LMFDB - Embedded newform 4026.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 21x^{5} + 39x^{4} + 89x^{3} - 100x^{2} - 96x + 8 $ x^7 - 2x^6 - 21x^5 + 39x^4 + 89x^3 - 100x^2 - 96x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.63928$ of defining polynomial
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.63928 q^{5} +1.00000 q^{6} -1.98112 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.63928 q^{5} +1.00000 q^{6} -1.98112 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.63928 q^{10} -1.00000 q^{11} -1.00000 q^{12} +3.77959 q^{13} +1.98112 q^{14} +1.63928 q^{15} +1.00000 q^{16} -4.54566 q^{17} -1.00000 q^{18} +3.03601 q^{19} -1.63928 q^{20} +1.98112 q^{21} +1.00000 q^{22} +0.766068 q^{23} +1.00000 q^{24} -2.31277 q^{25} -3.77959 q^{26} -1.00000 q^{27} -1.98112 q^{28} +4.56939 q^{29} -1.63928 q^{30} -1.32178 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.54566 q^{34} +3.24761 q^{35} +1.00000 q^{36} -1.58707 q^{37} -3.03601 q^{38} -3.77959 q^{39} +1.63928 q^{40} -6.07884 q^{41} -1.98112 q^{42} +12.8090 q^{43} -1.00000 q^{44} -1.63928 q^{45} -0.766068 q^{46} -5.55550 q^{47} -1.00000 q^{48} -3.07515 q^{49} +2.31277 q^{50} +4.54566 q^{51} +3.77959 q^{52} +12.3422 q^{53} +1.00000 q^{54} +1.63928 q^{55} +1.98112 q^{56} -3.03601 q^{57} -4.56939 q^{58} +7.18859 q^{59} +1.63928 q^{60} -1.00000 q^{61} +1.32178 q^{62} -1.98112 q^{63} +1.00000 q^{64} -6.19580 q^{65} -1.00000 q^{66} +8.89915 q^{67} -4.54566 q^{68} -0.766068 q^{69} -3.24761 q^{70} -9.98500 q^{71} -1.00000 q^{72} +9.46535 q^{73} +1.58707 q^{74} +2.31277 q^{75} +3.03601 q^{76} +1.98112 q^{77} +3.77959 q^{78} +5.54447 q^{79} -1.63928 q^{80} +1.00000 q^{81} +6.07884 q^{82} +2.25107 q^{83} +1.98112 q^{84} +7.45160 q^{85} -12.8090 q^{86} -4.56939 q^{87} +1.00000 q^{88} -7.45160 q^{89} +1.63928 q^{90} -7.48784 q^{91} +0.766068 q^{92} +1.32178 q^{93} +5.55550 q^{94} -4.97687 q^{95} +1.00000 q^{96} -1.35660 q^{97} +3.07515 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} - 2 q^{5} + 7 q^{6} + q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 - 2 * q^5 + 7 * q^6 + q^7 - 7 * q^8 + 7 * q^9	$ 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} - 2 q^{5} + 7 q^{6} + q^{7} - 7 q^{8} + 7 q^{9} + 2 q^{10} - 7 q^{11} - 7 q^{12} - q^{14} + 2 q^{15} + 7 q^{16} + 3 q^{17} - 7 q^{18} - 5 q^{19} - 2 q^{20} - q^{21} + 7 q^{22} - 3 q^{23} + 7 q^{24} + 11 q^{25} - 7 q^{27} + q^{28} - 14 q^{29} - 2 q^{30} + 5 q^{31} - 7 q^{32} + 7 q^{33} - 3 q^{34} - 9 q^{35} + 7 q^{36} + 14 q^{37} + 5 q^{38} + 2 q^{40} - 7 q^{41} + q^{42} + q^{43} - 7 q^{44} - 2 q^{45} + 3 q^{46} - 7 q^{48} - 11 q^{50} - 3 q^{51} - 3 q^{53} + 7 q^{54} + 2 q^{55} - q^{56} + 5 q^{57} + 14 q^{58} - 14 q^{59} + 2 q^{60} - 7 q^{61} - 5 q^{62} + q^{63} + 7 q^{64} - 10 q^{65} - 7 q^{66} + 3 q^{68} + 3 q^{69} + 9 q^{70} - 22 q^{71} - 7 q^{72} + q^{73} - 14 q^{74} - 11 q^{75} - 5 q^{76} - q^{77} + 10 q^{79} - 2 q^{80} + 7 q^{81} + 7 q^{82} - 17 q^{83} - q^{84} + 18 q^{85} - q^{86} + 14 q^{87} + 7 q^{88} - 18 q^{89} + 2 q^{90} + 21 q^{91} - 3 q^{92} - 5 q^{93} - 41 q^{95} + 7 q^{96} + 25 q^{97} - 7 q^{99}+O(q^{100}) $ 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 - 2 * q^5 + 7 * q^6 + q^7 - 7 * q^8 + 7 * q^9 + 2 * q^10 - 7 * q^11 - 7 * q^12 - q^14 + 2 * q^15 + 7 * q^16 + 3 * q^17 - 7 * q^18 - 5 * q^19 - 2 * q^20 - q^21 + 7 * q^22 - 3 * q^23 + 7 * q^24 + 11 * q^25 - 7 * q^27 + q^28 - 14 * q^29 - 2 * q^30 + 5 * q^31 - 7 * q^32 + 7 * q^33 - 3 * q^34 - 9 * q^35 + 7 * q^36 + 14 * q^37 + 5 * q^38 + 2 * q^40 - 7 * q^41 + q^42 + q^43 - 7 * q^44 - 2 * q^45 + 3 * q^46 - 7 * q^48 - 11 * q^50 - 3 * q^51 - 3 * q^53 + 7 * q^54 + 2 * q^55 - q^56 + 5 * q^57 + 14 * q^58 - 14 * q^59 + 2 * q^60 - 7 * q^61 - 5 * q^62 + q^63 + 7 * q^64 - 10 * q^65 - 7 * q^66 + 3 * q^68 + 3 * q^69 + 9 * q^70 - 22 * q^71 - 7 * q^72 + q^73 - 14 * q^74 - 11 * q^75 - 5 * q^76 - q^77 + 10 * q^79 - 2 * q^80 + 7 * q^81 + 7 * q^82 - 17 * q^83 - q^84 + 18 * q^85 - q^86 + 14 * q^87 + 7 * q^88 - 18 * q^89 + 2 * q^90 + 21 * q^91 - 3 * q^92 - 5 * q^93 - 41 * q^95 + 7 * q^96 + 25 * q^97 - 7 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.63928	−0.733107	−0.366553	−	0.930397i	$-0.619462\pi$
$5$	−1.63928	−0.733107	−0.366553	+	0.930397i	$0.619462\pi$
$6$	1.00000	0.408248
$7$	−1.98112	−0.748794	−0.374397	−	0.927269i	$-0.622150\pi$
$7$	−1.98112	−0.748794	−0.374397	+	0.927269i	$0.622150\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.63928	0.518385
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.00000	−0.288675
$13$	3.77959	1.04827	0.524135	−	0.851635i	$-0.324389\pi$
$13$	3.77959	1.04827	0.524135	+	0.851635i	$0.324389\pi$
$14$	1.98112	0.529477
$15$	1.63928	0.423259
$16$	1.00000	0.250000
$17$	−4.54566	−1.10249	−0.551243	−	0.834345i	$-0.685846\pi$
$17$	−4.54566	−1.10249	−0.551243	+	0.834345i	$0.685846\pi$
$18$	−1.00000	−0.235702
$19$	3.03601	0.696510	0.348255	−	0.937400i	$-0.386774\pi$
$19$	3.03601	0.696510	0.348255	+	0.937400i	$0.386774\pi$
$20$	−1.63928	−0.366553
$21$	1.98112	0.432316
$22$	1.00000	0.213201
$23$	0.766068	0.159736	0.0798681	−	0.996805i	$-0.474550\pi$
$23$	0.766068	0.159736	0.0798681	+	0.996805i	$0.474550\pi$
$24$	1.00000	0.204124
$25$	−2.31277	−0.462554
$26$	−3.77959	−0.741239
$27$	−1.00000	−0.192450
$28$	−1.98112	−0.374397
$29$	4.56939	0.848513	0.424257	−	0.905542i	$-0.360535\pi$
$29$	4.56939	0.848513	0.424257	+	0.905542i	$0.360535\pi$
$30$	−1.63928	−0.299290
$31$	−1.32178	−0.237398	−0.118699	−	0.992930i	$-0.537872\pi$
$31$	−1.32178	−0.237398	−0.118699	+	0.992930i	$0.537872\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	4.54566	0.779575
$35$	3.24761	0.548946
$36$	1.00000	0.166667
$37$	−1.58707	−0.260913	−0.130456	−	0.991454i	$-0.541644\pi$
$37$	−1.58707	−0.260913	−0.130456	+	0.991454i	$0.541644\pi$
$38$	−3.03601	−0.492507
$39$	−3.77959	−0.605219
$40$	1.63928	0.259192
$41$	−6.07884	−0.949355	−0.474678	−	0.880160i	$-0.657435\pi$
$41$	−6.07884	−0.949355	−0.474678	+	0.880160i	$0.657435\pi$
$42$	−1.98112	−0.305694
$43$	12.8090	1.95335	0.976677	−	0.214716i	$-0.0688826\pi$
$43$	12.8090	1.95335	0.976677	+	0.214716i	$0.0688826\pi$
$44$	−1.00000	−0.150756
$45$	−1.63928	−0.244369
$46$	−0.766068	−0.112951
$47$	−5.55550	−0.810353	−0.405177	−	0.914238i	$-0.632790\pi$
$47$	−5.55550	−0.810353	−0.405177	+	0.914238i	$0.632790\pi$
$48$	−1.00000	−0.144338
$49$	−3.07515	−0.439308
$50$	2.31277	0.327075
$51$	4.54566	0.636520
$52$	3.77959	0.524135
$53$	12.3422	1.69533	0.847664	−	0.530534i	$-0.178009\pi$
$53$	12.3422	1.69533	0.847664	+	0.530534i	$0.178009\pi$
$54$	1.00000	0.136083
$55$	1.63928	0.221040
$56$	1.98112	0.264739
$57$	−3.03601	−0.402130
$58$	−4.56939	−0.599990
$59$	7.18859	0.935875	0.467938	−	0.883762i	$-0.344997\pi$
$59$	7.18859	0.935875	0.467938	+	0.883762i	$0.344997\pi$
$60$	1.63928	0.211630
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	1.32178	0.167866
$63$	−1.98112	−0.249598
$64$	1.00000	0.125000
$65$	−6.19580	−0.768495
$66$	−1.00000	−0.123091
$67$	8.89915	1.08720	0.543602	−	0.839343i	$-0.317060\pi$
$67$	8.89915	1.08720	0.543602	+	0.839343i	$0.317060\pi$
$68$	−4.54566	−0.551243
$69$	−0.766068	−0.0922238
$70$	−3.24761	−0.388163
$71$	−9.98500	−1.18500	−0.592501	−	0.805570i	$-0.701859\pi$
$71$	−9.98500	−1.18500	−0.592501	+	0.805570i	$0.701859\pi$
$72$	−1.00000	−0.117851
$73$	9.46535	1.10784	0.553918	−	0.832571i	$-0.313132\pi$
$73$	9.46535	1.10784	0.553918	+	0.832571i	$0.313132\pi$
$74$	1.58707	0.184493
$75$	2.31277	0.267056
$76$	3.03601	0.348255
$77$	1.98112	0.225770
$78$	3.77959	0.427955
$79$	5.54447	0.623802	0.311901	−	0.950115i	$-0.399034\pi$
$79$	5.54447	0.623802	0.311901	+	0.950115i	$0.399034\pi$
$80$	−1.63928	−0.183277
$81$	1.00000	0.111111
$82$	6.07884	0.671296
$83$	2.25107	0.247087	0.123543	−	0.992339i	$-0.460574\pi$
$83$	2.25107	0.247087	0.123543	+	0.992339i	$0.460574\pi$
$84$	1.98112	0.216158
$85$	7.45160	0.808239
$86$	−12.8090	−1.38123
$87$	−4.56939	−0.489889
$88$	1.00000	0.106600
$89$	−7.45160	−0.789868	−0.394934	−	0.918710i	$-0.629233\pi$
$89$	−7.45160	−0.789868	−0.394934	+	0.918710i	$0.629233\pi$
$90$	1.63928	0.172795
$91$	−7.48784	−0.784939
$92$	0.766068	0.0798681
$93$	1.32178	0.137062
$94$	5.55550	0.573006
$95$	−4.97687	−0.510616
$96$	1.00000	0.102062
$97$	−1.35660	−0.137742	−0.0688710	−	0.997626i	$-0.521940\pi$
$97$	−1.35660	−0.137742	−0.0688710	+	0.997626i	$0.521940\pi$
$98$	3.07515	0.310637
$99$	−1.00000	−0.100504
$100$	−2.31277	−0.231277
$101$	−11.3934	−1.13368	−0.566842	−	0.823827i	$-0.691835\pi$
$101$	−11.3934	−1.13368	−0.566842	+	0.823827i	$0.691835\pi$
$102$	−4.54566	−0.450088
$103$	−11.6368	−1.14661	−0.573304	−	0.819343i	$-0.694339\pi$
$103$	−11.6368	−1.14661	−0.573304	+	0.819343i	$0.694339\pi$
$104$	−3.77959	−0.370620
$105$	−3.24761	−0.316934
$106$	−12.3422	−1.19878
$107$	−7.07862	−0.684316	−0.342158	−	0.939643i	$-0.611158\pi$
$107$	−7.07862	−0.684316	−0.342158	+	0.939643i	$0.611158\pi$
$108$	−1.00000	−0.0962250
$109$	16.9714	1.62557	0.812785	−	0.582564i	$-0.197951\pi$
$109$	16.9714	1.62557	0.812785	+	0.582564i	$0.197951\pi$
$110$	−1.63928	−0.156299
$111$	1.58707	0.150638
$112$	−1.98112	−0.187198
$113$	9.80228	0.922121	0.461060	−	0.887369i	$-0.347469\pi$
$113$	9.80228	0.922121	0.461060	+	0.887369i	$0.347469\pi$
$114$	3.03601	0.284349
$115$	−1.25580	−0.117104
$116$	4.56939	0.424257
$117$	3.77959	0.349424
$118$	−7.18859	−0.661764
$119$	9.00551	0.825534
$120$	−1.63928	−0.149645
$121$	1.00000	0.0909091
$122$	1.00000	0.0905357
$123$	6.07884	0.548110
$124$	−1.32178	−0.118699
$125$	11.9877	1.07221
$126$	1.98112	0.176492
$127$	−4.93054	−0.437514	−0.218757	−	0.975779i	$-0.570200\pi$
$127$	−4.93054	−0.437514	−0.218757	+	0.975779i	$0.570200\pi$
$128$	−1.00000	−0.0883883
$129$	−12.8090	−1.12777
$130$	6.19580	0.543408
$131$	−17.5319	−1.53177	−0.765883	−	0.642980i	$-0.777698\pi$
$131$	−17.5319	−1.53177	−0.765883	+	0.642980i	$0.777698\pi$
$132$	1.00000	0.0870388
$133$	−6.01472	−0.521542
$134$	−8.89915	−0.768769
$135$	1.63928	0.141086
$136$	4.54566	0.389787
$137$	6.77076	0.578465	0.289233	−	0.957259i	$-0.406600\pi$
$137$	6.77076	0.578465	0.289233	+	0.957259i	$0.406600\pi$
$138$	0.766068	0.0652121
$139$	−14.5061	−1.23039	−0.615194	−	0.788376i	$-0.710922\pi$
$139$	−14.5061	−1.23039	−0.615194	+	0.788376i	$0.710922\pi$
$140$	3.24761	0.274473
$141$	5.55550	0.467858
$142$	9.98500	0.837923
$143$	−3.77959	−0.316066
$144$	1.00000	0.0833333
$145$	−7.49049	−0.622051
$146$	−9.46535	−0.783358
$147$	3.07515	0.253634
$148$	−1.58707	−0.130456
$149$	2.52616	0.206951	0.103476	−	0.994632i	$-0.467004\pi$
$149$	2.52616	0.206951	0.103476	+	0.994632i	$0.467004\pi$
$150$	−2.31277	−0.188837
$151$	−14.6892	−1.19539	−0.597697	−	0.801722i	$-0.703917\pi$
$151$	−14.6892	−1.19539	−0.597697	+	0.801722i	$0.703917\pi$
$152$	−3.03601	−0.246253
$153$	−4.54566	−0.367495
$154$	−1.98112	−0.159643
$155$	2.16676	0.174038
$156$	−3.77959	−0.302610
$157$	8.21625	0.655728	0.327864	−	0.944725i	$-0.393671\pi$
$157$	8.21625	0.655728	0.327864	+	0.944725i	$0.393671\pi$
$158$	−5.54447	−0.441094
$159$	−12.3422	−0.978798
$160$	1.63928	0.129596
$161$	−1.51767	−0.119610
$162$	−1.00000	−0.0785674
$163$	−10.9111	−0.854620	−0.427310	−	0.904105i	$-0.640539\pi$
$163$	−10.9111	−0.854620	−0.427310	+	0.904105i	$0.640539\pi$
$164$	−6.07884	−0.474678
$165$	−1.63928	−0.127618
$166$	−2.25107	−0.174717
$167$	−4.48700	−0.347214	−0.173607	−	0.984815i	$-0.555542\pi$
$167$	−4.48700	−0.347214	−0.173607	+	0.984815i	$0.555542\pi$
$168$	−1.98112	−0.152847
$169$	1.28533	0.0988717
$170$	−7.45160	−0.571512
$171$	3.03601	0.232170
$172$	12.8090	0.976677
$173$	−11.2494	−0.855280	−0.427640	−	0.903949i	$-0.640655\pi$
$173$	−11.2494	−0.855280	−0.427640	+	0.903949i	$0.640655\pi$
$174$	4.56939	0.346404
$175$	4.58188	0.346358
$176$	−1.00000	−0.0753778
$177$	−7.18859	−0.540328
$178$	7.45160	0.558521
$179$	12.5429	0.937502	0.468751	−	0.883330i	$-0.344704\pi$
$179$	12.5429	0.937502	0.468751	+	0.883330i	$0.344704\pi$
$180$	−1.63928	−0.122184
$181$	−12.0377	−0.894752	−0.447376	−	0.894346i	$-0.647641\pi$
$181$	−12.0377	−0.894752	−0.447376	+	0.894346i	$0.647641\pi$
$182$	7.48784	0.555036
$183$	1.00000	0.0739221
$184$	−0.766068	−0.0564753
$185$	2.60165	0.191277
$186$	−1.32178	−0.0969174
$187$	4.54566	0.332412
$188$	−5.55550	−0.405177
$189$	1.98112	0.144105
$190$	4.97687	0.361060
$191$	−9.26525	−0.670410	−0.335205	−	0.942145i	$-0.608806\pi$
$191$	−9.26525	−0.670410	−0.335205	+	0.942145i	$0.608806\pi$
$192$	−1.00000	−0.0721688
$193$	−0.770410	−0.0554554	−0.0277277	−	0.999616i	$-0.508827\pi$
$193$	−0.770410	−0.0554554	−0.0277277	+	0.999616i	$0.508827\pi$
$194$	1.35660	0.0973983
$195$	6.19580	0.443691
$196$	−3.07515	−0.219654
$197$	−2.06632	−0.147219	−0.0736097	−	0.997287i	$-0.523452\pi$
$197$	−2.06632	−0.147219	−0.0736097	+	0.997287i	$0.523452\pi$
$198$	1.00000	0.0710669
$199$	−5.35476	−0.379589	−0.189794	−	0.981824i	$-0.560782\pi$
$199$	−5.35476	−0.379589	−0.189794	+	0.981824i	$0.560782\pi$
$200$	2.31277	0.163538
$201$	−8.89915	−0.627697
$202$	11.3934	0.801635
$203$	−9.05251	−0.635362
$204$	4.54566	0.318260
$205$	9.96490	0.695979
$206$	11.6368	0.810775
$207$	0.766068	0.0532454
$208$	3.77959	0.262068
$209$	−3.03601	−0.210006
$210$	3.24761	0.224106
$211$	−23.9778	−1.65070	−0.825348	−	0.564624i	$-0.809021\pi$
$211$	−23.9778	−1.65070	−0.825348	+	0.564624i	$0.809021\pi$
$212$	12.3422	0.847664
$213$	9.98500	0.684161
$214$	7.07862	0.483884
$215$	−20.9975	−1.43202
$216$	1.00000	0.0680414
$217$	2.61860	0.177762
$218$	−16.9714	−1.14945
$219$	−9.46535	−0.639609
$220$	1.63928	0.110520
$221$	−17.1808	−1.15570
$222$	−1.58707	−0.106517
$223$	3.54077	0.237107	0.118554	−	0.992948i	$-0.462174\pi$
$223$	3.54077	0.237107	0.118554	+	0.992948i	$0.462174\pi$
$224$	1.98112	0.132369
$225$	−2.31277	−0.154185
$226$	−9.80228	−0.652038
$227$	−9.60631	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$227$	−9.60631	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$228$	−3.03601	−0.201065
$229$	22.4259	1.48194	0.740972	−	0.671535i	$-0.234365\pi$
$229$	22.4259	1.48194	0.740972	+	0.671535i	$0.234365\pi$
$230$	1.25580	0.0828049
$231$	−1.98112	−0.130348
$232$	−4.56939	−0.299995
$233$	17.6993	1.15952	0.579759	−	0.814788i	$-0.303147\pi$
$233$	17.6993	1.15952	0.579759	+	0.814788i	$0.303147\pi$
$234$	−3.77959	−0.247080
$235$	9.10701	0.594076
$236$	7.18859	0.467938
$237$	−5.54447	−0.360152
$238$	−9.00551	−0.583741
$239$	−29.6363	−1.91701	−0.958505	−	0.285076i	$-0.907981\pi$
$239$	−29.6363	−1.91701	−0.958505	+	0.285076i	$0.907981\pi$
$240$	1.63928	0.105815
$241$	−7.68981	−0.495344	−0.247672	−	0.968844i	$-0.579666\pi$
$241$	−7.68981	−0.495344	−0.247672	+	0.968844i	$0.579666\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−1.00000	−0.0640184
$245$	5.04103	0.322060
$246$	−6.07884	−0.387573
$247$	11.4749	0.730131
$248$	1.32178	0.0839329
$249$	−2.25107	−0.142656
$250$	−11.9877	−0.758166
$251$	−13.2509	−0.836387	−0.418194	−	0.908358i	$-0.637337\pi$
$251$	−13.2509	−0.836387	−0.418194	+	0.908358i	$0.637337\pi$
$252$	−1.98112	−0.124799
$253$	−0.766068	−0.0481623
$254$	4.93054	0.309369
$255$	−7.45160	−0.466637
$256$	1.00000	0.0625000
$257$	−19.5882	−1.22188	−0.610940	−	0.791677i	$-0.709208\pi$
$257$	−19.5882	−1.22188	−0.610940	+	0.791677i	$0.709208\pi$
$258$	12.8090	0.797453
$259$	3.14418	0.195370
$260$	−6.19580	−0.384247
$261$	4.56939	0.282838
$262$	17.5319	1.08312
$263$	7.54730	0.465386	0.232693	−	0.972550i	$-0.425246\pi$
$263$	7.54730	0.465386	0.232693	+	0.972550i	$0.425246\pi$
$264$	−1.00000	−0.0615457
$265$	−20.2322	−1.24286
$266$	6.01472	0.368786
$267$	7.45160	0.456030
$268$	8.89915	0.543602
$269$	−10.5635	−0.644066	−0.322033	−	0.946728i	$-0.604366\pi$
$269$	−10.5635	−0.644066	−0.322033	+	0.946728i	$0.604366\pi$
$270$	−1.63928	−0.0997632
$271$	12.2232	0.742508	0.371254	−	0.928531i	$-0.378928\pi$
$271$	12.2232	0.742508	0.371254	+	0.928531i	$0.378928\pi$
$272$	−4.54566	−0.275621
$273$	7.48784	0.453185
$274$	−6.77076	−0.409037
$275$	2.31277	0.139465
$276$	−0.766068	−0.0461119
$277$	−7.88949	−0.474033	−0.237017	−	0.971506i	$-0.576170\pi$
$277$	−7.88949	−0.474033	−0.237017	+	0.971506i	$0.576170\pi$
$278$	14.5061	0.870016
$279$	−1.32178	−0.0791327
$280$	−3.24761	−0.194082
$281$	6.40968	0.382370	0.191185	−	0.981554i	$-0.438767\pi$
$281$	6.40968	0.382370	0.191185	+	0.981554i	$0.438767\pi$
$282$	−5.55550	−0.330825
$283$	13.2015	0.784747	0.392373	−	0.919806i	$-0.371654\pi$
$283$	13.2015	0.784747	0.392373	+	0.919806i	$0.371654\pi$
$284$	−9.98500	−0.592501
$285$	4.97687	0.294804
$286$	3.77959	0.223492
$287$	12.0429	0.710871
$288$	−1.00000	−0.0589256
$289$	3.66305	0.215473
$290$	7.49049	0.439857
$291$	1.35660	0.0795254
$292$	9.46535	0.553918
$293$	14.5648	0.850883	0.425442	−	0.904986i	$-0.360119\pi$
$293$	14.5648	0.850883	0.425442	+	0.904986i	$0.360119\pi$
$294$	−3.07515	−0.179347
$295$	−11.7841	−0.686097
$296$	1.58707	0.0922467
$297$	1.00000	0.0580259
$298$	−2.52616	−0.146337
$299$	2.89543	0.167447
$300$	2.31277	0.133528
$301$	−25.3762	−1.46266
$302$	14.6892	0.845271
$303$	11.3934	0.654532
$304$	3.03601	0.174127
$305$	1.63928	0.0938647
$306$	4.54566	0.259858
$307$	−7.66151	−0.437266	−0.218633	−	0.975807i	$-0.570160\pi$
$307$	−7.66151	−0.437266	−0.218633	+	0.975807i	$0.570160\pi$
$308$	1.98112	0.112885
$309$	11.6368	0.661995
$310$	−2.16676	−0.123064
$311$	11.7353	0.665448	0.332724	−	0.943024i	$-0.392032\pi$
$311$	11.7353	0.665448	0.332724	+	0.943024i	$0.392032\pi$
$312$	3.77959	0.213977
$313$	−7.29620	−0.412405	−0.206203	−	0.978509i	$-0.566111\pi$
$313$	−7.29620	−0.412405	−0.206203	+	0.978509i	$0.566111\pi$
$314$	−8.21625	−0.463670
$315$	3.24761	0.182982
$316$	5.54447	0.311901
$317$	−24.9132	−1.39926	−0.699632	−	0.714503i	$-0.746653\pi$
$317$	−24.9132	−1.39926	−0.699632	+	0.714503i	$0.746653\pi$
$318$	12.3422	0.692115
$319$	−4.56939	−0.255836
$320$	−1.63928	−0.0916384
$321$	7.07862	0.395090
$322$	1.51767	0.0845767
$323$	−13.8007	−0.767891
$324$	1.00000	0.0555556
$325$	−8.74134	−0.484882
$326$	10.9111	0.604307
$327$	−16.9714	−0.938523
$328$	6.07884	0.335648
$329$	11.0061	0.606788
$330$	1.63928	0.0902392
$331$	7.60932	0.418246	0.209123	−	0.977889i	$-0.432939\pi$
$331$	7.60932	0.418246	0.209123	+	0.977889i	$0.432939\pi$
$332$	2.25107	0.123543
$333$	−1.58707	−0.0869710
$334$	4.48700	0.245517
$335$	−14.5882	−0.797037
$336$	1.98112	0.108079
$337$	−14.5868	−0.794595	−0.397297	−	0.917690i	$-0.630052\pi$
$337$	−14.5868	−0.794595	−0.397297	+	0.917690i	$0.630052\pi$
$338$	−1.28533	−0.0699129
$339$	−9.80228	−0.532387
$340$	7.45160	0.404120
$341$	1.32178	0.0715782
$342$	−3.03601	−0.164169
$343$	19.9601	1.07774
$344$	−12.8090	−0.690615
$345$	1.25580	0.0676099
$346$	11.2494	0.604774
$347$	−21.0431	−1.12966	−0.564828	−	0.825209i	$-0.691057\pi$
$347$	−21.0431	−1.12966	−0.564828	+	0.825209i	$0.691057\pi$
$348$	−4.56939	−0.244945
$349$	−20.3305	−1.08827	−0.544133	−	0.838999i	$-0.683141\pi$
$349$	−20.3305	−1.08827	−0.544133	+	0.838999i	$0.683141\pi$
$350$	−4.58188	−0.244912
$351$	−3.77959	−0.201740
$352$	1.00000	0.0533002
$353$	5.61723	0.298975	0.149488	−	0.988764i	$-0.452238\pi$
$353$	5.61723	0.298975	0.149488	+	0.988764i	$0.452238\pi$
$354$	7.18859	0.382069
$355$	16.3682	0.868733
$356$	−7.45160	−0.394934
$357$	−9.00551	−0.476622
$358$	−12.5429	−0.662914
$359$	−29.2412	−1.54329	−0.771647	−	0.636051i	$-0.780567\pi$
$359$	−29.2412	−1.54329	−0.771647	+	0.636051i	$0.780567\pi$
$360$	1.63928	0.0863975
$361$	−9.78261	−0.514874
$362$	12.0377	0.632685
$363$	−1.00000	−0.0524864
$364$	−7.48784	−0.392469
$365$	−15.5163	−0.812162
$366$	−1.00000	−0.0522708
$367$	−5.23927	−0.273488	−0.136744	−	0.990606i	$-0.543664\pi$
$367$	−5.23927	−0.273488	−0.136744	+	0.990606i	$0.543664\pi$
$368$	0.766068	0.0399341
$369$	−6.07884	−0.316452
$370$	−2.60165	−0.135253
$371$	−24.4513	−1.26945
$372$	1.32178	0.0685309
$373$	−4.53851	−0.234995	−0.117498	−	0.993073i	$-0.537487\pi$
$373$	−4.53851	−0.234995	−0.117498	+	0.993073i	$0.537487\pi$
$374$	−4.54566	−0.235051
$375$	−11.9877	−0.619040
$376$	5.55550	0.286503
$377$	17.2704	0.889472
$378$	−1.98112	−0.101898
$379$	6.27341	0.322244	0.161122	−	0.986935i	$-0.448489\pi$
$379$	6.27341	0.322244	0.161122	+	0.986935i	$0.448489\pi$
$380$	−4.97687	−0.255308
$381$	4.93054	0.252599
$382$	9.26525	0.474052
$383$	−28.7571	−1.46942	−0.734710	−	0.678381i	$-0.762682\pi$
$383$	−28.7571	−1.46942	−0.734710	+	0.678381i	$0.762682\pi$
$384$	1.00000	0.0510310
$385$	−3.24761	−0.165513
$386$	0.770410	0.0392129
$387$	12.8090	0.651118
$388$	−1.35660	−0.0688710
$389$	−9.58124	−0.485788	−0.242894	−	0.970053i	$-0.578097\pi$
$389$	−9.58124	−0.485788	−0.242894	+	0.970053i	$0.578097\pi$
$390$	−6.19580	−0.313737
$391$	−3.48229	−0.176107
$392$	3.07515	0.155319
$393$	17.5319	0.884365
$394$	2.06632	0.104100
$395$	−9.08892	−0.457313
$396$	−1.00000	−0.0502519
$397$	29.2344	1.46723	0.733615	−	0.679565i	$-0.237832\pi$
$397$	29.2344	1.46723	0.733615	+	0.679565i	$0.237832\pi$
$398$	5.35476	0.268410
$399$	6.01472	0.301112
$400$	−2.31277	−0.115639
$401$	−16.4747	−0.822710	−0.411355	−	0.911475i	$-0.634944\pi$
$401$	−16.4747	−0.822710	−0.411355	+	0.911475i	$0.634944\pi$
$402$	8.89915	0.443849
$403$	−4.99578	−0.248858
$404$	−11.3934	−0.566842
$405$	−1.63928	−0.0814563
$406$	9.05251	0.449269
$407$	1.58707	0.0786682
$408$	−4.54566	−0.225044
$409$	13.4655	0.665827	0.332913	−	0.942957i	$-0.391968\pi$
$409$	13.4655	0.665827	0.332913	+	0.942957i	$0.391968\pi$
$410$	−9.96490	−0.492131
$411$	−6.77076	−0.333977
$412$	−11.6368	−0.573304
$413$	−14.2415	−0.700778
$414$	−0.766068	−0.0376502
$415$	−3.69013	−0.181141
$416$	−3.77959	−0.185310
$417$	14.5061	0.710365
$418$	3.03601	0.148496
$419$	−27.0108	−1.31957	−0.659783	−	0.751456i	$-0.729352\pi$
$419$	−27.0108	−1.31957	−0.659783	+	0.751456i	$0.729352\pi$
$420$	−3.24761	−0.158467
$421$	1.54761	0.0754260	0.0377130	−	0.999289i	$-0.487993\pi$
$421$	1.54761	0.0754260	0.0377130	+	0.999289i	$0.487993\pi$
$422$	23.9778	1.16722
$423$	−5.55550	−0.270118
$424$	−12.3422	−0.599389
$425$	10.5131	0.509959
$426$	−9.98500	−0.483775
$427$	1.98112	0.0958732
$428$	−7.07862	−0.342158
$429$	3.77959	0.182481
$430$	20.9975	1.01259
$431$	10.4212	0.501972	0.250986	−	0.967991i	$-0.419245\pi$
$431$	10.4212	0.501972	0.250986	+	0.967991i	$0.419245\pi$
$432$	−1.00000	−0.0481125
$433$	26.7354	1.28482	0.642411	−	0.766360i	$-0.277934\pi$
$433$	26.7354	1.28482	0.642411	+	0.766360i	$0.277934\pi$
$434$	−2.61860	−0.125697
$435$	7.49049	0.359141
$436$	16.9714	0.812785
$437$	2.32579	0.111258
$438$	9.46535	0.452272
$439$	6.41334	0.306092	0.153046	−	0.988219i	$-0.451092\pi$
$439$	6.41334	0.306092	0.153046	+	0.988219i	$0.451092\pi$
$440$	−1.63928	−0.0781495
$441$	−3.07515	−0.146436
$442$	17.1808	0.817205
$443$	15.6676	0.744390	0.372195	−	0.928155i	$-0.378605\pi$
$443$	15.6676	0.744390	0.372195	+	0.928155i	$0.378605\pi$
$444$	1.58707	0.0753191
$445$	12.2152	0.579058
$446$	−3.54077	−0.167660
$447$	−2.52616	−0.119483
$448$	−1.98112	−0.0935992
$449$	3.65900	0.172679	0.0863395	−	0.996266i	$-0.472483\pi$
$449$	3.65900	0.172679	0.0863395	+	0.996266i	$0.472483\pi$
$450$	2.31277	0.109025
$451$	6.07884	0.286241
$452$	9.80228	0.461060
$453$	14.6892	0.690160
$454$	9.60631	0.450846
$455$	12.2746	0.575444
$456$	3.03601	0.142174
$457$	−20.2055	−0.945176	−0.472588	−	0.881284i	$-0.656680\pi$
$457$	−20.2055	−0.945176	−0.472588	+	0.881284i	$0.656680\pi$
$458$	−22.4259	−1.04789
$459$	4.54566	0.212173
$460$	−1.25580	−0.0585519
$461$	−36.5097	−1.70043	−0.850213	−	0.526439i	$-0.823527\pi$
$461$	−36.5097	−1.70043	−0.850213	+	0.526439i	$0.823527\pi$
$462$	1.98112	0.0921702
$463$	11.3661	0.528227	0.264113	−	0.964492i	$-0.414921\pi$
$463$	11.3661	0.528227	0.264113	+	0.964492i	$0.414921\pi$
$464$	4.56939	0.212128
$465$	−2.16676	−0.100481
$466$	−17.6993	−0.819903
$467$	−30.8812	−1.42901	−0.714505	−	0.699630i	$-0.753348\pi$
$467$	−30.8812	−1.42901	−0.714505	+	0.699630i	$0.753348\pi$
$468$	3.77959	0.174712
$469$	−17.6303	−0.814092
$470$	−9.10701	−0.420075
$471$	−8.21625	−0.378585
$472$	−7.18859	−0.330882
$473$	−12.8090	−0.588958
$474$	5.54447	0.254666
$475$	−7.02161	−0.322173
$476$	9.00551	0.412767
$477$	12.3422	0.565109
$478$	29.6363	1.35553
$479$	−28.9860	−1.32440	−0.662202	−	0.749325i	$-0.730378\pi$
$479$	−28.9860	−1.32440	−0.662202	+	0.749325i	$0.730378\pi$
$480$	−1.63928	−0.0748224
$481$	−5.99849	−0.273507
$482$	7.68981	0.350261
$483$	1.51767	0.0690566
$484$	1.00000	0.0454545
$485$	2.22384	0.100980
$486$	1.00000	0.0453609
$487$	21.7513	0.985645	0.492823	−	0.870130i	$-0.335965\pi$
$487$	21.7513	0.985645	0.492823	+	0.870130i	$0.335965\pi$
$488$	1.00000	0.0452679
$489$	10.9111	0.493415
$490$	−5.04103	−0.227730
$491$	−38.5187	−1.73832	−0.869162	−	0.494527i	$-0.835341\pi$
$491$	−38.5187	−1.73832	−0.869162	+	0.494527i	$0.835341\pi$
$492$	6.07884	0.274055
$493$	−20.7709	−0.935473
$494$	−11.4749	−0.516280
$495$	1.63928	0.0736800
$496$	−1.32178	−0.0593495
$497$	19.7815	0.887322
$498$	2.25107	0.100873
$499$	−14.5378	−0.650799	−0.325400	−	0.945577i	$-0.605499\pi$
$499$	−14.5378	−0.650799	−0.325400	+	0.945577i	$0.605499\pi$
$500$	11.9877	0.536104
$501$	4.48700	0.200464
$502$	13.2509	0.591415
$503$	−43.9447	−1.95940	−0.979699	−	0.200473i	$-0.935752\pi$
$503$	−43.9447	−1.95940	−0.979699	+	0.200473i	$0.935752\pi$
$504$	1.98112	0.0882462
$505$	18.6769	0.831111
$506$	0.766068	0.0340559
$507$	−1.28533	−0.0570836
$508$	−4.93054	−0.218757
$509$	19.7157	0.873885	0.436942	−	0.899490i	$-0.356061\pi$
$509$	19.7157	0.873885	0.436942	+	0.899490i	$0.356061\pi$
$510$	7.45160	0.329962
$511$	−18.7520	−0.829540
$512$	−1.00000	−0.0441942
$513$	−3.03601	−0.134043
$514$	19.5882	0.863999
$515$	19.0759	0.840587
$516$	−12.8090	−0.563884
$517$	5.55550	0.244331
$518$	−3.14418	−0.138147
$519$	11.2494	0.493796
$520$	6.19580	0.271704
$521$	15.6554	0.685875	0.342938	−	0.939358i	$-0.388578\pi$
$521$	15.6554	0.685875	0.342938	+	0.939358i	$0.388578\pi$
$522$	−4.56939	−0.199997
$523$	−17.6891	−0.773489	−0.386744	−	0.922187i	$-0.626400\pi$
$523$	−17.6891	−0.773489	−0.386744	+	0.922187i	$0.626400\pi$
$524$	−17.5319	−0.765883
$525$	−4.58188	−0.199970
$526$	−7.54730	−0.329078
$527$	6.00835	0.261728
$528$	1.00000	0.0435194
$529$	−22.4131	−0.974484
$530$	20.2322	0.878832
$531$	7.18859	0.311958
$532$	−6.01472	−0.260771
$533$	−22.9755	−0.995181
$534$	−7.45160	−0.322462
$535$	11.6038	0.501676
$536$	−8.89915	−0.384385
$537$	−12.5429	−0.541267
$538$	10.5635	0.455423
$539$	3.07515	0.132456
$540$	1.63928	0.0705432
$541$	−34.6634	−1.49030	−0.745148	−	0.666899i	$-0.767621\pi$
$541$	−34.6634	−1.49030	−0.745148	+	0.666899i	$0.767621\pi$
$542$	−12.2232	−0.525033
$543$	12.0377	0.516586
$544$	4.54566	0.194894
$545$	−27.8209	−1.19172
$546$	−7.48784	−0.320450
$547$	−33.2747	−1.42272	−0.711362	−	0.702825i	$-0.751922\pi$
$547$	−33.2747	−1.42272	−0.711362	+	0.702825i	$0.751922\pi$
$548$	6.77076	0.289233
$549$	−1.00000	−0.0426790
$550$	−2.31277	−0.0986169
$551$	13.8727	0.590998
$552$	0.766068	0.0326060
$553$	−10.9843	−0.467099
$554$	7.88949	0.335192
$555$	−2.60165	−0.110434
$556$	−14.5061	−0.615194
$557$	−7.66265	−0.324677	−0.162338	−	0.986735i	$-0.551904\pi$
$557$	−7.66265	−0.324677	−0.162338	+	0.986735i	$0.551904\pi$
$558$	1.32178	0.0559553
$559$	48.4128	2.04764
$560$	3.24761	0.137236
$561$	−4.54566	−0.191918
$562$	−6.40968	−0.270376
$563$	29.7158	1.25237	0.626187	−	0.779673i	$-0.284615\pi$
$563$	29.7158	1.25237	0.626187	+	0.779673i	$0.284615\pi$
$564$	5.55550	0.233929
$565$	−16.0686	−0.676013
$566$	−13.2015	−0.554900
$567$	−1.98112	−0.0831993
$568$	9.98500	0.418961
$569$	43.6172	1.82853	0.914264	−	0.405120i	$-0.132770\pi$
$569$	43.6172	1.82853	0.914264	+	0.405120i	$0.132770\pi$
$570$	−4.97687	−0.208458
$571$	21.4803	0.898921	0.449461	−	0.893300i	$-0.351616\pi$
$571$	21.4803	0.898921	0.449461	+	0.893300i	$0.351616\pi$
$572$	−3.77959	−0.158033
$573$	9.26525	0.387061
$574$	−12.0429	−0.502662
$575$	−1.77174	−0.0738867
$576$	1.00000	0.0416667
$577$	29.4587	1.22638	0.613192	−	0.789934i	$-0.289885\pi$
$577$	29.4587	1.22638	0.613192	+	0.789934i	$0.289885\pi$
$578$	−3.66305	−0.152363
$579$	0.770410	0.0320172
$580$	−7.49049	−0.311026
$581$	−4.45964	−0.185017
$582$	−1.35660	−0.0562329
$583$	−12.3422	−0.511160
$584$	−9.46535	−0.391679
$585$	−6.19580	−0.256165
$586$	−14.5648	−0.601665
$587$	11.5491	0.476683	0.238341	−	0.971181i	$-0.423396\pi$
$587$	11.5491	0.476683	0.238341	+	0.971181i	$0.423396\pi$
$588$	3.07515	0.126817
$589$	−4.01293	−0.165350
$590$	11.7841	0.485144
$591$	2.06632	0.0849972
$592$	−1.58707	−0.0652282
$593$	36.8179	1.51193	0.755965	−	0.654612i	$-0.227168\pi$
$593$	36.8179	1.51193	0.755965	+	0.654612i	$0.227168\pi$
$594$	−1.00000	−0.0410305
$595$	−14.7625	−0.605205
$596$	2.52616	0.103476
$597$	5.35476	0.219156
$598$	−2.89543	−0.118403
$599$	−35.5206	−1.45133	−0.725667	−	0.688046i	$-0.758469\pi$
$599$	−35.5206	−1.45133	−0.725667	+	0.688046i	$0.758469\pi$
$600$	−2.31277	−0.0944185
$601$	−18.0348	−0.735657	−0.367828	−	0.929894i	$-0.619899\pi$
$601$	−18.0348	−0.735657	−0.367828	+	0.929894i	$0.619899\pi$
$602$	25.3762	1.03426
$603$	8.89915	0.362401
$604$	−14.6892	−0.597697
$605$	−1.63928	−0.0666461
$606$	−11.3934	−0.462824
$607$	11.1310	0.451794	0.225897	−	0.974151i	$-0.427469\pi$
$607$	11.1310	0.451794	0.225897	+	0.974151i	$0.427469\pi$
$608$	−3.03601	−0.123127
$609$	9.05251	0.366826
$610$	−1.63928	−0.0663724
$611$	−20.9975	−0.849470
$612$	−4.54566	−0.183748
$613$	−17.6512	−0.712925	−0.356463	−	0.934310i	$-0.616017\pi$
$613$	−17.6512	−0.712925	−0.356463	+	0.934310i	$0.616017\pi$
$614$	7.66151	0.309193
$615$	−9.96490	−0.401824
$616$	−1.98112	−0.0798217
$617$	3.44829	0.138823	0.0694115	−	0.997588i	$-0.477888\pi$
$617$	3.44829	0.138823	0.0694115	+	0.997588i	$0.477888\pi$
$618$	−11.6368	−0.468101
$619$	−21.8758	−0.879263	−0.439631	−	0.898178i	$-0.644891\pi$
$619$	−21.8758	−0.879263	−0.439631	+	0.898178i	$0.644891\pi$
$620$	2.16676	0.0870191
$621$	−0.766068	−0.0307413
$622$	−11.7353	−0.470543
$623$	14.7625	0.591448
$624$	−3.77959	−0.151305
$625$	−8.08723	−0.323489
$626$	7.29620	0.291615
$627$	3.03601	0.121247
$628$	8.21625	0.327864
$629$	7.21429	0.287653
$630$	−3.24761	−0.129388
$631$	35.4519	1.41132	0.705658	−	0.708553i	$-0.250652\pi$
$631$	35.4519	1.41132	0.705658	+	0.708553i	$0.250652\pi$
$632$	−5.54447	−0.220547
$633$	23.9778	0.953030
$634$	24.9132	0.989429
$635$	8.08252	0.320745
$636$	−12.3422	−0.489399
$637$	−11.6228	−0.460513
$638$	4.56939	0.180904
$639$	−9.98500	−0.395001
$640$	1.63928	0.0647981
$641$	45.3905	1.79282	0.896409	−	0.443228i	$-0.146167\pi$
$641$	45.3905	1.79282	0.896409	+	0.443228i	$0.146167\pi$
$642$	−7.07862	−0.279371
$643$	−22.0112	−0.868036	−0.434018	−	0.900904i	$-0.642905\pi$
$643$	−22.0112	−0.868036	−0.434018	+	0.900904i	$0.642905\pi$
$644$	−1.51767	−0.0598048
$645$	20.9975	0.826775
$646$	13.8007	0.542981
$647$	3.03521	0.119326	0.0596632	−	0.998219i	$-0.480997\pi$
$647$	3.03521	0.119326	0.0596632	+	0.998219i	$0.480997\pi$
$648$	−1.00000	−0.0392837
$649$	−7.18859	−0.282177
$650$	8.74134	0.342863
$651$	−2.61860	−0.102631
$652$	−10.9111	−0.427310
$653$	7.17214	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$653$	7.17214	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$654$	16.9714	0.663636
$655$	28.7396	1.12295
$656$	−6.07884	−0.237339
$657$	9.46535	0.369278
$658$	−11.0061	−0.429064
$659$	−2.92414	−0.113908	−0.0569541	−	0.998377i	$-0.518139\pi$
$659$	−2.92414	−0.113908	−0.0569541	+	0.998377i	$0.518139\pi$
$660$	−1.63928	−0.0638088
$661$	38.9022	1.51312	0.756561	−	0.653923i	$-0.226878\pi$
$661$	38.9022	1.51312	0.756561	+	0.653923i	$0.226878\pi$
$662$	−7.60932	−0.295745
$663$	17.1808	0.667245
$664$	−2.25107	−0.0873584
$665$	9.85979	0.382346
$666$	1.58707	0.0614978
$667$	3.50046	0.135538
$668$	−4.48700	−0.173607
$669$	−3.54077	−0.136894
$670$	14.5882	0.563590
$671$	1.00000	0.0386046
$672$	−1.98112	−0.0764235
$673$	−22.0101	−0.848427	−0.424214	−	0.905562i	$-0.639449\pi$
$673$	−22.0101	−0.848427	−0.424214	+	0.905562i	$0.639449\pi$
$674$	14.5868	0.561863
$675$	2.31277	0.0890186
$676$	1.28533	0.0494359
$677$	−3.18457	−0.122393	−0.0611965	−	0.998126i	$-0.519492\pi$
$677$	−3.18457	−0.122393	−0.0611965	+	0.998126i	$0.519492\pi$
$678$	9.80228	0.376454
$679$	2.68759	0.103140
$680$	−7.45160	−0.285756
$681$	9.60631	0.368114
$682$	−1.32178	−0.0506135
$683$	39.4688	1.51023	0.755116	−	0.655591i	$-0.227580\pi$
$683$	39.4688	1.51023	0.755116	+	0.655591i	$0.227580\pi$
$684$	3.03601	0.116085
$685$	−11.0992	−0.424077
$686$	−19.9601	−0.762081
$687$	−22.4259	−0.855601
$688$	12.8090	0.488338
$689$	46.6484	1.77716
$690$	−1.25580	−0.0478074
$691$	−30.6302	−1.16523	−0.582613	−	0.812750i	$-0.697970\pi$
$691$	−30.6302	−1.16523	−0.582613	+	0.812750i	$0.697970\pi$
$692$	−11.2494	−0.427640
$693$	1.98112	0.0752566
$694$	21.0431	0.798787
$695$	23.7795	0.902006
$696$	4.56939	0.173202
$697$	27.6324	1.04665
$698$	20.3305	0.769521
$699$	−17.6993	−0.669448
$700$	4.58188	0.173179
$701$	50.5981	1.91107	0.955533	−	0.294885i	$-0.0952813\pi$
$701$	50.5981	1.91107	0.955533	+	0.294885i	$0.0952813\pi$
$702$	3.77959	0.142652
$703$	−4.81837	−0.181728
$704$	−1.00000	−0.0376889
$705$	−9.10701	−0.342990
$706$	−5.61723	−0.211407
$707$	22.5717	0.848895
$708$	−7.18859	−0.270164
$709$	−16.2498	−0.610274	−0.305137	−	0.952309i	$-0.598702\pi$
$709$	−16.2498	−0.610274	−0.305137	+	0.952309i	$0.598702\pi$
$710$	−16.3682	−0.614287
$711$	5.54447	0.207934
$712$	7.45160	0.279260
$713$	−1.01257	−0.0379211
$714$	9.00551	0.337023
$715$	6.19580	0.231710
$716$	12.5429	0.468751
$717$	29.6363	1.10679
$718$	29.2412	1.09127
$719$	−30.0198	−1.11955	−0.559774	−	0.828645i	$-0.689112\pi$
$719$	−30.0198	−1.11955	−0.559774	+	0.828645i	$0.689112\pi$
$720$	−1.63928	−0.0610922
$721$	23.0539	0.858573
$722$	9.78261	0.364071
$723$	7.68981	0.285987
$724$	−12.0377	−0.447376
$725$	−10.5679	−0.392484
$726$	1.00000	0.0371135
$727$	−32.5269	−1.20635	−0.603177	−	0.797607i	$-0.706099\pi$
$727$	−32.5269	−1.20635	−0.603177	+	0.797607i	$0.706099\pi$
$728$	7.48784	0.277518
$729$	1.00000	0.0370370
$730$	15.5163	0.574285
$731$	−58.2254	−2.15354
$732$	1.00000	0.0369611
$733$	−12.6776	−0.468258	−0.234129	−	0.972206i	$-0.575224\pi$
$733$	−12.6776	−0.468258	−0.234129	+	0.972206i	$0.575224\pi$
$734$	5.23927	0.193385
$735$	−5.04103	−0.185941
$736$	−0.766068	−0.0282376
$737$	−8.89915	−0.327804
$738$	6.07884	0.223765
$739$	28.5871	1.05159	0.525797	−	0.850610i	$-0.323767\pi$
$739$	28.5871	1.05159	0.525797	+	0.850610i	$0.323767\pi$
$740$	2.60165	0.0956386
$741$	−11.4749	−0.421541
$742$	24.4513	0.897637
$743$	15.0576	0.552411	0.276205	−	0.961099i	$-0.410923\pi$
$743$	15.0576	0.552411	0.276205	+	0.961099i	$0.410923\pi$
$744$	−1.32178	−0.0484587
$745$	−4.14108	−0.151717
$746$	4.53851	0.166167
$747$	2.25107	0.0823623
$748$	4.54566	0.166206
$749$	14.0236	0.512411
$750$	11.9877	0.437727
$751$	6.38019	0.232816	0.116408	−	0.993201i	$-0.462862\pi$
$751$	6.38019	0.232816	0.116408	+	0.993201i	$0.462862\pi$
$752$	−5.55550	−0.202588
$753$	13.2509	0.482888
$754$	−17.2704	−0.628952
$755$	24.0797	0.876351
$756$	1.98112	0.0720527
$757$	−39.6369	−1.44063	−0.720314	−	0.693648i	$-0.756002\pi$
$757$	−39.6369	−1.44063	−0.720314	+	0.693648i	$0.756002\pi$
$758$	−6.27341	−0.227861
$759$	0.766068	0.0278065
$760$	4.97687	0.180530
$761$	−0.207321	−0.00751537	−0.00375769	−	0.999993i	$-0.501196\pi$
$761$	−0.207321	−0.00751537	−0.00375769	+	0.999993i	$0.501196\pi$
$762$	−4.93054	−0.178615
$763$	−33.6225	−1.21722
$764$	−9.26525	−0.335205
$765$	7.45160	0.269413
$766$	28.7571	1.03904
$767$	27.1700	0.981051
$768$	−1.00000	−0.0360844
$769$	1.83160	0.0660492	0.0330246	−	0.999455i	$-0.489486\pi$
$769$	1.83160	0.0660492	0.0330246	+	0.999455i	$0.489486\pi$
$770$	3.24761	0.117036
$771$	19.5882	0.705452
$772$	−0.770410	−0.0277277
$773$	−35.4656	−1.27561	−0.637805	−	0.770198i	$-0.720157\pi$
$773$	−35.4656	−1.27561	−0.637805	+	0.770198i	$0.720157\pi$
$774$	−12.8090	−0.460410
$775$	3.05697	0.109810
$776$	1.35660	0.0486991
$777$	−3.14418	−0.112797
$778$	9.58124	0.343504
$779$	−18.4554	−0.661235
$780$	6.19580	0.221845
$781$	9.98500	0.357291
$782$	3.48229	0.124526
$783$	−4.56939	−0.163296
$784$	−3.07515	−0.109827
$785$	−13.4687	−0.480719
$786$	−17.5319	−0.625341
$787$	−19.9711	−0.711892	−0.355946	−	0.934506i	$-0.615841\pi$
$787$	−19.9711	−0.711892	−0.355946	+	0.934506i	$0.615841\pi$
$788$	−2.06632	−0.0736097
$789$	−7.54730	−0.268691
$790$	9.08892	0.323369
$791$	−19.4195	−0.690478
$792$	1.00000	0.0355335
$793$	−3.77959	−0.134217
$794$	−29.2344	−1.03749
$795$	20.2322	0.717563
$796$	−5.35476	−0.189794
$797$	22.1152	0.783359	0.391679	−	0.920102i	$-0.371894\pi$
$797$	22.1152	0.783359	0.391679	+	0.920102i	$0.371894\pi$
$798$	−6.01472	−0.212919
$799$	25.2534	0.893403
$800$	2.31277	0.0817688
$801$	−7.45160	−0.263289
$802$	16.4747	0.581744
$803$	−9.46535	−0.334025
$804$	−8.89915	−0.313849
$805$	2.48789	0.0876866
$806$	4.99578	0.175969
$807$	10.5635	0.371852
$808$	11.3934	0.400818
$809$	−8.93216	−0.314038	−0.157019	−	0.987596i	$-0.550188\pi$
$809$	−8.93216	−0.314038	−0.157019	+	0.987596i	$0.550188\pi$
$810$	1.63928	0.0575983
$811$	38.9711	1.36846	0.684229	−	0.729267i	$-0.260139\pi$
$811$	38.9711	1.36846	0.684229	+	0.729267i	$0.260139\pi$
$812$	−9.05251	−0.317681
$813$	−12.2232	−0.428687
$814$	−1.58707	−0.0556268
$815$	17.8862	0.626528
$816$	4.54566	0.159130
$817$	38.8883	1.36053
$818$	−13.4655	−0.470811
$819$	−7.48784	−0.261646
$820$	9.96490	0.347989
$821$	20.5680	0.717828	0.358914	−	0.933371i	$-0.383147\pi$
$821$	20.5680	0.717828	0.358914	+	0.933371i	$0.383147\pi$
$822$	6.77076	0.236157
$823$	3.20187	0.111610	0.0558051	−	0.998442i	$-0.482227\pi$
$823$	3.20187	0.111610	0.0558051	+	0.998442i	$0.482227\pi$
$824$	11.6368	0.405387
$825$	−2.31277	−0.0805204
$826$	14.2415	0.495525
$827$	40.4033	1.40496	0.702480	−	0.711703i	$-0.252076\pi$
$827$	40.4033	1.40496	0.702480	+	0.711703i	$0.252076\pi$
$828$	0.766068	0.0266227
$829$	42.5005	1.47610	0.738052	−	0.674744i	$-0.235746\pi$
$829$	42.5005	1.47610	0.738052	+	0.674744i	$0.235746\pi$
$830$	3.69013	0.128086
$831$	7.88949	0.273683
$832$	3.77959	0.131034
$833$	13.9786	0.484330
$834$	−14.5061	−0.502304
$835$	7.35543	0.254545
$836$	−3.03601	−0.105003
$837$	1.32178	0.0456873
$838$	27.0108	0.933074
$839$	30.1130	1.03962	0.519808	−	0.854283i	$-0.326004\pi$
$839$	30.1130	1.03962	0.519808	+	0.854283i	$0.326004\pi$
$840$	3.24761	0.112053
$841$	−8.12072	−0.280025
$842$	−1.54761	−0.0533342
$843$	−6.40968	−0.220761
$844$	−23.9778	−0.825348
$845$	−2.10702	−0.0724835
$846$	5.55550	0.191002
$847$	−1.98112	−0.0680722
$848$	12.3422	0.423832
$849$	−13.2015	−0.453074
$850$	−10.5131	−0.360596
$851$	−1.21581	−0.0416773
$852$	9.98500	0.342080
$853$	26.8372	0.918889	0.459444	−	0.888207i	$-0.348049\pi$
$853$	26.8372	0.918889	0.459444	+	0.888207i	$0.348049\pi$
$854$	−1.98112	−0.0677926
$855$	−4.97687	−0.170205
$856$	7.07862	0.241942
$857$	−11.7185	−0.400298	−0.200149	−	0.979765i	$-0.564143\pi$
$857$	−11.7185	−0.400298	−0.200149	+	0.979765i	$0.564143\pi$
$858$	−3.77959	−0.129033
$859$	19.5870	0.668301	0.334150	−	0.942520i	$-0.391551\pi$
$859$	19.5870	0.668301	0.334150	+	0.942520i	$0.391551\pi$
$860$	−20.9975	−0.716008
$861$	−12.0429	−0.410422
$862$	−10.4212	−0.354948
$863$	29.4195	1.00145	0.500726	−	0.865606i	$-0.333067\pi$
$863$	29.4195	1.00145	0.500726	+	0.865606i	$0.333067\pi$
$864$	1.00000	0.0340207
$865$	18.4410	0.627011
$866$	−26.7354	−0.908506
$867$	−3.66305	−0.124404
$868$	2.61860	0.0888811
$869$	−5.54447	−0.188083
$870$	−7.49049	−0.253951
$871$	33.6352	1.13968
$872$	−16.9714	−0.574726
$873$	−1.35660	−0.0459140
$874$	−2.32579	−0.0786712
$875$	−23.7490	−0.802863
$876$	−9.46535	−0.319805
$877$	0.297849	0.0100576	0.00502882	−	0.999987i	$-0.498399\pi$
$877$	0.297849	0.0100576	0.00502882	+	0.999987i	$0.498399\pi$
$878$	−6.41334	−0.216440
$879$	−14.5648	−0.491258
$880$	1.63928	0.0552600
$881$	29.4516	0.992251	0.496126	−	0.868251i	$-0.334756\pi$
$881$	29.4516	0.992251	0.496126	+	0.868251i	$0.334756\pi$
$882$	3.07515	0.103546
$883$	6.16722	0.207543	0.103772	−	0.994601i	$-0.466909\pi$
$883$	6.16722	0.207543	0.103772	+	0.994601i	$0.466909\pi$
$884$	−17.1808	−0.577851
$885$	11.7841	0.396118
$886$	−15.6676	−0.526363
$887$	32.6998	1.09795	0.548976	−	0.835838i	$-0.315018\pi$
$887$	32.6998	1.09795	0.548976	+	0.835838i	$0.315018\pi$
$888$	−1.58707	−0.0532586
$889$	9.76800	0.327608
$890$	−12.2152	−0.409456
$891$	−1.00000	−0.0335013
$892$	3.54077	0.118554
$893$	−16.8666	−0.564419
$894$	2.52616	0.0844875
$895$	−20.5613	−0.687289
$896$	1.98112	0.0661847
$897$	−2.89543	−0.0966755
$898$	−3.65900	−0.122102
$899$	−6.03971	−0.201436
$900$	−2.31277	−0.0770924
$901$	−56.1033	−1.86907
$902$	−6.07884	−0.202403
$903$	25.3762	0.844466
$904$	−9.80228	−0.326019
$905$	19.7331	0.655949
$906$	−14.6892	−0.488017
$907$	6.09287	0.202311	0.101155	−	0.994871i	$-0.467746\pi$
$907$	6.09287	0.202311	0.101155	+	0.994871i	$0.467746\pi$
$908$	−9.60631	−0.318796
$909$	−11.3934	−0.377894
$910$	−12.2746	−0.406900
$911$	41.4743	1.37410	0.687052	−	0.726608i	$-0.258905\pi$
$911$	41.4743	1.37410	0.687052	+	0.726608i	$0.258905\pi$
$912$	−3.03601	−0.100532
$913$	−2.25107	−0.0744995
$914$	20.2055	0.668340
$915$	−1.63928	−0.0541928
$916$	22.4259	0.740972
$917$	34.7327	1.14698
$918$	−4.54566	−0.150029
$919$	−40.7353	−1.34373	−0.671866	−	0.740672i	$-0.734507\pi$
$919$	−40.7353	−1.34373	−0.671866	+	0.740672i	$0.734507\pi$
$920$	1.25580	0.0414024
$921$	7.66151	0.252455
$922$	36.5097	1.20238
$923$	−37.7393	−1.24220
$924$	−1.98112	−0.0651741
$925$	3.67053	0.120686
$926$	−11.3661	−0.373513
$927$	−11.6368	−0.382203
$928$	−4.56939	−0.149997
$929$	−46.2897	−1.51871	−0.759357	−	0.650674i	$-0.774487\pi$
$929$	−46.2897	−1.51871	−0.759357	+	0.650674i	$0.774487\pi$
$930$	2.16676	0.0710508
$931$	−9.33621	−0.305982
$932$	17.6993	0.579759
$933$	−11.7353	−0.384197
$934$	30.8812	1.01046
$935$	−7.45160	−0.243693
$936$	−3.77959	−0.123540
$937$	−47.1556	−1.54051	−0.770254	−	0.637738i	$-0.779870\pi$
$937$	−47.1556	−1.54051	−0.770254	+	0.637738i	$0.779870\pi$
$938$	17.6303	0.575650
$939$	7.29620	0.238102
$940$	9.10701	0.297038
$941$	−16.4360	−0.535799	−0.267899	−	0.963447i	$-0.586330\pi$
$941$	−16.4360	−0.535799	−0.267899	+	0.963447i	$0.586330\pi$
$942$	8.21625	0.267700
$943$	−4.65681	−0.151646
$944$	7.18859	0.233969
$945$	−3.24761	−0.105645
$946$	12.8090	0.416456
$947$	−8.12504	−0.264028	−0.132014	−	0.991248i	$-0.542144\pi$
$947$	−8.12504	−0.264028	−0.132014	+	0.991248i	$0.542144\pi$
$948$	−5.54447	−0.180076
$949$	35.7752	1.16131
$950$	7.02161	0.227811
$951$	24.9132	0.807865
$952$	−9.00551	−0.291870
$953$	−24.9318	−0.807622	−0.403811	−	0.914843i	$-0.632315\pi$
$953$	−24.9318	−0.807622	−0.403811	+	0.914843i	$0.632315\pi$
$954$	−12.3422	−0.399592
$955$	15.1883	0.491482
$956$	−29.6363	−0.958505
$957$	4.56939	0.147707
$958$	28.9860	0.936495
$959$	−13.4137	−0.433151
$960$	1.63928	0.0529074
$961$	−29.2529	−0.943642
$962$	5.99849	0.193399
$963$	−7.07862	−0.228105
$964$	−7.68981	−0.247672
$965$	1.26292	0.0406547
$966$	−1.51767	−0.0488304
$967$	26.0976	0.839244	0.419622	−	0.907699i	$-0.362163\pi$
$967$	26.0976	0.839244	0.419622	+	0.907699i	$0.362163\pi$
$968$	−1.00000	−0.0321412
$969$	13.8007	0.443342
$970$	−2.22384	−0.0714033
$971$	10.1457	0.325591	0.162796	−	0.986660i	$-0.447949\pi$
$971$	10.1457	0.325591	0.162796	+	0.986660i	$0.447949\pi$
$972$	−1.00000	−0.0320750
$973$	28.7383	0.921307
$974$	−21.7513	−0.696957
$975$	8.74134	0.279947
$976$	−1.00000	−0.0320092
$977$	−6.07879	−0.194478	−0.0972388	−	0.995261i	$-0.531001\pi$
$977$	−6.07879	−0.194478	−0.0972388	+	0.995261i	$0.531001\pi$
$978$	−10.9111	−0.348897
$979$	7.45160	0.238154
$980$	5.04103	0.161030
$981$	16.9714	0.541856
$982$	38.5187	1.22918
$983$	−27.6775	−0.882775	−0.441388	−	0.897316i	$-0.645514\pi$
$983$	−27.6775	−0.882775	−0.441388	+	0.897316i	$0.645514\pi$
$984$	−6.07884	−0.193786
$985$	3.38728	0.107928
$986$	20.7709	0.661480
$987$	−11.0061	−0.350329
$988$	11.4749	0.365065
$989$	9.81256	0.312021
$990$	−1.63928	−0.0520996
$991$	−25.6496	−0.814786	−0.407393	−	0.913253i	$-0.633562\pi$
$991$	−25.6496	−0.814786	−0.407393	+	0.913253i	$0.633562\pi$
$992$	1.32178	0.0419665
$993$	−7.60932	−0.241475
$994$	−19.7815	−0.627431
$995$	8.77794	0.278279
$996$	−2.25107	−0.0713279
$997$	5.74257	0.181869	0.0909345	−	0.995857i	$-0.471015\pi$
$997$	5.74257	0.181869	0.0909345	+	0.995857i	$0.471015\pi$
$998$	14.5378	0.460185
$999$	1.58707	0.0502127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.y.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.y.1.3	✓	7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.63928 q^{5} +1.00000 q^{6} -1.98112 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.63928 q^{5} +1.00000 q^{6} -1.98112 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.63928 q^{10} -1.00000 q^{11} -1.00000 q^{12} +3.77959 q^{13} +1.98112 q^{14} +1.63928 q^{15} +1.00000 q^{16} -4.54566 q^{17} -1.00000 q^{18} +3.03601 q^{19} -1.63928 q^{20} +1.98112 q^{21} +1.00000 q^{22} +0.766068 q^{23} +1.00000 q^{24} -2.31277 q^{25} -3.77959 q^{26} -1.00000 q^{27} -1.98112 q^{28} +4.56939 q^{29} -1.63928 q^{30} -1.32178 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.54566 q^{34} +3.24761 q^{35} +1.00000 q^{36} -1.58707 q^{37} -3.03601 q^{38} -3.77959 q^{39} +1.63928 q^{40} -6.07884 q^{41} -1.98112 q^{42} +12.8090 q^{43} -1.00000 q^{44} -1.63928 q^{45} -0.766068 q^{46} -5.55550 q^{47} -1.00000 q^{48} -3.07515 q^{49} +2.31277 q^{50} +4.54566 q^{51} +3.77959 q^{52} +12.3422 q^{53} +1.00000 q^{54} +1.63928 q^{55} +1.98112 q^{56} -3.03601 q^{57} -4.56939 q^{58} +7.18859 q^{59} +1.63928 q^{60} -1.00000 q^{61} +1.32178 q^{62} -1.98112 q^{63} +1.00000 q^{64} -6.19580 q^{65} -1.00000 q^{66} +8.89915 q^{67} -4.54566 q^{68} -0.766068 q^{69} -3.24761 q^{70} -9.98500 q^{71} -1.00000 q^{72} +9.46535 q^{73} +1.58707 q^{74} +2.31277 q^{75} +3.03601 q^{76} +1.98112 q^{77} +3.77959 q^{78} +5.54447 q^{79} -1.63928 q^{80} +1.00000 q^{81} +6.07884 q^{82} +2.25107 q^{83} +1.98112 q^{84} +7.45160 q^{85} -12.8090 q^{86} -4.56939 q^{87} +1.00000 q^{88} -7.45160 q^{89} +1.63928 q^{90} -7.48784 q^{91} +0.766068 q^{92} +1.32178 q^{93} +5.55550 q^{94} -4.97687 q^{95} +1.00000 q^{96} -1.35660 q^{97} +3.07515 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} - 2 q^{5} + 7 q^{6} + q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 - 2 * q^5 + 7 * q^6 + q^7 - 7 * q^8 + 7 * q^9	\( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} - 2 q^{5} + 7 q^{6} + q^{7} - 7 q^{8} + 7 q^{9} + 2 q^{10} - 7 q^{11} - 7 q^{12} - q^{14} + 2 q^{15} + 7 q^{16} + 3 q^{17} - 7 q^{18} - 5 q^{19} - 2 q^{20} - q^{21} + 7 q^{22} - 3 q^{23} + 7 q^{24} + 11 q^{25} - 7 q^{27} + q^{28} - 14 q^{29} - 2 q^{30} + 5 q^{31} - 7 q^{32} + 7 q^{33} - 3 q^{34} - 9 q^{35} + 7 q^{36} + 14 q^{37} + 5 q^{38} + 2 q^{40} - 7 q^{41} + q^{42} + q^{43} - 7 q^{44} - 2 q^{45} + 3 q^{46} - 7 q^{48} - 11 q^{50} - 3 q^{51} - 3 q^{53} + 7 q^{54} + 2 q^{55} - q^{56} + 5 q^{57} + 14 q^{58} - 14 q^{59} + 2 q^{60} - 7 q^{61} - 5 q^{62} + q^{63} + 7 q^{64} - 10 q^{65} - 7 q^{66} + 3 q^{68} + 3 q^{69} + 9 q^{70} - 22 q^{71} - 7 q^{72} + q^{73} - 14 q^{74} - 11 q^{75} - 5 q^{76} - q^{77} + 10 q^{79} - 2 q^{80} + 7 q^{81} + 7 q^{82} - 17 q^{83} - q^{84} + 18 q^{85} - q^{86} + 14 q^{87} + 7 q^{88} - 18 q^{89} + 2 q^{90} + 21 q^{91} - 3 q^{92} - 5 q^{93} - 41 q^{95} + 7 q^{96} + 25 q^{97} - 7 q^{99}+O(q^{100}) \) 7 * q - 7 * q^2 - 7 * q^3 + 7 * q^4 - 2 * q^5 + 7 * q^6 + q^7 - 7 * q^8 + 7 * q^9 + 2 * q^10 - 7 * q^11 - 7 * q^12 - q^14 + 2 * q^15 + 7 * q^16 + 3 * q^17 - 7 * q^18 - 5 * q^19 - 2 * q^20 - q^21 + 7 * q^22 - 3 * q^23 + 7 * q^24 + 11 * q^25 - 7 * q^27 + q^28 - 14 * q^29 - 2 * q^30 + 5 * q^31 - 7 * q^32 + 7 * q^33 - 3 * q^34 - 9 * q^35 + 7 * q^36 + 14 * q^37 + 5 * q^38 + 2 * q^40 - 7 * q^41 + q^42 + q^43 - 7 * q^44 - 2 * q^45 + 3 * q^46 - 7 * q^48 - 11 * q^50 - 3 * q^51 - 3 * q^53 + 7 * q^54 + 2 * q^55 - q^56 + 5 * q^57 + 14 * q^58 - 14 * q^59 + 2 * q^60 - 7 * q^61 - 5 * q^62 + q^63 + 7 * q^64 - 10 * q^65 - 7 * q^66 + 3 * q^68 + 3 * q^69 + 9 * q^70 - 22 * q^71 - 7 * q^72 + q^73 - 14 * q^74 - 11 * q^75 - 5 * q^76 - q^77 + 10 * q^79 - 2 * q^80 + 7 * q^81 + 7 * q^82 - 17 * q^83 - q^84 + 18 * q^85 - q^86 + 14 * q^87 + 7 * q^88 - 18 * q^89 + 2 * q^90 + 21 * q^91 - 3 * q^92 - 5 * q^93 - 41 * q^95 + 7 * q^96 + 25 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more