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Properties

Label 2312.2.b.i.577.4
Level $2312$
Weight $2$
Character 2312.577
Analytic conductor $18.461$
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] = [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,-4,0,0,0,-16,0,8] 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: \(\Q(\sqrt{-2 + \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.4
Root \(0.765367i\) of defining polynomial
Character \(\chi\) \(=\) 2312.577
Dual form 2312.2.b.i.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)
 
\(f(q)\) \(=\) \(q+2.61313i q^{3} -2.93015i q^{5} -1.08239i q^{7} -3.82843 q^{9} +4.14386i q^{11} -2.58579 q^{13} +7.65685 q^{15} +7.65685 q^{19} +2.82843 q^{21} -6.30864i q^{23} -3.58579 q^{25} -2.16478i q^{27} +5.54328i q^{29} +1.08239i q^{31} -10.8284 q^{33} -3.17157 q^{35} -10.7695i q^{37} -6.75699i q^{39} +9.23880i q^{41} +0.343146 q^{43} +11.2179i q^{45} +1.17157 q^{47} +5.82843 q^{49} +8.24264 q^{53} +12.1421 q^{55} +20.0083i q^{57} +7.17157 q^{59} +4.46088i q^{61} +4.14386i q^{63} +7.57675i q^{65} +2.82843 q^{67} +16.4853 q^{69} +9.10748i q^{71} +7.97069i q^{73} -9.37011i q^{75} +4.48528 q^{77} +4.14386i q^{79} -5.82843 q^{81} +12.8284 q^{83} -14.4853 q^{87} +4.24264 q^{89} +2.79884i q^{91} -2.82843 q^{93} -22.4357i q^{95} +4.46088i q^{97} -15.8645i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 16 q^{13} + 8 q^{15} + 8 q^{19} - 20 q^{25} - 32 q^{33} - 24 q^{35} + 24 q^{43} + 16 q^{47} + 12 q^{49} + 16 q^{53} - 8 q^{55} + 40 q^{59} + 32 q^{69} - 16 q^{77} - 12 q^{81} + 40 q^{83}+ \cdots - 24 q^{87}+O(q^{100}) \) 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
\(3\) 2.61313i 1.50869i 0.656479 + 0.754344i \(0.272045\pi\)
−0.656479 + 0.754344i \(0.727955\pi\)
\(4\) 0 0
\(5\) − 2.93015i − 1.31040i −0.755454 − 0.655202i \(-0.772584\pi\)
0.755454 − 0.655202i \(-0.227416\pi\)
\(6\) 0 0
\(7\) − 1.08239i − 0.409106i −0.978856 − 0.204553i \(-0.934426\pi\)
0.978856 − 0.204553i \(-0.0655740\pi\)
\(8\) 0 0
\(9\) −3.82843 −1.27614
\(10\) 0 0
\(11\) 4.14386i 1.24942i 0.780857 + 0.624710i \(0.214783\pi\)
−0.780857 + 0.624710i \(0.785217\pi\)
\(12\) 0 0
\(13\) −2.58579 −0.717168 −0.358584 − 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(14\) 0 0
\(15\) 7.65685 1.97699
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 7.65685 1.75660 0.878301 − 0.478107i \(-0.158677\pi\)
0.878301 + 0.478107i \(0.158677\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) − 6.30864i − 1.31544i −0.753261 − 0.657722i \(-0.771520\pi\)
0.753261 − 0.657722i \(-0.228480\pi\)
\(24\) 0 0
\(25\) −3.58579 −0.717157
\(26\) 0 0
\(27\) − 2.16478i − 0.416613i
\(28\) 0 0
\(29\) 5.54328i 1.02936i 0.857382 + 0.514680i \(0.172089\pi\)
−0.857382 + 0.514680i \(0.827911\pi\)
\(30\) 0 0
\(31\) 1.08239i 0.194403i 0.995265 + 0.0972017i \(0.0309892\pi\)
−0.995265 + 0.0972017i \(0.969011\pi\)
\(32\) 0 0
\(33\) −10.8284 −1.88499
\(34\) 0 0
\(35\) −3.17157 −0.536094
\(36\) 0 0
\(37\) − 10.7695i − 1.77050i −0.465116 − 0.885250i \(-0.653987\pi\)
0.465116 − 0.885250i \(-0.346013\pi\)
\(38\) 0 0
\(39\) − 6.75699i − 1.08198i
\(40\) 0 0
\(41\) 9.23880i 1.44286i 0.692489 + 0.721429i \(0.256514\pi\)
−0.692489 + 0.721429i \(0.743486\pi\)
\(42\) 0 0
\(43\) 0.343146 0.0523292 0.0261646 − 0.999658i \(-0.491671\pi\)
0.0261646 + 0.999658i \(0.491671\pi\)
\(44\) 0 0
\(45\) 11.2179i 1.67226i
\(46\) 0 0
\(47\) 1.17157 0.170891 0.0854457 − 0.996343i \(-0.472769\pi\)
0.0854457 + 0.996343i \(0.472769\pi\)
\(48\) 0 0
\(49\) 5.82843 0.832632
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.24264 1.13221 0.566107 − 0.824332i \(-0.308449\pi\)
0.566107 + 0.824332i \(0.308449\pi\)
\(54\) 0 0
\(55\) 12.1421 1.63725
\(56\) 0 0
\(57\) 20.0083i 2.65017i
\(58\) 0 0
\(59\) 7.17157 0.933659 0.466830 − 0.884347i \(-0.345396\pi\)
0.466830 + 0.884347i \(0.345396\pi\)
\(60\) 0 0
\(61\) 4.46088i 0.571158i 0.958355 + 0.285579i \(0.0921859\pi\)
−0.958355 + 0.285579i \(0.907814\pi\)
\(62\) 0 0
\(63\) 4.14386i 0.522077i
\(64\) 0 0
\(65\) 7.57675i 0.939780i
\(66\) 0 0
\(67\) 2.82843 0.345547 0.172774 − 0.984962i \(-0.444727\pi\)
0.172774 + 0.984962i \(0.444727\pi\)
\(68\) 0 0
\(69\) 16.4853 1.98459
\(70\) 0 0
\(71\) 9.10748i 1.08086i 0.841389 + 0.540429i \(0.181738\pi\)
−0.841389 + 0.540429i \(0.818262\pi\)
\(72\) 0 0
\(73\) 7.97069i 0.932899i 0.884548 + 0.466450i \(0.154467\pi\)
−0.884548 + 0.466450i \(0.845533\pi\)
\(74\) 0 0
\(75\) − 9.37011i − 1.08197i
\(76\) 0 0
\(77\) 4.48528 0.511145
\(78\) 0 0
\(79\) 4.14386i 0.466221i 0.972450 + 0.233110i \(0.0748903\pi\)
−0.972450 + 0.233110i \(0.925110\pi\)
\(80\) 0 0
\(81\) −5.82843 −0.647603
\(82\) 0 0
\(83\) 12.8284 1.40810 0.704051 − 0.710149i \(-0.251372\pi\)
0.704051 + 0.710149i \(0.251372\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −14.4853 −1.55299
\(88\) 0 0
\(89\) 4.24264 0.449719 0.224860 − 0.974391i \(-0.427808\pi\)
0.224860 + 0.974391i \(0.427808\pi\)
\(90\) 0 0
\(91\) 2.79884i 0.293398i
\(92\) 0 0
\(93\) −2.82843 −0.293294
\(94\) 0 0
\(95\) − 22.4357i − 2.30186i
\(96\) 0 0
\(97\) 4.46088i 0.452934i 0.974019 + 0.226467i \(0.0727176\pi\)
−0.974019 + 0.226467i \(0.927282\pi\)
\(98\) 0 0
\(99\) − 15.8645i − 1.59444i
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.i.577.4 4
17.4 even 4 2312.2.a.t.1.4 4
17.6 odd 16 136.2.n.b.49.1 yes 4
17.13 even 4 2312.2.a.t.1.1 4
17.14 odd 16 136.2.n.b.25.1 ✓ 4
17.16 even 2 inner 2312.2.b.i.577.1 4
51.14 even 16 1224.2.bq.b.433.1 4
51.23 even 16 1224.2.bq.b.865.1 4
68.23 even 16 272.2.v.a.49.1 4
68.31 even 16 272.2.v.a.161.1 4
68.47 odd 4 4624.2.a.bo.1.4 4
68.55 odd 4 4624.2.a.bo.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.n.b.25.1 ✓ 4 17.14 odd 16
136.2.n.b.49.1 yes 4 17.6 odd 16
272.2.v.a.49.1 4 68.23 even 16
272.2.v.a.161.1 4 68.31 even 16
1224.2.bq.b.433.1 4 51.14 even 16
1224.2.bq.b.865.1 4 51.23 even 16
2312.2.a.t.1.1 4 17.13 even 4
2312.2.a.t.1.4 4 17.4 even 4
2312.2.b.i.577.1 4 17.16 even 2 inner
2312.2.b.i.577.4 4 1.1 even 1 trivial
4624.2.a.bo.1.1 4 68.55 odd 4
4624.2.a.bo.1.4 4 68.47 odd 4