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

$q$-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 \(-0.781710\pi\)
0.773927 − 0.633275i \(-0.218290\pi\)
\(3\) 2.57070i 0.494732i 0.968922 + 0.247366i \(0.0795650\pi\)
−0.968922 + 0.247366i \(0.920435\pi\)
\(4\) −4.83317 −0.604147
\(5\) 0 0
\(6\) 9.20914 0.626603
\(7\) − 25.2782i − 1.36489i −0.730935 − 0.682447i \(-0.760916\pi\)
0.730935 − 0.682447i \(-0.239084\pi\)
\(8\) − 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 \(0.0732144\pi\)
−0.973664 + 0.227987i \(0.926786\pi\)
\(14\) −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 \(-0.989047\pi\)
0.999408 − 0.0344044i \(-0.0109534\pi\)
\(24\) 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 \(-0.606001\pi\)
0.326892 − 0.945062i \(-0.393999\pi\)
\(38\) − 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 \(-0.916086\pi\)
0.965452 − 0.260581i \(-0.0839141\pi\)
\(44\) −56.9980 −0.195290
\(45\) 0 0
\(46\) −27.1896 −0.0871497
\(47\) − 500.876i − 1.55447i −0.629208 − 0.777237i \(-0.716621\pi\)
0.629208 − 0.777237i \(-0.283379\pi\)
\(48\) − 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 \(-0.901723\pi\)
0.952715 − 0.303865i \(-0.0982773\pi\)
\(54\) 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 \(0.0559919\pi\)
−0.984569 + 0.174998i \(0.944008\pi\)
\(68\) 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 \(-0.883159\pi\)
0.933384 − 0.358879i \(-0.116841\pi\)
\(74\) −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 \(0.243144\pi\)
−0.722171 + 0.691715i \(0.756856\pi\)
\(84\) −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 \(-0.953731\pi\)
0.989454 − 0.144848i \(-0.0462693\pi\)
\(98\) 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.324.3 10
5.2 odd 4 85.4.a.g.1.4 ✓ 5
5.3 odd 4 425.4.a.i.1.2 5
5.4 even 2 inner 425.4.b.i.324.8 10
15.2 even 4 765.4.a.m.1.2 5
20.7 even 4 1360.4.a.w.1.2 5
85.67 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 5.2 odd 4
425.4.a.i.1.2 5 5.3 odd 4
425.4.b.i.324.3 10 1.1 even 1 trivial
425.4.b.i.324.8 10 5.4 even 2 inner
765.4.a.m.1.2 5 15.2 even 4
1360.4.a.w.1.2 5 20.7 even 4
1445.4.a.l.1.4 5 85.67 odd 4