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Properties

Label 17.5.e.a
Level $17$
Weight $5$
Character orbit 17.e
Analytic conductor $1.757$
Analytic rank $0$
Dimension $40$
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] = [17,5,Mod(3,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 17.e (of order \(16\), degree \(8\), 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: \(1.75728937243\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} + 376 q^{10} + 112 q^{11} - 776 q^{12} - 416 q^{13} - 776 q^{14} - 704 q^{15} + 256 q^{17} + 2032 q^{18} + 688 q^{19} + 2680 q^{20} + 2032 q^{21} + 760 q^{22} - 176 q^{23} + 1672 q^{24} - 2600 q^{26} - 2600 q^{27} - 7448 q^{28} - 3368 q^{29} - 9800 q^{30} - 3720 q^{31} - 2400 q^{32} + 4280 q^{34} + 4208 q^{35} + 11960 q^{36} + 7416 q^{37} + 16720 q^{38} + 15624 q^{39} + 20280 q^{40} + 2656 q^{41} - 6392 q^{42} - 7512 q^{43} - 31592 q^{44} - 23368 q^{45} - 25752 q^{46} - 10208 q^{47} - 14080 q^{48} - 3112 q^{49} + 3224 q^{51} + 12784 q^{52} + 24424 q^{53} + 51672 q^{54} + 26648 q^{55} + 40432 q^{56} + 10352 q^{57} - 4336 q^{58} - 3176 q^{59} - 38896 q^{60} - 24600 q^{61} - 39248 q^{62} - 55664 q^{63} - 45560 q^{64} - 37928 q^{65} - 29376 q^{66} + 34912 q^{68} + 46592 q^{69} + 59536 q^{70} + 21736 q^{71} + 59824 q^{72} + 28592 q^{73} + 15976 q^{74} + 46168 q^{75} - 9280 q^{76} + 2392 q^{77} - 6048 q^{78} - 15912 q^{79} - 47640 q^{80} - 48696 q^{81} - 58368 q^{82} - 27296 q^{83} + 18872 q^{85} + 74336 q^{86} + 38536 q^{87} + 55608 q^{88} - 1232 q^{89} - 24136 q^{90} - 7800 q^{91} - 27032 q^{92} - 32232 q^{93} - 37096 q^{94} - 49640 q^{95} - 79696 q^{96} - 12392 q^{97} - 76304 q^{98} + 6056 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −7.12052 + 2.94942i −0.829686 − 0.165035i 30.6890 − 30.6890i −14.8705 − 22.2553i 6.39455 − 1.27196i 16.8953 − 25.2857i −80.8164 + 195.108i −74.1731 − 30.7235i 171.526 + 114.610i
3.2 −2.92749 + 1.21261i 8.22545 + 1.63614i −4.21393 + 4.21393i 21.2448 + 31.7951i −26.0639 + 5.18443i 5.22138 − 7.81435i 26.6281 − 64.2859i −9.85322 − 4.08134i −100.749 − 67.3181i
3.3 −1.63167 + 0.675861i −11.4219 − 2.27196i −9.10814 + 9.10814i −6.20205 − 9.28203i 20.1724 − 4.01253i −12.6661 + 18.9562i 19.5194 − 47.1241i 50.4644 + 20.9030i 16.3931 + 10.9535i
3.4 3.43536 − 1.42297i 6.42184 + 1.27738i −1.53684 + 1.53684i −8.81924 − 13.1989i 23.8790 − 4.74983i 5.40438 − 8.08823i −25.8603 + 62.4323i −35.2259 − 14.5911i −49.0790 − 32.7936i
3.5 6.53721 − 2.70780i −10.8435 − 2.15691i 24.0892 − 24.0892i 16.7298 + 25.0379i −76.7267 + 15.2619i −31.8191 + 47.6207i 48.9226 − 118.110i 38.0949 + 15.7794i 177.164 + 118.377i
5.1 −2.55538 − 6.16923i −1.90204 − 2.84660i −20.2157 + 20.2157i −0.0751892 + 0.378002i −12.7009 + 19.0082i −11.8164 − 59.4051i 77.6662 + 32.1704i 26.5120 − 64.0055i 2.52411 − 0.502078i
5.2 −1.10044 − 2.65671i 8.44506 + 12.6389i 5.46660 − 5.46660i 3.31718 − 16.6766i 24.2846 − 36.3445i 6.66998 + 33.5323i −63.0461 − 26.1146i −57.4261 + 138.639i −47.9552 + 9.53888i
5.3 −0.210299 − 0.507707i −5.96248 − 8.92348i 11.1002 − 11.1002i 1.57331 − 7.90954i −3.27661 + 4.90379i 7.85546 + 39.4921i −16.0933 − 6.66606i −13.0800 + 31.5779i −4.34659 + 0.864591i
5.4 1.12676 + 2.72024i 2.50076 + 3.74265i 5.18357 − 5.18357i −7.34972 + 36.9495i −7.36316 + 11.0198i −15.5535 − 78.1927i 63.4651 + 26.2881i 23.2437 − 56.1153i −108.793 + 21.6403i
5.5 2.44646 + 5.90629i 0.660541 + 0.988570i −17.5853 + 17.5853i 5.91128 − 29.7180i −4.22279 + 6.31985i 9.51676 + 47.8440i −52.3853 − 21.6987i 30.4564 − 73.5283i 189.985 − 37.7903i
6.1 −7.12052 − 2.94942i −0.829686 + 0.165035i 30.6890 + 30.6890i −14.8705 + 22.2553i 6.39455 + 1.27196i 16.8953 + 25.2857i −80.8164 − 195.108i −74.1731 + 30.7235i 171.526 − 114.610i
6.2 −2.92749 − 1.21261i 8.22545 − 1.63614i −4.21393 − 4.21393i 21.2448 − 31.7951i −26.0639 − 5.18443i 5.22138 + 7.81435i 26.6281 + 64.2859i −9.85322 + 4.08134i −100.749 + 67.3181i
6.3 −1.63167 − 0.675861i −11.4219 + 2.27196i −9.10814 − 9.10814i −6.20205 + 9.28203i 20.1724 + 4.01253i −12.6661 − 18.9562i 19.5194 + 47.1241i 50.4644 − 20.9030i 16.3931 − 10.9535i
6.4 3.43536 + 1.42297i 6.42184 − 1.27738i −1.53684 − 1.53684i −8.81924 + 13.1989i 23.8790 + 4.74983i 5.40438 + 8.08823i −25.8603 − 62.4323i −35.2259 + 14.5911i −49.0790 + 32.7936i
6.5 6.53721 + 2.70780i −10.8435 + 2.15691i 24.0892 + 24.0892i 16.7298 − 25.0379i −76.7267 − 15.2619i −31.8191 − 47.6207i 48.9226 + 118.110i 38.0949 − 15.7794i 177.164 − 118.377i
7.1 −2.55538 + 6.16923i −1.90204 + 2.84660i −20.2157 − 20.2157i −0.0751892 − 0.378002i −12.7009 − 19.0082i −11.8164 + 59.4051i 77.6662 − 32.1704i 26.5120 + 64.0055i 2.52411 + 0.502078i
7.2 −1.10044 + 2.65671i 8.44506 − 12.6389i 5.46660 + 5.46660i 3.31718 + 16.6766i 24.2846 + 36.3445i 6.66998 − 33.5323i −63.0461 + 26.1146i −57.4261 − 138.639i −47.9552 − 9.53888i
7.3 −0.210299 + 0.507707i −5.96248 + 8.92348i 11.1002 + 11.1002i 1.57331 + 7.90954i −3.27661 − 4.90379i 7.85546 − 39.4921i −16.0933 + 6.66606i −13.0800 − 31.5779i −4.34659 − 0.864591i
7.4 1.12676 − 2.72024i 2.50076 − 3.74265i 5.18357 + 5.18357i −7.34972 − 36.9495i −7.36316 − 11.0198i −15.5535 + 78.1927i 63.4651 − 26.2881i 23.2437 + 56.1153i −108.793 − 21.6403i
7.5 2.44646 − 5.90629i 0.660541 − 0.988570i −17.5853 − 17.5853i 5.91128 + 29.7180i −4.22279 − 6.31985i 9.51676 − 47.8440i −52.3853 + 21.6987i 30.4564 + 73.5283i 189.985 + 37.7903i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.5.e.a ✓ 40
3.b odd 2 1 153.5.p.a 40
17.e odd 16 1 inner 17.5.e.a ✓ 40
51.i even 16 1 153.5.p.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.5.e.a ✓ 40 1.a even 1 1 trivial
17.5.e.a ✓ 40 17.e odd 16 1 inner
153.5.p.a 40 3.b odd 2 1
153.5.p.a 40 51.i even 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(17, [\chi])\).