LMFDB - Embedded newform 9025.2.a.bz.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.41289040.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5 $ x^6 - 3x^5 - 8x^4 + 21x^3 + 18x^2 - 25*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.4
Root			$-2.24415$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.311108 q^{2} +1.02983 q^{3} -1.90321 q^{4} +0.320390 q^{6} -3.28038 q^{7} -1.21432 q^{8} -1.93944 q^{9} -5.16792 q^{11} -1.95999 q^{12} +3.53471 q^{13} -1.02055 q^{14} +3.42864 q^{16} -1.00928 q^{17} -0.603375 q^{18} -3.37825 q^{21} -1.60778 q^{22} -7.66351 q^{23} -1.25055 q^{24} +1.09968 q^{26} -5.08681 q^{27} +6.24326 q^{28} -4.02606 q^{29} -4.60077 q^{31} +3.49532 q^{32} -5.32210 q^{33} -0.313995 q^{34} +3.69117 q^{36} -6.48831 q^{37} +3.64017 q^{39} -6.80553 q^{41} -1.05100 q^{42} +6.31022 q^{43} +9.83565 q^{44} -2.38418 q^{46} -3.84157 q^{47} +3.53093 q^{48} +3.76091 q^{49} -1.03939 q^{51} -6.72730 q^{52} -7.11892 q^{53} -1.58255 q^{54} +3.98343 q^{56} -1.25254 q^{58} +13.4730 q^{59} +6.13700 q^{61} -1.43134 q^{62} +6.36211 q^{63} -5.76986 q^{64} -1.65575 q^{66} -11.1979 q^{67} +1.92088 q^{68} -7.89215 q^{69} -0.455404 q^{71} +2.35510 q^{72} +4.13382 q^{73} -2.01856 q^{74} +16.9528 q^{77} +1.13248 q^{78} +2.88828 q^{79} +0.579752 q^{81} -2.11725 q^{82} -5.50061 q^{83} +6.42953 q^{84} +1.96316 q^{86} -4.14617 q^{87} +6.27551 q^{88} -7.12866 q^{89} -11.5952 q^{91} +14.5853 q^{92} -4.73803 q^{93} -1.19514 q^{94} +3.59960 q^{96} -10.8250 q^{97} +1.17005 q^{98} +10.0229 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 6 q^{8} + 7 q^{9} - q^{11} + 7 q^{12} + 5 q^{13} - 6 q^{14} - 6 q^{16} - 3 q^{17} + 7 q^{18} + 3 q^{21} + 9 q^{22} - 6 q^{23} + 11 q^{24} + 19 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - q^6 + 2 * q^7 + 6 * q^8 + 7 * q^9 - q^11 + 7 * q^12 + 5 * q^13 - 6 * q^14 - 6 * q^16 - 3 * q^17 + 7 * q^18 + 3 * q^21 + 9 * q^22 - 6 * q^23 + 11 * q^24 + 19 * q^26 + 18 * q^27 - 4 * q^28 + 3 * q^29 - 3 * q^31 - 6 * q^32 - 18 * q^33 - q^34 + 13 * q^36 - 6 * q^37 + 8 * q^39 + 11 * q^41 - 11 * q^42 + 13 * q^43 + 21 * q^44 - 12 * q^46 - 6 * q^47 - 19 * q^48 + 4 * q^49 - 17 * q^51 - q^52 + 18 * q^53 + 18 * q^54 + 4 * q^56 + 5 * q^58 + 4 * q^59 + 25 * q^61 - 21 * q^62 + 43 * q^63 - 22 * q^64 + 34 * q^66 + 6 * q^67 - q^68 + 13 * q^69 + 18 * q^71 + 13 * q^72 + q^73 - 6 * q^74 - 11 * q^77 + 72 * q^78 + 3 * q^79 + 2 * q^81 + 31 * q^82 - 23 * q^83 + 37 * q^84 + 9 * q^86 + 11 * q^87 + 11 * q^88 + 12 * q^89 - 11 * q^91 + 28 * q^92 - 13 * q^93 + 8 * q^94 - 13 * q^96 + 3 * q^97 - 22 * q^98 - 20 * q^99

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 $

$2$	0.311108	0.219986	0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108	0.219986	0.109993	+	0.993932i	$0.464917\pi$
$3$	1.02983	0.594575	0.297288	−	0.954788i	$-0.403918\pi$
$3$	1.02983	0.594575	0.297288	+	0.954788i	$0.403918\pi$
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	0.320390	0.130799
$7$	−3.28038	−1.23987	−0.619934	−	0.784654i	$-0.712841\pi$
$7$	−3.28038	−1.23987	−0.619934	+	0.784654i	$0.712841\pi$
$8$	−1.21432	−0.429327
$9$	−1.93944	−0.646480
$10$	0	0
$11$	−5.16792	−1.55819	−0.779093	−	0.626908i	$-0.784320\pi$
$11$	−5.16792	−1.55819	−0.779093	+	0.626908i	$0.784320\pi$
$12$	−1.95999	−0.565801
$13$	3.53471	0.980352	0.490176	−	0.871623i	$-0.336933\pi$
$13$	3.53471	0.980352	0.490176	+	0.871623i	$0.336933\pi$
$14$	−1.02055	−0.272754
$15$	0	0
$16$	3.42864	0.857160
$17$	−1.00928	−0.244787	−0.122393	−	0.992482i	$-0.539057\pi$
$17$	−1.00928	−0.244787	−0.122393	+	0.992482i	$0.539057\pi$
$18$	−0.603375	−0.142217
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−3.37825	−0.737195
$22$	−1.60778	−0.342780
$23$	−7.66351	−1.59795	−0.798976	−	0.601362i	$-0.794625\pi$
$23$	−7.66351	−1.59795	−0.798976	+	0.601362i	$0.794625\pi$
$24$	−1.25055	−0.255267
$25$	0	0
$26$	1.09968	0.215664
$27$	−5.08681	−0.978956
$28$	6.24326	1.17987
$29$	−4.02606	−0.747620	−0.373810	−	0.927505i	$-0.621949\pi$
$29$	−4.02606	−0.747620	−0.373810	+	0.927505i	$0.621949\pi$
$30$	0	0
$31$	−4.60077	−0.826323	−0.413162	−	0.910658i	$-0.635576\pi$
$31$	−4.60077	−0.826323	−0.413162	+	0.910658i	$0.635576\pi$
$32$	3.49532	0.617890
$33$	−5.32210	−0.926459
$34$	−0.313995	−0.0538498
$35$	0	0
$36$	3.69117	0.615194
$37$	−6.48831	−1.06667	−0.533336	−	0.845904i	$-0.679062\pi$
$37$	−6.48831	−1.06667	−0.533336	+	0.845904i	$0.679062\pi$
$38$	0	0
$39$	3.64017	0.582893
$40$	0	0
$41$	−6.80553	−1.06285	−0.531423	−	0.847107i	$-0.678342\pi$
$41$	−6.80553	−1.06285	−0.531423	+	0.847107i	$0.678342\pi$
$42$	−1.05100	−0.162173
$43$	6.31022	0.962299	0.481150	−	0.876639i	$-0.340219\pi$
$43$	6.31022	0.962299	0.481150	+	0.876639i	$0.340219\pi$
$44$	9.83565	1.48278
$45$	0	0
$46$	−2.38418	−0.351528
$47$	−3.84157	−0.560351	−0.280175	−	0.959949i	$-0.590393\pi$
$47$	−3.84157	−0.560351	−0.280175	+	0.959949i	$0.590393\pi$
$48$	3.53093	0.509646
$49$	3.76091	0.537273
$50$	0	0
$51$	−1.03939	−0.145544
$52$	−6.72730	−0.932909
$53$	−7.11892	−0.977858	−0.488929	−	0.872323i	$-0.662612\pi$
$53$	−7.11892	−0.977858	−0.488929	+	0.872323i	$0.662612\pi$
$54$	−1.58255	−0.215357
$55$	0	0
$56$	3.98343	0.532309
$57$	0	0
$58$	−1.25254	−0.164466
$59$	13.4730	1.75403	0.877016	−	0.480460i	$-0.159530\pi$
$59$	13.4730	1.75403	0.877016	+	0.480460i	$0.159530\pi$
$60$	0	0
$61$	6.13700	0.785763	0.392881	−	0.919589i	$-0.371478\pi$
$61$	6.13700	0.785763	0.392881	+	0.919589i	$0.371478\pi$
$62$	−1.43134	−0.181780
$63$	6.36211	0.801550
$64$	−5.76986	−0.721232
$65$	0	0
$66$	−1.65575	−0.203808
$67$	−11.1979	−1.36805	−0.684023	−	0.729460i	$-0.739771\pi$
$67$	−11.1979	−1.36805	−0.684023	+	0.729460i	$0.739771\pi$
$68$	1.92088	0.232941
$69$	−7.89215	−0.950103
$70$	0	0
$71$	−0.455404	−0.0540465	−0.0270233	−	0.999635i	$-0.508603\pi$
$71$	−0.455404	−0.0540465	−0.0270233	+	0.999635i	$0.508603\pi$
$72$	2.35510	0.277551
$73$	4.13382	0.483827	0.241914	−	0.970298i	$-0.422225\pi$
$73$	4.13382	0.483827	0.241914	+	0.970298i	$0.422225\pi$
$74$	−2.01856	−0.234653
$75$	0	0
$76$	0	0
$77$	16.9528	1.93195
$78$	1.13248	0.128229
$79$	2.88828	0.324957	0.162478	−	0.986712i	$-0.448051\pi$
$79$	2.88828	0.324957	0.162478	+	0.986712i	$0.448051\pi$
$80$	0	0
$81$	0.579752	0.0644169
$82$	−2.11725	−0.233812
$83$	−5.50061	−0.603770	−0.301885	−	0.953344i	$-0.597616\pi$
$83$	−5.50061	−0.603770	−0.301885	+	0.953344i	$0.597616\pi$
$84$	6.42953	0.701519
$85$	0	0
$86$	1.96316	0.211693
$87$	−4.14617	−0.444516
$88$	6.27551	0.668971
$89$	−7.12866	−0.755636	−0.377818	−	0.925880i	$-0.623326\pi$
$89$	−7.12866	−0.755636	−0.377818	+	0.925880i	$0.623326\pi$
$90$	0	0
$91$	−11.5952	−1.21551
$92$	14.5853	1.52062
$93$	−4.73803	−0.491311
$94$	−1.19514	−0.123270
$95$	0	0
$96$	3.59960	0.367382
$97$	−10.8250	−1.09912	−0.549558	−	0.835456i	$-0.685204\pi$
$97$	−10.8250	−1.09912	−0.549558	+	0.835456i	$0.685204\pi$
$98$	1.17005	0.118193
$99$	10.0229	1.00734
$100$	0	0
$101$	−6.18022	−0.614955	−0.307478	−	0.951555i	$-0.599485\pi$
$101$	−6.18022	−0.614955	−0.307478	+	0.951555i	$0.599485\pi$
$102$	−0.323363	−0.0320177
$103$	18.3217	1.80529	0.902644	−	0.430387i	$-0.141623\pi$
$103$	18.3217	1.80529	0.902644	+	0.430387i	$0.141623\pi$
$104$	−4.29227	−0.420891
$105$	0	0
$106$	−2.21475	−0.215116
$107$	8.73445	0.844391	0.422196	−	0.906505i	$-0.361260\pi$
$107$	8.73445	0.844391	0.422196	+	0.906505i	$0.361260\pi$
$108$	9.68127	0.931581
$109$	−6.09679	−0.583966	−0.291983	−	0.956423i	$-0.594315\pi$
$109$	−6.09679	−0.583966	−0.291983	+	0.956423i	$0.594315\pi$
$110$	0	0
$111$	−6.68188	−0.634216
$112$	−11.2473	−1.06277
$113$	10.3443	0.973111	0.486556	−	0.873650i	$-0.338253\pi$
$113$	10.3443	0.973111	0.486556	+	0.873650i	$0.338253\pi$
$114$	0	0
$115$	0	0
$116$	7.66244	0.711440
$117$	−6.85536	−0.633778
$118$	4.19155	0.385863
$119$	3.31083	0.303503
$120$	0	0
$121$	15.7074	1.42794
$122$	1.90927	0.172857
$123$	−7.00857	−0.631942
$124$	8.75625	0.786334
$125$	0	0
$126$	1.97930	0.176330
$127$	3.36713	0.298784	0.149392	−	0.988778i	$-0.452268\pi$
$127$	3.36713	0.298784	0.149392	+	0.988778i	$0.452268\pi$
$128$	−8.78568	−0.776552
$129$	6.49848	0.572159
$130$	0	0
$131$	2.42178	0.211592	0.105796	−	0.994388i	$-0.466261\pi$
$131$	2.42178	0.211592	0.105796	+	0.994388i	$0.466261\pi$
$132$	10.1291	0.881624
$133$	0	0
$134$	−3.48377	−0.300952
$135$	0	0
$136$	1.22559	0.105094
$137$	−11.5191	−0.984144	−0.492072	−	0.870555i	$-0.663760\pi$
$137$	−11.5191	−0.984144	−0.492072	+	0.870555i	$0.663760\pi$
$138$	−2.45531	−0.209010
$139$	7.07553	0.600138	0.300069	−	0.953917i	$-0.402990\pi$
$139$	7.07553	0.600138	0.300069	+	0.953917i	$0.402990\pi$
$140$	0	0
$141$	−3.95618	−0.333171
$142$	−0.141680	−0.0118895
$143$	−18.2671	−1.52757
$144$	−6.64964	−0.554137
$145$	0	0
$146$	1.28606	0.106435
$147$	3.87312	0.319449
$148$	12.3486	1.01505
$149$	12.7495	1.04448	0.522240	−	0.852798i	$-0.325096\pi$
$149$	12.7495	1.04448	0.522240	+	0.852798i	$0.325096\pi$
$150$	0	0
$151$	−19.6361	−1.59797	−0.798983	−	0.601353i	$-0.794628\pi$
$151$	−19.6361	−1.59797	−0.798983	+	0.601353i	$0.794628\pi$
$152$	0	0
$153$	1.95744	0.158250
$154$	5.27413	0.425002
$155$	0	0
$156$	−6.92801	−0.554685
$157$	3.08489	0.246201	0.123101	−	0.992394i	$-0.460716\pi$
$157$	3.08489	0.246201	0.123101	+	0.992394i	$0.460716\pi$
$158$	0.898567	0.0714861
$159$	−7.33131	−0.581410
$160$	0	0
$161$	25.1393	1.98125
$162$	0.180365	0.0141708
$163$	−22.4721	−1.76015	−0.880077	−	0.474831i	$-0.842509\pi$
$163$	−22.4721	−1.76015	−0.880077	+	0.474831i	$0.842509\pi$
$164$	12.9524	1.01141
$165$	0	0
$166$	−1.71128	−0.132821
$167$	10.5397	0.815590	0.407795	−	0.913074i	$-0.366298\pi$
$167$	10.5397	0.815590	0.407795	+	0.913074i	$0.366298\pi$
$168$	4.10228	0.316498
$169$	−0.505830	−0.0389100
$170$	0	0
$171$	0	0
$172$	−12.0097	−0.915730
$173$	22.9381	1.74395	0.871977	−	0.489547i	$-0.162838\pi$
$173$	22.9381	1.74395	0.871977	+	0.489547i	$0.162838\pi$
$174$	−1.28991	−0.0977876
$175$	0	0
$176$	−17.7189	−1.33561
$177$	13.8749	1.04290
$178$	−2.21778	−0.166230
$179$	23.2705	1.73932	0.869661	−	0.493649i	$-0.164337\pi$
$179$	23.2705	1.73932	0.869661	+	0.493649i	$0.164337\pi$
$180$	0	0
$181$	−15.9669	−1.18681	−0.593406	−	0.804904i	$-0.702217\pi$
$181$	−15.9669	−1.18681	−0.593406	+	0.804904i	$0.702217\pi$
$182$	−3.60736	−0.267395
$183$	6.32010	0.467195
$184$	9.30595	0.686044
$185$	0	0
$186$	−1.47404	−0.108082
$187$	5.21589	0.381423
$188$	7.31133	0.533233
$189$	16.6867	1.21378
$190$	0	0
$191$	−2.44600	−0.176986	−0.0884930	−	0.996077i	$-0.528205\pi$
$191$	−2.44600	−0.176986	−0.0884930	+	0.996077i	$0.528205\pi$
$192$	−5.94200	−0.428827
$193$	−19.0456	−1.37093	−0.685466	−	0.728104i	$-0.740402\pi$
$193$	−19.0456	−1.37093	−0.685466	+	0.728104i	$0.740402\pi$
$194$	−3.36775	−0.241791
$195$	0	0
$196$	−7.15782	−0.511273
$197$	−6.18524	−0.440680	−0.220340	−	0.975423i	$-0.570717\pi$
$197$	−6.18524	−0.440680	−0.220340	+	0.975423i	$0.570717\pi$
$198$	3.11819	0.221600
$199$	7.21050	0.511138	0.255569	−	0.966791i	$-0.417737\pi$
$199$	7.21050	0.511138	0.255569	+	0.966791i	$0.417737\pi$
$200$	0	0
$201$	−11.5320	−0.813407
$202$	−1.92272	−0.135282
$203$	13.2070	0.926950
$204$	1.97819	0.138501
$205$	0	0
$206$	5.70002	0.397139
$207$	14.8629	1.03304
$208$	12.1192	0.840318
$209$	0	0
$210$	0	0
$211$	−19.6467	−1.35253	−0.676266	−	0.736658i	$-0.736403\pi$
$211$	−19.6467	−1.35253	−0.676266	+	0.736658i	$0.736403\pi$
$212$	13.5488	0.930536
$213$	−0.468991	−0.0321347
$214$	2.71736	0.185755
$215$	0	0
$216$	6.17701	0.420292
$217$	15.0923	1.02453
$218$	−1.89676	−0.128465
$219$	4.25715	0.287672
$220$	0	0
$221$	−3.56752	−0.239977
$222$	−2.07879	−0.139519
$223$	−11.3338	−0.758967	−0.379484	−	0.925198i	$-0.623898\pi$
$223$	−11.3338	−0.758967	−0.379484	+	0.925198i	$0.623898\pi$
$224$	−11.4660	−0.766103
$225$	0	0
$226$	3.21820	0.214071
$227$	−11.1369	−0.739180	−0.369590	−	0.929195i	$-0.620502\pi$
$227$	−11.1369	−0.739180	−0.369590	+	0.929195i	$0.620502\pi$
$228$	0	0
$229$	3.51221	0.232093	0.116047	−	0.993244i	$-0.462978\pi$
$229$	3.51221	0.232093	0.116047	+	0.993244i	$0.462978\pi$
$230$	0	0
$231$	17.4585	1.14869
$232$	4.88892	0.320973
$233$	6.18998	0.405519	0.202759	−	0.979229i	$-0.435009\pi$
$233$	6.18998	0.405519	0.202759	+	0.979229i	$0.435009\pi$
$234$	−2.13276	−0.139423
$235$	0	0
$236$	−25.6419	−1.66915
$237$	2.97445	0.193211
$238$	1.03003	0.0667666
$239$	12.1221	0.784116	0.392058	−	0.919940i	$-0.371763\pi$
$239$	12.1221	0.784116	0.392058	+	0.919940i	$0.371763\pi$
$240$	0	0
$241$	−5.10204	−0.328651	−0.164326	−	0.986406i	$-0.552545\pi$
$241$	−5.10204	−0.328651	−0.164326	+	0.986406i	$0.552545\pi$
$242$	4.88669	0.314128
$243$	15.8575	1.01726
$244$	−11.6800	−0.747737
$245$	0	0
$246$	−2.18042	−0.139019
$247$	0	0
$248$	5.58681	0.354763
$249$	−5.66472	−0.358987
$250$	0	0
$251$	25.4244	1.60477	0.802385	−	0.596806i	$-0.203564\pi$
$251$	25.4244	1.60477	0.802385	+	0.596806i	$0.203564\pi$
$252$	−12.1084	−0.762760
$253$	39.6044	2.48991
$254$	1.04754	0.0657285
$255$	0	0
$256$	8.80642	0.550401
$257$	18.4499	1.15087	0.575436	−	0.817847i	$-0.304833\pi$
$257$	18.4499	1.15087	0.575436	+	0.817847i	$0.304833\pi$
$258$	2.02173	0.125867
$259$	21.2841	1.32253
$260$	0	0
$261$	7.80830	0.483322
$262$	0.753434	0.0465473
$263$	12.9506	0.798568	0.399284	−	0.916827i	$-0.369259\pi$
$263$	12.9506	0.798568	0.399284	+	0.916827i	$0.369259\pi$
$264$	6.46273	0.397754
$265$	0	0
$266$	0	0
$267$	−7.34134	−0.449283
$268$	21.3121	1.30184
$269$	0.156494	0.00954163	0.00477081	−	0.999989i	$-0.498481\pi$
$269$	0.156494	0.00954163	0.00477081	+	0.999989i	$0.498481\pi$
$270$	0	0
$271$	27.8191	1.68989	0.844944	−	0.534855i	$-0.179634\pi$
$271$	27.8191	1.68989	0.844944	+	0.534855i	$0.179634\pi$
$272$	−3.46046	−0.209821
$273$	−11.9411	−0.722711
$274$	−3.58368	−0.216498
$275$	0	0
$276$	15.0204	0.904124
$277$	−0.524062	−0.0314878	−0.0157439	−	0.999876i	$-0.505012\pi$
$277$	−0.524062	−0.0314878	−0.0157439	+	0.999876i	$0.505012\pi$
$278$	2.20125	0.132022
$279$	8.92293	0.534202
$280$	0	0
$281$	−4.88826	−0.291609	−0.145804	−	0.989313i	$-0.546577\pi$
$281$	−4.88826	−0.291609	−0.145804	+	0.989313i	$0.546577\pi$
$282$	−1.23080	−0.0732931
$283$	−17.8786	−1.06277	−0.531387	−	0.847129i	$-0.678329\pi$
$283$	−17.8786	−1.06277	−0.531387	+	0.847129i	$0.678329\pi$
$284$	0.866730	0.0514310
$285$	0	0
$286$	−5.68304	−0.336045
$287$	22.3248	1.31779
$288$	−6.77896	−0.399454
$289$	−15.9814	−0.940079
$290$	0	0
$291$	−11.1480	−0.653507
$292$	−7.86754	−0.460413
$293$	7.25399	0.423783	0.211891	−	0.977293i	$-0.432038\pi$
$293$	7.25399	0.423783	0.211891	+	0.977293i	$0.432038\pi$
$294$	1.20496	0.0702745
$295$	0	0
$296$	7.87888	0.457951
$297$	26.2882	1.52540
$298$	3.96647	0.229772
$299$	−27.0883	−1.56656
$300$	0	0
$301$	−20.6999	−1.19312
$302$	−6.10896	−0.351531
$303$	−6.36461	−0.365637
$304$	0	0
$305$	0	0
$306$	0.608976	0.0348128
$307$	14.3803	0.820727	0.410364	−	0.911922i	$-0.365402\pi$
$307$	14.3803	0.820727	0.410364	+	0.911922i	$0.365402\pi$
$308$	−32.2647	−1.83845
$309$	18.8683	1.07338
$310$	0	0
$311$	8.52590	0.483459	0.241730	−	0.970344i	$-0.422285\pi$
$311$	8.52590	0.483459	0.241730	+	0.970344i	$0.422285\pi$
$312$	−4.42033	−0.250252
$313$	−21.3619	−1.20744	−0.603722	−	0.797195i	$-0.706316\pi$
$313$	−21.3619	−1.20744	−0.603722	+	0.797195i	$0.706316\pi$
$314$	0.959733	0.0541609
$315$	0	0
$316$	−5.49701	−0.309231
$317$	28.0928	1.57785	0.788924	−	0.614490i	$-0.210638\pi$
$317$	28.0928	1.57785	0.788924	+	0.614490i	$0.210638\pi$
$318$	−2.28083	−0.127902
$319$	20.8063	1.16493
$320$	0	0
$321$	8.99504	0.502054
$322$	7.82102	0.435848
$323$	0	0
$324$	−1.10339	−0.0612995
$325$	0	0
$326$	−6.99126	−0.387210
$327$	−6.27868	−0.347212
$328$	8.26409	0.456308
$329$	12.6018	0.694761
$330$	0	0
$331$	15.1725	0.833957	0.416979	−	0.908916i	$-0.363089\pi$
$331$	15.1725	0.833957	0.416979	+	0.908916i	$0.363089\pi$
$332$	10.4688	0.574552
$333$	12.5837	0.689582
$334$	3.27900	0.179419
$335$	0	0
$336$	−11.5828	−0.631894
$337$	−17.5008	−0.953330	−0.476665	−	0.879085i	$-0.658155\pi$
$337$	−17.5008	−0.953330	−0.476665	+	0.879085i	$0.658155\pi$
$338$	−0.157368	−0.00855967
$339$	10.6529	0.578588
$340$	0	0
$341$	23.7764	1.28757
$342$	0	0
$343$	10.6254	0.573720
$344$	−7.66262	−0.413141
$345$	0	0
$346$	7.13623	0.383646
$347$	17.3687	0.932401	0.466201	−	0.884679i	$-0.345623\pi$
$347$	17.3687	0.932401	0.466201	+	0.884679i	$0.345623\pi$
$348$	7.89105	0.423004
$349$	−34.2563	−1.83370	−0.916850	−	0.399232i	$-0.869277\pi$
$349$	−34.2563	−1.83370	−0.916850	+	0.399232i	$0.869277\pi$
$350$	0	0
$351$	−17.9804	−0.959722
$352$	−18.0635	−0.962788
$353$	34.9287	1.85907	0.929534	−	0.368736i	$-0.120209\pi$
$353$	34.9287	1.85907	0.929534	+	0.368736i	$0.120209\pi$
$354$	4.31660	0.229425
$355$	0	0
$356$	13.5673	0.719068
$357$	3.40961	0.180456
$358$	7.23965	0.382627
$359$	−0.976546	−0.0515401	−0.0257701	−	0.999668i	$-0.508204\pi$
$359$	−0.976546	−0.0515401	−0.0257701	+	0.999668i	$0.508204\pi$
$360$	0	0
$361$	0	0
$362$	−4.96743	−0.261082
$363$	16.1760	0.849020
$364$	22.0681	1.15668
$365$	0	0
$366$	1.96623	0.102777
$367$	9.86635	0.515019	0.257510	−	0.966276i	$-0.417098\pi$
$367$	9.86635	0.515019	0.257510	+	0.966276i	$0.417098\pi$
$368$	−26.2754	−1.36970
$369$	13.1989	0.687109
$370$	0	0
$371$	23.3528	1.21242
$372$	9.01748	0.467535
$373$	−12.9521	−0.670634	−0.335317	−	0.942105i	$-0.608843\pi$
$373$	−12.9521	−0.670634	−0.335317	+	0.942105i	$0.608843\pi$
$374$	1.62270	0.0839080
$375$	0	0
$376$	4.66490	0.240574
$377$	−14.2309	−0.732931
$378$	5.19136	0.267015
$379$	5.21597	0.267926	0.133963	−	0.990986i	$-0.457230\pi$
$379$	5.21597	0.267926	0.133963	+	0.990986i	$0.457230\pi$
$380$	0	0
$381$	3.46758	0.177650
$382$	−0.760969	−0.0389345
$383$	33.5314	1.71337	0.856686	−	0.515838i	$-0.172519\pi$
$383$	33.5314	1.71337	0.856686	+	0.515838i	$0.172519\pi$
$384$	−9.04780	−0.461718
$385$	0	0
$386$	−5.92524	−0.301587
$387$	−12.2383	−0.622107
$388$	20.6023	1.04593
$389$	27.7406	1.40651	0.703253	−	0.710940i	$-0.251730\pi$
$389$	27.7406	1.40651	0.703253	+	0.710940i	$0.251730\pi$
$390$	0	0
$391$	7.73464	0.391158
$392$	−4.56695	−0.230666
$393$	2.49403	0.125807
$394$	−1.92428	−0.0969437
$395$	0	0
$396$	−19.0757	−0.958588
$397$	−5.49942	−0.276008	−0.138004	−	0.990432i	$-0.544069\pi$
$397$	−5.49942	−0.276008	−0.138004	+	0.990432i	$0.544069\pi$
$398$	2.24324	0.112444
$399$	0	0
$400$	0	0
$401$	−8.51309	−0.425124	−0.212562	−	0.977148i	$-0.568181\pi$
$401$	−8.51309	−0.425124	−0.212562	+	0.977148i	$0.568181\pi$
$402$	−3.58770	−0.178938
$403$	−16.2624	−0.810088
$404$	11.7623	0.585195
$405$	0	0
$406$	4.10880	0.203917
$407$	33.5311	1.66207
$408$	1.26216	0.0624860
$409$	−21.4706	−1.06165	−0.530826	−	0.847481i	$-0.678118\pi$
$409$	−21.4706	−1.06165	−0.530826	+	0.847481i	$0.678118\pi$
$410$	0	0
$411$	−11.8628	−0.585148
$412$	−34.8700	−1.71792
$413$	−44.1965	−2.17477
$414$	4.62397	0.227256
$415$	0	0
$416$	12.3549	0.605750
$417$	7.28662	0.356827
$418$	0	0
$419$	−5.09378	−0.248848	−0.124424	−	0.992229i	$-0.539708\pi$
$419$	−5.09378	−0.248848	−0.124424	+	0.992229i	$0.539708\pi$
$420$	0	0
$421$	5.65823	0.275765	0.137883	−	0.990449i	$-0.455970\pi$
$421$	5.65823	0.275765	0.137883	+	0.990449i	$0.455970\pi$
$422$	−6.11223	−0.297539
$423$	7.45050	0.362256
$424$	8.64464	0.419821
$425$	0	0
$426$	−0.145907	−0.00706920
$427$	−20.1317	−0.974242
$428$	−16.6235	−0.803528
$429$	−18.8121	−0.908256
$430$	0	0
$431$	−38.3668	−1.84807	−0.924033	−	0.382313i	$-0.875128\pi$
$431$	−38.3668	−1.84807	−0.924033	+	0.382313i	$0.875128\pi$
$432$	−17.4408	−0.839122
$433$	6.96250	0.334596	0.167298	−	0.985906i	$-0.446496\pi$
$433$	6.96250	0.334596	0.167298	+	0.985906i	$0.446496\pi$
$434$	4.69533	0.225383
$435$	0	0
$436$	11.6035	0.555706
$437$	0	0
$438$	1.32443	0.0632839
$439$	9.52692	0.454695	0.227347	−	0.973814i	$-0.426995\pi$
$439$	9.52692	0.454695	0.227347	+	0.973814i	$0.426995\pi$
$440$	0	0
$441$	−7.29407	−0.347337
$442$	−1.10988	−0.0527917
$443$	34.7038	1.64883	0.824414	−	0.565987i	$-0.191505\pi$
$443$	34.7038	1.64883	0.824414	+	0.565987i	$0.191505\pi$
$444$	12.7170	0.603524
$445$	0	0
$446$	−3.52603	−0.166963
$447$	13.1299	0.621023
$448$	18.9273	0.894233
$449$	−8.84228	−0.417293	−0.208646	−	0.977991i	$-0.566906\pi$
$449$	−8.84228	−0.417293	−0.208646	+	0.977991i	$0.566906\pi$
$450$	0	0
$451$	35.1704	1.65611
$452$	−19.6874	−0.926019
$453$	−20.2220	−0.950111
$454$	−3.46477	−0.162610
$455$	0	0
$456$	0	0
$457$	0.664998	0.0311073	0.0155536	−	0.999879i	$-0.495049\pi$
$457$	0.664998	0.0311073	0.0155536	+	0.999879i	$0.495049\pi$
$458$	1.09268	0.0510574
$459$	5.13402	0.239636
$460$	0	0
$461$	−39.5392	−1.84152	−0.920762	−	0.390124i	$-0.872432\pi$
$461$	−39.5392	−1.84152	−0.920762	+	0.390124i	$0.872432\pi$
$462$	5.43149	0.252696
$463$	−15.5782	−0.723981	−0.361991	−	0.932182i	$-0.617903\pi$
$463$	−15.5782	−0.723981	−0.361991	+	0.932182i	$0.617903\pi$
$464$	−13.8039	−0.640830
$465$	0	0
$466$	1.92575	0.0892086
$467$	41.0358	1.89891	0.949456	−	0.313901i	$-0.101636\pi$
$467$	41.0358	1.89891	0.949456	+	0.313901i	$0.101636\pi$
$468$	13.0472	0.603107
$469$	36.7335	1.69620
$470$	0	0
$471$	3.17693	0.146385
$472$	−16.3605	−0.753053
$473$	−32.6107	−1.49944
$474$	0.925375	0.0425039
$475$	0	0
$476$	−6.30121	−0.288816
$477$	13.8067	0.632166
$478$	3.77129	0.172495
$479$	−23.5587	−1.07643	−0.538213	−	0.842809i	$-0.680900\pi$
$479$	−23.5587	−1.07643	−0.538213	+	0.842809i	$0.680900\pi$
$480$	0	0
$481$	−22.9343	−1.04571
$482$	−1.58728	−0.0722988
$483$	25.8893	1.17800
$484$	−29.8945	−1.35884
$485$	0	0
$486$	4.93338	0.223783
$487$	−23.9166	−1.08377	−0.541883	−	0.840454i	$-0.682288\pi$
$487$	−23.9166	−1.08377	−0.541883	+	0.840454i	$0.682288\pi$
$488$	−7.45228	−0.337349
$489$	−23.1426	−1.04654
$490$	0	0
$491$	16.3475	0.737751	0.368875	−	0.929479i	$-0.379743\pi$
$491$	16.3475	0.737751	0.368875	+	0.929479i	$0.379743\pi$
$492$	13.3388	0.601360
$493$	4.06343	0.183007
$494$	0	0
$495$	0	0
$496$	−15.7744	−0.708291
$497$	1.49390	0.0670105
$498$	−1.76234	−0.0789723
$499$	−32.0708	−1.43569	−0.717844	−	0.696204i	$-0.754871\pi$
$499$	−32.0708	−1.43569	−0.717844	+	0.696204i	$0.754871\pi$
$500$	0	0
$501$	10.8542	0.484930
$502$	7.90972	0.353028
$503$	14.7254	0.656572	0.328286	−	0.944578i	$-0.393529\pi$
$503$	14.7254	0.656572	0.328286	+	0.944578i	$0.393529\pi$
$504$	−7.72563	−0.344127
$505$	0	0
$506$	12.3212	0.547746
$507$	−0.520921	−0.0231349
$508$	−6.40836	−0.284325
$509$	−22.0663	−0.978073	−0.489037	−	0.872263i	$-0.662652\pi$
$509$	−22.0663	−0.978073	−0.489037	+	0.872263i	$0.662652\pi$
$510$	0	0
$511$	−13.5605	−0.599882
$512$	20.3111	0.897633
$513$	0	0
$514$	5.73990	0.253176
$515$	0	0
$516$	−12.3680	−0.544470
$517$	19.8529	0.873131
$518$	6.62166	0.290939
$519$	23.6225	1.03691
$520$	0	0
$521$	−14.7781	−0.647442	−0.323721	−	0.946153i	$-0.604934\pi$
$521$	−14.7781	−0.647442	−0.323721	+	0.946153i	$0.604934\pi$
$522$	2.42922	0.106324
$523$	−12.7100	−0.555768	−0.277884	−	0.960615i	$-0.589633\pi$
$523$	−12.7100	−0.555768	−0.277884	+	0.960615i	$0.589633\pi$
$524$	−4.60916	−0.201352
$525$	0	0
$526$	4.02903	0.175674
$527$	4.64348	0.202273
$528$	−18.2476	−0.794124
$529$	35.7294	1.55345
$530$	0	0
$531$	−26.1301	−1.13395
$532$	0	0
$533$	−24.0556	−1.04196
$534$	−2.28395	−0.0988361
$535$	0	0
$536$	13.5979	0.587339
$537$	23.9648	1.03416
$538$	0.0486866	0.00209903
$539$	−19.4361	−0.837172
$540$	0	0
$541$	−30.2345	−1.29988	−0.649941	−	0.759985i	$-0.725206\pi$
$541$	−30.2345	−1.29988	−0.649941	+	0.759985i	$0.725206\pi$
$542$	8.65473	0.371752
$543$	−16.4433	−0.705649
$544$	−3.52776	−0.151251
$545$	0	0
$546$	−3.71498	−0.158987
$547$	−7.80744	−0.333822	−0.166911	−	0.985972i	$-0.553379\pi$
$547$	−7.80744	−0.333822	−0.166911	+	0.985972i	$0.553379\pi$
$548$	21.9233	0.936517
$549$	−11.9024	−0.507980
$550$	0	0
$551$	0	0
$552$	9.58359	0.407905
$553$	−9.47467	−0.402904
$554$	−0.163040	−0.00692689
$555$	0	0
$556$	−13.4662	−0.571095
$557$	2.41495	0.102325	0.0511623	−	0.998690i	$-0.483707\pi$
$557$	2.41495	0.102325	0.0511623	+	0.998690i	$0.483707\pi$
$558$	2.77599	0.117517
$559$	22.3048	0.943392
$560$	0	0
$561$	5.37150	0.226785
$562$	−1.52078	−0.0641500
$563$	25.3392	1.06792	0.533960	−	0.845510i	$-0.320703\pi$
$563$	25.3392	1.06792	0.533960	+	0.845510i	$0.320703\pi$
$564$	7.52946	0.317047
$565$	0	0
$566$	−5.56218	−0.233796
$567$	−1.90181	−0.0798685
$568$	0.553006	0.0232036
$569$	−45.1046	−1.89088	−0.945441	−	0.325793i	$-0.894369\pi$
$569$	−45.1046	−1.89088	−0.945441	+	0.325793i	$0.894369\pi$
$570$	0	0
$571$	−9.73299	−0.407313	−0.203656	−	0.979042i	$-0.565283\pi$
$571$	−9.73299	−0.407313	−0.203656	+	0.979042i	$0.565283\pi$
$572$	34.7661	1.45365
$573$	−2.51897	−0.105232
$574$	6.94541	0.289896
$575$	0	0
$576$	11.1903	0.466262
$577$	−8.83145	−0.367658	−0.183829	−	0.982958i	$-0.558849\pi$
$577$	−8.83145	−0.367658	−0.183829	+	0.982958i	$0.558849\pi$
$578$	−4.97192	−0.206805
$579$	−19.6138	−0.815123
$580$	0	0
$581$	18.0441	0.748596
$582$	−3.46823	−0.143763
$583$	36.7900	1.52369
$584$	−5.01978	−0.207720
$585$	0	0
$586$	2.25677	0.0932265
$587$	−26.9932	−1.11413	−0.557065	−	0.830469i	$-0.688073\pi$
$587$	−26.9932	−1.11413	−0.557065	+	0.830469i	$0.688073\pi$
$588$	−7.37137	−0.303990
$589$	0	0
$590$	0	0
$591$	−6.36978	−0.262018
$592$	−22.2461	−0.914308
$593$	−21.2339	−0.871974	−0.435987	−	0.899953i	$-0.643601\pi$
$593$	−21.2339	−0.871974	−0.435987	+	0.899953i	$0.643601\pi$
$594$	8.17847	0.335567
$595$	0	0
$596$	−24.2650	−0.993934
$597$	7.42562	0.303910
$598$	−8.42738	−0.344621
$599$	−30.3294	−1.23922	−0.619612	−	0.784908i	$-0.712710\pi$
$599$	−30.3294	−1.23922	−0.619612	+	0.784908i	$0.712710\pi$
$600$	0	0
$601$	−10.7285	−0.437624	−0.218812	−	0.975767i	$-0.570218\pi$
$601$	−10.7285	−0.437624	−0.218812	+	0.975767i	$0.570218\pi$
$602$	−6.43991	−0.262471
$603$	21.7177	0.884415
$604$	37.3717	1.52063
$605$	0	0
$606$	−1.98008	−0.0804352
$607$	8.81498	0.357789	0.178894	−	0.983868i	$-0.442748\pi$
$607$	8.81498	0.357789	0.178894	+	0.983868i	$0.442748\pi$
$608$	0	0
$609$	13.6010	0.551142
$610$	0	0
$611$	−13.5788	−0.549341
$612$	−3.72543	−0.150591
$613$	24.1307	0.974631	0.487315	−	0.873226i	$-0.337976\pi$
$613$	24.1307	0.974631	0.487315	+	0.873226i	$0.337976\pi$
$614$	4.47383	0.180549
$615$	0	0
$616$	−20.5861	−0.829436
$617$	−15.4650	−0.622595	−0.311298	−	0.950312i	$-0.600764\pi$
$617$	−15.4650	−0.622595	−0.311298	+	0.950312i	$0.600764\pi$
$618$	5.87008	0.236129
$619$	0.0390990	0.00157152	0.000785761	−	1.00000i	$-0.499750\pi$
$619$	0.0390990	0.00157152	0.000785761		1.00000i	$0.499750\pi$
$620$	0	0
$621$	38.9828	1.56433
$622$	2.65247	0.106354
$623$	23.3847	0.936889
$624$	12.4808	0.499633
$625$	0	0
$626$	−6.64584	−0.265621
$627$	0	0
$628$	−5.87120	−0.234286
$629$	6.54853	0.261107
$630$	0	0
$631$	6.73516	0.268122	0.134061	−	0.990973i	$-0.457198\pi$
$631$	6.73516	0.268122	0.134061	+	0.990973i	$0.457198\pi$
$632$	−3.50730	−0.139513
$633$	−20.2328	−0.804182
$634$	8.73989	0.347105
$635$	0	0
$636$	13.9530	0.553274
$637$	13.2937	0.526717
$638$	6.47301	0.256269
$639$	0.883229	0.0349400
$640$	0	0
$641$	27.5788	1.08930	0.544649	−	0.838664i	$-0.316663\pi$
$641$	27.5788	1.08930	0.544649	+	0.838664i	$0.316663\pi$
$642$	2.79843	0.110445
$643$	11.9081	0.469611	0.234806	−	0.972042i	$-0.424555\pi$
$643$	11.9081	0.469611	0.234806	+	0.972042i	$0.424555\pi$
$644$	−47.8453	−1.88537
$645$	0	0
$646$	0	0
$647$	−11.6648	−0.458590	−0.229295	−	0.973357i	$-0.573642\pi$
$647$	−11.6648	−0.458590	−0.229295	+	0.973357i	$0.573642\pi$
$648$	−0.704005	−0.0276559
$649$	−69.6273	−2.73311
$650$	0	0
$651$	15.5426	0.609161
$652$	42.7692	1.67497
$653$	−40.7527	−1.59477	−0.797387	−	0.603468i	$-0.793785\pi$
$653$	−40.7527	−1.59477	−0.797387	+	0.603468i	$0.793785\pi$
$654$	−1.95335	−0.0763819
$655$	0	0
$656$	−23.3337	−0.911029
$657$	−8.01730	−0.312785
$658$	3.92053	0.152838
$659$	−23.1446	−0.901585	−0.450793	−	0.892629i	$-0.648859\pi$
$659$	−23.1446	−0.901585	−0.450793	+	0.892629i	$0.648859\pi$
$660$	0	0
$661$	4.88970	0.190187	0.0950936	−	0.995468i	$-0.469685\pi$
$661$	4.88970	0.190187	0.0950936	+	0.995468i	$0.469685\pi$
$662$	4.72029	0.183459
$663$	−3.67395	−0.142685
$664$	6.67950	0.259215
$665$	0	0
$666$	3.91488	0.151699
$667$	30.8537	1.19466
$668$	−20.0594	−0.776120
$669$	−11.6719	−0.451263
$670$	0	0
$671$	−31.7155	−1.22436
$672$	−11.8081	−0.455506
$673$	29.5149	1.13771	0.568857	−	0.822436i	$-0.307386\pi$
$673$	29.5149	1.13771	0.568857	+	0.822436i	$0.307386\pi$
$674$	−5.44464	−0.209720
$675$	0	0
$676$	0.962701	0.0370270
$677$	−29.4248	−1.13089	−0.565443	−	0.824788i	$-0.691295\pi$
$677$	−29.4248	−1.13089	−0.565443	+	0.824788i	$0.691295\pi$
$678$	3.31421	0.127282
$679$	35.5103	1.36276
$680$	0	0
$681$	−11.4691	−0.439498
$682$	7.39703	0.283247
$683$	29.9433	1.14575	0.572875	−	0.819643i	$-0.305828\pi$
$683$	29.9433	1.14575	0.572875	+	0.819643i	$0.305828\pi$
$684$	0	0
$685$	0	0
$686$	3.30566	0.126211
$687$	3.61699	0.137997
$688$	21.6355	0.824844
$689$	−25.1633	−0.958645
$690$	0	0
$691$	−24.2225	−0.921467	−0.460733	−	0.887539i	$-0.652414\pi$
$691$	−24.2225	−0.921467	−0.460733	+	0.887539i	$0.652414\pi$
$692$	−43.6561	−1.65956
$693$	−32.8789	−1.24896
$694$	5.40354	0.205116
$695$	0	0
$696$	5.03478	0.190843
$697$	6.86870	0.260171
$698$	−10.6574	−0.403389
$699$	6.37465	0.241111
$700$	0	0
$701$	−14.8539	−0.561024	−0.280512	−	0.959851i	$-0.590504\pi$
$701$	−14.8539	−0.561024	−0.280512	+	0.959851i	$0.590504\pi$
$702$	−5.59384	−0.211126
$703$	0	0
$704$	29.8182	1.12381
$705$	0	0
$706$	10.8666	0.408970
$707$	20.2735	0.762463
$708$	−26.4070	−0.992434
$709$	−29.8009	−1.11920	−0.559598	−	0.828764i	$-0.689044\pi$
$709$	−29.8009	−1.11920	−0.559598	+	0.828764i	$0.689044\pi$
$710$	0	0
$711$	−5.60165	−0.210078
$712$	8.65647	0.324415
$713$	35.2581	1.32043
$714$	1.06076	0.0396978
$715$	0	0
$716$	−44.2888	−1.65515
$717$	12.4838	0.466216
$718$	−0.303811	−0.0113381
$719$	−25.8925	−0.965628	−0.482814	−	0.875723i	$-0.660385\pi$
$719$	−25.8925	−0.965628	−0.482814	+	0.875723i	$0.660385\pi$
$720$	0	0
$721$	−60.1021	−2.23832
$722$	0	0
$723$	−5.25426	−0.195408
$724$	30.3884	1.12938
$725$	0	0
$726$	5.03248	0.186773
$727$	9.34885	0.346729	0.173365	−	0.984858i	$-0.444536\pi$
$727$	9.34885	0.346729	0.173365	+	0.984858i	$0.444536\pi$
$728$	14.0803	0.521850
$729$	14.5913	0.540419
$730$	0	0
$731$	−6.36879	−0.235558
$732$	−12.0285	−0.444586
$733$	7.20434	0.266098	0.133049	−	0.991109i	$-0.457523\pi$
$733$	7.20434	0.266098	0.133049	+	0.991109i	$0.457523\pi$
$734$	3.06950	0.113297
$735$	0	0
$736$	−26.7864	−0.987360
$737$	57.8701	2.13167
$738$	4.10629	0.151155
$739$	12.9740	0.477258	0.238629	−	0.971111i	$-0.423302\pi$
$739$	12.9740	0.477258	0.238629	+	0.971111i	$0.423302\pi$
$740$	0	0
$741$	0	0
$742$	7.26523	0.266715
$743$	9.23100	0.338652	0.169326	−	0.985560i	$-0.445841\pi$
$743$	9.23100	0.338652	0.169326	+	0.985560i	$0.445841\pi$
$744$	5.75349	0.210933
$745$	0	0
$746$	−4.02950	−0.147530
$747$	10.6681	0.390326
$748$	−9.92694	−0.362965
$749$	−28.6523	−1.04693
$750$	0	0
$751$	14.9226	0.544534	0.272267	−	0.962222i	$-0.412227\pi$
$751$	14.9226	0.544534	0.272267	+	0.962222i	$0.412227\pi$
$752$	−13.1714	−0.480310
$753$	26.1829	0.954157
$754$	−4.42736	−0.161235
$755$	0	0
$756$	−31.7583	−1.15504
$757$	24.4976	0.890382	0.445191	−	0.895436i	$-0.353136\pi$
$757$	24.4976	0.890382	0.445191	+	0.895436i	$0.353136\pi$
$758$	1.62273	0.0589402
$759$	40.7860	1.48044
$760$	0	0
$761$	−15.8968	−0.576260	−0.288130	−	0.957591i	$-0.593034\pi$
$761$	−15.8968	−0.576260	−0.288130	+	0.957591i	$0.593034\pi$
$762$	1.07879	0.0390805
$763$	19.9998	0.724041
$764$	4.65525	0.168421
$765$	0	0
$766$	10.4319	0.376919
$767$	47.6231	1.71957
$768$	9.06916	0.327255
$769$	−29.5850	−1.06686	−0.533432	−	0.845843i	$-0.679098\pi$
$769$	−29.5850	−1.06686	−0.533432	+	0.845843i	$0.679098\pi$
$770$	0	0
$771$	19.0003	0.684280
$772$	36.2478	1.30459
$773$	14.1310	0.508257	0.254129	−	0.967170i	$-0.418211\pi$
$773$	14.1310	0.508257	0.254129	+	0.967170i	$0.418211\pi$
$774$	−3.80743	−0.136855
$775$	0	0
$776$	13.1451	0.471880
$777$	21.9191	0.786345
$778$	8.63032	0.309412
$779$	0	0
$780$	0	0
$781$	2.35349	0.0842145
$782$	2.40631	0.0860494
$783$	20.4798	0.731887
$784$	12.8948	0.460529
$785$	0	0
$786$	0.775912	0.0276759
$787$	24.2465	0.864293	0.432146	−	0.901803i	$-0.357756\pi$
$787$	24.2465	0.864293	0.432146	+	0.901803i	$0.357756\pi$
$788$	11.7718	0.419354
$789$	13.3370	0.474809
$790$	0	0
$791$	−33.9333	−1.20653
$792$	−12.1710	−0.432477
$793$	21.6925	0.770324
$794$	−1.71091	−0.0607180
$795$	0	0
$796$	−13.7231	−0.486402
$797$	−27.9258	−0.989184	−0.494592	−	0.869125i	$-0.664682\pi$
$797$	−27.9258	−0.989184	−0.494592	+	0.869125i	$0.664682\pi$
$798$	0	0
$799$	3.87723	0.137166
$800$	0	0
$801$	13.8256	0.488504
$802$	−2.64849	−0.0935214
$803$	−21.3633	−0.753893
$804$	21.9479	0.774043
$805$	0	0
$806$	−5.05936	−0.178208
$807$	0.161163	0.00567321
$808$	7.50476	0.264017
$809$	8.78914	0.309010	0.154505	−	0.987992i	$-0.450622\pi$
$809$	8.78914	0.309010	0.154505	+	0.987992i	$0.450622\pi$
$810$	0	0
$811$	−14.0993	−0.495092	−0.247546	−	0.968876i	$-0.579624\pi$
$811$	−14.0993	−0.495092	−0.247546	+	0.968876i	$0.579624\pi$
$812$	−25.1357	−0.882091
$813$	28.6490	1.00477
$814$	10.4318	0.365633
$815$	0	0
$816$	−3.56370	−0.124755
$817$	0	0
$818$	−6.67967	−0.233549
$819$	22.4882	0.785801
$820$	0	0
$821$	−30.5326	−1.06559	−0.532797	−	0.846243i	$-0.678859\pi$
$821$	−30.5326	−1.06559	−0.532797	+	0.846243i	$0.678859\pi$
$822$	−3.69060	−0.128725
$823$	−14.3583	−0.500498	−0.250249	−	0.968182i	$-0.580512\pi$
$823$	−14.3583	−0.500498	−0.250249	+	0.968182i	$0.580512\pi$
$824$	−22.2484	−0.775059
$825$	0	0
$826$	−13.7499	−0.478420
$827$	−8.27364	−0.287703	−0.143851	−	0.989599i	$-0.545949\pi$
$827$	−8.27364	−0.287703	−0.143851	+	0.989599i	$0.545949\pi$
$828$	−28.2873	−0.983052
$829$	13.1498	0.456710	0.228355	−	0.973578i	$-0.426665\pi$
$829$	13.1498	0.456710	0.228355	+	0.973578i	$0.426665\pi$
$830$	0	0
$831$	−0.539697	−0.0187219
$832$	−20.3948	−0.707062
$833$	−3.79582	−0.131517
$834$	2.26693	0.0784972
$835$	0	0
$836$	0	0
$837$	23.4032	0.808934
$838$	−1.58472	−0.0547431
$839$	−8.41534	−0.290530	−0.145265	−	0.989393i	$-0.546403\pi$
$839$	−8.41534	−0.290530	−0.145265	+	0.989393i	$0.546403\pi$
$840$	0	0
$841$	−12.7909	−0.441064
$842$	1.76032	0.0606646
$843$	−5.03410	−0.173384
$844$	37.3918	1.28708
$845$	0	0
$846$	2.31791	0.0796914
$847$	−51.5262	−1.77046
$848$	−24.4082	−0.838181
$849$	−18.4120	−0.631899
$850$	0	0
$851$	49.7232	1.70449
$852$	0.892589	0.0305796
$853$	52.0363	1.78169	0.890845	−	0.454308i	$-0.150113\pi$
$853$	52.0363	1.78169	0.890845	+	0.454308i	$0.150113\pi$
$854$	−6.26314	−0.214320
$855$	0	0
$856$	−10.6064	−0.362520
$857$	−21.4595	−0.733043	−0.366522	−	0.930410i	$-0.619451\pi$
$857$	−21.4595	−0.733043	−0.366522	+	0.930410i	$0.619451\pi$
$858$	−5.85259	−0.199804
$859$	18.1479	0.619199	0.309599	−	0.950867i	$-0.399805\pi$
$859$	18.1479	0.619199	0.309599	+	0.950867i	$0.399805\pi$
$860$	0	0
$861$	22.9908	0.783525
$862$	−11.9362	−0.406549
$863$	−13.7867	−0.469303	−0.234652	−	0.972080i	$-0.575395\pi$
$863$	−13.7867	−0.469303	−0.234652	+	0.972080i	$0.575395\pi$
$864$	−17.7800	−0.604888
$865$	0	0
$866$	2.16609	0.0736066
$867$	−16.4581	−0.558948
$868$	−28.7238	−0.974951
$869$	−14.9264	−0.506343
$870$	0	0
$871$	−39.5815	−1.34117
$872$	7.40345	0.250712
$873$	20.9945	0.710557
$874$	0	0
$875$	0	0
$876$	−8.10226	−0.273750
$877$	−21.1937	−0.715660	−0.357830	−	0.933787i	$-0.616483\pi$
$877$	−21.1937	−0.715660	−0.357830	+	0.933787i	$0.616483\pi$
$878$	2.96390	0.100027
$879$	7.47041	0.251971
$880$	0	0
$881$	44.5944	1.50242	0.751212	−	0.660061i	$-0.229469\pi$
$881$	44.5944	1.50242	0.751212	+	0.660061i	$0.229469\pi$
$882$	−2.26924	−0.0764093
$883$	1.69355	0.0569924	0.0284962	−	0.999594i	$-0.490928\pi$
$883$	1.69355	0.0569924	0.0284962	+	0.999594i	$0.490928\pi$
$884$	6.78974	0.228364
$885$	0	0
$886$	10.7966	0.362720
$887$	48.4354	1.62630	0.813151	−	0.582053i	$-0.197750\pi$
$887$	48.4354	1.62630	0.813151	+	0.582053i	$0.197750\pi$
$888$	8.11394	0.272286
$889$	−11.0455	−0.370453
$890$	0	0
$891$	−2.99611	−0.100374
$892$	21.5706	0.722238
$893$	0	0
$894$	4.08481	0.136617
$895$	0	0
$896$	28.8204	0.962822
$897$	−27.8965	−0.931436
$898$	−2.75090	−0.0917988
$899$	18.5230	0.617776
$900$	0	0
$901$	7.18499	0.239367
$902$	10.9418	0.364322
$903$	−21.3175	−0.709402
$904$	−12.5613	−0.417783
$905$	0	0
$906$	−6.29122	−0.209012
$907$	27.9388	0.927692	0.463846	−	0.885916i	$-0.346469\pi$
$907$	27.9388	0.927692	0.463846	+	0.885916i	$0.346469\pi$
$908$	21.1958	0.703408
$909$	11.9862	0.397556
$910$	0	0
$911$	45.0862	1.49377	0.746887	−	0.664951i	$-0.231548\pi$
$911$	45.0862	1.49377	0.746887	+	0.664951i	$0.231548\pi$
$912$	0	0
$913$	28.4267	0.940787
$914$	0.206886	0.00684318
$915$	0	0
$916$	−6.68448	−0.220861
$917$	−7.94436	−0.262346
$918$	1.59723	0.0527166
$919$	−5.86849	−0.193584	−0.0967918	−	0.995305i	$-0.530858\pi$
$919$	−5.86849	−0.193584	−0.0967918	+	0.995305i	$0.530858\pi$
$920$	0	0
$921$	14.8093	0.487984
$922$	−12.3010	−0.405111
$923$	−1.60972	−0.0529846
$924$	−33.2273	−1.09310
$925$	0	0
$926$	−4.84651	−0.159266
$927$	−35.5338	−1.16708
$928$	−14.0723	−0.461947
$929$	46.6314	1.52993	0.764963	−	0.644075i	$-0.222757\pi$
$929$	46.6314	1.52993	0.764963	+	0.644075i	$0.222757\pi$
$930$	0	0
$931$	0	0
$932$	−11.7808	−0.385894
$933$	8.78026	0.287453
$934$	12.7666	0.417735
$935$	0	0
$936$	8.32460	0.272098
$937$	−51.2940	−1.67570	−0.837850	−	0.545900i	$-0.816188\pi$
$937$	−51.2940	−1.67570	−0.837850	+	0.545900i	$0.816188\pi$
$938$	11.4281	0.373140
$939$	−21.9992	−0.717916
$940$	0	0
$941$	−11.3486	−0.369954	−0.184977	−	0.982743i	$-0.559221\pi$
$941$	−11.3486	−0.369954	−0.184977	+	0.982743i	$0.559221\pi$
$942$	0.988367	0.0322027
$943$	52.1543	1.69838
$944$	46.1940	1.50349
$945$	0	0
$946$	−10.1454	−0.329857
$947$	−7.20935	−0.234272	−0.117136	−	0.993116i	$-0.537371\pi$
$947$	−7.20935	−0.234272	−0.117136	+	0.993116i	$0.537371\pi$
$948$	−5.66101	−0.183861
$949$	14.6119	0.474321
$950$	0	0
$951$	28.9309	0.938150
$952$	−4.02041	−0.130302
$953$	20.5394	0.665338	0.332669	−	0.943044i	$-0.392051\pi$
$953$	20.5394	0.665338	0.332669	+	0.943044i	$0.392051\pi$
$954$	4.29538	0.139068
$955$	0	0
$956$	−23.0710	−0.746170
$957$	21.4271	0.692639
$958$	−7.32930	−0.236799
$959$	37.7871	1.22021
$960$	0	0
$961$	−9.83289	−0.317190
$962$	−7.13504	−0.230043
$963$	−16.9400	−0.545882
$964$	9.71026	0.312746
$965$	0	0
$966$	8.05436	0.259145
$967$	−17.0669	−0.548835	−0.274417	−	0.961611i	$-0.588485\pi$
$967$	−17.0669	−0.548835	−0.274417	+	0.961611i	$0.588485\pi$
$968$	−19.0738	−0.613055
$969$	0	0
$970$	0	0
$971$	−3.48171	−0.111733	−0.0558667	−	0.998438i	$-0.517792\pi$
$971$	−3.48171	−0.111733	−0.0558667	+	0.998438i	$0.517792\pi$
$972$	−30.1801	−0.968028
$973$	−23.2104	−0.744093
$974$	−7.44065	−0.238414
$975$	0	0
$976$	21.0416	0.673524
$977$	21.9600	0.702562	0.351281	−	0.936270i	$-0.385746\pi$
$977$	21.9600	0.702562	0.351281	+	0.936270i	$0.385746\pi$
$978$	−7.19984	−0.230226
$979$	36.8403	1.17742
$980$	0	0
$981$	11.8244	0.377523
$982$	5.08583	0.162295
$983$	−5.49704	−0.175328	−0.0876641	−	0.996150i	$-0.527940\pi$
$983$	−5.49704	−0.175328	−0.0876641	+	0.996150i	$0.527940\pi$
$984$	8.51065	0.271310
$985$	0	0
$986$	1.26416	0.0402592
$987$	12.9778	0.413088
$988$	0	0
$989$	−48.3584	−1.53771
$990$	0	0
$991$	−29.4970	−0.937003	−0.468502	−	0.883463i	$-0.655206\pi$
$991$	−29.4970	−0.937003	−0.468502	+	0.883463i	$0.655206\pi$
$992$	−16.0812	−0.510577
$993$	15.6252	0.495850
$994$	0.464764	0.0147414
$995$	0	0
$996$	10.7812	0.341614
$997$	−7.26449	−0.230069	−0.115034	−	0.993362i	$-0.536698\pi$
$997$	−7.26449	−0.230069	−0.115034	+	0.993362i	$0.536698\pi$
$998$	−9.97749	−0.315832
$999$	33.0048	1.04422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bz.1.4		6
5.4	even	2		9025.2.a.br.1.3		6
19.7	even	3		475.2.e.f.201.3	yes	12
19.11	even	3		475.2.e.f.26.3	✓	12
19.18	odd	2		9025.2.a.bs.1.3		6
95.7	odd	12		475.2.j.d.49.6		24
95.49	even	6		475.2.e.h.26.4	yes	12
95.64	even	6		475.2.e.h.201.4	yes	12
95.68	odd	12		475.2.j.d.349.6		24
95.83	odd	12		475.2.j.d.49.7		24
95.87	odd	12		475.2.j.d.349.7		24
95.94	odd	2		9025.2.a.by.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.e.f.26.3	✓	12	19.11	even	3
475.2.e.f.201.3	yes	12	19.7	even	3
475.2.e.h.26.4	yes	12	95.49	even	6
475.2.e.h.201.4	yes	12	95.64	even	6
475.2.j.d.49.6		24	95.7	odd	12
475.2.j.d.49.7		24	95.83	odd	12
475.2.j.d.349.6		24	95.68	odd	12
475.2.j.d.349.7		24	95.87	odd	12
9025.2.a.br.1.3		6	5.4	even	2
9025.2.a.bs.1.3		6	19.18	odd	2
9025.2.a.by.1.4		6	95.94	odd	2
9025.2.a.bz.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} +1.02983 q^{3} -1.90321 q^{4} +0.320390 q^{6} -3.28038 q^{7} -1.21432 q^{8} -1.93944 q^{9} -5.16792 q^{11} -1.95999 q^{12} +3.53471 q^{13} -1.02055 q^{14} +3.42864 q^{16} -1.00928 q^{17} -0.603375 q^{18} -3.37825 q^{21} -1.60778 q^{22} -7.66351 q^{23} -1.25055 q^{24} +1.09968 q^{26} -5.08681 q^{27} +6.24326 q^{28} -4.02606 q^{29} -4.60077 q^{31} +3.49532 q^{32} -5.32210 q^{33} -0.313995 q^{34} +3.69117 q^{36} -6.48831 q^{37} +3.64017 q^{39} -6.80553 q^{41} -1.05100 q^{42} +6.31022 q^{43} +9.83565 q^{44} -2.38418 q^{46} -3.84157 q^{47} +3.53093 q^{48} +3.76091 q^{49} -1.03939 q^{51} -6.72730 q^{52} -7.11892 q^{53} -1.58255 q^{54} +3.98343 q^{56} -1.25254 q^{58} +13.4730 q^{59} +6.13700 q^{61} -1.43134 q^{62} +6.36211 q^{63} -5.76986 q^{64} -1.65575 q^{66} -11.1979 q^{67} +1.92088 q^{68} -7.89215 q^{69} -0.455404 q^{71} +2.35510 q^{72} +4.13382 q^{73} -2.01856 q^{74} +16.9528 q^{77} +1.13248 q^{78} +2.88828 q^{79} +0.579752 q^{81} -2.11725 q^{82} -5.50061 q^{83} +6.42953 q^{84} +1.96316 q^{86} -4.14617 q^{87} +6.27551 q^{88} -7.12866 q^{89} -11.5952 q^{91} +14.5853 q^{92} -4.73803 q^{93} -1.19514 q^{94} +3.59960 q^{96} -10.8250 q^{97} +1.17005 q^{98} +10.0229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 6 q^{8} + 7 q^{9} - q^{11} + 7 q^{12} + 5 q^{13} - 6 q^{14} - 6 q^{16} - 3 q^{17} + 7 q^{18} + 3 q^{21} + 9 q^{22} - 6 q^{23} + 11 q^{24} + 19 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - q^6 + 2 * q^7 + 6 * q^8 + 7 * q^9 - q^11 + 7 * q^12 + 5 * q^13 - 6 * q^14 - 6 * q^16 - 3 * q^17 + 7 * q^18 + 3 * q^21 + 9 * q^22 - 6 * q^23 + 11 * q^24 + 19 * q^26 + 18 * q^27 - 4 * q^28 + 3 * q^29 - 3 * q^31 - 6 * q^32 - 18 * q^33 - q^34 + 13 * q^36 - 6 * q^37 + 8 * q^39 + 11 * q^41 - 11 * q^42 + 13 * q^43 + 21 * q^44 - 12 * q^46 - 6 * q^47 - 19 * q^48 + 4 * q^49 - 17 * q^51 - q^52 + 18 * q^53 + 18 * q^54 + 4 * q^56 + 5 * q^58 + 4 * q^59 + 25 * q^61 - 21 * q^62 + 43 * q^63 - 22 * q^64 + 34 * q^66 + 6 * q^67 - q^68 + 13 * q^69 + 18 * q^71 + 13 * q^72 + q^73 - 6 * q^74 - 11 * q^77 + 72 * q^78 + 3 * q^79 + 2 * q^81 + 31 * q^82 - 23 * q^83 + 37 * q^84 + 9 * q^86 + 11 * q^87 + 11 * q^88 + 12 * q^89 - 11 * q^91 + 28 * q^92 - 13 * q^93 + 8 * q^94 - 13 * q^96 + 3 * q^97 - 22 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

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Database

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