LMFDB - Embedded newform 4864.2.a.bo.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 48x^{4} - 68x^{3} - 62x^{2} + 32x + 16 $ x^8 - 2x^7 - 13x^6 + 24x^5 + 48x^4 - 68x^3 - 62x^2 + 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.932340$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.579017 q^{3} +2.10882 q^{5} +2.73436 q^{7} -2.66474 q^{9} +O(q^{10})$	$q+0.579017 q^{3} +2.10882 q^{5} +2.73436 q^{7} -2.66474 q^{9} -4.66474 q^{11} +4.47791 q^{13} +1.22104 q^{15} -6.85237 q^{17} -1.00000 q^{19} +1.58324 q^{21} +1.20416 q^{23} -0.552894 q^{25} -3.27998 q^{27} -9.57484 q^{29} -5.35842 q^{31} -2.70096 q^{33} +5.76625 q^{35} -1.09693 q^{37} +2.59279 q^{39} -7.33797 q^{41} -7.64408 q^{43} -5.61945 q^{45} +7.56486 q^{47} +0.476702 q^{49} -3.96764 q^{51} -3.11949 q^{53} -9.83708 q^{55} -0.579017 q^{57} +10.2442 q^{59} +0.722061 q^{61} -7.28634 q^{63} +9.44309 q^{65} -6.13698 q^{67} +0.697232 q^{69} -4.62247 q^{71} +6.19270 q^{73} -0.320135 q^{75} -12.7551 q^{77} -3.26723 q^{79} +6.09505 q^{81} -8.97934 q^{83} -14.4504 q^{85} -5.54400 q^{87} -0.620707 q^{89} +12.2442 q^{91} -3.10262 q^{93} -2.10882 q^{95} -1.67284 q^{97} +12.4303 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 + 4 * q^7 + 12 * q^9	$ 8 q - 8 q^{5} + 4 q^{7} + 12 q^{9} - 4 q^{11} - 8 q^{13} - 4 q^{17} - 8 q^{19} - 16 q^{21} + 12 q^{25} - 28 q^{29} + 8 q^{31} + 12 q^{35} - 4 q^{37} - 4 q^{39} - 8 q^{41} + 4 q^{43} - 24 q^{45} + 12 q^{47} + 12 q^{49} - 12 q^{51} - 32 q^{53} - 8 q^{55} - 12 q^{59} - 8 q^{61} - 16 q^{63} + 8 q^{65} + 4 q^{67} - 28 q^{69} - 24 q^{71} - 24 q^{77} - 24 q^{79} - 8 q^{81} - 40 q^{83} - 24 q^{85} + 24 q^{87} + 8 q^{89} + 4 q^{91} - 32 q^{93} + 8 q^{95} + 16 q^{97} + 76 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 + 4 * q^7 + 12 * q^9 - 4 * q^11 - 8 * q^13 - 4 * q^17 - 8 * q^19 - 16 * q^21 + 12 * q^25 - 28 * q^29 + 8 * q^31 + 12 * q^35 - 4 * q^37 - 4 * q^39 - 8 * q^41 + 4 * q^43 - 24 * q^45 + 12 * q^47 + 12 * q^49 - 12 * q^51 - 32 * q^53 - 8 * q^55 - 12 * q^59 - 8 * q^61 - 16 * q^63 + 8 * q^65 + 4 * q^67 - 28 * q^69 - 24 * q^71 - 24 * q^77 - 24 * q^79 - 8 * q^81 - 40 * q^83 - 24 * q^85 + 24 * q^87 + 8 * q^89 + 4 * q^91 - 32 * q^93 + 8 * q^95 + 16 * q^97 + 76 * 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
$3$	0.579017	0.334296	0.167148	−	0.985932i	$-0.446544\pi$
$3$	0.579017	0.334296	0.167148	+	0.985932i	$0.446544\pi$
$4$	0	0
$5$	2.10882	0.943091	0.471546	−	0.881842i	$-0.343696\pi$
$5$	2.10882	0.943091	0.471546	+	0.881842i	$0.343696\pi$
$6$	0	0
$7$	2.73436	1.03349	0.516745	−	0.856140i	$-0.327144\pi$
$7$	2.73436	1.03349	0.516745	+	0.856140i	$0.327144\pi$
$8$	0	0
$9$	−2.66474	−0.888246
$10$	0	0
$11$	−4.66474	−1.40647	−0.703236	−	0.710957i	$-0.748262\pi$
$11$	−4.66474	−1.40647	−0.703236	+	0.710957i	$0.748262\pi$
$12$	0	0
$13$	4.47791	1.24195	0.620974	−	0.783831i	$-0.286737\pi$
$13$	4.47791	1.24195	0.620974	+	0.783831i	$0.286737\pi$
$14$	0	0
$15$	1.22104	0.315271
$16$	0	0
$17$	−6.85237	−1.66194	−0.830972	−	0.556314i	$-0.812215\pi$
$17$	−6.85237	−1.66194	−0.830972	+	0.556314i	$0.812215\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.58324	0.345491
$22$	0	0
$23$	1.20416	0.251085	0.125543	−	0.992088i	$-0.459933\pi$
$23$	1.20416	0.251085	0.125543	+	0.992088i	$0.459933\pi$
$24$	0	0
$25$	−0.552894	−0.110579
$26$	0	0
$27$	−3.27998	−0.631233
$28$	0	0
$29$	−9.57484	−1.77800	−0.889002	−	0.457904i	$-0.848600\pi$
$29$	−9.57484	−1.77800	−0.889002	+	0.457904i	$0.848600\pi$
$30$	0	0
$31$	−5.35842	−0.962400	−0.481200	−	0.876611i	$-0.659799\pi$
$31$	−5.35842	−0.962400	−0.481200	+	0.876611i	$0.659799\pi$
$32$	0	0
$33$	−2.70096	−0.470178
$34$	0	0
$35$	5.76625	0.974675
$36$	0	0
$37$	−1.09693	−0.180335	−0.0901674	−	0.995927i	$-0.528740\pi$
$37$	−1.09693	−0.180335	−0.0901674	+	0.995927i	$0.528740\pi$
$38$	0	0
$39$	2.59279	0.415178
$40$	0	0
$41$	−7.33797	−1.14600	−0.573000	−	0.819556i	$-0.694220\pi$
$41$	−7.33797	−1.14600	−0.573000	+	0.819556i	$0.694220\pi$
$42$	0	0
$43$	−7.64408	−1.16571	−0.582856	−	0.812576i	$-0.698065\pi$
$43$	−7.64408	−1.16571	−0.582856	+	0.812576i	$0.698065\pi$
$44$	0	0
$45$	−5.61945	−0.837697
$46$	0	0
$47$	7.56486	1.10345	0.551724	−	0.834027i	$-0.313970\pi$
$47$	7.56486	1.10345	0.551724	+	0.834027i	$0.313970\pi$
$48$	0	0
$49$	0.476702	0.0681003
$50$	0	0
$51$	−3.96764	−0.555581
$52$	0	0
$53$	−3.11949	−0.428495	−0.214248	−	0.976779i	$-0.568730\pi$
$53$	−3.11949	−0.428495	−0.214248	+	0.976779i	$0.568730\pi$
$54$	0	0
$55$	−9.83708	−1.32643
$56$	0	0
$57$	−0.579017	−0.0766927
$58$	0	0
$59$	10.2442	1.33368	0.666840	−	0.745201i	$-0.267646\pi$
$59$	10.2442	1.33368	0.666840	+	0.745201i	$0.267646\pi$
$60$	0	0
$61$	0.722061	0.0924504	0.0462252	−	0.998931i	$-0.485281\pi$
$61$	0.722061	0.0924504	0.0462252	+	0.998931i	$0.485281\pi$
$62$	0	0
$63$	−7.28634	−0.917993
$64$	0	0
$65$	9.44309	1.17127
$66$	0	0
$67$	−6.13698	−0.749752	−0.374876	−	0.927075i	$-0.622315\pi$
$67$	−6.13698	−0.749752	−0.374876	+	0.927075i	$0.622315\pi$
$68$	0	0
$69$	0.697232	0.0839368
$70$	0	0
$71$	−4.62247	−0.548587	−0.274293	−	0.961646i	$-0.588444\pi$
$71$	−4.62247	−0.548587	−0.274293	+	0.961646i	$0.588444\pi$
$72$	0	0
$73$	6.19270	0.724801	0.362400	−	0.932022i	$-0.381957\pi$
$73$	6.19270	0.724801	0.362400	+	0.932022i	$0.381957\pi$
$74$	0	0
$75$	−0.320135	−0.0369660
$76$	0	0
$77$	−12.7551	−1.45357
$78$	0	0
$79$	−3.26723	−0.367592	−0.183796	−	0.982964i	$-0.558839\pi$
$79$	−3.26723	−0.367592	−0.183796	+	0.982964i	$0.558839\pi$
$80$	0	0
$81$	6.09505	0.677228
$82$	0	0
$83$	−8.97934	−0.985611	−0.492805	−	0.870140i	$-0.664029\pi$
$83$	−8.97934	−0.985611	−0.492805	+	0.870140i	$0.664029\pi$
$84$	0	0
$85$	−14.4504	−1.56736
$86$	0	0
$87$	−5.54400	−0.594379
$88$	0	0
$89$	−0.620707	−0.0657948	−0.0328974	−	0.999459i	$-0.510473\pi$
$89$	−0.620707	−0.0657948	−0.0328974	+	0.999459i	$0.510473\pi$
$90$	0	0
$91$	12.2442	1.28354
$92$	0	0
$93$	−3.10262	−0.321726
$94$	0	0
$95$	−2.10882	−0.216360
$96$	0	0
$97$	−1.67284	−0.169851	−0.0849257	−	0.996387i	$-0.527065\pi$
$97$	−1.67284	−0.169851	−0.0849257	+	0.996387i	$0.527065\pi$
$98$	0	0
$99$	12.4303	1.24929
$100$	0	0
$101$	−7.30571	−0.726946	−0.363473	−	0.931605i	$-0.618409\pi$
$101$	−7.30571	−0.726946	−0.363473	+	0.931605i	$0.618409\pi$
$102$	0	0
$103$	8.77332	0.864461	0.432231	−	0.901763i	$-0.357727\pi$
$103$	8.77332	0.864461	0.432231	+	0.901763i	$0.357727\pi$
$104$	0	0
$105$	3.33876	0.325830
$106$	0	0
$107$	9.91355	0.958379	0.479190	−	0.877711i	$-0.340931\pi$
$107$	9.91355	0.958379	0.479190	+	0.877711i	$0.340931\pi$
$108$	0	0
$109$	2.84589	0.272587	0.136293	−	0.990669i	$-0.456481\pi$
$109$	2.84589	0.272587	0.136293	+	0.990669i	$0.456481\pi$
$110$	0	0
$111$	−0.635143	−0.0602851
$112$	0	0
$113$	−1.58487	−0.149092	−0.0745462	−	0.997218i	$-0.523751\pi$
$113$	−1.58487	−0.149092	−0.0745462	+	0.997218i	$0.523751\pi$
$114$	0	0
$115$	2.53936	0.236797
$116$	0	0
$117$	−11.9325	−1.10316
$118$	0	0
$119$	−18.7368	−1.71760
$120$	0	0
$121$	10.7598	0.978163
$122$	0	0
$123$	−4.24881	−0.383103
$124$	0	0
$125$	−11.7100	−1.04738
$126$	0	0
$127$	14.2933	1.26833	0.634163	−	0.773199i	$-0.281345\pi$
$127$	14.2933	1.26833	0.634163	+	0.773199i	$0.281345\pi$
$128$	0	0
$129$	−4.42605	−0.389692
$130$	0	0
$131$	−0.609006	−0.0532091	−0.0266046	−	0.999646i	$-0.508469\pi$
$131$	−0.609006	−0.0532091	−0.0266046	+	0.999646i	$0.508469\pi$
$132$	0	0
$133$	−2.73436	−0.237099
$134$	0	0
$135$	−6.91688	−0.595310
$136$	0	0
$137$	−18.1589	−1.55142	−0.775710	−	0.631089i	$-0.782608\pi$
$137$	−18.1589	−1.55142	−0.775710	+	0.631089i	$0.782608\pi$
$138$	0	0
$139$	−1.93421	−0.164058	−0.0820289	−	0.996630i	$-0.526140\pi$
$139$	−1.93421	−0.164058	−0.0820289	+	0.996630i	$0.526140\pi$
$140$	0	0
$141$	4.38018	0.368878
$142$	0	0
$143$	−20.8883	−1.74677
$144$	0	0
$145$	−20.1916	−1.67682
$146$	0	0
$147$	0.276019	0.0227656
$148$	0	0
$149$	−15.5775	−1.27616	−0.638080	−	0.769970i	$-0.720271\pi$
$149$	−15.5775	−1.27616	−0.638080	+	0.769970i	$0.720271\pi$
$150$	0	0
$151$	19.9746	1.62551	0.812756	−	0.582605i	$-0.197966\pi$
$151$	19.9746	1.62551	0.812756	+	0.582605i	$0.197966\pi$
$152$	0	0
$153$	18.2598	1.47622
$154$	0	0
$155$	−11.2999	−0.907631
$156$	0	0
$157$	20.8872	1.66698	0.833489	−	0.552536i	$-0.186340\pi$
$157$	20.8872	1.66698	0.833489	+	0.552536i	$0.186340\pi$
$158$	0	0
$159$	−1.80624	−0.143244
$160$	0	0
$161$	3.29261	0.259494
$162$	0	0
$163$	2.54835	0.199602	0.0998010	−	0.995007i	$-0.468179\pi$
$163$	2.54835	0.199602	0.0998010	+	0.995007i	$0.468179\pi$
$164$	0	0
$165$	−5.69584	−0.443420
$166$	0	0
$167$	18.6644	1.44429	0.722146	−	0.691740i	$-0.243156\pi$
$167$	18.6644	1.44429	0.722146	+	0.691740i	$0.243156\pi$
$168$	0	0
$169$	7.05167	0.542436
$170$	0	0
$171$	2.66474	0.203778
$172$	0	0
$173$	8.42370	0.640442	0.320221	−	0.947343i	$-0.396243\pi$
$173$	8.42370	0.640442	0.320221	+	0.947343i	$0.396243\pi$
$174$	0	0
$175$	−1.51181	−0.114282
$176$	0	0
$177$	5.93157	0.445844
$178$	0	0
$179$	−16.6809	−1.24679	−0.623394	−	0.781908i	$-0.714247\pi$
$179$	−16.6809	−1.24679	−0.623394	+	0.781908i	$0.714247\pi$
$180$	0	0
$181$	−4.65888	−0.346292	−0.173146	−	0.984896i	$-0.555393\pi$
$181$	−4.65888	−0.346292	−0.173146	+	0.984896i	$0.555393\pi$
$182$	0	0
$183$	0.418086	0.0309058
$184$	0	0
$185$	−2.31323	−0.170072
$186$	0	0
$187$	31.9645	2.33748
$188$	0	0
$189$	−8.96864	−0.652372
$190$	0	0
$191$	20.0609	1.45155	0.725777	−	0.687930i	$-0.241480\pi$
$191$	20.0609	1.45155	0.725777	+	0.687930i	$0.241480\pi$
$192$	0	0
$193$	1.85117	0.133250	0.0666251	−	0.997778i	$-0.478777\pi$
$193$	1.85117	0.133250	0.0666251	+	0.997778i	$0.478777\pi$
$194$	0	0
$195$	5.46771	0.391551
$196$	0	0
$197$	−0.224883	−0.0160222	−0.00801112	−	0.999968i	$-0.502550\pi$
$197$	−0.224883	−0.0160222	−0.00801112	+	0.999968i	$0.502550\pi$
$198$	0	0
$199$	1.77685	0.125957	0.0629787	−	0.998015i	$-0.479940\pi$
$199$	1.77685	0.125957	0.0629787	+	0.998015i	$0.479940\pi$
$200$	0	0
$201$	−3.55342	−0.250639
$202$	0	0
$203$	−26.1810	−1.83755
$204$	0	0
$205$	−15.4744	−1.08078
$206$	0	0
$207$	−3.20878	−0.223026
$208$	0	0
$209$	4.66474	0.322667
$210$	0	0
$211$	17.0194	1.17166	0.585831	−	0.810433i	$-0.300768\pi$
$211$	17.0194	1.17166	0.585831	+	0.810433i	$0.300768\pi$
$212$	0	0
$213$	−2.67649	−0.183390
$214$	0	0
$215$	−16.1200	−1.09937
$216$	0	0
$217$	−14.6518	−0.994630
$218$	0	0
$219$	3.58568	0.242298
$220$	0	0
$221$	−30.6843	−2.06405
$222$	0	0
$223$	13.6634	0.914971	0.457486	−	0.889217i	$-0.348750\pi$
$223$	13.6634	0.914971	0.457486	+	0.889217i	$0.348750\pi$
$224$	0	0
$225$	1.47332	0.0982212
$226$	0	0
$227$	−5.46152	−0.362494	−0.181247	−	0.983438i	$-0.558013\pi$
$227$	−5.46152	−0.362494	−0.181247	+	0.983438i	$0.558013\pi$
$228$	0	0
$229$	−3.22791	−0.213306	−0.106653	−	0.994296i	$-0.534013\pi$
$229$	−3.22791	−0.213306	−0.106653	+	0.994296i	$0.534013\pi$
$230$	0	0
$231$	−7.38540	−0.485923
$232$	0	0
$233$	17.8344	1.16837	0.584185	−	0.811621i	$-0.301414\pi$
$233$	17.8344	1.16837	0.584185	+	0.811621i	$0.301414\pi$
$234$	0	0
$235$	15.9529	1.04065
$236$	0	0
$237$	−1.89178	−0.122884
$238$	0	0
$239$	1.82556	0.118086	0.0590428	−	0.998255i	$-0.481195\pi$
$239$	1.82556	0.118086	0.0590428	+	0.998255i	$0.481195\pi$
$240$	0	0
$241$	6.47608	0.417161	0.208581	−	0.978005i	$-0.433116\pi$
$241$	6.47608	0.417161	0.208581	+	0.978005i	$0.433116\pi$
$242$	0	0
$243$	13.3691	0.857627
$244$	0	0
$245$	1.00528	0.0642248
$246$	0	0
$247$	−4.47791	−0.284923
$248$	0	0
$249$	−5.19919	−0.329485
$250$	0	0
$251$	−3.10899	−0.196238	−0.0981189	−	0.995175i	$-0.531283\pi$
$251$	−3.10899	−0.196238	−0.0981189	+	0.995175i	$0.531283\pi$
$252$	0	0
$253$	−5.61711	−0.353145
$254$	0	0
$255$	−8.36702	−0.523963
$256$	0	0
$257$	5.75606	0.359053	0.179526	−	0.983753i	$-0.442543\pi$
$257$	5.75606	0.359053	0.179526	+	0.983753i	$0.442543\pi$
$258$	0	0
$259$	−2.99941	−0.186374
$260$	0	0
$261$	25.5145	1.57931
$262$	0	0
$263$	4.58595	0.282782	0.141391	−	0.989954i	$-0.454843\pi$
$263$	4.58595	0.282782	0.141391	+	0.989954i	$0.454843\pi$
$264$	0	0
$265$	−6.57844	−0.404110
$266$	0	0
$267$	−0.359400	−0.0219949
$268$	0	0
$269$	−11.3913	−0.694542	−0.347271	−	0.937765i	$-0.612892\pi$
$269$	−11.3913	−0.694542	−0.347271	+	0.937765i	$0.612892\pi$
$270$	0	0
$271$	25.1252	1.52625	0.763124	−	0.646252i	$-0.223664\pi$
$271$	25.1252	1.52625	0.763124	+	0.646252i	$0.223664\pi$
$272$	0	0
$273$	7.08960	0.429082
$274$	0	0
$275$	2.57910	0.155526
$276$	0	0
$277$	−18.2998	−1.09953	−0.549763	−	0.835321i	$-0.685282\pi$
$277$	−18.2998	−1.09953	−0.549763	+	0.835321i	$0.685282\pi$
$278$	0	0
$279$	14.2788	0.854848
$280$	0	0
$281$	18.0708	1.07802	0.539008	−	0.842301i	$-0.318799\pi$
$281$	18.0708	1.07802	0.539008	+	0.842301i	$0.318799\pi$
$282$	0	0
$283$	−15.3972	−0.915269	−0.457634	−	0.889140i	$-0.651303\pi$
$283$	−15.3972	−0.915269	−0.457634	+	0.889140i	$0.651303\pi$
$284$	0	0
$285$	−1.22104	−0.0723282
$286$	0	0
$287$	−20.0646	−1.18438
$288$	0	0
$289$	29.9550	1.76206
$290$	0	0
$291$	−0.968605	−0.0567806
$292$	0	0
$293$	−11.4407	−0.668374	−0.334187	−	0.942507i	$-0.608462\pi$
$293$	−11.4407	−0.668374	−0.334187	+	0.942507i	$0.608462\pi$
$294$	0	0
$295$	21.6031	1.25778
$296$	0	0
$297$	15.3003	0.887811
$298$	0	0
$299$	5.39214	0.311835
$300$	0	0
$301$	−20.9016	−1.20475
$302$	0	0
$303$	−4.23013	−0.243015
$304$	0	0
$305$	1.52269	0.0871892
$306$	0	0
$307$	−21.3453	−1.21824	−0.609122	−	0.793077i	$-0.708478\pi$
$307$	−21.3453	−1.21824	−0.609122	+	0.793077i	$0.708478\pi$
$308$	0	0
$309$	5.07991	0.288986
$310$	0	0
$311$	21.7295	1.23216	0.616082	−	0.787682i	$-0.288719\pi$
$311$	21.7295	1.23216	0.616082	+	0.787682i	$0.288719\pi$
$312$	0	0
$313$	−34.1009	−1.92750	−0.963748	−	0.266814i	$-0.914029\pi$
$313$	−34.1009	−1.92750	−0.963748	+	0.266814i	$0.914029\pi$
$314$	0	0
$315$	−15.3656	−0.865751
$316$	0	0
$317$	−22.2281	−1.24845	−0.624226	−	0.781244i	$-0.714586\pi$
$317$	−22.2281	−1.24845	−0.624226	+	0.781244i	$0.714586\pi$
$318$	0	0
$319$	44.6641	2.50071
$320$	0	0
$321$	5.74012	0.320382
$322$	0	0
$323$	6.85237	0.381276
$324$	0	0
$325$	−2.47581	−0.137333
$326$	0	0
$327$	1.64782	0.0911245
$328$	0	0
$329$	20.6850	1.14040
$330$	0	0
$331$	−24.1969	−1.32998	−0.664991	−	0.746852i	$-0.731564\pi$
$331$	−24.1969	−1.32998	−0.664991	+	0.746852i	$0.731564\pi$
$332$	0	0
$333$	2.92304	0.160182
$334$	0	0
$335$	−12.9418	−0.707084
$336$	0	0
$337$	−34.2735	−1.86700	−0.933498	−	0.358583i	$-0.883260\pi$
$337$	−34.2735	−1.86700	−0.933498	+	0.358583i	$0.883260\pi$
$338$	0	0
$339$	−0.917670	−0.0498410
$340$	0	0
$341$	24.9956	1.35359
$342$	0	0
$343$	−17.8370	−0.963108
$344$	0	0
$345$	1.47033	0.0791601
$346$	0	0
$347$	−26.9233	−1.44532	−0.722658	−	0.691206i	$-0.757080\pi$
$347$	−26.9233	−1.44532	−0.722658	+	0.691206i	$0.757080\pi$
$348$	0	0
$349$	8.65769	0.463436	0.231718	−	0.972783i	$-0.425565\pi$
$349$	8.65769	0.463436	0.231718	+	0.972783i	$0.425565\pi$
$350$	0	0
$351$	−14.6875	−0.783959
$352$	0	0
$353$	−9.11265	−0.485018	−0.242509	−	0.970149i	$-0.577970\pi$
$353$	−9.11265	−0.485018	−0.242509	+	0.970149i	$0.577970\pi$
$354$	0	0
$355$	−9.74795	−0.517367
$356$	0	0
$357$	−10.8489	−0.574187
$358$	0	0
$359$	−4.29978	−0.226934	−0.113467	−	0.993542i	$-0.536196\pi$
$359$	−4.29978	−0.226934	−0.113467	+	0.993542i	$0.536196\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.23010	0.326996
$364$	0	0
$365$	13.0593	0.683553
$366$	0	0
$367$	−21.2873	−1.11119	−0.555594	−	0.831454i	$-0.687509\pi$
$367$	−21.2873	−1.11119	−0.555594	+	0.831454i	$0.687509\pi$
$368$	0	0
$369$	19.5538	1.01793
$370$	0	0
$371$	−8.52980	−0.442845
$372$	0	0
$373$	11.1076	0.575129	0.287565	−	0.957761i	$-0.407154\pi$
$373$	11.1076	0.575129	0.287565	+	0.957761i	$0.407154\pi$
$374$	0	0
$375$	−6.78031	−0.350134
$376$	0	0
$377$	−42.8753	−2.20819
$378$	0	0
$379$	20.6908	1.06281	0.531407	−	0.847117i	$-0.321664\pi$
$379$	20.6908	1.06281	0.531407	+	0.847117i	$0.321664\pi$
$380$	0	0
$381$	8.27608	0.423996
$382$	0	0
$383$	−12.1869	−0.622720	−0.311360	−	0.950292i	$-0.600785\pi$
$383$	−12.1869	−0.622720	−0.311360	+	0.950292i	$0.600785\pi$
$384$	0	0
$385$	−26.8981	−1.37085
$386$	0	0
$387$	20.3695	1.03544
$388$	0	0
$389$	−26.1769	−1.32722	−0.663611	−	0.748078i	$-0.730977\pi$
$389$	−26.1769	−1.32722	−0.663611	+	0.748078i	$0.730977\pi$
$390$	0	0
$391$	−8.25137	−0.417290
$392$	0	0
$393$	−0.352625	−0.0177876
$394$	0	0
$395$	−6.88999	−0.346673
$396$	0	0
$397$	−14.8451	−0.745055	−0.372528	−	0.928021i	$-0.621509\pi$
$397$	−14.8451	−0.745055	−0.372528	+	0.928021i	$0.621509\pi$
$398$	0	0
$399$	−1.58324	−0.0792611
$400$	0	0
$401$	34.4111	1.71841	0.859205	−	0.511632i	$-0.170959\pi$
$401$	34.4111	1.71841	0.859205	+	0.511632i	$0.170959\pi$
$402$	0	0
$403$	−23.9945	−1.19525
$404$	0	0
$405$	12.8533	0.638688
$406$	0	0
$407$	5.11691	0.253636
$408$	0	0
$409$	−1.43283	−0.0708487	−0.0354243	−	0.999372i	$-0.511278\pi$
$409$	−1.43283	−0.0708487	−0.0354243	+	0.999372i	$0.511278\pi$
$410$	0	0
$411$	−10.5143	−0.518633
$412$	0	0
$413$	28.0113	1.37834
$414$	0	0
$415$	−18.9358	−0.929521
$416$	0	0
$417$	−1.11994	−0.0548438
$418$	0	0
$419$	33.6886	1.64579	0.822897	−	0.568190i	$-0.192356\pi$
$419$	33.6886	1.64579	0.822897	+	0.568190i	$0.192356\pi$
$420$	0	0
$421$	−0.237430	−0.0115716	−0.00578582	−	0.999983i	$-0.501842\pi$
$421$	−0.237430	−0.0115716	−0.00578582	+	0.999983i	$0.501842\pi$
$422$	0	0
$423$	−20.1584	−0.980134
$424$	0	0
$425$	3.78863	0.183776
$426$	0	0
$427$	1.97437	0.0955465
$428$	0	0
$429$	−12.0947	−0.583936
$430$	0	0
$431$	13.0743	0.629766	0.314883	−	0.949130i	$-0.398035\pi$
$431$	13.0743	0.629766	0.314883	+	0.949130i	$0.398035\pi$
$432$	0	0
$433$	−9.67718	−0.465056	−0.232528	−	0.972590i	$-0.574700\pi$
$433$	−9.67718	−0.465056	−0.232528	+	0.972590i	$0.574700\pi$
$434$	0	0
$435$	−11.6913	−0.560554
$436$	0	0
$437$	−1.20416	−0.0576030
$438$	0	0
$439$	−17.7983	−0.849468	−0.424734	−	0.905318i	$-0.639632\pi$
$439$	−17.7983	−0.849468	−0.424734	+	0.905318i	$0.639632\pi$
$440$	0	0
$441$	−1.27029	−0.0604898
$442$	0	0
$443$	3.30468	0.157010	0.0785050	−	0.996914i	$-0.474985\pi$
$443$	3.30468	0.157010	0.0785050	+	0.996914i	$0.474985\pi$
$444$	0	0
$445$	−1.30896	−0.0620505
$446$	0	0
$447$	−9.01966	−0.426615
$448$	0	0
$449$	−29.6543	−1.39947	−0.699735	−	0.714402i	$-0.746699\pi$
$449$	−29.6543	−1.39947	−0.699735	+	0.714402i	$0.746699\pi$
$450$	0	0
$451$	34.2297	1.61182
$452$	0	0
$453$	11.5656	0.543402
$454$	0	0
$455$	25.8208	1.21050
$456$	0	0
$457$	20.2310	0.946368	0.473184	−	0.880964i	$-0.343105\pi$
$457$	20.2310	0.946368	0.473184	+	0.880964i	$0.343105\pi$
$458$	0	0
$459$	22.4756	1.04907
$460$	0	0
$461$	−41.2429	−1.92087	−0.960436	−	0.278502i	$-0.910162\pi$
$461$	−41.2429	−1.92087	−0.960436	+	0.278502i	$0.910162\pi$
$462$	0	0
$463$	−16.5232	−0.767896	−0.383948	−	0.923355i	$-0.625436\pi$
$463$	−16.5232	−0.767896	−0.383948	+	0.923355i	$0.625436\pi$
$464$	0	0
$465$	−6.54285	−0.303417
$466$	0	0
$467$	−24.9967	−1.15671	−0.578355	−	0.815786i	$-0.696305\pi$
$467$	−24.9967	−1.15671	−0.578355	+	0.815786i	$0.696305\pi$
$468$	0	0
$469$	−16.7807	−0.774860
$470$	0	0
$471$	12.0940	0.557264
$472$	0	0
$473$	35.6576	1.63954
$474$	0	0
$475$	0.552894	0.0253685
$476$	0	0
$477$	8.31263	0.380609
$478$	0	0
$479$	10.0298	0.458273	0.229137	−	0.973394i	$-0.426410\pi$
$479$	10.0298	0.458273	0.229137	+	0.973394i	$0.426410\pi$
$480$	0	0
$481$	−4.91197	−0.223966
$482$	0	0
$483$	1.90648	0.0867478
$484$	0	0
$485$	−3.52772	−0.160185
$486$	0	0
$487$	−41.6248	−1.88620	−0.943101	−	0.332508i	$-0.892105\pi$
$487$	−41.6248	−1.88620	−0.943101	+	0.332508i	$0.892105\pi$
$488$	0	0
$489$	1.47554	0.0667261
$490$	0	0
$491$	−34.6939	−1.56572	−0.782858	−	0.622200i	$-0.786239\pi$
$491$	−34.6939	−1.56572	−0.782858	+	0.622200i	$0.786239\pi$
$492$	0	0
$493$	65.6104	2.95494
$494$	0	0
$495$	26.2132	1.17820
$496$	0	0
$497$	−12.6395	−0.566959
$498$	0	0
$499$	−14.1270	−0.632413	−0.316207	−	0.948690i	$-0.602409\pi$
$499$	−14.1270	−0.632413	−0.316207	+	0.948690i	$0.602409\pi$
$500$	0	0
$501$	10.8070	0.482821
$502$	0	0
$503$	−3.01157	−0.134279	−0.0671395	−	0.997744i	$-0.521387\pi$
$503$	−3.01157	−0.134279	−0.0671395	+	0.997744i	$0.521387\pi$
$504$	0	0
$505$	−15.4064	−0.685576
$506$	0	0
$507$	4.08304	0.181334
$508$	0	0
$509$	−2.38122	−0.105546	−0.0527730	−	0.998607i	$-0.516806\pi$
$509$	−2.38122	−0.105546	−0.0527730	+	0.998607i	$0.516806\pi$
$510$	0	0
$511$	16.9330	0.749074
$512$	0	0
$513$	3.27998	0.144815
$514$	0	0
$515$	18.5013	0.815266
$516$	0	0
$517$	−35.2881	−1.55197
$518$	0	0
$519$	4.87747	0.214097
$520$	0	0
$521$	−17.9133	−0.784798	−0.392399	−	0.919795i	$-0.628355\pi$
$521$	−17.9133	−0.784798	−0.392399	+	0.919795i	$0.628355\pi$
$522$	0	0
$523$	18.9602	0.829071	0.414536	−	0.910033i	$-0.363944\pi$
$523$	18.9602	0.829071	0.414536	+	0.910033i	$0.363944\pi$
$524$	0	0
$525$	−0.875363	−0.0382040
$526$	0	0
$527$	36.7178	1.59945
$528$	0	0
$529$	−21.5500	−0.936956
$530$	0	0
$531$	−27.2981	−1.18464
$532$	0	0
$533$	−32.8588	−1.42327
$534$	0	0
$535$	20.9059	0.903839
$536$	0	0
$537$	−9.65852	−0.416796
$538$	0	0
$539$	−2.22369	−0.0957811
$540$	0	0
$541$	−34.2027	−1.47049	−0.735245	−	0.677802i	$-0.762933\pi$
$541$	−34.2027	−1.47049	−0.735245	+	0.677802i	$0.762933\pi$
$542$	0	0
$543$	−2.69757	−0.115764
$544$	0	0
$545$	6.00145	0.257074
$546$	0	0
$547$	−0.645113	−0.0275830	−0.0137915	−	0.999905i	$-0.504390\pi$
$547$	−0.645113	−0.0275830	−0.0137915	+	0.999905i	$0.504390\pi$
$548$	0	0
$549$	−1.92410	−0.0821187
$550$	0	0
$551$	9.57484	0.407902
$552$	0	0
$553$	−8.93377	−0.379903
$554$	0	0
$555$	−1.33940	−0.0568544
$556$	0	0
$557$	10.2596	0.434714	0.217357	−	0.976092i	$-0.430256\pi$
$557$	10.2596	0.434714	0.217357	+	0.976092i	$0.430256\pi$
$558$	0	0
$559$	−34.2295	−1.44775
$560$	0	0
$561$	18.5080	0.781409
$562$	0	0
$563$	−12.9293	−0.544906	−0.272453	−	0.962169i	$-0.587835\pi$
$563$	−12.9293	−0.544906	−0.272453	+	0.962169i	$0.587835\pi$
$564$	0	0
$565$	−3.34221	−0.140608
$566$	0	0
$567$	16.6660	0.699908
$568$	0	0
$569$	−6.32637	−0.265215	−0.132608	−	0.991169i	$-0.542335\pi$
$569$	−6.32637	−0.265215	−0.132608	+	0.991169i	$0.542335\pi$
$570$	0	0
$571$	31.1997	1.30566	0.652832	−	0.757502i	$-0.273581\pi$
$571$	31.1997	1.30566	0.652832	+	0.757502i	$0.273581\pi$
$572$	0	0
$573$	11.6156	0.485248
$574$	0	0
$575$	−0.665774	−0.0277647
$576$	0	0
$577$	5.48996	0.228550	0.114275	−	0.993449i	$-0.463545\pi$
$577$	5.48996	0.228550	0.114275	+	0.993449i	$0.463545\pi$
$578$	0	0
$579$	1.07186	0.0445450
$580$	0	0
$581$	−24.5527	−1.01862
$582$	0	0
$583$	14.5516	0.602666
$584$	0	0
$585$	−25.1634	−1.04038
$586$	0	0
$587$	17.4576	0.720552	0.360276	−	0.932846i	$-0.382683\pi$
$587$	17.4576	0.720552	0.360276	+	0.932846i	$0.382683\pi$
$588$	0	0
$589$	5.35842	0.220790
$590$	0	0
$591$	−0.130211	−0.00535617
$592$	0	0
$593$	−16.7657	−0.688486	−0.344243	−	0.938881i	$-0.611864\pi$
$593$	−16.7657	−0.688486	−0.344243	+	0.938881i	$0.611864\pi$
$594$	0	0
$595$	−39.5125	−1.61985
$596$	0	0
$597$	1.02883	0.0421070
$598$	0	0
$599$	0.357982	0.0146267	0.00731337	−	0.999973i	$-0.497672\pi$
$599$	0.357982	0.0146267	0.00731337	+	0.999973i	$0.497672\pi$
$600$	0	0
$601$	−22.8949	−0.933903	−0.466952	−	0.884283i	$-0.654648\pi$
$601$	−22.8949	−0.933903	−0.466952	+	0.884283i	$0.654648\pi$
$602$	0	0
$603$	16.3535	0.665964
$604$	0	0
$605$	22.6904	0.922497
$606$	0	0
$607$	1.33917	0.0543551	0.0271775	−	0.999631i	$-0.491348\pi$
$607$	1.33917	0.0543551	0.0271775	+	0.999631i	$0.491348\pi$
$608$	0	0
$609$	−15.1593	−0.614284
$610$	0	0
$611$	33.8747	1.37043
$612$	0	0
$613$	29.8667	1.20631	0.603153	−	0.797625i	$-0.293911\pi$
$613$	29.8667	1.20631	0.603153	+	0.797625i	$0.293911\pi$
$614$	0	0
$615$	−8.95997	−0.361301
$616$	0	0
$617$	36.4935	1.46917	0.734587	−	0.678514i	$-0.237376\pi$
$617$	36.4935	1.46917	0.734587	+	0.678514i	$0.237376\pi$
$618$	0	0
$619$	8.88669	0.357186	0.178593	−	0.983923i	$-0.442845\pi$
$619$	8.88669	0.357186	0.178593	+	0.983923i	$0.442845\pi$
$620$	0	0
$621$	−3.94963	−0.158493
$622$	0	0
$623$	−1.69723	−0.0679982
$624$	0	0
$625$	−21.9298	−0.877194
$626$	0	0
$627$	2.70096	0.107866
$628$	0	0
$629$	7.51659	0.299706
$630$	0	0
$631$	28.1724	1.12153	0.560763	−	0.827977i	$-0.310508\pi$
$631$	28.1724	1.12153	0.560763	+	0.827977i	$0.310508\pi$
$632$	0	0
$633$	9.85452	0.391682
$634$	0	0
$635$	30.1420	1.19615
$636$	0	0
$637$	2.13463	0.0845771
$638$	0	0
$639$	12.3177	0.487280
$640$	0	0
$641$	3.58435	0.141573	0.0707867	−	0.997491i	$-0.477449\pi$
$641$	3.58435	0.141573	0.0707867	+	0.997491i	$0.477449\pi$
$642$	0	0
$643$	19.8837	0.784137	0.392069	−	0.919936i	$-0.371760\pi$
$643$	19.8837	0.784137	0.392069	+	0.919936i	$0.371760\pi$
$644$	0	0
$645$	−9.33373	−0.367515
$646$	0	0
$647$	−35.7712	−1.40631	−0.703156	−	0.711036i	$-0.748226\pi$
$647$	−35.7712	−1.40631	−0.703156	+	0.711036i	$0.748226\pi$
$648$	0	0
$649$	−47.7865	−1.87578
$650$	0	0
$651$	−8.48365	−0.332501
$652$	0	0
$653$	−8.57378	−0.335518	−0.167759	−	0.985828i	$-0.553653\pi$
$653$	−8.57378	−0.335518	−0.167759	+	0.985828i	$0.553653\pi$
$654$	0	0
$655$	−1.28428	−0.0501810
$656$	0	0
$657$	−16.5019	−0.643802
$658$	0	0
$659$	−22.9993	−0.895927	−0.447964	−	0.894052i	$-0.647851\pi$
$659$	−22.9993	−0.895927	−0.447964	+	0.894052i	$0.647851\pi$
$660$	0	0
$661$	28.4720	1.10743	0.553716	−	0.832706i	$-0.313209\pi$
$661$	28.4720	1.10743	0.553716	+	0.832706i	$0.313209\pi$
$662$	0	0
$663$	−17.7667	−0.690003
$664$	0	0
$665$	−5.76625	−0.223606
$666$	0	0
$667$	−11.5297	−0.446431
$668$	0	0
$669$	7.91136	0.305871
$670$	0	0
$671$	−3.36822	−0.130029
$672$	0	0
$673$	6.63685	0.255832	0.127916	−	0.991785i	$-0.459171\pi$
$673$	6.63685	0.255832	0.127916	+	0.991785i	$0.459171\pi$
$674$	0	0
$675$	1.81348	0.0698009
$676$	0	0
$677$	−29.0823	−1.11772	−0.558862	−	0.829261i	$-0.688762\pi$
$677$	−29.0823	−1.11772	−0.558862	+	0.829261i	$0.688762\pi$
$678$	0	0
$679$	−4.57415	−0.175540
$680$	0	0
$681$	−3.16231	−0.121180
$682$	0	0
$683$	−4.63939	−0.177521	−0.0887607	−	0.996053i	$-0.528291\pi$
$683$	−4.63939	−0.177521	−0.0887607	+	0.996053i	$0.528291\pi$
$684$	0	0
$685$	−38.2938	−1.46313
$686$	0	0
$687$	−1.86902	−0.0713074
$688$	0	0
$689$	−13.9688	−0.532169
$690$	0	0
$691$	5.11245	0.194487	0.0972434	−	0.995261i	$-0.468997\pi$
$691$	5.11245	0.194487	0.0972434	+	0.995261i	$0.468997\pi$
$692$	0	0
$693$	33.9889	1.29113
$694$	0	0
$695$	−4.07890	−0.154721
$696$	0	0
$697$	50.2825	1.90459
$698$	0	0
$699$	10.3264	0.390581
$700$	0	0
$701$	5.27938	0.199399	0.0996997	−	0.995018i	$-0.468212\pi$
$701$	5.27938	0.199399	0.0996997	+	0.995018i	$0.468212\pi$
$702$	0	0
$703$	1.09693	0.0413716
$704$	0	0
$705$	9.23700	0.347886
$706$	0	0
$707$	−19.9764	−0.751290
$708$	0	0
$709$	28.1441	1.05697	0.528486	−	0.848942i	$-0.322760\pi$
$709$	28.1441	1.05697	0.528486	+	0.848942i	$0.322760\pi$
$710$	0	0
$711$	8.70632	0.326512
$712$	0	0
$713$	−6.45241	−0.241645
$714$	0	0
$715$	−44.0495	−1.64736
$716$	0	0
$717$	1.05703	0.0394755
$718$	0	0
$719$	9.51160	0.354723	0.177361	−	0.984146i	$-0.443244\pi$
$719$	9.51160	0.354723	0.177361	+	0.984146i	$0.443244\pi$
$720$	0	0
$721$	23.9894	0.893412
$722$	0	0
$723$	3.74976	0.139455
$724$	0	0
$725$	5.29387	0.196609
$726$	0	0
$727$	−7.06196	−0.261914	−0.130957	−	0.991388i	$-0.541805\pi$
$727$	−7.06196	−0.261914	−0.130957	+	0.991388i	$0.541805\pi$
$728$	0	0
$729$	−10.5442	−0.390527
$730$	0	0
$731$	52.3801	1.93735
$732$	0	0
$733$	7.62443	0.281615	0.140807	−	0.990037i	$-0.455030\pi$
$733$	7.62443	0.281615	0.140807	+	0.990037i	$0.455030\pi$
$734$	0	0
$735$	0.582073	0.0214701
$736$	0	0
$737$	28.6274	1.05450
$738$	0	0
$739$	45.0131	1.65583	0.827917	−	0.560851i	$-0.189526\pi$
$739$	45.0131	1.65583	0.827917	+	0.560851i	$0.189526\pi$
$740$	0	0
$741$	−2.59279	−0.0952484
$742$	0	0
$743$	−15.3137	−0.561805	−0.280903	−	0.959736i	$-0.590634\pi$
$743$	−15.3137	−0.561805	−0.280903	+	0.959736i	$0.590634\pi$
$744$	0	0
$745$	−32.8501	−1.20354
$746$	0	0
$747$	23.9276	0.875465
$748$	0	0
$749$	27.1072	0.990475
$750$	0	0
$751$	4.09580	0.149458	0.0747290	−	0.997204i	$-0.476191\pi$
$751$	4.09580	0.149458	0.0747290	+	0.997204i	$0.476191\pi$
$752$	0	0
$753$	−1.80016	−0.0656015
$754$	0	0
$755$	42.1228	1.53301
$756$	0	0
$757$	9.51720	0.345909	0.172954	−	0.984930i	$-0.444669\pi$
$757$	9.51720	0.345909	0.172954	+	0.984930i	$0.444669\pi$
$758$	0	0
$759$	−3.25240	−0.118055
$760$	0	0
$761$	38.4841	1.39505	0.697523	−	0.716562i	$-0.254285\pi$
$761$	38.4841	1.39505	0.697523	+	0.716562i	$0.254285\pi$
$762$	0	0
$763$	7.78167	0.281715
$764$	0	0
$765$	38.5065	1.39221
$766$	0	0
$767$	45.8726	1.65636
$768$	0	0
$769$	15.0210	0.541670	0.270835	−	0.962626i	$-0.412700\pi$
$769$	15.0210	0.541670	0.270835	+	0.962626i	$0.412700\pi$
$770$	0	0
$771$	3.33286	0.120030
$772$	0	0
$773$	38.4471	1.38285	0.691423	−	0.722451i	$-0.256984\pi$
$773$	38.4471	1.38285	0.691423	+	0.722451i	$0.256984\pi$
$774$	0	0
$775$	2.96263	0.106421
$776$	0	0
$777$	−1.73671	−0.0623040
$778$	0	0
$779$	7.33797	0.262910
$780$	0	0
$781$	21.5626	0.771572
$782$	0	0
$783$	31.4053	1.12233
$784$	0	0
$785$	44.0472	1.57211
$786$	0	0
$787$	−30.8457	−1.09953	−0.549765	−	0.835319i	$-0.685283\pi$
$787$	−30.8457	−1.09953	−0.549765	+	0.835319i	$0.685283\pi$
$788$	0	0
$789$	2.65535	0.0945328
$790$	0	0
$791$	−4.33361	−0.154085
$792$	0	0
$793$	3.23332	0.114819
$794$	0	0
$795$	−3.80903	−0.135092
$796$	0	0
$797$	−34.1285	−1.20889	−0.604446	−	0.796646i	$-0.706606\pi$
$797$	−34.1285	−1.20889	−0.604446	+	0.796646i	$0.706606\pi$
$798$	0	0
$799$	−51.8372	−1.83387
$800$	0	0
$801$	1.65402	0.0584420
$802$	0	0
$803$	−28.8873	−1.01941
$804$	0	0
$805$	6.94351	0.244727
$806$	0	0
$807$	−6.59578	−0.232182
$808$	0	0
$809$	25.9170	0.911192	0.455596	−	0.890187i	$-0.349426\pi$
$809$	25.9170	0.911192	0.455596	+	0.890187i	$0.349426\pi$
$810$	0	0
$811$	−33.7874	−1.18644	−0.593219	−	0.805041i	$-0.702143\pi$
$811$	−33.7874	−1.18644	−0.593219	+	0.805041i	$0.702143\pi$
$812$	0	0
$813$	14.5479	0.510218
$814$	0	0
$815$	5.37400	0.188243
$816$	0	0
$817$	7.64408	0.267432
$818$	0	0
$819$	−32.6276	−1.14010
$820$	0	0
$821$	27.6475	0.964903	0.482451	−	0.875923i	$-0.339746\pi$
$821$	27.6475	0.964903	0.482451	+	0.875923i	$0.339746\pi$
$822$	0	0
$823$	12.4983	0.435665	0.217832	−	0.975986i	$-0.430101\pi$
$823$	12.4983	0.435665	0.217832	+	0.975986i	$0.430101\pi$
$824$	0	0
$825$	1.49335	0.0519916
$826$	0	0
$827$	−9.03084	−0.314033	−0.157017	−	0.987596i	$-0.550188\pi$
$827$	−9.03084	−0.314033	−0.157017	+	0.987596i	$0.550188\pi$
$828$	0	0
$829$	55.7684	1.93692	0.968458	−	0.249175i	$-0.0801595\pi$
$829$	55.7684	1.93692	0.968458	+	0.249175i	$0.0801595\pi$
$830$	0	0
$831$	−10.5959	−0.367567
$832$	0	0
$833$	−3.26654	−0.113179
$834$	0	0
$835$	39.3597	1.36210
$836$	0	0
$837$	17.5755	0.607498
$838$	0	0
$839$	14.3128	0.494134	0.247067	−	0.968998i	$-0.420533\pi$
$839$	14.3128	0.494134	0.247067	+	0.968998i	$0.420533\pi$
$840$	0	0
$841$	62.6776	2.16130
$842$	0	0
$843$	10.4633	0.360376
$844$	0	0
$845$	14.8707	0.511567
$846$	0	0
$847$	29.4211	1.01092
$848$	0	0
$849$	−8.91525	−0.305971
$850$	0	0
$851$	−1.32089	−0.0452794
$852$	0	0
$853$	38.0325	1.30221	0.651105	−	0.758988i	$-0.274306\pi$
$853$	38.0325	1.30221	0.651105	+	0.758988i	$0.274306\pi$
$854$	0	0
$855$	5.61945	0.192181
$856$	0	0
$857$	23.1869	0.792048	0.396024	−	0.918240i	$-0.370390\pi$
$857$	23.1869	0.792048	0.396024	+	0.918240i	$0.370390\pi$
$858$	0	0
$859$	−46.5870	−1.58953	−0.794764	−	0.606919i	$-0.792405\pi$
$859$	−46.5870	−1.58953	−0.794764	+	0.606919i	$0.792405\pi$
$860$	0	0
$861$	−11.6178	−0.395933
$862$	0	0
$863$	−54.3149	−1.84890	−0.924451	−	0.381301i	$-0.875476\pi$
$863$	−54.3149	−1.84890	−0.924451	+	0.381301i	$0.875476\pi$
$864$	0	0
$865$	17.7640	0.603995
$866$	0	0
$867$	17.3444	0.589048
$868$	0	0
$869$	15.2408	0.517008
$870$	0	0
$871$	−27.4808	−0.931153
$872$	0	0
$873$	4.45769	0.150870
$874$	0	0
$875$	−32.0194	−1.08245
$876$	0	0
$877$	1.80324	0.0608910	0.0304455	−	0.999536i	$-0.490307\pi$
$877$	1.80324	0.0608910	0.0304455	+	0.999536i	$0.490307\pi$
$878$	0	0
$879$	−6.62438	−0.223435
$880$	0	0
$881$	12.6372	0.425759	0.212880	−	0.977078i	$-0.431716\pi$
$881$	12.6372	0.425759	0.212880	+	0.977078i	$0.431716\pi$
$882$	0	0
$883$	39.9276	1.34367	0.671835	−	0.740700i	$-0.265506\pi$
$883$	39.9276	1.34367	0.671835	+	0.740700i	$0.265506\pi$
$884$	0	0
$885$	12.5086	0.420471
$886$	0	0
$887$	14.3642	0.482304	0.241152	−	0.970487i	$-0.422475\pi$
$887$	14.3642	0.482304	0.241152	+	0.970487i	$0.422475\pi$
$888$	0	0
$889$	39.0830	1.31080
$890$	0	0
$891$	−28.4318	−0.952502
$892$	0	0
$893$	−7.56486	−0.253148
$894$	0	0
$895$	−35.1769	−1.17584
$896$	0	0
$897$	3.12214	0.104245
$898$	0	0
$899$	51.3060	1.71115
$900$	0	0
$901$	21.3759	0.712135
$902$	0	0
$903$	−12.1024	−0.402743
$904$	0	0
$905$	−9.82473	−0.326585
$906$	0	0
$907$	3.32787	0.110500	0.0552501	−	0.998473i	$-0.482404\pi$
$907$	3.32787	0.110500	0.0552501	+	0.998473i	$0.482404\pi$
$908$	0	0
$909$	19.4678	0.645707
$910$	0	0
$911$	−35.8252	−1.18694	−0.593471	−	0.804855i	$-0.702243\pi$
$911$	−35.8252	−1.18694	−0.593471	+	0.804855i	$0.702243\pi$
$912$	0	0
$913$	41.8863	1.38623
$914$	0	0
$915$	0.881666	0.0291470
$916$	0	0
$917$	−1.66524	−0.0549910
$918$	0	0
$919$	48.3186	1.59388	0.796942	−	0.604055i	$-0.206449\pi$
$919$	48.3186	1.59388	0.796942	+	0.604055i	$0.206449\pi$
$920$	0	0
$921$	−12.3593	−0.407254
$922$	0	0
$923$	−20.6990	−0.681316
$924$	0	0
$925$	0.606487	0.0199412
$926$	0	0
$927$	−23.3786	−0.767855
$928$	0	0
$929$	−23.2127	−0.761585	−0.380792	−	0.924661i	$-0.624349\pi$
$929$	−23.2127	−0.761585	−0.380792	+	0.924661i	$0.624349\pi$
$930$	0	0
$931$	−0.476702	−0.0156233
$932$	0	0
$933$	12.5817	0.411908
$934$	0	0
$935$	67.4073	2.20445
$936$	0	0
$937$	−7.10467	−0.232099	−0.116050	−	0.993243i	$-0.537023\pi$
$937$	−7.10467	−0.232099	−0.116050	+	0.993243i	$0.537023\pi$
$938$	0	0
$939$	−19.7450	−0.644354
$940$	0	0
$941$	55.5352	1.81040	0.905199	−	0.424989i	$-0.139722\pi$
$941$	55.5352	1.81040	0.905199	+	0.424989i	$0.139722\pi$
$942$	0	0
$943$	−8.83612	−0.287744
$944$	0	0
$945$	−18.9132	−0.615247
$946$	0	0
$947$	−15.5988	−0.506894	−0.253447	−	0.967349i	$-0.581564\pi$
$947$	−15.5988	−0.506894	−0.253447	+	0.967349i	$0.581564\pi$
$948$	0	0
$949$	27.7304	0.900165
$950$	0	0
$951$	−12.8704	−0.417352
$952$	0	0
$953$	−21.1243	−0.684282	−0.342141	−	0.939649i	$-0.611152\pi$
$953$	−21.1243	−0.684282	−0.342141	+	0.939649i	$0.611152\pi$
$954$	0	0
$955$	42.3047	1.36895
$956$	0	0
$957$	25.8613	0.835977
$958$	0	0
$959$	−49.6529	−1.60338
$960$	0	0
$961$	−2.28738	−0.0737863
$962$	0	0
$963$	−26.4170	−0.851277
$964$	0	0
$965$	3.90378	0.125667
$966$	0	0
$967$	−46.1207	−1.48314	−0.741570	−	0.670875i	$-0.765919\pi$
$967$	−46.1207	−1.48314	−0.741570	+	0.670875i	$0.765919\pi$
$968$	0	0
$969$	3.96764	0.127459
$970$	0	0
$971$	−22.3253	−0.716454	−0.358227	−	0.933635i	$-0.616619\pi$
$971$	−22.3253	−0.716454	−0.358227	+	0.933635i	$0.616619\pi$
$972$	0	0
$973$	−5.28882	−0.169552
$974$	0	0
$975$	−1.43354	−0.0459099
$976$	0	0
$977$	14.2650	0.456379	0.228190	−	0.973617i	$-0.426719\pi$
$977$	14.2650	0.456379	0.228190	+	0.973617i	$0.426719\pi$
$978$	0	0
$979$	2.89544	0.0925385
$980$	0	0
$981$	−7.58355	−0.242124
$982$	0	0
$983$	−12.7242	−0.405838	−0.202919	−	0.979196i	$-0.565043\pi$
$983$	−12.7242	−0.405838	−0.202919	+	0.979196i	$0.565043\pi$
$984$	0	0
$985$	−0.474237	−0.0151104
$986$	0	0
$987$	11.9770	0.381232
$988$	0	0
$989$	−9.20472	−0.292693
$990$	0	0
$991$	−10.8169	−0.343611	−0.171806	−	0.985131i	$-0.554960\pi$
$991$	−10.8169	−0.343611	−0.171806	+	0.985131i	$0.554960\pi$
$992$	0	0
$993$	−14.0104	−0.444607
$994$	0	0
$995$	3.74705	0.118789
$996$	0	0
$997$	−5.39134	−0.170746	−0.0853728	−	0.996349i	$-0.527208\pi$
$997$	−5.39134	−0.170746	−0.0853728	+	0.996349i	$0.527208\pi$
$998$	0	0
$999$	3.59792	0.113833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bo.1.5		8
4.3	odd	2		4864.2.a.bn.1.4		8
8.3	odd	2		4864.2.a.bp.1.5		8
8.5	even	2		4864.2.a.bq.1.4		8
16.3	odd	4		608.2.c.b.305.8		16
16.5	even	4		152.2.c.b.77.16	yes	16
16.11	odd	4		608.2.c.b.305.9		16
16.13	even	4		152.2.c.b.77.15	✓	16
48.5	odd	4		1368.2.g.b.685.1		16
48.11	even	4		5472.2.g.b.2737.5		16
48.29	odd	4		1368.2.g.b.685.2		16
48.35	even	4		5472.2.g.b.2737.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.15	✓	16	16.13	even	4
152.2.c.b.77.16	yes	16	16.5	even	4
608.2.c.b.305.8		16	16.3	odd	4
608.2.c.b.305.9		16	16.11	odd	4
1368.2.g.b.685.1		16	48.5	odd	4
1368.2.g.b.685.2		16	48.29	odd	4
4864.2.a.bn.1.4		8	4.3	odd	2
4864.2.a.bo.1.5		8	1.1	even	1	trivial
4864.2.a.bp.1.5		8	8.3	odd	2
4864.2.a.bq.1.4		8	8.5	even	2
5472.2.g.b.2737.5		16	48.11	even	4
5472.2.g.b.2737.12		16	48.35	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+0.579017 q^{3} +2.10882 q^{5} +2.73436 q^{7} -2.66474 q^{9} +O(q^{10})\)	\(q+0.579017 q^{3} +2.10882 q^{5} +2.73436 q^{7} -2.66474 q^{9} -4.66474 q^{11} +4.47791 q^{13} +1.22104 q^{15} -6.85237 q^{17} -1.00000 q^{19} +1.58324 q^{21} +1.20416 q^{23} -0.552894 q^{25} -3.27998 q^{27} -9.57484 q^{29} -5.35842 q^{31} -2.70096 q^{33} +5.76625 q^{35} -1.09693 q^{37} +2.59279 q^{39} -7.33797 q^{41} -7.64408 q^{43} -5.61945 q^{45} +7.56486 q^{47} +0.476702 q^{49} -3.96764 q^{51} -3.11949 q^{53} -9.83708 q^{55} -0.579017 q^{57} +10.2442 q^{59} +0.722061 q^{61} -7.28634 q^{63} +9.44309 q^{65} -6.13698 q^{67} +0.697232 q^{69} -4.62247 q^{71} +6.19270 q^{73} -0.320135 q^{75} -12.7551 q^{77} -3.26723 q^{79} +6.09505 q^{81} -8.97934 q^{83} -14.4504 q^{85} -5.54400 q^{87} -0.620707 q^{89} +12.2442 q^{91} -3.10262 q^{93} -2.10882 q^{95} -1.67284 q^{97} +12.4303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 + 4 * q^7 + 12 * q^9	\( 8 q - 8 q^{5} + 4 q^{7} + 12 q^{9} - 4 q^{11} - 8 q^{13} - 4 q^{17} - 8 q^{19} - 16 q^{21} + 12 q^{25} - 28 q^{29} + 8 q^{31} + 12 q^{35} - 4 q^{37} - 4 q^{39} - 8 q^{41} + 4 q^{43} - 24 q^{45} + 12 q^{47} + 12 q^{49} - 12 q^{51} - 32 q^{53} - 8 q^{55} - 12 q^{59} - 8 q^{61} - 16 q^{63} + 8 q^{65} + 4 q^{67} - 28 q^{69} - 24 q^{71} - 24 q^{77} - 24 q^{79} - 8 q^{81} - 40 q^{83} - 24 q^{85} + 24 q^{87} + 8 q^{89} + 4 q^{91} - 32 q^{93} + 8 q^{95} + 16 q^{97} + 76 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 + 4 * q^7 + 12 * q^9 - 4 * q^11 - 8 * q^13 - 4 * q^17 - 8 * q^19 - 16 * q^21 + 12 * q^25 - 28 * q^29 + 8 * q^31 + 12 * q^35 - 4 * q^37 - 4 * q^39 - 8 * q^41 + 4 * q^43 - 24 * q^45 + 12 * q^47 + 12 * q^49 - 12 * q^51 - 32 * q^53 - 8 * q^55 - 12 * q^59 - 8 * q^61 - 16 * q^63 + 8 * q^65 + 4 * q^67 - 28 * q^69 - 24 * q^71 - 24 * q^77 - 24 * q^79 - 8 * q^81 - 40 * q^83 - 24 * q^85 + 24 * q^87 + 8 * q^89 + 4 * q^91 - 32 * q^93 + 8 * q^95 + 16 * q^97 + 76 * q^99

Introduction

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