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Properties

Label 15.11.d.c
Level $15$
Weight $11$
Character orbit 15.d
Analytic conductor $9.530$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [15,11,Mod(14,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.14");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 15.d (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: \(9.53035879011\)
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: \( x^{16} + 17880 x^{14} + 140656106 x^{12} + 568287997200 x^{10} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{32}\cdot 5^{10} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{6}) q^{3} + ( - \beta_{2} + 137) q^{4} + (\beta_{8} + \beta_{7} + 12 \beta_{6}) q^{5} + ( - 3 \beta_{2} - \beta_1 + 1346) q^{6} + ( - \beta_{12} - 3 \beta_{7}) q^{7} + ( - \beta_{13} + 2 \beta_{11} + \cdots + 112 \beta_{6}) q^{8}+ \cdots + (3 \beta_{9} + 3 \beta_{8} + \cdots - 3938) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{6}) q^{3} + ( - \beta_{2} + 137) q^{4} + (\beta_{8} + \beta_{7} + 12 \beta_{6}) q^{5} + ( - 3 \beta_{2} - \beta_1 + 1346) q^{6} + ( - \beta_{12} - 3 \beta_{7}) q^{7} + ( - \beta_{13} + 2 \beta_{11} + \cdots + 112 \beta_{6}) q^{8}+ \cdots + (28161 \beta_{10} + 343962 \beta_{9} + \cdots + 2686512234) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2184 q^{4} + 21516 q^{6} - 63000 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2184 q^{4} + 21516 q^{6} - 63000 q^{9} - 221680 q^{10} + 731640 q^{15} - 4218352 q^{16} + 1487600 q^{19} + 2444616 q^{21} + 28021368 q^{24} - 9324800 q^{25} - 35678700 q^{30} - 77667568 q^{31} + 251882368 q^{34} - 22344768 q^{36} - 70953984 q^{39} - 179966240 q^{40} - 484245000 q^{45} - 72018968 q^{46} + 89098816 q^{49} + 686556816 q^{51} + 441096084 q^{54} + 1424671200 q^{55} + 610091280 q^{60} - 1671368368 q^{61} - 4172730848 q^{64} + 2368350000 q^{66} - 2906594616 q^{69} + 2838634200 q^{70} - 1285884000 q^{75} + 4079367264 q^{76} - 3904999600 q^{79} - 11376681984 q^{81} + 13119851016 q^{84} + 8097594320 q^{85} - 15222287520 q^{90} + 19275224832 q^{91} - 9366481832 q^{94} - 5999937552 q^{96} + 42957756000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 17880 x^{14} + 140656106 x^{12} + 568287997200 x^{10} + \cdots + 16\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 84\!\cdots\!87 \nu^{14} + \cdots + 29\!\cdots\!96 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 75\!\cdots\!49 \nu^{14} + \cdots + 26\!\cdots\!80 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 79\!\cdots\!11 \nu^{14} + \cdots - 36\!\cdots\!88 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 90\!\cdots\!86 \nu^{14} + \cdots + 43\!\cdots\!88 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 62\!\cdots\!55 \nu^{14} + \cdots + 23\!\cdots\!36 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!52 \nu^{15} + \cdots + 11\!\cdots\!68 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!19 \nu^{15} + \cdots - 16\!\cdots\!64 \nu ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!67 \nu^{15} + \cdots - 11\!\cdots\!32 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\!\cdots\!67 \nu^{15} + \cdots - 11\!\cdots\!32 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!89 \nu^{15} + \cdots + 10\!\cdots\!64 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 88\!\cdots\!11 \nu^{15} + \cdots - 31\!\cdots\!96 \nu ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!53 \nu^{15} + \cdots - 48\!\cdots\!48 \nu ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 18\!\cdots\!84 \nu^{15} + \cdots + 11\!\cdots\!44 \nu ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 18\!\cdots\!98 \nu^{15} + \cdots + 10\!\cdots\!44 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 98\!\cdots\!88 \nu^{15} + \cdots - 10\!\cdots\!44 ) / 50\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{13} + 14\beta_{11} - 18\beta_{9} + 18\beta_{8} - 744\beta_{7} - 1042\beta_{6} ) / 3240 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 27 \beta_{10} - 570 \beta_{9} - 543 \beta_{8} - 57 \beta_{5} + 88 \beta_{4} + 97 \beta_{3} + \cdots - 3621603 ) / 1620 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1440 \beta_{15} - 3960 \beta_{14} - 4395 \beta_{13} - 13230 \beta_{12} - 27920 \beta_{11} + \cdots - 1260 \beta_{3} ) / 1620 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 34614 \beta_{10} - 500955 \beta_{9} - 535569 \beta_{8} + 88419 \beta_{5} - 107636 \beta_{4} + \cdots + 1295478291 ) / 270 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2426940 \beta_{15} - 1869390 \beta_{14} + 42754056 \beta_{13} - 36219015 \beta_{12} + \cdots - 2148165 \beta_{3} ) / 810 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 857977758 \beta_{10} + 1810978275 \beta_{9} + 2668956033 \beta_{8} - 1173139323 \beta_{5} + \cdots + 12540520618833 ) / 810 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2100441780 \beta_{15} - 28576551870 \beta_{14} - 50926279083 \beta_{13} + \cdots - 13238055045 \beta_{3} ) / 810 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 936358332048 \beta_{10} + 2233908239415 \beta_{9} + 1297549907367 \beta_{8} + 1773491336583 \beta_{5} + \cdots - 20\!\cdots\!33 ) / 270 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 55516895896140 \beta_{15} + 390193029414990 \beta_{14} - 56007845102991 \beta_{13} + \cdots + 222854962655565 \beta_{3} ) / 810 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 35\!\cdots\!64 \beta_{10} + \cdots + 81\!\cdots\!89 ) / 162 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 59\!\cdots\!80 \beta_{15} + \cdots - 63\!\cdots\!55 \beta_{3} ) / 810 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 34\!\cdots\!76 \beta_{10} + \cdots - 16\!\cdots\!49 ) / 90 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 87\!\cdots\!40 \beta_{15} + \cdots + 20\!\cdots\!15 \beta_{3} ) / 810 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 26\!\cdots\!48 \beta_{10} + \cdots - 30\!\cdots\!93 ) / 810 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 23\!\cdots\!40 \beta_{15} + \cdots + 28\!\cdots\!15 \beta_{3} ) / 810 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
−19.4690 + 50.0826i
−19.4690 − 50.0826i
6.53233 + 63.3725i
6.53233 − 63.3725i
−29.3450 + 80.7929i
−29.3450 − 80.7929i
61.4341 + 30.7435i
61.4341 − 30.7435i
−61.4341 + 30.7435i
−61.4341 − 30.7435i
29.3450 + 80.7929i
29.3450 − 80.7929i
−6.53233 + 63.3725i
−6.53233 − 63.3725i
19.4690 + 50.0826i
19.4690 − 50.0826i
−50.9058 −190.983 − 150.248i 1567.41 553.573 + 3075.58i 9722.16 + 7648.49i 8178.10i −27662.5 13900.2 + 57389.6i −28180.1 − 156565.i
14.2 −50.9058 −190.983 + 150.248i 1567.41 553.573 − 3075.58i 9722.16 − 7648.49i − 8178.10i −27662.5 13900.2 − 57389.6i −28180.1 + 156565.i
14.3 −36.7249 151.342 − 190.117i 324.720 2839.18 − 1305.64i −5558.02 + 6982.05i 17185.2i 25681.0 −13240.3 − 57545.5i −104269. + 47949.4i
14.4 −36.7249 151.342 + 190.117i 324.720 2839.18 + 1305.64i −5558.02 − 6982.05i − 17185.2i 25681.0 −13240.3 + 57545.5i −104269. − 47949.4i
14.5 −26.2772 −17.3641 − 242.379i −333.511 −3042.13 − 714.880i 456.279 + 6369.03i − 22118.4i 35671.5 −58446.0 + 8417.38i 79938.6 + 18785.0i
14.6 −26.2772 −17.3641 + 242.379i −333.511 −3042.13 + 714.880i 456.279 − 6369.03i 22118.4i 35671.5 −58446.0 − 8417.38i 79938.6 − 18785.0i
14.7 −3.37421 −224.817 − 92.2305i −1012.61 862.380 − 3003.65i 758.580 + 311.205i 16006.0i 6871.97 42036.1 + 41469.9i −2909.86 + 10135.0i
14.8 −3.37421 −224.817 + 92.2305i −1012.61 862.380 + 3003.65i 758.580 − 311.205i − 16006.0i 6871.97 42036.1 − 41469.9i −2909.86 − 10135.0i
14.9 3.37421 224.817 − 92.2305i −1012.61 −862.380 + 3003.65i 758.580 − 311.205i 16006.0i −6871.97 42036.1 − 41469.9i −2909.86 + 10135.0i
14.10 3.37421 224.817 + 92.2305i −1012.61 −862.380 − 3003.65i 758.580 + 311.205i − 16006.0i −6871.97 42036.1 + 41469.9i −2909.86 − 10135.0i
14.11 26.2772 17.3641 − 242.379i −333.511 3042.13 + 714.880i 456.279 − 6369.03i − 22118.4i −35671.5 −58446.0 − 8417.38i 79938.6 + 18785.0i
14.12 26.2772 17.3641 + 242.379i −333.511 3042.13 − 714.880i 456.279 + 6369.03i 22118.4i −35671.5 −58446.0 + 8417.38i 79938.6 − 18785.0i
14.13 36.7249 −151.342 − 190.117i 324.720 −2839.18 + 1305.64i −5558.02 − 6982.05i 17185.2i −25681.0 −13240.3 + 57545.5i −104269. + 47949.4i
14.14 36.7249 −151.342 + 190.117i 324.720 −2839.18 − 1305.64i −5558.02 + 6982.05i − 17185.2i −25681.0 −13240.3 − 57545.5i −104269. − 47949.4i
14.15 50.9058 190.983 − 150.248i 1567.41 −553.573 − 3075.58i 9722.16 − 7648.49i 8178.10i 27662.5 13900.2 − 57389.6i −28180.1 − 156565.i
14.16 50.9058 190.983 + 150.248i 1567.41 −553.573 + 3075.58i 9722.16 + 7648.49i − 8178.10i 27662.5 13900.2 + 57389.6i −28180.1 + 156565.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.11.d.c ✓ 16
3.b odd 2 1 inner 15.11.d.c ✓ 16
5.b even 2 1 inner 15.11.d.c ✓ 16
5.c odd 4 2 75.11.c.h 16
15.d odd 2 1 inner 15.11.d.c ✓ 16
15.e even 4 2 75.11.c.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.d.c ✓ 16 1.a even 1 1 trivial
15.11.d.c ✓ 16 3.b odd 2 1 inner
15.11.d.c ✓ 16 5.b even 2 1 inner
15.11.d.c ✓ 16 15.d odd 2 1 inner
75.11.c.h 16 5.c odd 4 2
75.11.c.h 16 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 4642T_{2}^{6} + 6268416T_{2}^{4} - 2484083200T_{2}^{2} + 27476377600 \) acting on \(S_{11}^{\mathrm{new}}(15, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4642 T^{6} + \cdots + 27476377600)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 82\!\cdots\!16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 89\!\cdots\!16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 64\!\cdots\!84)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 30\!\cdots\!84)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
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