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Properties

Label 245.3.m.b
Level $245$
Weight $3$
Character orbit 245.m
Analytic conductor $6.676$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,3,Mod(18,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.18");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.m (of order \(12\), degree \(4\), 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: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 2 q^{3} + 4 q^{5} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 2 q^{3} + 4 q^{5} - 36 q^{8} - 14 q^{10} - 24 q^{11} + 46 q^{12} + 8 q^{13} + 52 q^{15} + 20 q^{16} + 48 q^{17} - 4 q^{18} + 72 q^{20} + 104 q^{22} - 86 q^{23} - 16 q^{25} - 140 q^{26} - 76 q^{27} + 64 q^{30} - 120 q^{31} + 130 q^{32} - 116 q^{33} - 496 q^{36} + 44 q^{37} - 16 q^{38} + 158 q^{40} - 16 q^{41} - 196 q^{43} + 104 q^{45} - 148 q^{46} + 208 q^{47} + 52 q^{48} + 580 q^{50} - 160 q^{51} + 288 q^{52} - 72 q^{53} - 208 q^{55} + 656 q^{57} - 2 q^{58} + 262 q^{60} - 308 q^{61} - 176 q^{62} + 132 q^{65} - 316 q^{66} + 198 q^{67} - 332 q^{68} - 792 q^{71} + 308 q^{72} - 380 q^{73} + 450 q^{75} + 400 q^{76} - 720 q^{78} + 324 q^{80} - 352 q^{81} + 818 q^{82} + 460 q^{83} + 144 q^{85} - 336 q^{86} + 214 q^{87} - 288 q^{88} - 120 q^{90} + 1372 q^{92} - 68 q^{93} - 88 q^{95} - 816 q^{96} + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −3.54535 − 0.949975i 1.41502 − 0.379154i 8.20298 + 4.73599i −0.551807 − 4.96946i −5.37694 0 −14.2019 − 14.2019i −5.93570 + 3.42698i −2.76451 + 18.1427i
18.2 −1.89206 − 0.506976i −2.62470 + 0.703286i −0.141229 − 0.0815388i −4.09808 + 2.86456i 5.32264 0 5.76622 + 5.76622i −1.39979 + 0.808169i 9.20609 − 3.34229i
18.3 −1.23172 − 0.330037i 4.01589 − 1.07605i −2.05590 − 1.18698i 1.75247 + 4.68282i −5.30157 0 5.74725 + 5.74725i 7.17525 − 4.14263i −0.613038 − 6.34629i
18.4 0.585559 + 0.156900i −4.00038 + 1.07190i −3.14584 − 1.81625i 4.51288 − 2.15265i −2.51064 0 −3.27174 − 3.27174i 7.05981 − 4.07598i 2.98031 − 0.552433i
18.5 1.75904 + 0.471334i 0.195743 − 0.0524492i −0.592022 − 0.341804i −4.78371 − 1.45469i 0.369042 0 −6.03113 − 6.03113i −7.75866 + 4.47947i −7.72911 − 4.81359i
18.6 2.95850 + 0.792728i 2.36445 − 0.633552i 4.66022 + 2.69058i 4.16825 + 2.76146i 7.49746 0 2.99127 + 2.99127i −2.60501 + 1.50400i 10.1427 + 11.4741i
67.1 −0.792728 + 2.95850i −0.633552 − 2.36445i −4.66022 − 2.69058i 0.307372 + 4.99054i 7.49746 0 2.99127 − 2.99127i 2.60501 − 1.50400i −15.0082 − 3.04678i
67.2 −0.471334 + 1.75904i −0.0524492 − 0.195743i 0.592022 + 0.341804i 1.13206 − 4.87016i 0.369042 0 −6.03113 + 6.03113i 7.75866 − 4.47947i 8.03325 + 4.28681i
67.3 −0.156900 + 0.585559i 1.07190 + 4.00038i 3.14584 + 1.81625i −4.12069 + 2.83194i −2.51064 0 −3.27174 + 3.27174i −7.05981 + 4.07598i −1.01173 − 2.85724i
67.4 0.330037 − 1.23172i −1.07605 − 4.01589i 2.05590 + 1.18698i 3.17921 + 3.85910i −5.30157 0 5.74725 − 5.74725i −7.17525 + 4.14263i 5.80257 − 2.64224i
67.5 0.506976 − 1.89206i 0.703286 + 2.62470i 0.141229 + 0.0815388i 4.52982 − 2.11677i 5.32264 0 5.76622 − 5.76622i 1.39979 − 0.808169i −1.70854 − 9.64386i
67.6 0.949975 − 3.54535i −0.379154 − 1.41502i −8.20298 − 4.73599i −4.02777 − 2.96261i −5.37694 0 −14.2019 + 14.2019i 5.93570 − 3.42698i −14.3298 + 11.4655i
128.1 −0.792728 − 2.95850i −0.633552 + 2.36445i −4.66022 + 2.69058i 0.307372 − 4.99054i 7.49746 0 2.99127 + 2.99127i 2.60501 + 1.50400i −15.0082 + 3.04678i
128.2 −0.471334 − 1.75904i −0.0524492 + 0.195743i 0.592022 − 0.341804i 1.13206 + 4.87016i 0.369042 0 −6.03113 − 6.03113i 7.75866 + 4.47947i 8.03325 − 4.28681i
128.3 −0.156900 − 0.585559i 1.07190 − 4.00038i 3.14584 − 1.81625i −4.12069 − 2.83194i −2.51064 0 −3.27174 − 3.27174i −7.05981 − 4.07598i −1.01173 + 2.85724i
128.4 0.330037 + 1.23172i −1.07605 + 4.01589i 2.05590 − 1.18698i 3.17921 − 3.85910i −5.30157 0 5.74725 + 5.74725i −7.17525 − 4.14263i 5.80257 + 2.64224i
128.5 0.506976 + 1.89206i 0.703286 − 2.62470i 0.141229 − 0.0815388i 4.52982 + 2.11677i 5.32264 0 5.76622 + 5.76622i 1.39979 + 0.808169i −1.70854 + 9.64386i
128.6 0.949975 + 3.54535i −0.379154 + 1.41502i −8.20298 + 4.73599i −4.02777 + 2.96261i −5.37694 0 −14.2019 − 14.2019i 5.93570 + 3.42698i −14.3298 − 11.4655i
177.1 −3.54535 + 0.949975i 1.41502 + 0.379154i 8.20298 − 4.73599i −0.551807 + 4.96946i −5.37694 0 −14.2019 + 14.2019i −5.93570 − 3.42698i −2.76451 − 18.1427i
177.2 −1.89206 + 0.506976i −2.62470 − 0.703286i −0.141229 + 0.0815388i −4.09808 − 2.86456i 5.32264 0 5.76622 − 5.76622i −1.39979 − 0.808169i 9.20609 + 3.34229i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.3.m.b 24
5.c odd 4 1 inner 245.3.m.b 24
7.b odd 2 1 35.3.l.a ✓ 24
7.c even 3 1 245.3.g.b 12
7.c even 3 1 inner 245.3.m.b 24
7.d odd 6 1 35.3.l.a ✓ 24
7.d odd 6 1 245.3.g.c 12
21.c even 2 1 315.3.ca.a 24
21.g even 6 1 315.3.ca.a 24
35.c odd 2 1 175.3.p.c 24
35.f even 4 1 35.3.l.a ✓ 24
35.f even 4 1 175.3.p.c 24
35.i odd 6 1 175.3.p.c 24
35.k even 12 1 35.3.l.a ✓ 24
35.k even 12 1 175.3.p.c 24
35.k even 12 1 245.3.g.c 12
35.l odd 12 1 245.3.g.b 12
35.l odd 12 1 inner 245.3.m.b 24
105.k odd 4 1 315.3.ca.a 24
105.w odd 12 1 315.3.ca.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.l.a ✓ 24 7.b odd 2 1
35.3.l.a ✓ 24 7.d odd 6 1
35.3.l.a ✓ 24 35.f even 4 1
35.3.l.a ✓ 24 35.k even 12 1
175.3.p.c 24 35.c odd 2 1
175.3.p.c 24 35.f even 4 1
175.3.p.c 24 35.i odd 6 1
175.3.p.c 24 35.k even 12 1
245.3.g.b 12 7.c even 3 1
245.3.g.b 12 35.l odd 12 1
245.3.g.c 12 7.d odd 6 1
245.3.g.c 12 35.k even 12 1
245.3.m.b 24 1.a even 1 1 trivial
245.3.m.b 24 5.c odd 4 1 inner
245.3.m.b 24 7.c even 3 1 inner
245.3.m.b 24 35.l odd 12 1 inner
315.3.ca.a 24 21.c even 2 1
315.3.ca.a 24 21.g even 6 1
315.3.ca.a 24 105.k odd 4 1
315.3.ca.a 24 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{24} + 2 T_{2}^{23} + 2 T_{2}^{22} + 24 T_{2}^{21} - 103 T_{2}^{20} - 368 T_{2}^{19} + \cdots + 923521 \) Copy content Toggle raw display
\( T_{3}^{24} - 2 T_{3}^{23} + 2 T_{3}^{22} - 346 T_{3}^{20} + 682 T_{3}^{19} - 672 T_{3}^{18} + \cdots + 1336336 \) Copy content Toggle raw display