Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,8,Mod(49,144)] 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("144.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 144.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.9834436697\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{15} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 97.5
Root \(0.500000 - 6.17443i\) of defining polynomial
Character \(\chi\) \(=\) 144.97
Dual form 144.8.i.c.49.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(33.3118 + 32.8226i) q^{3} +(-246.026 - 426.130i) q^{5} +(382.311 - 662.182i) q^{7} +(32.3523 + 2186.76i) q^{9} +(-36.3512 + 62.9621i) q^{11} +(-3010.77 - 5214.80i) q^{13} +(5791.12 - 22270.4i) q^{15} -5989.93 q^{17} -18676.2 q^{19} +(34470.0 - 9510.03i) q^{21} +(12139.5 + 21026.3i) q^{23} +(-81995.2 + 142020. i) q^{25} +(-70697.5 + 73906.8i) q^{27} +(-43378.1 + 75133.0i) q^{29} +(105890. + 183406. i) q^{31} +(-3277.50 + 904.240i) q^{33} -376234. q^{35} -327978. q^{37} +(70869.3 - 272536. i) q^{39} +(-196036. - 339545. i) q^{41} +(-343611. + 595152. i) q^{43} +(923884. - 551787. i) q^{45} +(320755. - 555563. i) q^{47} +(119448. + 206890. i) q^{49} +(-199535. - 196605. i) q^{51} -814485. q^{53} +35773.3 q^{55} +(-622137. - 613001. i) q^{57} +(-1.25863e6 - 2.18002e6i) q^{59} +(221621. - 383858. i) q^{61} +(1.46040e6 + 814600. i) q^{63} +(-1.48145e6 + 2.56595e6i) q^{65} +(296048. + 512770. i) q^{67} +(-285748. + 1.09887e6i) q^{69} -1.48821e6 q^{71} -5.41341e6 q^{73} +(-7.39287e6 + 2.03964e6i) q^{75} +(27794.9 + 48142.2i) q^{77} +(-444736. + 770305. i) q^{79} +(-4.78088e6 + 141493. i) q^{81} +(-1.69323e6 + 2.93276e6i) q^{83} +(1.47368e6 + 2.55249e6i) q^{85} +(-3.91107e6 + 1.07904e6i) q^{87} +1.17388e6 q^{89} -4.60420e6 q^{91} +(-2.49250e6 + 9.58516e6i) q^{93} +(4.59483e6 + 7.95847e6i) q^{95} +(4.30014e6 - 7.44806e6i) q^{97} +(-138859. - 77454.3i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{3} - 180 q^{5} + 84 q^{7} + 990 q^{9} + 8460 q^{11} - 1848 q^{13} + 1188 q^{15} + 30564 q^{17} - 24432 q^{19} - 187224 q^{21} + 51588 q^{23} + 4746 q^{25} - 322272 q^{27} - 414648 q^{29} - 8196 q^{31}+ \cdots + 49382676 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 33.3118 + 32.8226i 0.712318 + 0.701857i
\(4\) 0 0
\(5\) −246.026 − 426.130i −0.880210 − 1.52457i −0.851107 − 0.524992i \(-0.824068\pi\)
−0.0291025 − 0.999576i \(-0.509265\pi\)
\(6\) 0 0
\(7\) 382.311 − 662.182i 0.421283 − 0.729683i −0.574783 − 0.818306i \(-0.694913\pi\)
0.996065 + 0.0886232i \(0.0282467\pi\)
\(8\) 0 0
\(9\) 32.3523 + 2186.76i 0.0147930 + 0.999891i
\(10\) 0 0
\(11\) −36.3512 + 62.9621i −0.00823463 + 0.0142628i −0.870113 − 0.492852i \(-0.835955\pi\)
0.861879 + 0.507114i \(0.169288\pi\)
\(12\) 0 0
\(13\) −3010.77 − 5214.80i −0.380080 − 0.658318i 0.610993 − 0.791636i \(-0.290770\pi\)
−0.991073 + 0.133318i \(0.957437\pi\)
\(14\) 0 0
\(15\) 5791.12 − 22270.4i 0.443040 − 1.70376i
\(16\) 0 0
\(17\) −5989.93 −0.295700 −0.147850 − 0.989010i \(-0.547235\pi\)
−0.147850 + 0.989010i \(0.547235\pi\)
\(18\) 0 0
\(19\) −18676.2 −0.624670 −0.312335 − 0.949972i \(-0.601111\pi\)
−0.312335 + 0.949972i \(0.601111\pi\)
\(20\) 0 0
\(21\) 34470.0 − 9510.03i 0.812220 − 0.224086i
\(22\) 0 0
\(23\) 12139.5 + 21026.3i 0.208043 + 0.360342i 0.951098 − 0.308889i \(-0.0999571\pi\)
−0.743055 + 0.669231i \(0.766624\pi\)
\(24\) 0 0
\(25\) −81995.2 + 142020.i −1.04954 + 1.81785i
\(26\) 0 0
\(27\) −70697.5 + 73906.8i −0.691243 + 0.722622i
\(28\) 0 0
\(29\) −43378.1 + 75133.0i −0.330276 + 0.572055i −0.982566 − 0.185914i \(-0.940475\pi\)
0.652290 + 0.757970i \(0.273809\pi\)
\(30\) 0 0
\(31\) 105890. + 183406.i 0.638392 + 1.10573i 0.985786 + 0.168008i \(0.0537334\pi\)
−0.347394 + 0.937719i \(0.612933\pi\)
\(32\) 0 0
\(33\) −3277.50 + 904.240i −0.0158761 + 0.00438011i
\(34\) 0 0
\(35\) −376234. −1.48327
\(36\) 0 0
\(37\) −327978. −1.06448 −0.532242 − 0.846592i \(-0.678650\pi\)
−0.532242 + 0.846592i \(0.678650\pi\)
\(38\) 0 0
\(39\) 70869.3 − 272536.i 0.191308 − 0.735694i
\(40\) 0 0
\(41\) −196036. − 339545.i −0.444214 − 0.769402i 0.553783 − 0.832661i \(-0.313184\pi\)
−0.997997 + 0.0632592i \(0.979851\pi\)
\(42\) 0 0
\(43\) −343611. + 595152.i −0.659064 + 1.14153i 0.321794 + 0.946810i \(0.395714\pi\)
−0.980858 + 0.194723i \(0.937619\pi\)
\(44\) 0 0
\(45\) 923884. − 551787.i 1.51138 − 0.902666i
\(46\) 0 0
\(47\) 320755. − 555563.i 0.450641 − 0.780533i −0.547785 − 0.836619i \(-0.684529\pi\)
0.998426 + 0.0560862i \(0.0178622\pi\)
\(48\) 0 0
\(49\) 119448. + 206890.i 0.145042 + 0.251220i
\(50\) 0 0
\(51\) −199535. − 196605.i −0.210632 − 0.207539i
\(52\) 0 0
\(53\) −814485. −0.751480 −0.375740 − 0.926725i \(-0.622611\pi\)
−0.375740 + 0.926725i \(0.622611\pi\)
\(54\) 0 0
\(55\) 35773.3 0.0289928
\(56\) 0 0
\(57\) −622137. − 613001.i −0.444963 − 0.438429i
\(58\) 0 0
\(59\) −1.25863e6 − 2.18002e6i −0.797843 − 1.38190i −0.921018 − 0.389519i \(-0.872641\pi\)
0.123176 − 0.992385i \(-0.460692\pi\)
\(60\) 0 0
\(61\) 221621. − 383858.i 0.125013 − 0.216529i −0.796725 − 0.604342i \(-0.793436\pi\)
0.921738 + 0.387813i \(0.126769\pi\)
\(62\) 0 0
\(63\) 1.46040e6 + 814600.i 0.735835 + 0.410442i
\(64\) 0 0
\(65\) −1.48145e6 + 2.56595e6i −0.669101 + 1.15892i
\(66\) 0 0
\(67\) 296048. + 512770.i 0.120254 + 0.208286i 0.919868 − 0.392228i \(-0.128296\pi\)
−0.799614 + 0.600515i \(0.794962\pi\)
\(68\) 0 0
\(69\) −285748. + 1.09887e6i −0.104715 + 0.402695i
\(70\) 0 0
\(71\) −1.48821e6 −0.493469 −0.246734 − 0.969083i \(-0.579357\pi\)
−0.246734 + 0.969083i \(0.579357\pi\)
\(72\) 0 0
\(73\) −5.41341e6 −1.62870 −0.814350 − 0.580374i \(-0.802906\pi\)
−0.814350 + 0.580374i \(0.802906\pi\)
\(74\) 0 0
\(75\) −7.39287e6 + 2.03964e6i −2.02348 + 0.558264i
\(76\) 0 0
\(77\) 27794.9 + 48142.2i 0.00693821 + 0.0120173i
\(78\) 0 0
\(79\) −444736. + 770305.i −0.101486 + 0.175779i −0.912297 − 0.409529i \(-0.865693\pi\)
0.810811 + 0.585308i \(0.199026\pi\)
\(80\) 0 0
\(81\) −4.78088e6 + 141493.i −0.999562 + 0.0295828i
\(82\) 0 0
\(83\) −1.69323e6 + 2.93276e6i −0.325044 + 0.562993i −0.981521 − 0.191353i \(-0.938713\pi\)
0.656477 + 0.754346i \(0.272046\pi\)
\(84\) 0 0
\(85\) 1.47368e6 + 2.55249e6i 0.260278 + 0.450814i
\(86\) 0 0
\(87\) −3.91107e6 + 1.07904e6i −0.636763 + 0.175678i
\(88\) 0 0
\(89\) 1.17388e6 0.176506 0.0882531 − 0.996098i \(-0.471872\pi\)
0.0882531 + 0.996098i \(0.471872\pi\)
\(90\) 0 0
\(91\) −4.60420e6 −0.640485
\(92\) 0 0
\(93\) −2.49250e6 + 9.58516e6i −0.321325 + 1.23569i
\(94\) 0 0
\(95\) 4.59483e6 + 7.95847e6i 0.549840 + 0.952351i
\(96\) 0 0
\(97\) 4.30014e6 − 7.44806e6i 0.478390 − 0.828595i −0.521303 − 0.853371i \(-0.674554\pi\)
0.999693 + 0.0247763i \(0.00788736\pi\)
\(98\) 0 0
\(99\) −138859. − 77454.3i −0.0143830 − 0.00802274i
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 144.8.i.c.97.5 12
3.2 odd 2 432.8.i.c.289.6 12
4.3 odd 2 9.8.c.a.7.2 yes 12
9.4 even 3 inner 144.8.i.c.49.5 12
9.5 odd 6 432.8.i.c.145.6 12
12.11 even 2 27.8.c.a.19.5 12
36.7 odd 6 81.8.a.e.1.5 6
36.11 even 6 81.8.a.c.1.2 6
36.23 even 6 27.8.c.a.10.5 12
36.31 odd 6 9.8.c.a.4.2 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.2 ✓ 12 36.31 odd 6
9.8.c.a.7.2 yes 12 4.3 odd 2
27.8.c.a.10.5 12 36.23 even 6
27.8.c.a.19.5 12 12.11 even 2
81.8.a.c.1.2 6 36.11 even 6
81.8.a.e.1.5 6 36.7 odd 6
144.8.i.c.49.5 12 9.4 even 3 inner
144.8.i.c.97.5 12 1.1 even 1 trivial
432.8.i.c.145.6 12 9.5 odd 6
432.8.i.c.289.6 12 3.2 odd 2