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Properties

Label 751.2.h.a
Level $751$
Weight $2$
Character orbit 751.h
Analytic conductor $5.997$
Analytic rank $0$
Dimension $1220$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [751,2,Mod(51,751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(751, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("751.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 751.h (of order \(25\), degree \(20\), minimal)

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: no
Analytic conductor: \(5.99676519180\)
Analytic rank: \(0\)
Dimension: \(1220\)
Relative dimension: \(61\) over \(\Q(\zeta_{25})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1220 q - 40 q^{2} - 15 q^{3} - 40 q^{4} - 20 q^{5} - 10 q^{6} - 20 q^{7} - 40 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1220 q - 40 q^{2} - 15 q^{3} - 40 q^{4} - 20 q^{5} - 10 q^{6} - 20 q^{7} - 40 q^{8} - 15 q^{9} + 10 q^{10} + 5 q^{11} - 120 q^{13} - 20 q^{14} - 40 q^{15} - 160 q^{16} - 20 q^{17} - 20 q^{18} - 20 q^{19} - 10 q^{21} - 30 q^{22} - 15 q^{23} - 10 q^{24} + 10 q^{25} + 25 q^{26} + 115 q^{28} - 20 q^{29} + 50 q^{30} - 180 q^{31} + 20 q^{32} + 45 q^{33} + 5 q^{34} - 20 q^{35} + 60 q^{36} + 35 q^{37} + 45 q^{38} - 160 q^{39} + 115 q^{40} + 55 q^{41} + 50 q^{42} - 20 q^{43} - 20 q^{44} + 55 q^{45} + 85 q^{46} + 10 q^{47} + 55 q^{48} - 20 q^{49} - 100 q^{50} - 105 q^{51} + 65 q^{52} - 75 q^{53} - 20 q^{54} + 15 q^{55} - 160 q^{56} - 160 q^{57} + 45 q^{58} + 25 q^{59} - 125 q^{60} - 15 q^{61} + 5 q^{62} - 125 q^{63} - 180 q^{64} + 105 q^{65} + 110 q^{66} - 40 q^{67} - 10 q^{68} + 55 q^{69} - 15 q^{70} - 300 q^{71} - 40 q^{72} + 10 q^{73} - 65 q^{74} + 20 q^{75} + 40 q^{76} - 40 q^{77} + 60 q^{78} - 30 q^{79} - 340 q^{80} + 75 q^{81} + 55 q^{82} - 20 q^{83} + 100 q^{84} - 40 q^{86} + 60 q^{87} - 350 q^{88} - 30 q^{89} + 20 q^{90} + 55 q^{91} - 490 q^{92} + 120 q^{93} + 50 q^{94} - 155 q^{95} + 25 q^{96} - 30 q^{97} + 5 q^{98} - 130 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −2.69696 − 0.340706i −0.302875 − 1.58773i 5.22037 + 1.34036i −1.11310 + 0.611930i 0.275896 + 4.38524i 0.358877 + 1.88130i −8.56747 − 3.39210i 0.360183 − 0.142606i 3.21047 − 1.27112i
51.2 −2.69653 − 0.340651i 0.292085 + 1.53116i 5.21805 + 1.33977i 1.77550 − 0.976089i −0.266023 − 4.22832i −0.228196 − 1.19624i −8.56005 − 3.38916i 0.530187 − 0.209916i −5.12019 + 2.02723i
51.3 −2.60532 − 0.329129i 0.565044 + 2.96206i 4.74221 + 1.21759i −3.75635 + 2.06507i −0.497221 − 7.90310i −0.746680 − 3.91423i −7.07100 − 2.79961i −5.66522 + 2.24302i 10.4662 − 4.14385i
51.4 −2.41250 − 0.304769i 0.534355 + 2.80119i 3.79009 + 0.973130i 0.941276 − 0.517471i −0.435414 − 6.92071i 0.651780 + 3.41675i −4.32518 − 1.71246i −4.77179 + 1.88928i −2.42853 + 0.961525i
51.5 −2.40185 − 0.303425i −0.353381 − 1.85249i 3.73967 + 0.960183i 2.41846 − 1.32956i 0.286678 + 4.55662i 0.389887 + 2.04386i −4.18892 − 1.65851i −0.517496 + 0.204891i −6.21220 + 2.45959i
51.6 −2.36489 − 0.298755i 0.139225 + 0.729841i 3.56626 + 0.915661i −2.66874 + 1.46715i −0.111207 − 1.76759i 0.599321 + 3.14175i −3.72767 − 1.47589i 2.27604 − 0.901150i 6.74959 − 2.67235i
51.7 −2.30796 − 0.291563i −0.460324 − 2.41310i 3.30451 + 0.848454i 3.22148 − 1.77103i 0.358838 + 5.70356i −0.689894 − 3.61655i −3.05341 − 1.20893i −2.82183 + 1.11724i −7.95143 + 3.14819i
51.8 −2.26863 − 0.286594i −0.516543 − 2.70782i 3.12737 + 0.802972i −2.30499 + 1.26718i 0.395800 + 6.29106i −0.592288 − 3.10488i −2.61255 − 1.03438i −4.27612 + 1.69303i 5.59233 − 2.21416i
51.9 −2.15886 − 0.272727i 0.0452989 + 0.237465i 2.64911 + 0.680176i 2.97647 − 1.63633i −0.0330307 − 0.525008i 0.0137700 + 0.0721847i −1.48713 − 0.588796i 2.73499 − 1.08286i −6.87203 + 2.72083i
51.10 −2.14572 − 0.271068i 0.305078 + 1.59928i 2.59348 + 0.665894i −1.29720 + 0.713142i −0.221101 − 3.51430i 0.498575 + 2.61362i −1.36260 − 0.539491i 0.324717 − 0.128565i 2.97674 − 1.17858i
51.11 −2.06700 − 0.261123i −0.0575597 − 0.301739i 2.26715 + 0.582106i −2.90452 + 1.59677i 0.0401852 + 0.638725i −0.292059 − 1.53103i −0.659959 − 0.261296i 2.70160 − 1.06964i 6.42060 − 2.54210i
51.12 −1.96836 − 0.248662i 0.357124 + 1.87211i 1.87546 + 0.481536i 0.118252 − 0.0650098i −0.237427 − 3.77380i −0.653982 − 3.42829i 0.117525 + 0.0465313i −0.587926 + 0.232776i −0.248929 + 0.0985581i
51.13 −1.80961 − 0.228607i 0.0286187 + 0.150024i 1.28526 + 0.329998i 0.189281 − 0.104058i −0.0174920 − 0.278028i −0.224680 − 1.17781i 1.14144 + 0.451927i 2.76764 − 1.09579i −0.366314 + 0.145034i
51.14 −1.74256 − 0.220137i −0.532804 − 2.79306i 1.05090 + 0.269826i 1.36661 − 0.751300i 0.313590 + 4.98437i 0.578844 + 3.03440i 1.49428 + 0.591625i −4.72796 + 1.87193i −2.54679 + 1.00835i
51.15 −1.65944 − 0.209636i −0.274554 − 1.43926i 0.772626 + 0.198377i 0.373662 − 0.205422i 0.153885 + 2.44593i 0.464948 + 2.43734i 1.86980 + 0.740307i 0.793228 − 0.314061i −0.663133 + 0.262553i
51.16 −1.62851 − 0.205729i 0.576595 + 3.02261i 0.672554 + 0.172682i 3.05446 − 1.67920i −0.317152 − 5.04098i −0.618766 − 3.24368i 1.99263 + 0.788940i −6.01441 + 2.38127i −5.31968 + 2.10621i
51.17 −1.54025 − 0.194579i −0.373572 − 1.95833i 0.397341 + 0.102020i −2.35449 + 1.29439i 0.194344 + 3.08901i −0.311535 − 1.63312i 2.29479 + 0.908570i −0.906184 + 0.358784i 3.87836 − 1.53555i
51.18 −1.37096 − 0.173192i 0.371021 + 1.94496i −0.0876440 − 0.0225032i 3.08357 − 1.69521i −0.171802 − 2.73071i 0.915626 + 4.79988i 2.68588 + 1.06342i −0.855882 + 0.338867i −4.52103 + 1.79000i
51.19 −1.14895 − 0.145147i −0.208842 − 1.09479i −0.638140 − 0.163846i 1.25442 − 0.689623i 0.0810450 + 1.28817i −0.845552 − 4.43254i 2.86293 + 1.13352i 1.63438 − 0.647099i −1.54137 + 0.610270i
51.20 −1.10347 − 0.139401i 0.467950 + 2.45308i −0.738944 − 0.189729i −1.21966 + 0.670512i −0.174409 − 2.77214i −0.102689 − 0.538315i 2.85723 + 1.13126i −3.00930 + 1.19147i 1.43933 − 0.569871i
See next 80 embeddings (of 1220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.61
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
751.h even 25 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 751.2.h.a ✓ 1220
751.h even 25 1 inner 751.2.h.a ✓ 1220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
751.2.h.a ✓ 1220 1.a even 1 1 trivial
751.2.h.a ✓ 1220 751.h even 25 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(751, [\chi])\).