LMFDB - Embedded newform 7728.2.a.bt.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.52892$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.52892 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.52892 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.86651 q^{11} +0.604574 q^{13} +1.52892 q^{15} -2.92434 q^{17} +1.86651 q^{19} -1.00000 q^{21} +1.00000 q^{23} -2.66241 q^{25} -1.00000 q^{27} +7.19133 q^{29} +5.86651 q^{31} +1.86651 q^{33} -1.52892 q^{35} -3.19133 q^{37} -0.604574 q^{39} -6.65736 q^{41} +5.66241 q^{43} -1.52892 q^{45} -10.7909 q^{47} +1.00000 q^{49} +2.92434 q^{51} -1.52892 q^{53} +2.85374 q^{55} -1.86651 q^{57} +6.31977 q^{59} -3.66241 q^{61} +1.00000 q^{63} -0.924344 q^{65} -5.37761 q^{67} -1.00000 q^{69} -15.2441 q^{71} +13.9822 q^{73} +2.66241 q^{75} -1.86651 q^{77} -9.19133 q^{79} +1.00000 q^{81} +0.924344 q^{83} +4.47108 q^{85} -7.19133 q^{87} +7.39543 q^{89} +0.604574 q^{91} -5.86651 q^{93} -2.85374 q^{95} +5.98218 q^{97} -1.86651 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 9 q^{13} - 3 q^{15} + 6 q^{17} + 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} + 6 q^{29} + 18 q^{31} + 6 q^{33} + 3 q^{35} + 6 q^{37} - 9 q^{39} - 6 q^{41} + 9 q^{43} + 3 q^{45} - 18 q^{47} + 3 q^{49} - 6 q^{51} + 3 q^{53} - 15 q^{55} - 6 q^{57} - 3 q^{59} - 3 q^{61} + 3 q^{63} + 12 q^{65} + 21 q^{67} - 3 q^{69} - 9 q^{71} + 12 q^{73} - 6 q^{77} - 12 q^{79} + 3 q^{81} - 12 q^{83} + 21 q^{85} - 6 q^{87} + 15 q^{89} + 9 q^{91} - 18 q^{93} + 15 q^{95} - 12 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 - 6 * q^11 + 9 * q^13 - 3 * q^15 + 6 * q^17 + 6 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 + 6 * q^29 + 18 * q^31 + 6 * q^33 + 3 * q^35 + 6 * q^37 - 9 * q^39 - 6 * q^41 + 9 * q^43 + 3 * q^45 - 18 * q^47 + 3 * q^49 - 6 * q^51 + 3 * q^53 - 15 * q^55 - 6 * q^57 - 3 * q^59 - 3 * q^61 + 3 * q^63 + 12 * q^65 + 21 * q^67 - 3 * q^69 - 9 * q^71 + 12 * q^73 - 6 * q^77 - 12 * q^79 + 3 * q^81 - 12 * q^83 + 21 * q^85 - 6 * q^87 + 15 * q^89 + 9 * q^91 - 18 * q^93 + 15 * q^95 - 12 * q^97 - 6 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.52892	−0.683753	−0.341876	−	0.939745i	$-0.611062\pi$
$5$	−1.52892	−0.683753	−0.341876	+	0.939745i	$0.611062\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.86651	−0.562773	−0.281387	−	0.959594i	$-0.590794\pi$
$11$	−1.86651	−0.562773	−0.281387	+	0.959594i	$0.590794\pi$
$12$	0	0
$13$	0.604574	0.167679	0.0838393	−	0.996479i	$-0.473282\pi$
$13$	0.604574	0.167679	0.0838393	+	0.996479i	$0.473282\pi$
$14$	0	0
$15$	1.52892	0.394765
$16$	0	0
$17$	−2.92434	−0.709258	−0.354629	−	0.935007i	$-0.615393\pi$
$17$	−2.92434	−0.709258	−0.354629	+	0.935007i	$0.615393\pi$
$18$	0	0
$19$	1.86651	0.428206	0.214103	−	0.976811i	$-0.431317\pi$
$19$	1.86651	0.428206	0.214103	+	0.976811i	$0.431317\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.66241	−0.532482
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.19133	1.33540	0.667698	−	0.744432i	$-0.267280\pi$
$29$	7.19133	1.33540	0.667698	+	0.744432i	$0.267280\pi$
$30$	0	0
$31$	5.86651	1.05366	0.526828	−	0.849972i	$-0.323381\pi$
$31$	5.86651	1.05366	0.526828	+	0.849972i	$0.323381\pi$
$32$	0	0
$33$	1.86651	0.324917
$34$	0	0
$35$	−1.52892	−0.258434
$36$	0	0
$37$	−3.19133	−0.524651	−0.262326	−	0.964979i	$-0.584489\pi$
$37$	−3.19133	−0.524651	−0.262326	+	0.964979i	$0.584489\pi$
$38$	0	0
$39$	−0.604574	−0.0968093
$40$	0	0
$41$	−6.65736	−1.03970	−0.519852	−	0.854256i	$-0.674013\pi$
$41$	−6.65736	−1.03970	−0.519852	+	0.854256i	$0.674013\pi$
$42$	0	0
$43$	5.66241	0.863509	0.431755	−	0.901991i	$-0.357895\pi$
$43$	5.66241	0.863509	0.431755	+	0.901991i	$0.357895\pi$
$44$	0	0
$45$	−1.52892	−0.227918
$46$	0	0
$47$	−10.7909	−1.57401	−0.787004	−	0.616948i	$-0.788369\pi$
$47$	−10.7909	−1.57401	−0.787004	+	0.616948i	$0.788369\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.92434	0.409490
$52$	0	0
$53$	−1.52892	−0.210013	−0.105007	−	0.994472i	$-0.533486\pi$
$53$	−1.52892	−0.210013	−0.105007	+	0.994472i	$0.533486\pi$
$54$	0	0
$55$	2.85374	0.384798
$56$	0	0
$57$	−1.86651	−0.247225
$58$	0	0
$59$	6.31977	0.822764	0.411382	−	0.911463i	$-0.365046\pi$
$59$	6.31977	0.822764	0.411382	+	0.911463i	$0.365046\pi$
$60$	0	0
$61$	−3.66241	−0.468924	−0.234462	−	0.972125i	$-0.575333\pi$
$61$	−3.66241	−0.468924	−0.234462	+	0.972125i	$0.575333\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.924344	−0.114651
$66$	0	0
$67$	−5.37761	−0.656979	−0.328490	−	0.944508i	$-0.606540\pi$
$67$	−5.37761	−0.656979	−0.328490	+	0.944508i	$0.606540\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−15.2441	−1.80914	−0.904572	−	0.426321i	$-0.859809\pi$
$71$	−15.2441	−1.80914	−0.904572	+	0.426321i	$0.859809\pi$
$72$	0	0
$73$	13.9822	1.63649	0.818245	−	0.574869i	$-0.194947\pi$
$73$	13.9822	1.63649	0.818245	+	0.574869i	$0.194947\pi$
$74$	0	0
$75$	2.66241	0.307429
$76$	0	0
$77$	−1.86651	−0.212708
$78$	0	0
$79$	−9.19133	−1.03411	−0.517053	−	0.855954i	$-0.672971\pi$
$79$	−9.19133	−1.03411	−0.517053	+	0.855954i	$0.672971\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.924344	0.101460	0.0507300	−	0.998712i	$-0.483845\pi$
$83$	0.924344	0.101460	0.0507300	+	0.998712i	$0.483845\pi$
$84$	0	0
$85$	4.47108	0.484957
$86$	0	0
$87$	−7.19133	−0.770991
$88$	0	0
$89$	7.39543	0.783914	0.391957	−	0.919984i	$-0.371798\pi$
$89$	7.39543	0.783914	0.391957	+	0.919984i	$0.371798\pi$
$90$	0	0
$91$	0.604574	0.0633766
$92$	0	0
$93$	−5.86651	−0.608329
$94$	0	0
$95$	−2.85374	−0.292787
$96$	0	0
$97$	5.98218	0.607398	0.303699	−	0.952768i	$-0.401778\pi$
$97$	5.98218	0.607398	0.303699	+	0.952768i	$0.401778\pi$
$98$	0	0
$99$	−1.86651	−0.187591
$100$	0	0
$101$	4.05279	0.403267	0.201634	−	0.979461i	$-0.435375\pi$
$101$	4.05279	0.403267	0.201634	+	0.979461i	$0.435375\pi$
$102$	0	0
$103$	6.11567	0.602595	0.301298	−	0.953530i	$-0.402580\pi$
$103$	6.11567	0.602595	0.301298	+	0.953530i	$0.402580\pi$
$104$	0	0
$105$	1.52892	0.149207
$106$	0	0
$107$	2.33759	0.225983	0.112992	−	0.993596i	$-0.463957\pi$
$107$	2.33759	0.225983	0.112992	+	0.993596i	$0.463957\pi$
$108$	0	0
$109$	−8.58675	−0.822462	−0.411231	−	0.911531i	$-0.634901\pi$
$109$	−8.58675	−0.822462	−0.411231	+	0.911531i	$0.634901\pi$
$110$	0	0
$111$	3.19133	0.302907
$112$	0	0
$113$	19.3776	1.82289	0.911446	−	0.411420i	$-0.134967\pi$
$113$	19.3776	1.82289	0.911446	+	0.411420i	$0.134967\pi$
$114$	0	0
$115$	−1.52892	−0.142572
$116$	0	0
$117$	0.604574	0.0558929
$118$	0	0
$119$	−2.92434	−0.268074
$120$	0	0
$121$	−7.51615	−0.683286
$122$	0	0
$123$	6.65736	0.600274
$124$	0	0
$125$	11.7152	1.04784
$126$	0	0
$127$	−4.98723	−0.442545	−0.221273	−	0.975212i	$-0.571021\pi$
$127$	−4.98723	−0.442545	−0.221273	+	0.975212i	$0.571021\pi$
$128$	0	0
$129$	−5.66241	−0.498547
$130$	0	0
$131$	1.86651	0.163078	0.0815388	−	0.996670i	$-0.474017\pi$
$131$	1.86651	0.163078	0.0815388	+	0.996670i	$0.474017\pi$
$132$	0	0
$133$	1.86651	0.161847
$134$	0	0
$135$	1.52892	0.131588
$136$	0	0
$137$	−15.1913	−1.29788	−0.648941	−	0.760838i	$-0.724788\pi$
$137$	−15.1913	−1.29788	−0.648941	+	0.760838i	$0.724788\pi$
$138$	0	0
$139$	12.4711	1.05778	0.528892	−	0.848689i	$-0.322608\pi$
$139$	12.4711	1.05778	0.528892	+	0.848689i	$0.322608\pi$
$140$	0	0
$141$	10.7909	0.908754
$142$	0	0
$143$	−1.12844	−0.0943651
$144$	0	0
$145$	−10.9950	−0.913081
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−0.523868	−0.0429170	−0.0214585	−	0.999770i	$-0.506831\pi$
$149$	−0.523868	−0.0429170	−0.0214585	+	0.999770i	$0.506831\pi$
$150$	0	0
$151$	11.3248	0.921601	0.460800	−	0.887504i	$-0.347562\pi$
$151$	11.3248	0.921601	0.460800	+	0.887504i	$0.347562\pi$
$152$	0	0
$153$	−2.92434	−0.236419
$154$	0	0
$155$	−8.96941	−0.720440
$156$	0	0
$157$	21.0222	1.67775	0.838877	−	0.544321i	$-0.183213\pi$
$157$	21.0222	1.67775	0.838877	+	0.544321i	$0.183213\pi$
$158$	0	0
$159$	1.52892	0.121251
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	13.3776	1.04781	0.523907	−	0.851775i	$-0.324474\pi$
$163$	13.3776	1.04781	0.523907	+	0.851775i	$0.324474\pi$
$164$	0	0
$165$	−2.85374	−0.222163
$166$	0	0
$167$	−20.5161	−1.58759	−0.793794	−	0.608187i	$-0.791897\pi$
$167$	−20.5161	−1.58759	−0.793794	+	0.608187i	$0.791897\pi$
$168$	0	0
$169$	−12.6345	−0.971884
$170$	0	0
$171$	1.86651	0.142735
$172$	0	0
$173$	−1.34264	−0.102079	−0.0510395	−	0.998697i	$-0.516253\pi$
$173$	−1.34264	−0.102079	−0.0510395	+	0.998697i	$0.516253\pi$
$174$	0	0
$175$	−2.66241	−0.201259
$176$	0	0
$177$	−6.31977	−0.475023
$178$	0	0
$179$	0.204098	0.0152550	0.00762751	−	0.999971i	$-0.497572\pi$
$179$	0.204098	0.0152550	0.00762751	+	0.999971i	$0.497572\pi$
$180$	0	0
$181$	17.4482	1.29692	0.648458	−	0.761251i	$-0.275414\pi$
$181$	17.4482	1.29692	0.648458	+	0.761251i	$0.275414\pi$
$182$	0	0
$183$	3.66241	0.270733
$184$	0	0
$185$	4.87928	0.358732
$186$	0	0
$187$	5.45831	0.399151
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−19.3248	−1.39829	−0.699147	−	0.714978i	$-0.746437\pi$
$191$	−19.3248	−1.39829	−0.699147	+	0.714978i	$0.746437\pi$
$192$	0	0
$193$	−1.32482	−0.0953626	−0.0476813	−	0.998863i	$-0.515183\pi$
$193$	−1.32482	−0.0953626	−0.0476813	+	0.998863i	$0.515183\pi$
$194$	0	0
$195$	0.924344	0.0661936
$196$	0	0
$197$	17.5111	1.24761	0.623807	−	0.781578i	$-0.285585\pi$
$197$	17.5111	1.24761	0.623807	+	0.781578i	$0.285585\pi$
$198$	0	0
$199$	22.1786	1.57220	0.786098	−	0.618102i	$-0.212098\pi$
$199$	22.1786	1.57220	0.786098	+	0.618102i	$0.212098\pi$
$200$	0	0
$201$	5.37761	0.379307
$202$	0	0
$203$	7.19133	0.504732
$204$	0	0
$205$	10.1786	0.710901
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.48385	−0.240983
$210$	0	0
$211$	−2.40048	−0.165256	−0.0826278	−	0.996580i	$-0.526331\pi$
$211$	−2.40048	−0.165256	−0.0826278	+	0.996580i	$0.526331\pi$
$212$	0	0
$213$	15.2441	1.04451
$214$	0	0
$215$	−8.65736	−0.590427
$216$	0	0
$217$	5.86651	0.398245
$218$	0	0
$219$	−13.9822	−0.944828
$220$	0	0
$221$	−1.76798	−0.118927
$222$	0	0
$223$	14.0528	0.941044	0.470522	−	0.882388i	$-0.344066\pi$
$223$	14.0528	0.941044	0.470522	+	0.882388i	$0.344066\pi$
$224$	0	0
$225$	−2.66241	−0.177494
$226$	0	0
$227$	−14.8537	−0.985877	−0.492939	−	0.870064i	$-0.664077\pi$
$227$	−14.8537	−0.985877	−0.492939	+	0.870064i	$0.664077\pi$
$228$	0	0
$229$	−23.1106	−1.52719	−0.763596	−	0.645694i	$-0.776568\pi$
$229$	−23.1106	−1.52719	−0.763596	+	0.645694i	$0.776568\pi$
$230$	0	0
$231$	1.86651	0.122807
$232$	0	0
$233$	−21.1207	−1.38366	−0.691832	−	0.722058i	$-0.743196\pi$
$233$	−21.1207	−1.38366	−0.691832	+	0.722058i	$0.743196\pi$
$234$	0	0
$235$	16.4983	1.07623
$236$	0	0
$237$	9.19133	0.597041
$238$	0	0
$239$	5.64459	0.365118	0.182559	−	0.983195i	$-0.441562\pi$
$239$	5.64459	0.365118	0.182559	+	0.983195i	$0.441562\pi$
$240$	0	0
$241$	−9.44821	−0.608613	−0.304306	−	0.952574i	$-0.598425\pi$
$241$	−9.44821	−0.608613	−0.304306	+	0.952574i	$0.598425\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.52892	−0.0976790
$246$	0	0
$247$	1.12844	0.0718011
$248$	0	0
$249$	−0.924344	−0.0585779
$250$	0	0
$251$	−0.639540	−0.0403674	−0.0201837	−	0.999796i	$-0.506425\pi$
$251$	−0.639540	−0.0403674	−0.0201837	+	0.999796i	$0.506425\pi$
$252$	0	0
$253$	−1.86651	−0.117346
$254$	0	0
$255$	−4.47108	−0.279990
$256$	0	0
$257$	−18.6395	−1.16270	−0.581351	−	0.813653i	$-0.697476\pi$
$257$	−18.6395	−1.16270	−0.581351	+	0.813653i	$0.697476\pi$
$258$	0	0
$259$	−3.19133	−0.198299
$260$	0	0
$261$	7.19133	0.445132
$262$	0	0
$263$	25.1557	1.55117	0.775583	−	0.631246i	$-0.217456\pi$
$263$	25.1557	1.55117	0.775583	+	0.631246i	$0.217456\pi$
$264$	0	0
$265$	2.33759	0.143597
$266$	0	0
$267$	−7.39543	−0.452593
$268$	0	0
$269$	−4.97713	−0.303461	−0.151730	−	0.988422i	$-0.548485\pi$
$269$	−4.97713	−0.303461	−0.151730	+	0.988422i	$0.548485\pi$
$270$	0	0
$271$	−4.92434	−0.299133	−0.149566	−	0.988752i	$-0.547788\pi$
$271$	−4.92434	−0.299133	−0.149566	+	0.988752i	$0.547788\pi$
$272$	0	0
$273$	−0.604574	−0.0365905
$274$	0	0
$275$	4.96941	0.299667
$276$	0	0
$277$	25.1029	1.50829	0.754144	−	0.656710i	$-0.228052\pi$
$277$	25.1029	1.50829	0.754144	+	0.656710i	$0.228052\pi$
$278$	0	0
$279$	5.86651	0.351219
$280$	0	0
$281$	26.3726	1.57325	0.786627	−	0.617428i	$-0.211825\pi$
$281$	26.3726	1.57325	0.786627	+	0.617428i	$0.211825\pi$
$282$	0	0
$283$	−12.4354	−0.739210	−0.369605	−	0.929189i	$-0.620507\pi$
$283$	−12.4354	−0.739210	−0.369605	+	0.929189i	$0.620507\pi$
$284$	0	0
$285$	2.85374	0.169041
$286$	0	0
$287$	−6.65736	−0.392972
$288$	0	0
$289$	−8.44821	−0.496954
$290$	0	0
$291$	−5.98218	−0.350682
$292$	0	0
$293$	24.3827	1.42445	0.712225	−	0.701951i	$-0.247688\pi$
$293$	24.3827	1.42445	0.712225	+	0.701951i	$0.247688\pi$
$294$	0	0
$295$	−9.66241	−0.562567
$296$	0	0
$297$	1.86651	0.108306
$298$	0	0
$299$	0.604574	0.0349634
$300$	0	0
$301$	5.66241	0.326376
$302$	0	0
$303$	−4.05279	−0.232826
$304$	0	0
$305$	5.59952	0.320628
$306$	0	0
$307$	6.13349	0.350057	0.175028	−	0.984563i	$-0.443998\pi$
$307$	6.13349	0.350057	0.175028	+	0.984563i	$0.443998\pi$
$308$	0	0
$309$	−6.11567	−0.347908
$310$	0	0
$311$	1.12844	0.0639881	0.0319940	−	0.999488i	$-0.489814\pi$
$311$	1.12844	0.0639881	0.0319940	+	0.999488i	$0.489814\pi$
$312$	0	0
$313$	25.4304	1.43741	0.718705	−	0.695315i	$-0.244735\pi$
$313$	25.4304	1.43741	0.718705	+	0.695315i	$0.244735\pi$
$314$	0	0
$315$	−1.52892	−0.0861448
$316$	0	0
$317$	−7.64459	−0.429363	−0.214681	−	0.976684i	$-0.568871\pi$
$317$	−7.64459	−0.429363	−0.214681	+	0.976684i	$0.568871\pi$
$318$	0	0
$319$	−13.4227	−0.751525
$320$	0	0
$321$	−2.33759	−0.130472
$322$	0	0
$323$	−5.45831	−0.303709
$324$	0	0
$325$	−1.60962	−0.0892859
$326$	0	0
$327$	8.58675	0.474849
$328$	0	0
$329$	−10.7909	−0.594919
$330$	0	0
$331$	10.1513	0.557967	0.278983	−	0.960296i	$-0.410003\pi$
$331$	10.1513	0.557967	0.278983	+	0.960296i	$0.410003\pi$
$332$	0	0
$333$	−3.19133	−0.174884
$334$	0	0
$335$	8.22192	0.449211
$336$	0	0
$337$	4.60457	0.250827	0.125414	−	0.992105i	$-0.459974\pi$
$337$	4.60457	0.250827	0.125414	+	0.992105i	$0.459974\pi$
$338$	0	0
$339$	−19.3776	−1.05245
$340$	0	0
$341$	−10.9499	−0.592970
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.52892	0.0823142
$346$	0	0
$347$	15.5740	0.836055	0.418028	−	0.908434i	$-0.362722\pi$
$347$	15.5740	0.836055	0.418028	+	0.908434i	$0.362722\pi$
$348$	0	0
$349$	25.2263	1.35033	0.675166	−	0.737666i	$-0.264072\pi$
$349$	25.2263	1.35033	0.675166	+	0.737666i	$0.264072\pi$
$350$	0	0
$351$	−0.604574	−0.0322698
$352$	0	0
$353$	28.5060	1.51722	0.758612	−	0.651543i	$-0.225878\pi$
$353$	28.5060	1.51722	0.758612	+	0.651543i	$0.225878\pi$
$354$	0	0
$355$	23.3070	1.23701
$356$	0	0
$357$	2.92434	0.154773
$358$	0	0
$359$	32.4711	1.71376	0.856879	−	0.515517i	$-0.172400\pi$
$359$	32.4711	1.71376	0.856879	+	0.515517i	$0.172400\pi$
$360$	0	0
$361$	−15.5161	−0.816639
$362$	0	0
$363$	7.51615	0.394495
$364$	0	0
$365$	−21.3776	−1.11896
$366$	0	0
$367$	6.87156	0.358692	0.179346	−	0.983786i	$-0.442602\pi$
$367$	6.87156	0.358692	0.179346	+	0.983786i	$0.442602\pi$
$368$	0	0
$369$	−6.65736	−0.346568
$370$	0	0
$371$	−1.52892	−0.0793775
$372$	0	0
$373$	−10.6574	−0.551817	−0.275909	−	0.961184i	$-0.588979\pi$
$373$	−10.6574	−0.551817	−0.275909	+	0.961184i	$0.588979\pi$
$374$	0	0
$375$	−11.7152	−0.604970
$376$	0	0
$377$	4.34769	0.223917
$378$	0	0
$379$	35.9287	1.84553	0.922767	−	0.385358i	$-0.125922\pi$
$379$	35.9287	1.84553	0.922767	+	0.385358i	$0.125922\pi$
$380$	0	0
$381$	4.98723	0.255504
$382$	0	0
$383$	0.657360	0.0335895	0.0167948	−	0.999859i	$-0.494654\pi$
$383$	0.657360	0.0335895	0.0167948	+	0.999859i	$0.494654\pi$
$384$	0	0
$385$	2.85374	0.145440
$386$	0	0
$387$	5.66241	0.287836
$388$	0	0
$389$	25.4126	1.28847	0.644234	−	0.764828i	$-0.277176\pi$
$389$	25.4126	1.28847	0.644234	+	0.764828i	$0.277176\pi$
$390$	0	0
$391$	−2.92434	−0.147890
$392$	0	0
$393$	−1.86651	−0.0941529
$394$	0	0
$395$	14.0528	0.707072
$396$	0	0
$397$	35.7075	1.79211	0.896053	−	0.443946i	$-0.146422\pi$
$397$	35.7075	1.79211	0.896053	+	0.443946i	$0.146422\pi$
$398$	0	0
$399$	−1.86651	−0.0934423
$400$	0	0
$401$	19.0299	0.950309	0.475154	−	0.879902i	$-0.342392\pi$
$401$	19.0299	0.950309	0.475154	+	0.879902i	$0.342392\pi$
$402$	0	0
$403$	3.54674	0.176676
$404$	0	0
$405$	−1.52892	−0.0759725
$406$	0	0
$407$	5.95664	0.295260
$408$	0	0
$409$	−4.77303	−0.236011	−0.118006	−	0.993013i	$-0.537650\pi$
$409$	−4.77303	−0.236011	−0.118006	+	0.993013i	$0.537650\pi$
$410$	0	0
$411$	15.1913	0.749333
$412$	0	0
$413$	6.31977	0.310976
$414$	0	0
$415$	−1.41325	−0.0693735
$416$	0	0
$417$	−12.4711	−0.610712
$418$	0	0
$419$	−17.6089	−0.860253	−0.430127	−	0.902769i	$-0.641531\pi$
$419$	−17.6089	−0.860253	−0.430127	+	0.902769i	$0.641531\pi$
$420$	0	0
$421$	40.4075	1.96934	0.984671	−	0.174422i	$-0.0558056\pi$
$421$	40.4075	1.96934	0.984671	+	0.174422i	$0.0558056\pi$
$422$	0	0
$423$	−10.7909	−0.524669
$424$	0	0
$425$	7.78580	0.377667
$426$	0	0
$427$	−3.66241	−0.177236
$428$	0	0
$429$	1.12844	0.0544817
$430$	0	0
$431$	28.0094	1.34917	0.674583	−	0.738199i	$-0.264323\pi$
$431$	28.0094	1.34917	0.674583	+	0.738199i	$0.264323\pi$
$432$	0	0
$433$	−30.3726	−1.45961	−0.729806	−	0.683654i	$-0.760390\pi$
$433$	−30.3726	−1.45961	−0.729806	+	0.683654i	$0.760390\pi$
$434$	0	0
$435$	10.9950	0.527168
$436$	0	0
$437$	1.86651	0.0892872
$438$	0	0
$439$	5.33254	0.254508	0.127254	−	0.991870i	$-0.459384\pi$
$439$	5.33254	0.254508	0.127254	+	0.991870i	$0.459384\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.1157	0.860702	0.430351	−	0.902662i	$-0.358390\pi$
$443$	18.1157	0.860702	0.430351	+	0.902662i	$0.358390\pi$
$444$	0	0
$445$	−11.3070	−0.536003
$446$	0	0
$447$	0.523868	0.0247781
$448$	0	0
$449$	−33.7603	−1.59325	−0.796623	−	0.604477i	$-0.793382\pi$
$449$	−33.7603	−1.59325	−0.796623	+	0.604477i	$0.793382\pi$
$450$	0	0
$451$	12.4260	0.585118
$452$	0	0
$453$	−11.3248	−0.532086
$454$	0	0
$455$	−0.924344	−0.0433339
$456$	0	0
$457$	−1.29757	−0.0606980	−0.0303490	−	0.999539i	$-0.509662\pi$
$457$	−1.29757	−0.0606980	−0.0303490	+	0.999539i	$0.509662\pi$
$458$	0	0
$459$	2.92434	0.136497
$460$	0	0
$461$	−27.3420	−1.27344	−0.636721	−	0.771094i	$-0.719710\pi$
$461$	−27.3420	−1.27344	−0.636721	+	0.771094i	$0.719710\pi$
$462$	0	0
$463$	24.6217	1.14427	0.572134	−	0.820160i	$-0.306116\pi$
$463$	24.6217	1.14427	0.572134	+	0.820160i	$0.306116\pi$
$464$	0	0
$465$	8.96941	0.415946
$466$	0	0
$467$	12.6574	0.585713	0.292856	−	0.956156i	$-0.405394\pi$
$467$	12.6574	0.585713	0.292856	+	0.956156i	$0.405394\pi$
$468$	0	0
$469$	−5.37761	−0.248315
$470$	0	0
$471$	−21.0222	−0.968652
$472$	0	0
$473$	−10.5689	−0.485960
$474$	0	0
$475$	−4.96941	−0.228012
$476$	0	0
$477$	−1.52892	−0.0700043
$478$	0	0
$479$	−7.18123	−0.328119	−0.164059	−	0.986450i	$-0.552459\pi$
$479$	−7.18123	−0.328119	−0.164059	+	0.986450i	$0.552459\pi$
$480$	0	0
$481$	−1.92939	−0.0879728
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−9.14626	−0.415310
$486$	0	0
$487$	20.8887	0.946558	0.473279	−	0.880913i	$-0.343070\pi$
$487$	20.8887	0.946558	0.473279	+	0.880913i	$0.343070\pi$
$488$	0	0
$489$	−13.3776	−0.604956
$490$	0	0
$491$	20.7202	0.935092	0.467546	−	0.883969i	$-0.345138\pi$
$491$	20.7202	0.935092	0.467546	+	0.883969i	$0.345138\pi$
$492$	0	0
$493$	−21.0299	−0.947140
$494$	0	0
$495$	2.85374	0.128266
$496$	0	0
$497$	−15.2441	−0.683792
$498$	0	0
$499$	7.22629	0.323493	0.161747	−	0.986832i	$-0.448287\pi$
$499$	7.22629	0.323493	0.161747	+	0.986832i	$0.448287\pi$
$500$	0	0
$501$	20.5161	0.916594
$502$	0	0
$503$	32.0272	1.42802	0.714012	−	0.700133i	$-0.246876\pi$
$503$	32.0272	1.42802	0.714012	+	0.700133i	$0.246876\pi$
$504$	0	0
$505$	−6.19638	−0.275735
$506$	0	0
$507$	12.6345	0.561117
$508$	0	0
$509$	38.3369	1.69925	0.849627	−	0.527384i	$-0.176827\pi$
$509$	38.3369	1.69925	0.849627	+	0.527384i	$0.176827\pi$
$510$	0	0
$511$	13.9822	0.618535
$512$	0	0
$513$	−1.86651	−0.0824083
$514$	0	0
$515$	−9.35036	−0.412026
$516$	0	0
$517$	20.1412	0.885810
$518$	0	0
$519$	1.34264	0.0589353
$520$	0	0
$521$	−31.2791	−1.37036	−0.685181	−	0.728373i	$-0.740277\pi$
$521$	−31.2791	−1.37036	−0.685181	+	0.728373i	$0.740277\pi$
$522$	0	0
$523$	29.0323	1.26949	0.634747	−	0.772720i	$-0.281104\pi$
$523$	29.0323	1.26949	0.634747	+	0.772720i	$0.281104\pi$
$524$	0	0
$525$	2.66241	0.116197
$526$	0	0
$527$	−17.1557	−0.747313
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.31977	0.274255
$532$	0	0
$533$	−4.02487	−0.174336
$534$	0	0
$535$	−3.57398	−0.154517
$536$	0	0
$537$	−0.204098	−0.00880749
$538$	0	0
$539$	−1.86651	−0.0803962
$540$	0	0
$541$	−21.8231	−0.938250	−0.469125	−	0.883132i	$-0.655431\pi$
$541$	−21.8231	−0.938250	−0.469125	+	0.883132i	$0.655431\pi$
$542$	0	0
$543$	−17.4482	−0.748774
$544$	0	0
$545$	13.1284	0.562361
$546$	0	0
$547$	−37.7502	−1.61408	−0.807040	−	0.590497i	$-0.798932\pi$
$547$	−37.7502	−1.61408	−0.807040	+	0.590497i	$0.798932\pi$
$548$	0	0
$549$	−3.66241	−0.156308
$550$	0	0
$551$	13.4227	0.571825
$552$	0	0
$553$	−9.19133	−0.390855
$554$	0	0
$555$	−4.87928	−0.207114
$556$	0	0
$557$	−13.8588	−0.587216	−0.293608	−	0.955926i	$-0.594856\pi$
$557$	−13.8588	−0.587216	−0.293608	+	0.955926i	$0.594856\pi$
$558$	0	0
$559$	3.42335	0.144792
$560$	0	0
$561$	−5.45831	−0.230450
$562$	0	0
$563$	36.5588	1.54077	0.770386	−	0.637578i	$-0.220064\pi$
$563$	36.5588	1.54077	0.770386	+	0.637578i	$0.220064\pi$
$564$	0	0
$565$	−29.6268	−1.24641
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	11.3147	0.474338	0.237169	−	0.971468i	$-0.423781\pi$
$569$	11.3147	0.474338	0.237169	+	0.971468i	$0.423781\pi$
$570$	0	0
$571$	−27.2714	−1.14127	−0.570635	−	0.821204i	$-0.693303\pi$
$571$	−27.2714	−1.14127	−0.570635	+	0.821204i	$0.693303\pi$
$572$	0	0
$573$	19.3248	0.807306
$574$	0	0
$575$	−2.66241	−0.111030
$576$	0	0
$577$	25.4660	1.06016	0.530082	−	0.847946i	$-0.322161\pi$
$577$	25.4660	1.06016	0.530082	+	0.847946i	$0.322161\pi$
$578$	0	0
$579$	1.32482	0.0550576
$580$	0	0
$581$	0.924344	0.0383483
$582$	0	0
$583$	2.85374	0.118190
$584$	0	0
$585$	−0.924344	−0.0382169
$586$	0	0
$587$	5.16408	0.213144	0.106572	−	0.994305i	$-0.466012\pi$
$587$	5.16408	0.213144	0.106572	+	0.994305i	$0.466012\pi$
$588$	0	0
$589$	10.9499	0.451182
$590$	0	0
$591$	−17.5111	−0.720310
$592$	0	0
$593$	−3.20915	−0.131784	−0.0658920	−	0.997827i	$-0.520989\pi$
$593$	−3.20915	−0.131784	−0.0658920	+	0.997827i	$0.520989\pi$
$594$	0	0
$595$	4.47108	0.183296
$596$	0	0
$597$	−22.1786	−0.907708
$598$	0	0
$599$	0.186278	0.00761112	0.00380556	−	0.999993i	$-0.498789\pi$
$599$	0.186278	0.00761112	0.00380556	+	0.999993i	$0.498789\pi$
$600$	0	0
$601$	14.1328	0.576490	0.288245	−	0.957557i	$-0.406928\pi$
$601$	14.1328	0.576490	0.288245	+	0.957557i	$0.406928\pi$
$602$	0	0
$603$	−5.37761	−0.218993
$604$	0	0
$605$	11.4916	0.467199
$606$	0	0
$607$	29.8938	1.21335	0.606675	−	0.794950i	$-0.292503\pi$
$607$	29.8938	1.21335	0.606675	+	0.794950i	$0.292503\pi$
$608$	0	0
$609$	−7.19133	−0.291407
$610$	0	0
$611$	−6.52387	−0.263927
$612$	0	0
$613$	−27.5282	−1.11186	−0.555928	−	0.831231i	$-0.687637\pi$
$613$	−27.5282	−1.11186	−0.555928	+	0.831231i	$0.687637\pi$
$614$	0	0
$615$	−10.1786	−0.410439
$616$	0	0
$617$	35.0572	1.41135	0.705674	−	0.708537i	$-0.250644\pi$
$617$	35.0572	1.41135	0.705674	+	0.708537i	$0.250644\pi$
$618$	0	0
$619$	−21.1284	−0.849224	−0.424612	−	0.905375i	$-0.639589\pi$
$619$	−21.1284	−0.849224	−0.424612	+	0.905375i	$0.639589\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	7.39543	0.296291
$624$	0	0
$625$	−4.59952	−0.183981
$626$	0	0
$627$	3.48385	0.139132
$628$	0	0
$629$	9.33254	0.372113
$630$	0	0
$631$	−37.0145	−1.47352	−0.736761	−	0.676153i	$-0.763646\pi$
$631$	−37.0145	−1.47352	−0.736761	+	0.676153i	$0.763646\pi$
$632$	0	0
$633$	2.40048	0.0954103
$634$	0	0
$635$	7.62506	0.302591
$636$	0	0
$637$	0.604574	0.0239541
$638$	0	0
$639$	−15.2441	−0.603048
$640$	0	0
$641$	−39.9993	−1.57988	−0.789939	−	0.613185i	$-0.789888\pi$
$641$	−39.9993	−1.57988	−0.789939	+	0.613185i	$0.789888\pi$
$642$	0	0
$643$	31.9193	1.25877	0.629387	−	0.777092i	$-0.283306\pi$
$643$	31.9193	1.25877	0.629387	+	0.777092i	$0.283306\pi$
$644$	0	0
$645$	8.65736	0.340883
$646$	0	0
$647$	19.1005	0.750919	0.375460	−	0.926839i	$-0.377485\pi$
$647$	19.1005	0.750919	0.375460	+	0.926839i	$0.377485\pi$
$648$	0	0
$649$	−11.7959	−0.463030
$650$	0	0
$651$	−5.86651	−0.229927
$652$	0	0
$653$	−26.3998	−1.03310	−0.516552	−	0.856256i	$-0.672785\pi$
$653$	−26.3998	−1.03310	−0.516552	+	0.856256i	$0.672785\pi$
$654$	0	0
$655$	−2.85374	−0.111505
$656$	0	0
$657$	13.9822	0.545497
$658$	0	0
$659$	−7.98218	−0.310942	−0.155471	−	0.987840i	$-0.549689\pi$
$659$	−7.98218	−0.310942	−0.155471	+	0.987840i	$0.549689\pi$
$660$	0	0
$661$	45.6796	1.77673	0.888364	−	0.459139i	$-0.151842\pi$
$661$	45.6796	1.77673	0.888364	+	0.459139i	$0.151842\pi$
$662$	0	0
$663$	1.76798	0.0686627
$664$	0	0
$665$	−2.85374	−0.110663
$666$	0	0
$667$	7.19133	0.278449
$668$	0	0
$669$	−14.0528	−0.543312
$670$	0	0
$671$	6.83592	0.263898
$672$	0	0
$673$	23.0501	0.888517	0.444258	−	0.895899i	$-0.353467\pi$
$673$	23.0501	0.888517	0.444258	+	0.895899i	$0.353467\pi$
$674$	0	0
$675$	2.66241	0.102476
$676$	0	0
$677$	24.0350	0.923739	0.461869	−	0.886948i	$-0.347179\pi$
$677$	24.0350	0.923739	0.461869	+	0.886948i	$0.347179\pi$
$678$	0	0
$679$	5.98218	0.229575
$680$	0	0
$681$	14.8537	0.569196
$682$	0	0
$683$	−23.5740	−0.902033	−0.451017	−	0.892516i	$-0.648939\pi$
$683$	−23.5740	−0.902033	−0.451017	+	0.892516i	$0.648939\pi$
$684$	0	0
$685$	23.2263	0.887431
$686$	0	0
$687$	23.1106	0.881725
$688$	0	0
$689$	−0.924344	−0.0352147
$690$	0	0
$691$	26.2663	0.999218	0.499609	−	0.866251i	$-0.333477\pi$
$691$	26.2663	0.999218	0.499609	+	0.866251i	$0.333477\pi$
$692$	0	0
$693$	−1.86651	−0.0709028
$694$	0	0
$695$	−19.0673	−0.723262
$696$	0	0
$697$	19.4684	0.737419
$698$	0	0
$699$	21.1207	0.798859
$700$	0	0
$701$	34.8282	1.31544	0.657721	−	0.753261i	$-0.271520\pi$
$701$	34.8282	1.31544	0.657721	+	0.753261i	$0.271520\pi$
$702$	0	0
$703$	−5.95664	−0.224659
$704$	0	0
$705$	−16.4983	−0.621363
$706$	0	0
$707$	4.05279	0.152421
$708$	0	0
$709$	10.4176	0.391242	0.195621	−	0.980680i	$-0.437328\pi$
$709$	10.4176	0.391242	0.195621	+	0.980680i	$0.437328\pi$
$710$	0	0
$711$	−9.19133	−0.344702
$712$	0	0
$713$	5.86651	0.219702
$714$	0	0
$715$	1.72530	0.0645224
$716$	0	0
$717$	−5.64459	−0.210801
$718$	0	0
$719$	−10.2593	−0.382606	−0.191303	−	0.981531i	$-0.561271\pi$
$719$	−10.2593	−0.382606	−0.191303	+	0.981531i	$0.561271\pi$
$720$	0	0
$721$	6.11567	0.227760
$722$	0	0
$723$	9.44821	0.351383
$724$	0	0
$725$	−19.1463	−0.711074
$726$	0	0
$727$	31.4328	1.16578	0.582888	−	0.812552i	$-0.301922\pi$
$727$	31.4328	1.16578	0.582888	+	0.812552i	$0.301922\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.5588	−0.612451
$732$	0	0
$733$	1.32482	0.0489333	0.0244667	−	0.999701i	$-0.492211\pi$
$733$	1.32482	0.0489333	0.0244667	+	0.999701i	$0.492211\pi$
$734$	0	0
$735$	1.52892	0.0563950
$736$	0	0
$737$	10.0373	0.369730
$738$	0	0
$739$	−1.33254	−0.0490183	−0.0245091	−	0.999700i	$-0.507802\pi$
$739$	−1.33254	−0.0490183	−0.0245091	+	0.999700i	$0.507802\pi$
$740$	0	0
$741$	−1.12844	−0.0414544
$742$	0	0
$743$	−8.06289	−0.295799	−0.147899	−	0.989002i	$-0.547251\pi$
$743$	−8.06289	−0.295799	−0.147899	+	0.989002i	$0.547251\pi$
$744$	0	0
$745$	0.800952	0.0293446
$746$	0	0
$747$	0.924344	0.0338200
$748$	0	0
$749$	2.33759	0.0854137
$750$	0	0
$751$	−8.39980	−0.306513	−0.153257	−	0.988186i	$-0.548976\pi$
$751$	−8.39980	−0.306513	−0.153257	+	0.988186i	$0.548976\pi$
$752$	0	0
$753$	0.639540	0.0233061
$754$	0	0
$755$	−17.3147	−0.630147
$756$	0	0
$757$	2.95998	0.107582	0.0537912	−	0.998552i	$-0.482869\pi$
$757$	2.95998	0.107582	0.0537912	+	0.998552i	$0.482869\pi$
$758$	0	0
$759$	1.86651	0.0677500
$760$	0	0
$761$	−47.7774	−1.73193	−0.865965	−	0.500105i	$-0.833295\pi$
$761$	−47.7774	−1.73193	−0.865965	+	0.500105i	$0.833295\pi$
$762$	0	0
$763$	−8.58675	−0.310861
$764$	0	0
$765$	4.47108	0.161652
$766$	0	0
$767$	3.82077	0.137960
$768$	0	0
$769$	44.7196	1.61263	0.806315	−	0.591487i	$-0.201459\pi$
$769$	44.7196	1.61263	0.806315	+	0.591487i	$0.201459\pi$
$770$	0	0
$771$	18.6395	0.671287
$772$	0	0
$773$	22.2848	0.801529	0.400764	−	0.916181i	$-0.368745\pi$
$773$	22.2848	0.801529	0.400764	+	0.916181i	$0.368745\pi$
$774$	0	0
$775$	−15.6190	−0.561053
$776$	0	0
$777$	3.19133	0.114488
$778$	0	0
$779$	−12.4260	−0.445208
$780$	0	0
$781$	28.4533	1.01814
$782$	0	0
$783$	−7.19133	−0.256997
$784$	0	0
$785$	−32.1412	−1.14717
$786$	0	0
$787$	13.9472	0.497164	0.248582	−	0.968611i	$-0.420035\pi$
$787$	13.9472	0.497164	0.248582	+	0.968611i	$0.420035\pi$
$788$	0	0
$789$	−25.1557	−0.895566
$790$	0	0
$791$	19.3776	0.688988
$792$	0	0
$793$	−2.21420	−0.0786285
$794$	0	0
$795$	−2.33759	−0.0829058
$796$	0	0
$797$	−26.3369	−0.932901	−0.466451	−	0.884547i	$-0.654468\pi$
$797$	−26.3369	−0.932901	−0.466451	+	0.884547i	$0.654468\pi$
$798$	0	0
$799$	31.5562	1.11638
$800$	0	0
$801$	7.39543	0.261305
$802$	0	0
$803$	−26.0979	−0.920973
$804$	0	0
$805$	−1.52892	−0.0538873
$806$	0	0
$807$	4.97713	0.175203
$808$	0	0
$809$	−36.7101	−1.29066	−0.645330	−	0.763904i	$-0.723280\pi$
$809$	−36.7101	−1.29066	−0.645330	+	0.763904i	$0.723280\pi$
$810$	0	0
$811$	−10.7730	−0.378292	−0.189146	−	0.981949i	$-0.560572\pi$
$811$	−10.7730	−0.378292	−0.189146	+	0.981949i	$0.560572\pi$
$812$	0	0
$813$	4.92434	0.172704
$814$	0	0
$815$	−20.4533	−0.716447
$816$	0	0
$817$	10.5689	0.369760
$818$	0	0
$819$	0.604574	0.0211255
$820$	0	0
$821$	−10.1258	−0.353392	−0.176696	−	0.984265i	$-0.556541\pi$
$821$	−10.1258	−0.353392	−0.176696	+	0.984265i	$0.556541\pi$
$822$	0	0
$823$	−29.6446	−1.03335	−0.516673	−	0.856183i	$-0.672830\pi$
$823$	−29.6446	−1.03335	−0.516673	+	0.856183i	$0.672830\pi$
$824$	0	0
$825$	−4.96941	−0.173013
$826$	0	0
$827$	−42.4253	−1.47527	−0.737637	−	0.675198i	$-0.764058\pi$
$827$	−42.4253	−1.47527	−0.737637	+	0.675198i	$0.764058\pi$
$828$	0	0
$829$	−3.97208	−0.137956	−0.0689780	−	0.997618i	$-0.521974\pi$
$829$	−3.97208	−0.137956	−0.0689780	+	0.997618i	$0.521974\pi$
$830$	0	0
$831$	−25.1029	−0.870810
$832$	0	0
$833$	−2.92434	−0.101323
$834$	0	0
$835$	31.3675	1.08552
$836$	0	0
$837$	−5.86651	−0.202776
$838$	0	0
$839$	21.8938	0.755856	0.377928	−	0.925835i	$-0.376637\pi$
$839$	21.8938	0.755856	0.377928	+	0.925835i	$0.376637\pi$
$840$	0	0
$841$	22.7152	0.783283
$842$	0	0
$843$	−26.3726	−0.908319
$844$	0	0
$845$	19.3171	0.664528
$846$	0	0
$847$	−7.51615	−0.258258
$848$	0	0
$849$	12.4354	0.426783
$850$	0	0
$851$	−3.19133	−0.109397
$852$	0	0
$853$	−1.02220	−0.0349993	−0.0174997	−	0.999847i	$-0.505571\pi$
$853$	−1.02220	−0.0349993	−0.0174997	+	0.999847i	$0.505571\pi$
$854$	0	0
$855$	−2.85374	−0.0975958
$856$	0	0
$857$	39.6718	1.35516	0.677582	−	0.735447i	$-0.263028\pi$
$857$	39.6718	1.35516	0.677582	+	0.735447i	$0.263028\pi$
$858$	0	0
$859$	−47.1278	−1.60798	−0.803989	−	0.594644i	$-0.797293\pi$
$859$	−47.1278	−1.60798	−0.803989	+	0.594644i	$0.797293\pi$
$860$	0	0
$861$	6.65736	0.226882
$862$	0	0
$863$	16.1258	0.548928	0.274464	−	0.961597i	$-0.411500\pi$
$863$	16.1258	0.548928	0.274464	+	0.961597i	$0.411500\pi$
$864$	0	0
$865$	2.05279	0.0697968
$866$	0	0
$867$	8.44821	0.286916
$868$	0	0
$869$	17.1557	0.581967
$870$	0	0
$871$	−3.25116	−0.110161
$872$	0	0
$873$	5.98218	0.202466
$874$	0	0
$875$	11.7152	0.396046
$876$	0	0
$877$	34.2670	1.15711	0.578557	−	0.815642i	$-0.303616\pi$
$877$	34.2670	1.15711	0.578557	+	0.815642i	$0.303616\pi$
$878$	0	0
$879$	−24.3827	−0.822407
$880$	0	0
$881$	−19.2091	−0.647173	−0.323586	−	0.946199i	$-0.604889\pi$
$881$	−19.2091	−0.647173	−0.323586	+	0.946199i	$0.604889\pi$
$882$	0	0
$883$	−46.4432	−1.56294	−0.781468	−	0.623945i	$-0.785529\pi$
$883$	−46.4432	−1.56294	−0.781468	+	0.623945i	$0.785529\pi$
$884$	0	0
$885$	9.66241	0.324798
$886$	0	0
$887$	−42.0350	−1.41140	−0.705698	−	0.708513i	$-0.749366\pi$
$887$	−42.0350	−1.41140	−0.705698	+	0.708513i	$0.749366\pi$
$888$	0	0
$889$	−4.98723	−0.167266
$890$	0	0
$891$	−1.86651	−0.0625304
$892$	0	0
$893$	−20.1412	−0.674000
$894$	0	0
$895$	−0.312049	−0.0104307
$896$	0	0
$897$	−0.604574	−0.0201861
$898$	0	0
$899$	42.1880	1.40705
$900$	0	0
$901$	4.47108	0.148953
$902$	0	0
$903$	−5.66241	−0.188433
$904$	0	0
$905$	−26.6769	−0.886770
$906$	0	0
$907$	−35.6547	−1.18389	−0.591947	−	0.805977i	$-0.701641\pi$
$907$	−35.6547	−1.18389	−0.591947	+	0.805977i	$0.701641\pi$
$908$	0	0
$909$	4.05279	0.134422
$910$	0	0
$911$	3.32482	0.110156	0.0550781	−	0.998482i	$-0.482459\pi$
$911$	3.32482	0.110156	0.0550781	+	0.998482i	$0.482459\pi$
$912$	0	0
$913$	−1.72530	−0.0570989
$914$	0	0
$915$	−5.59952	−0.185115
$916$	0	0
$917$	1.86651	0.0616375
$918$	0	0
$919$	29.8665	0.985205	0.492603	−	0.870254i	$-0.336046\pi$
$919$	29.8665	0.985205	0.492603	+	0.870254i	$0.336046\pi$
$920$	0	0
$921$	−6.13349	−0.202105
$922$	0	0
$923$	−9.21619	−0.303355
$924$	0	0
$925$	8.49662	0.279367
$926$	0	0
$927$	6.11567	0.200865
$928$	0	0
$929$	−5.15636	−0.169175	−0.0845874	−	0.996416i	$-0.526957\pi$
$929$	−5.15636	−0.169175	−0.0845874	+	0.996416i	$0.526957\pi$
$930$	0	0
$931$	1.86651	0.0611723
$932$	0	0
$933$	−1.12844	−0.0369435
$934$	0	0
$935$	−8.34531	−0.272921
$936$	0	0
$937$	−47.1200	−1.53934	−0.769672	−	0.638439i	$-0.779580\pi$
$937$	−47.1200	−1.53934	−0.769672	+	0.638439i	$0.779580\pi$
$938$	0	0
$939$	−25.4304	−0.829889
$940$	0	0
$941$	−10.7808	−0.351442	−0.175721	−	0.984440i	$-0.556226\pi$
$941$	−10.7808	−0.351442	−0.175721	+	0.984440i	$0.556226\pi$
$942$	0	0
$943$	−6.65736	−0.216793
$944$	0	0
$945$	1.52892	0.0497357
$946$	0	0
$947$	−42.0800	−1.36742	−0.683709	−	0.729755i	$-0.739634\pi$
$947$	−42.0800	−1.36742	−0.683709	+	0.729755i	$0.739634\pi$
$948$	0	0
$949$	8.45326	0.274404
$950$	0	0
$951$	7.64459	0.247893
$952$	0	0
$953$	−28.8716	−0.935241	−0.467621	−	0.883929i	$-0.654889\pi$
$953$	−28.8716	−0.935241	−0.467621	+	0.883929i	$0.654889\pi$
$954$	0	0
$955$	29.5461	0.956088
$956$	0	0
$957$	13.4227	0.433893
$958$	0	0
$959$	−15.1913	−0.490554
$960$	0	0
$961$	3.41592	0.110191
$962$	0	0
$963$	2.33759	0.0753278
$964$	0	0
$965$	2.02554	0.0652045
$966$	0	0
$967$	−30.8964	−0.993562	−0.496781	−	0.867876i	$-0.665485\pi$
$967$	−30.8964	−0.993562	−0.496781	+	0.867876i	$0.665485\pi$
$968$	0	0
$969$	5.45831	0.175346
$970$	0	0
$971$	24.4176	0.783599	0.391799	−	0.920051i	$-0.371853\pi$
$971$	24.4176	0.783599	0.391799	+	0.920051i	$0.371853\pi$
$972$	0	0
$973$	12.4711	0.399805
$974$	0	0
$975$	1.60962	0.0515492
$976$	0	0
$977$	6.86146	0.219518	0.109759	−	0.993958i	$-0.464992\pi$
$977$	6.86146	0.219518	0.109759	+	0.993958i	$0.464992\pi$
$978$	0	0
$979$	−13.8036	−0.441166
$980$	0	0
$981$	−8.58675	−0.274154
$982$	0	0
$983$	−44.9243	−1.43286	−0.716432	−	0.697657i	$-0.754226\pi$
$983$	−44.9243	−1.43286	−0.716432	+	0.697657i	$0.754226\pi$
$984$	0	0
$985$	−26.7730	−0.853060
$986$	0	0
$987$	10.7909	0.343477
$988$	0	0
$989$	5.66241	0.180054
$990$	0	0
$991$	−8.31205	−0.264041	−0.132020	−	0.991247i	$-0.542146\pi$
$991$	−8.31205	−0.264041	−0.132020	+	0.991247i	$0.542146\pi$
$992$	0	0
$993$	−10.1513	−0.322142
$994$	0	0
$995$	−33.9092	−1.07499
$996$	0	0
$997$	−22.3191	−0.706853	−0.353426	−	0.935462i	$-0.614984\pi$
$997$	−22.3191	−0.706853	−0.353426	+	0.935462i	$0.614984\pi$
$998$	0	0
$999$	3.19133	0.100969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bt.1.1		3
4.3	odd	2		483.2.a.h.1.3	✓	3
12.11	even	2		1449.2.a.l.1.1		3
28.27	even	2		3381.2.a.v.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.h.1.3	✓	3	4.3	odd	2
1449.2.a.l.1.1		3	12.11	even	2
3381.2.a.v.1.3		3	28.27	even	2
7728.2.a.bt.1.1		3	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.52892 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.52892 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.86651 q^{11} +0.604574 q^{13} +1.52892 q^{15} -2.92434 q^{17} +1.86651 q^{19} -1.00000 q^{21} +1.00000 q^{23} -2.66241 q^{25} -1.00000 q^{27} +7.19133 q^{29} +5.86651 q^{31} +1.86651 q^{33} -1.52892 q^{35} -3.19133 q^{37} -0.604574 q^{39} -6.65736 q^{41} +5.66241 q^{43} -1.52892 q^{45} -10.7909 q^{47} +1.00000 q^{49} +2.92434 q^{51} -1.52892 q^{53} +2.85374 q^{55} -1.86651 q^{57} +6.31977 q^{59} -3.66241 q^{61} +1.00000 q^{63} -0.924344 q^{65} -5.37761 q^{67} -1.00000 q^{69} -15.2441 q^{71} +13.9822 q^{73} +2.66241 q^{75} -1.86651 q^{77} -9.19133 q^{79} +1.00000 q^{81} +0.924344 q^{83} +4.47108 q^{85} -7.19133 q^{87} +7.39543 q^{89} +0.604574 q^{91} -5.86651 q^{93} -2.85374 q^{95} +5.98218 q^{97} -1.86651 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 9 q^{13} - 3 q^{15} + 6 q^{17} + 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} + 6 q^{29} + 18 q^{31} + 6 q^{33} + 3 q^{35} + 6 q^{37} - 9 q^{39} - 6 q^{41} + 9 q^{43} + 3 q^{45} - 18 q^{47} + 3 q^{49} - 6 q^{51} + 3 q^{53} - 15 q^{55} - 6 q^{57} - 3 q^{59} - 3 q^{61} + 3 q^{63} + 12 q^{65} + 21 q^{67} - 3 q^{69} - 9 q^{71} + 12 q^{73} - 6 q^{77} - 12 q^{79} + 3 q^{81} - 12 q^{83} + 21 q^{85} - 6 q^{87} + 15 q^{89} + 9 q^{91} - 18 q^{93} + 15 q^{95} - 12 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^7 + 3 * q^9 - 6 * q^11 + 9 * q^13 - 3 * q^15 + 6 * q^17 + 6 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 + 6 * q^29 + 18 * q^31 + 6 * q^33 + 3 * q^35 + 6 * q^37 - 9 * q^39 - 6 * q^41 + 9 * q^43 + 3 * q^45 - 18 * q^47 + 3 * q^49 - 6 * q^51 + 3 * q^53 - 15 * q^55 - 6 * q^57 - 3 * q^59 - 3 * q^61 + 3 * q^63 + 12 * q^65 + 21 * q^67 - 3 * q^69 - 9 * q^71 + 12 * q^73 - 6 * q^77 - 12 * q^79 + 3 * q^81 - 12 * q^83 + 21 * q^85 - 6 * q^87 + 15 * q^89 + 9 * q^91 - 18 * q^93 + 15 * q^95 - 12 * q^97 - 6 * q^99

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