LMFDB - Embedded newform 8048.2.a.v.1.17

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:	$28$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+0.596863 q^{3} +3.74289 q^{5} -1.10831 q^{7} -2.64375 q^{9} +O(q^{10})$	$q+0.596863 q^{3} +3.74289 q^{5} -1.10831 q^{7} -2.64375 q^{9} +4.05435 q^{11} -6.28166 q^{13} +2.23399 q^{15} +3.42667 q^{17} -4.55414 q^{19} -0.661511 q^{21} +7.12930 q^{23} +9.00922 q^{25} -3.36855 q^{27} -7.58996 q^{29} -10.1526 q^{31} +2.41989 q^{33} -4.14829 q^{35} -10.9132 q^{37} -3.74929 q^{39} +4.41002 q^{41} -9.01601 q^{43} -9.89528 q^{45} -0.789928 q^{47} -5.77164 q^{49} +2.04525 q^{51} -12.5917 q^{53} +15.1750 q^{55} -2.71820 q^{57} -3.67098 q^{59} -12.4825 q^{61} +2.93010 q^{63} -23.5116 q^{65} +8.33923 q^{67} +4.25522 q^{69} +16.5483 q^{71} +13.9080 q^{73} +5.37727 q^{75} -4.49349 q^{77} -1.08841 q^{79} +5.92070 q^{81} +8.85350 q^{83} +12.8256 q^{85} -4.53017 q^{87} -13.8450 q^{89} +6.96204 q^{91} -6.05974 q^{93} -17.0456 q^{95} +7.23701 q^{97} -10.7187 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	$ 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) $ 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * 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
$3$	0.596863	0.344599	0.172300	−	0.985045i	$-0.444880\pi$
$3$	0.596863	0.344599	0.172300	+	0.985045i	$0.444880\pi$
$4$	0	0
$5$	3.74289	1.67387	0.836936	−	0.547301i	$-0.184345\pi$
$5$	3.74289	1.67387	0.836936	+	0.547301i	$0.184345\pi$
$6$	0	0
$7$	−1.10831	−0.418903	−0.209451	−	0.977819i	$-0.567168\pi$
$7$	−1.10831	−0.418903	−0.209451	+	0.977819i	$0.567168\pi$
$8$	0	0
$9$	−2.64375	−0.881251
$10$	0	0
$11$	4.05435	1.22243	0.611216	−	0.791464i	$-0.290681\pi$
$11$	4.05435	1.22243	0.611216	+	0.791464i	$0.290681\pi$
$12$	0	0
$13$	−6.28166	−1.74222	−0.871110	−	0.491088i	$-0.836599\pi$
$13$	−6.28166	−1.74222	−0.871110	+	0.491088i	$0.836599\pi$
$14$	0	0
$15$	2.23399	0.576815
$16$	0	0
$17$	3.42667	0.831089	0.415545	−	0.909573i	$-0.363591\pi$
$17$	3.42667	0.831089	0.415545	+	0.909573i	$0.363591\pi$
$18$	0	0
$19$	−4.55414	−1.04479	−0.522395	−	0.852704i	$-0.674961\pi$
$19$	−4.55414	−1.04479	−0.522395	+	0.852704i	$0.674961\pi$
$20$	0	0
$21$	−0.661511	−0.144353
$22$	0	0
$23$	7.12930	1.48656	0.743281	−	0.668980i	$-0.233269\pi$
$23$	7.12930	1.48656	0.743281	+	0.668980i	$0.233269\pi$
$24$	0	0
$25$	9.00922	1.80184
$26$	0	0
$27$	−3.36855	−0.648278
$28$	0	0
$29$	−7.58996	−1.40942	−0.704710	−	0.709495i	$-0.748923\pi$
$29$	−7.58996	−1.40942	−0.704710	+	0.709495i	$0.748923\pi$
$30$	0	0
$31$	−10.1526	−1.82347	−0.911735	−	0.410779i	$-0.865257\pi$
$31$	−10.1526	−1.82347	−0.911735	+	0.410779i	$0.865257\pi$
$32$	0	0
$33$	2.41989	0.421249
$34$	0	0
$35$	−4.14829	−0.701189
$36$	0	0
$37$	−10.9132	−1.79413	−0.897063	−	0.441902i	$-0.854304\pi$
$37$	−10.9132	−1.79413	−0.897063	+	0.441902i	$0.854304\pi$
$38$	0	0
$39$	−3.74929	−0.600368
$40$	0	0
$41$	4.41002	0.688730	0.344365	−	0.938836i	$-0.388094\pi$
$41$	4.41002	0.688730	0.344365	+	0.938836i	$0.388094\pi$
$42$	0	0
$43$	−9.01601	−1.37493	−0.687464	−	0.726218i	$-0.741276\pi$
$43$	−9.01601	−1.37493	−0.687464	+	0.726218i	$0.741276\pi$
$44$	0	0
$45$	−9.89528	−1.47510
$46$	0	0
$47$	−0.789928	−0.115223	−0.0576115	−	0.998339i	$-0.518348\pi$
$47$	−0.789928	−0.115223	−0.0576115	+	0.998339i	$0.518348\pi$
$48$	0	0
$49$	−5.77164	−0.824521
$50$	0	0
$51$	2.04525	0.286393
$52$	0	0
$53$	−12.5917	−1.72960	−0.864799	−	0.502117i	$-0.832555\pi$
$53$	−12.5917	−1.72960	−0.864799	+	0.502117i	$0.832555\pi$
$54$	0	0
$55$	15.1750	2.04620
$56$	0	0
$57$	−2.71820	−0.360034
$58$	0	0
$59$	−3.67098	−0.477921	−0.238961	−	0.971029i	$-0.576807\pi$
$59$	−3.67098	−0.477921	−0.238961	+	0.971029i	$0.576807\pi$
$60$	0	0
$61$	−12.4825	−1.59823	−0.799113	−	0.601181i	$-0.794697\pi$
$61$	−12.4825	−1.59823	−0.799113	+	0.601181i	$0.794697\pi$
$62$	0	0
$63$	2.93010	0.369158
$64$	0	0
$65$	−23.5116	−2.91625
$66$	0	0
$67$	8.33923	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$67$	8.33923	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$68$	0	0
$69$	4.25522	0.512268
$70$	0	0
$71$	16.5483	1.96392	0.981961	−	0.189084i	$-0.0605517\pi$
$71$	16.5483	1.96392	0.981961	+	0.189084i	$0.0605517\pi$
$72$	0	0
$73$	13.9080	1.62780	0.813902	−	0.581002i	$-0.197339\pi$
$73$	13.9080	1.62780	0.813902	+	0.581002i	$0.197339\pi$
$74$	0	0
$75$	5.37727	0.620914
$76$	0	0
$77$	−4.49349	−0.512080
$78$	0	0
$79$	−1.08841	−0.122456	−0.0612280	−	0.998124i	$-0.519502\pi$
$79$	−1.08841	−0.122456	−0.0612280	+	0.998124i	$0.519502\pi$
$80$	0	0
$81$	5.92070	0.657855
$82$	0	0
$83$	8.85350	0.971798	0.485899	−	0.874015i	$-0.338492\pi$
$83$	8.85350	0.971798	0.485899	+	0.874015i	$0.338492\pi$
$84$	0	0
$85$	12.8256	1.39114
$86$	0	0
$87$	−4.53017	−0.485685
$88$	0	0
$89$	−13.8450	−1.46757	−0.733784	−	0.679383i	$-0.762248\pi$
$89$	−13.8450	−1.46757	−0.733784	+	0.679383i	$0.762248\pi$
$90$	0	0
$91$	6.96204	0.729820
$92$	0	0
$93$	−6.05974	−0.628366
$94$	0	0
$95$	−17.0456	−1.74884
$96$	0	0
$97$	7.23701	0.734807	0.367403	−	0.930062i	$-0.380247\pi$
$97$	7.23701	0.734807	0.367403	+	0.930062i	$0.380247\pi$
$98$	0	0
$99$	−10.7187	−1.07727
$100$	0	0
$101$	−12.3018	−1.22407	−0.612036	−	0.790830i	$-0.709649\pi$
$101$	−12.3018	−1.22407	−0.612036	+	0.790830i	$0.709649\pi$
$102$	0	0
$103$	4.52046	0.445414	0.222707	−	0.974885i	$-0.428511\pi$
$103$	4.52046	0.445414	0.222707	+	0.974885i	$0.428511\pi$
$104$	0	0
$105$	−2.47596	−0.241629
$106$	0	0
$107$	−16.6723	−1.61177	−0.805886	−	0.592071i	$-0.798310\pi$
$107$	−16.6723	−1.61177	−0.805886	+	0.592071i	$0.798310\pi$
$108$	0	0
$109$	−2.13728	−0.204714	−0.102357	−	0.994748i	$-0.532638\pi$
$109$	−2.13728	−0.204714	−0.102357	+	0.994748i	$0.532638\pi$
$110$	0	0
$111$	−6.51372	−0.618255
$112$	0	0
$113$	3.98985	0.375334	0.187667	−	0.982233i	$-0.439907\pi$
$113$	3.98985	0.375334	0.187667	+	0.982233i	$0.439907\pi$
$114$	0	0
$115$	26.6842	2.48831
$116$	0	0
$117$	16.6072	1.53533
$118$	0	0
$119$	−3.79782	−0.348145
$120$	0	0
$121$	5.43776	0.494342
$122$	0	0
$123$	2.63218	0.237336
$124$	0	0
$125$	15.0061	1.34218
$126$	0	0
$127$	10.6610	0.946008	0.473004	−	0.881060i	$-0.343170\pi$
$127$	10.6610	0.946008	0.473004	+	0.881060i	$0.343170\pi$
$128$	0	0
$129$	−5.38132	−0.473799
$130$	0	0
$131$	8.97324	0.783996	0.391998	−	0.919966i	$-0.371784\pi$
$131$	8.97324	0.783996	0.391998	+	0.919966i	$0.371784\pi$
$132$	0	0
$133$	5.04740	0.437665
$134$	0	0
$135$	−12.6081	−1.08513
$136$	0	0
$137$	−15.1837	−1.29723	−0.648614	−	0.761117i	$-0.724651\pi$
$137$	−15.1837	−1.29723	−0.648614	+	0.761117i	$0.724651\pi$
$138$	0	0
$139$	11.0272	0.935315	0.467657	−	0.883910i	$-0.345098\pi$
$139$	11.0272	0.935315	0.467657	+	0.883910i	$0.345098\pi$
$140$	0	0
$141$	−0.471479	−0.0397057
$142$	0	0
$143$	−25.4681	−2.12975
$144$	0	0
$145$	−28.4084	−2.35919
$146$	0	0
$147$	−3.44488	−0.284129
$148$	0	0
$149$	5.53804	0.453694	0.226847	−	0.973930i	$-0.427158\pi$
$149$	5.53804	0.453694	0.226847	+	0.973930i	$0.427158\pi$
$150$	0	0
$151$	7.05565	0.574181	0.287090	−	0.957903i	$-0.407312\pi$
$151$	7.05565	0.574181	0.287090	+	0.957903i	$0.407312\pi$
$152$	0	0
$153$	−9.05927	−0.732398
$154$	0	0
$155$	−38.0002	−3.05225
$156$	0	0
$157$	−14.7281	−1.17543	−0.587714	−	0.809069i	$-0.699972\pi$
$157$	−14.7281	−1.17543	−0.587714	+	0.809069i	$0.699972\pi$
$158$	0	0
$159$	−7.51551	−0.596018
$160$	0	0
$161$	−7.90149	−0.622724
$162$	0	0
$163$	−12.3357	−0.966206	−0.483103	−	0.875563i	$-0.660490\pi$
$163$	−12.3357	−0.966206	−0.483103	+	0.875563i	$0.660490\pi$
$164$	0	0
$165$	9.05739	0.705117
$166$	0	0
$167$	−8.22308	−0.636321	−0.318160	−	0.948037i	$-0.603065\pi$
$167$	−8.22308	−0.636321	−0.318160	+	0.948037i	$0.603065\pi$
$168$	0	0
$169$	26.4593	2.03533
$170$	0	0
$171$	12.0400	0.920723
$172$	0	0
$173$	−4.10789	−0.312317	−0.156158	−	0.987732i	$-0.549911\pi$
$173$	−4.10789	−0.312317	−0.156158	+	0.987732i	$0.549911\pi$
$174$	0	0
$175$	−9.98503	−0.754797
$176$	0	0
$177$	−2.19107	−0.164691
$178$	0	0
$179$	−11.4420	−0.855218	−0.427609	−	0.903964i	$-0.640644\pi$
$179$	−11.4420	−0.855218	−0.427609	+	0.903964i	$0.640644\pi$
$180$	0	0
$181$	−2.82949	−0.210315	−0.105157	−	0.994456i	$-0.533535\pi$
$181$	−2.82949	−0.210315	−0.105157	+	0.994456i	$0.533535\pi$
$182$	0	0
$183$	−7.45037	−0.550747
$184$	0	0
$185$	−40.8471	−3.00314
$186$	0	0
$187$	13.8929	1.01595
$188$	0	0
$189$	3.73340	0.271565
$190$	0	0
$191$	11.8684	0.858770	0.429385	−	0.903122i	$-0.358730\pi$
$191$	11.8684	0.858770	0.429385	+	0.903122i	$0.358730\pi$
$192$	0	0
$193$	2.58302	0.185930	0.0929649	−	0.995669i	$-0.470366\pi$
$193$	2.58302	0.185930	0.0929649	+	0.995669i	$0.470366\pi$
$194$	0	0
$195$	−14.0332	−1.00494
$196$	0	0
$197$	16.0693	1.14489	0.572444	−	0.819944i	$-0.305995\pi$
$197$	16.0693	1.14489	0.572444	+	0.819944i	$0.305995\pi$
$198$	0	0
$199$	−18.5167	−1.31261	−0.656305	−	0.754495i	$-0.727882\pi$
$199$	−18.5167	−1.31261	−0.656305	+	0.754495i	$0.727882\pi$
$200$	0	0
$201$	4.97738	0.351077
$202$	0	0
$203$	8.41204	0.590410
$204$	0	0
$205$	16.5062	1.15285
$206$	0	0
$207$	−18.8481	−1.31003
$208$	0	0
$209$	−18.4641	−1.27719
$210$	0	0
$211$	−12.0114	−0.826900	−0.413450	−	0.910527i	$-0.635676\pi$
$211$	−12.0114	−0.826900	−0.413450	+	0.910527i	$0.635676\pi$
$212$	0	0
$213$	9.87707	0.676766
$214$	0	0
$215$	−33.7459	−2.30145
$216$	0	0
$217$	11.2523	0.763856
$218$	0	0
$219$	8.30115	0.560940
$220$	0	0
$221$	−21.5252	−1.44794
$222$	0	0
$223$	4.53206	0.303489	0.151745	−	0.988420i	$-0.451511\pi$
$223$	4.53206	0.303489	0.151745	+	0.988420i	$0.451511\pi$
$224$	0	0
$225$	−23.8182	−1.58788
$226$	0	0
$227$	−8.77728	−0.582568	−0.291284	−	0.956637i	$-0.594083\pi$
$227$	−8.77728	−0.582568	−0.291284	+	0.956637i	$0.594083\pi$
$228$	0	0
$229$	−21.7649	−1.43827	−0.719133	−	0.694873i	$-0.755461\pi$
$229$	−21.7649	−1.43827	−0.719133	+	0.694873i	$0.755461\pi$
$230$	0	0
$231$	−2.68200	−0.176462
$232$	0	0
$233$	15.4379	1.01137	0.505685	−	0.862718i	$-0.331240\pi$
$233$	15.4379	1.01137	0.505685	+	0.862718i	$0.331240\pi$
$234$	0	0
$235$	−2.95662	−0.192868
$236$	0	0
$237$	−0.649634	−0.0421983
$238$	0	0
$239$	−5.89595	−0.381377	−0.190689	−	0.981651i	$-0.561072\pi$
$239$	−5.89595	−0.381377	−0.190689	+	0.981651i	$0.561072\pi$
$240$	0	0
$241$	17.6126	1.13453	0.567264	−	0.823536i	$-0.308002\pi$
$241$	17.6126	1.13453	0.567264	+	0.823536i	$0.308002\pi$
$242$	0	0
$243$	13.6395	0.874974
$244$	0	0
$245$	−21.6026	−1.38014
$246$	0	0
$247$	28.6075	1.82025
$248$	0	0
$249$	5.28433	0.334881
$250$	0	0
$251$	18.2226	1.15020	0.575101	−	0.818082i	$-0.304963\pi$
$251$	18.2226	1.15020	0.575101	+	0.818082i	$0.304963\pi$
$252$	0	0
$253$	28.9047	1.81722
$254$	0	0
$255$	7.65515	0.479384
$256$	0	0
$257$	2.00865	0.125296	0.0626479	−	0.998036i	$-0.480045\pi$
$257$	2.00865	0.125296	0.0626479	+	0.998036i	$0.480045\pi$
$258$	0	0
$259$	12.0953	0.751564
$260$	0	0
$261$	20.0660	1.24205
$262$	0	0
$263$	26.0383	1.60559	0.802794	−	0.596256i	$-0.203346\pi$
$263$	26.0383	1.60559	0.802794	+	0.596256i	$0.203346\pi$
$264$	0	0
$265$	−47.1292	−2.89513
$266$	0	0
$267$	−8.26358	−0.505723
$268$	0	0
$269$	25.7113	1.56764	0.783822	−	0.620985i	$-0.213267\pi$
$269$	25.7113	1.56764	0.783822	+	0.620985i	$0.213267\pi$
$270$	0	0
$271$	22.3375	1.35691	0.678453	−	0.734644i	$-0.262651\pi$
$271$	22.3375	1.35691	0.678453	+	0.734644i	$0.262651\pi$
$272$	0	0
$273$	4.15539	0.251496
$274$	0	0
$275$	36.5266	2.20263
$276$	0	0
$277$	12.0231	0.722396	0.361198	−	0.932489i	$-0.382368\pi$
$277$	12.0231	0.722396	0.361198	+	0.932489i	$0.382368\pi$
$278$	0	0
$279$	26.8411	1.60694
$280$	0	0
$281$	−0.165606	−0.00987924	−0.00493962	−	0.999988i	$-0.501572\pi$
$281$	−0.165606	−0.00987924	−0.00493962	+	0.999988i	$0.501572\pi$
$282$	0	0
$283$	9.10918	0.541484	0.270742	−	0.962652i	$-0.412731\pi$
$283$	9.10918	0.541484	0.270742	+	0.962652i	$0.412731\pi$
$284$	0	0
$285$	−10.1739	−0.602650
$286$	0	0
$287$	−4.88768	−0.288511
$288$	0	0
$289$	−5.25794	−0.309291
$290$	0	0
$291$	4.31950	0.253214
$292$	0	0
$293$	−4.00724	−0.234105	−0.117053	−	0.993126i	$-0.537345\pi$
$293$	−4.00724	−0.234105	−0.117053	+	0.993126i	$0.537345\pi$
$294$	0	0
$295$	−13.7401	−0.799979
$296$	0	0
$297$	−13.6573	−0.792476
$298$	0	0
$299$	−44.7838	−2.58992
$300$	0	0
$301$	9.99255	0.575961
$302$	0	0
$303$	−7.34248	−0.421814
$304$	0	0
$305$	−46.7208	−2.67522
$306$	0	0
$307$	5.30062	0.302522	0.151261	−	0.988494i	$-0.451667\pi$
$307$	5.30062	0.302522	0.151261	+	0.988494i	$0.451667\pi$
$308$	0	0
$309$	2.69809	0.153489
$310$	0	0
$311$	−11.9719	−0.678862	−0.339431	−	0.940631i	$-0.610235\pi$
$311$	−11.9719	−0.678862	−0.339431	+	0.940631i	$0.610235\pi$
$312$	0	0
$313$	−19.9937	−1.13011	−0.565056	−	0.825053i	$-0.691145\pi$
$313$	−19.9937	−1.13011	−0.565056	+	0.825053i	$0.691145\pi$
$314$	0	0
$315$	10.9671	0.617924
$316$	0	0
$317$	−9.46272	−0.531479	−0.265740	−	0.964045i	$-0.585616\pi$
$317$	−9.46272	−0.531479	−0.265740	+	0.964045i	$0.585616\pi$
$318$	0	0
$319$	−30.7724	−1.72292
$320$	0	0
$321$	−9.95108	−0.555415
$322$	0	0
$323$	−15.6055	−0.868314
$324$	0	0
$325$	−56.5929	−3.13921
$326$	0	0
$327$	−1.27567	−0.0705444
$328$	0	0
$329$	0.875487	0.0482672
$330$	0	0
$331$	−12.7619	−0.701459	−0.350730	−	0.936477i	$-0.614066\pi$
$331$	−12.7619	−0.701459	−0.350730	+	0.936477i	$0.614066\pi$
$332$	0	0
$333$	28.8519	1.58108
$334$	0	0
$335$	31.2128	1.70534
$336$	0	0
$337$	5.84959	0.318648	0.159324	−	0.987226i	$-0.449069\pi$
$337$	5.84959	0.318648	0.159324	+	0.987226i	$0.449069\pi$
$338$	0	0
$339$	2.38140	0.129340
$340$	0	0
$341$	−41.1624	−2.22907
$342$	0	0
$343$	14.1550	0.764296
$344$	0	0
$345$	15.9268	0.857470
$346$	0	0
$347$	10.7771	0.578548	0.289274	−	0.957246i	$-0.406586\pi$
$347$	10.7771	0.578548	0.289274	+	0.957246i	$0.406586\pi$
$348$	0	0
$349$	7.48714	0.400777	0.200389	−	0.979716i	$-0.435780\pi$
$349$	7.48714	0.400777	0.200389	+	0.979716i	$0.435780\pi$
$350$	0	0
$351$	21.1601	1.12944
$352$	0	0
$353$	−18.8339	−1.00243	−0.501215	−	0.865323i	$-0.667113\pi$
$353$	−18.8339	−1.00243	−0.501215	+	0.865323i	$0.667113\pi$
$354$	0	0
$355$	61.9385	3.28735
$356$	0	0
$357$	−2.26678	−0.119971
$358$	0	0
$359$	19.4047	1.02414	0.512070	−	0.858944i	$-0.328879\pi$
$359$	19.4047	1.02414	0.512070	+	0.858944i	$0.328879\pi$
$360$	0	0
$361$	1.74015	0.0915867
$362$	0	0
$363$	3.24560	0.170350
$364$	0	0
$365$	52.0560	2.72473
$366$	0	0
$367$	10.7684	0.562108	0.281054	−	0.959692i	$-0.409316\pi$
$367$	10.7684	0.562108	0.281054	+	0.959692i	$0.409316\pi$
$368$	0	0
$369$	−11.6590	−0.606944
$370$	0	0
$371$	13.9555	0.724533
$372$	0	0
$373$	−12.6677	−0.655910	−0.327955	−	0.944693i	$-0.606359\pi$
$373$	−12.6677	−0.655910	−0.327955	+	0.944693i	$0.606359\pi$
$374$	0	0
$375$	8.95658	0.462516
$376$	0	0
$377$	47.6776	2.45552
$378$	0	0
$379$	−27.0510	−1.38952	−0.694759	−	0.719243i	$-0.744489\pi$
$379$	−27.0510	−1.38952	−0.694759	+	0.719243i	$0.744489\pi$
$380$	0	0
$381$	6.36314	0.325993
$382$	0	0
$383$	−25.0428	−1.27963	−0.639813	−	0.768531i	$-0.720988\pi$
$383$	−25.0428	−1.27963	−0.639813	+	0.768531i	$0.720988\pi$
$384$	0	0
$385$	−16.8186	−0.857156
$386$	0	0
$387$	23.8361	1.21166
$388$	0	0
$389$	15.2476	0.773085	0.386543	−	0.922272i	$-0.373669\pi$
$389$	15.2476	0.773085	0.386543	+	0.922272i	$0.373669\pi$
$390$	0	0
$391$	24.4297	1.23546
$392$	0	0
$393$	5.35580	0.270164
$394$	0	0
$395$	−4.07381	−0.204976
$396$	0	0
$397$	11.9551	0.600007	0.300004	−	0.953938i	$-0.403012\pi$
$397$	11.9551	0.600007	0.300004	+	0.953938i	$0.403012\pi$
$398$	0	0
$399$	3.01261	0.150819
$400$	0	0
$401$	−4.83325	−0.241361	−0.120680	−	0.992691i	$-0.538508\pi$
$401$	−4.83325	−0.241361	−0.120680	+	0.992691i	$0.538508\pi$
$402$	0	0
$403$	63.7755	3.17689
$404$	0	0
$405$	22.1605	1.10117
$406$	0	0
$407$	−44.2461	−2.19320
$408$	0	0
$409$	−18.7528	−0.927264	−0.463632	−	0.886028i	$-0.653454\pi$
$409$	−18.7528	−0.927264	−0.463632	+	0.886028i	$0.653454\pi$
$410$	0	0
$411$	−9.06258	−0.447024
$412$	0	0
$413$	4.06859	0.200202
$414$	0	0
$415$	33.1377	1.62666
$416$	0	0
$417$	6.58173	0.322309
$418$	0	0
$419$	6.75926	0.330211	0.165106	−	0.986276i	$-0.447203\pi$
$419$	6.75926	0.330211	0.165106	+	0.986276i	$0.447203\pi$
$420$	0	0
$421$	−23.9070	−1.16515	−0.582577	−	0.812775i	$-0.697956\pi$
$421$	−23.9070	−1.16515	−0.582577	+	0.812775i	$0.697956\pi$
$422$	0	0
$423$	2.08838	0.101540
$424$	0	0
$425$	30.8716	1.49749
$426$	0	0
$427$	13.8346	0.669501
$428$	0	0
$429$	−15.2010	−0.733909
$430$	0	0
$431$	−22.0435	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$431$	−22.0435	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$432$	0	0
$433$	7.78620	0.374181	0.187090	−	0.982343i	$-0.440094\pi$
$433$	7.78620	0.374181	0.187090	+	0.982343i	$0.440094\pi$
$434$	0	0
$435$	−16.9559	−0.812974
$436$	0	0
$437$	−32.4678	−1.55314
$438$	0	0
$439$	−15.7303	−0.750766	−0.375383	−	0.926870i	$-0.622489\pi$
$439$	−15.7303	−0.750766	−0.375383	+	0.926870i	$0.622489\pi$
$440$	0	0
$441$	15.2588	0.726610
$442$	0	0
$443$	16.1674	0.768135	0.384068	−	0.923305i	$-0.374523\pi$
$443$	16.1674	0.768135	0.384068	+	0.923305i	$0.374523\pi$
$444$	0	0
$445$	−51.8204	−2.45652
$446$	0	0
$447$	3.30546	0.156343
$448$	0	0
$449$	37.0269	1.74741	0.873703	−	0.486460i	$-0.161712\pi$
$449$	37.0269	1.74741	0.873703	+	0.486460i	$0.161712\pi$
$450$	0	0
$451$	17.8798	0.841926
$452$	0	0
$453$	4.21126	0.197862
$454$	0	0
$455$	26.0582	1.22163
$456$	0	0
$457$	0.144604	0.00676429	0.00338214	−	0.999994i	$-0.498923\pi$
$457$	0.144604	0.00676429	0.00338214	+	0.999994i	$0.498923\pi$
$458$	0	0
$459$	−11.5429	−0.538777
$460$	0	0
$461$	−6.52018	−0.303675	−0.151837	−	0.988405i	$-0.548519\pi$
$461$	−6.52018	−0.303675	−0.151837	+	0.988405i	$0.548519\pi$
$462$	0	0
$463$	−26.2598	−1.22039	−0.610197	−	0.792250i	$-0.708910\pi$
$463$	−26.2598	−1.22039	−0.610197	+	0.792250i	$0.708910\pi$
$464$	0	0
$465$	−22.6810	−1.05180
$466$	0	0
$467$	−26.3872	−1.22105	−0.610527	−	0.791996i	$-0.709042\pi$
$467$	−26.3872	−1.22105	−0.610527	+	0.791996i	$0.709042\pi$
$468$	0	0
$469$	−9.24247	−0.426778
$470$	0	0
$471$	−8.79064	−0.405052
$472$	0	0
$473$	−36.5541	−1.68076
$474$	0	0
$475$	−41.0292	−1.88255
$476$	0	0
$477$	33.2893	1.52421
$478$	0	0
$479$	−22.9251	−1.04747	−0.523736	−	0.851880i	$-0.675462\pi$
$479$	−22.9251	−1.04747	−0.523736	+	0.851880i	$0.675462\pi$
$480$	0	0
$481$	68.5533	3.12576
$482$	0	0
$483$	−4.71611	−0.214590
$484$	0	0
$485$	27.0873	1.22997
$486$	0	0
$487$	−9.95964	−0.451314	−0.225657	−	0.974207i	$-0.572453\pi$
$487$	−9.95964	−0.451314	−0.225657	+	0.974207i	$0.572453\pi$
$488$	0	0
$489$	−7.36272	−0.332954
$490$	0	0
$491$	3.65342	0.164877	0.0824384	−	0.996596i	$-0.473729\pi$
$491$	3.65342	0.164877	0.0824384	+	0.996596i	$0.473729\pi$
$492$	0	0
$493$	−26.0083	−1.17135
$494$	0	0
$495$	−40.1189	−1.80321
$496$	0	0
$497$	−18.3407	−0.822692
$498$	0	0
$499$	−27.8117	−1.24502	−0.622511	−	0.782611i	$-0.713887\pi$
$499$	−27.8117	−1.24502	−0.622511	+	0.782611i	$0.713887\pi$
$500$	0	0
$501$	−4.90805	−0.219276
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−46.0442	−2.04894
$506$	0	0
$507$	15.7926	0.701373
$508$	0	0
$509$	−1.97937	−0.0877341	−0.0438671	−	0.999037i	$-0.513968\pi$
$509$	−1.97937	−0.0877341	−0.0438671	+	0.999037i	$0.513968\pi$
$510$	0	0
$511$	−15.4144	−0.681891
$512$	0	0
$513$	15.3408	0.677314
$514$	0	0
$515$	16.9196	0.745565
$516$	0	0
$517$	−3.20265	−0.140852
$518$	0	0
$519$	−2.45185	−0.107624
$520$	0	0
$521$	−7.35762	−0.322343	−0.161172	−	0.986926i	$-0.551527\pi$
$521$	−7.35762	−0.322343	−0.161172	+	0.986926i	$0.551527\pi$
$522$	0	0
$523$	−7.94830	−0.347555	−0.173778	−	0.984785i	$-0.555597\pi$
$523$	−7.94830	−0.347555	−0.173778	+	0.984785i	$0.555597\pi$
$524$	0	0
$525$	−5.95970	−0.260103
$526$	0	0
$527$	−34.7898	−1.51547
$528$	0	0
$529$	27.8269	1.20986
$530$	0	0
$531$	9.70517	0.421169
$532$	0	0
$533$	−27.7023	−1.19992
$534$	0	0
$535$	−62.4026	−2.69790
$536$	0	0
$537$	−6.82933	−0.294707
$538$	0	0
$539$	−23.4003	−1.00792
$540$	0	0
$541$	−19.4451	−0.836009	−0.418005	−	0.908445i	$-0.637270\pi$
$541$	−19.4451	−0.836009	−0.418005	+	0.908445i	$0.637270\pi$
$542$	0	0
$543$	−1.68882	−0.0724742
$544$	0	0
$545$	−7.99961	−0.342666
$546$	0	0
$547$	−13.4460	−0.574907	−0.287454	−	0.957795i	$-0.592809\pi$
$547$	−13.4460	−0.574907	−0.287454	+	0.957795i	$0.592809\pi$
$548$	0	0
$549$	33.0008	1.40844
$550$	0	0
$551$	34.5657	1.47255
$552$	0	0
$553$	1.20630	0.0512972
$554$	0	0
$555$	−24.3801	−1.03488
$556$	0	0
$557$	−20.3211	−0.861032	−0.430516	−	0.902583i	$-0.641668\pi$
$557$	−20.3211	−0.861032	−0.430516	+	0.902583i	$0.641668\pi$
$558$	0	0
$559$	56.6355	2.39543
$560$	0	0
$561$	8.29217	0.350096
$562$	0	0
$563$	16.8840	0.711576	0.355788	−	0.934567i	$-0.384212\pi$
$563$	16.8840	0.711576	0.355788	+	0.934567i	$0.384212\pi$
$564$	0	0
$565$	14.9336	0.628260
$566$	0	0
$567$	−6.56198	−0.275577
$568$	0	0
$569$	14.8114	0.620928	0.310464	−	0.950585i	$-0.399516\pi$
$569$	14.8114	0.620928	0.310464	+	0.950585i	$0.399516\pi$
$570$	0	0
$571$	−20.4136	−0.854284	−0.427142	−	0.904185i	$-0.640480\pi$
$571$	−20.4136	−0.854284	−0.427142	+	0.904185i	$0.640480\pi$
$572$	0	0
$573$	7.08384	0.295931
$574$	0	0
$575$	64.2294	2.67855
$576$	0	0
$577$	4.09712	0.170565	0.0852827	−	0.996357i	$-0.472821\pi$
$577$	4.09712	0.170565	0.0852827	+	0.996357i	$0.472821\pi$
$578$	0	0
$579$	1.54171	0.0640712
$580$	0	0
$581$	−9.81244	−0.407089
$582$	0	0
$583$	−51.0511	−2.11432
$584$	0	0
$585$	62.1588	2.56995
$586$	0	0
$587$	−12.5359	−0.517412	−0.258706	−	0.965956i	$-0.583296\pi$
$587$	−12.5359	−0.517412	−0.258706	+	0.965956i	$0.583296\pi$
$588$	0	0
$589$	46.2365	1.90514
$590$	0	0
$591$	9.59116	0.394528
$592$	0	0
$593$	17.2115	0.706793	0.353396	−	0.935474i	$-0.385027\pi$
$593$	17.2115	0.706793	0.353396	+	0.935474i	$0.385027\pi$
$594$	0	0
$595$	−14.2148	−0.582751
$596$	0	0
$597$	−11.0519	−0.452325
$598$	0	0
$599$	−6.04421	−0.246960	−0.123480	−	0.992347i	$-0.539405\pi$
$599$	−6.04421	−0.246960	−0.123480	+	0.992347i	$0.539405\pi$
$600$	0	0
$601$	43.0009	1.75404	0.877021	−	0.480452i	$-0.159527\pi$
$601$	43.0009	1.75404	0.877021	+	0.480452i	$0.159527\pi$
$602$	0	0
$603$	−22.0469	−0.897818
$604$	0	0
$605$	20.3529	0.827465
$606$	0	0
$607$	−9.34473	−0.379291	−0.189645	−	0.981853i	$-0.560734\pi$
$607$	−9.34473	−0.379291	−0.189645	+	0.981853i	$0.560734\pi$
$608$	0	0
$609$	5.02084	0.203455
$610$	0	0
$611$	4.96207	0.200744
$612$	0	0
$613$	23.2822	0.940360	0.470180	−	0.882570i	$-0.344189\pi$
$613$	23.2822	0.940360	0.470180	+	0.882570i	$0.344189\pi$
$614$	0	0
$615$	9.85196	0.397270
$616$	0	0
$617$	−32.8413	−1.32214	−0.661071	−	0.750324i	$-0.729898\pi$
$617$	−32.8413	−1.32214	−0.661071	+	0.750324i	$0.729898\pi$
$618$	0	0
$619$	−9.26759	−0.372496	−0.186248	−	0.982503i	$-0.559633\pi$
$619$	−9.26759	−0.372496	−0.186248	+	0.982503i	$0.559633\pi$
$620$	0	0
$621$	−24.0154	−0.963705
$622$	0	0
$623$	15.3446	0.614768
$624$	0	0
$625$	11.1200	0.444800
$626$	0	0
$627$	−11.0205	−0.440117
$628$	0	0
$629$	−37.3961	−1.49108
$630$	0	0
$631$	7.74843	0.308460	0.154230	−	0.988035i	$-0.450710\pi$
$631$	7.74843	0.308460	0.154230	+	0.988035i	$0.450710\pi$
$632$	0	0
$633$	−7.16917	−0.284949
$634$	0	0
$635$	39.9028	1.58349
$636$	0	0
$637$	36.2555	1.43650
$638$	0	0
$639$	−43.7496	−1.73071
$640$	0	0
$641$	−29.2774	−1.15639	−0.578193	−	0.815900i	$-0.696242\pi$
$641$	−29.2774	−1.15639	−0.578193	+	0.815900i	$0.696242\pi$
$642$	0	0
$643$	24.5922	0.969824	0.484912	−	0.874563i	$-0.338852\pi$
$643$	24.5922	0.969824	0.484912	+	0.874563i	$0.338852\pi$
$644$	0	0
$645$	−20.1417	−0.793079
$646$	0	0
$647$	22.5557	0.886758	0.443379	−	0.896334i	$-0.353780\pi$
$647$	22.5557	0.886758	0.443379	+	0.896334i	$0.353780\pi$
$648$	0	0
$649$	−14.8835	−0.584227
$650$	0	0
$651$	6.71609	0.263224
$652$	0	0
$653$	7.45203	0.291621	0.145810	−	0.989313i	$-0.453421\pi$
$653$	7.45203	0.291621	0.145810	+	0.989313i	$0.453421\pi$
$654$	0	0
$655$	33.5858	1.31231
$656$	0	0
$657$	−36.7692	−1.43450
$658$	0	0
$659$	−1.23818	−0.0482326	−0.0241163	−	0.999709i	$-0.507677\pi$
$659$	−1.23818	−0.0482326	−0.0241163	+	0.999709i	$0.507677\pi$
$660$	0	0
$661$	38.7457	1.50703	0.753517	−	0.657428i	$-0.228356\pi$
$661$	38.7457	1.50703	0.753517	+	0.657428i	$0.228356\pi$
$662$	0	0
$663$	−12.8476	−0.498959
$664$	0	0
$665$	18.8919	0.732595
$666$	0	0
$667$	−54.1111	−2.09519
$668$	0	0
$669$	2.70502	0.104582
$670$	0	0
$671$	−50.6086	−1.95372
$672$	0	0
$673$	−2.29534	−0.0884788	−0.0442394	−	0.999021i	$-0.514086\pi$
$673$	−2.29534	−0.0884788	−0.0442394	+	0.999021i	$0.514086\pi$
$674$	0	0
$675$	−30.3480	−1.16810
$676$	0	0
$677$	30.9949	1.19123	0.595615	−	0.803270i	$-0.296908\pi$
$677$	30.9949	1.19123	0.595615	+	0.803270i	$0.296908\pi$
$678$	0	0
$679$	−8.02086	−0.307812
$680$	0	0
$681$	−5.23884	−0.200753
$682$	0	0
$683$	12.6566	0.484290	0.242145	−	0.970240i	$-0.422149\pi$
$683$	12.6566	0.484290	0.242145	+	0.970240i	$0.422149\pi$
$684$	0	0
$685$	−56.8308	−2.17139
$686$	0	0
$687$	−12.9907	−0.495625
$688$	0	0
$689$	79.0966	3.01334
$690$	0	0
$691$	47.5327	1.80823	0.904114	−	0.427291i	$-0.140532\pi$
$691$	47.5327	1.80823	0.904114	+	0.427291i	$0.140532\pi$
$692$	0	0
$693$	11.8797	0.451271
$694$	0	0
$695$	41.2736	1.56560
$696$	0	0
$697$	15.1117	0.572396
$698$	0	0
$699$	9.21431	0.348517
$700$	0	0
$701$	−41.1521	−1.55429	−0.777147	−	0.629319i	$-0.783334\pi$
$701$	−41.1521	−1.55429	−0.777147	+	0.629319i	$0.783334\pi$
$702$	0	0
$703$	49.7004	1.87449
$704$	0	0
$705$	−1.76470	−0.0664623
$706$	0	0
$707$	13.6342	0.512767
$708$	0	0
$709$	5.55047	0.208452	0.104226	−	0.994554i	$-0.466763\pi$
$709$	5.55047	0.208452	0.104226	+	0.994554i	$0.466763\pi$
$710$	0	0
$711$	2.87750	0.107915
$712$	0	0
$713$	−72.3813	−2.71070
$714$	0	0
$715$	−95.3242	−3.56492
$716$	0	0
$717$	−3.51908	−0.131422
$718$	0	0
$719$	−14.0126	−0.522583	−0.261291	−	0.965260i	$-0.584148\pi$
$719$	−14.0126	−0.522583	−0.261291	+	0.965260i	$0.584148\pi$
$720$	0	0
$721$	−5.01007	−0.186585
$722$	0	0
$723$	10.5123	0.390958
$724$	0	0
$725$	−68.3796	−2.53956
$726$	0	0
$727$	0.112833	0.00418476	0.00209238	−	0.999998i	$-0.499334\pi$
$727$	0.112833	0.00418476	0.00209238	+	0.999998i	$0.499334\pi$
$728$	0	0
$729$	−9.62118	−0.356340
$730$	0	0
$731$	−30.8949	−1.14269
$732$	0	0
$733$	22.0094	0.812935	0.406467	−	0.913665i	$-0.366760\pi$
$733$	22.0094	0.812935	0.406467	+	0.913665i	$0.366760\pi$
$734$	0	0
$735$	−12.8938	−0.475596
$736$	0	0
$737$	33.8102	1.24541
$738$	0	0
$739$	4.05841	0.149291	0.0746455	−	0.997210i	$-0.476217\pi$
$739$	4.05841	0.149291	0.0746455	+	0.997210i	$0.476217\pi$
$740$	0	0
$741$	17.0748	0.627258
$742$	0	0
$743$	−18.6795	−0.685283	−0.342642	−	0.939466i	$-0.611322\pi$
$743$	−18.6795	−0.685283	−0.342642	+	0.939466i	$0.611322\pi$
$744$	0	0
$745$	20.7283	0.759426
$746$	0	0
$747$	−23.4065	−0.856398
$748$	0	0
$749$	18.4781	0.675175
$750$	0	0
$751$	37.8836	1.38239	0.691196	−	0.722667i	$-0.257084\pi$
$751$	37.8836	1.38239	0.691196	+	0.722667i	$0.257084\pi$
$752$	0	0
$753$	10.8764	0.396359
$754$	0	0
$755$	26.4085	0.961105
$756$	0	0
$757$	−26.3896	−0.959146	−0.479573	−	0.877502i	$-0.659208\pi$
$757$	−26.3896	−0.959146	−0.479573	+	0.877502i	$0.659208\pi$
$758$	0	0
$759$	17.2521	0.626213
$760$	0	0
$761$	47.4889	1.72147	0.860736	−	0.509052i	$-0.170004\pi$
$761$	47.4889	1.72147	0.860736	+	0.509052i	$0.170004\pi$
$762$	0	0
$763$	2.36878	0.0857554
$764$	0	0
$765$	−33.9078	−1.22594
$766$	0	0
$767$	23.0599	0.832644
$768$	0	0
$769$	−22.1250	−0.797849	−0.398924	−	0.916984i	$-0.630616\pi$
$769$	−22.1250	−0.797849	−0.398924	+	0.916984i	$0.630616\pi$
$770$	0	0
$771$	1.19889	0.0431769
$772$	0	0
$773$	−4.70020	−0.169054	−0.0845272	−	0.996421i	$-0.526938\pi$
$773$	−4.70020	−0.169054	−0.0845272	+	0.996421i	$0.526938\pi$
$774$	0	0
$775$	−91.4675	−3.28561
$776$	0	0
$777$	7.21923	0.258988
$778$	0	0
$779$	−20.0838	−0.719578
$780$	0	0
$781$	67.0926	2.40076
$782$	0	0
$783$	25.5672	0.913696
$784$	0	0
$785$	−55.1255	−1.96751
$786$	0	0
$787$	5.97274	0.212905	0.106453	−	0.994318i	$-0.466051\pi$
$787$	5.97274	0.212905	0.106453	+	0.994318i	$0.466051\pi$
$788$	0	0
$789$	15.5413	0.553284
$790$	0	0
$791$	−4.42200	−0.157228
$792$	0	0
$793$	78.4111	2.78446
$794$	0	0
$795$	−28.1297	−0.997658
$796$	0	0
$797$	−13.7237	−0.486120	−0.243060	−	0.970011i	$-0.578151\pi$
$797$	−13.7237	−0.486120	−0.243060	+	0.970011i	$0.578151\pi$
$798$	0	0
$799$	−2.70682	−0.0957605
$800$	0	0
$801$	36.6028	1.29330
$802$	0	0
$803$	56.3878	1.98988
$804$	0	0
$805$	−29.5744	−1.04236
$806$	0	0
$807$	15.3461	0.540209
$808$	0	0
$809$	−44.2563	−1.55597	−0.777985	−	0.628283i	$-0.783758\pi$
$809$	−44.2563	−1.55597	−0.777985	+	0.628283i	$0.783758\pi$
$810$	0	0
$811$	−47.3816	−1.66379	−0.831896	−	0.554932i	$-0.812744\pi$
$811$	−47.3816	−1.66379	−0.831896	+	0.554932i	$0.812744\pi$
$812$	0	0
$813$	13.3324	0.467589
$814$	0	0
$815$	−46.1712	−1.61730
$816$	0	0
$817$	41.0601	1.43651
$818$	0	0
$819$	−18.4059	−0.643155
$820$	0	0
$821$	−11.7955	−0.411666	−0.205833	−	0.978587i	$-0.565990\pi$
$821$	−11.7955	−0.411666	−0.205833	+	0.978587i	$0.565990\pi$
$822$	0	0
$823$	−2.96953	−0.103511	−0.0517557	−	0.998660i	$-0.516482\pi$
$823$	−2.96953	−0.103511	−0.0517557	+	0.998660i	$0.516482\pi$
$824$	0	0
$825$	21.8014	0.759026
$826$	0	0
$827$	−23.7782	−0.826850	−0.413425	−	0.910538i	$-0.635668\pi$
$827$	−23.7782	−0.826850	−0.413425	+	0.910538i	$0.635668\pi$
$828$	0	0
$829$	−14.6100	−0.507426	−0.253713	−	0.967280i	$-0.581652\pi$
$829$	−14.6100	−0.507426	−0.253713	+	0.967280i	$0.581652\pi$
$830$	0	0
$831$	7.17613	0.248937
$832$	0	0
$833$	−19.7775	−0.685250
$834$	0	0
$835$	−30.7781	−1.06512
$836$	0	0
$837$	34.1997	1.18211
$838$	0	0
$839$	22.2987	0.769836	0.384918	−	0.922951i	$-0.374230\pi$
$839$	22.2987	0.769836	0.384918	+	0.922951i	$0.374230\pi$
$840$	0	0
$841$	28.6075	0.986465
$842$	0	0
$843$	−0.0988443	−0.00340438
$844$	0	0
$845$	99.0342	3.40688
$846$	0	0
$847$	−6.02674	−0.207081
$848$	0	0
$849$	5.43693	0.186595
$850$	0	0
$851$	−77.8038	−2.66708
$852$	0	0
$853$	25.6780	0.879199	0.439599	−	0.898194i	$-0.355120\pi$
$853$	25.6780	0.879199	0.439599	+	0.898194i	$0.355120\pi$
$854$	0	0
$855$	45.0644	1.54117
$856$	0	0
$857$	−0.639804	−0.0218553	−0.0109276	−	0.999940i	$-0.503478\pi$
$857$	−0.639804	−0.0218553	−0.0109276	+	0.999940i	$0.503478\pi$
$858$	0	0
$859$	−15.2690	−0.520973	−0.260486	−	0.965478i	$-0.583883\pi$
$859$	−15.2690	−0.520973	−0.260486	+	0.965478i	$0.583883\pi$
$860$	0	0
$861$	−2.91728	−0.0994206
$862$	0	0
$863$	−9.05247	−0.308150	−0.154075	−	0.988059i	$-0.549240\pi$
$863$	−9.05247	−0.308150	−0.154075	+	0.988059i	$0.549240\pi$
$864$	0	0
$865$	−15.3754	−0.522778
$866$	0	0
$867$	−3.13827	−0.106581
$868$	0	0
$869$	−4.41281	−0.149694
$870$	0	0
$871$	−52.3843	−1.77497
$872$	0	0
$873$	−19.1329	−0.647549
$874$	0	0
$875$	−16.6314	−0.562245
$876$	0	0
$877$	−20.7534	−0.700793	−0.350397	−	0.936601i	$-0.613953\pi$
$877$	−20.7534	−0.700793	−0.350397	+	0.936601i	$0.613953\pi$
$878$	0	0
$879$	−2.39177	−0.0806725
$880$	0	0
$881$	−31.1931	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$881$	−31.1931	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$882$	0	0
$883$	−11.1829	−0.376333	−0.188167	−	0.982137i	$-0.560255\pi$
$883$	−11.1829	−0.376333	−0.188167	+	0.982137i	$0.560255\pi$
$884$	0	0
$885$	−8.20095	−0.275672
$886$	0	0
$887$	9.12495	0.306386	0.153193	−	0.988196i	$-0.451044\pi$
$887$	9.12495	0.306386	0.153193	+	0.988196i	$0.451044\pi$
$888$	0	0
$889$	−11.8157	−0.396285
$890$	0	0
$891$	24.0046	0.804184
$892$	0	0
$893$	3.59744	0.120384
$894$	0	0
$895$	−42.8263	−1.43152
$896$	0	0
$897$	−26.7298	−0.892483
$898$	0	0
$899$	77.0582	2.57003
$900$	0	0
$901$	−43.1475	−1.43745
$902$	0	0
$903$	5.96419	0.198476
$904$	0	0
$905$	−10.5905	−0.352039
$906$	0	0
$907$	−43.2911	−1.43746	−0.718728	−	0.695291i	$-0.755275\pi$
$907$	−43.2911	−1.43746	−0.718728	+	0.695291i	$0.755275\pi$
$908$	0	0
$909$	32.5229	1.07872
$910$	0	0
$911$	5.88679	0.195038	0.0975190	−	0.995234i	$-0.468909\pi$
$911$	5.88679	0.195038	0.0975190	+	0.995234i	$0.468909\pi$
$912$	0	0
$913$	35.8952	1.18796
$914$	0	0
$915$	−27.8859	−0.921880
$916$	0	0
$917$	−9.94515	−0.328418
$918$	0	0
$919$	52.5637	1.73392	0.866958	−	0.498380i	$-0.166072\pi$
$919$	52.5637	1.73392	0.866958	+	0.498380i	$0.166072\pi$
$920$	0	0
$921$	3.16374	0.104249
$922$	0	0
$923$	−103.951	−3.42158
$924$	0	0
$925$	−98.3199	−3.23274
$926$	0	0
$927$	−11.9510	−0.392521
$928$	0	0
$929$	38.5702	1.26545	0.632724	−	0.774378i	$-0.281937\pi$
$929$	38.5702	1.26545	0.632724	+	0.774378i	$0.281937\pi$
$930$	0	0
$931$	26.2848	0.861451
$932$	0	0
$933$	−7.14557	−0.233935
$934$	0	0
$935$	51.9997	1.70057
$936$	0	0
$937$	48.6182	1.58829	0.794144	−	0.607729i	$-0.207919\pi$
$937$	48.6182	1.58829	0.794144	+	0.607729i	$0.207919\pi$
$938$	0	0
$939$	−11.9335	−0.389436
$940$	0	0
$941$	30.3489	0.989346	0.494673	−	0.869079i	$-0.335288\pi$
$941$	30.3489	0.989346	0.494673	+	0.869079i	$0.335288\pi$
$942$	0	0
$943$	31.4404	1.02384
$944$	0	0
$945$	13.9737	0.454565
$946$	0	0
$947$	−50.4765	−1.64027	−0.820133	−	0.572172i	$-0.806101\pi$
$947$	−50.4765	−1.64027	−0.820133	+	0.572172i	$0.806101\pi$
$948$	0	0
$949$	−87.3651	−2.83599
$950$	0	0
$951$	−5.64795	−0.183147
$952$	0	0
$953$	5.77733	0.187146	0.0935731	−	0.995612i	$-0.470171\pi$
$953$	5.77733	0.187146	0.0935731	+	0.995612i	$0.470171\pi$
$954$	0	0
$955$	44.4223	1.43747
$956$	0	0
$957$	−18.3669	−0.593717
$958$	0	0
$959$	16.8282	0.543412
$960$	0	0
$961$	72.0763	2.32504
$962$	0	0
$963$	44.0774	1.42038
$964$	0	0
$965$	9.66795	0.311222
$966$	0	0
$967$	12.2393	0.393588	0.196794	−	0.980445i	$-0.436947\pi$
$967$	12.2393	0.393588	0.196794	+	0.980445i	$0.436947\pi$
$968$	0	0
$969$	−9.31436	−0.299220
$970$	0	0
$971$	12.9154	0.414476	0.207238	−	0.978291i	$-0.433553\pi$
$971$	12.9154	0.414476	0.207238	+	0.978291i	$0.433553\pi$
$972$	0	0
$973$	−12.2216	−0.391806
$974$	0	0
$975$	−33.7782	−1.08177
$976$	0	0
$977$	17.5065	0.560084	0.280042	−	0.959988i	$-0.409652\pi$
$977$	17.5065	0.560084	0.280042	+	0.959988i	$0.409652\pi$
$978$	0	0
$979$	−56.1325	−1.79400
$980$	0	0
$981$	5.65045	0.180405
$982$	0	0
$983$	−7.14983	−0.228044	−0.114022	−	0.993478i	$-0.536373\pi$
$983$	−7.14983	−0.228044	−0.114022	+	0.993478i	$0.536373\pi$
$984$	0	0
$985$	60.1455	1.91640
$986$	0	0
$987$	0.522546	0.0166328
$988$	0	0
$989$	−64.2778	−2.04392
$990$	0	0
$991$	−42.8989	−1.36273	−0.681364	−	0.731945i	$-0.738613\pi$
$991$	−42.8989	−1.36273	−0.681364	+	0.731945i	$0.738613\pi$
$992$	0	0
$993$	−7.61713	−0.241722
$994$	0	0
$995$	−69.3058	−2.19714
$996$	0	0
$997$	−32.1968	−1.01968	−0.509842	−	0.860268i	$-0.670296\pi$
$997$	−32.1968	−1.01968	−0.509842	+	0.860268i	$0.670296\pi$
$998$	0	0
$999$	36.7618	1.16309

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.v.1.17		28
4.3	odd	2		4024.2.a.d.1.12	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.d.1.12	✓	28	4.3	odd	2
8048.2.a.v.1.17		28	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+0.596863 q^{3} +3.74289 q^{5} -1.10831 q^{7} -2.64375 q^{9} +O(q^{10})\)	\(q+0.596863 q^{3} +3.74289 q^{5} -1.10831 q^{7} -2.64375 q^{9} +4.05435 q^{11} -6.28166 q^{13} +2.23399 q^{15} +3.42667 q^{17} -4.55414 q^{19} -0.661511 q^{21} +7.12930 q^{23} +9.00922 q^{25} -3.36855 q^{27} -7.58996 q^{29} -10.1526 q^{31} +2.41989 q^{33} -4.14829 q^{35} -10.9132 q^{37} -3.74929 q^{39} +4.41002 q^{41} -9.01601 q^{43} -9.89528 q^{45} -0.789928 q^{47} -5.77164 q^{49} +2.04525 q^{51} -12.5917 q^{53} +15.1750 q^{55} -2.71820 q^{57} -3.67098 q^{59} -12.4825 q^{61} +2.93010 q^{63} -23.5116 q^{65} +8.33923 q^{67} +4.25522 q^{69} +16.5483 q^{71} +13.9080 q^{73} +5.37727 q^{75} -4.49349 q^{77} -1.08841 q^{79} +5.92070 q^{81} +8.85350 q^{83} +12.8256 q^{85} -4.53017 q^{87} -13.8450 q^{89} +6.96204 q^{91} -6.05974 q^{93} -17.0456 q^{95} +7.23701 q^{97} -10.7187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9	\( 28 q - 2 q^{3} - 12 q^{5} + 18 q^{9} + 14 q^{11} - 31 q^{13} + 2 q^{15} - 9 q^{17} + 8 q^{19} - 28 q^{21} + 4 q^{23} + 22 q^{25} + 4 q^{27} - 47 q^{29} + 5 q^{31} - 26 q^{33} + 13 q^{35} - 67 q^{37} + 9 q^{39} - 28 q^{41} - 15 q^{43} - 57 q^{45} + 10 q^{47} + 20 q^{49} + 11 q^{51} - 58 q^{53} - 15 q^{55} - 31 q^{57} + 32 q^{59} - 55 q^{61} + 16 q^{63} - 44 q^{65} - 22 q^{67} - 44 q^{69} + 47 q^{71} - 5 q^{73} + 25 q^{75} - 50 q^{77} + 14 q^{79} - 28 q^{81} + 16 q^{83} - 78 q^{85} + 11 q^{87} - 20 q^{89} + 15 q^{91} - 83 q^{93} + 27 q^{95} - 8 q^{97} + 70 q^{99}+O(q^{100}) \) 28 * q - 2 * q^3 - 12 * q^5 + 18 * q^9 + 14 * q^11 - 31 * q^13 + 2 * q^15 - 9 * q^17 + 8 * q^19 - 28 * q^21 + 4 * q^23 + 22 * q^25 + 4 * q^27 - 47 * q^29 + 5 * q^31 - 26 * q^33 + 13 * q^35 - 67 * q^37 + 9 * q^39 - 28 * q^41 - 15 * q^43 - 57 * q^45 + 10 * q^47 + 20 * q^49 + 11 * q^51 - 58 * q^53 - 15 * q^55 - 31 * q^57 + 32 * q^59 - 55 * q^61 + 16 * q^63 - 44 * q^65 - 22 * q^67 - 44 * q^69 + 47 * q^71 - 5 * q^73 + 25 * q^75 - 50 * q^77 + 14 * q^79 - 28 * q^81 + 16 * q^83 - 78 * q^85 + 11 * q^87 - 20 * q^89 + 15 * q^91 - 83 * q^93 + 27 * q^95 - 8 * q^97 + 70 * q^99

Introduction

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