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Properties

Label 138.3.c.a
Level $138$
Weight $3$
Character orbit 138.c
Analytic conductor $3.760$
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] = [138,3,Mod(47,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 138.c (of order \(2\), degree \(1\), 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: \(3.76022764817\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_1 q^{3} - 2 q^{4} - \beta_{7} q^{5} + \beta_{11} q^{6} - \beta_{6} q^{7} + 2 \beta_{3} q^{8} + (\beta_{10} - \beta_{8} + \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_1 q^{3} - 2 q^{4} - \beta_{7} q^{5} + \beta_{11} q^{6} - \beta_{6} q^{7} + 2 \beta_{3} q^{8} + (\beta_{10} - \beta_{8} + \beta_{5} - 1) q^{9} + ( - \beta_{8} + \beta_{4} + \beta_{2} - 1) q^{10} + (\beta_{13} + \beta_{12} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} - 32 q^{4} - 8 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} - 32 q^{4} - 8 q^{6} - 4 q^{9} - 8 q^{12} - 8 q^{13} + 28 q^{15} + 64 q^{16} + 16 q^{18} + 40 q^{19} + 4 q^{21} + 16 q^{22} + 16 q^{24} - 192 q^{25} - 80 q^{27} - 24 q^{30} + 136 q^{31} - 84 q^{33} - 16 q^{34} + 8 q^{36} - 136 q^{37} + 156 q^{39} + 128 q^{42} + 72 q^{43} + 4 q^{45} + 16 q^{48} + 224 q^{49} - 4 q^{51} + 16 q^{52} - 176 q^{54} - 96 q^{55} - 160 q^{57} - 56 q^{60} + 48 q^{61} + 204 q^{63} - 128 q^{64} - 144 q^{66} - 304 q^{67} - 176 q^{70} - 32 q^{72} + 408 q^{73} + 68 q^{75} - 80 q^{76} + 328 q^{78} + 312 q^{79} + 164 q^{81} + 160 q^{82} - 8 q^{84} - 464 q^{85} - 268 q^{87} - 32 q^{88} + 32 q^{90} - 72 q^{91} - 108 q^{93} - 32 q^{96} + 168 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 10 x^{14} + 8 x^{13} - 119 x^{12} + 416 x^{11} - 774 x^{10} - 1284 x^{9} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 217 \nu^{15} + 3982 \nu^{14} + 2141 \nu^{13} - 71198 \nu^{12} + 44174 \nu^{11} + \cdots + 33461651124 ) / 5739562800 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1049 \nu^{15} - 2761 \nu^{14} - 563 \nu^{13} + 46859 \nu^{12} - 119312 \nu^{11} + \cdots - 31342795857 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2257 \nu^{15} - 24758 \nu^{14} + 36191 \nu^{13} - 18308 \nu^{12} + 241214 \nu^{11} + \cdots - 62628196086 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 449 \nu^{15} - 598 \nu^{14} - 6497 \nu^{13} - 17650 \nu^{12} + 70810 \nu^{11} + \cdots + 6782250042 ) / 3443737680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 329 \nu^{15} - 1708 \nu^{14} + 3703 \nu^{13} + 2120 \nu^{12} - 22220 \nu^{11} + \cdots - 10005971148 ) / 2295825120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6992 \nu^{15} - 36013 \nu^{14} + 58996 \nu^{13} - 326953 \nu^{12} + 578104 \nu^{11} + \cdots - 137983872681 ) / 34437376800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 667 \nu^{15} - 3082 \nu^{14} + 874 \nu^{13} - 4717 \nu^{12} + 31606 \nu^{11} + \cdots + 1095299901 ) / 2869781400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4039 \nu^{15} - 3041 \nu^{14} + 49277 \nu^{13} - 178841 \nu^{12} + 468248 \nu^{11} + \cdots + 72064993923 ) / 11479125600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6247 \nu^{15} - 15502 \nu^{14} + 37729 \nu^{13} + 59948 \nu^{12} - 164414 \nu^{11} + \cdots - 10120762404 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 773 \nu^{15} - 1103 \nu^{14} - 6139 \nu^{13} + 27127 \nu^{12} - 90736 \nu^{11} + \cdots - 5017334481 ) / 1913187600 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 559 \nu^{15} + 2326 \nu^{14} - 199 \nu^{13} - 6164 \nu^{12} + 75908 \nu^{11} + \cdots + 9795520512 ) / 1147912560 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13753 \nu^{15} - 64453 \nu^{14} + 112681 \nu^{13} + 104957 \nu^{12} - 1214876 \nu^{11} + \cdots - 214176568851 ) / 17218688400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3461 \nu^{15} + 18971 \nu^{14} - 24797 \nu^{13} - 41569 \nu^{12} + 437512 \nu^{11} + \cdots + 61746535467 ) / 3826375200 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 49627 \nu^{15} + 159457 \nu^{14} + 8801 \nu^{13} - 1090223 \nu^{12} + 5845844 \nu^{11} + \cdots + 673542477549 ) / 34437376800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{8} + \beta_{5} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{10} + \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{15} + 6 \beta_{13} + 9 \beta_{11} + \beta_{10} - 6 \beta_{9} - 4 \beta_{8} + \beta_{5} + \cdots + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{15} + 4 \beta_{14} + 11 \beta_{13} - 18 \beta_{12} + 6 \beta_{11} - 8 \beta_{10} + 4 \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{15} - 30 \beta_{14} - 36 \beta_{13} + 6 \beta_{12} - 90 \beta_{11} + 4 \beta_{10} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 84 \beta_{15} + 184 \beta_{14} + 2 \beta_{13} - 114 \beta_{12} - 60 \beta_{11} + 160 \beta_{10} + \cdots + 504 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 330 \beta_{15} + 552 \beta_{14} + 624 \beta_{13} + 408 \beta_{12} - 18 \beta_{11} - 281 \beta_{10} + \cdots + 485 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 373 \beta_{14} + 1001 \beta_{13} + 1311 \beta_{12} + 918 \beta_{11} + 1255 \beta_{10} + 373 \beta_{9} + \cdots - 7086 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 996 \beta_{15} - 1512 \beta_{14} + 564 \beta_{13} - 1512 \beta_{12} - 6201 \beta_{11} - 809 \beta_{10} + \cdots + 13763 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12360 \beta_{15} - 3632 \beta_{14} - 2275 \beta_{13} - 5220 \beta_{12} - 4836 \beta_{11} + 6904 \beta_{10} + \cdots + 16527 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 28260 \beta_{15} + 6468 \beta_{14} + 21384 \beta_{13} + 348 \beta_{12} + 29556 \beta_{11} + \cdots + 259997 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4488 \beta_{15} - 35240 \beta_{14} - 24460 \beta_{13} - 14388 \beta_{12} - 127104 \beta_{11} + \cdots - 404640 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 49764 \beta_{15} + 133464 \beta_{14} + 227136 \beta_{13} + 78888 \beta_{12} - 95004 \beta_{11} + \cdots - 1093681 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 46080 \beta_{15} + 1249717 \beta_{14} + 1304399 \beta_{13} - 491403 \beta_{12} + 430236 \beta_{11} + \cdots - 4803294 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−2.99749 − 0.122585i
−2.52762 − 1.61590i
−0.752365 − 2.90413i
−0.0515370 + 2.99956i
1.30755 + 2.70006i
1.62763 − 2.52009i
2.42141 − 1.77110i
2.97243 + 0.405752i
−2.99749 + 0.122585i
−2.52762 + 1.61590i
−0.752365 + 2.90413i
−0.0515370 − 2.99956i
1.30755 − 2.70006i
1.62763 + 2.52009i
2.42141 + 1.77110i
2.97243 − 0.405752i
− 1.41421i −2.99749 − 0.122585i −2.00000 4.36583i −0.173361 + 4.23910i 5.99969 2.82843i 8.96995 + 0.734895i 6.17421
47.2 − 1.41421i −2.52762 − 1.61590i −2.00000 − 7.30416i −2.28522 + 3.57460i −0.709880 2.82843i 3.77777 + 8.16875i −10.3296
47.3 − 1.41421i −0.752365 − 2.90413i −2.00000 9.12709i −4.10705 + 1.06400i −11.5474 2.82843i −7.86789 + 4.36992i 12.9076
47.4 − 1.41421i −0.0515370 + 2.99956i −2.00000 2.42657i 4.24201 + 0.0728843i −2.87957 2.82843i −8.99469 − 0.309176i 3.43169
47.5 − 1.41421i 1.30755 + 2.70006i −2.00000 − 8.14461i 3.81846 − 1.84916i 7.21035 2.82843i −5.58060 + 7.06094i −11.5182
47.6 − 1.41421i 1.62763 − 2.52009i −2.00000 − 1.81026i −3.56394 − 2.30181i 9.47484 2.82843i −3.70167 − 8.20351i −2.56010
47.7 − 1.41421i 2.42141 − 1.77110i −2.00000 − 4.98213i −2.50471 − 3.42438i −12.5950 2.82843i 2.72641 − 8.57710i −7.04580
47.8 − 1.41421i 2.97243 + 0.405752i −2.00000 6.32168i 0.573819 − 4.20366i 5.04690 2.82843i 8.67073 + 2.41214i 8.94020
47.9 1.41421i −2.99749 + 0.122585i −2.00000 − 4.36583i −0.173361 − 4.23910i 5.99969 − 2.82843i 8.96995 − 0.734895i 6.17421
47.10 1.41421i −2.52762 + 1.61590i −2.00000 7.30416i −2.28522 − 3.57460i −0.709880 − 2.82843i 3.77777 − 8.16875i −10.3296
47.11 1.41421i −0.752365 + 2.90413i −2.00000 − 9.12709i −4.10705 − 1.06400i −11.5474 − 2.82843i −7.86789 − 4.36992i 12.9076
47.12 1.41421i −0.0515370 − 2.99956i −2.00000 − 2.42657i 4.24201 − 0.0728843i −2.87957 − 2.82843i −8.99469 + 0.309176i 3.43169
47.13 1.41421i 1.30755 − 2.70006i −2.00000 8.14461i 3.81846 + 1.84916i 7.21035 − 2.82843i −5.58060 − 7.06094i −11.5182
47.14 1.41421i 1.62763 + 2.52009i −2.00000 1.81026i −3.56394 + 2.30181i 9.47484 − 2.82843i −3.70167 + 8.20351i −2.56010
47.15 1.41421i 2.42141 + 1.77110i −2.00000 4.98213i −2.50471 + 3.42438i −12.5950 − 2.82843i 2.72641 + 8.57710i −7.04580
47.16 1.41421i 2.97243 − 0.405752i −2.00000 − 6.32168i 0.573819 + 4.20366i 5.04690 − 2.82843i 8.67073 − 2.41214i 8.94020
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.3.c.a ✓ 16
3.b odd 2 1 inner 138.3.c.a ✓ 16
4.b odd 2 1 1104.3.g.c 16
12.b even 2 1 1104.3.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.3.c.a ✓ 16 1.a even 1 1 trivial
138.3.c.a ✓ 16 3.b odd 2 1 inner
1104.3.g.c 16 4.b odd 2 1
1104.3.g.c 16 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(138, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 107557761600 \) Copy content Toggle raw display
$7$ \( (T^{8} - 252 T^{6} + \cdots + 615000)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + \cdots + 960369700)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{8} - 20 T^{7} + \cdots - 780768)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} - 68 T^{7} + \cdots + 145917216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 68 T^{7} + \cdots + 44098389696)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 69\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 2868882576000)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 20431830161152)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 3918307649688)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 171604877785344)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 173075158110240)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} - 84 T^{7} + \cdots - 44369259840)^{2} \) Copy content Toggle raw display
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