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Properties

Label 26.4.e.a
Level $26$
Weight $4$
Character orbit 26.e
Analytic conductor $1.534$
Analytic rank $0$
Dimension $8$
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] = [26,4,Mod(17,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 26.e (of order \(6\), degree \(2\), 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: \(1.53404966015\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 122x^{6} + 5305x^{4} + 97056x^{2} + 627264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 - 2) q^{3} + 4 \beta_1 q^{4} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{5}+ \cdots + (3 \beta_{7} + 6 \beta_{3} + \cdots - 7 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 - 2) q^{3} + 4 \beta_1 q^{4} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{5}+ \cdots + (52 \beta_{7} + 6 \beta_{6} + \cdots - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} + 16 q^{4} + 18 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} + 16 q^{4} + 18 q^{7} - 22 q^{9} - 8 q^{10} - 18 q^{11} - 48 q^{12} - 130 q^{13} + 80 q^{14} - 192 q^{15} - 64 q^{16} + 112 q^{17} + 594 q^{19} + 72 q^{20} - 72 q^{22} - 230 q^{23} - 180 q^{25} - 184 q^{26} + 468 q^{27} + 72 q^{28} + 32 q^{29} + 328 q^{30} - 42 q^{33} - 128 q^{35} + 88 q^{36} - 768 q^{37} - 576 q^{38} - 230 q^{39} - 64 q^{40} - 564 q^{41} - 688 q^{42} - 114 q^{43} + 630 q^{45} + 576 q^{46} - 96 q^{48} - 110 q^{49} + 1968 q^{50} + 1300 q^{51} - 104 q^{52} + 36 q^{53} + 648 q^{54} + 1248 q^{55} + 160 q^{56} - 1848 q^{58} - 1110 q^{59} + 900 q^{61} + 1064 q^{62} - 1980 q^{63} - 512 q^{64} + 1870 q^{65} - 2400 q^{66} + 510 q^{67} - 448 q^{68} - 2402 q^{69} - 1470 q^{71} + 576 q^{72} - 680 q^{74} - 862 q^{75} + 2376 q^{76} + 2340 q^{77} + 1016 q^{78} + 784 q^{79} + 288 q^{80} + 1868 q^{81} + 704 q^{82} - 2136 q^{84} - 2898 q^{85} + 1598 q^{87} + 288 q^{88} - 4434 q^{89} - 2384 q^{90} - 886 q^{91} - 1840 q^{92} + 3108 q^{93} - 2568 q^{94} - 816 q^{95} + 1854 q^{97} + 4272 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 122x^{6} + 5305x^{4} + 97056x^{2} + 627264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 122\nu^{5} + 4513\nu^{3} + 48744\nu + 9504 ) / 19008 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\nu^{7} + 132\nu^{6} + 2746\nu^{5} + 12144\nu^{4} + 80981\nu^{3} + 325644\nu^{2} + 737208\nu + 2414016 ) / 123552 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29\nu^{7} - 132\nu^{6} + 2746\nu^{5} - 12144\nu^{4} + 80981\nu^{3} - 325644\nu^{2} + 737208\nu - 2414016 ) / 123552 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 92\nu^{4} + 2623\nu^{2} + 23124 ) / 156 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 105\nu^{4} + 3416\nu^{2} + 156\nu + 33420 ) / 156 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 105\nu^{4} - 3416\nu^{2} + 156\nu - 33420 ) / 156 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 119 \nu^{7} + 792 \nu^{6} - 10558 \nu^{5} + 72864 \nu^{4} - 266975 \nu^{3} + 2077416 \nu^{2} + \cdots + 18314208 ) / 247104 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 3\beta_{3} - 3\beta_{2} - 31 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24\beta_{7} - 31\beta_{6} - 31\beta_{5} - 12\beta_{4} + 30\beta_{3} + 30\beta_{2} - 48\beta _1 + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} + 6\beta_{5} - 73\beta_{4} - 183\beta_{3} + 183\beta_{2} + 1099 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1512 \beta_{7} + 1099 \beta_{6} + 1099 \beta_{5} + 756 \beta_{4} - 2046 \beta_{3} - 2046 \beta_{2} + \cdots - 2208 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 552\beta_{6} - 552\beta_{5} + 4249\beta_{4} + 8967\beta_{3} - 8967\beta_{2} - 42919 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 76152 \beta_{7} - 42919 \beta_{6} - 42919 \beta_{5} - 38076 \beta_{4} + 114222 \beta_{3} + 114222 \beta_{2} + \cdots + 142056 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
− 4.95620i
3.95620i
6.87513i
− 5.87513i
4.95620i
− 3.95620i
− 6.87513i
5.87513i
−1.73205 − 1.00000i −2.97810 + 5.15822i 2.00000 + 3.46410i 13.0327i 10.3164 − 5.95620i −3.11419 + 1.79798i − 8.00000i −4.23814 − 7.34068i 13.0327 − 22.5733i
17.2 −1.73205 − 1.00000i 1.47810 − 2.56014i 2.00000 + 3.46410i − 20.2288i −5.12028 + 2.95620i −1.04606 + 0.603945i − 8.00000i 9.13045 + 15.8144i −20.2288 + 35.0374i
17.3 1.73205 + 1.00000i −3.93757 + 6.82006i 2.00000 + 3.46410i − 3.30629i −13.6401 + 7.87513i 27.1849 − 15.6952i 8.00000i −17.5089 − 30.3262i 3.30629 − 5.72666i
17.4 1.73205 + 1.00000i 2.43757 − 4.22199i 2.00000 + 3.46410i 0.110135i 8.44398 − 4.87513i −14.0246 + 8.09712i 8.00000i 1.61655 + 2.79994i −0.110135 + 0.190760i
23.1 −1.73205 + 1.00000i −2.97810 − 5.15822i 2.00000 − 3.46410i − 13.0327i 10.3164 + 5.95620i −3.11419 − 1.79798i 8.00000i −4.23814 + 7.34068i 13.0327 + 22.5733i
23.2 −1.73205 + 1.00000i 1.47810 + 2.56014i 2.00000 − 3.46410i 20.2288i −5.12028 − 2.95620i −1.04606 − 0.603945i 8.00000i 9.13045 − 15.8144i −20.2288 − 35.0374i
23.3 1.73205 − 1.00000i −3.93757 − 6.82006i 2.00000 − 3.46410i 3.30629i −13.6401 − 7.87513i 27.1849 + 15.6952i − 8.00000i −17.5089 + 30.3262i 3.30629 + 5.72666i
23.4 1.73205 − 1.00000i 2.43757 + 4.22199i 2.00000 − 3.46410i − 0.110135i 8.44398 + 4.87513i −14.0246 − 8.09712i − 8.00000i 1.61655 − 2.79994i −0.110135 − 0.190760i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.4.e.a ✓ 8
3.b odd 2 1 234.4.l.b 8
4.b odd 2 1 208.4.w.d 8
13.b even 2 1 338.4.e.e 8
13.c even 3 1 338.4.b.g 8
13.c even 3 1 338.4.e.e 8
13.d odd 4 1 338.4.c.m 8
13.d odd 4 1 338.4.c.n 8
13.e even 6 1 inner 26.4.e.a ✓ 8
13.e even 6 1 338.4.b.g 8
13.f odd 12 1 338.4.a.l 4
13.f odd 12 1 338.4.a.m 4
13.f odd 12 1 338.4.c.m 8
13.f odd 12 1 338.4.c.n 8
39.h odd 6 1 234.4.l.b 8
52.i odd 6 1 208.4.w.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.e.a ✓ 8 1.a even 1 1 trivial
26.4.e.a ✓ 8 13.e even 6 1 inner
208.4.w.d 8 4.b odd 2 1
208.4.w.d 8 52.i odd 6 1
234.4.l.b 8 3.b odd 2 1
234.4.l.b 8 39.h odd 6 1
338.4.a.l 4 13.f odd 12 1
338.4.a.m 4 13.f odd 12 1
338.4.b.g 8 13.c even 3 1
338.4.b.g 8 13.e even 6 1
338.4.c.m 8 13.d odd 4 1
338.4.c.m 8 13.f odd 12 1
338.4.c.n 8 13.d odd 4 1
338.4.c.n 8 13.f odd 12 1
338.4.e.e 8 13.b even 2 1
338.4.e.e 8 13.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(26, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + \cdots + 456976 \) Copy content Toggle raw display
$5$ \( T^{8} + 590 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$7$ \( T^{8} - 18 T^{7} + \cdots + 4875264 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 482146919424 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 132748349592609 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 46\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 99\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} - 18 T^{3} + \cdots + 7709429988)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 28\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 79\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{4} - 392 T^{3} + \cdots + 40629076224)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
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