LMFDB - Embedded newform 1012.1.r.a.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1012,1,Mod(197,1012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1012, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 14]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1012.197");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1012 = 2^{2} \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1012.r (of order $22$, degree $10$, 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:	$0.505053792785$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

Embedding invariants

Embedding label			593.2
Root			$0.928368 - 0.371662i$ of defining polynomial
Character	$\chi$	$=$	1012.593
Dual form			1012.1.r.a.285.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.30379 - 1.50465i) q^{3} +(-0.0671040 + 0.466718i) q^{5} +(-0.421801 - 2.93369i) q^{9} +O(q^{10})$	$q+(1.30379 - 1.50465i) q^{3} +(-0.0671040 + 0.466718i) q^{5} +(-0.421801 - 2.93369i) q^{9} +(-0.959493 + 0.281733i) q^{11} +(0.614761 + 0.709472i) q^{15} +(0.235759 - 0.971812i) q^{23} +(0.746170 + 0.219095i) q^{25} +(-3.28924 - 2.11387i) q^{27} +(1.02951 + 1.18812i) q^{31} +(-0.827068 + 1.81103i) q^{33} +(0.252989 + 1.75958i) q^{37} +1.39751 q^{45} -0.284630 q^{47} +(-0.654861 + 0.755750i) q^{49} +(0.698939 + 1.53046i) q^{53} +(-0.0671040 - 0.466718i) q^{55} +(0.0395325 - 0.0865641i) q^{59} +(-1.11312 - 0.326842i) q^{67} +(-1.15486 - 1.62177i) q^{69} +(-1.38884 - 0.407799i) q^{71} +(1.30251 - 0.837074i) q^{75} +(-4.62533 + 1.35812i) q^{81} +(-0.759713 + 0.876756i) q^{89} +3.12998 q^{93} +(0.223734 - 1.55610i) q^{97} +(1.23123 + 2.69602i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} + 2 q^{5}+O(q^{10}) $ 20 * q + 2 * q^3 + 2 * q^5	$ 20 q + 2 q^{3} + 2 q^{5} - 2 q^{11} - 13 q^{15} + q^{23} - 2 q^{27} + 2 q^{31} - 9 q^{33} + 2 q^{37} - 4 q^{47} - 2 q^{49} - 4 q^{53} + 2 q^{55} + 2 q^{59} + 2 q^{67} - 12 q^{69} - 9 q^{71} + 22 q^{75} + 2 q^{81} + 2 q^{89} - 2 q^{93} - 9 q^{97}+O(q^{100}) $ 20 * q + 2 * q^3 + 2 * q^5 - 2 * q^11 - 13 * q^15 + q^23 - 2 * q^27 + 2 * q^31 - 9 * q^33 + 2 * q^37 - 4 * q^47 - 2 * q^49 - 4 * q^53 + 2 * q^55 + 2 * q^59 + 2 * q^67 - 12 * q^69 - 9 * q^71 + 22 * q^75 + 2 * q^81 + 2 * q^89 - 2 * q^93 - 9 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1012\mathbb{Z}\right)^\times$.

$n$	$277$	$507$	$925$
$\chi(n)$	$-1$	$1$	$e\left(\frac{6}{11}\right)$

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
$3$	1.30379	−	1.50465i	1.30379	−	1.50465i	0.580057	−	0.814576i	$-0.303030\pi$
$3$	1.30379	−	1.50465i	1.30379	−	1.50465i	0.723734	−	0.690079i	$-0.242424\pi$
$4$		0			0
$5$	−0.0671040	+	0.466718i	−0.0671040	+	0.466718i	0.928368	+	0.371662i	$0.121212\pi$
$5$	−0.0671040	+	0.466718i	−0.0671040	+	0.466718i	−0.995472	+	0.0950560i	$0.969697\pi$
$6$		0			0
$7$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$7$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$8$		0			0
$9$	−0.421801	−	2.93369i	−0.421801	−	2.93369i
$10$		0			0
$11$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$11$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$12$		0			0
$13$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$13$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$14$		0			0
$15$	0.614761	+	0.709472i	0.614761	+	0.709472i
$16$		0			0
$17$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$17$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$18$		0			0
$19$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$19$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.235759	−	0.971812i	0.235759	−	0.971812i
$23$	0.235759	−	0.971812i	0.235759	−	0.971812i
$24$		0			0
$25$	0.746170	+	0.219095i	0.746170	+	0.219095i
$26$		0			0
$27$	−3.28924	−	2.11387i	−3.28924	−	2.11387i
$28$		0			0
$29$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$29$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$30$		0			0
$31$	1.02951	+	1.18812i	1.02951	+	1.18812i	0.981929	+	0.189251i	$0.0606061\pi$
$31$	1.02951	+	1.18812i	1.02951	+	1.18812i	0.0475819	+	0.998867i	$0.484848\pi$
$32$		0			0
$33$	−0.827068	+	1.81103i	−0.827068	+	1.81103i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.252989	+	1.75958i	0.252989	+	1.75958i	0.580057	+	0.814576i	$0.303030\pi$
$37$	0.252989	+	1.75958i	0.252989	+	1.75958i	−0.327068	+	0.945001i	$0.606061\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$41$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$42$		0			0
$43$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$43$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$44$		0			0
$45$	1.39751			1.39751
$46$		0			0
$47$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$47$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$48$		0			0
$49$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$53$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$54$		0			0
$55$	−0.0671040	−	0.466718i	−0.0671040	−	0.466718i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	−0.888835	−	0.458227i	$-0.848485\pi$
$59$	0.0395325	−	0.0865641i	0.0395325	−	0.0865641i	0.928368	+	0.371662i	$0.121212\pi$
$60$		0			0
$61$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$61$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.11312	−	0.326842i	−1.11312	−	0.326842i	−0.327068	−	0.945001i	$-0.606061\pi$
$67$	−1.11312	−	0.326842i	−1.11312	−	0.326842i	−0.786053	+	0.618159i	$0.787879\pi$
$68$		0			0
$69$	−1.15486	−	1.62177i	−1.15486	−	1.62177i
$70$		0			0
$71$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.888835	+	0.458227i	$0.848485\pi$
$72$		0			0
$73$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$73$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$74$		0			0
$75$	1.30251	−	0.837074i	1.30251	−	0.837074i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$79$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$80$		0			0
$81$	−4.62533	+	1.35812i	−4.62533	+	1.35812i
$82$		0			0
$83$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$83$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	−0.995472	−	0.0950560i	$-0.969697\pi$
$89$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	0.235759	+	0.971812i	$0.424242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.12998			3.12998
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.223734	−	1.55610i	0.223734	−	1.55610i	−0.500000	−	0.866025i	$-0.666667\pi$
$97$	0.223734	−	1.55610i	0.223734	−	1.55610i	0.723734	−	0.690079i	$-0.242424\pi$
$98$		0			0
$99$	1.23123	+	2.69602i	1.23123	+	2.69602i
$100$		0			0
$101$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$101$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$102$		0			0
$103$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$103$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$107$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$108$		0			0
$109$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$109$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$110$		0			0
$111$	2.97740	+	1.91346i	2.97740	+	1.91346i
$112$		0			0
$113$	−1.78153	−	0.523103i	−1.78153	−	0.523103i	−0.786053	−	0.618159i	$-0.787879\pi$
$113$	−1.78153	−	0.523103i	−1.78153	−	0.523103i	−0.995472	+	0.0950560i	$0.969697\pi$
$114$		0			0
$115$	0.437742	+	0.175245i	0.437742	+	0.175245i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.841254	−	0.540641i	0.841254	−	0.540641i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.348202	+	0.762457i	−0.348202	+	0.762457i
$126$		0			0
$127$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$127$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$131$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.20730	−	1.39330i	1.20730	−	1.39330i
$136$		0			0
$137$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$137$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−0.371098	+	0.428269i	−0.371098	+	0.428269i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	0.283341	+	1.97068i	0.283341	+	1.97068i
$148$		0			0
$149$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$149$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$150$		0			0
$151$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$151$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.623601	+	0.400764i	−0.623601	+	0.400764i
$156$		0			0
$157$	−0.841254	−	0.540641i	−0.841254	−	0.540641i	0.0475819	−	0.998867i	$-0.484848\pi$
$157$	−0.841254	−	0.540641i	−0.841254	−	0.540641i	−0.888835	+	0.458227i	$0.848485\pi$
$158$		0			0
$159$	3.21409	+	0.943741i	3.21409	+	0.943741i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$163$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$164$		0			0
$165$	−0.789740	−	0.507535i	−0.789740	−	0.507535i
$166$		0			0
$167$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$167$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$168$		0			0
$169$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$173$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.0787070	−	0.172344i	−0.0787070	−	0.172344i
$178$		0			0
$179$	0.0930932	−	0.647478i	0.0930932	−	0.647478i	−0.888835	−	0.458227i	$-0.848485\pi$
$179$	0.0930932	−	0.647478i	0.0930932	−	0.647478i	0.981929	−	0.189251i	$-0.0606061\pi$
$180$		0			0
$181$	−1.28605	+	1.48418i	−1.28605	+	1.48418i	−0.500000	+	0.866025i	$0.666667\pi$
$181$	−1.28605	+	1.48418i	−1.28605	+	1.48418i	−0.786053	+	0.618159i	$0.787879\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.838204			−0.838204
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.827068	−	1.81103i	−0.827068	−	1.81103i	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−0.827068	−	1.81103i	−0.827068	−	1.81103i	−0.327068	−	0.945001i	$-0.606061\pi$
$192$		0			0
$193$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$193$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$197$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$198$		0			0
$199$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$199$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$200$		0			0
$201$	−1.94306	+	1.24873i	−1.94306	+	1.24873i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.95044	−	0.281733i	−2.95044	−	0.281733i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$211$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$212$		0			0
$213$	−2.42435	+	1.55803i	−2.42435	+	1.55803i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.271738	−	0.595023i	−0.271738	−	0.595023i	0.723734	−	0.690079i	$-0.242424\pi$
$223$	−0.271738	−	0.595023i	−0.271738	−	0.595023i	−0.995472	+	0.0950560i	$0.969697\pi$
$224$		0			0
$225$	0.328022	−	2.28145i	0.328022	−	2.28145i
$226$		0			0
$227$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$227$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$228$		0			0
$229$	1.85674			1.85674			0.928368	−	0.371662i	$-0.121212\pi$
$229$	1.85674			1.85674			0.928368	+	0.371662i	$0.121212\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$233$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$234$		0			0
$235$	0.0190998	−	0.132842i	0.0190998	−	0.132842i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$239$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$240$		0			0
$241$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$241$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$242$		0			0
$243$	−2.36271	+	5.17362i	−2.36271	+	5.17362i
$244$		0			0
$245$	−0.308779	−	0.356349i	−0.308779	−	0.356349i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.0913090	−	0.0268107i	−0.0913090	−	0.0268107i	0.235759	−	0.971812i	$-0.424242\pi$
$251$	−0.0913090	−	0.0268107i	−0.0913090	−	0.0268107i	−0.327068	+	0.945001i	$0.606061\pi$
$252$		0			0
$253$	0.0475819	+	0.998867i	0.0475819	+	0.998867i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.41542	+	0.909632i	1.41542	+	0.909632i	1.00000			$0$
$257$	1.41542	+	0.909632i	1.41542	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$263$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$264$		0			0
$265$	−0.761197	+	0.223507i	−0.761197	+	0.223507i
$266$		0			0
$267$	0.328708	+	2.28621i	0.328708	+	2.28621i
$268$		0			0
$269$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$269$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$270$		0			0
$271$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$271$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.777671			−0.777671
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	3.05132	−	3.52141i	3.05132	−	3.52141i
$280$		0			0
$281$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$281$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$282$		0			0
$283$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$283$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.415415	−	0.909632i	0.415415	−	0.909632i
$290$		0			0
$291$	−2.04970	−	2.36548i	−2.04970	−	2.36548i
$292$		0			0
$293$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$293$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$294$		0			0
$295$	0.0377483	+	0.0242593i	0.0377483	+	0.0242593i
$296$		0			0
$297$	3.75155	+	1.10155i	3.75155	+	1.10155i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$307$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$308$		0			0
$309$	1.08323	−	2.37194i	1.08323	−	2.37194i
$310$		0			0
$311$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$311$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$312$		0			0
$313$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	−0.786053	−	0.618159i	$-0.787879\pi$
$313$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.264241	+	1.83784i	−0.264241	+	1.83784i	0.235759	+	0.971812i	$0.424242\pi$
$317$	−0.264241	+	1.83784i	−0.264241	+	1.83784i	−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	−0.786053	−	0.618159i	$-0.787879\pi$
$331$	−0.205996	−	1.43273i	−0.205996	−	1.43273i	0.580057	−	0.814576i	$-0.303030\pi$
$332$		0			0
$333$	5.05534	−	1.48438i	5.05534	−	1.48438i
$334$		0			0
$335$	0.227238	−	0.497582i	0.227238	−	0.497582i
$336$		0			0
$337$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$337$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$338$		0			0
$339$	−3.10983	+	1.99856i	−3.10983	+	1.99856i
$340$		0			0
$341$	−1.32254	−	0.849945i	−1.32254	−	0.849945i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	0.834408	−	0.430167i	0.834408	−	0.430167i
$346$		0			0
$347$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$347$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$348$		0			0
$349$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$349$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.428368	+	0.494363i	0.428368	+	0.494363i	0.928368	−	0.371662i	$-0.121212\pi$
$353$	0.428368	+	0.494363i	0.428368	+	0.494363i	−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	0.283524	−	0.620830i	0.283524	−	0.620830i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$359$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$360$		0			0
$361$	0.415415	+	0.909632i	0.415415	+	0.909632i
$362$		0			0
$363$	0.283341	−	1.97068i	0.283341	−	1.97068i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.44747			1.44747			0.723734	−	0.690079i	$-0.242424\pi$
$367$	1.44747			1.44747			0.723734	+	0.690079i	$0.242424\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$373$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$374$		0			0
$375$	0.693251	+	1.51801i	0.693251	+	1.51801i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0913090	+	0.0268107i	−0.0913090	+	0.0268107i	−0.327068	−	0.945001i	$-0.606061\pi$
$379$	−0.0913090	+	0.0268107i	−0.0913090	+	0.0268107i	0.235759	+	0.971812i	$0.424242\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.28605	−	1.48418i	−1.28605	−	1.48418i	−0.786053	−	0.618159i	$-0.787879\pi$
$383$	−1.28605	−	1.48418i	−1.28605	−	1.48418i	−0.500000	−	0.866025i	$-0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.627639	+	0.184291i	0.627639	+	0.184291i	0.580057	−	0.814576i	$-0.303030\pi$
$389$	0.627639	+	0.184291i	0.627639	+	0.184291i	0.0475819	+	0.998867i	$0.484848\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$397$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$401$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.323481	−	2.24986i	−0.323481	−	2.24986i
$406$		0			0
$407$	−0.738471	−	1.61703i	−0.738471	−	1.61703i
$408$		0			0
$409$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$409$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$410$		0			0
$411$	2.56046	−	2.95493i	2.56046	−	2.95493i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$419$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$420$		0			0
$421$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$421$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$422$		0			0
$423$	0.120057	+	0.835015i	0.120057	+	0.835015i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$431$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$432$		0			0
$433$	−0.550294	−	0.353653i	−0.550294	−	0.353653i	0.235759	−	0.971812i	$-0.424242\pi$
$433$	−0.550294	−	0.353653i	−0.550294	−	0.353653i	−0.786053	+	0.618159i	$0.787879\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$439$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$440$		0			0
$441$	2.49336	+	1.60238i	2.49336	+	1.60238i
$442$		0			0
$443$	0.396666	−	0.254922i	0.396666	−	0.254922i	−0.327068	−	0.945001i	$-0.606061\pi$
$443$	0.396666	−	0.254922i	0.396666	−	0.254922i	0.723734	+	0.690079i	$0.242424\pi$
$444$		0			0
$445$	−0.358218	−	0.413406i	−0.358218	−	0.413406i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.50842	−	0.442913i	1.50842	−	0.442913i	0.580057	−	0.814576i	$-0.303030\pi$
$449$	1.50842	−	0.442913i	1.50842	−	0.442913i	0.928368	+	0.371662i	$0.121212\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$457$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.327068	+	0.945001i	$0.606061\pi$
$463$	−1.21590	+	1.40323i	−1.21590	+	1.40323i	−0.888835	+	0.458227i	$0.848485\pi$
$464$		0			0
$465$	−0.210034	+	1.46082i	−0.210034	+	1.46082i
$466$		0			0
$467$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.723734	+	0.690079i	$0.242424\pi$
$467$	0.771316	+	1.68895i	0.771316	+	1.68895i	0.0475819	+	0.998867i	$0.484848\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.91030	+	0.560914i	−1.91030	+	0.560914i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.19509	−	2.69602i	4.19509	−	2.69602i
$478$		0			0
$479$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$479$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.711249	+	0.208842i	0.711249	+	0.208842i
$486$		0			0
$487$	0.975950	+	0.627205i	0.975950	+	0.627205i	0.928368	−	0.371662i	$-0.121212\pi$
$487$	0.975950	+	0.627205i	0.975950	+	0.627205i	0.0475819	+	0.998867i	$0.484848\pi$
$488$		0			0
$489$	2.19364	−	1.40977i	2.19364	−	1.40977i
$490$		0			0
$491$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$491$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−1.34090	+	0.393725i	−1.34090	+	0.393725i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$499$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$503$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.99094			−1.99094
$508$		0			0
$509$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	−0.995472	−	0.0950560i	$-0.969697\pi$
$509$	−0.759713	+	0.876756i	−0.759713	+	0.876756i	0.235759	+	0.971812i	$0.424242\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.0878875	+	0.611271i	0.0878875	+	0.611271i
$516$		0			0
$517$	0.273100	−	0.0801894i	0.273100	−	0.0801894i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.0623191	−	0.0719200i	−0.0623191	−	0.0719200i	0.723734	−	0.690079i	$-0.242424\pi$
$521$	−0.0623191	−	0.0719200i	−0.0623191	−	0.0719200i	−0.786053	+	0.618159i	$0.787879\pi$
$522$		0			0
$523$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$523$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.888835	−	0.458227i	−0.888835	−	0.458227i
$530$		0			0
$531$	−0.270627	−	0.0794632i	−0.270627	−	0.0794632i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.852856	−	0.984249i	−0.852856	−	0.984249i
$538$		0			0
$539$	0.415415	−	0.909632i	0.415415	−	0.909632i
$540$		0			0
$541$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$541$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$542$		0			0
$543$	0.556441	+	3.87013i	0.556441	+	3.87013i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$547$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.09284	+	1.26121i	−1.09284	+	1.26121i
$556$		0			0
$557$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$557$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$563$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$564$		0			0
$565$	0.363689	−	0.796368i	0.363689	−	0.796368i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$569$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$570$		0			0
$571$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$571$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$572$		0			0
$573$	−3.80329	−	1.11675i	−3.80329	−	1.11675i
$574$		0			0
$575$	0.388835	−	0.673483i	0.388835	−	0.673483i
$576$		0			0
$577$	−0.0913090	−	0.0268107i	−0.0913090	−	0.0268107i	0.235759	−	0.971812i	$-0.424242\pi$
$577$	−0.0913090	−	0.0268107i	−0.0913090	−	0.0268107i	−0.327068	+	0.945001i	$0.606061\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$587$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$593$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.65414			−1.65414
$598$		0			0
$599$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$599$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$600$		0			0
$601$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$601$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$602$		0			0
$603$	−0.489337	+	3.40341i	−0.489337	+	3.40341i
$604$		0			0
$605$	0.195876	+	0.428908i	0.195876	+	0.428908i
$606$		0			0
$607$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$607$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$613$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.698939	+	0.449181i	0.698939	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$617$	0.698939	+	0.449181i	0.698939	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$618$		0			0
$619$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−1.38884	−	0.407799i	−1.38884	−	0.407799i	−0.888835	+	0.458227i	$0.848485\pi$
$620$		0			0
$621$	−2.82975	+	2.69816i	−2.82975	+	2.69816i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.321731	+	0.206764i	0.321731	+	0.206764i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.653077	+	1.43004i	−0.653077	+	1.43004i	0.235759	+	0.971812i	$0.424242\pi$
$631$	−0.653077	+	1.43004i	−0.653077	+	1.43004i	−0.888835	+	0.458227i	$0.848485\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−0.610543	+	4.24642i	−0.610543	+	4.24642i
$640$		0			0
$641$	1.16413	−	1.34347i	1.16413	−	1.34347i	0.235759	−	0.971812i	$-0.424242\pi$
$641$	1.16413	−	1.34347i	1.16413	−	1.34347i	0.928368	−	0.371662i	$-0.121212\pi$
$642$		0			0
$643$	−1.57211			−1.57211			−0.786053	−	0.618159i	$-0.787879\pi$
$643$	−1.57211			−1.57211			−0.786053	+	0.618159i	$0.787879\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.28605	+	1.48418i	−1.28605	+	1.48418i	−0.500000	+	0.866025i	$0.666667\pi$
$647$	−1.28605	+	1.48418i	−1.28605	+	1.48418i	−0.786053	+	0.618159i	$0.787879\pi$
$648$		0			0
$649$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.0930932	+	0.647478i	0.0930932	+	0.647478i	0.981929	+	0.189251i	$0.0606061\pi$
$653$	0.0930932	+	0.647478i	0.0930932	+	0.647478i	−0.888835	+	0.458227i	$0.848485\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$659$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$660$		0			0
$661$	−0.550294	+	0.353653i	−0.550294	+	0.353653i	−0.786053	−	0.618159i	$-0.787879\pi$
$661$	−0.550294	+	0.353653i	−0.550294	+	0.353653i	0.235759	+	0.971812i	$0.424242\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.24959	−	0.366914i	−1.24959	−	0.366914i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$673$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$674$		0			0
$675$	−1.99119	−	2.29796i	−1.99119	−	2.29796i
$676$		0			0
$677$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$677$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$683$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$684$		0			0
$685$	−0.131783	+	0.916569i	−0.131783	+	0.916569i
$686$		0			0
$687$	2.42080	−	2.79375i	2.42080	−	2.79375i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.96386			1.96386			0.981929	−	0.189251i	$-0.0606061\pi$
$691$	1.96386			1.96386			0.981929	+	0.189251i	$0.0606061\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$701$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−0.174979	−	0.201937i	−0.174979	−	0.201937i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.67489	−	1.07639i	−1.67489	−	1.07639i	−0.888835	−	0.458227i	$-0.848485\pi$
$709$	−1.67489	−	1.07639i	−1.67489	−	1.07639i	−0.786053	−	0.618159i	$-0.787879\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.39734	−	0.720381i	1.39734	−	0.720381i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.975950	−	0.627205i	0.975950	−	0.627205i	0.0475819	−	0.998867i	$-0.484848\pi$
$719$	0.975950	−	0.627205i	0.975950	−	0.627205i	0.928368	+	0.371662i	$0.121212\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.0135432	−	0.0941952i	−0.0135432	−	0.0941952i	0.981929	−	0.189251i	$-0.0606061\pi$
$727$	−0.0135432	−	0.0941952i	−0.0135432	−	0.0941952i	−0.995472	+	0.0950560i	$0.969697\pi$
$728$		0			0
$729$	2.70149	+	5.91543i	2.70149	+	5.91543i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$733$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$734$		0			0
$735$	−0.938766			−0.938766
$736$		0			0
$737$	1.16011			1.16011
$738$		0			0
$739$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$739$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$743$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.654861	+	0.755750i	0.654861	+	0.755750i	0.981929	−	0.189251i	$-0.0606061\pi$
$751$	0.654861	+	0.755750i	0.654861	+	0.755750i	−0.327068	+	0.945001i	$0.606061\pi$
$752$		0			0
$753$	−0.159389	+	0.102433i	−0.159389	+	0.102433i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$757$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$758$		0			0
$759$	1.56499	+	1.23072i	1.56499	+	1.23072i
$760$		0			0
$761$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$761$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$769$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$770$		0			0
$771$	3.21409	−	0.943741i	3.21409	−	0.943741i
$772$		0			0
$773$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$773$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$774$		0			0
$775$	0.507879	+	1.11210i	0.507879	+	1.11210i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	1.44747			1.44747
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.308779	−	0.356349i	0.308779	−	0.356349i
$786$		0			0
$787$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$787$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.656140	+	1.43675i	−0.656140	+	1.43675i
$796$		0			0
$797$	1.30379	+	1.50465i	1.30379	+	1.50465i	0.723734	+	0.690079i	$0.242424\pi$
$797$	1.30379	+	1.50465i	1.30379	+	1.50465i	0.580057	+	0.814576i	$0.303030\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	2.89258	+	1.85895i	2.89258	+	1.85895i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−3.66583	−	1.07639i	−3.66583	−	1.07639i
$808$		0			0
$809$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$809$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$810$		0			0
$811$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$811$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.256542	+	0.561750i	−0.256542	+	0.561750i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$821$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$822$		0			0
$823$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i	−0.995472	−	0.0950560i	$-0.969697\pi$
$823$	−0.0135432	+	0.0941952i	−0.0135432	+	0.0941952i	0.981929	+	0.189251i	$0.0606061\pi$
$824$		0			0
$825$	−1.01392	+	1.17013i	−1.01392	+	1.17013i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−0.654136			−0.654136			−0.327068	−	0.945001i	$-0.606061\pi$
$829$	−0.654136			−0.654136			−0.327068	+	0.945001i	$0.606061\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.874784	−	6.08426i	−0.874784	−	6.08426i
$838$		0			0
$839$	−1.78153	+	0.523103i	−1.78153	+	0.523103i	−0.995472	−	0.0950560i	$-0.969697\pi$
$839$	−1.78153	+	0.523103i	−1.78153	+	0.523103i	−0.786053	+	0.618159i	$0.787879\pi$
$840$		0			0
$841$	0.415415	−	0.909632i	0.415415	−	0.909632i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.396666	−	0.254922i	0.396666	−	0.254922i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.76962	+	0.168978i	1.76962	+	0.168978i
$852$		0			0
$853$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$853$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$857$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$858$		0			0
$859$	−0.759713	−	0.876756i	−0.759713	−	0.876756i	0.235759	−	0.971812i	$-0.424242\pi$
$859$	−0.759713	−	0.876756i	−0.759713	−	0.876756i	−0.995472	+	0.0950560i	$0.969697\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$863$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.827068	−	1.81103i	−0.827068	−	1.81103i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−4.65950			−4.65950
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$877$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.271738	−	0.595023i	−0.271738	−	0.595023i	0.723734	−	0.690079i	$-0.242424\pi$
$881$	−0.271738	−	0.595023i	−0.271738	−	0.595023i	−0.995472	+	0.0950560i	$0.969697\pi$
$882$		0			0
$883$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$883$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$884$		0			0
$885$	0.0857178	−	0.0251690i	0.0857178	−	0.0251690i
$886$		0			0
$887$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$887$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	4.05534	−	2.60621i	4.05534	−	2.60621i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	0.295943	+	0.0868967i	0.295943	+	0.0868967i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.606397	−	0.699819i	−0.606397	−	0.699819i
$906$		0			0
$907$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$907$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$911$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.196742	+	1.36837i	−0.196742	+	1.36837i
$926$		0			0
$927$	−1.61257	−	3.53103i	−1.61257	−	3.53103i
$928$		0			0
$929$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$929$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1.39155	+	3.04706i	−1.39155	+	3.04706i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$937$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$938$		0			0
$939$	−2.42435	−	1.55803i	−2.42435	−	1.55803i
$940$		0			0
$941$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$941$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.65210	+	1.06174i	1.65210	+	1.06174i	0.928368	+	0.371662i	$0.121212\pi$
$947$	1.65210	+	1.06174i	1.65210	+	1.06174i	0.723734	+	0.690079i	$0.242424\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	2.42080	+	2.79375i	2.42080	+	2.79375i
$952$		0			0
$953$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$953$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$954$		0			0
$955$	0.900739	−	0.264481i	0.900739	−	0.264481i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.209419	+	1.45654i	−0.209419	+	1.45654i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.279486	+	1.94387i	−0.279486	+	1.94387i	0.0475819	+	0.998867i	$0.484848\pi$
$971$	−0.279486	+	1.94387i	−0.279486	+	1.94387i	−0.327068	+	0.945001i	$0.606061\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.627639	−	0.184291i	0.627639	−	0.184291i	0.0475819	−	0.998867i	$-0.484848\pi$
$977$	0.627639	−	0.184291i	0.627639	−	0.184291i	0.580057	+	0.814576i	$0.303030\pi$
$978$		0			0
$979$	0.481929	−	1.05528i	0.481929	−	1.05528i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	−0.500000	−	0.866025i	$-0.666667\pi$
$983$	0.0800569	−	0.0514495i	0.0800569	−	0.0514495i	0.580057	+	0.814576i	$0.303030\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$991$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$992$		0			0
$993$	−2.42435	−	1.55803i	−2.42435	−	1.55803i
$994$		0			0
$995$	0.329562	−	0.211797i	0.329562	−	0.211797i
$996$		0			0
$997$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$997$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$998$		0			0
$999$	2.88737	−	6.32246i	2.88737	−	6.32246i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1012.1.r.a.593.2	yes	20
11.10	odd	2	CM	1012.1.r.a.593.2	yes	20
23.9	even	11	inner	1012.1.r.a.285.2	✓	20
253.32	odd	22	inner	1012.1.r.a.285.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1012.1.r.a.285.2	✓	20	23.9	even	11	inner
1012.1.r.a.285.2	✓	20	253.32	odd	22	inner
1012.1.r.a.593.2	yes	20	1.1	even	1	trivial
1012.1.r.a.593.2	yes	20	11.10	odd	2	CM

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:	\( N \)	\(=\)	\( 1012 = 2^{2} \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1012.r (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(1.30379 - 1.50465i) q^{3} +(-0.0671040 + 0.466718i) q^{5} +(-0.421801 - 2.93369i) q^{9} +O(q^{10})\)	\(q+(1.30379 - 1.50465i) q^{3} +(-0.0671040 + 0.466718i) q^{5} +(-0.421801 - 2.93369i) q^{9} +(-0.959493 + 0.281733i) q^{11} +(0.614761 + 0.709472i) q^{15} +(0.235759 - 0.971812i) q^{23} +(0.746170 + 0.219095i) q^{25} +(-3.28924 - 2.11387i) q^{27} +(1.02951 + 1.18812i) q^{31} +(-0.827068 + 1.81103i) q^{33} +(0.252989 + 1.75958i) q^{37} +1.39751 q^{45} -0.284630 q^{47} +(-0.654861 + 0.755750i) q^{49} +(0.698939 + 1.53046i) q^{53} +(-0.0671040 - 0.466718i) q^{55} +(0.0395325 - 0.0865641i) q^{59} +(-1.11312 - 0.326842i) q^{67} +(-1.15486 - 1.62177i) q^{69} +(-1.38884 - 0.407799i) q^{71} +(1.30251 - 0.837074i) q^{75} +(-4.62533 + 1.35812i) q^{81} +(-0.759713 + 0.876756i) q^{89} +3.12998 q^{93} +(0.223734 - 1.55610i) q^{97} +(1.23123 + 2.69602i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) 20 * q + 2 * q^3 + 2 * q^5	\( 20 q + 2 q^{3} + 2 q^{5} - 2 q^{11} - 13 q^{15} + q^{23} - 2 q^{27} + 2 q^{31} - 9 q^{33} + 2 q^{37} - 4 q^{47} - 2 q^{49} - 4 q^{53} + 2 q^{55} + 2 q^{59} + 2 q^{67} - 12 q^{69} - 9 q^{71} + 22 q^{75} + 2 q^{81} + 2 q^{89} - 2 q^{93} - 9 q^{97}+O(q^{100}) \) 20 * q + 2 * q^3 + 2 * q^5 - 2 * q^11 - 13 * q^15 + q^23 - 2 * q^27 + 2 * q^31 - 9 * q^33 + 2 * q^37 - 4 * q^47 - 2 * q^49 - 4 * q^53 + 2 * q^55 + 2 * q^59 + 2 * q^67 - 12 * q^69 - 9 * q^71 + 22 * q^75 + 2 * q^81 + 2 * q^89 - 2 * q^93 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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