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Properties

Label 775.2.bp.a
Level $775$
Weight $2$
Character orbit 775.bp
Analytic conductor $6.188$
Analytic rank $0$
Dimension $624$
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] = [775,2,Mod(231,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([12, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.231");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bp (of order \(15\), degree \(8\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(78\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 12 q^{2} - 7 q^{3} + 596 q^{4} - 3 q^{5} - 23 q^{6} - 14 q^{7} - 24 q^{8} + 67 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 624 q - 12 q^{2} - 7 q^{3} + 596 q^{4} - 3 q^{5} - 23 q^{6} - 14 q^{7} - 24 q^{8} + 67 q^{9} - 19 q^{10} - 16 q^{11} - 74 q^{12} - 3 q^{13} - 15 q^{14} + 11 q^{15} + 540 q^{16} - 3 q^{17} - 46 q^{18} - 8 q^{19} - 62 q^{20} - 13 q^{21} + 13 q^{22} - 4 q^{23} - 92 q^{24} - 13 q^{25} + 19 q^{26} - 37 q^{27} - 50 q^{28} + 14 q^{29} - 60 q^{30} - 16 q^{31} + 10 q^{32} - 19 q^{33} + 23 q^{34} - 42 q^{35} + 94 q^{36} + 28 q^{37} - 26 q^{38} + 16 q^{39} - 84 q^{40} - 32 q^{41} + 16 q^{42} + 4 q^{43} - 49 q^{44} - 91 q^{45} - 26 q^{46} - 7 q^{47} - 163 q^{48} + 60 q^{49} + 13 q^{50} - 43 q^{51} - 48 q^{52} + 14 q^{53} + 36 q^{54} - 158 q^{55} - 71 q^{56} - 52 q^{57} - 6 q^{58} - 13 q^{59} + 51 q^{60} + 12 q^{61} + 27 q^{62} - 72 q^{63} + 440 q^{64} + 96 q^{65} - 92 q^{66} + 44 q^{67} - 32 q^{68} - 53 q^{69} - 70 q^{70} - 71 q^{71} - 61 q^{72} + 56 q^{73} - 43 q^{74} + 60 q^{75} - 86 q^{76} - 2 q^{77} + 31 q^{78} - 28 q^{79} - 137 q^{80} + 38 q^{81} + 29 q^{82} - 275 q^{83} - 154 q^{84} - 41 q^{85} - 72 q^{86} - 57 q^{87} - 8 q^{88} + 29 q^{89} - 284 q^{90} + 16 q^{91} - 78 q^{92} + 34 q^{93} - 29 q^{94} - 10 q^{95} - 210 q^{96} - 176 q^{97} - 3 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
231.1 −2.73609 0.295384 − 2.81039i 5.48620 1.09949 + 1.94708i −0.808199 + 7.68950i −0.0994634 + 0.946331i −9.53856 −4.87662 − 1.03656i −3.00830 − 5.32739i
231.2 −2.70847 −0.00996890 + 0.0948478i 5.33581 2.16008 − 0.577983i 0.0270005 − 0.256892i −0.160553 + 1.52756i −9.03493 2.92555 + 0.621844i −5.85050 + 1.56545i
231.3 −2.70365 −0.105894 + 1.00752i 5.30974 −0.821565 + 2.07967i 0.286301 − 2.72397i 0.353359 − 3.36199i −8.94839 1.93057 + 0.410355i 2.22123 − 5.62271i
231.4 −2.61055 −0.0431630 + 0.410669i 4.81499 −1.24952 − 1.85437i 0.112679 − 1.07207i 0.0489286 − 0.465525i −7.34869 2.76766 + 0.588284i 3.26194 + 4.84094i
231.5 −2.55632 −0.345135 + 3.28374i 4.53477 −2.23389 + 0.0987643i 0.882276 − 8.39429i −0.151518 + 1.44160i −6.47968 −7.72940 − 1.64293i 5.71053 − 0.252473i
231.6 −2.47902 −0.272615 + 2.59376i 4.14552 1.36542 − 1.77077i 0.675818 − 6.42998i 0.367626 − 3.49773i −5.31879 −3.71884 − 0.790465i −3.38491 + 4.38976i
231.7 −2.46706 −0.226227 + 2.15241i 4.08639 1.28864 + 1.82741i 0.558116 − 5.31012i −0.353823 + 3.36640i −5.14725 −1.64723 − 0.350129i −3.17915 − 4.50832i
231.8 −2.40727 0.174278 − 1.65814i 3.79496 −2.20531 + 0.369581i −0.419534 + 3.99160i 0.157158 − 1.49526i −4.32095 0.215376 + 0.0457795i 5.30879 − 0.889683i
231.9 −2.30414 −0.0454028 + 0.431979i 3.30908 −1.48935 − 1.66788i 0.104615 − 0.995342i −0.495713 + 4.71640i −3.01631 2.74990 + 0.584509i 3.43168 + 3.84305i
231.10 −2.25692 0.218731 − 2.08108i 3.09370 2.23057 − 0.156730i −0.493658 + 4.69685i 0.538584 − 5.12428i −2.46839 −1.34863 − 0.286660i −5.03422 + 0.353726i
231.11 −2.10012 0.158767 − 1.51057i 2.41051 −1.11570 + 1.93784i −0.333431 + 3.17238i −0.376067 + 3.57804i −0.862125 0.677831 + 0.144077i 2.34311 − 4.06969i
231.12 −2.00182 −0.267480 + 2.54490i 2.00728 0.454897 − 2.18931i 0.535446 − 5.09443i −0.221236 + 2.10492i −0.0145755 −3.47052 − 0.737682i −0.910621 + 4.38260i
231.13 −1.98562 0.274980 − 2.61626i 1.94270 2.08696 − 0.802856i −0.546006 + 5.19490i −0.423627 + 4.03054i 0.113779 −3.83476 − 0.815102i −4.14393 + 1.59417i
231.14 −1.98285 0.0598819 − 0.569738i 1.93170 0.484558 + 2.18293i −0.118737 + 1.12971i −0.145453 + 1.38389i 0.135426 2.61343 + 0.555501i −0.960807 − 4.32844i
231.15 −1.90152 −0.163974 + 1.56011i 1.61579 0.259645 + 2.22094i 0.311800 − 2.96658i 0.409703 − 3.89806i 0.730586 0.527394 + 0.112101i −0.493721 − 4.22317i
231.16 −1.87362 0.0233295 − 0.221965i 1.51047 2.13667 + 0.659265i −0.0437106 + 0.415879i 0.0972966 − 0.925715i 0.917204 2.88572 + 0.613378i −4.00332 − 1.23521i
231.17 −1.84931 −0.00794772 + 0.0756175i 1.41994 −0.749142 − 2.10684i 0.0146978 − 0.139840i 0.449479 − 4.27651i 1.07271 2.92879 + 0.622533i 1.38539 + 3.89620i
231.18 −1.77260 0.199789 − 1.90086i 1.14211 0.371373 − 2.20501i −0.354146 + 3.36947i −0.131250 + 1.24876i 1.52069 −0.638924 − 0.135807i −0.658296 + 3.90861i
231.19 −1.70174 −0.237548 + 2.26012i 0.895935 −2.23431 + 0.0885277i 0.404246 − 3.84614i 0.280316 − 2.66703i 1.87884 −2.11725 − 0.450036i 3.80223 − 0.150652i
231.20 −1.64064 0.339638 − 3.23144i 0.691686 −0.166517 + 2.22986i −0.557222 + 5.30161i 0.192474 − 1.83127i 2.14647 −7.39238 − 1.57130i 0.273194 − 3.65839i
See next 80 embeddings (of 624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 231.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
775.bp even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bp.a yes 624
25.d even 5 1 775.2.bn.a ✓ 624
31.g even 15 1 775.2.bn.a ✓ 624
775.bp even 15 1 inner 775.2.bp.a yes 624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.bn.a ✓ 624 25.d even 5 1
775.2.bn.a ✓ 624 31.g even 15 1
775.2.bp.a yes 624 1.a even 1 1 trivial
775.2.bp.a yes 624 775.bp even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(775, [\chi])\).