LMFDB - Embedded newform 700.2.c.h.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:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			699.4
Root			$-0.228425 + 1.39564i$ of defining polynomial
Character	$\chi$	$=$	700.699
Dual form			700.2.c.h.699.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.228425 + 1.39564i) q^{2} +(-1.89564 - 0.637600i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} +O(q^{10})$	$q+(-0.228425 + 1.39564i) q^{2} +(-1.89564 - 0.637600i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} +6.10985i q^{11} +(0.604356 - 3.69253i) q^{14} +(3.18693 + 2.41733i) q^{16} +(-0.685275 + 4.18693i) q^{18} +(-8.52718 - 1.39564i) q^{22} -9.57395 q^{23} +(5.01540 + 1.68693i) q^{28} -10.1652 q^{29} +(-4.10170 + 3.89564i) q^{32} +(-5.68693 - 1.91280i) q^{36} +6.16515i q^{37} -7.74655 q^{43} +(3.89564 - 11.5821i) q^{44} +(2.18693 - 13.3618i) q^{46} +7.00000 q^{49} +10.0000i q^{53} +(-3.50000 + 6.61438i) q^{56} +(2.32198 - 14.1869i) q^{58} -7.93725 q^{63} +(-4.50000 - 6.61438i) q^{64} +11.4014 q^{67} +11.2107i q^{71} +(3.96863 - 7.50000i) q^{72} +(-8.60436 - 1.40828i) q^{74} -16.1652i q^{77} -1.00905i q^{79} +9.00000 q^{81} +(1.76951 - 10.8114i) q^{86} +(15.2746 + 8.08258i) q^{88} +(18.1488 + 6.10436i) q^{92} +(-1.59898 + 9.76951i) q^{98} +18.3296i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q - 6 * q^4 + 24 * q^9	$ 8 q - 6 q^{4} + 24 q^{9} + 14 q^{14} - 2 q^{16} - 8 q^{29} - 18 q^{36} + 22 q^{44} - 10 q^{46} + 56 q^{49} - 28 q^{56} - 36 q^{64} - 78 q^{74} + 72 q^{81} - 50 q^{86}+O(q^{100}) $ 8 * q - 6 * q^4 + 24 * q^9 + 14 * q^14 - 2 * q^16 - 8 * q^29 - 18 * q^36 + 22 * q^44 - 10 * q^46 + 56 * q^49 - 28 * q^56 - 36 * q^64 - 78 * q^74 + 72 * q^81 - 50 * 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 $

$2$	−0.228425	+	1.39564i	−0.161521	+	0.986869i
$2$	−0.228425	+	1.39564i	−0.161521	+	0.986869i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−1.89564	−	0.637600i	−0.947822	−	0.318800i
$5$		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
$11$			6.10985i			1.84219i	0.389338	+	0.921095i	$0.372704\pi$
$11$			6.10985i			1.84219i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	0.604356	−	3.69253i	0.161521	−	0.986869i
$15$		0			0
$16$	3.18693	+	2.41733i	0.796733	+	0.604332i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.685275	+	4.18693i	−0.161521	+	0.986869i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−8.52718	−	1.39564i	−1.81800	−	0.297552i
$23$	−9.57395			−1.99631			−0.998154	−	0.0607377i	$-0.980655\pi$
$23$	−9.57395			−1.99631			−0.998154	+	0.0607377i	$0.980655\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	5.01540	+	1.68693i	0.947822	+	0.318800i
$29$	−10.1652			−1.88762			−0.943811	−	0.330487i	$-0.892787\pi$
$29$	−10.1652			−1.88762			−0.943811	+	0.330487i	$0.892787\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−4.10170	+	3.89564i	−0.725085	+	0.688659i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.68693	−	1.91280i	−0.947822	−	0.318800i
$37$			6.16515i			1.01354i	0.862080	+	0.506772i	$0.169162\pi$
$37$			6.16515i			1.01354i	−0.862080	+	0.506772i	$0.830838\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−7.74655			−1.18134			−0.590669	−	0.806914i	$-0.701136\pi$
$43$	−7.74655			−1.18134			−0.590669	+	0.806914i	$0.701136\pi$
$44$	3.89564	−	11.5821i	0.587290	−	1.74607i
$45$		0			0
$46$	2.18693	−	13.3618i	0.322445	−	1.97009i
$47$		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	$0.240986\pi$
$53$			10.0000i			1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$		0			0
$55$		0			0
$56$	−3.50000	+	6.61438i	−0.467707	+	0.883883i
$57$		0			0
$58$	2.32198	−	14.1869i	0.304890	−	1.86284i
$59$		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
$67$	11.4014			1.39290			0.696449	−	0.717607i	$-0.254762\pi$
$67$	11.4014			1.39290			0.696449	+	0.717607i	$0.254762\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			11.2107i			1.33046i	0.746639	+	0.665230i	$0.231667\pi$
$71$			11.2107i			1.33046i	−0.746639	+	0.665230i	$0.768333\pi$
$72$	3.96863	−	7.50000i	0.467707	−	0.883883i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	−8.60436	−	1.40828i	−1.00024	−	0.163709i
$75$		0			0
$76$		0			0
$77$		−	16.1652i		−	1.84219i
$78$		0			0
$79$		−	1.00905i		−	0.113527i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$		−	1.00905i		−	0.113527i	0.998388	−	0.0567635i	$-0.0180781\pi$
$80$		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
$86$	1.76951	−	10.8114i	0.190811	−	1.16583i
$87$		0			0
$88$	15.2746	+	8.08258i	1.62828	+	0.861605i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	18.1488	+	6.10436i	1.89214	+	0.636423i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−1.59898	+	9.76951i	−0.161521	+	0.986869i
$99$			18.3296i			1.84219i
$100$		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
$106$	−13.9564	−	2.28425i	−1.35557	−	0.221866i
$107$	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
$109$	0.165151			0.0158186			0.00790932	−	0.999969i	$-0.497482\pi$
$109$	0.165151			0.0158186			0.00790932	+	0.999969i	$0.497482\pi$
$110$		0			0
$111$		0			0
$112$	−8.43183	−	6.39564i	−0.796733	−	0.604332i
$113$		−	17.3303i		−	1.63030i	−0.579252	−	0.815149i	$-0.696655\pi$
$113$		−	17.3303i		−	1.63030i	0.579252	−	0.815149i	$-0.303345\pi$
$114$		0			0
$115$		0			0
$116$	19.2695	+	6.48130i	1.78913	+	0.601774i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−26.3303			−2.39366
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.81307	−	11.0776i	0.161521	−	0.986869i
$127$	5.91915			0.525240			0.262620	−	0.964899i	$-0.415413\pi$
$127$	5.91915			0.525240			0.262620	+	0.964899i	$0.415413\pi$
$128$	10.2592	−	4.76951i	0.906796	−	0.421569i
$129$		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
$134$	−2.60436	+	15.9122i	−0.224982	+	1.37461i
$135$		0			0
$136$		0			0
$137$			10.0000i			0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$			10.0000i			0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−15.6461	−	2.56080i	−1.31299	−	0.214897i
$143$		0			0
$144$	9.56080	+	7.25198i	0.796733	+	0.604332i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	3.93090	−	11.6869i	0.323118	−	0.960660i
$149$	20.1652			1.65199			0.825997	−	0.563675i	$-0.190613\pi$
$149$	20.1652			1.65199			0.825997	+	0.563675i	$0.190613\pi$
$150$		0			0
$151$		−	23.4304i		−	1.90674i	−0.301811	−	0.953368i	$-0.597591\pi$
$151$		−	23.4304i		−	1.90674i	0.301811	−	0.953368i	$-0.402409\pi$
$152$		0			0
$153$		0			0
$154$	22.5608	+	3.69253i	1.81800	+	0.297552i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	1.40828	+	0.230493i	0.112036	+	0.0183370i
$159$		0			0
$160$		0			0
$161$	25.3303			1.99631
$162$	−2.05583	+	12.5608i	−0.161521	+	0.986869i
$163$	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
$172$	14.6847	+	4.93920i	1.11970	+	0.376611i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−14.7695	+	19.4717i	−1.11329	+	1.46773i
$177$		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
$184$	−12.6652	+	23.9349i	−0.933687	+	1.76450i
$185$		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	$0.406534\pi$
$191$			26.4575i			1.91440i	−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$			27.3303i			1.96728i	0.180150	+	0.983639i	$0.442342\pi$
$193$			27.3303i			1.96728i	−0.180150	+	0.983639i	$0.557658\pi$
$194$		0			0
$195$		0			0
$196$	−13.2695	−	4.46320i	−0.947822	−	0.318800i
$197$			3.83485i			0.273222i	0.990625	+	0.136611i	$0.0436210\pi$
$197$			3.83485i			0.273222i	−0.990625	+	0.136611i	$0.956379\pi$
$198$	−25.5815	−	4.18693i	−1.81800	−	0.297552i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	26.8945			1.88762
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−28.7219			−1.99631
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	26.4575i		−	1.82141i	−0.413057	−	0.910705i	$-0.635539\pi$
$211$		−	26.4575i		−	1.82141i	0.413057	−	0.910705i	$-0.364461\pi$
$212$	6.37600	−	18.9564i	0.437906	−	1.30193i
$213$		0			0
$214$	−1.20871	+	7.38505i	−0.0826259	+	0.504832i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.0377247	+	0.230493i	−0.00255504	+	0.0156109i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	10.8521	−	10.3069i	0.725085	−	0.688659i
$225$		0			0
$226$	24.1869	+	3.95868i	1.60889	+	0.263327i
$227$		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
$232$	−13.4472	+	25.4129i	−0.882854	+	1.66844i
$233$			7.33030i			0.480224i	0.970745	+	0.240112i	$0.0771842\pi$
$233$			7.33030i			0.480224i	−0.970745	+	0.240112i	$0.922816\pi$
$234$		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			$\pi$
$242$	6.01450	−	36.7477i	0.386627	−	2.36223i
$243$		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	$0.5\pi$
$252$	15.0462	+	5.06080i	0.947822	+	0.318800i
$253$		−	58.4955i		−	3.67758i
$254$	−1.35208	+	8.26103i	−0.0848372	+	0.518343i
$255$		0			0
$256$	4.31307	+	15.4077i	0.269567	+	0.962982i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		−	16.3115i		−	1.01354i
$260$		0			0
$261$	−30.4955			−1.88762
$262$		0			0
$263$	25.0671			1.54570			0.772851	−	0.634588i	$-0.218830\pi$
$263$	25.0671			1.54570			0.772851	+	0.634588i	$0.218830\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−21.6129	−	7.26951i	−1.32022	−	0.444056i
$269$		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
$274$	−13.9564	−	2.28425i	−0.843139	−	0.137997i
$275$		0			0
$276$		0			0
$277$		−	10.0000i		−	0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$		−	10.0000i		−	0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.33030			0.317979			0.158990	−	0.987280i	$-0.449176\pi$
$281$	5.33030			0.317979			0.158990	+	0.987280i	$0.449176\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	7.14792	−	21.2514i	0.424151	−	1.26104i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.3051	+	11.6869i	−0.725085	+	0.688659i
$289$	−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
$296$	15.4129	+	8.15573i	0.895855	+	0.474042i
$297$		0			0
$298$	−4.60623	+	28.1434i	−0.266832	+	1.63030i
$299$		0			0
$300$		0			0
$301$	20.4955			1.18134
$302$	32.7004	+	5.35208i	1.88170	+	0.307978i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−10.3069	+	30.6434i	−0.587290	+	1.74607i
$309$		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
$316$	−0.643371	+	1.91280i	−0.0361924	+	0.107603i
$317$			26.1652i			1.46958i	0.678294	+	0.734791i	$0.262720\pi$
$317$			26.1652i			1.46958i	−0.678294	+	0.734791i	$0.737280\pi$
$318$		0			0
$319$		−	62.1076i		−	3.47736i
$320$		0			0
$321$		0			0
$322$	−5.78608	+	35.3521i	−0.322445	+	1.97009i
$323$		0			0
$324$	−17.0608	−	5.73840i	−0.947822	−	0.318800i
$325$		0			0
$326$	−3.62614	+	22.1552i	−0.200833	+	1.22706i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	28.5312i		−	1.56821i	−0.620625	−	0.784107i	$-0.713121\pi$
$331$		−	28.5312i		−	1.56821i	0.620625	−	0.784107i	$-0.286879\pi$
$332$		0			0
$333$			18.4955i			1.01354i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	30.0000i		−	1.63420i	−0.576493	−	0.817102i	$-0.695579\pi$
$337$		−	30.0000i		−	1.63420i	0.576493	−	0.817102i	$-0.304421\pi$
$338$	2.96953	−	18.1434i	0.161521	−	0.986869i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.5203			−1.00000
$344$	−10.2477	+	19.3664i	−0.552520	+	1.04417i
$345$		0			0
$346$		0			0
$347$	15.0562			0.808257			0.404128	−	0.914702i	$-0.367575\pi$
$347$	15.0562			0.808257			0.404128	+	0.914702i	$0.367575\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−23.8018	−	25.0608i	−1.26864	−	1.33574i
$353$		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
$358$	−36.9253	−	6.04356i	−1.95156	−	0.319412i
$359$		−	25.4485i		−	1.34312i	−0.740951	−	0.671559i	$-0.765625\pi$
$359$		−	25.4485i		−	1.34312i	0.740951	−	0.671559i	$-0.234375\pi$
$360$		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			$\pi$
$368$	−30.5115	−	23.1434i	−1.59052	−	1.20643i
$369$		0			0
$370$		0			0
$371$		−	26.4575i		−	1.37361i
$372$		0			0
$373$			38.4955i			1.99322i	0.0822766	+	0.996610i	$0.473781\pi$
$373$			38.4955i			1.99322i	−0.0822766	+	0.996610i	$0.526219\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			8.12795i			0.417505i	0.977969	+	0.208752i	$0.0669403\pi$
$379$			8.12795i			0.417505i	−0.977969	+	0.208752i	$0.933060\pi$
$380$		0			0
$381$		0			0
$382$	−36.9253	−	6.04356i	−1.88926	−	0.309215i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−38.1434	−	6.24293i	−1.94145	−	0.317757i
$387$	−23.2397			−1.18134
$388$		0			0
$389$	9.83485			0.498647			0.249323	−	0.968420i	$-0.419792\pi$
$389$	9.83485			0.498647			0.249323	+	0.968420i	$0.419792\pi$
$390$		0			0
$391$		0			0
$392$	9.26013	−	17.5000i	0.467707	−	0.883883i
$393$		0			0
$394$	−5.35208	−	0.875976i	−0.269634	−	0.0441310i
$395$		0			0
$396$	11.6869	−	34.7463i	0.587290	−	1.74607i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−35.3303			−1.76431			−0.882156	−	0.470958i	$-0.843908\pi$
$401$	−35.3303			−1.76431			−0.882156	+	0.470958i	$0.843908\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−6.14337	+	37.5351i	−0.304890	+	1.86284i
$407$	−37.6682			−1.86714
$408$		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
$414$	6.56080	−	40.0855i	0.322445	−	1.97009i
$415$		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
$421$	40.4955			1.97363			0.986814	−	0.161859i	$-0.0517491\pi$
$421$	40.4955			1.97363			0.986814	+	0.161859i	$0.0517491\pi$
$422$	36.9253	+	6.04356i	1.79749	+	0.294196i
$423$		0			0
$424$	25.0000	+	13.2288i	1.21411	+	0.642445i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−10.0308	−	3.37386i	−0.484857	−	0.163082i
$429$		0			0
$430$		0			0
$431$			26.4575i			1.27441i	0.770693	+	0.637207i	$0.219910\pi$
$431$			26.4575i			1.27441i	−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−0.313068	−	0.105301i	−0.0149932	−	0.00504298i
$437$		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
$449$	−35.6606			−1.68293			−0.841464	−	0.540313i	$-0.818306\pi$
$449$	−35.6606			−1.68293			−0.841464	+	0.540313i	$0.818306\pi$
$450$		0			0
$451$		0			0
$452$	−11.0498	+	32.8521i	−0.519739	+	1.54523i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			39.6606i			1.85524i	0.373519	+	0.927622i	$0.378151\pi$
$457$			39.6606i			1.85524i	−0.373519	+	0.927622i	$0.621849\pi$
$458$		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	$0.620257\pi$
$464$	−32.3956	−	24.5725i	−1.50393	−	1.14075i
$465$		0			0
$466$	−10.2305	−	1.67443i	−0.473918	−	0.0775663i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−30.1652			−1.39290
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	47.3303i		−	2.17625i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			30.0000i			1.37361i
$478$	36.9253	+	6.04356i	1.68892	+	0.276426i
$479$		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
$484$	49.9129	+	16.7882i	2.26877	+	0.763100i
$485$		0			0
$486$		0			0
$487$	2.26435			0.102608			0.0513038	−	0.998683i	$-0.483662\pi$
$487$	2.26435			0.102608			0.0513038	+	0.998683i	$0.483662\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			40.7509i			1.83906i	0.393019	+	0.919530i	$0.371431\pi$
$491$			40.7509i			1.83906i	−0.393019	+	0.919530i	$0.628569\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	29.6606i		−	1.33046i
$498$		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		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	−10.5000	+	19.8431i	−0.467707	+	0.883883i
$505$		0			0
$506$	81.6388	+	13.3618i	3.62929	+	0.594006i
$507$		0			0
$508$	−11.2206	−	3.77405i	−0.497834	−	0.167447i
$509$		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
$518$	22.7650	+	3.72595i	1.00024	+	0.163709i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	6.96593	−	42.5608i	0.304890	−	1.86284i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	−5.72595	+	34.9847i	−0.249663	+	1.52541i
$527$		0			0
$528$		0			0
$529$	68.6606			2.98524
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	15.0826	−	28.5034i	0.651468	−	1.23116i
$537$		0			0
$538$		0			0
$539$			42.7690i			1.84219i
$540$		0			0
$541$	−10.4955			−0.451235			−0.225617	−	0.974216i	$-0.572440\pi$
$541$	−10.4955			−0.451235			−0.225617	+	0.974216i	$0.572440\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	46.0424			1.96863			0.984315	−	0.176421i	$-0.0564520\pi$
$547$	46.0424			1.96863			0.984315	+	0.176421i	$0.0564520\pi$
$548$	6.37600	−	18.9564i	0.272369	−	0.809779i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			2.66970i			0.113527i
$554$	13.9564	+	2.28425i	0.592952	+	0.0970485i
$555$		0			0
$556$		0			0
$557$		−	13.8348i		−	0.586201i	−0.956082	−	0.293101i	$-0.905313\pi$
$557$		−	13.8348i		−	0.586201i	0.956082	−	0.293101i	$-0.0946871\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.21758	+	7.43920i	−0.0513603	+	0.313804i
$563$		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
$568$	28.0266	+	14.8303i	1.17597	+	0.622266i
$569$	25.6606			1.07575			0.537874	−	0.843025i	$-0.319228\pi$
$569$	25.6606			1.07575			0.537874	+	0.843025i	$0.319228\pi$
$570$		0			0
$571$			20.3477i			0.851523i	0.904835	+	0.425762i	$0.139994\pi$
$571$			20.3477i			0.851523i	−0.904835	+	0.425762i	$0.860006\pi$
$572$		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	$0.5\pi$
$578$	3.88323	−	23.7259i	0.161521	−	0.986869i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−61.0985			−2.53044
$584$		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
$592$	−14.9032	+	19.6479i	−0.612517	+	0.807524i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−38.2259	−	12.8573i	−1.56580	−	0.526656i
$597$		0			0
$598$		0			0
$599$			9.19255i			0.375598i	0.982208	+	0.187799i	$0.0601353\pi$
$599$			9.19255i			0.375598i	−0.982208	+	0.187799i	$0.939865\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−4.68168	+	28.6044i	−0.190811	+	1.16583i
$603$	34.2041			1.39290
$604$	−14.9392	+	44.4156i	−0.607868	+	1.80725i
$605$		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
$613$		−	8.49545i		−	0.343128i	−0.985173	−	0.171564i	$-0.945118\pi$
$613$		−	8.49545i		−	0.343128i	0.985173	−	0.171564i	$-0.0548821\pi$
$614$		0			0
$615$		0			0
$616$	−40.4129	−	21.3845i	−1.62828	−	0.861605i
$617$		−	49.6606i		−	1.99926i	−0.0271876	−	0.999630i	$-0.508655\pi$
$617$		−	49.6606i		−	1.99926i	0.0271876	−	0.999630i	$-0.491345\pi$
$618$		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
$631$		−	37.6682i		−	1.49955i	−0.661695	−	0.749773i	$-0.730163\pi$
$631$		−	37.6682i		−	1.49955i	0.661695	−	0.749773i	$-0.269837\pi$
$632$	−2.52263	−	1.33485i	−0.100345	−	0.0530974i
$633$		0			0
$634$	−36.5172	−	5.97678i	−1.45028	−	0.237368i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	86.6801	+	14.1869i	3.43170	+	0.561666i
$639$			33.6320i			1.33046i
$640$		0			0
$641$	4.66970			0.184442			0.0922210	−	0.995739i	$-0.470603\pi$
$641$	4.66970			0.184442			0.0922210	+	0.995739i	$0.470603\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−48.0172	−	16.1506i	−1.89214	−	0.636423i
$645$		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
$652$	−30.0924	−	10.1216i	−1.17851	−	0.396392i
$653$			50.0000i			1.95665i	0.207072	+	0.978326i	$0.433606\pi$
$653$			50.0000i			1.95665i	−0.207072	+	0.978326i	$0.566394\pi$
$654$		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			$\pi$
$662$	39.8193	+	6.51723i	1.54762	+	0.253300i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−25.8131	−	4.22483i	−1.00024	−	0.163709i
$667$	97.3207			3.76827
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			30.0000i			1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$			30.0000i			1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$	41.8693	+	6.85275i	1.61275	+	0.263958i
$675$		0			0
$676$	24.6434	+	8.28880i	0.947822	+	0.318800i
$677$		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
$683$	−42.3876			−1.62192			−0.810958	−	0.585105i	$-0.801053\pi$
$683$	−42.3876			−1.62192			−0.810958	+	0.585105i	$0.801053\pi$
$684$		0			0
$685$		0			0
$686$	4.23049	−	25.8477i	0.161521	−	0.986869i
$687$		0			0
$688$	−24.6877	−	18.7259i	−0.941211	−	0.713920i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		−	48.4955i		−	1.84219i
$694$	−3.43920	+	21.0130i	−0.130550	+	0.797644i
$695$		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
$704$	40.4129	−	27.4943i	1.52312	−	1.03623i
$705$		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
$711$		−	3.02715i		−	0.113527i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	16.8693	−	50.1540i	0.630436	−	1.87434i
$717$		0			0
$718$	35.5170	+	5.81307i	1.32548	+	0.216942i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	4.34008	−	26.5172i	0.161521	−	0.986869i
$723$		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
$736$	39.2695	−	37.2967i	1.44749	−	1.37478i
$737$			69.6606i			2.56598i
$738$		0			0
$739$			52.9706i			1.94855i	0.225354	+	0.974277i	$0.427646\pi$
$739$			52.9706i			1.94855i	−0.225354	+	0.974277i	$0.572354\pi$
$740$		0			0
$741$		0			0
$742$	36.9253	+	6.04356i	1.35557	+	0.221866i
$743$	−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
$746$	−53.7259	−	8.79333i	−1.96705	−	0.321947i
$747$		0			0
$748$		0			0
$749$	−14.0000			−0.511549
$750$		0			0
$751$			26.4575i			0.965448i	0.875772	+	0.482724i	$0.160353\pi$
$751$			26.4575i			0.965448i	−0.875772	+	0.482724i	$0.839647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	36.1652i		−	1.31444i	−0.753697	−	0.657222i	$-0.771731\pi$
$757$		−	36.1652i		−	1.31444i	0.753697	−	0.657222i	$-0.228269\pi$
$758$	−11.3437	−	1.85663i	−0.412023	−	0.0674358i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−0.436950			−0.0158186
$764$	16.8693	−	50.1540i	0.610310	−	1.81451i
$765$		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
$772$	17.4258	−	51.8085i	0.627169	−	1.86463i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	5.30852	−	32.4343i	0.190811	−	1.16583i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−2.24653	+	13.7259i	−0.0805419	+	0.492099i
$779$		0			0
$780$		0			0
$781$	−68.4955			−2.45096
$782$		0			0
$783$		0			0
$784$	22.3085	+	16.9213i	0.796733	+	0.604332i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	2.44510	−	7.26951i	0.0871031	−	0.258965i
$789$		0			0
$790$		0			0
$791$			45.8517i			1.63030i
$792$	45.8239	+	24.2477i	1.62828	+	0.861605i
$793$		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
$802$	8.07033	−	49.3085i	0.284973	−	1.74114i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−55.6606			−1.95692			−0.978461	−	0.206430i	$-0.933815\pi$
$809$	−55.6606			−1.95692			−0.978461	+	0.206430i	$0.933815\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−50.9823	−	17.1479i	−1.78913	−	0.601774i
$813$		0			0
$814$	8.60436	−	52.5714i	0.301583	−	1.84262i
$815$		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
$823$	−51.5246			−1.79603			−0.898017	−	0.439961i	$-0.854992\pi$
$823$	−51.5246			−1.79603			−0.898017	+	0.439961i	$0.854992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−19.5849			−0.681032			−0.340516	−	0.940239i	$-0.610602\pi$
$827$	−19.5849			−0.681032			−0.340516	+	0.940239i	$0.610602\pi$
$828$	54.4464	+	18.3131i	1.89214	+	0.636423i
$829$		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
$841$	74.3303			2.56311
$842$	−9.25018	+	56.5172i	−0.318782	+	1.94771i
$843$		0			0
$844$	−16.8693	+	50.1540i	−0.580666	+	1.72637i
$845$		0			0
$846$		0			0
$847$	69.6634			2.39366
$848$	−24.1733	+	31.8693i	−0.830113	+	1.09440i
$849$		0			0
$850$		0			0
$851$		−	59.0249i		−	2.02335i
$852$		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
$862$	−36.9253	−	6.04356i	−1.25768	−	0.205844i
$863$	36.0315			1.22653			0.613263	−	0.789879i	$-0.289857\pi$
$863$	36.0315			1.22653			0.613263	+	0.789879i	$0.289857\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.16515			0.209138
$870$		0			0
$871$		0			0
$872$	0.218475	−	0.412878i	0.00739849	−	0.0139818i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	50.0000i		−	1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$		−	50.0000i		−	1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−4.79693	+	29.3085i	−0.161521	+	0.986869i
$883$	−18.7110			−0.629674			−0.314837	−	0.949146i	$-0.601950\pi$
$883$	−18.7110			−0.629674			−0.314837	+	0.949146i	$0.601950\pi$
$884$		0			0
$885$		0			0
$886$	−8.46099	+	51.6954i	−0.284252	+	1.73674i
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−15.6606			−0.525240
$890$		0			0
$891$			54.9887i			1.84219i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−27.1434	+	12.6189i	−0.906796	+	0.421569i
$897$		0			0
$898$	8.14578	−	49.7695i	0.271828	−	1.66083i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−43.3258	−	22.9258i	−1.44099	−	0.762502i
$905$		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
$911$			15.2469i			0.505151i	0.967577	+	0.252575i	$0.0812776\pi$
$911$			15.2469i			0.505151i	−0.967577	+	0.252575i	$0.918722\pi$
$912$		0			0
$913$		0			0
$914$	−55.3521	−	9.05948i	−1.83088	−	0.299661i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	60.0895i		−	1.98217i	−0.133235	−	0.991084i	$-0.542536\pi$
$919$		−	60.0895i		−	1.98217i	0.133235	−	0.991084i	$-0.457464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	3.62614	−	22.1552i	0.119162	−	0.728064i
$927$		0			0
$928$	41.6944	−	39.5998i	1.36869	−	1.29993i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	4.67380	−	13.8956i	0.153096	−	0.455167i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	6.89048	−	42.0998i	0.224982	−	1.37461i
$939$		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
$946$	66.0562	+	10.8114i	2.14767	+	0.351510i
$947$	−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
$953$			47.3303i			1.53318i	0.642138	+	0.766589i	$0.278048\pi$
$953$			47.3303i			1.53318i	−0.642138	+	0.766589i	$0.721952\pi$
$954$	−41.8693	−	6.85275i	−1.35557	−	0.221866i
$955$		0			0
$956$	−16.8693	+	50.1540i	−0.545593	+	1.62210i
$957$		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	$0.777624\pi$
$968$	−34.8317	+	65.8258i	−1.11953	+	2.11572i
$969$		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
$974$	−0.517235	+	3.16023i	−0.0165733	+	0.101260i
$975$		0			0
$976$		0			0
$977$			59.6606i			1.90871i	0.298672	+	0.954356i	$0.403456\pi$
$977$			59.6606i			1.90871i	−0.298672	+	0.954356i	$0.596544\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.495454			0.0158186
$982$	−56.8737	−	9.30852i	−1.81491	−	0.297047i
$983$		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
$989$	74.1652			2.35831
$990$		0			0
$991$			49.8879i			1.58474i	0.610040	+	0.792370i	$0.291153\pi$
$991$			49.8879i			1.58474i	−0.610040	+	0.792370i	$0.708847\pi$
$992$		0			0
$993$		0			0
$994$	41.3956	+	6.77523i	1.31299	+	0.214897i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	−36.9253	−	6.04356i	−1.16885	−	0.191306i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.c.h.699.4		8
4.3	odd	2	inner	700.2.c.h.699.6		8
5.2	odd	4		700.2.g.c.251.1	✓	4
5.3	odd	4		700.2.g.e.251.4	yes	4
5.4	even	2	inner	700.2.c.h.699.5		8
7.6	odd	2	CM	700.2.c.h.699.4		8
20.3	even	4		700.2.g.e.251.3	yes	4
20.7	even	4		700.2.g.c.251.2	yes	4
20.19	odd	2	inner	700.2.c.h.699.3		8
28.27	even	2	inner	700.2.c.h.699.6		8
35.13	even	4		700.2.g.e.251.4	yes	4
35.27	even	4		700.2.g.c.251.1	✓	4
35.34	odd	2	inner	700.2.c.h.699.5		8
140.27	odd	4		700.2.g.c.251.2	yes	4
140.83	odd	4		700.2.g.e.251.3	yes	4
140.139	even	2	inner	700.2.c.h.699.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.c.h.699.3		8	20.19	odd	2	inner
700.2.c.h.699.3		8	140.139	even	2	inner
700.2.c.h.699.4		8	1.1	even	1	trivial
700.2.c.h.699.4		8	7.6	odd	2	CM
700.2.c.h.699.5		8	5.4	even	2	inner
700.2.c.h.699.5		8	35.34	odd	2	inner
700.2.c.h.699.6		8	4.3	odd	2	inner
700.2.c.h.699.6		8	28.27	even	2	inner
700.2.g.c.251.1	✓	4	5.2	odd	4
700.2.g.c.251.1	✓	4	35.27	even	4
700.2.g.c.251.2	yes	4	20.7	even	4
700.2.g.c.251.2	yes	4	140.27	odd	4
700.2.g.e.251.3	yes	4	20.3	even	4
700.2.g.e.251.3	yes	4	140.83	odd	4
700.2.g.e.251.4	yes	4	5.3	odd	4
700.2.g.e.251.4	yes	4	35.13	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)

\(f(q)\)	\(=\)	\(q+(-0.228425 + 1.39564i) q^{2} +(-1.89564 - 0.637600i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+(-0.228425 + 1.39564i) q^{2} +(-1.89564 - 0.637600i) q^{4} -2.64575 q^{7} +(1.32288 - 2.50000i) q^{8} +3.00000 q^{9} +6.10985i q^{11} +(0.604356 - 3.69253i) q^{14} +(3.18693 + 2.41733i) q^{16} +(-0.685275 + 4.18693i) q^{18} +(-8.52718 - 1.39564i) q^{22} -9.57395 q^{23} +(5.01540 + 1.68693i) q^{28} -10.1652 q^{29} +(-4.10170 + 3.89564i) q^{32} +(-5.68693 - 1.91280i) q^{36} +6.16515i q^{37} -7.74655 q^{43} +(3.89564 - 11.5821i) q^{44} +(2.18693 - 13.3618i) q^{46} +7.00000 q^{49} +10.0000i q^{53} +(-3.50000 + 6.61438i) q^{56} +(2.32198 - 14.1869i) q^{58} -7.93725 q^{63} +(-4.50000 - 6.61438i) q^{64} +11.4014 q^{67} +11.2107i q^{71} +(3.96863 - 7.50000i) q^{72} +(-8.60436 - 1.40828i) q^{74} -16.1652i q^{77} -1.00905i q^{79} +9.00000 q^{81} +(1.76951 - 10.8114i) q^{86} +(15.2746 + 8.08258i) q^{88} +(18.1488 + 6.10436i) q^{92} +(-1.59898 + 9.76951i) q^{98} +18.3296i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q - 6 * q^4 + 24 * q^9	\( 8 q - 6 q^{4} + 24 q^{9} + 14 q^{14} - 2 q^{16} - 8 q^{29} - 18 q^{36} + 22 q^{44} - 10 q^{46} + 56 q^{49} - 28 q^{56} - 36 q^{64} - 78 q^{74} + 72 q^{81} - 50 q^{86}+O(q^{100}) \) 8 * q - 6 * q^4 + 24 * q^9 + 14 * q^14 - 2 * q^16 - 8 * q^29 - 18 * q^36 + 22 * q^44 - 10 * q^46 + 56 * q^49 - 28 * q^56 - 36 * q^64 - 78 * q^74 + 72 * q^81 - 50 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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