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Properties

Label 245.3.m.c
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 4 q^{3} - 8 q^{5} + 48 q^{6} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} - 4 q^{3} - 8 q^{5} + 48 q^{6} + 48 q^{8} + 28 q^{10} + 12 q^{11} + 16 q^{12} + 8 q^{13} - 128 q^{15} - 40 q^{16} - 12 q^{17} + 56 q^{18} - 120 q^{20} - 136 q^{22} + 16 q^{23} - 64 q^{25} - 56 q^{26} - 328 q^{27} + 76 q^{30} - 96 q^{31} - 32 q^{32} + 124 q^{33} + 464 q^{36} + 104 q^{37} + 80 q^{38} - 124 q^{40} + 416 q^{41} + 152 q^{43} + 92 q^{45} + 80 q^{46} - 164 q^{47} + 784 q^{48} - 104 q^{50} - 220 q^{51} + 216 q^{52} + 204 q^{53} - 232 q^{55} - 472 q^{57} - 356 q^{58} - 152 q^{60} + 280 q^{61} - 1136 q^{62} + 192 q^{65} - 544 q^{66} - 324 q^{67} + 184 q^{68} + 288 q^{71} + 440 q^{72} - 248 q^{73} + 108 q^{75} + 1264 q^{76} + 24 q^{78} + 60 q^{80} + 260 q^{81} - 376 q^{82} + 448 q^{83} - 648 q^{85} - 456 q^{86} + 244 q^{87} + 24 q^{88} - 1560 q^{90} - 848 q^{92} - 236 q^{93} - 52 q^{95} + 504 q^{96} - 1128 q^{97}+O(q^{100}) \) 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 −2.53905 − 0.680336i 0.223763 − 0.0599571i 2.51981 + 1.45481i −4.29356 − 2.56229i −0.608937 0 2.02669 + 2.02669i −7.74775 + 4.47317i 9.15835 + 9.42684i
18.2 −1.86467 − 0.499636i −5.16641 + 1.38434i −0.236760 − 0.136693i 2.12654 + 4.52524i 10.3253 0 5.83330 + 5.83330i 16.9812 − 9.80408i −1.70431 − 9.50056i
18.3 −0.149951 − 0.0401793i 2.28511 − 0.612292i −3.44323 − 1.98795i 3.83485 − 3.20841i −0.367256 0 0.875529 + 0.875529i −2.94742 + 1.70169i −0.703952 + 0.327024i
18.4 1.29808 + 0.347820i 0.368128 − 0.0986396i −1.90007 − 1.09700i −0.595274 + 4.96444i 0.512169 0 −5.88593 − 5.88593i −7.66844 + 4.42738i −2.49944 + 6.23720i
18.5 2.60699 + 0.698542i −4.51352 + 1.20939i 2.84435 + 1.64219i 1.33324 − 4.81897i −12.6115 0 −1.36573 − 1.36573i 11.1150 − 6.41723i 6.84199 − 11.6317i
18.6 3.38064 + 0.905840i 4.07088 − 1.09079i 7.14409 + 4.12464i −4.40579 − 2.36411i 14.7503 0 10.5161 + 10.5161i 7.58799 − 4.38093i −12.7529 − 11.9832i
67.1 −0.905840 + 3.38064i −1.09079 − 4.07088i −7.14409 − 4.12464i 0.155515 − 4.99758i 14.7503 0 10.5161 − 10.5161i −7.58799 + 4.38093i 16.7542 + 5.05275i
67.2 −0.698542 + 2.60699i 1.20939 + 4.51352i −2.84435 − 1.64219i −4.83997 − 1.25487i −12.6115 0 −1.36573 + 1.36573i −11.1150 + 6.41723i 6.65236 − 11.7412i
67.3 −0.347820 + 1.29808i −0.0986396 − 0.368128i 1.90007 + 1.09700i 4.59697 + 1.96670i 0.512169 0 −5.88593 + 5.88593i 7.66844 − 4.42738i −4.15185 + 5.28318i
67.4 0.0401793 − 0.149951i −0.612292 − 2.28511i 3.44323 + 1.98795i −4.69599 + 1.71687i −0.367256 0 0.875529 − 0.875529i 2.94742 − 1.70169i 0.0687652 + 0.773152i
67.5 0.499636 − 1.86467i 1.38434 + 5.16641i 0.236760 + 0.136693i 2.85571 + 4.10426i 10.3253 0 5.83330 − 5.83330i −16.9812 + 9.80408i 9.07988 − 3.27431i
67.6 0.680336 − 2.53905i −0.0599571 − 0.223763i −2.51981 − 1.45481i −0.0722266 − 4.99948i −0.608937 0 2.02669 − 2.02669i 7.74775 − 4.47317i −12.7431 − 3.21794i
128.1 −0.905840 − 3.38064i −1.09079 + 4.07088i −7.14409 + 4.12464i 0.155515 + 4.99758i 14.7503 0 10.5161 + 10.5161i −7.58799 − 4.38093i 16.7542 − 5.05275i
128.2 −0.698542 − 2.60699i 1.20939 − 4.51352i −2.84435 + 1.64219i −4.83997 + 1.25487i −12.6115 0 −1.36573 − 1.36573i −11.1150 − 6.41723i 6.65236 + 11.7412i
128.3 −0.347820 − 1.29808i −0.0986396 + 0.368128i 1.90007 − 1.09700i 4.59697 − 1.96670i 0.512169 0 −5.88593 − 5.88593i 7.66844 + 4.42738i −4.15185 − 5.28318i
128.4 0.0401793 + 0.149951i −0.612292 + 2.28511i 3.44323 − 1.98795i −4.69599 − 1.71687i −0.367256 0 0.875529 + 0.875529i 2.94742 + 1.70169i 0.0687652 − 0.773152i
128.5 0.499636 + 1.86467i 1.38434 − 5.16641i 0.236760 − 0.136693i 2.85571 − 4.10426i 10.3253 0 5.83330 + 5.83330i −16.9812 − 9.80408i 9.07988 + 3.27431i
128.6 0.680336 + 2.53905i −0.0599571 + 0.223763i −2.51981 + 1.45481i −0.0722266 + 4.99948i −0.608937 0 2.02669 + 2.02669i 7.74775 + 4.47317i −12.7431 + 3.21794i
177.1 −2.53905 + 0.680336i 0.223763 + 0.0599571i 2.51981 − 1.45481i −4.29356 + 2.56229i −0.608937 0 2.02669 − 2.02669i −7.74775 − 4.47317i 9.15835 − 9.42684i
177.2 −1.86467 + 0.499636i −5.16641 − 1.38434i −0.236760 + 0.136693i 2.12654 − 4.52524i 10.3253 0 5.83330 − 5.83330i 16.9812 + 9.80408i −1.70431 + 9.50056i
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.c 24
5.c odd 4 1 inner 245.3.m.c 24
7.b odd 2 1 245.3.m.d 24
7.c even 3 1 245.3.g.a 12
7.c even 3 1 inner 245.3.m.c 24
7.d odd 6 1 35.3.g.a ✓ 12
7.d odd 6 1 245.3.m.d 24
21.g even 6 1 315.3.o.a 12
28.f even 6 1 560.3.bh.e 12
35.f even 4 1 245.3.m.d 24
35.i odd 6 1 175.3.g.b 12
35.k even 12 1 35.3.g.a ✓ 12
35.k even 12 1 175.3.g.b 12
35.k even 12 1 245.3.m.d 24
35.l odd 12 1 245.3.g.a 12
35.l odd 12 1 inner 245.3.m.c 24
105.w odd 12 1 315.3.o.a 12
140.x odd 12 1 560.3.bh.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.g.a ✓ 12 7.d odd 6 1
35.3.g.a ✓ 12 35.k even 12 1
175.3.g.b 12 35.i odd 6 1
175.3.g.b 12 35.k even 12 1
245.3.g.a 12 7.c even 3 1
245.3.g.a 12 35.l odd 12 1
245.3.m.c 24 1.a even 1 1 trivial
245.3.m.c 24 5.c odd 4 1 inner
245.3.m.c 24 7.c even 3 1 inner
245.3.m.c 24 35.l odd 12 1 inner
245.3.m.d 24 7.b odd 2 1
245.3.m.d 24 7.d odd 6 1
245.3.m.d 24 35.f even 4 1
245.3.m.d 24 35.k even 12 1
315.3.o.a 12 21.g even 6 1
315.3.o.a 12 105.w odd 12 1
560.3.bh.e 12 28.f even 6 1
560.3.bh.e 12 140.x odd 12 1

Hecke kernels

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

\( T_{2}^{24} - 4 T_{2}^{23} + 8 T_{2}^{22} - 48 T_{2}^{21} + 26 T_{2}^{20} + 376 T_{2}^{19} - 560 T_{2}^{18} + \cdots + 10000 \) Copy content Toggle raw display
\( T_{3}^{24} + 4 T_{3}^{23} + 8 T_{3}^{22} + 168 T_{3}^{21} - 181 T_{3}^{20} - 3488 T_{3}^{19} + \cdots + 234256 \) Copy content Toggle raw display