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Properties

Label 26.8.e.a
Level $26$
Weight $8$
Character orbit 26.e
Analytic conductor $8.122$
Analytic rank $0$
Dimension $16$
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] = [26,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.17");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(8.12201066259\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 22180 x^{14} + 184473654 x^{12} + 707524481236 x^{10} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{34}\cdot 13^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{9} - \beta_{7}) q^{2} + \beta_{3} q^{3} + 64 \beta_1 q^{4} + (5 \beta_{9} - \beta_{8} - \beta_{3} + \cdots - 10) q^{5}+ \cdots + ( - \beta_{14} + 2 \beta_{12} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} - \beta_{7}) q^{2} + \beta_{3} q^{3} + 64 \beta_1 q^{4} + (5 \beta_{9} - \beta_{8} - \beta_{3} + \cdots - 10) q^{5}+ \cdots + (1371 \beta_{15} + 1371 \beta_{14} + \cdots - 3324159) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 512 q^{4} + 2520 q^{7} - 4684 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 512 q^{4} + 2520 q^{7} - 4684 q^{9} + 2432 q^{10} - 8496 q^{11} - 3620 q^{13} + 34560 q^{14} + 51648 q^{15} - 32768 q^{16} + 41520 q^{17} - 54432 q^{19} - 16128 q^{20} - 17280 q^{22} - 7560 q^{23} - 273960 q^{25} + 51072 q^{26} + 133920 q^{27} + 161280 q^{28} - 346056 q^{29} + 170368 q^{30} + 486300 q^{33} - 283248 q^{35} + 299776 q^{36} - 68280 q^{37} - 1059840 q^{38} - 1122680 q^{39} + 311296 q^{40} - 1288008 q^{41} + 408320 q^{42} + 285360 q^{43} + 5743980 q^{45} - 188928 q^{46} - 2131660 q^{49} + 85248 q^{50} + 957280 q^{51} + 354560 q^{52} + 3666600 q^{53} - 4074624 q^{54} - 1091472 q^{55} + 1105920 q^{56} + 3504000 q^{58} + 1799280 q^{59} - 4626000 q^{61} - 2770560 q^{62} - 8609760 q^{63} - 4194304 q^{64} - 11518020 q^{65} + 7941120 q^{66} - 7094400 q^{67} - 2657280 q^{68} + 18027316 q^{69} + 25958520 q^{71} + 4423680 q^{72} - 1142400 q^{74} - 10898032 q^{75} - 3483648 q^{76} - 20577480 q^{77} - 22023040 q^{78} + 34143488 q^{79} - 1032192 q^{80} - 15745864 q^{81} + 12202240 q^{82} + 17311488 q^{84} + 7123932 q^{85} - 10460440 q^{87} + 1105920 q^{88} + 10609452 q^{89} + 4070656 q^{90} - 38683952 q^{91} - 967680 q^{92} - 52747080 q^{93} - 14303616 q^{94} + 33837504 q^{95} + 83706300 q^{97} + 19031040 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 22180 x^{14} + 184473654 x^{12} + 707524481236 x^{10} + \cdots + 18\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 55444640623573 \nu^{15} + \cdots + 19\!\cdots\!56 ) / 39\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 59\!\cdots\!34 \nu^{15} + \cdots - 36\!\cdots\!24 ) / 26\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!34 \nu^{15} + \cdots + 36\!\cdots\!24 ) / 26\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 98\!\cdots\!94 \nu^{15} + \cdots - 76\!\cdots\!88 ) / 44\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 177385634706611 \nu^{15} + \cdots + 25\!\cdots\!00 \nu ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!97 \nu^{15} + \cdots - 75\!\cdots\!16 ) / 28\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!37 \nu^{15} + \cdots - 17\!\cdots\!40 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 24\!\cdots\!99 \nu^{15} + \cdots + 14\!\cdots\!40 ) / 28\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14312481878129 \nu^{15} + \cdots - 60\!\cdots\!20 \nu ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!77 \nu^{15} + \cdots + 27\!\cdots\!40 ) / 42\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 30\!\cdots\!87 \nu^{15} + \cdots - 31\!\cdots\!32 ) / 84\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 60\!\cdots\!93 \nu^{15} + \cdots - 42\!\cdots\!20 ) / 84\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!49 \nu^{15} + \cdots - 13\!\cdots\!84 ) / 84\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 24\!\cdots\!79 \nu^{15} + \cdots - 13\!\cdots\!84 ) / 84\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 24\!\cdots\!79 \nu^{15} + \cdots + 13\!\cdots\!84 ) / 84\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} - \beta_{12} + 2 \beta_{10} + 45 \beta_{9} + 2 \beta_{8} + 90 \beta_{7} + \cdots - 2776 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{15} + 6 \beta_{14} - 24 \beta_{13} + 600 \beta_{12} + 12 \beta_{11} + 12 \beta_{10} + \cdots + 170106 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7435 \beta_{15} + 7435 \beta_{14} + 4645 \beta_{12} - 270 \beta_{11} - 9560 \beta_{10} + \cdots + 15395674 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 149538 \beta_{15} + 149538 \beta_{14} - 27480 \beta_{13} - 4120440 \beta_{12} + 13740 \beta_{11} + \cdots - 1490551746 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 53388163 \beta_{15} - 53388163 \beta_{14} - 23453353 \beta_{12} + 3402972 \beta_{11} + \cdots - 95021531338 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2584157826 \beta_{15} - 2584157826 \beta_{14} + 854970024 \beta_{13} + 27401044920 \beta_{12} + \cdots + 11191787397186 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 375438752443 \beta_{15} + 375438752443 \beta_{14} + 127115388973 \beta_{12} - 27662390730 \beta_{11} + \cdots + 613209725436874 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 28843757071266 \beta_{15} + 28843757071266 \beta_{14} - 6788061036408 \beta_{13} - 187600864333560 \beta_{12} + \cdots - 82\!\cdots\!26 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26\!\cdots\!83 \beta_{15} + \cdots - 40\!\cdots\!70 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27\!\cdots\!38 \beta_{15} + \cdots + 61\!\cdots\!30 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18\!\cdots\!43 \beta_{15} + \cdots + 27\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 23\!\cdots\!38 \beta_{15} + \cdots - 46\!\cdots\!10 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12\!\cdots\!15 \beta_{15} + \cdots - 18\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 19\!\cdots\!26 \beta_{15} + \cdots + 34\!\cdots\!42 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\)
\(\chi(n)\) \(\beta_{1}\)

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
69.7232i
12.8548i
− 32.3527i
− 50.2253i
− 79.3738i
− 7.49290i
1.87085i
84.9959i
− 69.7232i
− 12.8548i
32.3527i
50.2253i
79.3738i
7.49290i
− 1.87085i
− 84.9959i
−6.92820 − 4.00000i −34.8616 + 60.3821i 32.0000 + 55.4256i 226.832i 483.056 − 278.893i −154.640 + 89.2812i − 512.000i −1337.16 − 2316.03i 907.328 − 1571.54i
17.2 −6.92820 − 4.00000i −6.42739 + 11.1326i 32.0000 + 55.4256i − 260.039i 89.0605 − 51.4191i −850.835 + 491.230i − 512.000i 1010.88 + 1750.89i −1040.15 + 1801.60i
17.3 −6.92820 − 4.00000i 16.1763 − 28.0182i 32.0000 + 55.4256i − 211.408i −224.146 + 129.411i 635.796 − 367.077i − 512.000i 570.153 + 987.534i −845.633 + 1464.68i
17.4 −6.92820 − 4.00000i 25.1127 − 43.4964i 32.0000 + 55.4256i 469.361i −347.971 + 200.901i 64.3718 − 37.1651i − 512.000i −167.792 − 290.624i 1877.44 − 3251.83i
17.5 6.92820 + 4.00000i −39.6869 + 68.7398i 32.0000 + 55.4256i − 12.5639i −549.918 + 317.495i −93.7198 + 54.1091i 512.000i −2056.60 − 3562.14i 50.2557 − 87.0454i
17.6 6.92820 + 4.00000i −3.74645 + 6.48904i 32.0000 + 55.4256i − 522.947i −51.9123 + 29.9716i 960.011 − 554.263i 512.000i 1065.43 + 1845.38i 2091.79 − 3623.08i
17.7 6.92820 + 4.00000i 0.935423 − 1.62020i 32.0000 + 55.4256i 218.203i 12.9616 − 7.48338i −361.682 + 208.817i 512.000i 1091.75 + 1890.97i −872.812 + 1511.75i
17.8 6.92820 + 4.00000i 42.4979 − 73.6086i 32.0000 + 55.4256i 238.054i 588.869 − 339.984i 1060.70 − 612.394i 512.000i −2518.65 − 4362.43i −952.217 + 1649.29i
23.1 −6.92820 + 4.00000i −34.8616 − 60.3821i 32.0000 − 55.4256i − 226.832i 483.056 + 278.893i −154.640 − 89.2812i 512.000i −1337.16 + 2316.03i 907.328 + 1571.54i
23.2 −6.92820 + 4.00000i −6.42739 − 11.1326i 32.0000 − 55.4256i 260.039i 89.0605 + 51.4191i −850.835 − 491.230i 512.000i 1010.88 − 1750.89i −1040.15 − 1801.60i
23.3 −6.92820 + 4.00000i 16.1763 + 28.0182i 32.0000 − 55.4256i 211.408i −224.146 − 129.411i 635.796 + 367.077i 512.000i 570.153 − 987.534i −845.633 − 1464.68i
23.4 −6.92820 + 4.00000i 25.1127 + 43.4964i 32.0000 − 55.4256i − 469.361i −347.971 − 200.901i 64.3718 + 37.1651i 512.000i −167.792 + 290.624i 1877.44 + 3251.83i
23.5 6.92820 − 4.00000i −39.6869 − 68.7398i 32.0000 − 55.4256i 12.5639i −549.918 − 317.495i −93.7198 − 54.1091i − 512.000i −2056.60 + 3562.14i 50.2557 + 87.0454i
23.6 6.92820 − 4.00000i −3.74645 − 6.48904i 32.0000 − 55.4256i 522.947i −51.9123 − 29.9716i 960.011 + 554.263i − 512.000i 1065.43 − 1845.38i 2091.79 + 3623.08i
23.7 6.92820 − 4.00000i 0.935423 + 1.62020i 32.0000 − 55.4256i − 218.203i 12.9616 + 7.48338i −361.682 − 208.817i − 512.000i 1091.75 − 1890.97i −872.812 − 1511.75i
23.8 6.92820 − 4.00000i 42.4979 + 73.6086i 32.0000 − 55.4256i − 238.054i 588.869 + 339.984i 1060.70 + 612.394i − 512.000i −2518.65 + 4362.43i −952.217 − 1649.29i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
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 26.8.e.a ✓ 16
3.b odd 2 1 234.8.l.c 16
4.b odd 2 1 208.8.w.b 16
13.c even 3 1 338.8.b.i 16
13.e even 6 1 inner 26.8.e.a ✓ 16
13.e even 6 1 338.8.b.i 16
13.f odd 12 1 338.8.a.m 8
13.f odd 12 1 338.8.a.n 8
39.h odd 6 1 234.8.l.c 16
52.i odd 6 1 208.8.w.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.e.a ✓ 16 1.a even 1 1 trivial
26.8.e.a ✓ 16 13.e even 6 1 inner
208.8.w.b 16 4.b odd 2 1
208.8.w.b 16 52.i odd 6 1
234.8.l.c 16 3.b odd 2 1
234.8.l.c 16 39.h odd 6 1
338.8.a.m 8 13.f odd 12 1
338.8.a.n 8 13.f odd 12 1
338.8.b.i 16 13.c even 3 1
338.8.b.i 16 13.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(26, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 64 T^{2} + 4096)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 59\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 96\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 86\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 46\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 36\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
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