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Properties

Label 26.8.b.a
Level $26$
Weight $8$
Character orbit 26.b
Analytic conductor $8.122$
Analytic rank $0$
Dimension $10$
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] = [26,8,Mod(25,26)]
 
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([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.25");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 26.b (of order \(2\), degree \(1\), 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: \(8.12201066259\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 17770x^{8} + 98320641x^{6} + 176057788072x^{4} + 109845194658832x^{2} + 14762086704451584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( - \beta_{2} + 5) q^{3} - 64 q^{4} + ( - \beta_{7} + 2 \beta_{6}) q^{5} + (5 \beta_{6} + \beta_1) q^{6} + (\beta_{9} - 6 \beta_{6} - \beta_1) q^{7} - 64 \beta_{6} q^{8} + ( - \beta_{3} + 1396) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + ( - \beta_{2} + 5) q^{3} - 64 q^{4} + ( - \beta_{7} + 2 \beta_{6}) q^{5} + (5 \beta_{6} + \beta_1) q^{6} + (\beta_{9} - 6 \beta_{6} - \beta_1) q^{7} - 64 \beta_{6} q^{8} + ( - \beta_{3} + 1396) q^{9} + ( - \beta_{5} + 12 \beta_{2} - 109) q^{10} + (\beta_{9} + \beta_{8} + \cdots + 3 \beta_1) q^{11}+ \cdots + ( - 315 \beta_{9} + 1629 \beta_{8} + \cdots - 2439 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 54 q^{3} - 640 q^{4} + 13960 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 54 q^{3} - 640 q^{4} + 13960 q^{9} - 1136 q^{10} - 3456 q^{12} + 3432 q^{13} + 3792 q^{14} + 40960 q^{16} - 6918 q^{17} + 17280 q^{22} + 94164 q^{23} - 330788 q^{25} + 118896 q^{26} - 53550 q^{27} - 131304 q^{29} - 434992 q^{30} + 873450 q^{35} - 893440 q^{36} + 602976 q^{38} - 1569460 q^{39} + 72704 q^{40} + 1800592 q^{42} + 2740774 q^{43} + 221184 q^{48} - 3083904 q^{49} + 1103186 q^{51} - 219648 q^{52} - 3605916 q^{53} + 3999840 q^{55} - 242688 q^{56} + 11727656 q^{61} - 4210560 q^{62} - 2621440 q^{64} - 7481034 q^{65} - 5557248 q^{66} + 442752 q^{68} + 2035796 q^{69} + 18145392 q^{74} - 32855720 q^{75} - 7147920 q^{77} - 8343632 q^{78} - 28449228 q^{79} + 49311466 q^{81} + 17868800 q^{82} + 50156320 q^{87} - 1105920 q^{88} - 23059744 q^{90} - 20225194 q^{91} - 6026496 q^{92} + 8240976 q^{94} + 16547484 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 17770x^{8} + 98320641x^{6} + 176057788072x^{4} + 109845194658832x^{2} + 14762086704451584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 8\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5867 \nu^{8} + 94017378 \nu^{6} + 429800329035 \nu^{4} + 434217718330916 \nu^{2} + 51\!\cdots\!64 ) / 999398686918320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5867 \nu^{8} + 94017378 \nu^{6} + 429800329035 \nu^{4} + 534157587022748 \nu^{2} + 40\!\cdots\!20 ) / 99939868691832 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1598873 \nu^{8} + 27672003788 \nu^{6} + 144604802041527 \nu^{4} + \cdots + 71\!\cdots\!92 ) / 54\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 86331697 \nu^{8} + 1442119842462 \nu^{6} + \cdots + 10\!\cdots\!68 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35664299 \nu^{9} - 574351546782 \nu^{7} + \cdots + 47\!\cdots\!36 \nu ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 539984434165 \nu^{9} + \cdots - 76\!\cdots\!48 \nu ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 558514318645 \nu^{9} + \cdots - 29\!\cdots\!12 \nu ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2389965867265 \nu^{9} + \cdots + 89\!\cdots\!32 \nu ) / 37\!\cdots\!40 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 10\beta_{2} - 3558 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 384\beta_{9} + 120\beta_{8} + 2712\beta_{7} + 36870\beta_{6} - 6283\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -702\beta_{5} + 744\beta_{4} - 8491\beta_{3} + 117940\beta_{2} + 23873508 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3363456\beta_{9} - 1541592\beta_{8} - 22621560\beta_{7} - 430498110\beta_{6} + 47081539\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9406854\beta_{5} - 7315968\beta_{4} + 66633091\beta_{3} - 1241764900\beta_{2} - 180093172884 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 27560562048 \beta_{9} + 15701549784 \beta_{8} + 166454354808 \beta_{7} + 4504976092350 \beta_{6} - 365618560051 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 99316161126 \beta_{5} + 62733370392 \beta_{4} - 519763359187 \beta_{3} + 12169491231340 \beta_{2} + 13\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 223674260832384 \beta_{9} - 148925022683736 \beta_{8} + \cdots + 28\!\cdots\!31 \beta_1 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
90.6609i
36.7123i
− 13.5100i
− 32.3570i
− 83.5063i
− 90.6609i
− 36.7123i
13.5100i
32.3570i
83.5063i
− 8.00000i −85.6609 −64.0000 197.476i 685.287i − 828.309i 512.000i 5150.80 1579.81
25.2 − 8.00000i −31.7123 −64.0000 − 319.050i 253.699i 1008.64i 512.000i −1181.33 −2552.40
25.3 − 8.00000i 18.5100 −64.0000 6.61448i − 148.080i − 1604.67i 512.000i −1844.38 52.9158
25.4 − 8.00000i 37.3570 −64.0000 477.073i − 298.856i 855.664i 512.000i −791.454 3816.58
25.5 − 8.00000i 88.5063 −64.0000 − 433.113i − 708.050i 805.670i 512.000i 5646.37 −3464.91
25.6 8.00000i −85.6609 −64.0000 − 197.476i − 685.287i 828.309i − 512.000i 5150.80 1579.81
25.7 8.00000i −31.7123 −64.0000 319.050i − 253.699i − 1008.64i − 512.000i −1181.33 −2552.40
25.8 8.00000i 18.5100 −64.0000 − 6.61448i 148.080i 1604.67i − 512.000i −1844.38 52.9158
25.9 8.00000i 37.3570 −64.0000 − 477.073i 298.856i − 855.664i − 512.000i −791.454 3816.58
25.10 8.00000i 88.5063 −64.0000 433.113i 708.050i − 805.670i − 512.000i 5646.37 −3464.91
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.10
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 26.8.b.a ✓ 10
3.b odd 2 1 234.8.b.c 10
4.b odd 2 1 208.8.f.c 10
13.b even 2 1 inner 26.8.b.a ✓ 10
13.d odd 4 1 338.8.a.k 5
13.d odd 4 1 338.8.a.l 5
39.d odd 2 1 234.8.b.c 10
52.b odd 2 1 208.8.f.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.b.a ✓ 10 1.a even 1 1 trivial
26.8.b.a ✓ 10 13.b even 2 1 inner
208.8.f.c 10 4.b odd 2 1
208.8.f.c 10 52.b odd 2 1
234.8.b.c 10 3.b odd 2 1
234.8.b.c 10 39.d odd 2 1
338.8.a.k 5 13.d odd 4 1
338.8.a.l 5 13.d odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(26, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 64)^{5} \) Copy content Toggle raw display
$3$ \( (T^{5} - 27 T^{4} + \cdots - 166250448)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 85\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 97\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 84\!\cdots\!72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 44\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 40\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 25\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 37\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
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