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Properties

Label 245.4.e.g
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (8 \zeta_{6} - 8) q^{4} - 5 \zeta_{6} q^{5} + 8 q^{6} + 23 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (8 \zeta_{6} - 8) q^{4} - 5 \zeta_{6} q^{5} + 8 q^{6} + 23 \zeta_{6} q^{9} + ( - 20 \zeta_{6} + 20) q^{10} + (32 \zeta_{6} - 32) q^{11} + 16 \zeta_{6} q^{12} + 38 q^{13} - 10 q^{15} + 64 \zeta_{6} q^{16} + ( - 26 \zeta_{6} + 26) q^{17} + (92 \zeta_{6} - 92) q^{18} + 100 \zeta_{6} q^{19} + 40 q^{20} - 128 q^{22} + 78 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 152 \zeta_{6} q^{26} + 100 q^{27} - 50 q^{29} - 40 \zeta_{6} q^{30} + (108 \zeta_{6} - 108) q^{31} + (256 \zeta_{6} - 256) q^{32} + 64 \zeta_{6} q^{33} + 104 q^{34} - 184 q^{36} - 266 \zeta_{6} q^{37} + (400 \zeta_{6} - 400) q^{38} + ( - 76 \zeta_{6} + 76) q^{39} - 22 q^{41} + 442 q^{43} - 256 \zeta_{6} q^{44} + ( - 115 \zeta_{6} + 115) q^{45} + (312 \zeta_{6} - 312) q^{46} - 514 \zeta_{6} q^{47} + 128 q^{48} - 100 q^{50} - 52 \zeta_{6} q^{51} + (304 \zeta_{6} - 304) q^{52} + (2 \zeta_{6} - 2) q^{53} + 400 \zeta_{6} q^{54} + 160 q^{55} + 200 q^{57} - 200 \zeta_{6} q^{58} + ( - 500 \zeta_{6} + 500) q^{59} + ( - 80 \zeta_{6} + 80) q^{60} - 518 \zeta_{6} q^{61} - 432 q^{62} - 512 q^{64} - 190 \zeta_{6} q^{65} + (256 \zeta_{6} - 256) q^{66} + (126 \zeta_{6} - 126) q^{67} + 208 \zeta_{6} q^{68} + 156 q^{69} + 412 q^{71} + (878 \zeta_{6} - 878) q^{73} + ( - 1064 \zeta_{6} + 1064) q^{74} + 50 \zeta_{6} q^{75} - 800 q^{76} + 304 q^{78} - 600 \zeta_{6} q^{79} + ( - 320 \zeta_{6} + 320) q^{80} + (421 \zeta_{6} - 421) q^{81} - 88 \zeta_{6} q^{82} - 282 q^{83} - 130 q^{85} + 1768 \zeta_{6} q^{86} + (100 \zeta_{6} - 100) q^{87} - 150 \zeta_{6} q^{89} + 460 q^{90} - 624 q^{92} + 216 \zeta_{6} q^{93} + ( - 2056 \zeta_{6} + 2056) q^{94} + ( - 500 \zeta_{6} + 500) q^{95} + 512 \zeta_{6} q^{96} - 386 q^{97} - 736 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 5 q^{5} + 16 q^{6} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} + 2 q^{3} - 8 q^{4} - 5 q^{5} + 16 q^{6} + 23 q^{9} + 20 q^{10} - 32 q^{11} + 16 q^{12} + 76 q^{13} - 20 q^{15} + 64 q^{16} + 26 q^{17} - 92 q^{18} + 100 q^{19} + 80 q^{20} - 256 q^{22} + 78 q^{23} - 25 q^{25} + 152 q^{26} + 200 q^{27} - 100 q^{29} - 40 q^{30} - 108 q^{31} - 256 q^{32} + 64 q^{33} + 208 q^{34} - 368 q^{36} - 266 q^{37} - 400 q^{38} + 76 q^{39} - 44 q^{41} + 884 q^{43} - 256 q^{44} + 115 q^{45} - 312 q^{46} - 514 q^{47} + 256 q^{48} - 200 q^{50} - 52 q^{51} - 304 q^{52} - 2 q^{53} + 400 q^{54} + 320 q^{55} + 400 q^{57} - 200 q^{58} + 500 q^{59} + 80 q^{60} - 518 q^{61} - 864 q^{62} - 1024 q^{64} - 190 q^{65} - 256 q^{66} - 126 q^{67} + 208 q^{68} + 312 q^{69} + 824 q^{71} - 878 q^{73} + 1064 q^{74} + 50 q^{75} - 1600 q^{76} + 608 q^{78} - 600 q^{79} + 320 q^{80} - 421 q^{81} - 88 q^{82} - 564 q^{83} - 260 q^{85} + 1768 q^{86} - 100 q^{87} - 150 q^{89} + 920 q^{90} - 1248 q^{92} + 216 q^{93} + 2056 q^{94} + 500 q^{95} + 512 q^{96} - 772 q^{97} - 1472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
2.00000 + 3.46410i 1.00000 − 1.73205i −4.00000 + 6.92820i −2.50000 − 4.33013i 8.00000 0 0 11.5000 + 19.9186i 10.0000 − 17.3205i
226.1 2.00000 − 3.46410i 1.00000 + 1.73205i −4.00000 − 6.92820i −2.50000 + 4.33013i 8.00000 0 0 11.5000 − 19.9186i 10.0000 + 17.3205i
\(n\): 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.g 2
7.b odd 2 1 245.4.e.f 2
7.c even 3 1 245.4.a.a 1
7.c even 3 1 inner 245.4.e.g 2
7.d odd 6 1 5.4.a.a ✓ 1
7.d odd 6 1 245.4.e.f 2
21.g even 6 1 45.4.a.d 1
21.h odd 6 1 2205.4.a.q 1
28.f even 6 1 80.4.a.d 1
35.i odd 6 1 25.4.a.c 1
35.j even 6 1 1225.4.a.k 1
35.k even 12 2 25.4.b.a 2
56.j odd 6 1 320.4.a.g 1
56.m even 6 1 320.4.a.h 1
63.i even 6 1 405.4.e.c 2
63.k odd 6 1 405.4.e.l 2
63.s even 6 1 405.4.e.c 2
63.t odd 6 1 405.4.e.l 2
77.i even 6 1 605.4.a.d 1
84.j odd 6 1 720.4.a.u 1
91.s odd 6 1 845.4.a.b 1
105.p even 6 1 225.4.a.b 1
105.w odd 12 2 225.4.b.c 2
112.v even 12 2 1280.4.d.l 2
112.x odd 12 2 1280.4.d.e 2
119.h odd 6 1 1445.4.a.a 1
133.o even 6 1 1805.4.a.h 1
140.s even 6 1 400.4.a.m 1
140.x odd 12 2 400.4.c.k 2
280.ba even 6 1 1600.4.a.s 1
280.bk odd 6 1 1600.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a ✓ 1 7.d odd 6 1
25.4.a.c 1 35.i odd 6 1
25.4.b.a 2 35.k even 12 2
45.4.a.d 1 21.g even 6 1
80.4.a.d 1 28.f even 6 1
225.4.a.b 1 105.p even 6 1
225.4.b.c 2 105.w odd 12 2
245.4.a.a 1 7.c even 3 1
245.4.e.f 2 7.b odd 2 1
245.4.e.f 2 7.d odd 6 1
245.4.e.g 2 1.a even 1 1 trivial
245.4.e.g 2 7.c even 3 1 inner
320.4.a.g 1 56.j odd 6 1
320.4.a.h 1 56.m even 6 1
400.4.a.m 1 140.s even 6 1
400.4.c.k 2 140.x odd 12 2
405.4.e.c 2 63.i even 6 1
405.4.e.c 2 63.s even 6 1
405.4.e.l 2 63.k odd 6 1
405.4.e.l 2 63.t odd 6 1
605.4.a.d 1 77.i even 6 1
720.4.a.u 1 84.j odd 6 1
845.4.a.b 1 91.s odd 6 1
1225.4.a.k 1 35.j even 6 1
1280.4.d.e 2 112.x odd 12 2
1280.4.d.l 2 112.v even 12 2
1445.4.a.a 1 119.h odd 6 1
1600.4.a.s 1 280.ba even 6 1
1600.4.a.bi 1 280.bk odd 6 1
1805.4.a.h 1 133.o even 6 1
2205.4.a.q 1 21.h odd 6 1

Hecke kernels

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

\( T_{2}^{2} - 4T_{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$13$ \( (T - 38)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 26T + 676 \) Copy content Toggle raw display
$19$ \( T^{2} - 100T + 10000 \) Copy content Toggle raw display
$23$ \( T^{2} - 78T + 6084 \) Copy content Toggle raw display
$29$ \( (T + 50)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 108T + 11664 \) Copy content Toggle raw display
$37$ \( T^{2} + 266T + 70756 \) Copy content Toggle raw display
$41$ \( (T + 22)^{2} \) Copy content Toggle raw display
$43$ \( (T - 442)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 514T + 264196 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 500T + 250000 \) Copy content Toggle raw display
$61$ \( T^{2} + 518T + 268324 \) Copy content Toggle raw display
$67$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$71$ \( (T - 412)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 878T + 770884 \) Copy content Toggle raw display
$79$ \( T^{2} + 600T + 360000 \) Copy content Toggle raw display
$83$ \( (T + 282)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 150T + 22500 \) Copy content Toggle raw display
$97$ \( (T + 386)^{2} \) Copy content Toggle raw display
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