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Properties

Label 432.7.q.b.305.1
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 305.1
Root \(-8.88570i\) of defining polynomial
Character \(\chi\) \(=\) 432.305
Dual form 432.7.q.b.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-202.253 + 116.771i) q^{5} +(-95.5752 + 165.541i) q^{7} +(-673.077 - 388.601i) q^{11} +(45.5802 + 78.9472i) q^{13} -7047.39i q^{17} -2731.10 q^{19} +(-17228.9 + 9947.14i) q^{23} +(19458.3 - 33702.7i) q^{25} +(-27104.3 - 15648.7i) q^{29} +(-6174.50 - 10694.6i) q^{31} -44641.5i q^{35} -27972.0 q^{37} +(37428.2 - 21609.2i) q^{41} +(-19256.1 + 33352.5i) q^{43} +(-143771. - 83006.2i) q^{47} +(40555.3 + 70243.8i) q^{49} +54741.5i q^{53} +181509. q^{55} +(-14102.1 + 8141.84i) q^{59} +(29443.7 - 50998.0i) q^{61} +(-18437.4 - 10644.9i) q^{65} +(147998. + 256341. i) q^{67} +157251. i q^{71} +80297.0 q^{73} +(128659. - 74281.3i) q^{77} +(-188424. + 326360. i) q^{79} +(733992. + 423771. i) q^{83} +(822929. + 1.42535e6i) q^{85} -1128.91i q^{89} -17425.3 q^{91} +(552371. - 318912. i) q^{95} +(675152. - 1.16940e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 432 q^{5} - 240 q^{7} + 378 q^{11} + 1680 q^{13} + 2820 q^{19} - 76248 q^{23} + 8094 q^{25} - 97092 q^{29} - 21480 q^{31} - 25536 q^{37} + 410562 q^{41} - 71430 q^{43} + 347652 q^{47} - 135954 q^{49}+ \cdots - 38874 q^{97}+O(q^{100}) \) 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
\(5\) −202.253 + 116.771i −1.61802 + 0.934165i −0.630591 + 0.776116i \(0.717187\pi\)
−0.987431 + 0.158050i \(0.949479\pi\)
\(6\) 0 0
\(7\) −95.5752 + 165.541i −0.278645 + 0.482627i −0.971048 − 0.238884i \(-0.923219\pi\)
0.692403 + 0.721511i \(0.256552\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −673.077 − 388.601i −0.505693 − 0.291962i 0.225369 − 0.974274i \(-0.427641\pi\)
−0.731061 + 0.682312i \(0.760975\pi\)
\(12\) 0 0
\(13\) 45.5802 + 78.9472i 0.0207466 + 0.0359341i 0.876212 − 0.481925i \(-0.160062\pi\)
−0.855466 + 0.517859i \(0.826729\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7047.39i − 1.43444i −0.696848 − 0.717219i \(-0.745415\pi\)
0.696848 − 0.717219i \(-0.254585\pi\)
\(18\) 0 0
\(19\) −2731.10 −0.398177 −0.199088 − 0.979982i \(-0.563798\pi\)
−0.199088 + 0.979982i \(0.563798\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −17228.9 + 9947.14i −1.41604 + 0.817550i −0.995948 − 0.0899317i \(-0.971335\pi\)
−0.420091 + 0.907482i \(0.638002\pi\)
\(24\) 0 0
\(25\) 19458.3 − 33702.7i 1.24533 − 2.15697i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −27104.3 − 15648.7i −1.11133 − 0.641629i −0.172159 − 0.985069i \(-0.555074\pi\)
−0.939175 + 0.343440i \(0.888408\pi\)
\(30\) 0 0
\(31\) −6174.50 − 10694.6i −0.207261 − 0.358986i 0.743590 − 0.668636i \(-0.233121\pi\)
−0.950851 + 0.309650i \(0.899788\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 44641.5i − 1.04120i
\(36\) 0 0
\(37\) −27972.0 −0.552228 −0.276114 − 0.961125i \(-0.589047\pi\)
−0.276114 + 0.961125i \(0.589047\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 37428.2 − 21609.2i 0.543059 − 0.313535i −0.203259 − 0.979125i \(-0.565153\pi\)
0.746318 + 0.665590i \(0.231820\pi\)
\(42\) 0 0
\(43\) −19256.1 + 33352.5i −0.242193 + 0.419491i −0.961339 − 0.275369i \(-0.911200\pi\)
0.719146 + 0.694860i \(0.244533\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −143771. − 83006.2i −1.38477 − 0.799497i −0.392050 − 0.919944i \(-0.628234\pi\)
−0.992720 + 0.120447i \(0.961567\pi\)
\(48\) 0 0
\(49\) 40555.3 + 70243.8i 0.344714 + 0.597062i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 54741.5i 0.367696i 0.982955 + 0.183848i \(0.0588554\pi\)
−0.982955 + 0.183848i \(0.941145\pi\)
\(54\) 0 0
\(55\) 181509. 1.09096
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14102.1 + 8141.84i −0.0686637 + 0.0396430i −0.533939 − 0.845523i \(-0.679289\pi\)
0.465275 + 0.885166i \(0.345955\pi\)
\(60\) 0 0
\(61\) 29443.7 − 50998.0i 0.129719 − 0.224680i −0.793849 − 0.608115i \(-0.791926\pi\)
0.923568 + 0.383436i \(0.125259\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −18437.4 − 10644.9i −0.0671368 − 0.0387614i
\(66\) 0 0
\(67\) 147998. + 256341.i 0.492076 + 0.852301i 0.999958 − 0.00912565i \(-0.00290482\pi\)
−0.507882 + 0.861427i \(0.669571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 157251.i 0.439358i 0.975572 + 0.219679i \(0.0705010\pi\)
−0.975572 + 0.219679i \(0.929499\pi\)
\(72\) 0 0
\(73\) 80297.0 0.206410 0.103205 − 0.994660i \(-0.467090\pi\)
0.103205 + 0.994660i \(0.467090\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 128659. − 74281.3i 0.281817 − 0.162707i
\(78\) 0 0
\(79\) −188424. + 326360.i −0.382169 + 0.661936i −0.991372 − 0.131078i \(-0.958156\pi\)
0.609203 + 0.793014i \(0.291489\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 733992. + 423771.i 1.28368 + 0.741134i 0.977519 − 0.210846i \(-0.0676218\pi\)
0.306162 + 0.951980i \(0.400955\pi\)
\(84\) 0 0
\(85\) 822929. + 1.42535e6i 1.34000 + 2.32095i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1128.91i − 0.00160136i −1.00000 0.000800679i \(-0.999745\pi\)
1.00000 0.000800679i \(-0.000254864\pi\)
\(90\) 0 0
\(91\) −17425.3 −0.0231237
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 552371. − 318912.i 0.644259 − 0.371963i
\(96\) 0 0
\(97\) 675152. − 1.16940e6i 0.739753 − 1.28129i −0.212854 − 0.977084i \(-0.568276\pi\)
0.952607 − 0.304205i \(-0.0983908\pi\)
\(98\) 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.305.1 12
3.2 odd 2 144.7.q.c.65.4 12
4.3 odd 2 54.7.d.a.35.1 12
9.4 even 3 144.7.q.c.113.4 12
9.5 odd 6 inner 432.7.q.b.17.1 12
12.11 even 2 18.7.d.a.11.5 yes 12
36.7 odd 6 162.7.b.c.161.7 12
36.11 even 6 162.7.b.c.161.6 12
36.23 even 6 54.7.d.a.17.1 12
36.31 odd 6 18.7.d.a.5.5 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.d.a.5.5 ✓ 12 36.31 odd 6
18.7.d.a.11.5 yes 12 12.11 even 2
54.7.d.a.17.1 12 36.23 even 6
54.7.d.a.35.1 12 4.3 odd 2
144.7.q.c.65.4 12 3.2 odd 2
144.7.q.c.113.4 12 9.4 even 3
162.7.b.c.161.6 12 36.11 even 6
162.7.b.c.161.7 12 36.7 odd 6
432.7.q.b.17.1 12 9.5 odd 6 inner
432.7.q.b.305.1 12 1.1 even 1 trivial