Properties

Label 17.16.c.a
Level $17$
Weight $16$
Character orbit 17.c
Analytic conductor $24.258$
Analytic rank $0$
Dimension $44$
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] = [17,16,Mod(4,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.4");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 17.c (of order \(4\), degree \(2\), 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: \(24.2578958670\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 5256 q^{3} - 720900 q^{4} - 252126 q^{5} - 1017326 q^{6} + 1854722 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 5256 q^{3} - 720900 q^{4} - 252126 q^{5} - 1017326 q^{6} + 1854722 q^{7} + 8409498 q^{10} - 9729300 q^{11} - 82056486 q^{12} + 146863304 q^{13} - 1773752796 q^{14} + 16082152452 q^{16} + 1961077578 q^{17} - 7034899972 q^{18} + 21230427174 q^{20} - 47057263940 q^{21} - 3442343910 q^{22} - 15842736534 q^{23} + 94553484290 q^{24} - 294640558020 q^{27} - 17774164556 q^{28} + 102726169530 q^{29} - 643189839240 q^{30} - 385043642942 q^{31} + 2378676720192 q^{33} + 158249007174 q^{34} + 408521183892 q^{35} + 1375134180422 q^{37} + 1543745804832 q^{38} + 603440813444 q^{39} - 8303553806742 q^{40} + 1443758220300 q^{41} - 4657636435362 q^{44} + 6106558559866 q^{45} + 9604057332832 q^{46} - 21302607302640 q^{47} + 25515630744334 q^{48} - 18255254395212 q^{50} + 19158460873088 q^{51} - 48159060495244 q^{52} - 78998282739512 q^{54} + 13393057779764 q^{55} + 137860752160332 q^{56} + 31257866131908 q^{57} - 69211880377042 q^{58} + 38643830633662 q^{61} + 238390785941076 q^{62} - 71567394446902 q^{63} - 498262566925124 q^{64} + 178535375770548 q^{65} + 57983772499388 q^{67} + 261849713539554 q^{68} - 115652283503244 q^{69} - 44377197355566 q^{71} + 93677094320796 q^{72} + 405163878377696 q^{73} - 405068893451898 q^{74} + 225803653508132 q^{75} - 802546741257484 q^{78} - 155549526248110 q^{79} - 124920621818622 q^{80} - 10\!\cdots\!28 q^{81}+ \cdots + 173739293641872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 348.759i 5015.99 5015.99i −88865.1 −20803.9 + 20803.9i −1.74937e6 1.74937e6i −1.51898e6 1.51898e6i 1.95644e7i 3.59714e7i 7.25557e6 + 7.25557e6i
4.2 338.865i −2552.04 + 2552.04i −82061.8 −182025. + 182025.i 864799. + 864799.i −103726. 103726.i 1.67040e7i 1.32307e6i 6.16820e7 + 6.16820e7i
4.3 302.124i −334.184 + 334.184i −58511.1 109767. 109767.i 100965. + 100965.i 1.54065e6 + 1.54065e6i 7.77762e6i 1.41255e7i −3.31634e7 3.31634e7i
4.4 250.655i −4190.42 + 4190.42i −30059.9 114890. 114890.i 1.05035e6 + 1.05035e6i −2.34763e6 2.34763e6i 678803.i 2.07703e7i −2.87976e7 2.87976e7i
4.5 235.953i 1501.32 1501.32i −22905.8 44470.0 44470.0i −354242. 354242.i −175577. 175577.i 2.32702e6i 9.84096e6i −1.04928e7 1.04928e7i
4.6 191.501i 1757.52 1757.52i −3904.64 −212618. + 212618.i −336567. 336567.i −620986. 620986.i 5.52736e6i 8.17116e6i 4.07165e7 + 4.07165e7i
4.7 155.544i −3969.60 + 3969.60i 8574.15 −108667. + 108667.i 617447. + 617447.i 2.10978e6 + 2.10978e6i 6.43051e6i 1.71666e7i 1.69025e7 + 1.69025e7i
4.8 133.029i 4742.62 4742.62i 15071.4 −14498.4 + 14498.4i −630905. 630905.i 2.54959e6 + 2.54959e6i 6.36401e6i 3.06360e7i 1.92870e6 + 1.92870e6i
4.9 80.4188i 3327.44 3327.44i 26300.8 211555. 211555.i −267589. 267589.i −2.55424e6 2.55424e6i 4.75024e6i 7.79484e6i −1.70130e7 1.70130e7i
4.10 66.3760i −1952.69 + 1952.69i 28362.2 −33212.7 + 33212.7i 129612. + 129612.i −1.64518e6 1.64518e6i 4.05758e6i 6.72291e6i 2.20452e6 + 2.20452e6i
4.11 53.4016i −2453.73 + 2453.73i 29916.3 225279. 225279.i 131033. + 131033.i 2.15818e6 + 2.15818e6i 3.34744e6i 2.30736e6i −1.20303e7 1.20303e7i
4.12 31.6064i −152.613 + 152.613i 31769.0 −69379.4 + 69379.4i −4823.54 4823.54i −162757. 162757.i 2.03978e6i 1.43023e7i −2.19284e6 2.19284e6i
4.13 39.3257i 2394.27 2394.27i 31221.5 13542.0 13542.0i 94156.4 + 94156.4i 1.69479e6 + 1.69479e6i 2.51643e6i 2.88386e6i 532547. + 532547.i
4.14 119.498i 4421.91 4421.91i 18488.2 −204259. + 204259.i 528410. + 528410.i −2.13755e6 2.13755e6i 6.12502e6i 2.47577e7i −2.44085e7 2.44085e7i
4.15 138.706i −4846.03 + 4846.03i 13528.7 −100507. + 100507.i −672173. 672173.i −478607. 478607.i 6.42162e6i 3.26192e7i −1.39409e7 1.39409e7i
4.16 167.241i −2780.51 + 2780.51i 4798.53 136750. 136750.i −465015. 465015.i −526443. 526443.i 6.28266e6i 1.11360e6i 2.28702e7 + 2.28702e7i
4.17 184.257i 3098.34 3098.34i −1182.81 93660.4 93660.4i 570892. + 570892.i −101141. 101141.i 5.81981e6i 4.85050e6i 1.72576e7 + 1.72576e7i
4.18 224.704i −688.326 + 688.326i −17723.8 −200306. + 200306.i −154669. 154669.i 2.77180e6 + 2.77180e6i 3.38049e6i 1.34013e7i −4.50094e7 4.50094e7i
4.19 274.663i 931.417 931.417i −42671.9 129422. 129422.i 255826. + 255826.i 151712. + 151712.i 2.72024e6i 1.26138e7i 3.55476e7 + 3.55476e7i
4.20 300.568i −890.573 + 890.573i −57572.8 −102815. + 102815.i −267677. 267677.i −2.55244e6 2.55244e6i 7.45553e6i 1.27627e7i −3.09027e7 3.09027e7i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.16.c.a 44
17.c even 4 1 inner 17.16.c.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.16.c.a 44 1.a even 1 1 trivial
17.16.c.a 44 17.c even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(17, [\chi])\).