LMFDB - Embedded newform 2645.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2645,2,Mod(1,2645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2645, 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("2645.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2645 = 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2645.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:	$21.1204313346$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4829696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 10x^{3} + 3x^{2} - 8x + 2 $ x^6 - 2x^5 - 5x^4 + 10x^3 + 3x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.02068$ of defining polynomial
Character	$\chi$	$=$	2645.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02068 q^{2} +3.15781 q^{3} +2.08316 q^{4} -1.00000 q^{5} -6.38093 q^{6} +0.246168 q^{7} -0.168045 q^{8} +6.97176 q^{9} +O(q^{10})$	\(q-2.02068 q^{2} +3.15781 q^{3} +2.08316 q^{4} -1.00000 q^{5} -6.38093 q^{6} +0.246168 q^{7} -0.168045 q^{8} +6.97176 q^{9} +2.02068 q^{10} -5.80372 q^{11} +6.57823 q^{12} -2.49571 q^{13} -0.497428 q^{14} -3.15781 q^{15} -3.82676 q^{16} -2.67062 q^{17} -14.0877 q^{18} +2.04137 q^{19} -2.08316 q^{20} +0.777352 q^{21} +11.7275 q^{22} -0.530656 q^{24} +1.00000 q^{25} +5.04304 q^{26} +12.5421 q^{27} +0.512808 q^{28} +6.71536 q^{29} +6.38093 q^{30} -10.0980 q^{31} +8.06876 q^{32} -18.3270 q^{33} +5.39647 q^{34} -0.246168 q^{35} +14.5233 q^{36} -8.79944 q^{37} -4.12496 q^{38} -7.88097 q^{39} +0.168045 q^{40} +2.55598 q^{41} -1.57078 q^{42} -4.86503 q^{43} -12.0901 q^{44} -6.97176 q^{45} -0.0941628 q^{47} -12.0842 q^{48} -6.93940 q^{49} -2.02068 q^{50} -8.43330 q^{51} -5.19897 q^{52} +9.27290 q^{53} -25.3435 q^{54} +5.80372 q^{55} -0.0413674 q^{56} +6.44625 q^{57} -13.5696 q^{58} -2.01965 q^{59} -6.57823 q^{60} -3.48554 q^{61} +20.4048 q^{62} +1.71622 q^{63} -8.65089 q^{64} +2.49571 q^{65} +37.0331 q^{66} -7.74434 q^{67} -5.56333 q^{68} +0.497428 q^{70} +0.526535 q^{71} -1.17157 q^{72} -13.1033 q^{73} +17.7809 q^{74} +3.15781 q^{75} +4.25250 q^{76} -1.42869 q^{77} +15.9249 q^{78} +0.337742 q^{79} +3.82676 q^{80} +18.6902 q^{81} -5.16483 q^{82} -3.87348 q^{83} +1.61935 q^{84} +2.67062 q^{85} +9.83068 q^{86} +21.2058 q^{87} +0.975288 q^{88} +8.80387 q^{89} +14.0877 q^{90} -0.614364 q^{91} -31.8874 q^{93} +0.190273 q^{94} -2.04137 q^{95} +25.4796 q^{96} -14.2661 q^{97} +14.0223 q^{98} -40.4621 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{3} + 2 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 14 q^{9}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 14 * q^9	\( 6 q + 2 q^{2} + 4 q^{3} + 2 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 14 q^{9} - 2 q^{10} - 8 q^{11} + 6 q^{12} + 4 q^{13} - 4 q^{14} - 4 q^{15} - 6 q^{16} - 14 q^{17} - 4 q^{18} - 16 q^{19} - 2 q^{20} - 8 q^{21} + 8 q^{22} - 8 q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} + 16 q^{28} + 6 q^{29} + 6 q^{30} + 6 q^{31} + 2 q^{32} - 4 q^{33} - 12 q^{34} + 6 q^{35} + 8 q^{36} - 14 q^{37} - 32 q^{38} - 28 q^{39} - 2 q^{41} - 32 q^{42} - 12 q^{43} - 16 q^{44} - 14 q^{45} + 12 q^{47} - 34 q^{48} + 4 q^{49} + 2 q^{50} + 12 q^{51} - 30 q^{52} + 2 q^{53} - 10 q^{54} + 8 q^{55} + 28 q^{56} + 4 q^{57} - 14 q^{58} - 2 q^{59} - 6 q^{60} - 32 q^{61} + 30 q^{62} - 38 q^{63} - 16 q^{64} - 4 q^{65} + 44 q^{66} - 34 q^{67} - 16 q^{68} + 4 q^{70} + 18 q^{71} - 24 q^{72} - 28 q^{73} + 28 q^{74} + 4 q^{75} - 12 q^{76} + 4 q^{77} - 6 q^{78} - 24 q^{79} + 6 q^{80} + 2 q^{81} - 10 q^{82} - 18 q^{83} - 40 q^{84} + 14 q^{85} - 32 q^{86} + 36 q^{87} - 4 q^{88} + 48 q^{89} + 4 q^{90} - 12 q^{91} + 6 q^{94} + 16 q^{95} + 18 q^{96} + 4 q^{97} + 46 q^{98} - 60 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 14 * q^9 - 2 * q^10 - 8 * q^11 + 6 * q^12 + 4 * q^13 - 4 * q^14 - 4 * q^15 - 6 * q^16 - 14 * q^17 - 4 * q^18 - 16 * q^19 - 2 * q^20 - 8 * q^21 + 8 * q^22 - 8 * q^24 + 6 * q^25 + 2 * q^26 + 4 * q^27 + 16 * q^28 + 6 * q^29 + 6 * q^30 + 6 * q^31 + 2 * q^32 - 4 * q^33 - 12 * q^34 + 6 * q^35 + 8 * q^36 - 14 * q^37 - 32 * q^38 - 28 * q^39 - 2 * q^41 - 32 * q^42 - 12 * q^43 - 16 * q^44 - 14 * q^45 + 12 * q^47 - 34 * q^48 + 4 * q^49 + 2 * q^50 + 12 * q^51 - 30 * q^52 + 2 * q^53 - 10 * q^54 + 8 * q^55 + 28 * q^56 + 4 * q^57 - 14 * q^58 - 2 * q^59 - 6 * q^60 - 32 * q^61 + 30 * q^62 - 38 * q^63 - 16 * q^64 - 4 * q^65 + 44 * q^66 - 34 * q^67 - 16 * q^68 + 4 * q^70 + 18 * q^71 - 24 * q^72 - 28 * q^73 + 28 * q^74 + 4 * q^75 - 12 * q^76 + 4 * q^77 - 6 * q^78 - 24 * q^79 + 6 * q^80 + 2 * q^81 - 10 * q^82 - 18 * q^83 - 40 * q^84 + 14 * q^85 - 32 * q^86 + 36 * q^87 - 4 * q^88 + 48 * q^89 + 4 * q^90 - 12 * q^91 + 6 * q^94 + 16 * q^95 + 18 * q^96 + 4 * q^97 + 46 * q^98 - 60 * 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 $

$2$	−2.02068	−1.42884	−0.714420	−	0.699718i	$-0.753309\pi$
$2$	−2.02068	−1.42884	−0.714420	+	0.699718i	$0.753309\pi$
$3$	3.15781	1.82316	0.911581	−	0.411120i	$-0.134862\pi$
$3$	3.15781	1.82316	0.911581	+	0.411120i	$0.134862\pi$
$4$	2.08316	1.04158
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−6.38093	−2.60501
$7$	0.246168	0.0930428	0.0465214	−	0.998917i	$-0.485186\pi$
$7$	0.246168	0.0930428	0.0465214	+	0.998917i	$0.485186\pi$
$8$	−0.168045	−0.0594130
$9$	6.97176	2.32392
$10$	2.02068	0.638996
$11$	−5.80372	−1.74989	−0.874943	−	0.484226i	$-0.839101\pi$
$11$	−5.80372	−1.74989	−0.874943	+	0.484226i	$0.839101\pi$
$12$	6.57823	1.89897
$13$	−2.49571	−0.692185	−0.346092	−	0.938200i	$-0.612492\pi$
$13$	−2.49571	−0.692185	−0.346092	+	0.938200i	$0.612492\pi$
$14$	−0.497428	−0.132943
$15$	−3.15781	−0.815343
$16$	−3.82676	−0.956690
$17$	−2.67062	−0.647720	−0.323860	−	0.946105i	$-0.604981\pi$
$17$	−2.67062	−0.647720	−0.323860	+	0.946105i	$0.604981\pi$
$18$	−14.0877	−3.32051
$19$	2.04137	0.468322	0.234161	−	0.972198i	$-0.424766\pi$
$19$	2.04137	0.468322	0.234161	+	0.972198i	$0.424766\pi$
$20$	−2.08316	−0.465809
$21$	0.777352	0.169632
$22$	11.7275	2.50031
$23$	0	0
$23$	0	0
$24$	−0.530656	−0.108320
$25$	1.00000	0.200000
$26$	5.04304	0.989021
$27$	12.5421	2.41372
$28$	0.512808	0.0969116
$29$	6.71536	1.24701	0.623505	−	0.781819i	$-0.285708\pi$
$29$	6.71536	1.24701	0.623505	+	0.781819i	$0.285708\pi$
$30$	6.38093	1.16499
$31$	−10.0980	−1.81365	−0.906824	−	0.421511i	$-0.861500\pi$
$31$	−10.0980	−1.81365	−0.906824	+	0.421511i	$0.861500\pi$
$32$	8.06876	1.42637
$33$	−18.3270	−3.19033
$34$	5.39647	0.925488
$35$	−0.246168	−0.0416100
$36$	14.5233	2.42055
$37$	−8.79944	−1.44662	−0.723309	−	0.690524i	$-0.757380\pi$
$37$	−8.79944	−1.44662	−0.723309	+	0.690524i	$0.757380\pi$
$38$	−4.12496	−0.669157
$39$	−7.88097	−1.26197
$40$	0.168045	0.0265703
$41$	2.55598	0.399177	0.199589	−	0.979880i	$-0.436039\pi$
$41$	2.55598	0.399177	0.199589	+	0.979880i	$0.436039\pi$
$42$	−1.57078	−0.242377
$43$	−4.86503	−0.741910	−0.370955	−	0.928651i	$-0.620970\pi$
$43$	−4.86503	−0.741910	−0.370955	+	0.928651i	$0.620970\pi$
$44$	−12.0901	−1.82265
$45$	−6.97176	−1.03929
$46$	0	0
$47$	−0.0941628	−0.0137351	−0.00686753	−	0.999976i	$-0.502186\pi$
$47$	−0.0941628	−0.0137351	−0.00686753	+	0.999976i	$0.502186\pi$
$48$	−12.0842	−1.74420
$49$	−6.93940	−0.991343
$50$	−2.02068	−0.285768
$51$	−8.43330	−1.18090
$52$	−5.19897	−0.720967
$53$	9.27290	1.27373	0.636866	−	0.770975i	$-0.280231\pi$
$53$	9.27290	1.27373	0.636866	+	0.770975i	$0.280231\pi$
$54$	−25.3435	−3.44882
$55$	5.80372	0.782573
$56$	−0.0413674	−0.00552796
$57$	6.44625	0.853827
$58$	−13.5696	−1.78178
$59$	−2.01965	−0.262937	−0.131468	−	0.991320i	$-0.541969\pi$
$59$	−2.01965	−0.262937	−0.131468	+	0.991320i	$0.541969\pi$
$60$	−6.57823	−0.849246
$61$	−3.48554	−0.446278	−0.223139	−	0.974787i	$-0.571630\pi$
$61$	−3.48554	−0.446278	−0.223139	+	0.974787i	$0.571630\pi$
$62$	20.4048	2.59141
$63$	1.71622	0.216224
$64$	−8.65089	−1.08136
$65$	2.49571	0.309554
$66$	37.0331	4.55846
$67$	−7.74434	−0.946122	−0.473061	−	0.881030i	$-0.656851\pi$
$67$	−7.74434	−0.946122	−0.473061	+	0.881030i	$0.656851\pi$
$68$	−5.56333	−0.674653
$69$	0	0
$70$	0.497428	0.0594540
$71$	0.526535	0.0624883	0.0312441	−	0.999512i	$-0.490053\pi$
$71$	0.526535	0.0624883	0.0312441	+	0.999512i	$0.490053\pi$
$72$	−1.17157	−0.138071
$73$	−13.1033	−1.53362	−0.766812	−	0.641871i	$-0.778158\pi$
$73$	−13.1033	−1.53362	−0.766812	+	0.641871i	$0.778158\pi$
$74$	17.7809	2.06699
$75$	3.15781	0.364632
$76$	4.25250	0.487795
$77$	−1.42869	−0.162814
$78$	15.9249	1.80315
$79$	0.337742	0.0379990	0.0189995	−	0.999819i	$-0.493952\pi$
$79$	0.337742	0.0379990	0.0189995	+	0.999819i	$0.493952\pi$
$80$	3.82676	0.427845
$81$	18.6902	2.07669
$82$	−5.16483	−0.570360
$83$	−3.87348	−0.425169	−0.212585	−	0.977143i	$-0.568188\pi$
$83$	−3.87348	−0.425169	−0.212585	+	0.977143i	$0.568188\pi$
$84$	1.61935	0.176686
$85$	2.67062	0.289669
$86$	9.83068	1.06007
$87$	21.2058	2.27350
$88$	0.975288	0.103966
$89$	8.80387	0.933208	0.466604	−	0.884466i	$-0.345477\pi$
$89$	8.80387	0.933208	0.466604	+	0.884466i	$0.345477\pi$
$90$	14.0877	1.48498
$91$	−0.614364	−0.0644028
$92$	0	0
$93$	−31.8874	−3.30657
$94$	0.190273	0.0196252
$95$	−2.04137	−0.209440
$96$	25.4796	2.60050
$97$	−14.2661	−1.44850	−0.724251	−	0.689536i	$-0.757814\pi$
$97$	−14.2661	−1.44850	−0.724251	+	0.689536i	$0.757814\pi$
$98$	14.0223	1.41647
$99$	−40.4621	−4.06660
$100$	2.08316	0.208316
$101$	−9.48457	−0.943750	−0.471875	−	0.881665i	$-0.656423\pi$
$101$	−9.48457	−0.943750	−0.471875	+	0.881665i	$0.656423\pi$
$102$	17.0410	1.68731
$103$	3.03841	0.299383	0.149692	−	0.988733i	$-0.452172\pi$
$103$	3.03841	0.299383	0.149692	+	0.988733i	$0.452172\pi$
$104$	0.419392	0.0411248
$105$	−0.777352	−0.0758618
$106$	−18.7376	−1.81996
$107$	−12.0490	−1.16482	−0.582409	−	0.812896i	$-0.697890\pi$
$107$	−12.0490	−1.16482	−0.582409	+	0.812896i	$0.697890\pi$
$108$	26.1272	2.51409
$109$	−2.36096	−0.226139	−0.113069	−	0.993587i	$-0.536068\pi$
$109$	−2.36096	−0.226139	−0.113069	+	0.993587i	$0.536068\pi$
$110$	−11.7275	−1.11817
$111$	−27.7870	−2.63742
$112$	−0.942026	−0.0890131
$113$	10.9149	1.02679	0.513394	−	0.858153i	$-0.328388\pi$
$113$	10.9149	1.02679	0.513394	+	0.858153i	$0.328388\pi$
$114$	−13.0258	−1.21998
$115$	0	0
$116$	13.9892	1.29886
$117$	−17.3995	−1.60858
$118$	4.08108	0.375694
$119$	−0.657421	−0.0602657
$120$	0.530656	0.0484420
$121$	22.6831	2.06210
$122$	7.04317	0.637659
$123$	8.07131	0.727765
$124$	−21.0357	−1.88906
$125$	−1.00000	−0.0894427
$126$	−3.46795	−0.308949
$127$	−0.934633	−0.0829353	−0.0414676	−	0.999140i	$-0.513203\pi$
$127$	−0.934633	−0.0829353	−0.0414676	+	0.999140i	$0.513203\pi$
$128$	1.34320	0.118723
$129$	−15.3628	−1.35262
$130$	−5.04304	−0.442304
$131$	−11.7679	−1.02817	−0.514084	−	0.857740i	$-0.671868\pi$
$131$	−11.7679	−1.02817	−0.514084	+	0.857740i	$0.671868\pi$
$132$	−38.1782	−3.32298
$133$	0.502520	0.0435740
$134$	15.6489	1.35186
$135$	−12.5421	−1.07945
$136$	0.448785	0.0384830
$137$	6.09615	0.520829	0.260415	−	0.965497i	$-0.416141\pi$
$137$	6.09615	0.520829	0.260415	+	0.965497i	$0.416141\pi$
$138$	0	0
$139$	−2.95158	−0.250350	−0.125175	−	0.992135i	$-0.539949\pi$
$139$	−2.95158	−0.250350	−0.125175	+	0.992135i	$0.539949\pi$
$140$	−0.512808	−0.0433402
$141$	−0.297348	−0.0250412
$142$	−1.06396	−0.0892857
$143$	14.4844	1.21124
$144$	−26.6792	−2.22327
$145$	−6.71536	−0.557680
$146$	26.4776	2.19130
$147$	−21.9133	−1.80738
$148$	−18.3307	−1.50677
$149$	−16.3754	−1.34153	−0.670765	−	0.741670i	$-0.734034\pi$
$149$	−16.3754	−1.34153	−0.670765	+	0.741670i	$0.734034\pi$
$150$	−6.38093	−0.521001
$151$	16.7455	1.36273	0.681367	−	0.731942i	$-0.261386\pi$
$151$	16.7455	1.36273	0.681367	+	0.731942i	$0.261386\pi$
$152$	−0.343043	−0.0278244
$153$	−18.6189	−1.50525
$154$	2.88693	0.232635
$155$	10.0980	0.811088
$156$	−16.4173	−1.31444
$157$	17.8247	1.42257	0.711285	−	0.702904i	$-0.248114\pi$
$157$	17.8247	1.42257	0.711285	+	0.702904i	$0.248114\pi$
$158$	−0.682471	−0.0542944
$159$	29.2821	2.32222
$160$	−8.06876	−0.637891
$161$	0	0
$162$	−37.7669	−2.96725
$163$	5.34626	0.418751	0.209376	−	0.977835i	$-0.432857\pi$
$163$	5.34626	0.418751	0.209376	+	0.977835i	$0.432857\pi$
$164$	5.32453	0.415776
$165$	18.3270	1.42676
$166$	7.82707	0.607499
$167$	−2.27812	−0.176286	−0.0881429	−	0.996108i	$-0.528093\pi$
$167$	−2.27812	−0.176286	−0.0881429	+	0.996108i	$0.528093\pi$
$168$	−0.130630	−0.0100784
$169$	−6.77144	−0.520880
$170$	−5.39647	−0.413891
$171$	14.2319	1.08834
$172$	−10.1346	−0.772759
$173$	1.49924	0.113985	0.0569924	−	0.998375i	$-0.481849\pi$
$173$	1.49924	0.113985	0.0569924	+	0.998375i	$0.481849\pi$
$174$	−42.8502	−3.24847
$175$	0.246168	0.0186086
$176$	22.2094	1.67410
$177$	−6.37768	−0.479376
$178$	−17.7898	−1.33340
$179$	−10.0729	−0.752885	−0.376442	−	0.926440i	$-0.622853\pi$
$179$	−10.0729	−0.752885	−0.376442	+	0.926440i	$0.622853\pi$
$180$	−14.5233	−1.08250
$181$	14.2974	1.06272	0.531360	−	0.847146i	$-0.321681\pi$
$181$	14.2974	1.06272	0.531360	+	0.847146i	$0.321681\pi$
$182$	1.24143	0.0920212
$183$	−11.0067	−0.813637
$184$	0	0
$185$	8.79944	0.646948
$186$	64.4344	4.72456
$187$	15.4995	1.13344
$188$	−0.196156	−0.0143062
$189$	3.08746	0.224579
$190$	4.12496	0.299256
$191$	−0.535086	−0.0387175	−0.0193587	−	0.999813i	$-0.506162\pi$
$191$	−0.535086	−0.0387175	−0.0193587	+	0.999813i	$0.506162\pi$
$192$	−27.3179	−1.97150
$193$	10.5663	0.760576	0.380288	−	0.924868i	$-0.375825\pi$
$193$	10.5663	0.760576	0.380288	+	0.924868i	$0.375825\pi$
$194$	28.8273	2.06968
$195$	7.88097	0.564368
$196$	−14.4559	−1.03256
$197$	−3.93988	−0.280705	−0.140352	−	0.990102i	$-0.544824\pi$
$197$	−3.93988	−0.280705	−0.140352	+	0.990102i	$0.544824\pi$
$198$	81.7611	5.81051
$199$	−5.93560	−0.420764	−0.210382	−	0.977619i	$-0.567471\pi$
$199$	−5.93560	−0.420764	−0.210382	+	0.977619i	$0.567471\pi$
$200$	−0.168045	−0.0118826
$201$	−24.4552	−1.72493
$202$	19.1653	1.34847
$203$	1.65311	0.116025
$204$	−17.5679	−1.23000
$205$	−2.55598	−0.178518
$206$	−6.13966	−0.427770
$207$	0	0
$208$	9.55047	0.662206
$209$	−11.8475	−0.819510
$210$	1.57078	0.108394
$211$	−14.7184	−1.01326	−0.506628	−	0.862165i	$-0.669108\pi$
$211$	−14.7184	−1.01326	−0.506628	+	0.862165i	$0.669108\pi$
$212$	19.3170	1.32669
$213$	1.66270	0.113926
$214$	24.3472	1.66434
$215$	4.86503	0.331792
$216$	−2.10764	−0.143407
$217$	−2.48580	−0.168747
$218$	4.77075	0.323116
$219$	−41.3777	−2.79605
$220$	12.0901	0.815113
$221$	6.66508	0.448342
$222$	56.1486	3.76845
$223$	−4.88720	−0.327271	−0.163636	−	0.986521i	$-0.552322\pi$
$223$	−4.88720	−0.327271	−0.163636	+	0.986521i	$0.552322\pi$
$224$	1.98627	0.132713
$225$	6.97176	0.464784
$226$	−22.0556	−1.46712
$227$	4.76425	0.316215	0.158107	−	0.987422i	$-0.449461\pi$
$227$	4.76425	0.316215	0.158107	+	0.987422i	$0.449461\pi$
$228$	13.4286	0.889330
$229$	27.5880	1.82307	0.911533	−	0.411226i	$-0.134899\pi$
$229$	27.5880	1.82307	0.911533	+	0.411226i	$0.134899\pi$
$230$	0	0
$231$	−4.51153	−0.296837
$232$	−1.12849	−0.0740887
$233$	21.6251	1.41670	0.708352	−	0.705859i	$-0.249439\pi$
$233$	21.6251	1.41670	0.708352	+	0.705859i	$0.249439\pi$
$234$	35.1588	2.29841
$235$	0.0941628	0.00614250
$236$	−4.20727	−0.273870
$237$	1.06653	0.0692783
$238$	1.32844	0.0861099
$239$	−11.9440	−0.772596	−0.386298	−	0.922374i	$-0.626246\pi$
$239$	−11.9440	−0.772596	−0.386298	+	0.922374i	$0.626246\pi$
$240$	12.0842	0.780030
$241$	−9.96982	−0.642213	−0.321106	−	0.947043i	$-0.604055\pi$
$241$	−9.96982	−0.642213	−0.321106	+	0.947043i	$0.604055\pi$
$242$	−45.8354	−2.94641
$243$	21.3938	1.37241
$244$	−7.26095	−0.464835
$245$	6.93940	0.443342
$246$	−16.3096	−1.03986
$247$	−5.09466	−0.324165
$248$	1.69692	0.107754
$249$	−12.2317	−0.775153
$250$	2.02068	0.127799
$251$	−18.4836	−1.16668	−0.583339	−	0.812229i	$-0.698254\pi$
$251$	−18.4836	−1.16668	−0.583339	+	0.812229i	$0.698254\pi$
$252$	3.57518	0.225215
$253$	0	0
$254$	1.88860	0.118501
$255$	8.43330	0.528114
$256$	14.5876	0.911725
$257$	17.2680	1.07715	0.538573	−	0.842579i	$-0.318964\pi$
$257$	17.2680	1.07715	0.538573	+	0.842579i	$0.318964\pi$
$258$	31.0434	1.93268
$259$	−2.16614	−0.134597
$260$	5.19897	0.322426
$261$	46.8179	2.89795
$262$	23.7793	1.46909
$263$	17.0142	1.04914	0.524571	−	0.851366i	$-0.324226\pi$
$263$	17.0142	1.04914	0.524571	+	0.851366i	$0.324226\pi$
$264$	3.07977	0.189547
$265$	−9.27290	−0.569630
$266$	−1.01543	−0.0622602
$267$	27.8009	1.70139
$268$	−16.1327	−0.985463
$269$	20.5691	1.25412	0.627060	−	0.778971i	$-0.284258\pi$
$269$	20.5691	1.25412	0.627060	+	0.778971i	$0.284258\pi$
$270$	25.3435	1.54236
$271$	−11.4435	−0.695142	−0.347571	−	0.937654i	$-0.612993\pi$
$271$	−11.4435	−0.695142	−0.347571	+	0.937654i	$0.612993\pi$
$272$	10.2198	0.619667
$273$	−1.94004	−0.117417
$274$	−12.3184	−0.744181
$275$	−5.80372	−0.349977
$276$	0	0
$277$	10.4035	0.625088	0.312544	−	0.949903i	$-0.398819\pi$
$277$	10.4035	0.625088	0.312544	+	0.949903i	$0.398819\pi$
$278$	5.96421	0.357709
$279$	−70.4006	−4.21477
$280$	0.0413674	0.00247218
$281$	−6.90368	−0.411839	−0.205920	−	0.978569i	$-0.566019\pi$
$281$	−6.90368	−0.411839	−0.205920	+	0.978569i	$0.566019\pi$
$282$	0.600846	0.0357799
$283$	−14.8213	−0.881037	−0.440519	−	0.897743i	$-0.645205\pi$
$283$	−14.8213	−0.881037	−0.440519	+	0.897743i	$0.645205\pi$
$284$	1.09686	0.0650866
$285$	−6.44625	−0.381843
$286$	−29.2683	−1.73067
$287$	0.629201	0.0371406
$288$	56.2535	3.31477
$289$	−9.86780	−0.580459
$290$	13.5696	0.796835
$291$	−45.0496	−2.64085
$292$	−27.2963	−1.59739
$293$	−1.35793	−0.0793308	−0.0396654	−	0.999213i	$-0.512629\pi$
$293$	−1.35793	−0.0793308	−0.0396654	+	0.999213i	$0.512629\pi$
$294$	44.2799	2.58245
$295$	2.01965	0.117589
$296$	1.47871	0.0859480
$297$	−72.7906	−4.22374
$298$	33.0896	1.91683
$299$	0	0
$300$	6.57823	0.379794
$301$	−1.19761	−0.0690293
$302$	−33.8375	−1.94713
$303$	−29.9505	−1.72061
$304$	−7.81182	−0.448039
$305$	3.48554	0.199581
$306$	37.6229	2.15076
$307$	31.7574	1.81249	0.906246	−	0.422751i	$-0.138936\pi$
$307$	31.7574	1.81249	0.906246	+	0.422751i	$0.138936\pi$
$308$	−2.97619	−0.169584
$309$	9.59471	0.545824
$310$	−20.4048	−1.15891
$311$	−1.04346	−0.0591694	−0.0295847	−	0.999562i	$-0.509418\pi$
$311$	−1.04346	−0.0591694	−0.0295847	+	0.999562i	$0.509418\pi$
$312$	1.32436	0.0749772
$313$	−22.0389	−1.24571	−0.622856	−	0.782336i	$-0.714028\pi$
$313$	−22.0389	−1.24571	−0.622856	+	0.782336i	$0.714028\pi$
$314$	−36.0182	−2.03262
$315$	−1.71622	−0.0966983
$316$	0.703572	0.0395790
$317$	−25.8653	−1.45274	−0.726370	−	0.687304i	$-0.758794\pi$
$317$	−25.8653	−1.45274	−0.726370	+	0.687304i	$0.758794\pi$
$318$	−59.1698	−3.31808
$319$	−38.9740	−2.18213
$320$	8.65089	0.483600
$321$	−38.0484	−2.12365
$322$	0	0
$323$	−5.45171	−0.303341
$324$	38.9347	2.16304
$325$	−2.49571	−0.138437
$326$	−10.8031	−0.598328
$327$	−7.45546	−0.412288
$328$	−0.429521	−0.0237164
$329$	−0.0231799	−0.00127795
$330$	−37.0331	−2.03861
$331$	−3.52183	−0.193577	−0.0967887	−	0.995305i	$-0.530857\pi$
$331$	−3.52183	−0.193577	−0.0967887	+	0.995305i	$0.530857\pi$
$332$	−8.06908	−0.442849
$333$	−61.3476	−3.36183
$334$	4.60335	0.251884
$335$	7.74434	0.423119
$336$	−2.97474	−0.162285
$337$	−23.4492	−1.27736	−0.638679	−	0.769473i	$-0.720519\pi$
$337$	−23.4492	−1.27736	−0.638679	+	0.769473i	$0.720519\pi$
$338$	13.6829	0.744254
$339$	34.4672	1.87200
$340$	5.56333	0.301714
$341$	58.6057	3.17368
$342$	−28.7582	−1.55507
$343$	−3.43144	−0.185280
$344$	0.817546	0.0440791
$345$	0	0
$346$	−3.02948	−0.162866
$347$	−11.3481	−0.609199	−0.304600	−	0.952480i	$-0.598523\pi$
$347$	−11.3481	−0.609199	−0.304600	+	0.952480i	$0.598523\pi$
$348$	44.1752	2.36804
$349$	−17.9996	−0.963496	−0.481748	−	0.876310i	$-0.659998\pi$
$349$	−17.9996	−0.963496	−0.481748	+	0.876310i	$0.659998\pi$
$350$	−0.497428	−0.0265886
$351$	−31.3013	−1.67074
$352$	−46.8288	−2.49598
$353$	22.3673	1.19049	0.595245	−	0.803544i	$-0.297055\pi$
$353$	22.3673	1.19049	0.595245	+	0.803544i	$0.297055\pi$
$354$	12.8873	0.684951
$355$	−0.526535	−0.0279456
$356$	18.3399	0.972013
$357$	−2.07601	−0.109874
$358$	20.3542	1.07575
$359$	−31.0653	−1.63956	−0.819782	−	0.572676i	$-0.805905\pi$
$359$	−31.0653	−1.63956	−0.819782	+	0.572676i	$0.805905\pi$
$360$	1.17157	0.0617473
$361$	−14.8328	−0.780675
$362$	−28.8906	−1.51845
$363$	71.6289	3.75954
$364$	−1.27982	−0.0670808
$365$	13.1033	0.685858
$366$	22.2410	1.16256
$367$	−4.17718	−0.218047	−0.109023	−	0.994039i	$-0.534772\pi$
$367$	−4.17718	−0.218047	−0.109023	+	0.994039i	$0.534772\pi$
$368$	0	0
$369$	17.8197	0.927657
$370$	−17.7809	−0.924384
$371$	2.28269	0.118512
$372$	−66.4267	−3.44406
$373$	−1.42779	−0.0739282	−0.0369641	−	0.999317i	$-0.511769\pi$
$373$	−1.42779	−0.0739282	−0.0369641	+	0.999317i	$0.511769\pi$
$374$	−31.3196	−1.61950
$375$	−3.15781	−0.163069
$376$	0.0158236	0.000816041 0
$377$	−16.7596	−0.863162
$378$	−6.23877	−0.320888
$379$	31.1895	1.60210	0.801050	−	0.598598i	$-0.204275\pi$
$379$	31.1895	1.60210	0.801050	+	0.598598i	$0.204275\pi$
$380$	−4.25250	−0.218149
$381$	−2.95139	−0.151204
$382$	1.08124	0.0553210
$383$	28.1174	1.43673	0.718365	−	0.695666i	$-0.244891\pi$
$383$	28.1174	1.43673	0.718365	+	0.695666i	$0.244891\pi$
$384$	4.24157	0.216452
$385$	1.42869	0.0728128
$386$	−21.3511	−1.08674
$387$	−33.9178	−1.72414
$388$	−29.7186	−1.50873
$389$	−2.17893	−0.110476	−0.0552381	−	0.998473i	$-0.517592\pi$
$389$	−2.17893	−0.110476	−0.0552381	+	0.998473i	$0.517592\pi$
$390$	−15.9249	−0.806391
$391$	0	0
$392$	1.16614	0.0588987
$393$	−37.1609	−1.87452
$394$	7.96125	0.401082
$395$	−0.337742	−0.0169937
$396$	−84.2892	−4.23569
$397$	17.7409	0.890390	0.445195	−	0.895434i	$-0.353134\pi$
$397$	17.7409	0.890390	0.445195	+	0.895434i	$0.353134\pi$
$398$	11.9940	0.601204
$399$	1.58686	0.0794424
$400$	−3.82676	−0.191338
$401$	7.80339	0.389683	0.194841	−	0.980835i	$-0.437581\pi$
$401$	7.80339	0.389683	0.194841	+	0.980835i	$0.437581\pi$
$402$	49.4162	2.46465
$403$	25.2016	1.25538
$404$	−19.7579	−0.982992
$405$	−18.6902	−0.928722
$406$	−3.34041	−0.165782
$407$	51.0694	2.53142
$408$	1.41718	0.0701608
$409$	−23.3300	−1.15360	−0.576798	−	0.816887i	$-0.695698\pi$
$409$	−23.3300	−1.15360	−0.576798	+	0.816887i	$0.695698\pi$
$410$	5.16483	0.255073
$411$	19.2505	0.949556
$412$	6.32950	0.311832
$413$	−0.497175	−0.0244644
$414$	0	0
$415$	3.87348	0.190142
$416$	−20.1373	−0.987311
$417$	−9.32052	−0.456428
$418$	23.9401	1.17095
$419$	11.4082	0.557326	0.278663	−	0.960389i	$-0.410109\pi$
$419$	11.4082	0.557326	0.278663	+	0.960389i	$0.410109\pi$
$420$	−1.61935	−0.0790162
$421$	2.91033	0.141841	0.0709205	−	0.997482i	$-0.477406\pi$
$421$	2.91033	0.141841	0.0709205	+	0.997482i	$0.477406\pi$
$422$	29.7412	1.44778
$423$	−0.656480	−0.0319192
$424$	−1.55827	−0.0756763
$425$	−2.67062	−0.129544
$426$	−3.35979	−0.162782
$427$	−0.858029	−0.0415229
$428$	−25.1000	−1.21325
$429$	45.7389	2.20830
$430$	−9.83068	−0.474077
$431$	7.00274	0.337310	0.168655	−	0.985675i	$-0.446058\pi$
$431$	7.00274	0.337310	0.168655	+	0.985675i	$0.446058\pi$
$432$	−47.9954	−2.30918
$433$	6.26000	0.300836	0.150418	−	0.988622i	$-0.451938\pi$
$433$	6.26000	0.300836	0.150418	+	0.988622i	$0.451938\pi$
$434$	5.02301	0.241112
$435$	−21.2058	−1.01674
$436$	−4.91826	−0.235542
$437$	0	0
$438$	83.6113	3.99510
$439$	−10.2835	−0.490805	−0.245403	−	0.969421i	$-0.578920\pi$
$439$	−10.2835	−0.490805	−0.245403	+	0.969421i	$0.578920\pi$
$440$	−0.975288	−0.0464950
$441$	−48.3798	−2.30380
$442$	−13.4680	−0.640608
$443$	−22.3570	−1.06221	−0.531107	−	0.847305i	$-0.678224\pi$
$443$	−22.3570	−1.06221	−0.531107	+	0.847305i	$0.678224\pi$
$444$	−57.8847	−2.74709
$445$	−8.80387	−0.417344
$446$	9.87548	0.467618
$447$	−51.7105	−2.44583
$448$	−2.12957	−0.100613
$449$	41.3075	1.94942	0.974712	−	0.223465i	$-0.0717370\pi$
$449$	41.3075	1.94942	0.974712	+	0.223465i	$0.0717370\pi$
$450$	−14.0877	−0.664102
$451$	−14.8342	−0.698515
$452$	22.7375	1.06948
$453$	52.8793	2.48448
$454$	−9.62705	−0.451820
$455$	0.614364	0.0288018
$456$	−1.08326	−0.0507284
$457$	−8.16486	−0.381936	−0.190968	−	0.981596i	$-0.561163\pi$
$457$	−8.16486	−0.381936	−0.190968	+	0.981596i	$0.561163\pi$
$458$	−55.7466	−2.60487
$459$	−33.4951	−1.56342
$460$	0	0
$461$	−11.4554	−0.533530	−0.266765	−	0.963762i	$-0.585955\pi$
$461$	−11.4554	−0.533530	−0.266765	+	0.963762i	$0.585955\pi$
$462$	9.11637	0.424132
$463$	19.1503	0.889992	0.444996	−	0.895533i	$-0.353205\pi$
$463$	19.1503	0.889992	0.444996	+	0.895533i	$0.353205\pi$
$464$	−25.6980	−1.19300
$465$	31.8874	1.47874
$466$	−43.6974	−2.02424
$467$	−24.9720	−1.15557	−0.577783	−	0.816191i	$-0.696082\pi$
$467$	−24.9720	−1.15557	−0.577783	+	0.816191i	$0.696082\pi$
$468$	−36.2459	−1.67547
$469$	−1.90641	−0.0880299
$470$	−0.190273	−0.00877665
$471$	56.2871	2.59357
$472$	0.339394	0.0156219
$473$	28.2352	1.29826
$474$	−2.15511	−0.0989876
$475$	2.04137	0.0936644
$476$	−1.36951	−0.0627716
$477$	64.6485	2.96005
$478$	24.1351	1.10392
$479$	−30.8704	−1.41050	−0.705252	−	0.708957i	$-0.749166\pi$
$479$	−30.8704	−1.41050	−0.705252	+	0.708957i	$0.749166\pi$
$480$	−25.4796	−1.16298
$481$	21.9608	1.00133
$482$	20.1459	0.917618
$483$	0	0
$484$	47.2526	2.14785
$485$	14.2661	0.647790
$486$	−43.2301	−1.96096
$487$	19.8715	0.900462	0.450231	−	0.892912i	$-0.351342\pi$
$487$	19.8715	0.900462	0.450231	+	0.892912i	$0.351342\pi$
$488$	0.585729	0.0265147
$489$	16.8825	0.763451
$490$	−14.0223	−0.633465
$491$	−10.4938	−0.473580	−0.236790	−	0.971561i	$-0.576095\pi$
$491$	−10.4938	−0.473580	−0.236790	+	0.971561i	$0.576095\pi$
$492$	16.8138	0.758027
$493$	−17.9341	−0.807713
$494$	10.2947	0.463180
$495$	40.4621	1.81864
$496$	38.6425	1.73510
$497$	0.129616	0.00581408
$498$	24.7164	1.10757
$499$	−18.7116	−0.837648	−0.418824	−	0.908067i	$-0.637558\pi$
$499$	−18.7116	−0.837648	−0.418824	+	0.908067i	$0.637558\pi$
$500$	−2.08316	−0.0931619
$501$	−7.19385	−0.321398
$502$	37.3496	1.66699
$503$	35.2581	1.57208	0.786040	−	0.618176i	$-0.212128\pi$
$503$	35.2581	1.57208	0.786040	+	0.618176i	$0.212128\pi$
$504$	−0.288404	−0.0128465
$505$	9.48457	0.422058
$506$	0	0
$507$	−21.3829	−0.949649
$508$	−1.94699	−0.0863838
$509$	−7.93554	−0.351736	−0.175868	−	0.984414i	$-0.556273\pi$
$509$	−7.93554	−0.351736	−0.175868	+	0.984414i	$0.556273\pi$
$510$	−17.0410	−0.754590
$511$	−3.22561	−0.142693
$512$	−32.1633	−1.42143
$513$	25.6030	1.13040
$514$	−34.8931	−1.53907
$515$	−3.03841	−0.133888
$516$	−32.0033	−1.40887
$517$	0.546494	0.0240348
$518$	4.37709	0.192318
$519$	4.73430	0.207813
$520$	−0.419392	−0.0183916
$521$	10.6701	0.467466	0.233733	−	0.972301i	$-0.424906\pi$
$521$	10.6701	0.467466	0.233733	+	0.972301i	$0.424906\pi$
$522$	−94.6041	−4.14071
$523$	32.1632	1.40640	0.703199	−	0.710993i	$-0.251754\pi$
$523$	32.1632	1.40640	0.703199	+	0.710993i	$0.251754\pi$
$524$	−24.5145	−1.07092
$525$	0.777352	0.0339264
$526$	−34.3804	−1.49906
$527$	26.9678	1.17474
$528$	70.1331	3.05215
$529$	0	0
$530$	18.7376	0.813910
$531$	−14.0805	−0.611044
$532$	1.04683	0.0453858
$533$	−6.37899	−0.276305
$534$	−56.1769	−2.43101
$535$	12.0490	0.520923
$536$	1.30140	0.0562120
$537$	−31.8083	−1.37263
$538$	−41.5636	−1.79193
$539$	40.2743	1.73474
$540$	−26.1272	−1.12433
$541$	−32.7670	−1.40877	−0.704383	−	0.709820i	$-0.748776\pi$
$541$	−32.7670	−1.40877	−0.704383	+	0.709820i	$0.748776\pi$
$542$	23.1237	0.993246
$543$	45.1485	1.93751
$544$	−21.5486	−0.923887
$545$	2.36096	0.101132
$546$	3.92021	0.167770
$547$	18.2743	0.781352	0.390676	−	0.920528i	$-0.372241\pi$
$547$	18.2743	0.781352	0.390676	+	0.920528i	$0.372241\pi$
$548$	12.6993	0.542486
$549$	−24.3004	−1.03711
$550$	11.7275	0.500061
$551$	13.7085	0.584002
$552$	0	0
$553$	0.0831414	0.00353553
$554$	−21.0223	−0.893150
$555$	27.7870	1.17949
$556$	−6.14862	−0.260759
$557$	−13.8110	−0.585191	−0.292596	−	0.956236i	$-0.594519\pi$
$557$	−13.8110	−0.585191	−0.292596	+	0.956236i	$0.594519\pi$
$558$	142.257	6.02223
$559$	12.1417	0.513539
$560$	0.942026	0.0398079
$561$	48.9445	2.06644
$562$	13.9502	0.588452
$563$	−39.4869	−1.66417	−0.832087	−	0.554646i	$-0.812854\pi$
$563$	−39.4869	−1.66417	−0.832087	+	0.554646i	$0.812854\pi$
$564$	−0.619424	−0.0260825
$565$	−10.9149	−0.459194
$566$	29.9492	1.25886
$567$	4.60092	0.193221
$568$	−0.0884819	−0.00371262
$569$	−13.0960	−0.549015	−0.274507	−	0.961585i	$-0.588515\pi$
$569$	−13.0960	−0.549015	−0.274507	+	0.961585i	$0.588515\pi$
$570$	13.0258	0.545592
$571$	−18.5349	−0.775663	−0.387832	−	0.921730i	$-0.626776\pi$
$571$	−18.5349	−0.775663	−0.387832	+	0.921730i	$0.626776\pi$
$572$	30.1733	1.26161
$573$	−1.68970	−0.0705882
$574$	−1.27142	−0.0530679
$575$	0	0
$576$	−60.3120	−2.51300
$577$	23.3254	0.971050	0.485525	−	0.874223i	$-0.338629\pi$
$577$	23.3254	0.971050	0.485525	+	0.874223i	$0.338629\pi$
$578$	19.9397	0.829382
$579$	33.3662	1.38665
$580$	−13.9892	−0.580869
$581$	−0.953527	−0.0395589
$582$	91.0310	3.77336
$583$	−53.8173	−2.22888
$584$	2.20195	0.0911173
$585$	17.3995	0.719380
$586$	2.74394	0.113351
$587$	43.4374	1.79285	0.896426	−	0.443193i	$-0.146154\pi$
$587$	43.4374	1.79285	0.896426	+	0.443193i	$0.146154\pi$
$588$	−45.6490	−1.88253
$589$	−20.6136	−0.849370
$590$	−4.08108	−0.168016
$591$	−12.4414	−0.511771
$592$	33.6733	1.38397
$593$	−16.6179	−0.682413	−0.341207	−	0.939988i	$-0.610836\pi$
$593$	−16.6179	−0.682413	−0.341207	+	0.939988i	$0.610836\pi$
$594$	147.087	6.03504
$595$	0.657421	0.0269516
$596$	−34.1127	−1.39731
$597$	−18.7435	−0.767121
$598$	0	0
$599$	−38.6068	−1.57743	−0.788715	−	0.614760i	$-0.789253\pi$
$599$	−38.6068	−1.57743	−0.788715	+	0.614760i	$0.789253\pi$
$600$	−0.530656	−0.0216639
$601$	−38.5324	−1.57177	−0.785884	−	0.618374i	$-0.787792\pi$
$601$	−38.5324	−1.57177	−0.785884	+	0.618374i	$0.787792\pi$
$602$	2.42000	0.0986318
$603$	−53.9917	−2.19871
$604$	34.8837	1.41940
$605$	−22.6831	−0.922200
$606$	60.5204	2.45847
$607$	6.53362	0.265191	0.132596	−	0.991170i	$-0.457669\pi$
$607$	6.53362	0.265191	0.132596	+	0.991170i	$0.457669\pi$
$608$	16.4713	0.668000
$609$	5.22020	0.211533
$610$	−7.04317	−0.285170
$611$	0.235003	0.00950719
$612$	−38.7862	−1.56784
$613$	12.7247	0.513947	0.256974	−	0.966418i	$-0.417275\pi$
$613$	12.7247	0.513947	0.256974	+	0.966418i	$0.417275\pi$
$614$	−64.1717	−2.58976
$615$	−8.07131	−0.325467
$616$	0.240085	0.00967329
$617$	29.1338	1.17288	0.586441	−	0.809992i	$-0.300528\pi$
$617$	29.1338	1.17288	0.586441	+	0.809992i	$0.300528\pi$
$618$	−19.3879	−0.779895
$619$	23.7588	0.954948	0.477474	−	0.878646i	$-0.341552\pi$
$619$	23.7588	0.954948	0.477474	+	0.878646i	$0.341552\pi$
$620$	21.0357	0.844814
$621$	0	0
$622$	2.10851	0.0845436
$623$	2.16723	0.0868283
$624$	30.1586	1.20731
$625$	1.00000	0.0400000
$626$	44.5336	1.77992
$627$	−37.4122	−1.49410
$628$	37.1318	1.48172
$629$	23.4999	0.937004
$630$	3.46795	0.138166
$631$	1.88614	0.0750860	0.0375430	−	0.999295i	$-0.488047\pi$
$631$	1.88614	0.0750860	0.0375430	+	0.999295i	$0.488047\pi$
$632$	−0.0567561	−0.00225764
$633$	−46.4779	−1.84733
$634$	52.2656	2.07573
$635$	0.934633	0.0370898
$636$	60.9993	2.41878
$637$	17.3187	0.686193
$638$	78.7542	3.11791
$639$	3.67088	0.145218
$640$	−1.34320	−0.0530947
$641$	−3.94428	−0.155790	−0.0778948	−	0.996962i	$-0.524820\pi$
$641$	−3.94428	−0.155790	−0.0778948	+	0.996962i	$0.524820\pi$
$642$	76.8837	3.03436
$643$	1.69794	0.0669603	0.0334802	−	0.999439i	$-0.489341\pi$
$643$	1.69794	0.0669603	0.0334802	+	0.999439i	$0.489341\pi$
$644$	0	0
$645$	15.3628	0.604911
$646$	11.0162	0.433426
$647$	12.0922	0.475394	0.237697	−	0.971339i	$-0.423607\pi$
$647$	12.0922	0.475394	0.237697	+	0.971339i	$0.423607\pi$
$648$	−3.14080	−0.123382
$649$	11.7215	0.460109
$650$	5.04304	0.197804
$651$	−7.84967	−0.307653
$652$	11.1371	0.436164
$653$	4.66026	0.182370	0.0911850	−	0.995834i	$-0.470935\pi$
$653$	4.66026	0.182370	0.0911850	+	0.995834i	$0.470935\pi$
$654$	15.0651	0.589093
$655$	11.7679	0.459811
$656$	−9.78113	−0.381889
$657$	−91.3530	−3.56402
$658$	0.0468392	0.00182598
$659$	30.2022	1.17651	0.588256	−	0.808675i	$-0.299815\pi$
$659$	30.2022	1.17651	0.588256	+	0.808675i	$0.299815\pi$
$660$	38.1782	1.48608
$661$	−1.14961	−0.0447145	−0.0223573	−	0.999750i	$-0.507117\pi$
$661$	−1.14961	−0.0447145	−0.0223573	+	0.999750i	$0.507117\pi$
$662$	7.11651	0.276591
$663$	21.0471	0.817400
$664$	0.650920	0.0252606
$665$	−0.502520	−0.0194869
$666$	123.964	4.80351
$667$	0	0
$668$	−4.74568	−0.183616
$669$	−15.4328	−0.596668
$670$	−15.6489	−0.604569
$671$	20.2291	0.780935
$672$	6.27227	0.241958
$673$	−2.55973	−0.0986702	−0.0493351	−	0.998782i	$-0.515710\pi$
$673$	−2.55973	−0.0986702	−0.0493351	+	0.998782i	$0.515710\pi$
$674$	47.3834	1.82514
$675$	12.5421	0.482744
$676$	−14.1060	−0.542539
$677$	27.9264	1.07330	0.536648	−	0.843806i	$-0.319690\pi$
$677$	27.9264	1.07330	0.536648	+	0.843806i	$0.319690\pi$
$678$	−69.6473	−2.67479
$679$	−3.51186	−0.134773
$680$	−0.448785	−0.0172101
$681$	15.0446	0.576510
$682$	−118.424	−4.53467
$683$	−20.3037	−0.776898	−0.388449	−	0.921470i	$-0.626989\pi$
$683$	−20.3037	−0.776898	−0.388449	+	0.921470i	$0.626989\pi$
$684$	29.6474	1.13360
$685$	−6.09615	−0.232922
$686$	6.93385	0.264735
$687$	87.1177	3.32375
$688$	18.6173	0.709777
$689$	−23.1425	−0.881657
$690$	0	0
$691$	−28.1055	−1.06918	−0.534591	−	0.845111i	$-0.679534\pi$
$691$	−28.1055	−1.06918	−0.534591	+	0.845111i	$0.679534\pi$
$692$	3.12315	0.118724
$693$	−9.96048	−0.378367
$694$	22.9310	0.870448
$695$	2.95158	0.111960
$696$	−3.56354	−0.135076
$697$	−6.82605	−0.258555
$698$	36.3715	1.37668
$699$	68.2878	2.58288
$700$	0.512808	0.0193823
$701$	49.1518	1.85644	0.928218	−	0.372037i	$-0.121340\pi$
$701$	49.1518	1.85644	0.928218	+	0.372037i	$0.121340\pi$
$702$	63.2501	2.38722
$703$	−17.9629	−0.677483
$704$	50.2073	1.89226
$705$	0.297348	0.0111988
$706$	−45.1972	−1.70102
$707$	−2.33480	−0.0878091
$708$	−13.2858	−0.499309
$709$	−47.6063	−1.78789	−0.893946	−	0.448175i	$-0.852074\pi$
$709$	−47.6063	−1.78789	−0.893946	+	0.448175i	$0.852074\pi$
$710$	1.06396	0.0399298
$711$	2.35466	0.0883066
$712$	−1.47945	−0.0554448
$713$	0	0
$714$	4.19496	0.156992
$715$	−14.4844	−0.541685
$716$	−20.9835	−0.784191
$717$	−37.7170	−1.40857
$718$	62.7732	2.34267
$719$	12.4361	0.463790	0.231895	−	0.972741i	$-0.425507\pi$
$719$	12.4361	0.463790	0.231895	+	0.972741i	$0.425507\pi$
$720$	26.6792	0.994277
$721$	0.747959	0.0278554
$722$	29.9724	1.11546
$723$	−31.4828	−1.17086
$724$	29.7839	1.10691
$725$	6.71536	0.249402
$726$	−144.739	−5.37178
$727$	36.3880	1.34955	0.674777	−	0.738021i	$-0.264240\pi$
$727$	36.3880	1.34955	0.674777	+	0.738021i	$0.264240\pi$
$728$	0.103241	0.00382637
$729$	11.4870	0.425445
$730$	−26.4776	−0.979980
$731$	12.9926	0.480550
$732$	−22.9287	−0.847469
$733$	24.8474	0.917761	0.458880	−	0.888498i	$-0.348251\pi$
$733$	24.8474	0.917761	0.458880	+	0.888498i	$0.348251\pi$
$734$	8.44075	0.311554
$735$	21.9133	0.808285
$736$	0	0
$737$	44.9460	1.65561
$738$	−36.0080	−1.32547
$739$	21.7108	0.798645	0.399323	−	0.916810i	$-0.369245\pi$
$739$	21.7108	0.798645	0.399323	+	0.916810i	$0.369245\pi$
$740$	18.3307	0.673849
$741$	−16.0880	−0.591006
$742$	−4.61260	−0.169334
$743$	11.8241	0.433784	0.216892	−	0.976196i	$-0.430408\pi$
$743$	11.8241	0.433784	0.216892	+	0.976196i	$0.430408\pi$
$744$	5.35854	0.196454
$745$	16.3754	0.599950
$746$	2.88511	0.105632
$747$	−27.0050	−0.988060
$748$	32.2880	1.18057
$749$	−2.96607	−0.108378
$750$	6.38093	0.232999
$751$	42.4363	1.54852	0.774262	−	0.632865i	$-0.218121\pi$
$751$	42.4363	1.54852	0.774262	+	0.632865i	$0.218121\pi$
$752$	0.360338	0.0131402
$753$	−58.3678	−2.12704
$754$	33.8658	1.23332
$755$	−16.7455	−0.609433
$756$	6.43167	0.233918
$757$	42.2851	1.53688	0.768439	−	0.639923i	$-0.221034\pi$
$757$	42.2851	1.53688	0.768439	+	0.639923i	$0.221034\pi$
$758$	−63.0242	−2.28914
$759$	0	0
$760$	0.343043	0.0124435
$761$	−12.7786	−0.463225	−0.231613	−	0.972808i	$-0.574400\pi$
$761$	−12.7786	−0.463225	−0.231613	+	0.972808i	$0.574400\pi$
$762$	5.96383	0.216047
$763$	−0.581193	−0.0210406
$764$	−1.11467	−0.0403274
$765$	18.6189	0.673168
$766$	−56.8163	−2.05286
$767$	5.04047	0.182001
$768$	46.0649	1.66222
$769$	−41.6560	−1.50215	−0.751077	−	0.660215i	$-0.770465\pi$
$769$	−41.6560	−1.50215	−0.751077	+	0.660215i	$0.770465\pi$
$770$	−2.88693	−0.104038
$771$	54.5290	1.96381
$772$	22.0112	0.792201
$773$	9.20158	0.330958	0.165479	−	0.986213i	$-0.447083\pi$
$773$	9.20158	0.330958	0.165479	+	0.986213i	$0.447083\pi$
$774$	68.5371	2.46352
$775$	−10.0980	−0.362729
$776$	2.39735	0.0860599
$777$	−6.84026	−0.245393
$778$	4.40293	0.157853
$779$	5.21770	0.186944
$780$	16.4173	0.587835
$781$	−3.05586	−0.109347
$782$	0	0
$783$	84.2244	3.00994
$784$	26.5554	0.948408
$785$	−17.8247	−0.636192
$786$	75.0904	2.67839
$787$	−47.2411	−1.68397	−0.841983	−	0.539505i	$-0.818611\pi$
$787$	−47.2411	−1.68397	−0.841983	+	0.539505i	$0.818611\pi$
$788$	−8.20741	−0.292377
$789$	53.7277	1.91276
$790$	0.682471	0.0242812
$791$	2.68690	0.0955353
$792$	6.79948	0.241609
$793$	8.69889	0.308907
$794$	−35.8487	−1.27222
$795$	−29.2821	−1.03853
$796$	−12.3648	−0.438260
$797$	49.6977	1.76038	0.880192	−	0.474618i	$-0.157414\pi$
$797$	49.6977	1.76038	0.880192	+	0.474618i	$0.157414\pi$
$798$	−3.20654	−0.113510
$799$	0.251473	0.00889647
$800$	8.06876	0.285274
$801$	61.3785	2.16870
$802$	−15.7682	−0.556794
$803$	76.0478	2.68367
$804$	−50.9441	−1.79666
$805$	0	0
$806$	−50.9244	−1.79373
$807$	64.9533	2.28646
$808$	1.59384	0.0560711
$809$	−20.9445	−0.736371	−0.368186	−	0.929752i	$-0.620021\pi$
$809$	−20.9445	−0.736371	−0.368186	+	0.929752i	$0.620021\pi$
$810$	37.7669	1.32699
$811$	41.5543	1.45917	0.729584	−	0.683891i	$-0.239714\pi$
$811$	41.5543	1.45917	0.729584	+	0.683891i	$0.239714\pi$
$812$	3.44369	0.120850
$813$	−36.1363	−1.26736
$814$	−103.195	−3.61699
$815$	−5.34626	−0.187271
$816$	32.2722	1.12975
$817$	−9.93131	−0.347452
$818$	47.1426	1.64830
$819$	−4.28320	−0.149667
$820$	−5.32453	−0.185941
$821$	44.9532	1.56888	0.784439	−	0.620206i	$-0.212951\pi$
$821$	44.9532	1.56888	0.784439	+	0.620206i	$0.212951\pi$
$822$	−38.8991	−1.35676
$823$	2.53862	0.0884908	0.0442454	−	0.999021i	$-0.485912\pi$
$823$	2.53862	0.0884908	0.0442454	+	0.999021i	$0.485912\pi$
$824$	−0.510590	−0.0177873
$825$	−18.3270	−0.638065
$826$	1.00463	0.0349556
$827$	42.0817	1.46332	0.731661	−	0.681668i	$-0.238745\pi$
$827$	42.0817	1.46332	0.731661	+	0.681668i	$0.238745\pi$
$828$	0	0
$829$	−32.9540	−1.14454	−0.572270	−	0.820066i	$-0.693937\pi$
$829$	−32.9540	−1.14454	−0.572270	+	0.820066i	$0.693937\pi$
$830$	−7.82707	−0.271682
$831$	32.8524	1.13964
$832$	21.5901	0.748502
$833$	18.5325	0.642113
$834$	18.8338	0.652162
$835$	2.27812	0.0788374
$836$	−24.6803	−0.853586
$837$	−126.649	−4.37764
$838$	−23.0523	−0.796329
$839$	42.2504	1.45865	0.729323	−	0.684169i	$-0.239835\pi$
$839$	42.2504	1.45865	0.729323	+	0.684169i	$0.239835\pi$
$840$	0.130630	0.00450718
$841$	16.0960	0.555035
$842$	−5.88086	−0.202668
$843$	−21.8005	−0.750850
$844$	−30.6608	−1.05539
$845$	6.77144	0.232945
$846$	1.32654	0.0456074
$847$	5.58386	0.191864
$848$	−35.4852	−1.21857
$849$	−46.8030	−1.60627
$850$	5.39647	0.185098
$851$	0	0
$852$	3.46367	0.118663
$853$	−4.95324	−0.169596	−0.0847978	−	0.996398i	$-0.527024\pi$
$853$	−4.95324	−0.169596	−0.0847978	+	0.996398i	$0.527024\pi$
$854$	1.73380	0.0593296
$855$	−14.2319	−0.486722
$856$	2.02478	0.0692054
$857$	−30.2562	−1.03353	−0.516766	−	0.856127i	$-0.672864\pi$
$857$	−30.2562	−1.03353	−0.516766	+	0.856127i	$0.672864\pi$
$858$	−92.4239	−3.15530
$859$	18.9391	0.646195	0.323098	−	0.946366i	$-0.395276\pi$
$859$	18.9391	0.646195	0.323098	+	0.946366i	$0.395276\pi$
$860$	10.1346	0.345588
$861$	1.98690	0.0677133
$862$	−14.1503	−0.481962
$863$	4.22018	0.143657	0.0718283	−	0.997417i	$-0.477117\pi$
$863$	4.22018	0.143657	0.0718283	+	0.997417i	$0.477117\pi$
$864$	101.199	3.44286
$865$	−1.49924	−0.0509755
$866$	−12.6495	−0.429847
$867$	−31.1606	−1.05827
$868$	−5.17832	−0.175763
$869$	−1.96016	−0.0664939
$870$	42.8502	1.45276
$871$	19.3276	0.654891
$872$	0.396748	0.0134356
$873$	−99.4598	−3.36620
$874$	0	0
$875$	−0.246168	−0.00832200
$876$	−86.1965	−2.91231
$877$	6.51400	0.219962	0.109981	−	0.993934i	$-0.464921\pi$
$877$	6.51400	0.219962	0.109981	+	0.993934i	$0.464921\pi$
$878$	20.7797	0.701282
$879$	−4.28807	−0.144633
$880$	−22.2094	−0.748679
$881$	−8.25548	−0.278134	−0.139067	−	0.990283i	$-0.544410\pi$
$881$	−8.25548	−0.278134	−0.139067	+	0.990283i	$0.544410\pi$
$882$	97.7604	3.29176
$883$	−32.0254	−1.07774	−0.538871	−	0.842389i	$-0.681149\pi$
$883$	−32.0254	−1.07774	−0.538871	+	0.842389i	$0.681149\pi$
$884$	13.8844	0.466985
$885$	6.37768	0.214384
$886$	45.1765	1.51773
$887$	32.6560	1.09648	0.548241	−	0.836320i	$-0.315298\pi$
$887$	32.6560	1.09648	0.548241	+	0.836320i	$0.315298\pi$
$888$	4.66947	0.156697
$889$	−0.230077	−0.00771653
$890$	17.7898	0.596317
$891$	−108.472	−3.63396
$892$	−10.1808	−0.340879
$893$	−0.192221	−0.00643242
$894$	104.491	3.49469
$895$	10.0729	0.336700
$896$	0.330653	0.0110464
$897$	0	0
$898$	−83.4695	−2.78541
$899$	−67.8114	−2.26164
$900$	14.5233	0.484110
$901$	−24.7644	−0.825021
$902$	29.9752	0.998066
$903$	−3.78184	−0.125852
$904$	−1.83420	−0.0610046
$905$	−14.2974	−0.475262
$906$	−106.852	−3.54993
$907$	−39.2241	−1.30241	−0.651207	−	0.758900i	$-0.725737\pi$
$907$	−39.2241	−1.30241	−0.651207	+	0.758900i	$0.725737\pi$
$908$	9.92471	0.329363
$909$	−66.1241	−2.19320
$910$	−1.24143	−0.0411532
$911$	−7.65935	−0.253766	−0.126883	−	0.991918i	$-0.540497\pi$
$911$	−7.65935	−0.253766	−0.126883	+	0.991918i	$0.540497\pi$
$912$	−24.6682	−0.816847
$913$	22.4806	0.743998
$914$	16.4986	0.545725
$915$	11.0067	0.363869
$916$	57.4703	1.89887
$917$	−2.89689	−0.0956637
$918$	67.6829	2.23387
$919$	38.5640	1.27211	0.636055	−	0.771644i	$-0.280565\pi$
$919$	38.5640	1.27211	0.636055	+	0.771644i	$0.280565\pi$
$920$	0	0
$921$	100.284	3.30447
$922$	23.1477	0.762328
$923$	−1.31408	−0.0432534
$924$	−9.39825	−0.309180
$925$	−8.79944	−0.289324
$926$	−38.6968	−1.27166
$927$	21.1830	0.695742
$928$	54.1846	1.77870
$929$	−40.3967	−1.32537	−0.662686	−	0.748898i	$-0.730583\pi$
$929$	−40.3967	−1.32537	−0.662686	+	0.748898i	$0.730583\pi$
$930$	−64.4344	−2.11289
$931$	−14.1659	−0.464268
$932$	45.0485	1.47561
$933$	−3.29506	−0.107875
$934$	50.4605	1.65112
$935$	−15.4995	−0.506888
$936$	2.92390	0.0955708
$937$	16.8833	0.551552	0.275776	−	0.961222i	$-0.411065\pi$
$937$	16.8833	0.551552	0.275776	+	0.961222i	$0.411065\pi$
$938$	3.85225	0.125781
$939$	−69.5946	−2.27114
$940$	0.196156	0.00639792
$941$	−57.2669	−1.86685	−0.933424	−	0.358775i	$-0.883195\pi$
$941$	−57.2669	−1.86685	−0.933424	+	0.358775i	$0.883195\pi$
$942$	−113.739	−3.70580
$943$	0	0
$944$	7.72873	0.251549
$945$	−3.08746	−0.100435
$946$	−57.0545	−1.85500
$947$	−14.9243	−0.484976	−0.242488	−	0.970154i	$-0.577963\pi$
$947$	−14.9243	−0.484976	−0.242488	+	0.970154i	$0.577963\pi$
$948$	2.22175	0.0721590
$949$	32.7020	1.06155
$950$	−4.12496	−0.133831
$951$	−81.6777	−2.64858
$952$	0.110477	0.00358057
$953$	29.8953	0.968402	0.484201	−	0.874957i	$-0.339110\pi$
$953$	29.8953	0.968402	0.484201	+	0.874957i	$0.339110\pi$
$954$	−130.634	−4.22944
$955$	0.535086	0.0173150
$956$	−24.8814	−0.804722
$957$	−123.073	−3.97837
$958$	62.3793	2.01538
$959$	1.50068	0.0484594
$960$	27.3179	0.881681
$961$	70.9688	2.28932
$962$	−44.3759	−1.43074
$963$	−84.0026	−2.70695
$964$	−20.7688	−0.668917
$965$	−10.5663	−0.340140
$966$	0	0
$967$	8.78265	0.282431	0.141215	−	0.989979i	$-0.454899\pi$
$967$	8.78265	0.282431	0.141215	+	0.989979i	$0.454899\pi$
$968$	−3.81179	−0.122516
$969$	−17.2155	−0.553040
$970$	−28.8273	−0.925588
$971$	46.3518	1.48750	0.743750	−	0.668458i	$-0.233045\pi$
$971$	46.3518	1.48750	0.743750	+	0.668458i	$0.233045\pi$
$972$	44.5667	1.42948
$973$	−0.726584	−0.0232932
$974$	−40.1539	−1.28662
$975$	−7.88097	−0.252393
$976$	13.3383	0.426949
$977$	−5.74893	−0.183924	−0.0919622	−	0.995762i	$-0.529314\pi$
$977$	−5.74893	−0.183924	−0.0919622	+	0.995762i	$0.529314\pi$
$978$	−34.1141	−1.09085
$979$	−51.0952	−1.63301
$980$	14.4559	0.461777
$981$	−16.4600	−0.525529
$982$	21.2047	0.676670
$983$	27.9830	0.892519	0.446259	−	0.894904i	$-0.352756\pi$
$983$	27.9830	0.892519	0.446259	+	0.894904i	$0.352756\pi$
$984$	−1.35635	−0.0432388
$985$	3.93988	0.125535
$986$	36.2392	1.15409
$987$	−0.0731976	−0.00232991
$988$	−10.6130	−0.337644
$989$	0	0
$990$	−81.7611	−2.59854
$991$	−12.1389	−0.385603	−0.192802	−	0.981238i	$-0.561757\pi$
$991$	−12.1389	−0.385603	−0.192802	+	0.981238i	$0.561757\pi$
$992$	−81.4780	−2.58693
$993$	−11.1213	−0.352923
$994$	−0.261913	−0.00830739
$995$	5.93560	0.188171
$996$	−25.4806	−0.807385
$997$	−2.30338	−0.0729487	−0.0364744	−	0.999335i	$-0.511613\pi$
$997$	−2.30338	−0.0729487	−0.0364744	+	0.999335i	$0.511613\pi$
$998$	37.8103	1.19686
$999$	−110.363	−3.49173

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2645.2.a.r.1.1	✓	6
23.22	odd	2		2645.2.a.s.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2645.2.a.r.1.1	✓	6	1.1	even	1	trivial
2645.2.a.s.1.1	yes	6	23.22	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:	\( N \)	\(=\)	\( 2645 = 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2645.a (trivial)

\(f(q)\)	\(=\)	\(q-2.02068 q^{2} +3.15781 q^{3} +2.08316 q^{4} -1.00000 q^{5} -6.38093 q^{6} +0.246168 q^{7} -0.168045 q^{8} +6.97176 q^{9} +O(q^{10})\)	\(q-2.02068 q^{2} +3.15781 q^{3} +2.08316 q^{4} -1.00000 q^{5} -6.38093 q^{6} +0.246168 q^{7} -0.168045 q^{8} +6.97176 q^{9} +2.02068 q^{10} -5.80372 q^{11} +6.57823 q^{12} -2.49571 q^{13} -0.497428 q^{14} -3.15781 q^{15} -3.82676 q^{16} -2.67062 q^{17} -14.0877 q^{18} +2.04137 q^{19} -2.08316 q^{20} +0.777352 q^{21} +11.7275 q^{22} -0.530656 q^{24} +1.00000 q^{25} +5.04304 q^{26} +12.5421 q^{27} +0.512808 q^{28} +6.71536 q^{29} +6.38093 q^{30} -10.0980 q^{31} +8.06876 q^{32} -18.3270 q^{33} +5.39647 q^{34} -0.246168 q^{35} +14.5233 q^{36} -8.79944 q^{37} -4.12496 q^{38} -7.88097 q^{39} +0.168045 q^{40} +2.55598 q^{41} -1.57078 q^{42} -4.86503 q^{43} -12.0901 q^{44} -6.97176 q^{45} -0.0941628 q^{47} -12.0842 q^{48} -6.93940 q^{49} -2.02068 q^{50} -8.43330 q^{51} -5.19897 q^{52} +9.27290 q^{53} -25.3435 q^{54} +5.80372 q^{55} -0.0413674 q^{56} +6.44625 q^{57} -13.5696 q^{58} -2.01965 q^{59} -6.57823 q^{60} -3.48554 q^{61} +20.4048 q^{62} +1.71622 q^{63} -8.65089 q^{64} +2.49571 q^{65} +37.0331 q^{66} -7.74434 q^{67} -5.56333 q^{68} +0.497428 q^{70} +0.526535 q^{71} -1.17157 q^{72} -13.1033 q^{73} +17.7809 q^{74} +3.15781 q^{75} +4.25250 q^{76} -1.42869 q^{77} +15.9249 q^{78} +0.337742 q^{79} +3.82676 q^{80} +18.6902 q^{81} -5.16483 q^{82} -3.87348 q^{83} +1.61935 q^{84} +2.67062 q^{85} +9.83068 q^{86} +21.2058 q^{87} +0.975288 q^{88} +8.80387 q^{89} +14.0877 q^{90} -0.614364 q^{91} -31.8874 q^{93} +0.190273 q^{94} -2.04137 q^{95} +25.4796 q^{96} -14.2661 q^{97} +14.0223 q^{98} -40.4621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{3} + 2 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 14 q^{9}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 14 * q^9	\( 6 q + 2 q^{2} + 4 q^{3} + 2 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 14 q^{9} - 2 q^{10} - 8 q^{11} + 6 q^{12} + 4 q^{13} - 4 q^{14} - 4 q^{15} - 6 q^{16} - 14 q^{17} - 4 q^{18} - 16 q^{19} - 2 q^{20} - 8 q^{21} + 8 q^{22} - 8 q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} + 16 q^{28} + 6 q^{29} + 6 q^{30} + 6 q^{31} + 2 q^{32} - 4 q^{33} - 12 q^{34} + 6 q^{35} + 8 q^{36} - 14 q^{37} - 32 q^{38} - 28 q^{39} - 2 q^{41} - 32 q^{42} - 12 q^{43} - 16 q^{44} - 14 q^{45} + 12 q^{47} - 34 q^{48} + 4 q^{49} + 2 q^{50} + 12 q^{51} - 30 q^{52} + 2 q^{53} - 10 q^{54} + 8 q^{55} + 28 q^{56} + 4 q^{57} - 14 q^{58} - 2 q^{59} - 6 q^{60} - 32 q^{61} + 30 q^{62} - 38 q^{63} - 16 q^{64} - 4 q^{65} + 44 q^{66} - 34 q^{67} - 16 q^{68} + 4 q^{70} + 18 q^{71} - 24 q^{72} - 28 q^{73} + 28 q^{74} + 4 q^{75} - 12 q^{76} + 4 q^{77} - 6 q^{78} - 24 q^{79} + 6 q^{80} + 2 q^{81} - 10 q^{82} - 18 q^{83} - 40 q^{84} + 14 q^{85} - 32 q^{86} + 36 q^{87} - 4 q^{88} + 48 q^{89} + 4 q^{90} - 12 q^{91} + 6 q^{94} + 16 q^{95} + 18 q^{96} + 4 q^{97} + 46 q^{98} - 60 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^3 + 2 * q^4 - 6 * q^5 - 6 * q^6 - 6 * q^7 + 14 * q^9 - 2 * q^10 - 8 * q^11 + 6 * q^12 + 4 * q^13 - 4 * q^14 - 4 * q^15 - 6 * q^16 - 14 * q^17 - 4 * q^18 - 16 * q^19 - 2 * q^20 - 8 * q^21 + 8 * q^22 - 8 * q^24 + 6 * q^25 + 2 * q^26 + 4 * q^27 + 16 * q^28 + 6 * q^29 + 6 * q^30 + 6 * q^31 + 2 * q^32 - 4 * q^33 - 12 * q^34 + 6 * q^35 + 8 * q^36 - 14 * q^37 - 32 * q^38 - 28 * q^39 - 2 * q^41 - 32 * q^42 - 12 * q^43 - 16 * q^44 - 14 * q^45 + 12 * q^47 - 34 * q^48 + 4 * q^49 + 2 * q^50 + 12 * q^51 - 30 * q^52 + 2 * q^53 - 10 * q^54 + 8 * q^55 + 28 * q^56 + 4 * q^57 - 14 * q^58 - 2 * q^59 - 6 * q^60 - 32 * q^61 + 30 * q^62 - 38 * q^63 - 16 * q^64 - 4 * q^65 + 44 * q^66 - 34 * q^67 - 16 * q^68 + 4 * q^70 + 18 * q^71 - 24 * q^72 - 28 * q^73 + 28 * q^74 + 4 * q^75 - 12 * q^76 + 4 * q^77 - 6 * q^78 - 24 * q^79 + 6 * q^80 + 2 * q^81 - 10 * q^82 - 18 * q^83 - 40 * q^84 + 14 * q^85 - 32 * q^86 + 36 * q^87 - 4 * q^88 + 48 * q^89 + 4 * q^90 - 12 * q^91 + 6 * q^94 + 16 * q^95 + 18 * q^96 + 4 * q^97 + 46 * q^98 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more