Properties

Label 245.4.b.f
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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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: \( x^{10} + 55x^{8} + 1007x^{6} + 6645x^{4} + 8636x^{2} + 3136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 3) q^{4} + \beta_{6} q^{5} + ( - \beta_{3} - \beta_{2} + 5) q^{6} + (\beta_{8} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_1) q^{8} + ( - \beta_{9} + \beta_{3} - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 3) q^{4} + \beta_{6} q^{5} + ( - \beta_{3} - \beta_{2} + 5) q^{6} + (\beta_{8} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_1) q^{8} + ( - \beta_{9} + \beta_{3} - 8) q^{9} + ( - \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_1 - 2) q^{10} + ( - 2 \beta_{9} - \beta_{6} - \beta_{5} - \beta_{3} - 2 \beta_{2} - 2) q^{11} + ( - 4 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + 5 \beta_1) q^{12} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 3 \beta_{4} - 2 \beta_1) q^{13} + ( - \beta_{9} + 4 \beta_{8} + \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{3} + \cdots - 7) q^{15}+ \cdots + ( - 10 \beta_{9} - 62 \beta_{6} - 62 \beta_{5} - 29 \beta_{3} + 14 \beta_{2} + \cdots + 740) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 30 q^{4} + 3 q^{5} + 48 q^{6} - 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 30 q^{4} + 3 q^{5} + 48 q^{6} - 82 q^{9} - 32 q^{10} - 36 q^{11} - 73 q^{15} + 22 q^{16} - 192 q^{19} - 292 q^{20} - 444 q^{24} + 187 q^{25} + 434 q^{26} + 130 q^{29} + 658 q^{30} + 834 q^{31} + 80 q^{34} + 258 q^{36} - 868 q^{39} + 674 q^{40} + 612 q^{41} - 542 q^{44} + 60 q^{45} + 1274 q^{46} - 1278 q^{50} - 986 q^{51} - 2808 q^{54} + 371 q^{55} - 2514 q^{59} + 204 q^{60} + 512 q^{61} + 3450 q^{64} + 946 q^{65} + 1396 q^{66} - 1532 q^{69} - 1472 q^{71} - 1590 q^{74} + 3003 q^{75} - 22 q^{76} - 46 q^{79} + 2304 q^{80} + 130 q^{81} - 2541 q^{85} + 1592 q^{86} - 5876 q^{89} + 1658 q^{90} - 3314 q^{94} + 2155 q^{95} + 3756 q^{96} + 6930 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 55x^{8} + 1007x^{6} + 6645x^{4} + 8636x^{2} + 3136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 58\nu^{6} + 1097\nu^{4} + 6968\nu^{2} + 4368 ) / 56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{9} - 163\nu^{7} - 2905\nu^{5} - 17741\nu^{3} - 11860\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 4 \nu^{8} + 53 \nu^{7} + 218 \nu^{6} + 919 \nu^{5} + 3912 \nu^{4} + 5515 \nu^{3} + 24162 \nu^{2} + 4472 \nu + 16464 ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{9} + 4 \nu^{8} - 53 \nu^{7} + 218 \nu^{6} - 919 \nu^{5} + 3912 \nu^{4} - 5515 \nu^{3} + 24162 \nu^{2} - 4472 \nu + 16464 ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{9} + 381\nu^{7} + 6817\nu^{5} + 42015\nu^{3} + 30116\nu ) / 112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} - 54\nu^{7} - 956\nu^{5} - 5800\nu^{3} - 3845\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{8} - 160\nu^{6} - 2787\nu^{4} - 16606\nu^{2} - 11256 ) / 28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{6} + \beta_{5} - 2\beta_{4} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 2\beta_{6} + 2\beta_{5} - 2\beta_{3} - 21\beta_{2} + 201 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -30\beta_{8} + 8\beta_{7} + 28\beta_{6} - 28\beta_{5} + 80\beta_{4} + 317\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -34\beta_{9} - 72\beta_{6} - 72\beta_{5} + 84\beta_{3} + 449\beta_{2} - 3991 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 811\beta_{8} - 296\beta_{7} - 695\beta_{6} + 695\beta_{5} - 2390\beta_{4} - 6257\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 875\beta_{9} + 1982\beta_{6} + 1982\beta_{5} - 2622\beta_{3} - 9973\beta_{2} + 83261 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -20928\beta_{8} + 8336\beta_{7} + 16562\beta_{6} - 16562\beta_{5} + 64180\beta_{4} + 129581\beta_1 \) Copy content Toggle raw display

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
4.87000i
4.06627i
3.63135i
0.915901i
0.850248i
0.850248i
0.915901i
3.63135i
4.06627i
4.87000i
4.87000i 8.01530i −15.7169 10.9686 2.16570i 39.0345 0 37.5814i −37.2450 −10.5470 53.4170i
99.2 4.06627i 4.54560i −8.53456 −0.119570 11.1797i −18.4836 0 2.17369i 6.33755 −45.4597 + 0.486203i
99.3 3.63135i 0.388534i −5.18672 −7.39622 + 8.38427i −1.41090 0 10.2160i 26.8490 30.4462 + 26.8583i
99.4 0.915901i 8.77417i 7.16112 −10.5240 + 3.77419i 8.03628 0 13.8861i −49.9861 3.45678 + 9.63899i
99.5 0.850248i 3.73570i 7.27708 8.57125 + 7.17869i −3.17627 0 12.9893i 13.0446 6.10367 7.28769i
99.6 0.850248i 3.73570i 7.27708 8.57125 7.17869i −3.17627 0 12.9893i 13.0446 6.10367 + 7.28769i
99.7 0.915901i 8.77417i 7.16112 −10.5240 3.77419i 8.03628 0 13.8861i −49.9861 3.45678 9.63899i
99.8 3.63135i 0.388534i −5.18672 −7.39622 8.38427i −1.41090 0 10.2160i 26.8490 30.4462 26.8583i
99.9 4.06627i 4.54560i −8.53456 −0.119570 + 11.1797i −18.4836 0 2.17369i 6.33755 −45.4597 0.486203i
99.10 4.87000i 8.01530i −15.7169 10.9686 + 2.16570i 39.0345 0 37.5814i −37.2450 −10.5470 + 53.4170i
\(n\): 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.f 10
5.b even 2 1 inner 245.4.b.f 10
5.c odd 4 2 1225.4.a.bq 10
7.b odd 2 1 245.4.b.e 10
7.c even 3 2 245.4.j.d 20
7.d odd 6 2 35.4.j.a 20
35.c odd 2 1 245.4.b.e 10
35.f even 4 2 1225.4.a.bp 10
35.i odd 6 2 35.4.j.a 20
35.j even 6 2 245.4.j.d 20
35.k even 12 4 175.4.e.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.j.a 20 7.d odd 6 2
35.4.j.a 20 35.i odd 6 2
175.4.e.g 20 35.k even 12 4
245.4.b.e 10 7.b odd 2 1
245.4.b.e 10 35.c odd 2 1
245.4.b.f 10 1.a even 1 1 trivial
245.4.b.f 10 5.b even 2 1 inner
245.4.j.d 20 7.c even 3 2
245.4.j.d 20 35.j even 6 2
1225.4.a.bp 10 35.f even 4 2
1225.4.a.bq 10 5.c odd 4 2

Hecke kernels

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

\( T_{2}^{10} + 55T_{2}^{8} + 1007T_{2}^{6} + 6645T_{2}^{4} + 8636T_{2}^{2} + 3136 \) Copy content Toggle raw display
\( T_{19}^{5} + 96T_{19}^{4} - 4015T_{19}^{3} - 505490T_{19}^{2} - 2712444T_{19} + 282876216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 55 T^{8} + 1007 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$3$ \( T^{10} + 176 T^{8} + 10150 T^{6} + \cdots + 215296 \) Copy content Toggle raw display
$5$ \( T^{10} - 3 T^{9} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} + 18 T^{4} - 4011 T^{3} + \cdots - 8279040)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 3445 T^{8} + \cdots + 3071819776 \) Copy content Toggle raw display
$17$ \( T^{10} + 24259 T^{8} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{5} + 96 T^{4} - 4015 T^{3} + \cdots + 282876216)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 34673 T^{8} + \cdots + 97\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{5} - 65 T^{4} - 31763 T^{3} + \cdots + 1210995800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 417 T^{4} + \cdots + 115393724432)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 164548 T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} - 306 T^{4} + \cdots - 30026757530)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 296445 T^{8} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{10} + 451664 T^{8} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + 1192144 T^{8} + \cdots + 77\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{5} + 1257 T^{4} + \cdots - 1162994045920)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 256 T^{4} + \cdots + 1787182936154)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 1943836 T^{8} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + 736 T^{4} + \cdots - 445136828288)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 1262423 T^{8} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{5} + 23 T^{4} + \cdots - 197075426883344)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 1878581 T^{8} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{5} + 2938 T^{4} + \cdots - 239387576587026)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 5233200 T^{8} + \cdots + 56\!\cdots\!24 \) Copy content Toggle raw display
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