Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,4,Mod(324,425)] 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("425.324"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-68,0,-10] 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: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 44x^{8} + 690x^{6} + 4708x^{4} + 14337x^{2} + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.5
Root \(4.05155i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.i.324.6

$q$-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)
 
\(f(q)\) \(=\) \(q-0.290753i q^{2} -7.70584i q^{3} +7.91546 q^{4} -2.24049 q^{6} -22.4661i q^{7} -4.62746i q^{8} -32.3800 q^{9} +39.6109 q^{11} -60.9953i q^{12} -6.70527i q^{13} -6.53208 q^{14} +61.9783 q^{16} -17.0000i q^{17} +9.41457i q^{18} -42.9490 q^{19} -173.120 q^{21} -11.5170i q^{22} -115.726i q^{23} -35.6585 q^{24} -1.94958 q^{26} +41.4574i q^{27} -177.830i q^{28} -184.009 q^{29} +201.848 q^{31} -55.0401i q^{32} -305.235i q^{33} -4.94280 q^{34} -256.303 q^{36} +189.885i q^{37} +12.4876i q^{38} -51.6697 q^{39} +297.987 q^{41} +50.3352i q^{42} +428.676i q^{43} +313.539 q^{44} -33.6476 q^{46} +311.134i q^{47} -477.595i q^{48} -161.725 q^{49} -130.999 q^{51} -53.0753i q^{52} -307.329i q^{53} +12.0538 q^{54} -103.961 q^{56} +330.959i q^{57} +53.5011i q^{58} -704.985 q^{59} -929.140 q^{61} -58.6879i q^{62} +727.452i q^{63} +479.823 q^{64} -88.7480 q^{66} -587.978i q^{67} -134.563i q^{68} -891.764 q^{69} +507.339 q^{71} +149.837i q^{72} -13.3237i q^{73} +55.2094 q^{74} -339.962 q^{76} -889.903i q^{77} +15.0231i q^{78} +143.573 q^{79} -554.796 q^{81} -86.6407i q^{82} +1017.25i q^{83} -1370.33 q^{84} +124.639 q^{86} +1417.94i q^{87} -183.298i q^{88} -772.337 q^{89} -150.641 q^{91} -916.023i q^{92} -1555.41i q^{93} +90.4630 q^{94} -424.130 q^{96} +1700.56i q^{97} +47.0221i q^{98} -1282.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 68 q^{4} - 10 q^{6} + 60 q^{9} + 252 q^{11} - 180 q^{14} + 644 q^{16} - 110 q^{19} + 12 q^{21} - 310 q^{24} - 790 q^{26} - 390 q^{29} + 194 q^{31} - 68 q^{34} - 2874 q^{36} - 746 q^{39} + 596 q^{41}+ \cdots + 1716 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.290753i − 0.102797i −0.998678 − 0.0513983i \(-0.983632\pi\)
0.998678 − 0.0513983i \(-0.0163678\pi\)
\(3\) − 7.70584i − 1.48299i −0.670958 − 0.741495i \(-0.734117\pi\)
0.670958 − 0.741495i \(-0.265883\pi\)
\(4\) 7.91546 0.989433
\(5\) 0 0
\(6\) −2.24049 −0.152446
\(7\) − 22.4661i − 1.21306i −0.795062 − 0.606528i \(-0.792562\pi\)
0.795062 − 0.606528i \(-0.207438\pi\)
\(8\) − 4.62746i − 0.204507i
\(9\) −32.3800 −1.19926
\(10\) 0 0
\(11\) 39.6109 1.08574 0.542870 − 0.839817i \(-0.317338\pi\)
0.542870 + 0.839817i \(0.317338\pi\)
\(12\) − 60.9953i − 1.46732i
\(13\) − 6.70527i − 0.143054i −0.997439 − 0.0715272i \(-0.977213\pi\)
0.997439 − 0.0715272i \(-0.0227873\pi\)
\(14\) −6.53208 −0.124698
\(15\) 0 0
\(16\) 61.9783 0.968410
\(17\) − 17.0000i − 0.242536i
\(18\) 9.41457i 0.123280i
\(19\) −42.9490 −0.518589 −0.259294 − 0.965798i \(-0.583490\pi\)
−0.259294 + 0.965798i \(0.583490\pi\)
\(20\) 0 0
\(21\) −173.120 −1.79895
\(22\) − 11.5170i − 0.111610i
\(23\) − 115.726i − 1.04915i −0.851364 − 0.524576i \(-0.824224\pi\)
0.851364 − 0.524576i \(-0.175776\pi\)
\(24\) −35.6585 −0.303282
\(25\) 0 0
\(26\) −1.94958 −0.0147055
\(27\) 41.4574i 0.295499i
\(28\) − 177.830i − 1.20024i
\(29\) −184.009 −1.17826 −0.589131 − 0.808037i \(-0.700530\pi\)
−0.589131 + 0.808037i \(0.700530\pi\)
\(30\) 0 0
\(31\) 201.848 1.16945 0.584726 − 0.811231i \(-0.301202\pi\)
0.584726 + 0.811231i \(0.301202\pi\)
\(32\) − 55.0401i − 0.304056i
\(33\) − 305.235i − 1.61014i
\(34\) −4.94280 −0.0249318
\(35\) 0 0
\(36\) −256.303 −1.18659
\(37\) 189.885i 0.843698i 0.906666 + 0.421849i \(0.138619\pi\)
−0.906666 + 0.421849i \(0.861381\pi\)
\(38\) 12.4876i 0.0533092i
\(39\) −51.6697 −0.212148
\(40\) 0 0
\(41\) 297.987 1.13507 0.567534 − 0.823350i \(-0.307897\pi\)
0.567534 + 0.823350i \(0.307897\pi\)
\(42\) 50.3352i 0.184926i
\(43\) 428.676i 1.52029i 0.649754 + 0.760144i \(0.274872\pi\)
−0.649754 + 0.760144i \(0.725128\pi\)
\(44\) 313.539 1.07427
\(45\) 0 0
\(46\) −33.6476 −0.107849
\(47\) 311.134i 0.965607i 0.875729 + 0.482803i \(0.160381\pi\)
−0.875729 + 0.482803i \(0.839619\pi\)
\(48\) − 477.595i − 1.43614i
\(49\) −161.725 −0.471503
\(50\) 0 0
\(51\) −130.999 −0.359678
\(52\) − 53.0753i − 0.141543i
\(53\) − 307.329i − 0.796507i −0.917275 − 0.398253i \(-0.869617\pi\)
0.917275 − 0.398253i \(-0.130383\pi\)
\(54\) 12.0538 0.0303763
\(55\) 0 0
\(56\) −103.961 −0.248078
\(57\) 330.959i 0.769062i
\(58\) 53.5011i 0.121121i
\(59\) −704.985 −1.55561 −0.777807 − 0.628503i \(-0.783668\pi\)
−0.777807 + 0.628503i \(0.783668\pi\)
\(60\) 0 0
\(61\) −929.140 −1.95023 −0.975117 − 0.221693i \(-0.928842\pi\)
−0.975117 + 0.221693i \(0.928842\pi\)
\(62\) − 58.6879i − 0.120216i
\(63\) 727.452i 1.45477i
\(64\) 479.823 0.937154
\(65\) 0 0
\(66\) −88.7480 −0.165517
\(67\) − 587.978i − 1.07213i −0.844175 − 0.536067i \(-0.819909\pi\)
0.844175 − 0.536067i \(-0.180091\pi\)
\(68\) − 134.563i − 0.239973i
\(69\) −891.764 −1.55588
\(70\) 0 0
\(71\) 507.339 0.848030 0.424015 − 0.905655i \(-0.360620\pi\)
0.424015 + 0.905655i \(0.360620\pi\)
\(72\) 149.837i 0.245257i
\(73\) − 13.3237i − 0.0213619i −0.999943 − 0.0106810i \(-0.996600\pi\)
0.999943 − 0.0106810i \(-0.00339992\pi\)
\(74\) 55.2094 0.0867293
\(75\) 0 0
\(76\) −339.962 −0.513109
\(77\) − 889.903i − 1.31706i
\(78\) 15.0231i 0.0218081i
\(79\) 143.573 0.204472 0.102236 − 0.994760i \(-0.467400\pi\)
0.102236 + 0.994760i \(0.467400\pi\)
\(80\) 0 0
\(81\) −554.796 −0.761037
\(82\) − 86.6407i − 0.116681i
\(83\) 1017.25i 1.34528i 0.739972 + 0.672638i \(0.234839\pi\)
−0.739972 + 0.672638i \(0.765161\pi\)
\(84\) −1370.33 −1.77994
\(85\) 0 0
\(86\) 124.639 0.156280
\(87\) 1417.94i 1.74735i
\(88\) − 183.298i − 0.222041i
\(89\) −772.337 −0.919860 −0.459930 − 0.887955i \(-0.652126\pi\)
−0.459930 + 0.887955i \(0.652126\pi\)
\(90\) 0 0
\(91\) −150.641 −0.173533
\(92\) − 916.023i − 1.03806i
\(93\) − 1555.41i − 1.73429i
\(94\) 90.4630 0.0992611
\(95\) 0 0
\(96\) −424.130 −0.450912
\(97\) 1700.56i 1.78006i 0.455899 + 0.890031i \(0.349318\pi\)
−0.455899 + 0.890031i \(0.650682\pi\)
\(98\) 47.0221i 0.0484689i
\(99\) −1282.60 −1.30208
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 425.4.b.i.324.5 10
5.2 odd 4 85.4.a.g.1.3 ✓ 5
5.3 odd 4 425.4.a.i.1.3 5
5.4 even 2 inner 425.4.b.i.324.6 10
15.2 even 4 765.4.a.m.1.3 5
20.7 even 4 1360.4.a.w.1.5 5
85.67 odd 4 1445.4.a.l.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.3 ✓ 5 5.2 odd 4
425.4.a.i.1.3 5 5.3 odd 4
425.4.b.i.324.5 10 1.1 even 1 trivial
425.4.b.i.324.6 10 5.4 even 2 inner
765.4.a.m.1.3 5 15.2 even 4
1360.4.a.w.1.5 5 20.7 even 4
1445.4.a.l.1.3 5 85.67 odd 4