LMFDB - Embedded newform 2601.2.a.bi.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, 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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3418281.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 $ x^6 - 9x^4 - 4x^3 + 18x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 867)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.73700$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.43915 q^{2} +0.0711653 q^{4} +2.31394 q^{5} -4.44173 q^{7} -2.77589 q^{8} +O(q^{10})$	$q+1.43915 q^{2} +0.0711653 q^{4} +2.31394 q^{5} -4.44173 q^{7} -2.77589 q^{8} +3.33012 q^{10} +4.52453 q^{11} +1.17466 q^{13} -6.39233 q^{14} -4.13727 q^{16} +4.86615 q^{19} +0.164673 q^{20} +6.51150 q^{22} +0.625397 q^{23} +0.354337 q^{25} +1.69052 q^{26} -0.316097 q^{28} +1.51032 q^{29} +8.73592 q^{31} -0.402383 q^{32} -10.2779 q^{35} +8.79254 q^{37} +7.00315 q^{38} -6.42326 q^{40} +0.464965 q^{41} -1.51327 q^{43} +0.321990 q^{44} +0.900043 q^{46} -6.01904 q^{47} +12.7289 q^{49} +0.509946 q^{50} +0.0835951 q^{52} -7.12324 q^{53} +10.4695 q^{55} +12.3297 q^{56} +2.17358 q^{58} +10.4025 q^{59} +2.55882 q^{61} +12.5723 q^{62} +7.69544 q^{64} +2.71810 q^{65} -3.71801 q^{67} -14.7915 q^{70} +15.2429 q^{71} +2.70186 q^{73} +12.6538 q^{74} +0.346301 q^{76} -20.0967 q^{77} +16.6972 q^{79} -9.57340 q^{80} +0.669156 q^{82} +4.45459 q^{83} -2.17782 q^{86} -12.5596 q^{88} -1.54030 q^{89} -5.21752 q^{91} +0.0445066 q^{92} -8.66233 q^{94} +11.2600 q^{95} -3.83886 q^{97} +18.3189 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 3 q^{7} - 12 q^{8}+O(q^{10}) $ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 3 * q^7 - 12 * q^8	$ 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 3 q^{7} - 12 q^{8} + 12 q^{10} + 9 q^{11} + 9 q^{13} + 6 q^{14} + 15 q^{16} + 9 q^{19} - 6 q^{20} - 18 q^{22} + 9 q^{23} + 15 q^{25} + 12 q^{26} - 15 q^{28} + 6 q^{29} + 24 q^{31} - 42 q^{32} - 3 q^{37} + 6 q^{38} - 3 q^{40} + 18 q^{41} + 3 q^{44} + 15 q^{46} - 24 q^{47} + 21 q^{49} - 12 q^{50} - 18 q^{52} - 24 q^{53} - 24 q^{55} + 54 q^{56} + 3 q^{58} + 9 q^{59} + 21 q^{61} + 30 q^{62} + 24 q^{64} - 9 q^{65} - 6 q^{67} - 3 q^{70} + 27 q^{71} + 18 q^{73} + 36 q^{74} - 3 q^{76} - 33 q^{77} + 24 q^{79} + 3 q^{80} - 15 q^{82} - 6 q^{83} - 6 q^{86} - 24 q^{88} + 39 q^{91} - 15 q^{94} + 42 q^{95} - 33 q^{97} + 57 q^{98}+O(q^{100}) $ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 3 * q^7 - 12 * q^8 + 12 * q^10 + 9 * q^11 + 9 * q^13 + 6 * q^14 + 15 * q^16 + 9 * q^19 - 6 * q^20 - 18 * q^22 + 9 * q^23 + 15 * q^25 + 12 * q^26 - 15 * q^28 + 6 * q^29 + 24 * q^31 - 42 * q^32 - 3 * q^37 + 6 * q^38 - 3 * q^40 + 18 * q^41 + 3 * q^44 + 15 * q^46 - 24 * q^47 + 21 * q^49 - 12 * q^50 - 18 * q^52 - 24 * q^53 - 24 * q^55 + 54 * q^56 + 3 * q^58 + 9 * q^59 + 21 * q^61 + 30 * q^62 + 24 * q^64 - 9 * q^65 - 6 * q^67 - 3 * q^70 + 27 * q^71 + 18 * q^73 + 36 * q^74 - 3 * q^76 - 33 * q^77 + 24 * q^79 + 3 * q^80 - 15 * q^82 - 6 * q^83 - 6 * q^86 - 24 * q^88 + 39 * q^91 - 15 * q^94 + 42 * q^95 - 33 * q^97 + 57 * 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 $

$2$	1.43915	1.01764	0.508818	−	0.860874i	$-0.330083\pi$
$2$	1.43915	1.01764	0.508818	+	0.860874i	$0.330083\pi$
$3$	0	0
$3$	0	0
$4$	0.0711653	0.0355827
$5$	2.31394	1.03483	0.517414	−	0.855735i	$-0.326895\pi$
$5$	2.31394	1.03483	0.517414	+	0.855735i	$0.326895\pi$
$6$	0	0
$7$	−4.44173	−1.67881	−0.839407	−	0.543503i	$-0.817098\pi$
$7$	−4.44173	−1.67881	−0.839407	+	0.543503i	$0.817098\pi$
$8$	−2.77589	−0.981426
$9$	0	0
$10$	3.33012	1.05308
$11$	4.52453	1.36420	0.682098	−	0.731260i	$-0.261068\pi$
$11$	4.52453	1.36420	0.682098	+	0.731260i	$0.261068\pi$
$12$	0	0
$13$	1.17466	0.325792	0.162896	−	0.986643i	$-0.447916\pi$
$13$	1.17466	0.325792	0.162896	+	0.986643i	$0.447916\pi$
$14$	−6.39233	−1.70842
$15$	0	0
$16$	−4.13727	−1.03432
$17$	0	0
$17$	0	0
$18$	0	0
$19$	4.86615	1.11637	0.558186	−	0.829716i	$-0.311498\pi$
$19$	4.86615	1.11637	0.558186	+	0.829716i	$0.311498\pi$
$20$	0.164673	0.0368219
$21$	0	0
$22$	6.51150	1.38826
$23$	0.625397	0.130404	0.0652022	−	0.997872i	$-0.479231\pi$
$23$	0.625397	0.130404	0.0652022	+	0.997872i	$0.479231\pi$
$24$	0	0
$25$	0.354337	0.0708674
$26$	1.69052	0.331538
$27$	0	0
$28$	−0.316097	−0.0597367
$29$	1.51032	0.280459	0.140230	−	0.990119i	$-0.455216\pi$
$29$	1.51032	0.280459	0.140230	+	0.990119i	$0.455216\pi$
$30$	0	0
$31$	8.73592	1.56902	0.784509	−	0.620117i	$-0.212915\pi$
$31$	8.73592	1.56902	0.784509	+	0.620117i	$0.212915\pi$
$32$	−0.402383	−0.0711319
$33$	0	0
$34$	0	0
$35$	−10.2779	−1.73728
$36$	0	0
$37$	8.79254	1.44549	0.722743	−	0.691117i	$-0.242881\pi$
$37$	8.79254	1.44549	0.722743	+	0.691117i	$0.242881\pi$
$38$	7.00315	1.13606
$39$	0	0
$40$	−6.42326	−1.01561
$41$	0.464965	0.0726153	0.0363077	−	0.999341i	$-0.488440\pi$
$41$	0.464965	0.0726153	0.0363077	+	0.999341i	$0.488440\pi$
$42$	0	0
$43$	−1.51327	−0.230771	−0.115385	−	0.993321i	$-0.536810\pi$
$43$	−1.51327	−0.230771	−0.115385	+	0.993321i	$0.536810\pi$
$44$	0.321990	0.0485418
$45$	0	0
$46$	0.900043	0.132704
$47$	−6.01904	−0.877968	−0.438984	−	0.898495i	$-0.644661\pi$
$47$	−6.01904	−0.877968	−0.438984	+	0.898495i	$0.644661\pi$
$48$	0	0
$49$	12.7289	1.81842
$50$	0.509946	0.0721172
$51$	0	0
$52$	0.0835951	0.0115925
$53$	−7.12324	−0.978452	−0.489226	−	0.872157i	$-0.662721\pi$
$53$	−7.12324	−0.978452	−0.489226	+	0.872157i	$0.662721\pi$
$54$	0	0
$55$	10.4695	1.41171
$56$	12.3297	1.64763
$57$	0	0
$58$	2.17358	0.285405
$59$	10.4025	1.35429	0.677143	−	0.735852i	$-0.263218\pi$
$59$	10.4025	1.35429	0.677143	+	0.735852i	$0.263218\pi$
$60$	0	0
$61$	2.55882	0.327624	0.163812	−	0.986492i	$-0.447621\pi$
$61$	2.55882	0.327624	0.163812	+	0.986492i	$0.447621\pi$
$62$	12.5723	1.59669
$63$	0	0
$64$	7.69544	0.961930
$65$	2.71810	0.337138
$66$	0	0
$67$	−3.71801	−0.454227	−0.227114	−	0.973868i	$-0.572929\pi$
$67$	−3.71801	−0.454227	−0.227114	+	0.973868i	$0.572929\pi$
$68$	0	0
$69$	0	0
$70$	−14.7915	−1.76792
$71$	15.2429	1.80900	0.904500	−	0.426473i	$-0.140244\pi$
$71$	15.2429	1.80900	0.904500	+	0.426473i	$0.140244\pi$
$72$	0	0
$73$	2.70186	0.316229	0.158114	−	0.987421i	$-0.449459\pi$
$73$	2.70186	0.316229	0.158114	+	0.987421i	$0.449459\pi$
$74$	12.6538	1.47098
$75$	0	0
$76$	0.346301	0.0397235
$77$	−20.0967	−2.29023
$78$	0	0
$79$	16.6972	1.87858	0.939291	−	0.343120i	$-0.111484\pi$
$79$	16.6972	1.87858	0.939291	+	0.343120i	$0.111484\pi$
$80$	−9.57340	−1.07034
$81$	0	0
$82$	0.669156	0.0738959
$83$	4.45459	0.488954	0.244477	−	0.969655i	$-0.421384\pi$
$83$	4.45459	0.488954	0.244477	+	0.969655i	$0.421384\pi$
$84$	0	0
$85$	0	0
$86$	−2.17782	−0.234841
$87$	0	0
$88$	−12.5596	−1.33886
$89$	−1.54030	−0.163272	−0.0816359	−	0.996662i	$-0.526014\pi$
$89$	−1.54030	−0.163272	−0.0816359	+	0.996662i	$0.526014\pi$
$90$	0	0
$91$	−5.21752	−0.546944
$92$	0.0445066	0.00464013
$93$	0	0
$94$	−8.66233	−0.893451
$95$	11.2600	1.15525
$96$	0	0
$97$	−3.83886	−0.389778	−0.194889	−	0.980825i	$-0.562435\pi$
$97$	−3.83886	−0.389778	−0.194889	+	0.980825i	$0.562435\pi$
$98$	18.3189	1.85049
$99$	0	0
$100$	0.0252165	0.00252165
$101$	−11.9158	−1.18567	−0.592833	−	0.805325i	$-0.701991\pi$
$101$	−11.9158	−1.18567	−0.592833	+	0.805325i	$0.701991\pi$
$102$	0	0
$103$	−1.41240	−0.139168	−0.0695838	−	0.997576i	$-0.522167\pi$
$103$	−1.41240	−0.139168	−0.0695838	+	0.997576i	$0.522167\pi$
$104$	−3.26073	−0.319741
$105$	0	0
$106$	−10.2514	−0.995708
$107$	−15.6540	−1.51333	−0.756665	−	0.653803i	$-0.773172\pi$
$107$	−15.6540	−1.51333	−0.756665	+	0.653803i	$0.773172\pi$
$108$	0	0
$109$	10.1437	0.971593	0.485796	−	0.874072i	$-0.338530\pi$
$109$	10.1437	0.971593	0.485796	+	0.874072i	$0.338530\pi$
$110$	15.0672	1.43660
$111$	0	0
$112$	18.3766	1.73643
$113$	0.798760	0.0751410	0.0375705	−	0.999294i	$-0.488038\pi$
$113$	0.798760	0.0751410	0.0375705	+	0.999294i	$0.488038\pi$
$114$	0	0
$115$	1.44713	0.134946
$116$	0.107482	0.00997949
$117$	0	0
$118$	14.9708	1.37817
$119$	0	0
$120$	0	0
$121$	9.47136	0.861033
$122$	3.68254	0.333402
$123$	0	0
$124$	0.621695	0.0558299
$125$	−10.7498	−0.961492
$126$	0	0
$127$	3.36696	0.298769	0.149385	−	0.988779i	$-0.452271\pi$
$127$	3.36696	0.298769	0.149385	+	0.988779i	$0.452271\pi$
$128$	11.8797	1.05003
$129$	0	0
$130$	3.91176	0.343084
$131$	1.43966	0.125783	0.0628916	−	0.998020i	$-0.479968\pi$
$131$	1.43966	0.125783	0.0628916	+	0.998020i	$0.479968\pi$
$132$	0	0
$133$	−21.6141	−1.87418
$134$	−5.35079	−0.462238
$135$	0	0
$136$	0	0
$137$	−7.86924	−0.672315	−0.336157	−	0.941806i	$-0.609127\pi$
$137$	−7.86924	−0.672315	−0.336157	+	0.941806i	$0.609127\pi$
$138$	0	0
$139$	9.82784	0.833586	0.416793	−	0.909001i	$-0.363154\pi$
$139$	9.82784	0.833586	0.416793	+	0.909001i	$0.363154\pi$
$140$	−0.731431	−0.0618172
$141$	0	0
$142$	21.9369	1.84090
$143$	5.31478	0.444444
$144$	0	0
$145$	3.49480	0.290227
$146$	3.88839	0.321806
$147$	0	0
$148$	0.625724	0.0514342
$149$	6.30050	0.516157	0.258078	−	0.966124i	$-0.416911\pi$
$149$	6.30050	0.516157	0.258078	+	0.966124i	$0.416911\pi$
$150$	0	0
$151$	−10.8279	−0.881159	−0.440580	−	0.897714i	$-0.645227\pi$
$151$	−10.8279	−0.881159	−0.440580	+	0.897714i	$0.645227\pi$
$152$	−13.5079	−1.09564
$153$	0	0
$154$	−28.9223	−2.33062
$155$	20.2144	1.62366
$156$	0	0
$157$	10.8250	0.863933	0.431966	−	0.901890i	$-0.357820\pi$
$157$	10.8250	0.863933	0.431966	+	0.901890i	$0.357820\pi$
$158$	24.0299	1.91171
$159$	0	0
$160$	−0.931092	−0.0736093
$161$	−2.77784	−0.218925
$162$	0	0
$163$	−11.2781	−0.883371	−0.441686	−	0.897170i	$-0.645619\pi$
$163$	−11.2781	−0.883371	−0.441686	+	0.897170i	$0.645619\pi$
$164$	0.0330894	0.00258385
$165$	0	0
$166$	6.41084	0.497577
$167$	−12.3669	−0.956980	−0.478490	−	0.878093i	$-0.658816\pi$
$167$	−12.3669	−0.956980	−0.478490	+	0.878093i	$0.658816\pi$
$168$	0	0
$169$	−11.6202	−0.893860
$170$	0	0
$171$	0	0
$172$	−0.107692	−0.00821145
$173$	−3.19740	−0.243094	−0.121547	−	0.992586i	$-0.538785\pi$
$173$	−3.19740	−0.243094	−0.121547	+	0.992586i	$0.538785\pi$
$174$	0	0
$175$	−1.57387	−0.118973
$176$	−18.7192	−1.41101
$177$	0	0
$178$	−2.21673	−0.166151
$179$	19.0135	1.42113	0.710567	−	0.703629i	$-0.248438\pi$
$179$	19.0135	1.42113	0.710567	+	0.703629i	$0.248438\pi$
$180$	0	0
$181$	−12.0196	−0.893409	−0.446705	−	0.894682i	$-0.647402\pi$
$181$	−12.0196	−0.893409	−0.446705	+	0.894682i	$0.647402\pi$
$182$	−7.50881	−0.556590
$183$	0	0
$184$	−1.73603	−0.127982
$185$	20.3455	1.49583
$186$	0	0
$187$	0	0
$188$	−0.428347	−0.0312404
$189$	0	0
$190$	16.2049	1.17563
$191$	−22.3102	−1.61431	−0.807154	−	0.590341i	$-0.798993\pi$
$191$	−22.3102	−1.61431	−0.807154	+	0.590341i	$0.798993\pi$
$192$	0	0
$193$	−3.49591	−0.251641	−0.125821	−	0.992053i	$-0.540156\pi$
$193$	−3.49591	−0.251641	−0.125821	+	0.992053i	$0.540156\pi$
$194$	−5.52472	−0.396652
$195$	0	0
$196$	0.905859	0.0647042
$197$	11.9973	0.854770	0.427385	−	0.904070i	$-0.359435\pi$
$197$	11.9973	0.854770	0.427385	+	0.904070i	$0.359435\pi$
$198$	0	0
$199$	4.02251	0.285148	0.142574	−	0.989784i	$-0.454462\pi$
$199$	4.02251	0.285148	0.142574	+	0.989784i	$0.454462\pi$
$200$	−0.983601	−0.0695511
$201$	0	0
$202$	−17.1487	−1.20658
$203$	−6.70843	−0.470839
$204$	0	0
$205$	1.07590	0.0751443
$206$	−2.03266	−0.141622
$207$	0	0
$208$	−4.85988	−0.336972
$209$	22.0170	1.52295
$210$	0	0
$211$	−15.9602	−1.09874	−0.549372	−	0.835578i	$-0.685133\pi$
$211$	−15.9602	−1.09874	−0.549372	+	0.835578i	$0.685133\pi$
$212$	−0.506928	−0.0348159
$213$	0	0
$214$	−22.5285	−1.54002
$215$	−3.50161	−0.238808
$216$	0	0
$217$	−38.8026	−2.63409
$218$	14.5984	0.988728
$219$	0	0
$220$	0.745066	0.0502323
$221$	0	0
$222$	0	0
$223$	5.35573	0.358646	0.179323	−	0.983790i	$-0.442609\pi$
$223$	5.35573	0.358646	0.179323	+	0.983790i	$0.442609\pi$
$224$	1.78728	0.119417
$225$	0	0
$226$	1.14954	0.0764662
$227$	−1.71274	−0.113679	−0.0568393	−	0.998383i	$-0.518102\pi$
$227$	−1.71274	−0.113679	−0.0568393	+	0.998383i	$0.518102\pi$
$228$	0	0
$229$	−0.380619	−0.0251520	−0.0125760	−	0.999921i	$-0.504003\pi$
$229$	−0.380619	−0.0251520	−0.0125760	+	0.999921i	$0.504003\pi$
$230$	2.08265	0.137326
$231$	0	0
$232$	−4.19248	−0.275250
$233$	8.96281	0.587173	0.293587	−	0.955933i	$-0.405151\pi$
$233$	8.96281	0.587173	0.293587	+	0.955933i	$0.405151\pi$
$234$	0	0
$235$	−13.9277	−0.908545
$236$	0.740295	0.0481891
$237$	0	0
$238$	0	0
$239$	6.70928	0.433987	0.216994	−	0.976173i	$-0.430375\pi$
$239$	6.70928	0.433987	0.216994	+	0.976173i	$0.430375\pi$
$240$	0	0
$241$	−8.83991	−0.569428	−0.284714	−	0.958612i	$-0.591899\pi$
$241$	−8.83991	−0.569428	−0.284714	+	0.958612i	$0.591899\pi$
$242$	13.6307	0.876218
$243$	0	0
$244$	0.182099	0.0116577
$245$	29.4540	1.88175
$246$	0	0
$247$	5.71607	0.363705
$248$	−24.2500	−1.53987
$249$	0	0
$250$	−15.4706	−0.978448
$251$	24.4193	1.54133	0.770667	−	0.637238i	$-0.219923\pi$
$251$	24.4193	1.54133	0.770667	+	0.637238i	$0.219923\pi$
$252$	0	0
$253$	2.82963	0.177897
$254$	4.84557	0.304038
$255$	0	0
$256$	1.70583	0.106614
$257$	−25.9327	−1.61764	−0.808819	−	0.588057i	$-0.799893\pi$
$257$	−25.9327	−1.61764	−0.808819	+	0.588057i	$0.799893\pi$
$258$	0	0
$259$	−39.0541	−2.42670
$260$	0.193434	0.0119963
$261$	0	0
$262$	2.07189	0.128002
$263$	−1.74151	−0.107386	−0.0536932	−	0.998557i	$-0.517099\pi$
$263$	−1.74151	−0.107386	−0.0536932	+	0.998557i	$0.517099\pi$
$264$	0	0
$265$	−16.4828	−1.01253
$266$	−31.1061	−1.90723
$267$	0	0
$268$	−0.264594	−0.0161626
$269$	−31.4782	−1.91926	−0.959630	−	0.281264i	$-0.909246\pi$
$269$	−31.4782	−1.91926	−0.959630	+	0.281264i	$0.909246\pi$
$270$	0	0
$271$	−19.8066	−1.20316	−0.601582	−	0.798811i	$-0.705463\pi$
$271$	−19.8066	−1.20316	−0.601582	+	0.798811i	$0.705463\pi$
$272$	0	0
$273$	0	0
$274$	−11.3251	−0.684172
$275$	1.60321	0.0966771
$276$	0	0
$277$	−2.33073	−0.140040	−0.0700201	−	0.997546i	$-0.522306\pi$
$277$	−2.33073	−0.140040	−0.0700201	+	0.997546i	$0.522306\pi$
$278$	14.1438	0.848287
$279$	0	0
$280$	28.5303	1.70501
$281$	−0.306147	−0.0182632	−0.00913161	−	0.999958i	$-0.502907\pi$
$281$	−0.306147	−0.0182632	−0.00913161	+	0.999958i	$0.502907\pi$
$282$	0	0
$283$	18.3142	1.08867	0.544333	−	0.838869i	$-0.316783\pi$
$283$	18.3142	1.08867	0.544333	+	0.838869i	$0.316783\pi$
$284$	1.08477	0.0643691
$285$	0	0
$286$	7.64879	0.452282
$287$	−2.06525	−0.121908
$288$	0	0
$289$	0	0
$290$	5.02955	0.295345
$291$	0	0
$292$	0.192279	0.0112523
$293$	−15.2652	−0.891801	−0.445900	−	0.895083i	$-0.647116\pi$
$293$	−15.2652	−0.891801	−0.445900	+	0.895083i	$0.647116\pi$
$294$	0	0
$295$	24.0707	1.40145
$296$	−24.4071	−1.41864
$297$	0	0
$298$	9.06739	0.525260
$299$	0.734629	0.0424847
$300$	0	0
$301$	6.72151	0.387422
$302$	−15.5830	−0.896699
$303$	0	0
$304$	−20.1326	−1.15468
$305$	5.92097	0.339034
$306$	0	0
$307$	29.9884	1.71153	0.855765	−	0.517365i	$-0.173087\pi$
$307$	29.9884	1.71153	0.855765	+	0.517365i	$0.173087\pi$
$308$	−1.43019	−0.0814926
$309$	0	0
$310$	29.0917	1.65230
$311$	−23.3246	−1.32261	−0.661307	−	0.750115i	$-0.729998\pi$
$311$	−23.3246	−1.32261	−0.661307	+	0.750115i	$0.729998\pi$
$312$	0	0
$313$	23.9568	1.35412	0.677059	−	0.735929i	$-0.263254\pi$
$313$	23.9568	1.35412	0.677059	+	0.735929i	$0.263254\pi$
$314$	15.5789	0.879169
$315$	0	0
$316$	1.18826	0.0668450
$317$	18.7844	1.05504	0.527519	−	0.849543i	$-0.323122\pi$
$317$	18.7844	1.05504	0.527519	+	0.849543i	$0.323122\pi$
$318$	0	0
$319$	6.83348	0.382602
$320$	17.8068	0.995432
$321$	0	0
$322$	−3.99775	−0.222786
$323$	0	0
$324$	0	0
$325$	0.416226	0.0230880
$326$	−16.2310	−0.898950
$327$	0	0
$328$	−1.29069	−0.0712665
$329$	26.7349	1.47395
$330$	0	0
$331$	23.6869	1.30195	0.650975	−	0.759099i	$-0.274360\pi$
$331$	23.6869	1.30195	0.650975	+	0.759099i	$0.274360\pi$
$332$	0.317012	0.0173983
$333$	0	0
$334$	−17.7979	−0.973857
$335$	−8.60327	−0.470047
$336$	0	0
$337$	−6.07072	−0.330693	−0.165347	−	0.986236i	$-0.552874\pi$
$337$	−6.07072	−0.330693	−0.165347	+	0.986236i	$0.552874\pi$
$338$	−16.7232	−0.909624
$339$	0	0
$340$	0	0
$341$	39.5259	2.14045
$342$	0	0
$343$	−25.4464	−1.37398
$344$	4.20066	0.226484
$345$	0	0
$346$	−4.60155	−0.247381
$347$	7.13506	0.383030	0.191515	−	0.981490i	$-0.438660\pi$
$347$	7.13506	0.383030	0.191515	+	0.981490i	$0.438660\pi$
$348$	0	0
$349$	15.7825	0.844819	0.422410	−	0.906405i	$-0.361184\pi$
$349$	15.7825	0.844819	0.422410	+	0.906405i	$0.361184\pi$
$350$	−2.26504	−0.121071
$351$	0	0
$352$	−1.82059	−0.0970380
$353$	−35.2308	−1.87515	−0.937573	−	0.347788i	$-0.886933\pi$
$353$	−35.2308	−1.87515	−0.937573	+	0.347788i	$0.886933\pi$
$354$	0	0
$355$	35.2712	1.87200
$356$	−0.109616	−0.00580965
$357$	0	0
$358$	27.3633	1.44620
$359$	25.7316	1.35806	0.679030	−	0.734110i	$-0.262400\pi$
$359$	25.7316	1.35806	0.679030	+	0.734110i	$0.262400\pi$
$360$	0	0
$361$	4.67944	0.246287
$362$	−17.2980	−0.909165
$363$	0	0
$364$	−0.371306	−0.0194617
$365$	6.25195	0.327242
$366$	0	0
$367$	−8.26095	−0.431218	−0.215609	−	0.976480i	$-0.569174\pi$
$367$	−8.26095	−0.431218	−0.215609	+	0.976480i	$0.569174\pi$
$368$	−2.58743	−0.134879
$369$	0	0
$370$	29.2802	1.52221
$371$	31.6395	1.64264
$372$	0	0
$373$	23.4360	1.21347	0.606734	−	0.794905i	$-0.292479\pi$
$373$	23.4360	1.21347	0.606734	+	0.794905i	$0.292479\pi$
$374$	0	0
$375$	0	0
$376$	16.7082	0.861660
$377$	1.77411	0.0913714
$378$	0	0
$379$	−15.0621	−0.773687	−0.386843	−	0.922145i	$-0.626435\pi$
$379$	−15.0621	−0.773687	−0.386843	+	0.922145i	$0.626435\pi$
$380$	0.801322	0.0411070
$381$	0	0
$382$	−32.1078	−1.64278
$383$	15.9980	0.817459	0.408729	−	0.912656i	$-0.365972\pi$
$383$	15.9980	0.817459	0.408729	+	0.912656i	$0.365972\pi$
$384$	0	0
$385$	−46.5027	−2.37000
$386$	−5.03116	−0.256079
$387$	0	0
$388$	−0.273194	−0.0138693
$389$	−2.98991	−0.151595	−0.0757973	−	0.997123i	$-0.524150\pi$
$389$	−2.98991	−0.151595	−0.0757973	+	0.997123i	$0.524150\pi$
$390$	0	0
$391$	0	0
$392$	−35.3341	−1.78464
$393$	0	0
$394$	17.2659	0.869844
$395$	38.6364	1.94401
$396$	0	0
$397$	−12.6527	−0.635019	−0.317509	−	0.948255i	$-0.602847\pi$
$397$	−12.6527	−0.635019	−0.317509	+	0.948255i	$0.602847\pi$
$398$	5.78902	0.290177
$399$	0	0
$400$	−1.46599	−0.0732994
$401$	−14.6689	−0.732528	−0.366264	−	0.930511i	$-0.619363\pi$
$401$	−14.6689	−0.732528	−0.366264	+	0.930511i	$0.619363\pi$
$402$	0	0
$403$	10.2617	0.511174
$404$	−0.847992	−0.0421892
$405$	0	0
$406$	−9.65446	−0.479143
$407$	39.7821	1.97193
$408$	0	0
$409$	−16.3219	−0.807067	−0.403534	−	0.914965i	$-0.632218\pi$
$409$	−16.3219	−0.807067	−0.403534	+	0.914965i	$0.632218\pi$
$410$	1.54839	0.0764695
$411$	0	0
$412$	−0.100514	−0.00495196
$413$	−46.2049	−2.27359
$414$	0	0
$415$	10.3077	0.505983
$416$	−0.472663	−0.0231742
$417$	0	0
$418$	31.6859	1.54981
$419$	15.7098	0.767473	0.383737	−	0.923443i	$-0.374637\pi$
$419$	15.7098	0.767473	0.383737	+	0.923443i	$0.374637\pi$
$420$	0	0
$421$	22.1035	1.07726	0.538630	−	0.842542i	$-0.318942\pi$
$421$	22.1035	1.07726	0.538630	+	0.842542i	$0.318942\pi$
$422$	−22.9692	−1.11812
$423$	0	0
$424$	19.7733	0.960278
$425$	0	0
$426$	0	0
$427$	−11.3656	−0.550019
$428$	−1.11402	−0.0538483
$429$	0	0
$430$	−5.03936	−0.243020
$431$	−4.15528	−0.200153	−0.100076	−	0.994980i	$-0.531909\pi$
$431$	−4.15528	−0.200153	−0.100076	+	0.994980i	$0.531909\pi$
$432$	0	0
$433$	−13.0766	−0.628420	−0.314210	−	0.949354i	$-0.601740\pi$
$433$	−13.0766	−0.628420	−0.314210	+	0.949354i	$0.601740\pi$
$434$	−55.8429	−2.68055
$435$	0	0
$436$	0.721882	0.0345719
$437$	3.04328	0.145580
$438$	0	0
$439$	11.5938	0.553342	0.276671	−	0.960965i	$-0.410769\pi$
$439$	11.5938	0.553342	0.276671	+	0.960965i	$0.410769\pi$
$440$	−29.0622	−1.38549
$441$	0	0
$442$	0	0
$443$	−17.6619	−0.839144	−0.419572	−	0.907722i	$-0.637820\pi$
$443$	−17.6619	−0.839144	−0.419572	+	0.907722i	$0.637820\pi$
$444$	0	0
$445$	−3.56418	−0.168958
$446$	7.70773	0.364971
$447$	0	0
$448$	−34.1810	−1.61490
$449$	−35.1414	−1.65843	−0.829213	−	0.558932i	$-0.811211\pi$
$449$	−35.1414	−1.65843	−0.829213	+	0.558932i	$0.811211\pi$
$450$	0	0
$451$	2.10375	0.0990616
$452$	0.0568440	0.00267372
$453$	0	0
$454$	−2.46490	−0.115683
$455$	−12.0730	−0.565993
$456$	0	0
$457$	24.6306	1.15217	0.576086	−	0.817389i	$-0.304580\pi$
$457$	24.6306	1.15217	0.576086	+	0.817389i	$0.304580\pi$
$458$	−0.547769	−0.0255956
$459$	0	0
$460$	0.102986	0.00480174
$461$	1.00413	0.0467668	0.0233834	−	0.999727i	$-0.492556\pi$
$461$	1.00413	0.0467668	0.0233834	+	0.999727i	$0.492556\pi$
$462$	0	0
$463$	−20.4600	−0.950856	−0.475428	−	0.879755i	$-0.657707\pi$
$463$	−20.4600	−0.950856	−0.475428	+	0.879755i	$0.657707\pi$
$464$	−6.24859	−0.290084
$465$	0	0
$466$	12.8989	0.597528
$467$	−35.1659	−1.62728	−0.813642	−	0.581366i	$-0.802519\pi$
$467$	−35.1659	−1.62728	−0.813642	+	0.581366i	$0.802519\pi$
$468$	0	0
$469$	16.5144	0.762564
$470$	−20.0442	−0.924568
$471$	0	0
$472$	−28.8761	−1.32913
$473$	−6.84682	−0.314817
$474$	0	0
$475$	1.72426	0.0791144
$476$	0	0
$477$	0	0
$478$	9.65570	0.441641
$479$	3.85721	0.176240	0.0881201	−	0.996110i	$-0.471914\pi$
$479$	3.85721	0.176240	0.0881201	+	0.996110i	$0.471914\pi$
$480$	0	0
$481$	10.3282	0.470927
$482$	−12.7220	−0.579471
$483$	0	0
$484$	0.674032	0.0306378
$485$	−8.88292	−0.403353
$486$	0	0
$487$	20.3689	0.923003	0.461502	−	0.887139i	$-0.347311\pi$
$487$	20.3689	0.923003	0.461502	+	0.887139i	$0.347311\pi$
$488$	−7.10301	−0.321538
$489$	0	0
$490$	42.3889	1.91494
$491$	6.38676	0.288231	0.144115	−	0.989561i	$-0.453966\pi$
$491$	6.38676	0.288231	0.144115	+	0.989561i	$0.453966\pi$
$492$	0	0
$493$	0	0
$494$	8.22631	0.370119
$495$	0	0
$496$	−36.1428	−1.62286
$497$	−67.7048	−3.03698
$498$	0	0
$499$	1.64385	0.0735887	0.0367944	−	0.999323i	$-0.488285\pi$
$499$	1.64385	0.0735887	0.0367944	+	0.999323i	$0.488285\pi$
$500$	−0.765013	−0.0342124
$501$	0	0
$502$	35.1432	1.56852
$503$	−26.6120	−1.18657	−0.593285	−	0.804993i	$-0.702169\pi$
$503$	−26.6120	−1.18657	−0.593285	+	0.804993i	$0.702169\pi$
$504$	0	0
$505$	−27.5725	−1.22696
$506$	4.07227	0.181035
$507$	0	0
$508$	0.239611	0.0106310
$509$	−14.0065	−0.620827	−0.310413	−	0.950602i	$-0.600467\pi$
$509$	−14.0065	−0.620827	−0.310413	+	0.950602i	$0.600467\pi$
$510$	0	0
$511$	−12.0009	−0.530890
$512$	−21.3044	−0.941532
$513$	0	0
$514$	−37.3212	−1.64617
$515$	−3.26821	−0.144014
$516$	0	0
$517$	−27.2333	−1.19772
$518$	−56.2048	−2.46950
$519$	0	0
$520$	−7.54514	−0.330876
$521$	−18.9615	−0.830717	−0.415358	−	0.909658i	$-0.636344\pi$
$521$	−18.9615	−0.830717	−0.415358	+	0.909658i	$0.636344\pi$
$522$	0	0
$523$	20.3838	0.891320	0.445660	−	0.895202i	$-0.352969\pi$
$523$	20.3838	0.891320	0.445660	+	0.895202i	$0.352969\pi$
$524$	0.102454	0.00447571
$525$	0	0
$526$	−2.50631	−0.109280
$527$	0	0
$528$	0	0
$529$	−22.6089	−0.982995
$530$	−23.7213	−1.03039
$531$	0	0
$532$	−1.53818	−0.0666884
$533$	0.546175	0.0236575
$534$	0	0
$535$	−36.2225	−1.56603
$536$	10.3208	0.445790
$537$	0	0
$538$	−45.3020	−1.95311
$539$	57.5924	2.48068
$540$	0	0
$541$	22.5413	0.969127	0.484563	−	0.874756i	$-0.338979\pi$
$541$	22.5413	0.969127	0.484563	+	0.874756i	$0.338979\pi$
$542$	−28.5047	−1.22438
$543$	0	0
$544$	0	0
$545$	23.4720	1.00543
$546$	0	0
$547$	−3.30730	−0.141410	−0.0707049	−	0.997497i	$-0.522525\pi$
$547$	−3.30730	−0.141410	−0.0707049	+	0.997497i	$0.522525\pi$
$548$	−0.560018	−0.0239228
$549$	0	0
$550$	2.30726	0.0983821
$551$	7.34945	0.313097
$552$	0	0
$553$	−74.1644	−3.15379
$554$	−3.35428	−0.142510
$555$	0	0
$556$	0.699401	0.0296612
$557$	−34.4355	−1.45908	−0.729540	−	0.683938i	$-0.760266\pi$
$557$	−34.4355	−1.45908	−0.729540	+	0.683938i	$0.760266\pi$
$558$	0	0
$559$	−1.77757	−0.0751833
$560$	42.5224	1.79690
$561$	0	0
$562$	−0.440593	−0.0185853
$563$	4.70471	0.198280	0.0991399	−	0.995074i	$-0.468391\pi$
$563$	4.70471	0.198280	0.0991399	+	0.995074i	$0.468391\pi$
$564$	0	0
$565$	1.84829	0.0777580
$566$	26.3570	1.10787
$567$	0	0
$568$	−42.3126	−1.77540
$569$	−37.2742	−1.56262	−0.781308	−	0.624145i	$-0.785447\pi$
$569$	−37.2742	−1.56262	−0.781308	+	0.624145i	$0.785447\pi$
$570$	0	0
$571$	20.3123	0.850043	0.425021	−	0.905183i	$-0.360267\pi$
$571$	20.3123	0.850043	0.425021	+	0.905183i	$0.360267\pi$
$572$	0.378228	0.0158145
$573$	0	0
$574$	−2.97221	−0.124058
$575$	0.221601	0.00924142
$576$	0	0
$577$	−33.5408	−1.39632	−0.698161	−	0.715941i	$-0.745998\pi$
$577$	−33.5408	−1.39632	−0.698161	+	0.715941i	$0.745998\pi$
$578$	0	0
$579$	0	0
$580$	0.248708	0.0103271
$581$	−19.7861	−0.820864
$582$	0	0
$583$	−32.2293	−1.33480
$584$	−7.50007	−0.310355
$585$	0	0
$586$	−21.9689	−0.907528
$587$	21.8364	0.901285	0.450642	−	0.892705i	$-0.351195\pi$
$587$	21.8364	0.901285	0.450642	+	0.892705i	$0.351195\pi$
$588$	0	0
$589$	42.5103	1.75161
$590$	34.6415	1.42617
$591$	0	0
$592$	−36.3771	−1.49509
$593$	24.6066	1.01047	0.505236	−	0.862981i	$-0.331406\pi$
$593$	24.6066	1.01047	0.505236	+	0.862981i	$0.331406\pi$
$594$	0	0
$595$	0	0
$596$	0.448377	0.0183662
$597$	0	0
$598$	1.05724	0.0432339
$599$	40.9057	1.67136	0.835681	−	0.549215i	$-0.185073\pi$
$599$	40.9057	1.67136	0.835681	+	0.549215i	$0.185073\pi$
$600$	0	0
$601$	−36.8554	−1.50336	−0.751682	−	0.659525i	$-0.770757\pi$
$601$	−36.8554	−1.50336	−0.751682	+	0.659525i	$0.770757\pi$
$602$	9.67330	0.394254
$603$	0	0
$604$	−0.770569	−0.0313540
$605$	21.9162	0.891020
$606$	0	0
$607$	48.2384	1.95794	0.978968	−	0.204015i	$-0.0653990\pi$
$607$	48.2384	1.95794	0.978968	+	0.204015i	$0.0653990\pi$
$608$	−1.95806	−0.0794097
$609$	0	0
$610$	8.52119	0.345013
$611$	−7.07033	−0.286035
$612$	0	0
$613$	26.2900	1.06184	0.530922	−	0.847421i	$-0.321846\pi$
$613$	26.2900	1.06184	0.530922	+	0.847421i	$0.321846\pi$
$614$	43.1580	1.74171
$615$	0	0
$616$	55.7863	2.24769
$617$	3.91276	0.157522	0.0787610	−	0.996894i	$-0.474904\pi$
$617$	3.91276	0.157522	0.0787610	+	0.996894i	$0.474904\pi$
$618$	0	0
$619$	8.50634	0.341899	0.170949	−	0.985280i	$-0.445317\pi$
$619$	8.50634	0.341899	0.170949	+	0.985280i	$0.445317\pi$
$620$	1.43857	0.0577743
$621$	0	0
$622$	−33.5676	−1.34594
$623$	6.84161	0.274103
$624$	0	0
$625$	−26.6461	−1.06585
$626$	34.4775	1.37800
$627$	0	0
$628$	0.770368	0.0307410
$629$	0	0
$630$	0	0
$631$	−7.26384	−0.289169	−0.144584	−	0.989492i	$-0.546185\pi$
$631$	−7.26384	−0.289169	−0.144584	+	0.989492i	$0.546185\pi$
$632$	−46.3496	−1.84369
$633$	0	0
$634$	27.0337	1.07364
$635$	7.79096	0.309175
$636$	0	0
$637$	14.9522	0.592427
$638$	9.83444	0.389349
$639$	0	0
$640$	27.4889	1.08660
$641$	−38.6207	−1.52542	−0.762712	−	0.646738i	$-0.776133\pi$
$641$	−38.6207	−1.52542	−0.762712	+	0.646738i	$0.776133\pi$
$642$	0	0
$643$	31.6378	1.24767	0.623836	−	0.781555i	$-0.285573\pi$
$643$	31.6378	1.24767	0.623836	+	0.781555i	$0.285573\pi$
$644$	−0.197686	−0.00778993
$645$	0	0
$646$	0	0
$647$	−2.64576	−0.104016	−0.0520078	−	0.998647i	$-0.516562\pi$
$647$	−2.64576	−0.104016	−0.0520078	+	0.998647i	$0.516562\pi$
$648$	0	0
$649$	47.0662	1.84751
$650$	0.599013	0.0234952
$651$	0	0
$652$	−0.802612	−0.0314327
$653$	−35.1366	−1.37500	−0.687502	−	0.726183i	$-0.741293\pi$
$653$	−35.1366	−1.37500	−0.687502	+	0.726183i	$0.741293\pi$
$654$	0	0
$655$	3.33128	0.130164
$656$	−1.92368	−0.0751072
$657$	0	0
$658$	38.4757	1.49994
$659$	−23.7941	−0.926888	−0.463444	−	0.886126i	$-0.653387\pi$
$659$	−23.7941	−0.926888	−0.463444	+	0.886126i	$0.653387\pi$
$660$	0	0
$661$	−35.3068	−1.37328	−0.686639	−	0.726999i	$-0.740914\pi$
$661$	−35.3068	−1.37328	−0.686639	+	0.726999i	$0.740914\pi$
$662$	34.0891	1.32491
$663$	0	0
$664$	−12.3654	−0.479872
$665$	−50.0139	−1.93945
$666$	0	0
$667$	0.944550	0.0365731
$668$	−0.880095	−0.0340519
$669$	0	0
$670$	−12.3814	−0.478337
$671$	11.5775	0.446943
$672$	0	0
$673$	23.3050	0.898341	0.449171	−	0.893446i	$-0.351720\pi$
$673$	23.3050	0.898341	0.449171	+	0.893446i	$0.351720\pi$
$674$	−8.73670	−0.336525
$675$	0	0
$676$	−0.826954	−0.0318059
$677$	10.3028	0.395967	0.197984	−	0.980205i	$-0.436561\pi$
$677$	10.3028	0.395967	0.197984	+	0.980205i	$0.436561\pi$
$678$	0	0
$679$	17.0512	0.654365
$680$	0	0
$681$	0	0
$682$	56.8839	2.17820
$683$	27.5907	1.05573	0.527865	−	0.849328i	$-0.322993\pi$
$683$	27.5907	1.05573	0.527865	+	0.849328i	$0.322993\pi$
$684$	0	0
$685$	−18.2090	−0.695730
$686$	−36.6213	−1.39821
$687$	0	0
$688$	6.26078	0.238690
$689$	−8.36738	−0.318772
$690$	0	0
$691$	−25.5242	−0.970985	−0.485493	−	0.874241i	$-0.661360\pi$
$691$	−25.5242	−0.970985	−0.485493	+	0.874241i	$0.661360\pi$
$692$	−0.227544	−0.00864992
$693$	0	0
$694$	10.2685	0.389785
$695$	22.7411	0.862618
$696$	0	0
$697$	0	0
$698$	22.7135	0.859718
$699$	0	0
$700$	−0.112005	−0.00423339
$701$	14.8327	0.560225	0.280113	−	0.959967i	$-0.409628\pi$
$701$	14.8327	0.560225	0.280113	+	0.959967i	$0.409628\pi$
$702$	0	0
$703$	42.7859	1.61370
$704$	34.8182	1.31226
$705$	0	0
$706$	−50.7026	−1.90822
$707$	52.9267	1.99051
$708$	0	0
$709$	4.69498	0.176324	0.0881618	−	0.996106i	$-0.471901\pi$
$709$	4.69498	0.176324	0.0881618	+	0.996106i	$0.471901\pi$
$710$	50.7607	1.90502
$711$	0	0
$712$	4.27571	0.160239
$713$	5.46342	0.204607
$714$	0	0
$715$	12.2981	0.459923
$716$	1.35310	0.0505678
$717$	0	0
$718$	37.0317	1.38201
$719$	−0.412832	−0.0153960	−0.00769802	−	0.999970i	$-0.502450\pi$
$719$	−0.412832	−0.0153960	−0.00769802	+	0.999970i	$0.502450\pi$
$720$	0	0
$721$	6.27348	0.233637
$722$	6.73444	0.250630
$723$	0	0
$724$	−0.855378	−0.0317899
$725$	0.535162	0.0198754
$726$	0	0
$727$	−32.9236	−1.22107	−0.610534	−	0.791990i	$-0.709045\pi$
$727$	−32.9236	−1.22107	−0.610534	+	0.791990i	$0.709045\pi$
$728$	14.4833	0.536785
$729$	0	0
$730$	8.99753	0.333013
$731$	0	0
$732$	0	0
$733$	36.5965	1.35172	0.675861	−	0.737029i	$-0.263772\pi$
$733$	36.5965	1.35172	0.675861	+	0.737029i	$0.263772\pi$
$734$	−11.8888	−0.438823
$735$	0	0
$736$	−0.251649	−0.00927591
$737$	−16.8222	−0.619655
$738$	0	0
$739$	−45.0262	−1.65632	−0.828158	−	0.560494i	$-0.810611\pi$
$739$	−45.0262	−1.65632	−0.828158	+	0.560494i	$0.810611\pi$
$740$	1.44789	0.0532255
$741$	0	0
$742$	45.5341	1.67161
$743$	19.7422	0.724271	0.362136	−	0.932125i	$-0.382048\pi$
$743$	19.7422	0.724271	0.362136	+	0.932125i	$0.382048\pi$
$744$	0	0
$745$	14.5790	0.534133
$746$	33.7280	1.23487
$747$	0	0
$748$	0	0
$749$	69.5308	2.54060
$750$	0	0
$751$	−23.2639	−0.848913	−0.424457	−	0.905448i	$-0.639535\pi$
$751$	−23.2639	−0.848913	−0.424457	+	0.905448i	$0.639535\pi$
$752$	24.9024	0.908097
$753$	0	0
$754$	2.55322	0.0929828
$755$	−25.0551	−0.911847
$756$	0	0
$757$	−9.82437	−0.357073	−0.178536	−	0.983933i	$-0.557136\pi$
$757$	−9.82437	−0.357073	−0.178536	+	0.983933i	$0.557136\pi$
$758$	−21.6767	−0.787332
$759$	0	0
$760$	−31.2565	−1.13379
$761$	38.2412	1.38624	0.693121	−	0.720822i	$-0.256235\pi$
$761$	38.2412	1.38624	0.693121	+	0.720822i	$0.256235\pi$
$762$	0	0
$763$	−45.0557	−1.63112
$764$	−1.58771	−0.0574414
$765$	0	0
$766$	23.0236	0.831876
$767$	12.2194	0.441215
$768$	0	0
$769$	44.2020	1.59396	0.796982	−	0.604002i	$-0.206428\pi$
$769$	44.2020	1.59396	0.796982	+	0.604002i	$0.206428\pi$
$770$	−66.9245	−2.41179
$771$	0	0
$772$	−0.248788	−0.00895407
$773$	−5.09149	−0.183128	−0.0915641	−	0.995799i	$-0.529187\pi$
$773$	−5.09149	−0.183128	−0.0915641	+	0.995799i	$0.529187\pi$
$774$	0	0
$775$	3.09546	0.111192
$776$	10.6563	0.382538
$777$	0	0
$778$	−4.30295	−0.154268
$779$	2.26259	0.0810657
$780$	0	0
$781$	68.9670	2.46783
$782$	0	0
$783$	0	0
$784$	−52.6630	−1.88082
$785$	25.0486	0.894021
$786$	0	0
$787$	9.27515	0.330623	0.165312	−	0.986241i	$-0.447137\pi$
$787$	9.27515	0.330623	0.165312	+	0.986241i	$0.447137\pi$
$788$	0.853789	0.0304150
$789$	0	0
$790$	55.6038	1.97829
$791$	−3.54787	−0.126148
$792$	0	0
$793$	3.00575	0.106737
$794$	−18.2091	−0.646218
$795$	0	0
$796$	0.286263	0.0101463
$797$	−24.7276	−0.875897	−0.437949	−	0.899000i	$-0.644295\pi$
$797$	−24.7276	−0.875897	−0.437949	+	0.899000i	$0.644295\pi$
$798$	0	0
$799$	0	0
$800$	−0.142579	−0.00504094
$801$	0	0
$802$	−21.1108	−0.745447
$803$	12.2246	0.431398
$804$	0	0
$805$	−6.42777	−0.226549
$806$	14.7682	0.520189
$807$	0	0
$808$	33.0770	1.16364
$809$	−29.7012	−1.04424	−0.522119	−	0.852873i	$-0.674858\pi$
$809$	−29.7012	−1.04424	−0.522119	+	0.852873i	$0.674858\pi$
$810$	0	0
$811$	−28.2770	−0.992940	−0.496470	−	0.868054i	$-0.665371\pi$
$811$	−28.2770	−0.992940	−0.496470	+	0.868054i	$0.665371\pi$
$812$	−0.477408	−0.0167537
$813$	0	0
$814$	57.2526	2.00670
$815$	−26.0970	−0.914137
$816$	0	0
$817$	−7.36378	−0.257626
$818$	−23.4898	−0.821301
$819$	0	0
$820$	0.0765670	0.00267383
$821$	42.8794	1.49650	0.748251	−	0.663415i	$-0.230894\pi$
$821$	42.8794	1.49650	0.748251	+	0.663415i	$0.230894\pi$
$822$	0	0
$823$	−29.1963	−1.01772	−0.508859	−	0.860850i	$-0.669932\pi$
$823$	−29.1963	−1.01772	−0.508859	+	0.860850i	$0.669932\pi$
$824$	3.92066	0.136583
$825$	0	0
$826$	−66.4960	−2.31369
$827$	−51.0303	−1.77450	−0.887249	−	0.461291i	$-0.847387\pi$
$827$	−51.0303	−1.77450	−0.887249	+	0.461291i	$0.847387\pi$
$828$	0	0
$829$	−16.2122	−0.563072	−0.281536	−	0.959551i	$-0.590844\pi$
$829$	−16.2122	−0.563072	−0.281536	+	0.959551i	$0.590844\pi$
$830$	14.8343	0.514907
$831$	0	0
$832$	9.03952	0.313389
$833$	0	0
$834$	0	0
$835$	−28.6163	−0.990309
$836$	1.56685	0.0541907
$837$	0	0
$838$	22.6088	0.781008
$839$	35.7772	1.23517	0.617583	−	0.786506i	$-0.288112\pi$
$839$	35.7772	1.23517	0.617583	+	0.786506i	$0.288112\pi$
$840$	0	0
$841$	−26.7189	−0.921343
$842$	31.8104	1.09626
$843$	0	0
$844$	−1.13581	−0.0390962
$845$	−26.8884	−0.924990
$846$	0	0
$847$	−42.0692	−1.44551
$848$	29.4707	1.01203
$849$	0	0
$850$	0	0
$851$	5.49883	0.188498
$852$	0	0
$853$	51.9313	1.77809	0.889047	−	0.457815i	$-0.151368\pi$
$853$	51.9313	1.77809	0.889047	+	0.457815i	$0.151368\pi$
$854$	−16.3568	−0.559720
$855$	0	0
$856$	43.4538	1.48522
$857$	−36.4929	−1.24657	−0.623287	−	0.781994i	$-0.714203\pi$
$857$	−36.4929	−1.24657	−0.623287	+	0.781994i	$0.714203\pi$
$858$	0	0
$859$	26.0367	0.888361	0.444181	−	0.895937i	$-0.353495\pi$
$859$	26.0367	0.888361	0.444181	+	0.895937i	$0.353495\pi$
$860$	−0.249193	−0.00849743
$861$	0	0
$862$	−5.98009	−0.203683
$863$	−37.5428	−1.27797	−0.638986	−	0.769218i	$-0.720646\pi$
$863$	−37.5428	−1.27797	−0.638986	+	0.769218i	$0.720646\pi$
$864$	0	0
$865$	−7.39860	−0.251560
$866$	−18.8192	−0.639502
$867$	0	0
$868$	−2.76140	−0.0937280
$869$	75.5470	2.56276
$870$	0	0
$871$	−4.36740	−0.147984
$872$	−28.1579	−0.953546
$873$	0	0
$874$	4.37975	0.148147
$875$	47.7477	1.61417
$876$	0	0
$877$	39.0738	1.31943	0.659714	−	0.751517i	$-0.270677\pi$
$877$	39.0738	1.31943	0.659714	+	0.751517i	$0.270677\pi$
$878$	16.6853	0.563101
$879$	0	0
$880$	−43.3151	−1.46015
$881$	44.2340	1.49028	0.745141	−	0.666907i	$-0.232382\pi$
$881$	44.2340	1.49028	0.745141	+	0.666907i	$0.232382\pi$
$882$	0	0
$883$	−21.8625	−0.735733	−0.367866	−	0.929879i	$-0.619912\pi$
$883$	−21.8625	−0.735733	−0.367866	+	0.929879i	$0.619912\pi$
$884$	0	0
$885$	0	0
$886$	−25.4182	−0.853943
$887$	−0.437004	−0.0146731	−0.00733657	−	0.999973i	$-0.502335\pi$
$887$	−0.437004	−0.0146731	−0.00733657	+	0.999973i	$0.502335\pi$
$888$	0	0
$889$	−14.9551	−0.501578
$890$	−5.12940	−0.171938
$891$	0	0
$892$	0.381143	0.0127616
$893$	−29.2896	−0.980139
$894$	0	0
$895$	43.9961	1.47063
$896$	−52.7664	−1.76280
$897$	0	0
$898$	−50.5739	−1.68767
$899$	13.1940	0.440046
$900$	0	0
$901$	0	0
$902$	3.02762	0.100809
$903$	0	0
$904$	−2.21727	−0.0737453
$905$	−27.8127	−0.924524
$906$	0	0
$907$	17.0070	0.564710	0.282355	−	0.959310i	$-0.408884\pi$
$907$	17.0070	0.564710	0.282355	+	0.959310i	$0.408884\pi$
$908$	−0.121888	−0.00404499
$909$	0	0
$910$	−17.3750	−0.575975
$911$	−19.1396	−0.634122	−0.317061	−	0.948405i	$-0.602696\pi$
$911$	−19.1396	−0.634122	−0.317061	+	0.948405i	$0.602696\pi$
$912$	0	0
$913$	20.1549	0.667030
$914$	35.4472	1.17249
$915$	0	0
$916$	−0.0270869	−0.000894975 0
$917$	−6.39456	−0.211167
$918$	0	0
$919$	−48.0934	−1.58645	−0.793227	−	0.608926i	$-0.791601\pi$
$919$	−48.0934	−1.58645	−0.793227	+	0.608926i	$0.791601\pi$
$920$	−4.01709	−0.132439
$921$	0	0
$922$	1.44509	0.0475916
$923$	17.9052	0.589358
$924$	0	0
$925$	3.11552	0.102438
$926$	−29.4451	−0.967625
$927$	0	0
$928$	−0.607727	−0.0199496
$929$	−5.22230	−0.171338	−0.0856691	−	0.996324i	$-0.527303\pi$
$929$	−5.22230	−0.171338	−0.0856691	+	0.996324i	$0.527303\pi$
$930$	0	0
$931$	61.9410	2.03003
$932$	0.637841	0.0208932
$933$	0	0
$934$	−50.6092	−1.65598
$935$	0	0
$936$	0	0
$937$	−6.53593	−0.213520	−0.106760	−	0.994285i	$-0.534048\pi$
$937$	−6.53593	−0.213520	−0.106760	+	0.994285i	$0.534048\pi$
$938$	23.7668	0.776012
$939$	0	0
$940$	−0.991172	−0.0323285
$941$	8.45197	0.275526	0.137763	−	0.990465i	$-0.456009\pi$
$941$	8.45197	0.275526	0.137763	+	0.990465i	$0.456009\pi$
$942$	0	0
$943$	0.290788	0.00946935
$944$	−43.0378	−1.40076
$945$	0	0
$946$	−9.85362	−0.320369
$947$	46.3276	1.50544	0.752722	−	0.658338i	$-0.228740\pi$
$947$	46.3276	1.50544	0.752722	+	0.658338i	$0.228740\pi$
$948$	0	0
$949$	3.17377	0.103025
$950$	2.48147	0.0805097
$951$	0	0
$952$	0	0
$953$	−34.8809	−1.12990	−0.564951	−	0.825124i	$-0.691105\pi$
$953$	−34.8809	−1.12990	−0.564951	+	0.825124i	$0.691105\pi$
$954$	0	0
$955$	−51.6245	−1.67053
$956$	0.477468	0.0154424
$957$	0	0
$958$	5.55111	0.179348
$959$	34.9530	1.12869
$960$	0	0
$961$	45.3164	1.46182
$962$	14.8639	0.479233
$963$	0	0
$964$	−0.629095	−0.0202618
$965$	−8.08934	−0.260405
$966$	0	0
$967$	45.1412	1.45164	0.725821	−	0.687884i	$-0.241460\pi$
$967$	45.1412	1.45164	0.725821	+	0.687884i	$0.241460\pi$
$968$	−26.2915	−0.845039
$969$	0	0
$970$	−12.7839	−0.410466
$971$	−42.2005	−1.35428	−0.677139	−	0.735855i	$-0.736780\pi$
$971$	−42.2005	−1.35428	−0.677139	+	0.735855i	$0.736780\pi$
$972$	0	0
$973$	−43.6526	−1.39944
$974$	29.3140	0.939281
$975$	0	0
$976$	−10.5865	−0.338867
$977$	38.0075	1.21597	0.607983	−	0.793950i	$-0.291979\pi$
$977$	38.0075	1.21597	0.607983	+	0.793950i	$0.291979\pi$
$978$	0	0
$979$	−6.96915	−0.222735
$980$	2.09611	0.0669577
$981$	0	0
$982$	9.19154	0.293314
$983$	43.3779	1.38354	0.691770	−	0.722118i	$-0.256831\pi$
$983$	43.3779	1.38354	0.691770	+	0.722118i	$0.256831\pi$
$984$	0	0
$985$	27.7610	0.884539
$986$	0	0
$987$	0	0
$988$	0.406786	0.0129416
$989$	−0.946392	−0.0300935
$990$	0	0
$991$	40.2588	1.27886	0.639431	−	0.768848i	$-0.279170\pi$
$991$	40.2588	1.27886	0.639431	+	0.768848i	$0.279170\pi$
$992$	−3.51519	−0.111607
$993$	0	0
$994$	−97.4377	−3.09054
$995$	9.30787	0.295079
$996$	0	0
$997$	−31.8282	−1.00801	−0.504005	−	0.863701i	$-0.668141\pi$
$997$	−31.8282	−1.00801	−0.504005	+	0.863701i	$0.668141\pi$
$998$	2.36575	0.0748865
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bi.1.5		6
3.2	odd	2		867.2.a.p.1.2	yes	6
17.16	even	2		2601.2.a.bh.1.5		6
51.2	odd	8		867.2.e.k.616.4		24
51.5	even	16		867.2.h.m.688.9		48
51.8	odd	8		867.2.e.k.829.9		24
51.11	even	16		867.2.h.m.733.3		48
51.14	even	16		867.2.h.m.757.3		48
51.20	even	16		867.2.h.m.757.4		48
51.23	even	16		867.2.h.m.733.4		48
51.26	odd	8		867.2.e.k.829.10		24
51.29	even	16		867.2.h.m.688.10		48
51.32	odd	8		867.2.e.k.616.3		24
51.38	odd	4		867.2.d.g.577.9		12
51.41	even	16		867.2.h.m.712.9		48
51.44	even	16		867.2.h.m.712.10		48
51.47	odd	4		867.2.d.g.577.10		12
51.50	odd	2		867.2.a.o.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
867.2.a.o.1.2	✓	6	51.50	odd	2
867.2.a.p.1.2	yes	6	3.2	odd	2
867.2.d.g.577.9		12	51.38	odd	4
867.2.d.g.577.10		12	51.47	odd	4
867.2.e.k.616.3		24	51.32	odd	8
867.2.e.k.616.4		24	51.2	odd	8
867.2.e.k.829.9		24	51.8	odd	8
867.2.e.k.829.10		24	51.26	odd	8
867.2.h.m.688.9		48	51.5	even	16
867.2.h.m.688.10		48	51.29	even	16
867.2.h.m.712.9		48	51.41	even	16
867.2.h.m.712.10		48	51.44	even	16
867.2.h.m.733.3		48	51.11	even	16
867.2.h.m.733.4		48	51.23	even	16
867.2.h.m.757.3		48	51.14	even	16
867.2.h.m.757.4		48	51.20	even	16
2601.2.a.bh.1.5		6	17.16	even	2
2601.2.a.bi.1.5		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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43915 q^{2} +0.0711653 q^{4} +2.31394 q^{5} -4.44173 q^{7} -2.77589 q^{8} +O(q^{10})\)	\(q+1.43915 q^{2} +0.0711653 q^{4} +2.31394 q^{5} -4.44173 q^{7} -2.77589 q^{8} +3.33012 q^{10} +4.52453 q^{11} +1.17466 q^{13} -6.39233 q^{14} -4.13727 q^{16} +4.86615 q^{19} +0.164673 q^{20} +6.51150 q^{22} +0.625397 q^{23} +0.354337 q^{25} +1.69052 q^{26} -0.316097 q^{28} +1.51032 q^{29} +8.73592 q^{31} -0.402383 q^{32} -10.2779 q^{35} +8.79254 q^{37} +7.00315 q^{38} -6.42326 q^{40} +0.464965 q^{41} -1.51327 q^{43} +0.321990 q^{44} +0.900043 q^{46} -6.01904 q^{47} +12.7289 q^{49} +0.509946 q^{50} +0.0835951 q^{52} -7.12324 q^{53} +10.4695 q^{55} +12.3297 q^{56} +2.17358 q^{58} +10.4025 q^{59} +2.55882 q^{61} +12.5723 q^{62} +7.69544 q^{64} +2.71810 q^{65} -3.71801 q^{67} -14.7915 q^{70} +15.2429 q^{71} +2.70186 q^{73} +12.6538 q^{74} +0.346301 q^{76} -20.0967 q^{77} +16.6972 q^{79} -9.57340 q^{80} +0.669156 q^{82} +4.45459 q^{83} -2.17782 q^{86} -12.5596 q^{88} -1.54030 q^{89} -5.21752 q^{91} +0.0445066 q^{92} -8.66233 q^{94} +11.2600 q^{95} -3.83886 q^{97} +18.3189 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 3 q^{7} - 12 q^{8}+O(q^{10}) \) 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 3 * q^7 - 12 * q^8	\( 6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 3 q^{7} - 12 q^{8} + 12 q^{10} + 9 q^{11} + 9 q^{13} + 6 q^{14} + 15 q^{16} + 9 q^{19} - 6 q^{20} - 18 q^{22} + 9 q^{23} + 15 q^{25} + 12 q^{26} - 15 q^{28} + 6 q^{29} + 24 q^{31} - 42 q^{32} - 3 q^{37} + 6 q^{38} - 3 q^{40} + 18 q^{41} + 3 q^{44} + 15 q^{46} - 24 q^{47} + 21 q^{49} - 12 q^{50} - 18 q^{52} - 24 q^{53} - 24 q^{55} + 54 q^{56} + 3 q^{58} + 9 q^{59} + 21 q^{61} + 30 q^{62} + 24 q^{64} - 9 q^{65} - 6 q^{67} - 3 q^{70} + 27 q^{71} + 18 q^{73} + 36 q^{74} - 3 q^{76} - 33 q^{77} + 24 q^{79} + 3 q^{80} - 15 q^{82} - 6 q^{83} - 6 q^{86} - 24 q^{88} + 39 q^{91} - 15 q^{94} + 42 q^{95} - 33 q^{97} + 57 q^{98}+O(q^{100}) \) 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 3 * q^7 - 12 * q^8 + 12 * q^10 + 9 * q^11 + 9 * q^13 + 6 * q^14 + 15 * q^16 + 9 * q^19 - 6 * q^20 - 18 * q^22 + 9 * q^23 + 15 * q^25 + 12 * q^26 - 15 * q^28 + 6 * q^29 + 24 * q^31 - 42 * q^32 - 3 * q^37 + 6 * q^38 - 3 * q^40 + 18 * q^41 + 3 * q^44 + 15 * q^46 - 24 * q^47 + 21 * q^49 - 12 * q^50 - 18 * q^52 - 24 * q^53 - 24 * q^55 + 54 * q^56 + 3 * q^58 + 9 * q^59 + 21 * q^61 + 30 * q^62 + 24 * q^64 - 9 * q^65 - 6 * q^67 - 3 * q^70 + 27 * q^71 + 18 * q^73 + 36 * q^74 - 3 * q^76 - 33 * q^77 + 24 * q^79 + 3 * q^80 - 15 * q^82 - 6 * q^83 - 6 * q^86 - 24 * q^88 + 39 * q^91 - 15 * q^94 + 42 * q^95 - 33 * q^97 + 57 * q^98

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