Properties

Label 26.8.a.a
Level $26$
Weight $8$
Character orbit 26.a
Self dual yes
Analytic conductor $8.122$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [26,8,Mod(1,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 26.a (trivial)

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: yes
Analytic conductor: \(8.12201066259\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 8 q^{2} - 39 q^{3} + 64 q^{4} + 385 q^{5} + 312 q^{6} - 293 q^{7} - 512 q^{8} - 666 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - 39 q^{3} + 64 q^{4} + 385 q^{5} + 312 q^{6} - 293 q^{7} - 512 q^{8} - 666 q^{9} - 3080 q^{10} - 5402 q^{11} - 2496 q^{12} + 2197 q^{13} + 2344 q^{14} - 15015 q^{15} + 4096 q^{16} - 21011 q^{17} + 5328 q^{18} - 27326 q^{19} + 24640 q^{20} + 11427 q^{21} + 43216 q^{22} - 63072 q^{23} + 19968 q^{24} + 70100 q^{25} - 17576 q^{26} + 111267 q^{27} - 18752 q^{28} + 122238 q^{29} + 120120 q^{30} - 208396 q^{31} - 32768 q^{32} + 210678 q^{33} + 168088 q^{34} - 112805 q^{35} - 42624 q^{36} - 442379 q^{37} + 218608 q^{38} - 85683 q^{39} - 197120 q^{40} + 58000 q^{41} - 91416 q^{42} - 202025 q^{43} - 345728 q^{44} - 256410 q^{45} + 504576 q^{46} + 588511 q^{47} - 159744 q^{48} - 737694 q^{49} - 560800 q^{50} + 819429 q^{51} + 140608 q^{52} + 1684336 q^{53} - 890136 q^{54} - 2079770 q^{55} + 150016 q^{56} + 1065714 q^{57} - 977904 q^{58} - 442630 q^{59} - 960960 q^{60} - 1083608 q^{61} + 1667168 q^{62} + 195138 q^{63} + 262144 q^{64} + 845845 q^{65} - 1685424 q^{66} + 3443486 q^{67} - 1344704 q^{68} + 2459808 q^{69} + 902440 q^{70} + 2084705 q^{71} + 340992 q^{72} + 5937890 q^{73} + 3539032 q^{74} - 2733900 q^{75} - 1748864 q^{76} + 1582786 q^{77} + 685464 q^{78} - 6609256 q^{79} + 1576960 q^{80} - 2882871 q^{81} - 464000 q^{82} - 142740 q^{83} + 731328 q^{84} - 8089235 q^{85} + 1616200 q^{86} - 4767282 q^{87} + 2765824 q^{88} - 6985286 q^{89} + 2051280 q^{90} - 643721 q^{91} - 4036608 q^{92} + 8127444 q^{93} - 4708088 q^{94} - 10520510 q^{95} + 1277952 q^{96} - 200762 q^{97} + 5901552 q^{98} + 3597732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−8.00000 −39.0000 64.0000 385.000 312.000 −293.000 −512.000 −666.000 −3080.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.8.a.a 1
3.b odd 2 1 234.8.a.d 1
4.b odd 2 1 208.8.a.c 1
13.b even 2 1 338.8.a.c 1
13.d odd 4 2 338.8.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.a 1 1.a even 1 1 trivial
208.8.a.c 1 4.b odd 2 1
234.8.a.d 1 3.b odd 2 1
338.8.a.c 1 13.b even 2 1
338.8.b.b 2 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 39 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(26))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T + 39 \) Copy content Toggle raw display
$5$ \( T - 385 \) Copy content Toggle raw display
$7$ \( T + 293 \) Copy content Toggle raw display
$11$ \( T + 5402 \) Copy content Toggle raw display
$13$ \( T - 2197 \) Copy content Toggle raw display
$17$ \( T + 21011 \) Copy content Toggle raw display
$19$ \( T + 27326 \) Copy content Toggle raw display
$23$ \( T + 63072 \) Copy content Toggle raw display
$29$ \( T - 122238 \) Copy content Toggle raw display
$31$ \( T + 208396 \) Copy content Toggle raw display
$37$ \( T + 442379 \) Copy content Toggle raw display
$41$ \( T - 58000 \) Copy content Toggle raw display
$43$ \( T + 202025 \) Copy content Toggle raw display
$47$ \( T - 588511 \) Copy content Toggle raw display
$53$ \( T - 1684336 \) Copy content Toggle raw display
$59$ \( T + 442630 \) Copy content Toggle raw display
$61$ \( T + 1083608 \) Copy content Toggle raw display
$67$ \( T - 3443486 \) Copy content Toggle raw display
$71$ \( T - 2084705 \) Copy content Toggle raw display
$73$ \( T - 5937890 \) Copy content Toggle raw display
$79$ \( T + 6609256 \) Copy content Toggle raw display
$83$ \( T + 142740 \) Copy content Toggle raw display
$89$ \( T + 6985286 \) Copy content Toggle raw display
$97$ \( T + 200762 \) Copy content Toggle raw display
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