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Properties

Label 26.5.f.a
Level $26$
Weight $5$
Character orbit 26.f
Analytic conductor $2.688$
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] = [26,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(2.68761904018\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.120336834816.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 46x^{5} + 445x^{4} + 68x^{3} + 32x^{2} + 1136x + 20164 \) 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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{2} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{3}+ \cdots + (9 \beta_{7} - 18 \beta_{6} + \cdots + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{2} + (\beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 1) q^{3}+ \cdots + (780 \beta_{7} + 3102 \beta_{6} + \cdots - 684) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 30 q^{5} + 36 q^{6} - 92 q^{7} - 128 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 30 q^{5} + 36 q^{6} - 92 q^{7} - 128 q^{8} + 90 q^{9} + 132 q^{10} + 192 q^{11} - 78 q^{13} - 344 q^{14} + 132 q^{15} + 256 q^{16} - 870 q^{17} - 360 q^{18} - 218 q^{19} - 144 q^{20} + 1914 q^{21} + 948 q^{22} + 264 q^{23} - 288 q^{24} - 520 q^{26} - 2268 q^{27} - 736 q^{28} - 252 q^{29} + 1728 q^{30} - 3376 q^{31} + 512 q^{32} + 1698 q^{33} + 1080 q^{34} + 6474 q^{35} + 2016 q^{36} + 2552 q^{37} - 5928 q^{39} - 960 q^{40} - 8472 q^{41} - 3828 q^{42} + 7290 q^{43} - 3792 q^{44} - 6282 q^{45} - 540 q^{46} + 4848 q^{47} + 9678 q^{49} + 2524 q^{50} - 832 q^{52} - 6840 q^{53} + 2268 q^{54} + 294 q^{55} + 4320 q^{56} - 4434 q^{57} - 7008 q^{58} - 4632 q^{59} + 9024 q^{60} + 8670 q^{61} + 3936 q^{62} + 8100 q^{63} - 2808 q^{65} - 984 q^{66} - 12446 q^{67} - 2160 q^{68} + 19206 q^{69} - 25896 q^{70} - 4554 q^{71} - 5472 q^{72} + 19294 q^{73} + 10112 q^{74} - 25992 q^{75} + 1744 q^{76} - 8580 q^{78} - 11328 q^{79} - 1152 q^{80} + 6048 q^{81} + 42672 q^{82} + 10344 q^{83} - 8592 q^{84} - 23178 q^{85} + 13536 q^{86} + 18372 q^{87} + 1440 q^{88} - 2538 q^{89} + 1456 q^{91} + 96 q^{92} - 162 q^{93} - 9696 q^{94} + 26568 q^{95} - 64282 q^{97} - 31676 q^{98} + 9504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 46x^{5} + 445x^{4} + 68x^{3} + 32x^{2} + 1136x + 20164 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 201 \nu^{7} + 130723 \nu^{6} - 912832 \nu^{5} + 2468434 \nu^{4} + 4181025 \nu^{3} + \cdots + 184831318 ) / 45882928 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 58751 \nu^{7} + 9302346 \nu^{6} - 31562976 \nu^{5} + 24393326 \nu^{4} + 823240165 \nu^{3} + \cdots + 3841960804 ) / 6515375776 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6832 \nu^{7} + 118318 \nu^{6} - 208275 \nu^{5} - 112561 \nu^{4} + 1229913 \nu^{3} + \cdots - 4888776 ) / 407210986 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1192317 \nu^{7} + 681628 \nu^{6} + 19291808 \nu^{5} - 187696102 \nu^{4} - 248385885 \nu^{3} + \cdots - 18281959832 ) / 6515375776 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -55\nu^{7} + 394\nu^{6} - 1104\nu^{5} - 1394\nu^{4} - 5731\nu^{3} + 57604\nu^{2} - 72618\nu + 51972 ) / 206752 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10142009 \nu^{7} + 32005550 \nu^{6} - 16506960 \nu^{5} - 778375774 \nu^{4} + \cdots - 46715131316 ) / 6515375776 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 6\beta_{6} - \beta_{5} + 13\beta_{4} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{7} - 31\beta_{6} - 24\beta_{5} + 3\beta_{4} + 15\beta_{3} + 4\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 35\beta_{7} - 42\beta_{6} - 229\beta_{5} - 97\beta_{4} + 42\beta_{3} - 35\beta_{2} - 7\beta _1 - 266 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 463\beta_{6} - 602\beta_{5} - 777\beta_{4} + 139\beta_{3} - 278\beta_{2} - 308\beta _1 - 777 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1049\beta_{7} + 5776\beta_{6} + 1049\beta_{5} - 5255\beta_{4} - 314\beta_{3} - 1049\beta_{2} - 1363\beta _1 - 1682 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8188\beta_{7} + 26335\beta_{6} + 26856\beta_{5} - 6287\beta_{4} - 7667\beta_{3} - 4094\beta _1 + 14475 \) 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
3.77568 + 3.77568i
−2.40965 − 2.40965i
1.80078 + 1.80078i
−2.16680 − 2.16680i
3.77568 − 3.77568i
−2.40965 + 2.40965i
1.80078 − 1.80078i
−2.16680 + 2.16680i
−2.73205 + 0.732051i −3.25971 + 5.64598i 6.92820 − 4.00000i −20.3978 − 20.3978i 4.77255 − 17.8114i −81.4882 − 21.8347i −16.0000 + 16.0000i 19.2486 + 33.3395i 70.6601 + 40.7956i
7.2 −2.73205 + 0.732051i 0.661634 − 1.14598i 6.92820 − 4.00000i 18.3715 + 18.3715i −0.968699 + 3.61523i 57.6222 + 15.4398i −16.0000 + 16.0000i 39.6245 + 68.6316i −63.6409 − 36.7431i
11.1 0.732051 − 2.73205i −3.39466 − 5.87972i −6.92820 − 4.00000i −10.7932 − 10.7932i −18.5487 + 4.97012i −4.14281 − 15.4612i −16.0000 + 16.0000i 17.4526 − 30.2288i −37.3887 + 21.5864i
11.2 0.732051 − 2.73205i 5.99273 + 10.3797i −6.92820 − 4.00000i 27.8195 + 27.8195i 32.7449 − 8.77397i −17.9912 − 67.1439i −16.0000 + 16.0000i −31.3257 + 54.2577i 96.3695 − 55.6389i
15.1 −2.73205 − 0.732051i −3.25971 − 5.64598i 6.92820 + 4.00000i −20.3978 + 20.3978i 4.77255 + 17.8114i −81.4882 + 21.8347i −16.0000 − 16.0000i 19.2486 − 33.3395i 70.6601 − 40.7956i
15.2 −2.73205 − 0.732051i 0.661634 + 1.14598i 6.92820 + 4.00000i 18.3715 − 18.3715i −0.968699 − 3.61523i 57.6222 − 15.4398i −16.0000 − 16.0000i 39.6245 − 68.6316i −63.6409 + 36.7431i
19.1 0.732051 + 2.73205i −3.39466 + 5.87972i −6.92820 + 4.00000i −10.7932 + 10.7932i −18.5487 − 4.97012i −4.14281 + 15.4612i −16.0000 − 16.0000i 17.4526 + 30.2288i −37.3887 − 21.5864i
19.2 0.732051 + 2.73205i 5.99273 − 10.3797i −6.92820 + 4.00000i 27.8195 − 27.8195i 32.7449 + 8.77397i −17.9912 + 67.1439i −16.0000 − 16.0000i −31.3257 − 54.2577i 96.3695 + 55.6389i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
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 26.5.f.a ✓ 8
3.b odd 2 1 234.5.bb.c 8
13.c even 3 1 338.5.d.h 8
13.e even 6 1 338.5.d.e 8
13.f odd 12 1 inner 26.5.f.a ✓ 8
13.f odd 12 1 338.5.d.e 8
13.f odd 12 1 338.5.d.h 8
39.k even 12 1 234.5.bb.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.5.f.a ✓ 8 1.a even 1 1 trivial
26.5.f.a ✓ 8 13.f odd 12 1 inner
234.5.bb.c 8 3.b odd 2 1
234.5.bb.c 8 39.k even 12 1
338.5.d.e 8 13.e even 6 1
338.5.d.e 8 13.f odd 12 1
338.5.d.h 8 13.c even 3 1
338.5.d.h 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 117T_{3}^{6} + 756T_{3}^{5} + 12987T_{3}^{4} + 44226T_{3}^{3} + 225018T_{3}^{2} - 265356T_{3} + 492804 \) acting on \(S_{5}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 117 T^{6} + \cdots + 492804 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 202570206084 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 31355833738384 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 71\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 1628446523236 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 78\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 39\!\cdots\!89 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 26\!\cdots\!89 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 30\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 117360089452212)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 86\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 285095913720288)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
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