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Properties

Label 17.10.c.a.4.7
Level $17$
Weight $10$
Character 17.4
Analytic conductor $8.756$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [17,10,Mod(4,17)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 17.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content 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: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 4.7
Character \(\chi\) \(=\) 17.4
Dual form 17.10.c.a.13.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.42963i q^{2} +(150.879 - 150.879i) q^{3} +456.801 q^{4} +(210.983 - 210.983i) q^{5} +(1120.97 + 1120.97i) q^{6} +(-5461.47 - 5461.47i) q^{7} +7197.83i q^{8} -25845.7i q^{9} +(1567.53 + 1567.53i) q^{10} +(6566.81 + 6566.81i) q^{11} +(68921.4 - 68921.4i) q^{12} +122491. q^{13} +(40576.7 - 40576.7i) q^{14} -63665.7i q^{15} +180405. q^{16} +(-343740. + 20755.1i) q^{17} +192024. q^{18} -464145. i q^{19} +(96377.3 - 96377.3i) q^{20} -1.64804e6 q^{21} +(-48789.0 + 48789.0i) q^{22} +(829781. + 829781. i) q^{23} +(1.08600e6 + 1.08600e6i) q^{24} +1.86410e6i q^{25} +910061. i q^{26} +(-929817. - 929817. i) q^{27} +(-2.49480e6 - 2.49480e6i) q^{28} +(929897. - 929897. i) q^{29} +473013. q^{30} +(-5.27129e6 + 5.27129e6i) q^{31} +5.02563e6i q^{32} +1.98158e6 q^{33} +(-154203. - 2.55386e6i) q^{34} -2.30456e6 q^{35} -1.18063e7i q^{36} +(1.11498e6 - 1.11498e6i) q^{37} +3.44843e6 q^{38} +(1.84812e7 - 1.84812e7i) q^{39} +(1.51862e6 + 1.51862e6i) q^{40} +(1.72411e7 + 1.72411e7i) q^{41} -1.22443e7i q^{42} -7.29437e6i q^{43} +(2.99972e6 + 2.99972e6i) q^{44} +(-5.45301e6 - 5.45301e6i) q^{45} +(-6.16497e6 + 6.16497e6i) q^{46} -5.46941e7 q^{47} +(2.72192e7 - 2.72192e7i) q^{48} +1.93017e7i q^{49} -1.38496e7 q^{50} +(-4.87315e7 + 5.49945e7i) q^{51} +5.59538e7 q^{52} +6.10696e7i q^{53} +(6.90820e6 - 6.90820e6i) q^{54} +2.77097e6 q^{55} +(3.93108e7 - 3.93108e7i) q^{56} +(-7.00296e7 - 7.00296e7i) q^{57} +(6.90879e6 + 6.90879e6i) q^{58} +8.90678e7i q^{59} -2.90825e7i q^{60} +(1.86288e7 + 1.86288e7i) q^{61} +(-3.91637e7 - 3.91637e7i) q^{62} +(-1.41155e8 + 1.41155e8i) q^{63} +5.50286e7 q^{64} +(2.58435e7 - 2.58435e7i) q^{65} +1.47224e7i q^{66} -2.68240e8 q^{67} +(-1.57021e8 + 9.48095e6i) q^{68} +2.50392e8 q^{69} -1.71220e7i q^{70} +(2.78042e8 - 2.78042e8i) q^{71} +1.86033e8 q^{72} +(1.98106e8 - 1.98106e8i) q^{73} +(8.28392e6 + 8.28392e6i) q^{74} +(2.81252e8 + 2.81252e8i) q^{75} -2.12022e8i q^{76} -7.17289e7i q^{77} +(1.37309e8 + 1.37309e8i) q^{78} +(-4.29542e8 - 4.29542e8i) q^{79} +(3.80624e7 - 3.80624e7i) q^{80} +2.28142e8 q^{81} +(-1.28095e8 + 1.28095e8i) q^{82} +5.22381e8i q^{83} -7.52825e8 q^{84} +(-6.81444e7 + 7.69024e7i) q^{85} +5.41945e7 q^{86} -2.80603e8i q^{87} +(-4.72668e7 + 4.72668e7i) q^{88} -2.71477e8 q^{89} +(4.05138e7 - 4.05138e7i) q^{90} +(-6.68980e8 - 6.68980e8i) q^{91} +(3.79045e8 + 3.79045e8i) q^{92} +1.59065e9i q^{93} -4.06357e8i q^{94} +(-9.79269e7 - 9.79269e7i) q^{95} +(7.58260e8 + 7.58260e8i) q^{96} +(6.26088e8 - 6.26088e8i) q^{97} -1.43405e8 q^{98} +(1.69724e8 - 1.69724e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 144 q^{3} - 5124 q^{4} - 1710 q^{5} - 8174 q^{6} + 3810 q^{7} + 21946 q^{10} + 18132 q^{11} - 95142 q^{12} + 244832 q^{13} - 341100 q^{14} - 279932 q^{16} - 98022 q^{17} + 888764 q^{18} + 1364262 q^{20}+ \cdots - 1224966600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.42963i 0.328346i 0.986432 + 0.164173i \(0.0524956\pi\)
−0.986432 + 0.164173i \(0.947504\pi\)
\(3\) 150.879 − 150.879i 1.07543 − 1.07543i 0.0785166 − 0.996913i \(-0.474982\pi\)
0.996913 − 0.0785166i \(-0.0250184\pi\)
\(4\) 456.801 0.892189
\(5\) 210.983 − 210.983i 0.150967 − 0.150967i −0.627583 − 0.778550i \(-0.715956\pi\)
0.778550 + 0.627583i \(0.215956\pi\)
\(6\) 1120.97 + 1120.97i 0.353113 + 0.353113i
\(7\) −5461.47 − 5461.47i −0.859743 − 0.859743i 0.131565 − 0.991308i \(-0.458000\pi\)
−0.991308 + 0.131565i \(0.958000\pi\)
\(8\) 7197.83i 0.621293i
\(9\) − 25845.7i − 1.31310i
\(10\) 1567.53 + 1567.53i 0.0495696 + 0.0495696i
\(11\) 6566.81 + 6566.81i 0.135234 + 0.135234i 0.771484 − 0.636249i \(-0.219515\pi\)
−0.636249 + 0.771484i \(0.719515\pi\)
\(12\) 68921.4 − 68921.4i 0.959486 − 0.959486i
\(13\) 122491. 1.18948 0.594741 − 0.803917i \(-0.297254\pi\)
0.594741 + 0.803917i \(0.297254\pi\)
\(14\) 40576.7 − 40576.7i 0.282293 − 0.282293i
\(15\) − 63665.7i − 0.324710i
\(16\) 180405. 0.688189
\(17\) −343740. + 20755.1i −0.998182 + 0.0602705i
\(18\) 192024. 0.431151
\(19\) − 464145.i − 0.817077i −0.912741 − 0.408538i \(-0.866039\pi\)
0.912741 − 0.408538i \(-0.133961\pi\)
\(20\) 96377.3 − 96377.3i 0.134691 − 0.134691i
\(21\) −1.64804e6 −1.84919
\(22\) −48789.0 + 48789.0i −0.0444037 + 0.0444037i
\(23\) 829781. + 829781.i 0.618285 + 0.618285i 0.945091 − 0.326807i \(-0.105973\pi\)
−0.326807 + 0.945091i \(0.605973\pi\)
\(24\) 1.08600e6 + 1.08600e6i 0.668157 + 0.668157i
\(25\) 1.86410e6i 0.954418i
\(26\) 910061.i 0.390562i
\(27\) −929817. − 929817.i −0.336714 − 0.336714i
\(28\) −2.49480e6 − 2.49480e6i −0.767053 − 0.767053i
\(29\) 929897. − 929897.i 0.244143 − 0.244143i −0.574419 − 0.818562i \(-0.694772\pi\)
0.818562 + 0.574419i \(0.194772\pi\)
\(30\) 473013. 0.106617
\(31\) −5.27129e6 + 5.27129e6i −1.02515 + 1.02515i −0.0254783 + 0.999675i \(0.508111\pi\)
−0.999675 + 0.0254783i \(0.991889\pi\)
\(32\) 5.02563e6i 0.847258i
\(33\) 1.98158e6 0.290870
\(34\) −154203. − 2.55386e6i −0.0197896 − 0.327749i
\(35\) −2.30456e6 −0.259586
\(36\) − 1.18063e7i − 1.17153i
\(37\) 1.11498e6 − 1.11498e6i 0.0978050 − 0.0978050i −0.656511 − 0.754316i \(-0.727969\pi\)
0.754316 + 0.656511i \(0.227969\pi\)
\(38\) 3.44843e6 0.268284
\(39\) 1.84812e7 − 1.84812e7i 1.27920 − 1.27920i
\(40\) 1.51862e6 + 1.51862e6i 0.0937950 + 0.0937950i
\(41\) 1.72411e7 + 1.72411e7i 0.952880 + 0.952880i 0.998939 − 0.0460589i \(-0.0146662\pi\)
−0.0460589 + 0.998939i \(0.514666\pi\)
\(42\) − 1.22443e7i − 0.607173i
\(43\) − 7.29437e6i − 0.325372i −0.986678 − 0.162686i \(-0.947984\pi\)
0.986678 − 0.162686i \(-0.0520157\pi\)
\(44\) 2.99972e6 + 2.99972e6i 0.120655 + 0.120655i
\(45\) −5.45301e6 − 5.45301e6i −0.198235 − 0.198235i
\(46\) −6.16497e6 + 6.16497e6i −0.203012 + 0.203012i
\(47\) −5.46941e7 −1.63493 −0.817466 − 0.575977i \(-0.804622\pi\)
−0.817466 + 0.575977i \(0.804622\pi\)
\(48\) 2.72192e7 − 2.72192e7i 0.740099 − 0.740099i
\(49\) 1.93017e7i 0.478315i
\(50\) −1.38496e7 −0.313380
\(51\) −4.87315e7 + 5.49945e7i −1.00866 + 1.13829i
\(52\) 5.59538e7 1.06124
\(53\) 6.10696e7i 1.06312i 0.847020 + 0.531561i \(0.178394\pi\)
−0.847020 + 0.531561i \(0.821606\pi\)
\(54\) 6.90820e6 − 6.90820e6i 0.110559 − 0.110559i
\(55\) 2.77097e6 0.0408320
\(56\) 3.93108e7 − 3.93108e7i 0.534152 − 0.534152i
\(57\) −7.00296e7 − 7.00296e7i −0.878708 − 0.878708i
\(58\) 6.90879e6 + 6.90879e6i 0.0801634 + 0.0801634i
\(59\) 8.90678e7i 0.956944i 0.878103 + 0.478472i \(0.158809\pi\)
−0.878103 + 0.478472i \(0.841191\pi\)
\(60\) − 2.90825e7i − 0.289702i
\(61\) 1.86288e7 + 1.86288e7i 0.172266 + 0.172266i 0.787974 − 0.615708i \(-0.211130\pi\)
−0.615708 + 0.787974i \(0.711130\pi\)
\(62\) −3.91637e7 − 3.91637e7i −0.336606 − 0.336606i
\(63\) −1.41155e8 + 1.41155e8i −1.12893 + 1.12893i
\(64\) 5.50286e7 0.409995
\(65\) 2.58435e7 − 2.58435e7i 0.179573 − 0.179573i
\(66\) 1.47224e7i 0.0955062i
\(67\) −2.68240e8 −1.62625 −0.813124 − 0.582091i \(-0.802235\pi\)
−0.813124 + 0.582091i \(0.802235\pi\)
\(68\) −1.57021e8 + 9.48095e6i −0.890567 + 0.0537727i
\(69\) 2.50392e8 1.32984
\(70\) − 1.71220e7i − 0.0852342i
\(71\) 2.78042e8 − 2.78042e8i 1.29852 − 1.29852i 0.369146 − 0.929371i \(-0.379650\pi\)
0.929371 − 0.369146i \(-0.120350\pi\)
\(72\) 1.86033e8 0.815818
\(73\) 1.98106e8 − 1.98106e8i 0.816481 − 0.816481i −0.169116 − 0.985596i \(-0.554091\pi\)
0.985596 + 0.169116i \(0.0540912\pi\)
\(74\) 8.28392e6 + 8.28392e6i 0.0321139 + 0.0321139i
\(75\) 2.81252e8 + 2.81252e8i 1.02641 + 1.02641i
\(76\) − 2.12022e8i − 0.728986i
\(77\) − 7.17289e7i − 0.232534i
\(78\) 1.37309e8 + 1.37309e8i 0.420022 + 0.420022i
\(79\) −4.29542e8 − 4.29542e8i −1.24075 − 1.24075i −0.959691 − 0.281058i \(-0.909315\pi\)
−0.281058 − 0.959691i \(-0.590685\pi\)
\(80\) 3.80624e7 − 3.80624e7i 0.103894 − 0.103894i
\(81\) 2.28142e8 0.588874
\(82\) −1.28095e8 + 1.28095e8i −0.312875 + 0.312875i
\(83\) 5.22381e8i 1.20819i 0.796912 + 0.604096i \(0.206466\pi\)
−0.796912 + 0.604096i \(0.793534\pi\)
\(84\) −7.52825e8 −1.64982
\(85\) −6.81444e7 + 7.69024e7i −0.141594 + 0.159792i
\(86\) 5.41945e7 0.106835
\(87\) − 2.80603e8i − 0.525117i
\(88\) −4.72668e7 + 4.72668e7i −0.0840203 + 0.0840203i
\(89\) −2.71477e8 −0.458647 −0.229324 − 0.973350i \(-0.573651\pi\)
−0.229324 + 0.973350i \(0.573651\pi\)
\(90\) 4.05138e7 − 4.05138e7i 0.0650897 − 0.0650897i
\(91\) −6.68980e8 − 6.68980e8i −1.02265 − 1.02265i
\(92\) 3.79045e8 + 3.79045e8i 0.551626 + 0.551626i
\(93\) 1.59065e9i 2.20496i
\(94\) − 4.06357e8i − 0.536824i
\(95\) −9.79269e7 − 9.79269e7i −0.123352 − 0.123352i
\(96\) 7.58260e8 + 7.58260e8i 0.911166 + 0.911166i
\(97\) 6.26088e8 − 6.26088e8i 0.718063 − 0.718063i −0.250145 − 0.968208i \(-0.580478\pi\)
0.968208 + 0.250145i \(0.0804783\pi\)
\(98\) −1.43405e8 −0.157053
\(99\) 1.69724e8 − 1.69724e8i 0.177576 − 0.177576i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 17.10.c.a.4.7 ✓ 24
17.8 even 8 289.10.a.f.1.13 24
17.9 even 8 289.10.a.f.1.14 24
17.13 even 4 inner 17.10.c.a.13.6 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.10.c.a.4.7 ✓ 24 1.1 even 1 trivial
17.10.c.a.13.6 yes 24 17.13 even 4 inner
289.10.a.f.1.13 24 17.8 even 8
289.10.a.f.1.14 24 17.9 even 8