LMFDB - Embedded newform 28.2.d.a.27.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,2,Mod(27,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("28.27");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	28.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.223581125660$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			27.2
Root			$0.500000 - 1.32288i$ of defining polynomial
Character	$\chi$	$=$	28.27
Dual form			28.2.d.a.27.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 - 1.32288i) q^{8} -3.00000 q^{9} +O(q^{10})$	$q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 - 1.32288i) q^{8} -3.00000 q^{9} +5.29150i q^{11} +(3.50000 + 1.32288i) q^{14} +(0.500000 + 3.96863i) q^{16} +(1.50000 - 3.96863i) q^{18} +(-7.00000 - 2.64575i) q^{22} -5.29150i q^{23} +5.00000 q^{25} +(-3.50000 + 3.96863i) q^{28} -2.00000 q^{29} +(-5.50000 - 1.32288i) q^{32} +(4.50000 + 3.96863i) q^{36} +6.00000 q^{37} +5.29150i q^{43} +(7.00000 - 7.93725i) q^{44} +(7.00000 + 2.64575i) q^{46} -7.00000 q^{49} +(-2.50000 + 6.61438i) q^{50} -10.0000 q^{53} +(-3.50000 - 6.61438i) q^{56} +(1.00000 - 2.64575i) q^{58} +7.93725i q^{63} +(4.50000 - 6.61438i) q^{64} -15.8745i q^{67} -5.29150i q^{71} +(-7.50000 + 3.96863i) q^{72} +(-3.00000 + 7.93725i) q^{74} +14.0000 q^{77} +15.8745i q^{79} +9.00000 q^{81} +(-7.00000 - 2.64575i) q^{86} +(7.00000 + 13.2288i) q^{88} +(-7.00000 + 7.93725i) q^{92} +(3.50000 - 9.26013i) q^{98} -15.8745i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 3 q^{4} + 5 q^{8} - 6 q^{9}+O(q^{10}) $ 2 * q - q^2 - 3 * q^4 + 5 * q^8 - 6 * q^9	$ 2 q - q^{2} - 3 q^{4} + 5 q^{8} - 6 q^{9} + 7 q^{14} + q^{16} + 3 q^{18} - 14 q^{22} + 10 q^{25} - 7 q^{28} - 4 q^{29} - 11 q^{32} + 9 q^{36} + 12 q^{37} + 14 q^{44} + 14 q^{46} - 14 q^{49} - 5 q^{50} - 20 q^{53} - 7 q^{56} + 2 q^{58} + 9 q^{64} - 15 q^{72} - 6 q^{74} + 28 q^{77} + 18 q^{81} - 14 q^{86} + 14 q^{88} - 14 q^{92} + 7 q^{98}+O(q^{100}) $ 2 * q - q^2 - 3 * q^4 + 5 * q^8 - 6 * q^9 + 7 * q^14 + q^16 + 3 * q^18 - 14 * q^22 + 10 * q^25 - 7 * q^28 - 4 * q^29 - 11 * q^32 + 9 * q^36 + 12 * q^37 + 14 * q^44 + 14 * q^46 - 14 * q^49 - 5 * q^50 - 20 * q^53 - 7 * q^56 + 2 * q^58 + 9 * q^64 - 15 * q^72 - 6 * q^74 + 28 * q^77 + 18 * q^81 - 14 * q^86 + 14 * q^88 - 14 * q^92 + 7 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/28\mathbb{Z}\right)^\times$.

$n$	$15$	$17$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−0.500000	+	1.32288i	−0.353553	+	0.935414i
$2$	−0.500000	+	1.32288i	−0.353553	+	0.935414i
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−1.50000	−	1.32288i	−0.750000	−	0.661438i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$	2.50000	−	1.32288i	0.883883	−	0.467707i
$9$	−3.00000			−1.00000
$10$		0			0
$11$			5.29150i			1.59545i	0.603023	+	0.797724i	$0.293963\pi$
$11$			5.29150i			1.59545i	−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	3.50000	+	1.32288i	0.935414	+	0.353553i
$15$		0			0
$16$	0.500000	+	3.96863i	0.125000	+	0.992157i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	1.50000	−	3.96863i	0.353553	−	0.935414i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−7.00000	−	2.64575i	−1.49241	−	0.564076i
$23$		−	5.29150i		−	1.10335i	−0.834058	−	0.551677i	$-0.813988\pi$
$23$		−	5.29150i		−	1.10335i	0.834058	−	0.551677i	$-0.186012\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$	−3.50000	+	3.96863i	−0.661438	+	0.750000i
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−5.50000	−	1.32288i	−0.972272	−	0.233854i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	4.50000	+	3.96863i	0.750000	+	0.661438i
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			5.29150i			0.806947i	0.914991	+	0.403473i	$0.132197\pi$
$43$			5.29150i			0.806947i	−0.914991	+	0.403473i	$0.867803\pi$
$44$	7.00000	−	7.93725i	1.05529	−	1.19659i
$45$		0			0
$46$	7.00000	+	2.64575i	1.03209	+	0.390095i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	−2.50000	+	6.61438i	−0.353553	+	0.935414i
$51$		0			0
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$	−3.50000	−	6.61438i	−0.467707	−	0.883883i
$57$		0			0
$58$	1.00000	−	2.64575i	0.131306	−	0.347404i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$			7.93725i			1.00000i
$64$	4.50000	−	6.61438i	0.562500	−	0.826797i
$65$		0			0
$66$		0			0
$67$		−	15.8745i		−	1.93938i	−0.244339	−	0.969690i	$-0.578571\pi$
$67$		−	15.8745i		−	1.93938i	0.244339	−	0.969690i	$-0.421429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.29150i		−	0.627986i	−0.949425	−	0.313993i	$-0.898333\pi$
$71$		−	5.29150i		−	0.627986i	0.949425	−	0.313993i	$-0.101667\pi$
$72$	−7.50000	+	3.96863i	−0.883883	+	0.467707i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−3.00000	+	7.93725i	−0.348743	+	0.922687i
$75$		0			0
$76$		0			0
$77$	14.0000			1.59545
$78$		0			0
$79$			15.8745i			1.78602i	0.450035	+	0.893011i	$0.351411\pi$
$79$			15.8745i			1.78602i	−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−7.00000	−	2.64575i	−0.754829	−	0.285299i
$87$		0			0
$88$	7.00000	+	13.2288i	0.746203	+	1.41019i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−7.00000	+	7.93725i	−0.729800	+	0.827516i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	3.50000	−	9.26013i	0.353553	−	0.935414i
$99$		−	15.8745i		−	1.59545i
$100$	−7.50000	−	6.61438i	−0.750000	−	0.661438i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	5.00000	−	13.2288i	0.485643	−	1.28489i
$107$			5.29150i			0.511549i	0.966736	+	0.255774i	$0.0823304\pi$
$107$			5.29150i			0.511549i	−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$	−18.0000			−1.72409			−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000			−1.72409			−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$	10.5000	−	1.32288i	0.992157	−	0.125000i
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$		0			0
$116$	3.00000	+	2.64575i	0.278543	+	0.245652i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−17.0000			−1.54545
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−10.5000	−	3.96863i	−0.935414	−	0.353553i
$127$			15.8745i			1.40863i	0.709885	+	0.704317i	$0.248747\pi$
$127$			15.8745i			1.40863i	−0.709885	+	0.704317i	$0.751253\pi$
$128$	6.50000	+	9.26013i	0.574524	+	0.818488i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	21.0000	+	7.93725i	1.81412	+	0.685674i
$135$		0			0
$136$		0			0
$137$	10.0000			0.854358			0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000			0.854358			0.427179	+	0.904167i	$0.359507\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	7.00000	+	2.64575i	0.587427	+	0.222027i
$143$		0			0
$144$	−1.50000	−	11.9059i	−0.125000	−	0.992157i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−9.00000	−	7.93725i	−0.739795	−	0.652438i
$149$	22.0000			1.80231			0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000			1.80231			0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$		−	5.29150i		−	0.430616i	−0.976546	−	0.215308i	$-0.930924\pi$
$151$		−	5.29150i		−	0.430616i	0.976546	−	0.215308i	$-0.0690756\pi$
$152$		0			0
$153$		0			0
$154$	−7.00000	+	18.5203i	−0.564076	+	1.49241i
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	−21.0000	−	7.93725i	−1.67067	−	0.631454i
$159$		0			0
$160$		0			0
$161$	−14.0000			−1.10335
$162$	−4.50000	+	11.9059i	−0.353553	+	0.935414i
$163$		−	15.8745i		−	1.24339i	−0.783260	−	0.621694i	$-0.786445\pi$
$163$		−	15.8745i		−	1.24339i	0.783260	−	0.621694i	$-0.213555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$	7.00000	−	7.93725i	0.533745	−	0.605210i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	13.2288i		−	1.00000i
$176$	−21.0000	+	2.64575i	−1.58293	+	0.199431i
$177$		0			0
$178$		0			0
$179$			26.4575i			1.97753i	0.149487	+	0.988764i	$0.452238\pi$
$179$			26.4575i			1.97753i	−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−7.00000	−	13.2288i	−0.516047	−	0.975237i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	26.4575i		−	1.91440i	−0.289430	−	0.957199i	$-0.593466\pi$
$191$		−	26.4575i		−	1.91440i	0.289430	−	0.957199i	$-0.406534\pi$
$192$		0			0
$193$	18.0000			1.29567			0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000			1.29567			0.647834	+	0.761781i	$0.275675\pi$
$194$		0			0
$195$		0			0
$196$	10.5000	+	9.26013i	0.750000	+	0.661438i
$197$	−26.0000			−1.85242			−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000			−1.85242			−0.926212	+	0.377004i	$0.876954\pi$
$198$	21.0000	+	7.93725i	1.49241	+	0.564076i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	12.5000	−	6.61438i	0.883883	−	0.467707i
$201$		0			0
$202$		0			0
$203$			5.29150i			0.371391i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			15.8745i			1.10335i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			26.4575i			1.82141i	0.413057	+	0.910705i	$0.364461\pi$
$211$			26.4575i			1.82141i	−0.413057	+	0.910705i	$0.635539\pi$
$212$	15.0000	+	13.2288i	1.03020	+	0.908555i
$213$		0			0
$214$	−7.00000	−	2.64575i	−0.478510	−	0.180860i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	9.00000	−	23.8118i	0.609557	−	1.61274i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−3.50000	+	14.5516i	−0.233854	+	0.972272i
$225$	−15.0000			−1.00000
$226$	−1.00000	+	2.64575i	−0.0665190	+	0.175993i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−5.00000	+	2.64575i	−0.328266	+	0.173702i
$233$	−22.0000			−1.44127			−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000			−1.44127			−0.720634	+	0.693316i	$0.756149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	26.4575i		−	1.71139i	−0.517477	−	0.855697i	$-0.673129\pi$
$239$		−	26.4575i		−	1.71139i	0.517477	−	0.855697i	$-0.326871\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	8.50000	−	22.4889i	0.546401	−	1.44564i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	10.5000	−	11.9059i	0.661438	−	0.750000i
$253$	28.0000			1.76034
$254$	−21.0000	−	7.93725i	−1.31766	−	0.498028i
$255$		0			0
$256$	−15.5000	+	3.96863i	−0.968750	+	0.248039i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		−	15.8745i		−	0.986394i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	5.29150i		−	0.326288i	−0.986602	−	0.163144i	$-0.947836\pi$
$263$		−	5.29150i		−	0.326288i	0.986602	−	0.163144i	$-0.0521635\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−21.0000	+	23.8118i	−1.28278	+	1.45453i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	−5.00000	+	13.2288i	−0.302061	+	0.799178i
$275$			26.4575i			1.59545i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	−7.00000	+	7.93725i	−0.415374	+	0.470989i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	16.5000	+	3.96863i	0.972272	+	0.233854i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	15.0000	−	7.93725i	0.871857	−	0.461344i
$297$		0			0
$298$	−11.0000	+	29.1033i	−0.637213	+	1.68591i
$299$		0			0
$300$		0			0
$301$	14.0000			0.806947
$302$	7.00000	+	2.64575i	0.402805	+	0.152246i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−21.0000	−	18.5203i	−1.19659	−	1.05529i
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	21.0000	−	23.8118i	1.18134	−	1.33952i
$317$	−34.0000			−1.90963			−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000			−1.90963			−0.954815	+	0.297200i	$0.903947\pi$
$318$		0			0
$319$		−	10.5830i		−	0.592535i
$320$		0			0
$321$		0			0
$322$	7.00000	−	18.5203i	0.390095	−	1.03209i
$323$		0			0
$324$	−13.5000	−	11.9059i	−0.750000	−	0.661438i
$325$		0			0
$326$	21.0000	+	7.93725i	1.16308	+	0.439604i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			5.29150i			0.290847i	0.989369	+	0.145424i	$0.0464545\pi$
$331$			5.29150i			0.290847i	−0.989369	+	0.145424i	$0.953545\pi$
$332$		0			0
$333$	−18.0000			−0.986394
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$	−6.50000	+	17.1974i	−0.353553	+	0.935414i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			18.5203i			1.00000i
$344$	7.00000	+	13.2288i	0.377415	+	0.713247i
$345$		0			0
$346$		0			0
$347$		−	37.0405i		−	1.98844i	−0.107366	−	0.994220i	$-0.534242\pi$
$347$		−	37.0405i		−	1.98844i	0.107366	−	0.994220i	$-0.465758\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	17.5000	+	6.61438i	0.935414	+	0.353553i
$351$		0			0
$352$	7.00000	−	29.1033i	0.373101	−	1.55121i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−35.0000	−	13.2288i	−1.84981	−	0.699162i
$359$			37.0405i			1.95492i	0.211112	+	0.977462i	$0.432292\pi$
$359$			37.0405i			1.95492i	−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	21.0000	−	2.64575i	1.09470	−	0.137919i
$369$		0			0
$370$		0			0
$371$			26.4575i			1.37361i
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	37.0405i		−	1.90264i	−0.308199	−	0.951322i	$-0.599726\pi$
$379$		−	37.0405i		−	1.90264i	0.308199	−	0.951322i	$-0.400274\pi$
$380$		0			0
$381$		0			0
$382$	35.0000	+	13.2288i	1.79076	+	0.676842i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	−9.00000	+	23.8118i	−0.458088	+	1.21199i
$387$		−	15.8745i		−	0.806947i
$388$		0			0
$389$	38.0000			1.92668			0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000			1.92668			0.963338	+	0.268290i	$0.0864585\pi$
$390$		0			0
$391$		0			0
$392$	−17.5000	+	9.26013i	−0.883883	+	0.467707i
$393$		0			0
$394$	13.0000	−	34.3948i	0.654931	−	1.73278i
$395$		0			0
$396$	−21.0000	+	23.8118i	−1.05529	+	1.19659i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	2.50000	+	19.8431i	0.125000	+	0.992157i
$401$	34.0000			1.69788			0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000			1.69788			0.848939	+	0.528490i	$0.177242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−7.00000	−	2.64575i	−0.347404	−	0.131306i
$407$			31.7490i			1.57374i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−21.0000	−	7.93725i	−1.03209	−	0.390095i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$	−35.0000	−	13.2288i	−1.70377	−	0.643966i
$423$		0			0
$424$	−25.0000	+	13.2288i	−1.21411	+	0.642445i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	7.00000	−	7.93725i	0.338358	−	0.383662i
$429$		0			0
$430$		0			0
$431$		−	26.4575i		−	1.27441i	−0.770693	−	0.637207i	$-0.780090\pi$
$431$		−	26.4575i		−	1.27441i	0.770693	−	0.637207i	$-0.219910\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	27.0000	+	23.8118i	1.29307	+	1.14038i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$		−	37.0405i		−	1.75985i	−0.475114	−	0.879924i	$-0.657593\pi$
$443$		−	37.0405i		−	1.75985i	0.475114	−	0.879924i	$-0.342407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−17.5000	−	11.9059i	−0.826797	−	0.562500i
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$	7.50000	−	19.8431i	0.353553	−	0.935414i
$451$		0			0
$452$	−3.00000	−	2.64575i	−0.141108	−	0.124446i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.00000			−0.280668			−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000			−0.280668			−0.140334	+	0.990104i	$0.544818\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$			15.8745i			0.737751i	0.929479	+	0.368875i	$0.120257\pi$
$463$			15.8745i			0.737751i	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−1.00000	−	7.93725i	−0.0464238	−	0.368478i
$465$		0			0
$466$	11.0000	−	29.1033i	0.509565	−	1.34818i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−42.0000			−1.93938
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−28.0000			−1.28744
$474$		0			0
$475$		0			0
$476$		0			0
$477$	30.0000			1.37361
$478$	35.0000	+	13.2288i	1.60086	+	0.605069i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	25.5000	+	22.4889i	1.15909	+	1.02222i
$485$		0			0
$486$		0			0
$487$			37.0405i			1.67847i	0.543772	+	0.839233i	$0.316996\pi$
$487$			37.0405i			1.67847i	−0.543772	+	0.839233i	$0.683004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			5.29150i			0.238802i	0.992846	+	0.119401i	$0.0380974\pi$
$491$			5.29150i			0.238802i	−0.992846	+	0.119401i	$0.961903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.0000			−0.627986
$498$		0			0
$499$			26.4575i			1.18440i	0.805791	+	0.592200i	$0.201741\pi$
$499$			26.4575i			1.18440i	−0.805791	+	0.592200i	$0.798259\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	10.5000	+	19.8431i	0.467707	+	0.883883i
$505$		0			0
$506$	−14.0000	+	37.0405i	−0.622376	+	1.64665i
$507$		0			0
$508$	21.0000	−	23.8118i	0.931724	−	1.05648i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	2.50000	−	22.4889i	0.110485	−	0.993878i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	21.0000	+	7.93725i	0.922687	+	0.348743i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−3.00000	+	7.93725i	−0.131306	+	0.347404i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	7.00000	+	2.64575i	0.305215	+	0.115360i
$527$		0			0
$528$		0			0
$529$	−5.00000			−0.217391
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−21.0000	−	39.6863i	−0.907062	−	1.71419i
$537$		0			0
$538$		0			0
$539$		−	37.0405i		−	1.59545i
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	15.8745i		−	0.678745i	−0.940652	−	0.339372i	$-0.889785\pi$
$547$		−	15.8745i		−	0.678745i	0.940652	−	0.339372i	$-0.110215\pi$
$548$	−15.0000	−	13.2288i	−0.640768	−	0.565104i
$549$		0			0
$550$	−35.0000	−	13.2288i	−1.49241	−	0.564076i
$551$		0			0
$552$		0			0
$553$	42.0000			1.78602
$554$	5.00000	−	13.2288i	0.212430	−	0.562036i
$555$		0			0
$556$		0			0
$557$	46.0000			1.94908			0.974541	−	0.224208i	$-0.0719796\pi$
$557$	46.0000			1.94908			0.974541	+	0.224208i	$0.0719796\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−13.0000	+	34.3948i	−0.548372	+	1.45086i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	23.8118i		−	1.00000i
$568$	−7.00000	−	13.2288i	−0.293713	−	0.555066i
$569$	−22.0000			−0.922288			−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000			−0.922288			−0.461144	+	0.887325i	$0.652561\pi$
$570$		0			0
$571$			47.6235i			1.99298i	0.0836974	+	0.996491i	$0.473327\pi$
$571$			47.6235i			1.99298i	−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	26.4575i		−	1.10335i
$576$	−13.5000	+	19.8431i	−0.562500	+	0.826797i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−8.50000	+	22.4889i	−0.353553	+	0.935414i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	52.9150i		−	2.19152i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	3.00000	+	23.8118i	0.123299	+	0.978657i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	−33.0000	−	29.1033i	−1.35173	−	1.19212i
$597$		0			0
$598$		0			0
$599$			37.0405i			1.51343i	0.653742	+	0.756717i	$0.273198\pi$
$599$			37.0405i			1.51343i	−0.653742	+	0.756717i	$0.726802\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−7.00000	+	18.5203i	−0.285299	+	0.754829i
$603$			47.6235i			1.93938i
$604$	−7.00000	+	7.93725i	−0.284826	+	0.322962i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$		0			0
$616$	35.0000	−	18.5203i	1.41019	−	0.746203i
$617$	26.0000			1.04672			0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000			1.04672			0.523360	+	0.852111i	$0.324678\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	47.6235i		−	1.89586i	−0.318475	−	0.947931i	$-0.603171\pi$
$631$		−	47.6235i		−	1.89586i	0.318475	−	0.947931i	$-0.396829\pi$
$632$	21.0000	+	39.6863i	0.835335	+	1.57864i
$633$		0			0
$634$	17.0000	−	44.9778i	0.675156	−	1.78630i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	14.0000	+	5.29150i	0.554265	+	0.209493i
$639$			15.8745i			0.627986i
$640$		0			0
$641$	−46.0000			−1.81689			−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000			−1.81689			−0.908445	+	0.418004i	$0.862730\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	21.0000	+	18.5203i	0.827516	+	0.729800i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	22.5000	−	11.9059i	0.883883	−	0.467707i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−21.0000	+	23.8118i	−0.822423	+	0.932541i
$653$	−50.0000			−1.95665			−0.978326	−	0.207072i	$-0.933606\pi$
$653$	−50.0000			−1.95665			−0.978326	+	0.207072i	$0.933606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			26.4575i			1.03064i	0.856998	+	0.515319i	$0.172327\pi$
$659$			26.4575i			1.03064i	−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−7.00000	−	2.64575i	−0.272063	−	0.102830i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	9.00000	−	23.8118i	0.348743	−	0.922687i
$667$			10.5830i			0.409776i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−30.0000			−1.15642			−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000			−1.15642			−0.578208	+	0.815890i	$0.696248\pi$
$674$	15.0000	−	39.6863i	0.577778	−	1.52866i
$675$		0			0
$676$	−19.5000	−	17.1974i	−0.750000	−	0.661438i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			5.29150i			0.202474i	0.994862	+	0.101237i	$0.0322800\pi$
$683$			5.29150i			0.202474i	−0.994862	+	0.101237i	$0.967720\pi$
$684$		0			0
$685$		0			0
$686$	−24.5000	−	9.26013i	−0.935414	−	0.353553i
$687$		0			0
$688$	−21.0000	+	2.64575i	−0.800617	+	0.100868i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$	−42.0000			−1.59545
$694$	49.0000	+	18.5203i	1.86001	+	0.703019i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−17.5000	+	19.8431i	−0.661438	+	0.750000i
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$		0			0
$704$	35.0000	+	23.8118i	1.31911	+	0.897440i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$		−	47.6235i		−	1.78602i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	35.0000	−	39.6863i	1.30801	−	1.48315i
$717$		0			0
$718$	−49.0000	−	18.5203i	−1.82866	−	0.691170i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	9.50000	−	25.1346i	0.353553	−	0.935414i
$723$		0			0
$724$		0			0
$725$	−10.0000			−0.371391
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−7.00000	+	29.1033i	−0.258023	+	1.07276i
$737$	84.0000			3.09418
$738$		0			0
$739$		−	15.8745i		−	0.583953i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$		−	15.8745i		−	0.583953i	0.956425	−	0.291977i	$-0.0943129\pi$
$740$		0			0
$741$		0			0
$742$	−35.0000	−	13.2288i	−1.28489	−	0.485643i
$743$			37.0405i			1.35888i	0.733729	+	0.679442i	$0.237778\pi$
$743$			37.0405i			1.35888i	−0.733729	+	0.679442i	$0.762222\pi$
$744$		0			0
$745$		0			0
$746$	−11.0000	+	29.1033i	−0.402739	+	1.06555i
$747$		0			0
$748$		0			0
$749$	14.0000			0.511549
$750$		0			0
$751$		−	26.4575i		−	0.965448i	−0.875772	−	0.482724i	$-0.839647\pi$
$751$		−	26.4575i		−	0.965448i	0.875772	−	0.482724i	$-0.160353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	54.0000			1.96266			0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000			1.96266			0.981332	+	0.192323i	$0.0616021\pi$
$758$	49.0000	+	18.5203i	1.77976	+	0.672686i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$			47.6235i			1.72409i
$764$	−35.0000	+	39.6863i	−1.26626	+	1.43580i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−27.0000	−	23.8118i	−0.971751	−	0.857004i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	21.0000	+	7.93725i	0.754829	+	0.285299i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−19.0000	+	50.2693i	−0.681183	+	1.80224i
$779$		0			0
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$		0			0
$784$	−3.50000	−	27.7804i	−0.125000	−	0.992157i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	39.0000	+	34.3948i	1.38932	+	1.22526i
$789$		0			0
$790$		0			0
$791$		−	5.29150i		−	0.188144i
$792$	−21.0000	−	39.6863i	−0.746203	−	1.41019i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−27.5000	−	6.61438i	−0.972272	−	0.233854i
$801$		0			0
$802$	−17.0000	+	44.9778i	−0.600291	+	1.58822i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−38.0000			−1.33601			−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000			−1.33601			−0.668004	+	0.744157i	$0.732851\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	7.00000	−	7.93725i	0.245652	−	0.278543i
$813$		0			0
$814$	−42.0000	−	15.8745i	−1.47210	−	0.556401i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$		0			0
$823$		−	47.6235i		−	1.66005i	−0.557725	−	0.830026i	$-0.688326\pi$
$823$		−	47.6235i		−	1.66005i	0.557725	−	0.830026i	$-0.311674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	37.0405i		−	1.28803i	−0.765015	−	0.644013i	$-0.777268\pi$
$827$		−	37.0405i		−	1.28803i	0.765015	−	0.644013i	$-0.222732\pi$
$828$	21.0000	−	23.8118i	0.729800	−	0.827516i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$	13.0000	−	34.3948i	0.448010	−	1.18532i
$843$		0			0
$844$	35.0000	−	39.6863i	1.20475	−	1.36606i
$845$		0			0
$846$		0			0
$847$			44.9778i			1.54545i
$848$	−5.00000	−	39.6863i	−0.171701	−	1.36283i
$849$		0			0
$850$		0			0
$851$		−	31.7490i		−	1.08834i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	7.00000	+	13.2288i	0.239255	+	0.452150i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	35.0000	+	13.2288i	1.19210	+	0.450573i
$863$			58.2065i			1.98137i	0.136162	+	0.990687i	$0.456523\pi$
$863$			58.2065i			1.98137i	−0.136162	+	0.990687i	$0.543477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−84.0000			−2.84950
$870$		0			0
$871$		0			0
$872$	−45.0000	+	23.8118i	−1.52389	+	0.806368i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−50.0000			−1.68838			−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000			−1.68838			−0.844190	+	0.536044i	$0.819918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−10.5000	+	27.7804i	−0.353553	+	0.935414i
$883$		−	58.2065i		−	1.95881i	−0.201916	−	0.979403i	$-0.564717\pi$
$883$		−	58.2065i		−	1.95881i	0.201916	−	0.979403i	$-0.435283\pi$
$884$		0			0
$885$		0			0
$886$	49.0000	+	18.5203i	1.64619	+	0.622200i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	42.0000			1.40863
$890$		0			0
$891$			47.6235i			1.59545i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	24.5000	−	17.1974i	0.818488	−	0.574524i
$897$		0			0
$898$	−1.00000	+	2.64575i	−0.0333704	+	0.0882899i
$899$		0			0
$900$	22.5000	+	19.8431i	0.750000	+	0.661438i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	5.00000	−	2.64575i	0.166298	−	0.0879964i
$905$		0			0
$906$		0			0
$907$			5.29150i			0.175701i	0.996134	+	0.0878507i	$0.0279999\pi$
$907$			5.29150i			0.175701i	−0.996134	+	0.0878507i	$0.972000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			58.2065i			1.92847i	0.265052	+	0.964234i	$0.414611\pi$
$911$			58.2065i			1.92847i	−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		0			0
$914$	3.00000	−	7.93725i	0.0992312	−	0.262541i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			37.0405i			1.22185i	0.791687	+	0.610927i	$0.209203\pi$
$919$			37.0405i			1.22185i	−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	30.0000			0.986394
$926$	−21.0000	−	7.93725i	−0.690103	−	0.260834i
$927$		0			0
$928$	11.0000	+	2.64575i	0.361093	+	0.0868510i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	33.0000	+	29.1033i	1.08095	+	0.953309i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	21.0000	−	55.5608i	0.685674	−	1.81412i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	14.0000	−	37.0405i	0.455179	−	1.20429i
$947$		−	58.2065i		−	1.89146i	−0.324956	−	0.945729i	$-0.605350\pi$
$947$		−	58.2065i		−	1.89146i	0.324956	−	0.945729i	$-0.394650\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	58.0000			1.87880			0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000			1.87880			0.939402	+	0.342817i	$0.111381\pi$
$954$	−15.0000	+	39.6863i	−0.485643	+	1.28489i
$955$		0			0
$956$	−35.0000	+	39.6863i	−1.13198	+	1.28355i
$957$		0			0
$958$		0			0
$959$		−	26.4575i		−	0.854358i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		−	15.8745i		−	0.511549i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	47.6235i		−	1.53147i	−0.643157	−	0.765735i	$-0.722376\pi$
$967$		−	47.6235i		−	1.53147i	0.643157	−	0.765735i	$-0.277624\pi$
$968$	−42.5000	+	22.4889i	−1.36600	+	0.722820i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$	−49.0000	−	18.5203i	−1.57006	−	0.593427i
$975$		0			0
$976$		0			0
$977$	−46.0000			−1.47167			−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000			−1.47167			−0.735835	+	0.677161i	$0.763210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	54.0000			1.72409
$982$	−7.00000	−	2.64575i	−0.223379	−	0.0844293i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	28.0000			0.890348
$990$		0			0
$991$			58.2065i			1.84899i	0.381193	+	0.924496i	$0.375513\pi$
$991$			58.2065i			1.84899i	−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		0			0
$994$	7.00000	−	18.5203i	0.222027	−	0.587427i
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−35.0000	−	13.2288i	−1.10791	−	0.418749i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.2.d.a.27.2	yes	2
3.2	odd	2		252.2.b.a.55.1		2
4.3	odd	2	inner	28.2.d.a.27.1	✓	2
5.2	odd	4		700.2.c.d.699.1		4
5.3	odd	4		700.2.c.d.699.4		4
5.4	even	2		700.2.g.a.251.1		2
7.2	even	3		196.2.f.b.31.2		4
7.3	odd	6		196.2.f.b.19.1		4
7.4	even	3		196.2.f.b.19.1		4
7.5	odd	6		196.2.f.b.31.2		4
7.6	odd	2	CM	28.2.d.a.27.2	yes	2
8.3	odd	2		448.2.f.b.447.2		2
8.5	even	2		448.2.f.b.447.1		2
12.11	even	2		252.2.b.a.55.2		2
16.3	odd	4		1792.2.e.b.895.1		4
16.5	even	4		1792.2.e.b.895.3		4
16.11	odd	4		1792.2.e.b.895.2		4
16.13	even	4		1792.2.e.b.895.4		4
20.3	even	4		700.2.c.d.699.2		4
20.7	even	4		700.2.c.d.699.3		4
20.19	odd	2		700.2.g.a.251.2		2
21.20	even	2		252.2.b.a.55.1		2
24.5	odd	2		4032.2.b.e.3583.1		2
24.11	even	2		4032.2.b.e.3583.2		2
28.3	even	6		196.2.f.b.19.2		4
28.11	odd	6		196.2.f.b.19.2		4
28.19	even	6		196.2.f.b.31.1		4
28.23	odd	6		196.2.f.b.31.1		4
28.27	even	2	inner	28.2.d.a.27.1	✓	2
35.13	even	4		700.2.c.d.699.4		4
35.27	even	4		700.2.c.d.699.1		4
35.34	odd	2		700.2.g.a.251.1		2
56.13	odd	2		448.2.f.b.447.1		2
56.27	even	2		448.2.f.b.447.2		2
84.83	odd	2		252.2.b.a.55.2		2
112.13	odd	4		1792.2.e.b.895.4		4
112.27	even	4		1792.2.e.b.895.2		4
112.69	odd	4		1792.2.e.b.895.3		4
112.83	even	4		1792.2.e.b.895.1		4
140.27	odd	4		700.2.c.d.699.3		4
140.83	odd	4		700.2.c.d.699.2		4
140.139	even	2		700.2.g.a.251.2		2
168.83	odd	2		4032.2.b.e.3583.2		2
168.125	even	2		4032.2.b.e.3583.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.d.a.27.1	✓	2	4.3	odd	2	inner
28.2.d.a.27.1	✓	2	28.27	even	2	inner
28.2.d.a.27.2	yes	2	1.1	even	1	trivial
28.2.d.a.27.2	yes	2	7.6	odd	2	CM
196.2.f.b.19.1		4	7.3	odd	6
196.2.f.b.19.1		4	7.4	even	3
196.2.f.b.19.2		4	28.3	even	6
196.2.f.b.19.2		4	28.11	odd	6
196.2.f.b.31.1		4	28.19	even	6
196.2.f.b.31.1		4	28.23	odd	6
196.2.f.b.31.2		4	7.2	even	3
196.2.f.b.31.2		4	7.5	odd	6
252.2.b.a.55.1		2	3.2	odd	2
252.2.b.a.55.1		2	21.20	even	2
252.2.b.a.55.2		2	12.11	even	2
252.2.b.a.55.2		2	84.83	odd	2
448.2.f.b.447.1		2	8.5	even	2
448.2.f.b.447.1		2	56.13	odd	2
448.2.f.b.447.2		2	8.3	odd	2
448.2.f.b.447.2		2	56.27	even	2
700.2.c.d.699.1		4	5.2	odd	4
700.2.c.d.699.1		4	35.27	even	4
700.2.c.d.699.2		4	20.3	even	4
700.2.c.d.699.2		4	140.83	odd	4
700.2.c.d.699.3		4	20.7	even	4
700.2.c.d.699.3		4	140.27	odd	4
700.2.c.d.699.4		4	5.3	odd	4
700.2.c.d.699.4		4	35.13	even	4
700.2.g.a.251.1		2	5.4	even	2
700.2.g.a.251.1		2	35.34	odd	2
700.2.g.a.251.2		2	20.19	odd	2
700.2.g.a.251.2		2	140.139	even	2
1792.2.e.b.895.1		4	16.3	odd	4
1792.2.e.b.895.1		4	112.83	even	4
1792.2.e.b.895.2		4	16.11	odd	4
1792.2.e.b.895.2		4	112.27	even	4
1792.2.e.b.895.3		4	16.5	even	4
1792.2.e.b.895.3		4	112.69	odd	4
1792.2.e.b.895.4		4	16.13	even	4
1792.2.e.b.895.4		4	112.13	odd	4
4032.2.b.e.3583.1		2	24.5	odd	2
4032.2.b.e.3583.1		2	168.125	even	2
4032.2.b.e.3583.2		2	24.11	even	2
4032.2.b.e.3583.2		2	168.83	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:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	28.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 - 1.32288i) q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+(-0.500000 + 1.32288i) q^{2} +(-1.50000 - 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 - 1.32288i) q^{8} -3.00000 q^{9} +5.29150i q^{11} +(3.50000 + 1.32288i) q^{14} +(0.500000 + 3.96863i) q^{16} +(1.50000 - 3.96863i) q^{18} +(-7.00000 - 2.64575i) q^{22} -5.29150i q^{23} +5.00000 q^{25} +(-3.50000 + 3.96863i) q^{28} -2.00000 q^{29} +(-5.50000 - 1.32288i) q^{32} +(4.50000 + 3.96863i) q^{36} +6.00000 q^{37} +5.29150i q^{43} +(7.00000 - 7.93725i) q^{44} +(7.00000 + 2.64575i) q^{46} -7.00000 q^{49} +(-2.50000 + 6.61438i) q^{50} -10.0000 q^{53} +(-3.50000 - 6.61438i) q^{56} +(1.00000 - 2.64575i) q^{58} +7.93725i q^{63} +(4.50000 - 6.61438i) q^{64} -15.8745i q^{67} -5.29150i q^{71} +(-7.50000 + 3.96863i) q^{72} +(-3.00000 + 7.93725i) q^{74} +14.0000 q^{77} +15.8745i q^{79} +9.00000 q^{81} +(-7.00000 - 2.64575i) q^{86} +(7.00000 + 13.2288i) q^{88} +(-7.00000 + 7.93725i) q^{92} +(3.50000 - 9.26013i) q^{98} -15.8745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 3 q^{4} + 5 q^{8} - 6 q^{9}+O(q^{10}) \) 2 * q - q^2 - 3 * q^4 + 5 * q^8 - 6 * q^9	\( 2 q - q^{2} - 3 q^{4} + 5 q^{8} - 6 q^{9} + 7 q^{14} + q^{16} + 3 q^{18} - 14 q^{22} + 10 q^{25} - 7 q^{28} - 4 q^{29} - 11 q^{32} + 9 q^{36} + 12 q^{37} + 14 q^{44} + 14 q^{46} - 14 q^{49} - 5 q^{50} - 20 q^{53} - 7 q^{56} + 2 q^{58} + 9 q^{64} - 15 q^{72} - 6 q^{74} + 28 q^{77} + 18 q^{81} - 14 q^{86} + 14 q^{88} - 14 q^{92} + 7 q^{98}+O(q^{100}) \) 2 * q - q^2 - 3 * q^4 + 5 * q^8 - 6 * q^9 + 7 * q^14 + q^16 + 3 * q^18 - 14 * q^22 + 10 * q^25 - 7 * q^28 - 4 * q^29 - 11 * q^32 + 9 * q^36 + 12 * q^37 + 14 * q^44 + 14 * q^46 - 14 * q^49 - 5 * q^50 - 20 * q^53 - 7 * q^56 + 2 * q^58 + 9 * q^64 - 15 * q^72 - 6 * q^74 + 28 * q^77 + 18 * q^81 - 14 * q^86 + 14 * q^88 - 14 * q^92 + 7 * q^98

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