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Properties

Label 17.7.e.a
Level $17$
Weight $7$
Character orbit 17.e
Analytic conductor $3.911$
Analytic rank $0$
Dimension $64$
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] = [17,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.3");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(3.91091942154\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} + 3736 q^{10} - 4488 q^{11} - 7688 q^{12} + 2872 q^{13} + 12152 q^{14} + 14776 q^{15} - 6408 q^{17} - 44816 q^{18} - 11528 q^{19} - 19720 q^{20} + 18232 q^{21} + 40312 q^{22} + 29112 q^{23} + 135304 q^{24} - 92840 q^{25} - 214536 q^{26} - 58328 q^{27} + 42472 q^{28} + 73832 q^{29} + 326584 q^{30} + 153784 q^{31} + 118304 q^{32} - 213320 q^{34} - 200400 q^{35} - 513736 q^{36} - 255368 q^{37} + 73872 q^{38} + 11064 q^{39} - 97736 q^{40} - 145768 q^{41} + 234376 q^{42} + 342952 q^{43} + 593048 q^{44} + 138152 q^{45} - 324632 q^{46} + 601912 q^{47} + 991424 q^{48} + 224248 q^{49} - 243208 q^{51} - 933904 q^{52} + 255280 q^{53} - 829992 q^{54} - 1389704 q^{55} - 2611408 q^{56} - 1982968 q^{57} - 909104 q^{58} + 463608 q^{59} + 3097040 q^{60} + 1713112 q^{61} + 3651696 q^{62} + 3783736 q^{63} + 1245576 q^{64} + 246112 q^{65} - 78336 q^{66} + 680096 q^{68} - 1859536 q^{69} - 4027696 q^{70} - 2283784 q^{71} - 8433296 q^{72} - 4507328 q^{73} - 2595128 q^{74} - 1608968 q^{75} + 1378048 q^{76} + 2587736 q^{77} + 8752032 q^{78} + 4152952 q^{79} + 9312744 q^{80} + 6663624 q^{81} + 4368928 q^{82} + 2470824 q^{83} - 4498640 q^{85} - 6837408 q^{86} - 7591592 q^{87} - 13258952 q^{88} - 5755208 q^{89} - 11377576 q^{90} - 4157576 q^{91} - 1636888 q^{92} + 2508024 q^{93} + 7744792 q^{94} + 8552600 q^{95} + 19873136 q^{96} + 6505336 q^{97} + 15189328 q^{98} + 14554736 q^{99}+O(q^{100}) \) 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 −11.7996 + 4.88755i 30.5590 + 6.07857i 70.0872 − 70.0872i 37.5749 + 56.2348i −390.293 + 77.6341i −304.501 + 455.719i −171.642 + 414.381i 223.396 + 92.5338i −718.218 − 479.898i
3.2 −10.8522 + 4.49513i −45.1411 − 8.97912i 52.3092 − 52.3092i 32.3776 + 48.4565i 530.242 − 105.472i −111.661 + 167.113i −44.8454 + 108.266i 1283.58 + 531.678i −569.186 − 380.318i
3.3 −7.60754 + 3.15115i 2.30914 + 0.459317i 2.69007 − 2.69007i −37.6489 − 56.3456i −19.0143 + 3.78217i 347.193 − 519.611i 189.685 − 457.941i −668.387 − 276.855i 463.969 + 310.014i
3.4 −0.477140 + 0.197638i −4.24307 − 0.844000i −45.0662 + 45.0662i −78.3099 − 117.199i 2.19135 − 0.435886i −283.582 + 424.411i 25.2449 − 60.9467i −656.217 − 271.814i 60.5278 + 40.4434i
3.5 2.00532 − 0.830632i 49.7424 + 9.89438i −41.9235 + 41.9235i −0.107593 − 0.161025i 107.968 − 21.4762i 95.0688 − 142.281i −102.408 + 247.234i 1702.90 + 705.364i −0.349511 − 0.233536i
3.6 2.06955 − 0.857237i −16.2910 − 3.24047i −41.7066 + 41.7066i 131.985 + 197.529i −36.4929 + 7.25888i 15.7762 − 23.6107i −105.425 + 254.518i −418.614 − 173.395i 442.479 + 295.655i
3.7 8.29774 − 3.43704i −47.0095 − 9.35076i 11.7845 − 11.7845i −78.7297 − 117.827i −422.211 + 83.9830i 182.983 − 273.853i −162.690 + 392.767i 1448.94 + 600.172i −1058.26 − 707.104i
3.8 11.4305 − 4.73466i 11.8403 + 2.35518i 62.9841 − 62.9841i 3.24672 + 4.85906i 146.491 − 29.1389i −20.3582 + 30.4682i 118.712 − 286.597i −538.863 − 223.204i 60.1176 + 40.1693i
5.1 −5.41645 − 13.0765i 23.8539 + 35.6999i −96.4011 + 96.4011i −36.9856 + 185.939i 337.625 − 505.292i −13.5689 − 68.2153i 945.844 + 391.781i −426.499 + 1029.66i 2631.76 − 523.489i
5.2 −4.91724 − 11.8713i −12.7379 − 19.0636i −71.4927 + 71.4927i 26.3800 − 132.621i −163.674 + 244.955i 112.838 + 567.276i 440.495 + 182.459i 77.8093 − 187.848i −1704.09 + 338.965i
5.3 −2.51960 − 6.08286i −22.9974 − 34.4181i 14.6021 − 14.6021i −44.8802 + 225.628i −151.416 + 226.610i −60.2072 − 302.682i −514.917 − 213.285i −376.746 + 909.545i 1485.54 − 295.493i
5.4 −2.24071 − 5.40954i 7.28688 + 10.9056i 21.0124 − 21.0124i 16.2048 − 81.4668i 42.6665 − 63.8549i −53.1046 − 266.975i −506.961 − 209.990i 213.143 − 514.573i −477.008 + 94.8828i
5.5 1.00461 + 2.42535i 11.3015 + 16.9140i 40.3818 − 40.3818i −20.7533 + 104.334i −29.6686 + 44.4022i 87.9970 + 442.391i 293.730 + 121.667i 120.619 − 291.201i −273.895 + 54.4811i
5.6 2.31799 + 5.59612i −17.3299 − 25.9361i 19.3113 − 19.3113i 20.3657 − 102.385i 104.971 − 157.100i −16.0286 − 80.5813i 510.984 + 211.656i −93.3769 + 225.432i 620.169 − 123.359i
5.7 4.04520 + 9.76597i 26.3029 + 39.3651i −33.7557 + 33.7557i 26.8660 − 135.064i −278.038 + 416.113i −82.0045 − 412.265i 158.816 + 65.7837i −578.791 + 1397.33i 1427.71 − 283.990i
5.8 5.26851 + 12.7193i −5.39946 − 8.08086i −88.7689 + 88.7689i −22.9945 + 115.601i 74.3359 − 111.252i 8.39465 + 42.2027i −782.723 − 324.214i 242.830 − 586.244i −1591.51 + 316.572i
6.1 −11.7996 − 4.88755i 30.5590 − 6.07857i 70.0872 + 70.0872i 37.5749 − 56.2348i −390.293 − 77.6341i −304.501 − 455.719i −171.642 − 414.381i 223.396 − 92.5338i −718.218 + 479.898i
6.2 −10.8522 − 4.49513i −45.1411 + 8.97912i 52.3092 + 52.3092i 32.3776 − 48.4565i 530.242 + 105.472i −111.661 − 167.113i −44.8454 − 108.266i 1283.58 − 531.678i −569.186 + 380.318i
6.3 −7.60754 − 3.15115i 2.30914 − 0.459317i 2.69007 + 2.69007i −37.6489 + 56.3456i −19.0143 − 3.78217i 347.193 + 519.611i 189.685 + 457.941i −668.387 + 276.855i 463.969 − 310.014i
6.4 −0.477140 − 0.197638i −4.24307 + 0.844000i −45.0662 − 45.0662i −78.3099 + 117.199i 2.19135 + 0.435886i −283.582 − 424.411i 25.2449 + 60.9467i −656.217 + 271.814i 60.5278 − 40.4434i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
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 17.7.e.a ✓ 64
17.e odd 16 1 inner 17.7.e.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.7.e.a ✓ 64 1.a even 1 1 trivial
17.7.e.a ✓ 64 17.e odd 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(17, [\chi])\).