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Properties

Label 322.2.g.a
Level $322$
Weight $2$
Character orbit 322.g
Analytic conductor $2.571$
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] = [322,2,Mod(45,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.g (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: \(2.57118294509\)
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: \( x^{16} + 18x^{14} + 226x^{12} + 1434x^{10} + 6585x^{8} + 14406x^{6} + 22423x^{4} + 8085x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 7 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_1 q^{3} + (\beta_{5} - 1) q^{4} + \beta_{6} q^{5} + ( - \beta_{5} - \beta_{3} + 1) q^{6} + \beta_{12} q^{7} + q^{8} + (\beta_{7} + 2 \beta_{5} + \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_1 q^{3} + (\beta_{5} - 1) q^{4} + \beta_{6} q^{5} + ( - \beta_{5} - \beta_{3} + 1) q^{6} + \beta_{12} q^{7} + q^{8} + (\beta_{7} + 2 \beta_{5} + \beta_{3} - 1) q^{9} - \beta_{13} q^{10} - \beta_{9} q^{11} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{12} + ( - \beta_{8} - \beta_{5} - \beta_{3} + \cdots + 1) q^{13}+ \cdots + ( - 2 \beta_{14} - 6 \beta_{13} + \cdots - 2 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{2} + 6 q^{3} - 8 q^{4} + 16 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{2} + 6 q^{3} - 8 q^{4} + 16 q^{8} + 10 q^{9} - 6 q^{12} - 8 q^{16} + 10 q^{18} - 4 q^{23} + 6 q^{24} + 2 q^{25} - 6 q^{26} + 16 q^{29} - 24 q^{31} - 8 q^{32} + 4 q^{35} - 20 q^{36} + 22 q^{39} - 4 q^{46} + 30 q^{47} - 58 q^{49} - 4 q^{50} + 6 q^{52} + 54 q^{54} - 8 q^{58} + 36 q^{59} + 16 q^{64} - 32 q^{70} - 12 q^{71} + 10 q^{72} - 24 q^{73} - 96 q^{75} - 38 q^{77} - 44 q^{78} - 36 q^{81} - 6 q^{82} + 24 q^{85} + 42 q^{87} + 8 q^{92} - 38 q^{93} - 30 q^{94} - 38 q^{95} - 6 q^{96} + 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 18x^{14} + 226x^{12} + 1434x^{10} + 6585x^{8} + 14406x^{6} + 22423x^{4} + 8085x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16016843 \nu^{14} - 357616089 \nu^{12} - 4992038792 \nu^{10} - 39156407581 \nu^{8} + \cdots - 650423475226 ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25899908 \nu^{14} - 838784719 \nu^{12} - 12366108886 \nu^{10} - 116812942191 \nu^{8} + \cdots - 2481148141444 ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31483524 \nu^{14} + 647630005 \nu^{12} + 8339521600 \nu^{10} + 58435186409 \nu^{8} + \cdots + 362977184262 ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46792861 \nu^{14} + 853307341 \nu^{12} + 11030105099 \nu^{10} + 71855612665 \nu^{8} + \cdots - 1235543183245 ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 68184274 \nu^{14} + 1196540914 \nu^{12} + 14913954672 \nu^{10} + 91738182609 \nu^{8} + \cdots + 506321481001 ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6975738 \nu^{15} - 112789235 \nu^{13} - 1368597087 \nu^{11} - 7411650444 \nu^{9} + \cdots + 234153598717 \nu ) / 58562202867 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 294128509 \nu^{14} - 5129397229 \nu^{12} - 63539668261 \nu^{10} - 386835300380 \nu^{8} + \cdots - 1717473487905 ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 379875222 \nu^{14} - 6640569708 \nu^{12} - 81813898297 \nu^{10} - 492908101323 \nu^{8} + \cdots + 671866451907 ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 500045446 \nu^{15} + 8793640344 \nu^{13} + 109886589690 \nu^{11} + 679206852725 \nu^{9} + \cdots - 2682037157653 \nu ) / 2869547940483 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 159118374 \nu^{15} + 2937898622 \nu^{13} + 36966138724 \nu^{11} + 239063115702 \nu^{9} + \cdots - 421218231742 \nu ) / 409935420069 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1482542225 \nu^{15} + 24724308082 \nu^{13} + 301479751327 \nu^{11} + \cdots - 9325440613244 \nu ) / 2869547940483 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1812976145 \nu^{15} + 33070488812 \nu^{13} + 418326213629 \nu^{11} + \cdots + 15022740886300 \nu ) / 2869547940483 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1849992808 \nu^{15} + 33041863886 \nu^{13} + 413665642185 \nu^{11} + \cdots + 14580373128671 \nu ) / 2869547940483 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2619126371 \nu^{15} - 45707452382 \nu^{13} - 565731629381 \nu^{11} + \cdots + 9891611938649 \nu ) / 2869547940483 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3550950679 \nu^{15} - 62041171232 \nu^{13} - 769038087803 \nu^{11} + \cdots + 2315919560941 \nu ) / 2869547940483 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - 2\beta_{14} - 3\beta_{11} - 3\beta_{10} - 2\beta_{9} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 5\beta_{5} - \beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -16\beta_{15} + 31\beta_{14} - 7\beta_{12} + 8\beta_{11} + 15\beta_{10} + 17\beta_{9} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 9\beta_{7} - 36\beta_{5} + 2\beta_{3} + 2\beta_{2} - 11\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 213 \beta_{15} - 253 \beta_{14} - 140 \beta_{13} + 182 \beta_{12} + 142 \beta_{11} + \cdots + 182 \beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{8} + 33\beta_{5} + 77\beta_{4} + 33\beta_{3} - 13\beta_{2} + 143\beta _1 + 289 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 659 \beta_{15} - 366 \beta_{14} + 952 \beta_{13} - 1001 \beta_{12} - 1928 \beta_{11} + \cdots - 1120 \beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -272\beta_{8} + 669\beta_{7} + 2052\beta_{5} - 669\beta_{4} - 796\beta_{3} - 136\beta_{2} - 398\beta _1 - 2052 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 11468 \beta_{15} + 28780 \beta_{14} + 10192 \beta_{13} - 10255 \beta_{12} + 6175 \beta_{11} + \cdots - 11074 \beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1339\beta_{8} - 5932\beta_{7} - 21473\beta_{5} + 4260\beta_{3} + 2678\beta_{2} - 10192\beta _1 - 4260 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 162599 \beta_{15} - 249145 \beta_{14} - 199997 \beta_{13} + 202328 \beta_{12} + 104708 \beta_{11} + \cdots + 202328 \beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12870\beta_{8} + 43023\beta_{5} + 53478\beta_{4} + 43023\beta_{3} - 12870\beta_{2} + 139524\beta _1 + 192335 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 542144 \beta_{15} - 228601 \beta_{14} + 855687 \beta_{13} - 976668 \beta_{12} + \cdots - 717801 \beta_{6} ) / 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 244482 \beta_{8} + 487880 \beta_{7} + 1326778 \beta_{5} - 487880 \beta_{4} - 841974 \beta_{3} + \cdots - 1326778 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 8908181 \beta_{15} + 24524552 \beta_{14} + 10931949 \beta_{13} - 9308978 \beta_{12} + \cdots - 12167484 \beta_{6} ) / 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(\beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
1.52769 + 2.64604i
−1.52769 − 2.64604i
−0.768626 − 1.33130i
0.768626 + 1.33130i
−0.306090 − 0.530164i
0.306090 + 0.530164i
1.21724 + 2.10833i
−1.21724 − 2.10833i
1.52769 − 2.64604i
−1.52769 + 2.64604i
−0.768626 + 1.33130i
0.768626 − 1.33130i
−0.306090 + 0.530164i
0.306090 − 0.530164i
1.21724 − 2.10833i
−1.21724 + 2.10833i
−0.500000 − 0.866025i −1.86037 − 1.07408i −0.500000 + 0.866025i −0.264227 − 0.457654i 2.14817i −1.26346 − 2.32458i 1.00000 0.807314 + 1.39831i −0.264227 + 0.457654i
45.2 −0.500000 − 0.866025i −1.86037 − 1.07408i −0.500000 + 0.866025i 0.264227 + 0.457654i 2.14817i 1.26346 + 2.32458i 1.00000 0.807314 + 1.39831i 0.264227 − 0.457654i
45.3 −0.500000 − 0.866025i −0.508912 − 0.293820i −0.500000 + 0.866025i −0.863449 − 1.49554i 0.587641i 1.63208 + 2.08239i 1.00000 −1.32734 − 2.29902i −0.863449 + 1.49554i
45.4 −0.500000 − 0.866025i −0.508912 − 0.293820i −0.500000 + 0.866025i 0.863449 + 1.49554i 0.587641i −1.63208 − 2.08239i 1.00000 −1.32734 − 2.29902i 0.863449 − 1.49554i
45.5 −0.500000 − 0.866025i 1.05590 + 0.609623i −0.500000 + 0.866025i −1.23610 − 2.14099i − 1.21925i 1.54219 − 2.14980i 1.00000 −0.756718 − 1.31067i −1.23610 + 2.14099i
45.6 −0.500000 − 0.866025i 1.05590 + 0.609623i −0.500000 + 0.866025i 1.23610 + 2.14099i − 1.21925i −1.54219 + 2.14980i 1.00000 −0.756718 − 1.31067i 1.23610 − 2.14099i
45.7 −0.500000 − 0.866025i 2.81338 + 1.62431i −0.500000 + 0.866025i −1.55135 − 2.68702i − 3.24861i 0.334108 + 2.62457i 1.00000 3.77674 + 6.54151i −1.55135 + 2.68702i
45.8 −0.500000 − 0.866025i 2.81338 + 1.62431i −0.500000 + 0.866025i 1.55135 + 2.68702i − 3.24861i −0.334108 − 2.62457i 1.00000 3.77674 + 6.54151i 1.55135 − 2.68702i
229.1 −0.500000 + 0.866025i −1.86037 + 1.07408i −0.500000 − 0.866025i −0.264227 + 0.457654i − 2.14817i −1.26346 + 2.32458i 1.00000 0.807314 − 1.39831i −0.264227 − 0.457654i
229.2 −0.500000 + 0.866025i −1.86037 + 1.07408i −0.500000 − 0.866025i 0.264227 − 0.457654i − 2.14817i 1.26346 − 2.32458i 1.00000 0.807314 − 1.39831i 0.264227 + 0.457654i
229.3 −0.500000 + 0.866025i −0.508912 + 0.293820i −0.500000 − 0.866025i −0.863449 + 1.49554i − 0.587641i 1.63208 − 2.08239i 1.00000 −1.32734 + 2.29902i −0.863449 − 1.49554i
229.4 −0.500000 + 0.866025i −0.508912 + 0.293820i −0.500000 − 0.866025i 0.863449 − 1.49554i − 0.587641i −1.63208 + 2.08239i 1.00000 −1.32734 + 2.29902i 0.863449 + 1.49554i
229.5 −0.500000 + 0.866025i 1.05590 − 0.609623i −0.500000 − 0.866025i −1.23610 + 2.14099i 1.21925i 1.54219 + 2.14980i 1.00000 −0.756718 + 1.31067i −1.23610 − 2.14099i
229.6 −0.500000 + 0.866025i 1.05590 − 0.609623i −0.500000 − 0.866025i 1.23610 − 2.14099i 1.21925i −1.54219 − 2.14980i 1.00000 −0.756718 + 1.31067i 1.23610 + 2.14099i
229.7 −0.500000 + 0.866025i 2.81338 − 1.62431i −0.500000 − 0.866025i −1.55135 + 2.68702i 3.24861i 0.334108 − 2.62457i 1.00000 3.77674 − 6.54151i −1.55135 − 2.68702i
229.8 −0.500000 + 0.866025i 2.81338 − 1.62431i −0.500000 − 0.866025i 1.55135 − 2.68702i 3.24861i −0.334108 + 2.62457i 1.00000 3.77674 − 6.54151i 1.55135 + 2.68702i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.g.a ✓ 16
7.c even 3 1 2254.2.c.c 16
7.d odd 6 1 inner 322.2.g.a ✓ 16
7.d odd 6 1 2254.2.c.c 16
23.b odd 2 1 inner 322.2.g.a ✓ 16
161.f odd 6 1 2254.2.c.c 16
161.g even 6 1 inner 322.2.g.a ✓ 16
161.g even 6 1 2254.2.c.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.g.a ✓ 16 1.a even 1 1 trivial
322.2.g.a ✓ 16 7.d odd 6 1 inner
322.2.g.a ✓ 16 23.b odd 2 1 inner
322.2.g.a ✓ 16 161.g even 6 1 inner
2254.2.c.c 16 7.c even 3 1
2254.2.c.c 16 7.d odd 6 1
2254.2.c.c 16 161.f odd 6 1
2254.2.c.c 16 161.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 3T_{3}^{7} - 4T_{3}^{6} + 21T_{3}^{5} + 35T_{3}^{4} - 63T_{3}^{3} - 8T_{3}^{2} + 45T_{3} + 25 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} - 3 T^{7} - 4 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 19 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( T^{16} + 29 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 44 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$13$ \( (T^{8} + 67 T^{6} + \cdots + 51529)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 671898241 \) Copy content Toggle raw display
$19$ \( T^{16} + 19 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} + \cdots - 335)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 12 T^{7} + \cdots + 561001)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1958569461121 \) Copy content Toggle raw display
$41$ \( (T^{8} + 67 T^{6} + \cdots + 51529)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 260 T^{6} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 15 T^{7} + \cdots + 14161)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13731656640625 \) Copy content Toggle raw display
$59$ \( (T^{8} - 18 T^{7} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 212558803681 \) Copy content Toggle raw display
$67$ \( T^{16} - 245 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$71$ \( (T^{4} + 3 T^{3} + \cdots + 693)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 12 T^{7} + \cdots + 561001)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 4408324359201 \) Copy content Toggle raw display
$83$ \( (T^{8} - 520 T^{6} + \cdots + 2941225)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{8} - 456 T^{6} + \cdots + 62236321)^{2} \) Copy content Toggle raw display
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