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Properties

Label 435.2.u.a
Level $435$
Weight $2$
Character orbit 435.u
Analytic conductor $3.473$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(16,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.u (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: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} - 5 q^{3} - 3 q^{4} + 5 q^{5} + q^{6} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q + q^{2} - 5 q^{3} - 3 q^{4} + 5 q^{5} + q^{6} - 5 q^{9} + 6 q^{10} - 18 q^{11} + 32 q^{12} - 5 q^{13} + 18 q^{14} + 5 q^{15} - 13 q^{16} + 44 q^{17} + q^{18} - 8 q^{19} + 10 q^{20} + 7 q^{21} - 19 q^{22} + 8 q^{23} + 14 q^{24} - 5 q^{25} - 34 q^{26} - 5 q^{27} - 102 q^{28} - 27 q^{29} - 8 q^{30} + 5 q^{31} - 2 q^{32} + 10 q^{33} - 5 q^{34} - 3 q^{36} + 9 q^{37} + 7 q^{38} + 2 q^{39} + 6 q^{41} - 17 q^{42} + 33 q^{43} + 5 q^{45} + 98 q^{46} - 30 q^{47} + q^{48} - 9 q^{49} + q^{50} - 26 q^{51} - 45 q^{52} + 38 q^{53} - 6 q^{54} - 10 q^{55} - 70 q^{56} - 8 q^{57} - 27 q^{58} + 102 q^{59} + 3 q^{60} + 8 q^{61} - 22 q^{62} + 30 q^{64} - 2 q^{65} - 40 q^{66} - 2 q^{67} + 23 q^{68} - 13 q^{69} + 24 q^{70} - 12 q^{71} + 14 q^{72} - 28 q^{73} - 65 q^{74} + 30 q^{75} - 41 q^{76} + 24 q^{77} + 71 q^{78} + 31 q^{79} - q^{80} - 5 q^{81} - 39 q^{82} - 17 q^{83} - 11 q^{84} - 2 q^{85} + 38 q^{86} - 27 q^{87} + 22 q^{88} + 25 q^{89} + 6 q^{90} + 90 q^{91} + 39 q^{92} + 5 q^{93} + 11 q^{94} + 8 q^{95} - 16 q^{96} - 27 q^{97} - 47 q^{98} + 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.70259 − 0.819922i 0.623490 − 0.781831i 0.979547 + 1.22831i 0.900969 + 0.433884i −1.70259 + 0.819922i −0.471718 + 0.591515i 0.180366 + 0.790235i −0.222521 − 0.974928i −1.17823 − 1.47745i
16.2 −1.57475 − 0.758361i 0.623490 − 0.781831i 0.657755 + 0.824799i 0.900969 + 0.433884i −1.57475 + 0.758361i −2.68227 + 3.36346i 0.367557 + 1.61037i −0.222521 − 0.974928i −1.08976 − 1.36652i
16.3 −0.212418 − 0.102295i 0.623490 − 0.781831i −1.21232 − 1.52020i 0.900969 + 0.433884i −0.212418 + 0.102295i 2.04210 − 2.56071i 0.206936 + 0.906644i −0.222521 − 0.974928i −0.146998 − 0.184330i
16.4 1.52210 + 0.733005i 0.623490 − 0.781831i 0.532514 + 0.667752i 0.900969 + 0.433884i 1.52210 − 0.733005i 1.30031 − 1.63054i −0.430781 − 1.88737i −0.222521 − 0.974928i 1.05333 + 1.32083i
16.5 2.19018 + 1.05473i 0.623490 − 0.781831i 2.43743 + 3.05645i 0.900969 + 0.433884i 2.19018 − 1.05473i −1.71288 + 2.14789i 1.03282 + 4.52507i −0.222521 − 0.974928i 1.51565 + 1.90056i
136.1 −1.70259 + 0.819922i 0.623490 + 0.781831i 0.979547 − 1.22831i 0.900969 − 0.433884i −1.70259 − 0.819922i −0.471718 − 0.591515i 0.180366 − 0.790235i −0.222521 + 0.974928i −1.17823 + 1.47745i
136.2 −1.57475 + 0.758361i 0.623490 + 0.781831i 0.657755 − 0.824799i 0.900969 − 0.433884i −1.57475 − 0.758361i −2.68227 − 3.36346i 0.367557 − 1.61037i −0.222521 + 0.974928i −1.08976 + 1.36652i
136.3 −0.212418 + 0.102295i 0.623490 + 0.781831i −1.21232 + 1.52020i 0.900969 − 0.433884i −0.212418 − 0.102295i 2.04210 + 2.56071i 0.206936 − 0.906644i −0.222521 + 0.974928i −0.146998 + 0.184330i
136.4 1.52210 − 0.733005i 0.623490 + 0.781831i 0.532514 − 0.667752i 0.900969 − 0.433884i 1.52210 + 0.733005i 1.30031 + 1.63054i −0.430781 + 1.88737i −0.222521 + 0.974928i 1.05333 − 1.32083i
136.5 2.19018 − 1.05473i 0.623490 + 0.781831i 2.43743 − 3.05645i 0.900969 − 0.433884i 2.19018 + 1.05473i −1.71288 − 2.14789i 1.03282 − 4.52507i −0.222521 + 0.974928i 1.51565 − 1.90056i
181.1 −0.525740 − 2.30342i −0.900969 − 0.433884i −3.22739 + 1.55423i 0.222521 + 0.974928i −0.525740 + 2.30342i 3.87321 + 1.86524i 2.33062 + 2.92251i 0.623490 + 0.781831i 2.12868 − 1.02512i
181.2 −0.410184 − 1.79714i −0.900969 − 0.433884i −1.25951 + 0.606547i 0.222521 + 0.974928i −0.410184 + 1.79714i −3.46070 − 1.66659i −0.691946 − 0.867673i 0.623490 + 0.781831i 1.66080 − 0.799801i
181.3 −0.224650 − 0.984254i −0.900969 − 0.433884i 0.883648 − 0.425543i 0.222521 + 0.974928i −0.224650 + 0.984254i 2.01432 + 0.970046i −1.87626 − 2.35276i 0.623490 + 0.781831i 0.909588 − 0.438034i
181.4 0.124254 + 0.544394i −0.900969 − 0.433884i 1.52101 − 0.732481i 0.222521 + 0.974928i 0.124254 − 0.544394i 1.04157 + 0.501593i 1.28406 + 1.61016i 0.623490 + 0.781831i −0.503096 + 0.242278i
181.5 0.412830 + 1.80872i −0.900969 − 0.433884i −1.29912 + 0.625623i 0.222521 + 0.974928i 0.412830 − 1.80872i −2.78995 − 1.34357i 0.645551 + 0.809496i 0.623490 + 0.781831i −1.67151 + 0.804958i
226.1 −1.44601 + 1.81324i −0.222521 + 0.974928i −0.751852 − 3.29408i −0.623490 + 0.781831i −1.44601 − 1.81324i 0.715709 − 3.13572i 2.88105 + 1.38744i −0.900969 − 0.433884i −0.516076 − 2.26107i
226.2 −0.703739 + 0.882461i −0.222521 + 0.974928i 0.161554 + 0.707812i −0.623490 + 0.781831i −0.703739 − 0.882461i 0.123775 − 0.542294i −2.77217 − 1.33501i −0.900969 − 0.433884i −0.251161 − 1.10041i
226.3 0.320185 − 0.401499i −0.222521 + 0.974928i 0.386359 + 1.69275i −0.623490 + 0.781831i 0.320185 + 0.401499i −0.169813 + 0.744000i 1.72870 + 0.832500i −0.900969 − 0.433884i 0.114273 + 0.500661i
226.4 1.12553 − 1.41136i −0.222521 + 0.974928i −0.280099 − 1.22719i −0.623490 + 0.781831i 1.12553 + 1.41136i −0.863563 + 3.78351i 1.20558 + 0.580579i −0.900969 − 0.433884i 0.401695 + 1.75994i
226.5 1.60501 − 2.01262i −0.222521 + 0.974928i −1.02953 − 4.51069i −0.623490 + 0.781831i 1.60501 + 2.01262i 1.03990 − 4.55611i −6.09209 − 2.93379i −0.900969 − 0.433884i 0.572821 + 2.50969i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.2.u.a ✓ 30
29.d even 7 1 inner 435.2.u.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.u.a ✓ 30 1.a even 1 1 trivial
435.2.u.a ✓ 30 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - T_{2}^{29} + 7 T_{2}^{28} - 8 T_{2}^{27} + 54 T_{2}^{26} - 13 T_{2}^{25} + 227 T_{2}^{24} + \cdots + 8281 \) acting on \(S_{2}^{\mathrm{new}}(435, [\chi])\). Copy content Toggle raw display