LMFDB - Embedded newform 418.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [418,2,Mod(1,418)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(418, 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("418.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 418 = 2 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	418.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:	$3.33774680449$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.772866$ of defining polynomial
Character	$\chi$	$=$	418.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} +0.772866 q^{6} +3.40268 q^{7} +1.00000 q^{8} -2.40268 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} +0.772866 q^{6} +3.40268 q^{7} +1.00000 q^{8} -2.40268 q^{9} +1.22713 q^{10} -1.00000 q^{11} +0.772866 q^{12} -1.40268 q^{13} +3.40268 q^{14} +0.948410 q^{15} +1.00000 q^{16} -4.80536 q^{17} -2.40268 q^{18} -1.00000 q^{19} +1.22713 q^{20} +2.62981 q^{21} -1.00000 q^{22} +6.80536 q^{23} +0.772866 q^{24} -3.49414 q^{25} -1.40268 q^{26} -4.17554 q^{27} +3.40268 q^{28} +8.03249 q^{29} +0.948410 q^{30} -4.94841 q^{31} +1.00000 q^{32} -0.772866 q^{33} -4.80536 q^{34} +4.17554 q^{35} -2.40268 q^{36} +2.35109 q^{37} -1.00000 q^{38} -1.08408 q^{39} +1.22713 q^{40} +2.94841 q^{41} +2.62981 q^{42} -9.12395 q^{43} -1.00000 q^{44} -2.94841 q^{45} +6.80536 q^{46} +5.25963 q^{47} +0.772866 q^{48} +4.57822 q^{49} -3.49414 q^{50} -3.71390 q^{51} -1.40268 q^{52} -4.45427 q^{53} -4.17554 q^{54} -1.22713 q^{55} +3.40268 q^{56} -0.772866 q^{57} +8.03249 q^{58} -14.5193 q^{59} +0.948410 q^{60} +2.00000 q^{61} -4.94841 q^{62} -8.17554 q^{63} +1.00000 q^{64} -1.72128 q^{65} -0.772866 q^{66} +3.40268 q^{67} -4.80536 q^{68} +5.25963 q^{69} +4.17554 q^{70} -7.57822 q^{71} -2.40268 q^{72} -10.0650 q^{73} +2.35109 q^{74} -2.70050 q^{75} -1.00000 q^{76} -3.40268 q^{77} -1.08408 q^{78} -3.71390 q^{79} +1.22713 q^{80} +3.98090 q^{81} +2.94841 q^{82} +10.6697 q^{83} +2.62981 q^{84} -5.89682 q^{85} -9.12395 q^{86} +6.20804 q^{87} -1.00000 q^{88} +14.9883 q^{89} -2.94841 q^{90} -4.77287 q^{91} +6.80536 q^{92} -3.82446 q^{93} +5.25963 q^{94} -1.22713 q^{95} +0.772866 q^{96} +2.35109 q^{97} +4.57822 q^{98} +2.40268 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9} + 5 q^{10} - 3 q^{11} + q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} + 4 q^{17} + 2 q^{18} - 3 q^{19} + 5 q^{20} - 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} - 2 q^{27} + q^{28} + 7 q^{29} - 9 q^{30} - 3 q^{31} + 3 q^{32} - q^{33} + 4 q^{34} + 2 q^{35} + 2 q^{36} - 14 q^{37} - 3 q^{38} + 2 q^{39} + 5 q^{40} - 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} + 2 q^{46} + q^{48} - 6 q^{49} + 4 q^{50} + 2 q^{51} + 5 q^{52} - 16 q^{53} - 2 q^{54} - 5 q^{55} + q^{56} - q^{57} + 7 q^{58} - 12 q^{59} - 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} + 8 q^{65} - q^{66} + q^{67} + 4 q^{68} + 2 q^{70} - 3 q^{71} + 2 q^{72} + 4 q^{73} - 14 q^{74} - 41 q^{75} - 3 q^{76} - q^{77} + 2 q^{78} + 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} - 5 q^{86} - 9 q^{87} - 3 q^{88} + 16 q^{89} + 3 q^{90} - 13 q^{91} + 2 q^{92} - 22 q^{93} - 5 q^{95} + q^{96} - 14 q^{97} - 6 q^{98} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9 + 5 * q^10 - 3 * q^11 + q^12 + 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 + 4 * q^17 + 2 * q^18 - 3 * q^19 + 5 * q^20 - 3 * q^22 + 2 * q^23 + q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 + q^28 + 7 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 - q^33 + 4 * q^34 + 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 + 2 * q^39 + 5 * q^40 - 3 * q^41 - 5 * q^43 - 3 * q^44 + 3 * q^45 + 2 * q^46 + q^48 - 6 * q^49 + 4 * q^50 + 2 * q^51 + 5 * q^52 - 16 * q^53 - 2 * q^54 - 5 * q^55 + q^56 - q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 + 6 * q^61 - 3 * q^62 - 14 * q^63 + 3 * q^64 + 8 * q^65 - q^66 + q^67 + 4 * q^68 + 2 * q^70 - 3 * q^71 + 2 * q^72 + 4 * q^73 - 14 * q^74 - 41 * q^75 - 3 * q^76 - q^77 + 2 * q^78 + 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 + 7 * q^83 + 6 * q^85 - 5 * q^86 - 9 * q^87 - 3 * q^88 + 16 * q^89 + 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 - 5 * q^95 + q^96 - 14 * q^97 - 6 * q^98 - 2 * 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
$3$	0.772866	0.446214	0.223107	−	0.974794i	$-0.428380\pi$
$3$	0.772866	0.446214	0.223107	+	0.974794i	$0.428380\pi$
$4$	1.00000	0.500000
$5$	1.22713	0.548791	0.274396	−	0.961617i	$-0.411522\pi$
$5$	1.22713	0.548791	0.274396	+	0.961617i	$0.411522\pi$
$6$	0.772866	0.315521
$7$	3.40268	1.28609	0.643046	−	0.765828i	$-0.277670\pi$
$7$	3.40268	1.28609	0.643046	+	0.765828i	$0.277670\pi$
$8$	1.00000	0.353553
$9$	−2.40268	−0.800893
$10$	1.22713	0.388054
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.772866	0.223107
$13$	−1.40268	−0.389033	−0.194517	−	0.980899i	$-0.562314\pi$
$13$	−1.40268	−0.389033	−0.194517	+	0.980899i	$0.562314\pi$
$14$	3.40268	0.909404
$15$	0.948410	0.244878
$16$	1.00000	0.250000
$17$	−4.80536	−1.16547	−0.582735	−	0.812662i	$-0.698018\pi$
$17$	−4.80536	−1.16547	−0.582735	+	0.812662i	$0.698018\pi$
$18$	−2.40268	−0.566317
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.22713	0.274396
$21$	2.62981	0.573872
$22$	−1.00000	−0.213201
$23$	6.80536	1.41902	0.709508	−	0.704698i	$-0.248917\pi$
$23$	6.80536	1.41902	0.709508	+	0.704698i	$0.248917\pi$
$24$	0.772866	0.157761
$25$	−3.49414	−0.698828
$26$	−1.40268	−0.275088
$27$	−4.17554	−0.803584
$28$	3.40268	0.643046
$29$	8.03249	1.49160	0.745798	−	0.666172i	$-0.232068\pi$
$29$	8.03249	1.49160	0.745798	+	0.666172i	$0.232068\pi$
$30$	0.948410	0.173155
$31$	−4.94841	−0.888761	−0.444380	−	0.895838i	$-0.646576\pi$
$31$	−4.94841	−0.888761	−0.444380	+	0.895838i	$0.646576\pi$
$32$	1.00000	0.176777
$33$	−0.772866	−0.134539
$34$	−4.80536	−0.824112
$35$	4.17554	0.705796
$36$	−2.40268	−0.400446
$37$	2.35109	0.386517	0.193258	−	0.981148i	$-0.438094\pi$
$37$	2.35109	0.386517	0.193258	+	0.981148i	$0.438094\pi$
$38$	−1.00000	−0.162221
$39$	−1.08408	−0.173592
$40$	1.22713	0.194027
$41$	2.94841	0.460464	0.230232	−	0.973136i	$-0.426051\pi$
$41$	2.94841	0.460464	0.230232	+	0.973136i	$0.426051\pi$
$42$	2.62981	0.405789
$43$	−9.12395	−1.39139	−0.695695	−	0.718337i	$-0.744903\pi$
$43$	−9.12395	−1.39139	−0.695695	+	0.718337i	$0.744903\pi$
$44$	−1.00000	−0.150756
$45$	−2.94841	−0.439523
$46$	6.80536	1.00340
$47$	5.25963	0.767195	0.383598	−	0.923500i	$-0.374685\pi$
$47$	5.25963	0.767195	0.383598	+	0.923500i	$0.374685\pi$
$48$	0.772866	0.111554
$49$	4.57822	0.654032
$50$	−3.49414	−0.494146
$51$	−3.71390	−0.520049
$52$	−1.40268	−0.194517
$53$	−4.45427	−0.611841	−0.305920	−	0.952057i	$-0.598964\pi$
$53$	−4.45427	−0.611841	−0.305920	+	0.952057i	$0.598964\pi$
$54$	−4.17554	−0.568220
$55$	−1.22713	−0.165467
$56$	3.40268	0.454702
$57$	−0.772866	−0.102369
$58$	8.03249	1.05472
$59$	−14.5193	−1.89025	−0.945123	−	0.326715i	$-0.894058\pi$
$59$	−14.5193	−1.89025	−0.945123	+	0.326715i	$0.894058\pi$
$60$	0.948410	0.122439
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−4.94841	−0.628449
$63$	−8.17554	−1.03002
$64$	1.00000	0.125000
$65$	−1.72128	−0.213498
$66$	−0.772866	−0.0951332
$67$	3.40268	0.415703	0.207852	−	0.978160i	$-0.433353\pi$
$67$	3.40268	0.415703	0.207852	+	0.978160i	$0.433353\pi$
$68$	−4.80536	−0.582735
$69$	5.25963	0.633185
$70$	4.17554	0.499073
$71$	−7.57822	−0.899370	−0.449685	−	0.893187i	$-0.648464\pi$
$71$	−7.57822	−0.899370	−0.449685	+	0.893187i	$0.648464\pi$
$72$	−2.40268	−0.283158
$73$	−10.0650	−1.17802	−0.589009	−	0.808127i	$-0.700482\pi$
$73$	−10.0650	−1.17802	−0.589009	+	0.808127i	$0.700482\pi$
$74$	2.35109	0.273309
$75$	−2.70050	−0.311827
$76$	−1.00000	−0.114708
$77$	−3.40268	−0.387771
$78$	−1.08408	−0.122748
$79$	−3.71390	−0.417846	−0.208923	−	0.977932i	$-0.566996\pi$
$79$	−3.71390	−0.417846	−0.208923	+	0.977932i	$0.566996\pi$
$80$	1.22713	0.137198
$81$	3.98090	0.442322
$82$	2.94841	0.325597
$83$	10.6697	1.17115	0.585575	−	0.810618i	$-0.300869\pi$
$83$	10.6697	1.17115	0.585575	+	0.810618i	$0.300869\pi$
$84$	2.62981	0.286936
$85$	−5.89682	−0.639600
$86$	−9.12395	−0.983861
$87$	6.20804	0.665571
$88$	−1.00000	−0.106600
$89$	14.9883	1.58875	0.794377	−	0.607425i	$-0.207797\pi$
$89$	14.9883	1.58875	0.794377	+	0.607425i	$0.207797\pi$
$90$	−2.94841	−0.310790
$91$	−4.77287	−0.500332
$92$	6.80536	0.709508
$93$	−3.82446	−0.396578
$94$	5.25963	0.542489
$95$	−1.22713	−0.125901
$96$	0.772866	0.0788803
$97$	2.35109	0.238717	0.119358	−	0.992851i	$-0.461916\pi$
$97$	2.35109	0.238717	0.119358	+	0.992851i	$0.461916\pi$
$98$	4.57822	0.462470
$99$	2.40268	0.241478
$100$	−3.49414	−0.349414
$101$	14.0650	1.39952	0.699759	−	0.714379i	$-0.253291\pi$
$101$	14.0650	1.39952	0.699759	+	0.714379i	$0.253291\pi$
$102$	−3.71390	−0.367730
$103$	−15.5782	−1.53497	−0.767484	−	0.641068i	$-0.778492\pi$
$103$	−15.5782	−1.53497	−0.767484	+	0.641068i	$0.778492\pi$
$104$	−1.40268	−0.137544
$105$	3.22713	0.314936
$106$	−4.45427	−0.432637
$107$	−4.35109	−0.420636	−0.210318	−	0.977633i	$-0.567450\pi$
$107$	−4.35109	−0.420636	−0.210318	+	0.977633i	$0.567450\pi$
$108$	−4.17554	−0.401792
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−1.22713	−0.117003
$111$	1.81708	0.172469
$112$	3.40268	0.321523
$113$	−19.9618	−1.87785	−0.938924	−	0.344124i	$-0.888176\pi$
$113$	−19.9618	−1.87785	−0.938924	+	0.344124i	$0.888176\pi$
$114$	−0.772866	−0.0723855
$115$	8.35109	0.778743
$116$	8.03249	0.745798
$117$	3.37019	0.311574
$118$	−14.5193	−1.33661
$119$	−16.3511	−1.49890
$120$	0.948410	0.0865776
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	2.27872	0.205466
$124$	−4.94841	−0.444380
$125$	−10.4235	−0.932302
$126$	−8.17554	−0.728335
$127$	11.7139	1.03944	0.519720	−	0.854337i	$-0.326036\pi$
$127$	11.7139	1.03944	0.519720	+	0.854337i	$0.326036\pi$
$128$	1.00000	0.0883883
$129$	−7.05159	−0.620858
$130$	−1.72128	−0.150966
$131$	1.85695	0.162242	0.0811211	−	0.996704i	$-0.474150\pi$
$131$	1.85695	0.162242	0.0811211	+	0.996704i	$0.474150\pi$
$132$	−0.772866	−0.0672693
$133$	−3.40268	−0.295050
$134$	3.40268	0.293947
$135$	−5.12395	−0.441000
$136$	−4.80536	−0.412056
$137$	10.5973	0.905390	0.452695	−	0.891665i	$-0.350463\pi$
$137$	10.5973	0.905390	0.452695	+	0.891665i	$0.350463\pi$
$138$	5.25963	0.447729
$139$	14.7347	1.24978	0.624889	−	0.780713i	$-0.285144\pi$
$139$	14.7347	1.24978	0.624889	+	0.780713i	$0.285144\pi$
$140$	4.17554	0.352898
$141$	4.06498	0.342333
$142$	−7.57822	−0.635950
$143$	1.40268	0.117298
$144$	−2.40268	−0.200223
$145$	9.85695	0.818575
$146$	−10.0650	−0.832984
$147$	3.53835	0.291838
$148$	2.35109	0.193258
$149$	18.3511	1.50338	0.751690	−	0.659517i	$-0.229239\pi$
$149$	18.3511	1.50338	0.751690	+	0.659517i	$0.229239\pi$
$150$	−2.70050	−0.220495
$151$	−7.64891	−0.622460	−0.311230	−	0.950335i	$-0.600741\pi$
$151$	−7.64891	−0.622460	−0.311230	+	0.950335i	$0.600741\pi$
$152$	−1.00000	−0.0811107
$153$	11.5457	0.933417
$154$	−3.40268	−0.274196
$155$	−6.07236	−0.487744
$156$	−1.08408	−0.0867960
$157$	8.38358	0.669083	0.334541	−	0.942381i	$-0.391419\pi$
$157$	8.38358	0.669083	0.334541	+	0.942381i	$0.391419\pi$
$158$	−3.71390	−0.295462
$159$	−3.44255	−0.273012
$160$	1.22713	0.0970135
$161$	23.1564	1.82498
$162$	3.98090	0.312769
$163$	−14.2479	−1.11598	−0.557991	−	0.829847i	$-0.688428\pi$
$163$	−14.2479	−1.11598	−0.557991	+	0.829847i	$0.688428\pi$
$164$	2.94841	0.230232
$165$	−0.948410	−0.0738336
$166$	10.6697	0.828128
$167$	−10.8703	−0.841172	−0.420586	−	0.907253i	$-0.638176\pi$
$167$	−10.8703	−0.841172	−0.420586	+	0.907253i	$0.638176\pi$
$168$	2.62981	0.202894
$169$	−11.0325	−0.848653
$170$	−5.89682	−0.452265
$171$	2.40268	0.183737
$172$	−9.12395	−0.695695
$173$	3.39530	0.258140	0.129070	−	0.991635i	$-0.458801\pi$
$173$	3.39530	0.258140	0.129070	+	0.991635i	$0.458801\pi$
$174$	6.20804	0.470630
$175$	−11.8894	−0.898757
$176$	−1.00000	−0.0753778
$177$	−11.2214	−0.843454
$178$	14.9883	1.12342
$179$	−0.311217	−0.0232614	−0.0116307	−	0.999932i	$-0.503702\pi$
$179$	−0.311217	−0.0232614	−0.0116307	+	0.999932i	$0.503702\pi$
$180$	−2.94841	−0.219762
$181$	−15.3246	−1.13907	−0.569535	−	0.821967i	$-0.692877\pi$
$181$	−15.3246	−1.13907	−0.569535	+	0.821967i	$0.692877\pi$
$182$	−4.77287	−0.353788
$183$	1.54573	0.114264
$184$	6.80536	0.501698
$185$	2.88510	0.212117
$186$	−3.82446	−0.280423
$187$	4.80536	0.351403
$188$	5.25963	0.383598
$189$	−14.2080	−1.03348
$190$	−1.22713	−0.0890257
$191$	26.8703	1.94427	0.972135	−	0.234422i	$-0.0753199\pi$
$191$	26.8703	1.94427	0.972135	+	0.234422i	$0.0753199\pi$
$192$	0.772866	0.0557768
$193$	18.5665	1.33645	0.668223	−	0.743961i	$-0.267055\pi$
$193$	18.5665	1.33645	0.668223	+	0.743961i	$0.267055\pi$
$194$	2.35109	0.168798
$195$	−1.33031	−0.0952658
$196$	4.57822	0.327016
$197$	14.9085	1.06219	0.531095	−	0.847312i	$-0.321781\pi$
$197$	14.9085	1.06219	0.531095	+	0.847312i	$0.321781\pi$
$198$	2.40268	0.170751
$199$	15.1564	1.07441	0.537206	−	0.843451i	$-0.319480\pi$
$199$	15.1564	1.07441	0.537206	+	0.843451i	$0.319480\pi$
$200$	−3.49414	−0.247073
$201$	2.62981	0.185493
$202$	14.0650	0.989609
$203$	27.3320	1.91833
$204$	−3.71390	−0.260025
$205$	3.61810	0.252699
$206$	−15.5782	−1.08539
$207$	−16.3511	−1.13648
$208$	−1.40268	−0.0972583
$209$	1.00000	0.0691714
$210$	3.22713	0.222693
$211$	11.1564	0.768041	0.384021	−	0.923324i	$-0.374539\pi$
$211$	11.1564	0.768041	0.384021	+	0.923324i	$0.374539\pi$
$212$	−4.45427	−0.305920
$213$	−5.85695	−0.401311
$214$	−4.35109	−0.297434
$215$	−11.1963	−0.763583
$216$	−4.17554	−0.284110
$217$	−16.8378	−1.14303
$218$	6.00000	0.406371
$219$	−7.77888	−0.525648
$220$	−1.22713	−0.0827334
$221$	6.74037	0.453407
$222$	1.81708	0.121954
$223$	1.61072	0.107861	0.0539307	−	0.998545i	$-0.482825\pi$
$223$	1.61072	0.107861	0.0539307	+	0.998545i	$0.482825\pi$
$224$	3.40268	0.227351
$225$	8.39530	0.559687
$226$	−19.9618	−1.32784
$227$	−16.7672	−1.11288	−0.556438	−	0.830889i	$-0.687832\pi$
$227$	−16.7672	−1.11288	−0.556438	+	0.830889i	$0.687832\pi$
$228$	−0.772866	−0.0511843
$229$	−8.55913	−0.565603	−0.282801	−	0.959178i	$-0.591264\pi$
$229$	−8.55913	−0.565603	−0.282801	+	0.959178i	$0.591264\pi$
$230$	8.35109	0.550654
$231$	−2.62981	−0.173029
$232$	8.03249	0.527359
$233$	5.44255	0.356553	0.178277	−	0.983980i	$-0.442948\pi$
$233$	5.44255	0.356553	0.178277	+	0.983980i	$0.442948\pi$
$234$	3.37019	0.220316
$235$	6.45427	0.421030
$236$	−14.5193	−0.945123
$237$	−2.87034	−0.186449
$238$	−16.3511	−1.05988
$239$	11.2921	0.730426	0.365213	−	0.930924i	$-0.380996\pi$
$239$	11.2921	0.730426	0.365213	+	0.930924i	$0.380996\pi$
$240$	0.948410	0.0612196
$241$	11.4101	0.734987	0.367493	−	0.930026i	$-0.380216\pi$
$241$	11.4101	0.734987	0.367493	+	0.930026i	$0.380216\pi$
$242$	1.00000	0.0642824
$243$	15.6033	1.00095
$244$	2.00000	0.128037
$245$	5.61810	0.358927
$246$	2.27872	0.145286
$247$	1.40268	0.0892503
$248$	−4.94841	−0.314224
$249$	8.24623	0.522584
$250$	−10.4235	−0.659237
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−8.17554	−0.515011
$253$	−6.80536	−0.427849
$254$	11.7139	0.734995
$255$	−4.55745	−0.285399
$256$	1.00000	0.0625000
$257$	24.2331	1.51162	0.755811	−	0.654790i	$-0.227243\pi$
$257$	24.2331	1.51162	0.755811	+	0.654790i	$0.227243\pi$
$258$	−7.05159	−0.439013
$259$	8.00000	0.497096
$260$	−1.72128	−0.106749
$261$	−19.2995	−1.19461
$262$	1.85695	0.114723
$263$	20.5665	1.26819	0.634093	−	0.773257i	$-0.281374\pi$
$263$	20.5665	1.26819	0.634093	+	0.773257i	$0.281374\pi$
$264$	−0.772866	−0.0475666
$265$	−5.46599	−0.335773
$266$	−3.40268	−0.208632
$267$	11.5839	0.708925
$268$	3.40268	0.207852
$269$	−1.36281	−0.0830918	−0.0415459	−	0.999137i	$-0.513228\pi$
$269$	−1.36281	−0.0830918	−0.0415459	+	0.999137i	$0.513228\pi$
$270$	−5.12395	−0.311834
$271$	−23.1963	−1.40908	−0.704538	−	0.709666i	$-0.748846\pi$
$271$	−23.1963	−1.40908	−0.704538	+	0.709666i	$0.748846\pi$
$272$	−4.80536	−0.291368
$273$	−3.68878	−0.223255
$274$	10.5973	0.640208
$275$	3.49414	0.210705
$276$	5.25963	0.316592
$277$	31.0385	1.86492	0.932462	−	0.361269i	$-0.117656\pi$
$277$	31.0385	1.86492	0.932462	+	0.361269i	$0.117656\pi$
$278$	14.7347	0.883727
$279$	11.8894	0.711802
$280$	4.17554	0.249537
$281$	−2.77287	−0.165415	−0.0827076	−	0.996574i	$-0.526357\pi$
$281$	−2.77287	−0.165415	−0.0827076	+	0.996574i	$0.526357\pi$
$282$	4.06498	0.242066
$283$	−9.23451	−0.548935	−0.274467	−	0.961596i	$-0.588502\pi$
$283$	−9.23451	−0.548935	−0.274467	+	0.961596i	$0.588502\pi$
$284$	−7.57822	−0.449685
$285$	−0.948410	−0.0561790
$286$	1.40268	0.0829421
$287$	10.0325	0.592199
$288$	−2.40268	−0.141579
$289$	6.09146	0.358321
$290$	9.85695	0.578820
$291$	1.81708	0.106519
$292$	−10.0650	−0.589009
$293$	17.7390	1.03632	0.518162	−	0.855283i	$-0.326616\pi$
$293$	17.7390	1.03632	0.518162	+	0.855283i	$0.326616\pi$
$294$	3.53835	0.206361
$295$	−17.8171	−1.03735
$296$	2.35109	0.136654
$297$	4.17554	0.242290
$298$	18.3511	1.06305
$299$	−9.54573	−0.552044
$300$	−2.70050	−0.155914
$301$	−31.0459	−1.78946
$302$	−7.64891	−0.440145
$303$	10.8703	0.624485
$304$	−1.00000	−0.0573539
$305$	2.45427	0.140531
$306$	11.5457	0.660026
$307$	20.9735	1.19702	0.598511	−	0.801115i	$-0.295759\pi$
$307$	20.9735	1.19702	0.598511	+	0.801115i	$0.295759\pi$
$308$	−3.40268	−0.193886
$309$	−12.0399	−0.684924
$310$	−6.07236	−0.344887
$311$	−3.36281	−0.190687	−0.0953436	−	0.995444i	$-0.530395\pi$
$311$	−3.36281	−0.190687	−0.0953436	+	0.995444i	$0.530395\pi$
$312$	−1.08408	−0.0613741
$313$	−4.94103	−0.279284	−0.139642	−	0.990202i	$-0.544595\pi$
$313$	−4.94103	−0.279284	−0.139642	+	0.990202i	$0.544595\pi$
$314$	8.38358	0.473113
$315$	−10.0325	−0.565267
$316$	−3.71390	−0.208923
$317$	14.2861	0.802388	0.401194	−	0.915993i	$-0.368595\pi$
$317$	14.2861	0.802388	0.401194	+	0.915993i	$0.368595\pi$
$318$	−3.44255	−0.193049
$319$	−8.03249	−0.449733
$320$	1.22713	0.0685989
$321$	−3.36281	−0.187694
$322$	23.1564	1.29046
$323$	4.80536	0.267377
$324$	3.98090	0.221161
$325$	4.90116	0.271867
$326$	−14.2479	−0.789119
$327$	4.63719	0.256437
$328$	2.94841	0.162799
$329$	17.8968	0.986684
$330$	−0.948410	−0.0522082
$331$	−31.9293	−1.75499	−0.877497	−	0.479582i	$-0.840788\pi$
$331$	−31.9293	−1.75499	−0.877497	+	0.479582i	$0.840788\pi$
$332$	10.6697	0.585575
$333$	−5.64891	−0.309558
$334$	−10.8703	−0.594799
$335$	4.17554	0.228134
$336$	2.62981	0.143468
$337$	−8.84523	−0.481830	−0.240915	−	0.970546i	$-0.577448\pi$
$337$	−8.84523	−0.481830	−0.240915	+	0.970546i	$0.577448\pi$
$338$	−11.0325	−0.600088
$339$	−15.4278	−0.837923
$340$	−5.89682	−0.319800
$341$	4.94841	0.267971
$342$	2.40268	0.129922
$343$	−8.24053	−0.444947
$344$	−9.12395	−0.491931
$345$	6.45427	0.347486
$346$	3.39530	0.182532
$347$	5.27439	0.283144	0.141572	−	0.989928i	$-0.454784\pi$
$347$	5.27439	0.283144	0.141572	+	0.989928i	$0.454784\pi$
$348$	6.20804	0.332786
$349$	−10.9085	−0.583921	−0.291960	−	0.956430i	$-0.594308\pi$
$349$	−10.9085	−0.583921	−0.291960	+	0.956430i	$0.594308\pi$
$350$	−11.8894	−0.635517
$351$	5.85695	0.312621
$352$	−1.00000	−0.0533002
$353$	−22.1300	−1.17786	−0.588930	−	0.808184i	$-0.700451\pi$
$353$	−22.1300	−1.17786	−0.588930	+	0.808184i	$0.700451\pi$
$354$	−11.2214	−0.596412
$355$	−9.29950	−0.493566
$356$	14.9883	0.794377
$357$	−12.6372	−0.668831
$358$	−0.311217	−0.0164483
$359$	−16.3910	−0.865082	−0.432541	−	0.901614i	$-0.642383\pi$
$359$	−16.3910	−0.865082	−0.432541	+	0.901614i	$0.642383\pi$
$360$	−2.94841	−0.155395
$361$	1.00000	0.0526316
$362$	−15.3246	−0.805444
$363$	0.772866	0.0405649
$364$	−4.77287	−0.250166
$365$	−12.3511	−0.646486
$366$	1.54573	0.0807967
$367$	−32.3511	−1.68871	−0.844357	−	0.535782i	$-0.820017\pi$
$367$	−32.3511	−1.68871	−0.844357	+	0.535782i	$0.820017\pi$
$368$	6.80536	0.354754
$369$	−7.08408	−0.368783
$370$	2.88510	0.149989
$371$	−15.1564	−0.786883
$372$	−3.82446	−0.198289
$373$	17.5782	0.910166	0.455083	−	0.890449i	$-0.349610\pi$
$373$	17.5782	0.910166	0.455083	+	0.890449i	$0.349610\pi$
$374$	4.80536	0.248479
$375$	−8.05593	−0.416006
$376$	5.25963	0.271245
$377$	−11.2670	−0.580280
$378$	−14.2080	−0.730783
$379$	8.93365	0.458891	0.229445	−	0.973322i	$-0.426309\pi$
$379$	8.93365	0.458891	0.229445	+	0.973322i	$0.426309\pi$
$380$	−1.22713	−0.0629507
$381$	9.05327	0.463813
$382$	26.8703	1.37481
$383$	−16.0399	−0.819599	−0.409800	−	0.912176i	$-0.634401\pi$
$383$	−16.0399	−0.819599	−0.409800	+	0.912176i	$0.634401\pi$
$384$	0.772866	0.0394401
$385$	−4.17554	−0.212805
$386$	18.5665	0.945010
$387$	21.9219	1.11435
$388$	2.35109	0.119358
$389$	17.2271	0.873450	0.436725	−	0.899595i	$-0.356138\pi$
$389$	17.2271	0.873450	0.436725	+	0.899595i	$0.356138\pi$
$390$	−1.33031	−0.0673631
$391$	−32.7022	−1.65382
$392$	4.57822	0.231235
$393$	1.43517	0.0723948
$394$	14.9085	0.751081
$395$	−4.55745	−0.229310
$396$	2.40268	0.120739
$397$	15.0784	0.756762	0.378381	−	0.925650i	$-0.376481\pi$
$397$	15.0784	0.756762	0.378381	+	0.925650i	$0.376481\pi$
$398$	15.1564	0.759724
$399$	−2.62981	−0.131655
$400$	−3.49414	−0.174707
$401$	−13.1564	−0.657002	−0.328501	−	0.944504i	$-0.606543\pi$
$401$	−13.1564	−0.657002	−0.328501	+	0.944504i	$0.606543\pi$
$402$	2.62981	0.131163
$403$	6.94103	0.345757
$404$	14.0650	0.699759
$405$	4.88510	0.242743
$406$	27.3320	1.35646
$407$	−2.35109	−0.116539
$408$	−3.71390	−0.183865
$409$	−16.3836	−0.810116	−0.405058	−	0.914291i	$-0.632749\pi$
$409$	−16.3836	−0.810116	−0.405058	+	0.914291i	$0.632749\pi$
$410$	3.61810	0.178685
$411$	8.19030	0.403998
$412$	−15.5782	−0.767484
$413$	−49.4044	−2.43103
$414$	−16.3511	−0.803612
$415$	13.0931	0.642717
$416$	−1.40268	−0.0687720
$417$	11.3879	0.557669
$418$	1.00000	0.0489116
$419$	1.12966	0.0551874	0.0275937	−	0.999619i	$-0.491216\pi$
$419$	1.12966	0.0551874	0.0275937	+	0.999619i	$0.491216\pi$
$420$	3.22713	0.157468
$421$	−2.63719	−0.128529	−0.0642645	−	0.997933i	$-0.520470\pi$
$421$	−2.63719	−0.128529	−0.0642645	+	0.997933i	$0.520470\pi$
$422$	11.1564	0.543087
$423$	−12.6372	−0.614441
$424$	−4.45427	−0.216318
$425$	16.7906	0.814464
$426$	−5.85695	−0.283770
$427$	6.80536	0.329334
$428$	−4.35109	−0.210318
$429$	1.08408	0.0523400
$430$	−11.1963	−0.539934
$431$	−28.4958	−1.37260	−0.686298	−	0.727321i	$-0.740765\pi$
$431$	−28.4958	−1.37260	−0.686298	+	0.727321i	$0.740765\pi$
$432$	−4.17554	−0.200896
$433$	−37.5075	−1.80250	−0.901249	−	0.433302i	$-0.857348\pi$
$433$	−37.5075	−1.80250	−0.901249	+	0.433302i	$0.857348\pi$
$434$	−16.8378	−0.808243
$435$	7.61810	0.365260
$436$	6.00000	0.287348
$437$	−6.80536	−0.325544
$438$	−7.77888	−0.371689
$439$	−32.1300	−1.53348	−0.766740	−	0.641958i	$-0.778122\pi$
$439$	−32.1300	−1.53348	−0.766740	+	0.641958i	$0.778122\pi$
$440$	−1.22713	−0.0585013
$441$	−11.0000	−0.523810
$442$	6.74037	0.320607
$443$	33.0385	1.56971	0.784853	−	0.619681i	$-0.212738\pi$
$443$	33.0385	1.56971	0.784853	+	0.619681i	$0.212738\pi$
$444$	1.81708	0.0862346
$445$	18.3926	0.871895
$446$	1.61072	0.0762696
$447$	14.1829	0.670829
$448$	3.40268	0.160761
$449$	7.88206	0.371977	0.185989	−	0.982552i	$-0.440451\pi$
$449$	7.88206	0.371977	0.185989	+	0.982552i	$0.440451\pi$
$450$	8.39530	0.395758
$451$	−2.94841	−0.138835
$452$	−19.9618	−0.938924
$453$	−5.91158	−0.277750
$454$	−16.7672	−0.786922
$455$	−5.85695	−0.274578
$456$	−0.772866	−0.0361927
$457$	24.9353	1.16643	0.583213	−	0.812320i	$-0.301795\pi$
$457$	24.9353	1.16643	0.583213	+	0.812320i	$0.301795\pi$
$458$	−8.55913	−0.399942
$459$	20.0650	0.936553
$460$	8.35109	0.389372
$461$	−26.6874	−1.24296	−0.621478	−	0.783431i	$-0.713468\pi$
$461$	−26.6874	−1.24296	−0.621478	+	0.783431i	$0.713468\pi$
$462$	−2.62981	−0.122350
$463$	−32.7022	−1.51980	−0.759900	−	0.650041i	$-0.774752\pi$
$463$	−32.7022	−1.51980	−0.759900	+	0.650041i	$0.774752\pi$
$464$	8.03249	0.372899
$465$	−4.69312	−0.217638
$466$	5.44255	0.252121
$467$	11.3776	0.526491	0.263246	−	0.964729i	$-0.415207\pi$
$467$	11.3776	0.526491	0.263246	+	0.964729i	$0.415207\pi$
$468$	3.37019	0.155787
$469$	11.5782	0.534633
$470$	6.45427	0.297713
$471$	6.47938	0.298554
$472$	−14.5193	−0.668303
$473$	9.12395	0.419520
$474$	−2.87034	−0.131839
$475$	3.49414	0.160322
$476$	−16.3511	−0.749451
$477$	10.7022	0.490019
$478$	11.2921	0.516489
$479$	−12.5973	−0.575586	−0.287793	−	0.957693i	$-0.592922\pi$
$479$	−12.5973	−0.575586	−0.287793	+	0.957693i	$0.592922\pi$
$480$	0.948410	0.0432888
$481$	−3.29782	−0.150368
$482$	11.4101	0.519714
$483$	17.8968	0.814334
$484$	1.00000	0.0454545
$485$	2.88510	0.131006
$486$	15.6033	0.707782
$487$	9.69616	0.439375	0.219688	−	0.975570i	$-0.429496\pi$
$487$	9.69616	0.439375	0.219688	+	0.975570i	$0.429496\pi$
$488$	2.00000	0.0905357
$489$	−11.0117	−0.497967
$490$	5.61810	0.253800
$491$	−11.7538	−0.530440	−0.265220	−	0.964188i	$-0.585445\pi$
$491$	−11.7538	−0.530440	−0.265220	+	0.964188i	$0.585445\pi$
$492$	2.27872	0.102733
$493$	−38.5990	−1.73841
$494$	1.40268	0.0631095
$495$	2.94841	0.132521
$496$	−4.94841	−0.222190
$497$	−25.7863	−1.15667
$498$	8.24623	0.369523
$499$	26.6640	1.19364	0.596822	−	0.802374i	$-0.296430\pi$
$499$	26.6640	1.19364	0.596822	+	0.802374i	$0.296430\pi$
$500$	−10.4235	−0.466151
$501$	−8.40131	−0.375343
$502$	−12.0000	−0.535586
$503$	11.1622	0.497696	0.248848	−	0.968543i	$-0.419948\pi$
$503$	11.1622	0.497696	0.248848	+	0.968543i	$0.419948\pi$
$504$	−8.17554	−0.364168
$505$	17.2596	0.768043
$506$	−6.80536	−0.302535
$507$	−8.52663	−0.378681
$508$	11.7139	0.519720
$509$	−35.9618	−1.59398	−0.796989	−	0.603993i	$-0.793575\pi$
$509$	−35.9618	−1.59398	−0.796989	+	0.603993i	$0.793575\pi$
$510$	−4.55745	−0.201807
$511$	−34.2479	−1.51504
$512$	1.00000	0.0441942
$513$	4.17554	0.184355
$514$	24.2331	1.06888
$515$	−19.1166	−0.842377
$516$	−7.05159	−0.310429
$517$	−5.25963	−0.231318
$518$	8.00000	0.351500
$519$	2.62411	0.115186
$520$	−1.72128	−0.0754829
$521$	10.1300	0.443802	0.221901	−	0.975069i	$-0.428774\pi$
$521$	10.1300	0.443802	0.221901	+	0.975069i	$0.428774\pi$
$522$	−19.2995	−0.844716
$523$	−7.93502	−0.346974	−0.173487	−	0.984836i	$-0.555503\pi$
$523$	−7.93502	−0.346974	−0.173487	+	0.984836i	$0.555503\pi$
$524$	1.85695	0.0811211
$525$	−9.18894	−0.401038
$526$	20.5665	0.896742
$527$	23.7789	1.03582
$528$	−0.772866	−0.0336347
$529$	23.3129	1.01360
$530$	−5.46599	−0.237427
$531$	34.8851	1.51388
$532$	−3.40268	−0.147525
$533$	−4.13567	−0.179136
$534$	11.5839	0.501286
$535$	−5.33937	−0.230841
$536$	3.40268	0.146973
$537$	−0.240529	−0.0103796
$538$	−1.36281	−0.0587548
$539$	−4.57822	−0.197198
$540$	−5.12395	−0.220500
$541$	10.8436	0.466201	0.233100	−	0.972453i	$-0.425113\pi$
$541$	10.8436	0.466201	0.233100	+	0.972453i	$0.425113\pi$
$542$	−23.1963	−0.996367
$543$	−11.8439	−0.508269
$544$	−4.80536	−0.206028
$545$	7.36281	0.315388
$546$	−3.68878	−0.157865
$547$	−10.3129	−0.440947	−0.220474	−	0.975393i	$-0.570760\pi$
$547$	−10.3129	−0.440947	−0.220474	+	0.975393i	$0.570760\pi$
$548$	10.5973	0.452695
$549$	−4.80536	−0.205088
$550$	3.49414	0.148991
$551$	−8.03249	−0.342196
$552$	5.25963	0.223865
$553$	−12.6372	−0.537388
$554$	31.0385	1.31870
$555$	2.22980	0.0946496
$556$	14.7347	0.624889
$557$	−14.4811	−0.613582	−0.306791	−	0.951777i	$-0.599255\pi$
$557$	−14.4811	−0.613582	−0.306791	+	0.951777i	$0.599255\pi$
$558$	11.8894	0.503320
$559$	12.7980	0.541297
$560$	4.17554	0.176449
$561$	3.71390	0.156801
$562$	−2.77287	−0.116966
$563$	10.5340	0.443956	0.221978	−	0.975052i	$-0.428749\pi$
$563$	10.5340	0.443956	0.221978	+	0.975052i	$0.428749\pi$
$564$	4.06498	0.171167
$565$	−24.4958	−1.03055
$566$	−9.23451	−0.388156
$567$	13.5457	0.568867
$568$	−7.57822	−0.317975
$569$	10.3762	0.434993	0.217496	−	0.976061i	$-0.430211\pi$
$569$	10.3762	0.434993	0.217496	+	0.976061i	$0.430211\pi$
$570$	−0.948410	−0.0397245
$571$	33.2847	1.39292	0.696461	−	0.717594i	$-0.254757\pi$
$571$	33.2847	1.39292	0.696461	+	0.717594i	$0.254757\pi$
$572$	1.40268	0.0586489
$573$	20.7672	0.867561
$574$	10.0325	0.418748
$575$	−23.7789	−0.991648
$576$	−2.40268	−0.100112
$577$	−2.93365	−0.122129	−0.0610647	−	0.998134i	$-0.519450\pi$
$577$	−2.93365	−0.122129	−0.0610647	+	0.998134i	$0.519450\pi$
$578$	6.09146	0.253371
$579$	14.3494	0.596341
$580$	9.85695	0.409287
$581$	36.3055	1.50621
$582$	1.81708	0.0753202
$583$	4.45427	0.184477
$584$	−10.0650	−0.416492
$585$	4.13567	0.170989
$586$	17.7390	0.732792
$587$	8.90854	0.367695	0.183847	−	0.982955i	$-0.441145\pi$
$587$	8.90854	0.367695	0.183847	+	0.982955i	$0.441145\pi$
$588$	3.53835	0.145919
$589$	4.94841	0.203896
$590$	−17.8171	−0.733517
$591$	11.5223	0.473964
$592$	2.35109	0.0966292
$593$	16.3129	0.669890	0.334945	−	0.942238i	$-0.391282\pi$
$593$	16.3129	0.669890	0.334945	+	0.942238i	$0.391282\pi$
$594$	4.17554	0.171325
$595$	−20.0650	−0.822584
$596$	18.3511	0.751690
$597$	11.7139	0.479418
$598$	−9.54573	−0.390354
$599$	40.9826	1.67450	0.837251	−	0.546818i	$-0.184161\pi$
$599$	40.9826	1.67450	0.837251	+	0.546818i	$0.184161\pi$
$600$	−2.70050	−0.110248
$601$	0.349413	0.0142528	0.00712642	−	0.999975i	$-0.497732\pi$
$601$	0.349413	0.0142528	0.00712642	+	0.999975i	$0.497732\pi$
$602$	−31.0459	−1.26534
$603$	−8.17554	−0.332934
$604$	−7.64891	−0.311230
$605$	1.22713	0.0498901
$606$	10.8703	0.441577
$607$	5.56049	0.225693	0.112847	−	0.993612i	$-0.464003\pi$
$607$	5.56049	0.225693	0.112847	+	0.993612i	$0.464003\pi$
$608$	−1.00000	−0.0405554
$609$	21.1240	0.855986
$610$	2.45427	0.0993704
$611$	−7.37757	−0.298464
$612$	11.5457	0.466709
$613$	−19.7407	−0.797319	−0.398659	−	0.917099i	$-0.630524\pi$
$613$	−19.7407	−0.797319	−0.398659	+	0.917099i	$0.630524\pi$
$614$	20.9735	0.846422
$615$	2.79630	0.112758
$616$	−3.40268	−0.137098
$617$	10.5973	0.426632	0.213316	−	0.976983i	$-0.431574\pi$
$617$	10.5973	0.426632	0.213316	+	0.976983i	$0.431574\pi$
$618$	−12.0399	−0.484315
$619$	4.22112	0.169661	0.0848306	−	0.996395i	$-0.472965\pi$
$619$	4.22112	0.169661	0.0848306	+	0.996395i	$0.472965\pi$
$620$	−6.07236	−0.243872
$621$	−28.4161	−1.14030
$622$	−3.36281	−0.134836
$623$	51.0003	2.04328
$624$	−1.08408	−0.0433980
$625$	4.67973	0.187189
$626$	−4.94103	−0.197483
$627$	0.772866	0.0308653
$628$	8.38358	0.334541
$629$	−11.2978	−0.450474
$630$	−10.0325	−0.399704
$631$	13.6905	0.545009	0.272504	−	0.962155i	$-0.412148\pi$
$631$	13.6905	0.545009	0.272504	+	0.962155i	$0.412148\pi$
$632$	−3.71390	−0.147731
$633$	8.62243	0.342711
$634$	14.2861	0.567374
$635$	14.3745	0.570436
$636$	−3.44255	−0.136506
$637$	−6.42178	−0.254440
$638$	−8.03249	−0.318009
$639$	18.2080	0.720299
$640$	1.22713	0.0485067
$641$	−42.9883	−1.69794	−0.848968	−	0.528445i	$-0.822775\pi$
$641$	−42.9883	−1.69794	−0.848968	+	0.528445i	$0.822775\pi$
$642$	−3.36281	−0.132719
$643$	16.3658	0.645406	0.322703	−	0.946500i	$-0.395408\pi$
$643$	16.3658	0.645406	0.322703	+	0.946500i	$0.395408\pi$
$644$	23.1564	0.912492
$645$	−8.65325	−0.340721
$646$	4.80536	0.189064
$647$	39.9886	1.57211	0.786057	−	0.618154i	$-0.212119\pi$
$647$	39.9886	1.57211	0.786057	+	0.618154i	$0.212119\pi$
$648$	3.98090	0.156385
$649$	14.5193	0.569931
$650$	4.90116	0.192239
$651$	−13.0134	−0.510035
$652$	−14.2479	−0.557991
$653$	−37.5976	−1.47131	−0.735655	−	0.677357i	$-0.763125\pi$
$653$	−37.5976	−1.47131	−0.735655	+	0.677357i	$0.763125\pi$
$654$	4.63719	0.181329
$655$	2.27872	0.0890371
$656$	2.94841	0.115116
$657$	24.1829	0.943466
$658$	17.8968	0.697691
$659$	−11.5075	−0.448270	−0.224135	−	0.974558i	$-0.571956\pi$
$659$	−11.5075	−0.448270	−0.224135	+	0.974558i	$0.571956\pi$
$660$	−0.948410	−0.0369168
$661$	−29.2362	−1.13716	−0.568578	−	0.822629i	$-0.692506\pi$
$661$	−29.2362	−1.13716	−0.568578	+	0.822629i	$0.692506\pi$
$662$	−31.9293	−1.24097
$663$	5.20940	0.202316
$664$	10.6697	0.414064
$665$	−4.17554	−0.161921
$666$	−5.64891	−0.218891
$667$	54.6640	2.11660
$668$	−10.8703	−0.420586
$669$	1.24487	0.0481293
$670$	4.17554	0.161315
$671$	−2.00000	−0.0772091
$672$	2.62981	0.101447
$673$	−47.0784	−1.81474	−0.907369	−	0.420335i	$-0.861913\pi$
$673$	−47.0784	−1.81474	−0.907369	+	0.420335i	$0.861913\pi$
$674$	−8.84523	−0.340706
$675$	14.5899	0.561567
$676$	−11.0325	−0.424327
$677$	43.2539	1.66238	0.831192	−	0.555986i	$-0.187659\pi$
$677$	43.2539	1.66238	0.831192	+	0.555986i	$0.187659\pi$
$678$	−15.4278	−0.592501
$679$	8.00000	0.307012
$680$	−5.89682	−0.226133
$681$	−12.9588	−0.496581
$682$	4.94841	0.189484
$683$	34.8556	1.33371	0.666856	−	0.745187i	$-0.267640\pi$
$683$	34.8556	1.33371	0.666856	+	0.745187i	$0.267640\pi$
$684$	2.40268	0.0918687
$685$	13.0043	0.496870
$686$	−8.24053	−0.314625
$687$	−6.61505	−0.252380
$688$	−9.12395	−0.347847
$689$	6.24791	0.238026
$690$	6.45427	0.245710
$691$	28.4811	1.08347	0.541735	−	0.840549i	$-0.317768\pi$
$691$	28.4811	1.08347	0.541735	+	0.840549i	$0.317768\pi$
$692$	3.39530	0.129070
$693$	8.17554	0.310563
$694$	5.27439	0.200213
$695$	18.0814	0.685867
$696$	6.20804	0.235315
$697$	−14.1682	−0.536657
$698$	−10.9085	−0.412894
$699$	4.20636	0.159099
$700$	−11.8894	−0.449379
$701$	12.8703	0.486106	0.243053	−	0.970013i	$-0.421851\pi$
$701$	12.8703	0.486106	0.243053	+	0.970013i	$0.421851\pi$
$702$	5.85695	0.221056
$703$	−2.35109	−0.0886730
$704$	−1.00000	−0.0376889
$705$	4.98828	0.187870
$706$	−22.1300	−0.832872
$707$	47.8586	1.79991
$708$	−11.2214	−0.421727
$709$	1.41744	0.0532330	0.0266165	−	0.999646i	$-0.491527\pi$
$709$	1.41744	0.0532330	0.0266165	+	0.999646i	$0.491527\pi$
$710$	−9.29950	−0.349004
$711$	8.92330	0.334650
$712$	14.9883	0.561710
$713$	−33.6757	−1.26116
$714$	−12.6372	−0.472935
$715$	1.72128	0.0643721
$716$	−0.311217	−0.0116307
$717$	8.72729	0.325927
$718$	−16.3910	−0.611705
$719$	−31.1564	−1.16194	−0.580970	−	0.813925i	$-0.697327\pi$
$719$	−31.1564	−1.16194	−0.580970	+	0.813925i	$0.697327\pi$
$720$	−2.94841	−0.109881
$721$	−53.0077	−1.97411
$722$	1.00000	0.0372161
$723$	8.81844	0.327961
$724$	−15.3246	−0.569535
$725$	−28.0667	−1.04237
$726$	0.772866	0.0286837
$727$	−8.57221	−0.317926	−0.158963	−	0.987285i	$-0.550815\pi$
$727$	−8.57221	−0.317926	−0.158963	+	0.987285i	$0.550815\pi$
$728$	−4.77287	−0.176894
$729$	0.116574	0.00431757
$730$	−12.3511	−0.457134
$731$	43.8439	1.62162
$732$	1.54573	0.0571319
$733$	44.6787	1.65025	0.825123	−	0.564952i	$-0.191105\pi$
$733$	44.6787	1.65025	0.825123	+	0.564952i	$0.191105\pi$
$734$	−32.3511	−1.19410
$735$	4.34203	0.160158
$736$	6.80536	0.250849
$737$	−3.40268	−0.125339
$738$	−7.08408	−0.260769
$739$	29.4898	1.08480	0.542400	−	0.840120i	$-0.317516\pi$
$739$	29.4898	1.08480	0.542400	+	0.840120i	$0.317516\pi$
$740$	2.88510	0.106058
$741$	1.08408	0.0398248
$742$	−15.1564	−0.556411
$743$	−13.2596	−0.486449	−0.243224	−	0.969970i	$-0.578205\pi$
$743$	−13.2596	−0.486449	−0.243224	+	0.969970i	$0.578205\pi$
$744$	−3.82446	−0.140211
$745$	22.5193	0.825042
$746$	17.5782	0.643584
$747$	−25.6358	−0.937966
$748$	4.80536	0.175701
$749$	−14.8054	−0.540976
$750$	−8.05593	−0.294161
$751$	−10.1829	−0.371580	−0.185790	−	0.982589i	$-0.559484\pi$
$751$	−10.1829	−0.371580	−0.185790	+	0.982589i	$0.559484\pi$
$752$	5.25963	0.191799
$753$	−9.27439	−0.337977
$754$	−11.2670	−0.410320
$755$	−9.38624	−0.341600
$756$	−14.2080	−0.516741
$757$	40.1122	1.45790	0.728952	−	0.684565i	$-0.240008\pi$
$757$	40.1122	1.45790	0.728952	+	0.684565i	$0.240008\pi$
$758$	8.93365	0.324485
$759$	−5.25963	−0.190912
$760$	−1.22713	−0.0445128
$761$	−38.7819	−1.40584	−0.702922	−	0.711267i	$-0.748122\pi$
$761$	−38.7819	−1.40584	−0.702922	+	0.711267i	$0.748122\pi$
$762$	9.05327	0.327965
$763$	20.4161	0.739111
$764$	26.8703	0.972135
$765$	14.1682	0.512251
$766$	−16.0399	−0.579544
$767$	20.3658	0.735368
$768$	0.772866	0.0278884
$769$	8.15340	0.294019	0.147010	−	0.989135i	$-0.453035\pi$
$769$	8.15340	0.294019	0.147010	+	0.989135i	$0.453035\pi$
$770$	−4.17554	−0.150476
$771$	18.7290	0.674507
$772$	18.5665	0.668223
$773$	−24.3893	−0.877222	−0.438611	−	0.898677i	$-0.644529\pi$
$773$	−24.3893	−0.877222	−0.438611	+	0.898677i	$0.644529\pi$
$774$	21.9219	0.787968
$775$	17.2904	0.621091
$776$	2.35109	0.0843992
$777$	6.18292	0.221811
$778$	17.2271	0.617623
$779$	−2.94841	−0.105638
$780$	−1.33031	−0.0476329
$781$	7.57822	0.271170
$782$	−32.7022	−1.16943
$783$	−33.5400	−1.19862
$784$	4.57822	0.163508
$785$	10.2878	0.367187
$786$	1.43517	0.0511909
$787$	−46.3779	−1.65319	−0.826596	−	0.562795i	$-0.809726\pi$
$787$	−46.3779	−1.65319	−0.826596	+	0.562795i	$0.809726\pi$
$788$	14.9085	0.531095
$789$	15.8951	0.565882
$790$	−4.55745	−0.162147
$791$	−67.9236	−2.41509
$792$	2.40268	0.0853755
$793$	−2.80536	−0.0996212
$794$	15.0784	0.535112
$795$	−4.22447	−0.149827
$796$	15.1564	0.537206
$797$	3.89682	0.138032	0.0690162	−	0.997616i	$-0.478014\pi$
$797$	3.89682	0.138032	0.0690162	+	0.997616i	$0.478014\pi$
$798$	−2.62981	−0.0930944
$799$	−25.2744	−0.894144
$800$	−3.49414	−0.123537
$801$	−36.0120	−1.27242
$802$	−13.1564	−0.464570
$803$	10.0650	0.355186
$804$	2.62981	0.0927464
$805$	28.4161	1.00153
$806$	6.94103	0.244487
$807$	−1.05327	−0.0370767
$808$	14.0650	0.494804
$809$	−43.3896	−1.52550	−0.762748	−	0.646695i	$-0.776151\pi$
$809$	−43.3896	−1.52550	−0.762748	+	0.646695i	$0.776151\pi$
$810$	4.88510	0.171645
$811$	−12.9233	−0.453798	−0.226899	−	0.973918i	$-0.572859\pi$
$811$	−12.9233	−0.453798	−0.226899	+	0.973918i	$0.572859\pi$
$812$	27.3320	0.959165
$813$	−17.9276	−0.628750
$814$	−2.35109	−0.0824056
$815$	−17.4841	−0.612441
$816$	−3.71390	−0.130012
$817$	9.12395	0.319207
$818$	−16.3836	−0.572838
$819$	11.4677	0.400713
$820$	3.61810	0.126349
$821$	−36.5842	−1.27680	−0.638399	−	0.769705i	$-0.720403\pi$
$821$	−36.5842	−1.27680	−0.638399	+	0.769705i	$0.720403\pi$
$822$	8.19030	0.285670
$823$	−21.6107	−0.753302	−0.376651	−	0.926355i	$-0.622924\pi$
$823$	−21.6107	−0.753302	−0.376651	+	0.926355i	$0.622924\pi$
$824$	−15.5782	−0.542693
$825$	2.70050	0.0940194
$826$	−49.4044	−1.71900
$827$	−34.8851	−1.21307	−0.606537	−	0.795055i	$-0.707442\pi$
$827$	−34.8851	−1.21307	−0.606537	+	0.795055i	$0.707442\pi$
$828$	−16.3511	−0.568240
$829$	31.7407	1.10240	0.551200	−	0.834373i	$-0.314170\pi$
$829$	31.7407	1.10240	0.551200	+	0.834373i	$0.314170\pi$
$830$	13.0931	0.454469
$831$	23.9886	0.832155
$832$	−1.40268	−0.0486291
$833$	−22.0000	−0.762255
$834$	11.3879	0.394331
$835$	−13.3394	−0.461628
$836$	1.00000	0.0345857
$837$	20.6623	0.714194
$838$	1.12966	0.0390234
$839$	13.4603	0.464701	0.232350	−	0.972632i	$-0.425358\pi$
$839$	13.4603	0.464701	0.232350	+	0.972632i	$0.425358\pi$
$840$	3.22713	0.111347
$841$	35.5209	1.22486
$842$	−2.63719	−0.0908837
$843$	−2.14305	−0.0738106
$844$	11.1564	0.384021
$845$	−13.5384	−0.465733
$846$	−12.6372	−0.434476
$847$	3.40268	0.116917
$848$	−4.45427	−0.152960
$849$	−7.13704	−0.244943
$850$	16.7906	0.575913
$851$	16.0000	0.548473
$852$	−5.85695	−0.200656
$853$	36.2981	1.24282	0.621412	−	0.783484i	$-0.286559\pi$
$853$	36.2981	1.24282	0.621412	+	0.783484i	$0.286559\pi$
$854$	6.80536	0.232875
$855$	2.94841	0.100833
$856$	−4.35109	−0.148717
$857$	−43.8569	−1.49812	−0.749062	−	0.662499i	$-0.769496\pi$
$857$	−43.8569	−1.49812	−0.749062	+	0.662499i	$0.769496\pi$
$858$	1.08408	0.0370100
$859$	−41.1980	−1.40566	−0.702829	−	0.711359i	$-0.748080\pi$
$859$	−41.1980	−1.40566	−0.702829	+	0.711359i	$0.748080\pi$
$860$	−11.1963	−0.381791
$861$	7.75377	0.264248
$862$	−28.4958	−0.970571
$863$	−24.8720	−0.846653	−0.423327	−	0.905977i	$-0.639138\pi$
$863$	−24.8720	−0.846653	−0.423327	+	0.905977i	$0.639138\pi$
$864$	−4.17554	−0.142055
$865$	4.16649	0.141665
$866$	−37.5075	−1.27456
$867$	4.70788	0.159888
$868$	−16.8378	−0.571514
$869$	3.71390	0.125985
$870$	7.61810	0.258278
$871$	−4.77287	−0.161722
$872$	6.00000	0.203186
$873$	−5.64891	−0.191187
$874$	−6.80536	−0.230195
$875$	−35.4677	−1.19903
$876$	−7.77888	−0.262824
$877$	39.8911	1.34703	0.673514	−	0.739175i	$-0.264784\pi$
$877$	39.8911	1.34703	0.673514	+	0.739175i	$0.264784\pi$
$878$	−32.1300	−1.08433
$879$	13.7099	0.462422
$880$	−1.22713	−0.0413667
$881$	−2.88343	−0.0971451	−0.0485725	−	0.998820i	$-0.515467\pi$
$881$	−2.88343	−0.0971451	−0.0485725	+	0.998820i	$0.515467\pi$
$882$	−11.0000	−0.370389
$883$	−4.27134	−0.143742	−0.0718711	−	0.997414i	$-0.522897\pi$
$883$	−4.27134	−0.143742	−0.0718711	+	0.997414i	$0.522897\pi$
$884$	6.74037	0.226703
$885$	−13.7702	−0.462880
$886$	33.0385	1.10995
$887$	22.3129	0.749194	0.374597	−	0.927188i	$-0.377781\pi$
$887$	22.3129	0.749194	0.374597	+	0.927188i	$0.377781\pi$
$888$	1.81708	0.0609771
$889$	39.8586	1.33682
$890$	18.3926	0.616523
$891$	−3.98090	−0.133365
$892$	1.61072	0.0539307
$893$	−5.25963	−0.176007
$894$	14.1829	0.474348
$895$	−0.381905	−0.0127657
$896$	3.40268	0.113676
$897$	−7.37757	−0.246330
$898$	7.88206	0.263028
$899$	−39.7481	−1.32567
$900$	8.39530	0.279843
$901$	21.4044	0.713082
$902$	−2.94841	−0.0981713
$903$	−23.9943	−0.798480
$904$	−19.9618	−0.663920
$905$	−18.8054	−0.625111
$906$	−5.91158	−0.196399
$907$	−35.5578	−1.18068	−0.590338	−	0.807156i	$-0.701006\pi$
$907$	−35.5578	−1.18068	−0.590338	+	0.807156i	$0.701006\pi$
$908$	−16.7672	−0.556438
$909$	−33.7936	−1.12086
$910$	−5.85695	−0.194156
$911$	−49.1980	−1.63000	−0.815001	−	0.579459i	$-0.803264\pi$
$911$	−49.1980	−1.63000	−0.815001	+	0.579459i	$0.803264\pi$
$912$	−0.772866	−0.0255921
$913$	−10.6697	−0.353115
$914$	24.9353	0.824787
$915$	1.89682	0.0627069
$916$	−8.55913	−0.282801
$917$	6.31860	0.208658
$918$	20.0650	0.662243
$919$	−8.47200	−0.279466	−0.139733	−	0.990189i	$-0.544624\pi$
$919$	−8.47200	−0.279466	−0.139733	+	0.990189i	$0.544624\pi$
$920$	8.35109	0.275327
$921$	16.2097	0.534128
$922$	−26.6874	−0.878903
$923$	10.6298	0.349885
$924$	−2.62981	−0.0865145
$925$	−8.21504	−0.270109
$926$	−32.7022	−1.07466
$927$	37.4295	1.22934
$928$	8.03249	0.263679
$929$	−16.0017	−0.524998	−0.262499	−	0.964932i	$-0.584547\pi$
$929$	−16.0017	−0.524998	−0.262499	+	0.964932i	$0.584547\pi$
$930$	−4.69312	−0.153894
$931$	−4.57822	−0.150045
$932$	5.44255	0.178277
$933$	−2.59900	−0.0850874
$934$	11.3776	0.372285
$935$	5.89682	0.192847
$936$	3.37019	0.110158
$937$	−15.9766	−0.521932	−0.260966	−	0.965348i	$-0.584041\pi$
$937$	−15.9766	−0.521932	−0.260966	+	0.965348i	$0.584041\pi$
$938$	11.5782	0.378042
$939$	−3.81875	−0.124620
$940$	6.45427	0.210515
$941$	−21.0915	−0.687562	−0.343781	−	0.939050i	$-0.611708\pi$
$941$	−21.0915	−0.687562	−0.343781	+	0.939050i	$0.611708\pi$
$942$	6.47938	0.211110
$943$	20.0650	0.653406
$944$	−14.5193	−0.472561
$945$	−17.4352	−0.567166
$946$	9.12395	0.296645
$947$	20.9735	0.681548	0.340774	−	0.940145i	$-0.389311\pi$
$947$	20.9735	0.681548	0.340774	+	0.940145i	$0.389311\pi$
$948$	−2.87034	−0.0932244
$949$	14.1179	0.458288
$950$	3.49414	0.113365
$951$	11.0412	0.358037
$952$	−16.3511	−0.529942
$953$	3.27439	0.106068	0.0530339	−	0.998593i	$-0.483111\pi$
$953$	3.27439	0.106068	0.0530339	+	0.998593i	$0.483111\pi$
$954$	10.7022	0.346496
$955$	32.9735	1.06700
$956$	11.2921	0.365213
$957$	−6.20804	−0.200677
$958$	−12.5973	−0.407001
$959$	36.0593	1.16441
$960$	0.948410	0.0306098
$961$	−6.51324	−0.210104
$962$	−3.29782	−0.106326
$963$	10.4543	0.336884
$964$	11.4101	0.367493
$965$	22.7836	0.733430
$966$	17.8968	0.575821
$967$	−29.2744	−0.941401	−0.470700	−	0.882293i	$-0.655999\pi$
$967$	−29.2744	−0.941401	−0.470700	+	0.882293i	$0.655999\pi$
$968$	1.00000	0.0321412
$969$	3.71390	0.119308
$970$	2.88510	0.0926350
$971$	−38.3836	−1.23179	−0.615894	−	0.787829i	$-0.711205\pi$
$971$	−38.3836	−1.23179	−0.615894	+	0.787829i	$0.711205\pi$
$972$	15.6033	0.500477
$973$	50.1373	1.60733
$974$	9.69616	0.310685
$975$	3.78794	0.121311
$976$	2.00000	0.0640184
$977$	10.9233	0.349467	0.174734	−	0.984616i	$-0.444094\pi$
$977$	10.9233	0.349467	0.174734	+	0.984616i	$0.444094\pi$
$978$	−11.0117	−0.352116
$979$	−14.9883	−0.479028
$980$	5.61810	0.179463
$981$	−14.4161	−0.460270
$982$	−11.7538	−0.375078
$983$	−32.1196	−1.02446	−0.512228	−	0.858849i	$-0.671180\pi$
$983$	−32.1196	−1.02446	−0.512228	+	0.858849i	$0.671180\pi$
$984$	2.27872	0.0726431
$985$	18.2948	0.582920
$986$	−38.5990	−1.22924
$987$	13.8318	0.440272
$988$	1.40268	0.0446252
$989$	−62.0918	−1.97440
$990$	2.94841	0.0937066
$991$	4.37620	0.139015	0.0695073	−	0.997581i	$-0.477857\pi$
$991$	4.37620	0.139015	0.0695073	+	0.997581i	$0.477857\pi$
$992$	−4.94841	−0.157112
$993$	−24.6771	−0.783103
$994$	−25.7863	−0.817890
$995$	18.5990	0.589628
$996$	8.24623	0.261292
$997$	−8.37788	−0.265330	−0.132665	−	0.991161i	$-0.542353\pi$
$997$	−8.37788	−0.265330	−0.132665	+	0.991161i	$0.542353\pi$
$998$	26.6640	0.844034
$999$	−9.81708	−0.310599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	418.2.a.h.1.2	✓	3
3.2	odd	2		3762.2.a.bd.1.2		3
4.3	odd	2		3344.2.a.p.1.2		3
11.10	odd	2		4598.2.a.bm.1.2		3
19.18	odd	2		7942.2.a.bc.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.h.1.2	✓	3	1.1	even	1	trivial
3344.2.a.p.1.2		3	4.3	odd	2
3762.2.a.bd.1.2		3	3.2	odd	2
4598.2.a.bm.1.2		3	11.10	odd	2
7942.2.a.bc.1.2		3	19.18	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:	\( N \)	\(=\)	\( 418 = 2 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	418.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} +0.772866 q^{6} +3.40268 q^{7} +1.00000 q^{8} -2.40268 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.772866 q^{3} +1.00000 q^{4} +1.22713 q^{5} +0.772866 q^{6} +3.40268 q^{7} +1.00000 q^{8} -2.40268 q^{9} +1.22713 q^{10} -1.00000 q^{11} +0.772866 q^{12} -1.40268 q^{13} +3.40268 q^{14} +0.948410 q^{15} +1.00000 q^{16} -4.80536 q^{17} -2.40268 q^{18} -1.00000 q^{19} +1.22713 q^{20} +2.62981 q^{21} -1.00000 q^{22} +6.80536 q^{23} +0.772866 q^{24} -3.49414 q^{25} -1.40268 q^{26} -4.17554 q^{27} +3.40268 q^{28} +8.03249 q^{29} +0.948410 q^{30} -4.94841 q^{31} +1.00000 q^{32} -0.772866 q^{33} -4.80536 q^{34} +4.17554 q^{35} -2.40268 q^{36} +2.35109 q^{37} -1.00000 q^{38} -1.08408 q^{39} +1.22713 q^{40} +2.94841 q^{41} +2.62981 q^{42} -9.12395 q^{43} -1.00000 q^{44} -2.94841 q^{45} +6.80536 q^{46} +5.25963 q^{47} +0.772866 q^{48} +4.57822 q^{49} -3.49414 q^{50} -3.71390 q^{51} -1.40268 q^{52} -4.45427 q^{53} -4.17554 q^{54} -1.22713 q^{55} +3.40268 q^{56} -0.772866 q^{57} +8.03249 q^{58} -14.5193 q^{59} +0.948410 q^{60} +2.00000 q^{61} -4.94841 q^{62} -8.17554 q^{63} +1.00000 q^{64} -1.72128 q^{65} -0.772866 q^{66} +3.40268 q^{67} -4.80536 q^{68} +5.25963 q^{69} +4.17554 q^{70} -7.57822 q^{71} -2.40268 q^{72} -10.0650 q^{73} +2.35109 q^{74} -2.70050 q^{75} -1.00000 q^{76} -3.40268 q^{77} -1.08408 q^{78} -3.71390 q^{79} +1.22713 q^{80} +3.98090 q^{81} +2.94841 q^{82} +10.6697 q^{83} +2.62981 q^{84} -5.89682 q^{85} -9.12395 q^{86} +6.20804 q^{87} -1.00000 q^{88} +14.9883 q^{89} -2.94841 q^{90} -4.77287 q^{91} +6.80536 q^{92} -3.82446 q^{93} +5.25963 q^{94} -1.22713 q^{95} +0.772866 q^{96} +2.35109 q^{97} +4.57822 q^{98} +2.40268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} + q^{6} + q^{7} + 3 q^{8} + 2 q^{9} + 5 q^{10} - 3 q^{11} + q^{12} + 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} + 4 q^{17} + 2 q^{18} - 3 q^{19} + 5 q^{20} - 3 q^{22} + 2 q^{23} + q^{24} + 4 q^{25} + 5 q^{26} - 2 q^{27} + q^{28} + 7 q^{29} - 9 q^{30} - 3 q^{31} + 3 q^{32} - q^{33} + 4 q^{34} + 2 q^{35} + 2 q^{36} - 14 q^{37} - 3 q^{38} + 2 q^{39} + 5 q^{40} - 3 q^{41} - 5 q^{43} - 3 q^{44} + 3 q^{45} + 2 q^{46} + q^{48} - 6 q^{49} + 4 q^{50} + 2 q^{51} + 5 q^{52} - 16 q^{53} - 2 q^{54} - 5 q^{55} + q^{56} - q^{57} + 7 q^{58} - 12 q^{59} - 9 q^{60} + 6 q^{61} - 3 q^{62} - 14 q^{63} + 3 q^{64} + 8 q^{65} - q^{66} + q^{67} + 4 q^{68} + 2 q^{70} - 3 q^{71} + 2 q^{72} + 4 q^{73} - 14 q^{74} - 41 q^{75} - 3 q^{76} - q^{77} + 2 q^{78} + 2 q^{79} + 5 q^{80} - 17 q^{81} - 3 q^{82} + 7 q^{83} + 6 q^{85} - 5 q^{86} - 9 q^{87} - 3 q^{88} + 16 q^{89} + 3 q^{90} - 13 q^{91} + 2 q^{92} - 22 q^{93} - 5 q^{95} + q^{96} - 14 q^{97} - 6 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 + q^6 + q^7 + 3 * q^8 + 2 * q^9 + 5 * q^10 - 3 * q^11 + q^12 + 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 + 4 * q^17 + 2 * q^18 - 3 * q^19 + 5 * q^20 - 3 * q^22 + 2 * q^23 + q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 + q^28 + 7 * q^29 - 9 * q^30 - 3 * q^31 + 3 * q^32 - q^33 + 4 * q^34 + 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 + 2 * q^39 + 5 * q^40 - 3 * q^41 - 5 * q^43 - 3 * q^44 + 3 * q^45 + 2 * q^46 + q^48 - 6 * q^49 + 4 * q^50 + 2 * q^51 + 5 * q^52 - 16 * q^53 - 2 * q^54 - 5 * q^55 + q^56 - q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 + 6 * q^61 - 3 * q^62 - 14 * q^63 + 3 * q^64 + 8 * q^65 - q^66 + q^67 + 4 * q^68 + 2 * q^70 - 3 * q^71 + 2 * q^72 + 4 * q^73 - 14 * q^74 - 41 * q^75 - 3 * q^76 - q^77 + 2 * q^78 + 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 + 7 * q^83 + 6 * q^85 - 5 * q^86 - 9 * q^87 - 3 * q^88 + 16 * q^89 + 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 - 5 * q^95 + q^96 - 14 * q^97 - 6 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more