LMFDB - Embedded newform 8048.2.a.q.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.74625$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74625 q^{3} -2.58578 q^{5} -0.635213 q^{7} +0.0494057 q^{9} +O(q^{10})$	$q-1.74625 q^{3} -2.58578 q^{5} -0.635213 q^{7} +0.0494057 q^{9} -3.51104 q^{11} +5.12735 q^{13} +4.51543 q^{15} -0.322021 q^{17} -3.38371 q^{19} +1.10924 q^{21} -3.77692 q^{23} +1.68625 q^{25} +5.15249 q^{27} +2.46547 q^{29} -1.98771 q^{31} +6.13117 q^{33} +1.64252 q^{35} -7.77317 q^{37} -8.95366 q^{39} +10.7578 q^{41} +9.96085 q^{43} -0.127752 q^{45} +3.93174 q^{47} -6.59651 q^{49} +0.562331 q^{51} +8.45466 q^{53} +9.07877 q^{55} +5.90881 q^{57} -11.9394 q^{59} +0.586889 q^{61} -0.0313831 q^{63} -13.2582 q^{65} +12.9100 q^{67} +6.59547 q^{69} +0.0667081 q^{71} -2.81224 q^{73} -2.94462 q^{75} +2.23026 q^{77} +4.39323 q^{79} -9.14578 q^{81} +16.7077 q^{83} +0.832675 q^{85} -4.30534 q^{87} -13.6156 q^{89} -3.25696 q^{91} +3.47104 q^{93} +8.74951 q^{95} -12.7052 q^{97} -0.173465 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * 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
$3$	−1.74625	−1.00820	−0.504100	−	0.863645i	$-0.668176\pi$
$3$	−1.74625	−1.00820	−0.504100	+	0.863645i	$0.668176\pi$
$4$	0	0
$5$	−2.58578	−1.15639	−0.578197	−	0.815897i	$-0.696244\pi$
$5$	−2.58578	−1.15639	−0.578197	+	0.815897i	$0.696244\pi$
$6$	0	0
$7$	−0.635213	−0.240088	−0.120044	−	0.992769i	$-0.538304\pi$
$7$	−0.635213	−0.240088	−0.120044	+	0.992769i	$0.538304\pi$
$8$	0	0
$9$	0.0494057	0.0164686
$10$	0	0
$11$	−3.51104	−1.05862	−0.529309	−	0.848429i	$-0.677549\pi$
$11$	−3.51104	−1.05862	−0.529309	+	0.848429i	$0.677549\pi$
$12$	0	0
$13$	5.12735	1.42207	0.711035	−	0.703156i	$-0.248227\pi$
$13$	5.12735	1.42207	0.711035	+	0.703156i	$0.248227\pi$
$14$	0	0
$15$	4.51543	1.16588
$16$	0	0
$17$	−0.322021	−0.0781016	−0.0390508	−	0.999237i	$-0.512433\pi$
$17$	−0.322021	−0.0781016	−0.0390508	+	0.999237i	$0.512433\pi$
$18$	0	0
$19$	−3.38371	−0.776276	−0.388138	−	0.921601i	$-0.626882\pi$
$19$	−3.38371	−0.776276	−0.388138	+	0.921601i	$0.626882\pi$
$20$	0	0
$21$	1.10924	0.242057
$22$	0	0
$23$	−3.77692	−0.787543	−0.393772	−	0.919208i	$-0.628830\pi$
$23$	−3.77692	−0.787543	−0.393772	+	0.919208i	$0.628830\pi$
$24$	0	0
$25$	1.68625	0.337249
$26$	0	0
$27$	5.15249	0.991597
$28$	0	0
$29$	2.46547	0.457826	0.228913	−	0.973447i	$-0.426483\pi$
$29$	2.46547	0.457826	0.228913	+	0.973447i	$0.426483\pi$
$30$	0	0
$31$	−1.98771	−0.357003	−0.178501	−	0.983940i	$-0.557125\pi$
$31$	−1.98771	−0.357003	−0.178501	+	0.983940i	$0.557125\pi$
$32$	0	0
$33$	6.13117	1.06730
$34$	0	0
$35$	1.64252	0.277636
$36$	0	0
$37$	−7.77317	−1.27790	−0.638951	−	0.769248i	$-0.720631\pi$
$37$	−7.77317	−1.27790	−0.638951	+	0.769248i	$0.720631\pi$
$38$	0	0
$39$	−8.95366	−1.43373
$40$	0	0
$41$	10.7578	1.68009	0.840045	−	0.542516i	$-0.182528\pi$
$41$	10.7578	1.68009	0.840045	+	0.542516i	$0.182528\pi$
$42$	0	0
$43$	9.96085	1.51902	0.759508	−	0.650498i	$-0.225440\pi$
$43$	9.96085	1.51902	0.759508	+	0.650498i	$0.225440\pi$
$44$	0	0
$45$	−0.127752	−0.0190442
$46$	0	0
$47$	3.93174	0.573503	0.286752	−	0.958005i	$-0.407425\pi$
$47$	3.93174	0.573503	0.286752	+	0.958005i	$0.407425\pi$
$48$	0	0
$49$	−6.59651	−0.942358
$50$	0	0
$51$	0.562331	0.0787421
$52$	0	0
$53$	8.45466	1.16134	0.580669	−	0.814140i	$-0.302791\pi$
$53$	8.45466	1.16134	0.580669	+	0.814140i	$0.302791\pi$
$54$	0	0
$55$	9.07877	1.22418
$56$	0	0
$57$	5.90881	0.782642
$58$	0	0
$59$	−11.9394	−1.55437	−0.777187	−	0.629270i	$-0.783354\pi$
$59$	−11.9394	−1.55437	−0.777187	+	0.629270i	$0.783354\pi$
$60$	0	0
$61$	0.586889	0.0751434	0.0375717	−	0.999294i	$-0.488038\pi$
$61$	0.586889	0.0751434	0.0375717	+	0.999294i	$0.488038\pi$
$62$	0	0
$63$	−0.0313831	−0.00395390
$64$	0	0
$65$	−13.2582	−1.64448
$66$	0	0
$67$	12.9100	1.57721	0.788605	−	0.614900i	$-0.210804\pi$
$67$	12.9100	1.57721	0.788605	+	0.614900i	$0.210804\pi$
$68$	0	0
$69$	6.59547	0.794002
$70$	0	0
$71$	0.0667081	0.00791679	0.00395840	−	0.999992i	$-0.498740\pi$
$71$	0.0667081	0.00791679	0.00395840	+	0.999992i	$0.498740\pi$
$72$	0	0
$73$	−2.81224	−0.329148	−0.164574	−	0.986365i	$-0.552625\pi$
$73$	−2.81224	−0.329148	−0.164574	+	0.986365i	$0.552625\pi$
$74$	0	0
$75$	−2.94462	−0.340015
$76$	0	0
$77$	2.23026	0.254161
$78$	0	0
$79$	4.39323	0.494277	0.247139	−	0.968980i	$-0.420510\pi$
$79$	4.39323	0.494277	0.247139	+	0.968980i	$0.420510\pi$
$80$	0	0
$81$	−9.14578	−1.01620
$82$	0	0
$83$	16.7077	1.83391	0.916955	−	0.398992i	$-0.130640\pi$
$83$	16.7077	1.83391	0.916955	+	0.398992i	$0.130640\pi$
$84$	0	0
$85$	0.832675	0.0903163
$86$	0	0
$87$	−4.30534	−0.461581
$88$	0	0
$89$	−13.6156	−1.44325	−0.721623	−	0.692286i	$-0.756604\pi$
$89$	−13.6156	−1.44325	−0.721623	+	0.692286i	$0.756604\pi$
$90$	0	0
$91$	−3.25696	−0.341422
$92$	0	0
$93$	3.47104	0.359930
$94$	0	0
$95$	8.74951	0.897681
$96$	0	0
$97$	−12.7052	−1.29002	−0.645008	−	0.764176i	$-0.723146\pi$
$97$	−12.7052	−1.29002	−0.645008	+	0.764176i	$0.723146\pi$
$98$	0	0
$99$	−0.173465	−0.0174339
$100$	0	0
$101$	9.15646	0.911101	0.455551	−	0.890210i	$-0.349442\pi$
$101$	9.15646	0.911101	0.455551	+	0.890210i	$0.349442\pi$
$102$	0	0
$103$	−1.18800	−0.117057	−0.0585285	−	0.998286i	$-0.518641\pi$
$103$	−1.18800	−0.117057	−0.0585285	+	0.998286i	$0.518641\pi$
$104$	0	0
$105$	−2.86826	−0.279913
$106$	0	0
$107$	10.5335	1.01831	0.509155	−	0.860675i	$-0.329958\pi$
$107$	10.5335	1.01831	0.509155	+	0.860675i	$0.329958\pi$
$108$	0	0
$109$	−7.51021	−0.719347	−0.359674	−	0.933078i	$-0.617112\pi$
$109$	−7.51021	−0.719347	−0.359674	+	0.933078i	$0.617112\pi$
$110$	0	0
$111$	13.5739	1.28838
$112$	0	0
$113$	14.2153	1.33726	0.668632	−	0.743594i	$-0.266880\pi$
$113$	14.2153	1.33726	0.668632	+	0.743594i	$0.266880\pi$
$114$	0	0
$115$	9.76629	0.910711
$116$	0	0
$117$	0.253320	0.0234195
$118$	0	0
$119$	0.204552	0.0187512
$120$	0	0
$121$	1.32741	0.120673
$122$	0	0
$123$	−18.7859	−1.69387
$124$	0	0
$125$	8.56863	0.766401
$126$	0	0
$127$	−20.3451	−1.80533	−0.902667	−	0.430340i	$-0.858394\pi$
$127$	−20.3451	−1.80533	−0.902667	+	0.430340i	$0.858394\pi$
$128$	0	0
$129$	−17.3942	−1.53147
$130$	0	0
$131$	19.8675	1.73583	0.867917	−	0.496710i	$-0.165459\pi$
$131$	19.8675	1.73583	0.867917	+	0.496710i	$0.165459\pi$
$132$	0	0
$133$	2.14937	0.186374
$134$	0	0
$135$	−13.3232	−1.14668
$136$	0	0
$137$	18.2376	1.55814	0.779072	−	0.626934i	$-0.215690\pi$
$137$	18.2376	1.55814	0.779072	+	0.626934i	$0.215690\pi$
$138$	0	0
$139$	−3.79729	−0.322082	−0.161041	−	0.986948i	$-0.551485\pi$
$139$	−3.79729	−0.322082	−0.161041	+	0.986948i	$0.551485\pi$
$140$	0	0
$141$	−6.86582	−0.578206
$142$	0	0
$143$	−18.0023	−1.50543
$144$	0	0
$145$	−6.37515	−0.529428
$146$	0	0
$147$	11.5192	0.950086
$148$	0	0
$149$	6.47248	0.530246	0.265123	−	0.964215i	$-0.414587\pi$
$149$	6.47248	0.530246	0.265123	+	0.964215i	$0.414587\pi$
$150$	0	0
$151$	20.4636	1.66530	0.832652	−	0.553797i	$-0.186822\pi$
$151$	20.4636	1.66530	0.832652	+	0.553797i	$0.186822\pi$
$152$	0	0
$153$	−0.0159097	−0.00128622
$154$	0	0
$155$	5.13977	0.412836
$156$	0	0
$157$	−14.4827	−1.15585	−0.577924	−	0.816090i	$-0.696137\pi$
$157$	−14.4827	−1.15585	−0.577924	+	0.816090i	$0.696137\pi$
$158$	0	0
$159$	−14.7640	−1.17086
$160$	0	0
$161$	2.39915	0.189079
$162$	0	0
$163$	3.01751	0.236349	0.118175	−	0.992993i	$-0.462296\pi$
$163$	3.01751	0.236349	0.118175	+	0.992993i	$0.462296\pi$
$164$	0	0
$165$	−15.8538	−1.23422
$166$	0	0
$167$	21.4972	1.66351	0.831753	−	0.555146i	$-0.187338\pi$
$167$	21.4972	1.66351	0.831753	+	0.555146i	$0.187338\pi$
$168$	0	0
$169$	13.2897	1.02229
$170$	0	0
$171$	−0.167174	−0.0127841
$172$	0	0
$173$	−9.82213	−0.746763	−0.373381	−	0.927678i	$-0.621802\pi$
$173$	−9.82213	−0.746763	−0.373381	+	0.927678i	$0.621802\pi$
$174$	0	0
$175$	−1.07113	−0.0809695
$176$	0	0
$177$	20.8492	1.56712
$178$	0	0
$179$	−6.49532	−0.485483	−0.242742	−	0.970091i	$-0.578047\pi$
$179$	−6.49532	−0.485483	−0.242742	+	0.970091i	$0.578047\pi$
$180$	0	0
$181$	4.08174	0.303393	0.151696	−	0.988427i	$-0.451526\pi$
$181$	4.08174	0.303393	0.151696	+	0.988427i	$0.451526\pi$
$182$	0	0
$183$	−1.02486	−0.0757596
$184$	0	0
$185$	20.0997	1.47776
$186$	0	0
$187$	1.13063	0.0826798
$188$	0	0
$189$	−3.27293	−0.238070
$190$	0	0
$191$	−23.4882	−1.69955	−0.849773	−	0.527148i	$-0.823261\pi$
$191$	−23.4882	−1.69955	−0.849773	+	0.527148i	$0.823261\pi$
$192$	0	0
$193$	9.86940	0.710415	0.355207	−	0.934788i	$-0.384410\pi$
$193$	9.86940	0.710415	0.355207	+	0.934788i	$0.384410\pi$
$194$	0	0
$195$	23.1522	1.65796
$196$	0	0
$197$	−1.66586	−0.118687	−0.0593437	−	0.998238i	$-0.518901\pi$
$197$	−1.66586	−0.118687	−0.0593437	+	0.998238i	$0.518901\pi$
$198$	0	0
$199$	−12.7759	−0.905663	−0.452831	−	0.891596i	$-0.649586\pi$
$199$	−12.7759	−0.905663	−0.452831	+	0.891596i	$0.649586\pi$
$200$	0	0
$201$	−22.5442	−1.59014
$202$	0	0
$203$	−1.56610	−0.109918
$204$	0	0
$205$	−27.8174	−1.94285
$206$	0	0
$207$	−0.186601	−0.0129697
$208$	0	0
$209$	11.8803	0.821780
$210$	0	0
$211$	−16.3237	−1.12377	−0.561885	−	0.827215i	$-0.689924\pi$
$211$	−16.3237	−1.12377	−0.561885	+	0.827215i	$0.689924\pi$
$212$	0	0
$213$	−0.116489	−0.00798171
$214$	0	0
$215$	−25.7565	−1.75658
$216$	0	0
$217$	1.26262	0.0857120
$218$	0	0
$219$	4.91089	0.331847
$220$	0	0
$221$	−1.65111	−0.111066
$222$	0	0
$223$	−8.02490	−0.537387	−0.268694	−	0.963226i	$-0.586592\pi$
$223$	−8.02490	−0.537387	−0.268694	+	0.963226i	$0.586592\pi$
$224$	0	0
$225$	0.0833102	0.00555401
$226$	0	0
$227$	−21.8320	−1.44904	−0.724520	−	0.689254i	$-0.757938\pi$
$227$	−21.8320	−1.44904	−0.724520	+	0.689254i	$0.757938\pi$
$228$	0	0
$229$	13.0772	0.864163	0.432082	−	0.901834i	$-0.357779\pi$
$229$	13.0772	0.864163	0.432082	+	0.901834i	$0.357779\pi$
$230$	0	0
$231$	−3.89460	−0.256246
$232$	0	0
$233$	−5.66551	−0.371160	−0.185580	−	0.982629i	$-0.559416\pi$
$233$	−5.66551	−0.371160	−0.185580	+	0.982629i	$0.559416\pi$
$234$	0	0
$235$	−10.1666	−0.663196
$236$	0	0
$237$	−7.67170	−0.498331
$238$	0	0
$239$	3.87186	0.250450	0.125225	−	0.992128i	$-0.460035\pi$
$239$	3.87186	0.250450	0.125225	+	0.992128i	$0.460035\pi$
$240$	0	0
$241$	−18.6798	−1.20327	−0.601636	−	0.798771i	$-0.705484\pi$
$241$	−18.6798	−1.20327	−0.601636	+	0.798771i	$0.705484\pi$
$242$	0	0
$243$	0.513387	0.0329338
$244$	0	0
$245$	17.0571	1.08974
$246$	0	0
$247$	−17.3494	−1.10392
$248$	0	0
$249$	−29.1759	−1.84895
$250$	0	0
$251$	18.0304	1.13807	0.569035	−	0.822313i	$-0.307317\pi$
$251$	18.0304	1.13807	0.569035	+	0.822313i	$0.307317\pi$
$252$	0	0
$253$	13.2609	0.833708
$254$	0	0
$255$	−1.45406	−0.0910570
$256$	0	0
$257$	−0.890250	−0.0555323	−0.0277661	−	0.999614i	$-0.508839\pi$
$257$	−0.890250	−0.0555323	−0.0277661	+	0.999614i	$0.508839\pi$
$258$	0	0
$259$	4.93762	0.306808
$260$	0	0
$261$	0.121808	0.00753973
$262$	0	0
$263$	−7.74170	−0.477373	−0.238687	−	0.971097i	$-0.576717\pi$
$263$	−7.74170	−0.477373	−0.238687	+	0.971097i	$0.576717\pi$
$264$	0	0
$265$	−21.8619	−1.34296
$266$	0	0
$267$	23.7762	1.45508
$268$	0	0
$269$	−27.9359	−1.70328	−0.851640	−	0.524127i	$-0.824392\pi$
$269$	−27.9359	−1.70328	−0.851640	+	0.524127i	$0.824392\pi$
$270$	0	0
$271$	−0.123443	−0.00749865	−0.00374932	−	0.999993i	$-0.501193\pi$
$271$	−0.123443	−0.00749865	−0.00374932	+	0.999993i	$0.501193\pi$
$272$	0	0
$273$	5.68748	0.344222
$274$	0	0
$275$	−5.92048	−0.357018
$276$	0	0
$277$	25.7507	1.54721	0.773606	−	0.633667i	$-0.218451\pi$
$277$	25.7507	1.54721	0.773606	+	0.633667i	$0.218451\pi$
$278$	0	0
$279$	−0.0982040	−0.00587932
$280$	0	0
$281$	−8.28626	−0.494317	−0.247158	−	0.968975i	$-0.579497\pi$
$281$	−8.28626	−0.494317	−0.247158	+	0.968975i	$0.579497\pi$
$282$	0	0
$283$	18.3897	1.09316	0.546578	−	0.837408i	$-0.315930\pi$
$283$	18.3897	1.09316	0.546578	+	0.837408i	$0.315930\pi$
$284$	0	0
$285$	−15.2789	−0.905043
$286$	0	0
$287$	−6.83351	−0.403369
$288$	0	0
$289$	−16.8963	−0.993900
$290$	0	0
$291$	22.1865	1.30059
$292$	0	0
$293$	−3.18472	−0.186053	−0.0930267	−	0.995664i	$-0.529654\pi$
$293$	−3.18472	−0.186053	−0.0930267	+	0.995664i	$0.529654\pi$
$294$	0	0
$295$	30.8726	1.79747
$296$	0	0
$297$	−18.0906	−1.04972
$298$	0	0
$299$	−19.3656	−1.11994
$300$	0	0
$301$	−6.32726	−0.364697
$302$	0	0
$303$	−15.9895	−0.918573
$304$	0	0
$305$	−1.51756	−0.0868955
$306$	0	0
$307$	1.54810	0.0883547	0.0441774	−	0.999024i	$-0.485933\pi$
$307$	1.54810	0.0883547	0.0441774	+	0.999024i	$0.485933\pi$
$308$	0	0
$309$	2.07455	0.118017
$310$	0	0
$311$	0.471497	0.0267361	0.0133681	−	0.999911i	$-0.495745\pi$
$311$	0.471497	0.0267361	0.0133681	+	0.999911i	$0.495745\pi$
$312$	0	0
$313$	−12.1078	−0.684373	−0.342187	−	0.939632i	$-0.611168\pi$
$313$	−12.1078	−0.684373	−0.342187	+	0.939632i	$0.611168\pi$
$314$	0	0
$315$	0.0811497	0.00457227
$316$	0	0
$317$	1.31831	0.0740439	0.0370219	−	0.999314i	$-0.488213\pi$
$317$	1.31831	0.0740439	0.0370219	+	0.999314i	$0.488213\pi$
$318$	0	0
$319$	−8.65636	−0.484663
$320$	0	0
$321$	−18.3941	−1.02666
$322$	0	0
$323$	1.08963	0.0606284
$324$	0	0
$325$	8.64598	0.479592
$326$	0	0
$327$	13.1147	0.725246
$328$	0	0
$329$	−2.49749	−0.137691
$330$	0	0
$331$	−30.5415	−1.67871	−0.839357	−	0.543580i	$-0.817069\pi$
$331$	−30.5415	−1.67871	−0.839357	+	0.543580i	$0.817069\pi$
$332$	0	0
$333$	−0.384039	−0.0210452
$334$	0	0
$335$	−33.3825	−1.82388
$336$	0	0
$337$	−20.9083	−1.13895	−0.569475	−	0.822009i	$-0.692853\pi$
$337$	−20.9083	−1.13895	−0.569475	+	0.822009i	$0.692853\pi$
$338$	0	0
$339$	−24.8235	−1.34823
$340$	0	0
$341$	6.97892	0.377930
$342$	0	0
$343$	8.63667	0.466336
$344$	0	0
$345$	−17.0544	−0.918179
$346$	0	0
$347$	12.4915	0.670577	0.335288	−	0.942116i	$-0.391166\pi$
$347$	12.4915	0.670577	0.335288	+	0.942116i	$0.391166\pi$
$348$	0	0
$349$	−22.3844	−1.19821	−0.599106	−	0.800670i	$-0.704477\pi$
$349$	−22.3844	−1.19821	−0.599106	+	0.800670i	$0.704477\pi$
$350$	0	0
$351$	26.4186	1.41012
$352$	0	0
$353$	−2.03469	−0.108296	−0.0541478	−	0.998533i	$-0.517244\pi$
$353$	−2.03469	−0.108296	−0.0541478	+	0.998533i	$0.517244\pi$
$354$	0	0
$355$	−0.172492	−0.00915494
$356$	0	0
$357$	−0.357200	−0.0189050
$358$	0	0
$359$	13.8523	0.731098	0.365549	−	0.930792i	$-0.380881\pi$
$359$	13.8523	0.731098	0.365549	+	0.930792i	$0.380881\pi$
$360$	0	0
$361$	−7.55053	−0.397396
$362$	0	0
$363$	−2.31799	−0.121663
$364$	0	0
$365$	7.27183	0.380625
$366$	0	0
$367$	−17.9620	−0.937608	−0.468804	−	0.883302i	$-0.655315\pi$
$367$	−17.9620	−0.937608	−0.468804	+	0.883302i	$0.655315\pi$
$368$	0	0
$369$	0.531498	0.0276687
$370$	0	0
$371$	−5.37051	−0.278823
$372$	0	0
$373$	−2.63034	−0.136194	−0.0680969	−	0.997679i	$-0.521693\pi$
$373$	−2.63034	−0.136194	−0.0680969	+	0.997679i	$0.521693\pi$
$374$	0	0
$375$	−14.9630	−0.772686
$376$	0	0
$377$	12.6413	0.651061
$378$	0	0
$379$	−34.5832	−1.77642	−0.888209	−	0.459439i	$-0.848051\pi$
$379$	−34.5832	−1.77642	−0.888209	+	0.459439i	$0.848051\pi$
$380$	0	0
$381$	35.5277	1.82014
$382$	0	0
$383$	1.34423	0.0686872	0.0343436	−	0.999410i	$-0.489066\pi$
$383$	1.34423	0.0686872	0.0343436	+	0.999410i	$0.489066\pi$
$384$	0	0
$385$	−5.76695	−0.293911
$386$	0	0
$387$	0.492122	0.0250160
$388$	0	0
$389$	10.2656	0.520485	0.260242	−	0.965543i	$-0.416198\pi$
$389$	10.2656	0.520485	0.260242	+	0.965543i	$0.416198\pi$
$390$	0	0
$391$	1.21625	0.0615084
$392$	0	0
$393$	−34.6938	−1.75007
$394$	0	0
$395$	−11.3599	−0.571580
$396$	0	0
$397$	−33.8549	−1.69913	−0.849564	−	0.527486i	$-0.823135\pi$
$397$	−33.8549	−1.69913	−0.849564	+	0.527486i	$0.823135\pi$
$398$	0	0
$399$	−3.75335	−0.187903
$400$	0	0
$401$	−10.4030	−0.519499	−0.259750	−	0.965676i	$-0.583640\pi$
$401$	−10.4030	−0.519499	−0.259750	+	0.965676i	$0.583640\pi$
$402$	0	0
$403$	−10.1917	−0.507683
$404$	0	0
$405$	23.6489	1.17513
$406$	0	0
$407$	27.2919	1.35281
$408$	0	0
$409$	15.9724	0.789787	0.394893	−	0.918727i	$-0.370782\pi$
$409$	15.9724	0.789787	0.394893	+	0.918727i	$0.370782\pi$
$410$	0	0
$411$	−31.8475	−1.57092
$412$	0	0
$413$	7.58404	0.373186
$414$	0	0
$415$	−43.2024	−2.12072
$416$	0	0
$417$	6.63104	0.324723
$418$	0	0
$419$	1.14327	0.0558524	0.0279262	−	0.999610i	$-0.491110\pi$
$419$	1.14327	0.0558524	0.0279262	+	0.999610i	$0.491110\pi$
$420$	0	0
$421$	15.0478	0.733386	0.366693	−	0.930342i	$-0.380490\pi$
$421$	15.0478	0.733386	0.366693	+	0.930342i	$0.380490\pi$
$422$	0	0
$423$	0.194250	0.00944477
$424$	0	0
$425$	−0.543007	−0.0263397
$426$	0	0
$427$	−0.372799	−0.0180410
$428$	0	0
$429$	31.4367	1.51778
$430$	0	0
$431$	−26.5403	−1.27840	−0.639201	−	0.769040i	$-0.720735\pi$
$431$	−26.5403	−1.27840	−0.639201	+	0.769040i	$0.720735\pi$
$432$	0	0
$433$	22.3094	1.07212	0.536061	−	0.844179i	$-0.319912\pi$
$433$	22.3094	1.07212	0.536061	+	0.844179i	$0.319912\pi$
$434$	0	0
$435$	11.1326	0.533769
$436$	0	0
$437$	12.7800	0.611351
$438$	0	0
$439$	−6.48397	−0.309463	−0.154732	−	0.987957i	$-0.549451\pi$
$439$	−6.48397	−0.309463	−0.154732	+	0.987957i	$0.549451\pi$
$440$	0	0
$441$	−0.325905	−0.0155193
$442$	0	0
$443$	−12.0749	−0.573694	−0.286847	−	0.957976i	$-0.592607\pi$
$443$	−12.0749	−0.573694	−0.286847	+	0.957976i	$0.592607\pi$
$444$	0	0
$445$	35.2068	1.66896
$446$	0	0
$447$	−11.3026	−0.534594
$448$	0	0
$449$	−11.4624	−0.540943	−0.270471	−	0.962728i	$-0.587180\pi$
$449$	−11.4624	−0.540943	−0.270471	+	0.962728i	$0.587180\pi$
$450$	0	0
$451$	−37.7712	−1.77857
$452$	0	0
$453$	−35.7346	−1.67896
$454$	0	0
$455$	8.42177	0.394818
$456$	0	0
$457$	1.55079	0.0725431	0.0362715	−	0.999342i	$-0.488452\pi$
$457$	1.55079	0.0725431	0.0362715	+	0.999342i	$0.488452\pi$
$458$	0	0
$459$	−1.65921	−0.0774453
$460$	0	0
$461$	21.0083	0.978454	0.489227	−	0.872157i	$-0.337279\pi$
$461$	21.0083	0.978454	0.489227	+	0.872157i	$0.337279\pi$
$462$	0	0
$463$	7.65014	0.355532	0.177766	−	0.984073i	$-0.443113\pi$
$463$	7.65014	0.355532	0.177766	+	0.984073i	$0.443113\pi$
$464$	0	0
$465$	−8.97534	−0.416222
$466$	0	0
$467$	23.9495	1.10825	0.554124	−	0.832434i	$-0.313053\pi$
$467$	23.9495	1.10825	0.554124	+	0.832434i	$0.313053\pi$
$468$	0	0
$469$	−8.20061	−0.378669
$470$	0	0
$471$	25.2906	1.16533
$472$	0	0
$473$	−34.9729	−1.60806
$474$	0	0
$475$	−5.70577	−0.261798
$476$	0	0
$477$	0.417708	0.0191255
$478$	0	0
$479$	7.57712	0.346207	0.173104	−	0.984904i	$-0.444620\pi$
$479$	7.57712	0.346207	0.173104	+	0.984904i	$0.444620\pi$
$480$	0	0
$481$	−39.8558	−1.81727
$482$	0	0
$483$	−4.18953	−0.190630
$484$	0	0
$485$	32.8528	1.49177
$486$	0	0
$487$	−9.92582	−0.449782	−0.224891	−	0.974384i	$-0.572203\pi$
$487$	−9.92582	−0.449782	−0.224891	+	0.974384i	$0.572203\pi$
$488$	0	0
$489$	−5.26933	−0.238287
$490$	0	0
$491$	−2.77590	−0.125275	−0.0626373	−	0.998036i	$-0.519951\pi$
$491$	−2.77590	−0.125275	−0.0626373	+	0.998036i	$0.519951\pi$
$492$	0	0
$493$	−0.793933	−0.0357570
$494$	0	0
$495$	0.448543	0.0201605
$496$	0	0
$497$	−0.0423738	−0.00190072
$498$	0	0
$499$	1.66429	0.0745039	0.0372519	−	0.999306i	$-0.488140\pi$
$499$	1.66429	0.0745039	0.0372519	+	0.999306i	$0.488140\pi$
$500$	0	0
$501$	−37.5396	−1.67715
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−23.6766	−1.05359
$506$	0	0
$507$	−23.2072	−1.03067
$508$	0	0
$509$	24.0760	1.06715	0.533574	−	0.845753i	$-0.320848\pi$
$509$	24.0760	1.06715	0.533574	+	0.845753i	$0.320848\pi$
$510$	0	0
$511$	1.78637	0.0790243
$512$	0	0
$513$	−17.4345	−0.769753
$514$	0	0
$515$	3.07190	0.135364
$516$	0	0
$517$	−13.8045	−0.607121
$518$	0	0
$519$	17.1519	0.752887
$520$	0	0
$521$	−21.4959	−0.941752	−0.470876	−	0.882199i	$-0.656062\pi$
$521$	−21.4959	−0.941752	−0.470876	+	0.882199i	$0.656062\pi$
$522$	0	0
$523$	1.97588	0.0863991	0.0431995	−	0.999066i	$-0.486245\pi$
$523$	1.97588	0.0863991	0.0431995	+	0.999066i	$0.486245\pi$
$524$	0	0
$525$	1.87046	0.0816335
$526$	0	0
$527$	0.640084	0.0278825
$528$	0	0
$529$	−8.73484	−0.379776
$530$	0	0
$531$	−0.589873	−0.0255983
$532$	0	0
$533$	55.1591	2.38921
$534$	0	0
$535$	−27.2372	−1.17757
$536$	0	0
$537$	11.3425	0.489464
$538$	0	0
$539$	23.1606	0.997598
$540$	0	0
$541$	24.5700	1.05635	0.528173	−	0.849137i	$-0.322877\pi$
$541$	24.5700	1.05635	0.528173	+	0.849137i	$0.322877\pi$
$542$	0	0
$543$	−7.12775	−0.305881
$544$	0	0
$545$	19.4197	0.831850
$546$	0	0
$547$	−28.0251	−1.19827	−0.599133	−	0.800649i	$-0.704488\pi$
$547$	−28.0251	−1.19827	−0.599133	+	0.800649i	$0.704488\pi$
$548$	0	0
$549$	0.0289956	0.00123750
$550$	0	0
$551$	−8.34242	−0.355399
$552$	0	0
$553$	−2.79064	−0.118670
$554$	0	0
$555$	−35.0992	−1.48988
$556$	0	0
$557$	19.6663	0.833288	0.416644	−	0.909070i	$-0.363206\pi$
$557$	19.6663	0.833288	0.416644	+	0.909070i	$0.363206\pi$
$558$	0	0
$559$	51.0727	2.16015
$560$	0	0
$561$	−1.97437	−0.0833578
$562$	0	0
$563$	−28.9537	−1.22025	−0.610126	−	0.792304i	$-0.708881\pi$
$563$	−28.9537	−1.22025	−0.610126	+	0.792304i	$0.708881\pi$
$564$	0	0
$565$	−36.7576	−1.54640
$566$	0	0
$567$	5.80951	0.243977
$568$	0	0
$569$	13.9319	0.584054	0.292027	−	0.956410i	$-0.405670\pi$
$569$	13.9319	0.584054	0.292027	+	0.956410i	$0.405670\pi$
$570$	0	0
$571$	−1.05250	−0.0440457	−0.0220228	−	0.999757i	$-0.507011\pi$
$571$	−1.05250	−0.0440457	−0.0220228	+	0.999757i	$0.507011\pi$
$572$	0	0
$573$	41.0164	1.71348
$574$	0	0
$575$	−6.36883	−0.265598
$576$	0	0
$577$	33.3043	1.38648	0.693239	−	0.720708i	$-0.256183\pi$
$577$	33.3043	1.38648	0.693239	+	0.720708i	$0.256183\pi$
$578$	0	0
$579$	−17.2345	−0.716241
$580$	0	0
$581$	−10.6129	−0.440299
$582$	0	0
$583$	−29.6847	−1.22941
$584$	0	0
$585$	−0.655029	−0.0270821
$586$	0	0
$587$	−9.86829	−0.407308	−0.203654	−	0.979043i	$-0.565282\pi$
$587$	−9.86829	−0.407308	−0.203654	+	0.979043i	$0.565282\pi$
$588$	0	0
$589$	6.72582	0.277132
$590$	0	0
$591$	2.90901	0.119661
$592$	0	0
$593$	−19.3262	−0.793632	−0.396816	−	0.917898i	$-0.629885\pi$
$593$	−19.3262	−0.793632	−0.396816	+	0.917898i	$0.629885\pi$
$594$	0	0
$595$	−0.528926	−0.0216838
$596$	0	0
$597$	22.3101	0.913090
$598$	0	0
$599$	−25.9927	−1.06203	−0.531016	−	0.847361i	$-0.678190\pi$
$599$	−25.9927	−1.06203	−0.531016	+	0.847361i	$0.678190\pi$
$600$	0	0
$601$	21.0730	0.859586	0.429793	−	0.902927i	$-0.358586\pi$
$601$	21.0730	0.859586	0.429793	+	0.902927i	$0.358586\pi$
$602$	0	0
$603$	0.637828	0.0259744
$604$	0	0
$605$	−3.43238	−0.139546
$606$	0	0
$607$	−23.2863	−0.945164	−0.472582	−	0.881287i	$-0.656678\pi$
$607$	−23.2863	−0.945164	−0.472582	+	0.881287i	$0.656678\pi$
$608$	0	0
$609$	2.73480	0.110820
$610$	0	0
$611$	20.1594	0.815562
$612$	0	0
$613$	17.9773	0.726096	0.363048	−	0.931770i	$-0.381736\pi$
$613$	17.9773	0.726096	0.363048	+	0.931770i	$0.381736\pi$
$614$	0	0
$615$	48.5762	1.95878
$616$	0	0
$617$	−12.6311	−0.508511	−0.254255	−	0.967137i	$-0.581830\pi$
$617$	−12.6311	−0.508511	−0.254255	+	0.967137i	$0.581830\pi$
$618$	0	0
$619$	−14.9543	−0.601064	−0.300532	−	0.953772i	$-0.597164\pi$
$619$	−14.9543	−0.601064	−0.300532	+	0.953772i	$0.597164\pi$
$620$	0	0
$621$	−19.4606	−0.780925
$622$	0	0
$623$	8.64877	0.346506
$624$	0	0
$625$	−30.5878	−1.22351
$626$	0	0
$627$	−20.7461	−0.828519
$628$	0	0
$629$	2.50313	0.0998061
$630$	0	0
$631$	19.1430	0.762071	0.381035	−	0.924561i	$-0.375568\pi$
$631$	19.1430	0.762071	0.381035	+	0.924561i	$0.375568\pi$
$632$	0	0
$633$	28.5053	1.13299
$634$	0	0
$635$	52.6078	2.08768
$636$	0	0
$637$	−33.8226	−1.34010
$638$	0	0
$639$	0.00329576	0.000130378 0
$640$	0	0
$641$	12.4900	0.493326	0.246663	−	0.969101i	$-0.420666\pi$
$641$	12.4900	0.493326	0.246663	+	0.969101i	$0.420666\pi$
$642$	0	0
$643$	10.7889	0.425473	0.212737	−	0.977110i	$-0.431762\pi$
$643$	10.7889	0.425473	0.212737	+	0.977110i	$0.431762\pi$
$644$	0	0
$645$	44.9775	1.77099
$646$	0	0
$647$	37.4908	1.47392	0.736958	−	0.675939i	$-0.236262\pi$
$647$	37.4908	1.47392	0.736958	+	0.675939i	$0.236262\pi$
$648$	0	0
$649$	41.9196	1.64549
$650$	0	0
$651$	−2.20485	−0.0864149
$652$	0	0
$653$	41.5290	1.62516	0.812578	−	0.582853i	$-0.198064\pi$
$653$	41.5290	1.62516	0.812578	+	0.582853i	$0.198064\pi$
$654$	0	0
$655$	−51.3730	−2.00731
$656$	0	0
$657$	−0.138941	−0.00542059
$658$	0	0
$659$	17.9805	0.700422	0.350211	−	0.936671i	$-0.386110\pi$
$659$	17.9805	0.700422	0.350211	+	0.936671i	$0.386110\pi$
$660$	0	0
$661$	−15.5956	−0.606599	−0.303300	−	0.952895i	$-0.598088\pi$
$661$	−15.5956	−0.606599	−0.303300	+	0.952895i	$0.598088\pi$
$662$	0	0
$663$	2.88327	0.111977
$664$	0	0
$665$	−5.55780	−0.215522
$666$	0	0
$667$	−9.31189	−0.360558
$668$	0	0
$669$	14.0135	0.541794
$670$	0	0
$671$	−2.06059	−0.0795482
$672$	0	0
$673$	−44.7748	−1.72594	−0.862972	−	0.505253i	$-0.831399\pi$
$673$	−44.7748	−1.72594	−0.862972	+	0.505253i	$0.831399\pi$
$674$	0	0
$675$	8.68837	0.334415
$676$	0	0
$677$	−4.40068	−0.169132	−0.0845659	−	0.996418i	$-0.526950\pi$
$677$	−4.40068	−0.169132	−0.0845659	+	0.996418i	$0.526950\pi$
$678$	0	0
$679$	8.07049	0.309717
$680$	0	0
$681$	38.1242	1.46092
$682$	0	0
$683$	−47.9219	−1.83368	−0.916840	−	0.399255i	$-0.869269\pi$
$683$	−47.9219	−1.83368	−0.916840	+	0.399255i	$0.869269\pi$
$684$	0	0
$685$	−47.1584	−1.80183
$686$	0	0
$687$	−22.8361	−0.871250
$688$	0	0
$689$	43.3500	1.65150
$690$	0	0
$691$	−3.83809	−0.146008	−0.0730039	−	0.997332i	$-0.523259\pi$
$691$	−3.83809	−0.146008	−0.0730039	+	0.997332i	$0.523259\pi$
$692$	0	0
$693$	0.110187	0.00418567
$694$	0	0
$695$	9.81895	0.372454
$696$	0	0
$697$	−3.46425	−0.131218
$698$	0	0
$699$	9.89343	0.374204
$700$	0	0
$701$	−32.9235	−1.24351	−0.621753	−	0.783214i	$-0.713579\pi$
$701$	−32.9235	−1.24351	−0.621753	+	0.783214i	$0.713579\pi$
$702$	0	0
$703$	26.3021	0.992004
$704$	0	0
$705$	17.7535	0.668635
$706$	0	0
$707$	−5.81630	−0.218744
$708$	0	0
$709$	5.77126	0.216744	0.108372	−	0.994110i	$-0.465436\pi$
$709$	5.77126	0.216744	0.108372	+	0.994110i	$0.465436\pi$
$710$	0	0
$711$	0.217051	0.00814003
$712$	0	0
$713$	7.50742	0.281155
$714$	0	0
$715$	46.5500	1.74087
$716$	0	0
$717$	−6.76126	−0.252504
$718$	0	0
$719$	12.1263	0.452234	0.226117	−	0.974100i	$-0.427397\pi$
$719$	12.1263	0.452234	0.226117	+	0.974100i	$0.427397\pi$
$720$	0	0
$721$	0.754632	0.0281040
$722$	0	0
$723$	32.6197	1.21314
$724$	0	0
$725$	4.15739	0.154402
$726$	0	0
$727$	−1.41480	−0.0524720	−0.0262360	−	0.999656i	$-0.508352\pi$
$727$	−1.41480	−0.0524720	−0.0262360	+	0.999656i	$0.508352\pi$
$728$	0	0
$729$	26.5408	0.982993
$730$	0	0
$731$	−3.20760	−0.118638
$732$	0	0
$733$	−45.8355	−1.69297	−0.846486	−	0.532411i	$-0.821286\pi$
$733$	−45.8355	−1.69297	−0.846486	+	0.532411i	$0.821286\pi$
$734$	0	0
$735$	−29.7860	−1.09867
$736$	0	0
$737$	−45.3276	−1.66966
$738$	0	0
$739$	−24.1465	−0.888245	−0.444123	−	0.895966i	$-0.646485\pi$
$739$	−24.1465	−0.888245	−0.444123	+	0.895966i	$0.646485\pi$
$740$	0	0
$741$	30.2966	1.11297
$742$	0	0
$743$	2.04962	0.0751931	0.0375966	−	0.999293i	$-0.488030\pi$
$743$	2.04962	0.0751931	0.0375966	+	0.999293i	$0.488030\pi$
$744$	0	0
$745$	−16.7364	−0.613174
$746$	0	0
$747$	0.825455	0.0302018
$748$	0	0
$749$	−6.69100	−0.244484
$750$	0	0
$751$	4.28315	0.156294	0.0781471	−	0.996942i	$-0.475100\pi$
$751$	4.28315	0.156294	0.0781471	+	0.996942i	$0.475100\pi$
$752$	0	0
$753$	−31.4857	−1.14740
$754$	0	0
$755$	−52.9143	−1.92575
$756$	0	0
$757$	−37.8734	−1.37653	−0.688266	−	0.725458i	$-0.741628\pi$
$757$	−37.8734	−1.37653	−0.688266	+	0.725458i	$0.741628\pi$
$758$	0	0
$759$	−23.1570	−0.840545
$760$	0	0
$761$	−34.5346	−1.25188	−0.625940	−	0.779872i	$-0.715284\pi$
$761$	−34.5346	−1.25188	−0.625940	+	0.779872i	$0.715284\pi$
$762$	0	0
$763$	4.77058	0.172706
$764$	0	0
$765$	0.0411389	0.00148738
$766$	0	0
$767$	−61.2173	−2.21043
$768$	0	0
$769$	−45.2000	−1.62995	−0.814977	−	0.579493i	$-0.803251\pi$
$769$	−45.2000	−1.62995	−0.814977	+	0.579493i	$0.803251\pi$
$770$	0	0
$771$	1.55460	0.0559877
$772$	0	0
$773$	3.76139	0.135288	0.0676439	−	0.997710i	$-0.478452\pi$
$773$	3.76139	0.135288	0.0676439	+	0.997710i	$0.478452\pi$
$774$	0	0
$775$	−3.35176	−0.120399
$776$	0	0
$777$	−8.62233	−0.309325
$778$	0	0
$779$	−36.4013	−1.30421
$780$	0	0
$781$	−0.234215	−0.00838086
$782$	0	0
$783$	12.7033	0.453979
$784$	0	0
$785$	37.4492	1.33662
$786$	0	0
$787$	10.7287	0.382438	0.191219	−	0.981547i	$-0.438756\pi$
$787$	10.7287	0.382438	0.191219	+	0.981547i	$0.438756\pi$
$788$	0	0
$789$	13.5190	0.481288
$790$	0	0
$791$	−9.02974	−0.321061
$792$	0	0
$793$	3.00918	0.106859
$794$	0	0
$795$	38.1764	1.35398
$796$	0	0
$797$	−0.766076	−0.0271358	−0.0135679	−	0.999908i	$-0.504319\pi$
$797$	−0.766076	−0.0271358	−0.0135679	+	0.999908i	$0.504319\pi$
$798$	0	0
$799$	−1.26610	−0.0447915
$800$	0	0
$801$	−0.672686	−0.0237682
$802$	0	0
$803$	9.87389	0.348442
$804$	0	0
$805$	−6.20367	−0.218651
$806$	0	0
$807$	48.7831	1.71725
$808$	0	0
$809$	30.0576	1.05677	0.528384	−	0.849005i	$-0.322798\pi$
$809$	30.0576	1.05677	0.528384	+	0.849005i	$0.322798\pi$
$810$	0	0
$811$	13.7785	0.483829	0.241914	−	0.970298i	$-0.422225\pi$
$811$	13.7785	0.483829	0.241914	+	0.970298i	$0.422225\pi$
$812$	0	0
$813$	0.215563	0.00756014
$814$	0	0
$815$	−7.80260	−0.273313
$816$	0	0
$817$	−33.7046	−1.17917
$818$	0	0
$819$	−0.160912	−0.00562272
$820$	0	0
$821$	21.8806	0.763639	0.381819	−	0.924237i	$-0.375298\pi$
$821$	21.8806	0.763639	0.381819	+	0.924237i	$0.375298\pi$
$822$	0	0
$823$	−38.1119	−1.32850	−0.664249	−	0.747511i	$-0.731249\pi$
$823$	−38.1119	−1.32850	−0.664249	+	0.747511i	$0.731249\pi$
$824$	0	0
$825$	10.3387	0.359946
$826$	0	0
$827$	12.0487	0.418975	0.209488	−	0.977811i	$-0.432820\pi$
$827$	12.0487	0.418975	0.209488	+	0.977811i	$0.432820\pi$
$828$	0	0
$829$	−5.44538	−0.189126	−0.0945629	−	0.995519i	$-0.530145\pi$
$829$	−5.44538	−0.189126	−0.0945629	+	0.995519i	$0.530145\pi$
$830$	0	0
$831$	−44.9673	−1.55990
$832$	0	0
$833$	2.12421	0.0735997
$834$	0	0
$835$	−55.5871	−1.92367
$836$	0	0
$837$	−10.2416	−0.354003
$838$	0	0
$839$	−40.8623	−1.41072	−0.705361	−	0.708848i	$-0.749215\pi$
$839$	−40.8623	−1.41072	−0.705361	+	0.708848i	$0.749215\pi$
$840$	0	0
$841$	−22.9215	−0.790395
$842$	0	0
$843$	14.4699	0.498370
$844$	0	0
$845$	−34.3642	−1.18217
$846$	0	0
$847$	−0.843185	−0.0289722
$848$	0	0
$849$	−32.1131	−1.10212
$850$	0	0
$851$	29.3587	1.00640
$852$	0	0
$853$	33.6213	1.15117	0.575586	−	0.817741i	$-0.304774\pi$
$853$	33.6213	1.15117	0.575586	+	0.817741i	$0.304774\pi$
$854$	0	0
$855$	0.432276	0.0147835
$856$	0	0
$857$	−38.0197	−1.29873	−0.649365	−	0.760477i	$-0.724965\pi$
$857$	−38.0197	−1.29873	−0.649365	+	0.760477i	$0.724965\pi$
$858$	0	0
$859$	17.6508	0.602237	0.301118	−	0.953587i	$-0.402640\pi$
$859$	17.6508	0.602237	0.301118	+	0.953587i	$0.402640\pi$
$860$	0	0
$861$	11.9330	0.406677
$862$	0	0
$863$	−21.0000	−0.714850	−0.357425	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714850	−0.357425	+	0.933942i	$0.616345\pi$
$864$	0	0
$865$	25.3978	0.863553
$866$	0	0
$867$	29.5052	1.00205
$868$	0	0
$869$	−15.4248	−0.523251
$870$	0	0
$871$	66.1942	2.24290
$872$	0	0
$873$	−0.627708	−0.0212447
$874$	0	0
$875$	−5.44290	−0.184004
$876$	0	0
$877$	13.3184	0.449731	0.224865	−	0.974390i	$-0.427806\pi$
$877$	13.3184	0.449731	0.224865	+	0.974390i	$0.427806\pi$
$878$	0	0
$879$	5.56133	0.187579
$880$	0	0
$881$	−23.1252	−0.779108	−0.389554	−	0.921004i	$-0.627371\pi$
$881$	−23.1252	−0.779108	−0.389554	+	0.921004i	$0.627371\pi$
$882$	0	0
$883$	50.9512	1.71465	0.857323	−	0.514780i	$-0.172126\pi$
$883$	50.9512	1.71465	0.857323	+	0.514780i	$0.172126\pi$
$884$	0	0
$885$	−53.9114	−1.81221
$886$	0	0
$887$	−24.3607	−0.817952	−0.408976	−	0.912545i	$-0.634114\pi$
$887$	−24.3607	−0.817952	−0.408976	+	0.912545i	$0.634114\pi$
$888$	0	0
$889$	12.9234	0.433438
$890$	0	0
$891$	32.1112	1.07577
$892$	0	0
$893$	−13.3039	−0.445196
$894$	0	0
$895$	16.7955	0.561410
$896$	0	0
$897$	33.8173	1.12913
$898$	0	0
$899$	−4.90063	−0.163445
$900$	0	0
$901$	−2.72258	−0.0907023
$902$	0	0
$903$	11.0490	0.367688
$904$	0	0
$905$	−10.5545	−0.350842
$906$	0	0
$907$	30.6192	1.01669	0.508347	−	0.861153i	$-0.330257\pi$
$907$	30.6192	1.01669	0.508347	+	0.861153i	$0.330257\pi$
$908$	0	0
$909$	0.452381	0.0150045
$910$	0	0
$911$	52.5031	1.73951	0.869753	−	0.493487i	$-0.164278\pi$
$911$	52.5031	1.73951	0.869753	+	0.493487i	$0.164278\pi$
$912$	0	0
$913$	−58.6614	−1.94141
$914$	0	0
$915$	2.65005	0.0876081
$916$	0	0
$917$	−12.6201	−0.416752
$918$	0	0
$919$	21.0907	0.695719	0.347860	−	0.937547i	$-0.386909\pi$
$919$	21.0907	0.695719	0.347860	+	0.937547i	$0.386909\pi$
$920$	0	0
$921$	−2.70338	−0.0890793
$922$	0	0
$923$	0.342036	0.0112582
$924$	0	0
$925$	−13.1075	−0.430971
$926$	0	0
$927$	−0.0586939	−0.00192776
$928$	0	0
$929$	36.4744	1.19668	0.598342	−	0.801240i	$-0.295826\pi$
$929$	36.4744	1.19668	0.598342	+	0.801240i	$0.295826\pi$
$930$	0	0
$931$	22.3206	0.731529
$932$	0	0
$933$	−0.823354	−0.0269554
$934$	0	0
$935$	−2.92356	−0.0956105
$936$	0	0
$937$	35.4964	1.15962	0.579809	−	0.814752i	$-0.303127\pi$
$937$	35.4964	1.15962	0.579809	+	0.814752i	$0.303127\pi$
$938$	0	0
$939$	21.1433	0.689986
$940$	0	0
$941$	10.3782	0.338321	0.169160	−	0.985589i	$-0.445894\pi$
$941$	10.3782	0.338321	0.169160	+	0.985589i	$0.445894\pi$
$942$	0	0
$943$	−40.6315	−1.32314
$944$	0	0
$945$	8.46306	0.275303
$946$	0	0
$947$	−10.9336	−0.355294	−0.177647	−	0.984094i	$-0.556848\pi$
$947$	−10.9336	−0.355294	−0.177647	+	0.984094i	$0.556848\pi$
$948$	0	0
$949$	−14.4193	−0.468071
$950$	0	0
$951$	−2.30211	−0.0746511
$952$	0	0
$953$	−46.0986	−1.49328	−0.746640	−	0.665228i	$-0.768334\pi$
$953$	−46.0986	−1.49328	−0.746640	+	0.665228i	$0.768334\pi$
$954$	0	0
$955$	60.7353	1.96535
$956$	0	0
$957$	15.1162	0.488638
$958$	0	0
$959$	−11.5848	−0.374092
$960$	0	0
$961$	−27.0490	−0.872549
$962$	0	0
$963$	0.520413	0.0167701
$964$	0	0
$965$	−25.5201	−0.821520
$966$	0	0
$967$	−27.3411	−0.879231	−0.439615	−	0.898186i	$-0.644885\pi$
$967$	−27.3411	−0.879231	−0.439615	+	0.898186i	$0.644885\pi$
$968$	0	0
$969$	−1.90276	−0.0611256
$970$	0	0
$971$	−18.5972	−0.596812	−0.298406	−	0.954439i	$-0.596455\pi$
$971$	−18.5972	−0.596812	−0.298406	+	0.954439i	$0.596455\pi$
$972$	0	0
$973$	2.41209	0.0773279
$974$	0	0
$975$	−15.0981	−0.483525
$976$	0	0
$977$	−43.2694	−1.38431	−0.692155	−	0.721749i	$-0.743339\pi$
$977$	−43.2694	−1.38431	−0.692155	+	0.721749i	$0.743339\pi$
$978$	0	0
$979$	47.8048	1.52785
$980$	0	0
$981$	−0.371047	−0.0118466
$982$	0	0
$983$	−54.4357	−1.73623	−0.868115	−	0.496363i	$-0.834669\pi$
$983$	−54.4357	−1.73623	−0.868115	+	0.496363i	$0.834669\pi$
$984$	0	0
$985$	4.30753	0.137249
$986$	0	0
$987$	4.36125	0.138820
$988$	0	0
$989$	−37.6214	−1.19629
$990$	0	0
$991$	36.0277	1.14446	0.572229	−	0.820094i	$-0.306079\pi$
$991$	36.0277	1.14446	0.572229	+	0.820094i	$0.306079\pi$
$992$	0	0
$993$	53.3333	1.69248
$994$	0	0
$995$	33.0358	1.04730
$996$	0	0
$997$	−56.7052	−1.79587	−0.897935	−	0.440128i	$-0.854933\pi$
$997$	−56.7052	−1.79587	−0.897935	+	0.440128i	$0.854933\pi$
$998$	0	0
$999$	−40.0512	−1.26716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.5		12
4.3	odd	2		1006.2.a.j.1.8	✓	12
12.11	even	2		9054.2.a.bi.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.8	✓	12	4.3	odd	2
8048.2.a.q.1.5		12	1.1	even	1	trivial
9054.2.a.bi.1.11		12	12.11	even	2

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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74625 q^{3} -2.58578 q^{5} -0.635213 q^{7} +0.0494057 q^{9} +O(q^{10})\)	\(q-1.74625 q^{3} -2.58578 q^{5} -0.635213 q^{7} +0.0494057 q^{9} -3.51104 q^{11} +5.12735 q^{13} +4.51543 q^{15} -0.322021 q^{17} -3.38371 q^{19} +1.10924 q^{21} -3.77692 q^{23} +1.68625 q^{25} +5.15249 q^{27} +2.46547 q^{29} -1.98771 q^{31} +6.13117 q^{33} +1.64252 q^{35} -7.77317 q^{37} -8.95366 q^{39} +10.7578 q^{41} +9.96085 q^{43} -0.127752 q^{45} +3.93174 q^{47} -6.59651 q^{49} +0.562331 q^{51} +8.45466 q^{53} +9.07877 q^{55} +5.90881 q^{57} -11.9394 q^{59} +0.586889 q^{61} -0.0313831 q^{63} -13.2582 q^{65} +12.9100 q^{67} +6.59547 q^{69} +0.0667081 q^{71} -2.81224 q^{73} -2.94462 q^{75} +2.23026 q^{77} +4.39323 q^{79} -9.14578 q^{81} +16.7077 q^{83} +0.832675 q^{85} -4.30534 q^{87} -13.6156 q^{89} -3.25696 q^{91} +3.47104 q^{93} +8.74951 q^{95} -12.7052 q^{97} -0.173465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more