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Properties

Label 1575.2.b.c.251.3
Level $1575$
Weight $2$
Character 1575.251
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
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] = [1575,2,Mod(251,1575)] 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("1575.251"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,0,12,0,-4,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.3
Root \(0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 1575.251
Dual form 1575.2.b.c.251.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.517638i q^{2} +1.73205 q^{4} +(-1.00000 - 2.44949i) q^{7} +1.93185i q^{8} +0.378937i q^{11} -1.79315i q^{13} +(1.26795 - 0.517638i) q^{14} +2.46410 q^{16} +3.46410 q^{17} -1.79315i q^{19} -0.196152 q^{22} +1.41421i q^{23} +0.928203 q^{26} +(-1.73205 - 4.24264i) q^{28} -1.41421i q^{29} -6.69213i q^{31} +5.13922i q^{32} +1.79315i q^{34} +1.46410 q^{37} +0.928203 q^{38} +10.3923 q^{41} +4.92820 q^{43} +0.656339i q^{44} -0.732051 q^{46} +9.46410 q^{47} +(-5.00000 + 4.89898i) q^{49} -3.10583i q^{52} -10.1769i q^{53} +(4.73205 - 1.93185i) q^{56} +0.732051 q^{58} -9.46410 q^{59} -13.3843i q^{61} +3.46410 q^{62} +2.26795 q^{64} +10.9282 q^{67} +6.00000 q^{68} +15.0759i q^{71} +1.79315i q^{73} +0.757875i q^{74} -3.10583i q^{76} +(0.928203 - 0.378937i) q^{77} -10.9282 q^{79} +5.37945i q^{82} -9.46410 q^{83} +2.55103i q^{86} -0.732051 q^{88} -12.9282 q^{89} +(-4.39230 + 1.79315i) q^{91} +2.44949i q^{92} +4.89898i q^{94} -15.1774i q^{97} +(-2.53590 - 2.58819i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} + 12 q^{14} - 4 q^{16} + 20 q^{22} - 24 q^{26} - 8 q^{37} - 24 q^{38} - 8 q^{43} + 4 q^{46} + 24 q^{47} - 20 q^{49} + 12 q^{56} - 4 q^{58} - 24 q^{59} + 16 q^{64} + 16 q^{67} + 24 q^{68} - 24 q^{77}+ \cdots - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.517638i 0.366025i 0.983111 + 0.183013i \(0.0585849\pi\)
−0.983111 + 0.183013i \(0.941415\pi\)
\(3\) 0 0
\(4\) 1.73205 0.866025
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 − 2.44949i −0.377964 − 0.925820i
\(8\) 1.93185i 0.683013i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.378937i 0.114254i 0.998367 + 0.0571270i \(0.0181940\pi\)
−0.998367 + 0.0571270i \(0.981806\pi\)
\(12\) 0 0
\(13\) − 1.79315i − 0.497331i −0.968589 − 0.248665i \(-0.920008\pi\)
0.968589 − 0.248665i \(-0.0799919\pi\)
\(14\) 1.26795 − 0.517638i 0.338874 − 0.138345i
\(15\) 0 0
\(16\) 2.46410 0.616025
\(17\) 3.46410 0.840168 0.420084 − 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) − 1.79315i − 0.411377i −0.978618 − 0.205689i \(-0.934057\pi\)
0.978618 − 0.205689i \(-0.0659434\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.196152 −0.0418198
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.928203 0.182036
\(27\) 0 0
\(28\) −1.73205 − 4.24264i −0.327327 − 0.801784i
\(29\) − 1.41421i − 0.262613i −0.991342 − 0.131306i \(-0.958083\pi\)
0.991342 − 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) − 6.69213i − 1.20194i −0.799271 − 0.600971i \(-0.794781\pi\)
0.799271 − 0.600971i \(-0.205219\pi\)
\(32\) 5.13922i 0.908494i
\(33\) 0 0
\(34\) 1.79315i 0.307523i
\(35\) 0 0
\(36\) 0 0
\(37\) 1.46410 0.240697 0.120348 − 0.992732i \(-0.461599\pi\)
0.120348 + 0.992732i \(0.461599\pi\)
\(38\) 0.928203 0.150574
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923 1.62301 0.811503 − 0.584349i \(-0.198650\pi\)
0.811503 + 0.584349i \(0.198650\pi\)
\(42\) 0 0
\(43\) 4.92820 0.751544 0.375772 − 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0.656339i 0.0989468i
\(45\) 0 0
\(46\) −0.732051 −0.107935
\(47\) 9.46410 1.38048 0.690241 − 0.723580i \(-0.257505\pi\)
0.690241 + 0.723580i \(0.257505\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) − 3.10583i − 0.430701i
\(53\) − 10.1769i − 1.39790i −0.715168 − 0.698952i \(-0.753650\pi\)
0.715168 − 0.698952i \(-0.246350\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.73205 − 1.93185i 0.632347 − 0.258155i
\(57\) 0 0
\(58\) 0.732051 0.0961230
\(59\) −9.46410 −1.23212 −0.616061 − 0.787699i \(-0.711272\pi\)
−0.616061 + 0.787699i \(0.711272\pi\)
\(60\) 0 0
\(61\) − 13.3843i − 1.71368i −0.515583 − 0.856840i \(-0.672425\pi\)
0.515583 − 0.856840i \(-0.327575\pi\)
\(62\) 3.46410 0.439941
\(63\) 0 0
\(64\) 2.26795 0.283494
\(65\) 0 0
\(66\) 0 0
\(67\) 10.9282 1.33509 0.667546 − 0.744568i \(-0.267345\pi\)
0.667546 + 0.744568i \(0.267345\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 15.0759i 1.78918i 0.446891 + 0.894589i \(0.352531\pi\)
−0.446891 + 0.894589i \(0.647469\pi\)
\(72\) 0 0
\(73\) 1.79315i 0.209872i 0.994479 + 0.104936i \(0.0334638\pi\)
−0.994479 + 0.104936i \(0.966536\pi\)
\(74\) 0.757875i 0.0881012i
\(75\) 0 0
\(76\) − 3.10583i − 0.356263i
\(77\) 0.928203 − 0.378937i 0.105779 − 0.0431839i
\(78\) 0 0
\(79\) −10.9282 −1.22952 −0.614759 − 0.788715i \(-0.710747\pi\)
−0.614759 + 0.788715i \(0.710747\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 5.37945i 0.594061i
\(83\) −9.46410 −1.03882 −0.519410 − 0.854525i \(-0.673848\pi\)
−0.519410 + 0.854525i \(0.673848\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.55103i 0.275084i
\(87\) 0 0
\(88\) −0.732051 −0.0780369
\(89\) −12.9282 −1.37039 −0.685193 − 0.728361i \(-0.740282\pi\)
−0.685193 + 0.728361i \(0.740282\pi\)
\(90\) 0 0
\(91\) −4.39230 + 1.79315i −0.460439 + 0.187973i
\(92\) 2.44949i 0.255377i
\(93\) 0 0
\(94\) 4.89898i 0.505291i
\(95\) 0 0
\(96\) 0 0
\(97\) − 15.1774i − 1.54103i −0.637420 − 0.770516i \(-0.719998\pi\)
0.637420 − 0.770516i \(-0.280002\pi\)
\(98\) −2.53590 − 2.58819i −0.256164 − 0.261447i
\(99\) 0 0
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 1575.2.b.c.251.3 4
3.2 odd 2 1575.2.b.b.251.2 4
5.2 odd 4 1575.2.g.a.1574.3 8
5.3 odd 4 1575.2.g.a.1574.6 8
5.4 even 2 315.2.b.b.251.2 yes 4
7.6 odd 2 1575.2.b.b.251.3 4
15.2 even 4 1575.2.g.c.1574.5 8
15.8 even 4 1575.2.g.c.1574.4 8
15.14 odd 2 315.2.b.a.251.3 yes 4
20.19 odd 2 5040.2.f.c.881.1 4
21.20 even 2 inner 1575.2.b.c.251.2 4
35.13 even 4 1575.2.g.c.1574.6 8
35.27 even 4 1575.2.g.c.1574.3 8
35.34 odd 2 315.2.b.a.251.2 ✓ 4
60.59 even 2 5040.2.f.a.881.2 4
105.62 odd 4 1575.2.g.a.1574.5 8
105.83 odd 4 1575.2.g.a.1574.4 8
105.104 even 2 315.2.b.b.251.3 yes 4
140.139 even 2 5040.2.f.a.881.3 4
420.419 odd 2 5040.2.f.c.881.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.2.b.a.251.2 ✓ 4 35.34 odd 2
315.2.b.a.251.3 yes 4 15.14 odd 2
315.2.b.b.251.2 yes 4 5.4 even 2
315.2.b.b.251.3 yes 4 105.104 even 2
1575.2.b.b.251.2 4 3.2 odd 2
1575.2.b.b.251.3 4 7.6 odd 2
1575.2.b.c.251.2 4 21.20 even 2 inner
1575.2.b.c.251.3 4 1.1 even 1 trivial
1575.2.g.a.1574.3 8 5.2 odd 4
1575.2.g.a.1574.4 8 105.83 odd 4
1575.2.g.a.1574.5 8 105.62 odd 4
1575.2.g.a.1574.6 8 5.3 odd 4
1575.2.g.c.1574.3 8 35.27 even 4
1575.2.g.c.1574.4 8 15.8 even 4
1575.2.g.c.1574.5 8 15.2 even 4
1575.2.g.c.1574.6 8 35.13 even 4
5040.2.f.a.881.2 4 60.59 even 2
5040.2.f.a.881.3 4 140.139 even 2
5040.2.f.c.881.1 4 20.19 odd 2
5040.2.f.c.881.4 4 420.419 odd 2