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Properties

Label 370.2.ba.a
Level $370$
Weight $2$
Character orbit 370.ba
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(17,370)]
 
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([9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.ba (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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 6 q^{3} + 6 q^{5} - 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 6 q^{3} + 6 q^{5} - 54 q^{8} - 12 q^{10} + 36 q^{11} - 6 q^{12} + 6 q^{13} + 12 q^{14} + 24 q^{15} + 12 q^{19} - 6 q^{20} - 42 q^{21} - 6 q^{22} - 6 q^{24} - 18 q^{25} - 6 q^{26} + 6 q^{27} - 12 q^{30} + 6 q^{33} - 54 q^{35} + 12 q^{37} + 48 q^{38} - 12 q^{40} + 48 q^{41} + 42 q^{42} + 6 q^{44} - 90 q^{45} + 6 q^{46} - 12 q^{47} - 12 q^{49} - 12 q^{50} - 12 q^{51} + 6 q^{52} + 36 q^{53} - 18 q^{54} + 36 q^{57} + 6 q^{58} + 24 q^{59} - 54 q^{60} - 36 q^{61} + 54 q^{62} - 96 q^{63} - 54 q^{64} - 18 q^{65} - 42 q^{67} - 96 q^{69} - 12 q^{70} - 48 q^{71} + 84 q^{73} + 42 q^{74} + 252 q^{75} - 6 q^{76} - 66 q^{77} - 24 q^{78} + 66 q^{79} + 6 q^{80} - 108 q^{81} + 36 q^{82} + 48 q^{83} - 36 q^{85} + 108 q^{87} - 36 q^{88} - 66 q^{89} + 6 q^{90} - 18 q^{91} - 12 q^{92} - 12 q^{93} + 18 q^{94} + 90 q^{95} + 12 q^{96} - 72 q^{97} - 24 q^{98} - 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0.173648 + 0.984808i −2.53220 + 1.77306i −0.939693 + 0.342020i 0.871338 − 2.05931i −2.18584 − 2.18584i −0.146400 − 1.67336i −0.500000 − 0.866025i 2.24221 − 6.16042i 2.17933 + 0.500504i
17.2 0.173648 + 0.984808i −1.63791 + 1.14688i −0.939693 + 0.342020i 0.410879 + 2.19799i −1.41388 − 1.41388i 0.220613 + 2.52162i −0.500000 − 0.866025i 0.341368 − 0.937902i −2.09325 + 0.786315i
17.3 0.173648 + 0.984808i −0.564256 + 0.395097i −0.939693 + 0.342020i −0.642179 − 2.14187i −0.487076 − 0.487076i 0.0586844 + 0.670766i −0.500000 − 0.866025i −0.863776 + 2.37321i 1.99782 − 1.00435i
17.4 0.173648 + 0.984808i −0.243511 + 0.170508i −0.939693 + 0.342020i −2.06463 − 0.858671i −0.210203 − 0.210203i −0.339652 − 3.88224i −0.500000 − 0.866025i −0.995836 + 2.73604i 0.487107 − 2.18237i
17.5 0.173648 + 0.984808i −0.0137859 + 0.00965302i −0.939693 + 0.342020i 1.95959 − 1.07704i −0.0119003 − 0.0119003i 0.106751 + 1.22016i −0.500000 − 0.866025i −1.02596 + 2.81881i 1.40096 + 1.74279i
17.6 0.173648 + 0.984808i 0.562923 − 0.394163i −0.939693 + 0.342020i −1.69379 + 1.45982i 0.485925 + 0.485925i 0.0264970 + 0.302862i −0.500000 − 0.866025i −0.864543 + 2.37531i −1.73177 − 1.41456i
17.7 0.173648 + 0.984808i 1.77919 − 1.24580i −0.939693 + 0.342020i 2.01987 + 0.959232i 1.53583 + 1.53583i 0.336121 + 3.84188i −0.500000 − 0.866025i 0.587426 − 1.61394i −0.593912 + 2.15575i
17.8 0.173648 + 0.984808i 2.02303 − 1.41654i −0.939693 + 0.342020i 0.955671 + 2.02156i 1.74632 + 1.74632i −0.403702 − 4.61433i −0.500000 − 0.866025i 1.06001 − 2.91234i −1.82489 + 1.29219i
17.9 0.173648 + 0.984808i 2.40138 − 1.68147i −0.939693 + 0.342020i −1.06638 − 1.96541i 2.07292 + 2.07292i −0.0272851 − 0.311870i −0.500000 − 0.866025i 1.91325 − 5.25661i 1.75038 − 1.39147i
87.1 0.173648 − 0.984808i −1.87571 + 2.67879i −0.939693 − 0.342020i −0.0437468 − 2.23564i 2.31238 + 2.31238i 1.88452 + 0.164874i −0.500000 + 0.866025i −2.63158 − 7.23020i −2.20927 − 0.345133i
87.2 0.173648 − 0.984808i −1.44027 + 2.05692i −0.939693 − 0.342020i −0.949056 + 2.02467i 1.77557 + 1.77557i 1.60039 + 0.140016i −0.500000 + 0.866025i −1.13048 − 3.10597i 1.82911 + 1.28622i
87.3 0.173648 − 0.984808i −0.691427 + 0.987459i −0.939693 − 0.342020i 0.955840 − 2.02148i 0.852393 + 0.852393i −4.04593 − 0.353973i −0.500000 + 0.866025i 0.529055 + 1.45357i −1.82479 − 1.29234i
87.4 0.173648 − 0.984808i −0.513008 + 0.732652i −0.939693 − 0.342020i −2.23092 − 0.151579i 0.632438 + 0.632438i 1.16632 + 0.102040i −0.500000 + 0.866025i 0.752459 + 2.06736i −0.536672 + 2.17071i
87.5 0.173648 − 0.984808i −0.0375421 + 0.0536157i −0.939693 − 0.342020i 0.0386192 + 2.23573i 0.0462820 + 0.0462820i −4.08757 − 0.357616i −0.500000 + 0.866025i 1.02460 + 2.81505i 2.20847 + 0.350199i
87.6 0.173648 − 0.984808i 0.0934022 − 0.133392i −0.939693 − 0.342020i 1.80470 − 1.32024i −0.115147 − 0.115147i 3.25079 + 0.284408i −0.500000 + 0.866025i 1.01699 + 2.79416i −0.986802 − 2.00654i
87.7 0.173648 − 0.984808i 0.131617 − 0.187969i −0.939693 − 0.342020i 0.561282 + 2.16448i −0.162258 − 0.162258i 2.06533 + 0.180693i −0.500000 + 0.866025i 1.00805 + 2.76960i 2.22906 − 0.176897i
87.8 0.173648 − 0.984808i 1.38744 − 1.98147i −0.939693 − 0.342020i −0.695479 − 2.12516i −1.71044 − 1.71044i −0.612411 − 0.0535790i −0.500000 + 0.866025i −0.975177 − 2.67928i −2.21364 + 0.315883i
87.9 0.173648 − 0.984808i 1.70273 − 2.43175i −0.939693 − 0.342020i 1.99318 + 1.01353i −2.09913 − 2.09913i −0.705781 − 0.0617478i −0.500000 + 0.866025i −1.98806 − 5.46214i 1.34424 − 1.78690i
113.1 0.766044 − 0.642788i −3.32389 + 0.290803i 0.173648 − 0.984808i −1.51072 − 1.64855i −2.35932 + 2.35932i −1.26501 + 2.71282i −0.500000 − 0.866025i 8.00925 − 1.41225i −2.21695 − 0.291792i
113.2 0.766044 − 0.642788i −2.16486 + 0.189401i 0.173648 − 0.984808i −0.482801 + 2.18332i −1.53664 + 1.53664i 0.959205 − 2.05702i −0.500000 − 0.866025i 1.69633 − 0.299109i 1.03357 + 1.98286i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.9
Significant digits:
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Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{108} + 6 T_{3}^{107} + 18 T_{3}^{106} + 34 T_{3}^{105} + 69 T_{3}^{104} + 348 T_{3}^{103} + \cdots + 1494904896 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display