↑

Properties

Label 2312.2.b.h.577.4
Level $2312$
Weight $2$
Character 2312.577
Analytic conductor $18.461$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2312,2,Mod(577,2312)] 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("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2312.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-10,0,0,0,10,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.4614129473\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 2312.577
Dual form 2312.2.b.h.577.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.30278i q^{3} -2.30278i q^{5} +2.30278i q^{7} -7.90833 q^{9} -5.60555i q^{11} +0.697224 q^{13} +7.60555 q^{15} +1.30278 q^{19} -7.60555 q^{21} -4.30278i q^{23} -0.302776 q^{25} -16.2111i q^{27} -0.697224i q^{29} -4.21110i q^{31} +18.5139 q^{33} +5.30278 q^{35} +8.60555i q^{37} +2.30278i q^{39} -6.00000i q^{41} +4.21110 q^{43} +18.2111i q^{45} +11.6056 q^{47} +1.69722 q^{49} -3.30278 q^{53} -12.9083 q^{55} +4.30278i q^{57} -3.21110 q^{59} +1.60555i q^{61} -18.2111i q^{63} -1.60555i q^{65} +0.605551 q^{67} +14.2111 q^{69} -12.4222i q^{71} +2.39445i q^{73} -1.00000i q^{75} +12.9083 q^{77} -10.0000i q^{79} +29.8167 q^{81} +8.69722 q^{83} +2.30278 q^{87} +7.21110 q^{89} +1.60555i q^{91} +13.9083 q^{93} -3.00000i q^{95} +16.1194i q^{97} +44.3305i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9} + 10 q^{13} + 16 q^{15} - 2 q^{19} - 16 q^{21} + 6 q^{25} + 38 q^{33} + 14 q^{35} - 12 q^{43} + 32 q^{47} + 14 q^{49} - 6 q^{53} - 30 q^{55} + 16 q^{59} - 12 q^{67} + 28 q^{69} + 30 q^{77}+ \cdots + 34 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.30278i 1.90686i 0.301617 + 0.953429i \(0.402474\pi\)
−0.301617 + 0.953429i \(0.597526\pi\)
\(4\) 0 0
\(5\) − 2.30278i − 1.02983i −0.857240 − 0.514916i \(-0.827823\pi\)
0.857240 − 0.514916i \(-0.172177\pi\)
\(6\) 0 0
\(7\) 2.30278i 0.870367i 0.900342 + 0.435184i \(0.143317\pi\)
−0.900342 + 0.435184i \(0.856683\pi\)
\(8\) 0 0
\(9\) −7.90833 −2.63611
\(10\) 0 0
\(11\) − 5.60555i − 1.69014i −0.534658 − 0.845069i \(-0.679559\pi\)
0.534658 − 0.845069i \(-0.320441\pi\)
\(12\) 0 0
\(13\) 0.697224 0.193375 0.0966876 − 0.995315i \(-0.469175\pi\)
0.0966876 + 0.995315i \(0.469175\pi\)
\(14\) 0 0
\(15\) 7.60555 1.96374
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 1.30278 0.298877 0.149439 − 0.988771i \(-0.452253\pi\)
0.149439 + 0.988771i \(0.452253\pi\)
\(20\) 0 0
\(21\) −7.60555 −1.65967
\(22\) 0 0
\(23\) − 4.30278i − 0.897191i −0.893735 − 0.448595i \(-0.851924\pi\)
0.893735 − 0.448595i \(-0.148076\pi\)
\(24\) 0 0
\(25\) −0.302776 −0.0605551
\(26\) 0 0
\(27\) − 16.2111i − 3.11983i
\(28\) 0 0
\(29\) − 0.697224i − 0.129471i −0.997902 − 0.0647357i \(-0.979380\pi\)
0.997902 − 0.0647357i \(-0.0206204\pi\)
\(30\) 0 0
\(31\) − 4.21110i − 0.756336i −0.925737 − 0.378168i \(-0.876554\pi\)
0.925737 − 0.378168i \(-0.123446\pi\)
\(32\) 0 0
\(33\) 18.5139 3.22285
\(34\) 0 0
\(35\) 5.30278 0.896333
\(36\) 0 0
\(37\) 8.60555i 1.41474i 0.706842 + 0.707372i \(0.250119\pi\)
−0.706842 + 0.707372i \(0.749881\pi\)
\(38\) 0 0
\(39\) 2.30278i 0.368739i
\(40\) 0 0
\(41\) − 6.00000i − 0.937043i −0.883452 − 0.468521i \(-0.844787\pi\)
0.883452 − 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 4.21110 0.642187 0.321094 − 0.947047i \(-0.395950\pi\)
0.321094 + 0.947047i \(0.395950\pi\)
\(44\) 0 0
\(45\) 18.2111i 2.71475i
\(46\) 0 0
\(47\) 11.6056 1.69284 0.846422 − 0.532513i \(-0.178752\pi\)
0.846422 + 0.532513i \(0.178752\pi\)
\(48\) 0 0
\(49\) 1.69722 0.242461
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.30278 −0.453671 −0.226836 − 0.973933i \(-0.572838\pi\)
−0.226836 + 0.973933i \(0.572838\pi\)
\(54\) 0 0
\(55\) −12.9083 −1.74056
\(56\) 0 0
\(57\) 4.30278i 0.569917i
\(58\) 0 0
\(59\) −3.21110 −0.418050 −0.209025 − 0.977910i \(-0.567029\pi\)
−0.209025 + 0.977910i \(0.567029\pi\)
\(60\) 0 0
\(61\) 1.60555i 0.205570i 0.994704 + 0.102785i \(0.0327753\pi\)
−0.994704 + 0.102785i \(0.967225\pi\)
\(62\) 0 0
\(63\) − 18.2111i − 2.29438i
\(64\) 0 0
\(65\) − 1.60555i − 0.199144i
\(66\) 0 0
\(67\) 0.605551 0.0739799 0.0369899 − 0.999316i \(-0.488223\pi\)
0.0369899 + 0.999316i \(0.488223\pi\)
\(68\) 0 0
\(69\) 14.2111 1.71082
\(70\) 0 0
\(71\) − 12.4222i − 1.47424i −0.675759 − 0.737122i \(-0.736184\pi\)
0.675759 − 0.737122i \(-0.263816\pi\)
\(72\) 0 0
\(73\) 2.39445i 0.280249i 0.990134 + 0.140125i \(0.0447503\pi\)
−0.990134 + 0.140125i \(0.955250\pi\)
\(74\) 0 0
\(75\) − 1.00000i − 0.115470i
\(76\) 0 0
\(77\) 12.9083 1.47104
\(78\) 0 0
\(79\) − 10.0000i − 1.12509i −0.826767 − 0.562544i \(-0.809823\pi\)
0.826767 − 0.562544i \(-0.190177\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 0 0
\(83\) 8.69722 0.954644 0.477322 − 0.878728i \(-0.341607\pi\)
0.477322 + 0.878728i \(0.341607\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.30278 0.246883
\(88\) 0 0
\(89\) 7.21110 0.764375 0.382188 − 0.924085i \(-0.375171\pi\)
0.382188 + 0.924085i \(0.375171\pi\)
\(90\) 0 0
\(91\) 1.60555i 0.168308i
\(92\) 0 0
\(93\) 13.9083 1.44223
\(94\) 0 0
\(95\) − 3.00000i − 0.307794i
\(96\) 0 0
\(97\) 16.1194i 1.63668i 0.574734 + 0.818340i \(0.305105\pi\)
−0.574734 + 0.818340i \(0.694895\pi\)
\(98\) 0 0
\(99\) 44.3305i 4.45539i
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 2312.2.b.h.577.4 4
17.4 even 4 2312.2.a.n.1.2 yes 2
17.13 even 4 2312.2.a.e.1.1 ✓ 2
17.16 even 2 inner 2312.2.b.h.577.1 4
68.47 odd 4 4624.2.a.y.1.2 2
68.55 odd 4 4624.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2312.2.a.e.1.1 ✓ 2 17.13 even 4
2312.2.a.n.1.2 yes 2 17.4 even 4
2312.2.b.h.577.1 4 17.16 even 2 inner
2312.2.b.h.577.4 4 1.1 even 1 trivial
4624.2.a.g.1.1 2 68.55 odd 4
4624.2.a.y.1.2 2 68.47 odd 4