LMFDB - Embedded newform 4680.2.l.f.2809.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, 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("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1698758656.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 $ x^8 + 18x^6 + 97x^4 + 176*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.6
Root			$0.692297i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.f.2809.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.19663 + 1.88893i) q^{5} +2.82843i q^{7} +O(q^{10})$	$q+(1.19663 + 1.88893i) q^{5} +2.82843i q^{7} -0.393270 q^{11} -1.00000i q^{13} +6.21302i q^{17} +7.34271 q^{19} -1.44383i q^{23} +(-2.13613 + 4.52072i) q^{25} -0.828427 q^{29} +1.65685 q^{31} +(-5.34271 + 3.38459i) q^{35} +6.78654i q^{37} -3.77786 q^{41} +2.51428i q^{43} -3.00868i q^{47} -1.00000 q^{49} -9.55573i q^{53} +(-0.470600 - 0.742860i) q^{55} +0.393270 q^{59} +11.0414 q^{61} +(1.88893 - 1.19663i) q^{65} +15.5557i q^{67} -6.56440 q^{71} -13.6150i q^{73} -1.11233i q^{77} +14.9403 q^{79} +16.4633i q^{83} +(-11.7360 + 7.43472i) q^{85} -9.67674 q^{89} +2.82843 q^{91} +(8.78654 + 13.8699i) q^{95} +0.0418875i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{11} + 24 q^{19} - 4 q^{25} + 16 q^{29} - 32 q^{31} - 8 q^{35} + 8 q^{41} - 8 q^{49} - 36 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{71} + 24 q^{79} - 40 q^{85} - 8 q^{89} + 32 q^{95}+O(q^{100}) $ 8 * q + 16 * q^11 + 24 * q^19 - 4 * q^25 + 16 * q^29 - 32 * q^31 - 8 * q^35 + 8 * q^41 - 8 * q^49 - 36 * q^55 - 16 * q^59 + 24 * q^61 - 4 * q^65 + 24 * q^71 + 24 * q^79 - 40 * q^85 - 8 * q^89 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.19663	+	1.88893i	0.535151	+	0.844756i
$5$	1.19663	+	1.88893i	0.535151	+	0.844756i
$6$		0			0
$7$			2.82843i			1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$			2.82843i			1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.393270			−0.118575			−0.0592877	−	0.998241i	$-0.518883\pi$
$11$	−0.393270			−0.118575			−0.0592877	+	0.998241i	$0.518883\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.21302i			1.50688i	0.657517	+	0.753440i	$0.271607\pi$
$17$			6.21302i			1.50688i	−0.657517	+	0.753440i	$0.728393\pi$
$18$		0			0
$19$	7.34271			1.68453			0.842266	−	0.539062i	$-0.181221\pi$
$19$	7.34271			1.68453			0.842266	+	0.539062i	$0.181221\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.44383i		−	0.301060i	−0.988605	−	0.150530i	$-0.951902\pi$
$23$		−	1.44383i		−	0.301060i	0.988605	−	0.150530i	$-0.0480980\pi$
$24$		0			0
$25$	−2.13613	+	4.52072i	−0.427226	+	0.904145i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.828427			−0.153835			−0.0769175	−	0.997037i	$-0.524508\pi$
$29$	−0.828427			−0.153835			−0.0769175	+	0.997037i	$0.524508\pi$
$30$		0			0
$31$	1.65685			0.297580			0.148790	−	0.988869i	$-0.452462\pi$
$31$	1.65685			0.297580			0.148790	+	0.988869i	$0.452462\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.34271	+	3.38459i	−0.903082	+	0.572101i
$36$		0			0
$37$			6.78654i			1.11570i	0.829942	+	0.557850i	$0.188374\pi$
$37$			6.78654i			1.11570i	−0.829942	+	0.557850i	$0.811626\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.77786			−0.590003			−0.295002	−	0.955497i	$-0.595320\pi$
$41$	−3.77786			−0.590003			−0.295002	+	0.955497i	$0.595320\pi$
$42$		0			0
$43$			2.51428i			0.383424i	0.981451	+	0.191712i	$0.0614040\pi$
$43$			2.51428i			0.383424i	−0.981451	+	0.191712i	$0.938596\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.00868i		−	0.438860i	−0.975628	−	0.219430i	$-0.929580\pi$
$47$		−	3.00868i		−	0.438860i	0.975628	−	0.219430i	$-0.0704198\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.55573i		−	1.31258i	−0.754509	−	0.656290i	$-0.772125\pi$
$53$		−	9.55573i		−	1.31258i	0.754509	−	0.656290i	$-0.227875\pi$
$54$		0			0
$55$	−0.470600	−	0.742860i	−0.0634557	−	0.100167i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.393270			0.0511994			0.0255997	−	0.999672i	$-0.491850\pi$
$59$	0.393270			0.0511994			0.0255997	+	0.999672i	$0.491850\pi$
$60$		0			0
$61$	11.0414			1.41371			0.706856	−	0.707357i	$-0.250113\pi$
$61$	11.0414			1.41371			0.706856	+	0.707357i	$0.250113\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.88893	−	1.19663i	0.234293	−	0.148424i
$66$		0			0
$67$			15.5557i			1.90043i	0.311588	+	0.950217i	$0.399139\pi$
$67$			15.5557i			1.90043i	−0.311588	+	0.950217i	$0.600861\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.56440			−0.779051			−0.389526	−	0.921016i	$-0.627361\pi$
$71$	−6.56440			−0.779051			−0.389526	+	0.921016i	$0.627361\pi$
$72$		0			0
$73$		−	13.6150i		−	1.59351i	−0.604302	−	0.796756i	$-0.706548\pi$
$73$		−	13.6150i		−	1.59351i	0.604302	−	0.796756i	$-0.293452\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1.11233i		−	0.126762i
$78$		0			0
$79$	14.9403			1.68092			0.840459	−	0.541875i	$-0.182286\pi$
$79$	14.9403			1.68092			0.840459	+	0.541875i	$0.182286\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			16.4633i			1.80708i	0.428504	+	0.903540i	$0.359041\pi$
$83$			16.4633i			1.80708i	−0.428504	+	0.903540i	$0.640959\pi$
$84$		0			0
$85$	−11.7360	+	7.43472i	−1.27295	+	0.806408i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.67674			−1.02573			−0.512866	−	0.858469i	$-0.671416\pi$
$89$	−9.67674			−1.02573			−0.512866	+	0.858469i	$0.671416\pi$
$90$		0			0
$91$	2.82843			0.296500
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.78654	+	13.8699i	0.901480	+	1.42302i
$96$		0			0
$97$			0.0418875i			0.00425303i	0.999998	+	0.00212652i	$0.000676892\pi$
$97$			0.0418875i			0.00425303i	−0.999998	+	0.00212652i	$0.999323\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.1421			−1.20819			−0.604094	−	0.796913i	$-0.706465\pi$
$101$	−12.1421			−1.20819			−0.604094	+	0.796913i	$0.706465\pi$
$102$		0			0
$103$			0.0173504i			0.00170958i	1.00000		0.000854792i	$0.000272089\pi$
$103$			0.0173504i			0.00170958i	−1.00000		0.000854792i	$0.999728\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	2.10113i		−	0.203123i	−0.994829	−	0.101562i	$-0.967616\pi$
$107$		−	2.10113i		−	0.203123i	0.994829	−	0.101562i	$-0.0323839\pi$
$108$		0			0
$109$	−6.21302			−0.595100			−0.297550	−	0.954706i	$-0.596169\pi$
$109$	−6.21302			−0.595100			−0.297550	+	0.954706i	$0.596169\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1.42648i			0.134192i	0.997747	+	0.0670961i	$0.0213734\pi$
$113$			1.42648i			0.134192i	−0.997747	+	0.0670961i	$0.978627\pi$
$114$		0			0
$115$	2.72730	−	1.72774i	0.254322	−	0.161113i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−17.5731			−1.61092
$120$		0			0
$121$	−10.8453			−0.985940
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0955	+	1.37465i	−0.992413	+	0.122953i
$126$		0			0
$127$			1.50307i			0.133376i	0.997774	+	0.0666880i	$0.0212432\pi$
$127$			1.50307i			0.133376i	−0.997774	+	0.0666880i	$0.978757\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.85699			−0.336987			−0.168493	−	0.985703i	$-0.553890\pi$
$131$	−3.85699			−0.336987			−0.168493	+	0.985703i	$0.553890\pi$
$132$		0			0
$133$			20.7683i			1.80084i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	16.5644i		−	1.41519i	−0.706617	−	0.707596i	$-0.749780\pi$
$137$		−	16.5644i		−	1.41519i	0.706617	−	0.707596i	$-0.250220\pi$
$138$		0			0
$139$	−10.6854			−0.906325			−0.453163	−	0.891428i	$-0.649704\pi$
$139$	−10.6854			−0.906325			−0.453163	+	0.891428i	$0.649704\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			0.393270i			0.0328869i
$144$		0			0
$145$	−0.991325	−	1.56484i	−0.0823250	−	0.129953i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.05012			0.331799			0.165900	−	0.986143i	$-0.446947\pi$
$149$	4.05012			0.331799			0.165900	+	0.986143i	$0.446947\pi$
$150$		0			0
$151$	5.02856			0.409218			0.204609	−	0.978844i	$-0.434408\pi$
$151$	5.02856			0.409218			0.204609	+	0.978844i	$0.434408\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.98265	+	3.12969i	0.159250	+	0.251382i
$156$		0			0
$157$			24.4961i			1.95500i	0.210940	+	0.977499i	$0.432347\pi$
$157$			24.4961i			1.95500i	−0.210940	+	0.977499i	$0.567653\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.08378			0.321847
$162$		0			0
$163$		−	15.3137i		−	1.19946i	−0.800202	−	0.599731i	$-0.795274\pi$
$163$		−	15.3137i		−	1.19946i	0.800202	−	0.599731i	$-0.204726\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.7770i			1.22086i	0.792070	+	0.610430i	$0.209003\pi$
$167$			15.7770i			1.22086i	−0.792070	+	0.610430i	$0.790997\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	14.9706i		−	1.13819i	−0.822272	−	0.569095i	$-0.807294\pi$
$173$		−	14.9706i		−	1.13819i	0.822272	−	0.569095i	$-0.192706\pi$
$174$		0			0
$175$	−12.7865	−	6.04189i	−0.966572	−	0.456724i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.84578			−0.362190			−0.181095	−	0.983466i	$-0.557964\pi$
$179$	−4.84578			−0.362190			−0.181095	+	0.983466i	$0.557964\pi$
$180$		0			0
$181$	−7.99912			−0.594570			−0.297285	−	0.954789i	$-0.596081\pi$
$181$	−7.99912			−0.594570			−0.297285	+	0.954789i	$0.596081\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.8193	+	8.12101i	−0.942495	+	0.597069i
$186$		0			0
$187$		−	2.44339i		−	0.178679i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.4260			1.47798			0.738988	−	0.673718i	$-0.235304\pi$
$191$	20.4260			1.47798			0.738988	+	0.673718i	$0.235304\pi$
$192$		0			0
$193$		−	24.0237i		−	1.72926i	−0.502408	−	0.864630i	$-0.667553\pi$
$193$		−	24.0237i		−	1.72926i	0.502408	−	0.864630i	$-0.332447\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.80642i			0.342444i	0.985233	+	0.171222i	$0.0547715\pi$
$197$			4.80642i			0.342444i	−0.985233	+	0.171222i	$0.945229\pi$
$198$		0			0
$199$	14.9403			1.05909			0.529546	−	0.848281i	$-0.322362\pi$
$199$	14.9403			1.05909			0.529546	+	0.848281i	$0.322362\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	2.34315i		−	0.164457i
$204$		0			0
$205$	−4.52072	−	7.13613i	−0.315741	−	0.498409i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.88767			−0.199744
$210$		0			0
$211$	−1.82799			−0.125844			−0.0629220	−	0.998018i	$-0.520042\pi$
$211$	−1.82799			−0.125844			−0.0629220	+	0.998018i	$0.520042\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.74930	+	3.00868i	−0.323900	+	0.205190i
$216$		0			0
$217$			4.68629i			0.318126i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.21302			0.417933
$222$		0			0
$223$			7.07045i			0.473472i	0.971574	+	0.236736i	$0.0760777\pi$
$223$			7.07045i			0.473472i	−0.971574	+	0.236736i	$0.923922\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	5.43560i		−	0.360773i	−0.983596	−	0.180387i	$-0.942265\pi$
$227$		−	5.43560i		−	0.360773i	0.983596	−	0.180387i	$-0.0577349\pi$
$228$		0			0
$229$	−24.8147			−1.63980			−0.819899	−	0.572508i	$-0.805971\pi$
$229$	−24.8147			−1.63980			−0.819899	+	0.572508i	$0.805971\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			23.4256i			1.53466i	0.641251	+	0.767331i	$0.278416\pi$
$233$			23.4256i			1.53466i	−0.641251	+	0.767331i	$0.721584\pi$
$234$		0			0
$235$	5.68318	−	3.60029i	0.370730	−	0.234857i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−21.7779			−1.40869			−0.704346	−	0.709856i	$-0.748760\pi$
$239$	−21.7779			−1.40869			−0.704346	+	0.709856i	$0.748760\pi$
$240$		0			0
$241$	−15.2126			−0.979929			−0.489964	−	0.871742i	$-0.662990\pi$
$241$	−15.2126			−0.979929			−0.489964	+	0.871742i	$0.662990\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.19663	−	1.88893i	−0.0764502	−	0.120679i
$246$		0			0
$247$		−	7.34271i		−	0.467205i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.9399			1.13236			0.566178	−	0.824283i	$-0.308422\pi$
$251$	17.9399			1.13236			0.566178	+	0.824283i	$0.308422\pi$
$252$		0			0
$253$			0.567816i			0.0356983i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	3.88810i		−	0.242533i	−0.992620	−	0.121267i	$-0.961304\pi$
$257$		−	3.88810i		−	0.242533i	0.992620	−	0.121267i	$-0.0386956\pi$
$258$		0			0
$259$	−19.1952			−1.19273
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	0.296797i		−	0.0183013i	−0.999958	−	0.00915064i	$-0.997087\pi$
$263$		−	0.296797i		−	0.0183013i	0.999958	−	0.00915064i	$-0.00291278\pi$
$264$		0			0
$265$	18.0501	−	11.4347i	1.10881	−	0.702429i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.25447			−0.0764864			−0.0382432	−	0.999268i	$-0.512176\pi$
$269$	−1.25447			−0.0764864			−0.0382432	+	0.999268i	$0.512176\pi$
$270$		0			0
$271$	9.81510			0.596225			0.298112	−	0.954531i	$-0.403643\pi$
$271$	9.81510			0.596225			0.298112	+	0.954531i	$0.403643\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.840075	−	1.77786i	0.0506585	−	0.107209i
$276$		0			0
$277$		−	4.35603i		−	0.261729i	−0.991400	−	0.130864i	$-0.958225\pi$
$277$		−	4.35603i		−	0.261729i	0.991400	−	0.130864i	$-0.0417752\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.45207			−0.205933			−0.102967	−	0.994685i	$-0.532833\pi$
$281$	−3.45207			−0.205933			−0.102967	+	0.994685i	$0.532833\pi$
$282$		0			0
$283$		−	2.25937i		−	0.134306i	−0.997743	−	0.0671528i	$-0.978608\pi$
$283$		−	2.25937i		−	0.134306i	0.997743	−	0.0671528i	$-0.0213915\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	10.6854i		−	0.630740i
$288$		0			0
$289$	−21.6016			−1.27068
$290$		0			0
$291$		0			0
$292$		0			0
$293$			19.5358i			1.14130i	0.821195	+	0.570648i	$0.193308\pi$
$293$			19.5358i			1.14130i	−0.821195	+	0.570648i	$0.806692\pi$
$294$		0			0
$295$	0.470600	+	0.742860i	0.0273994	+	0.0432510i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.44383			−0.0834990
$300$		0			0
$301$	−7.11146			−0.409898
$302$		0			0
$303$		0			0
$304$		0			0
$305$	13.2126	+	20.8565i	0.756550	+	1.19424i
$306$		0			0
$307$		−	27.8142i		−	1.58744i	−0.608282	−	0.793721i	$-0.708141\pi$
$307$		−	27.8142i		−	1.58744i	0.608282	−	0.793721i	$-0.291859\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.10113			−0.119144			−0.0595719	−	0.998224i	$-0.518974\pi$
$311$	−2.10113			−0.119144			−0.0595719	+	0.998224i	$0.518974\pi$
$312$		0			0
$313$		−	9.64397i		−	0.545109i	−0.962140	−	0.272555i	$-0.912131\pi$
$313$		−	9.64397i		−	0.545109i	0.962140	−	0.272555i	$-0.0878685\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			16.1201i			0.905397i	0.891664	+	0.452698i	$0.149539\pi$
$317$			16.1201i			0.905397i	−0.891664	+	0.452698i	$0.850461\pi$
$318$		0			0
$319$	0.325795			0.0182410
$320$		0			0
$321$		0			0
$322$		0			0
$323$			45.6204i			2.53839i
$324$		0			0
$325$	4.52072	+	2.13613i	0.250765	+	0.118491i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.50982			0.469161
$330$		0			0
$331$	13.9537			0.766962			0.383481	−	0.923549i	$-0.374725\pi$
$331$	13.9537			0.766962			0.383481	+	0.923549i	$0.374725\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−29.3837	+	18.6145i	−1.60540	+	1.01702i
$336$		0			0
$337$			9.66974i			0.526744i	0.964694	+	0.263372i	$0.0848347\pi$
$337$			9.66974i			0.526744i	−0.964694	+	0.263372i	$0.915165\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−0.651591			−0.0352856
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	10.2014i		−	0.547638i	−0.961781	−	0.273819i	$-0.911713\pi$
$347$		−	10.2014i		−	0.547638i	0.961781	−	0.273819i	$-0.0882870\pi$
$348$		0			0
$349$	−28.7402			−1.53843			−0.769214	−	0.638992i	$-0.779352\pi$
$349$	−28.7402			−1.53843			−0.769214	+	0.638992i	$0.779352\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			20.9067i			1.11275i	0.830931	+	0.556375i	$0.187808\pi$
$353$			20.9067i			1.11275i	−0.830931	+	0.556375i	$0.812192\pi$
$354$		0			0
$355$	−7.85519	−	12.3997i	−0.416910	−	0.658109i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	19.3162			1.01947			0.509736	−	0.860331i	$-0.329743\pi$
$359$	19.3162			1.01947			0.509736	+	0.860331i	$0.329743\pi$
$360$		0			0
$361$	34.9153			1.83765
$362$		0			0
$363$		0			0
$364$		0			0
$365$	25.7177	−	16.2921i	1.34613	−	0.852770i
$366$		0			0
$367$			37.6957i			1.96770i	0.178990	+	0.983851i	$0.442717\pi$
$367$			37.6957i			1.96770i	−0.178990	+	0.983851i	$0.557283\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	27.0277			1.40321
$372$		0			0
$373$		−	22.4607i		−	1.16297i	−0.813556	−	0.581487i	$-0.802471\pi$
$373$		−	22.4607i		−	1.16297i	0.813556	−	0.581487i	$-0.197529\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.828427i			0.0426662i
$378$		0			0
$379$	24.3133			1.24889			0.624444	−	0.781069i	$-0.285325\pi$
$379$	24.3133			1.24889			0.624444	+	0.781069i	$0.285325\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			37.2498i			1.90338i	0.307065	+	0.951688i	$0.400653\pi$
$383$			37.2498i			1.90338i	−0.307065	+	0.951688i	$0.599347\pi$
$384$		0			0
$385$	2.10113	−	1.33106i	0.107083	−	0.0678370i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.467931			−0.0237250			−0.0118625	−	0.999930i	$-0.503776\pi$
$389$	−0.467931			−0.0237250			−0.0118625	+	0.999930i	$0.503776\pi$
$390$		0			0
$391$	8.97056			0.453661
$392$		0			0
$393$		0			0
$394$		0			0
$395$	17.8781	+	28.2213i	0.899545	+	1.41997i
$396$		0			0
$397$			34.5263i			1.73282i	0.499329	+	0.866412i	$0.333580\pi$
$397$			34.5263i			1.73282i	−0.499329	+	0.866412i	$0.666420\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.8781			−1.59192			−0.795958	−	0.605351i	$-0.793033\pi$
$401$	−31.8781			−1.59192			−0.795958	+	0.605351i	$0.793033\pi$
$402$		0			0
$403$		−	1.65685i		−	0.0825338i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	2.66894i		−	0.132294i
$408$		0			0
$409$	8.01735			0.396432			0.198216	−	0.980158i	$-0.436485\pi$
$409$	8.01735			0.396432			0.198216	+	0.980158i	$0.436485\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.11233i			0.0547344i
$414$		0			0
$415$	−31.0980	+	19.7005i	−1.52654	+	0.967061i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.40151			0.410440			0.205220	−	0.978716i	$-0.434209\pi$
$419$	8.40151			0.410440			0.205220	+	0.978716i	$0.434209\pi$
$420$		0			0
$421$	−11.7514			−0.572728			−0.286364	−	0.958121i	$-0.592447\pi$
$421$	−11.7514			−0.572728			−0.286364	+	0.958121i	$0.592447\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−28.0874	−	13.2718i	−1.36244	−	0.643778i
$426$		0			0
$427$			31.2299i			1.51132i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	20.5818			0.991388			0.495694	−	0.868497i	$-0.334914\pi$
$431$	20.5818			0.991388			0.495694	+	0.868497i	$0.334914\pi$
$432$		0			0
$433$			5.12522i			0.246303i	0.992388	+	0.123151i	$0.0393000\pi$
$433$			5.12522i			0.246303i	−0.992388	+	0.123151i	$0.960700\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	10.6016i		−	0.507145i
$438$		0			0
$439$	12.8868			0.615053			0.307526	−	0.951540i	$-0.400499\pi$
$439$	12.8868			0.615053			0.307526	+	0.951540i	$0.400499\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	35.1279i		−	1.66898i	−0.551024	−	0.834489i	$-0.685763\pi$
$443$		−	35.1279i		−	1.66898i	0.551024	−	0.834489i	$-0.314237\pi$
$444$		0			0
$445$	−11.5795	−	18.2787i	−0.548922	−	0.866494i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	18.2213			0.859914			0.429957	−	0.902849i	$-0.358529\pi$
$449$	18.2213			0.859914			0.429957	+	0.902849i	$0.358529\pi$
$450$		0			0
$451$	1.48572			0.0699598
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.38459	+	5.34271i	0.158672	+	0.250470i
$456$		0			0
$457$		−	23.4118i		−	1.09516i	−0.836753	−	0.547580i	$-0.815549\pi$
$457$		−	23.4118i		−	1.09516i	0.836753	−	0.547580i	$-0.184451\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.2913			0.665611			0.332805	−	0.942996i	$-0.392005\pi$
$461$	14.2913			0.665611			0.332805	+	0.942996i	$0.392005\pi$
$462$		0			0
$463$			23.7152i			1.10214i	0.834459	+	0.551070i	$0.185780\pi$
$463$			23.7152i			1.10214i	−0.834459	+	0.551070i	$0.814220\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	21.8989i		−	1.01336i	−0.862134	−	0.506680i	$-0.830873\pi$
$467$		−	21.8989i		−	1.01336i	0.862134	−	0.506680i	$-0.169127\pi$
$468$		0			0
$469$	−43.9982			−2.03165
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	0.988790i		−	0.0454646i
$474$		0			0
$475$	−15.6850	+	33.1944i	−0.719676	+	1.52306i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.0916			−0.780934			−0.390467	−	0.920617i	$-0.627686\pi$
$479$	−17.0916			−0.780934			−0.390467	+	0.920617i	$0.627686\pi$
$480$		0			0
$481$	6.78654			0.309440
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.0791227	+	0.0501241i	−0.00359278	+	0.00227602i
$486$		0			0
$487$			1.87434i			0.0849343i	0.999098	+	0.0424672i	$0.0135218\pi$
$487$			1.87434i			0.0849343i	−0.999098	+	0.0424672i	$0.986478\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.7704			0.486063			0.243031	−	0.970018i	$-0.421858\pi$
$491$	10.7704			0.486063			0.243031	+	0.970018i	$0.421858\pi$
$492$		0			0
$493$		−	5.14704i		−	0.231811i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	18.5669i		−	0.832841i
$498$		0			0
$499$	−33.5259			−1.50082			−0.750412	−	0.660971i	$-0.770145\pi$
$499$	−33.5259			−1.50082			−0.750412	+	0.660971i	$0.770145\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7.58473i		−	0.338186i	−0.985600	−	0.169093i	$-0.945916\pi$
$503$		−	7.58473i		−	0.338186i	0.985600	−	0.169093i	$-0.0540839\pi$
$504$		0			0
$505$	−14.5297	−	22.9357i	−0.646563	−	1.02062i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	38.0145			1.68497			0.842483	−	0.538724i	$-0.181093\pi$
$509$	38.0145			1.68497			0.842483	+	0.538724i	$0.181093\pi$
$510$		0			0
$511$	38.5089			1.70354
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.0327737	+	0.0207621i	−0.00144418	+	0.000914886i
$516$		0			0
$517$			1.18322i			0.0520380i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.2420			0.799197			0.399599	−	0.916690i	$-0.369149\pi$
$521$	18.2420			0.799197			0.399599	+	0.916690i	$0.369149\pi$
$522$		0			0
$523$		−	10.1720i		−	0.444791i	−0.974957	−	0.222396i	$-0.928612\pi$
$523$		−	10.1720i		−	0.444791i	0.974957	−	0.222396i	$-0.0713876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			10.2941i			0.448417i
$528$		0			0
$529$	20.9153			0.909363
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3.77786i			0.163637i
$534$		0			0
$535$	3.96888	−	2.51428i	0.171590	−	0.108702i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.393270			0.0169393
$540$		0			0
$541$	1.98923			0.0855236			0.0427618	−	0.999085i	$-0.486384\pi$
$541$	1.98923			0.0855236			0.0427618	+	0.999085i	$0.486384\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.43472	−	11.7360i	−0.318468	−	0.502714i
$546$		0			0
$547$			1.31817i			0.0563609i	0.999603	+	0.0281804i	$0.00897130\pi$
$547$			1.31817i			0.0563609i	−0.999603	+	0.0281804i	$0.991029\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.08290			−0.259140
$552$		0			0
$553$			42.2576i			1.79698i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	12.0804i		−	0.511861i	−0.966695	−	0.255931i	$-0.917618\pi$
$557$		−	12.0804i		−	0.511861i	0.966695	−	0.255931i	$-0.0823819\pi$
$558$		0			0
$559$	2.51428			0.106343
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	22.9930i		−	0.969039i	−0.874781	−	0.484519i	$-0.838995\pi$
$563$		−	22.9930i		−	0.969039i	0.874781	−	0.484519i	$-0.161005\pi$
$564$		0			0
$565$	−2.69453	+	1.70698i	−0.113360	+	0.0718131i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	29.0370			1.21729			0.608647	−	0.793441i	$-0.291713\pi$
$569$	29.0370			1.21729			0.608647	+	0.793441i	$0.291713\pi$
$570$		0			0
$571$	5.56950			0.233076			0.116538	−	0.993186i	$-0.462820\pi$
$571$	5.56950			0.233076			0.116538	+	0.993186i	$0.462820\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6.52717	+	3.08421i	0.272202	+	0.128621i
$576$		0			0
$577$		−	20.2822i		−	0.844357i	−0.906513	−	0.422179i	$-0.861266\pi$
$577$		−	20.2822i		−	0.844357i	0.906513	−	0.422179i	$-0.138734\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−46.5652			−1.93185
$582$		0			0
$583$			3.75798i			0.155640i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	8.82377i		−	0.364196i	−0.983280	−	0.182098i	$-0.941711\pi$
$587$		−	8.82377i		−	0.364196i	0.983280	−	0.182098i	$-0.0582888\pi$
$588$		0			0
$589$	12.1658			0.501283
$590$		0			0
$591$		0			0
$592$		0			0
$593$			10.2213i			0.419737i	0.977730	+	0.209868i	$0.0673036\pi$
$593$			10.2213i			0.419737i	−0.977730	+	0.209868i	$0.932696\pi$
$594$		0			0
$595$	−21.0286	−	33.1944i	−0.862087	−	1.36084i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.4925			−0.755582			−0.377791	−	0.925891i	$-0.623316\pi$
$599$	−18.4925			−0.755582			−0.377791	+	0.925891i	$0.623316\pi$
$600$		0			0
$601$	26.0129			1.06109			0.530544	−	0.847657i	$-0.321988\pi$
$601$	26.0129			1.06109			0.530544	+	0.847657i	$0.321988\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.9779	−	20.4861i	−0.527627	−	0.832879i
$606$		0			0
$607$			24.1227i			0.979109i	0.871973	+	0.489554i	$0.162841\pi$
$607$			24.1227i			0.979109i	−0.871973	+	0.489554i	$0.837159\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.00868			−0.121718
$612$		0			0
$613$		−	34.2056i		−	1.38155i	−0.723070	−	0.690775i	$-0.757270\pi$
$613$		−	34.2056i		−	1.38155i	0.723070	−	0.690775i	$-0.242730\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.3622i			0.497682i	0.968544	+	0.248841i	$0.0800496\pi$
$617$			12.3622i			0.497682i	−0.968544	+	0.248841i	$0.919950\pi$
$618$		0			0
$619$	11.6685			0.468997			0.234498	−	0.972117i	$-0.424655\pi$
$619$	11.6685			0.468997			0.234498	+	0.972117i	$0.424655\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	27.3700i		−	1.09655i
$624$		0			0
$625$	−15.8739	−	19.3137i	−0.634956	−	0.772548i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−42.1649			−1.68123
$630$		0			0
$631$	3.79775			0.151186			0.0755930	−	0.997139i	$-0.475915\pi$
$631$	3.79775			0.151186			0.0755930	+	0.997139i	$0.475915\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.83920	+	1.79863i	−0.112670	+	0.0713763i
$636$		0			0
$637$			1.00000i			0.0396214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.91008			0.272932			0.136466	−	0.990645i	$-0.456426\pi$
$641$	6.91008			0.272932			0.136466	+	0.990645i	$0.456426\pi$
$642$		0			0
$643$			2.50139i			0.0986452i	0.998783	+	0.0493226i	$0.0157062\pi$
$643$			2.50139i			0.0986452i	−0.998783	+	0.0493226i	$0.984294\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.3124i			1.19170i	0.803095	+	0.595852i	$0.203185\pi$
$647$			30.3124i			1.19170i	−0.803095	+	0.595852i	$0.796815\pi$
$648$		0			0
$649$	−0.154661			−0.00607098
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	33.4537i		−	1.30915i	−0.755999	−	0.654573i	$-0.772849\pi$
$653$		−	33.4537i		−	1.30915i	0.755999	−	0.654573i	$-0.227151\pi$
$654$		0			0
$655$	−4.61541	−	7.28559i	−0.180339	−	0.284671i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−31.2942			−1.21905			−0.609525	−	0.792767i	$-0.708640\pi$
$659$	−31.2942			−1.21905			−0.609525	+	0.792767i	$0.708640\pi$
$660$		0			0
$661$	40.1119			1.56017			0.780086	−	0.625672i	$-0.215175\pi$
$661$	40.1119			1.56017			0.780086	+	0.625672i	$0.215175\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−39.2299	+	24.8521i	−1.52127	+	0.963723i
$666$		0			0
$667$			1.19611i			0.0463136i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.34227			−0.167631
$672$		0			0
$673$		−	4.99386i		−	0.192499i	−0.995357	−	0.0962496i	$-0.969315\pi$
$673$		−	4.99386i		−	0.192499i	0.995357	−	0.0962496i	$-0.0306847\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			16.0174i			0.615597i	0.951452	+	0.307798i	$0.0995922\pi$
$677$			16.0174i			0.615597i	−0.951452	+	0.307798i	$0.900408\pi$
$678$		0			0
$679$	−0.118476			−0.00454668
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	35.5400i		−	1.35990i	−0.733258	−	0.679951i	$-0.762001\pi$
$683$		−	35.5400i		−	1.35990i	0.733258	−	0.679951i	$-0.237999\pi$
$684$		0			0
$685$	31.2890	−	19.8215i	1.19549	−	0.757342i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.55573			−0.364044
$690$		0			0
$691$	13.0393			0.496040			0.248020	−	0.968755i	$-0.420220\pi$
$691$	13.0393			0.496040			0.248020	+	0.968755i	$0.420220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.7865	−	20.1840i	−0.485021	−	0.765624i
$696$		0			0
$697$		−	23.4720i		−	0.889064i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.502632			0.0189841			0.00949207	−	0.999955i	$-0.496979\pi$
$701$	0.502632			0.0189841			0.00949207	+	0.999955i	$0.496979\pi$
$702$		0			0
$703$			49.8316i			1.87943i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	34.3431i		−	1.29161i
$708$		0			0
$709$	−42.0712			−1.58002			−0.790009	−	0.613095i	$-0.789924\pi$
$709$	−42.0712			−1.58002			−0.790009	+	0.613095i	$0.789924\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2.39222i		−	0.0895893i
$714$		0			0
$715$	−0.742860	+	0.470600i	−0.0277814	+	0.0175995i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.7848			1.22267			0.611333	−	0.791373i	$-0.290634\pi$
$719$	32.7848			1.22267			0.611333	+	0.791373i	$0.290634\pi$
$720$		0			0
$721$	−0.0490743			−0.00182762
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.76963	−	3.74509i	0.0657223	−	0.139089i
$726$		0			0
$727$		−	49.5246i		−	1.83677i	−0.395692	−	0.918383i	$-0.629495\pi$
$727$		−	49.5246i		−	1.83677i	0.395692	−	0.918383i	$-0.370505\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15.6213			−0.577774
$732$		0			0
$733$			2.11848i			0.0782477i	0.999234	+	0.0391238i	$0.0124567\pi$
$733$			2.11848i			0.0782477i	−0.999234	+	0.0391238i	$0.987543\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.11760i		−	0.225345i
$738$		0			0
$739$	47.7014			1.75473			0.877363	−	0.479827i	$-0.159301\pi$
$739$	47.7014			1.75473			0.877363	+	0.479827i	$0.159301\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			32.8055i			1.20352i	0.798677	+	0.601759i	$0.205533\pi$
$743$			32.8055i			1.20352i	−0.798677	+	0.601759i	$0.794467\pi$
$744$		0			0
$745$	4.84652	+	7.65041i	0.177563	+	0.280289i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	5.94288			0.217148
$750$		0			0
$751$	−0.852087			−0.0310931			−0.0155465	−	0.999879i	$-0.504949\pi$
$751$	−0.852087			−0.0310931			−0.0155465	+	0.999879i	$0.504949\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.01735	+	9.49861i	0.218994	+	0.345690i
$756$		0			0
$757$		−	6.68454i		−	0.242954i	−0.992594	−	0.121477i	$-0.961237\pi$
$757$		−	6.68454i		−	0.242954i	0.992594	−	0.121477i	$-0.0387630\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.92578			0.106059			0.0530297	−	0.998593i	$-0.483112\pi$
$761$	2.92578			0.106059			0.0530297	+	0.998593i	$0.483112\pi$
$762$		0			0
$763$		−	17.5731i		−	0.636188i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	0.393270i		−	0.0142001i
$768$		0			0
$769$	12.6274			0.455356			0.227678	−	0.973736i	$-0.426887\pi$
$769$	12.6274			0.455356			0.227678	+	0.973736i	$0.426887\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			15.9454i			0.573517i	0.958003	+	0.286758i	$0.0925777\pi$
$773$			15.9454i			0.573517i	−0.958003	+	0.286758i	$0.907422\pi$
$774$		0			0
$775$	−3.53926	+	7.49018i	−0.127134	+	0.269055i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−27.7398			−0.993880
$780$		0			0
$781$	2.58158			0.0923763
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−46.2714	+	29.3128i	−1.65150	+	1.04622i
$786$		0			0
$787$			48.5825i			1.73178i	0.500234	+	0.865890i	$0.333247\pi$
$787$			48.5825i			1.73178i	−0.500234	+	0.865890i	$0.666753\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−4.03470			−0.143457
$792$		0			0
$793$		−	11.0414i		−	0.392093i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	19.8971i		−	0.704792i	−0.935851	−	0.352396i	$-0.885367\pi$
$797$		−	19.8971i		−	0.704792i	0.935851	−	0.352396i	$-0.114633\pi$
$798$		0			0
$799$	18.6930			0.661310
$800$		0			0
$801$		0			0
$802$		0			0
$803$			5.35436i			0.188951i
$804$		0			0
$805$	4.88679	+	7.71397i	0.172237	+	0.271882i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.5487			−0.441189			−0.220595	−	0.975366i	$-0.570800\pi$
$809$	−12.5487			−0.441189			−0.220595	+	0.975366i	$0.570800\pi$
$810$		0			0
$811$	30.9149			1.08557			0.542785	−	0.839872i	$-0.317370\pi$
$811$	30.9149			1.08557			0.542785	+	0.839872i	$0.317370\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	28.9266	−	18.3249i	1.01325	−	0.641894i
$816$		0			0
$817$			18.4616i			0.645890i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	29.9308			1.04459			0.522296	−	0.852765i	$-0.325076\pi$
$821$	29.9308			1.04459			0.522296	+	0.852765i	$0.325076\pi$
$822$		0			0
$823$			16.1831i			0.564109i	0.959398	+	0.282055i	$0.0910159\pi$
$823$			16.1831i			0.564109i	−0.959398	+	0.282055i	$0.908984\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	54.2022i		−	1.88479i	−0.334497	−	0.942397i	$-0.608566\pi$
$827$		−	54.2022i		−	1.88479i	0.334497	−	0.942397i	$-0.391434\pi$
$828$		0			0
$829$	42.2929			1.46889			0.734447	−	0.678666i	$-0.237442\pi$
$829$	42.2929			1.46889			0.734447	+	0.678666i	$0.237442\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	6.21302i		−	0.215268i
$834$		0			0
$835$	−29.8017	+	18.8793i	−1.03133	+	0.653345i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	36.7873			1.27004			0.635020	−	0.772496i	$-0.280992\pi$
$839$	36.7873			1.27004			0.635020	+	0.772496i	$0.280992\pi$
$840$		0			0
$841$	−28.3137			−0.976335
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.19663	−	1.88893i	−0.0411655	−	0.0649812i
$846$		0			0
$847$		−	30.6753i		−	1.05401i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	9.79863			0.335893
$852$		0			0
$853$			15.4148i			0.527794i	0.964551	+	0.263897i	$0.0850079\pi$
$853$			15.4148i			0.527794i	−0.964551	+	0.263897i	$0.914992\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	0.0628221i		−	0.00214596i	−0.999999	−	0.00107298i	$-0.999658\pi$
$857$		−	0.0628221i		−	0.00214596i	0.999999	−	0.00107298i	$-0.000341540\pi$
$858$		0			0
$859$	−28.6577			−0.977787			−0.488893	−	0.872344i	$-0.662599\pi$
$859$	−28.6577			−0.977787			−0.488893	+	0.872344i	$0.662599\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	5.96189i		−	0.202945i	−0.994838	−	0.101473i	$-0.967645\pi$
$863$		−	5.96189i		−	0.202945i	0.994838	−	0.101473i	$-0.0323554\pi$
$864$		0			0
$865$	28.2784	−	17.9143i	0.961494	−	0.609104i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.87558			−0.199315
$870$		0			0
$871$	15.5557			0.527086
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.88810	−	31.3828i	−0.131442	−	1.06093i
$876$		0			0
$877$		−	6.70276i		−	0.226336i	−0.993576	−	0.113168i	$-0.963900\pi$
$877$		−	6.70276i		−	0.226336i	0.993576	−	0.113168i	$-0.0360999\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−39.8624			−1.34300			−0.671500	−	0.741005i	$-0.734349\pi$
$881$	−39.8624			−1.34300			−0.671500	+	0.741005i	$0.734349\pi$
$882$		0			0
$883$		−	21.7131i		−	0.730704i	−0.930869	−	0.365352i	$-0.880949\pi$
$883$		−	21.7131i		−	0.730704i	0.930869	−	0.365352i	$-0.119051\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	51.3757i		−	1.72503i	−0.506035	−	0.862513i	$-0.668889\pi$
$887$		−	51.3757i		−	1.72503i	0.506035	−	0.862513i	$-0.331111\pi$
$888$		0			0
$889$	−4.25133			−0.142585
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	22.0918i		−	0.739275i
$894$		0			0
$895$	−5.79863	−	9.15335i	−0.193827	−	0.305963i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.37258			−0.0457782
$900$		0			0
$901$	59.3700			1.97790
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9.57203	−	15.1098i	−0.318185	−	0.502267i
$906$		0			0
$907$		−	21.3966i		−	0.710463i	−0.934778	−	0.355231i	$-0.884402\pi$
$907$		−	21.3966i		−	0.710463i	0.934778	−	0.355231i	$-0.115598\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	51.0044			1.68985			0.844925	−	0.534884i	$-0.179645\pi$
$911$	51.0044			1.68985			0.844925	+	0.534884i	$0.179645\pi$
$912$		0			0
$913$		−	6.47451i		−	0.214275i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	10.9092i		−	0.360254i
$918$		0			0
$919$	−11.3664			−0.374942			−0.187471	−	0.982270i	$-0.560029\pi$
$919$	−11.3664			−0.374942			−0.187471	+	0.982270i	$0.560029\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			6.56440i			0.216070i
$924$		0			0
$925$	−30.6801	−	14.4969i	−1.00875	−	0.476656i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	47.9344			1.57268			0.786338	−	0.617797i	$-0.211975\pi$
$929$	47.9344			1.57268			0.786338	+	0.617797i	$0.211975\pi$
$930$		0			0
$931$	−7.34271			−0.240648
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.61541	−	2.92385i	0.150940	−	0.0956201i
$936$		0			0
$937$		−	23.3266i		−	0.762047i	−0.924565	−	0.381023i	$-0.875572\pi$
$937$		−	23.3266i		−	0.762047i	0.924565	−	0.381023i	$-0.124428\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	39.6316			1.29195			0.645977	−	0.763357i	$-0.276450\pi$
$941$	39.6316			1.29195			0.645977	+	0.763357i	$0.276450\pi$
$942$		0			0
$943$			5.45460i			0.177626i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			19.7770i			0.642666i	0.946966	+	0.321333i	$0.104131\pi$
$947$			19.7770i			0.642666i	−0.946966	+	0.321333i	$0.895869\pi$
$948$		0			0
$949$	−13.6150			−0.441961
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	45.2847i		−	1.46692i	−0.679735	−	0.733458i	$-0.737905\pi$
$953$		−	45.2847i		−	1.46692i	0.679735	−	0.733458i	$-0.262095\pi$
$954$		0			0
$955$	24.4425	+	38.5834i	0.790941	+	1.24853i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	46.8512			1.51290
$960$		0			0
$961$	−28.2548			−0.911446
$962$		0			0
$963$		0			0
$964$		0			0
$965$	45.3791	−	28.7475i	1.46080	−	0.925416i
$966$		0			0
$967$			13.3903i			0.430603i	0.976548	+	0.215301i	$0.0690734\pi$
$967$			13.3903i			0.430603i	−0.976548	+	0.215301i	$0.930927\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	39.6150			1.27130			0.635652	−	0.771975i	$-0.280731\pi$
$971$	39.6150			1.27130			0.635652	+	0.771975i	$0.280731\pi$
$972$		0			0
$973$		−	30.2229i		−	0.968902i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			19.2100i			0.614584i	0.951615	+	0.307292i	$0.0994228\pi$
$977$			19.2100i			0.614584i	−0.951615	+	0.307292i	$0.900577\pi$
$978$		0			0
$979$	3.80557			0.121627
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	32.3960i		−	1.03327i	−0.856205	−	0.516636i	$-0.827184\pi$
$983$		−	32.3960i		−	1.03327i	0.856205	−	0.516636i	$-0.172816\pi$
$984$		0			0
$985$	−9.07901	+	5.75154i	−0.289281	+	0.183259i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.63020			0.115434
$990$		0			0
$991$	10.5107			0.333883			0.166942	−	0.985967i	$-0.446611\pi$
$991$	10.5107			0.333883			0.166942	+	0.985967i	$0.446611\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17.8781	+	28.2213i	0.566774	+	0.894674i
$996$		0			0
$997$			41.0753i			1.30087i	0.759563	+	0.650433i	$0.225413\pi$
$997$			41.0753i			1.30087i	−0.759563	+	0.650433i	$0.774587\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.f.2809.6		8
3.2	odd	2		1560.2.l.e.1249.6	yes	8
5.4	even	2	inner	4680.2.l.f.2809.5		8
12.11	even	2		3120.2.l.o.1249.2		8
15.2	even	4		7800.2.a.bw.1.2		4
15.8	even	4		7800.2.a.bv.1.4		4
15.14	odd	2		1560.2.l.e.1249.2	✓	8
60.59	even	2		3120.2.l.o.1249.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.2	✓	8	15.14	odd	2
1560.2.l.e.1249.6	yes	8	3.2	odd	2
3120.2.l.o.1249.2		8	12.11	even	2
3120.2.l.o.1249.6		8	60.59	even	2
4680.2.l.f.2809.5		8	5.4	even	2	inner
4680.2.l.f.2809.6		8	1.1	even	1	trivial
7800.2.a.bv.1.4		4	15.8	even	4
7800.2.a.bw.1.2		4	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.19663 + 1.88893i) q^{5} +2.82843i q^{7} +O(q^{10})\)	\(q+(1.19663 + 1.88893i) q^{5} +2.82843i q^{7} -0.393270 q^{11} -1.00000i q^{13} +6.21302i q^{17} +7.34271 q^{19} -1.44383i q^{23} +(-2.13613 + 4.52072i) q^{25} -0.828427 q^{29} +1.65685 q^{31} +(-5.34271 + 3.38459i) q^{35} +6.78654i q^{37} -3.77786 q^{41} +2.51428i q^{43} -3.00868i q^{47} -1.00000 q^{49} -9.55573i q^{53} +(-0.470600 - 0.742860i) q^{55} +0.393270 q^{59} +11.0414 q^{61} +(1.88893 - 1.19663i) q^{65} +15.5557i q^{67} -6.56440 q^{71} -13.6150i q^{73} -1.11233i q^{77} +14.9403 q^{79} +16.4633i q^{83} +(-11.7360 + 7.43472i) q^{85} -9.67674 q^{89} +2.82843 q^{91} +(8.78654 + 13.8699i) q^{95} +0.0418875i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{11} + 24 q^{19} - 4 q^{25} + 16 q^{29} - 32 q^{31} - 8 q^{35} + 8 q^{41} - 8 q^{49} - 36 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{71} + 24 q^{79} - 40 q^{85} - 8 q^{89} + 32 q^{95}+O(q^{100}) \) 8 * q + 16 * q^11 + 24 * q^19 - 4 * q^25 + 16 * q^29 - 32 * q^31 - 8 * q^35 + 8 * q^41 - 8 * q^49 - 36 * q^55 - 16 * q^59 + 24 * q^61 - 4 * q^65 + 24 * q^71 + 24 * q^79 - 40 * q^85 - 8 * q^89 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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