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Properties

Label 891.2.n.e
Level $891$
Weight $2$
Character orbit 891.n
Analytic conductor $7.115$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [891,2,Mod(136,891)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(891, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("891.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.n (of order \(15\), degree \(8\), not 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: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} - 15x^{12} + 116x^{10} + 69x^{8} - 814x^{6} + 2420x^{4} - 7986x^{2} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{11} - \beta_{9} + 2 \beta_{2}) q^{4} - \beta_{13} q^{5} + ( - 2 \beta_{9} + 3 \beta_{5} + \cdots + 2) q^{7}+ \cdots + ( - \beta_{10} - \beta_{8} - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{11} - \beta_{9} + 2 \beta_{2}) q^{4} - \beta_{13} q^{5} + ( - 2 \beta_{9} + 3 \beta_{5} + \cdots + 2) q^{7}+ \cdots + (8 \beta_{14} + 8 \beta_{13} + \cdots + 2 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{4} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{4} + 6 q^{7} - 56 q^{10} + 6 q^{13} - 2 q^{16} + 24 q^{19} - 28 q^{22} - 8 q^{25} + 72 q^{28} - 12 q^{31} + 16 q^{34} - 36 q^{37} + 16 q^{40} + 32 q^{43} - 84 q^{46} + 44 q^{49} - 6 q^{52} - 16 q^{55} - 28 q^{58} - 42 q^{61} + 92 q^{64} - 4 q^{67} - 26 q^{70} + 48 q^{73} + 32 q^{76} - 30 q^{79} + 20 q^{82} - 46 q^{85} + 14 q^{88} + 4 q^{91} + 26 q^{94} + 54 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} - 15x^{12} + 116x^{10} + 69x^{8} - 814x^{6} + 2420x^{4} - 7986x^{2} + 14641 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 694 \nu^{14} - 4125 \nu^{12} + 83325 \nu^{10} + 253828 \nu^{8} - 1394250 \nu^{6} + \cdots + 17569200 ) / 239877671 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33733 \nu^{14} + 510405 \nu^{12} + 2340324 \nu^{10} - 9424548 \nu^{8} + 19867463 \nu^{6} + \cdots + 71125978 ) / 2638654381 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 33733 \nu^{15} + 510405 \nu^{13} + 2340324 \nu^{11} - 9424548 \nu^{9} + 19867463 \nu^{7} + \cdots + 71125978 \nu ) / 2638654381 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2519552 \nu^{14} + 9695739 \nu^{12} - 29132320 \nu^{10} + 117316640 \nu^{8} + \cdots - 26414663968 ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2587648 \nu^{14} - 9957786 \nu^{12} + 29919680 \nu^{10} - 120487360 \nu^{8} + \cdots - 2681089202 ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2587648 \nu^{15} - 9957786 \nu^{13} + 29919680 \nu^{11} - 120487360 \nu^{9} + \cdots - 2681089202 \nu ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4184848 \nu^{15} - 9284300 \nu^{13} + 48387305 \nu^{11} - 194856985 \nu^{9} + \cdots + 14848218066 \nu ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23315 \nu^{14} + 9245 \nu^{12} - 186749 \nu^{10} + 2328065 \nu^{8} + 3124810 \nu^{6} + \cdots - 39376304 ) / 160360211 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23315 \nu^{15} + 9245 \nu^{13} - 186749 \nu^{11} + 2328065 \nu^{9} + 3124810 \nu^{7} + \cdots - 39376304 \nu ) / 160360211 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4555911 \nu^{14} - 3669845 \nu^{12} + 74130869 \nu^{10} - 298527013 \nu^{8} + \cdots + 15630603824 ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4750105 \nu^{14} - 6347305 \nu^{12} - 70578089 \nu^{10} + 569479805 \nu^{8} + \cdots - 37270003155 ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4750105 \nu^{15} + 6347305 \nu^{13} + 70578089 \nu^{11} - 569479805 \nu^{9} + \cdots + 37270003155 \nu ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5107200 \nu^{15} - 19653525 \nu^{13} + 59052000 \nu^{11} - 237804000 \nu^{9} + \cdots - 5291623425 \nu ) / 29025198191 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 7556746 \nu^{15} + 2765146 \nu^{13} + 115371648 \nu^{11} - 821179661 \nu^{9} + \cdots + 62341179681 \nu ) / 29025198191 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{9} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{15} + 4\beta_{14} + 4\beta_{13} - \beta_{10} - \beta_{8} - 4\beta_{7} + 3\beta_{4} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{12} - 7\beta_{9} + 10\beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{13} - 7\beta_{10} + 10\beta_{4} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -37\beta_{6} - 38\beta_{5} - 38 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 38\beta_{14} - 75\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 149\beta_{12} + 338\beta_{11} - 149\beta_{6} + 149\beta_{5} - 149\beta_{3} - 189\beta_{2} + 149 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 189\beta_{15} - 338\beta_{14} - 338\beta_{13} + 338\beta_{8} + 189\beta_{7} - 189\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -905\beta_{12} + 1541\beta_{9} - 905\beta_{5} - 905\beta_{2} - 1541 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 905\beta_{15} + 1541\beta_{10} - 905\beta_{4} - 1541\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4256\beta_{11} + 7069\beta_{5} - 4256\beta_{3} + 4256 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -7069\beta_{14} + 4256\beta_{8} + 7069\beta_{7} - 2813\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -19837\beta_{12} - 32532\beta_{11} + 19837\beta_{9} + 19837\beta_{6} - 19837\beta_{5} + 19837\beta_{3} - 19837 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 19837\beta_{14} + 19837\beta_{13} + 19837\beta_{10} - 32532\beta_{8} - 12695\beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(\beta_{5}\) \(-\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
−1.96317 + 0.874061i
1.96317 − 0.874061i
−1.96317 − 0.874061i
1.96317 + 0.874061i
−1.50964 − 0.320883i
1.50964 + 0.320883i
−0.224628 + 2.13719i
0.224628 − 2.13719i
−1.03271 − 1.14694i
1.03271 + 1.14694i
−1.03271 + 1.14694i
1.03271 − 1.14694i
−1.50964 + 0.320883i
1.50964 − 0.320883i
−0.224628 − 2.13719i
0.224628 + 2.13719i
−1.96317 + 0.874061i 0 1.75181 − 1.94558i 1.96317 + 0.874061i 0 4.14350 + 0.880728i −0.410415 + 1.26313i 0 −4.61803
136.2 1.96317 − 0.874061i 0 1.75181 − 1.94558i −1.96317 − 0.874061i 0 4.14350 + 0.880728i 0.410415 − 1.26313i 0 −4.61803
190.1 −1.96317 − 0.874061i 0 1.75181 + 1.94558i 1.96317 − 0.874061i 0 4.14350 − 0.880728i −0.410415 − 1.26313i 0 −4.61803
190.2 1.96317 + 0.874061i 0 1.75181 + 1.94558i −1.96317 + 0.874061i 0 4.14350 − 0.880728i 0.410415 + 1.26313i 0 −4.61803
379.1 −1.50964 − 0.320883i 0 0.348943 + 0.155360i 1.50964 − 0.320883i 0 −0.0246758 − 0.234775i 2.02029 + 1.46782i 0 −2.38197
379.2 1.50964 + 0.320883i 0 0.348943 + 0.155360i −1.50964 + 0.320883i 0 −0.0246758 − 0.234775i −2.02029 − 1.46782i 0 −2.38197
433.1 −0.224628 + 2.13719i 0 −2.56082 − 0.544320i 0.224628 + 2.13719i 0 −2.83448 + 3.14801i 0.410415 − 1.26313i 0 −4.61803
433.2 0.224628 − 2.13719i 0 −2.56082 − 0.544320i −0.224628 − 2.13719i 0 −2.83448 + 3.14801i −0.410415 + 1.26313i 0 −4.61803
460.1 −1.03271 − 1.14694i 0 −0.0399263 + 0.379874i 1.03271 − 1.14694i 0 0.215659 − 0.0960175i −2.02029 + 1.46782i 0 −2.38197
460.2 1.03271 + 1.14694i 0 −0.0399263 + 0.379874i −1.03271 + 1.14694i 0 0.215659 − 0.0960175i 2.02029 − 1.46782i 0 −2.38197
676.1 −1.03271 + 1.14694i 0 −0.0399263 − 0.379874i 1.03271 + 1.14694i 0 0.215659 + 0.0960175i −2.02029 − 1.46782i 0 −2.38197
676.2 1.03271 − 1.14694i 0 −0.0399263 − 0.379874i −1.03271 − 1.14694i 0 0.215659 + 0.0960175i 2.02029 + 1.46782i 0 −2.38197
757.1 −1.50964 + 0.320883i 0 0.348943 − 0.155360i 1.50964 + 0.320883i 0 −0.0246758 + 0.234775i 2.02029 − 1.46782i 0 −2.38197
757.2 1.50964 − 0.320883i 0 0.348943 − 0.155360i −1.50964 − 0.320883i 0 −0.0246758 + 0.234775i −2.02029 + 1.46782i 0 −2.38197
784.1 −0.224628 − 2.13719i 0 −2.56082 + 0.544320i 0.224628 − 2.13719i 0 −2.83448 − 3.14801i 0.410415 + 1.26313i 0 −4.61803
784.2 0.224628 + 2.13719i 0 −2.56082 + 0.544320i −0.224628 + 2.13719i 0 −2.83448 − 3.14801i −0.410415 − 1.26313i 0 −4.61803
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 136.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner
99.m even 15 1 inner
99.n odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.n.e 16
3.b odd 2 1 inner 891.2.n.e 16
9.c even 3 1 99.2.f.c ✓ 8
9.c even 3 1 inner 891.2.n.e 16
9.d odd 6 1 99.2.f.c ✓ 8
9.d odd 6 1 inner 891.2.n.e 16
11.c even 5 1 inner 891.2.n.e 16
33.h odd 10 1 inner 891.2.n.e 16
99.m even 15 1 99.2.f.c ✓ 8
99.m even 15 1 inner 891.2.n.e 16
99.m even 15 1 1089.2.a.v 4
99.n odd 30 1 99.2.f.c ✓ 8
99.n odd 30 1 inner 891.2.n.e 16
99.n odd 30 1 1089.2.a.v 4
99.o odd 30 1 1089.2.a.w 4
99.p even 30 1 1089.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.f.c ✓ 8 9.c even 3 1
99.2.f.c ✓ 8 9.d odd 6 1
99.2.f.c ✓ 8 99.m even 15 1
99.2.f.c ✓ 8 99.n odd 30 1
891.2.n.e 16 1.a even 1 1 trivial
891.2.n.e 16 3.b odd 2 1 inner
891.2.n.e 16 9.c even 3 1 inner
891.2.n.e 16 9.d odd 6 1 inner
891.2.n.e 16 11.c even 5 1 inner
891.2.n.e 16 33.h odd 10 1 inner
891.2.n.e 16 99.m even 15 1 inner
891.2.n.e 16 99.n odd 30 1 inner
1089.2.a.v 4 99.m even 15 1
1089.2.a.v 4 99.n odd 30 1
1089.2.a.w 4 99.o odd 30 1
1089.2.a.w 4 99.p even 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - T_{2}^{14} - 15T_{2}^{12} + 116T_{2}^{10} + 69T_{2}^{8} - 814T_{2}^{6} + 2420T_{2}^{4} - 7986T_{2}^{2} + 14641 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( (T^{8} - 3 T^{7} - 10 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 3 T^{7} - 10 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 34 T^{6} + \cdots + 1771561)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 72 T^{6} + \cdots + 793881)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 959512576 \) Copy content Toggle raw display
$31$ \( (T^{8} + 6 T^{7} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 9 T^{3} + \cdots + 121)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 5719140625 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots + 841)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} - 19 T^{14} + \cdots + 14641 \) Copy content Toggle raw display
$53$ \( (T^{8} + 130 T^{6} + \cdots + 75625)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 248656346483281 \) Copy content Toggle raw display
$61$ \( (T^{8} + 21 T^{7} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} + 12 T^{2} + \cdots + 121)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 25 T^{6} + \cdots + 47265625)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 12 T^{3} + \cdots + 256)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 15 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 3138428376721 \) Copy content Toggle raw display
$89$ \( (T^{4} - 360 T^{2} + 22275)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} - 27 T^{7} + \cdots + 519885601)^{2} \) Copy content Toggle raw display
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