Properties

Label 245.4.e.d
Level $245$
Weight $4$
Character orbit 245.e
Analytic conductor $14.455$
Analytic rank $0$
Dimension $2$
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(116,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.116");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - 6 \zeta_{6} + 6) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} - 15 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + ( - 6 \zeta_{6} + 6) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} - 15 q^{8} - 9 \zeta_{6} q^{9} + (5 \zeta_{6} - 5) q^{10} + ( - 44 \zeta_{6} + 44) q^{11} - 42 \zeta_{6} q^{12} - 6 q^{13} - 30 q^{15} - 41 \zeta_{6} q^{16} + (24 \zeta_{6} - 24) q^{17} + (9 \zeta_{6} - 9) q^{18} - 114 \zeta_{6} q^{19} - 35 q^{20} - 44 q^{22} + 52 \zeta_{6} q^{23} + (90 \zeta_{6} - 90) q^{24} + (25 \zeta_{6} - 25) q^{25} + 6 \zeta_{6} q^{26} + 108 q^{27} + 146 q^{29} + 30 \zeta_{6} q^{30} + (276 \zeta_{6} - 276) q^{31} + (161 \zeta_{6} - 161) q^{32} - 264 \zeta_{6} q^{33} + 24 q^{34} - 63 q^{36} + 210 \zeta_{6} q^{37} + (114 \zeta_{6} - 114) q^{38} + (36 \zeta_{6} - 36) q^{39} + 75 \zeta_{6} q^{40} - 444 q^{41} + 492 q^{43} - 308 \zeta_{6} q^{44} + (45 \zeta_{6} - 45) q^{45} + ( - 52 \zeta_{6} + 52) q^{46} - 612 \zeta_{6} q^{47} - 246 q^{48} + 25 q^{50} + 144 \zeta_{6} q^{51} + (42 \zeta_{6} - 42) q^{52} + (50 \zeta_{6} - 50) q^{53} - 108 \zeta_{6} q^{54} - 220 q^{55} - 684 q^{57} - 146 \zeta_{6} q^{58} + ( - 294 \zeta_{6} + 294) q^{59} + (210 \zeta_{6} - 210) q^{60} + 450 \zeta_{6} q^{61} + 276 q^{62} - 167 q^{64} + 30 \zeta_{6} q^{65} + (264 \zeta_{6} - 264) q^{66} + ( - 668 \zeta_{6} + 668) q^{67} + 168 \zeta_{6} q^{68} + 312 q^{69} - 308 q^{71} + 135 \zeta_{6} q^{72} + ( - 12 \zeta_{6} + 12) q^{73} + ( - 210 \zeta_{6} + 210) q^{74} + 150 \zeta_{6} q^{75} - 798 q^{76} + 36 q^{78} - 596 \zeta_{6} q^{79} + (205 \zeta_{6} - 205) q^{80} + ( - 891 \zeta_{6} + 891) q^{81} + 444 \zeta_{6} q^{82} + 966 q^{83} + 120 q^{85} - 492 \zeta_{6} q^{86} + ( - 876 \zeta_{6} + 876) q^{87} + (660 \zeta_{6} - 660) q^{88} - 408 \zeta_{6} q^{89} + 45 q^{90} + 364 q^{92} + 1656 \zeta_{6} q^{93} + (612 \zeta_{6} - 612) q^{94} + (570 \zeta_{6} - 570) q^{95} + 966 \zeta_{6} q^{96} + 1200 q^{97} - 396 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 6 q^{3} + 7 q^{4} - 5 q^{5} - 12 q^{6} - 30 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 6 q^{3} + 7 q^{4} - 5 q^{5} - 12 q^{6} - 30 q^{8} - 9 q^{9} - 5 q^{10} + 44 q^{11} - 42 q^{12} - 12 q^{13} - 60 q^{15} - 41 q^{16} - 24 q^{17} - 9 q^{18} - 114 q^{19} - 70 q^{20} - 88 q^{22} + 52 q^{23} - 90 q^{24} - 25 q^{25} + 6 q^{26} + 216 q^{27} + 292 q^{29} + 30 q^{30} - 276 q^{31} - 161 q^{32} - 264 q^{33} + 48 q^{34} - 126 q^{36} + 210 q^{37} - 114 q^{38} - 36 q^{39} + 75 q^{40} - 888 q^{41} + 984 q^{43} - 308 q^{44} - 45 q^{45} + 52 q^{46} - 612 q^{47} - 492 q^{48} + 50 q^{50} + 144 q^{51} - 42 q^{52} - 50 q^{53} - 108 q^{54} - 440 q^{55} - 1368 q^{57} - 146 q^{58} + 294 q^{59} - 210 q^{60} + 450 q^{61} + 552 q^{62} - 334 q^{64} + 30 q^{65} - 264 q^{66} + 668 q^{67} + 168 q^{68} + 624 q^{69} - 616 q^{71} + 135 q^{72} + 12 q^{73} + 210 q^{74} + 150 q^{75} - 1596 q^{76} + 72 q^{78} - 596 q^{79} - 205 q^{80} + 891 q^{81} + 444 q^{82} + 1932 q^{83} + 240 q^{85} - 492 q^{86} + 876 q^{87} - 660 q^{88} - 408 q^{89} + 90 q^{90} + 728 q^{92} + 1656 q^{93} - 612 q^{94} - 570 q^{95} + 966 q^{96} + 2400 q^{97} - 792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 3.00000 5.19615i 3.50000 6.06218i −2.50000 4.33013i −6.00000 0 −15.0000 −4.50000 7.79423i −2.50000 + 4.33013i
226.1 −0.500000 + 0.866025i 3.00000 + 5.19615i 3.50000 + 6.06218i −2.50000 + 4.33013i −6.00000 0 −15.0000 −4.50000 + 7.79423i −2.50000 4.33013i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.e.d 2
7.b odd 2 1 245.4.e.c 2
7.c even 3 1 245.4.a.b 1
7.c even 3 1 inner 245.4.e.d 2
7.d odd 6 1 245.4.a.c yes 1
7.d odd 6 1 245.4.e.c 2
21.g even 6 1 2205.4.a.n 1
21.h odd 6 1 2205.4.a.k 1
35.i odd 6 1 1225.4.a.f 1
35.j even 6 1 1225.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.b 1 7.c even 3 1
245.4.a.c yes 1 7.d odd 6 1
245.4.e.c 2 7.b odd 2 1
245.4.e.c 2 7.d odd 6 1
245.4.e.d 2 1.a even 1 1 trivial
245.4.e.d 2 7.c even 3 1 inner
1225.4.a.f 1 35.i odd 6 1
1225.4.a.g 1 35.j even 6 1
2205.4.a.k 1 21.h odd 6 1
2205.4.a.n 1 21.g even 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 6T_{3} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 44T + 1936 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 24T + 576 \) Copy content Toggle raw display
$19$ \( T^{2} + 114T + 12996 \) Copy content Toggle raw display
$23$ \( T^{2} - 52T + 2704 \) Copy content Toggle raw display
$29$ \( (T - 146)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 276T + 76176 \) Copy content Toggle raw display
$37$ \( T^{2} - 210T + 44100 \) Copy content Toggle raw display
$41$ \( (T + 444)^{2} \) Copy content Toggle raw display
$43$ \( (T - 492)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 612T + 374544 \) Copy content Toggle raw display
$53$ \( T^{2} + 50T + 2500 \) Copy content Toggle raw display
$59$ \( T^{2} - 294T + 86436 \) Copy content Toggle raw display
$61$ \( T^{2} - 450T + 202500 \) Copy content Toggle raw display
$67$ \( T^{2} - 668T + 446224 \) Copy content Toggle raw display
$71$ \( (T + 308)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$79$ \( T^{2} + 596T + 355216 \) Copy content Toggle raw display
$83$ \( (T - 966)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 408T + 166464 \) Copy content Toggle raw display
$97$ \( (T - 1200)^{2} \) Copy content Toggle raw display
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