LMFDB - Embedded newform 1305.2.d.a.811.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(811,1305)]

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, 1]))

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

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

chi := DirichletCharacter("1305.811");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.1
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	1305.811
Dual form			1305.2.d.a.811.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.93185i q^{2} -1.73205 q^{4} +1.00000 q^{5} -2.73205 q^{7} -0.517638i q^{8} -1.93185i q^{10} -0.378937i q^{11} +5.46410 q^{13} +5.27792i q^{14} -4.46410 q^{16} -3.48477i q^{17} -4.24264i q^{19} -1.73205 q^{20} -0.732051 q^{22} -2.19615 q^{23} +1.00000 q^{25} -10.5558i q^{26} +4.73205 q^{28} +(-5.19615 - 1.41421i) q^{29} -4.24264i q^{31} +7.58871i q^{32} -6.73205 q^{34} -2.73205 q^{35} +4.24264i q^{37} -8.19615 q^{38} -0.517638i q^{40} -5.93426i q^{41} -4.24264i q^{43} +0.656339i q^{44} +4.24264i q^{46} +11.2122i q^{47} +0.464102 q^{49} -1.93185i q^{50} -9.46410 q^{52} -0.378937i q^{55} +1.41421i q^{56} +(-2.73205 + 10.0382i) q^{58} -6.00000 q^{59} -11.5911i q^{61} -8.19615 q^{62} +5.73205 q^{64} +5.46410 q^{65} -13.1244 q^{67} +6.03579i q^{68} +5.27792i q^{70} -6.00000 q^{71} -15.8338i q^{73} +8.19615 q^{74} +7.34847i q^{76} +1.03528i q^{77} +1.13681i q^{79} -4.46410 q^{80} -11.4641 q^{82} +8.19615 q^{83} -3.48477i q^{85} -8.19615 q^{86} -0.196152 q^{88} -7.72741i q^{89} -14.9282 q^{91} +3.80385 q^{92} +21.6603 q^{94} -4.24264i q^{95} +7.34847i q^{97} -0.896575i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7} + 8 q^{13} - 4 q^{16} + 4 q^{22} + 12 q^{23} + 4 q^{25} + 12 q^{28} - 20 q^{34} - 4 q^{35} - 12 q^{38} - 12 q^{49} - 24 q^{52} - 4 q^{58} - 24 q^{59} - 12 q^{62} + 16 q^{64} + 8 q^{65}+ \cdots + 52 q^{94}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 + 8 * q^13 - 4 * q^16 + 4 * q^22 + 12 * q^23 + 4 * q^25 + 12 * q^28 - 20 * q^34 - 4 * q^35 - 12 * q^38 - 12 * q^49 - 24 * q^52 - 4 * q^58 - 24 * q^59 - 12 * q^62 + 16 * q^64 + 8 * q^65 - 4 * q^67 - 24 * q^71 + 12 * q^74 - 4 * q^80 - 32 * q^82 + 12 * q^83 - 12 * q^86 + 20 * q^88 - 32 * q^91 + 36 * q^92 + 52 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	1.93185i		−	1.36603i	−0.730406	−	0.683013i	$-0.760669\pi$
$2$		−	1.93185i		−	1.36603i	0.730406	−	0.683013i	$-0.239331\pi$
$3$		0			0
$3$		0			0
$4$	−1.73205			−0.866025
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−2.73205			−1.03262			−0.516309	−	0.856402i	$-0.672694\pi$
$7$	−2.73205			−1.03262			−0.516309	+	0.856402i	$0.672694\pi$
$8$		−	0.517638i		−	0.183013i
$9$		0			0
$10$		−	1.93185i		−	0.610905i
$11$		−	0.378937i		−	0.114254i	−0.998367	−	0.0571270i	$-0.981806\pi$
$11$		−	0.378937i		−	0.114254i	0.998367	−	0.0571270i	$-0.0181940\pi$
$12$		0			0
$13$	5.46410			1.51547			0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410			1.51547			0.757735	+	0.652563i	$0.226306\pi$
$14$			5.27792i			1.41058i
$15$		0			0
$16$	−4.46410			−1.11603
$17$		−	3.48477i		−	0.845180i	−0.906321	−	0.422590i	$-0.861121\pi$
$17$		−	3.48477i		−	0.845180i	0.906321	−	0.422590i	$-0.138879\pi$
$18$		0			0
$19$		−	4.24264i		−	0.973329i	−0.873589	−	0.486664i	$-0.838214\pi$
$19$		−	4.24264i		−	0.973329i	0.873589	−	0.486664i	$-0.161786\pi$
$20$	−1.73205			−0.387298
$21$		0			0
$22$	−0.732051			−0.156074
$23$	−2.19615			−0.457929			−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615			−0.457929			−0.228965	+	0.973435i	$0.573534\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	10.5558i		−	2.07017i
$27$		0			0
$28$	4.73205			0.894274
$29$	−5.19615	−	1.41421i	−0.964901	−	0.262613i
$29$	−5.19615	−	1.41421i	−0.964901	−	0.262613i
$30$		0			0
$31$		−	4.24264i		−	0.762001i	−0.924575	−	0.381000i	$-0.875580\pi$
$31$		−	4.24264i		−	0.762001i	0.924575	−	0.381000i	$-0.124420\pi$
$32$			7.58871i			1.34151i
$33$		0			0
$34$	−6.73205			−1.15454
$35$	−2.73205			−0.461801
$36$		0			0
$37$			4.24264i			0.697486i	0.937218	+	0.348743i	$0.113391\pi$
$37$			4.24264i			0.697486i	−0.937218	+	0.348743i	$0.886609\pi$
$38$	−8.19615			−1.32959
$39$		0			0
$40$		−	0.517638i		−	0.0818458i
$41$		−	5.93426i		−	0.926775i	−0.886156	−	0.463388i	$-0.846634\pi$
$41$		−	5.93426i		−	0.926775i	0.886156	−	0.463388i	$-0.153366\pi$
$42$		0			0
$43$		−	4.24264i		−	0.646997i	−0.946229	−	0.323498i	$-0.895141\pi$
$43$		−	4.24264i		−	0.646997i	0.946229	−	0.323498i	$-0.104859\pi$
$44$			0.656339i			0.0989468i
$45$		0			0
$46$			4.24264i			0.625543i
$47$			11.2122i			1.63546i	0.575600	+	0.817732i	$0.304769\pi$
$47$			11.2122i			1.63546i	−0.575600	+	0.817732i	$0.695231\pi$
$48$		0			0
$49$	0.464102			0.0663002
$50$		−	1.93185i		−	0.273205i
$51$		0			0
$52$	−9.46410			−1.31243
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		−	0.378937i		−	0.0510959i
$56$			1.41421i			0.188982i
$57$		0			0
$58$	−2.73205	+	10.0382i	−0.358736	+	1.31808i
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		−	11.5911i		−	1.48409i	−0.670350	−	0.742045i	$-0.733856\pi$
$61$		−	11.5911i		−	1.48409i	0.670350	−	0.742045i	$-0.266144\pi$
$62$	−8.19615			−1.04091
$63$		0			0
$64$	5.73205			0.716506
$65$	5.46410			0.677738
$66$		0			0
$67$	−13.1244			−1.60340			−0.801698	−	0.597730i	$-0.796070\pi$
$67$	−13.1244			−1.60340			−0.801698	+	0.597730i	$0.796070\pi$
$68$			6.03579i			0.731947i
$69$		0			0
$70$			5.27792i			0.630832i
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$		−	15.8338i		−	1.85320i	−0.376048	−	0.926600i	$-0.622717\pi$
$73$		−	15.8338i		−	1.85320i	0.376048	−	0.926600i	$-0.377283\pi$
$74$	8.19615			0.952783
$75$		0			0
$76$			7.34847i			0.842927i
$77$			1.03528i			0.117981i
$78$		0			0
$79$			1.13681i			0.127901i	0.997953	+	0.0639507i	$0.0203700\pi$
$79$			1.13681i			0.127901i	−0.997953	+	0.0639507i	$0.979630\pi$
$80$	−4.46410			−0.499102
$81$		0			0
$82$	−11.4641			−1.26600
$83$	8.19615			0.899645			0.449822	−	0.893118i	$-0.351487\pi$
$83$	8.19615			0.899645			0.449822	+	0.893118i	$0.351487\pi$
$84$		0			0
$85$		−	3.48477i		−	0.377976i
$86$	−8.19615			−0.883814
$87$		0			0
$88$	−0.196152			−0.0209099
$89$		−	7.72741i		−	0.819103i	−0.912287	−	0.409552i	$-0.865685\pi$
$89$		−	7.72741i		−	0.819103i	0.912287	−	0.409552i	$-0.134315\pi$
$90$		0			0
$91$	−14.9282			−1.56490
$92$	3.80385			0.396579
$93$		0			0
$94$	21.6603			2.23408
$95$		−	4.24264i		−	0.435286i
$96$		0			0
$97$			7.34847i			0.746124i	0.927806	+	0.373062i	$0.121692\pi$
$97$			7.34847i			0.746124i	−0.927806	+	0.373062i	$0.878308\pi$
$98$		−	0.896575i		−	0.0905678i
$99$		0			0
$100$	−1.73205			−0.173205
$101$			15.4548i			1.53781i	0.639362	+	0.768906i	$0.279198\pi$
$101$			15.4548i			1.53781i	−0.639362	+	0.768906i	$0.720802\pi$
$102$		0			0
$103$	10.1962			1.00466			0.502328	−	0.864677i	$-0.332477\pi$
$103$	10.1962			1.00466			0.502328	+	0.864677i	$0.332477\pi$
$104$		−	2.82843i		−	0.277350i
$105$		0			0
$106$		0			0
$107$	8.19615			0.792352			0.396176	−	0.918175i	$-0.370337\pi$
$107$	8.19615			0.792352			0.396176	+	0.918175i	$0.370337\pi$
$108$		0			0
$109$	5.46410			0.523366			0.261683	−	0.965154i	$-0.415723\pi$
$109$	5.46410			0.523366			0.261683	+	0.965154i	$0.415723\pi$
$110$	−0.732051			−0.0697983
$111$		0			0
$112$	12.1962			1.15243
$113$		−	8.86422i		−	0.833876i	−0.908935	−	0.416938i	$-0.863103\pi$
$113$		−	8.86422i		−	0.833876i	0.908935	−	0.416938i	$-0.136897\pi$
$114$		0			0
$115$	−2.19615			−0.204792
$116$	9.00000	+	2.44949i	0.835629	+	0.227429i
$117$		0			0
$118$			11.5911i			1.06705i
$119$			9.52056i			0.872748i
$120$		0			0
$121$	10.8564			0.986946
$122$	−22.3923			−2.02730
$123$		0			0
$124$			7.34847i			0.659912i
$125$	1.00000			0.0894427
$126$		0			0
$127$			12.7279i			1.12942i	0.825289	+	0.564710i	$0.191012\pi$
$127$			12.7279i			1.12942i	−0.825289	+	0.564710i	$0.808988\pi$
$128$			4.10394i			0.362740i
$129$		0			0
$130$		−	10.5558i		−	0.925808i
$131$		−	11.9700i		−	1.04583i	−0.852385	−	0.522914i	$-0.824845\pi$
$131$		−	11.9700i		−	1.04583i	0.852385	−	0.522914i	$-0.175155\pi$
$132$		0			0
$133$			11.5911i			1.00508i
$134$			25.3543i			2.19028i
$135$		0			0
$136$	−1.80385			−0.154679
$137$			9.89949i			0.845771i	0.906183	+	0.422885i	$0.138983\pi$
$137$			9.89949i			0.845771i	−0.906183	+	0.422885i	$0.861017\pi$
$138$		0			0
$139$	−6.53590			−0.554368			−0.277184	−	0.960817i	$-0.589401\pi$
$139$	−6.53590			−0.554368			−0.277184	+	0.960817i	$0.589401\pi$
$140$	4.73205			0.399931
$141$		0			0
$142$			11.5911i			0.972704i
$143$		−	2.07055i		−	0.173148i
$144$		0			0
$145$	−5.19615	−	1.41421i	−0.431517	−	0.117444i
$146$	−30.5885			−2.53152
$147$		0			0
$148$		−	7.34847i		−	0.604040i
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$	−2.39230			−0.194683			−0.0973415	−	0.995251i	$-0.531034\pi$
$151$	−2.39230			−0.194683			−0.0973415	+	0.995251i	$0.531034\pi$
$152$	−2.19615			−0.178131
$153$		0			0
$154$	2.00000			0.161165
$155$		−	4.24264i		−	0.340777i
$156$		0			0
$157$		−	7.34847i		−	0.586472i	−0.956040	−	0.293236i	$-0.905268\pi$
$157$		−	7.34847i		−	0.586472i	0.956040	−	0.293236i	$-0.0947321\pi$
$158$	2.19615			0.174717
$159$		0			0
$160$			7.58871i			0.599940i
$161$	6.00000			0.472866
$162$		0			0
$163$			4.24264i			0.332309i	0.986100	+	0.166155i	$0.0531351\pi$
$163$			4.24264i			0.332309i	−0.986100	+	0.166155i	$0.946865\pi$
$164$			10.2784i			0.802611i
$165$		0			0
$166$		−	15.8338i		−	1.22894i
$167$	−14.1962			−1.09853			−0.549266	−	0.835648i	$-0.685092\pi$
$167$	−14.1962			−1.09853			−0.549266	+	0.835648i	$0.685092\pi$
$168$		0			0
$169$	16.8564			1.29665
$170$	−6.73205			−0.516325
$171$		0			0
$172$			7.34847i			0.560316i
$173$	20.7846			1.58022			0.790112	−	0.612962i	$-0.210022\pi$
$173$	20.7846			1.58022			0.790112	+	0.612962i	$0.210022\pi$
$174$		0			0
$175$	−2.73205			−0.206524
$176$			1.69161i			0.127510i
$177$		0			0
$178$	−14.9282			−1.11892
$179$	4.39230			0.328296			0.164148	−	0.986436i	$-0.447512\pi$
$179$	4.39230			0.328296			0.164148	+	0.986436i	$0.447512\pi$
$180$		0			0
$181$	21.8564			1.62457			0.812287	−	0.583258i	$-0.198222\pi$
$181$	21.8564			1.62457			0.812287	+	0.583258i	$0.198222\pi$
$182$			28.8391i			2.13769i
$183$		0			0
$184$			1.13681i			0.0838069i
$185$			4.24264i			0.311925i
$186$		0			0
$187$	−1.32051			−0.0965651
$188$		−	19.4201i		−	1.41635i
$189$		0			0
$190$	−8.19615			−0.594611
$191$		−	24.8738i		−	1.79981i	−0.436089	−	0.899904i	$-0.643637\pi$
$191$		−	24.8738i		−	1.79981i	0.436089	−	0.899904i	$-0.356363\pi$
$192$		0			0
$193$			12.7279i			0.916176i	0.888907	+	0.458088i	$0.151466\pi$
$193$			12.7279i			0.916176i	−0.888907	+	0.458088i	$0.848534\pi$
$194$	14.1962			1.01922
$195$		0			0
$196$	−0.803848			−0.0574177
$197$	22.3923			1.59539			0.797693	−	0.603064i	$-0.206054\pi$
$197$	22.3923			1.59539			0.797693	+	0.603064i	$0.206054\pi$
$198$		0			0
$199$	−12.5359			−0.888646			−0.444323	−	0.895867i	$-0.646556\pi$
$199$	−12.5359			−0.888646			−0.444323	+	0.895867i	$0.646556\pi$
$200$		−	0.517638i		−	0.0366025i
$201$		0			0
$202$	29.8564			2.10069
$203$	14.1962	+	3.86370i	0.996375	+	0.271179i
$204$		0			0
$205$		−	5.93426i		−	0.414466i
$206$		−	19.6975i		−	1.37239i
$207$		0			0
$208$	−24.3923			−1.69130
$209$	−1.60770			−0.111207
$210$		0			0
$211$			22.0454i			1.51767i	0.651284	+	0.758834i	$0.274231\pi$
$211$			22.0454i			1.51767i	−0.651284	+	0.758834i	$0.725769\pi$
$212$		0			0
$213$		0			0
$214$		−	15.8338i		−	1.08237i
$215$		−	4.24264i		−	0.289346i
$216$		0			0
$217$			11.5911i			0.786856i
$218$		−	10.5558i		−	0.714931i
$219$		0			0
$220$			0.656339i			0.0442504i
$221$		−	19.0411i		−	1.28084i
$222$		0			0
$223$	7.66025			0.512969			0.256484	−	0.966548i	$-0.417436\pi$
$223$	7.66025			0.512969			0.256484	+	0.966548i	$0.417436\pi$
$224$		−	20.7327i		−	1.38526i
$225$		0			0
$226$	−17.1244			−1.13910
$227$	26.1962			1.73870			0.869350	−	0.494197i	$-0.164538\pi$
$227$	26.1962			1.73870			0.869350	+	0.494197i	$0.164538\pi$
$228$		0			0
$229$		−	25.4558i		−	1.68217i	−0.540903	−	0.841085i	$-0.681918\pi$
$229$		−	25.4558i		−	1.68217i	0.540903	−	0.841085i	$-0.318082\pi$
$230$			4.24264i			0.279751i
$231$		0			0
$232$	−0.732051	+	2.68973i	−0.0480615	+	0.176589i
$233$	−1.60770			−0.105324			−0.0526618	−	0.998612i	$-0.516771\pi$
$233$	−1.60770			−0.105324			−0.0526618	+	0.998612i	$0.516771\pi$
$234$		0			0
$235$			11.2122i			0.731401i
$236$	10.3923			0.676481
$237$		0			0
$238$	18.3923			1.19220
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	−4.92820			−0.317453			−0.158727	−	0.987323i	$-0.550739\pi$
$241$	−4.92820			−0.317453			−0.158727	+	0.987323i	$0.550739\pi$
$242$		−	20.9730i		−	1.34819i
$243$		0			0
$244$			20.0764i			1.28526i
$245$	0.464102			0.0296504
$246$		0			0
$247$		−	23.1822i		−	1.47505i
$248$	−2.19615			−0.139456
$249$		0			0
$250$		−	1.93185i		−	0.122181i
$251$		−	8.86422i		−	0.559505i	−0.960072	−	0.279752i	$-0.909748\pi$
$251$		−	8.86422i		−	0.559505i	0.960072	−	0.279752i	$-0.0902523\pi$
$252$		0			0
$253$			0.832204i			0.0523202i
$254$	24.5885			1.54282
$255$		0			0
$256$	19.3923			1.21202
$257$	−12.0000			−0.748539			−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000			−0.748539			−0.374270	+	0.927320i	$0.622107\pi$
$258$		0			0
$259$		−	11.5911i		−	0.720237i
$260$	−9.46410			−0.586939
$261$		0			0
$262$	−23.1244			−1.42863
$263$			24.5964i			1.51668i	0.651859	+	0.758341i	$0.273990\pi$
$263$			24.5964i			1.51668i	−0.651859	+	0.758341i	$0.726010\pi$
$264$		0			0
$265$		0			0
$266$	22.3923			1.37296
$267$		0			0
$268$	22.7321			1.38858
$269$			1.59008i			0.0969488i	0.998824	+	0.0484744i	$0.0154359\pi$
$269$			1.59008i			0.0969488i	−0.998824	+	0.0484744i	$0.984564\pi$
$270$		0			0
$271$		−	24.3190i		−	1.47728i	−0.674102	−	0.738638i	$-0.735469\pi$
$271$		−	24.3190i		−	1.47728i	0.674102	−	0.738638i	$-0.264531\pi$
$272$			15.5563i			0.943242i
$273$		0			0
$274$	19.1244			1.15534
$275$		−	0.378937i		−	0.0228508i
$276$		0			0
$277$	−10.9282			−0.656612			−0.328306	−	0.944571i	$-0.606478\pi$
$277$	−10.9282			−0.656612			−0.328306	+	0.944571i	$0.606478\pi$
$278$			12.6264i			0.757280i
$279$		0			0
$280$			1.41421i			0.0845154i
$281$	−16.3923			−0.977883			−0.488941	−	0.872317i	$-0.662617\pi$
$281$	−16.3923			−0.977883			−0.488941	+	0.872317i	$0.662617\pi$
$282$		0			0
$283$	30.9808			1.84162			0.920808	−	0.390017i	$-0.127531\pi$
$283$	30.9808			1.84162			0.920808	+	0.390017i	$0.127531\pi$
$284$	10.3923			0.616670
$285$		0			0
$286$	−4.00000			−0.236525
$287$			16.2127i			0.957005i
$288$		0			0
$289$	4.85641			0.285671
$290$	−2.73205	+	10.0382i	−0.160432	+	0.589463i
$291$		0			0
$292$			27.4249i			1.60492i
$293$			16.5916i			0.969293i	0.874710	+	0.484647i	$0.161052\pi$
$293$			16.5916i			0.969293i	−0.874710	+	0.484647i	$0.838948\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$	2.19615			0.127649
$297$		0			0
$298$		−	34.7733i		−	2.01436i
$299$	−12.0000			−0.693978
$300$		0			0
$301$			11.5911i			0.668100i
$302$			4.62158i			0.265942i
$303$		0			0
$304$			18.9396i			1.08626i
$305$		−	11.5911i		−	0.663705i
$306$		0			0
$307$		−	33.6365i		−	1.91974i	−0.280450	−	0.959869i	$-0.590484\pi$
$307$		−	33.6365i		−	1.91974i	0.280450	−	0.959869i	$-0.409516\pi$
$308$		−	1.79315i		−	0.102174i
$309$		0			0
$310$	−8.19615			−0.465510
$311$			19.6975i			1.11694i	0.829525	+	0.558470i	$0.188611\pi$
$311$			19.6975i			1.11694i	−0.829525	+	0.558470i	$0.811389\pi$
$312$		0			0
$313$	−18.5359			−1.04771			−0.523855	−	0.851807i	$-0.675507\pi$
$313$	−18.5359			−1.04771			−0.523855	+	0.851807i	$0.675507\pi$
$314$	−14.1962			−0.801135
$315$		0			0
$316$		−	1.96902i		−	0.110766i
$317$		−	18.1817i		−	1.02119i	−0.859822	−	0.510593i	$-0.829426\pi$
$317$		−	18.1817i		−	1.02119i	0.859822	−	0.510593i	$-0.170574\pi$
$318$		0			0
$319$	−0.535898	+	1.96902i	−0.0300045	+	0.110244i
$320$	5.73205			0.320431
$321$		0			0
$322$		−	11.5911i		−	0.645947i
$323$	−14.7846			−0.822638
$324$		0			0
$325$	5.46410			0.303094
$326$	8.19615			0.453943
$327$		0			0
$328$	−3.07180			−0.169612
$329$		−	30.6322i		−	1.68881i
$330$		0			0
$331$		−	10.4543i		−	0.574620i	−0.957838	−	0.287310i	$-0.907239\pi$
$331$		−	10.4543i		−	0.574620i	0.957838	−	0.287310i	$-0.0927610\pi$
$332$	−14.1962			−0.779115
$333$		0			0
$334$			27.4249i			1.50062i
$335$	−13.1244			−0.717060
$336$		0			0
$337$		−	32.8043i		−	1.78696i	−0.449098	−	0.893482i	$-0.648255\pi$
$337$		−	32.8043i		−	1.78696i	0.449098	−	0.893482i	$-0.351745\pi$
$338$		−	32.5641i		−	1.77125i
$339$		0			0
$340$			6.03579i			0.327337i
$341$	−1.60770			−0.0870616
$342$		0			0
$343$	17.8564			0.964155
$344$	−2.19615			−0.118409
$345$		0			0
$346$		−	40.1528i		−	2.15863i
$347$	−15.8038			−0.848395			−0.424197	−	0.905570i	$-0.639444\pi$
$347$	−15.8038			−0.848395			−0.424197	+	0.905570i	$0.639444\pi$
$348$		0			0
$349$	−2.39230			−0.128057			−0.0640286	−	0.997948i	$-0.520395\pi$
$349$	−2.39230			−0.128057			−0.0640286	+	0.997948i	$0.520395\pi$
$350$			5.27792i			0.282117i
$351$		0			0
$352$	2.87564			0.153272
$353$	8.78461			0.467558			0.233779	−	0.972290i	$-0.424891\pi$
$353$	8.78461			0.467558			0.233779	+	0.972290i	$0.424891\pi$
$354$		0			0
$355$	−6.00000			−0.318447
$356$			13.3843i			0.709364i
$357$		0			0
$358$		−	8.48528i		−	0.448461i
$359$		−	7.07107i		−	0.373197i	−0.982436	−	0.186598i	$-0.940254\pi$
$359$		−	7.07107i		−	0.373197i	0.982436	−	0.186598i	$-0.0597463\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		−	42.2233i		−	2.21921i
$363$		0			0
$364$	25.8564			1.35524
$365$		−	15.8338i		−	0.828776i
$366$		0			0
$367$			27.4249i			1.43157i	0.698323	+	0.715783i	$0.253930\pi$
$367$			27.4249i			1.43157i	−0.698323	+	0.715783i	$0.746070\pi$
$368$	9.80385			0.511061
$369$		0			0
$370$	8.19615			0.426098
$371$		0			0
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$			2.55103i			0.131910i
$375$		0			0
$376$	5.80385			0.299311
$377$	−28.3923	−	7.72741i	−1.46228	−	0.397982i
$378$		0			0
$379$			1.13681i			0.0583941i	0.999574	+	0.0291971i	$0.00929503\pi$
$379$			1.13681i			0.0583941i	−0.999574	+	0.0291971i	$0.990705\pi$
$380$			7.34847i			0.376969i
$381$		0			0
$382$	−48.0526			−2.45858
$383$	−8.19615			−0.418804			−0.209402	−	0.977830i	$-0.567152\pi$
$383$	−8.19615			−0.418804			−0.209402	+	0.977830i	$0.567152\pi$
$384$		0			0
$385$			1.03528i			0.0527626i
$386$	24.5885			1.25152
$387$		0			0
$388$		−	12.7279i		−	0.646162i
$389$			1.59008i			0.0806202i	0.999187	+	0.0403101i	$0.0128346\pi$
$389$			1.59008i			0.0806202i	−0.999187	+	0.0403101i	$0.987165\pi$
$390$		0			0
$391$			7.65308i			0.387033i
$392$		−	0.240237i		−	0.0121338i
$393$		0			0
$394$		−	43.2586i		−	2.17934i
$395$			1.13681i			0.0571992i
$396$		0			0
$397$	9.60770			0.482196			0.241098	−	0.970501i	$-0.422492\pi$
$397$	9.60770			0.482196			0.241098	+	0.970501i	$0.422492\pi$
$398$			24.2175i			1.21391i
$399$		0			0
$400$	−4.46410			−0.223205
$401$	32.7846			1.63719			0.818593	−	0.574375i	$-0.194755\pi$
$401$	32.7846			1.63719			0.818593	+	0.574375i	$0.194755\pi$
$402$		0			0
$403$		−	23.1822i		−	1.15479i
$404$		−	26.7685i		−	1.33178i
$405$		0			0
$406$	7.46410	−	27.4249i	0.370437	−	1.36107i
$407$	1.60770			0.0796905
$408$		0			0
$409$			26.2880i			1.29986i	0.759994	+	0.649930i	$0.225202\pi$
$409$			26.2880i			1.29986i	−0.759994	+	0.649930i	$0.774798\pi$
$410$	−11.4641			−0.566172
$411$		0			0
$412$	−17.6603			−0.870058
$413$	16.3923			0.806613
$414$		0			0
$415$	8.19615			0.402333
$416$			41.4655i			2.03301i
$417$		0			0
$418$			3.10583i			0.151911i
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$			26.2880i			1.28120i	0.767874	+	0.640601i	$0.221315\pi$
$421$			26.2880i			1.28120i	−0.767874	+	0.640601i	$0.778685\pi$
$422$	42.5885			2.07317
$423$		0			0
$424$		0			0
$425$		−	3.48477i		−	0.169036i
$426$		0			0
$427$			31.6675i			1.53250i
$428$	−14.1962			−0.686197
$429$		0			0
$430$	−8.19615			−0.395254
$431$	16.3923			0.789590			0.394795	−	0.918769i	$-0.370816\pi$
$431$	16.3923			0.789590			0.394795	+	0.918769i	$0.370816\pi$
$432$		0			0
$433$			12.7279i			0.611665i	0.952085	+	0.305832i	$0.0989347\pi$
$433$			12.7279i			0.611665i	−0.952085	+	0.305832i	$0.901065\pi$
$434$	22.3923			1.07487
$435$		0			0
$436$	−9.46410			−0.453248
$437$			9.31749i			0.445716i
$438$		0			0
$439$	−4.92820			−0.235210			−0.117605	−	0.993060i	$-0.537522\pi$
$439$	−4.92820			−0.235210			−0.117605	+	0.993060i	$0.537522\pi$
$440$	−0.196152			−0.00935120
$441$		0			0
$442$	−36.7846			−1.74967
$443$			5.00052i			0.237582i	0.992919	+	0.118791i	$0.0379018\pi$
$443$			5.00052i			0.237582i	−0.992919	+	0.118791i	$0.962098\pi$
$444$		0			0
$445$		−	7.72741i		−	0.366314i
$446$		−	14.7985i		−	0.700728i
$447$		0			0
$448$	−15.6603			−0.739877
$449$			6.13733i			0.289638i	0.989458	+	0.144819i	$0.0462601\pi$
$449$			6.13733i			0.289638i	−0.989458	+	0.144819i	$0.953740\pi$
$450$		0			0
$451$	−2.24871			−0.105888
$452$			15.3533i			0.722157i
$453$		0			0
$454$		−	50.6071i		−	2.37511i
$455$	−14.9282			−0.699845
$456$		0			0
$457$	13.0718			0.611473			0.305736	−	0.952116i	$-0.401097\pi$
$457$	13.0718			0.611473			0.305736	+	0.952116i	$0.401097\pi$
$458$	−49.1769			−2.29789
$459$		0			0
$460$	3.80385			0.177355
$461$			22.6274i			1.05386i	0.849907	+	0.526932i	$0.176658\pi$
$461$			22.6274i			1.05386i	−0.849907	+	0.526932i	$0.823342\pi$
$462$		0			0
$463$	−4.33975			−0.201685			−0.100843	−	0.994902i	$-0.532154\pi$
$463$	−4.33975			−0.201685			−0.100843	+	0.994902i	$0.532154\pi$
$464$	23.1962	+	6.31319i	1.07685	+	0.293083i
$465$		0			0
$466$			3.10583i			0.143875i
$467$			7.62587i			0.352883i	0.984311	+	0.176442i	$0.0564587\pi$
$467$			7.62587i			0.352883i	−0.984311	+	0.176442i	$0.943541\pi$
$468$		0			0
$469$	35.8564			1.65570
$470$	21.6603			0.999113
$471$		0			0
$472$			3.10583i			0.142957i
$473$	−1.60770			−0.0739219
$474$		0			0
$475$		−	4.24264i		−	0.194666i
$476$		−	16.4901i		−	0.755822i
$477$		0			0
$478$			11.5911i			0.530165i
$479$		−	10.1769i		−	0.464994i	−0.972597	−	0.232497i	$-0.925310\pi$
$479$		−	10.1769i		−	0.464994i	0.972597	−	0.232497i	$-0.0746896\pi$
$480$		0			0
$481$			23.1822i			1.05702i
$482$			9.52056i			0.433650i
$483$		0			0
$484$	−18.8038			−0.854720
$485$			7.34847i			0.333677i
$486$		0			0
$487$	8.58846			0.389180			0.194590	−	0.980885i	$-0.437662\pi$
$487$	8.58846			0.389180			0.194590	+	0.980885i	$0.437662\pi$
$488$	−6.00000			−0.271607
$489$		0			0
$490$		−	0.896575i		−	0.0405032i
$491$			1.41421i			0.0638226i	0.999491	+	0.0319113i	$0.0101594\pi$
$491$			1.41421i			0.0638226i	−0.999491	+	0.0319113i	$0.989841\pi$
$492$		0			0
$493$	−4.92820	+	18.1074i	−0.221955	+	0.815515i
$494$	−44.7846			−2.01495
$495$		0			0
$496$			18.9396i			0.850412i
$497$	16.3923			0.735295
$498$		0			0
$499$	21.8564			0.978427			0.489214	−	0.872164i	$-0.337284\pi$
$499$	21.8564			0.978427			0.489214	+	0.872164i	$0.337284\pi$
$500$	−1.73205			−0.0774597
$501$		0			0
$502$	−17.1244			−0.764297
$503$			5.00052i			0.222962i	0.993767	+	0.111481i	$0.0355594\pi$
$503$			5.00052i			0.222962i	−0.993767	+	0.111481i	$0.964441\pi$
$504$		0			0
$505$			15.4548i			0.687730i
$506$	1.60770			0.0714708
$507$		0			0
$508$		−	22.0454i		−	0.978107i
$509$	20.7846			0.921262			0.460631	−	0.887592i	$-0.347623\pi$
$509$	20.7846			0.921262			0.460631	+	0.887592i	$0.347623\pi$
$510$		0			0
$511$			43.2586i			1.91365i
$512$		−	29.2552i		−	1.29291i
$513$		0			0
$514$			23.1822i			1.02252i
$515$	10.1962			0.449296
$516$		0			0
$517$	4.24871			0.186858
$518$	−22.3923			−0.983861
$519$		0			0
$520$		−	2.82843i		−	0.124035i
$521$	−8.78461			−0.384861			−0.192430	−	0.981311i	$-0.561637\pi$
$521$	−8.78461			−0.384861			−0.192430	+	0.981311i	$0.561637\pi$
$522$		0			0
$523$	−4.33975			−0.189764			−0.0948819	−	0.995489i	$-0.530247\pi$
$523$	−4.33975			−0.189764			−0.0948819	+	0.995489i	$0.530247\pi$
$524$			20.7327i			0.905714i
$525$		0			0
$526$	47.5167			2.07182
$527$	−14.7846			−0.644028
$528$		0			0
$529$	−18.1769			−0.790301
$530$		0			0
$531$		0			0
$532$		−	20.0764i		−	0.870422i
$533$		−	32.4254i		−	1.40450i
$534$		0			0
$535$	8.19615			0.354351
$536$			6.79367i			0.293442i
$537$		0			0
$538$	3.07180			0.132435
$539$		−	0.175865i		−	0.00757506i
$540$		0			0
$541$		−	25.4558i		−	1.09443i	−0.836991	−	0.547216i	$-0.815688\pi$
$541$		−	25.4558i		−	1.09443i	0.836991	−	0.547216i	$-0.184312\pi$
$542$	−46.9808			−2.01800
$543$		0			0
$544$	26.4449			1.13381
$545$	5.46410			0.234056
$546$		0			0
$547$	−23.5167			−1.00550			−0.502750	−	0.864432i	$-0.667678\pi$
$547$	−23.5167			−1.00550			−0.502750	+	0.864432i	$0.667678\pi$
$548$		−	17.1464i		−	0.732459i
$549$		0			0
$550$	−0.732051			−0.0312148
$551$	−6.00000	+	22.0454i	−0.255609	+	0.939166i
$552$		0			0
$553$		−	3.10583i		−	0.132073i
$554$			21.1117i			0.896949i
$555$		0			0
$556$	11.3205			0.480096
$557$	1.60770			0.0681202			0.0340601	−	0.999420i	$-0.489156\pi$
$557$	1.60770			0.0681202			0.0340601	+	0.999420i	$0.489156\pi$
$558$		0			0
$559$		−	23.1822i		−	0.980503i
$560$	12.1962			0.515382
$561$		0			0
$562$			31.6675i			1.33581i
$563$			2.72689i			0.114925i	0.998348	+	0.0574624i	$0.0183009\pi$
$563$			2.72689i			0.114925i	−0.998348	+	0.0574624i	$0.981699\pi$
$564$		0			0
$565$		−	8.86422i		−	0.372920i
$566$		−	59.8502i		−	2.51569i
$567$		0			0
$568$			3.10583i			0.130318i
$569$			20.8343i			0.873418i	0.899603	+	0.436709i	$0.143856\pi$
$569$			20.8343i			0.873418i	−0.899603	+	0.436709i	$0.856144\pi$
$570$		0			0
$571$	−32.3923			−1.35558			−0.677788	−	0.735257i	$-0.737061\pi$
$571$	−32.3923			−1.35558			−0.677788	+	0.735257i	$0.737061\pi$
$572$			3.58630i			0.149951i
$573$		0			0
$574$	31.3205			1.30729
$575$	−2.19615			−0.0915859
$576$		0			0
$577$			15.0015i			0.624523i	0.949996	+	0.312261i	$0.101086\pi$
$577$			15.0015i			0.624523i	−0.949996	+	0.312261i	$0.898914\pi$
$578$		−	9.38186i		−	0.390234i
$579$		0			0
$580$	9.00000	+	2.44949i	0.373705	+	0.101710i
$581$	−22.3923			−0.928989
$582$		0			0
$583$		0			0
$584$	−8.19615			−0.339159
$585$		0			0
$586$	32.0526			1.32408
$587$	33.8038			1.39523			0.697617	−	0.716471i	$-0.254244\pi$
$587$	33.8038			1.39523			0.697617	+	0.716471i	$0.254244\pi$
$588$		0			0
$589$	−18.0000			−0.741677
$590$			11.5911i			0.477198i
$591$		0			0
$592$		−	18.9396i		−	0.778412i
$593$	40.3923			1.65871			0.829357	−	0.558720i	$-0.188707\pi$
$593$	40.3923			1.65871			0.829357	+	0.558720i	$0.188707\pi$
$594$		0			0
$595$			9.52056i			0.390305i
$596$	−31.1769			−1.27706
$597$		0			0
$598$			23.1822i			0.947991i
$599$		−	46.7434i		−	1.90988i	−0.296795	−	0.954941i	$-0.595918\pi$
$599$		−	46.7434i		−	1.90988i	0.296795	−	0.954941i	$-0.404082\pi$
$600$		0			0
$601$			5.37945i			0.219432i	0.993963	+	0.109716i	$0.0349942\pi$
$601$			5.37945i			0.219432i	−0.993963	+	0.109716i	$0.965006\pi$
$602$	22.3923			0.912642
$603$		0			0
$604$	4.14359			0.168600
$605$	10.8564			0.441376
$606$		0			0
$607$		−	12.7279i		−	0.516610i	−0.966063	−	0.258305i	$-0.916836\pi$
$607$		−	12.7279i		−	0.516610i	0.966063	−	0.258305i	$-0.0831640\pi$
$608$	32.1962			1.30573
$609$		0			0
$610$	−22.3923			−0.906638
$611$			61.2645i			2.47849i
$612$		0			0
$613$	19.0718			0.770303			0.385151	−	0.922853i	$-0.374149\pi$
$613$	19.0718			0.770303			0.385151	+	0.922853i	$0.374149\pi$
$614$	−64.9808			−2.62241
$615$		0			0
$616$	0.535898			0.0215920
$617$			21.4906i			0.865179i	0.901591	+	0.432590i	$0.142400\pi$
$617$			21.4906i			0.865179i	−0.901591	+	0.432590i	$0.857600\pi$
$618$		0			0
$619$			30.5307i			1.22713i	0.789643	+	0.613566i	$0.210266\pi$
$619$			30.5307i			1.22713i	−0.789643	+	0.613566i	$0.789734\pi$
$620$			7.34847i			0.295122i
$621$		0			0
$622$	38.0526			1.52577
$623$			21.1117i			0.845821i
$624$		0			0
$625$	1.00000			0.0400000
$626$			35.8086i			1.43120i
$627$		0			0
$628$			12.7279i			0.507899i
$629$	14.7846			0.589501
$630$		0			0
$631$	9.85641			0.392377			0.196189	−	0.980566i	$-0.437143\pi$
$631$	9.85641			0.392377			0.196189	+	0.980566i	$0.437143\pi$
$632$	0.588457			0.0234076
$633$		0			0
$634$	−35.1244			−1.39497
$635$			12.7279i			0.505092i
$636$		0			0
$637$	2.53590			0.100476
$638$	3.80385	+	1.03528i	0.150596	+	0.0409870i
$639$		0			0
$640$			4.10394i			0.162222i
$641$			20.8343i			0.822904i	0.911431	+	0.411452i	$0.134978\pi$
$641$			20.8343i			0.822904i	−0.911431	+	0.411452i	$0.865022\pi$
$642$		0			0
$643$	48.0526			1.89501			0.947504	−	0.319744i	$-0.103597\pi$
$643$	48.0526			1.89501			0.947504	+	0.319744i	$0.103597\pi$
$644$	−10.3923			−0.409514
$645$		0			0
$646$			28.5617i			1.12374i
$647$	−45.3731			−1.78380			−0.891900	−	0.452233i	$-0.850627\pi$
$647$	−45.3731			−1.78380			−0.891900	+	0.452233i	$0.850627\pi$
$648$		0			0
$649$			2.27362i			0.0892476i
$650$		−	10.5558i		−	0.414034i
$651$		0			0
$652$		−	7.34847i		−	0.287788i
$653$			1.89469i			0.0741448i	0.999313	+	0.0370724i	$0.0118032\pi$
$653$			1.89469i			0.0741448i	−0.999313	+	0.0370724i	$0.988197\pi$
$654$		0			0
$655$		−	11.9700i		−	0.467708i
$656$			26.4911i			1.03430i
$657$		0			0
$658$	−59.1769			−2.30696
$659$		−	20.4553i		−	0.796826i	−0.917206	−	0.398413i	$-0.869561\pi$
$659$		−	20.4553i		−	0.796826i	0.917206	−	0.398413i	$-0.130439\pi$
$660$		0			0
$661$	21.8564			0.850116			0.425058	−	0.905166i	$-0.360254\pi$
$661$	21.8564			0.850116			0.425058	+	0.905166i	$0.360254\pi$
$662$	−20.1962			−0.784946
$663$		0			0
$664$		−	4.24264i		−	0.164646i
$665$			11.5911i			0.449484i
$666$		0			0
$667$	11.4115	+	3.10583i	0.441857	+	0.120258i
$668$	24.5885			0.951356
$669$		0			0
$670$			25.3543i			0.979522i
$671$	−4.39230			−0.169563
$672$		0			0
$673$	−9.32051			−0.359279			−0.179640	−	0.983732i	$-0.557493\pi$
$673$	−9.32051			−0.359279			−0.179640	+	0.983732i	$0.557493\pi$
$674$	−63.3731			−2.44104
$675$		0			0
$676$	−29.1962			−1.12293
$677$		−	44.4698i		−	1.70911i	−0.519360	−	0.854556i	$-0.673830\pi$
$677$		−	44.4698i		−	1.70911i	0.519360	−	0.854556i	$-0.326170\pi$
$678$		0			0
$679$		−	20.0764i		−	0.770461i
$680$	−1.80385			−0.0691744
$681$		0			0
$682$			3.10583i			0.118928i
$683$	−21.8038			−0.834301			−0.417151	−	0.908837i	$-0.636971\pi$
$683$	−21.8038			−0.834301			−0.417151	+	0.908837i	$0.636971\pi$
$684$		0			0
$685$			9.89949i			0.378240i
$686$		−	34.4959i		−	1.31706i
$687$		0			0
$688$			18.9396i			0.722065i
$689$		0			0
$690$		0			0
$691$	−16.9282			−0.643979			−0.321990	−	0.946743i	$-0.604352\pi$
$691$	−16.9282			−0.643979			−0.321990	+	0.946743i	$0.604352\pi$
$692$	−36.0000			−1.36851
$693$		0			0
$694$			30.5307i			1.15893i
$695$	−6.53590			−0.247921
$696$		0			0
$697$	−20.6795			−0.783292
$698$			4.62158i			0.174929i
$699$		0			0
$700$	4.73205			0.178855
$701$	8.78461			0.331790			0.165895	−	0.986143i	$-0.446949\pi$
$701$	8.78461			0.331790			0.165895	+	0.986143i	$0.446949\pi$
$702$		0			0
$703$	18.0000			0.678883
$704$		−	2.17209i		−	0.0818637i
$705$		0			0
$706$		−	16.9706i		−	0.638696i
$707$		−	42.2233i		−	1.58797i
$708$		0			0
$709$	−48.7846			−1.83214			−0.916072	−	0.401013i	$-0.868658\pi$
$709$	−48.7846			−1.83214			−0.916072	+	0.401013i	$0.868658\pi$
$710$			11.5911i			0.435007i
$711$		0			0
$712$	−4.00000			−0.149906
$713$			9.31749i			0.348943i
$714$		0			0
$715$		−	2.07055i		−	0.0774343i
$716$	−7.60770			−0.284313
$717$		0			0
$718$	−13.6603			−0.509796
$719$	−10.3923			−0.387568			−0.193784	−	0.981044i	$-0.562076\pi$
$719$	−10.3923			−0.387568			−0.193784	+	0.981044i	$0.562076\pi$
$720$		0			0
$721$	−27.8564			−1.03743
$722$		−	1.93185i		−	0.0718961i
$723$		0			0
$724$	−37.8564			−1.40692
$725$	−5.19615	−	1.41421i	−0.192980	−	0.0525226i
$726$		0			0
$727$			8.18067i			0.303404i	0.988426	+	0.151702i	$0.0484755\pi$
$727$			8.18067i			0.303404i	−0.988426	+	0.151702i	$0.951525\pi$
$728$			7.72741i			0.286397i
$729$		0			0
$730$	−30.5885			−1.13213
$731$	−14.7846			−0.546829
$732$		0			0
$733$			44.3954i			1.63978i	0.572519	+	0.819891i	$0.305966\pi$
$733$			44.3954i			1.63978i	−0.572519	+	0.819891i	$0.694034\pi$
$734$	52.9808			1.95556
$735$		0			0
$736$		−	16.6660i		−	0.614315i
$737$			4.97331i			0.183194i
$738$		0			0
$739$			42.1218i			1.54948i	0.632283	+	0.774738i	$0.282118\pi$
$739$			42.1218i			1.54948i	−0.632283	+	0.774738i	$0.717882\pi$
$740$		−	7.34847i		−	0.270135i
$741$		0			0
$742$		0			0
$743$			19.6975i			0.722629i	0.932444	+	0.361315i	$0.117672\pi$
$743$			19.6975i			0.722629i	−0.932444	+	0.361315i	$0.882328\pi$
$744$		0			0
$745$	18.0000			0.659469
$746$			42.5007i			1.55606i
$747$		0			0
$748$	2.28719			0.0836278
$749$	−22.3923			−0.818197
$750$		0			0
$751$			10.4543i			0.381483i	0.981640	+	0.190741i	$0.0610892\pi$
$751$			10.4543i			0.381483i	−0.981640	+	0.190741i	$0.938911\pi$
$752$		−	50.0523i		−	1.82522i
$753$		0			0
$754$	−14.9282	+	54.8497i	−0.543653	+	1.99751i
$755$	−2.39230			−0.0870649
$756$		0			0
$757$			13.5601i			0.492851i	0.969162	+	0.246426i	$0.0792561\pi$
$757$			13.5601i			0.492851i	−0.969162	+	0.246426i	$0.920744\pi$
$758$	2.19615			0.0797678
$759$		0			0
$760$	−2.19615			−0.0796628
$761$	−10.3923			−0.376721			−0.188360	−	0.982100i	$-0.560317\pi$
$761$	−10.3923			−0.376721			−0.188360	+	0.982100i	$0.560317\pi$
$762$		0			0
$763$	−14.9282			−0.540437
$764$			43.0827i			1.55868i
$765$		0			0
$766$			15.8338i			0.572097i
$767$	−32.7846			−1.18378
$768$		0			0
$769$		−	14.6969i		−	0.529985i	−0.964250	−	0.264993i	$-0.914630\pi$
$769$		−	14.6969i		−	0.529985i	0.964250	−	0.264993i	$-0.0853695\pi$
$770$	2.00000			0.0720750
$771$		0			0
$772$		−	22.0454i		−	0.793432i
$773$			18.3848i			0.661254i	0.943761	+	0.330627i	$0.107260\pi$
$773$			18.3848i			0.661254i	−0.943761	+	0.330627i	$0.892740\pi$
$774$		0			0
$775$		−	4.24264i		−	0.152400i
$776$	3.80385			0.136550
$777$		0			0
$778$	3.07180			0.110129
$779$	−25.1769			−0.902057
$780$		0			0
$781$			2.27362i			0.0813567i
$782$	14.7846			0.528697
$783$		0			0
$784$	−2.07180			−0.0739927
$785$		−	7.34847i		−	0.262278i
$786$		0			0
$787$	7.41154			0.264193			0.132096	−	0.991237i	$-0.457829\pi$
$787$	7.41154			0.264193			0.132096	+	0.991237i	$0.457829\pi$
$788$	−38.7846			−1.38164
$789$		0			0
$790$	2.19615			0.0781356
$791$			24.2175i			0.861075i
$792$		0			0
$793$		−	63.3350i		−	2.24909i
$794$		−	18.5606i		−	0.658693i
$795$		0			0
$796$	21.7128			0.769590
$797$			21.4906i			0.761236i	0.924732	+	0.380618i	$0.124289\pi$
$797$			21.4906i			0.761236i	−0.924732	+	0.380618i	$0.875711\pi$
$798$		0			0
$799$	39.0718			1.38226
$800$			7.58871i			0.268301i
$801$		0			0
$802$		−	63.3350i		−	2.23644i
$803$	−6.00000			−0.211735
$804$		0			0
$805$	6.00000			0.211472
$806$	−44.7846			−1.57747
$807$		0			0
$808$	8.00000			0.281439
$809$		−	41.6685i		−	1.46499i	−0.680774	−	0.732494i	$-0.738356\pi$
$809$		−	41.6685i		−	1.46499i	0.680774	−	0.732494i	$-0.261644\pi$
$810$		0			0
$811$	−39.3205			−1.38073			−0.690365	−	0.723461i	$-0.742550\pi$
$811$	−39.3205			−1.38073			−0.690365	+	0.723461i	$0.742550\pi$
$812$	−24.5885	−	6.69213i	−0.862886	−	0.234848i
$813$		0			0
$814$		−	3.10583i		−	0.108859i
$815$			4.24264i			0.148613i
$816$		0			0
$817$	−18.0000			−0.629740
$818$	50.7846			1.77564
$819$		0			0
$820$			10.2784i			0.358938i
$821$	−14.7846			−0.515986			−0.257993	−	0.966147i	$-0.583061\pi$
$821$	−14.7846			−0.515986			−0.257993	+	0.966147i	$0.583061\pi$
$822$		0			0
$823$			8.18067i			0.285160i	0.989783	+	0.142580i	$0.0455399\pi$
$823$			8.18067i			0.285160i	−0.989783	+	0.142580i	$0.954460\pi$
$824$		−	5.27792i		−	0.183865i
$825$		0			0
$826$		−	31.6675i		−	1.10185i
$827$			3.68784i			0.128239i	0.997942	+	0.0641193i	$0.0204238\pi$
$827$			3.68784i			0.128239i	−0.997942	+	0.0641193i	$0.979576\pi$
$828$		0			0
$829$			30.8353i			1.07095i	0.844550	+	0.535477i	$0.179868\pi$
$829$			30.8353i			1.07095i	−0.844550	+	0.535477i	$0.820132\pi$
$830$		−	15.8338i		−	0.549598i
$831$		0			0
$832$	31.3205			1.08584
$833$		−	1.61729i		−	0.0560356i
$834$		0			0
$835$	−14.1962			−0.491278
$836$	2.78461			0.0963077
$837$		0			0
$838$		−	34.7733i		−	1.20122i
$839$		−	8.86422i		−	0.306027i	−0.988224	−	0.153013i	$-0.951102\pi$
$839$		−	8.86422i		−	0.306027i	0.988224	−	0.153013i	$-0.0488978\pi$
$840$		0			0
$841$	25.0000	+	14.6969i	0.862069	+	0.506791i
$842$	50.7846			1.75015
$843$		0			0
$844$		−	38.1838i		−	1.31434i
$845$	16.8564			0.579878
$846$		0			0
$847$	−29.6603			−1.01914
$848$		0			0
$849$		0			0
$850$	−6.73205			−0.230907
$851$		−	9.31749i		−	0.319399i
$852$		0			0
$853$		−	6.51626i		−	0.223113i	−0.993758	−	0.111556i	$-0.964416\pi$
$853$		−	6.51626i		−	0.223113i	0.993758	−	0.111556i	$-0.0355836\pi$
$854$	61.1769			2.09343
$855$		0			0
$856$		−	4.24264i		−	0.145010i
$857$	3.21539			0.109836			0.0549178	−	0.998491i	$-0.482510\pi$
$857$	3.21539			0.109836			0.0549178	+	0.998491i	$0.482510\pi$
$858$		0			0
$859$		−	39.0160i		−	1.33121i	−0.746305	−	0.665604i	$-0.768174\pi$
$859$		−	39.0160i		−	1.33121i	0.746305	−	0.665604i	$-0.231826\pi$
$860$			7.34847i			0.250581i
$861$		0			0
$862$		−	31.6675i		−	1.07860i
$863$	27.8038			0.946454			0.473227	−	0.880941i	$-0.343089\pi$
$863$	27.8038			0.946454			0.473227	+	0.880941i	$0.343089\pi$
$864$		0			0
$865$	20.7846			0.706698
$866$	24.5885			0.835550
$867$		0			0
$868$		−	20.0764i		−	0.681437i
$869$	0.430781			0.0146132
$870$		0			0
$871$	−71.7128			−2.42990
$872$		−	2.82843i		−	0.0957826i
$873$		0			0
$874$	18.0000			0.608859
$875$	−2.73205			−0.0923602
$876$		0			0
$877$	16.7846			0.566776			0.283388	−	0.959005i	$-0.408542\pi$
$877$	16.7846			0.566776			0.283388	+	0.959005i	$0.408542\pi$
$878$			9.52056i			0.321303i
$879$		0			0
$880$			1.69161i			0.0570243i
$881$			6.13733i			0.206772i	0.994641	+	0.103386i	$0.0329677\pi$
$881$			6.13733i			0.206772i	−0.994641	+	0.103386i	$0.967032\pi$
$882$		0			0
$883$	8.58846			0.289025			0.144512	−	0.989503i	$-0.453839\pi$
$883$	8.58846			0.289025			0.144512	+	0.989503i	$0.453839\pi$
$884$			32.9802i			1.10924i
$885$		0			0
$886$	9.66025			0.324543
$887$		−	18.1817i		−	0.610482i	−0.952275	−	0.305241i	$-0.901263\pi$
$887$		−	18.1817i		−	0.610482i	0.952275	−	0.305241i	$-0.0987370\pi$
$888$		0			0
$889$		−	34.7733i		−	1.16626i
$890$	−14.9282			−0.500395
$891$		0			0
$892$	−13.2679			−0.444244
$893$	47.5692			1.59184
$894$		0			0
$895$	4.39230			0.146819
$896$		−	11.2122i		−	0.374572i
$897$		0			0
$898$	11.8564			0.395653
$899$	−6.00000	+	22.0454i	−0.200111	+	0.735256i
$900$		0			0
$901$		0			0
$902$			4.34418i			0.144645i
$903$		0			0
$904$	−4.58846			−0.152610
$905$	21.8564			0.726532
$906$		0			0
$907$			6.51626i			0.216369i	0.994131	+	0.108185i	$0.0345037\pi$
$907$			6.51626i			0.216369i	−0.994131	+	0.108185i	$0.965496\pi$
$908$	−45.3731			−1.50576
$909$		0			0
$910$			28.8391i			0.956006i
$911$		−	18.1817i		−	0.602387i	−0.953563	−	0.301193i	$-0.902615\pi$
$911$		−	18.1817i		−	0.602387i	0.953563	−	0.301193i	$-0.0973850\pi$
$912$		0			0
$913$		−	3.10583i		−	0.102788i
$914$		−	25.2528i		−	0.835287i
$915$		0			0
$916$			44.0908i			1.45680i
$917$			32.7028i			1.07994i
$918$		0			0
$919$	3.85641			0.127211			0.0636056	−	0.997975i	$-0.479740\pi$
$919$	3.85641			0.127211			0.0636056	+	0.997975i	$0.479740\pi$
$920$			1.13681i			0.0374796i
$921$		0			0
$922$	43.7128			1.43960
$923$	−32.7846			−1.07912
$924$		0			0
$925$			4.24264i			0.139497i
$926$			8.38375i			0.275507i
$927$		0			0
$928$	10.7321	−	39.4321i	0.352297	−	1.29442i
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$		−	1.96902i		−	0.0645319i
$932$	2.78461			0.0912129
$933$		0			0
$934$	14.7321			0.482047
$935$	−1.32051			−0.0431852
$936$		0			0
$937$	2.24871			0.0734622			0.0367311	−	0.999325i	$-0.488305\pi$
$937$	2.24871			0.0734622			0.0367311	+	0.999325i	$0.488305\pi$
$938$		−	69.2693i		−	2.26172i
$939$		0			0
$940$		−	19.4201i		−	0.633412i
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$			13.0325i			0.424398i
$944$	26.7846			0.871765
$945$		0			0
$946$			3.10583i			0.100979i
$947$			0.453267i			0.0147292i	0.999973	+	0.00736460i	$0.00234425\pi$
$947$			0.453267i			0.0147292i	−0.999973	+	0.00736460i	$0.997656\pi$
$948$		0			0
$949$		−	86.5172i		−	2.80847i
$950$	−8.19615			−0.265918
$951$		0			0
$952$	4.92820			0.159724
$953$	−35.5692			−1.15220			−0.576100	−	0.817379i	$-0.695426\pi$
$953$	−35.5692			−1.15220			−0.576100	+	0.817379i	$0.695426\pi$
$954$		0			0
$955$		−	24.8738i		−	0.804898i
$956$	10.3923			0.336111
$957$		0			0
$958$	−19.6603			−0.635194
$959$		−	27.0459i		−	0.873358i
$960$		0			0
$961$	13.0000			0.419355
$962$	44.7846			1.44391
$963$		0			0
$964$	8.53590			0.274923
$965$			12.7279i			0.409726i
$966$		0			0
$967$			0.304608i			0.00979553i	0.999988	+	0.00489776i	$0.00155901\pi$
$967$			0.304608i			0.00979553i	−0.999988	+	0.00489776i	$0.998441\pi$
$968$		−	5.61969i		−	0.180624i
$969$		0			0
$970$	14.1962			0.455811
$971$			22.3228i			0.716373i	0.933650	+	0.358187i	$0.116605\pi$
$971$			22.3228i			0.716373i	−0.933650	+	0.358187i	$0.883395\pi$
$972$		0			0
$973$	17.8564			0.572450
$974$		−	16.5916i		−	0.531630i
$975$		0			0
$976$			51.7439i			1.65628i
$977$	−48.0000			−1.53566			−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000			−1.53566			−0.767828	+	0.640656i	$0.778662\pi$
$978$		0			0
$979$	−2.92820			−0.0935858
$980$	−0.803848			−0.0256780
$981$		0			0
$982$	2.73205			0.0871832
$983$		−	30.2533i		−	0.964930i	−0.875915	−	0.482465i	$-0.839742\pi$
$983$		−	30.2533i		−	0.964930i	0.875915	−	0.482465i	$-0.160258\pi$
$984$		0			0
$985$	22.3923			0.713478
$986$	34.9808	+	9.52056i	1.11401	+	0.303196i
$987$		0			0
$988$			40.1528i			1.27743i
$989$			9.31749i			0.296279i
$990$		0			0
$991$	26.6795			0.847502			0.423751	−	0.905779i	$-0.360713\pi$
$991$	26.6795			0.847502			0.423751	+	0.905779i	$0.360713\pi$
$992$	32.1962			1.02223
$993$		0			0
$994$		−	31.6675i		−	1.00443i
$995$	−12.5359			−0.397415
$996$		0			0
$997$		−	13.5601i		−	0.429454i	−0.976674	−	0.214727i	$-0.931114\pi$
$997$		−	13.5601i		−	0.429454i	0.976674	−	0.214727i	$-0.0688861\pi$
$998$		−	42.2233i		−	1.33656i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.d.a.811.1		4
3.2	odd	2		145.2.c.a.86.4	yes	4
12.11	even	2		2320.2.g.e.1681.2		4
15.2	even	4		725.2.d.b.724.1		8
15.8	even	4		725.2.d.b.724.8		8
15.14	odd	2		725.2.c.d.376.1		4
29.28	even	2	inner	1305.2.d.a.811.4		4
87.17	even	4		4205.2.a.g.1.4		4
87.41	even	4		4205.2.a.g.1.1		4
87.86	odd	2		145.2.c.a.86.1	✓	4
348.347	even	2		2320.2.g.e.1681.4		4
435.173	even	4		725.2.d.b.724.2		8
435.347	even	4		725.2.d.b.724.7		8
435.434	odd	2		725.2.c.d.376.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.c.a.86.1	✓	4	87.86	odd	2
145.2.c.a.86.4	yes	4	3.2	odd	2
725.2.c.d.376.1		4	15.14	odd	2
725.2.c.d.376.4		4	435.434	odd	2
725.2.d.b.724.1		8	15.2	even	4
725.2.d.b.724.2		8	435.173	even	4
725.2.d.b.724.7		8	435.347	even	4
725.2.d.b.724.8		8	15.8	even	4
1305.2.d.a.811.1		4	1.1	even	1	trivial
1305.2.d.a.811.4		4	29.28	even	2	inner
2320.2.g.e.1681.2		4	12.11	even	2
2320.2.g.e.1681.4		4	348.347	even	2
4205.2.a.g.1.1		4	87.41	even	4
4205.2.a.g.1.4		4	87.17	even	4

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.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.93185i q^{2} -1.73205 q^{4} +1.00000 q^{5} -2.73205 q^{7} -0.517638i q^{8} -1.93185i q^{10} -0.378937i q^{11} +5.46410 q^{13} +5.27792i q^{14} -4.46410 q^{16} -3.48477i q^{17} -4.24264i q^{19} -1.73205 q^{20} -0.732051 q^{22} -2.19615 q^{23} +1.00000 q^{25} -10.5558i q^{26} +4.73205 q^{28} +(-5.19615 - 1.41421i) q^{29} -4.24264i q^{31} +7.58871i q^{32} -6.73205 q^{34} -2.73205 q^{35} +4.24264i q^{37} -8.19615 q^{38} -0.517638i q^{40} -5.93426i q^{41} -4.24264i q^{43} +0.656339i q^{44} +4.24264i q^{46} +11.2122i q^{47} +0.464102 q^{49} -1.93185i q^{50} -9.46410 q^{52} -0.378937i q^{55} +1.41421i q^{56} +(-2.73205 + 10.0382i) q^{58} -6.00000 q^{59} -11.5911i q^{61} -8.19615 q^{62} +5.73205 q^{64} +5.46410 q^{65} -13.1244 q^{67} +6.03579i q^{68} +5.27792i q^{70} -6.00000 q^{71} -15.8338i q^{73} +8.19615 q^{74} +7.34847i q^{76} +1.03528i q^{77} +1.13681i q^{79} -4.46410 q^{80} -11.4641 q^{82} +8.19615 q^{83} -3.48477i q^{85} -8.19615 q^{86} -0.196152 q^{88} -7.72741i q^{89} -14.9282 q^{91} +3.80385 q^{92} +21.6603 q^{94} -4.24264i q^{95} +7.34847i q^{97} -0.896575i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7} + 8 q^{13} - 4 q^{16} + 4 q^{22} + 12 q^{23} + 4 q^{25} + 12 q^{28} - 20 q^{34} - 4 q^{35} - 12 q^{38} - 12 q^{49} - 24 q^{52} - 4 q^{58} - 24 q^{59} - 12 q^{62} + 16 q^{64} + 8 q^{65}+ \cdots + 52 q^{94}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 + 8 * q^13 - 4 * q^16 + 4 * q^22 + 12 * q^23 + 4 * q^25 + 12 * q^28 - 20 * q^34 - 4 * q^35 - 12 * q^38 - 12 * q^49 - 24 * q^52 - 4 * q^58 - 24 * q^59 - 12 * q^62 + 16 * q^64 + 8 * q^65 - 4 * q^67 - 24 * q^71 + 12 * q^74 - 4 * q^80 - 32 * q^82 + 12 * q^83 - 12 * q^86 + 20 * q^88 - 32 * q^91 + 36 * q^92 + 52 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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