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Properties

Label 195.2.bl.a
Level $195$
Weight $2$
Character orbit 195.bl
Analytic conductor $1.557$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [195,2,Mod(68,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.bl (of order \(12\), degree \(4\), 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: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} - 4 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 2 q^{3} - 4 q^{6} - 4 q^{7} - 12 q^{10} - 28 q^{12} - 24 q^{13} - 2 q^{15} + 16 q^{16} + 16 q^{18} - 8 q^{21} - 12 q^{22} - 16 q^{25} - 32 q^{27} - 44 q^{28} - 4 q^{30} + 16 q^{31} - 46 q^{33} - 36 q^{36} + 20 q^{37} - 56 q^{40} + 8 q^{42} - 16 q^{43} - 22 q^{45} + 64 q^{48} + 48 q^{51} - 76 q^{52} + 68 q^{57} - 20 q^{58} + 96 q^{60} - 18 q^{63} + 32 q^{66} - 52 q^{67} - 24 q^{70} + 6 q^{72} + 64 q^{73} + 36 q^{75} - 104 q^{76} + 144 q^{78} - 52 q^{82} - 108 q^{85} + 14 q^{87} + 84 q^{88} - 68 q^{90} + 64 q^{91} + 32 q^{93} - 240 q^{96} + 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −0.683455 + 2.55069i 1.73202 + 0.0104094i −4.30685 − 2.48656i −0.647175 + 2.14037i −1.21031 + 4.41073i 0.572219 + 2.13555i 5.55152 − 5.55152i 2.99978 + 0.0360587i −5.01709 − 3.11359i
68.2 −0.617961 + 2.30626i 0.693746 − 1.58705i −3.20491 − 1.85036i −0.343688 − 2.20950i 3.23144 + 2.58069i −0.570618 − 2.12957i 2.87132 − 2.87132i −2.03743 − 2.20201i 5.30806 + 0.572747i
68.3 −0.604143 + 2.25469i −1.72678 + 0.134968i −2.98660 − 1.72431i −2.18894 − 0.456647i 0.738913 − 3.97491i 0.0109257 + 0.0407754i 2.39103 − 2.39103i 2.96357 − 0.466121i 2.35203 − 4.65951i
68.4 −0.576369 + 2.15104i −1.00323 − 1.41192i −2.56272 − 1.47959i 1.91481 + 1.15477i 3.61533 − 1.34421i 0.598395 + 2.23324i 1.51038 − 1.51038i −0.987040 + 2.83298i −3.58760 + 3.45325i
68.5 −0.472349 + 1.76283i 0.528445 + 1.64947i −1.15241 − 0.665344i −2.02881 − 0.940166i −3.15735 + 0.152433i −0.0579742 − 0.216363i −0.863735 + 0.863735i −2.44149 + 1.74331i 2.61566 − 3.13237i
68.6 −0.402483 + 1.50209i 0.909499 + 1.47405i −0.362218 − 0.209127i 1.25653 + 1.84963i −2.58020 + 0.772867i −0.727252 − 2.71414i −1.73929 + 1.73929i −1.34562 + 2.68129i −3.28404 + 1.14297i
68.7 −0.395659 + 1.47662i −1.07285 + 1.35978i −0.291809 − 0.168476i 1.88815 − 1.19787i −1.58339 − 2.12220i 1.22324 + 4.56520i −1.79769 + 1.79769i −0.697981 − 2.91767i 1.02173 + 3.26203i
68.8 −0.283199 + 1.05691i 0.745396 − 1.56345i 0.695185 + 0.401365i −1.29528 + 1.82270i 1.44134 + 1.23059i 0.767714 + 2.86515i −2.16851 + 2.16851i −1.88877 − 2.33078i −1.55962 − 1.88519i
68.9 −0.200533 + 0.748401i −1.68649 − 0.394648i 1.21216 + 0.699841i 1.44969 − 1.70247i 0.633553 − 1.18303i −0.994250 − 3.71059i −1.86258 + 1.86258i 2.68851 + 1.33114i 0.983416 + 1.42635i
68.10 −0.181708 + 0.678144i 1.16593 − 1.28086i 1.30519 + 0.753551i 2.16466 + 0.560584i 0.656753 + 1.02341i −0.708309 − 2.64344i −1.74105 + 1.74105i −0.281230 − 2.98679i −0.773493 + 1.36609i
68.11 −0.151334 + 0.564785i 1.72850 − 0.110909i 1.43597 + 0.829058i −1.24888 − 1.85481i −0.198940 + 0.993014i 0.448316 + 1.67314i −1.51245 + 1.51245i 2.97540 − 0.383413i 1.23656 − 0.424654i
68.12 −0.115776 + 0.432084i −1.25778 + 1.19080i 1.55876 + 0.899950i −0.848645 + 2.06877i −0.368904 − 0.681331i −0.196382 − 0.732909i −1.20194 + 1.20194i 0.163999 − 2.99551i −0.795628 − 0.606201i
68.13 0.115776 − 0.432084i 0.493867 + 1.66015i 1.55876 + 0.899950i 0.848645 − 2.06877i 0.774502 − 0.0211855i −0.196382 − 0.732909i 1.20194 − 1.20194i −2.51219 + 1.63978i −0.795628 − 0.606201i
68.14 0.151334 − 0.564785i −1.44147 − 0.960299i 1.43597 + 0.829058i 1.24888 + 1.85481i −0.760505 + 0.668794i 0.448316 + 1.67314i 1.51245 − 1.51245i 1.15565 + 2.76848i 1.23656 − 0.424654i
68.15 0.181708 − 0.678144i −0.369290 − 1.69222i 1.30519 + 0.753551i −2.16466 − 0.560584i −1.21468 − 0.0570595i −0.708309 − 2.64344i 1.74105 − 1.74105i −2.72725 + 1.24984i −0.773493 + 1.36609i
68.16 0.200533 − 0.748401i 1.65787 + 0.501470i 1.21216 + 0.699841i −1.44969 + 1.70247i 0.707759 − 1.14019i −0.994250 − 3.71059i 1.86258 − 1.86258i 2.49706 + 1.66274i 0.983416 + 1.42635i
68.17 0.283199 − 1.05691i 0.136195 − 1.72669i 0.695185 + 0.401365i 1.29528 − 1.82270i −1.78639 − 0.632943i 0.767714 + 2.86515i 2.16851 − 2.16851i −2.96290 − 0.470332i −1.55962 − 1.88519i
68.18 0.395659 − 1.47662i 0.249228 + 1.71403i −0.291809 − 0.168476i −1.88815 + 1.19787i 2.62957 + 0.310154i 1.22324 + 4.56520i 1.79769 − 1.79769i −2.87577 + 0.854368i 1.02173 + 3.26203i
68.19 0.402483 − 1.50209i −1.52467 + 0.821812i −0.362218 − 0.209127i −1.25653 − 1.84963i 0.620778 + 2.62095i −0.727252 − 2.71414i 1.73929 − 1.73929i 1.64925 − 2.50599i −3.28404 + 1.14297i
68.20 0.472349 − 1.76283i −1.28238 + 1.16426i −1.15241 − 0.665344i 2.02881 + 0.940166i 1.44666 + 2.81056i −0.0579742 − 0.216363i 0.863735 − 0.863735i 0.289000 − 2.98605i 2.61566 − 3.13237i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.c even 3 1 inner
15.e even 4 1 inner
39.i odd 6 1 inner
65.q odd 12 1 inner
195.bl even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.bl.a ✓ 96
3.b odd 2 1 inner 195.2.bl.a ✓ 96
5.b even 2 1 975.2.bt.m 96
5.c odd 4 1 inner 195.2.bl.a ✓ 96
5.c odd 4 1 975.2.bt.m 96
13.c even 3 1 inner 195.2.bl.a ✓ 96
15.d odd 2 1 975.2.bt.m 96
15.e even 4 1 inner 195.2.bl.a ✓ 96
15.e even 4 1 975.2.bt.m 96
39.i odd 6 1 inner 195.2.bl.a ✓ 96
65.n even 6 1 975.2.bt.m 96
65.q odd 12 1 inner 195.2.bl.a ✓ 96
65.q odd 12 1 975.2.bt.m 96
195.x odd 6 1 975.2.bt.m 96
195.bl even 12 1 inner 195.2.bl.a ✓ 96
195.bl even 12 1 975.2.bt.m 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bl.a ✓ 96 1.a even 1 1 trivial
195.2.bl.a ✓ 96 3.b odd 2 1 inner
195.2.bl.a ✓ 96 5.c odd 4 1 inner
195.2.bl.a ✓ 96 13.c even 3 1 inner
195.2.bl.a ✓ 96 15.e even 4 1 inner
195.2.bl.a ✓ 96 39.i odd 6 1 inner
195.2.bl.a ✓ 96 65.q odd 12 1 inner
195.2.bl.a ✓ 96 195.bl even 12 1 inner
975.2.bt.m 96 5.b even 2 1
975.2.bt.m 96 5.c odd 4 1
975.2.bt.m 96 15.d odd 2 1
975.2.bt.m 96 15.e even 4 1
975.2.bt.m 96 65.n even 6 1
975.2.bt.m 96 65.q odd 12 1
975.2.bt.m 96 195.x odd 6 1
975.2.bt.m 96 195.bl even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(195, [\chi])\).