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Properties

Label 675.2.ba.c
Level $675$
Weight $2$
Character orbit 675.ba
Analytic conductor $5.390$
Analytic rank $0$
Dimension $288$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,2,Mod(32,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([10, 9])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.ba (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [288] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(24\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 24 q^{6} + 36 q^{11} + 48 q^{21} - 192 q^{36} - 180 q^{41} - 60 q^{51} - 288 q^{56} + 72 q^{61} + 144 q^{71} + 216 q^{76} + 24 q^{81} - 36 q^{91} - 168 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.231616 + 2.64738i 1.67242 + 0.450581i −4.98538 − 0.879058i 0 −1.58022 + 4.32317i −0.275047 + 0.392807i 2.10628 − 7.86074i 2.59395 + 1.50712i 0
32.2 −0.220056 + 2.51525i 0.915897 − 1.47008i −4.30845 − 0.759695i 0 3.49607 + 2.62721i 1.30850 − 1.86873i 1.55196 − 5.79199i −1.32227 − 2.69288i 0
32.3 −0.210285 + 2.40357i −1.72931 + 0.0973950i −3.76332 − 0.663575i 0 0.129553 − 4.17700i 0.492686 − 0.703628i 1.13739 − 4.24479i 2.98103 − 0.336852i 0
32.4 −0.197521 + 2.25768i −1.63814 − 0.562574i −3.08849 − 0.544583i 0 1.59368 − 3.58728i −2.77821 + 3.96769i 0.666412 − 2.48708i 2.36702 + 1.84315i 0
32.5 −0.162917 + 1.86215i 0.0781133 − 1.73029i −1.47143 − 0.259453i 0 3.20932 + 0.427351i −1.59729 + 2.28117i −0.244740 + 0.913382i −2.98780 − 0.270317i 0
32.6 −0.148243 + 1.69442i −0.193157 + 1.72125i −0.879474 − 0.155075i 0 −2.88788 − 0.582451i −0.203237 + 0.290253i −0.487310 + 1.81866i −2.92538 − 0.664941i 0
32.7 −0.140766 + 1.60897i 1.63374 + 0.575232i −0.599344 − 0.105680i 0 −1.15550 + 2.54766i 0.784107 − 1.11982i −0.581640 + 2.17071i 2.33822 + 1.87956i 0
32.8 −0.0884834 + 1.01137i 1.07761 + 1.35601i 0.954575 + 0.168317i 0 −1.46678 + 0.969879i −0.0546810 + 0.0780926i −0.780219 + 2.91182i −0.677513 + 2.92249i 0
32.9 −0.0683995 + 0.781810i 0.800191 − 1.53613i 1.36307 + 0.240346i 0 1.14623 + 0.730667i −1.68864 + 2.41162i −0.687378 + 2.56533i −1.71939 − 2.45839i 0
32.10 −0.0632765 + 0.723253i −1.46710 + 0.920668i 1.45052 + 0.255767i 0 −0.573043 − 1.11934i −2.68947 + 3.84096i −0.652582 + 2.43547i 1.30474 − 2.70142i 0
32.11 −0.0488346 + 0.558182i −1.33253 − 1.10651i 1.66043 + 0.292779i 0 0.682705 − 0.689760i 2.24269 − 3.20290i −0.534551 + 1.99497i 0.551290 + 2.94891i 0
32.12 −0.0304268 + 0.347779i −1.64956 + 0.528165i 1.84959 + 0.326133i 0 −0.133494 − 0.589753i 0.696364 − 0.994510i −0.350411 + 1.30775i 2.44208 − 1.74248i 0
32.13 0.0304268 − 0.347779i 1.64956 − 0.528165i 1.84959 + 0.326133i 0 −0.133494 − 0.589753i −0.696364 + 0.994510i 0.350411 − 1.30775i 2.44208 − 1.74248i 0
32.14 0.0488346 − 0.558182i 1.33253 + 1.10651i 1.66043 + 0.292779i 0 0.682705 − 0.689760i −2.24269 + 3.20290i 0.534551 − 1.99497i 0.551290 + 2.94891i 0
32.15 0.0632765 − 0.723253i 1.46710 − 0.920668i 1.45052 + 0.255767i 0 −0.573043 − 1.11934i 2.68947 − 3.84096i 0.652582 − 2.43547i 1.30474 − 2.70142i 0
32.16 0.0683995 − 0.781810i −0.800191 + 1.53613i 1.36307 + 0.240346i 0 1.14623 + 0.730667i 1.68864 − 2.41162i 0.687378 − 2.56533i −1.71939 − 2.45839i 0
32.17 0.0884834 − 1.01137i −1.07761 − 1.35601i 0.954575 + 0.168317i 0 −1.46678 + 0.969879i 0.0546810 − 0.0780926i 0.780219 − 2.91182i −0.677513 + 2.92249i 0
32.18 0.140766 − 1.60897i −1.63374 − 0.575232i −0.599344 − 0.105680i 0 −1.15550 + 2.54766i −0.784107 + 1.11982i 0.581640 − 2.17071i 2.33822 + 1.87956i 0
32.19 0.148243 − 1.69442i 0.193157 − 1.72125i −0.879474 − 0.155075i 0 −2.88788 − 0.582451i 0.203237 − 0.290253i 0.487310 − 1.81866i −2.92538 − 0.664941i 0
32.20 0.162917 − 1.86215i −0.0781133 + 1.73029i −1.47143 − 0.259453i 0 3.20932 + 0.427351i 1.59729 − 2.28117i 0.244740 − 0.913382i −2.98780 − 0.270317i 0
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
27.f odd 18 1 inner
135.n odd 18 1 inner
135.q even 36 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.ba.c ✓ 288
5.b even 2 1 inner 675.2.ba.c ✓ 288
5.c odd 4 2 inner 675.2.ba.c ✓ 288
27.f odd 18 1 inner 675.2.ba.c ✓ 288
135.n odd 18 1 inner 675.2.ba.c ✓ 288
135.q even 36 2 inner 675.2.ba.c ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
675.2.ba.c ✓ 288 1.a even 1 1 trivial
675.2.ba.c ✓ 288 5.b even 2 1 inner
675.2.ba.c ✓ 288 5.c odd 4 2 inner
675.2.ba.c ✓ 288 27.f odd 18 1 inner
675.2.ba.c ✓ 288 135.n odd 18 1 inner
675.2.ba.c ✓ 288 135.q even 36 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} + 468 T_{2}^{280} - 286956 T_{2}^{276} + 200934 T_{2}^{272} - 160478280 T_{2}^{268} + \cdots + 17\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display