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Properties

Label 1603.2.a.c
Level $1603$
Weight $2$
Character orbit 1603.a
Self dual yes
Analytic conductor $12.800$
Analytic rank $1$
Dimension $26$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1603,2,Mod(1,1603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1603, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1603.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1603 = 7 \cdot 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1603.a (trivial)

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: yes
Analytic conductor: \(12.8000194440\)
Analytic rank: \(1\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 6 q^{2} - 14 q^{3} + 24 q^{4} - 25 q^{5} - 9 q^{6} + 26 q^{7} - 18 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 6 q^{2} - 14 q^{3} + 24 q^{4} - 25 q^{5} - 9 q^{6} + 26 q^{7} - 18 q^{8} + 20 q^{9} - 9 q^{10} - 10 q^{11} - 28 q^{12} - 12 q^{13} - 6 q^{14} + 5 q^{15} + 20 q^{16} - 37 q^{17} + 2 q^{18} - 20 q^{19} - 37 q^{20} - 14 q^{21} - 12 q^{22} - 21 q^{23} - q^{24} + 31 q^{25} - 19 q^{26} - 50 q^{27} + 24 q^{28} - 28 q^{29} + 31 q^{30} - 5 q^{31} - 32 q^{32} - 35 q^{33} + 3 q^{34} - 25 q^{35} + 47 q^{36} - 26 q^{37} - 12 q^{38} + 8 q^{39} - 14 q^{40} - 37 q^{41} - 9 q^{42} - 10 q^{43} + q^{44} - 48 q^{45} + 36 q^{46} - 45 q^{47} - 48 q^{48} + 26 q^{49} + 6 q^{50} + 14 q^{51} + 10 q^{52} - 83 q^{53} - 3 q^{54} - 12 q^{55} - 18 q^{56} - 72 q^{59} + 45 q^{60} - 5 q^{61} - 62 q^{62} + 20 q^{63} - 6 q^{64} - 12 q^{65} + 23 q^{66} + 9 q^{67} - 89 q^{68} - 30 q^{69} - 9 q^{70} - 15 q^{71} - 37 q^{72} - 7 q^{73} - 6 q^{74} - 43 q^{75} - 61 q^{76} - 10 q^{77} - 55 q^{78} + 15 q^{79} - 29 q^{80} + 34 q^{81} + q^{82} - 77 q^{83} - 28 q^{84} - 12 q^{85} - 30 q^{86} - 6 q^{87} + 8 q^{88} - 52 q^{89} - 87 q^{90} - 12 q^{91} - 71 q^{92} + 12 q^{93} + 48 q^{94} + 15 q^{95} + 43 q^{96} - 31 q^{97} - 6 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.78100 −0.922103 5.73396 −0.418609 2.56437 1.00000 −10.3841 −2.14973 1.16415
1.2 −2.58159 2.54730 4.66460 −0.203796 −6.57608 1.00000 −6.87889 3.48874 0.526118
1.3 −2.56023 −3.32563 4.55479 1.57395 8.51439 1.00000 −6.54087 8.05983 −4.02968
1.4 −2.34684 −2.13414 3.50764 −4.26677 5.00848 1.00000 −3.53818 1.55456 10.0134
1.5 −2.08237 −0.260687 2.33625 −3.28394 0.542846 1.00000 −0.700185 −2.93204 6.83836
1.6 −2.00891 1.90355 2.03571 −2.44650 −3.82406 1.00000 −0.0717348 0.623509 4.91480
1.7 −1.93255 −0.485033 1.73476 1.17715 0.937351 1.00000 0.512599 −2.76474 −2.27490
1.8 −1.67926 1.21226 0.819912 2.03361 −2.03570 1.00000 1.98167 −1.53042 −3.41495
1.9 −1.42706 0.434569 0.0365114 −0.884629 −0.620158 1.00000 2.80202 −2.81115 1.26242
1.10 −1.30355 −2.83261 −0.300768 1.46194 3.69244 1.00000 2.99916 5.02367 −1.90570
1.11 −0.582652 2.44892 −1.66052 −4.02095 −1.42687 1.00000 2.13281 2.99721 2.34282
1.12 −0.504466 −0.151263 −1.74551 −3.81273 0.0763070 1.00000 1.88949 −2.97712 1.92339
1.13 −0.409532 −1.64756 −1.83228 2.66718 0.674731 1.00000 1.56944 −0.285530 −1.09230
1.14 −0.0917805 0.117509 −1.99158 3.06852 −0.0107851 1.00000 0.366349 −2.98619 −0.281631
1.15 0.0357644 −3.11617 −1.99872 −4.43704 −0.111448 1.00000 −0.143012 6.71052 −0.158688
1.16 0.222025 1.13299 −1.95070 −0.233687 0.251552 1.00000 −0.877155 −1.71634 −0.0518843
1.17 0.348843 −2.32956 −1.87831 −1.53944 −0.812652 1.00000 −1.35292 2.42685 −0.537025
1.18 0.830435 2.29544 −1.31038 −0.923601 1.90621 1.00000 −2.74905 2.26904 −0.766990
1.19 1.03724 −2.16289 −0.924130 −0.0122910 −2.24343 1.00000 −3.03303 1.67807 −0.0127488
1.20 1.50678 0.225482 0.270388 0.338776 0.339752 1.00000 −2.60615 −2.94916 0.510461
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(229\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1603.2.a.c ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1603.2.a.c ✓ 26 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} + 6 T_{2}^{25} - 20 T_{2}^{24} - 178 T_{2}^{23} + 83 T_{2}^{22} + 2250 T_{2}^{21} + 1139 T_{2}^{20} + \cdots + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1603))\). Copy content Toggle raw display