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Properties

Label 417.2.p.b
Level $417$
Weight $2$
Character orbit 417.p
Analytic conductor $3.330$
Analytic rank $0$
Dimension $1936$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [417,2,Mod(2,417)]
 
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([69, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("417.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 417 = 3 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 417.p (of order \(138\), 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: \(1936\)
Relative dimension: \(44\) over \(\Q(\zeta_{138})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{138}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1936 q - 40 q^{3} - 46 q^{4} - 36 q^{6} - 92 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1936 q - 40 q^{3} - 46 q^{4} - 36 q^{6} - 92 q^{7} - 48 q^{9} - 92 q^{10} - 52 q^{12} - 84 q^{13} - 64 q^{15} - 46 q^{16} - 55 q^{18} - 50 q^{19} - 100 q^{21} - 98 q^{22} + 7 q^{24} - 130 q^{25} - 115 q^{27} - 114 q^{28} - 73 q^{30} + 2 q^{31} - 46 q^{33} - 228 q^{34} + 36 q^{36} - 94 q^{37} - 46 q^{39} - 12 q^{40} + q^{42} - 180 q^{43} - 16 q^{45} - 152 q^{46} - 46 q^{48} - 52 q^{49} - 75 q^{51} - 112 q^{52} + 257 q^{54} - 120 q^{55} - 4 q^{57} + 16 q^{58} - 46 q^{60} - 62 q^{61} - 60 q^{63} - 520 q^{64} - 102 q^{66} - 98 q^{67} - 61 q^{69} - 98 q^{70} + 317 q^{72} - 122 q^{73} - 184 q^{75} - 92 q^{76} - 23 q^{78} - 76 q^{79} - 80 q^{81} - 92 q^{82} + 184 q^{84} - 272 q^{85} - 46 q^{87} + 178 q^{88} - 226 q^{90} - 40 q^{91} + 41 q^{93} - 506 q^{94} - 37 q^{96} - 108 q^{97} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.0632504 + 2.77791i 0.743130 + 1.56453i −5.71484 − 0.260378i −1.30629 − 0.532393i −4.39312 + 1.96539i 1.66042 + 3.59635i 0.705532 − 10.3145i −1.89552 + 2.32530i 1.56156 − 3.59507i
2.2 −0.0603037 + 2.64849i −0.215032 − 1.71865i −5.01293 − 0.228398i 2.61791 + 1.06696i 4.56480 − 0.465870i −1.93740 − 4.19626i 0.545637 − 7.97693i −2.90752 + 0.739131i −2.98370 + 6.86917i
2.3 −0.0575598 + 2.52798i 1.31851 − 1.12317i −4.38943 − 0.199990i −3.26334 − 1.33001i 2.76347 + 3.39782i −0.457408 − 0.990712i 0.413106 − 6.03939i 0.476958 − 2.96184i 3.55008 − 8.17310i
2.4 −0.0564765 + 2.48040i 1.72169 − 0.189158i −4.15126 − 0.189139i 1.58183 + 0.644693i 0.371952 + 4.28116i 0.0490429 + 0.106223i 0.364965 − 5.33559i 2.92844 − 0.651343i −1.68843 + 3.88716i
2.5 −0.0550867 + 2.41936i −1.72998 + 0.0846950i −3.85235 − 0.175520i 3.36633 + 1.37199i −0.109609 − 4.19011i 1.65806 + 3.59124i 0.306567 − 4.48185i 2.98565 − 0.293041i −3.50477 + 8.06880i
2.6 −0.0541593 + 2.37863i −1.39635 + 1.02480i −3.65703 − 0.166621i −0.903003 − 0.368029i −2.36200 − 3.37690i −0.814489 − 1.76412i 0.269661 − 3.94230i 0.899571 − 2.86195i 0.924312 − 2.12798i
2.7 −0.0473817 + 2.08097i 0.196862 − 1.72083i −2.33025 − 0.106170i −2.18429 − 0.890233i 3.57166 + 0.491198i 0.896911 + 1.94264i 0.0472542 − 0.690833i −2.92249 − 0.677530i 1.95604 − 4.50326i
2.8 −0.0465468 + 2.04430i 0.954422 + 1.44536i −2.17906 − 0.0992819i 3.14460 + 1.28162i −2.99918 + 1.88384i −0.766064 − 1.65924i 0.0253020 − 0.369902i −1.17816 + 2.75897i −2.76637 + 6.36883i
2.9 −0.0460949 + 2.02445i 0.894617 + 1.48313i −2.09834 − 0.0956044i −3.18383 − 1.29760i −3.04375 + 1.74274i −1.94674 − 4.21648i 0.0138910 − 0.203079i −1.39932 + 2.65366i 2.77369 − 6.38568i
2.10 −0.0447879 + 1.96705i −1.26618 − 1.18186i −1.86935 − 0.0851710i 0.266091 + 0.108448i 2.38149 − 2.43770i −0.352093 − 0.762606i −0.0172819 + 0.252653i 0.206405 + 2.99289i −0.225241 + 0.518556i
2.11 −0.0382912 + 1.68172i −0.422459 + 1.67974i −0.828783 − 0.0377608i −1.00613 − 0.410060i −2.80867 − 0.774776i 1.76412 + 3.82094i −0.134350 + 1.96413i −2.64306 − 1.41924i 0.728131 − 1.67633i
2.12 −0.0355833 + 1.56279i −1.55344 − 0.766037i −0.443114 − 0.0201891i −2.41354 − 0.983664i 1.25243 − 2.40045i 0.596511 + 1.29200i −0.166033 + 2.42732i 1.82637 + 2.37999i 1.62314 − 3.73684i
2.13 −0.0352091 + 1.54636i 1.41003 − 1.00590i −0.392051 − 0.0178626i 2.25874 + 0.920574i 1.50583 + 2.21582i 1.57526 + 3.41189i −0.169683 + 2.48068i 0.976345 − 2.83668i −1.50306 + 3.46040i
2.14 −0.0315653 + 1.38632i 0.693415 − 1.58719i 0.0770389 + 0.00351003i 1.05635 + 0.430526i 2.17847 + 1.01140i −0.431798 − 0.935242i −0.196559 + 2.87358i −2.03835 − 2.20116i −0.630190 + 1.45084i
2.15 −0.0299198 + 1.31405i −0.485769 + 1.66254i 0.272092 + 0.0123970i 1.46102 + 0.595456i −2.17012 − 0.688068i −0.232474 − 0.503521i −0.203826 + 2.97983i −2.52806 − 1.61522i −0.826173 + 1.90204i
2.16 −0.0274703 + 1.20647i 1.69898 + 0.336873i 0.543110 + 0.0247451i −2.68110 − 1.09271i −0.453099 + 2.04051i 0.997040 + 2.15951i −0.209481 + 3.06251i 2.77303 + 1.14468i 1.39198 − 3.20466i
2.17 −0.0184347 + 0.809638i 1.72108 − 0.194600i 1.34275 + 0.0611783i 0.745941 + 0.304017i 0.125828 + 1.39704i −2.00782 − 4.34878i −0.184818 + 2.70194i 2.92426 − 0.669846i −0.259895 + 0.598338i
2.18 −0.0141557 + 0.621707i −1.69183 + 0.371080i 1.61161 + 0.0734278i −3.40663 − 1.38841i −0.206754 − 1.05708i −0.779481 − 1.68830i −0.153340 + 2.24175i 2.72460 − 1.25561i 0.911409 − 2.09827i
2.19 −0.0115061 + 0.505337i −1.34643 + 1.08955i 1.74269 + 0.0794003i 3.23028 + 1.31654i −0.535097 − 0.692939i −0.392796 − 0.850766i −0.129164 + 1.88831i 0.625767 − 2.93401i −0.702463 + 1.61723i
2.20 −0.0102997 + 0.452354i −0.327040 − 1.70090i 1.79341 + 0.0817110i −1.99614 − 0.813551i 0.772775 − 0.130419i −1.28646 − 2.78639i −0.117189 + 1.71325i −2.78609 + 1.11252i 0.388572 − 0.894584i
See next 80 embeddings (of 1936 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
139.h odd 138 1 inner
417.p even 138 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 417.2.p.b ✓ 1936
3.b odd 2 1 inner 417.2.p.b ✓ 1936
139.h odd 138 1 inner 417.2.p.b ✓ 1936
417.p even 138 1 inner 417.2.p.b ✓ 1936
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
417.2.p.b ✓ 1936 1.a even 1 1 trivial
417.2.p.b ✓ 1936 3.b odd 2 1 inner
417.2.p.b ✓ 1936 139.h odd 138 1 inner
417.2.p.b ✓ 1936 417.p even 138 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1936} - 21 T_{2}^{1934} + 75 T_{2}^{1932} + 3320 T_{2}^{1930} - 54570 T_{2}^{1928} + \cdots + 15\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(417, [\chi])\). Copy content Toggle raw display