LMFDB - Embedded newform 1680.2.t.i.1009.5

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:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1009.5
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	1680.1009
Dual form			1680.2.t.i.1009.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.311108 - 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-0.311108 - 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +3.80642 q^{11} -0.622216i q^{13} +(2.21432 - 0.311108i) q^{15} -4.42864i q^{17} +0.622216 q^{19} +1.00000 q^{21} +2.62222i q^{23} +(-4.80642 + 1.37778i) q^{25} -1.00000i q^{27} -9.61285 q^{29} +0.622216 q^{31} +3.80642i q^{33} +(-2.21432 + 0.311108i) q^{35} -1.24443i q^{37} +0.622216 q^{39} +4.62222 q^{41} -4.85728i q^{43} +(0.311108 + 2.21432i) q^{45} -11.6128i q^{47} -1.00000 q^{49} +4.42864 q^{51} -13.4795i q^{53} +(-1.18421 - 8.42864i) q^{55} +0.622216i q^{57} +11.6128 q^{59} -8.10171 q^{61} +1.00000i q^{63} +(-1.37778 + 0.193576i) q^{65} -2.62222 q^{69} -2.56199 q^{71} -10.9906i q^{73} +(-1.37778 - 4.80642i) q^{75} -3.80642i q^{77} -6.75557 q^{79} +1.00000 q^{81} +11.6128i q^{83} +(-9.80642 + 1.37778i) q^{85} -9.61285i q^{87} +8.23506 q^{89} -0.622216 q^{91} +0.622216i q^{93} +(-0.193576 - 1.37778i) q^{95} +4.23506i q^{97} -3.80642 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^5 - 6 * q^9	$ 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 + 4 * q^31 + 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 + 20 * q^55 + 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 + 12 * q^71 - 8 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 - 4 * q^91 - 28 * 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
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$5$	−0.311108	−	2.21432i	−0.139132	−	0.990274i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.80642			1.14768			0.573840	−	0.818967i	$-0.305453\pi$
$11$	3.80642			1.14768			0.573840	+	0.818967i	$0.305453\pi$
$12$		0			0
$13$		−	0.622216i		−	0.172572i	−0.996270	−	0.0862858i	$-0.972500\pi$
$13$		−	0.622216i		−	0.172572i	0.996270	−	0.0862858i	$-0.0274998\pi$
$14$		0			0
$15$	2.21432	−	0.311108i	0.571735	−	0.0803277i
$16$		0			0
$17$		−	4.42864i		−	1.07410i	−0.843550	−	0.537051i	$-0.819538\pi$
$17$		−	4.42864i		−	1.07410i	0.843550	−	0.537051i	$-0.180462\pi$
$18$		0			0
$19$	0.622216			0.142746			0.0713730	−	0.997450i	$-0.477262\pi$
$19$	0.622216			0.142746			0.0713730	+	0.997450i	$0.477262\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$			2.62222i			0.546770i	0.961905	+	0.273385i	$0.0881433\pi$
$23$			2.62222i			0.546770i	−0.961905	+	0.273385i	$0.911857\pi$
$24$		0			0
$25$	−4.80642	+	1.37778i	−0.961285	+	0.275557i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−9.61285			−1.78506			−0.892531	−	0.450987i	$-0.851072\pi$
$29$	−9.61285			−1.78506			−0.892531	+	0.450987i	$0.851072\pi$
$30$		0			0
$31$	0.622216			0.111753			0.0558766	−	0.998438i	$-0.482205\pi$
$31$	0.622216			0.111753			0.0558766	+	0.998438i	$0.482205\pi$
$32$		0			0
$33$			3.80642i			0.662613i
$34$		0			0
$35$	−2.21432	+	0.311108i	−0.374288	+	0.0525868i
$36$		0			0
$37$		−	1.24443i		−	0.204583i	−0.994754	−	0.102292i	$-0.967383\pi$
$37$		−	1.24443i		−	0.204583i	0.994754	−	0.102292i	$-0.0326175\pi$
$38$		0			0
$39$	0.622216			0.0996342
$40$		0			0
$41$	4.62222			0.721869			0.360934	−	0.932591i	$-0.382458\pi$
$41$	4.62222			0.721869			0.360934	+	0.932591i	$0.382458\pi$
$42$		0			0
$43$		−	4.85728i		−	0.740728i	−0.928887	−	0.370364i	$-0.879233\pi$
$43$		−	4.85728i		−	0.740728i	0.928887	−	0.370364i	$-0.120767\pi$
$44$		0			0
$45$	0.311108	+	2.21432i	0.0463772	+	0.330091i
$46$		0			0
$47$		−	11.6128i		−	1.69391i	−0.531666	−	0.846954i	$-0.678434\pi$
$47$		−	11.6128i		−	1.69391i	0.531666	−	0.846954i	$-0.321566\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	4.42864			0.620134
$52$		0			0
$53$		−	13.4795i		−	1.85155i	−0.378074	−	0.925775i	$-0.623413\pi$
$53$		−	13.4795i		−	1.85155i	0.378074	−	0.925775i	$-0.376587\pi$
$54$		0			0
$55$	−1.18421	−	8.42864i	−0.159679	−	1.13652i
$56$		0			0
$57$			0.622216i			0.0824145i
$58$		0			0
$59$	11.6128			1.51186			0.755932	−	0.654650i	$-0.227184\pi$
$59$	11.6128			1.51186			0.755932	+	0.654650i	$0.227184\pi$
$60$		0			0
$61$	−8.10171			−1.03732			−0.518659	−	0.854981i	$-0.673569\pi$
$61$	−8.10171			−1.03732			−0.518659	+	0.854981i	$0.673569\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−1.37778	+	0.193576i	−0.170893	+	0.0240102i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−2.62222			−0.315678
$70$		0			0
$71$	−2.56199			−0.304053			−0.152026	−	0.988376i	$-0.548580\pi$
$71$	−2.56199			−0.304053			−0.152026	+	0.988376i	$0.548580\pi$
$72$		0			0
$73$		−	10.9906i		−	1.28636i	−0.765717	−	0.643178i	$-0.777615\pi$
$73$		−	10.9906i		−	1.28636i	0.765717	−	0.643178i	$-0.222385\pi$
$74$		0			0
$75$	−1.37778	−	4.80642i	−0.159093	−	0.554998i
$76$		0			0
$77$		−	3.80642i		−	0.433782i
$78$		0			0
$79$	−6.75557			−0.760061			−0.380030	−	0.924974i	$-0.624086\pi$
$79$	−6.75557			−0.760061			−0.380030	+	0.924974i	$0.624086\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			11.6128i			1.27468i	0.770585	+	0.637338i	$0.219964\pi$
$83$			11.6128i			1.27468i	−0.770585	+	0.637338i	$0.780036\pi$
$84$		0			0
$85$	−9.80642	+	1.37778i	−1.06366	+	0.149442i
$86$		0			0
$87$		−	9.61285i		−	1.03061i
$88$		0			0
$89$	8.23506			0.872915			0.436457	−	0.899725i	$-0.356233\pi$
$89$	8.23506			0.872915			0.436457	+	0.899725i	$0.356233\pi$
$90$		0			0
$91$	−0.622216			−0.0652259
$92$		0			0
$93$			0.622216i			0.0645208i
$94$		0			0
$95$	−0.193576	−	1.37778i	−0.0198605	−	0.141358i
$96$		0			0
$97$			4.23506i			0.430006i	0.976613	+	0.215003i	$0.0689761\pi$
$97$			4.23506i			0.430006i	−0.976613	+	0.215003i	$0.931024\pi$
$98$		0			0
$99$	−3.80642			−0.382560
$100$		0			0
$101$	18.7239			1.86310			0.931550	−	0.363613i	$-0.118457\pi$
$101$	18.7239			1.86310			0.931550	+	0.363613i	$0.118457\pi$
$102$		0			0
$103$		−	0.857279i		−	0.0844702i	−0.999108	−	0.0422351i	$-0.986552\pi$
$103$		−	0.857279i		−	0.0844702i	0.999108	−	0.0422351i	$-0.0134479\pi$
$104$		0			0
$105$	−0.311108	−	2.21432i	−0.0303610	−	0.216095i
$106$		0			0
$107$			11.0923i			1.07234i	0.844111	+	0.536169i	$0.180129\pi$
$107$			11.0923i			1.07234i	−0.844111	+	0.536169i	$0.819871\pi$
$108$		0			0
$109$	−5.61285			−0.537613			−0.268807	−	0.963194i	$-0.586629\pi$
$109$	−5.61285			−0.537613			−0.268807	+	0.963194i	$0.586629\pi$
$110$		0			0
$111$	1.24443			0.118116
$112$		0			0
$113$		−	16.2351i		−	1.52727i	−0.645650	−	0.763633i	$-0.723414\pi$
$113$		−	16.2351i		−	1.52727i	0.645650	−	0.763633i	$-0.276586\pi$
$114$		0			0
$115$	5.80642	−	0.815792i	0.541452	−	0.0760730i
$116$		0			0
$117$			0.622216i			0.0575239i
$118$		0			0
$119$	−4.42864			−0.405973
$120$		0			0
$121$	3.48886			0.317169
$122$		0			0
$123$			4.62222i			0.416771i
$124$		0			0
$125$	4.54617	+	10.2143i	0.406622	+	0.913597i
$126$		0			0
$127$		−	15.3461i		−	1.36175i	−0.732400	−	0.680875i	$-0.761600\pi$
$127$		−	15.3461i		−	1.36175i	0.732400	−	0.680875i	$-0.238400\pi$
$128$		0			0
$129$	4.85728			0.427660
$130$		0			0
$131$	4.00000			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$132$		0			0
$133$		−	0.622216i		−	0.0539529i
$134$		0			0
$135$	−2.21432	+	0.311108i	−0.190578	+	0.0267759i
$136$		0			0
$137$		−	17.0923i		−	1.46030i	−0.683288	−	0.730149i	$-0.739451\pi$
$137$		−	17.0923i		−	1.46030i	0.683288	−	0.730149i	$-0.260549\pi$
$138$		0			0
$139$	−13.4795			−1.14332			−0.571658	−	0.820492i	$-0.693700\pi$
$139$	−13.4795			−1.14332			−0.571658	+	0.820492i	$0.693700\pi$
$140$		0			0
$141$	11.6128			0.977978
$142$		0			0
$143$		−	2.36842i		−	0.198057i
$144$		0			0
$145$	2.99063	+	21.2859i	0.248358	+	1.76770i
$146$		0			0
$147$		−	1.00000i		−	0.0824786i
$148$		0			0
$149$	9.34614			0.765666			0.382833	−	0.923818i	$-0.374949\pi$
$149$	9.34614			0.765666			0.382833	+	0.923818i	$0.374949\pi$
$150$		0			0
$151$	7.14272			0.581266			0.290633	−	0.956835i	$-0.406134\pi$
$151$	7.14272			0.581266			0.290633	+	0.956835i	$0.406134\pi$
$152$		0			0
$153$			4.42864i			0.358034i
$154$		0			0
$155$	−0.193576	−	1.37778i	−0.0155484	−	0.110666i
$156$		0			0
$157$		−	6.99063i		−	0.557913i	−0.960304	−	0.278957i	$-0.910011\pi$
$157$		−	6.99063i		−	0.557913i	0.960304	−	0.278957i	$-0.0899886\pi$
$158$		0			0
$159$	13.4795			1.06899
$160$		0			0
$161$	2.62222			0.206660
$162$		0			0
$163$			15.6128i			1.22289i	0.791286	+	0.611446i	$0.209412\pi$
$163$			15.6128i			1.22289i	−0.791286	+	0.611446i	$0.790588\pi$
$164$		0			0
$165$	8.42864	−	1.18421i	0.656169	−	0.0921905i
$166$		0			0
$167$			1.51114i			0.116935i	0.998289	+	0.0584677i	$0.0186215\pi$
$167$			1.51114i			0.116935i	−0.998289	+	0.0584677i	$0.981379\pi$
$168$		0			0
$169$	12.6128			0.970219
$170$		0			0
$171$	−0.622216			−0.0475820
$172$		0			0
$173$			6.53035i			0.496493i	0.968697	+	0.248247i	$0.0798543\pi$
$173$			6.53035i			0.496493i	−0.968697	+	0.248247i	$0.920146\pi$
$174$		0			0
$175$	1.37778	+	4.80642i	0.104151	+	0.363331i
$176$		0			0
$177$			11.6128i			0.872875i
$178$		0			0
$179$	−6.29529			−0.470532			−0.235266	−	0.971931i	$-0.575596\pi$
$179$	−6.29529			−0.470532			−0.235266	+	0.971931i	$0.575596\pi$
$180$		0			0
$181$	−6.85728			−0.509698			−0.254849	−	0.966981i	$-0.582026\pi$
$181$	−6.85728			−0.509698			−0.254849	+	0.966981i	$0.582026\pi$
$182$		0			0
$183$		−	8.10171i		−	0.598896i
$184$		0			0
$185$	−2.75557	+	0.387152i	−0.202593	+	0.0284640i
$186$		0			0
$187$		−	16.8573i		−	1.23273i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	10.5620			0.764239			0.382119	−	0.924113i	$-0.375194\pi$
$191$	10.5620			0.764239			0.382119	+	0.924113i	$0.375194\pi$
$192$		0			0
$193$		−	5.24443i		−	0.377502i	−0.982025	−	0.188751i	$-0.939556\pi$
$193$		−	5.24443i		−	0.377502i	0.982025	−	0.188751i	$-0.0604440\pi$
$194$		0			0
$195$	−0.193576	−	1.37778i	−0.0138623	−	0.0986652i
$196$		0			0
$197$			17.7462i			1.26436i	0.774820	+	0.632182i	$0.217841\pi$
$197$			17.7462i			1.26436i	−0.774820	+	0.632182i	$0.782159\pi$
$198$		0			0
$199$	−20.2351			−1.43443			−0.717213	−	0.696854i	$-0.754582\pi$
$199$	−20.2351			−1.43443			−0.717213	+	0.696854i	$0.754582\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			9.61285i			0.674690i
$204$		0			0
$205$	−1.43801	−	10.2351i	−0.100435	−	0.714848i
$206$		0			0
$207$		−	2.62222i		−	0.182257i
$208$		0			0
$209$	2.36842			0.163827
$210$		0			0
$211$	−21.3274			−1.46824			−0.734120	−	0.679020i	$-0.762405\pi$
$211$	−21.3274			−1.46824			−0.734120	+	0.679020i	$0.762405\pi$
$212$		0			0
$213$		−	2.56199i		−	0.175545i
$214$		0			0
$215$	−10.7556	+	1.51114i	−0.733524	+	0.103059i
$216$		0			0
$217$		−	0.622216i		−	0.0422387i
$218$		0			0
$219$	10.9906			0.742678
$220$		0			0
$221$	−2.75557			−0.185360
$222$		0			0
$223$		−	9.71456i		−	0.650535i	−0.945622	−	0.325267i	$-0.894546\pi$
$223$		−	9.71456i		−	0.650535i	0.945622	−	0.325267i	$-0.105454\pi$
$224$		0			0
$225$	4.80642	−	1.37778i	0.320428	−	0.0918523i
$226$		0			0
$227$			11.3461i			0.753070i	0.926402	+	0.376535i	$0.122885\pi$
$227$			11.3461i			0.753070i	−0.926402	+	0.376535i	$0.877115\pi$
$228$		0			0
$229$	−1.34614			−0.0889555			−0.0444778	−	0.999010i	$-0.514162\pi$
$229$	−1.34614			−0.0889555			−0.0444778	+	0.999010i	$0.514162\pi$
$230$		0			0
$231$	3.80642			0.250444
$232$		0			0
$233$		−	15.3778i		−	1.00743i	−0.863869	−	0.503716i	$-0.831966\pi$
$233$		−	15.3778i		−	1.00743i	0.863869	−	0.503716i	$-0.168034\pi$
$234$		0			0
$235$	−25.7146	+	3.61285i	−1.67743	+	0.235676i
$236$		0			0
$237$		−	6.75557i		−	0.438821i
$238$		0			0
$239$	7.53972			0.487704			0.243852	−	0.969812i	$-0.421589\pi$
$239$	7.53972			0.487704			0.243852	+	0.969812i	$0.421589\pi$
$240$		0			0
$241$	23.9813			1.54477			0.772385	−	0.635155i	$-0.219064\pi$
$241$	23.9813			1.54477			0.772385	+	0.635155i	$0.219064\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	0.311108	+	2.21432i	0.0198759	+	0.141468i
$246$		0			0
$247$		−	0.387152i		−	0.0246339i
$248$		0			0
$249$	−11.6128			−0.735934
$250$		0			0
$251$	−14.1017			−0.890092			−0.445046	−	0.895508i	$-0.646813\pi$
$251$	−14.1017			−0.890092			−0.445046	+	0.895508i	$0.646813\pi$
$252$		0			0
$253$			9.98126i			0.627517i
$254$		0			0
$255$	−1.37778	−	9.80642i	−0.0862802	−	0.614102i
$256$		0			0
$257$			17.0192i			1.06163i	0.847488	+	0.530815i	$0.178114\pi$
$257$			17.0192i			1.06163i	−0.847488	+	0.530815i	$0.821886\pi$
$258$		0			0
$259$	−1.24443			−0.0773252
$260$		0			0
$261$	9.61285			0.595020
$262$		0			0
$263$			12.6035i			0.777164i	0.921414	+	0.388582i	$0.127035\pi$
$263$			12.6035i			0.777164i	−0.921414	+	0.388582i	$0.872965\pi$
$264$		0			0
$265$	−29.8479	+	4.19358i	−1.83354	+	0.257609i
$266$		0			0
$267$			8.23506i			0.503978i
$268$		0			0
$269$	−3.76494			−0.229552			−0.114776	−	0.993391i	$-0.536615\pi$
$269$	−3.76494			−0.229552			−0.114776	+	0.993391i	$0.536615\pi$
$270$		0			0
$271$	17.8666			1.08532			0.542661	−	0.839952i	$-0.317417\pi$
$271$	17.8666			1.08532			0.542661	+	0.839952i	$0.317417\pi$
$272$		0			0
$273$		−	0.622216i		−	0.0376582i
$274$		0			0
$275$	−18.2953	+	5.24443i	−1.10325	+	0.316251i
$276$		0			0
$277$		−	1.24443i		−	0.0747706i	−0.999301	−	0.0373853i	$-0.988097\pi$
$277$		−	1.24443i		−	0.0747706i	0.999301	−	0.0373853i	$-0.0119029\pi$
$278$		0			0
$279$	−0.622216			−0.0372511
$280$		0			0
$281$	−8.95899			−0.534448			−0.267224	−	0.963634i	$-0.586106\pi$
$281$	−8.95899			−0.534448			−0.267224	+	0.963634i	$0.586106\pi$
$282$		0			0
$283$			30.5718i			1.81731i	0.417551	+	0.908654i	$0.362889\pi$
$283$			30.5718i			1.81731i	−0.417551	+	0.908654i	$0.637111\pi$
$284$		0			0
$285$	1.37778	−	0.193576i	0.0816129	−	0.0114665i
$286$		0			0
$287$		−	4.62222i		−	0.272841i
$288$		0			0
$289$	−2.61285			−0.153697
$290$		0			0
$291$	−4.23506			−0.248264
$292$		0			0
$293$			5.67307i			0.331424i	0.986174	+	0.165712i	$0.0529923\pi$
$293$			5.67307i			0.331424i	−0.986174	+	0.165712i	$0.947008\pi$
$294$		0			0
$295$	−3.61285	−	25.7146i	−0.210348	−	1.49716i
$296$		0			0
$297$		−	3.80642i		−	0.220871i
$298$		0			0
$299$	1.63158			0.0943569
$300$		0			0
$301$	−4.85728			−0.279969
$302$		0			0
$303$			18.7239i			1.07566i
$304$		0			0
$305$	2.52051	+	17.9398i	0.144324	+	1.02723i
$306$		0			0
$307$			4.85728i			0.277220i	0.990347	+	0.138610i	$0.0442634\pi$
$307$			4.85728i			0.277220i	−0.990347	+	0.138610i	$0.955737\pi$
$308$		0			0
$309$	0.857279			0.0487689
$310$		0			0
$311$	34.5718			1.96039			0.980195	−	0.198037i	$-0.0634567\pi$
$311$	34.5718			1.96039			0.980195	+	0.198037i	$0.0634567\pi$
$312$		0			0
$313$			6.33677i			0.358176i	0.983833	+	0.179088i	$0.0573146\pi$
$313$			6.33677i			0.358176i	−0.983833	+	0.179088i	$0.942685\pi$
$314$		0			0
$315$	2.21432	−	0.311108i	0.124763	−	0.0175289i
$316$		0			0
$317$			15.9684i			0.896872i	0.893815	+	0.448436i	$0.148019\pi$
$317$			15.9684i			0.896872i	−0.893815	+	0.448436i	$0.851981\pi$
$318$		0			0
$319$	−36.5906			−2.04868
$320$		0			0
$321$	−11.0923			−0.619114
$322$		0			0
$323$		−	2.75557i		−	0.153324i
$324$		0			0
$325$	0.857279	+	2.99063i	0.0475533	+	0.165890i
$326$		0			0
$327$		−	5.61285i		−	0.310391i
$328$		0			0
$329$	−11.6128			−0.640237
$330$		0			0
$331$	27.6128			1.51774			0.758870	−	0.651243i	$-0.225752\pi$
$331$	27.6128			1.51774			0.758870	+	0.651243i	$0.225752\pi$
$332$		0			0
$333$			1.24443i			0.0681944i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		0			0
$339$	16.2351			0.881768
$340$		0			0
$341$	2.36842			0.128257
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	0.815792	+	5.80642i	0.0439208	+	0.312607i
$346$		0			0
$347$			11.8666i			0.637035i	0.947917	+	0.318517i	$0.103185\pi$
$347$			11.8666i			0.637035i	−0.947917	+	0.318517i	$0.896815\pi$
$348$		0			0
$349$	−21.8163			−1.16780			−0.583899	−	0.811826i	$-0.698474\pi$
$349$	−21.8163			−1.16780			−0.583899	+	0.811826i	$0.698474\pi$
$350$		0			0
$351$	−0.622216			−0.0332114
$352$		0			0
$353$			2.79706i			0.148872i	0.997226	+	0.0744361i	$0.0237157\pi$
$353$			2.79706i			0.148872i	−0.997226	+	0.0744361i	$0.976284\pi$
$354$		0			0
$355$	0.797056	+	5.67307i	0.0423033	+	0.301095i
$356$		0			0
$357$		−	4.42864i		−	0.234388i
$358$		0			0
$359$	−13.0509			−0.688798			−0.344399	−	0.938823i	$-0.611917\pi$
$359$	−13.0509			−0.688798			−0.344399	+	0.938823i	$0.611917\pi$
$360$		0			0
$361$	−18.6128			−0.979624
$362$		0			0
$363$			3.48886i			0.183118i
$364$		0			0
$365$	−24.3368	+	3.41927i	−1.27384	+	0.178973i
$366$		0			0
$367$			10.4889i			0.547514i	0.961799	+	0.273757i	$0.0882664\pi$
$367$			10.4889i			0.547514i	−0.961799	+	0.273757i	$0.911734\pi$
$368$		0			0
$369$	−4.62222			−0.240623
$370$		0			0
$371$	−13.4795			−0.699820
$372$		0			0
$373$		−	30.1847i		−	1.56290i	−0.623966	−	0.781452i	$-0.714479\pi$
$373$		−	30.1847i		−	1.56290i	0.623966	−	0.781452i	$-0.285521\pi$
$374$		0			0
$375$	−10.2143	+	4.54617i	−0.527465	+	0.234763i
$376$		0			0
$377$			5.98126i			0.308051i
$378$		0			0
$379$	−12.8573			−0.660434			−0.330217	−	0.943905i	$-0.607122\pi$
$379$	−12.8573			−0.660434			−0.330217	+	0.943905i	$0.607122\pi$
$380$		0			0
$381$	15.3461			0.786207
$382$		0			0
$383$		−	24.4701i		−	1.25037i	−0.780479	−	0.625183i	$-0.785025\pi$
$383$		−	24.4701i		−	1.25037i	0.780479	−	0.625183i	$-0.214975\pi$
$384$		0			0
$385$	−8.42864	+	1.18421i	−0.429563	+	0.0603528i
$386$		0			0
$387$			4.85728i			0.246909i
$388$		0			0
$389$	1.61285			0.0817746			0.0408873	−	0.999164i	$-0.486982\pi$
$389$	1.61285			0.0817746			0.0408873	+	0.999164i	$0.486982\pi$
$390$		0			0
$391$	11.6128			0.587287
$392$		0			0
$393$			4.00000i			0.201773i
$394$		0			0
$395$	2.10171	+	14.9590i	0.105749	+	0.752668i
$396$		0			0
$397$			22.2163i			1.11501i	0.830175	+	0.557503i	$0.188240\pi$
$397$			22.2163i			1.11501i	−0.830175	+	0.557503i	$0.811760\pi$
$398$		0			0
$399$	0.622216			0.0311497
$400$		0			0
$401$	19.9813			0.997817			0.498908	−	0.866655i	$-0.333734\pi$
$401$	19.9813			0.997817			0.498908	+	0.866655i	$0.333734\pi$
$402$		0			0
$403$		−	0.387152i		−	0.0192854i
$404$		0			0
$405$	−0.311108	−	2.21432i	−0.0154591	−	0.110030i
$406$		0			0
$407$		−	4.73683i		−	0.234796i
$408$		0			0
$409$	5.73329			0.283493			0.141747	−	0.989903i	$-0.454728\pi$
$409$	5.73329			0.283493			0.141747	+	0.989903i	$0.454728\pi$
$410$		0			0
$411$	17.0923			0.843103
$412$		0			0
$413$		−	11.6128i		−	0.571431i
$414$		0			0
$415$	25.7146	−	3.61285i	1.26228	−	0.177348i
$416$		0			0
$417$		−	13.4795i		−	0.660094i
$418$		0			0
$419$	26.3684			1.28818			0.644091	−	0.764949i	$-0.277236\pi$
$419$	26.3684			1.28818			0.644091	+	0.764949i	$0.277236\pi$
$420$		0			0
$421$	−19.3274			−0.941960			−0.470980	−	0.882144i	$-0.656100\pi$
$421$	−19.3274			−0.941960			−0.470980	+	0.882144i	$0.656100\pi$
$422$		0			0
$423$			11.6128i			0.564636i
$424$		0			0
$425$	6.10171	+	21.2859i	0.295976	+	1.03252i
$426$		0			0
$427$			8.10171i			0.392069i
$428$		0			0
$429$	2.36842			0.114348
$430$		0			0
$431$	26.9491			1.29809			0.649047	−	0.760748i	$-0.275168\pi$
$431$	26.9491			1.29809			0.649047	+	0.760748i	$0.275168\pi$
$432$		0			0
$433$			2.13335i			0.102522i	0.998685	+	0.0512612i	$0.0163241\pi$
$433$			2.13335i			0.102522i	−0.998685	+	0.0512612i	$0.983676\pi$
$434$		0			0
$435$	−21.2859	+	2.99063i	−1.02058	+	0.143390i
$436$		0			0
$437$			1.63158i			0.0780492i
$438$		0			0
$439$	−10.5205			−0.502116			−0.251058	−	0.967972i	$-0.580779\pi$
$439$	−10.5205			−0.502116			−0.251058	+	0.967972i	$0.580779\pi$
$440$		0			0
$441$	1.00000			0.0476190
$442$		0			0
$443$		−	6.88892i		−	0.327303i	−0.986518	−	0.163651i	$-0.947673\pi$
$443$		−	6.88892i		−	0.327303i	0.986518	−	0.163651i	$-0.0523272\pi$
$444$		0			0
$445$	−2.56199	−	18.2351i	−0.121450	−	0.864425i
$446$		0			0
$447$			9.34614i			0.442057i
$448$		0			0
$449$	−39.9180			−1.88385			−0.941923	−	0.335829i	$-0.890984\pi$
$449$	−39.9180			−1.88385			−0.941923	+	0.335829i	$0.890984\pi$
$450$		0			0
$451$	17.5941			0.828474
$452$		0			0
$453$			7.14272i			0.335594i
$454$		0			0
$455$	0.193576	+	1.37778i	0.00907499	+	0.0645915i
$456$		0			0
$457$		−	8.47013i		−	0.396216i	−0.980180	−	0.198108i	$-0.936520\pi$
$457$		−	8.47013i		−	0.396216i	0.980180	−	0.198108i	$-0.0634796\pi$
$458$		0			0
$459$	−4.42864			−0.206711
$460$		0			0
$461$	40.1146			1.86832			0.934162	−	0.356849i	$-0.116149\pi$
$461$	40.1146			1.86832			0.934162	+	0.356849i	$0.116149\pi$
$462$		0			0
$463$			33.5941i			1.56125i	0.624999	+	0.780625i	$0.285099\pi$
$463$			33.5941i			1.56125i	−0.624999	+	0.780625i	$0.714901\pi$
$464$		0			0
$465$	1.37778	−	0.193576i	0.0638932	−	0.00897688i
$466$		0			0
$467$		−	11.3461i		−	0.525037i	−0.964927	−	0.262518i	$-0.915447\pi$
$467$		−	11.3461i		−	0.525037i	0.964927	−	0.262518i	$-0.0845530\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	6.99063			0.322111
$472$		0			0
$473$		−	18.4889i		−	0.850119i
$474$		0			0
$475$	−2.99063	+	0.857279i	−0.137220	+	0.0393347i
$476$		0			0
$477$			13.4795i			0.617184i
$478$		0			0
$479$	36.2864			1.65797			0.828984	−	0.559273i	$-0.188919\pi$
$479$	36.2864			1.65797			0.828984	+	0.559273i	$0.188919\pi$
$480$		0			0
$481$	−0.774305			−0.0353053
$482$		0			0
$483$			2.62222i			0.119315i
$484$		0			0
$485$	9.37778	−	1.31756i	0.425823	−	0.0598274i
$486$		0			0
$487$			38.8385i			1.75994i	0.475027	+	0.879971i	$0.342438\pi$
$487$			38.8385i			1.75994i	−0.475027	+	0.879971i	$0.657562\pi$
$488$		0			0
$489$	−15.6128			−0.706037
$490$		0			0
$491$	28.7467			1.29732			0.648660	−	0.761079i	$-0.275330\pi$
$491$	28.7467			1.29732			0.648660	+	0.761079i	$0.275330\pi$
$492$		0			0
$493$			42.5718i			1.91734i
$494$		0			0
$495$	1.18421	+	8.42864i	0.0532262	+	0.378839i
$496$		0			0
$497$			2.56199i			0.114921i
$498$		0			0
$499$	−5.63158			−0.252104			−0.126052	−	0.992024i	$-0.540231\pi$
$499$	−5.63158			−0.252104			−0.126052	+	0.992024i	$0.540231\pi$
$500$		0			0
$501$	−1.51114			−0.0675126
$502$		0			0
$503$		−	34.9590i		−	1.55874i	−0.626561	−	0.779372i	$-0.715538\pi$
$503$		−	34.9590i		−	1.55874i	0.626561	−	0.779372i	$-0.284462\pi$
$504$		0			0
$505$	−5.82516	−	41.4608i	−0.259216	−	1.84498i
$506$		0			0
$507$			12.6128i			0.560156i
$508$		0			0
$509$	10.9906			0.487151			0.243576	−	0.969882i	$-0.421680\pi$
$509$	10.9906			0.487151			0.243576	+	0.969882i	$0.421680\pi$
$510$		0			0
$511$	−10.9906			−0.486197
$512$		0			0
$513$		−	0.622216i		−	0.0274715i
$514$		0			0
$515$	−1.89829	+	0.266706i	−0.0836486	+	0.0117525i
$516$		0			0
$517$		−	44.2034i		−	1.94406i
$518$		0			0
$519$	−6.53035			−0.286651
$520$		0			0
$521$	6.90766			0.302630			0.151315	−	0.988486i	$-0.451649\pi$
$521$	6.90766			0.302630			0.151315	+	0.988486i	$0.451649\pi$
$522$		0			0
$523$			37.7146i			1.64914i	0.565758	+	0.824571i	$0.308584\pi$
$523$			37.7146i			1.64914i	−0.565758	+	0.824571i	$0.691416\pi$
$524$		0			0
$525$	−4.80642	+	1.37778i	−0.209770	+	0.0601314i
$526$		0			0
$527$		−	2.75557i		−	0.120034i
$528$		0			0
$529$	16.1240			0.701043
$530$		0			0
$531$	−11.6128			−0.503955
$532$		0			0
$533$		−	2.87601i		−	0.124574i
$534$		0			0
$535$	24.5620	−	3.45091i	1.06191	−	0.149196i
$536$		0			0
$537$		−	6.29529i		−	0.271662i
$538$		0			0
$539$	−3.80642			−0.163954
$540$		0			0
$541$	−3.12399			−0.134311			−0.0671553	−	0.997743i	$-0.521392\pi$
$541$	−3.12399			−0.134311			−0.0671553	+	0.997743i	$0.521392\pi$
$542$		0			0
$543$		−	6.85728i		−	0.294274i
$544$		0			0
$545$	1.74620	+	12.4286i	0.0747990	+	0.532384i
$546$		0			0
$547$			5.51114i			0.235639i	0.993035	+	0.117820i	$0.0375905\pi$
$547$			5.51114i			0.235639i	−0.993035	+	0.117820i	$0.962410\pi$
$548$		0			0
$549$	8.10171			0.345773
$550$		0			0
$551$	−5.98126			−0.254810
$552$		0			0
$553$			6.75557i			0.287276i
$554$		0			0
$555$	−0.387152	−	2.75557i	−0.0164337	−	0.116967i
$556$		0			0
$557$			36.7052i			1.55525i	0.628729	+	0.777624i	$0.283575\pi$
$557$			36.7052i			1.55525i	−0.628729	+	0.777624i	$0.716425\pi$
$558$		0			0
$559$	−3.02227			−0.127829
$560$		0			0
$561$	16.8573			0.711715
$562$		0			0
$563$		−	27.4924i		−	1.15867i	−0.815091	−	0.579333i	$-0.803313\pi$
$563$		−	27.4924i		−	1.15867i	0.815091	−	0.579333i	$-0.196687\pi$
$564$		0			0
$565$	−35.9496	+	5.05086i	−1.51241	+	0.212491i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−23.2444			−0.974457			−0.487229	−	0.873274i	$-0.661992\pi$
$569$	−23.2444			−0.974457			−0.487229	+	0.873274i	$0.661992\pi$
$570$		0			0
$571$	25.5111			1.06761			0.533804	−	0.845608i	$-0.320762\pi$
$571$	25.5111			1.06761			0.533804	+	0.845608i	$0.320762\pi$
$572$		0			0
$573$			10.5620i			0.441234i
$574$		0			0
$575$	−3.61285	−	12.6035i	−0.150666	−	0.525601i
$576$		0			0
$577$			26.0701i			1.08531i	0.839955	+	0.542656i	$0.182581\pi$
$577$			26.0701i			1.08531i	−0.839955	+	0.542656i	$0.817419\pi$
$578$		0			0
$579$	5.24443			0.217951
$580$		0			0
$581$	11.6128			0.481782
$582$		0			0
$583$		−	51.3087i		−	2.12499i
$584$		0			0
$585$	1.37778	−	0.193576i	0.0569644	−	0.00800339i
$586$		0			0
$587$			12.2667i			0.506301i	0.967427	+	0.253151i	$0.0814668\pi$
$587$			12.2667i			0.506301i	−0.967427	+	0.253151i	$0.918533\pi$
$588$		0			0
$589$	0.387152			0.0159523
$590$		0			0
$591$	−17.7462			−0.729981
$592$		0			0
$593$			14.9175i			0.612588i	0.951937	+	0.306294i	$0.0990891\pi$
$593$			14.9175i			0.612588i	−0.951937	+	0.306294i	$0.900911\pi$
$594$		0			0
$595$	1.37778	+	9.80642i	0.0564837	+	0.402024i
$596$		0			0
$597$		−	20.2351i		−	0.828166i
$598$		0			0
$599$	−26.5620			−1.08529			−0.542647	−	0.839961i	$-0.682578\pi$
$599$	−26.5620			−1.08529			−0.542647	+	0.839961i	$0.682578\pi$
$600$		0			0
$601$	39.7146			1.61999			0.809995	−	0.586436i	$-0.199470\pi$
$601$	39.7146			1.61999			0.809995	+	0.586436i	$0.199470\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.08541	−	7.72546i	−0.0441283	−	0.314084i
$606$		0			0
$607$		−	6.28544i		−	0.255118i	−0.991831	−	0.127559i	$-0.959286\pi$
$607$		−	6.28544i		−	0.255118i	0.991831	−	0.127559i	$-0.0407143\pi$
$608$		0			0
$609$	−9.61285			−0.389532
$610$		0			0
$611$	−7.22570			−0.292320
$612$		0			0
$613$			45.7146i			1.84639i	0.384328	+	0.923197i	$0.374433\pi$
$613$			45.7146i			1.84639i	−0.384328	+	0.923197i	$0.625567\pi$
$614$		0			0
$615$	10.2351	−	1.43801i	0.412718	−	0.0579861i
$616$		0			0
$617$			4.88892i			0.196821i	0.995146	+	0.0984103i	$0.0313758\pi$
$617$			4.88892i			0.196821i	−0.995146	+	0.0984103i	$0.968624\pi$
$618$		0			0
$619$	12.2351			0.491769			0.245884	−	0.969299i	$-0.420922\pi$
$619$	12.2351			0.491769			0.245884	+	0.969299i	$0.420922\pi$
$620$		0			0
$621$	2.62222			0.105226
$622$		0			0
$623$		−	8.23506i		−	0.329931i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$			2.36842i			0.0945854i
$628$		0			0
$629$	−5.51114			−0.219743
$630$		0			0
$631$	−1.24443			−0.0495400			−0.0247700	−	0.999693i	$-0.507885\pi$
$631$	−1.24443			−0.0495400			−0.0247700	+	0.999693i	$0.507885\pi$
$632$		0			0
$633$		−	21.3274i		−	0.847688i
$634$		0			0
$635$	−33.9813	+	4.77430i	−1.34851	+	0.189462i
$636$		0			0
$637$			0.622216i			0.0246531i
$638$		0			0
$639$	2.56199			0.101351
$640$		0			0
$641$	−48.1847			−1.90318			−0.951590	−	0.307369i	$-0.900551\pi$
$641$	−48.1847			−1.90318			−0.951590	+	0.307369i	$0.900551\pi$
$642$		0			0
$643$			4.85728i			0.191552i	0.995403	+	0.0957762i	$0.0305333\pi$
$643$			4.85728i			0.191552i	−0.995403	+	0.0957762i	$0.969467\pi$
$644$		0			0
$645$	−1.51114	−	10.7556i	−0.0595010	−	0.423500i
$646$		0			0
$647$		−	0.203420i		−	0.00799728i	−0.999992	−	0.00399864i	$-0.998727\pi$
$647$		−	0.203420i		−	0.00799728i	0.999992	−	0.00399864i	$-0.00127281\pi$
$648$		0			0
$649$	44.2034			1.73514
$650$		0			0
$651$	0.622216			0.0243866
$652$		0			0
$653$			27.3145i			1.06890i	0.845200	+	0.534449i	$0.179481\pi$
$653$			27.3145i			1.06890i	−0.845200	+	0.534449i	$0.820519\pi$
$654$		0			0
$655$	−1.24443	−	8.85728i	−0.0486240	−	0.346083i
$656$		0			0
$657$			10.9906i			0.428785i
$658$		0			0
$659$	33.3176			1.29787			0.648934	−	0.760845i	$-0.275215\pi$
$659$	33.3176			1.29787			0.648934	+	0.760845i	$0.275215\pi$
$660$		0			0
$661$	14.5906			0.567508			0.283754	−	0.958897i	$-0.408420\pi$
$661$	14.5906			0.567508			0.283754	+	0.958897i	$0.408420\pi$
$662$		0			0
$663$		−	2.75557i		−	0.107017i
$664$		0			0
$665$	−1.37778	+	0.193576i	−0.0534282	+	0.00750656i
$666$		0			0
$667$		−	25.2070i		−	0.976017i
$668$		0			0
$669$	9.71456			0.375587
$670$		0			0
$671$	−30.8385			−1.19051
$672$		0			0
$673$		−	4.53341i		−	0.174750i	−0.996175	−	0.0873751i	$-0.972152\pi$
$673$		−	4.53341i		−	0.174750i	0.996175	−	0.0873751i	$-0.0278479\pi$
$674$		0			0
$675$	1.37778	+	4.80642i	0.0530309	+	0.184999i
$676$		0			0
$677$		−	27.2672i		−	1.04796i	−0.851730	−	0.523981i	$-0.824446\pi$
$677$		−	27.2672i		−	1.04796i	0.851730	−	0.523981i	$-0.175554\pi$
$678$		0			0
$679$	4.23506			0.162527
$680$		0			0
$681$	−11.3461			−0.434785
$682$		0			0
$683$		−	29.5812i		−	1.13189i	−0.824442	−	0.565947i	$-0.808511\pi$
$683$		−	29.5812i		−	1.13189i	0.824442	−	0.565947i	$-0.191489\pi$
$684$		0			0
$685$	−37.8479	+	5.31756i	−1.44609	+	0.203174i
$686$		0			0
$687$		−	1.34614i		−	0.0513585i
$688$		0			0
$689$	−8.38715			−0.319525
$690$		0			0
$691$	−2.99063			−0.113769			−0.0568845	−	0.998381i	$-0.518117\pi$
$691$	−2.99063			−0.113769			−0.0568845	+	0.998381i	$0.518117\pi$
$692$		0			0
$693$			3.80642i			0.144594i
$694$		0			0
$695$	4.19358	+	29.8479i	0.159071	+	1.13220i
$696$		0			0
$697$		−	20.4701i		−	0.775361i
$698$		0			0
$699$	15.3778			0.581641
$700$		0			0
$701$	−41.0420			−1.55013			−0.775067	−	0.631879i	$-0.782284\pi$
$701$	−41.0420			−1.55013			−0.775067	+	0.631879i	$0.782284\pi$
$702$		0			0
$703$		−	0.774305i		−	0.0292035i
$704$		0			0
$705$	−3.61285	−	25.7146i	−0.136068	−	0.968466i
$706$		0			0
$707$		−	18.7239i		−	0.704186i
$708$		0			0
$709$	41.4291			1.55590			0.777952	−	0.628324i	$-0.216259\pi$
$709$	41.4291			1.55590			0.777952	+	0.628324i	$0.216259\pi$
$710$		0			0
$711$	6.75557			0.253354
$712$		0			0
$713$			1.63158i			0.0611033i
$714$		0			0
$715$	−5.24443	+	0.736833i	−0.196131	+	0.0275560i
$716$		0			0
$717$			7.53972i			0.281576i
$718$		0			0
$719$	−18.9590			−0.707051			−0.353525	−	0.935425i	$-0.615017\pi$
$719$	−18.9590			−0.707051			−0.353525	+	0.935425i	$0.615017\pi$
$720$		0			0
$721$	−0.857279			−0.0319267
$722$		0			0
$723$			23.9813i			0.891873i
$724$		0			0
$725$	46.2034	−	13.2444i	1.71595	−	0.491886i
$726$		0			0
$727$		−	41.7975i		−	1.55018i	−0.631848	−	0.775092i	$-0.717703\pi$
$727$		−	41.7975i		−	1.55018i	0.631848	−	0.775092i	$-0.282297\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−21.5111			−0.795618
$732$		0			0
$733$		−	15.3145i		−	0.565654i	−0.959171	−	0.282827i	$-0.908728\pi$
$733$		−	15.3145i		−	0.565654i	0.959171	−	0.282827i	$-0.0912722\pi$
$734$		0			0
$735$	−2.21432	+	0.311108i	−0.0816764	+	0.0114754i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−32.7368			−1.20424			−0.602122	−	0.798404i	$-0.705678\pi$
$739$	−32.7368			−1.20424			−0.602122	+	0.798404i	$0.705678\pi$
$740$		0			0
$741$	0.387152			0.0142224
$742$		0			0
$743$		−	37.3778i		−	1.37126i	−0.727951	−	0.685629i	$-0.759527\pi$
$743$		−	37.3778i		−	1.37126i	0.727951	−	0.685629i	$-0.240473\pi$
$744$		0			0
$745$	−2.90766	−	20.6953i	−0.106528	−	0.758219i
$746$		0			0
$747$		−	11.6128i		−	0.424892i
$748$		0			0
$749$	11.0923			0.405305
$750$		0			0
$751$	−20.3497			−0.742570			−0.371285	−	0.928519i	$-0.621083\pi$
$751$	−20.3497			−0.742570			−0.371285	+	0.928519i	$0.621083\pi$
$752$		0			0
$753$		−	14.1017i		−	0.513895i
$754$		0			0
$755$	−2.22216	−	15.8163i	−0.0808725	−	0.575613i
$756$		0			0
$757$		−	6.95899i		−	0.252929i	−0.991971	−	0.126464i	$-0.959637\pi$
$757$		−	6.95899i		−	0.252929i	0.991971	−	0.126464i	$-0.0403629\pi$
$758$		0			0
$759$	−9.98126			−0.362297
$760$		0			0
$761$	48.6419			1.76327			0.881634	−	0.471934i	$-0.156444\pi$
$761$	48.6419			1.76327			0.881634	+	0.471934i	$0.156444\pi$
$762$		0			0
$763$			5.61285i			0.203199i
$764$		0			0
$765$	9.80642	−	1.37778i	0.354552	−	0.0498139i
$766$		0			0
$767$		−	7.22570i		−	0.260905i
$768$		0			0
$769$	24.6923			0.890426			0.445213	−	0.895425i	$-0.353128\pi$
$769$	24.6923			0.890426			0.445213	+	0.895425i	$0.353128\pi$
$770$		0			0
$771$	−17.0192			−0.612932
$772$		0			0
$773$		−	36.0415i		−	1.29632i	−0.761503	−	0.648161i	$-0.775538\pi$
$773$		−	36.0415i		−	1.29632i	0.761503	−	0.648161i	$-0.224462\pi$
$774$		0			0
$775$	−2.99063	+	0.857279i	−0.107427	+	0.0307944i
$776$		0			0
$777$		−	1.24443i		−	0.0446437i
$778$		0			0
$779$	2.87601			0.103044
$780$		0			0
$781$	−9.75203			−0.348955
$782$		0			0
$783$			9.61285i			0.343535i
$784$		0			0
$785$	−15.4795	+	2.17484i	−0.552487	+	0.0776234i
$786$		0			0
$787$			32.2034i			1.14793i	0.818881	+	0.573964i	$0.194595\pi$
$787$			32.2034i			1.14793i	−0.818881	+	0.573964i	$0.805405\pi$
$788$		0			0
$789$	−12.6035			−0.448696
$790$		0			0
$791$	−16.2351			−0.577252
$792$		0			0
$793$			5.04101i			0.179012i
$794$		0			0
$795$	−4.19358	−	29.8479i	−0.148731	−	1.05860i
$796$		0			0
$797$		−	17.5526i		−	0.621746i	−0.950451	−	0.310873i	$-0.899379\pi$
$797$		−	17.5526i		−	0.621746i	0.950451	−	0.310873i	$-0.100621\pi$
$798$		0			0
$799$	−51.4291			−1.81943
$800$		0			0
$801$	−8.23506			−0.290972
$802$		0			0
$803$		−	41.8350i		−	1.47633i
$804$		0			0
$805$	−0.815792	−	5.80642i	−0.0287529	−	0.204650i
$806$		0			0
$807$		−	3.76494i		−	0.132532i
$808$		0			0
$809$	−46.1659			−1.62311			−0.811554	−	0.584277i	$-0.801378\pi$
$809$	−46.1659			−1.62311			−0.811554	+	0.584277i	$0.801378\pi$
$810$		0			0
$811$	−18.5205			−0.650343			−0.325171	−	0.945655i	$-0.605422\pi$
$811$	−18.5205			−0.650343			−0.325171	+	0.945655i	$0.605422\pi$
$812$		0			0
$813$			17.8666i			0.626611i
$814$		0			0
$815$	34.5718	−	4.85728i	1.21100	−	0.170143i
$816$		0			0
$817$		−	3.02227i		−	0.105736i
$818$		0			0
$819$	0.622216			0.0217420
$820$		0			0
$821$	43.0607			1.50283			0.751414	−	0.659831i	$-0.229372\pi$
$821$	43.0607			1.50283			0.751414	+	0.659831i	$0.229372\pi$
$822$		0			0
$823$			14.5718i			0.507942i	0.967212	+	0.253971i	$0.0817368\pi$
$823$			14.5718i			0.507942i	−0.967212	+	0.253971i	$0.918263\pi$
$824$		0			0
$825$	−5.24443	−	18.2953i	−0.182588	−	0.636960i
$826$		0			0
$827$			55.8292i			1.94137i	0.240354	+	0.970685i	$0.422736\pi$
$827$			55.8292i			1.94137i	−0.240354	+	0.970685i	$0.577264\pi$
$828$		0			0
$829$	37.3087			1.29578			0.647892	−	0.761732i	$-0.275651\pi$
$829$	37.3087			1.29578			0.647892	+	0.761732i	$0.275651\pi$
$830$		0			0
$831$	1.24443			0.0431688
$832$		0			0
$833$			4.42864i			0.153443i
$834$		0			0
$835$	3.34614	−	0.470127i	0.115798	−	0.0162694i
$836$		0			0
$837$		−	0.622216i		−	0.0215069i
$838$		0			0
$839$	−51.0420			−1.76216			−0.881082	−	0.472963i	$-0.843184\pi$
$839$	−51.0420			−1.76216			−0.881082	+	0.472963i	$0.843184\pi$
$840$		0			0
$841$	63.4068			2.18644
$842$		0			0
$843$		−	8.95899i		−	0.308564i
$844$		0			0
$845$	−3.92396	−	27.9289i	−0.134988	−	0.960783i
$846$		0			0
$847$		−	3.48886i		−	0.119879i
$848$		0			0
$849$	−30.5718			−1.04922
$850$		0			0
$851$	3.26317			0.111860
$852$		0			0
$853$		−	26.4197i		−	0.904595i	−0.891867	−	0.452297i	$-0.850605\pi$
$853$		−	26.4197i		−	0.904595i	0.891867	−	0.452297i	$-0.149395\pi$
$854$		0			0
$855$	0.193576	+	1.37778i	0.00662016	+	0.0471192i
$856$		0			0
$857$		−	0.161933i		−	0.00553154i	−0.999996	−	0.00276577i	$-0.999120\pi$
$857$		−	0.161933i		−	0.00553154i	0.999996	−	0.00276577i	$-0.000880372\pi$
$858$		0			0
$859$	51.3403			1.75171			0.875854	−	0.482575i	$-0.160299\pi$
$859$	51.3403			1.75171			0.875854	+	0.482575i	$0.160299\pi$
$860$		0			0
$861$	4.62222			0.157525
$862$		0			0
$863$			12.8702i			0.438106i	0.975713	+	0.219053i	$0.0702968\pi$
$863$			12.8702i			0.438106i	−0.975713	+	0.219053i	$0.929703\pi$
$864$		0			0
$865$	14.4603	−	2.03164i	0.491664	−	0.0690779i
$866$		0			0
$867$		−	2.61285i		−	0.0887370i
$868$		0			0
$869$	−25.7146			−0.872307
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	4.23506i		−	0.143335i
$874$		0			0
$875$	10.2143	−	4.54617i	0.345307	−	0.153689i
$876$		0			0
$877$		−	43.2257i		−	1.45963i	−0.683646	−	0.729814i	$-0.739607\pi$
$877$		−	43.2257i		−	1.45963i	0.683646	−	0.729814i	$-0.260393\pi$
$878$		0			0
$879$	−5.67307			−0.191348
$880$		0			0
$881$	−16.5018			−0.555959			−0.277979	−	0.960587i	$-0.589665\pi$
$881$	−16.5018			−0.555959			−0.277979	+	0.960587i	$0.589665\pi$
$882$		0			0
$883$		−	46.1847i		−	1.55424i	−0.629353	−	0.777119i	$-0.716680\pi$
$883$		−	46.1847i		−	1.55424i	0.629353	−	0.777119i	$-0.283320\pi$
$884$		0			0
$885$	25.7146	−	3.61285i	0.864385	−	0.121445i
$886$		0			0
$887$			37.3274i			1.25333i	0.779288	+	0.626666i	$0.215581\pi$
$887$			37.3274i			1.25333i	−0.779288	+	0.626666i	$0.784419\pi$
$888$		0			0
$889$	−15.3461			−0.514693
$890$		0			0
$891$	3.80642			0.127520
$892$		0			0
$893$		−	7.22570i		−	0.241799i
$894$		0			0
$895$	1.95851	+	13.9398i	0.0654659	+	0.465955i
$896$		0			0
$897$			1.63158i			0.0544770i
$898$		0			0
$899$	−5.98126			−0.199486
$900$		0			0
$901$	−59.6958			−1.98876
$902$		0			0
$903$		−	4.85728i		−	0.161640i
$904$		0			0
$905$	2.13335	+	15.1842i	0.0709151	+	0.504740i
$906$		0			0
$907$		−	24.6735i		−	0.819272i	−0.912249	−	0.409636i	$-0.865656\pi$
$907$		−	24.6735i		−	0.819272i	0.912249	−	0.409636i	$-0.134344\pi$
$908$		0			0
$909$	−18.7239			−0.621033
$910$		0			0
$911$	−29.4380			−0.975325			−0.487662	−	0.873032i	$-0.662150\pi$
$911$	−29.4380			−0.975325			−0.487662	+	0.873032i	$0.662150\pi$
$912$		0			0
$913$			44.2034i			1.46292i
$914$		0			0
$915$	−17.9398	+	2.52051i	−0.593071	+	0.0833253i
$916$		0			0
$917$		−	4.00000i		−	0.132092i
$918$		0			0
$919$	40.0197			1.32013			0.660064	−	0.751209i	$-0.270529\pi$
$919$	40.0197			1.32013			0.660064	+	0.751209i	$0.270529\pi$
$920$		0			0
$921$	−4.85728			−0.160053
$922$		0			0
$923$			1.59411i			0.0524708i
$924$		0			0
$925$	1.71456	+	5.98126i	0.0563743	+	0.196663i
$926$		0			0
$927$			0.857279i			0.0281567i
$928$		0			0
$929$	22.4572			0.736797			0.368399	−	0.929668i	$-0.379906\pi$
$929$	22.4572			0.736797			0.368399	+	0.929668i	$0.379906\pi$
$930$		0			0
$931$	−0.622216			−0.0203923
$932$		0			0
$933$			34.5718i			1.13183i
$934$		0			0
$935$	−37.3274	+	5.24443i	−1.22074	+	0.171511i
$936$		0			0
$937$		−	33.8292i		−	1.10515i	−0.833463	−	0.552575i	$-0.813645\pi$
$937$		−	33.8292i		−	1.10515i	0.833463	−	0.552575i	$-0.186355\pi$
$938$		0			0
$939$	−6.33677			−0.206793
$940$		0			0
$941$	47.8479			1.55980			0.779899	−	0.625906i	$-0.215271\pi$
$941$	47.8479			1.55980			0.779899	+	0.625906i	$0.215271\pi$
$942$		0			0
$943$			12.1204i			0.394696i
$944$		0			0
$945$	0.311108	+	2.21432i	0.0101203	+	0.0720318i
$946$		0			0
$947$		−	44.8069i		−	1.45603i	−0.685562	−	0.728014i	$-0.740443\pi$
$947$		−	44.8069i		−	1.45603i	0.685562	−	0.728014i	$-0.259557\pi$
$948$		0			0
$949$	−6.83854			−0.221989
$950$		0			0
$951$	−15.9684			−0.517809
$952$		0			0
$953$		−	15.7017i		−	0.508626i	−0.967122	−	0.254313i	$-0.918151\pi$
$953$		−	15.7017i		−	0.508626i	0.967122	−	0.254313i	$-0.0818494\pi$
$954$		0			0
$955$	−3.28592	−	23.3876i	−0.106330	−	0.756806i
$956$		0			0
$957$		−	36.5906i		−	1.18281i
$958$		0			0
$959$	−17.0923			−0.551941
$960$		0			0
$961$	−30.6128			−0.987511
$962$		0			0
$963$		−	11.0923i		−	0.357446i
$964$		0			0
$965$	−11.6128	+	1.63158i	−0.373831	+	0.0525225i
$966$		0			0
$967$			46.9590i			1.51010i	0.655668	+	0.755050i	$0.272387\pi$
$967$			46.9590i			1.51010i	−0.655668	+	0.755050i	$0.727613\pi$
$968$		0			0
$969$	2.75557			0.0885216
$970$		0			0
$971$	18.7556			0.601895			0.300947	−	0.953641i	$-0.402697\pi$
$971$	18.7556			0.601895			0.300947	+	0.953641i	$0.402697\pi$
$972$		0			0
$973$			13.4795i			0.432133i
$974$		0			0
$975$	−2.99063	+	0.857279i	−0.0957769	+	0.0274549i
$976$		0			0
$977$			37.8292i			1.21026i	0.796126	+	0.605131i	$0.206879\pi$
$977$			37.8292i			1.21026i	−0.796126	+	0.605131i	$0.793121\pi$
$978$		0			0
$979$	31.3461			1.00183
$980$		0			0
$981$	5.61285			0.179204
$982$		0			0
$983$		−	17.6316i		−	0.562360i	−0.959655	−	0.281180i	$-0.909274\pi$
$983$		−	17.6316i		−	0.562360i	0.959655	−	0.281180i	$-0.0907258\pi$
$984$		0			0
$985$	39.2958	−	5.52098i	1.25207	−	0.175913i
$986$		0			0
$987$		−	11.6128i		−	0.369641i
$988$		0			0
$989$	12.7368			0.405008
$990$		0			0
$991$	37.1240			1.17928			0.589641	−	0.807665i	$-0.299269\pi$
$991$	37.1240			1.17928			0.589641	+	0.807665i	$0.299269\pi$
$992$		0			0
$993$			27.6128i			0.876267i
$994$		0			0
$995$	6.29529	+	44.8069i	0.199574	+	1.42047i
$996$		0			0
$997$			42.4197i			1.34345i	0.740802	+	0.671723i	$0.234446\pi$
$997$			42.4197i			1.34345i	−0.740802	+	0.671723i	$0.765554\pi$
$998$		0			0
$999$	−1.24443			−0.0393721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.t.i.1009.5		6
3.2	odd	2		5040.2.t.ba.1009.4		6
4.3	odd	2		840.2.t.e.169.2	✓	6
5.2	odd	4		8400.2.a.dk.1.3		3
5.3	odd	4		8400.2.a.dh.1.3		3
5.4	even	2	inner	1680.2.t.i.1009.2		6
12.11	even	2		2520.2.t.j.1009.4		6
15.14	odd	2		5040.2.t.ba.1009.3		6
20.3	even	4		4200.2.a.bq.1.1		3
20.7	even	4		4200.2.a.bo.1.1		3
20.19	odd	2		840.2.t.e.169.5	yes	6
60.59	even	2		2520.2.t.j.1009.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.t.e.169.2	✓	6	4.3	odd	2
840.2.t.e.169.5	yes	6	20.19	odd	2
1680.2.t.i.1009.2		6	5.4	even	2	inner
1680.2.t.i.1009.5		6	1.1	even	1	trivial
2520.2.t.j.1009.3		6	60.59	even	2
2520.2.t.j.1009.4		6	12.11	even	2
4200.2.a.bo.1.1		3	20.7	even	4
4200.2.a.bq.1.1		3	20.3	even	4
5040.2.t.ba.1009.3		6	15.14	odd	2
5040.2.t.ba.1009.4		6	3.2	odd	2
8400.2.a.dh.1.3		3	5.3	odd	4
8400.2.a.dk.1.3		3	5.2	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.311108 - 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-0.311108 - 2.21432i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +3.80642 q^{11} -0.622216i q^{13} +(2.21432 - 0.311108i) q^{15} -4.42864i q^{17} +0.622216 q^{19} +1.00000 q^{21} +2.62222i q^{23} +(-4.80642 + 1.37778i) q^{25} -1.00000i q^{27} -9.61285 q^{29} +0.622216 q^{31} +3.80642i q^{33} +(-2.21432 + 0.311108i) q^{35} -1.24443i q^{37} +0.622216 q^{39} +4.62222 q^{41} -4.85728i q^{43} +(0.311108 + 2.21432i) q^{45} -11.6128i q^{47} -1.00000 q^{49} +4.42864 q^{51} -13.4795i q^{53} +(-1.18421 - 8.42864i) q^{55} +0.622216i q^{57} +11.6128 q^{59} -8.10171 q^{61} +1.00000i q^{63} +(-1.37778 + 0.193576i) q^{65} -2.62222 q^{69} -2.56199 q^{71} -10.9906i q^{73} +(-1.37778 - 4.80642i) q^{75} -3.80642i q^{77} -6.75557 q^{79} +1.00000 q^{81} +11.6128i q^{83} +(-9.80642 + 1.37778i) q^{85} -9.61285i q^{87} +8.23506 q^{89} -0.622216 q^{91} +0.622216i q^{93} +(-0.193576 - 1.37778i) q^{95} +4.23506i q^{97} -3.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^5 - 6 * q^9	\( 6 q - 2 q^{5} - 6 q^{9} - 4 q^{11} + 4 q^{19} + 6 q^{21} - 2 q^{25} - 4 q^{29} + 4 q^{31} + 4 q^{39} + 28 q^{41} + 2 q^{45} - 6 q^{49} + 20 q^{55} + 16 q^{59} + 4 q^{61} - 8 q^{65} - 16 q^{69} + 12 q^{71} - 8 q^{75} - 40 q^{79} + 6 q^{81} - 32 q^{85} - 4 q^{89} - 4 q^{91} - 28 q^{95} + 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^5 - 6 * q^9 - 4 * q^11 + 4 * q^19 + 6 * q^21 - 2 * q^25 - 4 * q^29 + 4 * q^31 + 4 * q^39 + 28 * q^41 + 2 * q^45 - 6 * q^49 + 20 * q^55 + 16 * q^59 + 4 * q^61 - 8 * q^65 - 16 * q^69 + 12 * q^71 - 8 * q^75 - 40 * q^79 + 6 * q^81 - 32 * q^85 - 4 * q^89 - 4 * q^91 - 28 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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