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Properties

Label 633.2.v.a
Level $633$
Weight $2$
Character orbit 633.v
Analytic conductor $5.055$
Analytic rank $0$
Dimension $408$
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: \(408\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 408 q - 7 q^{2} - 17 q^{3} + 23 q^{4} - 2 q^{5} - 7 q^{6} + 8 q^{7} - q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 408 q - 7 q^{2} - 17 q^{3} + 23 q^{4} - 2 q^{5} - 7 q^{6} + 8 q^{7} - q^{8} + 17 q^{9} - 22 q^{10} - 14 q^{11} + 71 q^{12} - 6 q^{13} + 48 q^{15} + 69 q^{16} - 43 q^{17} - 16 q^{19} - 5 q^{20} - 12 q^{21} + 8 q^{22} + q^{24} + q^{25} + 49 q^{26} - 17 q^{27} + 20 q^{28} + 41 q^{29} - 27 q^{30} - 14 q^{31} + 38 q^{32} - 2 q^{33} + 57 q^{34} - 11 q^{35} + 9 q^{36} + 3 q^{37} - 77 q^{38} - q^{39} - 184 q^{40} - 12 q^{41} + 26 q^{42} + 51 q^{43} - 7 q^{44} - 2 q^{45} - 17 q^{46} + 97 q^{47} - 8 q^{48} - 203 q^{49} - 98 q^{50} + 8 q^{51} - 31 q^{52} - 34 q^{53} - 16 q^{55} + 41 q^{56} + 72 q^{57} + 45 q^{58} - 131 q^{59} - 16 q^{61} - 31 q^{62} - 12 q^{63} + 7 q^{64} - 145 q^{65} + 6 q^{66} - 95 q^{67} - 2 q^{68} - 28 q^{69} + 18 q^{70} + 74 q^{71} + 20 q^{72} + 50 q^{73} + 56 q^{74} - 15 q^{75} + 49 q^{76} + 99 q^{77} - 19 q^{78} - 6 q^{79} + 54 q^{80} + 17 q^{81} - 93 q^{82} + 11 q^{83} - 6 q^{84} - 47 q^{85} + 69 q^{86} - 22 q^{87} - 227 q^{88} + 40 q^{89} + 13 q^{90} + 4 q^{91} + 29 q^{92} + 29 q^{93} - 5 q^{94} - 209 q^{95} - 76 q^{96} - 60 q^{97} + 142 q^{98} + 42 q^{99}+O(q^{100}) \) 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 −1.53116 + 2.31960i 0.0448648 + 0.998993i −2.25007 − 5.26429i 0.126464 − 2.81594i −2.38596 − 1.42555i 0.950109 + 1.43935i 10.1868 + 1.84864i −0.995974 + 0.0896393i 6.33824 + 4.60500i
13.2 −1.31778 + 1.99635i 0.0448648 + 0.998993i −1.46281 − 3.42242i −0.0465609 + 1.03676i −2.05346 − 1.22688i −2.26543 − 3.43198i 4.05277 + 0.735469i −0.995974 + 0.0896393i −2.00837 − 1.45917i
13.3 −1.26803 + 1.92098i 0.0448648 + 0.998993i −1.29623 − 3.03267i 0.0313821 − 0.698776i −1.97594 − 1.18057i 1.02163 + 1.54770i 2.93984 + 0.533501i −0.995974 + 0.0896393i 1.30254 + 0.946354i
13.4 −0.937310 + 1.41996i 0.0448648 + 0.998993i −0.351697 − 0.822837i −0.0730682 + 1.62699i −1.46059 − 0.872660i 1.57177 + 2.38114i −1.85012 − 0.335747i −0.995974 + 0.0896393i −2.24178 − 1.62875i
13.5 −0.733009 + 1.11046i 0.0448648 + 0.998993i 0.0902273 + 0.211097i 0.0726321 − 1.61728i −1.14223 − 0.682451i −2.18117 − 3.30433i −2.91894 − 0.529709i −0.995974 + 0.0896393i 1.74269 + 1.26614i
13.6 −0.639979 + 0.969527i 0.0448648 + 0.998993i 0.255640 + 0.598100i 0.155637 − 3.46552i −0.997264 − 0.595837i 0.000753323 0.00114124i −3.02955 − 0.549782i −0.995974 + 0.0896393i 3.26031 + 2.36876i
13.7 −0.529774 + 0.802574i 0.0448648 + 0.998993i 0.422586 + 0.988689i −0.126452 + 2.81568i −0.825534 − 0.493234i 1.02668 + 1.55535i −2.90978 − 0.528047i −0.995974 + 0.0896393i −2.19280 − 1.59316i
13.8 −0.153584 + 0.232670i 0.0448648 + 0.998993i 0.755503 + 1.76759i −0.111187 + 2.47576i −0.239327 − 0.142991i −2.11169 − 3.19907i −1.07592 − 0.195250i −0.995974 + 0.0896393i −0.558961 − 0.406109i
13.9 0.0547055 − 0.0828753i 0.0448648 + 0.998993i 0.782174 + 1.82999i 0.115599 − 2.57401i 0.0852462 + 0.0509322i 2.53106 + 3.83439i 0.389864 + 0.0707498i −0.995974 + 0.0896393i −0.206998 − 0.150393i
13.10 0.249950 − 0.378658i 0.0448648 + 0.998993i 0.705143 + 1.64976i −0.0752217 + 1.67494i 0.389491 + 0.232710i −0.155156 − 0.235051i 1.69379 + 0.307378i −0.995974 + 0.0896393i 0.615428 + 0.447135i
13.11 0.554574 − 0.840144i 0.0448648 + 0.998993i 0.387761 + 0.907212i 0.0809254 − 1.80194i 0.864179 + 0.516323i −2.15485 − 3.26446i 2.95822 + 0.536839i −0.995974 + 0.0896393i −1.46901 − 1.06730i
13.12 0.796029 − 1.20593i 0.0448648 + 0.998993i −0.0345602 − 0.0808576i −0.0543133 + 1.20938i 1.24043 + 0.741123i 2.06228 + 3.12422i 2.71847 + 0.493330i −0.995974 + 0.0896393i 1.41520 + 1.02820i
13.13 0.901841 − 1.36623i 0.0448648 + 0.998993i −0.267218 − 0.625188i 0.0571785 − 1.27318i 1.40532 + 0.839637i 0.370735 + 0.561639i 2.12633 + 0.385871i −0.995974 + 0.0896393i −1.68789 − 1.22632i
13.14 0.989967 − 1.49974i 0.0448648 + 0.998993i −0.483123 − 1.13032i −0.153032 + 3.40751i 1.54264 + 0.921685i −1.31500 − 1.99213i 1.36280 + 0.247312i −0.995974 + 0.0896393i 4.95887 + 3.60283i
13.15 1.33322 − 2.01974i 0.0448648 + 0.998993i −1.51582 − 3.54645i 0.186438 − 4.15136i 2.07752 + 1.24126i 0.887842 + 1.34502i −4.42143 − 0.802372i −0.995974 + 0.0896393i −8.13609 − 5.91122i
13.16 1.40822 − 2.13336i 0.0448648 + 0.998993i −1.78208 − 4.16939i −0.107501 + 2.39370i 2.19439 + 1.31109i 1.50738 + 2.28359i −6.37407 − 1.15672i −0.995974 + 0.0896393i 4.95524 + 3.60019i
13.17 1.45118 − 2.19844i 0.0448648 + 0.998993i −1.94117 − 4.54159i 0.0108105 − 0.240714i 2.26133 + 1.35108i −2.16780 − 3.28408i −7.61762 − 1.38240i −0.995974 + 0.0896393i −0.513507 − 0.373085i
25.1 −1.78070 − 1.86246i −0.983930 − 0.178557i −0.208158 + 4.63499i 0.220954 − 0.0400972i 1.41953 + 2.15049i 0.384894 − 0.402568i 5.12221 − 4.47514i 0.936235 + 0.351375i −0.468132 − 0.340118i
25.2 −1.49821 − 1.56700i −0.983930 − 0.178557i −0.121141 + 2.69741i −1.22630 + 0.222540i 1.19433 + 1.80934i 1.21464 − 1.27041i 1.14306 − 0.998663i 0.936235 + 0.351375i 2.18597 + 1.58820i
25.3 −1.32138 − 1.38205i −0.983930 − 0.178557i −0.0743004 + 1.65443i −3.31195 + 0.601030i 1.05337 + 1.59578i −2.86794 + 2.99963i −0.495204 + 0.432646i 0.936235 + 0.351375i 5.20698 + 3.78309i
See next 80 embeddings (of 408 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.17
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.a ✓ 408
211.l even 35 1 inner 633.2.v.a ✓ 408
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.v.a ✓ 408 1.a even 1 1 trivial
633.2.v.a ✓ 408 211.l even 35 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{408} + 7 T_{2}^{407} - 4 T_{2}^{406} - 142 T_{2}^{405} - 307 T_{2}^{404} + 512 T_{2}^{403} + \cdots + 69418041399001 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display