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Properties

Label 325.2.bh.a
Level $325$
Weight $2$
Character orbit 325.bh
Analytic conductor $2.595$
Analytic rank $0$
Dimension $256$
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] = [325,2,Mod(4,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.bh (of order \(30\), 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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(32\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 15 q^{2} - 5 q^{3} + 25 q^{4} - 9 q^{6} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 15 q^{2} - 5 q^{3} + 25 q^{4} - 9 q^{6} - 27 q^{9} - q^{10} - 9 q^{11} - 20 q^{12} - 10 q^{13} - 16 q^{14} - 9 q^{15} + 9 q^{16} - 5 q^{17} - 9 q^{19} + 36 q^{20} + 20 q^{22} - 15 q^{23} - 18 q^{24} - 40 q^{25} + 6 q^{26} - 20 q^{27} - 15 q^{28} + 7 q^{29} + 5 q^{30} + 15 q^{33} - 36 q^{35} - 39 q^{36} - 135 q^{37} - 20 q^{38} - 8 q^{39} + 22 q^{40} - 72 q^{41} + 55 q^{42} - 27 q^{45} - 69 q^{46} + 40 q^{48} - 56 q^{49} - 138 q^{50} + 112 q^{51} + 75 q^{52} - 60 q^{53} + 105 q^{54} - 6 q^{55} - 18 q^{56} - 195 q^{58} + 9 q^{59} - 35 q^{61} + 45 q^{62} + 72 q^{64} + 79 q^{65} + 32 q^{66} - 15 q^{67} - 32 q^{69} - 9 q^{71} - 75 q^{72} - 82 q^{74} - 72 q^{75} - 6 q^{76} + 90 q^{77} + 180 q^{78} - 44 q^{79} + 252 q^{80} - 29 q^{81} + 66 q^{84} - 57 q^{85} - 15 q^{87} - 105 q^{88} + 36 q^{89} - 76 q^{90} - 11 q^{91} - 320 q^{92} + 49 q^{94} + 21 q^{95} - 15 q^{97} - 30 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.86826 − 2.07491i −0.176331 − 0.396047i −0.605810 + 5.76389i −2.23105 − 0.149752i −0.492329 + 1.10579i 1.37887 + 2.38827i 8.57371 − 6.22917i 1.88163 − 2.08976i 3.85745 + 4.90900i
4.2 −1.64792 − 1.83020i −0.996705 − 2.23864i −0.424939 + 4.04302i 1.78065 − 1.35252i −2.45467 + 5.51327i −1.04972 − 1.81817i 4.11496 − 2.98969i −2.01068 + 2.23309i −5.40975 − 1.03010i
4.3 −1.61939 − 1.79851i 0.873466 + 1.96184i −0.403171 + 3.83592i 0.0403030 − 2.23570i 2.11390 − 4.74791i −0.448100 − 0.776131i 3.63596 − 2.64168i −1.07847 + 1.19776i −4.08620 + 3.54798i
4.4 −1.59350 − 1.76976i 1.17479 + 2.63862i −0.383756 + 3.65119i −0.293055 + 2.21678i 2.79770 − 6.28374i 0.217221 + 0.376237i 3.21999 − 2.33946i −3.57480 + 3.97022i 4.39016 − 3.01380i
4.5 −1.45479 − 1.61571i −0.339847 − 0.763309i −0.285041 + 2.71199i 0.879916 + 2.05566i −0.738878 + 1.65955i −0.0845182 − 0.146390i 1.27860 − 0.928959i 1.54025 − 1.71062i 2.04126 − 4.41224i
4.6 −1.40001 − 1.55487i −0.467721 − 1.05052i −0.248534 + 2.36464i 1.87800 − 1.21373i −0.978607 + 2.19799i 2.27741 + 3.94459i 0.639272 − 0.464458i 1.12256 − 1.24673i −4.51641 − 1.22081i
4.7 −1.34779 − 1.49688i −0.0376103 − 0.0844741i −0.215034 + 2.04591i −2.21492 + 0.306795i −0.0757563 + 0.170151i −1.67890 − 2.90793i 0.0931792 − 0.0676987i 2.00167 − 2.22308i 3.44449 + 2.90197i
4.8 −1.07873 − 1.19805i 0.686364 + 1.54160i −0.0626109 + 0.595703i 2.22071 − 0.261600i 1.10651 − 2.48527i −2.31817 − 4.01518i −1.82727 + 1.32759i 0.101959 − 0.113237i −2.70896 − 2.37833i
4.9 −0.989176 − 1.09859i −1.01062 − 2.26989i −0.0193764 + 0.184354i −1.33681 − 1.79247i −1.49400 + 3.35558i −0.995241 − 1.72381i −2.17024 + 1.57677i −2.12366 + 2.35856i −0.646847 + 3.24167i
4.10 −0.947779 − 1.05262i 0.221668 + 0.497874i −0.000656824 0.00624927i 0.808457 + 2.08480i 0.313978 − 0.705205i 0.901071 + 1.56070i −2.28464 + 1.65989i 1.80865 − 2.00871i 1.42826 − 2.82692i
4.11 −0.773510 − 0.859070i −1.05082 − 2.36017i 0.0693736 − 0.660045i −1.94136 + 1.10956i −1.21473 + 2.72834i 2.58000 + 4.46870i −2.49112 + 1.80991i −2.45880 + 2.73077i 2.45485 + 0.809510i
4.12 −0.752903 − 0.836184i 1.22982 + 2.76223i 0.0767169 − 0.729913i −2.17745 − 0.508623i 1.38379 − 3.10805i 0.919692 + 1.59295i −2.48871 + 1.80815i −4.11004 + 4.56466i 1.21411 + 2.20370i
4.13 −0.496276 − 0.551171i 0.929036 + 2.08665i 0.151558 − 1.44198i 2.23606 + 0.00741972i 0.689041 − 1.54761i 1.61235 + 2.79268i −2.07004 + 1.50397i −1.48360 + 1.64771i −1.10561 − 1.23613i
4.14 −0.390483 − 0.433676i −0.969371 − 2.17724i 0.173460 − 1.65036i 1.38087 + 1.75875i −0.565694 + 1.27057i −1.56599 − 2.71237i −1.72769 + 1.25524i −1.79332 + 1.99168i 0.223519 − 1.28561i
4.15 −0.388240 − 0.431185i −0.181177 − 0.406931i 0.173867 − 1.65424i −0.255222 − 2.22145i −0.105122 + 0.236108i 0.437595 + 0.757938i −1.71959 + 1.24936i 1.87462 − 2.08198i −0.858770 + 0.972506i
4.16 −0.195887 − 0.217555i 0.0560401 + 0.125868i 0.200099 − 1.90381i −1.26486 + 1.84394i 0.0164057 − 0.0368478i 0.162325 + 0.281156i −0.927058 + 0.673547i 1.99469 − 2.21533i 0.648929 − 0.0860285i
4.17 0.0636668 + 0.0707092i 0.543693 + 1.22116i 0.208111 − 1.98004i −2.12528 − 0.695104i −0.0517317 + 0.116191i −1.98283 − 3.43436i 0.307210 − 0.223201i 0.811773 − 0.901566i −0.0861598 − 0.194532i
4.18 0.138701 + 0.154043i 1.02441 + 2.30086i 0.204566 − 1.94631i 0.589185 − 2.15705i −0.212345 + 0.476935i −0.659077 − 1.14156i 0.663583 − 0.482122i −2.23716 + 2.48462i 0.413999 − 0.208425i
4.19 0.253540 + 0.281585i −0.570987 − 1.28246i 0.194050 − 1.84626i 2.18272 − 0.485527i 0.216353 − 0.485936i 0.999863 + 1.73181i 1.18217 − 0.858894i 0.688718 − 0.764899i 0.690124 + 0.491520i
4.20 0.347293 + 0.385708i −1.00244 − 2.25153i 0.180899 − 1.72114i −1.55030 + 1.61139i 0.520291 − 1.16859i −1.60542 − 2.78067i 1.56648 − 1.13811i −2.05709 + 2.28463i −1.15993 − 0.0383361i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner
25.e even 10 1 inner
325.bh even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.bh.a ✓ 256
13.e even 6 1 inner 325.2.bh.a ✓ 256
25.e even 10 1 inner 325.2.bh.a ✓ 256
325.bh even 30 1 inner 325.2.bh.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.bh.a ✓ 256 1.a even 1 1 trivial
325.2.bh.a ✓ 256 13.e even 6 1 inner
325.2.bh.a ✓ 256 25.e even 10 1 inner
325.2.bh.a ✓ 256 325.bh even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(325, [\chi])\).