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Properties

Label 575.2.u.a
Level $575$
Weight $2$
Character orbit 575.u
Analytic conductor $4.591$
Analytic rank $0$
Dimension $2320$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [575,2,Mod(4,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([11, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.u (of order \(110\), degree \(40\), 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(2320\)
Relative dimension: \(58\) over \(\Q(\zeta_{110})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{110}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2320 q - 45 q^{2} - 45 q^{3} - 83 q^{4} - 46 q^{5} - 19 q^{6} - 45 q^{8} - 85 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2320 q - 45 q^{2} - 45 q^{3} - 83 q^{4} - 46 q^{5} - 19 q^{6} - 45 q^{8} - 85 q^{9} - 32 q^{10} - 25 q^{11} - 85 q^{12} - 45 q^{13} - 47 q^{14} - 75 q^{15} + 33 q^{16} - 35 q^{17} - 23 q^{19} - 94 q^{20} - 37 q^{21} - 110 q^{22} - 170 q^{23} - 156 q^{24} - 56 q^{25} - 44 q^{26} - 15 q^{27} - 85 q^{28} - 15 q^{29} - 104 q^{30} - 15 q^{31} - 25 q^{33} - 59 q^{34} - 48 q^{35} + 21 q^{36} - 45 q^{37} - 45 q^{38} + 9 q^{39} + 6 q^{40} - 27 q^{41} - 105 q^{42} - 115 q^{44} - 58 q^{45} - 18 q^{46} - 300 q^{47} - 105 q^{48} + 92 q^{49} + 68 q^{50} - 124 q^{51} + 515 q^{52} - 65 q^{53} + 17 q^{54} - 52 q^{55} - 17 q^{56} - 45 q^{58} + 3 q^{59} - 128 q^{60} - 11 q^{61} - 55 q^{62} - 125 q^{63} - 99 q^{64} - 121 q^{65} - 81 q^{66} - 75 q^{67} + 123 q^{69} - 192 q^{70} + 51 q^{71} - 220 q^{72} - 45 q^{73} + 16 q^{74} + 304 q^{75} - 218 q^{76} - 75 q^{77} - 45 q^{78} - 31 q^{79} - 63 q^{80} + 329 q^{81} + 65 q^{83} - 65 q^{84} + 26 q^{85} + 275 q^{86} - 65 q^{87} - 45 q^{88} - 31 q^{89} - 17 q^{90} - 266 q^{91} - 85 q^{92} - 59 q^{94} - 451 q^{95} - 559 q^{96} - 75 q^{97} - 25 q^{98} + 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.318009 − 2.77158i 0.227249 − 0.636910i −5.63248 + 1.30978i 0.957102 + 2.02088i −1.83751 − 0.427295i 3.52749 − 0.507177i 3.54633 + 9.93929i 1.96841 + 1.60956i 5.29666 − 3.29534i
4.2 −0.300657 − 2.62035i −0.913724 + 2.56089i −4.82782 + 1.12266i −0.294698 − 2.21656i 6.98515 + 1.62433i 1.67531 − 0.240874i 2.62060 + 7.34475i −3.40084 − 2.78085i −5.71957 + 1.43864i
4.3 −0.298531 − 2.60182i 0.959625 − 2.68954i −4.73234 + 1.10046i −1.17960 − 1.89962i −7.28418 − 1.69386i 0.568980 − 0.0818070i 2.51580 + 7.05101i −3.99031 − 3.26286i −4.59033 + 3.63620i
4.4 −0.292347 − 2.54793i −0.261528 + 0.732983i −4.45844 + 1.03677i 2.13250 − 0.672635i 1.94404 + 0.452068i −2.31740 + 0.333192i 2.22132 + 6.22569i 1.85356 + 1.51565i −2.33725 − 5.23681i
4.5 −0.285625 − 2.48934i −0.841695 + 2.35902i −4.16722 + 0.969045i −0.858688 + 2.06462i 6.11281 + 1.42147i −2.80836 + 0.403782i 1.91849 + 5.37693i −2.53408 − 2.07211i 5.38481 + 1.54786i
4.6 −0.282815 − 2.46485i 0.0839745 − 0.235355i −4.04749 + 0.941203i −2.22896 − 0.178115i −0.603865 − 0.140423i −1.09005 + 0.156726i 1.79712 + 5.03679i 2.27408 + 1.85951i 0.191357 + 5.54444i
4.7 −0.259807 − 2.26432i 0.811204 − 2.27356i −3.11164 + 0.723581i 2.07394 + 0.835920i −5.35883 − 1.24614i −4.66972 + 0.671405i 0.915014 + 2.56451i −2.18859 − 1.78960i 1.35397 − 4.91326i
4.8 −0.258045 − 2.24897i 0.496585 − 1.39178i −3.04324 + 0.707676i 1.76515 − 1.37268i −3.25820 − 0.757662i 3.48528 − 0.501108i 0.855389 + 2.39739i 0.631981 + 0.516768i −3.54261 − 3.61554i
4.9 −0.250895 − 2.18665i −0.248232 + 0.695720i −2.77048 + 0.644247i −2.23329 − 0.111372i 1.58358 + 0.368245i 3.27823 − 0.471339i 0.624555 + 1.75044i 1.90002 + 1.55364i 0.316790 + 4.91138i
4.10 −0.228388 − 1.99049i −0.950297 + 2.66339i −1.96187 + 0.456213i 1.98617 + 1.02720i 5.51850 + 1.28327i 3.05916 − 0.439841i 0.00957076 + 0.0268239i −3.86818 − 3.16299i 1.59102 − 4.18805i
4.11 −0.225996 − 1.96965i 1.12131 − 3.14270i −1.88043 + 0.437274i −0.418655 + 2.19653i −6.44344 − 1.49836i 2.42660 − 0.348892i −0.0462379 − 0.129591i −6.29680 − 5.14887i 4.42100 + 0.328198i
4.12 −0.208412 − 1.81640i 0.484788 − 1.35871i −1.30785 + 0.304127i 0.0219758 − 2.23596i −2.56900 − 0.597396i −1.83110 + 0.263273i −0.403822 − 1.13179i 0.711342 + 0.581661i −4.06598 + 0.426085i
4.13 −0.205218 − 1.78856i −0.459759 + 1.28857i −1.20879 + 0.281093i −1.58820 − 1.57405i 2.39902 + 0.557868i −3.39511 + 0.488143i −0.459157 − 1.28688i 0.873403 + 0.714178i −2.48935 + 3.16360i
4.14 −0.201986 − 1.76039i −0.421647 + 1.18175i −1.11015 + 0.258153i 0.186100 + 2.22831i 2.16551 + 0.503567i −2.83983 + 0.408306i −0.512234 − 1.43564i 1.10368 + 0.902474i 3.88510 − 0.777696i
4.15 −0.200083 − 1.74380i 0.246653 − 0.691293i −1.05280 + 0.244817i −0.207889 + 2.22638i −1.25483 − 0.291798i 2.32361 − 0.334084i −0.542139 − 1.51945i 1.90538 + 1.55802i 3.92397 − 0.0829436i
4.16 −0.148272 − 1.29225i 0.261474 − 0.732834i 0.300097 − 0.0697845i 1.52695 + 1.63354i −0.985774 − 0.229232i −2.61621 + 0.376154i −1.00889 − 2.82762i 1.85375 + 1.51580i 1.88453 − 2.21540i
4.17 −0.147747 − 1.28767i −0.873803 + 2.44900i 0.311753 − 0.0724950i 1.64551 − 1.51403i 3.28262 + 0.763339i −3.47060 + 0.498998i −1.01053 − 2.83221i −2.91166 − 2.38085i −2.19269 − 1.89519i
4.18 −0.143093 − 1.24711i 0.760042 − 2.13017i 0.413212 − 0.0960882i 2.23421 + 0.0910663i −2.76531 − 0.643046i 1.39999 − 0.201288i −1.02264 − 2.86615i −1.63752 − 1.33900i −0.206130 − 2.79934i
4.19 −0.141994 − 1.23753i −0.334448 + 0.937355i 0.436700 − 0.101550i −0.583697 − 2.15854i 1.20750 + 0.280791i 4.48251 − 0.644487i −1.02488 − 2.87243i 1.55564 + 1.27204i −2.58838 + 1.02884i
4.20 −0.133859 − 1.16664i −1.12328 + 3.14821i 0.604894 − 0.140662i −2.20916 + 0.345865i 3.82319 + 0.889044i 0.543112 − 0.0780877i −1.03431 − 2.89887i −6.32703 − 5.17359i 0.699216 + 2.53099i
See next 80 embeddings (of 2320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner
25.e even 10 1 inner
575.u even 110 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.u.a ✓ 2320
23.c even 11 1 inner 575.2.u.a ✓ 2320
25.e even 10 1 inner 575.2.u.a ✓ 2320
575.u even 110 1 inner 575.2.u.a ✓ 2320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.u.a ✓ 2320 1.a even 1 1 trivial
575.2.u.a ✓ 2320 23.c even 11 1 inner
575.2.u.a ✓ 2320 25.e even 10 1 inner
575.2.u.a ✓ 2320 575.u even 110 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(575, [\chi])\).