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Properties

Label 975.2.bw.b
Level $975$
Weight $2$
Character orbit 975.bw
Analytic conductor $7.785$
Analytic rank $0$
Dimension $288$
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] = [975,2,Mod(16,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 6, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bw (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 2 q^{2} + 36 q^{3} + 32 q^{4} - 4 q^{5} + 2 q^{6} - 12 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 2 q^{2} + 36 q^{3} + 32 q^{4} - 4 q^{5} + 2 q^{6} - 12 q^{8} + 36 q^{9} - 7 q^{10} - 2 q^{11} - 84 q^{12} + 20 q^{14} + 7 q^{15} + 44 q^{16} + 16 q^{18} - 6 q^{19} + 30 q^{20} + 21 q^{22} + 5 q^{23} - 24 q^{24} - 4 q^{25} + 66 q^{26} - 72 q^{27} - 13 q^{29} - 7 q^{30} - 76 q^{32} + 3 q^{33} + 94 q^{34} + 7 q^{35} + 42 q^{36} + 18 q^{37} - 52 q^{38} + 6 q^{39} + 24 q^{40} - 17 q^{41} - 10 q^{42} - 38 q^{43} + 12 q^{44} - 3 q^{45} + 40 q^{46} - 6 q^{47} + 44 q^{48} - 184 q^{49} + 16 q^{50} + 80 q^{51} - 38 q^{52} - 52 q^{53} + 2 q^{54} + 29 q^{55} - 50 q^{56} - 8 q^{57} - 8 q^{58} + 9 q^{59} - 10 q^{60} + 4 q^{61} - 36 q^{62} - 72 q^{64} - 31 q^{65} + 8 q^{66} - 98 q^{68} + 10 q^{69} - 144 q^{70} - 58 q^{71} + 6 q^{72} + 32 q^{73} + 24 q^{74} - 3 q^{75} + 102 q^{76} - 52 q^{77} + 48 q^{78} - 104 q^{79} + 37 q^{80} + 36 q^{81} + 6 q^{82} + 28 q^{83} - 97 q^{85} + 40 q^{86} + 22 q^{87} + 34 q^{88} - 42 q^{89} - 6 q^{90} - 70 q^{91} + 64 q^{92} - 14 q^{94} + 45 q^{95} - 88 q^{96} + 2 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100}) \) 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 −0.293814 + 2.79546i 0.669131 − 0.743145i −5.77196 − 1.22687i −1.74089 + 1.40332i 1.88083 + 2.08887i 1.69514 − 2.93607i 3.38834 − 10.4282i −0.104528 − 0.994522i −3.41142 − 5.27890i
16.2 −0.278534 + 2.65007i 0.669131 − 0.743145i −4.98900 − 1.06044i 1.82027 + 1.29870i 1.78301 + 1.98023i −2.04028 + 3.53387i 2.55300 − 7.85732i −0.104528 − 0.994522i −3.94865 + 4.46211i
16.3 −0.267095 + 2.54124i 0.669131 − 0.743145i −4.43026 − 0.941680i 0.451002 − 2.19011i 1.70979 + 1.89891i −0.424044 + 0.734465i 1.99711 − 6.14647i −0.104528 − 0.994522i 5.44514 + 1.73107i
16.4 −0.246392 + 2.34427i 0.669131 − 0.743145i −3.47858 − 0.739395i −2.14829 − 0.620378i 1.57726 + 1.75173i −1.42777 + 2.47297i 1.13362 − 3.48891i −0.104528 − 0.994522i 1.98365 − 4.88330i
16.5 −0.239907 + 2.28257i 0.669131 − 0.743145i −3.19626 − 0.679386i 2.14572 − 0.629210i 1.53575 + 1.70562i 2.09067 − 3.62114i 0.899077 − 2.76707i −0.104528 − 0.994522i 0.921442 + 5.04869i
16.6 −0.227886 + 2.16819i 0.669131 − 0.743145i −2.69281 − 0.572374i −1.78928 + 1.34107i 1.45879 + 1.62015i 0.461956 − 0.800131i 0.507272 − 1.56122i −0.104528 − 0.994522i −2.49993 − 4.18511i
16.7 −0.226555 + 2.15552i 0.669131 − 0.743145i −2.63865 − 0.560863i 1.76647 + 1.37098i 1.45027 + 1.61069i 1.31102 − 2.27076i 0.467229 − 1.43798i −0.104528 − 0.994522i −3.35537 + 3.49707i
16.8 −0.197682 + 1.88082i 0.669131 − 0.743145i −1.54210 − 0.327783i −2.16346 − 0.565200i 1.26544 + 1.40542i 1.13233 − 1.96126i −0.247468 + 0.761629i −0.104528 − 0.994522i 1.49071 − 3.95734i
16.9 −0.187308 + 1.78212i 0.669131 − 0.743145i −1.18457 − 0.251788i −0.920882 + 2.03764i 1.19904 + 1.33167i −1.85307 + 3.20960i −0.436881 + 1.34458i −0.104528 − 0.994522i −3.45883 − 2.02279i
16.10 −0.172067 + 1.63711i 0.669131 − 0.743145i −0.694212 − 0.147559i −0.0238169 + 2.23594i 1.10147 + 1.22331i −0.133211 + 0.230728i −0.656339 + 2.02000i −0.104528 − 0.994522i −3.65637 − 0.423722i
16.11 −0.162781 + 1.54876i 0.669131 − 0.743145i −0.415861 − 0.0883940i −0.200248 − 2.22708i 1.04203 + 1.15729i −1.53442 + 2.65770i −0.757863 + 2.33246i −0.104528 − 0.994522i 3.48181 + 0.0523916i
16.12 −0.145360 + 1.38301i 0.669131 − 0.743145i 0.0647137 + 0.0137553i −0.243168 − 2.22281i 0.930510 + 1.03344i 0.997022 − 1.72689i −0.887885 + 2.73263i −0.104528 − 0.994522i 3.10951 − 0.0131960i
16.13 −0.111724 + 1.06298i 0.669131 − 0.743145i 0.838855 + 0.178304i 1.92661 + 1.13497i 0.715189 + 0.794298i 0.0340425 − 0.0589633i −0.943828 + 2.90481i −0.104528 − 0.994522i −1.42170 + 1.92114i
16.14 −0.0786208 + 0.748027i 0.669131 − 0.743145i 1.40293 + 0.298202i −2.18446 − 0.477618i 0.503285 + 0.558954i 1.35132 − 2.34055i −0.798216 + 2.45666i −0.104528 − 0.994522i 0.529015 − 1.59649i
16.15 −0.0612663 + 0.582910i 0.669131 − 0.743145i 1.62026 + 0.344398i 2.14189 − 0.642112i 0.392191 + 0.435573i −0.870504 + 1.50776i −0.662263 + 2.03824i −0.104528 − 0.994522i 0.243068 + 1.28787i
16.16 −0.0522388 + 0.497019i 0.669131 − 0.743145i 1.71200 + 0.363896i 1.77246 − 1.36323i 0.334403 + 0.371392i 2.37614 − 4.11560i −0.579163 + 1.78248i −0.104528 − 0.994522i 0.584961 + 0.952159i
16.17 −0.0393684 + 0.374566i 0.669131 − 0.743145i 1.81755 + 0.386331i −1.94526 − 1.10270i 0.252014 + 0.279890i −1.77950 + 3.08219i −0.449030 + 1.38197i −0.104528 − 0.994522i 0.489617 − 0.685217i
16.18 −0.0271137 + 0.257969i 0.669131 − 0.743145i 1.89048 + 0.401834i 0.528691 + 2.17267i 0.173566 + 0.192765i −1.67663 + 2.90401i −0.315231 + 0.970181i −0.104528 − 0.994522i −0.574817 + 0.0774770i
16.19 −0.0206942 + 0.196892i 0.669131 − 0.743145i 1.91796 + 0.407674i −1.22511 + 1.87059i 0.132472 + 0.147125i 2.14124 − 3.70874i −0.242315 + 0.745768i −0.104528 − 0.994522i −0.342951 − 0.279926i
16.20 0.0292765 − 0.278547i 0.669131 − 0.743145i 1.87956 + 0.399514i 0.956544 − 2.02114i −0.187411 − 0.208141i −0.517439 + 0.896230i 0.339410 − 1.04460i −0.104528 − 0.994522i −0.534980 − 0.325615i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
25.d even 5 1 inner
325.y even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bw.b ✓ 288
13.c even 3 1 inner 975.2.bw.b ✓ 288
25.d even 5 1 inner 975.2.bw.b ✓ 288
325.y even 15 1 inner 975.2.bw.b ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bw.b ✓ 288 1.a even 1 1 trivial
975.2.bw.b ✓ 288 13.c even 3 1 inner
975.2.bw.b ✓ 288 25.d even 5 1 inner
975.2.bw.b ✓ 288 325.y even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} - 2 T_{2}^{287} - 50 T_{2}^{286} + 112 T_{2}^{285} + 1084 T_{2}^{284} + \cdots + 19\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\). Copy content Toggle raw display