LMFDB - Embedded newform 8048.2.a.s.1.7

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:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.93158 q^{3} -3.25684 q^{5} -2.49607 q^{7} +0.730999 q^{9} +O(q^{10})$	$q-1.93158 q^{3} -3.25684 q^{5} -2.49607 q^{7} +0.730999 q^{9} -5.22489 q^{11} -1.60713 q^{13} +6.29085 q^{15} -3.28603 q^{17} -0.318129 q^{19} +4.82135 q^{21} -0.844194 q^{23} +5.60702 q^{25} +4.38276 q^{27} +10.1373 q^{29} -5.96428 q^{31} +10.0923 q^{33} +8.12929 q^{35} +7.11538 q^{37} +3.10429 q^{39} +3.46821 q^{41} -6.45802 q^{43} -2.38075 q^{45} -6.34293 q^{47} -0.769654 q^{49} +6.34723 q^{51} -3.13271 q^{53} +17.0166 q^{55} +0.614491 q^{57} +10.5002 q^{59} +6.63652 q^{61} -1.82462 q^{63} +5.23416 q^{65} +3.17640 q^{67} +1.63063 q^{69} +3.43455 q^{71} +5.65907 q^{73} -10.8304 q^{75} +13.0417 q^{77} -6.10707 q^{79} -10.6586 q^{81} +13.3768 q^{83} +10.7021 q^{85} -19.5810 q^{87} -0.348778 q^{89} +4.01150 q^{91} +11.5205 q^{93} +1.03609 q^{95} +1.21946 q^{97} -3.81939 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	$ 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) $ 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * 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
$3$	−1.93158	−1.11520	−0.557599	−	0.830110i	$-0.688277\pi$
$3$	−1.93158	−1.11520	−0.557599	+	0.830110i	$0.688277\pi$
$4$	0	0
$5$	−3.25684	−1.45650	−0.728252	−	0.685309i	$-0.759667\pi$
$5$	−3.25684	−1.45650	−0.728252	+	0.685309i	$0.759667\pi$
$6$	0	0
$7$	−2.49607	−0.943424	−0.471712	−	0.881753i	$-0.656364\pi$
$7$	−2.49607	−0.943424	−0.471712	+	0.881753i	$0.656364\pi$
$8$	0	0
$9$	0.730999	0.243666
$10$	0	0
$11$	−5.22489	−1.57536	−0.787682	−	0.616082i	$-0.788719\pi$
$11$	−5.22489	−1.57536	−0.787682	+	0.616082i	$0.788719\pi$
$12$	0	0
$13$	−1.60713	−0.445737	−0.222869	−	0.974849i	$-0.571542\pi$
$13$	−1.60713	−0.445737	−0.222869	+	0.974849i	$0.571542\pi$
$14$	0	0
$15$	6.29085	1.62429
$16$	0	0
$17$	−3.28603	−0.796980	−0.398490	−	0.917173i	$-0.630466\pi$
$17$	−3.28603	−0.796980	−0.398490	+	0.917173i	$0.630466\pi$
$18$	0	0
$19$	−0.318129	−0.0729837	−0.0364918	−	0.999334i	$-0.511618\pi$
$19$	−0.318129	−0.0729837	−0.0364918	+	0.999334i	$0.511618\pi$
$20$	0	0
$21$	4.82135	1.05210
$22$	0	0
$23$	−0.844194	−0.176027	−0.0880133	−	0.996119i	$-0.528052\pi$
$23$	−0.844194	−0.176027	−0.0880133	+	0.996119i	$0.528052\pi$
$24$	0	0
$25$	5.60702	1.12140
$26$	0	0
$27$	4.38276	0.843462
$28$	0	0
$29$	10.1373	1.88245	0.941224	−	0.337783i	$-0.109677\pi$
$29$	10.1373	1.88245	0.941224	+	0.337783i	$0.109677\pi$
$30$	0	0
$31$	−5.96428	−1.07122	−0.535609	−	0.844466i	$-0.679918\pi$
$31$	−5.96428	−1.07122	−0.535609	+	0.844466i	$0.679918\pi$
$32$	0	0
$33$	10.0923	1.75684
$34$	0	0
$35$	8.12929	1.37410
$36$	0	0
$37$	7.11538	1.16976	0.584881	−	0.811119i	$-0.301141\pi$
$37$	7.11538	1.16976	0.584881	+	0.811119i	$0.301141\pi$
$38$	0	0
$39$	3.10429	0.497085
$40$	0	0
$41$	3.46821	0.541644	0.270822	−	0.962629i	$-0.412705\pi$
$41$	3.46821	0.541644	0.270822	+	0.962629i	$0.412705\pi$
$42$	0	0
$43$	−6.45802	−0.984839	−0.492420	−	0.870358i	$-0.663887\pi$
$43$	−6.45802	−0.984839	−0.492420	+	0.870358i	$0.663887\pi$
$44$	0	0
$45$	−2.38075	−0.354901
$46$	0	0
$47$	−6.34293	−0.925212	−0.462606	−	0.886564i	$-0.653085\pi$
$47$	−6.34293	−0.925212	−0.462606	+	0.886564i	$0.653085\pi$
$48$	0	0
$49$	−0.769654	−0.109951
$50$	0	0
$51$	6.34723	0.888791
$52$	0	0
$53$	−3.13271	−0.430310	−0.215155	−	0.976580i	$-0.569026\pi$
$53$	−3.13271	−0.430310	−0.215155	+	0.976580i	$0.569026\pi$
$54$	0	0
$55$	17.0166	2.29452
$56$	0	0
$57$	0.614491	0.0813913
$58$	0	0
$59$	10.5002	1.36701	0.683504	−	0.729946i	$-0.260455\pi$
$59$	10.5002	1.36701	0.683504	+	0.729946i	$0.260455\pi$
$60$	0	0
$61$	6.63652	0.849719	0.424860	−	0.905259i	$-0.360323\pi$
$61$	6.63652	0.849719	0.424860	+	0.905259i	$0.360323\pi$
$62$	0	0
$63$	−1.82462	−0.229881
$64$	0	0
$65$	5.23416	0.649218
$66$	0	0
$67$	3.17640	0.388060	0.194030	−	0.980996i	$-0.437844\pi$
$67$	3.17640	0.388060	0.194030	+	0.980996i	$0.437844\pi$
$68$	0	0
$69$	1.63063	0.196305
$70$	0	0
$71$	3.43455	0.407607	0.203803	−	0.979012i	$-0.434670\pi$
$71$	3.43455	0.407607	0.203803	+	0.979012i	$0.434670\pi$
$72$	0	0
$73$	5.65907	0.662344	0.331172	−	0.943570i	$-0.392556\pi$
$73$	5.65907	0.662344	0.331172	+	0.943570i	$0.392556\pi$
$74$	0	0
$75$	−10.8304	−1.25059
$76$	0	0
$77$	13.0417	1.48624
$78$	0	0
$79$	−6.10707	−0.687099	−0.343550	−	0.939134i	$-0.611629\pi$
$79$	−6.10707	−0.687099	−0.343550	+	0.939134i	$0.611629\pi$
$80$	0	0
$81$	−10.6586	−1.18429
$82$	0	0
$83$	13.3768	1.46830	0.734148	−	0.678990i	$-0.237582\pi$
$83$	13.3768	1.46830	0.734148	+	0.678990i	$0.237582\pi$
$84$	0	0
$85$	10.7021	1.16080
$86$	0	0
$87$	−19.5810	−2.09930
$88$	0	0
$89$	−0.348778	−0.0369704	−0.0184852	−	0.999829i	$-0.505884\pi$
$89$	−0.348778	−0.0369704	−0.0184852	+	0.999829i	$0.505884\pi$
$90$	0	0
$91$	4.01150	0.420519
$92$	0	0
$93$	11.5205	1.19462
$94$	0	0
$95$	1.03609	0.106301
$96$	0	0
$97$	1.21946	0.123818	0.0619089	−	0.998082i	$-0.480281\pi$
$97$	1.21946	0.123818	0.0619089	+	0.998082i	$0.480281\pi$
$98$	0	0
$99$	−3.81939	−0.383863
$100$	0	0
$101$	5.11687	0.509147	0.254574	−	0.967053i	$-0.418065\pi$
$101$	5.11687	0.509147	0.254574	+	0.967053i	$0.418065\pi$
$102$	0	0
$103$	−7.79945	−0.768503	−0.384251	−	0.923228i	$-0.625540\pi$
$103$	−7.79945	−0.768503	−0.384251	+	0.923228i	$0.625540\pi$
$104$	0	0
$105$	−15.7024	−1.53239
$106$	0	0
$107$	−8.42983	−0.814942	−0.407471	−	0.913218i	$-0.633589\pi$
$107$	−8.42983	−0.814942	−0.407471	+	0.913218i	$0.633589\pi$
$108$	0	0
$109$	2.59498	0.248554	0.124277	−	0.992248i	$-0.460339\pi$
$109$	2.59498	0.248554	0.124277	+	0.992248i	$0.460339\pi$
$110$	0	0
$111$	−13.7439	−1.30452
$112$	0	0
$113$	−1.26951	−0.119425	−0.0597126	−	0.998216i	$-0.519018\pi$
$113$	−1.26951	−0.119425	−0.0597126	+	0.998216i	$0.519018\pi$
$114$	0	0
$115$	2.74941	0.256384
$116$	0	0
$117$	−1.17481	−0.108611
$118$	0	0
$119$	8.20216	0.751890
$120$	0	0
$121$	16.2995	1.48177
$122$	0	0
$123$	−6.69913	−0.604040
$124$	0	0
$125$	−1.97697	−0.176826
$126$	0	0
$127$	2.80679	0.249063	0.124531	−	0.992216i	$-0.460257\pi$
$127$	2.80679	0.249063	0.124531	+	0.992216i	$0.460257\pi$
$128$	0	0
$129$	12.4742	1.09829
$130$	0	0
$131$	−9.39553	−0.820891	−0.410446	−	0.911885i	$-0.634627\pi$
$131$	−9.39553	−0.820891	−0.410446	+	0.911885i	$0.634627\pi$
$132$	0	0
$133$	0.794070	0.0688546
$134$	0	0
$135$	−14.2739	−1.22851
$136$	0	0
$137$	9.99367	0.853817	0.426908	−	0.904295i	$-0.359603\pi$
$137$	9.99367	0.853817	0.426908	+	0.904295i	$0.359603\pi$
$138$	0	0
$139$	10.0177	0.849688	0.424844	−	0.905266i	$-0.360329\pi$
$139$	10.0177	0.849688	0.424844	+	0.905266i	$0.360329\pi$
$140$	0	0
$141$	12.2519	1.03179
$142$	0	0
$143$	8.39707	0.702198
$144$	0	0
$145$	−33.0156	−2.74179
$146$	0	0
$147$	1.48665	0.122617
$148$	0	0
$149$	−13.7578	−1.12708	−0.563540	−	0.826089i	$-0.690561\pi$
$149$	−13.7578	−1.12708	−0.563540	+	0.826089i	$0.690561\pi$
$150$	0	0
$151$	−18.6834	−1.52044	−0.760218	−	0.649669i	$-0.774908\pi$
$151$	−18.6834	−1.52044	−0.760218	+	0.649669i	$0.774908\pi$
$152$	0	0
$153$	−2.40209	−0.194197
$154$	0	0
$155$	19.4247	1.56023
$156$	0	0
$157$	15.0437	1.20062	0.600310	−	0.799768i	$-0.295044\pi$
$157$	15.0437	1.20062	0.600310	+	0.799768i	$0.295044\pi$
$158$	0	0
$159$	6.05107	0.479881
$160$	0	0
$161$	2.10716	0.166068
$162$	0	0
$163$	−20.3881	−1.59692	−0.798460	−	0.602048i	$-0.794352\pi$
$163$	−20.3881	−1.59692	−0.798460	+	0.602048i	$0.794352\pi$
$164$	0	0
$165$	−32.8690	−2.55885
$166$	0	0
$167$	−11.7257	−0.907361	−0.453681	−	0.891164i	$-0.649889\pi$
$167$	−11.7257	−0.907361	−0.453681	+	0.891164i	$0.649889\pi$
$168$	0	0
$169$	−10.4171	−0.801319
$170$	0	0
$171$	−0.232552	−0.0177837
$172$	0	0
$173$	−8.39655	−0.638378	−0.319189	−	0.947691i	$-0.603410\pi$
$173$	−8.39655	−0.638378	−0.319189	+	0.947691i	$0.603410\pi$
$174$	0	0
$175$	−13.9955	−1.05796
$176$	0	0
$177$	−20.2820	−1.52449
$178$	0	0
$179$	20.1106	1.50314	0.751569	−	0.659654i	$-0.229297\pi$
$179$	20.1106	1.50314	0.751569	+	0.659654i	$0.229297\pi$
$180$	0	0
$181$	−25.2776	−1.87887	−0.939436	−	0.342724i	$-0.888650\pi$
$181$	−25.2776	−1.87887	−0.939436	+	0.342724i	$0.888650\pi$
$182$	0	0
$183$	−12.8190	−0.947605
$184$	0	0
$185$	−23.1737	−1.70376
$186$	0	0
$187$	17.1692	1.25553
$188$	0	0
$189$	−10.9396	−0.795742
$190$	0	0
$191$	14.8973	1.07793	0.538966	−	0.842328i	$-0.318815\pi$
$191$	14.8973	1.07793	0.538966	+	0.842328i	$0.318815\pi$
$192$	0	0
$193$	6.81893	0.490838	0.245419	−	0.969417i	$-0.421075\pi$
$193$	6.81893	0.490838	0.245419	+	0.969417i	$0.421075\pi$
$194$	0	0
$195$	−10.1102	−0.724006
$196$	0	0
$197$	6.38147	0.454661	0.227330	−	0.973818i	$-0.427000\pi$
$197$	6.38147	0.454661	0.227330	+	0.973818i	$0.427000\pi$
$198$	0	0
$199$	16.2779	1.15391	0.576957	−	0.816775i	$-0.304240\pi$
$199$	16.2779	1.15391	0.576957	+	0.816775i	$0.304240\pi$
$200$	0	0
$201$	−6.13548	−0.432763
$202$	0	0
$203$	−25.3034	−1.77595
$204$	0	0
$205$	−11.2954	−0.788906
$206$	0	0
$207$	−0.617105	−0.0428918
$208$	0	0
$209$	1.66219	0.114976
$210$	0	0
$211$	5.56207	0.382908	0.191454	−	0.981502i	$-0.438680\pi$
$211$	5.56207	0.382908	0.191454	+	0.981502i	$0.438680\pi$
$212$	0	0
$213$	−6.63411	−0.454562
$214$	0	0
$215$	21.0328	1.43442
$216$	0	0
$217$	14.8872	1.01061
$218$	0	0
$219$	−10.9309	−0.738644
$220$	0	0
$221$	5.28107	0.355244
$222$	0	0
$223$	−13.5443	−0.906993	−0.453496	−	0.891258i	$-0.649823\pi$
$223$	−13.5443	−0.906993	−0.453496	+	0.891258i	$0.649823\pi$
$224$	0	0
$225$	4.09873	0.273249
$226$	0	0
$227$	22.9245	1.52155	0.760776	−	0.649014i	$-0.224819\pi$
$227$	22.9245	1.52155	0.760776	+	0.649014i	$0.224819\pi$
$228$	0	0
$229$	20.2850	1.34047	0.670235	−	0.742149i	$-0.266193\pi$
$229$	20.2850	1.34047	0.670235	+	0.742149i	$0.266193\pi$
$230$	0	0
$231$	−25.1910	−1.65745
$232$	0	0
$233$	−15.5186	−1.01666	−0.508329	−	0.861163i	$-0.669737\pi$
$233$	−15.5186	−1.01666	−0.508329	+	0.861163i	$0.669737\pi$
$234$	0	0
$235$	20.6579	1.34757
$236$	0	0
$237$	11.7963	0.766252
$238$	0	0
$239$	15.2540	0.986697	0.493349	−	0.869832i	$-0.335773\pi$
$239$	15.2540	0.986697	0.493349	+	0.869832i	$0.335773\pi$
$240$	0	0
$241$	−9.92736	−0.639477	−0.319739	−	0.947506i	$-0.603595\pi$
$241$	−9.92736	−0.639477	−0.319739	+	0.947506i	$0.603595\pi$
$242$	0	0
$243$	7.43974	0.477260
$244$	0	0
$245$	2.50664	0.160143
$246$	0	0
$247$	0.511273	0.0325315
$248$	0	0
$249$	−25.8384	−1.63744
$250$	0	0
$251$	22.4811	1.41900	0.709498	−	0.704708i	$-0.248922\pi$
$251$	22.4811	1.41900	0.709498	+	0.704708i	$0.248922\pi$
$252$	0	0
$253$	4.41082	0.277306
$254$	0	0
$255$	−20.6719	−1.29453
$256$	0	0
$257$	−6.28947	−0.392327	−0.196163	−	0.980571i	$-0.562848\pi$
$257$	−6.28947	−0.392327	−0.196163	+	0.980571i	$0.562848\pi$
$258$	0	0
$259$	−17.7605	−1.10358
$260$	0	0
$261$	7.41035	0.458689
$262$	0	0
$263$	−3.02818	−0.186725	−0.0933627	−	0.995632i	$-0.529762\pi$
$263$	−3.02818	−0.186725	−0.0933627	+	0.995632i	$0.529762\pi$
$264$	0	0
$265$	10.2027	0.626749
$266$	0	0
$267$	0.673692	0.0412293
$268$	0	0
$269$	−9.89976	−0.603599	−0.301799	−	0.953371i	$-0.597587\pi$
$269$	−9.89976	−0.603599	−0.301799	+	0.953371i	$0.597587\pi$
$270$	0	0
$271$	19.3965	1.17825	0.589126	−	0.808041i	$-0.299472\pi$
$271$	19.3965	1.17825	0.589126	+	0.808041i	$0.299472\pi$
$272$	0	0
$273$	−7.74852	−0.468962
$274$	0	0
$275$	−29.2961	−1.76662
$276$	0	0
$277$	20.7842	1.24880	0.624400	−	0.781105i	$-0.285344\pi$
$277$	20.7842	1.24880	0.624400	+	0.781105i	$0.285344\pi$
$278$	0	0
$279$	−4.35989	−0.261020
$280$	0	0
$281$	−7.83071	−0.467141	−0.233570	−	0.972340i	$-0.575041\pi$
$281$	−7.83071	−0.467141	−0.233570	+	0.972340i	$0.575041\pi$
$282$	0	0
$283$	5.77094	0.343047	0.171523	−	0.985180i	$-0.445131\pi$
$283$	5.77094	0.343047	0.171523	+	0.985180i	$0.445131\pi$
$284$	0	0
$285$	−2.00130	−0.118547
$286$	0	0
$287$	−8.65689	−0.511000
$288$	0	0
$289$	−6.20199	−0.364823
$290$	0	0
$291$	−2.35549	−0.138081
$292$	0	0
$293$	24.5711	1.43546	0.717731	−	0.696321i	$-0.245181\pi$
$293$	24.5711	1.43546	0.717731	+	0.696321i	$0.245181\pi$
$294$	0	0
$295$	−34.1975	−1.99105
$296$	0	0
$297$	−22.8994	−1.32876
$298$	0	0
$299$	1.35673	0.0784616
$300$	0	0
$301$	16.1196	0.929121
$302$	0	0
$303$	−9.88364	−0.567800
$304$	0	0
$305$	−21.6141	−1.23762
$306$	0	0
$307$	2.93006	0.167227	0.0836136	−	0.996498i	$-0.473354\pi$
$307$	2.93006	0.167227	0.0836136	+	0.996498i	$0.473354\pi$
$308$	0	0
$309$	15.0653	0.857033
$310$	0	0
$311$	31.2827	1.77388	0.886940	−	0.461885i	$-0.152827\pi$
$311$	31.2827	1.77388	0.886940	+	0.461885i	$0.152827\pi$
$312$	0	0
$313$	8.49398	0.480108	0.240054	−	0.970760i	$-0.422835\pi$
$313$	8.49398	0.480108	0.240054	+	0.970760i	$0.422835\pi$
$314$	0	0
$315$	5.94251	0.334822
$316$	0	0
$317$	−22.8681	−1.28440	−0.642199	−	0.766538i	$-0.721978\pi$
$317$	−22.8681	−1.28440	−0.642199	+	0.766538i	$0.721978\pi$
$318$	0	0
$319$	−52.9663	−2.96554
$320$	0	0
$321$	16.2829	0.908822
$322$	0	0
$323$	1.04538	0.0581665
$324$	0	0
$325$	−9.01120	−0.499851
$326$	0	0
$327$	−5.01242	−0.277187
$328$	0	0
$329$	15.8324	0.872867
$330$	0	0
$331$	−34.9427	−1.92062	−0.960312	−	0.278928i	$-0.910021\pi$
$331$	−34.9427	−1.92062	−0.960312	+	0.278928i	$0.910021\pi$
$332$	0	0
$333$	5.20134	0.285032
$334$	0	0
$335$	−10.3450	−0.565210
$336$	0	0
$337$	−10.8691	−0.592080	−0.296040	−	0.955176i	$-0.595666\pi$
$337$	−10.8691	−0.592080	−0.296040	+	0.955176i	$0.595666\pi$
$338$	0	0
$339$	2.45215	0.133183
$340$	0	0
$341$	31.1627	1.68756
$342$	0	0
$343$	19.3936	1.04715
$344$	0	0
$345$	−5.31070	−0.285918
$346$	0	0
$347$	0.518032	0.0278094	0.0139047	−	0.999903i	$-0.495574\pi$
$347$	0.518032	0.0278094	0.0139047	+	0.999903i	$0.495574\pi$
$348$	0	0
$349$	16.0553	0.859423	0.429711	−	0.902966i	$-0.358615\pi$
$349$	16.0553	0.859423	0.429711	+	0.902966i	$0.358615\pi$
$350$	0	0
$351$	−7.04365	−0.375962
$352$	0	0
$353$	−29.9508	−1.59412	−0.797060	−	0.603900i	$-0.793613\pi$
$353$	−29.9508	−1.59412	−0.797060	+	0.603900i	$0.793613\pi$
$354$	0	0
$355$	−11.1858	−0.593681
$356$	0	0
$357$	−15.8431	−0.838507
$358$	0	0
$359$	17.2927	0.912676	0.456338	−	0.889807i	$-0.349161\pi$
$359$	17.2927	0.912676	0.456338	+	0.889807i	$0.349161\pi$
$360$	0	0
$361$	−18.8988	−0.994673
$362$	0	0
$363$	−31.4838	−1.65247
$364$	0	0
$365$	−18.4307	−0.964706
$366$	0	0
$367$	8.67168	0.452658	0.226329	−	0.974051i	$-0.427328\pi$
$367$	8.67168	0.452658	0.226329	+	0.974051i	$0.427328\pi$
$368$	0	0
$369$	2.53526	0.131980
$370$	0	0
$371$	7.81944	0.405965
$372$	0	0
$373$	−16.9910	−0.879759	−0.439879	−	0.898057i	$-0.644979\pi$
$373$	−16.9910	−0.879759	−0.439879	+	0.898057i	$0.644979\pi$
$374$	0	0
$375$	3.81868	0.197196
$376$	0	0
$377$	−16.2919	−0.839077
$378$	0	0
$379$	−12.1999	−0.626667	−0.313334	−	0.949643i	$-0.601446\pi$
$379$	−12.1999	−0.626667	−0.313334	+	0.949643i	$0.601446\pi$
$380$	0	0
$381$	−5.42154	−0.277754
$382$	0	0
$383$	29.0447	1.48412	0.742058	−	0.670335i	$-0.233850\pi$
$383$	29.0447	1.48412	0.742058	+	0.670335i	$0.233850\pi$
$384$	0	0
$385$	−42.4747	−2.16471
$386$	0	0
$387$	−4.72081	−0.239972
$388$	0	0
$389$	18.5970	0.942907	0.471453	−	0.881891i	$-0.343730\pi$
$389$	18.5970	0.942907	0.471453	+	0.881891i	$0.343730\pi$
$390$	0	0
$391$	2.77405	0.140290
$392$	0	0
$393$	18.1482	0.915456
$394$	0	0
$395$	19.8898	1.00076
$396$	0	0
$397$	−24.9790	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$397$	−24.9790	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$398$	0	0
$399$	−1.53381	−0.0767865
$400$	0	0
$401$	−15.7785	−0.787940	−0.393970	−	0.919123i	$-0.628899\pi$
$401$	−15.7785	−0.787940	−0.393970	+	0.919123i	$0.628899\pi$
$402$	0	0
$403$	9.58537	0.477481
$404$	0	0
$405$	34.7135	1.72493
$406$	0	0
$407$	−37.1771	−1.84280
$408$	0	0
$409$	−17.5303	−0.866820	−0.433410	−	0.901197i	$-0.642690\pi$
$409$	−17.5303	−0.866820	−0.433410	+	0.901197i	$0.642690\pi$
$410$	0	0
$411$	−19.3036	−0.952175
$412$	0	0
$413$	−26.2092	−1.28967
$414$	0	0
$415$	−43.5662	−2.13858
$416$	0	0
$417$	−19.3499	−0.947571
$418$	0	0
$419$	1.79975	0.0879234	0.0439617	−	0.999033i	$-0.486002\pi$
$419$	1.79975	0.0879234	0.0439617	+	0.999033i	$0.486002\pi$
$420$	0	0
$421$	2.92513	0.142562	0.0712810	−	0.997456i	$-0.477291\pi$
$421$	2.92513	0.142562	0.0712810	+	0.997456i	$0.477291\pi$
$422$	0	0
$423$	−4.63668	−0.225443
$424$	0	0
$425$	−18.4249	−0.893737
$426$	0	0
$427$	−16.5652	−0.801646
$428$	0	0
$429$	−16.2196	−0.783090
$430$	0	0
$431$	−5.45469	−0.262743	−0.131372	−	0.991333i	$-0.541938\pi$
$431$	−5.45469	−0.262743	−0.131372	+	0.991333i	$0.541938\pi$
$432$	0	0
$433$	−22.5873	−1.08548	−0.542738	−	0.839902i	$-0.682612\pi$
$433$	−22.5873	−1.08548	−0.542738	+	0.839902i	$0.682612\pi$
$434$	0	0
$435$	63.7722	3.05764
$436$	0	0
$437$	0.268562	0.0128471
$438$	0	0
$439$	1.26310	0.0602847	0.0301423	−	0.999546i	$-0.490404\pi$
$439$	1.26310	0.0602847	0.0301423	+	0.999546i	$0.490404\pi$
$440$	0	0
$441$	−0.562617	−0.0267913
$442$	0	0
$443$	−6.70758	−0.318687	−0.159343	−	0.987223i	$-0.550938\pi$
$443$	−6.70758	−0.318687	−0.159343	+	0.987223i	$0.550938\pi$
$444$	0	0
$445$	1.13591	0.0538475
$446$	0	0
$447$	26.5742	1.25692
$448$	0	0
$449$	31.9482	1.50773	0.753865	−	0.657029i	$-0.228187\pi$
$449$	31.9482	1.50773	0.753865	+	0.657029i	$0.228187\pi$
$450$	0	0
$451$	−18.1210	−0.853286
$452$	0	0
$453$	36.0885	1.69559
$454$	0	0
$455$	−13.0648	−0.612488
$456$	0	0
$457$	10.0022	0.467884	0.233942	−	0.972251i	$-0.424837\pi$
$457$	10.0022	0.467884	0.233942	+	0.972251i	$0.424837\pi$
$458$	0	0
$459$	−14.4019	−0.672222
$460$	0	0
$461$	9.98124	0.464873	0.232436	−	0.972612i	$-0.425330\pi$
$461$	9.98124	0.464873	0.232436	+	0.972612i	$0.425330\pi$
$462$	0	0
$463$	−3.90654	−0.181552	−0.0907762	−	0.995871i	$-0.528935\pi$
$463$	−3.90654	−0.181552	−0.0907762	+	0.995871i	$0.528935\pi$
$464$	0	0
$465$	−37.5204	−1.73997
$466$	0	0
$467$	−38.0908	−1.76263	−0.881317	−	0.472525i	$-0.843343\pi$
$467$	−38.0908	−1.76263	−0.881317	+	0.472525i	$0.843343\pi$
$468$	0	0
$469$	−7.92852	−0.366105
$470$	0	0
$471$	−29.0581	−1.33893
$472$	0	0
$473$	33.7425	1.55148
$474$	0	0
$475$	−1.78375	−0.0818442
$476$	0	0
$477$	−2.29001	−0.104852
$478$	0	0
$479$	36.4234	1.66423	0.832115	−	0.554604i	$-0.187130\pi$
$479$	36.4234	1.66423	0.832115	+	0.554604i	$0.187130\pi$
$480$	0	0
$481$	−11.4353	−0.521406
$482$	0	0
$483$	−4.07016	−0.185198
$484$	0	0
$485$	−3.97160	−0.180341
$486$	0	0
$487$	17.1748	0.778266	0.389133	−	0.921182i	$-0.372775\pi$
$487$	17.1748	0.778266	0.389133	+	0.921182i	$0.372775\pi$
$488$	0	0
$489$	39.3813	1.78088
$490$	0	0
$491$	−32.3296	−1.45901	−0.729507	−	0.683974i	$-0.760250\pi$
$491$	−32.3296	−1.45901	−0.729507	+	0.683974i	$0.760250\pi$
$492$	0	0
$493$	−33.3115	−1.50027
$494$	0	0
$495$	12.4392	0.559098
$496$	0	0
$497$	−8.57287	−0.384546
$498$	0	0
$499$	−2.80730	−0.125672	−0.0628359	−	0.998024i	$-0.520014\pi$
$499$	−2.80730	−0.125672	−0.0628359	+	0.998024i	$0.520014\pi$
$500$	0	0
$501$	22.6491	1.01189
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−16.6648	−0.741575
$506$	0	0
$507$	20.1215	0.893629
$508$	0	0
$509$	−26.5444	−1.17656	−0.588281	−	0.808657i	$-0.700195\pi$
$509$	−26.5444	−1.17656	−0.588281	+	0.808657i	$0.700195\pi$
$510$	0	0
$511$	−14.1254	−0.624871
$512$	0	0
$513$	−1.39428	−0.0615589
$514$	0	0
$515$	25.4016	1.11933
$516$	0	0
$517$	33.1411	1.45755
$518$	0	0
$519$	16.2186	0.711918
$520$	0	0
$521$	28.9337	1.26761	0.633803	−	0.773494i	$-0.281493\pi$
$521$	28.9337	1.26761	0.633803	+	0.773494i	$0.281493\pi$
$522$	0	0
$523$	−23.0490	−1.00786	−0.503932	−	0.863743i	$-0.668114\pi$
$523$	−23.0490	−1.00786	−0.503932	+	0.863743i	$0.668114\pi$
$524$	0	0
$525$	27.0334	1.17983
$526$	0	0
$527$	19.5988	0.853739
$528$	0	0
$529$	−22.2873	−0.969015
$530$	0	0
$531$	7.67563	0.333094
$532$	0	0
$533$	−5.57386	−0.241431
$534$	0	0
$535$	27.4546	1.18697
$536$	0	0
$537$	−38.8453	−1.67630
$538$	0	0
$539$	4.02136	0.173212
$540$	0	0
$541$	−1.44150	−0.0619748	−0.0309874	−	0.999520i	$-0.509865\pi$
$541$	−1.44150	−0.0619748	−0.0309874	+	0.999520i	$0.509865\pi$
$542$	0	0
$543$	48.8258	2.09531
$544$	0	0
$545$	−8.45145	−0.362020
$546$	0	0
$547$	11.2540	0.481185	0.240592	−	0.970626i	$-0.422658\pi$
$547$	11.2540	0.481185	0.240592	+	0.970626i	$0.422658\pi$
$548$	0	0
$549$	4.85129	0.207048
$550$	0	0
$551$	−3.22496	−0.137388
$552$	0	0
$553$	15.2437	0.648226
$554$	0	0
$555$	44.7618	1.90003
$556$	0	0
$557$	2.89368	0.122609	0.0613045	−	0.998119i	$-0.480474\pi$
$557$	2.89368	0.122609	0.0613045	+	0.998119i	$0.480474\pi$
$558$	0	0
$559$	10.3789	0.438979
$560$	0	0
$561$	−33.1636	−1.40017
$562$	0	0
$563$	−14.8085	−0.624103	−0.312052	−	0.950065i	$-0.601016\pi$
$563$	−14.8085	−0.624103	−0.312052	+	0.950065i	$0.601016\pi$
$564$	0	0
$565$	4.13458	0.173943
$566$	0	0
$567$	26.6047	1.11729
$568$	0	0
$569$	1.96531	0.0823900	0.0411950	−	0.999151i	$-0.486884\pi$
$569$	1.96531	0.0823900	0.0411950	+	0.999151i	$0.486884\pi$
$570$	0	0
$571$	−38.1249	−1.59548	−0.797739	−	0.603004i	$-0.793970\pi$
$571$	−38.1249	−1.59548	−0.797739	+	0.603004i	$0.793970\pi$
$572$	0	0
$573$	−28.7754	−1.20211
$574$	0	0
$575$	−4.73341	−0.197397
$576$	0	0
$577$	−4.22543	−0.175907	−0.0879534	−	0.996125i	$-0.528033\pi$
$577$	−4.22543	−0.175907	−0.0879534	+	0.996125i	$0.528033\pi$
$578$	0	0
$579$	−13.1713	−0.547381
$580$	0	0
$581$	−33.3894	−1.38523
$582$	0	0
$583$	16.3680	0.677895
$584$	0	0
$585$	3.82617	0.158193
$586$	0	0
$587$	−4.04087	−0.166784	−0.0833922	−	0.996517i	$-0.526575\pi$
$587$	−4.04087	−0.166784	−0.0833922	+	0.996517i	$0.526575\pi$
$588$	0	0
$589$	1.89741	0.0781814
$590$	0	0
$591$	−12.3263	−0.507037
$592$	0	0
$593$	39.4053	1.61818	0.809091	−	0.587683i	$-0.199960\pi$
$593$	39.4053	1.61818	0.809091	+	0.587683i	$0.199960\pi$
$594$	0	0
$595$	−26.7131	−1.09513
$596$	0	0
$597$	−31.4421	−1.28684
$598$	0	0
$599$	29.4858	1.20476	0.602378	−	0.798211i	$-0.294220\pi$
$599$	29.4858	1.20476	0.602378	+	0.798211i	$0.294220\pi$
$600$	0	0
$601$	−8.78475	−0.358337	−0.179169	−	0.983818i	$-0.557341\pi$
$601$	−8.78475	−0.358337	−0.179169	+	0.983818i	$0.557341\pi$
$602$	0	0
$603$	2.32195	0.0945571
$604$	0	0
$605$	−53.0849	−2.15821
$606$	0	0
$607$	−21.5398	−0.874272	−0.437136	−	0.899395i	$-0.644007\pi$
$607$	−21.5398	−0.874272	−0.437136	+	0.899395i	$0.644007\pi$
$608$	0	0
$609$	48.8754	1.98053
$610$	0	0
$611$	10.1939	0.412401
$612$	0	0
$613$	2.36894	0.0956804	0.0478402	−	0.998855i	$-0.484766\pi$
$613$	2.36894	0.0956804	0.0478402	+	0.998855i	$0.484766\pi$
$614$	0	0
$615$	21.8180	0.879787
$616$	0	0
$617$	4.06832	0.163784	0.0818921	−	0.996641i	$-0.473904\pi$
$617$	4.06832	0.163784	0.0818921	+	0.996641i	$0.473904\pi$
$618$	0	0
$619$	27.5252	1.10633	0.553165	−	0.833072i	$-0.313420\pi$
$619$	27.5252	1.10633	0.553165	+	0.833072i	$0.313420\pi$
$620$	0	0
$621$	−3.69990	−0.148472
$622$	0	0
$623$	0.870573	0.0348788
$624$	0	0
$625$	−21.5964	−0.863857
$626$	0	0
$627$	−3.21065	−0.128221
$628$	0	0
$629$	−23.3814	−0.932277
$630$	0	0
$631$	39.7768	1.58349	0.791746	−	0.610851i	$-0.209173\pi$
$631$	39.7768	1.58349	0.791746	+	0.610851i	$0.209173\pi$
$632$	0	0
$633$	−10.7436	−0.427019
$634$	0	0
$635$	−9.14128	−0.362761
$636$	0	0
$637$	1.23693	0.0490090
$638$	0	0
$639$	2.51066	0.0993200
$640$	0	0
$641$	−3.58262	−0.141505	−0.0707524	−	0.997494i	$-0.522540\pi$
$641$	−3.58262	−0.141505	−0.0707524	+	0.997494i	$0.522540\pi$
$642$	0	0
$643$	9.27629	0.365821	0.182911	−	0.983130i	$-0.441448\pi$
$643$	9.27629	0.365821	0.182911	+	0.983130i	$0.441448\pi$
$644$	0	0
$645$	−40.6264	−1.59966
$646$	0	0
$647$	12.0206	0.472577	0.236289	−	0.971683i	$-0.424069\pi$
$647$	12.0206	0.472577	0.236289	+	0.971683i	$0.424069\pi$
$648$	0	0
$649$	−54.8624	−2.15354
$650$	0	0
$651$	−28.7559	−1.12703
$652$	0	0
$653$	−26.9230	−1.05358	−0.526790	−	0.849995i	$-0.676605\pi$
$653$	−26.9230	−1.05358	−0.526790	+	0.849995i	$0.676605\pi$
$654$	0	0
$655$	30.5998	1.19563
$656$	0	0
$657$	4.13677	0.161391
$658$	0	0
$659$	−21.9253	−0.854088	−0.427044	−	0.904231i	$-0.640445\pi$
$659$	−21.9253	−0.854088	−0.427044	+	0.904231i	$0.640445\pi$
$660$	0	0
$661$	15.4481	0.600861	0.300430	−	0.953804i	$-0.402870\pi$
$661$	15.4481	0.600861	0.300430	+	0.953804i	$0.402870\pi$
$662$	0	0
$663$	−10.2008	−0.396167
$664$	0	0
$665$	−2.58616	−0.100287
$666$	0	0
$667$	−8.55784	−0.331361
$668$	0	0
$669$	26.1619	1.01148
$670$	0	0
$671$	−34.6751	−1.33862
$672$	0	0
$673$	−13.2391	−0.510331	−0.255165	−	0.966897i	$-0.582130\pi$
$673$	−13.2391	−0.510331	−0.255165	+	0.966897i	$0.582130\pi$
$674$	0	0
$675$	24.5742	0.945861
$676$	0	0
$677$	−3.48269	−0.133851	−0.0669253	−	0.997758i	$-0.521319\pi$
$677$	−3.48269	−0.133851	−0.0669253	+	0.997758i	$0.521319\pi$
$678$	0	0
$679$	−3.04386	−0.116813
$680$	0	0
$681$	−44.2805	−1.69683
$682$	0	0
$683$	20.1787	0.772115	0.386058	−	0.922475i	$-0.373837\pi$
$683$	20.1787	0.772115	0.386058	+	0.922475i	$0.373837\pi$
$684$	0	0
$685$	−32.5478	−1.24359
$686$	0	0
$687$	−39.1821	−1.49489
$688$	0	0
$689$	5.03466	0.191805
$690$	0	0
$691$	−11.3469	−0.431657	−0.215829	−	0.976431i	$-0.569245\pi$
$691$	−11.3469	−0.431657	−0.215829	+	0.976431i	$0.569245\pi$
$692$	0	0
$693$	9.53345	0.362146
$694$	0	0
$695$	−32.6260	−1.23757
$696$	0	0
$697$	−11.3967	−0.431679
$698$	0	0
$699$	29.9754	1.13378
$700$	0	0
$701$	38.5860	1.45737	0.728686	−	0.684848i	$-0.240131\pi$
$701$	38.5860	1.45737	0.728686	+	0.684848i	$0.240131\pi$
$702$	0	0
$703$	−2.26361	−0.0853735
$704$	0	0
$705$	−39.9024	−1.50281
$706$	0	0
$707$	−12.7720	−0.480342
$708$	0	0
$709$	4.66768	0.175298	0.0876491	−	0.996151i	$-0.472065\pi$
$709$	4.66768	0.175298	0.0876491	+	0.996151i	$0.472065\pi$
$710$	0	0
$711$	−4.46427	−0.167423
$712$	0	0
$713$	5.03501	0.188563
$714$	0	0
$715$	−27.3479	−1.02275
$716$	0	0
$717$	−29.4643	−1.10036
$718$	0	0
$719$	3.43973	0.128280	0.0641402	−	0.997941i	$-0.479570\pi$
$719$	3.43973	0.128280	0.0641402	+	0.997941i	$0.479570\pi$
$720$	0	0
$721$	19.4679	0.725024
$722$	0	0
$723$	19.1755	0.713144
$724$	0	0
$725$	56.8400	2.11098
$726$	0	0
$727$	−25.1334	−0.932148	−0.466074	−	0.884746i	$-0.654332\pi$
$727$	−25.1334	−0.932148	−0.466074	+	0.884746i	$0.654332\pi$
$728$	0	0
$729$	17.6055	0.652054
$730$	0	0
$731$	21.2213	0.784897
$732$	0	0
$733$	−46.9560	−1.73436	−0.867180	−	0.497994i	$-0.834070\pi$
$733$	−46.9560	−1.73436	−0.867180	+	0.497994i	$0.834070\pi$
$734$	0	0
$735$	−4.84178	−0.178592
$736$	0	0
$737$	−16.5964	−0.611335
$738$	0	0
$739$	10.3351	0.380181	0.190091	−	0.981767i	$-0.439122\pi$
$739$	10.3351	0.380181	0.190091	+	0.981767i	$0.439122\pi$
$740$	0	0
$741$	−0.987565	−0.0362791
$742$	0	0
$743$	−45.2593	−1.66040	−0.830201	−	0.557464i	$-0.811775\pi$
$743$	−45.2593	−1.66040	−0.830201	+	0.557464i	$0.811775\pi$
$744$	0	0
$745$	44.8069	1.64160
$746$	0	0
$747$	9.77844	0.357774
$748$	0	0
$749$	21.0414	0.768836
$750$	0	0
$751$	3.01822	0.110136	0.0550681	−	0.998483i	$-0.482462\pi$
$751$	3.01822	0.110136	0.0550681	+	0.998483i	$0.482462\pi$
$752$	0	0
$753$	−43.4241	−1.58246
$754$	0	0
$755$	60.8490	2.21452
$756$	0	0
$757$	49.4304	1.79658	0.898290	−	0.439404i	$-0.144810\pi$
$757$	49.4304	1.79658	0.898290	+	0.439404i	$0.144810\pi$
$758$	0	0
$759$	−8.51986	−0.309251
$760$	0	0
$761$	−20.6539	−0.748702	−0.374351	−	0.927287i	$-0.622134\pi$
$761$	−20.6539	−0.748702	−0.374351	+	0.927287i	$0.622134\pi$
$762$	0	0
$763$	−6.47725	−0.234492
$764$	0	0
$765$	7.82322	0.282849
$766$	0	0
$767$	−16.8752	−0.609326
$768$	0	0
$769$	31.4669	1.13472	0.567362	−	0.823468i	$-0.307964\pi$
$769$	31.4669	1.13472	0.567362	+	0.823468i	$0.307964\pi$
$770$	0	0
$771$	12.1486	0.437522
$772$	0	0
$773$	19.9296	0.716816	0.358408	−	0.933565i	$-0.383320\pi$
$773$	19.9296	0.716816	0.358408	+	0.933565i	$0.383320\pi$
$774$	0	0
$775$	−33.4419	−1.20127
$776$	0	0
$777$	34.3057	1.23071
$778$	0	0
$779$	−1.10334	−0.0395312
$780$	0	0
$781$	−17.9452	−0.642129
$782$	0	0
$783$	44.4293	1.58777
$784$	0	0
$785$	−48.9950	−1.74871
$786$	0	0
$787$	−50.7585	−1.80935	−0.904673	−	0.426107i	$-0.859885\pi$
$787$	−50.7585	−1.80935	−0.904673	+	0.426107i	$0.859885\pi$
$788$	0	0
$789$	5.84916	0.208236
$790$	0	0
$791$	3.16877	0.112669
$792$	0	0
$793$	−10.6657	−0.378751
$794$	0	0
$795$	−19.7074	−0.698949
$796$	0	0
$797$	42.9104	1.51997	0.759983	−	0.649943i	$-0.225207\pi$
$797$	42.9104	1.51997	0.759983	+	0.649943i	$0.225207\pi$
$798$	0	0
$799$	20.8431	0.737375
$800$	0	0
$801$	−0.254956	−0.00900844
$802$	0	0
$803$	−29.5680	−1.04343
$804$	0	0
$805$	−6.86270	−0.241878
$806$	0	0
$807$	19.1222	0.673132
$808$	0	0
$809$	−31.1611	−1.09557	−0.547784	−	0.836620i	$-0.684528\pi$
$809$	−31.1611	−1.09557	−0.547784	+	0.836620i	$0.684528\pi$
$810$	0	0
$811$	−6.89675	−0.242178	−0.121089	−	0.992642i	$-0.538639\pi$
$811$	−6.89675	−0.242178	−0.121089	+	0.992642i	$0.538639\pi$
$812$	0	0
$813$	−37.4658	−1.31398
$814$	0	0
$815$	66.4009	2.32592
$816$	0	0
$817$	2.05448	0.0718772
$818$	0	0
$819$	2.93240	0.102466
$820$	0	0
$821$	39.7112	1.38593	0.692966	−	0.720970i	$-0.256304\pi$
$821$	39.7112	1.38593	0.692966	+	0.720970i	$0.256304\pi$
$822$	0	0
$823$	7.59636	0.264792	0.132396	−	0.991197i	$-0.457733\pi$
$823$	7.59636	0.264792	0.132396	+	0.991197i	$0.457733\pi$
$824$	0	0
$825$	56.5877	1.97013
$826$	0	0
$827$	36.6977	1.27611	0.638053	−	0.769993i	$-0.279740\pi$
$827$	36.6977	1.27611	0.638053	+	0.769993i	$0.279740\pi$
$828$	0	0
$829$	50.1438	1.74157	0.870784	−	0.491666i	$-0.163612\pi$
$829$	50.1438	1.74157	0.870784	+	0.491666i	$0.163612\pi$
$830$	0	0
$831$	−40.1463	−1.39266
$832$	0	0
$833$	2.52911	0.0876284
$834$	0	0
$835$	38.1887	1.32158
$836$	0	0
$837$	−26.1400	−0.903531
$838$	0	0
$839$	−44.1212	−1.52323	−0.761616	−	0.648028i	$-0.775594\pi$
$839$	−44.1212	−1.52323	−0.761616	+	0.648028i	$0.775594\pi$
$840$	0	0
$841$	73.7647	2.54361
$842$	0	0
$843$	15.1256	0.520954
$844$	0	0
$845$	33.9270	1.16712
$846$	0	0
$847$	−40.6846	−1.39794
$848$	0	0
$849$	−11.1470	−0.382565
$850$	0	0
$851$	−6.00676	−0.205909
$852$	0	0
$853$	−22.7447	−0.778763	−0.389382	−	0.921076i	$-0.627311\pi$
$853$	−22.7447	−0.778763	−0.389382	+	0.921076i	$0.627311\pi$
$854$	0	0
$855$	0.757384	0.0259020
$856$	0	0
$857$	36.7269	1.25457	0.627284	−	0.778791i	$-0.284167\pi$
$857$	36.7269	1.25457	0.627284	+	0.778791i	$0.284167\pi$
$858$	0	0
$859$	−11.3178	−0.386160	−0.193080	−	0.981183i	$-0.561848\pi$
$859$	−11.3178	−0.386160	−0.193080	+	0.981183i	$0.561848\pi$
$860$	0	0
$861$	16.7215	0.569866
$862$	0	0
$863$	−46.2917	−1.57579	−0.787895	−	0.615810i	$-0.788829\pi$
$863$	−46.2917	−1.57579	−0.787895	+	0.615810i	$0.788829\pi$
$864$	0	0
$865$	27.3462	0.929800
$866$	0	0
$867$	11.9796	0.406850
$868$	0	0
$869$	31.9088	1.08243
$870$	0	0
$871$	−5.10489	−0.172973
$872$	0	0
$873$	0.891427	0.0301702
$874$	0	0
$875$	4.93465	0.166822
$876$	0	0
$877$	−5.50862	−0.186013	−0.0930064	−	0.995666i	$-0.529648\pi$
$877$	−5.50862	−0.186013	−0.0930064	+	0.995666i	$0.529648\pi$
$878$	0	0
$879$	−47.4611	−1.60082
$880$	0	0
$881$	55.3576	1.86505	0.932523	−	0.361111i	$-0.117602\pi$
$881$	55.3576	1.86505	0.932523	+	0.361111i	$0.117602\pi$
$882$	0	0
$883$	−18.5166	−0.623133	−0.311567	−	0.950224i	$-0.600854\pi$
$883$	−18.5166	−0.623133	−0.311567	+	0.950224i	$0.600854\pi$
$884$	0	0
$885$	66.0551	2.22042
$886$	0	0
$887$	6.90636	0.231893	0.115947	−	0.993255i	$-0.463010\pi$
$887$	6.90636	0.231893	0.115947	+	0.993255i	$0.463010\pi$
$888$	0	0
$889$	−7.00594	−0.234972
$890$	0	0
$891$	55.6902	1.86569
$892$	0	0
$893$	2.01787	0.0675254
$894$	0	0
$895$	−65.4971	−2.18933
$896$	0	0
$897$	−2.62063	−0.0875002
$898$	0	0
$899$	−60.4617	−2.01651
$900$	0	0
$901$	10.2942	0.342949
$902$	0	0
$903$	−31.1364	−1.03615
$904$	0	0
$905$	82.3253	2.73658
$906$	0	0
$907$	28.3324	0.940762	0.470381	−	0.882463i	$-0.344116\pi$
$907$	28.3324	0.940762	0.470381	+	0.882463i	$0.344116\pi$
$908$	0	0
$909$	3.74043	0.124062
$910$	0	0
$911$	−34.3942	−1.13953	−0.569765	−	0.821807i	$-0.692966\pi$
$911$	−34.3942	−1.13953	−0.569765	+	0.821807i	$0.692966\pi$
$912$	0	0
$913$	−69.8924	−2.31310
$914$	0	0
$915$	41.7494	1.38019
$916$	0	0
$917$	23.4519	0.774449
$918$	0	0
$919$	−18.5091	−0.610559	−0.305280	−	0.952263i	$-0.598750\pi$
$919$	−18.5091	−0.610559	−0.305280	+	0.952263i	$0.598750\pi$
$920$	0	0
$921$	−5.65964	−0.186491
$922$	0	0
$923$	−5.51977	−0.181685
$924$	0	0
$925$	39.8961	1.31178
$926$	0	0
$927$	−5.70139	−0.187258
$928$	0	0
$929$	−17.0365	−0.558951	−0.279475	−	0.960153i	$-0.590160\pi$
$929$	−17.0365	−0.558951	−0.279475	+	0.960153i	$0.590160\pi$
$930$	0	0
$931$	0.244849	0.00802460
$932$	0	0
$933$	−60.4250	−1.97823
$934$	0	0
$935$	−55.9173	−1.82869
$936$	0	0
$937$	−42.3399	−1.38318	−0.691592	−	0.722289i	$-0.743090\pi$
$937$	−42.3399	−1.38318	−0.691592	+	0.722289i	$0.743090\pi$
$938$	0	0
$939$	−16.4068	−0.535416
$940$	0	0
$941$	−49.9124	−1.62710	−0.813548	−	0.581497i	$-0.802467\pi$
$941$	−49.9124	−1.62710	−0.813548	+	0.581497i	$0.802467\pi$
$942$	0	0
$943$	−2.92784	−0.0953437
$944$	0	0
$945$	35.6287	1.15900
$946$	0	0
$947$	8.61036	0.279799	0.139900	−	0.990166i	$-0.455322\pi$
$947$	8.61036	0.279799	0.139900	+	0.990166i	$0.455322\pi$
$948$	0	0
$949$	−9.09484	−0.295231
$950$	0	0
$951$	44.1715	1.43236
$952$	0	0
$953$	24.1661	0.782817	0.391408	−	0.920217i	$-0.371988\pi$
$953$	24.1661	0.782817	0.391408	+	0.920217i	$0.371988\pi$
$954$	0	0
$955$	−48.5182	−1.57001
$956$	0	0
$957$	102.309	3.30716
$958$	0	0
$959$	−24.9449	−0.805512
$960$	0	0
$961$	4.57269	0.147506
$962$	0	0
$963$	−6.16220	−0.198574
$964$	0	0
$965$	−22.2082	−0.714907
$966$	0	0
$967$	48.4646	1.55852	0.779258	−	0.626704i	$-0.215596\pi$
$967$	48.4646	1.55852	0.779258	+	0.626704i	$0.215596\pi$
$968$	0	0
$969$	−2.01924	−0.0648672
$970$	0	0
$971$	52.8078	1.69468	0.847341	−	0.531048i	$-0.178202\pi$
$971$	52.8078	1.69468	0.847341	+	0.531048i	$0.178202\pi$
$972$	0	0
$973$	−25.0048	−0.801617
$974$	0	0
$975$	17.4058	0.557433
$976$	0	0
$977$	15.4177	0.493256	0.246628	−	0.969110i	$-0.420677\pi$
$977$	15.4177	0.493256	0.246628	+	0.969110i	$0.420677\pi$
$978$	0	0
$979$	1.82233	0.0582418
$980$	0	0
$981$	1.89693	0.0605643
$982$	0	0
$983$	23.9771	0.764750	0.382375	−	0.924007i	$-0.375106\pi$
$983$	23.9771	0.764750	0.382375	+	0.924007i	$0.375106\pi$
$984$	0	0
$985$	−20.7834	−0.662215
$986$	0	0
$987$	−30.5815	−0.973420
$988$	0	0
$989$	5.45182	0.173358
$990$	0	0
$991$	−9.05591	−0.287670	−0.143835	−	0.989602i	$-0.545944\pi$
$991$	−9.05591	−0.287670	−0.143835	+	0.989602i	$0.545944\pi$
$992$	0	0
$993$	67.4946	2.14188
$994$	0	0
$995$	−53.0147	−1.68068
$996$	0	0
$997$	−5.02994	−0.159300	−0.0796499	−	0.996823i	$-0.525380\pi$
$997$	−5.02994	−0.159300	−0.0796499	+	0.996823i	$0.525380\pi$
$998$	0	0
$999$	31.1850	0.986649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.s.1.7		21
4.3	odd	2		2012.2.a.b.1.15	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.b.1.15	✓	21	4.3	odd	2
8048.2.a.s.1.7		21	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)

\(f(q)\)	\(=\)	\(q-1.93158 q^{3} -3.25684 q^{5} -2.49607 q^{7} +0.730999 q^{9} +O(q^{10})\)	\(q-1.93158 q^{3} -3.25684 q^{5} -2.49607 q^{7} +0.730999 q^{9} -5.22489 q^{11} -1.60713 q^{13} +6.29085 q^{15} -3.28603 q^{17} -0.318129 q^{19} +4.82135 q^{21} -0.844194 q^{23} +5.60702 q^{25} +4.38276 q^{27} +10.1373 q^{29} -5.96428 q^{31} +10.0923 q^{33} +8.12929 q^{35} +7.11538 q^{37} +3.10429 q^{39} +3.46821 q^{41} -6.45802 q^{43} -2.38075 q^{45} -6.34293 q^{47} -0.769654 q^{49} +6.34723 q^{51} -3.13271 q^{53} +17.0166 q^{55} +0.614491 q^{57} +10.5002 q^{59} +6.63652 q^{61} -1.82462 q^{63} +5.23416 q^{65} +3.17640 q^{67} +1.63063 q^{69} +3.43455 q^{71} +5.65907 q^{73} -10.8304 q^{75} +13.0417 q^{77} -6.10707 q^{79} -10.6586 q^{81} +13.3768 q^{83} +10.7021 q^{85} -19.5810 q^{87} -0.348778 q^{89} +4.01150 q^{91} +11.5205 q^{93} +1.03609 q^{95} +1.21946 q^{97} -3.81939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9	\( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) \) 21 * q - 10 * q^3 + 3 * q^5 - 13 * q^7 + 21 * q^9 - 7 * q^11 + 12 * q^13 - 14 * q^15 + q^17 - 14 * q^19 + 14 * q^21 - 26 * q^23 + 18 * q^25 - 37 * q^27 + 9 * q^29 - 28 * q^31 + 3 * q^33 - 20 * q^35 + 31 * q^37 - 29 * q^39 + 4 * q^41 - 38 * q^43 + 24 * q^45 - 9 * q^47 + 16 * q^49 - 15 * q^51 + 22 * q^53 - 35 * q^55 - q^57 - 10 * q^59 + 22 * q^61 - 35 * q^63 - 14 * q^65 - 58 * q^67 + 15 * q^69 - 27 * q^71 - 6 * q^73 - 48 * q^75 + 16 * q^77 - 47 * q^79 + 29 * q^81 - 22 * q^83 + 14 * q^85 - 29 * q^87 + q^89 - 51 * q^91 + 34 * q^93 - 23 * q^95 - 2 * q^97 - 22 * q^99

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