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Properties

Label 1603.2.a.e
Level $1603$
Weight $2$
Character orbit 1603.a
Self dual yes
Analytic conductor $12.800$
Analytic rank $0$
Dimension $32$
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: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 7 q^{2} + 4 q^{3} + 39 q^{4} + 19 q^{5} - q^{6} - 32 q^{7} + 21 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 7 q^{2} + 4 q^{3} + 39 q^{4} + 19 q^{5} - q^{6} - 32 q^{7} + 21 q^{8} + 42 q^{9} + 9 q^{10} + 16 q^{11} + 12 q^{12} + 14 q^{13} - 7 q^{14} + 17 q^{15} + 53 q^{16} + 27 q^{17} + 21 q^{18} - 12 q^{19} + 33 q^{20} - 4 q^{21} + 12 q^{22} + 31 q^{23} + q^{24} + 53 q^{25} - q^{26} + 16 q^{27} - 39 q^{28} + 38 q^{29} + 3 q^{30} - 7 q^{31} + 39 q^{32} + 17 q^{33} + 3 q^{34} - 19 q^{35} + 58 q^{36} + 50 q^{37} + 18 q^{38} - 4 q^{39} + 40 q^{40} - 11 q^{41} + q^{42} + 24 q^{43} + 47 q^{44} + 60 q^{45} + 12 q^{46} + 9 q^{47} + 50 q^{48} + 32 q^{49} + 27 q^{50} + 10 q^{51} + 16 q^{52} + 125 q^{53} - 9 q^{54} + 14 q^{55} - 21 q^{56} + 32 q^{57} + 38 q^{58} - 51 q^{59} + 37 q^{60} + 25 q^{61} - 42 q^{63} + 63 q^{64} + 36 q^{65} - 69 q^{66} + 23 q^{67} + 69 q^{68} + 18 q^{69} - 9 q^{70} + 39 q^{71} - 28 q^{72} + 6 q^{73} + 48 q^{74} + 15 q^{75} - 43 q^{76} - 16 q^{77} - 15 q^{78} + 3 q^{79} + 11 q^{80} + 72 q^{81} + 23 q^{82} - 33 q^{83} - 12 q^{84} + 48 q^{85} - 22 q^{86} + 2 q^{87} + 8 q^{88} + 26 q^{89} - 59 q^{90} - 14 q^{91} + 81 q^{92} + 4 q^{93} - 44 q^{94} + 27 q^{95} + 21 q^{96} + 7 q^{97} + 7 q^{98} + 14 q^{99}+O(q^{100}) \) 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 −2.71550 3.09024 5.37393 1.23369 −8.39153 −1.00000 −9.16188 6.54957 −3.35007
1.2 −2.51467 −1.78753 4.32356 −1.66962 4.49504 −1.00000 −5.84299 0.195260 4.19855
1.3 −2.39755 −0.404767 3.74827 −0.559546 0.970450 −1.00000 −4.19156 −2.83616 1.34154
1.4 −2.27152 −2.75563 3.15979 2.64305 6.25945 −1.00000 −2.63447 4.59347 −6.00374
1.5 −2.16000 1.03291 2.66561 0.272902 −2.23109 −1.00000 −1.43773 −1.93309 −0.589470
1.6 −1.94471 2.68926 1.78190 3.20622 −5.22983 −1.00000 0.424137 4.23209 −6.23518
1.7 −1.66413 0.406389 0.769343 −3.87658 −0.676286 −1.00000 2.04798 −2.83485 6.45115
1.8 −1.45066 0.513850 0.104414 −1.29218 −0.745422 −1.00000 2.74985 −2.73596 1.87451
1.9 −1.18243 −0.310609 −0.601858 4.38981 0.367274 −1.00000 3.07652 −2.90352 −5.19065
1.10 −1.09887 2.04061 −0.792488 4.22405 −2.24236 −1.00000 3.06858 1.16407 −4.64168
1.11 −0.762228 −3.04375 −1.41901 2.22048 2.32003 −1.00000 2.60606 6.26444 −1.69251
1.12 −0.757778 −0.266919 −1.42577 2.07208 0.202266 −1.00000 2.59598 −2.92875 −1.57018
1.13 −0.653674 −2.92314 −1.57271 −1.30600 1.91078 −1.00000 2.33539 5.54476 0.853700
1.14 −0.404053 2.44288 −1.83674 −3.71801 −0.987054 −1.00000 1.55025 2.96766 1.50228
1.15 0.218664 3.26066 −1.95219 −0.264248 0.712988 −1.00000 −0.864200 7.63189 −0.0577814
1.16 0.228102 −1.36925 −1.94797 −0.814151 −0.312330 −1.00000 −0.900540 −1.12514 −0.185709
1.17 0.297584 1.90856 −1.91144 2.26984 0.567957 −1.00000 −1.16398 0.642601 0.675469
1.18 0.658838 −1.56579 −1.56593 2.09114 −1.03160 −1.00000 −2.34937 −0.548308 1.37772
1.19 0.738035 0.306551 −1.45530 −3.23700 0.226246 −1.00000 −2.55014 −2.90603 −2.38902
1.20 0.924035 −1.27076 −1.14616 1.38904 −1.17423 −1.00000 −2.90716 −1.38516 1.28352
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
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.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1603.2.a.e ✓ 32 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 7 T_{2}^{31} - 27 T_{2}^{30} + 287 T_{2}^{29} + 143 T_{2}^{28} - 5157 T_{2}^{27} + \cdots + 3704 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1603))\). Copy content Toggle raw display