LMFDB - Embedded newform 1680.2.f.b.881.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(881,1680)]

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, 1, 0, 1]))

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

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

chi := DirichletCharacter("1680.881");

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.f (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:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1680.881
Dual form			1680.2.f.b.881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -1.00000 q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -1.00000 q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000 q^{9} -3.46410i q^{11} -1.73205i q^{15} +6.00000 q^{17} +3.46410i q^{19} +(3.00000 - 3.46410i) q^{21} +3.46410i q^{23} +1.00000 q^{25} -5.19615i q^{27} +6.92820i q^{29} -3.46410i q^{31} +6.00000 q^{33} +(2.00000 + 1.73205i) q^{35} -2.00000 q^{37} +6.00000 q^{41} +8.00000 q^{43} +3.00000 q^{45} +12.0000 q^{47} +(1.00000 + 6.92820i) q^{49} +10.3923i q^{51} +3.46410i q^{55} -6.00000 q^{57} +12.0000 q^{59} +6.92820i q^{61} +(6.00000 + 5.19615i) q^{63} -8.00000 q^{67} -6.00000 q^{69} +3.46410i q^{71} -6.92820i q^{73} +1.73205i q^{75} +(-6.00000 + 6.92820i) q^{77} -8.00000 q^{79} +9.00000 q^{81} -6.00000 q^{85} -12.0000 q^{87} +6.00000 q^{89} +6.00000 q^{93} -3.46410i q^{95} -6.92820i q^{97} +10.3923i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^7 - 6 * q^9	$ 2 q - 2 q^{5} - 4 q^{7} - 6 q^{9} + 12 q^{17} + 6 q^{21} + 2 q^{25} + 12 q^{33} + 4 q^{35} - 4 q^{37} + 12 q^{41} + 16 q^{43} + 6 q^{45} + 24 q^{47} + 2 q^{49} - 12 q^{57} + 24 q^{59} + 12 q^{63} - 16 q^{67} - 12 q^{69} - 12 q^{77} - 16 q^{79} + 18 q^{81} - 12 q^{85} - 24 q^{87} + 12 q^{89} + 12 q^{93}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 - 6 * q^9 + 12 * q^17 + 6 * q^21 + 2 * q^25 + 12 * q^33 + 4 * q^35 - 4 * q^37 + 12 * q^41 + 16 * q^43 + 6 * q^45 + 24 * q^47 + 2 * q^49 - 12 * q^57 + 24 * q^59 + 12 * q^63 - 16 * q^67 - 12 * q^69 - 12 * q^77 - 16 * q^79 + 18 * q^81 - 12 * q^85 - 24 * q^87 + 12 * q^89 + 12 * q^93

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
$3$			1.73205i			1.00000i
$3$			1.73205i			1.00000i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		−	1.73205i		−	0.447214i
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	3.00000	−	3.46410i	0.654654	−	0.755929i
$22$		0			0
$23$			3.46410i			0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$			3.46410i			0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$			6.92820i			1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$			6.92820i			1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$	6.00000			1.04447
$34$		0			0
$35$	2.00000	+	1.73205i	0.338062	+	0.292770i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		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
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$	3.00000			0.447214
$46$		0			0
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$			10.3923i			1.45521i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			3.46410i			0.467099i
$56$		0			0
$57$	−6.00000			−0.794719
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	6.00000	+	5.19615i	0.755929	+	0.654654i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		0			0
$75$			1.73205i			0.200000i
$76$		0			0
$77$	−6.00000	+	6.92820i	−0.683763	+	0.789542i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$	−12.0000			−1.28654
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	6.00000			0.622171
$94$		0			0
$95$		−	3.46410i		−	0.355409i
$96$		0			0
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$		0			0
$99$			10.3923i			1.04447i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		−	3.46410i		−	0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$		−	3.46410i		−	0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$	−3.00000	+	3.46410i	−0.292770	+	0.338062i
$106$		0			0
$107$			10.3923i			1.00466i	0.864675	+	0.502331i	$0.167524\pi$
$107$			10.3923i			1.00466i	−0.864675	+	0.502331i	$0.832476\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$		−	3.46410i		−	0.328798i
$112$		0			0
$113$			6.92820i			0.651751i	0.945413	+	0.325875i	$0.105659\pi$
$113$			6.92820i			0.651751i	−0.945413	+	0.325875i	$0.894341\pi$
$114$		0			0
$115$		−	3.46410i		−	0.323029i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.0000	−	10.3923i	−1.10004	−	0.952661i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			10.3923i			0.937043i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	4.00000			0.354943			0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000			0.354943			0.177471	+	0.984126i	$0.443208\pi$
$128$		0			0
$129$			13.8564i			1.21999i
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	6.00000	−	6.92820i	0.520266	−	0.600751i
$134$		0			0
$135$			5.19615i			0.447214i
$136$		0			0
$137$		−	20.7846i		−	1.77575i	−0.460086	−	0.887875i	$-0.652181\pi$
$137$		−	20.7846i		−	1.77575i	0.460086	−	0.887875i	$-0.347819\pi$
$138$		0			0
$139$			17.3205i			1.46911i	0.678551	+	0.734553i	$0.262608\pi$
$139$			17.3205i			1.46911i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$			20.7846i			1.75038i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	6.92820i		−	0.575356i
$146$		0			0
$147$	−12.0000	+	1.73205i	−0.989743	+	0.142857i
$148$		0			0
$149$		−	6.92820i		−	0.567581i	−0.958886	−	0.283790i	$-0.908408\pi$
$149$		−	6.92820i		−	0.567581i	0.958886	−	0.283790i	$-0.0915919\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	−18.0000			−1.45521
$154$		0			0
$155$			3.46410i			0.278243i
$156$		0			0
$157$		−	13.8564i		−	1.10586i	−0.833227	−	0.552931i	$-0.813509\pi$
$157$		−	13.8564i		−	1.10586i	0.833227	−	0.552931i	$-0.186491\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000	−	6.92820i	0.472866	−	0.546019i
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$	−6.00000			−0.467099
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		−	10.3923i		−	0.794719i
$172$		0			0
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		0			0
$175$	−2.00000	−	1.73205i	−0.151186	−	0.130931i
$176$		0			0
$177$			20.7846i			1.56227i
$178$		0			0
$179$			10.3923i			0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$			10.3923i			0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$		0			0
$181$			20.7846i			1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$			20.7846i			1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$	2.00000			0.147043
$186$		0			0
$187$		−	20.7846i		−	1.51992i
$188$		0			0
$189$	−9.00000	+	10.3923i	−0.654654	+	0.755929i
$190$		0			0
$191$		−	10.3923i		−	0.751961i	−0.926628	−	0.375980i	$-0.877306\pi$
$191$		−	10.3923i		−	0.751961i	0.926628	−	0.375980i	$-0.122694\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			13.8564i			0.987228i	0.869681	+	0.493614i	$0.164324\pi$
$197$			13.8564i			0.987228i	−0.869681	+	0.493614i	$0.835676\pi$
$198$		0			0
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		0			0
$201$		−	13.8564i		−	0.977356i
$202$		0			0
$203$	12.0000	−	13.8564i	0.842235	−	0.972529i
$204$		0			0
$205$	−6.00000			−0.419058
$206$		0			0
$207$		−	10.3923i		−	0.722315i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	−6.00000			−0.411113
$214$		0			0
$215$	−8.00000			−0.545595
$216$		0			0
$217$	−6.00000	+	6.92820i	−0.407307	+	0.470317i
$218$		0			0
$219$	12.0000			0.810885
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	17.3205i		−	1.15987i	−0.814664	−	0.579934i	$-0.803079\pi$
$223$		−	17.3205i		−	1.15987i	0.814664	−	0.579934i	$-0.196921\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$		0			0
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$		0			0
$231$	−12.0000	−	10.3923i	−0.789542	−	0.683763i
$232$		0			0
$233$		−	6.92820i		−	0.453882i	−0.973909	−	0.226941i	$-0.927128\pi$
$233$		−	6.92820i		−	0.453882i	0.973909	−	0.226941i	$-0.0728724\pi$
$234$		0			0
$235$	−12.0000			−0.782794
$236$		0			0
$237$		−	13.8564i		−	0.900070i
$238$		0			0
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$			15.5885i			1.00000i
$244$		0			0
$245$	−1.00000	−	6.92820i	−0.0638877	−	0.442627i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$		−	10.3923i		−	0.650791i
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	4.00000	+	3.46410i	0.248548	+	0.215249i
$260$		0			0
$261$		−	20.7846i		−	1.28654i
$262$		0			0
$263$		−	24.2487i		−	1.49524i	−0.664127	−	0.747620i	$-0.731197\pi$
$263$		−	24.2487i		−	1.49524i	0.664127	−	0.747620i	$-0.268803\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			10.3923i			0.635999i
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$			24.2487i			1.47300i	0.676435	+	0.736502i	$0.263524\pi$
$271$			24.2487i			1.47300i	−0.676435	+	0.736502i	$0.736476\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	3.46410i		−	0.208893i
$276$		0			0
$277$	14.0000			0.841178			0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000			0.841178			0.420589	+	0.907251i	$0.361823\pi$
$278$		0			0
$279$			10.3923i			0.622171i
$280$		0			0
$281$		−	13.8564i		−	0.826604i	−0.910594	−	0.413302i	$-0.864375\pi$
$281$		−	13.8564i		−	0.826604i	0.910594	−	0.413302i	$-0.135625\pi$
$282$		0			0
$283$			17.3205i			1.02960i	0.857311	+	0.514799i	$0.172133\pi$
$283$			17.3205i			1.02960i	−0.857311	+	0.514799i	$0.827867\pi$
$284$		0			0
$285$	6.00000			0.355409
$286$		0			0
$287$	−12.0000	−	10.3923i	−0.708338	−	0.613438i
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$	12.0000			0.703452
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$	−12.0000			−0.698667
$296$		0			0
$297$	−18.0000			−1.04447
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−16.0000	−	13.8564i	−0.922225	−	0.798670i
$302$		0			0
$303$		−	10.3923i		−	0.597022i
$304$		0			0
$305$		−	6.92820i		−	0.396708i
$306$		0			0
$307$		−	24.2487i		−	1.38395i	−0.721923	−	0.691974i	$-0.756741\pi$
$307$		−	24.2487i		−	1.38395i	0.721923	−	0.691974i	$-0.243259\pi$
$308$		0			0
$309$	6.00000			0.341328
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			20.7846i			1.17482i	0.809291	+	0.587408i	$0.199852\pi$
$313$			20.7846i			1.17482i	−0.809291	+	0.587408i	$0.800148\pi$
$314$		0			0
$315$	−6.00000	−	5.19615i	−0.338062	−	0.292770i
$316$		0			0
$317$		−	27.7128i		−	1.55651i	−0.627950	−	0.778253i	$-0.716106\pi$
$317$		−	27.7128i		−	1.55651i	0.627950	−	0.778253i	$-0.283894\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$	−18.0000			−1.00466
$322$		0			0
$323$			20.7846i			1.15649i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	3.46410i		−	0.191565i
$328$		0			0
$329$	−24.0000	−	20.7846i	−1.32316	−	1.14589i
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		0			0
$333$	6.00000			0.328798
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	−12.0000			−0.651751
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$	6.00000			0.323029
$346$		0			0
$347$		−	17.3205i		−	0.929814i	−0.885360	−	0.464907i	$-0.846088\pi$
$347$		−	17.3205i		−	0.929814i	0.885360	−	0.464907i	$-0.153912\pi$
$348$		0			0
$349$		−	6.92820i		−	0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$		−	6.92820i		−	0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$		−	3.46410i		−	0.183855i
$356$		0			0
$357$	18.0000	−	20.7846i	0.952661	−	1.10004i
$358$		0			0
$359$			3.46410i			0.182828i	0.995813	+	0.0914141i	$0.0291387\pi$
$359$			3.46410i			0.182828i	−0.995813	+	0.0914141i	$0.970861\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$		−	1.73205i		−	0.0909091i
$364$		0			0
$365$			6.92820i			0.362639i
$366$		0			0
$367$			10.3923i			0.542474i	0.962513	+	0.271237i	$0.0874327\pi$
$367$			10.3923i			0.542474i	−0.962513	+	0.271237i	$0.912567\pi$
$368$		0			0
$369$	−18.0000			−0.937043
$370$		0			0
$371$		0			0
$372$		0			0
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		−	1.73205i		−	0.0894427i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$			6.92820i			0.354943i
$382$		0			0
$383$	−12.0000			−0.613171			−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000			−0.613171			−0.306586	+	0.951843i	$0.599187\pi$
$384$		0			0
$385$	6.00000	−	6.92820i	0.305788	−	0.353094i
$386$		0			0
$387$	−24.0000			−1.21999
$388$		0			0
$389$		−	6.92820i		−	0.351274i	−0.984455	−	0.175637i	$-0.943802\pi$
$389$		−	6.92820i		−	0.351274i	0.984455	−	0.175637i	$-0.0561985\pi$
$390$		0			0
$391$			20.7846i			1.05112i
$392$		0			0
$393$			20.7846i			1.04844i
$394$		0			0
$395$	8.00000			0.402524
$396$		0			0
$397$		−	13.8564i		−	0.695433i	−0.937600	−	0.347717i	$-0.886957\pi$
$397$		−	13.8564i		−	0.695433i	0.937600	−	0.347717i	$-0.113043\pi$
$398$		0			0
$399$	12.0000	+	10.3923i	0.600751	+	0.520266i
$400$		0			0
$401$			27.7128i			1.38391i	0.721940	+	0.691956i	$0.243251\pi$
$401$			27.7128i			1.38391i	−0.721940	+	0.691956i	$0.756749\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−9.00000			−0.447214
$406$		0			0
$407$			6.92820i			0.343418i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	36.0000			1.77575
$412$		0			0
$413$	−24.0000	−	20.7846i	−1.18096	−	1.02274i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−30.0000			−1.46911
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$	−36.0000			−1.75038
$424$		0			0
$425$	6.00000			0.291043
$426$		0			0
$427$	12.0000	−	13.8564i	0.580721	−	0.670559i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	10.3923i		−	0.500580i	−0.968171	−	0.250290i	$-0.919474\pi$
$431$		−	10.3923i		−	0.500580i	0.968171	−	0.250290i	$-0.0805259\pi$
$432$		0			0
$433$			6.92820i			0.332948i	0.986046	+	0.166474i	$0.0532382\pi$
$433$			6.92820i			0.332948i	−0.986046	+	0.166474i	$0.946762\pi$
$434$		0			0
$435$	12.0000			0.575356
$436$		0			0
$437$	−12.0000			−0.574038
$438$		0			0
$439$		−	17.3205i		−	0.826663i	−0.910581	−	0.413331i	$-0.864365\pi$
$439$		−	17.3205i		−	0.826663i	0.910581	−	0.413331i	$-0.135635\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$			10.3923i			0.493753i	0.969047	+	0.246877i	$0.0794043\pi$
$443$			10.3923i			0.493753i	−0.969047	+	0.246877i	$0.920596\pi$
$444$		0			0
$445$	−6.00000			−0.284427
$446$		0			0
$447$	12.0000			0.567581
$448$		0			0
$449$			13.8564i			0.653924i	0.945037	+	0.326962i	$0.106025\pi$
$449$			13.8564i			0.653924i	−0.945037	+	0.326962i	$0.893975\pi$
$450$		0			0
$451$		−	20.7846i		−	0.978709i
$452$		0			0
$453$			13.8564i			0.651031i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−38.0000			−1.77757			−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000			−1.77757			−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$		−	31.1769i		−	1.45521i
$460$		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
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$	−6.00000			−0.278243
$466$		0			0
$467$	−24.0000			−1.11059			−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000			−1.11059			−0.555294	+	0.831654i	$0.687394\pi$
$468$		0			0
$469$	16.0000	+	13.8564i	0.738811	+	0.639829i
$470$		0			0
$471$	24.0000			1.10586
$472$		0			0
$473$		−	27.7128i		−	1.27424i
$474$		0			0
$475$			3.46410i			0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	12.0000	+	10.3923i	0.546019	+	0.472866i
$484$		0			0
$485$			6.92820i			0.314594i
$486$		0			0
$487$	−4.00000			−0.181257			−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000			−0.181257			−0.0906287	+	0.995885i	$0.528888\pi$
$488$		0			0
$489$			27.7128i			1.25322i
$490$		0			0
$491$			24.2487i			1.09433i	0.837025	+	0.547165i	$0.184293\pi$
$491$			24.2487i			1.09433i	−0.837025	+	0.547165i	$0.815707\pi$
$492$		0			0
$493$			41.5692i			1.87218i
$494$		0			0
$495$		−	10.3923i		−	0.467099i
$496$		0			0
$497$	6.00000	−	6.92820i	0.269137	−	0.310772i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$		−	20.7846i		−	0.928588i
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$			22.5167i			1.00000i
$508$		0			0
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$	−12.0000	+	13.8564i	−0.530849	+	0.612971i
$512$		0			0
$513$	18.0000			0.794719
$514$		0			0
$515$			3.46410i			0.152647i
$516$		0			0
$517$		−	41.5692i		−	1.82821i
$518$		0			0
$519$			31.1769i			1.36851i
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$			17.3205i			0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$			17.3205i			0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$		0			0
$525$	3.00000	−	3.46410i	0.130931	−	0.151186i
$526$		0			0
$527$		−	20.7846i		−	0.905392i
$528$		0			0
$529$	11.0000			0.478261
$530$		0			0
$531$	−36.0000			−1.56227
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	10.3923i		−	0.449299i
$536$		0			0
$537$	−18.0000			−0.776757
$538$		0			0
$539$	24.0000	−	3.46410i	1.03375	−	0.149209i
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		0			0
$543$	−36.0000			−1.54491
$544$		0			0
$545$	2.00000			0.0856706
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$		−	20.7846i		−	0.887066i
$550$		0			0
$551$	−24.0000			−1.02243
$552$		0			0
$553$	16.0000	+	13.8564i	0.680389	+	0.589234i
$554$		0			0
$555$			3.46410i			0.147043i
$556$		0			0
$557$			27.7128i			1.17423i	0.809504	+	0.587115i	$0.199736\pi$
$557$			27.7128i			1.17423i	−0.809504	+	0.587115i	$0.800264\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	36.0000			1.51992
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		−	6.92820i		−	0.291472i
$566$		0			0
$567$	−18.0000	−	15.5885i	−0.755929	−	0.654654i
$568$		0			0
$569$			27.7128i			1.16178i	0.813982	+	0.580891i	$0.197296\pi$
$569$			27.7128i			1.16178i	−0.813982	+	0.580891i	$0.802704\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$	18.0000			0.751961
$574$		0			0
$575$			3.46410i			0.144463i
$576$		0			0
$577$		−	34.6410i		−	1.44212i	−0.692870	−	0.721062i	$-0.743654\pi$
$577$		−	34.6410i		−	1.44212i	0.692870	−	0.721062i	$-0.256346\pi$
$578$		0			0
$579$		−	24.2487i		−	1.00774i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$	−24.0000			−0.987228
$592$		0			0
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$	12.0000	+	10.3923i	0.491952	+	0.426043i
$596$		0			0
$597$	−18.0000			−0.736691
$598$		0			0
$599$		−	24.2487i		−	0.990775i	−0.868672	−	0.495388i	$-0.835026\pi$
$599$		−	24.2487i		−	0.990775i	0.868672	−	0.495388i	$-0.164974\pi$
$600$		0			0
$601$		−	13.8564i		−	0.565215i	−0.959236	−	0.282607i	$-0.908801\pi$
$601$		−	13.8564i		−	0.565215i	0.959236	−	0.282607i	$-0.0911993\pi$
$602$		0			0
$603$	24.0000			0.977356
$604$		0			0
$605$	1.00000			0.0406558
$606$		0			0
$607$		−	17.3205i		−	0.703018i	−0.936185	−	0.351509i	$-0.885669\pi$
$607$		−	17.3205i		−	0.703018i	0.936185	−	0.351509i	$-0.114331\pi$
$608$		0			0
$609$	24.0000	+	20.7846i	0.972529	+	0.842235i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		−	10.3923i		−	0.419058i
$616$		0			0
$617$			6.92820i			0.278919i	0.990228	+	0.139459i	$0.0445365\pi$
$617$			6.92820i			0.278919i	−0.990228	+	0.139459i	$0.955464\pi$
$618$		0			0
$619$			17.3205i			0.696170i	0.937463	+	0.348085i	$0.113168\pi$
$619$			17.3205i			0.696170i	−0.937463	+	0.348085i	$0.886832\pi$
$620$		0			0
$621$	18.0000			0.722315
$622$		0			0
$623$	−12.0000	−	10.3923i	−0.480770	−	0.416359i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			20.7846i			0.830057i
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$		0			0
$633$		−	6.92820i		−	0.275371i
$634$		0			0
$635$	−4.00000			−0.158735
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	10.3923i		−	0.411113i
$640$		0			0
$641$		−	27.7128i		−	1.09459i	−0.836940	−	0.547295i	$-0.815658\pi$
$641$		−	27.7128i		−	1.09459i	0.836940	−	0.547295i	$-0.184342\pi$
$642$		0			0
$643$			31.1769i			1.22950i	0.788723	+	0.614749i	$0.210743\pi$
$643$			31.1769i			1.22950i	−0.788723	+	0.614749i	$0.789257\pi$
$644$		0			0
$645$		−	13.8564i		−	0.545595i
$646$		0			0
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$		0			0
$649$		−	41.5692i		−	1.63173i
$650$		0			0
$651$	−12.0000	−	10.3923i	−0.470317	−	0.407307i
$652$		0			0
$653$			41.5692i			1.62673i	0.581754	+	0.813365i	$0.302367\pi$
$653$			41.5692i			1.62673i	−0.581754	+	0.813365i	$0.697633\pi$
$654$		0			0
$655$	−12.0000			−0.468879
$656$		0			0
$657$			20.7846i			0.810885i
$658$		0			0
$659$			10.3923i			0.404827i	0.979300	+	0.202413i	$0.0648785\pi$
$659$			10.3923i			0.404827i	−0.979300	+	0.202413i	$0.935122\pi$
$660$		0			0
$661$			48.4974i			1.88633i	0.332323	+	0.943166i	$0.392168\pi$
$661$			48.4974i			1.88633i	−0.332323	+	0.943166i	$0.607832\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000	+	6.92820i	−0.232670	+	0.268664i
$666$		0			0
$667$	−24.0000			−0.929284
$668$		0			0
$669$	30.0000			1.15987
$670$		0			0
$671$	24.0000			0.926510
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		0			0
$675$		−	5.19615i		−	0.200000i
$676$		0			0
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$	−12.0000	+	13.8564i	−0.460518	+	0.531760i
$680$		0			0
$681$			41.5692i			1.59294i
$682$		0			0
$683$		−	17.3205i		−	0.662751i	−0.943499	−	0.331375i	$-0.892487\pi$
$683$		−	17.3205i		−	0.662751i	0.943499	−	0.331375i	$-0.107513\pi$
$684$		0			0
$685$			20.7846i			0.794139i
$686$		0			0
$687$	−12.0000			−0.457829
$688$		0			0
$689$		0			0
$690$		0			0
$691$			31.1769i			1.18603i	0.805193	+	0.593013i	$0.202062\pi$
$691$			31.1769i			1.18603i	−0.805193	+	0.593013i	$0.797938\pi$
$692$		0			0
$693$	18.0000	−	20.7846i	0.683763	−	0.789542i
$694$		0			0
$695$		−	17.3205i		−	0.657004i
$696$		0			0
$697$	36.0000			1.36360
$698$		0			0
$699$	12.0000			0.453882
$700$		0			0
$701$			20.7846i			0.785024i	0.919747	+	0.392512i	$0.128394\pi$
$701$			20.7846i			0.785024i	−0.919747	+	0.392512i	$0.871606\pi$
$702$		0			0
$703$		−	6.92820i		−	0.261302i
$704$		0			0
$705$		−	20.7846i		−	0.782794i
$706$		0			0
$707$	12.0000	+	10.3923i	0.451306	+	0.390843i
$708$		0			0
$709$	22.0000			0.826227			0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000			0.826227			0.413114	+	0.910679i	$0.364441\pi$
$710$		0			0
$711$	24.0000			0.900070
$712$		0			0
$713$	12.0000			0.449404
$714$		0			0
$715$		0			0
$716$		0			0
$717$	18.0000			0.672222
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−6.00000	+	6.92820i	−0.223452	+	0.258020i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			6.92820i			0.257307i
$726$		0			0
$727$		−	3.46410i		−	0.128476i	−0.997935	−	0.0642382i	$-0.979538\pi$
$727$		−	3.46410i		−	0.128476i	0.997935	−	0.0642382i	$-0.0204617\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	48.0000			1.77534
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	12.0000	−	1.73205i	0.442627	−	0.0638877i
$736$		0			0
$737$			27.7128i			1.02081i
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31.1769i			1.14377i	0.820334	+	0.571885i	$0.193788\pi$
$743$			31.1769i			1.14377i	−0.820334	+	0.571885i	$0.806212\pi$
$744$		0			0
$745$			6.92820i			0.253830i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000	−	20.7846i	0.657706	−	0.759453i
$750$		0			0
$751$	−16.0000			−0.583848			−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000			−0.583848			−0.291924	+	0.956441i	$0.594295\pi$
$752$		0			0
$753$		−	20.7846i		−	0.757433i
$754$		0			0
$755$	−8.00000			−0.291150
$756$		0			0
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$		0			0
$759$			20.7846i			0.754434i
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$	4.00000	+	3.46410i	0.144810	+	0.125409i
$764$		0			0
$765$	18.0000			0.650791
$766$		0			0
$767$		0			0
$768$		0			0
$769$			41.5692i			1.49902i	0.661991	+	0.749512i	$0.269712\pi$
$769$			41.5692i			1.49902i	−0.661991	+	0.749512i	$0.730288\pi$
$770$		0			0
$771$			10.3923i			0.374270i
$772$		0			0
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$		0			0
$775$		−	3.46410i		−	0.124434i
$776$		0			0
$777$	−6.00000	+	6.92820i	−0.215249	+	0.248548i
$778$		0			0
$779$			20.7846i			0.744686i
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$	36.0000			1.28654
$784$		0			0
$785$			13.8564i			0.494556i
$786$		0			0
$787$		−	24.2487i		−	0.864373i	−0.901784	−	0.432187i	$-0.857742\pi$
$787$		−	24.2487i		−	0.864373i	0.901784	−	0.432187i	$-0.142258\pi$
$788$		0			0
$789$	42.0000			1.49524
$790$		0			0
$791$	12.0000	−	13.8564i	0.426671	−	0.492677i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$	72.0000			2.54718
$800$		0			0
$801$	−18.0000			−0.635999
$802$		0			0
$803$	−24.0000			−0.846942
$804$		0			0
$805$	−6.00000	+	6.92820i	−0.211472	+	0.244187i
$806$		0			0
$807$			31.1769i			1.09748i
$808$		0			0
$809$		−	55.4256i		−	1.94866i	−0.225122	−	0.974331i	$-0.572278\pi$
$809$		−	55.4256i		−	1.94866i	0.225122	−	0.974331i	$-0.427722\pi$
$810$		0			0
$811$		−	38.1051i		−	1.33805i	−0.743239	−	0.669026i	$-0.766712\pi$
$811$		−	38.1051i		−	1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$		0			0
$813$	−42.0000			−1.47300
$814$		0			0
$815$	−16.0000			−0.560456
$816$		0			0
$817$			27.7128i			0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.7846i		−	0.725388i	−0.931908	−	0.362694i	$-0.881857\pi$
$821$		−	20.7846i		−	0.725388i	0.931908	−	0.362694i	$-0.118143\pi$
$822$		0			0
$823$	−44.0000			−1.53374			−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000			−1.53374			−0.766872	+	0.641800i	$0.778188\pi$
$824$		0			0
$825$	6.00000			0.208893
$826$		0			0
$827$			38.1051i			1.32504i	0.749042	+	0.662522i	$0.230514\pi$
$827$			38.1051i			1.32504i	−0.749042	+	0.662522i	$0.769486\pi$
$828$		0			0
$829$		−	34.6410i		−	1.20313i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$		−	34.6410i		−	1.20313i	0.798823	−	0.601566i	$-0.205456\pi$
$830$		0			0
$831$			24.2487i			0.841178i
$832$		0			0
$833$	6.00000	+	41.5692i	0.207888	+	1.44029i
$834$		0			0
$835$	12.0000			0.415277
$836$		0			0
$837$	−18.0000			−0.622171
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$	24.0000			0.826604
$844$		0			0
$845$	−13.0000			−0.447214
$846$		0			0
$847$	2.00000	+	1.73205i	0.0687208	+	0.0595140i
$848$		0			0
$849$	−30.0000			−1.02960
$850$		0			0
$851$		−	6.92820i		−	0.237496i
$852$		0			0
$853$		−	41.5692i		−	1.42330i	−0.702533	−	0.711651i	$-0.747948\pi$
$853$		−	41.5692i		−	1.42330i	0.702533	−	0.711651i	$-0.252052\pi$
$854$		0			0
$855$			10.3923i			0.355409i
$856$		0			0
$857$	30.0000			1.02478			0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000			1.02478			0.512390	+	0.858753i	$0.328760\pi$
$858$		0			0
$859$		−	38.1051i		−	1.30013i	−0.759879	−	0.650065i	$-0.774742\pi$
$859$		−	38.1051i		−	1.30013i	0.759879	−	0.650065i	$-0.225258\pi$
$860$		0			0
$861$	18.0000	−	20.7846i	0.613438	−	0.708338i
$862$		0			0
$863$			17.3205i			0.589597i	0.955559	+	0.294798i	$0.0952525\pi$
$863$			17.3205i			0.589597i	−0.955559	+	0.294798i	$0.904747\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$		0			0
$867$			32.9090i			1.11765i
$868$		0			0
$869$			27.7128i			0.940093i
$870$		0			0
$871$		0			0
$872$		0			0
$873$			20.7846i			0.703452i
$874$		0			0
$875$	2.00000	+	1.73205i	0.0676123	+	0.0585540i
$876$		0			0
$877$	38.0000			1.28317			0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000			1.28317			0.641584	+	0.767052i	$0.278277\pi$
$878$		0			0
$879$		−	10.3923i		−	0.350524i
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$		−	20.7846i		−	0.698667i
$886$		0			0
$887$	36.0000			1.20876			0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000			1.20876			0.604381	+	0.796696i	$0.293421\pi$
$888$		0			0
$889$	−8.00000	−	6.92820i	−0.268311	−	0.232364i
$890$		0			0
$891$		−	31.1769i		−	1.04447i
$892$		0			0
$893$			41.5692i			1.39106i
$894$		0			0
$895$		−	10.3923i		−	0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24.0000			0.800445
$900$		0			0
$901$		0			0
$902$		0			0
$903$	24.0000	−	27.7128i	0.798670	−	0.922225i
$904$		0			0
$905$		−	20.7846i		−	0.690904i
$906$		0			0
$907$	32.0000			1.06254			0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000			1.06254			0.531271	+	0.847202i	$0.321714\pi$
$908$		0			0
$909$	18.0000			0.597022
$910$		0			0
$911$			17.3205i			0.573854i	0.957952	+	0.286927i	$0.0926337\pi$
$911$			17.3205i			0.573854i	−0.957952	+	0.286927i	$0.907366\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	12.0000			0.396708
$916$		0			0
$917$	−24.0000	−	20.7846i	−0.792550	−	0.686368i
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$	42.0000			1.38395
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.00000			−0.0657596
$926$		0			0
$927$			10.3923i			0.341328i
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$	−24.0000	+	3.46410i	−0.786568	+	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			20.7846i			0.679729i
$936$		0			0
$937$			6.92820i			0.226335i	0.993576	+	0.113167i	$0.0360996\pi$
$937$			6.92820i			0.226335i	−0.993576	+	0.113167i	$0.963900\pi$
$938$		0			0
$939$	−36.0000			−1.17482
$940$		0			0
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$			20.7846i			0.676840i
$944$		0			0
$945$	9.00000	−	10.3923i	0.292770	−	0.338062i
$946$		0			0
$947$		−	31.1769i		−	1.01311i	−0.862207	−	0.506557i	$-0.830918\pi$
$947$		−	31.1769i		−	1.01311i	0.862207	−	0.506557i	$-0.169082\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	48.0000			1.55651
$952$		0			0
$953$		−	34.6410i		−	1.12213i	−0.827771	−	0.561066i	$-0.810391\pi$
$953$		−	34.6410i		−	1.12213i	0.827771	−	0.561066i	$-0.189609\pi$
$954$		0			0
$955$			10.3923i			0.336287i
$956$		0			0
$957$			41.5692i			1.34374i
$958$		0			0
$959$	−36.0000	+	41.5692i	−1.16250	+	1.34234i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		−	31.1769i		−	1.00466i
$964$		0			0
$965$	14.0000			0.450676
$966$		0			0
$967$	−20.0000			−0.643157			−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000			−0.643157			−0.321578	+	0.946883i	$0.604213\pi$
$968$		0			0
$969$	−36.0000			−1.15649
$970$		0			0
$971$	−60.0000			−1.92549			−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000			−1.92549			−0.962746	+	0.270408i	$0.912841\pi$
$972$		0			0
$973$	30.0000	−	34.6410i	0.961756	−	1.11054i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	6.92820i		−	0.221653i	−0.993840	−	0.110826i	$-0.964650\pi$
$977$		−	6.92820i		−	0.221653i	0.993840	−	0.110826i	$-0.0353498\pi$
$978$		0			0
$979$		−	20.7846i		−	0.664279i
$980$		0			0
$981$	6.00000			0.191565
$982$		0			0
$983$	60.0000			1.91370			0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000			1.91370			0.956851	+	0.290578i	$0.0938475\pi$
$984$		0			0
$985$		−	13.8564i		−	0.441502i
$986$		0			0
$987$	36.0000	−	41.5692i	1.14589	−	1.32316i
$988$		0			0
$989$			27.7128i			0.881216i
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$		−	48.4974i		−	1.53902i
$994$		0			0
$995$		−	10.3923i		−	0.329458i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$			10.3923i			0.328798i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.f.b.881.2		2
3.2	odd	2		1680.2.f.c.881.2		2
4.3	odd	2		105.2.b.a.41.1	✓	2
7.6	odd	2		1680.2.f.c.881.1		2
12.11	even	2		105.2.b.b.41.2	yes	2
20.3	even	4		525.2.g.b.524.1		4
20.7	even	4		525.2.g.b.524.4		4
20.19	odd	2		525.2.b.a.251.2		2
21.20	even	2	inner	1680.2.f.b.881.1		2
28.3	even	6		735.2.s.f.656.1		2
28.11	odd	6		735.2.s.d.656.1		2
28.19	even	6		735.2.s.a.521.1		2
28.23	odd	6		735.2.s.b.521.1		2
28.27	even	2		105.2.b.b.41.1	yes	2
60.23	odd	4		525.2.g.c.524.3		4
60.47	odd	4		525.2.g.c.524.2		4
60.59	even	2		525.2.b.b.251.1		2
84.11	even	6		735.2.s.a.656.1		2
84.23	even	6		735.2.s.f.521.1		2
84.47	odd	6		735.2.s.d.521.1		2
84.59	odd	6		735.2.s.b.656.1		2
84.83	odd	2		105.2.b.a.41.2	yes	2
140.27	odd	4		525.2.g.c.524.4		4
140.83	odd	4		525.2.g.c.524.1		4
140.139	even	2		525.2.b.b.251.2		2
420.83	even	4		525.2.g.b.524.3		4
420.167	even	4		525.2.g.b.524.2		4
420.419	odd	2		525.2.b.a.251.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.b.a.41.1	✓	2	4.3	odd	2
105.2.b.a.41.2	yes	2	84.83	odd	2
105.2.b.b.41.1	yes	2	28.27	even	2
105.2.b.b.41.2	yes	2	12.11	even	2
525.2.b.a.251.1		2	420.419	odd	2
525.2.b.a.251.2		2	20.19	odd	2
525.2.b.b.251.1		2	60.59	even	2
525.2.b.b.251.2		2	140.139	even	2
525.2.g.b.524.1		4	20.3	even	4
525.2.g.b.524.2		4	420.167	even	4
525.2.g.b.524.3		4	420.83	even	4
525.2.g.b.524.4		4	20.7	even	4
525.2.g.c.524.1		4	140.83	odd	4
525.2.g.c.524.2		4	60.47	odd	4
525.2.g.c.524.3		4	60.23	odd	4
525.2.g.c.524.4		4	140.27	odd	4
735.2.s.a.521.1		2	28.19	even	6
735.2.s.a.656.1		2	84.11	even	6
735.2.s.b.521.1		2	28.23	odd	6
735.2.s.b.656.1		2	84.59	odd	6
735.2.s.d.521.1		2	84.47	odd	6
735.2.s.d.656.1		2	28.11	odd	6
735.2.s.f.521.1		2	84.23	even	6
735.2.s.f.656.1		2	28.3	even	6
1680.2.f.b.881.1		2	21.20	even	2	inner
1680.2.f.b.881.2		2	1.1	even	1	trivial
1680.2.f.c.881.1		2	7.6	odd	2
1680.2.f.c.881.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -1.00000 q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -1.00000 q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000 q^{9} -3.46410i q^{11} -1.73205i q^{15} +6.00000 q^{17} +3.46410i q^{19} +(3.00000 - 3.46410i) q^{21} +3.46410i q^{23} +1.00000 q^{25} -5.19615i q^{27} +6.92820i q^{29} -3.46410i q^{31} +6.00000 q^{33} +(2.00000 + 1.73205i) q^{35} -2.00000 q^{37} +6.00000 q^{41} +8.00000 q^{43} +3.00000 q^{45} +12.0000 q^{47} +(1.00000 + 6.92820i) q^{49} +10.3923i q^{51} +3.46410i q^{55} -6.00000 q^{57} +12.0000 q^{59} +6.92820i q^{61} +(6.00000 + 5.19615i) q^{63} -8.00000 q^{67} -6.00000 q^{69} +3.46410i q^{71} -6.92820i q^{73} +1.73205i q^{75} +(-6.00000 + 6.92820i) q^{77} -8.00000 q^{79} +9.00000 q^{81} -6.00000 q^{85} -12.0000 q^{87} +6.00000 q^{89} +6.00000 q^{93} -3.46410i q^{95} -6.92820i q^{97} +10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^7 - 6 * q^9	\( 2 q - 2 q^{5} - 4 q^{7} - 6 q^{9} + 12 q^{17} + 6 q^{21} + 2 q^{25} + 12 q^{33} + 4 q^{35} - 4 q^{37} + 12 q^{41} + 16 q^{43} + 6 q^{45} + 24 q^{47} + 2 q^{49} - 12 q^{57} + 24 q^{59} + 12 q^{63} - 16 q^{67} - 12 q^{69} - 12 q^{77} - 16 q^{79} + 18 q^{81} - 12 q^{85} - 24 q^{87} + 12 q^{89} + 12 q^{93}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 - 6 * q^9 + 12 * q^17 + 6 * q^21 + 2 * q^25 + 12 * q^33 + 4 * q^35 - 4 * q^37 + 12 * q^41 + 16 * q^43 + 6 * q^45 + 24 * q^47 + 2 * q^49 - 12 * q^57 + 24 * q^59 + 12 * q^63 - 16 * q^67 - 12 * q^69 - 12 * q^77 - 16 * q^79 + 18 * q^81 - 12 * q^85 - 24 * q^87 + 12 * q^89 + 12 * q^93

Introduction

L-functions

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Database

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