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Properties

Label 370.2.bd.a
Level $370$
Weight $2$
Character orbit 370.bd
Analytic conductor $2.954$
Analytic rank $0$
Dimension $108$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(13,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([27, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.bd (of order \(36\), degree \(12\), 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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(9\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 108 q + 6 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q + 6 q^{3} + 6 q^{10} + 36 q^{11} - 6 q^{12} - 12 q^{14} - 12 q^{17} - 12 q^{19} - 42 q^{21} - 36 q^{23} + 6 q^{24} - 6 q^{25} - 6 q^{26} - 6 q^{27} - 24 q^{30} + 6 q^{33} - 66 q^{35} + 48 q^{38} - 12 q^{40} + 48 q^{41} + 66 q^{42} - 6 q^{44} - 162 q^{45} + 6 q^{46} + 24 q^{47} + 12 q^{49} + 36 q^{50} - 12 q^{51} + 12 q^{53} + 18 q^{54} + 48 q^{57} + 6 q^{58} - 24 q^{59} - 36 q^{61} - 30 q^{62} + 102 q^{63} + 54 q^{64} + 54 q^{65} - 54 q^{67} + 96 q^{69} - 36 q^{70} - 48 q^{71} - 84 q^{73} - 42 q^{74} + 252 q^{75} - 6 q^{76} + 36 q^{77} - 36 q^{78} - 66 q^{79} - 6 q^{80} - 108 q^{81} + 6 q^{82} - 60 q^{83} - 18 q^{85} - 168 q^{87} + 12 q^{88} + 66 q^{89} + 12 q^{90} - 18 q^{91} - 18 q^{92} - 18 q^{94} - 18 q^{95} + 12 q^{96} - 36 q^{97} - 60 q^{98} + 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.984808 − 0.173648i −2.67879 − 1.87571i 0.939693 + 0.342020i −0.431297 + 2.19408i 2.31238 + 2.31238i 0.164874 − 1.88452i −0.866025 − 0.500000i 2.63158 + 7.23020i 0.805742 − 2.08585i
13.2 −0.984808 − 0.173648i −2.05692 − 1.44027i 0.939693 + 0.342020i −0.583057 − 2.15871i 1.77557 + 1.77557i 0.140016 − 1.60039i −0.866025 − 0.500000i 1.13048 + 3.10597i 0.199343 + 2.22716i
13.3 −0.984808 − 0.173648i −0.987459 − 0.691427i 0.939693 + 0.342020i 0.590293 + 2.15675i 0.852393 + 0.852393i −0.353973 + 4.04593i −0.866025 − 0.500000i −0.529055 − 1.45357i −0.206810 − 2.22648i
13.4 −0.984808 − 0.173648i −0.732652 − 0.513008i 0.939693 + 0.342020i −2.22335 − 0.238120i 0.632438 + 0.632438i 0.102040 − 1.16632i −0.866025 − 0.500000i −0.752459 − 2.06736i 2.14823 + 0.620583i
13.5 −0.984808 − 0.173648i −0.0536157 − 0.0375421i 0.939693 + 0.342020i 0.426264 − 2.19506i 0.0462820 + 0.0462820i −0.357616 + 4.08757i −0.866025 − 0.500000i −1.02460 − 2.81505i −0.800956 + 2.08769i
13.6 −0.984808 − 0.173648i 0.133392 + 0.0934022i 0.939693 + 0.342020i 1.54803 + 1.61357i −0.115147 − 0.115147i 0.284408 − 3.25079i −0.866025 − 0.500000i −1.01699 − 2.79416i −1.24432 − 1.85787i
13.7 −0.984808 − 0.173648i 0.187969 + 0.131617i 0.939693 + 0.342020i 0.928613 − 2.03413i −0.162258 − 0.162258i 0.180693 − 2.06533i −0.866025 − 0.500000i −1.00805 − 2.76960i −1.26773 + 1.84197i
13.8 −0.984808 − 0.173648i 1.98147 + 1.38744i 0.939693 + 0.342020i −1.05394 + 1.97211i −1.71044 − 1.71044i −0.0535790 + 0.612411i −0.866025 − 0.500000i 0.975177 + 2.67928i 1.38038 − 1.75913i
13.9 −0.984808 − 0.173648i 2.43175 + 1.70273i 0.939693 + 0.342020i 2.13889 − 0.652021i −2.09913 − 2.09913i −0.0617478 + 0.705781i −0.866025 − 0.500000i 1.98806 + 5.46214i −2.21962 + 0.270700i
57.1 −0.984808 + 0.173648i −2.67879 + 1.87571i 0.939693 − 0.342020i −0.431297 − 2.19408i 2.31238 − 2.31238i 0.164874 + 1.88452i −0.866025 + 0.500000i 2.63158 − 7.23020i 0.805742 + 2.08585i
57.2 −0.984808 + 0.173648i −2.05692 + 1.44027i 0.939693 − 0.342020i −0.583057 + 2.15871i 1.77557 − 1.77557i 0.140016 + 1.60039i −0.866025 + 0.500000i 1.13048 − 3.10597i 0.199343 − 2.22716i
57.3 −0.984808 + 0.173648i −0.987459 + 0.691427i 0.939693 − 0.342020i 0.590293 − 2.15675i 0.852393 − 0.852393i −0.353973 − 4.04593i −0.866025 + 0.500000i −0.529055 + 1.45357i −0.206810 + 2.22648i
57.4 −0.984808 + 0.173648i −0.732652 + 0.513008i 0.939693 − 0.342020i −2.22335 + 0.238120i 0.632438 − 0.632438i 0.102040 + 1.16632i −0.866025 + 0.500000i −0.752459 + 2.06736i 2.14823 − 0.620583i
57.5 −0.984808 + 0.173648i −0.0536157 + 0.0375421i 0.939693 − 0.342020i 0.426264 + 2.19506i 0.0462820 − 0.0462820i −0.357616 − 4.08757i −0.866025 + 0.500000i −1.02460 + 2.81505i −0.800956 − 2.08769i
57.6 −0.984808 + 0.173648i 0.133392 − 0.0934022i 0.939693 − 0.342020i 1.54803 − 1.61357i −0.115147 + 0.115147i 0.284408 + 3.25079i −0.866025 + 0.500000i −1.01699 + 2.79416i −1.24432 + 1.85787i
57.7 −0.984808 + 0.173648i 0.187969 − 0.131617i 0.939693 − 0.342020i 0.928613 + 2.03413i −0.162258 + 0.162258i 0.180693 + 2.06533i −0.866025 + 0.500000i −1.00805 + 2.76960i −1.26773 − 1.84197i
57.8 −0.984808 + 0.173648i 1.98147 − 1.38744i 0.939693 − 0.342020i −1.05394 − 1.97211i −1.71044 + 1.71044i −0.0535790 − 0.612411i −0.866025 + 0.500000i 0.975177 − 2.67928i 1.38038 + 1.75913i
57.9 −0.984808 + 0.173648i 2.43175 − 1.70273i 0.939693 − 0.342020i 2.13889 + 0.652021i −2.09913 + 2.09913i −0.0617478 − 0.705781i −0.866025 + 0.500000i 1.98806 − 5.46214i −2.21962 − 0.270700i
93.1 0.642788 − 0.766044i −0.238237 + 2.72306i −0.173648 − 0.984808i 1.97861 − 1.04168i 1.93285 + 1.93285i 2.60421 − 1.21436i −0.866025 − 0.500000i −4.40390 − 0.776526i 0.473855 − 2.18528i
93.2 0.642788 − 0.766044i −0.190747 + 2.18025i −0.173648 − 0.984808i −0.178132 + 2.22896i 1.54756 + 1.54756i −2.39110 + 1.11499i −0.866025 − 0.500000i −1.76267 − 0.310806i 1.59298 + 1.56921i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.bc even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.bd.a yes 108
5.c odd 4 1 370.2.ba.a ✓ 108
37.i odd 36 1 370.2.ba.a ✓ 108
185.bc even 36 1 inner 370.2.bd.a yes 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.ba.a ✓ 108 5.c odd 4 1
370.2.ba.a ✓ 108 37.i odd 36 1
370.2.bd.a yes 108 1.a even 1 1 trivial
370.2.bd.a yes 108 185.bc even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{108} - 6 T_{3}^{107} + 18 T_{3}^{106} - 34 T_{3}^{105} + 69 T_{3}^{104} - 384 T_{3}^{103} + \cdots + 1494904896 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display