Properties

Label 78.2.b.a
Level $78$
Weight $2$
Character orbit 78.b
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 2.00000i 1.00000i 2.00000i 1.00000i 1.00000 −2.00000
25.2 1.00000i 1.00000 −1.00000 2.00000i 1.00000i 2.00000i 1.00000i 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.b.a 2
3.b odd 2 1 234.2.b.a 2
4.b odd 2 1 624.2.c.a 2
5.b even 2 1 1950.2.b.c 2
5.c odd 4 1 1950.2.f.d 2
5.c odd 4 1 1950.2.f.g 2
7.b odd 2 1 3822.2.c.d 2
8.b even 2 1 2496.2.c.f 2
8.d odd 2 1 2496.2.c.m 2
12.b even 2 1 1872.2.c.b 2
13.b even 2 1 inner 78.2.b.a 2
13.c even 3 2 1014.2.i.c 4
13.d odd 4 1 1014.2.a.b 1
13.d odd 4 1 1014.2.a.g 1
13.e even 6 2 1014.2.i.c 4
13.f odd 12 2 1014.2.e.b 2
13.f odd 12 2 1014.2.e.e 2
39.d odd 2 1 234.2.b.a 2
39.f even 4 1 3042.2.a.c 1
39.f even 4 1 3042.2.a.n 1
52.b odd 2 1 624.2.c.a 2
52.f even 4 1 8112.2.a.g 1
52.f even 4 1 8112.2.a.j 1
65.d even 2 1 1950.2.b.c 2
65.h odd 4 1 1950.2.f.d 2
65.h odd 4 1 1950.2.f.g 2
91.b odd 2 1 3822.2.c.d 2
104.e even 2 1 2496.2.c.f 2
104.h odd 2 1 2496.2.c.m 2
156.h even 2 1 1872.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 1.a even 1 1 trivial
78.2.b.a 2 13.b even 2 1 inner
234.2.b.a 2 3.b odd 2 1
234.2.b.a 2 39.d odd 2 1
624.2.c.a 2 4.b odd 2 1
624.2.c.a 2 52.b odd 2 1
1014.2.a.b 1 13.d odd 4 1
1014.2.a.g 1 13.d odd 4 1
1014.2.e.b 2 13.f odd 12 2
1014.2.e.e 2 13.f odd 12 2
1014.2.i.c 4 13.c even 3 2
1014.2.i.c 4 13.e even 6 2
1872.2.c.b 2 12.b even 2 1
1872.2.c.b 2 156.h even 2 1
1950.2.b.c 2 5.b even 2 1
1950.2.b.c 2 65.d even 2 1
1950.2.f.d 2 5.c odd 4 1
1950.2.f.d 2 65.h odd 4 1
1950.2.f.g 2 5.c odd 4 1
1950.2.f.g 2 65.h odd 4 1
2496.2.c.f 2 8.b even 2 1
2496.2.c.f 2 104.e even 2 1
2496.2.c.m 2 8.d odd 2 1
2496.2.c.m 2 104.h odd 2 1
3042.2.a.c 1 39.f even 4 1
3042.2.a.n 1 39.f even 4 1
3822.2.c.d 2 7.b odd 2 1
3822.2.c.d 2 91.b odd 2 1
8112.2.a.g 1 52.f even 4 1
8112.2.a.j 1 52.f even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 6 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( -4 + T )^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( 100 + T^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( 100 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( 36 + T^{2} \)
$97$ \( 144 + T^{2} \)
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