LMFDB - Embedded newform 700.2.g.d.251.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,2,Mod(251,700)]

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, 0, 1]))

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

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

chi := DirichletCharacter("700.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	700.g (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:	$5.58952814149$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.3
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	700.251
Dual form			700.2.g.d.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -3.16228 q^{3} -2.00000 q^{4} -4.47214i q^{6} +(-1.58114 - 2.12132i) q^{7} -2.82843i q^{8} +7.00000 q^{9} +6.32456 q^{12} +(3.00000 - 2.23607i) q^{14} +4.00000 q^{16} +9.89949i q^{18} +(5.00000 + 6.70820i) q^{21} +1.41421i q^{23} +8.94427i q^{24} -12.6491 q^{27} +(3.16228 + 4.24264i) q^{28} -6.00000 q^{29} +5.65685i q^{32} -14.0000 q^{36} -4.47214i q^{41} +(-9.48683 + 7.07107i) q^{42} +12.7279i q^{43} -2.00000 q^{46} +9.48683 q^{47} -12.6491 q^{48} +(-2.00000 + 6.70820i) q^{49} -17.8885i q^{54} +(-6.00000 + 4.47214i) q^{56} -8.48528i q^{58} +13.4164i q^{61} +(-11.0680 - 14.8492i) q^{63} -8.00000 q^{64} +4.24264i q^{67} -4.47214i q^{69} -19.7990i q^{72} +19.0000 q^{81} +6.32456 q^{82} +9.48683 q^{83} +(-10.0000 - 13.4164i) q^{84} -18.0000 q^{86} +18.9737 q^{87} +17.8885i q^{89} -2.82843i q^{92} +13.4164i q^{94} -17.8885i q^{96} +(-9.48683 - 2.82843i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 28 q^{9} + 12 q^{14} + 16 q^{16} + 20 q^{21} - 24 q^{29} - 56 q^{36} - 8 q^{46} - 8 q^{49} - 24 q^{56} - 32 q^{64} + 76 q^{81} - 40 q^{84} - 72 q^{86}+O(q^{100}) $ 4 * q - 8 * q^4 + 28 * q^9 + 12 * q^14 + 16 * q^16 + 20 * q^21 - 24 * q^29 - 56 * q^36 - 8 * q^46 - 8 * q^49 - 24 * q^56 - 32 * q^64 + 76 * q^81 - 40 * q^84 - 72 * q^86

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 $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$	−3.16228			−1.82574			−0.912871	−	0.408248i	$-0.866140\pi$
$3$	−3.16228			−1.82574			−0.912871	+	0.408248i	$0.866140\pi$
$4$	−2.00000			−1.00000
$5$		0			0
$5$		0			0
$6$		−	4.47214i		−	1.82574i
$7$	−1.58114	−	2.12132i	−0.597614	−	0.801784i
$7$	−1.58114	−	2.12132i	−0.597614	−	0.801784i
$8$		−	2.82843i		−	1.00000i
$9$	7.00000			2.33333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	6.32456			1.82574
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	3.00000	−	2.23607i	0.801784	−	0.597614i
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$			9.89949i			2.33333i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	5.00000	+	6.70820i	1.09109	+	1.46385i
$22$		0			0
$23$			1.41421i			0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$			1.41421i			0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$			8.94427i			1.82574i
$25$		0			0
$26$		0			0
$27$	−12.6491			−2.43432
$28$	3.16228	+	4.24264i	0.597614	+	0.801784i
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$			5.65685i			1.00000i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−14.0000			−2.33333
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.47214i		−	0.698430i	−0.937043	−	0.349215i	$-0.886448\pi$
$41$		−	4.47214i		−	0.698430i	0.937043	−	0.349215i	$-0.113552\pi$
$42$	−9.48683	+	7.07107i	−1.46385	+	1.09109i
$43$			12.7279i			1.94099i	0.241121	+	0.970495i	$0.422485\pi$
$43$			12.7279i			1.94099i	−0.241121	+	0.970495i	$0.577515\pi$
$44$		0			0
$45$		0			0
$46$	−2.00000			−0.294884
$47$	9.48683			1.38380			0.691898	−	0.721995i	$-0.256775\pi$
$47$	9.48683			1.38380			0.691898	+	0.721995i	$0.256775\pi$
$48$	−12.6491			−1.82574
$49$	−2.00000	+	6.70820i	−0.285714	+	0.958315i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		−	17.8885i		−	2.43432i
$55$		0			0
$56$	−6.00000	+	4.47214i	−0.801784	+	0.597614i
$57$		0			0
$58$		−	8.48528i		−	1.11417i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	−11.0680	−	14.8492i	−1.39443	−	1.87083i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$			4.24264i			0.518321i	0.965834	+	0.259161i	$0.0834459\pi$
$67$			4.24264i			0.518321i	−0.965834	+	0.259161i	$0.916554\pi$
$68$		0			0
$69$		−	4.47214i		−	0.538382i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		−	19.7990i		−	2.33333i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	19.0000			2.11111
$82$	6.32456			0.698430
$83$	9.48683			1.04132			0.520658	−	0.853766i	$-0.325687\pi$
$83$	9.48683			1.04132			0.520658	+	0.853766i	$0.325687\pi$
$84$	−10.0000	−	13.4164i	−1.09109	−	1.46385i
$85$		0			0
$86$	−18.0000			−1.94099
$87$	18.9737			2.03419
$88$		0			0
$89$			17.8885i			1.89618i	0.317999	+	0.948091i	$0.396989\pi$
$89$			17.8885i			1.89618i	−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$		−	2.82843i		−	0.294884i
$93$		0			0
$94$			13.4164i			1.38380i
$95$		0			0
$96$		−	17.8885i		−	1.82574i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−9.48683	−	2.82843i	−0.958315	−	0.285714i
$99$		0			0
$100$		0			0
$101$		−	8.94427i		−	0.889988i	−0.895533	−	0.444994i	$-0.853206\pi$
$101$		−	8.94427i		−	0.889988i	0.895533	−	0.444994i	$-0.146794\pi$
$102$		0			0
$103$	15.8114			1.55794			0.778971	−	0.627060i	$-0.215742\pi$
$103$	15.8114			1.55794			0.778971	+	0.627060i	$0.215742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			18.3848i			1.77732i	0.458563	+	0.888662i	$0.348364\pi$
$107$			18.3848i			1.77732i	−0.458563	+	0.888662i	$0.651636\pi$
$108$	25.2982			2.43432
$109$	16.0000			1.53252			0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000			1.53252			0.766261	+	0.642529i	$0.222115\pi$
$110$		0			0
$111$		0			0
$112$	−6.32456	−	8.48528i	−0.597614	−	0.801784i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	12.0000			1.11417
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−18.9737			−1.71780
$123$			14.1421i			1.27515i
$124$		0			0
$125$		0			0
$126$	21.0000	−	15.6525i	1.87083	−	1.39443i
$127$		−	4.24264i		−	0.376473i	−0.982124	−	0.188237i	$-0.939723\pi$
$127$		−	4.24264i		−	0.376473i	0.982124	−	0.188237i	$-0.0602772\pi$
$128$		−	11.3137i		−	1.00000i
$129$		−	40.2492i		−	3.54375i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	−6.00000			−0.518321
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	6.32456			0.538382
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−30.0000			−2.52646
$142$		0			0
$143$		0			0
$144$	28.0000			2.33333
$145$		0			0
$146$		0			0
$147$	6.32456	−	21.2132i	0.521641	−	1.74964i
$148$		0			0
$149$	−24.0000			−1.96616			−0.983078	−	0.183186i	$-0.941359\pi$
$149$	−24.0000			−1.96616			−0.983078	+	0.183186i	$0.941359\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.00000	−	2.23607i	0.236433	−	0.176227i
$162$			26.8701i			2.11111i
$163$		−	12.7279i		−	0.996928i	−0.866910	−	0.498464i	$-0.833898\pi$
$163$		−	12.7279i		−	0.996928i	0.866910	−	0.498464i	$-0.166102\pi$
$164$			8.94427i			0.698430i
$165$		0			0
$166$			13.4164i			1.04132i
$167$	−9.48683			−0.734113			−0.367057	−	0.930199i	$-0.619634\pi$
$167$	−9.48683			−0.734113			−0.367057	+	0.930199i	$0.619634\pi$
$168$	18.9737	−	14.1421i	1.46385	−	1.09109i
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		−	25.4558i		−	1.94099i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$			26.8328i			2.03419i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−25.2982			−1.89618
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			26.8328i			1.99447i	0.0743294	+	0.997234i	$0.476318\pi$
$181$			26.8328i			1.99447i	−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		−	42.4264i		−	3.13625i
$184$	4.00000			0.294884
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−18.9737			−1.38380
$189$	20.0000	+	26.8328i	1.45479	+	1.95180i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	25.2982			1.82574
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	4.00000	−	13.4164i	0.285714	−	0.958315i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	13.4164i		−	0.946320i
$202$	12.6491			0.889988
$203$	9.48683	+	12.7279i	0.665845	+	0.893325i
$204$		0			0
$205$		0			0
$206$			22.3607i			1.55794i
$207$			9.89949i			0.688062i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	−26.0000			−1.77732
$215$		0			0
$216$			35.7771i			2.43432i
$217$		0			0
$218$			22.6274i			1.53252i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	3.16228			0.211762			0.105881	−	0.994379i	$-0.466234\pi$
$223$	3.16228			0.211762			0.105881	+	0.994379i	$0.466234\pi$
$224$	12.0000	−	8.94427i	0.801784	−	0.597614i
$225$		0			0
$226$		0			0
$227$	−28.4605			−1.88899			−0.944495	−	0.328526i	$-0.893448\pi$
$227$	−28.4605			−1.88899			−0.944495	+	0.328526i	$0.893448\pi$
$228$		0			0
$229$			26.8328i			1.77316i	0.462573	+	0.886581i	$0.346926\pi$
$229$			26.8328i			1.77316i	−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$			16.9706i			1.11417i
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	13.4164i		−	0.864227i	−0.901819	−	0.432113i	$-0.857768\pi$
$241$		−	13.4164i		−	0.864227i	0.901819	−	0.432113i	$-0.142232\pi$
$242$			15.5563i			1.00000i
$243$	−22.1359			−1.42002
$244$		−	26.8328i		−	1.71780i
$245$		0			0
$246$	−20.0000			−1.27515
$247$		0			0
$248$		0			0
$249$	−30.0000			−1.90117
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	22.1359	+	29.6985i	1.39443	+	1.87083i
$253$		0			0
$254$	6.00000			0.376473
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	56.9210			3.54375
$259$		0			0
$260$		0			0
$261$	−42.0000			−2.59973
$262$		0			0
$263$			15.5563i			0.959246i	0.877475	+	0.479623i	$0.159226\pi$
$263$			15.5563i			0.959246i	−0.877475	+	0.479623i	$0.840774\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	56.5685i		−	3.46194i
$268$		−	8.48528i		−	0.518321i
$269$		−	22.3607i		−	1.36335i	−0.731653	−	0.681677i	$-0.761251\pi$
$269$		−	22.3607i		−	1.36335i	0.731653	−	0.681677i	$-0.238749\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$			8.94427i			0.538382i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		−	42.4264i		−	2.52646i
$283$	−15.8114			−0.939889			−0.469945	−	0.882696i	$-0.655726\pi$
$283$	−15.8114			−0.939889			−0.469945	+	0.882696i	$0.655726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.48683	+	7.07107i	−0.559990	+	0.417392i
$288$			39.5980i			2.33333i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	30.0000	+	8.94427i	1.74964	+	0.521641i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		−	33.9411i		−	1.96616i
$299$		0			0
$300$		0			0
$301$	27.0000	−	20.1246i	1.55625	−	1.15996i
$302$		0			0
$303$			28.2843i			1.62489i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	34.7851			1.98529			0.992644	−	0.121070i	$-0.0386326\pi$
$307$	34.7851			1.98529			0.992644	+	0.121070i	$0.0386326\pi$
$308$		0			0
$309$	−50.0000			−2.84440
$310$		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
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		−	58.1378i		−	3.24493i
$322$	3.16228	+	4.24264i	0.176227	+	0.236433i
$323$		0			0
$324$	−38.0000			−2.11111
$325$		0			0
$326$	18.0000			0.996928
$327$	−50.5964			−2.79799
$328$	−12.6491			−0.698430
$329$	−15.0000	−	20.1246i	−0.826977	−	1.10951i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−18.9737			−1.04132
$333$		0			0
$334$		−	13.4164i		−	0.734113i
$335$		0			0
$336$	20.0000	+	26.8328i	1.09109	+	1.46385i
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$			18.3848i			1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.3925	−	6.36396i	0.939108	−	0.343622i
$344$	36.0000			1.94099
$345$		0			0
$346$		0			0
$347$			24.0416i			1.29062i	0.763920	+	0.645311i	$0.223272\pi$
$347$			24.0416i			1.29062i	−0.763920	+	0.645311i	$0.776728\pi$
$348$	−37.9473			−2.03419
$349$		−	26.8328i		−	1.43633i	−0.695874	−	0.718164i	$-0.744983\pi$
$349$		−	26.8328i		−	1.43633i	0.695874	−	0.718164i	$-0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		−	35.7771i		−	1.89618i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$	−37.9473			−1.99447
$363$	−34.7851			−1.82574
$364$		0			0
$365$		0			0
$366$	60.0000			3.13625
$367$	3.16228			0.165070			0.0825348	−	0.996588i	$-0.473698\pi$
$367$	3.16228			0.165070			0.0825348	+	0.996588i	$0.473698\pi$
$368$			5.65685i			0.294884i
$369$		−	31.3050i		−	1.62967i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		−	26.8328i		−	1.38380i
$377$		0			0
$378$	−37.9473	+	28.2843i	−1.95180	+	1.45479i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$			13.4164i			0.687343i
$382$		0			0
$383$	−28.4605			−1.45426			−0.727132	−	0.686498i	$-0.759147\pi$
$383$	−28.4605			−1.45426			−0.727132	+	0.686498i	$0.759147\pi$
$384$			35.7771i			1.82574i
$385$		0			0
$386$		0			0
$387$			89.0955i			4.52898i
$388$		0			0
$389$	24.0000			1.21685			0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000			1.21685			0.608424	+	0.793612i	$0.291802\pi$
$390$		0			0
$391$		0			0
$392$	18.9737	+	5.65685i	0.958315	+	0.285714i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$	18.9737			0.946320
$403$		0			0
$404$			17.8885i			0.889988i
$405$		0			0
$406$	−18.0000	+	13.4164i	−0.893325	+	0.665845i
$407$		0			0
$408$		0			0
$409$			40.2492i			1.99020i	0.0988936	+	0.995098i	$0.468470\pi$
$409$			40.2492i			1.99020i	−0.0988936	+	0.995098i	$0.531530\pi$
$410$		0			0
$411$		0			0
$412$	−31.6228			−1.55794
$413$		0			0
$414$	−14.0000			−0.688062
$415$		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
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	66.4078			3.22886
$424$		0			0
$425$		0			0
$426$		0			0
$427$	28.4605	−	21.2132i	1.37730	−	1.02658i
$428$		−	36.7696i		−	1.77732i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−50.5964			−2.43432
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−32.0000			−1.53252
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−14.0000	+	46.9574i	−0.666667	+	2.23607i
$442$		0			0
$443$			41.0122i			1.94855i	0.225367	+	0.974274i	$0.427642\pi$
$443$			41.0122i			1.94855i	−0.225367	+	0.974274i	$0.572358\pi$
$444$		0			0
$445$		0			0
$446$			4.47214i			0.211762i
$447$	75.8947			3.58969
$448$	12.6491	+	16.9706i	0.597614	+	0.801784i
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	40.2492i		−	1.88899i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	−37.9473			−1.77316
$459$		0			0
$460$		0			0
$461$			8.94427i			0.416576i	0.978068	+	0.208288i	$0.0667892\pi$
$461$			8.94427i			0.416576i	−0.978068	+	0.208288i	$0.933211\pi$
$462$		0			0
$463$			12.7279i			0.591517i	0.955263	+	0.295758i	$0.0955723\pi$
$463$			12.7279i			0.591517i	−0.955263	+	0.295758i	$0.904428\pi$
$464$	−24.0000			−1.11417
$465$		0			0
$466$		0			0
$467$	28.4605			1.31699			0.658497	−	0.752583i	$-0.271192\pi$
$467$	28.4605			1.31699			0.658497	+	0.752583i	$0.271192\pi$
$468$		0			0
$469$	9.00000	−	6.70820i	0.415581	−	0.309756i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	18.9737			0.864227
$483$	−9.48683	+	7.07107i	−0.431666	+	0.321745i
$484$	−22.0000			−1.00000
$485$		0			0
$486$		−	31.3050i		−	1.42002i
$487$		−	38.1838i		−	1.73027i	−0.501538	−	0.865136i	$-0.667232\pi$
$487$		−	38.1838i		−	1.73027i	0.501538	−	0.865136i	$-0.332768\pi$
$488$	37.9473			1.71780
$489$			40.2492i			1.82013i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		−	28.2843i		−	1.27515i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		−	42.4264i		−	1.90117i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	30.0000			1.34030
$502$		0			0
$503$	9.48683			0.422997			0.211498	−	0.977378i	$-0.432166\pi$
$503$	9.48683			0.422997			0.211498	+	0.977378i	$0.432166\pi$
$504$	−42.0000	+	31.3050i	−1.87083	+	1.39443i
$505$		0			0
$506$		0			0
$507$	−41.1096			−1.82574
$508$			8.48528i			0.376473i
$509$		−	44.7214i		−	1.98224i	−0.132973	−	0.991120i	$-0.542452\pi$
$509$		−	44.7214i		−	1.98224i	0.132973	−	0.991120i	$-0.457548\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$			80.4984i			3.54375i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.8885i			0.783711i	0.920027	+	0.391856i	$0.128167\pi$
$521$			17.8885i			0.783711i	−0.920027	+	0.391856i	$0.871833\pi$
$522$		−	59.3970i		−	2.59973i
$523$	34.7851			1.52104			0.760522	−	0.649312i	$-0.224943\pi$
$523$	34.7851			1.52104			0.760522	+	0.649312i	$0.224943\pi$
$524$		0			0
$525$		0			0
$526$	−22.0000			−0.959246
$527$		0			0
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	80.0000			3.46194
$535$		0			0
$536$	12.0000			0.518321
$537$		0			0
$538$	31.6228			1.36335
$539$		0			0
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$		−	84.8528i		−	3.64138i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			46.6690i			1.99542i	0.0676046	+	0.997712i	$0.478464\pi$
$547$			46.6690i			1.99542i	−0.0676046	+	0.997712i	$0.521536\pi$
$548$		0			0
$549$			93.9149i			4.00819i
$550$		0			0
$551$		0			0
$552$	−12.6491			−0.538382
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		−	16.9706i		−	0.715860i
$563$	−47.4342			−1.99911			−0.999556	−	0.0298010i	$-0.990513\pi$
$563$	−47.4342			−1.99911			−0.999556	+	0.0298010i	$0.990513\pi$
$564$	60.0000			2.52646
$565$		0			0
$566$		−	22.3607i		−	0.939889i
$567$	−30.0416	−	40.3051i	−1.26163	−	1.69265i
$568$		0			0
$569$	−36.0000			−1.50920			−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000			−1.50920			−0.754599	+	0.656186i	$0.772169\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	−10.0000	−	13.4164i	−0.417392	−	0.559990i
$575$		0			0
$576$	−56.0000			−2.33333
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			24.0416i			1.00000i
$579$		0			0
$580$		0			0
$581$	−15.0000	−	20.1246i	−0.622305	−	0.834910i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−47.4342			−1.95782			−0.978909	−	0.204298i	$-0.934509\pi$
$587$	−47.4342			−1.95782			−0.978909	+	0.204298i	$0.934509\pi$
$588$	−12.6491	+	42.4264i	−0.521641	+	1.74964i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	48.0000			1.96616
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			40.2492i			1.64180i	0.571072	+	0.820900i	$0.306528\pi$
$601$			40.2492i			1.64180i	−0.571072	+	0.820900i	$0.693472\pi$
$602$	28.4605	+	38.1838i	1.15996	+	1.55625i
$603$			29.6985i			1.20942i
$604$		0			0
$605$		0			0
$606$	−40.0000			−1.62489
$607$	−15.8114			−0.641764			−0.320882	−	0.947119i	$-0.603979\pi$
$607$	−15.8114			−0.641764			−0.320882	+	0.947119i	$0.603979\pi$
$608$		0			0
$609$	−30.0000	−	40.2492i	−1.21566	−	1.63098i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$			49.1935i			1.98529i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		−	70.7107i		−	2.84440i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		−	17.8885i		−	0.717843i
$622$		0			0
$623$	37.9473	−	28.2843i	1.52033	−	1.13319i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$	82.2192			3.24493
$643$	−41.1096			−1.62120			−0.810602	−	0.585597i	$-0.800860\pi$
$643$	−41.1096			−1.62120			−0.810602	+	0.585597i	$0.800860\pi$
$644$	−6.00000	+	4.47214i	−0.236433	+	0.176227i
$645$		0			0
$646$		0			0
$647$	47.4342			1.86483			0.932415	−	0.361390i	$-0.117698\pi$
$647$	47.4342			1.86483			0.932415	+	0.361390i	$0.117698\pi$
$648$		−	53.7401i		−	2.11111i
$649$		0			0
$650$		0			0
$651$		0			0
$652$			25.4558i			0.996928i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		−	71.5542i		−	2.79799i
$655$		0			0
$656$		−	17.8885i		−	0.698430i
$657$		0			0
$658$	28.4605	−	21.2132i	1.10951	−	0.826977i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	40.2492i		−	1.56551i	−0.622328	−	0.782757i	$-0.713813\pi$
$661$		−	40.2492i		−	1.56551i	0.622328	−	0.782757i	$-0.286187\pi$
$662$		0			0
$663$		0			0
$664$		−	26.8328i		−	1.04132i
$665$		0			0
$666$		0			0
$667$		−	8.48528i		−	0.328551i
$668$	18.9737			0.734113
$669$	−10.0000			−0.386622
$670$		0			0
$671$		0			0
$672$	−37.9473	+	28.2843i	−1.46385	+	1.09109i
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	−26.0000			−1.00000
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	90.0000			3.44881
$682$		0			0
$683$		−	43.8406i		−	1.67751i	−0.544505	−	0.838757i	$-0.683283\pi$
$683$		−	43.8406i		−	1.67751i	0.544505	−	0.838757i	$-0.316717\pi$
$684$		0			0
$685$		0			0
$686$	9.00000	+	24.5967i	0.343622	+	0.939108i
$687$		−	84.8528i		−	3.23734i
$688$			50.9117i			1.94099i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	−34.0000			−1.29062
$695$		0			0
$696$		−	53.6656i		−	2.03419i
$697$		0			0
$698$	37.9473			1.43633
$699$		0			0
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18.9737	+	14.1421i	−0.713578	+	0.531870i
$708$		0			0
$709$	46.0000			1.72757			0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000			1.72757			0.863783	+	0.503864i	$0.168089\pi$
$710$		0			0
$711$		0			0
$712$	50.5964			1.89618
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−25.0000	−	33.5410i	−0.931049	−	1.24913i
$722$		−	26.8701i		−	1.00000i
$723$			42.4264i			1.57786i
$724$		−	53.6656i		−	1.99447i
$725$		0			0
$726$		−	49.1935i		−	1.82574i
$727$	−53.7587			−1.99380			−0.996900	−	0.0786754i	$-0.974931\pi$
$727$	−53.7587			−1.99380			−0.996900	+	0.0786754i	$0.974931\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$		0			0
$732$			84.8528i			3.13625i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$			4.47214i			0.165070i
$735$		0			0
$736$	−8.00000			−0.294884
$737$		0			0
$738$	44.2719			1.62967
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	26.8701i		−	0.985767i	−0.870095	−	0.492883i	$-0.835943\pi$
$743$		−	26.8701i		−	0.985767i	0.870095	−	0.492883i	$-0.164057\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	66.4078			2.42974
$748$		0			0
$749$	39.0000	−	29.0689i	1.42503	−	1.06215i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	37.9473			1.38380
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−40.0000	−	53.6656i	−1.45479	−	1.95180i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			35.7771i			1.29692i	0.761249	+	0.648459i	$0.224586\pi$
$761$			35.7771i			1.29692i	−0.761249	+	0.648459i	$0.775414\pi$
$762$	−18.9737			−0.687343
$763$	−25.2982	−	33.9411i	−0.915857	−	1.22875i
$764$		0			0
$765$		0			0
$766$		−	40.2492i		−	1.45426i
$767$		0			0
$768$	−50.5964			−1.82574
$769$			53.6656i			1.93523i	0.252426	+	0.967616i	$0.418771\pi$
$769$			53.6656i			1.93523i	−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	−126.000			−4.52898
$775$		0			0
$776$		0			0
$777$		0			0
$778$			33.9411i			1.21685i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	75.8947			2.71225
$784$	−8.00000	+	26.8328i	−0.285714	+	0.958315i
$785$		0			0
$786$		0			0
$787$	−41.1096			−1.46540			−0.732700	−	0.680552i	$-0.761740\pi$
$787$	−41.1096			−1.46540			−0.732700	+	0.680552i	$0.761740\pi$
$788$		0			0
$789$		−	49.1935i		−	1.75133i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		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
$800$		0			0
$801$			125.220i			4.42442i
$802$		−	25.4558i		−	0.898877i
$803$		0			0
$804$			26.8328i			0.946320i
$805$		0			0
$806$		0			0
$807$			70.7107i			2.48913i
$808$	−25.2982			−0.889988
$809$	−54.0000			−1.89854			−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000			−1.89854			−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−18.9737	−	25.4558i	−0.665845	−	0.893325i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−56.9210			−1.99020
$819$		0			0
$820$		0			0
$821$	−48.0000			−1.67521			−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000			−1.67521			−0.837606	+	0.546275i	$0.816045\pi$
$822$		0			0
$823$			55.1543i			1.92256i	0.275575	+	0.961280i	$0.411132\pi$
$823$			55.1543i			1.92256i	−0.275575	+	0.961280i	$0.588868\pi$
$824$		−	44.7214i		−	1.55794i
$825$		0			0
$826$		0			0
$827$		−	32.5269i		−	1.13107i	−0.824724	−	0.565536i	$-0.808669\pi$
$827$		−	32.5269i		−	1.13107i	0.824724	−	0.565536i	$-0.191331\pi$
$828$		−	19.7990i		−	0.688062i
$829$			13.4164i			0.465971i	0.972480	+	0.232986i	$0.0748495\pi$
$829$			13.4164i			0.465971i	−0.972480	+	0.232986i	$0.925151\pi$
$830$		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
$841$	7.00000			0.241379
$842$		−	11.3137i		−	0.389896i
$843$	37.9473			1.30698
$844$		0			0
$845$		0			0
$846$			93.9149i			3.22886i
$847$	−17.3925	−	23.3345i	−0.597614	−	0.801784i
$848$		0			0
$849$	50.0000			1.71600
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$	30.0000	+	40.2492i	1.02658	+	1.37730i
$855$		0			0
$856$	52.0000			1.77732
$857$		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
$861$	30.0000	−	22.3607i	1.02240	−	0.762050i
$862$		0			0
$863$		−	57.9828i		−	1.97376i	−0.161468	−	0.986878i	$-0.551623\pi$
$863$		−	57.9828i		−	1.97376i	0.161468	−	0.986878i	$-0.448377\pi$
$864$		−	71.5542i		−	2.43432i
$865$		0			0
$866$		0			0
$867$	−53.7587			−1.82574
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	45.2548i		−	1.53252i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			58.1378i			1.95871i	0.202145	+	0.979356i	$0.435209\pi$
$881$			58.1378i			1.95871i	−0.202145	+	0.979356i	$0.564791\pi$
$882$	−66.4078	−	19.7990i	−2.23607	−	0.666667i
$883$		−	55.1543i		−	1.85609i	−0.372467	−	0.928045i	$-0.621488\pi$
$883$		−	55.1543i		−	1.85609i	0.372467	−	0.928045i	$-0.378512\pi$
$884$		0			0
$885$		0			0
$886$	−58.0000			−1.94855
$887$	28.4605			0.955610			0.477805	−	0.878466i	$-0.341433\pi$
$887$	28.4605			0.955610			0.477805	+	0.878466i	$0.341433\pi$
$888$		0			0
$889$	−9.00000	+	6.70820i	−0.301850	+	0.224986i
$890$		0			0
$891$		0			0
$892$	−6.32456			−0.211762
$893$		0			0
$894$			107.331i			3.58969i
$895$		0			0
$896$	−24.0000	+	17.8885i	−0.801784	+	0.597614i
$897$		0			0
$898$			50.9117i			1.69895i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−85.3815	+	63.6396i	−2.84132	+	2.11779i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			4.24264i			0.140875i	0.997516	+	0.0704373i	$0.0224395\pi$
$907$			4.24264i			0.140875i	−0.997516	+	0.0704373i	$0.977561\pi$
$908$	56.9210			1.88899
$909$		−	62.6099i		−	2.07664i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	53.6656i		−	1.77316i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−110.000			−3.62462
$922$	−12.6491			−0.416576
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−18.0000			−0.591517
$927$	110.680			3.63520
$928$		−	33.9411i		−	1.11417i
$929$			49.1935i			1.61399i	0.590561	+	0.806993i	$0.298907\pi$
$929$			49.1935i			1.61399i	−0.590561	+	0.806993i	$0.701093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$			40.2492i			1.31699i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	9.48683	+	12.7279i	0.309756	+	0.415581i
$939$		0			0
$940$		0			0
$941$		−	44.7214i		−	1.45787i	−0.684580	−	0.728937i	$-0.740015\pi$
$941$		−	44.7214i		−	1.45787i	0.684580	−	0.728937i	$-0.259985\pi$
$942$		0			0
$943$	6.32456			0.205956
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	60.8112i		−	1.97610i	−0.154140	−	0.988049i	$-0.549261\pi$
$947$		−	60.8112i		−	1.97610i	0.154140	−	0.988049i	$-0.450739\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$			128.693i			4.14709i
$964$			26.8328i			0.864227i
$965$		0			0
$966$	−10.0000	−	13.4164i	−0.321745	−	0.431666i
$967$			46.6690i			1.50078i	0.660998	+	0.750388i	$0.270133\pi$
$967$			46.6690i			1.50078i	−0.660998	+	0.750388i	$0.729867\pi$
$968$		−	31.1127i		−	1.00000i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	44.2719			1.42002
$973$		0			0
$974$	54.0000			1.73027
$975$		0			0
$976$			53.6656i			1.71780i
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	−56.9210			−1.82013
$979$		0			0
$980$		0			0
$981$	112.000			3.57588
$982$		0			0
$983$	−47.4342			−1.51291			−0.756457	−	0.654043i	$-0.773072\pi$
$983$	−47.4342			−1.51291			−0.756457	+	0.654043i	$0.773072\pi$
$984$	40.0000			1.27515
$985$		0			0
$986$		0			0
$987$	47.4342	+	63.6396i	1.50985	+	2.02567i
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	60.0000			1.90117
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.g.d.251.3		4
4.3	odd	2	inner	700.2.g.d.251.2		4
5.2	odd	4		140.2.c.a.139.2	yes	4
5.3	odd	4		140.2.c.a.139.3	yes	4
5.4	even	2	inner	700.2.g.d.251.2		4
7.6	odd	2	inner	700.2.g.d.251.4		4
20.3	even	4		140.2.c.a.139.2	yes	4
20.7	even	4		140.2.c.a.139.3	yes	4
20.19	odd	2	CM	700.2.g.d.251.3		4
28.27	even	2	inner	700.2.g.d.251.1		4
35.2	odd	12		980.2.s.b.619.3		8
35.3	even	12		980.2.s.b.19.2		8
35.12	even	12		980.2.s.b.619.4		8
35.13	even	4		140.2.c.a.139.4	yes	4
35.17	even	12		980.2.s.b.19.3		8
35.18	odd	12		980.2.s.b.19.1		8
35.23	odd	12		980.2.s.b.619.2		8
35.27	even	4		140.2.c.a.139.1	✓	4
35.32	odd	12		980.2.s.b.19.4		8
35.33	even	12		980.2.s.b.619.1		8
35.34	odd	2	inner	700.2.g.d.251.1		4
40.3	even	4		2240.2.e.a.2239.1		4
40.13	odd	4		2240.2.e.a.2239.3		4
40.27	even	4		2240.2.e.a.2239.3		4
40.37	odd	4		2240.2.e.a.2239.1		4
140.3	odd	12		980.2.s.b.19.3		8
140.23	even	12		980.2.s.b.619.3		8
140.27	odd	4		140.2.c.a.139.4	yes	4
140.47	odd	12		980.2.s.b.619.1		8
140.67	even	12		980.2.s.b.19.1		8
140.83	odd	4		140.2.c.a.139.1	✓	4
140.87	odd	12		980.2.s.b.19.2		8
140.103	odd	12		980.2.s.b.619.4		8
140.107	even	12		980.2.s.b.619.2		8
140.123	even	12		980.2.s.b.19.4		8
140.139	even	2	inner	700.2.g.d.251.4		4
280.13	even	4		2240.2.e.a.2239.2		4
280.27	odd	4		2240.2.e.a.2239.2		4
280.83	odd	4		2240.2.e.a.2239.4		4
280.237	even	4		2240.2.e.a.2239.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.c.a.139.1	✓	4	35.27	even	4
140.2.c.a.139.1	✓	4	140.83	odd	4
140.2.c.a.139.2	yes	4	5.2	odd	4
140.2.c.a.139.2	yes	4	20.3	even	4
140.2.c.a.139.3	yes	4	5.3	odd	4
140.2.c.a.139.3	yes	4	20.7	even	4
140.2.c.a.139.4	yes	4	35.13	even	4
140.2.c.a.139.4	yes	4	140.27	odd	4
700.2.g.d.251.1		4	28.27	even	2	inner
700.2.g.d.251.1		4	35.34	odd	2	inner
700.2.g.d.251.2		4	4.3	odd	2	inner
700.2.g.d.251.2		4	5.4	even	2	inner
700.2.g.d.251.3		4	1.1	even	1	trivial
700.2.g.d.251.3		4	20.19	odd	2	CM
700.2.g.d.251.4		4	7.6	odd	2	inner
700.2.g.d.251.4		4	140.139	even	2	inner
980.2.s.b.19.1		8	35.18	odd	12
980.2.s.b.19.1		8	140.67	even	12
980.2.s.b.19.2		8	35.3	even	12
980.2.s.b.19.2		8	140.87	odd	12
980.2.s.b.19.3		8	35.17	even	12
980.2.s.b.19.3		8	140.3	odd	12
980.2.s.b.19.4		8	35.32	odd	12
980.2.s.b.19.4		8	140.123	even	12
980.2.s.b.619.1		8	35.33	even	12
980.2.s.b.619.1		8	140.47	odd	12
980.2.s.b.619.2		8	35.23	odd	12
980.2.s.b.619.2		8	140.107	even	12
980.2.s.b.619.3		8	35.2	odd	12
980.2.s.b.619.3		8	140.23	even	12
980.2.s.b.619.4		8	35.12	even	12
980.2.s.b.619.4		8	140.103	odd	12
2240.2.e.a.2239.1		4	40.3	even	4
2240.2.e.a.2239.1		4	40.37	odd	4
2240.2.e.a.2239.2		4	280.13	even	4
2240.2.e.a.2239.2		4	280.27	odd	4
2240.2.e.a.2239.3		4	40.13	odd	4
2240.2.e.a.2239.3		4	40.27	even	4
2240.2.e.a.2239.4		4	280.83	odd	4
2240.2.e.a.2239.4		4	280.237	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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -3.16228 q^{3} -2.00000 q^{4} -4.47214i q^{6} +(-1.58114 - 2.12132i) q^{7} -2.82843i q^{8} +7.00000 q^{9} +6.32456 q^{12} +(3.00000 - 2.23607i) q^{14} +4.00000 q^{16} +9.89949i q^{18} +(5.00000 + 6.70820i) q^{21} +1.41421i q^{23} +8.94427i q^{24} -12.6491 q^{27} +(3.16228 + 4.24264i) q^{28} -6.00000 q^{29} +5.65685i q^{32} -14.0000 q^{36} -4.47214i q^{41} +(-9.48683 + 7.07107i) q^{42} +12.7279i q^{43} -2.00000 q^{46} +9.48683 q^{47} -12.6491 q^{48} +(-2.00000 + 6.70820i) q^{49} -17.8885i q^{54} +(-6.00000 + 4.47214i) q^{56} -8.48528i q^{58} +13.4164i q^{61} +(-11.0680 - 14.8492i) q^{63} -8.00000 q^{64} +4.24264i q^{67} -4.47214i q^{69} -19.7990i q^{72} +19.0000 q^{81} +6.32456 q^{82} +9.48683 q^{83} +(-10.0000 - 13.4164i) q^{84} -18.0000 q^{86} +18.9737 q^{87} +17.8885i q^{89} -2.82843i q^{92} +13.4164i q^{94} -17.8885i q^{96} +(-9.48683 - 2.82843i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 28 q^{9} + 12 q^{14} + 16 q^{16} + 20 q^{21} - 24 q^{29} - 56 q^{36} - 8 q^{46} - 8 q^{49} - 24 q^{56} - 32 q^{64} + 76 q^{81} - 40 q^{84} - 72 q^{86}+O(q^{100}) \) 4 * q - 8 * q^4 + 28 * q^9 + 12 * q^14 + 16 * q^16 + 20 * q^21 - 24 * q^29 - 56 * q^36 - 8 * q^46 - 8 * q^49 - 24 * q^56 - 32 * q^64 + 76 * q^81 - 40 * q^84 - 72 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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