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Properties

Label 503.2.a.f
Level $503$
Weight $2$
Character orbit 503.a
Self dual yes
Analytic conductor $4.016$
Analytic rank $0$
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] = [503,2,Mod(1,503)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(503, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("503.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.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: \(4.01647522167\)
Analytic rank: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 4 q^{2} + 4 q^{3} + 36 q^{4} + 9 q^{5} - 4 q^{6} + 11 q^{7} + 18 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 4 q^{2} + 4 q^{3} + 36 q^{4} + 9 q^{5} - 4 q^{6} + 11 q^{7} + 18 q^{8} + 42 q^{9} + 4 q^{10} - 17 q^{11} + 19 q^{12} + 14 q^{13} + q^{14} + 18 q^{15} + 48 q^{16} + 17 q^{17} - 10 q^{18} - 22 q^{19} - 19 q^{20} - 16 q^{21} + 38 q^{22} + 27 q^{23} - 9 q^{24} + 93 q^{25} + q^{26} + 31 q^{27} - 9 q^{28} + 13 q^{29} - 28 q^{30} + 26 q^{31} + 5 q^{32} + 6 q^{33} - 32 q^{34} - 22 q^{35} + 52 q^{36} + 55 q^{37} - 24 q^{38} - 15 q^{39} - 7 q^{40} + 24 q^{41} - 50 q^{42} + 20 q^{43} - 27 q^{44} - 8 q^{45} + 6 q^{46} - 25 q^{47} + 29 q^{48} + 65 q^{49} - 16 q^{50} + 7 q^{51} + 32 q^{52} + 30 q^{53} - 82 q^{54} + 25 q^{55} + 3 q^{56} + 9 q^{57} + 58 q^{58} - 26 q^{59} - 68 q^{60} + 15 q^{61} - 12 q^{62} - 19 q^{63} + 44 q^{64} + 20 q^{65} - 55 q^{66} - 20 q^{67} - 4 q^{68} - 27 q^{69} + 2 q^{70} - 35 q^{71} - 26 q^{72} + 38 q^{73} - 59 q^{74} + 2 q^{75} - 42 q^{76} - 6 q^{77} - 47 q^{78} + 21 q^{79} - 100 q^{80} + 70 q^{81} - 59 q^{82} - 48 q^{83} - 116 q^{84} + 6 q^{85} - 7 q^{86} - 9 q^{87} + 106 q^{88} - 5 q^{89} - 118 q^{90} - 24 q^{91} + 26 q^{92} - 8 q^{93} - 22 q^{94} + 43 q^{95} - 100 q^{96} + 142 q^{97} - 38 q^{98} - 50 q^{99}+O(q^{100}) \) 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 −2.73065 0.249551 5.45647 −3.41933 −0.681437 −3.14640 −9.43843 −2.93772 9.33700
1.2 −2.69742 3.34519 5.27606 2.36922 −9.02336 −0.269938 −8.83689 8.19027 −6.39078
1.3 −2.26823 −1.83158 3.14489 −3.37084 4.15444 3.93290 −2.59688 0.354669 7.64586
1.4 −2.16310 −2.85594 2.67899 1.91587 6.17767 1.96852 −1.46872 5.15637 −4.14421
1.5 −1.89206 1.34769 1.57988 4.23270 −2.54991 −3.43664 0.794895 −1.18373 −8.00851
1.6 −1.79231 2.94588 1.21237 −4.09318 −5.27994 3.85397 1.41167 5.67824 7.33625
1.7 −1.44158 −1.65492 0.0781455 1.16845 2.38569 −5.22170 2.77050 −0.261242 −1.68442
1.8 −1.43135 2.32813 0.0487521 1.62138 −3.33236 3.43484 2.79291 2.42020 −2.32076
1.9 −1.28997 −1.08298 −0.335977 3.81229 1.39701 4.19464 3.01334 −1.82716 −4.91774
1.10 −0.462244 3.26409 −1.78633 1.15565 −1.50881 −1.19216 1.75021 7.65430 −0.534193
1.11 −0.425199 −1.16576 −1.81921 −3.04539 0.495677 0.946556 1.62392 −1.64102 1.29490
1.12 −0.349081 1.74581 −1.87814 1.36534 −0.609428 −0.0430977 1.35379 0.0478420 −0.476616
1.13 −0.328801 −2.39019 −1.89189 −3.16539 0.785896 −3.22876 1.27966 2.71299 1.04078
1.14 0.342177 −2.49378 −1.88292 4.14215 −0.853311 −0.810419 −1.32864 3.21891 1.41735
1.15 0.477293 −0.705033 −1.77219 −0.869095 −0.336507 5.04945 −1.80044 −2.50293 −0.414813
1.16 0.738482 2.27911 −1.45464 3.79116 1.68308 4.03115 −2.55119 2.19436 2.79970
1.17 1.26316 −3.11338 −0.404427 −4.34050 −3.93270 1.70318 −3.03718 6.69315 −5.48274
1.18 1.61798 2.76613 0.617860 3.34245 4.47554 −5.14301 −2.23628 4.65146 5.40802
1.19 1.81497 0.577924 1.29413 2.10638 1.04892 1.73516 −1.28114 −2.66600 3.82302
1.20 1.88946 −1.08803 1.57004 3.67682 −2.05579 1.62501 −0.812389 −1.81619 6.94719
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
\(503\) \(-1\)

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 503.2.a.f ✓ 26
3.b odd 2 1 4527.2.a.o 26
4.b odd 2 1 8048.2.a.u 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
503.2.a.f ✓ 26 1.a even 1 1 trivial
4527.2.a.o 26 3.b odd 2 1
8048.2.a.u 26 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(503))\):

\( T_{2}^{26} - 4 T_{2}^{25} - 36 T_{2}^{24} + 154 T_{2}^{23} + 554 T_{2}^{22} - 2577 T_{2}^{21} + \cdots - 1583 \) Copy content Toggle raw display
\( T_{3}^{26} - 4 T_{3}^{25} - 52 T_{3}^{24} + 211 T_{3}^{23} + 1175 T_{3}^{22} - 4814 T_{3}^{21} + \cdots - 113513 \) Copy content Toggle raw display