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Properties

Label 409.2.m.a
Level $409$
Weight $2$
Character orbit 409.m
Analytic conductor $3.266$
Analytic rank $0$
Dimension $1056$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [409,2,Mod(4,409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(409, base_ring=CyclotomicField(102)) chi = DirichletCharacter(H, H._module([41])) 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("409.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 409 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 409.m (of order \(102\), degree \(32\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.26588144267\)
Analytic rank: \(0\)
Dimension: \(1056\)
Relative dimension: \(33\) over \(\Q(\zeta_{102})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{102}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1056 q - 33 q^{2} - 31 q^{3} + 29 q^{4} - 22 q^{5} - 44 q^{6} - 57 q^{7} - 22 q^{8} + 2 q^{9} - 30 q^{10} + 17 q^{11} + 67 q^{12} - 34 q^{13} - 68 q^{14} - 32 q^{15} + 21 q^{16} - 27 q^{17} + 22 q^{18}+ \cdots - 229 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.71461 + 0.336142i 0.641913 − 2.93106i 5.31655 − 1.33717i −1.78410 + 3.58296i −0.757292 + 8.17248i 1.56632 − 0.904315i −8.88163 + 3.44076i −5.45368 − 2.50909i 3.63877 − 10.3261i
4.2 −2.59763 + 0.321657i −0.362240 + 1.65404i 4.70464 − 1.18327i 0.734587 − 1.47525i 0.408934 − 4.41310i −1.81212 + 1.04623i −6.95890 + 2.69589i 0.120770 + 0.0555630i −1.43366 + 4.06844i
4.3 −2.44555 + 0.302825i 0.203668 − 0.929975i 3.94943 − 0.993322i 0.799625 − 1.60586i −0.216460 + 2.33598i 2.44108 − 1.40936i −4.76209 + 1.84484i 1.90202 + 0.875070i −1.46923 + 4.16937i
4.4 −2.44024 + 0.302168i −0.527261 + 2.40755i 3.92389 − 0.986898i −1.22273 + 2.45557i 0.559162 − 6.03432i 1.16589 − 0.673128i −4.69136 + 1.81744i −2.79288 − 1.28493i 2.24176 − 6.36166i
4.5 −2.38232 + 0.294995i 0.583884 − 2.66609i 3.64884 − 0.917721i 1.56697 − 3.14691i −0.604514 + 6.52374i −3.39230 + 1.95855i −3.94517 + 1.52837i −4.04174 − 1.85950i −2.80471 + 7.95920i
4.6 −2.01647 + 0.249693i 0.0951539 − 0.434486i 2.06420 − 0.519168i −0.964566 + 1.93711i −0.0833868 + 0.899887i −1.20954 + 0.698327i −0.243446 + 0.0943116i 2.54567 + 1.17119i 1.46133 − 4.14696i
4.7 −1.81839 + 0.225166i 0.0538242 − 0.245769i 1.31626 − 0.331054i −0.121124 + 0.243249i −0.0425349 + 0.459024i 3.86720 − 2.23273i 1.09816 − 0.425430i 2.66789 + 1.22742i 0.165479 − 0.469596i
4.8 −1.62587 + 0.201326i 0.446735 − 2.03985i 0.663313 − 0.166830i −0.650392 + 1.30616i −0.315655 + 3.40647i −1.92598 + 1.11196i 2.01044 − 0.778847i −1.23603 − 0.568664i 0.794486 − 2.25459i
4.9 −1.52819 + 0.189231i −0.292770 + 1.33683i 0.359969 − 0.0905358i 1.10345 − 2.21603i 0.194439 − 2.09833i −2.32698 + 1.34348i 2.33879 − 0.906052i 1.02400 + 0.471117i −1.26695 + 3.59533i
4.10 −1.36768 + 0.169355i −0.738030 + 3.36995i −0.0977346 + 0.0245812i −0.477318 + 0.958584i 0.438669 − 4.73399i −3.78786 + 2.18692i 2.69963 − 1.04584i −8.08647 − 3.72037i 0.490476 − 1.39187i
4.11 −1.12461 + 0.139257i 0.660209 − 3.01461i −0.694241 + 0.174609i 0.846864 − 1.70073i −0.322672 + 3.48219i 3.49702 − 2.01900i 2.86979 − 1.11176i −5.92657 − 2.72666i −0.715552 + 2.03059i
4.12 −1.09959 + 0.136159i −0.601511 + 2.74659i −0.749037 + 0.188390i 0.354399 − 0.711729i 0.287444 − 3.10202i 3.88313 − 2.24192i 2.86432 − 1.10964i −4.45652 − 2.05032i −0.292785 + 0.830864i
4.13 −1.09740 + 0.135888i 0.168622 − 0.769951i −0.753764 + 0.189579i 1.14792 − 2.30534i −0.0804192 + 0.867861i −2.29327 + 1.32402i 2.86365 − 1.10938i 2.16100 + 0.994220i −0.946468 + 2.68588i
4.14 −0.660303 + 0.0817632i −0.201422 + 0.919720i −1.51028 + 0.379850i 0.0742276 − 0.149069i 0.0578001 − 0.623763i 1.49636 − 0.863923i 2.20702 − 0.855003i 1.92008 + 0.883378i −0.0368243 + 0.104500i
4.15 −0.577700 + 0.0715348i −0.295059 + 1.34728i −1.61097 + 0.405176i −1.46049 + 2.93305i 0.0740781 − 0.799430i −1.41038 + 0.814285i 1.98728 − 0.769877i 0.997297 + 0.458829i 0.633907 − 1.79890i
4.16 −0.382682 + 0.0473863i 0.393250 − 1.79563i −1.79539 + 0.451560i −1.01508 + 2.03855i −0.0654011 + 0.705791i −0.224957 + 0.129879i 1.38480 − 0.536473i −0.344257 − 0.158383i 0.291852 − 0.828215i
4.17 −0.0467833 + 0.00579302i 0.297826 − 1.35991i −1.93744 + 0.487285i 1.58600 − 3.18511i −0.00605525 + 0.0653466i 2.56040 − 1.47825i 0.175731 − 0.0680787i 0.964728 + 0.443845i −0.0557467 + 0.158198i
4.18 0.275196 − 0.0340767i 0.564527 − 2.57771i −1.86502 + 0.469072i −0.964382 + 1.93674i 0.0675159 − 0.728612i −0.713507 + 0.411944i −1.01441 + 0.392984i −3.60049 − 1.65649i −0.199397 + 0.565847i
4.19 0.430306 − 0.0532834i −0.344272 + 1.57199i −1.75727 + 0.441971i 1.70873 − 3.43158i −0.0643811 + 0.694782i −0.624407 + 0.360502i −1.54124 + 0.597079i 0.372752 + 0.171493i 0.552428 − 1.56768i
4.20 0.689284 − 0.0853519i 0.0642285 − 0.293276i −1.47177 + 0.370164i −0.212539 + 0.426836i 0.0192400 − 0.207633i 2.30590 − 1.33131i −2.27817 + 0.882566i 2.64351 + 1.21621i −0.110069 + 0.312352i
See next 80 embeddings (of 1056 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.33
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
409.m even 102 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 409.2.m.a ✓ 1056
409.m even 102 1 inner 409.2.m.a ✓ 1056
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
409.2.m.a ✓ 1056 1.a even 1 1 trivial
409.2.m.a ✓ 1056 409.m even 102 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(409, [\chi])\).