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Properties

Label 432.7.q.b.305.5
Level $432$
Weight $7$
Character 432.305
Analytic conductor $99.383$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [432,7,Mod(17,432)] 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("432.17"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 432.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-432] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(99.3833641238\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
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: \( x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{20} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 305.5
Root \(-7.20150i\) of defining polynomial
Character \(\chi\) \(=\) 432.305
Dual form 432.7.q.b.17.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(39.5602 - 22.8401i) q^{5} +(245.097 - 424.521i) q^{7} +(-873.336 - 504.221i) q^{11} +(466.801 + 808.523i) q^{13} +8090.59i q^{17} +7727.36 q^{19} +(-11848.0 + 6840.45i) q^{23} +(-6769.16 + 11724.5i) q^{25} +(1964.70 + 1134.32i) q^{29} +(17062.6 + 29553.2i) q^{31} -22392.2i q^{35} +92058.0 q^{37} +(31021.6 - 17910.3i) q^{41} +(-34570.9 + 59878.5i) q^{43} +(13211.1 + 7627.46i) q^{47} +(-61321.0 - 106211. i) q^{49} -236591. i q^{53} -46065.9 q^{55} +(-221890. + 128108. i) q^{59} +(-19919.6 + 34501.8i) q^{61} +(36933.5 + 21323.6i) q^{65} +(160204. + 277482. i) q^{67} -404593. i q^{71} +393719. q^{73} +(-428105. + 247167. i) q^{77} +(449184. - 778010. i) q^{79} +(154916. + 89441.0i) q^{83} +(184790. + 320066. i) q^{85} +826458. i q^{89} +457647. q^{91} +(305696. - 176494. i) q^{95} +(-317981. + 550760. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 432 q^{5} - 240 q^{7} + 378 q^{11} + 1680 q^{13} + 2820 q^{19} - 76248 q^{23} + 8094 q^{25} - 97092 q^{29} - 21480 q^{31} - 25536 q^{37} + 410562 q^{41} - 71430 q^{43} + 347652 q^{47} - 135954 q^{49}+ \cdots - 38874 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 39.5602 − 22.8401i 0.316482 − 0.182721i −0.333341 − 0.942806i \(-0.608176\pi\)
0.649823 + 0.760085i \(0.274843\pi\)
\(6\) 0 0
\(7\) 245.097 − 424.521i 0.714570 − 1.23767i −0.248556 − 0.968618i \(-0.579956\pi\)
0.963125 − 0.269053i \(-0.0867108\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −873.336 − 504.221i −0.656151 − 0.378829i 0.134658 − 0.990892i \(-0.457006\pi\)
−0.790809 + 0.612063i \(0.790340\pi\)
\(12\) 0 0
\(13\) 466.801 + 808.523i 0.212472 + 0.368012i 0.952488 − 0.304577i \(-0.0985152\pi\)
−0.740016 + 0.672590i \(0.765182\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8090.59i 1.64677i 0.567482 + 0.823386i \(0.307918\pi\)
−0.567482 + 0.823386i \(0.692082\pi\)
\(18\) 0 0
\(19\) 7727.36 1.12660 0.563301 − 0.826252i \(-0.309531\pi\)
0.563301 + 0.826252i \(0.309531\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −11848.0 + 6840.45i −0.973782 + 0.562213i −0.900387 − 0.435090i \(-0.856717\pi\)
−0.0733950 + 0.997303i \(0.523383\pi\)
\(24\) 0 0
\(25\) −6769.16 + 11724.5i −0.433226 + 0.750370i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1964.70 + 1134.32i 0.0805570 + 0.0465096i 0.539737 − 0.841833i \(-0.318524\pi\)
−0.459180 + 0.888343i \(0.651857\pi\)
\(30\) 0 0
\(31\) 17062.6 + 29553.2i 0.572742 + 0.992019i 0.996283 + 0.0861417i \(0.0274538\pi\)
−0.423541 + 0.905877i \(0.639213\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 22392.2i − 0.522267i
\(36\) 0 0
\(37\) 92058.0 1.81743 0.908713 − 0.417422i \(-0.137066\pi\)
0.908713 + 0.417422i \(0.137066\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 31021.6 − 17910.3i 0.450103 − 0.259867i −0.257771 − 0.966206i \(-0.582988\pi\)
0.707874 + 0.706339i \(0.249655\pi\)
\(42\) 0 0
\(43\) −34570.9 + 59878.5i −0.434816 + 0.753123i −0.997281 − 0.0736985i \(-0.976520\pi\)
0.562465 + 0.826821i \(0.309853\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13211.1 + 7627.46i 0.127247 + 0.0734659i 0.562272 − 0.826952i \(-0.309927\pi\)
−0.435025 + 0.900418i \(0.643261\pi\)
\(48\) 0 0
\(49\) −61321.0 − 106211.i −0.521220 − 0.902779i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 236591.i − 1.58917i −0.607153 − 0.794585i \(-0.707688\pi\)
0.607153 − 0.794585i \(-0.292312\pi\)
\(54\) 0 0
\(55\) −46065.9 −0.276880
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −221890. + 128108.i −1.08039 + 0.623766i −0.931003 − 0.365012i \(-0.881065\pi\)
−0.149392 + 0.988778i \(0.547732\pi\)
\(60\) 0 0
\(61\) −19919.6 + 34501.8i −0.0877589 + 0.152003i −0.906563 − 0.422069i \(-0.861304\pi\)
0.818805 + 0.574072i \(0.194637\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 36933.5 + 21323.6i 0.134487 + 0.0776462i
\(66\) 0 0
\(67\) 160204. + 277482.i 0.532660 + 0.922593i 0.999273 + 0.0381319i \(0.0121407\pi\)
−0.466613 + 0.884461i \(0.654526\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 404593.i − 1.13043i −0.824944 − 0.565215i \(-0.808793\pi\)
0.824944 − 0.565215i \(-0.191207\pi\)
\(72\) 0 0
\(73\) 393719. 1.01209 0.506044 − 0.862508i \(-0.331107\pi\)
0.506044 + 0.862508i \(0.331107\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −428105. + 247167.i −0.937731 + 0.541399i
\(78\) 0 0
\(79\) 449184. − 778010.i 0.911052 − 1.57799i 0.0984709 − 0.995140i \(-0.468605\pi\)
0.812581 − 0.582848i \(-0.198062\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 154916. + 89441.0i 0.270934 + 0.156424i 0.629312 − 0.777153i \(-0.283337\pi\)
−0.358378 + 0.933576i \(0.616670\pi\)
\(84\) 0 0
\(85\) 184790. + 320066.i 0.300900 + 0.521173i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 826458.i 1.17233i 0.810191 + 0.586166i \(0.199364\pi\)
−0.810191 + 0.586166i \(0.800636\pi\)
\(90\) 0 0
\(91\) 457647. 0.607304
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 305696. − 176494.i 0.356549 − 0.205854i
\(96\) 0 0
\(97\) −317981. + 550760.i −0.348406 + 0.603458i −0.985967 − 0.166943i \(-0.946610\pi\)
0.637560 + 0.770401i \(0.279944\pi\)
\(98\) 0 0
\(99\) 0 0
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 432.7.q.b.305.5 12
3.2 odd 2 144.7.q.c.65.3 12
4.3 odd 2 54.7.d.a.35.6 12
9.4 even 3 144.7.q.c.113.3 12
9.5 odd 6 inner 432.7.q.b.17.5 12
12.11 even 2 18.7.d.a.11.2 yes 12
36.7 odd 6 162.7.b.c.161.5 12
36.11 even 6 162.7.b.c.161.8 12
36.23 even 6 54.7.d.a.17.6 12
36.31 odd 6 18.7.d.a.5.2 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.2 ✓ 12 36.31 odd 6
18.7.d.a.11.2 yes 12 12.11 even 2
54.7.d.a.17.6 12 36.23 even 6
54.7.d.a.35.6 12 4.3 odd 2
144.7.q.c.65.3 12 3.2 odd 2
144.7.q.c.113.3 12 9.4 even 3
162.7.b.c.161.5 12 36.7 odd 6
162.7.b.c.161.8 12 36.11 even 6
432.7.q.b.17.5 12 9.5 odd 6 inner
432.7.q.b.305.5 12 1.1 even 1 trivial