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Properties

Label 340.2.bf.a.11.9
Level $340$
Weight $2$
Character 340.11
Analytic conductor $2.715$
Analytic rank $0$
Dimension $288$
Inner twists $4$

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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] = [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 11.9
Character \(\chi\) \(=\) 340.11
Dual form 340.2.bf.a.31.9

$q$-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+(-0.956424 - 1.04175i) q^{2} +(-0.113145 + 0.0225060i) q^{3} +(-0.170507 + 1.99272i) q^{4} +(-0.831470 - 0.555570i) q^{5} +(0.131661 + 0.0963444i) q^{6} +(1.29866 + 1.94358i) q^{7} +(2.23900 - 1.72826i) q^{8} +(-2.75934 + 1.14296i) q^{9} +(0.216469 + 1.39755i) q^{10} +(0.268664 - 1.35066i) q^{11} +(-0.0255561 - 0.229304i) q^{12} +(-3.67119 + 3.67119i) q^{13} +(0.782667 - 3.21177i) q^{14} +(0.106581 + 0.0441471i) q^{15} +(-3.94186 - 0.679543i) q^{16} +(1.55336 + 3.81930i) q^{17} +(3.82978 + 1.78141i) q^{18} +(-1.89163 + 4.56679i) q^{19} +(1.24887 - 1.56216i) q^{20} +(-0.190679 - 0.190679i) q^{21} +(-1.66402 + 1.01193i) q^{22} +(9.11497 + 1.81308i) q^{23} +(-0.214436 + 0.245935i) q^{24} +(0.382683 + 0.923880i) q^{25} +(7.33569 + 0.313266i) q^{26} +(0.574243 - 0.383697i) q^{27} +(-4.09444 + 2.25647i) q^{28} +(-3.61732 + 5.41370i) q^{29} +(-0.0559457 - 0.153254i) q^{30} +(-1.36742 - 6.87450i) q^{31} +(3.06217 + 4.75638i) q^{32} +0.158868i q^{33} +(2.49311 - 5.27109i) q^{34} -2.33753i q^{35} +(-1.80711 - 5.69348i) q^{36} +(-0.224781 - 1.13005i) q^{37} +(6.56667 - 2.39718i) q^{38} +(0.332754 - 0.498001i) q^{39} +(-2.82183 + 0.193072i) q^{40} +(-2.14439 + 1.43283i) q^{41} +(-0.0162709 + 0.381012i) q^{42} +(4.81491 + 11.6242i) q^{43} +(2.64568 + 0.765668i) q^{44} +(2.92930 + 0.582675i) q^{45} +(-6.82899 - 11.2296i) q^{46} +(0.415634 + 0.415634i) q^{47} +(0.461296 - 0.0118283i) q^{48} +(0.587790 - 1.41905i) q^{49} +(0.596448 - 1.28228i) q^{50} +(-0.261712 - 0.397176i) q^{51} +(-6.68968 - 7.94160i) q^{52} +(-6.23827 - 2.58398i) q^{53} +(-0.948938 - 0.231244i) q^{54} +(-0.973774 + 0.973774i) q^{55} +(6.26671 + 2.10726i) q^{56} +(0.111248 - 0.559283i) q^{57} +(9.09943 - 1.40943i) q^{58} +(9.07703 - 3.75983i) q^{59} +(-0.106145 + 0.204858i) q^{60} +(-0.482152 - 0.721592i) q^{61} +(-5.85371 + 7.99946i) q^{62} +(-5.80488 - 3.87870i) q^{63} +(2.02625 - 7.73914i) q^{64} +(5.09208 - 1.01288i) q^{65} +(0.165501 - 0.151945i) q^{66} -7.28579 q^{67} +(-7.87565 + 2.44419i) q^{68} -1.07212 q^{69} +(-2.43513 + 2.23567i) q^{70} +(-10.4081 + 2.07029i) q^{71} +(-4.20285 + 7.32794i) q^{72} +(1.79299 + 1.19804i) q^{73} +(-0.962250 + 1.31497i) q^{74} +(-0.0640916 - 0.0959199i) q^{75} +(-8.77779 - 4.54815i) q^{76} +(2.97403 - 1.23188i) q^{77} +(-0.837048 + 0.129652i) q^{78} +(-1.28460 + 6.45814i) q^{79} +(2.90000 + 2.75500i) q^{80} +(6.27939 - 6.27939i) q^{81} +(3.54360 + 0.863529i) q^{82} +(-13.4065 - 5.55315i) q^{83} +(0.412483 - 0.347458i) q^{84} +(0.830320 - 4.03863i) q^{85} +(7.50449 - 16.1336i) q^{86} +(0.287442 - 0.693945i) q^{87} +(-1.73276 - 3.48846i) q^{88} +(-5.91159 - 5.91159i) q^{89} +(-2.19465 - 3.60890i) q^{90} +(-11.9029 - 2.36763i) q^{91} +(-5.16712 + 17.8544i) q^{92} +(0.309435 + 0.747042i) q^{93} +(0.0354665 - 0.830511i) q^{94} +(4.11000 - 2.74622i) q^{95} +(-0.453517 - 0.469245i) q^{96} +(-2.26540 + 3.39040i) q^{97} +(-2.04048 + 0.744881i) q^{98} +(0.802416 + 4.03402i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.956424 − 1.04175i −0.676294 − 0.736632i
\(3\) −0.113145 + 0.0225060i −0.0653244 + 0.0129938i −0.227644 − 0.973744i \(-0.573102\pi\)
0.162320 + 0.986738i \(0.448102\pi\)
\(4\) −0.170507 + 1.99272i −0.0852533 + 0.996359i
\(5\) −0.831470 − 0.555570i −0.371845 − 0.248459i
\(6\) 0.131661 + 0.0963444i 0.0537502 + 0.0393324i
\(7\) 1.29866 + 1.94358i 0.490847 + 0.734605i 0.991366 − 0.131120i \(-0.0418575\pi\)
−0.500519 + 0.865726i \(0.666857\pi\)
\(8\) 2.23900 − 1.72826i 0.791606 − 0.611031i
\(9\) −2.75934 + 1.14296i −0.919781 + 0.380986i
\(10\) 0.216469 + 1.39755i 0.0684536 + 0.441944i
\(11\) 0.268664 − 1.35066i 0.0810052 − 0.407240i −0.918913 − 0.394460i \(-0.870932\pi\)
0.999918 − 0.0127805i \(-0.00406828\pi\)
\(12\) −0.0255561 − 0.229304i −0.00737741 − 0.0661944i
\(13\) −3.67119 + 3.67119i −1.01820 + 1.01820i −0.0183724 + 0.999831i \(0.505848\pi\)
−0.999831 + 0.0183724i \(0.994152\pi\)
\(14\) 0.782667 − 3.21177i 0.209176 − 0.858383i
\(15\) 0.106581 + 0.0441471i 0.0275190 + 0.0113987i
\(16\) −3.94186 − 0.679543i −0.985464 − 0.169886i
\(17\) 1.55336 + 3.81930i 0.376745 + 0.926317i
\(18\) 3.82978 + 1.78141i 0.902689 + 0.419882i
\(19\) −1.89163 + 4.56679i −0.433969 + 1.04769i 0.544027 + 0.839068i \(0.316899\pi\)
−0.977996 + 0.208625i \(0.933101\pi\)
\(20\) 1.24887 − 1.56216i 0.279255 − 0.349309i
\(21\) −0.190679 − 0.190679i −0.0416097 − 0.0416097i
\(22\) −1.66402 + 1.01193i −0.354770 + 0.215743i
\(23\) 9.11497 + 1.81308i 1.90060 + 0.378053i 0.998562 − 0.0535999i \(-0.0170696\pi\)
0.902040 + 0.431653i \(0.142070\pi\)
\(24\) −0.214436 + 0.245935i −0.0437716 + 0.0502013i
\(25\) 0.382683 + 0.923880i 0.0765367 + 0.184776i
\(26\) 7.33569 + 0.313266i 1.43865 + 0.0614365i
\(27\) 0.574243 − 0.383697i 0.110513 − 0.0738425i
\(28\) −4.09444 + 2.25647i −0.773777 + 0.426433i
\(29\) −3.61732 + 5.41370i −0.671719 + 1.00530i 0.326474 + 0.945206i \(0.394139\pi\)
−0.998193 + 0.0600922i \(0.980861\pi\)
\(30\) −0.0559457 − 0.153254i −0.0102142 − 0.0279802i
\(31\) −1.36742 − 6.87450i −0.245596 − 1.23470i −0.884914 − 0.465754i \(-0.845783\pi\)
0.639318 − 0.768943i \(-0.279217\pi\)
\(32\) 3.06217 + 4.75638i 0.541320 + 0.840817i
\(33\) 0.158868i 0.0276553i
\(34\) 2.49311 − 5.27109i 0.427564 − 0.903985i
\(35\) − 2.33753i − 0.395114i
\(36\) −1.80711 − 5.69348i −0.301184 − 0.948913i
\(37\) −0.224781 − 1.13005i −0.0369538 − 0.185779i 0.957900 − 0.287103i \(-0.0926922\pi\)
−0.994853 + 0.101324i \(0.967692\pi\)
\(38\) 6.56667 − 2.39718i 1.06525 − 0.388873i
\(39\) 0.332754 − 0.498001i 0.0532832 − 0.0797440i
\(40\) −2.82183 + 0.193072i −0.446170 + 0.0305273i
\(41\) −2.14439 + 1.43283i −0.334897 + 0.223771i −0.711636 − 0.702549i \(-0.752045\pi\)
0.376739 + 0.926319i \(0.377045\pi\)
\(42\) −0.0162709 + 0.381012i −0.00251065 + 0.0587914i
\(43\) 4.81491 + 11.6242i 0.734267 + 1.77268i 0.627821 + 0.778358i \(0.283947\pi\)
0.106446 + 0.994319i \(0.466053\pi\)
\(44\) 2.64568 + 0.765668i 0.398852 + 0.115429i
\(45\) 2.92930 + 0.582675i 0.436675 + 0.0868600i
\(46\) −6.82899 − 11.2296i −1.00688 − 1.65572i
\(47\) 0.415634 + 0.415634i 0.0606265 + 0.0606265i 0.736770 − 0.676143i \(-0.236350\pi\)
−0.676143 + 0.736770i \(0.736350\pi\)
\(48\) 0.461296 − 0.0118283i 0.0665823 − 0.00170727i
\(49\) 0.587790 − 1.41905i 0.0839700 − 0.202722i
\(50\) 0.596448 − 1.28228i 0.0843505 − 0.181342i
\(51\) −0.261712 − 0.397176i −0.0366471 − 0.0556158i
\(52\) −6.68968 − 7.94160i −0.927692 − 1.10130i
\(53\) −6.23827 − 2.58398i −0.856892 − 0.354936i −0.0894010 − 0.995996i \(-0.528495\pi\)
−0.767491 + 0.641059i \(0.778495\pi\)
\(54\) −0.948938 − 0.231244i −0.129134 − 0.0314683i
\(55\) −0.973774 + 0.973774i −0.131304 + 0.131304i
\(56\) 6.26671 + 2.10726i 0.837425 + 0.281595i
\(57\) 0.111248 − 0.559283i 0.0147352 − 0.0740789i
\(58\) 9.09943 − 1.40943i 1.19481 − 0.185068i
\(59\) 9.07703 − 3.75983i 1.18173 − 0.489488i 0.296676 − 0.954978i \(-0.404122\pi\)
0.885053 + 0.465490i \(0.154122\pi\)
\(60\) −0.106145 + 0.204858i −0.0137033 + 0.0264470i
\(61\) −0.482152 − 0.721592i −0.0617333 − 0.0923904i 0.799319 − 0.600907i \(-0.205194\pi\)
−0.861052 + 0.508517i \(0.830194\pi\)
\(62\) −5.85371 + 7.99946i −0.743422 + 1.01593i
\(63\) −5.80488 − 3.87870i −0.731346 − 0.488670i
\(64\) 2.02625 − 7.73914i 0.253281 − 0.967393i
\(65\) 5.09208 − 1.01288i 0.631595 − 0.125632i
\(66\) 0.165501 − 0.151945i 0.0203718 − 0.0187031i
\(67\) −7.28579 −0.890101 −0.445051 − 0.895505i \(-0.646814\pi\)
−0.445051 + 0.895505i \(0.646814\pi\)
\(68\) −7.87565 + 2.44419i −0.955063 + 0.296402i
\(69\) −1.07212 −0.129068
\(70\) −2.43513 + 2.23567i −0.291054 + 0.267213i
\(71\) −10.4081 + 2.07029i −1.23521 + 0.245699i −0.769158 − 0.639058i \(-0.779324\pi\)
−0.466053 + 0.884757i \(0.654324\pi\)
\(72\) −4.20285 + 7.32794i −0.495310 + 0.863606i
\(73\) 1.79299 + 1.19804i 0.209853 + 0.140219i 0.656055 − 0.754713i \(-0.272224\pi\)
−0.446202 + 0.894932i \(0.647224\pi\)
\(74\) −0.962250 + 1.31497i −0.111859 + 0.152863i
\(75\) −0.0640916 − 0.0959199i −0.00740067 − 0.0110759i
\(76\) −8.77779 − 4.54815i −1.00688 − 0.521708i
\(77\) 2.97403 − 1.23188i 0.338922 − 0.140386i
\(78\) −0.837048 + 0.129652i −0.0947771 + 0.0146802i
\(79\) −1.28460 + 6.45814i −0.144529 + 0.726597i 0.838753 + 0.544511i \(0.183285\pi\)
−0.983283 + 0.182086i \(0.941715\pi\)
\(80\) 2.90000 + 2.75500i 0.324230 + 0.308018i
\(81\) 6.27939 − 6.27939i 0.697710 − 0.697710i
\(82\) 3.54360 + 0.863529i 0.391325 + 0.0953608i
\(83\) −13.4065 − 5.55315i −1.47155 − 0.609538i −0.504341 − 0.863504i \(-0.668265\pi\)
−0.967213 + 0.253967i \(0.918265\pi\)
\(84\) 0.412483 − 0.347458i 0.0450055 − 0.0379108i
\(85\) 0.830320 − 4.03863i 0.0900608 − 0.438051i
\(86\) 7.50449 − 16.1336i 0.809230 − 1.73973i
\(87\) 0.287442 − 0.693945i 0.0308170 − 0.0743988i
\(88\) −1.73276 − 3.48846i −0.184712 − 0.371871i
\(89\) −5.91159 − 5.91159i −0.626627 − 0.626627i 0.320591 − 0.947218i \(-0.396119\pi\)
−0.947218 + 0.320591i \(0.896119\pi\)
\(90\) −2.19465 − 3.60890i −0.231337 − 0.380411i
\(91\) −11.9029 − 2.36763i −1.24776 − 0.248195i
\(92\) −5.16712 + 17.8544i −0.538709 + 1.86145i
\(93\) 0.309435 + 0.747042i 0.0320869 + 0.0774646i
\(94\) 0.0354665 − 0.830511i 0.00365809 − 0.0856607i
\(95\) 4.11000 − 2.74622i 0.421677 − 0.281756i
\(96\) −0.453517 − 0.469245i −0.0462869 − 0.0478921i
\(97\) −2.26540 + 3.39040i −0.230016 + 0.344243i −0.928467 − 0.371414i \(-0.878873\pi\)
0.698451 + 0.715658i \(0.253873\pi\)
\(98\) −2.04048 + 0.744881i −0.206120 + 0.0752443i
\(99\) 0.802416 + 4.03402i 0.0806458 + 0.405434i
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.11.9 yes 288
4.3 odd 2 inner 340.2.bf.a.11.1 ✓ 288
17.14 odd 16 inner 340.2.bf.a.31.1 yes 288
68.31 even 16 inner 340.2.bf.a.31.9 yes 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bf.a.11.1 ✓ 288 4.3 odd 2 inner
340.2.bf.a.11.9 yes 288 1.1 even 1 trivial
340.2.bf.a.31.1 yes 288 17.14 odd 16 inner
340.2.bf.a.31.9 yes 288 68.31 even 16 inner