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Properties

Label 26.3.f.b
Level $26$
Weight $3$
Character orbit 26.f
Analytic conductor $0.708$
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] = [26,3,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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.612074651904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3}) q^{2} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + 2 \beta_{5} q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{5} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_1) q^{6} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 2 \beta_{5} + 2 \beta_{4} - 2) q^{8} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 11 \beta_{3} - 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3}) q^{2} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{3} + 2 \beta_{5} q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{5} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_1) q^{6} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} + ( - 2 \beta_{5} + 2 \beta_{4} - 2) q^{8} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 11 \beta_{3} - 11) q^{9} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{10} + (\beta_{7} - \beta_{6} - \beta_{5} + 6 \beta_{4} - 6 \beta_{3} - \beta_{2} - \beta_1) q^{11} + (2 \beta_{6} - 2 \beta_{3} + 2) q^{12} + (2 \beta_{7} - \beta_{6} - 9 \beta_{5} + 6 \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{13} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{14} + ( - \beta_{7} + 2 \beta_{6} + 15 \beta_{5} - 15 \beta_{4} - 15 \beta_{3} + \cdots + 17) q^{15}+ \cdots + (2 \beta_{7} - 4 \beta_{6} + 52 \beta_{5} - 14 \beta_{4} + 38 \beta_{3} + 4 \beta_{2} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 6 q^{5} + 6 q^{6} - 2 q^{7} - 16 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 6 q^{5} + 6 q^{6} - 2 q^{7} - 16 q^{8} - 42 q^{9} - 18 q^{10} - 18 q^{11} + 36 q^{13} + 20 q^{14} + 66 q^{15} + 16 q^{16} - 42 q^{17} + 84 q^{18} + 46 q^{19} + 24 q^{20} - 102 q^{21} - 42 q^{22} - 36 q^{23} - 12 q^{24} + 40 q^{26} + 72 q^{27} - 4 q^{28} - 6 q^{29} - 192 q^{30} + 32 q^{31} + 16 q^{32} + 42 q^{33} - 60 q^{34} - 78 q^{35} - 48 q^{36} - 106 q^{37} + 12 q^{39} - 24 q^{40} + 132 q^{41} + 102 q^{42} - 108 q^{43} + 84 q^{44} + 240 q^{45} + 90 q^{46} + 60 q^{47} + 258 q^{49} + 194 q^{50} + 32 q^{52} - 132 q^{53} - 270 q^{54} - 162 q^{55} - 12 q^{56} - 294 q^{57} - 24 q^{58} + 18 q^{59} - 120 q^{60} + 36 q^{61} - 12 q^{62} - 72 q^{63} - 300 q^{65} + 108 q^{66} - 74 q^{67} + 60 q^{68} + 258 q^{69} + 156 q^{70} - 174 q^{71} + 132 q^{72} + 166 q^{73} - 32 q^{74} + 6 q^{75} - 92 q^{76} + 126 q^{78} - 96 q^{79} + 48 q^{80} - 12 q^{81} - 252 q^{82} - 240 q^{83} - 132 q^{84} - 24 q^{85} + 132 q^{86} + 360 q^{87} - 12 q^{88} + 294 q^{89} + 298 q^{91} - 216 q^{92} + 270 q^{93} - 60 q^{94} + 714 q^{95} - 58 q^{97} - 250 q^{98} + 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 74x^{6} + 2067x^{4} - 25778x^{2} + 121801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - 37\nu^{2} + 4\nu + 349 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 37\nu^{2} + 4\nu - 349 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 74\nu^{5} + 1718\nu^{3} - 12865\nu + 1396 ) / 2792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\nu^{7} - 349\nu^{6} - 2262\nu^{5} + 19195\nu^{4} + 44290\nu^{3} - 345859\nu^{2} - 296126\nu + 2024898 ) / 86552 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -40\nu^{7} - 349\nu^{6} + 2262\nu^{5} + 19195\nu^{4} - 44290\nu^{3} - 345859\nu^{2} + 296126\nu + 2024898 ) / 86552 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 55\nu^{4} + 1053\nu^{2} - 6980 ) / 62 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 318 \nu^{7} + 698 \nu^{6} + 16901 \nu^{5} - 38390 \nu^{4} - 292601 \nu^{3} + 734994 \nu^{2} + 1626083 \nu - 4828764 ) / 86552 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{5} + 2\beta_{4} + 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - 2\beta_{6} - 19\beta_{5} + 19\beta_{4} - 8\beta_{3} + 18\beta_{2} + 18\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 37\beta_{6} + 74\beta_{5} + 74\beta_{4} - 4\beta_{2} + 4\beta _1 + 354 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 140\beta_{7} - 70\beta_{6} - 727\beta_{5} + 727\beta_{4} - 440\beta_{3} + 317\beta_{2} + 317\beta _1 + 150 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1044\beta_{6} + 1964\beta_{5} + 1964\beta_{4} - 220\beta_{2} + 220\beta _1 + 6443 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3488 \beta_{7} - 1744 \beta_{6} - 21156 \beta_{5} + 21156 \beta_{4} - 16024 \beta_{3} + 5399 \beta_{2} + 5399 \beta _1 + 6268 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(\beta_{5}\)

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
4.71318 + 0.500000i
−4.71318 + 0.500000i
3.90972 + 0.500000i
−3.90972 + 0.500000i
4.71318 − 0.500000i
−4.71318 − 0.500000i
3.90972 − 0.500000i
−3.90972 − 0.500000i
−1.36603 + 0.366025i −2.78960 + 4.83174i 1.73205 − 1.00000i 0.323893 + 0.323893i 2.04213 − 7.62134i 7.67890 + 2.05755i −2.00000 + 2.00000i −11.0638 − 19.1630i −0.560999 − 0.323893i
7.2 −1.36603 + 0.366025i 1.92358 − 3.33174i 1.73205 − 1.00000i 3.77418 + 3.77418i −1.40816 + 5.25532i −9.91095 − 2.65563i −2.00000 + 2.00000i −2.90031 − 5.02349i −6.53708 − 3.77418i
11.1 0.366025 − 1.36603i −1.52185 − 2.63592i −1.73205 − 1.00000i 4.79174 + 4.79174i −4.15776 + 1.11407i 1.13983 + 4.25390i −2.00000 + 2.00000i −0.132034 + 0.228689i 8.29953 − 4.79174i
11.2 0.366025 − 1.36603i 2.38787 + 4.13592i −1.73205 − 1.00000i −5.88981 − 5.88981i 6.52379 − 1.74804i 0.0922225 + 0.344179i −2.00000 + 2.00000i −6.90386 + 11.9578i −10.2015 + 5.88981i
15.1 −1.36603 − 0.366025i −2.78960 − 4.83174i 1.73205 + 1.00000i 0.323893 − 0.323893i 2.04213 + 7.62134i 7.67890 − 2.05755i −2.00000 − 2.00000i −11.0638 + 19.1630i −0.560999 + 0.323893i
15.2 −1.36603 − 0.366025i 1.92358 + 3.33174i 1.73205 + 1.00000i 3.77418 − 3.77418i −1.40816 − 5.25532i −9.91095 + 2.65563i −2.00000 − 2.00000i −2.90031 + 5.02349i −6.53708 + 3.77418i
19.1 0.366025 + 1.36603i −1.52185 + 2.63592i −1.73205 + 1.00000i 4.79174 − 4.79174i −4.15776 − 1.11407i 1.13983 − 4.25390i −2.00000 − 2.00000i −0.132034 − 0.228689i 8.29953 + 4.79174i
19.2 0.366025 + 1.36603i 2.38787 − 4.13592i −1.73205 + 1.00000i −5.88981 + 5.88981i 6.52379 + 1.74804i 0.0922225 − 0.344179i −2.00000 − 2.00000i −6.90386 − 11.9578i −10.2015 − 5.88981i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
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.b ✓ 8
3.b odd 2 1 234.3.bb.f 8
4.b odd 2 1 208.3.bd.f 8
13.b even 2 1 338.3.f.i 8
13.c even 3 1 338.3.d.g 8
13.c even 3 1 338.3.f.h 8
13.d odd 4 1 338.3.f.h 8
13.d odd 4 1 338.3.f.j 8
13.e even 6 1 338.3.d.f 8
13.e even 6 1 338.3.f.j 8
13.f odd 12 1 inner 26.3.f.b ✓ 8
13.f odd 12 1 338.3.d.f 8
13.f odd 12 1 338.3.d.g 8
13.f odd 12 1 338.3.f.i 8
39.k even 12 1 234.3.bb.f 8
52.l even 12 1 208.3.bd.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.b ✓ 8 1.a even 1 1 trivial
26.3.f.b ✓ 8 13.f odd 12 1 inner
208.3.bd.f 8 4.b odd 2 1
208.3.bd.f 8 52.l even 12 1
234.3.bb.f 8 3.b odd 2 1
234.3.bb.f 8 39.k even 12 1
338.3.d.f 8 13.e even 6 1
338.3.d.f 8 13.f odd 12 1
338.3.d.g 8 13.c even 3 1
338.3.d.g 8 13.f odd 12 1
338.3.f.h 8 13.c even 3 1
338.3.f.h 8 13.d odd 4 1
338.3.f.i 8 13.b even 2 1
338.3.f.i 8 13.f odd 12 1
338.3.f.j 8 13.d odd 4 1
338.3.f.j 8 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 39T_{3}^{6} - 24T_{3}^{5} + 1209T_{3}^{4} - 468T_{3}^{3} + 12312T_{3}^{2} + 3744T_{3} + 97344 \) acting on \(S_{3}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 39 T^{6} - 24 T^{5} + \cdots + 97344 \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 19044 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} - 127 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{7} + 105 T^{6} + \cdots + 389376 \) Copy content Toggle raw display
$13$ \( T^{8} - 36 T^{7} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{8} + 42 T^{7} + 570 T^{6} + \cdots + 471969 \) Copy content Toggle raw display
$19$ \( T^{8} - 46 T^{7} + \cdots + 1228362304 \) Copy content Toggle raw display
$23$ \( T^{8} + 36 T^{7} + \cdots + 2508807744 \) Copy content Toggle raw display
$29$ \( T^{8} + 6 T^{7} + \cdots + 25455883401 \) Copy content Toggle raw display
$31$ \( T^{8} - 32 T^{7} + \cdots + 8111524096 \) Copy content Toggle raw display
$37$ \( T^{8} + 106 T^{7} + \cdots + 321419829721 \) Copy content Toggle raw display
$41$ \( T^{8} - 132 T^{7} + \cdots + 326485389321 \) Copy content Toggle raw display
$43$ \( T^{8} + 108 T^{7} + \cdots + 325666531584 \) Copy content Toggle raw display
$47$ \( T^{8} - 60 T^{7} + \cdots + 1853819136 \) Copy content Toggle raw display
$53$ \( (T^{4} + 66 T^{3} - 5319 T^{2} + \cdots - 5234376)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 18 T^{7} + \cdots + 70410089309184 \) Copy content Toggle raw display
$61$ \( T^{8} - 36 T^{7} + \cdots + 313453297161 \) Copy content Toggle raw display
$67$ \( T^{8} + 74 T^{7} + \cdots + 2456391674944 \) Copy content Toggle raw display
$71$ \( T^{8} + 174 T^{7} + \cdots + 950999436864 \) Copy content Toggle raw display
$73$ \( T^{8} - 166 T^{7} + \cdots + 3554348548804 \) Copy content Toggle raw display
$79$ \( (T^{4} + 48 T^{3} - 14880 T^{2} + \cdots + 2312448)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 154848357540864 \) Copy content Toggle raw display
$89$ \( T^{8} - 294 T^{7} + \cdots + 14950765690884 \) Copy content Toggle raw display
$97$ \( T^{8} + 58 T^{7} + \cdots + 9988090235236 \) Copy content Toggle raw display
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