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Properties

Label 633.2.v.b
Level $633$
Weight $2$
Character orbit 633.v
Analytic conductor $5.055$
Analytic rank $0$
Dimension $456$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [633,2,Mod(13,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([0, 48]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.v (of order \(35\), degree \(24\), 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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(456\)
Relative dimension: \(19\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 456 q - 3 q^{2} + 19 q^{3} + 9 q^{4} + 2 q^{5} - 3 q^{6} - 10 q^{7} - 9 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 456 q - 3 q^{2} + 19 q^{3} + 9 q^{4} + 2 q^{5} - 3 q^{6} - 10 q^{7} - 9 q^{8} + 19 q^{9} - 18 q^{10} - 10 q^{11} - 87 q^{12} + 14 q^{13} + 24 q^{14} + 48 q^{15} + 35 q^{16} + 33 q^{17} - 6 q^{18} - 62 q^{19} + 17 q^{20} - 60 q^{21} - 16 q^{22} - 28 q^{23} + q^{24} + 19 q^{25} - 33 q^{26} + 19 q^{27} - 134 q^{28} + 11 q^{29} + 51 q^{30} + 14 q^{31} - 50 q^{32} - 14 q^{33} - 19 q^{34} - 21 q^{35} + 23 q^{36} - 27 q^{37} + 17 q^{38} - 21 q^{39} + 120 q^{40} + 16 q^{41} + 6 q^{42} - 67 q^{43} + 11 q^{44} + 2 q^{45} - 53 q^{46} + 29 q^{47} - 2 q^{48} + 173 q^{49} - 2 q^{50} - 12 q^{51} + 29 q^{52} + 16 q^{53} + 4 q^{54} - 28 q^{55} - 107 q^{56} + 52 q^{57} + 29 q^{58} + 97 q^{59} - 18 q^{60} - 16 q^{61} - 115 q^{62} - 55 q^{64} + 59 q^{65} - 18 q^{66} - 33 q^{67} + 20 q^{68} - 78 q^{70} + 32 q^{71} - 6 q^{72} - 104 q^{73} - 72 q^{74} + 13 q^{75} + 71 q^{76} + 119 q^{77} - 31 q^{78} + 34 q^{79} - 184 q^{80} + 19 q^{81} - 45 q^{82} - 49 q^{83} - 8 q^{84} - 91 q^{85} + 45 q^{86} + 4 q^{87} + 125 q^{88} - 4 q^{89} - 19 q^{90} - 52 q^{91} + 233 q^{92} - 51 q^{93} - 149 q^{94} + 59 q^{95} + 46 q^{96} + 2 q^{97} - 272 q^{98} + 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.49211 + 2.26044i −0.0448648 − 0.998993i −2.09717 − 4.90658i −0.107597 + 2.39584i 2.32511 + 1.38919i −0.800573 − 1.21282i 8.89031 + 1.61335i −0.995974 + 0.0896393i −5.25510 − 3.81806i
13.2 −1.32185 + 2.00252i −0.0448648 − 0.998993i −1.47674 − 3.45500i 0.0551171 − 1.22728i 2.05981 + 1.23068i 1.45562 + 2.20517i 4.14894 + 0.752922i −0.995974 + 0.0896393i 2.38479 + 1.73265i
13.3 −1.29351 + 1.95959i −0.0448648 − 0.998993i −1.38076 − 3.23046i 0.179002 − 3.98578i 2.01565 + 1.20430i −2.51265 − 3.80650i 3.49586 + 0.634404i −0.995974 + 0.0896393i 7.57896 + 5.50644i
13.4 −0.952947 + 1.44365i −0.0448648 − 0.998993i −0.389977 − 0.912395i −0.158041 + 3.51907i 1.48495 + 0.887218i −1.24965 − 1.89314i −1.71521 − 0.311265i −0.995974 + 0.0896393i −4.92971 − 3.58164i
13.5 −0.751271 + 1.13813i −0.0448648 − 0.998993i 0.0551249 + 0.128971i 0.0157259 − 0.350165i 1.17069 + 0.699453i −0.462051 − 0.699977i −2.87182 − 0.521158i −0.995974 + 0.0896393i 0.386717 + 0.280967i
13.6 −0.625045 + 0.946903i −0.0448648 − 0.998993i 0.280106 + 0.655341i 0.0854003 − 1.90158i 0.973992 + 0.581933i 2.26270 + 3.42785i −3.02835 − 0.549564i −0.995974 + 0.0896393i 1.74724 + 1.26944i
13.7 −0.332101 + 0.503112i −0.0448648 − 0.998993i 0.643220 + 1.50489i −0.161038 + 3.58578i 0.517505 + 0.309195i 1.64739 + 2.49568i −2.15704 − 0.391445i −0.995974 + 0.0896393i −1.75057 − 1.27186i
13.8 −0.299463 + 0.453666i −0.0448648 − 0.998993i 0.669915 + 1.56734i −0.00982972 + 0.218876i 0.466645 + 0.278807i −2.14892 − 3.25548i −1.98138 − 0.359567i −0.995974 + 0.0896393i −0.0963529 − 0.0700045i
13.9 0.00485002 − 0.00734746i −0.0448648 − 0.998993i 0.786020 + 1.83898i 0.172162 − 3.83349i −0.00755766 − 0.00451549i −0.557643 − 0.844794i 0.0346488 + 0.00628784i −0.995974 + 0.0896393i −0.0273315 − 0.0198575i
13.10 0.190889 − 0.289184i −0.0448648 − 0.998993i 0.738861 + 1.72865i −0.0341054 + 0.759415i −0.297457 − 0.177722i −0.266564 − 0.403828i 1.32281 + 0.240055i −0.995974 + 0.0896393i 0.213100 + 0.154826i
13.11 0.264523 − 0.400735i −0.0448648 − 0.998993i 0.695434 + 1.62705i 0.148575 − 3.30828i −0.412200 − 0.246278i 1.38356 + 2.09600i 1.78088 + 0.323181i −0.995974 + 0.0896393i −1.28644 − 0.934655i
13.12 0.290506 − 0.440097i −0.0448648 − 0.998993i 0.676758 + 1.58335i −0.126125 + 2.80839i −0.452688 − 0.270468i 0.344240 + 0.521501i 1.93115 + 0.350452i −0.995974 + 0.0896393i 1.19932 + 0.871360i
13.13 0.747253 − 1.13204i −0.0448648 − 0.998993i 0.0629232 + 0.147216i −0.0969906 + 2.15966i −1.16443 − 0.695712i −2.37953 − 3.60484i 2.88294 + 0.523176i −0.995974 + 0.0896393i 2.37235 + 1.72361i
13.14 0.863649 − 1.30837i −0.0448648 − 0.998993i −0.179899 − 0.420895i 0.0370537 − 0.825064i −1.34580 − 0.804080i −1.91536 − 2.90165i 2.37898 + 0.431722i −0.995974 + 0.0896393i −1.04749 − 0.761046i
13.15 0.941842 − 1.42683i −0.0448648 − 0.998993i −0.362728 − 0.848643i 0.0118301 − 0.263418i −1.46765 − 0.876880i 2.35499 + 3.56766i 1.81185 + 0.328803i −0.995974 + 0.0896393i −0.364710 − 0.264977i
13.16 1.12078 − 1.69791i −0.0448648 − 0.998993i −0.840703 − 1.96692i −0.166025 + 3.69683i −1.74649 − 1.04348i 1.68372 + 2.55073i −0.278360 − 0.0505148i −0.995974 + 0.0896393i 6.09081 + 4.42523i
13.17 1.21703 − 1.84372i −0.0448648 − 0.998993i −1.13210 − 2.64868i 0.128317 − 2.85720i −1.89647 − 1.13309i 0.374569 + 0.567447i −1.91388 − 0.347318i −0.995974 + 0.0896393i −5.11173 − 3.71389i
13.18 1.42252 − 2.15503i −0.0448648 − 0.998993i −1.83453 − 4.29209i 0.0723156 − 1.61023i −2.21668 − 1.32440i −0.796547 − 1.20672i −6.77783 − 1.22999i −0.995974 + 0.0896393i −3.36722 − 2.44643i
13.19 1.49957 − 2.27176i −0.0448648 − 0.998993i −2.12610 − 4.97427i −0.135477 + 3.01663i −2.33675 − 1.39614i −0.880990 − 1.33464i −9.13194 − 1.65720i −0.995974 + 0.0896393i 6.64990 + 4.83143i
25.1 −1.88247 − 1.96891i 0.983930 + 0.178557i −0.243180 + 5.41482i 4.28194 − 0.777057i −1.50065 − 2.27339i −1.29277 + 1.35213i 7.01630 − 6.12995i 0.936235 + 0.351375i −9.59056 − 6.96795i
See next 80 embeddings (of 456 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.l even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.v.b ✓ 456
211.l even 35 1 inner 633.2.v.b ✓ 456
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.v.b ✓ 456 1.a even 1 1 trivial
633.2.v.b ✓ 456 211.l even 35 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{456} + 3 T_{2}^{455} - 19 T_{2}^{454} - 55 T_{2}^{453} + 103 T_{2}^{452} + 268 T_{2}^{451} + \cdots + 15\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display