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Properties

Label 465.2.g.e.464.5
Level $465$
Weight $2$
Character 465.464
Analytic conductor $3.713$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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(464,465)] 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("465.464"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 464.5
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 465.464
Dual form 465.2.g.e.464.6

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +(-1.41421 - 1.00000i) q^{3} +1.00000 q^{4} +(-1.73205 + 1.41421i) q^{5} +(-2.44949 - 1.73205i) q^{6} +2.44949i q^{7} -1.73205 q^{8} +(1.00000 + 2.82843i) q^{9} +(-3.00000 + 2.44949i) q^{10} +(-1.41421 - 1.00000i) q^{12} -4.24264 q^{13} +4.24264i q^{14} +(3.86370 - 0.267949i) q^{15} -5.00000 q^{16} +6.00000i q^{17} +(1.73205 + 4.89898i) q^{18} +4.00000 q^{19} +(-1.73205 + 1.41421i) q^{20} +(2.44949 - 3.46410i) q^{21} +(2.44949 + 1.73205i) q^{24} +(1.00000 - 4.89898i) q^{25} -7.34847 q^{26} +(1.41421 - 5.00000i) q^{27} +2.44949i q^{28} -7.34847 q^{29} +(6.69213 - 0.464102i) q^{30} +(2.00000 + 5.19615i) q^{31} -5.19615 q^{32} +10.3923i q^{34} +(-3.46410 - 4.24264i) q^{35} +(1.00000 + 2.82843i) q^{36} +4.24264 q^{37} +6.92820 q^{38} +(6.00000 + 4.24264i) q^{39} +(3.00000 - 2.44949i) q^{40} -2.82843i q^{41} +(4.24264 - 6.00000i) q^{42} +(-5.73205 - 3.48477i) q^{45} -10.3923 q^{47} +(7.07107 + 5.00000i) q^{48} +1.00000 q^{49} +(1.73205 - 8.48528i) q^{50} +(6.00000 - 8.48528i) q^{51} -4.24264 q^{52} -6.00000i q^{53} +(2.44949 - 8.66025i) q^{54} -4.24264i q^{56} +(-5.65685 - 4.00000i) q^{57} -12.7279 q^{58} +9.89949i q^{59} +(3.86370 - 0.267949i) q^{60} +(3.46410 + 9.00000i) q^{62} +(-6.92820 + 2.44949i) q^{63} +1.00000 q^{64} +(7.34847 - 6.00000i) q^{65} -7.34847i q^{67} +6.00000i q^{68} +(-6.00000 - 7.34847i) q^{70} -1.41421i q^{71} +(-1.73205 - 4.89898i) q^{72} +12.7279 q^{73} +7.34847 q^{74} +(-6.31319 + 5.92820i) q^{75} +4.00000 q^{76} +(10.3923 + 7.34847i) q^{78} +(8.66025 - 7.07107i) q^{80} +(-7.00000 + 5.65685i) q^{81} -4.89898i q^{82} +12.0000i q^{83} +(2.44949 - 3.46410i) q^{84} +(-8.48528 - 10.3923i) q^{85} +(10.3923 + 7.34847i) q^{87} -7.34847 q^{89} +(-9.92820 - 6.03579i) q^{90} -10.3923i q^{91} +(2.36773 - 9.34847i) q^{93} -18.0000 q^{94} +(-6.92820 + 5.65685i) q^{95} +(7.34847 + 5.19615i) q^{96} +4.89898i q^{97} +1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 8 q^{9} - 24 q^{10} - 40 q^{16} + 32 q^{19} + 8 q^{25} + 16 q^{31} + 8 q^{36} + 48 q^{39} + 24 q^{40} - 32 q^{45} + 8 q^{49} + 48 q^{51} + 8 q^{64} - 48 q^{70} + 32 q^{76} - 56 q^{81} - 24 q^{90}+ \cdots - 144 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 − 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) −1.41421 − 1.00000i −0.816497 − 0.577350i
\(4\) 1.00000 0.500000
\(5\) −1.73205 + 1.41421i −0.774597 + 0.632456i
\(6\) −2.44949 − 1.73205i −1.00000 − 0.707107i
\(7\) 2.44949i 0.925820i 0.886405 + 0.462910i \(0.153195\pi\)
−0.886405 + 0.462910i \(0.846805\pi\)
\(8\) −1.73205 −0.612372
\(9\) 1.00000 + 2.82843i 0.333333 + 0.942809i
\(10\) −3.00000 + 2.44949i −0.948683 + 0.774597i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.41421 − 1.00000i −0.408248 − 0.288675i
\(13\) −4.24264 −1.17670 −0.588348 − 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 4.24264i 1.13389i
\(15\) 3.86370 − 0.267949i 0.997604 − 0.0691842i
\(16\) −5.00000 −1.25000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 1.73205 + 4.89898i 0.408248 + 1.15470i
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.73205 + 1.41421i −0.387298 + 0.316228i
\(21\) 2.44949 − 3.46410i 0.534522 − 0.755929i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 2.44949 + 1.73205i 0.500000 + 0.353553i
\(25\) 1.00000 − 4.89898i 0.200000 − 0.979796i
\(26\) −7.34847 −1.44115
\(27\) 1.41421 − 5.00000i 0.272166 − 0.962250i
\(28\) 2.44949i 0.462910i
\(29\) −7.34847 −1.36458 −0.682288 − 0.731083i \(-0.739015\pi\)
−0.682288 + 0.731083i \(0.739015\pi\)
\(30\) 6.69213 − 0.464102i 1.22181 − 0.0847330i
\(31\) 2.00000 + 5.19615i 0.359211 + 0.933257i
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) 10.3923i 1.78227i
\(35\) −3.46410 − 4.24264i −0.585540 − 0.717137i
\(36\) 1.00000 + 2.82843i 0.166667 + 0.471405i
\(37\) 4.24264 0.697486 0.348743 − 0.937218i \(-0.386609\pi\)
0.348743 + 0.937218i \(0.386609\pi\)
\(38\) 6.92820 1.12390
\(39\) 6.00000 + 4.24264i 0.960769 + 0.679366i
\(40\) 3.00000 − 2.44949i 0.474342 − 0.387298i
\(41\) − 2.82843i − 0.441726i −0.975305 − 0.220863i \(-0.929113\pi\)
0.975305 − 0.220863i \(-0.0708874\pi\)
\(42\) 4.24264 − 6.00000i 0.654654 − 0.925820i
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −5.73205 − 3.48477i −0.854484 − 0.519478i
\(46\) 0 0
\(47\) −10.3923 −1.51587 −0.757937 − 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) 7.07107 + 5.00000i 1.02062 + 0.721688i
\(49\) 1.00000 0.142857
\(50\) 1.73205 − 8.48528i 0.244949 − 1.20000i
\(51\) 6.00000 − 8.48528i 0.840168 − 1.18818i
\(52\) −4.24264 −0.588348
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 2.44949 − 8.66025i 0.333333 − 1.17851i
\(55\) 0 0
\(56\) − 4.24264i − 0.566947i
\(57\) −5.65685 − 4.00000i −0.749269 − 0.529813i
\(58\) −12.7279 −1.67126
\(59\) 9.89949i 1.28880i 0.764687 + 0.644402i \(0.222894\pi\)
−0.764687 + 0.644402i \(0.777106\pi\)
\(60\) 3.86370 − 0.267949i 0.498802 − 0.0345921i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 3.46410 + 9.00000i 0.439941 + 1.14300i
\(63\) −6.92820 + 2.44949i −0.872872 + 0.308607i
\(64\) 1.00000 0.125000
\(65\) 7.34847 − 6.00000i 0.911465 − 0.744208i
\(66\) 0 0
\(67\) − 7.34847i − 0.897758i −0.893592 − 0.448879i \(-0.851823\pi\)
0.893592 − 0.448879i \(-0.148177\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) −6.00000 − 7.34847i −0.717137 − 0.878310i
\(71\) − 1.41421i − 0.167836i −0.996473 − 0.0839181i \(-0.973257\pi\)
0.996473 − 0.0839181i \(-0.0267434\pi\)
\(72\) −1.73205 − 4.89898i −0.204124 − 0.577350i
\(73\) 12.7279 1.48969 0.744845 − 0.667237i \(-0.232523\pi\)
0.744845 + 0.667237i \(0.232523\pi\)
\(74\) 7.34847 0.854242
\(75\) −6.31319 + 5.92820i −0.728985 + 0.684530i
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 10.3923 + 7.34847i 1.17670 + 0.832050i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.66025 − 7.07107i 0.968246 − 0.790569i
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) − 4.89898i − 0.541002i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 2.44949 − 3.46410i 0.267261 − 0.377964i
\(85\) −8.48528 − 10.3923i −0.920358 − 1.12720i
\(86\) 0 0
\(87\) 10.3923 + 7.34847i 1.11417 + 0.787839i
\(88\) 0 0
\(89\) −7.34847 −0.778936 −0.389468 − 0.921040i \(-0.627341\pi\)
−0.389468 + 0.921040i \(0.627341\pi\)
\(90\) −9.92820 − 6.03579i −1.04652 − 0.636228i
\(91\) − 10.3923i − 1.08941i
\(92\) 0 0
\(93\) 2.36773 − 9.34847i 0.245522 − 0.969391i
\(94\) −18.0000 −1.85656
\(95\) −6.92820 + 5.65685i −0.710819 + 0.580381i
\(96\) 7.34847 + 5.19615i 0.750000 + 0.530330i
\(97\) 4.89898i 0.497416i 0.968579 + 0.248708i \(0.0800060\pi\)
−0.968579 + 0.248708i \(0.919994\pi\)
\(98\) 1.73205 0.174964
\(99\) 0 0
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.g.e.464.5 yes 8
3.2 odd 2 inner 465.2.g.e.464.2 yes 8
5.4 even 2 inner 465.2.g.e.464.4 yes 8
15.14 odd 2 inner 465.2.g.e.464.7 yes 8
31.30 odd 2 inner 465.2.g.e.464.8 yes 8
93.92 even 2 inner 465.2.g.e.464.3 yes 8
155.154 odd 2 inner 465.2.g.e.464.1 ✓ 8
465.464 even 2 inner 465.2.g.e.464.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.g.e.464.1 ✓ 8 155.154 odd 2 inner
465.2.g.e.464.2 yes 8 3.2 odd 2 inner
465.2.g.e.464.3 yes 8 93.92 even 2 inner
465.2.g.e.464.4 yes 8 5.4 even 2 inner
465.2.g.e.464.5 yes 8 1.1 even 1 trivial
465.2.g.e.464.6 yes 8 465.464 even 2 inner
465.2.g.e.464.7 yes 8 15.14 odd 2 inner
465.2.g.e.464.8 yes 8 31.30 odd 2 inner