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Properties

Label 931.2.f.p
Level $931$
Weight $2$
Character orbit 931.f
Analytic conductor $7.434$
Analytic rank $0$
Dimension $14$
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] = [931,2,Mod(324,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.f (of order \(3\), degree \(2\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} + 15 x^{12} - 18 x^{11} + 86 x^{10} - 96 x^{9} + 310 x^{8} - 68 x^{7} + 220 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} - \beta_{7} q^{3} + ( - \beta_{12} - \beta_{9} - \beta_{4} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{11} + 2 \beta_{4} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} - \beta_{7} q^{3} + ( - \beta_{12} - \beta_{9} - \beta_{4} + \cdots - 1) q^{4}+ \cdots + ( - 2 \beta_{13} - \beta_{12} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 2 q^{3} - 10 q^{4} - 2 q^{5} + 8 q^{6} + 24 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} - 2 q^{3} - 10 q^{4} - 2 q^{5} + 8 q^{6} + 24 q^{8} - 15 q^{9} - 7 q^{11} - 22 q^{12} - 12 q^{13} + 4 q^{15} - 24 q^{16} + 19 q^{17} + 12 q^{18} + 7 q^{19} + 16 q^{20} - 12 q^{22} + q^{23} + 20 q^{24} + 3 q^{25} + 12 q^{26} + 28 q^{27} + 48 q^{29} + 20 q^{30} - 26 q^{32} - 14 q^{33} - 12 q^{34} + 92 q^{36} - 8 q^{37} + 2 q^{38} - 16 q^{39} - 10 q^{40} + 8 q^{41} + 8 q^{43} - 26 q^{44} + 14 q^{45} + 16 q^{46} + 5 q^{47} + 56 q^{48} + 32 q^{50} + 4 q^{51} + 42 q^{52} - 20 q^{53} - 24 q^{54} + 60 q^{55} - 4 q^{57} - 16 q^{59} + 44 q^{60} + 5 q^{61} - 48 q^{62} + 64 q^{64} - 26 q^{65} + 68 q^{66} + 4 q^{67} + 22 q^{68} + 72 q^{69} + 24 q^{71} + 3 q^{73} + 4 q^{74} + 18 q^{75} - 20 q^{76} - 28 q^{78} + 20 q^{79} - 4 q^{80} - 27 q^{81} - 48 q^{82} + 22 q^{83} + 52 q^{85} - 36 q^{86} - 16 q^{87} + 32 q^{88} - 10 q^{89} - 64 q^{90} - 60 q^{92} + 4 q^{93} + 16 q^{94} + 2 q^{95} - 12 q^{96} + 8 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + 15 x^{12} - 18 x^{11} + 86 x^{10} - 96 x^{9} + 310 x^{8} - 68 x^{7} + 220 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 455887 \nu^{13} + 10810324 \nu^{12} - 30588081 \nu^{11} + 133735751 \nu^{10} + \cdots + 90688665 ) / 140898768 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9052729 \nu^{13} - 11551136 \nu^{12} + 94384895 \nu^{11} + 59200723 \nu^{10} + 564427011 \nu^{9} + \cdots + 183122713 ) / 140898768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4589237 \nu^{13} - 20657433 \nu^{12} + 83266376 \nu^{11} - 166633057 \nu^{10} + 423863356 \nu^{9} + \cdots - 60671948 ) / 70449384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6889722 \nu^{13} + 14427821 \nu^{12} - 84026791 \nu^{11} + 29188974 \nu^{10} - 473155759 \nu^{9} + \cdots - 75038621 ) / 70449384 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15607051 \nu^{13} - 41406040 \nu^{12} + 222149845 \nu^{11} - 214107683 \nu^{10} + 1310938801 \nu^{9} + \cdots - 9052729 ) / 140898768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16982533 \nu^{13} - 42193334 \nu^{12} + 225639633 \nu^{11} - 173306357 \nu^{10} + \cdots - 438044805 ) / 140898768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24434527 \nu^{13} - 54943322 \nu^{12} + 303609363 \nu^{11} - 141368879 \nu^{10} + \cdots + 13630233 ) / 140898768 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8941589 \nu^{13} - 22036638 \nu^{12} + 117554333 \nu^{11} - 82223061 \nu^{10} + 648843457 \nu^{9} + \cdots + 65661043 ) / 46966256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13688965 \nu^{13} - 55339709 \nu^{12} + 250761696 \nu^{11} - 467596355 \nu^{10} + 1470533040 \nu^{9} + \cdots - 40896666 ) / 70449384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32010647 \nu^{13} + 81721644 \nu^{12} - 428971229 \nu^{11} + 332443339 \nu^{10} + \cdots - 211856059 ) / 140898768 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12193313 \nu^{13} + 23556200 \nu^{12} - 144685831 \nu^{11} + 25691285 \nu^{10} + \cdots - 156534833 ) / 46966256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 103171355 \nu^{13} - 310248186 \nu^{12} + 1539171695 \nu^{11} - 1840528555 \nu^{10} + \cdots - 265186067 ) / 140898768 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 34398638 \nu^{13} + 106343273 \nu^{12} - 522590557 \nu^{11} + 657680818 \nu^{10} + \cdots + 62115805 ) / 35224692 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{4} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + \beta_{12} + \beta_{9} - 2\beta_{7} - 2\beta_{5} - 5\beta_{4} + \beta_{3} + \beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{13} + 3\beta_{12} - 4\beta_{10} + 4\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{11} - 13\beta_{10} + 14\beta_{8} + 33\beta_{4} - 11\beta_{3} - 9\beta_{2} + 18\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 59 \beta_{13} - 41 \beta_{12} + 22 \beta_{11} - 67 \beta_{9} + 20 \beta_{7} - 22 \beta_{6} + \cdots + 69 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 54 \beta_{13} - 38 \beta_{12} + 71 \beta_{10} - 71 \beta_{9} - 50 \beta_{8} + 50 \beta_{7} + \cdots + 127 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -214\beta_{11} + 575\beta_{10} - 184\beta_{8} - 609\beta_{4} + 461\beta_{3} + 301\beta_{2} - 364\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1021 \beta_{13} + 657 \beta_{12} - 578 \beta_{11} + 1415 \beta_{9} - 762 \beta_{7} + 578 \beta_{6} + \cdots - 2077 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1894 \beta_{13} + 1179 \beta_{12} - 2517 \beta_{10} + 2517 \beta_{9} + 839 \beta_{8} - 839 \beta_{7} + \cdots - 2718 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5690 \beta_{11} - 13483 \beta_{10} + 6146 \beta_{8} + 17645 \beta_{4} - 9471 \beta_{3} + \cdots + 12834 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 32263 \beta_{13} - 19429 \beta_{12} + 18524 \beta_{11} - 44641 \beta_{9} + 15260 \beta_{7} + \cdots + 48813 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 43464 \beta_{13} - 25798 \beta_{12} + 62762 \beta_{10} - 62762 \beta_{9} - 25846 \beta_{8} + \cdots + 76823 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 169356 \beta_{11} + 399001 \beta_{10} - 138524 \beta_{8} - 439741 \beta_{4} + 281337 \beta_{3} + \cdots - 330116 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\) \(248\) \(344\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
−0.175365 + 0.303740i
−1.14699 + 1.98665i
1.13568 − 1.96706i
0.136852 − 0.237035i
1.50352 − 2.60418i
0.431499 − 0.747378i
−0.385202 + 0.667190i
−0.175365 − 0.303740i
−1.14699 − 1.98665i
1.13568 + 1.96706i
0.136852 + 0.237035i
1.50352 + 2.60418i
0.431499 + 0.747378i
−0.385202 − 0.667190i
−1.34182 + 2.32410i 0.558381 + 0.967144i −2.60097 − 4.50501i −1.25024 + 2.16547i −2.99699 0 8.59284 0.876421 − 1.51801i −3.35519 − 5.81136i
324.2 −1.29710 + 2.24665i −1.44963 − 2.51083i −2.36495 − 4.09622i 0.929029 − 1.60913i 7.52126 0 7.08194 −2.70284 + 4.68145i 2.41009 + 4.17440i
324.3 −0.567531 + 0.982993i −1.09726 − 1.90052i 0.355816 + 0.616292i −0.915553 + 1.58578i 2.49093 0 −3.07787 −0.907972 + 1.57265i −1.03921 − 1.79996i
324.4 −0.134831 + 0.233534i 0.699343 + 1.21130i 0.963641 + 1.66908i 1.68994 − 2.92706i −0.377172 0 −1.05904 0.521837 − 0.903849i 0.455711 + 0.789315i
324.5 0.406326 − 0.703777i 0.207727 + 0.359794i 0.669798 + 1.16012i −1.33725 + 2.31618i 0.337619 0 2.71393 1.41370 − 2.44860i 1.08672 + 1.88225i
324.6 0.703252 − 1.21807i 1.59752 + 2.76699i 0.0108746 + 0.0188353i 0.147876 − 0.256129i 4.49383 0 2.84360 −3.60414 + 6.24256i −0.207988 − 0.360246i
324.7 1.23171 − 2.13338i −1.51608 − 2.62593i −2.03421 − 3.52336i −0.263807 + 0.456927i −7.46948 0 −5.09539 −3.09701 + 5.36417i 0.649866 + 1.12560i
704.1 −1.34182 − 2.32410i 0.558381 − 0.967144i −2.60097 + 4.50501i −1.25024 − 2.16547i −2.99699 0 8.59284 0.876421 + 1.51801i −3.35519 + 5.81136i
704.2 −1.29710 − 2.24665i −1.44963 + 2.51083i −2.36495 + 4.09622i 0.929029 + 1.60913i 7.52126 0 7.08194 −2.70284 − 4.68145i 2.41009 − 4.17440i
704.3 −0.567531 − 0.982993i −1.09726 + 1.90052i 0.355816 − 0.616292i −0.915553 − 1.58578i 2.49093 0 −3.07787 −0.907972 − 1.57265i −1.03921 + 1.79996i
704.4 −0.134831 − 0.233534i 0.699343 − 1.21130i 0.963641 − 1.66908i 1.68994 + 2.92706i −0.377172 0 −1.05904 0.521837 + 0.903849i 0.455711 − 0.789315i
704.5 0.406326 + 0.703777i 0.207727 − 0.359794i 0.669798 − 1.16012i −1.33725 − 2.31618i 0.337619 0 2.71393 1.41370 + 2.44860i 1.08672 − 1.88225i
704.6 0.703252 + 1.21807i 1.59752 − 2.76699i 0.0108746 − 0.0188353i 0.147876 + 0.256129i 4.49383 0 2.84360 −3.60414 − 6.24256i −0.207988 + 0.360246i
704.7 1.23171 + 2.13338i −1.51608 + 2.62593i −2.03421 + 3.52336i −0.263807 − 0.456927i −7.46948 0 −5.09539 −3.09701 − 5.36417i 0.649866 − 1.12560i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.f.p 14
7.b odd 2 1 133.2.f.d ✓ 14
7.c even 3 1 931.2.a.o 7
7.c even 3 1 inner 931.2.f.p 14
7.d odd 6 1 133.2.f.d ✓ 14
7.d odd 6 1 931.2.a.n 7
21.c even 2 1 1197.2.j.l 14
21.g even 6 1 1197.2.j.l 14
21.g even 6 1 8379.2.a.cl 7
21.h odd 6 1 8379.2.a.ck 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.d ✓ 14 7.b odd 2 1
133.2.f.d ✓ 14 7.d odd 6 1
931.2.a.n 7 7.d odd 6 1
931.2.a.o 7 7.c even 3 1
931.2.f.p 14 1.a even 1 1 trivial
931.2.f.p 14 7.c even 3 1 inner
1197.2.j.l 14 21.c even 2 1
1197.2.j.l 14 21.g even 6 1
8379.2.a.ck 7 21.h odd 6 1
8379.2.a.cl 7 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{14} + 2 T_{2}^{13} + 14 T_{2}^{12} + 12 T_{2}^{11} + 105 T_{2}^{10} + 76 T_{2}^{9} + 442 T_{2}^{8} + \cdots + 36 \) Copy content Toggle raw display
\( T_{3}^{14} + 2 T_{3}^{13} + 20 T_{3}^{12} + 20 T_{3}^{11} + 236 T_{3}^{10} + 180 T_{3}^{9} + 1468 T_{3}^{8} + \cdots + 1600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{14} + 2 T^{13} + \cdots + 1600 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 7 T^{13} + \cdots + 18225 \) Copy content Toggle raw display
$13$ \( (T^{7} + 6 T^{6} + \cdots + 1208)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} - 19 T^{13} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{14} - T^{13} + \cdots + 10017225 \) Copy content Toggle raw display
$29$ \( (T^{7} - 24 T^{6} + \cdots + 12480)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 96 T^{12} + \cdots + 74511424 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 24266162176 \) Copy content Toggle raw display
$41$ \( (T^{7} - 4 T^{6} + \cdots + 768)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 4 T^{6} + \cdots + 303284)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 3157653249 \) Copy content Toggle raw display
$53$ \( T^{14} + 20 T^{13} + \cdots + 166464 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 15383937024 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 1642357587025 \) Copy content Toggle raw display
$67$ \( T^{14} - 4 T^{13} + \cdots + 67108864 \) Copy content Toggle raw display
$71$ \( (T^{7} - 12 T^{6} + \cdots + 6648)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 26994161401 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 23004395584 \) Copy content Toggle raw display
$83$ \( (T^{7} - 11 T^{6} + \cdots - 142371)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 109495296 \) Copy content Toggle raw display
$97$ \( (T^{7} - 4 T^{6} - 258 T^{5} + \cdots - 32)^{2} \) Copy content Toggle raw display
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