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Properties

Label 294.2.j.a.41.7
Level $294$
Weight $2$
Character 294.41
Analytic conductor $2.348$
Analytic rank $0$
Dimension $120$
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(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 41.7
Character \(\chi\) \(=\) 294.41
Dual form 294.2.j.a.251.7

$q$-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} +(1.06385 + 1.36683i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-2.67532 + 1.28836i) q^{5} +(-1.68396 - 0.405333i) q^{6} +(-1.08169 - 2.41453i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-0.736454 + 2.90820i) q^{9} +(1.28836 - 2.67532i) q^{10} +(-3.64839 + 2.90950i) q^{11} +(1.56929 - 0.733027i) q^{12} +(-2.21443 + 1.76595i) q^{13} +(2.35113 + 1.21333i) q^{14} +(-4.60711 - 2.28608i) q^{15} +(-0.900969 - 0.433884i) q^{16} +(-1.10828 - 4.85571i) q^{17} +(-1.23745 - 2.73289i) q^{18} +6.81113i q^{19} +(0.660748 + 2.89493i) q^{20} +(2.14950 - 4.04718i) q^{21} +(1.03839 - 4.54947i) q^{22} +(2.40771 + 0.549543i) q^{23} +(-0.769886 + 1.55154i) q^{24} +(2.37998 - 2.98440i) q^{25} +(0.630260 - 2.76135i) q^{26} +(-4.75849 + 2.08728i) q^{27} +(-2.59469 + 0.517287i) q^{28} +(5.53364 - 1.26302i) q^{29} +(5.02733 - 1.08515i) q^{30} -7.19891i q^{31} +(0.974928 - 0.222521i) q^{32} +(-7.85813 - 1.89147i) q^{33} +(3.89398 + 3.10535i) q^{34} +(6.00465 + 5.06601i) q^{35} +(2.67141 + 1.36512i) q^{36} +(1.90513 + 8.34691i) q^{37} +(-4.24667 - 5.32516i) q^{38} +(-4.76958 - 1.14805i) q^{39} +(-2.32155 - 1.85138i) q^{40} +(-7.72614 + 3.72071i) q^{41} +(0.842830 + 4.50440i) q^{42} +(-2.96889 - 1.42974i) q^{43} +(2.02471 + 4.20435i) q^{44} +(-1.77658 - 8.72918i) q^{45} +(-2.22505 + 1.07153i) q^{46} +(2.98949 + 3.74870i) q^{47} +(-0.365448 - 1.69306i) q^{48} +(-4.65989 + 5.22354i) q^{49} +3.81720i q^{50} +(5.45789 - 6.68058i) q^{51} +(1.22892 + 2.55187i) q^{52} +(9.35169 + 2.13446i) q^{53} +(2.41894 - 4.59877i) q^{54} +(6.01211 - 12.4843i) q^{55} +(1.70609 - 2.02219i) q^{56} +(-9.30967 + 7.24601i) q^{57} +(-3.53890 + 4.43764i) q^{58} +(-0.790974 - 0.380913i) q^{59} +(-3.25394 + 3.98290i) q^{60} +(-5.12665 + 1.17012i) q^{61} +(4.48845 + 5.62834i) q^{62} +(7.81855 - 1.36758i) q^{63} +(-0.623490 + 0.781831i) q^{64} +(3.64912 - 7.57747i) q^{65} +(7.32305 - 3.42065i) q^{66} +2.52179 q^{67} -4.98059 q^{68} +(1.81030 + 3.87556i) q^{69} +(-7.85324 - 0.216929i) q^{70} +(7.21235 + 1.64617i) q^{71} +(-2.93973 + 0.598300i) q^{72} +(13.1553 + 10.4910i) q^{73} +(-6.69370 - 5.33805i) q^{74} +(6.61112 + 0.0780810i) q^{75} +(6.64036 + 1.51562i) q^{76} +(10.9715 + 5.66198i) q^{77} +(4.44480 - 2.07620i) q^{78} +0.959342 q^{79} +2.96938 q^{80} +(-7.91527 - 4.28351i) q^{81} +(3.72071 - 7.72614i) q^{82} +(-0.996870 + 1.25004i) q^{83} +(-3.46740 - 2.99619i) q^{84} +(9.22094 + 11.5627i) q^{85} +(3.21260 - 0.733254i) q^{86} +(7.61329 + 6.21989i) q^{87} +(-4.20435 - 2.02471i) q^{88} +(-4.90125 + 6.14597i) q^{89} +(6.83154 + 5.71707i) q^{90} +(6.65927 + 3.43660i) q^{91} +(1.07153 - 2.22505i) q^{92} +(9.83970 - 7.65855i) q^{93} +(-4.67456 - 1.06694i) q^{94} +(-8.77522 - 18.2219i) q^{95} +(1.34132 + 1.09583i) q^{96} +2.75461i q^{97} +(0.386425 - 6.98933i) q^{98} +(-5.77453 - 12.7530i) 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
\(3\) 1.06385 + 1.36683i 0.614213 + 0.789140i
\(4\) 0.222521 − 0.974928i 0.111260 − 0.487464i
\(5\) −2.67532 + 1.28836i −1.19644 + 0.576174i −0.922658 − 0.385619i \(-0.873988\pi\)
−0.273779 + 0.961793i \(0.588274\pi\)
\(6\) −1.68396 − 0.405333i −0.687472 − 0.165476i
\(7\) −1.08169 − 2.41453i −0.408840 − 0.912606i
\(8\) 0.433884 + 0.900969i 0.153401 + 0.318541i
\(9\) −0.736454 + 2.90820i −0.245485 + 0.969401i
\(10\) 1.28836 − 2.67532i 0.407416 − 0.846009i
\(11\) −3.64839 + 2.90950i −1.10003 + 0.877247i −0.993129 − 0.117023i \(-0.962665\pi\)
−0.106903 + 0.994269i \(0.534093\pi\)
\(12\) 1.56929 − 0.733027i 0.453015 − 0.211607i
\(13\) −2.21443 + 1.76595i −0.614173 + 0.489787i −0.880468 − 0.474105i \(-0.842772\pi\)
0.266295 + 0.963891i \(0.414200\pi\)
\(14\) 2.35113 + 1.21333i 0.628367 + 0.324276i
\(15\) −4.60711 − 2.28608i −1.18955 − 0.590263i
\(16\) −0.900969 − 0.433884i −0.225242 − 0.108471i
\(17\) −1.10828 − 4.85571i −0.268799 − 1.17768i −0.911413 − 0.411494i \(-0.865007\pi\)
0.642614 − 0.766190i \(-0.277850\pi\)
\(18\) −1.23745 − 2.73289i −0.291670 − 0.644149i
\(19\) 6.81113i 1.56258i 0.624168 + 0.781290i \(0.285438\pi\)
−0.624168 + 0.781290i \(0.714562\pi\)
\(20\) 0.660748 + 2.89493i 0.147748 + 0.647326i
\(21\) 2.14950 − 4.04718i 0.469059 − 0.883167i
\(22\) 1.03839 − 4.54947i 0.221385 − 0.969951i
\(23\) 2.40771 + 0.549543i 0.502041 + 0.114588i 0.466039 − 0.884764i \(-0.345681\pi\)
0.0360021 + 0.999352i \(0.488538\pi\)
\(24\) −0.769886 + 1.55154i −0.157152 + 0.316707i
\(25\) 2.37998 − 2.98440i 0.475996 − 0.596881i
\(26\) 0.630260 − 2.76135i 0.123604 − 0.541546i
\(27\) −4.75849 + 2.08728i −0.915773 + 0.401697i
\(28\) −2.59469 + 0.517287i −0.490350 + 0.0977580i
\(29\) 5.53364 − 1.26302i 1.02757 − 0.234537i 0.324673 − 0.945826i \(-0.394746\pi\)
0.702899 + 0.711290i \(0.251889\pi\)
\(30\) 5.02733 − 1.08515i 0.917860 − 0.198121i
\(31\) − 7.19891i − 1.29296i −0.762930 − 0.646482i \(-0.776240\pi\)
0.762930 − 0.646482i \(-0.223760\pi\)
\(32\) 0.974928 − 0.222521i 0.172345 − 0.0393365i
\(33\) −7.85813 − 1.89147i −1.36792 − 0.329263i
\(34\) 3.89398 + 3.10535i 0.667812 + 0.532562i
\(35\) 6.00465 + 5.06601i 1.01497 + 0.856313i
\(36\) 2.67141 + 1.36512i 0.445235 + 0.227521i
\(37\) 1.90513 + 8.34691i 0.313201 + 1.37222i 0.849230 + 0.528024i \(0.177067\pi\)
−0.536028 + 0.844200i \(0.680076\pi\)
\(38\) −4.24667 − 5.32516i −0.688901 − 0.863855i
\(39\) −4.76958 − 1.14805i −0.763743 − 0.183835i
\(40\) −2.32155 − 1.85138i −0.367070 − 0.292728i
\(41\) −7.72614 + 3.72071i −1.20662 + 0.581078i −0.925556 − 0.378611i \(-0.876402\pi\)
−0.281064 + 0.959689i \(0.590687\pi\)
\(42\) 0.842830 + 4.50440i 0.130051 + 0.695044i
\(43\) −2.96889 − 1.42974i −0.452751 − 0.218033i 0.193589 − 0.981083i \(-0.437987\pi\)
−0.646340 + 0.763049i \(0.723701\pi\)
\(44\) 2.02471 + 4.20435i 0.305236 + 0.633829i
\(45\) −1.77658 − 8.72918i −0.264836 − 1.30127i
\(46\) −2.22505 + 1.07153i −0.328066 + 0.157988i
\(47\) 2.98949 + 3.74870i 0.436062 + 0.546805i 0.950501 − 0.310723i \(-0.100571\pi\)
−0.514438 + 0.857527i \(0.672000\pi\)
\(48\) −0.365448 − 1.69306i −0.0527479 − 0.244372i
\(49\) −4.65989 + 5.22354i −0.665699 + 0.746220i
\(50\) 3.81720i 0.539833i
\(51\) 5.45789 − 6.68058i 0.764258 − 0.935468i
\(52\) 1.22892 + 2.55187i 0.170420 + 0.353881i
\(53\) 9.35169 + 2.13446i 1.28455 + 0.293191i 0.809684 − 0.586866i \(-0.199638\pi\)
0.474869 + 0.880057i \(0.342496\pi\)
\(54\) 2.41894 − 4.59877i 0.329177 − 0.625814i
\(55\) 6.01211 − 12.4843i 0.810673 − 1.68338i
\(56\) 1.70609 − 2.02219i 0.227985 − 0.270227i
\(57\) −9.30967 + 7.24601i −1.23310 + 0.959758i
\(58\) −3.53890 + 4.43764i −0.464680 + 0.582690i
\(59\) −0.790974 − 0.380913i −0.102976 − 0.0495906i 0.381687 − 0.924292i \(-0.375343\pi\)
−0.484663 + 0.874701i \(0.661058\pi\)
\(60\) −3.25394 + 3.98290i −0.420082 + 0.514190i
\(61\) −5.12665 + 1.17012i −0.656400 + 0.149819i −0.537733 − 0.843115i \(-0.680719\pi\)
−0.118667 + 0.992934i \(0.537862\pi\)
\(62\) 4.48845 + 5.62834i 0.570034 + 0.714800i
\(63\) 7.81855 − 1.36758i 0.985045 − 0.172299i
\(64\) −0.623490 + 0.781831i −0.0779362 + 0.0977289i
\(65\) 3.64912 − 7.57747i 0.452617 − 0.939869i
\(66\) 7.32305 − 3.42065i 0.901405 − 0.421053i
\(67\) 2.52179 0.308086 0.154043 − 0.988064i \(-0.450771\pi\)
0.154043 + 0.988064i \(0.450771\pi\)
\(68\) −4.98059 −0.603985
\(69\) 1.81030 + 3.87556i 0.217935 + 0.466562i
\(70\) −7.85324 − 0.216929i −0.938641 − 0.0259280i
\(71\) 7.21235 + 1.64617i 0.855948 + 0.195365i 0.627908 − 0.778287i \(-0.283911\pi\)
0.228040 + 0.973652i \(0.426768\pi\)
\(72\) −2.93973 + 0.598300i −0.346451 + 0.0705103i
\(73\) 13.1553 + 10.4910i 1.53972 + 1.22788i 0.878558 + 0.477635i \(0.158506\pi\)
0.661157 + 0.750247i \(0.270066\pi\)
\(74\) −6.69370 − 5.33805i −0.778127 − 0.620536i
\(75\) 6.61112 + 0.0780810i 0.763386 + 0.00901602i
\(76\) 6.64036 + 1.51562i 0.761702 + 0.173853i
\(77\) 10.9715 + 5.66198i 1.25032 + 0.645242i
\(78\) 4.44480 − 2.07620i 0.503275 − 0.235083i
\(79\) 0.959342 0.107934 0.0539672 − 0.998543i \(-0.482813\pi\)
0.0539672 + 0.998543i \(0.482813\pi\)
\(80\) 2.96938 0.331986
\(81\) −7.91527 − 4.28351i −0.879475 − 0.475946i
\(82\) 3.72071 − 7.72614i 0.410884 − 0.853209i
\(83\) −0.996870 + 1.25004i −0.109421 + 0.137209i −0.833526 − 0.552480i \(-0.813682\pi\)
0.724105 + 0.689689i \(0.242253\pi\)
\(84\) −3.46740 − 2.99619i −0.378324 − 0.326911i
\(85\) 9.22094 + 11.5627i 1.00015 + 1.25415i
\(86\) 3.21260 − 0.733254i 0.346423 − 0.0790689i
\(87\) 7.61329 + 6.21989i 0.816230 + 0.666843i
\(88\) −4.20435 − 2.02471i −0.448185 − 0.215834i
\(89\) −4.90125 + 6.14597i −0.519532 + 0.651472i −0.970509 − 0.241063i \(-0.922504\pi\)
0.450978 + 0.892535i \(0.351075\pi\)
\(90\) 6.83154 + 5.71707i 0.720107 + 0.602632i
\(91\) 6.65927 + 3.43660i 0.698081 + 0.360253i
\(92\) 1.07153 − 2.22505i 0.111715 − 0.231978i
\(93\) 9.83970 − 7.65855i 1.02033 − 0.794155i
\(94\) −4.67456 − 1.06694i −0.482144 − 0.110046i
\(95\) −8.77522 − 18.2219i −0.900318 − 1.86953i
\(96\) 1.34132 + 1.09583i 0.136898 + 0.111843i
\(97\) 2.75461i 0.279689i 0.990174 + 0.139844i \(0.0446602\pi\)
−0.990174 + 0.139844i \(0.955340\pi\)
\(98\) 0.386425 − 6.98933i 0.0390349 − 0.706029i
\(99\) −5.77453 − 12.7530i −0.580362 − 1.28172i
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.41.7 ✓ 120
3.2 odd 2 inner 294.2.j.a.41.17 yes 120
49.6 odd 14 inner 294.2.j.a.251.17 yes 120
147.104 even 14 inner 294.2.j.a.251.7 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.j.a.41.7 ✓ 120 1.1 even 1 trivial
294.2.j.a.41.17 yes 120 3.2 odd 2 inner
294.2.j.a.251.7 yes 120 147.104 even 14 inner
294.2.j.a.251.17 yes 120 49.6 odd 14 inner