↑

Properties

Label 294.2.j.a.41.4
Level $294$
Weight $2$
Character 294.41
Analytic conductor $2.348$
Analytic rank $0$
Dimension $120$
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] = [294,2,Mod(41,294)] 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("294.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.j (of order \(14\), degree \(6\), 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: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 41.4
Character \(\chi\) \(=\) 294.41
Dual form 294.2.j.a.251.4

$q$-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+(-0.781831 + 0.623490i) q^{2} +(-0.580532 - 1.63186i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-3.14996 + 1.51694i) q^{5} +(1.47133 + 0.913887i) q^{6} +(2.47808 + 0.926885i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-2.32596 + 1.89470i) q^{9} +(1.51694 - 3.14996i) q^{10} +(1.47826 - 1.17888i) q^{11} +(-1.72013 + 0.202853i) q^{12} +(-0.731696 + 0.583508i) q^{13} +(-2.51534 + 0.820391i) q^{14} +(4.30409 + 4.25967i) q^{15} +(-0.900969 - 0.433884i) q^{16} +(1.24123 + 5.43819i) q^{17} +(0.637185 - 2.93155i) q^{18} +6.71507i q^{19} +(0.777975 + 3.40853i) q^{20} +(0.0739439 - 4.58198i) q^{21} +(-0.420736 + 1.84336i) q^{22} +(5.84179 + 1.33335i) q^{23} +(1.21838 - 1.23108i) q^{24} +(4.50368 - 5.64743i) q^{25} +(0.208252 - 0.912410i) q^{26} +(4.44219 + 2.69572i) q^{27} +(1.45507 - 2.20970i) q^{28} +(0.519155 - 0.118494i) q^{29} +(-6.02094 - 0.646788i) q^{30} +5.34122i q^{31} +(0.974928 - 0.222521i) q^{32} +(-2.78195 - 1.72795i) q^{33} +(-4.36109 - 3.47785i) q^{34} +(-9.21188 + 0.839452i) q^{35} +(1.32962 + 2.68926i) q^{36} +(0.612916 + 2.68536i) q^{37} +(-4.18678 - 5.25005i) q^{38} +(1.37698 + 0.855284i) q^{39} +(-2.73343 - 2.17984i) q^{40} +(-1.61381 + 0.777171i) q^{41} +(2.79901 + 3.62844i) q^{42} +(-9.41518 - 4.53411i) q^{43} +(-0.820375 - 1.70353i) q^{44} +(4.45254 - 9.49657i) q^{45} +(-5.39863 + 2.59984i) q^{46} +(-7.08465 - 8.88387i) q^{47} +(-0.184998 + 1.72214i) q^{48} +(5.28177 + 4.59379i) q^{49} +7.22334i q^{50} +(8.15381 - 5.18257i) q^{51} +(0.406061 + 0.843194i) q^{52} +(-5.46957 - 1.24839i) q^{53} +(-5.15380 + 0.662060i) q^{54} +(-2.86818 + 5.95585i) q^{55} +(0.240105 + 2.63483i) q^{56} +(10.9581 - 3.89832i) q^{57} +(-0.332012 + 0.416330i) q^{58} +(0.981235 + 0.472538i) q^{59} +(5.11062 - 3.24831i) q^{60} +(4.27189 - 0.975031i) q^{61} +(-3.33020 - 4.17593i) q^{62} +(-7.52010 + 2.53932i) q^{63} +(-0.623490 + 0.781831i) q^{64} +(1.41966 - 2.94796i) q^{65} +(3.25237 - 0.383549i) q^{66} -9.15032 q^{67} +5.57804 q^{68} +(-1.21550 - 10.3071i) q^{69} +(6.67875 - 6.39982i) q^{70} +(5.70374 + 1.30184i) q^{71} +(-2.71626 - 1.27354i) q^{72} +(1.01364 + 0.808350i) q^{73} +(-2.15349 - 1.71735i) q^{74} +(-11.8304 - 4.07087i) q^{75} +(6.54671 + 1.49424i) q^{76} +(4.75594 - 1.55117i) q^{77} +(-1.60983 + 0.189845i) q^{78} +15.3224 q^{79} +3.49619 q^{80} +(1.82022 - 8.81401i) q^{81} +(0.777171 - 1.61381i) q^{82} +(-9.92846 + 12.4499i) q^{83} +(-4.45065 - 1.09168i) q^{84} +(-12.1592 - 15.2472i) q^{85} +(10.1881 - 2.32536i) q^{86} +(-0.494752 - 0.778401i) q^{87} +(1.70353 + 0.820375i) q^{88} +(-6.59006 + 8.26368i) q^{89} +(2.43988 + 10.2008i) q^{90} +(-2.35405 + 0.767783i) q^{91} +(2.59984 - 5.39863i) q^{92} +(8.71615 - 3.10075i) q^{93} +(11.0780 + 2.52848i) q^{94} +(-10.1864 - 21.1522i) q^{95} +(-0.929101 - 1.46177i) q^{96} -7.02844i q^{97} +(-6.99364 - 0.298440i) q^{98} +(-1.20477 + 5.54289i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 20 q^{4} - 14 q^{6} + 4 q^{7} - 10 q^{9} + 12 q^{15} - 20 q^{16} - 8 q^{18} - 12 q^{21} - 10 q^{22} - 24 q^{25} + 42 q^{27} - 4 q^{28} + 8 q^{30} + 10 q^{36} - 58 q^{37} + 8 q^{39} - 14 q^{40} - 2 q^{42}+ \cdots - 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.781831 + 0.623490i −0.552838 + 0.440874i
\(3\) −0.580532 − 1.63186i −0.335171 − 0.942157i
\(4\) 0.222521 − 0.974928i 0.111260 − 0.487464i
\(5\) −3.14996 + 1.51694i −1.40870 + 0.678396i −0.974907 − 0.222614i \(-0.928541\pi\)
−0.433797 + 0.901010i \(0.642827\pi\)
\(6\) 1.47133 + 0.913887i 0.600668 + 0.373093i
\(7\) 2.47808 + 0.926885i 0.936627 + 0.350329i
\(8\) 0.433884 + 0.900969i 0.153401 + 0.318541i
\(9\) −2.32596 + 1.89470i −0.775321 + 0.631567i
\(10\) 1.51694 − 3.14996i 0.479698 − 0.996104i
\(11\) 1.47826 − 1.17888i 0.445713 − 0.355445i −0.374767 − 0.927119i \(-0.622277\pi\)
0.820481 + 0.571674i \(0.193706\pi\)
\(12\) −1.72013 + 0.202853i −0.496559 + 0.0585587i
\(13\) −0.731696 + 0.583508i −0.202936 + 0.161836i −0.719685 − 0.694301i \(-0.755714\pi\)
0.516749 + 0.856137i \(0.327142\pi\)
\(14\) −2.51534 + 0.820391i −0.672254 + 0.219259i
\(15\) 4.30409 + 4.25967i 1.11131 + 1.09984i
\(16\) −0.900969 − 0.433884i −0.225242 − 0.108471i
\(17\) 1.24123 + 5.43819i 0.301043 + 1.31895i 0.868556 + 0.495591i \(0.165048\pi\)
−0.567513 + 0.823364i \(0.692094\pi\)
\(18\) 0.637185 − 2.93155i 0.150186 − 0.690973i
\(19\) 6.71507i 1.54054i 0.637716 + 0.770271i \(0.279879\pi\)
−0.637716 + 0.770271i \(0.720121\pi\)
\(20\) 0.777975 + 3.40853i 0.173961 + 0.762171i
\(21\) 0.0739439 − 4.58198i 0.0161359 − 0.999870i
\(22\) −0.420736 + 1.84336i −0.0897012 + 0.393007i
\(23\) 5.84179 + 1.33335i 1.21810 + 0.278023i 0.782806 − 0.622266i \(-0.213788\pi\)
0.435292 + 0.900289i \(0.356645\pi\)
\(24\) 1.21838 − 1.23108i 0.248700 − 0.251293i
\(25\) 4.50368 − 5.64743i 0.900736 − 1.12949i
\(26\) 0.208252 − 0.912410i 0.0408415 − 0.178938i
\(27\) 4.44219 + 2.69572i 0.854900 + 0.518792i
\(28\) 1.45507 − 2.20970i 0.274982 − 0.417594i
\(29\) 0.519155 − 0.118494i 0.0964047 − 0.0220037i −0.174046 − 0.984737i \(-0.555684\pi\)
0.270451 + 0.962734i \(0.412827\pi\)
\(30\) −6.02094 − 0.646788i −1.09927 − 0.118087i
\(31\) 5.34122i 0.959311i 0.877457 + 0.479656i \(0.159238\pi\)
−0.877457 + 0.479656i \(0.840762\pi\)
\(32\) 0.974928 − 0.222521i 0.172345 − 0.0393365i
\(33\) −2.78195 − 1.72795i −0.484275 − 0.300798i
\(34\) −4.36109 − 3.47785i −0.747921 − 0.596447i
\(35\) −9.21188 + 0.839452i −1.55709 + 0.141893i
\(36\) 1.32962 + 2.68926i 0.221603 + 0.448210i
\(37\) 0.612916 + 2.68536i 0.100763 + 0.441470i 0.999992 + 0.00399278i \(0.00127095\pi\)
−0.899229 + 0.437478i \(0.855872\pi\)
\(38\) −4.18678 − 5.25005i −0.679185 − 0.851671i
\(39\) 1.37698 + 0.855284i 0.220493 + 0.136955i
\(40\) −2.73343 − 2.17984i −0.432193 − 0.344663i
\(41\) −1.61381 + 0.777171i −0.252035 + 0.121374i −0.555636 − 0.831425i \(-0.687525\pi\)
0.303601 + 0.952799i \(0.401811\pi\)
\(42\) 2.79901 + 3.62844i 0.431896 + 0.559880i
\(43\) −9.41518 − 4.53411i −1.43580 − 0.691445i −0.455735 − 0.890116i \(-0.650623\pi\)
−0.980066 + 0.198670i \(0.936338\pi\)
\(44\) −0.820375 − 1.70353i −0.123676 − 0.256816i
\(45\) 4.45254 − 9.49657i 0.663746 − 1.41567i
\(46\) −5.39863 + 2.59984i −0.795984 + 0.383326i
\(47\) −7.08465 − 8.88387i −1.03340 − 1.29584i −0.954259 − 0.298982i \(-0.903353\pi\)
−0.0791429 − 0.996863i \(-0.525218\pi\)
\(48\) −0.184998 + 1.72214i −0.0267021 + 0.248570i
\(49\) 5.28177 + 4.59379i 0.754539 + 0.656256i
\(50\) 7.22334i 1.02153i
\(51\) 8.15381 − 5.18257i 1.14176 − 0.725705i
\(52\) 0.406061 + 0.843194i 0.0563105 + 0.116930i
\(53\) −5.46957 − 1.24839i −0.751304 − 0.171480i −0.170313 − 0.985390i \(-0.554478\pi\)
−0.580991 + 0.813910i \(0.697335\pi\)
\(54\) −5.15380 + 0.662060i −0.701344 + 0.0900950i
\(55\) −2.86818 + 5.95585i −0.386746 + 0.803086i
\(56\) 0.240105 + 2.63483i 0.0320854 + 0.352094i
\(57\) 10.9581 − 3.89832i 1.45143 − 0.516345i
\(58\) −0.332012 + 0.416330i −0.0435953 + 0.0546668i
\(59\) 0.981235 + 0.472538i 0.127746 + 0.0615192i 0.496665 − 0.867942i \(-0.334558\pi\)
−0.368919 + 0.929462i \(0.620272\pi\)
\(60\) 5.11062 − 3.24831i 0.659779 − 0.419356i
\(61\) 4.27189 − 0.975031i 0.546960 − 0.124840i 0.0598930 − 0.998205i \(-0.480924\pi\)
0.487067 + 0.873365i \(0.338067\pi\)
\(62\) −3.33020 − 4.17593i −0.422935 − 0.530344i
\(63\) −7.52010 + 2.53932i −0.947443 + 0.319924i
\(64\) −0.623490 + 0.781831i −0.0779362 + 0.0977289i
\(65\) 1.41966 − 2.94796i 0.176088 − 0.365650i
\(66\) 3.25237 − 0.383549i 0.400339 − 0.0472116i
\(67\) −9.15032 −1.11789 −0.558945 − 0.829205i \(-0.688794\pi\)
−0.558945 + 0.829205i \(0.688794\pi\)
\(68\) 5.57804 0.676437
\(69\) −1.21550 − 10.3071i −0.146329 − 1.24083i
\(70\) 6.67875 − 6.39982i 0.798263 − 0.764925i
\(71\) 5.70374 + 1.30184i 0.676909 + 0.154500i 0.547136 − 0.837044i \(-0.315718\pi\)
0.129773 + 0.991544i \(0.458575\pi\)
\(72\) −2.71626 − 1.27354i −0.320115 − 0.150088i
\(73\) 1.01364 + 0.808350i 0.118637 + 0.0946102i 0.681011 − 0.732273i \(-0.261540\pi\)
−0.562374 + 0.826883i \(0.690112\pi\)
\(74\) −2.15349 − 1.71735i −0.250338 − 0.199638i
\(75\) −11.8304 − 4.07087i −1.36605 − 0.470064i
\(76\) 6.54671 + 1.49424i 0.750959 + 0.171402i
\(77\) 4.75594 − 1.55117i 0.541990 − 0.176772i
\(78\) −1.60983 + 0.189845i −0.182277 + 0.0214957i
\(79\) 15.3224 1.72390 0.861950 − 0.506993i \(-0.169243\pi\)
0.861950 + 0.506993i \(0.169243\pi\)
\(80\) 3.49619 0.390886
\(81\) 1.82022 − 8.81401i 0.202247 − 0.979335i
\(82\) 0.777171 − 1.61381i 0.0858242 − 0.178216i
\(83\) −9.92846 + 12.4499i −1.08979 + 1.36655i −0.164904 + 0.986310i \(0.552731\pi\)
−0.924886 + 0.380244i \(0.875840\pi\)
\(84\) −4.45065 − 1.09168i −0.485605 − 0.119112i
\(85\) −12.1592 − 15.2472i −1.31885 − 1.65379i
\(86\) 10.1881 − 2.32536i 1.09861 − 0.250750i
\(87\) −0.494752 − 0.778401i −0.0530430 − 0.0834534i
\(88\) 1.70353 + 0.820375i 0.181596 + 0.0874522i
\(89\) −6.59006 + 8.26368i −0.698545 + 0.875948i −0.996914 − 0.0785028i \(-0.974986\pi\)
0.298369 + 0.954451i \(0.403557\pi\)
\(90\) 2.43988 + 10.2008i 0.257186 + 1.07526i
\(91\) −2.35405 + 0.767783i −0.246771 + 0.0804855i
\(92\) 2.59984 − 5.39863i 0.271052 − 0.562846i
\(93\) 8.71615 − 3.10075i 0.903822 − 0.321533i
\(94\) 11.0780 + 2.52848i 1.14261 + 0.260793i
\(95\) −10.1864 − 21.1522i −1.04510 − 2.17017i
\(96\) −0.929101 − 1.46177i −0.0948260 − 0.149191i
\(97\) − 7.02844i − 0.713630i −0.934175 − 0.356815i \(-0.883863\pi\)
0.934175 − 0.356815i \(-0.116137\pi\)
\(98\) −6.99364 − 0.298440i −0.706464 − 0.0301470i
\(99\) −1.20477 + 5.54289i −0.121084 + 0.557082i
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.j.a.41.4 ✓ 120
3.2 odd 2 inner 294.2.j.a.41.14 yes 120
49.6 odd 14 inner 294.2.j.a.251.14 yes 120
147.104 even 14 inner 294.2.j.a.251.4 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.j.a.41.4 ✓ 120 1.1 even 1 trivial
294.2.j.a.41.14 yes 120 3.2 odd 2 inner
294.2.j.a.251.4 yes 120 147.104 even 14 inner
294.2.j.a.251.14 yes 120 49.6 odd 14 inner