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Properties

Label 3013.2.a.a
Level $3013$
Weight $2$
Character orbit 3013.a
Self dual yes
Analytic conductor $24.059$
Analytic rank $1$
Dimension $51$
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] = [3013,2,Mod(1,3013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3013, 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("3013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3013 = 23 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3013.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: \(24.0589261290\)
Analytic rank: \(1\)
Dimension: \(51\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 51 q - 14 q^{2} - 20 q^{3} + 40 q^{4} - 7 q^{5} - 5 q^{7} - 36 q^{8} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 51 q - 14 q^{2} - 20 q^{3} + 40 q^{4} - 7 q^{5} - 5 q^{7} - 36 q^{8} + 33 q^{9} - 11 q^{10} - 27 q^{11} - 28 q^{12} - 24 q^{13} - 11 q^{14} - 14 q^{15} + 22 q^{16} - 25 q^{17} - 18 q^{18} - 8 q^{19} - 14 q^{20} + 5 q^{21} + 4 q^{22} + 51 q^{23} - 2 q^{24} + 26 q^{25} + 2 q^{26} - 65 q^{27} + 21 q^{28} - 41 q^{29} + 5 q^{30} - 30 q^{31} - 56 q^{32} - q^{33} + 30 q^{34} - 53 q^{35} + 22 q^{36} - 26 q^{37} - 6 q^{38} - 39 q^{39} - 31 q^{40} - 22 q^{41} - 11 q^{42} - 50 q^{43} - 55 q^{44} - 28 q^{45} - 14 q^{46} - 57 q^{47} - 3 q^{48} + 22 q^{49} - 37 q^{50} - 35 q^{51} - 44 q^{52} - 87 q^{53} - 15 q^{54} - 9 q^{55} - 54 q^{56} + 6 q^{57} - 38 q^{58} - 82 q^{59} - 29 q^{60} + 8 q^{61} - 36 q^{62} - 44 q^{63} + 30 q^{64} - 20 q^{65} - 60 q^{66} - 43 q^{67} - 102 q^{68} - 20 q^{69} - 7 q^{70} - 63 q^{71} - 2 q^{72} - 42 q^{73} - 46 q^{74} - 68 q^{75} - 8 q^{76} - 69 q^{77} + 19 q^{78} - 27 q^{79} + 10 q^{80} + 15 q^{81} - 57 q^{83} + 37 q^{84} - 38 q^{85} - q^{86} + 6 q^{87} - 15 q^{88} - 41 q^{89} + 47 q^{90} - 39 q^{91} + 40 q^{92} - 22 q^{93} + 53 q^{94} - 91 q^{95} + 55 q^{96} - 22 q^{97} - 152 q^{98} - 58 q^{99}+O(q^{100}) \) 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 −2.75693 −1.20889 5.60067 4.15965 3.33284 4.10880 −9.92679 −1.53857 −11.4679
1.2 −2.75560 0.865417 5.59331 −1.60214 −2.38474 3.06257 −9.90172 −2.25105 4.41485
1.3 −2.71654 −1.67209 5.37957 0.875752 4.54229 −1.08483 −9.18073 −0.204114 −2.37901
1.4 −2.56393 −3.31762 4.57373 −1.39989 8.50614 −3.71527 −6.59885 8.00662 3.58921
1.5 −2.53715 2.16755 4.43711 −3.51293 −5.49940 3.28950 −6.18331 1.69828 8.91282
1.6 −2.37699 1.82547 3.65008 2.13128 −4.33912 −0.868073 −3.92221 0.332339 −5.06603
1.7 −2.21391 0.0837544 2.90141 1.46814 −0.185425 2.83122 −1.99565 −2.99299 −3.25034
1.8 −2.19015 1.77399 2.79676 1.50414 −3.88529 −2.98767 −1.74502 0.147024 −3.29428
1.9 −2.18332 −1.07340 2.76690 1.41360 2.34357 −4.73017 −1.67440 −1.84782 −3.08635
1.10 −2.18286 −1.53307 2.76487 −4.06876 3.34647 2.66466 −1.66960 −0.649701 8.88152
1.11 −2.13535 0.368575 2.55974 −2.41155 −0.787038 −3.42151 −1.19524 −2.86415 5.14952
1.12 −1.77039 −3.07639 1.13427 0.742331 5.44640 0.598707 1.53267 6.46416 −1.31421
1.13 −1.75002 −2.57150 1.06255 3.57916 4.50016 −1.41941 1.64054 3.61259 −6.26358
1.14 −1.62017 −3.32090 0.624956 −1.83008 5.38043 5.24363 2.22781 8.02839 2.96504
1.15 −1.58283 2.65709 0.505353 1.33359 −4.20572 −2.63004 2.36577 4.06013 −2.11085
1.16 −1.57867 1.18812 0.492193 −0.155064 −1.87565 2.98606 2.38033 −1.58837 0.244794
1.17 −1.50360 3.01598 0.260827 −2.20627 −4.53485 0.404946 2.61503 6.09616 3.31736
1.18 −1.48046 −2.17750 0.191750 −2.12112 3.22369 −3.53201 2.67703 1.74151 3.14022
1.19 −1.31013 −0.752825 −0.283553 −0.797197 0.986301 0.816274 2.99176 −2.43325 1.04443
1.20 −1.08656 −0.209341 −0.819395 4.03869 0.227461 −0.0612382 3.06343 −2.95618 −4.38827
See all 51 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.51
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(23\) \(-1\)
\(131\) \(-1\)

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 3013.2.a.a ✓ 51
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3013.2.a.a ✓ 51 1.a even 1 1 trivial