LMFDB - Embedded newform 1700.2.o.a.1101.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1700,2,Mod(701,1700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1700.701"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1700 = 2^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1700.o (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$13.5745683436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1101.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1700.1101
Dual form			1700.2.o.a.701.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +(-2.00000 + 2.00000i) q^{11} +2.00000 q^{13} +(-4.00000 - 1.00000i) q^{17} +(4.00000 - 4.00000i) q^{23} +(4.00000 - 4.00000i) q^{27} +(5.00000 + 5.00000i) q^{29} +(4.00000 + 4.00000i) q^{31} +8.00000 q^{33} +(5.00000 + 5.00000i) q^{37} +(-4.00000 - 4.00000i) q^{39} +(-5.00000 + 5.00000i) q^{41} -4.00000i q^{43} +8.00000 q^{47} +7.00000i q^{49} +(6.00000 + 10.0000i) q^{51} -6.00000i q^{53} -12.0000i q^{59} +(3.00000 - 3.00000i) q^{61} -4.00000 q^{67} -16.0000 q^{69} +(-8.00000 - 8.00000i) q^{71} +(1.00000 + 1.00000i) q^{73} +(12.0000 - 12.0000i) q^{79} -1.00000 q^{81} -4.00000i q^{83} -20.0000i q^{87} +16.0000 q^{89} -16.0000i q^{93} +(-5.00000 - 5.00000i) q^{97} +(-10.0000 - 10.0000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 4 q^{11} + 4 q^{13} - 8 q^{17} + 8 q^{23} + 8 q^{27} + 10 q^{29} + 8 q^{31} + 16 q^{33} + 10 q^{37} - 8 q^{39} - 10 q^{41} + 16 q^{47} + 12 q^{51} + 6 q^{61} - 8 q^{67} - 32 q^{69} - 16 q^{71}+ \cdots - 20 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 4 * q^11 + 4 * q^13 - 8 * q^17 + 8 * q^23 + 8 * q^27 + 10 * q^29 + 8 * q^31 + 16 * q^33 + 10 * q^37 - 8 * q^39 - 10 * q^41 + 16 * q^47 + 12 * q^51 + 6 * q^61 - 8 * q^67 - 32 * q^69 - 16 * q^71 + 2 * q^73 + 24 * q^79 - 2 * q^81 + 32 * q^89 - 10 * q^97 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times$.

$n$	$477$	$851$	$1601$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.169102	−	0.985599i	$-0.554087\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$			5.00000i			1.66667i
$10$		0			0
$11$	−2.00000	+	2.00000i	−0.603023	+	0.603023i	−0.941113	−	0.338091i	$-0.890219\pi$
$11$	−2.00000	+	2.00000i	−0.603023	+	0.603023i	0.338091	+	0.941113i	$0.390219\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.00000	−	4.00000i	0.834058	−	0.834058i	−0.154011	−	0.988069i	$-0.549219\pi$
$23$	4.00000	−	4.00000i	0.834058	−	0.834058i	0.988069	+	0.154011i	$0.0492193\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$		0			0
$29$	5.00000	+	5.00000i	0.928477	+	0.928477i	0.997608	−	0.0691309i	$-0.0220226\pi$
$29$	5.00000	+	5.00000i	0.928477	+	0.928477i	−0.0691309	+	0.997608i	$0.522023\pi$
$30$		0			0
$31$	4.00000	+	4.00000i	0.718421	+	0.718421i	0.968282	−	0.249861i	$-0.0803848\pi$
$31$	4.00000	+	4.00000i	0.718421	+	0.718421i	−0.249861	+	0.968282i	$0.580385\pi$
$32$		0			0
$33$	8.00000			1.39262
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	−4.00000	−	4.00000i	−0.640513	−	0.640513i
$40$		0			0
$41$	−5.00000	+	5.00000i	−0.780869	+	0.780869i	−0.979977	−	0.199109i	$-0.936195\pi$
$41$	−5.00000	+	5.00000i	−0.780869	+	0.780869i	0.199109	+	0.979977i	$0.436195\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$	6.00000	+	10.0000i	0.840168	+	1.40028i
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.0000i		−	1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$		−	12.0000i		−	1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$		0			0
$61$	3.00000	−	3.00000i	0.384111	−	0.384111i	−0.488470	−	0.872581i	$-0.662445\pi$
$61$	3.00000	−	3.00000i	0.384111	−	0.384111i	0.872581	+	0.488470i	$0.162445\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$	−16.0000			−1.92617
$70$		0			0
$71$	−8.00000	−	8.00000i	−0.949425	−	0.949425i	0.0493559	−	0.998781i	$-0.484283\pi$
$71$	−8.00000	−	8.00000i	−0.949425	−	0.949425i	−0.998781	+	0.0493559i	$0.984283\pi$
$72$		0			0
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.0000	−	12.0000i	1.35011	−	1.35011i	0.464568	−	0.885537i	$-0.346210\pi$
$79$	12.0000	−	12.0000i	1.35011	−	1.35011i	0.885537	−	0.464568i	$-0.153790\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	20.0000i		−	2.14423i
$88$		0			0
$89$	16.0000			1.69600			0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000			1.69600			0.847998	+	0.529999i	$0.177808\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	16.0000i		−	1.65912i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	0.406138	−	0.913812i	$-0.366875\pi$
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$	−10.0000	−	10.0000i	−1.00504	−	1.00504i
$100$		0			0
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$		0			0
$103$	12.0000			1.18240			0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000			1.18240			0.591198	+	0.806527i	$0.298655\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.0000	−	14.0000i	−1.35343	−	1.35343i	−0.881791	−	0.471640i	$-0.843662\pi$
$107$	−14.0000	−	14.0000i	−1.35343	−	1.35343i	−0.471640	−	0.881791i	$-0.656338\pi$
$108$		0			0
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	−0.836031	−	0.548683i	$-0.815129\pi$
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	0.548683	+	0.836031i	$0.315129\pi$
$110$		0			0
$111$		−	20.0000i		−	1.89832i
$112$		0			0
$113$	−11.0000	+	11.0000i	−1.03479	+	1.03479i	−0.0354205	+	0.999372i	$0.511277\pi$
$113$	−11.0000	+	11.0000i	−1.03479	+	1.03479i	−0.999372	+	0.0354205i	$0.988723\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			10.0000i			0.924500i
$118$		0			0
$119$		0			0
$120$		0			0
$121$			3.00000i			0.272727i
$122$		0			0
$123$	20.0000			1.80334
$124$		0			0
$125$		0			0
$126$		0			0
$127$			20.0000i			1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$			20.0000i			1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		0			0
$129$	−8.00000	+	8.00000i	−0.704361	+	0.704361i
$130$		0			0
$131$	−6.00000	−	6.00000i	−0.524222	−	0.524222i	0.394621	−	0.918844i	$-0.370876\pi$
$131$	−6.00000	−	6.00000i	−0.524222	−	0.524222i	−0.918844	+	0.394621i	$0.870876\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$	−2.00000	−	2.00000i	−0.169638	−	0.169638i	0.617182	−	0.786820i	$-0.288274\pi$
$139$	−2.00000	−	2.00000i	−0.169638	−	0.169638i	−0.786820	+	0.617182i	$0.788274\pi$
$140$		0			0
$141$	−16.0000	−	16.0000i	−1.34744	−	1.34744i
$142$		0			0
$143$	−4.00000	+	4.00000i	−0.334497	+	0.334497i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	14.0000	−	14.0000i	1.15470	−	1.15470i
$148$		0			0
$149$	22.0000			1.80231			0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000			1.80231			0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$		0			0
$153$	5.00000	−	20.0000i	0.404226	−	1.61690i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	16.0000			1.27694			0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000			1.27694			0.638470	+	0.769647i	$0.279568\pi$
$158$		0			0
$159$	−12.0000	+	12.0000i	−0.951662	+	0.951662i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	−	6.00000i	0.469956	−	0.469956i	−0.431944	−	0.901900i	$-0.642172\pi$
$163$	6.00000	−	6.00000i	0.469956	−	0.469956i	0.901900	+	0.431944i	$0.142172\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.00000	+	8.00000i	0.619059	+	0.619059i	0.945290	−	0.326231i	$-0.105779\pi$
$167$	8.00000	+	8.00000i	0.619059	+	0.619059i	−0.326231	+	0.945290i	$0.605779\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.00000	−	7.00000i	−0.532200	−	0.532200i	0.389026	−	0.921227i	$-0.372811\pi$
$173$	−7.00000	−	7.00000i	−0.532200	−	0.532200i	−0.921227	+	0.389026i	$0.872811\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−24.0000	+	24.0000i	−1.80395	+	1.80395i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	17.0000	−	17.0000i	1.26360	−	1.26360i	0.314265	−	0.949335i	$-0.398242\pi$
$181$	17.0000	−	17.0000i	1.26360	−	1.26360i	0.949335	−	0.314265i	$-0.101758\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$		0			0
$186$		0			0
$187$	10.0000	−	6.00000i	0.731272	−	0.438763i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.00000			0.289430			0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000			0.289430			0.144715	+	0.989473i	$0.453773\pi$
$192$		0			0
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	−0.506188	−	0.862423i	$-0.668946\pi$
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	0.862423	+	0.506188i	$0.168946\pi$
$198$		0			0
$199$	12.0000	+	12.0000i	0.850657	+	0.850657i	0.990214	−	0.139557i	$-0.0445677\pi$
$199$	12.0000	+	12.0000i	0.850657	+	0.850657i	−0.139557	+	0.990214i	$0.544568\pi$
$200$		0			0
$201$	8.00000	+	8.00000i	0.564276	+	0.564276i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	20.0000	+	20.0000i	1.39010	+	1.39010i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−10.0000	+	10.0000i	−0.688428	+	0.688428i	−0.961884	−	0.273456i	$-0.911833\pi$
$211$	−10.0000	+	10.0000i	−0.688428	+	0.688428i	0.273456	+	0.961884i	$0.411833\pi$
$212$		0			0
$213$			32.0000i			2.19260i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	−8.00000	−	2.00000i	−0.538138	−	0.134535i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	−0.770357	−	0.637613i	$-0.779922\pi$
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	0.637613	+	0.770357i	$0.279922\pi$
$228$		0			0
$229$		−	10.0000i		−	0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$		−	10.0000i		−	0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	0.739104	−	0.673592i	$-0.235249\pi$
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	−0.673592	+	0.739104i	$0.735249\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−48.0000			−3.11794
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	7.00000	+	7.00000i	0.450910	+	0.450910i	0.895656	−	0.444747i	$-0.146706\pi$
$241$	7.00000	+	7.00000i	0.450910	+	0.450910i	−0.444747	+	0.895656i	$0.646706\pi$
$242$		0			0
$243$	−10.0000	−	10.0000i	−0.641500	−	0.641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−8.00000	+	8.00000i	−0.506979	+	0.506979i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	24.0000i		−	1.49708i	−0.663090	−	0.748539i	$-0.730755\pi$
$257$		−	24.0000i		−	1.49708i	0.663090	−	0.748539i	$-0.269245\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−25.0000	+	25.0000i	−1.54746	+	1.54746i
$262$		0			0
$263$			12.0000i			0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$			12.0000i			0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−32.0000	−	32.0000i	−1.95837	−	1.95837i
$268$		0			0
$269$	5.00000	+	5.00000i	0.304855	+	0.304855i	0.842910	−	0.538055i	$-0.180841\pi$
$269$	5.00000	+	5.00000i	0.304855	+	0.304855i	−0.538055	+	0.842910i	$0.680841\pi$
$270$		0			0
$271$	4.00000			0.242983			0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000			0.242983			0.121491	+	0.992592i	$0.461232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	0.611213	−	0.791466i	$-0.290682\pi$
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	−0.791466	+	0.611213i	$0.790682\pi$
$278$		0			0
$279$	−20.0000	+	20.0000i	−1.19737	+	1.19737i
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$		0			0
$283$	−10.0000	+	10.0000i	−0.594438	+	0.594438i	−0.938827	−	0.344389i	$-0.888086\pi$
$283$	−10.0000	+	10.0000i	−0.594438	+	0.594438i	0.344389	+	0.938827i	$0.388086\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.0000	+	8.00000i	0.882353	+	0.470588i
$290$		0			0
$291$			20.0000i			1.17242i
$292$		0			0
$293$	16.0000			0.934730			0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000			0.934730			0.467365	+	0.884064i	$0.345203\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			16.0000i			0.928414i
$298$		0			0
$299$	8.00000	−	8.00000i	0.462652	−	0.462652i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−16.0000	−	16.0000i	−0.919176	−	0.919176i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	32.0000			1.82634			0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000			1.82634			0.913168	+	0.407583i	$0.133628\pi$
$308$		0			0
$309$	−24.0000	−	24.0000i	−1.36531	−	1.36531i
$310$		0			0
$311$	12.0000	+	12.0000i	0.680458	+	0.680458i	0.960103	−	0.279646i	$-0.0902170\pi$
$311$	12.0000	+	12.0000i	0.680458	+	0.680458i	−0.279646	+	0.960103i	$0.590217\pi$
$312$		0			0
$313$	−17.0000	+	17.0000i	−0.960897	+	0.960897i	−0.999264	−	0.0383669i	$-0.987784\pi$
$313$	−17.0000	+	17.0000i	−0.960897	+	0.960897i	0.0383669	+	0.999264i	$0.487784\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.0000	+	17.0000i	−0.954815	+	0.954815i	−0.999022	−	0.0442073i	$-0.985924\pi$
$317$	−17.0000	+	17.0000i	−0.954815	+	0.954815i	0.0442073	+	0.999022i	$0.485924\pi$
$318$		0			0
$319$	−20.0000			−1.11979
$320$		0			0
$321$			56.0000i			3.12562i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	12.0000			0.663602
$328$		0			0
$329$		0			0
$330$		0			0
$331$			4.00000i			0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$			4.00000i			0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$		0			0
$333$	−25.0000	+	25.0000i	−1.36999	+	1.36999i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.0000	+	13.0000i	0.708155	+	0.708155i	0.966147	−	0.257992i	$-0.0830608\pi$
$337$	13.0000	+	13.0000i	0.708155	+	0.708155i	−0.257992	+	0.966147i	$0.583061\pi$
$338$		0			0
$339$	44.0000			2.38975
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.0000	+	14.0000i	−0.751559	+	0.751559i	−0.974770	−	0.223211i	$-0.928346\pi$
$347$	−14.0000	+	14.0000i	−0.751559	+	0.751559i	0.223211	+	0.974770i	$0.428346\pi$
$348$		0			0
$349$		−	24.0000i		−	1.28469i	−0.766415	−	0.642345i	$-0.777962\pi$
$349$		−	24.0000i		−	1.28469i	0.766415	−	0.642345i	$-0.222038\pi$
$350$		0			0
$351$	8.00000	−	8.00000i	0.427008	−	0.427008i
$352$		0			0
$353$	−24.0000			−1.27739			−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000			−1.27739			−0.638696	+	0.769460i	$0.720526\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			20.0000i			1.05556i	0.849381	+	0.527780i	$0.176975\pi$
$359$			20.0000i			1.05556i	−0.849381	+	0.527780i	$0.823025\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	6.00000	−	6.00000i	0.314918	−	0.314918i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	8.00000	−	8.00000i	0.417597	−	0.417597i	−0.466778	−	0.884375i	$-0.654585\pi$
$367$	8.00000	−	8.00000i	0.417597	−	0.417597i	0.884375	+	0.466778i	$0.154585\pi$
$368$		0			0
$369$	−25.0000	−	25.0000i	−1.30145	−	1.30145i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10.0000	+	10.0000i	0.515026	+	0.515026i
$378$		0			0
$379$	−2.00000	−	2.00000i	−0.102733	−	0.102733i	0.653872	−	0.756605i	$-0.273143\pi$
$379$	−2.00000	−	2.00000i	−0.102733	−	0.102733i	−0.756605	+	0.653872i	$0.773143\pi$
$380$		0			0
$381$	40.0000	−	40.0000i	2.04926	−	2.04926i
$382$		0			0
$383$			4.00000i			0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$			4.00000i			0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	20.0000			1.01666
$388$		0			0
$389$		−	6.00000i		−	0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$		−	6.00000i		−	0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$		0			0
$391$	−20.0000	+	12.0000i	−1.01144	+	0.606866i
$392$		0			0
$393$			24.0000i			1.21064i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.00000	+	7.00000i	−0.351320	+	0.351320i	−0.860601	−	0.509281i	$-0.829912\pi$
$397$	−7.00000	+	7.00000i	−0.351320	+	0.351320i	0.509281	+	0.860601i	$0.329912\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.00000	−	9.00000i	0.449439	−	0.449439i	−0.445729	−	0.895168i	$-0.647056\pi$
$401$	9.00000	−	9.00000i	0.449439	−	0.449439i	0.895168	+	0.445729i	$0.147056\pi$
$402$		0			0
$403$	8.00000	+	8.00000i	0.398508	+	0.398508i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.0000			−0.991363
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$	−12.0000	−	12.0000i	−0.591916	−	0.591916i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			8.00000i			0.391762i
$418$		0			0
$419$	−26.0000	+	26.0000i	−1.27018	+	1.27018i	−0.324192	+	0.945991i	$0.605092\pi$
$419$	−26.0000	+	26.0000i	−1.27018	+	1.27018i	−0.945991	+	0.324192i	$0.894908\pi$
$420$		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
$423$			40.0000i			1.94487i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	16.0000			0.772487
$430$		0			0
$431$	−4.00000	+	4.00000i	−0.192673	+	0.192673i	−0.796850	−	0.604177i	$-0.793502\pi$
$431$	−4.00000	+	4.00000i	−0.192673	+	0.192673i	0.604177	+	0.796850i	$0.293502\pi$
$432$		0			0
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	28.0000	+	28.0000i	1.33637	+	1.33637i	0.899551	+	0.436816i	$0.143894\pi$
$439$	28.0000	+	28.0000i	1.33637	+	1.33637i	0.436816	+	0.899551i	$0.356106\pi$
$440$		0			0
$441$	−35.0000			−1.66667
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−44.0000	−	44.0000i	−2.08113	−	2.08113i
$448$		0			0
$449$	23.0000	−	23.0000i	1.08544	−	1.08544i	0.0894454	−	0.995992i	$-0.471491\pi$
$449$	23.0000	−	23.0000i	1.08544	−	1.08544i	0.995992	−	0.0894454i	$-0.0285095\pi$
$450$		0			0
$451$		−	20.0000i		−	0.941763i
$452$		0			0
$453$	−16.0000	+	16.0000i	−0.751746	+	0.751746i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			32.0000i			1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$			32.0000i			1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$		0			0
$459$	−20.0000	+	12.0000i	−0.933520	+	0.560112i
$460$		0			0
$461$		−	24.0000i		−	1.11779i	−0.829238	−	0.558896i	$-0.811225\pi$
$461$		−	24.0000i		−	1.11779i	0.829238	−	0.558896i	$-0.188775\pi$
$462$		0			0
$463$	−12.0000			−0.557687			−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000			−0.557687			−0.278844	+	0.960337i	$0.589951\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−32.0000	−	32.0000i	−1.47448	−	1.47448i
$472$		0			0
$473$	8.00000	+	8.00000i	0.367840	+	0.367840i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	30.0000			1.37361
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$	10.0000	+	10.0000i	0.455961	+	0.455961i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.00000	+	8.00000i	−0.362515	+	0.362515i	−0.864738	−	0.502223i	$-0.832516\pi$
$487$	−8.00000	+	8.00000i	−0.362515	+	0.362515i	0.502223	+	0.864738i	$0.332516\pi$
$488$		0			0
$489$	−24.0000			−1.08532
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−15.0000	−	25.0000i	−0.675566	−	1.12594i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.00000	−	6.00000i	0.268597	−	0.268597i	−0.559938	−	0.828535i	$-0.689175\pi$
$499$	6.00000	−	6.00000i	0.268597	−	0.268597i	0.828535	+	0.559938i	$0.189175\pi$
$500$		0			0
$501$		−	32.0000i		−	1.42965i
$502$		0			0
$503$	8.00000	−	8.00000i	0.356702	−	0.356702i	−0.505894	−	0.862596i	$-0.668837\pi$
$503$	8.00000	−	8.00000i	0.356702	−	0.356702i	0.862596	+	0.505894i	$0.168837\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	18.0000	+	18.0000i	0.799408	+	0.799408i
$508$		0			0
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−16.0000	+	16.0000i	−0.703679	+	0.703679i
$518$		0			0
$519$			28.0000i			1.22906i
$520$		0			0
$521$	29.0000	−	29.0000i	1.27051	−	1.27051i	0.324694	−	0.945819i	$-0.394739\pi$
$521$	29.0000	−	29.0000i	1.27051	−	1.27051i	0.945819	−	0.324694i	$-0.105261\pi$
$522$		0			0
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000	−	20.0000i	−0.522728	−	0.871214i
$528$		0			0
$529$		−	9.00000i		−	0.391304i
$530$		0			0
$531$	60.0000			2.60378
$532$		0			0
$533$	−10.0000	+	10.0000i	−0.433148	+	0.433148i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14.0000	−	14.0000i	−0.603023	−	0.603023i
$540$		0			0
$541$	−27.0000	−	27.0000i	−1.16082	−	1.16082i	−0.984296	−	0.176524i	$-0.943515\pi$
$541$	−27.0000	−	27.0000i	−1.16082	−	1.16082i	−0.176524	−	0.984296i	$-0.556485\pi$
$542$		0			0
$543$	−68.0000			−2.91816
$544$		0			0
$545$		0			0
$546$		0			0
$547$	22.0000	+	22.0000i	0.940652	+	0.940652i	0.998335	−	0.0576829i	$-0.0183712\pi$
$547$	22.0000	+	22.0000i	0.940652	+	0.940652i	−0.0576829	+	0.998335i	$0.518371\pi$
$548$		0			0
$549$	15.0000	+	15.0000i	0.640184	+	0.640184i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$	−32.0000	−	8.00000i	−1.35104	−	0.337760i
$562$		0			0
$563$		−	44.0000i		−	1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$		−	44.0000i		−	1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−2.00000	+	2.00000i	−0.0836974	+	0.0836974i	−0.747716	−	0.664019i	$-0.768850\pi$
$571$	−2.00000	+	2.00000i	−0.0836974	+	0.0836974i	0.664019	+	0.747716i	$0.268850\pi$
$572$		0			0
$573$	−8.00000	−	8.00000i	−0.334205	−	0.334205i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$	20.0000			0.831172
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.0000	+	12.0000i	0.496989	+	0.496989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	24.0000i		−	0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$		−	24.0000i		−	0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−20.0000			−0.822690
$592$		0			0
$593$		−	22.0000i		−	0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$		−	22.0000i		−	0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	48.0000i		−	1.96451i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	3.00000	−	3.00000i	0.122373	−	0.122373i	−0.643268	−	0.765641i	$-0.722422\pi$
$601$	3.00000	−	3.00000i	0.122373	−	0.122373i	0.765641	+	0.643268i	$0.222422\pi$
$602$		0			0
$603$		−	20.0000i		−	0.814463i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	16.0000	+	16.0000i	0.649420	+	0.649420i	0.952853	−	0.303433i	$-0.0981328\pi$
$607$	16.0000	+	16.0000i	0.649420	+	0.649420i	−0.303433	+	0.952853i	$0.598133\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16.0000			0.647291
$612$		0			0
$613$	16.0000			0.646234			0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000			0.646234			0.323117	+	0.946359i	$0.395269\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23.0000	−	23.0000i	−0.925945	−	0.925945i	0.0714958	−	0.997441i	$-0.477223\pi$
$617$	−23.0000	−	23.0000i	−0.925945	−	0.925945i	−0.997441	+	0.0714958i	$0.977223\pi$
$618$		0			0
$619$	14.0000	−	14.0000i	0.562708	−	0.562708i	−0.367368	−	0.930076i	$-0.619741\pi$
$619$	14.0000	−	14.0000i	0.562708	−	0.562708i	0.930076	+	0.367368i	$0.119741\pi$
$620$		0			0
$621$		−	32.0000i		−	1.28412i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.0000	−	25.0000i	−0.598089	−	0.996815i
$630$		0			0
$631$			4.00000i			0.159237i	0.996825	+	0.0796187i	$0.0253703\pi$
$631$			4.00000i			0.159237i	−0.996825	+	0.0796187i	$0.974630\pi$
$632$		0			0
$633$	40.0000			1.58986
$634$		0			0
$635$		0			0
$636$		0			0
$637$			14.0000i			0.554700i
$638$		0			0
$639$	40.0000	−	40.0000i	1.58238	−	1.58238i
$640$		0			0
$641$	−15.0000	−	15.0000i	−0.592464	−	0.592464i	0.345832	−	0.938296i	$-0.387597\pi$
$641$	−15.0000	−	15.0000i	−0.592464	−	0.592464i	−0.938296	+	0.345832i	$0.887597\pi$
$642$		0			0
$643$	22.0000	+	22.0000i	0.867595	+	0.867595i	0.992206	−	0.124610i	$-0.0397681\pi$
$643$	22.0000	+	22.0000i	0.867595	+	0.867595i	−0.124610	+	0.992206i	$0.539768\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.0000			−0.629025			−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000			−0.629025			−0.314512	+	0.949253i	$0.601841\pi$
$648$		0			0
$649$	24.0000	+	24.0000i	0.942082	+	0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.00000	−	7.00000i	0.273931	−	0.273931i	−0.556749	−	0.830681i	$-0.687952\pi$
$653$	7.00000	−	7.00000i	0.273931	−	0.273931i	0.830681	+	0.556749i	$0.187952\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−5.00000	+	5.00000i	−0.195069	+	0.195069i
$658$		0			0
$659$	−40.0000			−1.55818			−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000			−1.55818			−0.779089	+	0.626913i	$0.784318\pi$
$660$		0			0
$661$			24.0000i			0.933492i	0.884391	+	0.466746i	$0.154574\pi$
$661$			24.0000i			0.933492i	−0.884391	+	0.466746i	$0.845426\pi$
$662$		0			0
$663$	12.0000	+	20.0000i	0.466041	+	0.776736i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	40.0000			1.54881
$668$		0			0
$669$		0			0
$670$		0			0
$671$			12.0000i			0.463255i
$672$		0			0
$673$	−19.0000	+	19.0000i	−0.732396	+	0.732396i	−0.971094	−	0.238698i	$-0.923279\pi$
$673$	−19.0000	+	19.0000i	−0.732396	+	0.732396i	0.238698	+	0.971094i	$0.423279\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.00000	+	5.00000i	0.192166	+	0.192166i	0.796631	−	0.604466i	$-0.206613\pi$
$677$	5.00000	+	5.00000i	0.192166	+	0.192166i	−0.604466	+	0.796631i	$0.706613\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	8.00000			0.306561
$682$		0			0
$683$	22.0000	+	22.0000i	0.841807	+	0.841807i	0.989094	−	0.147287i	$-0.0470541\pi$
$683$	22.0000	+	22.0000i	0.841807	+	0.841807i	−0.147287	+	0.989094i	$0.547054\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−20.0000	+	20.0000i	−0.763048	+	0.763048i
$688$		0			0
$689$		−	12.0000i		−	0.457164i
$690$		0			0
$691$	−26.0000	+	26.0000i	−0.989087	+	0.989087i	−0.999941	−	0.0108545i	$-0.996545\pi$
$691$	−26.0000	+	26.0000i	−0.989087	+	0.989087i	0.0108545	+	0.999941i	$0.496545\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	25.0000	−	15.0000i	0.946943	−	0.568166i
$698$		0			0
$699$		−	4.00000i		−	0.151294i
$700$		0			0
$701$	8.00000			0.302156			0.151078	−	0.988522i	$-0.451726\pi$
$701$	8.00000			0.302156			0.151078	+	0.988522i	$0.451726\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−15.0000	−	15.0000i	−0.563337	−	0.563337i	0.366917	−	0.930254i	$-0.380413\pi$
$709$	−15.0000	−	15.0000i	−0.563337	−	0.563337i	−0.930254	+	0.366917i	$0.880413\pi$
$710$		0			0
$711$	60.0000	+	60.0000i	2.25018	+	2.25018i
$712$		0			0
$713$	32.0000			1.19841
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−24.0000	−	24.0000i	−0.896296	−	0.896296i
$718$		0			0
$719$	−4.00000	−	4.00000i	−0.149175	−	0.149175i	0.628575	−	0.777749i	$-0.283639\pi$
$719$	−4.00000	−	4.00000i	−0.149175	−	0.149175i	−0.777749	+	0.628575i	$0.783639\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	28.0000i		−	1.04133i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$			43.0000i			1.59259i
$730$		0			0
$731$	−4.00000	+	16.0000i	−0.147945	+	0.591781i
$732$		0			0
$733$		−	34.0000i		−	1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$		−	34.0000i		−	1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.00000	−	8.00000i	0.294684	−	0.294684i
$738$		0			0
$739$			40.0000i			1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$			40.0000i			1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.0000	−	12.0000i	−0.440237	−	0.440237i	0.451854	−	0.892092i	$-0.350763\pi$
$743$	−12.0000	−	12.0000i	−0.440237	−	0.440237i	−0.892092	+	0.451854i	$0.850763\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	20.0000			0.731762
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.00000	+	4.00000i	0.145962	+	0.145962i	0.776312	−	0.630349i	$-0.217088\pi$
$751$	4.00000	+	4.00000i	0.145962	+	0.145962i	−0.630349	+	0.776312i	$0.717088\pi$
$752$		0			0
$753$	−24.0000	−	24.0000i	−0.874609	−	0.874609i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			16.0000i			0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$			16.0000i			0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$		0			0
$759$	32.0000	−	32.0000i	1.16153	−	1.16153i
$760$		0			0
$761$	−8.00000			−0.290000			−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000			−0.290000			−0.145000	+	0.989432i	$0.546318\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	24.0000i		−	0.866590i
$768$		0			0
$769$	−24.0000			−0.865462			−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000			−0.865462			−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$	−48.0000	+	48.0000i	−1.72868	+	1.72868i
$772$		0			0
$773$			32.0000i			1.15096i	0.817816	+	0.575480i	$0.195185\pi$
$773$			32.0000i			1.15096i	−0.817816	+	0.575480i	$0.804815\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	32.0000			1.14505
$782$		0			0
$783$	40.0000			1.42948
$784$		0			0
$785$		0			0
$786$		0			0
$787$	14.0000	+	14.0000i	0.499046	+	0.499046i	0.911141	−	0.412095i	$-0.135203\pi$
$787$	14.0000	+	14.0000i	0.499046	+	0.499046i	−0.412095	+	0.911141i	$0.635203\pi$
$788$		0			0
$789$	24.0000	−	24.0000i	0.854423	−	0.854423i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.00000	−	6.00000i	0.213066	−	0.213066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			30.0000i			1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$			30.0000i			1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$		0			0
$799$	−32.0000	−	8.00000i	−1.13208	−	0.283020i
$800$		0			0
$801$			80.0000i			2.82666i
$802$		0			0
$803$	−4.00000			−0.141157
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	20.0000i		−	0.704033i
$808$		0			0
$809$	−27.0000	+	27.0000i	−0.949269	+	0.949269i	−0.998774	−	0.0495045i	$-0.984236\pi$
$809$	−27.0000	+	27.0000i	−0.949269	+	0.949269i	0.0495045	+	0.998774i	$0.484236\pi$
$810$		0			0
$811$	−38.0000	−	38.0000i	−1.33436	−	1.33436i	−0.901425	−	0.432936i	$-0.857478\pi$
$811$	−38.0000	−	38.0000i	−1.33436	−	1.33436i	−0.432936	−	0.901425i	$-0.642522\pi$
$812$		0			0
$813$	−8.00000	−	8.00000i	−0.280572	−	0.280572i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.0000	+	25.0000i	0.872506	+	0.872506i	0.992745	−	0.120239i	$-0.0383661\pi$
$821$	25.0000	+	25.0000i	0.872506	+	0.872506i	−0.120239	+	0.992745i	$0.538366\pi$
$822$		0			0
$823$	−20.0000	+	20.0000i	−0.697156	+	0.697156i	−0.963796	−	0.266640i	$-0.914087\pi$
$823$	−20.0000	+	20.0000i	−0.697156	+	0.697156i	0.266640	+	0.963796i	$0.414087\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.00000	+	6.00000i	−0.208640	+	0.208640i	−0.803689	−	0.595049i	$-0.797133\pi$
$827$	−6.00000	+	6.00000i	−0.208640	+	0.208640i	0.595049	+	0.803689i	$0.297133\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$			12.0000i			0.416275i
$832$		0			0
$833$	7.00000	−	28.0000i	0.242536	−	0.970143i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	32.0000			1.10608
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$			21.0000i			0.724138i
$842$		0			0
$843$	48.0000	−	48.0000i	1.65321	−	1.65321i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	40.0000			1.37280
$850$		0			0
$851$	40.0000			1.37118
$852$		0			0
$853$	−21.0000	−	21.0000i	−0.719026	−	0.719026i	0.249380	−	0.968406i	$-0.419773\pi$
$853$	−21.0000	−	21.0000i	−0.719026	−	0.719026i	−0.968406	+	0.249380i	$0.919773\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.00000	−	9.00000i	0.307434	−	0.307434i	−0.536479	−	0.843913i	$-0.680246\pi$
$857$	9.00000	−	9.00000i	0.307434	−	0.307434i	0.843913	+	0.536479i	$0.180246\pi$
$858$		0			0
$859$		−	36.0000i		−	1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$		−	36.0000i		−	1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.0000			−0.408485			−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000			−0.408485			−0.204242	+	0.978920i	$0.565473\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−14.0000	−	46.0000i	−0.475465	−	1.56224i
$868$		0			0
$869$			48.0000i			1.62829i
$870$		0			0
$871$	−8.00000			−0.271070
$872$		0			0
$873$	25.0000	−	25.0000i	0.846122	−	0.846122i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	33.0000	−	33.0000i	1.11433	−	1.11433i	0.121773	−	0.992558i	$-0.461142\pi$
$877$	33.0000	−	33.0000i	1.11433	−	1.11433i	0.992558	−	0.121773i	$-0.0388579\pi$
$878$		0			0
$879$	−32.0000	−	32.0000i	−1.07933	−	1.07933i
$880$		0			0
$881$	23.0000	+	23.0000i	0.774890	+	0.774890i	0.978957	−	0.204067i	$-0.0654161\pi$
$881$	23.0000	+	23.0000i	0.774890	+	0.774890i	−0.204067	+	0.978957i	$0.565416\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−4.00000	−	4.00000i	−0.134307	−	0.134307i	0.636757	−	0.771064i	$-0.280275\pi$
$887$	−4.00000	−	4.00000i	−0.134307	−	0.134307i	−0.771064	+	0.636757i	$0.780275\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.00000	−	2.00000i	0.0670025	−	0.0670025i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−32.0000			−1.06845
$898$		0			0
$899$			40.0000i			1.33407i
$900$		0			0
$901$	−6.00000	+	24.0000i	−0.199889	+	0.799556i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.0000	+	22.0000i	−0.730498	+	0.730498i	−0.970718	−	0.240220i	$-0.922780\pi$
$907$	−22.0000	+	22.0000i	−0.730498	+	0.730498i	0.240220	+	0.970718i	$0.422780\pi$
$908$		0			0
$909$			40.0000i			1.32672i
$910$		0			0
$911$	16.0000	−	16.0000i	0.530104	−	0.530104i	−0.390499	−	0.920603i	$-0.627698\pi$
$911$	16.0000	−	16.0000i	0.530104	−	0.530104i	0.920603	+	0.390499i	$0.127698\pi$
$912$		0			0
$913$	8.00000	+	8.00000i	0.264761	+	0.264761i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−12.0000			−0.395843			−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000			−0.395843			−0.197922	+	0.980218i	$0.563419\pi$
$920$		0			0
$921$	−64.0000	−	64.0000i	−2.10887	−	2.10887i
$922$		0			0
$923$	−16.0000	−	16.0000i	−0.526646	−	0.526646i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			60.0000i			1.97066i
$928$		0			0
$929$	9.00000	−	9.00000i	0.295280	−	0.295280i	−0.543882	−	0.839162i	$-0.683046\pi$
$929$	9.00000	−	9.00000i	0.295280	−	0.295280i	0.839162	+	0.543882i	$0.183046\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	48.0000i		−	1.57145i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			10.0000i			0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$			10.0000i			0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$		0			0
$939$	68.0000			2.21910
$940$		0			0
$941$	−21.0000	+	21.0000i	−0.684580	+	0.684580i	−0.961029	−	0.276448i	$-0.910843\pi$
$941$	−21.0000	+	21.0000i	−0.684580	+	0.684580i	0.276448	+	0.961029i	$0.410843\pi$
$942$		0			0
$943$			40.0000i			1.30258i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.0000	−	30.0000i	−0.974869	−	0.974869i	0.0248229	−	0.999692i	$-0.492098\pi$
$947$	−30.0000	−	30.0000i	−0.974869	−	0.974869i	−0.999692	+	0.0248229i	$0.992098\pi$
$948$		0			0
$949$	2.00000	+	2.00000i	0.0649227	+	0.0649227i
$950$		0			0
$951$	68.0000			2.20505
$952$		0			0
$953$	−30.0000			−0.971795			−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000			−0.971795			−0.485898	+	0.874016i	$0.661507\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	40.0000	+	40.0000i	1.29302	+	1.29302i
$958$		0			0
$959$		0			0
$960$		0			0
$961$			1.00000i			0.0322581i
$962$		0			0
$963$	70.0000	−	70.0000i	2.25572	−	2.25572i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	48.0000i		−	1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$		−	48.0000i		−	1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			4.00000i			0.128366i	0.997938	+	0.0641831i	$0.0204442\pi$
$971$			4.00000i			0.128366i	−0.997938	+	0.0641831i	$0.979556\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$		0			0
$979$	−32.0000	+	32.0000i	−1.02272	+	1.02272i
$980$		0			0
$981$	−15.0000	−	15.0000i	−0.478913	−	0.478913i
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000	−	16.0000i	−0.508770	−	0.508770i
$990$		0			0
$991$	−36.0000	−	36.0000i	−1.14358	−	1.14358i	−0.987791	−	0.155787i	$-0.950209\pi$
$991$	−36.0000	−	36.0000i	−1.14358	−	1.14358i	−0.155787	−	0.987791i	$-0.549791\pi$
$992$		0			0
$993$	8.00000	−	8.00000i	0.253872	−	0.253872i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.00000	+	3.00000i	−0.0950110	+	0.0950110i	−0.753015	−	0.658004i	$-0.771401\pi$
$997$	−3.00000	+	3.00000i	−0.0950110	+	0.0950110i	0.658004	+	0.753015i	$0.271401\pi$
$998$		0			0
$999$	40.0000			1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1700.2.o.a.1101.1		2
5.2	odd	4		340.2.m.a.149.1	yes	2
5.3	odd	4		340.2.m.b.149.1	yes	2
5.4	even	2		1700.2.o.b.1101.1		2
15.2	even	4		3060.2.z.b.829.1		2
15.8	even	4		3060.2.z.a.829.1		2
17.4	even	4	inner	1700.2.o.a.701.1		2
85.4	even	4		1700.2.o.b.701.1		2
85.38	odd	4		340.2.m.a.89.1	✓	2
85.72	odd	4		340.2.m.b.89.1	yes	2
255.38	even	4		3060.2.z.b.2809.1		2
255.242	even	4		3060.2.z.a.2809.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.m.a.89.1	✓	2	85.38	odd	4
340.2.m.a.149.1	yes	2	5.2	odd	4
340.2.m.b.89.1	yes	2	85.72	odd	4
340.2.m.b.149.1	yes	2	5.3	odd	4
1700.2.o.a.701.1		2	17.4	even	4	inner
1700.2.o.a.1101.1		2	1.1	even	1	trivial
1700.2.o.b.701.1		2	85.4	even	4
1700.2.o.b.1101.1		2	5.4	even	2
3060.2.z.a.829.1		2	15.8	even	4
3060.2.z.a.2809.1		2	255.242	even	4
3060.2.z.b.829.1		2	15.2	even	4
3060.2.z.b.2809.1		2	255.38	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1700.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 2.00000i) q^{3} +5.00000i q^{9} +(-2.00000 + 2.00000i) q^{11} +2.00000 q^{13} +(-4.00000 - 1.00000i) q^{17} +(4.00000 - 4.00000i) q^{23} +(4.00000 - 4.00000i) q^{27} +(5.00000 + 5.00000i) q^{29} +(4.00000 + 4.00000i) q^{31} +8.00000 q^{33} +(5.00000 + 5.00000i) q^{37} +(-4.00000 - 4.00000i) q^{39} +(-5.00000 + 5.00000i) q^{41} -4.00000i q^{43} +8.00000 q^{47} +7.00000i q^{49} +(6.00000 + 10.0000i) q^{51} -6.00000i q^{53} -12.0000i q^{59} +(3.00000 - 3.00000i) q^{61} -4.00000 q^{67} -16.0000 q^{69} +(-8.00000 - 8.00000i) q^{71} +(1.00000 + 1.00000i) q^{73} +(12.0000 - 12.0000i) q^{79} -1.00000 q^{81} -4.00000i q^{83} -20.0000i q^{87} +16.0000 q^{89} -16.0000i q^{93} +(-5.00000 - 5.00000i) q^{97} +(-10.0000 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 4 q^{11} + 4 q^{13} - 8 q^{17} + 8 q^{23} + 8 q^{27} + 10 q^{29} + 8 q^{31} + 16 q^{33} + 10 q^{37} - 8 q^{39} - 10 q^{41} + 16 q^{47} + 12 q^{51} + 6 q^{61} - 8 q^{67} - 32 q^{69} - 16 q^{71}+ \cdots - 20 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 4 * q^11 + 4 * q^13 - 8 * q^17 + 8 * q^23 + 8 * q^27 + 10 * q^29 + 8 * q^31 + 16 * q^33 + 10 * q^37 - 8 * q^39 - 10 * q^41 + 16 * q^47 + 12 * q^51 + 6 * q^61 - 8 * q^67 - 32 * q^69 - 16 * q^71 + 2 * q^73 + 24 * q^79 - 2 * q^81 + 32 * q^89 - 10 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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