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Properties

Label 49.8.c.h.18.2
Level $49$
Weight $8$
Character 49.18
Analytic conductor $15.307$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,8,Mod(18,49)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("49.18"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-30,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.3068662487\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 1124 x^{14} - 4480 x^{13} + 503818 x^{12} + 3794560 x^{11} - 106136536 x^{10} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{12}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 18.2
Root \(4.20346 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.18
Dual form 49.8.c.h.30.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-10.2199 - 17.7014i) q^{2} +(30.4918 - 52.8133i) q^{3} +(-144.894 + 250.963i) q^{4} +(-133.959 - 232.023i) q^{5} -1246.49 q^{6} +3306.90 q^{8} +(-765.996 - 1326.74i) q^{9} +(-2738.10 + 4742.52i) q^{10} +(-3429.34 + 5939.80i) q^{11} +(8836.12 + 15304.6i) q^{12} +553.232 q^{13} -16338.6 q^{15} +(-15249.9 - 26413.6i) q^{16} +(-9391.19 + 16266.0i) q^{17} +(-15656.8 + 27118.4i) q^{18} +(-13263.1 - 22972.3i) q^{19} +77639.0 q^{20} +140190. q^{22} +(7103.02 + 12302.8i) q^{23} +(100833. - 174649. i) q^{24} +(3172.62 - 5495.14i) q^{25} +(-5653.98 - 9792.99i) q^{26} +39944.7 q^{27} -218240. q^{29} +(166979. + 289216. i) q^{30} +(88811.2 - 153825. i) q^{31} +(-100064. + 173316. i) q^{32} +(209133. + 362230. i) q^{33} +383909. q^{34} +443952. q^{36} +(-83324.7 - 144323. i) q^{37} +(-271095. + 469550. i) q^{38} +(16869.0 - 29218.0i) q^{39} +(-442989. - 767279. i) q^{40} -392711. q^{41} -349297. q^{43} +(-993780. - 1.72128e6i) q^{44} +(-205224. + 355458. i) q^{45} +(145185. - 251467. i) q^{46} +(499256. + 864737. i) q^{47} -1.85999e6 q^{48} -129696. q^{50} +(572708. + 991959. i) q^{51} +(-80159.7 + 138841. i) q^{52} +(-279447. + 484016. i) q^{53} +(-408232. - 707078. i) q^{54} +1.83756e6 q^{55} -1.61766e6 q^{57} +(2.23040e6 + 3.86316e6i) q^{58} +(-1.43110e6 + 2.47874e6i) q^{59} +(2.36735e6 - 4.10037e6i) q^{60} +(402096. + 696451. i) q^{61} -3.63057e6 q^{62} +186612. q^{64} +(-74110.2 - 128363. i) q^{65} +(4.27466e6 - 7.40392e6i) q^{66} +(-505758. + 875998. i) q^{67} +(-2.72145e6 - 4.71368e6i) q^{68} +866335. q^{69} +1.38526e6 q^{71} +(-2.53308e6 - 4.38742e6i) q^{72} +(292236. - 506167. i) q^{73} +(-1.70314e6 + 2.94993e6i) q^{74} +(-193478. - 335113. i) q^{75} +7.68693e6 q^{76} -689600. q^{78} +(-1.89279e6 - 3.27841e6i) q^{79} +(-4.08572e6 + 7.07668e6i) q^{80} +(2.89322e6 - 5.01120e6i) q^{81} +(4.01347e6 + 6.95154e6i) q^{82} -3.66559e6 q^{83} +5.03213e6 q^{85} +(3.56979e6 + 6.18306e6i) q^{86} +(-6.65453e6 + 1.15260e7i) q^{87} +(-1.13405e7 + 1.96423e7i) q^{88} +(4.60887e6 + 7.98279e6i) q^{89} +8.38948e6 q^{90} -4.11673e6 q^{92} +(-5.41602e6 - 9.38082e6i) q^{93} +(1.02047e7 - 1.76751e7i) q^{94} +(-3.55340e6 + 6.15468e6i) q^{95} +(6.10227e6 + 1.05694e7i) q^{96} -8.21720e6 q^{97} +1.05075e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 30 q^{2} - 458 q^{4} + 8580 q^{8} - 7600 q^{9} - 17760 q^{11} + 58432 q^{15} - 33762 q^{16} - 244850 q^{18} + 861320 q^{22} - 96000 q^{23} - 468992 q^{25} + 472560 q^{29} + 193752 q^{30} - 1064370 q^{32}+ \cdots + 72185600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.2199 − 17.7014i −0.903322 − 1.56460i −0.823154 − 0.567818i \(-0.807788\pi\)
−0.0801680 − 0.996781i \(-0.525546\pi\)
\(3\) 30.4918 − 52.8133i 0.652016 − 1.12932i −0.330617 − 0.943765i \(-0.607257\pi\)
0.982633 − 0.185560i \(-0.0594098\pi\)
\(4\) −144.894 + 250.963i −1.13198 + 1.96065i
\(5\) −133.959 − 232.023i −0.479265 − 0.830112i 0.520452 − 0.853891i \(-0.325763\pi\)
−0.999717 + 0.0237791i \(0.992430\pi\)
\(6\) −1246.49 −2.35592
\(7\) 0 0
\(8\) 3306.90 2.28353
\(9\) −765.996 − 1326.74i −0.350250 − 0.606650i
\(10\) −2738.10 + 4742.52i −0.865862 + 1.49972i
\(11\) −3429.34 + 5939.80i −0.776849 + 1.34554i 0.156901 + 0.987614i \(0.449850\pi\)
−0.933749 + 0.357927i \(0.883484\pi\)
\(12\) 8836.12 + 15304.6i 1.47614 + 2.55675i
\(13\) 553.232 0.0698402 0.0349201 − 0.999390i \(-0.488882\pi\)
0.0349201 + 0.999390i \(0.488882\pi\)
\(14\) 0 0
\(15\) −16338.6 −1.24995
\(16\) −15249.9 − 26413.6i −0.930782 − 1.61216i
\(17\) −9391.19 + 16266.0i −0.463606 + 0.802990i −0.999137 − 0.0415257i \(-0.986778\pi\)
0.535531 + 0.844516i \(0.320111\pi\)
\(18\) −15656.8 + 27118.4i −0.632777 + 1.09600i
\(19\) −13263.1 − 22972.3i −0.443615 − 0.768363i 0.554340 − 0.832290i \(-0.312971\pi\)
−0.997955 + 0.0639272i \(0.979637\pi\)
\(20\) 77639.0 2.17008
\(21\) 0 0
\(22\) 140190. 2.80698
\(23\) 7103.02 + 12302.8i 0.121730 + 0.210842i 0.920450 − 0.390861i \(-0.127823\pi\)
−0.798720 + 0.601702i \(0.794489\pi\)
\(24\) 100833. − 174649.i 1.48890 − 2.57885i
\(25\) 3172.62 − 5495.14i 0.0406095 − 0.0703378i
\(26\) −5653.98 − 9792.99i −0.0630882 − 0.109272i
\(27\) 39944.7 0.390558
\(28\) 0 0
\(29\) −218240. −1.66166 −0.830830 − 0.556527i \(-0.812134\pi\)
−0.830830 + 0.556527i \(0.812134\pi\)
\(30\) 166979. + 289216.i 1.12911 + 1.95568i
\(31\) 88811.2 − 153825.i 0.535429 − 0.927390i −0.463714 − 0.885985i \(-0.653483\pi\)
0.999142 − 0.0414045i \(-0.0131832\pi\)
\(32\) −100064. + 173316.i −0.539826 + 0.935006i
\(33\) 209133. + 362230.i 1.01304 + 1.75463i
\(34\) 383909. 1.67514
\(35\) 0 0
\(36\) 443952. 1.58590
\(37\) −83324.7 − 144323.i −0.270438 − 0.468413i 0.698536 − 0.715575i \(-0.253835\pi\)
−0.968974 + 0.247162i \(0.920502\pi\)
\(38\) −271095. + 469550.i −0.801454 + 1.38816i
\(39\) 16869.0 − 29218.0i 0.0455369 − 0.0788722i
\(40\) −442989. − 767279.i −1.09442 − 1.89559i
\(41\) −392711. −0.889875 −0.444938 − 0.895562i \(-0.646774\pi\)
−0.444938 + 0.895562i \(0.646774\pi\)
\(42\) 0 0
\(43\) −349297. −0.669971 −0.334985 − 0.942223i \(-0.608731\pi\)
−0.334985 + 0.942223i \(0.608731\pi\)
\(44\) −993780. − 1.72128e6i −1.75876 − 3.04626i
\(45\) −205224. + 355458.i −0.335725 + 0.581493i
\(46\) 145185. − 251467.i 0.219922 − 0.380916i
\(47\) 499256. + 864737.i 0.701425 + 1.21490i 0.967966 + 0.251080i \(0.0807857\pi\)
−0.266542 + 0.963823i \(0.585881\pi\)
\(48\) −1.85999e6 −2.42754
\(49\) 0 0
\(50\) −129696. −0.146734
\(51\) 572708. + 991959.i 0.604558 + 1.04712i
\(52\) −80159.7 + 138841.i −0.0790578 + 0.136932i
\(53\) −279447. + 484016.i −0.257830 + 0.446575i −0.965660 − 0.259808i \(-0.916341\pi\)
0.707830 + 0.706383i \(0.249674\pi\)
\(54\) −408232. − 707078.i −0.352800 − 0.611067i
\(55\) 1.83756e6 1.48927
\(56\) 0 0
\(57\) −1.61766e6 −1.15698
\(58\) 2.23040e6 + 3.86316e6i 1.50101 + 2.59983i
\(59\) −1.43110e6 + 2.47874e6i −0.907169 + 1.57126i −0.0891895 + 0.996015i \(0.528428\pi\)
−0.817979 + 0.575248i \(0.804906\pi\)
\(60\) 2.36735e6 − 4.10037e6i 1.41493 − 2.45072i
\(61\) 402096. + 696451.i 0.226817 + 0.392858i 0.956863 − 0.290540i \(-0.0938348\pi\)
−0.730046 + 0.683398i \(0.760501\pi\)
\(62\) −3.63057e6 −1.93466
\(63\) 0 0
\(64\) 186612. 0.0889836
\(65\) −74110.2 − 128363.i −0.0334720 − 0.0579752i
\(66\) 4.27466e6 − 7.40392e6i 1.83019 − 3.16999i
\(67\) −505758. + 875998.i −0.205438 + 0.355829i −0.950272 − 0.311421i \(-0.899195\pi\)
0.744834 + 0.667250i \(0.232529\pi\)
\(68\) −2.72145e6 − 4.71368e6i −1.04959 − 1.81794i
\(69\) 866335. 0.317478
\(70\) 0 0
\(71\) 1.38526e6 0.459333 0.229666 − 0.973269i \(-0.426236\pi\)
0.229666 + 0.973269i \(0.426236\pi\)
\(72\) −2.53308e6 − 4.38742e6i −0.799806 − 1.38530i
\(73\) 292236. − 506167.i 0.0879231 − 0.152287i −0.818710 − 0.574207i \(-0.805310\pi\)
0.906633 + 0.421920i \(0.138644\pi\)
\(74\) −1.70314e6 + 2.94993e6i −0.488585 + 0.846255i
\(75\) −193478. − 335113.i −0.0529561 − 0.0917227i
\(76\) 7.68693e6 2.00865
\(77\) 0 0
\(78\) −689600. −0.164538
\(79\) −1.89279e6 − 3.27841e6i −0.431925 − 0.748116i 0.565114 − 0.825013i \(-0.308832\pi\)
−0.997039 + 0.0768969i \(0.975499\pi\)
\(80\) −4.08572e6 + 7.07668e6i −0.892183 + 1.54531i
\(81\) 2.89322e6 − 5.01120e6i 0.604900 − 1.04772i
\(82\) 4.01347e6 + 6.95154e6i 0.803844 + 1.39230i
\(83\) −3.66559e6 −0.703673 −0.351837 − 0.936061i \(-0.614443\pi\)
−0.351837 + 0.936061i \(0.614443\pi\)
\(84\) 0 0
\(85\) 5.03213e6 0.888762
\(86\) 3.56979e6 + 6.18306e6i 0.605199 + 1.04824i
\(87\) −6.65453e6 + 1.15260e7i −1.08343 + 1.87655i
\(88\) −1.13405e7 + 1.96423e7i −1.77396 + 3.07258i
\(89\) 4.60887e6 + 7.98279e6i 0.692993 + 1.20030i 0.970852 + 0.239678i \(0.0770419\pi\)
−0.277859 + 0.960622i \(0.589625\pi\)
\(90\) 8.38948e6 1.21307
\(91\) 0 0
\(92\) −4.11673e6 −0.551182
\(93\) −5.41602e6 − 9.38082e6i −0.698216 − 1.20935i
\(94\) 1.02047e7 − 1.76751e7i 1.26722 − 2.19490i
\(95\) −3.55340e6 + 6.15468e6i −0.425218 + 0.736500i
\(96\) 6.10227e6 + 1.05694e7i 0.703950 + 1.21928i
\(97\) −8.21720e6 −0.914161 −0.457081 − 0.889425i \(-0.651105\pi\)
−0.457081 + 0.889425i \(0.651105\pi\)
\(98\) 0 0
\(99\) 1.05075e7 1.08836
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 49.8.c.h.18.2 16
7.2 even 3 inner 49.8.c.h.30.2 16
7.3 odd 6 49.8.a.g.1.8 yes 8
7.4 even 3 49.8.a.g.1.7 ✓ 8
7.5 odd 6 inner 49.8.c.h.30.1 16
7.6 odd 2 inner 49.8.c.h.18.1 16
21.11 odd 6 441.8.a.ba.1.1 8
21.17 even 6 441.8.a.ba.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.8.a.g.1.7 ✓ 8 7.4 even 3
49.8.a.g.1.8 yes 8 7.3 odd 6
49.8.c.h.18.1 16 7.6 odd 2 inner
49.8.c.h.18.2 16 1.1 even 1 trivial
49.8.c.h.30.1 16 7.5 odd 6 inner
49.8.c.h.30.2 16 7.2 even 3 inner
441.8.a.ba.1.1 8 21.11 odd 6
441.8.a.ba.1.2 8 21.17 even 6