Properties

Label 26.7.f.a
Level $26$
Weight $7$
Character orbit 26.f
Analytic conductor $5.981$
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] = [26,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.7");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(5.98140617412\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 5712 x^{10} + 12204858 x^{8} + 11783014920 x^{6} + 4783763172609 x^{4} + 532462287988296 x^{2} + 172998936244836 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \beta_{3} + 4 \beta_{2} - 4) q^{2} + ( - \beta_{11} - \beta_{7}) q^{3} + 32 \beta_{4} q^{4} + (\beta_{11} + \beta_{8} - \beta_{7} + \beta_{5} + 5 \beta_{4} - 30 \beta_{3} + 34 \beta_{2} - \beta_1 - 4) q^{5} + (4 \beta_{8} + 4 \beta_{7} + 4 \beta_1) q^{6} + ( - 5 \beta_{11} + 4 \beta_{9} - 4 \beta_{8} - \beta_{7} - 99 \beta_{4} + 36 \beta_{3} + \cdots - 40) q^{7}+ \cdots + ( - 7 \beta_{11} + 4 \beta_{10} + 7 \beta_{9} + 6 \beta_{8} + 3 \beta_{6} - 7 \beta_{5} + \cdots - 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \beta_{3} + 4 \beta_{2} - 4) q^{2} + ( - \beta_{11} - \beta_{7}) q^{3} + 32 \beta_{4} q^{4} + (\beta_{11} + \beta_{8} - \beta_{7} + \beta_{5} + 5 \beta_{4} - 30 \beta_{3} + 34 \beta_{2} - \beta_1 - 4) q^{5} + (4 \beta_{8} + 4 \beta_{7} + 4 \beta_1) q^{6} + ( - 5 \beta_{11} + 4 \beta_{9} - 4 \beta_{8} - \beta_{7} - 99 \beta_{4} + 36 \beta_{3} + \cdots - 40) q^{7}+ \cdots + (13516 \beta_{11} + 1890 \beta_{10} + 4538 \beta_{8} + \cdots - 469848) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{2} + 150 q^{5} - 1026 q^{7} - 1536 q^{8} - 1338 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{2} + 150 q^{5} - 1026 q^{7} - 1536 q^{8} - 1338 q^{9} - 2472 q^{10} - 414 q^{11} - 2586 q^{13} + 11040 q^{14} - 19494 q^{15} + 6144 q^{16} - 11592 q^{17} + 10704 q^{18} + 2100 q^{19} + 12288 q^{20} + 23076 q^{21} - 5376 q^{22} + 23124 q^{23} - 11064 q^{26} + 83844 q^{27} - 32832 q^{28} + 8118 q^{29} + 101760 q^{30} + 32496 q^{31} + 24576 q^{32} - 139596 q^{33} - 55344 q^{34} - 223188 q^{35} - 27264 q^{36} - 36816 q^{37} - 53364 q^{39} - 38400 q^{40} + 186168 q^{41} - 92304 q^{42} + 64290 q^{43} + 43008 q^{44} - 46614 q^{45} - 106416 q^{46} - 211932 q^{47} + 48744 q^{49} - 273312 q^{50} - 54144 q^{52} + 826032 q^{53} + 159552 q^{54} + 120276 q^{55} + 86016 q^{56} + 720600 q^{57} - 61176 q^{58} - 140202 q^{59} - 433536 q^{60} - 511572 q^{61} - 648528 q^{62} - 378180 q^{63} + 1436640 q^{65} + 1161984 q^{66} + 1837104 q^{67} + 221376 q^{68} - 1103952 q^{69} + 1785504 q^{70} - 173352 q^{71} + 280320 q^{72} - 1877034 q^{73} - 1089480 q^{74} - 4264686 q^{75} - 67200 q^{76} + 1108080 q^{78} - 794880 q^{79} + 393216 q^{80} + 1086270 q^{81} - 2141544 q^{82} + 2905464 q^{83} + 368256 q^{84} - 855852 q^{85} - 944400 q^{86} - 3153372 q^{87} - 66048 q^{88} - 2794182 q^{89} + 2486574 q^{91} + 222720 q^{92} + 3523116 q^{93} + 847728 q^{94} - 3175842 q^{95} + 4229166 q^{97} - 659496 q^{98} - 3795372 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 5712 x^{10} + 12204858 x^{8} + 11783014920 x^{6} + 4783763172609 x^{4} + 532462287988296 x^{2} + 172998936244836 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5041 \nu^{11} + 8137167 \nu^{9} - 24715335957 \nu^{7} - 60154208982819 \nu^{5} + \cdots + 15\!\cdots\!96 ) / 30\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 473951376997 \nu^{11} - 16377229949334 \nu^{10} + \cdots - 25\!\cdots\!48 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 473951376997 \nu^{11} + 16377229949334 \nu^{10} + \cdots + 25\!\cdots\!48 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 122263262666039 \nu^{11} + 724355636007390 \nu^{10} + \cdots - 18\!\cdots\!00 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20456202340506 \nu^{11} + 123455477659454 \nu^{10} + \cdots - 30\!\cdots\!72 ) / 48\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5459076649778 \nu^{11} + 509983434063397 \nu^{10} + \cdots + 27\!\cdots\!64 ) / 96\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2295225 \nu^{10} - 9582225015 \nu^{8} - 13283598577171 \nu^{6} + \cdots - 96\!\cdots\!64 ) / 34\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18710298795073 \nu^{11} + \cdots + 25\!\cdots\!20 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1519695618173 \nu^{11} + 169777730744787 \nu^{10} + \cdots - 20\!\cdots\!96 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5424241 \nu^{10} - 22485891789 \nu^{8} - 30976701041091 \nu^{6} + \cdots - 29\!\cdots\!92 ) / 51\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 14 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 12 \beta_{8} - 4 \beta_{6} - 4 \beta_{5} - 47 \beta_{4} + 48 \beta_{3} + 6 \beta _1 - 952 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -71\beta_{11} - 142\beta_{7} - 6987\beta_{4} - 6987\beta_{3} - 9090\beta_{2} - 1428\beta _1 + 4545 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 26482 \beta_{11} + 5065 \beta_{10} - 5065 \beta_{9} - 24664 \beta_{8} + 5641 \beta_{6} + 5641 \beta_{5} + 134992 \beta_{4} - 135568 \beta_{3} - 12332 \beta _1 + 1369538 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 139893 \beta_{11} - 29034 \beta_{10} - 29034 \beta_{9} + 279786 \beta_{7} + 3459 \beta_{6} - 3459 \beta_{5} + 15041160 \beta_{4} + 15015585 \beta_{3} + 19963560 \beta_{2} + 2154906 \beta _1 - 9956205 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 45359568 \beta_{11} - 8393457 \beta_{10} + 8393457 \beta_{9} + 44167344 \beta_{8} - 8014452 \beta_{6} - 8014452 \beta_{5} - 226894455 \beta_{4} + 226515450 \beta_{3} + \cdots - 2095283700 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 209739777 \beta_{11} + 77815920 \beta_{10} + 77815920 \beta_{9} - 419479554 \beta_{7} - 6247254 \beta_{6} + 6247254 \beta_{5} - 27111617247 \beta_{4} + \cdots + 18384643719 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75387853074 \beta_{11} + 13353581829 \beta_{10} - 13353581829 \beta_{9} - 76388569464 \beta_{8} + 11978215617 \beta_{6} + 11978215617 \beta_{5} + \cdots + 3281498434230 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 271749750993 \beta_{11} - 157063640058 \beta_{10} - 157063640058 \beta_{9} + 543499501986 \beta_{7} + 5759555139 \beta_{6} - 5759555139 \beta_{5} + \cdots - 32206591303713 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 123879710762640 \beta_{11} - 20763765050985 \beta_{10} + 20763765050985 \beta_{9} + 130259389293504 \beta_{8} - 18525632781096 \beta_{6} + \cdots - 51\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 300277409195673 \beta_{11} + 288590035861080 \beta_{10} + 288590035861080 \beta_{9} - 600554818391346 \beta_{7} + 1349725340430 \beta_{6} + \cdots + 55\!\cdots\!43 \) Copy content Toggle raw display

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(\beta_{4}\)

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
39.0939i
0.570839i
39.6647i
40.8506i
13.1145i
27.7361i
39.0939i
0.570839i
39.6647i
40.8506i
13.1145i
27.7361i
−5.46410 + 1.46410i −19.5470 + 33.8563i 27.7128 16.0000i 163.787 + 163.787i 57.2374 213.613i −593.183 158.943i −128.000 + 128.000i −399.667 692.243i −1134.75 655.146i
7.2 −5.46410 + 1.46410i −0.285419 + 0.494361i 27.7128 16.0000i −20.2585 20.2585i 0.835766 3.11912i 144.179 + 38.6327i −128.000 + 128.000i 364.337 + 631.050i 140.355 + 81.0338i
7.3 −5.46410 + 1.46410i 19.8324 34.3507i 27.7128 16.0000i −16.8275 16.8275i −58.0732 + 216.732i −154.773 41.4712i −128.000 + 128.000i −422.146 731.178i 116.584 + 67.3098i
11.1 1.46410 5.46410i −20.4253 35.3776i −27.7128 16.0000i −4.28106 4.28106i −223.212 + 59.8094i 50.6889 + 189.174i −128.000 + 128.000i −469.884 + 813.864i −29.6601 + 17.1243i
11.2 1.46410 5.46410i 6.55725 + 11.3575i −27.7128 16.0000i −97.4789 97.4789i 71.6589 19.2009i −87.6994 327.299i −128.000 + 128.000i 278.505 482.385i −675.354 + 389.916i
11.3 1.46410 5.46410i 13.8680 + 24.0201i −27.7128 16.0000i 50.0594 + 50.0594i 151.553 40.6084i 127.787 + 476.906i −128.000 + 128.000i −20.1450 + 34.8922i 346.821 200.237i
15.1 −5.46410 1.46410i −19.5470 33.8563i 27.7128 + 16.0000i 163.787 163.787i 57.2374 + 213.613i −593.183 + 158.943i −128.000 128.000i −399.667 + 692.243i −1134.75 + 655.146i
15.2 −5.46410 1.46410i −0.285419 0.494361i 27.7128 + 16.0000i −20.2585 + 20.2585i 0.835766 + 3.11912i 144.179 38.6327i −128.000 128.000i 364.337 631.050i 140.355 81.0338i
15.3 −5.46410 1.46410i 19.8324 + 34.3507i 27.7128 + 16.0000i −16.8275 + 16.8275i −58.0732 216.732i −154.773 + 41.4712i −128.000 128.000i −422.146 + 731.178i 116.584 67.3098i
19.1 1.46410 + 5.46410i −20.4253 + 35.3776i −27.7128 + 16.0000i −4.28106 + 4.28106i −223.212 59.8094i 50.6889 189.174i −128.000 128.000i −469.884 813.864i −29.6601 17.1243i
19.2 1.46410 + 5.46410i 6.55725 11.3575i −27.7128 + 16.0000i −97.4789 + 97.4789i 71.6589 + 19.2009i −87.6994 + 327.299i −128.000 128.000i 278.505 + 482.385i −675.354 389.916i
19.3 1.46410 + 5.46410i 13.8680 24.0201i −27.7128 + 16.0000i 50.0594 50.0594i 151.553 + 40.6084i 127.787 476.906i −128.000 128.000i −20.1450 34.8922i 346.821 + 200.237i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
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 26.7.f.a 12
13.f odd 12 1 inner 26.7.f.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.7.f.a 12 1.a even 1 1 trivial
26.7.f.a 12 13.f odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 2856 T_{3}^{10} - 27948 T_{3}^{9} + 6132675 T_{3}^{8} - 61800786 T_{3}^{7} + 5949685080 T_{3}^{6} - 96757403490 T_{3}^{5} + 4365163653093 T_{3}^{4} + \cdots + 172998936244836 \) acting on \(S_{7}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 8 T^{3} + 32 T^{2} + 256 T + 1024)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 172998936244836 \) Copy content Toggle raw display
$5$ \( T^{12} - 150 T^{11} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + 1026 T^{11} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{12} + 414 T^{11} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{12} + 2586 T^{11} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{12} + 11592 T^{11} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{12} - 2100 T^{11} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{12} - 23124 T^{11} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{12} - 8118 T^{11} + \cdots + 28\!\cdots\!29 \) Copy content Toggle raw display
$31$ \( T^{12} - 32496 T^{11} + \cdots + 75\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{12} + 36816 T^{11} + \cdots + 19\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{12} - 186168 T^{11} + \cdots + 55\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} - 64290 T^{11} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + 211932 T^{11} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{6} - 413016 T^{5} + \cdots - 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 140202 T^{11} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{12} + 511572 T^{11} + \cdots + 28\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{12} - 1837104 T^{11} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{12} + 173352 T^{11} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{12} + 1877034 T^{11} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + 397440 T^{5} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 2905464 T^{11} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + 2794182 T^{11} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{12} - 4229166 T^{11} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
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