LMFDB - Embedded newform 1840.2.e.e.369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (of order $2$, degree $1$, not 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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.11574317056.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 45x^{4} + 4 $ x^8 + 45*x^4 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.2
Root			$1.83051 - 1.83051i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.e.369.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40815i q^{3} +(1.83051 + 1.28422i) q^{5} -0.706585i q^{7} -2.79917 q^{9} +O(q^{10})$	$q-2.40815i q^{3} +(1.83051 + 1.28422i) q^{5} -0.706585i q^{7} -2.79917 q^{9} -0.747120 q^{11} -1.29341i q^{13} +(3.09259 - 4.40815i) q^{15} -5.50074i q^{17} +2.44868 q^{19} -1.70156 q^{21} +1.00000i q^{23} +(1.70156 + 4.70156i) q^{25} -0.483617i q^{27} -5.72371 q^{29} -7.52288 q^{31} +1.79917i q^{33} +(0.907411 - 1.29341i) q^{35} -5.07420i q^{37} -3.11473 q^{39} +10.0876 q^{41} -5.34045i q^{43} +(-5.12393 - 3.59475i) q^{45} -7.90888i q^{47} +6.50074 q^{49} -13.2466 q^{51} -5.84621i q^{53} +(-1.36761 - 0.959466i) q^{55} -5.89679i q^{57} +12.4146 q^{59} +1.57346 q^{61} +1.97786i q^{63} +(1.66103 - 2.36761i) q^{65} -5.25938i q^{67} +2.40815 q^{69} -2.68444 q^{71} -10.4589i q^{73} +(11.3221 - 4.09761i) q^{75} +0.527904i q^{77} -6.55005 q^{79} -9.56214 q^{81} +12.7068i q^{83} +(7.06415 - 10.0692i) q^{85} +13.7835i q^{87} -13.6426 q^{89} -0.913908 q^{91} +18.1162i q^{93} +(4.48235 + 3.14464i) q^{95} +1.50074i q^{97} +2.09132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 14 q^{9}+O(q^{10}) $ 8 * q - 14 * q^9	$ 8 q - 14 q^{9} - 10 q^{11} + 16 q^{15} - 2 q^{19} + 12 q^{21} - 12 q^{25} - 4 q^{29} - 10 q^{31} + 16 q^{35} + 46 q^{41} - 26 q^{45} + 18 q^{49} - 14 q^{51} + 18 q^{55} + 32 q^{59} + 18 q^{61} - 16 q^{65} - 6 q^{69} - 38 q^{71} + 32 q^{75} - 12 q^{79} + 32 q^{81} - 24 q^{85} - 60 q^{89} + 26 q^{91} - 18 q^{95} - 54 q^{99}+O(q^{100}) $ 8 * q - 14 * q^9 - 10 * q^11 + 16 * q^15 - 2 * q^19 + 12 * q^21 - 12 * q^25 - 4 * q^29 - 10 * q^31 + 16 * q^35 + 46 * q^41 - 26 * q^45 + 18 * q^49 - 14 * q^51 + 18 * q^55 + 32 * q^59 + 18 * q^61 - 16 * q^65 - 6 * q^69 - 38 * q^71 + 32 * q^75 - 12 * q^79 + 32 * q^81 - 24 * q^85 - 60 * q^89 + 26 * q^91 - 18 * q^95 - 54 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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			0
$2$		0			0
$3$		−	2.40815i		−	1.39034i	−0.718843	−	0.695172i	$-0.755328\pi$
$3$		−	2.40815i		−	1.39034i	0.718843	−	0.695172i	$-0.244672\pi$
$4$		0			0
$5$	1.83051	+	1.28422i	0.818631	+	0.574320i
$5$	1.83051	+	1.28422i	0.818631	+	0.574320i
$6$		0			0
$7$		−	0.706585i		−	0.267064i	−0.991044	−	0.133532i	$-0.957368\pi$
$7$		−	0.706585i		−	0.267064i	0.991044	−	0.133532i	$-0.0426319\pi$
$8$		0			0
$9$	−2.79917			−0.933058
$10$		0			0
$11$	−0.747120			−0.225265			−0.112633	−	0.993637i	$-0.535928\pi$
$11$	−0.747120			−0.225265			−0.112633	+	0.993637i	$0.535928\pi$
$12$		0			0
$13$		−	1.29341i		−	0.358729i	−0.983783	−	0.179364i	$-0.942596\pi$
$13$		−	1.29341i		−	0.358729i	0.983783	−	0.179364i	$-0.0574041\pi$
$14$		0			0
$15$	3.09259	−	4.40815i	0.798503	−	1.13818i
$16$		0			0
$17$		−	5.50074i		−	1.33412i	−0.745002	−	0.667062i	$-0.767551\pi$
$17$		−	5.50074i		−	1.33412i	0.745002	−	0.667062i	$-0.232449\pi$
$18$		0			0
$19$	2.44868			0.561766			0.280883	−	0.959742i	$-0.409373\pi$
$19$	2.44868			0.561766			0.280883	+	0.959742i	$0.409373\pi$
$20$		0			0
$21$	−1.70156			−0.371311
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	1.70156	+	4.70156i	0.340312	+	0.940312i
$26$		0			0
$27$		−	0.483617i		−	0.0930721i
$28$		0			0
$29$	−5.72371			−1.06287			−0.531433	−	0.847100i	$-0.678346\pi$
$29$	−5.72371			−1.06287			−0.531433	+	0.847100i	$0.678346\pi$
$30$		0			0
$31$	−7.52288			−1.35115			−0.675575	−	0.737292i	$-0.736104\pi$
$31$	−7.52288			−1.35115			−0.675575	+	0.737292i	$0.736104\pi$
$32$		0			0
$33$			1.79917i			0.313196i
$34$		0			0
$35$	0.907411	−	1.29341i	0.153380	−	0.218627i
$36$		0			0
$37$		−	5.07420i		−	0.834193i	−0.908862	−	0.417097i	$-0.863048\pi$
$37$		−	5.07420i		−	0.834193i	0.908862	−	0.417097i	$-0.136952\pi$
$38$		0			0
$39$	−3.11473			−0.498756
$40$		0			0
$41$	10.0876			1.57541			0.787707	−	0.616051i	$-0.211268\pi$
$41$	10.0876			1.57541			0.787707	+	0.616051i	$0.211268\pi$
$42$		0			0
$43$		−	5.34045i		−	0.814410i	−0.913337	−	0.407205i	$-0.866503\pi$
$43$		−	5.34045i		−	0.814410i	0.913337	−	0.407205i	$-0.133497\pi$
$44$		0			0
$45$	−5.12393	−	3.59475i	−0.763830	−	0.535874i
$46$		0			0
$47$		−	7.90888i		−	1.15363i	−0.816875	−	0.576815i	$-0.804295\pi$
$47$		−	7.90888i		−	1.15363i	0.816875	−	0.576815i	$-0.195705\pi$
$48$		0			0
$49$	6.50074			0.928677
$50$		0			0
$51$	−13.2466			−1.85489
$52$		0			0
$53$		−	5.84621i		−	0.803038i	−0.915851	−	0.401519i	$-0.868482\pi$
$53$		−	5.84621i		−	0.803038i	0.915851	−	0.401519i	$-0.131518\pi$
$54$		0			0
$55$	−1.36761	−	0.959466i	−0.184409	−	0.129374i
$56$		0			0
$57$		−	5.89679i		−	0.781049i
$58$		0			0
$59$	12.4146			1.61625			0.808125	−	0.589012i	$-0.200483\pi$
$59$	12.4146			1.61625			0.808125	+	0.589012i	$0.200483\pi$
$60$		0			0
$61$	1.57346			0.201461			0.100731	−	0.994914i	$-0.467882\pi$
$61$	1.57346			0.201461			0.100731	+	0.994914i	$0.467882\pi$
$62$		0			0
$63$			1.97786i			0.249186i
$64$		0			0
$65$	1.66103	−	2.36761i	0.206025	−	0.293666i
$66$		0			0
$67$		−	5.25938i		−	0.642535i	−0.946988	−	0.321268i	$-0.895891\pi$
$67$		−	5.25938i		−	0.642535i	0.946988	−	0.321268i	$-0.104109\pi$
$68$		0			0
$69$	2.40815			0.289907
$70$		0			0
$71$	−2.68444			−0.318585			−0.159292	−	0.987231i	$-0.550921\pi$
$71$	−2.68444			−0.318585			−0.159292	+	0.987231i	$0.550921\pi$
$72$		0			0
$73$		−	10.4589i		−	1.22413i	−0.790809	−	0.612063i	$-0.790340\pi$
$73$		−	10.4589i		−	1.22413i	0.790809	−	0.612063i	$-0.209660\pi$
$74$		0			0
$75$	11.3221	−	4.09761i	1.30736	−	0.473152i
$76$		0			0
$77$			0.527904i			0.0601602i
$78$		0			0
$79$	−6.55005			−0.736938			−0.368469	−	0.929640i	$-0.620118\pi$
$79$	−6.55005			−0.736938			−0.368469	+	0.929640i	$0.620118\pi$
$80$		0			0
$81$	−9.56214			−1.06246
$82$		0			0
$83$			12.7068i			1.39475i	0.716706	+	0.697376i	$0.245649\pi$
$83$			12.7068i			1.39475i	−0.716706	+	0.697376i	$0.754351\pi$
$84$		0			0
$85$	7.06415	−	10.0692i	0.766215	−	1.09216i
$86$		0			0
$87$			13.7835i			1.47775i
$88$		0			0
$89$	−13.6426			−1.44612			−0.723058	−	0.690787i	$-0.757264\pi$
$89$	−13.6426			−1.44612			−0.723058	+	0.690787i	$0.757264\pi$
$90$		0			0
$91$	−0.913908			−0.0958036
$92$		0			0
$93$			18.1162i			1.87856i
$94$		0			0
$95$	4.48235	+	3.14464i	0.459879	+	0.322634i
$96$		0			0
$97$			1.50074i			0.152377i	0.997093	+	0.0761884i	$0.0242750\pi$
$97$			1.50074i			0.152377i	−0.997093	+	0.0761884i	$0.975725\pi$
$98$		0			0
$99$	2.09132			0.210185
$100$		0			0
$101$	−16.0472			−1.59676			−0.798380	−	0.602154i	$-0.794309\pi$
$101$	−16.0472			−1.59676			−0.798380	+	0.602154i	$0.794309\pi$
$102$		0			0
$103$			9.49069i			0.935145i	0.883955	+	0.467573i	$0.154871\pi$
$103$			9.49069i			0.935145i	−0.883955	+	0.467573i	$0.845129\pi$
$104$		0			0
$105$	−3.11473	−	2.18518i	−0.303967	−	0.213252i
$106$		0			0
$107$		−	1.38473i		−	0.133867i	−0.997757	−	0.0669336i	$-0.978678\pi$
$107$		−	1.38473i		−	0.133867i	0.997757	−	0.0669336i	$-0.0213216\pi$
$108$		0			0
$109$	14.0370			1.34450			0.672250	−	0.740325i	$-0.265328\pi$
$109$	14.0370			1.34450			0.672250	+	0.740325i	$0.265328\pi$
$110$		0			0
$111$	−12.2194			−1.15982
$112$		0			0
$113$			16.3105i			1.53437i	0.641428	+	0.767183i	$0.278342\pi$
$113$			16.3105i			1.53437i	−0.641428	+	0.767183i	$0.721658\pi$
$114$		0			0
$115$	−1.28422	+	1.83051i	−0.119754	+	0.170696i
$116$		0			0
$117$			3.62049i			0.334715i
$118$		0			0
$119$	−3.88674			−0.356297
$120$		0			0
$121$	−10.4418			−0.949256
$122$		0			0
$123$		−	24.2923i		−	2.19037i
$124$		0			0
$125$	−2.92310	+	10.7915i	−0.261450	+	0.965217i
$126$		0			0
$127$			2.73523i			0.242712i	0.992609	+	0.121356i	$0.0387243\pi$
$127$			2.73523i			0.242712i	−0.992609	+	0.121356i	$0.961276\pi$
$128$		0			0
$129$	−12.8606			−1.13231
$130$		0			0
$131$	8.21942			0.718134			0.359067	−	0.933312i	$-0.383095\pi$
$131$	8.21942			0.718134			0.359067	+	0.933312i	$0.383095\pi$
$132$		0			0
$133$		−	1.73020i		−	0.150028i
$134$		0			0
$135$	0.621070	−	0.885267i	0.0534532	−	0.0761916i
$136$		0			0
$137$		−	22.8449i		−	1.95177i	−0.218275	−	0.975887i	$-0.570043\pi$
$137$		−	22.8449i		−	1.95177i	0.218275	−	0.975887i	$-0.429957\pi$
$138$		0			0
$139$	3.76049			0.318960			0.159480	−	0.987201i	$-0.449018\pi$
$139$	3.76049			0.318960			0.159480	+	0.987201i	$0.449018\pi$
$140$		0			0
$141$	−19.0458			−1.60394
$142$		0			0
$143$			0.966336i			0.0808090i
$144$		0			0
$145$	−10.4773	−	7.35049i	−0.870094	−	0.610425i
$146$		0			0
$147$		−	15.6547i		−	1.29118i
$148$		0			0
$149$	1.90614			0.156157			0.0780785	−	0.996947i	$-0.475122\pi$
$149$	1.90614			0.156157			0.0780785	+	0.996947i	$0.475122\pi$
$150$		0			0
$151$	9.41967			0.766562			0.383281	−	0.923632i	$-0.374794\pi$
$151$	9.41967			0.766562			0.383281	+	0.923632i	$0.374794\pi$
$152$		0			0
$153$			15.3975i			1.24482i
$154$		0			0
$155$	−13.7707	−	9.66103i	−1.10609	−	0.775992i
$156$		0			0
$157$			4.43304i			0.353795i	0.984229	+	0.176897i	$0.0566061\pi$
$157$			4.43304i			0.353795i	−0.984229	+	0.176897i	$0.943394\pi$
$158$		0			0
$159$	−14.0785			−1.11650
$160$		0			0
$161$	0.706585			0.0556867
$162$		0			0
$163$			9.92453i			0.777349i	0.921375	+	0.388675i	$0.127067\pi$
$163$			9.92453i			0.777349i	−0.921375	+	0.388675i	$0.872933\pi$
$164$		0			0
$165$	−2.31053	+	3.29341i	−0.179875	+	0.256392i
$166$		0			0
$167$			11.9089i			0.921537i	0.887520	+	0.460769i	$0.152426\pi$
$167$			11.9089i			0.921537i	−0.887520	+	0.460769i	$0.847574\pi$
$168$		0			0
$169$	11.3271			0.871314
$170$		0			0
$171$	−6.85429			−0.524161
$172$		0			0
$173$		−	2.91391i		−	0.221540i	−0.993846	−	0.110770i	$-0.964668\pi$
$173$		−	2.91391i		−	0.221540i	0.993846	−	0.110770i	$-0.0353317\pi$
$174$		0			0
$175$	3.32206	−	1.20230i	0.251124	−	0.0908853i
$176$		0			0
$177$		−	29.8963i		−	2.24714i
$178$		0			0
$179$	19.3663			1.44751			0.723754	−	0.690058i	$-0.242415\pi$
$179$	19.3663			1.44751			0.723754	+	0.690058i	$0.242415\pi$
$180$		0			0
$181$	17.5986			1.30809			0.654045	−	0.756456i	$-0.273071\pi$
$181$	17.5986			1.30809			0.654045	+	0.756456i	$0.273071\pi$
$182$		0			0
$183$		−	3.78913i		−	0.280100i
$184$		0			0
$185$	6.51638	−	9.28839i	0.479094	−	0.682896i
$186$		0			0
$187$			4.10971i			0.300532i
$188$		0			0
$189$	−0.341716			−0.0248562
$190$		0			0
$191$	−20.3578			−1.47304			−0.736518	−	0.676418i	$-0.763531\pi$
$191$	−20.3578			−1.47304			−0.736518	+	0.676418i	$0.763531\pi$
$192$		0			0
$193$			1.41317i			0.101722i	0.998706	+	0.0508611i	$0.0161966\pi$
$193$			1.41317i			0.101722i	−0.998706	+	0.0508611i	$0.983803\pi$
$194$		0			0
$195$	−5.70156	−	4.00000i	−0.408297	−	0.286446i
$196$		0			0
$197$			4.92395i			0.350817i	0.984496	+	0.175409i	$0.0561247\pi$
$197$			4.92395i			0.350817i	−0.984496	+	0.175409i	$0.943875\pi$
$198$		0			0
$199$	−10.3105			−0.730894			−0.365447	−	0.930832i	$-0.619084\pi$
$199$	−10.3105			−0.730894			−0.365447	+	0.930832i	$0.619084\pi$
$200$		0			0
$201$	−12.6654			−0.893345
$202$		0			0
$203$			4.04429i			0.283853i
$204$		0			0
$205$	18.4654	+	12.9546i	1.28968	+	0.904792i
$206$		0			0
$207$		−	2.79917i		−	0.194556i
$208$		0			0
$209$	−1.82946			−0.126546
$210$		0			0
$211$	−11.4161			−0.785918			−0.392959	−	0.919556i	$-0.628549\pi$
$211$	−11.4161			−0.785918			−0.392959	+	0.919556i	$0.628549\pi$
$212$		0			0
$213$			6.46453i			0.442942i
$214$		0			0
$215$	6.85830	−	9.77576i	0.467732	−	0.666701i
$216$		0			0
$217$			5.31556i			0.360844i
$218$		0			0
$219$	−25.1867			−1.70196
$220$		0			0
$221$	−7.11473			−0.478589
$222$		0			0
$223$		−	18.2094i		−	1.21939i	−0.792636	−	0.609695i	$-0.791292\pi$
$223$		−	18.2094i		−	1.21939i	0.792636	−	0.609695i	$-0.208708\pi$
$224$		0			0
$225$	−4.76297	−	13.1605i	−0.317531	−	0.877366i
$226$		0			0
$227$			18.9363i			1.25684i	0.777873	+	0.628422i	$0.216299\pi$
$227$			18.9363i			1.25684i	−0.777873	+	0.628422i	$0.783701\pi$
$228$		0			0
$229$	8.46433			0.559339			0.279669	−	0.960096i	$-0.409775\pi$
$229$	8.46433			0.559339			0.279669	+	0.960096i	$0.409775\pi$
$230$		0			0
$231$	1.27127			0.0836435
$232$		0			0
$233$		−	16.6341i		−	1.08973i	−0.838523	−	0.544867i	$-0.816580\pi$
$233$		−	16.6341i		−	1.08973i	0.838523	−	0.544867i	$-0.183420\pi$
$234$		0			0
$235$	10.1567	−	14.4773i	0.662553	−	0.944396i
$236$		0			0
$237$			15.7735i			1.02460i
$238$		0			0
$239$	−16.0840			−1.04039			−0.520194	−	0.854048i	$-0.674140\pi$
$239$	−16.0840			−1.04039			−0.520194	+	0.854048i	$0.674140\pi$
$240$		0			0
$241$	−28.1283			−1.81190			−0.905952	−	0.423381i	$-0.860843\pi$
$241$	−28.1283			−1.81190			−0.905952	+	0.423381i	$0.860843\pi$
$242$		0			0
$243$			21.5762i			1.38411i
$244$		0			0
$245$	11.8997	+	8.34837i	0.760243	+	0.533358i
$246$		0			0
$247$		−	3.16716i		−	0.201522i
$248$		0			0
$249$	30.5998			1.93919
$250$		0			0
$251$	20.5770			1.29881			0.649404	−	0.760444i	$-0.275018\pi$
$251$	20.5770			1.29881			0.649404	+	0.760444i	$0.275018\pi$
$252$		0			0
$253$		−	0.747120i		−	0.0469710i
$254$		0			0
$255$	−24.2481	−	17.0115i	−1.51847	−	1.06530i
$256$		0			0
$257$			16.8505i			1.05111i	0.850760	+	0.525554i	$0.176142\pi$
$257$			16.8505i			1.05111i	−0.850760	+	0.525554i	$0.823858\pi$
$258$		0			0
$259$	−3.58536			−0.222783
$260$		0			0
$261$	16.0217			0.991715
$262$		0			0
$263$			24.7630i			1.52695i	0.645837	+	0.763475i	$0.276508\pi$
$263$			24.7630i			1.52695i	−0.645837	+	0.763475i	$0.723492\pi$
$264$		0			0
$265$	7.50781	−	10.7016i	0.461201	−	0.657392i
$266$		0			0
$267$			32.8535i			2.01060i
$268$		0			0
$269$	3.55152			0.216540			0.108270	−	0.994122i	$-0.465469\pi$
$269$	3.55152			0.216540			0.108270	+	0.994122i	$0.465469\pi$
$270$		0			0
$271$	27.4761			1.66905			0.834526	−	0.550969i	$-0.185742\pi$
$271$	27.4761			1.66905			0.834526	+	0.550969i	$0.185742\pi$
$272$		0			0
$273$			2.20083i			0.133200i
$274$		0			0
$275$	−1.27127	−	3.51263i	−0.0766605	−	0.211820i
$276$		0			0
$277$			11.1001i			0.666940i	0.942761	+	0.333470i	$0.108220\pi$
$277$			11.1001i			0.666940i	−0.942761	+	0.333470i	$0.891780\pi$
$278$		0			0
$279$	21.0579			1.26070
$280$		0			0
$281$	13.0458			0.778245			0.389122	−	0.921186i	$-0.372778\pi$
$281$	13.0458			0.778245			0.389122	+	0.921186i	$0.372778\pi$
$282$		0			0
$283$		−	3.35049i		−	0.199166i	−0.995029	−	0.0995831i	$-0.968249\pi$
$283$		−	3.35049i		−	0.199166i	0.995029	−	0.0995831i	$-0.0317509\pi$
$284$		0			0
$285$	7.57277	−	10.7942i	0.448572	−	0.639390i
$286$		0			0
$287$		−	7.12773i		−	0.420736i
$288$		0			0
$289$	−13.2581			−0.779889
$290$		0			0
$291$	3.61400			0.211856
$292$		0			0
$293$			8.89197i			0.519474i	0.965679	+	0.259737i	$0.0836359\pi$
$293$			8.89197i			0.519474i	−0.965679	+	0.259737i	$0.916364\pi$
$294$		0			0
$295$	22.7252	+	15.9431i	1.32311	+	0.928245i
$296$		0			0
$297$			0.361320i			0.0209659i
$298$		0			0
$299$	1.29341			0.0748001
$300$		0			0
$301$	−3.77348			−0.217500
$302$		0			0
$303$			38.6441i			2.22005i
$304$		0			0
$305$	2.88024	+	2.02067i	0.164922	+	0.115703i
$306$		0			0
$307$		−	28.4111i		−	1.62151i	−0.585388	−	0.810753i	$-0.699058\pi$
$307$		−	28.4111i		−	1.62151i	0.585388	−	0.810753i	$-0.300942\pi$
$308$		0			0
$309$	22.8550			1.30017
$310$		0			0
$311$	28.0105			1.58833			0.794164	−	0.607704i	$-0.207909\pi$
$311$	28.0105			1.58833			0.794164	+	0.607704i	$0.207909\pi$
$312$		0			0
$313$			7.42882i			0.419902i	0.977712	+	0.209951i	$0.0673304\pi$
$313$			7.42882i			0.419902i	−0.977712	+	0.209951i	$0.932670\pi$
$314$		0			0
$315$	−2.54000	+	3.62049i	−0.143113	+	0.203992i
$316$		0			0
$317$			13.1480i			0.738463i	0.929337	+	0.369232i	$0.120379\pi$
$317$			13.1480i			0.738463i	−0.929337	+	0.369232i	$0.879621\pi$
$318$		0			0
$319$	4.27629			0.239427
$320$		0			0
$321$	−3.33464			−0.186122
$322$		0			0
$323$		−	13.4696i		−	0.749466i
$324$		0			0
$325$	6.08107	−	2.20083i	0.337317	−	0.122080i
$326$		0			0
$327$		−	33.8031i		−	1.86932i
$328$		0			0
$329$	−5.58830			−0.308093
$330$		0			0
$331$	15.6225			0.858693			0.429346	−	0.903140i	$-0.358744\pi$
$331$	15.6225			0.858693			0.429346	+	0.903140i	$0.358744\pi$
$332$		0			0
$333$			14.2036i			0.778351i
$334$		0			0
$335$	6.75419	−	9.62736i	0.369021	−	0.525999i
$336$		0			0
$337$		−	16.6965i		−	0.909518i	−0.890614	−	0.454759i	$-0.849725\pi$
$337$		−	16.6965i		−	0.909518i	0.890614	−	0.454759i	$-0.150275\pi$
$338$		0			0
$339$	39.2782			2.13330
$340$		0			0
$341$	5.62049			0.304367
$342$		0			0
$343$		−	9.53942i		−	0.515081i
$344$		0			0
$345$	4.40815	+	3.09259i	0.237327	+	0.166499i
$346$		0			0
$347$		−	16.3981i		−	0.880296i	−0.897925	−	0.440148i	$-0.854926\pi$
$347$		−	16.3981i		−	0.880296i	0.897925	−	0.440148i	$-0.145074\pi$
$348$		0			0
$349$	1.86165			0.0996518			0.0498259	−	0.998758i	$-0.484133\pi$
$349$	1.86165			0.0996518			0.0498259	+	0.998758i	$0.484133\pi$
$350$		0			0
$351$	−0.625517			−0.0333876
$352$		0			0
$353$			25.8408i			1.37537i	0.726010	+	0.687684i	$0.241372\pi$
$353$			25.8408i			1.37537i	−0.726010	+	0.687684i	$0.758628\pi$
$354$		0			0
$355$	−4.91391	−	3.44741i	−0.260803	−	0.182970i
$356$		0			0
$357$			9.35985i			0.495376i
$358$		0			0
$359$	8.72223			0.460342			0.230171	−	0.973150i	$-0.426071\pi$
$359$	8.72223			0.460342			0.230171	+	0.973150i	$0.426071\pi$
$360$		0			0
$361$	−13.0040			−0.684419
$362$		0			0
$363$			25.1454i			1.31979i
$364$		0			0
$365$	13.4316	−	19.1452i	0.703040	−	1.00211i
$366$		0			0
$367$		−	25.5958i		−	1.33609i	−0.744121	−	0.668045i	$-0.767131\pi$
$367$		−	25.5958i		−	1.33609i	0.744121	−	0.668045i	$-0.232869\pi$
$368$		0			0
$369$	−28.2369			−1.46995
$370$		0			0
$371$	−4.13084			−0.214463
$372$		0			0
$373$			17.2125i			0.891232i	0.895224	+	0.445616i	$0.147015\pi$
$373$			17.2125i			0.891232i	−0.895224	+	0.445616i	$0.852985\pi$
$374$		0			0
$375$	25.9874	+	7.03926i	1.34198	+	0.363506i
$376$		0			0
$377$			7.40312i			0.381280i
$378$		0			0
$379$	23.0339			1.18317			0.591585	−	0.806243i	$-0.298502\pi$
$379$	23.0339			1.18317			0.591585	+	0.806243i	$0.298502\pi$
$380$		0			0
$381$	6.58683			0.337453
$382$		0			0
$383$			3.81777i			0.195079i	0.995232	+	0.0975394i	$0.0310972\pi$
$383$			3.81777i			0.195079i	−0.995232	+	0.0975394i	$0.968903\pi$
$384$		0			0
$385$	−0.677944	+	0.966336i	−0.0345513	+	0.0492490i
$386$		0			0
$387$			14.9488i			0.759892i
$388$		0			0
$389$	8.25435			0.418512			0.209256	−	0.977861i	$-0.432896\pi$
$389$	8.25435			0.418512			0.209256	+	0.977861i	$0.432896\pi$
$390$		0			0
$391$	5.50074			0.278184
$392$		0			0
$393$		−	19.7936i		−	0.998454i
$394$		0			0
$395$	−11.9900	−	8.41170i	−0.603280	−	0.423238i
$396$		0			0
$397$			12.4967i			0.627193i	0.949556	+	0.313596i	$0.101534\pi$
$397$			12.4967i			0.627193i	−0.949556	+	0.313596i	$0.898466\pi$
$398$		0			0
$399$	−4.16658			−0.208590
$400$		0			0
$401$	34.5660			1.72614			0.863072	−	0.505082i	$-0.168538\pi$
$401$	34.5660			1.72614			0.863072	+	0.505082i	$0.168538\pi$
$402$		0			0
$403$			9.73020i			0.484696i
$404$		0			0
$405$	−17.5036	−	12.2799i	−0.869763	−	0.610193i
$406$		0			0
$407$			3.79103i			0.187915i
$408$		0			0
$409$	−12.0499			−0.595829			−0.297914	−	0.954593i	$-0.596291\pi$
$409$	−12.0499			−0.595829			−0.297914	+	0.954593i	$0.596291\pi$
$410$		0			0
$411$	−55.0140			−2.71364
$412$		0			0
$413$		−	8.77201i		−	0.431642i
$414$		0			0
$415$	−16.3183	+	23.2600i	−0.801034	+	1.14179i
$416$		0			0
$417$		−	9.05581i		−	0.443465i
$418$		0			0
$419$	38.6157			1.88650			0.943250	−	0.332085i	$-0.107752\pi$
$419$	38.6157			1.88650			0.943250	+	0.332085i	$0.107752\pi$
$420$		0			0
$421$	32.7384			1.59557			0.797785	−	0.602941i	$-0.206005\pi$
$421$	32.7384			1.59557			0.797785	+	0.602941i	$0.206005\pi$
$422$		0			0
$423$			22.1384i			1.07640i
$424$		0			0
$425$	25.8621	−	9.35985i	1.25449	−	0.454019i
$426$		0			0
$427$		−	1.11179i		−	0.0538031i
$428$		0			0
$429$	2.32708			0.112352
$430$		0			0
$431$	35.7739			1.72317			0.861584	−	0.507615i	$-0.169473\pi$
$431$	35.7739			1.72317			0.861584	+	0.507615i	$0.169473\pi$
$432$		0			0
$433$			10.5682i			0.507877i	0.967220	+	0.253938i	$0.0817261\pi$
$433$			10.5682i			0.507877i	−0.967220	+	0.253938i	$0.918274\pi$
$434$		0			0
$435$	−17.7011	+	25.2309i	−0.848701	+	1.20973i
$436$		0			0
$437$			2.44868i			0.117136i
$438$		0			0
$439$	−27.6642			−1.32034			−0.660170	−	0.751117i	$-0.729516\pi$
$439$	−27.6642			−1.32034			−0.660170	+	0.751117i	$0.729516\pi$
$440$		0			0
$441$	−18.1967			−0.866509
$442$		0			0
$443$			36.8082i			1.74881i	0.485198	+	0.874404i	$0.338747\pi$
$443$			36.8082i			1.74881i	−0.485198	+	0.874404i	$0.661253\pi$
$444$		0			0
$445$	−24.9730	−	17.5201i	−1.18384	−	0.830534i
$446$		0			0
$447$		−	4.59027i		−	0.217112i
$448$		0			0
$449$	−18.2711			−0.862267			−0.431133	−	0.902288i	$-0.641886\pi$
$449$	−18.2711			−0.862267			−0.431133	+	0.902288i	$0.641886\pi$
$450$		0			0
$451$	−7.53662			−0.354886
$452$		0			0
$453$		−	22.6840i		−	1.06579i
$454$		0			0
$455$	−1.67292	−	1.17366i	−0.0784278	−	0.0550219i
$456$		0			0
$457$		−	27.6326i		−	1.29260i	−0.763084	−	0.646299i	$-0.776316\pi$
$457$		−	27.6326i		−	1.29260i	0.763084	−	0.646299i	$-0.223684\pi$
$458$		0			0
$459$	−2.66025			−0.124170
$460$		0			0
$461$	35.5515			1.65580			0.827900	−	0.560876i	$-0.189536\pi$
$461$	35.5515			1.65580			0.827900	+	0.560876i	$0.189536\pi$
$462$		0			0
$463$		−	23.9331i		−	1.11226i	−0.831094	−	0.556132i	$-0.812285\pi$
$463$		−	23.9331i		−	1.11226i	0.831094	−	0.556132i	$-0.187715\pi$
$464$		0			0
$465$	−23.2652	+	33.1620i	−1.07890	+	1.53785i
$466$		0			0
$467$			19.9035i			0.921024i	0.887654	+	0.460512i	$0.152334\pi$
$467$			19.9035i			0.921024i	−0.887654	+	0.460512i	$0.847666\pi$
$468$		0			0
$469$	−3.71620			−0.171598
$470$		0			0
$471$	10.6754			0.491897
$472$		0			0
$473$			3.98995i			0.183458i
$474$		0			0
$475$	4.16658	+	11.5126i	0.191176	+	0.528236i
$476$		0			0
$477$			16.3646i			0.749281i
$478$		0			0
$479$	−3.86206			−0.176462			−0.0882309	−	0.996100i	$-0.528121\pi$
$479$	−3.86206			−0.176462			−0.0882309	+	0.996100i	$0.528121\pi$
$480$		0			0
$481$	−6.56304			−0.299249
$482$		0			0
$483$		−	1.70156i		−	0.0774238i
$484$		0			0
$485$	−1.92728	+	2.74712i	−0.0875131	+	0.124740i
$486$		0			0
$487$			15.1499i			0.686506i	0.939243	+	0.343253i	$0.111529\pi$
$487$			15.1499i			0.686506i	−0.939243	+	0.343253i	$0.888471\pi$
$488$		0			0
$489$	23.8997			1.08078
$490$		0			0
$491$	6.86312			0.309728			0.154864	−	0.987936i	$-0.450506\pi$
$491$	6.86312			0.309728			0.154864	+	0.987936i	$0.450506\pi$
$492$		0			0
$493$			31.4846i			1.41800i
$494$		0			0
$495$	3.82819	+	2.68571i	0.172064	+	0.120714i
$496$		0			0
$497$			1.89679i			0.0850826i
$498$		0			0
$499$	−18.6809			−0.836272			−0.418136	−	0.908385i	$-0.637316\pi$
$499$	−18.6809			−0.836272			−0.418136	+	0.908385i	$0.637316\pi$
$500$		0			0
$501$	28.6784			1.28125
$502$		0			0
$503$			23.1886i			1.03393i	0.856008	+	0.516963i	$0.172938\pi$
$503$			23.1886i			1.03393i	−0.856008	+	0.516963i	$0.827062\pi$
$504$		0			0
$505$	−29.3747	−	20.6082i	−1.30716	−	0.917051i
$506$		0			0
$507$		−	27.2773i		−	1.21143i
$508$		0			0
$509$	−11.5385			−0.511436			−0.255718	−	0.966751i	$-0.582312\pi$
$509$	−11.5385			−0.511436			−0.255718	+	0.966751i	$0.582312\pi$
$510$		0			0
$511$	−7.39013			−0.326920
$512$		0			0
$513$		−	1.18422i		−	0.0522847i
$514$		0			0
$515$	−12.1881	+	17.3728i	−0.537073	+	0.765539i
$516$		0			0
$517$			5.90888i			0.259872i
$518$		0			0
$519$	−7.01712			−0.308017
$520$		0			0
$521$	−16.2792			−0.713207			−0.356603	−	0.934256i	$-0.616065\pi$
$521$	−16.2792			−0.713207			−0.356603	+	0.934256i	$0.616065\pi$
$522$		0			0
$523$			30.8347i			1.34831i	0.738591	+	0.674153i	$0.235491\pi$
$523$			30.8347i			1.34831i	−0.738591	+	0.674153i	$0.764509\pi$
$524$		0			0
$525$	−2.89531	−	8.00000i	−0.126362	−	0.349149i
$526$		0			0
$527$			41.3814i			1.80260i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−34.7508			−1.50805
$532$		0			0
$533$		−	13.0474i		−	0.565146i
$534$		0			0
$535$	1.77830	−	2.53477i	0.0768827	−	0.109588i
$536$		0			0
$537$		−	46.6370i		−	2.01254i
$538$		0			0
$539$	−4.85683			−0.209198
$540$		0			0
$541$	−30.5199			−1.31215			−0.656077	−	0.754694i	$-0.727785\pi$
$541$	−30.5199			−1.31215			−0.656077	+	0.754694i	$0.727785\pi$
$542$		0			0
$543$		−	42.3799i		−	1.81870i
$544$		0			0
$545$	25.6949	+	18.0266i	1.10065	+	0.772173i
$546$		0			0
$547$		−	28.0416i		−	1.19897i	−0.800385	−	0.599487i	$-0.795371\pi$
$547$		−	28.0416i		−	1.19897i	0.800385	−	0.599487i	$-0.204629\pi$
$548$		0			0
$549$	−4.40439			−0.187975
$550$		0			0
$551$	−14.0155			−0.597082
$552$		0			0
$553$			4.62817i			0.196810i
$554$		0			0
$555$	−22.3678	−	15.6924i	−0.949461	−	0.666106i
$556$		0			0
$557$		−	4.15674i		−	0.176127i	−0.996115	−	0.0880634i	$-0.971932\pi$
$557$		−	4.15674i		−	0.176127i	0.996115	−	0.0880634i	$-0.0280678\pi$
$558$		0			0
$559$	−6.90741			−0.292152
$560$		0			0
$561$	9.89679			0.417843
$562$		0			0
$563$		−	21.3480i		−	0.899709i	−0.893102	−	0.449854i	$-0.851476\pi$
$563$		−	21.3480i		−	0.899709i	0.893102	−	0.449854i	$-0.148524\pi$
$564$		0			0
$565$	−20.9463	+	29.8567i	−0.881218	+	1.25608i
$566$		0			0
$567$			6.75647i			0.283745i
$568$		0			0
$569$	25.6430			1.07501			0.537506	−	0.843260i	$-0.319366\pi$
$569$	25.6430			1.07501			0.537506	+	0.843260i	$0.319366\pi$
$570$		0			0
$571$	7.10743			0.297437			0.148718	−	0.988880i	$-0.452485\pi$
$571$	7.10743			0.297437			0.148718	+	0.988880i	$0.452485\pi$
$572$		0			0
$573$			49.0245i			2.04803i
$574$		0			0
$575$	−4.70156	+	1.70156i	−0.196069	+	0.0709600i
$576$		0			0
$577$		−	22.7226i		−	0.945956i	−0.881074	−	0.472978i	$-0.843179\pi$
$577$		−	22.7226i		−	0.945956i	0.881074	−	0.472978i	$-0.156821\pi$
$578$		0			0
$579$	3.40312			0.141429
$580$		0			0
$581$	8.97843			0.372488
$582$		0			0
$583$			4.36782i			0.180896i
$584$		0			0
$585$	−4.64951	+	6.62736i	−0.192233	+	0.274008i
$586$		0			0
$587$			7.06600i			0.291645i	0.989311	+	0.145822i	$0.0465828\pi$
$587$			7.06600i			0.291645i	−0.989311	+	0.145822i	$0.953417\pi$
$588$		0			0
$589$	−18.4211			−0.759030
$590$		0			0
$591$	11.8576			0.487757
$592$		0			0
$593$			14.9914i			0.615624i	0.951447	+	0.307812i	$0.0995968\pi$
$593$			14.9914i			0.615624i	−0.951447	+	0.307812i	$0.900403\pi$
$594$		0			0
$595$	−7.11473	−	4.99143i	−0.291676	−	0.204629i
$596$		0			0
$597$			24.8293i			1.01620i
$598$		0			0
$599$	−4.03069			−0.164690			−0.0823448	−	0.996604i	$-0.526241\pi$
$599$	−4.03069			−0.164690			−0.0823448	+	0.996604i	$0.526241\pi$
$600$		0			0
$601$	13.9320			0.568300			0.284150	−	0.958780i	$-0.408289\pi$
$601$	13.9320			0.568300			0.284150	+	0.958780i	$0.408289\pi$
$602$		0			0
$603$			14.7219i			0.599523i
$604$		0			0
$605$	−19.1139	−	13.4096i	−0.777090	−	0.545177i
$606$		0			0
$607$		−	31.0588i		−	1.26064i	−0.776337	−	0.630318i	$-0.782925\pi$
$607$		−	31.0588i		−	1.26064i	0.776337	−	0.630318i	$-0.217075\pi$
$608$		0			0
$609$	9.73924			0.394654
$610$		0			0
$611$	−10.2295			−0.413840
$612$		0			0
$613$			19.6722i			0.794554i	0.917699	+	0.397277i	$0.130045\pi$
$613$			19.6722i			0.794554i	−0.917699	+	0.397277i	$0.869955\pi$
$614$		0			0
$615$	31.1967	−	44.4675i	1.25797	−	1.79310i
$616$		0			0
$617$			40.9043i			1.64674i	0.567502	+	0.823372i	$0.307910\pi$
$617$			40.9043i			1.64674i	−0.567502	+	0.823372i	$0.692090\pi$
$618$		0			0
$619$	−3.58556			−0.144116			−0.0720579	−	0.997400i	$-0.522957\pi$
$619$	−3.58556			−0.144116			−0.0720579	+	0.997400i	$0.522957\pi$
$620$		0			0
$621$	0.483617			0.0194069
$622$		0			0
$623$			9.63969i			0.386206i
$624$		0			0
$625$	−19.2094	+	16.0000i	−0.768375	+	0.640000i
$626$		0			0
$627$			4.40561i			0.175943i
$628$		0			0
$629$	−27.9118			−1.11292
$630$		0			0
$631$	7.19670			0.286496			0.143248	−	0.989687i	$-0.454245\pi$
$631$	7.19670			0.286496			0.143248	+	0.989687i	$0.454245\pi$
$632$		0			0
$633$			27.4917i			1.09270i
$634$		0			0
$635$	−3.51263	+	5.00687i	−0.139394	+	0.198692i
$636$		0			0
$637$		−	8.40815i		−	0.333143i
$638$		0			0
$639$	7.51422			0.297258
$640$		0			0
$641$	−16.3436			−0.645534			−0.322767	−	0.946478i	$-0.604613\pi$
$641$	−16.3436			−0.645534			−0.322767	+	0.946478i	$0.604613\pi$
$642$		0			0
$643$			6.55839i			0.258638i	0.991603	+	0.129319i	$0.0412791\pi$
$643$			6.55839i			0.258638i	−0.991603	+	0.129319i	$0.958721\pi$
$644$		0			0
$645$	−23.5415	−	16.5158i	−0.926945	−	0.650309i
$646$		0			0
$647$			16.5455i			0.650470i	0.945633	+	0.325235i	$0.105443\pi$
$647$			16.5455i			0.650470i	−0.945633	+	0.325235i	$0.894557\pi$
$648$		0			0
$649$	−9.27523			−0.364085
$650$		0			0
$651$	12.8006			0.501697
$652$		0			0
$653$		−	12.7408i		−	0.498587i	−0.968428	−	0.249294i	$-0.919802\pi$
$653$		−	12.7408i		−	0.498587i	0.968428	−	0.249294i	$-0.0801984\pi$
$654$		0			0
$655$	15.0458	+	10.5555i	0.587887	+	0.412439i
$656$		0			0
$657$			29.2764i			1.14218i
$658$		0			0
$659$	−23.8094			−0.927484			−0.463742	−	0.885970i	$-0.653493\pi$
$659$	−23.8094			−0.927484			−0.463742	+	0.885970i	$0.653493\pi$
$660$		0			0
$661$	24.9323			0.969754			0.484877	−	0.874582i	$-0.338864\pi$
$661$	24.9323			0.969754			0.484877	+	0.874582i	$0.338864\pi$
$662$		0			0
$663$			17.1333i			0.665403i
$664$		0			0
$665$	2.22196	−	3.16716i	0.0861639	−	0.122817i
$666$		0			0
$667$		−	5.72371i		−	0.221623i
$668$		0			0
$669$	−43.8509			−1.69537
$670$		0			0
$671$	−1.17556			−0.0453822
$672$		0			0
$673$			5.48050i			0.211258i	0.994406	+	0.105629i	$0.0336856\pi$
$673$			5.48050i			0.211258i	−0.994406	+	0.105629i	$0.966314\pi$
$674$		0			0
$675$	2.27375	−	0.822904i	0.0875168	−	0.0316736i
$676$		0			0
$677$		−	1.75255i		−	0.0673560i	−0.999433	−	0.0336780i	$-0.989278\pi$
$677$		−	1.75255i		−	0.0673560i	0.999433	−	0.0336780i	$-0.0107221\pi$
$678$		0			0
$679$	1.06040			0.0406944
$680$		0			0
$681$	45.6013			1.74745
$682$		0			0
$683$		−	23.5091i		−	0.899552i	−0.893141	−	0.449776i	$-0.851504\pi$
$683$		−	23.5091i		−	0.899552i	0.893141	−	0.449776i	$-0.148496\pi$
$684$		0			0
$685$	29.3379	−	41.8180i	1.12094	−	1.59778i
$686$		0			0
$687$		−	20.3834i		−	0.777673i
$688$		0			0
$689$	−7.56157			−0.288073
$690$		0			0
$691$	21.4834			0.817269			0.408634	−	0.912698i	$-0.366005\pi$
$691$	21.4834			0.817269			0.408634	+	0.912698i	$0.366005\pi$
$692$		0			0
$693$		−	1.47770i		−	0.0561330i
$694$		0			0
$695$	6.88362	+	4.82929i	0.261111	+	0.183185i
$696$		0			0
$697$		−	55.4890i		−	2.10180i
$698$		0			0
$699$	−40.0573			−1.51511
$700$		0			0
$701$	9.62985			0.363714			0.181857	−	0.983325i	$-0.441789\pi$
$701$	9.62985			0.363714			0.181857	+	0.983325i	$0.441789\pi$
$702$		0			0
$703$		−	12.4251i		−	0.468621i
$704$		0			0
$705$	−34.8635	−	24.4589i	−1.31304	−	0.921177i
$706$		0			0
$707$			11.3387i			0.426437i
$708$		0			0
$709$	−11.8970			−0.446801			−0.223400	−	0.974727i	$-0.571716\pi$
$709$	−11.8970			−0.446801			−0.223400	+	0.974727i	$0.571716\pi$
$710$		0			0
$711$	18.3347			0.687606
$712$		0			0
$713$		−	7.52288i		−	0.281734i
$714$		0			0
$715$	−1.24099	+	1.76889i	−0.0464103	+	0.0661528i
$716$		0			0
$717$			38.7327i			1.44650i
$718$		0			0
$719$	24.2949			0.906046			0.453023	−	0.891499i	$-0.350345\pi$
$719$	24.2949			0.906046			0.453023	+	0.891499i	$0.350345\pi$
$720$		0			0
$721$	6.70598			0.249744
$722$		0			0
$723$			67.7371i			2.51917i
$724$		0			0
$725$	−9.73924	−	26.9104i	−0.361706	−	0.999426i
$726$		0			0
$727$		−	6.25721i		−	0.232067i	−0.993245	−	0.116034i	$-0.962982\pi$
$727$		−	6.25721i		−	0.232067i	0.993245	−	0.116034i	$-0.0370180\pi$
$728$		0			0
$729$	23.2723			0.861935
$730$		0			0
$731$	−29.3764			−1.08653
$732$		0			0
$733$			37.3304i			1.37883i	0.724367	+	0.689415i	$0.242132\pi$
$733$			37.3304i			1.37883i	−0.724367	+	0.689415i	$0.757868\pi$
$734$		0			0
$735$	20.1041	−	28.6562i	0.741551	−	1.05700i
$736$		0			0
$737$			3.92939i			0.144741i
$738$		0			0
$739$	−26.9889			−0.992802			−0.496401	−	0.868093i	$-0.665345\pi$
$739$	−26.9889			−0.992802			−0.496401	+	0.868093i	$0.665345\pi$
$740$		0			0
$741$	−7.62699			−0.280185
$742$		0			0
$743$		−	32.1544i		−	1.17963i	−0.807538	−	0.589815i	$-0.799201\pi$
$743$		−	32.1544i		−	1.17963i	0.807538	−	0.589815i	$-0.200799\pi$
$744$		0			0
$745$	3.48922	+	2.44790i	0.127835	+	0.0896842i
$746$		0			0
$747$		−	35.5685i		−	1.30138i
$748$		0			0
$749$	−0.978433			−0.0357512
$750$		0			0
$751$	−39.7865			−1.45183			−0.725915	−	0.687785i	$-0.758583\pi$
$751$	−39.7865			−1.45183			−0.725915	+	0.687785i	$0.758583\pi$
$752$		0			0
$753$		−	49.5524i		−	1.80579i
$754$		0			0
$755$	17.2428	+	12.0969i	0.627531	+	0.440252i
$756$		0			0
$757$		−	26.9961i		−	0.981189i	−0.871388	−	0.490595i	$-0.836780\pi$
$757$		−	26.9961i		−	0.981189i	0.871388	−	0.490595i	$-0.163220\pi$
$758$		0			0
$759$	−1.79917			−0.0653059
$760$		0			0
$761$	−7.21529			−0.261554			−0.130777	−	0.991412i	$-0.541747\pi$
$761$	−7.21529			−0.261554			−0.130777	+	0.991412i	$0.541747\pi$
$762$		0			0
$763$		−	9.91833i		−	0.359068i
$764$		0			0
$765$	−19.7738	+	28.1854i	−0.714923	+	1.01904i
$766$		0			0
$767$		−	16.0573i		−	0.579795i
$768$		0			0
$769$	−6.40916			−0.231120			−0.115560	−	0.993300i	$-0.536866\pi$
$769$	−6.40916			−0.231120			−0.115560	+	0.993300i	$0.536866\pi$
$770$		0			0
$771$	40.5786			1.46140
$772$		0			0
$773$			50.2483i			1.80730i	0.428267	+	0.903652i	$0.359124\pi$
$773$			50.2483i			1.80730i	−0.428267	+	0.903652i	$0.640876\pi$
$774$		0			0
$775$	−12.8006	−	35.3693i	−0.459813	−	1.27050i
$776$		0			0
$777$			8.63406i			0.309745i
$778$		0			0
$779$	24.7012			0.885014
$780$		0			0
$781$	2.00560			0.0717660
$782$		0			0
$783$			2.76808i			0.0989231i
$784$		0			0
$785$	−5.69299	+	8.11473i	−0.203192	+	0.289627i
$786$		0			0
$787$			2.70933i			0.0965772i	0.998833	+	0.0482886i	$0.0153767\pi$
$787$			2.70933i			0.0965772i	−0.998833	+	0.0482886i	$0.984623\pi$
$788$		0			0
$789$	59.6329			2.12299
$790$		0			0
$791$	11.5248			0.409774
$792$		0			0
$793$		−	2.03514i		−	0.0722699i
$794$		0			0
$795$	−25.7709	−	18.0799i	−0.914001	−	0.641229i
$796$		0			0
$797$			15.1452i			0.536471i	0.963353	+	0.268236i	$0.0864406\pi$
$797$			15.1452i			0.536471i	−0.963353	+	0.268236i	$0.913559\pi$
$798$		0			0
$799$	−43.5047			−1.53909
$800$		0			0
$801$	38.1881			1.34931
$802$		0			0
$803$			7.81408i			0.275753i
$804$		0			0
$805$	1.29341	+	0.907411i	0.0455869	+	0.0319820i
$806$		0			0
$807$		−	8.55259i		−	0.301065i
$808$		0			0
$809$	3.94373			0.138654			0.0693270	−	0.997594i	$-0.477915\pi$
$809$	3.94373			0.138654			0.0693270	+	0.997594i	$0.477915\pi$
$810$		0			0
$811$	14.1521			0.496947			0.248474	−	0.968639i	$-0.420071\pi$
$811$	14.1521			0.496947			0.248474	+	0.968639i	$0.420071\pi$
$812$		0			0
$813$		−	66.1664i		−	2.32056i
$814$		0			0
$815$	−12.7453	+	18.1670i	−0.446447	+	0.636362i
$816$		0			0
$817$		−	13.0771i		−	0.457508i
$818$		0			0
$819$	2.55819			0.0893903
$820$		0			0
$821$	37.0094			1.29164			0.645818	−	0.763491i	$-0.276516\pi$
$821$	37.0094			1.29164			0.645818	+	0.763491i	$0.276516\pi$
$822$		0			0
$823$			18.1208i			0.631651i	0.948817	+	0.315826i	$0.102281\pi$
$823$			18.1208i			0.631651i	−0.948817	+	0.315826i	$0.897719\pi$
$824$		0			0
$825$	−8.45893	+	3.06141i	−0.294502	+	0.106585i
$826$		0			0
$827$		−	41.5398i		−	1.44448i	−0.691643	−	0.722240i	$-0.743113\pi$
$827$		−	41.5398i		−	1.44448i	0.691643	−	0.722240i	$-0.256887\pi$
$828$		0			0
$829$	−21.3976			−0.743171			−0.371585	−	0.928399i	$-0.621186\pi$
$829$	−21.3976			−0.743171			−0.371585	+	0.928399i	$0.621186\pi$
$830$		0			0
$831$	26.7307			0.927277
$832$		0			0
$833$		−	35.7588i		−	1.23897i
$834$		0			0
$835$	−15.2936	+	21.7994i	−0.529257	+	0.754399i
$836$		0			0
$837$			3.63819i			0.125754i
$838$		0			0
$839$	−41.3437			−1.42734			−0.713672	−	0.700480i	$-0.752969\pi$
$839$	−41.3437			−1.42734			−0.713672	+	0.700480i	$0.752969\pi$
$840$		0			0
$841$	3.76081			0.129683
$842$		0			0
$843$		−	31.4161i		−	1.08203i
$844$		0			0
$845$	20.7344	+	14.5465i	0.713284	+	0.500413i
$846$		0			0
$847$			7.37803i			0.253512i
$848$		0			0
$849$	−8.06848			−0.276910
$850$		0			0
$851$	5.07420			0.173941
$852$		0			0
$853$		−	56.0373i		−	1.91868i	−0.282252	−	0.959340i	$-0.591081\pi$
$853$		−	56.0373i		−	1.91868i	0.282252	−	0.959340i	$-0.408919\pi$
$854$		0			0
$855$	−12.5469	−	8.80241i	−0.429094	−	0.301036i
$856$		0			0
$857$			43.4734i			1.48502i	0.669833	+	0.742512i	$0.266366\pi$
$857$			43.4734i			1.48502i	−0.669833	+	0.742512i	$0.733634\pi$
$858$		0			0
$859$	11.1599			0.380771			0.190386	−	0.981709i	$-0.439026\pi$
$859$	11.1599			0.380771			0.190386	+	0.981709i	$0.439026\pi$
$860$		0			0
$861$	−17.1646			−0.584969
$862$		0			0
$863$		−	32.2767i		−	1.09871i	−0.835589	−	0.549356i	$-0.814873\pi$
$863$		−	32.2767i		−	1.09871i	0.835589	−	0.549356i	$-0.185127\pi$
$864$		0			0
$865$	3.74210	−	5.33395i	0.127235	−	0.181360i
$866$		0			0
$867$			31.9275i			1.08431i
$868$		0			0
$869$	4.89367			0.166006
$870$		0			0
$871$	−6.80256			−0.230496
$872$		0			0
$873$		−	4.20083i		−	0.142176i
$874$		0			0
$875$	7.62508	+	2.06542i	0.257775	+	0.0698240i
$876$		0			0
$877$			9.52493i			0.321634i	0.986984	+	0.160817i	$0.0514129\pi$
$877$			9.52493i			0.321634i	−0.986984	+	0.160817i	$0.948587\pi$
$878$		0			0
$879$	21.4132			0.722248
$880$		0			0
$881$	49.1585			1.65619			0.828096	−	0.560586i	$-0.189424\pi$
$881$	49.1585			1.65619			0.828096	+	0.560586i	$0.189424\pi$
$882$		0			0
$883$		−	33.1379i		−	1.11518i	−0.830117	−	0.557590i	$-0.811726\pi$
$883$		−	33.1379i		−	1.11518i	0.830117	−	0.557590i	$-0.188274\pi$
$884$		0			0
$885$	38.3934	−	54.7256i	1.29058	−	1.83958i
$886$		0			0
$887$		−	1.59084i		−	0.0534153i	−0.999643	−	0.0267076i	$-0.991498\pi$
$887$		−	1.59084i		−	0.0534153i	0.999643	−	0.0267076i	$-0.00850232\pi$
$888$		0			0
$889$	1.93267			0.0648197
$890$		0			0
$891$	7.14407			0.239335
$892$		0			0
$893$		−	19.3663i		−	0.648070i
$894$		0			0
$895$	35.4504	+	24.8706i	1.18497	+	0.831333i
$896$		0			0
$897$		−	3.11473i		−	0.103998i
$898$		0			0
$899$	43.0588			1.43609
$900$		0			0
$901$	−32.1584			−1.07135
$902$		0			0
$903$			9.08710i			0.302400i
$904$		0			0
$905$	32.2144	+	22.6004i	1.07084	+	0.751263i
$906$		0			0
$907$			17.4658i			0.579942i	0.957035	+	0.289971i	$0.0936457\pi$
$907$			17.4658i			0.579942i	−0.957035	+	0.289971i	$0.906354\pi$
$908$		0			0
$909$	44.9190			1.48987
$910$		0			0
$911$	21.6050			0.715805			0.357903	−	0.933759i	$-0.383492\pi$
$911$	21.6050			0.715805			0.357903	+	0.933759i	$0.383492\pi$
$912$		0			0
$913$		−	9.49349i		−	0.314189i
$914$		0			0
$915$	4.86607	−	6.93605i	0.160867	−	0.229299i
$916$		0			0
$917$		−	5.80772i		−	0.191788i
$918$		0			0
$919$	−41.7396			−1.37686			−0.688432	−	0.725301i	$-0.741701\pi$
$919$	−41.7396			−1.37686			−0.688432	+	0.725301i	$0.741701\pi$
$920$		0			0
$921$	−68.4181			−2.25445
$922$		0			0
$923$			3.47210i			0.114285i
$924$		0			0
$925$	23.8567	−	8.63406i	0.784402	−	0.283886i
$926$		0			0
$927$		−	26.5661i		−	0.872545i
$928$		0			0
$929$	−38.8505			−1.27464			−0.637322	−	0.770597i	$-0.719958\pi$
$929$	−38.8505			−1.27464			−0.637322	+	0.770597i	$0.719958\pi$
$930$		0			0
$931$	15.9182			0.521699
$932$		0			0
$933$		−	67.4533i		−	2.20832i
$934$		0			0
$935$	−5.27777	+	7.52288i	−0.172601	+	0.246025i
$936$		0			0
$937$		−	33.1555i		−	1.08314i	−0.840655	−	0.541571i	$-0.817830\pi$
$937$		−	33.1555i		−	1.08314i	0.840655	−	0.541571i	$-0.182170\pi$
$938$		0			0
$939$	17.8897			0.583808
$940$		0			0
$941$	−33.8004			−1.10186			−0.550931	−	0.834551i	$-0.685727\pi$
$941$	−33.8004			−1.10186			−0.550931	+	0.834551i	$0.685727\pi$
$942$		0			0
$943$			10.0876i			0.328496i
$944$		0			0
$945$	−0.625517	−	0.438839i	−0.0203481	−	0.0142754i
$946$		0			0
$947$		−	20.2406i		−	0.657730i	−0.944377	−	0.328865i	$-0.893334\pi$
$947$		−	20.2406i		−	0.657730i	0.944377	−	0.328865i	$-0.106666\pi$
$948$		0			0
$949$	−13.5277			−0.439129
$950$		0			0
$951$	31.6622			1.02672
$952$		0			0
$953$			28.3885i			0.919592i	0.888024	+	0.459796i	$0.152078\pi$
$953$			28.3885i			0.919592i	−0.888024	+	0.459796i	$0.847922\pi$
$954$		0			0
$955$	−37.2652	−	26.1438i	−1.20587	−	0.845995i
$956$		0			0
$957$		−	10.2979i		−	0.332885i
$958$		0			0
$959$	−16.1419			−0.521249
$960$		0			0
$961$	25.5937			0.825604
$962$		0			0
$963$			3.87611i			0.124906i
$964$		0			0
$965$	−1.81482	+	2.58683i	−0.0584212	+	0.0832730i
$966$		0			0
$967$			57.6163i			1.85282i	0.376522	+	0.926408i	$0.377120\pi$
$967$			57.6163i			1.85282i	−0.376522	+	0.926408i	$0.622880\pi$
$968$		0			0
$969$	−32.4367			−1.04202
$970$		0			0
$971$	1.47686			0.0473946			0.0236973	−	0.999719i	$-0.492456\pi$
$971$	1.47686			0.0473946			0.0236973	+	0.999719i	$0.492456\pi$
$972$		0			0
$973$		−	2.65711i		−	0.0851829i
$974$		0			0
$975$	−5.29991	−	14.6441i	−0.169733	−	0.468987i
$976$		0			0
$977$		−	11.5709i		−	0.370185i	−0.982721	−	0.185092i	$-0.940742\pi$
$977$		−	11.5709i		−	0.370185i	0.982721	−	0.185092i	$-0.0592584\pi$
$978$		0			0
$979$	10.1927			0.325760
$980$		0			0
$981$	−39.2920			−1.25450
$982$		0			0
$983$			37.5275i			1.19694i	0.801145	+	0.598470i	$0.204225\pi$
$983$			37.5275i			1.19694i	−0.801145	+	0.598470i	$0.795775\pi$
$984$		0			0
$985$	−6.32344	+	9.01337i	−0.201481	+	0.287190i
$986$		0			0
$987$			13.4575i			0.428356i
$988$		0			0
$989$	5.34045			0.169816
$990$		0			0
$991$	−9.87860			−0.313804			−0.156902	−	0.987614i	$-0.550151\pi$
$991$	−9.87860			−0.313804			−0.156902	+	0.987614i	$0.550151\pi$
$992$		0			0
$993$		−	37.6214i		−	1.19388i
$994$		0			0
$995$	−18.8736	−	13.2410i	−0.598333	−	0.419768i
$996$		0			0
$997$		−	53.6506i		−	1.69913i	−0.527484	−	0.849565i	$-0.676865\pi$
$997$		−	53.6506i		−	1.69913i	0.527484	−	0.849565i	$-0.323135\pi$
$998$		0			0
$999$	−2.45397			−0.0776401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.e.369.2		8
4.3	odd	2		230.2.b.b.139.8	yes	8
5.2	odd	4		9200.2.a.cr.1.1		4
5.3	odd	4		9200.2.a.cj.1.4		4
5.4	even	2	inner	1840.2.e.e.369.7		8
12.11	even	2		2070.2.d.f.829.1		8
20.3	even	4		1150.2.a.s.1.1		4
20.7	even	4		1150.2.a.r.1.4		4
20.19	odd	2		230.2.b.b.139.1	✓	8
60.59	even	2		2070.2.d.f.829.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.b.139.1	✓	8	20.19	odd	2
230.2.b.b.139.8	yes	8	4.3	odd	2
1150.2.a.r.1.4		4	20.7	even	4
1150.2.a.s.1.1		4	20.3	even	4
1840.2.e.e.369.2		8	1.1	even	1	trivial
1840.2.e.e.369.7		8	5.4	even	2	inner
2070.2.d.f.829.1		8	12.11	even	2
2070.2.d.f.829.5		8	60.59	even	2
9200.2.a.cj.1.4		4	5.3	odd	4
9200.2.a.cr.1.1		4	5.2	odd	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.40815i q^{3} +(1.83051 + 1.28422i) q^{5} -0.706585i q^{7} -2.79917 q^{9} +O(q^{10})\)	\(q-2.40815i q^{3} +(1.83051 + 1.28422i) q^{5} -0.706585i q^{7} -2.79917 q^{9} -0.747120 q^{11} -1.29341i q^{13} +(3.09259 - 4.40815i) q^{15} -5.50074i q^{17} +2.44868 q^{19} -1.70156 q^{21} +1.00000i q^{23} +(1.70156 + 4.70156i) q^{25} -0.483617i q^{27} -5.72371 q^{29} -7.52288 q^{31} +1.79917i q^{33} +(0.907411 - 1.29341i) q^{35} -5.07420i q^{37} -3.11473 q^{39} +10.0876 q^{41} -5.34045i q^{43} +(-5.12393 - 3.59475i) q^{45} -7.90888i q^{47} +6.50074 q^{49} -13.2466 q^{51} -5.84621i q^{53} +(-1.36761 - 0.959466i) q^{55} -5.89679i q^{57} +12.4146 q^{59} +1.57346 q^{61} +1.97786i q^{63} +(1.66103 - 2.36761i) q^{65} -5.25938i q^{67} +2.40815 q^{69} -2.68444 q^{71} -10.4589i q^{73} +(11.3221 - 4.09761i) q^{75} +0.527904i q^{77} -6.55005 q^{79} -9.56214 q^{81} +12.7068i q^{83} +(7.06415 - 10.0692i) q^{85} +13.7835i q^{87} -13.6426 q^{89} -0.913908 q^{91} +18.1162i q^{93} +(4.48235 + 3.14464i) q^{95} +1.50074i q^{97} +2.09132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 14 q^{9}+O(q^{10}) \) 8 * q - 14 * q^9	\( 8 q - 14 q^{9} - 10 q^{11} + 16 q^{15} - 2 q^{19} + 12 q^{21} - 12 q^{25} - 4 q^{29} - 10 q^{31} + 16 q^{35} + 46 q^{41} - 26 q^{45} + 18 q^{49} - 14 q^{51} + 18 q^{55} + 32 q^{59} + 18 q^{61} - 16 q^{65} - 6 q^{69} - 38 q^{71} + 32 q^{75} - 12 q^{79} + 32 q^{81} - 24 q^{85} - 60 q^{89} + 26 q^{91} - 18 q^{95} - 54 q^{99}+O(q^{100}) \) 8 * q - 14 * q^9 - 10 * q^11 + 16 * q^15 - 2 * q^19 + 12 * q^21 - 12 * q^25 - 4 * q^29 - 10 * q^31 + 16 * q^35 + 46 * q^41 - 26 * q^45 + 18 * q^49 - 14 * q^51 + 18 * q^55 + 32 * q^59 + 18 * q^61 - 16 * q^65 - 6 * q^69 - 38 * q^71 + 32 * q^75 - 12 * q^79 + 32 * q^81 - 24 * q^85 - 60 * q^89 + 26 * q^91 - 18 * q^95 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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