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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,12,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-266] 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: \(86.0544362227\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 101803 x^{6} + 6576400 x^{5} + 8539617914 x^{4} + 333205096780 x^{3} + \cdots + 33\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.4
Root \(152.619 - 264.344i\) of defining polynomial
Character \(\chi\) \(=\) 112.81
Dual form 112.12.i.a.65.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(271.738 - 470.665i) q^{3} +(3667.11 + 6351.63i) q^{5} +(28136.3 + 34433.6i) q^{7} +(-59110.0 - 102381. i) q^{9} +(24115.9 - 41769.9i) q^{11} -1.82289e6 q^{13} +3.98598e6 q^{15} +(3.35793e6 - 5.81611e6i) q^{17} +(8.95659e6 + 1.55133e7i) q^{19} +(2.38524e7 - 3.88585e6i) q^{21} +(-1.04379e7 - 1.80789e7i) q^{23} +(-2.48139e6 + 4.29789e6i) q^{25} +3.20255e7 q^{27} +2.38808e7 q^{29} +(-5.14984e7 + 8.91978e7i) q^{31} +(-1.31064e7 - 2.27010e7i) q^{33} +(-1.15530e8 + 3.04984e8i) q^{35} +(2.60667e8 + 4.51488e8i) q^{37} +(-4.95349e8 + 8.57970e8i) q^{39} -1.25858e9 q^{41} +1.03048e9 q^{43} +(4.33526e8 - 7.50889e8i) q^{45} +(7.99740e8 + 1.38519e9i) q^{47} +(-3.94019e8 + 1.93767e9i) q^{49} +(-1.82496e9 - 3.16092e9i) q^{51} +(-2.29236e9 + 3.97049e9i) q^{53} +3.53743e8 q^{55} +9.73540e9 q^{57} +(-2.88236e9 + 4.99240e9i) q^{59} +(2.91537e9 + 5.04957e9i) q^{61} +(1.86222e9 - 4.91601e9i) q^{63} +(-6.68475e9 - 1.15783e10i) q^{65} +(7.79178e9 - 1.34958e10i) q^{67} -1.13455e10 q^{69} +7.49669e8 q^{71} +(5.51048e9 - 9.54443e9i) q^{73} +(1.34858e9 + 2.33580e9i) q^{75} +(2.11682e9 - 3.44857e8i) q^{77} +(1.78570e10 + 3.09292e10i) q^{79} +(1.91737e10 - 3.32098e10i) q^{81} +3.77864e10 q^{83} +4.92557e10 q^{85} +(6.48933e9 - 1.12398e10i) q^{87} +(8.88627e9 + 1.53915e10i) q^{89} +(-5.12895e10 - 6.27687e10i) q^{91} +(2.79882e10 + 4.84769e10i) q^{93} +(-6.56897e10 + 1.13778e11i) q^{95} -8.79572e10 q^{97} -5.70196e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 266 q^{3} + 7504 q^{5} + 42224 q^{7} - 123520 q^{9} - 213026 q^{11} - 2609712 q^{13} - 2275500 q^{15} + 8854244 q^{17} - 7232806 q^{19} + 49275128 q^{21} + 10649134 q^{23} - 119407256 q^{25} - 19012196 q^{27}+ \cdots - 393415805736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 271.738 − 470.665i 0.645631 − 1.11826i −0.338525 − 0.940957i \(-0.609928\pi\)
0.984156 − 0.177307i \(-0.0567387\pi\)
\(4\) 0 0
\(5\) 3667.11 + 6351.63i 0.524795 + 0.908971i 0.999583 + 0.0288710i \(0.00919121\pi\)
−0.474789 + 0.880100i \(0.657475\pi\)
\(6\) 0 0
\(7\) 28136.3 + 34433.6i 0.632745 + 0.774361i
\(8\) 0 0
\(9\) −59110.0 − 102381.i −0.333678 − 0.577946i
\(10\) 0 0
\(11\) 24115.9 − 41769.9i 0.0451485 − 0.0781995i −0.842568 − 0.538590i \(-0.818957\pi\)
0.887717 + 0.460390i \(0.152291\pi\)
\(12\) 0 0
\(13\) −1.82289e6 −1.36167 −0.680836 − 0.732436i \(-0.738383\pi\)
−0.680836 + 0.732436i \(0.738383\pi\)
\(14\) 0 0
\(15\) 3.98598e6 1.35529
\(16\) 0 0
\(17\) 3.35793e6 − 5.81611e6i 0.573592 − 0.993490i −0.422601 − 0.906316i \(-0.638883\pi\)
0.996193 − 0.0871746i \(-0.0277838\pi\)
\(18\) 0 0
\(19\) 8.95659e6 + 1.55133e7i 0.829847 + 1.43734i 0.898158 + 0.439673i \(0.144906\pi\)
−0.0683108 + 0.997664i \(0.521761\pi\)
\(20\) 0 0
\(21\) 2.38524e7 − 3.88585e6i 1.27446 − 0.207625i
\(22\) 0 0
\(23\) −1.04379e7 − 1.80789e7i −0.338150 − 0.585693i 0.645935 − 0.763393i \(-0.276468\pi\)
−0.984085 + 0.177700i \(0.943134\pi\)
\(24\) 0 0
\(25\) −2.48139e6 + 4.29789e6i −0.0508188 + 0.0880207i
\(26\) 0 0
\(27\) 3.20255e7 0.429531
\(28\) 0 0
\(29\) 2.38808e7 0.216202 0.108101 − 0.994140i \(-0.465523\pi\)
0.108101 + 0.994140i \(0.465523\pi\)
\(30\) 0 0
\(31\) −5.14984e7 + 8.91978e7i −0.323076 + 0.559583i −0.981121 − 0.193395i \(-0.938050\pi\)
0.658045 + 0.752978i \(0.271383\pi\)
\(32\) 0 0
\(33\) −1.31064e7 − 2.27010e7i −0.0582985 − 0.100976i
\(34\) 0 0
\(35\) −1.15530e8 + 3.04984e8i −0.371810 + 0.981527i
\(36\) 0 0
\(37\) 2.60667e8 + 4.51488e8i 0.617982 + 1.07038i 0.989853 + 0.142092i \(0.0453830\pi\)
−0.371871 + 0.928284i \(0.621284\pi\)
\(38\) 0 0
\(39\) −4.95349e8 + 8.57970e8i −0.879136 + 1.52271i
\(40\) 0 0
\(41\) −1.25858e9 −1.69656 −0.848281 − 0.529546i \(-0.822362\pi\)
−0.848281 + 0.529546i \(0.822362\pi\)
\(42\) 0 0
\(43\) 1.03048e9 1.06896 0.534480 − 0.845181i \(-0.320508\pi\)
0.534480 + 0.845181i \(0.320508\pi\)
\(44\) 0 0
\(45\) 4.33526e8 − 7.50889e8i 0.350224 − 0.606606i
\(46\) 0 0
\(47\) 7.99740e8 + 1.38519e9i 0.508640 + 0.880991i 0.999950 + 0.0100058i \(0.00318499\pi\)
−0.491310 + 0.870985i \(0.663482\pi\)
\(48\) 0 0
\(49\) −3.94019e8 + 1.93767e9i −0.199268 + 0.979945i
\(50\) 0 0
\(51\) −1.82496e9 − 3.16092e9i −0.740657 − 1.28286i
\(52\) 0 0
\(53\) −2.29236e9 + 3.97049e9i −0.752949 + 1.30415i 0.193438 + 0.981113i \(0.438036\pi\)
−0.946387 + 0.323034i \(0.895297\pi\)
\(54\) 0 0
\(55\) 3.53743e8 0.0947748
\(56\) 0 0
\(57\) 9.73540e9 2.14310
\(58\) 0 0
\(59\) −2.88236e9 + 4.99240e9i −0.524883 + 0.909124i 0.474697 + 0.880149i \(0.342558\pi\)
−0.999580 + 0.0289746i \(0.990776\pi\)
\(60\) 0 0
\(61\) 2.91537e9 + 5.04957e9i 0.441957 + 0.765491i 0.997835 − 0.0657725i \(-0.0209512\pi\)
−0.555878 + 0.831264i \(0.687618\pi\)
\(62\) 0 0
\(63\) 1.86222e9 − 4.91601e9i 0.236406 − 0.624079i
\(64\) 0 0
\(65\) −6.68475e9 − 1.15783e10i −0.714598 − 1.23772i
\(66\) 0 0
\(67\) 7.79178e9 − 1.34958e10i 0.705059 − 1.22120i −0.261611 − 0.965173i \(-0.584254\pi\)
0.966670 − 0.256025i \(-0.0824128\pi\)
\(68\) 0 0
\(69\) −1.13455e10 −0.873279
\(70\) 0 0
\(71\) 7.49669e8 0.0493116 0.0246558 − 0.999696i \(-0.492151\pi\)
0.0246558 + 0.999696i \(0.492151\pi\)
\(72\) 0 0
\(73\) 5.51048e9 − 9.54443e9i 0.311110 − 0.538858i −0.667493 − 0.744616i \(-0.732633\pi\)
0.978603 + 0.205758i \(0.0659660\pi\)
\(74\) 0 0
\(75\) 1.34858e9 + 2.33580e9i 0.0656203 + 0.113658i
\(76\) 0 0
\(77\) 2.11682e9 − 3.44857e8i 0.0891221 − 0.0145191i
\(78\) 0 0
\(79\) 1.78570e10 + 3.09292e10i 0.652918 + 1.13089i 0.982411 + 0.186730i \(0.0597889\pi\)
−0.329493 + 0.944158i \(0.606878\pi\)
\(80\) 0 0
\(81\) 1.91737e10 − 3.32098e10i 0.610996 − 1.05828i
\(82\) 0 0
\(83\) 3.77864e10 1.05295 0.526473 − 0.850192i \(-0.323514\pi\)
0.526473 + 0.850192i \(0.323514\pi\)
\(84\) 0 0
\(85\) 4.92557e10 1.20407
\(86\) 0 0
\(87\) 6.48933e9 − 1.12398e10i 0.139587 − 0.241771i
\(88\) 0 0
\(89\) 8.88627e9 + 1.53915e10i 0.168684 + 0.292170i 0.937957 − 0.346750i \(-0.112715\pi\)
−0.769273 + 0.638920i \(0.779382\pi\)
\(90\) 0 0
\(91\) −5.12895e10 − 6.27687e10i −0.861590 − 1.05442i
\(92\) 0 0
\(93\) 2.79882e10 + 4.84769e10i 0.417175 + 0.722568i
\(94\) 0 0
\(95\) −6.56897e10 + 1.13778e11i −0.870999 + 1.50861i
\(96\) 0 0
\(97\) −8.79572e10 −1.03998 −0.519992 − 0.854171i \(-0.674065\pi\)
−0.519992 + 0.854171i \(0.674065\pi\)
\(98\) 0 0
\(99\) −5.70196e9 −0.0602602
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 112.12.i.a.81.4 8
4.3 odd 2 14.12.c.a.11.1 yes 8
7.2 even 3 inner 112.12.i.a.65.4 8
12.11 even 2 126.12.g.e.109.2 8
28.3 even 6 98.12.a.l.1.1 4
28.11 odd 6 98.12.a.j.1.4 4
28.19 even 6 98.12.c.l.79.4 8
28.23 odd 6 14.12.c.a.9.1 ✓ 8
28.27 even 2 98.12.c.l.67.4 8
84.23 even 6 126.12.g.e.37.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.12.c.a.9.1 ✓ 8 28.23 odd 6
14.12.c.a.11.1 yes 8 4.3 odd 2
98.12.a.j.1.4 4 28.11 odd 6
98.12.a.l.1.1 4 28.3 even 6
98.12.c.l.67.4 8 28.27 even 2
98.12.c.l.79.4 8 28.19 even 6
112.12.i.a.65.4 8 7.2 even 3 inner
112.12.i.a.81.4 8 1.1 even 1 trivial
126.12.g.e.37.2 8 84.23 even 6
126.12.g.e.109.2 8 12.11 even 2