Properties

Label 26.7.d.b
Level $26$
Weight $7$
Character orbit 26.d
Analytic conductor $5.981$
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] = [26,7,Mod(5,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.5");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 26.d (of order \(4\), 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: \(5.98140617412\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5226x^{6} + 6977529x^{4} + 486151524x^{2} + 5607014400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (4 \beta_{3} - 4) q^{2} - \beta_{2} q^{3} - 32 \beta_{3} q^{4} + ( - \beta_{4} - 10 \beta_{3} - \beta_{2} + \beta_1 + 11) q^{5} + (4 \beta_{2} - 4 \beta_1) q^{6} + (\beta_{7} + \beta_{5} - 29 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 29) q^{7} + (128 \beta_{3} + 128) q^{8} + ( - 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 9 \beta_{2} + \cdots + 576) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \beta_{3} - 4) q^{2} - \beta_{2} q^{3} - 32 \beta_{3} q^{4} + ( - \beta_{4} - 10 \beta_{3} - \beta_{2} + \beta_1 + 11) q^{5} + (4 \beta_{2} - 4 \beta_1) q^{6} + (\beta_{7} + \beta_{5} - 29 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 29) q^{7} + (128 \beta_{3} + 128) q^{8} + ( - 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 9 \beta_{2} + \cdots + 576) q^{9}+ \cdots + ( - 381 \beta_{7} - 11226 \beta_{5} - 378708 \beta_{3} + \cdots - 378708) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{2} + 84 q^{5} - 230 q^{7} + 1024 q^{8} + 4620 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{2} + 84 q^{5} - 230 q^{7} + 1024 q^{8} + 4620 q^{9} - 246 q^{11} + 4050 q^{13} + 1840 q^{14} + 13800 q^{15} - 8192 q^{16} - 18480 q^{18} + 6358 q^{19} - 2688 q^{20} - 41052 q^{21} + 1968 q^{22} - 40864 q^{26} - 58428 q^{27} - 7360 q^{28} + 31932 q^{29} + 69374 q^{31} + 32768 q^{32} + 66696 q^{33} - 29856 q^{34} - 58344 q^{35} + 108500 q^{37} - 40560 q^{39} + 21504 q^{40} + 91152 q^{41} + 328416 q^{42} - 7872 q^{44} + 171684 q^{45} + 2016 q^{46} - 609198 q^{47} - 103856 q^{50} + 197312 q^{52} - 309792 q^{53} + 233712 q^{54} + 1126428 q^{55} - 850260 q^{57} - 127728 q^{58} - 277602 q^{59} - 441600 q^{60} - 427032 q^{61} - 1624350 q^{63} + 1166808 q^{65} - 533568 q^{66} + 908626 q^{67} + 238848 q^{68} + 233376 q^{70} - 1084182 q^{71} + 591360 q^{72} - 425948 q^{73} - 868000 q^{74} - 203456 q^{76} + 327600 q^{78} + 770700 q^{79} - 86016 q^{80} + 8105040 q^{81} + 1160634 q^{83} - 1313664 q^{84} + 266916 q^{85} - 1194672 q^{86} - 3116856 q^{87} - 2041320 q^{89} + 2163010 q^{91} - 16128 q^{92} - 526908 q^{93} + 4873584 q^{94} - 2254048 q^{97} - 311840 q^{98} - 3073806 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5226x^{6} + 6977529x^{4} + 486151524x^{2} + 5607014400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 2613\nu^{2} + 74880 ) / 9738 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 5226\nu^{5} + 6902649\nu^{3} + 290490084\nu ) / 729181440 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11 \nu^{7} - 960 \nu^{6} + 62286 \nu^{5} - 4881600 \nu^{4} + 97820019 \nu^{3} - 6226104960 \nu^{2} + 26664257484 \nu - 184093585920 ) / 1626635520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11 \nu^{7} + 960 \nu^{6} + 62286 \nu^{5} + 4881600 \nu^{4} + 97820019 \nu^{3} + 6226104960 \nu^{2} + 26664257484 \nu + 185720221440 ) / 1626635520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1609 \nu^{7} + 3120 \nu^{6} - 8333754 \nu^{5} + 16679520 \nu^{4} - 10941083361 \nu^{3} + 23243753520 \nu^{2} - 544827067236 \nu + 1811751701760 ) / 5286565440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1609 \nu^{7} + 3120 \nu^{6} + 8333754 \nu^{5} + 16679520 \nu^{4} + 10941083361 \nu^{3} + 23243753520 \nu^{2} + 544827067236 \nu + 1811751701760 ) / 5286565440 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{7} + 3\beta_{6} - 3\beta_{5} + 3\beta_{4} - 9\beta_{2} - 1305 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 15\beta_{7} - 15\beta_{6} + 72\beta_{5} + 72\beta_{4} - 7368\beta_{3} - 2517\beta _1 - 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7839\beta_{7} - 7839\beta_{6} + 7839\beta_{5} - 7839\beta_{4} + 33255\beta_{2} + 3335085 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 68409 \beta_{7} + 68409 \beta_{6} - 158922 \beta_{5} - 158922 \beta_{4} + 31931460 \beta_{3} + 6589683 \beta _1 + 158922 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20404737 \beta_{7} + 20404737 \beta_{6} - 19557531 \beta_{5} + 19557531 \beta_{4} - 110731941 \beta_{2} - 8687907153 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 253965699 \beta_{7} - 253965699 \beta_{6} + 333535644 \beta_{5} + 333535644 \beta_{4} - 115285910688 \beta_{3} - 17354205909 \beta _1 - 333535644 \) Copy content Toggle raw display

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
52.6400i
3.81198i
7.64583i
48.8062i
52.6400i
3.81198i
7.64583i
48.8062i
−4.00000 + 4.00000i −52.6400 32.0000i −48.5302 + 48.5302i 210.560 210.560i 3.91130 + 3.91130i 128.000 + 128.000i 2041.97 388.242i
5.2 −4.00000 + 4.00000i −3.81198 32.0000i 65.3747 65.3747i 15.2479 15.2479i 324.917 + 324.917i 128.000 + 128.000i −714.469 522.998i
5.3 −4.00000 + 4.00000i 7.64583 32.0000i −81.7980 + 81.7980i −30.5833 + 30.5833i −62.6828 62.6828i 128.000 + 128.000i −670.541 654.384i
5.4 −4.00000 + 4.00000i 48.8062 32.0000i 106.954 106.954i −195.225 + 195.225i −381.146 381.146i 128.000 + 128.000i 1653.04 855.628i
21.1 −4.00000 4.00000i −52.6400 32.0000i −48.5302 48.5302i 210.560 + 210.560i 3.91130 3.91130i 128.000 128.000i 2041.97 388.242i
21.2 −4.00000 4.00000i −3.81198 32.0000i 65.3747 + 65.3747i 15.2479 + 15.2479i 324.917 324.917i 128.000 128.000i −714.469 522.998i
21.3 −4.00000 4.00000i 7.64583 32.0000i −81.7980 81.7980i −30.5833 30.5833i −62.6828 + 62.6828i 128.000 128.000i −670.541 654.384i
21.4 −4.00000 4.00000i 48.8062 32.0000i 106.954 + 106.954i −195.225 195.225i −381.146 + 381.146i 128.000 128.000i 1653.04 855.628i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.7.d.b 8
3.b odd 2 1 234.7.i.b 8
4.b odd 2 1 208.7.t.b 8
13.d odd 4 1 inner 26.7.d.b 8
39.f even 4 1 234.7.i.b 8
52.f even 4 1 208.7.t.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.7.d.b 8 1.a even 1 1 trivial
26.7.d.b 8 13.d odd 4 1 inner
208.7.t.b 8 4.b odd 2 1
208.7.t.b 8 52.f even 4 1
234.7.i.b 8 3.b odd 2 1
234.7.i.b 8 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2613T_{3}^{2} + 9738T_{3} + 74880 \) acting on \(S_{7}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8 T + 32)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2613 T^{2} + 9738 T + 74880)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 84 T^{7} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + 230 T^{7} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{8} + 246 T^{7} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{8} - 4050 T^{7} + \cdots + 54\!\cdots\!61 \) Copy content Toggle raw display
$17$ \( T^{8} + 121018950 T^{6} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{8} - 6358 T^{7} + \cdots + 52\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{8} + 704959452 T^{6} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{4} - 15966 T^{3} + \cdots - 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 69374 T^{7} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{8} - 108500 T^{7} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{8} - 91152 T^{7} + \cdots + 61\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{8} + 10542744582 T^{6} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{8} + 609198 T^{7} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + 154896 T^{3} + \cdots + 71\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 277602 T^{7} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + 213516 T^{3} + \cdots + 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 908626 T^{7} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{8} + 1084182 T^{7} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} + 425948 T^{7} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} - 385350 T^{3} + \cdots - 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 1160634 T^{7} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + 2041320 T^{7} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + 2254048 T^{7} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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