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Properties

Label 225.10.a.x
Level $225$
Weight $10$
Character orbit 225.a
Self dual yes
Analytic conductor $115.883$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [225,10,Mod(1,225)] 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([0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,144,0,0,0] 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: yes
Analytic conductor: \(115.883063137\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} - 3492 x^{6} + 10490 x^{5} + 3539940 x^{4} - 7097368 x^{3} - 848796093 x^{2} + \cdots + 43654030340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{10} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_1 + 18) q^{4} + \beta_{2} q^{7} + (8 \beta_{5} - 220 \beta_{3}) q^{8} - \beta_{6} q^{11} + (\beta_{4} - 3 \beta_{2}) q^{13} + ( - \beta_{7} - 3 \beta_{6}) q^{14} + (476 \beta_1 - 126136) q^{16}+ \cdots + (910560 \beta_{5} + 53014073 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 144 q^{4} - 1009088 q^{16} + 2079872 q^{19} + 12077344 q^{31} - 25286480 q^{34} + 3119840 q^{46} + 174619144 q^{49} - 165845744 q^{61} - 91038464 q^{64} + 1551352896 q^{76} + 1098932768 q^{79} - 1465128000 q^{91}+ \cdots + 4117219360 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} - 3492 x^{6} + 10490 x^{5} + 3539940 x^{4} - 7097368 x^{3} - 848796093 x^{2} + \cdots + 43654030340 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 904 \nu^{6} - 2712 \nu^{5} - 2367356 \nu^{4} + 4739232 \nu^{3} + 670443556 \nu^{2} + \cdots + 623604735190 ) / 1732747475 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 214204 \nu^{6} + 642612 \nu^{5} + 468943856 \nu^{4} - 938958732 \nu^{3} + \cdots + 20555349935060 ) / 1732747475 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1416624 \nu^{7} - 4958184 \nu^{6} + 7516854900 \nu^{5} - 18779741790 \nu^{4} + \cdots - 14\!\cdots\!20 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 559816 \nu^{6} + 1679448 \nu^{5} + 3858133424 \nu^{4} - 7719065928 \nu^{3} + \cdots + 692556397895240 ) / 1732747475 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 691508768 \nu^{7} - 2420280688 \nu^{6} - 2154503985700 \nu^{5} + 5392310665970 \nu^{4} + \cdots + 12\!\cdots\!10 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 348880456 \nu^{7} - 1221081596 \nu^{6} - 1910620208900 \nu^{5} + 4779603226240 \nu^{4} + \cdots + 61\!\cdots\!20 ) / 232266135286375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 283528888 \nu^{7} + 992351108 \nu^{6} + 1029673276700 \nu^{5} - 2576664069520 \nu^{4} + \cdots - 19\!\cdots\!60 ) / 46453227057275 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 7\beta_{7} + 5\beta_{6} + 60\beta_{5} - 420\beta_{3} + 9000 ) / 18000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{7} + 5\beta_{6} + 60\beta_{5} - 5\beta_{4} - 420\beta_{3} - 130\beta_{2} - 33900\beta _1 + 15750000 ) / 18000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13889 \beta_{7} + 19435 \beta_{6} + 88320 \beta_{5} - 15 \beta_{4} + 2450760 \beta_{3} + \cdots + 47241000 ) / 36000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6941 \beta_{7} + 9715 \beta_{6} + 44130 \beta_{5} + 1555 \beta_{4} + 1225590 \beta_{3} + \cdots + 11569806000 ) / 9000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20648829 \beta_{7} + 31645935 \beta_{6} + 115239120 \beta_{5} + 15575 \beta_{4} + \cdots + 115619328000 ) / 36000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 15469271 \beta_{7} + 23710165 \beta_{6} + 86319030 \beta_{5} + 5957610 \beta_{4} + \cdots + 18277776641250 ) / 9000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 16335718672 \beta_{7} + 25361828030 \beta_{6} + 114802674960 \beta_{5} + 41676005 \beta_{4} + \cdots + 127742130220500 ) / 18000 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−40.0274
39.7535
16.1499
−7.91265
−15.1499
8.91265
41.0274
−38.7535
−30.1771 0 398.657 0 0 −10271.7 3420.35 0 0
1.2 −30.1771 0 398.657 0 0 10271.7 3420.35 0 0
1.3 −12.2206 0 −362.657 0 0 −4342.19 10688.8 0 0
1.4 −12.2206 0 −362.657 0 0 4342.19 10688.8 0 0
1.5 12.2206 0 −362.657 0 0 −4342.19 −10688.8 0 0
1.6 12.2206 0 −362.657 0 0 4342.19 −10688.8 0 0
1.7 30.1771 0 398.657 0 0 −10271.7 −3420.35 0 0
1.8 30.1771 0 398.657 0 0 10271.7 −3420.35 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.10.a.x 8
3.b odd 2 1 inner 225.10.a.x 8
5.b even 2 1 inner 225.10.a.x 8
5.c odd 4 2 45.10.b.d ✓ 8
15.d odd 2 1 inner 225.10.a.x 8
15.e even 4 2 45.10.b.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.10.b.d ✓ 8 5.c odd 4 2
45.10.b.d ✓ 8 15.e even 4 2
225.10.a.x 8 1.a even 1 1 trivial
225.10.a.x 8 3.b odd 2 1 inner
225.10.a.x 8 5.b even 2 1 inner
225.10.a.x 8 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{4} - 1060T_{2}^{2} + 136000 \) Copy content Toggle raw display
\( T_{7}^{4} - 124362000T_{7}^{2} + 1989298362240000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 1060 T^{2} + 136000)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 519968 T - 179554976144)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 39095996869376)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 709545978526076)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 13\!\cdots\!16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
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