LMFDB - Embedded newform 2448.2.c.f.577.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2448,2,Mod(577,2448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2448.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,-2,0,0,0,-8,0,-10,0,0,0,0,0,-8,0,0,0, 0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)

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

Self dual:	no
Analytic conductor:	$19.5473784148$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2448.577
Dual form			2448.2.c.f.577.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{5} +2.00000i q^{7} +5.00000i q^{11} -1.00000 q^{13} +(-4.00000 + 1.00000i) q^{17} -5.00000 q^{19} +1.00000i q^{23} -4.00000 q^{25} +6.00000i q^{29} -10.0000i q^{31} -6.00000 q^{35} -2.00000i q^{37} -5.00000i q^{41} +1.00000 q^{43} -2.00000 q^{47} +3.00000 q^{49} +6.00000 q^{53} -15.0000 q^{55} -10.0000i q^{61} -3.00000i q^{65} +12.0000 q^{67} +6.00000i q^{73} -10.0000 q^{77} -4.00000i q^{79} -6.00000 q^{83} +(-3.00000 - 12.0000i) q^{85} +10.0000 q^{89} -2.00000i q^{91} -15.0000i q^{95} +8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{13} - 8 q^{17} - 10 q^{19} - 8 q^{25} - 12 q^{35} + 2 q^{43} - 4 q^{47} + 6 q^{49} + 12 q^{53} - 30 q^{55} + 24 q^{67} - 20 q^{77} - 12 q^{83} - 6 q^{85} + 20 q^{89}+O(q^{100}) $ 2 * q - 2 * q^13 - 8 * q^17 - 10 * q^19 - 8 * q^25 - 12 * q^35 + 2 * q^43 - 4 * q^47 + 6 * q^49 + 12 * q^53 - 30 * q^55 + 24 * q^67 - 20 * q^77 - 12 * q^83 - 6 * q^85 + 20 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2448\mathbb{Z}\right)^\times$.

$n$	$613$	$1361$	$1873$	$2143$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			3.00000i			1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$			3.00000i			1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.00000i			1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$			5.00000i			1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$		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
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$			1.00000i			0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			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
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000			−1.01419
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.00000i		−	0.780869i	−0.920631	−	0.390434i	$-0.872325\pi$
$41$		−	5.00000i		−	0.780869i	0.920631	−	0.390434i	$-0.127675\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000			−0.291730			−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000			−0.291730			−0.145865	+	0.989305i	$0.546597\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−15.0000			−2.02260
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	10.0000i		−	1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$		−	10.0000i		−	1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	3.00000i		−	0.372104i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10.0000			−1.13961
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	−3.00000	−	12.0000i	−0.325396	−	1.30158i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$		−	2.00000i		−	0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	15.0000i		−	1.53897i
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$	−9.00000			−0.886796			−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000			−0.886796			−0.443398	+	0.896325i	$0.646227\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		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
$112$		0			0
$113$		−	11.0000i		−	1.03479i	−0.855746	−	0.517396i	$-0.826901\pi$
$113$		−	11.0000i		−	1.03479i	0.855746	−	0.517396i	$-0.173099\pi$
$114$		0			0
$115$	−3.00000			−0.279751
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.00000	−	8.00000i	−0.183340	−	0.733359i
$120$		0			0
$121$	−14.0000			−1.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.00000i			0.268328i
$126$		0			0
$127$	7.00000			0.621150			0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000			0.621150			0.310575	+	0.950549i	$0.399478\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			5.00000i			0.436852i	0.975854	+	0.218426i	$0.0700922\pi$
$131$			5.00000i			0.436852i	−0.975854	+	0.218426i	$0.929908\pi$
$132$		0			0
$133$		−	10.0000i		−	0.867110i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.00000			−0.683486			−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000			−0.683486			−0.341743	+	0.939793i	$0.611017\pi$
$138$		0			0
$139$			16.0000i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$			16.0000i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	5.00000i		−	0.418121i
$144$		0			0
$145$	−18.0000			−1.49482
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	30.0000			2.40966
$156$		0			0
$157$	−17.0000			−1.35675			−0.678374	−	0.734717i	$-0.737315\pi$
$157$	−17.0000			−1.35675			−0.678374	+	0.734717i	$0.737315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000			−0.157622
$162$		0			0
$163$			14.0000i			1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.00000i		−	0.0760286i	−0.999277	−	0.0380143i	$-0.987897\pi$
$173$		−	1.00000i		−	0.0760286i	0.999277	−	0.0380143i	$-0.0121032\pi$
$174$		0			0
$175$		−	8.00000i		−	0.604743i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$			20.0000i			1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$			20.0000i			1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.00000			0.441129
$186$		0			0
$187$	−5.00000	−	20.0000i	−0.365636	−	1.46254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$			6.00000i			0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$			6.00000i			0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	23.0000i		−	1.63868i	−0.573306	−	0.819341i	$-0.694340\pi$
$197$		−	23.0000i		−	1.63868i	0.573306	−	0.819341i	$-0.305660\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.0000			−0.842235
$204$		0			0
$205$	15.0000			1.04765
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	25.0000i		−	1.72929i
$210$		0			0
$211$			10.0000i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$			10.0000i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			3.00000i			0.204598i
$216$		0			0
$217$	20.0000			1.35769
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.00000	−	1.00000i	0.269069	−	0.0672673i
$222$		0			0
$223$	11.0000			0.736614			0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000			0.736614			0.368307	+	0.929704i	$0.379937\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			3.00000i			0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$			3.00000i			0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			29.0000i			1.89985i	0.312473	+	0.949927i	$0.398843\pi$
$233$			29.0000i			1.89985i	−0.312473	+	0.949927i	$0.601157\pi$
$234$		0			0
$235$		−	6.00000i		−	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.0000			−1.29369			−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000			−1.29369			−0.646846	+	0.762620i	$0.723912\pi$
$240$		0			0
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			9.00000i			0.574989i
$246$		0			0
$247$	5.00000			0.318142
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$	−5.00000			−0.314347
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000			−0.369976			−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000			−0.369976			−0.184988	+	0.982741i	$0.559225\pi$
$264$		0			0
$265$			18.0000i			1.10573i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			21.0000i			1.28039i	0.768211	+	0.640196i	$0.221147\pi$
$269$			21.0000i			1.28039i	−0.768211	+	0.640196i	$0.778853\pi$
$270$		0			0
$271$	−27.0000			−1.64013			−0.820067	−	0.572268i	$-0.806064\pi$
$271$	−27.0000			−1.64013			−0.820067	+	0.572268i	$0.806064\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	20.0000i		−	1.20605i
$276$		0			0
$277$		−	22.0000i		−	1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$		−	22.0000i		−	1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.0000			0.590281
$288$		0			0
$289$	15.0000	−	8.00000i	0.882353	−	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.0000			−0.817889			−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000			−0.817889			−0.408944	+	0.912559i	$0.634103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	1.00000i		−	0.0578315i
$300$		0			0
$301$			2.00000i			0.115278i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	30.0000			1.71780
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			22.0000i			1.23564i	0.786318	+	0.617822i	$0.211985\pi$
$317$			22.0000i			1.23564i	−0.786318	+	0.617822i	$0.788015\pi$
$318$		0			0
$319$	−30.0000			−1.67968
$320$		0			0
$321$		0			0
$322$		0			0
$323$	20.0000	−	5.00000i	1.11283	−	0.278207i
$324$		0			0
$325$	4.00000			0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	4.00000i		−	0.220527i
$330$		0			0
$331$	23.0000			1.26419			0.632097	−	0.774889i	$-0.282194\pi$
$331$	23.0000			1.26419			0.632097	+	0.774889i	$0.282194\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			36.0000i			1.96689i
$336$		0			0
$337$		−	32.0000i		−	1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$		−	32.0000i		−	1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	50.0000			2.70765
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			8.00000i			0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$			8.00000i			0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$		0			0
$349$	−5.00000			−0.267644			−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000			−0.267644			−0.133822	+	0.991005i	$0.542725\pi$
$350$		0			0
$351$		0			0
$352$		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
$359$	−10.0000			−0.527780			−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000			−0.527780			−0.263890	+	0.964553i	$0.585006\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18.0000			−0.942163
$366$		0			0
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			12.0000i			0.623009i
$372$		0			0
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6.00000i		−	0.309016i
$378$		0			0
$379$		−	34.0000i		−	1.74646i	−0.487306	−	0.873231i	$-0.662020\pi$
$379$		−	34.0000i		−	1.74646i	0.487306	−	0.873231i	$-0.337980\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.0000			−0.817562			−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000			−0.817562			−0.408781	+	0.912633i	$0.634046\pi$
$384$		0			0
$385$		−	30.0000i		−	1.52894i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−1.00000	−	4.00000i	−0.0505722	−	0.202289i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000			0.603786
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			15.0000i			0.749064i	0.927214	+	0.374532i	$0.122197\pi$
$401$			15.0000i			0.749064i	−0.927214	+	0.374532i	$0.877803\pi$
$402$		0			0
$403$			10.0000i			0.498135i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	10.0000			0.495682
$408$		0			0
$409$	25.0000			1.23617			0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000			1.23617			0.618085	+	0.786111i	$0.287909\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	18.0000i		−	0.883585i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	16.0000i		−	0.781651i	−0.920465	−	0.390826i	$-0.872190\pi$
$419$		−	16.0000i		−	0.781651i	0.920465	−	0.390826i	$-0.127810\pi$
$420$		0			0
$421$	−13.0000			−0.633581			−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000			−0.633581			−0.316791	+	0.948495i	$0.602605\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	16.0000	−	4.00000i	0.776114	−	0.194029i
$426$		0			0
$427$	20.0000			0.967868
$428$		0			0
$429$		0			0
$430$		0			0
$431$			20.0000i			0.963366i	0.876346	+	0.481683i	$0.159974\pi$
$431$			20.0000i			0.963366i	−0.876346	+	0.481683i	$0.840026\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	5.00000i		−	0.239182i
$438$		0			0
$439$		−	24.0000i		−	1.14546i	−0.819745	−	0.572729i	$-0.805885\pi$
$439$		−	24.0000i		−	1.14546i	0.819745	−	0.572729i	$-0.194115\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	34.0000			1.61539			0.807694	−	0.589601i	$-0.200715\pi$
$443$	34.0000			1.61539			0.807694	+	0.589601i	$0.200715\pi$
$444$		0			0
$445$			30.0000i			1.42214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	34.0000i		−	1.60456i	−0.596948	−	0.802280i	$-0.703620\pi$
$449$		−	34.0000i		−	1.60456i	0.596948	−	0.802280i	$-0.296380\pi$
$450$		0			0
$451$	25.0000			1.17720
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6.00000			0.281284
$456$		0			0
$457$	23.0000			1.07589			0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000			1.07589			0.537947	+	0.842978i	$0.319200\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−32.0000			−1.49039			−0.745194	−	0.666847i	$-0.767643\pi$
$461$	−32.0000			−1.49039			−0.745194	+	0.666847i	$0.767643\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$			24.0000i			1.10822i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5.00000i			0.229900i
$474$		0			0
$475$	20.0000			0.917663
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	21.0000i		−	0.959514i	−0.877401	−	0.479757i	$-0.840725\pi$
$479$		−	21.0000i		−	0.959514i	0.877401	−	0.479757i	$-0.159275\pi$
$480$		0			0
$481$			2.00000i			0.0911922i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0000			−1.08978
$486$		0			0
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−38.0000			−1.71492			−0.857458	−	0.514554i	$-0.827958\pi$
$491$	−38.0000			−1.71492			−0.857458	+	0.514554i	$0.827958\pi$
$492$		0			0
$493$	−6.00000	−	24.0000i	−0.270226	−	1.08091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			6.00000i			0.268597i	0.990941	+	0.134298i	$0.0428781\pi$
$499$			6.00000i			0.268597i	−0.990941	+	0.134298i	$0.957122\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	9.00000i		−	0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$		−	9.00000i		−	0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$		0			0
$505$		−	36.0000i		−	1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−12.0000			−0.530849
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	27.0000i		−	1.18976i
$516$		0			0
$517$		−	10.0000i		−	0.439799i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15.0000i			0.657162i	0.944476	+	0.328581i	$0.106570\pi$
$521$			15.0000i			0.657162i	−0.944476	+	0.328581i	$0.893430\pi$
$522$		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
$527$	10.0000	+	40.0000i	0.435607	+	1.74243i
$528$		0			0
$529$	22.0000			0.956522
$530$		0			0
$531$		0			0
$532$		0			0
$533$			5.00000i			0.216574i
$534$		0			0
$535$	−9.00000			−0.389104
$536$		0			0
$537$		0			0
$538$		0			0
$539$			15.0000i			0.646096i
$540$		0			0
$541$		−	10.0000i		−	0.429934i	−0.976621	−	0.214967i	$-0.931036\pi$
$541$		−	10.0000i		−	0.429934i	0.976621	−	0.214967i	$-0.0689643\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000			−0.514024
$546$		0			0
$547$		−	18.0000i		−	0.769624i	−0.922995	−	0.384812i	$-0.874266\pi$
$547$		−	18.0000i		−	0.769624i	0.922995	−	0.384812i	$-0.125734\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	30.0000i		−	1.27804i
$552$		0			0
$553$	8.00000			0.340195
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$		0			0
$559$	−1.00000			−0.0422955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.0000			0.590030			0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000			0.590030			0.295015	+	0.955493i	$0.404675\pi$
$564$		0			0
$565$	33.0000			1.38832
$566$		0			0
$567$		0			0
$568$		0			0
$569$	40.0000			1.67689			0.838444	−	0.544988i	$-0.183466\pi$
$569$	40.0000			1.67689			0.838444	+	0.544988i	$0.183466\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4.00000i		−	0.166812i
$576$		0			0
$577$	−27.0000			−1.12402			−0.562012	−	0.827129i	$-0.689973\pi$
$577$	−27.0000			−1.12402			−0.562012	+	0.827129i	$0.689973\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	12.0000i		−	0.497844i
$582$		0			0
$583$			30.0000i			1.24247i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$			50.0000i			2.06021i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−14.0000			−0.574911			−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000			−0.574911			−0.287456	+	0.957794i	$0.592809\pi$
$594$		0			0
$595$	24.0000	−	6.00000i	0.983904	−	0.245976i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−20.0000			−0.817178			−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000			−0.817178			−0.408589	+	0.912719i	$0.633979\pi$
$600$		0			0
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	42.0000i		−	1.70754i
$606$		0			0
$607$		−	28.0000i		−	1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$		−	28.0000i		−	1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.00000			0.0809113
$612$		0			0
$613$	−21.0000			−0.848182			−0.424091	−	0.905620i	$-0.639406\pi$
$613$	−21.0000			−0.848182			−0.424091	+	0.905620i	$0.639406\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			42.0000i			1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$			42.0000i			1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$		0			0
$619$			36.0000i			1.44696i	0.690344	+	0.723481i	$0.257459\pi$
$619$			36.0000i			1.44696i	−0.690344	+	0.723481i	$0.742541\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.00000	+	8.00000i	0.0797452	+	0.318981i
$630$		0			0
$631$	23.0000			0.915616			0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000			0.915616			0.457808	+	0.889051i	$0.348635\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			21.0000i			0.833360i
$636$		0			0
$637$	−3.00000			−0.118864
$638$		0			0
$639$		0			0
$640$		0			0
$641$			35.0000i			1.38242i	0.722655	+	0.691208i	$0.242921\pi$
$641$			35.0000i			1.38242i	−0.722655	+	0.691208i	$0.757079\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.00000			0.314512			0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000			0.314512			0.157256	+	0.987558i	$0.449735\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	11.0000i		−	0.430463i	−0.976563	−	0.215232i	$-0.930949\pi$
$653$		−	11.0000i		−	0.430463i	0.976563	−	0.215232i	$-0.0690506\pi$
$654$		0			0
$655$	−15.0000			−0.586098
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	27.0000			1.05018			0.525089	−	0.851047i	$-0.324032\pi$
$661$	27.0000			1.05018			0.525089	+	0.851047i	$0.324032\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	30.0000			1.16335
$666$		0			0
$667$	−6.00000			−0.232321
$668$		0			0
$669$		0			0
$670$		0			0
$671$	50.0000			1.93023
$672$		0			0
$673$			46.0000i			1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$			46.0000i			1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	3.00000i		−	0.115299i	−0.998337	−	0.0576497i	$-0.981639\pi$
$677$		−	3.00000i		−	0.115299i	0.998337	−	0.0576497i	$-0.0183606\pi$
$678$		0			0
$679$	−16.0000			−0.614024
$680$		0			0
$681$		0			0
$682$		0			0
$683$			21.0000i			0.803543i	0.915740	+	0.401771i	$0.131605\pi$
$683$			21.0000i			0.803543i	−0.915740	+	0.401771i	$0.868395\pi$
$684$		0			0
$685$		−	24.0000i		−	0.916993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6.00000			−0.228582
$690$		0			0
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−48.0000			−1.82074
$696$		0			0
$697$	5.00000	+	20.0000i	0.189389	+	0.757554i
$698$		0			0
$699$		0			0
$700$		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
$703$			10.0000i			0.377157i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	24.0000i		−	0.902613i
$708$		0			0
$709$			34.0000i			1.27690i	0.769665	+	0.638448i	$0.220423\pi$
$709$			34.0000i			1.27690i	−0.769665	+	0.638448i	$0.779577\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.0000			0.374503
$714$		0			0
$715$	15.0000			0.560968
$716$		0			0
$717$		0			0
$718$		0			0
$719$			9.00000i			0.335643i	0.985817	+	0.167822i	$0.0536733\pi$
$719$			9.00000i			0.335643i	−0.985817	+	0.167822i	$0.946327\pi$
$720$		0			0
$721$		−	18.0000i		−	0.670355i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	24.0000i		−	0.891338i
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4.00000	+	1.00000i	−0.147945	+	0.0369863i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			60.0000i			2.21013i
$738$		0			0
$739$	−5.00000			−0.183928			−0.0919640	−	0.995762i	$-0.529314\pi$
$739$	−5.00000			−0.183928			−0.0919640	+	0.995762i	$0.529314\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.00000			−0.219235
$750$		0			0
$751$			10.0000i			0.364905i	0.983215	+	0.182453i	$0.0584036\pi$
$751$			10.0000i			0.364905i	−0.983215	+	0.182453i	$0.941596\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	36.0000i		−	1.31017i
$756$		0			0
$757$	43.0000			1.56286			0.781431	−	0.623992i	$-0.214490\pi$
$757$	43.0000			1.56286			0.781431	+	0.623992i	$0.214490\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$	−8.00000			−0.289619
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	5.00000			0.180305			0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000			0.180305			0.0901523	+	0.995928i	$0.471265\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	36.0000			1.29483			0.647415	−	0.762138i	$-0.275850\pi$
$773$	36.0000			1.29483			0.647415	+	0.762138i	$0.275850\pi$
$774$		0			0
$775$			40.0000i			1.43684i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			25.0000i			0.895718i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	51.0000i		−	1.82027i
$786$		0			0
$787$			12.0000i			0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$			12.0000i			0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22.0000			0.782230
$792$		0			0
$793$			10.0000i			0.355110i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.0000			0.425062			0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000			0.425062			0.212531	+	0.977154i	$0.431829\pi$
$798$		0			0
$799$	8.00000	−	2.00000i	0.283020	−	0.0707549i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−30.0000			−1.05868
$804$		0			0
$805$		−	6.00000i		−	0.211472i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			11.0000i			0.386739i	0.981126	+	0.193370i	$0.0619417\pi$
$809$			11.0000i			0.386739i	−0.981126	+	0.193370i	$0.938058\pi$
$810$		0			0
$811$		−	50.0000i		−	1.75574i	−0.478901	−	0.877869i	$-0.658965\pi$
$811$		−	50.0000i		−	1.75574i	0.478901	−	0.877869i	$-0.341035\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−42.0000			−1.47120
$816$		0			0
$817$	−5.00000			−0.174928
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	25.0000i		−	0.872506i	−0.899824	−	0.436253i	$-0.856305\pi$
$821$		−	25.0000i		−	0.872506i	0.899824	−	0.436253i	$-0.143695\pi$
$822$		0			0
$823$		−	46.0000i		−	1.60346i	−0.597687	−	0.801730i	$-0.703913\pi$
$823$		−	46.0000i		−	1.60346i	0.597687	−	0.801730i	$-0.296087\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.00000i			0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$			3.00000i			0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$		0			0
$829$	−50.0000			−1.73657			−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000			−1.73657			−0.868286	+	0.496064i	$0.834778\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.0000	+	3.00000i	−0.415775	+	0.103944i
$834$		0			0
$835$	−9.00000			−0.311458
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	41.0000i		−	1.41548i	−0.706474	−	0.707739i	$-0.749715\pi$
$839$		−	41.0000i		−	1.41548i	0.706474	−	0.707739i	$-0.250285\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	36.0000i		−	1.23844i
$846$		0			0
$847$		−	28.0000i		−	0.962091i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.00000			0.0685591
$852$		0			0
$853$			36.0000i			1.23262i	0.787505	+	0.616308i	$0.211372\pi$
$853$			36.0000i			1.23262i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			2.00000i			0.0683187i	0.999416	+	0.0341593i	$0.0108754\pi$
$857$			2.00000i			0.0683187i	−0.999416	+	0.0341593i	$0.989125\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	3.00000			0.102003
$866$		0			0
$867$		0			0
$868$		0			0
$869$	20.0000			0.678454
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.00000			−0.202837
$876$		0			0
$877$		−	32.0000i		−	1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$		−	32.0000i		−	1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	30.0000i		−	1.01073i	−0.862907	−	0.505363i	$-0.831359\pi$
$881$		−	30.0000i		−	1.01073i	0.862907	−	0.505363i	$-0.168641\pi$
$882$		0			0
$883$	−19.0000			−0.639401			−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000			−0.639401			−0.319700	+	0.947519i	$0.603582\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			13.0000i			0.436497i	0.975893	+	0.218249i	$0.0700344\pi$
$887$			13.0000i			0.436497i	−0.975893	+	0.218249i	$0.929966\pi$
$888$		0			0
$889$			14.0000i			0.469545i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	10.0000			0.334637
$894$		0			0
$895$			30.0000i			1.00279i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	60.0000			2.00111
$900$		0			0
$901$	−24.0000	+	6.00000i	−0.799556	+	0.199889i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−60.0000			−1.99447
$906$		0			0
$907$			32.0000i			1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$			32.0000i			1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			5.00000i			0.165657i	0.996564	+	0.0828287i	$0.0263954\pi$
$911$			5.00000i			0.165657i	−0.996564	+	0.0828287i	$0.973605\pi$
$912$		0			0
$913$		−	30.0000i		−	0.992855i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.0000			−0.330229
$918$		0			0
$919$	−15.0000			−0.494804			−0.247402	−	0.968913i	$-0.579577\pi$
$919$	−15.0000			−0.494804			−0.247402	+	0.968913i	$0.579577\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			8.00000i			0.263038i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			21.0000i			0.688988i	0.938789	+	0.344494i	$0.111949\pi$
$929$			21.0000i			0.688988i	−0.938789	+	0.344494i	$0.888051\pi$
$930$		0			0
$931$	−15.0000			−0.491605
$932$		0			0
$933$		0			0
$934$		0			0
$935$	60.0000	−	15.0000i	1.96221	−	0.490552i
$936$		0			0
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	30.0000i		−	0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$		−	30.0000i		−	0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$		0			0
$943$	5.00000			0.162822
$944$		0			0
$945$		0			0
$946$		0			0
$947$			48.0000i			1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$			48.0000i			1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$		0			0
$949$		−	6.00000i		−	0.194768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	36.0000			1.16615			0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000			1.16615			0.583077	+	0.812417i	$0.301849\pi$
$954$		0			0
$955$		−	24.0000i		−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	16.0000i		−	0.516667i
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.0000			−0.579441
$966$		0			0
$967$	57.0000			1.83300			0.916498	−	0.400039i	$-0.131003\pi$
$967$	57.0000			1.83300			0.916498	+	0.400039i	$0.131003\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	42.0000			1.34784			0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000			1.34784			0.673922	+	0.738802i	$0.264608\pi$
$972$		0			0
$973$	−32.0000			−1.02587
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.00000			−0.255943			−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000			−0.255943			−0.127971	+	0.991778i	$0.540847\pi$
$978$		0			0
$979$			50.0000i			1.59801i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	19.0000i		−	0.606006i	−0.952990	−	0.303003i	$-0.902011\pi$
$983$		−	19.0000i		−	0.606006i	0.952990	−	0.303003i	$-0.0979892\pi$
$984$		0			0
$985$	69.0000			2.19852
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.00000i			0.0317982i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−48.0000			−1.52170
$996$		0			0
$997$			28.0000i			0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$			28.0000i			0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.c.f.577.2		2
3.2	odd	2		816.2.c.b.577.2		2
4.3	odd	2		153.2.d.c.118.2		2
12.11	even	2		51.2.d.a.16.1	✓	2
17.16	even	2	inner	2448.2.c.f.577.1		2
24.5	odd	2		3264.2.c.h.577.1		2
24.11	even	2		3264.2.c.g.577.2		2
51.50	odd	2		816.2.c.b.577.1		2
60.23	odd	4		1275.2.d.c.424.2		2
60.47	odd	4		1275.2.d.a.424.1		2
60.59	even	2		1275.2.g.b.526.2		2
68.47	odd	4		2601.2.a.a.1.1		1
68.55	odd	4		2601.2.a.c.1.1		1
68.67	odd	2		153.2.d.c.118.1		2
204.11	odd	16		867.2.h.e.712.2		8
204.23	odd	16		867.2.h.e.712.1		8
204.47	even	4		867.2.a.e.1.1		1
204.59	even	8		867.2.e.a.616.1		4
204.71	odd	16		867.2.h.e.688.2		8
204.83	even	8		867.2.e.a.829.2		4
204.95	odd	16		867.2.h.e.733.2		8
204.107	odd	16		867.2.h.e.757.2		8
204.131	odd	16		867.2.h.e.757.1		8
204.143	odd	16		867.2.h.e.733.1		8
204.155	even	8		867.2.e.a.829.1		4
204.167	odd	16		867.2.h.e.688.1		8
204.179	even	8		867.2.e.a.616.2		4
204.191	even	4		867.2.a.d.1.1		1
204.203	even	2		51.2.d.a.16.2	yes	2
408.101	odd	2		3264.2.c.h.577.2		2
408.203	even	2		3264.2.c.g.577.1		2
1020.203	odd	4		1275.2.d.a.424.2		2
1020.407	odd	4		1275.2.d.c.424.1		2
1020.1019	even	2		1275.2.g.b.526.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.d.a.16.1	✓	2	12.11	even	2
51.2.d.a.16.2	yes	2	204.203	even	2
153.2.d.c.118.1		2	68.67	odd	2
153.2.d.c.118.2		2	4.3	odd	2
816.2.c.b.577.1		2	51.50	odd	2
816.2.c.b.577.2		2	3.2	odd	2
867.2.a.d.1.1		1	204.191	even	4
867.2.a.e.1.1		1	204.47	even	4
867.2.e.a.616.1		4	204.59	even	8
867.2.e.a.616.2		4	204.179	even	8
867.2.e.a.829.1		4	204.155	even	8
867.2.e.a.829.2		4	204.83	even	8
867.2.h.e.688.1		8	204.167	odd	16
867.2.h.e.688.2		8	204.71	odd	16
867.2.h.e.712.1		8	204.23	odd	16
867.2.h.e.712.2		8	204.11	odd	16
867.2.h.e.733.1		8	204.143	odd	16
867.2.h.e.733.2		8	204.95	odd	16
867.2.h.e.757.1		8	204.131	odd	16
867.2.h.e.757.2		8	204.107	odd	16
1275.2.d.a.424.1		2	60.47	odd	4
1275.2.d.a.424.2		2	1020.203	odd	4
1275.2.d.c.424.1		2	1020.407	odd	4
1275.2.d.c.424.2		2	60.23	odd	4
1275.2.g.b.526.1		2	1020.1019	even	2
1275.2.g.b.526.2		2	60.59	even	2
2448.2.c.f.577.1		2	17.16	even	2	inner
2448.2.c.f.577.2		2	1.1	even	1	trivial
2601.2.a.a.1.1		1	68.47	odd	4
2601.2.a.c.1.1		1	68.55	odd	4
3264.2.c.g.577.1		2	408.203	even	2
3264.2.c.g.577.2		2	24.11	even	2
3264.2.c.h.577.1		2	24.5	odd	2
3264.2.c.h.577.2		2	408.101	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{5} +2.00000i q^{7} +5.00000i q^{11} -1.00000 q^{13} +(-4.00000 + 1.00000i) q^{17} -5.00000 q^{19} +1.00000i q^{23} -4.00000 q^{25} +6.00000i q^{29} -10.0000i q^{31} -6.00000 q^{35} -2.00000i q^{37} -5.00000i q^{41} +1.00000 q^{43} -2.00000 q^{47} +3.00000 q^{49} +6.00000 q^{53} -15.0000 q^{55} -10.0000i q^{61} -3.00000i q^{65} +12.0000 q^{67} +6.00000i q^{73} -10.0000 q^{77} -4.00000i q^{79} -6.00000 q^{83} +(-3.00000 - 12.0000i) q^{85} +10.0000 q^{89} -2.00000i q^{91} -15.0000i q^{95} +8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{13} - 8 q^{17} - 10 q^{19} - 8 q^{25} - 12 q^{35} + 2 q^{43} - 4 q^{47} + 6 q^{49} + 12 q^{53} - 30 q^{55} + 24 q^{67} - 20 q^{77} - 12 q^{83} - 6 q^{85} + 20 q^{89}+O(q^{100}) \) 2 * q - 2 * q^13 - 8 * q^17 - 10 * q^19 - 8 * q^25 - 12 * q^35 + 2 * q^43 - 4 * q^47 + 6 * q^49 + 12 * q^53 - 30 * q^55 + 24 * q^67 - 20 * q^77 - 12 * q^83 - 6 * q^85 + 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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