LMFDB - Embedded newform 640.2.c.a.129.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [640,2,Mod(129,640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(640, 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("640.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	640.c (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:	$5.11042572936$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.3
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	640.129
Dual form			640.2.c.a.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.806063i q^{3} +(1.48119 + 1.67513i) q^{5} +2.15633i q^{7} +2.35026 q^{9} +O(q^{10})$	$q-0.806063i q^{3} +(1.48119 + 1.67513i) q^{5} +2.15633i q^{7} +2.35026 q^{9} -0.387873 q^{11} +0.962389i q^{13} +(1.35026 - 1.19394i) q^{15} +1.61213i q^{17} -6.31265 q^{19} +1.73813 q^{21} +6.15633i q^{23} +(-0.612127 + 4.96239i) q^{25} -4.31265i q^{27} +2.00000 q^{29} +9.92478 q^{31} +0.312650i q^{33} +(-3.61213 + 3.19394i) q^{35} +6.57452i q^{37} +0.775746 q^{39} +4.57452 q^{41} -11.5066i q^{43} +(3.48119 + 3.93700i) q^{45} -4.54420i q^{47} +2.35026 q^{49} +1.29948 q^{51} -8.96239i q^{53} +(-0.574515 - 0.649738i) q^{55} +5.08840i q^{57} +6.31265 q^{59} -0.261865 q^{61} +5.06793i q^{63} +(-1.61213 + 1.42548i) q^{65} +9.89446i q^{67} +4.96239 q^{69} -10.7005 q^{71} -13.0884i q^{73} +(4.00000 + 0.493413i) q^{75} -0.836381i q^{77} +1.92478 q^{79} +3.57452 q^{81} -2.88129i q^{83} +(-2.70052 + 2.38787i) q^{85} -1.61213i q^{87} -10.6253 q^{89} -2.07522 q^{91} -8.00000i q^{93} +(-9.35026 - 10.5745i) q^{95} +9.61213i q^{97} -0.911603 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^5 - 6 * q^9	$ 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} - 12 q^{15} + 4 q^{19} - 8 q^{21} - 2 q^{25} + 12 q^{29} + 16 q^{31} - 20 q^{35} + 8 q^{39} + 4 q^{41} + 10 q^{45} - 6 q^{49} + 48 q^{51} + 20 q^{55} - 4 q^{59} - 20 q^{61} - 8 q^{65} + 8 q^{69} - 24 q^{71} + 24 q^{75} - 32 q^{79} - 2 q^{81} + 24 q^{85} + 20 q^{89} - 56 q^{91} - 36 q^{95} - 44 q^{99}+O(q^{100}) $ 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 - 12 * q^15 + 4 * q^19 - 8 * q^21 - 2 * q^25 + 12 * q^29 + 16 * q^31 - 20 * q^35 + 8 * q^39 + 4 * q^41 + 10 * q^45 - 6 * q^49 + 48 * q^51 + 20 * q^55 - 4 * q^59 - 20 * q^61 - 8 * q^65 + 8 * q^69 - 24 * q^71 + 24 * q^75 - 32 * q^79 - 2 * q^81 + 24 * q^85 + 20 * q^89 - 56 * q^91 - 36 * q^95 - 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$1$	$1$

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			0
$2$		0			0
$3$		−	0.806063i		−	0.465381i	−0.972551	−	0.232690i	$-0.925247\pi$
$3$		−	0.806063i		−	0.465381i	0.972551	−	0.232690i	$-0.0747529\pi$
$4$		0			0
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$5$	1.48119	+	1.67513i	0.662410	+	0.749141i
$6$		0			0
$7$			2.15633i			0.815014i	0.913202	+	0.407507i	$0.133602\pi$
$7$			2.15633i			0.815014i	−0.913202	+	0.407507i	$0.866398\pi$
$8$		0			0
$9$	2.35026			0.783421
$10$		0			0
$11$	−0.387873			−0.116948			−0.0584741	−	0.998289i	$-0.518623\pi$
$11$	−0.387873			−0.116948			−0.0584741	+	0.998289i	$0.518623\pi$
$12$		0			0
$13$			0.962389i			0.266919i	0.991054	+	0.133459i	$0.0426085\pi$
$13$			0.962389i			0.266919i	−0.991054	+	0.133459i	$0.957391\pi$
$14$		0			0
$15$	1.35026	−	1.19394i	0.348636	−	0.308273i
$16$		0			0
$17$			1.61213i			0.390998i	0.980704	+	0.195499i	$0.0626327\pi$
$17$			1.61213i			0.390998i	−0.980704	+	0.195499i	$0.937367\pi$
$18$		0			0
$19$	−6.31265			−1.44822			−0.724111	−	0.689684i	$-0.757750\pi$
$19$	−6.31265			−1.44822			−0.724111	+	0.689684i	$0.757750\pi$
$20$		0			0
$21$	1.73813			0.379292
$22$		0			0
$23$			6.15633i			1.28368i	0.766838	+	0.641841i	$0.221829\pi$
$23$			6.15633i			1.28368i	−0.766838	+	0.641841i	$0.778171\pi$
$24$		0			0
$25$	−0.612127	+	4.96239i	−0.122425	+	0.992478i
$26$		0			0
$27$		−	4.31265i		−	0.829970i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	9.92478			1.78254			0.891271	−	0.453470i	$-0.149814\pi$
$31$	9.92478			1.78254			0.891271	+	0.453470i	$0.149814\pi$
$32$		0			0
$33$			0.312650i			0.0544254i
$34$		0			0
$35$	−3.61213	+	3.19394i	−0.610561	+	0.539874i
$36$		0			0
$37$			6.57452i			1.08084i	0.841394	+	0.540422i	$0.181735\pi$
$37$			6.57452i			1.08084i	−0.841394	+	0.540422i	$0.818265\pi$
$38$		0			0
$39$	0.775746			0.124219
$40$		0			0
$41$	4.57452			0.714419			0.357210	−	0.934024i	$-0.383728\pi$
$41$	4.57452			0.714419			0.357210	+	0.934024i	$0.383728\pi$
$42$		0			0
$43$		−	11.5066i		−	1.75474i	−0.479816	−	0.877369i	$-0.659297\pi$
$43$		−	11.5066i		−	1.75474i	0.479816	−	0.877369i	$-0.340703\pi$
$44$		0			0
$45$	3.48119	+	3.93700i	0.518946	+	0.586893i
$46$		0			0
$47$		−	4.54420i		−	0.662839i	−0.943483	−	0.331420i	$-0.892472\pi$
$47$		−	4.54420i		−	0.662839i	0.943483	−	0.331420i	$-0.107528\pi$
$48$		0			0
$49$	2.35026			0.335752
$50$		0			0
$51$	1.29948			0.181963
$52$		0			0
$53$		−	8.96239i		−	1.23108i	−0.788106	−	0.615539i	$-0.788938\pi$
$53$		−	8.96239i		−	1.23108i	0.788106	−	0.615539i	$-0.211062\pi$
$54$		0			0
$55$	−0.574515	−	0.649738i	−0.0774677	−	0.0876107i
$56$		0			0
$57$			5.08840i			0.673975i
$58$		0			0
$59$	6.31265			0.821837			0.410919	−	0.911672i	$-0.365208\pi$
$59$	6.31265			0.821837			0.410919	+	0.911672i	$0.365208\pi$
$60$		0			0
$61$	−0.261865			−0.0335284			−0.0167642	−	0.999859i	$-0.505336\pi$
$61$	−0.261865			−0.0335284			−0.0167642	+	0.999859i	$0.505336\pi$
$62$		0			0
$63$			5.06793i			0.638499i
$64$		0			0
$65$	−1.61213	+	1.42548i	−0.199960	+	0.176810i
$66$		0			0
$67$			9.89446i			1.20880i	0.796681	+	0.604400i	$0.206587\pi$
$67$			9.89446i			1.20880i	−0.796681	+	0.604400i	$0.793413\pi$
$68$		0			0
$69$	4.96239			0.597401
$70$		0			0
$71$	−10.7005			−1.26992			−0.634959	−	0.772546i	$-0.718983\pi$
$71$	−10.7005			−1.26992			−0.634959	+	0.772546i	$0.718983\pi$
$72$		0			0
$73$		−	13.0884i		−	1.53188i	−0.642911	−	0.765940i	$-0.722274\pi$
$73$		−	13.0884i		−	1.53188i	0.642911	−	0.765940i	$-0.277726\pi$
$74$		0			0
$75$	4.00000	+	0.493413i	0.461880	+	0.0569744i
$76$		0			0
$77$		−	0.836381i		−	0.0953144i
$78$		0			0
$79$	1.92478			0.216554			0.108277	−	0.994121i	$-0.465467\pi$
$79$	1.92478			0.216554			0.108277	+	0.994121i	$0.465467\pi$
$80$		0			0
$81$	3.57452			0.397168
$82$		0			0
$83$		−	2.88129i		−	0.316262i	−0.987418	−	0.158131i	$-0.949453\pi$
$83$		−	2.88129i		−	0.316262i	0.987418	−	0.158131i	$-0.0505469\pi$
$84$		0			0
$85$	−2.70052	+	2.38787i	−0.292913	+	0.259001i
$86$		0			0
$87$		−	1.61213i		−	0.172838i
$88$		0			0
$89$	−10.6253			−1.12628			−0.563140	−	0.826362i	$-0.690407\pi$
$89$	−10.6253			−1.12628			−0.563140	+	0.826362i	$0.690407\pi$
$90$		0			0
$91$	−2.07522			−0.217542
$92$		0			0
$93$		−	8.00000i		−	0.829561i
$94$		0			0
$95$	−9.35026	−	10.5745i	−0.959317	−	1.08492i
$96$		0			0
$97$			9.61213i			0.975964i	0.872854	+	0.487982i	$0.162267\pi$
$97$			9.61213i			0.975964i	−0.872854	+	0.487982i	$0.837733\pi$
$98$		0			0
$99$	−0.911603			−0.0916196
$100$		0			0
$101$	−11.4010			−1.13445			−0.567223	−	0.823564i	$-0.691982\pi$
$101$	−11.4010			−1.13445			−0.567223	+	0.823564i	$0.691982\pi$
$102$		0			0
$103$			10.9321i			1.07717i	0.842572	+	0.538585i	$0.181041\pi$
$103$			10.9321i			1.07717i	−0.842572	+	0.538585i	$0.818959\pi$
$104$		0			0
$105$	2.57452	+	2.91160i	0.251247	+	0.284143i
$106$		0			0
$107$			11.0435i			1.06761i	0.845606	+	0.533807i	$0.179239\pi$
$107$			11.0435i			1.06761i	−0.845606	+	0.533807i	$0.820761\pi$
$108$		0			0
$109$	−5.03761			−0.482516			−0.241258	−	0.970461i	$-0.577560\pi$
$109$	−5.03761			−0.482516			−0.241258	+	0.970461i	$0.577560\pi$
$110$		0			0
$111$	5.29948			0.503004
$112$		0			0
$113$		−	19.8496i		−	1.86729i	−0.358201	−	0.933645i	$-0.616610\pi$
$113$		−	19.8496i		−	1.86729i	0.358201	−	0.933645i	$-0.383390\pi$
$114$		0			0
$115$	−10.3127	+	9.11871i	−0.961660	+	0.850324i
$116$		0			0
$117$			2.26187i			0.209110i
$118$		0			0
$119$	−3.47627			−0.318669
$120$		0			0
$121$	−10.8496			−0.986323
$122$		0			0
$123$		−	3.68735i		−	0.332477i
$124$		0			0
$125$	−9.21933	+	6.32487i	−0.824602	+	0.565713i
$126$		0			0
$127$		−	11.7685i		−	1.04428i	−0.852859	−	0.522141i	$-0.825134\pi$
$127$		−	11.7685i		−	1.04428i	0.852859	−	0.522141i	$-0.174866\pi$
$128$		0			0
$129$	−9.27504			−0.816622
$130$		0			0
$131$	15.0884			1.31828			0.659140	−	0.752021i	$-0.270921\pi$
$131$	15.0884			1.31828			0.659140	+	0.752021i	$0.270921\pi$
$132$		0			0
$133$		−	13.6121i		−	1.18032i
$134$		0			0
$135$	7.22425	−	6.38787i	0.621765	−	0.549781i
$136$		0			0
$137$		−	18.5501i		−	1.58484i	−0.609976	−	0.792420i	$-0.708821\pi$
$137$		−	18.5501i		−	1.58484i	0.609976	−	0.792420i	$-0.291179\pi$
$138$		0			0
$139$	7.61213			0.645652			0.322826	−	0.946458i	$-0.395367\pi$
$139$	7.61213			0.645652			0.322826	+	0.946458i	$0.395367\pi$
$140$		0			0
$141$	−3.66291			−0.308473
$142$		0			0
$143$		−	0.373285i		−	0.0312156i
$144$		0			0
$145$	2.96239	+	3.35026i	0.246013	+	0.278224i
$146$		0			0
$147$		−	1.89446i		−	0.156252i
$148$		0			0
$149$	−13.6629			−1.11931			−0.559655	−	0.828726i	$-0.689066\pi$
$149$	−13.6629			−1.11931			−0.559655	+	0.828726i	$0.689066\pi$
$150$		0			0
$151$	−15.2243			−1.23893			−0.619466	−	0.785023i	$-0.712651\pi$
$151$	−15.2243			−1.23893			−0.619466	+	0.785023i	$0.712651\pi$
$152$		0			0
$153$			3.78892i			0.306316i
$154$		0			0
$155$	14.7005	+	16.6253i	1.18077	+	1.33538i
$156$		0			0
$157$			1.42548i			0.113766i	0.998381	+	0.0568830i	$0.0181162\pi$
$157$			1.42548i			0.113766i	−0.998381	+	0.0568830i	$0.981884\pi$
$158$		0			0
$159$	−7.22425			−0.572921
$160$		0			0
$161$	−13.2750			−1.04622
$162$		0			0
$163$		−	8.80606i		−	0.689744i	−0.938650	−	0.344872i	$-0.887922\pi$
$163$		−	8.80606i		−	0.689744i	0.938650	−	0.344872i	$-0.112078\pi$
$164$		0			0
$165$	−0.523730	+	0.463096i	−0.0407723	+	0.0360520i
$166$		0			0
$167$		−	10.4690i		−	0.810114i	−0.914292	−	0.405057i	$-0.867252\pi$
$167$		−	10.4690i		−	0.810114i	0.914292	−	0.405057i	$-0.132748\pi$
$168$		0			0
$169$	12.0738			0.928754
$170$		0			0
$171$	−14.8364			−1.13457
$172$		0			0
$173$		−	19.9756i		−	1.51871i	−0.650674	−	0.759357i	$-0.725514\pi$
$173$		−	19.9756i		−	1.51871i	0.650674	−	0.759357i	$-0.274486\pi$
$174$		0			0
$175$	−10.7005	−	1.31994i	−0.808884	−	0.0997784i
$176$		0			0
$177$		−	5.08840i		−	0.382467i
$178$		0			0
$179$	12.2374			0.914668			0.457334	−	0.889295i	$-0.348804\pi$
$179$	12.2374			0.914668			0.457334	+	0.889295i	$0.348804\pi$
$180$		0			0
$181$	13.8496			1.02943			0.514715	−	0.857362i	$-0.327898\pi$
$181$	13.8496			1.02943			0.514715	+	0.857362i	$0.327898\pi$
$182$		0			0
$183$			0.211080i			0.0156035i
$184$		0			0
$185$	−11.0132	+	9.73813i	−0.809705	+	0.715962i
$186$		0			0
$187$		−	0.625301i		−	0.0457265i
$188$		0			0
$189$	9.29948			0.676437
$190$		0			0
$191$	−3.84955			−0.278544			−0.139272	−	0.990254i	$-0.544476\pi$
$191$	−3.84955			−0.278544			−0.139272	+	0.990254i	$0.544476\pi$
$192$		0			0
$193$		−	9.61213i		−	0.691896i	−0.938254	−	0.345948i	$-0.887557\pi$
$193$		−	9.61213i		−	0.691896i	0.938254	−	0.345948i	$-0.112443\pi$
$194$		0			0
$195$	1.14903	+	1.29948i	0.0822838	+	0.0930574i
$196$		0			0
$197$		−	9.58769i		−	0.683095i	−0.939865	−	0.341547i	$-0.889049\pi$
$197$		−	9.58769i		−	0.683095i	0.939865	−	0.341547i	$-0.110951\pi$
$198$		0			0
$199$	5.29948			0.375670			0.187835	−	0.982201i	$-0.439853\pi$
$199$	5.29948			0.375670			0.187835	+	0.982201i	$0.439853\pi$
$200$		0			0
$201$	7.97556			0.562553
$202$		0			0
$203$			4.31265i			0.302689i
$204$		0			0
$205$	6.77575	+	7.66291i	0.473239	+	0.535201i
$206$		0			0
$207$			14.4690i			1.00566i
$208$		0			0
$209$	2.44851			0.169367
$210$		0			0
$211$	15.0884			1.03873			0.519364	−	0.854553i	$-0.326169\pi$
$211$	15.0884			1.03873			0.519364	+	0.854553i	$0.326169\pi$
$212$		0			0
$213$			8.62530i			0.590996i
$214$		0			0
$215$	19.2750	−	17.0435i	1.31455	−	1.16236i
$216$		0			0
$217$			21.4010i			1.45280i
$218$		0			0
$219$	−10.5501			−0.712908
$220$		0			0
$221$	−1.55149			−0.104365
$222$		0			0
$223$			22.5296i			1.50869i	0.656476	+	0.754347i	$0.272046\pi$
$223$			22.5296i			1.50869i	−0.656476	+	0.754347i	$0.727954\pi$
$224$		0			0
$225$	−1.43866	+	11.6629i	−0.0959106	+	0.777527i
$226$		0			0
$227$		−	2.35756i		−	0.156476i	−0.996935	−	0.0782382i	$-0.975071\pi$
$227$		−	2.35756i		−	0.156476i	0.996935	−	0.0782382i	$-0.0249295\pi$
$228$		0			0
$229$	3.55149			0.234689			0.117345	−	0.993091i	$-0.462562\pi$
$229$	3.55149			0.234689			0.117345	+	0.993091i	$0.462562\pi$
$230$		0			0
$231$	−0.674176			−0.0443575
$232$		0			0
$233$		−	13.0884i		−	0.857449i	−0.903435	−	0.428725i	$-0.858963\pi$
$233$		−	13.0884i		−	0.857449i	0.903435	−	0.428725i	$-0.141037\pi$
$234$		0			0
$235$	7.61213	−	6.73084i	0.496560	−	0.439072i
$236$		0			0
$237$		−	1.55149i		−	0.100780i
$238$		0			0
$239$	−15.3258			−0.991345			−0.495673	−	0.868509i	$-0.665078\pi$
$239$	−15.3258			−0.991345			−0.495673	+	0.868509i	$0.665078\pi$
$240$		0			0
$241$	2.12601			0.136948			0.0684741	−	0.997653i	$-0.478187\pi$
$241$	2.12601			0.136948			0.0684741	+	0.997653i	$0.478187\pi$
$242$		0			0
$243$		−	15.8192i		−	1.01480i
$244$		0			0
$245$	3.48119	+	3.93700i	0.222405	+	0.251525i
$246$		0			0
$247$		−	6.07522i		−	0.386557i
$248$		0			0
$249$	−2.32250			−0.147182
$250$		0			0
$251$	−26.6859			−1.68440			−0.842201	−	0.539164i	$-0.818740\pi$
$251$	−26.6859			−1.68440			−0.842201	+	0.539164i	$0.818740\pi$
$252$		0			0
$253$		−	2.38787i		−	0.150124i
$254$		0			0
$255$	1.92478	+	2.17679i	0.120534	+	0.136316i
$256$		0			0
$257$			5.40105i			0.336908i	0.985710	+	0.168454i	$0.0538775\pi$
$257$			5.40105i			0.336908i	−0.985710	+	0.168454i	$0.946123\pi$
$258$		0			0
$259$	−14.1768			−0.880903
$260$		0			0
$261$	4.70052			0.290955
$262$		0			0
$263$		−	14.4690i		−	0.892195i	−0.894984	−	0.446098i	$-0.852813\pi$
$263$		−	14.4690i		−	0.892195i	0.894984	−	0.446098i	$-0.147187\pi$
$264$		0			0
$265$	15.0132	−	13.2750i	0.922252	−	0.815479i
$266$		0			0
$267$			8.56467i			0.524149i
$268$		0			0
$269$	−28.1114			−1.71398			−0.856992	−	0.515330i	$-0.827669\pi$
$269$	−28.1114			−1.71398			−0.856992	+	0.515330i	$0.827669\pi$
$270$		0			0
$271$	24.8773			1.51119			0.755595	−	0.655039i	$-0.227348\pi$
$271$	24.8773			1.51119			0.755595	+	0.655039i	$0.227348\pi$
$272$		0			0
$273$			1.67276i			0.101240i
$274$		0			0
$275$	0.237428	−	1.92478i	0.0143174	−	0.116068i
$276$		0			0
$277$			19.9756i			1.20022i	0.799919	+	0.600108i	$0.204876\pi$
$277$			19.9756i			1.20022i	−0.799919	+	0.600108i	$0.795124\pi$
$278$		0			0
$279$	23.3258			1.39648
$280$		0			0
$281$	−14.4993			−0.864955			−0.432478	−	0.901645i	$-0.642361\pi$
$281$	−14.4993			−0.864955			−0.432478	+	0.901645i	$0.642361\pi$
$282$		0			0
$283$		−	8.80606i		−	0.523466i	−0.965140	−	0.261733i	$-0.915706\pi$
$283$		−	8.80606i		−	0.523466i	0.965140	−	0.261733i	$-0.0842940\pi$
$284$		0			0
$285$	−8.52373	+	7.53690i	−0.504902	+	0.446448i
$286$		0			0
$287$			9.86414i			0.582262i
$288$		0			0
$289$	14.4010			0.847120
$290$		0			0
$291$	7.74798			0.454195
$292$		0			0
$293$			3.35026i			0.195724i	0.995200	+	0.0978622i	$0.0312004\pi$
$293$			3.35026i			0.195724i	−0.995200	+	0.0978622i	$0.968800\pi$
$294$		0			0
$295$	9.35026	+	10.5745i	0.544393	+	0.615672i
$296$		0			0
$297$			1.67276i			0.0970634i
$298$		0			0
$299$	−5.92478			−0.342639
$300$		0			0
$301$	24.8119			1.43014
$302$		0			0
$303$			9.18997i			0.527950i
$304$		0			0
$305$	−0.387873	−	0.438658i	−0.0222096	−	0.0251175i
$306$		0			0
$307$		−	1.95509i		−	0.111583i	−0.998442	−	0.0557916i	$-0.982232\pi$
$307$		−	1.95509i		−	0.111583i	0.998442	−	0.0557916i	$-0.0177682\pi$
$308$		0			0
$309$	8.81194			0.501294
$310$		0			0
$311$	−8.77575			−0.497627			−0.248813	−	0.968551i	$-0.580041\pi$
$311$	−8.77575			−0.497627			−0.248813	+	0.968551i	$0.580041\pi$
$312$		0			0
$313$			18.5501i			1.04851i	0.851561	+	0.524256i	$0.175657\pi$
$313$			18.5501i			1.04851i	−0.851561	+	0.524256i	$0.824343\pi$
$314$		0			0
$315$	−8.48944	+	7.50659i	−0.478326	+	0.422948i
$316$		0			0
$317$			9.58769i			0.538498i	0.963071	+	0.269249i	$0.0867755\pi$
$317$			9.58769i			0.538498i	−0.963071	+	0.269249i	$0.913224\pi$
$318$		0			0
$319$	−0.775746			−0.0434335
$320$		0			0
$321$	8.90175			0.496847
$322$		0			0
$323$		−	10.1768i		−	0.566252i
$324$		0			0
$325$	−4.77575	−	0.589104i	−0.264911	−	0.0326776i
$326$		0			0
$327$			4.06063i			0.224554i
$328$		0			0
$329$	9.79877			0.540224
$330$		0			0
$331$	−23.0884			−1.26905			−0.634527	−	0.772901i	$-0.718805\pi$
$331$	−23.0884			−1.26905			−0.634527	+	0.772901i	$0.718805\pi$
$332$		0			0
$333$			15.4518i			0.846755i
$334$		0			0
$335$	−16.5745	+	14.6556i	−0.905563	+	0.800722i
$336$		0			0
$337$			4.77575i			0.260151i	0.991504	+	0.130076i	$0.0415220\pi$
$337$			4.77575i			0.260151i	−0.991504	+	0.130076i	$0.958478\pi$
$338$		0			0
$339$	−16.0000			−0.869001
$340$		0			0
$341$	−3.84955			−0.208465
$342$		0			0
$343$			20.1622i			1.08866i
$344$		0			0
$345$	7.35026	+	8.31265i	0.395725	+	0.447538i
$346$		0			0
$347$			1.89446i			0.101700i	0.998706	+	0.0508500i	$0.0161930\pi$
$347$			1.89446i			0.101700i	−0.998706	+	0.0508500i	$0.983807\pi$
$348$		0			0
$349$	31.4010			1.68086			0.840430	−	0.541921i	$-0.182303\pi$
$349$	31.4010			1.68086			0.840430	+	0.541921i	$0.182303\pi$
$350$		0			0
$351$	4.15045			0.221534
$352$		0			0
$353$			12.7757i			0.679984i	0.940428	+	0.339992i	$0.110424\pi$
$353$			12.7757i			0.679984i	−0.940428	+	0.339992i	$0.889576\pi$
$354$		0			0
$355$	−15.8496	−	17.9248i	−0.841207	−	0.951348i
$356$		0			0
$357$			2.80209i			0.148303i
$358$		0			0
$359$	16.4749			0.869510			0.434755	−	0.900549i	$-0.356835\pi$
$359$	16.4749			0.869510			0.434755	+	0.900549i	$0.356835\pi$
$360$		0			0
$361$	20.8496			1.09734
$362$		0			0
$363$			8.74543i			0.459016i
$364$		0			0
$365$	21.9248	−	19.3865i	1.14760	−	1.01473i
$366$		0			0
$367$		−	10.0957i		−	0.526991i	−0.964661	−	0.263495i	$-0.915125\pi$
$367$		−	10.0957i		−	0.526991i	0.964661	−	0.263495i	$-0.0848754\pi$
$368$		0			0
$369$	10.7513			0.559691
$370$		0			0
$371$	19.3258			1.00335
$372$		0			0
$373$		−	6.82653i		−	0.353464i	−0.984259	−	0.176732i	$-0.943447\pi$
$373$		−	6.82653i		−	0.353464i	0.984259	−	0.176732i	$-0.0565527\pi$
$374$		0			0
$375$	5.09825	+	7.43136i	0.263272	+	0.383754i
$376$		0			0
$377$			1.92478i			0.0991311i
$378$		0			0
$379$	19.4617			0.999679			0.499840	−	0.866118i	$-0.333392\pi$
$379$	19.4617			0.999679			0.499840	+	0.866118i	$0.333392\pi$
$380$		0			0
$381$	−9.48612			−0.485989
$382$		0			0
$383$			5.90431i			0.301696i	0.988557	+	0.150848i	$0.0482004\pi$
$383$			5.90431i			0.301696i	−0.988557	+	0.150848i	$0.951800\pi$
$384$		0			0
$385$	1.40105	−	1.23884i	0.0714040	−	0.0631372i
$386$		0			0
$387$		−	27.0435i		−	1.37470i
$388$		0			0
$389$	14.1866			0.719291			0.359646	−	0.933089i	$-0.382898\pi$
$389$	14.1866			0.719291			0.359646	+	0.933089i	$0.382898\pi$
$390$		0			0
$391$	−9.92478			−0.501918
$392$		0			0
$393$		−	12.1622i		−	0.613502i
$394$		0			0
$395$	2.85097	+	3.22425i	0.143448	+	0.162230i
$396$		0			0
$397$			28.1866i			1.41465i	0.706890	+	0.707324i	$0.250098\pi$
$397$			28.1866i			1.41465i	−0.706890	+	0.707324i	$0.749902\pi$
$398$		0			0
$399$	−10.9722			−0.549299
$400$		0			0
$401$	6.62530			0.330852			0.165426	−	0.986222i	$-0.447100\pi$
$401$	6.62530			0.330852			0.165426	+	0.986222i	$0.447100\pi$
$402$		0			0
$403$			9.55149i			0.475794i
$404$		0			0
$405$	5.29455	+	5.98778i	0.263088	+	0.297535i
$406$		0			0
$407$		−	2.55008i		−	0.126403i
$408$		0			0
$409$	−12.6761			−0.626792			−0.313396	−	0.949623i	$-0.601467\pi$
$409$	−12.6761			−0.626792			−0.313396	+	0.949623i	$0.601467\pi$
$410$		0			0
$411$	−14.9525			−0.737554
$412$		0			0
$413$			13.6121i			0.669809i
$414$		0			0
$415$	4.82653	−	4.26774i	0.236925	−	0.209495i
$416$		0			0
$417$		−	6.13586i		−	0.300474i
$418$		0			0
$419$	14.8364			0.724805			0.362402	−	0.932022i	$-0.381957\pi$
$419$	14.8364			0.724805			0.362402	+	0.932022i	$0.381957\pi$
$420$		0			0
$421$	−0.261865			−0.0127625			−0.00638126	−	0.999980i	$-0.502031\pi$
$421$	−0.261865			−0.0127625			−0.00638126	+	0.999980i	$0.502031\pi$
$422$		0			0
$423$		−	10.6801i		−	0.519282i
$424$		0			0
$425$	−8.00000	−	0.986826i	−0.388057	−	0.0478681i
$426$		0			0
$427$		−	0.564666i		−	0.0273261i
$428$		0			0
$429$	−0.300891			−0.0145272
$430$		0			0
$431$	4.52373			0.217900			0.108950	−	0.994047i	$-0.465251\pi$
$431$	4.52373			0.217900			0.108950	+	0.994047i	$0.465251\pi$
$432$		0			0
$433$			15.6385i			0.751537i	0.926714	+	0.375769i	$0.122621\pi$
$433$			15.6385i			0.751537i	−0.926714	+	0.375769i	$0.877379\pi$
$434$		0			0
$435$	2.70052	−	2.38787i	0.129480	−	0.114490i
$436$		0			0
$437$		−	38.8627i		−	1.85906i
$438$		0			0
$439$	10.3272			0.492892			0.246446	−	0.969156i	$-0.420737\pi$
$439$	10.3272			0.492892			0.246446	+	0.969156i	$0.420737\pi$
$440$		0			0
$441$	5.52373			0.263035
$442$		0			0
$443$			15.1939i			0.721886i	0.932588	+	0.360943i	$0.117545\pi$
$443$			15.1939i			0.721886i	−0.932588	+	0.360943i	$0.882455\pi$
$444$		0			0
$445$	−15.7381	−	17.7988i	−0.746059	−	0.843743i
$446$		0			0
$447$			11.0132i			0.520905i
$448$		0			0
$449$	−24.6761			−1.16454			−0.582268	−	0.812997i	$-0.697835\pi$
$449$	−24.6761			−1.16454			−0.582268	+	0.812997i	$0.697835\pi$
$450$		0			0
$451$	−1.77433			−0.0835500
$452$		0			0
$453$			12.2717i			0.576575i
$454$		0			0
$455$	−3.07381	−	3.47627i	−0.144102	−	0.162970i
$456$		0			0
$457$			8.37328i			0.391686i	0.980635	+	0.195843i	$0.0627443\pi$
$457$			8.37328i			0.391686i	−0.980635	+	0.195843i	$0.937256\pi$
$458$		0			0
$459$	6.95254			0.324517
$460$		0			0
$461$	−8.29806			−0.386479			−0.193240	−	0.981152i	$-0.561899\pi$
$461$	−8.29806			−0.386479			−0.193240	+	0.981152i	$0.561899\pi$
$462$		0			0
$463$		−	39.4676i		−	1.83421i	−0.398642	−	0.917107i	$-0.630518\pi$
$463$		−	39.4676i		−	1.83421i	0.398642	−	0.917107i	$-0.369482\pi$
$464$		0			0
$465$	13.4010	−	11.8496i	0.621459	−	0.549510i
$466$		0			0
$467$			2.41819i			0.111901i	0.998434	+	0.0559503i	$0.0178188\pi$
$467$			2.41819i			0.111901i	−0.998434	+	0.0559503i	$0.982181\pi$
$468$		0			0
$469$	−21.3357			−0.985190
$470$		0			0
$471$	1.14903			0.0529446
$472$		0			0
$473$			4.46310i			0.205213i
$474$		0			0
$475$	3.86414	−	31.3258i	0.177299	−	1.43733i
$476$		0			0
$477$		−	21.0640i		−	0.964452i
$478$		0			0
$479$	6.44851			0.294640			0.147320	−	0.989089i	$-0.452935\pi$
$479$	6.44851			0.294640			0.147320	+	0.989089i	$0.452935\pi$
$480$		0			0
$481$	−6.32724			−0.288497
$482$		0			0
$483$			10.7005i			0.486891i
$484$		0			0
$485$	−16.1016	+	14.2374i	−0.731135	+	0.646488i
$486$		0			0
$487$			12.3331i			0.558867i	0.960165	+	0.279433i	$0.0901466\pi$
$487$			12.3331i			0.558867i	−0.960165	+	0.279433i	$0.909853\pi$
$488$		0			0
$489$	−7.09825			−0.320994
$490$		0			0
$491$	14.3127			0.645921			0.322960	−	0.946412i	$-0.395322\pi$
$491$	14.3127			0.645921			0.322960	+	0.946412i	$0.395322\pi$
$492$		0			0
$493$			3.22425i			0.145213i
$494$		0			0
$495$	−1.35026	−	1.52705i	−0.0606898	−	0.0686360i
$496$		0			0
$497$		−	23.0738i		−	1.03500i
$498$		0			0
$499$	−14.0606			−0.629440			−0.314720	−	0.949184i	$-0.601911\pi$
$499$	−14.0606			−0.629440			−0.314720	+	0.949184i	$0.601911\pi$
$500$		0			0
$501$	−8.43866			−0.377011
$502$		0			0
$503$		−	31.8700i		−	1.42101i	−0.703690	−	0.710507i	$-0.748466\pi$
$503$		−	31.8700i		−	1.42101i	0.703690	−	0.710507i	$-0.251534\pi$
$504$		0			0
$505$	−16.8872	−	19.0982i	−0.751469	−	0.849861i
$506$		0			0
$507$		−	9.73226i		−	0.432225i
$508$		0			0
$509$	26.0000			1.15243			0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000			1.15243			0.576215	+	0.817298i	$0.304529\pi$
$510$		0			0
$511$	28.2228			1.24850
$512$		0			0
$513$			27.2243i			1.20198i
$514$		0			0
$515$	−18.3127	+	16.1925i	−0.806952	+	0.713528i
$516$		0			0
$517$			1.76257i			0.0775178i
$518$		0			0
$519$	−16.1016			−0.706780
$520$		0			0
$521$	−19.4010			−0.849975			−0.424988	−	0.905199i	$-0.639722\pi$
$521$	−19.4010			−0.849975			−0.424988	+	0.905199i	$0.639722\pi$
$522$		0			0
$523$		−	7.75860i		−	0.339260i	−0.985508	−	0.169630i	$-0.945743\pi$
$523$		−	7.75860i		−	0.339260i	0.985508	−	0.169630i	$-0.0542573\pi$
$524$		0			0
$525$	−1.06396	+	8.62530i	−0.0464350	+	0.376439i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	−14.9003			−0.647841
$530$		0			0
$531$	14.8364			0.643844
$532$		0			0
$533$			4.40246i			0.190692i
$534$		0			0
$535$	−18.4993	+	16.3576i	−0.799794	+	0.707199i
$536$		0			0
$537$		−	9.86414i		−	0.425669i
$538$		0			0
$539$	−0.911603			−0.0392655
$540$		0			0
$541$	−25.8496			−1.11136			−0.555680	−	0.831397i	$-0.687542\pi$
$541$	−25.8496			−1.11136			−0.555680	+	0.831397i	$0.687542\pi$
$542$		0			0
$543$		−	11.1636i		−	0.479077i
$544$		0			0
$545$	−7.46168	−	8.43866i	−0.319623	−	0.361472i
$546$		0			0
$547$			38.2677i			1.63621i	0.575068	+	0.818105i	$0.304975\pi$
$547$			38.2677i			1.63621i	−0.575068	+	0.818105i	$0.695025\pi$
$548$		0			0
$549$	−0.615452			−0.0262668
$550$		0			0
$551$	−12.6253			−0.537856
$552$		0			0
$553$			4.15045i			0.176495i
$554$		0			0
$555$	7.84955	+	8.87732i	0.333195	+	0.376821i
$556$		0			0
$557$		−	37.5271i		−	1.59007i	−0.606562	−	0.795036i	$-0.707452\pi$
$557$		−	37.5271i		−	1.59007i	0.606562	−	0.795036i	$-0.292548\pi$
$558$		0			0
$559$	11.0738			0.468372
$560$		0			0
$561$	−0.504032			−0.0212802
$562$		0			0
$563$			43.6688i			1.84042i	0.391425	+	0.920210i	$0.371982\pi$
$563$			43.6688i			1.84042i	−0.391425	+	0.920210i	$0.628018\pi$
$564$		0			0
$565$	33.2506	−	29.4010i	1.39886	−	1.23691i
$566$		0			0
$567$			7.70782i			0.323698i
$568$		0			0
$569$	8.42407			0.353155			0.176578	−	0.984287i	$-0.443497\pi$
$569$	8.42407			0.353155			0.176578	+	0.984287i	$0.443497\pi$
$570$		0			0
$571$	24.8627			1.04047			0.520236	−	0.854022i	$-0.325844\pi$
$571$	24.8627			1.04047			0.520236	+	0.854022i	$0.325844\pi$
$572$		0			0
$573$			3.10299i			0.129629i
$574$		0			0
$575$	−30.5501	−	3.76845i	−1.27403	−	0.157155i
$576$		0			0
$577$		−	14.0263i		−	0.583924i	−0.956430	−	0.291962i	$-0.905692\pi$
$577$		−	14.0263i		−	0.583924i	0.956430	−	0.291962i	$-0.0943082\pi$
$578$		0			0
$579$	−7.74798			−0.321995
$580$		0			0
$581$	6.21299			0.257758
$582$		0			0
$583$			3.47627i			0.143972i
$584$		0			0
$585$	−3.78892	+	3.35026i	−0.156653	+	0.138516i
$586$		0			0
$587$		−	34.5804i		−	1.42729i	−0.700510	−	0.713643i	$-0.747044\pi$
$587$		−	34.5804i		−	1.42729i	0.700510	−	0.713643i	$-0.252956\pi$
$588$		0			0
$589$	−62.6516			−2.58152
$590$		0			0
$591$	−7.72829			−0.317899
$592$		0			0
$593$			8.00000i			0.328521i	0.986417	+	0.164260i	$0.0525237\pi$
$593$			8.00000i			0.328521i	−0.986417	+	0.164260i	$0.947476\pi$
$594$		0			0
$595$	−5.14903	−	5.82321i	−0.211090	−	0.238728i
$596$		0			0
$597$		−	4.27171i		−	0.174830i
$598$		0			0
$599$	−42.3996			−1.73240			−0.866201	−	0.499696i	$-0.833445\pi$
$599$	−42.3996			−1.73240			−0.866201	+	0.499696i	$0.833445\pi$
$600$		0			0
$601$	2.75131			0.112228			0.0561141	−	0.998424i	$-0.482129\pi$
$601$	2.75131			0.112228			0.0561141	+	0.998424i	$0.482129\pi$
$602$		0			0
$603$			23.2546i			0.946999i
$604$		0			0
$605$	−16.0703	−	18.1744i	−0.653351	−	0.738895i
$606$		0			0
$607$		−	37.7948i		−	1.53404i	−0.641621	−	0.767022i	$-0.721738\pi$
$607$		−	37.7948i		−	1.53404i	0.641621	−	0.767022i	$-0.278262\pi$
$608$		0			0
$609$	3.47627			0.140866
$610$		0			0
$611$	4.37328			0.176924
$612$		0			0
$613$			18.7659i			0.757947i	0.925407	+	0.378974i	$0.123723\pi$
$613$			18.7659i			0.757947i	−0.925407	+	0.378974i	$0.876277\pi$
$614$		0			0
$615$	6.17679	−	5.46168i	0.249072	−	0.220236i
$616$		0			0
$617$		−	18.6107i		−	0.749239i	−0.927179	−	0.374620i	$-0.877773\pi$
$617$		−	18.6107i		−	0.749239i	0.927179	−	0.374620i	$-0.122227\pi$
$618$		0			0
$619$	19.2097			0.772102			0.386051	−	0.922478i	$-0.373839\pi$
$619$	19.2097			0.772102			0.386051	+	0.922478i	$0.373839\pi$
$620$		0			0
$621$	26.5501			1.06542
$622$		0			0
$623$		−	22.9116i		−	0.917934i
$624$		0			0
$625$	−24.2506	−	6.07522i	−0.970024	−	0.243009i
$626$		0			0
$627$		−	1.97365i		−	0.0788201i
$628$		0			0
$629$	−10.5990			−0.422608
$630$		0			0
$631$	−26.0263			−1.03609			−0.518046	−	0.855353i	$-0.673341\pi$
$631$	−26.0263			−1.03609			−0.518046	+	0.855353i	$0.673341\pi$
$632$		0			0
$633$		−	12.1622i		−	0.483404i
$634$		0			0
$635$	19.7137	−	17.4314i	0.782314	−	0.691743i
$636$		0			0
$637$			2.26187i			0.0896184i
$638$		0			0
$639$	−25.1490			−0.994880
$640$		0			0
$641$	11.1735			0.441325			0.220663	−	0.975350i	$-0.429178\pi$
$641$	11.1735			0.441325			0.220663	+	0.975350i	$0.429178\pi$
$642$		0			0
$643$			35.1451i			1.38599i	0.720944	+	0.692993i	$0.243708\pi$
$643$			35.1451i			1.38599i	−0.720944	+	0.692993i	$0.756292\pi$
$644$		0			0
$645$	−13.7381	−	15.5369i	−0.540939	−	0.611765i
$646$		0			0
$647$			6.93207i			0.272528i	0.990673	+	0.136264i	$0.0435095\pi$
$647$			6.93207i			0.272528i	−0.990673	+	0.136264i	$0.956490\pi$
$648$		0			0
$649$	−2.44851			−0.0961123
$650$		0			0
$651$	17.2506			0.676104
$652$		0			0
$653$			29.7381i			1.16374i	0.813281	+	0.581872i	$0.197679\pi$
$653$			29.7381i			1.16374i	−0.813281	+	0.581872i	$0.802321\pi$
$654$		0			0
$655$	22.3488	+	25.2750i	0.873242	+	0.987577i
$656$		0			0
$657$		−	30.7612i		−	1.20011i
$658$		0			0
$659$	24.3879			0.950017			0.475008	−	0.879981i	$-0.342445\pi$
$659$	24.3879			0.950017			0.475008	+	0.879981i	$0.342445\pi$
$660$		0			0
$661$	−34.1378			−1.32781			−0.663903	−	0.747819i	$-0.731101\pi$
$661$	−34.1378			−1.32781			−0.663903	+	0.747819i	$0.731101\pi$
$662$		0			0
$663$			1.25060i			0.0485693i
$664$		0			0
$665$	22.8021	−	20.1622i	0.884227	−	0.781857i
$666$		0			0
$667$			12.3127i			0.476748i
$668$		0			0
$669$	18.1603			0.702118
$670$		0			0
$671$	0.101570			0.00392108
$672$		0			0
$673$		−	23.6385i		−	0.911196i	−0.890186	−	0.455598i	$-0.849425\pi$
$673$		−	23.6385i		−	0.911196i	0.890186	−	0.455598i	$-0.150575\pi$
$674$		0			0
$675$	21.4010	+	2.63989i	0.823727	+	0.101609i
$676$		0			0
$677$			2.88717i			0.110963i	0.998460	+	0.0554814i	$0.0176694\pi$
$677$			2.88717i			0.110963i	−0.998460	+	0.0554814i	$0.982331\pi$
$678$		0			0
$679$	−20.7269			−0.795424
$680$		0			0
$681$	−1.90034			−0.0728212
$682$		0			0
$683$			13.7440i			0.525900i	0.964809	+	0.262950i	$0.0846954\pi$
$683$			13.7440i			0.525900i	−0.964809	+	0.262950i	$0.915305\pi$
$684$		0			0
$685$	31.0738	−	27.4763i	1.18727	−	1.04981i
$686$		0			0
$687$		−	2.86273i		−	0.109220i
$688$		0			0
$689$	8.62530			0.328598
$690$		0			0
$691$	−31.5633			−1.20072			−0.600361	−	0.799729i	$-0.704977\pi$
$691$	−31.5633			−1.20072			−0.600361	+	0.799729i	$0.704977\pi$
$692$		0			0
$693$		−	1.96571i		−	0.0746713i
$694$		0			0
$695$	11.2750	+	12.7513i	0.427687	+	0.483685i
$696$		0			0
$697$			7.37470i			0.279337i
$698$		0			0
$699$	−10.5501			−0.399041
$700$		0			0
$701$	−23.8397			−0.900413			−0.450207	−	0.892924i	$-0.648650\pi$
$701$	−23.8397			−0.900413			−0.450207	+	0.892924i	$0.648650\pi$
$702$		0			0
$703$		−	41.5026i		−	1.56530i
$704$		0			0
$705$	−5.42548	−	6.13586i	−0.204336	−	0.231090i
$706$		0			0
$707$		−	24.5844i		−	0.924590i
$708$		0			0
$709$	21.8496			0.820577			0.410289	−	0.911956i	$-0.365428\pi$
$709$	21.8496			0.820577			0.410289	+	0.911956i	$0.365428\pi$
$710$		0			0
$711$	4.52373			0.169653
$712$		0			0
$713$			61.1002i			2.28822i
$714$		0			0
$715$	0.625301	−	0.552907i	0.0233849	−	0.0206776i
$716$		0			0
$717$			12.3536i			0.461353i
$718$		0			0
$719$	33.9248			1.26518			0.632590	−	0.774487i	$-0.281992\pi$
$719$	33.9248			1.26518			0.632590	+	0.774487i	$0.281992\pi$
$720$		0			0
$721$	−23.5731			−0.877908
$722$		0			0
$723$		−	1.71370i		−	0.0637331i
$724$		0			0
$725$	−1.22425	+	9.92478i	−0.0454676	+	0.368597i
$726$		0			0
$727$			2.78163i			0.103165i	0.998669	+	0.0515824i	$0.0164265\pi$
$727$			2.78163i			0.103165i	−0.998669	+	0.0515824i	$0.983574\pi$
$728$		0			0
$729$	−2.02776			−0.0751023
$730$		0			0
$731$	18.5501			0.686099
$732$		0			0
$733$		−	4.90175i		−	0.181050i	−0.995894	−	0.0905252i	$-0.971145\pi$
$733$		−	4.90175i		−	0.181050i	0.995894	−	0.0905252i	$-0.0288546\pi$
$734$		0			0
$735$	3.17347	−	2.80606i	0.117055	−	0.103503i
$736$		0			0
$737$		−	3.83780i		−	0.141367i
$738$		0			0
$739$	−25.9102			−0.953122			−0.476561	−	0.879141i	$-0.658117\pi$
$739$	−25.9102			−0.953122			−0.476561	+	0.879141i	$0.658117\pi$
$740$		0			0
$741$	−4.89701			−0.179896
$742$		0			0
$743$		−	9.06793i		−	0.332670i	−0.986069	−	0.166335i	$-0.946807\pi$
$743$		−	9.06793i		−	0.332670i	0.986069	−	0.166335i	$-0.0531933\pi$
$744$		0			0
$745$	−20.2374	−	22.8872i	−0.741442	−	0.838521i
$746$		0			0
$747$		−	6.77178i		−	0.247766i
$748$		0			0
$749$	−23.8134			−0.870121
$750$		0			0
$751$	−11.8496			−0.432396			−0.216198	−	0.976349i	$-0.569366\pi$
$751$	−11.8496			−0.432396			−0.216198	+	0.976349i	$0.569366\pi$
$752$		0			0
$753$			21.5106i			0.783888i
$754$		0			0
$755$	−22.5501	−	25.5026i	−0.820681	−	0.928135i
$756$		0			0
$757$		−	2.05079i		−	0.0745371i	−0.999305	−	0.0372685i	$-0.988134\pi$
$757$		−	2.05079i		−	0.0745371i	0.999305	−	0.0372685i	$-0.0118657\pi$
$758$		0			0
$759$	−1.92478			−0.0698650
$760$		0			0
$761$	25.0738			0.908925			0.454462	−	0.890766i	$-0.349831\pi$
$761$	25.0738			0.908925			0.454462	+	0.890766i	$0.349831\pi$
$762$		0			0
$763$		−	10.8627i		−	0.393257i
$764$		0			0
$765$	−6.34694	+	5.61213i	−0.229474	+	0.202907i
$766$		0			0
$767$			6.07522i			0.219364i
$768$		0			0
$769$	14.7466			0.531775			0.265887	−	0.964004i	$-0.414335\pi$
$769$	14.7466			0.531775			0.265887	+	0.964004i	$0.414335\pi$
$770$		0			0
$771$	4.35359			0.156791
$772$		0			0
$773$			10.8872i			0.391584i	0.980645	+	0.195792i	$0.0627278\pi$
$773$			10.8872i			0.391584i	−0.980645	+	0.195792i	$0.937272\pi$
$774$		0			0
$775$	−6.07522	+	49.2506i	−0.218228	+	1.76913i
$776$		0			0
$777$			11.4274i			0.409955i
$778$		0			0
$779$	−28.8773			−1.03464
$780$		0			0
$781$	4.15045			0.148515
$782$		0			0
$783$		−	8.62530i		−	0.308243i
$784$		0			0
$785$	−2.38787	+	2.11142i	−0.0852268	+	0.0753598i
$786$		0			0
$787$			36.8178i			1.31241i	0.754581	+	0.656207i	$0.227840\pi$
$787$			36.8178i			1.31241i	−0.754581	+	0.656207i	$0.772160\pi$
$788$		0			0
$789$	−11.6629			−0.415211
$790$		0			0
$791$	42.8021			1.52187
$792$		0			0
$793$		−	0.252016i		−	0.00894935i
$794$		0			0
$795$	−10.7005	−	12.1016i	−0.379508	−	0.429198i
$796$		0			0
$797$			12.8119i			0.453822i	0.973915	+	0.226911i	$0.0728627\pi$
$797$			12.8119i			0.453822i	−0.973915	+	0.226911i	$0.927137\pi$
$798$		0			0
$799$	7.32582			0.259169
$800$		0			0
$801$	−24.9722			−0.882351
$802$		0			0
$803$			5.07664i			0.179151i
$804$		0			0
$805$	−19.6629	−	22.2374i	−0.693027	−	0.783766i
$806$		0			0
$807$			22.6596i			0.797655i
$808$		0			0
$809$	−16.2981			−0.573009			−0.286505	−	0.958079i	$-0.592493\pi$
$809$	−16.2981			−0.573009			−0.286505	+	0.958079i	$0.592493\pi$
$810$		0			0
$811$	2.21108			0.0776415			0.0388208	−	0.999246i	$-0.487640\pi$
$811$	2.21108			0.0776415			0.0388208	+	0.999246i	$0.487640\pi$
$812$		0			0
$813$		−	20.0527i		−	0.703279i
$814$		0			0
$815$	14.7513	−	13.0435i	0.516716	−	0.456894i
$816$		0			0
$817$			72.6371i			2.54125i
$818$		0			0
$819$	−4.87732			−0.170427
$820$		0			0
$821$	50.6615			1.76810			0.884049	−	0.467394i	$-0.154807\pi$
$821$	50.6615			1.76810			0.884049	+	0.467394i	$0.154807\pi$
$822$		0			0
$823$		−	0.917483i		−	0.0319814i	−0.999872	−	0.0159907i	$-0.994910\pi$
$823$		−	0.917483i		−	0.0319814i	0.999872	−	0.0159907i	$-0.00509022\pi$
$824$		0			0
$825$	−1.55149	−	0.191382i	−0.0540160	−	0.00666305i
$826$		0			0
$827$			10.5198i			0.365808i	0.983131	+	0.182904i	$0.0585497\pi$
$827$			10.5198i			0.365808i	−0.983131	+	0.182904i	$0.941450\pi$
$828$		0			0
$829$	38.6907			1.34378			0.671891	−	0.740650i	$-0.265482\pi$
$829$	38.6907			1.34378			0.671891	+	0.740650i	$0.265482\pi$
$830$		0			0
$831$	16.1016			0.558557
$832$		0			0
$833$			3.78892i			0.131278i
$834$		0			0
$835$	17.5369	−	15.5066i	0.606890	−	0.536628i
$836$		0			0
$837$		−	42.8021i		−	1.47946i
$838$		0			0
$839$	10.0263			0.346148			0.173074	−	0.984909i	$-0.444630\pi$
$839$	10.0263			0.346148			0.173074	+	0.984909i	$0.444630\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			11.6873i			0.402534i
$844$		0			0
$845$	17.8837	+	20.2252i	0.615216	+	0.695768i
$846$		0			0
$847$		−	23.3952i		−	0.803867i
$848$		0			0
$849$	−7.09825			−0.243611
$850$		0			0
$851$	−40.4749			−1.38746
$852$		0			0
$853$			49.5388i			1.69618i	0.529855	+	0.848088i	$0.322246\pi$
$853$			49.5388i			1.69618i	−0.529855	+	0.848088i	$0.677754\pi$
$854$		0			0
$855$	−21.9756	−	24.8529i	−0.751548	−	0.849951i
$856$		0			0
$857$		−	43.4763i		−	1.48512i	−0.669779	−	0.742561i	$-0.733611\pi$
$857$		−	43.4763i		−	1.48512i	0.669779	−	0.742561i	$-0.266389\pi$
$858$		0			0
$859$	23.5633			0.803968			0.401984	−	0.915647i	$-0.368321\pi$
$859$	23.5633			0.803968			0.401984	+	0.915647i	$0.368321\pi$
$860$		0			0
$861$	7.95112			0.270974
$862$		0			0
$863$			13.3317i			0.453816i	0.973916	+	0.226908i	$0.0728617\pi$
$863$			13.3317i			0.453816i	−0.973916	+	0.226908i	$0.927138\pi$
$864$		0			0
$865$	33.4617	−	29.5877i	1.13773	−	1.00601i
$866$		0			0
$867$		−	11.6082i		−	0.394234i
$868$		0			0
$869$	−0.746569			−0.0253256
$870$		0			0
$871$	−9.52232			−0.322651
$872$		0			0
$873$			22.5910i			0.764590i
$874$		0			0
$875$	−13.6385	−	19.8799i	−0.461065	−	0.672062i
$876$		0			0
$877$		−	13.6483i		−	0.460871i	−0.973088	−	0.230436i	$-0.925985\pi$
$877$		−	13.6483i		−	0.460871i	0.973088	−	0.230436i	$-0.0740151\pi$
$878$		0			0
$879$	2.70052			0.0910864
$880$		0			0
$881$	16.3028			0.549255			0.274628	−	0.961551i	$-0.411445\pi$
$881$	16.3028			0.549255			0.274628	+	0.961551i	$0.411445\pi$
$882$		0			0
$883$		−	13.5818i		−	0.457064i	−0.973536	−	0.228532i	$-0.926607\pi$
$883$		−	13.5818i		−	0.457064i	0.973536	−	0.228532i	$-0.0733926\pi$
$884$		0			0
$885$	8.52373	−	7.53690i	0.286522	−	0.253350i
$886$		0			0
$887$			37.0797i			1.24501i	0.782614	+	0.622507i	$0.213886\pi$
$887$			37.0797i			1.24501i	−0.782614	+	0.622507i	$0.786114\pi$
$888$		0			0
$889$	25.3766			0.851104
$890$		0			0
$891$	−1.38646			−0.0464481
$892$		0			0
$893$			28.6859i			0.959938i
$894$		0			0
$895$	18.1260	+	20.4993i	0.605886	+	0.685216i
$896$		0			0
$897$			4.77575i			0.159458i
$898$		0			0
$899$	19.8496			0.662020
$900$		0			0
$901$	14.4485			0.481350
$902$		0			0
$903$		−	20.0000i		−	0.665558i
$904$		0			0
$905$	20.5139	+	23.1998i	0.681904	+	0.771188i
$906$		0			0
$907$		−	35.7294i		−	1.18638i	−0.805064	−	0.593188i	$-0.797869\pi$
$907$		−	35.7294i		−	1.18638i	0.805064	−	0.593188i	$-0.202131\pi$
$908$		0			0
$909$	−26.7954			−0.888749
$910$		0			0
$911$	−20.5237			−0.679982			−0.339991	−	0.940429i	$-0.610424\pi$
$911$	−20.5237			−0.679982			−0.339991	+	0.940429i	$0.610424\pi$
$912$		0			0
$913$			1.11757i			0.0369863i
$914$		0			0
$915$	−0.353586	+	0.312650i	−0.0116892	+	0.0103359i
$916$		0			0
$917$			32.5355i			1.07442i
$918$		0			0
$919$	20.9986			0.692679			0.346340	−	0.938109i	$-0.387424\pi$
$919$	20.9986			0.692679			0.346340	+	0.938109i	$0.387424\pi$
$920$		0			0
$921$	−1.57593			−0.0519287
$922$		0			0
$923$		−	10.2981i		−	0.338965i
$924$		0			0
$925$	−32.6253	−	4.02444i	−1.07271	−	0.132323i
$926$		0			0
$927$			25.6932i			0.843876i
$928$		0			0
$929$	31.5271			1.03437			0.517185	−	0.855874i	$-0.326980\pi$
$929$	31.5271			1.03437			0.517185	+	0.855874i	$0.326980\pi$
$930$		0			0
$931$	−14.8364			−0.486243
$932$		0			0
$933$			7.07381i			0.231586i
$934$		0			0
$935$	1.04746	−	0.926192i	0.0342556	−	0.0302897i
$936$		0			0
$937$			17.8641i			0.583596i	0.956480	+	0.291798i	$0.0942535\pi$
$937$			17.8641i			0.583596i	−0.956480	+	0.291798i	$0.905746\pi$
$938$		0			0
$939$	14.9525			0.487958
$940$		0			0
$941$	45.5487			1.48484			0.742422	−	0.669933i	$-0.233677\pi$
$941$	45.5487			1.48484			0.742422	+	0.669933i	$0.233677\pi$
$942$		0			0
$943$			28.1622i			0.917088i
$944$		0			0
$945$	13.7743	+	15.5778i	0.448079	+	0.506747i
$946$		0			0
$947$		−	43.2057i		−	1.40400i	−0.712179	−	0.701998i	$-0.752291\pi$
$947$		−	43.2057i		−	1.40400i	0.712179	−	0.701998i	$-0.247709\pi$
$948$		0			0
$949$	12.5961			0.408887
$950$		0			0
$951$	7.72829			0.250607
$952$		0			0
$953$			10.9722i			0.355426i	0.984082	+	0.177713i	$0.0568698\pi$
$953$			10.9722i			0.355426i	−0.984082	+	0.177713i	$0.943130\pi$
$954$		0			0
$955$	−5.70194	−	6.44851i	−0.184510	−	0.208669i
$956$		0			0
$957$			0.625301i			0.0202131i
$958$		0			0
$959$	40.0000			1.29167
$960$		0			0
$961$	67.5012			2.17746
$962$		0			0
$963$			25.9551i			0.836391i
$964$		0			0
$965$	16.1016	−	14.2374i	0.518328	−	0.458319i
$966$		0			0
$967$		−	44.1417i		−	1.41950i	−0.704452	−	0.709751i	$-0.748807\pi$
$967$		−	44.1417i		−	1.41950i	0.704452	−	0.709751i	$-0.251193\pi$
$968$		0			0
$969$	−8.20314			−0.263523
$970$		0			0
$971$	−20.1886			−0.647881			−0.323941	−	0.946077i	$-0.605008\pi$
$971$	−20.1886			−0.647881			−0.323941	+	0.946077i	$0.605008\pi$
$972$		0			0
$973$			16.4142i			0.526216i
$974$		0			0
$975$	−0.474855	+	3.84955i	−0.0152075	+	0.123284i
$976$		0			0
$977$			1.73340i			0.0554562i	0.999616	+	0.0277281i	$0.00882727\pi$
$977$			1.73340i			0.0554562i	−0.999616	+	0.0277281i	$0.991173\pi$
$978$		0			0
$979$	4.12127			0.131716
$980$		0			0
$981$	−11.8397			−0.378013
$982$		0			0
$983$			14.0059i			0.446718i	0.974736	+	0.223359i	$0.0717023\pi$
$983$			14.0059i			0.446718i	−0.974736	+	0.223359i	$0.928298\pi$
$984$		0			0
$985$	16.0606	−	14.2012i	0.511734	−	0.452489i
$986$		0			0
$987$		−	7.89843i		−	0.251410i
$988$		0			0
$989$	70.8383			2.25253
$990$		0			0
$991$	−42.8021			−1.35965			−0.679827	−	0.733373i	$-0.737945\pi$
$991$	−42.8021			−1.35965			−0.679827	+	0.733373i	$0.737945\pi$
$992$		0			0
$993$			18.6107i			0.590593i
$994$		0			0
$995$	7.84955	+	8.87732i	0.248848	+	0.281430i
$996$		0			0
$997$			42.8872i			1.35825i	0.734023	+	0.679125i	$0.237641\pi$
$997$			42.8872i			1.35825i	−0.734023	+	0.679125i	$0.762359\pi$
$998$		0			0
$999$	28.3536			0.897068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	640.2.c.a.129.3	✓	6
4.3	odd	2		640.2.c.b.129.4	yes	6
5.2	odd	4		3200.2.a.bq.1.2		3
5.3	odd	4		3200.2.a.bs.1.2		3
5.4	even	2	inner	640.2.c.a.129.4	yes	6
8.3	odd	2		640.2.c.c.129.3	yes	6
8.5	even	2		640.2.c.d.129.4	yes	6
16.3	odd	4		1280.2.f.j.129.3		6
16.5	even	4		1280.2.f.i.129.4		6
16.11	odd	4		1280.2.f.k.129.4		6
16.13	even	4		1280.2.f.l.129.3		6
20.3	even	4		3200.2.a.br.1.2		3
20.7	even	4		3200.2.a.bt.1.2		3
20.19	odd	2		640.2.c.b.129.3	yes	6
40.3	even	4		3200.2.a.bu.1.2		3
40.13	odd	4		3200.2.a.bp.1.2		3
40.19	odd	2		640.2.c.c.129.4	yes	6
40.27	even	4		3200.2.a.bo.1.2		3
40.29	even	2		640.2.c.d.129.3	yes	6
40.37	odd	4		3200.2.a.bv.1.2		3
80.19	odd	4		1280.2.f.k.129.3		6
80.29	even	4		1280.2.f.i.129.3		6
80.59	odd	4		1280.2.f.j.129.4		6
80.69	even	4		1280.2.f.l.129.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.c.a.129.3	✓	6	1.1	even	1	trivial
640.2.c.a.129.4	yes	6	5.4	even	2	inner
640.2.c.b.129.3	yes	6	20.19	odd	2
640.2.c.b.129.4	yes	6	4.3	odd	2
640.2.c.c.129.3	yes	6	8.3	odd	2
640.2.c.c.129.4	yes	6	40.19	odd	2
640.2.c.d.129.3	yes	6	40.29	even	2
640.2.c.d.129.4	yes	6	8.5	even	2
1280.2.f.i.129.3		6	80.29	even	4
1280.2.f.i.129.4		6	16.5	even	4
1280.2.f.j.129.3		6	16.3	odd	4
1280.2.f.j.129.4		6	80.59	odd	4
1280.2.f.k.129.3		6	80.19	odd	4
1280.2.f.k.129.4		6	16.11	odd	4
1280.2.f.l.129.3		6	16.13	even	4
1280.2.f.l.129.4		6	80.69	even	4
3200.2.a.bo.1.2		3	40.27	even	4
3200.2.a.bp.1.2		3	40.13	odd	4
3200.2.a.bq.1.2		3	5.2	odd	4
3200.2.a.br.1.2		3	20.3	even	4
3200.2.a.bs.1.2		3	5.3	odd	4
3200.2.a.bt.1.2		3	20.7	even	4
3200.2.a.bu.1.2		3	40.3	even	4
3200.2.a.bv.1.2		3	40.37	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:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	640.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.806063i q^{3} +(1.48119 + 1.67513i) q^{5} +2.15633i q^{7} +2.35026 q^{9} +O(q^{10})\)	\(q-0.806063i q^{3} +(1.48119 + 1.67513i) q^{5} +2.15633i q^{7} +2.35026 q^{9} -0.387873 q^{11} +0.962389i q^{13} +(1.35026 - 1.19394i) q^{15} +1.61213i q^{17} -6.31265 q^{19} +1.73813 q^{21} +6.15633i q^{23} +(-0.612127 + 4.96239i) q^{25} -4.31265i q^{27} +2.00000 q^{29} +9.92478 q^{31} +0.312650i q^{33} +(-3.61213 + 3.19394i) q^{35} +6.57452i q^{37} +0.775746 q^{39} +4.57452 q^{41} -11.5066i q^{43} +(3.48119 + 3.93700i) q^{45} -4.54420i q^{47} +2.35026 q^{49} +1.29948 q^{51} -8.96239i q^{53} +(-0.574515 - 0.649738i) q^{55} +5.08840i q^{57} +6.31265 q^{59} -0.261865 q^{61} +5.06793i q^{63} +(-1.61213 + 1.42548i) q^{65} +9.89446i q^{67} +4.96239 q^{69} -10.7005 q^{71} -13.0884i q^{73} +(4.00000 + 0.493413i) q^{75} -0.836381i q^{77} +1.92478 q^{79} +3.57452 q^{81} -2.88129i q^{83} +(-2.70052 + 2.38787i) q^{85} -1.61213i q^{87} -10.6253 q^{89} -2.07522 q^{91} -8.00000i q^{93} +(-9.35026 - 10.5745i) q^{95} +9.61213i q^{97} -0.911603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} - 12 q^{15} + 4 q^{19} - 8 q^{21} - 2 q^{25} + 12 q^{29} + 16 q^{31} - 20 q^{35} + 8 q^{39} + 4 q^{41} + 10 q^{45} - 6 q^{49} + 48 q^{51} + 20 q^{55} - 4 q^{59} - 20 q^{61} - 8 q^{65} + 8 q^{69} - 24 q^{71} + 24 q^{75} - 32 q^{79} - 2 q^{81} + 24 q^{85} + 20 q^{89} - 56 q^{91} - 36 q^{95} - 44 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 - 12 * q^15 + 4 * q^19 - 8 * q^21 - 2 * q^25 + 12 * q^29 + 16 * q^31 - 20 * q^35 + 8 * q^39 + 4 * q^41 + 10 * q^45 - 6 * q^49 + 48 * q^51 + 20 * q^55 - 4 * q^59 - 20 * q^61 - 8 * q^65 + 8 * q^69 - 24 * q^71 + 24 * q^75 - 32 * q^79 - 2 * q^81 + 24 * q^85 + 20 * q^89 - 56 * q^91 - 36 * q^95 - 44 * q^99

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