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Properties

Label 975.2.cm.a
Level $975$
Weight $2$
Character orbit 975.cm
Analytic conductor $7.785$
Analytic rank $0$
Dimension $544$
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] = [975,2,Mod(94,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 27, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.94");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.cm (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 544 q - 64 q^{4} - 4 q^{5} - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 544 q - 64 q^{4} - 4 q^{5} - 68 q^{9} + 4 q^{10} + 2 q^{15} + 68 q^{16} - 12 q^{19} + 36 q^{20} - 50 q^{22} - 20 q^{23} + 36 q^{25} + 48 q^{26} - 8 q^{29} + 4 q^{30} + 10 q^{33} + 24 q^{34} - 10 q^{35} + 64 q^{36} - 10 q^{37} - 132 q^{40} - 14 q^{41} - 2 q^{45} + 20 q^{46} + 208 q^{49} + 70 q^{50} - 128 q^{51} - 40 q^{52} - 100 q^{53} - 50 q^{55} + 36 q^{56} - 60 q^{58} - 12 q^{59} - 48 q^{60} - 28 q^{61} - 160 q^{62} + 176 q^{64} + 56 q^{65} + 48 q^{66} + 12 q^{69} - 24 q^{70} - 32 q^{71} - 160 q^{73} - 8 q^{74} - 16 q^{75} - 12 q^{76} + 80 q^{77} - 96 q^{79} + 4 q^{80} + 68 q^{81} + 64 q^{85} - 48 q^{86} + 40 q^{87} + 20 q^{88} - 60 q^{89} + 8 q^{90} - 64 q^{91} - 360 q^{92} + 4 q^{94} - 132 q^{95} + 60 q^{96} - 360 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1 −1.99699 − 1.79810i −0.406737 + 0.913545i 0.545758 + 5.19254i −1.08747 − 1.95382i 2.45490 − 1.09299i 2.07059 + 1.19546i 5.08781 − 7.00278i −0.669131 − 0.743145i −1.34149 + 5.85714i
94.2 −1.99070 − 1.79244i 0.406737 − 0.913545i 0.541010 + 5.14737i −1.30942 + 1.81257i −2.44716 + 1.08955i −0.942985 − 0.544433i 5.00027 − 6.88228i −0.669131 − 0.743145i 5.85559 − 1.26123i
94.3 −1.94605 − 1.75223i 0.406737 − 0.913545i 0.507739 + 4.83082i 1.75569 + 1.38476i −2.39227 + 1.06511i 0.986402 + 0.569500i 4.39819 − 6.05359i −0.669131 − 0.743145i −0.990229 − 5.77118i
94.4 −1.82368 − 1.64205i 0.406737 − 0.913545i 0.420427 + 4.00009i −1.89949 − 1.17981i −2.24184 + 0.998133i 1.16377 + 0.671901i 2.91677 − 4.01459i −0.669131 − 0.743145i 1.52676 + 5.27064i
94.5 −1.81734 − 1.63634i −0.406737 + 0.913545i 0.416060 + 3.95855i −0.0126617 − 2.23603i 2.23405 − 0.994665i −2.20640 − 1.27387i 2.84659 − 3.91800i −0.669131 − 0.743145i −3.63591 + 4.08436i
94.6 −1.81102 − 1.63065i −0.406737 + 0.913545i 0.411720 + 3.91726i 2.05372 − 0.884434i 2.22628 − 0.991206i −0.778403 − 0.449411i 2.77722 − 3.82251i −0.669131 − 0.743145i −5.16154 − 1.74718i
94.7 −1.80657 − 1.62664i −0.406737 + 0.913545i 0.408671 + 3.88824i −1.46579 + 1.68863i 2.22081 − 0.988768i 3.78010 + 2.18244i 2.72870 − 3.75574i −0.669131 − 0.743145i 5.39484 − 0.666321i
94.8 −1.79173 − 1.61328i −0.406737 + 0.913545i 0.398561 + 3.79206i 0.319263 + 2.21316i 2.20256 − 0.980644i −3.92236 − 2.26457i 2.56922 − 3.53622i −0.669131 − 0.743145i 2.99841 − 4.48043i
94.9 −1.74234 − 1.56881i 0.406737 − 0.913545i 0.365527 + 3.47775i 1.90511 + 1.17071i −2.14185 + 0.953614i −0.872867 − 0.503950i 2.06288 − 2.83931i −0.669131 − 0.743145i −1.48271 − 5.02853i
94.10 −1.67574 − 1.50884i −0.406737 + 0.913545i 0.322438 + 3.06779i −2.19711 − 0.415572i 2.05998 − 0.917161i −1.75259 − 1.01186i 1.43766 − 1.97877i −0.669131 − 0.743145i 3.05475 + 4.01148i
94.11 −1.49517 − 1.34626i 0.406737 − 0.913545i 0.214069 + 2.03673i −1.35179 + 1.78120i −1.83801 + 0.818334i −3.15793 − 1.82323i 0.0566996 − 0.0780403i −0.669131 − 0.743145i 4.41911 − 0.843348i
94.12 −1.46496 − 1.31905i 0.406737 − 0.913545i 0.197141 + 1.87567i 0.616631 − 2.14936i −1.80087 + 0.801798i 1.57339 + 0.908399i −0.132091 + 0.181808i −0.669131 − 0.743145i −3.73846 + 2.33536i
94.13 −1.44805 − 1.30383i 0.406737 − 0.913545i 0.187817 + 1.78696i −2.22938 + 0.172767i −1.78008 + 0.792542i 2.83076 + 1.63434i −0.232726 + 0.320319i −0.669131 − 0.743145i 3.45351 + 2.65656i
94.14 −1.41288 − 1.27217i −0.406737 + 0.913545i 0.168777 + 1.60581i 2.21554 − 0.302312i 1.73685 − 0.773298i 2.32666 + 1.34330i −0.430629 + 0.592710i −0.669131 − 0.743145i −3.51489 − 2.39140i
94.15 −1.38036 − 1.24289i 0.406737 − 0.913545i 0.151584 + 1.44223i 0.923922 − 2.03626i −1.69688 + 0.755499i −0.901717 − 0.520607i −0.600296 + 0.826236i −0.669131 − 0.743145i −3.80619 + 1.66246i
94.16 −1.23998 − 1.11648i 0.406737 − 0.913545i 0.0819571 + 0.779769i 2.04698 − 0.899935i −1.52430 + 0.678661i −3.74064 − 2.15966i −1.19253 + 1.64137i −0.669131 − 0.743145i −3.54296 − 1.16951i
94.17 −1.14245 − 1.02867i −0.406737 + 0.913545i 0.0379803 + 0.361358i 1.40835 + 1.73683i 1.40441 − 0.625284i −0.833373 − 0.481148i −1.47890 + 2.03553i −0.669131 − 0.743145i 0.177650 − 3.43296i
94.18 −1.11255 − 1.00174i −0.406737 + 0.913545i 0.0252173 + 0.239927i 0.957367 − 2.02075i 1.36765 − 0.608917i 0.830351 + 0.479404i −1.54763 + 2.13014i −0.669131 − 0.743145i −3.08939 + 1.28915i
94.19 −1.09363 − 0.984706i −0.406737 + 0.913545i 0.0173167 + 0.164757i −1.64143 − 1.51846i 1.34439 − 0.598562i 2.29078 + 1.32259i −1.58669 + 2.18390i −0.669131 − 0.743145i 0.299876 + 3.27695i
94.20 −1.07680 − 0.969553i −0.406737 + 0.913545i 0.0104032 + 0.0989797i −0.872595 + 2.05878i 1.32370 − 0.589351i 0.414921 + 0.239555i −1.61861 + 2.22782i −0.669131 − 0.743145i 2.93570 − 1.37086i
See next 80 embeddings (of 544 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
25.e even 10 1 inner
325.bf even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.cm.a ✓ 544
13.c even 3 1 inner 975.2.cm.a ✓ 544
25.e even 10 1 inner 975.2.cm.a ✓ 544
325.bf even 30 1 inner 975.2.cm.a ✓ 544
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.cm.a ✓ 544 1.a even 1 1 trivial
975.2.cm.a ✓ 544 13.c even 3 1 inner
975.2.cm.a ✓ 544 25.e even 10 1 inner
975.2.cm.a ✓ 544 325.bf even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(975, [\chi])\).