	$-0.691300\pi$
$997$	−35.7090	−1.13091	−0.565457	+	0.824778i	$0.691300\pi$
$998$	−38.0870	−1.20562
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bd.1.3		3
3.2	odd	2		1425.2.a.x.1.1		3
5.2	odd	4		855.2.c.e.514.5		6
5.3	odd	4		855.2.c.e.514.2		6
5.4	even	2		4275.2.a.bi.1.1		3
15.2	even	4		285.2.c.a.229.2	✓	6
15.8	even	4		285.2.c.a.229.5	yes	6
15.14	odd	2		1425.2.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.c.a.229.2	✓	6	15.2	even	4
285.2.c.a.229.5	yes	6	15.8	even	4
855.2.c.e.514.2		6	5.3	odd	4
855.2.c.e.514.5		6	5.2	odd	4
1425.2.a.s.1.3		3	15.14	odd	2
1425.2.a.x.1.1		3	3.2	odd	2
4275.2.a.bd.1.3		3	1.1	even	1	trivial
4275.2.a.bi.1.1		3	5.4	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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} +0.193937 q^{4} +4.48119 q^{7} -2.67513 q^{8} -3.67513 q^{11} +2.86907 q^{13} +6.63752 q^{14} -4.35026 q^{16} -6.15633 q^{17} -1.00000 q^{19} -5.44358 q^{22} -8.15633 q^{23} +4.24965 q^{26} +0.869067 q^{28} -4.63752 q^{29} -2.80606 q^{31} -1.09332 q^{32} -9.11871 q^{34} -3.44358 q^{37} -1.48119 q^{38} +5.59991 q^{41} -10.1441 q^{43} -0.712742 q^{44} -12.0811 q^{46} -3.84367 q^{47} +13.0811 q^{49} +0.556417 q^{52} +3.89446 q^{53} -11.9878 q^{56} -6.86907 q^{58} +9.53690 q^{59} -9.89446 q^{61} -4.15633 q^{62} +7.08110 q^{64} +8.70052 q^{67} -1.19394 q^{68} +2.00000 q^{71} -3.08840 q^{73} -5.10062 q^{74} -0.193937 q^{76} -16.4690 q^{77} +4.96239 q^{79} +8.29455 q^{82} -6.73084 q^{83} -15.0254 q^{86} +9.83146 q^{88} -11.4739 q^{89} +12.8568 q^{91} -1.58181 q^{92} -5.69323 q^{94} +9.63023 q^{97} +19.3757 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + 8 q^{7} - 3 q^{8} - 6 q^{11} + 4 q^{13} + 4 q^{14} - 3 q^{16} - 8 q^{17} - 3 q^{19} - 14 q^{23} - 4 q^{26} - 2 q^{28} + 2 q^{29} - 8 q^{31} + 3 q^{32} - 6 q^{34} + 6 q^{37} + q^{38}+ \cdots + 33 q^{98}+O(q^{100}) \) 3 * q - q^2 + q^4 + 8 * q^7 - 3 * q^8 - 6 * q^11 + 4 * q^13 + 4 * q^14 - 3 * q^16 - 8 * q^17 - 3 * q^19 - 14 * q^23 - 4 * q^26 - 2 * q^28 + 2 * q^29 - 8 * q^31 + 3 * q^32 - 6 * q^34 + 6 * q^37 + q^38 - 10 * q^41 + 6 * q^43 - 8 * q^44 - 4 * q^46 - 22 * q^47 + 7 * q^49 + 18 * q^52 - 8 * q^53 - 10 * q^56 - 16 * q^58 + 6 * q^59 - 10 * q^61 - 2 * q^62 - 11 * q^64 + 6 * q^67 - 4 * q^68 + 6 * q^71 + 10 * q^73 - 22 * q^74 - q^76 - 18 * q^77 + 4 * q^79 + 32 * q^82 + 2 * q^83 - 30 * q^86 + 14 * q^88 - 16 * q^89 + 8 * q^91 - 6 * q^92 + 16 * q^94 + 33 * q^98

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