LMFDB - Embedded newform 7623.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.a (trivial)

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:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{2} +3.30278 q^{4} -3.60555 q^{5} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})$	$q+2.30278 q^{2} +3.30278 q^{4} -3.60555 q^{5} +1.00000 q^{7} +3.00000 q^{8} -8.30278 q^{10} -6.60555 q^{13} +2.30278 q^{14} +0.302776 q^{16} +2.69722 q^{17} +3.00000 q^{19} -11.9083 q^{20} +2.69722 q^{23} +8.00000 q^{25} -15.2111 q^{26} +3.30278 q^{28} +4.69722 q^{29} +1.00000 q^{31} -5.30278 q^{32} +6.21110 q^{34} -3.60555 q^{35} -5.21110 q^{37} +6.90833 q^{38} -10.8167 q^{40} +7.00000 q^{41} -1.69722 q^{43} +6.21110 q^{46} +1.90833 q^{47} +1.00000 q^{49} +18.4222 q^{50} -21.8167 q^{52} +12.9083 q^{53} +3.00000 q^{56} +10.8167 q^{58} -6.69722 q^{59} +4.30278 q^{61} +2.30278 q^{62} -12.8167 q^{64} +23.8167 q^{65} +8.51388 q^{67} +8.90833 q^{68} -8.30278 q^{70} +4.30278 q^{71} +5.00000 q^{73} -12.0000 q^{74} +9.90833 q^{76} +8.30278 q^{79} -1.09167 q^{80} +16.1194 q^{82} +3.00000 q^{83} -9.72498 q^{85} -3.90833 q^{86} +14.7250 q^{89} -6.60555 q^{91} +8.90833 q^{92} +4.39445 q^{94} -10.8167 q^{95} -3.60555 q^{97} +2.30278 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8	$ 2 q + q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8} - 13 q^{10} - 6 q^{13} + q^{14} - 3 q^{16} + 9 q^{17} + 6 q^{19} - 13 q^{20} + 9 q^{23} + 16 q^{25} - 16 q^{26} + 3 q^{28} + 13 q^{29} + 2 q^{31} - 7 q^{32} - 2 q^{34} + 4 q^{37} + 3 q^{38} + 14 q^{41} - 7 q^{43} - 2 q^{46} - 7 q^{47} + 2 q^{49} + 8 q^{50} - 22 q^{52} + 15 q^{53} + 6 q^{56} - 17 q^{59} + 5 q^{61} + q^{62} - 4 q^{64} + 26 q^{65} - q^{67} + 7 q^{68} - 13 q^{70} + 5 q^{71} + 10 q^{73} - 24 q^{74} + 9 q^{76} + 13 q^{79} - 13 q^{80} + 7 q^{82} + 6 q^{83} + 13 q^{85} + 3 q^{86} - 3 q^{89} - 6 q^{91} + 7 q^{92} + 16 q^{94} + q^{98}+O(q^{100}) $ 2 * q + q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8 - 13 * q^10 - 6 * q^13 + q^14 - 3 * q^16 + 9 * q^17 + 6 * q^19 - 13 * q^20 + 9 * q^23 + 16 * q^25 - 16 * q^26 + 3 * q^28 + 13 * q^29 + 2 * q^31 - 7 * q^32 - 2 * q^34 + 4 * q^37 + 3 * q^38 + 14 * q^41 - 7 * q^43 - 2 * q^46 - 7 * q^47 + 2 * q^49 + 8 * q^50 - 22 * q^52 + 15 * q^53 + 6 * q^56 - 17 * q^59 + 5 * q^61 + q^62 - 4 * q^64 + 26 * q^65 - q^67 + 7 * q^68 - 13 * q^70 + 5 * q^71 + 10 * q^73 - 24 * q^74 + 9 * q^76 + 13 * q^79 - 13 * q^80 + 7 * q^82 + 6 * q^83 + 13 * q^85 + 3 * q^86 - 3 * q^89 - 6 * q^91 + 7 * q^92 + 16 * q^94 + q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.30278	1.62831	0.814154	−	0.580649i	$-0.197201\pi$
$2$	2.30278	1.62831	0.814154	+	0.580649i	$0.197201\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	−3.60555	−1.61245	−0.806226	−	0.591608i	$-0.798493\pi$
$5$	−3.60555	−1.61245	−0.806226	+	0.591608i	$0.798493\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.00000	1.06066
$9$	0	0
$10$	−8.30278	−2.62557
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.60555	−1.83205	−0.916025	−	0.401121i	$-0.868621\pi$
$13$	−6.60555	−1.83205	−0.916025	+	0.401121i	$0.868621\pi$
$14$	2.30278	0.615443
$15$	0	0
$16$	0.302776	0.0756939
$17$	2.69722	0.654173	0.327086	−	0.944994i	$-0.393933\pi$
$17$	2.69722	0.654173	0.327086	+	0.944994i	$0.393933\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	−11.9083	−2.66278
$21$	0	0
$22$	0	0
$23$	2.69722	0.562410	0.281205	−	0.959648i	$-0.409266\pi$
$23$	2.69722	0.562410	0.281205	+	0.959648i	$0.409266\pi$
$24$	0	0
$25$	8.00000	1.60000
$26$	−15.2111	−2.98314
$27$	0	0
$28$	3.30278	0.624166
$29$	4.69722	0.872253	0.436126	−	0.899885i	$-0.356350\pi$
$29$	4.69722	0.872253	0.436126	+	0.899885i	$0.356350\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−5.30278	−0.937407
$33$	0	0
$34$	6.21110	1.06520
$35$	−3.60555	−0.609449
$36$	0	0
$37$	−5.21110	−0.856700	−0.428350	−	0.903613i	$-0.640905\pi$
$37$	−5.21110	−0.856700	−0.428350	+	0.903613i	$0.640905\pi$
$38$	6.90833	1.12068
$39$	0	0
$40$	−10.8167	−1.71026
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−1.69722	−0.258824	−0.129412	−	0.991591i	$-0.541309\pi$
$43$	−1.69722	−0.258824	−0.129412	+	0.991591i	$0.541309\pi$
$44$	0	0
$45$	0	0
$46$	6.21110	0.915777
$47$	1.90833	0.278358	0.139179	−	0.990267i	$-0.455554\pi$
$47$	1.90833	0.278358	0.139179	+	0.990267i	$0.455554\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	18.4222	2.60529
$51$	0	0
$52$	−21.8167	−3.02543
$53$	12.9083	1.77310	0.886548	−	0.462638i	$-0.153097\pi$
$53$	12.9083	1.77310	0.886548	+	0.462638i	$0.153097\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	10.8167	1.42030
$59$	−6.69722	−0.871904	−0.435952	−	0.899970i	$-0.643588\pi$
$59$	−6.69722	−0.871904	−0.435952	+	0.899970i	$0.643588\pi$
$60$	0	0
$61$	4.30278	0.550914	0.275457	−	0.961313i	$-0.411171\pi$
$61$	4.30278	0.550914	0.275457	+	0.961313i	$0.411171\pi$
$62$	2.30278	0.292453
$63$	0	0
$64$	−12.8167	−1.60208
$65$	23.8167	2.95409
$66$	0	0
$67$	8.51388	1.04014	0.520068	−	0.854125i	$-0.325907\pi$
$67$	8.51388	1.04014	0.520068	+	0.854125i	$0.325907\pi$
$68$	8.90833	1.08029
$69$	0	0
$70$	−8.30278	−0.992371
$71$	4.30278	0.510646	0.255323	−	0.966856i	$-0.417818\pi$
$71$	4.30278	0.510646	0.255323	+	0.966856i	$0.417818\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	−12.0000	−1.39497
$75$	0	0
$76$	9.90833	1.13656
$77$	0	0
$78$	0	0
$79$	8.30278	0.934135	0.467068	−	0.884222i	$-0.345310\pi$
$79$	8.30278	0.934135	0.467068	+	0.884222i	$0.345310\pi$
$80$	−1.09167	−0.122053
$81$	0	0
$82$	16.1194	1.78009
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	−9.72498	−1.05482
$86$	−3.90833	−0.421446
$87$	0	0
$88$	0	0
$89$	14.7250	1.56084	0.780422	−	0.625253i	$-0.215004\pi$
$89$	14.7250	1.56084	0.780422	+	0.625253i	$0.215004\pi$
$90$	0	0
$91$	−6.60555	−0.692450
$92$	8.90833	0.928757
$93$	0	0
$94$	4.39445	0.453253
$95$	−10.8167	−1.10977
$96$	0	0
$97$	−3.60555	−0.366088	−0.183044	−	0.983105i	$-0.558595\pi$
$97$	−3.60555	−0.366088	−0.183044	+	0.983105i	$0.558595\pi$
$98$	2.30278	0.232615
$99$	0	0
$100$	26.4222	2.64222
$101$	−5.30278	−0.527646	−0.263823	−	0.964571i	$-0.584983\pi$
$101$	−5.30278	−0.527646	−0.263823	+	0.964571i	$0.584983\pi$
$102$	0	0
$103$	−14.3028	−1.40929	−0.704647	−	0.709558i	$-0.748895\pi$
$103$	−14.3028	−1.40929	−0.704647	+	0.709558i	$0.748895\pi$
$104$	−19.8167	−1.94318
$105$	0	0
$106$	29.7250	2.88715
$107$	12.3944	1.19822	0.599108	−	0.800668i	$-0.295522\pi$
$107$	12.3944	1.19822	0.599108	+	0.800668i	$0.295522\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	0	0
$112$	0.302776	0.0286096
$113$	10.6972	1.00631	0.503155	−	0.864196i	$-0.332172\pi$
$113$	10.6972	1.00631	0.503155	+	0.864196i	$0.332172\pi$
$114$	0	0
$115$	−9.72498	−0.906859
$116$	15.5139	1.44043
$117$	0	0
$118$	−15.4222	−1.41973
$119$	2.69722	0.247254
$120$	0	0
$121$	0	0
$122$	9.90833	0.897058
$123$	0	0
$124$	3.30278	0.296598
$125$	−10.8167	−0.967471
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−18.9083	−1.67128
$129$	0	0
$130$	54.8444	4.81017
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	19.6056	1.69366
$135$	0	0
$136$	8.09167	0.693855
$137$	10.1194	0.864561	0.432281	−	0.901739i	$-0.357709\pi$
$137$	10.1194	0.864561	0.432281	+	0.901739i	$0.357709\pi$
$138$	0	0
$139$	−21.6056	−1.83256	−0.916279	−	0.400540i	$-0.868823\pi$
$139$	−21.6056	−1.83256	−0.916279	+	0.400540i	$0.868823\pi$
$140$	−11.9083	−1.00644
$141$	0	0
$142$	9.90833	0.831488
$143$	0	0
$144$	0	0
$145$	−16.9361	−1.40647
$146$	11.5139	0.952895
$147$	0	0
$148$	−17.2111	−1.41474
$149$	−4.90833	−0.402106	−0.201053	−	0.979580i	$-0.564436\pi$
$149$	−4.90833	−0.402106	−0.201053	+	0.979580i	$0.564436\pi$
$150$	0	0
$151$	0.211103	0.0171793	0.00858964	−	0.999963i	$-0.497266\pi$
$151$	0.211103	0.0171793	0.00858964	+	0.999963i	$0.497266\pi$
$152$	9.00000	0.729996
$153$	0	0
$154$	0	0
$155$	−3.60555	−0.289605
$156$	0	0
$157$	7.21110	0.575509	0.287754	−	0.957704i	$-0.407091\pi$
$157$	7.21110	0.575509	0.287754	+	0.957704i	$0.407091\pi$
$158$	19.1194	1.52106
$159$	0	0
$160$	19.1194	1.51152
$161$	2.69722	0.212571
$162$	0	0
$163$	2.81665	0.220617	0.110309	−	0.993897i	$-0.464816\pi$
$163$	2.81665	0.220617	0.110309	+	0.993897i	$0.464816\pi$
$164$	23.1194	1.80532
$165$	0	0
$166$	6.90833	0.536190
$167$	−5.21110	−0.403247	−0.201624	−	0.979463i	$-0.564622\pi$
$167$	−5.21110	−0.403247	−0.201624	+	0.979463i	$0.564622\pi$
$168$	0	0
$169$	30.6333	2.35641
$170$	−22.3944	−1.71758
$171$	0	0
$172$	−5.60555	−0.427419
$173$	−1.30278	−0.0990482	−0.0495241	−	0.998773i	$-0.515770\pi$
$173$	−1.30278	−0.0990482	−0.0495241	+	0.998773i	$0.515770\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	0	0
$178$	33.9083	2.54154
$179$	6.39445	0.477944	0.238972	−	0.971027i	$-0.423190\pi$
$179$	6.39445	0.477944	0.238972	+	0.971027i	$0.423190\pi$
$180$	0	0
$181$	−25.2111	−1.87393	−0.936963	−	0.349428i	$-0.886376\pi$
$181$	−25.2111	−1.87393	−0.936963	+	0.349428i	$0.886376\pi$
$182$	−15.2111	−1.12752
$183$	0	0
$184$	8.09167	0.596526
$185$	18.7889	1.38139
$186$	0	0
$187$	0	0
$188$	6.30278	0.459677
$189$	0	0
$190$	−24.9083	−1.80704
$191$	26.8167	1.94038	0.970192	−	0.242336i	$-0.0779135\pi$
$191$	26.8167	1.94038	0.970192	+	0.242336i	$0.0779135\pi$
$192$	0	0
$193$	2.11943	0.152560	0.0762799	−	0.997086i	$-0.475696\pi$
$193$	2.11943	0.152560	0.0762799	+	0.997086i	$0.475696\pi$
$194$	−8.30278	−0.596105
$195$	0	0
$196$	3.30278	0.235913
$197$	−10.6056	−0.755614	−0.377807	−	0.925884i	$-0.623322\pi$
$197$	−10.6056	−0.755614	−0.377807	+	0.925884i	$0.623322\pi$
$198$	0	0
$199$	19.4222	1.37680	0.688402	−	0.725330i	$-0.258313\pi$
$199$	19.4222	1.37680	0.688402	+	0.725330i	$0.258313\pi$
$200$	24.0000	1.69706
$201$	0	0
$202$	−12.2111	−0.859170
$203$	4.69722	0.329681
$204$	0	0
$205$	−25.2389	−1.76276
$206$	−32.9361	−2.29477
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−15.5139	−1.06802	−0.534010	−	0.845478i	$-0.679315\pi$
$211$	−15.5139	−1.06802	−0.534010	+	0.845478i	$0.679315\pi$
$212$	42.6333	2.92807
$213$	0	0
$214$	28.5416	1.95107
$215$	6.11943	0.417342
$216$	0	0
$217$	1.00000	0.0678844
$218$	18.4222	1.24771
$219$	0	0
$220$	0	0
$221$	−17.8167	−1.19848
$222$	0	0
$223$	13.9083	0.931370	0.465685	−	0.884950i	$-0.345808\pi$
$223$	13.9083	0.931370	0.465685	+	0.884950i	$0.345808\pi$
$224$	−5.30278	−0.354307
$225$	0	0
$226$	24.6333	1.63858
$227$	6.51388	0.432341	0.216171	−	0.976356i	$-0.430643\pi$
$227$	6.51388	0.432341	0.216171	+	0.976356i	$0.430643\pi$
$228$	0	0
$229$	−2.60555	−0.172180	−0.0860898	−	0.996287i	$-0.527437\pi$
$229$	−2.60555	−0.172180	−0.0860898	+	0.996287i	$0.527437\pi$
$230$	−22.3944	−1.47665
$231$	0	0
$232$	14.0917	0.925164
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−6.88057	−0.448839
$236$	−22.1194	−1.43985
$237$	0	0
$238$	6.21110	0.402606
$239$	26.3305	1.70318	0.851590	−	0.524208i	$-0.175639\pi$
$239$	26.3305	1.70318	0.851590	+	0.524208i	$0.175639\pi$
$240$	0	0
$241$	−0.486122	−0.0313139	−0.0156569	−	0.999877i	$-0.504984\pi$
$241$	−0.486122	−0.0313139	−0.0156569	+	0.999877i	$0.504984\pi$
$242$	0	0
$243$	0	0
$244$	14.2111	0.909773
$245$	−3.60555	−0.230350
$246$	0	0
$247$	−19.8167	−1.26090
$248$	3.00000	0.190500
$249$	0	0
$250$	−24.9083	−1.57534
$251$	−4.39445	−0.277375	−0.138688	−	0.990336i	$-0.544288\pi$
$251$	−4.39445	−0.277375	−0.138688	+	0.990336i	$0.544288\pi$
$252$	0	0
$253$	0	0
$254$	−4.60555	−0.288978
$255$	0	0
$256$	−17.9083	−1.11927
$257$	−6.48612	−0.404593	−0.202297	−	0.979324i	$-0.564840\pi$
$257$	−6.48612	−0.404593	−0.202297	+	0.979324i	$0.564840\pi$
$258$	0	0
$259$	−5.21110	−0.323802
$260$	78.6611	4.87835
$261$	0	0
$262$	−13.8167	−0.853596
$263$	20.2389	1.24798	0.623991	−	0.781432i	$-0.285510\pi$
$263$	20.2389	1.24798	0.623991	+	0.781432i	$0.285510\pi$
$264$	0	0
$265$	−46.5416	−2.85903
$266$	6.90833	0.423577
$267$	0	0
$268$	28.1194	1.71767
$269$	−16.1194	−0.982819	−0.491409	−	0.870929i	$-0.663518\pi$
$269$	−16.1194	−0.982819	−0.491409	+	0.870929i	$0.663518\pi$
$270$	0	0
$271$	28.5139	1.73209	0.866047	−	0.499962i	$-0.166653\pi$
$271$	28.5139	1.73209	0.866047	+	0.499962i	$0.166653\pi$
$272$	0.816654	0.0495169
$273$	0	0
$274$	23.3028	1.40777
$275$	0	0
$276$	0	0
$277$	−29.9361	−1.79868	−0.899342	−	0.437245i	$-0.855954\pi$
$277$	−29.9361	−1.79868	−0.899342	+	0.437245i	$0.855954\pi$
$278$	−49.7527	−2.98397
$279$	0	0
$280$	−10.8167	−0.646419
$281$	6.39445	0.381461	0.190730	−	0.981642i	$-0.438914\pi$
$281$	6.39445	0.381461	0.190730	+	0.981642i	$0.438914\pi$
$282$	0	0
$283$	−4.39445	−0.261223	−0.130611	−	0.991434i	$-0.541694\pi$
$283$	−4.39445	−0.261223	−0.130611	+	0.991434i	$0.541694\pi$
$284$	14.2111	0.843274
$285$	0	0
$286$	0	0
$287$	7.00000	0.413197
$288$	0	0
$289$	−9.72498	−0.572058
$290$	−39.0000	−2.29016
$291$	0	0
$292$	16.5139	0.966402
$293$	4.81665	0.281392	0.140696	−	0.990053i	$-0.455066\pi$
$293$	4.81665	0.281392	0.140696	+	0.990053i	$0.455066\pi$
$294$	0	0
$295$	24.1472	1.40590
$296$	−15.6333	−0.908668
$297$	0	0
$298$	−11.3028	−0.654752
$299$	−17.8167	−1.03036
$300$	0	0
$301$	−1.69722	−0.0978264
$302$	0.486122	0.0279732
$303$	0	0
$304$	0.908327	0.0520961
$305$	−15.5139	−0.888322
$306$	0	0
$307$	16.6333	0.949313	0.474657	−	0.880171i	$-0.342572\pi$
$307$	16.6333	0.949313	0.474657	+	0.880171i	$0.342572\pi$
$308$	0	0
$309$	0	0
$310$	−8.30278	−0.471566
$311$	−6.39445	−0.362596	−0.181298	−	0.983428i	$-0.558030\pi$
$311$	−6.39445	−0.362596	−0.181298	+	0.983428i	$0.558030\pi$
$312$	0	0
$313$	15.3028	0.864964	0.432482	−	0.901643i	$-0.357638\pi$
$313$	15.3028	0.864964	0.432482	+	0.901643i	$0.357638\pi$
$314$	16.6056	0.937105
$315$	0	0
$316$	27.4222	1.54262
$317$	−17.3028	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$317$	−17.3028	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$318$	0	0
$319$	0	0
$320$	46.2111	2.58328
$321$	0	0
$322$	6.21110	0.346131
$323$	8.09167	0.450233
$324$	0	0
$325$	−52.8444	−2.93128
$326$	6.48612	0.359233
$327$	0	0
$328$	21.0000	1.15953
$329$	1.90833	0.105209
$330$	0	0
$331$	23.8167	1.30908	0.654541	−	0.756027i	$-0.272862\pi$
$331$	23.8167	1.30908	0.654541	+	0.756027i	$0.272862\pi$
$332$	9.90833	0.543790
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−30.6972	−1.67717
$336$	0	0
$337$	−11.7889	−0.642182	−0.321091	−	0.947048i	$-0.604050\pi$
$337$	−11.7889	−0.642182	−0.321091	+	0.947048i	$0.604050\pi$
$338$	70.5416	3.83696
$339$	0	0
$340$	−32.1194	−1.74192
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.09167	−0.274525
$345$	0	0
$346$	−3.00000	−0.161281
$347$	9.60555	0.515653	0.257827	−	0.966191i	$-0.416994\pi$
$347$	9.60555	0.515653	0.257827	+	0.966191i	$0.416994\pi$
$348$	0	0
$349$	4.69722	0.251437	0.125718	−	0.992066i	$-0.459876\pi$
$349$	4.69722	0.251437	0.125718	+	0.992066i	$0.459876\pi$
$350$	18.4222	0.984708
$351$	0	0
$352$	0	0
$353$	−5.09167	−0.271002	−0.135501	−	0.990777i	$-0.543264\pi$
$353$	−5.09167	−0.271002	−0.135501	+	0.990777i	$0.543264\pi$
$354$	0	0
$355$	−15.5139	−0.823391
$356$	48.6333	2.57756
$357$	0	0
$358$	14.7250	0.778239
$359$	−33.9361	−1.79108	−0.895539	−	0.444983i	$-0.853210\pi$
$359$	−33.9361	−1.79108	−0.895539	+	0.444983i	$0.853210\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−58.0555	−3.05133
$363$	0	0
$364$	−21.8167	−1.14350
$365$	−18.0278	−0.943616
$366$	0	0
$367$	−17.6333	−0.920451	−0.460226	−	0.887802i	$-0.652231\pi$
$367$	−17.6333	−0.920451	−0.460226	+	0.887802i	$0.652231\pi$
$368$	0.816654	0.0425710
$369$	0	0
$370$	43.2666	2.24932
$371$	12.9083	0.670167
$372$	0	0
$373$	20.1194	1.04174	0.520872	−	0.853635i	$-0.325607\pi$
$373$	20.1194	1.04174	0.520872	+	0.853635i	$0.325607\pi$
$374$	0	0
$375$	0	0
$376$	5.72498	0.295243
$377$	−31.0278	−1.59801
$378$	0	0
$379$	−15.8167	−0.812447	−0.406223	−	0.913774i	$-0.633155\pi$
$379$	−15.8167	−0.812447	−0.406223	+	0.913774i	$0.633155\pi$
$380$	−35.7250	−1.83265
$381$	0	0
$382$	61.7527	3.15954
$383$	−38.6333	−1.97407	−0.987035	−	0.160506i	$-0.948687\pi$
$383$	−38.6333	−1.97407	−0.987035	+	0.160506i	$0.948687\pi$
$384$	0	0
$385$	0	0
$386$	4.88057	0.248414
$387$	0	0
$388$	−11.9083	−0.604554
$389$	7.81665	0.396320	0.198160	−	0.980170i	$-0.436503\pi$
$389$	7.81665	0.396320	0.198160	+	0.980170i	$0.436503\pi$
$390$	0	0
$391$	7.27502	0.367914
$392$	3.00000	0.151523
$393$	0	0
$394$	−24.4222	−1.23037
$395$	−29.9361	−1.50625
$396$	0	0
$397$	30.2111	1.51625	0.758126	−	0.652108i	$-0.226115\pi$
$397$	30.2111	1.51625	0.758126	+	0.652108i	$0.226115\pi$
$398$	44.7250	2.24186
$399$	0	0
$400$	2.42221	0.121110
$401$	21.2111	1.05923	0.529616	−	0.848238i	$-0.322336\pi$
$401$	21.2111	1.05923	0.529616	+	0.848238i	$0.322336\pi$
$402$	0	0
$403$	−6.60555	−0.329046
$404$	−17.5139	−0.871348
$405$	0	0
$406$	10.8167	0.536822
$407$	0	0
$408$	0	0
$409$	−20.6056	−1.01888	−0.509439	−	0.860506i	$-0.670147\pi$
$409$	−20.6056	−1.01888	−0.509439	+	0.860506i	$0.670147\pi$
$410$	−58.1194	−2.87031
$411$	0	0
$412$	−47.2389	−2.32729
$413$	−6.69722	−0.329549
$414$	0	0
$415$	−10.8167	−0.530969
$416$	35.0278	1.71738
$417$	0	0
$418$	0	0
$419$	38.4222	1.87705	0.938524	−	0.345215i	$-0.112194\pi$
$419$	38.4222	1.87705	0.938524	+	0.345215i	$0.112194\pi$
$420$	0	0
$421$	−21.8167	−1.06328	−0.531639	−	0.846971i	$-0.678424\pi$
$421$	−21.8167	−1.06328	−0.531639	+	0.846971i	$0.678424\pi$
$422$	−35.7250	−1.73906
$423$	0	0
$424$	38.7250	1.88065
$425$	21.5778	1.04668
$426$	0	0
$427$	4.30278	0.208226
$428$	40.9361	1.97872
$429$	0	0
$430$	14.0917	0.679561
$431$	−3.11943	−0.150258	−0.0751288	−	0.997174i	$-0.523937\pi$
$431$	−3.11943	−0.150258	−0.0751288	+	0.997174i	$0.523937\pi$
$432$	0	0
$433$	29.1472	1.40072	0.700362	−	0.713788i	$-0.253022\pi$
$433$	29.1472	1.40072	0.700362	+	0.713788i	$0.253022\pi$
$434$	2.30278	0.110537
$435$	0	0
$436$	26.4222	1.26539
$437$	8.09167	0.387077
$438$	0	0
$439$	−27.7250	−1.32324	−0.661621	−	0.749839i	$-0.730131\pi$
$439$	−27.7250	−1.32324	−0.661621	+	0.749839i	$0.730131\pi$
$440$	0	0
$441$	0	0
$442$	−41.0278	−1.95149
$443$	−26.9361	−1.27977	−0.639886	−	0.768470i	$-0.721018\pi$
$443$	−26.9361	−1.27977	−0.639886	+	0.768470i	$0.721018\pi$
$444$	0	0
$445$	−53.0917	−2.51679
$446$	32.0278	1.51656
$447$	0	0
$448$	−12.8167	−0.605530
$449$	−40.3305	−1.90332	−0.951658	−	0.307160i	$-0.900621\pi$
$449$	−40.3305	−1.90332	−0.951658	+	0.307160i	$0.900621\pi$
$450$	0	0
$451$	0	0
$452$	35.3305	1.66181
$453$	0	0
$454$	15.0000	0.703985
$455$	23.8167	1.11654
$456$	0	0
$457$	−21.1194	−0.987925	−0.493963	−	0.869483i	$-0.664452\pi$
$457$	−21.1194	−0.987925	−0.493963	+	0.869483i	$0.664452\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	−32.1194	−1.49758
$461$	8.09167	0.376867	0.188433	−	0.982086i	$-0.439659\pi$
$461$	8.09167	0.376867	0.188433	+	0.982086i	$0.439659\pi$
$462$	0	0
$463$	−30.8167	−1.43217	−0.716086	−	0.698012i	$-0.754068\pi$
$463$	−30.8167	−1.43217	−0.716086	+	0.698012i	$0.754068\pi$
$464$	1.42221	0.0660242
$465$	0	0
$466$	41.4500	1.92013
$467$	13.5416	0.626632	0.313316	−	0.949649i	$-0.398560\pi$
$467$	13.5416	0.626632	0.313316	+	0.949649i	$0.398560\pi$
$468$	0	0
$469$	8.51388	0.393134
$470$	−15.8444	−0.730848
$471$	0	0
$472$	−20.0917	−0.924794
$473$	0	0
$474$	0	0
$475$	24.0000	1.10120
$476$	8.90833	0.408312
$477$	0	0
$478$	60.6333	2.77330
$479$	−28.6333	−1.30829	−0.654145	−	0.756370i	$-0.726971\pi$
$479$	−28.6333	−1.30829	−0.654145	+	0.756370i	$0.726971\pi$
$480$	0	0
$481$	34.4222	1.56952
$482$	−1.11943	−0.0509886
$483$	0	0
$484$	0	0
$485$	13.0000	0.590300
$486$	0	0
$487$	−2.81665	−0.127635	−0.0638174	−	0.997962i	$-0.520328\pi$
$487$	−2.81665	−0.127635	−0.0638174	+	0.997962i	$0.520328\pi$
$488$	12.9083	0.584333
$489$	0	0
$490$	−8.30278	−0.375081
$491$	−12.6972	−0.573018	−0.286509	−	0.958078i	$-0.592495\pi$
$491$	−12.6972	−0.573018	−0.286509	+	0.958078i	$0.592495\pi$
$492$	0	0
$493$	12.6695	0.570604
$494$	−45.6333	−2.05314
$495$	0	0
$496$	0.302776	0.0135950
$497$	4.30278	0.193006
$498$	0	0
$499$	2.90833	0.130195	0.0650973	−	0.997879i	$-0.479264\pi$
$499$	2.90833	0.130195	0.0650973	+	0.997879i	$0.479264\pi$
$500$	−35.7250	−1.59767
$501$	0	0
$502$	−10.1194	−0.451652
$503$	5.57779	0.248702	0.124351	−	0.992238i	$-0.460315\pi$
$503$	5.57779	0.248702	0.124351	+	0.992238i	$0.460315\pi$
$504$	0	0
$505$	19.1194	0.850803
$506$	0	0
$507$	0	0
$508$	−6.60555	−0.293074
$509$	−22.3028	−0.988553	−0.494277	−	0.869305i	$-0.664567\pi$
$509$	−22.3028	−0.988553	−0.494277	+	0.869305i	$0.664567\pi$
$510$	0	0
$511$	5.00000	0.221187
$512$	−3.42221	−0.151242
$513$	0	0
$514$	−14.9361	−0.658802
$515$	51.5694	2.27242
$516$	0	0
$517$	0	0
$518$	−12.0000	−0.527250
$519$	0	0
$520$	71.4500	3.13329
$521$	−7.42221	−0.325173	−0.162586	−	0.986694i	$-0.551984\pi$
$521$	−7.42221	−0.325173	−0.162586	+	0.986694i	$0.551984\pi$
$522$	0	0
$523$	44.5416	1.94767	0.973835	−	0.227257i	$-0.0729757\pi$
$523$	44.5416	1.94767	0.973835	+	0.227257i	$0.0729757\pi$
$524$	−19.8167	−0.865695
$525$	0	0
$526$	46.6056	2.03210
$527$	2.69722	0.117493
$528$	0	0
$529$	−15.7250	−0.683695
$530$	−107.175	−4.65538
$531$	0	0
$532$	9.90833	0.429580
$533$	−46.2389	−2.00283
$534$	0	0
$535$	−44.6888	−1.93207
$536$	25.5416	1.10323
$537$	0	0
$538$	−37.1194	−1.60033
$539$	0	0
$540$	0	0
$541$	32.3944	1.39275	0.696373	−	0.717680i	$-0.254796\pi$
$541$	32.3944	1.39275	0.696373	+	0.717680i	$0.254796\pi$
$542$	65.6611	2.82038
$543$	0	0
$544$	−14.3028	−0.613226
$545$	−28.8444	−1.23556
$546$	0	0
$547$	29.9361	1.27997	0.639987	−	0.768386i	$-0.278940\pi$
$547$	29.9361	1.27997	0.639987	+	0.768386i	$0.278940\pi$
$548$	33.4222	1.42773
$549$	0	0
$550$	0	0
$551$	14.0917	0.600325
$552$	0	0
$553$	8.30278	0.353070
$554$	−68.9361	−2.92881
$555$	0	0
$556$	−71.3583	−3.02627
$557$	7.23886	0.306720	0.153360	−	0.988170i	$-0.450991\pi$
$557$	7.23886	0.306720	0.153360	+	0.988170i	$0.450991\pi$
$558$	0	0
$559$	11.2111	0.474179
$560$	−1.09167	−0.0461316
$561$	0	0
$562$	14.7250	0.621136
$563$	−0.302776	−0.0127605	−0.00638024	−	0.999980i	$-0.502031\pi$
$563$	−0.302776	−0.0127605	−0.00638024	+	0.999980i	$0.502031\pi$
$564$	0	0
$565$	−38.5694	−1.62263
$566$	−10.1194	−0.425351
$567$	0	0
$568$	12.9083	0.541621
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−25.8444	−1.08155	−0.540777	−	0.841166i	$-0.681870\pi$
$571$	−25.8444	−1.08155	−0.540777	+	0.841166i	$0.681870\pi$
$572$	0	0
$573$	0	0
$574$	16.1194	0.672812
$575$	21.5778	0.899856
$576$	0	0
$577$	2.21110	0.0920494	0.0460247	−	0.998940i	$-0.485345\pi$
$577$	2.21110	0.0920494	0.0460247	+	0.998940i	$0.485345\pi$
$578$	−22.3944	−0.931486
$579$	0	0
$580$	−55.9361	−2.32262
$581$	3.00000	0.124461
$582$	0	0
$583$	0	0
$584$	15.0000	0.620704
$585$	0	0
$586$	11.0917	0.458193
$587$	13.6056	0.561561	0.280781	−	0.959772i	$-0.409407\pi$
$587$	13.6056	0.561561	0.280781	+	0.959772i	$0.409407\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	55.6056	2.28924
$591$	0	0
$592$	−1.57779	−0.0648470
$593$	22.6056	0.928299	0.464149	−	0.885757i	$-0.346360\pi$
$593$	22.6056	0.928299	0.464149	+	0.885757i	$0.346360\pi$
$594$	0	0
$595$	−9.72498	−0.398685
$596$	−16.2111	−0.664033
$597$	0	0
$598$	−41.0278	−1.67775
$599$	14.7889	0.604258	0.302129	−	0.953267i	$-0.402303\pi$
$599$	14.7889	0.604258	0.302129	+	0.953267i	$0.402303\pi$
$600$	0	0
$601$	5.81665	0.237266	0.118633	−	0.992938i	$-0.462149\pi$
$601$	5.81665	0.237266	0.118633	+	0.992938i	$0.462149\pi$
$602$	−3.90833	−0.159292
$603$	0	0
$604$	0.697224	0.0283697
$605$	0	0
$606$	0	0
$607$	20.5416	0.833759	0.416880	−	0.908962i	$-0.363124\pi$
$607$	20.5416	0.833759	0.416880	+	0.908962i	$0.363124\pi$
$608$	−15.9083	−0.645168
$609$	0	0
$610$	−35.7250	−1.44646
$611$	−12.6056	−0.509966
$612$	0	0
$613$	33.0555	1.33510	0.667550	−	0.744565i	$-0.267343\pi$
$613$	33.0555	1.33510	0.667550	+	0.744565i	$0.267343\pi$
$614$	38.3028	1.54577
$615$	0	0
$616$	0	0
$617$	−21.6333	−0.870924	−0.435462	−	0.900207i	$-0.643415\pi$
$617$	−21.6333	−0.870924	−0.435462	+	0.900207i	$0.643415\pi$
$618$	0	0
$619$	39.8167	1.60037	0.800183	−	0.599756i	$-0.204736\pi$
$619$	39.8167	1.60037	0.800183	+	0.599756i	$0.204736\pi$
$620$	−11.9083	−0.478250
$621$	0	0
$622$	−14.7250	−0.590418
$623$	14.7250	0.589944
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	35.2389	1.40843
$627$	0	0
$628$	23.8167	0.950388
$629$	−14.0555	−0.560430
$630$	0	0
$631$	−31.0555	−1.23630	−0.618150	−	0.786060i	$-0.712118\pi$
$631$	−31.0555	−1.23630	−0.618150	+	0.786060i	$0.712118\pi$
$632$	24.9083	0.990800
$633$	0	0
$634$	−39.8444	−1.58242
$635$	7.21110	0.286164
$636$	0	0
$637$	−6.60555	−0.261721
$638$	0	0
$639$	0	0
$640$	68.1749	2.69485
$641$	28.8167	1.13819	0.569095	−	0.822272i	$-0.307294\pi$
$641$	28.8167	1.13819	0.569095	+	0.822272i	$0.307294\pi$
$642$	0	0
$643$	−1.23886	−0.0488558	−0.0244279	−	0.999702i	$-0.507776\pi$
$643$	−1.23886	−0.0488558	−0.0244279	+	0.999702i	$0.507776\pi$
$644$	8.90833	0.351037
$645$	0	0
$646$	18.6333	0.733118
$647$	−10.6056	−0.416947	−0.208474	−	0.978028i	$-0.566850\pi$
$647$	−10.6056	−0.416947	−0.208474	+	0.978028i	$0.566850\pi$
$648$	0	0
$649$	0	0
$650$	−121.689	−4.77303
$651$	0	0
$652$	9.30278	0.364325
$653$	6.81665	0.266756	0.133378	−	0.991065i	$-0.457418\pi$
$653$	6.81665	0.266756	0.133378	+	0.991065i	$0.457418\pi$
$654$	0	0
$655$	21.6333	0.845283
$656$	2.11943	0.0827498
$657$	0	0
$658$	4.39445	0.171313
$659$	4.63331	0.180488	0.0902440	−	0.995920i	$-0.471235\pi$
$659$	4.63331	0.180488	0.0902440	+	0.995920i	$0.471235\pi$
$660$	0	0
$661$	−18.3028	−0.711895	−0.355948	−	0.934506i	$-0.615842\pi$
$661$	−18.3028	−0.711895	−0.355948	+	0.934506i	$0.615842\pi$
$662$	54.8444	2.13159
$663$	0	0
$664$	9.00000	0.349268
$665$	−10.8167	−0.419452
$666$	0	0
$667$	12.6695	0.490564
$668$	−17.2111	−0.665918
$669$	0	0
$670$	−70.6888	−2.73095
$671$	0	0
$672$	0	0
$673$	8.42221	0.324652	0.162326	−	0.986737i	$-0.448100\pi$
$673$	8.42221	0.324652	0.162326	+	0.986737i	$0.448100\pi$
$674$	−27.1472	−1.04567
$675$	0	0
$676$	101.175	3.89134
$677$	10.3028	0.395968	0.197984	−	0.980205i	$-0.436561\pi$
$677$	10.3028	0.395968	0.197984	+	0.980205i	$0.436561\pi$
$678$	0	0
$679$	−3.60555	−0.138368
$680$	−29.1749	−1.11881
$681$	0	0
$682$	0	0
$683$	−15.6972	−0.600638	−0.300319	−	0.953839i	$-0.597093\pi$
$683$	−15.6972	−0.600638	−0.300319	+	0.953839i	$0.597093\pi$
$684$	0	0
$685$	−36.4861	−1.39406
$686$	2.30278	0.0879204
$687$	0	0
$688$	−0.513878	−0.0195914
$689$	−85.2666	−3.24840
$690$	0	0
$691$	−17.0278	−0.647766	−0.323883	−	0.946097i	$-0.604988\pi$
$691$	−17.0278	−0.647766	−0.323883	+	0.946097i	$0.604988\pi$
$692$	−4.30278	−0.163567
$693$	0	0
$694$	22.1194	0.839642
$695$	77.8999	2.95491
$696$	0	0
$697$	18.8806	0.715153
$698$	10.8167	0.409416
$699$	0	0
$700$	26.4222	0.998665
$701$	−31.1194	−1.17536	−0.587682	−	0.809092i	$-0.699960\pi$
$701$	−31.1194	−1.17536	−0.587682	+	0.809092i	$0.699960\pi$
$702$	0	0
$703$	−15.6333	−0.589621
$704$	0	0
$705$	0	0
$706$	−11.7250	−0.441275
$707$	−5.30278	−0.199431
$708$	0	0
$709$	25.3305	0.951308	0.475654	−	0.879632i	$-0.342211\pi$
$709$	25.3305	0.951308	0.475654	+	0.879632i	$0.342211\pi$
$710$	−35.7250	−1.34073
$711$	0	0
$712$	44.1749	1.65553
$713$	2.69722	0.101012
$714$	0	0
$715$	0	0
$716$	21.1194	0.789270
$717$	0	0
$718$	−78.1472	−2.91643
$719$	−21.2389	−0.792076	−0.396038	−	0.918234i	$-0.629615\pi$
$719$	−21.2389	−0.792076	−0.396038	+	0.918234i	$0.629615\pi$
$720$	0	0
$721$	−14.3028	−0.532663
$722$	−23.0278	−0.857004
$723$	0	0
$724$	−83.2666	−3.09458
$725$	37.5778	1.39560
$726$	0	0
$727$	24.1194	0.894540	0.447270	−	0.894399i	$-0.352396\pi$
$727$	24.1194	0.894540	0.447270	+	0.894399i	$0.352396\pi$
$728$	−19.8167	−0.734454
$729$	0	0
$730$	−41.5139	−1.53650
$731$	−4.57779	−0.169316
$732$	0	0
$733$	6.78890	0.250754	0.125377	−	0.992109i	$-0.459986\pi$
$733$	6.78890	0.250754	0.125377	+	0.992109i	$0.459986\pi$
$734$	−40.6056	−1.49878
$735$	0	0
$736$	−14.3028	−0.527207
$737$	0	0
$738$	0	0
$739$	43.1194	1.58617	0.793087	−	0.609108i	$-0.208473\pi$
$739$	43.1194	1.58617	0.793087	+	0.609108i	$0.208473\pi$
$740$	62.0555	2.28121
$741$	0	0
$742$	29.7250	1.09124
$743$	44.9361	1.64855	0.824273	−	0.566193i	$-0.191584\pi$
$743$	44.9361	1.64855	0.824273	+	0.566193i	$0.191584\pi$
$744$	0	0
$745$	17.6972	0.648376
$746$	46.3305	1.69628
$747$	0	0
$748$	0	0
$749$	12.3944	0.452883
$750$	0	0
$751$	7.63331	0.278543	0.139272	−	0.990254i	$-0.455524\pi$
$751$	7.63331	0.278543	0.139272	+	0.990254i	$0.455524\pi$
$752$	0.577795	0.0210700
$753$	0	0
$754$	−71.4500	−2.60205
$755$	−0.761141	−0.0277008
$756$	0	0
$757$	−23.8167	−0.865631	−0.432816	−	0.901483i	$-0.642480\pi$
$757$	−23.8167	−0.865631	−0.432816	+	0.901483i	$0.642480\pi$
$758$	−36.4222	−1.32291
$759$	0	0
$760$	−32.4500	−1.17708
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	88.5694	3.20433
$765$	0	0
$766$	−88.9638	−3.21439
$767$	44.2389	1.59737
$768$	0	0
$769$	39.3305	1.41830	0.709148	−	0.705060i	$-0.249080\pi$
$769$	39.3305	1.41830	0.709148	+	0.705060i	$0.249080\pi$
$770$	0	0
$771$	0	0
$772$	7.00000	0.251936
$773$	−30.6333	−1.10180	−0.550902	−	0.834570i	$-0.685716\pi$
$773$	−30.6333	−1.10180	−0.550902	+	0.834570i	$0.685716\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	−10.8167	−0.388295
$777$	0	0
$778$	18.0000	0.645331
$779$	21.0000	0.752403
$780$	0	0
$781$	0	0
$782$	16.7527	0.599077
$783$	0	0
$784$	0.302776	0.0108134
$785$	−26.0000	−0.927980
$786$	0	0
$787$	22.4861	0.801544	0.400772	−	0.916178i	$-0.368742\pi$
$787$	22.4861	0.801544	0.400772	+	0.916178i	$0.368742\pi$
$788$	−35.0278	−1.24781
$789$	0	0
$790$	−68.9361	−2.45264
$791$	10.6972	0.380350
$792$	0	0
$793$	−28.4222	−1.00930
$794$	69.5694	2.46893
$795$	0	0
$796$	64.1472	2.27364
$797$	28.6333	1.01424	0.507122	−	0.861874i	$-0.330709\pi$
$797$	28.6333	1.01424	0.507122	+	0.861874i	$0.330709\pi$
$798$	0	0
$799$	5.14719	0.182094
$800$	−42.4222	−1.49985
$801$	0	0
$802$	48.8444	1.72476
$803$	0	0
$804$	0	0
$805$	−9.72498	−0.342761
$806$	−15.2111	−0.535788
$807$	0	0
$808$	−15.9083	−0.559653
$809$	8.09167	0.284488	0.142244	−	0.989832i	$-0.454568\pi$
$809$	8.09167	0.284488	0.142244	+	0.989832i	$0.454568\pi$
$810$	0	0
$811$	40.8444	1.43424	0.717121	−	0.696949i	$-0.245460\pi$
$811$	40.8444	1.43424	0.717121	+	0.696949i	$0.245460\pi$
$812$	15.5139	0.544430
$813$	0	0
$814$	0	0
$815$	−10.1556	−0.355735
$816$	0	0
$817$	−5.09167	−0.178135
$818$	−47.4500	−1.65905
$819$	0	0
$820$	−83.3583	−2.91100
$821$	−27.4222	−0.957042	−0.478521	−	0.878076i	$-0.658827\pi$
$821$	−27.4222	−0.957042	−0.478521	+	0.878076i	$0.658827\pi$
$822$	0	0
$823$	35.2111	1.22738	0.613691	−	0.789546i	$-0.289684\pi$
$823$	35.2111	1.22738	0.613691	+	0.789546i	$0.289684\pi$
$824$	−42.9083	−1.49478
$825$	0	0
$826$	−15.4222	−0.536607
$827$	−14.1833	−0.493203	−0.246602	−	0.969117i	$-0.579314\pi$
$827$	−14.1833	−0.493203	−0.246602	+	0.969117i	$0.579314\pi$
$828$	0	0
$829$	13.2750	0.461060	0.230530	−	0.973065i	$-0.425954\pi$
$829$	13.2750	0.461060	0.230530	+	0.973065i	$0.425954\pi$
$830$	−24.9083	−0.864581
$831$	0	0
$832$	84.6611	2.93509
$833$	2.69722	0.0934533
$834$	0	0
$835$	18.7889	0.650217
$836$	0	0
$837$	0	0
$838$	88.4777	3.05641
$839$	−42.2111	−1.45729	−0.728645	−	0.684892i	$-0.759849\pi$
$839$	−42.2111	−1.45729	−0.728645	+	0.684892i	$0.759849\pi$
$840$	0	0
$841$	−6.93608	−0.239175
$842$	−50.2389	−1.73135
$843$	0	0
$844$	−51.2389	−1.76371
$845$	−110.450	−3.79959
$846$	0	0
$847$	0	0
$848$	3.90833	0.134212
$849$	0	0
$850$	49.6888	1.70431
$851$	−14.0555	−0.481817
$852$	0	0
$853$	45.7250	1.56559	0.782797	−	0.622277i	$-0.213792\pi$
$853$	45.7250	1.56559	0.782797	+	0.622277i	$0.213792\pi$
$854$	9.90833	0.339056
$855$	0	0
$856$	37.1833	1.27090
$857$	35.3583	1.20782	0.603908	−	0.797054i	$-0.293609\pi$
$857$	35.3583	1.20782	0.603908	+	0.797054i	$0.293609\pi$
$858$	0	0
$859$	−25.3028	−0.863320	−0.431660	−	0.902036i	$-0.642072\pi$
$859$	−25.3028	−0.863320	−0.431660	+	0.902036i	$0.642072\pi$
$860$	20.2111	0.689193
$861$	0	0
$862$	−7.18335	−0.244666
$863$	−31.4222	−1.06962	−0.534812	−	0.844971i	$-0.679618\pi$
$863$	−31.4222	−1.06962	−0.534812	+	0.844971i	$0.679618\pi$
$864$	0	0
$865$	4.69722	0.159710
$866$	67.1194	2.28081
$867$	0	0
$868$	3.30278	0.112104
$869$	0	0
$870$	0	0
$871$	−56.2389	−1.90558
$872$	24.0000	0.812743
$873$	0	0
$874$	18.6333	0.630281
$875$	−10.8167	−0.365670
$876$	0	0
$877$	−47.0555	−1.58895	−0.794476	−	0.607296i	$-0.792254\pi$
$877$	−47.0555	−1.58895	−0.794476	+	0.607296i	$0.792254\pi$
$878$	−63.8444	−2.15464
$879$	0	0
$880$	0	0
$881$	−34.3305	−1.15663	−0.578313	−	0.815815i	$-0.696289\pi$
$881$	−34.3305	−1.15663	−0.578313	+	0.815815i	$0.696289\pi$
$882$	0	0
$883$	−7.72498	−0.259966	−0.129983	−	0.991516i	$-0.541492\pi$
$883$	−7.72498	−0.259966	−0.129983	+	0.991516i	$0.541492\pi$
$884$	−58.8444	−1.97915
$885$	0	0
$886$	−62.0278	−2.08386
$887$	44.1194	1.48139	0.740693	−	0.671844i	$-0.234497\pi$
$887$	44.1194	1.48139	0.740693	+	0.671844i	$0.234497\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	−122.258	−4.09810
$891$	0	0
$892$	45.9361	1.53805
$893$	5.72498	0.191579
$894$	0	0
$895$	−23.0555	−0.770661
$896$	−18.9083	−0.631683
$897$	0	0
$898$	−92.8722	−3.09918
$899$	4.69722	0.156661
$900$	0	0
$901$	34.8167	1.15991
$902$	0	0
$903$	0	0
$904$	32.0917	1.06735
$905$	90.8999	3.02162
$906$	0	0
$907$	0.394449	0.0130975	0.00654873	−	0.999979i	$-0.497915\pi$
$907$	0.394449	0.0130975	0.00654873	+	0.999979i	$0.497915\pi$
$908$	21.5139	0.713963
$909$	0	0
$910$	54.8444	1.81807
$911$	50.4500	1.67148	0.835741	−	0.549124i	$-0.185039\pi$
$911$	50.4500	1.67148	0.835741	+	0.549124i	$0.185039\pi$
$912$	0	0
$913$	0	0
$914$	−48.6333	−1.60865
$915$	0	0
$916$	−8.60555	−0.284335
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−55.4777	−1.83004	−0.915021	−	0.403407i	$-0.867826\pi$
$919$	−55.4777	−1.83004	−0.915021	+	0.403407i	$0.867826\pi$
$920$	−29.1749	−0.961869
$921$	0	0
$922$	18.6333	0.613655
$923$	−28.4222	−0.935528
$924$	0	0
$925$	−41.6888	−1.37072
$926$	−70.9638	−2.33202
$927$	0	0
$928$	−24.9083	−0.817656
$929$	27.6333	0.906619	0.453310	−	0.891353i	$-0.350243\pi$
$929$	27.6333	0.906619	0.453310	+	0.891353i	$0.350243\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	59.4500	1.94735
$933$	0	0
$934$	31.1833	1.02035
$935$	0	0
$936$	0	0
$937$	−31.6056	−1.03251	−0.516254	−	0.856435i	$-0.672674\pi$
$937$	−31.6056	−1.03251	−0.516254	+	0.856435i	$0.672674\pi$
$938$	19.6056	0.640144
$939$	0	0
$940$	−22.7250	−0.741207
$941$	37.4222	1.21993	0.609965	−	0.792429i	$-0.291184\pi$
$941$	37.4222	1.21993	0.609965	+	0.792429i	$0.291184\pi$
$942$	0	0
$943$	18.8806	0.614836
$944$	−2.02776	−0.0659978
$945$	0	0
$946$	0	0
$947$	−46.6611	−1.51628	−0.758140	−	0.652091i	$-0.773892\pi$
$947$	−46.6611	−1.51628	−0.758140	+	0.652091i	$0.773892\pi$
$948$	0	0
$949$	−33.0278	−1.07213
$950$	55.2666	1.79309
$951$	0	0
$952$	8.09167	0.262253
$953$	−6.02776	−0.195258	−0.0976291	−	0.995223i	$-0.531126\pi$
$953$	−6.02776	−0.195258	−0.0976291	+	0.995223i	$0.531126\pi$
$954$	0	0
$955$	−96.6888	−3.12878
$956$	86.9638	2.81261
$957$	0	0
$958$	−65.9361	−2.13030
$959$	10.1194	0.326773
$960$	0	0
$961$	−30.0000	−0.967742
$962$	79.2666	2.55566
$963$	0	0
$964$	−1.60555	−0.0517113
$965$	−7.64171	−0.245995
$966$	0	0
$967$	−40.5139	−1.30284	−0.651419	−	0.758718i	$-0.725826\pi$
$967$	−40.5139	−1.30284	−0.651419	+	0.758718i	$0.725826\pi$
$968$	0	0
$969$	0	0
$970$	29.9361	0.961190
$971$	6.36669	0.204317	0.102158	−	0.994768i	$-0.467425\pi$
$971$	6.36669	0.204317	0.102158	+	0.994768i	$0.467425\pi$
$972$	0	0
$973$	−21.6056	−0.692642
$974$	−6.48612	−0.207829
$975$	0	0
$976$	1.30278	0.0417008
$977$	−40.0278	−1.28060	−0.640301	−	0.768124i	$-0.721190\pi$
$977$	−40.0278	−1.28060	−0.640301	+	0.768124i	$0.721190\pi$
$978$	0	0
$979$	0	0
$980$	−11.9083	−0.380398
$981$	0	0
$982$	−29.2389	−0.933049
$983$	9.27502	0.295827	0.147914	−	0.989000i	$-0.452744\pi$
$983$	9.27502	0.295827	0.147914	+	0.989000i	$0.452744\pi$
$984$	0	0
$985$	38.2389	1.21839
$986$	29.1749	0.929119
$987$	0	0
$988$	−65.4500	−2.08224
$989$	−4.57779	−0.145565
$990$	0	0
$991$	−43.7250	−1.38897	−0.694485	−	0.719507i	$-0.744368\pi$
$991$	−43.7250	−1.38897	−0.694485	+	0.719507i	$0.744368\pi$
$992$	−5.30278	−0.168363
$993$	0	0
$994$	9.90833	0.314273
$995$	−70.0278	−2.22003
$996$	0	0
$997$	−45.6972	−1.44725	−0.723623	−	0.690196i	$-0.757524\pi$
$997$	−45.6972	−1.44725	−0.723623	+	0.690196i	$0.757524\pi$
$998$	6.69722	0.211997
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bs.1.2		2
3.2	odd	2		847.2.a.e.1.1	✓	2
11.10	odd	2		7623.2.a.bc.1.1		2
21.20	even	2		5929.2.a.k.1.1		2
33.2	even	10		847.2.f.o.323.1		8
33.5	odd	10		847.2.f.r.729.2		8
33.8	even	10		847.2.f.o.372.2		8
33.14	odd	10		847.2.f.r.372.1		8
33.17	even	10		847.2.f.o.729.1		8
33.20	odd	10		847.2.f.r.323.2		8
33.26	odd	10		847.2.f.r.148.1		8
33.29	even	10		847.2.f.o.148.2		8
33.32	even	2		847.2.a.g.1.2	yes	2
231.230	odd	2		5929.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.e.1.1	✓	2	3.2	odd	2
847.2.a.g.1.2	yes	2	33.32	even	2
847.2.f.o.148.2		8	33.29	even	10
847.2.f.o.323.1		8	33.2	even	10
847.2.f.o.372.2		8	33.8	even	10
847.2.f.o.729.1		8	33.17	even	10
847.2.f.r.148.1		8	33.26	odd	10
847.2.f.r.323.2		8	33.20	odd	10
847.2.f.r.372.1		8	33.14	odd	10
847.2.f.r.729.2		8	33.5	odd	10
5929.2.a.k.1.1		2	21.20	even	2
5929.2.a.p.1.2		2	231.230	odd	2
7623.2.a.bc.1.1		2	11.10	odd	2
7623.2.a.bs.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{2} +3.30278 q^{4} -3.60555 q^{5} +1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q+2.30278 q^{2} +3.30278 q^{4} -3.60555 q^{5} +1.00000 q^{7} +3.00000 q^{8} -8.30278 q^{10} -6.60555 q^{13} +2.30278 q^{14} +0.302776 q^{16} +2.69722 q^{17} +3.00000 q^{19} -11.9083 q^{20} +2.69722 q^{23} +8.00000 q^{25} -15.2111 q^{26} +3.30278 q^{28} +4.69722 q^{29} +1.00000 q^{31} -5.30278 q^{32} +6.21110 q^{34} -3.60555 q^{35} -5.21110 q^{37} +6.90833 q^{38} -10.8167 q^{40} +7.00000 q^{41} -1.69722 q^{43} +6.21110 q^{46} +1.90833 q^{47} +1.00000 q^{49} +18.4222 q^{50} -21.8167 q^{52} +12.9083 q^{53} +3.00000 q^{56} +10.8167 q^{58} -6.69722 q^{59} +4.30278 q^{61} +2.30278 q^{62} -12.8167 q^{64} +23.8167 q^{65} +8.51388 q^{67} +8.90833 q^{68} -8.30278 q^{70} +4.30278 q^{71} +5.00000 q^{73} -12.0000 q^{74} +9.90833 q^{76} +8.30278 q^{79} -1.09167 q^{80} +16.1194 q^{82} +3.00000 q^{83} -9.72498 q^{85} -3.90833 q^{86} +14.7250 q^{89} -6.60555 q^{91} +8.90833 q^{92} +4.39445 q^{94} -10.8167 q^{95} -3.60555 q^{97} +2.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8	\( 2 q + q^{2} + 3 q^{4} + 2 q^{7} + 6 q^{8} - 13 q^{10} - 6 q^{13} + q^{14} - 3 q^{16} + 9 q^{17} + 6 q^{19} - 13 q^{20} + 9 q^{23} + 16 q^{25} - 16 q^{26} + 3 q^{28} + 13 q^{29} + 2 q^{31} - 7 q^{32} - 2 q^{34} + 4 q^{37} + 3 q^{38} + 14 q^{41} - 7 q^{43} - 2 q^{46} - 7 q^{47} + 2 q^{49} + 8 q^{50} - 22 q^{52} + 15 q^{53} + 6 q^{56} - 17 q^{59} + 5 q^{61} + q^{62} - 4 q^{64} + 26 q^{65} - q^{67} + 7 q^{68} - 13 q^{70} + 5 q^{71} + 10 q^{73} - 24 q^{74} + 9 q^{76} + 13 q^{79} - 13 q^{80} + 7 q^{82} + 6 q^{83} + 13 q^{85} + 3 q^{86} - 3 q^{89} - 6 q^{91} + 7 q^{92} + 16 q^{94} + q^{98}+O(q^{100}) \) 2 * q + q^2 + 3 * q^4 + 2 * q^7 + 6 * q^8 - 13 * q^10 - 6 * q^13 + q^14 - 3 * q^16 + 9 * q^17 + 6 * q^19 - 13 * q^20 + 9 * q^23 + 16 * q^25 - 16 * q^26 + 3 * q^28 + 13 * q^29 + 2 * q^31 - 7 * q^32 - 2 * q^34 + 4 * q^37 + 3 * q^38 + 14 * q^41 - 7 * q^43 - 2 * q^46 - 7 * q^47 + 2 * q^49 + 8 * q^50 - 22 * q^52 + 15 * q^53 + 6 * q^56 - 17 * q^59 + 5 * q^61 + q^62 - 4 * q^64 + 26 * q^65 - q^67 + 7 * q^68 - 13 * q^70 + 5 * q^71 + 10 * q^73 - 24 * q^74 + 9 * q^76 + 13 * q^79 - 13 * q^80 + 7 * q^82 + 6 * q^83 + 13 * q^85 + 3 * q^86 - 3 * q^89 - 6 * q^91 + 7 * q^92 + 16 * q^94 + q^98

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