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Properties

Label 1225.4.a.bo
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,4,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

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: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 45x^{6} + 26x^{5} + 566x^{4} - 137x^{3} - 2154x^{2} + 376x + 1224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{5} + 2 \beta_1 - 1) q^{6} + (\beta_{6} + \beta_{4} - \beta_{3} + \cdots + 4) q^{8}+ \cdots + (\beta_{7} + \beta_{5} + \beta_{4} + \cdots + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{2} + \beta_1 + 3) q^{4} + ( - \beta_{5} + 2 \beta_1 - 1) q^{6} + (\beta_{6} + \beta_{4} - \beta_{3} + \cdots + 4) q^{8}+ \cdots + (26 \beta_{7} + 12 \beta_{6} + \cdots + 525) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 6 q^{3} + 27 q^{4} - 4 q^{6} + 42 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 6 q^{3} + 27 q^{4} - 4 q^{6} + 42 q^{8} + 42 q^{9} + 103 q^{12} + 138 q^{13} + 191 q^{16} + 138 q^{17} - 81 q^{18} + 42 q^{19} - 127 q^{22} - 124 q^{23} + 292 q^{24} - 23 q^{26} + 204 q^{27} + 288 q^{29} - 18 q^{31} + 562 q^{32} + 236 q^{33} + 488 q^{34} - 711 q^{36} - 220 q^{37} + 1104 q^{38} + 22 q^{39} - 168 q^{41} - 60 q^{43} - 880 q^{44} - 98 q^{46} + 1098 q^{47} + 338 q^{48} + 628 q^{51} + 1370 q^{52} + 734 q^{53} - 2163 q^{54} - 914 q^{57} - 924 q^{58} - 558 q^{59} + 1496 q^{61} + 801 q^{62} + 2436 q^{64} + 682 q^{66} + 1166 q^{67} + 2455 q^{68} - 822 q^{69} - 724 q^{71} + 2191 q^{72} + 892 q^{73} + 235 q^{74} + 1509 q^{76} - 2122 q^{78} - 840 q^{79} + 2392 q^{81} + 4154 q^{82} + 1146 q^{83} + 1816 q^{86} + 4286 q^{87} + 788 q^{88} - 2678 q^{89} - 6932 q^{92} + 1986 q^{93} - 3625 q^{94} + 6601 q^{96} + 664 q^{97} + 4086 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 45x^{6} + 26x^{5} + 566x^{4} - 137x^{3} - 2154x^{2} + 376x + 1224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} + 156\nu^{5} - 543\nu^{4} - 582\nu^{3} + 2731\nu^{2} - 3035\nu + 1090 ) / 518 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\nu^{7} + 13\nu^{6} - 911\nu^{5} - 830\nu^{4} + 12690\nu^{3} + 10061\nu^{2} - 52958\nu - 10728 ) / 1036 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{7} - 69\nu^{6} - 413\nu^{5} + 2248\nu^{4} + 2046\nu^{3} - 13805\nu^{2} + 3488\nu + 5602 ) / 518 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{7} + 23\nu^{6} + 1223\nu^{5} - 256\nu^{4} - 12818\nu^{3} - 5635\nu^{2} + 26168\nu + 20160 ) / 1036 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 34\nu^{7} - 115\nu^{6} - 1231\nu^{5} + 3759\nu^{4} + 9478\nu^{3} - 25142\nu^{2} - 10183\nu + 21374 ) / 518 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 21\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 2\beta_{6} - 3\beta_{5} + 28\beta_{2} + 37\beta _1 + 219 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 37\beta_{6} - 7\beta_{5} + 33\beta_{4} - 56\beta_{3} + 38\beta_{2} + 529\beta _1 + 192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 75\beta_{7} + 87\beta_{6} - 134\beta_{5} + 13\beta_{4} - 66\beta_{3} + 764\beta_{2} + 1221\beta _1 + 5339 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 84\beta_{7} + 1134\beta_{6} - 375\beta_{5} + 960\beta_{4} - 1972\beta_{3} + 1325\beta_{2} + 14377\beta _1 + 7188 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.99064
−2.92095
−2.71286
−0.731462
0.935685
2.21093
3.73931
5.46999
−4.99064 0.142407 16.9065 0 −0.710702 0 −44.4491 −26.9797 0
1.2 −2.92095 10.0483 0.531934 0 −29.3505 0 21.8138 73.9678 0
1.3 −2.71286 −1.49314 −0.640387 0 4.05067 0 23.4402 −24.7705 0
1.4 −0.731462 −8.29374 −7.46496 0 6.06655 0 11.3120 41.7861 0
1.5 0.935685 1.25282 −7.12449 0 1.17224 0 −14.1518 −25.4304 0
1.6 2.21093 5.79264 −3.11181 0 12.8071 0 −24.5674 6.55470 0
1.7 3.73931 −5.71572 5.98245 0 −21.3729 0 −7.54426 5.66951 0
1.8 5.46999 4.26646 21.9208 0 23.3375 0 76.1465 −8.79734 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
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.bo 8
5.b even 2 1 1225.4.a.bk 8
7.b odd 2 1 1225.4.a.bn 8
7.d odd 6 2 175.4.e.e ✓ 16
35.c odd 2 1 1225.4.a.bl 8
35.i odd 6 2 175.4.e.f yes 16
35.k even 12 4 175.4.k.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.e.e ✓ 16 7.d odd 6 2
175.4.e.f yes 16 35.i odd 6 2
175.4.k.e 32 35.k even 12 4
1225.4.a.bk 8 5.b even 2 1
1225.4.a.bl 8 35.c odd 2 1
1225.4.a.bn 8 7.b odd 2 1
1225.4.a.bo 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{8} - T_{2}^{7} - 45T_{2}^{6} + 26T_{2}^{5} + 566T_{2}^{4} - 137T_{2}^{3} - 2154T_{2}^{2} + 376T_{2} + 1224 \) Copy content Toggle raw display
\( T_{3}^{8} - 6T_{3}^{7} - 111T_{3}^{6} + 562T_{3}^{5} + 2744T_{3}^{4} - 12626T_{3}^{3} - 5735T_{3}^{2} + 23086T_{3} - 3136 \) Copy content Toggle raw display
\( T_{19}^{8} - 42 T_{19}^{7} - 35056 T_{19}^{6} + 1856624 T_{19}^{5} + 359705509 T_{19}^{4} + \cdots - 19\!\cdots\!52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 1224 \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{7} + \cdots - 3136 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 349799277600 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 20900355319536 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11760645333969 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 19\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 28\!\cdots\!79 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 93\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 75\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 41\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 40\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 28\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 86\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 74\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 77\!\cdots\!87 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
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