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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,-12,0,0,0,0,0,8] 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: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 2312.577
Dual form 2312.2.b.g.577.4

$q$-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)
 
\(f(q)\) \(=\) \(q-3.23607i q^{3} +2.00000i q^{5} -3.23607i q^{7} -7.47214 q^{9} -3.23607i q^{11} -4.47214 q^{13} +6.47214 q^{15} -2.47214 q^{19} -10.4721 q^{21} -3.23607i q^{23} +1.00000 q^{25} +14.4721i q^{27} +2.00000i q^{29} -3.23607i q^{31} -10.4721 q^{33} +6.47214 q^{35} +6.94427i q^{37} +14.4721i q^{39} -2.00000i q^{41} +10.4721 q^{43} -14.9443i q^{45} -4.94427 q^{47} -3.47214 q^{49} +2.00000 q^{53} +6.47214 q^{55} +8.00000i q^{57} -5.52786 q^{59} +10.9443i q^{61} +24.1803i q^{63} -8.94427i q^{65} -12.0000 q^{67} -10.4721 q^{69} +4.76393i q^{71} -2.94427i q^{73} -3.23607i q^{75} -10.4721 q^{77} +1.70820i q^{79} +24.4164 q^{81} -10.4721 q^{83} +6.47214 q^{87} -16.4721 q^{89} +14.4721i q^{91} -10.4721 q^{93} -4.94427i q^{95} +2.00000i q^{97} +24.1803i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 8 q^{15} + 8 q^{19} - 24 q^{21} + 4 q^{25} - 24 q^{33} + 8 q^{35} + 24 q^{43} + 16 q^{47} + 4 q^{49} + 8 q^{53} + 8 q^{55} - 40 q^{59} - 48 q^{67} - 24 q^{69} - 24 q^{77} + 44 q^{81} - 24 q^{83}+ \cdots - 24 q^{93}+O(q^{100}) \) 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
\(3\) − 3.23607i − 1.86834i −0.356822 − 0.934172i \(-0.616140\pi\)
0.356822 − 0.934172i \(-0.383860\pi\)
\(4\) 0 0
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) − 3.23607i − 1.22312i −0.791199 − 0.611559i \(-0.790543\pi\)
0.791199 − 0.611559i \(-0.209457\pi\)
\(8\) 0 0
\(9\) −7.47214 −2.49071
\(10\) 0 0
\(11\) − 3.23607i − 0.975711i −0.872924 − 0.487856i \(-0.837779\pi\)
0.872924 − 0.487856i \(-0.162221\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 − 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 6.47214 1.67110
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) −2.47214 −0.567147 −0.283573 − 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) 0 0
\(21\) −10.4721 −2.28521
\(22\) 0 0
\(23\) − 3.23607i − 0.674767i −0.941367 − 0.337383i \(-0.890458\pi\)
0.941367 − 0.337383i \(-0.109542\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 14.4721i 2.78516i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) − 3.23607i − 0.581215i −0.956842 − 0.290607i \(-0.906143\pi\)
0.956842 − 0.290607i \(-0.0938574\pi\)
\(32\) 0 0
\(33\) −10.4721 −1.82296
\(34\) 0 0
\(35\) 6.47214 1.09399
\(36\) 0 0
\(37\) 6.94427i 1.14163i 0.821078 + 0.570816i \(0.193373\pi\)
−0.821078 + 0.570816i \(0.806627\pi\)
\(38\) 0 0
\(39\) 14.4721i 2.31740i
\(40\) 0 0
\(41\) − 2.00000i − 0.312348i −0.987730 − 0.156174i \(-0.950084\pi\)
0.987730 − 0.156174i \(-0.0499160\pi\)
\(42\) 0 0
\(43\) 10.4721 1.59699 0.798493 − 0.602004i \(-0.205631\pi\)
0.798493 + 0.602004i \(0.205631\pi\)
\(44\) 0 0
\(45\) − 14.9443i − 2.22776i
\(46\) 0 0
\(47\) −4.94427 −0.721196 −0.360598 − 0.932721i \(-0.617427\pi\)
−0.360598 + 0.932721i \(0.617427\pi\)
\(48\) 0 0
\(49\) −3.47214 −0.496019
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 6.47214 0.872703
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) −5.52786 −0.719667 −0.359833 − 0.933017i \(-0.617166\pi\)
−0.359833 + 0.933017i \(0.617166\pi\)
\(60\) 0 0
\(61\) 10.9443i 1.40127i 0.713520 + 0.700635i \(0.247100\pi\)
−0.713520 + 0.700635i \(0.752900\pi\)
\(62\) 0 0
\(63\) 24.1803i 3.04644i
\(64\) 0 0
\(65\) − 8.94427i − 1.10940i
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −10.4721 −1.26070
\(70\) 0 0
\(71\) 4.76393i 0.565375i 0.959212 + 0.282687i \(0.0912259\pi\)
−0.959212 + 0.282687i \(0.908774\pi\)
\(72\) 0 0
\(73\) − 2.94427i − 0.344601i −0.985044 − 0.172300i \(-0.944880\pi\)
0.985044 − 0.172300i \(-0.0551200\pi\)
\(74\) 0 0
\(75\) − 3.23607i − 0.373669i
\(76\) 0 0
\(77\) −10.4721 −1.19341
\(78\) 0 0
\(79\) 1.70820i 0.192188i 0.995372 + 0.0960940i \(0.0306349\pi\)
−0.995372 + 0.0960940i \(0.969365\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) −10.4721 −1.14947 −0.574733 − 0.818341i \(-0.694894\pi\)
−0.574733 + 0.818341i \(0.694894\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.47214 0.693886
\(88\) 0 0
\(89\) −16.4721 −1.74604 −0.873021 − 0.487682i \(-0.837843\pi\)
−0.873021 + 0.487682i \(0.837843\pi\)
\(90\) 0 0
\(91\) 14.4721i 1.51709i
\(92\) 0 0
\(93\) −10.4721 −1.08591
\(94\) 0 0
\(95\) − 4.94427i − 0.507272i
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) 24.1803i 2.43022i
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.g.577.1 4
17.4 even 4 136.2.a.c.1.1 ✓ 2
17.13 even 4 2312.2.a.m.1.2 2
17.16 even 2 inner 2312.2.b.g.577.4 4
51.38 odd 4 1224.2.a.i.1.2 2
68.47 odd 4 4624.2.a.h.1.1 2
68.55 odd 4 272.2.a.f.1.2 2
85.4 even 4 3400.2.a.i.1.2 2
85.38 odd 4 3400.2.e.f.2449.1 4
85.72 odd 4 3400.2.e.f.2449.4 4
119.55 odd 4 6664.2.a.i.1.2 2
136.21 even 4 1088.2.a.s.1.2 2
136.123 odd 4 1088.2.a.o.1.1 2
204.191 even 4 2448.2.a.u.1.1 2
340.259 odd 4 6800.2.a.bd.1.1 2
408.293 odd 4 9792.2.a.db.1.2 2
408.395 even 4 9792.2.a.da.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.a.c.1.1 ✓ 2 17.4 even 4
272.2.a.f.1.2 2 68.55 odd 4
1088.2.a.o.1.1 2 136.123 odd 4
1088.2.a.s.1.2 2 136.21 even 4
1224.2.a.i.1.2 2 51.38 odd 4
2312.2.a.m.1.2 2 17.13 even 4
2312.2.b.g.577.1 4 1.1 even 1 trivial
2312.2.b.g.577.4 4 17.16 even 2 inner
2448.2.a.u.1.1 2 204.191 even 4
3400.2.a.i.1.2 2 85.4 even 4
3400.2.e.f.2449.1 4 85.38 odd 4
3400.2.e.f.2449.4 4 85.72 odd 4
4624.2.a.h.1.1 2 68.47 odd 4
6664.2.a.i.1.2 2 119.55 odd 4
6800.2.a.bd.1.1 2 340.259 odd 4
9792.2.a.da.1.1 2 408.395 even 4
9792.2.a.db.1.2 2 408.293 odd 4