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Properties

Label 93.3.i.a
Level $93$
Weight $3$
Character orbit 93.i
Analytic conductor $2.534$
Analytic rank $0$
Dimension $10$
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] = [93,3,Mod(37,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 93.i (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: \(2.53406645855\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 21x^{8} + 135x^{6} + 327x^{4} + 324x^{2} + 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{9} - \beta_{8} - 1) q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{6} - \beta_{5} - \beta_1 + 1) q^{4} + ( - \beta_{9} + \beta_{7} + \cdots - \beta_1) q^{5}+ \cdots + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} - \beta_{8} - 1) q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{6} - \beta_{5} - \beta_1 + 1) q^{4} + ( - \beta_{9} + \beta_{7} + \cdots - \beta_1) q^{5}+ \cdots + (6 \beta_{7} - 3 \beta_{4} - 6 \beta_{3} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{2} + 15 q^{3} + 14 q^{4} - 9 q^{6} - q^{7} - 18 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6 q^{2} + 15 q^{3} + 14 q^{4} - 9 q^{6} - q^{7} - 18 q^{8} + 15 q^{9} + 18 q^{10} + 21 q^{11} + 21 q^{12} + 36 q^{13} + 9 q^{14} + 46 q^{16} + 6 q^{17} - 9 q^{18} - 14 q^{19} - 66 q^{20} - 3 q^{21} - 63 q^{22} - 27 q^{24} + 17 q^{25} - 108 q^{26} - 29 q^{28} + 89 q^{31} - 258 q^{32} + 42 q^{33} - 216 q^{34} - 96 q^{35} + 21 q^{36} - 36 q^{37} + 72 q^{39} - 6 q^{40} + 12 q^{41} + 27 q^{42} + 24 q^{43} + 159 q^{44} + 192 q^{47} + 69 q^{48} - 18 q^{49} - 87 q^{50} + 6 q^{51} + 126 q^{52} - 27 q^{53} + 180 q^{55} + 69 q^{56} - 42 q^{57} - 57 q^{59} + 75 q^{62} - 6 q^{63} + 206 q^{64} - 162 q^{65} - 126 q^{66} + 124 q^{67} + 534 q^{68} + 42 q^{69} + 48 q^{70} - 24 q^{71} - 27 q^{72} - 186 q^{73} + 306 q^{74} + 51 q^{75} + 248 q^{76} - 216 q^{78} + 120 q^{79} - 18 q^{80} - 45 q^{81} - 180 q^{82} + 57 q^{83} - 87 q^{84} + 468 q^{86} + 129 q^{87} - 891 q^{88} - 54 q^{90} + 3 q^{93} - 672 q^{94} - 48 q^{95} - 387 q^{96} - 670 q^{97} + 246 q^{98} + 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 21x^{8} + 135x^{6} + 327x^{4} + 324x^{2} + 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 18\nu^{4} - 81\nu^{2} - 72 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{9} + 135\nu^{7} + 711\nu^{5} + 1029\nu^{3} + 342\nu + 36 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7 \nu^{9} - 12 \nu^{8} - 135 \nu^{7} - 234 \nu^{6} - 711 \nu^{5} - 1260 \nu^{4} - 1029 \nu^{3} + \cdots - 792 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{9} - 12 \nu^{8} + 135 \nu^{7} - 234 \nu^{6} + 711 \nu^{5} - 1260 \nu^{4} + 1029 \nu^{3} + \cdots - 792 ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8 \nu^{9} + 6 \nu^{8} + 153 \nu^{7} + 117 \nu^{6} + 792 \nu^{5} + 630 \nu^{4} + 1095 \nu^{3} + \cdots + 306 ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{9} - 6 \nu^{8} + 153 \nu^{7} - 117 \nu^{6} + 792 \nu^{5} - 630 \nu^{4} + 1095 \nu^{3} + \cdots - 306 ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{9} - 177\nu^{7} - 2\nu^{6} - 981\nu^{5} - 36\nu^{4} - 1671\nu^{3} - 162\nu^{2} - 846\nu - 144 ) / 24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25 \nu^{9} + 18 \nu^{8} + 495 \nu^{7} + 354 \nu^{6} + 2781 \nu^{5} + 1962 \nu^{4} + 4845 \nu^{3} + \cdots + 1584 ) / 72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25 \nu^{9} + 18 \nu^{8} - 495 \nu^{7} + 354 \nu^{6} - 2781 \nu^{5} + 1962 \nu^{4} - 4845 \nu^{3} + \cdots + 1584 ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} - 2\beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - 2\beta_{7} - \beta_{6} - \beta_{5} - 3\beta_{4} + 3\beta_{3} + 10\beta_{2} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + 2\beta_{8} - 11\beta_{6} + 11\beta_{5} + 14\beta_{4} + 14\beta_{3} + \beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{9} + 11 \beta_{8} + 20 \beta_{7} + 15 \beta_{6} + 15 \beta_{5} + 32 \beta_{4} - 32 \beta_{3} + \cdots + 67 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -36\beta_{9} - 36\beta_{8} + 117\beta_{6} - 117\beta_{5} - 171\beta_{4} - 171\beta_{3} - 24\beta _1 - 477 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 113 \beta_{9} - 113 \beta_{8} - 194 \beta_{7} - 191 \beta_{6} - 191 \beta_{5} - 359 \beta_{4} + \cdots - 825 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 492\beta_{9} + 492\beta_{8} - 1287\beta_{6} + 1287\beta_{5} + 2022\beta_{4} + 2022\beta_{3} + 363\beta _1 + 5313 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1209 \beta_{9} + 1209 \beta_{8} + 2004 \beta_{7} + 2307 \beta_{6} + 2307 \beta_{5} + 4098 \beta_{4} + \cdots + 9819 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(34\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.836644i
3.40208i
− 1.30912i
− 2.37290i
1.17535i
− 0.836644i
− 3.40208i
1.30912i
2.37290i
− 1.17535i
−3.70300 1.50000 − 0.866025i 9.71222 −1.74453 + 3.02161i −5.55450 + 3.20689i −1.18612 − 2.05443i −21.1524 1.50000 − 2.59808i 6.45999 − 11.1890i
37.2 −1.59268 1.50000 − 0.866025i −1.46337 0.153524 − 0.265911i −2.38902 + 1.37930i −5.15518 − 8.92903i 8.70140 1.50000 − 2.59808i −0.244514 + 0.423511i
37.3 −1.36194 1.50000 − 0.866025i −2.14513 −1.77228 + 3.06968i −2.04291 + 1.17947i 6.12167 + 10.6031i 8.36928 1.50000 − 2.59808i 2.41373 − 4.18071i
37.4 0.778290 1.50000 − 0.866025i −3.39427 4.43266 − 7.67760i 1.16743 − 0.674019i −0.495870 − 0.858872i −5.75488 1.50000 − 2.59808i 3.44990 − 5.97540i
37.5 2.87933 1.50000 − 0.866025i 4.29054 −1.06938 + 1.85222i 4.31899 − 2.49357i 0.215500 + 0.373257i 0.836560 1.50000 − 2.59808i −3.07910 + 5.33316i
88.1 −3.70300 1.50000 + 0.866025i 9.71222 −1.74453 − 3.02161i −5.55450 − 3.20689i −1.18612 + 2.05443i −21.1524 1.50000 + 2.59808i 6.45999 + 11.1890i
88.2 −1.59268 1.50000 + 0.866025i −1.46337 0.153524 + 0.265911i −2.38902 − 1.37930i −5.15518 + 8.92903i 8.70140 1.50000 + 2.59808i −0.244514 − 0.423511i
88.3 −1.36194 1.50000 + 0.866025i −2.14513 −1.77228 − 3.06968i −2.04291 − 1.17947i 6.12167 − 10.6031i 8.36928 1.50000 + 2.59808i 2.41373 + 4.18071i
88.4 0.778290 1.50000 + 0.866025i −3.39427 4.43266 + 7.67760i 1.16743 + 0.674019i −0.495870 + 0.858872i −5.75488 1.50000 + 2.59808i 3.44990 + 5.97540i
88.5 2.87933 1.50000 + 0.866025i 4.29054 −1.06938 − 1.85222i 4.31899 + 2.49357i 0.215500 − 0.373257i 0.836560 1.50000 + 2.59808i −3.07910 − 5.33316i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.3.i.a ✓ 10
3.b odd 2 1 279.3.u.f 10
31.e odd 6 1 inner 93.3.i.a ✓ 10
93.g even 6 1 279.3.u.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.3.i.a ✓ 10 1.a even 1 1 trivial
93.3.i.a ✓ 10 31.e odd 6 1 inner
279.3.u.f 10 3.b odd 2 1
279.3.u.f 10 93.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 3T_{2}^{4} - 9T_{2}^{3} - 25T_{2}^{2} + 18 \) acting on \(S_{3}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + 3 T^{4} - 9 T^{3} + \cdots + 18)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + 54 T^{8} + \cdots + 5184 \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 1305669132 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 918330048 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 769468730112 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 1519908196 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 277350153408 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 69078706969068 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 819628286980801 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 716898078278028 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 89924785505424 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 586117892507328 \) Copy content Toggle raw display
$47$ \( (T^{5} - 96 T^{4} + \cdots + 23973156)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 91\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 49\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( (T^{5} + 335 T^{4} + \cdots - 2783190041)^{2} \) Copy content Toggle raw display
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