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Properties

Label 575.2.k.d
Level $575$
Weight $2$
Character orbit 575.k
Analytic conductor $4.591$
Analytic rank $0$
Dimension $50$
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(26,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.k (of order \(11\), degree \(10\), 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: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 5 q^{2} + 2 q^{3} - 11 q^{4} - 11 q^{6} + 5 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 5 q^{2} + 2 q^{3} - 11 q^{4} - 11 q^{6} + 5 q^{7} + 2 q^{8} + 3 q^{9} - 16 q^{11} + 9 q^{12} + 14 q^{13} - 12 q^{14} + 27 q^{16} - 38 q^{17} + 42 q^{18} - 5 q^{19} - 9 q^{21} - 6 q^{22} + 8 q^{23} + 102 q^{24} - 19 q^{26} - 7 q^{27} + 34 q^{28} - 38 q^{29} + 2 q^{31} - 49 q^{32} + 2 q^{33} - 31 q^{34} - 59 q^{36} + 35 q^{37} - 30 q^{38} + 32 q^{39} - 11 q^{41} + 102 q^{42} - 6 q^{43} - 55 q^{44} + 153 q^{46} + 10 q^{47} - 84 q^{48} - 20 q^{51} + 97 q^{52} + 29 q^{53} + 19 q^{54} + 77 q^{56} + 49 q^{57} + 12 q^{58} - 50 q^{59} + 4 q^{61} - 126 q^{62} - 74 q^{63} - 44 q^{64} - 144 q^{66} + 43 q^{67} - 54 q^{68} - 50 q^{69} - 25 q^{71} + 14 q^{72} + 20 q^{73} - 47 q^{74} - 26 q^{76} - 150 q^{77} - 174 q^{78} + 72 q^{79} - 71 q^{81} + 11 q^{82} - 36 q^{83} + 100 q^{84} - 20 q^{86} - 85 q^{87} + 45 q^{88} - 24 q^{89} + 38 q^{91} - 74 q^{92} - 100 q^{93} + 150 q^{94} - 169 q^{96} + 14 q^{97} + 44 q^{98} + 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −0.395474 − 2.75058i −0.0237855 + 0.0520830i −5.49030 + 1.61210i 0 0.152665 + 0.0448264i −1.15709 + 0.743614i 4.29671 + 9.40848i 1.96244 + 2.26477i 0
26.2 −0.185934 − 1.29320i 0.426013 − 0.932839i 0.281191 − 0.0825651i 0 −1.28556 − 0.377474i 1.03230 − 0.663422i −1.24453 − 2.72515i 1.27588 + 1.47244i 0
26.3 0.0393306 + 0.273550i −0.955665 + 2.09261i 1.84570 − 0.541947i 0 −0.610022 − 0.179119i 2.63800 − 1.69534i 0.450453 + 0.986355i −1.50116 − 1.73243i 0
26.4 0.166938 + 1.16108i 0.526900 − 1.15375i 0.598752 − 0.175809i 0 1.42755 + 0.419168i −4.23310 + 2.72045i 1.27866 + 2.79988i 0.911065 + 1.05142i 0
26.5 0.362593 + 2.52189i −0.804293 + 1.76116i −4.30948 + 1.26538i 0 −4.73308 − 1.38976i −0.564436 + 0.362741i −2.63693 − 5.77406i −0.490203 − 0.565724i 0
101.1 −1.91337 + 0.561816i 0.0122495 + 0.0141367i 1.66284 − 1.06864i 0 −0.0313800 − 0.0201667i 1.60166 − 3.50714i 0.0305297 − 0.0352332i 0.426895 − 2.96912i 0
101.2 −1.29680 + 0.380775i 1.87580 + 2.16478i −0.145804 + 0.0937028i 0 −3.25683 − 2.09304i −1.66529 + 3.64647i 1.92355 − 2.21990i −0.740735 + 5.15193i 0
101.3 −0.0847124 + 0.0248738i −1.89380 − 2.18556i −1.67595 + 1.07707i 0 0.214792 + 0.138038i −0.565342 + 1.23793i 0.230817 − 0.266377i −0.763258 + 5.30858i 0
101.4 2.01612 − 0.591987i 2.09058 + 2.41265i 2.03179 − 1.30575i 0 5.64312 + 3.62661i 1.44918 − 3.17325i 0.571316 − 0.659333i −1.02344 + 7.11821i 0
101.5 2.59594 − 0.762237i −0.775100 − 0.894513i 4.47539 − 2.87616i 0 −2.69394 − 1.73129i −1.26928 + 2.77933i 5.88204 − 6.78823i 0.227571 − 1.58279i 0
151.1 −1.29801 − 1.49799i −0.749526 − 0.481691i −0.274499 + 1.90918i 0 0.251328 + 1.74802i 3.44343 + 1.01108i −0.118700 + 0.0762838i −0.916482 − 2.00682i 0
151.2 −0.588656 − 0.679345i 1.58059 + 1.01578i 0.169636 − 1.17984i 0 −0.240356 − 1.67171i −2.75076 − 0.807695i −2.41379 + 1.55125i 0.220205 + 0.482181i 0
151.3 0.553310 + 0.638554i −2.80987 − 1.80579i 0.183031 − 1.27301i 0 −0.401632 − 2.79341i 3.04231 + 0.893304i 2.33575 − 1.50110i 3.38822 + 7.41917i 0
151.4 1.08398 + 1.25098i 1.71985 + 1.10528i −0.105306 + 0.732422i 0 0.481599 + 3.34959i 1.10450 + 0.324311i 1.75463 − 1.12763i 0.489993 + 1.07293i 0
151.5 1.81966 + 2.10000i −1.42355 − 0.914863i −0.814204 + 5.66292i 0 −0.669173 − 4.65420i −3.21768 − 0.944795i −8.69851 + 5.59019i −0.0567125 − 0.124183i 0
301.1 −2.10791 − 1.35467i 0.430654 + 2.99526i 1.77733 + 3.89181i 0 3.14983 − 6.89716i 0.630632 − 0.727789i 0.812483 − 5.65095i −5.90767 + 1.73465i 0
301.2 −1.52164 − 0.977902i −0.153622 − 1.06846i 0.528281 + 1.15677i 0 −0.811095 + 1.77605i 1.14141 − 1.31726i −0.187478 + 1.30394i 1.76046 − 0.516918i 0
301.3 0.00779572 + 0.00501001i 0.0396403 + 0.275704i −0.830794 − 1.81919i 0 −0.00107226 + 0.00234791i −2.74348 + 3.16615i 0.00527509 − 0.0366891i 2.80404 − 0.823340i 0
301.4 0.777284 + 0.499530i 0.318648 + 2.21624i −0.476190 − 1.04271i 0 −0.859400 + 1.88182i 2.14533 − 2.47584i 0.413717 − 2.87746i −1.93171 + 0.567203i 0
301.5 1.54373 + 0.992097i −0.350690 − 2.43910i 0.568025 + 1.24380i 0 1.87845 − 4.11324i 1.89250 − 2.18406i 0.165214 − 1.14909i −2.94776 + 0.865541i 0
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.5
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 575.2.k.d 50
5.b even 2 1 115.2.g.c ✓ 50
5.c odd 4 2 575.2.p.d 100
23.c even 11 1 inner 575.2.k.d 50
115.i odd 22 1 2645.2.a.x 25
115.j even 22 1 115.2.g.c ✓ 50
115.j even 22 1 2645.2.a.y 25
115.k odd 44 2 575.2.p.d 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.c ✓ 50 5.b even 2 1
115.2.g.c ✓ 50 115.j even 22 1
575.2.k.d 50 1.a even 1 1 trivial
575.2.k.d 50 23.c even 11 1 inner
575.2.p.d 100 5.c odd 4 2
575.2.p.d 100 115.k odd 44 2
2645.2.a.x 25 115.i odd 22 1
2645.2.a.y 25 115.j even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} - 5 T_{2}^{49} + 23 T_{2}^{48} - 66 T_{2}^{47} + 179 T_{2}^{46} - 399 T_{2}^{45} + 965 T_{2}^{44} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display