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Properties

Label 400.4.c.m.49.2
Level $400$
Weight $4$
Character 400.49
Analytic conductor $23.601$
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] = [400,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,52,0,38] 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: \(23.6007640023\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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 400.4.c.m.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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +6.00000i q^{7} +26.0000 q^{9} +19.0000 q^{11} -12.0000i q^{13} -75.0000i q^{17} -91.0000 q^{19} -6.00000 q^{21} +174.000i q^{23} +53.0000i q^{27} +272.000 q^{29} +230.000 q^{31} +19.0000i q^{33} -182.000i q^{37} +12.0000 q^{39} +117.000 q^{41} +372.000i q^{43} +52.0000i q^{47} +307.000 q^{49} +75.0000 q^{51} +402.000i q^{53} -91.0000i q^{57} +312.000 q^{59} +170.000 q^{61} +156.000i q^{63} -763.000i q^{67} -174.000 q^{69} +52.0000 q^{71} +981.000i q^{73} +114.000i q^{77} +1054.00 q^{79} +649.000 q^{81} +351.000i q^{83} +272.000i q^{87} -799.000 q^{89} +72.0000 q^{91} +230.000i q^{93} +962.000i q^{97} +494.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 52 q^{9} + 38 q^{11} - 182 q^{19} - 12 q^{21} + 544 q^{29} + 460 q^{31} + 24 q^{39} + 234 q^{41} + 614 q^{49} + 150 q^{51} + 624 q^{59} + 340 q^{61} - 348 q^{69} + 104 q^{71} + 2108 q^{79} + 1298 q^{81}+ \cdots + 988 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000i 0.192450i 0.995360 + 0.0962250i \(0.0306768\pi\)
−0.995360 + 0.0962250i \(0.969323\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.00000i 0.323970i 0.986793 + 0.161985i \(0.0517895\pi\)
−0.986793 + 0.161985i \(0.948210\pi\)
\(8\) 0 0
\(9\) 26.0000 0.962963
\(10\) 0 0
\(11\) 19.0000 0.520792 0.260396 − 0.965502i \(-0.416147\pi\)
0.260396 + 0.965502i \(0.416147\pi\)
\(12\) 0 0
\(13\) − 12.0000i − 0.256015i −0.991773 − 0.128008i \(-0.959142\pi\)
0.991773 − 0.128008i \(-0.0408582\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 75.0000i − 1.07001i −0.844849 − 0.535005i \(-0.820310\pi\)
0.844849 − 0.535005i \(-0.179690\pi\)
\(18\) 0 0
\(19\) −91.0000 −1.09878 −0.549390 − 0.835566i \(-0.685140\pi\)
−0.549390 + 0.835566i \(0.685140\pi\)
\(20\) 0 0
\(21\) −6.00000 −0.0623480
\(22\) 0 0
\(23\) 174.000i 1.57746i 0.614742 + 0.788728i \(0.289260\pi\)
−0.614742 + 0.788728i \(0.710740\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 53.0000i 0.377772i
\(28\) 0 0
\(29\) 272.000 1.74169 0.870847 − 0.491554i \(-0.163571\pi\)
0.870847 + 0.491554i \(0.163571\pi\)
\(30\) 0 0
\(31\) 230.000 1.33256 0.666278 − 0.745704i \(-0.267887\pi\)
0.666278 + 0.745704i \(0.267887\pi\)
\(32\) 0 0
\(33\) 19.0000i 0.100227i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 182.000i − 0.808665i −0.914612 − 0.404333i \(-0.867504\pi\)
0.914612 − 0.404333i \(-0.132496\pi\)
\(38\) 0 0
\(39\) 12.0000 0.0492702
\(40\) 0 0
\(41\) 117.000 0.445667 0.222833 − 0.974857i \(-0.428469\pi\)
0.222833 + 0.974857i \(0.428469\pi\)
\(42\) 0 0
\(43\) 372.000i 1.31929i 0.751577 + 0.659645i \(0.229293\pi\)
−0.751577 + 0.659645i \(0.770707\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 52.0000i 0.161383i 0.996739 + 0.0806913i \(0.0257128\pi\)
−0.996739 + 0.0806913i \(0.974287\pi\)
\(48\) 0 0
\(49\) 307.000 0.895044
\(50\) 0 0
\(51\) 75.0000 0.205924
\(52\) 0 0
\(53\) 402.000i 1.04187i 0.853597 + 0.520933i \(0.174416\pi\)
−0.853597 + 0.520933i \(0.825584\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 91.0000i − 0.211460i
\(58\) 0 0
\(59\) 312.000 0.688457 0.344228 − 0.938886i \(-0.388141\pi\)
0.344228 + 0.938886i \(0.388141\pi\)
\(60\) 0 0
\(61\) 170.000 0.356824 0.178412 − 0.983956i \(-0.442904\pi\)
0.178412 + 0.983956i \(0.442904\pi\)
\(62\) 0 0
\(63\) 156.000i 0.311971i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 763.000i − 1.39127i −0.718394 − 0.695636i \(-0.755122\pi\)
0.718394 − 0.695636i \(-0.244878\pi\)
\(68\) 0 0
\(69\) −174.000 −0.303582
\(70\) 0 0
\(71\) 52.0000 0.0869192 0.0434596 − 0.999055i \(-0.486162\pi\)
0.0434596 + 0.999055i \(0.486162\pi\)
\(72\) 0 0
\(73\) 981.000i 1.57284i 0.617692 + 0.786420i \(0.288068\pi\)
−0.617692 + 0.786420i \(0.711932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 114.000i 0.168721i
\(78\) 0 0
\(79\) 1054.00 1.50107 0.750533 − 0.660833i \(-0.229797\pi\)
0.750533 + 0.660833i \(0.229797\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 0 0
\(83\) 351.000i 0.464184i 0.972694 + 0.232092i \(0.0745570\pi\)
−0.972694 + 0.232092i \(0.925443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 272.000i 0.335189i
\(88\) 0 0
\(89\) −799.000 −0.951616 −0.475808 − 0.879549i \(-0.657844\pi\)
−0.475808 + 0.879549i \(0.657844\pi\)
\(90\) 0 0
\(91\) 72.0000 0.0829412
\(92\) 0 0
\(93\) 230.000i 0.256450i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 962.000i 1.00697i 0.864003 + 0.503486i \(0.167949\pi\)
−0.864003 + 0.503486i \(0.832051\pi\)
\(98\) 0 0
\(99\) 494.000 0.501504
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 400.4.c.m.49.2 2
4.3 odd 2 200.4.c.g.49.1 2
5.2 odd 4 400.4.a.k.1.1 1
5.3 odd 4 400.4.a.j.1.1 1
5.4 even 2 inner 400.4.c.m.49.1 2
12.11 even 2 1800.4.f.p.649.1 2
20.3 even 4 200.4.a.f.1.1 yes 1
20.7 even 4 200.4.a.e.1.1 ✓ 1
20.19 odd 2 200.4.c.g.49.2 2
40.3 even 4 1600.4.a.w.1.1 1
40.13 odd 4 1600.4.a.be.1.1 1
40.27 even 4 1600.4.a.bf.1.1 1
40.37 odd 4 1600.4.a.v.1.1 1
60.23 odd 4 1800.4.a.l.1.1 1
60.47 odd 4 1800.4.a.w.1.1 1
60.59 even 2 1800.4.f.p.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 ✓ 1 20.7 even 4
200.4.a.f.1.1 yes 1 20.3 even 4
200.4.c.g.49.1 2 4.3 odd 2
200.4.c.g.49.2 2 20.19 odd 2
400.4.a.j.1.1 1 5.3 odd 4
400.4.a.k.1.1 1 5.2 odd 4
400.4.c.m.49.1 2 5.4 even 2 inner
400.4.c.m.49.2 2 1.1 even 1 trivial
1600.4.a.v.1.1 1 40.37 odd 4
1600.4.a.w.1.1 1 40.3 even 4
1600.4.a.be.1.1 1 40.13 odd 4
1600.4.a.bf.1.1 1 40.27 even 4
1800.4.a.l.1.1 1 60.23 odd 4
1800.4.a.w.1.1 1 60.47 odd 4
1800.4.f.p.649.1 2 12.11 even 2
1800.4.f.p.649.2 2 60.59 even 2