LMFDB - Embedded newform 1425.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.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:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14510 q^{2} +1.00000 q^{3} +2.60147 q^{4} +2.14510 q^{6} +2.74657 q^{7} +1.29021 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.14510 q^{2} +1.00000 q^{3} +2.60147 q^{4} +2.14510 q^{6} +2.74657 q^{7} +1.29021 q^{8} +1.00000 q^{9} -3.74657 q^{11} +2.60147 q^{12} +6.29021 q^{13} +5.89167 q^{14} -2.43531 q^{16} +2.14510 q^{18} +1.00000 q^{19} +2.74657 q^{21} -8.03677 q^{22} +0.543637 q^{23} +1.29021 q^{24} +13.4931 q^{26} +1.00000 q^{27} +7.14510 q^{28} +3.00000 q^{29} +1.45636 q^{31} -7.80440 q^{32} -3.74657 q^{33} +2.60147 q^{36} +5.20293 q^{37} +2.14510 q^{38} +6.29021 q^{39} -12.5299 q^{41} +5.89167 q^{42} +8.00000 q^{43} -9.74657 q^{44} +1.16616 q^{46} -11.7833 q^{47} -2.43531 q^{48} +0.543637 q^{49} +16.3638 q^{52} -5.58041 q^{53} +2.14510 q^{54} +3.54364 q^{56} +1.00000 q^{57} +6.43531 q^{58} +5.25343 q^{59} -8.49314 q^{61} +3.12405 q^{62} +2.74657 q^{63} -11.8706 q^{64} -8.03677 q^{66} -2.83384 q^{67} +0.543637 q^{69} +7.83384 q^{71} +1.29021 q^{72} +7.58041 q^{73} +11.1608 q^{74} +2.60147 q^{76} -10.2902 q^{77} +13.4931 q^{78} -7.52991 q^{79} +1.00000 q^{81} -26.8779 q^{82} -2.25343 q^{83} +7.14510 q^{84} +17.1608 q^{86} +3.00000 q^{87} -4.83384 q^{88} +4.49314 q^{89} +17.2765 q^{91} +1.41425 q^{92} +1.45636 q^{93} -25.2765 q^{94} -7.80440 q^{96} -16.8706 q^{97} +1.16616 q^{98} -3.74657 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9	$ 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 6 q^{13} + 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} - 3 q^{23} - 9 q^{24} + 24 q^{26} + 3 q^{27} + 15 q^{28} + 9 q^{29} + 9 q^{31} - 18 q^{32} - 3 q^{33} + 6 q^{36} + 12 q^{37} + 6 q^{39} + 3 q^{42} + 24 q^{43} - 21 q^{44} + 21 q^{46} - 6 q^{47} + 12 q^{48} - 3 q^{49} - 6 q^{52} + 9 q^{53} + 6 q^{56} + 3 q^{57} + 24 q^{59} - 9 q^{61} - 21 q^{62} + 3 q^{64} - 3 q^{66} + 9 q^{67} - 3 q^{69} + 6 q^{71} - 9 q^{72} - 3 q^{73} - 18 q^{74} + 6 q^{76} - 18 q^{77} + 24 q^{78} + 15 q^{79} + 3 q^{81} - 33 q^{82} - 15 q^{83} + 15 q^{84} + 9 q^{87} + 3 q^{88} - 3 q^{89} + 6 q^{91} - 39 q^{92} + 9 q^{93} - 30 q^{94} - 18 q^{96} - 12 q^{97} + 21 q^{98} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9 - 3 * q^11 + 6 * q^12 + 6 * q^13 + 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 - 3 * q^23 - 9 * q^24 + 24 * q^26 + 3 * q^27 + 15 * q^28 + 9 * q^29 + 9 * q^31 - 18 * q^32 - 3 * q^33 + 6 * q^36 + 12 * q^37 + 6 * q^39 + 3 * q^42 + 24 * q^43 - 21 * q^44 + 21 * q^46 - 6 * q^47 + 12 * q^48 - 3 * q^49 - 6 * q^52 + 9 * q^53 + 6 * q^56 + 3 * q^57 + 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 3 * q^66 + 9 * q^67 - 3 * q^69 + 6 * q^71 - 9 * q^72 - 3 * q^73 - 18 * q^74 + 6 * q^76 - 18 * q^77 + 24 * q^78 + 15 * q^79 + 3 * q^81 - 33 * q^82 - 15 * q^83 + 15 * q^84 + 9 * q^87 + 3 * q^88 - 3 * q^89 + 6 * q^91 - 39 * q^92 + 9 * q^93 - 30 * q^94 - 18 * q^96 - 12 * q^97 + 21 * q^98 - 3 * q^99

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 $

$2$	2.14510	1.51682	0.758408	−	0.651780i	$-0.225977\pi$
$2$	2.14510	1.51682	0.758408	+	0.651780i	$0.225977\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.60147	1.30073
$5$	0	0
$5$	0	0
$6$	2.14510	0.875735
$7$	2.74657	1.03811	0.519053	−	0.854742i	$-0.326285\pi$
$7$	2.74657	1.03811	0.519053	+	0.854742i	$0.326285\pi$
$8$	1.29021	0.456156
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.74657	−1.12963	−0.564816	−	0.825217i	$-0.691053\pi$
$11$	−3.74657	−1.12963	−0.564816	+	0.825217i	$0.691053\pi$
$12$	2.60147	0.750978
$13$	6.29021	1.74459	0.872295	−	0.488981i	$-0.162631\pi$
$13$	6.29021	1.74459	0.872295	+	0.488981i	$0.162631\pi$
$14$	5.89167	1.57462
$15$	0	0
$16$	−2.43531	−0.608827
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	2.14510	0.505606
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.74657	0.599350
$22$	−8.03677	−1.71345
$23$	0.543637	0.113356	0.0566781	−	0.998393i	$-0.481949\pi$
$23$	0.543637	0.113356	0.0566781	+	0.998393i	$0.481949\pi$
$24$	1.29021	0.263362
$25$	0	0
$26$	13.4931	2.64622
$27$	1.00000	0.192450
$28$	7.14510	1.35030
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	1.45636	0.261570	0.130785	−	0.991411i	$-0.458250\pi$
$31$	1.45636	0.261570	0.130785	+	0.991411i	$0.458250\pi$
$32$	−7.80440	−1.37964
$33$	−3.74657	−0.652194
$34$	0	0
$35$	0	0
$36$	2.60147	0.433578
$37$	5.20293	0.855357	0.427678	−	0.903931i	$-0.359332\pi$
$37$	5.20293	0.855357	0.427678	+	0.903931i	$0.359332\pi$
$38$	2.14510	0.347982
$39$	6.29021	1.00724
$40$	0	0
$41$	−12.5299	−1.95684	−0.978422	−	0.206618i	$-0.933754\pi$
$41$	−12.5299	−1.95684	−0.978422	+	0.206618i	$0.933754\pi$
$42$	5.89167	0.909105
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−9.74657	−1.46935
$45$	0	0
$46$	1.16616	0.171941
$47$	−11.7833	−1.71878	−0.859389	−	0.511323i	$-0.829156\pi$
$47$	−11.7833	−1.71878	−0.859389	+	0.511323i	$0.829156\pi$
$48$	−2.43531	−0.351506
$49$	0.543637	0.0776624
$50$	0	0
$51$	0	0
$52$	16.3638	2.26924
$53$	−5.58041	−0.766528	−0.383264	−	0.923639i	$-0.625200\pi$
$53$	−5.58041	−0.766528	−0.383264	+	0.923639i	$0.625200\pi$
$54$	2.14510	0.291912
$55$	0	0
$56$	3.54364	0.473538
$57$	1.00000	0.132453
$58$	6.43531	0.844997
$59$	5.25343	0.683939	0.341969	−	0.939711i	$-0.388906\pi$
$59$	5.25343	0.683939	0.341969	+	0.939711i	$0.388906\pi$
$60$	0	0
$61$	−8.49314	−1.08743	−0.543717	−	0.839268i	$-0.682984\pi$
$61$	−8.49314	−1.08743	−0.543717	+	0.839268i	$0.682984\pi$
$62$	3.12405	0.396754
$63$	2.74657	0.346035
$64$	−11.8706	−1.48383
$65$	0	0
$66$	−8.03677	−0.989258
$67$	−2.83384	−0.346209	−0.173104	−	0.984903i	$-0.555380\pi$
$67$	−2.83384	−0.346209	−0.173104	+	0.984903i	$0.555380\pi$
$68$	0	0
$69$	0.543637	0.0654462
$70$	0	0
$71$	7.83384	0.929706	0.464853	−	0.885388i	$-0.346107\pi$
$71$	7.83384	0.929706	0.464853	+	0.885388i	$0.346107\pi$
$72$	1.29021	0.152052
$73$	7.58041	0.887220	0.443610	−	0.896220i	$-0.353698\pi$
$73$	7.58041	0.887220	0.443610	+	0.896220i	$0.353698\pi$
$74$	11.1608	1.29742
$75$	0	0
$76$	2.60147	0.298409
$77$	−10.2902	−1.17268
$78$	13.4931	1.52780
$79$	−7.52991	−0.847181	−0.423591	−	0.905854i	$-0.639230\pi$
$79$	−7.52991	−0.847181	−0.423591	+	0.905854i	$0.639230\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−26.8779	−2.96817
$83$	−2.25343	−0.247346	−0.123673	−	0.992323i	$-0.539467\pi$
$83$	−2.25343	−0.247346	−0.123673	+	0.992323i	$0.539467\pi$
$84$	7.14510	0.779595
$85$	0	0
$86$	17.1608	1.85050
$87$	3.00000	0.321634
$88$	−4.83384	−0.515289
$89$	4.49314	0.476272	0.238136	−	0.971232i	$-0.423464\pi$
$89$	4.49314	0.476272	0.238136	+	0.971232i	$0.423464\pi$
$90$	0	0
$91$	17.2765	1.81107
$92$	1.41425	0.147446
$93$	1.45636	0.151018
$94$	−25.2765	−2.60707
$95$	0	0
$96$	−7.80440	−0.796533
$97$	−16.8706	−1.71295	−0.856476	−	0.516187i	$-0.827351\pi$
$97$	−16.8706	−1.71295	−0.856476	+	0.516187i	$0.827351\pi$
$98$	1.16616	0.117800
$99$	−3.74657	−0.376544
$100$	0	0
$101$	−13.2765	−1.32106	−0.660529	−	0.750800i	$-0.729668\pi$
$101$	−13.2765	−1.32106	−0.660529	+	0.750800i	$0.729668\pi$
$102$	0	0
$103$	−0.253432	−0.0249714	−0.0124857	−	0.999922i	$-0.503974\pi$
$103$	−0.253432	−0.0249714	−0.0124857	+	0.999922i	$0.503974\pi$
$104$	8.11566	0.795806
$105$	0	0
$106$	−11.9706	−1.16268
$107$	2.23970	0.216520	0.108260	−	0.994123i	$-0.465472\pi$
$107$	2.23970	0.216520	0.108260	+	0.994123i	$0.465472\pi$
$108$	2.60147	0.250326
$109$	−1.20293	−0.115220	−0.0576100	−	0.998339i	$-0.518348\pi$
$109$	−1.20293	−0.115220	−0.0576100	+	0.998339i	$0.518348\pi$
$110$	0	0
$111$	5.20293	0.493840
$112$	−6.68874	−0.632027
$113$	−12.5436	−1.18001	−0.590003	−	0.807401i	$-0.700873\pi$
$113$	−12.5436	−1.18001	−0.590003	+	0.807401i	$0.700873\pi$
$114$	2.14510	0.200907
$115$	0	0
$116$	7.80440	0.724620
$117$	6.29021	0.581530
$118$	11.2692	1.03741
$119$	0	0
$120$	0	0
$121$	3.03677	0.276070
$122$	−18.2186	−1.64944
$123$	−12.5299	−1.12978
$124$	3.78868	0.340233
$125$	0	0
$126$	5.89167	0.524872
$127$	−12.4426	−1.10411	−0.552053	−	0.833809i	$-0.686155\pi$
$127$	−12.4426	−1.10411	−0.552053	+	0.833809i	$0.686155\pi$
$128$	−9.85490	−0.871058
$129$	8.00000	0.704361
$130$	0	0
$131$	18.9495	1.65563	0.827813	−	0.561005i	$-0.189585\pi$
$131$	18.9495	1.65563	0.827813	+	0.561005i	$0.189585\pi$
$132$	−9.74657	−0.848330
$133$	2.74657	0.238158
$134$	−6.07888	−0.525136
$135$	0	0
$136$	0	0
$137$	21.3921	1.82765	0.913827	−	0.406104i	$-0.133113\pi$
$137$	21.3921	1.82765	0.913827	+	0.406104i	$0.133113\pi$
$138$	1.16616	0.0992699
$139$	20.3133	1.72295	0.861474	−	0.507802i	$-0.169542\pi$
$139$	20.3133	1.72295	0.861474	+	0.507802i	$0.169542\pi$
$140$	0	0
$141$	−11.7833	−0.992336
$142$	16.8044	1.41019
$143$	−23.5667	−1.97075
$144$	−2.43531	−0.202942
$145$	0	0
$146$	16.2608	1.34575
$147$	0.543637	0.0448384
$148$	13.5352	1.11259
$149$	4.69607	0.384717	0.192358	−	0.981325i	$-0.438386\pi$
$149$	4.69607	0.384717	0.192358	+	0.981325i	$0.438386\pi$
$150$	0	0
$151$	1.59414	0.129729	0.0648645	−	0.997894i	$-0.479338\pi$
$151$	1.59414	0.129729	0.0648645	+	0.997894i	$0.479338\pi$
$152$	1.29021	0.104649
$153$	0	0
$154$	−22.0735	−1.77874
$155$	0	0
$156$	16.3638	1.31015
$157$	−13.1103	−1.04632	−0.523159	−	0.852235i	$-0.675247\pi$
$157$	−13.1103	−1.04632	−0.523159	+	0.852235i	$0.675247\pi$
$158$	−16.1524	−1.28502
$159$	−5.58041	−0.442555
$160$	0	0
$161$	1.49314	0.117676
$162$	2.14510	0.168535
$163$	−7.73284	−0.605683	−0.302841	−	0.953041i	$-0.597935\pi$
$163$	−7.73284	−0.605683	−0.302841	+	0.953041i	$0.597935\pi$
$164$	−32.5961	−2.54533
$165$	0	0
$166$	−4.83384	−0.375179
$167$	−9.32698	−0.721743	−0.360872	−	0.932615i	$-0.617521\pi$
$167$	−9.32698	−0.721743	−0.360872	+	0.932615i	$0.617521\pi$
$168$	3.54364	0.273398
$169$	26.5667	2.04359
$170$	0	0
$171$	1.00000	0.0764719
$172$	20.8117	1.58688
$173$	−5.17455	−0.393414	−0.196707	−	0.980462i	$-0.563025\pi$
$173$	−5.17455	−0.393414	−0.196707	+	0.980462i	$0.563025\pi$
$174$	6.43531	0.487859
$175$	0	0
$176$	9.12405	0.687751
$177$	5.25343	0.394872
$178$	9.63824	0.722417
$179$	15.3270	1.14559	0.572796	−	0.819698i	$-0.305859\pi$
$179$	15.3270	1.14559	0.572796	+	0.819698i	$0.305859\pi$
$180$	0	0
$181$	−10.4059	−0.773462	−0.386731	−	0.922193i	$-0.626396\pi$
$181$	−10.4059	−0.773462	−0.386731	+	0.922193i	$0.626396\pi$
$182$	37.0598	2.74706
$183$	−8.49314	−0.627831
$184$	0.701404	0.0517082
$185$	0	0
$186$	3.12405	0.229066
$187$	0	0
$188$	−30.6540	−2.23567
$189$	2.74657	0.199783
$190$	0	0
$191$	−6.54364	−0.473481	−0.236740	−	0.971573i	$-0.576079\pi$
$191$	−6.54364	−0.473481	−0.236740	+	0.971573i	$0.576079\pi$
$192$	−11.8706	−0.856688
$193$	−24.9863	−1.79855	−0.899276	−	0.437382i	$-0.855906\pi$
$193$	−24.9863	−1.79855	−0.899276	+	0.437382i	$0.855906\pi$
$194$	−36.1892	−2.59823
$195$	0	0
$196$	1.41425	0.101018
$197$	8.79707	0.626765	0.313383	−	0.949627i	$-0.398538\pi$
$197$	8.79707	0.626765	0.313383	+	0.949627i	$0.398538\pi$
$198$	−8.03677	−0.571149
$199$	8.74657	0.620028	0.310014	−	0.950732i	$-0.399666\pi$
$199$	8.74657	0.620028	0.310014	+	0.950732i	$0.399666\pi$
$200$	0	0
$201$	−2.83384	−0.199884
$202$	−28.4794	−2.00380
$203$	8.23970	0.578314
$204$	0	0
$205$	0	0
$206$	−0.543637	−0.0378770
$207$	0.543637	0.0377854
$208$	−15.3186	−1.06215
$209$	−3.74657	−0.259156
$210$	0	0
$211$	−12.7182	−0.875556	−0.437778	−	0.899083i	$-0.644234\pi$
$211$	−12.7182	−0.875556	−0.437778	+	0.899083i	$0.644234\pi$
$212$	−14.5172	−0.997049
$213$	7.83384	0.536766
$214$	4.80440	0.328422
$215$	0	0
$216$	1.29021	0.0877874
$217$	4.00000	0.271538
$218$	−2.58041	−0.174767
$219$	7.58041	0.512237
$220$	0	0
$221$	0	0
$222$	11.1608	0.749065
$223$	−10.5436	−0.706054	−0.353027	−	0.935613i	$-0.614848\pi$
$223$	−10.5436	−0.706054	−0.353027	+	0.935613i	$0.614848\pi$
$224$	−21.4353	−1.43221
$225$	0	0
$226$	−26.9074	−1.78985
$227$	7.83384	0.519950	0.259975	−	0.965615i	$-0.416286\pi$
$227$	7.83384	0.519950	0.259975	+	0.965615i	$0.416286\pi$
$228$	2.60147	0.172286
$229$	−23.6035	−1.55976	−0.779880	−	0.625929i	$-0.784720\pi$
$229$	−23.6035	−1.55976	−0.779880	+	0.625929i	$0.784720\pi$
$230$	0	0
$231$	−10.2902	−0.677046
$232$	3.87062	0.254118
$233$	−23.7833	−1.55810	−0.779049	−	0.626963i	$-0.784298\pi$
$233$	−23.7833	−1.55810	−0.779049	+	0.626963i	$0.784298\pi$
$234$	13.4931	0.882074
$235$	0	0
$236$	13.6666	0.889621
$237$	−7.52991	−0.489120
$238$	0	0
$239$	−1.27648	−0.0825685	−0.0412843	−	0.999147i	$-0.513145\pi$
$239$	−1.27648	−0.0825685	−0.0412843	+	0.999147i	$0.513145\pi$
$240$	0	0
$241$	21.4931	1.38449	0.692247	−	0.721660i	$-0.256621\pi$
$241$	21.4931	1.38449	0.692247	+	0.721660i	$0.256621\pi$
$242$	6.51419	0.418748
$243$	1.00000	0.0641500
$244$	−22.0946	−1.41446
$245$	0	0
$246$	−26.8779	−1.71368
$247$	6.29021	0.400236
$248$	1.87901	0.119317
$249$	−2.25343	−0.142805
$250$	0	0
$251$	−29.9725	−1.89185	−0.945925	−	0.324385i	$-0.894843\pi$
$251$	−29.9725	−1.89185	−0.945925	+	0.324385i	$0.894843\pi$
$252$	7.14510	0.450099
$253$	−2.03677	−0.128051
$254$	−26.6907	−1.67473
$255$	0	0
$256$	2.60147	0.162592
$257$	19.0735	1.18978	0.594888	−	0.803809i	$-0.297197\pi$
$257$	19.0735	1.18978	0.594888	+	0.803809i	$0.297197\pi$
$258$	17.1608	1.06839
$259$	14.2902	0.887950
$260$	0	0
$261$	3.00000	0.185695
$262$	40.6486	2.51128
$263$	11.6456	0.718096	0.359048	−	0.933319i	$-0.383101\pi$
$263$	11.6456	0.718096	0.359048	+	0.933319i	$0.383101\pi$
$264$	−4.83384	−0.297502
$265$	0	0
$266$	5.89167	0.361242
$267$	4.49314	0.274975
$268$	−7.37214	−0.450325
$269$	26.1471	1.59422	0.797108	−	0.603836i	$-0.206362\pi$
$269$	26.1471	1.59422	0.797108	+	0.603836i	$0.206362\pi$
$270$	0	0
$271$	15.8338	0.961837	0.480919	−	0.876765i	$-0.340303\pi$
$271$	15.8338	0.961837	0.480919	+	0.876765i	$0.340303\pi$
$272$	0	0
$273$	17.2765	1.04562
$274$	45.8883	2.77222
$275$	0	0
$276$	1.41425	0.0851280
$277$	20.6677	1.24180	0.620900	−	0.783889i	$-0.286767\pi$
$277$	20.6677	1.24180	0.620900	+	0.783889i	$0.286767\pi$
$278$	43.5740	2.61340
$279$	1.45636	0.0871902
$280$	0	0
$281$	1.35536	0.0808541	0.0404270	−	0.999182i	$-0.487128\pi$
$281$	1.35536	0.0808541	0.0404270	+	0.999182i	$0.487128\pi$
$282$	−25.2765	−1.50519
$283$	28.1471	1.67317	0.836586	−	0.547836i	$-0.184548\pi$
$283$	28.1471	1.67317	0.836586	+	0.547836i	$0.184548\pi$
$284$	20.3795	1.20930
$285$	0	0
$286$	−50.5530	−2.98926
$287$	−34.4143	−2.03141
$288$	−7.80440	−0.459878
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−16.8706	−0.988973
$292$	19.7202	1.15404
$293$	−8.47009	−0.494828	−0.247414	−	0.968910i	$-0.579581\pi$
$293$	−8.47009	−0.494828	−0.247414	+	0.968910i	$0.579581\pi$
$294$	1.16616	0.0680117
$295$	0	0
$296$	6.71285	0.390176
$297$	−3.74657	−0.217398
$298$	10.0735	0.583545
$299$	3.41959	0.197760
$300$	0	0
$301$	21.9725	1.26648
$302$	3.41959	0.196775
$303$	−13.2765	−0.762714
$304$	−2.43531	−0.139674
$305$	0	0
$306$	0	0
$307$	−19.3133	−1.10227	−0.551133	−	0.834418i	$-0.685804\pi$
$307$	−19.3133	−1.10227	−0.551133	+	0.834418i	$0.685804\pi$
$308$	−26.7696	−1.52534
$309$	−0.253432	−0.0144172
$310$	0	0
$311$	20.5804	1.16701	0.583504	−	0.812110i	$-0.301681\pi$
$311$	20.5804	1.16701	0.583504	+	0.812110i	$0.301681\pi$
$312$	8.11566	0.459459
$313$	22.4426	1.26853	0.634266	−	0.773115i	$-0.281302\pi$
$313$	22.4426	1.26853	0.634266	+	0.773115i	$0.281302\pi$
$314$	−28.1230	−1.58707
$315$	0	0
$316$	−19.5888	−1.10196
$317$	11.1745	0.627625	0.313813	−	0.949485i	$-0.398394\pi$
$317$	11.1745	0.627625	0.313813	+	0.949485i	$0.398394\pi$
$318$	−11.9706	−0.671275
$319$	−11.2397	−0.629303
$320$	0	0
$321$	2.23970	0.125008
$322$	3.20293	0.178492
$323$	0	0
$324$	2.60147	0.144526
$325$	0	0
$326$	−16.5877	−0.918710
$327$	−1.20293	−0.0665222
$328$	−16.1662	−0.892627
$329$	−32.3638	−1.78427
$330$	0	0
$331$	−25.9358	−1.42556	−0.712779	−	0.701388i	$-0.752564\pi$
$331$	−25.9358	−1.42556	−0.712779	+	0.701388i	$0.752564\pi$
$332$	−5.86223	−0.321731
$333$	5.20293	0.285119
$334$	−20.0073	−1.09475
$335$	0	0
$336$	−6.68874	−0.364901
$337$	15.0873	0.821856	0.410928	−	0.911668i	$-0.365205\pi$
$337$	15.0873	0.821856	0.410928	+	0.911668i	$0.365205\pi$
$338$	56.9883	3.09975
$339$	−12.5436	−0.681277
$340$	0	0
$341$	−5.45636	−0.295479
$342$	2.14510	0.115994
$343$	−17.7328	−0.957483
$344$	10.3216	0.556506
$345$	0	0
$346$	−11.0999	−0.596736
$347$	−6.62252	−0.355516	−0.177758	−	0.984074i	$-0.556884\pi$
$347$	−6.62252	−0.355516	−0.177758	+	0.984074i	$0.556884\pi$
$348$	7.80440	0.418360
$349$	9.07355	0.485696	0.242848	−	0.970064i	$-0.421918\pi$
$349$	9.07355	0.485696	0.242848	+	0.970064i	$0.421918\pi$
$350$	0	0
$351$	6.29021	0.335746
$352$	29.2397	1.55848
$353$	20.1471	1.07232	0.536161	−	0.844116i	$-0.319874\pi$
$353$	20.1471	1.07232	0.536161	+	0.844116i	$0.319874\pi$
$354$	11.2692	0.598949
$355$	0	0
$356$	11.6887	0.619502
$357$	0	0
$358$	32.8779	1.73765
$359$	15.8843	0.838344	0.419172	−	0.907907i	$-0.362321\pi$
$359$	15.8843	0.838344	0.419172	+	0.907907i	$0.362321\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−22.3216	−1.17320
$363$	3.03677	0.159389
$364$	44.9442	2.35571
$365$	0	0
$366$	−18.2186	−0.952304
$367$	24.8853	1.29900	0.649500	−	0.760361i	$-0.274978\pi$
$367$	24.8853	1.29900	0.649500	+	0.760361i	$0.274978\pi$
$368$	−1.32392	−0.0690143
$369$	−12.5299	−0.652281
$370$	0	0
$371$	−15.3270	−0.795737
$372$	3.78868	0.196434
$373$	11.6677	0.604130	0.302065	−	0.953287i	$-0.402324\pi$
$373$	11.6677	0.604130	0.302065	+	0.953287i	$0.402324\pi$
$374$	0	0
$375$	0	0
$376$	−15.2029	−0.784031
$377$	18.8706	0.971886
$378$	5.89167	0.303035
$379$	6.29021	0.323106	0.161553	−	0.986864i	$-0.448350\pi$
$379$	6.29021	0.323106	0.161553	+	0.986864i	$0.448350\pi$
$380$	0	0
$381$	−12.4426	−0.637456
$382$	−14.0368	−0.718184
$383$	12.0652	0.616501	0.308250	−	0.951305i	$-0.400257\pi$
$383$	12.0652	0.616501	0.308250	+	0.951305i	$0.400257\pi$
$384$	−9.85490	−0.502906
$385$	0	0
$386$	−53.5981	−2.72807
$387$	8.00000	0.406663
$388$	−43.8883	−2.22809
$389$	−14.3638	−0.728271	−0.364136	−	0.931346i	$-0.618636\pi$
$389$	−14.3638	−0.728271	−0.364136	+	0.931346i	$0.618636\pi$
$390$	0	0
$391$	0	0
$392$	0.701404	0.0354262
$393$	18.9495	0.955876
$394$	18.8706	0.950688
$395$	0	0
$396$	−9.74657	−0.489783
$397$	−31.0966	−1.56069	−0.780347	−	0.625347i	$-0.784957\pi$
$397$	−31.0966	−1.56069	−0.780347	+	0.625347i	$0.784957\pi$
$398$	18.7623	0.940468
$399$	2.74657	0.137500
$400$	0	0
$401$	−0.543637	−0.0271479	−0.0135740	−	0.999908i	$-0.504321\pi$
$401$	−0.543637	−0.0271479	−0.0135740	+	0.999908i	$0.504321\pi$
$402$	−6.07888	−0.303187
$403$	9.16082	0.456333
$404$	−34.5383	−1.71834
$405$	0	0
$406$	17.6750	0.877196
$407$	−19.4931	−0.966239
$408$	0	0
$409$	26.4648	1.30860	0.654299	−	0.756236i	$-0.272964\pi$
$409$	26.4648	1.30860	0.654299	+	0.756236i	$0.272964\pi$
$410$	0	0
$411$	21.3921	1.05520
$412$	−0.659294	−0.0324811
$413$	14.4289	0.710000
$414$	1.16616	0.0573135
$415$	0	0
$416$	−49.0913	−2.40690
$417$	20.3133	0.994744
$418$	−8.03677	−0.393091
$419$	−6.18920	−0.302362	−0.151181	−	0.988506i	$-0.548308\pi$
$419$	−6.18920	−0.302362	−0.151181	+	0.988506i	$0.548308\pi$
$420$	0	0
$421$	15.2765	0.744530	0.372265	−	0.928126i	$-0.378581\pi$
$421$	15.2765	0.744530	0.372265	+	0.928126i	$0.378581\pi$
$422$	−27.2818	−1.32806
$423$	−11.7833	−0.572926
$424$	−7.19988	−0.349657
$425$	0	0
$426$	16.8044	0.814176
$427$	−23.3270	−1.12887
$428$	5.82651	0.281635
$429$	−23.5667	−1.13781
$430$	0	0
$431$	13.5583	0.653080	0.326540	−	0.945183i	$-0.394117\pi$
$431$	13.5583	0.653080	0.326540	+	0.945183i	$0.394117\pi$
$432$	−2.43531	−0.117169
$433$	−29.6823	−1.42644	−0.713221	−	0.700939i	$-0.752764\pi$
$433$	−29.6823	−1.42644	−0.713221	+	0.700939i	$0.752764\pi$
$434$	8.58041	0.411873
$435$	0	0
$436$	−3.12938	−0.149870
$437$	0.543637	0.0260057
$438$	16.2608	0.776969
$439$	34.0093	1.62318	0.811588	−	0.584230i	$-0.198603\pi$
$439$	34.0093	1.62318	0.811588	+	0.584230i	$0.198603\pi$
$440$	0	0
$441$	0.543637	0.0258875
$442$	0	0
$443$	−16.3416	−0.776414	−0.388207	−	0.921572i	$-0.626906\pi$
$443$	−16.3416	−0.776414	−0.388207	+	0.921572i	$0.626906\pi$
$444$	13.5352	0.642354
$445$	0	0
$446$	−22.6172	−1.07095
$447$	4.69607	0.222116
$448$	−32.6035	−1.54037
$449$	16.8990	0.797513	0.398757	−	0.917057i	$-0.369442\pi$
$449$	16.8990	0.797513	0.398757	+	0.917057i	$0.369442\pi$
$450$	0	0
$451$	46.9442	2.21051
$452$	−32.6318	−1.53487
$453$	1.59414	0.0748991
$454$	16.8044	0.788669
$455$	0	0
$456$	1.29021	0.0604194
$457$	16.8622	0.788782	0.394391	−	0.918943i	$-0.370956\pi$
$457$	16.8622	0.788782	0.394391	+	0.918943i	$0.370956\pi$
$458$	−50.6318	−2.36587
$459$	0	0
$460$	0	0
$461$	−24.4648	−1.13944	−0.569719	−	0.821840i	$-0.692948\pi$
$461$	−24.4648	−1.13944	−0.569719	+	0.821840i	$0.692948\pi$
$462$	−22.0735	−1.02695
$463$	−14.7550	−0.685721	−0.342861	−	0.939386i	$-0.611396\pi$
$463$	−14.7550	−0.685721	−0.342861	+	0.939386i	$0.611396\pi$
$464$	−7.30592	−0.339169
$465$	0	0
$466$	−51.0177	−2.36335
$467$	12.3270	0.570425	0.285212	−	0.958464i	$-0.407936\pi$
$467$	12.3270	0.570425	0.285212	+	0.958464i	$0.407936\pi$
$468$	16.3638	0.756415
$469$	−7.78334	−0.359401
$470$	0	0
$471$	−13.1103	−0.604092
$472$	6.77801	0.311983
$473$	−29.9725	−1.37814
$474$	−16.1524	−0.741906
$475$	0	0
$476$	0	0
$477$	−5.58041	−0.255509
$478$	−2.73818	−0.125241
$479$	−19.0084	−0.868516	−0.434258	−	0.900789i	$-0.642989\pi$
$479$	−19.0084	−0.868516	−0.434258	+	0.900789i	$0.642989\pi$
$480$	0	0
$481$	32.7275	1.49225
$482$	46.1050	2.10002
$483$	1.49314	0.0679401
$484$	7.90006	0.359094
$485$	0	0
$486$	2.14510	0.0973038
$487$	33.2765	1.50790	0.753951	−	0.656931i	$-0.228146\pi$
$487$	33.2765	1.50790	0.753951	+	0.656931i	$0.228146\pi$
$488$	−10.9579	−0.496040
$489$	−7.73284	−0.349691
$490$	0	0
$491$	28.7275	1.29645	0.648227	−	0.761447i	$-0.275511\pi$
$491$	28.7275	1.29645	0.648227	+	0.761447i	$0.275511\pi$
$492$	−32.5961	−1.46955
$493$	0	0
$494$	13.4931	0.607085
$495$	0	0
$496$	−3.54669	−0.159251
$497$	21.5162	0.965133
$498$	−4.83384	−0.216610
$499$	−24.4878	−1.09622	−0.548112	−	0.836405i	$-0.684653\pi$
$499$	−24.4878	−1.09622	−0.548112	+	0.836405i	$0.684653\pi$
$500$	0	0
$501$	−9.32698	−0.416699
$502$	−64.2942	−2.86959
$503$	−7.30393	−0.325666	−0.162833	−	0.986654i	$-0.552063\pi$
$503$	−7.30393	−0.325666	−0.162833	+	0.986654i	$0.552063\pi$
$504$	3.54364	0.157846
$505$	0	0
$506$	−4.36909	−0.194230
$507$	26.5667	1.17987
$508$	−32.3691	−1.43615
$509$	33.3784	1.47947	0.739736	−	0.672897i	$-0.234950\pi$
$509$	33.3784	1.47947	0.739736	+	0.672897i	$0.234950\pi$
$510$	0	0
$511$	20.8201	0.921028
$512$	25.2902	1.11768
$513$	1.00000	0.0441511
$514$	40.9147	1.80467
$515$	0	0
$516$	20.8117	0.916185
$517$	44.1471	1.94159
$518$	30.6540	1.34686
$519$	−5.17455	−0.227137
$520$	0	0
$521$	−4.08727	−0.179067	−0.0895334	−	0.995984i	$-0.528538\pi$
$521$	−4.08727	−0.179067	−0.0895334	+	0.995984i	$0.528538\pi$
$522$	6.43531	0.281666
$523$	27.9304	1.22131	0.610656	−	0.791896i	$-0.290906\pi$
$523$	27.9304	1.22131	0.610656	+	0.791896i	$0.290906\pi$
$524$	49.2965	2.15353
$525$	0	0
$526$	24.9809	1.08922
$527$	0	0
$528$	9.12405	0.397073
$529$	−22.7045	−0.987150
$530$	0	0
$531$	5.25343	0.227980
$532$	7.14510	0.309779
$533$	−78.8157	−3.41389
$534$	9.63824	0.417087
$535$	0	0
$536$	−3.65624	−0.157925
$537$	15.3270	0.661408
$538$	56.0882	2.41813
$539$	−2.03677	−0.0877301
$540$	0	0
$541$	26.7917	1.15187	0.575933	−	0.817497i	$-0.304639\pi$
$541$	26.7917	1.15187	0.575933	+	0.817497i	$0.304639\pi$
$542$	33.9652	1.45893
$543$	−10.4059	−0.446558
$544$	0	0
$545$	0	0
$546$	37.0598	1.58601
$547$	−9.26716	−0.396235	−0.198118	−	0.980178i	$-0.563483\pi$
$547$	−9.26716	−0.396235	−0.198118	+	0.980178i	$0.563483\pi$
$548$	55.6509	2.37729
$549$	−8.49314	−0.362478
$550$	0	0
$551$	3.00000	0.127804
$552$	0.701404	0.0298537
$553$	−20.6814	−0.879463
$554$	44.3343	1.88358
$555$	0	0
$556$	52.8442	2.24109
$557$	−32.5804	−1.38048	−0.690238	−	0.723582i	$-0.742494\pi$
$557$	−32.5804	−1.38048	−0.690238	+	0.723582i	$0.742494\pi$
$558$	3.12405	0.132251
$559$	50.3216	2.12838
$560$	0	0
$561$	0	0
$562$	2.90739	0.122641
$563$	33.3270	1.40456	0.702282	−	0.711899i	$-0.252164\pi$
$563$	33.3270	1.40456	0.702282	+	0.711899i	$0.252164\pi$
$564$	−30.6540	−1.29076
$565$	0	0
$566$	60.3784	2.53789
$567$	2.74657	0.115345
$568$	10.1073	0.424091
$569$	−27.3859	−1.14808	−0.574038	−	0.818829i	$-0.694624\pi$
$569$	−27.3859	−1.14808	−0.574038	+	0.818829i	$0.694624\pi$
$570$	0	0
$571$	−43.2995	−1.81203	−0.906014	−	0.423247i	$-0.860890\pi$
$571$	−43.2995	−1.81203	−0.906014	+	0.423247i	$0.860890\pi$
$572$	−61.3079	−2.56341
$573$	−6.54364	−0.273364
$574$	−73.8221	−3.08128
$575$	0	0
$576$	−11.8706	−0.494609
$577$	47.3721	1.97213	0.986064	−	0.166366i	$-0.0532033\pi$
$577$	47.3721	1.97213	0.986064	+	0.166366i	$0.0532033\pi$
$578$	−36.4667	−1.51682
$579$	−24.9863	−1.03839
$580$	0	0
$581$	−6.18920	−0.256771
$582$	−36.1892	−1.50009
$583$	20.9074	0.865896
$584$	9.78029	0.404711
$585$	0	0
$586$	−18.1692	−0.750563
$587$	−29.0230	−1.19791	−0.598955	−	0.800783i	$-0.704417\pi$
$587$	−29.0230	−1.19791	−0.598955	+	0.800783i	$0.704417\pi$
$588$	1.41425	0.0583228
$589$	1.45636	0.0600084
$590$	0	0
$591$	8.79707	0.361863
$592$	−12.6707	−0.520764
$593$	30.8706	1.26770	0.633852	−	0.773454i	$-0.281473\pi$
$593$	30.8706	1.26770	0.633852	+	0.773454i	$0.281473\pi$
$594$	−8.03677	−0.329753
$595$	0	0
$596$	12.2167	0.500414
$597$	8.74657	0.357973
$598$	7.33537	0.299966
$599$	17.1608	0.701172	0.350586	−	0.936531i	$-0.385982\pi$
$599$	17.1608	0.701172	0.350586	+	0.936531i	$0.385982\pi$
$600$	0	0
$601$	−13.2029	−0.538559	−0.269279	−	0.963062i	$-0.586786\pi$
$601$	−13.2029	−0.538559	−0.269279	+	0.963062i	$0.586786\pi$
$602$	47.1334	1.92101
$603$	−2.83384	−0.115403
$604$	4.14709	0.168743
$605$	0	0
$606$	−28.4794	−1.15690
$607$	−16.7603	−0.680279	−0.340140	−	0.940375i	$-0.610474\pi$
$607$	−16.7603	−0.680279	−0.340140	+	0.940375i	$0.610474\pi$
$608$	−7.80440	−0.316510
$609$	8.23970	0.333890
$610$	0	0
$611$	−74.1196	−2.99856
$612$	0	0
$613$	32.7780	1.32389	0.661946	−	0.749552i	$-0.269731\pi$
$613$	32.7780	1.32389	0.661946	+	0.749552i	$0.269731\pi$
$614$	−41.4289	−1.67193
$615$	0	0
$616$	−13.2765	−0.534925
$617$	−33.4510	−1.34669	−0.673344	−	0.739329i	$-0.735143\pi$
$617$	−33.4510	−1.34669	−0.673344	+	0.739329i	$0.735143\pi$
$618$	−0.543637	−0.0218683
$619$	39.3731	1.58254	0.791269	−	0.611469i	$-0.209421\pi$
$619$	39.3731	1.58254	0.791269	+	0.611469i	$0.209421\pi$
$620$	0	0
$621$	0.543637	0.0218154
$622$	44.1471	1.77014
$623$	12.3407	0.494420
$624$	−15.3186	−0.613234
$625$	0	0
$626$	48.1418	1.92413
$627$	−3.74657	−0.149624
$628$	−34.1060	−1.36098
$629$	0	0
$630$	0	0
$631$	40.5804	1.61548	0.807740	−	0.589538i	$-0.200690\pi$
$631$	40.5804	1.61548	0.807740	+	0.589538i	$0.200690\pi$
$632$	−9.71513	−0.386447
$633$	−12.7182	−0.505502
$634$	23.9706	0.951992
$635$	0	0
$636$	−14.5172	−0.575646
$637$	3.41959	0.135489
$638$	−24.1103	−0.954537
$639$	7.83384	0.309902
$640$	0	0
$641$	−6.81172	−0.269047	−0.134523	−	0.990910i	$-0.542950\pi$
$641$	−6.81172	−0.269047	−0.134523	+	0.990910i	$0.542950\pi$
$642$	4.80440	0.189614
$643$	−14.3868	−0.567360	−0.283680	−	0.958919i	$-0.591555\pi$
$643$	−14.3868	−0.567360	−0.283680	+	0.958919i	$0.591555\pi$
$644$	3.88434	0.153065
$645$	0	0
$646$	0	0
$647$	0.137775	0.00541649	0.00270825	−	0.999996i	$-0.499138\pi$
$647$	0.137775	0.00541649	0.00270825	+	0.999996i	$0.499138\pi$
$648$	1.29021	0.0506841
$649$	−19.6823	−0.772599
$650$	0	0
$651$	4.00000	0.156772
$652$	−20.1167	−0.787832
$653$	−11.1019	−0.434452	−0.217226	−	0.976121i	$-0.569701\pi$
$653$	−11.1019	−0.434452	−0.217226	+	0.976121i	$0.569701\pi$
$654$	−2.58041	−0.100902
$655$	0	0
$656$	30.5142	1.19138
$657$	7.58041	0.295740
$658$	−69.4236	−2.70641
$659$	−29.9725	−1.16756	−0.583782	−	0.811910i	$-0.698428\pi$
$659$	−29.9725	−1.16756	−0.583782	+	0.811910i	$0.698428\pi$
$660$	0	0
$661$	−42.5530	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$661$	−42.5530	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$662$	−55.6349	−2.16231
$663$	0	0
$664$	−2.90739	−0.112829
$665$	0	0
$666$	11.1608	0.432473
$667$	1.63091	0.0631491
$668$	−24.2638	−0.938795
$669$	−10.5436	−0.407641
$670$	0	0
$671$	31.8201	1.22840
$672$	−21.4353	−0.826885
$673$	12.6961	0.489397	0.244699	−	0.969599i	$-0.421311\pi$
$673$	12.6961	0.489397	0.244699	+	0.969599i	$0.421311\pi$
$674$	32.3638	1.24661
$675$	0	0
$676$	69.1123	2.65817
$677$	−21.3784	−0.821639	−0.410819	−	0.911717i	$-0.634757\pi$
$677$	−21.3784	−0.821639	−0.410819	+	0.911717i	$0.634757\pi$
$678$	−26.9074	−1.03337
$679$	−46.3363	−1.77822
$680$	0	0
$681$	7.83384	0.300193
$682$	−11.7045	−0.448187
$683$	24.0652	0.920828	0.460414	−	0.887704i	$-0.347701\pi$
$683$	24.0652	0.920828	0.460414	+	0.887704i	$0.347701\pi$
$684$	2.60147	0.0994695
$685$	0	0
$686$	−38.0388	−1.45233
$687$	−23.6035	−0.900528
$688$	−19.4825	−0.742762
$689$	−35.1019	−1.33728
$690$	0	0
$691$	29.6677	1.12861	0.564306	−	0.825566i	$-0.309144\pi$
$691$	29.6677	1.12861	0.564306	+	0.825566i	$0.309144\pi$
$692$	−13.4614	−0.511726
$693$	−10.2902	−0.390893
$694$	−14.2060	−0.539252
$695$	0	0
$696$	3.87062	0.146715
$697$	0	0
$698$	19.4637	0.736712
$699$	−23.7833	−0.899569
$700$	0	0
$701$	6.65396	0.251317	0.125658	−	0.992074i	$-0.459896\pi$
$701$	6.65396	0.251317	0.125658	+	0.992074i	$0.459896\pi$
$702$	13.4931	0.509266
$703$	5.20293	0.196232
$704$	44.4741	1.67618
$705$	0	0
$706$	43.2176	1.62652
$707$	−36.4648	−1.37140
$708$	13.6666	0.513623
$709$	−29.0735	−1.09188	−0.545940	−	0.837824i	$-0.683827\pi$
$709$	−29.0735	−1.09188	−0.545940	+	0.837824i	$0.683827\pi$
$710$	0	0
$711$	−7.52991	−0.282394
$712$	5.79707	0.217254
$713$	0.791733	0.0296506
$714$	0	0
$715$	0	0
$716$	39.8726	1.49011
$717$	−1.27648	−0.0476710
$718$	34.0735	1.27161
$719$	12.6739	0.472659	0.236329	−	0.971673i	$-0.424056\pi$
$719$	12.6739	0.472659	0.236329	+	0.971673i	$0.424056\pi$
$720$	0	0
$721$	−0.696068	−0.0259229
$722$	2.14510	0.0798325
$723$	21.4931	0.799338
$724$	−27.0705	−1.00607
$725$	0	0
$726$	6.51419	0.241764
$727$	13.2260	0.490524	0.245262	−	0.969457i	$-0.421126\pi$
$727$	13.2260	0.490524	0.245262	+	0.969457i	$0.421126\pi$
$728$	22.2902	0.826130
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−22.0946	−0.816640
$733$	33.0735	1.22160	0.610800	−	0.791785i	$-0.290848\pi$
$733$	33.0735	1.22160	0.610800	+	0.791785i	$0.290848\pi$
$734$	53.3815	1.97035
$735$	0	0
$736$	−4.24276	−0.156390
$737$	10.6172	0.391089
$738$	−26.8779	−0.989391
$739$	−27.2260	−1.00152	−0.500762	−	0.865585i	$-0.666947\pi$
$739$	−27.2260	−1.00152	−0.500762	+	0.865585i	$0.666947\pi$
$740$	0	0
$741$	6.29021	0.231076
$742$	−32.8779	−1.20699
$743$	51.5751	1.89211	0.946053	−	0.324012i	$-0.105032\pi$
$743$	51.5751	1.89211	0.946053	+	0.324012i	$0.105032\pi$
$744$	1.87901	0.0688877
$745$	0	0
$746$	25.0284	0.916354
$747$	−2.25343	−0.0824488
$748$	0	0
$749$	6.15150	0.224771
$750$	0	0
$751$	16.7971	0.612934	0.306467	−	0.951881i	$-0.400853\pi$
$751$	16.7971	0.612934	0.306467	+	0.951881i	$0.400853\pi$
$752$	28.6961	1.04644
$753$	−29.9725	−1.09226
$754$	40.4794	1.47417
$755$	0	0
$756$	7.14510	0.259865
$757$	−34.6402	−1.25902	−0.629510	−	0.776992i	$-0.716744\pi$
$757$	−34.6402	−1.25902	−0.629510	+	0.776992i	$0.716744\pi$
$758$	13.4931	0.490093
$759$	−2.03677	−0.0739302
$760$	0	0
$761$	33.8294	1.22632	0.613158	−	0.789960i	$-0.289899\pi$
$761$	33.8294	1.22632	0.613158	+	0.789960i	$0.289899\pi$
$762$	−26.6907	−0.966903
$763$	−3.30393	−0.119610
$764$	−17.0230	−0.615872
$765$	0	0
$766$	25.8810	0.935119
$767$	33.0452	1.19319
$768$	2.60147	0.0938723
$769$	21.7550	0.784504	0.392252	−	0.919858i	$-0.371696\pi$
$769$	21.7550	0.784504	0.392252	+	0.919858i	$0.371696\pi$
$770$	0	0
$771$	19.0735	0.686917
$772$	−65.0009	−2.33943
$773$	−1.77495	−0.0638405	−0.0319203	−	0.999490i	$-0.510162\pi$
$773$	−1.77495	−0.0638405	−0.0319203	+	0.999490i	$0.510162\pi$
$774$	17.1608	0.616833
$775$	0	0
$776$	−21.7666	−0.781374
$777$	14.2902	0.512658
$778$	−30.8117	−1.10465
$779$	−12.5299	−0.448931
$780$	0	0
$781$	−29.3500	−1.05023
$782$	0	0
$783$	3.00000	0.107211
$784$	−1.32392	−0.0472830
$785$	0	0
$786$	40.6486	1.44989
$787$	19.8937	0.709132	0.354566	−	0.935031i	$-0.384629\pi$
$787$	19.8937	0.709132	0.354566	+	0.935031i	$0.384629\pi$
$788$	22.8853	0.815254
$789$	11.6456	0.414593
$790$	0	0
$791$	−34.4520	−1.22497
$792$	−4.83384	−0.171763
$793$	−53.4236	−1.89713
$794$	−66.7054	−2.36729
$795$	0	0
$796$	22.7539	0.806490
$797$	−7.36909	−0.261027	−0.130513	−	0.991447i	$-0.541663\pi$
$797$	−7.36909	−0.261027	−0.130513	+	0.991447i	$0.541663\pi$
$798$	5.89167	0.208563
$799$	0	0
$800$	0	0
$801$	4.49314	0.158757
$802$	−1.16616	−0.0411785
$803$	−28.4005	−1.00223
$804$	−7.37214	−0.259995
$805$	0	0
$806$	19.6509	0.692174
$807$	26.1471	0.920421
$808$	−17.1294	−0.602610
$809$	−10.4205	−0.366366	−0.183183	−	0.983079i	$-0.558640\pi$
$809$	−10.4205	−0.366366	−0.183183	+	0.983079i	$0.558640\pi$
$810$	0	0
$811$	46.9074	1.64714	0.823571	−	0.567214i	$-0.191979\pi$
$811$	46.9074	1.64714	0.823571	+	0.567214i	$0.191979\pi$
$812$	21.4353	0.752232
$813$	15.8338	0.555317
$814$	−41.8148	−1.46561
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	56.7696	1.98490
$819$	17.2765	0.603689
$820$	0	0
$821$	35.5392	1.24033	0.620164	−	0.784472i	$-0.287066\pi$
$821$	35.5392	1.24033	0.620164	+	0.784472i	$0.287066\pi$
$822$	45.8883	1.60054
$823$	17.7603	0.619085	0.309542	−	0.950886i	$-0.399824\pi$
$823$	17.7603	0.619085	0.309542	+	0.950886i	$0.399824\pi$
$824$	−0.326979	−0.0113909
$825$	0	0
$826$	30.9515	1.07694
$827$	−30.7843	−1.07047	−0.535237	−	0.844702i	$-0.679778\pi$
$827$	−30.7843	−1.07047	−0.535237	+	0.844702i	$0.679778\pi$
$828$	1.41425	0.0491487
$829$	−23.7098	−0.823475	−0.411738	−	0.911302i	$-0.635078\pi$
$829$	−23.7098	−0.823475	−0.411738	+	0.911302i	$0.635078\pi$
$830$	0	0
$831$	20.6677	0.716954
$832$	−74.6686	−2.58867
$833$	0	0
$834$	43.5740	1.50884
$835$	0	0
$836$	−9.74657	−0.337092
$837$	1.45636	0.0503393
$838$	−13.2765	−0.458628
$839$	34.1662	1.17955	0.589773	−	0.807569i	$-0.299217\pi$
$839$	34.1662	1.17955	0.589773	+	0.807569i	$0.299217\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	32.7696	1.12932
$843$	1.35536	0.0466811
$844$	−33.0859	−1.13886
$845$	0	0
$846$	−25.2765	−0.869023
$847$	8.34071	0.286590
$848$	13.5900	0.466683
$849$	28.1471	0.966006
$850$	0	0
$851$	2.82851	0.0969600
$852$	20.3795	0.698189
$853$	−28.7780	−0.985340	−0.492670	−	0.870216i	$-0.663979\pi$
$853$	−28.7780	−0.985340	−0.492670	+	0.870216i	$0.663979\pi$
$854$	−50.0388	−1.71229
$855$	0	0
$856$	2.88968	0.0987672
$857$	−47.2849	−1.61522	−0.807610	−	0.589717i	$-0.799239\pi$
$857$	−47.2849	−1.61522	−0.807610	+	0.589717i	$0.799239\pi$
$858$	−50.5530	−1.72585
$859$	−23.0221	−0.785505	−0.392752	−	0.919644i	$-0.628477\pi$
$859$	−23.0221	−0.785505	−0.392752	+	0.919644i	$0.628477\pi$
$860$	0	0
$861$	−34.4143	−1.17283
$862$	29.0839	0.990603
$863$	56.0545	1.90812	0.954058	−	0.299621i	$-0.0968601\pi$
$863$	56.0545	1.90812	0.954058	+	0.299621i	$0.0968601\pi$
$864$	−7.80440	−0.265511
$865$	0	0
$866$	−63.6717	−2.16365
$867$	−17.0000	−0.577350
$868$	10.4059	0.353198
$869$	28.2113	0.957004
$870$	0	0
$871$	−17.8255	−0.603992
$872$	−1.55203	−0.0525583
$873$	−16.8706	−0.570984
$874$	1.16616	0.0394459
$875$	0	0
$876$	19.7202	0.666283
$877$	−3.37748	−0.114049	−0.0570247	−	0.998373i	$-0.518161\pi$
$877$	−3.37748	−0.114049	−0.0570247	+	0.998373i	$0.518161\pi$
$878$	72.9535	2.46206
$879$	−8.47009	−0.285689
$880$	0	0
$881$	36.0314	1.21393	0.606965	−	0.794729i	$-0.292387\pi$
$881$	36.0314	1.21393	0.606965	+	0.794729i	$0.292387\pi$
$882$	1.16616	0.0392666
$883$	−6.23970	−0.209983	−0.104991	−	0.994473i	$-0.533482\pi$
$883$	−6.23970	−0.209983	−0.104991	+	0.994473i	$0.533482\pi$
$884$	0	0
$885$	0	0
$886$	−35.0545	−1.17768
$887$	40.2942	1.35295	0.676473	−	0.736467i	$-0.263507\pi$
$887$	40.2942	1.35295	0.676473	+	0.736467i	$0.263507\pi$
$888$	6.71285	0.225268
$889$	−34.1745	−1.14618
$890$	0	0
$891$	−3.74657	−0.125515
$892$	−27.4289	−0.918388
$893$	−11.7833	−0.394315
$894$	10.0735	0.336910
$895$	0	0
$896$	−27.0671	−0.904250
$897$	3.41959	0.114177
$898$	36.2501	1.20968
$899$	4.36909	0.145717
$900$	0	0
$901$	0	0
$902$	100.700	3.35294
$903$	21.9725	0.731201
$904$	−16.1839	−0.538267
$905$	0	0
$906$	3.41959	0.113608
$907$	1.59414	0.0529325	0.0264662	−	0.999650i	$-0.491575\pi$
$907$	1.59414	0.0529325	0.0264662	+	0.999650i	$0.491575\pi$
$908$	20.3795	0.676317
$909$	−13.2765	−0.440353
$910$	0	0
$911$	−6.58880	−0.218297	−0.109148	−	0.994025i	$-0.534812\pi$
$911$	−6.58880	−0.218297	−0.109148	+	0.994025i	$0.534812\pi$
$912$	−2.43531	−0.0806411
$913$	8.44264	0.279410
$914$	36.1712	1.19644
$915$	0	0
$916$	−61.4036	−2.02883
$917$	52.0461	1.71871
$918$	0	0
$919$	−13.3270	−0.439616	−0.219808	−	0.975543i	$-0.570543\pi$
$919$	−13.3270	−0.439616	−0.219808	+	0.975543i	$0.570543\pi$
$920$	0	0
$921$	−19.3133	−0.636393
$922$	−52.4794	−1.72832
$923$	49.2765	1.62196
$924$	−26.7696	−0.880656
$925$	0	0
$926$	−31.6509	−1.04011
$927$	−0.253432	−0.00832379
$928$	−23.4132	−0.768576
$929$	−22.0735	−0.724210	−0.362105	−	0.932137i	$-0.617942\pi$
$929$	−22.0735	−0.724210	−0.362105	+	0.932137i	$0.617942\pi$
$930$	0	0
$931$	0.543637	0.0178170
$932$	−61.8715	−2.02667
$933$	20.5804	0.673772
$934$	26.4426	0.865229
$935$	0	0
$936$	8.11566	0.265269
$937$	−13.5436	−0.442451	−0.221226	−	0.975223i	$-0.571006\pi$
$937$	−13.5436	−0.442451	−0.221226	+	0.975223i	$0.571006\pi$
$938$	−16.6961	−0.545146
$939$	22.4426	0.732388
$940$	0	0
$941$	3.96323	0.129197	0.0645987	−	0.997911i	$-0.479423\pi$
$941$	3.96323	0.129197	0.0645987	+	0.997911i	$0.479423\pi$
$942$	−28.1230	−0.916296
$943$	−6.81172	−0.221820
$944$	−12.7937	−0.416400
$945$	0	0
$946$	−64.2942	−2.09038
$947$	24.3784	0.792192	0.396096	−	0.918209i	$-0.370365\pi$
$947$	24.3784	0.792192	0.396096	+	0.918209i	$0.370365\pi$
$948$	−19.5888	−0.636215
$949$	47.6823	1.54783
$950$	0	0
$951$	11.1745	0.362360
$952$	0	0
$953$	−41.4226	−1.34181	−0.670906	−	0.741543i	$-0.734094\pi$
$953$	−41.4226	−1.34181	−0.670906	+	0.741543i	$0.734094\pi$
$954$	−11.9706	−0.387561
$955$	0	0
$956$	−3.32071	−0.107400
$957$	−11.2397	−0.363328
$958$	−40.7750	−1.31738
$959$	58.7550	1.89730
$960$	0	0
$961$	−28.8790	−0.931581
$962$	70.2039	2.26346
$963$	2.23970	0.0721735
$964$	55.9137	1.80086
$965$	0	0
$966$	3.20293	0.103053
$967$	−34.5888	−1.11230	−0.556150	−	0.831082i	$-0.687722\pi$
$967$	−34.5888	−1.11230	−0.556150	+	0.831082i	$0.687722\pi$
$968$	3.91806	0.125931
$969$	0	0
$970$	0	0
$971$	9.48475	0.304380	0.152190	−	0.988351i	$-0.451367\pi$
$971$	9.48475	0.304380	0.152190	+	0.988351i	$0.451367\pi$
$972$	2.60147	0.0834420
$973$	55.7917	1.78860
$974$	71.3815	2.28721
$975$	0	0
$976$	20.6834	0.662060
$977$	−46.7275	−1.49495	−0.747473	−	0.664293i	$-0.768733\pi$
$977$	−46.7275	−1.49495	−0.747473	+	0.664293i	$0.768733\pi$
$978$	−16.5877	−0.530417
$979$	−16.8338	−0.538012
$980$	0	0
$981$	−1.20293	−0.0384066
$982$	61.6234	1.96648
$983$	−21.2344	−0.677271	−0.338636	−	0.940918i	$-0.609965\pi$
$983$	−21.2344	−0.677271	−0.338636	+	0.940918i	$0.609965\pi$
$984$	−16.1662	−0.515358
$985$	0	0
$986$	0	0
$987$	−32.3638	−1.03015
$988$	16.3638	0.520600
$989$	4.34910	0.138293
$990$	0	0
$991$	29.7466	0.944931	0.472465	−	0.881349i	$-0.343364\pi$
$991$	29.7466	0.944931	0.472465	+	0.881349i	$0.343364\pi$
$992$	−11.3660	−0.360872
$993$	−25.9358	−0.823047
$994$	46.1544	1.46393
$995$	0	0
$996$	−5.86223	−0.185752
$997$	4.84850	0.153553	0.0767767	−	0.997048i	$-0.475537\pi$
$997$	4.84850	0.153553	0.0767767	+	0.997048i	$0.475537\pi$
$998$	−52.5288	−1.66277
$999$	5.20293	0.164613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.a.v.1.3	yes	3
3.2	odd	2		4275.2.a.bh.1.1		3
5.2	odd	4		1425.2.c.p.799.5		6
5.3	odd	4		1425.2.c.p.799.2		6
5.4	even	2		1425.2.a.u.1.1	✓	3
15.14	odd	2		4275.2.a.be.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.1	✓	3	5.4	even	2
1425.2.a.v.1.3	yes	3	1.1	even	1	trivial
1425.2.c.p.799.2		6	5.3	odd	4
1425.2.c.p.799.5		6	5.2	odd	4
4275.2.a.be.1.3		3	15.14	odd	2
4275.2.a.bh.1.1		3	3.2	odd	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:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14510 q^{2} +1.00000 q^{3} +2.60147 q^{4} +2.14510 q^{6} +2.74657 q^{7} +1.29021 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.14510 q^{2} +1.00000 q^{3} +2.60147 q^{4} +2.14510 q^{6} +2.74657 q^{7} +1.29021 q^{8} +1.00000 q^{9} -3.74657 q^{11} +2.60147 q^{12} +6.29021 q^{13} +5.89167 q^{14} -2.43531 q^{16} +2.14510 q^{18} +1.00000 q^{19} +2.74657 q^{21} -8.03677 q^{22} +0.543637 q^{23} +1.29021 q^{24} +13.4931 q^{26} +1.00000 q^{27} +7.14510 q^{28} +3.00000 q^{29} +1.45636 q^{31} -7.80440 q^{32} -3.74657 q^{33} +2.60147 q^{36} +5.20293 q^{37} +2.14510 q^{38} +6.29021 q^{39} -12.5299 q^{41} +5.89167 q^{42} +8.00000 q^{43} -9.74657 q^{44} +1.16616 q^{46} -11.7833 q^{47} -2.43531 q^{48} +0.543637 q^{49} +16.3638 q^{52} -5.58041 q^{53} +2.14510 q^{54} +3.54364 q^{56} +1.00000 q^{57} +6.43531 q^{58} +5.25343 q^{59} -8.49314 q^{61} +3.12405 q^{62} +2.74657 q^{63} -11.8706 q^{64} -8.03677 q^{66} -2.83384 q^{67} +0.543637 q^{69} +7.83384 q^{71} +1.29021 q^{72} +7.58041 q^{73} +11.1608 q^{74} +2.60147 q^{76} -10.2902 q^{77} +13.4931 q^{78} -7.52991 q^{79} +1.00000 q^{81} -26.8779 q^{82} -2.25343 q^{83} +7.14510 q^{84} +17.1608 q^{86} +3.00000 q^{87} -4.83384 q^{88} +4.49314 q^{89} +17.2765 q^{91} +1.41425 q^{92} +1.45636 q^{93} -25.2765 q^{94} -7.80440 q^{96} -16.8706 q^{97} +1.16616 q^{98} -3.74657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9	\( 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 6 q^{13} + 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} - 3 q^{23} - 9 q^{24} + 24 q^{26} + 3 q^{27} + 15 q^{28} + 9 q^{29} + 9 q^{31} - 18 q^{32} - 3 q^{33} + 6 q^{36} + 12 q^{37} + 6 q^{39} + 3 q^{42} + 24 q^{43} - 21 q^{44} + 21 q^{46} - 6 q^{47} + 12 q^{48} - 3 q^{49} - 6 q^{52} + 9 q^{53} + 6 q^{56} + 3 q^{57} + 24 q^{59} - 9 q^{61} - 21 q^{62} + 3 q^{64} - 3 q^{66} + 9 q^{67} - 3 q^{69} + 6 q^{71} - 9 q^{72} - 3 q^{73} - 18 q^{74} + 6 q^{76} - 18 q^{77} + 24 q^{78} + 15 q^{79} + 3 q^{81} - 33 q^{82} - 15 q^{83} + 15 q^{84} + 9 q^{87} + 3 q^{88} - 3 q^{89} + 6 q^{91} - 39 q^{92} + 9 q^{93} - 30 q^{94} - 18 q^{96} - 12 q^{97} + 21 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9 - 3 * q^11 + 6 * q^12 + 6 * q^13 + 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 - 3 * q^23 - 9 * q^24 + 24 * q^26 + 3 * q^27 + 15 * q^28 + 9 * q^29 + 9 * q^31 - 18 * q^32 - 3 * q^33 + 6 * q^36 + 12 * q^37 + 6 * q^39 + 3 * q^42 + 24 * q^43 - 21 * q^44 + 21 * q^46 - 6 * q^47 + 12 * q^48 - 3 * q^49 - 6 * q^52 + 9 * q^53 + 6 * q^56 + 3 * q^57 + 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 3 * q^66 + 9 * q^67 - 3 * q^69 + 6 * q^71 - 9 * q^72 - 3 * q^73 - 18 * q^74 + 6 * q^76 - 18 * q^77 + 24 * q^78 + 15 * q^79 + 3 * q^81 - 33 * q^82 - 15 * q^83 + 15 * q^84 + 9 * q^87 + 3 * q^88 - 3 * q^89 + 6 * q^91 - 39 * q^92 + 9 * q^93 - 30 * q^94 - 18 * q^96 - 12 * q^97 + 21 * q^98 - 3 * q^99

Introduction

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