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Properties

Label 78.2.i.b
Level $78$
Weight $2$
Character orbit 78.i
Analytic conductor $0.623$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i − 1.73205i −0.866025 + 0.500000i 1.09808 − 0.633975i − 1.00000i −0.500000 − 0.866025i −0.866025 + 1.50000i
43.2 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i − 1.73205i 0.866025 − 0.500000i −4.09808 + 2.36603i 1.00000i −0.500000 − 0.866025i 0.866025 − 1.50000i
49.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 1.73205i −0.866025 − 0.500000i 1.09808 + 0.633975i 1.00000i −0.500000 + 0.866025i −0.866025 − 1.50000i
49.2 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 1.73205i 0.866025 + 0.500000i −4.09808 − 2.36603i − 1.00000i −0.500000 + 0.866025i 0.866025 + 1.50000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.i.b ✓ 4
3.b odd 2 1 234.2.l.a 4
4.b odd 2 1 624.2.bv.d 4
5.b even 2 1 1950.2.bc.c 4
5.c odd 4 1 1950.2.y.a 4
5.c odd 4 1 1950.2.y.h 4
12.b even 2 1 1872.2.by.k 4
13.b even 2 1 1014.2.i.f 4
13.c even 3 1 1014.2.b.d 4
13.c even 3 1 1014.2.i.f 4
13.d odd 4 1 1014.2.e.h 4
13.d odd 4 1 1014.2.e.j 4
13.e even 6 1 inner 78.2.i.b ✓ 4
13.e even 6 1 1014.2.b.d 4
13.f odd 12 1 1014.2.a.h 2
13.f odd 12 1 1014.2.a.j 2
13.f odd 12 1 1014.2.e.h 4
13.f odd 12 1 1014.2.e.j 4
39.h odd 6 1 234.2.l.a 4
39.h odd 6 1 3042.2.b.l 4
39.i odd 6 1 3042.2.b.l 4
39.k even 12 1 3042.2.a.s 2
39.k even 12 1 3042.2.a.v 2
52.i odd 6 1 624.2.bv.d 4
52.l even 12 1 8112.2.a.bq 2
52.l even 12 1 8112.2.a.bx 2
65.l even 6 1 1950.2.bc.c 4
65.r odd 12 1 1950.2.y.a 4
65.r odd 12 1 1950.2.y.h 4
156.r even 6 1 1872.2.by.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b ✓ 4 1.a even 1 1 trivial
78.2.i.b ✓ 4 13.e even 6 1 inner
234.2.l.a 4 3.b odd 2 1
234.2.l.a 4 39.h odd 6 1
624.2.bv.d 4 4.b odd 2 1
624.2.bv.d 4 52.i odd 6 1
1014.2.a.h 2 13.f odd 12 1
1014.2.a.j 2 13.f odd 12 1
1014.2.b.d 4 13.c even 3 1
1014.2.b.d 4 13.e even 6 1
1014.2.e.h 4 13.d odd 4 1
1014.2.e.h 4 13.f odd 12 1
1014.2.e.j 4 13.d odd 4 1
1014.2.e.j 4 13.f odd 12 1
1014.2.i.f 4 13.b even 2 1
1014.2.i.f 4 13.c even 3 1
1872.2.by.k 4 12.b even 2 1
1872.2.by.k 4 156.r even 6 1
1950.2.y.a 4 5.c odd 4 1
1950.2.y.a 4 65.r odd 12 1
1950.2.y.h 4 5.c odd 4 1
1950.2.y.h 4 65.r odd 12 1
1950.2.bc.c 4 5.b even 2 1
1950.2.bc.c 4 65.l even 6 1
3042.2.a.s 2 39.k even 12 1
3042.2.a.v 2 39.k even 12 1
3042.2.b.l 4 39.h odd 6 1
3042.2.b.l 4 39.i odd 6 1
8112.2.a.bq 2 52.l even 12 1
8112.2.a.bx 2 52.l even 12 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$37$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676 \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T - 3)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 24 T + 192)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} - 66 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 4356 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 24 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$97$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
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