LMFDB - Embedded newform 9054.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9054,2,Mod(1,9054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9054, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9054 = 2 \cdot 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9054.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:	$72.2965539901$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
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	$\chi$	$=$	9054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.135598 q^{5} +4.34834 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.135598 q^{5} +4.34834 q^{7} +1.00000 q^{8} +0.135598 q^{10} +6.26839 q^{11} -0.927120 q^{13} +4.34834 q^{14} +1.00000 q^{16} +4.33042 q^{17} -0.490327 q^{19} +0.135598 q^{20} +6.26839 q^{22} +5.32761 q^{23} -4.98161 q^{25} -0.927120 q^{26} +4.34834 q^{28} +0.478108 q^{29} -3.55167 q^{31} +1.00000 q^{32} +4.33042 q^{34} +0.589625 q^{35} +3.72948 q^{37} -0.490327 q^{38} +0.135598 q^{40} -11.9236 q^{41} +5.93796 q^{43} +6.26839 q^{44} +5.32761 q^{46} +8.77304 q^{47} +11.9080 q^{49} -4.98161 q^{50} -0.927120 q^{52} +7.12594 q^{53} +0.849979 q^{55} +4.34834 q^{56} +0.478108 q^{58} +0.455180 q^{59} +1.44028 q^{61} -3.55167 q^{62} +1.00000 q^{64} -0.125715 q^{65} -7.56319 q^{67} +4.33042 q^{68} +0.589625 q^{70} -3.70352 q^{71} -9.39254 q^{73} +3.72948 q^{74} -0.490327 q^{76} +27.2571 q^{77} -3.18770 q^{79} +0.135598 q^{80} -11.9236 q^{82} -11.4455 q^{83} +0.587196 q^{85} +5.93796 q^{86} +6.26839 q^{88} +9.72360 q^{89} -4.03143 q^{91} +5.32761 q^{92} +8.77304 q^{94} -0.0664872 q^{95} -6.57307 q^{97} +11.9080 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8} + q^{10} + 11 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 20 q^{17} - 4 q^{19} + q^{20} + 11 q^{22} + 4 q^{23} - 6 q^{25} + 5 q^{26} + 3 q^{28} + 6 q^{29} - 3 q^{31} + 5 q^{32} + 20 q^{34} + 3 q^{35} - 10 q^{37} - 4 q^{38} + q^{40} + 6 q^{41} + 11 q^{43} + 11 q^{44} + 4 q^{46} + 9 q^{47} - 4 q^{49} - 6 q^{50} + 5 q^{52} + 22 q^{53} + 14 q^{55} + 3 q^{56} + 6 q^{58} + 10 q^{59} - 11 q^{61} - 3 q^{62} + 5 q^{64} + 12 q^{65} + 14 q^{67} + 20 q^{68} + 3 q^{70} + 26 q^{71} - 7 q^{73} - 10 q^{74} - 4 q^{76} + 26 q^{77} - 15 q^{79} + q^{80} + 6 q^{82} + 12 q^{83} + 12 q^{85} + 11 q^{86} + 11 q^{88} + 5 q^{89} - 22 q^{91} + 4 q^{92} + 9 q^{94} + 10 q^{95} + 6 q^{97} - 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8 + q^10 + 11 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 20 * q^17 - 4 * q^19 + q^20 + 11 * q^22 + 4 * q^23 - 6 * q^25 + 5 * q^26 + 3 * q^28 + 6 * q^29 - 3 * q^31 + 5 * q^32 + 20 * q^34 + 3 * q^35 - 10 * q^37 - 4 * q^38 + q^40 + 6 * q^41 + 11 * q^43 + 11 * q^44 + 4 * q^46 + 9 * q^47 - 4 * q^49 - 6 * q^50 + 5 * q^52 + 22 * q^53 + 14 * q^55 + 3 * q^56 + 6 * q^58 + 10 * q^59 - 11 * q^61 - 3 * q^62 + 5 * q^64 + 12 * q^65 + 14 * q^67 + 20 * q^68 + 3 * q^70 + 26 * q^71 - 7 * q^73 - 10 * q^74 - 4 * q^76 + 26 * q^77 - 15 * q^79 + q^80 + 6 * q^82 + 12 * q^83 + 12 * q^85 + 11 * q^86 + 11 * q^88 + 5 * q^89 - 22 * q^91 + 4 * q^92 + 9 * q^94 + 10 * q^95 + 6 * q^97 - 4 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.135598	0.0606411	0.0303206	−	0.999540i	$-0.490347\pi$
$5$	0.135598	0.0606411	0.0303206	+	0.999540i	$0.490347\pi$
$6$	0	0
$7$	4.34834	1.64352	0.821758	−	0.569836i	$-0.192993\pi$
$7$	4.34834	1.64352	0.821758	+	0.569836i	$0.192993\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.135598	0.0428798
$11$	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	$0.541038\pi$
$14$	4.34834	1.16214
$15$	0	0
$16$	1.00000	0.250000
$17$	4.33042	1.05028	0.525141	−	0.851015i	$-0.324013\pi$
$17$	4.33042	1.05028	0.525141	+	0.851015i	$0.324013\pi$
$18$	0	0
$19$	−0.490327	−0.112489	−0.0562444	−	0.998417i	$-0.517913\pi$
$19$	−0.490327	−0.112489	−0.0562444	+	0.998417i	$0.517913\pi$
$20$	0.135598	0.0303206
$21$	0	0
$22$	6.26839	1.33642
$23$	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
$26$	−0.927120	−0.181823
$27$	0	0
$28$	4.34834	0.821758
$29$	0.478108	0.0887824	0.0443912	−	0.999014i	$-0.485865\pi$
$29$	0.478108	0.0887824	0.0443912	+	0.999014i	$0.485865\pi$
$30$	0	0
$31$	−3.55167	−0.637898	−0.318949	−	0.947772i	$-0.603330\pi$
$31$	−3.55167	−0.637898	−0.318949	+	0.947772i	$0.603330\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.33042	0.742662
$35$	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	$0.400821\pi$
$38$	−0.490327	−0.0795416
$39$	0	0
$40$	0.135598	0.0214399
$41$	−11.9236	−1.86216	−0.931078	−	0.364821i	$-0.881130\pi$
$41$	−11.9236	−1.86216	−0.931078	+	0.364821i	$0.881130\pi$
$42$	0	0
$43$	5.93796	0.905531	0.452765	−	0.891630i	$-0.350438\pi$
$43$	5.93796	0.905531	0.452765	+	0.891630i	$0.350438\pi$
$44$	6.26839	0.944995
$45$	0	0
$46$	5.32761	0.785514
$47$	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
$50$	−4.98161	−0.704507
$51$	0	0
$52$	−0.927120	−0.128568
$53$	7.12594	0.978823	0.489412	−	0.872053i	$-0.337212\pi$
$53$	7.12594	0.978823	0.489412	+	0.872053i	$0.337212\pi$
$54$	0	0
$55$	0.849979	0.114611
$56$	4.34834	0.581071
$57$	0	0
$58$	0.478108	0.0627787
$59$	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	$0.470609\pi$
$62$	−3.55167	−0.451062
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.125715	−0.0155931
$66$	0	0
$67$	−7.56319	−0.923991	−0.461995	−	0.886882i	$-0.652866\pi$
$67$	−7.56319	−0.923991	−0.461995	+	0.886882i	$0.652866\pi$
$68$	4.33042	0.525141
$69$	0	0
$70$	0.589625	0.0704736
$71$	−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	$0.685241\pi$
$74$	3.72948	0.433544
$75$	0	0
$76$	−0.490327	−0.0562444
$77$	27.2571	3.10623
$78$	0	0
$79$	−3.18770	−0.358644	−0.179322	−	0.983790i	$-0.557390\pi$
$79$	−3.18770	−0.358644	−0.179322	+	0.983790i	$0.557390\pi$
$80$	0.135598	0.0151603
$81$	0	0
$82$	−11.9236	−1.31674
$83$	−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
$86$	5.93796	0.640307
$87$	0	0
$88$	6.26839	0.668212
$89$	9.72360	1.03070	0.515350	−	0.856980i	$-0.327662\pi$
$89$	9.72360	1.03070	0.515350	+	0.856980i	$0.327662\pi$
$90$	0	0
$91$	−4.03143	−0.422609
$92$	5.32761	0.555442
$93$	0	0
$94$	8.77304	0.904870
$95$	−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	$0.608296\pi$
$98$	11.9080	1.20289
$99$	0	0
$100$	−4.98161	−0.498161
$101$	−6.97683	−0.694220	−0.347110	−	0.937824i	$-0.612837\pi$
$101$	−6.97683	−0.694220	−0.347110	+	0.937824i	$0.612837\pi$
$102$	0	0
$103$	−6.14762	−0.605743	−0.302872	−	0.953031i	$-0.597945\pi$
$103$	−6.14762	−0.605743	−0.302872	+	0.953031i	$0.597945\pi$
$104$	−0.927120	−0.0909116
$105$	0	0
$106$	7.12594	0.692132
$107$	−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	$0.626275\pi$
$110$	0.849979	0.0810423
$111$	0	0
$112$	4.34834	0.410879
$113$	19.1797	1.80428	0.902138	−	0.431447i	$-0.141997\pi$
$113$	19.1797	1.80428	0.902138	+	0.431447i	$0.141997\pi$
$114$	0	0
$115$	0.722412	0.0673653
$116$	0.478108	0.0443912
$117$	0	0
$118$	0.455180	0.0419027
$119$	18.8301	1.72616
$120$	0	0
$121$	28.2927	2.57206
$122$	1.44028	0.130397
$123$	0	0
$124$	−3.55167	−0.318949
$125$	−1.35348	−0.121059
$126$	0	0
$127$	1.25290	0.111177	0.0555885	−	0.998454i	$-0.482297\pi$
$127$	1.25290	0.111177	0.0555885	+	0.998454i	$0.482297\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.125715	−0.0110260
$131$	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
$134$	−7.56319	−0.653360
$135$	0	0
$136$	4.33042	0.371331
$137$	−17.0323	−1.45517	−0.727585	−	0.686018i	$-0.759357\pi$
$137$	−17.0323	−1.45517	−0.727585	+	0.686018i	$0.759357\pi$
$138$	0	0
$139$	−19.9120	−1.68891	−0.844456	−	0.535625i	$-0.820076\pi$
$139$	−19.9120	−1.68891	−0.844456	+	0.535625i	$0.820076\pi$
$140$	0.589625	0.0498324
$141$	0	0
$142$	−3.70352	−0.310793
$143$	−5.81155	−0.485986
$144$	0	0
$145$	0.0648304	0.00538387
$146$	−9.39254	−0.777332
$147$	0	0
$148$	3.72948	0.306562
$149$	−0.985577	−0.0807416	−0.0403708	−	0.999185i	$-0.512854\pi$
$149$	−0.985577	−0.0807416	−0.0403708	+	0.999185i	$0.512854\pi$
$150$	0	0
$151$	−18.0186	−1.46633	−0.733165	−	0.680050i	$-0.761958\pi$
$151$	−18.0186	−1.46633	−0.733165	+	0.680050i	$0.761958\pi$
$152$	−0.490327	−0.0397708
$153$	0	0
$154$	27.2571	2.19644
$155$	−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	$0.687902\pi$
$158$	−3.18770	−0.253600
$159$	0	0
$160$	0.135598	0.0107199
$161$	23.1663	1.82576
$162$	0	0
$163$	21.8557	1.71187	0.855937	−	0.517080i	$-0.172981\pi$
$163$	21.8557	1.71187	0.855937	+	0.517080i	$0.172981\pi$
$164$	−11.9236	−0.931078
$165$	0	0
$166$	−11.4455	−0.888343
$167$	−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
$170$	0.587196	0.0450358
$171$	0	0
$172$	5.93796	0.452765
$173$	−6.93774	−0.527466	−0.263733	−	0.964596i	$-0.584954\pi$
$173$	−6.93774	−0.527466	−0.263733	+	0.964596i	$0.584954\pi$
$174$	0	0
$175$	−21.6617	−1.63747
$176$	6.26839	0.472497
$177$	0	0
$178$	9.72360	0.728814
$179$	−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	$0.670613\pi$
$182$	−4.03143	−0.298830
$183$	0	0
$184$	5.32761	0.392757
$185$	0.505710	0.0371805
$186$	0	0
$187$	27.1448	1.98502
$188$	8.77304	0.639840
$189$	0	0
$190$	−0.0664872	−0.00482349
$191$	−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	$0.361601\pi$
$194$	−6.57307	−0.471919
$195$	0	0
$196$	11.9080	0.850574
$197$	4.76975	0.339830	0.169915	−	0.985459i	$-0.445651\pi$
$197$	4.76975	0.339830	0.169915	+	0.985459i	$0.445651\pi$
$198$	0	0
$199$	4.27433	0.302999	0.151500	−	0.988457i	$-0.451590\pi$
$199$	4.27433	0.302999	0.151500	+	0.988457i	$0.451590\pi$
$200$	−4.98161	−0.352253
$201$	0	0
$202$	−6.97683	−0.490888
$203$	2.07897	0.145915
$204$	0	0
$205$	−1.61681	−0.112923
$206$	−6.14762	−0.428325
$207$	0	0
$208$	−0.927120	−0.0642842
$209$	−3.07356	−0.212603
$210$	0	0
$211$	0.0640960	0.00441255	0.00220628	−	0.999998i	$-0.499298\pi$
$211$	0.0640960	0.00441255	0.00220628	+	0.999998i	$0.499298\pi$
$212$	7.12594	0.489412
$213$	0	0
$214$	−15.3500	−1.04930
$215$	0.805174	0.0549124
$216$	0	0
$217$	−15.4438	−1.04840
$218$	−8.06787	−0.546425
$219$	0	0
$220$	0.849979	0.0573056
$221$	−4.01482	−0.270066
$222$	0	0
$223$	−13.3505	−0.894017	−0.447009	−	0.894530i	$-0.647511\pi$
$223$	−13.3505	−0.894017	−0.447009	+	0.894530i	$0.647511\pi$
$224$	4.34834	0.290535
$225$	0	0
$226$	19.1797	1.27582
$227$	−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	$0.205080\pi$
$230$	0.722412	0.0476345
$231$	0	0
$232$	0.478108	0.0313893
$233$	18.0849	1.18478	0.592392	−	0.805650i	$-0.298184\pi$
$233$	18.0849	1.18478	0.592392	+	0.805650i	$0.298184\pi$
$234$	0	0
$235$	1.18960	0.0776013
$236$	0.455180	0.0296297
$237$	0	0
$238$	18.8301	1.22058
$239$	−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	$0.470461\pi$
$242$	28.2927	1.81872
$243$	0	0
$244$	1.44028	0.0922047
$245$	1.61470	0.103160
$246$	0	0
$247$	0.454592	0.0289250
$248$	−3.55167	−0.225531
$249$	0	0
$250$	−1.35348	−0.0856018
$251$	−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
$254$	1.25290	0.0786140
$255$	0	0
$256$	1.00000	0.0625000
$257$	25.5981	1.59677	0.798384	−	0.602149i	$-0.205689\pi$
$257$	25.5981	1.59677	0.798384	+	0.602149i	$0.205689\pi$
$258$	0	0
$259$	16.2171	1.00768
$260$	−0.125715	−0.00779654
$261$	0	0
$262$	5.81298	0.359127
$263$	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
$266$	−2.13211	−0.130728
$267$	0	0
$268$	−7.56319	−0.461995
$269$	27.1739	1.65682	0.828411	−	0.560121i	$-0.189245\pi$
$269$	27.1739	1.65682	0.828411	+	0.560121i	$0.189245\pi$
$270$	0	0
$271$	1.65441	0.100498	0.0502492	−	0.998737i	$-0.483998\pi$
$271$	1.65441	0.100498	0.0502492	+	0.998737i	$0.483998\pi$
$272$	4.33042	0.262571
$273$	0	0
$274$	−17.0323	−1.02896
$275$	−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	$0.307122\pi$
$278$	−19.9120	−1.19424
$279$	0	0
$280$	0.589625	0.0352368
$281$	17.6353	1.05204	0.526018	−	0.850473i	$-0.323684\pi$
$281$	17.6353	1.05204	0.526018	+	0.850473i	$0.323684\pi$
$282$	0	0
$283$	27.1354	1.61303	0.806515	−	0.591214i	$-0.201351\pi$
$283$	27.1354	1.61303	0.806515	+	0.591214i	$0.201351\pi$
$284$	−3.70352	−0.219764
$285$	0	0
$286$	−5.81155	−0.343644
$287$	−51.8479	−3.06048
$288$	0	0
$289$	1.75257	0.103092
$290$	0.0648304	0.00380697
$291$	0	0
$292$	−9.39254	−0.549657
$293$	−12.6631	−0.739788	−0.369894	−	0.929074i	$-0.620606\pi$
$293$	−12.6631	−0.739788	−0.369894	+	0.929074i	$0.620606\pi$
$294$	0	0
$295$	0.0617214	0.00359356
$296$	3.72948	0.216772
$297$	0	0
$298$	−0.985577	−0.0570929
$299$	−4.93934	−0.285649
$300$	0	0
$301$	25.8203	1.48826
$302$	−18.0186	−1.03685
$303$	0	0
$304$	−0.490327	−0.0281222
$305$	0.195299	0.0111828
$306$	0	0
$307$	19.0168	1.08534	0.542672	−	0.839945i	$-0.317413\pi$
$307$	19.0168	1.08534	0.542672	+	0.839945i	$0.317413\pi$
$308$	27.2571	1.55311
$309$	0	0
$310$	−0.481598	−0.0273529
$311$	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	$0.549247\pi$
$314$	−13.9489	−0.787181
$315$	0	0
$316$	−3.18770	−0.179322
$317$	−22.2744	−1.25106	−0.625528	−	0.780202i	$-0.715116\pi$
$317$	−22.2744	−1.25106	−0.625528	+	0.780202i	$0.715116\pi$
$318$	0	0
$319$	2.99697	0.167798
$320$	0.135598	0.00758014
$321$	0	0
$322$	23.1663	1.29101
$323$	−2.12332	−0.118145
$324$	0	0
$325$	4.61856	0.256191
$326$	21.8557	1.21048
$327$	0	0
$328$	−11.9236	−0.658371
$329$	38.1482	2.10318
$330$	0	0
$331$	11.6341	0.639467	0.319734	−	0.947507i	$-0.396407\pi$
$331$	11.6341	0.639467	0.319734	+	0.947507i	$0.396407\pi$
$332$	−11.4455	−0.628154
$333$	0	0
$334$	−19.5905	−1.07195
$335$	−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	$0.382275\pi$
$338$	−12.1404	−0.660353
$339$	0	0
$340$	0.587196	0.0318452
$341$	−22.2632	−1.20562
$342$	0	0
$343$	21.3418	1.15235
$344$	5.93796	0.320153
$345$	0	0
$346$	−6.93774	−0.372975
$347$	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	$0.800170\pi$
$350$	−21.6617	−1.15787
$351$	0	0
$352$	6.26839	0.334106
$353$	12.7468	0.678444	0.339222	−	0.940706i	$-0.389836\pi$
$353$	12.7468	0.678444	0.339222	+	0.940706i	$0.389836\pi$
$354$	0	0
$355$	−0.502189	−0.0266534
$356$	9.72360	0.515350
$357$	0	0
$358$	−3.51035	−0.185528
$359$	−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
$362$	−13.7415	−0.722235
$363$	0	0
$364$	−4.03143	−0.211304
$365$	−1.27361	−0.0666636
$366$	0	0
$367$	−4.01777	−0.209726	−0.104863	−	0.994487i	$-0.533440\pi$
$367$	−4.01777	−0.209726	−0.104863	+	0.994487i	$0.533440\pi$
$368$	5.32761	0.277721
$369$	0	0
$370$	0.505710	0.0262906
$371$	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	$0.458284\pi$
$374$	27.1448	1.40362
$375$	0	0
$376$	8.77304	0.452435
$377$	−0.443264	−0.0228292
$378$	0	0
$379$	1.19492	0.0613788	0.0306894	−	0.999529i	$-0.490230\pi$
$379$	1.19492	0.0613788	0.0306894	+	0.999529i	$0.490230\pi$
$380$	−0.0664872	−0.00341072
$381$	0	0
$382$	−11.1243	−0.569168
$383$	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
$386$	11.7036	0.595698
$387$	0	0
$388$	−6.57307	−0.333697
$389$	−28.2349	−1.43157	−0.715783	−	0.698322i	$-0.753930\pi$
$389$	−28.2349	−1.43157	−0.715783	+	0.698322i	$0.753930\pi$
$390$	0	0
$391$	23.0708	1.16674
$392$	11.9080	0.601447
$393$	0	0
$394$	4.76975	0.240296
$395$	−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	$0.628797\pi$
$398$	4.27433	0.214253
$399$	0	0
$400$	−4.98161	−0.249081
$401$	17.2093	0.859391	0.429696	−	0.902974i	$-0.358621\pi$
$401$	17.2093	0.859391	0.429696	+	0.902974i	$0.358621\pi$
$402$	0	0
$403$	3.29282	0.164027
$404$	−6.97683	−0.347110
$405$	0	0
$406$	2.07897	0.103178
$407$	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	$0.574961\pi$
$410$	−1.61681	−0.0798488
$411$	0	0
$412$	−6.14762	−0.302872
$413$	1.97928	0.0973938
$414$	0	0
$415$	−1.55198	−0.0761839
$416$	−0.927120	−0.0454558
$417$	0	0
$418$	−3.07356	−0.150333
$419$	−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	$0.776222\pi$
$422$	0.0640960	0.00312015
$423$	0	0
$424$	7.12594	0.346066
$425$	−21.5725	−1.04642
$426$	0	0
$427$	6.26284	0.303080
$428$	−15.3500	−0.741969
$429$	0	0
$430$	0.805174	0.0388289
$431$	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	$0.718468\pi$
$434$	−15.4438	−0.741328
$435$	0	0
$436$	−8.06787	−0.386381
$437$	−2.61227	−0.124962
$438$	0	0
$439$	−0.766258	−0.0365715	−0.0182857	−	0.999833i	$-0.505821\pi$
$439$	−0.766258	−0.0365715	−0.0182857	+	0.999833i	$0.505821\pi$
$440$	0.849979	0.0405211
$441$	0	0
$442$	−4.01482	−0.190966
$443$	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
$446$	−13.3505	−0.632166
$447$	0	0
$448$	4.34834	0.205440
$449$	−26.1149	−1.23244	−0.616218	−	0.787575i	$-0.711336\pi$
$449$	−26.1149	−1.23244	−0.616218	+	0.787575i	$0.711336\pi$
$450$	0	0
$451$	−74.7418	−3.51945
$452$	19.1797	0.902138
$453$	0	0
$454$	−13.3815	−0.628027
$455$	−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	$0.455202\pi$
$458$	24.1983	1.13071
$459$	0	0
$460$	0.722412	0.0336826
$461$	33.9088	1.57929	0.789644	−	0.613565i	$-0.210265\pi$
$461$	33.9088	1.57929	0.789644	+	0.613565i	$0.210265\pi$
$462$	0	0
$463$	23.7318	1.10291	0.551455	−	0.834205i	$-0.314073\pi$
$463$	23.7318	1.10291	0.551455	+	0.834205i	$0.314073\pi$
$464$	0.478108	0.0221956
$465$	0	0
$466$	18.0849	0.837769
$467$	−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
$470$	1.18960	0.0548724
$471$	0	0
$472$	0.455180	0.0209514
$473$	37.2214	1.71144
$474$	0	0
$475$	2.44262	0.112075
$476$	18.8301	0.863078
$477$	0	0
$478$	−21.8920	−1.00132
$479$	−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
$482$	2.87717	0.131051
$483$	0	0
$484$	28.2927	1.28603
$485$	−0.891293	−0.0404716
$486$	0	0
$487$	13.8268	0.626553	0.313277	−	0.949662i	$-0.398573\pi$
$487$	13.8268	0.626553	0.313277	+	0.949662i	$0.398573\pi$
$488$	1.44028	0.0651986
$489$	0	0
$490$	1.61470	0.0729448
$491$	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
$494$	0.454592	0.0204531
$495$	0	0
$496$	−3.55167	−0.159475
$497$	−16.1042	−0.722370
$498$	0	0
$499$	8.78566	0.393300	0.196650	−	0.980474i	$-0.436994\pi$
$499$	8.78566	0.393300	0.196650	+	0.980474i	$0.436994\pi$
$500$	−1.35348	−0.0605296
$501$	0	0
$502$	−17.2483	−0.769828
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−0.946042	−0.0420983
$506$	33.3955	1.48461
$507$	0	0
$508$	1.25290	0.0555885
$509$	9.17462	0.406658	0.203329	−	0.979110i	$-0.434824\pi$
$509$	9.17462	0.406658	0.203329	+	0.979110i	$0.434824\pi$
$510$	0	0
$511$	−40.8419	−1.80674
$512$	1.00000	0.0441942
$513$	0	0
$514$	25.5981	1.12909
$515$	−0.833604	−0.0367330
$516$	0	0
$517$	54.9928	2.41858
$518$	16.2171	0.712537
$519$	0	0
$520$	−0.125715	−0.00551299
$521$	−2.41914	−0.105985	−0.0529923	−	0.998595i	$-0.516876\pi$
$521$	−2.41914	−0.105985	−0.0529923	+	0.998595i	$0.516876\pi$
$522$	0	0
$523$	19.2074	0.839882	0.419941	−	0.907551i	$-0.362051\pi$
$523$	19.2074	0.839882	0.419941	+	0.907551i	$0.362051\pi$
$524$	5.81298	0.253941
$525$	0	0
$526$	24.2567	1.05764
$527$	−15.3802	−0.669973
$528$	0	0
$529$	5.38347	0.234064
$530$	0.966261	0.0419717
$531$	0	0
$532$	−2.13211	−0.0924386
$533$	11.0546	0.478829
$534$	0	0
$535$	−2.08142	−0.0899877
$536$	−7.56319	−0.326680
$537$	0	0
$538$	27.1739	1.17155
$539$	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	$0.413007\pi$
$542$	1.65441	0.0710630
$543$	0	0
$544$	4.33042	0.185665
$545$	−1.09398	−0.0468611
$546$	0	0
$547$	−5.65918	−0.241969	−0.120985	−	0.992654i	$-0.538605\pi$
$547$	−5.65918	−0.241969	−0.120985	+	0.992654i	$0.538605\pi$
$548$	−17.0323	−0.727585
$549$	0	0
$550$	−31.2267	−1.33151
$551$	−0.234429	−0.00998703
$552$	0	0
$553$	−13.8612	−0.589438
$554$	18.9580	0.805449
$555$	0	0
$556$	−19.9120	−0.844456
$557$	−1.04296	−0.0441915	−0.0220958	−	0.999756i	$-0.507034\pi$
$557$	−1.04296	−0.0441915	−0.0220958	+	0.999756i	$0.507034\pi$
$558$	0	0
$559$	−5.50521	−0.232845
$560$	0.589625	0.0249162
$561$	0	0
$562$	17.6353	0.743902
$563$	−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
$566$	27.1354	1.14058
$567$	0	0
$568$	−3.70352	−0.155396
$569$	18.8856	0.791727	0.395864	−	0.918309i	$-0.370445\pi$
$569$	18.8856	0.791727	0.395864	+	0.918309i	$0.370445\pi$
$570$	0	0
$571$	5.61412	0.234944	0.117472	−	0.993076i	$-0.462521\pi$
$571$	5.61412	0.234944	0.117472	+	0.993076i	$0.462521\pi$
$572$	−5.81155	−0.242993
$573$	0	0
$574$	−51.8479	−2.16409
$575$	−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	$0.269455\pi$
$578$	1.75257	0.0728973
$579$	0	0
$580$	0.0648304	0.00269193
$581$	−49.7689	−2.06476
$582$	0	0
$583$	44.6681	1.84997
$584$	−9.39254	−0.388666
$585$	0	0
$586$	−12.6631	−0.523109
$587$	−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
$590$	0.0617214	0.00254103
$591$	0	0
$592$	3.72948	0.153281
$593$	−3.66746	−0.150604	−0.0753022	−	0.997161i	$-0.523992\pi$
$593$	−3.66746	−0.150604	−0.0753022	+	0.997161i	$0.523992\pi$
$594$	0	0
$595$	2.55332	0.104676
$596$	−0.985577	−0.0403708
$597$	0	0
$598$	−4.93934	−0.201985
$599$	−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	$0.119455\pi$
$602$	25.8203	1.05236
$603$	0	0
$604$	−18.0186	−0.733165
$605$	3.83642	0.155973
$606$	0	0
$607$	26.9627	1.09438	0.547192	−	0.837007i	$-0.315697\pi$
$607$	26.9627	1.09438	0.547192	+	0.837007i	$0.315697\pi$
$608$	−0.490327	−0.0198854
$609$	0	0
$610$	0.195299	0.00790743
$611$	−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	$0.578955\pi$
$614$	19.0168	0.767454
$615$	0	0
$616$	27.2571	1.09822
$617$	32.1456	1.29413	0.647067	−	0.762433i	$-0.275995\pi$
$617$	32.1456	1.29413	0.647067	+	0.762433i	$0.275995\pi$
$618$	0	0
$619$	−41.7277	−1.67718	−0.838588	−	0.544766i	$-0.816618\pi$
$619$	−41.7277	−1.67718	−0.838588	+	0.544766i	$0.816618\pi$
$620$	−0.481598	−0.0193414
$621$	0	0
$622$	11.1020	0.445148
$623$	42.2815	1.69397
$624$	0	0
$625$	24.7245	0.988981
$626$	−5.45250	−0.217926
$627$	0	0
$628$	−13.9489	−0.556621
$629$	16.1502	0.643953
$630$	0	0
$631$	−20.2155	−0.804767	−0.402383	−	0.915471i	$-0.631818\pi$
$631$	−20.2155	−0.804767	−0.402383	+	0.915471i	$0.631818\pi$
$632$	−3.18770	−0.126800
$633$	0	0
$634$	−22.2744	−0.884630
$635$	0.169891	0.00674190
$636$	0	0
$637$	−11.0402	−0.437428
$638$	2.99697	0.118651
$639$	0	0
$640$	0.135598	0.00535997
$641$	2.13732	0.0844189	0.0422095	−	0.999109i	$-0.486560\pi$
$641$	2.13732	0.0844189	0.0422095	+	0.999109i	$0.486560\pi$
$642$	0	0
$643$	−42.3794	−1.67128	−0.835641	−	0.549277i	$-0.814903\pi$
$643$	−42.3794	−1.67128	−0.835641	+	0.549277i	$0.814903\pi$
$644$	23.1663	0.912879
$645$	0	0
$646$	−2.12332	−0.0835411
$647$	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
$650$	4.61856	0.181155
$651$	0	0
$652$	21.8557	0.855937
$653$	40.4529	1.58304	0.791522	−	0.611140i	$-0.209289\pi$
$653$	40.4529	1.58304	0.791522	+	0.611140i	$0.209289\pi$
$654$	0	0
$655$	0.788227	0.0307986
$656$	−11.9236	−0.465539
$657$	0	0
$658$	38.1482	1.48717
$659$	−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	$0.669951\pi$
$662$	11.6341	0.452172
$663$	0	0
$664$	−11.4455	−0.444172
$665$	−0.289109	−0.0112112
$666$	0	0
$667$	2.54717	0.0986270
$668$	−19.5905	−0.757980
$669$	0	0
$670$	−1.02555	−0.0396205
$671$	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	$0.803560\pi$
$674$	13.2714	0.511197
$675$	0	0
$676$	−12.1404	−0.466940
$677$	−26.3289	−1.01190	−0.505950	−	0.862563i	$-0.668858\pi$
$677$	−26.3289	−1.01190	−0.505950	+	0.862563i	$0.668858\pi$
$678$	0	0
$679$	−28.5819	−1.09687
$680$	0.587196	0.0225179
$681$	0	0
$682$	−22.2632	−0.852503
$683$	−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
$686$	21.3418	0.814833
$687$	0	0
$688$	5.93796	0.226383
$689$	−6.60660	−0.251692
$690$	0	0
$691$	−11.8568	−0.451052	−0.225526	−	0.974237i	$-0.572410\pi$
$691$	−11.8568	−0.451052	−0.225526	+	0.974237i	$0.572410\pi$
$692$	−6.93774	−0.263733
$693$	0	0
$694$	27.0757	1.02778
$695$	−2.70002	−0.102418
$696$	0	0
$697$	−51.6343	−1.95579
$698$	−30.2391	−1.14457
$699$	0	0
$700$	−21.6617	−0.818737
$701$	1.93296	0.0730069	0.0365035	−	0.999334i	$-0.488378\pi$
$701$	1.93296	0.0730069	0.0365035	+	0.999334i	$0.488378\pi$
$702$	0	0
$703$	−1.82867	−0.0689695
$704$	6.26839	0.236249
$705$	0	0
$706$	12.7468	0.479733
$707$	−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	$0.465207\pi$
$710$	−0.502189	−0.0188468
$711$	0	0
$712$	9.72360	0.364407
$713$	−18.9219	−0.708631
$714$	0	0
$715$	−0.788033	−0.0294708
$716$	−3.51035	−0.131188
$717$	0	0
$718$	−12.4924	−0.466214
$719$	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
$722$	−18.7596	−0.698159
$723$	0	0
$724$	−13.7415	−0.510697
$725$	−2.38175	−0.0884559
$726$	0	0
$727$	−8.81895	−0.327077	−0.163538	−	0.986537i	$-0.552291\pi$
$727$	−8.81895	−0.327077	−0.163538	+	0.986537i	$0.552291\pi$
$728$	−4.03143	−0.149415
$729$	0	0
$730$	−1.27361	−0.0471383
$731$	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	$0.369095\pi$
$734$	−4.01777	−0.148299
$735$	0	0
$736$	5.32761	0.196378
$737$	−47.4090	−1.74633
$738$	0	0
$739$	2.22552	0.0818672	0.0409336	−	0.999162i	$-0.486967\pi$
$739$	2.22552	0.0818672	0.0409336	+	0.999162i	$0.486967\pi$
$740$	0.505710	0.0185903
$741$	0	0
$742$	30.9860	1.13753
$743$	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
$746$	5.04769	0.184809
$747$	0	0
$748$	27.1448	0.992511
$749$	−66.7469	−2.43888
$750$	0	0
$751$	7.67058	0.279903	0.139952	−	0.990158i	$-0.455305\pi$
$751$	7.67058	0.279903	0.139952	+	0.990158i	$0.455305\pi$
$752$	8.77304	0.319920
$753$	0	0
$754$	−0.443264	−0.0161427
$755$	−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	$0.202485\pi$
$758$	1.19492	0.0434014
$759$	0	0
$760$	−0.0664872	−0.00241175
$761$	48.6407	1.76322	0.881612	−	0.471974i	$-0.156458\pi$
$761$	48.6407	1.76322	0.881612	+	0.471974i	$0.156458\pi$
$762$	0	0
$763$	−35.0818	−1.27005
$764$	−11.1243	−0.402462
$765$	0	0
$766$	35.7173	1.29052
$767$	−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	$0.407195\pi$
$770$	3.69599	0.133194
$771$	0	0
$772$	11.7036	0.421222
$773$	51.9972	1.87021	0.935104	−	0.354373i	$-0.115306\pi$
$773$	51.9972	1.87021	0.935104	+	0.354373i	$0.115306\pi$
$774$	0	0
$775$	17.6930	0.635553
$776$	−6.57307	−0.235959
$777$	0	0
$778$	−28.2349	−1.01227
$779$	5.84647	0.209472
$780$	0	0
$781$	−23.2151	−0.830702
$782$	23.0708	0.825011
$783$	0	0
$784$	11.9080	0.425287
$785$	−1.89144	−0.0675082
$786$	0	0
$787$	−3.49306	−0.124514	−0.0622571	−	0.998060i	$-0.519830\pi$
$787$	−3.49306	−0.124514	−0.0622571	+	0.998060i	$0.519830\pi$
$788$	4.76975	0.169915
$789$	0	0
$790$	−0.432245	−0.0153786
$791$	83.3999	2.96536
$792$	0	0
$793$	−1.33532	−0.0474185
$794$	−15.6879	−0.556744
$795$	0	0
$796$	4.27433	0.151500
$797$	−0.522094	−0.0184935	−0.00924677	−	0.999957i	$-0.502943\pi$
$797$	−0.522094	−0.0184935	−0.00924677	+	0.999957i	$0.502943\pi$
$798$	0	0
$799$	37.9910	1.34402
$800$	−4.98161	−0.176127
$801$	0	0
$802$	17.2093	0.607681
$803$	−58.8760	−2.07769
$804$	0	0
$805$	3.14129	0.110716
$806$	3.29282	0.115985
$807$	0	0
$808$	−6.97683	−0.245444
$809$	−23.6408	−0.831166	−0.415583	−	0.909555i	$-0.636423\pi$
$809$	−23.6408	−0.831166	−0.415583	+	0.909555i	$0.636423\pi$
$810$	0	0
$811$	−30.4387	−1.06885	−0.534424	−	0.845217i	$-0.679471\pi$
$811$	−30.4387	−1.06885	−0.534424	+	0.845217i	$0.679471\pi$
$812$	2.07897	0.0729577
$813$	0	0
$814$	23.3778	0.819393
$815$	2.96359	0.103810
$816$	0	0
$817$	−2.91154	−0.101862
$818$	−9.43745	−0.329973
$819$	0	0
$820$	−1.61681	−0.0564616
$821$	13.0982	0.457129	0.228564	−	0.973529i	$-0.426597\pi$
$821$	13.0982	0.457129	0.228564	+	0.973529i	$0.426597\pi$
$822$	0	0
$823$	−27.1349	−0.945865	−0.472932	−	0.881099i	$-0.656805\pi$
$823$	−27.1349	−0.945865	−0.472932	+	0.881099i	$0.656805\pi$
$824$	−6.14762	−0.214163
$825$	0	0
$826$	1.97928	0.0688678
$827$	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	$0.780048\pi$
$830$	−1.55198	−0.0538702
$831$	0	0
$832$	−0.927120	−0.0321421
$833$	51.5668	1.78669
$834$	0	0
$835$	−2.65643	−0.0919295
$836$	−3.07356	−0.106301
$837$	0	0
$838$	−2.07274	−0.0716015
$839$	−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
$842$	−31.3066	−1.07890
$843$	0	0
$844$	0.0640960	0.00220628
$845$	−1.64622	−0.0566316
$846$	0	0
$847$	123.026	4.22722
$848$	7.12594	0.244706
$849$	0	0
$850$	−21.5725	−0.739931
$851$	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	$0.169870\pi$
$854$	6.26284	0.214310
$855$	0	0
$856$	−15.3500	−0.524651
$857$	−14.0977	−0.481569	−0.240785	−	0.970579i	$-0.577405\pi$
$857$	−14.0977	−0.481569	−0.240785	+	0.970579i	$0.577405\pi$
$858$	0	0
$859$	3.15309	0.107582	0.0537910	−	0.998552i	$-0.482870\pi$
$859$	3.15309	0.107582	0.0537910	+	0.998552i	$0.482870\pi$
$860$	0.805174	0.0274562
$861$	0	0
$862$	7.55623	0.257366
$863$	−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
$866$	−26.3732	−0.896199
$867$	0	0
$868$	−15.4438	−0.524198
$869$	−19.9817	−0.677834
$870$	0	0
$871$	7.01199	0.237592
$872$	−8.06787	−0.273212
$873$	0	0
$874$	−2.61227	−0.0883615
$875$	−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	$0.446918\pi$
$878$	−0.766258	−0.0258600
$879$	0	0
$880$	0.849979	0.0286528
$881$	−14.1391	−0.476357	−0.238179	−	0.971221i	$-0.576550\pi$
$881$	−14.1391	−0.476357	−0.238179	+	0.971221i	$0.576550\pi$
$882$	0	0
$883$	−1.25437	−0.0422128	−0.0211064	−	0.999777i	$-0.506719\pi$
$883$	−1.25437	−0.0422128	−0.0211064	+	0.999777i	$0.506719\pi$
$884$	−4.01482	−0.135033
$885$	0	0
$886$	2.32539	0.0781228
$887$	−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
$890$	1.31850	0.0441961
$891$	0	0
$892$	−13.3505	−0.447009
$893$	−4.30166	−0.143950
$894$	0	0
$895$	−0.475996	−0.0159108
$896$	4.34834	0.145268
$897$	0	0
$898$	−26.1149	−0.871464
$899$	−1.69808	−0.0566342
$900$	0	0
$901$	30.8583	1.02804
$902$	−74.7418	−2.48863
$903$	0	0
$904$	19.1797	0.637908
$905$	−1.86331	−0.0619385
$906$	0	0
$907$	18.7722	0.623322	0.311661	−	0.950193i	$-0.399115\pi$
$907$	18.7722	0.623322	0.311661	+	0.950193i	$0.399115\pi$
$908$	−13.3815	−0.444082
$909$	0	0
$910$	−0.546653	−0.0181214
$911$	−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
$914$	5.99736	0.198375
$915$	0	0
$916$	24.1983	0.799534
$917$	25.2768	0.834713
$918$	0	0
$919$	38.5096	1.27031	0.635157	−	0.772383i	$-0.280935\pi$
$919$	38.5096	1.27031	0.635157	+	0.772383i	$0.280935\pi$
$920$	0.722412	0.0238172
$921$	0	0
$922$	33.9088	1.11673
$923$	3.43361	0.113019
$924$	0	0
$925$	−18.5788	−0.610869
$926$	23.7318	0.779875
$927$	0	0
$928$	0.478108	0.0156947
$929$	24.9635	0.819026	0.409513	−	0.912304i	$-0.365699\pi$
$929$	24.9635	0.819026	0.409513	+	0.912304i	$0.365699\pi$
$930$	0	0
$931$	−5.83883	−0.191360
$932$	18.0849	0.592392
$933$	0	0
$934$	−6.25402	−0.204638
$935$	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	$0.591346\pi$
$938$	−32.8873	−1.07381
$939$	0	0
$940$	1.18960	0.0388006
$941$	8.15568	0.265867	0.132934	−	0.991125i	$-0.457560\pi$
$941$	8.15568	0.265867	0.132934	+	0.991125i	$0.457560\pi$
$942$	0	0
$943$	−63.5244	−2.06864
$944$	0.455180	0.0148148
$945$	0	0
$946$	37.2214	1.21017
$947$	−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
$950$	2.44262	0.0792491
$951$	0	0
$952$	18.8301	0.610288
$953$	−58.0120	−1.87919	−0.939597	−	0.342284i	$-0.888800\pi$
$953$	−58.0120	−1.87919	−0.939597	+	0.342284i	$0.888800\pi$
$954$	0	0
$955$	−1.50843	−0.0488116
$956$	−21.8920	−0.708039
$957$	0	0
$958$	−15.8682	−0.512677
$959$	−74.0623	−2.39160
$960$	0	0
$961$	−18.3857	−0.593086
$962$	−3.45768	−0.111480
$963$	0	0
$964$	2.87717	0.0926674
$965$	1.58698	0.0510868
$966$	0	0
$967$	1.32798	0.0427049	0.0213524	−	0.999772i	$-0.493203\pi$
$967$	1.32798	0.0427049	0.0213524	+	0.999772i	$0.493203\pi$
$968$	28.2927	0.909360
$969$	0	0
$970$	−0.891293	−0.0286177
$971$	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
$974$	13.8268	0.443040
$975$	0	0
$976$	1.44028	0.0461023
$977$	52.8527	1.69091	0.845453	−	0.534050i	$-0.179330\pi$
$977$	52.8527	1.69091	0.845453	+	0.534050i	$0.179330\pi$
$978$	0	0
$979$	60.9512	1.94801
$980$	1.61470	0.0515798
$981$	0	0
$982$	23.8037	0.759606
$983$	−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
$986$	2.07041	0.0659353
$987$	0	0
$988$	0.454592	0.0144625
$989$	31.6352	1.00594
$990$	0	0
$991$	−16.2704	−0.516848	−0.258424	−	0.966032i	$-0.583203\pi$
$991$	−16.2704	−0.516848	−0.258424	+	0.966032i	$0.583203\pi$
$992$	−3.55167	−0.112766
$993$	0	0
$994$	−16.1042	−0.510793
$995$	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	$0.210983\pi$
$998$	8.78566	0.278105
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.g.1.3	✓	5	3.2	odd	2
8048.2.a.n.1.3		5	12.11	even	2
9054.2.a.bb.1.3		5	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:	\( N \)	\(=\)	\( 9054 = 2 \cdot 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9054.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.135598 q^{5} +4.34834 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.135598 q^{5} +4.34834 q^{7} +1.00000 q^{8} +0.135598 q^{10} +6.26839 q^{11} -0.927120 q^{13} +4.34834 q^{14} +1.00000 q^{16} +4.33042 q^{17} -0.490327 q^{19} +0.135598 q^{20} +6.26839 q^{22} +5.32761 q^{23} -4.98161 q^{25} -0.927120 q^{26} +4.34834 q^{28} +0.478108 q^{29} -3.55167 q^{31} +1.00000 q^{32} +4.33042 q^{34} +0.589625 q^{35} +3.72948 q^{37} -0.490327 q^{38} +0.135598 q^{40} -11.9236 q^{41} +5.93796 q^{43} +6.26839 q^{44} +5.32761 q^{46} +8.77304 q^{47} +11.9080 q^{49} -4.98161 q^{50} -0.927120 q^{52} +7.12594 q^{53} +0.849979 q^{55} +4.34834 q^{56} +0.478108 q^{58} +0.455180 q^{59} +1.44028 q^{61} -3.55167 q^{62} +1.00000 q^{64} -0.125715 q^{65} -7.56319 q^{67} +4.33042 q^{68} +0.589625 q^{70} -3.70352 q^{71} -9.39254 q^{73} +3.72948 q^{74} -0.490327 q^{76} +27.2571 q^{77} -3.18770 q^{79} +0.135598 q^{80} -11.9236 q^{82} -11.4455 q^{83} +0.587196 q^{85} +5.93796 q^{86} +6.26839 q^{88} +9.72360 q^{89} -4.03143 q^{91} +5.32761 q^{92} +8.77304 q^{94} -0.0664872 q^{95} -6.57307 q^{97} +11.9080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} + q^{5} + 3 q^{7} + 5 q^{8} + q^{10} + 11 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 20 q^{17} - 4 q^{19} + q^{20} + 11 q^{22} + 4 q^{23} - 6 q^{25} + 5 q^{26} + 3 q^{28} + 6 q^{29} - 3 q^{31} + 5 q^{32} + 20 q^{34} + 3 q^{35} - 10 q^{37} - 4 q^{38} + q^{40} + 6 q^{41} + 11 q^{43} + 11 q^{44} + 4 q^{46} + 9 q^{47} - 4 q^{49} - 6 q^{50} + 5 q^{52} + 22 q^{53} + 14 q^{55} + 3 q^{56} + 6 q^{58} + 10 q^{59} - 11 q^{61} - 3 q^{62} + 5 q^{64} + 12 q^{65} + 14 q^{67} + 20 q^{68} + 3 q^{70} + 26 q^{71} - 7 q^{73} - 10 q^{74} - 4 q^{76} + 26 q^{77} - 15 q^{79} + q^{80} + 6 q^{82} + 12 q^{83} + 12 q^{85} + 11 q^{86} + 11 q^{88} + 5 q^{89} - 22 q^{91} + 4 q^{92} + 9 q^{94} + 10 q^{95} + 6 q^{97} - 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + q^5 + 3 * q^7 + 5 * q^8 + q^10 + 11 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 20 * q^17 - 4 * q^19 + q^20 + 11 * q^22 + 4 * q^23 - 6 * q^25 + 5 * q^26 + 3 * q^28 + 6 * q^29 - 3 * q^31 + 5 * q^32 + 20 * q^34 + 3 * q^35 - 10 * q^37 - 4 * q^38 + q^40 + 6 * q^41 + 11 * q^43 + 11 * q^44 + 4 * q^46 + 9 * q^47 - 4 * q^49 - 6 * q^50 + 5 * q^52 + 22 * q^53 + 14 * q^55 + 3 * q^56 + 6 * q^58 + 10 * q^59 - 11 * q^61 - 3 * q^62 + 5 * q^64 + 12 * q^65 + 14 * q^67 + 20 * q^68 + 3 * q^70 + 26 * q^71 - 7 * q^73 - 10 * q^74 - 4 * q^76 + 26 * q^77 - 15 * q^79 + q^80 + 6 * q^82 + 12 * q^83 + 12 * q^85 + 11 * q^86 + 11 * q^88 + 5 * q^89 - 22 * q^91 + 4 * q^92 + 9 * q^94 + 10 * q^95 + 6 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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