LMFDB - Embedded newform 1800.2.k.g.901.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(901,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("1800.901");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.k (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:	$14.3730723638$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			901.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1800.901
Dual form			1800.2.k.g.901.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} -2.00000 q^{7} +(-2.00000 - 2.00000i) q^{8} -4.00000i q^{11} +(-2.00000 + 2.00000i) q^{14} -4.00000 q^{16} -6.00000 q^{17} +4.00000i q^{19} +(-4.00000 - 4.00000i) q^{22} -4.00000 q^{23} +4.00000i q^{28} +6.00000i q^{29} +10.0000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-6.00000 + 6.00000i) q^{34} -4.00000i q^{37} +(4.00000 + 4.00000i) q^{38} -10.0000 q^{41} +4.00000i q^{43} -8.00000 q^{44} +(-4.00000 + 4.00000i) q^{46} -4.00000 q^{47} -3.00000 q^{49} -10.0000i q^{53} +(4.00000 + 4.00000i) q^{56} +(6.00000 + 6.00000i) q^{58} -8.00000i q^{59} -8.00000i q^{61} +(10.0000 - 10.0000i) q^{62} +8.00000i q^{64} +12.0000i q^{67} +12.0000i q^{68} +4.00000 q^{71} -10.0000 q^{73} +(-4.00000 - 4.00000i) q^{74} +8.00000 q^{76} +8.00000i q^{77} -14.0000 q^{79} +(-10.0000 + 10.0000i) q^{82} +(4.00000 + 4.00000i) q^{86} +(-8.00000 + 8.00000i) q^{88} -14.0000 q^{89} +8.00000i q^{92} +(-4.00000 + 4.00000i) q^{94} +10.0000 q^{97} +(-3.00000 + 3.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{7} - 4 q^{8} - 4 q^{14} - 8 q^{16} - 12 q^{17} - 8 q^{22} - 8 q^{23} + 20 q^{31} - 8 q^{32} - 12 q^{34} + 8 q^{38} - 20 q^{41} - 16 q^{44} - 8 q^{46} - 8 q^{47} - 6 q^{49} + 8 q^{56}+ \cdots - 6 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^7 - 4 * q^8 - 4 * q^14 - 8 * q^16 - 12 * q^17 - 8 * q^22 - 8 * q^23 + 20 * q^31 - 8 * q^32 - 12 * q^34 + 8 * q^38 - 20 * q^41 - 16 * q^44 - 8 * q^46 - 8 * q^47 - 6 * q^49 + 8 * q^56 + 12 * q^58 + 20 * q^62 + 8 * q^71 - 20 * q^73 - 8 * q^74 + 16 * q^76 - 28 * q^79 - 20 * q^82 + 8 * q^86 - 16 * q^88 - 28 * q^89 - 8 * q^94 + 20 * q^97 - 6 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−2.00000	+	2.00000i	−0.534522	+	0.534522i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$	−4.00000	−	4.00000i	−0.852803	−	0.852803i
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$			4.00000i			0.755929i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$	10.0000			1.79605			0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000			1.79605			0.898027	+	0.439941i	$0.145001\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$	−6.00000	+	6.00000i	−1.02899	+	1.02899i
$35$		0			0
$36$		0			0
$37$		−	4.00000i		−	0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$		−	4.00000i		−	0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	4.00000	+	4.00000i	0.648886	+	0.648886i
$39$		0			0
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−8.00000			−1.20605
$45$		0			0
$46$	−4.00000	+	4.00000i	−0.589768	+	0.589768i
$47$	−4.00000			−0.583460			−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000			−0.583460			−0.291730	+	0.956501i	$0.594231\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$		0			0
$55$		0			0
$56$	4.00000	+	4.00000i	0.534522	+	0.534522i
$57$		0			0
$58$	6.00000	+	6.00000i	0.787839	+	0.787839i
$59$		−	8.00000i		−	1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$		−	8.00000i		−	1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$		0			0
$61$		−	8.00000i		−	1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$		−	8.00000i		−	1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	10.0000	−	10.0000i	1.27000	−	1.27000i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$			12.0000i			1.45521i
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$	−4.00000	−	4.00000i	−0.464991	−	0.464991i
$75$		0			0
$76$	8.00000			0.917663
$77$			8.00000i			0.911685i
$78$		0			0
$79$	−14.0000			−1.57512			−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000			−1.57512			−0.787562	+	0.616236i	$0.788657\pi$
$80$		0			0
$81$		0			0
$82$	−10.0000	+	10.0000i	−1.10432	+	1.10432i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	4.00000	+	4.00000i	0.431331	+	0.431331i
$87$		0			0
$88$	−8.00000	+	8.00000i	−0.852803	+	0.852803i
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$		0			0
$92$			8.00000i			0.834058i
$93$		0			0
$94$	−4.00000	+	4.00000i	−0.412568	+	0.412568i
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$	−3.00000	+	3.00000i	−0.303046	+	0.303046i
$99$		0			0
$100$		0			0
$101$		−	14.0000i		−	1.39305i	−0.717532	−	0.696526i	$-0.754728\pi$
$101$		−	14.0000i		−	1.39305i	0.717532	−	0.696526i	$-0.245272\pi$
$102$		0			0
$103$	2.00000			0.197066			0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000			0.197066			0.0985329	+	0.995134i	$0.468585\pi$
$104$		0			0
$105$		0			0
$106$	−10.0000	−	10.0000i	−0.971286	−	0.971286i
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		0			0
$112$	8.00000			0.755929
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$	12.0000			1.11417
$117$		0			0
$118$	−8.00000	−	8.00000i	−0.736460	−	0.736460i
$119$	12.0000			1.10004
$120$		0			0
$121$	−5.00000			−0.454545
$122$	−8.00000	−	8.00000i	−0.724286	−	0.724286i
$123$		0			0
$124$		−	20.0000i		−	1.79605i
$125$		0			0
$126$		0			0
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$	12.0000	+	12.0000i	1.03664	+	1.03664i
$135$		0			0
$136$	12.0000	+	12.0000i	1.02899	+	1.02899i
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$		0			0
$142$	4.00000	−	4.00000i	0.335673	−	0.335673i
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−10.0000	+	10.0000i	−0.827606	+	0.827606i
$147$		0			0
$148$	−8.00000			−0.657596
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$	8.00000	−	8.00000i	0.648886	−	0.648886i
$153$		0			0
$154$	8.00000	+	8.00000i	0.644658	+	0.644658i
$155$		0			0
$156$		0			0
$157$		−	20.0000i		−	1.59617i	−0.602542	−	0.798087i	$-0.705846\pi$
$157$		−	20.0000i		−	1.59617i	0.602542	−	0.798087i	$-0.294154\pi$
$158$	−14.0000	+	14.0000i	−1.11378	+	1.11378i
$159$		0			0
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$			20.0000i			1.56174i
$165$		0			0
$166$		0			0
$167$	24.0000			1.85718			0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000			1.85718			0.928588	+	0.371113i	$0.121024\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$	8.00000			0.609994
$173$		−	10.0000i		−	0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$		−	10.0000i		−	0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$		0			0
$175$		0			0
$176$			16.0000i			1.20605i
$177$		0			0
$178$	−14.0000	+	14.0000i	−1.04934	+	1.04934i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	20.0000i		−	1.48659i	−0.668965	−	0.743294i	$-0.733262\pi$
$181$		−	20.0000i		−	1.48659i	0.668965	−	0.743294i	$-0.266738\pi$
$182$		0			0
$183$		0			0
$184$	8.00000	+	8.00000i	0.589768	+	0.589768i
$185$		0			0
$186$		0			0
$187$			24.0000i			1.75505i
$188$			8.00000i			0.583460i
$189$		0			0
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$	10.0000	−	10.0000i	0.717958	−	0.717958i
$195$		0			0
$196$			6.00000i			0.428571i
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		0			0
$199$	−6.00000			−0.425329			−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000			−0.425329			−0.212664	+	0.977125i	$0.568214\pi$
$200$		0			0
$201$		0			0
$202$	−14.0000	−	14.0000i	−0.985037	−	0.985037i
$203$		−	12.0000i		−	0.842235i
$204$		0			0
$205$		0			0
$206$	2.00000	−	2.00000i	0.139347	−	0.139347i
$207$		0			0
$208$		0			0
$209$	16.0000			1.10674
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	−20.0000			−1.37361
$213$		0			0
$214$	4.00000	+	4.00000i	0.273434	+	0.273434i
$215$		0			0
$216$		0			0
$217$	−20.0000			−1.35769
$218$	4.00000	+	4.00000i	0.270914	+	0.270914i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−10.0000			−0.669650			−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000			−0.669650			−0.334825	+	0.942280i	$0.608677\pi$
$224$	8.00000	−	8.00000i	0.534522	−	0.534522i
$225$		0			0
$226$	6.00000	−	6.00000i	0.399114	−	0.399114i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$	12.0000	−	12.0000i	0.787839	−	0.787839i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		0			0
$236$	−16.0000			−1.04151
$237$		0			0
$238$	12.0000	−	12.0000i	0.777844	−	0.777844i
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−30.0000			−1.93247			−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000			−1.93247			−0.966235	+	0.257663i	$0.917048\pi$
$242$	−5.00000	+	5.00000i	−0.321412	+	0.321412i
$243$		0			0
$244$	−16.0000			−1.02430
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−20.0000	−	20.0000i	−1.27000	−	1.27000i
$249$		0			0
$250$		0			0
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$		0			0
$253$			16.0000i			1.00591i
$254$	6.00000	−	6.00000i	0.376473	−	0.376473i
$255$		0			0
$256$	16.0000			1.00000
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$			8.00000i			0.497096i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$	−8.00000	−	8.00000i	−0.490511	−	0.490511i
$267$		0			0
$268$	24.0000			1.46603
$269$		−	10.0000i		−	0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$		−	10.0000i		−	0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$	24.0000			1.45521
$273$		0			0
$274$	−2.00000	+	2.00000i	−0.120824	+	0.120824i
$275$		0			0
$276$		0			0
$277$			16.0000i			0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$			16.0000i			0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$	−4.00000	−	4.00000i	−0.239904	−	0.239904i
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		−	8.00000i		−	0.474713i
$285$		0			0
$286$		0			0
$287$	20.0000			1.18056
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$			20.0000i			1.17041i
$293$		−	2.00000i		−	0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$		−	2.00000i		−	0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$		0			0
$295$		0			0
$296$	−8.00000	+	8.00000i	−0.464991	+	0.464991i
$297$		0			0
$298$	6.00000	+	6.00000i	0.347571	+	0.347571i
$299$		0			0
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$	−2.00000	+	2.00000i	−0.115087	+	0.115087i
$303$		0			0
$304$		−	16.0000i		−	0.917663i
$305$		0			0
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	16.0000			0.911685
$309$		0			0
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$	−20.0000	−	20.0000i	−1.12867	−	1.12867i
$315$		0			0
$316$			28.0000i			1.57512i
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$		0			0
$322$	8.00000	−	8.00000i	0.445823	−	0.445823i
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$		0			0
$326$	−20.0000	−	20.0000i	−1.10770	−	1.10770i
$327$		0			0
$328$	20.0000	+	20.0000i	1.10432	+	1.10432i
$329$	8.00000			0.441054
$330$		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
$333$		0			0
$334$	24.0000	−	24.0000i	1.31322	−	1.31322i
$335$		0			0
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$	13.0000	−	13.0000i	0.707107	−	0.707107i
$339$		0			0
$340$		0			0
$341$		−	40.0000i		−	2.16612i
$342$		0			0
$343$	20.0000			1.07990
$344$	8.00000	−	8.00000i	0.431331	−	0.431331i
$345$		0			0
$346$	−10.0000	−	10.0000i	−0.537603	−	0.537603i
$347$			16.0000i			0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$			16.0000i			0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$		0			0
$349$			8.00000i			0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$349$			8.00000i			0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$350$		0			0
$351$		0			0
$352$	16.0000	+	16.0000i	0.852803	+	0.852803i
$353$	−18.0000			−0.958043			−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000			−0.958043			−0.479022	+	0.877803i	$0.659008\pi$
$354$		0			0
$355$		0			0
$356$			28.0000i			1.48400i
$357$		0			0
$358$		0			0
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−20.0000	−	20.0000i	−1.05118	−	1.05118i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$	16.0000			0.834058
$369$		0			0
$370$		0			0
$371$			20.0000i			1.03835i
$372$		0			0
$373$		−	20.0000i		−	1.03556i	−0.855514	−	0.517780i	$-0.826758\pi$
$373$		−	20.0000i		−	1.03556i	0.855514	−	0.517780i	$-0.173242\pi$
$374$	24.0000	+	24.0000i	1.24101	+	1.24101i
$375$		0			0
$376$	8.00000	+	8.00000i	0.412568	+	0.412568i
$377$		0			0
$378$		0			0
$379$		−	36.0000i		−	1.84920i	−0.380945	−	0.924598i	$-0.624401\pi$
$379$		−	36.0000i		−	1.84920i	0.380945	−	0.924598i	$-0.375599\pi$
$380$		0			0
$381$		0			0
$382$	8.00000	−	8.00000i	0.409316	−	0.409316i
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$		0			0
$386$	14.0000	−	14.0000i	0.712581	−	0.712581i
$387$		0			0
$388$		−	20.0000i		−	1.01535i
$389$		−	10.0000i		−	0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$389$		−	10.0000i		−	0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$	6.00000	+	6.00000i	0.303046	+	0.303046i
$393$		0			0
$394$	−10.0000	−	10.0000i	−0.503793	−	0.503793i
$395$		0			0
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	−6.00000	+	6.00000i	−0.300753	+	0.300753i
$399$		0			0
$400$		0			0
$401$	−30.0000			−1.49813			−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000			−1.49813			−0.749064	+	0.662497i	$0.769497\pi$
$402$		0			0
$403$		0			0
$404$	−28.0000			−1.39305
$405$		0			0
$406$	−12.0000	−	12.0000i	−0.595550	−	0.595550i
$407$	−16.0000			−0.793091
$408$		0			0
$409$	−6.00000			−0.296681			−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000			−0.296681			−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$		0			0
$412$		−	4.00000i		−	0.197066i
$413$			16.0000i			0.787309i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	16.0000	−	16.0000i	0.782586	−	0.782586i
$419$			4.00000i			0.195413i	0.995215	+	0.0977064i	$0.0311506\pi$
$419$			4.00000i			0.195413i	−0.995215	+	0.0977064i	$0.968849\pi$
$420$		0			0
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$	12.0000	+	12.0000i	0.584151	+	0.584151i
$423$		0			0
$424$	−20.0000	+	20.0000i	−0.971286	+	0.971286i
$425$		0			0
$426$		0			0
$427$			16.0000i			0.774294i
$428$	8.00000			0.386695
$429$		0			0
$430$		0			0
$431$	36.0000			1.73406			0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000			1.73406			0.867029	+	0.498257i	$0.166026\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$	−20.0000	+	20.0000i	−0.960031	+	0.960031i
$435$		0			0
$436$	8.00000			0.383131
$437$		−	16.0000i		−	0.765384i
$438$		0			0
$439$	2.00000			0.0954548			0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000			0.0954548			0.0477274	+	0.998860i	$0.484802\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			16.0000i			0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$			16.0000i			0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$		0			0
$445$		0			0
$446$	−10.0000	+	10.0000i	−0.473514	+	0.473514i
$447$		0			0
$448$		−	16.0000i		−	0.755929i
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$			40.0000i			1.88353i
$452$		−	12.0000i		−	0.564433i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18.0000			−0.842004			−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000			−0.842004			−0.421002	+	0.907060i	$0.638322\pi$
$458$	−4.00000	−	4.00000i	−0.186908	−	0.186908i
$459$		0			0
$460$		0			0
$461$			34.0000i			1.58354i	0.610821	+	0.791769i	$0.290840\pi$
$461$			34.0000i			1.58354i	−0.610821	+	0.791769i	$0.709160\pi$
$462$		0			0
$463$	22.0000			1.02243			0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000			1.02243			0.511213	+	0.859454i	$0.329196\pi$
$464$		−	24.0000i		−	1.11417i
$465$		0			0
$466$	−6.00000	+	6.00000i	−0.277945	+	0.277945i
$467$		−	8.00000i		−	0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$		−	8.00000i		−	0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$		0			0
$469$		−	24.0000i		−	1.10822i
$470$		0			0
$471$		0			0
$472$	−16.0000	+	16.0000i	−0.736460	+	0.736460i
$473$	16.0000			0.735681
$474$		0			0
$475$		0			0
$476$		−	24.0000i		−	1.10004i
$477$		0			0
$478$	−16.0000	+	16.0000i	−0.731823	+	0.731823i
$479$	−4.00000			−0.182765			−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000			−0.182765			−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$		0			0
$482$	−30.0000	+	30.0000i	−1.36646	+	1.36646i
$483$		0			0
$484$			10.0000i			0.454545i
$485$		0			0
$486$		0			0
$487$	30.0000			1.35943			0.679715	−	0.733476i	$-0.262104\pi$
$487$	30.0000			1.35943			0.679715	+	0.733476i	$0.262104\pi$
$488$	−16.0000	+	16.0000i	−0.724286	+	0.724286i
$489$		0			0
$490$		0			0
$491$			40.0000i			1.80517i	0.430507	+	0.902587i	$0.358335\pi$
$491$			40.0000i			1.80517i	−0.430507	+	0.902587i	$0.641665\pi$
$492$		0			0
$493$		−	36.0000i		−	1.62136i
$494$		0			0
$495$		0			0
$496$	−40.0000			−1.79605
$497$	−8.00000			−0.358849
$498$		0			0
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$		0			0
$502$	−12.0000	−	12.0000i	−0.535586	−	0.535586i
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$		0			0
$506$	16.0000	+	16.0000i	0.711287	+	0.711287i
$507$		0			0
$508$		−	12.0000i		−	0.532414i
$509$		−	18.0000i		−	0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$		−	18.0000i		−	0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$		0			0
$511$	20.0000			0.884748
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	−2.00000	+	2.00000i	−0.0882162	+	0.0882162i
$515$		0			0
$516$		0			0
$517$			16.0000i			0.703679i
$518$	8.00000	+	8.00000i	0.351500	+	0.351500i
$519$		0			0
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$			20.0000i			0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$			20.0000i			0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−60.0000			−2.61364
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$	−16.0000			−0.693688
$533$		0			0
$534$		0			0
$535$		0			0
$536$	24.0000	−	24.0000i	1.03664	−	1.03664i
$537$		0			0
$538$	−10.0000	−	10.0000i	−0.431131	−	0.431131i
$539$			12.0000i			0.516877i
$540$		0			0
$541$		−	44.0000i		−	1.89171i	−0.324593	−	0.945854i	$-0.605227\pi$
$541$		−	44.0000i		−	1.89171i	0.324593	−	0.945854i	$-0.394773\pi$
$542$	−2.00000	+	2.00000i	−0.0859074	+	0.0859074i
$543$		0			0
$544$	24.0000	−	24.0000i	1.02899	−	1.02899i
$545$		0			0
$546$		0			0
$547$			4.00000i			0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$			4.00000i			0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$			4.00000i			0.170872i
$549$		0			0
$550$		0			0
$551$	−24.0000			−1.02243
$552$		0			0
$553$	28.0000			1.19068
$554$	16.0000	+	16.0000i	0.679775	+	0.679775i
$555$		0			0
$556$	−8.00000			−0.339276
$557$		−	42.0000i		−	1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$		−	42.0000i		−	1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	10.0000	−	10.0000i	0.421825	−	0.421825i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$	4.00000	+	4.00000i	0.168133	+	0.168133i
$567$		0			0
$568$	−8.00000	−	8.00000i	−0.335673	−	0.335673i
$569$	−2.00000			−0.0838444			−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000			−0.0838444			−0.0419222	+	0.999121i	$0.513348\pi$
$570$		0			0
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$		0			0
$573$		0			0
$574$	20.0000	−	20.0000i	0.834784	−	0.834784i
$575$		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	$0.486745\pi$
$578$	19.0000	−	19.0000i	0.790296	−	0.790296i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−40.0000			−1.65663
$584$	20.0000	+	20.0000i	0.827606	+	0.827606i
$585$		0			0
$586$	−2.00000	−	2.00000i	−0.0826192	−	0.0826192i
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$			40.0000i			1.64817i
$590$		0			0
$591$		0			0
$592$			16.0000i			0.657596i
$593$	22.0000			0.903432			0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000			0.903432			0.451716	+	0.892162i	$0.350812\pi$
$594$		0			0
$595$		0			0
$596$	12.0000			0.491539
$597$		0			0
$598$		0			0
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$	−8.00000	−	8.00000i	−0.326056	−	0.326056i
$603$		0			0
$604$			4.00000i			0.162758i
$605$		0			0
$606$		0			0
$607$	−2.00000			−0.0811775			−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000			−0.0811775			−0.0405887	+	0.999176i	$0.512923\pi$
$608$	−16.0000	−	16.0000i	−0.648886	−	0.648886i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			20.0000i			0.807792i	0.914805	+	0.403896i	$0.132344\pi$
$613$			20.0000i			0.807792i	−0.914805	+	0.403896i	$0.867656\pi$
$614$	−28.0000	−	28.0000i	−1.12999	−	1.12999i
$615$		0			0
$616$	16.0000	−	16.0000i	0.644658	−	0.644658i
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$			20.0000i			0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$			20.0000i			0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$		0			0
$621$		0			0
$622$	−8.00000	+	8.00000i	−0.320771	+	0.320771i
$623$	28.0000			1.12180
$624$		0			0
$625$		0			0
$626$	−6.00000	+	6.00000i	−0.239808	+	0.239808i
$627$		0			0
$628$	−40.0000			−1.59617
$629$			24.0000i			0.956943i
$630$		0			0
$631$	−18.0000			−0.716569			−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000			−0.716569			−0.358284	+	0.933613i	$0.616638\pi$
$632$	28.0000	+	28.0000i	1.11378	+	1.11378i
$633$		0			0
$634$	18.0000	+	18.0000i	0.714871	+	0.714871i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	24.0000	−	24.0000i	0.950169	−	0.950169i
$639$		0			0
$640$		0			0
$641$	−14.0000			−0.552967			−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000			−0.552967			−0.276483	+	0.961019i	$0.589169\pi$
$642$		0			0
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$		−	16.0000i		−	0.630488i
$645$		0			0
$646$	−24.0000	−	24.0000i	−0.944267	−	0.944267i
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$	−32.0000			−1.25611
$650$		0			0
$651$		0			0
$652$	−40.0000			−1.56652
$653$			30.0000i			1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$			30.0000i			1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$		0			0
$655$		0			0
$656$	40.0000			1.56174
$657$		0			0
$658$	8.00000	−	8.00000i	0.311872	−	0.311872i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			32.0000i			1.24466i	0.782757	+	0.622328i	$0.213813\pi$
$661$			32.0000i			1.24466i	−0.782757	+	0.622328i	$0.786187\pi$
$662$	4.00000	+	4.00000i	0.155464	+	0.155464i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	24.0000i		−	0.929284i
$668$		−	48.0000i		−	1.85718i
$669$		0			0
$670$		0			0
$671$	−32.0000			−1.23535
$672$		0			0
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$	−30.0000	+	30.0000i	−1.15556	+	1.15556i
$675$		0			0
$676$		−	26.0000i		−	1.00000i
$677$		−	38.0000i		−	1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$		−	38.0000i		−	1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$		0			0
$679$	−20.0000			−0.767530
$680$		0			0
$681$		0			0
$682$	−40.0000	−	40.0000i	−1.53168	−	1.53168i
$683$			48.0000i			1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$			48.0000i			1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$		0			0
$685$		0			0
$686$	20.0000	−	20.0000i	0.763604	−	0.763604i
$687$		0			0
$688$		−	16.0000i		−	0.609994i
$689$		0			0
$690$		0			0
$691$			12.0000i			0.456502i	0.973602	+	0.228251i	$0.0733006\pi$
$691$			12.0000i			0.456502i	−0.973602	+	0.228251i	$0.926699\pi$
$692$	−20.0000			−0.760286
$693$		0			0
$694$	16.0000	+	16.0000i	0.607352	+	0.607352i
$695$		0			0
$696$		0			0
$697$	60.0000			2.27266
$698$	8.00000	+	8.00000i	0.302804	+	0.302804i
$699$		0			0
$700$		0			0
$701$			22.0000i			0.830929i	0.909610	+	0.415464i	$0.136381\pi$
$701$			22.0000i			0.830929i	−0.909610	+	0.415464i	$0.863619\pi$
$702$		0			0
$703$	16.0000			0.603451
$704$	32.0000			1.20605
$705$		0			0
$706$	−18.0000	+	18.0000i	−0.677439	+	0.677439i
$707$			28.0000i			1.05305i
$708$		0			0
$709$			44.0000i			1.65245i	0.563337	+	0.826227i	$0.309517\pi$
$709$			44.0000i			1.65245i	−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$	28.0000	+	28.0000i	1.04934	+	1.04934i
$713$	−40.0000			−1.49801
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−12.0000	+	12.0000i	−0.447836	+	0.447836i
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$		0			0
$721$	−4.00000			−0.148968
$722$	3.00000	−	3.00000i	0.111648	−	0.111648i
$723$		0			0
$724$	−40.0000			−1.48659
$725$		0			0
$726$		0			0
$727$	14.0000			0.519231			0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000			0.519231			0.259616	+	0.965712i	$0.416404\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	24.0000i		−	0.887672i
$732$		0			0
$733$			24.0000i			0.886460i	0.896408	+	0.443230i	$0.146168\pi$
$733$			24.0000i			0.886460i	−0.896408	+	0.443230i	$0.853832\pi$
$734$	18.0000	−	18.0000i	0.664392	−	0.664392i
$735$		0			0
$736$	16.0000	−	16.0000i	0.589768	−	0.589768i
$737$	48.0000			1.76810
$738$		0			0
$739$			12.0000i			0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$			12.0000i			0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$		0			0
$741$		0			0
$742$	20.0000	+	20.0000i	0.734223	+	0.734223i
$743$	−48.0000			−1.76095			−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000			−1.76095			−0.880475	+	0.474093i	$0.842776\pi$
$744$		0			0
$745$		0			0
$746$	−20.0000	−	20.0000i	−0.732252	−	0.732252i
$747$		0			0
$748$	48.0000			1.75505
$749$		−	8.00000i		−	0.292314i
$750$		0			0
$751$	26.0000			0.948753			0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000			0.948753			0.474377	+	0.880322i	$0.342673\pi$
$752$	16.0000			0.583460
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	−36.0000	−	36.0000i	−1.30758	−	1.30758i
$759$		0			0
$760$		0			0
$761$	−10.0000			−0.362500			−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000			−0.362500			−0.181250	+	0.983437i	$0.558014\pi$
$762$		0			0
$763$		−	8.00000i		−	0.289619i
$764$		−	16.0000i		−	0.578860i
$765$		0			0
$766$	24.0000	−	24.0000i	0.867155	−	0.867155i
$767$		0			0
$768$		0			0
$769$	−6.00000			−0.216366			−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000			−0.216366			−0.108183	+	0.994131i	$0.534503\pi$
$770$		0			0
$771$		0			0
$772$		−	28.0000i		−	1.00774i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		0			0
$775$		0			0
$776$	−20.0000	−	20.0000i	−0.717958	−	0.717958i
$777$		0			0
$778$	−10.0000	−	10.0000i	−0.358517	−	0.358517i
$779$		−	40.0000i		−	1.43315i
$780$		0			0
$781$		−	16.0000i		−	0.572525i
$782$	24.0000	−	24.0000i	0.858238	−	0.858238i
$783$		0			0
$784$	12.0000			0.428571
$785$		0			0
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	−20.0000			−0.712470
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$		0			0
$794$	−20.0000	−	20.0000i	−0.709773	−	0.709773i
$795$		0			0
$796$			12.0000i			0.425329i
$797$			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
$799$	24.0000			0.849059
$800$		0			0
$801$		0			0
$802$	−30.0000	+	30.0000i	−1.05934	+	1.05934i
$803$			40.0000i			1.41157i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−28.0000	+	28.0000i	−0.985037	+	0.985037i
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$		−	4.00000i		−	0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$		−	4.00000i		−	0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$	−24.0000			−0.842235
$813$		0			0
$814$	−16.0000	+	16.0000i	−0.560800	+	0.560800i
$815$		0			0
$816$		0			0
$817$	−16.0000			−0.559769
$818$	−6.00000	+	6.00000i	−0.209785	+	0.209785i
$819$		0			0
$820$		0			0
$821$			10.0000i			0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$			10.0000i			0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$		0			0
$823$	−42.0000			−1.46403			−0.732014	−	0.681290i	$-0.761419\pi$
$823$	−42.0000			−1.46403			−0.732014	+	0.681290i	$0.761419\pi$
$824$	−4.00000	−	4.00000i	−0.139347	−	0.139347i
$825$		0			0
$826$	16.0000	+	16.0000i	0.556711	+	0.556711i
$827$		−	4.00000i		−	0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$		−	4.00000i		−	0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$		0			0
$829$		−	44.0000i		−	1.52818i	−0.645108	−	0.764092i	$-0.723188\pi$
$829$		−	44.0000i		−	1.52818i	0.645108	−	0.764092i	$-0.276812\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$		0			0
$836$		−	32.0000i		−	1.10674i
$837$		0			0
$838$	4.00000	+	4.00000i	0.138178	+	0.138178i
$839$	−4.00000			−0.138095			−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000			−0.138095			−0.0690477	+	0.997613i	$0.521996\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$	20.0000	+	20.0000i	0.689246	+	0.689246i
$843$		0			0
$844$	24.0000			0.826114
$845$		0			0
$846$		0			0
$847$	10.0000			0.343604
$848$			40.0000i			1.37361i
$849$		0			0
$850$		0			0
$851$			16.0000i			0.548473i
$852$		0			0
$853$			44.0000i			1.50653i	0.657716	+	0.753266i	$0.271523\pi$
$853$			44.0000i			1.50653i	−0.657716	+	0.753266i	$0.728477\pi$
$854$	16.0000	+	16.0000i	0.547509	+	0.547509i
$855$		0			0
$856$	8.00000	−	8.00000i	0.273434	−	0.273434i
$857$	6.00000			0.204956			0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000			0.204956			0.102478	+	0.994735i	$0.467323\pi$
$858$		0			0
$859$		−	44.0000i		−	1.50126i	−0.660722	−	0.750630i	$-0.729750\pi$
$859$		−	44.0000i		−	1.50126i	0.660722	−	0.750630i	$-0.270250\pi$
$860$		0			0
$861$		0			0
$862$	36.0000	−	36.0000i	1.22616	−	1.22616i
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$		0			0
$866$	−14.0000	+	14.0000i	−0.475739	+	0.475739i
$867$		0			0
$868$			40.0000i			1.35769i
$869$			56.0000i			1.89967i
$870$		0			0
$871$		0			0
$872$	8.00000	−	8.00000i	0.270914	−	0.270914i
$873$		0			0
$874$	−16.0000	−	16.0000i	−0.541208	−	0.541208i
$875$		0			0
$876$		0			0
$877$		−	44.0000i		−	1.48577i	−0.669417	−	0.742887i	$-0.733456\pi$
$877$		−	44.0000i		−	1.48577i	0.669417	−	0.742887i	$-0.266544\pi$
$878$	2.00000	−	2.00000i	0.0674967	−	0.0674967i
$879$		0			0
$880$		0			0
$881$	6.00000			0.202145			0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000			0.202145			0.101073	+	0.994879i	$0.467773\pi$
$882$		0			0
$883$			28.0000i			0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$			28.0000i			0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$		0			0
$885$		0			0
$886$	16.0000	+	16.0000i	0.537531	+	0.537531i
$887$	16.0000			0.537227			0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000			0.537227			0.268614	+	0.963248i	$0.413434\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$		0			0
$891$		0			0
$892$			20.0000i			0.669650i
$893$		−	16.0000i		−	0.535420i
$894$		0			0
$895$		0			0
$896$	−16.0000	−	16.0000i	−0.534522	−	0.534522i
$897$		0			0
$898$	18.0000	−	18.0000i	0.600668	−	0.600668i
$899$			60.0000i			2.00111i
$900$		0			0
$901$			60.0000i			1.99889i
$902$	40.0000	+	40.0000i	1.33185	+	1.33185i
$903$		0			0
$904$	−12.0000	−	12.0000i	−0.399114	−	0.399114i
$905$		0			0
$906$		0			0
$907$			52.0000i			1.72663i	0.504664	+	0.863316i	$0.331616\pi$
$907$			52.0000i			1.72663i	−0.504664	+	0.863316i	$0.668384\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$		0			0
$914$	−18.0000	+	18.0000i	−0.595387	+	0.595387i
$915$		0			0
$916$	−8.00000			−0.264327
$917$		0			0
$918$		0			0
$919$	−26.0000			−0.857661			−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000			−0.857661			−0.428830	+	0.903385i	$0.641074\pi$
$920$		0			0
$921$		0			0
$922$	34.0000	+	34.0000i	1.11973	+	1.11973i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	22.0000	−	22.0000i	0.722965	−	0.722965i
$927$		0			0
$928$	−24.0000	−	24.0000i	−0.787839	−	0.787839i
$929$	−46.0000			−1.50921			−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000			−1.50921			−0.754606	+	0.656179i	$0.772172\pi$
$930$		0			0
$931$		−	12.0000i		−	0.393284i
$932$			12.0000i			0.393073i
$933$		0			0
$934$	−8.00000	−	8.00000i	−0.261768	−	0.261768i
$935$		0			0
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$	−24.0000	−	24.0000i	−0.783628	−	0.783628i
$939$		0			0
$940$		0			0
$941$			26.0000i			0.847576i	0.905761	+	0.423788i	$0.139300\pi$
$941$			26.0000i			0.847576i	−0.905761	+	0.423788i	$0.860700\pi$
$942$		0			0
$943$	40.0000			1.30258
$944$			32.0000i			1.04151i
$945$		0			0
$946$	16.0000	−	16.0000i	0.520205	−	0.520205i
$947$		−	36.0000i		−	1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$		−	36.0000i		−	1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−24.0000	−	24.0000i	−0.777844	−	0.777844i
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$		0			0
$956$			32.0000i			1.03495i
$957$		0			0
$958$	−4.00000	+	4.00000i	−0.129234	+	0.129234i
$959$	4.00000			0.129167
$960$		0			0
$961$	69.0000			2.22581
$962$		0			0
$963$		0			0
$964$			60.0000i			1.93247i
$965$		0			0
$966$		0			0
$967$	−46.0000			−1.47926			−0.739630	−	0.673014i	$-0.765000\pi$
$967$	−46.0000			−1.47926			−0.739630	+	0.673014i	$0.765000\pi$
$968$	10.0000	+	10.0000i	0.321412	+	0.321412i
$969$		0			0
$970$		0			0
$971$		−	12.0000i		−	0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$		−	12.0000i		−	0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$	30.0000	−	30.0000i	0.961262	−	0.961262i
$975$		0			0
$976$			32.0000i			1.02430i
$977$	−22.0000			−0.703842			−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000			−0.703842			−0.351921	+	0.936030i	$0.614471\pi$
$978$		0			0
$979$			56.0000i			1.78977i
$980$		0			0
$981$		0			0
$982$	40.0000	+	40.0000i	1.27645	+	1.27645i
$983$	−32.0000			−1.02064			−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000			−1.02064			−0.510321	+	0.859984i	$0.670473\pi$
$984$		0			0
$985$		0			0
$986$	−36.0000	−	36.0000i	−1.14647	−	1.14647i
$987$		0			0
$988$		0			0
$989$		−	16.0000i		−	0.508770i
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$	−40.0000	+	40.0000i	−1.27000	+	1.27000i
$993$		0			0
$994$	−8.00000	+	8.00000i	−0.253745	+	0.253745i
$995$		0			0
$996$		0			0
$997$			28.0000i			0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$			28.0000i			0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	−20.0000	−	20.0000i	−0.633089	−	0.633089i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.k.g.901.1		2
3.2	odd	2		600.2.k.a.301.2		2
4.3	odd	2		7200.2.k.f.3601.2		2
5.2	odd	4		1800.2.d.h.1549.2		2
5.3	odd	4		1800.2.d.c.1549.1		2
5.4	even	2		360.2.k.b.181.2		2
8.3	odd	2		7200.2.k.f.3601.1		2
8.5	even	2	inner	1800.2.k.g.901.2		2
12.11	even	2		2400.2.k.b.1201.2		2
15.2	even	4		600.2.d.a.349.1		2
15.8	even	4		600.2.d.d.349.2		2
15.14	odd	2		120.2.k.a.61.1	✓	2
20.3	even	4		7200.2.d.e.2449.1		2
20.7	even	4		7200.2.d.f.2449.2		2
20.19	odd	2		1440.2.k.a.721.2		2
24.5	odd	2		600.2.k.a.301.1		2
24.11	even	2		2400.2.k.b.1201.1		2
40.3	even	4		7200.2.d.f.2449.1		2
40.13	odd	4		1800.2.d.h.1549.1		2
40.19	odd	2		1440.2.k.a.721.1		2
40.27	even	4		7200.2.d.e.2449.2		2
40.29	even	2		360.2.k.b.181.1		2
40.37	odd	4		1800.2.d.c.1549.2		2
60.23	odd	4		2400.2.d.a.49.1		2
60.47	odd	4		2400.2.d.d.49.2		2
60.59	even	2		480.2.k.a.241.1		2
120.29	odd	2		120.2.k.a.61.2	yes	2
120.53	even	4		600.2.d.a.349.2		2
120.59	even	2		480.2.k.a.241.2		2
120.77	even	4		600.2.d.d.349.1		2
120.83	odd	4		2400.2.d.d.49.1		2
120.107	odd	4		2400.2.d.a.49.2		2
240.29	odd	4		3840.2.a.d.1.1		1
240.59	even	4		3840.2.a.m.1.1		1
240.149	odd	4		3840.2.a.w.1.1		1
240.179	even	4		3840.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.a.61.1	✓	2	15.14	odd	2
120.2.k.a.61.2	yes	2	120.29	odd	2
360.2.k.b.181.1		2	40.29	even	2
360.2.k.b.181.2		2	5.4	even	2
480.2.k.a.241.1		2	60.59	even	2
480.2.k.a.241.2		2	120.59	even	2
600.2.d.a.349.1		2	15.2	even	4
600.2.d.a.349.2		2	120.53	even	4
600.2.d.d.349.1		2	120.77	even	4
600.2.d.d.349.2		2	15.8	even	4
600.2.k.a.301.1		2	24.5	odd	2
600.2.k.a.301.2		2	3.2	odd	2
1440.2.k.a.721.1		2	40.19	odd	2
1440.2.k.a.721.2		2	20.19	odd	2
1800.2.d.c.1549.1		2	5.3	odd	4
1800.2.d.c.1549.2		2	40.37	odd	4
1800.2.d.h.1549.1		2	40.13	odd	4
1800.2.d.h.1549.2		2	5.2	odd	4
1800.2.k.g.901.1		2	1.1	even	1	trivial
1800.2.k.g.901.2		2	8.5	even	2	inner
2400.2.d.a.49.1		2	60.23	odd	4
2400.2.d.a.49.2		2	120.107	odd	4
2400.2.d.d.49.1		2	120.83	odd	4
2400.2.d.d.49.2		2	60.47	odd	4
2400.2.k.b.1201.1		2	24.11	even	2
2400.2.k.b.1201.2		2	12.11	even	2
3840.2.a.d.1.1		1	240.29	odd	4
3840.2.a.m.1.1		1	240.59	even	4
3840.2.a.r.1.1		1	240.179	even	4
3840.2.a.w.1.1		1	240.149	odd	4
7200.2.d.e.2449.1		2	20.3	even	4
7200.2.d.e.2449.2		2	40.27	even	4
7200.2.d.f.2449.1		2	40.3	even	4
7200.2.d.f.2449.2		2	20.7	even	4
7200.2.k.f.3601.1		2	8.3	odd	2
7200.2.k.f.3601.2		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} -2.00000 q^{7} +(-2.00000 - 2.00000i) q^{8} -4.00000i q^{11} +(-2.00000 + 2.00000i) q^{14} -4.00000 q^{16} -6.00000 q^{17} +4.00000i q^{19} +(-4.00000 - 4.00000i) q^{22} -4.00000 q^{23} +4.00000i q^{28} +6.00000i q^{29} +10.0000 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-6.00000 + 6.00000i) q^{34} -4.00000i q^{37} +(4.00000 + 4.00000i) q^{38} -10.0000 q^{41} +4.00000i q^{43} -8.00000 q^{44} +(-4.00000 + 4.00000i) q^{46} -4.00000 q^{47} -3.00000 q^{49} -10.0000i q^{53} +(4.00000 + 4.00000i) q^{56} +(6.00000 + 6.00000i) q^{58} -8.00000i q^{59} -8.00000i q^{61} +(10.0000 - 10.0000i) q^{62} +8.00000i q^{64} +12.0000i q^{67} +12.0000i q^{68} +4.00000 q^{71} -10.0000 q^{73} +(-4.00000 - 4.00000i) q^{74} +8.00000 q^{76} +8.00000i q^{77} -14.0000 q^{79} +(-10.0000 + 10.0000i) q^{82} +(4.00000 + 4.00000i) q^{86} +(-8.00000 + 8.00000i) q^{88} -14.0000 q^{89} +8.00000i q^{92} +(-4.00000 + 4.00000i) q^{94} +10.0000 q^{97} +(-3.00000 + 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{7} - 4 q^{8} - 4 q^{14} - 8 q^{16} - 12 q^{17} - 8 q^{22} - 8 q^{23} + 20 q^{31} - 8 q^{32} - 12 q^{34} + 8 q^{38} - 20 q^{41} - 16 q^{44} - 8 q^{46} - 8 q^{47} - 6 q^{49} + 8 q^{56}+ \cdots - 6 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^7 - 4 * q^8 - 4 * q^14 - 8 * q^16 - 12 * q^17 - 8 * q^22 - 8 * q^23 + 20 * q^31 - 8 * q^32 - 12 * q^34 + 8 * q^38 - 20 * q^41 - 16 * q^44 - 8 * q^46 - 8 * q^47 - 6 * q^49 + 8 * q^56 + 12 * q^58 + 20 * q^62 + 8 * q^71 - 20 * q^73 - 8 * q^74 + 16 * q^76 - 28 * q^79 - 20 * q^82 + 8 * q^86 - 16 * q^88 - 28 * q^89 - 8 * q^94 + 20 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

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