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Properties

Label 17.7.e.a.3.2
Level $17$
Weight $7$
Character 17.3
Analytic conductor $3.911$
Analytic rank $0$
Dimension $64$
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] = [17,7,Mod(3,17)] 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("17.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 17.e (of order \(16\), degree \(8\), 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: \(3.91091942154\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 3.2
Character \(\chi\) \(=\) 17.3
Dual form 17.7.e.a.6.2

$q$-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+(-10.8522 + 4.49513i) q^{2} +(-45.1411 - 8.97912i) q^{3} +(52.3092 - 52.3092i) q^{4} +(32.3776 + 48.4565i) q^{5} +(530.242 - 105.472i) q^{6} +(-111.661 + 167.113i) q^{7} +(-44.8454 + 108.266i) q^{8} +(1283.58 + 531.678i) q^{9} +(-569.186 - 380.318i) q^{10} +(-445.595 - 2240.16i) q^{11} +(-2830.99 + 1891.60i) q^{12} +(1583.21 + 1583.21i) q^{13} +(460.576 - 2315.47i) q^{14} +(-1026.46 - 2478.10i) q^{15} +3357.98i q^{16} +(-4239.98 + 2481.97i) q^{17} -16319.7 q^{18} +(9453.89 - 3915.93i) q^{19} +(4228.37 + 841.075i) q^{20} +(6541.03 - 6541.03i) q^{21} +(14905.5 + 22307.6i) q^{22} +(10490.6 - 2086.70i) q^{23} +(2996.50 - 4484.59i) q^{24} +(4679.71 - 11297.8i) q^{25} +(-24298.0 - 10064.6i) q^{26} +(-25270.5 - 16885.2i) q^{27} +(2900.63 + 14582.4i) q^{28} +(4380.57 - 2927.01i) q^{29} +(22278.8 + 22278.8i) q^{30} +(-305.311 + 1534.90i) q^{31} +(-17964.6 - 43370.5i) q^{32} +105124. i q^{33} +(34856.3 - 45994.0i) q^{34} -11713.0 q^{35} +(94955.0 - 39331.6i) q^{36} +(-7295.49 - 1451.16i) q^{37} +(-84992.9 + 84992.9i) q^{38} +(-57251.9 - 85683.5i) q^{39} +(-6698.19 + 1332.35i) q^{40} +(33256.9 - 49772.4i) q^{41} +(-41581.8 + 100387. i) q^{42} +(23297.9 + 9650.30i) q^{43} +(-140490. - 93872.1i) q^{44} +(15796.1 + 79412.4i) q^{45} +(-104466. + 69801.7i) q^{46} +(-72800.7 - 72800.7i) q^{47} +(30151.7 - 151583. i) q^{48} +(29563.9 + 71373.5i) q^{49} +143642. i q^{50} +(213683. - 73967.3i) q^{51} +165633. q^{52} +(203140. - 84143.4i) q^{53} +(350141. + 69647.4i) q^{54} +(94122.8 - 94122.8i) q^{55} +(-13085.2 - 19583.4i) q^{56} +(-461921. + 91881.7i) q^{57} +(-34381.6 + 51455.7i) q^{58} +(6681.62 - 16130.9i) q^{59} +(-183321. - 75934.0i) q^{60} +(216097. + 144391. i) q^{61} +(-3586.29 - 18029.5i) q^{62} +(-232177. + 155135. i) q^{63} +(237947. + 237947. i) q^{64} +(-25456.2 + 127977. i) q^{65} +(-472546. - 1.14083e6i) q^{66} +159194. i q^{67} +(-91960.3 + 351620. i) q^{68} -492292. q^{69} +(127112. - 52651.5i) q^{70} +(60556.9 + 12045.5i) q^{71} +(-115126. + 115126. i) q^{72} +(-228451. - 341901. i) q^{73} +(85695.3 - 17045.9i) q^{74} +(-312691. + 467976. i) q^{75} +(289687. - 699365. i) q^{76} +(424114. + 175674. i) q^{77} +(1.00647e6 + 672500. i) q^{78} +(-71060.1 - 357243. i) q^{79} +(-162716. + 108723. i) q^{80} +(272943. + 272943. i) q^{81} +(-137177. + 689634. i) q^{82} +(51056.8 + 123262. i) q^{83} -684312. i q^{84} +(-257548. - 125094. i) q^{85} -296213. q^{86} +(-224026. + 92794.5i) q^{87} +(262516. + 52217.7i) q^{88} +(365459. - 365459. i) q^{89} +(-528392. - 790794. i) q^{90} +(-441357. + 87791.3i) q^{91} +(439599. - 657907. i) q^{92} +(27564.2 - 66545.8i) q^{93} +(1.11730e6 + 462799. i) q^{94} +(495846. + 331314. i) q^{95} +(421515. + 2.11910e6i) q^{96} +(101419. - 67765.7i) q^{97} +(-641666. - 641666. i) q^{98} +(619083. - 3.11234e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9} + 3736 q^{10} - 4488 q^{11} - 7688 q^{12} + 2872 q^{13} + 12152 q^{14} + 14776 q^{15} - 6408 q^{17} - 44816 q^{18} - 11528 q^{19}+ \cdots + 14554736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{16}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.8522 + 4.49513i −1.35653 + 0.561891i −0.938101 − 0.346361i \(-0.887417\pi\)
−0.418424 + 0.908252i \(0.637417\pi\)
\(3\) −45.1411 − 8.97912i −1.67189 − 0.332560i −0.733911 − 0.679246i \(-0.762307\pi\)
−0.937981 + 0.346686i \(0.887307\pi\)
\(4\) 52.3092 − 52.3092i 0.817332 − 0.817332i
\(5\) 32.3776 + 48.4565i 0.259021 + 0.387652i 0.938073 − 0.346438i \(-0.112609\pi\)
−0.679052 + 0.734090i \(0.737609\pi\)
\(6\) 530.242 − 105.472i 2.45483 − 0.488295i
\(7\) −111.661 + 167.113i −0.325543 + 0.487209i −0.957755 − 0.287586i \(-0.907147\pi\)
0.632212 + 0.774795i \(0.282147\pi\)
\(8\) −44.8454 + 108.266i −0.0875886 + 0.211458i
\(9\) 1283.58 + 531.678i 1.76075 + 0.729325i
\(10\) −569.186 − 380.318i −0.569186 − 0.380318i
\(11\) −445.595 − 2240.16i −0.334782 − 1.68306i −0.671142 − 0.741329i \(-0.734196\pi\)
0.336360 − 0.941733i \(-0.390804\pi\)
\(12\) −2830.99 + 1891.60i −1.63830 + 1.09468i
\(13\) 1583.21 + 1583.21i 0.720622 + 0.720622i 0.968732 − 0.248110i \(-0.0798094\pi\)
−0.248110 + 0.968732i \(0.579809\pi\)
\(14\) 460.576 − 2315.47i 0.167848 − 0.843831i
\(15\) −1026.46 − 2478.10i −0.304137 − 0.734252i
\(16\) 3357.98i 0.819819i
\(17\) −4239.98 + 2481.97i −0.863012 + 0.505183i
\(18\) −16319.7 −2.79830
\(19\) 9453.89 − 3915.93i 1.37832 − 0.570919i 0.434288 − 0.900774i \(-0.357000\pi\)
0.944031 + 0.329855i \(0.107000\pi\)
\(20\) 4228.37 + 841.075i 0.528546 + 0.105134i
\(21\) 6541.03 − 6541.03i 0.706298 − 0.706298i
\(22\) 14905.5 + 22307.6i 1.39984 + 2.09500i
\(23\) 10490.6 − 2086.70i 0.862214 − 0.171505i 0.255873 − 0.966710i \(-0.417637\pi\)
0.606340 + 0.795205i \(0.292637\pi\)
\(24\) 2996.50 − 4484.59i 0.216761 − 0.324406i
\(25\) 4679.71 − 11297.8i 0.299501 − 0.723060i
\(26\) −24298.0 − 10064.6i −1.38245 − 0.572631i
\(27\) −25270.5 − 16885.2i −1.28387 − 0.857856i
\(28\) 2900.63 + 14582.4i 0.132135 + 0.664288i
\(29\) 4380.57 − 2927.01i 0.179613 − 0.120013i −0.462514 − 0.886612i \(-0.653052\pi\)
0.642126 + 0.766599i \(0.278052\pi\)
\(30\) 22278.8 + 22278.8i 0.825139 + 0.825139i
\(31\) −305.311 + 1534.90i −0.0102484 + 0.0515224i −0.985571 − 0.169262i \(-0.945862\pi\)
0.975323 + 0.220784i \(0.0708617\pi\)
\(32\) −17964.6 − 43370.5i −0.548238 − 1.32356i
\(33\) 105124.i 2.92523i
\(34\) 34856.3 − 45994.0i 0.886839 − 1.17021i
\(35\) −11713.0 −0.273190
\(36\) 94955.0 − 39331.6i 2.03521 − 0.843014i
\(37\) −7295.49 − 1451.16i −0.144029 − 0.0286491i 0.122549 − 0.992462i \(-0.460893\pi\)
−0.266578 + 0.963813i \(0.585893\pi\)
\(38\) −84992.9 + 84992.9i −1.54893 + 1.54893i
\(39\) −57251.9 − 85683.5i −0.965152 − 1.44445i
\(40\) −6698.19 + 1332.35i −0.104659 + 0.0208180i
\(41\) 33256.9 − 49772.4i 0.482536 − 0.722166i −0.507705 − 0.861531i \(-0.669506\pi\)
0.990241 + 0.139365i \(0.0445060\pi\)
\(42\) −41581.8 + 100387.i −0.561249 + 1.35497i
\(43\) 23297.9 + 9650.30i 0.293029 + 0.121377i 0.524355 − 0.851500i \(-0.324306\pi\)
−0.231326 + 0.972876i \(0.574306\pi\)
\(44\) −140490. − 93872.1i −1.64925 − 1.10199i
\(45\) 15796.1 + 79412.4i 0.173346 + 0.871467i
\(46\) −104466. + 69801.7i −1.07325 + 0.717121i
\(47\) −72800.7 − 72800.7i −0.701200 − 0.701200i 0.263468 − 0.964668i \(-0.415134\pi\)
−0.964668 + 0.263468i \(0.915134\pi\)
\(48\) 30151.7 − 151583.i 0.272639 − 1.37065i
\(49\) 29563.9 + 71373.5i 0.251289 + 0.606665i
\(50\) 143642.i 1.14914i
\(51\) 213683. − 73967.3i 1.61087 − 0.557609i
\(52\) 165633. 1.17797
\(53\) 203140. − 84143.4i 1.36448 − 0.565187i 0.424196 − 0.905570i \(-0.360557\pi\)
0.940287 + 0.340383i \(0.110557\pi\)
\(54\) 350141. + 69647.4i 2.22363 + 0.442307i
\(55\) 94122.8 − 94122.8i 0.565727 − 0.565727i
\(56\) −13085.2 − 19583.4i −0.0745102 − 0.111512i
\(57\) −461921. + 91881.7i −2.49427 + 0.496140i
\(58\) −34381.6 + 51455.7i −0.176215 + 0.263724i
\(59\) 6681.62 − 16130.9i 0.0325331 − 0.0785419i −0.906778 − 0.421609i \(-0.861466\pi\)
0.939311 + 0.343067i \(0.111466\pi\)
\(60\) −183321. − 75934.0i −0.848708 − 0.351546i
\(61\) 216097. + 144391.i 0.952047 + 0.636137i 0.931535 − 0.363652i \(-0.118470\pi\)
0.0205120 + 0.999790i \(0.493470\pi\)
\(62\) −3586.29 − 18029.5i −0.0150477 − 0.0756499i
\(63\) −232177. + 155135.i −0.928532 + 0.620425i
\(64\) 237947. + 237947.i 0.907697 + 0.907697i
\(65\) −25456.2 + 127977.i −0.0926945 + 0.466007i
\(66\) −472546. − 1.14083e6i −1.64366 − 3.96815i
\(67\) 159194.i 0.529300i 0.964345 + 0.264650i \(0.0852564\pi\)
−0.964345 + 0.264650i \(0.914744\pi\)
\(68\) −91960.3 + 351620.i −0.292465 + 1.11827i
\(69\) −492292. −1.49856
\(70\) 127112. − 52651.5i 0.370589 − 0.153503i
\(71\) 60556.9 + 12045.5i 0.169195 + 0.0336551i 0.278961 − 0.960302i \(-0.410010\pi\)
−0.109765 + 0.993958i \(0.535010\pi\)
\(72\) −115126. + 115126.i −0.308443 + 0.308443i
\(73\) −228451. − 341901.i −0.587252 − 0.878884i 0.412230 − 0.911080i \(-0.364750\pi\)
−0.999482 + 0.0321958i \(0.989750\pi\)
\(74\) 85695.3 − 17045.9i 0.211476 − 0.0420653i
\(75\) −312691. + 467976.i −0.741194 + 1.10928i
\(76\) 289687. − 699365.i 0.659915 − 1.59317i
\(77\) 424114. + 175674.i 0.928989 + 0.384800i
\(78\) 1.00647e6 + 672500.i 2.12088 + 1.41713i
\(79\) −71060.1 − 357243.i −0.144127 − 0.724574i −0.983484 − 0.180994i \(-0.942068\pi\)
0.839357 − 0.543580i \(-0.182932\pi\)
\(80\) −162716. + 108723.i −0.317804 + 0.212350i
\(81\) 272943. + 272943.i 0.513589 + 0.513589i
\(82\) −137177. + 689634.i −0.248794 + 1.25077i
\(83\) 51056.8 + 123262.i 0.0892934 + 0.215573i 0.962217 − 0.272283i \(-0.0877788\pi\)
−0.872924 + 0.487857i \(0.837779\pi\)
\(84\) − 684312.i − 1.15456i
\(85\) −257548. − 125094.i −0.419373 − 0.203695i
\(86\) −296213. −0.465702
\(87\) −224026. + 92794.5i −0.340205 + 0.140917i
\(88\) 262516. + 52217.7i 0.385219 + 0.0766249i
\(89\) 365459. − 365459.i 0.518405 − 0.518405i −0.398684 − 0.917088i \(-0.630533\pi\)
0.917088 + 0.398684i \(0.130533\pi\)
\(90\) −528392. − 790794.i −0.724817 − 1.08477i
\(91\) −441357. + 87791.3i −0.585687 + 0.116500i
\(92\) 439599. − 657907.i 0.564538 − 0.844891i
\(93\) 27564.2 − 66545.8i 0.0342686 − 0.0827316i
\(94\) 1.11730e6 + 462799.i 1.34519 + 0.557197i
\(95\) 495846. + 331314.i 0.578331 + 0.386428i
\(96\) 421515. + 2.11910e6i 0.476430 + 2.39518i
\(97\) 101419. − 67765.7i 0.111123 − 0.0742497i −0.498767 − 0.866736i \(-0.666214\pi\)
0.609890 + 0.792486i \(0.291214\pi\)
\(98\) −641666. − 641666.i −0.681759 − 0.681759i
\(99\) 619083. − 3.11234e6i 0.638033 − 3.20761i
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 17.7.e.a.3.2 ✓ 64
17.6 odd 16 inner 17.7.e.a.6.2 yes 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.7.e.a.3.2 ✓ 64 1.1 even 1 trivial
17.7.e.a.6.2 yes 64 17.6 odd 16 inner