LMFDB - Embedded newform 3025.2.a.bi.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 $ x^8 - 3x^7 - 7x^6 + 18x^5 + 16x^4 - 30x^3 - 12x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.860597$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.139403 q^{2} +2.98582 q^{3} -1.98057 q^{4} -0.416233 q^{6} -3.56959 q^{7} +0.554904 q^{8} +5.91512 q^{9} +O(q^{10})$	\(q-0.139403 q^{2} +2.98582 q^{3} -1.98057 q^{4} -0.416233 q^{6} -3.56959 q^{7} +0.554904 q^{8} +5.91512 q^{9} -5.91361 q^{12} -1.19562 q^{13} +0.497612 q^{14} +3.88378 q^{16} -5.43544 q^{17} -0.824586 q^{18} +4.80260 q^{19} -10.6581 q^{21} -1.82203 q^{23} +1.65684 q^{24} +0.166673 q^{26} +8.70401 q^{27} +7.06980 q^{28} -4.14137 q^{29} +1.28688 q^{31} -1.65122 q^{32} +0.757717 q^{34} -11.7153 q^{36} -3.16092 q^{37} -0.669497 q^{38} -3.56990 q^{39} -6.40712 q^{41} +1.48578 q^{42} +4.23557 q^{43} +0.253997 q^{46} -7.02813 q^{47} +11.5963 q^{48} +5.74195 q^{49} -16.2292 q^{51} +2.36800 q^{52} -10.8109 q^{53} -1.21337 q^{54} -1.98078 q^{56} +14.3397 q^{57} +0.577320 q^{58} -4.01607 q^{59} -14.4814 q^{61} -0.179395 q^{62} -21.1145 q^{63} -7.53737 q^{64} +8.14233 q^{67} +10.7652 q^{68} -5.44026 q^{69} +8.51253 q^{71} +3.28232 q^{72} -8.54108 q^{73} +0.440643 q^{74} -9.51187 q^{76} +0.497655 q^{78} +0.393997 q^{79} +8.24324 q^{81} +0.893172 q^{82} -1.56363 q^{83} +21.1092 q^{84} -0.590452 q^{86} -12.3654 q^{87} -6.90425 q^{89} +4.26786 q^{91} +3.60866 q^{92} +3.84240 q^{93} +0.979744 q^{94} -4.93024 q^{96} -9.31179 q^{97} -0.800446 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	$ 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} - 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} + q^{19} + 5 q^{21} - 2 q^{23} + q^{24} - 2 q^{26} + 2 q^{27} - 9 q^{28} + 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} + 8 q^{37} + 37 q^{38} - q^{39} - 41 q^{42} - 14 q^{43} - 20 q^{46} - 11 q^{47} + 27 q^{48} - 12 q^{49} - 25 q^{51} + 7 q^{52} - 11 q^{53} + 30 q^{54} - 10 q^{56} + 2 q^{57} - 27 q^{58} + 17 q^{59} - 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} - 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} - 6 q^{74} - 31 q^{76} + 17 q^{78} + 23 q^{79} - 4 q^{81} + 17 q^{82} - 41 q^{83} + 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} - 33 q^{92} + 59 q^{93} + 50 q^{94} - 61 q^{96} - 2 q^{97} - 26 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 - 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 + q^19 + 5 * q^21 - 2 * q^23 + q^24 - 2 * q^26 + 2 * q^27 - 9 * q^28 + 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 + 8 * q^37 + 37 * q^38 - q^39 - 41 * q^42 - 14 * q^43 - 20 * q^46 - 11 * q^47 + 27 * q^48 - 12 * q^49 - 25 * q^51 + 7 * q^52 - 11 * q^53 + 30 * q^54 - 10 * q^56 + 2 * q^57 - 27 * q^58 + 17 * q^59 - 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 - 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 - 6 * q^74 - 31 * q^76 + 17 * q^78 + 23 * q^79 - 4 * q^81 + 17 * q^82 - 41 * q^83 + 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 - 33 * q^92 + 59 * q^93 + 50 * q^94 - 61 * q^96 - 2 * q^97 - 26 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−0.139403	−0.0985729	−0.0492865	−	0.998785i	$-0.515695\pi$
$2$	−0.139403	−0.0985729	−0.0492865	+	0.998785i	$0.515695\pi$
$3$	2.98582	1.72386	0.861932	−	0.507024i	$-0.169255\pi$
$3$	2.98582	1.72386	0.861932	+	0.507024i	$0.169255\pi$
$4$	−1.98057	−0.990283
$5$	0	0
$5$	0	0
$6$	−0.416233	−0.169926
$7$	−3.56959	−1.34918	−0.674588	−	0.738194i	$-0.735679\pi$
$7$	−3.56959	−1.34918	−0.674588	+	0.738194i	$0.735679\pi$
$8$	0.554904	0.196188
$9$	5.91512	1.97171
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−5.91361	−1.70711
$13$	−1.19562	−0.331604	−0.165802	−	0.986159i	$-0.553021\pi$
$13$	−1.19562	−0.331604	−0.165802	+	0.986159i	$0.553021\pi$
$14$	0.497612	0.132992
$15$	0	0
$16$	3.88378	0.970945
$17$	−5.43544	−1.31829	−0.659143	−	0.752017i	$-0.729081\pi$
$17$	−5.43544	−1.31829	−0.659143	+	0.752017i	$0.729081\pi$
$18$	−0.824586	−0.194357
$19$	4.80260	1.10179	0.550896	−	0.834574i	$-0.314286\pi$
$19$	4.80260	1.10179	0.550896	+	0.834574i	$0.314286\pi$
$20$	0	0
$21$	−10.6581	−2.32580
$22$	0	0
$23$	−1.82203	−0.379920	−0.189960	−	0.981792i	$-0.560836\pi$
$23$	−1.82203	−0.379920	−0.189960	+	0.981792i	$0.560836\pi$
$24$	1.65684	0.338201
$25$	0	0
$26$	0.166673	0.0326872
$27$	8.70401	1.67509
$28$	7.06980	1.33607
$29$	−4.14137	−0.769033	−0.384517	−	0.923118i	$-0.625632\pi$
$29$	−4.14137	−0.769033	−0.384517	+	0.923118i	$0.625632\pi$
$30$	0	0
$31$	1.28688	0.231131	0.115565	−	0.993300i	$-0.463132\pi$
$31$	1.28688	0.231131	0.115565	+	0.993300i	$0.463132\pi$
$32$	−1.65122	−0.291897
$33$	0	0
$34$	0.757717	0.129947
$35$	0	0
$36$	−11.7153	−1.95255
$37$	−3.16092	−0.519653	−0.259826	−	0.965655i	$-0.583665\pi$
$37$	−3.16092	−0.519653	−0.259826	+	0.965655i	$0.583665\pi$
$38$	−0.669497	−0.108607
$39$	−3.56990	−0.571641
$40$	0	0
$41$	−6.40712	−1.00062	−0.500312	−	0.865845i	$-0.666781\pi$
$41$	−6.40712	−1.00062	−0.500312	+	0.865845i	$0.666781\pi$
$42$	1.48578	0.229261
$43$	4.23557	0.645919	0.322959	−	0.946413i	$-0.395322\pi$
$43$	4.23557	0.645919	0.322959	+	0.946413i	$0.395322\pi$
$44$	0	0
$45$	0	0
$46$	0.253997	0.0374498
$47$	−7.02813	−1.02516	−0.512579	−	0.858640i	$-0.671310\pi$
$47$	−7.02813	−1.02516	−0.512579	+	0.858640i	$0.671310\pi$
$48$	11.5963	1.67378
$49$	5.74195	0.820278
$50$	0	0
$51$	−16.2292	−2.27255
$52$	2.36800	0.328382
$53$	−10.8109	−1.48500	−0.742498	−	0.669848i	$-0.766359\pi$
$53$	−10.8109	−1.48500	−0.742498	+	0.669848i	$0.766359\pi$
$54$	−1.21337	−0.165118
$55$	0	0
$56$	−1.98078	−0.264692
$57$	14.3397	1.89934
$58$	0.577320	0.0758058
$59$	−4.01607	−0.522848	−0.261424	−	0.965224i	$-0.584192\pi$
$59$	−4.01607	−0.522848	−0.261424	+	0.965224i	$0.584192\pi$
$60$	0	0
$61$	−14.4814	−1.85415	−0.927076	−	0.374873i	$-0.877686\pi$
$61$	−14.4814	−1.85415	−0.927076	+	0.374873i	$0.877686\pi$
$62$	−0.179395	−0.0227832
$63$	−21.1145	−2.66018
$64$	−7.53737	−0.942171
$65$	0	0
$66$	0	0
$67$	8.14233	0.994744	0.497372	−	0.867537i	$-0.334298\pi$
$67$	8.14233	0.994744	0.497372	+	0.867537i	$0.334298\pi$
$68$	10.7652	1.30548
$69$	−5.44026	−0.654930
$70$	0	0
$71$	8.51253	1.01025	0.505125	−	0.863046i	$-0.331446\pi$
$71$	8.51253	1.01025	0.505125	+	0.863046i	$0.331446\pi$
$72$	3.28232	0.386825
$73$	−8.54108	−0.999657	−0.499829	−	0.866124i	$-0.666604\pi$
$73$	−8.54108	−0.999657	−0.499829	+	0.866124i	$0.666604\pi$
$74$	0.440643	0.0512237
$75$	0	0
$76$	−9.51187	−1.09109
$77$	0	0
$78$	0.497655	0.0563483
$79$	0.393997	0.0443282	0.0221641	−	0.999754i	$-0.492944\pi$
$79$	0.393997	0.0443282	0.0221641	+	0.999754i	$0.492944\pi$
$80$	0	0
$81$	8.24324	0.915916
$82$	0.893172	0.0986344
$83$	−1.56363	−0.171631	−0.0858155	−	0.996311i	$-0.527350\pi$
$83$	−1.56363	−0.171631	−0.0858155	+	0.996311i	$0.527350\pi$
$84$	21.1092	2.30320
$85$	0	0
$86$	−0.590452	−0.0636701
$87$	−12.3654	−1.32571
$88$	0	0
$89$	−6.90425	−0.731849	−0.365925	−	0.930645i	$-0.619247\pi$
$89$	−6.90425	−0.731849	−0.365925	+	0.930645i	$0.619247\pi$
$90$	0	0
$91$	4.26786	0.447393
$92$	3.60866	0.376229
$93$	3.84240	0.398438
$94$	0.979744	0.101053
$95$	0	0
$96$	−4.93024	−0.503190
$97$	−9.31179	−0.945469	−0.472735	−	0.881205i	$-0.656733\pi$
$97$	−9.31179	−0.945469	−0.472735	+	0.881205i	$0.656733\pi$
$98$	−0.800446	−0.0808572
$99$	0	0
$100$	0	0
$101$	−2.62204	−0.260903	−0.130452	−	0.991455i	$-0.541643\pi$
$101$	−2.62204	−0.260903	−0.130452	+	0.991455i	$0.541643\pi$
$102$	2.26241	0.224012
$103$	6.60559	0.650869	0.325434	−	0.945565i	$-0.394490\pi$
$103$	6.60559	0.650869	0.325434	+	0.945565i	$0.394490\pi$
$104$	−0.663452	−0.0650568
$105$	0	0
$106$	1.50708	0.146380
$107$	−14.0622	−1.35945	−0.679723	−	0.733469i	$-0.737900\pi$
$107$	−14.0622	−1.35945	−0.679723	+	0.733469i	$0.737900\pi$
$108$	−17.2389	−1.65881
$109$	12.3514	1.18305	0.591525	−	0.806286i	$-0.298526\pi$
$109$	12.3514	1.18305	0.591525	+	0.806286i	$0.298526\pi$
$110$	0	0
$111$	−9.43795	−0.895810
$112$	−13.8635	−1.30998
$113$	−3.56866	−0.335711	−0.167855	−	0.985812i	$-0.553684\pi$
$113$	−3.56866	−0.335711	−0.167855	+	0.985812i	$0.553684\pi$
$114$	−1.99900	−0.187223
$115$	0	0
$116$	8.20226	0.761561
$117$	−7.07221	−0.653826
$118$	0.559853	0.0515387
$119$	19.4023	1.77860
$120$	0	0
$121$	0	0
$122$	2.01875	0.182769
$123$	−19.1305	−1.72494
$124$	−2.54875	−0.228885
$125$	0	0
$126$	2.94343	0.262222
$127$	6.27949	0.557215	0.278608	−	0.960405i	$-0.410127\pi$
$127$	6.27949	0.557215	0.278608	+	0.960405i	$0.410127\pi$
$128$	4.35317	0.384769
$129$	12.6467	1.11348
$130$	0	0
$131$	18.4755	1.61421	0.807105	−	0.590408i	$-0.201033\pi$
$131$	18.4755	1.61421	0.807105	+	0.590408i	$0.201033\pi$
$132$	0	0
$133$	−17.1433	−1.48651
$134$	−1.13507	−0.0980548
$135$	0	0
$136$	−3.01614	−0.258632
$137$	−4.17657	−0.356829	−0.178414	−	0.983955i	$-0.557097\pi$
$137$	−4.17657	−0.356829	−0.178414	+	0.983955i	$0.557097\pi$
$138$	0.758389	0.0645584
$139$	−2.73081	−0.231624	−0.115812	−	0.993271i	$-0.536947\pi$
$139$	−2.73081	−0.231624	−0.115812	+	0.993271i	$0.536947\pi$
$140$	0	0
$141$	−20.9847	−1.76723
$142$	−1.18667	−0.0995834
$143$	0	0
$144$	22.9730	1.91442
$145$	0	0
$146$	1.19065	0.0985391
$147$	17.1444	1.41405
$148$	6.26042	0.514604
$149$	−17.7487	−1.45403	−0.727016	−	0.686620i	$-0.759094\pi$
$149$	−17.7487	−1.45403	−0.727016	+	0.686620i	$0.759094\pi$
$150$	0	0
$151$	−1.45380	−0.118309	−0.0591543	−	0.998249i	$-0.518840\pi$
$151$	−1.45380	−0.118309	−0.0591543	+	0.998249i	$0.518840\pi$
$152$	2.66498	0.216158
$153$	−32.1512	−2.59927
$154$	0	0
$155$	0	0
$156$	7.07042	0.566086
$157$	3.35698	0.267916	0.133958	−	0.990987i	$-0.457231\pi$
$157$	3.35698	0.267916	0.133958	+	0.990987i	$0.457231\pi$
$158$	−0.0549245	−0.00436955
$159$	−32.2795	−2.55993
$160$	0	0
$161$	6.50390	0.512579
$162$	−1.14913	−0.0902845
$163$	6.80969	0.533376	0.266688	−	0.963783i	$-0.414071\pi$
$163$	6.80969	0.533376	0.266688	+	0.963783i	$0.414071\pi$
$164$	12.6897	0.990901
$165$	0	0
$166$	0.217975	0.0169182
$167$	−19.6417	−1.51992	−0.759960	−	0.649970i	$-0.774781\pi$
$167$	−19.6417	−1.51992	−0.759960	+	0.649970i	$0.774781\pi$
$168$	−5.91424	−0.456293
$169$	−11.5705	−0.890038
$170$	0	0
$171$	28.4079	2.17241
$172$	−8.38883	−0.639643
$173$	0.234158	0.0178027	0.00890134	−	0.999960i	$-0.497167\pi$
$173$	0.234158	0.0178027	0.00890134	+	0.999960i	$0.497167\pi$
$174$	1.72377	0.130679
$175$	0	0
$176$	0	0
$177$	−11.9913	−0.901319
$178$	0.962474	0.0721405
$179$	16.9855	1.26956	0.634778	−	0.772694i	$-0.281091\pi$
$179$	16.9855	1.26956	0.634778	+	0.772694i	$0.281091\pi$
$180$	0	0
$181$	−17.3454	−1.28928	−0.644638	−	0.764488i	$-0.722992\pi$
$181$	−17.3454	−1.28928	−0.644638	+	0.764488i	$0.722992\pi$
$182$	−0.594953	−0.0441008
$183$	−43.2388	−3.19630
$184$	−1.01105	−0.0745358
$185$	0	0
$186$	−0.535642	−0.0392752
$187$	0	0
$188$	13.9197	1.01520
$189$	−31.0697	−2.25999
$190$	0	0
$191$	20.7695	1.50283	0.751416	−	0.659829i	$-0.229371\pi$
$191$	20.7695	1.50283	0.751416	+	0.659829i	$0.229371\pi$
$192$	−22.5052	−1.62417
$193$	−8.51249	−0.612742	−0.306371	−	0.951912i	$-0.599115\pi$
$193$	−8.51249	−0.612742	−0.306371	+	0.951912i	$0.599115\pi$
$194$	1.29809	0.0931976
$195$	0	0
$196$	−11.3723	−0.812308
$197$	2.60251	0.185421	0.0927106	−	0.995693i	$-0.470447\pi$
$197$	2.60251	0.185421	0.0927106	+	0.995693i	$0.470447\pi$
$198$	0	0
$199$	−8.77731	−0.622207	−0.311104	−	0.950376i	$-0.600699\pi$
$199$	−8.77731	−0.622207	−0.311104	+	0.950376i	$0.600699\pi$
$200$	0	0
$201$	24.3115	1.71480
$202$	0.365521	0.0257180
$203$	14.7830	1.03756
$204$	32.1431	2.25046
$205$	0	0
$206$	−0.920841	−0.0641580
$207$	−10.7775	−0.749090
$208$	−4.64351	−0.321970
$209$	0	0
$210$	0	0
$211$	10.8092	0.744139	0.372070	−	0.928205i	$-0.378648\pi$
$211$	10.8092	0.744139	0.372070	+	0.928205i	$0.378648\pi$
$212$	21.4118	1.47057
$213$	25.4169	1.74153
$214$	1.96032	0.134005
$215$	0	0
$216$	4.82988	0.328632
$217$	−4.59363	−0.311836
$218$	−1.72183	−0.116617
$219$	−25.5021	−1.72327
$220$	0	0
$221$	6.49870	0.437150
$222$	1.31568	0.0883026
$223$	−8.04861	−0.538975	−0.269487	−	0.963004i	$-0.586854\pi$
$223$	−8.04861	−0.538975	−0.269487	+	0.963004i	$0.586854\pi$
$224$	5.89417	0.393820
$225$	0	0
$226$	0.497482	0.0330920
$227$	20.7195	1.37520	0.687601	−	0.726089i	$-0.258664\pi$
$227$	20.7195	1.37520	0.687601	+	0.726089i	$0.258664\pi$
$228$	−28.4007	−1.88088
$229$	8.01239	0.529474	0.264737	−	0.964321i	$-0.414715\pi$
$229$	8.01239	0.529474	0.264737	+	0.964321i	$0.414715\pi$
$230$	0	0
$231$	0	0
$232$	−2.29806	−0.150875
$233$	25.4593	1.66790	0.833948	−	0.551842i	$-0.186075\pi$
$233$	25.4593	1.66790	0.833948	+	0.551842i	$0.186075\pi$
$234$	0.985888	0.0644495
$235$	0	0
$236$	7.95410	0.517768
$237$	1.17640	0.0764157
$238$	−2.70474	−0.175322
$239$	0.547059	0.0353863	0.0176931	−	0.999843i	$-0.494368\pi$
$239$	0.547059	0.0353863	0.0176931	+	0.999843i	$0.494368\pi$
$240$	0	0
$241$	21.9600	1.41457	0.707285	−	0.706929i	$-0.249920\pi$
$241$	21.9600	1.41457	0.707285	+	0.706929i	$0.249920\pi$
$242$	0	0
$243$	−1.49919	−0.0961728
$244$	28.6814	1.83614
$245$	0	0
$246$	2.66685	0.170032
$247$	−5.74207	−0.365359
$248$	0.714095	0.0453451
$249$	−4.66872	−0.295868
$250$	0	0
$251$	6.05276	0.382047	0.191023	−	0.981585i	$-0.438819\pi$
$251$	6.05276	0.382047	0.191023	+	0.981585i	$0.438819\pi$
$252$	41.8187	2.63433
$253$	0	0
$254$	−0.875381	−0.0549263
$255$	0	0
$256$	14.4679	0.904244
$257$	−3.05765	−0.190731	−0.0953656	−	0.995442i	$-0.530402\pi$
$257$	−3.05765	−0.190731	−0.0953656	+	0.995442i	$0.530402\pi$
$258$	−1.76298	−0.109759
$259$	11.2832	0.701104
$260$	0	0
$261$	−24.4967	−1.51631
$262$	−2.57554	−0.159117
$263$	−18.1365	−1.11835	−0.559174	−	0.829050i	$-0.688882\pi$
$263$	−18.1365	−1.11835	−0.559174	+	0.829050i	$0.688882\pi$
$264$	0	0
$265$	0	0
$266$	2.38983	0.146530
$267$	−20.6148	−1.26161
$268$	−16.1264	−0.985079
$269$	17.4420	1.06346	0.531729	−	0.846915i	$-0.321543\pi$
$269$	17.4420	1.06346	0.531729	+	0.846915i	$0.321543\pi$
$270$	0	0
$271$	−7.04566	−0.427993	−0.213997	−	0.976834i	$-0.568648\pi$
$271$	−7.04566	−0.427993	−0.213997	+	0.976834i	$0.568648\pi$
$272$	−21.1100	−1.27998
$273$	12.7431	0.771245
$274$	0.582227	0.0351736
$275$	0	0
$276$	10.7748	0.648567
$277$	18.5247	1.11304	0.556521	−	0.830833i	$-0.312136\pi$
$277$	18.5247	1.11304	0.556521	+	0.830833i	$0.312136\pi$
$278$	0.380684	0.0228319
$279$	7.61205	0.455722
$280$	0	0
$281$	21.0170	1.25377	0.626885	−	0.779112i	$-0.284330\pi$
$281$	21.0170	1.25377	0.626885	+	0.779112i	$0.284330\pi$
$282$	2.92534	0.174201
$283$	−1.62479	−0.0965840	−0.0482920	−	0.998833i	$-0.515378\pi$
$283$	−1.62479	−0.0965840	−0.0482920	+	0.998833i	$0.515378\pi$
$284$	−16.8596	−1.00043
$285$	0	0
$286$	0	0
$287$	22.8708	1.35002
$288$	−9.76714	−0.575535
$289$	12.5440	0.737880
$290$	0	0
$291$	−27.8033	−1.62986
$292$	16.9162	0.989944
$293$	12.6068	0.736496	0.368248	−	0.929728i	$-0.379958\pi$
$293$	12.6068	0.736496	0.368248	+	0.929728i	$0.379958\pi$
$294$	−2.38999	−0.139387
$295$	0	0
$296$	−1.75401	−0.101950
$297$	0	0
$298$	2.47423	0.143328
$299$	2.17845	0.125983
$300$	0	0
$301$	−15.1192	−0.871459
$302$	0.202664	0.0116620
$303$	−7.82895	−0.449762
$304$	18.6522	1.06978
$305$	0	0
$306$	4.48198	0.256218
$307$	−22.6055	−1.29017	−0.645083	−	0.764112i	$-0.723177\pi$
$307$	−22.6055	−1.29017	−0.645083	+	0.764112i	$0.723177\pi$
$308$	0	0
$309$	19.7231	1.12201
$310$	0	0
$311$	−18.6534	−1.05774	−0.528870	−	0.848703i	$-0.677384\pi$
$311$	−18.6534	−1.05774	−0.528870	+	0.848703i	$0.677384\pi$
$312$	−1.98095	−0.112149
$313$	17.4844	0.988276	0.494138	−	0.869383i	$-0.335484\pi$
$313$	17.4844	0.988276	0.494138	+	0.869383i	$0.335484\pi$
$314$	−0.467973	−0.0264093
$315$	0	0
$316$	−0.780338	−0.0438974
$317$	−11.1847	−0.628197	−0.314098	−	0.949390i	$-0.601702\pi$
$317$	−11.1847	−0.628197	−0.314098	+	0.949390i	$0.601702\pi$
$318$	4.49987	0.252340
$319$	0	0
$320$	0	0
$321$	−41.9872	−2.34350
$322$	−0.906664	−0.0505264
$323$	−26.1042	−1.45248
$324$	−16.3263	−0.907016
$325$	0	0
$326$	−0.949293	−0.0525765
$327$	36.8791	2.03942
$328$	−3.55533	−0.196310
$329$	25.0875	1.38312
$330$	0	0
$331$	−25.7621	−1.41602	−0.708008	−	0.706205i	$-0.750406\pi$
$331$	−25.7621	−1.41602	−0.708008	+	0.706205i	$0.750406\pi$
$332$	3.09688	0.169963
$333$	−18.6972	−1.02460
$334$	2.73811	0.149823
$335$	0	0
$336$	−41.3938	−2.25822
$337$	11.4981	0.626342	0.313171	−	0.949697i	$-0.398609\pi$
$337$	11.4981	0.626342	0.313171	+	0.949697i	$0.398609\pi$
$338$	1.61296	0.0877337
$339$	−10.6554	−0.578720
$340$	0	0
$341$	0	0
$342$	−3.96015	−0.214141
$343$	4.49073	0.242476
$344$	2.35033	0.126722
$345$	0	0
$346$	−0.0326423	−0.00175486
$347$	0.865880	0.0464829	0.0232414	−	0.999730i	$-0.492601\pi$
$347$	0.865880	0.0464829	0.0232414	+	0.999730i	$0.492601\pi$
$348$	24.4905	1.31283
$349$	17.7916	0.952361	0.476180	−	0.879348i	$-0.342021\pi$
$349$	17.7916	0.952361	0.476180	+	0.879348i	$0.342021\pi$
$350$	0	0
$351$	−10.4067	−0.555466
$352$	0	0
$353$	−1.11550	−0.0593723	−0.0296862	−	0.999559i	$-0.509451\pi$
$353$	−1.11550	−0.0593723	−0.0296862	+	0.999559i	$0.509451\pi$
$354$	1.67162	0.0888457
$355$	0	0
$356$	13.6743	0.724738
$357$	57.9316	3.06607
$358$	−2.36783	−0.125144
$359$	1.51968	0.0802058	0.0401029	−	0.999196i	$-0.487231\pi$
$359$	1.51968	0.0802058	0.0401029	+	0.999196i	$0.487231\pi$
$360$	0	0
$361$	4.06496	0.213945
$362$	2.41801	0.127088
$363$	0	0
$364$	−8.45278	−0.443046
$365$	0	0
$366$	6.02763	0.315069
$367$	10.6261	0.554678	0.277339	−	0.960772i	$-0.410547\pi$
$367$	10.6261	0.554678	0.277339	+	0.960772i	$0.410547\pi$
$368$	−7.07637	−0.368881
$369$	−37.8988	−1.97293
$370$	0	0
$371$	38.5906	2.00352
$372$	−7.61012	−0.394566
$373$	27.9435	1.44686	0.723429	−	0.690399i	$-0.242565\pi$
$373$	27.9435	1.44686	0.723429	+	0.690399i	$0.242565\pi$
$374$	0	0
$375$	0	0
$376$	−3.89993	−0.201124
$377$	4.95149	0.255015
$378$	4.33121	0.222774
$379$	18.6711	0.959071	0.479535	−	0.877523i	$-0.340805\pi$
$379$	18.6711	0.959071	0.479535	+	0.877523i	$0.340805\pi$
$380$	0	0
$381$	18.7494	0.960563
$382$	−2.89534	−0.148138
$383$	−21.6620	−1.10688	−0.553439	−	0.832890i	$-0.686685\pi$
$383$	−21.6620	−1.10688	−0.553439	+	0.832890i	$0.686685\pi$
$384$	12.9978	0.663290
$385$	0	0
$386$	1.18667	0.0603998
$387$	25.0539	1.27356
$388$	18.4426	0.936282
$389$	−37.4805	−1.90034	−0.950168	−	0.311737i	$-0.899089\pi$
$389$	−37.4805	−1.90034	−0.950168	+	0.311737i	$0.899089\pi$
$390$	0	0
$391$	9.90354	0.500844
$392$	3.18623	0.160929
$393$	55.1644	2.78268
$394$	−0.362798	−0.0182775
$395$	0	0
$396$	0	0
$397$	−18.3084	−0.918873	−0.459437	−	0.888211i	$-0.651949\pi$
$397$	−18.3084	−0.918873	−0.459437	+	0.888211i	$0.651949\pi$
$398$	1.22359	0.0613328
$399$	−51.1868	−2.56254
$400$	0	0
$401$	13.0878	0.653571	0.326786	−	0.945098i	$-0.394034\pi$
$401$	13.0878	0.653571	0.326786	+	0.945098i	$0.394034\pi$
$402$	−3.38910	−0.169033
$403$	−1.53862	−0.0766440
$404$	5.19314	0.258368
$405$	0	0
$406$	−2.06079	−0.102275
$407$	0	0
$408$	−9.00566	−0.445846
$409$	21.2641	1.05144	0.525721	−	0.850657i	$-0.323796\pi$
$409$	21.2641	1.05144	0.525721	+	0.850657i	$0.323796\pi$
$410$	0	0
$411$	−12.4705	−0.615124
$412$	−13.0828	−0.644544
$413$	14.3357	0.705415
$414$	1.50242	0.0738400
$415$	0	0
$416$	1.97422	0.0967943
$417$	−8.15371	−0.399289
$418$	0	0
$419$	−37.5627	−1.83506	−0.917528	−	0.397670i	$-0.869819\pi$
$419$	−37.5627	−1.83506	−0.917528	+	0.397670i	$0.869819\pi$
$420$	0	0
$421$	−9.16346	−0.446600	−0.223300	−	0.974750i	$-0.571683\pi$
$421$	−9.16346	−0.446600	−0.223300	+	0.974750i	$0.571683\pi$
$422$	−1.50684	−0.0733520
$423$	−41.5722	−2.02131
$424$	−5.99903	−0.291339
$425$	0	0
$426$	−3.54319	−0.171668
$427$	51.6926	2.50158
$428$	27.8512	1.34624
$429$	0	0
$430$	0	0
$431$	27.9373	1.34569	0.672847	−	0.739781i	$-0.265071\pi$
$431$	27.9373	1.34569	0.672847	+	0.739781i	$0.265071\pi$
$432$	33.8044	1.62642
$433$	20.3509	0.978003	0.489001	−	0.872283i	$-0.337361\pi$
$433$	20.3509	0.978003	0.489001	+	0.872283i	$0.337361\pi$
$434$	0.640367	0.0307386
$435$	0	0
$436$	−24.4628	−1.17156
$437$	−8.75049	−0.418593
$438$	3.55507	0.169868
$439$	−5.87262	−0.280285	−0.140142	−	0.990131i	$-0.544756\pi$
$439$	−5.87262	−0.280285	−0.140142	+	0.990131i	$0.544756\pi$
$440$	0	0
$441$	33.9643	1.61735
$442$	−0.905939	−0.0430911
$443$	36.1202	1.71612	0.858062	−	0.513546i	$-0.171668\pi$
$443$	36.1202	1.71612	0.858062	+	0.513546i	$0.171668\pi$
$444$	18.6925	0.887106
$445$	0	0
$446$	1.12200	0.0531283
$447$	−52.9945	−2.50655
$448$	26.9053	1.27116
$449$	6.85217	0.323374	0.161687	−	0.986842i	$-0.448306\pi$
$449$	6.85217	0.323374	0.161687	+	0.986842i	$0.448306\pi$
$450$	0	0
$451$	0	0
$452$	7.06796	0.332449
$453$	−4.34078	−0.203948
$454$	−2.88836	−0.135558
$455$	0	0
$456$	7.95715	0.372627
$457$	−21.3124	−0.996952	−0.498476	−	0.866903i	$-0.666107\pi$
$457$	−21.3124	−0.996952	−0.498476	+	0.866903i	$0.666107\pi$
$458$	−1.11695	−0.0521918
$459$	−47.3101	−2.20825
$460$	0	0
$461$	−29.0126	−1.35125	−0.675626	−	0.737244i	$-0.736127\pi$
$461$	−29.0126	−1.35125	−0.675626	+	0.737244i	$0.736127\pi$
$462$	0	0
$463$	−17.8392	−0.829057	−0.414528	−	0.910036i	$-0.636053\pi$
$463$	−17.8392	−0.829057	−0.414528	+	0.910036i	$0.636053\pi$
$464$	−16.0842	−0.746689
$465$	0	0
$466$	−3.54911	−0.164409
$467$	−4.00229	−0.185204	−0.0926019	−	0.995703i	$-0.529518\pi$
$467$	−4.00229	−0.185204	−0.0926019	+	0.995703i	$0.529518\pi$
$468$	14.0070	0.647473
$469$	−29.0648	−1.34209
$470$	0	0
$471$	10.0233	0.461851
$472$	−2.22853	−0.102577
$473$	0	0
$474$	−0.163994	−0.00753252
$475$	0	0
$476$	−38.4275	−1.76132
$477$	−63.9480	−2.92798
$478$	−0.0762617	−0.00348813
$479$	−1.20388	−0.0550065	−0.0275032	−	0.999622i	$-0.508756\pi$
$479$	−1.20388	−0.0550065	−0.0275032	+	0.999622i	$0.508756\pi$
$480$	0	0
$481$	3.77925	0.172319
$482$	−3.06130	−0.139438
$483$	19.4195	0.883617
$484$	0	0
$485$	0	0
$486$	0.208991	0.00948003
$487$	−42.2614	−1.91505	−0.957523	−	0.288358i	$-0.906891\pi$
$487$	−42.2614	−1.91505	−0.957523	+	0.288358i	$0.906891\pi$
$488$	−8.03577	−0.363762
$489$	20.3325	0.919468
$490$	0	0
$491$	18.2825	0.825076	0.412538	−	0.910940i	$-0.364642\pi$
$491$	18.2825	0.825076	0.412538	+	0.910940i	$0.364642\pi$
$492$	37.8892	1.70818
$493$	22.5102	1.01381
$494$	0.800462	0.0360145
$495$	0	0
$496$	4.99796	0.224415
$497$	−30.3862	−1.36301
$498$	0.650835	0.0291646
$499$	22.6846	1.01550	0.507750	−	0.861504i	$-0.330477\pi$
$499$	22.6846	1.01550	0.507750	+	0.861504i	$0.330477\pi$
$500$	0	0
$501$	−58.6465	−2.62013
$502$	−0.843774	−0.0376595
$503$	−32.2748	−1.43906	−0.719531	−	0.694461i	$-0.755643\pi$
$503$	−32.2748	−1.43906	−0.719531	+	0.694461i	$0.755643\pi$
$504$	−11.7165	−0.521895
$505$	0	0
$506$	0	0
$507$	−34.5474	−1.53430
$508$	−12.4370	−0.551801
$509$	22.0973	0.979447	0.489724	−	0.871878i	$-0.337098\pi$
$509$	22.0973	0.979447	0.489724	+	0.871878i	$0.337098\pi$
$510$	0	0
$511$	30.4881	1.34871
$512$	−10.7232	−0.473903
$513$	41.8019	1.84560
$514$	0.426247	0.0188009
$515$	0	0
$516$	−25.0475	−1.10266
$517$	0	0
$518$	−1.57291	−0.0691098
$519$	0.699153	0.0306894
$520$	0	0
$521$	−17.0464	−0.746818	−0.373409	−	0.927667i	$-0.621811\pi$
$521$	−17.0464	−0.746818	−0.373409	+	0.927667i	$0.621811\pi$
$522$	3.41491	0.149467
$523$	−19.9933	−0.874247	−0.437124	−	0.899401i	$-0.644003\pi$
$523$	−19.9933	−0.874247	−0.437124	+	0.899401i	$0.644003\pi$
$524$	−36.5919	−1.59852
$525$	0	0
$526$	2.52829	0.110239
$527$	−6.99476	−0.304697
$528$	0	0
$529$	−19.6802	−0.855661
$530$	0	0
$531$	−23.7555	−1.03090
$532$	33.9534	1.47207
$533$	7.66046	0.331811
$534$	2.87377	0.124360
$535$	0	0
$536$	4.51821	0.195157
$537$	50.7156	2.18854
$538$	−2.43147	−0.104828
$539$	0	0
$540$	0	0
$541$	−7.40374	−0.318312	−0.159156	−	0.987253i	$-0.550877\pi$
$541$	−7.40374	−0.318312	−0.159156	+	0.987253i	$0.550877\pi$
$542$	0.982187	0.0421886
$543$	−51.7903	−2.22253
$544$	8.97509	0.384804
$545$	0	0
$546$	−1.77642	−0.0760238
$547$	19.6038	0.838196	0.419098	−	0.907941i	$-0.362346\pi$
$547$	19.6038	0.838196	0.419098	+	0.907941i	$0.362346\pi$
$548$	8.27198	0.353362
$549$	−85.6591	−3.65584
$550$	0	0
$551$	−19.8893	−0.847314
$552$	−3.01882	−0.128489
$553$	−1.40641	−0.0598065
$554$	−2.58240	−0.109716
$555$	0	0
$556$	5.40855	0.229374
$557$	−16.9060	−0.716332	−0.358166	−	0.933658i	$-0.616598\pi$
$557$	−16.9060	−0.716332	−0.358166	+	0.933658i	$0.616598\pi$
$558$	−1.06114	−0.0449218
$559$	−5.06412	−0.214189
$560$	0	0
$561$	0	0
$562$	−2.92984	−0.123588
$563$	−19.1013	−0.805026	−0.402513	−	0.915414i	$-0.631863\pi$
$563$	−19.1013	−0.805026	−0.402513	+	0.915414i	$0.631863\pi$
$564$	41.5617	1.75006
$565$	0	0
$566$	0.226501	0.00952057
$567$	−29.4250	−1.23573
$568$	4.72363	0.198199
$569$	−17.6993	−0.741995	−0.370997	−	0.928634i	$-0.620984\pi$
$569$	−17.6993	−0.741995	−0.370997	+	0.928634i	$0.620984\pi$
$570$	0	0
$571$	−5.15632	−0.215785	−0.107893	−	0.994163i	$-0.534410\pi$
$571$	−5.15632	−0.215785	−0.107893	+	0.994163i	$0.534410\pi$
$572$	0	0
$573$	62.0141	2.59068
$574$	−3.18826	−0.133075
$575$	0	0
$576$	−44.5844	−1.85768
$577$	−20.1707	−0.839717	−0.419859	−	0.907590i	$-0.637920\pi$
$577$	−20.1707	−0.839717	−0.419859	+	0.907590i	$0.637920\pi$
$578$	−1.74867	−0.0727350
$579$	−25.4167	−1.05628
$580$	0	0
$581$	5.58152	0.231560
$582$	3.87587	0.160660
$583$	0	0
$584$	−4.73947	−0.196121
$585$	0	0
$586$	−1.75742	−0.0725985
$587$	−35.4185	−1.46188	−0.730938	−	0.682444i	$-0.760917\pi$
$587$	−35.4185	−1.46188	−0.730938	+	0.682444i	$0.760917\pi$
$588$	−33.9557	−1.40031
$589$	6.18038	0.254658
$590$	0	0
$591$	7.77063	0.319641
$592$	−12.2763	−0.504554
$593$	38.2459	1.57057	0.785285	−	0.619134i	$-0.212516\pi$
$593$	38.2459	1.57057	0.785285	+	0.619134i	$0.212516\pi$
$594$	0	0
$595$	0	0
$596$	35.1525	1.43990
$597$	−26.2075	−1.07260
$598$	−0.303683	−0.0124185
$599$	23.7075	0.968661	0.484331	−	0.874885i	$-0.339063\pi$
$599$	23.7075	0.968661	0.484331	+	0.874885i	$0.339063\pi$
$600$	0	0
$601$	9.80242	0.399849	0.199924	−	0.979811i	$-0.435930\pi$
$601$	9.80242	0.399849	0.199924	+	0.979811i	$0.435930\pi$
$602$	2.10767	0.0859022
$603$	48.1628	1.96134
$604$	2.87935	0.117159
$605$	0	0
$606$	1.09138	0.0443343
$607$	−2.11199	−0.0857229	−0.0428614	−	0.999081i	$-0.513647\pi$
$607$	−2.11199	−0.0857229	−0.0428614	+	0.999081i	$0.513647\pi$
$608$	−7.93014	−0.321610
$609$	44.1393	1.78861
$610$	0	0
$611$	8.40295	0.339947
$612$	63.6777	2.57402
$613$	2.83477	0.114495	0.0572476	−	0.998360i	$-0.481768\pi$
$613$	2.83477	0.114495	0.0572476	+	0.998360i	$0.481768\pi$
$614$	3.15128	0.127175
$615$	0	0
$616$	0	0
$617$	43.2099	1.73957	0.869783	−	0.493434i	$-0.164259\pi$
$617$	43.2099	1.73957	0.869783	+	0.493434i	$0.164259\pi$
$618$	−2.74946	−0.110600
$619$	−46.8566	−1.88333	−0.941663	−	0.336557i	$-0.890738\pi$
$619$	−46.8566	−1.88333	−0.941663	+	0.336557i	$0.890738\pi$
$620$	0	0
$621$	−15.8590	−0.636399
$622$	2.60035	0.104264
$623$	24.6453	0.987394
$624$	−13.8647	−0.555032
$625$	0	0
$626$	−2.43738	−0.0974173
$627$	0	0
$628$	−6.64872	−0.265313
$629$	17.1810	0.685051
$630$	0	0
$631$	5.14581	0.204851	0.102426	−	0.994741i	$-0.467340\pi$
$631$	5.14581	0.204851	0.102426	+	0.994741i	$0.467340\pi$
$632$	0.218630	0.00869665
$633$	32.2744	1.28279
$634$	1.55919	0.0619232
$635$	0	0
$636$	63.9317	2.53506
$637$	−6.86517	−0.272008
$638$	0	0
$639$	50.3526	1.99192
$640$	0	0
$641$	16.6669	0.658305	0.329152	−	0.944277i	$-0.393237\pi$
$641$	16.6669	0.658305	0.329152	+	0.944277i	$0.393237\pi$
$642$	5.85315	0.231005
$643$	−35.7955	−1.41164	−0.705819	−	0.708392i	$-0.749421\pi$
$643$	−35.7955	−1.41164	−0.705819	+	0.708392i	$0.749421\pi$
$644$	−12.8814	−0.507599
$645$	0	0
$646$	3.63901	0.143175
$647$	−31.8601	−1.25255	−0.626274	−	0.779603i	$-0.715421\pi$
$647$	−31.8601	−1.25255	−0.626274	+	0.779603i	$0.715421\pi$
$648$	4.57421	0.179692
$649$	0	0
$650$	0	0
$651$	−13.7158	−0.537563
$652$	−13.4871	−0.528194
$653$	−2.02576	−0.0792740	−0.0396370	−	0.999214i	$-0.512620\pi$
$653$	−2.02576	−0.0792740	−0.0396370	+	0.999214i	$0.512620\pi$
$654$	−5.14106	−0.201031
$655$	0	0
$656$	−24.8838	−0.971550
$657$	−50.5215	−1.97103
$658$	−3.49728	−0.136338
$659$	−14.3580	−0.559308	−0.279654	−	0.960101i	$-0.590220\pi$
$659$	−14.3580	−0.559308	−0.279654	+	0.960101i	$0.590220\pi$
$660$	0	0
$661$	28.0721	1.09188	0.545938	−	0.837825i	$-0.316173\pi$
$661$	28.0721	1.09188	0.545938	+	0.837825i	$0.316173\pi$
$662$	3.59132	0.139581
$663$	19.4039	0.753586
$664$	−0.867665	−0.0336719
$665$	0	0
$666$	2.60645	0.100998
$667$	7.54571	0.292171
$668$	38.9017	1.50515
$669$	−24.0317	−0.929118
$670$	0	0
$671$	0	0
$672$	17.5989	0.678893
$673$	39.2324	1.51230	0.756149	−	0.654400i	$-0.227079\pi$
$673$	39.2324	1.51230	0.756149	+	0.654400i	$0.227079\pi$
$674$	−1.60287	−0.0617403
$675$	0	0
$676$	22.9161	0.881390
$677$	−49.3853	−1.89803	−0.949015	−	0.315230i	$-0.897918\pi$
$677$	−49.3853	−1.89803	−0.949015	+	0.315230i	$0.897918\pi$
$678$	1.48539	0.0570461
$679$	33.2392	1.27561
$680$	0	0
$681$	61.8647	2.37066
$682$	0	0
$683$	29.7973	1.14016	0.570081	−	0.821589i	$-0.306912\pi$
$683$	29.7973	1.14016	0.570081	+	0.821589i	$0.306912\pi$
$684$	−56.2638	−2.15130
$685$	0	0
$686$	−0.626021	−0.0239016
$687$	23.9236	0.912741
$688$	16.4500	0.627151
$689$	12.9257	0.492432
$690$	0	0
$691$	32.1783	1.22412	0.612060	−	0.790812i	$-0.290341\pi$
$691$	32.1783	1.22412	0.612060	+	0.790812i	$0.290341\pi$
$692$	−0.463765	−0.0176297
$693$	0	0
$694$	−0.120706	−0.00458195
$695$	0	0
$696$	−6.86159	−0.260088
$697$	34.8255	1.31911
$698$	−2.48020	−0.0938770
$699$	76.0170	2.87523
$700$	0	0
$701$	13.4188	0.506821	0.253411	−	0.967359i	$-0.418448\pi$
$701$	13.4188	0.506821	0.253411	+	0.967359i	$0.418448\pi$
$702$	1.45072	0.0547539
$703$	−15.1807	−0.572549
$704$	0	0
$705$	0	0
$706$	0.155505	0.00585250
$707$	9.35962	0.352005
$708$	23.7495	0.892561
$709$	−22.5831	−0.848125	−0.424062	−	0.905633i	$-0.639396\pi$
$709$	−22.5831	−0.848125	−0.424062	+	0.905633i	$0.639396\pi$
$710$	0	0
$711$	2.33054	0.0874020
$712$	−3.83119	−0.143580
$713$	−2.34474	−0.0878112
$714$	−8.07585	−0.302231
$715$	0	0
$716$	−33.6409	−1.25722
$717$	1.63342	0.0610011
$718$	−0.211848	−0.00790611
$719$	−26.6894	−0.995347	−0.497674	−	0.867364i	$-0.665812\pi$
$719$	−26.6894	−0.995347	−0.497674	+	0.867364i	$0.665812\pi$
$720$	0	0
$721$	−23.5792	−0.878137
$722$	−0.566668	−0.0210892
$723$	65.5686	2.43852
$724$	34.3538	1.27675
$725$	0	0
$726$	0	0
$727$	10.6151	0.393693	0.196846	−	0.980434i	$-0.436930\pi$
$727$	10.6151	0.393693	0.196846	+	0.980434i	$0.436930\pi$
$728$	2.36825	0.0877732
$729$	−29.2060	−1.08170
$730$	0	0
$731$	−23.0222	−0.851506
$732$	85.6373	3.16525
$733$	12.8410	0.474292	0.237146	−	0.971474i	$-0.423788\pi$
$733$	12.8410	0.474292	0.237146	+	0.971474i	$0.423788\pi$
$734$	−1.48131	−0.0546763
$735$	0	0
$736$	3.00857	0.110897
$737$	0	0
$738$	5.28322	0.194478
$739$	33.9561	1.24910	0.624548	−	0.780987i	$-0.285283\pi$
$739$	33.9561	1.24910	0.624548	+	0.780987i	$0.285283\pi$
$740$	0	0
$741$	−17.1448	−0.629829
$742$	−5.37965	−0.197493
$743$	−46.1042	−1.69140	−0.845700	−	0.533658i	$-0.820817\pi$
$743$	−46.1042	−1.69140	−0.845700	+	0.533658i	$0.820817\pi$
$744$	2.13216	0.0781687
$745$	0	0
$746$	−3.89541	−0.142621
$747$	−9.24906	−0.338406
$748$	0	0
$749$	50.1963	1.83413
$750$	0	0
$751$	41.8818	1.52829	0.764145	−	0.645044i	$-0.223161\pi$
$751$	41.8818	1.52829	0.764145	+	0.645044i	$0.223161\pi$
$752$	−27.2957	−0.995372
$753$	18.0724	0.658596
$754$	−0.690254	−0.0251376
$755$	0	0
$756$	61.5356	2.23803
$757$	−23.2778	−0.846045	−0.423022	−	0.906119i	$-0.639031\pi$
$757$	−23.2778	−0.846045	−0.423022	+	0.906119i	$0.639031\pi$
$758$	−2.60281	−0.0945384
$759$	0	0
$760$	0	0
$761$	20.9612	0.759844	0.379922	−	0.925019i	$-0.375951\pi$
$761$	20.9612	0.759844	0.379922	+	0.925019i	$0.375951\pi$
$762$	−2.61373	−0.0946854
$763$	−44.0894	−1.59614
$764$	−41.1355	−1.48823
$765$	0	0
$766$	3.01975	0.109108
$767$	4.80169	0.173379
$768$	43.1985	1.55879
$769$	24.9924	0.901249	0.450624	−	0.892714i	$-0.351201\pi$
$769$	24.9924	0.901249	0.450624	+	0.892714i	$0.351201\pi$
$770$	0	0
$771$	−9.12960	−0.328795
$772$	16.8595	0.606788
$773$	−21.8160	−0.784666	−0.392333	−	0.919823i	$-0.628332\pi$
$773$	−21.8160	−0.784666	−0.392333	+	0.919823i	$0.628332\pi$
$774$	−3.49259	−0.125539
$775$	0	0
$776$	−5.16715	−0.185490
$777$	33.6896	1.20861
$778$	5.22490	0.187322
$779$	−30.7708	−1.10248
$780$	0	0
$781$	0	0
$782$	−1.38058	−0.0493696
$783$	−36.0465	−1.28820
$784$	22.3005	0.796445
$785$	0	0
$786$	−7.69009	−0.274296
$787$	23.7831	0.847774	0.423887	−	0.905715i	$-0.360665\pi$
$787$	23.7831	0.847774	0.423887	+	0.905715i	$0.360665\pi$
$788$	−5.15445	−0.183620
$789$	−54.1525	−1.92788
$790$	0	0
$791$	12.7386	0.452933
$792$	0	0
$793$	17.3142	0.614845
$794$	2.55225	0.0905760
$795$	0	0
$796$	17.3841	0.616162
$797$	−35.7368	−1.26586	−0.632932	−	0.774208i	$-0.718149\pi$
$797$	−35.7368	−1.26586	−0.632932	+	0.774208i	$0.718149\pi$
$798$	7.13560	0.252597
$799$	38.2010	1.35145
$800$	0	0
$801$	−40.8394	−1.44299
$802$	−1.82447	−0.0644244
$803$	0	0
$804$	−48.1506	−1.69814
$805$	0	0
$806$	0.214488	0.00755502
$807$	52.0787	1.83326
$808$	−1.45498	−0.0511861
$809$	−41.6038	−1.46271	−0.731356	−	0.681996i	$-0.761112\pi$
$809$	−41.6038	−1.46271	−0.731356	+	0.681996i	$0.761112\pi$
$810$	0	0
$811$	24.1765	0.848950	0.424475	−	0.905440i	$-0.360459\pi$
$811$	24.1765	0.848950	0.424475	+	0.905440i	$0.360459\pi$
$812$	−29.2787	−1.02748
$813$	−21.0371	−0.737802
$814$	0	0
$815$	0	0
$816$	−63.0307	−2.20652
$817$	20.3418	0.711668
$818$	−2.96428	−0.103644
$819$	25.2449	0.882127
$820$	0	0
$821$	34.0377	1.18792	0.593962	−	0.804493i	$-0.297563\pi$
$821$	34.0377	1.18792	0.593962	+	0.804493i	$0.297563\pi$
$822$	1.73843	0.0606346
$823$	5.54052	0.193130	0.0965652	−	0.995327i	$-0.469214\pi$
$823$	5.54052	0.193130	0.0965652	+	0.995327i	$0.469214\pi$
$824$	3.66547	0.127693
$825$	0	0
$826$	−1.99845	−0.0695348
$827$	−25.6712	−0.892675	−0.446337	−	0.894865i	$-0.647272\pi$
$827$	−25.6712	−0.892675	−0.446337	+	0.894865i	$0.647272\pi$
$828$	21.3456	0.741812
$829$	41.7568	1.45027	0.725136	−	0.688606i	$-0.241777\pi$
$829$	41.7568	1.45027	0.725136	+	0.688606i	$0.241777\pi$
$830$	0	0
$831$	55.3115	1.91873
$832$	9.01181	0.312428
$833$	−31.2100	−1.08136
$834$	1.13665	0.0393590
$835$	0	0
$836$	0	0
$837$	11.2010	0.387164
$838$	5.23636	0.180887
$839$	10.2571	0.354114	0.177057	−	0.984201i	$-0.443342\pi$
$839$	10.2571	0.354114	0.177057	+	0.984201i	$0.443342\pi$
$840$	0	0
$841$	−11.8491	−0.408588
$842$	1.27742	0.0440226
$843$	62.7530	2.16133
$844$	−21.4084	−0.736909
$845$	0	0
$846$	5.79530	0.199246
$847$	0	0
$848$	−41.9873	−1.44185
$849$	−4.85134	−0.166498
$850$	0	0
$851$	5.75931	0.197427
$852$	−50.3398	−1.72461
$853$	28.8449	0.987630	0.493815	−	0.869567i	$-0.335602\pi$
$853$	28.8449	0.987630	0.493815	+	0.869567i	$0.335602\pi$
$854$	−7.20611	−0.246588
$855$	0	0
$856$	−7.80317	−0.266707
$857$	−20.7640	−0.709283	−0.354642	−	0.935002i	$-0.615397\pi$
$857$	−20.7640	−0.709283	−0.354642	+	0.935002i	$0.615397\pi$
$858$	0	0
$859$	12.1297	0.413859	0.206929	−	0.978356i	$-0.433653\pi$
$859$	12.1297	0.413859	0.206929	+	0.978356i	$0.433653\pi$
$860$	0	0
$861$	68.2879	2.32725
$862$	−3.89455	−0.132649
$863$	37.4148	1.27361	0.636807	−	0.771023i	$-0.280255\pi$
$863$	37.4148	1.27361	0.636807	+	0.771023i	$0.280255\pi$
$864$	−14.3722	−0.488953
$865$	0	0
$866$	−2.83698	−0.0964046
$867$	37.4540	1.27200
$868$	9.09800	0.308806
$869$	0	0
$870$	0	0
$871$	−9.73511	−0.329862
$872$	6.85384	0.232100
$873$	−55.0803	−1.86419
$874$	1.21985	0.0412619
$875$	0	0
$876$	50.5086	1.70653
$877$	52.3856	1.76894	0.884468	−	0.466601i	$-0.154521\pi$
$877$	52.3856	1.76894	0.884468	+	0.466601i	$0.154521\pi$
$878$	0.818661	0.0276285
$879$	37.6416	1.26962
$880$	0	0
$881$	11.0403	0.371956	0.185978	−	0.982554i	$-0.440455\pi$
$881$	11.0403	0.371956	0.185978	+	0.982554i	$0.440455\pi$
$882$	−4.73473	−0.159427
$883$	−24.1889	−0.814022	−0.407011	−	0.913423i	$-0.633429\pi$
$883$	−24.1889	−0.814022	−0.407011	+	0.913423i	$0.633429\pi$
$884$	−12.8711	−0.432902
$885$	0	0
$886$	−5.03528	−0.169163
$887$	−17.4310	−0.585275	−0.292637	−	0.956223i	$-0.594533\pi$
$887$	−17.4310	−0.585275	−0.292637	+	0.956223i	$0.594533\pi$
$888$	−5.23715	−0.175747
$889$	−22.4152	−0.751782
$890$	0	0
$891$	0	0
$892$	15.9408	0.533738
$893$	−33.7533	−1.12951
$894$	7.38760	0.247078
$895$	0	0
$896$	−15.5390	−0.519122
$897$	6.50447	0.217178
$898$	−0.955214	−0.0318759
$899$	−5.32945	−0.177747
$900$	0	0
$901$	58.7622	1.95765
$902$	0	0
$903$	−45.1433	−1.50228
$904$	−1.98026	−0.0658625
$905$	0	0
$906$	0.605119	0.0201037
$907$	12.9734	0.430775	0.215388	−	0.976529i	$-0.430899\pi$
$907$	12.9734	0.430775	0.215388	+	0.976529i	$0.430899\pi$
$908$	−41.0364	−1.36184
$909$	−15.5097	−0.514424
$910$	0	0
$911$	−20.3447	−0.674051	−0.337026	−	0.941496i	$-0.609421\pi$
$911$	−20.3447	−0.674051	−0.337026	+	0.941496i	$0.609421\pi$
$912$	55.6922	1.84415
$913$	0	0
$914$	2.97102	0.0982725
$915$	0	0
$916$	−15.8691	−0.524329
$917$	−65.9498	−2.17785
$918$	6.59517	0.217673
$919$	8.83489	0.291436	0.145718	−	0.989326i	$-0.453451\pi$
$919$	8.83489	0.291436	0.145718	+	0.989326i	$0.453451\pi$
$920$	0	0
$921$	−67.4960	−2.22407
$922$	4.04445	0.133197
$923$	−10.1777	−0.335004
$924$	0	0
$925$	0	0
$926$	2.48684	0.0817226
$927$	39.0729	1.28332
$928$	6.83830	0.224478
$929$	15.4629	0.507322	0.253661	−	0.967293i	$-0.418365\pi$
$929$	15.4629	0.507322	0.253661	+	0.967293i	$0.418365\pi$
$930$	0	0
$931$	27.5763	0.903776
$932$	−50.4239	−1.65169
$933$	−55.6958	−1.82340
$934$	0.557931	0.0182561
$935$	0	0
$936$	−3.92439	−0.128273
$937$	−41.6461	−1.36052	−0.680260	−	0.732971i	$-0.738133\pi$
$937$	−41.6461	−1.36052	−0.680260	+	0.732971i	$0.738133\pi$
$938$	4.05172	0.132293
$939$	52.2052	1.70365
$940$	0	0
$941$	47.3644	1.54404	0.772018	−	0.635601i	$-0.219247\pi$
$941$	47.3644	1.54404	0.772018	+	0.635601i	$0.219247\pi$
$942$	−1.39728	−0.0455260
$943$	11.6740	0.380157
$944$	−15.5975	−0.507657
$945$	0	0
$946$	0	0
$947$	−41.5599	−1.35052	−0.675258	−	0.737581i	$-0.735968\pi$
$947$	−41.5599	−1.35052	−0.675258	+	0.737581i	$0.735968\pi$
$948$	−2.32995	−0.0756732
$949$	10.2119	0.331491
$950$	0	0
$951$	−33.3956	−1.08293
$952$	10.7664	0.348940
$953$	−28.1561	−0.912066	−0.456033	−	0.889963i	$-0.650730\pi$
$953$	−28.1561	−0.912066	−0.456033	+	0.889963i	$0.650730\pi$
$954$	8.91455	0.288619
$955$	0	0
$956$	−1.08349	−0.0350425
$957$	0	0
$958$	0.167824	0.00542215
$959$	14.9086	0.481425
$960$	0	0
$961$	−29.3439	−0.946579
$962$	−0.526840	−0.0169860
$963$	−83.1796	−2.68043
$964$	−43.4933	−1.40082
$965$	0	0
$966$	−2.70714	−0.0871007
$967$	53.8069	1.73031	0.865157	−	0.501502i	$-0.167219\pi$
$967$	53.8069	1.73031	0.865157	+	0.501502i	$0.167219\pi$
$968$	0	0
$969$	−77.9425	−2.50387
$970$	0	0
$971$	−11.8692	−0.380899	−0.190450	−	0.981697i	$-0.560995\pi$
$971$	−11.8692	−0.380899	−0.190450	+	0.981697i	$0.560995\pi$
$972$	2.96924	0.0952383
$973$	9.74786	0.312502
$974$	5.89137	0.188772
$975$	0	0
$976$	−56.2425	−1.80028
$977$	34.6894	1.10981	0.554907	−	0.831913i	$-0.312754\pi$
$977$	34.6894	1.10981	0.554907	+	0.831913i	$0.312754\pi$
$978$	−2.83442	−0.0906346
$979$	0	0
$980$	0	0
$981$	73.0600	2.33263
$982$	−2.54863	−0.0813301
$983$	56.9285	1.81574	0.907868	−	0.419256i	$-0.137709\pi$
$983$	56.9285	1.81574	0.907868	+	0.419256i	$0.137709\pi$
$984$	−10.6156	−0.338412
$985$	0	0
$986$	−3.13799	−0.0999338
$987$	74.9068	2.38431
$988$	11.3726	0.361809
$989$	−7.71735	−0.245397
$990$	0	0
$991$	36.4470	1.15778	0.578889	−	0.815407i	$-0.303487\pi$
$991$	36.4470	1.15778	0.578889	+	0.815407i	$0.303487\pi$
$992$	−2.12492	−0.0674663
$993$	−76.9211	−2.44102
$994$	4.23593	0.134356
$995$	0	0
$996$	9.24672	0.292993
$997$	−24.0878	−0.762868	−0.381434	−	0.924396i	$-0.624570\pi$
$997$	−24.0878	−0.762868	−0.381434	+	0.924396i	$0.624570\pi$
$998$	−3.16230	−0.100101
$999$	−27.5127	−0.870464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bi.1.5		8
5.4	even	2		3025.2.a.bn.1.4		8
11.3	even	5		275.2.h.c.251.3	yes	16
11.4	even	5		275.2.h.c.126.3	✓	16
11.10	odd	2		3025.2.a.bm.1.4		8
55.3	odd	20		275.2.z.c.174.5		32
55.4	even	10		275.2.h.e.126.2	yes	16
55.14	even	10		275.2.h.e.251.2	yes	16
55.37	odd	20		275.2.z.c.49.5		32
55.47	odd	20		275.2.z.c.174.4		32
55.48	odd	20		275.2.z.c.49.4		32
55.54	odd	2		3025.2.a.bj.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.h.c.126.3	✓	16	11.4	even	5
275.2.h.c.251.3	yes	16	11.3	even	5
275.2.h.e.126.2	yes	16	55.4	even	10
275.2.h.e.251.2	yes	16	55.14	even	10
275.2.z.c.49.4		32	55.48	odd	20
275.2.z.c.49.5		32	55.37	odd	20
275.2.z.c.174.4		32	55.47	odd	20
275.2.z.c.174.5		32	55.3	odd	20
3025.2.a.bi.1.5		8	1.1	even	1	trivial
3025.2.a.bj.1.5		8	55.54	odd	2
3025.2.a.bm.1.4		8	11.10	odd	2
3025.2.a.bn.1.4		8	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.139403 q^{2} +2.98582 q^{3} -1.98057 q^{4} -0.416233 q^{6} -3.56959 q^{7} +0.554904 q^{8} +5.91512 q^{9} +O(q^{10})\)	\(q-0.139403 q^{2} +2.98582 q^{3} -1.98057 q^{4} -0.416233 q^{6} -3.56959 q^{7} +0.554904 q^{8} +5.91512 q^{9} -5.91361 q^{12} -1.19562 q^{13} +0.497612 q^{14} +3.88378 q^{16} -5.43544 q^{17} -0.824586 q^{18} +4.80260 q^{19} -10.6581 q^{21} -1.82203 q^{23} +1.65684 q^{24} +0.166673 q^{26} +8.70401 q^{27} +7.06980 q^{28} -4.14137 q^{29} +1.28688 q^{31} -1.65122 q^{32} +0.757717 q^{34} -11.7153 q^{36} -3.16092 q^{37} -0.669497 q^{38} -3.56990 q^{39} -6.40712 q^{41} +1.48578 q^{42} +4.23557 q^{43} +0.253997 q^{46} -7.02813 q^{47} +11.5963 q^{48} +5.74195 q^{49} -16.2292 q^{51} +2.36800 q^{52} -10.8109 q^{53} -1.21337 q^{54} -1.98078 q^{56} +14.3397 q^{57} +0.577320 q^{58} -4.01607 q^{59} -14.4814 q^{61} -0.179395 q^{62} -21.1145 q^{63} -7.53737 q^{64} +8.14233 q^{67} +10.7652 q^{68} -5.44026 q^{69} +8.51253 q^{71} +3.28232 q^{72} -8.54108 q^{73} +0.440643 q^{74} -9.51187 q^{76} +0.497655 q^{78} +0.393997 q^{79} +8.24324 q^{81} +0.893172 q^{82} -1.56363 q^{83} +21.1092 q^{84} -0.590452 q^{86} -12.3654 q^{87} -6.90425 q^{89} +4.26786 q^{91} +3.60866 q^{92} +3.84240 q^{93} +0.979744 q^{94} -4.93024 q^{96} -9.31179 q^{97} -0.800446 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - q^{3} + 9 q^{4} + q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} - 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} + q^{19} + 5 q^{21} - 2 q^{23} + q^{24} - 2 q^{26} + 2 q^{27} - 9 q^{28} + 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} + 8 q^{37} + 37 q^{38} - q^{39} - 41 q^{42} - 14 q^{43} - 20 q^{46} - 11 q^{47} + 27 q^{48} - 12 q^{49} - 25 q^{51} + 7 q^{52} - 11 q^{53} + 30 q^{54} - 10 q^{56} + 2 q^{57} - 27 q^{58} + 17 q^{59} - 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} - 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} - 6 q^{74} - 31 q^{76} + 17 q^{78} + 23 q^{79} - 4 q^{81} + 17 q^{82} - 41 q^{83} + 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} - 33 q^{92} + 59 q^{93} + 50 q^{94} - 61 q^{96} - 2 q^{97} - 26 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 - q^3 + 9 * q^4 + q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 - 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 + q^19 + 5 * q^21 - 2 * q^23 + q^24 - 2 * q^26 + 2 * q^27 - 9 * q^28 + 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 + 8 * q^37 + 37 * q^38 - q^39 - 41 * q^42 - 14 * q^43 - 20 * q^46 - 11 * q^47 + 27 * q^48 - 12 * q^49 - 25 * q^51 + 7 * q^52 - 11 * q^53 + 30 * q^54 - 10 * q^56 + 2 * q^57 - 27 * q^58 + 17 * q^59 - 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 - 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 - 6 * q^74 - 31 * q^76 + 17 * q^78 + 23 * q^79 - 4 * q^81 + 17 * q^82 - 41 * q^83 + 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 - 33 * q^92 + 59 * q^93 + 50 * q^94 - 61 * q^96 - 2 * q^97 - 26 * q^98

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