LMFDB - Embedded newform 2646.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(1,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2646.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 378)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	2646.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.64575 q^{5} +1.00000 q^{8} +3.64575 q^{10} +3.64575 q^{11} +4.64575 q^{13} +1.00000 q^{16} -3.64575 q^{17} -2.00000 q^{19} +3.64575 q^{20} +3.64575 q^{22} -1.29150 q^{23} +8.29150 q^{25} +4.64575 q^{26} +2.35425 q^{29} +4.64575 q^{31} +1.00000 q^{32} -3.64575 q^{34} -11.9373 q^{37} -2.00000 q^{38} +3.64575 q^{40} -10.9373 q^{41} +5.00000 q^{43} +3.64575 q^{44} -1.29150 q^{46} -4.93725 q^{47} +8.29150 q^{50} +4.64575 q^{52} +6.00000 q^{53} +13.2915 q^{55} +2.35425 q^{58} -8.35425 q^{59} -7.35425 q^{61} +4.64575 q^{62} +1.00000 q^{64} +16.9373 q^{65} -2.29150 q^{67} -3.64575 q^{68} +15.6458 q^{71} -10.5830 q^{73} -11.9373 q^{74} -2.00000 q^{76} +11.2288 q^{79} +3.64575 q^{80} -10.9373 q^{82} +13.2915 q^{83} -13.2915 q^{85} +5.00000 q^{86} +3.64575 q^{88} -4.93725 q^{89} -1.29150 q^{92} -4.93725 q^{94} -7.29150 q^{95} -13.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 6 q^{25} + 4 q^{26} + 10 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{34}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^20 + 2 * q^22 + 8 * q^23 + 6 * q^25 + 4 * q^26 + 10 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^34 - 8 * q^37 - 4 * q^38 + 2 * q^40 - 6 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 6 * q^47 + 6 * q^50 + 4 * q^52 + 12 * q^53 + 16 * q^55 + 10 * q^58 - 22 * q^59 - 20 * q^61 + 4 * q^62 + 2 * q^64 + 18 * q^65 + 6 * q^67 - 2 * q^68 + 26 * q^71 - 8 * q^74 - 4 * q^76 - 4 * q^79 + 2 * q^80 - 6 * q^82 + 16 * q^83 - 16 * q^85 + 10 * q^86 + 2 * q^88 + 6 * q^89 + 8 * q^92 + 6 * q^94 - 4 * q^95 - 6 * q^97

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.64575	1.63043	0.815215	−	0.579159i	$-0.196619\pi$
$5$	3.64575	1.63043	0.815215	+	0.579159i	$0.196619\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.64575	1.15289
$11$	3.64575	1.09924	0.549618	−	0.835416i	$-0.314773\pi$
$11$	3.64575	1.09924	0.549618	+	0.835416i	$0.314773\pi$
$12$	0	0
$13$	4.64575	1.28850	0.644250	−	0.764815i	$-0.277170\pi$
$13$	4.64575	1.28850	0.644250	+	0.764815i	$0.277170\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.64575	−0.884225	−0.442112	−	0.896960i	$-0.645771\pi$
$17$	−3.64575	−0.884225	−0.442112	+	0.896960i	$0.645771\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	3.64575	0.815215
$21$	0	0
$22$	3.64575	0.777277
$23$	−1.29150	−0.269297	−0.134648	−	0.990893i	$-0.542991\pi$
$23$	−1.29150	−0.269297	−0.134648	+	0.990893i	$0.542991\pi$
$24$	0	0
$25$	8.29150	1.65830
$26$	4.64575	0.911107
$27$	0	0
$28$	0	0
$29$	2.35425	0.437173	0.218587	−	0.975818i	$-0.429855\pi$
$29$	2.35425	0.437173	0.218587	+	0.975818i	$0.429855\pi$
$30$	0	0
$31$	4.64575	0.834402	0.417201	−	0.908814i	$-0.363011\pi$
$31$	4.64575	0.834402	0.417201	+	0.908814i	$0.363011\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.64575	−0.625241
$35$	0	0
$36$	0	0
$37$	−11.9373	−1.96247	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−11.9373	−1.96247	−0.981236	+	0.192809i	$0.938240\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	3.64575	0.576444
$41$	−10.9373	−1.70811	−0.854056	−	0.520181i	$-0.825864\pi$
$41$	−10.9373	−1.70811	−0.854056	+	0.520181i	$0.825864\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	3.64575	0.549618
$45$	0	0
$46$	−1.29150	−0.190422
$47$	−4.93725	−0.720173	−0.360086	−	0.932919i	$-0.617253\pi$
$47$	−4.93725	−0.720173	−0.360086	+	0.932919i	$0.617253\pi$
$48$	0	0
$49$	0	0
$50$	8.29150	1.17260
$51$	0	0
$52$	4.64575	0.644250
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	13.2915	1.79223
$56$	0	0
$57$	0	0
$58$	2.35425	0.309128
$59$	−8.35425	−1.08763	−0.543815	−	0.839205i	$-0.683021\pi$
$59$	−8.35425	−1.08763	−0.543815	+	0.839205i	$0.683021\pi$
$60$	0	0
$61$	−7.35425	−0.941615	−0.470808	−	0.882236i	$-0.656037\pi$
$61$	−7.35425	−0.941615	−0.470808	+	0.882236i	$0.656037\pi$
$62$	4.64575	0.590011
$63$	0	0
$64$	1.00000	0.125000
$65$	16.9373	2.10081
$66$	0	0
$67$	−2.29150	−0.279952	−0.139976	−	0.990155i	$-0.544702\pi$
$67$	−2.29150	−0.279952	−0.139976	+	0.990155i	$0.544702\pi$
$68$	−3.64575	−0.442112
$69$	0	0
$70$	0	0
$71$	15.6458	1.85681	0.928405	−	0.371571i	$-0.121181\pi$
$71$	15.6458	1.85681	0.928405	+	0.371571i	$0.121181\pi$
$72$	0	0
$73$	−10.5830	−1.23865	−0.619324	−	0.785136i	$-0.712593\pi$
$73$	−10.5830	−1.23865	−0.619324	+	0.785136i	$0.712593\pi$
$74$	−11.9373	−1.38768
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	11.2288	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$79$	11.2288	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$80$	3.64575	0.407607
$81$	0	0
$82$	−10.9373	−1.20782
$83$	13.2915	1.45893	0.729466	−	0.684017i	$-0.239769\pi$
$83$	13.2915	1.45893	0.729466	+	0.684017i	$0.239769\pi$
$84$	0	0
$85$	−13.2915	−1.44167
$86$	5.00000	0.539164
$87$	0	0
$88$	3.64575	0.388638
$89$	−4.93725	−0.523348	−0.261674	−	0.965156i	$-0.584275\pi$
$89$	−4.93725	−0.523348	−0.261674	+	0.965156i	$0.584275\pi$
$90$	0	0
$91$	0	0
$92$	−1.29150	−0.134648
$93$	0	0
$94$	−4.93725	−0.509239
$95$	−7.29150	−0.748092
$96$	0	0
$97$	−13.5830	−1.37915	−0.689573	−	0.724217i	$-0.742202\pi$
$97$	−13.5830	−1.37915	−0.689573	+	0.724217i	$0.742202\pi$
$98$	0	0
$99$	0	0
$100$	8.29150	0.829150
$101$	−8.35425	−0.831279	−0.415639	−	0.909529i	$-0.636442\pi$
$101$	−8.35425	−0.831279	−0.415639	+	0.909529i	$0.636442\pi$
$102$	0	0
$103$	−3.93725	−0.387949	−0.193975	−	0.981007i	$-0.562138\pi$
$103$	−3.93725	−0.387949	−0.193975	+	0.981007i	$0.562138\pi$
$104$	4.64575	0.455553
$105$	0	0
$106$	6.00000	0.582772
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−3.35425	−0.321279	−0.160639	−	0.987013i	$-0.551356\pi$
$109$	−3.35425	−0.321279	−0.160639	+	0.987013i	$0.551356\pi$
$110$	13.2915	1.26730
$111$	0	0
$112$	0	0
$113$	2.35425	0.221469	0.110735	−	0.993850i	$-0.464680\pi$
$113$	2.35425	0.221469	0.110735	+	0.993850i	$0.464680\pi$
$114$	0	0
$115$	−4.70850	−0.439070
$116$	2.35425	0.218587
$117$	0	0
$118$	−8.35425	−0.769071
$119$	0	0
$120$	0	0
$121$	2.29150	0.208318
$122$	−7.35425	−0.665822
$123$	0	0
$124$	4.64575	0.417201
$125$	12.0000	1.07331
$126$	0	0
$127$	1.35425	0.120170	0.0600851	−	0.998193i	$-0.480863\pi$
$127$	1.35425	0.120170	0.0600851	+	0.998193i	$0.480863\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	16.9373	1.48550
$131$	14.5830	1.27412	0.637062	−	0.770813i	$-0.280150\pi$
$131$	14.5830	1.27412	0.637062	+	0.770813i	$0.280150\pi$
$132$	0	0
$133$	0	0
$134$	−2.29150	−0.197956
$135$	0	0
$136$	−3.64575	−0.312621
$137$	1.29150	0.110341	0.0551703	−	0.998477i	$-0.482430\pi$
$137$	1.29150	0.110341	0.0551703	+	0.998477i	$0.482430\pi$
$138$	0	0
$139$	−1.58301	−0.134269	−0.0671344	−	0.997744i	$-0.521386\pi$
$139$	−1.58301	−0.134269	−0.0671344	+	0.997744i	$0.521386\pi$
$140$	0	0
$141$	0	0
$142$	15.6458	1.31296
$143$	16.9373	1.41636
$144$	0	0
$145$	8.58301	0.712780
$146$	−10.5830	−0.875856
$147$	0	0
$148$	−11.9373	−0.981236
$149$	22.9373	1.87909	0.939547	−	0.342421i	$-0.111247\pi$
$149$	22.9373	1.87909	0.939547	+	0.342421i	$0.111247\pi$
$150$	0	0
$151$	−19.2288	−1.56481	−0.782407	−	0.622767i	$-0.786008\pi$
$151$	−19.2288	−1.56481	−0.782407	+	0.622767i	$0.786008\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	16.9373	1.36043
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	11.2288	0.893312
$159$	0	0
$160$	3.64575	0.288222
$161$	0	0
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−10.9373	−0.854056
$165$	0	0
$166$	13.2915	1.03162
$167$	8.58301	0.664173	0.332086	−	0.943249i	$-0.392247\pi$
$167$	8.58301	0.664173	0.332086	+	0.943249i	$0.392247\pi$
$168$	0	0
$169$	8.58301	0.660231
$170$	−13.2915	−1.01941
$171$	0	0
$172$	5.00000	0.381246
$173$	14.5830	1.10873	0.554363	−	0.832275i	$-0.312962\pi$
$173$	14.5830	1.10873	0.554363	+	0.832275i	$0.312962\pi$
$174$	0	0
$175$	0	0
$176$	3.64575	0.274809
$177$	0	0
$178$	−4.93725	−0.370063
$179$	16.9373	1.26595	0.632975	−	0.774172i	$-0.281834\pi$
$179$	16.9373	1.26595	0.632975	+	0.774172i	$0.281834\pi$
$180$	0	0
$181$	2.70850	0.201321	0.100661	−	0.994921i	$-0.467904\pi$
$181$	2.70850	0.201321	0.100661	+	0.994921i	$0.467904\pi$
$182$	0	0
$183$	0	0
$184$	−1.29150	−0.0952108
$185$	−43.5203	−3.19967
$186$	0	0
$187$	−13.2915	−0.971971
$188$	−4.93725	−0.360086
$189$	0	0
$190$	−7.29150	−0.528981
$191$	−20.8118	−1.50589	−0.752943	−	0.658086i	$-0.771366\pi$
$191$	−20.8118	−1.50589	−0.752943	+	0.658086i	$0.771366\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	−13.5830	−0.975203
$195$	0	0
$196$	0	0
$197$	4.70850	0.335467	0.167733	−	0.985832i	$-0.446355\pi$
$197$	4.70850	0.335467	0.167733	+	0.985832i	$0.446355\pi$
$198$	0	0
$199$	−6.06275	−0.429777	−0.214888	−	0.976639i	$-0.568939\pi$
$199$	−6.06275	−0.429777	−0.214888	+	0.976639i	$0.568939\pi$
$200$	8.29150	0.586298
$201$	0	0
$202$	−8.35425	−0.587803
$203$	0	0
$204$	0	0
$205$	−39.8745	−2.78496
$206$	−3.93725	−0.274321
$207$	0	0
$208$	4.64575	0.322125
$209$	−7.29150	−0.504364
$210$	0	0
$211$	14.8745	1.02400	0.512002	−	0.858984i	$-0.328904\pi$
$211$	14.8745	1.02400	0.512002	+	0.858984i	$0.328904\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−6.00000	−0.410152
$215$	18.2288	1.24319
$216$	0	0
$217$	0	0
$218$	−3.35425	−0.227178
$219$	0	0
$220$	13.2915	0.896113
$221$	−16.9373	−1.13932
$222$	0	0
$223$	13.8745	0.929106	0.464553	−	0.885545i	$-0.346215\pi$
$223$	13.8745	0.929106	0.464553	+	0.885545i	$0.346215\pi$
$224$	0	0
$225$	0	0
$226$	2.35425	0.156602
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	9.35425	0.618146	0.309073	−	0.951038i	$-0.399981\pi$
$229$	9.35425	0.618146	0.309073	+	0.951038i	$0.399981\pi$
$230$	−4.70850	−0.310469
$231$	0	0
$232$	2.35425	0.154564
$233$	−19.2915	−1.26383	−0.631914	−	0.775038i	$-0.717730\pi$
$233$	−19.2915	−1.26383	−0.631914	+	0.775038i	$0.717730\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	−8.35425	−0.543815
$237$	0	0
$238$	0	0
$239$	−10.9373	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$239$	−10.9373	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	2.29150	0.147303
$243$	0	0
$244$	−7.35425	−0.470808
$245$	0	0
$246$	0	0
$247$	−9.29150	−0.591204
$248$	4.64575	0.295006
$249$	0	0
$250$	12.0000	0.758947
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−4.70850	−0.296021
$254$	1.35425	0.0849731
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.8745	−0.990225	−0.495112	−	0.868829i	$-0.664873\pi$
$257$	−15.8745	−0.990225	−0.495112	+	0.868829i	$0.664873\pi$
$258$	0	0
$259$	0	0
$260$	16.9373	1.05040
$261$	0	0
$262$	14.5830	0.900941
$263$	−4.93725	−0.304444	−0.152222	−	0.988346i	$-0.548643\pi$
$263$	−4.93725	−0.304444	−0.152222	+	0.988346i	$0.548643\pi$
$264$	0	0
$265$	21.8745	1.34374
$266$	0	0
$267$	0	0
$268$	−2.29150	−0.139976
$269$	16.7085	1.01874	0.509368	−	0.860549i	$-0.329879\pi$
$269$	16.7085	1.01874	0.509368	+	0.860549i	$0.329879\pi$
$270$	0	0
$271$	−5.22876	−0.317624	−0.158812	−	0.987309i	$-0.550766\pi$
$271$	−5.22876	−0.317624	−0.158812	+	0.987309i	$0.550766\pi$
$272$	−3.64575	−0.221056
$273$	0	0
$274$	1.29150	0.0780225
$275$	30.2288	1.82286
$276$	0	0
$277$	−2.06275	−0.123938	−0.0619692	−	0.998078i	$-0.519738\pi$
$277$	−2.06275	−0.123938	−0.0619692	+	0.998078i	$0.519738\pi$
$278$	−1.58301	−0.0949423
$279$	0	0
$280$	0	0
$281$	19.5203	1.16448	0.582241	−	0.813017i	$-0.302176\pi$
$281$	19.5203	1.16448	0.582241	+	0.813017i	$0.302176\pi$
$282$	0	0
$283$	2.29150	0.136216	0.0681078	−	0.997678i	$-0.478304\pi$
$283$	2.29150	0.136216	0.0681078	+	0.997678i	$0.478304\pi$
$284$	15.6458	0.928405
$285$	0	0
$286$	16.9373	1.00152
$287$	0	0
$288$	0	0
$289$	−3.70850	−0.218147
$290$	8.58301	0.504011
$291$	0	0
$292$	−10.5830	−0.619324
$293$	−7.06275	−0.412610	−0.206305	−	0.978488i	$-0.566144\pi$
$293$	−7.06275	−0.412610	−0.206305	+	0.978488i	$0.566144\pi$
$294$	0	0
$295$	−30.4575	−1.77330
$296$	−11.9373	−0.693839
$297$	0	0
$298$	22.9373	1.32872
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	−19.2288	−1.10649
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−26.8118	−1.53524
$306$	0	0
$307$	−7.58301	−0.432785	−0.216392	−	0.976306i	$-0.569429\pi$
$307$	−7.58301	−0.432785	−0.216392	+	0.976306i	$0.569429\pi$
$308$	0	0
$309$	0	0
$310$	16.9373	0.961971
$311$	13.5203	0.766664	0.383332	−	0.923611i	$-0.374777\pi$
$311$	13.5203	0.766664	0.383332	+	0.923611i	$0.374777\pi$
$312$	0	0
$313$	−12.7085	−0.718327	−0.359163	−	0.933275i	$-0.616938\pi$
$313$	−12.7085	−0.718327	−0.359163	+	0.933275i	$0.616938\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	11.2288	0.631667
$317$	26.8118	1.50590	0.752949	−	0.658079i	$-0.228631\pi$
$317$	26.8118	1.50590	0.752949	+	0.658079i	$0.228631\pi$
$318$	0	0
$319$	8.58301	0.480556
$320$	3.64575	0.203804
$321$	0	0
$322$	0	0
$323$	7.29150	0.405710
$324$	0	0
$325$	38.5203	2.13672
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	−10.9373	−0.603909
$329$	0	0
$330$	0	0
$331$	−31.8745	−1.75198	−0.875991	−	0.482328i	$-0.839791\pi$
$331$	−31.8745	−1.75198	−0.875991	+	0.482328i	$0.839791\pi$
$332$	13.2915	0.729466
$333$	0	0
$334$	8.58301	0.469641
$335$	−8.35425	−0.456441
$336$	0	0
$337$	−30.5830	−1.66596	−0.832981	−	0.553301i	$-0.813368\pi$
$337$	−30.5830	−1.66596	−0.832981	+	0.553301i	$0.813368\pi$
$338$	8.58301	0.466854
$339$	0	0
$340$	−13.2915	−0.720833
$341$	16.9373	0.917204
$342$	0	0
$343$	0	0
$344$	5.00000	0.269582
$345$	0	0
$346$	14.5830	0.783987
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	1.22876	0.0657738	0.0328869	−	0.999459i	$-0.489530\pi$
$349$	1.22876	0.0657738	0.0328869	+	0.999459i	$0.489530\pi$
$350$	0	0
$351$	0	0
$352$	3.64575	0.194319
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	57.0405	3.02740
$356$	−4.93725	−0.261674
$357$	0	0
$358$	16.9373	0.895162
$359$	11.1660	0.589319	0.294660	−	0.955602i	$-0.404794\pi$
$359$	11.1660	0.589319	0.294660	+	0.955602i	$0.404794\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.70850	0.142355
$363$	0	0
$364$	0	0
$365$	−38.5830	−2.01953
$366$	0	0
$367$	−29.8745	−1.55944	−0.779718	−	0.626130i	$-0.784638\pi$
$367$	−29.8745	−1.55944	−0.779718	+	0.626130i	$0.784638\pi$
$368$	−1.29150	−0.0673242
$369$	0	0
$370$	−43.5203	−2.26251
$371$	0	0
$372$	0	0
$373$	4.58301	0.237299	0.118650	−	0.992936i	$-0.462144\pi$
$373$	4.58301	0.237299	0.118650	+	0.992936i	$0.462144\pi$
$374$	−13.2915	−0.687287
$375$	0	0
$376$	−4.93725	−0.254619
$377$	10.9373	0.563297
$378$	0	0
$379$	25.5830	1.31411	0.657055	−	0.753842i	$-0.271802\pi$
$379$	25.5830	1.31411	0.657055	+	0.753842i	$0.271802\pi$
$380$	−7.29150	−0.374046
$381$	0	0
$382$	−20.8118	−1.06482
$383$	−20.5830	−1.05174	−0.525871	−	0.850564i	$-0.676261\pi$
$383$	−20.5830	−1.05174	−0.525871	+	0.850564i	$0.676261\pi$
$384$	0	0
$385$	0	0
$386$	−7.00000	−0.356291
$387$	0	0
$388$	−13.5830	−0.689573
$389$	19.2915	0.978118	0.489059	−	0.872251i	$-0.337340\pi$
$389$	19.2915	0.978118	0.489059	+	0.872251i	$0.337340\pi$
$390$	0	0
$391$	4.70850	0.238119
$392$	0	0
$393$	0	0
$394$	4.70850	0.237211
$395$	40.9373	2.05978
$396$	0	0
$397$	16.6458	0.835426	0.417713	−	0.908579i	$-0.362832\pi$
$397$	16.6458	0.835426	0.417713	+	0.908579i	$0.362832\pi$
$398$	−6.06275	−0.303898
$399$	0	0
$400$	8.29150	0.414575
$401$	−20.8118	−1.03929	−0.519645	−	0.854382i	$-0.673936\pi$
$401$	−20.8118	−1.03929	−0.519645	+	0.854382i	$0.673936\pi$
$402$	0	0
$403$	21.5830	1.07513
$404$	−8.35425	−0.415639
$405$	0	0
$406$	0	0
$407$	−43.5203	−2.15722
$408$	0	0
$409$	−14.8745	−0.735497	−0.367749	−	0.929925i	$-0.619871\pi$
$409$	−14.8745	−0.735497	−0.367749	+	0.929925i	$0.619871\pi$
$410$	−39.8745	−1.96926
$411$	0	0
$412$	−3.93725	−0.193975
$413$	0	0
$414$	0	0
$415$	48.4575	2.37869
$416$	4.64575	0.227777
$417$	0	0
$418$	−7.29150	−0.356639
$419$	−28.9373	−1.41368	−0.706839	−	0.707375i	$-0.749879\pi$
$419$	−28.9373	−1.41368	−0.706839	+	0.707375i	$0.749879\pi$
$420$	0	0
$421$	−14.7085	−0.716848	−0.358424	−	0.933559i	$-0.616686\pi$
$421$	−14.7085	−0.716848	−0.358424	+	0.933559i	$0.616686\pi$
$422$	14.8745	0.724080
$423$	0	0
$424$	6.00000	0.291386
$425$	−30.2288	−1.46631
$426$	0	0
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	18.2288	0.879069
$431$	38.8118	1.86950	0.934748	−	0.355310i	$-0.115625\pi$
$431$	38.8118	1.86950	0.934748	+	0.355310i	$0.115625\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	0	0
$436$	−3.35425	−0.160639
$437$	2.58301	0.123562
$438$	0	0
$439$	−25.1660	−1.20111	−0.600554	−	0.799584i	$-0.705053\pi$
$439$	−25.1660	−1.20111	−0.600554	+	0.799584i	$0.705053\pi$
$440$	13.2915	0.633648
$441$	0	0
$442$	−16.9373	−0.805623
$443$	−19.2915	−0.916567	−0.458283	−	0.888806i	$-0.651536\pi$
$443$	−19.2915	−0.916567	−0.458283	+	0.888806i	$0.651536\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	13.8745	0.656977
$447$	0	0
$448$	0	0
$449$	2.35425	0.111104	0.0555519	−	0.998456i	$-0.482308\pi$
$449$	2.35425	0.111104	0.0555519	+	0.998456i	$0.482308\pi$
$450$	0	0
$451$	−39.8745	−1.87762
$452$	2.35425	0.110735
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	−8.29150	−0.387860	−0.193930	−	0.981015i	$-0.562123\pi$
$457$	−8.29150	−0.387860	−0.193930	+	0.981015i	$0.562123\pi$
$458$	9.35425	0.437095
$459$	0	0
$460$	−4.70850	−0.219535
$461$	−24.2288	−1.12845	−0.564223	−	0.825623i	$-0.690824\pi$
$461$	−24.2288	−1.12845	−0.564223	+	0.825623i	$0.690824\pi$
$462$	0	0
$463$	18.7085	0.869458	0.434729	−	0.900561i	$-0.356844\pi$
$463$	18.7085	0.869458	0.434729	+	0.900561i	$0.356844\pi$
$464$	2.35425	0.109293
$465$	0	0
$466$	−19.2915	−0.893662
$467$	−0.228757	−0.0105856	−0.00529280	−	0.999986i	$-0.501685\pi$
$467$	−0.228757	−0.0105856	−0.00529280	+	0.999986i	$0.501685\pi$
$468$	0	0
$469$	0	0
$470$	−18.0000	−0.830278
$471$	0	0
$472$	−8.35425	−0.384535
$473$	18.2288	0.838159
$474$	0	0
$475$	−16.5830	−0.760880
$476$	0	0
$477$	0	0
$478$	−10.9373	−0.500258
$479$	−3.64575	−0.166579	−0.0832893	−	0.996525i	$-0.526543\pi$
$479$	−3.64575	−0.166579	−0.0832893	+	0.996525i	$0.526543\pi$
$480$	0	0
$481$	−55.4575	−2.52864
$482$	−5.00000	−0.227744
$483$	0	0
$484$	2.29150	0.104159
$485$	−49.5203	−2.24860
$486$	0	0
$487$	23.8745	1.08186	0.540929	−	0.841069i	$-0.318073\pi$
$487$	23.8745	1.08186	0.540929	+	0.841069i	$0.318073\pi$
$488$	−7.35425	−0.332911
$489$	0	0
$490$	0	0
$491$	25.7490	1.16204	0.581018	−	0.813890i	$-0.302654\pi$
$491$	25.7490	1.16204	0.581018	+	0.813890i	$0.302654\pi$
$492$	0	0
$493$	−8.58301	−0.386559
$494$	−9.29150	−0.418044
$495$	0	0
$496$	4.64575	0.208600
$497$	0	0
$498$	0	0
$499$	−6.16601	−0.276029	−0.138014	−	0.990430i	$-0.544072\pi$
$499$	−6.16601	−0.276029	−0.138014	+	0.990430i	$0.544072\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	18.0000	0.803379
$503$	−3.87451	−0.172756	−0.0863779	−	0.996262i	$-0.527529\pi$
$503$	−3.87451	−0.172756	−0.0863779	+	0.996262i	$0.527529\pi$
$504$	0	0
$505$	−30.4575	−1.35534
$506$	−4.70850	−0.209318
$507$	0	0
$508$	1.35425	0.0600851
$509$	−8.12549	−0.360156	−0.180078	−	0.983652i	$-0.557635\pi$
$509$	−8.12549	−0.360156	−0.180078	+	0.983652i	$0.557635\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.8745	−0.700195
$515$	−14.3542	−0.632524
$516$	0	0
$517$	−18.0000	−0.791639
$518$	0	0
$519$	0	0
$520$	16.9373	0.742748
$521$	8.12549	0.355984	0.177992	−	0.984032i	$-0.443040\pi$
$521$	8.12549	0.355984	0.177992	+	0.984032i	$0.443040\pi$
$522$	0	0
$523$	1.00000	0.0437269	0.0218635	−	0.999761i	$-0.493040\pi$
$523$	1.00000	0.0437269	0.0218635	+	0.999761i	$0.493040\pi$
$524$	14.5830	0.637062
$525$	0	0
$526$	−4.93725	−0.215275
$527$	−16.9373	−0.737798
$528$	0	0
$529$	−21.3320	−0.927479
$530$	21.8745	0.950168
$531$	0	0
$532$	0	0
$533$	−50.8118	−2.20090
$534$	0	0
$535$	−21.8745	−0.945717
$536$	−2.29150	−0.0989778
$537$	0	0
$538$	16.7085	0.720354
$539$	0	0
$540$	0	0
$541$	1.16601	0.0501307	0.0250654	−	0.999686i	$-0.492021\pi$
$541$	1.16601	0.0501307	0.0250654	+	0.999686i	$0.492021\pi$
$542$	−5.22876	−0.224594
$543$	0	0
$544$	−3.64575	−0.156310
$545$	−12.2288	−0.523822
$546$	0	0
$547$	18.2915	0.782088	0.391044	−	0.920372i	$-0.372114\pi$
$547$	18.2915	0.782088	0.391044	+	0.920372i	$0.372114\pi$
$548$	1.29150	0.0551703
$549$	0	0
$550$	30.2288	1.28896
$551$	−4.70850	−0.200589
$552$	0	0
$553$	0	0
$554$	−2.06275	−0.0876377
$555$	0	0
$556$	−1.58301	−0.0671344
$557$	−5.77124	−0.244535	−0.122268	−	0.992497i	$-0.539017\pi$
$557$	−5.77124	−0.244535	−0.122268	+	0.992497i	$0.539017\pi$
$558$	0	0
$559$	23.2288	0.982472
$560$	0	0
$561$	0	0
$562$	19.5203	0.823412
$563$	−18.4575	−0.777891	−0.388946	−	0.921261i	$-0.627161\pi$
$563$	−18.4575	−0.777891	−0.388946	+	0.921261i	$0.627161\pi$
$564$	0	0
$565$	8.58301	0.361090
$566$	2.29150	0.0963190
$567$	0	0
$568$	15.6458	0.656481
$569$	−16.9373	−0.710047	−0.355023	−	0.934857i	$-0.615527\pi$
$569$	−16.9373	−0.710047	−0.355023	+	0.934857i	$0.615527\pi$
$570$	0	0
$571$	9.29150	0.388837	0.194419	−	0.980919i	$-0.437718\pi$
$571$	9.29150	0.388837	0.194419	+	0.980919i	$0.437718\pi$
$572$	16.9373	0.708182
$573$	0	0
$574$	0	0
$575$	−10.7085	−0.446575
$576$	0	0
$577$	32.2915	1.34431	0.672156	−	0.740409i	$-0.265368\pi$
$577$	32.2915	1.34431	0.672156	+	0.740409i	$0.265368\pi$
$578$	−3.70850	−0.154253
$579$	0	0
$580$	8.58301	0.356390
$581$	0	0
$582$	0	0
$583$	21.8745	0.905950
$584$	−10.5830	−0.437928
$585$	0	0
$586$	−7.06275	−0.291759
$587$	−11.7712	−0.485851	−0.242926	−	0.970045i	$-0.578107\pi$
$587$	−11.7712	−0.485851	−0.242926	+	0.970045i	$0.578107\pi$
$588$	0	0
$589$	−9.29150	−0.382850
$590$	−30.4575	−1.25392
$591$	0	0
$592$	−11.9373	−0.490618
$593$	−40.9373	−1.68109	−0.840546	−	0.541741i	$-0.817766\pi$
$593$	−40.9373	−1.68109	−0.840546	+	0.541741i	$0.817766\pi$
$594$	0	0
$595$	0	0
$596$	22.9373	0.939547
$597$	0	0
$598$	−6.00000	−0.245358
$599$	9.87451	0.403461	0.201731	−	0.979441i	$-0.435343\pi$
$599$	9.87451	0.403461	0.201731	+	0.979441i	$0.435343\pi$
$600$	0	0
$601$	−26.8745	−1.09623	−0.548117	−	0.836402i	$-0.684655\pi$
$601$	−26.8745	−1.09623	−0.548117	+	0.836402i	$0.684655\pi$
$602$	0	0
$603$	0	0
$604$	−19.2288	−0.782407
$605$	8.35425	0.339649
$606$	0	0
$607$	−5.41699	−0.219869	−0.109935	−	0.993939i	$-0.535064\pi$
$607$	−5.41699	−0.219869	−0.109935	+	0.993939i	$0.535064\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−26.8118	−1.08558
$611$	−22.9373	−0.927942
$612$	0	0
$613$	−42.3948	−1.71231	−0.856154	−	0.516720i	$-0.827153\pi$
$613$	−42.3948	−1.71231	−0.856154	+	0.516720i	$0.827153\pi$
$614$	−7.58301	−0.306025
$615$	0	0
$616$	0	0
$617$	1.52026	0.0612033	0.0306017	−	0.999532i	$-0.490258\pi$
$617$	1.52026	0.0612033	0.0306017	+	0.999532i	$0.490258\pi$
$618$	0	0
$619$	8.29150	0.333264	0.166632	−	0.986019i	$-0.446711\pi$
$619$	8.29150	0.333264	0.166632	+	0.986019i	$0.446711\pi$
$620$	16.9373	0.680216
$621$	0	0
$622$	13.5203	0.542113
$623$	0	0
$624$	0	0
$625$	2.29150	0.0916601
$626$	−12.7085	−0.507934
$627$	0	0
$628$	4.00000	0.159617
$629$	43.5203	1.73527
$630$	0	0
$631$	6.06275	0.241354	0.120677	−	0.992692i	$-0.461493\pi$
$631$	6.06275	0.241354	0.120677	+	0.992692i	$0.461493\pi$
$632$	11.2288	0.446656
$633$	0	0
$634$	26.8118	1.06483
$635$	4.93725	0.195929
$636$	0	0
$637$	0	0
$638$	8.58301	0.339804
$639$	0	0
$640$	3.64575	0.144111
$641$	−18.2288	−0.719993	−0.359996	−	0.932954i	$-0.617222\pi$
$641$	−18.2288	−0.719993	−0.359996	+	0.932954i	$0.617222\pi$
$642$	0	0
$643$	3.12549	0.123257	0.0616287	−	0.998099i	$-0.480371\pi$
$643$	3.12549	0.123257	0.0616287	+	0.998099i	$0.480371\pi$
$644$	0	0
$645$	0	0
$646$	7.29150	0.286880
$647$	−9.87451	−0.388207	−0.194103	−	0.980981i	$-0.562180\pi$
$647$	−9.87451	−0.388207	−0.194103	+	0.980981i	$0.562180\pi$
$648$	0	0
$649$	−30.4575	−1.19556
$650$	38.5203	1.51089
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	53.1660	2.07737
$656$	−10.9373	−0.427028
$657$	0	0
$658$	0	0
$659$	−24.2288	−0.943818	−0.471909	−	0.881647i	$-0.656435\pi$
$659$	−24.2288	−0.943818	−0.471909	+	0.881647i	$0.656435\pi$
$660$	0	0
$661$	3.16601	0.123144	0.0615718	−	0.998103i	$-0.480389\pi$
$661$	3.16601	0.123144	0.0615718	+	0.998103i	$0.480389\pi$
$662$	−31.8745	−1.23884
$663$	0	0
$664$	13.2915	0.515810
$665$	0	0
$666$	0	0
$667$	−3.04052	−0.117729
$668$	8.58301	0.332086
$669$	0	0
$670$	−8.35425	−0.322753
$671$	−26.8118	−1.03506
$672$	0	0
$673$	15.7490	0.607080	0.303540	−	0.952819i	$-0.401831\pi$
$673$	15.7490	0.607080	0.303540	+	0.952819i	$0.401831\pi$
$674$	−30.5830	−1.17801
$675$	0	0
$676$	8.58301	0.330116
$677$	45.8745	1.76310	0.881550	−	0.472090i	$-0.156500\pi$
$677$	45.8745	1.76310	0.881550	+	0.472090i	$0.156500\pi$
$678$	0	0
$679$	0	0
$680$	−13.2915	−0.509706
$681$	0	0
$682$	16.9373	0.648561
$683$	−26.8118	−1.02592	−0.512962	−	0.858411i	$-0.671452\pi$
$683$	−26.8118	−1.02592	−0.512962	+	0.858411i	$0.671452\pi$
$684$	0	0
$685$	4.70850	0.179902
$686$	0	0
$687$	0	0
$688$	5.00000	0.190623
$689$	27.8745	1.06193
$690$	0	0
$691$	38.7490	1.47408	0.737041	−	0.675848i	$-0.236222\pi$
$691$	38.7490	1.47408	0.737041	+	0.675848i	$0.236222\pi$
$692$	14.5830	0.554363
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−5.77124	−0.218916
$696$	0	0
$697$	39.8745	1.51035
$698$	1.22876	0.0465091
$699$	0	0
$700$	0	0
$701$	−26.5830	−1.00403	−0.502013	−	0.864860i	$-0.667407\pi$
$701$	−26.5830	−1.00403	−0.502013	+	0.864860i	$0.667407\pi$
$702$	0	0
$703$	23.8745	0.900444
$704$	3.64575	0.137404
$705$	0	0
$706$	12.0000	0.451626
$707$	0	0
$708$	0	0
$709$	37.8118	1.42005	0.710025	−	0.704176i	$-0.248683\pi$
$709$	37.8118	1.42005	0.710025	+	0.704176i	$0.248683\pi$
$710$	57.0405	2.14069
$711$	0	0
$712$	−4.93725	−0.185031
$713$	−6.00000	−0.224702
$714$	0	0
$715$	61.7490	2.30928
$716$	16.9373	0.632975
$717$	0	0
$718$	11.1660	0.416712
$719$	−22.9373	−0.855415	−0.427708	−	0.903917i	$-0.640679\pi$
$719$	−22.9373	−0.855415	−0.427708	+	0.903917i	$0.640679\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	0	0
$724$	2.70850	0.100661
$725$	19.5203	0.724964
$726$	0	0
$727$	1.22876	0.0455721	0.0227860	−	0.999740i	$-0.492746\pi$
$727$	1.22876	0.0455721	0.0227860	+	0.999740i	$0.492746\pi$
$728$	0	0
$729$	0	0
$730$	−38.5830	−1.42802
$731$	−18.2288	−0.674215
$732$	0	0
$733$	−35.2288	−1.30120	−0.650602	−	0.759419i	$-0.725483\pi$
$733$	−35.2288	−1.30120	−0.650602	+	0.759419i	$0.725483\pi$
$734$	−29.8745	−1.10269
$735$	0	0
$736$	−1.29150	−0.0476054
$737$	−8.35425	−0.307733
$738$	0	0
$739$	−31.4575	−1.15718	−0.578592	−	0.815617i	$-0.696397\pi$
$739$	−31.4575	−1.15718	−0.578592	+	0.815617i	$0.696397\pi$
$740$	−43.5203	−1.59984
$741$	0	0
$742$	0	0
$743$	−28.9373	−1.06160	−0.530802	−	0.847496i	$-0.678109\pi$
$743$	−28.9373	−1.06160	−0.530802	+	0.847496i	$0.678109\pi$
$744$	0	0
$745$	83.6235	3.06373
$746$	4.58301	0.167796
$747$	0	0
$748$	−13.2915	−0.485985
$749$	0	0
$750$	0	0
$751$	−52.4575	−1.91420	−0.957101	−	0.289755i	$-0.906426\pi$
$751$	−52.4575	−1.91420	−0.957101	+	0.289755i	$0.906426\pi$
$752$	−4.93725	−0.180043
$753$	0	0
$754$	10.9373	0.398311
$755$	−70.1033	−2.55132
$756$	0	0
$757$	48.9778	1.78013	0.890064	−	0.455836i	$-0.150660\pi$
$757$	48.9778	1.78013	0.890064	+	0.455836i	$0.150660\pi$
$758$	25.5830	0.929217
$759$	0	0
$760$	−7.29150	−0.264491
$761$	−2.58301	−0.0936339	−0.0468169	−	0.998903i	$-0.514908\pi$
$761$	−2.58301	−0.0936339	−0.0468169	+	0.998903i	$0.514908\pi$
$762$	0	0
$763$	0	0
$764$	−20.8118	−0.752943
$765$	0	0
$766$	−20.5830	−0.743694
$767$	−38.8118	−1.40141
$768$	0	0
$769$	−40.5830	−1.46346	−0.731730	−	0.681594i	$-0.761287\pi$
$769$	−40.5830	−1.46346	−0.731730	+	0.681594i	$0.761287\pi$
$770$	0	0
$771$	0	0
$772$	−7.00000	−0.251936
$773$	46.3320	1.66645	0.833223	−	0.552936i	$-0.186493\pi$
$773$	46.3320	1.66645	0.833223	+	0.552936i	$0.186493\pi$
$774$	0	0
$775$	38.5203	1.38369
$776$	−13.5830	−0.487601
$777$	0	0
$778$	19.2915	0.691634
$779$	21.8745	0.783736
$780$	0	0
$781$	57.0405	2.04107
$782$	4.70850	0.168376
$783$	0	0
$784$	0	0
$785$	14.5830	0.520490
$786$	0	0
$787$	11.7085	0.417363	0.208681	−	0.977984i	$-0.433083\pi$
$787$	11.7085	0.417363	0.208681	+	0.977984i	$0.433083\pi$
$788$	4.70850	0.167733
$789$	0	0
$790$	40.9373	1.45648
$791$	0	0
$792$	0	0
$793$	−34.1660	−1.21327
$794$	16.6458	0.590736
$795$	0	0
$796$	−6.06275	−0.214888
$797$	14.8118	0.524660	0.262330	−	0.964978i	$-0.415509\pi$
$797$	14.8118	0.524660	0.262330	+	0.964978i	$0.415509\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	8.29150	0.293149
$801$	0	0
$802$	−20.8118	−0.734889
$803$	−38.5830	−1.36156
$804$	0	0
$805$	0	0
$806$	21.5830	0.760229
$807$	0	0
$808$	−8.35425	−0.293901
$809$	15.4170	0.542033	0.271016	−	0.962575i	$-0.412640\pi$
$809$	15.4170	0.542033	0.271016	+	0.962575i	$0.412640\pi$
$810$	0	0
$811$	29.2915	1.02856	0.514282	−	0.857621i	$-0.328059\pi$
$811$	29.2915	1.02856	0.514282	+	0.857621i	$0.328059\pi$
$812$	0	0
$813$	0	0
$814$	−43.5203	−1.52538
$815$	−3.64575	−0.127705
$816$	0	0
$817$	−10.0000	−0.349856
$818$	−14.8745	−0.520075
$819$	0	0
$820$	−39.8745	−1.39248
$821$	20.5830	0.718352	0.359176	−	0.933270i	$-0.383058\pi$
$821$	20.5830	0.718352	0.359176	+	0.933270i	$0.383058\pi$
$822$	0	0
$823$	−35.1033	−1.22362	−0.611811	−	0.791004i	$-0.709559\pi$
$823$	−35.1033	−1.22362	−0.611811	+	0.791004i	$0.709559\pi$
$824$	−3.93725	−0.137161
$825$	0	0
$826$	0	0
$827$	−43.7490	−1.52130	−0.760651	−	0.649161i	$-0.775120\pi$
$827$	−43.7490	−1.52130	−0.760651	+	0.649161i	$0.775120\pi$
$828$	0	0
$829$	29.2915	1.01734	0.508668	−	0.860963i	$-0.330138\pi$
$829$	29.2915	1.01734	0.508668	+	0.860963i	$0.330138\pi$
$830$	48.4575	1.68198
$831$	0	0
$832$	4.64575	0.161062
$833$	0	0
$834$	0	0
$835$	31.2915	1.08289
$836$	−7.29150	−0.252182
$837$	0	0
$838$	−28.9373	−0.999621
$839$	49.2915	1.70173	0.850866	−	0.525383i	$-0.176078\pi$
$839$	49.2915	1.70173	0.850866	+	0.525383i	$0.176078\pi$
$840$	0	0
$841$	−23.4575	−0.808880
$842$	−14.7085	−0.506888
$843$	0	0
$844$	14.8745	0.512002
$845$	31.2915	1.07646
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−30.2288	−1.03684
$851$	15.4170	0.528488
$852$	0	0
$853$	−47.8745	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$853$	−47.8745	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$854$	0	0
$855$	0	0
$856$	−6.00000	−0.205076
$857$	26.8118	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$857$	26.8118	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$858$	0	0
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	18.2288	0.621595
$861$	0	0
$862$	38.8118	1.32193
$863$	29.1660	0.992823	0.496411	−	0.868087i	$-0.334651\pi$
$863$	29.1660	0.992823	0.496411	+	0.868087i	$0.334651\pi$
$864$	0	0
$865$	53.1660	1.80770
$866$	19.0000	0.645646
$867$	0	0
$868$	0	0
$869$	40.9373	1.38870
$870$	0	0
$871$	−10.6458	−0.360718
$872$	−3.35425	−0.113589
$873$	0	0
$874$	2.58301	0.0873715
$875$	0	0
$876$	0	0
$877$	26.6458	0.899763	0.449882	−	0.893088i	$-0.351466\pi$
$877$	26.6458	0.899763	0.449882	+	0.893088i	$0.351466\pi$
$878$	−25.1660	−0.849312
$879$	0	0
$880$	13.2915	0.448056
$881$	3.87451	0.130535	0.0652677	−	0.997868i	$-0.479210\pi$
$881$	3.87451	0.130535	0.0652677	+	0.997868i	$0.479210\pi$
$882$	0	0
$883$	−19.8745	−0.668830	−0.334415	−	0.942426i	$-0.608539\pi$
$883$	−19.8745	−0.668830	−0.334415	+	0.942426i	$0.608539\pi$
$884$	−16.9373	−0.569661
$885$	0	0
$886$	−19.2915	−0.648111
$887$	−15.8745	−0.533014	−0.266507	−	0.963833i	$-0.585870\pi$
$887$	−15.8745	−0.533014	−0.266507	+	0.963833i	$0.585870\pi$
$888$	0	0
$889$	0	0
$890$	−18.0000	−0.603361
$891$	0	0
$892$	13.8745	0.464553
$893$	9.87451	0.330438
$894$	0	0
$895$	61.7490	2.06404
$896$	0	0
$897$	0	0
$898$	2.35425	0.0785623
$899$	10.9373	0.364778
$900$	0	0
$901$	−21.8745	−0.728746
$902$	−39.8745	−1.32768
$903$	0	0
$904$	2.35425	0.0783011
$905$	9.87451	0.328240
$906$	0	0
$907$	36.2915	1.20504	0.602520	−	0.798104i	$-0.294163\pi$
$907$	36.2915	1.20504	0.602520	+	0.798104i	$0.294163\pi$
$908$	−6.00000	−0.199117
$909$	0	0
$910$	0	0
$911$	28.9373	0.958734	0.479367	−	0.877615i	$-0.340866\pi$
$911$	28.9373	0.958734	0.479367	+	0.877615i	$0.340866\pi$
$912$	0	0
$913$	48.4575	1.60371
$914$	−8.29150	−0.274259
$915$	0	0
$916$	9.35425	0.309073
$917$	0	0
$918$	0	0
$919$	28.7712	0.949076	0.474538	−	0.880235i	$-0.342615\pi$
$919$	28.7712	0.949076	0.474538	+	0.880235i	$0.342615\pi$
$920$	−4.70850	−0.155235
$921$	0	0
$922$	−24.2288	−0.797932
$923$	72.6863	2.39250
$924$	0	0
$925$	−98.9778	−3.25437
$926$	18.7085	0.614799
$927$	0	0
$928$	2.35425	0.0772820
$929$	16.9373	0.555693	0.277847	−	0.960625i	$-0.410379\pi$
$929$	16.9373	0.555693	0.277847	+	0.960625i	$0.410379\pi$
$930$	0	0
$931$	0	0
$932$	−19.2915	−0.631914
$933$	0	0
$934$	−0.228757	−0.00748514
$935$	−48.4575	−1.58473
$936$	0	0
$937$	44.7490	1.46189	0.730943	−	0.682438i	$-0.239080\pi$
$937$	44.7490	1.46189	0.730943	+	0.682438i	$0.239080\pi$
$938$	0	0
$939$	0	0
$940$	−18.0000	−0.587095
$941$	53.3948	1.74062	0.870310	−	0.492505i	$-0.163919\pi$
$941$	53.3948	1.74062	0.870310	+	0.492505i	$0.163919\pi$
$942$	0	0
$943$	14.1255	0.459989
$944$	−8.35425	−0.271908
$945$	0	0
$946$	18.2288	0.592668
$947$	10.4797	0.340546	0.170273	−	0.985397i	$-0.445535\pi$
$947$	10.4797	0.340546	0.170273	+	0.985397i	$0.445535\pi$
$948$	0	0
$949$	−49.1660	−1.59600
$950$	−16.5830	−0.538024
$951$	0	0
$952$	0	0
$953$	52.3320	1.69520	0.847600	−	0.530635i	$-0.178047\pi$
$953$	52.3320	1.69520	0.847600	+	0.530635i	$0.178047\pi$
$954$	0	0
$955$	−75.8745	−2.45524
$956$	−10.9373	−0.353736
$957$	0	0
$958$	−3.64575	−0.117789
$959$	0	0
$960$	0	0
$961$	−9.41699	−0.303774
$962$	−55.4575	−1.78802
$963$	0	0
$964$	−5.00000	−0.161039
$965$	−25.5203	−0.821526
$966$	0	0
$967$	36.0627	1.15970	0.579850	−	0.814723i	$-0.303111\pi$
$967$	36.0627	1.15970	0.579850	+	0.814723i	$0.303111\pi$
$968$	2.29150	0.0736517
$969$	0	0
$970$	−49.5203	−1.59000
$971$	9.87451	0.316888	0.158444	−	0.987368i	$-0.449352\pi$
$971$	9.87451	0.316888	0.158444	+	0.987368i	$0.449352\pi$
$972$	0	0
$973$	0	0
$974$	23.8745	0.764989
$975$	0	0
$976$	−7.35425	−0.235404
$977$	−51.8745	−1.65961	−0.829806	−	0.558052i	$-0.811549\pi$
$977$	−51.8745	−1.65961	−0.829806	+	0.558052i	$0.811549\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	25.7490	0.821684
$983$	−20.8118	−0.663792	−0.331896	−	0.943316i	$-0.607688\pi$
$983$	−20.8118	−0.663792	−0.331896	+	0.943316i	$0.607688\pi$
$984$	0	0
$985$	17.1660	0.546955
$986$	−8.58301	−0.273339
$987$	0	0
$988$	−9.29150	−0.295602
$989$	−6.45751	−0.205337
$990$	0	0
$991$	−26.0627	−0.827910	−0.413955	−	0.910297i	$-0.635853\pi$
$991$	−26.0627	−0.827910	−0.413955	+	0.910297i	$0.635853\pi$
$992$	4.64575	0.147503
$993$	0	0
$994$	0	0
$995$	−22.1033	−0.700721
$996$	0	0
$997$	−23.6863	−0.750152	−0.375076	−	0.926994i	$-0.622383\pi$
$997$	−23.6863	−0.750152	−0.375076	+	0.926994i	$0.622383\pi$
$998$	−6.16601	−0.195182
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.a.bo.1.2		2
3.2	odd	2		2646.2.a.bf.1.1		2
7.3	odd	6		378.2.g.g.163.2	yes	4
7.5	odd	6		378.2.g.g.109.2	✓	4
7.6	odd	2		2646.2.a.bl.1.1		2
21.5	even	6		378.2.g.h.109.1	yes	4
21.17	even	6		378.2.g.h.163.1	yes	4
21.20	even	2		2646.2.a.bi.1.2		2
63.5	even	6		1134.2.e.q.865.1		4
63.31	odd	6		1134.2.h.q.541.1		4
63.38	even	6		1134.2.e.q.919.1		4
63.40	odd	6		1134.2.e.t.865.2		4
63.47	even	6		1134.2.h.t.109.2		4
63.52	odd	6		1134.2.e.t.919.2		4
63.59	even	6		1134.2.h.t.541.2		4
63.61	odd	6		1134.2.h.q.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.g.g.109.2	✓	4	7.5	odd	6
378.2.g.g.163.2	yes	4	7.3	odd	6
378.2.g.h.109.1	yes	4	21.5	even	6
378.2.g.h.163.1	yes	4	21.17	even	6
1134.2.e.q.865.1		4	63.5	even	6
1134.2.e.q.919.1		4	63.38	even	6
1134.2.e.t.865.2		4	63.40	odd	6
1134.2.e.t.919.2		4	63.52	odd	6
1134.2.h.q.109.1		4	63.61	odd	6
1134.2.h.q.541.1		4	63.31	odd	6
1134.2.h.t.109.2		4	63.47	even	6
1134.2.h.t.541.2		4	63.59	even	6
2646.2.a.bf.1.1		2	3.2	odd	2
2646.2.a.bi.1.2		2	21.20	even	2
2646.2.a.bl.1.1		2	7.6	odd	2
2646.2.a.bo.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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.64575 q^{5} +1.00000 q^{8} +3.64575 q^{10} +3.64575 q^{11} +4.64575 q^{13} +1.00000 q^{16} -3.64575 q^{17} -2.00000 q^{19} +3.64575 q^{20} +3.64575 q^{22} -1.29150 q^{23} +8.29150 q^{25} +4.64575 q^{26} +2.35425 q^{29} +4.64575 q^{31} +1.00000 q^{32} -3.64575 q^{34} -11.9373 q^{37} -2.00000 q^{38} +3.64575 q^{40} -10.9373 q^{41} +5.00000 q^{43} +3.64575 q^{44} -1.29150 q^{46} -4.93725 q^{47} +8.29150 q^{50} +4.64575 q^{52} +6.00000 q^{53} +13.2915 q^{55} +2.35425 q^{58} -8.35425 q^{59} -7.35425 q^{61} +4.64575 q^{62} +1.00000 q^{64} +16.9373 q^{65} -2.29150 q^{67} -3.64575 q^{68} +15.6458 q^{71} -10.5830 q^{73} -11.9373 q^{74} -2.00000 q^{76} +11.2288 q^{79} +3.64575 q^{80} -10.9373 q^{82} +13.2915 q^{83} -13.2915 q^{85} +5.00000 q^{86} +3.64575 q^{88} -4.93725 q^{89} -1.29150 q^{92} -4.93725 q^{94} -7.29150 q^{95} -13.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 6 q^{25} + 4 q^{26} + 10 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{34}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^20 + 2 * q^22 + 8 * q^23 + 6 * q^25 + 4 * q^26 + 10 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^34 - 8 * q^37 - 4 * q^38 + 2 * q^40 - 6 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 6 * q^47 + 6 * q^50 + 4 * q^52 + 12 * q^53 + 16 * q^55 + 10 * q^58 - 22 * q^59 - 20 * q^61 + 4 * q^62 + 2 * q^64 + 18 * q^65 + 6 * q^67 - 2 * q^68 + 26 * q^71 - 8 * q^74 - 4 * q^76 - 4 * q^79 + 2 * q^80 - 6 * q^82 + 16 * q^83 - 16 * q^85 + 10 * q^86 + 2 * q^88 + 6 * q^89 + 8 * q^92 + 6 * q^94 - 4 * q^95 - 6 * q^97

Introduction

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