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Properties

Label 417.2.m.b
Level $417$
Weight $2$
Character orbit 417.m
Analytic conductor $3.330$
Analytic rank $0$
Dimension $528$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [417,2,Mod(4,417)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(417, base_ring=CyclotomicField(138))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("417.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 417 = 3 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 417.m (of order \(69\), degree \(44\), 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: \(3.32976176429\)
Analytic rank: \(0\)
Dimension: \(528\)
Relative dimension: \(12\) over \(\Q(\zeta_{69})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{69}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 528 q - q^{2} + 12 q^{3} + 9 q^{4} + q^{5} + 2 q^{6} - 2 q^{7} - 40 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 528 q - q^{2} + 12 q^{3} + 9 q^{4} + q^{5} + 2 q^{6} - 2 q^{7} - 40 q^{8} + 12 q^{9} + 6 q^{10} - 49 q^{11} + 9 q^{12} + 8 q^{13} - 32 q^{14} + q^{15} + 11 q^{16} - 44 q^{17} - q^{18} + 10 q^{19} - 7 q^{20} - 2 q^{21} + 8 q^{22} + 14 q^{23} - 3 q^{24} + 7 q^{25} - 123 q^{26} - 24 q^{27} + 11 q^{28} + 2 q^{29} - 95 q^{30} - 37 q^{31} + 259 q^{32} + 6 q^{33} - 90 q^{34} + 2 q^{35} - 18 q^{36} + 2 q^{37} - 110 q^{38} - 16 q^{39} + 27 q^{40} - 36 q^{41} - 7 q^{42} - 91 q^{43} - 30 q^{44} - 2 q^{45} - 44 q^{46} + 186 q^{47} - 22 q^{48} + 10 q^{49} + 21 q^{50} - 21 q^{51} - 256 q^{52} - 27 q^{53} - q^{54} + 130 q^{55} + 15 q^{56} - 66 q^{57} - 189 q^{58} - 124 q^{59} + 14 q^{60} + 15 q^{61} + 16 q^{62} + 4 q^{63} - 190 q^{64} + 14 q^{65} - 38 q^{66} - 152 q^{67} - 54 q^{68} - 99 q^{69} + 193 q^{70} + 56 q^{71} - 3 q^{72} + 3 q^{73} - 248 q^{74} - 37 q^{75} - 12 q^{76} - 54 q^{77} + 107 q^{78} - 66 q^{79} + 48 q^{80} + 12 q^{81} - 266 q^{82} - 90 q^{83} - 22 q^{84} - 70 q^{85} + 395 q^{86} - 73 q^{87} - 127 q^{88} - 140 q^{89} - 49 q^{90} - 46 q^{91} - 154 q^{92} + 9 q^{93} + 190 q^{94} + 50 q^{95} - 17 q^{96} - 91 q^{97} - 98 q^{98} - 3 q^{99}+O(q^{100}) \) 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 −2.53322 − 0.115418i 0.291646 − 0.956526i 4.41219 + 0.402891i −1.99778 − 1.95281i −0.849205 + 2.38943i −0.188719 + 0.221471i −6.10610 − 0.839265i −0.829885 − 0.557934i 4.83544 + 5.17749i
4.2 −2.26647 − 0.103265i 0.291646 − 0.956526i 3.13453 + 0.286223i 1.97279 + 1.92838i −0.759783 + 2.13782i 1.51815 − 1.78162i −2.57938 − 0.354527i −0.829885 − 0.557934i −4.27214 − 4.57435i
4.3 −1.99577 − 0.0909309i 0.291646 − 0.956526i 1.98312 + 0.181085i 1.68590 + 1.64795i −0.669036 + 1.88249i −3.17368 + 3.72447i 0.0170824 + 0.00234792i −0.829885 − 0.557934i −3.21482 − 3.44223i
4.4 −1.22247 − 0.0556981i 0.291646 − 0.956526i −0.500378 − 0.0456911i −2.91058 − 2.84506i −0.409806 + 1.15308i 0.355241 − 0.416894i 3.03384 + 0.416991i −0.829885 − 0.557934i 3.39964 + 3.64012i
4.5 −1.13695 − 0.0518017i 0.291646 − 0.956526i −0.701731 − 0.0640773i −0.640657 − 0.626236i −0.381138 + 1.07242i 1.88605 − 2.21338i 3.04958 + 0.419156i −0.829885 − 0.557934i 0.695958 + 0.745189i
4.6 −0.735921 − 0.0335299i 0.291646 − 0.956526i −1.45126 − 0.132519i −0.161505 − 0.157869i −0.246701 + 0.694149i −2.34590 + 2.75303i 2.52321 + 0.346808i −0.829885 − 0.557934i 0.113561 + 0.121595i
4.7 0.312936 + 0.0142580i 0.291646 − 0.956526i −1.89399 − 0.172946i −0.491960 − 0.480886i 0.104905 − 0.295174i −0.143353 + 0.168232i −1.21092 − 0.166437i −0.829885 − 0.557934i −0.147096 − 0.157501i
4.8 0.391213 + 0.0178244i 0.291646 − 0.956526i −1.83898 − 0.167923i 1.73513 + 1.69607i 0.131145 − 0.369007i 1.83313 − 2.15127i −1.49238 − 0.205123i −0.829885 − 0.557934i 0.648573 + 0.694452i
4.9 1.46370 + 0.0666888i 0.291646 − 0.956526i 0.146253 + 0.0133548i −3.16040 − 3.08926i 0.490671 − 1.38062i −1.91987 + 2.25306i −2.68996 − 0.369727i −0.829885 − 0.557934i −4.41985 − 4.73250i
4.10 2.10379 + 0.0958527i 0.291646 − 0.956526i 2.42505 + 0.221439i −0.661774 − 0.646877i 0.705249 − 1.98438i 1.40576 − 1.64973i 0.907852 + 0.124781i −0.829885 − 0.557934i −1.33023 − 1.42433i
4.11 2.12753 + 0.0969340i 0.291646 − 0.956526i 2.52526 + 0.230590i 2.79377 + 2.73088i 0.713205 − 2.00677i 0.682261 − 0.800669i 1.13041 + 0.155372i −0.829885 − 0.557934i 5.67910 + 6.08083i
4.12 2.73759 + 0.124730i 0.291646 − 0.956526i 5.48714 + 0.501048i −0.470257 − 0.459671i 0.917715 − 2.58220i −1.51005 + 1.77212i 9.52923 + 1.30976i −0.829885 − 0.557934i −1.23004 − 1.31705i
7.1 −1.04123 − 2.25523i 0.898128 − 0.439735i −2.70474 + 3.17415i 0.806168 + 0.441285i −1.92686 − 1.56762i 1.96197 + 0.748010i 5.19096 + 1.45444i 0.613267 − 0.789876i 0.155791 − 2.27758i
7.2 −0.778569 − 1.68632i 0.898128 − 0.439735i −0.940347 + 1.10355i −3.10541 − 1.69985i −1.44079 − 1.17217i 3.31265 + 1.26296i −0.983944 − 0.275688i 0.613267 − 0.789876i −0.448728 + 6.56017i
7.3 −0.701393 − 1.51916i 0.898128 − 0.439735i −0.518744 + 0.608773i −1.84724 − 1.01115i −1.29797 − 1.05598i −3.03486 − 1.15705i −1.93376 − 0.541814i 0.613267 − 0.789876i −0.240465 + 3.51548i
7.4 −0.474674 − 1.02811i 0.898128 − 0.439735i 0.465473 − 0.546257i 1.47169 + 0.805580i −0.878413 − 0.714642i −2.47752 − 0.944566i −2.96337 − 0.830297i 0.613267 − 0.789876i 0.129652 − 1.89544i
7.5 −0.321964 − 0.697350i 0.898128 − 0.439735i 0.914528 − 1.07325i 2.96289 + 1.62184i −0.595814 − 0.484731i 0.727298 + 0.277286i −2.52208 − 0.706654i 0.613267 − 0.789876i 0.177048 − 2.58835i
7.6 0.0240257 + 0.0520379i 0.898128 − 0.439735i 1.29503 − 1.51979i −0.468615 − 0.256513i 0.0444610 + 0.0361717i 4.78923 + 1.82591i 0.220583 + 0.0618043i 0.613267 − 0.789876i 0.00208958 − 0.0305486i
7.7 0.0876704 + 0.189888i 0.898128 − 0.439735i 1.26879 − 1.48899i 0.137497 + 0.0752637i 0.162239 + 0.131992i −0.366488 − 0.139725i 0.796764 + 0.223243i 0.613267 − 0.789876i −0.00223724 + 0.0327073i
7.8 0.205256 + 0.444568i 0.898128 − 0.439735i 1.14165 − 1.33979i −3.24271 − 1.77501i 0.379838 + 0.309021i −4.52575 − 1.72546i 1.77297 + 0.496763i 0.613267 − 0.789876i 0.123530 − 1.80594i
See next 80 embeddings (of 528 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
139.g even 69 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 417.2.m.b ✓ 528
139.g even 69 1 inner 417.2.m.b ✓ 528
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
417.2.m.b ✓ 528 1.a even 1 1 trivial
417.2.m.b ✓ 528 139.g even 69 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{528} + T_{2}^{527} - 16 T_{2}^{526} + 25 T_{2}^{525} + 129 T_{2}^{524} - 832 T_{2}^{523} + \cdots + 67\!\cdots\!84 \) acting on \(S_{2}^{\mathrm{new}}(417, [\chi])\). Copy content Toggle raw display