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Properties

Label 751.2.o.a
Level $751$
Weight $2$
Character orbit 751.o
Analytic conductor $5.997$
Analytic rank $0$
Dimension $12400$
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(2,751)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(751, base_ring=CyclotomicField(750))
 
chi = DirichletCharacter(H, H._module([416]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("751.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 751.o (of order \(375\), degree \(200\), 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: \(12400\)
Relative dimension: \(62\) over \(\Q(\zeta_{375})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{375}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 12400 q - 225 q^{2} - 200 q^{3} - 225 q^{4} - 200 q^{5} - 200 q^{6} - 200 q^{7} - 150 q^{8} - 200 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 12400 q - 225 q^{2} - 200 q^{3} - 225 q^{4} - 200 q^{5} - 200 q^{6} - 200 q^{7} - 150 q^{8} - 200 q^{9} - 200 q^{10} - 200 q^{11} - 200 q^{12} - 75 q^{13} - 200 q^{14} - 250 q^{15} - 425 q^{16} - 200 q^{17} - 200 q^{18} - 200 q^{19} - 200 q^{20} - 200 q^{21} - 100 q^{22} - 200 q^{23} - 200 q^{24} - 200 q^{25} - 200 q^{26} - 200 q^{27} - 200 q^{28} - 200 q^{29} - 200 q^{30} - 225 q^{31} - 100 q^{32} - 200 q^{33} - 200 q^{34} - 200 q^{35} - 200 q^{36} - 200 q^{37} - 200 q^{38} + 200 q^{39} - 200 q^{40} - 200 q^{41} + 200 q^{42} - 200 q^{43} - 300 q^{44} - 200 q^{45} - 200 q^{46} - 200 q^{47} - 200 q^{48} - 200 q^{49} - 300 q^{50} - 400 q^{51} - 200 q^{52} + 200 q^{53} - 200 q^{54} - 200 q^{55} + 300 q^{56} - 425 q^{57} - 200 q^{58} - 200 q^{59} + 600 q^{60} - 200 q^{61} - 200 q^{62} - 400 q^{63} - 50 q^{64} - 200 q^{65} - 200 q^{66} - 300 q^{67} - 200 q^{68} - 200 q^{69} + 200 q^{70} - 625 q^{71} - 100 q^{72} + 300 q^{73} + 200 q^{74} - 200 q^{75} - 225 q^{76} - 400 q^{77} - 325 q^{78} - 650 q^{80} - 200 q^{81} - 200 q^{82} - 375 q^{83} - 200 q^{84} - 275 q^{85} - 375 q^{86} - 200 q^{87} + 800 q^{88} - 300 q^{89} - 200 q^{90} - 200 q^{91} - 975 q^{92} - 200 q^{94} + 800 q^{95} - 200 q^{96} - 100 q^{97} - 300 q^{98} - 550 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.89436 + 2.00043i −0.0346530 + 0.199788i −0.304266 − 5.58202i −2.60355 − 2.36407i −0.334016 − 0.447790i 1.50558 + 0.427170i 7.54168 + 6.40034i 2.78605 + 0.996453i 9.66121 − 0.729822i
2.2 −1.88888 + 1.99464i 0.335543 − 1.93453i −0.301878 − 5.53822i −0.210427 − 0.191072i 3.22490 + 4.32338i −3.37950 − 0.958852i 7.42798 + 6.30384i −0.805069 − 0.287939i 0.778590 − 0.0588158i
2.3 −1.76818 + 1.86719i 0.189877 − 1.09471i −0.251066 − 4.60603i 2.01281 + 1.82767i 1.70830 + 2.29019i 3.67248 + 1.04198i 5.12295 + 4.34765i 1.66242 + 0.594579i −6.97162 + 0.526646i
2.4 −1.73343 + 1.83049i −0.259368 + 1.49535i −0.237058 − 4.34903i 0.443717 + 0.402903i −2.28764 − 3.06687i −0.624190 − 0.177099i 4.52755 + 3.84236i 0.655952 + 0.234607i −1.50667 + 0.113816i
2.5 −1.73031 + 1.82720i −0.456585 + 2.63239i −0.235814 − 4.32621i 0.743767 + 0.675353i −4.01986 − 5.38913i 4.49866 + 1.27639i 4.47553 + 3.79822i −3.89624 − 1.39352i −2.52095 + 0.190436i
2.6 −1.70597 + 1.80149i 0.566630 − 3.26684i −0.226184 − 4.14954i 1.52253 + 1.38248i 4.91853 + 6.59390i 0.295921 + 0.0839605i 4.07788 + 3.46074i −7.52640 − 2.69188i −5.08791 + 0.384348i
2.7 −1.63333 + 1.72478i 0.172311 − 0.993441i −0.198259 − 3.63723i 1.29977 + 1.18021i 1.43203 + 1.91981i 2.53510 + 0.719273i 2.97500 + 2.52477i 1.86753 + 0.667937i −4.15855 + 0.314143i
2.8 −1.57142 + 1.65940i −0.517508 + 2.98363i −0.175418 − 3.21819i 2.01391 + 1.82867i −4.13783 − 5.54728i −2.27558 − 0.645640i 2.13099 + 1.80849i −5.80948 − 2.07781i −6.19919 + 0.468296i
2.9 −1.56655 + 1.65426i −0.261795 + 1.50935i −0.173660 − 3.18594i −1.57919 − 1.43394i −2.08675 − 2.79755i −4.79691 − 1.36101i 2.06828 + 1.75527i 0.615167 + 0.220019i 4.84599 − 0.366073i
2.10 −1.54831 + 1.63500i 0.310830 − 1.79206i −0.167119 − 3.06594i −1.98713 − 1.80435i 2.44875 + 3.28286i 0.554361 + 0.157286i 1.83786 + 1.55973i −0.290082 − 0.103750i 6.02682 − 0.455274i
2.11 −1.42083 + 1.50039i −0.228513 + 1.31747i −0.123547 − 2.26658i −1.84771 − 1.67775i −1.65203 − 2.21476i 1.97831 + 0.561298i 0.425300 + 0.360936i 1.14127 + 0.408184i 5.14258 − 0.388478i
2.12 −1.40409 + 1.48271i 0.0969875 − 0.559170i −0.118103 − 2.16670i 0.738795 + 0.670839i 0.692909 + 0.928932i −2.97890 − 0.845191i 0.264549 + 0.224513i 2.52150 + 0.901835i −2.03200 + 0.153500i
2.13 −1.26598 + 1.33687i −0.484960 + 2.79598i −0.0756496 − 1.38786i −2.24013 − 2.03408i −3.12390 − 4.18798i 0.646788 + 0.183510i −0.856426 − 0.726816i −4.75755 − 1.70158i 5.55526 − 0.419652i
2.14 −1.21421 + 1.28220i −0.213812 + 1.23271i −0.0608694 − 1.11670i 2.96457 + 2.69188i −1.32097 − 1.77093i 0.455589 + 0.129262i −1.18703 − 1.00739i 1.35091 + 0.483163i −7.05115 + 0.532654i
2.15 −1.11025 + 1.17242i 0.494010 − 2.84816i −0.0330496 − 0.606324i −0.403124 − 0.366044i 2.79076 + 3.74136i 1.47564 + 0.418677i −1.71466 − 1.45516i −5.04320 − 1.80374i 0.876725 − 0.0662291i
2.16 −1.03102 + 1.08875i 0.366931 − 2.11550i −0.0135196 − 0.248030i 2.87189 + 2.60773i 1.92494 + 2.58062i −4.61909 − 1.31055i −2.00253 − 1.69947i −1.51592 − 0.542180i −5.80017 + 0.438153i
2.17 −0.976869 + 1.03157i 0.182472 − 1.05202i −0.00100160 − 0.0183753i −1.35426 − 1.22969i 0.906976 + 1.21592i 4.42762 + 1.25623i −2.14648 − 1.82163i 1.75132 + 0.626373i 2.59145 − 0.195762i
2.18 −0.975086 + 1.02968i −0.128543 + 0.741098i −0.000600861 − 0.0110233i 0.420606 + 0.381918i −0.637756 − 0.854992i 0.0582991 + 0.0165410i −2.15052 − 1.82506i 2.29206 + 0.819774i −0.803382 + 0.0606886i
2.19 −0.874248 + 0.923199i −0.154714 + 0.891987i 0.0208671 + 0.382825i 1.07765 + 0.978527i −0.688223 − 0.922650i 1.44060 + 0.408734i −2.31049 − 1.96083i 2.05306 + 0.734293i −1.84551 + 0.139412i
2.20 −0.807326 + 0.852530i 0.491411 − 2.83317i 0.0338222 + 0.620497i 1.26165 + 1.14560i 2.01863 + 2.70623i −0.128797 − 0.0365430i −2.34671 − 1.99156i −4.96061 − 1.77420i −1.99522 + 0.150722i
See next 80 embeddings (of 12400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
751.o even 375 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 751.2.o.a ✓ 12400
751.o even 375 1 inner 751.2.o.a ✓ 12400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
751.2.o.a ✓ 12400 1.a even 1 1 trivial
751.2.o.a ✓ 12400 751.o even 375 1 inner

Hecke kernels

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