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Properties

Label 225.9.c.d
Level $225$
Weight $9$
Character orbit 225.c
Analytic conductor $91.660$
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] = [225,9,Mod(26,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 9, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.26"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 225.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-2524,0,0,6928] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(91.6601872638\)
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: \( x^{12} + 698x^{10} + 179931x^{8} + 20356724x^{6} + 872357011x^{4} + 2973132090x^{2} + 1458246969 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{25}\cdot 5^{12} \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_1 - 210) q^{4} + (\beta_{5} + 577) q^{7} + (\beta_{11} + \beta_{10} + \cdots + 165 \beta_{6}) q^{8} + (2 \beta_{10} + \beta_{9} + \cdots + 258 \beta_{6}) q^{11} + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots - 10370) q^{13}+ \cdots + ( - 1864 \beta_{11} + \cdots - 1893753 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2524 q^{4} + 6928 q^{7} - 124408 q^{13} + 297572 q^{16} + 124960 q^{19} + 1370512 q^{22} - 1114496 q^{28} - 620968 q^{31} + 7486496 q^{34} + 11533176 q^{37} - 14405296 q^{43} - 7161768 q^{46} + 21010156 q^{49}+ \cdots + 302816184 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 698x^{10} + 179931x^{8} + 20356724x^{6} + 872357011x^{4} + 2973132090x^{2} + 1458246969 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 63573929 \nu^{10} + 40853487373 \nu^{8} + 9568072289826 \nu^{6} + 971611084177930 \nu^{4} + \cdots + 33\!\cdots\!81 ) / 127458257734080 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2785687 \nu^{10} + 1149588939 \nu^{8} + 156236636113 \nu^{6} + 10260692423625 \nu^{4} + \cdots - 49\!\cdots\!02 ) / 2655380369460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 450814433 \nu^{10} - 221092018741 \nu^{8} - 33385825615122 \nu^{6} + \cdots - 56\!\cdots\!77 ) / 254916515468160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 230770343 \nu^{10} + 262632245371 \nu^{8} + 93967884514722 \nu^{6} + \cdots + 15\!\cdots\!67 ) / 63729128867040 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 958623863 \nu^{10} - 565038638851 \nu^{8} - 117196983011982 \nu^{6} + \cdots - 39\!\cdots\!27 ) / 254916515468160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47803205353 \nu^{11} - 33390916669181 \nu^{9} + \cdots - 43\!\cdots\!57 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 652419 \nu^{11} - 474851103 \nu^{9} - 127579878486 \nu^{7} - 15021653375790 \nu^{5} + \cdots - 31\!\cdots\!71 \nu ) / 18417104700800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3202987051669 \nu^{11} + \cdots + 10\!\cdots\!41 \nu ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 80932409602 \nu^{11} + 57864326024669 \nu^{9} + \cdots + 10\!\cdots\!23 \nu ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29753765681707 \nu^{11} + \cdots + 56\!\cdots\!83 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 55993762879469 \nu^{11} + \cdots + 11\!\cdots\!01 \nu ) / 81\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 23\beta_{11} - 63\beta_{10} - 74\beta_{9} + 79\beta_{8} + 106\beta_{7} + 8352\beta_{6} ) / 40500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -93\beta_{5} + 33\beta_{4} - 167\beta_{3} - 18\beta_{2} - 1495\beta _1 - 942818 ) / 8100 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5383\beta_{11} + 13965\beta_{10} + 13504\beta_{9} - 14609\beta_{8} - 149276\beta_{7} - 1293894\beta_{6} ) / 40500 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6945\beta_{5} - 1683\beta_{4} + 6367\beta_{3} + 345\beta_{2} + 86429\beta _1 + 43006096 ) / 2025 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1207553 \beta_{11} - 3023817 \beta_{10} - 2704214 \beta_{9} + 2790769 \beta_{8} + \cdots + 179827416 \beta_{6} ) / 40500 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 6879711 \beta_{5} + 1427907 \beta_{4} - 4109693 \beta_{3} + 330414 \beta_{2} - 77501893 \beta _1 - 32914442018 ) / 8100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 271314433 \beta_{11} + 643797459 \beta_{10} + 566352904 \beta_{9} - 527402759 \beta_{8} + \cdots - 24096509658 \beta_{6} ) / 40500 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 398926374 \beta_{5} - 76952802 \beta_{4} + 177975298 \beta_{3} - 51387426 \beta_{2} + \cdots + 1625436495415 ) / 2025 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 61142307623 \beta_{11} - 137174804031 \beta_{10} - 121762354874 \beta_{9} + \cdots + 2918148143760 \beta_{6} ) / 40500 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 72212944641 \beta_{5} + 13359894885 \beta_{4} - 26421275155 \beta_{3} + 14325214974 \beta_{2} + \cdots - 263067656398354 ) / 1620 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13811124010423 \beta_{11} + 29475838399773 \beta_{10} + 26571719989024 \beta_{9} + \cdots - 258705616825062 \beta_{6} ) / 40500 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
− 1.75534i
12.9148i
10.5572i
− 0.769310i
13.8483i
− 14.9768i
14.9768i
− 13.8483i
0.769310i
− 10.5572i
− 12.9148i
1.75534i
− 31.8503i 0 −758.442 0 0 1917.57 16002.9i 0 0
26.2 − 25.3359i 0 −385.909 0 0 2907.18 3291.35i 0 0
26.3 − 23.2704i 0 −285.513 0 0 −4399.16 686.784i 0 0
26.4 − 19.1733i 0 −111.617 0 0 −367.611 − 2768.31i 0 0
26.5 − 13.3807i 0 76.9563 0 0 −256.182 − 4455.20i 0 0
26.6 − 7.31273i 0 202.524 0 0 3662.21 − 3353.06i 0 0
26.7 7.31273i 0 202.524 0 0 3662.21 3353.06i 0 0
26.8 13.3807i 0 76.9563 0 0 −256.182 4455.20i 0 0
26.9 19.1733i 0 −111.617 0 0 −367.611 2768.31i 0 0
26.10 23.2704i 0 −285.513 0 0 −4399.16 − 686.784i 0 0
26.11 25.3359i 0 −385.909 0 0 2907.18 − 3291.35i 0 0
26.12 31.8503i 0 −758.442 0 0 1917.57 − 16002.9i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.9.c.d 12
3.b odd 2 1 inner 225.9.c.d 12
5.b even 2 1 45.9.c.a ✓ 12
5.c odd 4 2 225.9.d.c 24
15.d odd 2 1 45.9.c.a ✓ 12
15.e even 4 2 225.9.d.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.9.c.a ✓ 12 5.b even 2 1
45.9.c.a ✓ 12 15.d odd 2 1
225.9.c.d 12 1.a even 1 1 trivial
225.9.c.d 12 3.b odd 2 1 inner
225.9.d.c 24 5.c odd 4 2
225.9.d.c 24 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{12} + 2798 T_{2}^{10} + 2962185 T_{2}^{8} + 1494134240 T_{2}^{6} + 366509587840 T_{2}^{4} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
\( T_{7}^{6} - 3464 T_{7}^{5} - 16547294 T_{7}^{4} + 72662823072 T_{7}^{3} - 40089172083156 T_{7}^{2} + \cdots - 84\!\cdots\!56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 84\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 11\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 91\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 33\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 98\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 92\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 40\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 91\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 96\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 41\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 93\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 68\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
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