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Properties

Label 945.2.p.a
Level $945$
Weight $2$
Character orbit 945.p
Analytic conductor $7.546$
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] = [945,2,Mod(433,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.p (of order \(4\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 36x^{14} + 534x^{12} - 4068x^{10} + 17755x^{8} - 42084x^{6} + 44538x^{4} + 1152x^{2} + 32761 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{10} - \beta_{9} - 3 \beta_{3}) q^{4} + ( - 2 \beta_{13} - \beta_{8}) q^{5} + ( - \beta_{12} + \beta_{10}) q^{7} + (\beta_{13} + \beta_{8} + \cdots - 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{10} - \beta_{9} - 3 \beta_{3}) q^{4} + ( - 2 \beta_{13} - \beta_{8}) q^{5} + ( - \beta_{12} + \beta_{10}) q^{7} + (\beta_{13} + \beta_{8} + \cdots - 2 \beta_{2}) q^{8}+ \cdots + (4 \beta_{15} + 8 \beta_{13} + \cdots - \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{7} - 104 q^{16} + 64 q^{25} + 64 q^{28} + 8 q^{37} + 72 q^{43} - 40 q^{46} - 40 q^{58} + 108 q^{70} - 32 q^{85} + 72 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 36x^{14} + 534x^{12} - 4068x^{10} + 17755x^{8} - 42084x^{6} + 44538x^{4} + 1152x^{2} + 32761 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 254 \nu^{14} + 5773 \nu^{12} - 34409 \nu^{10} - 116069 \nu^{8} + 994049 \nu^{6} + \cdots + 1571456 ) / 5197500 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 478 \nu^{14} + 13436 \nu^{12} - 132688 \nu^{10} + 397817 \nu^{8} + 360418 \nu^{6} + \cdots + 3271117 ) / 5197500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{14} + 67\nu^{12} - 875\nu^{10} + 5179\nu^{8} - 13789\nu^{6} + 7319\nu^{4} + 26387\nu^{2} + 14186 ) / 20790 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7254 \nu^{15} - 40544 \nu^{14} - 262773 \nu^{13} + 1387003 \nu^{12} + 4748409 \nu^{11} + \cdots + 307638641 ) / 1881495000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13847 \nu^{15} + 90500 \nu^{14} + 146411 \nu^{13} - 3031750 \nu^{12} - 11331238 \nu^{11} + \cdots - 1582664000 ) / 1881495000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17633 \nu^{15} + 104799 \nu^{14} - 411796 \nu^{13} - 4081188 \nu^{12} + 2907443 \nu^{11} + \cdots + 4605792789 ) / 1881495000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17633 \nu^{15} - 860293 \nu^{14} + 411796 \nu^{13} + 29500466 \nu^{12} - 2907443 \nu^{11} + \cdots + 7444026277 ) / 1881495000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10643 \nu^{15} - 469666 \nu^{13} + 8115278 \nu^{11} - 67312252 \nu^{9} + 260971342 \nu^{7} + \cdots + 730849198 \nu ) / 940747500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51581 \nu^{15} - 129234 \nu^{14} + 1680622 \nu^{13} + 3559908 \nu^{12} - 20455751 \nu^{11} + \cdots - 3494269074 ) / 1881495000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51581 \nu^{15} + 235662 \nu^{14} - 1680622 \nu^{13} - 6683244 \nu^{12} + 20455751 \nu^{11} + \cdots + 3491638782 ) / 1881495000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 56029 \nu^{15} + 90500 \nu^{14} + 3222323 \nu^{13} - 3031750 \nu^{12} - 67342684 \nu^{11} + \cdots - 1582664000 ) / 1881495000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28276 \nu^{15} + 881462 \nu^{13} - 11022721 \nu^{11} + 69204989 \nu^{9} + \cdots + 867188089 \nu ) / 940747500 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28276 \nu^{15} + 881462 \nu^{13} - 11022721 \nu^{11} + 69204989 \nu^{9} - 288699119 \nu^{7} + \cdots - 73559411 \nu ) / 940747500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 8873 \nu^{15} + 284676 \nu^{13} - 3497608 \nu^{11} + 18559722 \nu^{9} - 40331212 \nu^{7} + \cdots + 152529522 \nu ) / 104527500 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 343272 \nu^{15} + 40544 \nu^{14} - 11586189 \nu^{13} - 1387003 \nu^{12} + 155979687 \nu^{11} + \cdots - 307638641 ) / 1881495000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{13} + \beta_{12} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + \beta_{12} - \beta_{10} + \beta_{9} - \beta_{3} - 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{15} - 3\beta_{14} - 15\beta_{13} + 6\beta_{12} - \beta_{11} + \beta_{8} - \beta_{5} - 3\beta_{4} - \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{14} + 9 \beta_{12} - 15 \beta_{10} + 3 \beta_{9} + 4 \beta_{7} + 4 \beta_{6} - 30 \beta_{3} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 55 \beta_{15} - 60 \beta_{14} - 169 \beta_{13} + 38 \beta_{12} - 19 \beta_{11} + 50 \beta_{8} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 55 \beta_{14} - 20 \beta_{13} + 55 \beta_{12} - 190 \beta_{10} - 80 \beta_{9} - 20 \beta_{8} + \cdots + 176 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 784 \beta_{15} - 896 \beta_{14} - 1652 \beta_{13} + 111 \beta_{12} - 224 \beta_{11} + 1189 \beta_{8} + \cdots - 35 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 99 \beta_{14} - 504 \beta_{13} - 99 \beta_{12} - 1833 \beta_{10} - 2031 \beta_{9} - 504 \beta_{8} + \cdots - 494 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9489 \beta_{15} - 11052 \beta_{14} - 12682 \beta_{13} - 2993 \beta_{12} - 1833 \beta_{11} + \cdots + 603 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10646 \beta_{14} - 8280 \beta_{13} - 10646 \beta_{12} - 11359 \beta_{10} - 32651 \beta_{9} + \cdots - 38281 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 96459 \beta_{15} - 113025 \beta_{14} - 40557 \beta_{13} - 86727 \beta_{12} - 6121 \beta_{11} + \cdots + 18926 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 228150 \beta_{14} - 109692 \beta_{13} - 228150 \beta_{12} + 28170 \beta_{10} - 428130 \beta_{9} + \cdots - 809145 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 742690 \beta_{15} - 871494 \beta_{14} + 1009373 \beta_{13} - 1534771 \beta_{12} + 144974 \beta_{11} + \cdots + 337842 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3574829 \beta_{14} - 1219400 \beta_{13} - 3574829 \beta_{12} + 2408573 \beta_{10} - 4741085 \beta_{9} + \cdots - 12657241 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2086633 \beta_{15} - 2457839 \beta_{14} + 29283907 \beta_{13} - 21762114 \beta_{12} + 4249657 \beta_{11} + \cdots + 4794229 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
−3.45326 + 0.707107i
3.45326 − 0.707107i
−1.91484 + 0.707107i
1.91484 − 0.707107i
0.500626 + 0.707107i
−0.500626 − 0.707107i
2.03905 + 0.707107i
−2.03905 − 0.707107i
−3.45326 − 0.707107i
3.45326 + 0.707107i
−1.91484 − 0.707107i
1.91484 + 0.707107i
0.500626 − 0.707107i
−0.500626 + 0.707107i
2.03905 − 0.707107i
−2.03905 + 0.707107i
−1.94183 + 1.94183i 0 − 5.54138i −2.12132 + 0.707107i 0 0.102388 + 2.64377i 6.87675 + 6.87675i 0 2.74616 − 5.49232i
433.2 −1.94183 + 1.94183i 0 − 5.54138i 2.12132 − 0.707107i 0 −2.64377 − 0.102388i 6.87675 + 6.87675i 0 −2.74616 + 5.49232i
433.3 −0.853996 + 0.853996i 0 0.541381i −2.12132 + 0.707107i 0 2.37456 − 1.16682i −2.17033 − 2.17033i 0 1.20773 − 2.41547i
433.4 −0.853996 + 0.853996i 0 0.541381i 2.12132 − 0.707107i 0 1.16682 − 2.37456i −2.17033 − 2.17033i 0 −1.20773 + 2.41547i
433.5 0.853996 − 0.853996i 0 0.541381i −2.12132 + 0.707107i 0 1.16682 − 2.37456i 2.17033 + 2.17033i 0 −1.20773 + 2.41547i
433.6 0.853996 − 0.853996i 0 0.541381i 2.12132 − 0.707107i 0 2.37456 − 1.16682i 2.17033 + 2.17033i 0 1.20773 − 2.41547i
433.7 1.94183 − 1.94183i 0 − 5.54138i −2.12132 + 0.707107i 0 −2.64377 − 0.102388i −6.87675 − 6.87675i 0 −2.74616 + 5.49232i
433.8 1.94183 − 1.94183i 0 − 5.54138i 2.12132 − 0.707107i 0 0.102388 + 2.64377i −6.87675 − 6.87675i 0 2.74616 − 5.49232i
622.1 −1.94183 − 1.94183i 0 5.54138i −2.12132 − 0.707107i 0 0.102388 − 2.64377i 6.87675 − 6.87675i 0 2.74616 + 5.49232i
622.2 −1.94183 − 1.94183i 0 5.54138i 2.12132 + 0.707107i 0 −2.64377 + 0.102388i 6.87675 − 6.87675i 0 −2.74616 − 5.49232i
622.3 −0.853996 − 0.853996i 0 − 0.541381i −2.12132 − 0.707107i 0 2.37456 + 1.16682i −2.17033 + 2.17033i 0 1.20773 + 2.41547i
622.4 −0.853996 − 0.853996i 0 − 0.541381i 2.12132 + 0.707107i 0 1.16682 + 2.37456i −2.17033 + 2.17033i 0 −1.20773 − 2.41547i
622.5 0.853996 + 0.853996i 0 − 0.541381i −2.12132 − 0.707107i 0 1.16682 + 2.37456i 2.17033 − 2.17033i 0 −1.20773 − 2.41547i
622.6 0.853996 + 0.853996i 0 − 0.541381i 2.12132 + 0.707107i 0 2.37456 + 1.16682i 2.17033 − 2.17033i 0 1.20773 + 2.41547i
622.7 1.94183 + 1.94183i 0 5.54138i −2.12132 − 0.707107i 0 −2.64377 + 0.102388i −6.87675 + 6.87675i 0 −2.74616 − 5.49232i
622.8 1.94183 + 1.94183i 0 5.54138i 2.12132 + 0.707107i 0 0.102388 − 2.64377i −6.87675 + 6.87675i 0 2.74616 + 5.49232i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.p.a ✓ 16
3.b odd 2 1 inner 945.2.p.a ✓ 16
5.c odd 4 1 inner 945.2.p.a ✓ 16
7.b odd 2 1 inner 945.2.p.a ✓ 16
15.e even 4 1 inner 945.2.p.a ✓ 16
21.c even 2 1 inner 945.2.p.a ✓ 16
35.f even 4 1 inner 945.2.p.a ✓ 16
105.k odd 4 1 inner 945.2.p.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.p.a ✓ 16 1.a even 1 1 trivial
945.2.p.a ✓ 16 3.b odd 2 1 inner
945.2.p.a ✓ 16 5.c odd 4 1 inner
945.2.p.a ✓ 16 7.b odd 2 1 inner
945.2.p.a ✓ 16 15.e even 4 1 inner
945.2.p.a ✓ 16 21.c even 2 1 inner
945.2.p.a ✓ 16 35.f even 4 1 inner
945.2.p.a ✓ 16 105.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 59T_{2}^{4} + 121 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 59 T^{4} + 121)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{2} + 25)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 236 T^{4} + 1936)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 64 T^{2} + 99)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1226 T^{4} + 290521)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 96 T^{2} + 2156)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 104 T^{2} + 891)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 324)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 62 T^{2} + 36)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 18 T^{3} + \cdots + 484)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 6896 T^{4} + 614656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 3898 T^{4} + 9801)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 212 T^{2} + 1764)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 152 T^{2} + 4851)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 5476)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 94 T^{2} + 396)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 28556 T^{4} + 28344976)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 106 T^{2} + 441)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 10354 T^{4} + 194481)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 248 T^{2} + 576)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 10828 T^{4} + 12702096)^{2} \) Copy content Toggle raw display
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