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Properties

Label 294.2.p.a.5.4
Level $294$
Weight $2$
Character 294.5
Analytic conductor $2.348$
Analytic rank $0$
Dimension $216$
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] = [294,2,Mod(5,294)] 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("294.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 29])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.p (of order \(42\), degree \(12\), 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: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 5.4
Character \(\chi\) \(=\) 294.5
Dual form 294.2.p.a.59.4

$q$-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+(-0.149042 - 0.988831i) q^{2} +(-0.406691 + 1.68363i) q^{3} +(-0.955573 + 0.294755i) q^{4} +(-0.0790178 + 1.05442i) q^{5} +(1.72544 + 0.151217i) q^{6} +(-2.58461 - 0.565480i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-2.66920 - 1.36943i) q^{9} +(1.05442 - 0.0790178i) q^{10} +(3.39470 + 1.33232i) q^{11} +(-0.107635 - 1.72870i) q^{12} +(-4.20767 + 3.35551i) q^{13} +(-0.173948 + 2.64003i) q^{14} +(-1.74311 - 0.561859i) q^{15} +(0.826239 - 0.563320i) q^{16} +(-4.60639 + 4.27411i) q^{17} +(-0.956313 + 2.84350i) q^{18} +(-4.49613 - 2.59584i) q^{19} +(-0.235288 - 1.03086i) q^{20} +(2.00320 - 4.12155i) q^{21} +(0.811486 - 3.55535i) q^{22} +(-3.42372 + 3.68990i) q^{23} +(-1.69335 + 0.364083i) q^{24} +(3.83860 + 0.578576i) q^{25} +(3.94515 + 3.66056i) q^{26} +(3.39116 - 3.93701i) q^{27} +(2.63647 - 0.221471i) q^{28} +(6.65392 - 1.51871i) q^{29} +(-0.295786 + 1.80738i) q^{30} +(-7.28580 + 4.20646i) q^{31} +(-0.680173 - 0.733052i) q^{32} +(-3.62372 + 5.17356i) q^{33} +(4.91292 + 3.91792i) q^{34} +(0.800484 - 2.68058i) q^{35} +(2.95427 + 0.521830i) q^{36} +(3.36622 + 1.03834i) q^{37} +(-1.89674 + 4.83281i) q^{38} +(-3.93820 - 8.44881i) q^{39} +(-0.984283 + 0.386303i) q^{40} +(9.91464 - 4.77464i) q^{41} +(-4.37408 - 1.36654i) q^{42} +(-2.06966 - 0.996693i) q^{43} +(-3.63659 - 0.272525i) q^{44} +(1.65487 - 2.70625i) q^{45} +(4.15896 + 2.83553i) q^{46} +(6.97372 - 1.05112i) q^{47} +(0.612397 + 1.62018i) q^{48} +(6.36046 + 2.92310i) q^{49} -3.88196i q^{50} +(-5.32263 - 9.49369i) q^{51} +(3.03168 - 4.44666i) q^{52} +(1.97402 + 6.39960i) q^{53} +(-4.39846 - 2.76650i) q^{54} +(-1.67306 + 3.47416i) q^{55} +(-0.611942 - 2.57401i) q^{56} +(6.19897 - 6.51411i) q^{57} +(-2.49347 - 6.35325i) q^{58} +(-0.332296 - 4.43419i) q^{59} +(1.83128 + 0.0231057i) q^{60} +(-0.977226 + 3.16809i) q^{61} +(5.24537 + 6.57749i) q^{62} +(6.12448 + 5.04884i) q^{63} +(-0.623490 + 0.781831i) q^{64} +(-3.20563 - 4.70179i) q^{65} +(5.65587 + 2.81217i) q^{66} +(-2.72185 - 4.71438i) q^{67} +(3.14193 - 5.44198i) q^{68} +(-4.82002 - 7.26493i) q^{69} +(-2.76995 - 0.392023i) q^{70} +(-9.94200 - 2.26920i) q^{71} +(0.0756914 - 2.99904i) q^{72} +(-0.116401 + 0.772270i) q^{73} +(0.525035 - 3.48338i) q^{74} +(-2.53523 + 6.22747i) q^{75} +(5.06152 + 1.15526i) q^{76} +(-8.02058 - 5.36317i) q^{77} +(-7.76748 + 5.15344i) q^{78} +(-1.32947 + 2.30272i) q^{79} +(0.528688 + 0.915714i) q^{80} +(5.24931 + 7.31059i) q^{81} +(-6.19901 - 9.09228i) q^{82} +(-2.33922 + 2.93328i) q^{83} +(-0.699352 + 4.52890i) q^{84} +(-4.14271 - 5.19480i) q^{85} +(-0.677095 + 2.19509i) q^{86} +(-0.149140 + 11.8204i) q^{87} +(0.272525 + 3.63659i) q^{88} +(6.45187 + 16.4391i) q^{89} +(-2.92267 - 1.23304i) q^{90} +(12.7727 - 6.29333i) q^{91} +(2.18400 - 4.53513i) q^{92} +(-4.11904 - 13.9773i) q^{93} +(-2.07876 - 6.73917i) q^{94} +(3.09238 - 4.53569i) q^{95} +(1.51081 - 0.847032i) q^{96} -1.61706i q^{97} +(1.94247 - 6.72509i) q^{98} +(-7.23662 - 8.20504i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q - 18 q^{4} + 14 q^{6} + 10 q^{7} + 16 q^{9} + 6 q^{10} - 12 q^{15} + 18 q^{16} - 4 q^{18} - 12 q^{19} + 10 q^{22} - 6 q^{24} + 16 q^{25} - 42 q^{27} + 2 q^{28} + 4 q^{30} + 6 q^{31} + 18 q^{33} - 10 q^{36}+ \cdots + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(e\left(\frac{29}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.149042 − 0.988831i −0.105389 − 0.699209i
\(3\) −0.406691 + 1.68363i −0.234803 + 0.972043i
\(4\) −0.955573 + 0.294755i −0.477786 + 0.147378i
\(5\) −0.0790178 + 1.05442i −0.0353378 + 0.471550i 0.951261 + 0.308386i \(0.0997890\pi\)
−0.986599 + 0.163164i \(0.947830\pi\)
\(6\) 1.72544 + 0.151217i 0.704407 + 0.0617340i
\(7\) −2.58461 − 0.565480i −0.976892 − 0.213732i
\(8\) 0.433884 + 0.900969i 0.153401 + 0.318541i
\(9\) −2.66920 − 1.36943i −0.889735 − 0.456477i
\(10\) 1.05442 − 0.0790178i 0.333436 − 0.0249876i
\(11\) 3.39470 + 1.33232i 1.02354 + 0.401710i 0.816944 − 0.576717i \(-0.195666\pi\)
0.206595 + 0.978426i \(0.433762\pi\)
\(12\) −0.107635 − 1.72870i −0.0310716 − 0.499034i
\(13\) −4.20767 + 3.35551i −1.16700 + 0.930650i −0.998484 − 0.0550493i \(-0.982468\pi\)
−0.168514 + 0.985699i \(0.553897\pi\)
\(14\) −0.173948 + 2.64003i −0.0464895 + 0.705577i
\(15\) −1.74311 − 0.561859i −0.450070 − 0.145071i
\(16\) 0.826239 − 0.563320i 0.206560 − 0.140830i
\(17\) −4.60639 + 4.27411i −1.11721 + 1.03662i −0.118167 + 0.992994i \(0.537702\pi\)
−0.999048 + 0.0436299i \(0.986108\pi\)
\(18\) −0.956313 + 2.84350i −0.225405 + 0.670218i
\(19\) −4.49613 − 2.59584i −1.03148 − 0.595528i −0.114075 − 0.993472i \(-0.536390\pi\)
−0.917409 + 0.397945i \(0.869724\pi\)
\(20\) −0.235288 − 1.03086i −0.0526120 − 0.230508i
\(21\) 2.00320 − 4.12155i 0.437134 − 0.899397i
\(22\) 0.811486 − 3.55535i 0.173009 − 0.758004i
\(23\) −3.42372 + 3.68990i −0.713896 + 0.769397i −0.980945 − 0.194286i \(-0.937761\pi\)
0.267049 + 0.963683i \(0.413951\pi\)
\(24\) −1.69335 + 0.364083i −0.345654 + 0.0743181i
\(25\) 3.83860 + 0.578576i 0.767720 + 0.115715i
\(26\) 3.94515 + 3.66056i 0.773707 + 0.717895i
\(27\) 3.39116 − 3.93701i 0.652628 − 0.757678i
\(28\) 2.63647 − 0.221471i 0.498245 − 0.0418540i
\(29\) 6.65392 − 1.51871i 1.23560 − 0.282018i 0.445661 − 0.895202i \(-0.352969\pi\)
0.789940 + 0.613184i \(0.210111\pi\)
\(30\) −0.295786 + 1.80738i −0.0540029 + 0.329982i
\(31\) −7.28580 + 4.20646i −1.30857 + 0.755503i −0.981857 − 0.189621i \(-0.939274\pi\)
−0.326712 + 0.945124i \(0.605941\pi\)
\(32\) −0.680173 − 0.733052i −0.120239 − 0.129586i
\(33\) −3.62372 + 5.17356i −0.630809 + 0.900602i
\(34\) 4.91292 + 3.91792i 0.842558 + 0.671918i
\(35\) 0.800484 − 2.68058i 0.135306 − 0.453101i
\(36\) 2.95427 + 0.521830i 0.492378 + 0.0869717i
\(37\) 3.36622 + 1.03834i 0.553403 + 0.170702i 0.558829 − 0.829283i \(-0.311251\pi\)
−0.00542577 + 0.999985i \(0.501727\pi\)
\(38\) −1.89674 + 4.83281i −0.307691 + 0.783985i
\(39\) −3.93820 − 8.44881i −0.630617 − 1.35289i
\(40\) −0.984283 + 0.386303i −0.155629 + 0.0610798i
\(41\) 9.91464 − 4.77464i 1.54841 − 0.745674i 0.552285 − 0.833655i \(-0.313756\pi\)
0.996122 + 0.0879819i \(0.0280418\pi\)
\(42\) −4.37408 − 1.36654i −0.674935 − 0.210861i
\(43\) −2.06966 − 0.996693i −0.315619 − 0.151994i 0.269367 − 0.963038i \(-0.413186\pi\)
−0.584986 + 0.811043i \(0.698900\pi\)
\(44\) −3.63659 − 0.272525i −0.548236 − 0.0410846i
\(45\) 1.65487 − 2.70625i 0.246693 − 0.403424i
\(46\) 4.15896 + 2.83553i 0.613206 + 0.418077i
\(47\) 6.97372 − 1.05112i 1.01722 − 0.153322i 0.380796 − 0.924659i \(-0.375650\pi\)
0.636426 + 0.771338i \(0.280412\pi\)
\(48\) 0.612397 + 1.62018i 0.0883920 + 0.233852i
\(49\) 6.36046 + 2.92310i 0.908638 + 0.417585i
\(50\) − 3.88196i − 0.548992i
\(51\) −5.32263 − 9.49369i −0.745317 − 1.32938i
\(52\) 3.03168 − 4.44666i 0.420419 − 0.616641i
\(53\) 1.97402 + 6.39960i 0.271152 + 0.879053i 0.984066 + 0.177802i \(0.0568987\pi\)
−0.712914 + 0.701251i \(0.752625\pi\)
\(54\) −4.39846 − 2.76650i −0.598555 − 0.376473i
\(55\) −1.67306 + 3.47416i −0.225596 + 0.468455i
\(56\) −0.611942 − 2.57401i −0.0817742 − 0.343967i
\(57\) 6.19897 − 6.51411i 0.821074 − 0.862815i
\(58\) −2.49347 − 6.35325i −0.327408 − 0.834222i
\(59\) −0.332296 − 4.43419i −0.0432613 − 0.577282i −0.976147 − 0.217110i \(-0.930337\pi\)
0.932886 − 0.360172i \(-0.117282\pi\)
\(60\) 1.83128 + 0.0231057i 0.236418 + 0.00298293i
\(61\) −0.977226 + 3.16809i −0.125121 + 0.405632i −0.996362 − 0.0852234i \(-0.972840\pi\)
0.871241 + 0.490856i \(0.163316\pi\)
\(62\) 5.24537 + 6.57749i 0.666163 + 0.835342i
\(63\) 6.12448 + 5.04884i 0.771612 + 0.636094i
\(64\) −0.623490 + 0.781831i −0.0779362 + 0.0977289i
\(65\) −3.20563 − 4.70179i −0.397609 − 0.583186i
\(66\) 5.65587 + 2.81217i 0.696189 + 0.346154i
\(67\) −2.72185 − 4.71438i −0.332527 − 0.575953i 0.650480 − 0.759524i \(-0.274568\pi\)
−0.983007 + 0.183570i \(0.941235\pi\)
\(68\) 3.14193 − 5.44198i 0.381015 − 0.659937i
\(69\) −4.82002 − 7.26493i −0.580262 − 0.874594i
\(70\) −2.76995 − 0.392023i −0.331072 − 0.0468557i
\(71\) −9.94200 − 2.26920i −1.17990 − 0.269304i −0.412793 − 0.910825i \(-0.635447\pi\)
−0.767106 + 0.641521i \(0.778304\pi\)
\(72\) 0.0756914 − 2.99904i 0.00892031 − 0.353441i
\(73\) −0.116401 + 0.772270i −0.0136237 + 0.0903874i −0.994609 − 0.103693i \(-0.966934\pi\)
0.980986 + 0.194080i \(0.0621722\pi\)
\(74\) 0.525035 − 3.48338i 0.0610340 − 0.404934i
\(75\) −2.53523 + 6.22747i −0.292743 + 0.719086i
\(76\) 5.06152 + 1.15526i 0.580596 + 0.132517i
\(77\) −8.02058 − 5.36317i −0.914030 − 0.611190i
\(78\) −7.76748 + 5.15344i −0.879494 + 0.583513i
\(79\) −1.32947 + 2.30272i −0.149578 + 0.259076i −0.931071 − 0.364837i \(-0.881125\pi\)
0.781494 + 0.623913i \(0.214458\pi\)
\(80\) 0.528688 + 0.915714i 0.0591091 + 0.102380i
\(81\) 5.24931 + 7.31059i 0.583257 + 0.812288i
\(82\) −6.19901 − 9.09228i −0.684566 − 1.00407i
\(83\) −2.33922 + 2.93328i −0.256762 + 0.321970i −0.893459 − 0.449144i \(-0.851729\pi\)
0.636697 + 0.771114i \(0.280300\pi\)
\(84\) −0.699352 + 4.52890i −0.0763056 + 0.494143i
\(85\) −4.14271 − 5.19480i −0.449340 − 0.563455i
\(86\) −0.677095 + 2.19509i −0.0730130 + 0.236702i
\(87\) −0.149140 + 11.8204i −0.0159895 + 1.26728i
\(88\) 0.272525 + 3.63659i 0.0290512 + 0.387662i
\(89\) 6.45187 + 16.4391i 0.683897 + 1.74254i 0.669690 + 0.742641i \(0.266427\pi\)
0.0142070 + 0.999899i \(0.495478\pi\)
\(90\) −2.92267 − 1.23304i −0.308076 − 0.129974i
\(91\) 12.7727 − 6.29333i 1.33894 − 0.659721i
\(92\) 2.18400 − 4.53513i 0.227698 − 0.472820i
\(93\) −4.11904 − 13.9773i −0.427125 − 1.44938i
\(94\) −2.07876 − 6.73917i −0.214408 − 0.695092i
\(95\) 3.09238 − 4.53569i 0.317272 − 0.465352i
\(96\) 1.51081 − 0.847032i 0.154196 − 0.0864499i
\(97\) − 1.61706i − 0.164187i −0.996625 − 0.0820936i \(-0.973839\pi\)
0.996625 − 0.0820936i \(-0.0261607\pi\)
\(98\) 1.94247 − 6.72509i 0.196219 − 0.679336i
\(99\) −7.23662 − 8.20504i −0.727308 − 0.824638i
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 294.2.p.a.5.4 ✓ 216
3.2 odd 2 inner 294.2.p.a.5.12 yes 216
49.10 odd 42 inner 294.2.p.a.59.12 yes 216
147.59 even 42 inner 294.2.p.a.59.4 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.p.a.5.4 ✓ 216 1.1 even 1 trivial
294.2.p.a.5.12 yes 216 3.2 odd 2 inner
294.2.p.a.59.4 yes 216 147.59 even 42 inner
294.2.p.a.59.12 yes 216 49.10 odd 42 inner