Properties

Label 78.2.g.a
Level $78$
Weight $2$
Character orbit 78.g
Analytic conductor $0.623$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.58498535041007616.52
Defining polynomial: \( x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{4} q^{3} + \beta_{8} q^{4} + ( - \beta_{11} - \beta_1) q^{5} + \beta_{11} q^{6} + ( - \beta_{9} - \beta_{8} - \beta_{5} - 1) q^{7} - \beta_{6} q^{8} + (\beta_{9} - \beta_{7} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{4} q^{3} + \beta_{8} q^{4} + ( - \beta_{11} - \beta_1) q^{5} + \beta_{11} q^{6} + ( - \beta_{9} - \beta_{8} - \beta_{5} - 1) q^{7} - \beta_{6} q^{8} + (\beta_{9} - \beta_{7} + \beta_1) q^{9} + (\beta_{10} - \beta_{3}) q^{10} + (\beta_{10} + \beta_{7} + \beta_{4} + \beta_{3}) q^{11} - \beta_{10} q^{12} + ( - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} + \beta_{3}) q^{13} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{2}) q^{14} + (\beta_{7} - \beta_{3} - 3 \beta_{2} + \beta_1) q^{15} - q^{16} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{5} - \beta_{3} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{3} - \beta_1) q^{18} + ( - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} - 1) q^{19} + ( - \beta_{9} + \beta_{5}) q^{20} + (\beta_{10} - \beta_{9} - \beta_{7} + 3 \beta_{6} + \beta_{4} - \beta_{3}) q^{21} + ( - \beta_{11} + \beta_{9} + \beta_{5} + \beta_1) q^{22} + ( - \beta_{11} + \beta_{9} - \beta_{5} - \beta_1) q^{23} - \beta_{5} q^{24} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} - \beta_{3} - \beta_1) q^{25} + (\beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4}) q^{26} + ( - 3 \beta_{6} + 3 \beta_{2} - 3) q^{27} + ( - \beta_{11} - \beta_{8} + \beta_1 + 1) q^{28} + ( - \beta_{11} - \beta_{9} + \beta_{5} - \beta_1) q^{29} + ( - \beta_{9} - 3 \beta_{8} + \beta_{3} + \beta_1) q^{30} + (2 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{31} - \beta_{2} q^{32} + ( - 2 \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{3} + 3) q^{33} + ( - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3}) q^{34} + (2 \beta_{11} + 2 \beta_{9} - \beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_1) q^{35} + (\beta_{9} - \beta_{3} - \beta_1) q^{36} + ( - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{37} + ( - \beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} - \beta_{2} - \beta_1) q^{38} + (\beta_{10} + 2 \beta_{9} + 3 \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_1) q^{39} + (\beta_{7} - \beta_{4}) q^{40} + (2 \beta_{11} + 6 \beta_{2} + 2 \beta_1) q^{41} + ( - \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} - \beta_1 + 3) q^{42} + ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{43} + (\beta_{10} - \beta_{7} - \beta_{4} + \beta_{3}) q^{44} + ( - 3 \beta_{11} - 3 \beta_{8} + 3 \beta_{2} + 3) q^{45} + (\beta_{10} - \beta_{7} + \beta_{4} - \beta_{3}) q^{46} + ( - \beta_{10} - \beta_{9} - \beta_{7} - 6 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3}) q^{47} + \beta_{4} q^{48} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} + \beta_{3} - \beta_1) q^{49} + ( - \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3}) q^{50} + (2 \beta_{9} - 3 \beta_{8} + 3 \beta_{6} + \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{51} + (\beta_{11} - \beta_{7} + \beta_{4} - \beta_1 + 1) q^{52} + (\beta_{11} + \beta_{9} - 2 \beta_{7} + 6 \beta_{6} - \beta_{5} - 2 \beta_{4} - 6 \beta_{2} + \beta_1) q^{53} + (3 \beta_{8} - 3 \beta_{2} - 3) q^{54} + (\beta_{11} - \beta_{9} - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_1) q^{55} + (\beta_{10} + \beta_{6} + \beta_{3} + \beta_{2}) q^{56} + ( - \beta_{10} + 3 \beta_{8} + \beta_{7} + \beta_{4} - \beta_{3} - 2 \beta_1 - 3) q^{57} + (\beta_{10} + \beta_{7} - \beta_{4} - \beta_{3}) q^{58} + ( - 2 \beta_{9} + 2 \beta_{5}) q^{59} + (\beta_{9} + \beta_{7} + 3 \beta_{6} + \beta_{3}) q^{60} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{5} + 2 \beta_1) q^{61} + (\beta_{11} - 2 \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{62} + ( - 2 \beta_{9} - 3 \beta_{8} + \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + \beta_{3} - 3) q^{63} - \beta_{8} q^{64} + ( - \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{7} + 6 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots - \beta_1) q^{65}+ \cdots + ( - 3 \beta_{10} + 3 \beta_{8} + 6 \beta_{6} - 3 \beta_{4} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 12 q^{16} - 12 q^{19} - 36 q^{27} + 12 q^{28} + 12 q^{31} + 36 q^{33} + 12 q^{37} + 36 q^{42} + 36 q^{45} + 12 q^{52} - 36 q^{54} - 36 q^{57} - 36 q^{63} - 12 q^{67} - 12 q^{73} - 12 q^{76} - 36 q^{78} + 72 q^{79} - 72 q^{85} - 12 q^{91} + 36 q^{93} - 72 q^{94} - 60 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 3\nu^{6} + 18\nu^{3} - 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 3\nu^{7} + 18\nu^{4} - 81\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + 3\nu^{8} - 45\nu^{5} + 162\nu^{2} ) / 243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 12\nu^{8} - 72\nu^{5} + 324\nu^{2} ) / 243 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{9} + 15\nu^{6} - 63\nu^{3} + 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{10} + 15\nu^{7} - 63\nu^{4} + 243\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2\nu^{9} - 15\nu^{6} + 90\nu^{3} - 324 ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2\nu^{10} - 15\nu^{7} + 90\nu^{4} - 324\nu ) / 81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} + 6\nu^{8} - 27\nu^{5} + 135\nu^{2} ) / 81 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4\nu^{11} - 30\nu^{8} + 153\nu^{5} - 486\nu^{2} ) / 243 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{8} + 3\beta_{6} + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 3\beta_{7} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{11} + 6\beta_{10} - 6\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{8} + 18\beta_{6} + 18\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{9} + 18\beta_{7} + 18\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -9\beta_{11} + 9\beta_{5} - 45\beta_{4} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -27\beta_{8} + 135\beta_{2} + 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -27\beta_{9} + 135\beta_{3} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -108\beta_{10} + 189\beta_{5} - 108\beta_{4} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(-1\) \(-\beta_{8}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.54662 + 0.779723i
−1.44857 + 0.949550i
−0.0980500 1.72927i
−0.779723 1.54662i
−0.949550 + 1.44857i
1.72927 + 0.0980500i
1.54662 0.779723i
−1.44857 0.949550i
−0.0980500 + 1.72927i
−0.779723 + 1.54662i
−0.949550 1.44857i
1.72927 0.0980500i
−0.707107 0.707107i −1.64497 0.542278i 1.00000i −2.32634 2.32634i 0.779723 + 1.54662i −1.76690 1.76690i 0.707107 0.707107i 2.41187 + 1.78406i 3.28995i
5.2 −0.707107 0.707107i 0.352860 + 1.69573i 1.00000i 0.499019 + 0.499019i 0.949550 1.44857i 1.39812 + 1.39812i 0.707107 0.707107i −2.75098 + 1.19671i 0.705720i
5.3 −0.707107 0.707107i 1.29211 1.15345i 1.00000i 1.82732 + 1.82732i −1.72927 0.0980500i −2.63122 2.63122i 0.707107 0.707107i 0.339111 2.98077i 2.58423i
5.4 0.707107 + 0.707107i −1.64497 + 0.542278i 1.00000i 2.32634 + 2.32634i −1.54662 0.779723i −1.76690 1.76690i −0.707107 + 0.707107i 2.41187 1.78406i 3.28995i
5.5 0.707107 + 0.707107i 0.352860 1.69573i 1.00000i −0.499019 0.499019i 1.44857 0.949550i 1.39812 + 1.39812i −0.707107 + 0.707107i −2.75098 1.19671i 0.705720i
5.6 0.707107 + 0.707107i 1.29211 + 1.15345i 1.00000i −1.82732 1.82732i 0.0980500 + 1.72927i −2.63122 2.63122i −0.707107 + 0.707107i 0.339111 + 2.98077i 2.58423i
47.1 −0.707107 + 0.707107i −1.64497 + 0.542278i 1.00000i −2.32634 + 2.32634i 0.779723 1.54662i −1.76690 + 1.76690i 0.707107 + 0.707107i 2.41187 1.78406i 3.28995i
47.2 −0.707107 + 0.707107i 0.352860 1.69573i 1.00000i 0.499019 0.499019i 0.949550 + 1.44857i 1.39812 1.39812i 0.707107 + 0.707107i −2.75098 1.19671i 0.705720i
47.3 −0.707107 + 0.707107i 1.29211 + 1.15345i 1.00000i 1.82732 1.82732i −1.72927 + 0.0980500i −2.63122 + 2.63122i 0.707107 + 0.707107i 0.339111 + 2.98077i 2.58423i
47.4 0.707107 0.707107i −1.64497 0.542278i 1.00000i 2.32634 2.32634i −1.54662 + 0.779723i −1.76690 + 1.76690i −0.707107 0.707107i 2.41187 + 1.78406i 3.28995i
47.5 0.707107 0.707107i 0.352860 + 1.69573i 1.00000i −0.499019 + 0.499019i 1.44857 + 0.949550i 1.39812 1.39812i −0.707107 0.707107i −2.75098 + 1.19671i 0.705720i
47.6 0.707107 0.707107i 1.29211 1.15345i 1.00000i −1.82732 + 1.82732i 0.0980500 1.72927i −2.63122 + 2.63122i −0.707107 0.707107i 0.339111 2.98077i 2.58423i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.g.a 12
3.b odd 2 1 inner 78.2.g.a 12
4.b odd 2 1 624.2.bf.f 12
12.b even 2 1 624.2.bf.f 12
13.b even 2 1 1014.2.g.b 12
13.d odd 4 1 inner 78.2.g.a 12
13.d odd 4 1 1014.2.g.b 12
39.d odd 2 1 1014.2.g.b 12
39.f even 4 1 inner 78.2.g.a 12
39.f even 4 1 1014.2.g.b 12
52.f even 4 1 624.2.bf.f 12
156.l odd 4 1 624.2.bf.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.g.a 12 1.a even 1 1 trivial
78.2.g.a 12 3.b odd 2 1 inner
78.2.g.a 12 13.d odd 4 1 inner
78.2.g.a 12 39.f even 4 1 inner
624.2.bf.f 12 4.b odd 2 1
624.2.bf.f 12 12.b even 2 1
624.2.bf.f 12 52.f even 4 1
624.2.bf.f 12 156.l odd 4 1
1014.2.g.b 12 13.b even 2 1
1014.2.g.b 12 13.d odd 4 1
1014.2.g.b 12 39.d odd 2 1
1014.2.g.b 12 39.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(78, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} + 6 T^{3} + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 162 T^{8} + 5265 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T^{6} + 6 T^{5} + 18 T^{4} + 8 T^{3} + \cdots + 338)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 648 T^{8} + 63504 T^{4} + \cdots + 331776 \) Copy content Toggle raw display
$13$ \( (T^{6} + 21 T^{4} + 24 T^{3} + 273 T^{2} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 54 T^{4} + 729 T^{2} - 1152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + 18 T^{4} - 16 T^{3} + \cdots + 1568)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 36 T^{4} + 324 T^{2} - 288)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 36 T^{4} + 324 T^{2} + 288)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 6 T^{5} + 18 T^{4} + 256 T^{3} + \cdots + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + 18 T^{4} + 28 T^{3} + \cdots + 4802)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 18576 T^{8} + \cdots + 5308416 \) Copy content Toggle raw display
$43$ \( (T^{6} + 162 T^{4} + 6561 T^{2} + \cdots + 324)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 15714 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$53$ \( (T^{6} + 324 T^{4} + 24516 T^{2} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 2592 T^{8} + \cdots + 5308416 \) Copy content Toggle raw display
$61$ \( (T^{3} - 72 T + 192)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 6 T^{5} + 18 T^{4} - 208 T^{3} + \cdots + 392)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 13986 T^{8} + 37517985 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + 18 T^{4} - 16 T^{3} + \cdots + 1568)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 18 T^{2} + 54 T + 12)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + 78408 T^{8} + \cdots + 21743271936 \) Copy content Toggle raw display
$89$ \( T^{12} + 31536 T^{8} + \cdots + 27710263296 \) Copy content Toggle raw display
$97$ \( (T^{6} + 30 T^{5} + 450 T^{4} + 2368 T^{3} + \cdots + 392)^{2} \) Copy content Toggle raw display
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