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Properties

Label 425.2.bg.a
Level $425$
Weight $2$
Character orbit 425.bg
Analytic conductor $3.394$
Analytic rank $0$
Dimension $1376$
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] = [425,2,Mod(12,425)] 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("425.12"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(80)) chi = DirichletCharacter(H, H._module([36, 65])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.bg (of order \(80\), 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: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(1376\)
Relative dimension: \(43\) over \(\Q(\zeta_{80})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{80}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1376 q - 32 q^{2} - 32 q^{3} - 40 q^{4} - 32 q^{5} - 24 q^{6} - 32 q^{7} - 16 q^{8} - 40 q^{9} - 56 q^{10} - 24 q^{11} - 56 q^{12} - 24 q^{13} - 40 q^{14} - 56 q^{15} - 32 q^{17} - 64 q^{18} - 40 q^{19}+ \cdots + 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −2.28369 + 1.39945i −1.68488 + 0.199419i 2.34881 − 4.60979i −0.133441 + 2.23208i 3.56866 − 2.81331i −1.32910 − 0.888076i 0.666928 + 8.47412i −0.118062 + 0.0283441i −2.81894 − 5.28413i
12.2 −2.26679 + 1.38909i −1.24485 + 0.147338i 2.30078 − 4.51554i −1.10693 − 1.94286i 2.61716 − 2.06320i 3.05530 + 2.04149i 0.639929 + 8.13107i −1.38916 + 0.333507i 5.20799 + 2.86643i
12.3 −2.08885 + 1.28005i 1.69651 − 0.200795i 1.81678 − 3.56563i −1.83367 + 1.27971i −3.28672 + 2.59104i −1.16964 − 0.781531i 0.384773 + 4.88900i −0.0792901 + 0.0190359i 2.19217 − 5.02030i
12.4 −1.97620 + 1.21102i −3.03773 + 0.359540i 1.53083 − 3.00442i 2.22181 − 0.252114i 5.56777 − 4.38927i 0.479616 + 0.320469i 0.249485 + 3.17000i 6.18145 − 1.48403i −4.08543 + 3.18888i
12.5 −1.96202 + 1.20233i 0.639872 − 0.0757338i 1.49595 − 2.93597i 1.75473 − 1.38598i −1.16438 + 0.917927i −1.07125 − 0.715784i 0.233825 + 2.97103i −2.51341 + 0.603416i −1.77641 + 4.82908i
12.6 −1.84695 + 1.13181i −1.50364 + 0.177968i 1.22225 − 2.39881i −1.53054 − 1.63017i 2.57573 − 2.03055i −4.02539 − 2.68968i 0.117650 + 1.49489i −0.687837 + 0.165135i 4.67188 + 1.27857i
12.7 −1.65934 + 1.01684i 0.875841 − 0.103663i 0.811449 − 1.59256i 0.473042 + 2.18546i −1.34791 + 1.06260i −0.0206660 − 0.0138086i −0.0324667 − 0.412529i −2.16076 + 0.518752i −3.00721 − 3.14540i
12.8 −1.63873 + 1.00422i 2.00511 − 0.237320i 0.769013 − 1.50927i −2.19271 − 0.438180i −3.04751 + 2.40247i 1.64146 + 1.09679i −0.0461587 − 0.586502i 1.04703 − 0.251369i 4.03330 − 1.48390i
12.9 −1.62562 + 0.996181i 3.33464 − 0.394681i 0.742281 − 1.45681i 2.08481 + 0.808425i −5.02769 + 3.96351i −3.70614 − 2.47636i −0.0545973 − 0.693724i 8.04697 − 1.93191i −4.19445 + 0.762660i
12.10 −1.53819 + 0.942601i 2.89857 − 0.343068i 0.569537 − 1.11778i 0.851038 − 2.06778i −4.13516 + 3.25990i 4.13132 + 2.76046i −0.105518 − 1.34073i 5.36690 − 1.28848i 0.640042 + 3.98283i
12.11 −1.36065 + 0.833806i −0.932150 + 0.110327i 0.248148 − 0.487018i 2.01135 + 0.976963i 1.17634 − 0.927349i 3.74353 + 2.50135i −0.181974 − 2.31220i −2.06038 + 0.494653i −3.55134 + 0.347776i
12.12 −1.20350 + 0.737506i −1.94136 + 0.229775i −0.00348201 + 0.00683383i 0.361817 + 2.20660i 2.16697 − 1.70830i −1.98054 − 1.32336i −0.222339 − 2.82509i 0.798962 − 0.191814i −2.06283 − 2.38880i
12.13 −1.16787 + 0.715673i 0.160647 − 0.0190138i −0.0562432 + 0.110383i 0.677584 − 2.13093i −0.174007 + 0.137176i 0.135728 + 0.0906908i −0.228247 − 2.90015i −2.89166 + 0.694227i 0.733721 + 2.97359i
12.14 −1.06617 + 0.653351i −1.00626 + 0.119099i −0.198125 + 0.388842i −2.22932 + 0.173527i 0.995035 − 0.784422i 0.420888 + 0.281228i −0.239031 − 3.03718i −1.91873 + 0.460647i 2.26347 − 1.64154i
12.15 −0.898015 + 0.550304i −2.80409 + 0.331886i −0.404385 + 0.793650i −0.751660 − 2.10595i 2.33548 − 1.84114i 1.72519 + 1.15274i −0.238874 − 3.03518i 4.83568 − 1.16094i 1.83391 + 1.47753i
12.16 −0.632361 + 0.387511i 2.32066 − 0.274669i −0.658266 + 1.29192i −0.0477319 + 2.23556i −1.36106 + 1.07297i 1.40332 + 0.937670i −0.200750 − 2.55077i 2.39292 − 0.574490i −0.836120 − 1.43218i
12.17 −0.585685 + 0.358909i 1.55360 − 0.183880i −0.693769 + 1.36160i −0.252319 − 2.22179i −0.843923 + 0.665295i −2.06829 − 1.38199i −0.190147 − 2.41605i −0.537260 + 0.128985i 0.945197 + 1.21071i
12.18 −0.537595 + 0.329439i −2.35099 + 0.278258i −0.727502 + 1.42780i 2.23408 − 0.0942119i 1.17221 − 0.924098i −3.56252 − 2.38040i −0.178210 − 2.26437i 2.53263 − 0.608030i −1.16999 + 0.786641i
12.19 −0.325631 + 0.199547i −0.668941 + 0.0791743i −0.841765 + 1.65206i −1.88116 + 1.20881i 0.202029 − 0.159267i −0.192671 − 0.128739i −0.115486 − 1.46739i −2.47590 + 0.594410i 0.371350 − 0.769006i
12.20 −0.0637156 + 0.0390450i −2.83200 + 0.335189i −0.905446 + 1.77704i −0.0425797 + 2.23566i 0.167355 − 0.131932i 3.87393 + 2.58847i −0.0234194 − 0.297572i 4.99075 − 1.19817i −0.0845784 − 0.144109i
See next 80 embeddings (of 1376 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.43
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
425.bg even 80 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.bg.a ✓ 1376
17.e odd 16 1 425.2.bj.a yes 1376
25.f odd 20 1 425.2.bj.a yes 1376
425.bg even 80 1 inner 425.2.bg.a ✓ 1376
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.bg.a ✓ 1376 1.a even 1 1 trivial
425.2.bg.a ✓ 1376 425.bg even 80 1 inner
425.2.bj.a yes 1376 17.e odd 16 1
425.2.bj.a yes 1376 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(425, [\chi])\).