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Properties

Label 17.4.d.a.15.1
Level $17$
Weight $4$
Character 17.15
Analytic conductor $1.003$
Analytic rank $0$
Dimension $12$
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] = [17,4,Mod(2,17)] 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("17.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.d (of order \(8\), degree \(4\), 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: \(1.00303247010\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 54x^{10} + 1085x^{8} + 9836x^{6} + 38276x^{4} + 49664x^{2} + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 15.1
Root \(3.68604i\) of defining polynomial
Character \(\chi\) \(=\) 17.15
Dual form 17.4.d.a.8.1

$q$-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+(-1.89932 - 1.89932i) q^{2} +(-1.65755 - 4.00167i) q^{3} -0.785167i q^{4} +(1.92782 - 0.798529i) q^{5} +(-4.45224 + 10.7487i) q^{6} +(23.0956 + 9.56650i) q^{7} +(-16.6858 + 16.6858i) q^{8} +(5.82599 - 5.82599i) q^{9} +(-5.17821 - 2.14488i) q^{10} +(1.46245 - 3.53068i) q^{11} +(-3.14198 + 1.30145i) q^{12} -17.6726i q^{13} +(-25.6960 - 62.0357i) q^{14} +(-6.39089 - 6.39089i) q^{15} +57.1022 q^{16} +(-69.7847 - 6.56433i) q^{17} -22.1309 q^{18} +(113.784 + 113.784i) q^{19} +(-0.626979 - 1.51366i) q^{20} -108.278i q^{21} +(-9.48355 + 3.92822i) q^{22} +(-38.2560 + 92.3581i) q^{23} +(94.4287 + 39.1137i) q^{24} +(-85.3095 + 85.3095i) q^{25} +(-33.5659 + 33.5659i) q^{26} +(-141.016 - 58.4106i) q^{27} +(7.51131 - 18.1339i) q^{28} +(185.315 - 76.7600i) q^{29} +24.2767i q^{30} +(-29.2899 - 70.7121i) q^{31} +(25.0315 + 25.0315i) q^{32} -16.5527 q^{33} +(120.076 + 145.011i) q^{34} +52.1632 q^{35} +(-4.57438 - 4.57438i) q^{36} +(-93.6650 - 226.127i) q^{37} -432.224i q^{38} +(-70.7198 + 29.2931i) q^{39} +(-18.8432 + 45.4914i) q^{40} +(-49.9941 - 20.7082i) q^{41} +(-205.654 + 205.654i) q^{42} +(-100.471 + 100.471i) q^{43} +(-2.77217 - 1.14827i) q^{44} +(6.57924 - 15.8837i) q^{45} +(248.078 - 102.757i) q^{46} +468.451i q^{47} +(-94.6494 - 228.504i) q^{48} +(199.350 + 199.350i) q^{49} +324.060 q^{50} +(89.4031 + 290.136i) q^{51} -13.8759 q^{52} +(68.2834 + 68.2834i) q^{53} +(156.893 + 378.774i) q^{54} -7.97432i q^{55} +(-544.994 + 225.744i) q^{56} +(266.723 - 643.927i) q^{57} +(-497.764 - 206.181i) q^{58} +(257.729 - 257.729i) q^{59} +(-5.01792 + 5.01792i) q^{60} +(-653.988 - 270.891i) q^{61} +(-78.6740 + 189.936i) q^{62} +(190.289 - 78.8203i) q^{63} -551.903i q^{64} +(-14.1121 - 34.0695i) q^{65} +(31.4388 + 31.4388i) q^{66} +304.454 q^{67} +(-5.15410 + 54.7927i) q^{68} +432.997 q^{69} +(-99.0747 - 99.0747i) q^{70} +(-179.862 - 434.226i) q^{71} +194.423i q^{72} +(-131.243 + 54.3626i) q^{73} +(-251.588 + 607.388i) q^{74} +(482.785 + 199.976i) q^{75} +(89.3393 - 89.3393i) q^{76} +(67.5525 - 67.5525i) q^{77} +(189.956 + 78.6825i) q^{78} +(-274.715 + 663.221i) q^{79} +(110.083 - 45.5977i) q^{80} +438.657i q^{81} +(55.6232 + 134.286i) q^{82} +(-259.960 - 259.960i) q^{83} -85.0162 q^{84} +(-139.774 + 43.0703i) q^{85} +381.653 q^{86} +(-614.336 - 614.336i) q^{87} +(34.5100 + 83.3146i) q^{88} +1042.28i q^{89} +(-42.6643 + 17.6721i) q^{90} +(169.065 - 408.158i) q^{91} +(72.5165 + 30.0373i) q^{92} +(-234.417 + 234.417i) q^{93} +(889.738 - 889.738i) q^{94} +(310.214 + 128.495i) q^{95} +(58.6768 - 141.658i) q^{96} +(-834.757 + 345.768i) q^{97} -757.260i q^{98} +(-12.0495 - 29.0899i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 4 q^{3} - 20 q^{5} + 20 q^{6} - 4 q^{7} + 28 q^{8} - 64 q^{9} - 116 q^{10} + 40 q^{11} + 56 q^{12} - 132 q^{14} + 244 q^{15} + 184 q^{16} + 52 q^{17} - 12 q^{19} + 572 q^{20} - 620 q^{22}+ \cdots + 3280 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.89932 − 1.89932i −0.671511 − 0.671511i 0.286553 − 0.958064i \(-0.407490\pi\)
−0.958064 + 0.286553i \(0.907490\pi\)
\(3\) −1.65755 − 4.00167i −0.318995 − 0.770121i −0.999308 − 0.0371998i \(-0.988156\pi\)
0.680313 − 0.732922i \(-0.261844\pi\)
\(4\) − 0.785167i − 0.0981459i
\(5\) 1.92782 − 0.798529i 0.172429 − 0.0714226i −0.294799 − 0.955559i \(-0.595253\pi\)
0.467228 + 0.884137i \(0.345253\pi\)
\(6\) −4.45224 + 10.7487i −0.302936 + 0.731353i
\(7\) 23.0956 + 9.56650i 1.24704 + 0.516543i 0.905909 − 0.423472i \(-0.139189\pi\)
0.341135 + 0.940014i \(0.389189\pi\)
\(8\) −16.6858 + 16.6858i −0.737417 + 0.737417i
\(9\) 5.82599 − 5.82599i 0.215778 − 0.215778i
\(10\) −5.17821 − 2.14488i −0.163749 − 0.0678272i
\(11\) 1.46245 − 3.53068i 0.0400860 − 0.0967763i −0.902568 − 0.430546i \(-0.858321\pi\)
0.942655 + 0.333770i \(0.108321\pi\)
\(12\) −3.14198 + 1.30145i −0.0755843 + 0.0313080i
\(13\) − 17.6726i − 0.377038i −0.982070 − 0.188519i \(-0.939631\pi\)
0.982070 − 0.188519i \(-0.0603687\pi\)
\(14\) −25.6960 − 62.0357i −0.490540 − 1.18427i
\(15\) −6.39089 − 6.39089i −0.110008 − 0.110008i
\(16\) 57.1022 0.892221
\(17\) −69.7847 − 6.56433i −0.995605 − 0.0936520i
\(18\) −22.1309 −0.289794
\(19\) 113.784 + 113.784i 1.37388 + 1.37388i 0.854605 + 0.519279i \(0.173799\pi\)
0.519279 + 0.854605i \(0.326201\pi\)
\(20\) −0.626979 − 1.51366i −0.00700984 − 0.0169232i
\(21\) − 108.278i − 1.12515i
\(22\) −9.48355 + 3.92822i −0.0919046 + 0.0380681i
\(23\) −38.2560 + 92.3581i −0.346823 + 0.837304i 0.650169 + 0.759790i \(0.274698\pi\)
−0.996991 + 0.0775138i \(0.975302\pi\)
\(24\) 94.4287 + 39.1137i 0.803133 + 0.332668i
\(25\) −85.3095 + 85.3095i −0.682476 + 0.682476i
\(26\) −33.5659 + 33.5659i −0.253185 + 0.253185i
\(27\) −141.016 − 58.4106i −1.00513 − 0.416338i
\(28\) 7.51131 − 18.1339i 0.0506966 − 0.122392i
\(29\) 185.315 − 76.7600i 1.18662 − 0.491516i 0.299970 − 0.953949i \(-0.403023\pi\)
0.886655 + 0.462432i \(0.153023\pi\)
\(30\) 24.2767i 0.147743i
\(31\) −29.2899 − 70.7121i −0.169697 − 0.409686i 0.816036 − 0.578001i \(-0.196167\pi\)
−0.985733 + 0.168316i \(0.946167\pi\)
\(32\) 25.0315 + 25.0315i 0.138281 + 0.138281i
\(33\) −16.5527 −0.0873167
\(34\) 120.076 + 145.011i 0.605671 + 0.731448i
\(35\) 52.1632 0.251920
\(36\) −4.57438 − 4.57438i −0.0211777 − 0.0211777i
\(37\) −93.6650 − 226.127i −0.416174 − 1.00473i −0.983446 − 0.181202i \(-0.942001\pi\)
0.567272 − 0.823531i \(-0.307999\pi\)
\(38\) − 432.224i − 1.84516i
\(39\) −70.7198 + 29.2931i −0.290365 + 0.120273i
\(40\) −18.8432 + 45.4914i −0.0744841 + 0.179821i
\(41\) −49.9941 − 20.7082i −0.190433 − 0.0788800i 0.285429 − 0.958400i \(-0.407864\pi\)
−0.475862 + 0.879520i \(0.657864\pi\)
\(42\) −205.654 + 205.654i −0.755550 + 0.755550i
\(43\) −100.471 + 100.471i −0.356318 + 0.356318i −0.862454 − 0.506136i \(-0.831073\pi\)
0.506136 + 0.862454i \(0.331073\pi\)
\(44\) −2.77217 − 1.14827i −0.00949820 − 0.00393428i
\(45\) 6.57924 − 15.8837i 0.0217950 − 0.0526178i
\(46\) 248.078 − 102.757i 0.795154 − 0.329363i
\(47\) 468.451i 1.45384i 0.686721 + 0.726921i \(0.259049\pi\)
−0.686721 + 0.726921i \(0.740951\pi\)
\(48\) −94.6494 − 228.504i −0.284614 − 0.687119i
\(49\) 199.350 + 199.350i 0.581196 + 0.581196i
\(50\) 324.060 0.916580
\(51\) 89.4031 + 290.136i 0.245469 + 0.796611i
\(52\) −13.8759 −0.0370047
\(53\) 68.2834 + 68.2834i 0.176971 + 0.176971i 0.790034 − 0.613063i \(-0.210063\pi\)
−0.613063 + 0.790034i \(0.710063\pi\)
\(54\) 156.893 + 378.774i 0.395379 + 0.954530i
\(55\) − 7.97432i − 0.0195501i
\(56\) −544.994 + 225.744i −1.30050 + 0.538684i
\(57\) 266.723 − 643.927i 0.619796 − 1.49632i
\(58\) −497.764 − 206.181i −1.12689 − 0.466773i
\(59\) 257.729 − 257.729i 0.568703 − 0.568703i −0.363062 − 0.931765i \(-0.618269\pi\)
0.931765 + 0.363062i \(0.118269\pi\)
\(60\) −5.01792 + 5.01792i −0.0107968 + 0.0107968i
\(61\) −653.988 − 270.891i −1.37270 − 0.568590i −0.430179 − 0.902743i \(-0.641550\pi\)
−0.942519 + 0.334153i \(0.891550\pi\)
\(62\) −78.6740 + 189.936i −0.161155 + 0.389062i
\(63\) 190.289 − 78.8203i 0.380542 − 0.157626i
\(64\) − 551.903i − 1.07794i
\(65\) −14.1121 − 34.0695i −0.0269290 − 0.0650124i
\(66\) 31.4388 + 31.4388i 0.0586341 + 0.0586341i
\(67\) 304.454 0.555150 0.277575 − 0.960704i \(-0.410469\pi\)
0.277575 + 0.960704i \(0.410469\pi\)
\(68\) −5.15410 + 54.7927i −0.00919156 + 0.0977146i
\(69\) 432.997 0.755460
\(70\) −99.0747 − 99.0747i −0.169167 − 0.169167i
\(71\) −179.862 − 434.226i −0.300644 − 0.725819i −0.999940 − 0.0109864i \(-0.996503\pi\)
0.699296 − 0.714833i \(-0.253497\pi\)
\(72\) 194.423i 0.318236i
\(73\) −131.243 + 54.3626i −0.210422 + 0.0871597i −0.485405 − 0.874289i \(-0.661328\pi\)
0.274983 + 0.961449i \(0.411328\pi\)
\(74\) −251.588 + 607.388i −0.395224 + 0.954155i
\(75\) 482.785 + 199.976i 0.743296 + 0.307883i
\(76\) 89.3393 − 89.3393i 0.134841 − 0.134841i
\(77\) 67.5525 − 67.5525i 0.0999781 − 0.0999781i
\(78\) 189.956 + 78.6825i 0.275748 + 0.114219i
\(79\) −274.715 + 663.221i −0.391239 + 0.944534i 0.598432 + 0.801174i \(0.295791\pi\)
−0.989671 + 0.143360i \(0.954209\pi\)
\(80\) 110.083 − 45.5977i 0.153845 − 0.0637248i
\(81\) 438.657i 0.601725i
\(82\) 55.6232 + 134.286i 0.0749092 + 0.180847i
\(83\) −259.960 − 259.960i −0.343787 − 0.343787i 0.514002 − 0.857789i \(-0.328162\pi\)
−0.857789 + 0.514002i \(0.828162\pi\)
\(84\) −85.0162 −0.110429
\(85\) −139.774 + 43.0703i −0.178360 + 0.0549603i
\(86\) 381.653 0.478542
\(87\) −614.336 − 614.336i −0.757054 − 0.757054i
\(88\) 34.5100 + 83.3146i 0.0418043 + 0.100925i
\(89\) 1042.28i 1.24137i 0.784061 + 0.620684i \(0.213145\pi\)
−0.784061 + 0.620684i \(0.786855\pi\)
\(90\) −42.6643 + 17.6721i −0.0499690 + 0.0206978i
\(91\) 169.065 − 408.158i 0.194756 − 0.470183i
\(92\) 72.5165 + 30.0373i 0.0821780 + 0.0340392i
\(93\) −234.417 + 234.417i −0.261375 + 0.261375i
\(94\) 889.738 − 889.738i 0.976271 − 0.976271i
\(95\) 310.214 + 128.495i 0.335024 + 0.138772i
\(96\) 58.6768 − 141.658i 0.0623820 − 0.150604i
\(97\) −834.757 + 345.768i −0.873781 + 0.361932i −0.774082 − 0.633086i \(-0.781788\pi\)
−0.0996991 + 0.995018i \(0.531788\pi\)
\(98\) − 757.260i − 0.780559i
\(99\) −12.0495 − 29.0899i −0.0122325 − 0.0295318i
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 17.4.d.a.15.1 yes 12
3.2 odd 2 153.4.l.a.100.3 12
17.3 odd 16 289.4.b.e.288.3 12
17.5 odd 16 289.4.a.g.1.10 12
17.8 even 8 inner 17.4.d.a.8.1 ✓ 12
17.12 odd 16 289.4.a.g.1.9 12
17.14 odd 16 289.4.b.e.288.4 12
51.8 odd 8 153.4.l.a.127.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.8.1 ✓ 12 17.8 even 8 inner
17.4.d.a.15.1 yes 12 1.1 even 1 trivial
153.4.l.a.100.3 12 3.2 odd 2
153.4.l.a.127.3 12 51.8 odd 8
289.4.a.g.1.9 12 17.12 odd 16
289.4.a.g.1.10 12 17.5 odd 16
289.4.b.e.288.3 12 17.3 odd 16
289.4.b.e.288.4 12 17.14 odd 16