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Properties

Label 400.3.p.g.257.1
Level $400$
Weight $3$
Character 400.257
Analytic conductor $10.899$
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] = [400,3,Mod(193,400)] 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("400.193"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6,0,0,0,-6,0,0,0,-24] 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: \(10.8992105744\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.257
Dual form 400.3.p.g.193.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.00000 - 3.00000i) q^{3} +(-3.00000 - 3.00000i) q^{7} -9.00000i q^{9} -12.0000 q^{11} +(12.0000 - 12.0000i) q^{13} +(-12.0000 - 12.0000i) q^{17} -20.0000i q^{19} -18.0000 q^{21} +(3.00000 - 3.00000i) q^{23} -30.0000i q^{29} +8.00000 q^{31} +(-36.0000 + 36.0000i) q^{33} +(48.0000 + 48.0000i) q^{37} -72.0000i q^{39} -48.0000 q^{41} +(-27.0000 + 27.0000i) q^{43} +(27.0000 + 27.0000i) q^{47} -31.0000i q^{49} -72.0000 q^{51} +(12.0000 - 12.0000i) q^{53} +(-60.0000 - 60.0000i) q^{57} +60.0000i q^{59} +32.0000 q^{61} +(-27.0000 + 27.0000i) q^{63} +(-3.00000 - 3.00000i) q^{67} -18.0000i q^{69} +48.0000 q^{71} +(12.0000 - 12.0000i) q^{73} +(36.0000 + 36.0000i) q^{77} -40.0000i q^{79} +81.0000 q^{81} +(93.0000 - 93.0000i) q^{83} +(-90.0000 - 90.0000i) q^{87} +30.0000i q^{89} -72.0000 q^{91} +(24.0000 - 24.0000i) q^{93} +(-12.0000 - 12.0000i) q^{97} +108.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 6 q^{7} - 24 q^{11} + 24 q^{13} - 24 q^{17} - 36 q^{21} + 6 q^{23} + 16 q^{31} - 72 q^{33} + 96 q^{37} - 96 q^{41} - 54 q^{43} + 54 q^{47} - 144 q^{51} + 24 q^{53} - 120 q^{57} + 64 q^{61}+ \cdots - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 3.00000 − 3.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.00000 − 3.00000i −0.428571 − 0.428571i 0.459570 − 0.888142i \(-0.348004\pi\)
−0.888142 + 0.459570i \(0.848004\pi\)
\(8\) 0 0
\(9\) − 9.00000i − 1.00000i
\(10\) 0 0
\(11\) −12.0000 −1.09091 −0.545455 − 0.838140i \(-0.683643\pi\)
−0.545455 + 0.838140i \(0.683643\pi\)
\(12\) 0 0
\(13\) 12.0000 − 12.0000i 0.923077 − 0.923077i −0.0741688 − 0.997246i \(-0.523630\pi\)
0.997246 + 0.0741688i \(0.0236304\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −12.0000 − 12.0000i −0.705882 − 0.705882i 0.259784 − 0.965667i \(-0.416349\pi\)
−0.965667 + 0.259784i \(0.916349\pi\)
\(18\) 0 0
\(19\) − 20.0000i − 1.05263i −0.850289 − 0.526316i \(-0.823573\pi\)
0.850289 − 0.526316i \(-0.176427\pi\)
\(20\) 0 0
\(21\) −18.0000 −0.857143
\(22\) 0 0
\(23\) 3.00000 − 3.00000i 0.130435 − 0.130435i −0.638875 − 0.769310i \(-0.720600\pi\)
0.769310 + 0.638875i \(0.220600\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 30.0000i − 1.03448i −0.855840 − 0.517241i \(-0.826959\pi\)
0.855840 − 0.517241i \(-0.173041\pi\)
\(30\) 0 0
\(31\) 8.00000 0.258065 0.129032 − 0.991640i \(-0.458813\pi\)
0.129032 + 0.991640i \(0.458813\pi\)
\(32\) 0 0
\(33\) −36.0000 + 36.0000i −1.09091 + 1.09091i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 48.0000 + 48.0000i 1.29730 + 1.29730i 0.930171 + 0.367126i \(0.119658\pi\)
0.367126 + 0.930171i \(0.380342\pi\)
\(38\) 0 0
\(39\) − 72.0000i − 1.84615i
\(40\) 0 0
\(41\) −48.0000 −1.17073 −0.585366 − 0.810769i \(-0.699049\pi\)
−0.585366 + 0.810769i \(0.699049\pi\)
\(42\) 0 0
\(43\) −27.0000 + 27.0000i −0.627907 + 0.627907i −0.947541 − 0.319634i \(-0.896440\pi\)
0.319634 + 0.947541i \(0.396440\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 27.0000 + 27.0000i 0.574468 + 0.574468i 0.933374 − 0.358906i \(-0.116850\pi\)
−0.358906 + 0.933374i \(0.616850\pi\)
\(48\) 0 0
\(49\) − 31.0000i − 0.632653i
\(50\) 0 0
\(51\) −72.0000 −1.41176
\(52\) 0 0
\(53\) 12.0000 − 12.0000i 0.226415 − 0.226415i −0.584778 − 0.811193i \(-0.698818\pi\)
0.811193 + 0.584778i \(0.198818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −60.0000 − 60.0000i −1.05263 − 1.05263i
\(58\) 0 0
\(59\) 60.0000i 1.01695i 0.861077 + 0.508475i \(0.169790\pi\)
−0.861077 + 0.508475i \(0.830210\pi\)
\(60\) 0 0
\(61\) 32.0000 0.524590 0.262295 − 0.964988i \(-0.415521\pi\)
0.262295 + 0.964988i \(0.415521\pi\)
\(62\) 0 0
\(63\) −27.0000 + 27.0000i −0.428571 + 0.428571i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 − 3.00000i −0.0447761 − 0.0447761i 0.684364 − 0.729140i \(-0.260080\pi\)
−0.729140 + 0.684364i \(0.760080\pi\)
\(68\) 0 0
\(69\) − 18.0000i − 0.260870i
\(70\) 0 0
\(71\) 48.0000 0.676056 0.338028 − 0.941136i \(-0.390240\pi\)
0.338028 + 0.941136i \(0.390240\pi\)
\(72\) 0 0
\(73\) 12.0000 − 12.0000i 0.164384 − 0.164384i −0.620122 − 0.784505i \(-0.712917\pi\)
0.784505 + 0.620122i \(0.212917\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 36.0000 + 36.0000i 0.467532 + 0.467532i
\(78\) 0 0
\(79\) − 40.0000i − 0.506329i −0.967423 − 0.253165i \(-0.918529\pi\)
0.967423 − 0.253165i \(-0.0814714\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 93.0000 − 93.0000i 1.12048 − 1.12048i 0.128813 − 0.991669i \(-0.458883\pi\)
0.991669 − 0.128813i \(-0.0411167\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −90.0000 − 90.0000i −1.03448 − 1.03448i
\(88\) 0 0
\(89\) 30.0000i 0.337079i 0.985695 + 0.168539i \(0.0539050\pi\)
−0.985695 + 0.168539i \(0.946095\pi\)
\(90\) 0 0
\(91\) −72.0000 −0.791209
\(92\) 0 0
\(93\) 24.0000 − 24.0000i 0.258065 − 0.258065i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 − 12.0000i −0.123711 − 0.123711i 0.642540 − 0.766252i \(-0.277880\pi\)
−0.766252 + 0.642540i \(0.777880\pi\)
\(98\) 0 0
\(99\) 108.000i 1.09091i
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 400.3.p.g.257.1 2
4.3 odd 2 50.3.c.b.7.1 yes 2
5.2 odd 4 400.3.p.a.193.1 2
5.3 odd 4 inner 400.3.p.g.193.1 2
5.4 even 2 400.3.p.a.257.1 2
12.11 even 2 450.3.g.c.307.1 2
20.3 even 4 50.3.c.b.43.1 yes 2
20.7 even 4 50.3.c.a.43.1 yes 2
20.19 odd 2 50.3.c.a.7.1 ✓ 2
60.23 odd 4 450.3.g.c.343.1 2
60.47 odd 4 450.3.g.e.343.1 2
60.59 even 2 450.3.g.e.307.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.3.c.a.7.1 ✓ 2 20.19 odd 2
50.3.c.a.43.1 yes 2 20.7 even 4
50.3.c.b.7.1 yes 2 4.3 odd 2
50.3.c.b.43.1 yes 2 20.3 even 4
400.3.p.a.193.1 2 5.2 odd 4
400.3.p.a.257.1 2 5.4 even 2
400.3.p.g.193.1 2 5.3 odd 4 inner
400.3.p.g.257.1 2 1.1 even 1 trivial
450.3.g.c.307.1 2 12.11 even 2
450.3.g.c.343.1 2 60.23 odd 4
450.3.g.e.307.1 2 60.59 even 2
450.3.g.e.343.1 2 60.47 odd 4