Properties

Label 78.2.e.a
Level $78$
Weight $2$
Character orbit 78.e
Analytic conductor $0.623$
Analytic rank $0$
Dimension $2$
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.622833135766\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 3.00000 0.500000 0.866025i −1.00000 + 1.73205i −1.00000 −0.500000 + 0.866025i 1.50000 + 2.59808i
61.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 3.00000 0.500000 + 0.866025i −1.00000 1.73205i −1.00000 −0.500000 0.866025i 1.50000 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.e.a 2
3.b odd 2 1 234.2.h.a 2
4.b odd 2 1 624.2.q.g 2
5.b even 2 1 1950.2.i.m 2
5.c odd 4 2 1950.2.z.g 4
12.b even 2 1 1872.2.t.c 2
13.b even 2 1 1014.2.e.a 2
13.c even 3 1 inner 78.2.e.a 2
13.c even 3 1 1014.2.a.c 1
13.d odd 4 2 1014.2.i.b 4
13.e even 6 1 1014.2.a.f 1
13.e even 6 1 1014.2.e.a 2
13.f odd 12 2 1014.2.b.c 2
13.f odd 12 2 1014.2.i.b 4
39.h odd 6 1 3042.2.a.h 1
39.i odd 6 1 234.2.h.a 2
39.i odd 6 1 3042.2.a.i 1
39.k even 12 2 3042.2.b.h 2
52.i odd 6 1 8112.2.a.c 1
52.j odd 6 1 624.2.q.g 2
52.j odd 6 1 8112.2.a.m 1
65.n even 6 1 1950.2.i.m 2
65.q odd 12 2 1950.2.z.g 4
156.p even 6 1 1872.2.t.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 1.a even 1 1 trivial
78.2.e.a 2 13.c even 3 1 inner
234.2.h.a 2 3.b odd 2 1
234.2.h.a 2 39.i odd 6 1
624.2.q.g 2 4.b odd 2 1
624.2.q.g 2 52.j odd 6 1
1014.2.a.c 1 13.c even 3 1
1014.2.a.f 1 13.e even 6 1
1014.2.b.c 2 13.f odd 12 2
1014.2.e.a 2 13.b even 2 1
1014.2.e.a 2 13.e even 6 1
1014.2.i.b 4 13.d odd 4 2
1014.2.i.b 4 13.f odd 12 2
1872.2.t.c 2 12.b even 2 1
1872.2.t.c 2 156.p even 6 1
1950.2.i.m 2 5.b even 2 1
1950.2.i.m 2 65.n even 6 1
1950.2.z.g 4 5.c odd 4 2
1950.2.z.g 4 65.q odd 12 2
3042.2.a.h 1 39.h odd 6 1
3042.2.a.i 1 39.i odd 6 1
3042.2.b.h 2 39.k even 12 2
8112.2.a.c 1 52.i odd 6 1
8112.2.a.m 1 52.j odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 13)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
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