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Properties

Label 9300.2.g.t
Level $9300$
Weight $2$
Character orbit 9300.g
Analytic conductor $74.261$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9300,2,Mod(3349,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9300.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-12,0,-6,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(74.2608738798\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 6 x^{9} + 44 x^{8} - 164 x^{7} + 322 x^{6} + 216 x^{5} + 304 x^{4} + \cdots + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{6} + \beta_{4}) q^{7} - q^{9} + ( - \beta_{7} - 1) q^{11} + \beta_1 q^{13} + ( - \beta_{9} - \beta_{6} - \beta_1) q^{17} + (\beta_{5} + \beta_{2}) q^{19} + (\beta_{3} + 1) q^{21}+ \cdots + (\beta_{7} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9} - 6 q^{11} + 6 q^{19} + 8 q^{21} - 12 q^{31} - 4 q^{39} + 36 q^{41} - 28 q^{49} + 8 q^{51} + 24 q^{59} + 56 q^{61} - 10 q^{69} + 50 q^{71} - 6 q^{79} + 12 q^{81} - 8 q^{89} + 32 q^{91} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 6 x^{9} + 44 x^{8} - 164 x^{7} + 322 x^{6} + 216 x^{5} + 304 x^{4} + \cdots + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6478273 \nu^{11} + 132311070 \nu^{10} - 220398322 \nu^{9} + 75672456 \nu^{8} + \cdots + 20893865136 ) / 38189747704 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 154180051 \nu^{11} - 809309509 \nu^{10} + 2653952840 \nu^{9} - 4053394910 \nu^{8} + \cdots - 2866652452352 ) / 878364197192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 338205766 \nu^{11} + 1168797349 \nu^{10} - 2346158288 \nu^{9} - 1721440044 \nu^{8} + \cdots - 1330362012392 ) / 878364197192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4708310575 \nu^{11} + 20862476896 \nu^{10} - 44679268694 \nu^{9} - 14172913722 \nu^{8} + \cdots - 3629033624784 ) / 5270185183152 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1010335503 \nu^{11} - 2975879751 \nu^{10} + 4395339980 \nu^{9} + 11100775106 \nu^{8} + \cdots + 3849278921880 ) / 878364197192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6737545171 \nu^{11} - 27875260990 \nu^{10} + 58756218422 \nu^{9} + 24501553986 \nu^{8} + \cdots + 6341020515984 ) / 5270185183152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2633308943 \nu^{11} + 4368820614 \nu^{10} + 9004817936 \nu^{9} - 86355094646 \nu^{8} + \cdots - 11065088153760 ) / 1756728394384 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3293933167 \nu^{11} + 9640177244 \nu^{10} - 14754068936 \nu^{9} - 19027616118 \nu^{8} + \cdots + 252782391072 ) / 1756728394384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8402003852 \nu^{11} - 38218561745 \nu^{10} + 82320062524 \nu^{9} + 23181507312 \nu^{8} + \cdots + 5902073803968 ) / 2635092591576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7114160191 \nu^{11} - 29655261262 \nu^{10} + 63426899086 \nu^{9} + 24178313946 \nu^{8} + \cdots + 6235021704144 ) / 1756728394384 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31661996147 \nu^{11} + 136172264066 \nu^{10} - 304984632166 \nu^{9} + \cdots - 23471222820432 ) / 5270185183152 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{6} + 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + 4\beta_{6} - \beta_{5} + 5\beta_{4} - 4\beta_{3} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + \beta_{7} - 3\beta_{5} - 13\beta_{3} - 11\beta_{2} - 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{10} - 17 \beta_{9} - 2 \beta_{8} - 39 \beta_{6} - 14 \beta_{5} - 53 \beta_{4} - 39 \beta_{3} + \cdots - 53 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{11} + 52\beta_{10} - 116\beta_{9} - 154\beta_{6} - 328\beta_{4} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{11} + 165 \beta_{10} - 219 \beta_{9} + 39 \beta_{8} - 3 \beta_{7} - 409 \beta_{6} + \cdots + 586 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 240\beta_{8} - 84\beta_{7} + 684\beta_{5} + 1766\beta_{3} + 1238\beta_{2} + 3154 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 60 \beta_{11} - 1862 \beta_{10} + 2594 \beta_{9} + 540 \beta_{8} - 60 \beta_{7} + 4424 \beta_{6} + \cdots + 6550 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 722\beta_{11} - 8190\beta_{10} + 13414\beta_{9} + 19946\beta_{6} + 33890\beta_{4} + 2882\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 852 \beta_{11} - 20752 \beta_{10} + 29710 \beta_{9} - 6616 \beta_{8} + 852 \beta_{7} + 48522 \beta_{6} + \cdots - 73258 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\) \(4651\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3349.1
−1.70537 − 1.70537i
−1.16146 − 1.16146i
−0.324975 − 0.324975i
1.23991 + 1.23991i
1.59451 + 1.59451i
2.35739 + 2.35739i
2.35739 − 2.35739i
1.59451 − 1.59451i
1.23991 − 1.23991i
−0.324975 + 0.324975i
−1.16146 + 1.16146i
−1.70537 + 1.70537i
0 − 1.00000i 0 0 0 − 3.41074i 0 −1.00000 0
3349.2 0 − 1.00000i 0 0 0 − 2.32292i 0 −1.00000 0
3349.3 0 − 1.00000i 0 0 0 − 0.649950i 0 −1.00000 0
3349.4 0 − 1.00000i 0 0 0 2.47982i 0 −1.00000 0
3349.5 0 − 1.00000i 0 0 0 3.18902i 0 −1.00000 0
3349.6 0 − 1.00000i 0 0 0 4.71477i 0 −1.00000 0
3349.7 0 1.00000i 0 0 0 − 4.71477i 0 −1.00000 0
3349.8 0 1.00000i 0 0 0 − 3.18902i 0 −1.00000 0
3349.9 0 1.00000i 0 0 0 − 2.47982i 0 −1.00000 0
3349.10 0 1.00000i 0 0 0 0.649950i 0 −1.00000 0
3349.11 0 1.00000i 0 0 0 2.32292i 0 −1.00000 0
3349.12 0 1.00000i 0 0 0 3.41074i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3349.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9300.2.g.t 12
5.b even 2 1 inner 9300.2.g.t 12
5.c odd 4 1 9300.2.a.y ✓ 6
5.c odd 4 1 9300.2.a.ba yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9300.2.a.y ✓ 6 5.c odd 4 1
9300.2.a.ba yes 6 5.c odd 4 1
9300.2.g.t 12 1.a even 1 1 trivial
9300.2.g.t 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(9300, [\chi])\):

\( T_{7}^{12} + 56T_{7}^{10} + 1168T_{7}^{8} + 11536T_{7}^{6} + 55040T_{7}^{4} + 108544T_{7}^{2} + 36864 \) Copy content Toggle raw display
\( T_{11}^{6} + 3T_{11}^{5} - 39T_{11}^{4} - 67T_{11}^{3} + 210T_{11}^{2} - 84T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{12} + 92T_{13}^{10} + 3040T_{13}^{8} + 43408T_{13}^{6} + 250112T_{13}^{4} + 342016T_{13}^{2} + 36864 \) Copy content Toggle raw display
\( T_{17}^{12} + 150T_{17}^{10} + 9195T_{17}^{8} + 294200T_{17}^{6} + 5167059T_{17}^{4} + 47038434T_{17}^{2} + 172318129 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 56 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{5} - 39 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 92 T^{10} + \cdots + 36864 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 172318129 \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} + \cdots + 5476)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 460703296 \) Copy content Toggle raw display
$29$ \( (T^{6} - 69 T^{4} + \cdots + 531)^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 664402176 \) Copy content Toggle raw display
$41$ \( (T^{6} - 18 T^{5} + \cdots + 9424)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 696537664 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 425019456 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 27459141264 \) Copy content Toggle raw display
$59$ \( (T^{6} - 12 T^{5} + \cdots - 8496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 28 T^{5} + \cdots - 17184)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 387 T^{10} + \cdots + 67963536 \) Copy content Toggle raw display
$71$ \( (T^{6} - 25 T^{5} + \cdots - 51336)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$79$ \( (T^{6} + 3 T^{5} + \cdots + 18108)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 192643743744 \) Copy content Toggle raw display
$89$ \( (T^{6} + 4 T^{5} + \cdots - 86211)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 206 T^{10} + \cdots + 31640625 \) Copy content Toggle raw display
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