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Properties

Label 213.2.f.a
Level $213$
Weight $2$
Character orbit 213.f
Analytic conductor $1.701$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,2,Mod(37,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 213.f (of order \(7\), degree \(6\), 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: \(1.70081356305\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} - 6 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 12 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} - 6 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{7} - 12 q^{8} - 6 q^{9} + 3 q^{10} - 10 q^{11} - 5 q^{12} - 8 q^{13} + 6 q^{14} + 5 q^{15} + 16 q^{16} + 8 q^{17} + 5 q^{18} - 8 q^{19} - 4 q^{20} - 4 q^{21} - 32 q^{22} - 10 q^{23} + 23 q^{24} + 10 q^{25} - 12 q^{26} - 6 q^{27} - 30 q^{28} + 25 q^{29} - 4 q^{30} - 13 q^{31} + 59 q^{32} - 3 q^{33} - 40 q^{34} - 16 q^{35} - 12 q^{36} - 16 q^{37} - 34 q^{38} - 8 q^{39} + 26 q^{40} - 21 q^{41} + 13 q^{42} + 11 q^{43} + 86 q^{44} - 2 q^{45} + 88 q^{46} - 3 q^{47} - 5 q^{48} - 32 q^{49} - 46 q^{50} - 13 q^{51} + 9 q^{52} - 22 q^{53} - 2 q^{54} + 18 q^{55} + 21 q^{56} + 34 q^{57} + 43 q^{58} - 42 q^{59} + 17 q^{60} - 9 q^{61} - 9 q^{62} - 4 q^{63} - 8 q^{64} - 42 q^{65} + 10 q^{66} - 36 q^{67} + 33 q^{68} - 10 q^{69} - 84 q^{70} + 14 q^{71} - 12 q^{72} + 6 q^{73} + 68 q^{74} + 24 q^{75} + 102 q^{76} - 12 q^{77} - 26 q^{78} + 42 q^{79} - 6 q^{81} - 65 q^{82} - 32 q^{83} - 23 q^{84} + 114 q^{85} + 5 q^{86} - 3 q^{87} - 48 q^{88} - 46 q^{89} - 4 q^{90} + 42 q^{91} + 14 q^{92} + 8 q^{93} - 2 q^{94} - 31 q^{95} - 46 q^{96} + 23 q^{97} - 10 q^{98} - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.50357 + 1.88542i −0.222521 − 0.974928i −0.849036 − 3.71987i −1.26831 2.17272 + 1.04633i 2.99971 + 3.76152i 3.94465 + 1.89964i −0.900969 + 0.433884i 1.90700 − 2.39130i
37.2 −0.793428 + 0.994928i −0.222521 − 0.974928i 0.0846894 + 0.371049i −1.86735 1.14654 + 0.552143i −1.93132 − 2.42180i −2.72944 − 1.31443i −0.900969 + 0.433884i 1.48161 − 1.85788i
37.3 −0.498602 + 0.625227i −0.222521 − 0.974928i 0.302737 + 1.32638i 3.17413 0.720500 + 0.346975i 0.0583236 + 0.0731355i −2.42123 − 1.16600i −0.900969 + 0.433884i −1.58263 + 1.98455i
37.4 0.176821 − 0.221726i −0.222521 − 0.974928i 0.427145 + 1.87144i −1.47598 −0.255513 − 0.123049i 2.76512 + 3.46735i 1.00150 + 0.482299i −0.900969 + 0.433884i −0.260984 + 0.327264i
37.5 1.02950 − 1.29096i −0.222521 − 0.974928i −0.161649 − 0.708230i 2.91347 −1.48767 − 0.716426i 0.462930 + 0.580495i 1.89464 + 0.912410i −0.900969 + 0.433884i 2.99943 − 3.76117i
37.6 1.31180 − 1.64494i −0.222521 − 0.974928i −0.539981 − 2.36581i −1.92100 −1.89560 − 0.912875i −1.86080 − 2.33337i −0.808768 − 0.389482i −0.900969 + 0.433884i −2.51997 + 3.15994i
91.1 −0.412266 − 1.80626i −0.900969 + 0.433884i −1.29066 + 0.621549i 2.48759 1.15514 + 1.44850i 0.0523269 − 0.229259i −0.655517 − 0.821993i 0.623490 − 0.781831i −1.02555 − 4.49323i
91.2 −0.283049 − 1.24012i −0.900969 + 0.433884i 0.344159 − 0.165738i −3.70486 0.793086 + 0.994498i −0.967010 + 4.23675i −1.88912 − 2.36888i 0.623490 − 0.781831i 1.04866 + 4.59446i
91.3 −0.00164779 − 0.00721946i −0.900969 + 0.433884i 1.80189 − 0.867744i −0.629549 0.00461701 + 0.00578955i 0.949387 − 4.15953i −0.0184678 − 0.0231579i 0.623490 − 0.781831i 0.00103737 + 0.00454500i
91.4 0.104752 + 0.458947i −0.900969 + 0.433884i 1.60228 − 0.771617i 0.271680 −0.293507 − 0.368047i −0.697076 + 3.05409i 1.10899 + 1.39063i 0.623490 − 0.781831i 0.0284589 + 0.124686i
91.5 0.395798 + 1.73411i −0.900969 + 0.433884i −1.04853 + 0.504945i 1.30530 −1.10900 − 1.39065i −0.619894 + 2.71593i 0.927370 + 1.16288i 0.623490 − 0.781831i 0.516634 + 2.26352i
91.6 0.597382 + 2.61730i −0.900969 + 0.433884i −4.69146 + 2.25929i −1.53210 −1.67383 − 2.09891i 0.392183 − 1.71827i −5.36818 − 6.73148i 0.623490 − 0.781831i −0.915248 − 4.00996i
103.1 −0.412266 + 1.80626i −0.900969 − 0.433884i −1.29066 − 0.621549i 2.48759 1.15514 − 1.44850i 0.0523269 + 0.229259i −0.655517 + 0.821993i 0.623490 + 0.781831i −1.02555 + 4.49323i
103.2 −0.283049 + 1.24012i −0.900969 − 0.433884i 0.344159 + 0.165738i −3.70486 0.793086 − 0.994498i −0.967010 − 4.23675i −1.88912 + 2.36888i 0.623490 + 0.781831i 1.04866 − 4.59446i
103.3 −0.00164779 + 0.00721946i −0.900969 − 0.433884i 1.80189 + 0.867744i −0.629549 0.00461701 − 0.00578955i 0.949387 + 4.15953i −0.0184678 + 0.0231579i 0.623490 + 0.781831i 0.00103737 − 0.00454500i
103.4 0.104752 − 0.458947i −0.900969 − 0.433884i 1.60228 + 0.771617i 0.271680 −0.293507 + 0.368047i −0.697076 − 3.05409i 1.10899 − 1.39063i 0.623490 + 0.781831i 0.0284589 − 0.124686i
103.5 0.395798 − 1.73411i −0.900969 − 0.433884i −1.04853 − 0.504945i 1.30530 −1.10900 + 1.39065i −0.619894 − 2.71593i 0.927370 − 1.16288i 0.623490 + 0.781831i 0.516634 − 2.26352i
103.6 0.597382 − 2.61730i −0.900969 − 0.433884i −4.69146 − 2.25929i −1.53210 −1.67383 + 2.09891i 0.392183 + 1.71827i −5.36818 + 6.73148i 0.623490 + 0.781831i −0.915248 + 4.00996i
172.1 −2.33695 − 1.12542i 0.623490 + 0.781831i 2.94780 + 3.69642i −2.03105 −0.577179 − 2.52879i −2.52085 + 1.21398i −1.57449 − 6.89828i −0.222521 + 0.974928i 4.74646 + 2.28578i
172.2 −1.71355 − 0.825202i 0.623490 + 0.781831i 1.00832 + 1.26439i 4.32389 −0.423212 − 1.85421i −1.77940 + 0.856913i 0.162000 + 0.709768i −0.222521 + 0.974928i −7.40920 − 3.56808i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.2.f.a ✓ 36
3.b odd 2 1 639.2.j.e 36
71.d even 7 1 inner 213.2.f.a ✓ 36
213.k odd 14 1 639.2.j.e 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.2.f.a ✓ 36 1.a even 1 1 trivial
213.2.f.a ✓ 36 71.d even 7 1 inner
639.2.j.e 36 3.b odd 2 1
639.2.j.e 36 213.k odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 2 T_{2}^{35} + 7 T_{2}^{34} + 18 T_{2}^{33} + 52 T_{2}^{32} + 41 T_{2}^{31} + 240 T_{2}^{30} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(213, [\chi])\). Copy content Toggle raw display