LMFDB - Embedded newform 8048.2.a.n.1.3

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:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.410375$ 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)

$f(q)$	$=$	$q-0.330424 q^{3} -0.135598 q^{5} -4.34834 q^{7} -2.89082 q^{9} +O(q^{10})$	$q-0.330424 q^{3} -0.135598 q^{5} -4.34834 q^{7} -2.89082 q^{9} +6.26839 q^{11} -0.927120 q^{13} +0.0448047 q^{15} -4.33042 q^{17} +0.490327 q^{19} +1.43679 q^{21} +5.32761 q^{23} -4.98161 q^{25} +1.94647 q^{27} -0.478108 q^{29} +3.55167 q^{31} -2.07122 q^{33} +0.589625 q^{35} +3.72948 q^{37} +0.306343 q^{39} +11.9236 q^{41} -5.93796 q^{43} +0.391989 q^{45} +8.77304 q^{47} +11.9080 q^{49} +1.43087 q^{51} -7.12594 q^{53} -0.849979 q^{55} -0.162016 q^{57} +0.455180 q^{59} +1.44028 q^{61} +12.5703 q^{63} +0.125715 q^{65} +7.56319 q^{67} -1.76037 q^{69} -3.70352 q^{71} -9.39254 q^{73} +1.64604 q^{75} -27.2571 q^{77} +3.18770 q^{79} +8.02930 q^{81} -11.4455 q^{83} +0.587196 q^{85} +0.157978 q^{87} -9.72360 q^{89} +4.03143 q^{91} -1.17355 q^{93} -0.0664872 q^{95} -6.57307 q^{97} -18.1208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10}) $ 5 * q - q^5 - 3 * q^7 - 3 * q^9	$ 5 q - q^{5} - 3 q^{7} - 3 q^{9} + 11 q^{11} + 5 q^{13} + 8 q^{15} - 20 q^{17} + 4 q^{19} - 4 q^{21} + 4 q^{23} - 6 q^{25} - 3 q^{27} - 6 q^{29} + 3 q^{31} - 7 q^{33} + 3 q^{35} - 10 q^{37} - 4 q^{39} - 6 q^{41} - 11 q^{43} - 17 q^{45} + 9 q^{47} - 4 q^{49} + 12 q^{51} - 22 q^{53} - 14 q^{55} + 10 q^{57} + 10 q^{59} - 11 q^{61} + 3 q^{63} - 12 q^{65} - 14 q^{67} - 3 q^{69} + 26 q^{71} - 7 q^{73} - 15 q^{75} - 26 q^{77} + 15 q^{79} - 7 q^{81} + 12 q^{83} + 12 q^{85} + 18 q^{87} - 5 q^{89} + 22 q^{91} - 21 q^{93} + 10 q^{95} + 6 q^{97} - 8 q^{99}+O(q^{100}) $ 5 * q - q^5 - 3 * q^7 - 3 * q^9 + 11 * q^11 + 5 * q^13 + 8 * q^15 - 20 * q^17 + 4 * q^19 - 4 * q^21 + 4 * q^23 - 6 * q^25 - 3 * q^27 - 6 * q^29 + 3 * q^31 - 7 * q^33 + 3 * q^35 - 10 * q^37 - 4 * q^39 - 6 * q^41 - 11 * q^43 - 17 * q^45 + 9 * q^47 - 4 * q^49 + 12 * q^51 - 22 * q^53 - 14 * q^55 + 10 * q^57 + 10 * q^59 - 11 * q^61 + 3 * q^63 - 12 * q^65 - 14 * q^67 - 3 * q^69 + 26 * q^71 - 7 * q^73 - 15 * q^75 - 26 * q^77 + 15 * q^79 - 7 * q^81 + 12 * q^83 + 12 * q^85 + 18 * q^87 - 5 * q^89 + 22 * q^91 - 21 * q^93 + 10 * q^95 + 6 * q^97 - 8 * 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
$3$	−0.330424	−0.190770	−0.0953851	−	0.995440i	$-0.530408\pi$
$3$	−0.330424	−0.190770	−0.0953851	+	0.995440i	$0.530408\pi$
$4$	0	0
$5$	−0.135598	−0.0606411	−0.0303206	−	0.999540i	$-0.509653\pi$
$5$	−0.135598	−0.0606411	−0.0303206	+	0.999540i	$0.509653\pi$
$6$	0	0
$7$	−4.34834	−1.64352	−0.821758	−	0.569836i	$-0.807007\pi$
$7$	−4.34834	−1.64352	−0.821758	+	0.569836i	$0.807007\pi$
$8$	0	0
$9$	−2.89082	−0.963607
$10$	0	0
$11$	6.26839	1.88999	0.944995	−	0.327086i	$-0.106067\pi$
$11$	6.26839	1.88999	0.944995	+	0.327086i	$0.106067\pi$
$12$	0	0
$13$	−0.927120	−0.257137	−0.128568	−	0.991701i	$-0.541038\pi$
$13$	−0.927120	−0.257137	−0.128568	+	0.991701i	$0.541038\pi$
$14$	0	0
$15$	0.0448047	0.0115685
$16$	0	0
$17$	−4.33042	−1.05028	−0.525141	−	0.851015i	$-0.675987\pi$
$17$	−4.33042	−1.05028	−0.525141	+	0.851015i	$0.675987\pi$
$18$	0	0
$19$	0.490327	0.112489	0.0562444	−	0.998417i	$-0.482087\pi$
$19$	0.490327	0.112489	0.0562444	+	0.998417i	$0.482087\pi$
$20$	0	0
$21$	1.43679	0.313534
$22$	0	0
$23$	5.32761	1.11088	0.555442	−	0.831555i	$-0.312549\pi$
$23$	5.32761	1.11088	0.555442	+	0.831555i	$0.312549\pi$
$24$	0	0
$25$	−4.98161	−0.996323
$26$	0	0
$27$	1.94647	0.374598
$28$	0	0
$29$	−0.478108	−0.0887824	−0.0443912	−	0.999014i	$-0.514135\pi$
$29$	−0.478108	−0.0887824	−0.0443912	+	0.999014i	$0.514135\pi$
$30$	0	0
$31$	3.55167	0.637898	0.318949	−	0.947772i	$-0.396670\pi$
$31$	3.55167	0.637898	0.318949	+	0.947772i	$0.396670\pi$
$32$	0	0
$33$	−2.07122	−0.360554
$34$	0	0
$35$	0.589625	0.0996648
$36$	0	0
$37$	3.72948	0.613123	0.306562	−	0.951851i	$-0.400821\pi$
$37$	3.72948	0.613123	0.306562	+	0.951851i	$0.400821\pi$
$38$	0	0
$39$	0.306343	0.0490541
$40$	0	0
$41$	11.9236	1.86216	0.931078	−	0.364821i	$-0.118870\pi$
$41$	11.9236	1.86216	0.931078	+	0.364821i	$0.118870\pi$
$42$	0	0
$43$	−5.93796	−0.905531	−0.452765	−	0.891630i	$-0.649562\pi$
$43$	−5.93796	−0.905531	−0.452765	+	0.891630i	$0.649562\pi$
$44$	0	0
$45$	0.391989	0.0584342
$46$	0	0
$47$	8.77304	1.27968	0.639840	−	0.768508i	$-0.279001\pi$
$47$	8.77304	1.27968	0.639840	+	0.768508i	$0.279001\pi$
$48$	0	0
$49$	11.9080	1.70115
$50$	0	0
$51$	1.43087	0.200362
$52$	0	0
$53$	−7.12594	−0.978823	−0.489412	−	0.872053i	$-0.662788\pi$
$53$	−7.12594	−0.978823	−0.489412	+	0.872053i	$0.662788\pi$
$54$	0	0
$55$	−0.849979	−0.114611
$56$	0	0
$57$	−0.162016	−0.0214595
$58$	0	0
$59$	0.455180	0.0592594	0.0296297	−	0.999561i	$-0.490567\pi$
$59$	0.455180	0.0592594	0.0296297	+	0.999561i	$0.490567\pi$
$60$	0	0
$61$	1.44028	0.184409	0.0922047	−	0.995740i	$-0.470609\pi$
$61$	1.44028	0.184409	0.0922047	+	0.995740i	$0.470609\pi$
$62$	0	0
$63$	12.5703	1.58370
$64$	0	0
$65$	0.125715	0.0155931
$66$	0	0
$67$	7.56319	0.923991	0.461995	−	0.886882i	$-0.347134\pi$
$67$	7.56319	0.923991	0.461995	+	0.886882i	$0.347134\pi$
$68$	0	0
$69$	−1.76037	−0.211924
$70$	0	0
$71$	−3.70352	−0.439527	−0.219764	−	0.975553i	$-0.570529\pi$
$71$	−3.70352	−0.439527	−0.219764	+	0.975553i	$0.570529\pi$
$72$	0	0
$73$	−9.39254	−1.09931	−0.549657	−	0.835391i	$-0.685241\pi$
$73$	−9.39254	−1.09931	−0.549657	+	0.835391i	$0.685241\pi$
$74$	0	0
$75$	1.64604	0.190069
$76$	0	0
$77$	−27.2571	−3.10623
$78$	0	0
$79$	3.18770	0.358644	0.179322	−	0.983790i	$-0.442610\pi$
$79$	3.18770	0.358644	0.179322	+	0.983790i	$0.442610\pi$
$80$	0	0
$81$	8.02930	0.892145
$82$	0	0
$83$	−11.4455	−1.25631	−0.628154	−	0.778089i	$-0.716189\pi$
$83$	−11.4455	−1.25631	−0.628154	+	0.778089i	$0.716189\pi$
$84$	0	0
$85$	0.587196	0.0636903
$86$	0	0
$87$	0.157978	0.0169370
$88$	0	0
$89$	−9.72360	−1.03070	−0.515350	−	0.856980i	$-0.672338\pi$
$89$	−9.72360	−1.03070	−0.515350	+	0.856980i	$0.672338\pi$
$90$	0	0
$91$	4.03143	0.422609
$92$	0	0
$93$	−1.17355	−0.121692
$94$	0	0
$95$	−0.0664872	−0.00682145
$96$	0	0
$97$	−6.57307	−0.667394	−0.333697	−	0.942680i	$-0.608296\pi$
$97$	−6.57307	−0.667394	−0.333697	+	0.942680i	$0.608296\pi$
$98$	0	0
$99$	−18.1208	−1.82121
$100$	0	0
$101$	6.97683	0.694220	0.347110	−	0.937824i	$-0.387163\pi$
$101$	6.97683	0.694220	0.347110	+	0.937824i	$0.387163\pi$
$102$	0	0
$103$	6.14762	0.605743	0.302872	−	0.953031i	$-0.402055\pi$
$103$	6.14762	0.605743	0.302872	+	0.953031i	$0.402055\pi$
$104$	0	0
$105$	−0.194826	−0.0190131
$106$	0	0
$107$	−15.3500	−1.48394	−0.741969	−	0.670434i	$-0.766108\pi$
$107$	−15.3500	−1.48394	−0.741969	+	0.670434i	$0.766108\pi$
$108$	0	0
$109$	−8.06787	−0.772761	−0.386381	−	0.922339i	$-0.626275\pi$
$109$	−8.06787	−0.772761	−0.386381	+	0.922339i	$0.626275\pi$
$110$	0	0
$111$	−1.23231	−0.116966
$112$	0	0
$113$	−19.1797	−1.80428	−0.902138	−	0.431447i	$-0.858003\pi$
$113$	−19.1797	−1.80428	−0.902138	+	0.431447i	$0.858003\pi$
$114$	0	0
$115$	−0.722412	−0.0673653
$116$	0	0
$117$	2.68014	0.247779
$118$	0	0
$119$	18.8301	1.72616
$120$	0	0
$121$	28.2927	2.57206
$122$	0	0
$123$	−3.93984	−0.355244
$124$	0	0
$125$	1.35348	0.121059
$126$	0	0
$127$	−1.25290	−0.111177	−0.0555885	−	0.998454i	$-0.517703\pi$
$127$	−1.25290	−0.111177	−0.0555885	+	0.998454i	$0.517703\pi$
$128$	0	0
$129$	1.96204	0.172748
$130$	0	0
$131$	5.81298	0.507882	0.253941	−	0.967220i	$-0.418273\pi$
$131$	5.81298	0.507882	0.253941	+	0.967220i	$0.418273\pi$
$132$	0	0
$133$	−2.13211	−0.184877
$134$	0	0
$135$	−0.263936	−0.0227160
$136$	0	0
$137$	17.0323	1.45517	0.727585	−	0.686018i	$-0.240643\pi$
$137$	17.0323	1.45517	0.727585	+	0.686018i	$0.240643\pi$
$138$	0	0
$139$	19.9120	1.68891	0.844456	−	0.535625i	$-0.179924\pi$
$139$	19.9120	1.68891	0.844456	+	0.535625i	$0.179924\pi$
$140$	0	0
$141$	−2.89882	−0.244125
$142$	0	0
$143$	−5.81155	−0.485986
$144$	0	0
$145$	0.0648304	0.00538387
$146$	0	0
$147$	−3.93470	−0.324528
$148$	0	0
$149$	0.985577	0.0807416	0.0403708	−	0.999185i	$-0.487146\pi$
$149$	0.985577	0.0807416	0.0403708	+	0.999185i	$0.487146\pi$
$150$	0	0
$151$	18.0186	1.46633	0.733165	−	0.680050i	$-0.238042\pi$
$151$	18.0186	1.46633	0.733165	+	0.680050i	$0.238042\pi$
$152$	0	0
$153$	12.5185	1.01206
$154$	0	0
$155$	−0.481598	−0.0386829
$156$	0	0
$157$	−13.9489	−1.11324	−0.556621	−	0.830767i	$-0.687902\pi$
$157$	−13.9489	−1.11324	−0.556621	+	0.830767i	$0.687902\pi$
$158$	0	0
$159$	2.35458	0.186730
$160$	0	0
$161$	−23.1663	−1.82576
$162$	0	0
$163$	−21.8557	−1.71187	−0.855937	−	0.517080i	$-0.827019\pi$
$163$	−21.8557	−1.71187	−0.855937	+	0.517080i	$0.827019\pi$
$164$	0	0
$165$	0.280853	0.0218644
$166$	0	0
$167$	−19.5905	−1.51596	−0.757980	−	0.652278i	$-0.773813\pi$
$167$	−19.5905	−1.51596	−0.757980	+	0.652278i	$0.773813\pi$
$168$	0	0
$169$	−12.1404	−0.933881
$170$	0	0
$171$	−1.41745	−0.108395
$172$	0	0
$173$	6.93774	0.527466	0.263733	−	0.964596i	$-0.415046\pi$
$173$	6.93774	0.527466	0.263733	+	0.964596i	$0.415046\pi$
$174$	0	0
$175$	21.6617	1.63747
$176$	0	0
$177$	−0.150402	−0.0113049
$178$	0	0
$179$	−3.51035	−0.262376	−0.131188	−	0.991358i	$-0.541879\pi$
$179$	−3.51035	−0.262376	−0.131188	+	0.991358i	$0.541879\pi$
$180$	0	0
$181$	−13.7415	−1.02139	−0.510697	−	0.859761i	$-0.670613\pi$
$181$	−13.7415	−1.02139	−0.510697	+	0.859761i	$0.670613\pi$
$182$	0	0
$183$	−0.475904	−0.0351798
$184$	0	0
$185$	−0.505710	−0.0371805
$186$	0	0
$187$	−27.1448	−1.98502
$188$	0	0
$189$	−8.46389	−0.615657
$190$	0	0
$191$	−11.1243	−0.804925	−0.402462	−	0.915436i	$-0.631846\pi$
$191$	−11.1243	−0.804925	−0.402462	+	0.915436i	$0.631846\pi$
$192$	0	0
$193$	11.7036	0.842445	0.421222	−	0.906957i	$-0.361601\pi$
$193$	11.7036	0.842445	0.421222	+	0.906957i	$0.361601\pi$
$194$	0	0
$195$	−0.0415393	−0.00297469
$196$	0	0
$197$	−4.76975	−0.339830	−0.169915	−	0.985459i	$-0.554349\pi$
$197$	−4.76975	−0.339830	−0.169915	+	0.985459i	$0.554349\pi$
$198$	0	0
$199$	−4.27433	−0.302999	−0.151500	−	0.988457i	$-0.548410\pi$
$199$	−4.27433	−0.302999	−0.151500	+	0.988457i	$0.548410\pi$
$200$	0	0
$201$	−2.49906	−0.176270
$202$	0	0
$203$	2.07897	0.145915
$204$	0	0
$205$	−1.61681	−0.112923
$206$	0	0
$207$	−15.4012	−1.07046
$208$	0	0
$209$	3.07356	0.212603
$210$	0	0
$211$	−0.0640960	−0.00441255	−0.00220628	−	0.999998i	$-0.500702\pi$
$211$	−0.0640960	−0.00441255	−0.00220628	+	0.999998i	$0.500702\pi$
$212$	0	0
$213$	1.22373	0.0838487
$214$	0	0
$215$	0.805174	0.0549124
$216$	0	0
$217$	−15.4438	−1.04840
$218$	0	0
$219$	3.10352	0.209716
$220$	0	0
$221$	4.01482	0.270066
$222$	0	0
$223$	13.3505	0.894017	0.447009	−	0.894530i	$-0.352489\pi$
$223$	13.3505	0.894017	0.447009	+	0.894530i	$0.352489\pi$
$224$	0	0
$225$	14.4009	0.960063
$226$	0	0
$227$	−13.3815	−0.888165	−0.444082	−	0.895986i	$-0.646470\pi$
$227$	−13.3815	−0.888165	−0.444082	+	0.895986i	$0.646470\pi$
$228$	0	0
$229$	24.1983	1.59907	0.799534	−	0.600621i	$-0.205080\pi$
$229$	24.1983	1.59907	0.799534	+	0.600621i	$0.205080\pi$
$230$	0	0
$231$	9.00637	0.592576
$232$	0	0
$233$	−18.0849	−1.18478	−0.592392	−	0.805650i	$-0.701816\pi$
$233$	−18.0849	−1.18478	−0.592392	+	0.805650i	$0.701816\pi$
$234$	0	0
$235$	−1.18960	−0.0776013
$236$	0	0
$237$	−1.05329	−0.0684186
$238$	0	0
$239$	−21.8920	−1.41608	−0.708039	−	0.706173i	$-0.750420\pi$
$239$	−21.8920	−1.41608	−0.708039	+	0.706173i	$0.750420\pi$
$240$	0	0
$241$	2.87717	0.185335	0.0926674	−	0.995697i	$-0.470461\pi$
$241$	2.87717	0.185335	0.0926674	+	0.995697i	$0.470461\pi$
$242$	0	0
$243$	−8.49247	−0.544792
$244$	0	0
$245$	−1.61470	−0.103160
$246$	0	0
$247$	−0.454592	−0.0289250
$248$	0	0
$249$	3.78187	0.239666
$250$	0	0
$251$	−17.2483	−1.08870	−0.544351	−	0.838858i	$-0.683224\pi$
$251$	−17.2483	−1.08870	−0.544351	+	0.838858i	$0.683224\pi$
$252$	0	0
$253$	33.3955	2.09956
$254$	0	0
$255$	−0.194023	−0.0121502
$256$	0	0
$257$	−25.5981	−1.59677	−0.798384	−	0.602149i	$-0.794311\pi$
$257$	−25.5981	−1.59677	−0.798384	+	0.602149i	$0.794311\pi$
$258$	0	0
$259$	−16.2171	−1.00768
$260$	0	0
$261$	1.38212	0.0855513
$262$	0	0
$263$	24.2567	1.49573	0.747866	−	0.663850i	$-0.231079\pi$
$263$	24.2567	1.49573	0.747866	+	0.663850i	$0.231079\pi$
$264$	0	0
$265$	0.966261	0.0593570
$266$	0	0
$267$	3.21291	0.196627
$268$	0	0
$269$	−27.1739	−1.65682	−0.828411	−	0.560121i	$-0.810755\pi$
$269$	−27.1739	−1.65682	−0.828411	+	0.560121i	$0.810755\pi$
$270$	0	0
$271$	−1.65441	−0.100498	−0.0502492	−	0.998737i	$-0.516002\pi$
$271$	−1.65441	−0.100498	−0.0502492	+	0.998737i	$0.516002\pi$
$272$	0	0
$273$	−1.33208	−0.0806212
$274$	0	0
$275$	−31.2267	−1.88304
$276$	0	0
$277$	18.9580	1.13908	0.569538	−	0.821965i	$-0.307122\pi$
$277$	18.9580	1.13908	0.569538	+	0.821965i	$0.307122\pi$
$278$	0	0
$279$	−10.2672	−0.614683
$280$	0	0
$281$	−17.6353	−1.05204	−0.526018	−	0.850473i	$-0.676316\pi$
$281$	−17.6353	−1.05204	−0.526018	+	0.850473i	$0.676316\pi$
$282$	0	0
$283$	−27.1354	−1.61303	−0.806515	−	0.591214i	$-0.798649\pi$
$283$	−27.1354	−1.61303	−0.806515	+	0.591214i	$0.798649\pi$
$284$	0	0
$285$	0.0219690	0.00130133
$286$	0	0
$287$	−51.8479	−3.06048
$288$	0	0
$289$	1.75257	0.103092
$290$	0	0
$291$	2.17190	0.127319
$292$	0	0
$293$	12.6631	0.739788	0.369894	−	0.929074i	$-0.379394\pi$
$293$	12.6631	0.739788	0.369894	+	0.929074i	$0.379394\pi$
$294$	0	0
$295$	−0.0617214	−0.00359356
$296$	0	0
$297$	12.2012	0.707985
$298$	0	0
$299$	−4.93934	−0.285649
$300$	0	0
$301$	25.8203	1.48826
$302$	0	0
$303$	−2.30531	−0.132437
$304$	0	0
$305$	−0.195299	−0.0111828
$306$	0	0
$307$	−19.0168	−1.08534	−0.542672	−	0.839945i	$-0.682587\pi$
$307$	−19.0168	−1.08534	−0.542672	+	0.839945i	$0.682587\pi$
$308$	0	0
$309$	−2.03132	−0.115558
$310$	0	0
$311$	11.1020	0.629534	0.314767	−	0.949169i	$-0.398074\pi$
$311$	11.1020	0.629534	0.314767	+	0.949169i	$0.398074\pi$
$312$	0	0
$313$	−5.45250	−0.308194	−0.154097	−	0.988056i	$-0.549247\pi$
$313$	−5.45250	−0.308194	−0.154097	+	0.988056i	$0.549247\pi$
$314$	0	0
$315$	−1.70450	−0.0960376
$316$	0	0
$317$	22.2744	1.25106	0.625528	−	0.780202i	$-0.284884\pi$
$317$	22.2744	1.25106	0.625528	+	0.780202i	$0.284884\pi$
$318$	0	0
$319$	−2.99697	−0.167798
$320$	0	0
$321$	5.07199	0.283091
$322$	0	0
$323$	−2.12332	−0.118145
$324$	0	0
$325$	4.61856	0.256191
$326$	0	0
$327$	2.66581	0.147420
$328$	0	0
$329$	−38.1482	−2.10318
$330$	0	0
$331$	−11.6341	−0.639467	−0.319734	−	0.947507i	$-0.603593\pi$
$331$	−11.6341	−0.639467	−0.319734	+	0.947507i	$0.603593\pi$
$332$	0	0
$333$	−10.7813	−0.590810
$334$	0	0
$335$	−1.02555	−0.0560318
$336$	0	0
$337$	13.2714	0.722941	0.361471	−	0.932383i	$-0.382275\pi$
$337$	13.2714	0.722941	0.361471	+	0.932383i	$0.382275\pi$
$338$	0	0
$339$	6.33743	0.344202
$340$	0	0
$341$	22.2632	1.20562
$342$	0	0
$343$	−21.3418	−1.15235
$344$	0	0
$345$	0.238702	0.0128513
$346$	0	0
$347$	27.0757	1.45350	0.726750	−	0.686902i	$-0.241030\pi$
$347$	27.0757	1.45350	0.726750	+	0.686902i	$0.241030\pi$
$348$	0	0
$349$	−30.2391	−1.61866	−0.809330	−	0.587354i	$-0.800170\pi$
$349$	−30.2391	−1.61866	−0.809330	+	0.587354i	$0.800170\pi$
$350$	0	0
$351$	−1.80461	−0.0963229
$352$	0	0
$353$	−12.7468	−0.678444	−0.339222	−	0.940706i	$-0.610164\pi$
$353$	−12.7468	−0.678444	−0.339222	+	0.940706i	$0.610164\pi$
$354$	0	0
$355$	0.502189	0.0266534
$356$	0	0
$357$	−6.22192	−0.329299
$358$	0	0
$359$	−12.4924	−0.659326	−0.329663	−	0.944099i	$-0.606935\pi$
$359$	−12.4924	−0.659326	−0.329663	+	0.944099i	$0.606935\pi$
$360$	0	0
$361$	−18.7596	−0.987346
$362$	0	0
$363$	−9.34856	−0.490672
$364$	0	0
$365$	1.27361	0.0666636
$366$	0	0
$367$	4.01777	0.209726	0.104863	−	0.994487i	$-0.466560\pi$
$367$	4.01777	0.209726	0.104863	+	0.994487i	$0.466560\pi$
$368$	0	0
$369$	−34.4690	−1.79439
$370$	0	0
$371$	30.9860	1.60871
$372$	0	0
$373$	5.04769	0.261359	0.130680	−	0.991425i	$-0.458284\pi$
$373$	5.04769	0.261359	0.130680	+	0.991425i	$0.458284\pi$
$374$	0	0
$375$	−0.447223	−0.0230945
$376$	0	0
$377$	0.443264	0.0228292
$378$	0	0
$379$	−1.19492	−0.0613788	−0.0306894	−	0.999529i	$-0.509770\pi$
$379$	−1.19492	−0.0613788	−0.0306894	+	0.999529i	$0.509770\pi$
$380$	0	0
$381$	0.413988	0.0212093
$382$	0	0
$383$	35.7173	1.82507	0.912534	−	0.409002i	$-0.134123\pi$
$383$	35.7173	1.82507	0.912534	+	0.409002i	$0.134123\pi$
$384$	0	0
$385$	3.69599	0.188365
$386$	0	0
$387$	17.1656	0.872575
$388$	0	0
$389$	28.2349	1.43157	0.715783	−	0.698322i	$-0.246070\pi$
$389$	28.2349	1.43157	0.715783	+	0.698322i	$0.246070\pi$
$390$	0	0
$391$	−23.0708	−1.16674
$392$	0	0
$393$	−1.92075	−0.0968888
$394$	0	0
$395$	−0.432245	−0.0217486
$396$	0	0
$397$	−15.6879	−0.787354	−0.393677	−	0.919249i	$-0.628797\pi$
$397$	−15.6879	−0.787354	−0.393677	+	0.919249i	$0.628797\pi$
$398$	0	0
$399$	0.704499	0.0352691
$400$	0	0
$401$	−17.2093	−0.859391	−0.429696	−	0.902974i	$-0.641379\pi$
$401$	−17.2093	−0.859391	−0.429696	+	0.902974i	$0.641379\pi$
$402$	0	0
$403$	−3.29282	−0.164027
$404$	0	0
$405$	−1.08876	−0.0541007
$406$	0	0
$407$	23.3778	1.15880
$408$	0	0
$409$	−9.43745	−0.466652	−0.233326	−	0.972399i	$-0.574961\pi$
$409$	−9.43745	−0.466652	−0.233326	+	0.972399i	$0.574961\pi$
$410$	0	0
$411$	−5.62788	−0.277603
$412$	0	0
$413$	−1.97928	−0.0973938
$414$	0	0
$415$	1.55198	0.0761839
$416$	0	0
$417$	−6.57939	−0.322194
$418$	0	0
$419$	−2.07274	−0.101260	−0.0506299	−	0.998717i	$-0.516123\pi$
$419$	−2.07274	−0.101260	−0.0506299	+	0.998717i	$0.516123\pi$
$420$	0	0
$421$	−31.3066	−1.52579	−0.762894	−	0.646523i	$-0.776222\pi$
$421$	−31.3066	−1.52579	−0.762894	+	0.646523i	$0.776222\pi$
$422$	0	0
$423$	−25.3613	−1.23311
$424$	0	0
$425$	21.5725	1.04642
$426$	0	0
$427$	−6.26284	−0.303080
$428$	0	0
$429$	1.92027	0.0927116
$430$	0	0
$431$	7.55623	0.363971	0.181985	−	0.983301i	$-0.441748\pi$
$431$	7.55623	0.363971	0.181985	+	0.983301i	$0.441748\pi$
$432$	0	0
$433$	−26.3732	−1.26742	−0.633708	−	0.773572i	$-0.718468\pi$
$433$	−26.3732	−1.26742	−0.633708	+	0.773572i	$0.718468\pi$
$434$	0	0
$435$	−0.0214215	−0.00102708
$436$	0	0
$437$	2.61227	0.124962
$438$	0	0
$439$	0.766258	0.0365715	0.0182857	−	0.999833i	$-0.494179\pi$
$439$	0.766258	0.0365715	0.0182857	+	0.999833i	$0.494179\pi$
$440$	0	0
$441$	−34.4240	−1.63924
$442$	0	0
$443$	2.32539	0.110482	0.0552412	−	0.998473i	$-0.482407\pi$
$443$	2.32539	0.110482	0.0552412	+	0.998473i	$0.482407\pi$
$444$	0	0
$445$	1.31850	0.0625028
$446$	0	0
$447$	−0.325658	−0.0154031
$448$	0	0
$449$	26.1149	1.23244	0.616218	−	0.787575i	$-0.288664\pi$
$449$	26.1149	1.23244	0.616218	+	0.787575i	$0.288664\pi$
$450$	0	0
$451$	74.7418	3.51945
$452$	0	0
$453$	−5.95376	−0.279732
$454$	0	0
$455$	−0.546653	−0.0256275
$456$	0	0
$457$	5.99736	0.280544	0.140272	−	0.990113i	$-0.455202\pi$
$457$	5.99736	0.280544	0.140272	+	0.990113i	$0.455202\pi$
$458$	0	0
$459$	−8.42902	−0.393433
$460$	0	0
$461$	−33.9088	−1.57929	−0.789644	−	0.613565i	$-0.789735\pi$
$461$	−33.9088	−1.57929	−0.789644	+	0.613565i	$0.789735\pi$
$462$	0	0
$463$	−23.7318	−1.10291	−0.551455	−	0.834205i	$-0.685927\pi$
$463$	−23.7318	−1.10291	−0.551455	+	0.834205i	$0.685927\pi$
$464$	0	0
$465$	0.159131	0.00737954
$466$	0	0
$467$	−6.25402	−0.289402	−0.144701	−	0.989475i	$-0.546222\pi$
$467$	−6.25402	−0.289402	−0.144701	+	0.989475i	$0.546222\pi$
$468$	0	0
$469$	−32.8873	−1.51859
$470$	0	0
$471$	4.60904	0.212373
$472$	0	0
$473$	−37.2214	−1.71144
$474$	0	0
$475$	−2.44262	−0.112075
$476$	0	0
$477$	20.5998	0.943201
$478$	0	0
$479$	−15.8682	−0.725035	−0.362518	−	0.931977i	$-0.618083\pi$
$479$	−15.8682	−0.725035	−0.362518	+	0.931977i	$0.618083\pi$
$480$	0	0
$481$	−3.45768	−0.157657
$482$	0	0
$483$	7.65468	0.348300
$484$	0	0
$485$	0.891293	0.0404716
$486$	0	0
$487$	−13.8268	−0.626553	−0.313277	−	0.949662i	$-0.601427\pi$
$487$	−13.8268	−0.626553	−0.313277	+	0.949662i	$0.601427\pi$
$488$	0	0
$489$	7.22166	0.326575
$490$	0	0
$491$	23.8037	1.07425	0.537123	−	0.843504i	$-0.319511\pi$
$491$	23.8037	1.07425	0.537123	+	0.843504i	$0.319511\pi$
$492$	0	0
$493$	2.07041	0.0932466
$494$	0	0
$495$	2.45714	0.110440
$496$	0	0
$497$	16.1042	0.722370
$498$	0	0
$499$	−8.78566	−0.393300	−0.196650	−	0.980474i	$-0.563006\pi$
$499$	−8.78566	−0.393300	−0.196650	+	0.980474i	$0.563006\pi$
$500$	0	0
$501$	6.47317	0.289200
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−0.946042	−0.0420983
$506$	0	0
$507$	4.01149	0.178157
$508$	0	0
$509$	−9.17462	−0.406658	−0.203329	−	0.979110i	$-0.565176\pi$
$509$	−9.17462	−0.406658	−0.203329	+	0.979110i	$0.565176\pi$
$510$	0	0
$511$	40.8419	1.80674
$512$	0	0
$513$	0.954405	0.0421380
$514$	0	0
$515$	−0.833604	−0.0367330
$516$	0	0
$517$	54.9928	2.41858
$518$	0	0
$519$	−2.29239	−0.100625
$520$	0	0
$521$	2.41914	0.105985	0.0529923	−	0.998595i	$-0.483124\pi$
$521$	2.41914	0.105985	0.0529923	+	0.998595i	$0.483124\pi$
$522$	0	0
$523$	−19.2074	−0.839882	−0.419941	−	0.907551i	$-0.637949\pi$
$523$	−19.2074	−0.839882	−0.419941	+	0.907551i	$0.637949\pi$
$524$	0	0
$525$	−7.15755	−0.312381
$526$	0	0
$527$	−15.3802	−0.669973
$528$	0	0
$529$	5.38347	0.234064
$530$	0	0
$531$	−1.31584	−0.0571028
$532$	0	0
$533$	−11.0546	−0.478829
$534$	0	0
$535$	2.08142	0.0899877
$536$	0	0
$537$	1.15990	0.0500535
$538$	0	0
$539$	74.6442	3.21515
$540$	0	0
$541$	12.5558	0.539816	0.269908	−	0.962886i	$-0.413007\pi$
$541$	12.5558	0.539816	0.269908	+	0.962886i	$0.413007\pi$
$542$	0	0
$543$	4.54050	0.194852
$544$	0	0
$545$	1.09398	0.0468611
$546$	0	0
$547$	5.65918	0.241969	0.120985	−	0.992654i	$-0.461395\pi$
$547$	5.65918	0.241969	0.120985	+	0.992654i	$0.461395\pi$
$548$	0	0
$549$	−4.16360	−0.177698
$550$	0	0
$551$	−0.234429	−0.00998703
$552$	0	0
$553$	−13.8612	−0.589438
$554$	0	0
$555$	0.167098	0.00709293
$556$	0	0
$557$	1.04296	0.0441915	0.0220958	−	0.999756i	$-0.492966\pi$
$557$	1.04296	0.0441915	0.0220958	+	0.999756i	$0.492966\pi$
$558$	0	0
$559$	5.50521	0.232845
$560$	0	0
$561$	8.96927	0.378683
$562$	0	0
$563$	−7.84218	−0.330509	−0.165254	−	0.986251i	$-0.552844\pi$
$563$	−7.84218	−0.330509	−0.165254	+	0.986251i	$0.552844\pi$
$564$	0	0
$565$	2.60073	0.109413
$566$	0	0
$567$	−34.9141	−1.46625
$568$	0	0
$569$	−18.8856	−0.791727	−0.395864	−	0.918309i	$-0.629555\pi$
$569$	−18.8856	−0.791727	−0.395864	+	0.918309i	$0.629555\pi$
$570$	0	0
$571$	−5.61412	−0.234944	−0.117472	−	0.993076i	$-0.537479\pi$
$571$	−5.61412	−0.234944	−0.117472	+	0.993076i	$0.537479\pi$
$572$	0	0
$573$	3.67573	0.153556
$574$	0	0
$575$	−26.5401	−1.10680
$576$	0	0
$577$	31.8322	1.32519	0.662596	−	0.748977i	$-0.269455\pi$
$577$	31.8322	1.32519	0.662596	+	0.748977i	$0.269455\pi$
$578$	0	0
$579$	−3.86715	−0.160713
$580$	0	0
$581$	49.7689	2.06476
$582$	0	0
$583$	−44.6681	−1.84997
$584$	0	0
$585$	−0.363421	−0.0150256
$586$	0	0
$587$	−19.4404	−0.802391	−0.401195	−	0.915992i	$-0.631405\pi$
$587$	−19.4404	−0.802391	−0.401195	+	0.915992i	$0.631405\pi$
$588$	0	0
$589$	1.74148	0.0717564
$590$	0	0
$591$	1.57604	0.0648295
$592$	0	0
$593$	3.66746	0.150604	0.0753022	−	0.997161i	$-0.476008\pi$
$593$	3.66746	0.150604	0.0753022	+	0.997161i	$0.476008\pi$
$594$	0	0
$595$	−2.55332	−0.104676
$596$	0	0
$597$	1.41234	0.0578032
$598$	0	0
$599$	−30.7454	−1.25622	−0.628112	−	0.778123i	$-0.716172\pi$
$599$	−30.7454	−1.25622	−0.628112	+	0.778123i	$0.716172\pi$
$600$	0	0
$601$	45.6183	1.86081	0.930405	−	0.366532i	$-0.119455\pi$
$601$	45.6183	1.86081	0.930405	+	0.366532i	$0.119455\pi$
$602$	0	0
$603$	−21.8638	−0.890364
$604$	0	0
$605$	−3.83642	−0.155973
$606$	0	0
$607$	−26.9627	−1.09438	−0.547192	−	0.837007i	$-0.684303\pi$
$607$	−26.9627	−1.09438	−0.547192	+	0.837007i	$0.684303\pi$
$608$	0	0
$609$	−0.686942	−0.0278363
$610$	0	0
$611$	−8.13367	−0.329053
$612$	0	0
$613$	−12.1570	−0.491015	−0.245508	−	0.969395i	$-0.578955\pi$
$613$	−12.1570	−0.491015	−0.245508	+	0.969395i	$0.578955\pi$
$614$	0	0
$615$	0.534234	0.0215424
$616$	0	0
$617$	−32.1456	−1.29413	−0.647067	−	0.762433i	$-0.724005\pi$
$617$	−32.1456	−1.29413	−0.647067	+	0.762433i	$0.724005\pi$
$618$	0	0
$619$	41.7277	1.67718	0.838588	−	0.544766i	$-0.183382\pi$
$619$	41.7277	1.67718	0.838588	+	0.544766i	$0.183382\pi$
$620$	0	0
$621$	10.3700	0.416135
$622$	0	0
$623$	42.2815	1.69397
$624$	0	0
$625$	24.7245	0.988981
$626$	0	0
$627$	−1.01558	−0.0405582
$628$	0	0
$629$	−16.1502	−0.643953
$630$	0	0
$631$	20.2155	0.804767	0.402383	−	0.915471i	$-0.368182\pi$
$631$	20.2155	0.804767	0.402383	+	0.915471i	$0.368182\pi$
$632$	0	0
$633$	0.0211788	0.000841784 0
$634$	0	0
$635$	0.169891	0.00674190
$636$	0	0
$637$	−11.0402	−0.437428
$638$	0	0
$639$	10.7062	0.423531
$640$	0	0
$641$	−2.13732	−0.0844189	−0.0422095	−	0.999109i	$-0.513440\pi$
$641$	−2.13732	−0.0844189	−0.0422095	+	0.999109i	$0.513440\pi$
$642$	0	0
$643$	42.3794	1.67128	0.835641	−	0.549277i	$-0.185097\pi$
$643$	42.3794	1.67128	0.835641	+	0.549277i	$0.185097\pi$
$644$	0	0
$645$	−0.266049	−0.0104757
$646$	0	0
$647$	1.59232	0.0626006	0.0313003	−	0.999510i	$-0.490035\pi$
$647$	1.59232	0.0626006	0.0313003	+	0.999510i	$0.490035\pi$
$648$	0	0
$649$	2.85324	0.112000
$650$	0	0
$651$	5.10301	0.200003
$652$	0	0
$653$	−40.4529	−1.58304	−0.791522	−	0.611140i	$-0.790711\pi$
$653$	−40.4529	−1.58304	−0.791522	+	0.611140i	$0.790711\pi$
$654$	0	0
$655$	−0.788227	−0.0307986
$656$	0	0
$657$	27.1521	1.05931
$658$	0	0
$659$	−23.0110	−0.896382	−0.448191	−	0.893938i	$-0.647931\pi$
$659$	−23.0110	−0.896382	−0.448191	+	0.893938i	$0.647931\pi$
$660$	0	0
$661$	−26.1681	−1.01782	−0.508910	−	0.860820i	$-0.669951\pi$
$661$	−26.1681	−1.01782	−0.508910	+	0.860820i	$0.669951\pi$
$662$	0	0
$663$	−1.32659	−0.0515206
$664$	0	0
$665$	0.289109	0.0112112
$666$	0	0
$667$	−2.54717	−0.0986270
$668$	0	0
$669$	−4.41133	−0.170552
$670$	0	0
$671$	9.02825	0.348532
$672$	0	0
$673$	−42.3139	−1.63108	−0.815540	−	0.578700i	$-0.803560\pi$
$673$	−42.3139	−1.63108	−0.815540	+	0.578700i	$0.803560\pi$
$674$	0	0
$675$	−9.69654	−0.373220
$676$	0	0
$677$	26.3289	1.01190	0.505950	−	0.862563i	$-0.331142\pi$
$677$	26.3289	1.01190	0.505950	+	0.862563i	$0.331142\pi$
$678$	0	0
$679$	28.5819	1.09687
$680$	0	0
$681$	4.42158	0.169435
$682$	0	0
$683$	−0.518560	−0.0198422	−0.00992108	−	0.999951i	$-0.503158\pi$
$683$	−0.518560	−0.0198422	−0.00992108	+	0.999951i	$0.503158\pi$
$684$	0	0
$685$	−2.30954	−0.0882431
$686$	0	0
$687$	−7.99568	−0.305054
$688$	0	0
$689$	6.60660	0.251692
$690$	0	0
$691$	11.8568	0.451052	0.225526	−	0.974237i	$-0.427590\pi$
$691$	11.8568	0.451052	0.225526	+	0.974237i	$0.427590\pi$
$692$	0	0
$693$	78.7952	2.99318
$694$	0	0
$695$	−2.70002	−0.102418
$696$	0	0
$697$	−51.6343	−1.95579
$698$	0	0
$699$	5.97569	0.226022
$700$	0	0
$701$	−1.93296	−0.0730069	−0.0365035	−	0.999334i	$-0.511622\pi$
$701$	−1.93296	−0.0730069	−0.0365035	+	0.999334i	$0.511622\pi$
$702$	0	0
$703$	1.82867	0.0689695
$704$	0	0
$705$	0.393074	0.0148040
$706$	0	0
$707$	−30.3376	−1.14096
$708$	0	0
$709$	5.80940	0.218176	0.109088	−	0.994032i	$-0.465207\pi$
$709$	5.80940	0.218176	0.109088	+	0.994032i	$0.465207\pi$
$710$	0	0
$711$	−9.21507	−0.345592
$712$	0	0
$713$	18.9219	0.708631
$714$	0	0
$715$	0.788033	0.0294708
$716$	0	0
$717$	7.23364	0.270145
$718$	0	0
$719$	0.347674	0.0129660	0.00648302	−	0.999979i	$-0.497936\pi$
$719$	0.347674	0.0129660	0.00648302	+	0.999979i	$0.497936\pi$
$720$	0	0
$721$	−26.7319	−0.995549
$722$	0	0
$723$	−0.950685	−0.0353563
$724$	0	0
$725$	2.38175	0.0884559
$726$	0	0
$727$	8.81895	0.327077	0.163538	−	0.986537i	$-0.447709\pi$
$727$	8.81895	0.327077	0.163538	+	0.986537i	$0.447709\pi$
$728$	0	0
$729$	−21.2818	−0.788215
$730$	0	0
$731$	25.7139	0.951063
$732$	0	0
$733$	21.6460	0.799514	0.399757	−	0.916621i	$-0.369095\pi$
$733$	21.6460	0.799514	0.399757	+	0.916621i	$0.369095\pi$
$734$	0	0
$735$	0.533536	0.0196798
$736$	0	0
$737$	47.4090	1.74633
$738$	0	0
$739$	−2.22552	−0.0818672	−0.0409336	−	0.999162i	$-0.513033\pi$
$739$	−2.22552	−0.0818672	−0.0409336	+	0.999162i	$0.513033\pi$
$740$	0	0
$741$	0.150208	0.00551803
$742$	0	0
$743$	49.7029	1.82342	0.911711	−	0.410833i	$-0.134762\pi$
$743$	49.7029	1.82342	0.911711	+	0.410833i	$0.134762\pi$
$744$	0	0
$745$	−0.133642	−0.00489626
$746$	0	0
$747$	33.0869	1.21059
$748$	0	0
$749$	66.7469	2.43888
$750$	0	0
$751$	−7.67058	−0.279903	−0.139952	−	0.990158i	$-0.544695\pi$
$751$	−7.67058	−0.279903	−0.139952	+	0.990158i	$0.544695\pi$
$752$	0	0
$753$	5.69924	0.207692
$754$	0	0
$755$	−2.44328	−0.0889200
$756$	0	0
$757$	44.2641	1.60881	0.804403	−	0.594083i	$-0.202485\pi$
$757$	44.2641	1.60881	0.804403	+	0.594083i	$0.202485\pi$
$758$	0	0
$759$	−11.0347	−0.400533
$760$	0	0
$761$	−48.6407	−1.76322	−0.881612	−	0.471974i	$-0.843542\pi$
$761$	−48.6407	−1.76322	−0.881612	+	0.471974i	$0.843542\pi$
$762$	0	0
$763$	35.0818	1.27005
$764$	0	0
$765$	−1.69748	−0.0613724
$766$	0	0
$767$	−0.422007	−0.0152378
$768$	0	0
$769$	15.9421	0.574886	0.287443	−	0.957798i	$-0.407195\pi$
$769$	15.9421	0.574886	0.287443	+	0.957798i	$0.407195\pi$
$770$	0	0
$771$	8.45823	0.304616
$772$	0	0
$773$	−51.9972	−1.87021	−0.935104	−	0.354373i	$-0.884694\pi$
$773$	−51.9972	−1.87021	−0.935104	+	0.354373i	$0.884694\pi$
$774$	0	0
$775$	−17.6930	−0.635553
$776$	0	0
$777$	5.35850	0.192235
$778$	0	0
$779$	5.84647	0.209472
$780$	0	0
$781$	−23.2151	−0.830702
$782$	0	0
$783$	−0.930621	−0.0332577
$784$	0	0
$785$	1.89144	0.0675082
$786$	0	0
$787$	3.49306	0.124514	0.0622571	−	0.998060i	$-0.480170\pi$
$787$	3.49306	0.124514	0.0622571	+	0.998060i	$0.480170\pi$
$788$	0	0
$789$	−8.01499	−0.285341
$790$	0	0
$791$	83.3999	2.96536
$792$	0	0
$793$	−1.33532	−0.0474185
$794$	0	0
$795$	−0.319276	−0.0113235
$796$	0	0
$797$	0.522094	0.0184935	0.00924677	−	0.999957i	$-0.497057\pi$
$797$	0.522094	0.0184935	0.00924677	+	0.999957i	$0.497057\pi$
$798$	0	0
$799$	−37.9910	−1.34402
$800$	0	0
$801$	28.1092	0.993189
$802$	0	0
$803$	−58.8760	−2.07769
$804$	0	0
$805$	3.14129	0.110716
$806$	0	0
$807$	8.97890	0.316072
$808$	0	0
$809$	23.6408	0.831166	0.415583	−	0.909555i	$-0.363577\pi$
$809$	23.6408	0.831166	0.415583	+	0.909555i	$0.363577\pi$
$810$	0	0
$811$	30.4387	1.06885	0.534424	−	0.845217i	$-0.320529\pi$
$811$	30.4387	1.06885	0.534424	+	0.845217i	$0.320529\pi$
$812$	0	0
$813$	0.546657	0.0191721
$814$	0	0
$815$	2.96359	0.103810
$816$	0	0
$817$	−2.91154	−0.101862
$818$	0	0
$819$	−11.6541	−0.407229
$820$	0	0
$821$	−13.0982	−0.457129	−0.228564	−	0.973529i	$-0.573403\pi$
$821$	−13.0982	−0.457129	−0.228564	+	0.973529i	$0.573403\pi$
$822$	0	0
$823$	27.1349	0.945865	0.472932	−	0.881099i	$-0.343195\pi$
$823$	27.1349	0.945865	0.472932	+	0.881099i	$0.343195\pi$
$824$	0	0
$825$	10.3180	0.359228
$826$	0	0
$827$	31.0209	1.07870	0.539351	−	0.842081i	$-0.318669\pi$
$827$	31.0209	1.07870	0.539351	+	0.842081i	$0.318669\pi$
$828$	0	0
$829$	−44.3754	−1.54122	−0.770610	−	0.637307i	$-0.780048\pi$
$829$	−44.3754	−1.54122	−0.770610	+	0.637307i	$0.780048\pi$
$830$	0	0
$831$	−6.26418	−0.217302
$832$	0	0
$833$	−51.5668	−1.78669
$834$	0	0
$835$	2.65643	0.0919295
$836$	0	0
$837$	6.91320	0.238955
$838$	0	0
$839$	−32.2700	−1.11408	−0.557041	−	0.830485i	$-0.688064\pi$
$839$	−32.2700	−1.11408	−0.557041	+	0.830485i	$0.688064\pi$
$840$	0	0
$841$	−28.7714	−0.992118
$842$	0	0
$843$	5.82714	0.200697
$844$	0	0
$845$	1.64622	0.0566316
$846$	0	0
$847$	−123.026	−4.22722
$848$	0	0
$849$	8.96616	0.307718
$850$	0	0
$851$	19.8693	0.681109
$852$	0	0
$853$	50.2901	1.72190	0.860950	−	0.508689i	$-0.169870\pi$
$853$	50.2901	1.72190	0.860950	+	0.508689i	$0.169870\pi$
$854$	0	0
$855$	0.192203	0.00657319
$856$	0	0
$857$	14.0977	0.481569	0.240785	−	0.970579i	$-0.422595\pi$
$857$	14.0977	0.481569	0.240785	+	0.970579i	$0.422595\pi$
$858$	0	0
$859$	−3.15309	−0.107582	−0.0537910	−	0.998552i	$-0.517130\pi$
$859$	−3.15309	−0.107582	−0.0537910	+	0.998552i	$0.517130\pi$
$860$	0	0
$861$	17.1318	0.583849
$862$	0	0
$863$	−55.1445	−1.87714	−0.938570	−	0.345088i	$-0.887849\pi$
$863$	−55.1445	−1.87714	−0.938570	+	0.345088i	$0.887849\pi$
$864$	0	0
$865$	−0.940741	−0.0319862
$866$	0	0
$867$	−0.579090	−0.0196669
$868$	0	0
$869$	19.9817	0.677834
$870$	0	0
$871$	−7.01199	−0.237592
$872$	0	0
$873$	19.0016	0.643106
$874$	0	0
$875$	−5.88540	−0.198963
$876$	0	0
$877$	9.83130	0.331979	0.165990	−	0.986127i	$-0.446918\pi$
$877$	9.83130	0.331979	0.165990	+	0.986127i	$0.446918\pi$
$878$	0	0
$879$	−4.18420	−0.141129
$880$	0	0
$881$	14.1391	0.476357	0.238179	−	0.971221i	$-0.423450\pi$
$881$	14.1391	0.476357	0.238179	+	0.971221i	$0.423450\pi$
$882$	0	0
$883$	1.25437	0.0422128	0.0211064	−	0.999777i	$-0.493281\pi$
$883$	1.25437	0.0422128	0.0211064	+	0.999777i	$0.493281\pi$
$884$	0	0
$885$	0.0203942	0.000685544 0
$886$	0	0
$887$	−28.4847	−0.956423	−0.478211	−	0.878245i	$-0.658715\pi$
$887$	−28.4847	−0.956423	−0.478211	+	0.878245i	$0.658715\pi$
$888$	0	0
$889$	5.44804	0.182721
$890$	0	0
$891$	50.3308	1.68614
$892$	0	0
$893$	4.30166	0.143950
$894$	0	0
$895$	0.475996	0.0159108
$896$	0	0
$897$	1.63207	0.0544934
$898$	0	0
$899$	−1.69808	−0.0566342
$900$	0	0
$901$	30.8583	1.02804
$902$	0	0
$903$	−8.53162	−0.283915
$904$	0	0
$905$	1.86331	0.0619385
$906$	0	0
$907$	−18.7722	−0.623322	−0.311661	−	0.950193i	$-0.600885\pi$
$907$	−18.7722	−0.623322	−0.311661	+	0.950193i	$0.600885\pi$
$908$	0	0
$909$	−20.1688	−0.668956
$910$	0	0
$911$	−23.3371	−0.773192	−0.386596	−	0.922249i	$-0.626349\pi$
$911$	−23.3371	−0.773192	−0.386596	+	0.922249i	$0.626349\pi$
$912$	0	0
$913$	−71.7448	−2.37441
$914$	0	0
$915$	0.0645315	0.00213334
$916$	0	0
$917$	−25.2768	−0.834713
$918$	0	0
$919$	−38.5096	−1.27031	−0.635157	−	0.772383i	$-0.719065\pi$
$919$	−38.5096	−1.27031	−0.635157	+	0.772383i	$0.719065\pi$
$920$	0	0
$921$	6.28359	0.207051
$922$	0	0
$923$	3.43361	0.113019
$924$	0	0
$925$	−18.5788	−0.610869
$926$	0	0
$927$	−17.7717	−0.583698
$928$	0	0
$929$	−24.9635	−0.819026	−0.409513	−	0.912304i	$-0.634301\pi$
$929$	−24.9635	−0.819026	−0.409513	+	0.912304i	$0.634301\pi$
$930$	0	0
$931$	5.83883	0.191360
$932$	0	0
$933$	−3.66835	−0.120096
$934$	0	0
$935$	3.68077	0.120374
$936$	0	0
$937$	−17.3284	−0.566096	−0.283048	−	0.959106i	$-0.591346\pi$
$937$	−17.3284	−0.566096	−0.283048	+	0.959106i	$0.591346\pi$
$938$	0	0
$939$	1.80164	0.0587942
$940$	0	0
$941$	−8.15568	−0.265867	−0.132934	−	0.991125i	$-0.542440\pi$
$941$	−8.15568	−0.265867	−0.132934	+	0.991125i	$0.542440\pi$
$942$	0	0
$943$	63.5244	2.06864
$944$	0	0
$945$	1.14768	0.0373342
$946$	0	0
$947$	−50.5391	−1.64230	−0.821150	−	0.570713i	$-0.806667\pi$
$947$	−50.5391	−1.64230	−0.821150	+	0.570713i	$0.806667\pi$
$948$	0	0
$949$	8.70801	0.282674
$950$	0	0
$951$	−7.35999	−0.238664
$952$	0	0
$953$	58.0120	1.87919	0.939597	−	0.342284i	$-0.111200\pi$
$953$	58.0120	1.87919	0.939597	+	0.342284i	$0.111200\pi$
$954$	0	0
$955$	1.50843	0.0488116
$956$	0	0
$957$	0.990268	0.0320108
$958$	0	0
$959$	−74.0623	−2.39160
$960$	0	0
$961$	−18.3857	−0.593086
$962$	0	0
$963$	44.3740	1.42993
$964$	0	0
$965$	−1.58698	−0.0510868
$966$	0	0
$967$	−1.32798	−0.0427049	−0.0213524	−	0.999772i	$-0.506797\pi$
$967$	−1.32798	−0.0427049	−0.0213524	+	0.999772i	$0.506797\pi$
$968$	0	0
$969$	0.701596	0.0225385
$970$	0	0
$971$	0.0356780	0.00114496	0.000572481	−	1.00000i	$-0.499818\pi$
$971$	0.0356780	0.00114496	0.000572481		1.00000i	$0.499818\pi$
$972$	0	0
$973$	−86.5840	−2.77576
$974$	0	0
$975$	−1.52608	−0.0488737
$976$	0	0
$977$	−52.8527	−1.69091	−0.845453	−	0.534050i	$-0.820670\pi$
$977$	−52.8527	−1.69091	−0.845453	+	0.534050i	$0.820670\pi$
$978$	0	0
$979$	−60.9512	−1.94801
$980$	0	0
$981$	23.3227	0.744638
$982$	0	0
$983$	−53.1201	−1.69427	−0.847135	−	0.531378i	$-0.821674\pi$
$983$	−53.1201	−1.69427	−0.847135	+	0.531378i	$0.821674\pi$
$984$	0	0
$985$	0.646767	0.0206077
$986$	0	0
$987$	12.6051	0.401223
$988$	0	0
$989$	−31.6352	−1.00594
$990$	0	0
$991$	16.2704	0.516848	0.258424	−	0.966032i	$-0.416797\pi$
$991$	16.2704	0.516848	0.258424	+	0.966032i	$0.416797\pi$
$992$	0	0
$993$	3.84418	0.121991
$994$	0	0
$995$	0.579589	0.0183742
$996$	0	0
$997$	49.7790	1.57652	0.788258	−	0.615345i	$-0.210983\pi$
$997$	49.7790	1.57652	0.788258	+	0.615345i	$0.210983\pi$
$998$	0	0
$999$	7.25932	0.229675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.n.1.3		5
4.3	odd	2		1006.2.a.g.1.3	✓	5
12.11	even	2		9054.2.a.bb.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.g.1.3	✓	5	4.3	odd	2
8048.2.a.n.1.3		5	1.1	even	1	trivial
9054.2.a.bb.1.3		5	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)

\(f(q)\)	\(=\)	\(q-0.330424 q^{3} -0.135598 q^{5} -4.34834 q^{7} -2.89082 q^{9} +O(q^{10})\)	\(q-0.330424 q^{3} -0.135598 q^{5} -4.34834 q^{7} -2.89082 q^{9} +6.26839 q^{11} -0.927120 q^{13} +0.0448047 q^{15} -4.33042 q^{17} +0.490327 q^{19} +1.43679 q^{21} +5.32761 q^{23} -4.98161 q^{25} +1.94647 q^{27} -0.478108 q^{29} +3.55167 q^{31} -2.07122 q^{33} +0.589625 q^{35} +3.72948 q^{37} +0.306343 q^{39} +11.9236 q^{41} -5.93796 q^{43} +0.391989 q^{45} +8.77304 q^{47} +11.9080 q^{49} +1.43087 q^{51} -7.12594 q^{53} -0.849979 q^{55} -0.162016 q^{57} +0.455180 q^{59} +1.44028 q^{61} +12.5703 q^{63} +0.125715 q^{65} +7.56319 q^{67} -1.76037 q^{69} -3.70352 q^{71} -9.39254 q^{73} +1.64604 q^{75} -27.2571 q^{77} +3.18770 q^{79} +8.02930 q^{81} -11.4455 q^{83} +0.587196 q^{85} +0.157978 q^{87} -9.72360 q^{89} +4.03143 q^{91} -1.17355 q^{93} -0.0664872 q^{95} -6.57307 q^{97} -18.1208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{5} - 3 q^{7} - 3 q^{9}+O(q^{10}) \) 5 * q - q^5 - 3 * q^7 - 3 * q^9	\( 5 q - q^{5} - 3 q^{7} - 3 q^{9} + 11 q^{11} + 5 q^{13} + 8 q^{15} - 20 q^{17} + 4 q^{19} - 4 q^{21} + 4 q^{23} - 6 q^{25} - 3 q^{27} - 6 q^{29} + 3 q^{31} - 7 q^{33} + 3 q^{35} - 10 q^{37} - 4 q^{39} - 6 q^{41} - 11 q^{43} - 17 q^{45} + 9 q^{47} - 4 q^{49} + 12 q^{51} - 22 q^{53} - 14 q^{55} + 10 q^{57} + 10 q^{59} - 11 q^{61} + 3 q^{63} - 12 q^{65} - 14 q^{67} - 3 q^{69} + 26 q^{71} - 7 q^{73} - 15 q^{75} - 26 q^{77} + 15 q^{79} - 7 q^{81} + 12 q^{83} + 12 q^{85} + 18 q^{87} - 5 q^{89} + 22 q^{91} - 21 q^{93} + 10 q^{95} + 6 q^{97} - 8 q^{99}+O(q^{100}) \) 5 * q - q^5 - 3 * q^7 - 3 * q^9 + 11 * q^11 + 5 * q^13 + 8 * q^15 - 20 * q^17 + 4 * q^19 - 4 * q^21 + 4 * q^23 - 6 * q^25 - 3 * q^27 - 6 * q^29 + 3 * q^31 - 7 * q^33 + 3 * q^35 - 10 * q^37 - 4 * q^39 - 6 * q^41 - 11 * q^43 - 17 * q^45 + 9 * q^47 - 4 * q^49 + 12 * q^51 - 22 * q^53 - 14 * q^55 + 10 * q^57 + 10 * q^59 - 11 * q^61 + 3 * q^63 - 12 * q^65 - 14 * q^67 - 3 * q^69 + 26 * q^71 - 7 * q^73 - 15 * q^75 - 26 * q^77 + 15 * q^79 - 7 * q^81 + 12 * q^83 + 12 * q^85 + 18 * q^87 - 5 * q^89 + 22 * q^91 - 21 * q^93 + 10 * q^95 + 6 * q^97 - 8 * q^99

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