LMFDB - Embedded newform 4026.2.a.u.1.5

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.9176805.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 12x^{3} + 7x^{2} + 30x - 20 $ x^5 - x^4 - 12x^3 + 7x^2 + 30*x - 20
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			1.5
Root			$-2.48204$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.48204 q^{5} -1.00000 q^{6} +1.57766 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.48204 q^{5} -1.00000 q^{6} +1.57766 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.48204 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.97551 q^{13} -1.57766 q^{14} +2.48204 q^{15} +1.00000 q^{16} +7.22020 q^{17} -1.00000 q^{18} +5.46325 q^{19} +2.48204 q^{20} +1.57766 q^{21} +1.00000 q^{22} -1.75531 q^{23} -1.00000 q^{24} +1.16050 q^{25} +4.97551 q^{26} +1.00000 q^{27} +1.57766 q^{28} +2.32153 q^{29} -2.48204 q^{30} -2.23734 q^{31} -1.00000 q^{32} -1.00000 q^{33} -7.22020 q^{34} +3.91581 q^{35} +1.00000 q^{36} +3.65450 q^{37} -5.46325 q^{38} -4.97551 q^{39} -2.48204 q^{40} -0.0841890 q^{41} -1.57766 q^{42} +3.92316 q^{43} -1.00000 q^{44} +2.48204 q^{45} +1.75531 q^{46} -5.61805 q^{47} +1.00000 q^{48} -4.51099 q^{49} -1.16050 q^{50} +7.22020 q^{51} -4.97551 q^{52} +0.738164 q^{53} -1.00000 q^{54} -2.48204 q^{55} -1.57766 q^{56} +5.46325 q^{57} -2.32153 q^{58} +6.61123 q^{59} +2.48204 q^{60} -1.00000 q^{61} +2.23734 q^{62} +1.57766 q^{63} +1.00000 q^{64} -12.3494 q^{65} +1.00000 q^{66} +7.08992 q^{67} +7.22020 q^{68} -1.75531 q^{69} -3.91581 q^{70} +1.20142 q^{71} -1.00000 q^{72} +12.1189 q^{73} -3.65450 q^{74} +1.16050 q^{75} +5.46325 q^{76} -1.57766 q^{77} +4.97551 q^{78} +10.4861 q^{79} +2.48204 q^{80} +1.00000 q^{81} +0.0841890 q^{82} +0.359100 q^{83} +1.57766 q^{84} +17.9208 q^{85} -3.92316 q^{86} +2.32153 q^{87} +1.00000 q^{88} +4.63572 q^{89} -2.48204 q^{90} -7.84966 q^{91} -1.75531 q^{92} -2.23734 q^{93} +5.61805 q^{94} +13.5600 q^{95} -1.00000 q^{96} +5.94529 q^{97} +4.51099 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} - 3 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - q^5 - 5 * q^6 - 3 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} - 3 q^{7} - 5 q^{8} + 5 q^{9} + q^{10} - 5 q^{11} + 5 q^{12} + 2 q^{13} + 3 q^{14} - q^{15} + 5 q^{16} + 6 q^{17} - 5 q^{18} + 7 q^{19} - q^{20} - 3 q^{21} + 5 q^{22} - 12 q^{23} - 5 q^{24} - 2 q^{26} + 5 q^{27} - 3 q^{28} + 4 q^{29} + q^{30} - q^{31} - 5 q^{32} - 5 q^{33} - 6 q^{34} + 17 q^{35} + 5 q^{36} + 3 q^{37} - 7 q^{38} + 2 q^{39} + q^{40} - 3 q^{41} + 3 q^{42} + 24 q^{43} - 5 q^{44} - q^{45} + 12 q^{46} + 18 q^{47} + 5 q^{48} + 26 q^{49} + 6 q^{51} + 2 q^{52} - 13 q^{53} - 5 q^{54} + q^{55} + 3 q^{56} + 7 q^{57} - 4 q^{58} - 16 q^{59} - q^{60} - 5 q^{61} + q^{62} - 3 q^{63} + 5 q^{64} + 4 q^{65} + 5 q^{66} + 18 q^{67} + 6 q^{68} - 12 q^{69} - 17 q^{70} - 31 q^{71} - 5 q^{72} + 8 q^{73} - 3 q^{74} + 7 q^{76} + 3 q^{77} - 2 q^{78} + 32 q^{79} - q^{80} + 5 q^{81} + 3 q^{82} + 8 q^{83} - 3 q^{84} + 29 q^{85} - 24 q^{86} + 4 q^{87} + 5 q^{88} + q^{89} + q^{90} - 3 q^{91} - 12 q^{92} - q^{93} - 18 q^{94} + 33 q^{95} - 5 q^{96} - 4 q^{97} - 26 q^{98} - 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - q^5 - 5 * q^6 - 3 * q^7 - 5 * q^8 + 5 * q^9 + q^10 - 5 * q^11 + 5 * q^12 + 2 * q^13 + 3 * q^14 - q^15 + 5 * q^16 + 6 * q^17 - 5 * q^18 + 7 * q^19 - q^20 - 3 * q^21 + 5 * q^22 - 12 * q^23 - 5 * q^24 - 2 * q^26 + 5 * q^27 - 3 * q^28 + 4 * q^29 + q^30 - q^31 - 5 * q^32 - 5 * q^33 - 6 * q^34 + 17 * q^35 + 5 * q^36 + 3 * q^37 - 7 * q^38 + 2 * q^39 + q^40 - 3 * q^41 + 3 * q^42 + 24 * q^43 - 5 * q^44 - q^45 + 12 * q^46 + 18 * q^47 + 5 * q^48 + 26 * q^49 + 6 * q^51 + 2 * q^52 - 13 * q^53 - 5 * q^54 + q^55 + 3 * q^56 + 7 * q^57 - 4 * q^58 - 16 * q^59 - q^60 - 5 * q^61 + q^62 - 3 * q^63 + 5 * q^64 + 4 * q^65 + 5 * q^66 + 18 * q^67 + 6 * q^68 - 12 * q^69 - 17 * q^70 - 31 * q^71 - 5 * q^72 + 8 * q^73 - 3 * q^74 + 7 * q^76 + 3 * q^77 - 2 * q^78 + 32 * q^79 - q^80 + 5 * q^81 + 3 * q^82 + 8 * q^83 - 3 * q^84 + 29 * q^85 - 24 * q^86 + 4 * q^87 + 5 * q^88 + q^89 + q^90 - 3 * q^91 - 12 * q^92 - q^93 - 18 * q^94 + 33 * q^95 - 5 * q^96 - 4 * q^97 - 26 * q^98 - 5 * 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
$5$	2.48204	1.11000	0.555000	−	0.831850i	$-0.312718\pi$
$5$	2.48204	1.11000	0.555000	+	0.831850i	$0.312718\pi$
$6$	−1.00000	−0.408248
$7$	1.57766	0.596300	0.298150	−	0.954519i	$-0.403630\pi$
$7$	1.57766	0.596300	0.298150	+	0.954519i	$0.403630\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.48204	−0.784889
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−4.97551	−1.37996	−0.689979	−	0.723830i	$-0.742380\pi$
$13$	−4.97551	−1.37996	−0.689979	+	0.723830i	$0.742380\pi$
$14$	−1.57766	−0.421648
$15$	2.48204	0.640859
$16$	1.00000	0.250000
$17$	7.22020	1.75116	0.875578	−	0.483077i	$-0.160481\pi$
$17$	7.22020	1.75116	0.875578	+	0.483077i	$0.160481\pi$
$18$	−1.00000	−0.235702
$19$	5.46325	1.25336	0.626678	−	0.779278i	$-0.284414\pi$
$19$	5.46325	1.25336	0.626678	+	0.779278i	$0.284414\pi$
$20$	2.48204	0.555000
$21$	1.57766	0.344274
$22$	1.00000	0.213201
$23$	−1.75531	−0.366007	−0.183004	−	0.983112i	$-0.558582\pi$
$23$	−1.75531	−0.366007	−0.183004	+	0.983112i	$0.558582\pi$
$24$	−1.00000	−0.204124
$25$	1.16050	0.232101
$26$	4.97551	0.975777
$27$	1.00000	0.192450
$28$	1.57766	0.298150
$29$	2.32153	0.431098	0.215549	−	0.976493i	$-0.430846\pi$
$29$	2.32153	0.431098	0.215549	+	0.976493i	$0.430846\pi$
$30$	−2.48204	−0.453156
$31$	−2.23734	−0.401839	−0.200919	−	0.979608i	$-0.564393\pi$
$31$	−2.23734	−0.401839	−0.200919	+	0.979608i	$0.564393\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−7.22020	−1.23825
$35$	3.91581	0.661893
$36$	1.00000	0.166667
$37$	3.65450	0.600796	0.300398	−	0.953814i	$-0.402880\pi$
$37$	3.65450	0.600796	0.300398	+	0.953814i	$0.402880\pi$
$38$	−5.46325	−0.886257
$39$	−4.97551	−0.796719
$40$	−2.48204	−0.392444
$41$	−0.0841890	−0.0131481	−0.00657406	−	0.999978i	$-0.502093\pi$
$41$	−0.0841890	−0.0131481	−0.00657406	+	0.999978i	$0.502093\pi$
$42$	−1.57766	−0.243438
$43$	3.92316	0.598276	0.299138	−	0.954210i	$-0.403301\pi$
$43$	3.92316	0.598276	0.299138	+	0.954210i	$0.403301\pi$
$44$	−1.00000	−0.150756
$45$	2.48204	0.370000
$46$	1.75531	0.258806
$47$	−5.61805	−0.819476	−0.409738	−	0.912203i	$-0.634380\pi$
$47$	−5.61805	−0.819476	−0.409738	+	0.912203i	$0.634380\pi$
$48$	1.00000	0.144338
$49$	−4.51099	−0.644427
$50$	−1.16050	−0.164120
$51$	7.22020	1.01103
$52$	−4.97551	−0.689979
$53$	0.738164	0.101395	0.0506973	−	0.998714i	$-0.483856\pi$
$53$	0.738164	0.101395	0.0506973	+	0.998714i	$0.483856\pi$
$54$	−1.00000	−0.136083
$55$	−2.48204	−0.334678
$56$	−1.57766	−0.210824
$57$	5.46325	0.723625
$58$	−2.32153	−0.304832
$59$	6.61123	0.860708	0.430354	−	0.902660i	$-0.358389\pi$
$59$	6.61123	0.860708	0.430354	+	0.902660i	$0.358389\pi$
$60$	2.48204	0.320429
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	2.23734	0.284143
$63$	1.57766	0.198767
$64$	1.00000	0.125000
$65$	−12.3494	−1.53175
$66$	1.00000	0.123091
$67$	7.08992	0.866171	0.433086	−	0.901353i	$-0.357425\pi$
$67$	7.08992	0.866171	0.433086	+	0.901353i	$0.357425\pi$
$68$	7.22020	0.875578
$69$	−1.75531	−0.211314
$70$	−3.91581	−0.468029
$71$	1.20142	0.142582	0.0712909	−	0.997456i	$-0.477288\pi$
$71$	1.20142	0.142582	0.0712909	+	0.997456i	$0.477288\pi$
$72$	−1.00000	−0.117851
$73$	12.1189	1.41841	0.709203	−	0.705004i	$-0.249055\pi$
$73$	12.1189	1.41841	0.709203	+	0.705004i	$0.249055\pi$
$74$	−3.65450	−0.424827
$75$	1.16050	0.134003
$76$	5.46325	0.626678
$77$	−1.57766	−0.179791
$78$	4.97551	0.563365
$79$	10.4861	1.17978	0.589891	−	0.807483i	$-0.299171\pi$
$79$	10.4861	1.17978	0.589891	+	0.807483i	$0.299171\pi$
$80$	2.48204	0.277500
$81$	1.00000	0.111111
$82$	0.0841890	0.00929712
$83$	0.359100	0.0394164	0.0197082	−	0.999806i	$-0.493726\pi$
$83$	0.359100	0.0394164	0.0197082	+	0.999806i	$0.493726\pi$
$84$	1.57766	0.172137
$85$	17.9208	1.94378
$86$	−3.92316	−0.423045
$87$	2.32153	0.248894
$88$	1.00000	0.106600
$89$	4.63572	0.491385	0.245693	−	0.969348i	$-0.420985\pi$
$89$	4.63572	0.491385	0.245693	+	0.969348i	$0.420985\pi$
$90$	−2.48204	−0.261630
$91$	−7.84966	−0.822868
$92$	−1.75531	−0.183004
$93$	−2.23734	−0.232002
$94$	5.61805	0.579457
$95$	13.5600	1.39123
$96$	−1.00000	−0.102062
$97$	5.94529	0.603653	0.301826	−	0.953363i	$-0.402404\pi$
$97$	5.94529	0.603653	0.301826	+	0.953363i	$0.402404\pi$
$98$	4.51099	0.455678
$99$	−1.00000	−0.100504
$100$	1.16050	0.116050
$101$	−12.6243	−1.25616	−0.628082	−	0.778147i	$-0.716160\pi$
$101$	−12.6243	−1.25616	−0.628082	+	0.778147i	$0.716160\pi$
$102$	−7.22020	−0.714906
$103$	10.1900	1.00405	0.502024	−	0.864854i	$-0.332589\pi$
$103$	10.1900	1.00405	0.502024	+	0.864854i	$0.332589\pi$
$104$	4.97551	0.487889
$105$	3.91581	0.382144
$106$	−0.738164	−0.0716968
$107$	−1.35801	−0.131283	−0.0656417	−	0.997843i	$-0.520909\pi$
$107$	−1.35801	−0.131283	−0.0656417	+	0.997843i	$0.520909\pi$
$108$	1.00000	0.0962250
$109$	6.58337	0.630572	0.315286	−	0.948997i	$-0.397899\pi$
$109$	6.58337	0.630572	0.315286	+	0.948997i	$0.397899\pi$
$110$	2.48204	0.236653
$111$	3.65450	0.346870
$112$	1.57766	0.149075
$113$	−4.58448	−0.431272	−0.215636	−	0.976474i	$-0.569182\pi$
$113$	−4.58448	−0.431272	−0.215636	+	0.976474i	$0.569182\pi$
$114$	−5.46325	−0.511680
$115$	−4.35674	−0.406268
$116$	2.32153	0.215549
$117$	−4.97551	−0.459986
$118$	−6.61123	−0.608613
$119$	11.3910	1.04421
$120$	−2.48204	−0.226578
$121$	1.00000	0.0909091
$122$	1.00000	0.0905357
$123$	−0.0841890	−0.00759107
$124$	−2.23734	−0.200919
$125$	−9.52977	−0.852369
$126$	−1.57766	−0.140549
$127$	−6.83090	−0.606144	−0.303072	−	0.952968i	$-0.598012\pi$
$127$	−6.83090	−0.606144	−0.303072	+	0.952968i	$0.598012\pi$
$128$	−1.00000	−0.0883883
$129$	3.92316	0.345415
$130$	12.3494	1.08311
$131$	6.39323	0.558579	0.279290	−	0.960207i	$-0.409901\pi$
$131$	6.39323	0.558579	0.279290	+	0.960207i	$0.409901\pi$
$132$	−1.00000	−0.0870388
$133$	8.61916	0.747376
$134$	−7.08992	−0.612475
$135$	2.48204	0.213620
$136$	−7.22020	−0.619127
$137$	−2.37624	−0.203016	−0.101508	−	0.994835i	$-0.532367\pi$
$137$	−2.37624	−0.203016	−0.101508	+	0.994835i	$0.532367\pi$
$138$	1.75531	0.149422
$139$	6.12294	0.519341	0.259670	−	0.965697i	$-0.416386\pi$
$139$	6.12294	0.519341	0.259670	+	0.965697i	$0.416386\pi$
$140$	3.91581	0.330946
$141$	−5.61805	−0.473125
$142$	−1.20142	−0.100821
$143$	4.97551	0.416073
$144$	1.00000	0.0833333
$145$	5.76213	0.478519
$146$	−12.1189	−1.00296
$147$	−4.51099	−0.372060
$148$	3.65450	0.300398
$149$	14.0581	1.15168	0.575841	−	0.817562i	$-0.304675\pi$
$149$	14.0581	1.15168	0.575841	+	0.817562i	$0.304675\pi$
$150$	−1.16050	−0.0947547
$151$	−10.2538	−0.834440	−0.417220	−	0.908806i	$-0.636995\pi$
$151$	−10.2538	−0.834440	−0.417220	+	0.908806i	$0.636995\pi$
$152$	−5.46325	−0.443128
$153$	7.22020	0.583719
$154$	1.57766	0.127132
$155$	−5.55317	−0.446041
$156$	−4.97551	−0.398359
$157$	−8.82517	−0.704325	−0.352163	−	0.935939i	$-0.614554\pi$
$157$	−8.82517	−0.704325	−0.352163	+	0.935939i	$0.614554\pi$
$158$	−10.4861	−0.834231
$159$	0.738164	0.0585402
$160$	−2.48204	−0.196222
$161$	−2.76928	−0.218250
$162$	−1.00000	−0.0785674
$163$	−11.7260	−0.918452	−0.459226	−	0.888320i	$-0.651873\pi$
$163$	−11.7260	−0.918452	−0.459226	+	0.888320i	$0.651873\pi$
$164$	−0.0841890	−0.00657406
$165$	−2.48204	−0.193226
$166$	−0.359100	−0.0278716
$167$	3.73191	0.288784	0.144392	−	0.989521i	$-0.453877\pi$
$167$	3.73191	0.288784	0.144392	+	0.989521i	$0.453877\pi$
$168$	−1.57766	−0.121719
$169$	11.7557	0.904283
$170$	−17.9208	−1.37446
$171$	5.46325	0.417785
$172$	3.92316	0.299138
$173$	−15.7364	−1.19641	−0.598207	−	0.801342i	$-0.704120\pi$
$173$	−15.7364	−1.19641	−0.598207	+	0.801342i	$0.704120\pi$
$174$	−2.32153	−0.175995
$175$	1.83088	0.138402
$176$	−1.00000	−0.0753778
$177$	6.61123	0.496930
$178$	−4.63572	−0.347462
$179$	−13.1480	−0.982726	−0.491363	−	0.870955i	$-0.663501\pi$
$179$	−13.1480	−0.982726	−0.491363	+	0.870955i	$0.663501\pi$
$180$	2.48204	0.185000
$181$	−23.7767	−1.76731	−0.883655	−	0.468139i	$-0.844925\pi$
$181$	−23.7767	−1.76731	−0.883655	+	0.468139i	$0.844925\pi$
$182$	7.84966	0.581856
$183$	−1.00000	−0.0739221
$184$	1.75531	0.129403
$185$	9.07061	0.666884
$186$	2.23734	0.164050
$187$	−7.22020	−0.527993
$188$	−5.61805	−0.409738
$189$	1.57766	0.114758
$190$	−13.5600	−0.983745
$191$	−22.9311	−1.65924	−0.829619	−	0.558329i	$-0.811443\pi$
$191$	−22.9311	−1.65924	−0.829619	+	0.558329i	$0.811443\pi$
$192$	1.00000	0.0721688
$193$	15.2772	1.09967	0.549837	−	0.835272i	$-0.314690\pi$
$193$	15.2772	1.09967	0.549837	+	0.835272i	$0.314690\pi$
$194$	−5.94529	−0.426847
$195$	−12.3494	−0.884358
$196$	−4.51099	−0.322213
$197$	−9.38641	−0.668754	−0.334377	−	0.942439i	$-0.608526\pi$
$197$	−9.38641	−0.668754	−0.334377	+	0.942439i	$0.608526\pi$
$198$	1.00000	0.0710669
$199$	11.3483	0.804461	0.402230	−	0.915538i	$-0.368235\pi$
$199$	11.3483	0.804461	0.402230	+	0.915538i	$0.368235\pi$
$200$	−1.16050	−0.0820600
$201$	7.08992	0.500084
$202$	12.6243	0.888243
$203$	3.66259	0.257064
$204$	7.22020	0.505515
$205$	−0.208960	−0.0145944
$206$	−10.1900	−0.709970
$207$	−1.75531	−0.122002
$208$	−4.97551	−0.344989
$209$	−5.46325	−0.377901
$210$	−3.91581	−0.270217
$211$	19.7222	1.35773	0.678866	−	0.734262i	$-0.262472\pi$
$211$	19.7222	1.35773	0.678866	+	0.734262i	$0.262472\pi$
$212$	0.738164	0.0506973
$213$	1.20142	0.0823197
$214$	1.35801	0.0928314
$215$	9.73742	0.664087
$216$	−1.00000	−0.0680414
$217$	−3.52977	−0.239616
$218$	−6.58337	−0.445882
$219$	12.1189	0.818917
$220$	−2.48204	−0.167339
$221$	−35.9242	−2.41652
$222$	−3.65450	−0.245274
$223$	−12.0039	−0.803843	−0.401921	−	0.915674i	$-0.631657\pi$
$223$	−12.0039	−0.803843	−0.401921	+	0.915674i	$0.631657\pi$
$224$	−1.57766	−0.105412
$225$	1.16050	0.0773669
$226$	4.58448	0.304955
$227$	3.72526	0.247254	0.123627	−	0.992329i	$-0.460547\pi$
$227$	3.72526	0.247254	0.123627	+	0.992329i	$0.460547\pi$
$228$	5.46325	0.361813
$229$	23.6720	1.56429	0.782146	−	0.623095i	$-0.214125\pi$
$229$	23.6720	1.56429	0.782146	+	0.623095i	$0.214125\pi$
$230$	4.35674	0.287275
$231$	−1.57766	−0.103802
$232$	−2.32153	−0.152416
$233$	−10.8622	−0.711607	−0.355804	−	0.934561i	$-0.615793\pi$
$233$	−10.8622	−0.711607	−0.355804	+	0.934561i	$0.615793\pi$
$234$	4.97551	0.325259
$235$	−13.9442	−0.909619
$236$	6.61123	0.430354
$237$	10.4861	0.681147
$238$	−11.3910	−0.738371
$239$	1.82533	0.118071	0.0590353	−	0.998256i	$-0.481198\pi$
$239$	1.82533	0.118071	0.0590353	+	0.998256i	$0.481198\pi$
$240$	2.48204	0.160215
$241$	17.4546	1.12435	0.562176	−	0.827018i	$-0.309964\pi$
$241$	17.4546	1.12435	0.562176	+	0.827018i	$0.309964\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−1.00000	−0.0640184
$245$	−11.1964	−0.715314
$246$	0.0841890	0.00536769
$247$	−27.1825	−1.72958
$248$	2.23734	0.142071
$249$	0.359100	0.0227571
$250$	9.52977	0.602716
$251$	20.2679	1.27930	0.639650	−	0.768666i	$-0.279079\pi$
$251$	20.2679	1.27930	0.639650	+	0.768666i	$0.279079\pi$
$252$	1.57766	0.0993833
$253$	1.75531	0.110355
$254$	6.83090	0.428609
$255$	17.9208	1.12224
$256$	1.00000	0.0625000
$257$	7.60217	0.474210	0.237105	−	0.971484i	$-0.423801\pi$
$257$	7.60217	0.474210	0.237105	+	0.971484i	$0.423801\pi$
$258$	−3.92316	−0.244245
$259$	5.76556	0.358255
$260$	−12.3494	−0.765877
$261$	2.32153	0.143699
$262$	−6.39323	−0.394975
$263$	4.33404	0.267248	0.133624	−	0.991032i	$-0.457338\pi$
$263$	4.33404	0.267248	0.133624	+	0.991032i	$0.457338\pi$
$264$	1.00000	0.0615457
$265$	1.83215	0.112548
$266$	−8.61916	−0.528475
$267$	4.63572	0.283701
$268$	7.08992	0.433086
$269$	−11.4364	−0.697289	−0.348645	−	0.937255i	$-0.613358\pi$
$269$	−11.4364	−0.697289	−0.348645	+	0.937255i	$0.613358\pi$
$270$	−2.48204	−0.151052
$271$	6.09101	0.370002	0.185001	−	0.982738i	$-0.440771\pi$
$271$	6.09101	0.370002	0.185001	+	0.982738i	$0.440771\pi$
$272$	7.22020	0.437789
$273$	−7.84966	−0.475083
$274$	2.37624	0.143554
$275$	−1.16050	−0.0699810
$276$	−1.75531	−0.105657
$277$	9.24397	0.555416	0.277708	−	0.960665i	$-0.410425\pi$
$277$	9.24397	0.555416	0.277708	+	0.960665i	$0.410425\pi$
$278$	−6.12294	−0.367229
$279$	−2.23734	−0.133946
$280$	−3.91581	−0.234014
$281$	4.42234	0.263815	0.131907	−	0.991262i	$-0.457890\pi$
$281$	4.42234	0.263815	0.131907	+	0.991262i	$0.457890\pi$
$282$	5.61805	0.334550
$283$	−12.6077	−0.749449	−0.374725	−	0.927136i	$-0.622263\pi$
$283$	−12.6077	−0.749449	−0.374725	+	0.927136i	$0.622263\pi$
$284$	1.20142	0.0712909
$285$	13.5600	0.803224
$286$	−4.97551	−0.294208
$287$	−0.132822	−0.00784022
$288$	−1.00000	−0.0589256
$289$	35.1313	2.06655
$290$	−5.76213	−0.338364
$291$	5.94529	0.348519
$292$	12.1189	0.709203
$293$	5.90312	0.344864	0.172432	−	0.985021i	$-0.444837\pi$
$293$	5.90312	0.344864	0.172432	+	0.985021i	$0.444837\pi$
$294$	4.51099	0.263086
$295$	16.4093	0.955386
$296$	−3.65450	−0.212414
$297$	−1.00000	−0.0580259
$298$	−14.0581	−0.814362
$299$	8.73355	0.505074
$300$	1.16050	0.0670017
$301$	6.18941	0.356752
$302$	10.2538	0.590038
$303$	−12.6243	−0.725247
$304$	5.46325	0.313339
$305$	−2.48204	−0.142121
$306$	−7.22020	−0.412751
$307$	18.1342	1.03497	0.517487	−	0.855691i	$-0.326868\pi$
$307$	18.1342	1.03497	0.517487	+	0.855691i	$0.326868\pi$
$308$	−1.57766	−0.0898956
$309$	10.1900	0.579688
$310$	5.55317	0.315399
$311$	−14.9799	−0.849435	−0.424717	−	0.905326i	$-0.639627\pi$
$311$	−14.9799	−0.849435	−0.424717	+	0.905326i	$0.639627\pi$
$312$	4.97551	0.281683
$313$	−12.9832	−0.733855	−0.366928	−	0.930249i	$-0.619590\pi$
$313$	−12.9832	−0.733855	−0.366928	+	0.930249i	$0.619590\pi$
$314$	8.82517	0.498033
$315$	3.91581	0.220631
$316$	10.4861	0.589891
$317$	−29.4666	−1.65501	−0.827505	−	0.561458i	$-0.810241\pi$
$317$	−29.4666	−1.65501	−0.827505	+	0.561458i	$0.810241\pi$
$318$	−0.738164	−0.0413942
$319$	−2.32153	−0.129981
$320$	2.48204	0.138750
$321$	−1.35801	−0.0757966
$322$	2.76928	0.154326
$323$	39.4458	2.19482
$324$	1.00000	0.0555556
$325$	−5.77409	−0.320289
$326$	11.7260	0.649443
$327$	6.58337	0.364061
$328$	0.0841890	0.00464856
$329$	−8.86337	−0.488654
$330$	2.48204	0.136632
$331$	−13.0564	−0.717642	−0.358821	−	0.933406i	$-0.616821\pi$
$331$	−13.0564	−0.717642	−0.358821	+	0.933406i	$0.616821\pi$
$332$	0.359100	0.0197082
$333$	3.65450	0.200265
$334$	−3.73191	−0.204201
$335$	17.5974	0.961450
$336$	1.57766	0.0860685
$337$	5.16212	0.281199	0.140599	−	0.990067i	$-0.455097\pi$
$337$	5.16212	0.281199	0.140599	+	0.990067i	$0.455097\pi$
$338$	−11.7557	−0.639425
$339$	−4.58448	−0.248995
$340$	17.9208	0.971892
$341$	2.23734	0.121159
$342$	−5.46325	−0.295419
$343$	−18.1604	−0.980571
$344$	−3.92316	−0.211523
$345$	−4.35674	−0.234559
$346$	15.7364	0.845992
$347$	12.2368	0.656905	0.328452	−	0.944521i	$-0.393473\pi$
$347$	12.2368	0.656905	0.328452	+	0.944521i	$0.393473\pi$
$348$	2.32153	0.124447
$349$	−18.5418	−0.992521	−0.496260	−	0.868174i	$-0.665294\pi$
$349$	−18.5418	−0.992521	−0.496260	+	0.868174i	$0.665294\pi$
$350$	−1.83088	−0.0978647
$351$	−4.97551	−0.265573
$352$	1.00000	0.0533002
$353$	1.97315	0.105020	0.0525100	−	0.998620i	$-0.483278\pi$
$353$	1.97315	0.105020	0.0525100	+	0.998620i	$0.483278\pi$
$354$	−6.61123	−0.351383
$355$	2.98196	0.158266
$356$	4.63572	0.245693
$357$	11.3910	0.602877
$358$	13.1480	0.694892
$359$	−9.23490	−0.487399	−0.243700	−	0.969851i	$-0.578361\pi$
$359$	−9.23490	−0.487399	−0.243700	+	0.969851i	$0.578361\pi$
$360$	−2.48204	−0.130815
$361$	10.8471	0.570901
$362$	23.7767	1.24968
$363$	1.00000	0.0524864
$364$	−7.84966	−0.411434
$365$	30.0795	1.57443
$366$	1.00000	0.0522708
$367$	23.3975	1.22134	0.610670	−	0.791885i	$-0.290900\pi$
$367$	23.3975	1.22134	0.610670	+	0.791885i	$0.290900\pi$
$368$	−1.75531	−0.0915018
$369$	−0.0841890	−0.00438270
$370$	−9.07061	−0.471558
$371$	1.16457	0.0604616
$372$	−2.23734	−0.116001
$373$	6.44001	0.333451	0.166726	−	0.986003i	$-0.446681\pi$
$373$	6.44001	0.333451	0.166726	+	0.986003i	$0.446681\pi$
$374$	7.22020	0.373348
$375$	−9.52977	−0.492115
$376$	5.61805	0.289729
$377$	−11.5508	−0.594897
$378$	−1.57766	−0.0811461
$379$	13.5104	0.693985	0.346992	−	0.937868i	$-0.387203\pi$
$379$	13.5104	0.693985	0.346992	+	0.937868i	$0.387203\pi$
$380$	13.5600	0.695613
$381$	−6.83090	−0.349958
$382$	22.9311	1.17326
$383$	−16.5203	−0.844146	−0.422073	−	0.906562i	$-0.638697\pi$
$383$	−16.5203	−0.844146	−0.422073	+	0.906562i	$0.638697\pi$
$384$	−1.00000	−0.0510310
$385$	−3.91581	−0.199568
$386$	−15.2772	−0.777587
$387$	3.92316	0.199425
$388$	5.94529	0.301826
$389$	−11.1952	−0.567617	−0.283809	−	0.958881i	$-0.591598\pi$
$389$	−11.1952	−0.567617	−0.283809	+	0.958881i	$0.591598\pi$
$390$	12.3494	0.625336
$391$	−12.6737	−0.640935
$392$	4.51099	0.227839
$393$	6.39323	0.322496
$394$	9.38641	0.472881
$395$	26.0269	1.30956
$396$	−1.00000	−0.0502519
$397$	−21.6990	−1.08904	−0.544520	−	0.838748i	$-0.683288\pi$
$397$	−21.6990	−1.08904	−0.544520	+	0.838748i	$0.683288\pi$
$398$	−11.3483	−0.568840
$399$	8.61916	0.431498
$400$	1.16050	0.0580251
$401$	−18.4153	−0.919618	−0.459809	−	0.888018i	$-0.652082\pi$
$401$	−18.4153	−0.919618	−0.459809	+	0.888018i	$0.652082\pi$
$402$	−7.08992	−0.353613
$403$	11.1319	0.554521
$404$	−12.6243	−0.628082
$405$	2.48204	0.123333
$406$	−3.66259	−0.181771
$407$	−3.65450	−0.181147
$408$	−7.22020	−0.357453
$409$	4.30492	0.212864	0.106432	−	0.994320i	$-0.466057\pi$
$409$	4.30492	0.212864	0.106432	+	0.994320i	$0.466057\pi$
$410$	0.208960	0.0103198
$411$	−2.37624	−0.117212
$412$	10.1900	0.502024
$413$	10.4303	0.513240
$414$	1.75531	0.0862687
$415$	0.891300	0.0437522
$416$	4.97551	0.243944
$417$	6.12294	0.299841
$418$	5.46325	0.267216
$419$	−7.35619	−0.359373	−0.179687	−	0.983724i	$-0.557508\pi$
$419$	−7.35619	−0.359373	−0.179687	+	0.983724i	$0.557508\pi$
$420$	3.91581	0.191072
$421$	26.5246	1.29273	0.646365	−	0.763028i	$-0.276288\pi$
$421$	26.5246	1.29273	0.646365	+	0.763028i	$0.276288\pi$
$422$	−19.7222	−0.960062
$423$	−5.61805	−0.273159
$424$	−0.738164	−0.0358484
$425$	8.37906	0.406444
$426$	−1.20142	−0.0582088
$427$	−1.57766	−0.0763484
$428$	−1.35801	−0.0656417
$429$	4.97551	0.240220
$430$	−9.73742	−0.469580
$431$	22.2025	1.06946	0.534729	−	0.845024i	$-0.320414\pi$
$431$	22.2025	1.06946	0.534729	+	0.845024i	$0.320414\pi$
$432$	1.00000	0.0481125
$433$	16.1513	0.776183	0.388091	−	0.921621i	$-0.373134\pi$
$433$	16.1513	0.776183	0.388091	+	0.921621i	$0.373134\pi$
$434$	3.52977	0.169434
$435$	5.76213	0.276273
$436$	6.58337	0.315286
$437$	−9.58969	−0.458737
$438$	−12.1189	−0.579062
$439$	3.76764	0.179820	0.0899099	−	0.995950i	$-0.471342\pi$
$439$	3.76764	0.179820	0.0899099	+	0.995950i	$0.471342\pi$
$440$	2.48204	0.118326
$441$	−4.51099	−0.214809
$442$	35.9242	1.70874
$443$	−28.9033	−1.37324	−0.686618	−	0.727018i	$-0.740906\pi$
$443$	−28.9033	−1.37324	−0.686618	+	0.727018i	$0.740906\pi$
$444$	3.65450	0.173435
$445$	11.5060	0.545438
$446$	12.0039	0.568403
$447$	14.0581	0.664924
$448$	1.57766	0.0745375
$449$	−17.3249	−0.817612	−0.408806	−	0.912621i	$-0.634055\pi$
$449$	−17.3249	−0.817612	−0.408806	+	0.912621i	$0.634055\pi$
$450$	−1.16050	−0.0547066
$451$	0.0841890	0.00396431
$452$	−4.58448	−0.215636
$453$	−10.2538	−0.481764
$454$	−3.72526	−0.174835
$455$	−19.4831	−0.913384
$456$	−5.46325	−0.255840
$457$	−27.9527	−1.30757	−0.653787	−	0.756679i	$-0.726821\pi$
$457$	−27.9527	−1.30757	−0.653787	+	0.756679i	$0.726821\pi$
$458$	−23.6720	−1.10612
$459$	7.22020	0.337010
$460$	−4.35674	−0.203134
$461$	−19.7301	−0.918922	−0.459461	−	0.888198i	$-0.651957\pi$
$461$	−19.7301	−0.918922	−0.459461	+	0.888198i	$0.651957\pi$
$462$	1.57766	0.0733994
$463$	−35.0083	−1.62698	−0.813488	−	0.581582i	$-0.802434\pi$
$463$	−35.0083	−1.62698	−0.813488	+	0.581582i	$0.802434\pi$
$464$	2.32153	0.107774
$465$	−5.55317	−0.257522
$466$	10.8622	0.503182
$467$	2.95172	0.136589	0.0682947	−	0.997665i	$-0.478244\pi$
$467$	2.95172	0.136589	0.0682947	+	0.997665i	$0.478244\pi$
$468$	−4.97551	−0.229993
$469$	11.1855	0.516498
$470$	13.9442	0.643198
$471$	−8.82517	−0.406642
$472$	−6.61123	−0.304306
$473$	−3.92316	−0.180387
$474$	−10.4861	−0.481644
$475$	6.34012	0.290905
$476$	11.3910	0.522107
$477$	0.738164	0.0337982
$478$	−1.82533	−0.0834886
$479$	13.6000	0.621398	0.310699	−	0.950508i	$-0.399437\pi$
$479$	13.6000	0.621398	0.310699	+	0.950508i	$0.399437\pi$
$480$	−2.48204	−0.113289
$481$	−18.1830	−0.829074
$482$	−17.4546	−0.795037
$483$	−2.76928	−0.126007
$484$	1.00000	0.0454545
$485$	14.7564	0.670055
$486$	−1.00000	−0.0453609
$487$	29.5866	1.34070	0.670348	−	0.742047i	$-0.266145\pi$
$487$	29.5866	1.34070	0.670348	+	0.742047i	$0.266145\pi$
$488$	1.00000	0.0452679
$489$	−11.7260	−0.530268
$490$	11.1964	0.505803
$491$	−7.69301	−0.347180	−0.173590	−	0.984818i	$-0.555537\pi$
$491$	−7.69301	−0.347180	−0.173590	+	0.984818i	$0.555537\pi$
$492$	−0.0841890	−0.00379553
$493$	16.7619	0.754919
$494$	27.1825	1.22300
$495$	−2.48204	−0.111559
$496$	−2.23734	−0.100460
$497$	1.89543	0.0850215
$498$	−0.359100	−0.0160917
$499$	−1.53213	−0.0685877	−0.0342938	−	0.999412i	$-0.510918\pi$
$499$	−1.53213	−0.0685877	−0.0342938	+	0.999412i	$0.510918\pi$
$500$	−9.52977	−0.426184
$501$	3.73191	0.166729
$502$	−20.2679	−0.904602
$503$	11.5692	0.515846	0.257923	−	0.966165i	$-0.416962\pi$
$503$	11.5692	0.515846	0.257923	+	0.966165i	$0.416962\pi$
$504$	−1.57766	−0.0702746
$505$	−31.3340	−1.39434
$506$	−1.75531	−0.0780330
$507$	11.7557	0.522088
$508$	−6.83090	−0.303072
$509$	7.65272	0.339201	0.169601	−	0.985513i	$-0.445752\pi$
$509$	7.65272	0.339201	0.169601	+	0.985513i	$0.445752\pi$
$510$	−17.9208	−0.793546
$511$	19.1195	0.845795
$512$	−1.00000	−0.0441942
$513$	5.46325	0.241208
$514$	−7.60217	−0.335317
$515$	25.2919	1.11449
$516$	3.92316	0.172707
$517$	5.61805	0.247081
$518$	−5.76556	−0.253324
$519$	−15.7364	−0.690750
$520$	12.3494	0.541557
$521$	−13.2638	−0.581099	−0.290549	−	0.956860i	$-0.593838\pi$
$521$	−13.2638	−0.581099	−0.290549	+	0.956860i	$0.593838\pi$
$522$	−2.32153	−0.101611
$523$	12.6709	0.554061	0.277030	−	0.960861i	$-0.410650\pi$
$523$	12.6709	0.554061	0.277030	+	0.960861i	$0.410650\pi$
$524$	6.39323	0.279290
$525$	1.83088	0.0799062
$526$	−4.33404	−0.188973
$527$	−16.1541	−0.703682
$528$	−1.00000	−0.0435194
$529$	−19.9189	−0.866039
$530$	−1.83215	−0.0795835
$531$	6.61123	0.286903
$532$	8.61916	0.373688
$533$	0.418883	0.0181438
$534$	−4.63572	−0.200607
$535$	−3.37062	−0.145725
$536$	−7.08992	−0.306238
$537$	−13.1480	−0.567377
$538$	11.4364	0.493058
$539$	4.51099	0.194302
$540$	2.48204	0.106810
$541$	−1.05233	−0.0452432	−0.0226216	−	0.999744i	$-0.507201\pi$
$541$	−1.05233	−0.0452432	−0.0226216	+	0.999744i	$0.507201\pi$
$542$	−6.09101	−0.261631
$543$	−23.7767	−1.02036
$544$	−7.22020	−0.309564
$545$	16.3402	0.699936
$546$	7.84966	0.335935
$547$	27.8952	1.19271	0.596357	−	0.802720i	$-0.296614\pi$
$547$	27.8952	1.19271	0.596357	+	0.802720i	$0.296614\pi$
$548$	−2.37624	−0.101508
$549$	−1.00000	−0.0426790
$550$	1.16050	0.0494840
$551$	12.6831	0.540319
$552$	1.75531	0.0747109
$553$	16.5435	0.703503
$554$	−9.24397	−0.392739
$555$	9.07061	0.385026
$556$	6.12294	0.259670
$557$	12.6051	0.534097	0.267049	−	0.963683i	$-0.413952\pi$
$557$	12.6051	0.534097	0.267049	+	0.963683i	$0.413952\pi$
$558$	2.23734	0.0947143
$559$	−19.5197	−0.825596
$560$	3.91581	0.165473
$561$	−7.22020	−0.304837
$562$	−4.42234	−0.186545
$563$	33.6327	1.41745	0.708724	−	0.705486i	$-0.249271\pi$
$563$	33.6327	1.41745	0.708724	+	0.705486i	$0.249271\pi$
$564$	−5.61805	−0.236562
$565$	−11.3788	−0.478712
$566$	12.6077	0.529940
$567$	1.57766	0.0662555
$568$	−1.20142	−0.0504103
$569$	0.260216	0.0109088	0.00545440	−	0.999985i	$-0.498264\pi$
$569$	0.260216	0.0109088	0.00545440	+	0.999985i	$0.498264\pi$
$570$	−13.5600	−0.567965
$571$	6.25904	0.261933	0.130966	−	0.991387i	$-0.458192\pi$
$571$	6.25904	0.261933	0.130966	+	0.991387i	$0.458192\pi$
$572$	4.97551	0.208036
$573$	−22.9311	−0.957962
$574$	0.132822	0.00554387
$575$	−2.03704	−0.0849504
$576$	1.00000	0.0416667
$577$	−41.0482	−1.70886	−0.854429	−	0.519568i	$-0.826093\pi$
$577$	−41.0482	−1.70886	−0.854429	+	0.519568i	$0.826093\pi$
$578$	−35.1313	−1.46127
$579$	15.2772	0.634897
$580$	5.76213	0.239259
$581$	0.566539	0.0235040
$582$	−5.94529	−0.246440
$583$	−0.738164	−0.0305716
$584$	−12.1189	−0.501482
$585$	−12.3494	−0.510584
$586$	−5.90312	−0.243856
$587$	−44.9987	−1.85730	−0.928649	−	0.370961i	$-0.879028\pi$
$587$	−44.9987	−1.85730	−0.928649	+	0.370961i	$0.879028\pi$
$588$	−4.51099	−0.186030
$589$	−12.2232	−0.503647
$590$	−16.4093	−0.675560
$591$	−9.38641	−0.386105
$592$	3.65450	0.150199
$593$	−38.2024	−1.56878	−0.784391	−	0.620266i	$-0.787025\pi$
$593$	−38.2024	−1.56878	−0.784391	+	0.620266i	$0.787025\pi$
$594$	1.00000	0.0410305
$595$	28.2729	1.15908
$596$	14.0581	0.575841
$597$	11.3483	0.464456
$598$	−8.73355	−0.357141
$599$	15.2200	0.621874	0.310937	−	0.950431i	$-0.399357\pi$
$599$	15.2200	0.621874	0.310937	+	0.950431i	$0.399357\pi$
$600$	−1.16050	−0.0473773
$601$	−15.8396	−0.646112	−0.323056	−	0.946380i	$-0.604710\pi$
$601$	−15.8396	−0.646112	−0.323056	+	0.946380i	$0.604710\pi$
$602$	−6.18941	−0.252262
$603$	7.08992	0.288724
$604$	−10.2538	−0.417220
$605$	2.48204	0.100909
$606$	12.6243	0.512827
$607$	15.1823	0.616229	0.308115	−	0.951349i	$-0.400302\pi$
$607$	15.1823	0.616229	0.308115	+	0.951349i	$0.400302\pi$
$608$	−5.46325	−0.221564
$609$	3.66259	0.148416
$610$	2.48204	0.100495
$611$	27.9526	1.13084
$612$	7.22020	0.291859
$613$	3.87932	0.156684	0.0783421	−	0.996927i	$-0.475037\pi$
$613$	3.87932	0.156684	0.0783421	+	0.996927i	$0.475037\pi$
$614$	−18.1342	−0.731837
$615$	−0.208960	−0.00842609
$616$	1.57766	0.0635658
$617$	19.1740	0.771916	0.385958	−	0.922516i	$-0.373871\pi$
$617$	19.1740	0.771916	0.385958	+	0.922516i	$0.373871\pi$
$618$	−10.1900	−0.409901
$619$	7.44720	0.299328	0.149664	−	0.988737i	$-0.452181\pi$
$619$	7.44720	0.299328	0.149664	+	0.988737i	$0.452181\pi$
$620$	−5.55317	−0.223021
$621$	−1.75531	−0.0704381
$622$	14.9799	0.600641
$623$	7.31359	0.293013
$624$	−4.97551	−0.199180
$625$	−29.4557	−1.17823
$626$	12.9832	0.518914
$627$	−5.46325	−0.218181
$628$	−8.82517	−0.352163
$629$	26.3862	1.05209
$630$	−3.91581	−0.156010
$631$	4.60480	0.183314	0.0916570	−	0.995791i	$-0.470784\pi$
$631$	4.60480	0.183314	0.0916570	+	0.995791i	$0.470784\pi$
$632$	−10.4861	−0.417116
$633$	19.7222	0.783887
$634$	29.4666	1.17027
$635$	−16.9545	−0.672820
$636$	0.738164	0.0292701
$637$	22.4444	0.889281
$638$	2.32153	0.0919104
$639$	1.20142	0.0475273
$640$	−2.48204	−0.0981111
$641$	−3.00269	−0.118599	−0.0592996	−	0.998240i	$-0.518887\pi$
$641$	−3.00269	−0.118599	−0.0592996	+	0.998240i	$0.518887\pi$
$642$	1.35801	0.0535963
$643$	−18.0116	−0.710310	−0.355155	−	0.934807i	$-0.615572\pi$
$643$	−18.0116	−0.710310	−0.355155	+	0.934807i	$0.615572\pi$
$644$	−2.76928	−0.109125
$645$	9.73742	0.383411
$646$	−39.4458	−1.55197
$647$	49.1170	1.93099	0.965493	−	0.260427i	$-0.0838634\pi$
$647$	49.1170	1.93099	0.965493	+	0.260427i	$0.0838634\pi$
$648$	−1.00000	−0.0392837
$649$	−6.61123	−0.259513
$650$	5.77409	0.226479
$651$	−3.52977	−0.138343
$652$	−11.7260	−0.459226
$653$	17.1394	0.670718	0.335359	−	0.942090i	$-0.391142\pi$
$653$	17.1394	0.670718	0.335359	+	0.942090i	$0.391142\pi$
$654$	−6.58337	−0.257430
$655$	15.8682	0.620023
$656$	−0.0841890	−0.00328703
$657$	12.1189	0.472802
$658$	8.86337	0.345530
$659$	2.75457	0.107303	0.0536513	−	0.998560i	$-0.482914\pi$
$659$	2.75457	0.107303	0.0536513	+	0.998560i	$0.482914\pi$
$660$	−2.48204	−0.0966131
$661$	32.8559	1.27795	0.638973	−	0.769230i	$-0.279360\pi$
$661$	32.8559	1.27795	0.638973	+	0.769230i	$0.279360\pi$
$662$	13.0564	0.507449
$663$	−35.9242	−1.39518
$664$	−0.359100	−0.0139358
$665$	21.3931	0.829587
$666$	−3.65450	−0.141609
$667$	−4.07501	−0.157785
$668$	3.73191	0.144392
$669$	−12.0039	−0.464099
$670$	−17.5974	−0.679848
$671$	1.00000	0.0386046
$672$	−1.57766	−0.0608596
$673$	−2.74514	−0.105817	−0.0529087	−	0.998599i	$-0.516849\pi$
$673$	−2.74514	−0.105817	−0.0529087	+	0.998599i	$0.516849\pi$
$674$	−5.16212	−0.198838
$675$	1.16050	0.0446678
$676$	11.7557	0.452141
$677$	−21.9923	−0.845234	−0.422617	−	0.906308i	$-0.638888\pi$
$677$	−21.9923	−0.845234	−0.422617	+	0.906308i	$0.638888\pi$
$678$	4.58448	0.176066
$679$	9.37965	0.359958
$680$	−17.9208	−0.687231
$681$	3.72526	0.142752
$682$	−2.23734	−0.0856723
$683$	−35.2564	−1.34905	−0.674525	−	0.738252i	$-0.735651\pi$
$683$	−35.2564	−1.34905	−0.674525	+	0.738252i	$0.735651\pi$
$684$	5.46325	0.208893
$685$	−5.89792	−0.225348
$686$	18.1604	0.693369
$687$	23.6720	0.903144
$688$	3.92316	0.149569
$689$	−3.67274	−0.139920
$690$	4.35674	0.165858
$691$	28.0494	1.06705	0.533524	−	0.845785i	$-0.320867\pi$
$691$	28.0494	1.06705	0.533524	+	0.845785i	$0.320867\pi$
$692$	−15.7364	−0.598207
$693$	−1.57766	−0.0599304
$694$	−12.2368	−0.464502
$695$	15.1973	0.576468
$696$	−2.32153	−0.0879975
$697$	−0.607861	−0.0230244
$698$	18.5418	0.701818
$699$	−10.8622	−0.410847
$700$	1.83088	0.0692008
$701$	2.85019	0.107650	0.0538251	−	0.998550i	$-0.482859\pi$
$701$	2.85019	0.107650	0.0538251	+	0.998550i	$0.482859\pi$
$702$	4.97551	0.187788
$703$	19.9655	0.753012
$704$	−1.00000	−0.0376889
$705$	−13.9442	−0.525169
$706$	−1.97315	−0.0742603
$707$	−19.9169	−0.749051
$708$	6.61123	0.248465
$709$	17.2618	0.648279	0.324140	−	0.946009i	$-0.394925\pi$
$709$	17.2618	0.648279	0.324140	+	0.946009i	$0.394925\pi$
$710$	−2.98196	−0.111911
$711$	10.4861	0.393260
$712$	−4.63572	−0.173731
$713$	3.92723	0.147076
$714$	−11.3910	−0.426298
$715$	12.3494	0.461841
$716$	−13.1480	−0.491363
$717$	1.82533	0.0681681
$718$	9.23490	0.344643
$719$	37.7255	1.40692	0.703462	−	0.710733i	$-0.251637\pi$
$719$	37.7255	1.40692	0.703462	+	0.710733i	$0.251637\pi$
$720$	2.48204	0.0925000
$721$	16.0763	0.598714
$722$	−10.8471	−0.403688
$723$	17.4546	0.649145
$724$	−23.7767	−0.883655
$725$	2.69415	0.100058
$726$	−1.00000	−0.0371135
$727$	−30.4200	−1.12821	−0.564107	−	0.825702i	$-0.690780\pi$
$727$	−30.4200	−1.12821	−0.564107	+	0.825702i	$0.690780\pi$
$728$	7.84966	0.290928
$729$	1.00000	0.0370370
$730$	−30.0795	−1.11329
$731$	28.3260	1.04767
$732$	−1.00000	−0.0369611
$733$	21.8473	0.806949	0.403475	−	0.914991i	$-0.367802\pi$
$733$	21.8473	0.806949	0.403475	+	0.914991i	$0.367802\pi$
$734$	−23.3975	−0.863617
$735$	−11.1964	−0.412987
$736$	1.75531	0.0647015
$737$	−7.08992	−0.261160
$738$	0.0841890	0.00309904
$739$	3.62541	0.133363	0.0666815	−	0.997774i	$-0.478759\pi$
$739$	3.62541	0.133363	0.0666815	+	0.997774i	$0.478759\pi$
$740$	9.07061	0.333442
$741$	−27.1825	−0.998572
$742$	−1.16457	−0.0427528
$743$	−28.6055	−1.04944	−0.524718	−	0.851276i	$-0.675829\pi$
$743$	−28.6055	−1.04944	−0.524718	+	0.851276i	$0.675829\pi$
$744$	2.23734	0.0820250
$745$	34.8927	1.27837
$746$	−6.44001	−0.235786
$747$	0.359100	0.0131388
$748$	−7.22020	−0.263997
$749$	−2.14247	−0.0782843
$750$	9.52977	0.347978
$751$	4.90312	0.178918	0.0894588	−	0.995991i	$-0.471486\pi$
$751$	4.90312	0.178918	0.0894588	+	0.995991i	$0.471486\pi$
$752$	−5.61805	−0.204869
$753$	20.2679	0.738605
$754$	11.5508	0.420656
$755$	−25.4502	−0.926228
$756$	1.57766	0.0573790
$757$	45.3746	1.64917	0.824583	−	0.565741i	$-0.191410\pi$
$757$	45.3746	1.64917	0.824583	+	0.565741i	$0.191410\pi$
$758$	−13.5104	−0.490721
$759$	1.75531	0.0637136
$760$	−13.5600	−0.491873
$761$	−32.0786	−1.16285	−0.581424	−	0.813601i	$-0.697504\pi$
$761$	−32.0786	−1.16285	−0.581424	+	0.813601i	$0.697504\pi$
$762$	6.83090	0.247457
$763$	10.3863	0.376010
$764$	−22.9311	−0.829619
$765$	17.9208	0.647928
$766$	16.5203	0.596901
$767$	−32.8942	−1.18774
$768$	1.00000	0.0360844
$769$	−22.0409	−0.794815	−0.397407	−	0.917642i	$-0.630090\pi$
$769$	−22.0409	−0.794815	−0.397407	+	0.917642i	$0.630090\pi$
$770$	3.91581	0.141116
$771$	7.60217	0.273786
$772$	15.2772	0.549837
$773$	1.84767	0.0664562	0.0332281	−	0.999448i	$-0.489421\pi$
$773$	1.84767	0.0664562	0.0332281	+	0.999448i	$0.489421\pi$
$774$	−3.92316	−0.141015
$775$	−2.59644	−0.0932670
$776$	−5.94529	−0.213423
$777$	5.76556	0.206838
$778$	11.1952	0.401366
$779$	−0.459946	−0.0164793
$780$	−12.3494	−0.442179
$781$	−1.20142	−0.0429901
$782$	12.6737	0.453210
$783$	2.32153	0.0829648
$784$	−4.51099	−0.161107
$785$	−21.9044	−0.781801
$786$	−6.39323	−0.228039
$787$	−35.6316	−1.27013	−0.635064	−	0.772459i	$-0.719026\pi$
$787$	−35.6316	−1.27013	−0.635064	+	0.772459i	$0.719026\pi$
$788$	−9.38641	−0.334377
$789$	4.33404	0.154296
$790$	−26.0269	−0.925997
$791$	−7.23276	−0.257167
$792$	1.00000	0.0355335
$793$	4.97551	0.176685
$794$	21.6990	0.770068
$795$	1.83215	0.0649796
$796$	11.3483	0.402230
$797$	−7.69723	−0.272650	−0.136325	−	0.990664i	$-0.543529\pi$
$797$	−7.69723	−0.272650	−0.136325	+	0.990664i	$0.543529\pi$
$798$	−8.61916	−0.305115
$799$	−40.5634	−1.43503
$800$	−1.16050	−0.0410300
$801$	4.63572	0.163795
$802$	18.4153	0.650268
$803$	−12.1189	−0.427666
$804$	7.08992	0.250042
$805$	−6.87345	−0.242257
$806$	−11.1319	−0.392105
$807$	−11.4364	−0.402580
$808$	12.6243	0.444121
$809$	−25.4627	−0.895222	−0.447611	−	0.894228i	$-0.647725\pi$
$809$	−25.4627	−0.895222	−0.447611	+	0.894228i	$0.647725\pi$
$810$	−2.48204	−0.0872099
$811$	−46.0069	−1.61552	−0.807760	−	0.589511i	$-0.799321\pi$
$811$	−46.0069	−1.61552	−0.807760	+	0.589511i	$0.799321\pi$
$812$	3.66259	0.128532
$813$	6.09101	0.213621
$814$	3.65450	0.128090
$815$	−29.1044	−1.01948
$816$	7.22020	0.252758
$817$	21.4332	0.749853
$818$	−4.30492	−0.150518
$819$	−7.84966	−0.274289
$820$	−0.208960	−0.00729720
$821$	−55.5046	−1.93712	−0.968562	−	0.248771i	$-0.919973\pi$
$821$	−55.5046	−1.93712	−0.968562	+	0.248771i	$0.919973\pi$
$822$	2.37624	0.0828810
$823$	15.8969	0.554130	0.277065	−	0.960851i	$-0.410638\pi$
$823$	15.8969	0.554130	0.277065	+	0.960851i	$0.410638\pi$
$824$	−10.1900	−0.354985
$825$	−1.16050	−0.0404035
$826$	−10.4303	−0.362916
$827$	31.5467	1.09699	0.548493	−	0.836155i	$-0.315202\pi$
$827$	31.5467	1.09699	0.548493	+	0.836155i	$0.315202\pi$
$828$	−1.75531	−0.0610012
$829$	23.0397	0.800203	0.400101	−	0.916471i	$-0.368975\pi$
$829$	23.0397	0.800203	0.400101	+	0.916471i	$0.368975\pi$
$830$	−0.891300	−0.0309375
$831$	9.24397	0.320670
$832$	−4.97551	−0.172495
$833$	−32.5702	−1.12849
$834$	−6.12294	−0.212020
$835$	9.26273	0.320550
$836$	−5.46325	−0.188951
$837$	−2.23734	−0.0773339
$838$	7.35619	0.254115
$839$	−18.9841	−0.655405	−0.327703	−	0.944781i	$-0.606274\pi$
$839$	−18.9841	−0.655405	−0.327703	+	0.944781i	$0.606274\pi$
$840$	−3.91581	−0.135108
$841$	−23.6105	−0.814155
$842$	−26.5246	−0.914098
$843$	4.42234	0.152313
$844$	19.7222	0.678866
$845$	29.1780	1.00375
$846$	5.61805	0.193152
$847$	1.57766	0.0542091
$848$	0.738164	0.0253486
$849$	−12.6077	−0.432695
$850$	−8.37906	−0.287400
$851$	−6.41478	−0.219896
$852$	1.20142	0.0411598
$853$	−17.8157	−0.610000	−0.305000	−	0.952352i	$-0.598656\pi$
$853$	−17.8157	−0.610000	−0.305000	+	0.952352i	$0.598656\pi$
$854$	1.57766	0.0539864
$855$	13.5600	0.463742
$856$	1.35801	0.0464157
$857$	−23.7992	−0.812966	−0.406483	−	0.913658i	$-0.633245\pi$
$857$	−23.7992	−0.812966	−0.406483	+	0.913658i	$0.633245\pi$
$858$	−4.97551	−0.169861
$859$	−36.6484	−1.25043	−0.625214	−	0.780453i	$-0.714988\pi$
$859$	−36.6484	−1.25043	−0.625214	+	0.780453i	$0.714988\pi$
$860$	9.73742	0.332043
$861$	−0.132822	−0.00452655
$862$	−22.2025	−0.756221
$863$	2.33478	0.0794770	0.0397385	−	0.999210i	$-0.487348\pi$
$863$	2.33478	0.0794770	0.0397385	+	0.999210i	$0.487348\pi$
$864$	−1.00000	−0.0340207
$865$	−39.0582	−1.32802
$866$	−16.1513	−0.548844
$867$	35.1313	1.19312
$868$	−3.52977	−0.119808
$869$	−10.4861	−0.355717
$870$	−5.76213	−0.195354
$871$	−35.2759	−1.19528
$872$	−6.58337	−0.222941
$873$	5.94529	0.201218
$874$	9.58969	0.324376
$875$	−15.0347	−0.508267
$876$	12.1189	0.409459
$877$	−30.5720	−1.03234	−0.516172	−	0.856485i	$-0.672644\pi$
$877$	−30.5720	−1.03234	−0.516172	+	0.856485i	$0.672644\pi$
$878$	−3.76764	−0.127152
$879$	5.90312	0.199107
$880$	−2.48204	−0.0836694
$881$	39.6527	1.33593	0.667966	−	0.744192i	$-0.267165\pi$
$881$	39.6527	1.33593	0.667966	+	0.744192i	$0.267165\pi$
$882$	4.51099	0.151893
$883$	24.9384	0.839246	0.419623	−	0.907699i	$-0.362162\pi$
$883$	24.9384	0.839246	0.419623	+	0.907699i	$0.362162\pi$
$884$	−35.9242	−1.20826
$885$	16.4093	0.551593
$886$	28.9033	0.971025
$887$	−35.3662	−1.18748	−0.593740	−	0.804657i	$-0.702349\pi$
$887$	−35.3662	−1.18748	−0.593740	+	0.804657i	$0.702349\pi$
$888$	−3.65450	−0.122637
$889$	−10.7768	−0.361444
$890$	−11.5060	−0.385683
$891$	−1.00000	−0.0335013
$892$	−12.0039	−0.401921
$893$	−30.6928	−1.02710
$894$	−14.0581	−0.470172
$895$	−32.6337	−1.09083
$896$	−1.57766	−0.0527059
$897$	8.73355	0.291605
$898$	17.3249	0.578139
$899$	−5.19407	−0.173232
$900$	1.16050	0.0386834
$901$	5.32969	0.177558
$902$	−0.0841890	−0.00280319
$903$	6.18941	0.205971
$904$	4.58448	0.152478
$905$	−59.0147	−1.96171
$906$	10.2538	0.340659
$907$	−24.1847	−0.803041	−0.401520	−	0.915850i	$-0.631518\pi$
$907$	−24.1847	−0.803041	−0.401520	+	0.915850i	$0.631518\pi$
$908$	3.72526	0.123627
$909$	−12.6243	−0.418722
$910$	19.4831	0.645860
$911$	−30.1740	−0.999710	−0.499855	−	0.866109i	$-0.666613\pi$
$911$	−30.1740	−0.999710	−0.499855	+	0.866109i	$0.666613\pi$
$912$	5.46325	0.180906
$913$	−0.359100	−0.0118845
$914$	27.9527	0.924594
$915$	−2.48204	−0.0820536
$916$	23.6720	0.782146
$917$	10.0864	0.333081
$918$	−7.22020	−0.238302
$919$	44.9120	1.48151	0.740755	−	0.671775i	$-0.234468\pi$
$919$	44.9120	1.48151	0.740755	+	0.671775i	$0.234468\pi$
$920$	4.35674	0.143637
$921$	18.1342	0.597543
$922$	19.7301	0.649776
$923$	−5.97766	−0.196757
$924$	−1.57766	−0.0519012
$925$	4.24106	0.139445
$926$	35.0083	1.15045
$927$	10.1900	0.334683
$928$	−2.32153	−0.0762081
$929$	−58.7619	−1.92791	−0.963957	−	0.266057i	$-0.914279\pi$
$929$	−58.7619	−1.92791	−0.963957	+	0.266057i	$0.914279\pi$
$930$	5.55317	0.182096
$931$	−24.6447	−0.807696
$932$	−10.8622	−0.355804
$933$	−14.9799	−0.490421
$934$	−2.95172	−0.0965832
$935$	−17.9208	−0.586073
$936$	4.97551	0.162630
$937$	−40.8754	−1.33534	−0.667671	−	0.744457i	$-0.732709\pi$
$937$	−40.8754	−1.33534	−0.667671	+	0.744457i	$0.732709\pi$
$938$	−11.1855	−0.365219
$939$	−12.9832	−0.423692
$940$	−13.9442	−0.454809
$941$	2.90275	0.0946271	0.0473135	−	0.998880i	$-0.484934\pi$
$941$	2.90275	0.0946271	0.0473135	+	0.998880i	$0.484934\pi$
$942$	8.82517	0.287540
$943$	0.147778	0.00481230
$944$	6.61123	0.215177
$945$	3.91581	0.127381
$946$	3.92316	0.127553
$947$	16.0737	0.522324	0.261162	−	0.965295i	$-0.415894\pi$
$947$	16.0737	0.522324	0.261162	+	0.965295i	$0.415894\pi$
$948$	10.4861	0.340573
$949$	−60.2975	−1.95734
$950$	−6.34012	−0.205701
$951$	−29.4666	−0.955521
$952$	−11.3910	−0.369185
$953$	49.3712	1.59929	0.799645	−	0.600473i	$-0.205021\pi$
$953$	49.3712	1.59929	0.799645	+	0.600473i	$0.205021\pi$
$954$	−0.738164	−0.0238989
$955$	−56.9159	−1.84176
$956$	1.82533	0.0590353
$957$	−2.32153	−0.0750445
$958$	−13.6000	−0.439395
$959$	−3.74891	−0.121059
$960$	2.48204	0.0801074
$961$	−25.9943	−0.838526
$962$	18.1830	0.586244
$963$	−1.35801	−0.0437612
$964$	17.4546	0.562176
$965$	37.9185	1.22064
$966$	2.76928	0.0891001
$967$	32.1599	1.03419	0.517096	−	0.855927i	$-0.327013\pi$
$967$	32.1599	1.03419	0.517096	+	0.855927i	$0.327013\pi$
$968$	−1.00000	−0.0321412
$969$	39.4458	1.26718
$970$	−14.7564	−0.473800
$971$	−8.44075	−0.270877	−0.135438	−	0.990786i	$-0.543244\pi$
$971$	−8.44075	−0.270877	−0.135438	+	0.990786i	$0.543244\pi$
$972$	1.00000	0.0320750
$973$	9.65992	0.309683
$974$	−29.5866	−0.948015
$975$	−5.77409	−0.184919
$976$	−1.00000	−0.0320092
$977$	−57.5522	−1.84126	−0.920629	−	0.390439i	$-0.872323\pi$
$977$	−57.5522	−1.84126	−0.920629	+	0.390439i	$0.872323\pi$
$978$	11.7260	0.374956
$979$	−4.63572	−0.148158
$980$	−11.1964	−0.357657
$981$	6.58337	0.210191
$982$	7.69301	0.245494
$983$	−12.4597	−0.397404	−0.198702	−	0.980060i	$-0.563673\pi$
$983$	−12.4597	−0.397404	−0.198702	+	0.980060i	$0.563673\pi$
$984$	0.0841890	0.00268385
$985$	−23.2974	−0.742317
$986$	−16.7619	−0.533809
$987$	−8.86337	−0.282124
$988$	−27.1825	−0.864789
$989$	−6.88635	−0.218973
$990$	2.48204	0.0788843
$991$	−27.3577	−0.869046	−0.434523	−	0.900661i	$-0.643083\pi$
$991$	−27.3577	−0.869046	−0.434523	+	0.900661i	$0.643083\pi$
$992$	2.23734	0.0710357
$993$	−13.0564	−0.414331
$994$	−1.89543	−0.0601193
$995$	28.1669	0.892952
$996$	0.359100	0.0113785
$997$	−49.6451	−1.57228	−0.786138	−	0.618051i	$-0.787922\pi$
$997$	−49.6451	−1.57228	−0.786138	+	0.618051i	$0.787922\pi$
$998$	1.53213	0.0484988
$999$	3.65450	0.115623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.u.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.u.1.5	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.48204 q^{5} -1.00000 q^{6} +1.57766 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.48204 q^{5} -1.00000 q^{6} +1.57766 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.48204 q^{10} -1.00000 q^{11} +1.00000 q^{12} -4.97551 q^{13} -1.57766 q^{14} +2.48204 q^{15} +1.00000 q^{16} +7.22020 q^{17} -1.00000 q^{18} +5.46325 q^{19} +2.48204 q^{20} +1.57766 q^{21} +1.00000 q^{22} -1.75531 q^{23} -1.00000 q^{24} +1.16050 q^{25} +4.97551 q^{26} +1.00000 q^{27} +1.57766 q^{28} +2.32153 q^{29} -2.48204 q^{30} -2.23734 q^{31} -1.00000 q^{32} -1.00000 q^{33} -7.22020 q^{34} +3.91581 q^{35} +1.00000 q^{36} +3.65450 q^{37} -5.46325 q^{38} -4.97551 q^{39} -2.48204 q^{40} -0.0841890 q^{41} -1.57766 q^{42} +3.92316 q^{43} -1.00000 q^{44} +2.48204 q^{45} +1.75531 q^{46} -5.61805 q^{47} +1.00000 q^{48} -4.51099 q^{49} -1.16050 q^{50} +7.22020 q^{51} -4.97551 q^{52} +0.738164 q^{53} -1.00000 q^{54} -2.48204 q^{55} -1.57766 q^{56} +5.46325 q^{57} -2.32153 q^{58} +6.61123 q^{59} +2.48204 q^{60} -1.00000 q^{61} +2.23734 q^{62} +1.57766 q^{63} +1.00000 q^{64} -12.3494 q^{65} +1.00000 q^{66} +7.08992 q^{67} +7.22020 q^{68} -1.75531 q^{69} -3.91581 q^{70} +1.20142 q^{71} -1.00000 q^{72} +12.1189 q^{73} -3.65450 q^{74} +1.16050 q^{75} +5.46325 q^{76} -1.57766 q^{77} +4.97551 q^{78} +10.4861 q^{79} +2.48204 q^{80} +1.00000 q^{81} +0.0841890 q^{82} +0.359100 q^{83} +1.57766 q^{84} +17.9208 q^{85} -3.92316 q^{86} +2.32153 q^{87} +1.00000 q^{88} +4.63572 q^{89} -2.48204 q^{90} -7.84966 q^{91} -1.75531 q^{92} -2.23734 q^{93} +5.61805 q^{94} +13.5600 q^{95} -1.00000 q^{96} +5.94529 q^{97} +4.51099 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} - 3 q^{7} - 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - q^5 - 5 * q^6 - 3 * q^7 - 5 * q^8 + 5 * q^9	\( 5 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - q^{5} - 5 q^{6} - 3 q^{7} - 5 q^{8} + 5 q^{9} + q^{10} - 5 q^{11} + 5 q^{12} + 2 q^{13} + 3 q^{14} - q^{15} + 5 q^{16} + 6 q^{17} - 5 q^{18} + 7 q^{19} - q^{20} - 3 q^{21} + 5 q^{22} - 12 q^{23} - 5 q^{24} - 2 q^{26} + 5 q^{27} - 3 q^{28} + 4 q^{29} + q^{30} - q^{31} - 5 q^{32} - 5 q^{33} - 6 q^{34} + 17 q^{35} + 5 q^{36} + 3 q^{37} - 7 q^{38} + 2 q^{39} + q^{40} - 3 q^{41} + 3 q^{42} + 24 q^{43} - 5 q^{44} - q^{45} + 12 q^{46} + 18 q^{47} + 5 q^{48} + 26 q^{49} + 6 q^{51} + 2 q^{52} - 13 q^{53} - 5 q^{54} + q^{55} + 3 q^{56} + 7 q^{57} - 4 q^{58} - 16 q^{59} - q^{60} - 5 q^{61} + q^{62} - 3 q^{63} + 5 q^{64} + 4 q^{65} + 5 q^{66} + 18 q^{67} + 6 q^{68} - 12 q^{69} - 17 q^{70} - 31 q^{71} - 5 q^{72} + 8 q^{73} - 3 q^{74} + 7 q^{76} + 3 q^{77} - 2 q^{78} + 32 q^{79} - q^{80} + 5 q^{81} + 3 q^{82} + 8 q^{83} - 3 q^{84} + 29 q^{85} - 24 q^{86} + 4 q^{87} + 5 q^{88} + q^{89} + q^{90} - 3 q^{91} - 12 q^{92} - q^{93} - 18 q^{94} + 33 q^{95} - 5 q^{96} - 4 q^{97} - 26 q^{98} - 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - q^5 - 5 * q^6 - 3 * q^7 - 5 * q^8 + 5 * q^9 + q^10 - 5 * q^11 + 5 * q^12 + 2 * q^13 + 3 * q^14 - q^15 + 5 * q^16 + 6 * q^17 - 5 * q^18 + 7 * q^19 - q^20 - 3 * q^21 + 5 * q^22 - 12 * q^23 - 5 * q^24 - 2 * q^26 + 5 * q^27 - 3 * q^28 + 4 * q^29 + q^30 - q^31 - 5 * q^32 - 5 * q^33 - 6 * q^34 + 17 * q^35 + 5 * q^36 + 3 * q^37 - 7 * q^38 + 2 * q^39 + q^40 - 3 * q^41 + 3 * q^42 + 24 * q^43 - 5 * q^44 - q^45 + 12 * q^46 + 18 * q^47 + 5 * q^48 + 26 * q^49 + 6 * q^51 + 2 * q^52 - 13 * q^53 - 5 * q^54 + q^55 + 3 * q^56 + 7 * q^57 - 4 * q^58 - 16 * q^59 - q^60 - 5 * q^61 + q^62 - 3 * q^63 + 5 * q^64 + 4 * q^65 + 5 * q^66 + 18 * q^67 + 6 * q^68 - 12 * q^69 - 17 * q^70 - 31 * q^71 - 5 * q^72 + 8 * q^73 - 3 * q^74 + 7 * q^76 + 3 * q^77 - 2 * q^78 + 32 * q^79 - q^80 + 5 * q^81 + 3 * q^82 + 8 * q^83 - 3 * q^84 + 29 * q^85 - 24 * q^86 + 4 * q^87 + 5 * q^88 + q^89 + q^90 - 3 * q^91 - 12 * q^92 - q^93 - 18 * q^94 + 33 * q^95 - 5 * q^96 - 4 * q^97 - 26 * q^98 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more