Properties

Label 78.3.l.a
Level $78$
Weight $3$
Character orbit 78.l
Analytic conductor $2.125$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 78.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.12534606201\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{10} + (10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 8 \zeta_{12} - 2) q^{11} + (4 \zeta_{12}^{2} - 2) q^{12} + 13 \zeta_{12} q^{13} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12} - 5) q^{14} + ( - 6 \zeta_{12} - 6) q^{15} + 4 \zeta_{12}^{2} q^{16} + (4 \zeta_{12}^{2} - 12 \zeta_{12} + 4) q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{18} + (12 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 19 \zeta_{12} + 19) q^{19} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12} - 8) q^{20} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 10) q^{21} + ( - 4 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 2 \zeta_{12} + 18) q^{22} + ( - 18 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 18 \zeta_{12} + 16) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + (\zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{25} + ( - 13 \zeta_{12}^{3} - 13) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10) q^{28} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{29} + (6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{30} + ( - 29 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 29) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + (18 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 12 \zeta_{12} - 12) q^{33} + (16 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 16) q^{34} + ( - 20 \zeta_{12}^{3} + 10 \zeta_{12}) q^{35} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + (23 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 13 \zeta_{12} - 36) q^{37} + (26 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{38} + (26 \zeta_{12}^{2} - 13) q^{39} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 12) q^{40} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 20 \zeta_{12} - 28) q^{41} + ( - 5 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 5 \zeta_{12}) q^{42} + (5 \zeta_{12}^{2} + 24 \zeta_{12} + 5) q^{43} + (20 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 20) q^{44} + ( - 6 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + (16 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 26 \zeta_{12} - 26) q^{46} + (18 \zeta_{12}^{3} + 30 \zeta_{12}^{2} + 30 \zeta_{12} + 18) q^{47} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{48} + ( - \zeta_{12}^{3} + 25 \zeta_{12}^{2} + \zeta_{12} - 50) q^{49} + (11 \zeta_{12}^{3} + 11 \zeta_{12}^{2} + 13 \zeta_{12} - 24) q^{50} + (12 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{51} + 26 \zeta_{12}^{2} q^{52} + ( - 46 \zeta_{12}^{3} + 92 \zeta_{12} + 18) q^{53} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 6) q^{54} + ( - 24 \zeta_{12}^{3} - 60 \zeta_{12}^{2} - 24 \zeta_{12}) q^{55} + (10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{56} + (5 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 26 \zeta_{12} - 5) q^{57} + (20 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{58} + (2 \zeta_{12}^{3} - 46 \zeta_{12}^{2} + 44 \zeta_{12} + 44) q^{59} + ( - 12 \zeta_{12}^{2} - 12 \zeta_{12}) q^{60} + (50 \zeta_{12}^{3} + 36 \zeta_{12}^{2} - 25 \zeta_{12} - 36) q^{61} + (53 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 53 \zeta_{12} - 10) q^{62} + ( - 15 \zeta_{12} + 15) q^{63} + 8 \zeta_{12}^{3} q^{64} + (26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} - 52 \zeta_{12} - 52) q^{65} + ( - 18 \zeta_{12}^{3} + 36 \zeta_{12} + 6) q^{66} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 31 \zeta_{12} - 43) q^{67} + (8 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 8 \zeta_{12}) q^{68} + (18 \zeta_{12}^{2} + 24 \zeta_{12} + 18) q^{69} + (10 \zeta_{12}^{3} + 20 \zeta_{12}^{2} - 20 \zeta_{12} - 10) q^{70} + (22 \zeta_{12}^{3} - 46 \zeta_{12}^{2} - 68 \zeta_{12} + 68) q^{71} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + ( - 59 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 23 \zeta_{12} - 59) q^{73} + ( - 72 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 36 \zeta_{12} + 10) q^{74} + ( - 36 \zeta_{12}^{3} + \zeta_{12}^{2} + 36 \zeta_{12} - 2) q^{75} + ( - 14 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 38 \zeta_{12} - 24) q^{76} + (30 \zeta_{12}^{3} - 80 \zeta_{12}^{2} + 40) q^{77} + ( - 13 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 13 \zeta_{12} + 26) q^{78} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} + 24) q^{79} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 16 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + (28 \zeta_{12}^{2} + 12 \zeta_{12} + 28) q^{82} + ( - 40 \zeta_{12}^{3} + 52 \zeta_{12}^{2} - 52 \zeta_{12} + 40) q^{83} + (10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 20 \zeta_{12} + 20) q^{84} + ( - 24 \zeta_{12}^{2} + 24 \zeta_{12} + 24) q^{85} + ( - 19 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 10 \zeta_{12} - 19) q^{86} + ( - 30 \zeta_{12}^{2} + 30) q^{87} + ( - 36 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 36 \zeta_{12} + 8) q^{88} + (64 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 104 \zeta_{12} + 40) q^{89} + (18 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{90} + ( - 65 \zeta_{12}^{3} + 65 \zeta_{12}^{2} - 65) q^{91} + ( - 16 \zeta_{12}^{3} + 32 \zeta_{12} + 36) q^{92} + (19 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 34 \zeta_{12} + 53) q^{93} + ( - 30 \zeta_{12}^{3} - 66 \zeta_{12}^{2} - 30 \zeta_{12}) q^{94} + ( - 10 \zeta_{12}^{2} + 42 \zeta_{12} - 10) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (71 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 84 \zeta_{12} + 84) q^{97} + ( - 50 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 24 \zeta_{12} + 24) q^{98} + ( - 24 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} - 24) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 12 q^{5} - 6 q^{6} - 10 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 12 q^{5} - 6 q^{6} - 10 q^{7} - 8 q^{8} - 6 q^{9} + 12 q^{10} + 12 q^{11} - 20 q^{14} - 24 q^{15} + 8 q^{16} + 24 q^{17} + 12 q^{18} + 62 q^{19} - 24 q^{20} - 30 q^{21} + 36 q^{22} + 48 q^{23} + 12 q^{24} - 52 q^{26} - 20 q^{28} + 36 q^{30} + 106 q^{31} + 8 q^{32} - 60 q^{33} + 48 q^{34} - 98 q^{37} + 48 q^{40} - 96 q^{41} + 30 q^{42} + 30 q^{43} - 72 q^{44} - 84 q^{46} + 132 q^{47} - 150 q^{49} - 74 q^{50} + 52 q^{52} + 72 q^{53} + 18 q^{54} - 120 q^{55} + 60 q^{56} - 72 q^{57} + 60 q^{58} + 84 q^{59} - 24 q^{60} - 72 q^{61} - 30 q^{62} + 60 q^{63} - 156 q^{65} + 24 q^{66} - 148 q^{67} - 48 q^{68} + 108 q^{69} + 180 q^{71} + 12 q^{72} - 190 q^{73} + 20 q^{74} - 6 q^{75} - 124 q^{76} + 78 q^{78} + 96 q^{79} - 48 q^{80} - 18 q^{81} + 168 q^{82} + 264 q^{83} + 60 q^{84} + 48 q^{85} - 96 q^{86} + 60 q^{87} + 24 q^{88} + 288 q^{89} - 130 q^{91} + 144 q^{92} + 174 q^{93} - 132 q^{94} - 60 q^{95} + 310 q^{97} + 148 q^{98} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.36603 + 0.366025i 0.866025 1.50000i 1.73205 1.00000i −4.73205 4.73205i −0.633975 + 2.36603i −2.50000 0.669873i −2.00000 + 2.00000i −1.50000 2.59808i 8.19615 + 4.73205i
19.1 0.366025 + 1.36603i −0.866025 + 1.50000i −1.73205 + 1.00000i −1.26795 + 1.26795i −2.36603 0.633975i −2.50000 + 9.33013i −2.00000 2.00000i −1.50000 2.59808i −2.19615 1.26795i
37.1 0.366025 1.36603i −0.866025 1.50000i −1.73205 1.00000i −1.26795 1.26795i −2.36603 + 0.633975i −2.50000 9.33013i −2.00000 + 2.00000i −1.50000 + 2.59808i −2.19615 + 1.26795i
67.1 −1.36603 0.366025i 0.866025 + 1.50000i 1.73205 + 1.00000i −4.73205 + 4.73205i −0.633975 2.36603i −2.50000 + 0.669873i −2.00000 2.00000i −1.50000 + 2.59808i 8.19615 4.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.3.l.a 4
3.b odd 2 1 234.3.bb.c 4
13.c even 3 1 1014.3.f.e 4
13.e even 6 1 1014.3.f.d 4
13.f odd 12 1 inner 78.3.l.a 4
13.f odd 12 1 1014.3.f.d 4
13.f odd 12 1 1014.3.f.e 4
39.k even 12 1 234.3.bb.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.3.l.a 4 1.a even 1 1 trivial
78.3.l.a 4 13.f odd 12 1 inner
234.3.bb.c 4 3.b odd 2 1
234.3.bb.c 4 39.k even 12 1
1014.3.f.d 4 13.e even 6 1
1014.3.f.d 4 13.f odd 12 1
1014.3.f.e 4 13.c even 3 1
1014.3.f.e 4 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 12T_{5}^{3} + 72T_{5}^{2} + 144T_{5} + 144 \) acting on \(S_{3}^{\mathrm{new}}(78, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{4} + 10 T^{3} + 125 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + 180 T^{2} + \cdots + 24336 \) Copy content Toggle raw display
$13$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$17$ \( T^{4} - 24 T^{3} + 96 T^{2} + \cdots + 9216 \) Copy content Toggle raw display
$19$ \( T^{4} - 62 T^{3} + 986 T^{2} + \cdots + 14884 \) Copy content Toggle raw display
$23$ \( T^{4} - 48 T^{3} + 636 T^{2} + \cdots + 17424 \) Copy content Toggle raw display
$29$ \( T^{4} + 300 T^{2} + 90000 \) Copy content Toggle raw display
$31$ \( T^{4} - 106 T^{3} + 5618 T^{2} + \cdots + 1868689 \) Copy content Toggle raw display
$37$ \( T^{4} + 98 T^{3} + 5882 T^{2} + \cdots + 3587236 \) Copy content Toggle raw display
$41$ \( T^{4} + 96 T^{3} + 3600 T^{2} + \cdots + 1218816 \) Copy content Toggle raw display
$43$ \( T^{4} - 30 T^{3} - 201 T^{2} + \cdots + 251001 \) Copy content Toggle raw display
$47$ \( T^{4} - 132 T^{3} + 8712 T^{2} + \cdots + 685584 \) Copy content Toggle raw display
$53$ \( (T^{2} - 36 T - 6024)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 84 T^{3} + 4068 T^{2} + \cdots + 16353936 \) Copy content Toggle raw display
$61$ \( T^{4} + 72 T^{3} + 5763 T^{2} + \cdots + 335241 \) Copy content Toggle raw display
$67$ \( T^{4} + 148 T^{3} + 8501 T^{2} + \cdots + 6723649 \) Copy content Toggle raw display
$71$ \( T^{4} - 180 T^{3} + \cdots + 33315984 \) Copy content Toggle raw display
$73$ \( T^{4} + 190 T^{3} + \cdots + 13830961 \) Copy content Toggle raw display
$79$ \( (T^{2} - 48 T + 429)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 264 T^{3} + \cdots + 21678336 \) Copy content Toggle raw display
$89$ \( T^{4} - 288 T^{3} + \cdots + 137170944 \) Copy content Toggle raw display
$97$ \( T^{4} - 310 T^{3} + 27389 T^{2} + \cdots + 8162449 \) Copy content Toggle raw display
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