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Properties

Label 49.8.c.h.18.3
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 18.3
Root \(-10.4785 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 49.18
Dual form 49.8.c.h.30.3

$q$-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+(-4.85070 - 8.40166i) q^{2} +(-45.3030 + 78.4672i) q^{3} +(16.9415 - 29.3435i) q^{4} +(-115.940 - 200.814i) q^{5} +879.006 q^{6} -1570.49 q^{8} +(-3011.23 - 5215.61i) q^{9} +(-1124.78 + 1948.18i) q^{10} +(-548.900 + 950.723i) q^{11} +(1535.00 + 2658.70i) q^{12} +3168.45 q^{13} +21009.7 q^{15} +(5449.47 + 9438.75i) q^{16} +(-349.111 + 604.678i) q^{17} +(-29213.2 + 50598.7i) q^{18} +(20788.8 + 36007.3i) q^{19} -7856.77 q^{20} +10650.2 q^{22} +(12885.3 + 22318.1i) q^{23} +(71148.0 - 123232. i) q^{24} +(12178.3 - 21093.5i) q^{25} +(-15369.2 - 26620.2i) q^{26} +347516. q^{27} +89397.4 q^{29} +(-101912. - 176517. i) q^{30} +(-36608.1 + 63407.1i) q^{31} +(-47643.9 + 82521.7i) q^{32} +(-49733.7 - 86141.3i) q^{33} +6773.73 q^{34} -204059. q^{36} +(134513. + 232984. i) q^{37} +(201681. - 349321. i) q^{38} +(-143541. + 248620. i) q^{39} +(182083. + 315376. i) q^{40} -581002. q^{41} +627120. q^{43} +(18598.3 + 32213.3i) q^{44} +(-698245. + 1.20939e6i) q^{45} +(125006. - 216517. i) q^{46} +(435674. + 754610. i) q^{47} -987510. q^{48} -236294. q^{50} +(-31631.6 - 54787.6i) q^{51} +(53678.2 - 92973.4i) q^{52} +(776663. - 1.34522e6i) q^{53} +(-1.68570e6 - 2.91971e6i) q^{54} +254558. q^{55} -3.76719e6 q^{57} +(-433640. - 751086. i) q^{58} +(226017. - 391473. i) q^{59} +(355936. - 616499. i) q^{60} +(218868. + 379090. i) q^{61} +710300. q^{62} +2.31949e6 q^{64} +(-367350. - 636269. i) q^{65} +(-482486. + 835691. i) q^{66} +(590115. - 1.02211e6i) q^{67} +(11828.9 + 20488.3i) q^{68} -2.33498e6 q^{69} +859481. q^{71} +(4.72911e6 + 8.19106e6i) q^{72} +(1.30428e6 - 2.25908e6i) q^{73} +(1.30497e6 - 2.26027e6i) q^{74} +(1.10343e6 + 1.91120e6i) q^{75} +1.40877e6 q^{76} +2.78509e6 q^{78} +(-3.14300e6 - 5.44383e6i) q^{79} +(1.26362e6 - 2.18866e6i) q^{80} +(-9.15799e6 + 1.58621e7i) q^{81} +(2.81827e6 + 4.88138e6i) q^{82} -2.61378e6 q^{83} +161904. q^{85} +(-3.04197e6 - 5.26885e6i) q^{86} +(-4.04998e6 + 7.01477e6i) q^{87} +(862042. - 1.49310e6i) q^{88} +(-930123. - 1.61102e6i) q^{89} +1.35479e7 q^{90} +873187. q^{92} +(-3.31692e6 - 5.74507e6i) q^{93} +(4.22665e6 - 7.32077e6i) q^{94} +(4.82051e6 - 8.34937e6i) q^{95} +(-4.31683e6 - 7.47697e6i) q^{96} +1.43368e7 q^{97} +6.61146e6 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.85070 − 8.40166i −0.428745 − 0.742608i 0.568017 − 0.823017i \(-0.307711\pi\)
−0.996762 + 0.0804085i \(0.974378\pi\)
\(3\) −45.3030 + 78.4672i −0.968731 + 1.67789i −0.269492 + 0.963003i \(0.586856\pi\)
−0.699239 + 0.714888i \(0.746478\pi\)
\(4\) 16.9415 − 29.3435i 0.132355 − 0.229246i
\(5\) −115.940 − 200.814i −0.414800 − 0.718454i 0.580608 − 0.814183i \(-0.302815\pi\)
−0.995407 + 0.0957295i \(0.969482\pi\)
\(6\) 879.006 1.66135
\(7\) 0 0
\(8\) −1570.49 −1.08448
\(9\) −3011.23 − 5215.61i −1.37688 − 2.38482i
\(10\) −1124.78 + 1948.18i −0.355687 + 0.616067i
\(11\) −548.900 + 950.723i −0.124342 + 0.215367i −0.921476 − 0.388436i \(-0.873015\pi\)
0.797133 + 0.603803i \(0.206349\pi\)
\(12\) 1535.00 + 2658.70i 0.256433 + 0.444155i
\(13\) 3168.45 0.399987 0.199993 − 0.979797i \(-0.435908\pi\)
0.199993 + 0.979797i \(0.435908\pi\)
\(14\) 0 0
\(15\) 21009.7 1.60732
\(16\) 5449.47 + 9438.75i 0.332609 + 0.576096i
\(17\) −349.111 + 604.678i −0.0172343 + 0.0298506i −0.874514 − 0.485000i \(-0.838819\pi\)
0.857280 + 0.514851i \(0.172153\pi\)
\(18\) −29213.2 + 50598.7i −1.18066 + 2.04496i
\(19\) 20788.8 + 36007.3i 0.695332 + 1.20435i 0.970069 + 0.242831i \(0.0780759\pi\)
−0.274737 + 0.961519i \(0.588591\pi\)
\(20\) −7856.77 −0.219603
\(21\) 0 0
\(22\) 10650.2 0.213245
\(23\) 12885.3 + 22318.1i 0.220825 + 0.382481i 0.955059 − 0.296416i \(-0.0957916\pi\)
−0.734234 + 0.678897i \(0.762458\pi\)
\(24\) 71148.0 − 123232.i 1.05057 − 1.81963i
\(25\) 12178.3 − 21093.5i 0.155883 − 0.269997i
\(26\) −15369.2 − 26620.2i −0.171492 − 0.297033i
\(27\) 347516. 3.39783
\(28\) 0 0
\(29\) 89397.4 0.680663 0.340331 − 0.940306i \(-0.389461\pi\)
0.340331 + 0.940306i \(0.389461\pi\)
\(30\) −101912. − 176517.i −0.689129 − 1.19361i
\(31\) −36608.1 + 63407.1i −0.220705 + 0.382272i −0.955022 − 0.296535i \(-0.904169\pi\)
0.734318 + 0.678806i \(0.237502\pi\)
\(32\) −47643.9 + 82521.7i −0.257029 + 0.445188i
\(33\) −49733.7 − 86141.3i −0.240908 − 0.417266i
\(34\) 6773.73 0.0295564
\(35\) 0 0
\(36\) −204059. −0.728948
\(37\) 134513. + 232984.i 0.436575 + 0.756170i 0.997423 − 0.0717489i \(-0.0228580\pi\)
−0.560848 + 0.827919i \(0.689525\pi\)
\(38\) 201681. − 349321.i 0.596240 − 1.03272i
\(39\) −143541. + 248620.i −0.387479 + 0.671134i
\(40\) 182083. + 315376.i 0.449841 + 0.779147i
\(41\) −581002. −1.31654 −0.658270 − 0.752782i \(-0.728712\pi\)
−0.658270 + 0.752782i \(0.728712\pi\)
\(42\) 0 0
\(43\) 627120. 1.20285 0.601425 − 0.798929i \(-0.294600\pi\)
0.601425 + 0.798929i \(0.294600\pi\)
\(44\) 18598.3 + 32213.3i 0.0329147 + 0.0570099i
\(45\) −698245. + 1.20939e6i −1.14226 + 1.97845i
\(46\) 125006. − 216517.i 0.189356 − 0.327973i
\(47\) 435674. + 754610.i 0.612096 + 1.06018i 0.990887 + 0.134699i \(0.0430067\pi\)
−0.378791 + 0.925482i \(0.623660\pi\)
\(48\) −987510. −1.28883
\(49\) 0 0
\(50\) −236294. −0.267336
\(51\) −31631.6 − 54787.6i −0.0333907 − 0.0578344i
\(52\) 53678.2 − 92973.4i 0.0529403 − 0.0916953i
\(53\) 776663. − 1.34522e6i 0.716584 − 1.24116i −0.245761 − 0.969330i \(-0.579038\pi\)
0.962345 − 0.271830i \(-0.0876288\pi\)
\(54\) −1.68570e6 − 2.91971e6i −1.45681 − 2.52326i
\(55\) 254558. 0.206309
\(56\) 0 0
\(57\) −3.76719e6 −2.69436
\(58\) −433640. − 751086.i −0.291831 − 0.505466i
\(59\) 226017. − 391473.i 0.143271 − 0.248153i −0.785455 − 0.618918i \(-0.787571\pi\)
0.928727 + 0.370765i \(0.120905\pi\)
\(60\) 355936. − 616499.i 0.212737 − 0.368471i
\(61\) 218868. + 379090.i 0.123460 + 0.213840i 0.921130 − 0.389255i \(-0.127267\pi\)
−0.797670 + 0.603095i \(0.793934\pi\)
\(62\) 710300. 0.378504
\(63\) 0 0
\(64\) 2.31949e6 1.10602
\(65\) −367350. − 636269.i −0.165914 − 0.287372i
\(66\) −482486. + 835691.i −0.206577 + 0.357801i
\(67\) 590115. − 1.02211e6i 0.239704 − 0.415179i −0.720926 − 0.693012i \(-0.756283\pi\)
0.960629 + 0.277834i \(0.0896164\pi\)
\(68\) 11828.9 + 20488.3i 0.00456209 + 0.00790177i
\(69\) −2.33498e6 −0.855681
\(70\) 0 0
\(71\) 859481. 0.284992 0.142496 − 0.989795i \(-0.454487\pi\)
0.142496 + 0.989795i \(0.454487\pi\)
\(72\) 4.72911e6 + 8.19106e6i 1.49319 + 2.58629i
\(73\) 1.30428e6 − 2.25908e6i 0.392411 − 0.679676i −0.600356 − 0.799733i \(-0.704974\pi\)
0.992767 + 0.120057i \(0.0383078\pi\)
\(74\) 1.30497e6 − 2.26027e6i 0.374359 − 0.648408i
\(75\) 1.10343e6 + 1.91120e6i 0.302017 + 0.523108i
\(76\) 1.40877e6 0.368123
\(77\) 0 0
\(78\) 2.78509e6 0.664519
\(79\) −3.14300e6 − 5.44383e6i −0.717215 − 1.24225i −0.962099 − 0.272700i \(-0.912083\pi\)
0.244884 − 0.969552i \(-0.421250\pi\)
\(80\) 1.26362e6 − 2.18866e6i 0.275932 − 0.477929i
\(81\) −9.15799e6 + 1.58621e7i −1.91471 + 3.31637i
\(82\) 2.81827e6 + 4.88138e6i 0.564461 + 0.977674i
\(83\) −2.61378e6 −0.501760 −0.250880 − 0.968018i \(-0.580720\pi\)
−0.250880 + 0.968018i \(0.580720\pi\)
\(84\) 0 0
\(85\) 161904. 0.0285951
\(86\) −3.04197e6 − 5.26885e6i −0.515716 − 0.893246i
\(87\) −4.04998e6 + 7.01477e6i −0.659379 + 1.14208i
\(88\) 862042. − 1.49310e6i 0.134846 − 0.233561i
\(89\) −930123. − 1.61102e6i −0.139854 − 0.242234i 0.787587 − 0.616203i \(-0.211330\pi\)
−0.927441 + 0.373969i \(0.877997\pi\)
\(90\) 1.35479e7 1.95895
\(91\) 0 0
\(92\) 873187. 0.116909
\(93\) −3.31692e6 − 5.74507e6i −0.427607 − 0.740636i
\(94\) 4.22665e6 − 7.32077e6i 0.524866 − 0.909095i
\(95\) 4.82051e6 − 8.34937e6i 0.576847 − 0.999128i
\(96\) −4.31683e6 − 7.47697e6i −0.497984 − 0.862534i
\(97\) 1.43368e7 1.59497 0.797485 − 0.603339i \(-0.206164\pi\)
0.797485 + 0.603339i \(0.206164\pi\)
\(98\) 0 0
\(99\) 6.61146e6 0.684817
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.18.3 16
7.2 even 3 inner 49.8.c.h.30.3 16
7.3 odd 6 49.8.a.g.1.5 ✓ 8
7.4 even 3 49.8.a.g.1.6 yes 8
7.5 odd 6 inner 49.8.c.h.30.4 16
7.6 odd 2 inner 49.8.c.h.18.4 16
21.11 odd 6 441.8.a.ba.1.3 8
21.17 even 6 441.8.a.ba.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.8.a.g.1.5 ✓ 8 7.3 odd 6
49.8.a.g.1.6 yes 8 7.4 even 3
49.8.c.h.18.3 16 1.1 even 1 trivial
49.8.c.h.18.4 16 7.6 odd 2 inner
49.8.c.h.30.3 16 7.2 even 3 inner
49.8.c.h.30.4 16 7.5 odd 6 inner
441.8.a.ba.1.3 8 21.11 odd 6
441.8.a.ba.1.4 8 21.17 even 6