Properties

Label 26.2.b.a
Level 26
Weight 2
Character orbit 26.b
Analytic conductor 0.208
Analytic rank 0
Dimension 2
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 26 = 2 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 26.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.207611045255\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{3} - q^{4} -3 i q^{5} -i q^{6} + 3 i q^{7} -i q^{8} -2 q^{9} +O(q^{10})\) \( q + i q^{2} - q^{3} - q^{4} -3 i q^{5} -i q^{6} + 3 i q^{7} -i q^{8} -2 q^{9} + 3 q^{10} + q^{12} + ( 2 + 3 i ) q^{13} -3 q^{14} + 3 i q^{15} + q^{16} + 3 q^{17} -2 i q^{18} -6 i q^{19} + 3 i q^{20} -3 i q^{21} -6 q^{23} + i q^{24} -4 q^{25} + ( -3 + 2 i ) q^{26} + 5 q^{27} -3 i q^{28} -3 q^{30} + i q^{32} + 3 i q^{34} + 9 q^{35} + 2 q^{36} + 3 i q^{37} + 6 q^{38} + ( -2 - 3 i ) q^{39} -3 q^{40} + 3 q^{42} - q^{43} + 6 i q^{45} -6 i q^{46} + 3 i q^{47} - q^{48} -2 q^{49} -4 i q^{50} -3 q^{51} + ( -2 - 3 i ) q^{52} -6 q^{53} + 5 i q^{54} + 3 q^{56} + 6 i q^{57} -6 i q^{59} -3 i q^{60} -8 q^{61} -6 i q^{63} - q^{64} + ( 9 - 6 i ) q^{65} -12 i q^{67} -3 q^{68} + 6 q^{69} + 9 i q^{70} + 15 i q^{71} + 2 i q^{72} + 6 i q^{73} -3 q^{74} + 4 q^{75} + 6 i q^{76} + ( 3 - 2 i ) q^{78} + 10 q^{79} -3 i q^{80} + q^{81} + 6 i q^{83} + 3 i q^{84} -9 i q^{85} -i q^{86} -6 i q^{89} -6 q^{90} + ( -9 + 6 i ) q^{91} + 6 q^{92} -3 q^{94} -18 q^{95} -i q^{96} -12 i q^{97} -2 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} - 4q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} - 4q^{9} + 6q^{10} + 2q^{12} + 4q^{13} - 6q^{14} + 2q^{16} + 6q^{17} - 12q^{23} - 8q^{25} - 6q^{26} + 10q^{27} - 6q^{30} + 18q^{35} + 4q^{36} + 12q^{38} - 4q^{39} - 6q^{40} + 6q^{42} - 2q^{43} - 2q^{48} - 4q^{49} - 6q^{51} - 4q^{52} - 12q^{53} + 6q^{56} - 16q^{61} - 2q^{64} + 18q^{65} - 6q^{68} + 12q^{69} - 6q^{74} + 8q^{75} + 6q^{78} + 20q^{79} + 2q^{81} - 12q^{90} - 18q^{91} + 12q^{92} - 6q^{94} - 36q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(-1\)

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 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
25.2 1.00000i −1.00000 −1.00000 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
\(n\): 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 26.2.b.a 2
3.b odd 2 1 234.2.b.b 2
4.b odd 2 1 208.2.f.a 2
5.b even 2 1 650.2.d.b 2
5.c odd 4 1 650.2.c.a 2
5.c odd 4 1 650.2.c.d 2
7.b odd 2 1 1274.2.d.c 2
7.c even 3 2 1274.2.n.d 4
7.d odd 6 2 1274.2.n.c 4
8.b even 2 1 832.2.f.d 2
8.d odd 2 1 832.2.f.b 2
12.b even 2 1 1872.2.c.f 2
13.b even 2 1 inner 26.2.b.a 2
13.c even 3 2 338.2.e.c 4
13.d odd 4 1 338.2.a.b 1
13.d odd 4 1 338.2.a.d 1
13.e even 6 2 338.2.e.c 4
13.f odd 12 2 338.2.c.b 2
13.f odd 12 2 338.2.c.f 2
39.d odd 2 1 234.2.b.b 2
39.f even 4 1 3042.2.a.g 1
39.f even 4 1 3042.2.a.j 1
52.b odd 2 1 208.2.f.a 2
52.f even 4 1 2704.2.a.j 1
52.f even 4 1 2704.2.a.k 1
65.d even 2 1 650.2.d.b 2
65.g odd 4 1 8450.2.a.h 1
65.g odd 4 1 8450.2.a.u 1
65.h odd 4 1 650.2.c.a 2
65.h odd 4 1 650.2.c.d 2
91.b odd 2 1 1274.2.d.c 2
91.r even 6 2 1274.2.n.d 4
91.s odd 6 2 1274.2.n.c 4
104.e even 2 1 832.2.f.d 2
104.h odd 2 1 832.2.f.b 2
156.h even 2 1 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 1.a even 1 1 trivial
26.2.b.a 2 13.b even 2 1 inner
208.2.f.a 2 4.b odd 2 1
208.2.f.a 2 52.b odd 2 1
234.2.b.b 2 3.b odd 2 1
234.2.b.b 2 39.d odd 2 1
338.2.a.b 1 13.d odd 4 1
338.2.a.d 1 13.d odd 4 1
338.2.c.b 2 13.f odd 12 2
338.2.c.f 2 13.f odd 12 2
338.2.e.c 4 13.c even 3 2
338.2.e.c 4 13.e even 6 2
650.2.c.a 2 5.c odd 4 1
650.2.c.a 2 65.h odd 4 1
650.2.c.d 2 5.c odd 4 1
650.2.c.d 2 65.h odd 4 1
650.2.d.b 2 5.b even 2 1
650.2.d.b 2 65.d even 2 1
832.2.f.b 2 8.d odd 2 1
832.2.f.b 2 104.h odd 2 1
832.2.f.d 2 8.b even 2 1
832.2.f.d 2 104.e even 2 1
1274.2.d.c 2 7.b odd 2 1
1274.2.d.c 2 91.b odd 2 1
1274.2.n.c 4 7.d odd 6 2
1274.2.n.c 4 91.s odd 6 2
1274.2.n.d 4 7.c even 3 2
1274.2.n.d 4 91.r even 6 2
1872.2.c.f 2 12.b even 2 1
1872.2.c.f 2 156.h even 2 1
2704.2.a.j 1 52.f even 4 1
2704.2.a.k 1 52.f even 4 1
3042.2.a.g 1 39.f even 4 1
3042.2.a.j 1 39.f even 4 1
8450.2.a.h 1 65.g odd 4 1
8450.2.a.u 1 65.g odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(26, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + T + 3 T^{2} )^{2} \)
$5$ \( 1 - T^{2} + 25 T^{4} \)
$7$ \( 1 - 5 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( ( 1 - 3 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 2 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 6 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{2} \)
$37$ \( 1 - 65 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 - 85 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 82 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( 1 + 83 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( 1 - 142 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 50 T^{2} + 9409 T^{4} \)
show more
show less