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Properties

Label 26.6.e.a
Level $26$
Weight $6$
Character orbit 26.e
Analytic conductor $4.170$
Analytic rank $0$
Dimension $12$
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] = [26,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(4.16997931514\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2164x^{10} + 1693662x^{8} + 565924828x^{6} + 68470588153x^{4} + 28868949144x^{2} + 2602224144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{3} q^{3} + 16 \beta_{7} q^{4} + ( - \beta_{11} - 10 \beta_{7} + \cdots + 5) q^{5}+ \cdots + ( - 6 \beta_{11} - 3 \beta_{10} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{3} q^{3} + 16 \beta_{7} q^{4} + ( - \beta_{11} - 10 \beta_{7} + \cdots + 5) q^{5}+ \cdots + (540 \beta_{11} + 81 \beta_{9} + \cdots - 23613) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 96 q^{4} + 360 q^{7} - 706 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 96 q^{4} + 360 q^{7} - 706 q^{9} - 368 q^{10} + 504 q^{11} + 3202 q^{13} - 2272 q^{14} + 2112 q^{15} - 1536 q^{16} - 1910 q^{17} - 6888 q^{19} + 1440 q^{20} + 2160 q^{22} + 5920 q^{23} - 4672 q^{25} - 2320 q^{26} - 2736 q^{27} + 5760 q^{28} + 7922 q^{29} - 11696 q^{30} - 18780 q^{33} + 16864 q^{35} + 11296 q^{36} - 882 q^{37} + 26496 q^{38} + 48304 q^{39} - 11776 q^{40} + 44346 q^{41} + 23648 q^{42} - 41920 q^{43} - 158850 q^{45} - 33408 q^{46} + 4682 q^{49} + 11040 q^{50} + 133264 q^{51} + 43520 q^{52} - 187812 q^{53} + 1296 q^{54} + 17472 q^{55} - 18176 q^{56} - 73488 q^{58} + 74832 q^{59} + 27934 q^{61} - 4336 q^{62} + 156528 q^{63} - 49152 q^{64} + 69646 q^{65} + 6336 q^{66} + 85104 q^{67} + 30560 q^{68} - 77348 q^{69} - 106320 q^{71} - 78336 q^{72} + 96208 q^{74} + 192080 q^{75} - 110208 q^{76} - 151944 q^{77} + 9392 q^{78} + 30464 q^{79} + 23040 q^{80} - 72310 q^{81} - 87040 q^{82} + 53952 q^{84} - 148986 q^{85} - 121936 q^{87} - 34560 q^{88} + 93108 q^{89} + 282016 q^{90} - 56776 q^{91} + 189440 q^{92} + 107616 q^{93} + 158256 q^{94} + 232560 q^{95} + 147156 q^{97} + 46752 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2164x^{10} + 1693662x^{8} + 565924828x^{6} + 68470588153x^{4} + 28868949144x^{2} + 2602224144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1809429 \nu^{11} + 4295018 \nu^{10} - 3912590728 \nu^{9} + 11212860152 \nu^{8} + \cdots + 13\!\cdots\!68 ) / 21\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3409520 \nu^{11} - 1506814 \nu^{10} + 7374789907 \nu^{9} - 2481960576 \nu^{8} + \cdots - 23\!\cdots\!88 ) / 21\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3409520 \nu^{11} + 1506814 \nu^{10} + 7374789907 \nu^{9} + 2481960576 \nu^{8} + \cdots + 23\!\cdots\!88 ) / 21\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1809429 \nu^{11} + 6819040 \nu^{10} - 3912590728 \nu^{9} + 14749579814 \nu^{8} + \cdots + 98\!\cdots\!00 ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7028378 \nu^{11} - 1506814 \nu^{10} - 15199971363 \nu^{9} - 2481960576 \nu^{8} + \cdots - 23\!\cdots\!88 ) / 21\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 867 \nu^{11} - 1874744 \nu^{9} - 1466026458 \nu^{7} - 489440484828 \nu^{5} + \cdots - 36479917489860 \nu ) / 2597481252288 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 289 \nu^{11} - 625387 \nu^{9} - 489442083 \nu^{7} - 163517329849 \nu^{5} + \cdots + 398593892736 ) / 797187785472 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5179610743 \nu^{11} + 9907217520 \nu^{10} - 11218093837156 \nu^{9} + \cdots + 13\!\cdots\!84 ) / 74\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5179610743 \nu^{11} - 9907217520 \nu^{10} - 11218093837156 \nu^{9} + \cdots - 13\!\cdots\!84 ) / 74\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21895167121 \nu^{11} + 1452556104 \nu^{10} - 47374793364289 \nu^{9} + \cdots - 72\!\cdots\!68 ) / 14\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21895167121 \nu^{11} + 47374793364289 \nu^{9} + \cdots + 33\!\cdots\!20 \nu ) / 74\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + 2\beta_{5} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 3 \beta_{11} - 6 \beta_{10} - \beta_{9} + \beta_{8} + 17 \beta_{6} - \beta_{5} - 32 \beta_{4} + \cdots - 361 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -144\beta_{7} + 313\beta_{6} - 1082\beta_{5} - 1490\beta_{3} - 408\beta_{2} + 72 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1827 \beta_{11} + 3654 \beta_{10} + 1000 \beta_{9} - 1000 \beta_{8} - 18551 \beta_{6} + 532 \beta_{5} + \cdots + 216262 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2472 \beta_{11} - 1440 \beta_{9} - 1440 \beta_{8} + 440880 \beta_{7} - 110569 \beta_{6} + 647570 \beta_{5} + \cdots - 220440 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1192407 \beta_{11} - 2384814 \beta_{10} - 820531 \beta_{9} + 820531 \beta_{8} + 15617633 \beta_{6} + \cdots - 139658131 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4437264 \beta_{11} + 1810224 \beta_{9} + 1810224 \beta_{8} - 579744048 \beta_{7} + 24821785 \beta_{6} + \cdots + 289872024 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 812655795 \beta_{11} + 1625311590 \beta_{10} + 626435434 \beta_{9} - 626435434 \beta_{8} + \cdots + 94590116020 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5262625368 \beta_{11} - 1808447472 \beta_{9} - 1808447472 \beta_{8} + 599010370128 \beta_{7} + \cdots - 299505185064 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 567665946855 \beta_{11} - 1135331893710 \beta_{10} - 463388806477 \beta_{9} + 463388806477 \beta_{8} + \cdots - 65853562042873 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5272002060288 \beta_{11} + 1662438369888 \beta_{9} + 1662438369888 \beta_{8} - 554014945414032 \beta_{7} + \cdots + 277007472707016 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(\beta_{7}\)

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
− 26.6069i
− 0.361203i
26.9681i
18.8114i
0.540666i
− 19.3520i
26.6069i
0.361203i
− 26.9681i
− 18.8114i
− 0.540666i
19.3520i
−3.46410 − 2.00000i −13.3035 + 23.0423i 8.00000 + 13.8564i − 96.2889i 92.1691 − 53.2138i 44.1160 − 25.4704i − 64.0000i −232.464 − 402.640i −192.578 + 333.554i
17.2 −3.46410 − 2.00000i −0.180601 + 0.312811i 8.00000 + 13.8564i 41.9261i 1.25124 − 0.722405i 186.503 − 107.678i − 64.0000i 121.435 + 210.331i 83.8523 − 145.236i
17.3 −3.46410 − 2.00000i 13.4841 − 23.3551i 8.00000 + 13.8564i − 17.6180i −93.4203 + 53.9362i −17.6436 + 10.1865i − 64.0000i −242.140 − 419.398i −35.2360 + 61.0306i
17.4 3.46410 + 2.00000i −9.40569 + 16.2911i 8.00000 + 13.8564i − 10.8158i −65.1645 + 37.6228i −167.247 + 96.5603i 64.0000i −55.4339 − 96.0144i 21.6316 − 37.4670i
17.5 3.46410 + 2.00000i −0.270333 + 0.468230i 8.00000 + 13.8564i 83.0706i −1.87292 + 1.08133i 117.712 − 67.9611i 64.0000i 121.354 + 210.191i −166.141 + 287.765i
17.6 3.46410 + 2.00000i 9.67602 − 16.7594i 8.00000 + 13.8564i − 52.2356i 67.0374 − 38.7041i 16.5596 − 9.56069i 64.0000i −65.7507 − 113.884i 104.471 − 180.949i
23.1 −3.46410 + 2.00000i −13.3035 − 23.0423i 8.00000 − 13.8564i 96.2889i 92.1691 + 53.2138i 44.1160 + 25.4704i 64.0000i −232.464 + 402.640i −192.578 − 333.554i
23.2 −3.46410 + 2.00000i −0.180601 − 0.312811i 8.00000 − 13.8564i − 41.9261i 1.25124 + 0.722405i 186.503 + 107.678i 64.0000i 121.435 − 210.331i 83.8523 + 145.236i
23.3 −3.46410 + 2.00000i 13.4841 + 23.3551i 8.00000 − 13.8564i 17.6180i −93.4203 − 53.9362i −17.6436 − 10.1865i 64.0000i −242.140 + 419.398i −35.2360 − 61.0306i
23.4 3.46410 − 2.00000i −9.40569 − 16.2911i 8.00000 − 13.8564i 10.8158i −65.1645 − 37.6228i −167.247 − 96.5603i − 64.0000i −55.4339 + 96.0144i 21.6316 + 37.4670i
23.5 3.46410 − 2.00000i −0.270333 − 0.468230i 8.00000 − 13.8564i − 83.0706i −1.87292 − 1.08133i 117.712 + 67.9611i − 64.0000i 121.354 − 210.191i −166.141 − 287.765i
23.6 3.46410 − 2.00000i 9.67602 + 16.7594i 8.00000 − 13.8564i 52.2356i 67.0374 + 38.7041i 16.5596 + 9.56069i − 64.0000i −65.7507 + 113.884i 104.471 + 180.949i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
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 26.6.e.a ✓ 12
3.b odd 2 1 234.6.l.c 12
4.b odd 2 1 208.6.w.c 12
13.c even 3 1 338.6.b.g 12
13.e even 6 1 inner 26.6.e.a ✓ 12
13.e even 6 1 338.6.b.g 12
13.f odd 12 1 338.6.a.m 6
13.f odd 12 1 338.6.a.o 6
39.h odd 6 1 234.6.l.c 12
52.i odd 6 1 208.6.w.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.6.e.a ✓ 12 1.a even 1 1 trivial
26.6.e.a ✓ 12 13.e even 6 1 inner
208.6.w.c 12 4.b odd 2 1
208.6.w.c 12 52.i odd 6 1
234.6.l.c 12 3.b odd 2 1
234.6.l.c 12 39.h odd 6 1
338.6.a.m 6 13.f odd 12 1
338.6.a.o 6 13.f odd 12 1
338.6.b.g 12 13.c even 3 1
338.6.b.g 12 13.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(26, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 16 T^{2} + 256)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 2602224144 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 26\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 70\!\cdots\!49 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 51\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 16\!\cdots\!29 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 45\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 12\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 40\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
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