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Properties

Label 465.2.i.b.346.2
Level $465$
Weight $2$
Character 465.346
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 346.2
Root \(2.71221 + 4.69769i\) of defining polynomial
Character \(\chi\) \(=\) 465.346
Dual form 465.2.i.b.211.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} -2.00000 q^{4} +(0.500000 + 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(2.71221 + 4.69769i) q^{11} +(-1.00000 + 1.73205i) q^{12} +(2.21221 + 3.83167i) q^{13} +1.00000 q^{15} +4.00000 q^{16} +(2.00000 - 3.46410i) q^{17} +(4.21221 - 7.29577i) q^{19} +(-1.00000 - 1.73205i) q^{20} +2.00000 q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +3.42443 q^{29} +(-4.71221 + 2.96564i) q^{31} +5.42443 q^{33} +(1.00000 + 1.73205i) q^{36} +(-4.21221 + 7.29577i) q^{37} +4.42443 q^{39} +(0.712214 + 1.23359i) q^{41} +(-3.21221 + 5.56372i) q^{43} +(-5.42443 - 9.39539i) q^{44} +(0.500000 - 0.866025i) q^{45} +8.84886 q^{47} +(2.00000 - 3.46410i) q^{48} +(3.50000 + 6.06218i) q^{49} +(-2.00000 - 3.46410i) q^{51} +(-4.42443 - 7.66334i) q^{52} +(-5.42443 - 9.39539i) q^{53} +(-2.71221 + 4.69769i) q^{55} +(-4.21221 - 7.29577i) q^{57} +(2.71221 - 4.69769i) q^{59} -2.00000 q^{60} +4.57557 q^{61} -8.00000 q^{64} +(-2.21221 + 3.83167i) q^{65} +(-3.42443 - 5.93128i) q^{67} +(-4.00000 + 6.92820i) q^{68} +(1.00000 - 1.73205i) q^{69} +(3.71221 + 6.42974i) q^{71} +(0.787786 + 1.36448i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-8.42443 + 14.5915i) q^{76} +(-6.13664 + 10.6290i) q^{79} +(2.00000 + 3.46410i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-3.00000 - 5.19615i) q^{83} +4.00000 q^{85} +(1.71221 - 2.96564i) q^{87} -13.4244 q^{89} -4.00000 q^{92} +(0.212214 + 5.56372i) q^{93} +8.42443 q^{95} -12.4244 q^{97} +(2.71221 - 4.69769i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 8 q^{4} + 2 q^{5} - 2 q^{9} + q^{11} - 4 q^{12} - q^{13} + 4 q^{15} + 16 q^{16} + 8 q^{17} + 7 q^{19} - 4 q^{20} + 8 q^{23} - 2 q^{25} - 4 q^{27} - 6 q^{29} - 9 q^{31} + 2 q^{33} + 4 q^{36}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 2.71221 + 4.69769i 0.817763 + 1.41641i 0.907327 + 0.420427i \(0.138120\pi\)
−0.0895631 + 0.995981i \(0.528547\pi\)
\(12\) −1.00000 + 1.73205i −0.288675 + 0.500000i
\(13\) 2.21221 + 3.83167i 0.613558 + 1.06271i 0.990636 + 0.136532i \(0.0435956\pi\)
−0.377078 + 0.926182i \(0.623071\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) 2.00000 − 3.46410i 0.485071 − 0.840168i −0.514782 − 0.857321i \(-0.672127\pi\)
0.999853 + 0.0171533i \(0.00546033\pi\)
\(18\) 0 0
\(19\) 4.21221 − 7.29577i 0.966348 − 1.67376i 0.260400 − 0.965501i \(-0.416146\pi\)
0.705948 − 0.708264i \(-0.250521\pi\)
\(20\) −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
\(29\) 3.42443 0.635900 0.317950 − 0.948107i \(-0.397006\pi\)
0.317950 + 0.948107i \(0.397006\pi\)
\(30\) 0 0
\(31\) −4.71221 + 2.96564i −0.846339 + 0.532645i
\(32\) 0 0
\(33\) 5.42443 0.944272
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 + 1.73205i 0.166667 + 0.288675i
\(37\) −4.21221 + 7.29577i −0.692484 + 1.19942i 0.278538 + 0.960425i \(0.410150\pi\)
−0.971022 + 0.238992i \(0.923183\pi\)
\(38\) 0 0
\(39\) 4.42443 0.708476
\(40\) 0 0
\(41\) 0.712214 + 1.23359i 0.111229 + 0.192655i 0.916266 − 0.400570i \(-0.131188\pi\)
−0.805037 + 0.593225i \(0.797855\pi\)
\(42\) 0 0
\(43\) −3.21221 + 5.56372i −0.489858 + 0.848459i −0.999932 − 0.0116715i \(-0.996285\pi\)
0.510074 + 0.860131i \(0.329618\pi\)
\(44\) −5.42443 − 9.39539i −0.817763 − 1.41641i
\(45\) 0.500000 − 0.866025i 0.0745356 − 0.129099i
\(46\) 0 0
\(47\) 8.84886 1.29074 0.645369 − 0.763871i \(-0.276704\pi\)
0.645369 + 0.763871i \(0.276704\pi\)
\(48\) 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
\(52\) −4.42443 − 7.66334i −0.613558 − 1.06271i
\(53\) −5.42443 − 9.39539i −0.745103 − 1.29056i −0.950147 − 0.311803i \(-0.899067\pi\)
0.205044 − 0.978753i \(-0.434266\pi\)
\(54\) 0 0
\(55\) −2.71221 + 4.69769i −0.365715 + 0.633437i
\(56\) 0 0
\(57\) −4.21221 − 7.29577i −0.557921 − 0.966348i
\(58\) 0 0
\(59\) 2.71221 − 4.69769i 0.353100 − 0.611588i −0.633691 − 0.773587i \(-0.718461\pi\)
0.986791 + 0.161999i \(0.0517941\pi\)
\(60\) −2.00000 −0.258199
\(61\) 4.57557 0.585842 0.292921 − 0.956137i \(-0.405373\pi\)
0.292921 + 0.956137i \(0.405373\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −2.21221 + 3.83167i −0.274391 + 0.475260i
\(66\) 0 0
\(67\) −3.42443 − 5.93128i −0.418361 − 0.724622i 0.577414 − 0.816451i \(-0.304062\pi\)
−0.995775 + 0.0918296i \(0.970728\pi\)
\(68\) −4.00000 + 6.92820i −0.485071 + 0.840168i
\(69\) 1.00000 − 1.73205i 0.120386 − 0.208514i
\(70\) 0 0
\(71\) 3.71221 + 6.42974i 0.440559 + 0.763070i 0.997731 − 0.0673268i \(-0.0214470\pi\)
−0.557172 + 0.830397i \(0.688114\pi\)
\(72\) 0 0
\(73\) 0.787786 + 1.36448i 0.0922033 + 0.159701i 0.908438 − 0.418020i \(-0.137276\pi\)
−0.816235 + 0.577721i \(0.803942\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) −8.42443 + 14.5915i −0.966348 + 1.67376i
\(77\) 0 0
\(78\) 0 0
\(79\) −6.13664 + 10.6290i −0.690426 + 1.19585i 0.281272 + 0.959628i \(0.409244\pi\)
−0.971698 + 0.236225i \(0.924090\pi\)
\(80\) 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 0.653079 − 0.757290i \(-0.273477\pi\)
−0.982372 + 0.186938i \(0.940144\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 1.71221 − 2.96564i 0.183569 − 0.317950i
\(88\) 0 0
\(89\) −13.4244 −1.42299 −0.711493 − 0.702693i \(-0.751981\pi\)
−0.711493 + 0.702693i \(0.751981\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0.212214 + 5.56372i 0.0220056 + 0.576931i
\(94\) 0 0
\(95\) 8.42443 0.864328
\(96\) 0 0
\(97\) −12.4244 −1.26151 −0.630755 − 0.775982i \(-0.717255\pi\)
−0.630755 + 0.775982i \(0.717255\pi\)
\(98\) 0 0
\(99\) 2.71221 − 4.69769i 0.272588 − 0.472136i
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.346.2 yes 4
31.25 even 3 inner 465.2.i.b.211.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.i.b.211.2 ✓ 4 31.25 even 3 inner
465.2.i.b.346.2 yes 4 1.1 even 1 trivial