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 324.8
Root \(-1.68729i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.i.324.3

$q$-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)
 
\(f(q)\) \(=\) \(q+3.58234i q^{2} -2.57070i q^{3} -4.83317 q^{4} +9.20914 q^{6} +25.2782i q^{7} +11.3447i q^{8} +20.3915 q^{9} +11.7931 q^{11} +12.4247i q^{12} -21.3725i q^{13} -90.5551 q^{14} -79.3058 q^{16} +17.0000i q^{17} +73.0493i q^{18} +148.641 q^{19} +64.9827 q^{21} +42.2468i q^{22} +7.58989i q^{23} +29.1637 q^{24} +76.5636 q^{26} -121.829i q^{27} -122.174i q^{28} +5.09866 q^{29} -157.775 q^{31} -193.343i q^{32} -30.3165i q^{33} -60.8998 q^{34} -98.5556 q^{36} +425.395i q^{37} +532.484i q^{38} -54.9423 q^{39} -381.427 q^{41} +232.790i q^{42} +146.952i q^{43} -56.9980 q^{44} -27.1896 q^{46} +500.876i q^{47} +203.872i q^{48} -295.986 q^{49} +43.7020 q^{51} +103.297i q^{52} +234.490i q^{53} +436.435 q^{54} -286.772 q^{56} -382.113i q^{57} +18.2651i q^{58} -425.699 q^{59} +309.393 q^{61} -565.205i q^{62} +515.460i q^{63} +58.1754 q^{64} +108.604 q^{66} -191.944i q^{67} -82.1640i q^{68} +19.5114 q^{69} +768.770 q^{71} +231.334i q^{72} +447.674i q^{73} -1523.91 q^{74} -718.410 q^{76} +298.108i q^{77} -196.822i q^{78} -1344.69 q^{79} +237.382 q^{81} -1366.40i q^{82} -1046.10i q^{83} -314.073 q^{84} -526.432 q^{86} -13.1071i q^{87} +133.788i q^{88} +313.947 q^{89} +540.258 q^{91} -36.6833i q^{92} +405.594i q^{93} -1794.31 q^{94} -497.029 q^{96} +276.758i q^{97} -1060.32i q^{98} +240.478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 68 q^{4} - 10 q^{6} + 60 q^{9} + 252 q^{11} - 180 q^{14} + 644 q^{16} - 110 q^{19} + 12 q^{21} - 310 q^{24} - 790 q^{26} - 390 q^{29} + 194 q^{31} - 68 q^{34} - 2874 q^{36} - 746 q^{39} + 596 q^{41}+ \cdots + 1716 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.58234i 1.26655i 0.773927 + 0.633275i \(0.218290\pi\)
−0.773927 + 0.633275i \(0.781710\pi\)
\(3\) − 2.57070i − 0.494732i −0.968922 − 0.247366i \(-0.920435\pi\)
0.968922 − 0.247366i \(-0.0795650\pi\)
\(4\) −4.83317 −0.604147
\(5\) 0 0
\(6\) 9.20914 0.626603
\(7\) 25.2782i 1.36489i 0.730935 + 0.682447i \(0.239084\pi\)
−0.730935 + 0.682447i \(0.760916\pi\)
\(8\) 11.3447i 0.501368i
\(9\) 20.3915 0.755240
\(10\) 0 0
\(11\) 11.7931 0.323250 0.161625 − 0.986852i \(-0.448327\pi\)
0.161625 + 0.986852i \(0.448327\pi\)
\(12\) 12.4247i 0.298891i
\(13\) − 21.3725i − 0.455974i −0.973664 − 0.227987i \(-0.926786\pi\)
0.973664 − 0.227987i \(-0.0732144\pi\)
\(14\) −90.5551 −1.72870
\(15\) 0 0
\(16\) −79.3058 −1.23915
\(17\) 17.0000i 0.242536i
\(18\) 73.0493i 0.956549i
\(19\) 148.641 1.79477 0.897386 − 0.441246i \(-0.145463\pi\)
0.897386 + 0.441246i \(0.145463\pi\)
\(20\) 0 0
\(21\) 64.9827 0.675257
\(22\) 42.2468i 0.409412i
\(23\) 7.58989i 0.0688087i 0.999408 + 0.0344044i \(0.0109534\pi\)
−0.999408 + 0.0344044i \(0.989047\pi\)
\(24\) 29.1637 0.248043
\(25\) 0 0
\(26\) 76.5636 0.577513
\(27\) − 121.829i − 0.868374i
\(28\) − 122.174i − 0.824596i
\(29\) 5.09866 0.0326482 0.0163241 − 0.999867i \(-0.494804\pi\)
0.0163241 + 0.999867i \(0.494804\pi\)
\(30\) 0 0
\(31\) −157.775 −0.914106 −0.457053 − 0.889439i \(-0.651095\pi\)
−0.457053 + 0.889439i \(0.651095\pi\)
\(32\) − 193.343i − 1.06808i
\(33\) − 30.3165i − 0.159922i
\(34\) −60.8998 −0.307183
\(35\) 0 0
\(36\) −98.5556 −0.456276
\(37\) 425.395i 1.89012i 0.326892 + 0.945062i \(0.393999\pi\)
−0.326892 + 0.945062i \(0.606001\pi\)
\(38\) 532.484i 2.27317i
\(39\) −54.9423 −0.225585
\(40\) 0 0
\(41\) −381.427 −1.45290 −0.726450 − 0.687220i \(-0.758831\pi\)
−0.726450 + 0.687220i \(0.758831\pi\)
\(42\) 232.790i 0.855246i
\(43\) 146.952i 0.521162i 0.965452 + 0.260581i \(0.0839141\pi\)
−0.965452 + 0.260581i \(0.916086\pi\)
\(44\) −56.9980 −0.195290
\(45\) 0 0
\(46\) −27.1896 −0.0871497
\(47\) 500.876i 1.55447i 0.629208 + 0.777237i \(0.283379\pi\)
−0.629208 + 0.777237i \(0.716621\pi\)
\(48\) 203.872i 0.613049i
\(49\) −295.986 −0.862934
\(50\) 0 0
\(51\) 43.7020 0.119990
\(52\) 103.297i 0.275475i
\(53\) 234.490i 0.607730i 0.952715 + 0.303865i \(0.0982773\pi\)
−0.952715 + 0.303865i \(0.901723\pi\)
\(54\) 436.435 1.09984
\(55\) 0 0
\(56\) −286.772 −0.684313
\(57\) − 382.113i − 0.887932i
\(58\) 18.2651i 0.0413505i
\(59\) −425.699 −0.939344 −0.469672 − 0.882841i \(-0.655628\pi\)
−0.469672 + 0.882841i \(0.655628\pi\)
\(60\) 0 0
\(61\) 309.393 0.649405 0.324703 − 0.945816i \(-0.394736\pi\)
0.324703 + 0.945816i \(0.394736\pi\)
\(62\) − 565.205i − 1.15776i
\(63\) 515.460i 1.03082i
\(64\) 58.1754 0.113624
\(65\) 0 0
\(66\) 108.604 0.202549
\(67\) − 191.944i − 0.349996i −0.984569 − 0.174998i \(-0.944008\pi\)
0.984569 − 0.174998i \(-0.0559919\pi\)
\(68\) − 82.1640i − 0.146527i
\(69\) 19.5114 0.0340419
\(70\) 0 0
\(71\) 768.770 1.28502 0.642509 − 0.766278i \(-0.277894\pi\)
0.642509 + 0.766278i \(0.277894\pi\)
\(72\) 231.334i 0.378653i
\(73\) 447.674i 0.717758i 0.933384 + 0.358879i \(0.116841\pi\)
−0.933384 + 0.358879i \(0.883159\pi\)
\(74\) −1523.91 −2.39393
\(75\) 0 0
\(76\) −718.410 −1.08431
\(77\) 298.108i 0.441201i
\(78\) − 196.822i − 0.285715i
\(79\) −1344.69 −1.91506 −0.957529 − 0.288337i \(-0.906898\pi\)
−0.957529 + 0.288337i \(0.906898\pi\)
\(80\) 0 0
\(81\) 237.382 0.325628
\(82\) − 1366.40i − 1.84017i
\(83\) − 1046.10i − 1.38343i −0.722171 − 0.691715i \(-0.756856\pi\)
0.722171 − 0.691715i \(-0.243144\pi\)
\(84\) −314.073 −0.407954
\(85\) 0 0
\(86\) −526.432 −0.660077
\(87\) − 13.1071i − 0.0161521i
\(88\) 133.788i 0.162067i
\(89\) 313.947 0.373914 0.186957 − 0.982368i \(-0.440137\pi\)
0.186957 + 0.982368i \(0.440137\pi\)
\(90\) 0 0
\(91\) 540.258 0.622356
\(92\) − 36.6833i − 0.0415706i
\(93\) 405.594i 0.452238i
\(94\) −1794.31 −1.96882
\(95\) 0 0
\(96\) −497.029 −0.528414
\(97\) 276.758i 0.289696i 0.989454 + 0.144848i \(0.0462693\pi\)
−0.989454 + 0.144848i \(0.953731\pi\)
\(98\) − 1060.32i − 1.09295i
\(99\) 240.478 0.244131
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.324.8 10
5.2 odd 4 425.4.a.i.1.2 5
5.3 odd 4 85.4.a.g.1.4 ✓ 5
5.4 even 2 inner 425.4.b.i.324.3 10
15.8 even 4 765.4.a.m.1.2 5
20.3 even 4 1360.4.a.w.1.2 5
85.33 odd 4 1445.4.a.l.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.4 ✓ 5 5.3 odd 4
425.4.a.i.1.2 5 5.2 odd 4
425.4.b.i.324.3 10 5.4 even 2 inner
425.4.b.i.324.8 10 1.1 even 1 trivial
765.4.a.m.1.2 5 15.8 even 4
1360.4.a.w.1.2 5 20.3 even 4
1445.4.a.l.1.4 5 85.33 odd 4