↑

Properties

Label 340.2.bf.a.11.8
Level $340$
Weight $2$
Character 340.11
Analytic conductor $2.715$
Analytic rank $0$
Dimension $288$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [340,2,Mod(11,340)] 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("340.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bf (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: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 11.8
Character \(\chi\) \(=\) 340.11
Dual form 340.2.bf.a.31.8

$q$-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+(-1.15606 + 0.814564i) q^{2} +(-0.177801 + 0.0353667i) q^{3} +(0.672971 - 1.88338i) q^{4} +(0.831470 + 0.555570i) q^{5} +(0.176741 - 0.185716i) q^{6} +(-2.15819 - 3.22996i) q^{7} +(0.756133 + 2.72548i) q^{8} +(-2.74128 + 1.13547i) q^{9} +(-1.41378 + 0.0350101i) q^{10} +(1.02675 - 5.16184i) q^{11} +(-0.0530458 + 0.358666i) q^{12} +(-2.69222 + 2.69222i) q^{13} +(5.12601 + 1.97606i) q^{14} +(-0.167485 - 0.0693744i) q^{15} +(-3.09422 - 2.53492i) q^{16} +(-3.27391 - 2.50629i) q^{17} +(2.24418 - 3.54563i) q^{18} +(-1.66081 + 4.00954i) q^{19} +(1.60590 - 1.19209i) q^{20} +(0.497960 + 0.497960i) q^{21} +(3.01766 + 6.80378i) q^{22} +(-5.09948 - 1.01435i) q^{23} +(-0.230832 - 0.457851i) q^{24} +(0.382683 + 0.923880i) q^{25} +(0.919396 - 5.30537i) q^{26} +(0.899439 - 0.600986i) q^{27} +(-7.53563 + 1.89102i) q^{28} +(5.25962 - 7.87158i) q^{29} +(0.250133 - 0.0562256i) q^{30} +(-1.36864 - 6.88061i) q^{31} +(5.64197 + 0.410088i) q^{32} +0.954092i q^{33} +(5.82639 + 0.230627i) q^{34} -3.88464i q^{35} +(0.293726 + 5.92700i) q^{36} +(-0.277878 - 1.39699i) q^{37} +(-1.34603 - 5.98812i) q^{38} +(0.383464 - 0.573894i) q^{39} +(-0.885496 + 2.68624i) q^{40} +(3.90589 - 2.60983i) q^{41} +(-0.981295 - 0.170054i) q^{42} +(-0.294987 - 0.712162i) q^{43} +(-9.03072 - 5.40754i) q^{44} +(-2.91012 - 0.578860i) q^{45} +(6.72159 - 2.98120i) q^{46} +(4.88545 + 4.88545i) q^{47} +(0.639806 + 0.341277i) q^{48} +(-3.09606 + 7.47455i) q^{49} +(-1.19497 - 0.756344i) q^{50} +(0.670743 + 0.329833i) q^{51} +(3.25868 + 6.88226i) q^{52} +(-6.84389 - 2.83483i) q^{53} +(-0.550268 + 1.42743i) q^{54} +(3.72148 - 3.72148i) q^{55} +(7.17132 - 8.32438i) q^{56} +(0.153488 - 0.771636i) q^{57} +(0.331443 + 13.3844i) q^{58} +(0.336624 - 0.139434i) q^{59} +(-0.243370 + 0.268750i) q^{60} +(5.68633 + 8.51020i) q^{61} +(7.18693 + 6.83959i) q^{62} +(9.58372 + 6.40364i) q^{63} +(-6.85653 + 4.12166i) q^{64} +(-3.73422 + 0.742783i) q^{65} +(-0.777169 - 1.10299i) q^{66} -1.60911 q^{67} +(-6.92354 + 4.47935i) q^{68} +0.942566 q^{69} +(3.16428 + 4.49089i) q^{70} +(-15.4107 + 3.06538i) q^{71} +(-5.16749 - 6.61274i) q^{72} +(2.24376 + 1.49923i) q^{73} +(1.45918 + 1.38866i) q^{74} +(-0.100716 - 0.150732i) q^{75} +(6.43381 + 5.82623i) q^{76} +(-18.8885 + 7.82386i) q^{77} +(0.0241645 + 0.975815i) q^{78} +(0.100781 - 0.506660i) q^{79} +(-1.16443 - 3.82676i) q^{80} +(6.15558 - 6.15558i) q^{81} +(-2.38958 + 6.19873i) q^{82} +(-2.37736 - 0.984736i) q^{83} +(1.27296 - 0.602734i) q^{84} +(-1.32974 - 3.90279i) q^{85} +(0.921126 + 0.583020i) q^{86} +(-0.656772 + 1.58559i) q^{87} +(14.8449 - 1.10464i) q^{88} +(10.2434 + 10.2434i) q^{89} +(3.83581 - 1.70128i) q^{90} +(14.5061 + 2.88544i) q^{91} +(-5.34221 + 8.92163i) q^{92} +(0.486690 + 1.17497i) q^{93} +(-9.62740 - 1.66838i) q^{94} +(-3.60849 + 2.41112i) q^{95} +(-1.01765 + 0.126624i) q^{96} +(-3.06593 + 4.58848i) q^{97} +(-2.50925 - 11.1630i) q^{98} +(3.04652 + 15.3159i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q - 16 q^{24} - 32 q^{26} - 96 q^{28} - 80 q^{32} - 64 q^{34} - 64 q^{36} - 80 q^{38} - 96 q^{42} - 32 q^{44} - 16 q^{46} + 80 q^{54} + 80 q^{56} - 160 q^{57} - 64 q^{61} + 112 q^{62} + 96 q^{64} + 208 q^{66}+ \cdots - 272 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{7}{16}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15606 + 0.814564i −0.817461 + 0.575984i
\(3\) −0.177801 + 0.0353667i −0.102653 + 0.0204190i −0.246150 − 0.969232i \(-0.579165\pi\)
0.143496 + 0.989651i \(0.454165\pi\)
\(4\) 0.672971 − 1.88338i 0.336486 − 0.941689i
\(5\) 0.831470 + 0.555570i 0.371845 + 0.248459i
\(6\) 0.176741 − 0.185716i 0.0721540 − 0.0758183i
\(7\) −2.15819 − 3.22996i −0.815718 − 1.22081i −0.972437 − 0.233166i \(-0.925092\pi\)
0.156719 − 0.987643i \(-0.449908\pi\)
\(8\) 0.756133 + 2.72548i 0.267333 + 0.963604i
\(9\) −2.74128 + 1.13547i −0.913759 + 0.378491i
\(10\) −1.41378 + 0.0350101i −0.447077 + 0.0110712i
\(11\) 1.02675 − 5.16184i 0.309578 − 1.55635i −0.442187 − 0.896923i \(-0.645797\pi\)
0.751765 − 0.659431i \(-0.229203\pi\)
\(12\) −0.0530458 + 0.358666i −0.0153130 + 0.103538i
\(13\) −2.69222 + 2.69222i −0.746689 + 0.746689i −0.973856 − 0.227167i \(-0.927054\pi\)
0.227167 + 0.973856i \(0.427054\pi\)
\(14\) 5.12601 + 1.97606i 1.36998 + 0.528123i
\(15\) −0.167485 − 0.0693744i −0.0432443 − 0.0179124i
\(16\) −3.09422 − 2.53492i −0.773555 − 0.633729i
\(17\) −3.27391 − 2.50629i −0.794040 − 0.607865i
\(18\) 2.24418 − 3.54563i 0.528958 − 0.835712i
\(19\) −1.66081 + 4.00954i −0.381015 + 0.919852i 0.610755 + 0.791820i \(0.290866\pi\)
−0.991770 + 0.128032i \(0.959134\pi\)
\(20\) 1.60590 − 1.19209i 0.359091 − 0.266559i
\(21\) 0.497960 + 0.497960i 0.108664 + 0.108664i
\(22\) 3.01766 + 6.80378i 0.643367 + 1.45057i
\(23\) −5.09948 − 1.01435i −1.06332 − 0.211507i −0.367718 − 0.929937i \(-0.619861\pi\)
−0.695598 + 0.718431i \(0.744861\pi\)
\(24\) −0.230832 − 0.457851i −0.0471185 − 0.0934584i
\(25\) 0.382683 + 0.923880i 0.0765367 + 0.184776i
\(26\) 0.919396 − 5.30537i 0.180308 − 1.04047i
\(27\) 0.899439 − 0.600986i 0.173097 − 0.115660i
\(28\) −7.53563 + 1.89102i −1.42410 + 0.357368i
\(29\) 5.25962 − 7.87158i 0.976687 − 1.46172i 0.0918738 − 0.995771i \(-0.470714\pi\)
0.884814 − 0.465945i \(-0.154286\pi\)
\(30\) 0.250133 − 0.0562256i 0.0456678 − 0.0102653i
\(31\) −1.36864 − 6.88061i −0.245815 − 1.23579i −0.884578 − 0.466391i \(-0.845554\pi\)
0.638764 − 0.769403i \(-0.279446\pi\)
\(32\) 5.64197 + 0.410088i 0.997369 + 0.0724939i
\(33\) 0.954092i 0.166086i
\(34\) 5.82639 + 0.230627i 0.999218 + 0.0395522i
\(35\) − 3.88464i − 0.656623i
\(36\) 0.293726 + 5.92700i 0.0489543 + 0.987833i
\(37\) −0.277878 − 1.39699i −0.0456828 − 0.229663i 0.951198 − 0.308580i \(-0.0998538\pi\)
−0.996881 + 0.0789170i \(0.974854\pi\)
\(38\) −1.34603 − 5.98812i −0.218355 − 0.971402i
\(39\) 0.383464 − 0.573894i 0.0614034 − 0.0918966i
\(40\) −0.885496 + 2.68624i −0.140009 + 0.424732i
\(41\) 3.90589 − 2.60983i 0.609997 − 0.407587i −0.211844 − 0.977303i \(-0.567947\pi\)
0.821841 + 0.569717i \(0.192947\pi\)
\(42\) −0.981295 − 0.170054i −0.151417 − 0.0262399i
\(43\) −0.294987 − 0.712162i −0.0449851 − 0.108604i 0.899790 − 0.436324i \(-0.143720\pi\)
−0.944775 + 0.327720i \(0.893720\pi\)
\(44\) −9.03072 − 5.40754i −1.36143 − 0.815217i
\(45\) −2.91012 − 0.578860i −0.433816 − 0.0862913i
\(46\) 6.72159 − 2.98120i 0.991044 − 0.439554i
\(47\) 4.88545 + 4.88545i 0.712616 + 0.712616i 0.967082 − 0.254466i \(-0.0818997\pi\)
−0.254466 + 0.967082i \(0.581900\pi\)
\(48\) 0.639806 + 0.341277i 0.0923480 + 0.0492591i
\(49\) −3.09606 + 7.47455i −0.442294 + 1.06779i
\(50\) −1.19497 − 0.756344i −0.168994 − 0.106963i
\(51\) 0.670743 + 0.329833i 0.0939228 + 0.0461859i
\(52\) 3.25868 + 6.88226i 0.451898 + 0.954398i
\(53\) −6.84389 − 2.83483i −0.940080 − 0.389394i −0.140586 − 0.990068i \(-0.544899\pi\)
−0.799494 + 0.600674i \(0.794899\pi\)
\(54\) −0.550268 + 1.42743i −0.0748820 + 0.194249i
\(55\) 3.72148 − 3.72148i 0.501804 − 0.501804i
\(56\) 7.17132 − 8.32438i 0.958308 − 1.11239i
\(57\) 0.153488 − 0.771636i 0.0203300 − 0.102206i
\(58\) 0.331443 + 13.3844i 0.0435206 + 1.75745i
\(59\) 0.336624 − 0.139434i 0.0438247 − 0.0181528i −0.360663 − 0.932696i \(-0.617450\pi\)
0.404488 + 0.914543i \(0.367450\pi\)
\(60\) −0.243370 + 0.268750i −0.0314190 + 0.0346954i
\(61\) 5.68633 + 8.51020i 0.728060 + 1.08962i 0.992142 + 0.125119i \(0.0399313\pi\)
−0.264081 + 0.964500i \(0.585069\pi\)
\(62\) 7.18693 + 6.83959i 0.912741 + 0.868629i
\(63\) 9.58372 + 6.40364i 1.20744 + 0.806783i
\(64\) −6.85653 + 4.12166i −0.857066 + 0.515207i
\(65\) −3.73422 + 0.742783i −0.463173 + 0.0921309i
\(66\) −0.777169 − 1.10299i −0.0956629 − 0.135769i
\(67\) −1.60911 −0.196584 −0.0982918 − 0.995158i \(-0.531338\pi\)
−0.0982918 + 0.995158i \(0.531338\pi\)
\(68\) −6.92354 + 4.47935i −0.839603 + 0.543201i
\(69\) 0.942566 0.113472
\(70\) 3.16428 + 4.49089i 0.378204 + 0.536764i
\(71\) −15.4107 + 3.06538i −1.82892 + 0.363794i −0.984982 − 0.172655i \(-0.944766\pi\)
−0.843933 + 0.536449i \(0.819766\pi\)
\(72\) −5.16749 − 6.61274i −0.608994 − 0.779318i
\(73\) 2.24376 + 1.49923i 0.262612 + 0.175472i 0.679905 − 0.733300i \(-0.262021\pi\)
−0.417293 + 0.908772i \(0.637021\pi\)
\(74\) 1.45918 + 1.38866i 0.169626 + 0.161428i
\(75\) −0.100716 − 0.150732i −0.0116297 − 0.0174050i
\(76\) 6.43381 + 5.82623i 0.738008 + 0.668315i
\(77\) −18.8885 + 7.82386i −2.15254 + 0.891611i
\(78\) 0.0241645 + 0.975815i 0.00273610 + 0.110489i
\(79\) 0.100781 − 0.506660i 0.0113387 − 0.0570037i −0.974702 − 0.223510i \(-0.928249\pi\)
0.986040 + 0.166506i \(0.0532485\pi\)
\(80\) −1.16443 − 3.82676i −0.130187 − 0.427845i
\(81\) 6.15558 − 6.15558i 0.683953 − 0.683953i
\(82\) −2.38958 + 6.19873i −0.263885 + 0.684535i
\(83\) −2.37736 − 0.984736i −0.260950 − 0.108089i 0.248373 − 0.968664i \(-0.420104\pi\)
−0.509323 + 0.860576i \(0.670104\pi\)
\(84\) 1.27296 − 0.602734i 0.138891 − 0.0657637i
\(85\) −1.32974 − 3.90279i −0.144230 − 0.423317i
\(86\) 0.921126 + 0.583020i 0.0993276 + 0.0628686i
\(87\) −0.656772 + 1.58559i −0.0704133 + 0.169993i
\(88\) 14.8449 − 1.10464i 1.58247 − 0.117755i
\(89\) 10.2434 + 10.2434i 1.08579 + 1.08579i 0.995957 + 0.0898364i \(0.0286344\pi\)
0.0898364 + 0.995957i \(0.471366\pi\)
\(90\) 3.83581 − 1.70128i 0.404330 − 0.179331i
\(91\) 14.5061 + 2.88544i 1.52065 + 0.302476i
\(92\) −5.34221 + 8.92163i −0.556964 + 0.930144i
\(93\) 0.486690 + 1.17497i 0.0504674 + 0.121839i
\(94\) −9.62740 − 1.66838i −0.992991 − 0.172081i
\(95\) −3.60849 + 2.41112i −0.370224 + 0.247375i
\(96\) −1.01765 + 0.126624i −0.103863 + 0.0129235i
\(97\) −3.06593 + 4.58848i −0.311298 + 0.465890i −0.953821 − 0.300376i \(-0.902888\pi\)
0.642523 + 0.766266i \(0.277888\pi\)
\(98\) −2.50925 − 11.1630i −0.253473 − 1.12763i
\(99\) 3.04652 + 15.3159i 0.306187 + 1.53930i
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 340.2.bf.a.11.8 ✓ 288
4.3 odd 2 inner 340.2.bf.a.11.16 yes 288
17.14 odd 16 inner 340.2.bf.a.31.16 yes 288
68.31 even 16 inner 340.2.bf.a.31.8 yes 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bf.a.11.8 ✓ 288 1.1 even 1 trivial
340.2.bf.a.11.16 yes 288 4.3 odd 2 inner
340.2.bf.a.31.8 yes 288 68.31 even 16 inner
340.2.bf.a.31.16 yes 288 17.14 odd 16 inner