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Properties

Label 465.2.i.b.211.1
Level $465$
Weight $2$
Character 465.211
Analytic conductor $3.713$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [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 = [4] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{97})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 25x^{2} + 24x + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 211.1
Root \(-2.21221 + 3.83167i\) of defining polynomial
Character \(\chi\) \(=\) 465.211
Dual form 465.2.i.b.346.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} -2.00000 q^{4} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-2.21221 + 3.83167i) q^{11} +(-1.00000 - 1.73205i) q^{12} +(-2.71221 + 4.69769i) q^{13} +1.00000 q^{15} +4.00000 q^{16} +(2.00000 + 3.46410i) q^{17} +(-0.712214 - 1.23359i) q^{19} +(-1.00000 + 1.73205i) q^{20} +2.00000 q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -6.42443 q^{29} +(0.212214 + 5.56372i) q^{31} -4.42443 q^{33} +(1.00000 - 1.73205i) q^{36} +(0.712214 + 1.23359i) q^{37} -5.42443 q^{39} +(-4.21221 + 7.29577i) q^{41} +(1.71221 + 2.96564i) q^{43} +(4.42443 - 7.66334i) q^{44} +(0.500000 + 0.866025i) q^{45} -10.8489 q^{47} +(2.00000 + 3.46410i) q^{48} +(3.50000 - 6.06218i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(5.42443 - 9.39539i) q^{52} +(4.42443 - 7.66334i) q^{53} +(2.21221 + 3.83167i) q^{55} +(0.712214 - 1.23359i) q^{57} +(-2.21221 - 3.83167i) q^{59} -2.00000 q^{60} +14.4244 q^{61} -8.00000 q^{64} +(2.71221 + 4.69769i) q^{65} +(6.42443 - 11.1274i) q^{67} +(-4.00000 - 6.92820i) q^{68} +(1.00000 + 1.73205i) q^{69} +(-1.21221 + 2.09962i) q^{71} +(5.71221 - 9.89385i) q^{73} +(0.500000 - 0.866025i) q^{75} +(1.42443 + 2.46718i) q^{76} +(8.63664 + 14.9591i) q^{79} +(2.00000 - 3.46410i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-3.00000 + 5.19615i) q^{83} +4.00000 q^{85} +(-3.21221 - 5.56372i) q^{87} -3.57557 q^{89} -4.00000 q^{92} +(-4.71221 + 2.96564i) q^{93} -1.42443 q^{95} -2.57557 q^{97} +(-2.21221 - 3.83167i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 8 q^{4} + 2 q^{5} - 2 q^{9} + q^{11} - 4 q^{12} - q^{13} + 4 q^{15} + 16 q^{16} + 8 q^{17} + 7 q^{19} - 4 q^{20} + 8 q^{23} - 2 q^{25} - 4 q^{27} - 6 q^{29} - 9 q^{31} + 2 q^{33} + 4 q^{36}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) −2.00000 −1.00000
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 0 0
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −2.21221 + 3.83167i −0.667008 + 1.15529i 0.311729 + 0.950171i \(0.399092\pi\)
−0.978737 + 0.205120i \(0.934241\pi\)
\(12\) −1.00000 − 1.73205i −0.288675 − 0.500000i
\(13\) −2.71221 + 4.69769i −0.752233 + 1.30291i 0.194505 + 0.980901i \(0.437690\pi\)
−0.946738 + 0.322004i \(0.895643\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) 2.00000 + 3.46410i 0.485071 + 0.840168i 0.999853 − 0.0171533i \(-0.00546033\pi\)
−0.514782 + 0.857321i \(0.672127\pi\)
\(18\) 0 0
\(19\) −0.712214 − 1.23359i −0.163393 − 0.283005i 0.772690 − 0.634783i \(-0.218911\pi\)
−0.936084 + 0.351778i \(0.885577\pi\)
\(20\) −1.00000 + 1.73205i −0.223607 + 0.387298i
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.42443 −1.19299 −0.596493 − 0.802618i \(-0.703440\pi\)
−0.596493 + 0.802618i \(0.703440\pi\)
\(30\) 0 0
\(31\) 0.212214 + 5.56372i 0.0381148 + 0.999273i
\(32\) 0 0
\(33\) −4.42443 −0.770194
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 − 1.73205i 0.166667 − 0.288675i
\(37\) 0.712214 + 1.23359i 0.117087 + 0.202801i 0.918612 − 0.395160i \(-0.129311\pi\)
−0.801525 + 0.597961i \(0.795978\pi\)
\(38\) 0 0
\(39\) −5.42443 −0.868604
\(40\) 0 0
\(41\) −4.21221 + 7.29577i −0.657837 + 1.13941i 0.323337 + 0.946284i \(0.395195\pi\)
−0.981174 + 0.193124i \(0.938138\pi\)
\(42\) 0 0
\(43\) 1.71221 + 2.96564i 0.261110 + 0.452256i 0.966537 − 0.256526i \(-0.0825781\pi\)
−0.705427 + 0.708783i \(0.749245\pi\)
\(44\) 4.42443 − 7.66334i 0.667008 − 1.15529i
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) 0 0
\(47\) −10.8489 −1.58247 −0.791234 − 0.611513i \(-0.790561\pi\)
−0.791234 + 0.611513i \(0.790561\pi\)
\(48\) 2.00000 + 3.46410i 0.288675 + 0.500000i
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) 0 0
\(51\) −2.00000 + 3.46410i −0.280056 + 0.485071i
\(52\) 5.42443 − 9.39539i 0.752233 − 1.30291i
\(53\) 4.42443 − 7.66334i 0.607742 − 1.05264i −0.383870 − 0.923387i \(-0.625409\pi\)
0.991612 − 0.129253i \(-0.0412579\pi\)
\(54\) 0 0
\(55\) 2.21221 + 3.83167i 0.298295 + 0.516662i
\(56\) 0 0
\(57\) 0.712214 − 1.23359i 0.0943351 − 0.163393i
\(58\) 0 0
\(59\) −2.21221 − 3.83167i −0.288006 − 0.498841i 0.685328 − 0.728235i \(-0.259659\pi\)
−0.973334 + 0.229394i \(0.926326\pi\)
\(60\) −2.00000 −0.258199
\(61\) 14.4244 1.84686 0.923429 − 0.383768i \(-0.125374\pi\)
0.923429 + 0.383768i \(0.125374\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 2.71221 + 4.69769i 0.336409 + 0.582677i
\(66\) 0 0
\(67\) 6.42443 − 11.1274i 0.784869 − 1.35943i −0.144208 − 0.989547i \(-0.546064\pi\)
0.929077 − 0.369885i \(-0.120603\pi\)
\(68\) −4.00000 − 6.92820i −0.485071 − 0.840168i
\(69\) 1.00000 + 1.73205i 0.120386 + 0.208514i
\(70\) 0 0
\(71\) −1.21221 + 2.09962i −0.143863 + 0.249179i −0.928948 − 0.370209i \(-0.879286\pi\)
0.785085 + 0.619388i \(0.212619\pi\)
\(72\) 0 0
\(73\) 5.71221 − 9.89385i 0.668564 − 1.15799i −0.309742 − 0.950821i \(-0.600243\pi\)
0.978306 − 0.207166i \(-0.0664241\pi\)
\(74\) 0 0
\(75\) 0.500000 − 0.866025i 0.0577350 − 0.100000i
\(76\) 1.42443 + 2.46718i 0.163393 + 0.283005i
\(77\) 0 0
\(78\) 0 0
\(79\) 8.63664 + 14.9591i 0.971698 + 1.68303i 0.690426 + 0.723403i \(0.257423\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 2.00000 − 3.46410i 0.223607 − 0.387298i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −3.00000 + 5.19615i −0.329293 + 0.570352i −0.982372 − 0.186938i \(-0.940144\pi\)
0.653079 + 0.757290i \(0.273477\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −3.21221 − 5.56372i −0.344386 − 0.596493i
\(88\) 0 0
\(89\) −3.57557 −0.379010 −0.189505 − 0.981880i \(-0.560688\pi\)
−0.189505 + 0.981880i \(0.560688\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −4.71221 + 2.96564i −0.488634 + 0.307523i
\(94\) 0 0
\(95\) −1.42443 −0.146143
\(96\) 0 0
\(97\) −2.57557 −0.261510 −0.130755 − 0.991415i \(-0.541740\pi\)
−0.130755 + 0.991415i \(0.541740\pi\)
\(98\) 0 0
\(99\) −2.21221 − 3.83167i −0.222336 − 0.385097i
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.b.211.1 ✓ 4
31.5 even 3 inner 465.2.i.b.346.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.i.b.211.1 ✓ 4 1.1 even 1 trivial
465.2.i.b.346.1 yes 4 31.5 even 3 inner