LMFDB - Embedded newform 840.2.t.b.169.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(169,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("840.169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.t (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.70743376979$
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			169.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	840.169
Dual form			840.2.t.b.169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +2.00000i q^{13} +(2.00000 - 1.00000i) q^{15} +6.00000 q^{19} -1.00000 q^{21} +(-3.00000 + 4.00000i) q^{25} -1.00000i q^{27} +6.00000 q^{29} +10.0000 q^{31} +2.00000i q^{33} +(2.00000 - 1.00000i) q^{35} -2.00000 q^{39} +6.00000 q^{41} +8.00000i q^{43} +(1.00000 + 2.00000i) q^{45} -12.0000i q^{47} -1.00000 q^{49} +6.00000i q^{53} +(-2.00000 - 4.00000i) q^{55} +6.00000i q^{57} -6.00000 q^{61} -1.00000i q^{63} +(4.00000 - 2.00000i) q^{65} +4.00000i q^{67} +6.00000 q^{71} +14.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{87} +6.00000 q^{89} -2.00000 q^{91} +10.0000i q^{93} +(-6.00000 - 12.0000i) q^{95} -2.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^9	$ 2 q - 2 q^{5} - 2 q^{9} + 4 q^{11} + 4 q^{15} + 12 q^{19} - 2 q^{21} - 6 q^{25} + 12 q^{29} + 20 q^{31} + 4 q^{35} - 4 q^{39} + 12 q^{41} + 2 q^{45} - 2 q^{49} - 4 q^{55} - 12 q^{61} + 8 q^{65} + 12 q^{71} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{89} - 4 q^{91} - 12 q^{95} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 + 4 * q^11 + 4 * q^15 + 12 * q^19 - 2 * q^21 - 6 * q^25 + 12 * q^29 + 20 * q^31 + 4 * q^35 - 4 * q^39 + 12 * q^41 + 2 * q^45 - 2 * q^49 - 4 * q^55 - 12 * q^61 + 8 * q^65 + 12 * q^71 - 8 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^89 - 4 * q^91 - 12 * q^95 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$	2.00000	−	1.00000i	0.516398	−	0.258199i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$		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
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	10.0000			1.79605			0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000			1.79605			0.898027	+	0.439941i	$0.145001\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	2.00000	−	1.00000i	0.338062	−	0.169031i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−2.00000			−0.320256
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$			8.00000i			1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$			8.00000i			1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	−2.00000	−	4.00000i	−0.269680	−	0.539360i
$56$		0			0
$57$			6.00000i			0.794719i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	4.00000	−	2.00000i	0.496139	−	0.248069i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$			2.00000i			0.227921i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			6.00000i			0.643268i
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$			10.0000i			1.03695i
$94$		0			0
$95$	−6.00000	−	12.0000i	−0.615587	−	1.23117i
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$		0			0
$105$	1.00000	+	2.00000i	0.0975900	+	0.195180i
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	10.0000i		−	0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$		−	10.0000i		−	0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	2.00000i		−	0.184900i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$			6.00000i			0.541002i
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$		−	4.00000i		−	0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$		−	4.00000i		−	0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$		0			0
$129$	−8.00000			−0.704361
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$			6.00000i			0.520266i
$134$		0			0
$135$	−2.00000	+	1.00000i	−0.172133	+	0.0860663i
$136$		0			0
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		0			0
$139$	6.00000			0.508913			0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000			0.508913			0.254457	+	0.967084i	$0.418103\pi$
$140$		0			0
$141$	12.0000			1.01058
$142$		0			0
$143$			4.00000i			0.334497i
$144$		0			0
$145$	−6.00000	−	12.0000i	−0.498273	−	0.996546i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	−14.0000			−1.14692			−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000			−1.14692			−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.0000	−	20.0000i	−0.803219	−	1.60644i
$156$		0			0
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$		0			0
$162$		0			0
$163$			16.0000i			1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$			16.0000i			1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$		0			0
$165$	4.00000	−	2.00000i	0.311400	−	0.155700i
$166$		0			0
$167$		−	24.0000i		−	1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$		−	24.0000i		−	1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	−6.00000			−0.458831
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	−4.00000	−	3.00000i	−0.302372	−	0.226779i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$	−26.0000			−1.93256			−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000			−1.93256			−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$		−	6.00000i		−	0.443533i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	10.0000			0.723575			0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000			0.723575			0.361787	+	0.932261i	$0.382167\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	2.00000	+	4.00000i	0.143223	+	0.286446i
$196$		0			0
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$	22.0000			1.55954			0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000			1.55954			0.779769	+	0.626067i	$0.215336\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$			6.00000i			0.421117i
$204$		0			0
$205$	−6.00000	−	12.0000i	−0.419058	−	0.838116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$			6.00000i			0.411113i
$214$		0			0
$215$	16.0000	−	8.00000i	1.09119	−	0.545595i
$216$		0			0
$217$			10.0000i			0.678844i
$218$		0			0
$219$	−14.0000			−0.946032
$220$		0			0
$221$		0			0
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$	3.00000	−	4.00000i	0.200000	−	0.266667i
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$	−2.00000			−0.131590
$232$		0			0
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$	−24.0000	+	12.0000i	−1.56559	+	0.782794i
$236$		0			0
$237$		−	4.00000i		−	0.259828i
$238$		0			0
$239$	−30.0000			−1.94054			−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000			−1.94054			−0.970269	+	0.242028i	$0.922188\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	1.00000	+	2.00000i	0.0638877	+	0.127775i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	20.0000i		−	1.24757i	−0.781598	−	0.623783i	$-0.785595\pi$
$257$		−	20.0000i		−	1.24757i	0.781598	−	0.623783i	$-0.214405\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$			8.00000i			0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$			8.00000i			0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$		0			0
$265$	12.0000	−	6.00000i	0.737154	−	0.368577i
$266$		0			0
$267$			6.00000i			0.367194i
$268$		0			0
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	30.0000			1.82237			0.911185	−	0.411997i	$-0.135169\pi$
$271$	30.0000			1.82237			0.911185	+	0.411997i	$0.135169\pi$
$272$		0			0
$273$		−	2.00000i		−	0.121046i
$274$		0			0
$275$	−6.00000	+	8.00000i	−0.361814	+	0.482418i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−10.0000			−0.598684
$280$		0			0
$281$	−26.0000			−1.55103			−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000			−1.55103			−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$			12.0000i			0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$			12.0000i			0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$		0			0
$285$	12.0000	−	6.00000i	0.710819	−	0.355409i
$286$		0			0
$287$			6.00000i			0.354169i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	2.00000			0.117242
$292$		0			0
$293$			16.0000i			0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$			16.0000i			0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$		−	10.0000i		−	0.574485i
$304$		0			0
$305$	6.00000	+	12.0000i	0.343559	+	0.687118i
$306$		0			0
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$	16.0000			0.910208
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$		0			0
$315$	−2.00000	+	1.00000i	−0.112687	+	0.0563436i
$316$		0			0
$317$			26.0000i			1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$			26.0000i			1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−8.00000	−	6.00000i	−0.443760	−	0.332820i
$326$		0			0
$327$		−	10.0000i		−	0.553001i
$328$		0			0
$329$	12.0000			0.661581
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.00000	−	4.00000i	0.437087	−	0.218543i
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$	10.0000			0.543125
$340$		0			0
$341$	20.0000			1.08306
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			4.00000i			0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$			4.00000i			0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$		0			0
$355$	−6.00000	−	12.0000i	−0.318447	−	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	22.0000			1.16112			0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000			1.16112			0.580558	+	0.814219i	$0.302835\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		−	7.00000i		−	0.367405i
$364$		0			0
$365$	28.0000	−	14.0000i	1.46559	−	0.732793i
$366$		0			0
$367$			16.0000i			0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$			16.0000i			0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$		0			0
$369$	−6.00000			−0.312348
$370$		0			0
$371$	−6.00000			−0.311504
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$	−2.00000	+	11.0000i	−0.103280	+	0.568038i
$376$		0			0
$377$			12.0000i			0.618031i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$	4.00000			0.204926
$382$		0			0
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$		0			0
$385$	4.00000	−	2.00000i	0.203859	−	0.101929i
$386$		0			0
$387$		−	8.00000i		−	0.406663i
$388$		0			0
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			12.0000i			0.605320i
$394$		0			0
$395$	4.00000	+	8.00000i	0.201262	+	0.402524i
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		0			0
$399$	−6.00000			−0.300376
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$			20.0000i			0.996271i
$404$		0			0
$405$	−1.00000	−	2.00000i	−0.0496904	−	0.0993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	6.00000			0.296681			0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000			0.296681			0.148340	+	0.988936i	$0.452607\pi$
$410$		0			0
$411$	2.00000			0.0986527
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			6.00000i			0.293821i
$418$		0			0
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$			12.0000i			0.583460i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	6.00000i		−	0.290360i
$428$		0			0
$429$	−4.00000			−0.193122
$430$		0			0
$431$	14.0000			0.674356			0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000			0.674356			0.337178	+	0.941441i	$0.390528\pi$
$432$		0			0
$433$		−	30.0000i		−	1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$		−	30.0000i		−	1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$		0			0
$435$	12.0000	−	6.00000i	0.575356	−	0.287678i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			28.0000i			1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$			28.0000i			1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$		0			0
$445$	−6.00000	−	12.0000i	−0.284427	−	0.568855i
$446$		0			0
$447$		−	14.0000i		−	0.662177i
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000	+	4.00000i	0.0937614	+	0.187523i
$456$		0			0
$457$			12.0000i			0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$			12.0000i			0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$		0			0
$465$	20.0000	−	10.0000i	0.927478	−	0.463739i
$466$		0			0
$467$			32.0000i			1.48078i	0.672176	+	0.740392i	$0.265360\pi$
$467$			32.0000i			1.48078i	−0.672176	+	0.740392i	$0.734640\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$	22.0000			1.01371
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$	−18.0000	+	24.0000i	−0.825897	+	1.10120i
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.00000	+	2.00000i	−0.181631	+	0.0908153i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$	−16.0000			−0.723545
$490$		0			0
$491$	−22.0000			−0.992846			−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000			−0.992846			−0.496423	+	0.868081i	$0.665354\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	2.00000	+	4.00000i	0.0898933	+	0.179787i
$496$		0			0
$497$			6.00000i			0.269137i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	24.0000			1.07224
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$	10.0000	+	20.0000i	0.444994	+	0.889988i
$506$		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
$511$	−14.0000			−0.619324
$512$		0			0
$513$		−	6.00000i		−	0.264906i
$514$		0			0
$515$	−32.0000	+	16.0000i	−1.41009	+	0.705044i
$516$		0			0
$517$		−	24.0000i		−	1.05552i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$		−	20.0000i		−	0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$		−	20.0000i		−	0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$		0			0
$525$	3.00000	−	4.00000i	0.130931	−	0.174574i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12.0000i			0.519778i
$534$		0			0
$535$	8.00000	−	4.00000i	0.345870	−	0.172935i
$536$		0			0
$537$		−	18.0000i		−	0.776757i
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$		−	26.0000i		−	1.11577i
$544$		0			0
$545$	10.0000	+	20.0000i	0.428353	+	0.856706i
$546$		0			0
$547$			36.0000i			1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$			36.0000i			1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$		0			0
$549$	6.00000			0.256074
$550$		0			0
$551$	36.0000			1.53365
$552$		0			0
$553$		−	4.00000i		−	0.170097i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			2.00000i			0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$			2.00000i			0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$			20.0000i			0.842900i	0.906852	+	0.421450i	$0.138479\pi$
$563$			20.0000i			0.842900i	−0.906852	+	0.421450i	$0.861521\pi$
$564$		0			0
$565$	−20.0000	+	10.0000i	−0.841406	+	0.420703i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	−34.0000			−1.42535			−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000			−1.42535			−0.712677	+	0.701492i	$0.752517\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$			10.0000i			0.417756i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			22.0000i			0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$			22.0000i			0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$		0			0
$579$	−16.0000			−0.664937
$580$		0			0
$581$		0			0
$582$		0			0
$583$			12.0000i			0.496989i
$584$		0			0
$585$	−4.00000	+	2.00000i	−0.165380	+	0.0826898i
$586$		0			0
$587$		−	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
$590$		0			0
$591$	18.0000			0.740421
$592$		0			0
$593$		−	40.0000i		−	1.64260i	−0.570494	−	0.821302i	$-0.693248\pi$
$593$		−	40.0000i		−	1.64260i	0.570494	−	0.821302i	$-0.306752\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			22.0000i			0.900400i
$598$		0			0
$599$	−42.0000			−1.71607			−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000			−1.71607			−0.858037	+	0.513588i	$0.828316\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		−	4.00000i		−	0.162893i
$604$		0			0
$605$	7.00000	+	14.0000i	0.284590	+	0.569181i
$606$		0			0
$607$		−	32.0000i		−	1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$		−	32.0000i		−	1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$		0			0
$609$	−6.00000			−0.243132
$610$		0			0
$611$	24.0000			0.970936
$612$		0			0
$613$			24.0000i			0.969351i	0.874694	+	0.484675i	$0.161062\pi$
$613$			24.0000i			0.969351i	−0.874694	+	0.484675i	$0.838938\pi$
$614$		0			0
$615$	12.0000	−	6.00000i	0.483887	−	0.241943i
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$	22.0000			0.884255			0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000			0.884255			0.442127	+	0.896952i	$0.354224\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			6.00000i			0.240385i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			12.0000i			0.479234i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$		0			0
$633$		−	8.00000i		−	0.317971i
$634$		0			0
$635$	−8.00000	+	4.00000i	−0.317470	+	0.158735i
$636$		0			0
$637$		−	2.00000i		−	0.0792429i
$638$		0			0
$639$	−6.00000			−0.237356
$640$		0			0
$641$	46.0000			1.81689			0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000			1.81689			0.908445	+	0.418004i	$0.137270\pi$
$642$		0			0
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$		0			0
$645$	8.00000	+	16.0000i	0.315000	+	0.629999i
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−10.0000			−0.391931
$652$		0			0
$653$		−	38.0000i		−	1.48705i	−0.668705	−	0.743527i	$-0.733151\pi$
$653$		−	38.0000i		−	1.48705i	0.668705	−	0.743527i	$-0.266849\pi$
$654$		0			0
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$		0			0
$657$		−	14.0000i		−	0.546192i
$658$		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
$661$	−38.0000			−1.47803			−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000			−1.47803			−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.0000	−	6.00000i	0.465340	−	0.232670i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−16.0000			−0.618596
$670$		0			0
$671$	−12.0000			−0.463255
$672$		0			0
$673$			12.0000i			0.462566i	0.972887	+	0.231283i	$0.0742923\pi$
$673$			12.0000i			0.462566i	−0.972887	+	0.231283i	$0.925708\pi$
$674$		0			0
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$		0			0
$677$			48.0000i			1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$			48.0000i			1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$		0			0
$679$	2.00000			0.0767530
$680$		0			0
$681$	−8.00000			−0.306561
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$	−4.00000	+	2.00000i	−0.152832	+	0.0764161i
$686$		0			0
$687$			6.00000i			0.228914i
$688$		0			0
$689$	−12.0000			−0.457164
$690$		0			0
$691$	−10.0000			−0.380418			−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000			−0.380418			−0.190209	+	0.981744i	$0.560917\pi$
$692$		0			0
$693$		−	2.00000i		−	0.0759737i
$694$		0			0
$695$	−6.00000	−	12.0000i	−0.227593	−	0.455186i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−6.00000			−0.226941
$700$		0			0
$701$	−46.0000			−1.73740			−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000			−1.73740			−0.868698	+	0.495342i	$0.835043\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−12.0000	−	24.0000i	−0.451946	−	0.903892i
$706$		0			0
$707$		−	10.0000i		−	0.376089i
$708$		0			0
$709$	−26.0000			−0.976450			−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000			−0.976450			−0.488225	+	0.872718i	$0.662356\pi$
$710$		0			0
$711$	4.00000			0.150012
$712$		0			0
$713$		0			0
$714$		0			0
$715$	8.00000	−	4.00000i	0.299183	−	0.149592i
$716$		0			0
$717$		−	30.0000i		−	1.12037i
$718$		0			0
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		−	26.0000i		−	0.966950i
$724$		0			0
$725$	−18.0000	+	24.0000i	−0.668503	+	0.891338i
$726$		0			0
$727$		−	16.0000i		−	0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$		−	16.0000i		−	0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		0			0
$735$	−2.00000	+	1.00000i	−0.0737711	+	0.0368856i
$736$		0			0
$737$			8.00000i			0.294684i
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$	−12.0000			−0.440831
$742$		0			0
$743$		−	16.0000i		−	0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$		0			0
$745$	14.0000	+	28.0000i	0.512920	+	1.02584i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.00000			−0.146157
$750$		0			0
$751$	4.00000			0.145962			0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000			0.145962			0.0729810	+	0.997333i	$0.476749\pi$
$752$		0			0
$753$		−	8.00000i		−	0.291536i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			12.0000i			0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$			12.0000i			0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.0000			−0.362500			−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000			−0.362500			−0.181250	+	0.983437i	$0.558014\pi$
$762$		0			0
$763$		−	10.0000i		−	0.362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	42.0000			1.51456			0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000			1.51456			0.757279	+	0.653091i	$0.226528\pi$
$770$		0			0
$771$	20.0000			0.720282
$772$		0			0
$773$		−	32.0000i		−	1.15096i	−0.817816	−	0.575480i	$-0.804815\pi$
$773$		−	32.0000i		−	1.15096i	0.817816	−	0.575480i	$-0.195185\pi$
$774$		0			0
$775$	−30.0000	+	40.0000i	−1.07763	+	1.43684i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	36.0000			1.28983
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		−	6.00000i		−	0.214423i
$784$		0			0
$785$	−44.0000	+	22.0000i	−1.57043	+	0.785214i
$786$		0			0
$787$		−	12.0000i		−	0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$		−	12.0000i		−	0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$		0			0
$789$	−8.00000			−0.284808
$790$		0			0
$791$	10.0000			0.355559
$792$		0			0
$793$		−	12.0000i		−	0.426132i
$794$		0			0
$795$	6.00000	+	12.0000i	0.212798	+	0.425596i
$796$		0			0
$797$			48.0000i			1.70025i	0.526583	+	0.850124i	$0.323473\pi$
$797$			48.0000i			1.70025i	−0.526583	+	0.850124i	$0.676527\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$			28.0000i			0.988099i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	14.0000i		−	0.492823i
$808$		0			0
$809$	−54.0000			−1.89854			−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000			−1.89854			−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$	−38.0000			−1.33436			−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000			−1.33436			−0.667180	+	0.744896i	$0.732499\pi$
$812$		0			0
$813$			30.0000i			1.05215i
$814$		0			0
$815$	32.0000	−	16.0000i	1.12091	−	0.560456i
$816$		0			0
$817$			48.0000i			1.67931i
$818$		0			0
$819$	2.00000			0.0698857
$820$		0			0
$821$	26.0000			0.907406			0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000			0.907406			0.453703	+	0.891153i	$0.350103\pi$
$822$		0			0
$823$		−	44.0000i		−	1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$		−	44.0000i		−	1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$		0			0
$825$	−8.00000	−	6.00000i	−0.278524	−	0.208893i
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−48.0000	+	24.0000i	−1.66111	+	0.830554i
$836$		0			0
$837$		−	10.0000i		−	0.345651i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	26.0000i		−	0.895488i
$844$		0			0
$845$	−9.00000	−	18.0000i	−0.309609	−	0.619219i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$	−12.0000			−0.411839
$850$		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
$855$	6.00000	+	12.0000i	0.205196	+	0.410391i
$856$		0			0
$857$		−	28.0000i		−	0.956462i	−0.878234	−	0.478231i	$-0.841278\pi$
$857$		−	28.0000i		−	0.956462i	0.878234	−	0.478231i	$-0.158722\pi$
$858$		0			0
$859$	2.00000			0.0682391			0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000			0.0682391			0.0341196	+	0.999418i	$0.489137\pi$
$860$		0			0
$861$	−6.00000			−0.204479
$862$		0			0
$863$		−	32.0000i		−	1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$		−	32.0000i		−	1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			17.0000i			0.577350i
$868$		0			0
$869$	−8.00000			−0.271381
$870$		0			0
$871$	−8.00000			−0.271070
$872$		0			0
$873$			2.00000i			0.0676897i
$874$		0			0
$875$	−2.00000	+	11.0000i	−0.0676123	+	0.371868i
$876$		0			0
$877$		−	56.0000i		−	1.89099i	−0.325643	−	0.945493i	$-0.605581\pi$
$877$		−	56.0000i		−	1.89099i	0.325643	−	0.945493i	$-0.394419\pi$
$878$		0			0
$879$	−16.0000			−0.539667
$880$		0			0
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$	4.00000			0.134156
$890$		0			0
$891$	2.00000			0.0670025
$892$		0			0
$893$		−	72.0000i		−	2.40939i
$894$		0			0
$895$	18.0000	+	36.0000i	0.601674	+	1.20335i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	60.0000			2.00111
$900$		0			0
$901$		0			0
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$		0			0
$905$	26.0000	+	52.0000i	0.864269	+	1.72854i
$906$		0			0
$907$		−	28.0000i		−	0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$		−	28.0000i		−	0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$		0			0
$909$	10.0000			0.331679
$910$		0			0
$911$	−30.0000			−0.993944			−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000			−0.993944			−0.496972	+	0.867766i	$0.665555\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−12.0000	+	6.00000i	−0.396708	+	0.198354i
$916$		0			0
$917$			12.0000i			0.396275i
$918$		0			0
$919$	−44.0000			−1.45143			−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000			−1.45143			−0.725713	+	0.687998i	$0.758490\pi$
$920$		0			0
$921$	4.00000			0.131804
$922$		0			0
$923$			12.0000i			0.394985i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			16.0000i			0.525509i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	−6.00000			−0.196642
$932$		0			0
$933$		−	8.00000i		−	0.261908i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			18.0000i			0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$			18.0000i			0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$		0			0
$939$	14.0000			0.456873
$940$		0			0
$941$	−6.00000			−0.195594			−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000			−0.195594			−0.0977972	+	0.995206i	$0.531180\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−1.00000	−	2.00000i	−0.0325300	−	0.0650600i
$946$		0			0
$947$		−	4.00000i		−	0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$		−	4.00000i		−	0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$		0			0
$949$	−28.0000			−0.908918
$950$		0			0
$951$	−26.0000			−0.843108
$952$		0			0
$953$		−	18.0000i		−	0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$		−	18.0000i		−	0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$		0			0
$955$	−10.0000	−	20.0000i	−0.323592	−	0.647185i
$956$		0			0
$957$			12.0000i			0.387905i
$958$		0			0
$959$	2.00000			0.0645834
$960$		0			0
$961$	69.0000			2.22581
$962$		0			0
$963$		−	4.00000i		−	0.128898i
$964$		0			0
$965$	32.0000	−	16.0000i	1.03012	−	0.515058i
$966$		0			0
$967$			40.0000i			1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$			40.0000i			1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32.0000			1.02693			0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000			1.02693			0.513464	+	0.858111i	$0.328362\pi$
$972$		0			0
$973$			6.00000i			0.192351i
$974$		0			0
$975$	6.00000	−	8.00000i	0.192154	−	0.256205i
$976$		0			0
$977$		−	30.0000i		−	0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$		−	30.0000i		−	0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$		0			0
$979$	12.0000			0.383522
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$		−	28.0000i		−	0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$		−	28.0000i		−	0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$		0			0
$985$	−36.0000	+	18.0000i	−1.14706	+	0.573528i
$986$		0			0
$987$			12.0000i			0.381964i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.0000	−	44.0000i	−0.697447	−	1.39489i
$996$		0			0
$997$			22.0000i			0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$			22.0000i			0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.t.b.169.2	yes	2
3.2	odd	2		2520.2.t.e.1009.2		2
4.3	odd	2		1680.2.t.b.1009.1		2
5.2	odd	4		4200.2.a.w.1.1		1
5.3	odd	4		4200.2.a.j.1.1		1
5.4	even	2	inner	840.2.t.b.169.1	✓	2
12.11	even	2		5040.2.t.l.1009.2		2
15.14	odd	2		2520.2.t.e.1009.1		2
20.3	even	4		8400.2.a.bs.1.1		1
20.7	even	4		8400.2.a.v.1.1		1
20.19	odd	2		1680.2.t.b.1009.2		2
60.59	even	2		5040.2.t.l.1009.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.b.169.1	✓	2	5.4	even	2	inner
840.2.t.b.169.2	yes	2	1.1	even	1	trivial
1680.2.t.b.1009.1		2	4.3	odd	2
1680.2.t.b.1009.2		2	20.19	odd	2
2520.2.t.e.1009.1		2	15.14	odd	2
2520.2.t.e.1009.2		2	3.2	odd	2
4200.2.a.j.1.1		1	5.3	odd	4
4200.2.a.w.1.1		1	5.2	odd	4
5040.2.t.l.1009.1		2	60.59	even	2
5040.2.t.l.1009.2		2	12.11	even	2
8400.2.a.v.1.1		1	20.7	even	4
8400.2.a.bs.1.1		1	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.t (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-1.00000 - 2.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} +2.00000i q^{13} +(2.00000 - 1.00000i) q^{15} +6.00000 q^{19} -1.00000 q^{21} +(-3.00000 + 4.00000i) q^{25} -1.00000i q^{27} +6.00000 q^{29} +10.0000 q^{31} +2.00000i q^{33} +(2.00000 - 1.00000i) q^{35} -2.00000 q^{39} +6.00000 q^{41} +8.00000i q^{43} +(1.00000 + 2.00000i) q^{45} -12.0000i q^{47} -1.00000 q^{49} +6.00000i q^{53} +(-2.00000 - 4.00000i) q^{55} +6.00000i q^{57} -6.00000 q^{61} -1.00000i q^{63} +(4.00000 - 2.00000i) q^{65} +4.00000i q^{67} +6.00000 q^{71} +14.0000i q^{73} +(-4.00000 - 3.00000i) q^{75} +2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} +6.00000i q^{87} +6.00000 q^{89} -2.00000 q^{91} +10.0000i q^{93} +(-6.00000 - 12.0000i) q^{95} -2.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^9	\( 2 q - 2 q^{5} - 2 q^{9} + 4 q^{11} + 4 q^{15} + 12 q^{19} - 2 q^{21} - 6 q^{25} + 12 q^{29} + 20 q^{31} + 4 q^{35} - 4 q^{39} + 12 q^{41} + 2 q^{45} - 2 q^{49} - 4 q^{55} - 12 q^{61} + 8 q^{65} + 12 q^{71} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{89} - 4 q^{91} - 12 q^{95} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 + 4 * q^11 + 4 * q^15 + 12 * q^19 - 2 * q^21 - 6 * q^25 + 12 * q^29 + 20 * q^31 + 4 * q^35 - 4 * q^39 + 12 * q^41 + 2 * q^45 - 2 * q^49 - 4 * q^55 - 12 * q^61 + 8 * q^65 + 12 * q^71 - 8 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^89 - 4 * q^91 - 12 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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