Properties

Label 26.3.f.a
Level $26$
Weight $3$
Character orbit 26.f
Analytic conductor $0.708$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [26,3,Mod(7,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 26.f (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.708448687337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9}+O(q^{10}) \) 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} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + (2 \zeta_{12}^{2} - 4) q^{10} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 8 \zeta_{12} - 9) q^{13} + (7 \zeta_{12}^{3} - 14 \zeta_{12} - 1) q^{14} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{15} + 4 \zeta_{12}^{2} q^{16} + (13 \zeta_{12}^{2} - 6 \zeta_{12} + 13) q^{17} + ( - 6 \zeta_{12}^{3} + 6) q^{18} + ( - 17 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{19} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{20} + (10 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 10) q^{21} + ( - 2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} + \zeta_{12} + 9) q^{22} + ( - 21 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + 19 \zeta_{12}^{3} q^{25} + (15 \zeta_{12}^{3} + \zeta_{12}^{2} + 5 \zeta_{12} - 4) q^{26} + (15 \zeta_{12}^{3} - 30 \zeta_{12}) q^{27} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 8 \zeta_{12} - 14) q^{28} + ( - 16 \zeta_{12}^{3} - 27 \zeta_{12}^{2} - 16 \zeta_{12}) q^{29} + 6 \zeta_{12} q^{30} + (37 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 37) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{33} + ( - 19 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 26 \zeta_{12} - 19) q^{34} + ( - 2 \zeta_{12}^{3} - 21 \zeta_{12}^{2} + \zeta_{12} + 21) q^{35} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{36} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 13 \zeta_{12} + 7) q^{37} + (7 \zeta_{12}^{3} - 34 \zeta_{12}^{2} + 17) q^{38} + ( - 15 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 21 \zeta_{12} + 4) q^{39} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{40} + ( - 26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 13 \zeta_{12} - 13) q^{41} + (\zeta_{12}^{3} + 21 \zeta_{12}^{2} + \zeta_{12}) q^{42} + (20 \zeta_{12}^{2} - 39 \zeta_{12} + 20) q^{43} + ( - 10 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 10) q^{44} + (12 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + ( - 4 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 23 \zeta_{12} + 23) q^{46} + (33 \zeta_{12}^{3} + 33) q^{47} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - 25 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 25 \zeta_{12} + 14) q^{49} + (19 \zeta_{12}^{3} + 19 \zeta_{12}^{2} - 19 \zeta_{12}) q^{50} + ( - 39 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{51} + (24 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 18 \zeta_{12} + 4) q^{52} + ( - 22 \zeta_{12}^{3} + 44 \zeta_{12} - 42) q^{53} + ( - 15 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 15 \zeta_{12} - 30) q^{54} + (9 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 9 \zeta_{12}) q^{55} + ( - 14 \zeta_{12}^{2} - 2 \zeta_{12} - 14) q^{56} + (22 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12} - 22) q^{57} + ( - 32 \zeta_{12}^{3} - 43 \zeta_{12}^{2} - 11 \zeta_{12} + 11) q^{58} + ( - 13 \zeta_{12}^{3} + 23 \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{59} + (6 \zeta_{12}^{3} + 6) q^{60} + ( - 12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} + 6 \zeta_{12} + 39) q^{61} + (64 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 64 \zeta_{12} - 20) q^{62} + (24 \zeta_{12}^{3} + 24 \zeta_{12}^{2} + 18 \zeta_{12} - 42) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 7 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 2 \zeta_{12} - 25) q^{65} + (9 \zeta_{12}^{3} - 18 \zeta_{12} - 3) q^{66} + (23 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 33) q^{67} + (26 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 26 \zeta_{12}) q^{68} + ( - 21 \zeta_{12}^{2} - 6 \zeta_{12} - 21) q^{69} + ( - 22 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 22) q^{70} + (25 \zeta_{12}^{3} + 29 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{71} + ( - 12 \zeta_{12}^{2} + 12 \zeta_{12} + 12) q^{72} + ( - 7 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} - 7) q^{73} + ( - 14 \zeta_{12}^{3} + 19 \zeta_{12}^{2} + 7 \zeta_{12} - 19) q^{74} + ( - 19 \zeta_{12}^{2} + 38) q^{75} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 24 \zeta_{12} + 34) q^{76} + (15 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 32) q^{77} + (2 \zeta_{12}^{3} - 25 \zeta_{12}^{2} + 5 \zeta_{12} + 35) q^{78} + ( - 36 \zeta_{12}^{3} + 72 \zeta_{12} + 48) q^{79} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + ( - 13 \zeta_{12}^{2} + 39 \zeta_{12} - 13) q^{82} + ( - 25 \zeta_{12}^{3} + 46 \zeta_{12}^{2} - 46 \zeta_{12} + 25) q^{83} + (2 \zeta_{12}^{3} + 22 \zeta_{12}^{2} + 20 \zeta_{12} - 20) q^{84} + ( - 12 \zeta_{12}^{3} + 45 \zeta_{12}^{2} - 33 \zeta_{12} - 33) q^{85} + ( - 59 \zeta_{12}^{3} + 40 \zeta_{12}^{2} + 40 \zeta_{12} - 59) q^{86} + (54 \zeta_{12}^{3} + 48 \zeta_{12}^{2} - 27 \zeta_{12} - 48) q^{87} + ( - 18 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 18 \zeta_{12} + 4) q^{88} + ( - 56 \zeta_{12}^{3} - 56 \zeta_{12}^{2} + 43 \zeta_{12} + 13) q^{89} + (24 \zeta_{12}^{2} - 12) q^{90} + ( - 37 \zeta_{12}^{3} - 25 \zeta_{12}^{2} - 86 \zeta_{12} + 22) q^{91} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 42) q^{92} + (17 \zeta_{12}^{3} - 17 \zeta_{12}^{2} + 47 \zeta_{12} + 64) q^{93} + 66 \zeta_{12}^{2} q^{94} + ( - 7 \zeta_{12}^{2} + 51 \zeta_{12} - 7) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (69 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 71 \zeta_{12} + 71) q^{97} + ( - 14 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 32 \zeta_{12} + 32) q^{98} + ( - 24 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} - 24) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9} - 12 q^{10} - 6 q^{11} - 12 q^{13} - 4 q^{14} + 6 q^{15} + 8 q^{16} + 78 q^{17} + 24 q^{18} - 58 q^{19} - 12 q^{20} + 42 q^{21} + 18 q^{22} + 12 q^{23} + 12 q^{24} - 14 q^{26} - 44 q^{28} - 54 q^{29} - 128 q^{31} - 8 q^{32} - 30 q^{33} - 24 q^{34} + 42 q^{35} + 40 q^{37} - 12 q^{39} + 42 q^{42} + 120 q^{43} + 36 q^{44} + 36 q^{45} + 54 q^{46} + 132 q^{47} + 42 q^{49} + 38 q^{50} + 40 q^{52} - 168 q^{53} - 90 q^{54} + 6 q^{55} - 84 q^{56} - 102 q^{57} - 42 q^{58} + 6 q^{59} + 24 q^{60} + 78 q^{61} - 60 q^{62} - 120 q^{63} - 120 q^{65} - 12 q^{66} + 86 q^{67} - 24 q^{68} - 126 q^{69} + 84 q^{70} + 42 q^{71} + 24 q^{72} - 136 q^{73} - 38 q^{74} + 114 q^{75} + 116 q^{76} + 90 q^{78} + 192 q^{79} - 24 q^{80} - 18 q^{81} - 78 q^{82} + 192 q^{83} - 36 q^{84} - 42 q^{85} - 156 q^{86} - 96 q^{87} + 12 q^{88} - 60 q^{89} + 38 q^{91} + 168 q^{92} + 222 q^{93} + 132 q^{94} - 42 q^{95} + 280 q^{97} + 92 q^{98} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(\zeta_{12}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 −1.73205 1.73205i −0.633975 + 2.36603i −8.96410 2.40192i 2.00000 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
11.1 −0.366025 + 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i 1.73205 + 1.73205i −2.36603 + 0.633975i −2.03590 7.59808i 2.00000 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
15.1 1.36603 + 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i −1.73205 + 1.73205i −0.633975 2.36603i −8.96410 + 2.40192i 2.00000 + 2.00000i 3.00000 5.19615i −3.00000 + 1.73205i
19.1 −0.366025 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i 1.73205 1.73205i −2.36603 0.633975i −2.03590 + 7.59808i 2.00000 + 2.00000i 3.00000 + 5.19615i −3.00000 1.73205i
\(n\): 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 26.3.f.a 4
3.b odd 2 1 234.3.bb.b 4
4.b odd 2 1 208.3.bd.c 4
13.b even 2 1 338.3.f.d 4
13.c even 3 1 338.3.d.d 4
13.c even 3 1 338.3.f.f 4
13.d odd 4 1 338.3.f.c 4
13.d odd 4 1 338.3.f.f 4
13.e even 6 1 338.3.d.e 4
13.e even 6 1 338.3.f.c 4
13.f odd 12 1 inner 26.3.f.a 4
13.f odd 12 1 338.3.d.d 4
13.f odd 12 1 338.3.d.e 4
13.f odd 12 1 338.3.f.d 4
39.k even 12 1 234.3.bb.b 4
52.l even 12 1 208.3.bd.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a 4 1.a even 1 1 trivial
26.3.f.a 4 13.f odd 12 1 inner
208.3.bd.c 4 4.b odd 2 1
208.3.bd.c 4 52.l even 12 1
234.3.bb.b 4 3.b odd 2 1
234.3.bb.b 4 39.k even 12 1
338.3.d.d 4 13.c even 3 1
338.3.d.d 4 13.f odd 12 1
338.3.d.e 4 13.e even 6 1
338.3.d.e 4 13.f odd 12 1
338.3.f.c 4 13.d odd 4 1
338.3.f.c 4 13.e even 6 1
338.3.f.d 4 13.b even 2 1
338.3.f.d 4 13.f odd 12 1
338.3.f.f 4 13.c even 3 1
338.3.f.f 4 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(26, [\chi])\). 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
$5$ \( T^{4} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} + 22 T^{3} + 221 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 45 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + 182 T^{2} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{4} - 78 T^{3} + 2499 T^{2} + \cdots + 221841 \) Copy content Toggle raw display
$19$ \( T^{4} + 58 T^{3} + 1325 T^{2} + \cdots + 167281 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} - 381 T^{2} + \cdots + 184041 \) Copy content Toggle raw display
$29$ \( T^{4} + 54 T^{3} + 2955 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$31$ \( T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3602404 \) Copy content Toggle raw display
$37$ \( T^{4} - 40 T^{3} + 401 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$41$ \( T^{4} + 1521 T^{2} - 39546 T + 257049 \) Copy content Toggle raw display
$43$ \( T^{4} - 120 T^{3} + 4479 T^{2} + \cdots + 103041 \) Copy content Toggle raw display
$47$ \( (T^{2} - 66 T + 2178)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 84 T + 312)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 1305 T^{2} + \cdots + 84681 \) Copy content Toggle raw display
$61$ \( T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569 \) Copy content Toggle raw display
$67$ \( T^{4} - 86 T^{3} + 4985 T^{2} + \cdots + 2399401 \) Copy content Toggle raw display
$71$ \( T^{4} - 42 T^{3} + 3357 T^{2} + \cdots + 154449 \) Copy content Toggle raw display
$73$ \( T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844 \) Copy content Toggle raw display
$79$ \( (T^{2} - 96 T - 1584)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 2056356 \) Copy content Toggle raw display
$89$ \( T^{4} + 60 T^{3} + 5661 T^{2} + \cdots + 21594609 \) Copy content Toggle raw display
$97$ \( T^{4} - 280 T^{3} + \cdots + 20043529 \) Copy content Toggle raw display
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