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Properties

Label 400.8.c.j.49.2
Level $400$
Weight $8$
Character 400.49
Analytic conductor $124.954$
Analytic rank $0$
Dimension $2$
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] = [400,8,Mod(49,400)] 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("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,4086] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(124.954010194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.8.c.j.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+12.0000i q^{3} -1016.00i q^{7} +2043.00 q^{9} -1092.00 q^{11} -1382.00i q^{13} +14706.0i q^{17} -39940.0 q^{19} +12192.0 q^{21} +68712.0i q^{23} +50760.0i q^{27} +102570. q^{29} -227552. q^{31} -13104.0i q^{33} +160526. i q^{37} +16584.0 q^{39} +10842.0 q^{41} -630748. i q^{43} -472656. i q^{47} -208713. q^{49} -176472. q^{51} +1.49402e6i q^{53} -479280. i q^{57} +2.64066e6 q^{59} +827702. q^{61} -2.07569e6i q^{63} +126004. i q^{67} -824544. q^{69} +1.41473e6 q^{71} -980282. i q^{73} +1.10947e6i q^{77} -3.56680e6 q^{79} +3.85892e6 q^{81} +5.67289e6i q^{83} +1.23084e6i q^{87} +1.19512e7 q^{89} -1.40411e6 q^{91} -2.73062e6i q^{93} +8.68215e6i q^{97} -2.23096e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4086 q^{9} - 2184 q^{11} - 79880 q^{19} + 24384 q^{21} + 205140 q^{29} - 455104 q^{31} + 33168 q^{39} + 21684 q^{41} - 417426 q^{49} - 352944 q^{51} + 5281320 q^{59} + 1655404 q^{61} - 1649088 q^{69}+ \cdots - 4461912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 12.0000i 0.256600i 0.991735 + 0.128300i \(0.0409521\pi\)
−0.991735 + 0.128300i \(0.959048\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1016.00i − 1.11957i −0.828638 − 0.559784i \(-0.810884\pi\)
0.828638 − 0.559784i \(-0.189116\pi\)
\(8\) 0 0
\(9\) 2043.00 0.934156
\(10\) 0 0
\(11\) −1092.00 −0.247371 −0.123685 − 0.992321i \(-0.539471\pi\)
−0.123685 + 0.992321i \(0.539471\pi\)
\(12\) 0 0
\(13\) − 1382.00i − 0.174464i −0.996188 − 0.0872321i \(-0.972198\pi\)
0.996188 − 0.0872321i \(-0.0278022\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 14706.0i 0.725978i 0.931793 + 0.362989i \(0.118244\pi\)
−0.931793 + 0.362989i \(0.881756\pi\)
\(18\) 0 0
\(19\) −39940.0 −1.33589 −0.667945 − 0.744211i \(-0.732826\pi\)
−0.667945 + 0.744211i \(0.732826\pi\)
\(20\) 0 0
\(21\) 12192.0 0.287281
\(22\) 0 0
\(23\) 68712.0i 1.17757i 0.808291 + 0.588783i \(0.200393\pi\)
−0.808291 + 0.588783i \(0.799607\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 50760.0i 0.496305i
\(28\) 0 0
\(29\) 102570. 0.780957 0.390479 − 0.920612i \(-0.372310\pi\)
0.390479 + 0.920612i \(0.372310\pi\)
\(30\) 0 0
\(31\) −227552. −1.37188 −0.685938 − 0.727660i \(-0.740608\pi\)
−0.685938 + 0.727660i \(0.740608\pi\)
\(32\) 0 0
\(33\) − 13104.0i − 0.0634753i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 160526.i 0.521002i 0.965474 + 0.260501i \(0.0838877\pi\)
−0.965474 + 0.260501i \(0.916112\pi\)
\(38\) 0 0
\(39\) 16584.0 0.0447675
\(40\) 0 0
\(41\) 10842.0 0.0245678 0.0122839 − 0.999925i \(-0.496090\pi\)
0.0122839 + 0.999925i \(0.496090\pi\)
\(42\) 0 0
\(43\) − 630748.i − 1.20981i −0.796299 − 0.604904i \(-0.793212\pi\)
0.796299 − 0.604904i \(-0.206788\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 472656.i − 0.664053i −0.943270 − 0.332026i \(-0.892268\pi\)
0.943270 − 0.332026i \(-0.107732\pi\)
\(48\) 0 0
\(49\) −208713. −0.253433
\(50\) 0 0
\(51\) −176472. −0.186286
\(52\) 0 0
\(53\) 1.49402e6i 1.37845i 0.724548 + 0.689224i \(0.242048\pi\)
−0.724548 + 0.689224i \(0.757952\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 479280.i − 0.342789i
\(58\) 0 0
\(59\) 2.64066e6 1.67390 0.836952 − 0.547277i \(-0.184335\pi\)
0.836952 + 0.547277i \(0.184335\pi\)
\(60\) 0 0
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) 0 0
\(63\) − 2.07569e6i − 1.04585i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 126004.i 0.0511826i 0.999672 + 0.0255913i \(0.00814686\pi\)
−0.999672 + 0.0255913i \(0.991853\pi\)
\(68\) 0 0
\(69\) −824544. −0.302164
\(70\) 0 0
\(71\) 1.41473e6 0.469104 0.234552 − 0.972104i \(-0.424638\pi\)
0.234552 + 0.972104i \(0.424638\pi\)
\(72\) 0 0
\(73\) − 980282.i − 0.294931i −0.989067 − 0.147466i \(-0.952888\pi\)
0.989067 − 0.147466i \(-0.0471116\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.10947e6i 0.276948i
\(78\) 0 0
\(79\) −3.56680e6 −0.813924 −0.406962 − 0.913445i \(-0.633412\pi\)
−0.406962 + 0.913445i \(0.633412\pi\)
\(80\) 0 0
\(81\) 3.85892e6 0.806805
\(82\) 0 0
\(83\) 5.67289e6i 1.08901i 0.838758 + 0.544504i \(0.183282\pi\)
−0.838758 + 0.544504i \(0.816718\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.23084e6i 0.200394i
\(88\) 0 0
\(89\) 1.19512e7 1.79699 0.898496 − 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) 0 0
\(91\) −1.40411e6 −0.195325
\(92\) 0 0
\(93\) − 2.73062e6i − 0.352023i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.68215e6i 0.965886i 0.875652 + 0.482943i \(0.160432\pi\)
−0.875652 + 0.482943i \(0.839568\pi\)
\(98\) 0 0
\(99\) −2.23096e6 −0.231083
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 400.8.c.j.49.2 2
4.3 odd 2 50.8.b.c.49.1 2
5.2 odd 4 400.8.a.l.1.1 1
5.3 odd 4 16.8.a.b.1.1 1
5.4 even 2 inner 400.8.c.j.49.1 2
12.11 even 2 450.8.c.g.199.2 2
15.8 even 4 144.8.a.i.1.1 1
20.3 even 4 2.8.a.a.1.1 ✓ 1
20.7 even 4 50.8.a.g.1.1 1
20.19 odd 2 50.8.b.c.49.2 2
40.3 even 4 64.8.a.c.1.1 1
40.13 odd 4 64.8.a.e.1.1 1
60.23 odd 4 18.8.a.b.1.1 1
60.47 odd 4 450.8.a.c.1.1 1
60.59 even 2 450.8.c.g.199.1 2
80.3 even 4 256.8.b.b.129.2 2
80.13 odd 4 256.8.b.f.129.1 2
80.43 even 4 256.8.b.b.129.1 2
80.53 odd 4 256.8.b.f.129.2 2
120.53 even 4 576.8.a.f.1.1 1
120.83 odd 4 576.8.a.g.1.1 1
140.3 odd 12 98.8.c.e.79.1 2
140.23 even 12 98.8.c.d.67.1 2
140.83 odd 4 98.8.a.a.1.1 1
140.103 odd 12 98.8.c.e.67.1 2
140.123 even 12 98.8.c.d.79.1 2
180.23 odd 12 162.8.c.a.55.1 2
180.43 even 12 162.8.c.l.109.1 2
180.83 odd 12 162.8.c.a.109.1 2
180.103 even 12 162.8.c.l.55.1 2
220.43 odd 4 242.8.a.e.1.1 1
260.83 odd 4 338.8.b.d.337.2 2
260.103 even 4 338.8.a.d.1.1 1
260.203 odd 4 338.8.b.d.337.1 2
340.203 even 4 578.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 20.3 even 4
16.8.a.b.1.1 1 5.3 odd 4
18.8.a.b.1.1 1 60.23 odd 4
50.8.a.g.1.1 1 20.7 even 4
50.8.b.c.49.1 2 4.3 odd 2
50.8.b.c.49.2 2 20.19 odd 2
64.8.a.c.1.1 1 40.3 even 4
64.8.a.e.1.1 1 40.13 odd 4
98.8.a.a.1.1 1 140.83 odd 4
98.8.c.d.67.1 2 140.23 even 12
98.8.c.d.79.1 2 140.123 even 12
98.8.c.e.67.1 2 140.103 odd 12
98.8.c.e.79.1 2 140.3 odd 12
144.8.a.i.1.1 1 15.8 even 4
162.8.c.a.55.1 2 180.23 odd 12
162.8.c.a.109.1 2 180.83 odd 12
162.8.c.l.55.1 2 180.103 even 12
162.8.c.l.109.1 2 180.43 even 12
242.8.a.e.1.1 1 220.43 odd 4
256.8.b.b.129.1 2 80.43 even 4
256.8.b.b.129.2 2 80.3 even 4
256.8.b.f.129.1 2 80.13 odd 4
256.8.b.f.129.2 2 80.53 odd 4
338.8.a.d.1.1 1 260.103 even 4
338.8.b.d.337.1 2 260.203 odd 4
338.8.b.d.337.2 2 260.83 odd 4
400.8.a.l.1.1 1 5.2 odd 4
400.8.c.j.49.1 2 5.4 even 2 inner
400.8.c.j.49.2 2 1.1 even 1 trivial
450.8.a.c.1.1 1 60.47 odd 4
450.8.c.g.199.1 2 60.59 even 2
450.8.c.g.199.2 2 12.11 even 2
576.8.a.f.1.1 1 120.53 even 4
576.8.a.g.1.1 1 120.83 odd 4
578.8.a.b.1.1 1 340.203 even 4