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Properties

Label 931.2.p.g
Level $931$
Weight $2$
Character orbit 931.p
Analytic conductor $7.434$
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] = [931,2,Mod(293,931)]
 
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([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.p (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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 26x^{14} + 265x^{12} + 1335x^{10} + 3450x^{8} + 4344x^{6} + 2376x^{4} + 423x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{9} + \beta_{6} - 1) q^{3} + (\beta_{15} + \beta_{14} + \beta_{12} + \cdots - 2) q^{4}+ \cdots + (\beta_{14} + \beta_{12} - \beta_{11} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{9} + \beta_{6} - 1) q^{3} + (\beta_{15} + \beta_{14} + \beta_{12} + \cdots - 2) q^{4}+ \cdots + (2 \beta_{14} + 2 \beta_{13} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 10 q^{4} + 9 q^{5} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 10 q^{4} + 9 q^{5} + 6 q^{6} - 8 q^{9} + 2 q^{10} + 18 q^{11} - 34 q^{12} - 15 q^{13} - 12 q^{15} - 22 q^{16} + 18 q^{17} - 8 q^{19} - 27 q^{22} - 15 q^{23} + 9 q^{24} + 13 q^{25} + 38 q^{27} - 18 q^{29} + 64 q^{30} - 54 q^{31} - 27 q^{32} + 12 q^{33} + 16 q^{34} + 4 q^{36} + 27 q^{38} + 18 q^{39} - 9 q^{40} - 9 q^{41} - 9 q^{43} - 9 q^{44} + 36 q^{47} - 35 q^{48} + 21 q^{51} - 18 q^{53} - 18 q^{54} - 22 q^{57} + 40 q^{58} + 9 q^{59} - 60 q^{60} - 57 q^{61} + 45 q^{62} - 22 q^{64} - 45 q^{66} - 18 q^{67} - 30 q^{69} - 9 q^{71} - 81 q^{72} + 63 q^{73} - 9 q^{74} + 20 q^{75} - 98 q^{76} + 54 q^{78} - 21 q^{79} + 27 q^{80} - 20 q^{81} + 12 q^{82} + 31 q^{85} + 63 q^{86} + 12 q^{89} - 28 q^{90} + 54 q^{92} + 6 q^{93} - 98 q^{94} + 33 q^{95} - q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 26x^{14} + 265x^{12} + 1335x^{10} + 3450x^{8} + 4344x^{6} + 2376x^{4} + 423x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11 \nu^{14} - 270 \nu^{12} - 2494 \nu^{10} - 10605 \nu^{8} - 19980 \nu^{6} - 12276 \nu^{4} + \cdots - 243 ) / 1866 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{15} + 238 \nu^{13} + 1505 \nu^{11} + 1504 \nu^{9} - 19086 \nu^{7} - 72444 \nu^{5} + \cdots - 24417 \nu ) / 2799 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{14} - 659\nu^{12} - 5896\nu^{10} - 24450\nu^{8} - 43479\nu^{6} - 13521\nu^{4} + 27270\nu^{2} + 7524 ) / 2799 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12 \nu^{15} - 96 \nu^{14} - 238 \nu^{13} - 2526 \nu^{12} - 1505 \nu^{11} - 25413 \nu^{10} + \cdots + 5004 ) / 5598 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27 \nu^{15} + 691 \nu^{13} + 6885 \nu^{11} + 33551 \nu^{9} + 82545 \nu^{7} + 97308 \nu^{5} + \cdots + 933 ) / 1866 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 211 \nu^{15} + 25 \nu^{14} - 5377 \nu^{13} + 755 \nu^{12} - 53183 \nu^{11} + 8863 \nu^{10} + \cdots + 7677 ) / 5598 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 211 \nu^{15} - 25 \nu^{14} - 5377 \nu^{13} - 755 \nu^{12} - 53183 \nu^{11} - 8863 \nu^{10} + \cdots - 7677 ) / 5598 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 225 \nu^{15} + 28 \nu^{14} + 5862 \nu^{13} + 659 \nu^{12} + 60174 \nu^{11} + 5896 \nu^{10} + \cdots - 10323 ) / 5598 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 234 \nu^{15} + 3 \nu^{14} + 6196 \nu^{13} - 96 \nu^{12} + 64646 \nu^{11} - 2967 \nu^{10} + \cdots - 4005 ) / 5598 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 234 \nu^{15} - 3 \nu^{14} + 6196 \nu^{13} + 96 \nu^{12} + 64646 \nu^{11} + 2967 \nu^{10} + \cdots + 4005 ) / 5598 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 246 \nu^{15} - 51 \nu^{14} + 6434 \nu^{13} - 1167 \nu^{12} + 66151 \nu^{11} - 10206 \nu^{10} + \cdots - 13086 ) / 5598 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 308 \nu^{15} + 109 \nu^{14} - 7871 \nu^{13} + 2732 \nu^{12} - 78229 \nu^{11} + 26551 \nu^{10} + \cdots - 3699 ) / 5598 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 308 \nu^{15} - 109 \nu^{14} - 7871 \nu^{13} - 2732 \nu^{12} - 78229 \nu^{11} - 26551 \nu^{10} + \cdots + 3699 ) / 5598 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 296 \nu^{15} + 190 \nu^{14} - 7633 \nu^{13} + 4805 \nu^{12} - 76724 \nu^{11} + 47206 \nu^{10} + \cdots + 19719 ) / 5598 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} + \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{13} - \beta_{12} + \beta_{11} + \beta_{3} - \beta_{2} - 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} + 8 \beta_{11} - 7 \beta_{10} + 7 \beta_{8} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{15} - \beta_{14} - 10 \beta_{13} + 9 \beta_{12} - 8 \beta_{11} + \beta_{10} - 2 \beta_{9} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{15} - 10 \beta_{14} + 19 \beta_{13} + 9 \beta_{12} - 57 \beta_{11} + 48 \beta_{10} + \cdots - 134 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 73 \beta_{15} + 8 \beta_{14} + 81 \beta_{13} - 73 \beta_{12} + 58 \beta_{11} - 15 \beta_{10} + \cdots + 105 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 66 \beta_{15} + 79 \beta_{14} - 145 \beta_{13} - 66 \beta_{12} + 399 \beta_{11} - 333 \beta_{10} + \cdots + 860 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 574 \beta_{15} - 38 \beta_{14} - 612 \beta_{13} + 574 \beta_{12} - 418 \beta_{11} + 156 \beta_{10} + \cdots - 933 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 456 \beta_{15} - 583 \beta_{14} + 1039 \beta_{13} + 456 \beta_{12} - 2787 \beta_{11} + 2331 \beta_{10} + \cdots - 5684 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4432 \beta_{15} + 56 \beta_{14} + 4488 \beta_{13} - 4432 \beta_{12} + 3022 \beta_{11} - 1410 \beta_{10} + \cdots + 7902 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3078 \beta_{15} + 4201 \beta_{14} - 7279 \beta_{13} - 3078 \beta_{12} + 19497 \beta_{11} - 16419 \beta_{10} + \cdots + 38252 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 33784 \beta_{15} + 1330 \beta_{14} - 32454 \beta_{13} + 33784 \beta_{12} - 21892 \beta_{11} + \cdots - 64695 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 20562 \beta_{15} - 30049 \beta_{14} + 50611 \beta_{13} + 20562 \beta_{12} - 136752 \beta_{11} + \cdots - 260537 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 255067 \beta_{15} - 21952 \beta_{14} + 233115 \beta_{13} - 255067 \beta_{12} + 158674 \beta_{11} + \cdots + 516756 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(248\) \(344\)
\(\chi(n)\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
− 2.53791i
− 2.43648i
− 1.24034i
0.156525i
0.525923i
0.918115i
1.92321i
2.69097i
2.53791i
2.43648i
1.24034i
− 0.156525i
− 0.525923i
− 0.918115i
− 1.92321i
− 2.69097i
−2.19790 − 1.26896i −1.44400 + 2.50108i 2.22050 + 3.84603i 2.26105 + 1.30542i 6.34752 − 3.66474i 0 − 6.19507i −2.67026 − 4.62502i −3.31303 − 5.73834i
293.2 −2.11006 − 1.21824i 0.259752 − 0.449904i 1.96822 + 3.40907i −2.56167 − 1.47898i −1.09618 + 0.632882i 0 − 4.71813i 1.36506 + 2.36435i 3.60351 + 6.24146i
293.3 −1.07417 − 0.620172i −0.434588 + 0.752729i −0.230773 − 0.399710i 2.73585 + 1.57954i 0.933643 − 0.539039i 0 3.05316i 1.12227 + 1.94382i −1.95918 − 3.39339i
293.4 0.135555 + 0.0782626i 1.22336 − 2.11893i −0.987750 − 1.71083i 1.55540 + 0.898012i 0.331666 − 0.191487i 0 − 0.622266i −1.49324 − 2.58637i 0.140561 + 0.243460i
293.5 0.455463 + 0.262961i −0.0476887 + 0.0825992i −0.861703 − 1.49251i −3.27538 − 1.89104i −0.0434408 + 0.0250806i 0 − 1.95822i 1.49545 + 2.59020i −0.994541 − 1.72259i
293.6 0.795111 + 0.459058i −1.62407 + 2.81297i −0.578532 − 1.00205i 1.25521 + 0.724696i −2.58264 + 1.49109i 0 − 2.89855i −3.77522 − 6.53887i 0.665355 + 1.15243i
293.7 1.66554 + 0.961603i 0.905328 − 1.56807i 0.849360 + 1.47113i 2.41986 + 1.39711i 3.01573 − 1.74113i 0 − 0.579425i −0.139237 − 0.241166i 2.68692 + 4.65388i
293.8 2.33045 + 1.34549i −0.838099 + 1.45163i 2.62067 + 4.53913i 0.109680 + 0.0633240i −3.90630 + 2.25530i 0 8.72235i 0.0951799 + 0.164856i 0.170403 + 0.295147i
734.1 −2.19790 + 1.26896i −1.44400 − 2.50108i 2.22050 − 3.84603i 2.26105 − 1.30542i 6.34752 + 3.66474i 0 6.19507i −2.67026 + 4.62502i −3.31303 + 5.73834i
734.2 −2.11006 + 1.21824i 0.259752 + 0.449904i 1.96822 − 3.40907i −2.56167 + 1.47898i −1.09618 − 0.632882i 0 4.71813i 1.36506 − 2.36435i 3.60351 − 6.24146i
734.3 −1.07417 + 0.620172i −0.434588 − 0.752729i −0.230773 + 0.399710i 2.73585 − 1.57954i 0.933643 + 0.539039i 0 − 3.05316i 1.12227 − 1.94382i −1.95918 + 3.39339i
734.4 0.135555 − 0.0782626i 1.22336 + 2.11893i −0.987750 + 1.71083i 1.55540 − 0.898012i 0.331666 + 0.191487i 0 0.622266i −1.49324 + 2.58637i 0.140561 − 0.243460i
734.5 0.455463 − 0.262961i −0.0476887 − 0.0825992i −0.861703 + 1.49251i −3.27538 + 1.89104i −0.0434408 − 0.0250806i 0 1.95822i 1.49545 − 2.59020i −0.994541 + 1.72259i
734.6 0.795111 − 0.459058i −1.62407 − 2.81297i −0.578532 + 1.00205i 1.25521 − 0.724696i −2.58264 − 1.49109i 0 2.89855i −3.77522 + 6.53887i 0.665355 − 1.15243i
734.7 1.66554 − 0.961603i 0.905328 + 1.56807i 0.849360 − 1.47113i 2.41986 − 1.39711i 3.01573 + 1.74113i 0 0.579425i −0.139237 + 0.241166i 2.68692 − 4.65388i
734.8 2.33045 − 1.34549i −0.838099 − 1.45163i 2.62067 − 4.53913i 0.109680 − 0.0633240i −3.90630 − 2.25530i 0 − 8.72235i 0.0951799 − 0.164856i 0.170403 − 0.295147i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.p.g 16
7.b odd 2 1 931.2.p.h 16
7.c even 3 1 133.2.i.d ✓ 16
7.c even 3 1 931.2.s.g 16
7.d odd 6 1 133.2.s.d yes 16
7.d odd 6 1 931.2.i.g 16
19.d odd 6 1 931.2.p.h 16
133.i even 6 1 931.2.s.g 16
133.j odd 6 1 133.2.s.d yes 16
133.n odd 6 1 931.2.i.g 16
133.p even 6 1 inner 931.2.p.g 16
133.s even 6 1 133.2.i.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.i.d ✓ 16 7.c even 3 1
133.2.i.d ✓ 16 133.s even 6 1
133.2.s.d yes 16 7.d odd 6 1
133.2.s.d yes 16 133.j odd 6 1
931.2.i.g 16 7.d odd 6 1
931.2.i.g 16 133.n odd 6 1
931.2.p.g 16 1.a even 1 1 trivial
931.2.p.g 16 133.p even 6 1 inner
931.2.p.h 16 7.b odd 2 1
931.2.p.h 16 19.d odd 6 1
931.2.s.g 16 7.c even 3 1
931.2.s.g 16 133.i even 6 1

Hecke kernels

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

\( T_{2}^{16} - 13 T_{2}^{14} + 121 T_{2}^{12} - 9 T_{2}^{11} - 537 T_{2}^{10} + 99 T_{2}^{9} + 1731 T_{2}^{8} + \cdots + 9 \) Copy content Toggle raw display
\( T_{3}^{16} + 4 T_{3}^{15} + 24 T_{3}^{14} + 46 T_{3}^{13} + 211 T_{3}^{12} + 333 T_{3}^{11} + 1269 T_{3}^{10} + \cdots + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 13 T^{14} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{16} - 9 T^{15} + \cdots + 7225 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 9 T^{7} - 3 T^{6} + \cdots + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 15 T^{15} + \cdots + 18225 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 179051161 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10126195641 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 1548501201 \) Copy content Toggle raw display
$31$ \( (T^{8} + 27 T^{7} + \cdots - 1323)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 270372249 \) Copy content Toggle raw display
$41$ \( T^{16} + 9 T^{15} + \cdots + 49154121 \) Copy content Toggle raw display
$43$ \( T^{16} + 9 T^{15} + \cdots + 10093329 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 27397201441 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 4685904443025 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 149597361 \) Copy content Toggle raw display
$61$ \( T^{16} + 57 T^{15} + \cdots + 13329801 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 3534336080289 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 4065985912041 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 9244245609 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 3602134489041 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 337915503025 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 390743759025 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 1688515726041 \) Copy content Toggle raw display
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