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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [525,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-136,0,180] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(84.2015054018\)
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 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.6.d.a.274.1

$q$-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)
 
\(f(q)\) \(=\) \(q+10.0000i q^{2} -9.00000i q^{3} -68.0000 q^{4} +90.0000 q^{6} -49.0000i q^{7} -360.000i q^{8} -81.0000 q^{9} +92.0000 q^{11} +612.000i q^{12} -670.000i q^{13} +490.000 q^{14} +1424.00 q^{16} -222.000i q^{17} -810.000i q^{18} +908.000 q^{19} -441.000 q^{21} +920.000i q^{22} +1176.00i q^{23} -3240.00 q^{24} +6700.00 q^{26} +729.000i q^{27} +3332.00i q^{28} -1118.00 q^{29} +3696.00 q^{31} +2720.00i q^{32} -828.000i q^{33} +2220.00 q^{34} +5508.00 q^{36} +4182.00i q^{37} +9080.00i q^{38} -6030.00 q^{39} -6662.00 q^{41} -4410.00i q^{42} +3700.00i q^{43} -6256.00 q^{44} -11760.0 q^{46} -7056.00i q^{47} -12816.0i q^{48} -2401.00 q^{49} -1998.00 q^{51} +45560.0i q^{52} +37578.0i q^{53} -7290.00 q^{54} -17640.0 q^{56} -8172.00i q^{57} -11180.0i q^{58} -32700.0 q^{59} -10802.0 q^{61} +36960.0i q^{62} +3969.00i q^{63} +18368.0 q^{64} +8280.00 q^{66} +64996.0i q^{67} +15096.0i q^{68} +10584.0 q^{69} -61320.0 q^{71} +29160.0i q^{72} -38922.0i q^{73} -41820.0 q^{74} -61744.0 q^{76} -4508.00i q^{77} -60300.0i q^{78} +88096.0 q^{79} +6561.00 q^{81} -66620.0i q^{82} -71892.0i q^{83} +29988.0 q^{84} -37000.0 q^{86} +10062.0i q^{87} -33120.0i q^{88} -111818. q^{89} -32830.0 q^{91} -79968.0i q^{92} -33264.0i q^{93} +70560.0 q^{94} +24480.0 q^{96} -150846. i q^{97} -24010.0i q^{98} -7452.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 136 q^{4} + 180 q^{6} - 162 q^{9} + 184 q^{11} + 980 q^{14} + 2848 q^{16} + 1816 q^{19} - 882 q^{21} - 6480 q^{24} + 13400 q^{26} - 2236 q^{29} + 7392 q^{31} + 4440 q^{34} + 11016 q^{36} - 12060 q^{39}+ \cdots - 14904 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0000i 1.76777i 0.467707 + 0.883883i \(0.345080\pi\)
−0.467707 + 0.883883i \(0.654920\pi\)
\(3\) − 9.00000i − 0.577350i
\(4\) −68.0000 −2.12500
\(5\) 0 0
\(6\) 90.0000 1.02062
\(7\) − 49.0000i − 0.377964i
\(8\) − 360.000i − 1.98874i
\(9\) −81.0000 −0.333333
\(10\) 0 0
\(11\) 92.0000 0.229248 0.114624 − 0.993409i \(-0.463434\pi\)
0.114624 + 0.993409i \(0.463434\pi\)
\(12\) 612.000i 1.22687i
\(13\) − 670.000i − 1.09955i −0.835312 − 0.549777i \(-0.814713\pi\)
0.835312 − 0.549777i \(-0.185287\pi\)
\(14\) 490.000 0.668153
\(15\) 0 0
\(16\) 1424.00 1.39062
\(17\) − 222.000i − 0.186308i −0.995652 − 0.0931538i \(-0.970305\pi\)
0.995652 − 0.0931538i \(-0.0296948\pi\)
\(18\) − 810.000i − 0.589256i
\(19\) 908.000 0.577035 0.288517 − 0.957475i \(-0.406838\pi\)
0.288517 + 0.957475i \(0.406838\pi\)
\(20\) 0 0
\(21\) −441.000 −0.218218
\(22\) 920.000i 0.405258i
\(23\) 1176.00i 0.463541i 0.972771 + 0.231770i \(0.0744518\pi\)
−0.972771 + 0.231770i \(0.925548\pi\)
\(24\) −3240.00 −1.14820
\(25\) 0 0
\(26\) 6700.00 1.94375
\(27\) 729.000i 0.192450i
\(28\) 3332.00i 0.803175i
\(29\) −1118.00 −0.246858 −0.123429 − 0.992353i \(-0.539389\pi\)
−0.123429 + 0.992353i \(0.539389\pi\)
\(30\) 0 0
\(31\) 3696.00 0.690761 0.345380 − 0.938463i \(-0.387750\pi\)
0.345380 + 0.938463i \(0.387750\pi\)
\(32\) 2720.00i 0.469563i
\(33\) − 828.000i − 0.132357i
\(34\) 2220.00 0.329348
\(35\) 0 0
\(36\) 5508.00 0.708333
\(37\) 4182.00i 0.502203i 0.967961 + 0.251102i \(0.0807929\pi\)
−0.967961 + 0.251102i \(0.919207\pi\)
\(38\) 9080.00i 1.02006i
\(39\) −6030.00 −0.634828
\(40\) 0 0
\(41\) −6662.00 −0.618935 −0.309467 − 0.950910i \(-0.600151\pi\)
−0.309467 + 0.950910i \(0.600151\pi\)
\(42\) − 4410.00i − 0.385758i
\(43\) 3700.00i 0.305162i 0.988291 + 0.152581i \(0.0487585\pi\)
−0.988291 + 0.152581i \(0.951241\pi\)
\(44\) −6256.00 −0.487153
\(45\) 0 0
\(46\) −11760.0 −0.819432
\(47\) − 7056.00i − 0.465923i −0.972486 − 0.232961i \(-0.925158\pi\)
0.972486 − 0.232961i \(-0.0748416\pi\)
\(48\) − 12816.0i − 0.802878i
\(49\) −2401.00 −0.142857
\(50\) 0 0
\(51\) −1998.00 −0.107565
\(52\) 45560.0i 2.33655i
\(53\) 37578.0i 1.83757i 0.394758 + 0.918785i \(0.370828\pi\)
−0.394758 + 0.918785i \(0.629172\pi\)
\(54\) −7290.00 −0.340207
\(55\) 0 0
\(56\) −17640.0 −0.751672
\(57\) − 8172.00i − 0.333151i
\(58\) − 11180.0i − 0.436387i
\(59\) −32700.0 −1.22298 −0.611488 − 0.791254i \(-0.709429\pi\)
−0.611488 + 0.791254i \(0.709429\pi\)
\(60\) 0 0
\(61\) −10802.0 −0.371689 −0.185844 − 0.982579i \(-0.559502\pi\)
−0.185844 + 0.982579i \(0.559502\pi\)
\(62\) 36960.0i 1.22110i
\(63\) 3969.00i 0.125988i
\(64\) 18368.0 0.560547
\(65\) 0 0
\(66\) 8280.00 0.233976
\(67\) 64996.0i 1.76889i 0.466649 + 0.884443i \(0.345461\pi\)
−0.466649 + 0.884443i \(0.654539\pi\)
\(68\) 15096.0i 0.395904i
\(69\) 10584.0 0.267625
\(70\) 0 0
\(71\) −61320.0 −1.44363 −0.721816 − 0.692085i \(-0.756692\pi\)
−0.721816 + 0.692085i \(0.756692\pi\)
\(72\) 29160.0i 0.662913i
\(73\) − 38922.0i − 0.854846i −0.904052 − 0.427423i \(-0.859421\pi\)
0.904052 − 0.427423i \(-0.140579\pi\)
\(74\) −41820.0 −0.887779
\(75\) 0 0
\(76\) −61744.0 −1.22620
\(77\) − 4508.00i − 0.0866477i
\(78\) − 60300.0i − 1.12223i
\(79\) 88096.0 1.58814 0.794069 − 0.607827i \(-0.207959\pi\)
0.794069 + 0.607827i \(0.207959\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) − 66620.0i − 1.09413i
\(83\) − 71892.0i − 1.14547i −0.819739 − 0.572737i \(-0.805882\pi\)
0.819739 − 0.572737i \(-0.194118\pi\)
\(84\) 29988.0 0.463713
\(85\) 0 0
\(86\) −37000.0 −0.539455
\(87\) 10062.0i 0.142523i
\(88\) − 33120.0i − 0.455915i
\(89\) −111818. −1.49636 −0.748181 − 0.663495i \(-0.769073\pi\)
−0.748181 + 0.663495i \(0.769073\pi\)
\(90\) 0 0
\(91\) −32830.0 −0.415592
\(92\) − 79968.0i − 0.985024i
\(93\) − 33264.0i − 0.398811i
\(94\) 70560.0 0.823643
\(95\) 0 0
\(96\) 24480.0 0.271102
\(97\) − 150846.i − 1.62781i −0.580996 − 0.813906i \(-0.697337\pi\)
0.580996 − 0.813906i \(-0.302663\pi\)
\(98\) − 24010.0i − 0.252538i
\(99\) −7452.00 −0.0764161
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 525.6.d.a.274.2 2
5.2 odd 4 525.6.a.a.1.1 1
5.3 odd 4 21.6.a.d.1.1 ✓ 1
5.4 even 2 inner 525.6.d.a.274.1 2
15.8 even 4 63.6.a.a.1.1 1
20.3 even 4 336.6.a.a.1.1 1
35.3 even 12 147.6.e.b.79.1 2
35.13 even 4 147.6.a.g.1.1 1
35.18 odd 12 147.6.e.a.79.1 2
35.23 odd 12 147.6.e.a.67.1 2
35.33 even 12 147.6.e.b.67.1 2
60.23 odd 4 1008.6.a.bc.1.1 1
105.83 odd 4 441.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.d.1.1 ✓ 1 5.3 odd 4
63.6.a.a.1.1 1 15.8 even 4
147.6.a.g.1.1 1 35.13 even 4
147.6.e.a.67.1 2 35.23 odd 12
147.6.e.a.79.1 2 35.18 odd 12
147.6.e.b.67.1 2 35.33 even 12
147.6.e.b.79.1 2 35.3 even 12
336.6.a.a.1.1 1 20.3 even 4
441.6.a.b.1.1 1 105.83 odd 4
525.6.a.a.1.1 1 5.2 odd 4
525.6.d.a.274.1 2 5.4 even 2 inner
525.6.d.a.274.2 2 1.1 even 1 trivial
1008.6.a.bc.1.1 1 60.23 odd 4