LMFDB - Embedded newform 1305.2.d.a.811.2

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			811.2
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	1305.811
Dual form			1305.2.d.a.811.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.517638i q^{2} +1.73205 q^{4} +1.00000 q^{5} +0.732051 q^{7} -1.93185i q^{8} -0.517638i q^{10} +5.27792i q^{11} -1.46410 q^{13} -0.378937i q^{14} +2.46410 q^{16} -6.31319i q^{17} +4.24264i q^{19} +1.73205 q^{20} +2.73205 q^{22} +8.19615 q^{23} +1.00000 q^{25} +0.757875i q^{26} +1.26795 q^{28} +(5.19615 + 1.41421i) q^{29} +4.24264i q^{31} -5.13922i q^{32} -3.26795 q^{34} +0.732051 q^{35} -4.24264i q^{37} +2.19615 q^{38} -1.93185i q^{40} -8.76268i q^{41} +4.24264i q^{43} +9.14162i q^{44} -4.24264i q^{46} +8.38375i q^{47} -6.46410 q^{49} -0.517638i q^{50} -2.53590 q^{52} +5.27792i q^{55} -1.41421i q^{56} +(0.732051 - 2.68973i) q^{58} -6.00000 q^{59} -3.10583i q^{61} +2.19615 q^{62} +2.26795 q^{64} -1.46410 q^{65} +11.1244 q^{67} -10.9348i q^{68} -0.378937i q^{70} -6.00000 q^{71} +1.13681i q^{73} -2.19615 q^{74} +7.34847i q^{76} +3.86370i q^{77} -15.8338i q^{79} +2.46410 q^{80} -4.53590 q^{82} -2.19615 q^{83} -6.31319i q^{85} +2.19615 q^{86} +10.1962 q^{88} -2.07055i q^{89} -1.07180 q^{91} +14.1962 q^{92} +4.33975 q^{94} +4.24264i q^{95} +7.34847i q^{97} +3.34607i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	0.517638i		−	0.366025i	−0.983111	−	0.183013i	$-0.941415\pi$
$2$		−	0.517638i		−	0.366025i	0.983111	−	0.183013i	$-0.0585849\pi$
$3$		0			0
$3$		0			0
$4$	1.73205			0.866025
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	0.732051			0.276689			0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051			0.276689			0.138345	+	0.990384i	$0.455822\pi$
$8$		−	1.93185i		−	0.683013i
$9$		0			0
$10$		−	0.517638i		−	0.163692i
$11$			5.27792i			1.59135i	0.605723	+	0.795676i	$0.292884\pi$
$11$			5.27792i			1.59135i	−0.605723	+	0.795676i	$0.707116\pi$
$12$		0			0
$13$	−1.46410			−0.406069			−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410			−0.406069			−0.203034	+	0.979172i	$0.565080\pi$
$14$		−	0.378937i		−	0.101275i
$15$		0			0
$16$	2.46410			0.616025
$17$		−	6.31319i		−	1.53117i	−0.643332	−	0.765587i	$-0.722449\pi$
$17$		−	6.31319i		−	1.53117i	0.643332	−	0.765587i	$-0.277551\pi$
$18$		0			0
$19$			4.24264i			0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$			4.24264i			0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$	1.73205			0.387298
$21$		0			0
$22$	2.73205			0.582475
$23$	8.19615			1.70902			0.854508	−	0.519438i	$-0.173859\pi$
$23$	8.19615			1.70902			0.854508	+	0.519438i	$0.173859\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			0.757875i			0.148631i
$27$		0			0
$28$	1.26795			0.239620
$29$	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	$0.124420\pi$
$31$			4.24264i			0.762001i	−0.924575	+	0.381000i	$0.875580\pi$
$32$		−	5.13922i		−	0.908494i
$33$		0			0
$34$	−3.26795			−0.560449
$35$	0.732051			0.123739
$36$		0			0
$37$		−	4.24264i		−	0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$		−	4.24264i		−	0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$	2.19615			0.356263
$39$		0			0
$40$		−	1.93185i		−	0.305453i
$41$		−	8.76268i		−	1.36850i	−0.729247	−	0.684251i	$-0.760129\pi$
$41$		−	8.76268i		−	1.36850i	0.729247	−	0.684251i	$-0.239871\pi$
$42$		0			0
$43$			4.24264i			0.646997i	0.946229	+	0.323498i	$0.104859\pi$
$43$			4.24264i			0.646997i	−0.946229	+	0.323498i	$0.895141\pi$
$44$			9.14162i			1.37815i
$45$		0			0
$46$		−	4.24264i		−	0.625543i
$47$			8.38375i			1.22289i	0.791285	+	0.611447i	$0.209412\pi$
$47$			8.38375i			1.22289i	−0.791285	+	0.611447i	$0.790588\pi$
$48$		0			0
$49$	−6.46410			−0.923443
$50$		−	0.517638i		−	0.0732051i
$51$		0			0
$52$	−2.53590			−0.351666
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$			5.27792i			0.711674i
$56$		−	1.41421i		−	0.188982i
$57$		0			0
$58$	0.732051	−	2.68973i	0.0961230	−	0.353178i
$59$	−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
$61$		−	3.10583i		−	0.397661i	−0.980034	−	0.198830i	$-0.936286\pi$
$61$		−	3.10583i		−	0.397661i	0.980034	−	0.198830i	$-0.0637143\pi$
$62$	2.19615			0.278912
$63$		0			0
$64$	2.26795			0.283494
$65$	−1.46410			−0.181599
$66$		0			0
$67$	11.1244			1.35906			0.679528	−	0.733649i	$-0.262185\pi$
$67$	11.1244			1.35906			0.679528	+	0.733649i	$0.262185\pi$
$68$		−	10.9348i		−	1.32604i
$69$		0			0
$70$		−	0.378937i		−	0.0452917i
$71$	−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
$73$			1.13681i			0.133054i	0.997785	+	0.0665269i	$0.0211918\pi$
$73$			1.13681i			0.133054i	−0.997785	+	0.0665269i	$0.978808\pi$
$74$	−2.19615			−0.255298
$75$		0			0
$76$			7.34847i			0.842927i
$77$			3.86370i			0.440310i
$78$		0			0
$79$		−	15.8338i		−	1.78144i	−0.454556	−	0.890718i	$-0.650202\pi$
$79$		−	15.8338i		−	1.78144i	0.454556	−	0.890718i	$-0.349798\pi$
$80$	2.46410			0.275495
$81$		0			0
$82$	−4.53590			−0.500906
$83$	−2.19615			−0.241059			−0.120530	−	0.992710i	$-0.538459\pi$
$83$	−2.19615			−0.241059			−0.120530	+	0.992710i	$0.538459\pi$
$84$		0			0
$85$		−	6.31319i		−	0.684762i
$86$	2.19615			0.236817
$87$		0			0
$88$	10.1962			1.08691
$89$		−	2.07055i		−	0.219478i	−0.993960	−	0.109739i	$-0.964998\pi$
$89$		−	2.07055i		−	0.219478i	0.993960	−	0.109739i	$-0.0350015\pi$
$90$		0			0
$91$	−1.07180			−0.112355
$92$	14.1962			1.48005
$93$		0			0
$94$	4.33975			0.447611
$95$			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	$0.878308\pi$
$98$			3.34607i			0.338004i
$99$		0			0
$100$	1.73205			0.173205
$101$			4.14110i			0.412055i	0.978546	+	0.206028i	$0.0660537\pi$
$101$			4.14110i			0.412055i	−0.978546	+	0.206028i	$0.933946\pi$
$102$		0			0
$103$	−0.196152			−0.0193275			−0.00966374	−	0.999953i	$-0.503076\pi$
$103$	−0.196152			−0.0193275			−0.00966374	+	0.999953i	$0.503076\pi$
$104$			2.82843i			0.277350i
$105$		0			0
$106$		0			0
$107$	−2.19615			−0.212310			−0.106155	−	0.994350i	$-0.533854\pi$
$107$	−2.19615			−0.212310			−0.106155	+	0.994350i	$0.533854\pi$
$108$		0			0
$109$	−1.46410			−0.140236			−0.0701178	−	0.997539i	$-0.522338\pi$
$109$	−1.46410			−0.140236			−0.0701178	+	0.997539i	$0.522338\pi$
$110$	2.73205			0.260491
$111$		0			0
$112$	1.80385			0.170448
$113$			13.7632i			1.29473i	0.762179	+	0.647366i	$0.224130\pi$
$113$			13.7632i			1.29473i	−0.762179	+	0.647366i	$0.775870\pi$
$114$		0			0
$115$	8.19615			0.764295
$116$	9.00000	+	2.44949i	0.835629	+	0.227429i
$117$		0			0
$118$			3.10583i			0.285915i
$119$		−	4.62158i		−	0.423659i
$120$		0			0
$121$	−16.8564			−1.53240
$122$	−1.60770			−0.145554
$123$		0			0
$124$			7.34847i			0.659912i
$125$	1.00000			0.0894427
$126$		0			0
$127$		−	12.7279i		−	1.12942i	−0.825289	−	0.564710i	$-0.808988\pi$
$127$		−	12.7279i		−	1.12942i	0.825289	−	0.564710i	$-0.191012\pi$
$128$		−	11.4524i		−	1.01226i
$129$		0			0
$130$			0.757875i			0.0664700i
$131$			2.17209i			0.189776i	0.995488	+	0.0948881i	$0.0302493\pi$
$131$			2.17209i			0.189776i	−0.995488	+	0.0948881i	$0.969751\pi$
$132$		0			0
$133$			3.10583i			0.269309i
$134$		−	5.75839i		−	0.497449i
$135$		0			0
$136$	−12.1962			−1.04581
$137$		−	9.89949i		−	0.845771i	−0.906183	−	0.422885i	$-0.861017\pi$
$137$		−	9.89949i		−	0.845771i	0.906183	−	0.422885i	$-0.138983\pi$
$138$		0			0
$139$	−13.4641			−1.14201			−0.571005	−	0.820947i	$-0.693446\pi$
$139$	−13.4641			−1.14201			−0.571005	+	0.820947i	$0.693446\pi$
$140$	1.26795			0.107161
$141$		0			0
$142$			3.10583i			0.260635i
$143$		−	7.72741i		−	0.646198i
$144$		0			0
$145$	5.19615	+	1.41421i	0.431517	+	0.117444i
$146$	0.588457			0.0487011
$147$		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
$151$	18.3923			1.49674			0.748372	−	0.663279i	$-0.230836\pi$
$151$	18.3923			1.49674			0.748372	+	0.663279i	$0.230836\pi$
$152$	8.19615			0.664796
$153$		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	$-0.0947321\pi$
$158$	−8.19615			−0.652051
$159$		0			0
$160$		−	5.13922i		−	0.406291i
$161$	6.00000			0.472866
$162$		0			0
$163$		−	4.24264i		−	0.332309i	−0.986100	−	0.166155i	$-0.946865\pi$
$163$		−	4.24264i		−	0.332309i	0.986100	−	0.166155i	$-0.0531351\pi$
$164$		−	15.1774i		−	1.18516i
$165$		0			0
$166$			1.13681i			0.0882337i
$167$	−3.80385			−0.294351			−0.147175	−	0.989110i	$-0.547018\pi$
$167$	−3.80385			−0.294351			−0.147175	+	0.989110i	$0.547018\pi$
$168$		0			0
$169$	−10.8564			−0.835108
$170$	−3.26795			−0.250640
$171$		0			0
$172$			7.34847i			0.560316i
$173$	−20.7846			−1.58022			−0.790112	−	0.612962i	$-0.789978\pi$
$173$	−20.7846			−1.58022			−0.790112	+	0.612962i	$0.789978\pi$
$174$		0			0
$175$	0.732051			0.0553378
$176$			13.0053i			0.980313i
$177$		0			0
$178$	−1.07180			−0.0803346
$179$	−16.3923			−1.22522			−0.612609	−	0.790386i	$-0.709880\pi$
$179$	−16.3923			−1.22522			−0.612609	+	0.790386i	$0.709880\pi$
$180$		0			0
$181$	−5.85641			−0.435303			−0.217652	−	0.976027i	$-0.569840\pi$
$181$	−5.85641			−0.435303			−0.217652	+	0.976027i	$0.569840\pi$
$182$			0.554803i			0.0411247i
$183$		0			0
$184$		−	15.8338i		−	1.16728i
$185$		−	4.24264i		−	0.311925i
$186$		0			0
$187$	33.3205			2.43664
$188$			14.5211i			1.05906i
$189$		0			0
$190$	2.19615			0.159326
$191$		−	19.2170i		−	1.39049i	−0.718772	−	0.695246i	$-0.755295\pi$
$191$		−	19.2170i		−	1.39049i	0.718772	−	0.695246i	$-0.244705\pi$
$192$		0			0
$193$		−	12.7279i		−	0.916176i	−0.888907	−	0.458088i	$-0.848534\pi$
$193$		−	12.7279i		−	0.916176i	0.888907	−	0.458088i	$-0.151466\pi$
$194$	3.80385			0.273100
$195$		0			0
$196$	−11.1962			−0.799725
$197$	1.60770			0.114544			0.0572718	−	0.998359i	$-0.481760\pi$
$197$	1.60770			0.114544			0.0572718	+	0.998359i	$0.481760\pi$
$198$		0			0
$199$	−19.4641			−1.37977			−0.689887	−	0.723917i	$-0.742340\pi$
$199$	−19.4641			−1.37977			−0.689887	+	0.723917i	$0.742340\pi$
$200$		−	1.93185i		−	0.136603i
$201$		0			0
$202$	2.14359			0.150823
$203$	3.80385	+	1.03528i	0.266978	+	0.0726621i
$204$		0			0
$205$		−	8.76268i		−	0.612012i
$206$			0.101536i			0.00707435i
$207$		0			0
$208$	−3.60770			−0.250149
$209$	−22.3923			−1.54891
$210$		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
$214$			1.13681i			0.0777109i
$215$			4.24264i			0.289346i
$216$		0			0
$217$			3.10583i			0.210837i
$218$			0.757875i			0.0513298i
$219$		0			0
$220$			9.14162i			0.616328i
$221$			9.24316i			0.621762i
$222$		0			0
$223$	−9.66025			−0.646898			−0.323449	−	0.946246i	$-0.604842\pi$
$223$	−9.66025			−0.646898			−0.323449	+	0.946246i	$0.604842\pi$
$224$		−	3.76217i		−	0.251370i
$225$		0			0
$226$	7.12436			0.473905
$227$	15.8038			1.04894			0.524469	−	0.851429i	$-0.324264\pi$
$227$	15.8038			1.04894			0.524469	+	0.851429i	$0.324264\pi$
$228$		0			0
$229$			25.4558i			1.68217i	0.540903	+	0.841085i	$0.318082\pi$
$229$			25.4558i			1.68217i	−0.540903	+	0.841085i	$0.681918\pi$
$230$		−	4.24264i		−	0.279751i
$231$		0			0
$232$	2.73205	−	10.0382i	0.179368	−	0.659040i
$233$	−22.3923			−1.46697			−0.733484	−	0.679706i	$-0.762107\pi$
$233$	−22.3923			−1.46697			−0.733484	+	0.679706i	$0.762107\pi$
$234$		0			0
$235$			8.38375i			0.546895i
$236$	−10.3923			−0.676481
$237$		0			0
$238$	−2.39230			−0.155070
$239$	−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
$241$	8.92820			0.575116			0.287558	−	0.957763i	$-0.407157\pi$
$241$	8.92820			0.575116			0.287558	+	0.957763i	$0.407157\pi$
$242$			8.72552i			0.560898i
$243$		0			0
$244$		−	5.37945i		−	0.344384i
$245$	−6.46410			−0.412976
$246$		0			0
$247$		−	6.21166i		−	0.395238i
$248$	8.19615			0.520456
$249$		0			0
$250$		−	0.517638i		−	0.0327383i
$251$			13.7632i			0.868725i	0.900738	+	0.434363i	$0.143026\pi$
$251$			13.7632i			0.868725i	−0.900738	+	0.434363i	$0.856974\pi$
$252$		0			0
$253$			43.2586i			2.71965i
$254$	−6.58846			−0.413397
$255$		0			0
$256$	−1.39230			−0.0870191
$257$	−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
$259$		−	3.10583i		−	0.192987i
$260$	−2.53590			−0.157270
$261$		0			0
$262$	1.12436			0.0694629
$263$			4.79744i			0.295823i	0.989001	+	0.147912i	$0.0472551\pi$
$263$			4.79744i			0.295823i	−0.989001	+	0.147912i	$0.952745\pi$
$264$		0			0
$265$		0			0
$266$	1.60770			0.0985741
$267$		0			0
$268$	19.2679			1.17698
$269$			32.7028i			1.99392i	0.0778925	+	0.996962i	$0.475181\pi$
$269$			32.7028i			1.99392i	−0.0778925	+	0.996962i	$0.524819\pi$
$270$		0			0
$271$			9.62209i			0.584501i	0.956342	+	0.292250i	$0.0944040\pi$
$271$			9.62209i			0.584501i	−0.956342	+	0.292250i	$0.905596\pi$
$272$		−	15.5563i		−	0.943242i
$273$		0			0
$274$	−5.12436			−0.309574
$275$			5.27792i			0.318270i
$276$		0			0
$277$	2.92820			0.175939			0.0879693	−	0.996123i	$-0.471962\pi$
$277$	2.92820			0.175939			0.0879693	+	0.996123i	$0.471962\pi$
$278$			6.96953i			0.418005i
$279$		0			0
$280$		−	1.41421i		−	0.0845154i
$281$	4.39230			0.262023			0.131011	−	0.991381i	$-0.458178\pi$
$281$	4.39230			0.262023			0.131011	+	0.991381i	$0.458178\pi$
$282$		0			0
$283$	−20.9808			−1.24718			−0.623588	−	0.781753i	$-0.714326\pi$
$283$	−20.9808			−1.24718			−0.623588	+	0.781753i	$0.714326\pi$
$284$	−10.3923			−0.616670
$285$		0			0
$286$	−4.00000			−0.236525
$287$		−	6.41473i		−	0.378649i
$288$		0			0
$289$	−22.8564			−1.34449
$290$	0.732051	−	2.68973i	0.0429875	−	0.157946i
$291$		0			0
$292$			1.96902i			0.115228i
$293$		−	11.6926i		−	0.683092i	−0.939865	−	0.341546i	$-0.889050\pi$
$293$		−	11.6926i		−	0.683092i	0.939865	−	0.341546i	$-0.110950\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$	−8.19615			−0.476392
$297$		0			0
$298$		−	9.31749i		−	0.539747i
$299$	−12.0000			−0.693978
$300$		0			0
$301$			3.10583i			0.179017i
$302$		−	9.52056i		−	0.547847i
$303$		0			0
$304$			10.4543i			0.599595i
$305$		−	3.10583i		−	0.177839i
$306$		0			0
$307$		−	25.1512i		−	1.43546i	−0.696323	−	0.717728i	$-0.745182\pi$
$307$		−	25.1512i		−	1.43546i	0.696323	−	0.717728i	$-0.254818\pi$
$308$			6.69213i			0.381320i
$309$		0			0
$310$	2.19615			0.124733
$311$		−	0.101536i		−	0.00575758i	−0.999996	−	0.00287879i	$-0.999084\pi$
$311$		−	0.101536i		−	0.00575758i	0.999996	−	0.00287879i	$-0.000916348\pi$
$312$		0			0
$313$	−25.4641			−1.43932			−0.719658	−	0.694329i	$-0.755701\pi$
$313$	−25.4641			−1.43932			−0.719658	+	0.694329i	$0.755701\pi$
$314$	−3.80385			−0.214664
$315$		0			0
$316$		−	27.4249i		−	1.54277i
$317$		−	21.0101i		−	1.18005i	−0.807386	−	0.590023i	$-0.799119\pi$
$317$		−	21.0101i		−	1.18005i	0.807386	−	0.590023i	$-0.200881\pi$
$318$		0			0
$319$	−7.46410	+	27.4249i	−0.417909	+	1.53550i
$320$	2.26795			0.126782
$321$		0			0
$322$		−	3.10583i		−	0.173081i
$323$	26.7846			1.49034
$324$		0			0
$325$	−1.46410			−0.0812137
$326$	−2.19615			−0.121634
$327$		0			0
$328$	−16.9282			−0.934704
$329$			6.13733i			0.338362i
$330$		0			0
$331$		−	18.9396i		−	1.04101i	−0.853858	−	0.520507i	$-0.825743\pi$
$331$		−	18.9396i		−	1.04101i	0.853858	−	0.520507i	$-0.174257\pi$
$332$	−3.80385			−0.208763
$333$		0			0
$334$			1.96902i			0.107740i
$335$	11.1244			0.607788
$336$		0			0
$337$			18.1074i			0.986372i	0.869924	+	0.493186i	$0.164168\pi$
$337$			18.1074i			0.986372i	−0.869924	+	0.493186i	$0.835832\pi$
$338$			5.61969i			0.305671i
$339$		0			0
$340$		−	10.9348i		−	0.593021i
$341$	−22.3923			−1.21261
$342$		0			0
$343$	−9.85641			−0.532196
$344$	8.19615			0.441907
$345$		0			0
$346$			10.7589i			0.578402i
$347$	−26.1962			−1.40628			−0.703142	−	0.711050i	$-0.748220\pi$
$347$	−26.1962			−1.40628			−0.703142	+	0.711050i	$0.748220\pi$
$348$		0			0
$349$	18.3923			0.984518			0.492259	−	0.870449i	$-0.336171\pi$
$349$	18.3923			0.984518			0.492259	+	0.870449i	$0.336171\pi$
$350$		−	0.378937i		−	0.0202551i
$351$		0			0
$352$	27.1244			1.44573
$353$	−32.7846			−1.74495			−0.872474	−	0.488660i	$-0.837486\pi$
$353$	−32.7846			−1.74495			−0.872474	+	0.488660i	$0.837486\pi$
$354$		0			0
$355$	−6.00000			−0.318447
$356$		−	3.58630i		−	0.190074i
$357$		0			0
$358$			8.48528i			0.448461i
$359$			7.07107i			0.373197i	0.982436	+	0.186598i	$0.0597463\pi$
$359$			7.07107i			0.373197i	−0.982436	+	0.186598i	$0.940254\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$			3.03150i			0.159332i
$363$		0			0
$364$	−1.85641			−0.0973021
$365$			1.13681i			0.0595035i
$366$		0			0
$367$			1.96902i			0.102782i	0.998679	+	0.0513909i	$0.0163654\pi$
$367$			1.96902i			0.102782i	−0.998679	+	0.0513909i	$0.983635\pi$
$368$	20.1962			1.05280
$369$		0			0
$370$	−2.19615			−0.114173
$371$		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	$0.692886\pi$
$374$		−	17.2480i		−	0.891871i
$375$		0			0
$376$	16.1962			0.835253
$377$	−7.60770	−	2.07055i	−0.391816	−	0.106639i
$378$		0			0
$379$		−	15.8338i		−	0.813325i	−0.913578	−	0.406663i	$-0.866692\pi$
$379$		−	15.8338i		−	0.813325i	0.913578	−	0.406663i	$-0.133308\pi$
$380$			7.34847i			0.376969i
$381$		0			0
$382$	−9.94744			−0.508955
$383$	2.19615			0.112218			0.0561091	−	0.998425i	$-0.482131\pi$
$383$	2.19615			0.112218			0.0561091	+	0.998425i	$0.482131\pi$
$384$		0			0
$385$			3.86370i			0.196913i
$386$	−6.58846			−0.335344
$387$		0			0
$388$			12.7279i			0.646162i
$389$			32.7028i			1.65810i	0.559177	+	0.829048i	$0.311117\pi$
$389$			32.7028i			1.65810i	−0.559177	+	0.829048i	$0.688883\pi$
$390$		0			0
$391$		−	51.7439i		−	2.61680i
$392$			12.4877i			0.630723i
$393$		0			0
$394$		−	0.832204i		−	0.0419258i
$395$		−	15.8338i		−	0.796682i
$396$		0			0
$397$	30.3923			1.52535			0.762673	−	0.646784i	$-0.223887\pi$
$397$	30.3923			1.52535			0.762673	+	0.646784i	$0.223887\pi$
$398$			10.0754i			0.505032i
$399$		0			0
$400$	2.46410			0.123205
$401$	−8.78461			−0.438682			−0.219341	−	0.975648i	$-0.570391\pi$
$401$	−8.78461			−0.438682			−0.219341	+	0.975648i	$0.570391\pi$
$402$		0			0
$403$		−	6.21166i		−	0.309425i
$404$			7.17260i			0.356850i
$405$		0			0
$406$	0.535898	−	1.96902i	0.0265962	−	0.0977206i
$407$	22.3923			1.10995
$408$		0			0
$409$			17.8028i			0.880290i	0.897927	+	0.440145i	$0.145073\pi$
$409$			17.8028i			0.880290i	−0.897927	+	0.440145i	$0.854927\pi$
$410$	−4.53590			−0.224012
$411$		0			0
$412$	−0.339746			−0.0167381
$413$	−4.39230			−0.216131
$414$		0			0
$415$	−2.19615			−0.107805
$416$			7.52433i			0.368911i
$417$		0			0
$418$			11.5911i			0.566940i
$419$	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
$421$			17.8028i			0.867654i	0.900996	+	0.433827i	$0.142837\pi$
$421$			17.8028i			0.867654i	−0.900996	+	0.433827i	$0.857163\pi$
$422$	11.4115			0.555505
$423$		0			0
$424$		0			0
$425$		−	6.31319i		−	0.306235i
$426$		0			0
$427$		−	2.27362i		−	0.110028i
$428$	−3.80385			−0.183866
$429$		0			0
$430$	2.19615			0.105908
$431$	−4.39230			−0.211570			−0.105785	−	0.994389i	$-0.533736\pi$
$431$	−4.39230			−0.211570			−0.105785	+	0.994389i	$0.533736\pi$
$432$		0			0
$433$		−	12.7279i		−	0.611665i	−0.952085	−	0.305832i	$-0.901065\pi$
$433$		−	12.7279i		−	0.611665i	0.952085	−	0.305832i	$-0.0989347\pi$
$434$	1.60770			0.0771718
$435$		0			0
$436$	−2.53590			−0.121448
$437$			34.7733i			1.66343i
$438$		0			0
$439$	8.92820			0.426120			0.213060	−	0.977039i	$-0.431657\pi$
$439$	8.92820			0.426120			0.213060	+	0.977039i	$0.431657\pi$
$440$	10.1962			0.486082
$441$		0			0
$442$	4.78461			0.227581
$443$		−	14.7985i		−	0.703097i	−0.936170	−	0.351548i	$-0.885655\pi$
$443$		−	14.7985i		−	0.703097i	0.936170	−	0.351548i	$-0.114345\pi$
$444$		0			0
$445$		−	2.07055i		−	0.0981536i
$446$			5.00052i			0.236781i
$447$		0			0
$448$	1.66025			0.0784396
$449$		−	30.6322i		−	1.44562i	−0.691045	−	0.722812i	$-0.742849\pi$
$449$		−	30.6322i		−	1.44562i	0.691045	−	0.722812i	$-0.257151\pi$
$450$		0			0
$451$	46.2487			2.17777
$452$			23.8386i			1.12127i
$453$		0			0
$454$		−	8.18067i		−	0.383938i
$455$	−1.07180			−0.0502466
$456$		0			0
$457$	26.9282			1.25965			0.629824	−	0.776738i	$-0.283127\pi$
$457$	26.9282			1.25965			0.629824	+	0.776738i	$0.283127\pi$
$458$	13.1769			0.615717
$459$		0			0
$460$	14.1962			0.661899
$461$		−	22.6274i		−	1.05386i	−0.849907	−	0.526932i	$-0.823342\pi$
$461$		−	22.6274i		−	1.05386i	0.849907	−	0.526932i	$-0.176658\pi$
$462$		0			0
$463$	−21.6603			−1.00664			−0.503319	−	0.864101i	$-0.667888\pi$
$463$	−21.6603			−1.00664			−0.503319	+	0.864101i	$0.667888\pi$
$464$	12.8038	+	3.48477i	0.594404	+	0.161776i
$465$		0			0
$466$			11.5911i			0.536948i
$467$			21.7680i			1.00730i	0.863907	+	0.503652i	$0.168010\pi$
$467$			21.7680i			1.00730i	−0.863907	+	0.503652i	$0.831990\pi$
$468$		0			0
$469$	8.14359			0.376036
$470$	4.33975			0.200178
$471$		0			0
$472$			11.5911i			0.533524i
$473$	−22.3923			−1.02960
$474$		0			0
$475$			4.24264i			0.194666i
$476$		−	8.00481i		−	0.366900i
$477$		0			0
$478$			3.10583i			0.142057i
$479$		−	4.52004i		−	0.206526i	−0.994654	−	0.103263i	$-0.967072\pi$
$479$		−	4.52004i		−	0.206526i	0.994654	−	0.103263i	$-0.0329283\pi$
$480$		0			0
$481$			6.21166i			0.283227i
$482$		−	4.62158i		−	0.210507i
$483$		0			0
$484$	−29.1962			−1.32710
$485$			7.34847i			0.333677i
$486$		0			0
$487$	−22.5885			−1.02358			−0.511790	−	0.859110i	$-0.671018\pi$
$487$	−22.5885			−1.02358			−0.511790	+	0.859110i	$0.671018\pi$
$488$	−6.00000			−0.271607
$489$		0			0
$490$			3.34607i			0.151160i
$491$		−	1.41421i		−	0.0638226i	−0.999491	−	0.0319113i	$-0.989841\pi$
$491$		−	1.41421i		−	0.0638226i	0.999491	−	0.0319113i	$-0.0101594\pi$
$492$		0			0
$493$	8.92820	−	32.8043i	0.402106	−	1.47743i
$494$	−3.21539			−0.144667
$495$		0			0
$496$			10.4543i			0.469412i
$497$	−4.39230			−0.197022
$498$		0			0
$499$	−5.85641			−0.262169			−0.131084	−	0.991371i	$-0.541846\pi$
$499$	−5.85641			−0.262169			−0.131084	+	0.991371i	$0.541846\pi$
$500$	1.73205			0.0774597
$501$		0			0
$502$	7.12436			0.317976
$503$		−	14.7985i		−	0.659831i	−0.944010	−	0.329916i	$-0.892980\pi$
$503$		−	14.7985i		−	0.659831i	0.944010	−	0.329916i	$-0.107020\pi$
$504$		0			0
$505$			4.14110i			0.184277i
$506$	22.3923			0.995459
$507$		0			0
$508$		−	22.0454i		−	0.978107i
$509$	−20.7846			−0.921262			−0.460631	−	0.887592i	$-0.652377\pi$
$509$	−20.7846			−0.921262			−0.460631	+	0.887592i	$0.652377\pi$
$510$		0			0
$511$			0.832204i			0.0368145i
$512$		−	22.1841i		−	0.980408i
$513$		0			0
$514$			6.21166i			0.273984i
$515$	−0.196152			−0.00864351
$516$		0			0
$517$	−44.2487			−1.94606
$518$	−1.60770			−0.0706381
$519$		0			0
$520$			2.82843i			0.124035i
$521$	32.7846			1.43632			0.718160	−	0.695878i	$-0.244985\pi$
$521$	32.7846			1.43632			0.718160	+	0.695878i	$0.244985\pi$
$522$		0			0
$523$	−21.6603			−0.947137			−0.473568	−	0.880757i	$-0.657034\pi$
$523$	−21.6603			−0.947137			−0.473568	+	0.880757i	$0.657034\pi$
$524$			3.76217i			0.164351i
$525$		0			0
$526$	2.48334			0.108279
$527$	26.7846			1.16676
$528$		0			0
$529$	44.1769			1.92074
$530$		0			0
$531$		0			0
$532$			5.37945i			0.233229i
$533$			12.8295i			0.555706i
$534$		0			0
$535$	−2.19615			−0.0949479
$536$		−	21.4906i		−	0.928253i
$537$		0			0
$538$	16.9282			0.729827
$539$		−	34.1170i		−	1.46952i
$540$		0			0
$541$			25.4558i			1.09443i	0.836991	+	0.547216i	$0.184312\pi$
$541$			25.4558i			1.09443i	−0.836991	+	0.547216i	$0.815688\pi$
$542$	4.98076			0.213942
$543$		0			0
$544$	−32.4449			−1.39106
$545$	−1.46410			−0.0627152
$546$		0			0
$547$	21.5167			0.919986			0.459993	−	0.887923i	$-0.347852\pi$
$547$	21.5167			0.919986			0.459993	+	0.887923i	$0.347852\pi$
$548$		−	17.1464i		−	0.732459i
$549$		0			0
$550$	2.73205			0.116495
$551$	−6.00000	+	22.0454i	−0.255609	+	0.939166i
$552$		0			0
$553$		−	11.5911i		−	0.492904i
$554$		−	1.51575i		−	0.0643980i
$555$		0			0
$556$	−23.3205			−0.989010
$557$	22.3923			0.948792			0.474396	−	0.880311i	$-0.342666\pi$
$557$	22.3923			0.948792			0.474396	+	0.880311i	$0.342666\pi$
$558$		0			0
$559$		−	6.21166i		−	0.262725i
$560$	1.80385			0.0762265
$561$		0			0
$562$		−	2.27362i		−	0.0959071i
$563$			16.8690i			0.710945i	0.934687	+	0.355472i	$0.115680\pi$
$563$			16.8690i			0.710945i	−0.934687	+	0.355472i	$0.884320\pi$
$564$		0			0
$565$			13.7632i			0.579022i
$566$			10.8604i			0.456498i
$567$		0			0
$568$			11.5911i			0.486352i
$569$		−	15.9353i		−	0.668042i	−0.942566	−	0.334021i	$-0.891594\pi$
$569$		−	15.9353i		−	0.668042i	0.942566	−	0.334021i	$-0.108406\pi$
$570$		0			0
$571$	−11.6077			−0.485767			−0.242883	−	0.970055i	$-0.578093\pi$
$571$	−11.6077			−0.485767			−0.242883	+	0.970055i	$0.578093\pi$
$572$		−	13.3843i		−	0.559624i
$573$		0			0
$574$	−3.32051			−0.138595
$575$	8.19615			0.341803
$576$		0			0
$577$		−	44.3954i		−	1.84821i	−0.382144	−	0.924103i	$-0.624814\pi$
$577$		−	44.3954i		−	1.84821i	0.382144	−	0.924103i	$-0.375186\pi$
$578$			11.8313i			0.492119i
$579$		0			0
$580$	9.00000	+	2.44949i	0.373705	+	0.101710i
$581$	−1.60770			−0.0666984
$582$		0			0
$583$		0			0
$584$	2.19615			0.0908774
$585$		0			0
$586$	−6.05256			−0.250029
$587$	44.1962			1.82417			0.912085	−	0.410001i	$-0.134472\pi$
$587$	44.1962			1.82417			0.912085	+	0.410001i	$0.134472\pi$
$588$		0			0
$589$	−18.0000			−0.741677
$590$			3.10583i			0.127865i
$591$		0			0
$592$		−	10.4543i		−	0.429669i
$593$	19.6077			0.805192			0.402596	−	0.915378i	$-0.368108\pi$
$593$	19.6077			0.805192			0.402596	+	0.915378i	$0.368108\pi$
$594$		0			0
$595$		−	4.62158i		−	0.189466i
$596$	31.1769			1.27706
$597$		0			0
$598$			6.21166i			0.254014i
$599$		−	7.14540i		−	0.291953i	−0.989288	−	0.145977i	$-0.953368\pi$
$599$		−	7.14540i		−	0.291953i	0.989288	−	0.145977i	$-0.0466324\pi$
$600$		0			0
$601$		−	20.0764i		−	0.818933i	−0.912325	−	0.409467i	$-0.865715\pi$
$601$		−	20.0764i		−	0.818933i	0.912325	−	0.409467i	$-0.134285\pi$
$602$	1.60770			0.0655248
$603$		0			0
$604$	31.8564			1.29622
$605$	−16.8564			−0.685310
$606$		0			0
$607$			12.7279i			0.516610i	0.966063	+	0.258305i	$0.0831640\pi$
$607$			12.7279i			0.516610i	−0.966063	+	0.258305i	$0.916836\pi$
$608$	21.8038			0.884263
$609$		0			0
$610$	−1.60770			−0.0650937
$611$		−	12.2747i		−	0.496579i
$612$		0			0
$613$	32.9282			1.32996			0.664979	−	0.746862i	$-0.268441\pi$
$613$	32.9282			1.32996			0.664979	+	0.746862i	$0.268441\pi$
$614$	−13.0192			−0.525414
$615$		0			0
$616$	7.46410			0.300737
$617$		−	6.79367i		−	0.273503i	−0.990605	−	0.136751i	$-0.956334\pi$
$617$		−	6.79367i		−	0.273503i	0.990605	−	0.136751i	$-0.0436661\pi$
$618$		0			0
$619$			13.5601i			0.545027i	0.962152	+	0.272514i	$0.0878550\pi$
$619$			13.5601i			0.545027i	−0.962152	+	0.272514i	$0.912145\pi$
$620$			7.34847i			0.295122i
$621$		0			0
$622$	−0.0525589			−0.00210742
$623$		−	1.51575i		−	0.0607272i
$624$		0			0
$625$	1.00000			0.0400000
$626$			13.1812i			0.526826i
$627$		0			0
$628$		−	12.7279i		−	0.507899i
$629$	−26.7846			−1.06797
$630$		0			0
$631$	−17.8564			−0.710852			−0.355426	−	0.934704i	$-0.615664\pi$
$631$	−17.8564			−0.710852			−0.355426	+	0.934704i	$0.615664\pi$
$632$	−30.5885			−1.21674
$633$		0			0
$634$	−10.8756			−0.431927
$635$		−	12.7279i		−	0.505092i
$636$		0			0
$637$	9.46410			0.374981
$638$	14.1962	+	3.86370i	0.562031	+	0.152965i
$639$		0			0
$640$		−	11.4524i		−	0.452696i
$641$		−	15.9353i		−	0.629406i	−0.949190	−	0.314703i	$-0.898095\pi$
$641$		−	15.9353i		−	0.629406i	0.949190	−	0.314703i	$-0.101905\pi$
$642$		0			0
$643$	9.94744			0.392289			0.196144	−	0.980575i	$-0.437158\pi$
$643$	9.94744			0.392289			0.196144	+	0.980575i	$0.437158\pi$
$644$	10.3923			0.409514
$645$		0			0
$646$		−	13.8647i		−	0.545501i
$647$	27.3731			1.07615			0.538073	−	0.842898i	$-0.319152\pi$
$647$	27.3731			1.07615			0.538073	+	0.842898i	$0.319152\pi$
$648$		0			0
$649$		−	31.6675i		−	1.24306i
$650$			0.757875i			0.0297263i
$651$		0			0
$652$		−	7.34847i		−	0.287788i
$653$		−	26.3896i		−	1.03270i	−0.856376	−	0.516352i	$-0.827290\pi$
$653$		−	26.3896i		−	1.03270i	0.856376	−	0.516352i	$-0.172710\pi$
$654$		0			0
$655$			2.17209i			0.0848705i
$656$		−	21.5921i		−	0.843031i
$657$		0			0
$658$	3.17691			0.123849
$659$			10.6574i			0.415152i	0.978219	+	0.207576i	$0.0665575\pi$
$659$			10.6574i			0.415152i	−0.978219	+	0.207576i	$0.933443\pi$
$660$		0			0
$661$	−5.85641			−0.227788			−0.113894	−	0.993493i	$-0.536332\pi$
$661$	−5.85641			−0.227788			−0.113894	+	0.993493i	$0.536332\pi$
$662$	−9.80385			−0.381037
$663$		0			0
$664$			4.24264i			0.164646i
$665$			3.10583i			0.120439i
$666$		0			0
$667$	42.5885	+	11.5911i	1.64903	+	0.448810i
$668$	−6.58846			−0.254915
$669$		0			0
$670$		−	5.75839i		−	0.222466i
$671$	16.3923			0.632818
$672$		0			0
$673$	25.3205			0.976034			0.488017	−	0.872834i	$-0.337720\pi$
$673$	25.3205			0.976034			0.488017	+	0.872834i	$0.337720\pi$
$674$	9.37307			0.361037
$675$		0			0
$676$	−18.8038			−0.723225
$677$		−	38.8129i		−	1.49170i	−0.666113	−	0.745850i	$-0.732043\pi$
$677$		−	38.8129i		−	1.49170i	0.666113	−	0.745850i	$-0.267957\pi$
$678$		0			0
$679$			5.37945i			0.206444i
$680$	−12.1962			−0.467701
$681$		0			0
$682$			11.5911i			0.443847i
$683$	−32.1962			−1.23195			−0.615976	−	0.787765i	$-0.711238\pi$
$683$	−32.1962			−1.23195			−0.615976	+	0.787765i	$0.711238\pi$
$684$		0			0
$685$		−	9.89949i		−	0.378240i
$686$			5.10205i			0.194797i
$687$		0			0
$688$			10.4543i			0.398566i
$689$		0			0
$690$		0			0
$691$	−3.07180			−0.116857			−0.0584283	−	0.998292i	$-0.518609\pi$
$691$	−3.07180			−0.116857			−0.0584283	+	0.998292i	$0.518609\pi$
$692$	−36.0000			−1.36851
$693$		0			0
$694$			13.5601i			0.514735i
$695$	−13.4641			−0.510722
$696$		0			0
$697$	−55.3205			−2.09541
$698$		−	9.52056i		−	0.360358i
$699$		0			0
$700$	1.26795			0.0479240
$701$	−32.7846			−1.23826			−0.619129	−	0.785289i	$-0.712514\pi$
$701$	−32.7846			−1.23826			−0.619129	+	0.785289i	$0.712514\pi$
$702$		0			0
$703$	18.0000			0.678883
$704$			11.9700i			0.451138i
$705$		0			0
$706$			16.9706i			0.638696i
$707$			3.03150i			0.114011i
$708$		0			0
$709$	−7.21539			−0.270980			−0.135490	−	0.990779i	$-0.543261\pi$
$709$	−7.21539			−0.270980			−0.135490	+	0.990779i	$0.543261\pi$
$710$			3.10583i			0.116560i
$711$		0			0
$712$	−4.00000			−0.149906
$713$			34.7733i			1.30227i
$714$		0			0
$715$		−	7.72741i		−	0.288989i
$716$	−28.3923			−1.06107
$717$		0			0
$718$	3.66025			0.136599
$719$	10.3923			0.387568			0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923			0.387568			0.193784	+	0.981044i	$0.437924\pi$
$720$		0			0
$721$	−0.143594			−0.00534770
$722$		−	0.517638i		−	0.0192645i
$723$		0			0
$724$	−10.1436			−0.376984
$725$	5.19615	+	1.41421i	0.192980	+	0.0525226i
$726$		0			0
$727$			50.6071i			1.87691i	0.345398	+	0.938456i	$0.387744\pi$
$727$			50.6071i			1.87691i	−0.345398	+	0.938456i	$0.612256\pi$
$728$			2.07055i			0.0767398i
$729$		0			0
$730$	0.588457			0.0217798
$731$	26.7846			0.990665
$732$		0			0
$733$		−	15.0015i		−	0.554095i	−0.960856	−	0.277047i	$-0.910644\pi$
$733$		−	15.0015i		−	0.554095i	0.960856	−	0.277047i	$-0.0893559\pi$
$734$	1.01924			0.0376208
$735$		0			0
$736$		−	42.1218i		−	1.55263i
$737$			58.7134i			2.16274i
$738$		0			0
$739$			16.6660i			0.613067i	0.951860	+	0.306534i	$0.0991692\pi$
$739$			16.6660i			0.613067i	−0.951860	+	0.306534i	$0.900831\pi$
$740$		−	7.34847i		−	0.270135i
$741$		0			0
$742$		0			0
$743$		−	0.101536i		−	0.00372499i	−0.999998	−	0.00186250i	$-0.999407\pi$
$743$		−	0.101536i		−	0.00372499i	0.999998	−	0.00186250i	$-0.000592851\pi$
$744$		0			0
$745$	18.0000			0.659469
$746$			11.3880i			0.416946i
$747$		0			0
$748$	57.7128			2.11019
$749$	−1.60770			−0.0587439
$750$		0			0
$751$			18.9396i			0.691115i	0.938398	+	0.345558i	$0.112310\pi$
$751$			18.9396i			0.691115i	−0.938398	+	0.345558i	$0.887690\pi$
$752$			20.6584i			0.753334i
$753$		0			0
$754$	−1.07180	+	3.93803i	−0.0390325	+	0.143415i
$755$	18.3923			0.669365
$756$		0			0
$757$			30.5307i			1.10966i	0.831965	+	0.554828i	$0.187216\pi$
$757$			30.5307i			1.10966i	−0.831965	+	0.554828i	$0.812784\pi$
$758$	−8.19615			−0.297698
$759$		0			0
$760$	8.19615			0.297306
$761$	10.3923			0.376721			0.188360	−	0.982100i	$-0.439683\pi$
$761$	10.3923			0.376721			0.188360	+	0.982100i	$0.439683\pi$
$762$		0			0
$763$	−1.07180			−0.0388016
$764$		−	33.2848i		−	1.20420i
$765$		0			0
$766$		−	1.13681i		−	0.0410747i
$767$	8.78461			0.317194
$768$		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	$-0.892740\pi$
$773$		−	18.3848i		−	0.661254i	0.943761	−	0.330627i	$-0.107260\pi$
$774$		0			0
$775$			4.24264i			0.152400i
$776$	14.1962			0.509612
$777$		0			0
$778$	16.9282			0.606905
$779$	37.1769			1.33200
$780$		0			0
$781$		−	31.6675i		−	1.13315i
$782$	−26.7846			−0.957816
$783$		0			0
$784$	−15.9282			−0.568864
$785$		−	7.34847i		−	0.262278i
$786$		0			0
$787$	38.5885			1.37553			0.687765	−	0.725933i	$-0.258592\pi$
$787$	38.5885			1.37553			0.687765	+	0.725933i	$0.258592\pi$
$788$	2.78461			0.0991976
$789$		0			0
$790$	−8.19615			−0.291606
$791$			10.0754i			0.358239i
$792$		0			0
$793$			4.54725i			0.161478i
$794$		−	15.7322i		−	0.558315i
$795$		0			0
$796$	−33.7128			−1.19492
$797$		−	6.79367i		−	0.240644i	−0.992735	−	0.120322i	$-0.961607\pi$
$797$		−	6.79367i		−	0.240644i	0.992735	−	0.120322i	$-0.0383927\pi$
$798$		0			0
$799$	52.9282			1.87247
$800$		−	5.13922i		−	0.181699i
$801$		0			0
$802$			4.54725i			0.160569i
$803$	−6.00000			−0.211735
$804$		0			0
$805$	6.00000			0.211472
$806$	−3.21539			−0.113257
$807$		0			0
$808$	8.00000			0.281439
$809$			31.8706i			1.12051i	0.828320	+	0.560255i	$0.189297\pi$
$809$			31.8706i			1.12051i	−0.828320	+	0.560255i	$0.810703\pi$
$810$		0			0
$811$	−4.67949			−0.164319			−0.0821596	−	0.996619i	$-0.526182\pi$
$811$	−4.67949			−0.164319			−0.0821596	+	0.996619i	$0.526182\pi$
$812$	6.58846	+	1.79315i	0.231210	+	0.0629273i
$813$		0			0
$814$		−	11.5911i		−	0.406268i
$815$		−	4.24264i		−	0.148613i
$816$		0			0
$817$	−18.0000			−0.629740
$818$	9.21539			0.322209
$819$		0			0
$820$		−	15.1774i		−	0.530018i
$821$	26.7846			0.934789			0.467395	−	0.884049i	$-0.345193\pi$
$821$	26.7846			0.934789			0.467395	+	0.884049i	$0.345193\pi$
$822$		0			0
$823$			50.6071i			1.76405i	0.471201	+	0.882026i	$0.343821\pi$
$823$			50.6071i			1.76405i	−0.471201	+	0.882026i	$0.656179\pi$
$824$			0.378937i			0.0132009i
$825$		0			0
$826$			2.27362i			0.0791095i
$827$		−	33.0817i		−	1.15036i	−0.818025	−	0.575182i	$-0.804931\pi$
$827$		−	33.0817i		−	1.15036i	0.818025	−	0.575182i	$-0.195069\pi$
$828$		0			0
$829$		−	45.5322i		−	1.58140i	−0.612204	−	0.790700i	$-0.709717\pi$
$829$		−	45.5322i		−	1.58140i	0.612204	−	0.790700i	$-0.290283\pi$
$830$			1.13681i			0.0394593i
$831$		0			0
$832$	−3.32051			−0.115118
$833$			40.8091i			1.41395i
$834$		0			0
$835$	−3.80385			−0.131638
$836$	−38.7846			−1.34139
$837$		0			0
$838$		−	9.31749i		−	0.321867i
$839$			13.7632i			0.475158i	0.971368	+	0.237579i	$0.0763539\pi$
$839$			13.7632i			0.475158i	−0.971368	+	0.237579i	$0.923646\pi$
$840$		0			0
$841$	25.0000	+	14.6969i	0.862069	+	0.506791i
$842$	9.21539			0.317583
$843$		0			0
$844$			38.1838i			1.31434i
$845$	−10.8564			−0.373472
$846$		0			0
$847$	−12.3397			−0.423999
$848$		0			0
$849$		0			0
$850$	−3.26795			−0.112090
$851$		−	34.7733i		−	1.19201i
$852$		0			0
$853$			35.9101i			1.22954i	0.788707	+	0.614770i	$0.210751\pi$
$853$			35.9101i			1.22954i	−0.788707	+	0.614770i	$0.789249\pi$
$854$	−1.17691			−0.0402732
$855$		0			0
$856$			4.24264i			0.145010i
$857$	44.7846			1.52981			0.764907	−	0.644141i	$-0.222785\pi$
$857$	44.7846			1.52981			0.764907	+	0.644141i	$0.222785\pi$
$858$		0			0
$859$		−	5.07484i		−	0.173151i	−0.996245	−	0.0865757i	$-0.972408\pi$
$859$		−	5.07484i		−	0.173151i	0.996245	−	0.0865757i	$-0.0275925\pi$
$860$			7.34847i			0.250581i
$861$		0			0
$862$			2.27362i			0.0774400i
$863$	38.1962			1.30021			0.650106	−	0.759843i	$-0.274724\pi$
$863$	38.1962			1.30021			0.650106	+	0.759843i	$0.274724\pi$
$864$		0			0
$865$	−20.7846			−0.706698
$866$	−6.58846			−0.223885
$867$		0			0
$868$			5.37945i			0.182591i
$869$	83.5692			2.83489
$870$		0			0
$871$	−16.2872			−0.551870
$872$			2.82843i			0.0957826i
$873$		0			0
$874$	18.0000			0.608859
$875$	0.732051			0.0247478
$876$		0			0
$877$	−24.7846			−0.836917			−0.418458	−	0.908236i	$-0.637429\pi$
$877$	−24.7846			−0.836917			−0.418458	+	0.908236i	$0.637429\pi$
$878$		−	4.62158i		−	0.155971i
$879$		0			0
$880$			13.0053i			0.438409i
$881$		−	30.6322i		−	1.03203i	−0.856581	−	0.516013i	$-0.827416\pi$
$881$		−	30.6322i		−	1.03203i	0.856581	−	0.516013i	$-0.172584\pi$
$882$		0			0
$883$	−22.5885			−0.760162			−0.380081	−	0.924953i	$-0.624104\pi$
$883$	−22.5885			−0.760162			−0.380081	+	0.924953i	$0.624104\pi$
$884$			16.0096i			0.538462i
$885$		0			0
$886$	−7.66025			−0.257351
$887$		−	21.0101i		−	0.705451i	−0.935727	−	0.352726i	$-0.885255\pi$
$887$		−	21.0101i		−	0.705451i	0.935727	−	0.352726i	$-0.114745\pi$
$888$		0			0
$889$		−	9.31749i		−	0.312498i
$890$	−1.07180			−0.0359267
$891$		0			0
$892$	−16.7321			−0.560230
$893$	−35.5692			−1.19028
$894$		0			0
$895$	−16.3923			−0.547934
$896$		−	8.38375i		−	0.280081i
$897$		0			0
$898$	−15.8564			−0.529135
$899$	−6.00000	+	22.0454i	−0.200111	+	0.735256i
$900$		0			0
$901$		0			0
$902$		−	23.9401i		−	0.797118i
$903$		0			0
$904$	26.5885			0.884319
$905$	−5.85641			−0.194674
$906$		0			0
$907$		−	35.9101i		−	1.19238i	−0.802845	−	0.596188i	$-0.796681\pi$
$907$		−	35.9101i		−	1.19238i	0.802845	−	0.596188i	$-0.203319\pi$
$908$	27.3731			0.908407
$909$		0			0
$910$			0.554803i			0.0183915i
$911$		−	21.0101i		−	0.696097i	−0.937477	−	0.348048i	$-0.886844\pi$
$911$		−	21.0101i		−	0.696097i	0.937477	−	0.348048i	$-0.113156\pi$
$912$		0			0
$913$		−	11.5911i		−	0.383610i
$914$		−	13.9391i		−	0.461063i
$915$		0			0
$916$			44.0908i			1.45680i
$917$			1.59008i			0.0525090i
$918$		0			0
$919$	−23.8564			−0.786950			−0.393475	−	0.919335i	$-0.628727\pi$
$919$	−23.8564			−0.786950			−0.393475	+	0.919335i	$0.628727\pi$
$920$		−	15.8338i		−	0.522023i
$921$		0			0
$922$	−11.7128			−0.385741
$923$	8.78461			0.289149
$924$		0			0
$925$		−	4.24264i		−	0.139497i
$926$			11.2122i			0.368455i
$927$		0			0
$928$	7.26795	−	26.7042i	0.238582	−	0.876607i
$929$	−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
$931$		−	27.4249i		−	0.898814i
$932$	−38.7846			−1.27043
$933$		0			0
$934$	11.2679			0.368699
$935$	33.3205			1.08970
$936$		0			0
$937$	−46.2487			−1.51088			−0.755440	−	0.655218i	$-0.772577\pi$
$937$	−46.2487			−1.51088			−0.755440	+	0.655218i	$0.772577\pi$
$938$		−	4.21543i		−	0.137639i
$939$		0			0
$940$			14.5211i			0.473625i
$941$	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
$943$		−	71.8203i		−	2.33879i
$944$	−14.7846			−0.481198
$945$		0			0
$946$			11.5911i			0.376859i
$947$			48.5365i			1.57723i	0.614890	+	0.788613i	$0.289200\pi$
$947$			48.5365i			1.57723i	−0.614890	+	0.788613i	$0.710800\pi$
$948$		0			0
$949$		−	1.66441i		−	0.0540290i
$950$	2.19615			0.0712526
$951$		0			0
$952$	−8.92820			−0.289365
$953$	47.5692			1.54092			0.770459	−	0.637489i	$-0.220027\pi$
$953$	47.5692			1.54092			0.770459	+	0.637489i	$0.220027\pi$
$954$		0			0
$955$		−	19.2170i		−	0.621847i
$956$	−10.3923			−0.336111
$957$		0			0
$958$	−2.33975			−0.0755938
$959$		−	7.24693i		−	0.234016i
$960$		0			0
$961$	13.0000			0.419355
$962$	3.21539			0.103668
$963$		0			0
$964$	15.4641			0.498065
$965$		−	12.7279i		−	0.409726i
$966$		0			0
$967$		−	59.0924i		−	1.90028i	−0.311820	−	0.950141i	$-0.600939\pi$
$967$		−	59.0924i		−	1.90028i	0.311820	−	0.950141i	$-0.399061\pi$
$968$			32.5641i			1.04665i
$969$		0			0
$970$	3.80385			0.122134
$971$			36.4649i			1.17022i	0.810955	+	0.585108i	$0.198948\pi$
$971$			36.4649i			1.17022i	−0.810955	+	0.585108i	$0.801052\pi$
$972$		0			0
$973$	−9.85641			−0.315982
$974$			11.6926i			0.374657i
$975$		0			0
$976$		−	7.65308i		−	0.244969i
$977$	−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
$979$	10.9282			0.349267
$980$	−11.1962			−0.357648
$981$		0			0
$982$	−0.732051			−0.0233607
$983$			0.859411i			0.0274109i	0.999906	+	0.0137055i	$0.00436272\pi$
$983$			0.859411i			0.0274109i	−0.999906	+	0.0137055i	$0.995637\pi$
$984$		0			0
$985$	1.60770			0.0512254
$986$	−16.9808	−	4.62158i	−0.540778	−	0.147181i
$987$		0			0
$988$		−	10.7589i		−	0.342286i
$989$			34.7733i			1.10573i
$990$		0			0
$991$	61.3205			1.94791			0.973955	−	0.226741i	$-0.0728071\pi$
$991$	61.3205			1.94791			0.973955	+	0.226741i	$0.0728071\pi$
$992$	21.8038			0.692273
$993$		0			0
$994$			2.27362i			0.0721150i
$995$	−19.4641			−0.617054
$996$		0			0
$997$		−	30.5307i		−	0.966917i	−0.875367	−	0.483458i	$-0.839380\pi$
$997$		−	30.5307i		−	0.966917i	0.875367	−	0.483458i	$-0.160620\pi$
$998$			3.03150i			0.0959604i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.d.a.811.2		4
3.2	odd	2		145.2.c.a.86.3	yes	4
12.11	even	2		2320.2.g.e.1681.3		4
15.2	even	4		725.2.d.b.724.4		8
15.8	even	4		725.2.d.b.724.5		8
15.14	odd	2		725.2.c.d.376.2		4
29.28	even	2	inner	1305.2.d.a.811.3		4
87.17	even	4		4205.2.a.g.1.3		4
87.41	even	4		4205.2.a.g.1.2		4
87.86	odd	2		145.2.c.a.86.2	✓	4
348.347	even	2		2320.2.g.e.1681.1		4
435.173	even	4		725.2.d.b.724.3		8
435.347	even	4		725.2.d.b.724.6		8
435.434	odd	2		725.2.c.d.376.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.c.a.86.2	✓	4	87.86	odd	2
145.2.c.a.86.3	yes	4	3.2	odd	2
725.2.c.d.376.2		4	15.14	odd	2
725.2.c.d.376.3		4	435.434	odd	2
725.2.d.b.724.3		8	435.173	even	4
725.2.d.b.724.4		8	15.2	even	4
725.2.d.b.724.5		8	15.8	even	4
725.2.d.b.724.6		8	435.347	even	4
1305.2.d.a.811.2		4	1.1	even	1	trivial
1305.2.d.a.811.3		4	29.28	even	2	inner
2320.2.g.e.1681.1		4	348.347	even	2
2320.2.g.e.1681.3		4	12.11	even	2
4205.2.a.g.1.2		4	87.41	even	4
4205.2.a.g.1.3		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)

\(f(q)\)	\(=\)	\(q-0.517638i q^{2} +1.73205 q^{4} +1.00000 q^{5} +0.732051 q^{7} -1.93185i q^{8} -0.517638i q^{10} +5.27792i q^{11} -1.46410 q^{13} -0.378937i q^{14} +2.46410 q^{16} -6.31319i q^{17} +4.24264i q^{19} +1.73205 q^{20} +2.73205 q^{22} +8.19615 q^{23} +1.00000 q^{25} +0.757875i q^{26} +1.26795 q^{28} +(5.19615 + 1.41421i) q^{29} +4.24264i q^{31} -5.13922i q^{32} -3.26795 q^{34} +0.732051 q^{35} -4.24264i q^{37} +2.19615 q^{38} -1.93185i q^{40} -8.76268i q^{41} +4.24264i q^{43} +9.14162i q^{44} -4.24264i q^{46} +8.38375i q^{47} -6.46410 q^{49} -0.517638i q^{50} -2.53590 q^{52} +5.27792i q^{55} -1.41421i q^{56} +(0.732051 - 2.68973i) q^{58} -6.00000 q^{59} -3.10583i q^{61} +2.19615 q^{62} +2.26795 q^{64} -1.46410 q^{65} +11.1244 q^{67} -10.9348i q^{68} -0.378937i q^{70} -6.00000 q^{71} +1.13681i q^{73} -2.19615 q^{74} +7.34847i q^{76} +3.86370i q^{77} -15.8338i q^{79} +2.46410 q^{80} -4.53590 q^{82} -2.19615 q^{83} -6.31319i q^{85} +2.19615 q^{86} +10.1962 q^{88} -2.07055i q^{89} -1.07180 q^{91} +14.1962 q^{92} +4.33975 q^{94} +4.24264i q^{95} +7.34847i q^{97} +3.34607i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7} + 8 q^{13} - 4 q^{16} + 4 q^{22} + 12 q^{23} + 4 q^{25} + 12 q^{28} - 20 q^{34} - 4 q^{35} - 12 q^{38} - 12 q^{49} - 24 q^{52} - 4 q^{58} - 24 q^{59} - 12 q^{62} + 16 q^{64} + 8 q^{65}+ \cdots + 52 q^{94}+O(q^{100}) \) 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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