Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3400,2,Mod(2449,3400)] 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("3400.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3400 = 2^{3} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3400.e (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,-12,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.1491366872\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 3400.2449
Dual form 3400.2.e.f.2449.4

$q$-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)
 
\(f(q)\) \(=\) \(q-3.23607i q^{3} -3.23607i q^{7} -7.47214 q^{9} +3.23607 q^{11} -4.47214i q^{13} -1.00000i q^{17} -2.47214 q^{19} -10.4721 q^{21} +3.23607i q^{23} +14.4721i q^{27} -2.00000 q^{29} -3.23607 q^{31} -10.4721i q^{33} -6.94427i q^{37} -14.4721 q^{39} +2.00000 q^{41} -10.4721i q^{43} +4.94427i q^{47} -3.47214 q^{49} -3.23607 q^{51} -2.00000i q^{53} +8.00000i q^{57} -5.52786 q^{59} -10.9443 q^{61} +24.1803i q^{63} +12.0000i q^{67} +10.4721 q^{69} +4.76393 q^{71} -2.94427i q^{73} -10.4721i q^{77} +1.70820 q^{79} +24.4164 q^{81} +10.4721i q^{83} +6.47214i q^{87} +16.4721 q^{89} -14.4721 q^{91} +10.4721i q^{93} -2.00000i q^{97} -24.1803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 4 q^{11} + 8 q^{19} - 24 q^{21} - 8 q^{29} - 4 q^{31} - 40 q^{39} + 8 q^{41} + 4 q^{49} - 4 q^{51} - 40 q^{59} - 8 q^{61} + 24 q^{69} + 28 q^{71} - 20 q^{79} + 44 q^{81} + 48 q^{89} - 40 q^{91}+ \cdots - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3400\mathbb{Z}\right)^\times\).

\(n\) \(1601\) \(1701\) \(2177\) \(2551\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 0 0
\(6\) 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
\(11\) 3.23607 0.975711 0.487856 − 0.872924i \(-0.337779\pi\)
0.487856 + 0.872924i \(0.337779\pi\)
\(12\) 0 0
\(13\) − 4.47214i − 1.24035i −0.784465 − 0.620174i \(-0.787062\pi\)
0.784465 − 0.620174i \(-0.212938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.00000i − 0.242536i
\(18\) 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 \(0.109542\pi\)
−0.941367 + 0.337383i \(0.890458\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.4721i 2.78516i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −3.23607 −0.581215 −0.290607 − 0.956842i \(-0.593857\pi\)
−0.290607 + 0.956842i \(0.593857\pi\)
\(32\) 0 0
\(33\) − 10.4721i − 1.82296i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.94427i − 1.14163i −0.821078 − 0.570816i \(-0.806627\pi\)
0.821078 − 0.570816i \(-0.193373\pi\)
\(38\) 0 0
\(39\) −14.4721 −2.31740
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 10.4721i − 1.59699i −0.602004 − 0.798493i \(-0.705631\pi\)
0.602004 − 0.798493i \(-0.294369\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.94427i 0.721196i 0.932721 + 0.360598i \(0.117427\pi\)
−0.932721 + 0.360598i \(0.882573\pi\)
\(48\) 0 0
\(49\) −3.47214 −0.496019
\(50\) 0 0
\(51\) −3.23607 −0.453140
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 − 0.137361i \(-0.956138\pi\)
0.990521 − 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(61\) −10.9443 −1.40127 −0.700635 − 0.713520i \(-0.747100\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(62\) 0 0
\(63\) 24.1803i 3.04644i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) 10.4721 1.26070
\(70\) 0 0
\(71\) 4.76393 0.565375 0.282687 − 0.959212i \(-0.408774\pi\)
0.282687 + 0.959212i \(0.408774\pi\)
\(72\) 0 0
\(73\) − 2.94427i − 0.344601i −0.985044 − 0.172300i \(-0.944880\pi\)
0.985044 − 0.172300i \(-0.0551200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 10.4721i − 1.19341i
\(78\) 0 0
\(79\) 1.70820 0.192188 0.0960940 − 0.995372i \(-0.469365\pi\)
0.0960940 + 0.995372i \(0.469365\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) 10.4721i 1.14947i 0.818341 + 0.574733i \(0.194894\pi\)
−0.818341 + 0.574733i \(0.805106\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.47214i 0.693886i
\(88\) 0 0
\(89\) 16.4721 1.74604 0.873021 − 0.487682i \(-0.162157\pi\)
0.873021 + 0.487682i \(0.162157\pi\)
\(90\) 0 0
\(91\) −14.4721 −1.51709
\(92\) 0 0
\(93\) 10.4721i 1.08591i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) −24.1803 −2.43022
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 3400.2.e.f.2449.1 4
5.2 odd 4 136.2.a.c.1.1 ✓ 2
5.3 odd 4 3400.2.a.i.1.2 2
5.4 even 2 inner 3400.2.e.f.2449.4 4
15.2 even 4 1224.2.a.i.1.2 2
20.3 even 4 6800.2.a.bd.1.1 2
20.7 even 4 272.2.a.f.1.2 2
35.27 even 4 6664.2.a.i.1.2 2
40.27 even 4 1088.2.a.o.1.1 2
40.37 odd 4 1088.2.a.s.1.2 2
60.47 odd 4 2448.2.a.u.1.1 2
85.47 odd 4 2312.2.b.g.577.1 4
85.67 odd 4 2312.2.a.m.1.2 2
85.72 odd 4 2312.2.b.g.577.4 4
120.77 even 4 9792.2.a.db.1.2 2
120.107 odd 4 9792.2.a.da.1.1 2
340.67 even 4 4624.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.a.c.1.1 ✓ 2 5.2 odd 4
272.2.a.f.1.2 2 20.7 even 4
1088.2.a.o.1.1 2 40.27 even 4
1088.2.a.s.1.2 2 40.37 odd 4
1224.2.a.i.1.2 2 15.2 even 4
2312.2.a.m.1.2 2 85.67 odd 4
2312.2.b.g.577.1 4 85.47 odd 4
2312.2.b.g.577.4 4 85.72 odd 4
2448.2.a.u.1.1 2 60.47 odd 4
3400.2.a.i.1.2 2 5.3 odd 4
3400.2.e.f.2449.1 4 1.1 even 1 trivial
3400.2.e.f.2449.4 4 5.4 even 2 inner
4624.2.a.h.1.1 2 340.67 even 4
6664.2.a.i.1.2 2 35.27 even 4
6800.2.a.bd.1.1 2 20.3 even 4
9792.2.a.da.1.1 2 120.107 odd 4
9792.2.a.db.1.2 2 120.77 even 4