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Properties

Label 945.2.j.i
Level $945$
Weight $2$
Character orbit 945.j
Analytic conductor $7.546$
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] = [945,2,Mod(541,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.j (of order \(3\), 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: \(14\)
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: \( x^{14} + 12 x^{12} - 2 x^{11} + 104 x^{10} - 17 x^{9} + 419 x^{8} - 86 x^{7} + 1233 x^{6} - 66 x^{5} + \cdots + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{2} + (\beta_{10} - \beta_{7} + \beta_{3}) q^{4} + (\beta_{7} - 1) q^{5} + ( - \beta_{11} - \beta_{8}) q^{7} + (\beta_{12} + \beta_{11} + \beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{2} + (\beta_{10} - \beta_{7} + \beta_{3}) q^{4} + (\beta_{7} - 1) q^{5} + ( - \beta_{11} - \beta_{8}) q^{7} + (\beta_{12} + \beta_{11} + \beta_{2} + 1) q^{8} + \beta_1 q^{10} + (\beta_{13} - \beta_{4}) q^{11} + ( - \beta_{6} + \beta_{2}) q^{13} + (\beta_{12} + \beta_{10} + \cdots + \beta_{3}) q^{14}+ \cdots + (\beta_{12} - 2 \beta_{9} - \beta_{8} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 10 q^{4} - 7 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 10 q^{4} - 7 q^{5} + 2 q^{7} + 6 q^{8} + 2 q^{11} - 7 q^{14} - 12 q^{16} - 12 q^{17} - 8 q^{19} + 20 q^{20} - 14 q^{22} - 2 q^{23} - 7 q^{25} - 27 q^{26} + 4 q^{28} + 2 q^{29} + q^{31} - 11 q^{32} - 8 q^{34} + 2 q^{35} + 8 q^{37} - 22 q^{38} - 3 q^{40} + 52 q^{41} - 2 q^{43} + 25 q^{44} - 14 q^{46} - 9 q^{47} + 30 q^{49} + 5 q^{52} - 12 q^{53} - 4 q^{55} - 66 q^{56} + 16 q^{58} + 4 q^{59} + 6 q^{61} + 26 q^{62} + 10 q^{64} + 2 q^{67} - 58 q^{68} + 8 q^{70} + 14 q^{71} - q^{73} - 4 q^{74} + 20 q^{76} - 25 q^{77} + 8 q^{79} - 12 q^{80} - 28 q^{82} + 72 q^{83} + 24 q^{85} + 58 q^{86} + 34 q^{88} - 46 q^{89} - 39 q^{91} + 18 q^{92} + 19 q^{94} - 8 q^{95} - 32 q^{97} - 25 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 12 x^{12} - 2 x^{11} + 104 x^{10} - 17 x^{9} + 419 x^{8} - 86 x^{7} + 1233 x^{6} - 66 x^{5} + \cdots + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 109808140667 \nu^{13} - 132399056377 \nu^{12} - 867866192798 \nu^{11} + \cdots + 78273637840782 ) / 413668841947377 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 132399056377 \nu^{13} + 449831495206 \nu^{12} - 1450215829834 \nu^{11} + \cdots + 12\!\cdots\!43 ) / 413668841947377 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 654351987077 \nu^{13} + 12078032462915 \nu^{12} - 14497175071730 \nu^{11} + \cdots + 55\!\cdots\!68 ) / 827337683894754 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 930193030042 \nu^{13} + 3092485728001 \nu^{12} + 11882690998253 \nu^{11} + \cdots + 94121766203916 ) / 827337683894754 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 537216032405 \nu^{13} - 584417781539 \nu^{12} + 5277606606802 \nu^{11} + \cdots + 14611653529288 ) / 275779227964918 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4348535435599 \nu^{13} - 219616281334 \nu^{12} - 52447223339942 \nu^{11} + \cdots - 37008568697580 ) / 827337683894754 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4430331067045 \nu^{13} + 1755428716526 \nu^{12} + 53323884301919 \nu^{11} + \cdots + 26\!\cdots\!34 ) / 827337683894754 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9678529493056 \nu^{13} - 2252190691240 \nu^{12} - 115632122062917 \nu^{11} + \cdots - 27\!\cdots\!58 ) / 827337683894754 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4348535435599 \nu^{13} - 219616281334 \nu^{12} - 52447223339942 \nu^{11} + \cdots - 864346252592334 ) / 275779227964918 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15979252065210 \nu^{13} + 4697122082371 \nu^{12} - 191790856426710 \nu^{11} + \cdots - 21\!\cdots\!34 ) / 827337683894754 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17077333471880 \nu^{13} - 3373131518601 \nu^{12} + 200469518354690 \nu^{11} + \cdots + 584250828108360 ) / 827337683894754 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12822938962766 \nu^{13} + 58664099283 \nu^{12} + 152078153409809 \nu^{11} + \cdots + 19\!\cdots\!36 ) / 275779227964918 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + 3\beta_{7} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{11} + 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} + 7\beta_{10} - \beta_{8} - 15\beta_{7} - \beta_{6} + \beta_{5} + 7\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 9 \beta_{12} - 8 \beta_{11} + \beta_{10} - 8 \beta_{9} - 9 \beta_{8} + 10 \beta_{7} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{12} + \beta_{11} - 10\beta_{9} + 10\beta_{8} + 11\beta_{6} - \beta_{4} - 46\beta_{3} + 9\beta_{2} + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{13} + 13 \beta_{12} - 13 \beta_{11} - 10 \beta_{10} + 68 \beta_{9} + 55 \beta_{8} + \cdots + 180 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{13} - 16 \beta_{12} + 65 \beta_{11} - 302 \beta_{10} + 65 \beta_{9} - 16 \beta_{8} + \cdots - 542 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 367 \beta_{12} + 488 \beta_{11} - 121 \beta_{9} + 121 \beta_{8} + 18 \beta_{6} - 105 \beta_{4} + \cdots + 513 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 123 \beta_{13} + 611 \beta_{12} - 611 \beta_{11} + 1996 \beta_{10} + 169 \beta_{9} - 442 \beta_{8} + \cdots + 468 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 819 \beta_{13} - 3426 \beta_{12} - 2438 \beta_{11} + 501 \beta_{10} - 2438 \beta_{9} - 3426 \beta_{8} + \cdots - 3523 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2939 \beta_{12} + 1513 \beta_{11} - 4452 \beta_{9} + 4452 \beta_{8} + 5026 \beta_{6} - 1026 \beta_{4} + \cdots + 23184 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6052 \beta_{13} + 7565 \beta_{12} - 7565 \beta_{11} - 3112 \beta_{10} + 23783 \beta_{9} + \cdots + 50768 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1 + \beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
−1.31681 − 2.28079i
−1.07328 − 1.85897i
−0.386571 − 0.669561i
−0.106565 − 0.184577i
0.691606 + 1.19790i
0.904296 + 1.56629i
1.28733 + 2.22972i
−1.31681 + 2.28079i
−1.07328 + 1.85897i
−0.386571 + 0.669561i
−0.106565 + 0.184577i
0.691606 − 1.19790i
0.904296 − 1.56629i
1.28733 − 2.22972i
−1.31681 + 2.28079i 0 −2.46800 − 4.27469i −0.500000 + 0.866025i 0 −2.47386 + 0.938090i 7.73231 0 −1.31681 − 2.28079i
541.2 −1.07328 + 1.85897i 0 −1.30386 − 2.25834i −0.500000 + 0.866025i 0 2.60849 + 0.442489i 1.30448 0 −1.07328 − 1.85897i
541.3 −0.386571 + 0.669561i 0 0.701125 + 1.21438i −0.500000 + 0.866025i 0 2.37137 − 1.17329i −2.63042 0 −0.386571 − 0.669561i
541.4 −0.106565 + 0.184577i 0 0.977288 + 1.69271i −0.500000 + 0.866025i 0 −1.59727 − 2.10920i −0.842843 0 −0.106565 − 0.184577i
541.5 0.691606 − 1.19790i 0 0.0433621 + 0.0751054i −0.500000 + 0.866025i 0 0.984150 + 2.45590i 2.88638 0 0.691606 + 1.19790i
541.6 0.904296 − 1.56629i 0 −0.635503 − 1.10072i −0.500000 + 0.866025i 0 −2.62978 − 0.290255i 1.31845 0 0.904296 + 1.56629i
541.7 1.28733 − 2.22972i 0 −2.31442 − 4.00870i −0.500000 + 0.866025i 0 1.73691 − 1.99578i −6.76836 0 1.28733 + 2.22972i
676.1 −1.31681 − 2.28079i 0 −2.46800 + 4.27469i −0.500000 − 0.866025i 0 −2.47386 − 0.938090i 7.73231 0 −1.31681 + 2.28079i
676.2 −1.07328 − 1.85897i 0 −1.30386 + 2.25834i −0.500000 − 0.866025i 0 2.60849 − 0.442489i 1.30448 0 −1.07328 + 1.85897i
676.3 −0.386571 − 0.669561i 0 0.701125 − 1.21438i −0.500000 − 0.866025i 0 2.37137 + 1.17329i −2.63042 0 −0.386571 + 0.669561i
676.4 −0.106565 − 0.184577i 0 0.977288 − 1.69271i −0.500000 − 0.866025i 0 −1.59727 + 2.10920i −0.842843 0 −0.106565 + 0.184577i
676.5 0.691606 + 1.19790i 0 0.0433621 − 0.0751054i −0.500000 − 0.866025i 0 0.984150 − 2.45590i 2.88638 0 0.691606 − 1.19790i
676.6 0.904296 + 1.56629i 0 −0.635503 + 1.10072i −0.500000 − 0.866025i 0 −2.62978 + 0.290255i 1.31845 0 0.904296 − 1.56629i
676.7 1.28733 + 2.22972i 0 −2.31442 + 4.00870i −0.500000 − 0.866025i 0 1.73691 + 1.99578i −6.76836 0 1.28733 − 2.22972i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.7
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 945.2.j.i ✓ 14
3.b odd 2 1 945.2.j.j yes 14
7.c even 3 1 inner 945.2.j.i ✓ 14
7.c even 3 1 6615.2.a.bx 7
7.d odd 6 1 6615.2.a.bv 7
21.g even 6 1 6615.2.a.bw 7
21.h odd 6 1 945.2.j.j yes 14
21.h odd 6 1 6615.2.a.bu 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.j.i ✓ 14 1.a even 1 1 trivial
945.2.j.i ✓ 14 7.c even 3 1 inner
945.2.j.j yes 14 3.b odd 2 1
945.2.j.j yes 14 21.h odd 6 1
6615.2.a.bu 7 21.h odd 6 1
6615.2.a.bv 7 7.d odd 6 1
6615.2.a.bw 7 21.g even 6 1
6615.2.a.bx 7 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 12 T_{2}^{12} - 2 T_{2}^{11} + 104 T_{2}^{10} - 17 T_{2}^{9} + 419 T_{2}^{8} - 86 T_{2}^{7} + \cdots + 36 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 12 T^{12} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} - 2 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} - 2 T^{13} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( (T^{7} - 68 T^{5} + 7 T^{4} + \cdots - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + 12 T^{13} + \cdots + 14379264 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 1257127936 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 796594176 \) Copy content Toggle raw display
$29$ \( (T^{7} - T^{6} + \cdots - 73728)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 81817737444 \) Copy content Toggle raw display
$37$ \( T^{14} - 8 T^{13} + \cdots + 1296 \) Copy content Toggle raw display
$41$ \( (T^{7} - 26 T^{6} + \cdots - 93888)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + T^{6} - 135 T^{5} + \cdots + 9056)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + 9 T^{13} + \cdots + 147456 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 162357420096 \) Copy content Toggle raw display
$59$ \( T^{14} - 4 T^{13} + \cdots + 3326976 \) Copy content Toggle raw display
$61$ \( T^{14} - 6 T^{13} + \cdots + 24860196 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 10274660496 \) Copy content Toggle raw display
$71$ \( (T^{7} - 7 T^{6} + \cdots - 21888)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 26717863936 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 235776282624 \) Copy content Toggle raw display
$83$ \( (T^{7} - 36 T^{6} + \cdots - 442368)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 4608544149504 \) Copy content Toggle raw display
$97$ \( (T^{7} + 16 T^{6} + \cdots - 4678)^{2} \) Copy content Toggle raw display
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