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Properties

Label 525.4.d.g.274.3
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
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] = [525,4,Mod(274,525)] 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("525.274"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-34,0,-18,0,0,-36,0,-12] 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: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{57})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 29x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.3
Root \(3.27492i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.g.274.2

$q$-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)
 
\(f(q)\) \(=\) \(q+2.27492i q^{2} -3.00000i q^{3} +2.82475 q^{4} +6.82475 q^{6} +7.00000i q^{7} +24.6254i q^{8} -9.00000 q^{9} -40.7492 q^{11} -8.47425i q^{12} -53.2990i q^{13} -15.9244 q^{14} -33.4228 q^{16} +4.54983i q^{17} -20.4743i q^{18} -122.598 q^{19} +21.0000 q^{21} -92.7010i q^{22} -131.347i q^{23} +73.8762 q^{24} +121.251 q^{26} +27.0000i q^{27} +19.7733i q^{28} +216.598 q^{29} -251.794 q^{31} +120.969i q^{32} +122.248i q^{33} -10.3505 q^{34} -25.4228 q^{36} +11.8970i q^{37} -278.900i q^{38} -159.897 q^{39} -111.752 q^{41} +47.7733i q^{42} -369.196i q^{43} -115.106 q^{44} +298.804 q^{46} -262.694i q^{47} +100.268i q^{48} -49.0000 q^{49} +13.6495 q^{51} -150.556i q^{52} +567.100i q^{53} -61.4228 q^{54} -172.378 q^{56} +367.794i q^{57} +492.743i q^{58} -839.890 q^{59} -485.794 q^{61} -572.811i q^{62} -63.0000i q^{63} -542.577 q^{64} -278.103 q^{66} -333.691i q^{67} +12.8522i q^{68} -394.042 q^{69} +590.248 q^{71} -221.629i q^{72} -490.701i q^{73} -27.0647 q^{74} -346.309 q^{76} -285.244i q^{77} -363.752i q^{78} -121.691 q^{79} +81.0000 q^{81} -254.228i q^{82} -609.608i q^{83} +59.3198 q^{84} +839.890 q^{86} -649.794i q^{87} -1003.47i q^{88} -719.038 q^{89} +373.093 q^{91} -371.023i q^{92} +755.382i q^{93} +597.608 q^{94} +362.908 q^{96} -637.877i q^{97} -111.471i q^{98} +366.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 34 q^{4} - 18 q^{6} - 36 q^{9} - 12 q^{11} + 42 q^{14} + 274 q^{16} - 128 q^{19} + 84 q^{21} + 522 q^{24} + 636 q^{26} + 504 q^{29} + 80 q^{31} - 132 q^{34} + 306 q^{36} - 96 q^{39} - 900 q^{41} - 1608 q^{44}+ \cdots + 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.27492i 0.804305i 0.915573 + 0.402152i \(0.131738\pi\)
−0.915573 + 0.402152i \(0.868262\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 2.82475 0.353094
\(5\) 0 0
\(6\) 6.82475 0.464366
\(7\) 7.00000i 0.377964i
\(8\) 24.6254i 1.08830i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −40.7492 −1.11694 −0.558470 − 0.829525i \(-0.688611\pi\)
−0.558470 + 0.829525i \(0.688611\pi\)
\(12\) − 8.47425i − 0.203859i
\(13\) − 53.2990i − 1.13711i −0.822644 − 0.568557i \(-0.807502\pi\)
0.822644 − 0.568557i \(-0.192498\pi\)
\(14\) −15.9244 −0.303999
\(15\) 0 0
\(16\) −33.4228 −0.522231
\(17\) 4.54983i 0.0649116i 0.999473 + 0.0324558i \(0.0103328\pi\)
−0.999473 + 0.0324558i \(0.989667\pi\)
\(18\) − 20.4743i − 0.268102i
\(19\) −122.598 −1.48031 −0.740156 − 0.672436i \(-0.765248\pi\)
−0.740156 + 0.672436i \(0.765248\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) − 92.7010i − 0.898360i
\(23\) − 131.347i − 1.19077i −0.803439 − 0.595387i \(-0.796999\pi\)
0.803439 − 0.595387i \(-0.203001\pi\)
\(24\) 73.8762 0.628330
\(25\) 0 0
\(26\) 121.251 0.914586
\(27\) 27.0000i 0.192450i
\(28\) 19.7733i 0.133457i
\(29\) 216.598 1.38694 0.693470 − 0.720486i \(-0.256081\pi\)
0.693470 + 0.720486i \(0.256081\pi\)
\(30\) 0 0
\(31\) −251.794 −1.45882 −0.729412 − 0.684075i \(-0.760206\pi\)
−0.729412 + 0.684075i \(0.760206\pi\)
\(32\) 120.969i 0.668267i
\(33\) 122.248i 0.644865i
\(34\) −10.3505 −0.0522087
\(35\) 0 0
\(36\) −25.4228 −0.117698
\(37\) 11.8970i 0.0528610i 0.999651 + 0.0264305i \(0.00841407\pi\)
−0.999651 + 0.0264305i \(0.991586\pi\)
\(38\) − 278.900i − 1.19062i
\(39\) −159.897 −0.656513
\(40\) 0 0
\(41\) −111.752 −0.425678 −0.212839 − 0.977087i \(-0.568271\pi\)
−0.212839 + 0.977087i \(0.568271\pi\)
\(42\) 47.7733i 0.175514i
\(43\) − 369.196i − 1.30935i −0.755912 − 0.654673i \(-0.772806\pi\)
0.755912 − 0.654673i \(-0.227194\pi\)
\(44\) −115.106 −0.394385
\(45\) 0 0
\(46\) 298.804 0.957744
\(47\) − 262.694i − 0.815275i −0.913144 − 0.407637i \(-0.866353\pi\)
0.913144 − 0.407637i \(-0.133647\pi\)
\(48\) 100.268i 0.301510i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 13.6495 0.0374767
\(52\) − 150.556i − 0.401508i
\(53\) 567.100i 1.46976i 0.678199 + 0.734879i \(0.262761\pi\)
−0.678199 + 0.734879i \(0.737239\pi\)
\(54\) −61.4228 −0.154789
\(55\) 0 0
\(56\) −172.378 −0.411339
\(57\) 367.794i 0.854658i
\(58\) 492.743i 1.11552i
\(59\) −839.890 −1.85330 −0.926648 − 0.375931i \(-0.877323\pi\)
−0.926648 + 0.375931i \(0.877323\pi\)
\(60\) 0 0
\(61\) −485.794 −1.01966 −0.509832 − 0.860274i \(-0.670293\pi\)
−0.509832 + 0.860274i \(0.670293\pi\)
\(62\) − 572.811i − 1.17334i
\(63\) − 63.0000i − 0.125988i
\(64\) −542.577 −1.05972
\(65\) 0 0
\(66\) −278.103 −0.518668
\(67\) − 333.691i − 0.608460i −0.952599 − 0.304230i \(-0.901601\pi\)
0.952599 − 0.304230i \(-0.0983992\pi\)
\(68\) 12.8522i 0.0229199i
\(69\) −394.042 −0.687493
\(70\) 0 0
\(71\) 590.248 0.986613 0.493306 − 0.869856i \(-0.335788\pi\)
0.493306 + 0.869856i \(0.335788\pi\)
\(72\) − 221.629i − 0.362767i
\(73\) − 490.701i − 0.786743i −0.919380 − 0.393371i \(-0.871309\pi\)
0.919380 − 0.393371i \(-0.128691\pi\)
\(74\) −27.0647 −0.0425164
\(75\) 0 0
\(76\) −346.309 −0.522689
\(77\) − 285.244i − 0.422164i
\(78\) − 363.752i − 0.528037i
\(79\) −121.691 −0.173308 −0.0866539 − 0.996238i \(-0.527617\pi\)
−0.0866539 + 0.996238i \(0.527617\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 254.228i − 0.342375i
\(83\) − 609.608i − 0.806183i −0.915160 − 0.403091i \(-0.867936\pi\)
0.915160 − 0.403091i \(-0.132064\pi\)
\(84\) 59.3198 0.0770514
\(85\) 0 0
\(86\) 839.890 1.05311
\(87\) − 649.794i − 0.800750i
\(88\) − 1003.47i − 1.21557i
\(89\) −719.038 −0.856381 −0.428190 − 0.903689i \(-0.640849\pi\)
−0.428190 + 0.903689i \(0.640849\pi\)
\(90\) 0 0
\(91\) 373.093 0.429789
\(92\) − 371.023i − 0.420455i
\(93\) 755.382i 0.842252i
\(94\) 597.608 0.655729
\(95\) 0 0
\(96\) 362.908 0.385824
\(97\) − 637.877i − 0.667697i −0.942627 − 0.333849i \(-0.891653\pi\)
0.942627 − 0.333849i \(-0.108347\pi\)
\(98\) − 111.471i − 0.114901i
\(99\) 366.743 0.372313
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 525.4.d.g.274.3 4
5.2 odd 4 525.4.a.n.1.1 2
5.3 odd 4 21.4.a.c.1.2 ✓ 2
5.4 even 2 inner 525.4.d.g.274.2 4
15.2 even 4 1575.4.a.p.1.2 2
15.8 even 4 63.4.a.e.1.1 2
20.3 even 4 336.4.a.m.1.1 2
35.3 even 12 147.4.e.m.79.1 4
35.13 even 4 147.4.a.i.1.2 2
35.18 odd 12 147.4.e.l.79.1 4
35.23 odd 12 147.4.e.l.67.1 4
35.33 even 12 147.4.e.m.67.1 4
40.3 even 4 1344.4.a.bo.1.2 2
40.13 odd 4 1344.4.a.bg.1.2 2
60.23 odd 4 1008.4.a.ba.1.2 2
105.23 even 12 441.4.e.q.361.2 4
105.38 odd 12 441.4.e.p.226.2 4
105.53 even 12 441.4.e.q.226.2 4
105.68 odd 12 441.4.e.p.361.2 4
105.83 odd 4 441.4.a.r.1.1 2
140.83 odd 4 2352.4.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 ✓ 2 5.3 odd 4
63.4.a.e.1.1 2 15.8 even 4
147.4.a.i.1.2 2 35.13 even 4
147.4.e.l.67.1 4 35.23 odd 12
147.4.e.l.79.1 4 35.18 odd 12
147.4.e.m.67.1 4 35.33 even 12
147.4.e.m.79.1 4 35.3 even 12
336.4.a.m.1.1 2 20.3 even 4
441.4.a.r.1.1 2 105.83 odd 4
441.4.e.p.226.2 4 105.38 odd 12
441.4.e.p.361.2 4 105.68 odd 12
441.4.e.q.226.2 4 105.53 even 12
441.4.e.q.361.2 4 105.23 even 12
525.4.a.n.1.1 2 5.2 odd 4
525.4.d.g.274.2 4 5.4 even 2 inner
525.4.d.g.274.3 4 1.1 even 1 trivial
1008.4.a.ba.1.2 2 60.23 odd 4
1344.4.a.bg.1.2 2 40.13 odd 4
1344.4.a.bo.1.2 2 40.3 even 4
1575.4.a.p.1.2 2 15.2 even 4
2352.4.a.bz.1.2 2 140.83 odd 4