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Properties

Label 672.2.bd.a.431.8
Level $672$
Weight $2$
Character 672.431
Analytic conductor $5.366$
Analytic rank $0$
Dimension $56$
Inner twists $8$

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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] = [672,2,Mod(431,672)] 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("672.431"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bd (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 431.8
Character \(\chi\) \(=\) 672.431
Dual form 672.2.bd.a.527.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.03943 - 1.38549i) q^{3} +(0.692094 + 1.19874i) q^{5} +(-2.08863 - 1.62408i) q^{7} +(-0.839162 + 2.88024i) q^{9} +(3.82316 + 2.20731i) q^{11} -6.43609i q^{13} +(0.941460 - 2.20490i) q^{15} +(-2.52866 - 1.45992i) q^{17} +(-1.58928 - 2.75271i) q^{19} +(-0.0791630 + 4.58189i) q^{21} +(-1.84696 - 3.19902i) q^{23} +(1.54201 - 2.67084i) q^{25} +(4.86280 - 1.83117i) q^{27} -6.67884 q^{29} +(-2.17580 - 1.25620i) q^{31} +(-0.915722 - 7.59130i) q^{33} +(0.501330 - 3.62774i) q^{35} +(-5.00149 + 2.88761i) q^{37} +(-8.91713 + 6.68987i) q^{39} +0.497427i q^{41} -0.865898 q^{43} +(-4.03345 + 0.987462i) q^{45} +(-1.59001 - 2.75398i) q^{47} +(1.72472 + 6.78420i) q^{49} +(0.605663 + 5.02092i) q^{51} +(4.12906 - 7.15175i) q^{53} +6.11065i q^{55} +(-2.16191 + 5.06319i) q^{57} +(-6.62279 - 3.82367i) q^{59} +(2.99321 - 1.72813i) q^{61} +(6.43045 - 4.65289i) q^{63} +(7.71521 - 4.45438i) q^{65} +(3.36806 - 5.83364i) q^{67} +(-2.51243 + 5.88411i) q^{69} -1.90437 q^{71} +(3.23515 - 5.60345i) q^{73} +(-5.30324 + 0.639719i) q^{75} +(-4.40032 - 10.8194i) q^{77} +(-1.65073 + 0.953048i) q^{79} +(-7.59162 - 4.83398i) q^{81} +7.00064i q^{83} -4.04161i q^{85} +(6.94220 + 9.25347i) q^{87} +(8.22776 - 4.75030i) q^{89} +(-10.4527 + 13.4426i) q^{91} +(0.521147 + 4.32029i) q^{93} +(2.19986 - 3.81027i) q^{95} +1.19214 q^{97} +(-9.56583 + 9.15936i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 2 q^{3} - 2 q^{9} + 4 q^{19} - 16 q^{25} + 8 q^{27} - 14 q^{33} + 16 q^{43} - 16 q^{49} + 34 q^{51} + 4 q^{57} + 36 q^{67} + 4 q^{73} - 10 q^{81} - 72 q^{91} - 32 q^{97} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.03943 − 1.38549i −0.600116 − 0.799913i
\(4\) 0 0
\(5\) 0.692094 + 1.19874i 0.309514 + 0.536094i 0.978256 − 0.207401i \(-0.0665004\pi\)
−0.668742 + 0.743494i \(0.733167\pi\)
\(6\) 0 0
\(7\) −2.08863 − 1.62408i −0.789426 − 0.613845i
\(8\) 0 0
\(9\) −0.839162 + 2.88024i −0.279721 + 0.960081i
\(10\) 0 0
\(11\) 3.82316 + 2.20731i 1.15273 + 0.665528i 0.949550 − 0.313614i \(-0.101540\pi\)
0.203177 + 0.979142i \(0.434873\pi\)
\(12\) 0 0
\(13\) − 6.43609i − 1.78505i −0.450999 − 0.892525i \(-0.648932\pi\)
0.450999 − 0.892525i \(-0.351068\pi\)
\(14\) 0 0
\(15\) 0.941460 − 2.20490i 0.243084 − 0.569303i
\(16\) 0 0
\(17\) −2.52866 − 1.45992i −0.613289 − 0.354083i 0.160962 − 0.986961i \(-0.448540\pi\)
−0.774252 + 0.632878i \(0.781874\pi\)
\(18\) 0 0
\(19\) −1.58928 − 2.75271i −0.364606 − 0.631515i 0.624107 − 0.781339i \(-0.285463\pi\)
−0.988713 + 0.149823i \(0.952130\pi\)
\(20\) 0 0
\(21\) −0.0791630 + 4.58189i −0.0172748 + 0.999851i
\(22\) 0 0
\(23\) −1.84696 − 3.19902i −0.385117 − 0.667043i 0.606668 − 0.794955i \(-0.292506\pi\)
−0.991785 + 0.127912i \(0.959172\pi\)
\(24\) 0 0
\(25\) 1.54201 − 2.67084i 0.308402 − 0.534169i
\(26\) 0 0
\(27\) 4.86280 − 1.83117i 0.935846 − 0.352409i
\(28\) 0 0
\(29\) −6.67884 −1.24023 −0.620115 − 0.784511i \(-0.712914\pi\)
−0.620115 + 0.784511i \(0.712914\pi\)
\(30\) 0 0
\(31\) −2.17580 − 1.25620i −0.390786 − 0.225620i 0.291715 − 0.956505i \(-0.405774\pi\)
−0.682500 + 0.730885i \(0.739107\pi\)
\(32\) 0 0
\(33\) −0.915722 − 7.59130i −0.159407 − 1.32148i
\(34\) 0 0
\(35\) 0.501330 − 3.62774i 0.0847402 − 0.613200i
\(36\) 0 0
\(37\) −5.00149 + 2.88761i −0.822240 + 0.474720i −0.851188 − 0.524861i \(-0.824117\pi\)
0.0289485 + 0.999581i \(0.490784\pi\)
\(38\) 0 0
\(39\) −8.91713 + 6.68987i −1.42788 + 1.07124i
\(40\) 0 0
\(41\) 0.497427i 0.0776850i 0.999245 + 0.0388425i \(0.0123671\pi\)
−0.999245 + 0.0388425i \(0.987633\pi\)
\(42\) 0 0
\(43\) −0.865898 −0.132048 −0.0660241 − 0.997818i \(-0.521031\pi\)
−0.0660241 + 0.997818i \(0.521031\pi\)
\(44\) 0 0
\(45\) −4.03345 + 0.987462i −0.601271 + 0.147202i
\(46\) 0 0
\(47\) −1.59001 − 2.75398i −0.231927 − 0.401710i 0.726448 − 0.687221i \(-0.241170\pi\)
−0.958375 + 0.285512i \(0.907836\pi\)
\(48\) 0 0
\(49\) 1.72472 + 6.78420i 0.246388 + 0.969171i
\(50\) 0 0
\(51\) 0.605663 + 5.02092i 0.0848097 + 0.703069i
\(52\) 0 0
\(53\) 4.12906 − 7.15175i 0.567171 − 0.982368i −0.429673 − 0.902984i \(-0.641371\pi\)
0.996844 − 0.0793841i \(-0.0252954\pi\)
\(54\) 0 0
\(55\) 6.11065i 0.823960i
\(56\) 0 0
\(57\) −2.16191 + 5.06319i −0.286351 + 0.670635i
\(58\) 0 0
\(59\) −6.62279 − 3.82367i −0.862214 − 0.497799i 0.00253919 − 0.999997i \(-0.499192\pi\)
−0.864753 + 0.502197i \(0.832525\pi\)
\(60\) 0 0
\(61\) 2.99321 − 1.72813i 0.383241 − 0.221265i −0.295986 − 0.955192i \(-0.595648\pi\)
0.679228 + 0.733928i \(0.262315\pi\)
\(62\) 0 0
\(63\) 6.43045 − 4.65289i 0.810160 − 0.586209i
\(64\) 0 0
\(65\) 7.71521 − 4.45438i 0.956954 − 0.552497i
\(66\) 0 0
\(67\) 3.36806 − 5.83364i 0.411473 − 0.712693i −0.583578 − 0.812057i \(-0.698348\pi\)
0.995051 + 0.0993644i \(0.0316810\pi\)
\(68\) 0 0
\(69\) −2.51243 + 5.88411i −0.302461 + 0.708363i
\(70\) 0 0
\(71\) −1.90437 −0.226007 −0.113004 − 0.993595i \(-0.536047\pi\)
−0.113004 + 0.993595i \(0.536047\pi\)
\(72\) 0 0
\(73\) 3.23515 − 5.60345i 0.378646 − 0.655834i −0.612219 − 0.790688i \(-0.709723\pi\)
0.990866 + 0.134854i \(0.0430564\pi\)
\(74\) 0 0
\(75\) −5.30324 + 0.639719i −0.612365 + 0.0738684i
\(76\) 0 0
\(77\) −4.40032 − 10.8194i −0.501463 − 1.23298i
\(78\) 0 0
\(79\) −1.65073 + 0.953048i −0.185721 + 0.107226i −0.589978 − 0.807419i \(-0.700864\pi\)
0.404257 + 0.914646i \(0.367530\pi\)
\(80\) 0 0
\(81\) −7.59162 − 4.83398i −0.843513 − 0.537109i
\(82\) 0 0
\(83\) 7.00064i 0.768420i 0.923246 + 0.384210i \(0.125526\pi\)
−0.923246 + 0.384210i \(0.874474\pi\)
\(84\) 0 0
\(85\) − 4.04161i − 0.438374i
\(86\) 0 0
\(87\) 6.94220 + 9.25347i 0.744282 + 0.992076i
\(88\) 0 0
\(89\) 8.22776 − 4.75030i 0.872141 − 0.503531i 0.00408165 − 0.999992i \(-0.498701\pi\)
0.868059 + 0.496461i \(0.165367\pi\)
\(90\) 0 0
\(91\) −10.4527 + 13.4426i −1.09574 + 1.40916i
\(92\) 0 0
\(93\) 0.521147 + 4.32029i 0.0540404 + 0.447993i
\(94\) 0 0
\(95\) 2.19986 − 3.81027i 0.225701 − 0.390926i
\(96\) 0 0
\(97\) 1.19214 0.121044 0.0605218 − 0.998167i \(-0.480724\pi\)
0.0605218 + 0.998167i \(0.480724\pi\)
\(98\) 0 0
\(99\) −9.56583 + 9.15936i −0.961402 + 0.920551i
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 672.2.bd.a.431.8 56
3.2 odd 2 inner 672.2.bd.a.431.27 56
4.3 odd 2 168.2.v.a.11.19 yes 56
7.2 even 3 inner 672.2.bd.a.527.28 56
8.3 odd 2 inner 672.2.bd.a.431.7 56
8.5 even 2 168.2.v.a.11.28 yes 56
12.11 even 2 168.2.v.a.11.10 yes 56
21.2 odd 6 inner 672.2.bd.a.527.7 56
24.5 odd 2 168.2.v.a.11.1 ✓ 56
24.11 even 2 inner 672.2.bd.a.431.28 56
28.23 odd 6 168.2.v.a.107.1 yes 56
56.37 even 6 168.2.v.a.107.10 yes 56
56.51 odd 6 inner 672.2.bd.a.527.27 56
84.23 even 6 168.2.v.a.107.28 yes 56
168.107 even 6 inner 672.2.bd.a.527.8 56
168.149 odd 6 168.2.v.a.107.19 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.v.a.11.1 ✓ 56 24.5 odd 2
168.2.v.a.11.10 yes 56 12.11 even 2
168.2.v.a.11.19 yes 56 4.3 odd 2
168.2.v.a.11.28 yes 56 8.5 even 2
168.2.v.a.107.1 yes 56 28.23 odd 6
168.2.v.a.107.10 yes 56 56.37 even 6
168.2.v.a.107.19 yes 56 168.149 odd 6
168.2.v.a.107.28 yes 56 84.23 even 6
672.2.bd.a.431.7 56 8.3 odd 2 inner
672.2.bd.a.431.8 56 1.1 even 1 trivial
672.2.bd.a.431.27 56 3.2 odd 2 inner
672.2.bd.a.431.28 56 24.11 even 2 inner
672.2.bd.a.527.7 56 21.2 odd 6 inner
672.2.bd.a.527.8 56 168.107 even 6 inner
672.2.bd.a.527.27 56 56.51 odd 6 inner
672.2.bd.a.527.28 56 7.2 even 3 inner