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Properties

Label 245.4.b.d
Level $245$
Weight $4$
Character orbit 245.b
Analytic conductor $14.455$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(99,245)]
 
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([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{6} q^{3} + (\beta_1 - 4) q^{4} + (\beta_{8} - 1) q^{5} + ( - \beta_{3} - 1) q^{6} + (\beta_{9} + \beta_{8} + \cdots - 3 \beta_{2}) q^{8}+ \cdots + (\beta_{8} - \beta_{7} - \beta_{4} + \cdots - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{6} q^{3} + (\beta_1 - 4) q^{4} + (\beta_{8} - 1) q^{5} + ( - \beta_{3} - 1) q^{6} + (\beta_{9} + \beta_{8} + \cdots - 3 \beta_{2}) q^{8}+ \cdots + (44 \beta_{8} - 44 \beta_{7} + \cdots - 536) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 36 q^{4} - 6 q^{5} - 12 q^{6} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 36 q^{4} - 6 q^{5} - 12 q^{6} - 46 q^{9} + 16 q^{10} + 84 q^{11} + 8 q^{15} + 148 q^{16} - 72 q^{19} + 68 q^{20} - 72 q^{24} - 362 q^{25} + 620 q^{26} + 88 q^{29} + 52 q^{30} - 120 q^{31} - 964 q^{34} - 420 q^{36} + 212 q^{39} - 1396 q^{40} + 852 q^{41} - 1424 q^{44} + 510 q^{45} + 176 q^{46} + 1644 q^{50} + 1276 q^{51} + 996 q^{54} + 1136 q^{55} - 864 q^{59} - 1692 q^{60} + 884 q^{61} - 2724 q^{64} + 520 q^{65} - 1148 q^{66} - 4808 q^{69} + 880 q^{71} + 3024 q^{74} - 720 q^{75} + 2672 q^{76} - 3580 q^{79} + 2316 q^{80} - 302 q^{81} + 1572 q^{85} + 2432 q^{86} + 1492 q^{89} + 7748 q^{90} - 1652 q^{94} - 1628 q^{95} - 4080 q^{96} - 5304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 46\nu^{6} + 611\nu^{4} + 2506\nu^{2} + 4740 ) / 420 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - 25\nu^{7} + 397\nu^{5} + 11921\nu^{3} + 74220\nu ) / 25200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{8} - 251\nu^{6} - 3895\nu^{4} - 18725\nu^{2} - 11268 ) / 672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -59\nu^{8} - 2945\nu^{6} - 42937\nu^{4} - 159551\nu^{2} - 20940 ) / 3360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -151\nu^{9} - 9445\nu^{7} - 199613\nu^{5} - 1603819\nu^{3} - 3635580\nu ) / 100800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{9} - 1535\nu^{7} - 25567\nu^{5} - 140561\nu^{3} - 169620\nu ) / 14400 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 89 \nu^{9} - 30 \nu^{8} - 4619 \nu^{7} - 2010 \nu^{6} - 72019 \nu^{5} - 43530 \nu^{4} + \cdots - 585720 ) / 40320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 89 \nu^{9} + 30 \nu^{8} - 4619 \nu^{7} + 2010 \nu^{6} - 72019 \nu^{5} + 43530 \nu^{4} + \cdots + 585720 ) / 40320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 733\nu^{9} + 37855\nu^{7} + 608639\nu^{5} + 3305617\nu^{3} + 6152340\nu ) / 100800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + 13\beta_{2} ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} - 2\beta_{7} + 2\beta_{3} + 5\beta _1 - 81 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{9} - 4\beta_{8} - 4\beta_{7} + 61\beta_{6} - 25\beta_{5} - 241\beta_{2} ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -48\beta_{8} + 48\beta_{7} + 10\beta_{4} - 70\beta_{3} - 115\beta _1 + 1581 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -204\beta_{9} + 228\beta_{8} + 228\beta_{7} - 1793\beta_{6} + 609\beta_{5} + 5257\beta_{2} ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1330\beta_{8} - 1330\beta_{7} - 400\beta_{4} + 1986\beta_{3} + 2425\beta _1 - 35201 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4536\beta_{9} - 8036\beta_{8} - 8036\beta_{7} + 48077\beta_{6} - 15737\beta_{5} - 120609\beta_{2} ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -36864\beta_{8} + 36864\beta_{7} + 12290\beta_{4} - 53598\beta_{3} - 50875\beta _1 + 823061 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -99020\beta_{9} + 243732\beta_{8} + 243732\beta_{7} - 1260785\beta_{6} + 411393\beta_{5} + 2841353\beta_{2} ) / 14 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
− 4.31366i
− 5.04851i
− 1.85474i
− 2.67516i
− 0.555276i
0.555276i
2.67516i
1.85474i
5.04851i
4.31366i
− 5.31366i − 1.93939i −20.2350 2.32771 + 10.9353i −10.3052 0 65.0123i 23.2388 58.1067 − 12.3686i
99.2 − 4.04851i 6.52749i −8.39045 −6.15172 − 9.33576i 26.4266 0 1.58074i −15.6081 −37.7959 + 24.9053i
99.3 − 2.85474i − 8.98858i −0.149548 −3.91321 − 10.4731i −25.6601 0 − 22.4110i −53.7945 −29.8981 + 11.1712i
99.4 − 1.67516i − 2.49396i 5.19383 −6.35505 + 9.19855i −4.17779 0 − 22.1018i 20.7802 15.4091 + 10.6457i
99.5 − 1.55528i 4.96149i 5.58112 11.0923 + 1.40060i 7.71648 0 − 21.1224i 2.38365 2.17831 − 17.2515i
99.6 1.55528i − 4.96149i 5.58112 11.0923 − 1.40060i 7.71648 0 21.1224i 2.38365 2.17831 + 17.2515i
99.7 1.67516i 2.49396i 5.19383 −6.35505 − 9.19855i −4.17779 0 22.1018i 20.7802 15.4091 − 10.6457i
99.8 2.85474i 8.98858i −0.149548 −3.91321 + 10.4731i −25.6601 0 22.4110i −53.7945 −29.8981 − 11.1712i
99.9 4.04851i − 6.52749i −8.39045 −6.15172 + 9.33576i 26.4266 0 − 1.58074i −15.6081 −37.7959 − 24.9053i
99.10 5.31366i 1.93939i −20.2350 2.32771 − 10.9353i −10.3052 0 − 65.0123i 23.2388 58.1067 + 12.3686i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.10
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 245.4.b.d 10
5.b even 2 1 inner 245.4.b.d 10
5.c odd 4 1 1225.4.a.be 5
5.c odd 4 1 1225.4.a.bh 5
7.b odd 2 1 35.4.b.a ✓ 10
7.c even 3 2 245.4.j.f 20
7.d odd 6 2 245.4.j.e 20
21.c even 2 1 315.4.d.c 10
28.d even 2 1 560.4.g.f 10
35.c odd 2 1 35.4.b.a ✓ 10
35.f even 4 1 175.4.a.i 5
35.f even 4 1 175.4.a.j 5
35.i odd 6 2 245.4.j.e 20
35.j even 6 2 245.4.j.f 20
105.g even 2 1 315.4.d.c 10
105.k odd 4 1 1575.4.a.bn 5
105.k odd 4 1 1575.4.a.bq 5
140.c even 2 1 560.4.g.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.b.a ✓ 10 7.b odd 2 1
35.4.b.a ✓ 10 35.c odd 2 1
175.4.a.i 5 35.f even 4 1
175.4.a.j 5 35.f even 4 1
245.4.b.d 10 1.a even 1 1 trivial
245.4.b.d 10 5.b even 2 1 inner
245.4.j.e 20 7.d odd 6 2
245.4.j.e 20 35.i odd 6 2
245.4.j.f 20 7.c even 3 2
245.4.j.f 20 35.j even 6 2
315.4.d.c 10 21.c even 2 1
315.4.d.c 10 105.g even 2 1
560.4.g.f 10 28.d even 2 1
560.4.g.f 10 140.c even 2 1
1225.4.a.be 5 5.c odd 4 1
1225.4.a.bh 5 5.c odd 4 1
1575.4.a.bn 5 105.k odd 4 1
1575.4.a.bq 5 105.k odd 4 1

Hecke kernels

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

\( T_{2}^{10} + 58T_{2}^{8} + 1109T_{2}^{6} + 8448T_{2}^{4} + 25316T_{2}^{2} + 25600 \) Copy content Toggle raw display
\( T_{19}^{5} + 36T_{19}^{4} - 17350T_{19}^{3} - 559016T_{19}^{2} + 25860480T_{19} - 20133120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 58 T^{8} + \cdots + 25600 \) Copy content Toggle raw display
$3$ \( T^{10} + 158 T^{8} + \cdots + 1982464 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} - 42 T^{4} + \cdots + 11139072)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{5} + 36 T^{4} + \cdots - 20133120)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{5} - 44 T^{4} + \cdots - 126081243400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 60 T^{4} + \cdots + 34433580800)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{5} - 426 T^{4} + \cdots - 109849343072)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 5009454255200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 442 T^{4} + \cdots + 650238065792)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 7767441797120)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 88\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{5} + 1790 T^{4} + \cdots - 201540544400)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 78\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 68855772276480)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
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