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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(211,465)] 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("465.211"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 346.1
Root \(0.500000 - 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 465.346
Dual form 465.2.i.c.211.1

$q$-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-2.58836 q^{2} +(0.500000 - 0.866025i) q^{3} +4.69963 q^{4} +(0.500000 + 0.866025i) q^{5} +(-1.29418 + 2.24159i) q^{6} +(-1.29418 + 2.24159i) q^{7} -6.98762 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.29418 - 2.24159i) q^{10} +(-3.19963 - 5.54192i) q^{11} +(2.34981 - 4.07000i) q^{12} +(0.555632 + 0.962383i) q^{13} +(3.34981 - 5.80205i) q^{14} +1.00000 q^{15} +8.68725 q^{16} +(0.849814 - 1.47192i) q^{17} +(1.29418 + 2.24159i) q^{18} +(2.84362 - 4.92530i) q^{19} +(2.34981 + 4.07000i) q^{20} +(1.29418 + 2.24159i) q^{21} +(8.28180 + 14.3445i) q^{22} -4.98762 q^{23} +(-3.49381 + 6.05146i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(-1.43818 - 2.49100i) q^{26} -1.00000 q^{27} +(-6.08217 + 10.5346i) q^{28} +7.58836 q^{29} -2.58836 q^{30} +(-1.76145 - 5.28179i) q^{31} -8.51052 q^{32} -6.39926 q^{33} +(-2.19963 + 3.80987i) q^{34} -2.58836 q^{35} +(-2.34981 - 4.07000i) q^{36} +(5.58217 - 9.66861i) q^{37} +(-7.36033 + 12.7485i) q^{38} +1.11126 q^{39} +(-3.49381 - 6.05146i) q^{40} +(1.88874 + 3.27139i) q^{41} +(-3.34981 - 5.80205i) q^{42} +(1.65019 - 2.85821i) q^{43} +(-15.0371 - 26.0450i) q^{44} +(0.500000 - 0.866025i) q^{45} +12.9098 q^{46} +4.06546 q^{47} +(4.34362 - 7.52338i) q^{48} +(0.150186 + 0.260130i) q^{49} +(1.29418 - 2.24159i) q^{50} +(-0.849814 - 1.47192i) q^{51} +(2.61126 + 4.52284i) q^{52} +(-4.84362 - 8.38940i) q^{53} +2.58836 q^{54} +(3.19963 - 5.54192i) q^{55} +(9.04325 - 15.6634i) q^{56} +(-2.84362 - 4.92530i) q^{57} -19.6414 q^{58} +(0.666896 - 1.15510i) q^{59} +4.69963 q^{60} -4.18911 q^{61} +(4.55927 + 13.6712i) q^{62} +2.58836 q^{63} +4.65383 q^{64} +(-0.555632 + 0.962383i) q^{65} +16.5636 q^{66} +(-6.56615 - 11.3729i) q^{67} +(3.99381 - 6.91748i) q^{68} +(-2.49381 + 4.31941i) q^{69} +6.69963 q^{70} +(-3.49381 - 6.05146i) q^{71} +(3.49381 + 6.05146i) q^{72} +(-2.97710 - 5.15649i) q^{73} +(-14.4487 + 25.0259i) q^{74} +(0.500000 + 0.866025i) q^{75} +(13.3640 - 23.1471i) q^{76} +16.5636 q^{77} -2.87636 q^{78} +(-5.95489 + 10.3142i) q^{79} +(4.34362 + 7.52338i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-4.88874 - 8.46754i) q^{82} +(3.84362 + 6.65735i) q^{83} +(6.08217 + 10.5346i) q^{84} +1.69963 q^{85} +(-4.27128 + 7.39808i) q^{86} +(3.79418 - 6.57172i) q^{87} +(22.3578 + 38.7248i) q^{88} -3.24219 q^{89} +(-1.29418 + 2.24159i) q^{90} -2.87636 q^{91} -23.4400 q^{92} +(-5.45489 - 1.11543i) q^{93} -10.5229 q^{94} +5.68725 q^{95} +(-4.25526 + 7.37033i) q^{96} -15.3535 q^{97} +(-0.388736 - 0.673310i) q^{98} +(-3.19963 + 5.54192i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{2} + 3 q^{3} + 16 q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} - 6 q^{8} - 3 q^{9} - 2 q^{10} - 7 q^{11} + 8 q^{12} + 3 q^{13} + 14 q^{14} + 6 q^{15} + 4 q^{16} - q^{17} + 2 q^{18} - 7 q^{19} + 8 q^{20}+ \cdots - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58836 −1.83025 −0.915125 − 0.403171i \(-0.867908\pi\)
−0.915125 + 0.403171i \(0.867908\pi\)
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i
\(4\) 4.69963 2.34981
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) −1.29418 + 2.24159i −0.528348 + 0.915125i
\(7\) −1.29418 + 2.24159i −0.489155 + 0.847241i −0.999922 − 0.0124780i \(-0.996028\pi\)
0.510767 + 0.859719i \(0.329361\pi\)
\(8\) −6.98762 −2.47050
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −1.29418 − 2.24159i −0.409256 − 0.708853i
\(11\) −3.19963 − 5.54192i −0.964724 − 1.67095i −0.710354 − 0.703844i \(-0.751465\pi\)
−0.254370 − 0.967107i \(-0.581868\pi\)
\(12\) 2.34981 − 4.07000i 0.678333 − 1.17491i
\(13\) 0.555632 + 0.962383i 0.154105 + 0.266917i 0.932733 − 0.360569i \(-0.117417\pi\)
−0.778628 + 0.627486i \(0.784084\pi\)
\(14\) 3.34981 − 5.80205i 0.895275 − 1.55066i
\(15\) 1.00000 0.258199
\(16\) 8.68725 2.17181
\(17\) 0.849814 − 1.47192i 0.206110 − 0.356993i −0.744376 − 0.667761i \(-0.767253\pi\)
0.950486 + 0.310768i \(0.100586\pi\)
\(18\) 1.29418 + 2.24159i 0.305042 + 0.528348i
\(19\) 2.84362 − 4.92530i 0.652372 − 1.12994i −0.330174 − 0.943920i \(-0.607107\pi\)
0.982546 − 0.186021i \(-0.0595594\pi\)
\(20\) 2.34981 + 4.07000i 0.525434 + 0.910079i
\(21\) 1.29418 + 2.24159i 0.282414 + 0.489155i
\(22\) 8.28180 + 14.3445i 1.76569 + 3.05826i
\(23\) −4.98762 −1.03999 −0.519995 − 0.854169i \(-0.674066\pi\)
−0.519995 + 0.854169i \(0.674066\pi\)
\(24\) −3.49381 + 6.05146i −0.713171 + 1.23525i
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) −1.43818 − 2.49100i −0.282050 − 0.488525i
\(27\) −1.00000 −0.192450
\(28\) −6.08217 + 10.5346i −1.14942 + 1.99086i
\(29\) 7.58836 1.40912 0.704562 − 0.709643i \(-0.251144\pi\)
0.704562 + 0.709643i \(0.251144\pi\)
\(30\) −2.58836 −0.472568
\(31\) −1.76145 − 5.28179i −0.316366 − 0.948637i
\(32\) −8.51052 −1.50446
\(33\) −6.39926 −1.11397
\(34\) −2.19963 + 3.80987i −0.377233 + 0.653387i
\(35\) −2.58836 −0.437513
\(36\) −2.34981 − 4.07000i −0.391636 − 0.678333i
\(37\) 5.58217 − 9.66861i 0.917704 − 1.58951i 0.114810 − 0.993387i \(-0.463374\pi\)
0.802894 − 0.596122i \(-0.203293\pi\)
\(38\) −7.36033 + 12.7485i −1.19400 + 2.06808i
\(39\) 1.11126 0.177945
\(40\) −3.49381 − 6.05146i −0.552420 − 0.956819i
\(41\) 1.88874 + 3.27139i 0.294971 + 0.510905i 0.974978 − 0.222301i \(-0.0713567\pi\)
−0.680007 + 0.733205i \(0.738023\pi\)
\(42\) −3.34981 − 5.80205i −0.516888 − 0.895275i
\(43\) 1.65019 − 2.85821i 0.251651 − 0.435872i −0.712330 − 0.701845i \(-0.752360\pi\)
0.963981 + 0.265973i \(0.0856932\pi\)
\(44\) −15.0371 − 26.0450i −2.26692 − 3.92642i
\(45\) 0.500000 − 0.866025i 0.0745356 − 0.129099i
\(46\) 12.9098 1.90344
\(47\) 4.06546 0.593009 0.296504 − 0.955031i \(-0.404179\pi\)
0.296504 + 0.955031i \(0.404179\pi\)
\(48\) 4.34362 − 7.52338i 0.626948 − 1.08591i
\(49\) 0.150186 + 0.260130i 0.0214551 + 0.0371614i
\(50\) 1.29418 − 2.24159i 0.183025 − 0.317009i
\(51\) −0.849814 − 1.47192i −0.118998 − 0.206110i
\(52\) 2.61126 + 4.52284i 0.362117 + 0.627205i
\(53\) −4.84362 − 8.38940i −0.665323 − 1.15237i −0.979198 − 0.202909i \(-0.934961\pi\)
0.313875 − 0.949464i \(-0.398373\pi\)
\(54\) 2.58836 0.352232
\(55\) 3.19963 − 5.54192i 0.431438 − 0.747272i
\(56\) 9.04325 − 15.6634i 1.20846 − 2.09311i
\(57\) −2.84362 − 4.92530i −0.376647 − 0.652372i
\(58\) −19.6414 −2.57905
\(59\) 0.666896 − 1.15510i 0.0868225 − 0.150381i −0.819344 − 0.573302i \(-0.805662\pi\)
0.906166 + 0.422921i \(0.138995\pi\)
\(60\) 4.69963 0.606719
\(61\) −4.18911 −0.536360 −0.268180 − 0.963369i \(-0.586422\pi\)
−0.268180 + 0.963369i \(0.586422\pi\)
\(62\) 4.55927 + 13.6712i 0.579028 + 1.73624i
\(63\) 2.58836 0.326103
\(64\) 4.65383 0.581728
\(65\) −0.555632 + 0.962383i −0.0689177 + 0.119369i
\(66\) 16.5636 2.03884
\(67\) −6.56615 − 11.3729i −0.802183 − 1.38942i −0.918176 − 0.396172i \(-0.870338\pi\)
0.115993 − 0.993250i \(-0.462995\pi\)
\(68\) 3.99381 − 6.91748i 0.484321 − 0.838868i
\(69\) −2.49381 + 4.31941i −0.300219 + 0.519995i
\(70\) 6.69963 0.800759
\(71\) −3.49381 − 6.05146i −0.414639 − 0.718176i 0.580752 − 0.814081i \(-0.302759\pi\)
−0.995390 + 0.0959051i \(0.969425\pi\)
\(72\) 3.49381 + 6.05146i 0.411749 + 0.713171i
\(73\) −2.97710 − 5.15649i −0.348443 − 0.603521i 0.637530 − 0.770426i \(-0.279956\pi\)
−0.985973 + 0.166904i \(0.946623\pi\)
\(74\) −14.4487 + 25.0259i −1.67963 + 2.90920i
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 13.3640 − 23.1471i 1.53295 − 2.65515i
\(77\) 16.5636 1.88760
\(78\) −2.87636 −0.325683
\(79\) −5.95489 + 10.3142i −0.669977 + 1.16043i 0.307933 + 0.951408i \(0.400363\pi\)
−0.977910 + 0.209027i \(0.932970\pi\)
\(80\) 4.34362 + 7.52338i 0.485632 + 0.841139i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −4.88874 − 8.46754i −0.539871 − 0.935083i
\(83\) 3.84362 + 6.65735i 0.421893 + 0.730739i 0.996125 − 0.0879532i \(-0.0280326\pi\)
−0.574232 + 0.818693i \(0.694699\pi\)
\(84\) 6.08217 + 10.5346i 0.663620 + 1.14942i
\(85\) 1.69963 0.184351
\(86\) −4.27128 + 7.39808i −0.460584 + 0.797755i
\(87\) 3.79418 − 6.57172i 0.406779 − 0.704562i
\(88\) 22.3578 + 38.7248i 2.38335 + 4.12808i
\(89\) −3.24219 −0.343672 −0.171836 − 0.985126i \(-0.554970\pi\)
−0.171836 + 0.985126i \(0.554970\pi\)
\(90\) −1.29418 + 2.24159i −0.136419 + 0.236284i
\(91\) −2.87636 −0.301524
\(92\) −23.4400 −2.44378
\(93\) −5.45489 − 1.11543i −0.565646 − 0.115665i
\(94\) −10.5229 −1.08535
\(95\) 5.68725 0.583499
\(96\) −4.25526 + 7.37033i −0.434301 + 0.752231i
\(97\) −15.3535 −1.55891 −0.779454 − 0.626460i \(-0.784503\pi\)
−0.779454 + 0.626460i \(0.784503\pi\)
\(98\) −0.388736 − 0.673310i −0.0392683 − 0.0680146i
\(99\) −3.19963 + 5.54192i −0.321575 + 0.556984i
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 465.2.i.c.346.1 yes 6
31.25 even 3 inner 465.2.i.c.211.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.i.c.211.1 ✓ 6 31.25 even 3 inner
465.2.i.c.346.1 yes 6 1.1 even 1 trivial