LMFDB - Embedded newform 3822.2.c.d.883.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3822,2,Mod(883,3822)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3822.883"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3822, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,-2,0,0,0,0,2,4,0,2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$30.5188236525$
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 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3822.883
Dual form			3822.2.c.d.883.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +(3.00000 + 2.00000i) q^{13} +2.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +2.00000i q^{20} +4.00000 q^{23} +1.00000i q^{24} +1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} -10.0000i q^{31} +1.00000i q^{32} +2.00000i q^{34} -1.00000 q^{36} +8.00000i q^{37} -6.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} -2.00000 q^{40} -10.0000i q^{41} +4.00000 q^{43} -2.00000i q^{45} +4.00000i q^{46} +12.0000i q^{47} -1.00000 q^{48} +1.00000i q^{50} -2.00000 q^{51} +(-3.00000 - 2.00000i) q^{52} -6.00000 q^{53} -1.00000i q^{54} -6.00000i q^{57} -10.0000i q^{58} -4.00000i q^{59} -2.00000i q^{60} -2.00000 q^{61} +10.0000 q^{62} -1.00000 q^{64} +(4.00000 - 6.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{68} -4.00000 q^{69} -1.00000i q^{72} +4.00000i q^{73} -8.00000 q^{74} -1.00000 q^{75} -6.00000i q^{76} +(2.00000 - 3.00000i) q^{78} -2.00000i q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000i q^{83} -4.00000i q^{85} +4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} +2.00000 q^{90} -4.00000 q^{92} +10.0000i q^{93} -12.0000 q^{94} +12.0000 q^{95} -1.00000i q^{96} +12.0000i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 20 q^{29} - 4 q^{30} - 2 q^{36} - 12 q^{38} - 6 q^{39} - 4 q^{40} + 8 q^{43}+ \cdots + 24 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 4 * q^10 + 2 * q^12 + 6 * q^13 + 2 * q^16 + 4 * q^17 + 8 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 20 * q^29 - 4 * q^30 - 2 * q^36 - 12 * q^38 - 6 * q^39 - 4 * q^40 + 8 * q^43 - 2 * q^48 - 4 * q^51 - 6 * q^52 - 12 * q^53 - 4 * q^61 + 20 * q^62 - 2 * q^64 + 8 * q^65 - 4 * q^68 - 8 * q^69 - 16 * q^74 - 2 * q^75 + 4 * q^78 + 2 * q^81 + 20 * q^82 + 20 * q^87 + 4 * q^90 - 8 * q^92 - 24 * q^94 + 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times$.

$n$	$1471$	$2549$	$3433$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−1.00000			−0.500000
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		−	1.00000i		−	0.408248i
$7$		0			0
$7$		0			0
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$	2.00000			0.632456
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	1.00000			0.288675
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$		0			0
$15$			2.00000i			0.516398i
$16$	1.00000			0.250000
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$			1.00000i			0.235702i
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$			2.00000i			0.447214i
$21$		0			0
$22$		0			0
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$			1.00000i			0.204124i
$25$	1.00000			0.200000
$26$	−2.00000	+	3.00000i	−0.392232	+	0.588348i
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−10.0000			−1.85695			−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000			−1.85695			−0.928477	+	0.371391i	$0.878881\pi$
$30$	−2.00000			−0.365148
$31$		−	10.0000i		−	1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$		−	10.0000i		−	1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			2.00000i			0.342997i
$35$		0			0
$36$	−1.00000			−0.166667
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	−6.00000			−0.973329
$39$	−3.00000	−	2.00000i	−0.480384	−	0.320256i
$40$	−2.00000			−0.316228
$41$		−	10.0000i		−	1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$		−	10.0000i		−	1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		−	2.00000i		−	0.298142i
$46$			4.00000i			0.589768i
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	−1.00000			−0.144338
$49$		0			0
$50$			1.00000i			0.141421i
$51$	−2.00000			−0.280056
$52$	−3.00000	−	2.00000i	−0.416025	−	0.277350i
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		−	10.0000i		−	1.31306i
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		−	2.00000i		−	0.258199i
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$	10.0000			1.27000
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.00000	−	6.00000i	0.496139	−	0.744208i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	−2.00000			−0.242536
$69$	−4.00000			−0.481543
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		−	1.00000i		−	0.117851i
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−8.00000			−0.929981
$75$	−1.00000			−0.115470
$76$		−	6.00000i		−	0.688247i
$77$		0			0
$78$	2.00000	−	3.00000i	0.226455	−	0.339683i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		−	2.00000i		−	0.223607i
$81$	1.00000			0.111111
$82$	10.0000			1.10432
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$		−	4.00000i		−	0.433861i
$86$			4.00000i			0.431331i
$87$	10.0000			1.07211
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	2.00000			0.210819
$91$		0			0
$92$	−4.00000			−0.417029
$93$			10.0000i			1.03695i
$94$	−12.0000			−1.23771
$95$	12.0000			1.23117
$96$		−	1.00000i		−	0.102062i
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$		−	2.00000i		−	0.198030i
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$	2.00000	−	3.00000i	0.196116	−	0.294174i
$105$		0			0
$106$		−	6.00000i		−	0.582772i
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000			0.0962250
$109$			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
$111$		−	8.00000i		−	0.759326i
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$	6.00000			0.561951
$115$		−	8.00000i		−	0.746004i
$116$	10.0000			0.928477
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$	4.00000			0.368230
$119$		0			0
$120$	2.00000			0.182574
$121$	11.0000			1.00000
$122$		−	2.00000i		−	0.181071i
$123$			10.0000i			0.901670i
$124$			10.0000i			0.898027i
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−4.00000			−0.352180
$130$	6.00000	+	4.00000i	0.526235	+	0.350823i
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$		0			0
$134$	2.00000			0.172774
$135$			2.00000i			0.172133i
$136$		−	2.00000i		−	0.171499i
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		−	4.00000i		−	0.340503i
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$		−	12.0000i		−	1.01058i
$142$		0			0
$143$		0			0
$144$	1.00000			0.0833333
$145$			20.0000i			1.66091i
$146$	−4.00000			−0.331042
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		−	1.00000i		−	0.0816497i
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	6.00000			0.486664
$153$	2.00000			0.161690
$154$		0			0
$155$	−20.0000			−1.60644
$156$	3.00000	+	2.00000i	0.240192	+	0.160128i
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	6.00000			0.475831
$160$	2.00000			0.158114
$161$		0			0
$162$			1.00000i			0.0785674i
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$			10.0000i			0.780869i
$165$		0			0
$166$	−4.00000			−0.310460
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$	4.00000			0.306786
$171$			6.00000i			0.458831i
$172$	−4.00000			−0.304997
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$			10.0000i			0.758098i
$175$		0			0
$176$		0			0
$177$			4.00000i			0.300658i
$178$	−6.00000			−0.449719
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$			2.00000i			0.149071i
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		−	4.00000i		−	0.294884i
$185$	16.0000			1.17634
$186$	−10.0000			−0.733236
$187$		0			0
$188$		−	12.0000i		−	0.875190i
$189$		0			0
$190$			12.0000i			0.870572i
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000			0.0721688
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−12.0000			−0.861550
$195$	−4.00000	+	6.00000i	−0.286446	+	0.429669i
$196$		0			0
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		−	1.00000i		−	0.0707107i
$201$			2.00000i			0.141069i
$202$		−	2.00000i		−	0.140720i
$203$		0			0
$204$	2.00000			0.140028
$205$	−20.0000			−1.39686
$206$			16.0000i			1.11477i
$207$	4.00000			0.278019
$208$	3.00000	+	2.00000i	0.208013	+	0.138675i
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	6.00000			0.412082
$213$		0			0
$214$			8.00000i			0.546869i
$215$		−	8.00000i		−	0.545595i
$216$			1.00000i			0.0680414i
$217$		0			0
$218$	−4.00000			−0.270914
$219$		−	4.00000i		−	0.270295i
$220$		0			0
$221$	6.00000	+	4.00000i	0.403604	+	0.269069i
$222$	8.00000			0.536925
$223$			14.0000i			0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$			14.0000i			0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$			14.0000i			0.931266i
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$			6.00000i			0.397360i
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$	8.00000			0.527504
$231$		0			0
$232$			10.0000i			0.656532i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	+	3.00000i	−0.130744	+	0.196116i
$235$	24.0000			1.56559
$236$			4.00000i			0.260378i
$237$		0			0
$238$		0			0
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$			2.00000i			0.129099i
$241$		−	20.0000i		−	1.28831i	−0.764894	−	0.644157i	$-0.777208\pi$
$241$		−	20.0000i		−	1.28831i	0.764894	−	0.644157i	$-0.222792\pi$
$242$			11.0000i			0.707107i
$243$	−1.00000			−0.0641500
$244$	2.00000			0.128037
$245$		0			0
$246$	−10.0000			−0.637577
$247$	−12.0000	+	18.0000i	−0.763542	+	1.14531i
$248$	−10.0000			−0.635001
$249$		−	4.00000i		−	0.253490i
$250$	12.0000			0.758947
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$		0			0
$253$		0			0
$254$			8.00000i			0.501965i
$255$			4.00000i			0.250490i
$256$	1.00000			0.0625000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$		−	4.00000i		−	0.249029i
$259$		0			0
$260$	−4.00000	+	6.00000i	−0.248069	+	0.372104i
$261$	−10.0000			−0.618984
$262$			8.00000i			0.494242i
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$			12.0000i			0.737154i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$			2.00000i			0.122169i
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$	−2.00000			−0.121716
$271$			10.0000i			0.607457i	0.952759	+	0.303728i	$0.0982315\pi$
$271$			10.0000i			0.607457i	−0.952759	+	0.303728i	$0.901768\pi$
$272$	2.00000			0.121268
$273$		0			0
$274$	2.00000			0.120824
$275$		0			0
$276$	4.00000			0.240772
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$			20.0000i			1.19952i
$279$		−	10.0000i		−	0.598684i
$280$		0			0
$281$		−	10.0000i		−	0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$	12.0000			0.714590
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$		0			0
$285$	−12.0000			−0.710819
$286$		0			0
$287$		0			0
$288$			1.00000i			0.0589256i
$289$	−13.0000			−0.764706
$290$	−20.0000			−1.17444
$291$		−	12.0000i		−	0.703452i
$292$		−	4.00000i		−	0.234082i
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$	8.00000			0.464991
$297$		0			0
$298$	−14.0000			−0.810998
$299$	12.0000	+	8.00000i	0.693978	+	0.462652i
$300$	1.00000			0.0577350
$301$		0			0
$302$	−10.0000			−0.575435
$303$	2.00000			0.114897
$304$			6.00000i			0.344124i
$305$			4.00000i			0.229039i
$306$			2.00000i			0.114332i
$307$			2.00000i			0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$			2.00000i			0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		−	20.0000i		−	1.13592i
$311$	28.0000			1.58773			0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000			1.58773			0.793867	+	0.608091i	$0.208065\pi$
$312$	−2.00000	+	3.00000i	−0.113228	+	0.169842i
$313$	26.0000			1.46961			0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000			1.46961			0.734803	+	0.678280i	$0.237274\pi$
$314$			2.00000i			0.112867i
$315$		0			0
$316$		0			0
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$			6.00000i			0.336463i
$319$		0			0
$320$			2.00000i			0.111803i
$321$	−8.00000			−0.446516
$322$		0			0
$323$			12.0000i			0.667698i
$324$	−1.00000			−0.0555556
$325$	3.00000	+	2.00000i	0.166410	+	0.110940i
$326$	14.0000			0.775388
$327$		−	4.00000i		−	0.221201i
$328$	−10.0000			−0.552158
$329$		0			0
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$		−	4.00000i		−	0.219529i
$333$			8.00000i			0.438397i
$334$	−12.0000			−0.656611
$335$	−4.00000			−0.218543
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−12.0000	+	5.00000i	−0.652714	+	0.271964i
$339$	−14.0000			−0.760376
$340$			4.00000i			0.216930i
$341$		0			0
$342$	−6.00000			−0.324443
$343$		0			0
$344$		−	4.00000i		−	0.215666i
$345$			8.00000i			0.430706i
$346$			6.00000i			0.322562i
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$	−10.0000			−0.536056
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$	−3.00000	−	2.00000i	−0.160128	−	0.106752i
$352$		0			0
$353$		−	26.0000i		−	1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$		−	26.0000i		−	1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	−4.00000			−0.212598
$355$		0			0
$356$		−	6.00000i		−	0.317999i
$357$		0			0
$358$		0			0
$359$			4.00000i			0.211112i	0.994413	+	0.105556i	$0.0336622\pi$
$359$			4.00000i			0.211112i	−0.994413	+	0.105556i	$0.966338\pi$
$360$	−2.00000			−0.105409
$361$	−17.0000			−0.894737
$362$		−	22.0000i		−	1.15629i
$363$	−11.0000			−0.577350
$364$		0			0
$365$	8.00000			0.418739
$366$			2.00000i			0.104542i
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$	4.00000			0.208514
$369$		−	10.0000i		−	0.520579i
$370$			16.0000i			0.831800i
$371$		0			0
$372$		−	10.0000i		−	0.518476i
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$	12.0000			0.618853
$377$	−30.0000	−	20.0000i	−1.54508	−	1.03005i
$378$		0			0
$379$			34.0000i			1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$			34.0000i			1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$	−12.0000			−0.615587
$381$	−8.00000			−0.409852
$382$			12.0000i			0.613973i
$383$			4.00000i			0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$			4.00000i			0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$	−16.0000			−0.814379
$387$	4.00000			0.203331
$388$		−	12.0000i		−	0.609208i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$	−6.00000	−	4.00000i	−0.303822	−	0.202548i
$391$	8.00000			0.404577
$392$		0			0
$393$	−8.00000			−0.403547
$394$	22.0000			1.10834
$395$		0			0
$396$		0			0
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			0.0500000
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$	−2.00000			−0.0997509
$403$	20.0000	−	30.0000i	0.996271	−	1.49441i
$404$	2.00000			0.0995037
$405$		−	2.00000i		−	0.0993808i
$406$		0			0
$407$		0			0
$408$			2.00000i			0.0990148i
$409$		−	4.00000i		−	0.197787i	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$		−	4.00000i		−	0.197787i	0.995098	−	0.0988936i	$-0.0315304\pi$
$410$		−	20.0000i		−	0.987730i
$411$			2.00000i			0.0986527i
$412$	−16.0000			−0.788263
$413$		0			0
$414$			4.00000i			0.196589i
$415$	8.00000			0.392705
$416$	−2.00000	+	3.00000i	−0.0980581	+	0.147087i
$417$	−20.0000			−0.979404
$418$		0			0
$419$	−40.0000			−1.95413			−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000			−1.95413			−0.977064	+	0.212946i	$0.931694\pi$
$420$		0			0
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$			12.0000i			0.584151i
$423$			12.0000i			0.583460i
$424$			6.00000i			0.291386i
$425$	2.00000			0.0970143
$426$		0			0
$427$		0			0
$428$	−8.00000			−0.386695
$429$		0			0
$430$	8.00000			0.385794
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$	−1.00000			−0.0481125
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$		0			0
$435$		−	20.0000i		−	0.958927i
$436$		−	4.00000i		−	0.191565i
$437$			24.0000i			1.14808i
$438$	4.00000			0.191127
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$	−4.00000	+	6.00000i	−0.190261	+	0.285391i
$443$	−16.0000			−0.760183			−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000			−0.760183			−0.380091	+	0.924949i	$0.624107\pi$
$444$			8.00000i			0.379663i
$445$	12.0000			0.568855
$446$	−14.0000			−0.662919
$447$		−	14.0000i		−	0.662177i
$448$		0			0
$449$		−	6.00000i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$			1.00000i			0.0471405i
$451$		0			0
$452$	−14.0000			−0.658505
$453$		−	10.0000i		−	0.469841i
$454$	8.00000			0.375459
$455$		0			0
$456$	−6.00000			−0.280976
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	4.00000			0.186908
$459$	−2.00000			−0.0933520
$460$			8.00000i			0.373002i
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	−10.0000			−0.464238
$465$	20.0000			0.927478
$466$		−	6.00000i		−	0.277945i
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$	−3.00000	−	2.00000i	−0.138675	−	0.0924500i
$469$		0			0
$470$			24.0000i			1.10704i
$471$	−2.00000			−0.0921551
$472$	−4.00000			−0.184115
$473$		0			0
$474$		0			0
$475$			6.00000i			0.275299i
$476$		0			0
$477$	−6.00000			−0.274721
$478$	16.0000			0.731823
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$	−2.00000			−0.0912871
$481$	−16.0000	+	24.0000i	−0.729537	+	1.09431i
$482$	20.0000			0.910975
$483$		0			0
$484$	−11.0000			−0.500000
$485$	24.0000			1.08978
$486$		−	1.00000i		−	0.0453609i
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$			2.00000i			0.0905357i
$489$			14.0000i			0.633102i
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		−	10.0000i		−	0.450835i
$493$	−20.0000			−0.900755
$494$	−18.0000	−	12.0000i	−0.809858	−	0.539906i
$495$		0			0
$496$		−	10.0000i		−	0.449013i
$497$		0			0
$498$	4.00000			0.179244
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$			12.0000i			0.536656i
$501$		−	12.0000i		−	0.536120i
$502$			28.0000i			1.24970i
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$			4.00000i			0.177998i
$506$		0			0
$507$	−5.00000	−	12.0000i	−0.222058	−	0.532939i
$508$	−8.00000			−0.354943
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$	−4.00000			−0.177123
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		−	6.00000i		−	0.264906i
$514$		−	18.0000i		−	0.793946i
$515$		−	32.0000i		−	1.41009i
$516$	4.00000			0.176090
$517$		0			0
$518$		0			0
$519$	−6.00000			−0.263371
$520$	−6.00000	−	4.00000i	−0.263117	−	0.175412i
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		−	10.0000i		−	0.437688i
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−8.00000			−0.349482
$525$		0			0
$526$			24.0000i			1.04645i
$527$		−	20.0000i		−	0.871214i
$528$		0			0
$529$	−7.00000			−0.304348
$530$	−12.0000			−0.521247
$531$		−	4.00000i		−	0.173585i
$532$		0			0
$533$	20.0000	−	30.0000i	0.866296	−	1.29944i
$534$	6.00000			0.259645
$535$		−	16.0000i		−	0.691740i
$536$	−2.00000			−0.0863868
$537$		0			0
$538$			10.0000i			0.431131i
$539$		0			0
$540$		−	2.00000i		−	0.0860663i
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$	−10.0000			−0.429537
$543$	22.0000			0.944110
$544$			2.00000i			0.0857493i
$545$	8.00000			0.342682
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$			2.00000i			0.0854358i
$549$	−2.00000			−0.0853579
$550$		0			0
$551$		−	60.0000i		−	2.55609i
$552$			4.00000i			0.170251i
$553$		0			0
$554$		−	2.00000i		−	0.0849719i
$555$	−16.0000			−0.679162
$556$	−20.0000			−0.848189
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	10.0000			0.423334
$559$	12.0000	+	8.00000i	0.507546	+	0.338364i
$560$		0			0
$561$		0			0
$562$	10.0000			0.421825
$563$	16.0000			0.674320			0.337160	−	0.941447i	$-0.390534\pi$
$563$	16.0000			0.674320			0.337160	+	0.941447i	$0.390534\pi$
$564$			12.0000i			0.505291i
$565$		−	28.0000i		−	1.17797i
$566$		−	4.00000i		−	0.168133i
$567$		0			0
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		−	12.0000i		−	0.502625i
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$	−12.0000			−0.501307
$574$		0			0
$575$	4.00000			0.166812
$576$	−1.00000			−0.0416667
$577$		−	8.00000i		−	0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$		−	13.0000i		−	0.540729i
$579$		−	16.0000i		−	0.664937i
$580$		−	20.0000i		−	0.830455i
$581$		0			0
$582$	12.0000			0.497416
$583$		0			0
$584$	4.00000			0.165521
$585$	4.00000	−	6.00000i	0.165380	−	0.248069i
$586$	−14.0000			−0.578335
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$	60.0000			2.47226
$590$		−	8.00000i		−	0.329355i
$591$			22.0000i			0.904959i
$592$			8.00000i			0.328798i
$593$		−	26.0000i		−	1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$		−	26.0000i		−	1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$		0			0
$595$		0			0
$596$		−	14.0000i		−	0.573462i
$597$		0			0
$598$	−8.00000	+	12.0000i	−0.327144	+	0.490716i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$			1.00000i			0.0408248i
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		−	10.0000i		−	0.406894i
$605$		−	22.0000i		−	0.894427i
$606$			2.00000i			0.0812444i
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$	−6.00000			−0.243332
$609$		0			0
$610$	−4.00000			−0.161955
$611$	−24.0000	+	36.0000i	−0.970936	+	1.45640i
$612$	−2.00000			−0.0808452
$613$			16.0000i			0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$			16.0000i			0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	−2.00000			−0.0807134
$615$	20.0000			0.806478
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		−	16.0000i		−	0.643614i
$619$			26.0000i			1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$			26.0000i			1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$	20.0000			0.803219
$621$	−4.00000			−0.160514
$622$			28.0000i			1.12270i
$623$		0			0
$624$	−3.00000	−	2.00000i	−0.120096	−	0.0800641i
$625$	−19.0000			−0.760000
$626$			26.0000i			1.03917i
$627$		0			0
$628$	−2.00000			−0.0798087
$629$			16.0000i			0.637962i
$630$		0			0
$631$			10.0000i			0.398094i	0.979990	+	0.199047i	$0.0637846\pi$
$631$			10.0000i			0.398094i	−0.979990	+	0.199047i	$0.936215\pi$
$632$		0			0
$633$	−12.0000			−0.476957
$634$	−18.0000			−0.714871
$635$		−	16.0000i		−	0.634941i
$636$	−6.00000			−0.237915
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−2.00000			−0.0790569
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		−	8.00000i		−	0.315735i
$643$		−	6.00000i		−	0.236617i	−0.992977	−	0.118308i	$-0.962253\pi$
$643$		−	6.00000i		−	0.236617i	0.992977	−	0.118308i	$-0.0377472\pi$
$644$		0			0
$645$			8.00000i			0.315000i
$646$	−12.0000			−0.472134
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		0			0
$650$	−2.00000	+	3.00000i	−0.0784465	+	0.117670i
$651$		0			0
$652$			14.0000i			0.548282i
$653$	−26.0000			−1.01746			−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000			−1.01746			−0.508729	+	0.860927i	$0.669885\pi$
$654$	4.00000			0.156412
$655$		−	16.0000i		−	0.625172i
$656$		−	10.0000i		−	0.390434i
$657$			4.00000i			0.156055i
$658$		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
$661$			40.0000i			1.55582i	0.628376	+	0.777910i	$0.283720\pi$
$661$			40.0000i			1.55582i	−0.628376	+	0.777910i	$0.716280\pi$
$662$	10.0000			0.388661
$663$	−6.00000	−	4.00000i	−0.233021	−	0.155347i
$664$	4.00000			0.155230
$665$		0			0
$666$	−8.00000			−0.309994
$667$	−40.0000			−1.54881
$668$		−	12.0000i		−	0.464294i
$669$		−	14.0000i		−	0.541271i
$670$		−	4.00000i		−	0.154533i
$671$		0			0
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		−	2.00000i		−	0.0770371i
$675$	−1.00000			−0.0384900
$676$	−5.00000	−	12.0000i	−0.192308	−	0.461538i
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		−	14.0000i		−	0.537667i
$679$		0			0
$680$	−4.00000			−0.153393
$681$			8.00000i			0.306561i
$682$		0			0
$683$		−	24.0000i		−	0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$		−	24.0000i		−	0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$		−	6.00000i		−	0.229416i
$685$	−4.00000			−0.152832
$686$		0			0
$687$			4.00000i			0.152610i
$688$	4.00000			0.152499
$689$	−18.0000	−	12.0000i	−0.685745	−	0.457164i
$690$	−8.00000			−0.304555
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$		−	12.0000i		−	0.455514i
$695$		−	40.0000i		−	1.51729i
$696$		−	10.0000i		−	0.379049i
$697$		−	20.0000i		−	0.757554i
$698$	−16.0000			−0.605609
$699$	6.00000			0.226941
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$	2.00000	−	3.00000i	0.0754851	−	0.113228i
$703$	−48.0000			−1.81035
$704$		0			0
$705$	−24.0000			−0.903892
$706$	26.0000			0.978523
$707$		0			0
$708$		−	4.00000i		−	0.150329i
$709$		−	36.0000i		−	1.35201i	−0.736898	−	0.676004i	$-0.763710\pi$
$709$		−	36.0000i		−	1.35201i	0.736898	−	0.676004i	$-0.236290\pi$
$710$		0			0
$711$		0			0
$712$	6.00000			0.224860
$713$		−	40.0000i		−	1.49801i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			16.0000i			0.597531i
$718$	−4.00000			−0.149279
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		−	2.00000i		−	0.0745356i
$721$		0			0
$722$		−	17.0000i		−	0.632674i
$723$			20.0000i			0.743808i
$724$	22.0000			0.817624
$725$	−10.0000			−0.371391
$726$		−	11.0000i		−	0.408248i
$727$	−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
$729$	1.00000			0.0370370
$730$			8.00000i			0.296093i
$731$	8.00000			0.295891
$732$	−2.00000			−0.0739221
$733$			44.0000i			1.62518i	0.582838	+	0.812589i	$0.301942\pi$
$733$			44.0000i			1.62518i	−0.582838	+	0.812589i	$0.698058\pi$
$734$		−	8.00000i		−	0.295285i
$735$		0			0
$736$			4.00000i			0.147442i
$737$		0			0
$738$	10.0000			0.368105
$739$		−	26.0000i		−	0.956425i	−0.878244	−	0.478213i	$-0.841285\pi$
$739$		−	26.0000i		−	0.956425i	0.878244	−	0.478213i	$-0.158715\pi$
$740$	−16.0000			−0.588172
$741$	12.0000	−	18.0000i	0.440831	−	0.661247i
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	10.0000			0.366618
$745$	28.0000			1.02584
$746$		−	6.00000i		−	0.219676i
$747$			4.00000i			0.146352i
$748$		0			0
$749$		0			0
$750$	−12.0000			−0.438178
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$			12.0000i			0.437595i
$753$	−28.0000			−1.02038
$754$	20.0000	−	30.0000i	0.728357	−	1.09254i
$755$	20.0000			0.727875
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$	−34.0000			−1.23494
$759$		0			0
$760$		−	12.0000i		−	0.435286i
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$		−	8.00000i		−	0.289809i
$763$		0			0
$764$	−12.0000			−0.434145
$765$		−	4.00000i		−	0.144620i
$766$	−4.00000			−0.144526
$767$	8.00000	−	12.0000i	0.288863	−	0.433295i
$768$	−1.00000			−0.0360844
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		−	16.0000i		−	0.575853i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$			4.00000i			0.143777i
$775$		−	10.0000i		−	0.359211i
$776$	12.0000			0.430775
$777$		0			0
$778$			30.0000i			1.07555i
$779$	60.0000			2.14972
$780$	4.00000	−	6.00000i	0.143223	−	0.214834i
$781$		0			0
$782$			8.00000i			0.286079i
$783$	10.0000			0.357371
$784$		0			0
$785$		−	4.00000i		−	0.142766i
$786$		−	8.00000i		−	0.285351i
$787$		−	38.0000i		−	1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$		−	38.0000i		−	1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$			22.0000i			0.783718i
$789$	−24.0000			−0.854423
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−6.00000	−	4.00000i	−0.213066	−	0.142044i
$794$	8.00000			0.283909
$795$		−	12.0000i		−	0.425596i
$796$		0			0
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$			1.00000i			0.0353553i
$801$			6.00000i			0.212000i
$802$	30.0000			1.05934
$803$		0			0
$804$		−	2.00000i		−	0.0705346i
$805$		0			0
$806$	30.0000	+	20.0000i	1.05670	+	0.704470i
$807$	−10.0000			−0.352017
$808$			2.00000i			0.0703598i
$809$	−50.0000			−1.75791			−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000			−1.75791			−0.878953	+	0.476908i	$0.841757\pi$
$810$	2.00000			0.0702728
$811$		−	10.0000i		−	0.351147i	−0.984466	−	0.175574i	$-0.943822\pi$
$811$		−	10.0000i		−	0.351147i	0.984466	−	0.175574i	$-0.0561780\pi$
$812$		0			0
$813$		−	10.0000i		−	0.350715i
$814$		0			0
$815$	−28.0000			−0.980797
$816$	−2.00000			−0.0700140
$817$			24.0000i			0.839654i
$818$	4.00000			0.139857
$819$		0			0
$820$	20.0000			0.698430
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	−2.00000			−0.0697580
$823$	24.0000			0.836587			0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000			0.836587			0.418294	+	0.908312i	$0.362628\pi$
$824$		−	16.0000i		−	0.557386i
$825$		0			0
$826$		0			0
$827$			48.0000i			1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$			48.0000i			1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	−4.00000			−0.139010
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$			8.00000i			0.277684i
$831$	2.00000			0.0693792
$832$	−3.00000	−	2.00000i	−0.104006	−	0.0693375i
$833$		0			0
$834$		−	20.0000i		−	0.692543i
$835$	24.0000			0.830554
$836$		0			0
$837$			10.0000i			0.345651i
$838$		−	40.0000i		−	1.38178i
$839$			16.0000i			0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$			16.0000i			0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$	−20.0000			−0.689246
$843$			10.0000i			0.344418i
$844$	−12.0000			−0.413057
$845$	24.0000	−	10.0000i	0.825625	−	0.344010i
$846$	−12.0000			−0.412568
$847$		0			0
$848$	−6.00000			−0.206041
$849$	4.00000			0.137280
$850$			2.00000i			0.0685994i
$851$			32.0000i			1.09695i
$852$		0			0
$853$		−	56.0000i		−	1.91740i	−0.284413	−	0.958702i	$-0.591799\pi$
$853$		−	56.0000i		−	1.91740i	0.284413	−	0.958702i	$-0.408201\pi$
$854$		0			0
$855$	12.0000			0.410391
$856$		−	8.00000i		−	0.273434i
$857$	22.0000			0.751506			0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000			0.751506			0.375753	+	0.926720i	$0.377384\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$			8.00000i			0.272798i
$861$		0			0
$862$	20.0000			0.681203
$863$		−	44.0000i		−	1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$		−	44.0000i		−	1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		−	12.0000i		−	0.408012i
$866$			26.0000i			0.883516i
$867$	13.0000			0.441503
$868$		0			0
$869$		0			0
$870$	20.0000			0.678064
$871$	4.00000	−	6.00000i	0.135535	−	0.203302i
$872$	4.00000			0.135457
$873$			12.0000i			0.406138i
$874$	−24.0000			−0.811812
$875$		0			0
$876$			4.00000i			0.135147i
$877$			8.00000i			0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$			8.00000i			0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$		0			0
$879$		−	14.0000i		−	0.472208i
$880$		0			0
$881$	−42.0000			−1.41502			−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000			−1.41502			−0.707508	+	0.706705i	$0.750181\pi$
$882$		0			0
$883$	−36.0000			−1.21150			−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000			−1.21150			−0.605748	+	0.795656i	$0.707126\pi$
$884$	−6.00000	−	4.00000i	−0.201802	−	0.134535i
$885$	8.00000			0.268917
$886$		−	16.0000i		−	0.537531i
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$	−8.00000			−0.268462
$889$		0			0
$890$			12.0000i			0.402241i
$891$		0			0
$892$		−	14.0000i		−	0.468755i
$893$	−72.0000			−2.40939
$894$	14.0000			0.468230
$895$		0			0
$896$		0			0
$897$	−12.0000	−	8.00000i	−0.400668	−	0.267112i
$898$	6.00000			0.200223
$899$			100.000i			3.33519i
$900$	−1.00000			−0.0333333
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		−	14.0000i		−	0.465633i
$905$			44.0000i			1.46261i
$906$	10.0000			0.332228
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$			8.00000i			0.265489i
$909$	−2.00000			−0.0663358
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		−	6.00000i		−	0.198680i
$913$		0			0
$914$	−28.0000			−0.926158
$915$		−	4.00000i		−	0.132236i
$916$			4.00000i			0.132164i
$917$		0			0
$918$		−	2.00000i		−	0.0660098i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−8.00000			−0.263752
$921$		−	2.00000i		−	0.0659022i
$922$	−30.0000			−0.987997
$923$		0			0
$924$		0			0
$925$			8.00000i			0.263038i
$926$	−6.00000			−0.197172
$927$	16.0000			0.525509
$928$		−	10.0000i		−	0.328266i
$929$			6.00000i			0.196854i	0.995144	+	0.0984268i	$0.0313810\pi$
$929$			6.00000i			0.196854i	−0.995144	+	0.0984268i	$0.968619\pi$
$930$			20.0000i			0.655826i
$931$		0			0
$932$	6.00000			0.196537
$933$	−28.0000			−0.916679
$934$			12.0000i			0.392652i
$935$		0			0
$936$	2.00000	−	3.00000i	0.0653720	−	0.0980581i
$937$	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
$939$	−26.0000			−0.848478
$940$	−24.0000			−0.782794
$941$			10.0000i			0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$			10.0000i			0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$		−	2.00000i		−	0.0651635i
$943$		−	40.0000i		−	1.30258i
$944$		−	4.00000i		−	0.130189i
$945$		0			0
$946$		0			0
$947$		−	52.0000i		−	1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$		−	52.0000i		−	1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$		0			0
$949$	−8.00000	+	12.0000i	−0.259691	+	0.389536i
$950$	−6.00000			−0.194666
$951$		−	18.0000i		−	0.583690i
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		−	6.00000i		−	0.194257i
$955$		−	24.0000i		−	0.776622i
$956$			16.0000i			0.517477i
$957$		0			0
$958$	24.0000			0.775405
$959$		0			0
$960$		−	2.00000i		−	0.0645497i
$961$	−69.0000			−2.22581
$962$	−24.0000	−	16.0000i	−0.773791	−	0.515861i
$963$	8.00000			0.257796
$964$			20.0000i			0.644157i
$965$	32.0000			1.03012
$966$		0			0
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$		−	11.0000i		−	0.353553i
$969$		−	12.0000i		−	0.385496i
$970$			24.0000i			0.770594i
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000			0.0320750
$973$		0			0
$974$	−18.0000			−0.576757
$975$	−3.00000	−	2.00000i	−0.0960769	−	0.0640513i
$976$	−2.00000			−0.0640184
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	−14.0000			−0.447671
$979$		0			0
$980$		0			0
$981$			4.00000i			0.127710i
$982$		−	28.0000i		−	0.893516i
$983$			24.0000i			0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$			24.0000i			0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	10.0000			0.318788
$985$	−44.0000			−1.40196
$986$		−	20.0000i		−	0.636930i
$987$		0			0
$988$	12.0000	−	18.0000i	0.381771	−	0.572656i
$989$	16.0000			0.508770
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$	10.0000			0.317500
$993$			10.0000i			0.317340i
$994$		0			0
$995$		0			0
$996$			4.00000i			0.126745i
$997$	42.0000			1.33015			0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000			1.33015			0.665077	+	0.746775i	$0.268399\pi$
$998$	−14.0000			−0.443162
$999$		−	8.00000i		−	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.c.d.883.2		2
7.6	odd	2		78.2.b.a.25.2	yes	2
13.12	even	2	inner	3822.2.c.d.883.1		2
21.20	even	2		234.2.b.a.181.1		2
28.27	even	2		624.2.c.a.337.2		2
35.13	even	4		1950.2.f.g.649.2		2
35.27	even	4		1950.2.f.d.649.1		2
35.34	odd	2		1950.2.b.c.1351.1		2
56.13	odd	2		2496.2.c.f.961.1		2
56.27	even	2		2496.2.c.m.961.1		2
84.83	odd	2		1872.2.c.b.1585.1		2
91.6	even	12		1014.2.e.b.991.1		2
91.20	even	12		1014.2.e.e.991.1		2
91.34	even	4		1014.2.a.b.1.1		1
91.41	even	12		1014.2.e.b.529.1		2
91.48	odd	6		1014.2.i.c.361.2		4
91.55	odd	6		1014.2.i.c.823.1		4
91.62	odd	6		1014.2.i.c.823.2		4
91.69	odd	6		1014.2.i.c.361.1		4
91.76	even	12		1014.2.e.e.529.1		2
91.83	even	4		1014.2.a.g.1.1		1
91.90	odd	2		78.2.b.a.25.1	✓	2
273.83	odd	4		3042.2.a.c.1.1		1
273.125	odd	4		3042.2.a.n.1.1		1
273.272	even	2		234.2.b.a.181.2		2
364.83	odd	4		8112.2.a.j.1.1		1
364.307	odd	4		8112.2.a.g.1.1		1
364.363	even	2		624.2.c.a.337.1		2
455.272	even	4		1950.2.f.g.649.1		2
455.363	even	4		1950.2.f.d.649.2		2
455.454	odd	2		1950.2.b.c.1351.2		2
728.181	odd	2		2496.2.c.f.961.2		2
728.363	even	2		2496.2.c.m.961.2		2
1092.1091	odd	2		1872.2.c.b.1585.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.b.a.25.1	✓	2	91.90	odd	2
78.2.b.a.25.2	yes	2	7.6	odd	2
234.2.b.a.181.1		2	21.20	even	2
234.2.b.a.181.2		2	273.272	even	2
624.2.c.a.337.1		2	364.363	even	2
624.2.c.a.337.2		2	28.27	even	2
1014.2.a.b.1.1		1	91.34	even	4
1014.2.a.g.1.1		1	91.83	even	4
1014.2.e.b.529.1		2	91.41	even	12
1014.2.e.b.991.1		2	91.6	even	12
1014.2.e.e.529.1		2	91.76	even	12
1014.2.e.e.991.1		2	91.20	even	12
1014.2.i.c.361.1		4	91.69	odd	6
1014.2.i.c.361.2		4	91.48	odd	6
1014.2.i.c.823.1		4	91.55	odd	6
1014.2.i.c.823.2		4	91.62	odd	6
1872.2.c.b.1585.1		2	84.83	odd	2
1872.2.c.b.1585.2		2	1092.1091	odd	2
1950.2.b.c.1351.1		2	35.34	odd	2
1950.2.b.c.1351.2		2	455.454	odd	2
1950.2.f.d.649.1		2	35.27	even	4
1950.2.f.d.649.2		2	455.363	even	4
1950.2.f.g.649.1		2	455.272	even	4
1950.2.f.g.649.2		2	35.13	even	4
2496.2.c.f.961.1		2	56.13	odd	2
2496.2.c.f.961.2		2	728.181	odd	2
2496.2.c.m.961.1		2	56.27	even	2
2496.2.c.m.961.2		2	728.363	even	2
3042.2.a.c.1.1		1	273.83	odd	4
3042.2.a.n.1.1		1	273.125	odd	4
3822.2.c.d.883.1		2	13.12	even	2	inner
3822.2.c.d.883.2		2	1.1	even	1	trivial
8112.2.a.g.1.1		1	364.307	odd	4
8112.2.a.j.1.1		1	364.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +(3.00000 + 2.00000i) q^{13} +2.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +2.00000i q^{20} +4.00000 q^{23} +1.00000i q^{24} +1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} -10.0000i q^{31} +1.00000i q^{32} +2.00000i q^{34} -1.00000 q^{36} +8.00000i q^{37} -6.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} -2.00000 q^{40} -10.0000i q^{41} +4.00000 q^{43} -2.00000i q^{45} +4.00000i q^{46} +12.0000i q^{47} -1.00000 q^{48} +1.00000i q^{50} -2.00000 q^{51} +(-3.00000 - 2.00000i) q^{52} -6.00000 q^{53} -1.00000i q^{54} -6.00000i q^{57} -10.0000i q^{58} -4.00000i q^{59} -2.00000i q^{60} -2.00000 q^{61} +10.0000 q^{62} -1.00000 q^{64} +(4.00000 - 6.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{68} -4.00000 q^{69} -1.00000i q^{72} +4.00000i q^{73} -8.00000 q^{74} -1.00000 q^{75} -6.00000i q^{76} +(2.00000 - 3.00000i) q^{78} -2.00000i q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000i q^{83} -4.00000i q^{85} +4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} +2.00000 q^{90} -4.00000 q^{92} +10.0000i q^{93} -12.0000 q^{94} +12.0000 q^{95} -1.00000i q^{96} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 20 q^{29} - 4 q^{30} - 2 q^{36} - 12 q^{38} - 6 q^{39} - 4 q^{40} + 8 q^{43}+ \cdots + 24 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 4 * q^10 + 2 * q^12 + 6 * q^13 + 2 * q^16 + 4 * q^17 + 8 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 20 * q^29 - 4 * q^30 - 2 * q^36 - 12 * q^38 - 6 * q^39 - 4 * q^40 + 8 * q^43 - 2 * q^48 - 4 * q^51 - 6 * q^52 - 12 * q^53 - 4 * q^61 + 20 * q^62 - 2 * q^64 + 8 * q^65 - 4 * q^68 - 8 * q^69 - 16 * q^74 - 2 * q^75 + 4 * q^78 + 2 * q^81 + 20 * q^82 + 20 * q^87 + 4 * q^90 - 8 * q^92 - 24 * q^94 + 24 * q^95

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