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Properties

Label 575.2.g.b
Level $575$
Weight $2$
Character orbit 575.g
Analytic conductor $4.591$
Analytic rank $0$
Dimension $108$
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] = [575,2,Mod(116,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.g (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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 4 q^{2} + q^{3} - 20 q^{4} + 3 q^{5} + 8 q^{6} + 24 q^{7} + 13 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 4 q^{2} + q^{3} - 20 q^{4} + 3 q^{5} + 8 q^{6} + 24 q^{7} + 13 q^{8} - 36 q^{9} - 24 q^{10} + 9 q^{11} - 20 q^{12} - 18 q^{13} - 31 q^{14} - 5 q^{15} - 20 q^{16} - 28 q^{17} + 34 q^{18} - q^{19} - 34 q^{20} - q^{21} - 35 q^{22} - 27 q^{23} + 178 q^{24} - 13 q^{25} + 14 q^{26} - 14 q^{27} - 3 q^{28} - 12 q^{29} - 35 q^{30} + 26 q^{31} + 22 q^{32} - 33 q^{33} - 15 q^{34} - 42 q^{35} - 49 q^{36} - 21 q^{37} - 11 q^{38} - 52 q^{39} + 25 q^{40} + 29 q^{41} + 6 q^{42} + 22 q^{43} - 58 q^{44} + 16 q^{45} + q^{46} - q^{47} - 83 q^{48} + 220 q^{49} - 16 q^{50} - 70 q^{51} - 76 q^{52} - 33 q^{53} - 106 q^{54} + 26 q^{55} + 124 q^{56} + 122 q^{57} + 46 q^{58} - 11 q^{59} + 20 q^{60} + 14 q^{61} - 40 q^{62} - 47 q^{63} - 87 q^{64} + 69 q^{65} + 47 q^{66} - 3 q^{67} + 154 q^{68} + q^{69} - 79 q^{70} - 51 q^{71} + 18 q^{72} - 26 q^{73} + 118 q^{74} - 55 q^{75} - 142 q^{76} - 5 q^{77} - 78 q^{78} - 7 q^{79} + 168 q^{80} + 29 q^{81} + 88 q^{82} + 7 q^{83} - 116 q^{84} + 9 q^{85} - 36 q^{86} - 130 q^{87} + 34 q^{88} + 9 q^{89} + 187 q^{90} + 58 q^{91} - 20 q^{92} + 26 q^{93} - 7 q^{94} - 86 q^{95} - 71 q^{96} - 50 q^{97} - 82 q^{98} + 124 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −2.19573 − 1.59529i 1.01539 + 3.12505i 1.65823 + 5.10351i 0.189641 − 2.22801i 2.75584 − 8.48159i −1.13790 2.82317 − 8.68881i −6.30788 + 4.58294i −3.97072 + 4.58957i
116.2 −2.19270 − 1.59309i 0.203471 + 0.626220i 1.65197 + 5.08423i −1.98470 + 1.03004i 0.551473 − 1.69726i 3.97503 2.80230 − 8.62458i 2.07630 − 1.50852i 5.99279 + 0.903245i
116.3 −2.05972 − 1.49648i 0.307396 + 0.946069i 1.38498 + 4.26254i 2.22584 + 0.213596i 0.782618 − 2.40865i 3.28601 1.95262 − 6.00954i 1.62650 − 1.18172i −4.26498 − 3.77087i
116.4 −1.76352 − 1.28127i −0.200393 − 0.616747i 0.850316 + 2.61700i −1.76374 − 1.37450i −0.436824 + 1.34441i −0.389050 0.506336 − 1.55834i 2.08683 − 1.51617i 1.34928 + 4.68378i
116.5 −1.70439 − 1.23831i 0.128999 + 0.397017i 0.753502 + 2.31904i 0.575645 − 2.16070i 0.271768 − 0.836415i −3.81619 0.285397 − 0.878362i 2.28607 − 1.66093i −3.65676 + 2.96986i
116.6 −1.62357 − 1.17959i −1.06461 − 3.27654i 0.626505 + 1.92818i 1.50900 + 1.65013i −2.13651 + 6.57550i 4.84819 0.0169997 − 0.0523197i −7.17528 + 5.21315i −0.503493 − 4.45910i
116.7 −1.57115 − 1.14151i −0.813827 − 2.50470i 0.547445 + 1.68486i −2.20505 + 0.371141i −1.58050 + 4.86426i −3.60929 −0.137087 + 0.421911i −3.18417 + 2.31343i 3.88814 + 1.93397i
116.8 −1.43650 − 1.04368i 0.856391 + 2.63570i 0.356232 + 1.09637i −2.12850 − 0.685202i 1.52062 − 4.67998i 0.873118 −0.464858 + 1.43069i −3.78647 + 2.75103i 2.34245 + 3.20576i
116.9 −1.18778 − 0.862976i 0.788130 + 2.42562i 0.0480709 + 0.147947i −0.443530 + 2.19164i 1.15712 − 3.56125i 2.11933 −0.836810 + 2.57544i −2.83541 + 2.06005i 2.41815 − 2.22044i
116.10 −0.910596 − 0.661587i −0.829740 − 2.55368i −0.226546 − 0.697236i 0.967467 − 2.01594i −0.933921 + 2.87431i −0.376418 −0.950624 + 2.92572i −3.40574 + 2.47442i −2.21469 + 1.19564i
116.11 −0.849358 − 0.617095i 0.0365978 + 0.112636i −0.277431 − 0.853845i 1.65887 + 1.49939i 0.0384227 − 0.118253i −2.34652 −0.940116 + 2.89338i 2.41570 − 1.75511i −0.483706 − 2.29719i
116.12 −0.729699 − 0.530157i 0.0188366 + 0.0579731i −0.366640 − 1.12840i 2.22947 − 0.171635i 0.0169898 − 0.0522892i 2.47822 −0.888134 + 2.73340i 2.42404 − 1.76117i −1.71784 − 1.05673i
116.13 −0.114174 − 0.0829525i −0.173523 − 0.534050i −0.611879 − 1.88317i −1.62285 − 1.53829i −0.0244889 + 0.0753690i 4.41024 −0.173574 + 0.534206i 2.17195 − 1.57802i 0.0576831 + 0.310253i
116.14 0.122324 + 0.0888736i 0.732175 + 2.25340i −0.610969 − 1.88037i 2.00192 − 0.996145i −0.110705 + 0.340716i 3.35558 0.185826 − 0.571914i −2.11469 + 1.53641i 0.333414 + 0.0560656i
116.15 0.141651 + 0.102916i −0.481619 − 1.48227i −0.608561 − 1.87296i −0.503993 − 2.17853i 0.0843271 − 0.259532i −3.72840 0.214765 − 0.660980i 0.461878 − 0.335574i 0.152814 − 0.360460i
116.16 0.408201 + 0.296575i 0.753427 + 2.31881i −0.539363 − 1.65999i 0.551873 + 2.16690i −0.380152 + 1.16999i −4.79502 0.583981 − 1.79731i −2.38217 + 1.73075i −0.417373 + 1.04820i
116.17 0.546376 + 0.396965i −0.655118 − 2.01624i −0.477089 − 1.46833i 2.06765 + 0.851375i 0.442439 − 1.36169i 1.65352 0.739600 − 2.27625i −1.20901 + 0.878399i 0.791746 + 1.28595i
116.18 0.692607 + 0.503208i 0.806129 + 2.48101i −0.391548 − 1.20506i −1.83535 − 1.27730i −0.690135 + 2.12402i −4.00896 0.864313 − 2.66008i −3.07852 + 2.23667i −0.628426 − 1.80823i
116.19 1.08563 + 0.788755i 0.515975 + 1.58801i −0.0615794 − 0.189522i 1.64162 − 1.51825i −0.692392 + 2.13096i −1.60785 0.911981 − 2.80679i 0.171512 − 0.124611i 2.97972 − 0.353410i
116.20 1.20788 + 0.877574i 0.126910 + 0.390589i 0.0707967 + 0.217890i −1.46445 + 1.68979i −0.189479 + 0.583156i 4.65328 0.817035 − 2.51458i 2.29060 − 1.66422i −3.25179 + 0.755891i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.g.b ✓ 108
25.d even 5 1 inner 575.2.g.b ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.g.b ✓ 108 1.a even 1 1 trivial
575.2.g.b ✓ 108 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{108} + 4 T_{2}^{107} + 45 T_{2}^{106} + 153 T_{2}^{105} + 1080 T_{2}^{104} + 3275 T_{2}^{103} + \cdots + 88755241 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display