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Properties

Label 325.3.bm.a
Level $325$
Weight $3$
Character orbit 325.bm
Analytic conductor $8.856$
Analytic rank $0$
Dimension $1088$
Inner twists $4$

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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] = [325,3,Mod(6,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.6"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([24, 25])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 325.bm (of order \(60\), degree \(16\), 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: \(8.85560859171\)
Analytic rank: \(0\)
Dimension: \(1088\)
Relative dimension: \(68\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1088 q - 16 q^{2} - 6 q^{3} - 18 q^{4} - 6 q^{5} - 28 q^{6} - 32 q^{7} - 44 q^{8} + 378 q^{9} - 54 q^{10} - 12 q^{11} + 44 q^{13} - 24 q^{14} + 54 q^{15} - 382 q^{16} - 66 q^{17} - 500 q^{18} + 48 q^{19}+ \cdots - 792 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −3.23555 + 2.10119i 0.162560 + 1.54666i 4.42684 − 9.94285i −4.43110 + 2.31632i −3.77580 − 4.66272i −1.64843 + 6.15203i 4.15449 + 26.2304i 6.43760 − 1.36835i 9.47003 − 16.8052i
6.2 −3.21795 + 2.08976i −0.0637219 − 0.606273i 4.36114 − 9.79528i 4.96740 − 0.570062i 1.47202 + 1.81779i −0.216885 + 0.809426i 4.03492 + 25.4755i 8.43982 − 1.79394i −14.7935 + 12.2151i
6.3 −3.11118 + 2.02042i 0.597107 + 5.68109i 3.97039 − 8.91764i 2.21104 − 4.48456i −13.3359 − 16.4685i −1.68960 + 6.30567i 3.34354 + 21.1103i −23.1150 + 4.91324i 2.18179 + 18.4195i
6.4 −3.07034 + 1.99390i 0.152494 + 1.45089i 3.82440 − 8.58974i −3.70151 − 3.36137i −3.36113 − 4.15066i 3.50562 − 13.0832i 3.09410 + 19.5354i 6.72151 − 1.42870i 18.0671 + 2.94010i
6.5 −2.90131 + 1.88413i −0.428913 − 4.08083i 3.24070 − 7.27873i −2.87751 + 4.08900i 8.93325 + 11.0316i 3.13792 − 11.7109i 2.14713 + 13.5565i −7.66591 + 1.62944i 0.644328 − 17.2851i
6.6 −2.89753 + 1.88168i −0.433688 − 4.12627i 3.22801 − 7.25023i 2.76127 + 4.16838i 9.02093 + 11.1399i −1.31106 + 4.89295i 2.12747 + 13.4323i −8.03469 + 1.70783i −15.8444 − 6.88219i
6.7 −2.77924 + 1.80486i 0.382223 + 3.63661i 2.83972 − 6.37812i 3.47687 + 3.59324i −7.62586 − 9.41715i 2.76221 − 10.3087i 1.54573 + 9.75936i −4.27548 + 0.908781i −16.1484 − 3.71121i
6.8 −2.70600 + 1.75730i 0.363225 + 3.45585i 2.60740 − 5.85632i 1.84247 + 4.64815i −7.05584 − 8.71324i −1.38389 + 5.16475i 1.21670 + 7.68194i −3.00764 + 0.639294i −13.1539 − 9.34014i
6.9 −2.69978 + 1.75326i −0.408696 − 3.88848i 2.58796 − 5.81264i −4.96786 − 0.565971i 7.92090 + 9.78150i −1.09939 + 4.10296i 1.18983 + 7.51227i −6.14992 + 1.30721i 14.4044 − 7.18195i
6.10 −2.62906 + 1.70733i −0.233137 − 2.21815i 2.37002 − 5.32316i 1.42208 − 4.79350i 4.40005 + 5.43361i 0.539336 − 2.01283i 0.895905 + 5.65652i 3.93748 − 0.836938i 4.44536 + 15.0304i
6.11 −2.52003 + 1.63653i 0.0597045 + 0.568050i 2.04539 − 4.59403i −1.44158 − 4.78768i −1.08009 − 1.33380i −2.53391 + 9.45669i 0.483586 + 3.05324i 8.48421 − 1.80337i 11.4680 + 9.70592i
6.12 −2.30000 + 1.49364i 0.0606957 + 0.577481i 1.43209 − 3.21654i −2.69645 + 4.21060i −1.00215 − 1.23755i −0.410482 + 1.53194i −0.205524 − 1.29763i 8.47353 − 1.80110i −0.0872631 − 13.7119i
6.13 −2.20368 + 1.43109i 0.404487 + 3.84843i 1.18125 − 2.65312i −4.37972 − 2.41206i −6.39879 − 7.90185i 0.471172 − 1.75844i −0.450421 − 2.84385i −5.84349 + 1.24207i 13.1034 − 0.952364i
6.14 −2.09137 + 1.35815i −0.608207 − 5.78670i 0.902312 − 2.02663i 3.88765 − 3.14423i 9.13122 + 11.2761i −2.13644 + 7.97331i −0.694990 − 4.38799i −24.3127 + 5.16782i −3.86019 + 11.8558i
6.15 −2.02248 + 1.31341i 0.388850 + 3.69966i 0.738411 − 1.65850i 3.25935 − 3.79165i −5.64561 − 6.97175i 1.30936 − 4.88658i −0.824111 − 5.20323i −4.73294 + 1.00602i −1.61196 + 11.9494i
6.16 −1.96882 + 1.27857i −0.268630 − 2.55584i 0.614567 − 1.38034i 4.97524 + 0.496976i 3.79670 + 4.68853i 2.43747 − 9.09675i −0.914064 − 5.77117i 2.34315 − 0.498053i −10.4308 + 5.38271i
6.17 −1.77771 + 1.15446i 0.0867468 + 0.825341i 0.200542 − 0.450424i 4.99853 + 0.121176i −1.10704 − 1.36707i −3.54203 + 13.2190i −1.16287 − 7.34210i 8.12967 − 1.72801i −9.02586 + 5.55520i
6.18 −1.71380 + 1.11296i −0.0422985 − 0.402443i 0.0715016 − 0.160595i −1.92025 + 4.61656i 0.520393 + 0.642632i 1.55360 − 5.79810i −1.22248 − 7.71846i 8.64316 − 1.83716i −1.84710 − 10.0490i
6.19 −1.65077 + 1.07203i 0.498194 + 4.73999i −0.0511289 + 0.114837i −4.96527 + 0.588284i −5.90380 − 7.29059i −1.98843 + 7.42093i −1.27036 − 8.02074i −13.4160 + 2.85166i 7.56589 − 6.29402i
6.20 −1.51092 + 0.981204i −0.486696 − 4.63060i −0.306824 + 0.689137i −3.35256 − 3.70950i 5.27892 + 6.51893i 1.63781 − 6.11239i −1.33991 − 8.45984i −12.4023 + 2.63618i 8.70522 + 2.31522i
See next 80 embeddings (of 1088 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.68
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner
25.d even 5 1 inner
325.bm odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.3.bm.a ✓ 1088
13.f odd 12 1 inner 325.3.bm.a ✓ 1088
25.d even 5 1 inner 325.3.bm.a ✓ 1088
325.bm odd 60 1 inner 325.3.bm.a ✓ 1088
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.3.bm.a ✓ 1088 1.a even 1 1 trivial
325.3.bm.a ✓ 1088 13.f odd 12 1 inner
325.3.bm.a ✓ 1088 25.d even 5 1 inner
325.3.bm.a ✓ 1088 325.bm odd 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(325, [\chi])\).