LMFDB - Embedded newform 2040.2.h.e.1801.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,2,Mod(1801,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("2040.1801"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2040.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,0,0,-4,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2894820123$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1801.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2040.1801
Dual form			2040.2.h.e.1801.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000i q^{7} -1.00000 q^{9} +5.00000i q^{11} -2.00000 q^{13} +1.00000 q^{15} +(1.00000 - 4.00000i) q^{17} +1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{25} +1.00000i q^{27} +7.00000i q^{29} +5.00000 q^{33} +1.00000 q^{35} +5.00000i q^{37} +2.00000i q^{39} +9.00000i q^{41} -12.0000 q^{43} -1.00000i q^{45} +1.00000 q^{47} +6.00000 q^{49} +(-4.00000 - 1.00000i) q^{51} +3.00000 q^{53} -5.00000 q^{55} -1.00000i q^{57} -8.00000 q^{59} +8.00000i q^{61} +1.00000i q^{63} -2.00000i q^{65} -2.00000 q^{67} +12.0000i q^{71} +11.0000i q^{73} +1.00000i q^{75} +5.00000 q^{77} -10.0000i q^{79} +1.00000 q^{81} +8.00000 q^{83} +(4.00000 + 1.00000i) q^{85} +7.00000 q^{87} +16.0000 q^{89} +2.00000i q^{91} +1.00000i q^{95} -10.0000i q^{97} -5.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 4 q^{13} + 2 q^{15} + 2 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{25} + 10 q^{33} + 2 q^{35} - 24 q^{43} + 2 q^{47} + 12 q^{49} - 8 q^{51} + 6 q^{53} - 10 q^{55} - 16 q^{59} - 4 q^{67} + 10 q^{77}+ \cdots + 32 q^{89}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^13 + 2 * q^15 + 2 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^25 + 10 * q^33 + 2 * q^35 - 24 * q^43 + 2 * q^47 + 12 * q^49 - 8 * q^51 + 6 * q^53 - 10 * q^55 - 16 * q^59 - 4 * q^67 + 10 * q^77 + 2 * q^81 + 16 * q^83 + 8 * q^85 + 14 * q^87 + 32 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		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
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	1.00000	−	4.00000i	0.242536	−	0.970143i
$17$	1.00000	−	4.00000i	0.242536	−	0.970143i
$18$		0			0
$19$	1.00000			0.229416			0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000			0.229416			0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$			7.00000i			1.29987i	0.759991	+	0.649934i	$0.225203\pi$
$29$			7.00000i			1.29987i	−0.759991	+	0.649934i	$0.774797\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	5.00000			0.870388
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$			5.00000i			0.821995i	0.911636	+	0.410997i	$0.134819\pi$
$37$			5.00000i			0.821995i	−0.911636	+	0.410997i	$0.865181\pi$
$38$		0			0
$39$			2.00000i			0.320256i
$40$		0			0
$41$			9.00000i			1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$41$			9.00000i			1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$42$		0			0
$43$	−12.0000			−1.82998			−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000			−1.82998			−0.914991	+	0.403473i	$0.867803\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$	1.00000			0.145865			0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000			0.145865			0.0729325	+	0.997337i	$0.476764\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$	−4.00000	−	1.00000i	−0.560112	−	0.140028i
$52$		0			0
$53$	3.00000			0.412082			0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000			0.412082			0.206041	+	0.978543i	$0.433942\pi$
$54$		0			0
$55$	−5.00000			−0.674200
$56$		0			0
$57$		−	1.00000i		−	0.132453i
$58$		0			0
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$			8.00000i			1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$			8.00000i			1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$		−	2.00000i		−	0.248069i
$66$		0			0
$67$	−2.00000			−0.244339			−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000			−0.244339			−0.122169	+	0.992509i	$0.538985\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$			11.0000i			1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$			11.0000i			1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	5.00000			0.569803
$78$		0			0
$79$		−	10.0000i		−	1.12509i	−0.826767	−	0.562544i	$-0.809823\pi$
$79$		−	10.0000i		−	1.12509i	0.826767	−	0.562544i	$-0.190177\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	8.00000			0.878114			0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000			0.878114			0.439057	+	0.898459i	$0.355313\pi$
$84$		0			0
$85$	4.00000	+	1.00000i	0.433861	+	0.108465i
$86$		0			0
$87$	7.00000			0.750479
$88$		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
$91$			2.00000i			0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1.00000i			0.102598i
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$		0			0
$99$		−	5.00000i		−	0.502519i
$100$		0			0
$101$	−4.00000			−0.398015			−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000			−0.398015			−0.199007	+	0.979998i	$0.563772\pi$
$102$		0			0
$103$	−2.00000			−0.197066			−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000			−0.197066			−0.0985329	+	0.995134i	$0.531415\pi$
$104$		0			0
$105$		−	1.00000i		−	0.0975900i
$106$		0			0
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$		0			0
$109$			20.0000i			1.91565i	0.287348	+	0.957826i	$0.407226\pi$
$109$			20.0000i			1.91565i	−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$	5.00000			0.474579
$112$		0			0
$113$			8.00000i			0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$			8.00000i			0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$	−4.00000	−	1.00000i	−0.366679	−	0.0916698i
$120$		0			0
$121$	−14.0000			−1.27273
$122$		0			0
$123$	9.00000			0.811503
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$			12.0000i			1.05654i
$130$		0			0
$131$		−	20.0000i		−	1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$		−	20.0000i		−	1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$		0			0
$133$		−	1.00000i		−	0.0867110i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	3.00000			0.256307			0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000			0.256307			0.128154	+	0.991754i	$0.459095\pi$
$138$		0			0
$139$			10.0000i			0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$			10.0000i			0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$		0			0
$141$		−	1.00000i		−	0.0842152i
$142$		0			0
$143$		−	10.0000i		−	0.836242i
$144$		0			0
$145$	−7.00000			−0.581318
$146$		0			0
$147$		−	6.00000i		−	0.494872i
$148$		0			0
$149$	20.0000			1.63846			0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000			1.63846			0.819232	+	0.573462i	$0.194400\pi$
$150$		0			0
$151$	−7.00000			−0.569652			−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000			−0.569652			−0.284826	+	0.958579i	$0.591936\pi$
$152$		0			0
$153$	−1.00000	+	4.00000i	−0.0808452	+	0.323381i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−18.0000			−1.43656			−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000			−1.43656			−0.718278	+	0.695756i	$0.755069\pi$
$158$		0			0
$159$		−	3.00000i		−	0.237915i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			25.0000i			1.95815i	0.203497	+	0.979076i	$0.434769\pi$
$163$			25.0000i			1.95815i	−0.203497	+	0.979076i	$0.565231\pi$
$164$		0			0
$165$			5.00000i			0.389249i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−1.00000			−0.0764719
$172$		0			0
$173$		−	2.00000i		−	0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$		−	2.00000i		−	0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		0			0
$185$	−5.00000			−0.367607
$186$		0			0
$187$	20.0000	+	5.00000i	1.46254	+	0.365636i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		−	2.00000i		−	0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$		−	2.00000i		−	0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$		0			0
$195$	−2.00000			−0.143223
$196$		0			0
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$			4.00000i			0.283552i	0.989899	+	0.141776i	$0.0452813\pi$
$199$			4.00000i			0.283552i	−0.989899	+	0.141776i	$0.954719\pi$
$200$		0			0
$201$			2.00000i			0.141069i
$202$		0			0
$203$	7.00000			0.491304
$204$		0			0
$205$	−9.00000			−0.628587
$206$		0			0
$207$		0			0
$208$		0			0
$209$			5.00000i			0.345857i
$210$		0			0
$211$		−	22.0000i		−	1.51454i	−0.653101	−	0.757271i	$-0.726532\pi$
$211$		−	22.0000i		−	1.51454i	0.653101	−	0.757271i	$-0.273468\pi$
$212$		0			0
$213$	12.0000			0.822226
$214$		0			0
$215$		−	12.0000i		−	0.818393i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	11.0000			0.743311
$220$		0			0
$221$	−2.00000	+	8.00000i	−0.134535	+	0.538138i
$222$		0			0
$223$	−6.00000			−0.401790			−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000			−0.401790			−0.200895	+	0.979613i	$0.564385\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$		−	10.0000i		−	0.663723i	−0.943328	−	0.331862i	$-0.892323\pi$
$227$		−	10.0000i		−	0.663723i	0.943328	−	0.331862i	$-0.107677\pi$
$228$		0			0
$229$	9.00000			0.594737			0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000			0.594737			0.297368	+	0.954763i	$0.403891\pi$
$230$		0			0
$231$		−	5.00000i		−	0.328976i
$232$		0			0
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$		0			0
$235$			1.00000i			0.0652328i
$236$		0			0
$237$	−10.0000			−0.649570
$238$		0			0
$239$	22.0000			1.42306			0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000			1.42306			0.711531	+	0.702655i	$0.248002\pi$
$240$		0			0
$241$		−	14.0000i		−	0.901819i	−0.892570	−	0.450910i	$-0.851100\pi$
$241$		−	14.0000i		−	0.901819i	0.892570	−	0.450910i	$-0.148900\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			6.00000i			0.383326i
$246$		0			0
$247$	−2.00000			−0.127257
$248$		0			0
$249$		−	8.00000i		−	0.506979i
$250$		0			0
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	1.00000	−	4.00000i	0.0626224	−	0.250490i
$256$		0			0
$257$	30.0000			1.87135			0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000			1.87135			0.935674	+	0.352865i	$0.114792\pi$
$258$		0			0
$259$	5.00000			0.310685
$260$		0			0
$261$		−	7.00000i		−	0.433289i
$262$		0			0
$263$	−15.0000			−0.924940			−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000			−0.924940			−0.462470	+	0.886635i	$0.653037\pi$
$264$		0			0
$265$			3.00000i			0.184289i
$266$		0			0
$267$		−	16.0000i		−	0.979184i
$268$		0			0
$269$			1.00000i			0.0609711i	0.999535	+	0.0304855i	$0.00970535\pi$
$269$			1.00000i			0.0609711i	−0.999535	+	0.0304855i	$0.990295\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$		0			0
$273$	2.00000			0.121046
$274$		0			0
$275$		−	5.00000i		−	0.301511i
$276$		0			0
$277$			6.00000i			0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$			6.00000i			0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.00000			−0.119310			−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000			−0.119310			−0.0596550	+	0.998219i	$0.519000\pi$
$282$		0			0
$283$		−	9.00000i		−	0.534994i	−0.963559	−	0.267497i	$-0.913803\pi$
$283$		−	9.00000i		−	0.534994i	0.963559	−	0.267497i	$-0.0861966\pi$
$284$		0			0
$285$	1.00000			0.0592349
$286$		0			0
$287$	9.00000			0.531253
$288$		0			0
$289$	−15.0000	−	8.00000i	−0.882353	−	0.470588i
$290$		0			0
$291$	−10.0000			−0.586210
$292$		0			0
$293$	9.00000			0.525786			0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000			0.525786			0.262893	+	0.964825i	$0.415323\pi$
$294$		0			0
$295$		−	8.00000i		−	0.465778i
$296$		0			0
$297$	−5.00000			−0.290129
$298$		0			0
$299$		0			0
$300$		0			0
$301$			12.0000i			0.691669i
$302$		0			0
$303$			4.00000i			0.229794i
$304$		0			0
$305$	−8.00000			−0.458079
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$			2.00000i			0.113776i
$310$		0			0
$311$			23.0000i			1.30421i	0.758129	+	0.652105i	$0.226114\pi$
$311$			23.0000i			1.30421i	−0.758129	+	0.652105i	$0.773886\pi$
$312$		0			0
$313$			5.00000i			0.282617i	0.989966	+	0.141308i	$0.0451309\pi$
$313$			5.00000i			0.282617i	−0.989966	+	0.141308i	$0.954869\pi$
$314$		0			0
$315$	−1.00000			−0.0563436
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$	−35.0000			−1.95962
$320$		0			0
$321$	2.00000			0.111629
$322$		0			0
$323$	1.00000	−	4.00000i	0.0556415	−	0.222566i
$324$		0			0
$325$	2.00000			0.110940
$326$		0			0
$327$	20.0000			1.10600
$328$		0			0
$329$		−	1.00000i		−	0.0551318i
$330$		0			0
$331$	−23.0000			−1.26419			−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000			−1.26419			−0.632097	+	0.774889i	$0.717806\pi$
$332$		0			0
$333$		−	5.00000i		−	0.273998i
$334$		0			0
$335$		−	2.00000i		−	0.109272i
$336$		0			0
$337$		−	17.0000i		−	0.926049i	−0.886345	−	0.463025i	$-0.846764\pi$
$337$		−	17.0000i		−	0.926049i	0.886345	−	0.463025i	$-0.153236\pi$
$338$		0			0
$339$	8.00000			0.434500
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		0			0
$349$	−19.0000			−1.01705			−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000			−1.01705			−0.508523	+	0.861048i	$0.669808\pi$
$350$		0			0
$351$		−	2.00000i		−	0.106752i
$352$		0			0
$353$	19.0000			1.01127			0.505634	−	0.862748i	$-0.331259\pi$
$353$	19.0000			1.01127			0.505634	+	0.862748i	$0.331259\pi$
$354$		0			0
$355$	−12.0000			−0.636894
$356$		0			0
$357$	−1.00000	+	4.00000i	−0.0529256	+	0.211702i
$358$		0			0
$359$	−34.0000			−1.79445			−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000			−1.79445			−0.897226	+	0.441572i	$0.854421\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$		0			0
$363$			14.0000i			0.734809i
$364$		0			0
$365$	−11.0000			−0.575766
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		0			0
$369$		−	9.00000i		−	0.468521i
$370$		0			0
$371$		−	3.00000i		−	0.155752i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$		−	14.0000i		−	0.721037i
$378$		0			0
$379$		−	26.0000i		−	1.33553i	−0.744372	−	0.667765i	$-0.767251\pi$
$379$		−	26.0000i		−	1.33553i	0.744372	−	0.667765i	$-0.232749\pi$
$380$		0			0
$381$		−	12.0000i		−	0.614779i
$382$		0			0
$383$	−31.0000			−1.58403			−0.792013	−	0.610504i	$-0.790967\pi$
$383$	−31.0000			−1.58403			−0.792013	+	0.610504i	$0.790967\pi$
$384$		0			0
$385$			5.00000i			0.254824i
$386$		0			0
$387$	12.0000			0.609994
$388$		0			0
$389$	26.0000			1.31825			0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000			1.31825			0.659126	+	0.752032i	$0.270926\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−20.0000			−1.00887
$394$		0			0
$395$	10.0000			0.503155
$396$		0			0
$397$		−	9.00000i		−	0.451697i	−0.974162	−	0.225849i	$-0.927485\pi$
$397$		−	9.00000i		−	0.451697i	0.974162	−	0.225849i	$-0.0725154\pi$
$398$		0			0
$399$	−1.00000			−0.0500626
$400$		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
$403$		0			0
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$	−25.0000			−1.23920
$408$		0			0
$409$	17.0000			0.840596			0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000			0.840596			0.420298	+	0.907386i	$0.361926\pi$
$410$		0			0
$411$		−	3.00000i		−	0.147979i
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$			8.00000i			0.392705i
$416$		0			0
$417$	10.0000			0.489702
$418$		0			0
$419$		−	9.00000i		−	0.439679i	−0.975536	−	0.219839i	$-0.929447\pi$
$419$		−	9.00000i		−	0.439679i	0.975536	−	0.219839i	$-0.0705533\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$	−1.00000			−0.0486217
$424$		0			0
$425$	−1.00000	+	4.00000i	−0.0485071	+	0.194029i
$426$		0			0
$427$	8.00000			0.387147
$428$		0			0
$429$	−10.0000			−0.482805
$430$		0			0
$431$			23.0000i			1.10787i	0.832560	+	0.553936i	$0.186875\pi$
$431$			23.0000i			1.10787i	−0.832560	+	0.553936i	$0.813125\pi$
$432$		0			0
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		0			0
$435$			7.00000i			0.335624i
$436$		0			0
$437$		0			0
$438$		0			0
$439$			8.00000i			0.381819i	0.981608	+	0.190910i	$0.0611437\pi$
$439$			8.00000i			0.381819i	−0.981608	+	0.190910i	$0.938856\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$			16.0000i			0.758473i
$446$		0			0
$447$		−	20.0000i		−	0.945968i
$448$		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
$451$	−45.0000			−2.11897
$452$		0			0
$453$			7.00000i			0.328889i
$454$		0			0
$455$	−2.00000			−0.0937614
$456$		0			0
$457$	−24.0000			−1.12267			−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000			−1.12267			−0.561336	+	0.827588i	$0.689713\pi$
$458$		0			0
$459$	4.00000	+	1.00000i	0.186704	+	0.0466760i
$460$		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
$467$	21.0000			0.971764			0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000			0.971764			0.485882	+	0.874024i	$0.338498\pi$
$468$		0			0
$469$			2.00000i			0.0923514i
$470$		0			0
$471$			18.0000i			0.829396i
$472$		0			0
$473$		−	60.0000i		−	2.75880i
$474$		0			0
$475$	−1.00000			−0.0458831
$476$		0			0
$477$	−3.00000			−0.137361
$478$		0			0
$479$		−	32.0000i		−	1.46212i	−0.682315	−	0.731059i	$-0.739027\pi$
$479$		−	32.0000i		−	1.46212i	0.682315	−	0.731059i	$-0.260973\pi$
$480$		0			0
$481$		−	10.0000i		−	0.455961i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.0000			0.454077
$486$		0			0
$487$		−	32.0000i		−	1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$		−	32.0000i		−	1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$		0			0
$489$	25.0000			1.13054
$490$		0			0
$491$	4.00000			0.180517			0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000			0.180517			0.0902587	+	0.995918i	$0.471231\pi$
$492$		0			0
$493$	28.0000	+	7.00000i	1.26106	+	0.315264i
$494$		0			0
$495$	5.00000			0.224733
$496$		0			0
$497$	12.0000			0.538274
$498$		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
$503$		−	14.0000i		−	0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$		−	14.0000i		−	0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$		0			0
$505$		−	4.00000i		−	0.177998i
$506$		0			0
$507$			9.00000i			0.399704i
$508$		0			0
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$	11.0000			0.486611
$512$		0			0
$513$			1.00000i			0.0441511i
$514$		0			0
$515$		−	2.00000i		−	0.0881305i
$516$		0			0
$517$			5.00000i			0.219900i
$518$		0			0
$519$	−2.00000			−0.0877903
$520$		0			0
$521$		−	33.0000i		−	1.44576i	−0.690976	−	0.722878i	$-0.742819\pi$
$521$		−	33.0000i		−	1.44576i	0.690976	−	0.722878i	$-0.257181\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$		0			0
$525$	1.00000			0.0436436
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	8.00000			0.347170
$532$		0			0
$533$		−	18.0000i		−	0.779667i
$534$		0			0
$535$	−2.00000			−0.0864675
$536$		0			0
$537$		0			0
$538$		0			0
$539$			30.0000i			1.29219i
$540$		0			0
$541$			8.00000i			0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$			8.00000i			0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$		0			0
$543$	10.0000			0.429141
$544$		0			0
$545$	−20.0000			−0.856706
$546$		0			0
$547$		−	19.0000i		−	0.812381i	−0.913788	−	0.406191i	$-0.866857\pi$
$547$		−	19.0000i		−	0.812381i	0.913788	−	0.406191i	$-0.133143\pi$
$548$		0			0
$549$		−	8.00000i		−	0.341432i
$550$		0			0
$551$			7.00000i			0.298210i
$552$		0			0
$553$	−10.0000			−0.425243
$554$		0			0
$555$			5.00000i			0.212238i
$556$		0			0
$557$	30.0000			1.27114			0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000			1.27114			0.635570	+	0.772043i	$0.280765\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$	5.00000	−	20.0000i	0.211100	−	0.844401i
$562$		0			0
$563$	−9.00000			−0.379305			−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000			−0.379305			−0.189652	+	0.981851i	$0.560736\pi$
$564$		0			0
$565$	−8.00000			−0.336563
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$		0			0
$571$		−	2.00000i		−	0.0836974i	−0.999124	−	0.0418487i	$-0.986675\pi$
$571$		−	2.00000i		−	0.0836974i	0.999124	−	0.0418487i	$-0.0133247\pi$
$572$		0			0
$573$			24.0000i			1.00261i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		0			0
$579$	−2.00000			−0.0831172
$580$		0			0
$581$		−	8.00000i		−	0.331896i
$582$		0			0
$583$			15.0000i			0.621237i
$584$		0			0
$585$			2.00000i			0.0826898i
$586$		0			0
$587$	37.0000			1.52715			0.763577	−	0.645717i	$-0.223441\pi$
$587$	37.0000			1.52715			0.763577	+	0.645717i	$0.223441\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	10.0000			0.411345
$592$		0			0
$593$	−11.0000			−0.451716			−0.225858	−	0.974160i	$-0.572519\pi$
$593$	−11.0000			−0.451716			−0.225858	+	0.974160i	$0.572519\pi$
$594$		0			0
$595$	1.00000	−	4.00000i	0.0409960	−	0.163984i
$596$		0			0
$597$	4.00000			0.163709
$598$		0			0
$599$	6.00000			0.245153			0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000			0.245153			0.122577	+	0.992459i	$0.460884\pi$
$600$		0			0
$601$			8.00000i			0.326327i	0.986599	+	0.163163i	$0.0521698\pi$
$601$			8.00000i			0.326327i	−0.986599	+	0.163163i	$0.947830\pi$
$602$		0			0
$603$	2.00000			0.0814463
$604$		0			0
$605$		−	14.0000i		−	0.569181i
$606$		0			0
$607$			13.0000i			0.527654i	0.964570	+	0.263827i	$0.0849848\pi$
$607$			13.0000i			0.527654i	−0.964570	+	0.263827i	$0.915015\pi$
$608$		0			0
$609$		−	7.00000i		−	0.283654i
$610$		0			0
$611$	−2.00000			−0.0809113
$612$		0			0
$613$	40.0000			1.61558			0.807792	−	0.589467i	$-0.200662\pi$
$613$	40.0000			1.61558			0.807792	+	0.589467i	$0.200662\pi$
$614$		0			0
$615$			9.00000i			0.362915i
$616$		0			0
$617$			32.0000i			1.28827i	0.764911	+	0.644136i	$0.222783\pi$
$617$			32.0000i			1.28827i	−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$		−	14.0000i		−	0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$		−	14.0000i		−	0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	16.0000i		−	0.641026i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	5.00000			0.199681
$628$		0			0
$629$	20.0000	+	5.00000i	0.797452	+	0.199363i
$630$		0			0
$631$	−7.00000			−0.278666			−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000			−0.278666			−0.139333	+	0.990246i	$0.544496\pi$
$632$		0			0
$633$	−22.0000			−0.874421
$634$		0			0
$635$			12.0000i			0.476205i
$636$		0			0
$637$	−12.0000			−0.475457
$638$		0			0
$639$		−	12.0000i		−	0.474713i
$640$		0			0
$641$			1.00000i			0.0394976i	0.999805	+	0.0197488i	$0.00628665\pi$
$641$			1.00000i			0.0394976i	−0.999805	+	0.0197488i	$0.993713\pi$
$642$		0			0
$643$			9.00000i			0.354925i	0.984128	+	0.177463i	$0.0567889\pi$
$643$			9.00000i			0.354925i	−0.984128	+	0.177463i	$0.943211\pi$
$644$		0			0
$645$	−12.0000			−0.472500
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$		−	40.0000i		−	1.57014i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$	20.0000			0.781465
$656$		0			0
$657$		−	11.0000i		−	0.429151i
$658$		0			0
$659$	−22.0000			−0.856998			−0.428499	−	0.903542i	$-0.640958\pi$
$659$	−22.0000			−0.856998			−0.428499	+	0.903542i	$0.640958\pi$
$660$		0			0
$661$	−11.0000			−0.427850			−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000			−0.427850			−0.213925	+	0.976850i	$0.568625\pi$
$662$		0			0
$663$	8.00000	+	2.00000i	0.310694	+	0.0776736i
$664$		0			0
$665$	1.00000			0.0387783
$666$		0			0
$667$		0			0
$668$		0			0
$669$			6.00000i			0.231973i
$670$		0			0
$671$	−40.0000			−1.54418
$672$		0			0
$673$		−	46.0000i		−	1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$		−	46.0000i		−	1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$			32.0000i			1.22986i	0.788582	+	0.614930i	$0.210816\pi$
$677$			32.0000i			1.22986i	−0.788582	+	0.614930i	$0.789184\pi$
$678$		0			0
$679$	−10.0000			−0.383765
$680$		0			0
$681$	−10.0000			−0.383201
$682$		0			0
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$		0			0
$685$			3.00000i			0.114624i
$686$		0			0
$687$		−	9.00000i		−	0.343371i
$688$		0			0
$689$	−6.00000			−0.228582
$690$		0			0
$691$		−	6.00000i		−	0.228251i	−0.993466	−	0.114125i	$-0.963593\pi$
$691$		−	6.00000i		−	0.228251i	0.993466	−	0.114125i	$-0.0364066\pi$
$692$		0			0
$693$	−5.00000			−0.189934
$694$		0			0
$695$	−10.0000			−0.379322
$696$		0			0
$697$	36.0000	+	9.00000i	1.36360	+	0.340899i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	32.0000			1.20862			0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000			1.20862			0.604312	+	0.796748i	$0.293448\pi$
$702$		0			0
$703$			5.00000i			0.188579i
$704$		0			0
$705$	1.00000			0.0376622
$706$		0			0
$707$			4.00000i			0.150435i
$708$		0			0
$709$		−	14.0000i		−	0.525781i	−0.964826	−	0.262891i	$-0.915324\pi$
$709$		−	14.0000i		−	0.525781i	0.964826	−	0.262891i	$-0.0846758\pi$
$710$		0			0
$711$			10.0000i			0.375029i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	10.0000			0.373979
$716$		0			0
$717$		−	22.0000i		−	0.821605i
$718$		0			0
$719$			5.00000i			0.186469i	0.995644	+	0.0932343i	$0.0297206\pi$
$719$			5.00000i			0.186469i	−0.995644	+	0.0932343i	$0.970279\pi$
$720$		0			0
$721$			2.00000i			0.0744839i
$722$		0			0
$723$	−14.0000			−0.520666
$724$		0			0
$725$		−	7.00000i		−	0.259973i
$726$		0			0
$727$	42.0000			1.55769			0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000			1.55769			0.778847	+	0.627214i	$0.215805\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−12.0000	+	48.0000i	−0.443836	+	1.77534i
$732$		0			0
$733$	12.0000			0.443230			0.221615	−	0.975134i	$-0.428867\pi$
$733$	12.0000			0.443230			0.221615	+	0.975134i	$0.428867\pi$
$734$		0			0
$735$	6.00000			0.221313
$736$		0			0
$737$		−	10.0000i		−	0.368355i
$738$		0			0
$739$	−12.0000			−0.441427			−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000			−0.441427			−0.220714	+	0.975339i	$0.570839\pi$
$740$		0			0
$741$			2.00000i			0.0734718i
$742$		0			0
$743$			22.0000i			0.807102i	0.914957	+	0.403551i	$0.132224\pi$
$743$			22.0000i			0.807102i	−0.914957	+	0.403551i	$0.867776\pi$
$744$		0			0
$745$			20.0000i			0.732743i
$746$		0			0
$747$	−8.00000			−0.292705
$748$		0			0
$749$	2.00000			0.0730784
$750$		0			0
$751$			2.00000i			0.0729810i	0.999334	+	0.0364905i	$0.0116179\pi$
$751$			2.00000i			0.0729810i	−0.999334	+	0.0364905i	$0.988382\pi$
$752$		0			0
$753$		−	6.00000i		−	0.218652i
$754$		0			0
$755$		−	7.00000i		−	0.254756i
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$	20.0000			0.724049
$764$		0			0
$765$	−4.00000	−	1.00000i	−0.144620	−	0.0361551i
$766$		0			0
$767$	16.0000			0.577727
$768$		0			0
$769$	35.0000			1.26213			0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000			1.26213			0.631066	+	0.775729i	$0.282618\pi$
$770$		0			0
$771$		−	30.0000i		−	1.08042i
$772$		0			0
$773$	9.00000			0.323708			0.161854	−	0.986815i	$-0.448253\pi$
$773$	9.00000			0.323708			0.161854	+	0.986815i	$0.448253\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	5.00000i		−	0.179374i
$778$		0			0
$779$			9.00000i			0.322458i
$780$		0			0
$781$	−60.0000			−2.14697
$782$		0			0
$783$	−7.00000			−0.250160
$784$		0			0
$785$		−	18.0000i		−	0.642448i
$786$		0			0
$787$			25.0000i			0.891154i	0.895244	+	0.445577i	$0.147001\pi$
$787$			25.0000i			0.891154i	−0.895244	+	0.445577i	$0.852999\pi$
$788$		0			0
$789$			15.0000i			0.534014i
$790$		0			0
$791$	8.00000			0.284447
$792$		0			0
$793$		−	16.0000i		−	0.568177i
$794$		0			0
$795$	3.00000			0.106399
$796$		0			0
$797$	−47.0000			−1.66483			−0.832413	−	0.554156i	$-0.813041\pi$
$797$	−47.0000			−1.66483			−0.832413	+	0.554156i	$0.813041\pi$
$798$		0			0
$799$	1.00000	−	4.00000i	0.0353775	−	0.141510i
$800$		0			0
$801$	−16.0000			−0.565332
$802$		0			0
$803$	−55.0000			−1.94091
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.00000			0.0352017
$808$		0			0
$809$		−	6.00000i		−	0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$		−	6.00000i		−	0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$			8.00000i			0.280572i
$814$		0			0
$815$	−25.0000			−0.875712
$816$		0			0
$817$	−12.0000			−0.419827
$818$		0			0
$819$		−	2.00000i		−	0.0698857i
$820$		0			0
$821$			42.0000i			1.46581i	0.680331	+	0.732905i	$0.261836\pi$
$821$			42.0000i			1.46581i	−0.680331	+	0.732905i	$0.738164\pi$
$822$		0			0
$823$			17.0000i			0.592583i	0.955098	+	0.296291i	$0.0957499\pi$
$823$			17.0000i			0.592583i	−0.955098	+	0.296291i	$0.904250\pi$
$824$		0			0
$825$	−5.00000			−0.174078
$826$		0			0
$827$			36.0000i			1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$			36.0000i			1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$		0			0
$829$	5.00000			0.173657			0.0868286	−	0.996223i	$-0.472327\pi$
$829$	5.00000			0.173657			0.0868286	+	0.996223i	$0.472327\pi$
$830$		0			0
$831$	6.00000			0.208138
$832$		0			0
$833$	6.00000	−	24.0000i	0.207888	−	0.831551i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			27.0000i			0.932144i	0.884747	+	0.466072i	$0.154331\pi$
$839$			27.0000i			0.932144i	−0.884747	+	0.466072i	$0.845669\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$		0			0
$843$			2.00000i			0.0688837i
$844$		0			0
$845$		−	9.00000i		−	0.309609i
$846$		0			0
$847$			14.0000i			0.481046i
$848$		0			0
$849$	−9.00000			−0.308879
$850$		0			0
$851$		0			0
$852$		0			0
$853$			10.0000i			0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$			10.0000i			0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$		0			0
$855$		−	1.00000i		−	0.0341993i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−25.0000			−0.852989			−0.426494	−	0.904490i	$-0.640252\pi$
$859$	−25.0000			−0.852989			−0.426494	+	0.904490i	$0.640252\pi$
$860$		0			0
$861$		−	9.00000i		−	0.306719i
$862$		0			0
$863$	3.00000			0.102121			0.0510606	−	0.998696i	$-0.483740\pi$
$863$	3.00000			0.102121			0.0510606	+	0.998696i	$0.483740\pi$
$864$		0			0
$865$	2.00000			0.0680020
$866$		0			0
$867$	−8.00000	+	15.0000i	−0.271694	+	0.509427i
$868$		0			0
$869$	50.0000			1.69613
$870$		0			0
$871$	4.00000			0.135535
$872$		0			0
$873$			10.0000i			0.338449i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$			11.0000i			0.371444i	0.982602	+	0.185722i	$0.0594623\pi$
$877$			11.0000i			0.371444i	−0.982602	+	0.185722i	$0.940538\pi$
$878$		0			0
$879$		−	9.00000i		−	0.303562i
$880$		0			0
$881$		−	49.0000i		−	1.65085i	−0.564510	−	0.825426i	$-0.690935\pi$
$881$		−	49.0000i		−	1.65085i	0.564510	−	0.825426i	$-0.309065\pi$
$882$		0			0
$883$	−10.0000			−0.336527			−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000			−0.336527			−0.168263	+	0.985742i	$0.553816\pi$
$884$		0			0
$885$	−8.00000			−0.268917
$886$		0			0
$887$			58.0000i			1.94745i	0.227728	+	0.973725i	$0.426870\pi$
$887$			58.0000i			1.94745i	−0.227728	+	0.973725i	$0.573130\pi$
$888$		0			0
$889$		−	12.0000i		−	0.402467i
$890$		0			0
$891$			5.00000i			0.167506i
$892$		0			0
$893$	1.00000			0.0334637
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	3.00000	−	12.0000i	0.0999445	−	0.399778i
$902$		0			0
$903$	12.0000			0.399335
$904$		0			0
$905$	−10.0000			−0.332411
$906$		0			0
$907$			47.0000i			1.56061i	0.625400	+	0.780305i	$0.284936\pi$
$907$			47.0000i			1.56061i	−0.625400	+	0.780305i	$0.715064\pi$
$908$		0			0
$909$	4.00000			0.132672
$910$		0			0
$911$			21.0000i			0.695761i	0.937539	+	0.347881i	$0.113099\pi$
$911$			21.0000i			0.695761i	−0.937539	+	0.347881i	$0.886901\pi$
$912$		0			0
$913$			40.0000i			1.32381i
$914$		0			0
$915$			8.00000i			0.264472i
$916$		0			0
$917$	−20.0000			−0.660458
$918$		0			0
$919$	39.0000			1.28649			0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000			1.28649			0.643246	+	0.765660i	$0.277587\pi$
$920$		0			0
$921$		−	16.0000i		−	0.527218i
$922$		0			0
$923$		−	24.0000i		−	0.789970i
$924$		0			0
$925$		−	5.00000i		−	0.164399i
$926$		0			0
$927$	2.00000			0.0656886
$928$		0			0
$929$			41.0000i			1.34517i	0.740022	+	0.672583i	$0.234815\pi$
$929$			41.0000i			1.34517i	−0.740022	+	0.672583i	$0.765185\pi$
$930$		0			0
$931$	6.00000			0.196642
$932$		0			0
$933$	23.0000			0.752986
$934$		0			0
$935$	−5.00000	+	20.0000i	−0.163517	+	0.654070i
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$	5.00000			0.163169
$940$		0			0
$941$			50.0000i			1.62995i	0.579494	+	0.814977i	$0.303250\pi$
$941$			50.0000i			1.62995i	−0.579494	+	0.814977i	$0.696750\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$			1.00000i			0.0325300i
$946$		0			0
$947$		−	6.00000i		−	0.194974i	−0.995237	−	0.0974869i	$-0.968920\pi$
$947$		−	6.00000i		−	0.194974i	0.995237	−	0.0974869i	$-0.0310804\pi$
$948$		0			0
$949$		−	22.0000i		−	0.714150i
$950$		0			0
$951$	6.00000			0.194563
$952$		0			0
$953$	31.0000			1.00419			0.502094	−	0.864813i	$-0.332563\pi$
$953$	31.0000			1.00419			0.502094	+	0.864813i	$0.332563\pi$
$954$		0			0
$955$		−	24.0000i		−	0.776622i
$956$		0			0
$957$			35.0000i			1.13139i
$958$		0			0
$959$		−	3.00000i		−	0.0968751i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		−	2.00000i		−	0.0644491i
$964$		0			0
$965$	2.00000			0.0643823
$966$		0			0
$967$	26.0000			0.836104			0.418052	−	0.908423i	$-0.362713\pi$
$967$	26.0000			0.836104			0.418052	+	0.908423i	$0.362713\pi$
$968$		0			0
$969$	−4.00000	−	1.00000i	−0.128499	−	0.0321246i
$970$		0			0
$971$	−6.00000			−0.192549			−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000			−0.192549			−0.0962746	+	0.995355i	$0.530693\pi$
$972$		0			0
$973$	10.0000			0.320585
$974$		0			0
$975$		−	2.00000i		−	0.0640513i
$976$		0			0
$977$	−42.0000			−1.34370			−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000			−1.34370			−0.671850	+	0.740688i	$0.734500\pi$
$978$		0			0
$979$			80.0000i			2.55681i
$980$		0			0
$981$		−	20.0000i		−	0.638551i
$982$		0			0
$983$		−	32.0000i		−	1.02064i	−0.859984	−	0.510321i	$-0.829527\pi$
$983$		−	32.0000i		−	1.02064i	0.859984	−	0.510321i	$-0.170473\pi$
$984$		0			0
$985$	−10.0000			−0.318626
$986$		0			0
$987$	−1.00000			−0.0318304
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	38.0000i		−	1.20711i	−0.797321	−	0.603555i	$-0.793750\pi$
$991$		−	38.0000i		−	1.20711i	0.797321	−	0.603555i	$-0.206250\pi$
$992$		0			0
$993$			23.0000i			0.729883i
$994$		0			0
$995$	−4.00000			−0.126809
$996$		0			0
$997$		−	27.0000i		−	0.855099i	−0.903992	−	0.427549i	$-0.859377\pi$
$997$		−	27.0000i		−	0.855099i	0.903992	−	0.427549i	$-0.140623\pi$
$998$		0			0
$999$	−5.00000			−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.2.h.e.1801.1	✓	2
3.2	odd	2		6120.2.h.c.1801.1		2
4.3	odd	2		4080.2.h.e.3841.2		2
17.16	even	2	inner	2040.2.h.e.1801.2	yes	2
51.50	odd	2		6120.2.h.c.1801.2		2
68.67	odd	2		4080.2.h.e.3841.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.h.e.1801.1	✓	2	1.1	even	1	trivial
2040.2.h.e.1801.2	yes	2	17.16	even	2	inner
4080.2.h.e.3841.1		2	68.67	odd	2
4080.2.h.e.3841.2		2	4.3	odd	2
6120.2.h.c.1801.1		2	3.2	odd	2
6120.2.h.c.1801.2		2	51.50	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:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2040.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000i q^{7} -1.00000 q^{9} +5.00000i q^{11} -2.00000 q^{13} +1.00000 q^{15} +(1.00000 - 4.00000i) q^{17} +1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{25} +1.00000i q^{27} +7.00000i q^{29} +5.00000 q^{33} +1.00000 q^{35} +5.00000i q^{37} +2.00000i q^{39} +9.00000i q^{41} -12.0000 q^{43} -1.00000i q^{45} +1.00000 q^{47} +6.00000 q^{49} +(-4.00000 - 1.00000i) q^{51} +3.00000 q^{53} -5.00000 q^{55} -1.00000i q^{57} -8.00000 q^{59} +8.00000i q^{61} +1.00000i q^{63} -2.00000i q^{65} -2.00000 q^{67} +12.0000i q^{71} +11.0000i q^{73} +1.00000i q^{75} +5.00000 q^{77} -10.0000i q^{79} +1.00000 q^{81} +8.00000 q^{83} +(4.00000 + 1.00000i) q^{85} +7.00000 q^{87} +16.0000 q^{89} +2.00000i q^{91} +1.00000i q^{95} -10.0000i q^{97} -5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 4 q^{13} + 2 q^{15} + 2 q^{17} + 2 q^{19} - 2 q^{21} - 2 q^{25} + 10 q^{33} + 2 q^{35} - 24 q^{43} + 2 q^{47} + 12 q^{49} - 8 q^{51} + 6 q^{53} - 10 q^{55} - 16 q^{59} - 4 q^{67} + 10 q^{77}+ \cdots + 32 q^{89}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^13 + 2 * q^15 + 2 * q^17 + 2 * q^19 - 2 * q^21 - 2 * q^25 + 10 * q^33 + 2 * q^35 - 24 * q^43 + 2 * q^47 + 12 * q^49 - 8 * q^51 + 6 * q^53 - 10 * q^55 - 16 * q^59 - 4 * q^67 + 10 * q^77 + 2 * q^81 + 16 * q^83 + 8 * q^85 + 14 * q^87 + 32 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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