LMFDB - Embedded newform 1680.2.t.d.1009.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1009,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, 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("1680.1009");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.t (of order $2$, degree $1$, not 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:	$13.4148675396$
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:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1680.1009
Dual form			1680.2.t.d.1009.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +8.00000i q^{17} -2.00000 q^{19} -1.00000 q^{21} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +6.00000 q^{29} -6.00000 q^{31} -2.00000i q^{33} +(2.00000 + 1.00000i) q^{35} +8.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +8.00000i q^{43} +(1.00000 - 2.00000i) q^{45} -4.00000i q^{47} -1.00000 q^{49} +8.00000 q^{51} +2.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} +2.00000i q^{57} -8.00000 q^{59} +10.0000 q^{61} +1.00000i q^{63} +(4.00000 + 2.00000i) q^{65} +12.0000i q^{67} +14.0000 q^{71} +10.0000i q^{73} +(-4.00000 + 3.00000i) q^{75} -2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +16.0000i q^{83} +(-16.0000 - 8.00000i) q^{85} -6.00000i q^{87} -10.0000 q^{89} -2.00000 q^{91} +6.00000i q^{93} +(2.00000 - 4.00000i) q^{95} +10.0000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{19} - 2 q^{21} - 6 q^{25} + 12 q^{29} - 12 q^{31} + 4 q^{35} - 4 q^{39} + 12 q^{41} + 2 q^{45} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 16 q^{59} + 20 q^{61} + 8 q^{65} + 28 q^{71} - 8 q^{75} + 8 q^{79} + 2 q^{81} - 32 q^{85} - 20 q^{89} - 4 q^{91} + 4 q^{95} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 + 4 * q^11 + 4 * q^15 - 4 * q^19 - 2 * q^21 - 6 * q^25 + 12 * q^29 - 12 * q^31 + 4 * q^35 - 4 * q^39 + 12 * q^41 + 2 * q^45 - 2 * q^49 + 16 * q^51 - 4 * q^55 - 16 * q^59 + 20 * q^61 + 8 * q^65 + 28 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 - 32 * q^85 - 20 * q^89 - 4 * q^91 + 4 * q^95 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$1$	$-1$	$1$	$1$	$1$

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	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$	2.00000	+	1.00000i	0.516398	+	0.258199i
$16$		0			0
$17$			8.00000i			1.94029i	0.242536	+	0.970143i	$0.422021\pi$
$17$			8.00000i			1.94029i	−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		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
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$		0			0
$33$		−	2.00000i		−	0.348155i
$34$		0			0
$35$	2.00000	+	1.00000i	0.338062	+	0.169031i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		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
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	8.00000			1.12022
$52$		0			0
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		0			0
$55$	−2.00000	+	4.00000i	−0.269680	+	0.539360i
$56$		0			0
$57$			2.00000i			0.264906i
$58$		0			0
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$			10.0000i			1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$			10.0000i			1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$		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	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			16.0000i			1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$			16.0000i			1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$		0			0
$85$	−16.0000	−	8.00000i	−1.73544	−	0.867722i
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$		0			0
$93$			6.00000i			0.622171i
$94$		0			0
$95$	2.00000	−	4.00000i	0.205196	−	0.410391i
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$		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
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	1.00000	−	2.00000i	0.0975900	−	0.195180i
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$	8.00000			0.759326
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			2.00000i			0.184900i
$118$		0			0
$119$	8.00000			0.733359
$120$		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
$127$		−	12.0000i		−	1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$		−	12.0000i		−	1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		0			0
$131$	20.0000			1.74741			0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000			1.74741			0.873704	+	0.486458i	$0.161711\pi$
$132$		0			0
$133$			2.00000i			0.173422i
$134$		0			0
$135$	−2.00000	−	1.00000i	−0.172133	−	0.0860663i
$136$		0			0
$137$		−	14.0000i		−	1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$		−	14.0000i		−	1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$		0			0
$139$	−2.00000			−0.169638			−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000			−0.169638			−0.0848189	+	0.996396i	$0.527031\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		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
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		−	8.00000i		−	0.646762i
$154$		0			0
$155$	6.00000	−	12.0000i	0.481932	−	0.963863i
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$		0			0
$159$	2.00000			0.158610
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	4.00000	+	2.00000i	0.311400	+	0.155700i
$166$		0			0
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	2.00000			0.152944
$172$		0			0
$173$			16.0000i			1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$173$			16.0000i			1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	−2.00000			−0.149487			−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000			−0.149487			−0.0747435	+	0.997203i	$0.523814\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$	−16.0000	−	8.00000i	−1.17634	−	0.588172i
$186$		0			0
$187$			16.0000i			1.17004i
$188$		0			0
$189$	1.00000			0.0727393
$190$		0			0
$191$	−14.0000			−1.01300			−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000			−1.01300			−0.506502	+	0.862239i	$0.669062\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	2.00000	−	4.00000i	0.143223	−	0.286446i
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	−26.0000			−1.84309			−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000			−1.84309			−0.921546	+	0.388270i	$0.873073\pi$
$200$		0			0
$201$	12.0000			0.846415
$202$		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
$209$	−4.00000			−0.276686
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$		−	14.0000i		−	0.959264i
$214$		0			0
$215$	−16.0000	−	8.00000i	−1.09119	−	0.545595i
$216$		0			0
$217$			6.00000i			0.407307i
$218$		0			0
$219$	10.0000			0.675737
$220$		0			0
$221$	16.0000			1.07628
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		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
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$	−2.00000			−0.131590
$232$		0			0
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$		0			0
$235$	8.00000	+	4.00000i	0.521862	+	0.260931i
$236$		0			0
$237$		−	4.00000i		−	0.259828i
$238$		0			0
$239$	−22.0000			−1.42306			−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000			−1.42306			−0.711531	+	0.702655i	$0.751998\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	1.00000	−	2.00000i	0.0638877	−	0.127775i
$246$		0			0
$247$			4.00000i			0.254514i
$248$		0			0
$249$	16.0000			1.01396
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−8.00000	+	16.0000i	−0.500979	+	1.00196i
$256$		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
$259$	8.00000			0.497096
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−4.00000	−	2.00000i	−0.245718	−	0.122859i
$266$		0			0
$267$			10.0000i			0.611990i
$268$		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
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$		0			0
$273$			2.00000i			0.121046i
$274$		0			0
$275$	−6.00000	−	8.00000i	−0.361814	−	0.482418i
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		0			0
$279$	6.00000			0.359211
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		0			0
$285$	−4.00000	−	2.00000i	−0.236940	−	0.118470i
$286$		0			0
$287$		−	6.00000i		−	0.354169i
$288$		0			0
$289$	−47.0000			−2.76471
$290$		0			0
$291$	10.0000			0.586210
$292$		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
$295$	8.00000	−	16.0000i	0.465778	−	0.931556i
$296$		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
$305$	−10.0000	+	20.0000i	−0.572598	+	1.14520i
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		0			0
$315$	−2.00000	−	1.00000i	−0.112687	−	0.0563436i
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	−8.00000	+	6.00000i	−0.443760	+	0.332820i
$326$		0			0
$327$		−	6.00000i		−	0.331801i
$328$		0			0
$329$	−4.00000			−0.220527
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		−	8.00000i		−	0.438397i
$334$		0			0
$335$	−24.0000	−	12.0000i	−1.31126	−	0.655630i
$336$		0			0
$337$			32.0000i			1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$			32.0000i			1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$	−6.00000			−0.325875
$340$		0			0
$341$	−12.0000			−0.649836
$342$		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	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$		0			0
$349$	−18.0000			−0.963518			−0.481759	−	0.876304i	$-0.660002\pi$
$349$	−18.0000			−0.963518			−0.481759	+	0.876304i	$0.660002\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	−14.0000	+	28.0000i	−0.743043	+	1.48609i
$356$		0			0
$357$		−	8.00000i		−	0.423405i
$358$		0			0
$359$	14.0000			0.738892			0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000			0.738892			0.369446	+	0.929252i	$0.379548\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$			7.00000i			0.367405i
$364$		0			0
$365$	−20.0000	−	10.0000i	−1.04685	−	0.523424i
$366$		0			0
$367$			32.0000i			1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$			32.0000i			1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$		0			0
$369$	−6.00000			−0.312348
$370$		0			0
$371$	2.00000			0.103835
$372$		0			0
$373$		−	36.0000i		−	1.86401i	−0.362446	−	0.932005i	$-0.618058\pi$
$373$		−	36.0000i		−	1.86401i	0.362446	−	0.932005i	$-0.381942\pi$
$374$		0			0
$375$	−2.00000	−	11.0000i	−0.103280	−	0.568038i
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	4.00000	+	2.00000i	0.203859	+	0.101929i
$386$		0			0
$387$		−	8.00000i		−	0.406663i
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	20.0000i		−	1.00887i
$394$		0			0
$395$	−4.00000	+	8.00000i	−0.201262	+	0.402524i
$396$		0			0
$397$			2.00000i			0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$			2.00000i			0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$		0			0
$399$	2.00000			0.100125
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$			12.0000i			0.597763i
$404$		0			0
$405$	−1.00000	+	2.00000i	−0.0496904	+	0.0993808i
$406$		0			0
$407$			16.0000i			0.793091i
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$	−14.0000			−0.690569
$412$		0			0
$413$			8.00000i			0.393654i
$414$		0			0
$415$	−32.0000	−	16.0000i	−1.57082	−	0.785409i
$416$		0			0
$417$			2.00000i			0.0979404i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		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
$423$			4.00000i			0.194487i
$424$		0			0
$425$	32.0000	−	24.0000i	1.55223	−	1.16417i
$426$		0			0
$427$		−	10.0000i		−	0.483934i
$428$		0			0
$429$	−4.00000			−0.193122
$430$		0			0
$431$	6.00000			0.289010			0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000			0.289010			0.144505	+	0.989504i	$0.453841\pi$
$432$		0			0
$433$		−	26.0000i		−	1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$		−	26.0000i		−	1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$		0			0
$435$	12.0000	+	6.00000i	0.575356	+	0.287678i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	6.00000			0.286364			0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000			0.286364			0.143182	+	0.989696i	$0.454267\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$			36.0000i			1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$			36.0000i			1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$		0			0
$445$	10.0000	−	20.0000i	0.474045	−	0.948091i
$446$		0			0
$447$		−	18.0000i		−	0.851371i
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$	2.00000	−	4.00000i	0.0937614	−	0.187523i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		0			0
$459$	−8.00000			−0.373408
$460$		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
$463$		−	12.0000i		−	0.557687i	−0.960337	−	0.278844i	$-0.910049\pi$
$463$		−	12.0000i		−	0.557687i	0.960337	−	0.278844i	$-0.0899511\pi$
$464$		0			0
$465$	−12.0000	−	6.00000i	−0.556487	−	0.278243i
$466$		0			0
$467$		−	16.0000i		−	0.740392i	−0.928954	−	0.370196i	$-0.879291\pi$
$467$		−	16.0000i		−	0.740392i	0.928954	−	0.370196i	$-0.120709\pi$
$468$		0			0
$469$	12.0000			0.554109
$470$		0			0
$471$	−10.0000			−0.460776
$472$		0			0
$473$			16.0000i			0.735681i
$474$		0			0
$475$	6.00000	+	8.00000i	0.275299	+	0.367065i
$476$		0			0
$477$		−	2.00000i		−	0.0915737i
$478$		0			0
$479$	8.00000			0.365529			0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000			0.365529			0.182765	+	0.983157i	$0.441495\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.0000	−	10.0000i	−0.908153	−	0.454077i
$486$		0			0
$487$		−	36.0000i		−	1.63132i	−0.578535	−	0.815658i	$-0.696375\pi$
$487$		−	36.0000i		−	1.63132i	0.578535	−	0.815658i	$-0.303625\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	42.0000			1.89543			0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000			1.89543			0.947717	+	0.319113i	$0.103385\pi$
$492$		0			0
$493$			48.0000i			2.16181i
$494$		0			0
$495$	2.00000	−	4.00000i	0.0898933	−	0.179787i
$496$		0			0
$497$		−	14.0000i		−	0.627986i
$498$		0			0
$499$	−28.0000			−1.25345			−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000			−1.25345			−0.626726	+	0.779240i	$0.715605\pi$
$500$		0			0
$501$	8.00000			0.357414
$502$		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
$509$	−2.00000			−0.0886484			−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000			−0.0886484			−0.0443242	+	0.999017i	$0.514113\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$		0			0
$513$		−	2.00000i		−	0.0883022i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	8.00000i		−	0.351840i
$518$		0			0
$519$	16.0000			0.702322
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$		−	12.0000i		−	0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$		−	12.0000i		−	0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$		0			0
$525$	3.00000	+	4.00000i	0.130931	+	0.174574i
$526$		0			0
$527$		−	48.0000i		−	2.09091i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	8.00000			0.347170
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−24.0000	−	12.0000i	−1.03761	−	0.518805i
$536$		0			0
$537$			2.00000i			0.0863064i
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$		−	22.0000i		−	0.944110i
$544$		0			0
$545$	−6.00000	+	12.0000i	−0.257012	+	0.514024i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		0			0
$549$	−10.0000			−0.426790
$550$		0			0
$551$	−12.0000			−0.511217
$552$		0			0
$553$		−	4.00000i		−	0.170097i
$554$		0			0
$555$	−8.00000	+	16.0000i	−0.339581	+	0.679162i
$556$		0			0
$557$			6.00000i			0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$			6.00000i			0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$	16.0000			0.675521
$562$		0			0
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$	12.0000	+	6.00000i	0.504844	+	0.252422i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−18.0000			−0.754599			−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000			−0.754599			−0.377300	+	0.926091i	$0.623147\pi$
$570$		0			0
$571$	44.0000			1.84134			0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000			1.84134			0.920671	+	0.390339i	$0.127642\pi$
$572$		0			0
$573$			14.0000i			0.584858i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			18.0000i			0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$			18.0000i			0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	16.0000			0.663792
$582$		0			0
$583$			4.00000i			0.165663i
$584$		0			0
$585$	−4.00000	−	2.00000i	−0.165380	−	0.0826898i
$586$		0			0
$587$		−	4.00000i		−	0.165098i	−0.996587	−	0.0825488i	$-0.973694\pi$
$587$		−	4.00000i		−	0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$	−6.00000			−0.246807
$592$		0			0
$593$			32.0000i			1.31408i	0.753855	+	0.657041i	$0.228192\pi$
$593$			32.0000i			1.31408i	−0.753855	+	0.657041i	$0.771808\pi$
$594$		0			0
$595$	−8.00000	+	16.0000i	−0.327968	+	0.655936i
$596$		0			0
$597$			26.0000i			1.06411i
$598$		0			0
$599$	14.0000			0.572024			0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000			0.572024			0.286012	+	0.958226i	$0.407670\pi$
$600$		0			0
$601$	−42.0000			−1.71322			−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000			−1.71322			−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$		−	12.0000i		−	0.488678i
$604$		0			0
$605$	7.00000	−	14.0000i	0.284590	−	0.569181i
$606$		0			0
$607$			16.0000i			0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$			16.0000i			0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$		0			0
$609$	−6.00000			−0.243132
$610$		0			0
$611$	−8.00000			−0.323645
$612$		0			0
$613$		−	32.0000i		−	1.29247i	−0.763139	−	0.646234i	$-0.776343\pi$
$613$		−	32.0000i		−	1.29247i	0.763139	−	0.646234i	$-0.223657\pi$
$614$		0			0
$615$	12.0000	+	6.00000i	0.483887	+	0.241943i
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$	−18.0000			−0.723481			−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000			−0.723481			−0.361741	+	0.932279i	$0.617817\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			10.0000i			0.400642i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			4.00000i			0.159745i
$628$		0			0
$629$	−64.0000			−2.55185
$630$		0			0
$631$	−12.0000			−0.477712			−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000			−0.477712			−0.238856	+	0.971055i	$0.576772\pi$
$632$		0			0
$633$		−	8.00000i		−	0.317971i
$634$		0			0
$635$	24.0000	+	12.0000i	0.952411	+	0.476205i
$636$		0			0
$637$			2.00000i			0.0792429i
$638$		0			0
$639$	−14.0000			−0.553831
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$		−	20.0000i		−	0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$		−	20.0000i		−	0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$		0			0
$645$	−8.00000	+	16.0000i	−0.315000	+	0.629999i
$646$		0			0
$647$		−	16.0000i		−	0.629025i	−0.949253	−	0.314512i	$-0.898159\pi$
$647$		−	16.0000i		−	0.629025i	0.949253	−	0.314512i	$-0.101841\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$	6.00000			0.235159
$652$		0			0
$653$			14.0000i			0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$			14.0000i			0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$		0			0
$655$	−20.0000	+	40.0000i	−0.781465	+	1.56293i
$656$		0			0
$657$		−	10.0000i		−	0.390137i
$658$		0			0
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	10.0000			0.388955			0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000			0.388955			0.194477	+	0.980907i	$0.437699\pi$
$662$		0			0
$663$		−	16.0000i		−	0.621389i
$664$		0			0
$665$	−4.00000	−	2.00000i	−0.155113	−	0.0775567i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−16.0000			−0.618596
$670$		0			0
$671$	20.0000			0.772091
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$		0			0
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	10.0000			0.383765
$680$		0			0
$681$	8.00000			0.306561
$682$		0			0
$683$			28.0000i			1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$			28.0000i			1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$		0			0
$685$	28.0000	+	14.0000i	1.06983	+	0.534913i
$686$		0			0
$687$			26.0000i			0.991962i
$688$		0			0
$689$	4.00000			0.152388
$690$		0			0
$691$	14.0000			0.532585			0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000			0.532585			0.266293	+	0.963892i	$0.414201\pi$
$692$		0			0
$693$			2.00000i			0.0759737i
$694$		0			0
$695$	2.00000	−	4.00000i	0.0758643	−	0.151729i
$696$		0			0
$697$			48.0000i			1.81813i
$698$		0			0
$699$	−6.00000			−0.226941
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$		−	16.0000i		−	0.603451i
$704$		0			0
$705$	4.00000	−	8.00000i	0.150649	−	0.301297i
$706$		0			0
$707$			10.0000i			0.376089i
$708$		0			0
$709$	38.0000			1.42712			0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000			1.42712			0.713560	+	0.700594i	$0.247082\pi$
$710$		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
$717$			22.0000i			0.821605i
$718$		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
$721$		0			0
$722$		0			0
$723$			10.0000i			0.371904i
$724$		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
$731$	−64.0000			−2.36713
$732$		0			0
$733$			30.0000i			1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$			30.0000i			1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$		0			0
$735$	−2.00000	−	1.00000i	−0.0737711	−	0.0368856i
$736$		0			0
$737$			24.0000i			0.884051i
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$		0			0
$741$	4.00000			0.146944
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−18.0000	+	36.0000i	−0.659469	+	1.31894i
$746$		0			0
$747$		−	16.0000i		−	0.585409i
$748$		0			0
$749$	12.0000			0.438470
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	4.00000i		−	0.145382i	−0.997354	−	0.0726912i	$-0.976841\pi$
$757$		−	4.00000i		−	0.145382i	0.997354	−	0.0726912i	$-0.0231588\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$		−	6.00000i		−	0.217215i
$764$		0			0
$765$	16.0000	+	8.00000i	0.578481	+	0.289241i
$766$		0			0
$767$			16.0000i			0.577727i
$768$		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	$0.195185\pi$
$773$			32.0000i			1.15096i	−0.817816	+	0.575480i	$0.804815\pi$
$774$		0			0
$775$	18.0000	+	24.0000i	0.646579	+	0.862105i
$776$		0			0
$777$		−	8.00000i		−	0.286998i
$778$		0			0
$779$	−12.0000			−0.429945
$780$		0			0
$781$	28.0000			1.00192
$782$		0			0
$783$			6.00000i			0.214423i
$784$		0			0
$785$	20.0000	+	10.0000i	0.713831	+	0.356915i
$786$		0			0
$787$			44.0000i			1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$			44.0000i			1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$		−	20.0000i		−	0.710221i
$794$		0			0
$795$	−2.00000	+	4.00000i	−0.0709327	+	0.141865i
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	32.0000			1.13208
$800$		0			0
$801$	10.0000			0.353333
$802$		0			0
$803$			20.0000i			0.705785i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			14.0000i			0.492823i
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	−14.0000			−0.491606			−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000			−0.491606			−0.245803	+	0.969320i	$0.579052\pi$
$812$		0			0
$813$			2.00000i			0.0701431i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	16.0000i		−	0.559769i
$818$		0			0
$819$	2.00000			0.0698857
$820$		0			0
$821$	−6.00000			−0.209401			−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000			−0.209401			−0.104701	+	0.994504i	$0.533388\pi$
$822$		0			0
$823$		−	4.00000i		−	0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$		−	4.00000i		−	0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$		0			0
$825$	−8.00000	+	6.00000i	−0.278524	+	0.208893i
$826$		0			0
$827$			36.0000i			1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$			36.0000i			1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$		0			0
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$	−8.00000			−0.277517
$832$		0			0
$833$		−	8.00000i		−	0.277184i
$834$		0			0
$835$	−16.0000	−	8.00000i	−0.553703	−	0.276851i
$836$		0			0
$837$		−	6.00000i		−	0.207390i
$838$		0			0
$839$	44.0000			1.51905			0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000			1.51905			0.759524	+	0.650479i	$0.225432\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	22.0000i		−	0.757720i
$844$		0			0
$845$	−9.00000	+	18.0000i	−0.309609	+	0.619219i
$846$		0			0
$847$			7.00000i			0.240523i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	10.0000i		−	0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$		−	10.0000i		−	0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$		0			0
$855$	−2.00000	+	4.00000i	−0.0683986	+	0.136797i
$856$		0			0
$857$			20.0000i			0.683187i	0.939848	+	0.341593i	$0.110967\pi$
$857$			20.0000i			0.683187i	−0.939848	+	0.341593i	$0.889033\pi$
$858$		0			0
$859$	−6.00000			−0.204717			−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000			−0.204717			−0.102359	+	0.994748i	$0.532639\pi$
$860$		0			0
$861$	−6.00000			−0.204479
$862$		0			0
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$		0			0
$865$	−32.0000	−	16.0000i	−1.08803	−	0.544016i
$866$		0			0
$867$			47.0000i			1.59620i
$868$		0			0
$869$	8.00000			0.271381
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$		−	10.0000i		−	0.338449i
$874$		0			0
$875$	−2.00000	−	11.0000i	−0.0676123	−	0.371868i
$876$		0			0
$877$		−	48.0000i		−	1.62084i	−0.585846	−	0.810422i	$-0.699238\pi$
$877$		−	48.0000i		−	1.62084i	0.585846	−	0.810422i	$-0.300762\pi$
$878$		0			0
$879$	16.0000			0.539667
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$			52.0000i			1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$			52.0000i			1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$		0			0
$885$	−16.0000	−	8.00000i	−0.537834	−	0.268917i
$886$		0			0
$887$		−	36.0000i		−	1.20876i	−0.796696	−	0.604381i	$-0.793421\pi$
$887$		−	36.0000i		−	1.20876i	0.796696	−	0.604381i	$-0.206579\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$		0			0
$891$	2.00000			0.0670025
$892$		0			0
$893$			8.00000i			0.267710i
$894$		0			0
$895$	2.00000	−	4.00000i	0.0668526	−	0.133705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.0000			−1.20067
$900$		0			0
$901$	−16.0000			−0.533037
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$		0			0
$905$	−22.0000	+	44.0000i	−0.731305	+	1.46261i
$906$		0			0
$907$			44.0000i			1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$			44.0000i			1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$		0			0
$909$	10.0000			0.331679
$910$		0			0
$911$	−6.00000			−0.198789			−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000			−0.198789			−0.0993944	+	0.995048i	$0.531691\pi$
$912$		0			0
$913$			32.0000i			1.05905i
$914$		0			0
$915$	20.0000	+	10.0000i	0.661180	+	0.330590i
$916$		0			0
$917$		−	20.0000i		−	0.660458i
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$		0			0
$921$	−28.0000			−0.922631
$922$		0			0
$923$		−	28.0000i		−	0.921631i
$924$		0			0
$925$	32.0000	−	24.0000i	1.05215	−	0.789115i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$	2.00000			0.0655474
$932$		0			0
$933$		−	8.00000i		−	0.261908i
$934$		0			0
$935$	−32.0000	−	16.0000i	−1.04651	−	0.523256i
$936$		0			0
$937$		−	26.0000i		−	0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$		−	26.0000i		−	0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$		0			0
$939$	−10.0000			−0.326338
$940$		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
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	20.0000			0.649227
$950$		0			0
$951$	−2.00000			−0.0648544
$952$		0			0
$953$			18.0000i			0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$			18.0000i			0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$		0			0
$955$	14.0000	−	28.0000i	0.453029	−	0.906059i
$956$		0			0
$957$		−	12.0000i		−	0.387905i
$958$		0			0
$959$	−14.0000			−0.452084
$960$		0			0
$961$	5.00000			0.161290
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			24.0000i			0.771788i	0.922543	+	0.385894i	$0.126107\pi$
$967$			24.0000i			0.771788i	−0.922543	+	0.385894i	$0.873893\pi$
$968$		0			0
$969$	−16.0000			−0.513994
$970$		0			0
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$			2.00000i			0.0641171i
$974$		0			0
$975$	6.00000	+	8.00000i	0.192154	+	0.256205i
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$	−20.0000			−0.639203
$980$		0			0
$981$	−6.00000			−0.191565
$982$		0			0
$983$			12.0000i			0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$			12.0000i			0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$		0			0
$985$	12.0000	+	6.00000i	0.382352	+	0.191176i
$986$		0			0
$987$			4.00000i			0.127321i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	60.0000			1.90596			0.952981	−	0.303029i	$-0.0979978\pi$
$991$	60.0000			1.90596			0.952981	+	0.303029i	$0.0979978\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	26.0000	−	52.0000i	0.824255	−	1.64851i
$996$		0			0
$997$		−	38.0000i		−	1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$		−	38.0000i		−	1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$		0			0
$999$	−8.00000			−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.d.1009.1		2
3.2	odd	2		5040.2.t.k.1009.1		2
4.3	odd	2		210.2.g.a.169.2	yes	2
5.2	odd	4		8400.2.a.bd.1.1		1
5.3	odd	4		8400.2.a.ca.1.1		1
5.4	even	2	inner	1680.2.t.d.1009.2		2
12.11	even	2		630.2.g.d.379.1		2
15.14	odd	2		5040.2.t.k.1009.2		2
20.3	even	4		1050.2.a.m.1.1		1
20.7	even	4		1050.2.a.g.1.1		1
20.19	odd	2		210.2.g.a.169.1	✓	2
28.3	even	6		1470.2.n.c.79.1		4
28.11	odd	6		1470.2.n.g.79.1		4
28.19	even	6		1470.2.n.c.949.2		4
28.23	odd	6		1470.2.n.g.949.2		4
28.27	even	2		1470.2.g.e.589.2		2
60.23	odd	4		3150.2.a.q.1.1		1
60.47	odd	4		3150.2.a.be.1.1		1
60.59	even	2		630.2.g.d.379.2		2
140.19	even	6		1470.2.n.c.949.1		4
140.27	odd	4		7350.2.a.g.1.1		1
140.39	odd	6		1470.2.n.g.79.2		4
140.59	even	6		1470.2.n.c.79.2		4
140.79	odd	6		1470.2.n.g.949.1		4
140.83	odd	4		7350.2.a.co.1.1		1
140.139	even	2		1470.2.g.e.589.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.g.a.169.1	✓	2	20.19	odd	2
210.2.g.a.169.2	yes	2	4.3	odd	2
630.2.g.d.379.1		2	12.11	even	2
630.2.g.d.379.2		2	60.59	even	2
1050.2.a.g.1.1		1	20.7	even	4
1050.2.a.m.1.1		1	20.3	even	4
1470.2.g.e.589.1		2	140.139	even	2
1470.2.g.e.589.2		2	28.27	even	2
1470.2.n.c.79.1		4	28.3	even	6
1470.2.n.c.79.2		4	140.59	even	6
1470.2.n.c.949.1		4	140.19	even	6
1470.2.n.c.949.2		4	28.19	even	6
1470.2.n.g.79.1		4	28.11	odd	6
1470.2.n.g.79.2		4	140.39	odd	6
1470.2.n.g.949.1		4	140.79	odd	6
1470.2.n.g.949.2		4	28.23	odd	6
1680.2.t.d.1009.1		2	1.1	even	1	trivial
1680.2.t.d.1009.2		2	5.4	even	2	inner
3150.2.a.q.1.1		1	60.23	odd	4
3150.2.a.be.1.1		1	60.47	odd	4
5040.2.t.k.1009.1		2	3.2	odd	2
5040.2.t.k.1009.2		2	15.14	odd	2
7350.2.a.g.1.1		1	140.27	odd	4
7350.2.a.co.1.1		1	140.83	odd	4
8400.2.a.bd.1.1		1	5.2	odd	4
8400.2.a.ca.1.1		1	5.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.t (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +8.00000i q^{17} -2.00000 q^{19} -1.00000 q^{21} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +6.00000 q^{29} -6.00000 q^{31} -2.00000i q^{33} +(2.00000 + 1.00000i) q^{35} +8.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +8.00000i q^{43} +(1.00000 - 2.00000i) q^{45} -4.00000i q^{47} -1.00000 q^{49} +8.00000 q^{51} +2.00000i q^{53} +(-2.00000 + 4.00000i) q^{55} +2.00000i q^{57} -8.00000 q^{59} +10.0000 q^{61} +1.00000i q^{63} +(4.00000 + 2.00000i) q^{65} +12.0000i q^{67} +14.0000 q^{71} +10.0000i q^{73} +(-4.00000 + 3.00000i) q^{75} -2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +16.0000i q^{83} +(-16.0000 - 8.00000i) q^{85} -6.00000i q^{87} -10.0000 q^{89} -2.00000 q^{91} +6.00000i q^{93} +(2.00000 - 4.00000i) q^{95} +10.0000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^9	\( 2 q - 2 q^{5} - 2 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{19} - 2 q^{21} - 6 q^{25} + 12 q^{29} - 12 q^{31} + 4 q^{35} - 4 q^{39} + 12 q^{41} + 2 q^{45} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 16 q^{59} + 20 q^{61} + 8 q^{65} + 28 q^{71} - 8 q^{75} + 8 q^{79} + 2 q^{81} - 32 q^{85} - 20 q^{89} - 4 q^{91} + 4 q^{95} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 + 4 * q^11 + 4 * q^15 - 4 * q^19 - 2 * q^21 - 6 * q^25 + 12 * q^29 - 12 * q^31 + 4 * q^35 - 4 * q^39 + 12 * q^41 + 2 * q^45 - 2 * q^49 + 16 * q^51 - 4 * q^55 - 16 * q^59 + 20 * q^61 + 8 * q^65 + 28 * q^71 - 8 * q^75 + 8 * q^79 + 2 * q^81 - 32 * q^85 - 20 * q^89 - 4 * q^91 + 4 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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