LMFDB - Embedded newform 7098.2.a.co.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, 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("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.27743$ of defining polynomial
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.332808 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.332808 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.332808 q^{10} +2.61023 q^{11} -1.00000 q^{12} +1.00000 q^{14} -0.332808 q^{15} +1.00000 q^{16} -3.88766 q^{17} +1.00000 q^{18} +5.61181 q^{19} +0.332808 q^{20} -1.00000 q^{21} +2.61023 q^{22} +4.21257 q^{23} -1.00000 q^{24} -4.88924 q^{25} -1.00000 q^{27} +1.00000 q^{28} -1.18667 q^{29} -0.332808 q^{30} +7.07434 q^{31} +1.00000 q^{32} -2.61023 q^{33} -3.88766 q^{34} +0.332808 q^{35} +1.00000 q^{36} +0.576440 q^{37} +5.61181 q^{38} +0.332808 q^{40} -0.521059 q^{41} -1.00000 q^{42} +3.07434 q^{43} +2.61023 q^{44} +0.332808 q^{45} +4.21257 q^{46} +12.0528 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.88924 q^{50} +3.88766 q^{51} -9.71047 q^{53} -1.00000 q^{54} +0.868706 q^{55} +1.00000 q^{56} -5.61181 q^{57} -1.18667 q^{58} -10.4752 q^{59} -0.332808 q^{60} +7.43978 q^{61} +7.07434 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.61023 q^{66} -11.9203 q^{67} -3.88766 q^{68} -4.21257 q^{69} +0.332808 q^{70} -0.944620 q^{71} +1.00000 q^{72} +4.66562 q^{73} +0.576440 q^{74} +4.88924 q^{75} +5.61181 q^{76} +2.61023 q^{77} -0.943042 q^{79} +0.332808 q^{80} +1.00000 q^{81} -0.521059 q^{82} +13.7172 q^{83} -1.00000 q^{84} -1.29384 q^{85} +3.07434 q^{86} +1.18667 q^{87} +2.61023 q^{88} +3.38029 q^{89} +0.332808 q^{90} +4.21257 q^{92} -7.07434 q^{93} +12.0528 q^{94} +1.86765 q^{95} -1.00000 q^{96} -5.81587 q^{97} +1.00000 q^{98} +2.61023 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9} + 6 q^{10} + 4 q^{11} - 4 q^{12} + 4 q^{14} - 6 q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} + 2 q^{19} + 6 q^{20} - 4 q^{21} + 4 q^{22} + 8 q^{23} - 4 q^{24} + 12 q^{25} - 4 q^{27} + 4 q^{28} - 2 q^{29} - 6 q^{30} + 8 q^{31} + 4 q^{32} - 4 q^{33} + 2 q^{34} + 6 q^{35} + 4 q^{36} + 6 q^{37} + 2 q^{38} + 6 q^{40} + 10 q^{41} - 4 q^{42} - 8 q^{43} + 4 q^{44} + 6 q^{45} + 8 q^{46} + 2 q^{47} - 4 q^{48} + 4 q^{49} + 12 q^{50} - 2 q^{51} - 6 q^{53} - 4 q^{54} + 22 q^{55} + 4 q^{56} - 2 q^{57} - 2 q^{58} + 4 q^{59} - 6 q^{60} + 8 q^{61} + 8 q^{62} + 4 q^{63} + 4 q^{64} - 4 q^{66} - 24 q^{67} + 2 q^{68} - 8 q^{69} + 6 q^{70} + 12 q^{71} + 4 q^{72} + 28 q^{73} + 6 q^{74} - 12 q^{75} + 2 q^{76} + 4 q^{77} - 2 q^{79} + 6 q^{80} + 4 q^{81} + 10 q^{82} + 22 q^{83} - 4 q^{84} - 6 q^{85} - 8 q^{86} + 2 q^{87} + 4 q^{88} + 38 q^{89} + 6 q^{90} + 8 q^{92} - 8 q^{93} + 2 q^{94} - 50 q^{95} - 4 q^{96} + 22 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9 + 6 * q^10 + 4 * q^11 - 4 * q^12 + 4 * q^14 - 6 * q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 + 2 * q^19 + 6 * q^20 - 4 * q^21 + 4 * q^22 + 8 * q^23 - 4 * q^24 + 12 * q^25 - 4 * q^27 + 4 * q^28 - 2 * q^29 - 6 * q^30 + 8 * q^31 + 4 * q^32 - 4 * q^33 + 2 * q^34 + 6 * q^35 + 4 * q^36 + 6 * q^37 + 2 * q^38 + 6 * q^40 + 10 * q^41 - 4 * q^42 - 8 * q^43 + 4 * q^44 + 6 * q^45 + 8 * q^46 + 2 * q^47 - 4 * q^48 + 4 * q^49 + 12 * q^50 - 2 * q^51 - 6 * q^53 - 4 * q^54 + 22 * q^55 + 4 * q^56 - 2 * q^57 - 2 * q^58 + 4 * q^59 - 6 * q^60 + 8 * q^61 + 8 * q^62 + 4 * q^63 + 4 * q^64 - 4 * q^66 - 24 * q^67 + 2 * q^68 - 8 * q^69 + 6 * q^70 + 12 * q^71 + 4 * q^72 + 28 * q^73 + 6 * q^74 - 12 * q^75 + 2 * q^76 + 4 * q^77 - 2 * q^79 + 6 * q^80 + 4 * q^81 + 10 * q^82 + 22 * q^83 - 4 * q^84 - 6 * q^85 - 8 * q^86 + 2 * q^87 + 4 * q^88 + 38 * q^89 + 6 * q^90 + 8 * q^92 - 8 * q^93 + 2 * q^94 - 50 * q^95 - 4 * q^96 + 22 * q^97 + 4 * q^98 + 4 * 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
$5$	0.332808	0.148836	0.0744180	−	0.997227i	$-0.476290\pi$
$5$	0.332808	0.148836	0.0744180	+	0.997227i	$0.476290\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0.332808	0.105243
$11$	2.61023	0.787015	0.393508	−	0.919321i	$-0.371261\pi$
$11$	2.61023	0.787015	0.393508	+	0.919321i	$0.371261\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	−0.332808	−0.0859306
$16$	1.00000	0.250000
$17$	−3.88766	−0.942897	−0.471448	−	0.881894i	$-0.656269\pi$
$17$	−3.88766	−0.942897	−0.471448	+	0.881894i	$0.656269\pi$
$18$	1.00000	0.235702
$19$	5.61181	1.28744	0.643719	−	0.765262i	$-0.277391\pi$
$19$	5.61181	1.28744	0.643719	+	0.765262i	$0.277391\pi$
$20$	0.332808	0.0744180
$21$	−1.00000	−0.218218
$22$	2.61023	0.556504
$23$	4.21257	0.878381	0.439191	−	0.898394i	$-0.355265\pi$
$23$	4.21257	0.878381	0.439191	+	0.898394i	$0.355265\pi$
$24$	−1.00000	−0.204124
$25$	−4.88924	−0.977848
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−1.18667	−0.220360	−0.110180	−	0.993912i	$-0.535143\pi$
$29$	−1.18667	−0.220360	−0.110180	+	0.993912i	$0.535143\pi$
$30$	−0.332808	−0.0607621
$31$	7.07434	1.27059	0.635294	−	0.772270i	$-0.280879\pi$
$31$	7.07434	1.27059	0.635294	+	0.772270i	$0.280879\pi$
$32$	1.00000	0.176777
$33$	−2.61023	−0.454384
$34$	−3.88766	−0.666729
$35$	0.332808	0.0562548
$36$	1.00000	0.166667
$37$	0.576440	0.0947661	0.0473831	−	0.998877i	$-0.484912\pi$
$37$	0.576440	0.0947661	0.0473831	+	0.998877i	$0.484912\pi$
$38$	5.61181	0.910356
$39$	0	0
$40$	0.332808	0.0526215
$41$	−0.521059	−0.0813758	−0.0406879	−	0.999172i	$-0.512955\pi$
$41$	−0.521059	−0.0813758	−0.0406879	+	0.999172i	$0.512955\pi$
$42$	−1.00000	−0.154303
$43$	3.07434	0.468832	0.234416	−	0.972136i	$-0.424682\pi$
$43$	3.07434	0.468832	0.234416	+	0.972136i	$0.424682\pi$
$44$	2.61023	0.393508
$45$	0.332808	0.0496120
$46$	4.21257	0.621109
$47$	12.0528	1.75807	0.879037	−	0.476753i	$-0.158186\pi$
$47$	12.0528	1.75807	0.879037	+	0.476753i	$0.158186\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−4.88924	−0.691443
$51$	3.88766	0.544382
$52$	0	0
$53$	−9.71047	−1.33384	−0.666918	−	0.745132i	$-0.732387\pi$
$53$	−9.71047	−1.33384	−0.666918	+	0.745132i	$0.732387\pi$
$54$	−1.00000	−0.136083
$55$	0.868706	0.117136
$56$	1.00000	0.133631
$57$	−5.61181	−0.743303
$58$	−1.18667	−0.155818
$59$	−10.4752	−1.36375	−0.681875	−	0.731469i	$-0.738835\pi$
$59$	−10.4752	−1.36375	−0.681875	+	0.731469i	$0.738835\pi$
$60$	−0.332808	−0.0429653
$61$	7.43978	0.952567	0.476283	−	0.879292i	$-0.341984\pi$
$61$	7.43978	0.952567	0.476283	+	0.879292i	$0.341984\pi$
$62$	7.07434	0.898442
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.61023	−0.321298
$67$	−11.9203	−1.45630	−0.728148	−	0.685420i	$-0.759619\pi$
$67$	−11.9203	−1.45630	−0.728148	+	0.685420i	$0.759619\pi$
$68$	−3.88766	−0.471448
$69$	−4.21257	−0.507134
$70$	0.332808	0.0397781
$71$	−0.944620	−0.112106	−0.0560529	−	0.998428i	$-0.517852\pi$
$71$	−0.944620	−0.112106	−0.0560529	+	0.998428i	$0.517852\pi$
$72$	1.00000	0.117851
$73$	4.66562	0.546069	0.273034	−	0.962004i	$-0.411973\pi$
$73$	4.66562	0.546069	0.273034	+	0.962004i	$0.411973\pi$
$74$	0.576440	0.0670098
$75$	4.88924	0.564561
$76$	5.61181	0.643719
$77$	2.61023	0.297464
$78$	0	0
$79$	−0.943042	−0.106101	−0.0530503	−	0.998592i	$-0.516894\pi$
$79$	−0.943042	−0.106101	−0.0530503	+	0.998592i	$0.516894\pi$
$80$	0.332808	0.0372090
$81$	1.00000	0.111111
$82$	−0.521059	−0.0575414
$83$	13.7172	1.50566	0.752830	−	0.658215i	$-0.228688\pi$
$83$	13.7172	1.50566	0.752830	+	0.658215i	$0.228688\pi$
$84$	−1.00000	−0.109109
$85$	−1.29384	−0.140337
$86$	3.07434	0.331514
$87$	1.18667	0.127225
$88$	2.61023	0.278252
$89$	3.38029	0.358310	0.179155	−	0.983821i	$-0.442664\pi$
$89$	3.38029	0.358310	0.179155	+	0.983821i	$0.442664\pi$
$90$	0.332808	0.0350810
$91$	0	0
$92$	4.21257	0.439191
$93$	−7.07434	−0.733575
$94$	12.0528	1.24315
$95$	1.86765	0.191617
$96$	−1.00000	−0.102062
$97$	−5.81587	−0.590512	−0.295256	−	0.955418i	$-0.595405\pi$
$97$	−5.81587	−0.590512	−0.295256	+	0.955418i	$0.595405\pi$
$98$	1.00000	0.101015
$99$	2.61023	0.262338
$100$	−4.88924	−0.488924
$101$	−15.9703	−1.58911	−0.794553	−	0.607195i	$-0.792295\pi$
$101$	−15.9703	−1.58911	−0.794553	+	0.607195i	$0.792295\pi$
$102$	3.88766	0.384936
$103$	7.50641	0.739629	0.369814	−	0.929106i	$-0.379421\pi$
$103$	7.50641	0.739629	0.369814	+	0.929106i	$0.379421\pi$
$104$	0	0
$105$	−0.332808	−0.0324787
$106$	−9.71047	−0.943164
$107$	8.65078	0.836302	0.418151	−	0.908378i	$-0.362678\pi$
$107$	8.65078	0.836302	0.418151	+	0.908378i	$0.362678\pi$
$108$	−1.00000	−0.0962250
$109$	18.6144	1.78293	0.891466	−	0.453088i	$-0.149678\pi$
$109$	18.6144	1.78293	0.891466	+	0.453088i	$0.149678\pi$
$110$	0.868706	0.0828279
$111$	−0.576440	−0.0547132
$112$	1.00000	0.0944911
$113$	9.15025	0.860783	0.430392	−	0.902642i	$-0.358375\pi$
$113$	9.15025	0.860783	0.430392	+	0.902642i	$0.358375\pi$
$114$	−5.61181	−0.525594
$115$	1.40197	0.130735
$116$	−1.18667	−0.110180
$117$	0	0
$118$	−10.4752	−0.964316
$119$	−3.88766	−0.356381
$120$	−0.332808	−0.0303810
$121$	−4.18667	−0.380607
$122$	7.43978	0.673566
$123$	0.521059	0.0469823
$124$	7.07434	0.635294
$125$	−3.29121	−0.294375
$126$	1.00000	0.0890871
$127$	6.77953	0.601586	0.300793	−	0.953689i	$-0.402749\pi$
$127$	6.77953	0.601586	0.300793	+	0.953689i	$0.402749\pi$
$128$	1.00000	0.0883883
$129$	−3.07434	−0.270680
$130$	0	0
$131$	7.22879	0.631583	0.315791	−	0.948829i	$-0.397730\pi$
$131$	7.22879	0.631583	0.315791	+	0.948829i	$0.397730\pi$
$132$	−2.61023	−0.227192
$133$	5.61181	0.486606
$134$	−11.9203	−1.02976
$135$	−0.332808	−0.0286435
$136$	−3.88766	−0.333364
$137$	11.8892	1.01577	0.507883	−	0.861426i	$-0.330428\pi$
$137$	11.8892	1.01577	0.507883	+	0.861426i	$0.330428\pi$
$138$	−4.21257	−0.358598
$139$	14.3211	1.21470	0.607351	−	0.794434i	$-0.292232\pi$
$139$	14.3211	1.21470	0.607351	+	0.794434i	$0.292232\pi$
$140$	0.332808	0.0281274
$141$	−12.0528	−1.01502
$142$	−0.944620	−0.0792707
$143$	0	0
$144$	1.00000	0.0833333
$145$	−0.394934	−0.0327975
$146$	4.66562	0.386129
$147$	−1.00000	−0.0824786
$148$	0.576440	0.0473831
$149$	−6.05696	−0.496205	−0.248103	−	0.968734i	$-0.579807\pi$
$149$	−6.05696	−0.496205	−0.248103	+	0.968734i	$0.579807\pi$
$150$	4.88924	0.399205
$151$	−21.4953	−1.74926	−0.874630	−	0.484791i	$-0.838896\pi$
$151$	−21.4953	−1.74926	−0.874630	+	0.484791i	$0.838896\pi$
$152$	5.61181	0.455178
$153$	−3.88766	−0.314299
$154$	2.61023	0.210339
$155$	2.35439	0.189109
$156$	0	0
$157$	2.29227	0.182943	0.0914714	−	0.995808i	$-0.470843\pi$
$157$	2.29227	0.182943	0.0914714	+	0.995808i	$0.470843\pi$
$158$	−0.943042	−0.0750244
$159$	9.71047	0.770090
$160$	0.332808	0.0263108
$161$	4.21257	0.331997
$162$	1.00000	0.0785674
$163$	−3.34071	−0.261664	−0.130832	−	0.991405i	$-0.541765\pi$
$163$	−3.34071	−0.261664	−0.130832	+	0.991405i	$0.541765\pi$
$164$	−0.521059	−0.0406879
$165$	−0.868706	−0.0676287
$166$	13.7172	1.06466
$167$	6.44094	0.498415	0.249207	−	0.968450i	$-0.419830\pi$
$167$	6.44094	0.498415	0.249207	+	0.968450i	$0.419830\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	−1.29384	−0.0992333
$171$	5.61181	0.429146
$172$	3.07434	0.234416
$173$	−18.6144	−1.41522	−0.707611	−	0.706602i	$-0.750227\pi$
$173$	−18.6144	−1.41522	−0.707611	+	0.706602i	$0.750227\pi$
$174$	1.18667	0.0899616
$175$	−4.88924	−0.369592
$176$	2.61023	0.196754
$177$	10.4752	0.787361
$178$	3.38029	0.253363
$179$	−22.2163	−1.66052	−0.830261	−	0.557375i	$-0.811809\pi$
$179$	−22.2163	−1.66052	−0.830261	+	0.557375i	$0.811809\pi$
$180$	0.332808	0.0248060
$181$	4.05853	0.301669	0.150834	−	0.988559i	$-0.451804\pi$
$181$	4.05853	0.301669	0.150834	+	0.988559i	$0.451804\pi$
$182$	0	0
$183$	−7.43978	−0.549965
$184$	4.21257	0.310555
$185$	0.191844	0.0141046
$186$	−7.07434	−0.518716
$187$	−10.1477	−0.742074
$188$	12.0528	0.879037
$189$	−1.00000	−0.0727393
$190$	1.86765	0.135494
$191$	−11.7204	−0.848056	−0.424028	−	0.905649i	$-0.639384\pi$
$191$	−11.7204	−0.848056	−0.424028	+	0.905649i	$0.639384\pi$
$192$	−1.00000	−0.0721688
$193$	−23.8010	−1.71324	−0.856618	−	0.515952i	$-0.827438\pi$
$193$	−23.8010	−1.71324	−0.856618	+	0.515952i	$0.827438\pi$
$194$	−5.81587	−0.417555
$195$	0	0
$196$	1.00000	0.0714286
$197$	−10.0827	−0.718360	−0.359180	−	0.933268i	$-0.616944\pi$
$197$	−10.0827	−0.718360	−0.359180	+	0.933268i	$0.616944\pi$
$198$	2.61023	0.185501
$199$	13.8501	0.981806	0.490903	−	0.871214i	$-0.336667\pi$
$199$	13.8501	0.981806	0.490903	+	0.871214i	$0.336667\pi$
$200$	−4.88924	−0.345721
$201$	11.9203	0.840793
$202$	−15.9703	−1.12367
$203$	−1.18667	−0.0832882
$204$	3.88766	0.272191
$205$	−0.173412	−0.0121117
$206$	7.50641	0.522997
$207$	4.21257	0.292794
$208$	0	0
$209$	14.6481	1.01323
$210$	−0.332808	−0.0229659
$211$	4.55485	0.313569	0.156785	−	0.987633i	$-0.449887\pi$
$211$	4.55485	0.313569	0.156785	+	0.987633i	$0.449887\pi$
$212$	−9.71047	−0.666918
$213$	0.944620	0.0647243
$214$	8.65078	0.591355
$215$	1.02316	0.0697791
$216$	−1.00000	−0.0680414
$217$	7.07434	0.480237
$218$	18.6144	1.26072
$219$	−4.66562	−0.315273
$220$	0.868706	0.0585681
$221$	0	0
$222$	−0.576440	−0.0386881
$223$	8.43346	0.564746	0.282373	−	0.959305i	$-0.408878\pi$
$223$	8.43346	0.564746	0.282373	+	0.959305i	$0.408878\pi$
$224$	1.00000	0.0668153
$225$	−4.88924	−0.325949
$226$	9.15025	0.608666
$227$	27.9893	1.85771	0.928857	−	0.370439i	$-0.120793\pi$
$227$	27.9893	1.85771	0.928857	+	0.370439i	$0.120793\pi$
$228$	−5.61181	−0.371651
$229$	28.7785	1.90174	0.950868	−	0.309598i	$-0.100194\pi$
$229$	28.7785	1.90174	0.950868	+	0.309598i	$0.100194\pi$
$230$	1.40197	0.0924435
$231$	−2.61023	−0.171741
$232$	−1.18667	−0.0779090
$233$	18.8877	1.23737	0.618686	−	0.785638i	$-0.287665\pi$
$233$	18.8877	1.23737	0.618686	+	0.785638i	$0.287665\pi$
$234$	0	0
$235$	4.01125	0.261665
$236$	−10.4752	−0.681875
$237$	0.943042	0.0612572
$238$	−3.88766	−0.252000
$239$	11.0933	0.717565	0.358783	−	0.933421i	$-0.383192\pi$
$239$	11.0933	0.717565	0.358783	+	0.933421i	$0.383192\pi$
$240$	−0.332808	−0.0214826
$241$	8.11392	0.522663	0.261332	−	0.965249i	$-0.415838\pi$
$241$	8.11392	0.522663	0.261332	+	0.965249i	$0.415838\pi$
$242$	−4.18667	−0.269130
$243$	−1.00000	−0.0641500
$244$	7.43978	0.476283
$245$	0.332808	0.0212623
$246$	0.521059	0.0332215
$247$	0	0
$248$	7.07434	0.449221
$249$	−13.7172	−0.869293
$250$	−3.29121	−0.208155
$251$	−3.16930	−0.200044	−0.100022	−	0.994985i	$-0.531891\pi$
$251$	−3.16930	−0.200044	−0.100022	+	0.994985i	$0.531891\pi$
$252$	1.00000	0.0629941
$253$	10.9958	0.691300
$254$	6.77953	0.425386
$255$	1.29384	0.0810236
$256$	1.00000	0.0625000
$257$	−0.193421	−0.0120653	−0.00603263	−	0.999982i	$-0.501920\pi$
$257$	−0.193421	−0.0120653	−0.00603263	+	0.999982i	$0.501920\pi$
$258$	−3.07434	−0.191400
$259$	0.576440	0.0358182
$260$	0	0
$261$	−1.18667	−0.0734533
$262$	7.22879	0.446596
$263$	12.0437	0.742646	0.371323	−	0.928504i	$-0.378904\pi$
$263$	12.0437	0.742646	0.371323	+	0.928504i	$0.378904\pi$
$264$	−2.61023	−0.160649
$265$	−3.23172	−0.198523
$266$	5.61181	0.344082
$267$	−3.38029	−0.206870
$268$	−11.9203	−0.728148
$269$	17.2097	1.04930	0.524648	−	0.851319i	$-0.324197\pi$
$269$	17.2097	1.04930	0.524648	+	0.851319i	$0.324197\pi$
$270$	−0.332808	−0.0202540
$271$	−17.1419	−1.04130	−0.520649	−	0.853771i	$-0.674310\pi$
$271$	−17.1419	−1.04130	−0.520649	+	0.853771i	$0.674310\pi$
$272$	−3.88766	−0.235724
$273$	0	0
$274$	11.8892	0.718255
$275$	−12.7621	−0.769581
$276$	−4.21257	−0.253567
$277$	−15.7132	−0.944115	−0.472057	−	0.881568i	$-0.656488\pi$
$277$	−15.7132	−0.944115	−0.472057	+	0.881568i	$0.656488\pi$
$278$	14.3211	0.858924
$279$	7.07434	0.423529
$280$	0.332808	0.0198891
$281$	4.07337	0.242997	0.121499	−	0.992592i	$-0.461230\pi$
$281$	4.07337	0.242997	0.121499	+	0.992592i	$0.461230\pi$
$282$	−12.0528	−0.717731
$283$	21.5348	1.28011	0.640057	−	0.768328i	$-0.278911\pi$
$283$	21.5348	1.28011	0.640057	+	0.768328i	$0.278911\pi$
$284$	−0.944620	−0.0560529
$285$	−1.86765	−0.110630
$286$	0	0
$287$	−0.521059	−0.0307572
$288$	1.00000	0.0589256
$289$	−1.88608	−0.110946
$290$	−0.394934	−0.0231913
$291$	5.81587	0.340932
$292$	4.66562	0.273034
$293$	−0.176042	−0.0102845	−0.00514224	−	0.999987i	$-0.501637\pi$
$293$	−0.176042	−0.0102845	−0.00514224	+	0.999987i	$0.501637\pi$
$294$	−1.00000	−0.0583212
$295$	−3.48621	−0.202975
$296$	0.576440	0.0335049
$297$	−2.61023	−0.151461
$298$	−6.05696	−0.350870
$299$	0	0
$300$	4.88924	0.282280
$301$	3.07434	0.177202
$302$	−21.4953	−1.23691
$303$	15.9703	0.917471
$304$	5.61181	0.321859
$305$	2.47602	0.141776
$306$	−3.88766	−0.222243
$307$	20.7405	1.18372	0.591861	−	0.806040i	$-0.298394\pi$
$307$	20.7405	1.18372	0.591861	+	0.806040i	$0.298394\pi$
$308$	2.61023	0.148732
$309$	−7.50641	−0.427025
$310$	2.35439	0.133721
$311$	−10.9678	−0.621926	−0.310963	−	0.950422i	$-0.600652\pi$
$311$	−10.9678	−0.621926	−0.310963	+	0.950422i	$0.600652\pi$
$312$	0	0
$313$	16.2189	0.916746	0.458373	−	0.888760i	$-0.348432\pi$
$313$	16.2189	0.916746	0.458373	+	0.888760i	$0.348432\pi$
$314$	2.29227	0.129360
$315$	0.332808	0.0187516
$316$	−0.943042	−0.0530503
$317$	10.5358	0.591750	0.295875	−	0.955227i	$-0.404389\pi$
$317$	10.5358	0.591750	0.295875	+	0.955227i	$0.404389\pi$
$318$	9.71047	0.544536
$319$	−3.09750	−0.173427
$320$	0.332808	0.0186045
$321$	−8.65078	−0.482839
$322$	4.21257	0.234757
$323$	−21.8168	−1.21392
$324$	1.00000	0.0555556
$325$	0	0
$326$	−3.34071	−0.185025
$327$	−18.6144	−1.02938
$328$	−0.521059	−0.0287707
$329$	12.0528	0.664490
$330$	−0.868706	−0.0478207
$331$	−16.1297	−0.886569	−0.443285	−	0.896381i	$-0.646187\pi$
$331$	−16.1297	−0.886569	−0.443285	+	0.896381i	$0.646187\pi$
$332$	13.7172	0.752830
$333$	0.576440	0.0315887
$334$	6.44094	0.352433
$335$	−3.96717	−0.216750
$336$	−1.00000	−0.0545545
$337$	−1.78785	−0.0973906	−0.0486953	−	0.998814i	$-0.515506\pi$
$337$	−1.78785	−0.0973906	−0.0486953	+	0.998814i	$0.515506\pi$
$338$	0	0
$339$	−9.15025	−0.496973
$340$	−1.29384	−0.0701685
$341$	18.4657	0.999973
$342$	5.61181	0.303452
$343$	1.00000	0.0539949
$344$	3.07434	0.165757
$345$	−1.40197	−0.0754798
$346$	−18.6144	−1.00071
$347$	25.4061	1.36387	0.681935	−	0.731413i	$-0.261139\pi$
$347$	25.4061	1.36387	0.681935	+	0.731413i	$0.261139\pi$
$348$	1.18667	0.0636124
$349$	−28.3385	−1.51693	−0.758463	−	0.651717i	$-0.774049\pi$
$349$	−28.3385	−1.51693	−0.758463	+	0.651717i	$0.774049\pi$
$350$	−4.88924	−0.261341
$351$	0	0
$352$	2.61023	0.139126
$353$	1.95314	0.103955	0.0519774	−	0.998648i	$-0.483448\pi$
$353$	1.95314	0.103955	0.0519774	+	0.998648i	$0.483448\pi$
$354$	10.4752	0.556748
$355$	−0.314377	−0.0166854
$356$	3.38029	0.179155
$357$	3.88766	0.205757
$358$	−22.2163	−1.17417
$359$	−27.6565	−1.45965	−0.729826	−	0.683633i	$-0.760399\pi$
$359$	−27.6565	−1.45965	−0.729826	+	0.683633i	$0.760399\pi$
$360$	0.332808	0.0175405
$361$	12.4924	0.657496
$362$	4.05853	0.213312
$363$	4.18667	0.219743
$364$	0	0
$365$	1.55275	0.0812748
$366$	−7.43978	−0.388884
$367$	5.75775	0.300552	0.150276	−	0.988644i	$-0.451984\pi$
$367$	5.75775	0.300552	0.150276	+	0.988644i	$0.451984\pi$
$368$	4.21257	0.219595
$369$	−0.521059	−0.0271253
$370$	0.191844	0.00997347
$371$	−9.71047	−0.504142
$372$	−7.07434	−0.366787
$373$	27.8487	1.44195	0.720975	−	0.692961i	$-0.243694\pi$
$373$	27.8487	1.44195	0.720975	+	0.692961i	$0.243694\pi$
$374$	−10.1477	−0.524726
$375$	3.29121	0.169958
$376$	12.0528	0.621573
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	12.4518	0.639606	0.319803	−	0.947484i	$-0.396383\pi$
$379$	12.4518	0.639606	0.319803	+	0.947484i	$0.396383\pi$
$380$	1.86765	0.0958086
$381$	−6.77953	−0.347326
$382$	−11.7204	−0.599666
$383$	−9.72573	−0.496961	−0.248481	−	0.968637i	$-0.579931\pi$
$383$	−9.72573	−0.496961	−0.248481	+	0.968637i	$0.579931\pi$
$384$	−1.00000	−0.0510310
$385$	0.868706	0.0442734
$386$	−23.8010	−1.21144
$387$	3.07434	0.156277
$388$	−5.81587	−0.295256
$389$	13.8910	0.704302	0.352151	−	0.935943i	$-0.385450\pi$
$389$	13.8910	0.704302	0.352151	+	0.935943i	$0.385450\pi$
$390$	0	0
$391$	−16.3770	−0.828223
$392$	1.00000	0.0505076
$393$	−7.22879	−0.364644
$394$	−10.0827	−0.507957
$395$	−0.313852	−0.0157916
$396$	2.61023	0.131169
$397$	−6.77848	−0.340202	−0.170101	−	0.985427i	$-0.554409\pi$
$397$	−6.77848	−0.340202	−0.170101	+	0.985427i	$0.554409\pi$
$398$	13.8501	0.694242
$399$	−5.61181	−0.280942
$400$	−4.88924	−0.244462
$401$	14.4630	0.722250	0.361125	−	0.932517i	$-0.382393\pi$
$401$	14.4630	0.722250	0.361125	+	0.932517i	$0.382393\pi$
$402$	11.9203	0.594531
$403$	0	0
$404$	−15.9703	−0.794553
$405$	0.332808	0.0165373
$406$	−1.18667	−0.0588937
$407$	1.50464	0.0745824
$408$	3.88766	0.192468
$409$	23.3286	1.15353	0.576763	−	0.816912i	$-0.304316\pi$
$409$	23.3286	1.15353	0.576763	+	0.816912i	$0.304316\pi$
$410$	−0.173412	−0.00856423
$411$	−11.8892	−0.586453
$412$	7.50641	0.369814
$413$	−10.4752	−0.515449
$414$	4.21257	0.207036
$415$	4.56519	0.224096
$416$	0	0
$417$	−14.3211	−0.701308
$418$	14.6481	0.716464
$419$	−22.5795	−1.10308	−0.551540	−	0.834148i	$-0.685960\pi$
$419$	−22.5795	−1.10308	−0.551540	+	0.834148i	$0.685960\pi$
$420$	−0.332808	−0.0162393
$421$	34.1708	1.66538	0.832691	−	0.553738i	$-0.186799\pi$
$421$	34.1708	1.66538	0.832691	+	0.553738i	$0.186799\pi$
$422$	4.55485	0.221727
$423$	12.0528	0.586025
$424$	−9.71047	−0.471582
$425$	19.0077	0.922009
$426$	0.944620	0.0457670
$427$	7.43978	0.360036
$428$	8.65078	0.418151
$429$	0	0
$430$	1.02316	0.0493413
$431$	−5.23689	−0.252252	−0.126126	−	0.992014i	$-0.540254\pi$
$431$	−5.23689	−0.252252	−0.126126	+	0.992014i	$0.540254\pi$
$432$	−1.00000	−0.0481125
$433$	16.8206	0.808346	0.404173	−	0.914682i	$-0.367559\pi$
$433$	16.8206	0.808346	0.404173	+	0.914682i	$0.367559\pi$
$434$	7.07434	0.339579
$435$	0.394934	0.0189356
$436$	18.6144	0.891466
$437$	23.6401	1.13086
$438$	−4.66562	−0.222932
$439$	−14.7256	−0.702816	−0.351408	−	0.936222i	$-0.614297\pi$
$439$	−14.7256	−0.702816	−0.351408	+	0.936222i	$0.614297\pi$
$440$	0.868706	0.0414139
$441$	1.00000	0.0476190
$442$	0	0
$443$	36.2393	1.72178	0.860891	−	0.508789i	$-0.169907\pi$
$443$	36.2393	1.72178	0.860891	+	0.508789i	$0.169907\pi$
$444$	−0.576440	−0.0273566
$445$	1.12499	0.0533294
$446$	8.43346	0.399336
$447$	6.05696	0.286484
$448$	1.00000	0.0472456
$449$	24.2893	1.14628	0.573140	−	0.819457i	$-0.305725\pi$
$449$	24.2893	1.14628	0.573140	+	0.819457i	$0.305725\pi$
$450$	−4.88924	−0.230481
$451$	−1.36009	−0.0640440
$452$	9.15025	0.430392
$453$	21.4953	1.00994
$454$	27.9893	1.31360
$455$	0	0
$456$	−5.61181	−0.262797
$457$	33.3412	1.55964	0.779819	−	0.626006i	$-0.215311\pi$
$457$	33.3412	1.55964	0.779819	+	0.626006i	$0.215311\pi$
$458$	28.7785	1.34473
$459$	3.88766	0.181461
$460$	1.40197	0.0653674
$461$	−29.5970	−1.37847	−0.689234	−	0.724539i	$-0.742053\pi$
$461$	−29.5970	−1.37847	−0.689234	+	0.724539i	$0.742053\pi$
$462$	−2.61023	−0.121439
$463$	−36.6027	−1.70107	−0.850535	−	0.525918i	$-0.823722\pi$
$463$	−36.6027	−1.70107	−0.850535	+	0.525918i	$0.823722\pi$
$464$	−1.18667	−0.0550900
$465$	−2.35439	−0.109182
$466$	18.8877	0.874954
$467$	6.21635	0.287658	0.143829	−	0.989603i	$-0.454058\pi$
$467$	6.21635	0.287658	0.143829	+	0.989603i	$0.454058\pi$
$468$	0	0
$469$	−11.9203	−0.550428
$470$	4.01125	0.185025
$471$	−2.29227	−0.105622
$472$	−10.4752	−0.482158
$473$	8.02474	0.368978
$474$	0.943042	0.0433154
$475$	−27.4375	−1.25892
$476$	−3.88766	−0.178191
$477$	−9.71047	−0.444612
$478$	11.0933	0.507395
$479$	−33.5737	−1.53402	−0.767011	−	0.641634i	$-0.778257\pi$
$479$	−33.5737	−1.53402	−0.767011	+	0.641634i	$0.778257\pi$
$480$	−0.332808	−0.0151905
$481$	0	0
$482$	8.11392	0.369579
$483$	−4.21257	−0.191679
$484$	−4.18667	−0.190303
$485$	−1.93556	−0.0878894
$486$	−1.00000	−0.0453609
$487$	20.2095	0.915781	0.457890	−	0.889009i	$-0.348605\pi$
$487$	20.2095	0.915781	0.457890	+	0.889009i	$0.348605\pi$
$488$	7.43978	0.336783
$489$	3.34071	0.151072
$490$	0.332808	0.0150347
$491$	−6.77953	−0.305956	−0.152978	−	0.988230i	$-0.548886\pi$
$491$	−6.77953	−0.305956	−0.152978	+	0.988230i	$0.548886\pi$
$492$	0.521059	0.0234912
$493$	4.61339	0.207777
$494$	0	0
$495$	0.868706	0.0390454
$496$	7.07434	0.317647
$497$	−0.944620	−0.0423720
$498$	−13.7172	−0.614683
$499$	−11.0813	−0.496066	−0.248033	−	0.968752i	$-0.579784\pi$
$499$	−11.0813	−0.496066	−0.248033	+	0.968752i	$0.579784\pi$
$500$	−3.29121	−0.147188
$501$	−6.44094	−0.287760
$502$	−3.16930	−0.141453
$503$	−29.9916	−1.33726	−0.668629	−	0.743596i	$-0.733119\pi$
$503$	−29.9916	−1.33726	−0.668629	+	0.743596i	$0.733119\pi$
$504$	1.00000	0.0445435
$505$	−5.31504	−0.236516
$506$	10.9958	0.488823
$507$	0	0
$508$	6.77953	0.300793
$509$	−31.4804	−1.39535	−0.697673	−	0.716417i	$-0.745781\pi$
$509$	−31.4804	−1.39535	−0.697673	+	0.716417i	$0.745781\pi$
$510$	1.29384	0.0572924
$511$	4.66562	0.206395
$512$	1.00000	0.0441942
$513$	−5.61181	−0.247768
$514$	−0.193421	−0.00853142
$515$	2.49819	0.110083
$516$	−3.07434	−0.135340
$517$	31.4605	1.38363
$518$	0.576440	0.0253273
$519$	18.6144	0.817079
$520$	0	0
$521$	17.8010	0.779877	0.389939	−	0.920841i	$-0.372496\pi$
$521$	17.8010	0.779877	0.389939	+	0.920841i	$0.372496\pi$
$522$	−1.18667	−0.0519393
$523$	32.9404	1.44038	0.720192	−	0.693775i	$-0.244054\pi$
$523$	32.9404	1.44038	0.720192	+	0.693775i	$0.244054\pi$
$524$	7.22879	0.315791
$525$	4.88924	0.213384
$526$	12.0437	0.525130
$527$	−27.5026	−1.19803
$528$	−2.61023	−0.113596
$529$	−5.25426	−0.228446
$530$	−3.23172	−0.140377
$531$	−10.4752	−0.454583
$532$	5.61181	0.243303
$533$	0	0
$534$	−3.38029	−0.146279
$535$	2.87904	0.124472
$536$	−11.9203	−0.514879
$537$	22.2163	0.958703
$538$	17.2097	0.741965
$539$	2.61023	0.112431
$540$	−0.332808	−0.0143218
$541$	−15.5204	−0.667276	−0.333638	−	0.942701i	$-0.608276\pi$
$541$	−15.5204	−0.667276	−0.333638	+	0.942701i	$0.608276\pi$
$542$	−17.1419	−0.736309
$543$	−4.05853	−0.174168
$544$	−3.88766	−0.166682
$545$	6.19500	0.265365
$546$	0	0
$547$	22.9529	0.981397	0.490698	−	0.871329i	$-0.336742\pi$
$547$	22.9529	0.981397	0.490698	+	0.871329i	$0.336742\pi$
$548$	11.8892	0.507883
$549$	7.43978	0.317522
$550$	−12.7621	−0.544176
$551$	−6.65939	−0.283700
$552$	−4.21257	−0.179299
$553$	−0.943042	−0.0401022
$554$	−15.7132	−0.667590
$555$	−0.191844	−0.00814330
$556$	14.3211	0.607351
$557$	−24.0363	−1.01845	−0.509226	−	0.860633i	$-0.670068\pi$
$557$	−24.0363	−1.01845	−0.509226	+	0.860633i	$0.670068\pi$
$558$	7.07434	0.299481
$559$	0	0
$560$	0.332808	0.0140637
$561$	10.1477	0.428437
$562$	4.07337	0.171825
$563$	−37.8759	−1.59628	−0.798139	−	0.602473i	$-0.794182\pi$
$563$	−37.8759	−1.59628	−0.798139	+	0.602473i	$0.794182\pi$
$564$	−12.0528	−0.507512
$565$	3.04527	0.128116
$566$	21.5348	0.905177
$567$	1.00000	0.0419961
$568$	−0.944620	−0.0396354
$569$	−27.3665	−1.14726	−0.573632	−	0.819113i	$-0.694466\pi$
$569$	−27.3665	−1.14726	−0.573632	+	0.819113i	$0.694466\pi$
$570$	−1.86765	−0.0782274
$571$	−17.6639	−0.739213	−0.369607	−	0.929188i	$-0.620508\pi$
$571$	−17.6639	−0.739213	−0.369607	+	0.929188i	$0.620508\pi$
$572$	0	0
$573$	11.7204	0.489625
$574$	−0.521059	−0.0217486
$575$	−20.5963	−0.858923
$576$	1.00000	0.0416667
$577$	−11.3804	−0.473772	−0.236886	−	0.971537i	$-0.576127\pi$
$577$	−11.3804	−0.473772	−0.236886	+	0.971537i	$0.576127\pi$
$578$	−1.88608	−0.0784508
$579$	23.8010	0.989137
$580$	−0.394934	−0.0163988
$581$	13.7172	0.569086
$582$	5.81587	0.241075
$583$	−25.3466	−1.04975
$584$	4.66562	0.193065
$585$	0	0
$586$	−0.176042	−0.00727222
$587$	−18.4694	−0.762313	−0.381156	−	0.924511i	$-0.624474\pi$
$587$	−18.4694	−0.762313	−0.381156	+	0.924511i	$0.624474\pi$
$588$	−1.00000	−0.0412393
$589$	39.6998	1.63580
$590$	−3.48621	−0.143525
$591$	10.0827	0.414745
$592$	0.576440	0.0236915
$593$	−19.3561	−0.794859	−0.397429	−	0.917633i	$-0.630098\pi$
$593$	−19.3561	−0.794859	−0.397429	+	0.917633i	$0.630098\pi$
$594$	−2.61023	−0.107099
$595$	−1.29384	−0.0530424
$596$	−6.05696	−0.248103
$597$	−13.8501	−0.566846
$598$	0	0
$599$	21.1818	0.865466	0.432733	−	0.901522i	$-0.357549\pi$
$599$	21.1818	0.865466	0.432733	+	0.901522i	$0.357549\pi$
$600$	4.88924	0.199602
$601$	22.1441	0.903277	0.451639	−	0.892201i	$-0.350840\pi$
$601$	22.1441	0.903277	0.451639	+	0.892201i	$0.350840\pi$
$602$	3.07434	0.125301
$603$	−11.9203	−0.485432
$604$	−21.4953	−0.874630
$605$	−1.39336	−0.0566480
$606$	15.9703	0.648750
$607$	34.2879	1.39170	0.695851	−	0.718186i	$-0.255027\pi$
$607$	34.2879	1.39170	0.695851	+	0.718186i	$0.255027\pi$
$608$	5.61181	0.227589
$609$	1.18667	0.0480865
$610$	2.47602	0.100251
$611$	0	0
$612$	−3.88766	−0.157149
$613$	−17.0097	−0.687014	−0.343507	−	0.939150i	$-0.611615\pi$
$613$	−17.0097	−0.687014	−0.343507	+	0.939150i	$0.611615\pi$
$614$	20.7405	0.837018
$615$	0.173412	0.00699267
$616$	2.61023	0.105169
$617$	17.3210	0.697319	0.348659	−	0.937250i	$-0.386637\pi$
$617$	17.3210	0.697319	0.348659	+	0.937250i	$0.386637\pi$
$618$	−7.50641	−0.301952
$619$	0.621482	0.0249795	0.0124897	−	0.999922i	$-0.496024\pi$
$619$	0.621482	0.0249795	0.0124897	+	0.999922i	$0.496024\pi$
$620$	2.35439	0.0945547
$621$	−4.21257	−0.169045
$622$	−10.9678	−0.439768
$623$	3.38029	0.135428
$624$	0	0
$625$	23.3509	0.934034
$626$	16.2189	0.648237
$627$	−14.6481	−0.584991
$628$	2.29227	0.0914714
$629$	−2.24100	−0.0893546
$630$	0.332808	0.0132594
$631$	−13.5763	−0.540466	−0.270233	−	0.962795i	$-0.587101\pi$
$631$	−13.5763	−0.540466	−0.270233	+	0.962795i	$0.587101\pi$
$632$	−0.943042	−0.0375122
$633$	−4.55485	−0.181039
$634$	10.5358	0.418430
$635$	2.25628	0.0895377
$636$	9.71047	0.385045
$637$	0	0
$638$	−3.09750	−0.122631
$639$	−0.944620	−0.0373686
$640$	0.332808	0.0131554
$641$	−20.5117	−0.810163	−0.405081	−	0.914281i	$-0.632757\pi$
$641$	−20.5117	−0.810163	−0.405081	+	0.914281i	$0.632757\pi$
$642$	−8.65078	−0.341419
$643$	−42.7531	−1.68602	−0.843009	−	0.537899i	$-0.819218\pi$
$643$	−42.7531	−1.68602	−0.843009	+	0.537899i	$0.819218\pi$
$644$	4.21257	0.165998
$645$	−1.02316	−0.0402870
$646$	−21.8168	−0.858372
$647$	−26.2230	−1.03093	−0.515466	−	0.856910i	$-0.672381\pi$
$647$	−26.2230	−1.03093	−0.515466	+	0.856910i	$0.672381\pi$
$648$	1.00000	0.0392837
$649$	−27.3426	−1.07329
$650$	0	0
$651$	−7.07434	−0.277265
$652$	−3.34071	−0.130832
$653$	−8.73871	−0.341972	−0.170986	−	0.985273i	$-0.554695\pi$
$653$	−8.73871	−0.341972	−0.170986	+	0.985273i	$0.554695\pi$
$654$	−18.6144	−0.727879
$655$	2.40580	0.0940023
$656$	−0.521059	−0.0203439
$657$	4.66562	0.182023
$658$	12.0528	0.469865
$659$	0.502103	0.0195592	0.00977958	−	0.999952i	$-0.496887\pi$
$659$	0.502103	0.0195592	0.00977958	+	0.999952i	$0.496887\pi$
$660$	−0.868706	−0.0338143
$661$	31.2995	1.21741	0.608705	−	0.793396i	$-0.291689\pi$
$661$	31.2995	1.21741	0.608705	+	0.793396i	$0.291689\pi$
$662$	−16.1297	−0.626899
$663$	0	0
$664$	13.7172	0.532331
$665$	1.86765	0.0724245
$666$	0.576440	0.0223366
$667$	−4.99895	−0.193560
$668$	6.44094	0.249207
$669$	−8.43346	−0.326056
$670$	−3.96717	−0.153265
$671$	19.4196	0.749685
$672$	−1.00000	−0.0385758
$673$	−23.7170	−0.914224	−0.457112	−	0.889409i	$-0.651116\pi$
$673$	−23.7170	−0.914224	−0.457112	+	0.889409i	$0.651116\pi$
$674$	−1.78785	−0.0688656
$675$	4.88924	0.188187
$676$	0	0
$677$	17.8136	0.684631	0.342315	−	0.939585i	$-0.388789\pi$
$677$	17.8136	0.684631	0.342315	+	0.939585i	$0.388789\pi$
$678$	−9.15025	−0.351413
$679$	−5.81587	−0.223192
$680$	−1.29384	−0.0496166
$681$	−27.9893	−1.07255
$682$	18.4657	0.707087
$683$	−11.9135	−0.455856	−0.227928	−	0.973678i	$-0.573195\pi$
$683$	−11.9135	−0.455856	−0.227928	+	0.973678i	$0.573195\pi$
$684$	5.61181	0.214573
$685$	3.95683	0.151183
$686$	1.00000	0.0381802
$687$	−28.7785	−1.09797
$688$	3.07434	0.117208
$689$	0	0
$690$	−1.40197	−0.0533723
$691$	19.9861	0.760308	0.380154	−	0.924923i	$-0.375871\pi$
$691$	19.9861	0.760308	0.380154	+	0.924923i	$0.375871\pi$
$692$	−18.6144	−0.707611
$693$	2.61023	0.0991546
$694$	25.4061	0.964402
$695$	4.76618	0.180791
$696$	1.18667	0.0449808
$697$	2.02570	0.0767289
$698$	−28.3385	−1.07263
$699$	−18.8877	−0.714397
$700$	−4.88924	−0.184796
$701$	23.9244	0.903613	0.451806	−	0.892116i	$-0.350780\pi$
$701$	23.9244	0.903613	0.451806	+	0.892116i	$0.350780\pi$
$702$	0	0
$703$	3.23487	0.122005
$704$	2.61023	0.0983769
$705$	−4.01125	−0.151072
$706$	1.95314	0.0735072
$707$	−15.9703	−0.600626
$708$	10.4752	0.393680
$709$	−35.4962	−1.33309	−0.666544	−	0.745465i	$-0.732227\pi$
$709$	−35.4962	−1.33309	−0.666544	+	0.745465i	$0.732227\pi$
$710$	−0.314377	−0.0117983
$711$	−0.943042	−0.0353669
$712$	3.38029	0.126682
$713$	29.8011	1.11606
$714$	3.88766	0.145492
$715$	0	0
$716$	−22.2163	−0.830261
$717$	−11.0933	−0.414287
$718$	−27.6565	−1.03213
$719$	−6.18194	−0.230548	−0.115274	−	0.993334i	$-0.536775\pi$
$719$	−6.18194	−0.230548	−0.115274	+	0.993334i	$0.536775\pi$
$720$	0.332808	0.0124030
$721$	7.50641	0.279553
$722$	12.4924	0.464920
$723$	−8.11392	−0.301760
$724$	4.05853	0.150834
$725$	5.80194	0.215478
$726$	4.18667	0.155382
$727$	43.2387	1.60363	0.801817	−	0.597569i	$-0.203867\pi$
$727$	43.2387	1.60363	0.801817	+	0.597569i	$0.203867\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.55275	0.0574699
$731$	−11.9520	−0.442060
$732$	−7.43978	−0.274982
$733$	32.0306	1.18308	0.591538	−	0.806277i	$-0.298521\pi$
$733$	32.0306	1.18308	0.591538	+	0.806277i	$0.298521\pi$
$734$	5.75775	0.212523
$735$	−0.332808	−0.0122758
$736$	4.21257	0.155277
$737$	−31.1148	−1.14613
$738$	−0.521059	−0.0191805
$739$	−25.4678	−0.936848	−0.468424	−	0.883504i	$-0.655178\pi$
$739$	−25.4678	−0.936848	−0.468424	+	0.883504i	$0.655178\pi$
$740$	0.191844	0.00705231
$741$	0	0
$742$	−9.71047	−0.356482
$743$	14.4953	0.531780	0.265890	−	0.964003i	$-0.414334\pi$
$743$	14.4953	0.531780	0.265890	+	0.964003i	$0.414334\pi$
$744$	−7.07434	−0.259358
$745$	−2.01580	−0.0738533
$746$	27.8487	1.01961
$747$	13.7172	0.501887
$748$	−10.1477	−0.371037
$749$	8.65078	0.316092
$750$	3.29121	0.120178
$751$	29.1265	1.06284	0.531420	−	0.847108i	$-0.321659\pi$
$751$	29.1265	1.06284	0.531420	+	0.847108i	$0.321659\pi$
$752$	12.0528	0.439519
$753$	3.16930	0.115496
$754$	0	0
$755$	−7.15379	−0.260353
$756$	−1.00000	−0.0363696
$757$	−20.0460	−0.728584	−0.364292	−	0.931285i	$-0.618689\pi$
$757$	−20.0460	−0.728584	−0.364292	+	0.931285i	$0.618689\pi$
$758$	12.4518	0.452270
$759$	−10.9958	−0.399122
$760$	1.86765	0.0677469
$761$	−19.0472	−0.690460	−0.345230	−	0.938518i	$-0.612199\pi$
$761$	−19.0472	−0.690460	−0.345230	+	0.938518i	$0.612199\pi$
$762$	−6.77953	−0.245596
$763$	18.6144	0.673885
$764$	−11.7204	−0.424028
$765$	−1.29384	−0.0467790
$766$	−9.72573	−0.351405
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−15.2484	−0.549870	−0.274935	−	0.961463i	$-0.588656\pi$
$769$	−15.2484	−0.549870	−0.274935	+	0.961463i	$0.588656\pi$
$770$	0.868706	0.0313060
$771$	0.193421	0.00696588
$772$	−23.8010	−0.856618
$773$	−6.66562	−0.239746	−0.119873	−	0.992789i	$-0.538249\pi$
$773$	−6.66562	−0.239746	−0.119873	+	0.992789i	$0.538249\pi$
$774$	3.07434	0.110505
$775$	−34.5881	−1.24244
$776$	−5.81587	−0.208777
$777$	−0.576440	−0.0206797
$778$	13.8910	0.498017
$779$	−2.92409	−0.104766
$780$	0	0
$781$	−2.46568	−0.0882289
$782$	−16.3770	−0.585642
$783$	1.18667	0.0424083
$784$	1.00000	0.0357143
$785$	0.762884	0.0272285
$786$	−7.22879	−0.257843
$787$	−5.84280	−0.208273	−0.104137	−	0.994563i	$-0.533208\pi$
$787$	−5.84280	−0.208273	−0.104137	+	0.994563i	$0.533208\pi$
$788$	−10.0827	−0.359180
$789$	−12.0437	−0.428767
$790$	−0.313852	−0.0111663
$791$	9.15025	0.325345
$792$	2.61023	0.0927507
$793$	0	0
$794$	−6.77848	−0.240559
$795$	3.23172	0.114617
$796$	13.8501	0.490903
$797$	27.3749	0.969670	0.484835	−	0.874606i	$-0.338880\pi$
$797$	27.3749	0.969670	0.484835	+	0.874606i	$0.338880\pi$
$798$	−5.61181	−0.198656
$799$	−46.8570	−1.65768
$800$	−4.88924	−0.172861
$801$	3.38029	0.119437
$802$	14.4630	0.510708
$803$	12.1784	0.429765
$804$	11.9203	0.420397
$805$	1.40197	0.0494131
$806$	0	0
$807$	−17.2097	−0.605812
$808$	−15.9703	−0.561834
$809$	−14.9961	−0.527235	−0.263618	−	0.964627i	$-0.584916\pi$
$809$	−14.9961	−0.527235	−0.263618	+	0.964627i	$0.584916\pi$
$810$	0.332808	0.0116937
$811$	−21.1826	−0.743820	−0.371910	−	0.928269i	$-0.621297\pi$
$811$	−21.1826	−0.743820	−0.371910	+	0.928269i	$0.621297\pi$
$812$	−1.18667	−0.0416441
$813$	17.1419	0.601194
$814$	1.50464	0.0527377
$815$	−1.11181	−0.0389451
$816$	3.88766	0.136095
$817$	17.2526	0.603592
$818$	23.3286	0.815665
$819$	0	0
$820$	−0.173412	−0.00605583
$821$	23.5985	0.823595	0.411798	−	0.911275i	$-0.364901\pi$
$821$	23.5985	0.823595	0.411798	+	0.911275i	$0.364901\pi$
$822$	−11.8892	−0.414685
$823$	26.5982	0.927156	0.463578	−	0.886056i	$-0.346565\pi$
$823$	26.5982	0.927156	0.463578	+	0.886056i	$0.346565\pi$
$824$	7.50641	0.261498
$825$	12.7621	0.444318
$826$	−10.4752	−0.364477
$827$	−7.76986	−0.270185	−0.135092	−	0.990833i	$-0.543133\pi$
$827$	−7.76986	−0.270185	−0.135092	+	0.990833i	$0.543133\pi$
$828$	4.21257	0.146397
$829$	−11.3814	−0.395291	−0.197645	−	0.980274i	$-0.563329\pi$
$829$	−11.3814	−0.395291	−0.197645	+	0.980274i	$0.563329\pi$
$830$	4.56519	0.158460
$831$	15.7132	0.545085
$832$	0	0
$833$	−3.88766	−0.134700
$834$	−14.3211	−0.495900
$835$	2.14359	0.0741821
$836$	14.6481	0.506617
$837$	−7.07434	−0.244525
$838$	−22.5795	−0.779996
$839$	−27.8271	−0.960699	−0.480349	−	0.877077i	$-0.659490\pi$
$839$	−27.8271	−0.960699	−0.480349	+	0.877077i	$0.659490\pi$
$840$	−0.332808	−0.0114830
$841$	−27.5918	−0.951442
$842$	34.1708	1.17760
$843$	−4.07337	−0.140294
$844$	4.55485	0.156785
$845$	0	0
$846$	12.0528	0.414382
$847$	−4.18667	−0.143856
$848$	−9.71047	−0.333459
$849$	−21.5348	−0.739074
$850$	19.0077	0.651959
$851$	2.42829	0.0832408
$852$	0.944620	0.0323621
$853$	−22.0318	−0.754354	−0.377177	−	0.926141i	$-0.623105\pi$
$853$	−22.0318	−0.754354	−0.377177	+	0.926141i	$0.623105\pi$
$854$	7.43978	0.254584
$855$	1.86765	0.0638724
$856$	8.65078	0.295677
$857$	−56.1169	−1.91692	−0.958458	−	0.285233i	$-0.907929\pi$
$857$	−56.1169	−1.91692	−0.958458	+	0.285233i	$0.907929\pi$
$858$	0	0
$859$	−31.4521	−1.07313	−0.536566	−	0.843858i	$-0.680279\pi$
$859$	−31.4521	−1.07313	−0.536566	+	0.843858i	$0.680279\pi$
$860$	1.02316	0.0348896
$861$	0.521059	0.0177577
$862$	−5.23689	−0.178369
$863$	−20.0297	−0.681818	−0.340909	−	0.940096i	$-0.610735\pi$
$863$	−20.0297	−0.681818	−0.340909	+	0.940096i	$0.610735\pi$
$864$	−1.00000	−0.0340207
$865$	−6.19500	−0.210636
$866$	16.8206	0.571587
$867$	1.88608	0.0640548
$868$	7.07434	0.240119
$869$	−2.46156	−0.0835028
$870$	0.394934	0.0133895
$871$	0	0
$872$	18.6144	0.630361
$873$	−5.81587	−0.196837
$874$	23.6401	0.799640
$875$	−3.29121	−0.111263
$876$	−4.66562	−0.157637
$877$	−53.6565	−1.81185	−0.905925	−	0.423438i	$-0.860823\pi$
$877$	−53.6565	−1.81185	−0.905925	+	0.423438i	$0.860823\pi$
$878$	−14.7256	−0.496966
$879$	0.176042	0.00593775
$880$	0.868706	0.0292841
$881$	−38.8312	−1.30826	−0.654129	−	0.756383i	$-0.726965\pi$
$881$	−38.8312	−1.30826	−0.654129	+	0.756383i	$0.726965\pi$
$882$	1.00000	0.0336718
$883$	−48.7852	−1.64175	−0.820877	−	0.571105i	$-0.806515\pi$
$883$	−48.7852	−1.64175	−0.820877	+	0.571105i	$0.806515\pi$
$884$	0	0
$885$	3.48621	0.117188
$886$	36.2393	1.21748
$887$	23.1026	0.775709	0.387854	−	0.921721i	$-0.373216\pi$
$887$	23.1026	0.775709	0.387854	+	0.921721i	$0.373216\pi$
$888$	−0.576440	−0.0193440
$889$	6.77953	0.227378
$890$	1.12499	0.0377096
$891$	2.61023	0.0874462
$892$	8.43346	0.282373
$893$	67.6378	2.26341
$894$	6.05696	0.202575
$895$	−7.39374	−0.247146
$896$	1.00000	0.0334077
$897$	0	0
$898$	24.2893	0.810543
$899$	−8.39493	−0.279987
$900$	−4.88924	−0.162975
$901$	37.7510	1.25767
$902$	−1.36009	−0.0452859
$903$	−3.07434	−0.102308
$904$	9.15025	0.304333
$905$	1.35071	0.0448992
$906$	21.4953	0.714132
$907$	54.0157	1.79356	0.896781	−	0.442474i	$-0.145899\pi$
$907$	54.0157	1.79356	0.896781	+	0.442474i	$0.145899\pi$
$908$	27.9893	0.928857
$909$	−15.9703	−0.529702
$910$	0	0
$911$	−21.3318	−0.706753	−0.353376	−	0.935481i	$-0.614966\pi$
$911$	−21.3318	−0.706753	−0.353376	+	0.935481i	$0.614966\pi$
$912$	−5.61181	−0.185826
$913$	35.8051	1.18498
$914$	33.3412	1.10283
$915$	−2.47602	−0.0818546
$916$	28.7785	0.950868
$917$	7.22879	0.238716
$918$	3.88766	0.128312
$919$	−51.4923	−1.69857	−0.849286	−	0.527932i	$-0.822967\pi$
$919$	−51.4923	−1.69857	−0.849286	+	0.527932i	$0.822967\pi$
$920$	1.40197	0.0462217
$921$	−20.7405	−0.683422
$922$	−29.5970	−0.974724
$923$	0	0
$924$	−2.61023	−0.0858704
$925$	−2.81835	−0.0926668
$926$	−36.6027	−1.20284
$927$	7.50641	0.246543
$928$	−1.18667	−0.0389545
$929$	−19.6100	−0.643385	−0.321692	−	0.946844i	$-0.604252\pi$
$929$	−19.6100	−0.643385	−0.321692	+	0.946844i	$0.604252\pi$
$930$	−2.35439	−0.0772036
$931$	5.61181	0.183920
$932$	18.8877	0.618686
$933$	10.9678	0.359069
$934$	6.21635	0.203405
$935$	−3.37724	−0.110447
$936$	0	0
$937$	−7.16637	−0.234115	−0.117058	−	0.993125i	$-0.537346\pi$
$937$	−7.16637	−0.234115	−0.117058	+	0.993125i	$0.537346\pi$
$938$	−11.9203	−0.389212
$939$	−16.2189	−0.529284
$940$	4.01125	0.130832
$941$	−40.4744	−1.31943	−0.659714	−	0.751516i	$-0.729323\pi$
$941$	−40.4744	−1.31943	−0.659714	+	0.751516i	$0.729323\pi$
$942$	−2.29227	−0.0746861
$943$	−2.19500	−0.0714790
$944$	−10.4752	−0.340937
$945$	−0.332808	−0.0108262
$946$	8.02474	0.260907
$947$	−25.8793	−0.840966	−0.420483	−	0.907301i	$-0.638139\pi$
$947$	−25.8793	−0.840966	−0.420483	+	0.907301i	$0.638139\pi$
$948$	0.943042	0.0306286
$949$	0	0
$950$	−27.4375	−0.890190
$951$	−10.5358	−0.341647
$952$	−3.88766	−0.126000
$953$	−10.0132	−0.324358	−0.162179	−	0.986761i	$-0.551852\pi$
$953$	−10.0132	−0.324358	−0.162179	+	0.986761i	$0.551852\pi$
$954$	−9.71047	−0.314388
$955$	−3.90063	−0.126221
$956$	11.0933	0.358783
$957$	3.09750	0.100128
$958$	−33.5737	−1.08472
$959$	11.8892	0.383924
$960$	−0.332808	−0.0107413
$961$	19.0462	0.614395
$962$	0	0
$963$	8.65078	0.278767
$964$	8.11392	0.261332
$965$	−7.92116	−0.254991
$966$	−4.21257	−0.135537
$967$	−32.1063	−1.03247	−0.516235	−	0.856447i	$-0.672667\pi$
$967$	−32.1063	−1.03247	−0.516235	+	0.856447i	$0.672667\pi$
$968$	−4.18667	−0.134565
$969$	21.8168	0.700857
$970$	−1.93556	−0.0621472
$971$	−1.29954	−0.0417041	−0.0208521	−	0.999783i	$-0.506638\pi$
$971$	−1.29954	−0.0417041	−0.0208521	+	0.999783i	$0.506638\pi$
$972$	−1.00000	−0.0320750
$973$	14.3211	0.459114
$974$	20.2095	0.647555
$975$	0	0
$976$	7.43978	0.238142
$977$	−29.3197	−0.938019	−0.469010	−	0.883193i	$-0.655389\pi$
$977$	−29.3197	−0.938019	−0.469010	+	0.883193i	$0.655389\pi$
$978$	3.34071	0.106824
$979$	8.82334	0.281995
$980$	0.332808	0.0106311
$981$	18.6144	0.594311
$982$	−6.77953	−0.216344
$983$	41.6812	1.32942	0.664712	−	0.747100i	$-0.268554\pi$
$983$	41.6812	1.32942	0.664712	+	0.747100i	$0.268554\pi$
$984$	0.521059	0.0166108
$985$	−3.35559	−0.106918
$986$	4.61339	0.146920
$987$	−12.0528	−0.383643
$988$	0	0
$989$	12.9509	0.411813
$990$	0.868706	0.0276093
$991$	45.3520	1.44065	0.720327	−	0.693635i	$-0.243992\pi$
$991$	45.3520	1.44065	0.720327	+	0.693635i	$0.243992\pi$
$992$	7.07434	0.224610
$993$	16.1297	0.511861
$994$	−0.944620	−0.0299615
$995$	4.60941	0.146128
$996$	−13.7172	−0.434646
$997$	−8.57917	−0.271705	−0.135853	−	0.990729i	$-0.543377\pi$
$997$	−8.57917	−0.271705	−0.135853	+	0.990729i	$0.543377\pi$
$998$	−11.0813	−0.350772
$999$	−0.576440	−0.0182377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.co.1.2		4
13.6	odd	12		546.2.s.e.127.2	yes	8
13.11	odd	12		546.2.s.e.43.1	✓	8
13.12	even	2		7098.2.a.cn.1.3		4
39.11	even	12		1638.2.bj.f.1135.4		8
39.32	even	12		1638.2.bj.f.127.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.s.e.43.1	✓	8	13.11	odd	12
546.2.s.e.127.2	yes	8	13.6	odd	12
1638.2.bj.f.127.3		8	39.32	even	12
1638.2.bj.f.1135.4		8	39.11	even	12
7098.2.a.cn.1.3		4	13.12	even	2
7098.2.a.co.1.2		4	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:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.332808 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.332808 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.332808 q^{10} +2.61023 q^{11} -1.00000 q^{12} +1.00000 q^{14} -0.332808 q^{15} +1.00000 q^{16} -3.88766 q^{17} +1.00000 q^{18} +5.61181 q^{19} +0.332808 q^{20} -1.00000 q^{21} +2.61023 q^{22} +4.21257 q^{23} -1.00000 q^{24} -4.88924 q^{25} -1.00000 q^{27} +1.00000 q^{28} -1.18667 q^{29} -0.332808 q^{30} +7.07434 q^{31} +1.00000 q^{32} -2.61023 q^{33} -3.88766 q^{34} +0.332808 q^{35} +1.00000 q^{36} +0.576440 q^{37} +5.61181 q^{38} +0.332808 q^{40} -0.521059 q^{41} -1.00000 q^{42} +3.07434 q^{43} +2.61023 q^{44} +0.332808 q^{45} +4.21257 q^{46} +12.0528 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.88924 q^{50} +3.88766 q^{51} -9.71047 q^{53} -1.00000 q^{54} +0.868706 q^{55} +1.00000 q^{56} -5.61181 q^{57} -1.18667 q^{58} -10.4752 q^{59} -0.332808 q^{60} +7.43978 q^{61} +7.07434 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.61023 q^{66} -11.9203 q^{67} -3.88766 q^{68} -4.21257 q^{69} +0.332808 q^{70} -0.944620 q^{71} +1.00000 q^{72} +4.66562 q^{73} +0.576440 q^{74} +4.88924 q^{75} +5.61181 q^{76} +2.61023 q^{77} -0.943042 q^{79} +0.332808 q^{80} +1.00000 q^{81} -0.521059 q^{82} +13.7172 q^{83} -1.00000 q^{84} -1.29384 q^{85} +3.07434 q^{86} +1.18667 q^{87} +2.61023 q^{88} +3.38029 q^{89} +0.332808 q^{90} +4.21257 q^{92} -7.07434 q^{93} +12.0528 q^{94} +1.86765 q^{95} -1.00000 q^{96} -5.81587 q^{97} +1.00000 q^{98} +2.61023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} + 4 q^{7} + 4 q^{8} + 4 q^{9} + 6 q^{10} + 4 q^{11} - 4 q^{12} + 4 q^{14} - 6 q^{15} + 4 q^{16} + 2 q^{17} + 4 q^{18} + 2 q^{19} + 6 q^{20} - 4 q^{21} + 4 q^{22} + 8 q^{23} - 4 q^{24} + 12 q^{25} - 4 q^{27} + 4 q^{28} - 2 q^{29} - 6 q^{30} + 8 q^{31} + 4 q^{32} - 4 q^{33} + 2 q^{34} + 6 q^{35} + 4 q^{36} + 6 q^{37} + 2 q^{38} + 6 q^{40} + 10 q^{41} - 4 q^{42} - 8 q^{43} + 4 q^{44} + 6 q^{45} + 8 q^{46} + 2 q^{47} - 4 q^{48} + 4 q^{49} + 12 q^{50} - 2 q^{51} - 6 q^{53} - 4 q^{54} + 22 q^{55} + 4 q^{56} - 2 q^{57} - 2 q^{58} + 4 q^{59} - 6 q^{60} + 8 q^{61} + 8 q^{62} + 4 q^{63} + 4 q^{64} - 4 q^{66} - 24 q^{67} + 2 q^{68} - 8 q^{69} + 6 q^{70} + 12 q^{71} + 4 q^{72} + 28 q^{73} + 6 q^{74} - 12 q^{75} + 2 q^{76} + 4 q^{77} - 2 q^{79} + 6 q^{80} + 4 q^{81} + 10 q^{82} + 22 q^{83} - 4 q^{84} - 6 q^{85} - 8 q^{86} + 2 q^{87} + 4 q^{88} + 38 q^{89} + 6 q^{90} + 8 q^{92} - 8 q^{93} + 2 q^{94} - 50 q^{95} - 4 q^{96} + 22 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 + 4 * q^7 + 4 * q^8 + 4 * q^9 + 6 * q^10 + 4 * q^11 - 4 * q^12 + 4 * q^14 - 6 * q^15 + 4 * q^16 + 2 * q^17 + 4 * q^18 + 2 * q^19 + 6 * q^20 - 4 * q^21 + 4 * q^22 + 8 * q^23 - 4 * q^24 + 12 * q^25 - 4 * q^27 + 4 * q^28 - 2 * q^29 - 6 * q^30 + 8 * q^31 + 4 * q^32 - 4 * q^33 + 2 * q^34 + 6 * q^35 + 4 * q^36 + 6 * q^37 + 2 * q^38 + 6 * q^40 + 10 * q^41 - 4 * q^42 - 8 * q^43 + 4 * q^44 + 6 * q^45 + 8 * q^46 + 2 * q^47 - 4 * q^48 + 4 * q^49 + 12 * q^50 - 2 * q^51 - 6 * q^53 - 4 * q^54 + 22 * q^55 + 4 * q^56 - 2 * q^57 - 2 * q^58 + 4 * q^59 - 6 * q^60 + 8 * q^61 + 8 * q^62 + 4 * q^63 + 4 * q^64 - 4 * q^66 - 24 * q^67 + 2 * q^68 - 8 * q^69 + 6 * q^70 + 12 * q^71 + 4 * q^72 + 28 * q^73 + 6 * q^74 - 12 * q^75 + 2 * q^76 + 4 * q^77 - 2 * q^79 + 6 * q^80 + 4 * q^81 + 10 * q^82 + 22 * q^83 - 4 * q^84 - 6 * q^85 - 8 * q^86 + 2 * q^87 + 4 * q^88 + 38 * q^89 + 6 * q^90 + 8 * q^92 - 8 * q^93 + 2 * q^94 - 50 * q^95 - 4 * q^96 + 22 * q^97 + 4 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more