LMFDB - Embedded newform 1305.2.a.s.1.5

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			1.5
Root			$-0.897436$ 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)

$f(q)$	$=$	\(q+0.897436 q^{2} -1.19461 q^{4} -1.00000 q^{5} +3.83036 q^{7} -2.86696 q^{8} -0.897436 q^{10} -2.82632 q^{11} +0.989088 q^{13} +3.43750 q^{14} -0.183696 q^{16} -1.01091 q^{17} +5.13268 q^{19} +1.19461 q^{20} -2.53645 q^{22} +7.78936 q^{23} +1.00000 q^{25} +0.887643 q^{26} -4.57577 q^{28} +1.00000 q^{29} -1.13268 q^{31} +5.56906 q^{32} -0.907229 q^{34} -3.83036 q^{35} +10.7648 q^{37} +4.60625 q^{38} +2.86696 q^{40} +2.58070 q^{41} -4.24141 q^{43} +3.37635 q^{44} +6.99045 q^{46} +0.252519 q^{47} +7.67162 q^{49} +0.897436 q^{50} -1.18157 q^{52} +5.70648 q^{53} +2.82632 q^{55} -10.9815 q^{56} +0.897436 q^{58} -3.66071 q^{59} -10.3820 q^{61} -1.01651 q^{62} +5.36527 q^{64} -0.989088 q^{65} +3.75939 q^{67} +1.20764 q^{68} -3.43750 q^{70} -6.26536 q^{71} +10.7545 q^{73} +9.66071 q^{74} -6.13154 q^{76} -10.8258 q^{77} +10.7853 q^{79} +0.183696 q^{80} +2.31601 q^{82} +1.06004 q^{83} +1.01091 q^{85} -3.80639 q^{86} +8.10295 q^{88} +14.3024 q^{89} +3.78856 q^{91} -9.30523 q^{92} +0.226620 q^{94} -5.13268 q^{95} +13.8941 q^{97} +6.88479 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.897436	0.634583	0.317292	−	0.948328i	$-0.397227\pi$
$2$	0.897436	0.634583	0.317292	+	0.948328i	$0.397227\pi$
$3$	0	0
$3$	0	0
$4$	−1.19461	−0.597304
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.83036	1.44774	0.723869	−	0.689937i	$-0.242362\pi$
$7$	3.83036	1.44774	0.723869	+	0.689937i	$0.242362\pi$
$8$	−2.86696	−1.01362
$9$	0	0
$10$	−0.897436	−0.283794
$11$	−2.82632	−0.852169	−0.426084	−	0.904683i	$-0.640107\pi$
$11$	−2.82632	−0.852169	−0.426084	+	0.904683i	$0.640107\pi$
$12$	0	0
$13$	0.989088	0.274324	0.137162	−	0.990549i	$-0.456202\pi$
$13$	0.989088	0.274324	0.137162	+	0.990549i	$0.456202\pi$
$14$	3.43750	0.918711
$15$	0	0
$16$	−0.183696	−0.0459240
$17$	−1.01091	−0.245182	−0.122591	−	0.992457i	$-0.539120\pi$
$17$	−1.01091	−0.245182	−0.122591	+	0.992457i	$0.539120\pi$
$18$	0	0
$19$	5.13268	1.17752	0.588759	−	0.808309i	$-0.299617\pi$
$19$	5.13268	1.17752	0.588759	+	0.808309i	$0.299617\pi$
$20$	1.19461	0.267122
$21$	0	0
$22$	−2.53645	−0.540772
$23$	7.78936	1.62419	0.812097	−	0.583523i	$-0.198326\pi$
$23$	7.78936	1.62419	0.812097	+	0.583523i	$0.198326\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.887643	0.174081
$27$	0	0
$28$	−4.57577	−0.864740
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.13268	−0.203435	−0.101718	−	0.994813i	$-0.532434\pi$
$31$	−1.13268	−0.203435	−0.101718	+	0.994813i	$0.532434\pi$
$32$	5.56906	0.984480
$33$	0	0
$34$	−0.907229	−0.155589
$35$	−3.83036	−0.647448
$36$	0	0
$37$	10.7648	1.76972	0.884860	−	0.465857i	$-0.154254\pi$
$37$	10.7648	1.76972	0.884860	+	0.465857i	$0.154254\pi$
$38$	4.60625	0.747233
$39$	0	0
$40$	2.86696	0.453306
$41$	2.58070	0.403037	0.201519	−	0.979485i	$-0.435412\pi$
$41$	2.58070	0.403037	0.201519	+	0.979485i	$0.435412\pi$
$42$	0	0
$43$	−4.24141	−0.646809	−0.323404	−	0.946261i	$-0.604827\pi$
$43$	−4.24141	−0.646809	−0.323404	+	0.946261i	$0.604827\pi$
$44$	3.37635	0.509004
$45$	0	0
$46$	6.99045	1.03069
$47$	0.252519	0.0368337	0.0184168	−	0.999830i	$-0.494137\pi$
$47$	0.252519	0.0368337	0.0184168	+	0.999830i	$0.494137\pi$
$48$	0	0
$49$	7.67162	1.09595
$50$	0.897436	0.126917
$51$	0	0
$52$	−1.18157	−0.163855
$53$	5.70648	0.783846	0.391923	−	0.919998i	$-0.371810\pi$
$53$	5.70648	0.783846	0.391923	+	0.919998i	$0.371810\pi$
$54$	0	0
$55$	2.82632	0.381101
$56$	−10.9815	−1.46746
$57$	0	0
$58$	0.897436	0.117839
$59$	−3.66071	−0.476584	−0.238292	−	0.971194i	$-0.576588\pi$
$59$	−3.66071	−0.476584	−0.238292	+	0.971194i	$0.576588\pi$
$60$	0	0
$61$	−10.3820	−1.32927	−0.664637	−	0.747166i	$-0.731414\pi$
$61$	−10.3820	−1.32927	−0.664637	+	0.747166i	$0.731414\pi$
$62$	−1.01651	−0.129096
$63$	0	0
$64$	5.36527	0.670659
$65$	−0.989088	−0.122681
$66$	0	0
$67$	3.75939	0.459283	0.229641	−	0.973275i	$-0.426245\pi$
$67$	3.75939	0.459283	0.229641	+	0.973275i	$0.426245\pi$
$68$	1.20764	0.146448
$69$	0	0
$70$	−3.43750	−0.410860
$71$	−6.26536	−0.743561	−0.371781	−	0.928321i	$-0.621253\pi$
$71$	−6.26536	−0.743561	−0.371781	+	0.928321i	$0.621253\pi$
$72$	0	0
$73$	10.7545	1.25872	0.629359	−	0.777115i	$-0.283318\pi$
$73$	10.7545	1.25872	0.629359	+	0.777115i	$0.283318\pi$
$74$	9.66071	1.12303
$75$	0	0
$76$	−6.13154	−0.703336
$77$	−10.8258	−1.23372
$78$	0	0
$79$	10.7853	1.21344	0.606722	−	0.794914i	$-0.292484\pi$
$79$	10.7853	1.21344	0.606722	+	0.794914i	$0.292484\pi$
$80$	0.183696	0.0205378
$81$	0	0
$82$	2.31601	0.255761
$83$	1.06004	0.116355	0.0581773	−	0.998306i	$-0.481471\pi$
$83$	1.06004	0.116355	0.0581773	+	0.998306i	$0.481471\pi$
$84$	0	0
$85$	1.01091	0.109649
$86$	−3.80639	−0.410454
$87$	0	0
$88$	8.10295	0.863777
$89$	14.3024	1.51606	0.758028	−	0.652222i	$-0.226163\pi$
$89$	14.3024	1.51606	0.758028	+	0.652222i	$0.226163\pi$
$90$	0	0
$91$	3.78856	0.397149
$92$	−9.30523	−0.970137
$93$	0	0
$94$	0.226620	0.0233740
$95$	−5.13268	−0.526602
$96$	0	0
$97$	13.8941	1.41073	0.705364	−	0.708845i	$-0.250784\pi$
$97$	13.8941	1.41073	0.705364	+	0.708845i	$0.250784\pi$
$98$	6.88479	0.695469
$99$	0	0
$100$	−1.19461	−0.119461
$101$	−14.4729	−1.44011	−0.720053	−	0.693919i	$-0.755883\pi$
$101$	−14.4729	−1.44011	−0.720053	+	0.693919i	$0.755883\pi$
$102$	0	0
$103$	8.70934	0.858157	0.429079	−	0.903267i	$-0.358838\pi$
$103$	8.70934	0.858157	0.429079	+	0.903267i	$0.358838\pi$
$104$	−2.83567	−0.278061
$105$	0	0
$106$	5.12121	0.497415
$107$	−1.89530	−0.183226	−0.0916129	−	0.995795i	$-0.529202\pi$
$107$	−1.89530	−0.183226	−0.0916129	+	0.995795i	$0.529202\pi$
$108$	0	0
$109$	6.62461	0.634522	0.317261	−	0.948338i	$-0.397237\pi$
$109$	6.62461	0.634522	0.317261	+	0.948338i	$0.397237\pi$
$110$	2.53645	0.241841
$111$	0	0
$112$	−0.703621	−0.0664859
$113$	1.00061	0.0941294	0.0470647	−	0.998892i	$-0.485013\pi$
$113$	1.00061	0.0941294	0.0470647	+	0.998892i	$0.485013\pi$
$114$	0	0
$115$	−7.78936	−0.726361
$116$	−1.19461	−0.110917
$117$	0	0
$118$	−3.28526	−0.302432
$119$	−3.87215	−0.354960
$120$	0	0
$121$	−3.01190	−0.273809
$122$	−9.31715	−0.843535
$123$	0	0
$124$	1.35311	0.121513
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.89406	−0.877955	−0.438978	−	0.898498i	$-0.644659\pi$
$127$	−9.89406	−0.877955	−0.438978	+	0.898498i	$0.644659\pi$
$128$	−6.32313	−0.558891
$129$	0	0
$130$	−0.887643	−0.0778515
$131$	13.1169	1.14603	0.573013	−	0.819546i	$-0.305774\pi$
$131$	13.1169	1.14603	0.573013	+	0.819546i	$0.305774\pi$
$132$	0	0
$133$	19.6600	1.70474
$134$	3.37381	0.291453
$135$	0	0
$136$	2.89824	0.248522
$137$	0.0709655	0.00606299	0.00303149	−	0.999995i	$-0.499035\pi$
$137$	0.0709655	0.00606299	0.00303149	+	0.999995i	$0.499035\pi$
$138$	0	0
$139$	−12.2462	−1.03871	−0.519354	−	0.854559i	$-0.673827\pi$
$139$	−12.2462	−1.03871	−0.519354	+	0.854559i	$0.673827\pi$
$140$	4.57577	0.386723
$141$	0	0
$142$	−5.62276	−0.471851
$143$	−2.79548	−0.233770
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	9.65147	0.798761
$147$	0	0
$148$	−12.8597	−1.05706
$149$	−7.32142	−0.599794	−0.299897	−	0.953972i	$-0.596952\pi$
$149$	−7.32142	−0.599794	−0.299897	+	0.953972i	$0.596952\pi$
$150$	0	0
$151$	−23.3800	−1.90264	−0.951318	−	0.308211i	$-0.900270\pi$
$151$	−23.3800	−1.90264	−0.951318	+	0.308211i	$0.900270\pi$
$152$	−14.7152	−1.19356
$153$	0	0
$154$	−9.71549	−0.782896
$155$	1.13268	0.0909789
$156$	0	0
$157$	−0.931394	−0.0743333	−0.0371667	−	0.999309i	$-0.511833\pi$
$157$	−0.931394	−0.0743333	−0.0371667	+	0.999309i	$0.511833\pi$
$158$	9.67914	0.770031
$159$	0	0
$160$	−5.56906	−0.440273
$161$	29.8360	2.35141
$162$	0	0
$163$	−1.03087	−0.0807441	−0.0403721	−	0.999185i	$-0.512854\pi$
$163$	−1.03087	−0.0807441	−0.0403721	+	0.999185i	$0.512854\pi$
$164$	−3.08292	−0.240736
$165$	0	0
$166$	0.951319	0.0738366
$167$	−22.4729	−1.73900	−0.869502	−	0.493930i	$-0.835560\pi$
$167$	−22.4729	−1.73900	−0.869502	+	0.493930i	$0.835560\pi$
$168$	0	0
$169$	−12.0217	−0.924747
$170$	0.907229	0.0695813
$171$	0	0
$172$	5.06682	0.386341
$173$	−21.2142	−1.61289	−0.806444	−	0.591310i	$-0.798611\pi$
$173$	−21.2142	−1.61289	−0.806444	+	0.591310i	$0.798611\pi$
$174$	0	0
$175$	3.83036	0.289548
$176$	0.519184	0.0391350
$177$	0	0
$178$	12.8355	0.962064
$179$	10.8003	0.807249	0.403625	−	0.914925i	$-0.367750\pi$
$179$	10.8003	0.807249	0.403625	+	0.914925i	$0.367750\pi$
$180$	0	0
$181$	−0.438278	−0.0325770	−0.0162885	−	0.999867i	$-0.505185\pi$
$181$	−0.438278	−0.0325770	−0.0162885	+	0.999867i	$0.505185\pi$
$182$	3.39999	0.252024
$183$	0	0
$184$	−22.3318	−1.64632
$185$	−10.7648	−0.791443
$186$	0	0
$187$	2.85716	0.208937
$188$	−0.301661	−0.0220009
$189$	0	0
$190$	−4.60625	−0.334173
$191$	2.64477	0.191369	0.0956844	−	0.995412i	$-0.469496\pi$
$191$	2.64477	0.191369	0.0956844	+	0.995412i	$0.469496\pi$
$192$	0	0
$193$	−16.4494	−1.18406	−0.592028	−	0.805917i	$-0.701673\pi$
$193$	−16.4494	−1.18406	−0.592028	+	0.805917i	$0.701673\pi$
$194$	12.4690	0.895224
$195$	0	0
$196$	−9.16458	−0.654613
$197$	−22.1732	−1.57977	−0.789887	−	0.613253i	$-0.789861\pi$
$197$	−22.1732	−1.57977	−0.789887	+	0.613253i	$0.789861\pi$
$198$	0	0
$199$	11.4778	0.813638	0.406819	−	0.913509i	$-0.366638\pi$
$199$	11.4778	0.813638	0.406819	+	0.913509i	$0.366638\pi$
$200$	−2.86696	−0.202724
$201$	0	0
$202$	−12.9885	−0.913867
$203$	3.83036	0.268838
$204$	0	0
$205$	−2.58070	−0.180244
$206$	7.81608	0.544572
$207$	0	0
$208$	−0.181691	−0.0125980
$209$	−14.5066	−1.00344
$210$	0	0
$211$	10.2941	0.708673	0.354337	−	0.935118i	$-0.384707\pi$
$211$	10.2941	0.708673	0.354337	+	0.935118i	$0.384707\pi$
$212$	−6.81701	−0.468194
$213$	0	0
$214$	−1.70091	−0.116272
$215$	4.24141	0.289262
$216$	0	0
$217$	−4.33856	−0.294521
$218$	5.94516	0.402657
$219$	0	0
$220$	−3.37635	−0.227633
$221$	−0.999881	−0.0672593
$222$	0	0
$223$	−13.3163	−0.891724	−0.445862	−	0.895102i	$-0.647103\pi$
$223$	−13.3163	−0.891724	−0.445862	+	0.895102i	$0.647103\pi$
$224$	21.3315	1.42527
$225$	0	0
$226$	0.897984	0.0597330
$227$	−24.5678	−1.63062	−0.815312	−	0.579022i	$-0.803434\pi$
$227$	−24.5678	−1.63062	−0.815312	+	0.579022i	$0.803434\pi$
$228$	0	0
$229$	−25.5488	−1.68831	−0.844157	−	0.536096i	$-0.819898\pi$
$229$	−25.5488	−1.68831	−0.844157	+	0.536096i	$0.819898\pi$
$230$	−6.99045	−0.460937
$231$	0	0
$232$	−2.86696	−0.188225
$233$	−5.77745	−0.378493	−0.189247	−	0.981930i	$-0.560605\pi$
$233$	−5.77745	−0.378493	−0.189247	+	0.981930i	$0.560605\pi$
$234$	0	0
$235$	−0.252519	−0.0164725
$236$	4.37311	0.284666
$237$	0	0
$238$	−3.47501	−0.225251
$239$	−4.16575	−0.269460	−0.134730	−	0.990882i	$-0.543017\pi$
$239$	−4.16575	−0.269460	−0.134730	+	0.990882i	$0.543017\pi$
$240$	0	0
$241$	11.3115	0.728637	0.364319	−	0.931274i	$-0.381302\pi$
$241$	11.3115	0.728637	0.364319	+	0.931274i	$0.381302\pi$
$242$	−2.70298	−0.173754
$243$	0	0
$244$	12.4024	0.793981
$245$	−7.67162	−0.490122
$246$	0	0
$247$	5.07667	0.323021
$248$	3.24734	0.206206
$249$	0	0
$250$	−0.897436	−0.0567589
$251$	6.70959	0.423505	0.211753	−	0.977323i	$-0.432083\pi$
$251$	6.70959	0.423505	0.211753	+	0.977323i	$0.432083\pi$
$252$	0	0
$253$	−22.0152	−1.38409
$254$	−8.87929	−0.557136
$255$	0	0
$256$	−16.4051	−1.02532
$257$	20.0804	1.25258	0.626290	−	0.779590i	$-0.284573\pi$
$257$	20.0804	1.25258	0.626290	+	0.779590i	$0.284573\pi$
$258$	0	0
$259$	41.2330	2.56209
$260$	1.18157	0.0732780
$261$	0	0
$262$	11.7716	0.727249
$263$	25.4859	1.57153	0.785765	−	0.618526i	$-0.212270\pi$
$263$	25.4859	1.57153	0.785765	+	0.618526i	$0.212270\pi$
$264$	0	0
$265$	−5.70648	−0.350546
$266$	17.6436	1.08180
$267$	0	0
$268$	−4.49100	−0.274331
$269$	26.1794	1.59618	0.798092	−	0.602535i	$-0.205843\pi$
$269$	26.1794	1.59618	0.798092	+	0.602535i	$0.205843\pi$
$270$	0	0
$271$	3.60583	0.219039	0.109519	−	0.993985i	$-0.465069\pi$
$271$	3.60583	0.219039	0.109519	+	0.993985i	$0.465069\pi$
$272$	0.185700	0.0112597
$273$	0	0
$274$	0.0636870	0.00384747
$275$	−2.82632	−0.170434
$276$	0	0
$277$	15.4340	0.927336	0.463668	−	0.886009i	$-0.346533\pi$
$277$	15.4340	0.927336	0.463668	+	0.886009i	$0.346533\pi$
$278$	−10.9902	−0.659147
$279$	0	0
$280$	10.9815	0.656268
$281$	−9.13957	−0.545221	−0.272611	−	0.962124i	$-0.587887\pi$
$281$	−9.13957	−0.545221	−0.272611	+	0.962124i	$0.587887\pi$
$282$	0	0
$283$	13.9174	0.827302	0.413651	−	0.910436i	$-0.364253\pi$
$283$	13.9174	0.827302	0.413651	+	0.910436i	$0.364253\pi$
$284$	7.48464	0.444132
$285$	0	0
$286$	−2.50877	−0.148347
$287$	9.88499	0.583493
$288$	0	0
$289$	−15.9781	−0.939886
$290$	−0.897436	−0.0526993
$291$	0	0
$292$	−12.8474	−0.751837
$293$	20.2400	1.18244	0.591218	−	0.806512i	$-0.298647\pi$
$293$	20.2400	1.18244	0.591218	+	0.806512i	$0.298647\pi$
$294$	0	0
$295$	3.66071	0.213135
$296$	−30.8622	−1.79383
$297$	0	0
$298$	−6.57051	−0.380619
$299$	7.70436	0.445555
$300$	0	0
$301$	−16.2461	−0.936410
$302$	−20.9820	−1.20738
$303$	0	0
$304$	−0.942852	−0.0540763
$305$	10.3820	0.594470
$306$	0	0
$307$	−2.57263	−0.146828	−0.0734140	−	0.997302i	$-0.523389\pi$
$307$	−2.57263	−0.146828	−0.0734140	+	0.997302i	$0.523389\pi$
$308$	12.9326	0.736904
$309$	0	0
$310$	1.01651	0.0577337
$311$	19.0082	1.07785	0.538927	−	0.842352i	$-0.318830\pi$
$311$	19.0082	1.07785	0.538927	+	0.842352i	$0.318830\pi$
$312$	0	0
$313$	0.555016	0.0313713	0.0156857	−	0.999877i	$-0.495007\pi$
$313$	0.555016	0.0313713	0.0156857	+	0.999877i	$0.495007\pi$
$314$	−0.835867	−0.0471707
$315$	0	0
$316$	−12.8842	−0.724795
$317$	28.5513	1.60360	0.801801	−	0.597592i	$-0.203876\pi$
$317$	28.5513	1.60360	0.801801	+	0.597592i	$0.203876\pi$
$318$	0	0
$319$	−2.82632	−0.158244
$320$	−5.36527	−0.299928
$321$	0	0
$322$	26.7759	1.49216
$323$	−5.18869	−0.288706
$324$	0	0
$325$	0.989088	0.0548647
$326$	−0.925142	−0.0512389
$327$	0	0
$328$	−7.39875	−0.408528
$329$	0.967237	0.0533255
$330$	0	0
$331$	3.88679	0.213637	0.106819	−	0.994279i	$-0.465934\pi$
$331$	3.88679	0.213637	0.106819	+	0.994279i	$0.465934\pi$
$332$	−1.26633	−0.0694990
$333$	0	0
$334$	−20.1680	−1.10354
$335$	−3.75939	−0.205397
$336$	0	0
$337$	7.48393	0.407676	0.203838	−	0.979005i	$-0.434658\pi$
$337$	7.48393	0.407676	0.203838	+	0.979005i	$0.434658\pi$
$338$	−10.7887	−0.586829
$339$	0	0
$340$	−1.20764	−0.0654937
$341$	3.20132	0.173361
$342$	0	0
$343$	2.57255	0.138905
$344$	12.1599	0.655620
$345$	0	0
$346$	−19.0384	−1.02351
$347$	−34.8424	−1.87044	−0.935220	−	0.354068i	$-0.884798\pi$
$347$	−34.8424	−1.87044	−0.935220	+	0.354068i	$0.884798\pi$
$348$	0	0
$349$	16.1457	0.864257	0.432128	−	0.901812i	$-0.357763\pi$
$349$	16.1457	0.864257	0.432128	+	0.901812i	$0.357763\pi$
$350$	3.43750	0.183742
$351$	0	0
$352$	−15.7400	−0.838943
$353$	−2.02732	−0.107903	−0.0539516	−	0.998544i	$-0.517182\pi$
$353$	−2.02732	−0.107903	−0.0539516	+	0.998544i	$0.517182\pi$
$354$	0	0
$355$	6.26536	0.332531
$356$	−17.0858	−0.905546
$357$	0	0
$358$	9.69254	0.512267
$359$	9.77827	0.516077	0.258039	−	0.966135i	$-0.416924\pi$
$359$	9.77827	0.516077	0.258039	+	0.966135i	$0.416924\pi$
$360$	0	0
$361$	7.34439	0.386547
$362$	−0.393327	−0.0206728
$363$	0	0
$364$	−4.52584	−0.237219
$365$	−10.7545	−0.562915
$366$	0	0
$367$	9.26435	0.483595	0.241797	−	0.970327i	$-0.422263\pi$
$367$	9.26435	0.483595	0.241797	+	0.970327i	$0.422263\pi$
$368$	−1.43087	−0.0745894
$369$	0	0
$370$	−9.66071	−0.502236
$371$	21.8579	1.13480
$372$	0	0
$373$	−19.3544	−1.00213	−0.501067	−	0.865408i	$-0.667059\pi$
$373$	−19.3544	−1.00213	−0.501067	+	0.865408i	$0.667059\pi$
$374$	2.56412	0.132588
$375$	0	0
$376$	−0.723961	−0.0373354
$377$	0.989088	0.0509406
$378$	0	0
$379$	15.8048	0.811838	0.405919	−	0.913909i	$-0.366952\pi$
$379$	15.8048	0.811838	0.405919	+	0.913909i	$0.366952\pi$
$380$	6.13154	0.314541
$381$	0	0
$382$	2.37351	0.121439
$383$	2.60067	0.132888	0.0664440	−	0.997790i	$-0.478835\pi$
$383$	2.60067	0.132888	0.0664440	+	0.997790i	$0.478835\pi$
$384$	0	0
$385$	10.8258	0.551735
$386$	−14.7623	−0.751383
$387$	0	0
$388$	−16.5979	−0.842633
$389$	−13.4068	−0.679750	−0.339875	−	0.940471i	$-0.610385\pi$
$389$	−13.4068	−0.679750	−0.339875	+	0.940471i	$0.610385\pi$
$390$	0	0
$391$	−7.87436	−0.398223
$392$	−21.9942	−1.11088
$393$	0	0
$394$	−19.8990	−1.00250
$395$	−10.7853	−0.542669
$396$	0	0
$397$	−33.8760	−1.70019	−0.850093	−	0.526633i	$-0.823454\pi$
$397$	−33.8760	−1.70019	−0.850093	+	0.526633i	$0.823454\pi$
$398$	10.3006	0.516321
$399$	0	0
$400$	−0.183696	−0.00918479
$401$	25.8578	1.29128	0.645639	−	0.763643i	$-0.276591\pi$
$401$	25.8578	1.29128	0.645639	+	0.763643i	$0.276591\pi$
$402$	0	0
$403$	−1.12032	−0.0558070
$404$	17.2894	0.860181
$405$	0	0
$406$	3.43750	0.170600
$407$	−30.4248	−1.50810
$408$	0	0
$409$	−12.3900	−0.612646	−0.306323	−	0.951928i	$-0.599099\pi$
$409$	−12.3900	−0.612646	−0.306323	+	0.951928i	$0.599099\pi$
$410$	−2.31601	−0.114380
$411$	0	0
$412$	−10.4043	−0.512581
$413$	−14.0218	−0.689969
$414$	0	0
$415$	−1.06004	−0.0520353
$416$	5.50829	0.270066
$417$	0	0
$418$	−13.0188	−0.636768
$419$	32.4545	1.58551	0.792753	−	0.609543i	$-0.208647\pi$
$419$	32.4545	1.58551	0.792753	+	0.609543i	$0.208647\pi$
$420$	0	0
$421$	13.2642	0.646458	0.323229	−	0.946321i	$-0.395232\pi$
$421$	13.2642	0.646458	0.323229	+	0.946321i	$0.395232\pi$
$422$	9.23828	0.449712
$423$	0	0
$424$	−16.3602	−0.794524
$425$	−1.01091	−0.0490364
$426$	0	0
$427$	−39.7666	−1.92444
$428$	2.26414	0.109441
$429$	0	0
$430$	3.80639	0.183561
$431$	15.5050	0.746849	0.373425	−	0.927661i	$-0.378183\pi$
$431$	15.5050	0.746849	0.373425	+	0.927661i	$0.378183\pi$
$432$	0	0
$433$	1.39425	0.0670034	0.0335017	−	0.999439i	$-0.489334\pi$
$433$	1.39425	0.0670034	0.0335017	+	0.999439i	$0.489334\pi$
$434$	−3.89358	−0.186898
$435$	0	0
$436$	−7.91381	−0.379003
$437$	39.9803	1.91252
$438$	0	0
$439$	20.2277	0.965416	0.482708	−	0.875781i	$-0.339653\pi$
$439$	20.2277	0.965416	0.482708	+	0.875781i	$0.339653\pi$
$440$	−8.10295	−0.386293
$441$	0	0
$442$	−0.897330	−0.0426816
$443$	−15.1065	−0.717731	−0.358865	−	0.933389i	$-0.616836\pi$
$443$	−15.1065	−0.717731	−0.358865	+	0.933389i	$0.616836\pi$
$444$	0	0
$445$	−14.3024	−0.678001
$446$	−11.9505	−0.565873
$447$	0	0
$448$	20.5509	0.970938
$449$	38.3771	1.81113	0.905564	−	0.424210i	$-0.139448\pi$
$449$	38.3771	1.81113	0.905564	+	0.424210i	$0.139448\pi$
$450$	0	0
$451$	−7.29389	−0.343456
$452$	−1.19534	−0.0562239
$453$	0	0
$454$	−22.0481	−1.03477
$455$	−3.78856	−0.177610
$456$	0	0
$457$	−22.2230	−1.03955	−0.519775	−	0.854303i	$-0.673984\pi$
$457$	−22.2230	−1.03955	−0.519775	+	0.854303i	$0.673984\pi$
$458$	−22.9284	−1.07138
$459$	0	0
$460$	9.30523	0.433858
$461$	−5.81739	−0.270943	−0.135472	−	0.990781i	$-0.543255\pi$
$461$	−5.81739	−0.270943	−0.135472	+	0.990781i	$0.543255\pi$
$462$	0	0
$463$	26.7453	1.24296	0.621479	−	0.783431i	$-0.286532\pi$
$463$	26.7453	1.24296	0.621479	+	0.783431i	$0.286532\pi$
$464$	−0.183696	−0.00852787
$465$	0	0
$466$	−5.18489	−0.240185
$467$	13.7869	0.637980	0.318990	−	0.947758i	$-0.396656\pi$
$467$	13.7869	0.637980	0.318990	+	0.947758i	$0.396656\pi$
$468$	0	0
$469$	14.3998	0.664921
$470$	−0.226620	−0.0104532
$471$	0	0
$472$	10.4951	0.483076
$473$	11.9876	0.551190
$474$	0	0
$475$	5.13268	0.235503
$476$	4.62570	0.212019
$477$	0	0
$478$	−3.73849	−0.170995
$479$	−15.6990	−0.717304	−0.358652	−	0.933471i	$-0.616764\pi$
$479$	−15.6990	−0.717304	−0.358652	+	0.933471i	$0.616764\pi$
$480$	0	0
$481$	10.6473	0.485476
$482$	10.1513	0.462381
$483$	0	0
$484$	3.59803	0.163547
$485$	−13.8941	−0.630897
$486$	0	0
$487$	30.0638	1.36232	0.681160	−	0.732135i	$-0.261476\pi$
$487$	30.0638	1.36232	0.681160	+	0.732135i	$0.261476\pi$
$488$	29.7646	1.34738
$489$	0	0
$490$	−6.88479	−0.311023
$491$	−35.3955	−1.59738	−0.798689	−	0.601744i	$-0.794473\pi$
$491$	−35.3955	−1.59738	−0.798689	+	0.601744i	$0.794473\pi$
$492$	0	0
$493$	−1.01091	−0.0455292
$494$	4.55599	0.204984
$495$	0	0
$496$	0.208068	0.00934254
$497$	−23.9985	−1.07648
$498$	0	0
$499$	8.84462	0.395940	0.197970	−	0.980208i	$-0.436565\pi$
$499$	8.84462	0.395940	0.197970	+	0.980208i	$0.436565\pi$
$500$	1.19461	0.0534245
$501$	0	0
$502$	6.02143	0.268749
$503$	−24.2714	−1.08221	−0.541104	−	0.840956i	$-0.681993\pi$
$503$	−24.2714	−1.08221	−0.541104	+	0.840956i	$0.681993\pi$
$504$	0	0
$505$	14.4729	0.644035
$506$	−19.7573	−0.878318
$507$	0	0
$508$	11.8195	0.524406
$509$	−7.45863	−0.330598	−0.165299	−	0.986243i	$-0.552859\pi$
$509$	−7.45863	−0.330598	−0.165299	+	0.986243i	$0.552859\pi$
$510$	0	0
$511$	41.1935	1.82229
$512$	−2.07631	−0.0917608
$513$	0	0
$514$	18.0209	0.794866
$515$	−8.70934	−0.383780
$516$	0	0
$517$	−0.713700	−0.0313885
$518$	37.0040	1.62586
$519$	0	0
$520$	2.83567	0.124352
$521$	−30.8783	−1.35280	−0.676401	−	0.736533i	$-0.736461\pi$
$521$	−30.8783	−1.35280	−0.676401	+	0.736533i	$0.736461\pi$
$522$	0	0
$523$	−39.1262	−1.71087	−0.855435	−	0.517910i	$-0.826710\pi$
$523$	−39.1262	−1.71087	−0.855435	+	0.517910i	$0.826710\pi$
$524$	−15.6695	−0.684526
$525$	0	0
$526$	22.8720	0.997266
$527$	1.14504	0.0498786
$528$	0	0
$529$	37.6741	1.63800
$530$	−5.12121	−0.222451
$531$	0	0
$532$	−23.4860	−1.01825
$533$	2.55254	0.110563
$534$	0	0
$535$	1.89530	0.0819410
$536$	−10.7780	−0.465539
$537$	0	0
$538$	23.4943	1.01291
$539$	−21.6825	−0.933931
$540$	0	0
$541$	−27.9986	−1.20375	−0.601877	−	0.798589i	$-0.705580\pi$
$541$	−27.9986	−1.20375	−0.601877	+	0.798589i	$0.705580\pi$
$542$	3.23601	0.138998
$543$	0	0
$544$	−5.62983	−0.241377
$545$	−6.62461	−0.283767
$546$	0	0
$547$	7.05192	0.301518	0.150759	−	0.988571i	$-0.451828\pi$
$547$	7.05192	0.301518	0.150759	+	0.988571i	$0.451828\pi$
$548$	−0.0847759	−0.00362145
$549$	0	0
$550$	−2.53645	−0.108154
$551$	5.13268	0.218659
$552$	0	0
$553$	41.3116	1.75675
$554$	13.8510	0.588472
$555$	0	0
$556$	14.6294	0.620424
$557$	38.4865	1.63073	0.815363	−	0.578949i	$-0.196537\pi$
$557$	38.4865	1.63073	0.815363	+	0.578949i	$0.196537\pi$
$558$	0	0
$559$	−4.19513	−0.177435
$560$	0.703621	0.0297334
$561$	0	0
$562$	−8.20218	−0.345988
$563$	−45.6728	−1.92488	−0.962439	−	0.271498i	$-0.912481\pi$
$563$	−45.6728	−1.92488	−0.962439	+	0.271498i	$0.912481\pi$
$564$	0	0
$565$	−1.00061	−0.0420960
$566$	12.4900	0.524992
$567$	0	0
$568$	17.9625	0.753690
$569$	44.9498	1.88439	0.942196	−	0.335062i	$-0.108757\pi$
$569$	44.9498	1.88439	0.942196	+	0.335062i	$0.108757\pi$
$570$	0	0
$571$	−1.65737	−0.0693587	−0.0346793	−	0.999398i	$-0.511041\pi$
$571$	−1.65737	−0.0693587	−0.0346793	+	0.999398i	$0.511041\pi$
$572$	3.33951	0.139632
$573$	0	0
$574$	8.87115	0.370275
$575$	7.78936	0.324839
$576$	0	0
$577$	−9.23893	−0.384622	−0.192311	−	0.981334i	$-0.561598\pi$
$577$	−9.23893	−0.384622	−0.192311	+	0.981334i	$0.561598\pi$
$578$	−14.3393	−0.596436
$579$	0	0
$580$	1.19461	0.0496034
$581$	4.06033	0.168451
$582$	0	0
$583$	−16.1284	−0.667969
$584$	−30.8326	−1.27586
$585$	0	0
$586$	18.1641	0.750354
$587$	10.6148	0.438120	0.219060	−	0.975711i	$-0.429701\pi$
$587$	10.6148	0.438120	0.219060	+	0.975711i	$0.429701\pi$
$588$	0	0
$589$	−5.81367	−0.239548
$590$	3.28526	0.135252
$591$	0	0
$592$	−1.97745	−0.0812726
$593$	−39.7432	−1.63206	−0.816029	−	0.578011i	$-0.803829\pi$
$593$	−39.7432	−1.63206	−0.816029	+	0.578011i	$0.803829\pi$
$594$	0	0
$595$	3.87215	0.158743
$596$	8.74623	0.358259
$597$	0	0
$598$	6.91417	0.282742
$599$	7.31224	0.298770	0.149385	−	0.988779i	$-0.452271\pi$
$599$	7.31224	0.298770	0.149385	+	0.988779i	$0.452271\pi$
$600$	0	0
$601$	−21.4162	−0.873585	−0.436793	−	0.899562i	$-0.643886\pi$
$601$	−21.4162	−0.873585	−0.436793	+	0.899562i	$0.643886\pi$
$602$	−14.5798	−0.594230
$603$	0	0
$604$	27.9299	1.13645
$605$	3.01190	0.122451
$606$	0	0
$607$	19.3315	0.784639	0.392320	−	0.919829i	$-0.371673\pi$
$607$	19.3315	0.784639	0.392320	+	0.919829i	$0.371673\pi$
$608$	28.5842	1.15924
$609$	0	0
$610$	9.31715	0.377240
$611$	0.249763	0.0101043
$612$	0	0
$613$	−27.5848	−1.11414	−0.557069	−	0.830466i	$-0.688074\pi$
$613$	−27.5848	−1.11414	−0.557069	+	0.830466i	$0.688074\pi$
$614$	−2.30878	−0.0931746
$615$	0	0
$616$	31.0372	1.25052
$617$	9.03573	0.363765	0.181882	−	0.983320i	$-0.441781\pi$
$617$	9.03573	0.363765	0.181882	+	0.983320i	$0.441781\pi$
$618$	0	0
$619$	−41.7871	−1.67956	−0.839782	−	0.542923i	$-0.817318\pi$
$619$	−41.7871	−1.67956	−0.839782	+	0.542923i	$0.817318\pi$
$620$	−1.35311	−0.0543421
$621$	0	0
$622$	17.0586	0.683989
$623$	54.7835	2.19485
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.498091	0.0199077
$627$	0	0
$628$	1.11265	0.0443996
$629$	−10.8823	−0.433904
$630$	0	0
$631$	44.6303	1.77670	0.888351	−	0.459164i	$-0.151851\pi$
$631$	44.6303	1.77670	0.888351	+	0.459164i	$0.151851\pi$
$632$	−30.9211	−1.22997
$633$	0	0
$634$	25.6230	1.01762
$635$	9.89406	0.392634
$636$	0	0
$637$	7.58791	0.300644
$638$	−2.53645	−0.100419
$639$	0	0
$640$	6.32313	0.249944
$641$	−38.4320	−1.51797	−0.758986	−	0.651107i	$-0.774305\pi$
$641$	−38.4320	−1.51797	−0.758986	+	0.651107i	$0.774305\pi$
$642$	0	0
$643$	−3.33226	−0.131411	−0.0657057	−	0.997839i	$-0.520930\pi$
$643$	−3.33226	−0.131411	−0.0657057	+	0.997839i	$0.520930\pi$
$644$	−35.6423	−1.40450
$645$	0	0
$646$	−4.65652	−0.183208
$647$	−25.0180	−0.983559	−0.491779	−	0.870720i	$-0.663653\pi$
$647$	−25.0180	−0.983559	−0.491779	+	0.870720i	$0.663653\pi$
$648$	0	0
$649$	10.3464	0.406130
$650$	0.887643	0.0348162
$651$	0	0
$652$	1.23149	0.0482288
$653$	−24.9697	−0.977142	−0.488571	−	0.872524i	$-0.662482\pi$
$653$	−24.9697	−0.977142	−0.488571	+	0.872524i	$0.662482\pi$
$654$	0	0
$655$	−13.1169	−0.512519
$656$	−0.474064	−0.0185091
$657$	0	0
$658$	0.868034	0.0338395
$659$	37.4164	1.45754	0.728769	−	0.684760i	$-0.240093\pi$
$659$	37.4164	1.45754	0.728769	+	0.684760i	$0.240093\pi$
$660$	0	0
$661$	−39.5042	−1.53653	−0.768267	−	0.640130i	$-0.778881\pi$
$661$	−39.5042	−1.53653	−0.768267	+	0.640130i	$0.778881\pi$
$662$	3.48814	0.135571
$663$	0	0
$664$	−3.03909	−0.117940
$665$	−19.6600	−0.762381
$666$	0	0
$667$	7.78936	0.301605
$668$	26.8463	1.03871
$669$	0	0
$670$	−3.37381	−0.130342
$671$	29.3428	1.13277
$672$	0	0
$673$	−49.7546	−1.91790	−0.958950	−	0.283576i	$-0.908479\pi$
$673$	−49.7546	−1.91790	−0.958950	+	0.283576i	$0.908479\pi$
$674$	6.71635	0.258704
$675$	0	0
$676$	14.3612	0.552355
$677$	−2.62339	−0.100825	−0.0504126	−	0.998728i	$-0.516054\pi$
$677$	−2.62339	−0.100825	−0.0504126	+	0.998728i	$0.516054\pi$
$678$	0	0
$679$	53.2192	2.04236
$680$	−2.89824	−0.111142
$681$	0	0
$682$	2.87298	0.110012
$683$	18.9784	0.726188	0.363094	−	0.931752i	$-0.381720\pi$
$683$	18.9784	0.726188	0.363094	+	0.931752i	$0.381720\pi$
$684$	0	0
$685$	−0.0709655	−0.00271145
$686$	2.30870	0.0881467
$687$	0	0
$688$	0.779129	0.0297040
$689$	5.64421	0.215027
$690$	0	0
$691$	33.6342	1.27951	0.639753	−	0.768580i	$-0.279037\pi$
$691$	33.6342	1.27951	0.639753	+	0.768580i	$0.279037\pi$
$692$	25.3427	0.963385
$693$	0	0
$694$	−31.2689	−1.18695
$695$	12.2462	0.464524
$696$	0	0
$697$	−2.60886	−0.0988176
$698$	14.4897	0.548443
$699$	0	0
$700$	−4.57577	−0.172948
$701$	29.6166	1.11861	0.559303	−	0.828964i	$-0.311069\pi$
$701$	29.6166	1.11861	0.559303	+	0.828964i	$0.311069\pi$
$702$	0	0
$703$	55.2522	2.08388
$704$	−15.1640	−0.571514
$705$	0	0
$706$	−1.81939	−0.0684736
$707$	−55.4363	−2.08490
$708$	0	0
$709$	−35.4031	−1.32959	−0.664795	−	0.747026i	$-0.731481\pi$
$709$	−35.4031	−1.32959	−0.664795	+	0.747026i	$0.731481\pi$
$710$	5.62276	0.211018
$711$	0	0
$712$	−41.0045	−1.53671
$713$	−8.82284	−0.330418
$714$	0	0
$715$	2.79548	0.104545
$716$	−12.9021	−0.482173
$717$	0	0
$718$	8.77538	0.327494
$719$	−7.78651	−0.290388	−0.145194	−	0.989403i	$-0.546381\pi$
$719$	−7.78651	−0.290388	−0.145194	+	0.989403i	$0.546381\pi$
$720$	0	0
$721$	33.3599	1.24239
$722$	6.59112	0.245296
$723$	0	0
$724$	0.523571	0.0194584
$725$	1.00000	0.0371391
$726$	0	0
$727$	−7.78963	−0.288901	−0.144451	−	0.989512i	$-0.546142\pi$
$727$	−7.78963	−0.288901	−0.144451	+	0.989512i	$0.546142\pi$
$728$	−10.8616	−0.402559
$729$	0	0
$730$	−9.65147	−0.357217
$731$	4.28769	0.158586
$732$	0	0
$733$	−0.518655	−0.0191570	−0.00957848	−	0.999954i	$-0.503049\pi$
$733$	−0.518655	−0.0191570	−0.00957848	+	0.999954i	$0.503049\pi$
$734$	8.31416	0.306881
$735$	0	0
$736$	43.3794	1.59899
$737$	−10.6253	−0.391386
$738$	0	0
$739$	30.7195	1.13003	0.565017	−	0.825079i	$-0.308870\pi$
$739$	30.7195	1.13003	0.565017	+	0.825079i	$0.308870\pi$
$740$	12.8597	0.472732
$741$	0	0
$742$	19.6160	0.720127
$743$	−21.7734	−0.798790	−0.399395	−	0.916779i	$-0.630780\pi$
$743$	−21.7734	−0.798790	−0.399395	+	0.916779i	$0.630780\pi$
$744$	0	0
$745$	7.32142	0.268236
$746$	−17.3694	−0.635938
$747$	0	0
$748$	−3.41319	−0.124799
$749$	−7.25968	−0.265263
$750$	0	0
$751$	24.9369	0.909961	0.454981	−	0.890501i	$-0.349646\pi$
$751$	24.9369	0.909961	0.454981	+	0.890501i	$0.349646\pi$
$752$	−0.0463867	−0.00169155
$753$	0	0
$754$	0.887643	0.0323261
$755$	23.3800	0.850885
$756$	0	0
$757$	1.27972	0.0465123	0.0232561	−	0.999730i	$-0.492597\pi$
$757$	1.27972	0.0465123	0.0232561	+	0.999730i	$0.492597\pi$
$758$	14.1838	0.515179
$759$	0	0
$760$	14.7152	0.533775
$761$	1.95635	0.0709177	0.0354588	−	0.999371i	$-0.488711\pi$
$761$	1.95635	0.0709177	0.0354588	+	0.999371i	$0.488711\pi$
$762$	0	0
$763$	25.3746	0.918622
$764$	−3.15946	−0.114305
$765$	0	0
$766$	2.33394	0.0843286
$767$	−3.62077	−0.130738
$768$	0	0
$769$	−22.1931	−0.800302	−0.400151	−	0.916449i	$-0.631042\pi$
$769$	−22.1931	−0.800302	−0.400151	+	0.916449i	$0.631042\pi$
$770$	9.71549	0.350122
$771$	0	0
$772$	19.6506	0.707242
$773$	15.6896	0.564317	0.282158	−	0.959368i	$-0.408950\pi$
$773$	15.6896	0.564317	0.282158	+	0.959368i	$0.408950\pi$
$774$	0	0
$775$	−1.13268	−0.0406870
$776$	−39.8337	−1.42995
$777$	0	0
$778$	−12.0317	−0.431358
$779$	13.2459	0.474583
$780$	0	0
$781$	17.7079	0.633639
$782$	−7.06673	−0.252706
$783$	0	0
$784$	−1.40925	−0.0503302
$785$	0.931394	0.0332429
$786$	0	0
$787$	−8.82023	−0.314407	−0.157204	−	0.987566i	$-0.550248\pi$
$787$	−8.82023	−0.314407	−0.157204	+	0.987566i	$0.550248\pi$
$788$	26.4882	0.943605
$789$	0	0
$790$	−9.67914	−0.344368
$791$	3.83269	0.136275
$792$	0	0
$793$	−10.2687	−0.364651
$794$	−30.4015	−1.07891
$795$	0	0
$796$	−13.7114	−0.485989
$797$	14.5755	0.516290	0.258145	−	0.966106i	$-0.416889\pi$
$797$	14.5755	0.516290	0.258145	+	0.966106i	$0.416889\pi$
$798$	0	0
$799$	−0.255274	−0.00903096
$800$	5.56906	0.196896
$801$	0	0
$802$	23.2058	0.819424
$803$	−30.3957	−1.07264
$804$	0	0
$805$	−29.8360	−1.05158
$806$	−1.00541	−0.0354142
$807$	0	0
$808$	41.4932	1.45972
$809$	10.4702	0.368111	0.184056	−	0.982916i	$-0.441077\pi$
$809$	10.4702	0.368111	0.184056	+	0.982916i	$0.441077\pi$
$810$	0	0
$811$	−9.74921	−0.342341	−0.171171	−	0.985241i	$-0.554755\pi$
$811$	−9.74921	−0.342341	−0.171171	+	0.985241i	$0.554755\pi$
$812$	−4.57577	−0.160578
$813$	0	0
$814$	−27.3043	−0.957015
$815$	1.03087	0.0361099
$816$	0	0
$817$	−21.7698	−0.761628
$818$	−11.1192	−0.388775
$819$	0	0
$820$	3.08292	0.107660
$821$	−40.9219	−1.42819	−0.714093	−	0.700051i	$-0.753160\pi$
$821$	−40.9219	−1.42819	−0.714093	+	0.700051i	$0.753160\pi$
$822$	0	0
$823$	−4.87322	−0.169870	−0.0849348	−	0.996387i	$-0.527068\pi$
$823$	−4.87322	−0.169870	−0.0849348	+	0.996387i	$0.527068\pi$
$824$	−24.9693	−0.869847
$825$	0	0
$826$	−12.5837	−0.437843
$827$	−15.6198	−0.543153	−0.271577	−	0.962417i	$-0.587545\pi$
$827$	−15.6198	−0.543153	−0.271577	+	0.962417i	$0.587545\pi$
$828$	0	0
$829$	−51.5007	−1.78869	−0.894346	−	0.447375i	$-0.852359\pi$
$829$	−51.5007	−1.78869	−0.894346	+	0.447375i	$0.852359\pi$
$830$	−0.951319	−0.0330207
$831$	0	0
$832$	5.30672	0.183977
$833$	−7.75534	−0.268706
$834$	0	0
$835$	22.4729	0.777706
$836$	17.3297	0.599361
$837$	0	0
$838$	29.1258	1.00614
$839$	−38.1841	−1.31826	−0.659130	−	0.752029i	$-0.729075\pi$
$839$	−38.1841	−1.31826	−0.659130	+	0.752029i	$0.729075\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	11.9038	0.410232
$843$	0	0
$844$	−12.2974	−0.423293
$845$	12.0217	0.413559
$846$	0	0
$847$	−11.5366	−0.396403
$848$	−1.04826	−0.0359973
$849$	0	0
$850$	−0.907229	−0.0311177
$851$	83.8508	2.87437
$852$	0	0
$853$	−38.0575	−1.30306	−0.651531	−	0.758622i	$-0.725873\pi$
$853$	−38.0575	−1.30306	−0.651531	+	0.758622i	$0.725873\pi$
$854$	−35.6880	−1.22122
$855$	0	0
$856$	5.43375	0.185722
$857$	−21.8229	−0.745455	−0.372728	−	0.927941i	$-0.621577\pi$
$857$	−21.8229	−0.745455	−0.372728	+	0.927941i	$0.621577\pi$
$858$	0	0
$859$	−55.1041	−1.88013	−0.940065	−	0.340996i	$-0.889236\pi$
$859$	−55.1041	−1.88013	−0.940065	+	0.340996i	$0.889236\pi$
$860$	−5.06682	−0.172777
$861$	0	0
$862$	13.9147	0.473938
$863$	10.5055	0.357612	0.178806	−	0.983884i	$-0.442777\pi$
$863$	10.5055	0.357612	0.178806	+	0.983884i	$0.442777\pi$
$864$	0	0
$865$	21.2142	0.721306
$866$	1.25125	0.0425192
$867$	0	0
$868$	5.18288	0.175918
$869$	−30.4828	−1.03406
$870$	0	0
$871$	3.71837	0.125992
$872$	−18.9925	−0.643166
$873$	0	0
$874$	35.8797	1.21365
$875$	−3.83036	−0.129490
$876$	0	0
$877$	−12.0379	−0.406492	−0.203246	−	0.979128i	$-0.565149\pi$
$877$	−12.0379	−0.406492	−0.203246	+	0.979128i	$0.565149\pi$
$878$	18.1531	0.612637
$879$	0	0
$880$	−0.519184	−0.0175017
$881$	−40.8416	−1.37599	−0.687994	−	0.725716i	$-0.741509\pi$
$881$	−40.8416	−1.37599	−0.687994	+	0.725716i	$0.741509\pi$
$882$	0	0
$883$	41.6187	1.40058	0.700290	−	0.713858i	$-0.253054\pi$
$883$	41.6187	1.40058	0.700290	+	0.713858i	$0.253054\pi$
$884$	1.19447	0.0401742
$885$	0	0
$886$	−13.5571	−0.455460
$887$	17.0891	0.573797	0.286898	−	0.957961i	$-0.407376\pi$
$887$	17.0891	0.573797	0.286898	+	0.957961i	$0.407376\pi$
$888$	0	0
$889$	−37.8978	−1.27105
$890$	−12.8355	−0.430248
$891$	0	0
$892$	15.9077	0.532630
$893$	1.29610	0.0433723
$894$	0	0
$895$	−10.8003	−0.361013
$896$	−24.2198	−0.809128
$897$	0	0
$898$	34.4410	1.14931
$899$	−1.13268	−0.0377769
$900$	0	0
$901$	−5.76875	−0.192185
$902$	−6.54580	−0.217951
$903$	0	0
$904$	−2.86871	−0.0954117
$905$	0.438278	0.0145689
$906$	0	0
$907$	22.8887	0.760006	0.380003	−	0.924985i	$-0.375923\pi$
$907$	22.8887	0.760006	0.380003	+	0.924985i	$0.375923\pi$
$908$	29.3489	0.973978
$909$	0	0
$910$	−3.39999	−0.112709
$911$	36.0929	1.19581	0.597906	−	0.801566i	$-0.296000\pi$
$911$	36.0929	1.19581	0.597906	+	0.801566i	$0.296000\pi$
$912$	0	0
$913$	−2.99602	−0.0991537
$914$	−19.9437	−0.659681
$915$	0	0
$916$	30.5208	1.00844
$917$	50.2423	1.65915
$918$	0	0
$919$	9.94627	0.328097	0.164049	−	0.986452i	$-0.447545\pi$
$919$	9.94627	0.328097	0.164049	+	0.986452i	$0.447545\pi$
$920$	22.3318	0.736256
$921$	0	0
$922$	−5.22074	−0.171936
$923$	−6.19699	−0.203976
$924$	0	0
$925$	10.7648	0.353944
$926$	24.0022	0.788760
$927$	0	0
$928$	5.56906	0.182813
$929$	30.2310	0.991846	0.495923	−	0.868366i	$-0.334830\pi$
$929$	30.2310	0.991846	0.495923	+	0.868366i	$0.334830\pi$
$930$	0	0
$931$	39.3760	1.29050
$932$	6.90179	0.226076
$933$	0	0
$934$	12.3728	0.404852
$935$	−2.85716	−0.0934393
$936$	0	0
$937$	28.4312	0.928808	0.464404	−	0.885624i	$-0.346269\pi$
$937$	28.4312	0.928808	0.464404	+	0.885624i	$0.346269\pi$
$938$	12.9229	0.421948
$939$	0	0
$940$	0.301661	0.00983910
$941$	7.18322	0.234166	0.117083	−	0.993122i	$-0.462646\pi$
$941$	7.18322	0.234166	0.117083	+	0.993122i	$0.462646\pi$
$942$	0	0
$943$	20.1020	0.654610
$944$	0.672458	0.0218866
$945$	0	0
$946$	10.7581	0.349776
$947$	44.1518	1.43474	0.717370	−	0.696692i	$-0.245346\pi$
$947$	44.1518	1.43474	0.717370	+	0.696692i	$0.245346\pi$
$948$	0	0
$949$	10.6371	0.345296
$950$	4.60625	0.149447
$951$	0	0
$952$	11.1013	0.359795
$953$	−21.5308	−0.697452	−0.348726	−	0.937225i	$-0.613386\pi$
$953$	−21.5308	−0.697452	−0.348726	+	0.937225i	$0.613386\pi$
$954$	0	0
$955$	−2.64477	−0.0855827
$956$	4.97644	0.160949
$957$	0	0
$958$	−14.0888	−0.455189
$959$	0.271823	0.00877762
$960$	0	0
$961$	−29.7170	−0.958614
$962$	9.55529	0.308075
$963$	0	0
$964$	−13.5128	−0.435218
$965$	16.4494	0.529526
$966$	0	0
$967$	−25.0483	−0.805498	−0.402749	−	0.915310i	$-0.631945\pi$
$967$	−25.0483	−0.805498	−0.402749	+	0.915310i	$0.631945\pi$
$968$	8.63498	0.277539
$969$	0	0
$970$	−12.4690	−0.400356
$971$	52.1718	1.67427	0.837136	−	0.546995i	$-0.184228\pi$
$971$	52.1718	1.67427	0.837136	+	0.546995i	$0.184228\pi$
$972$	0	0
$973$	−46.9072	−1.50378
$974$	26.9803	0.864506
$975$	0	0
$976$	1.90712	0.0610456
$977$	−21.1021	−0.675115	−0.337557	−	0.941305i	$-0.609601\pi$
$977$	−21.1021	−0.675115	−0.337557	+	0.941305i	$0.609601\pi$
$978$	0	0
$979$	−40.4233	−1.29194
$980$	9.16458	0.292752
$981$	0	0
$982$	−31.7652	−1.01367
$983$	19.1395	0.610457	0.305228	−	0.952279i	$-0.401267\pi$
$983$	19.1395	0.610457	0.305228	+	0.952279i	$0.401267\pi$
$984$	0	0
$985$	22.1732	0.706496
$986$	−0.907229	−0.0288921
$987$	0	0
$988$	−6.06463	−0.192942
$989$	−33.0378	−1.05054
$990$	0	0
$991$	−19.2995	−0.613069	−0.306535	−	0.951860i	$-0.599170\pi$
$991$	−19.2995	−0.613069	−0.306535	+	0.951860i	$0.599170\pi$
$992$	−6.30795	−0.200278
$993$	0	0
$994$	−21.5372	−0.683117
$995$	−11.4778	−0.363870
$996$	0	0
$997$	27.8422	0.881771	0.440886	−	0.897563i	$-0.354664\pi$
$997$	27.8422	0.881771	0.440886	+	0.897563i	$0.354664\pi$
$998$	7.93749	0.251257
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.s.1.5	✓	7
3.2	odd	2		1305.2.a.t.1.3	yes	7
5.4	even	2		6525.2.a.bw.1.3		7
15.14	odd	2		6525.2.a.bv.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.a.s.1.5	✓	7	1.1	even	1	trivial
1305.2.a.t.1.3	yes	7	3.2	odd	2
6525.2.a.bv.1.5		7	15.14	odd	2
6525.2.a.bw.1.3		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)

\(f(q)\)	\(=\)	\(q+0.897436 q^{2} -1.19461 q^{4} -1.00000 q^{5} +3.83036 q^{7} -2.86696 q^{8} -0.897436 q^{10} -2.82632 q^{11} +0.989088 q^{13} +3.43750 q^{14} -0.183696 q^{16} -1.01091 q^{17} +5.13268 q^{19} +1.19461 q^{20} -2.53645 q^{22} +7.78936 q^{23} +1.00000 q^{25} +0.887643 q^{26} -4.57577 q^{28} +1.00000 q^{29} -1.13268 q^{31} +5.56906 q^{32} -0.907229 q^{34} -3.83036 q^{35} +10.7648 q^{37} +4.60625 q^{38} +2.86696 q^{40} +2.58070 q^{41} -4.24141 q^{43} +3.37635 q^{44} +6.99045 q^{46} +0.252519 q^{47} +7.67162 q^{49} +0.897436 q^{50} -1.18157 q^{52} +5.70648 q^{53} +2.82632 q^{55} -10.9815 q^{56} +0.897436 q^{58} -3.66071 q^{59} -10.3820 q^{61} -1.01651 q^{62} +5.36527 q^{64} -0.989088 q^{65} +3.75939 q^{67} +1.20764 q^{68} -3.43750 q^{70} -6.26536 q^{71} +10.7545 q^{73} +9.66071 q^{74} -6.13154 q^{76} -10.8258 q^{77} +10.7853 q^{79} +0.183696 q^{80} +2.31601 q^{82} +1.06004 q^{83} +1.01091 q^{85} -3.80639 q^{86} +8.10295 q^{88} +14.3024 q^{89} +3.78856 q^{91} -9.30523 q^{92} +0.226620 q^{94} -5.13268 q^{95} +13.8941 q^{97} +6.88479 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 13 q^{4} - 7 q^{5} + 10 q^{7} + q^{10} + 3 q^{11} + 6 q^{13} + 9 q^{14} + 21 q^{16} - 8 q^{17} + 10 q^{19} - 13 q^{20} + 9 q^{22} - 11 q^{23} + 7 q^{25} - 3 q^{26} + 25 q^{28} + 7 q^{29}+ \cdots + 20 q^{98}+O(q^{100}) \) 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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