LMFDB - Embedded newform 2640.2.k.c.1871.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2640,2,Mod(1871,2640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2640, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2640.1871");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2640.k (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:	$21.0805061336$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.14786560000.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 43x^{4} + 361 $ x^8 + 43*x^4 + 361
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1871.7
Root			$-1.30038 + 1.30038i$ of defining polynomial
Character	$\chi$	$=$	2640.1871
Dual form			2640.2.k.c.1871.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 + 1.61803i) q^{3} +1.00000i q^{5} -4.20811i q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})$	\(q+(0.618034 + 1.61803i) q^{3} +1.00000i q^{5} -4.20811i q^{7} +(-2.23607 + 2.00000i) q^{9} -1.00000 q^{11} -3.83682 q^{13} +(-1.61803 + 0.618034i) q^{15} +6.20811i q^{17} +4.84342i q^{19} +(6.80887 - 2.60076i) q^{21} +5.20151 q^{23} -1.00000 q^{25} +(-4.61803 - 2.38197i) q^{27} -2.84342i q^{29} -3.20151i q^{31} +(-0.618034 - 1.61803i) q^{33} +4.20811 q^{35} -6.45078 q^{37} +(-2.37129 - 6.20811i) q^{39} -6.84342i q^{41} +7.46554i q^{43} +(-2.00000 - 2.23607i) q^{45} -6.43758 q^{47} -10.7082 q^{49} +(-10.0449 + 3.83682i) q^{51} -10.4376i q^{53} -1.00000i q^{55} +(-7.83682 + 2.99340i) q^{57} -7.96544 q^{59} -11.2015 q^{61} +(8.41622 + 9.40962i) q^{63} -3.83682i q^{65} +8.41622i q^{67} +(3.21471 + 8.41622i) q^{69} -5.18016 q^{71} -7.05154 q^{73} +(-0.618034 - 1.61803i) q^{75} +4.20811i q^{77} +15.5728i q^{79} +(1.00000 - 8.94427i) q^{81} +5.05154 q^{83} -6.20811 q^{85} +(4.60076 - 1.75733i) q^{87} -16.5620i q^{89} +16.1458i q^{91} +(5.18016 - 1.97864i) q^{93} -4.84342 q^{95} -1.29198 q^{97} +(2.23607 - 2.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3}+O(q^{10}) $ 8 * q - 4 * q^3	$ 8 q - 4 q^{3} - 8 q^{11} + 8 q^{13} - 4 q^{15} - 8 q^{25} - 28 q^{27} + 4 q^{33} - 8 q^{37} - 24 q^{39} - 16 q^{45} + 8 q^{47} - 32 q^{49} - 8 q^{51} - 24 q^{57} - 40 q^{59} - 48 q^{61} + 8 q^{71} + 8 q^{73} + 4 q^{75} + 8 q^{81} - 24 q^{83} - 16 q^{85} + 16 q^{87} - 8 q^{93} - 8 q^{95} - 24 q^{97}+O(q^{100}) $ 8 * q - 4 * q^3 - 8 * q^11 + 8 * q^13 - 4 * q^15 - 8 * q^25 - 28 * q^27 + 4 * q^33 - 8 * q^37 - 24 * q^39 - 16 * q^45 + 8 * q^47 - 32 * q^49 - 8 * q^51 - 24 * q^57 - 40 * q^59 - 48 * q^61 + 8 * q^71 + 8 * q^73 + 4 * q^75 + 8 * q^81 - 24 * q^83 - 16 * q^85 + 16 * q^87 - 8 * q^93 - 8 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$3$	0.618034	+	1.61803i	0.356822	+	0.934172i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	4.20811i		−	1.59052i	−0.606271	−	0.795258i	$-0.707335\pi$
$7$		−	4.20811i		−	1.59052i	0.606271	−	0.795258i	$-0.292665\pi$
$8$		0			0
$9$	−2.23607	+	2.00000i	−0.745356	+	0.666667i
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$	−3.83682			−1.06414			−0.532072	−	0.846699i	$-0.678586\pi$
$13$	−3.83682			−1.06414			−0.532072	+	0.846699i	$0.678586\pi$
$14$		0			0
$15$	−1.61803	+	0.618034i	−0.417775	+	0.159576i
$16$		0			0
$17$			6.20811i			1.50569i	0.658199	+	0.752844i	$0.271319\pi$
$17$			6.20811i			1.50569i	−0.658199	+	0.752844i	$0.728681\pi$
$18$		0			0
$19$			4.84342i			1.11116i	0.831464	+	0.555579i	$0.187503\pi$
$19$			4.84342i			1.11116i	−0.831464	+	0.555579i	$0.812497\pi$
$20$		0			0
$21$	6.80887	−	2.60076i	1.48582	−	0.567531i
$22$		0			0
$23$	5.20151			1.08459			0.542295	−	0.840188i	$-0.317555\pi$
$23$	5.20151			1.08459			0.542295	+	0.840188i	$0.317555\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.61803	−	2.38197i	−0.888741	−	0.458410i
$28$		0			0
$29$		−	2.84342i		−	0.528011i	−0.964521	−	0.264005i	$-0.914956\pi$
$29$		−	2.84342i		−	0.528011i	0.964521	−	0.264005i	$-0.0850436\pi$
$30$		0			0
$31$		−	3.20151i		−	0.575009i	−0.957779	−	0.287504i	$-0.907174\pi$
$31$		−	3.20151i		−	0.575009i	0.957779	−	0.287504i	$-0.0928256\pi$
$32$		0			0
$33$	−0.618034	−	1.61803i	−0.107586	−	0.281664i
$34$		0			0
$35$	4.20811			0.711301
$36$		0			0
$37$	−6.45078			−1.06050			−0.530251	−	0.847841i	$-0.677902\pi$
$37$	−6.45078			−1.06050			−0.530251	+	0.847841i	$0.677902\pi$
$38$		0			0
$39$	−2.37129	−	6.20811i	−0.379710	−	0.994093i
$40$		0			0
$41$		−	6.84342i		−	1.06876i	−0.845243	−	0.534382i	$-0.820545\pi$
$41$		−	6.84342i		−	1.06876i	0.845243	−	0.534382i	$-0.179455\pi$
$42$		0			0
$43$			7.46554i			1.13848i	0.822170	+	0.569242i	$0.192763\pi$
$43$			7.46554i			1.13848i	−0.822170	+	0.569242i	$0.807237\pi$
$44$		0			0
$45$	−2.00000	−	2.23607i	−0.298142	−	0.333333i
$46$		0			0
$47$	−6.43758			−0.939018			−0.469509	−	0.882928i	$-0.655569\pi$
$47$	−6.43758			−0.939018			−0.469509	+	0.882928i	$0.655569\pi$
$48$		0			0
$49$	−10.7082			−1.52974
$50$		0			0
$51$	−10.0449	+	3.83682i	−1.40657	+	0.537263i
$52$		0			0
$53$		−	10.4376i		−	1.43371i	−0.697221	−	0.716856i	$-0.745581\pi$
$53$		−	10.4376i		−	1.43371i	0.697221	−	0.716856i	$-0.254419\pi$
$54$		0			0
$55$		−	1.00000i		−	0.134840i
$56$		0			0
$57$	−7.83682	+	2.99340i	−1.03801	+	0.396486i
$58$		0			0
$59$	−7.96544			−1.03701			−0.518506	−	0.855074i	$-0.673512\pi$
$59$	−7.96544			−1.03701			−0.518506	+	0.855074i	$0.673512\pi$
$60$		0			0
$61$	−11.2015			−1.43421			−0.717103	−	0.696967i	$-0.754532\pi$
$61$	−11.2015			−1.43421			−0.717103	+	0.696967i	$0.754532\pi$
$62$		0			0
$63$	8.41622	+	9.40962i	1.06034	+	1.18550i
$64$		0			0
$65$		−	3.83682i		−	0.475899i
$66$		0			0
$67$			8.41622i			1.02821i	0.857729	+	0.514103i	$0.171875\pi$
$67$			8.41622i			1.02821i	−0.857729	+	0.514103i	$0.828125\pi$
$68$		0			0
$69$	3.21471	+	8.41622i	0.387006	+	1.01319i
$70$		0			0
$71$	−5.18016			−0.614771			−0.307386	−	0.951585i	$-0.599454\pi$
$71$	−5.18016			−0.614771			−0.307386	+	0.951585i	$0.599454\pi$
$72$		0			0
$73$	−7.05154			−0.825320			−0.412660	−	0.910885i	$-0.635400\pi$
$73$	−7.05154			−0.825320			−0.412660	+	0.910885i	$0.635400\pi$
$74$		0			0
$75$	−0.618034	−	1.61803i	−0.0713644	−	0.186834i
$76$		0			0
$77$			4.20811i			0.479559i
$78$		0			0
$79$			15.5728i			1.75208i	0.482241	+	0.876038i	$0.339823\pi$
$79$			15.5728i			1.75208i	−0.482241	+	0.876038i	$0.660177\pi$
$80$		0			0
$81$	1.00000	−	8.94427i	0.111111	−	0.993808i
$82$		0			0
$83$	5.05154			0.554478			0.277239	−	0.960801i	$-0.410581\pi$
$83$	5.05154			0.554478			0.277239	+	0.960801i	$0.410581\pi$
$84$		0			0
$85$	−6.20811			−0.673364
$86$		0			0
$87$	4.60076	−	1.75733i	0.493253	−	0.188406i
$88$		0			0
$89$		−	16.5620i		−	1.75557i	−0.479055	−	0.877785i	$-0.659021\pi$
$89$		−	16.5620i		−	1.75557i	0.479055	−	0.877785i	$-0.340979\pi$
$90$		0			0
$91$			16.1458i			1.69254i
$92$		0			0
$93$	5.18016	−	1.97864i	0.537157	−	0.205176i
$94$		0			0
$95$	−4.84342			−0.496925
$96$		0			0
$97$	−1.29198			−0.131181			−0.0655904	−	0.997847i	$-0.520893\pi$
$97$	−1.29198			−0.131181			−0.0655904	+	0.997847i	$0.520893\pi$
$98$		0			0
$99$	2.23607	−	2.00000i	0.224733	−	0.201008i
$100$		0			0
$101$			11.5169i			1.14597i	0.819565	+	0.572987i	$0.194215\pi$
$101$			11.5169i			1.14597i	−0.819565	+	0.572987i	$0.805785\pi$
$102$		0			0
$103$		−	8.94427i		−	0.881305i	−0.897678	−	0.440653i	$-0.854747\pi$
$103$		−	8.94427i		−	0.881305i	0.897678	−	0.440653i	$-0.145253\pi$
$104$		0			0
$105$	2.60076	+	6.80887i	0.253808	+	0.664477i
$106$		0			0
$107$	−17.9958			−1.73972			−0.869860	−	0.493298i	$-0.835791\pi$
$107$	−17.9958			−1.73972			−0.869860	+	0.493298i	$0.835791\pi$
$108$		0			0
$109$	−11.6868			−1.11940			−0.559698	−	0.828696i	$-0.689083\pi$
$109$	−11.6868			−1.11940			−0.559698	+	0.828696i	$0.689083\pi$
$110$		0			0
$111$	−3.98680	−	10.4376i	−0.378410	−	0.990691i
$112$		0			0
$113$			7.43740i			0.699651i	0.936815	+	0.349826i	$0.113759\pi$
$113$			7.43740i			0.699651i	−0.936815	+	0.349826i	$0.886241\pi$
$114$		0			0
$115$			5.20151i			0.485044i
$116$		0			0
$117$	8.57940	−	7.67365i	0.793166	−	0.709429i
$118$		0			0
$119$	26.1244			2.39482
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$	11.0729	−	4.22947i	0.998409	−	0.381358i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	12.2081i		−	1.08329i	−0.840606	−	0.541647i	$-0.817801\pi$
$127$		−	12.2081i		−	1.08329i	0.840606	−	0.541647i	$-0.182199\pi$
$128$		0			0
$129$	−12.0795	+	4.61396i	−1.06354	+	0.406236i
$130$		0			0
$131$	2.05591			0.179626			0.0898130	−	0.995959i	$-0.471373\pi$
$131$	2.05591			0.179626			0.0898130	+	0.995959i	$0.471373\pi$
$132$		0			0
$133$	20.3817			1.76731
$134$		0			0
$135$	2.38197	−	4.61803i	0.205007	−	0.397457i
$136$		0			0
$137$		−	4.40302i		−	0.376176i	−0.982152	−	0.188088i	$-0.939771\pi$
$137$		−	4.40302i		−	0.376176i	0.982152	−	0.188088i	$-0.0602290\pi$
$138$		0			0
$139$			4.42720i			0.375510i	0.982216	+	0.187755i	$0.0601211\pi$
$139$			4.42720i			0.375510i	−0.982216	+	0.187755i	$0.939879\pi$
$140$		0			0
$141$	−3.97864	−	10.4162i	−0.335062	−	0.877204i
$142$		0			0
$143$	3.83682			0.320851
$144$		0			0
$145$	2.84342			0.236133
$146$		0			0
$147$	−6.61803	−	17.3262i	−0.545846	−	1.42904i
$148$		0			0
$149$			6.88614i			0.564134i	0.959395	+	0.282067i	$0.0910201\pi$
$149$			6.88614i			0.564134i	−0.959395	+	0.282067i	$0.908980\pi$
$150$		0			0
$151$		−	8.51707i		−	0.693109i	−0.938030	−	0.346555i	$-0.887351\pi$
$151$		−	8.51707i		−	0.693109i	0.938030	−	0.346555i	$-0.112649\pi$
$152$		0			0
$153$	−12.4162	−	13.8818i	−1.00379	−	1.12227i
$154$		0			0
$155$	3.20151			0.257152
$156$		0			0
$157$	−8.14578			−0.650104			−0.325052	−	0.945696i	$-0.605382\pi$
$157$	−8.14578			−0.650104			−0.325052	+	0.945696i	$0.605382\pi$
$158$		0			0
$159$	16.8884	−	6.45078i	1.33933	−	0.511580i
$160$		0			0
$161$		−	21.8885i		−	1.72506i
$162$		0			0
$163$			12.0000i			0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$			12.0000i			0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$		0			0
$165$	1.61803	−	0.618034i	0.125964	−	0.0481139i
$166$		0			0
$167$	−2.78110			−0.215208			−0.107604	−	0.994194i	$-0.534318\pi$
$167$	−2.78110			−0.215208			−0.107604	+	0.994194i	$0.534318\pi$
$168$		0			0
$169$	1.72122			0.132401
$170$		0			0
$171$	−9.68685	−	10.8302i	−0.740772	−	0.828208i
$172$		0			0
$173$			20.3539i			1.54748i	0.633504	+	0.773739i	$0.281616\pi$
$173$			20.3539i			1.54748i	−0.633504	+	0.773739i	$0.718384\pi$
$174$		0			0
$175$			4.20811i			0.318103i
$176$		0			0
$177$	−4.92292	−	12.8884i	−0.370029	−	0.968748i
$178$		0			0
$179$	11.7296			0.876709			0.438354	−	0.898802i	$-0.355562\pi$
$179$	11.7296			0.876709			0.438354	+	0.898802i	$0.355562\pi$
$180$		0			0
$181$	−19.1242			−1.42149			−0.710747	−	0.703448i	$-0.751643\pi$
$181$	−19.1242			−1.42149			−0.710747	+	0.703448i	$0.751643\pi$
$182$		0			0
$183$	−6.92292	−	18.1244i	−0.511757	−	1.33980i
$184$		0			0
$185$		−	6.45078i		−	0.474271i
$186$		0			0
$187$		−	6.20811i		−	0.453982i
$188$		0			0
$189$	−10.0236	+	19.4332i	−0.729108	+	1.41356i
$190$		0			0
$191$	15.6177			1.13006			0.565030	−	0.825070i	$-0.308865\pi$
$191$	15.6177			1.13006			0.565030	+	0.825070i	$0.308865\pi$
$192$		0			0
$193$	−11.4206			−0.822073			−0.411036	−	0.911619i	$-0.634833\pi$
$193$	−11.4206			−0.822073			−0.411036	+	0.911619i	$0.634833\pi$
$194$		0			0
$195$	6.20811	−	2.37129i	0.444572	−	0.169811i
$196$		0			0
$197$			9.99358i			0.712013i	0.934483	+	0.356007i	$0.115862\pi$
$197$			9.99358i			0.712013i	−0.934483	+	0.356007i	$0.884138\pi$
$198$		0			0
$199$		−	1.68685i		−	0.119577i	−0.998211	−	0.0597887i	$-0.980957\pi$
$199$		−	1.68685i		−	0.119577i	0.998211	−	0.0597887i	$-0.0190427\pi$
$200$		0			0
$201$	−13.6177	+	5.20151i	−0.960521	+	0.366886i
$202$		0			0
$203$	−11.9654			−0.839810
$204$		0			0
$205$	6.84342			0.477965
$206$		0			0
$207$	−11.6309	+	10.4030i	−0.808406	+	0.723060i
$208$		0			0
$209$		−	4.84342i		−	0.335027i
$210$		0			0
$211$		−	20.1907i		−	1.38999i	−0.719016	−	0.694993i	$-0.755407\pi$
$211$		−	20.1907i		−	1.38999i	0.719016	−	0.694993i	$-0.244593\pi$
$212$		0			0
$213$	−3.20151	−	8.38167i	−0.219364	−	0.574302i
$214$		0			0
$215$	−7.46554			−0.509145
$216$		0			0
$217$	−13.4723			−0.914561
$218$		0			0
$219$	−4.35809	−	11.4096i	−0.294492	−	0.770991i
$220$		0			0
$221$		−	23.8194i		−	1.60227i
$222$		0			0
$223$			7.72140i			0.517063i	0.966003	+	0.258532i	$0.0832386\pi$
$223$			7.72140i			0.517063i	−0.966003	+	0.258532i	$0.916761\pi$
$224$		0			0
$225$	2.23607	−	2.00000i	0.149071	−	0.133333i
$226$		0			0
$227$	6.05135			0.401642			0.200821	−	0.979628i	$-0.435639\pi$
$227$	6.05135			0.401642			0.200821	+	0.979628i	$0.435639\pi$
$228$		0			0
$229$	24.8752			1.64380			0.821898	−	0.569634i	$-0.192915\pi$
$229$	24.8752			1.64380			0.821898	+	0.569634i	$0.192915\pi$
$230$		0			0
$231$	−6.80887	+	2.60076i	−0.447991	+	0.171117i
$232$		0			0
$233$			22.2980i			1.46079i	0.683025	+	0.730395i	$0.260664\pi$
$233$			22.2980i			1.46079i	−0.683025	+	0.730395i	$0.739336\pi$
$234$		0			0
$235$		−	6.43758i		−	0.419941i
$236$		0			0
$237$	−25.1973	+	9.62452i	−1.63674	+	0.625180i
$238$		0			0
$239$	−4.41622			−0.285662			−0.142831	−	0.989747i	$-0.545620\pi$
$239$	−4.41622			−0.285662			−0.142831	+	0.989747i	$0.545620\pi$
$240$		0			0
$241$	18.5061			1.19208			0.596041	−	0.802954i	$-0.296739\pi$
$241$	18.5061			1.19208			0.596041	+	0.802954i	$0.296739\pi$
$242$		0			0
$243$	15.0902	−	3.90983i	0.968035	−	0.250816i
$244$		0			0
$245$		−	10.7082i		−	0.684122i
$246$		0			0
$247$		−	18.5834i		−	1.18243i
$248$		0			0
$249$	3.12202	+	8.17356i	0.197850	+	0.517978i
$250$		0			0
$251$	28.4847			1.79794			0.898970	−	0.438009i	$-0.144316\pi$
$251$	28.4847			1.79794			0.898970	+	0.438009i	$0.144316\pi$
$252$		0			0
$253$	−5.20151			−0.327016
$254$		0			0
$255$	−3.83682	−	10.0449i	−0.240271	−	0.629038i
$256$		0			0
$257$			24.8192i			1.54818i	0.633074	+	0.774091i	$0.281793\pi$
$257$			24.8192i			1.54818i	−0.633074	+	0.774091i	$0.718207\pi$
$258$		0			0
$259$			27.1456i			1.68675i
$260$		0			0
$261$	5.68685	+	6.35809i	0.352007	+	0.393556i
$262$		0			0
$263$	23.5237			1.45053			0.725266	−	0.688469i	$-0.241717\pi$
$263$	23.5237			1.45053			0.725266	+	0.688469i	$0.241717\pi$
$264$		0			0
$265$	10.4376			0.641175
$266$		0			0
$267$	26.7979	−	10.2359i	1.64000	−	0.626426i
$268$		0			0
$269$		−	15.9309i		−	0.971323i	−0.874147	−	0.485662i	$-0.838579\pi$
$269$		−	15.9309i		−	0.971323i	0.874147	−	0.485662i	$-0.161421\pi$
$270$		0			0
$271$			21.4478i			1.30286i	0.758709	+	0.651430i	$0.225830\pi$
$271$			21.4478i			1.30286i	−0.758709	+	0.651430i	$0.774170\pi$
$272$		0			0
$273$	−26.1244	+	9.97864i	−1.58112	+	0.603935i
$274$		0			0
$275$	1.00000			0.0603023
$276$		0			0
$277$	16.3222			0.980703			0.490352	−	0.871525i	$-0.336868\pi$
$277$	16.3222			0.980703			0.490352	+	0.871525i	$0.336868\pi$
$278$		0			0
$279$	6.40302	+	7.15880i	0.383339	+	0.428586i
$280$		0			0
$281$		−	5.61551i		−	0.334993i	−0.985873	−	0.167497i	$-0.946432\pi$
$281$		−	5.61551i		−	0.334993i	0.985873	−	0.167497i	$-0.0535684\pi$
$282$		0			0
$283$			3.37566i			0.200662i	0.994954	+	0.100331i	$0.0319902\pi$
$283$			3.37566i			0.200662i	−0.994954	+	0.100331i	$0.968010\pi$
$284$		0			0
$285$	−2.99340	−	7.83682i	−0.177314	−	0.464213i
$286$		0			0
$287$	−28.7979			−1.69989
$288$		0			0
$289$	−21.5407			−1.26710
$290$		0			0
$291$	−0.798488	−	2.09047i	−0.0468082	−	0.122545i
$292$		0			0
$293$			10.6934i			0.624718i	0.949964	+	0.312359i	$0.101119\pi$
$293$			10.6934i			0.624718i	−0.949964	+	0.312359i	$0.898881\pi$
$294$		0			0
$295$		−	7.96544i		−	0.463766i
$296$		0			0
$297$	4.61803	+	2.38197i	0.267966	+	0.138216i
$298$		0			0
$299$	−19.9573			−1.15416
$300$		0			0
$301$	31.4158			1.81078
$302$		0			0
$303$	−18.6347	+	7.11783i	−1.07054	+	0.408909i
$304$		0			0
$305$		−	11.2015i		−	0.641397i
$306$		0			0
$307$			13.9377i			0.795465i	0.917501	+	0.397732i	$0.130203\pi$
$307$			13.9377i			0.795465i	−0.917501	+	0.397732i	$0.869797\pi$
$308$		0			0
$309$	14.4721	−	5.52786i	0.823291	−	0.314469i
$310$		0			0
$311$	29.8540			1.69286			0.846432	−	0.532497i	$-0.178746\pi$
$311$	29.8540			1.69286			0.846432	+	0.532497i	$0.178746\pi$
$312$		0			0
$313$	20.7847			1.17482			0.587410	−	0.809289i	$-0.300148\pi$
$313$	20.7847			1.17482			0.587410	+	0.809289i	$0.300148\pi$
$314$		0			0
$315$	−9.40962	+	8.41622i	−0.530172	+	0.474200i
$316$		0			0
$317$		−	5.90953i		−	0.331912i	−0.986133	−	0.165956i	$-0.946929\pi$
$317$		−	5.90953i		−	0.331912i	0.986133	−	0.165956i	$-0.0530710\pi$
$318$		0			0
$319$			2.84342i			0.159201i
$320$		0			0
$321$	−11.1220	−	29.1178i	−0.620771	−	1.62520i
$322$		0			0
$323$	−30.0685			−1.67306
$324$		0			0
$325$	3.83682			0.212829
$326$		0			0
$327$	−7.22287	−	18.9097i	−0.399426	−	1.04571i
$328$		0			0
$329$			27.0901i			1.49352i
$330$		0			0
$331$			6.52805i			0.358814i	0.983775	+	0.179407i	$0.0574179\pi$
$331$			6.52805i			0.358814i	−0.983775	+	0.179407i	$0.942582\pi$
$332$		0			0
$333$	14.4244	−	12.9016i	0.790451	−	0.707001i
$334$		0			0
$335$	−8.41622			−0.459827
$336$		0			0
$337$	−4.78110			−0.260443			−0.130221	−	0.991485i	$-0.541569\pi$
$337$	−4.78110			−0.260443			−0.130221	+	0.991485i	$0.541569\pi$
$338$		0			0
$339$	−12.0340	+	4.59656i	−0.653595	+	0.249651i
$340$		0			0
$341$			3.20151i			0.173372i
$342$		0			0
$343$			15.6045i			0.842566i
$344$		0			0
$345$	−8.41622	+	3.21471i	−0.453114	+	0.173074i
$346$		0			0
$347$	29.3986			1.57820			0.789101	−	0.614263i	$-0.210547\pi$
$347$	29.3986			1.57820			0.789101	+	0.614263i	$0.210547\pi$
$348$		0			0
$349$	7.42961			0.397698			0.198849	−	0.980030i	$-0.436280\pi$
$349$	7.42961			0.397698			0.198849	+	0.980030i	$0.436280\pi$
$350$		0			0
$351$	17.7186	+	9.13918i	0.945748	+	0.487814i
$352$		0			0
$353$		−	30.7555i		−	1.63695i	−0.574540	−	0.818476i	$-0.694819\pi$
$353$		−	30.7555i		−	1.63695i	0.574540	−	0.818476i	$-0.305181\pi$
$354$		0			0
$355$		−	5.18016i		−	0.274934i
$356$		0			0
$357$	16.1458	+	42.2702i	0.854526	+	2.23718i
$358$		0			0
$359$	−1.31334			−0.0693153			−0.0346576	−	0.999399i	$-0.511034\pi$
$359$	−1.31334			−0.0693153			−0.0346576	+	0.999399i	$0.511034\pi$
$360$		0			0
$361$	−4.45875			−0.234671
$362$		0			0
$363$	0.618034	+	1.61803i	0.0324384	+	0.0849248i
$364$		0			0
$365$		−	7.05154i		−	0.369094i
$366$		0			0
$367$			23.5832i			1.23103i	0.788124	+	0.615516i	$0.211052\pi$
$367$			23.5832i			1.23103i	−0.788124	+	0.615516i	$0.788948\pi$
$368$		0			0
$369$	13.6868	+	15.3024i	0.712509	+	0.796609i
$370$		0			0
$371$	−43.9225			−2.28034
$372$		0			0
$373$	−35.0723			−1.81597			−0.907987	−	0.418998i	$-0.862382\pi$
$373$	−35.0723			−1.81597			−0.907987	+	0.418998i	$0.862382\pi$
$374$		0			0
$375$	1.61803	−	0.618034i	0.0835549	−	0.0319151i
$376$		0			0
$377$			10.9097i			0.561879i
$378$		0			0
$379$		−	16.8752i		−	0.866819i	−0.901197	−	0.433409i	$-0.857310\pi$
$379$		−	16.8752i		−	0.866819i	0.901197	−	0.433409i	$-0.142690\pi$
$380$		0			0
$381$	19.7531	−	7.54503i	1.01198	−	0.386543i
$382$		0			0
$383$	−28.8752			−1.47545			−0.737726	−	0.675100i	$-0.764100\pi$
$383$	−28.8752			−1.47545			−0.737726	+	0.675100i	$0.764100\pi$
$384$		0			0
$385$	−4.20811			−0.214465
$386$		0			0
$387$	−14.9311	−	16.6934i	−0.758989	−	0.848576i
$388$		0			0
$389$		−	12.6735i		−	0.642570i	−0.946983	−	0.321285i	$-0.895885\pi$
$389$		−	12.6735i		−	0.642570i	0.946983	−	0.321285i	$-0.104115\pi$
$390$		0			0
$391$			32.2916i			1.63305i
$392$		0			0
$393$	1.27062	+	3.32654i	0.0640945	+	0.167802i
$394$		0			0
$395$	−15.5728			−0.783553
$396$		0			0
$397$	4.34771			0.218205			0.109103	−	0.994030i	$-0.465202\pi$
$397$	4.34771			0.218205			0.109103	+	0.994030i	$0.465202\pi$
$398$		0			0
$399$	12.5966	+	32.9782i	0.630617	+	1.65098i
$400$		0			0
$401$		−	29.3046i		−	1.46340i	−0.681626	−	0.731701i	$-0.738727\pi$
$401$		−	29.3046i		−	1.46340i	0.681626	−	0.731701i	$-0.261273\pi$
$402$		0			0
$403$			12.2836i			0.611892i
$404$		0			0
$405$	8.94427	+	1.00000i	0.444444	+	0.0496904i
$406$		0			0
$407$	6.45078			0.319753
$408$		0			0
$409$	−11.8885			−0.587851			−0.293925	−	0.955828i	$-0.594962\pi$
$409$	−11.8885			−0.587851			−0.293925	+	0.955828i	$0.594962\pi$
$410$		0			0
$411$	7.12424	−	2.72122i	0.351413	−	0.134228i
$412$		0			0
$413$			33.5195i			1.64939i
$414$		0			0
$415$			5.05154i			0.247970i
$416$		0			0
$417$	−7.16336	+	2.73616i	−0.350791	+	0.133990i
$418$		0			0
$419$	−11.6604			−0.569650			−0.284825	−	0.958580i	$-0.591936\pi$
$419$	−11.6604			−0.569650			−0.284825	+	0.958580i	$0.591936\pi$
$420$		0			0
$421$	31.5967			1.53993			0.769966	−	0.638085i	$-0.220273\pi$
$421$	31.5967			1.53993			0.769966	+	0.638085i	$0.220273\pi$
$422$		0			0
$423$	14.3949	−	12.8752i	0.699902	−	0.626012i
$424$		0			0
$425$		−	6.20811i		−	0.301138i
$426$		0			0
$427$			47.1372i			2.28113i
$428$		0			0
$429$	2.37129	+	6.20811i	0.114487	+	0.299730i
$430$		0			0
$431$	5.68685			0.273926			0.136963	−	0.990576i	$-0.456266\pi$
$431$	5.68685			0.273926			0.136963	+	0.990576i	$0.456266\pi$
$432$		0			0
$433$	−34.8878			−1.67660			−0.838299	−	0.545210i	$-0.816450\pi$
$433$	−34.8878			−1.67660			−0.838299	+	0.545210i	$0.816450\pi$
$434$		0			0
$435$	1.75733	+	4.60076i	0.0842576	+	0.220589i
$436$		0			0
$437$			25.1931i			1.20515i
$438$		0			0
$439$			27.0583i			1.29142i	0.763581	+	0.645712i	$0.223439\pi$
$439$			27.0583i			1.29142i	−0.763581	+	0.645712i	$0.776561\pi$
$440$		0			0
$441$	23.9443	−	21.4164i	1.14020	−	1.01983i
$442$		0			0
$443$	−26.2485			−1.24710			−0.623552	−	0.781782i	$-0.714311\pi$
$443$	−26.2485			−1.24710			−0.623552	+	0.781782i	$0.714311\pi$
$444$		0			0
$445$	16.5620			0.785114
$446$		0			0
$447$	−11.1420	+	4.25587i	−0.526999	+	0.201296i
$448$		0			0
$449$			27.0341i			1.27582i	0.770111	+	0.637910i	$0.220201\pi$
$449$			27.0341i			1.27582i	−0.770111	+	0.637910i	$0.779799\pi$
$450$		0			0
$451$			6.84342i			0.322244i
$452$		0			0
$453$	13.7809	−	5.26384i	0.647484	−	0.247317i
$454$		0			0
$455$	−16.1458			−0.756926
$456$		0			0
$457$	−27.9531			−1.30759			−0.653795	−	0.756671i	$-0.726824\pi$
$457$	−27.9531			−1.30759			−0.653795	+	0.756671i	$0.726824\pi$
$458$		0			0
$459$	14.7875	−	28.6693i	0.690222	−	1.33817i
$460$		0			0
$461$			26.1216i			1.21660i	0.793705	+	0.608302i	$0.208149\pi$
$461$			26.1216i			1.21660i	−0.793705	+	0.608302i	$0.791851\pi$
$462$		0			0
$463$		−	14.7080i		−	0.683540i	−0.939784	−	0.341770i	$-0.888974\pi$
$463$		−	14.7080i		−	0.683540i	0.939784	−	0.341770i	$-0.111026\pi$
$464$		0			0
$465$	1.97864	+	5.18016i	0.0917574	+	0.240224i
$466$		0			0
$467$	14.0767			0.651391			0.325695	−	0.945475i	$-0.394402\pi$
$467$	14.0767			0.651391			0.325695	+	0.945475i	$0.394402\pi$
$468$		0			0
$469$	35.4164			1.63538
$470$		0			0
$471$	−5.03437	−	13.1802i	−0.231972	−	0.607310i
$472$		0			0
$473$		−	7.46554i		−	0.343266i
$474$		0			0
$475$		−	4.84342i		−	0.222232i
$476$		0			0
$477$	20.8752	+	23.3391i	0.955808	+	1.06863i
$478$		0			0
$479$	16.5061			0.754183			0.377091	−	0.926176i	$-0.376924\pi$
$479$	16.5061			0.754183			0.377091	+	0.926176i	$0.376924\pi$
$480$		0			0
$481$	24.7505			1.12853
$482$		0			0
$483$	35.4164	−	13.5279i	1.61150	−	0.615539i
$484$		0			0
$485$		−	1.29198i		−	0.0586658i
$486$		0			0
$487$		−	0.999816i		−	0.0453060i	−0.999743	−	0.0226530i	$-0.992789\pi$
$487$		−	0.999816i		−	0.0453060i	0.999743	−	0.0226530i	$-0.00721129\pi$
$488$		0			0
$489$	−19.4164	+	7.41641i	−0.878040	+	0.335382i
$490$		0			0
$491$	7.93089			0.357916			0.178958	−	0.983857i	$-0.442727\pi$
$491$	7.93089			0.357916			0.178958	+	0.983857i	$0.442727\pi$
$492$		0			0
$493$	17.6523			0.795019
$494$		0			0
$495$	2.00000	+	2.23607i	0.0898933	+	0.100504i
$496$		0			0
$497$			21.7987i			0.977804i
$498$		0			0
$499$		−	27.4636i		−	1.22944i	−0.788746	−	0.614719i	$-0.789269\pi$
$499$		−	27.4636i		−	1.22944i	0.788746	−	0.614719i	$-0.210731\pi$
$500$		0			0
$501$	−1.71881	−	4.49991i	−0.0767908	−	0.201041i
$502$		0			0
$503$	−29.1206			−1.29843			−0.649213	−	0.760607i	$-0.724902\pi$
$503$	−29.1206			−1.29843			−0.649213	+	0.760607i	$0.724902\pi$
$504$		0			0
$505$	−11.5169			−0.512495
$506$		0			0
$507$	1.06377	+	2.78499i	0.0472438	+	0.123686i
$508$		0			0
$509$		−	20.8060i		−	0.922212i	−0.887345	−	0.461106i	$-0.847453\pi$
$509$		−	20.8060i		−	0.922212i	0.887345	−	0.461106i	$-0.152547\pi$
$510$		0			0
$511$			29.6736i			1.31268i
$512$		0			0
$513$	11.5369	−	22.3671i	0.509365	−	0.987531i
$514$		0			0
$515$	8.94427			0.394132
$516$		0			0
$517$	6.43758			0.283124
$518$		0			0
$519$	−32.9333	+	12.5794i	−1.44561	+	0.552174i
$520$		0			0
$521$			3.38964i			0.148503i	0.997240	+	0.0742514i	$0.0236567\pi$
$521$			3.38964i			0.148503i	−0.997240	+	0.0742514i	$0.976343\pi$
$522$		0			0
$523$		−	3.39642i		−	0.148515i	−0.997239	−	0.0742576i	$-0.976341\pi$
$523$		−	3.39642i		−	0.148515i	0.997239	−	0.0742576i	$-0.0236587\pi$
$524$		0			0
$525$	−6.80887	+	2.60076i	−0.297163	+	0.113506i
$526$		0			0
$527$	19.8753			0.865784
$528$		0			0
$529$	4.05573			0.176336
$530$		0			0
$531$	17.8113	−	15.9309i	0.772943	−	0.691342i
$532$		0			0
$533$			26.2570i			1.13732i
$534$		0			0
$535$		−	17.9958i		−	0.778027i
$536$		0			0
$537$	7.24927	+	18.9788i	0.312829	+	0.818997i
$538$		0			0
$539$	10.7082			0.461235
$540$		0			0
$541$	15.7296			0.676267			0.338133	−	0.941098i	$-0.390205\pi$
$541$	15.7296			0.676267			0.338133	+	0.941098i	$0.390205\pi$
$542$		0			0
$543$	−11.8194	−	30.9437i	−0.507220	−	1.32792i
$544$		0			0
$545$		−	11.6868i		−	0.500610i
$546$		0			0
$547$			12.0491i			0.515184i	0.966254	+	0.257592i	$0.0829290\pi$
$547$			12.0491i			0.515184i	−0.966254	+	0.257592i	$0.917071\pi$
$548$		0			0
$549$	25.0473	−	22.4030i	1.06899	−	0.956138i
$550$		0			0
$551$	13.7719			0.586703
$552$		0			0
$553$	65.5321			2.78671
$554$		0			0
$555$	10.4376	−	3.98680i	0.443051	−	0.169230i
$556$		0			0
$557$		−	24.3803i		−	1.03303i	−0.856279	−	0.516513i	$-0.827230\pi$
$557$		−	24.3803i		−	1.03303i	0.856279	−	0.516513i	$-0.172770\pi$
$558$		0			0
$559$		−	28.6439i		−	1.21151i
$560$		0			0
$561$	10.0449	−	3.83682i	0.424098	−	0.161991i
$562$		0			0
$563$	−12.6353			−0.532515			−0.266257	−	0.963902i	$-0.585787\pi$
$563$	−12.6353			−0.532515			−0.266257	+	0.963902i	$0.585787\pi$
$564$		0			0
$565$	−7.43740			−0.312894
$566$		0			0
$567$	−37.6385	−	4.20811i	−1.58067	−	0.176724i
$568$		0			0
$569$			14.6289i			0.613275i	0.951826	+	0.306638i	$0.0992040\pi$
$569$			14.6289i			0.613275i	−0.951826	+	0.306638i	$0.900796\pi$
$570$		0			0
$571$		−	14.4140i		−	0.603207i	−0.953433	−	0.301604i	$-0.902478\pi$
$571$		−	14.4140i		−	0.603207i	0.953433	−	0.301604i	$-0.0975219\pi$
$572$		0			0
$573$	9.65229	+	25.2700i	0.403230	+	1.05567i
$574$		0			0
$575$	−5.20151			−0.216918
$576$		0			0
$577$	−35.9648			−1.49724			−0.748618	−	0.663001i	$-0.769282\pi$
$577$	−35.9648			−1.49724			−0.748618	+	0.663001i	$0.769282\pi$
$578$		0			0
$579$	−7.05832	−	18.4789i	−0.293334	−	0.767958i
$580$		0			0
$581$		−	21.2574i		−	0.881907i
$582$		0			0
$583$			10.4376i			0.432280i
$584$		0			0
$585$	7.67365	+	8.57940i	0.317266	+	0.354715i
$586$		0			0
$587$	−11.6309			−0.480060			−0.240030	−	0.970765i	$-0.577157\pi$
$587$	−11.6309			−0.480060			−0.240030	+	0.970765i	$0.577157\pi$
$588$		0			0
$589$	15.5063			0.638925
$590$		0			0
$591$	−16.1700	+	6.17638i	−0.665143	+	0.254062i
$592$		0			0
$593$			4.81847i			0.197871i	0.995094	+	0.0989354i	$0.0315437\pi$
$593$			4.81847i			0.197871i	−0.995094	+	0.0989354i	$0.968456\pi$
$594$		0			0
$595$			26.1244i			1.07100i
$596$		0			0
$597$	2.72938	−	1.04253i	0.111706	−	0.0426679i
$598$		0			0
$599$	20.0000			0.817178			0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000			0.817178			0.408589	+	0.912719i	$0.366021\pi$
$600$		0			0
$601$	5.00018			0.203962			0.101981	−	0.994786i	$-0.467482\pi$
$601$	5.00018			0.203962			0.101981	+	0.994786i	$0.467482\pi$
$602$		0			0
$603$	−16.8324	−	18.8192i	−0.685470	−	0.766379i
$604$		0			0
$605$			1.00000i			0.0406558i
$606$		0			0
$607$		−	23.0274i		−	0.934652i	−0.884085	−	0.467326i	$-0.845217\pi$
$607$		−	23.0274i		−	0.934652i	0.884085	−	0.467326i	$-0.154783\pi$
$608$		0			0
$609$	−7.39505	−	19.3605i	−0.299663	−	0.784527i
$610$		0			0
$611$	24.6999			0.999249
$612$		0			0
$613$	−21.6691			−0.875206			−0.437603	−	0.899168i	$-0.644172\pi$
$613$	−21.6691			−0.875206			−0.437603	+	0.899168i	$0.644172\pi$
$614$		0			0
$615$	4.22947	+	11.0729i	0.170549	+	0.446502i
$616$		0			0
$617$			21.3523i			0.859613i	0.902921	+	0.429806i	$0.141418\pi$
$617$			21.3523i			0.859613i	−0.902921	+	0.429806i	$0.858582\pi$
$618$		0			0
$619$			22.8887i			0.919976i	0.887925	+	0.459988i	$0.152146\pi$
$619$			22.8887i			0.919976i	−0.887925	+	0.459988i	$0.847854\pi$
$620$		0			0
$621$	−24.0208	−	12.3898i	−0.963920	−	0.497187i
$622$		0			0
$623$	−69.6948			−2.79226
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	7.83682	−	2.99340i	0.312973	−	0.119545i
$628$		0			0
$629$		−	40.0472i		−	1.59678i
$630$		0			0
$631$			12.8117i			0.510025i	0.966938	+	0.255013i	$0.0820796\pi$
$631$			12.8117i			0.510025i	−0.966938	+	0.255013i	$0.917920\pi$
$632$		0			0
$633$	32.6693	−	12.4786i	1.29849	−	0.495978i
$634$		0			0
$635$	12.2081			0.484464
$636$		0			0
$637$	41.0855			1.62787
$638$		0			0
$639$	11.5832	−	10.3603i	0.458224	−	0.409848i
$640$		0			0
$641$		−	37.5490i		−	1.48310i	−0.670900	−	0.741548i	$-0.734092\pi$
$641$		−	37.5490i		−	1.48310i	0.670900	−	0.741548i	$-0.265908\pi$
$642$		0			0
$643$			9.45875i			0.373017i	0.982453	+	0.186508i	$0.0597172\pi$
$643$			9.45875i			0.373017i	−0.982453	+	0.186508i	$0.940283\pi$
$644$		0			0
$645$	−4.61396	−	12.0795i	−0.181674	−	0.475630i
$646$		0			0
$647$	17.8194			0.700554			0.350277	−	0.936646i	$-0.386087\pi$
$647$	17.8194			0.700554			0.350277	+	0.936646i	$0.386087\pi$
$648$		0			0
$649$	7.96544			0.312671
$650$		0			0
$651$	−8.32635	−	21.7987i	−0.326335	−	0.854357i
$652$		0			0
$653$		−	13.4028i		−	0.524494i	−0.965001	−	0.262247i	$-0.915537\pi$
$653$		−	13.4028i		−	0.524494i	0.965001	−	0.262247i	$-0.0844635\pi$
$654$		0			0
$655$			2.05591i			0.0803312i
$656$		0			0
$657$	15.7677	−	14.1031i	0.615157	−	0.550213i
$658$		0			0
$659$	25.7076			1.00143			0.500713	−	0.865613i	$-0.333071\pi$
$659$	25.7076			1.00143			0.500713	+	0.865613i	$0.333071\pi$
$660$		0			0
$661$	−29.7500			−1.15714			−0.578569	−	0.815633i	$-0.696389\pi$
$661$	−29.7500			−1.15714			−0.578569	+	0.815633i	$0.696389\pi$
$662$		0			0
$663$	38.5407	−	14.7212i	1.49679	−	0.571725i
$664$		0			0
$665$			20.3817i			0.790367i
$666$		0			0
$667$		−	14.7901i		−	0.572675i
$668$		0			0
$669$	−12.4935	+	4.77209i	−0.483026	+	0.184500i
$670$		0			0
$671$	11.2015			0.432430
$672$		0			0
$673$	−14.3354			−0.552587			−0.276294	−	0.961073i	$-0.589106\pi$
$673$	−14.3354			−0.552587			−0.276294	+	0.961073i	$0.589106\pi$
$674$		0			0
$675$	4.61803	+	2.38197i	0.177748	+	0.0916819i
$676$		0			0
$677$			24.3675i			0.936518i	0.883591	+	0.468259i	$0.155119\pi$
$677$			24.3675i			0.936518i	−0.883591	+	0.468259i	$0.844881\pi$
$678$		0			0
$679$			5.43680i			0.208645i
$680$		0			0
$681$	3.73994	+	9.79129i	0.143315	+	0.375203i
$682$		0			0
$683$	46.7128			1.78742			0.893708	−	0.448648i	$-0.148094\pi$
$683$	46.7128			1.78742			0.893708	+	0.448648i	$0.148094\pi$
$684$		0			0
$685$	4.40302			0.168231
$686$		0			0
$687$	15.3737	+	40.2489i	0.586543	+	1.53559i
$688$		0			0
$689$			40.0472i			1.52568i
$690$		0			0
$691$		−	8.90156i		−	0.338631i	−0.985562	−	0.169316i	$-0.945844\pi$
$691$		−	8.90156i		−	0.338631i	0.985562	−	0.169316i	$-0.0541557\pi$
$692$		0			0
$693$	−8.41622	−	9.40962i	−0.319706	−	0.357442i
$694$		0			0
$695$	−4.42720			−0.167933
$696$		0			0
$697$	42.4847			1.60922
$698$		0			0
$699$	−36.0789	+	13.7809i	−1.36463	+	0.521242i
$700$		0			0
$701$		−	2.01098i		−	0.0759535i	−0.999279	−	0.0379768i	$-0.987909\pi$
$701$		−	2.01098i		−	0.0759535i	0.999279	−	0.0379768i	$-0.0120913\pi$
$702$		0			0
$703$		−	31.2439i		−	1.17838i
$704$		0			0
$705$	10.4162	−	3.97864i	0.392298	−	0.149844i
$706$		0			0
$707$	48.4643			1.82269
$708$		0			0
$709$	−37.4585			−1.40678			−0.703392	−	0.710802i	$-0.748332\pi$
$709$	−37.4585			−1.40678			−0.703392	+	0.710802i	$0.748332\pi$
$710$		0			0
$711$	−31.1456	−	34.8218i	−1.16805	−	1.30592i
$712$		0			0
$713$		−	16.6527i		−	0.623649i
$714$		0			0
$715$			3.83682i			0.143489i
$716$		0			0
$717$	−2.72938	−	7.14560i	−0.101930	−	0.266857i
$718$		0			0
$719$	−42.3387			−1.57897			−0.789484	−	0.613771i	$-0.789652\pi$
$719$	−42.3387			−1.57897			−0.789484	+	0.613771i	$0.789652\pi$
$720$		0			0
$721$	−37.6385			−1.40173
$722$		0			0
$723$	11.4374	+	29.9435i	0.425361	+	1.11361i
$724$		0			0
$725$			2.84342i			0.105602i
$726$		0			0
$727$			16.7212i			0.620156i	0.950711	+	0.310078i	$0.100355\pi$
$727$			16.7212i			0.620156i	−0.950711	+	0.310078i	$0.899645\pi$
$728$		0			0
$729$	15.6525	+	22.0000i	0.579721	+	0.814815i
$730$		0			0
$731$	−46.3469			−1.71420
$732$		0			0
$733$	29.8927			1.10411			0.552057	−	0.833807i	$-0.313843\pi$
$733$	29.8927			1.10411			0.552057	+	0.833807i	$0.313843\pi$
$734$		0			0
$735$	17.3262	−	6.61803i	0.639088	−	0.244110i
$736$		0			0
$737$		−	8.41622i		−	0.310016i
$738$		0			0
$739$			2.84305i			0.104583i	0.998632	+	0.0522917i	$0.0166526\pi$
$739$			2.84305i			0.104583i	−0.998632	+	0.0522917i	$0.983347\pi$
$740$		0			0
$741$	30.0685	−	11.4852i	1.10459	−	0.421918i
$742$		0			0
$743$	35.8444			1.31500			0.657501	−	0.753453i	$-0.271613\pi$
$743$	35.8444			1.31500			0.657501	+	0.753453i	$0.271613\pi$
$744$		0			0
$745$	−6.88614			−0.252289
$746$		0			0
$747$	−11.2956	+	10.1031i	−0.413284	+	0.369652i
$748$		0			0
$749$			75.7284i			2.76705i
$750$		0			0
$751$			5.03414i			0.183698i	0.995773	+	0.0918492i	$0.0292778\pi$
$751$			5.03414i			0.183698i	−0.995773	+	0.0918492i	$0.970722\pi$
$752$		0			0
$753$	17.6045	+	46.0893i	0.641545	+	1.67959i
$754$		0			0
$755$	8.51707			0.309968
$756$		0			0
$757$	0.841201			0.0305740			0.0152870	−	0.999883i	$-0.495134\pi$
$757$	0.841201			0.0305740			0.0152870	+	0.999883i	$0.495134\pi$
$758$		0			0
$759$	−3.21471	−	8.41622i	−0.116687	−	0.305490i
$760$		0			0
$761$		−	28.8083i		−	1.04430i	−0.852854	−	0.522149i	$-0.825130\pi$
$761$		−	28.8083i		−	1.04430i	0.852854	−	0.522149i	$-0.174870\pi$
$762$		0			0
$763$			49.1796i			1.78042i
$764$		0			0
$765$	13.8818	−	12.4162i	0.501896	−	0.448910i
$766$		0			0
$767$	30.5620			1.10353
$768$		0			0
$769$	−14.6870			−0.529628			−0.264814	−	0.964300i	$-0.585311\pi$
$769$	−14.6870			−0.529628			−0.264814	+	0.964300i	$0.585311\pi$
$770$		0			0
$771$	−40.1584	+	15.3391i	−1.44627	+	0.552426i
$772$		0			0
$773$			32.3475i			1.16346i	0.813383	+	0.581729i	$0.197624\pi$
$773$			32.3475i			1.16346i	−0.813383	+	0.581729i	$0.802376\pi$
$774$		0			0
$775$			3.20151i			0.115002i
$776$		0			0
$777$	−43.9225	+	16.7769i	−1.57571	+	0.601868i
$778$		0			0
$779$	33.1456			1.18756
$780$		0			0
$781$	5.18016			0.185361
$782$		0			0
$783$	−6.77294	+	13.1310i	−0.242045	+	0.469265i
$784$		0			0
$785$		−	8.14578i		−	0.290735i
$786$		0			0
$787$		−	25.7703i		−	0.918612i	−0.888278	−	0.459306i	$-0.848098\pi$
$787$		−	25.7703i		−	0.918612i	0.888278	−	0.459306i	$-0.151902\pi$
$788$		0			0
$789$	14.5384	+	38.0621i	0.517582	+	1.35505i
$790$		0			0
$791$	31.2974			1.11281
$792$		0			0
$793$	42.9782			1.52620
$794$		0			0
$795$	6.45078	+	16.8884i	0.228786	+	0.598968i
$796$		0			0
$797$			44.7851i			1.58637i	0.608982	+	0.793184i	$0.291578\pi$
$797$			44.7851i			1.58637i	−0.608982	+	0.793184i	$0.708422\pi$
$798$		0			0
$799$		−	39.9652i		−	1.41387i
$800$		0			0
$801$	33.1240	+	37.0338i	1.17038	+	1.30852i
$802$		0			0
$803$	7.05154			0.248843
$804$		0			0
$805$	21.8885			0.771470
$806$		0			0
$807$	25.7767	−	9.84583i	0.907383	−	0.346590i
$808$		0			0
$809$			24.6421i			0.866370i	0.901305	+	0.433185i	$0.142610\pi$
$809$			24.6421i			0.866370i	−0.901305	+	0.433185i	$0.857390\pi$
$810$		0			0
$811$			9.35790i			0.328600i	0.986410	+	0.164300i	$0.0525366\pi$
$811$			9.35790i			0.328600i	−0.986410	+	0.164300i	$0.947463\pi$
$812$		0			0
$813$	−34.7032	+	13.2555i	−1.21710	+	0.464889i
$814$		0			0
$815$	−12.0000			−0.420342
$816$		0			0
$817$	−36.1588			−1.26503
$818$		0			0
$819$	−32.2916	−	36.1031i	−1.12836	−	1.26154i
$820$		0			0
$821$		−	18.8434i		−	0.657640i	−0.944393	−	0.328820i	$-0.893349\pi$
$821$		−	18.8434i		−	0.657640i	0.944393	−	0.328820i	$-0.106651\pi$
$822$		0			0
$823$		−	10.5412i		−	0.367445i	−0.982978	−	0.183722i	$-0.941185\pi$
$823$		−	10.5412i		−	0.367445i	0.982978	−	0.183722i	$-0.0588147\pi$
$824$		0			0
$825$	0.618034	+	1.61803i	0.0215172	+	0.0563327i
$826$		0			0
$827$	−19.7681			−0.687404			−0.343702	−	0.939079i	$-0.611681\pi$
$827$	−19.7681			−0.687404			−0.343702	+	0.939079i	$0.611681\pi$
$828$		0			0
$829$	−34.1376			−1.18565			−0.592824	−	0.805332i	$-0.701987\pi$
$829$	−34.1376			−1.18565			−0.592824	+	0.805332i	$0.701987\pi$
$830$		0			0
$831$	10.0876	+	26.4098i	0.349937	+	0.916146i
$832$		0			0
$833$		−	66.4777i		−	2.30332i
$834$		0			0
$835$		−	2.78110i		−	0.0962438i
$836$		0			0
$837$	−7.62589	+	14.7847i	−0.263589	+	0.511034i
$838$		0			0
$839$	−9.33877			−0.322410			−0.161205	−	0.986921i	$-0.551538\pi$
$839$	−9.33877			−0.322410			−0.161205	+	0.986921i	$0.551538\pi$
$840$		0			0
$841$	20.9149			0.721205
$842$		0			0
$843$	9.08609	−	3.47058i	0.312942	−	0.119533i
$844$		0			0
$845$			1.72122i			0.0592117i
$846$		0			0
$847$		−	4.20811i		−	0.144592i
$848$		0			0
$849$	−5.46194	+	2.08628i	−0.187453	+	0.0716008i
$850$		0			0
$851$	−33.5538			−1.15021
$852$		0			0
$853$	24.2738			0.831119			0.415560	−	0.909566i	$-0.363586\pi$
$853$	24.2738			0.831119			0.415560	+	0.909566i	$0.363586\pi$
$854$		0			0
$855$	10.8302	−	9.68685i	0.370386	−	0.331283i
$856$		0			0
$857$			44.6583i			1.52550i	0.646694	+	0.762749i	$0.276151\pi$
$857$			44.6583i			1.52550i	−0.646694	+	0.762749i	$0.723849\pi$
$858$		0			0
$859$			6.37351i			0.217461i	0.994071	+	0.108731i	$0.0346786\pi$
$859$			6.37351i			0.217461i	−0.994071	+	0.108731i	$0.965321\pi$
$860$		0			0
$861$	−17.7981	−	46.5960i	−0.606557	−	1.58799i
$862$		0			0
$863$	−19.4428			−0.661841			−0.330920	−	0.943659i	$-0.607359\pi$
$863$	−19.4428			−0.661841			−0.330920	+	0.943659i	$0.607359\pi$
$864$		0			0
$865$	−20.3539			−0.692053
$866$		0			0
$867$	−13.3129	−	34.8535i	−0.452128	−	1.18369i
$868$		0			0
$869$		−	15.5728i		−	0.528271i
$870$		0			0
$871$		−	32.2916i		−	1.09416i
$872$		0			0
$873$	2.88896	−	2.58396i	0.0977764	−	0.0874538i
$874$		0			0
$875$	−4.20811			−0.142260
$876$		0			0
$877$	−45.2316			−1.52736			−0.763682	−	0.645592i	$-0.776610\pi$
$877$	−45.2316			−1.52736			−0.763682	+	0.645592i	$0.776610\pi$
$878$		0			0
$879$	−17.3024	+	6.60891i	−0.583594	+	0.222913i
$880$		0			0
$881$		−	17.1033i		−	0.576223i	−0.957597	−	0.288112i	$-0.906973\pi$
$881$		−	17.1033i		−	0.576223i	0.957597	−	0.288112i	$-0.0930274\pi$
$882$		0			0
$883$			41.8879i			1.40964i	0.709385	+	0.704821i	$0.248973\pi$
$883$			41.8879i			1.40964i	−0.709385	+	0.704821i	$0.751027\pi$
$884$		0			0
$885$	12.8884	−	4.92292i	0.433237	−	0.165482i
$886$		0			0
$887$	−38.0857			−1.27879			−0.639396	−	0.768878i	$-0.720816\pi$
$887$	−38.0857			−1.27879			−0.639396	+	0.768878i	$0.720816\pi$
$888$		0			0
$889$	−51.3731			−1.72300
$890$		0			0
$891$	−1.00000	+	8.94427i	−0.0335013	+	0.299644i
$892$		0			0
$893$		−	31.1799i		−	1.04340i
$894$		0			0
$895$			11.7296i			0.392076i
$896$		0			0
$897$	−12.3343	−	32.2916i	−0.411830	−	1.07818i
$898$		0			0
$899$	−9.10326			−0.303611
$900$		0			0
$901$	64.7977			2.15872
$902$		0			0
$903$	19.4160	+	50.8318i	0.646125	+	1.69158i
$904$		0			0
$905$		−	19.1242i		−	0.635711i
$906$		0			0
$907$			14.6816i			0.487495i	0.969839	+	0.243748i	$0.0783769\pi$
$907$			14.6816i			0.487495i	−0.969839	+	0.243748i	$0.921623\pi$
$908$		0			0
$909$	−23.0338	−	25.7525i	−0.763982	−	0.854158i
$910$		0			0
$911$	21.6872			0.718530			0.359265	−	0.933236i	$-0.383028\pi$
$911$	21.6872			0.718530			0.359265	+	0.933236i	$0.383028\pi$
$912$		0			0
$913$	−5.05154			−0.167181
$914$		0			0
$915$	18.1244	−	6.92292i	0.599175	−	0.228865i
$916$		0			0
$917$		−	8.65151i		−	0.285698i
$918$		0			0
$919$		−	52.0525i		−	1.71705i	−0.512769	−	0.858527i	$-0.671380\pi$
$919$		−	52.0525i		−	1.71705i	0.512769	−	0.858527i	$-0.328620\pi$
$920$		0			0
$921$	−22.5516	+	8.61396i	−0.743101	+	0.283839i
$922$		0			0
$923$	19.8753			0.654205
$924$		0			0
$925$	6.45078			0.212100
$926$		0			0
$927$	17.8885	+	20.0000i	0.587537	+	0.656886i
$928$		0			0
$929$			23.3945i			0.767547i	0.923427	+	0.383774i	$0.125376\pi$
$929$			23.3945i			0.767547i	−0.923427	+	0.383774i	$0.874624\pi$
$930$		0			0
$931$		−	51.8644i		−	1.69979i
$932$		0			0
$933$	18.4508	+	48.3048i	0.604051	+	1.58143i
$934$		0			0
$935$	6.20811			0.203027
$936$		0			0
$937$	48.5873			1.58728			0.793639	−	0.608388i	$-0.208184\pi$
$937$	48.5873			1.58728			0.793639	+	0.608388i	$0.208184\pi$
$938$		0			0
$939$	12.8456	+	33.6303i	0.419202	+	1.09748i
$940$		0			0
$941$		−	21.2936i		−	0.694152i	−0.937837	−	0.347076i	$-0.887175\pi$
$941$		−	21.2936i		−	0.694152i	0.937837	−	0.347076i	$-0.112825\pi$
$942$		0			0
$943$		−	35.5962i		−	1.15917i
$944$		0			0
$945$	−19.4332	−	10.0236i	−0.632162	−	0.326067i
$946$		0			0
$947$	47.8116			1.55367			0.776835	−	0.629704i	$-0.216824\pi$
$947$	47.8116			1.55367			0.776835	+	0.629704i	$0.216824\pi$
$948$		0			0
$949$	27.0555			0.878259
$950$		0			0
$951$	9.56182	−	3.65229i	0.310063	−	0.118434i
$952$		0			0
$953$		−	4.42301i		−	0.143275i	−0.997431	−	0.0716376i	$-0.977177\pi$
$953$		−	4.42301i		−	0.143275i	0.997431	−	0.0716376i	$-0.0228225\pi$
$954$		0			0
$955$			15.6177i			0.505378i
$956$		0			0
$957$	−4.60076	+	1.75733i	−0.148721	+	0.0568065i
$958$		0			0
$959$	−18.5284			−0.598314
$960$		0			0
$961$	20.7503			0.669365
$962$		0			0
$963$	40.2398	−	35.9916i	1.29671	−	1.15981i
$964$		0			0
$965$		−	11.4206i		−	0.367642i
$966$		0			0
$967$			40.4441i			1.30060i	0.759679	+	0.650298i	$0.225356\pi$
$967$			40.4441i			1.30060i	−0.759679	+	0.650298i	$0.774644\pi$
$968$		0			0
$969$	−18.5834	−	48.6519i	−0.596984	−	1.56292i
$970$		0			0
$971$	−2.31760			−0.0743752			−0.0371876	−	0.999308i	$-0.511840\pi$
$971$	−2.31760			−0.0743752			−0.0371876	+	0.999308i	$0.511840\pi$
$972$		0			0
$973$	18.6302			0.597255
$974$		0			0
$975$	2.37129	+	6.20811i	0.0759420	+	0.198819i
$976$		0			0
$977$			30.1106i			0.963324i	0.876357	+	0.481662i	$0.159967\pi$
$977$			30.1106i			0.963324i	−0.876357	+	0.481662i	$0.840033\pi$
$978$		0			0
$979$			16.5620i			0.529324i
$980$		0			0
$981$	26.1326	−	23.3737i	0.834349	−	0.746265i
$982$		0			0
$983$	−19.5282			−0.622854			−0.311427	−	0.950270i	$-0.600807\pi$
$983$	−19.5282			−0.622854			−0.311427	+	0.950270i	$0.600807\pi$
$984$		0			0
$985$	−9.99358			−0.318422
$986$		0			0
$987$	−43.8326	+	16.7426i	−1.39521	+	0.532922i
$988$		0			0
$989$			38.8321i			1.23479i
$990$		0			0
$991$		−	36.0340i		−	1.14466i	−0.820025	−	0.572328i	$-0.806040\pi$
$991$		−	36.0340i		−	1.14466i	0.820025	−	0.572328i	$-0.193960\pi$
$992$		0			0
$993$	−10.5626	+	4.03456i	−0.335194	+	0.128033i
$994$		0			0
$995$	1.68685			0.0534767
$996$		0			0
$997$	15.4278			0.488603			0.244302	−	0.969699i	$-0.421441\pi$
$997$	15.4278			0.488603			0.244302	+	0.969699i	$0.421441\pi$
$998$		0			0
$999$	29.7899	+	15.3655i	0.942511	+	0.486144i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.k.c.1871.7	yes	8
3.2	odd	2		2640.2.k.e.1871.3	yes	8
4.3	odd	2		2640.2.k.e.1871.2	yes	8
12.11	even	2	inner	2640.2.k.c.1871.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2640.2.k.c.1871.6	✓	8	12.11	even	2	inner
2640.2.k.c.1871.7	yes	8	1.1	even	1	trivial
2640.2.k.e.1871.2	yes	8	4.3	odd	2
2640.2.k.e.1871.3	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2640.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 + 1.61803i) q^{3} +1.00000i q^{5} -4.20811i q^{7} +(-2.23607 + 2.00000i) q^{9} +O(q^{10})\)	\(q+(0.618034 + 1.61803i) q^{3} +1.00000i q^{5} -4.20811i q^{7} +(-2.23607 + 2.00000i) q^{9} -1.00000 q^{11} -3.83682 q^{13} +(-1.61803 + 0.618034i) q^{15} +6.20811i q^{17} +4.84342i q^{19} +(6.80887 - 2.60076i) q^{21} +5.20151 q^{23} -1.00000 q^{25} +(-4.61803 - 2.38197i) q^{27} -2.84342i q^{29} -3.20151i q^{31} +(-0.618034 - 1.61803i) q^{33} +4.20811 q^{35} -6.45078 q^{37} +(-2.37129 - 6.20811i) q^{39} -6.84342i q^{41} +7.46554i q^{43} +(-2.00000 - 2.23607i) q^{45} -6.43758 q^{47} -10.7082 q^{49} +(-10.0449 + 3.83682i) q^{51} -10.4376i q^{53} -1.00000i q^{55} +(-7.83682 + 2.99340i) q^{57} -7.96544 q^{59} -11.2015 q^{61} +(8.41622 + 9.40962i) q^{63} -3.83682i q^{65} +8.41622i q^{67} +(3.21471 + 8.41622i) q^{69} -5.18016 q^{71} -7.05154 q^{73} +(-0.618034 - 1.61803i) q^{75} +4.20811i q^{77} +15.5728i q^{79} +(1.00000 - 8.94427i) q^{81} +5.05154 q^{83} -6.20811 q^{85} +(4.60076 - 1.75733i) q^{87} -16.5620i q^{89} +16.1458i q^{91} +(5.18016 - 1.97864i) q^{93} -4.84342 q^{95} -1.29198 q^{97} +(2.23607 - 2.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3}+O(q^{10}) \) 8 * q - 4 * q^3	\( 8 q - 4 q^{3} - 8 q^{11} + 8 q^{13} - 4 q^{15} - 8 q^{25} - 28 q^{27} + 4 q^{33} - 8 q^{37} - 24 q^{39} - 16 q^{45} + 8 q^{47} - 32 q^{49} - 8 q^{51} - 24 q^{57} - 40 q^{59} - 48 q^{61} + 8 q^{71} + 8 q^{73} + 4 q^{75} + 8 q^{81} - 24 q^{83} - 16 q^{85} + 16 q^{87} - 8 q^{93} - 8 q^{95} - 24 q^{97}+O(q^{100}) \) 8 * q - 4 * q^3 - 8 * q^11 + 8 * q^13 - 4 * q^15 - 8 * q^25 - 28 * q^27 + 4 * q^33 - 8 * q^37 - 24 * q^39 - 16 * q^45 + 8 * q^47 - 32 * q^49 - 8 * q^51 - 24 * q^57 - 40 * q^59 - 48 * q^61 + 8 * q^71 + 8 * q^73 + 4 * q^75 + 8 * q^81 - 24 * q^83 - 16 * q^85 + 16 * q^87 - 8 * q^93 - 8 * q^95 - 24 * q^97

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