LMFDB - Embedded newform 1587.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1587,2,Mod(1,1587)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1587, 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("1587.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1587 = 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1587.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:	$12.6722588008$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.830830$ of defining polynomial
Character	$\chi$	$=$	1587.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.59435 q^{2} -1.00000 q^{3} +0.541956 q^{4} -0.521109 q^{5} +1.59435 q^{6} -1.32463 q^{7} +2.32463 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.59435 q^{2} -1.00000 q^{3} +0.541956 q^{4} -0.521109 q^{5} +1.59435 q^{6} -1.32463 q^{7} +2.32463 q^{8} +1.00000 q^{9} +0.830830 q^{10} +0.228707 q^{11} -0.541956 q^{12} +2.68472 q^{13} +2.11193 q^{14} +0.521109 q^{15} -4.79020 q^{16} -5.85326 q^{17} -1.59435 q^{18} -1.99223 q^{19} -0.282418 q^{20} +1.32463 q^{21} -0.364640 q^{22} -2.32463 q^{24} -4.72845 q^{25} -4.28038 q^{26} -1.00000 q^{27} -0.717893 q^{28} +6.40370 q^{29} -0.830830 q^{30} +9.92260 q^{31} +2.98798 q^{32} -0.228707 q^{33} +9.33215 q^{34} +0.690279 q^{35} +0.541956 q^{36} +8.79937 q^{37} +3.17631 q^{38} -2.68472 q^{39} -1.21139 q^{40} +0.730658 q^{41} -2.11193 q^{42} +10.4088 q^{43} +0.123949 q^{44} -0.521109 q^{45} -2.04380 q^{47} +4.79020 q^{48} -5.24534 q^{49} +7.53880 q^{50} +5.85326 q^{51} +1.45500 q^{52} -6.71353 q^{53} +1.59435 q^{54} -0.119181 q^{55} -3.07929 q^{56} +1.99223 q^{57} -10.2098 q^{58} -6.52352 q^{59} +0.282418 q^{60} -3.11933 q^{61} -15.8201 q^{62} -1.32463 q^{63} +4.81650 q^{64} -1.39903 q^{65} +0.364640 q^{66} -11.1620 q^{67} -3.17221 q^{68} -1.10055 q^{70} -5.70855 q^{71} +2.32463 q^{72} +2.34127 q^{73} -14.0293 q^{74} +4.72845 q^{75} -1.07970 q^{76} -0.302954 q^{77} +4.28038 q^{78} +9.74760 q^{79} +2.49621 q^{80} +1.00000 q^{81} -1.16493 q^{82} -0.360535 q^{83} +0.717893 q^{84} +3.05018 q^{85} -16.5953 q^{86} -6.40370 q^{87} +0.531661 q^{88} -14.8046 q^{89} +0.830830 q^{90} -3.55627 q^{91} -9.92260 q^{93} +3.25853 q^{94} +1.03817 q^{95} -2.98798 q^{96} +4.78837 q^{97} +8.36292 q^{98} +0.228707 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 2 * q^2 - 5 * q^3 + 4 * q^4 - 3 * q^5 + 2 * q^6 - 4 * q^7 + 9 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 5 q^{9} - q^{10} - 13 q^{11} - 4 q^{12} + 7 q^{13} - 16 q^{14} + 3 q^{15} + 6 q^{16} - q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{20} + 4 q^{21} + 3 q^{22} - 9 q^{24} - 10 q^{25} - 27 q^{26} - 5 q^{27} - 12 q^{28} - 7 q^{29} + q^{30} + 8 q^{31} + 6 q^{32} + 13 q^{33} + 18 q^{34} + 9 q^{35} + 4 q^{36} - 7 q^{37} + 9 q^{38} - 7 q^{39} - 12 q^{40} - 7 q^{41} + 16 q^{42} + 10 q^{43} - 17 q^{44} - 3 q^{45} + 13 q^{47} - 6 q^{48} - q^{49} + 4 q^{50} + q^{51} - q^{52} - 17 q^{53} + 2 q^{54} + 10 q^{55} - 38 q^{56} - 5 q^{57} - 17 q^{58} - 10 q^{59} - 2 q^{60} - 18 q^{61} - q^{62} - 4 q^{63} - 21 q^{64} - 24 q^{65} - 3 q^{66} - 9 q^{67} - 25 q^{68} + 3 q^{70} + 37 q^{71} + 9 q^{72} - 2 q^{73} + 5 q^{74} + 10 q^{75} + 4 q^{76} + 6 q^{77} + 27 q^{78} + 32 q^{79} - 8 q^{80} + 5 q^{81} - 17 q^{82} - 13 q^{83} + 12 q^{84} + 5 q^{85} - 26 q^{86} + 7 q^{87} - 19 q^{88} - 30 q^{89} - q^{90} + 23 q^{91} - 8 q^{93} + 8 q^{94} - 14 q^{95} - 6 q^{96} - 20 q^{97} + 51 q^{98} - 13 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 4 * q^4 - 3 * q^5 + 2 * q^6 - 4 * q^7 + 9 * q^8 + 5 * q^9 - q^10 - 13 * q^11 - 4 * q^12 + 7 * q^13 - 16 * q^14 + 3 * q^15 + 6 * q^16 - q^17 - 2 * q^18 + 5 * q^19 + 2 * q^20 + 4 * q^21 + 3 * q^22 - 9 * q^24 - 10 * q^25 - 27 * q^26 - 5 * q^27 - 12 * q^28 - 7 * q^29 + q^30 + 8 * q^31 + 6 * q^32 + 13 * q^33 + 18 * q^34 + 9 * q^35 + 4 * q^36 - 7 * q^37 + 9 * q^38 - 7 * q^39 - 12 * q^40 - 7 * q^41 + 16 * q^42 + 10 * q^43 - 17 * q^44 - 3 * q^45 + 13 * q^47 - 6 * q^48 - q^49 + 4 * q^50 + q^51 - q^52 - 17 * q^53 + 2 * q^54 + 10 * q^55 - 38 * q^56 - 5 * q^57 - 17 * q^58 - 10 * q^59 - 2 * q^60 - 18 * q^61 - q^62 - 4 * q^63 - 21 * q^64 - 24 * q^65 - 3 * q^66 - 9 * q^67 - 25 * q^68 + 3 * q^70 + 37 * q^71 + 9 * q^72 - 2 * q^73 + 5 * q^74 + 10 * q^75 + 4 * q^76 + 6 * q^77 + 27 * q^78 + 32 * q^79 - 8 * q^80 + 5 * q^81 - 17 * q^82 - 13 * q^83 + 12 * q^84 + 5 * q^85 - 26 * q^86 + 7 * q^87 - 19 * q^88 - 30 * q^89 - q^90 + 23 * q^91 - 8 * q^93 + 8 * q^94 - 14 * q^95 - 6 * q^96 - 20 * q^97 + 51 * q^98 - 13 * 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 $

$2$	−1.59435	−1.12738	−0.563688	−	0.825988i	$-0.690618\pi$
$2$	−1.59435	−1.12738	−0.563688	+	0.825988i	$0.690618\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.541956	0.270978
$5$	−0.521109	−0.233047	−0.116523	−	0.993188i	$-0.537175\pi$
$5$	−0.521109	−0.233047	−0.116523	+	0.993188i	$0.537175\pi$
$6$	1.59435	0.650891
$7$	−1.32463	−0.500665	−0.250332	−	0.968160i	$-0.580540\pi$
$7$	−1.32463	−0.500665	−0.250332	+	0.968160i	$0.580540\pi$
$8$	2.32463	0.821883
$9$	1.00000	0.333333
$10$	0.830830	0.262732
$11$	0.228707	0.0689579	0.0344789	−	0.999405i	$-0.489023\pi$
$11$	0.228707	0.0689579	0.0344789	+	0.999405i	$0.489023\pi$
$12$	−0.541956	−0.156449
$13$	2.68472	0.744607	0.372304	−	0.928111i	$-0.378568\pi$
$13$	2.68472	0.744607	0.372304	+	0.928111i	$0.378568\pi$
$14$	2.11193	0.564438
$15$	0.521109	0.134550
$16$	−4.79020	−1.19755
$17$	−5.85326	−1.41962	−0.709812	−	0.704391i	$-0.751220\pi$
$17$	−5.85326	−1.41962	−0.709812	+	0.704391i	$0.751220\pi$
$18$	−1.59435	−0.375792
$19$	−1.99223	−0.457049	−0.228524	−	0.973538i	$-0.573390\pi$
$19$	−1.99223	−0.457049	−0.228524	+	0.973538i	$0.573390\pi$
$20$	−0.282418	−0.0631505
$21$	1.32463	0.289059
$22$	−0.364640	−0.0777415
$23$	0	0
$23$	0	0
$24$	−2.32463	−0.474514
$25$	−4.72845	−0.945689
$26$	−4.28038	−0.839453
$27$	−1.00000	−0.192450
$28$	−0.717893	−0.135669
$29$	6.40370	1.18914	0.594569	−	0.804045i	$-0.297323\pi$
$29$	6.40370	1.18914	0.594569	+	0.804045i	$0.297323\pi$
$30$	−0.830830	−0.151688
$31$	9.92260	1.78215	0.891076	−	0.453854i	$-0.149951\pi$
$31$	9.92260	1.78215	0.891076	+	0.453854i	$0.149951\pi$
$32$	2.98798	0.528206
$33$	−0.228707	−0.0398129
$34$	9.33215	1.60045
$35$	0.690279	0.116678
$36$	0.541956	0.0903259
$37$	8.79937	1.44661	0.723304	−	0.690530i	$-0.242623\pi$
$37$	8.79937	1.44661	0.723304	+	0.690530i	$0.242623\pi$
$38$	3.17631	0.515266
$39$	−2.68472	−0.429899
$40$	−1.21139	−0.191537
$41$	0.730658	0.114110	0.0570548	−	0.998371i	$-0.481829\pi$
$41$	0.730658	0.114110	0.0570548	+	0.998371i	$0.481829\pi$
$42$	−2.11193	−0.325878
$43$	10.4088	1.58733	0.793665	−	0.608356i	$-0.208171\pi$
$43$	10.4088	1.58733	0.793665	+	0.608356i	$0.208171\pi$
$44$	0.123949	0.0186861
$45$	−0.521109	−0.0776823
$46$	0	0
$47$	−2.04380	−0.298119	−0.149059	−	0.988828i	$-0.547625\pi$
$47$	−2.04380	−0.298119	−0.149059	+	0.988828i	$0.547625\pi$
$48$	4.79020	0.691405
$49$	−5.24534	−0.749335
$50$	7.53880	1.06615
$51$	5.85326	0.819620
$52$	1.45500	0.201772
$53$	−6.71353	−0.922175	−0.461087	−	0.887355i	$-0.652541\pi$
$53$	−6.71353	−0.922175	−0.461087	+	0.887355i	$0.652541\pi$
$54$	1.59435	0.216964
$55$	−0.119181	−0.0160704
$56$	−3.07929	−0.411488
$57$	1.99223	0.263877
$58$	−10.2098	−1.34061
$59$	−6.52352	−0.849289	−0.424645	−	0.905360i	$-0.639601\pi$
$59$	−6.52352	−0.849289	−0.424645	+	0.905360i	$0.639601\pi$
$60$	0.282418	0.0364600
$61$	−3.11933	−0.399389	−0.199695	−	0.979858i	$-0.563995\pi$
$61$	−3.11933	−0.399389	−0.199695	+	0.979858i	$0.563995\pi$
$62$	−15.8201	−2.00916
$63$	−1.32463	−0.166888
$64$	4.81650	0.602062
$65$	−1.39903	−0.173528
$66$	0.364640	0.0448841
$67$	−11.1620	−1.36365	−0.681826	−	0.731514i	$-0.738814\pi$
$67$	−11.1620	−1.36365	−0.681826	+	0.731514i	$0.738814\pi$
$68$	−3.17221	−0.384687
$69$	0	0
$70$	−1.10055	−0.131540
$71$	−5.70855	−0.677480	−0.338740	−	0.940880i	$-0.610001\pi$
$71$	−5.70855	−0.677480	−0.338740	+	0.940880i	$0.610001\pi$
$72$	2.32463	0.273961
$73$	2.34127	0.274025	0.137012	−	0.990569i	$-0.456250\pi$
$73$	2.34127	0.274025	0.137012	+	0.990569i	$0.456250\pi$
$74$	−14.0293	−1.63087
$75$	4.72845	0.545994
$76$	−1.07970	−0.123850
$77$	−0.302954	−0.0345248
$78$	4.28038	0.484658
$79$	9.74760	1.09669	0.548346	−	0.836252i	$-0.315258\pi$
$79$	9.74760	1.09669	0.548346	+	0.836252i	$0.315258\pi$
$80$	2.49621	0.279085
$81$	1.00000	0.111111
$82$	−1.16493	−0.128644
$83$	−0.360535	−0.0395738	−0.0197869	−	0.999804i	$-0.506299\pi$
$83$	−0.360535	−0.0395738	−0.0197869	+	0.999804i	$0.506299\pi$
$84$	0.717893	0.0783286
$85$	3.05018	0.330839
$86$	−16.5953	−1.78952
$87$	−6.40370	−0.686549
$88$	0.531661	0.0566753
$89$	−14.8046	−1.56929	−0.784643	−	0.619948i	$-0.787154\pi$
$89$	−14.8046	−1.56929	−0.784643	+	0.619948i	$0.787154\pi$
$90$	0.830830	0.0875772
$91$	−3.55627	−0.372799
$92$	0	0
$93$	−9.92260	−1.02893
$94$	3.25853	0.336092
$95$	1.03817	0.106514
$96$	−2.98798	−0.304960
$97$	4.78837	0.486185	0.243092	−	0.970003i	$-0.421838\pi$
$97$	4.78837	0.486185	0.243092	+	0.970003i	$0.421838\pi$
$98$	8.36292	0.844782
$99$	0.228707	0.0229860
$100$	−2.56261	−0.256261
$101$	5.41254	0.538567	0.269284	−	0.963061i	$-0.413213\pi$
$101$	5.41254	0.538567	0.269284	+	0.963061i	$0.413213\pi$
$102$	−9.33215	−0.924021
$103$	−4.82547	−0.475467	−0.237734	−	0.971330i	$-0.576405\pi$
$103$	−4.82547	−0.475467	−0.237734	+	0.971330i	$0.576405\pi$
$104$	6.24099	0.611980
$105$	−0.690279	−0.0673643
$106$	10.7037	1.03964
$107$	−14.3020	−1.38262	−0.691311	−	0.722557i	$-0.742966\pi$
$107$	−14.3020	−1.38262	−0.691311	+	0.722557i	$0.742966\pi$
$108$	−0.541956	−0.0521497
$109$	2.77491	0.265788	0.132894	−	0.991130i	$-0.457573\pi$
$109$	2.77491	0.265788	0.132894	+	0.991130i	$0.457573\pi$
$110$	0.190017	0.0181174
$111$	−8.79937	−0.835199
$112$	6.34526	0.599571
$113$	−11.2901	−1.06208	−0.531041	−	0.847346i	$-0.678199\pi$
$113$	−11.2901	−1.06208	−0.531041	+	0.847346i	$0.678199\pi$
$114$	−3.17631	−0.297489
$115$	0	0
$116$	3.47052	0.322230
$117$	2.68472	0.248202
$118$	10.4008	0.957469
$119$	7.75343	0.710756
$120$	1.21139	0.110584
$121$	−10.9477	−0.995245
$122$	4.97331	0.450262
$123$	−0.730658	−0.0658812
$124$	5.37761	0.482924
$125$	5.06958	0.453437
$126$	2.11193	0.188146
$127$	−15.5814	−1.38263	−0.691314	−	0.722555i	$-0.742968\pi$
$127$	−15.5814	−1.38263	−0.691314	+	0.722555i	$0.742968\pi$
$128$	−13.6552	−1.20696
$129$	−10.4088	−0.916445
$130$	2.23055	0.195632
$131$	11.2451	0.982489	0.491244	−	0.871022i	$-0.336542\pi$
$131$	11.2451	0.982489	0.491244	+	0.871022i	$0.336542\pi$
$132$	−0.123949	−0.0107884
$133$	2.63898	0.228828
$134$	17.7961	1.53735
$135$	0.521109	0.0448499
$136$	−13.6067	−1.16676
$137$	−20.3976	−1.74269	−0.871343	−	0.490674i	$-0.836751\pi$
$137$	−20.3976	−1.74269	−0.871343	+	0.490674i	$0.836751\pi$
$138$	0	0
$139$	12.3293	1.04576	0.522879	−	0.852407i	$-0.324858\pi$
$139$	12.3293	1.04576	0.522879	+	0.852407i	$0.324858\pi$
$140$	0.374100	0.0316172
$141$	2.04380	0.172119
$142$	9.10144	0.763776
$143$	0.614015	0.0513465
$144$	−4.79020	−0.399183
$145$	−3.33703	−0.277125
$146$	−3.73281	−0.308929
$147$	5.24534	0.432629
$148$	4.76887	0.391999
$149$	−23.5339	−1.92797	−0.963986	−	0.265953i	$-0.914313\pi$
$149$	−23.5339	−1.92797	−0.963986	+	0.265953i	$0.914313\pi$
$150$	−7.53880	−0.615541
$151$	8.80838	0.716816	0.358408	−	0.933565i	$-0.383320\pi$
$151$	8.80838	0.716816	0.358408	+	0.933565i	$0.383320\pi$
$152$	−4.63120	−0.375640
$153$	−5.85326	−0.473208
$154$	0.483015	0.0389224
$155$	−5.17075	−0.415325
$156$	−1.45500	−0.116493
$157$	−23.5333	−1.87816	−0.939081	−	0.343696i	$-0.888321\pi$
$157$	−23.5333	−1.87816	−0.939081	+	0.343696i	$0.888321\pi$
$158$	−15.5411	−1.23638
$159$	6.71353	0.532418
$160$	−1.55706	−0.123097
$161$	0	0
$162$	−1.59435	−0.125264
$163$	4.99615	0.391329	0.195665	−	0.980671i	$-0.437314\pi$
$163$	4.99615	0.391329	0.195665	+	0.980671i	$0.437314\pi$
$164$	0.395984	0.0309212
$165$	0.119181	0.00927826
$166$	0.574819	0.0446146
$167$	−13.2135	−1.02249	−0.511245	−	0.859435i	$-0.670815\pi$
$167$	−13.2135	−1.02249	−0.511245	+	0.859435i	$0.670815\pi$
$168$	3.07929	0.237573
$169$	−5.79228	−0.445560
$170$	−4.86306	−0.372980
$171$	−1.99223	−0.152350
$172$	5.64111	0.430131
$173$	6.95856	0.529049	0.264525	−	0.964379i	$-0.414785\pi$
$173$	6.95856	0.529049	0.264525	+	0.964379i	$0.414785\pi$
$174$	10.2098	0.773999
$175$	6.26346	0.473473
$176$	−1.09555	−0.0825804
$177$	6.52352	0.490337
$178$	23.6038	1.76918
$179$	−2.40953	−0.180097	−0.0900484	−	0.995937i	$-0.528702\pi$
$179$	−2.40953	−0.180097	−0.0900484	+	0.995937i	$0.528702\pi$
$180$	−0.282418	−0.0210502
$181$	6.39518	0.475350	0.237675	−	0.971345i	$-0.423615\pi$
$181$	6.39518	0.475350	0.237675	+	0.971345i	$0.423615\pi$
$182$	5.66995	0.420284
$183$	3.11933	0.230588
$184$	0	0
$185$	−4.58543	−0.337127
$186$	15.8201	1.15999
$187$	−1.33868	−0.0978943
$188$	−1.10765	−0.0807836
$189$	1.32463	0.0963530
$190$	−1.65520	−0.120081
$191$	10.9138	0.789696	0.394848	−	0.918746i	$-0.370797\pi$
$191$	10.9138	0.789696	0.394848	+	0.918746i	$0.370797\pi$
$192$	−4.81650	−0.347601
$193$	−5.36709	−0.386331	−0.193166	−	0.981166i	$-0.561875\pi$
$193$	−5.36709	−0.386331	−0.193166	+	0.981166i	$0.561875\pi$
$194$	−7.63434	−0.548114
$195$	1.39903	0.100187
$196$	−2.84274	−0.203053
$197$	−5.18162	−0.369175	−0.184587	−	0.982816i	$-0.559095\pi$
$197$	−5.18162	−0.369175	−0.184587	+	0.982816i	$0.559095\pi$
$198$	−0.364640	−0.0259138
$199$	6.60209	0.468010	0.234005	−	0.972235i	$-0.424817\pi$
$199$	6.60209	0.468010	0.234005	+	0.972235i	$0.424817\pi$
$200$	−10.9919	−0.777245
$201$	11.1620	0.787305
$202$	−8.62948	−0.607168
$203$	−8.48257	−0.595360
$204$	3.17221	0.222099
$205$	−0.380752	−0.0265929
$206$	7.69349	0.536031
$207$	0	0
$208$	−12.8603	−0.891703
$209$	−0.455637	−0.0315171
$210$	1.10055	0.0759449
$211$	−18.7549	−1.29114	−0.645569	−	0.763702i	$-0.723380\pi$
$211$	−18.7549	−1.29114	−0.645569	+	0.763702i	$0.723380\pi$
$212$	−3.63844	−0.249889
$213$	5.70855	0.391144
$214$	22.8023	1.55874
$215$	−5.42412	−0.369922
$216$	−2.32463	−0.158171
$217$	−13.1438	−0.892261
$218$	−4.42418	−0.299643
$219$	−2.34127	−0.158208
$220$	−0.0645910	−0.00435473
$221$	−15.7144	−1.05706
$222$	14.0293	0.941584
$223$	19.6713	1.31729	0.658644	−	0.752455i	$-0.271130\pi$
$223$	19.6713	1.31729	0.658644	+	0.752455i	$0.271130\pi$
$224$	−3.95799	−0.264454
$225$	−4.72845	−0.315230
$226$	18.0004	1.19737
$227$	−21.5550	−1.43065	−0.715327	−	0.698790i	$-0.753722\pi$
$227$	−21.5550	−1.43065	−0.715327	+	0.698790i	$0.753722\pi$
$228$	1.07970	0.0715048
$229$	−2.58372	−0.170737	−0.0853686	−	0.996349i	$-0.527207\pi$
$229$	−2.58372	−0.170737	−0.0853686	+	0.996349i	$0.527207\pi$
$230$	0	0
$231$	0.302954	0.0199329
$232$	14.8863	0.977332
$233$	−6.35834	−0.416549	−0.208274	−	0.978070i	$-0.566785\pi$
$233$	−6.35834	−0.416549	−0.208274	+	0.978070i	$0.566785\pi$
$234$	−4.28038	−0.279818
$235$	1.06504	0.0694756
$236$	−3.53546	−0.230139
$237$	−9.74760	−0.633175
$238$	−12.3617	−0.801289
$239$	−18.1915	−1.17671	−0.588355	−	0.808603i	$-0.700224\pi$
$239$	−18.1915	−1.17671	−0.588355	+	0.808603i	$0.700224\pi$
$240$	−2.49621	−0.161130
$241$	−12.1136	−0.780307	−0.390153	−	0.920750i	$-0.627578\pi$
$241$	−12.1136	−0.780307	−0.390153	+	0.920750i	$0.627578\pi$
$242$	17.4545	1.12202
$243$	−1.00000	−0.0641500
$244$	−1.69054	−0.108226
$245$	2.73339	0.174630
$246$	1.16493	0.0742729
$247$	−5.34857	−0.340322
$248$	23.0664	1.46472
$249$	0.360535	0.0228480
$250$	−8.08268	−0.511194
$251$	3.15480	0.199129	0.0995647	−	0.995031i	$-0.468255\pi$
$251$	3.15480	0.199129	0.0995647	+	0.995031i	$0.468255\pi$
$252$	−0.717893	−0.0452230
$253$	0	0
$254$	24.8423	1.55874
$255$	−3.05018	−0.191010
$256$	12.1381	0.758632
$257$	16.8444	1.05072	0.525362	−	0.850879i	$-0.323930\pi$
$257$	16.8444	1.05072	0.525362	+	0.850879i	$0.323930\pi$
$258$	16.5953	1.03318
$259$	−11.6560	−0.724266
$260$	−0.758212	−0.0470223
$261$	6.40370	0.396379
$262$	−17.9286	−1.10763
$263$	15.1543	0.934456	0.467228	−	0.884137i	$-0.345253\pi$
$263$	15.1543	0.934456	0.467228	+	0.884137i	$0.345253\pi$
$264$	−0.531661	−0.0327215
$265$	3.49848	0.214910
$266$	−4.20745	−0.257976
$267$	14.8046	0.906028
$268$	−6.04930	−0.369520
$269$	11.4319	0.697017	0.348508	−	0.937306i	$-0.386688\pi$
$269$	11.4319	0.697017	0.348508	+	0.937306i	$0.386688\pi$
$270$	−0.830830	−0.0505627
$271$	−3.53528	−0.214753	−0.107376	−	0.994218i	$-0.534245\pi$
$271$	−3.53528	−0.214753	−0.107376	+	0.994218i	$0.534245\pi$
$272$	28.0383	1.70007
$273$	3.55627	0.215235
$274$	32.5210	1.96466
$275$	−1.08143	−0.0652127
$276$	0	0
$277$	3.41155	0.204980	0.102490	−	0.994734i	$-0.467319\pi$
$277$	3.41155	0.204980	0.102490	+	0.994734i	$0.467319\pi$
$278$	−19.6572	−1.17896
$279$	9.92260	0.594051
$280$	1.60465	0.0958959
$281$	−12.4591	−0.743245	−0.371622	−	0.928384i	$-0.621198\pi$
$281$	−12.4591	−0.743245	−0.371622	+	0.928384i	$0.621198\pi$
$282$	−3.25853	−0.194043
$283$	−8.83313	−0.525075	−0.262538	−	0.964922i	$-0.584559\pi$
$283$	−8.83313	−0.525075	−0.262538	+	0.964922i	$0.584559\pi$
$284$	−3.09378	−0.183582
$285$	−1.03817	−0.0614957
$286$	−0.978956	−0.0578869
$287$	−0.967855	−0.0571307
$288$	2.98798	0.176069
$289$	17.2606	1.01533
$290$	5.32039	0.312424
$291$	−4.78837	−0.280699
$292$	1.26886	0.0742547
$293$	19.1798	1.12050	0.560249	−	0.828324i	$-0.310705\pi$
$293$	19.1798	1.12050	0.560249	+	0.828324i	$0.310705\pi$
$294$	−8.36292	−0.487735
$295$	3.39946	0.197924
$296$	20.4553	1.18894
$297$	−0.228707	−0.0132710
$298$	37.5213	2.17355
$299$	0	0
$300$	2.56261	0.147952
$301$	−13.7879	−0.794720
$302$	−14.0437	−0.808121
$303$	−5.41254	−0.310942
$304$	9.54316	0.547338
$305$	1.62551	0.0930764
$306$	9.33215	0.533484
$307$	17.7269	1.01173	0.505865	−	0.862613i	$-0.331173\pi$
$307$	17.7269	1.01173	0.505865	+	0.862613i	$0.331173\pi$
$308$	−0.164188	−0.00935545
$309$	4.82547	0.274511
$310$	8.24399	0.468227
$311$	18.3846	1.04249	0.521247	−	0.853406i	$-0.325467\pi$
$311$	18.3846	1.04249	0.521247	+	0.853406i	$0.325467\pi$
$312$	−6.24099	−0.353327
$313$	−34.4133	−1.94515	−0.972576	−	0.232584i	$-0.925282\pi$
$313$	−34.4133	−1.94515	−0.972576	+	0.232584i	$0.925282\pi$
$314$	37.5203	2.11740
$315$	0.690279	0.0388928
$316$	5.28277	0.297179
$317$	0.482123	0.0270787	0.0135394	−	0.999908i	$-0.495690\pi$
$317$	0.482123	0.0270787	0.0135394	+	0.999908i	$0.495690\pi$
$318$	−10.7037	−0.600235
$319$	1.46457	0.0820004
$320$	−2.50992	−0.140309
$321$	14.3020	0.798257
$322$	0	0
$323$	11.6610	0.648837
$324$	0.541956	0.0301086
$325$	−12.6945	−0.704167
$326$	−7.96562	−0.441175
$327$	−2.77491	−0.153453
$328$	1.69851	0.0937847
$329$	2.70729	0.149258
$330$	−0.190017	−0.0104601
$331$	−4.15512	−0.228386	−0.114193	−	0.993459i	$-0.536428\pi$
$331$	−4.15512	−0.228386	−0.114193	+	0.993459i	$0.536428\pi$
$332$	−0.195394	−0.0107236
$333$	8.79937	0.482203
$334$	21.0669	1.15273
$335$	5.81660	0.317795
$336$	−6.34526	−0.346162
$337$	−4.05040	−0.220639	−0.110320	−	0.993896i	$-0.535187\pi$
$337$	−4.05040	−0.220639	−0.110320	+	0.993896i	$0.535187\pi$
$338$	9.23493	0.502314
$339$	11.2901	0.613193
$340$	1.65306	0.0896500
$341$	2.26937	0.122893
$342$	3.17631	0.171755
$343$	16.2206	0.875830
$344$	24.1967	1.30460
$345$	0	0
$346$	−11.0944	−0.596438
$347$	−24.6380	−1.32264	−0.661318	−	0.750106i	$-0.730002\pi$
$347$	−24.6380	−1.32264	−0.661318	+	0.750106i	$0.730002\pi$
$348$	−3.47052	−0.186040
$349$	−32.6758	−1.74909	−0.874547	−	0.484940i	$-0.838841\pi$
$349$	−32.6758	−1.74909	−0.874547	+	0.484940i	$0.838841\pi$
$350$	−9.98616	−0.533783
$351$	−2.68472	−0.143300
$352$	0.683374	0.0364240
$353$	−27.6142	−1.46976	−0.734878	−	0.678199i	$-0.762761\pi$
$353$	−27.6142	−1.46976	−0.734878	+	0.678199i	$0.762761\pi$
$354$	−10.4008	−0.552795
$355$	2.97478	0.157885
$356$	−8.02345	−0.425242
$357$	−7.75343	−0.410355
$358$	3.84164	0.203037
$359$	3.35230	0.176927	0.0884637	−	0.996079i	$-0.471804\pi$
$359$	3.35230	0.176927	0.0884637	+	0.996079i	$0.471804\pi$
$360$	−1.21139	−0.0638457
$361$	−15.0310	−0.791107
$362$	−10.1962	−0.535898
$363$	10.9477	0.574605
$364$	−1.92734	−0.101020
$365$	−1.22006	−0.0638606
$366$	−4.97331	−0.259959
$367$	−2.87657	−0.150156	−0.0750780	−	0.997178i	$-0.523921\pi$
$367$	−2.87657	−0.150156	−0.0750780	+	0.997178i	$0.523921\pi$
$368$	0	0
$369$	0.730658	0.0380365
$370$	7.31078	0.380069
$371$	8.89298	0.461700
$372$	−5.37761	−0.278816
$373$	−3.56237	−0.184453	−0.0922263	−	0.995738i	$-0.529398\pi$
$373$	−3.56237	−0.184453	−0.0922263	+	0.995738i	$0.529398\pi$
$374$	2.13433	0.110364
$375$	−5.06958	−0.261792
$376$	−4.75109	−0.245019
$377$	17.1921	0.885441
$378$	−2.11193	−0.108626
$379$	18.1277	0.931158	0.465579	−	0.885006i	$-0.345846\pi$
$379$	18.1277	0.931158	0.465579	+	0.885006i	$0.345846\pi$
$380$	0.562641	0.0288629
$381$	15.5814	0.798260
$382$	−17.4005	−0.890285
$383$	−29.0439	−1.48407	−0.742036	−	0.670360i	$-0.766140\pi$
$383$	−29.0439	−1.48407	−0.742036	+	0.670360i	$0.766140\pi$
$384$	13.6552	0.696837
$385$	0.157872	0.00804589
$386$	8.55702	0.435541
$387$	10.4088	0.529110
$388$	2.59508	0.131745
$389$	38.4893	1.95148	0.975742	−	0.218922i	$-0.0702539\pi$
$389$	38.4893	1.95148	0.975742	+	0.218922i	$0.0702539\pi$
$390$	−2.23055	−0.112948
$391$	0	0
$392$	−12.1935	−0.615865
$393$	−11.2451	−0.567240
$394$	8.26132	0.416199
$395$	−5.07956	−0.255580
$396$	0.123949	0.00622868
$397$	26.4245	1.32621	0.663103	−	0.748528i	$-0.269239\pi$
$397$	26.4245	1.32621	0.663103	+	0.748528i	$0.269239\pi$
$398$	−10.5261	−0.527623
$399$	−2.63898	−0.132114
$400$	22.6502	1.13251
$401$	14.3749	0.717849	0.358924	−	0.933367i	$-0.383144\pi$
$401$	14.3749	0.717849	0.358924	+	0.933367i	$0.383144\pi$
$402$	−17.7961	−0.887589
$403$	26.6394	1.32700
$404$	2.93335	0.145940
$405$	−0.521109	−0.0258941
$406$	13.5242	0.671195
$407$	2.01248	0.0997550
$408$	13.6067	0.673632
$409$	−26.7455	−1.32248	−0.661241	−	0.750174i	$-0.729970\pi$
$409$	−26.7455	−1.32248	−0.661241	+	0.750174i	$0.729970\pi$
$410$	0.607052	0.0299802
$411$	20.3976	1.00614
$412$	−2.61519	−0.128841
$413$	8.64128	0.425209
$414$	0	0
$415$	0.187878	0.00922255
$416$	8.02190	0.393306
$417$	−12.3293	−0.603769
$418$	0.726446	0.0355316
$419$	9.01662	0.440491	0.220245	−	0.975445i	$-0.429314\pi$
$419$	9.01662	0.440491	0.220245	+	0.975445i	$0.429314\pi$
$420$	−0.374100	−0.0182542
$421$	−31.3164	−1.52627	−0.763135	−	0.646239i	$-0.776341\pi$
$421$	−31.3164	−1.52627	−0.763135	+	0.646239i	$0.776341\pi$
$422$	29.9019	1.45560
$423$	−2.04380	−0.0993729
$424$	−15.6065	−0.757919
$425$	27.6768	1.34252
$426$	−9.10144	−0.440966
$427$	4.13197	0.199960
$428$	−7.75102	−0.374660
$429$	−0.614015	−0.0296449
$430$	8.64795	0.417041
$431$	−0.989003	−0.0476386	−0.0238193	−	0.999716i	$-0.507583\pi$
$431$	−0.989003	−0.0476386	−0.0238193	+	0.999716i	$0.507583\pi$
$432$	4.79020	0.230468
$433$	−5.36009	−0.257589	−0.128795	−	0.991671i	$-0.541111\pi$
$433$	−5.36009	−0.257589	−0.128795	+	0.991671i	$0.541111\pi$
$434$	20.9559	1.00591
$435$	3.33703	0.159998
$436$	1.50388	0.0720226
$437$	0	0
$438$	3.73281	0.178360
$439$	5.22012	0.249143	0.124571	−	0.992211i	$-0.460244\pi$
$439$	5.22012	0.249143	0.124571	+	0.992211i	$0.460244\pi$
$440$	−0.277053	−0.0132080
$441$	−5.24534	−0.249778
$442$	25.0542	1.19171
$443$	−27.9486	−1.32788	−0.663940	−	0.747786i	$-0.731117\pi$
$443$	−27.9486	−1.32788	−0.663940	+	0.747786i	$0.731117\pi$
$444$	−4.76887	−0.226320
$445$	7.71481	0.365717
$446$	−31.3630	−1.48508
$447$	23.5339	1.11312
$448$	−6.38010	−0.301431
$449$	−9.17481	−0.432986	−0.216493	−	0.976284i	$-0.569462\pi$
$449$	−9.17481	−0.432986	−0.216493	+	0.976284i	$0.569462\pi$
$450$	7.53880	0.355383
$451$	0.167107	0.00786876
$452$	−6.11872	−0.287801
$453$	−8.80838	−0.413854
$454$	34.3662	1.61289
$455$	1.85320	0.0868795
$456$	4.63120	0.216876
$457$	8.08404	0.378155	0.189078	−	0.981962i	$-0.439450\pi$
$457$	8.08404	0.378155	0.189078	+	0.981962i	$0.439450\pi$
$458$	4.11936	0.192485
$459$	5.85326	0.273207
$460$	0	0
$461$	20.4511	0.952505	0.476252	−	0.879309i	$-0.341995\pi$
$461$	20.4511	0.952505	0.476252	+	0.879309i	$0.341995\pi$
$462$	−0.483015	−0.0224719
$463$	1.60056	0.0743844	0.0371922	−	0.999308i	$-0.488159\pi$
$463$	1.60056	0.0743844	0.0371922	+	0.999308i	$0.488159\pi$
$464$	−30.6750	−1.42405
$465$	5.17075	0.239788
$466$	10.1374	0.469608
$467$	9.49713	0.439475	0.219737	−	0.975559i	$-0.429480\pi$
$467$	9.49713	0.439475	0.219737	+	0.975559i	$0.429480\pi$
$468$	1.45500	0.0672573
$469$	14.7855	0.682733
$470$	−1.69805	−0.0783252
$471$	23.5333	1.08436
$472$	−15.1648	−0.698016
$473$	2.38057	0.109459
$474$	15.5411	0.713827
$475$	9.42014	0.432226
$476$	4.20202	0.192599
$477$	−6.71353	−0.307392
$478$	29.0036	1.32660
$479$	30.5722	1.39688	0.698439	−	0.715670i	$-0.253879\pi$
$479$	30.5722	1.39688	0.698439	+	0.715670i	$0.253879\pi$
$480$	1.55706	0.0710699
$481$	23.6238	1.07715
$482$	19.3134	0.879700
$483$	0	0
$484$	−5.93316	−0.269689
$485$	−2.49526	−0.113304
$486$	1.59435	0.0723212
$487$	−11.7341	−0.531725	−0.265862	−	0.964011i	$-0.585657\pi$
$487$	−11.7341	−0.531725	−0.265862	+	0.964011i	$0.585657\pi$
$488$	−7.25130	−0.328251
$489$	−4.99615	−0.225934
$490$	−4.35799	−0.196874
$491$	25.6492	1.15753	0.578766	−	0.815494i	$-0.303535\pi$
$491$	25.6492	1.15753	0.578766	+	0.815494i	$0.303535\pi$
$492$	−0.395984	−0.0178523
$493$	−37.4825	−1.68813
$494$	8.52750	0.383671
$495$	−0.119181	−0.00535681
$496$	−47.5312	−2.13421
$497$	7.56175	0.339191
$498$	−0.574819	−0.0257582
$499$	3.07715	0.137752	0.0688761	−	0.997625i	$-0.478059\pi$
$499$	3.07715	0.137752	0.0688761	+	0.997625i	$0.478059\pi$
$500$	2.74749	0.122871
$501$	13.2135	0.590335
$502$	−5.02986	−0.224494
$503$	18.3365	0.817583	0.408791	−	0.912628i	$-0.365950\pi$
$503$	18.3365	0.817583	0.408791	+	0.912628i	$0.365950\pi$
$504$	−3.07929	−0.137163
$505$	−2.82052	−0.125511
$506$	0	0
$507$	5.79228	0.257244
$508$	−8.44444	−0.374661
$509$	10.6762	0.473214	0.236607	−	0.971605i	$-0.423965\pi$
$509$	10.6762	0.473214	0.236607	+	0.971605i	$0.423965\pi$
$510$	4.86306	0.215340
$511$	−3.10133	−0.137195
$512$	7.95788	0.351692
$513$	1.99223	0.0879590
$514$	−26.8559	−1.18456
$515$	2.51459	0.110806
$516$	−5.64111	−0.248336
$517$	−0.467432	−0.0205576
$518$	18.5837	0.816520
$519$	−6.95856	−0.305447
$520$	−3.25223	−0.142620
$521$	−2.58274	−0.113152	−0.0565759	−	0.998398i	$-0.518018\pi$
$521$	−2.58274	−0.113152	−0.0565759	+	0.998398i	$0.518018\pi$
$522$	−10.2098	−0.446869
$523$	−32.3911	−1.41636	−0.708182	−	0.706030i	$-0.750485\pi$
$523$	−32.3911	−1.41636	−0.708182	+	0.706030i	$0.750485\pi$
$524$	6.09434	0.266233
$525$	−6.26346	−0.273360
$526$	−24.1613	−1.05348
$527$	−58.0796	−2.52999
$528$	1.09555	0.0476778
$529$	0	0
$530$	−5.57780	−0.242284
$531$	−6.52352	−0.283096
$532$	1.43021	0.0620074
$533$	1.96161	0.0849668
$534$	−23.6038	−1.02143
$535$	7.45287	0.322216
$536$	−25.9475	−1.12076
$537$	2.40953	0.103979
$538$	−18.2265	−0.785801
$539$	−1.19965	−0.0516725
$540$	0.282418	0.0121533
$541$	11.4637	0.492863	0.246431	−	0.969160i	$-0.420742\pi$
$541$	11.4637	0.492863	0.246431	+	0.969160i	$0.420742\pi$
$542$	5.63647	0.242107
$543$	−6.39518	−0.274443
$544$	−17.4894	−0.749854
$545$	−1.44603	−0.0619410
$546$	−5.66995	−0.242651
$547$	−27.4199	−1.17239	−0.586194	−	0.810171i	$-0.699374\pi$
$547$	−27.4199	−1.17239	−0.586194	+	0.810171i	$0.699374\pi$
$548$	−11.0546	−0.472229
$549$	−3.11933	−0.133130
$550$	1.72418	0.0735193
$551$	−12.7576	−0.543494
$552$	0	0
$553$	−12.9120	−0.549075
$554$	−5.43921	−0.231090
$555$	4.58543	0.194641
$556$	6.68194	0.283377
$557$	−31.6734	−1.34204	−0.671022	−	0.741437i	$-0.734144\pi$
$557$	−31.6734	−1.34204	−0.671022	+	0.741437i	$0.734144\pi$
$558$	−15.8201	−0.669719
$559$	27.9447	1.18194
$560$	−3.30657	−0.139728
$561$	1.33868	0.0565193
$562$	19.8641	0.837917
$563$	−38.1891	−1.60948	−0.804739	−	0.593629i	$-0.797695\pi$
$563$	−38.1891	−1.60948	−0.804739	+	0.593629i	$0.797695\pi$
$564$	1.10765	0.0466404
$565$	5.88336	0.247515
$566$	14.0831	0.591957
$567$	−1.32463	−0.0556294
$568$	−13.2703	−0.556809
$569$	26.2707	1.10132	0.550662	−	0.834728i	$-0.314375\pi$
$569$	26.2707	1.10132	0.550662	+	0.834728i	$0.314375\pi$
$570$	1.65520	0.0693288
$571$	21.5002	0.899755	0.449877	−	0.893090i	$-0.351468\pi$
$571$	21.5002	0.899755	0.449877	+	0.893090i	$0.351468\pi$
$572$	0.332769	0.0139138
$573$	−10.9138	−0.455931
$574$	1.54310	0.0644078
$575$	0	0
$576$	4.81650	0.200687
$577$	41.8107	1.74060	0.870300	−	0.492522i	$-0.163925\pi$
$577$	41.8107	1.74060	0.870300	+	0.492522i	$0.163925\pi$
$578$	−27.5195	−1.14466
$579$	5.36709	0.223049
$580$	−1.80852	−0.0750947
$581$	0.477577	0.0198132
$582$	7.63434	0.316454
$583$	−1.53543	−0.0635912
$584$	5.44260	0.225216
$585$	−1.39903	−0.0578428
$586$	−30.5794	−1.26322
$587$	−40.8539	−1.68622	−0.843109	−	0.537742i	$-0.819277\pi$
$587$	−40.8539	−1.68622	−0.843109	+	0.537742i	$0.819277\pi$
$588$	2.84274	0.117233
$589$	−19.7681	−0.814530
$590$	−5.41993	−0.223135
$591$	5.18162	0.213143
$592$	−42.1507	−1.73238
$593$	6.23908	0.256208	0.128104	−	0.991761i	$-0.459111\pi$
$593$	6.23908	0.256208	0.128104	+	0.991761i	$0.459111\pi$
$594$	0.364640	0.0149614
$595$	−4.04038	−0.165639
$596$	−12.7543	−0.522438
$597$	−6.60209	−0.270206
$598$	0	0
$599$	4.84725	0.198053	0.0990267	−	0.995085i	$-0.468427\pi$
$599$	4.84725	0.198053	0.0990267	+	0.995085i	$0.468427\pi$
$600$	10.9919	0.448743
$601$	−20.5904	−0.839900	−0.419950	−	0.907547i	$-0.637952\pi$
$601$	−20.5904	−0.839900	−0.419950	+	0.907547i	$0.637952\pi$
$602$	21.9827	0.895949
$603$	−11.1620	−0.454551
$604$	4.77375	0.194241
$605$	5.70494	0.231939
$606$	8.62948	0.350549
$607$	38.4523	1.56073	0.780365	−	0.625324i	$-0.215033\pi$
$607$	38.4523	1.56073	0.780365	+	0.625324i	$0.215033\pi$
$608$	−5.95275	−0.241416
$609$	8.48257	0.343731
$610$	−2.59163	−0.104932
$611$	−5.48703	−0.221981
$612$	−3.17221	−0.128229
$613$	19.6779	0.794783	0.397392	−	0.917649i	$-0.369915\pi$
$613$	19.6779	0.794783	0.397392	+	0.917649i	$0.369915\pi$
$614$	−28.2630	−1.14060
$615$	0.380752	0.0153534
$616$	−0.704257	−0.0283753
$617$	−38.0280	−1.53095	−0.765475	−	0.643466i	$-0.777496\pi$
$617$	−38.0280	−1.53095	−0.765475	+	0.643466i	$0.777496\pi$
$618$	−7.69349	−0.309477
$619$	37.0927	1.49088	0.745440	−	0.666573i	$-0.232240\pi$
$619$	37.0927	1.49088	0.745440	+	0.666573i	$0.232240\pi$
$620$	−2.80232	−0.112544
$621$	0	0
$622$	−29.3115	−1.17528
$623$	19.6107	0.785687
$624$	12.8603	0.514825
$625$	21.0004	0.840017
$626$	54.8668	2.19292
$627$	0.455637	0.0181964
$628$	−12.7540	−0.508940
$629$	−51.5050	−2.05364
$630$	−1.10055	−0.0438468
$631$	−15.6619	−0.623490	−0.311745	−	0.950166i	$-0.600913\pi$
$631$	−15.6619	−0.623490	−0.311745	+	0.950166i	$0.600913\pi$
$632$	22.6596	0.901351
$633$	18.7549	0.745439
$634$	−0.768673	−0.0305279
$635$	8.11961	0.322217
$636$	3.63844	0.144273
$637$	−14.0823	−0.557960
$638$	−2.33505	−0.0924454
$639$	−5.70855	−0.225827
$640$	7.11582	0.281277
$641$	−22.3962	−0.884596	−0.442298	−	0.896868i	$-0.645837\pi$
$641$	−22.3962	−0.884596	−0.442298	+	0.896868i	$0.645837\pi$
$642$	−22.8023	−0.899936
$643$	−24.9262	−0.982993	−0.491496	−	0.870880i	$-0.663550\pi$
$643$	−24.9262	−0.982993	−0.491496	+	0.870880i	$0.663550\pi$
$644$	0	0
$645$	5.42412	0.213575
$646$	−18.5918	−0.731484
$647$	38.9154	1.52992	0.764961	−	0.644077i	$-0.222758\pi$
$647$	38.9154	1.52992	0.764961	+	0.644077i	$0.222758\pi$
$648$	2.32463	0.0913203
$649$	−1.49198	−0.0585652
$650$	20.2396	0.793861
$651$	13.1438	0.515147
$652$	2.70769	0.106041
$653$	−23.9969	−0.939070	−0.469535	−	0.882914i	$-0.655578\pi$
$653$	−23.9969	−0.939070	−0.469535	+	0.882914i	$0.655578\pi$
$654$	4.42418	0.172999
$655$	−5.85992	−0.228966
$656$	−3.49999	−0.136652
$657$	2.34127	0.0913416
$658$	−4.31637	−0.168270
$659$	20.5773	0.801579	0.400790	−	0.916170i	$-0.368736\pi$
$659$	20.5773	0.801579	0.400790	+	0.916170i	$0.368736\pi$
$660$	0.0645910	0.00251420
$661$	−23.2133	−0.902893	−0.451447	−	0.892298i	$-0.649092\pi$
$661$	−23.2133	−0.902893	−0.451447	+	0.892298i	$0.649092\pi$
$662$	6.62472	0.257477
$663$	15.7144	0.610295
$664$	−0.838111	−0.0325250
$665$	−1.37519	−0.0533277
$666$	−14.0293	−0.543624
$667$	0	0
$668$	−7.16112	−0.277072
$669$	−19.6713	−0.760536
$670$	−9.27371	−0.358275
$671$	−0.713414	−0.0275410
$672$	3.95799	0.152683
$673$	13.2375	0.510268	0.255134	−	0.966906i	$-0.417880\pi$
$673$	13.2375	0.510268	0.255134	+	0.966906i	$0.417880\pi$
$674$	6.45776	0.248744
$675$	4.72845	0.181998
$676$	−3.13916	−0.120737
$677$	−0.668555	−0.0256947	−0.0128473	−	0.999917i	$-0.504090\pi$
$677$	−0.668555	−0.0256947	−0.0128473	+	0.999917i	$0.504090\pi$
$678$	−18.0004	−0.691300
$679$	−6.34284	−0.243416
$680$	7.09056	0.271911
$681$	21.5550	0.825989
$682$	−3.61818	−0.138547
$683$	15.0699	0.576635	0.288318	−	0.957535i	$-0.406904\pi$
$683$	15.0699	0.576635	0.288318	+	0.957535i	$0.406904\pi$
$684$	−1.07970	−0.0412833
$685$	10.6294	0.406128
$686$	−25.8613	−0.987391
$687$	2.58372	0.0985752
$688$	−49.8603	−1.90090
$689$	−18.0239	−0.686658
$690$	0	0
$691$	21.5100	0.818279	0.409140	−	0.912472i	$-0.365829\pi$
$691$	21.5100	0.818279	0.409140	+	0.912472i	$0.365829\pi$
$692$	3.77123	0.143361
$693$	−0.302954	−0.0115083
$694$	39.2816	1.49111
$695$	−6.42491	−0.243711
$696$	−14.8863	−0.564263
$697$	−4.27673	−0.161993
$698$	52.0967	1.97189
$699$	6.35834	0.240495
$700$	3.39452	0.128301
$701$	9.14733	0.345490	0.172745	−	0.984967i	$-0.444736\pi$
$701$	9.14733	0.345490	0.172745	+	0.984967i	$0.444736\pi$
$702$	4.28038	0.161553
$703$	−17.5304	−0.661170
$704$	1.10157	0.0415169
$705$	−1.06504	−0.0401118
$706$	44.0268	1.65697
$707$	−7.16963	−0.269642
$708$	3.53546	0.132871
$709$	−5.37322	−0.201796	−0.100898	−	0.994897i	$-0.532171\pi$
$709$	−5.37322	−0.201796	−0.100898	+	0.994897i	$0.532171\pi$
$710$	−4.74284	−0.177995
$711$	9.74760	0.365564
$712$	−34.4153	−1.28977
$713$	0	0
$714$	12.3617	0.462625
$715$	−0.319969	−0.0119661
$716$	−1.30586	−0.0488022
$717$	18.1915	0.679374
$718$	−5.34474	−0.199464
$719$	−25.0678	−0.934870	−0.467435	−	0.884028i	$-0.654822\pi$
$719$	−25.0678	−0.934870	−0.467435	+	0.884028i	$0.654822\pi$
$720$	2.49621	0.0930283
$721$	6.39198	0.238050
$722$	23.9647	0.891875
$723$	12.1136	0.450510
$724$	3.46590	0.128809
$725$	−30.2796	−1.12456
$726$	−17.4545	−0.647796
$727$	42.2088	1.56544	0.782718	−	0.622376i	$-0.213833\pi$
$727$	42.2088	1.56544	0.782718	+	0.622376i	$0.213833\pi$
$728$	−8.26703	−0.306397
$729$	1.00000	0.0370370
$730$	1.94520	0.0719950
$731$	−60.9255	−2.25341
$732$	1.69054	0.0624841
$733$	−36.7742	−1.35829	−0.679143	−	0.734006i	$-0.737648\pi$
$733$	−36.7742	−1.35829	−0.679143	+	0.734006i	$0.737648\pi$
$734$	4.58627	0.169282
$735$	−2.73339	−0.100823
$736$	0	0
$737$	−2.55283	−0.0940346
$738$	−1.16493	−0.0428815
$739$	17.9776	0.661316	0.330658	−	0.943751i	$-0.392729\pi$
$739$	17.9776	0.661316	0.330658	+	0.943751i	$0.392729\pi$
$740$	−2.48510	−0.0913540
$741$	5.34857	0.196485
$742$	−14.1785	−0.520510
$743$	13.9900	0.513242	0.256621	−	0.966512i	$-0.417391\pi$
$743$	13.9900	0.513242	0.256621	+	0.966512i	$0.417391\pi$
$744$	−23.0664	−0.845656
$745$	12.2637	0.449308
$746$	5.67967	0.207948
$747$	−0.360535	−0.0131913
$748$	−0.725507	−0.0265272
$749$	18.9449	0.692230
$750$	8.08268	0.295138
$751$	−25.3788	−0.926087	−0.463043	−	0.886336i	$-0.653243\pi$
$751$	−25.3788	−0.926087	−0.463043	+	0.886336i	$0.653243\pi$
$752$	9.79020	0.357012
$753$	−3.15480	−0.114967
$754$	−27.4103	−0.998225
$755$	−4.59012	−0.167052
$756$	0.717893	0.0261095
$757$	−40.4420	−1.46989	−0.734944	−	0.678127i	$-0.762792\pi$
$757$	−40.4420	−1.46989	−0.734944	+	0.678127i	$0.762792\pi$
$758$	−28.9019	−1.04977
$759$	0	0
$760$	2.41336	0.0875418
$761$	25.9618	0.941115	0.470558	−	0.882369i	$-0.344053\pi$
$761$	25.9618	0.941115	0.470558	+	0.882369i	$0.344053\pi$
$762$	−24.8423	−0.899940
$763$	−3.67574	−0.133071
$764$	5.91480	0.213990
$765$	3.05018	0.110280
$766$	46.3061	1.67311
$767$	−17.5138	−0.632387
$768$	−12.1381	−0.437997
$769$	32.7263	1.18014	0.590070	−	0.807352i	$-0.299100\pi$
$769$	32.7263	1.18014	0.590070	+	0.807352i	$0.299100\pi$
$770$	−0.251703	−0.00907075
$771$	−16.8444	−0.606635
$772$	−2.90872	−0.104687
$773$	−12.3906	−0.445660	−0.222830	−	0.974857i	$-0.571529\pi$
$773$	−12.3906	−0.445660	−0.222830	+	0.974857i	$0.571529\pi$
$774$	−16.5953	−0.596506
$775$	−46.9185	−1.68536
$776$	11.1312	0.399587
$777$	11.6560	0.418155
$778$	−61.3655	−2.20006
$779$	−1.45564	−0.0521536
$780$	0.758212	0.0271483
$781$	−1.30559	−0.0467176
$782$	0	0
$783$	−6.40370	−0.228850
$784$	25.1262	0.897365
$785$	12.2634	0.437700
$786$	17.9286	0.639493
$787$	1.72236	0.0613954	0.0306977	−	0.999529i	$-0.490227\pi$
$787$	1.72236	0.0613954	0.0306977	+	0.999529i	$0.490227\pi$
$788$	−2.80821	−0.100038
$789$	−15.1543	−0.539508
$790$	8.09860	0.288135
$791$	14.9552	0.531747
$792$	0.531661	0.0188918
$793$	−8.37453	−0.297388
$794$	−42.1299	−1.49513
$795$	−3.49848	−0.124078
$796$	3.57804	0.126820
$797$	−14.9761	−0.530481	−0.265240	−	0.964182i	$-0.585451\pi$
$797$	−14.9761	−0.530481	−0.265240	+	0.964182i	$0.585451\pi$
$798$	4.20745	0.148942
$799$	11.9629	0.423217
$800$	−14.1285	−0.499519
$801$	−14.8046	−0.523095
$802$	−22.9187	−0.809286
$803$	0.535466	0.0188962
$804$	6.04930	0.213342
$805$	0	0
$806$	−42.4726	−1.49603
$807$	−11.4319	−0.402423
$808$	12.5822	0.442639
$809$	13.5579	0.476670	0.238335	−	0.971183i	$-0.423398\pi$
$809$	13.5579	0.476670	0.238335	+	0.971183i	$0.423398\pi$
$810$	0.830830	0.0291924
$811$	−10.1160	−0.355222	−0.177611	−	0.984101i	$-0.556837\pi$
$811$	−10.1160	−0.355222	−0.177611	+	0.984101i	$0.556837\pi$
$812$	−4.59718	−0.161329
$813$	3.53528	0.123988
$814$	−3.20860	−0.112461
$815$	−2.60354	−0.0911980
$816$	−28.0383	−0.981535
$817$	−20.7367	−0.725487
$818$	42.6418	1.49093
$819$	−3.55627	−0.124266
$820$	−0.206351	−0.00720608
$821$	−9.24778	−0.322750	−0.161375	−	0.986893i	$-0.551593\pi$
$821$	−9.24778	−0.322750	−0.161375	+	0.986893i	$0.551593\pi$
$822$	−32.5210	−1.13430
$823$	−11.1320	−0.388037	−0.194019	−	0.980998i	$-0.562152\pi$
$823$	−11.1320	−0.388037	−0.194019	+	0.980998i	$0.562152\pi$
$824$	−11.2174	−0.390778
$825$	1.08143	0.0376506
$826$	−13.7772	−0.479371
$827$	−28.2453	−0.982186	−0.491093	−	0.871107i	$-0.663402\pi$
$827$	−28.2453	−0.982186	−0.491093	+	0.871107i	$0.663402\pi$
$828$	0	0
$829$	−13.3377	−0.463237	−0.231618	−	0.972807i	$-0.574402\pi$
$829$	−13.3377	−0.463237	−0.231618	+	0.972807i	$0.574402\pi$
$830$	−0.299543	−0.0103973
$831$	−3.41155	−0.118345
$832$	12.9309	0.448300
$833$	30.7024	1.06377
$834$	19.6572	0.680675
$835$	6.88566	0.238288
$836$	−0.246935	−0.00854043
$837$	−9.92260	−0.342975
$838$	−14.3757	−0.496599
$839$	27.6192	0.953520	0.476760	−	0.879034i	$-0.341811\pi$
$839$	27.6192	0.953520	0.476760	+	0.879034i	$0.341811\pi$
$840$	−1.60465	−0.0553655
$841$	12.0074	0.414049
$842$	49.9294	1.72068
$843$	12.4591	0.429113
$844$	−10.1643	−0.349870
$845$	3.01841	0.103836
$846$	3.25853	0.112031
$847$	14.5017	0.498284
$848$	32.1591	1.10435
$849$	8.83313	0.303152
$850$	−44.1266	−1.51353
$851$	0	0
$852$	3.09378	0.105991
$853$	−20.1102	−0.688559	−0.344280	−	0.938867i	$-0.611877\pi$
$853$	−20.1102	−0.688559	−0.344280	+	0.938867i	$0.611877\pi$
$854$	−6.58782	−0.225430
$855$	1.03817	0.0355046
$856$	−33.2468	−1.13635
$857$	37.0866	1.26685	0.633426	−	0.773803i	$-0.281648\pi$
$857$	37.0866	1.26685	0.633426	+	0.773803i	$0.281648\pi$
$858$	0.978956	0.0334210
$859$	−14.6128	−0.498581	−0.249291	−	0.968429i	$-0.580197\pi$
$859$	−14.6128	−0.498581	−0.249291	+	0.968429i	$0.580197\pi$
$860$	−2.93963	−0.100241
$861$	0.967855	0.0329844
$862$	1.57682	0.0537067
$863$	−23.2130	−0.790181	−0.395091	−	0.918642i	$-0.629287\pi$
$863$	−23.2130	−0.790181	−0.395091	+	0.918642i	$0.629287\pi$
$864$	−2.98798	−0.101653
$865$	−3.62616	−0.123293
$866$	8.54586	0.290400
$867$	−17.2606	−0.586202
$868$	−7.12337	−0.241783
$869$	2.22935	0.0756255
$870$	−5.32039	−0.180378
$871$	−29.9668	−1.01539
$872$	6.45065	0.218446
$873$	4.78837	0.162062
$874$	0	0
$875$	−6.71534	−0.227020
$876$	−1.26886	−0.0428710
$877$	−18.2995	−0.617929	−0.308965	−	0.951073i	$-0.599983\pi$
$877$	−18.2995	−0.617929	−0.308965	+	0.951073i	$0.599983\pi$
$878$	−8.32270	−0.280878
$879$	−19.1798	−0.646920
$880$	0.570902	0.0192451
$881$	12.7417	0.429278	0.214639	−	0.976693i	$-0.431142\pi$
$881$	12.7417	0.429278	0.214639	+	0.976693i	$0.431142\pi$
$882$	8.36292	0.281594
$883$	−34.1979	−1.15085	−0.575426	−	0.817854i	$-0.695164\pi$
$883$	−34.1979	−1.15085	−0.575426	+	0.817854i	$0.695164\pi$
$884$	−8.51648	−0.286440
$885$	−3.39946	−0.114272
$886$	44.5600	1.49702
$887$	−46.1073	−1.54813	−0.774066	−	0.633105i	$-0.781780\pi$
$887$	−46.1073	−1.54813	−0.774066	+	0.633105i	$0.781780\pi$
$888$	−20.4553	−0.686436
$889$	20.6397	0.692233
$890$	−12.3001	−0.412301
$891$	0.228707	0.00766199
$892$	10.6610	0.356956
$893$	4.07172	0.136255
$894$	−37.5213	−1.25490
$895$	1.25563	0.0419710
$896$	18.0881	0.604281
$897$	0	0
$898$	14.6279	0.488138
$899$	63.5414	2.11922
$900$	−2.56261	−0.0854203
$901$	39.2960	1.30914
$902$	−0.266427	−0.00887105
$903$	13.7879	0.458832
$904$	−26.2453	−0.872906
$905$	−3.33258	−0.110779
$906$	14.0437	0.466569
$907$	−0.0237283	−0.000787885 0	−0.000393943	−	1.00000i	$-0.500125\pi$
$907$	−0.0237283	−0.000787885 0	−0.000393943		1.00000i	$0.500125\pi$
$908$	−11.6818	−0.387676
$909$	5.41254	0.179522
$910$	−2.95466	−0.0979460
$911$	27.1775	0.900431	0.450215	−	0.892920i	$-0.351347\pi$
$911$	27.1775	0.900431	0.450215	+	0.892920i	$0.351347\pi$
$912$	−9.54316	−0.316006
$913$	−0.0824569	−0.00272893
$914$	−12.8888	−0.426324
$915$	−1.62551	−0.0537377
$916$	−1.40026	−0.0462660
$917$	−14.8956	−0.491898
$918$	−9.33215	−0.308007
$919$	30.8038	1.01612	0.508061	−	0.861321i	$-0.330362\pi$
$919$	30.8038	1.01612	0.508061	+	0.861321i	$0.330362\pi$
$920$	0	0
$921$	−17.7269	−0.584123
$922$	−32.6063	−1.07383
$923$	−15.3259	−0.504457
$924$	0.164188	0.00540137
$925$	−41.6073	−1.36804
$926$	−2.55186	−0.0838592
$927$	−4.82547	−0.158489
$928$	19.1342	0.628110
$929$	52.4536	1.72095	0.860473	−	0.509496i	$-0.170168\pi$
$929$	52.4536	1.72095	0.860473	+	0.509496i	$0.170168\pi$
$930$	−8.24399	−0.270331
$931$	10.4499	0.342482
$932$	−3.44594	−0.112876
$933$	−18.3846	−0.601885
$934$	−15.1418	−0.495454
$935$	0.697600	0.0228139
$936$	6.24099	0.203993
$937$	3.84750	0.125692	0.0628461	−	0.998023i	$-0.479982\pi$
$937$	3.84750	0.125692	0.0628461	+	0.998023i	$0.479982\pi$
$938$	−23.5733	−0.769697
$939$	34.4133	1.12303
$940$	0.577205	0.0188264
$941$	−7.32866	−0.238907	−0.119454	−	0.992840i	$-0.538114\pi$
$941$	−7.32866	−0.238907	−0.119454	+	0.992840i	$0.538114\pi$
$942$	−37.5203	−1.22248
$943$	0	0
$944$	31.2489	1.01707
$945$	−0.690279	−0.0224548
$946$	−3.79547	−0.123401
$947$	38.6408	1.25566	0.627828	−	0.778352i	$-0.283944\pi$
$947$	38.6408	1.25566	0.627828	+	0.778352i	$0.283944\pi$
$948$	−5.28277	−0.171576
$949$	6.28565	0.204041
$950$	−15.0190	−0.487281
$951$	−0.482123	−0.0156339
$952$	18.0239	0.584158
$953$	−16.0774	−0.520797	−0.260399	−	0.965501i	$-0.583854\pi$
$953$	−16.0774	−0.520797	−0.260399	+	0.965501i	$0.583854\pi$
$954$	10.7037	0.346546
$955$	−5.68728	−0.184036
$956$	−9.85898	−0.318862
$957$	−1.46457	−0.0473430
$958$	−48.7427	−1.57481
$959$	27.0194	0.872502
$960$	2.50992	0.0810072
$961$	67.4580	2.17606
$962$	−37.6647	−1.21436
$963$	−14.3020	−0.460874
$964$	−6.56504	−0.211446
$965$	2.79683	0.0900333
$966$	0	0
$967$	24.1742	0.777391	0.388695	−	0.921366i	$-0.372926\pi$
$967$	24.1742	0.777391	0.388695	+	0.921366i	$0.372926\pi$
$968$	−25.4494	−0.817974
$969$	−11.6610	−0.374606
$970$	3.97832	0.127736
$971$	47.3446	1.51936	0.759680	−	0.650297i	$-0.225356\pi$
$971$	47.3446	1.51936	0.759680	+	0.650297i	$0.225356\pi$
$972$	−0.541956	−0.0173832
$973$	−16.3318	−0.523575
$974$	18.7083	0.599454
$975$	12.6945	0.406551
$976$	14.9422	0.478288
$977$	−9.05625	−0.289735	−0.144868	−	0.989451i	$-0.546276\pi$
$977$	−9.05625	−0.289735	−0.144868	+	0.989451i	$0.546276\pi$
$978$	7.96562	0.254713
$979$	−3.38593	−0.108215
$980$	1.48138	0.0473209
$981$	2.77491	0.0885960
$982$	−40.8938	−1.30497
$983$	10.0893	0.321798	0.160899	−	0.986971i	$-0.448561\pi$
$983$	10.0893	0.321798	0.160899	+	0.986971i	$0.448561\pi$
$984$	−1.69851	−0.0541466
$985$	2.70018	0.0860350
$986$	59.7603	1.90316
$987$	−2.70729	−0.0861739
$988$	−2.89869	−0.0922196
$989$	0	0
$990$	0.190017	0.00603914
$991$	−30.8782	−0.980877	−0.490438	−	0.871476i	$-0.663163\pi$
$991$	−30.8782	−0.980877	−0.490438	+	0.871476i	$0.663163\pi$
$992$	29.6486	0.941343
$993$	4.15512	0.131859
$994$	−12.0561	−0.382396
$995$	−3.44041	−0.109068
$996$	0.195394	0.00619129
$997$	10.3698	0.328415	0.164207	−	0.986426i	$-0.447493\pi$
$997$	10.3698	0.328415	0.164207	+	0.986426i	$0.447493\pi$
$998$	−4.90606	−0.155299
$999$	−8.79937	−0.278400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1587.2.a.o.1.2		5
3.2	odd	2		4761.2.a.br.1.4		5
23.2	even	11		69.2.e.a.4.1	✓	10
23.12	even	11		69.2.e.a.52.1	yes	10
23.22	odd	2		1587.2.a.p.1.2		5
69.2	odd	22		207.2.i.b.73.1		10
69.35	odd	22		207.2.i.b.190.1		10
69.68	even	2		4761.2.a.bq.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.e.a.4.1	✓	10	23.2	even	11
69.2.e.a.52.1	yes	10	23.12	even	11
207.2.i.b.73.1		10	69.2	odd	22
207.2.i.b.190.1		10	69.35	odd	22
1587.2.a.o.1.2		5	1.1	even	1	trivial
1587.2.a.p.1.2		5	23.22	odd	2
4761.2.a.bq.1.4		5	69.68	even	2
4761.2.a.br.1.4		5	3.2	odd	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:	\( N \)	\(=\)	\( 1587 = 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1587.a (trivial)

\(f(q)\)	\(=\)	\(q-1.59435 q^{2} -1.00000 q^{3} +0.541956 q^{4} -0.521109 q^{5} +1.59435 q^{6} -1.32463 q^{7} +2.32463 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.59435 q^{2} -1.00000 q^{3} +0.541956 q^{4} -0.521109 q^{5} +1.59435 q^{6} -1.32463 q^{7} +2.32463 q^{8} +1.00000 q^{9} +0.830830 q^{10} +0.228707 q^{11} -0.541956 q^{12} +2.68472 q^{13} +2.11193 q^{14} +0.521109 q^{15} -4.79020 q^{16} -5.85326 q^{17} -1.59435 q^{18} -1.99223 q^{19} -0.282418 q^{20} +1.32463 q^{21} -0.364640 q^{22} -2.32463 q^{24} -4.72845 q^{25} -4.28038 q^{26} -1.00000 q^{27} -0.717893 q^{28} +6.40370 q^{29} -0.830830 q^{30} +9.92260 q^{31} +2.98798 q^{32} -0.228707 q^{33} +9.33215 q^{34} +0.690279 q^{35} +0.541956 q^{36} +8.79937 q^{37} +3.17631 q^{38} -2.68472 q^{39} -1.21139 q^{40} +0.730658 q^{41} -2.11193 q^{42} +10.4088 q^{43} +0.123949 q^{44} -0.521109 q^{45} -2.04380 q^{47} +4.79020 q^{48} -5.24534 q^{49} +7.53880 q^{50} +5.85326 q^{51} +1.45500 q^{52} -6.71353 q^{53} +1.59435 q^{54} -0.119181 q^{55} -3.07929 q^{56} +1.99223 q^{57} -10.2098 q^{58} -6.52352 q^{59} +0.282418 q^{60} -3.11933 q^{61} -15.8201 q^{62} -1.32463 q^{63} +4.81650 q^{64} -1.39903 q^{65} +0.364640 q^{66} -11.1620 q^{67} -3.17221 q^{68} -1.10055 q^{70} -5.70855 q^{71} +2.32463 q^{72} +2.34127 q^{73} -14.0293 q^{74} +4.72845 q^{75} -1.07970 q^{76} -0.302954 q^{77} +4.28038 q^{78} +9.74760 q^{79} +2.49621 q^{80} +1.00000 q^{81} -1.16493 q^{82} -0.360535 q^{83} +0.717893 q^{84} +3.05018 q^{85} -16.5953 q^{86} -6.40370 q^{87} +0.531661 q^{88} -14.8046 q^{89} +0.830830 q^{90} -3.55627 q^{91} -9.92260 q^{93} +3.25853 q^{94} +1.03817 q^{95} -2.98798 q^{96} +4.78837 q^{97} +8.36292 q^{98} +0.228707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 2 * q^2 - 5 * q^3 + 4 * q^4 - 3 * q^5 + 2 * q^6 - 4 * q^7 + 9 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 4 q^{4} - 3 q^{5} + 2 q^{6} - 4 q^{7} + 9 q^{8} + 5 q^{9} - q^{10} - 13 q^{11} - 4 q^{12} + 7 q^{13} - 16 q^{14} + 3 q^{15} + 6 q^{16} - q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{20} + 4 q^{21} + 3 q^{22} - 9 q^{24} - 10 q^{25} - 27 q^{26} - 5 q^{27} - 12 q^{28} - 7 q^{29} + q^{30} + 8 q^{31} + 6 q^{32} + 13 q^{33} + 18 q^{34} + 9 q^{35} + 4 q^{36} - 7 q^{37} + 9 q^{38} - 7 q^{39} - 12 q^{40} - 7 q^{41} + 16 q^{42} + 10 q^{43} - 17 q^{44} - 3 q^{45} + 13 q^{47} - 6 q^{48} - q^{49} + 4 q^{50} + q^{51} - q^{52} - 17 q^{53} + 2 q^{54} + 10 q^{55} - 38 q^{56} - 5 q^{57} - 17 q^{58} - 10 q^{59} - 2 q^{60} - 18 q^{61} - q^{62} - 4 q^{63} - 21 q^{64} - 24 q^{65} - 3 q^{66} - 9 q^{67} - 25 q^{68} + 3 q^{70} + 37 q^{71} + 9 q^{72} - 2 q^{73} + 5 q^{74} + 10 q^{75} + 4 q^{76} + 6 q^{77} + 27 q^{78} + 32 q^{79} - 8 q^{80} + 5 q^{81} - 17 q^{82} - 13 q^{83} + 12 q^{84} + 5 q^{85} - 26 q^{86} + 7 q^{87} - 19 q^{88} - 30 q^{89} - q^{90} + 23 q^{91} - 8 q^{93} + 8 q^{94} - 14 q^{95} - 6 q^{96} - 20 q^{97} + 51 q^{98} - 13 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 4 * q^4 - 3 * q^5 + 2 * q^6 - 4 * q^7 + 9 * q^8 + 5 * q^9 - q^10 - 13 * q^11 - 4 * q^12 + 7 * q^13 - 16 * q^14 + 3 * q^15 + 6 * q^16 - q^17 - 2 * q^18 + 5 * q^19 + 2 * q^20 + 4 * q^21 + 3 * q^22 - 9 * q^24 - 10 * q^25 - 27 * q^26 - 5 * q^27 - 12 * q^28 - 7 * q^29 + q^30 + 8 * q^31 + 6 * q^32 + 13 * q^33 + 18 * q^34 + 9 * q^35 + 4 * q^36 - 7 * q^37 + 9 * q^38 - 7 * q^39 - 12 * q^40 - 7 * q^41 + 16 * q^42 + 10 * q^43 - 17 * q^44 - 3 * q^45 + 13 * q^47 - 6 * q^48 - q^49 + 4 * q^50 + q^51 - q^52 - 17 * q^53 + 2 * q^54 + 10 * q^55 - 38 * q^56 - 5 * q^57 - 17 * q^58 - 10 * q^59 - 2 * q^60 - 18 * q^61 - q^62 - 4 * q^63 - 21 * q^64 - 24 * q^65 - 3 * q^66 - 9 * q^67 - 25 * q^68 + 3 * q^70 + 37 * q^71 + 9 * q^72 - 2 * q^73 + 5 * q^74 + 10 * q^75 + 4 * q^76 + 6 * q^77 + 27 * q^78 + 32 * q^79 - 8 * q^80 + 5 * q^81 - 17 * q^82 - 13 * q^83 + 12 * q^84 + 5 * q^85 - 26 * q^86 + 7 * q^87 - 19 * q^88 - 30 * q^89 - q^90 + 23 * q^91 - 8 * q^93 + 8 * q^94 - 14 * q^95 - 6 * q^96 - 20 * q^97 + 51 * q^98 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more