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Properties

Label 751.2.a.b
Level $751$
Weight $2$
Character orbit 751.a
Self dual yes
Analytic conductor $5.997$
Analytic rank $0$
Dimension $38$
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] = [751,2,Mod(1,751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(751, 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("751.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 751.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: \(5.99676519180\)
Analytic rank: \(0\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 4 q^{2} + 6 q^{3} + 40 q^{4} + 18 q^{5} + 13 q^{6} + q^{7} + 9 q^{8} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 4 q^{2} + 6 q^{3} + 40 q^{4} + 18 q^{5} + 13 q^{6} + q^{7} + 9 q^{8} + 62 q^{9} + 20 q^{11} + 10 q^{12} + 10 q^{13} + 17 q^{14} - 7 q^{15} + 32 q^{16} + 61 q^{17} - q^{18} + 7 q^{19} + 27 q^{20} + 13 q^{21} - 10 q^{22} + 4 q^{23} + 38 q^{24} + 52 q^{25} + 25 q^{26} + 18 q^{27} - 22 q^{28} + 53 q^{29} - 13 q^{30} - 5 q^{31} + 12 q^{32} + 8 q^{33} - 7 q^{34} + 24 q^{35} + 53 q^{36} + q^{38} - 29 q^{40} + 117 q^{41} - 22 q^{42} - 5 q^{43} + 45 q^{44} + 19 q^{45} - 24 q^{46} + 32 q^{47} - 4 q^{48} + 71 q^{49} + 3 q^{50} + q^{51} - 3 q^{52} + 27 q^{53} + 46 q^{54} - 7 q^{55} + 21 q^{56} + 11 q^{57} - 40 q^{58} + 26 q^{59} - 68 q^{60} + 37 q^{61} + 34 q^{62} - 22 q^{63} + q^{64} + 61 q^{65} - 34 q^{66} - 25 q^{67} + 88 q^{68} - 5 q^{69} - 66 q^{70} + 3 q^{71} - 52 q^{72} + 24 q^{73} + 63 q^{74} + 18 q^{75} - 28 q^{76} + 21 q^{77} - 49 q^{78} - 29 q^{79} + 38 q^{80} + 106 q^{81} - 37 q^{82} + 9 q^{83} - 31 q^{84} + 3 q^{85} - 26 q^{86} + 5 q^{87} - 82 q^{88} + 120 q^{89} - 63 q^{90} - 33 q^{91} - 40 q^{92} - 41 q^{93} - 17 q^{94} + 22 q^{95} + q^{96} + 12 q^{97} - 34 q^{98} - 2 q^{99}+O(q^{100}) \) 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 −2.67852 −3.16360 5.17449 0.156255 8.47377 −3.48512 −8.50294 7.00834 −0.418532
1.2 −2.63240 1.20498 4.92952 2.54563 −3.17199 2.36553 −7.71167 −1.54802 −6.70110
1.3 −2.50694 −0.00328776 4.28477 3.68299 0.00824224 −3.83764 −5.72780 −2.99999 −9.23304
1.4 −2.24981 2.52205 3.06163 1.61977 −5.67413 1.54437 −2.38846 3.36074 −3.64416
1.5 −2.21566 −1.45462 2.90917 −1.02556 3.22295 −2.33783 −2.01441 −0.884086 2.27229
1.6 −2.11718 3.25885 2.48245 −2.93813 −6.89957 −3.35892 −1.02144 7.62008 6.22054
1.7 −1.92832 −0.517639 1.71843 −1.25816 0.998174 1.84493 0.542966 −2.73205 2.42614
1.8 −1.92671 −3.31257 1.71220 1.70672 6.38236 4.71179 0.554502 7.97314 −3.28834
1.9 −1.66925 −0.140212 0.786384 −3.96863 0.234049 −3.81057 2.02582 −2.98034 6.62462
1.10 −1.40329 2.89419 −0.0307656 0.110745 −4.06141 3.03090 2.84976 5.37636 −0.155408
1.11 −1.35659 −0.419181 −0.159654 3.86140 0.568659 2.27050 2.92977 −2.82429 −5.23835
1.12 −1.17799 −2.90601 −0.612340 2.74964 3.42325 −3.04840 3.07731 5.44488 −3.23904
1.13 −1.05794 1.20557 −0.880762 −0.468265 −1.27542 0.388950 3.04767 −1.54661 0.495397
1.14 −1.00377 −2.36646 −0.992439 −0.876534 2.37539 3.50946 3.00373 2.60012 0.879841
1.15 −0.725964 3.23925 −1.47298 3.95139 −2.35158 −1.12873 2.52126 7.49271 −2.86857
1.16 −0.388416 −0.0678888 −1.84913 −2.62178 0.0263690 −3.98683 1.49506 −2.99539 1.01834
1.17 −0.162284 2.02627 −1.97366 1.79917 −0.328831 4.95919 0.644861 1.10577 −0.291975
1.18 −0.0548569 1.65187 −1.99699 3.46473 −0.0906165 −2.18165 0.219263 −0.271326 −0.190064
1.19 0.0238621 −1.19462 −1.99943 0.471955 −0.0285060 −3.78975 −0.0954348 −1.57289 0.0112618
1.20 0.0762461 1.82153 −1.99419 −4.10784 0.138884 0.929351 −0.304541 0.317957 −0.313207
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.38
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(751\) \(-1\)

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 751.2.a.b ✓ 38
3.b odd 2 1 6759.2.a.f 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
751.2.a.b ✓ 38 1.a even 1 1 trivial
6759.2.a.f 38 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{38} - 4 T_{2}^{37} - 50 T_{2}^{36} + 213 T_{2}^{35} + 1118 T_{2}^{34} - 5158 T_{2}^{33} + \cdots + 36 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(751))\). Copy content Toggle raw display