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Properties

Label 775.2.m.a
Level $775$
Weight $2$
Character orbit 775.m
Analytic conductor $6.188$
Analytic rank $0$
Dimension $312$
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(16,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.m (of order \(5\), degree \(4\), 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: \(312\)
Relative dimension: \(78\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 6 q^{2} - 2 q^{3} + 298 q^{4} - 6 q^{5} - 4 q^{6} + 2 q^{7} - 12 q^{8} - 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 6 q^{2} - 2 q^{3} + 298 q^{4} - 6 q^{5} - 4 q^{6} + 2 q^{7} - 12 q^{8} - 76 q^{9} - 17 q^{10} + 7 q^{11} - 25 q^{12} - 6 q^{13} - 12 q^{14} - 8 q^{15} + 270 q^{16} - 6 q^{17} + 4 q^{18} - q^{19} - 10 q^{20} + 4 q^{21} - 4 q^{22} + q^{23} - 13 q^{24} + 22 q^{25} - 106 q^{26} + 13 q^{27} - 7 q^{28} + 19 q^{29} + 54 q^{30} + 7 q^{31} - 130 q^{32} + 13 q^{33} - 50 q^{34} - 27 q^{35} - 97 q^{36} - q^{37} + 41 q^{38} - 19 q^{39} - 24 q^{40} - 4 q^{41} - 55 q^{42} + 8 q^{43} - 14 q^{44} - 2 q^{45} + 11 q^{46} + 40 q^{47} - 71 q^{48} - 48 q^{49} + 47 q^{50} + 10 q^{51} - 96 q^{52} + 37 q^{53} - 27 q^{54} - 28 q^{55} - q^{56} - 8 q^{57} - 33 q^{58} - 26 q^{59} + 12 q^{60} + 6 q^{61} - 54 q^{62} + 54 q^{63} + 220 q^{64} - 27 q^{65} + 8 q^{66} - 11 q^{67} + 20 q^{68} - 61 q^{69} + 88 q^{70} - 19 q^{71} - 14 q^{72} - 41 q^{73} + q^{74} + 27 q^{75} + 38 q^{76} - q^{77} - 88 q^{78} + 55 q^{79} + 8 q^{80} - 74 q^{81} - 47 q^{82} - 100 q^{83} + 31 q^{84} - 52 q^{85} - 9 q^{86} + 48 q^{87} - 19 q^{88} + 40 q^{89} - 151 q^{90} + 29 q^{91} + 27 q^{92} - 82 q^{93} + 5 q^{94} + 31 q^{95} - 6 q^{96} + 50 q^{97} + 114 q^{98} - 152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.80082 −0.180171 + 0.554510i 5.84460 −0.279222 − 2.21857i 0.504628 − 1.55308i 1.13576 − 3.49550i −10.7680 2.15203 + 1.56354i 0.782049 + 6.21381i
16.2 −2.75945 0.978439 − 3.01133i 5.61457 −1.86584 − 1.23232i −2.69996 + 8.30961i −1.31582 + 4.04968i −9.97423 −5.68369 − 4.12945i 5.14871 + 3.40054i
16.3 −2.64375 −1.02426 + 3.15236i 4.98939 2.23182 + 0.137704i 2.70790 − 8.33405i 0.0818416 − 0.251883i −7.90319 −6.46123 − 4.69435i −5.90037 − 0.364055i
16.4 −2.62547 −0.659165 + 2.02870i 4.89310 −1.96816 + 1.06130i 1.73062 − 5.32630i −0.490385 + 1.50925i −7.59576 −1.25408 − 0.911141i 5.16735 − 2.78640i
16.5 −2.59591 0.450015 − 1.38500i 4.73874 2.12397 − 0.699105i −1.16820 + 3.59535i 0.433147 − 1.33309i −7.10951 0.711326 + 0.516809i −5.51363 + 1.81481i
16.6 −2.56413 0.0989937 − 0.304671i 4.57475 0.514678 + 2.17603i −0.253833 + 0.781216i −1.39116 + 4.28155i −6.60200 2.34403 + 1.70303i −1.31970 − 5.57962i
16.7 −2.46191 0.582193 − 1.79181i 4.06099 −0.00909330 + 2.23605i −1.43331 + 4.41126i 0.538451 − 1.65718i −5.07397 −0.444573 − 0.323001i 0.0223869 − 5.50495i
16.8 −2.40945 0.627411 − 1.93097i 3.80544 2.22555 − 0.216671i −1.51171 + 4.65258i −0.393353 + 1.21061i −4.35010 −0.907958 − 0.659670i −5.36233 + 0.522058i
16.9 −2.35348 −0.364898 + 1.12304i 3.53887 1.04967 + 1.97438i 0.858780 − 2.64305i 0.936902 − 2.88349i −3.62170 1.29898 + 0.943767i −2.47038 − 4.64667i
16.10 −2.30949 −0.387131 + 1.19147i 3.33373 1.01420 − 1.99284i 0.894074 − 2.75168i −1.26337 + 3.88824i −3.08022 1.15733 + 0.840847i −2.34229 + 4.60243i
16.11 −2.30783 0.0308203 − 0.0948551i 3.32606 −2.16800 − 0.547527i −0.0711278 + 0.218909i 0.0792364 − 0.243865i −3.06031 2.41900 + 1.75751i 5.00336 + 1.26360i
16.12 −2.23151 0.720398 − 2.21716i 2.97963 −2.17853 + 0.503973i −1.60757 + 4.94761i 1.22557 − 3.77191i −2.18605 −1.96977 − 1.43112i 4.86142 − 1.12462i
16.13 −1.99312 −0.807820 + 2.48621i 1.97253 −1.11387 − 1.93889i 1.61008 − 4.95532i 1.26026 − 3.87869i 0.0547567 −3.10164 − 2.25347i 2.22008 + 3.86444i
16.14 −1.91464 0.654645 − 2.01479i 1.66584 −0.629476 − 2.14564i −1.25341 + 3.85759i 0.642899 − 1.97864i 0.639803 −1.20376 − 0.874586i 1.20522 + 4.10812i
16.15 −1.86199 −0.449366 + 1.38301i 1.46699 1.71063 − 1.44005i 0.836713 − 2.57514i 0.540807 − 1.66443i 0.992452 0.716277 + 0.520406i −3.18518 + 2.68135i
16.16 −1.75722 0.943451 − 2.90364i 1.08783 −0.390967 + 2.20162i −1.65785 + 5.10235i −0.447925 + 1.37857i 1.60288 −5.11400 − 3.71554i 0.687016 − 3.86874i
16.17 −1.75359 −0.799073 + 2.45929i 1.07508 −1.99311 − 1.01367i 1.40125 − 4.31260i −0.938389 + 2.88807i 1.62193 −2.98255 − 2.16695i 3.49510 + 1.77756i
16.18 −1.74763 −0.841139 + 2.58876i 1.05422 1.07352 + 1.96152i 1.47000 − 4.52420i −0.786106 + 2.41939i 1.65288 −3.56712 − 2.59166i −1.87612 − 3.42801i
16.19 −1.68901 −0.0320786 + 0.0987278i 0.852766 1.69987 + 1.45273i 0.0541812 − 0.166753i 0.0695779 − 0.214139i 1.93769 2.41833 + 1.75702i −2.87111 − 2.45368i
16.20 −1.65125 0.823301 − 2.53386i 0.726619 1.22100 − 1.87328i −1.35947 + 4.18403i −0.613789 + 1.88905i 2.10267 −3.31557 − 2.40891i −2.01617 + 3.09324i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
775.m even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.m.a yes 312
25.d even 5 1 775.2.i.a ✓ 312
31.d even 5 1 775.2.i.a ✓ 312
775.m even 5 1 inner 775.2.m.a yes 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.i.a ✓ 312 25.d even 5 1
775.2.i.a ✓ 312 31.d even 5 1
775.2.m.a yes 312 1.a even 1 1 trivial
775.2.m.a yes 312 775.m even 5 1 inner

Hecke kernels

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