LMFDB - Embedded newform 1305.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.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:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18 $ x^7 - x^6 - 13x^5 + 12x^4 + 47x^3 - 37x^2 - 35*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.57501$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57501 q^{2} +4.63069 q^{4} -1.00000 q^{5} -2.24826 q^{7} -6.77405 q^{8} +O(q^{10})$	\(q-2.57501 q^{2} +4.63069 q^{4} -1.00000 q^{5} -2.24826 q^{7} -6.77405 q^{8} +2.57501 q^{10} -3.75744 q^{11} -1.55119 q^{13} +5.78930 q^{14} +8.18188 q^{16} -3.55119 q^{17} -1.18909 q^{19} -4.63069 q^{20} +9.67546 q^{22} -3.67991 q^{23} +1.00000 q^{25} +3.99434 q^{26} -10.4110 q^{28} +1.00000 q^{29} +5.18909 q^{31} -7.52034 q^{32} +9.14436 q^{34} +2.24826 q^{35} +0.969518 q^{37} +3.06193 q^{38} +6.77405 q^{40} +11.3809 q^{41} -0.884393 q^{43} -17.3995 q^{44} +9.47581 q^{46} +11.2251 q^{47} -1.94533 q^{49} -2.57501 q^{50} -7.18308 q^{52} -10.6567 q^{53} +3.75744 q^{55} +15.2298 q^{56} -2.57501 q^{58} +8.49652 q^{59} -10.8743 q^{61} -13.3620 q^{62} +3.00121 q^{64} +1.55119 q^{65} -4.05179 q^{67} -16.4445 q^{68} -5.78930 q^{70} +6.37818 q^{71} -3.26017 q^{73} -2.49652 q^{74} -5.50631 q^{76} +8.44770 q^{77} +6.32579 q^{79} -8.18188 q^{80} -29.3060 q^{82} -9.03930 q^{83} +3.55119 q^{85} +2.27732 q^{86} +25.4531 q^{88} +1.46717 q^{89} +3.48748 q^{91} -17.0405 q^{92} -28.9047 q^{94} +1.18909 q^{95} +12.3993 q^{97} +5.00924 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7}+O(q^{10}) $ 7 * q - q^2 + 13 * q^4 - 7 * q^5 + 10 * q^7	$ 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7} + q^{10} + 3 q^{11} + 6 q^{13} + 9 q^{14} + 21 q^{16} - 8 q^{17} + 10 q^{19} - 13 q^{20} + 9 q^{22} - 11 q^{23} + 7 q^{25} - 3 q^{26} + 25 q^{28} + 7 q^{29} + 18 q^{31} - q^{32} - q^{34} - 10 q^{35} + 13 q^{37} - 12 q^{38} + 13 q^{41} + 9 q^{43} + 37 q^{44} - 8 q^{46} - 2 q^{47} + 21 q^{49} - q^{50} - q^{52} - 5 q^{53} - 3 q^{55} + 30 q^{56} - q^{58} + 8 q^{59} + 14 q^{61} + 8 q^{62} + 8 q^{64} - 6 q^{65} + 14 q^{67} - 27 q^{68} - 9 q^{70} + 8 q^{71} + 3 q^{73} + 34 q^{74} + 4 q^{76} + 28 q^{77} + 4 q^{79} - 21 q^{80} - 20 q^{82} - 17 q^{83} + 8 q^{85} - 4 q^{86} + 26 q^{88} + 20 q^{89} + 12 q^{91} - 60 q^{92} - 21 q^{94} - 10 q^{95} + 13 q^{97} + 20 q^{98}+O(q^{100}) $ 7 * q - q^2 + 13 * q^4 - 7 * q^5 + 10 * q^7 + q^10 + 3 * q^11 + 6 * q^13 + 9 * q^14 + 21 * q^16 - 8 * q^17 + 10 * q^19 - 13 * q^20 + 9 * q^22 - 11 * q^23 + 7 * q^25 - 3 * q^26 + 25 * q^28 + 7 * q^29 + 18 * q^31 - q^32 - q^34 - 10 * q^35 + 13 * q^37 - 12 * q^38 + 13 * q^41 + 9 * q^43 + 37 * q^44 - 8 * q^46 - 2 * q^47 + 21 * q^49 - q^50 - q^52 - 5 * q^53 - 3 * q^55 + 30 * q^56 - q^58 + 8 * q^59 + 14 * q^61 + 8 * q^62 + 8 * q^64 - 6 * q^65 + 14 * q^67 - 27 * q^68 - 9 * q^70 + 8 * q^71 + 3 * q^73 + 34 * q^74 + 4 * q^76 + 28 * q^77 + 4 * q^79 - 21 * q^80 - 20 * q^82 - 17 * q^83 + 8 * q^85 - 4 * q^86 + 26 * q^88 + 20 * q^89 + 12 * q^91 - 60 * q^92 - 21 * q^94 - 10 * q^95 + 13 * q^97 + 20 * 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 $

$2$	−2.57501	−1.82081	−0.910404	−	0.413720i	$-0.864229\pi$
$2$	−2.57501	−1.82081	−0.910404	+	0.413720i	$0.864229\pi$
$3$	0	0
$3$	0	0
$4$	4.63069	2.31534
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.24826	−0.849762	−0.424881	−	0.905249i	$-0.639684\pi$
$7$	−2.24826	−0.849762	−0.424881	+	0.905249i	$0.639684\pi$
$8$	−6.77405	−2.39499
$9$	0	0
$10$	2.57501	0.814290
$11$	−3.75744	−1.13291	−0.566456	−	0.824092i	$-0.691686\pi$
$11$	−3.75744	−1.13291	−0.566456	+	0.824092i	$0.691686\pi$
$12$	0	0
$13$	−1.55119	−0.430223	−0.215112	−	0.976589i	$-0.569012\pi$
$13$	−1.55119	−0.430223	−0.215112	+	0.976589i	$0.569012\pi$
$14$	5.78930	1.54725
$15$	0	0
$16$	8.18188	2.04547
$17$	−3.55119	−0.861291	−0.430645	−	0.902521i	$-0.641714\pi$
$17$	−3.55119	−0.861291	−0.430645	+	0.902521i	$0.641714\pi$
$18$	0	0
$19$	−1.18909	−0.272796	−0.136398	−	0.990654i	$-0.543553\pi$
$19$	−1.18909	−0.272796	−0.136398	+	0.990654i	$0.543553\pi$
$20$	−4.63069	−1.03545
$21$	0	0
$22$	9.67546	2.06281
$23$	−3.67991	−0.767314	−0.383657	−	0.923476i	$-0.625336\pi$
$23$	−3.67991	−0.767314	−0.383657	+	0.923476i	$0.625336\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.99434	0.783354
$27$	0	0
$28$	−10.4110	−1.96749
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.18909	0.931988	0.465994	−	0.884788i	$-0.345697\pi$
$31$	5.18909	0.931988	0.465994	+	0.884788i	$0.345697\pi$
$32$	−7.52034	−1.32942
$33$	0	0
$34$	9.14436	1.56825
$35$	2.24826	0.380025
$36$	0	0
$37$	0.969518	0.159388	0.0796939	−	0.996819i	$-0.474606\pi$
$37$	0.969518	0.159388	0.0796939	+	0.996819i	$0.474606\pi$
$38$	3.06193	0.496710
$39$	0	0
$40$	6.77405	1.07107
$41$	11.3809	1.77740	0.888700	−	0.458489i	$-0.151609\pi$
$41$	11.3809	1.77740	0.888700	+	0.458489i	$0.151609\pi$
$42$	0	0
$43$	−0.884393	−0.134869	−0.0674343	−	0.997724i	$-0.521481\pi$
$43$	−0.884393	−0.134869	−0.0674343	+	0.997724i	$0.521481\pi$
$44$	−17.3995	−2.62308
$45$	0	0
$46$	9.47581	1.39713
$47$	11.2251	1.63734	0.818672	−	0.574262i	$-0.194711\pi$
$47$	11.2251	1.63734	0.818672	+	0.574262i	$0.194711\pi$
$48$	0	0
$49$	−1.94533	−0.277904
$50$	−2.57501	−0.364162
$51$	0	0
$52$	−7.18308	−0.996115
$53$	−10.6567	−1.46381	−0.731906	−	0.681406i	$-0.761369\pi$
$53$	−10.6567	−1.46381	−0.731906	+	0.681406i	$0.761369\pi$
$54$	0	0
$55$	3.75744	0.506653
$56$	15.2298	2.03517
$57$	0	0
$58$	−2.57501	−0.338116
$59$	8.49652	1.10615	0.553076	−	0.833131i	$-0.313454\pi$
$59$	8.49652	1.10615	0.553076	+	0.833131i	$0.313454\pi$
$60$	0	0
$61$	−10.8743	−1.39231	−0.696154	−	0.717892i	$-0.745107\pi$
$61$	−10.8743	−1.39231	−0.696154	+	0.717892i	$0.745107\pi$
$62$	−13.3620	−1.69697
$63$	0	0
$64$	3.00121	0.375151
$65$	1.55119	0.192402
$66$	0	0
$67$	−4.05179	−0.495005	−0.247502	−	0.968887i	$-0.579610\pi$
$67$	−4.05179	−0.495005	−0.247502	+	0.968887i	$0.579610\pi$
$68$	−16.4445	−1.99418
$69$	0	0
$70$	−5.78930	−0.691953
$71$	6.37818	0.756951	0.378476	−	0.925611i	$-0.376448\pi$
$71$	6.37818	0.756951	0.378476	+	0.925611i	$0.376448\pi$
$72$	0	0
$73$	−3.26017	−0.381573	−0.190787	−	0.981632i	$-0.561104\pi$
$73$	−3.26017	−0.381573	−0.190787	+	0.981632i	$0.561104\pi$
$74$	−2.49652	−0.290215
$75$	0	0
$76$	−5.50631	−0.631617
$77$	8.44770	0.962705
$78$	0	0
$79$	6.32579	0.711707	0.355854	−	0.934542i	$-0.384190\pi$
$79$	6.32579	0.711707	0.355854	+	0.934542i	$0.384190\pi$
$80$	−8.18188	−0.914762
$81$	0	0
$82$	−29.3060	−3.23630
$83$	−9.03930	−0.992193	−0.496096	−	0.868268i	$-0.665234\pi$
$83$	−9.03930	−0.992193	−0.496096	+	0.868268i	$0.665234\pi$
$84$	0	0
$85$	3.55119	0.385181
$86$	2.27732	0.245570
$87$	0	0
$88$	25.4531	2.71331
$89$	1.46717	0.155520	0.0777599	−	0.996972i	$-0.475223\pi$
$89$	1.46717	0.155520	0.0777599	+	0.996972i	$0.475223\pi$
$90$	0	0
$91$	3.48748	0.365588
$92$	−17.0405	−1.77660
$93$	0	0
$94$	−28.9047	−2.98129
$95$	1.18909	0.121998
$96$	0	0
$97$	12.3993	1.25896	0.629478	−	0.777018i	$-0.283269\pi$
$97$	12.3993	1.25896	0.629478	+	0.777018i	$0.283269\pi$
$98$	5.00924	0.506010
$99$	0	0
$100$	4.63069	0.463069
$101$	3.45196	0.343483	0.171742	−	0.985142i	$-0.445061\pi$
$101$	3.45196	0.343483	0.171742	+	0.985142i	$0.445061\pi$
$102$	0	0
$103$	18.1975	1.79306	0.896528	−	0.442988i	$-0.146082\pi$
$103$	18.1975	1.79306	0.896528	+	0.442988i	$0.146082\pi$
$104$	10.5078	1.03038
$105$	0	0
$106$	27.4412	2.66532
$107$	8.07919	0.781044	0.390522	−	0.920594i	$-0.372294\pi$
$107$	8.07919	0.781044	0.390522	+	0.920594i	$0.372294\pi$
$108$	0	0
$109$	−14.0114	−1.34205	−0.671026	−	0.741434i	$-0.734146\pi$
$109$	−14.0114	−1.34205	−0.671026	+	0.741434i	$0.734146\pi$
$110$	−9.67546	−0.922518
$111$	0	0
$112$	−18.3950	−1.73816
$113$	−0.678491	−0.0638270	−0.0319135	−	0.999491i	$-0.510160\pi$
$113$	−0.678491	−0.0638270	−0.0319135	+	0.999491i	$0.510160\pi$
$114$	0	0
$115$	3.67991	0.343153
$116$	4.63069	0.429948
$117$	0	0
$118$	−21.8786	−2.01409
$119$	7.98400	0.731892
$120$	0	0
$121$	3.11836	0.283488
$122$	28.0014	2.53513
$123$	0	0
$124$	24.0291	2.15787
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.39928	−0.745315	−0.372658	−	0.927969i	$-0.621553\pi$
$127$	−8.39928	−0.745315	−0.372658	+	0.927969i	$0.621553\pi$
$128$	7.31254	0.646343
$129$	0	0
$130$	−3.99434	−0.350327
$131$	7.34298	0.641559	0.320779	−	0.947154i	$-0.396055\pi$
$131$	7.34298	0.641559	0.320779	+	0.947154i	$0.396055\pi$
$132$	0	0
$133$	2.67339	0.231812
$134$	10.4334	0.901308
$135$	0	0
$136$	24.0559	2.06278
$137$	1.80353	0.154086	0.0770429	−	0.997028i	$-0.475452\pi$
$137$	1.80353	0.154086	0.0770429	+	0.997028i	$0.475452\pi$
$138$	0	0
$139$	0.707451	0.0600052	0.0300026	−	0.999550i	$-0.490448\pi$
$139$	0.707451	0.0600052	0.0300026	+	0.999550i	$0.490448\pi$
$140$	10.4110	0.879889
$141$	0	0
$142$	−16.4239	−1.37826
$143$	5.82851	0.487405
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	8.39496	0.694772
$147$	0	0
$148$	4.48953	0.369037
$149$	16.9930	1.39212	0.696062	−	0.717982i	$-0.254934\pi$
$149$	16.9930	1.39212	0.696062	+	0.717982i	$0.254934\pi$
$150$	0	0
$151$	7.86646	0.640163	0.320082	−	0.947390i	$-0.396290\pi$
$151$	7.86646	0.640163	0.320082	+	0.947390i	$0.396290\pi$
$152$	8.05497	0.653344
$153$	0	0
$154$	−21.7529	−1.75290
$155$	−5.18909	−0.416798
$156$	0	0
$157$	9.85591	0.786588	0.393294	−	0.919413i	$-0.371336\pi$
$157$	9.85591	0.786588	0.393294	+	0.919413i	$0.371336\pi$
$158$	−16.2890	−1.29588
$159$	0	0
$160$	7.52034	0.594535
$161$	8.27340	0.652035
$162$	0	0
$163$	16.5786	1.29853	0.649267	−	0.760560i	$-0.275076\pi$
$163$	16.5786	1.29853	0.649267	+	0.760560i	$0.275076\pi$
$164$	52.7014	4.11529
$165$	0	0
$166$	23.2763	1.80659
$167$	4.74360	0.367071	0.183536	−	0.983013i	$-0.441246\pi$
$167$	4.74360	0.367071	0.183536	+	0.983013i	$0.441246\pi$
$168$	0	0
$169$	−10.5938	−0.814908
$170$	−9.14436	−0.701341
$171$	0	0
$172$	−4.09534	−0.312267
$173$	19.8201	1.50689	0.753446	−	0.657510i	$-0.228390\pi$
$173$	19.8201	1.50689	0.753446	+	0.657510i	$0.228390\pi$
$174$	0	0
$175$	−2.24826	−0.169952
$176$	−30.7429	−2.31734
$177$	0	0
$178$	−3.77798	−0.283172
$179$	−7.42036	−0.554624	−0.277312	−	0.960780i	$-0.589443\pi$
$179$	−7.42036	−0.554624	−0.277312	+	0.960780i	$0.589443\pi$
$180$	0	0
$181$	19.8411	1.47478	0.737390	−	0.675468i	$-0.236058\pi$
$181$	19.8411	1.47478	0.737390	+	0.675468i	$0.236058\pi$
$182$	−8.98031	−0.665665
$183$	0	0
$184$	24.9279	1.83771
$185$	−0.969518	−0.0712804
$186$	0	0
$187$	13.3434	0.975766
$188$	51.9797	3.79101
$189$	0	0
$190$	−3.06193	−0.222136
$191$	−5.66409	−0.409839	−0.204920	−	0.978779i	$-0.565693\pi$
$191$	−5.66409	−0.409839	−0.204920	+	0.978779i	$0.565693\pi$
$192$	0	0
$193$	14.7896	1.06458	0.532289	−	0.846563i	$-0.321332\pi$
$193$	14.7896	1.06458	0.532289	+	0.846563i	$0.321332\pi$
$194$	−31.9283	−2.29232
$195$	0	0
$196$	−9.00820	−0.643443
$197$	−27.1349	−1.93328	−0.966640	−	0.256139i	$-0.917550\pi$
$197$	−27.1349	−1.93328	−0.966640	+	0.256139i	$0.917550\pi$
$198$	0	0
$199$	−9.53523	−0.675934	−0.337967	−	0.941158i	$-0.609739\pi$
$199$	−9.53523	−0.675934	−0.337967	+	0.941158i	$0.609739\pi$
$200$	−6.77405	−0.478997
$201$	0	0
$202$	−8.88885	−0.625417
$203$	−2.24826	−0.157797
$204$	0	0
$205$	−11.3809	−0.794877
$206$	−46.8588	−3.26481
$207$	0	0
$208$	−12.6917	−0.880009
$209$	4.46794	0.309054
$210$	0	0
$211$	21.5727	1.48513	0.742564	−	0.669775i	$-0.233609\pi$
$211$	21.5727	1.48513	0.742564	+	0.669775i	$0.233609\pi$
$212$	−49.3479	−3.38923
$213$	0	0
$214$	−20.8040	−1.42213
$215$	0.884393	0.0603151
$216$	0	0
$217$	−11.6664	−0.791969
$218$	36.0796	2.44362
$219$	0	0
$220$	17.3995	1.17308
$221$	5.50858	0.370547
$222$	0	0
$223$	22.5140	1.50765	0.753824	−	0.657076i	$-0.228207\pi$
$223$	22.5140	1.50765	0.753824	+	0.657076i	$0.228207\pi$
$224$	16.9077	1.12969
$225$	0	0
$226$	1.74712	0.116217
$227$	−8.38063	−0.556242	−0.278121	−	0.960546i	$-0.589712\pi$
$227$	−8.38063	−0.556242	−0.278121	+	0.960546i	$0.589712\pi$
$228$	0	0
$229$	−11.5492	−0.763192	−0.381596	−	0.924329i	$-0.624625\pi$
$229$	−11.5492	−0.763192	−0.381596	+	0.924329i	$0.624625\pi$
$230$	−9.47581	−0.624817
$231$	0	0
$232$	−6.77405	−0.444738
$233$	8.85319	0.579991	0.289996	−	0.957028i	$-0.406346\pi$
$233$	8.85319	0.579991	0.289996	+	0.957028i	$0.406346\pi$
$234$	0	0
$235$	−11.2251	−0.732242
$236$	39.3447	2.56112
$237$	0	0
$238$	−20.5589	−1.33264
$239$	−13.9536	−0.902584	−0.451292	−	0.892376i	$-0.649037\pi$
$239$	−13.9536	−0.902584	−0.451292	+	0.892376i	$0.649037\pi$
$240$	0	0
$241$	−24.2138	−1.55975	−0.779874	−	0.625937i	$-0.784717\pi$
$241$	−24.2138	−1.55975	−0.779874	+	0.625937i	$0.784717\pi$
$242$	−8.02983	−0.516177
$243$	0	0
$244$	−50.3554	−3.22367
$245$	1.94533	0.124282
$246$	0	0
$247$	1.84451	0.117363
$248$	−35.1512	−2.23210
$249$	0	0
$250$	2.57501	0.162858
$251$	10.1141	0.638397	0.319198	−	0.947688i	$-0.396586\pi$
$251$	10.1141	0.638397	0.319198	+	0.947688i	$0.396586\pi$
$252$	0	0
$253$	13.8270	0.869299
$254$	21.6282	1.35708
$255$	0	0
$256$	−24.8323	−1.55202
$257$	18.2290	1.13709	0.568546	−	0.822652i	$-0.307506\pi$
$257$	18.2290	1.13709	0.568546	+	0.822652i	$0.307506\pi$
$258$	0	0
$259$	−2.17973	−0.135442
$260$	7.18308	0.445476
$261$	0	0
$262$	−18.9083	−1.16816
$263$	−4.26573	−0.263036	−0.131518	−	0.991314i	$-0.541985\pi$
$263$	−4.26573	−0.263036	−0.131518	+	0.991314i	$0.541985\pi$
$264$	0	0
$265$	10.6567	0.654637
$266$	−6.88401	−0.422086
$267$	0	0
$268$	−18.7626	−1.14611
$269$	−8.10868	−0.494395	−0.247197	−	0.968965i	$-0.579510\pi$
$269$	−8.10868	−0.494395	−0.247197	+	0.968965i	$0.579510\pi$
$270$	0	0
$271$	26.9259	1.63563	0.817815	−	0.575481i	$-0.195185\pi$
$271$	26.9259	1.63563	0.817815	+	0.575481i	$0.195185\pi$
$272$	−29.0554	−1.76174
$273$	0	0
$274$	−4.64410	−0.280561
$275$	−3.75744	−0.226582
$276$	0	0
$277$	29.1380	1.75073	0.875367	−	0.483459i	$-0.160620\pi$
$277$	29.1380	1.75073	0.875367	+	0.483459i	$0.160620\pi$
$278$	−1.82169	−0.109258
$279$	0	0
$280$	−15.2298	−0.910156
$281$	−21.6594	−1.29209	−0.646047	−	0.763298i	$-0.723579\pi$
$281$	−21.6594	−1.29209	−0.646047	+	0.763298i	$0.723579\pi$
$282$	0	0
$283$	−16.1608	−0.960660	−0.480330	−	0.877088i	$-0.659483\pi$
$283$	−16.1608	−0.960660	−0.480330	+	0.877088i	$0.659483\pi$
$284$	29.5354	1.75260
$285$	0	0
$286$	−15.0085	−0.887471
$287$	−25.5872	−1.51037
$288$	0	0
$289$	−4.38903	−0.258178
$290$	2.57501	0.151210
$291$	0	0
$292$	−15.0968	−0.883473
$293$	−16.5348	−0.965975	−0.482988	−	0.875627i	$-0.660448\pi$
$293$	−16.5348	−0.965975	−0.482988	+	0.875627i	$0.660448\pi$
$294$	0	0
$295$	−8.49652	−0.494687
$296$	−6.56756	−0.381732
$297$	0	0
$298$	−43.7573	−2.53479
$299$	5.70825	0.330117
$300$	0	0
$301$	1.98834	0.114606
$302$	−20.2562	−1.16561
$303$	0	0
$304$	−9.72901	−0.557997
$305$	10.8743	0.622659
$306$	0	0
$307$	−25.3923	−1.44922	−0.724608	−	0.689161i	$-0.757979\pi$
$307$	−25.3923	−1.44922	−0.724608	+	0.689161i	$0.757979\pi$
$308$	39.1187	2.22899
$309$	0	0
$310$	13.3620	0.758909
$311$	−16.8950	−0.958030	−0.479015	−	0.877807i	$-0.659006\pi$
$311$	−16.8950	−0.958030	−0.479015	+	0.877807i	$0.659006\pi$
$312$	0	0
$313$	−22.1978	−1.25469	−0.627347	−	0.778740i	$-0.715859\pi$
$313$	−22.1978	−1.25469	−0.627347	+	0.778740i	$0.715859\pi$
$314$	−25.3791	−1.43223
$315$	0	0
$316$	29.2927	1.64785
$317$	26.1161	1.46683	0.733414	−	0.679783i	$-0.237926\pi$
$317$	26.1161	1.46683	0.733414	+	0.679783i	$0.237926\pi$
$318$	0	0
$319$	−3.75744	−0.210376
$320$	−3.00121	−0.167773
$321$	0	0
$322$	−21.3041	−1.18723
$323$	4.22270	0.234957
$324$	0	0
$325$	−1.55119	−0.0860447
$326$	−42.6900	−2.36438
$327$	0	0
$328$	−77.0948	−4.25685
$329$	−25.2369	−1.39135
$330$	0	0
$331$	24.3439	1.33806	0.669030	−	0.743236i	$-0.266710\pi$
$331$	24.3439	1.33806	0.669030	+	0.743236i	$0.266710\pi$
$332$	−41.8582	−2.29727
$333$	0	0
$334$	−12.2148	−0.668366
$335$	4.05179	0.221373
$336$	0	0
$337$	−23.5099	−1.28067	−0.640333	−	0.768098i	$-0.721204\pi$
$337$	−23.5099	−1.28067	−0.640333	+	0.768098i	$0.721204\pi$
$338$	27.2792	1.48379
$339$	0	0
$340$	16.4445	0.891826
$341$	−19.4977	−1.05586
$342$	0	0
$343$	20.1114	1.08591
$344$	5.99092	0.323009
$345$	0	0
$346$	−51.0369	−2.74376
$347$	29.9492	1.60776	0.803879	−	0.594793i	$-0.202766\pi$
$347$	29.9492	1.60776	0.803879	+	0.594793i	$0.202766\pi$
$348$	0	0
$349$	−17.0927	−0.914951	−0.457476	−	0.889222i	$-0.651246\pi$
$349$	−17.0927	−0.914951	−0.457476	+	0.889222i	$0.651246\pi$
$350$	5.78930	0.309451
$351$	0	0
$352$	28.2572	1.50612
$353$	6.40124	0.340704	0.170352	−	0.985383i	$-0.445510\pi$
$353$	6.40124	0.340704	0.170352	+	0.985383i	$0.445510\pi$
$354$	0	0
$355$	−6.37818	−0.338519
$356$	6.79400	0.360082
$357$	0	0
$358$	19.1075	1.00986
$359$	29.2509	1.54380	0.771902	−	0.635741i	$-0.219306\pi$
$359$	29.2509	1.54380	0.771902	+	0.635741i	$0.219306\pi$
$360$	0	0
$361$	−17.5861	−0.925582
$362$	−51.0911	−2.68529
$363$	0	0
$364$	16.1494	0.846461
$365$	3.26017	0.170645
$366$	0	0
$367$	−13.2919	−0.693832	−0.346916	−	0.937896i	$-0.612771\pi$
$367$	−13.2919	−0.693832	−0.346916	+	0.937896i	$0.612771\pi$
$368$	−30.1086	−1.56952
$369$	0	0
$370$	2.49652	0.129788
$371$	23.9591	1.24389
$372$	0	0
$373$	−19.7309	−1.02163	−0.510814	−	0.859691i	$-0.670656\pi$
$373$	−19.7309	−1.02163	−0.510814	+	0.859691i	$0.670656\pi$
$374$	−34.3594	−1.77668
$375$	0	0
$376$	−76.0391	−3.92142
$377$	−1.55119	−0.0798905
$378$	0	0
$379$	25.4806	1.30885	0.654424	−	0.756128i	$-0.272911\pi$
$379$	25.4806	1.30885	0.654424	+	0.756128i	$0.272911\pi$
$380$	5.50631	0.282468
$381$	0	0
$382$	14.5851	0.746239
$383$	0.542784	0.0277350	0.0138675	−	0.999904i	$-0.495586\pi$
$383$	0.542784	0.0277350	0.0138675	+	0.999904i	$0.495586\pi$
$384$	0	0
$385$	−8.44770	−0.430535
$386$	−38.0833	−1.93839
$387$	0	0
$388$	57.4171	2.91491
$389$	−19.9301	−1.01050	−0.505249	−	0.862974i	$-0.668599\pi$
$389$	−19.9301	−1.01050	−0.505249	+	0.862974i	$0.668599\pi$
$390$	0	0
$391$	13.0681	0.660881
$392$	13.1777	0.665576
$393$	0	0
$394$	69.8726	3.52013
$395$	−6.32579	−0.318285
$396$	0	0
$397$	9.40286	0.471916	0.235958	−	0.971763i	$-0.424177\pi$
$397$	9.40286	0.471916	0.235958	+	0.971763i	$0.424177\pi$
$398$	24.5533	1.23075
$399$	0	0
$400$	8.18188	0.409094
$401$	9.37578	0.468204	0.234102	−	0.972212i	$-0.424785\pi$
$401$	9.37578	0.468204	0.234102	+	0.972212i	$0.424785\pi$
$402$	0	0
$403$	−8.04928	−0.400963
$404$	15.9850	0.795282
$405$	0	0
$406$	5.78930	0.287318
$407$	−3.64291	−0.180572
$408$	0	0
$409$	19.7204	0.975111	0.487556	−	0.873092i	$-0.337889\pi$
$409$	19.7204	0.975111	0.487556	+	0.873092i	$0.337889\pi$
$410$	29.3060	1.44732
$411$	0	0
$412$	84.2670	4.15154
$413$	−19.1024	−0.939967
$414$	0	0
$415$	9.03930	0.443722
$416$	11.6655	0.571948
$417$	0	0
$418$	−11.5050	−0.562728
$419$	−6.24026	−0.304857	−0.152428	−	0.988315i	$-0.548709\pi$
$419$	−6.24026	−0.304857	−0.152428	+	0.988315i	$0.548709\pi$
$420$	0	0
$421$	6.31722	0.307882	0.153941	−	0.988080i	$-0.450803\pi$
$421$	6.31722	0.307882	0.153941	+	0.988080i	$0.450803\pi$
$422$	−55.5500	−2.70413
$423$	0	0
$424$	72.1891	3.50581
$425$	−3.55119	−0.172258
$426$	0	0
$427$	24.4482	1.18313
$428$	37.4122	1.80839
$429$	0	0
$430$	−2.27732	−0.109822
$431$	−37.8006	−1.82079	−0.910396	−	0.413737i	$-0.864223\pi$
$431$	−37.8006	−1.82079	−0.910396	+	0.413737i	$0.864223\pi$
$432$	0	0
$433$	37.8447	1.81870	0.909350	−	0.416032i	$-0.136580\pi$
$433$	37.8447	1.81870	0.909350	+	0.416032i	$0.136580\pi$
$434$	30.0412	1.44202
$435$	0	0
$436$	−64.8825	−3.10731
$437$	4.37575	0.209321
$438$	0	0
$439$	−15.5318	−0.741291	−0.370646	−	0.928774i	$-0.620864\pi$
$439$	−15.5318	−0.741291	−0.370646	+	0.928774i	$0.620864\pi$
$440$	−25.4531	−1.21343
$441$	0	0
$442$	−14.1847	−0.674696
$443$	−5.24223	−0.249066	−0.124533	−	0.992215i	$-0.539743\pi$
$443$	−5.24223	−0.249066	−0.124533	+	0.992215i	$0.539743\pi$
$444$	0	0
$445$	−1.46717	−0.0695505
$446$	−57.9738	−2.74514
$447$	0	0
$448$	−6.74749	−0.318789
$449$	−28.0123	−1.32198	−0.660992	−	0.750393i	$-0.729864\pi$
$449$	−28.0123	−1.32198	−0.660992	+	0.750393i	$0.729864\pi$
$450$	0	0
$451$	−42.7631	−2.01364
$452$	−3.14188	−0.147781
$453$	0	0
$454$	21.5802	1.01281
$455$	−3.48748	−0.163496
$456$	0	0
$457$	1.90391	0.0890610	0.0445305	−	0.999008i	$-0.485821\pi$
$457$	1.90391	0.0890610	0.0445305	+	0.999008i	$0.485821\pi$
$458$	29.7393	1.38963
$459$	0	0
$460$	17.0405	0.794518
$461$	30.5294	1.42190	0.710949	−	0.703244i	$-0.248266\pi$
$461$	30.5294	1.42190	0.710949	+	0.703244i	$0.248266\pi$
$462$	0	0
$463$	32.9230	1.53006	0.765030	−	0.643995i	$-0.222724\pi$
$463$	32.9230	1.53006	0.765030	+	0.643995i	$0.222724\pi$
$464$	8.18188	0.379834
$465$	0	0
$466$	−22.7971	−1.05605
$467$	−4.42775	−0.204892	−0.102446	−	0.994739i	$-0.532667\pi$
$467$	−4.42775	−0.204892	−0.102446	+	0.994739i	$0.532667\pi$
$468$	0	0
$469$	9.10947	0.420636
$470$	28.9047	1.33327
$471$	0	0
$472$	−57.5558	−2.64922
$473$	3.32305	0.152794
$474$	0	0
$475$	−1.18909	−0.0545593
$476$	36.9714	1.69458
$477$	0	0
$478$	35.9307	1.64343
$479$	−8.27071	−0.377898	−0.188949	−	0.981987i	$-0.560508\pi$
$479$	−8.27071	−0.377898	−0.188949	+	0.981987i	$0.560508\pi$
$480$	0	0
$481$	−1.50391	−0.0685723
$482$	62.3508	2.84000
$483$	0	0
$484$	14.4402	0.656371
$485$	−12.3993	−0.563022
$486$	0	0
$487$	39.9284	1.80933	0.904665	−	0.426124i	$-0.140121\pi$
$487$	39.9284	1.80933	0.904665	+	0.426124i	$0.140121\pi$
$488$	73.6628	3.33456
$489$	0	0
$490$	−5.00924	−0.226294
$491$	−33.3896	−1.50685	−0.753427	−	0.657532i	$-0.771601\pi$
$491$	−33.3896	−1.50685	−0.753427	+	0.657532i	$0.771601\pi$
$492$	0	0
$493$	−3.55119	−0.159938
$494$	−4.74964	−0.213696
$495$	0	0
$496$	42.4565	1.90635
$497$	−14.3398	−0.643229
$498$	0	0
$499$	−3.12737	−0.140000	−0.0700002	−	0.997547i	$-0.522300\pi$
$499$	−3.12737	−0.140000	−0.0700002	+	0.997547i	$0.522300\pi$
$500$	−4.63069	−0.207091
$501$	0	0
$502$	−26.0439	−1.16240
$503$	−17.2563	−0.769419	−0.384710	−	0.923038i	$-0.625698\pi$
$503$	−17.2563	−0.769419	−0.384710	+	0.923038i	$0.625698\pi$
$504$	0	0
$505$	−3.45196	−0.153610
$506$	−35.6048	−1.58283
$507$	0	0
$508$	−38.8944	−1.72566
$509$	−4.71879	−0.209157	−0.104578	−	0.994517i	$-0.533349\pi$
$509$	−4.71879	−0.209157	−0.104578	+	0.994517i	$0.533349\pi$
$510$	0	0
$511$	7.32970	0.324247
$512$	49.3183	2.17958
$513$	0	0
$514$	−46.9398	−2.07043
$515$	−18.1975	−0.801879
$516$	0	0
$517$	−42.1775	−1.85496
$518$	5.61283	0.246613
$519$	0	0
$520$	−10.5078	−0.460800
$521$	40.0385	1.75412	0.877060	−	0.480381i	$-0.159502\pi$
$521$	40.0385	1.75412	0.877060	+	0.480381i	$0.159502\pi$
$522$	0	0
$523$	−5.38694	−0.235555	−0.117777	−	0.993040i	$-0.537577\pi$
$523$	−5.38694	−0.235555	−0.117777	+	0.993040i	$0.537577\pi$
$524$	34.0030	1.48543
$525$	0	0
$526$	10.9843	0.478939
$527$	−18.4275	−0.802713
$528$	0	0
$529$	−9.45826	−0.411229
$530$	−27.4412	−1.19197
$531$	0	0
$532$	12.3796	0.536725
$533$	−17.6540	−0.764679
$534$	0	0
$535$	−8.07919	−0.349294
$536$	27.4470	1.18553
$537$	0	0
$538$	20.8799	0.900198
$539$	7.30945	0.314840
$540$	0	0
$541$	35.9892	1.54730	0.773649	−	0.633614i	$-0.218429\pi$
$541$	35.9892	1.54730	0.773649	+	0.633614i	$0.218429\pi$
$542$	−69.3345	−2.97817
$543$	0	0
$544$	26.7062	1.14502
$545$	14.0114	0.600184
$546$	0	0
$547$	24.2745	1.03790	0.518951	−	0.854804i	$-0.326323\pi$
$547$	24.2745	1.03790	0.518951	+	0.854804i	$0.326323\pi$
$548$	8.35157	0.356761
$549$	0	0
$550$	9.67546	0.412563
$551$	−1.18909	−0.0506570
$552$	0	0
$553$	−14.2220	−0.604782
$554$	−75.0307	−3.18775
$555$	0	0
$556$	3.27598	0.138933
$557$	−4.93097	−0.208932	−0.104466	−	0.994528i	$-0.533313\pi$
$557$	−4.93097	−0.208932	−0.104466	+	0.994528i	$0.533313\pi$
$558$	0	0
$559$	1.37186	0.0580236
$560$	18.3950	0.777330
$561$	0	0
$562$	55.7733	2.35266
$563$	−34.7020	−1.46252	−0.731258	−	0.682101i	$-0.761067\pi$
$563$	−34.7020	−1.46252	−0.731258	+	0.682101i	$0.761067\pi$
$564$	0	0
$565$	0.678491	0.0285443
$566$	41.6142	1.74918
$567$	0	0
$568$	−43.2061	−1.81289
$569$	19.9633	0.836903	0.418452	−	0.908239i	$-0.362573\pi$
$569$	19.9633	0.836903	0.418452	+	0.908239i	$0.362573\pi$
$570$	0	0
$571$	−7.22542	−0.302375	−0.151187	−	0.988505i	$-0.548310\pi$
$571$	−7.22542	−0.302375	−0.151187	+	0.988505i	$0.548310\pi$
$572$	26.9900	1.12851
$573$	0	0
$574$	65.8875	2.75009
$575$	−3.67991	−0.153463
$576$	0	0
$577$	17.6693	0.735581	0.367791	−	0.929909i	$-0.380114\pi$
$577$	17.6693	0.735581	0.367791	+	0.929909i	$0.380114\pi$
$578$	11.3018	0.470093
$579$	0	0
$580$	−4.63069	−0.192279
$581$	20.3227	0.843128
$582$	0	0
$583$	40.0420	1.65837
$584$	22.0845	0.913864
$585$	0	0
$586$	42.5774	1.75886
$587$	−13.1365	−0.542203	−0.271102	−	0.962551i	$-0.587388\pi$
$587$	−13.1365	−0.542203	−0.271102	+	0.962551i	$0.587388\pi$
$588$	0	0
$589$	−6.17031	−0.254243
$590$	21.8786	0.900729
$591$	0	0
$592$	7.93248	0.326023
$593$	−11.3675	−0.466807	−0.233403	−	0.972380i	$-0.574986\pi$
$593$	−11.3675	−0.466807	−0.233403	+	0.972380i	$0.574986\pi$
$594$	0	0
$595$	−7.98400	−0.327312
$596$	78.6894	3.22324
$597$	0	0
$598$	−14.6988	−0.601079
$599$	8.98828	0.367251	0.183626	−	0.982996i	$-0.441217\pi$
$599$	8.98828	0.367251	0.183626	+	0.982996i	$0.441217\pi$
$600$	0	0
$601$	−5.15704	−0.210360	−0.105180	−	0.994453i	$-0.533542\pi$
$601$	−5.15704	−0.210360	−0.105180	+	0.994453i	$0.533542\pi$
$602$	−5.12001	−0.208676
$603$	0	0
$604$	36.4271	1.48220
$605$	−3.11836	−0.126780
$606$	0	0
$607$	−36.3436	−1.47514	−0.737570	−	0.675271i	$-0.764027\pi$
$607$	−36.3436	−1.47514	−0.737570	+	0.675271i	$0.764027\pi$
$608$	8.94238	0.362661
$609$	0	0
$610$	−28.0014	−1.13374
$611$	−17.4122	−0.704423
$612$	0	0
$613$	18.9943	0.767171	0.383585	−	0.923505i	$-0.374689\pi$
$613$	18.9943	0.767171	0.383585	+	0.923505i	$0.374689\pi$
$614$	65.3855	2.63874
$615$	0	0
$616$	−57.2251	−2.30567
$617$	24.5060	0.986574	0.493287	−	0.869867i	$-0.335795\pi$
$617$	24.5060	0.986574	0.493287	+	0.869867i	$0.335795\pi$
$618$	0	0
$619$	12.5314	0.503680	0.251840	−	0.967769i	$-0.418964\pi$
$619$	12.5314	0.503680	0.251840	+	0.967769i	$0.418964\pi$
$620$	−24.0291	−0.965030
$621$	0	0
$622$	43.5049	1.74439
$623$	−3.29858	−0.132155
$624$	0	0
$625$	1.00000	0.0400000
$626$	57.1596	2.28456
$627$	0	0
$628$	45.6396	1.82122
$629$	−3.44294	−0.137279
$630$	0	0
$631$	17.8019	0.708681	0.354340	−	0.935116i	$-0.384705\pi$
$631$	17.8019	0.708681	0.354340	+	0.935116i	$0.384705\pi$
$632$	−42.8512	−1.70453
$633$	0	0
$634$	−67.2493	−2.67081
$635$	8.39928	0.333315
$636$	0	0
$637$	3.01758	0.119561
$638$	9.67546	0.383055
$639$	0	0
$640$	−7.31254	−0.289053
$641$	−20.3973	−0.805644	−0.402822	−	0.915278i	$-0.631971\pi$
$641$	−20.3973	−0.805644	−0.402822	+	0.915278i	$0.631971\pi$
$642$	0	0
$643$	−23.6531	−0.932787	−0.466394	−	0.884577i	$-0.654447\pi$
$643$	−23.6531	−0.932787	−0.466394	+	0.884577i	$0.654447\pi$
$644$	38.3115	1.50968
$645$	0	0
$646$	−10.8735	−0.427812
$647$	17.5789	0.691096	0.345548	−	0.938401i	$-0.387693\pi$
$647$	17.5789	0.691096	0.345548	+	0.938401i	$0.387693\pi$
$648$	0	0
$649$	−31.9252	−1.25317
$650$	3.99434	0.156671
$651$	0	0
$652$	76.7702	3.00655
$653$	−10.4035	−0.407121	−0.203560	−	0.979062i	$-0.565251\pi$
$653$	−10.4035	−0.407121	−0.203560	+	0.979062i	$0.565251\pi$
$654$	0	0
$655$	−7.34298	−0.286914
$656$	93.1172	3.63562
$657$	0	0
$658$	64.9852	2.53339
$659$	43.5638	1.69700	0.848502	−	0.529192i	$-0.177505\pi$
$659$	43.5638	1.69700	0.848502	+	0.529192i	$0.177505\pi$
$660$	0	0
$661$	18.2404	0.709469	0.354734	−	0.934967i	$-0.384571\pi$
$661$	18.2404	0.709469	0.354734	+	0.934967i	$0.384571\pi$
$662$	−62.6857	−2.43635
$663$	0	0
$664$	61.2327	2.37629
$665$	−2.67339	−0.103670
$666$	0	0
$667$	−3.67991	−0.142487
$668$	21.9661	0.849895
$669$	0	0
$670$	−10.4334	−0.403077
$671$	40.8595	1.57736
$672$	0	0
$673$	−25.8626	−0.996929	−0.498464	−	0.866910i	$-0.666103\pi$
$673$	−25.8626	−0.996929	−0.498464	+	0.866910i	$0.666103\pi$
$674$	60.5383	2.33185
$675$	0	0
$676$	−49.0566	−1.88679
$677$	−30.3287	−1.16563	−0.582813	−	0.812606i	$-0.698048\pi$
$677$	−30.3287	−1.16563	−0.582813	+	0.812606i	$0.698048\pi$
$678$	0	0
$679$	−27.8768	−1.06981
$680$	−24.0559	−0.922503
$681$	0	0
$682$	50.2068	1.92252
$683$	−39.4637	−1.51004	−0.755018	−	0.655704i	$-0.772372\pi$
$683$	−39.4637	−1.51004	−0.755018	+	0.655704i	$0.772372\pi$
$684$	0	0
$685$	−1.80353	−0.0689092
$686$	−51.7871	−1.97724
$687$	0	0
$688$	−7.23599	−0.275870
$689$	16.5306	0.629766
$690$	0	0
$691$	−24.5853	−0.935269	−0.467635	−	0.883922i	$-0.654894\pi$
$691$	−24.5853	−0.935269	−0.467635	+	0.883922i	$0.654894\pi$
$692$	91.7805	3.48897
$693$	0	0
$694$	−77.1196	−2.92742
$695$	−0.707451	−0.0268351
$696$	0	0
$697$	−40.4158	−1.53086
$698$	44.0139	1.66595
$699$	0	0
$700$	−10.4110	−0.393498
$701$	−34.8635	−1.31678	−0.658388	−	0.752678i	$-0.728762\pi$
$701$	−34.8635	−1.31678	−0.658388	+	0.752678i	$0.728762\pi$
$702$	0	0
$703$	−1.15285	−0.0434804
$704$	−11.2769	−0.425012
$705$	0	0
$706$	−16.4833	−0.620356
$707$	−7.76091	−0.291879
$708$	0	0
$709$	−24.4622	−0.918698	−0.459349	−	0.888256i	$-0.651917\pi$
$709$	−24.4622	−0.918698	−0.459349	+	0.888256i	$0.651917\pi$
$710$	16.4239	0.616378
$711$	0	0
$712$	−9.93868	−0.372468
$713$	−19.0954	−0.715128
$714$	0	0
$715$	−5.82851	−0.217974
$716$	−34.3613	−1.28414
$717$	0	0
$718$	−75.3214	−2.81097
$719$	−27.1333	−1.01190	−0.505951	−	0.862562i	$-0.668858\pi$
$719$	−27.1333	−1.01190	−0.505951	+	0.862562i	$0.668858\pi$
$720$	0	0
$721$	−40.9128	−1.52367
$722$	45.2843	1.68531
$723$	0	0
$724$	91.8780	3.41462
$725$	1.00000	0.0371391
$726$	0	0
$727$	33.9854	1.26045	0.630224	−	0.776413i	$-0.282963\pi$
$727$	33.9854	1.26045	0.630224	+	0.776413i	$0.282963\pi$
$728$	−23.6244	−0.875578
$729$	0	0
$730$	−8.39496	−0.310712
$731$	3.14065	0.116161
$732$	0	0
$733$	17.2131	0.635779	0.317889	−	0.948128i	$-0.397026\pi$
$733$	17.2131	0.635779	0.317889	+	0.948128i	$0.397026\pi$
$734$	34.2268	1.26333
$735$	0	0
$736$	27.6742	1.02008
$737$	15.2244	0.560796
$738$	0	0
$739$	−12.5603	−0.462039	−0.231019	−	0.972949i	$-0.574206\pi$
$739$	−12.5603	−0.462039	−0.231019	+	0.972949i	$0.574206\pi$
$740$	−4.48953	−0.165038
$741$	0	0
$742$	−61.6949	−2.26489
$743$	43.0323	1.57870	0.789350	−	0.613943i	$-0.210418\pi$
$743$	43.0323	1.57870	0.789350	+	0.613943i	$0.210418\pi$
$744$	0	0
$745$	−16.9930	−0.622577
$746$	50.8073	1.86019
$747$	0	0
$748$	61.7891	2.25923
$749$	−18.1641	−0.663702
$750$	0	0
$751$	−12.4133	−0.452969	−0.226485	−	0.974015i	$-0.572723\pi$
$751$	−12.4133	−0.452969	−0.226485	+	0.974015i	$0.572723\pi$
$752$	91.8421	3.34914
$753$	0	0
$754$	3.99434	0.145465
$755$	−7.86646	−0.286290
$756$	0	0
$757$	18.0439	0.655815	0.327907	−	0.944710i	$-0.393657\pi$
$757$	18.0439	0.655815	0.327907	+	0.944710i	$0.393657\pi$
$758$	−65.6127	−2.38316
$759$	0	0
$760$	−8.05497	−0.292184
$761$	−8.20477	−0.297423	−0.148711	−	0.988881i	$-0.547513\pi$
$761$	−8.20477	−0.297423	−0.148711	+	0.988881i	$0.547513\pi$
$762$	0	0
$763$	31.5013	1.14043
$764$	−26.2286	−0.948919
$765$	0	0
$766$	−1.39768	−0.0505001
$767$	−13.1797	−0.475893
$768$	0	0
$769$	−23.5524	−0.849320	−0.424660	−	0.905353i	$-0.639606\pi$
$769$	−23.5524	−0.849320	−0.424660	+	0.905353i	$0.639606\pi$
$770$	21.7529	0.783921
$771$	0	0
$772$	68.4859	2.46486
$773$	−27.2326	−0.979487	−0.489743	−	0.871867i	$-0.662910\pi$
$773$	−27.2326	−0.979487	−0.489743	+	0.871867i	$0.662910\pi$
$774$	0	0
$775$	5.18909	0.186398
$776$	−83.9933	−3.01518
$777$	0	0
$778$	51.3203	1.83992
$779$	−13.5330	−0.484868
$780$	0	0
$781$	−23.9657	−0.857559
$782$	−33.6504	−1.20334
$783$	0	0
$784$	−15.9164	−0.568444
$785$	−9.85591	−0.351773
$786$	0	0
$787$	20.2585	0.722137	0.361069	−	0.932539i	$-0.382412\pi$
$787$	20.2585	0.722137	0.361069	+	0.932539i	$0.382412\pi$
$788$	−125.653	−4.47621
$789$	0	0
$790$	16.2890	0.579536
$791$	1.52542	0.0542378
$792$	0	0
$793$	16.8681	0.599004
$794$	−24.2125	−0.859269
$795$	0	0
$796$	−44.1546	−1.56502
$797$	27.8574	0.986761	0.493381	−	0.869814i	$-0.335761\pi$
$797$	27.8574	0.986761	0.493381	+	0.869814i	$0.335761\pi$
$798$	0	0
$799$	−39.8624	−1.41023
$800$	−7.52034	−0.265884
$801$	0	0
$802$	−24.1427	−0.852510
$803$	12.2499	0.432289
$804$	0	0
$805$	−8.27340	−0.291599
$806$	20.7270	0.730077
$807$	0	0
$808$	−23.3838	−0.822638
$809$	−17.5060	−0.615477	−0.307739	−	0.951471i	$-0.599572\pi$
$809$	−17.5060	−0.615477	−0.307739	+	0.951471i	$0.599572\pi$
$810$	0	0
$811$	39.1690	1.37541	0.687704	−	0.725991i	$-0.258619\pi$
$811$	39.1690	1.37541	0.687704	+	0.725991i	$0.258619\pi$
$812$	−10.4110	−0.365354
$813$	0	0
$814$	9.38053	0.328787
$815$	−16.5786	−0.580722
$816$	0	0
$817$	1.05162	0.0367917
$818$	−50.7803	−1.77549
$819$	0	0
$820$	−52.7014	−1.84041
$821$	19.8338	0.692203	0.346101	−	0.938197i	$-0.387505\pi$
$821$	19.8338	0.692203	0.346101	+	0.938197i	$0.387505\pi$
$822$	0	0
$823$	10.3727	0.361568	0.180784	−	0.983523i	$-0.442136\pi$
$823$	10.3727	0.361568	0.180784	+	0.983523i	$0.442136\pi$
$824$	−123.271	−4.29434
$825$	0	0
$826$	49.1889	1.71150
$827$	−3.35272	−0.116586	−0.0582928	−	0.998300i	$-0.518566\pi$
$827$	−3.35272	−0.116586	−0.0582928	+	0.998300i	$0.518566\pi$
$828$	0	0
$829$	−43.0572	−1.49544	−0.747719	−	0.664015i	$-0.768851\pi$
$829$	−43.0572	−1.49544	−0.747719	+	0.664015i	$0.768851\pi$
$830$	−23.2763	−0.807933
$831$	0	0
$832$	−4.65545	−0.161399
$833$	6.90823	0.239356
$834$	0	0
$835$	−4.74360	−0.164159
$836$	20.6896	0.715566
$837$	0	0
$838$	16.0688	0.555086
$839$	39.3736	1.35933	0.679664	−	0.733523i	$-0.262125\pi$
$839$	39.3736	1.35933	0.679664	+	0.733523i	$0.262125\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−16.2669	−0.560595
$843$	0	0
$844$	99.8965	3.43858
$845$	10.5938	0.364438
$846$	0	0
$847$	−7.01089	−0.240897
$848$	−87.1919	−2.99418
$849$	0	0
$850$	9.14436	0.313649
$851$	−3.56774	−0.122301
$852$	0	0
$853$	−45.2743	−1.55016	−0.775081	−	0.631862i	$-0.782291\pi$
$853$	−45.2743	−1.55016	−0.775081	+	0.631862i	$0.782291\pi$
$854$	−62.9544	−2.15426
$855$	0	0
$856$	−54.7288	−1.87059
$857$	−24.1619	−0.825353	−0.412677	−	0.910878i	$-0.635406\pi$
$857$	−24.1619	−0.825353	−0.412677	+	0.910878i	$0.635406\pi$
$858$	0	0
$859$	−22.3680	−0.763185	−0.381593	−	0.924331i	$-0.624624\pi$
$859$	−22.3680	−0.763185	−0.381593	+	0.924331i	$0.624624\pi$
$860$	4.09534	0.139650
$861$	0	0
$862$	97.3371	3.31531
$863$	2.98467	0.101599	0.0507997	−	0.998709i	$-0.483823\pi$
$863$	2.98467	0.101599	0.0507997	+	0.998709i	$0.483823\pi$
$864$	0	0
$865$	−19.8201	−0.673902
$866$	−97.4505	−3.31150
$867$	0	0
$868$	−54.0236	−1.83368
$869$	−23.7688	−0.806301
$870$	0	0
$871$	6.28510	0.212963
$872$	94.9141	3.21420
$873$	0	0
$874$	−11.2676	−0.381133
$875$	2.24826	0.0760051
$876$	0	0
$877$	29.5037	0.996269	0.498135	−	0.867100i	$-0.334019\pi$
$877$	29.5037	0.996269	0.498135	+	0.867100i	$0.334019\pi$
$878$	39.9945	1.34975
$879$	0	0
$880$	30.7429	1.03634
$881$	52.0937	1.75508	0.877541	−	0.479502i	$-0.159183\pi$
$881$	52.0937	1.75508	0.877541	+	0.479502i	$0.159183\pi$
$882$	0	0
$883$	22.2221	0.747832	0.373916	−	0.927463i	$-0.378015\pi$
$883$	22.2221	0.747832	0.373916	+	0.927463i	$0.378015\pi$
$884$	25.5085	0.857944
$885$	0	0
$886$	13.4988	0.453502
$887$	38.6220	1.29680	0.648400	−	0.761300i	$-0.275438\pi$
$887$	38.6220	1.29680	0.648400	+	0.761300i	$0.275438\pi$
$888$	0	0
$889$	18.8838	0.633341
$890$	3.77798	0.126638
$891$	0	0
$892$	104.255	3.49072
$893$	−13.3476	−0.446662
$894$	0	0
$895$	7.42036	0.248035
$896$	−16.4405	−0.549238
$897$	0	0
$898$	72.1321	2.40708
$899$	5.18909	0.173066
$900$	0	0
$901$	37.8440	1.26077
$902$	110.116	3.66645
$903$	0	0
$904$	4.59613	0.152865
$905$	−19.8411	−0.659541
$906$	0	0
$907$	−29.7861	−0.989030	−0.494515	−	0.869169i	$-0.664654\pi$
$907$	−29.7861	−0.989030	−0.494515	+	0.869169i	$0.664654\pi$
$908$	−38.8080	−1.28789
$909$	0	0
$910$	8.98031	0.297694
$911$	53.7163	1.77970	0.889850	−	0.456254i	$-0.150809\pi$
$911$	53.7163	1.77970	0.889850	+	0.456254i	$0.150809\pi$
$912$	0	0
$913$	33.9647	1.12407
$914$	−4.90258	−0.162163
$915$	0	0
$916$	−53.4807	−1.76705
$917$	−16.5089	−0.545173
$918$	0	0
$919$	52.0358	1.71650	0.858251	−	0.513231i	$-0.171551\pi$
$919$	52.0358	1.71650	0.858251	+	0.513231i	$0.171551\pi$
$920$	−24.9279	−0.821848
$921$	0	0
$922$	−78.6137	−2.58900
$923$	−9.89379	−0.325658
$924$	0	0
$925$	0.969518	0.0318775
$926$	−84.7770	−2.78595
$927$	0	0
$928$	−7.52034	−0.246867
$929$	−52.5081	−1.72273	−0.861367	−	0.507982i	$-0.830391\pi$
$929$	−52.5081	−1.72273	−0.861367	+	0.507982i	$0.830391\pi$
$930$	0	0
$931$	2.31317	0.0758112
$932$	40.9963	1.34288
$933$	0	0
$934$	11.4015	0.373068
$935$	−13.3434	−0.436376
$936$	0	0
$937$	46.1947	1.50912	0.754558	−	0.656233i	$-0.227851\pi$
$937$	46.1947	1.50912	0.754558	+	0.656233i	$0.227851\pi$
$938$	−23.4570	−0.765898
$939$	0	0
$940$	−51.9797	−1.69539
$941$	29.8642	0.973545	0.486773	−	0.873529i	$-0.338174\pi$
$941$	29.8642	0.973545	0.486773	+	0.873529i	$0.338174\pi$
$942$	0	0
$943$	−41.8807	−1.36382
$944$	69.5175	2.26260
$945$	0	0
$946$	−8.55690	−0.278209
$947$	11.7523	0.381898	0.190949	−	0.981600i	$-0.438843\pi$
$947$	11.7523	0.381898	0.190949	+	0.981600i	$0.438843\pi$
$948$	0	0
$949$	5.05714	0.164162
$950$	3.06193	0.0993420
$951$	0	0
$952$	−54.0840	−1.75287
$953$	−17.1653	−0.556040	−0.278020	−	0.960575i	$-0.589678\pi$
$953$	−17.1653	−0.556040	−0.278020	+	0.960575i	$0.589678\pi$
$954$	0	0
$955$	5.66409	0.183286
$956$	−64.6147	−2.08979
$957$	0	0
$958$	21.2972	0.688081
$959$	−4.05480	−0.130936
$960$	0	0
$961$	−4.07332	−0.131397
$962$	3.87258	0.124857
$963$	0	0
$964$	−112.126	−3.61135
$965$	−14.7896	−0.476093
$966$	0	0
$967$	20.2076	0.649831	0.324916	−	0.945743i	$-0.394664\pi$
$967$	20.2076	0.649831	0.324916	+	0.945743i	$0.394664\pi$
$968$	−21.1239	−0.678949
$969$	0	0
$970$	31.9283	1.02516
$971$	24.7827	0.795315	0.397658	−	0.917534i	$-0.369823\pi$
$971$	24.7827	0.795315	0.397658	+	0.917534i	$0.369823\pi$
$972$	0	0
$973$	−1.59053	−0.0509902
$974$	−102.816	−3.29444
$975$	0	0
$976$	−88.9720	−2.84792
$977$	38.5423	1.23308	0.616539	−	0.787324i	$-0.288534\pi$
$977$	38.5423	1.23308	0.616539	+	0.787324i	$0.288534\pi$
$978$	0	0
$979$	−5.51281	−0.176190
$980$	9.00820	0.287756
$981$	0	0
$982$	85.9787	2.74369
$983$	13.0762	0.417065	0.208532	−	0.978015i	$-0.433131\pi$
$983$	13.0762	0.417065	0.208532	+	0.978015i	$0.433131\pi$
$984$	0	0
$985$	27.1349	0.864589
$986$	9.14436	0.291216
$987$	0	0
$988$	8.54135	0.271737
$989$	3.25449	0.103487
$990$	0	0
$991$	−36.8186	−1.16958	−0.584791	−	0.811184i	$-0.698823\pi$
$991$	−36.8186	−1.16958	−0.584791	+	0.811184i	$0.698823\pi$
$992$	−39.0237	−1.23900
$993$	0	0
$994$	36.9252	1.17120
$995$	9.53523	0.302287
$996$	0	0
$997$	−18.5570	−0.587707	−0.293854	−	0.955850i	$-0.594938\pi$
$997$	−18.5570	−0.587707	−0.293854	+	0.955850i	$0.594938\pi$
$998$	8.05302	0.254914
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.s.1.1	✓	7
3.2	odd	2		1305.2.a.t.1.7	yes	7
5.4	even	2		6525.2.a.bw.1.7		7
15.14	odd	2		6525.2.a.bv.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.a.s.1.1	✓	7	1.1	even	1	trivial
1305.2.a.t.1.7	yes	7	3.2	odd	2
6525.2.a.bv.1.1		7	15.14	odd	2
6525.2.a.bw.1.7		7	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57501 q^{2} +4.63069 q^{4} -1.00000 q^{5} -2.24826 q^{7} -6.77405 q^{8} +O(q^{10})\)	\(q-2.57501 q^{2} +4.63069 q^{4} -1.00000 q^{5} -2.24826 q^{7} -6.77405 q^{8} +2.57501 q^{10} -3.75744 q^{11} -1.55119 q^{13} +5.78930 q^{14} +8.18188 q^{16} -3.55119 q^{17} -1.18909 q^{19} -4.63069 q^{20} +9.67546 q^{22} -3.67991 q^{23} +1.00000 q^{25} +3.99434 q^{26} -10.4110 q^{28} +1.00000 q^{29} +5.18909 q^{31} -7.52034 q^{32} +9.14436 q^{34} +2.24826 q^{35} +0.969518 q^{37} +3.06193 q^{38} +6.77405 q^{40} +11.3809 q^{41} -0.884393 q^{43} -17.3995 q^{44} +9.47581 q^{46} +11.2251 q^{47} -1.94533 q^{49} -2.57501 q^{50} -7.18308 q^{52} -10.6567 q^{53} +3.75744 q^{55} +15.2298 q^{56} -2.57501 q^{58} +8.49652 q^{59} -10.8743 q^{61} -13.3620 q^{62} +3.00121 q^{64} +1.55119 q^{65} -4.05179 q^{67} -16.4445 q^{68} -5.78930 q^{70} +6.37818 q^{71} -3.26017 q^{73} -2.49652 q^{74} -5.50631 q^{76} +8.44770 q^{77} +6.32579 q^{79} -8.18188 q^{80} -29.3060 q^{82} -9.03930 q^{83} +3.55119 q^{85} +2.27732 q^{86} +25.4531 q^{88} +1.46717 q^{89} +3.48748 q^{91} -17.0405 q^{92} -28.9047 q^{94} +1.18909 q^{95} +12.3993 q^{97} +5.00924 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7}+O(q^{10}) \) 7 * q - q^2 + 13 * q^4 - 7 * q^5 + 10 * q^7	\( 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7} + q^{10} + 3 q^{11} + 6 q^{13} + 9 q^{14} + 21 q^{16} - 8 q^{17} + 10 q^{19} - 13 q^{20} + 9 q^{22} - 11 q^{23} + 7 q^{25} - 3 q^{26} + 25 q^{28} + 7 q^{29} + 18 q^{31} - q^{32} - q^{34} - 10 q^{35} + 13 q^{37} - 12 q^{38} + 13 q^{41} + 9 q^{43} + 37 q^{44} - 8 q^{46} - 2 q^{47} + 21 q^{49} - q^{50} - q^{52} - 5 q^{53} - 3 q^{55} + 30 q^{56} - q^{58} + 8 q^{59} + 14 q^{61} + 8 q^{62} + 8 q^{64} - 6 q^{65} + 14 q^{67} - 27 q^{68} - 9 q^{70} + 8 q^{71} + 3 q^{73} + 34 q^{74} + 4 q^{76} + 28 q^{77} + 4 q^{79} - 21 q^{80} - 20 q^{82} - 17 q^{83} + 8 q^{85} - 4 q^{86} + 26 q^{88} + 20 q^{89} + 12 q^{91} - 60 q^{92} - 21 q^{94} - 10 q^{95} + 13 q^{97} + 20 q^{98}+O(q^{100}) \) 7 * q - q^2 + 13 * q^4 - 7 * q^5 + 10 * q^7 + q^10 + 3 * q^11 + 6 * q^13 + 9 * q^14 + 21 * q^16 - 8 * q^17 + 10 * q^19 - 13 * q^20 + 9 * q^22 - 11 * q^23 + 7 * q^25 - 3 * q^26 + 25 * q^28 + 7 * q^29 + 18 * q^31 - q^32 - q^34 - 10 * q^35 + 13 * q^37 - 12 * q^38 + 13 * q^41 + 9 * q^43 + 37 * q^44 - 8 * q^46 - 2 * q^47 + 21 * q^49 - q^50 - q^52 - 5 * q^53 - 3 * q^55 + 30 * q^56 - q^58 + 8 * q^59 + 14 * q^61 + 8 * q^62 + 8 * q^64 - 6 * q^65 + 14 * q^67 - 27 * q^68 - 9 * q^70 + 8 * q^71 + 3 * q^73 + 34 * q^74 + 4 * q^76 + 28 * q^77 + 4 * q^79 - 21 * q^80 - 20 * q^82 - 17 * q^83 + 8 * q^85 - 4 * q^86 + 26 * q^88 + 20 * q^89 + 12 * q^91 - 60 * q^92 - 21 * q^94 - 10 * q^95 + 13 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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