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Properties

Label 17.10.c.a
Level $17$
Weight $10$
Character orbit 17.c
Analytic conductor $8.756$
Analytic rank $0$
Dimension $24$
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] = [17,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(8.75560921479\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 144 q^{3} - 5124 q^{4} - 1710 q^{5} - 8174 q^{6} + 3810 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 144 q^{3} - 5124 q^{4} - 1710 q^{5} - 8174 q^{6} + 3810 q^{7} + 21946 q^{10} + 18132 q^{11} - 95142 q^{12} + 244832 q^{13} - 341100 q^{14} - 279932 q^{16} - 98022 q^{17} + 888764 q^{18} + 1364262 q^{20} - 775748 q^{21} - 4573190 q^{22} - 377526 q^{23} - 307054 q^{24} + 3996108 q^{27} - 1024780 q^{28} + 2160042 q^{29} + 39084792 q^{30} - 585086 q^{31} - 30349992 q^{33} + 4441318 q^{34} - 25532364 q^{35} + 1515390 q^{37} + 13171392 q^{38} - 25687084 q^{39} - 63240118 q^{40} - 72707928 q^{41} - 3606450 q^{44} + 156729418 q^{45} + 118549536 q^{46} + 153365328 q^{47} + 26098270 q^{48} - 236105676 q^{50} + 256903592 q^{51} - 209898380 q^{52} - 187411976 q^{54} + 255767540 q^{55} - 107638596 q^{56} - 443390988 q^{57} - 281433730 q^{58} + 52149382 q^{61} + 176928228 q^{62} + 620111642 q^{63} + 1006108924 q^{64} - 489714396 q^{65} - 26076868 q^{67} - 308011950 q^{68} - 751973532 q^{69} + 149240178 q^{71} + 1171736028 q^{72} - 175256556 q^{73} - 37609818 q^{74} - 233543476 q^{75} - 1202722924 q^{78} - 469987182 q^{79} + 481103586 q^{80} + 347156560 q^{81} - 984422080 q^{82} - 935672904 q^{84} - 725828294 q^{85} + 690159588 q^{86} + 992281114 q^{88} - 191594460 q^{89} + 914946474 q^{90} + 926854816 q^{91} + 3064830552 q^{92} + 1213246572 q^{95} + 3594247550 q^{96} - 2200567348 q^{97} - 4413444720 q^{98} - 1224966600 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 − 38.1697i −95.6730 + 95.6730i −944.927 −23.8207 + 23.8207i 3651.81 + 3651.81i 3405.47 + 3405.47i 16524.7i 1376.34i 909.229 + 909.229i
4.2 − 36.7615i 151.130 − 151.130i −839.411 −1546.81 + 1546.81i −5555.78 − 5555.78i −1021.13 − 1021.13i 12036.1i − 25997.7i 56863.1 + 56863.1i
4.3 − 31.4159i 42.7255 − 42.7255i −474.957 1118.79 − 1118.79i −1342.26 − 1342.26i −7416.51 − 7416.51i − 1163.73i 16032.1i −35147.8 − 35147.8i
4.4 − 17.1503i 97.3786 − 97.3786i 217.866 882.390 − 882.390i −1670.08 − 1670.08i 7460.78 + 7460.78i − 12517.4i 717.820i −15133.3 − 15133.3i
4.5 − 12.8587i −144.765 + 144.765i 346.654 −85.3199 + 85.3199i 1861.49 + 1861.49i −3200.21 − 3200.21i − 11041.2i − 22230.9i 1097.10 + 1097.10i
4.6 − 9.57683i −2.66069 + 2.66069i 420.284 −1324.59 + 1324.59i 25.4810 + 25.4810i 1098.57 + 1098.57i − 8928.33i 19668.8i 12685.3 + 12685.3i
4.7 7.42963i 150.879 − 150.879i 456.801 210.983 − 210.983i 1120.97 + 1120.97i −5461.47 − 5461.47i 7197.83i − 25845.7i 1567.53 + 1567.53i
4.8 12.9147i −71.7165 + 71.7165i 345.210 1469.78 − 1469.78i −926.198 − 926.198i 1528.49 + 1528.49i 11070.6i 9396.49i 18981.8 + 18981.8i
4.9 21.4252i 22.6833 − 22.6833i 52.9591 −1056.12 + 1056.12i 485.995 + 485.995i −862.967 − 862.967i 12104.4i 18653.9i −22627.6 − 22627.6i
4.10 28.8329i −179.389 + 179.389i −319.336 −1133.17 + 1133.17i −5172.29 − 5172.29i 4922.20 + 4922.20i 5555.06i − 44677.5i −32672.5 − 32672.5i
4.11 35.4517i 137.825 − 137.825i −744.825 178.043 − 178.043i 4886.12 + 4886.12i 7491.11 + 7491.11i − 8254.04i − 18308.3i 6311.91 + 6311.91i
4.12 39.8788i −36.4169 + 36.4169i −1078.32 454.832 − 454.832i −1452.26 − 1452.26i −6039.34 − 6039.34i − 22584.0i 17030.6i 18138.1 + 18138.1i
13.1 − 39.8788i −36.4169 − 36.4169i −1078.32 454.832 + 454.832i −1452.26 + 1452.26i −6039.34 + 6039.34i 22584.0i − 17030.6i 18138.1 − 18138.1i
13.2 − 35.4517i 137.825 + 137.825i −744.825 178.043 + 178.043i 4886.12 − 4886.12i 7491.11 − 7491.11i 8254.04i 18308.3i 6311.91 − 6311.91i
13.3 − 28.8329i −179.389 − 179.389i −319.336 −1133.17 − 1133.17i −5172.29 + 5172.29i 4922.20 − 4922.20i − 5555.06i 44677.5i −32672.5 + 32672.5i
13.4 − 21.4252i 22.6833 + 22.6833i 52.9591 −1056.12 − 1056.12i 485.995 − 485.995i −862.967 + 862.967i − 12104.4i − 18653.9i −22627.6 + 22627.6i
13.5 − 12.9147i −71.7165 − 71.7165i 345.210 1469.78 + 1469.78i −926.198 + 926.198i 1528.49 − 1528.49i − 11070.6i − 9396.49i 18981.8 − 18981.8i
13.6 − 7.42963i 150.879 + 150.879i 456.801 210.983 + 210.983i 1120.97 − 1120.97i −5461.47 + 5461.47i − 7197.83i 25845.7i 1567.53 − 1567.53i
13.7 9.57683i −2.66069 − 2.66069i 420.284 −1324.59 − 1324.59i 25.4810 − 25.4810i 1098.57 − 1098.57i 8928.33i − 19668.8i 12685.3 − 12685.3i
13.8 12.8587i −144.765 − 144.765i 346.654 −85.3199 − 85.3199i 1861.49 − 1861.49i −3200.21 + 3200.21i 11041.2i 22230.9i 1097.10 − 1097.10i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
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 17.10.c.a ✓ 24
17.c even 4 1 inner 17.10.c.a ✓ 24
17.d even 8 2 289.10.a.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.c.a ✓ 24 1.a even 1 1 trivial
17.10.c.a ✓ 24 17.c even 4 1 inner
289.10.a.f 24 17.d even 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(17, [\chi])\).