LMFDB - Embedded newform 700.2.c.d.699.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,2,Mod(699,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("700.699");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	700.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.58952814149$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			699.4
Root			$1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	700.699
Dual form			700.2.c.d.699.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.32288	+	0.500000i	0.935414	+	0.353553i
$2$	1.32288	+	0.500000i	0.935414	+	0.353553i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	1.50000	+	1.32288i	0.750000	+	0.661438i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.64575			−1.00000
$7$	−2.64575			−1.00000
$8$	1.32288	+	2.50000i	0.467707	+	0.883883i
$9$	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			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−3.50000	−	1.32288i	−0.935414	−	0.353553i
$15$		0			0
$16$	0.500000	+	3.96863i	0.125000	+	0.992157i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	3.96863	+	1.50000i	0.935414	+	0.353553i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−2.64575	+	7.00000i	−0.564076	+	1.49241i
$23$	5.29150			1.10335			0.551677	−	0.834058i	$-0.313988\pi$
$23$	5.29150			1.10335			0.551677	+	0.834058i	$0.313988\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	−3.96863	−	3.50000i	−0.750000	−	0.661438i
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−1.32288	+	5.50000i	−0.233854	+	0.972272i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	4.50000	+	3.96863i	0.750000	+	0.661438i
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−5.29150			−0.806947			−0.403473	−	0.914991i	$-0.632197\pi$
$43$	−5.29150			−0.806947			−0.403473	+	0.914991i	$0.632197\pi$
$44$	−7.00000	+	7.93725i	−1.05529	+	1.19659i
$45$		0			0
$46$	7.00000	+	2.64575i	1.03209	+	0.390095i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$		0			0
$55$		0			0
$56$	−3.50000	−	6.61438i	−0.467707	−	0.883883i
$57$		0			0
$58$	2.64575	+	1.00000i	0.347404	+	0.131306i
$59$		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
$63$	−7.93725			−1.00000
$64$	−4.50000	+	6.61438i	−0.562500	+	0.826797i
$65$		0			0
$66$		0			0
$67$	−15.8745			−1.93938			−0.969690	−	0.244339i	$-0.921429\pi$
$67$	−15.8745			−1.93938			−0.969690	+	0.244339i	$0.921429\pi$
$68$		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	$-0.101667\pi$
$72$	3.96863	+	7.50000i	0.467707	+	0.883883i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	3.00000	−	7.93725i	0.348743	−	0.922687i
$75$		0			0
$76$		0			0
$77$		−	14.0000i		−	1.59545i
$78$		0			0
$79$		−	15.8745i		−	1.78602i	−0.450035	−	0.893011i	$-0.648589\pi$
$79$		−	15.8745i		−	1.78602i	0.450035	−	0.893011i	$-0.351411\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	−7.00000	−	2.64575i	−0.754829	−	0.285299i
$87$		0			0
$88$	−13.2288	+	7.00000i	−1.41019	+	0.746203i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	7.93725	+	7.00000i	0.827516	+	0.729800i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	9.26013	+	3.50000i	0.935414	+	0.353553i
$99$			15.8745i			1.59545i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	5.00000	−	13.2288i	0.485643	−	1.28489i
$107$	5.29150			0.511549			0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150			0.511549			0.255774	+	0.966736i	$0.417670\pi$
$108$		0			0
$109$	18.0000			1.72409			0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000			1.72409			0.862044	+	0.506834i	$0.169184\pi$
$110$		0			0
$111$		0			0
$112$	−1.32288	−	10.5000i	−0.125000	−	0.992157i
$113$			2.00000i			0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$			2.00000i			0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$		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
$127$	15.8745			1.40863			0.704317	−	0.709885i	$-0.251253\pi$
$127$	15.8745			1.40863			0.704317	+	0.709885i	$0.251253\pi$
$128$	−9.26013	+	6.50000i	−0.818488	+	0.574524i
$129$		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
$137$		−	10.0000i		−	0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$		−	10.0000i		−	0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	2.64575	−	7.00000i	0.222027	−	0.587427i
$143$		0			0
$144$	1.50000	+	11.9059i	0.125000	+	0.992157i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	7.93725	−	9.00000i	0.652438	−	0.739795i
$149$	−22.0000			−1.80231			−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000			−1.80231			−0.901155	+	0.433497i	$0.857280\pi$
$150$		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			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	7.93725	−	21.0000i	0.631454	−	1.67067i
$159$		0			0
$160$		0			0
$161$	−14.0000			−1.10335
$162$	11.9059	+	4.50000i	0.935414	+	0.353553i
$163$	15.8745			1.24339			0.621694	−	0.783260i	$-0.286445\pi$
$163$	15.8745			1.24339			0.621694	+	0.783260i	$0.286445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$	−7.93725	−	7.00000i	−0.605210	−	0.533745i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−21.0000	+	2.64575i	−1.58293	+	0.199431i
$177$		0			0
$178$		0			0
$179$		−	26.4575i		−	1.97753i	−0.149487	−	0.988764i	$-0.547762\pi$
$179$		−	26.4575i		−	1.97753i	0.149487	−	0.988764i	$-0.452238\pi$
$180$		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
$193$			18.0000i			1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$			18.0000i			1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$		0			0
$195$		0			0
$196$	10.5000	+	9.26013i	0.750000	+	0.661438i
$197$			26.0000i			1.85242i	0.377004	+	0.926212i	$0.376954\pi$
$197$			26.0000i			1.85242i	−0.377004	+	0.926212i	$0.623046\pi$
$198$	−7.93725	+	21.0000i	−0.564076	+	1.49241i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.29150			−0.371391
$204$		0			0
$205$		0			0
$206$		0			0
$207$	15.8745			1.10335
$208$		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	$0.635539\pi$
$212$	13.2288	−	15.0000i	0.908555	−	1.03020i
$213$		0			0
$214$	7.00000	+	2.64575i	0.478510	+	0.180860i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	23.8118	+	9.00000i	1.61274	+	0.609557i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	3.50000	−	14.5516i	0.233854	−	0.972272i
$225$		0			0
$226$	−1.00000	+	2.64575i	−0.0665190	+	0.175993i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	2.64575	+	5.00000i	0.173702	+	0.328266i
$233$		−	22.0000i		−	1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$		−	22.0000i		−	1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			26.4575i			1.71139i	0.517477	+	0.855697i	$0.326871\pi$
$239$			26.4575i			1.71139i	−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−22.4889	−	8.50000i	−1.44564	−	0.546401i
$243$		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	$0.5\pi$
$252$	−11.9059	−	10.5000i	−0.750000	−	0.661438i
$253$			28.0000i			1.76034i
$254$	21.0000	+	7.93725i	1.31766	+	0.498028i
$255$		0			0
$256$	−15.5000	+	3.96863i	−0.968750	+	0.248039i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$			15.8745i			0.986394i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$	5.29150			0.326288			0.163144	−	0.986602i	$-0.447836\pi$
$263$	5.29150			0.326288			0.163144	+	0.986602i	$0.447836\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−23.8118	−	21.0000i	−1.45453	−	1.28278i
$269$		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
$275$		0			0
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$		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		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	7.00000	−	7.93725i	0.415374	−	0.470989i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−3.96863	+	16.5000i	−0.233854	+	0.972272i
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	15.0000	−	7.93725i	0.871857	−	0.461344i
$297$		0			0
$298$	−29.1033	−	11.0000i	−1.68591	−	0.637213i
$299$		0			0
$300$		0			0
$301$	14.0000			0.806947
$302$	2.64575	−	7.00000i	0.152246	−	0.402805i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	18.5203	−	21.0000i	1.05529	−	1.19659i
$309$		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			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	21.0000	−	23.8118i	1.18134	−	1.33952i
$317$			34.0000i			1.90963i	0.297200	+	0.954815i	$0.403947\pi$
$317$			34.0000i			1.90963i	−0.297200	+	0.954815i	$0.596053\pi$
$318$		0			0
$319$			10.5830i			0.592535i
$320$		0			0
$321$		0			0
$322$	−18.5203	−	7.00000i	−1.03209	−	0.390095i
$323$		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
$333$		−	18.0000i		−	0.986394i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			30.0000i			1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$			30.0000i			1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$	−17.1974	−	6.50000i	−0.935414	−	0.353553i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.5203			−1.00000
$344$	−7.00000	−	13.2288i	−0.377415	−	0.713247i
$345$		0			0
$346$		0			0
$347$	−37.0405			−1.98844			−0.994220	−	0.107366i	$-0.965758\pi$
$347$	−37.0405			−1.98844			−0.994220	+	0.107366i	$0.965758\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−29.1033	−	7.00000i	−1.55121	−	0.373101i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	13.2288	−	35.0000i	0.699162	−	1.84981i
$359$		−	37.0405i		−	1.95492i	−0.211112	−	0.977462i	$-0.567708\pi$
$359$		−	37.0405i		−	1.95492i	0.211112	−	0.977462i	$-0.432292\pi$
$360$		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		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	2.64575	+	21.0000i	0.137919	+	1.09470i
$369$		0			0
$370$		0			0
$371$			26.4575i			1.37361i
$372$		0			0
$373$			22.0000i			1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$			22.0000i			1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			37.0405i			1.90264i	0.308199	+	0.951322i	$0.400274\pi$
$379$			37.0405i			1.90264i	−0.308199	+	0.951322i	$0.599726\pi$
$380$		0			0
$381$		0			0
$382$	13.2288	−	35.0000i	0.676842	−	1.79076i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−9.00000	+	23.8118i	−0.458088	+	1.21199i
$387$	−15.8745			−0.806947
$388$		0			0
$389$	−38.0000			−1.92668			−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000			−1.92668			−0.963338	+	0.268290i	$0.913542\pi$
$390$		0			0
$391$		0			0
$392$	9.26013	+	17.5000i	0.467707	+	0.883883i
$393$		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			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	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
$407$	31.7490			1.57374
$408$		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	$0.718416\pi$
$422$	−13.2288	+	35.0000i	−0.643966	+	1.70377i
$423$		0			0
$424$	25.0000	−	13.2288i	1.21411	−	0.642445i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	7.93725	+	7.00000i	0.383662	+	0.338358i
$429$		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			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		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
$443$	37.0405			1.75985			0.879924	−	0.475114i	$-0.157593\pi$
$443$	37.0405			1.75985			0.879924	+	0.475114i	$0.157593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	11.9059	−	17.5000i	0.562500	−	0.826797i
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$		0			0
$452$	−2.64575	+	3.00000i	−0.124446	+	0.141108i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			6.00000i			0.280668i	0.990104	+	0.140334i	$0.0448177\pi$
$457$			6.00000i			0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−15.8745			−0.737751			−0.368875	−	0.929479i	$-0.620257\pi$
$463$	−15.8745			−0.737751			−0.368875	+	0.929479i	$0.620257\pi$
$464$	1.00000	+	7.93725i	0.0464238	+	0.368478i
$465$		0			0
$466$	11.0000	−	29.1033i	0.509565	−	1.34818i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	42.0000			1.93938
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	28.0000i		−	1.28744i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	30.0000i		−	1.37361i
$478$	−13.2288	+	35.0000i	−0.605069	+	1.60086i
$479$		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
$487$	37.0405			1.67847			0.839233	−	0.543772i	$-0.183004\pi$
$487$	37.0405			1.67847			0.839233	+	0.543772i	$0.183004\pi$
$488$		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
$497$			14.0000i			0.627986i
$498$		0			0
$499$		−	26.4575i		−	1.18440i	−0.805791	−	0.592200i	$-0.798259\pi$
$499$		−	26.4575i		−	1.18440i	0.805791	−	0.592200i	$-0.201741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	−10.5000	−	19.8431i	−0.467707	−	0.883883i
$505$		0			0
$506$	−14.0000	+	37.0405i	−0.622376	+	1.64665i
$507$		0			0
$508$	23.8118	+	21.0000i	1.05648	+	0.931724i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−22.4889	−	2.50000i	−0.993878	−	0.110485i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−7.93725	+	21.0000i	−0.348743	+	0.922687i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	7.93725	+	3.00000i	0.347404	+	0.131306i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		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
$547$	−15.8745			−0.678745			−0.339372	−	0.940652i	$-0.610215\pi$
$547$	−15.8745			−0.678745			−0.339372	+	0.940652i	$0.610215\pi$
$548$	13.2288	−	15.0000i	0.565104	−	0.640768i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			42.0000i			1.78602i
$554$	−5.00000	+	13.2288i	−0.212430	+	0.562036i
$555$		0			0
$556$		0			0
$557$		−	46.0000i		−	1.94908i	−0.224208	−	0.974541i	$-0.571980\pi$
$557$		−	46.0000i		−	1.94908i	0.224208	−	0.974541i	$-0.428020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	34.3948	+	13.0000i	1.45086	+	0.548372i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23.8118			−1.00000
$568$	13.2288	−	7.00000i	0.555066	−	0.293713i
$569$	22.0000			0.922288			0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000			0.922288			0.461144	+	0.887325i	$0.347439\pi$
$570$		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
$575$		0			0
$576$	−13.5000	+	19.8431i	−0.562500	+	0.826797i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−22.4889	−	8.50000i	−0.935414	−	0.353553i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	52.9150			2.19152
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	23.8118	−	3.00000i	0.978657	−	0.123299i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		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	$-0.726802\pi$
$599$		−	37.0405i		−	1.51343i	0.653742	−	0.756717i	$-0.273198\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	18.5203	+	7.00000i	0.754829	+	0.285299i
$603$	−47.6235			−1.93938
$604$	7.00000	−	7.93725i	0.284826	−	0.322962i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			38.0000i			1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$			38.0000i			1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$		0			0
$615$		0			0
$616$	35.0000	−	18.5203i	1.41019	−	0.746203i
$617$		−	26.0000i		−	1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$		−	26.0000i		−	1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$		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
$625$		0			0
$626$		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	$-0.396829\pi$
$632$	39.6863	−	21.0000i	1.57864	−	0.835335i
$633$		0			0
$634$	−17.0000	+	44.9778i	−0.675156	+	1.78630i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−5.29150	+	14.0000i	−0.209493	+	0.554265i
$639$		−	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		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−21.0000	−	18.5203i	−0.827516	−	0.729800i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	11.9059	+	22.5000i	0.467707	+	0.883883i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	23.8118	+	21.0000i	0.932541	+	0.822423i
$653$		−	50.0000i		−	1.95665i	−0.207072	−	0.978326i	$-0.566394\pi$
$653$		−	50.0000i		−	1.95665i	0.207072	−	0.978326i	$-0.433606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	26.4575i		−	1.03064i	−0.856998	−	0.515319i	$-0.827673\pi$
$659$		−	26.4575i		−	1.03064i	0.856998	−	0.515319i	$-0.172327\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−2.64575	+	7.00000i	−0.102830	+	0.272063i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	9.00000	−	23.8118i	0.348743	−	0.922687i
$667$	10.5830			0.409776
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	30.0000i		−	1.15642i	−0.815890	−	0.578208i	$-0.803752\pi$
$673$		−	30.0000i		−	1.15642i	0.815890	−	0.578208i	$-0.196248\pi$
$674$	−15.0000	+	39.6863i	−0.577778	+	1.52866i
$675$		0			0
$676$	−19.5000	−	17.1974i	−0.750000	−	0.661438i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.29150			−0.202474			−0.101237	−	0.994862i	$-0.532280\pi$
$683$	−5.29150			−0.202474			−0.101237	+	0.994862i	$0.532280\pi$
$684$		0			0
$685$		0			0
$686$	−24.5000	−	9.26013i	−0.935414	−	0.353553i
$687$		0			0
$688$	−2.64575	−	21.0000i	−0.100868	−	0.800617i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		−	42.0000i		−	1.59545i
$694$	−49.0000	−	18.5203i	−1.86001	−	0.703019i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−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	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		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
$718$	18.5203	−	49.0000i	0.691170	−	1.82866i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−25.1346	−	9.50000i	−0.935414	−	0.353553i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	−7.00000	+	29.1033i	−0.258023	+	1.07276i
$737$		−	84.0000i		−	3.09418i
$738$		0			0
$739$			15.8745i			0.583953i	0.956425	+	0.291977i	$0.0943129\pi$
$739$			15.8745i			0.583953i	−0.956425	+	0.291977i	$0.905687\pi$
$740$		0			0
$741$		0			0
$742$	−13.2288	+	35.0000i	−0.485643	+	1.28489i
$743$	−37.0405			−1.35888			−0.679442	−	0.733729i	$-0.737778\pi$
$743$	−37.0405			−1.35888			−0.679442	+	0.733729i	$0.737778\pi$
$744$		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
$757$		−	54.0000i		−	1.96266i	−0.192323	−	0.981332i	$-0.561602\pi$
$757$		−	54.0000i		−	1.96266i	0.192323	−	0.981332i	$-0.438398\pi$
$758$	−18.5203	+	49.0000i	−0.672686	+	1.77976i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−47.6235			−1.72409
$764$	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
$772$	−23.8118	+	27.0000i	−0.857004	+	0.971751i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−21.0000	−	7.93725i	−0.754829	−	0.285299i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−50.2693	−	19.0000i	−1.80224	−	0.681183i
$779$		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		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−34.3948	+	39.0000i	−1.22526	+	1.38932i
$789$		0			0
$790$		0			0
$791$		−	5.29150i		−	0.188144i
$792$	−39.6863	+	21.0000i	−1.41019	+	0.746203i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	44.9778	+	17.0000i	1.58822	+	0.600291i
$803$		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	$-0.267149\pi$
$809$	38.0000			1.33601			0.668004	+	0.744157i	$0.267149\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−7.93725	−	7.00000i	−0.278543	−	0.245652i
$813$		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
$823$	47.6235			1.66005			0.830026	−	0.557725i	$-0.188326\pi$
$823$	47.6235			1.66005			0.830026	+	0.557725i	$0.188326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−37.0405			−1.28803			−0.644013	−	0.765015i	$-0.722732\pi$
$827$	−37.0405			−1.28803			−0.644013	+	0.765015i	$0.722732\pi$
$828$	23.8118	+	21.0000i	0.827516	+	0.729800i
$829$		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
$842$	−34.3948	−	13.0000i	−1.18532	−	0.448010i
$843$		0			0
$844$	−35.0000	+	39.6863i	−1.20475	+	1.36606i
$845$		0			0
$846$		0			0
$847$	44.9778			1.54545
$848$	39.6863	−	5.00000i	1.36283	−	0.171701i
$849$		0			0
$850$		0			0
$851$		−	31.7490i		−	1.08834i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	7.00000	+	13.2288i	0.239255	+	0.452150i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	13.2288	−	35.0000i	0.450573	−	1.19210i
$863$	−58.2065			−1.98137			−0.990687	−	0.136162i	$-0.956523\pi$
$863$	−58.2065			−1.98137			−0.990687	+	0.136162i	$0.956523\pi$
$864$		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
$872$	23.8118	+	45.0000i	0.806368	+	1.52389i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			50.0000i			1.68838i	0.536044	+	0.844190i	$0.319918\pi$
$877$			50.0000i			1.68838i	−0.536044	+	0.844190i	$0.680082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	27.7804	+	10.5000i	0.935414	+	0.353553i
$883$	58.2065			1.95881			0.979403	−	0.201916i	$-0.0647168\pi$
$883$	58.2065			1.95881			0.979403	+	0.201916i	$0.0647168\pi$
$884$		0			0
$885$		0			0
$886$	49.0000	+	18.5203i	1.64619	+	0.622200i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		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
$898$	−2.64575	−	1.00000i	−0.0882899	−	0.0333704i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−5.00000	+	2.64575i	−0.166298	+	0.0879964i
$905$		0			0
$906$		0			0
$907$	5.29150			0.175701			0.0878507	−	0.996134i	$-0.472000\pi$
$907$	5.29150			0.175701			0.0878507	+	0.996134i	$0.472000\pi$
$908$		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	$-0.790797\pi$
$919$		−	37.0405i		−	1.22185i	0.791687	−	0.610927i	$-0.209203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−21.0000	−	7.93725i	−0.690103	−	0.260834i
$927$		0			0
$928$	−2.64575	+	11.0000i	−0.0868510	+	0.361093i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	29.1033	−	33.0000i	0.953309	−	1.08095i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	55.5608	+	21.0000i	1.81412	+	0.685674i
$939$		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
$947$	−58.2065			−1.89146			−0.945729	−	0.324956i	$-0.894650\pi$
$947$	−58.2065			−1.89146			−0.945729	+	0.324956i	$0.894650\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			58.0000i			1.87880i	0.342817	+	0.939402i	$0.388619\pi$
$953$			58.0000i			1.87880i	−0.342817	+	0.939402i	$0.611381\pi$
$954$	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
$963$	15.8745			0.511549
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−47.6235			−1.53147			−0.765735	−	0.643157i	$-0.777624\pi$
$967$	−47.6235			−1.53147			−0.765735	+	0.643157i	$0.777624\pi$
$968$	−22.4889	−	42.5000i	−0.722820	−	1.36600i
$969$		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
$977$			46.0000i			1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$977$			46.0000i			1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	54.0000			1.72409
$982$	−2.64575	+	7.00000i	−0.0844293	+	0.223379i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		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			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	13.2288	−	35.0000i	0.418749	−	1.10791i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.c.d.699.4		4
4.3	odd	2	inner	700.2.c.d.699.2		4
5.2	odd	4		28.2.d.a.27.2	yes	2
5.3	odd	4		700.2.g.a.251.1		2
5.4	even	2	inner	700.2.c.d.699.1		4
7.6	odd	2	CM	700.2.c.d.699.4		4
15.2	even	4		252.2.b.a.55.1		2
20.3	even	4		700.2.g.a.251.2		2
20.7	even	4		28.2.d.a.27.1	✓	2
20.19	odd	2	inner	700.2.c.d.699.3		4
28.27	even	2	inner	700.2.c.d.699.2		4
35.2	odd	12		196.2.f.b.31.2		4
35.12	even	12		196.2.f.b.31.2		4
35.13	even	4		700.2.g.a.251.1		2
35.17	even	12		196.2.f.b.19.1		4
35.27	even	4		28.2.d.a.27.2	yes	2
35.32	odd	12		196.2.f.b.19.1		4
35.34	odd	2	inner	700.2.c.d.699.1		4
40.27	even	4		448.2.f.b.447.2		2
40.37	odd	4		448.2.f.b.447.1		2
60.47	odd	4		252.2.b.a.55.2		2
80.27	even	4		1792.2.e.b.895.2		4
80.37	odd	4		1792.2.e.b.895.3		4
80.67	even	4		1792.2.e.b.895.1		4
80.77	odd	4		1792.2.e.b.895.4		4
105.62	odd	4		252.2.b.a.55.1		2
120.77	even	4		4032.2.b.e.3583.1		2
120.107	odd	4		4032.2.b.e.3583.2		2
140.27	odd	4		28.2.d.a.27.1	✓	2
140.47	odd	12		196.2.f.b.31.1		4
140.67	even	12		196.2.f.b.19.2		4
140.83	odd	4		700.2.g.a.251.2		2
140.87	odd	12		196.2.f.b.19.2		4
140.107	even	12		196.2.f.b.31.1		4
140.139	even	2	inner	700.2.c.d.699.3		4
280.27	odd	4		448.2.f.b.447.2		2
280.237	even	4		448.2.f.b.447.1		2
420.167	even	4		252.2.b.a.55.2		2
560.27	odd	4		1792.2.e.b.895.2		4
560.237	even	4		1792.2.e.b.895.4		4
560.307	odd	4		1792.2.e.b.895.1		4
560.517	even	4		1792.2.e.b.895.3		4
840.587	even	4		4032.2.b.e.3583.2		2
840.797	odd	4		4032.2.b.e.3583.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.d.a.27.1	✓	2	20.7	even	4
28.2.d.a.27.1	✓	2	140.27	odd	4
28.2.d.a.27.2	yes	2	5.2	odd	4
28.2.d.a.27.2	yes	2	35.27	even	4
196.2.f.b.19.1		4	35.17	even	12
196.2.f.b.19.1		4	35.32	odd	12
196.2.f.b.19.2		4	140.67	even	12
196.2.f.b.19.2		4	140.87	odd	12
196.2.f.b.31.1		4	140.47	odd	12
196.2.f.b.31.1		4	140.107	even	12
196.2.f.b.31.2		4	35.2	odd	12
196.2.f.b.31.2		4	35.12	even	12
252.2.b.a.55.1		2	15.2	even	4
252.2.b.a.55.1		2	105.62	odd	4
252.2.b.a.55.2		2	60.47	odd	4
252.2.b.a.55.2		2	420.167	even	4
448.2.f.b.447.1		2	40.37	odd	4
448.2.f.b.447.1		2	280.237	even	4
448.2.f.b.447.2		2	40.27	even	4
448.2.f.b.447.2		2	280.27	odd	4
700.2.c.d.699.1		4	5.4	even	2	inner
700.2.c.d.699.1		4	35.34	odd	2	inner
700.2.c.d.699.2		4	4.3	odd	2	inner
700.2.c.d.699.2		4	28.27	even	2	inner
700.2.c.d.699.3		4	20.19	odd	2	inner
700.2.c.d.699.3		4	140.139	even	2	inner
700.2.c.d.699.4		4	1.1	even	1	trivial
700.2.c.d.699.4		4	7.6	odd	2	CM
700.2.g.a.251.1		2	5.3	odd	4
700.2.g.a.251.1		2	35.13	even	4
700.2.g.a.251.2		2	20.3	even	4
700.2.g.a.251.2		2	140.83	odd	4
1792.2.e.b.895.1		4	80.67	even	4
1792.2.e.b.895.1		4	560.307	odd	4
1792.2.e.b.895.2		4	80.27	even	4
1792.2.e.b.895.2		4	560.27	odd	4
1792.2.e.b.895.3		4	80.37	odd	4
1792.2.e.b.895.3		4	560.517	even	4
1792.2.e.b.895.4		4	80.77	odd	4
1792.2.e.b.895.4		4	560.237	even	4
4032.2.b.e.3583.1		2	120.77	even	4
4032.2.b.e.3583.1		2	840.797	odd	4
4032.2.b.e.3583.2		2	120.107	odd	4
4032.2.b.e.3583.2		2	840.587	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.c (of order \(2\), degree \(1\), not minimal)

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

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