Properties

Label 245.4.a.p
Level $245$
Weight $4$
Character orbit 245.a
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $6$
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] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, 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("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1163891200.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 7 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + ( - 6 \beta_{5} + 3 \beta_{4} + \cdots + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + (26 \beta_{5} + 244 \beta_{4} + \cdots - 536) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9} + 10 q^{10} - 16 q^{11} + 160 q^{12} + 168 q^{13} - 80 q^{15} + 298 q^{16} - 4 q^{17} + 354 q^{18} + 308 q^{19} - 70 q^{20} - 236 q^{22} - 336 q^{23} - 92 q^{24} + 150 q^{25} + 56 q^{26} + 964 q^{27} + 176 q^{29} - 120 q^{30} + 392 q^{31} - 770 q^{32} + 188 q^{33} + 812 q^{34} + 230 q^{36} - 140 q^{37} + 20 q^{38} + 140 q^{39} + 330 q^{40} + 656 q^{41} - 388 q^{43} - 160 q^{44} - 350 q^{45} - 388 q^{46} + 628 q^{47} + 1396 q^{48} - 50 q^{50} + 744 q^{51} + 1520 q^{52} - 676 q^{53} + 2284 q^{54} + 80 q^{55} + 1468 q^{57} - 2012 q^{58} + 996 q^{59} - 800 q^{60} + 740 q^{61} - 364 q^{62} + 1426 q^{64} - 840 q^{65} - 3620 q^{66} + 1768 q^{67} - 2940 q^{68} - 1048 q^{69} - 224 q^{71} + 2858 q^{72} + 2640 q^{73} + 928 q^{74} + 400 q^{75} - 1340 q^{76} + 8 q^{78} + 1636 q^{79} - 1490 q^{80} + 4442 q^{81} - 1756 q^{82} + 140 q^{83} + 20 q^{85} + 1180 q^{86} + 1940 q^{87} - 5652 q^{88} - 1904 q^{89} - 1770 q^{90} - 1952 q^{92} - 1592 q^{93} - 3332 q^{94} - 1540 q^{95} - 6460 q^{96} + 516 q^{97} - 2804 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 23\nu^{3} + 8\nu^{2} - 86\nu - 64 ) / 26 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 23\nu^{3} + 18\nu^{2} + 60\nu - 144 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 26\nu^{4} + 89\nu^{3} - 298\nu^{2} - 638\nu - 164 ) / 26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 49\nu^{3} + 8\nu^{2} - 476\nu - 428 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{5} - 26\nu^{4} - 155\nu^{3} + 240\nu^{2} + 800\nu + 264 ) / 26 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 56 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 15\beta_{5} - 8\beta_{4} + 15\beta_{3} + 15\beta_{2} + 83\beta _1 + 98 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36\beta_{5} - 29\beta_{4} + 43\beta_{3} + 127\beta_{2} + 265\beta _1 + 784 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 267\beta_{5} - 106\beta_{4} + 267\beta_{3} + 323\beta_{2} + 1315\beta _1 + 2254 ) / 7 \) Copy content Toggle raw display

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
4.10376
4.29508
−1.04490
−2.05886
−0.241849
−3.05323
−5.51797 3.86039 22.4480 −5.00000 −21.3015 0 −79.7239 −12.0974 27.5899
1.2 −2.88087 −2.89052 0.299392 −5.00000 8.32721 0 22.1844 −18.6449 14.4043
1.3 −0.369315 9.74070 −7.86361 −5.00000 −3.59738 0 5.85867 67.8812 1.84657
1.4 0.644648 −4.18687 −7.58443 −5.00000 −2.69906 0 −10.0465 −9.47008 −3.22324
1.5 1.65606 −0.332888 −5.25746 −5.00000 −0.551283 0 −21.9552 −26.8892 −8.28031
1.6 4.46745 9.80920 11.9581 −5.00000 43.8221 0 17.6824 69.2204 −22.3372
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 245.4.a.p yes 6
3.b odd 2 1 2205.4.a.ca 6
5.b even 2 1 1225.4.a.bi 6
7.b odd 2 1 245.4.a.o 6
7.c even 3 2 245.4.e.p 12
7.d odd 6 2 245.4.e.q 12
21.c even 2 1 2205.4.a.bz 6
35.c odd 2 1 1225.4.a.bj 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o 6 7.b odd 2 1
245.4.a.p yes 6 1.a even 1 1 trivial
245.4.e.p 12 7.c even 3 2
245.4.e.q 12 7.d odd 6 2
1225.4.a.bi 6 5.b even 2 1
1225.4.a.bj 6 35.c odd 2 1
2205.4.a.bz 6 21.c even 2 1
2205.4.a.ca 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{6} + 2T_{2}^{5} - 29T_{2}^{4} - 28T_{2}^{3} + 134T_{2}^{2} - 24T_{2} - 28 \) Copy content Toggle raw display
\( T_{3}^{6} - 16T_{3}^{5} + 12T_{3}^{4} + 564T_{3}^{3} - 355T_{3}^{2} - 4644T_{3} - 1486 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots - 28 \) Copy content Toggle raw display
$3$ \( T^{6} - 16 T^{5} + \cdots - 1486 \) Copy content Toggle raw display
$5$ \( (T + 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 9225436100 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 12513937372 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 3807091762 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 1617325344 \) Copy content Toggle raw display
$23$ \( T^{6} + 336 T^{5} + \cdots + 298897696 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 544215793700 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 16539893268192 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 271258136464 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 144691772208184 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 440374360000 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 251564448569400 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 590408333736048 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 842000334839552 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 885207397312 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 85\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 510868966648482 \) Copy content Toggle raw display
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