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(424,425)] 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("425.424"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 833x^{4} + 71824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 424.2
Root \(-3.68218 - 3.68218i\) of defining polynomial
Character \(\chi\) \(=\) 425.424
Dual form 425.4.c.c.424.8

$q$-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)
 
\(f(q)\) \(=\) \(q-3.37228i q^{2} +7.36435 q^{3} -3.37228 q^{4} -24.8347i q^{6} +17.4703 q^{7} -15.6060i q^{8} +27.2337 q^{9} +51.5505i q^{11} -24.8347 q^{12} -75.2119i q^{13} -58.9148i q^{14} -79.6060 q^{16} +(69.0208 - 12.2119i) q^{17} -91.8397i q^{18} +28.0000 q^{19} +128.658 q^{21} +173.843 q^{22} +19.1913 q^{23} -114.928i q^{24} -253.636 q^{26} +1.72096 q^{27} -58.9148 q^{28} +70.7417i q^{29} -41.4445i q^{31} +143.606i q^{32} +379.636i q^{33} +(-41.1821 - 232.757i) q^{34} -91.8397 q^{36} +135.460 q^{37} -94.4239i q^{38} -553.887i q^{39} -288.771i q^{41} -433.870i q^{42} -88.2934i q^{43} -173.843i q^{44} -64.7184i q^{46} +157.576i q^{47} -586.246 q^{48} -37.7881 q^{49} +(508.293 - 89.9330i) q^{51} +253.636i q^{52} -120.250i q^{53} -5.80356i q^{54} -272.641i q^{56} +206.202 q^{57} +238.561 q^{58} +696.119 q^{59} +683.544i q^{61} -139.763 q^{62} +475.781 q^{63} -152.568 q^{64} +1280.24 q^{66} +123.826i q^{67} +(-232.757 + 41.1821i) q^{68} +141.331 q^{69} -225.393i q^{71} -425.008i q^{72} -919.423 q^{73} -456.810i q^{74} -94.4239 q^{76} +900.603i q^{77} -1867.86 q^{78} -354.830i q^{79} -722.636 q^{81} -973.815 q^{82} +955.272i q^{83} -433.870 q^{84} -297.750 q^{86} +520.967i q^{87} +804.495 q^{88} -617.636 q^{89} -1313.98i q^{91} -64.7184 q^{92} -305.212i q^{93} +531.391 q^{94} +1057.56i q^{96} -428.533 q^{97} +127.432i q^{98} +1403.91i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} + 80 q^{9} - 476 q^{16} + 224 q^{19} + 248 q^{21} - 1064 q^{26} - 812 q^{34} - 436 q^{36} - 624 q^{49} + 2320 q^{51} + 2352 q^{59} - 2852 q^{64} + 3808 q^{66} - 2408 q^{69} - 112 q^{76} - 4816 q^{81}+ \cdots - 896 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.37228i − 1.19228i −0.802880 − 0.596141i \(-0.796700\pi\)
0.802880 − 0.596141i \(-0.203300\pi\)
\(3\) 7.36435 1.41727 0.708635 − 0.705575i \(-0.249311\pi\)
0.708635 + 0.705575i \(0.249311\pi\)
\(4\) −3.37228 −0.421535
\(5\) 0 0
\(6\) − 24.8347i − 1.68979i
\(7\) 17.4703 0.943308 0.471654 − 0.881784i \(-0.343657\pi\)
0.471654 + 0.881784i \(0.343657\pi\)
\(8\) − 15.6060i − 0.689693i
\(9\) 27.2337 1.00866
\(10\) 0 0
\(11\) 51.5505i 1.41300i 0.707711 + 0.706502i \(0.249728\pi\)
−0.707711 + 0.706502i \(0.750272\pi\)
\(12\) −24.8347 −0.597429
\(13\) − 75.2119i − 1.60462i −0.596909 − 0.802309i \(-0.703605\pi\)
0.596909 − 0.802309i \(-0.296395\pi\)
\(14\) − 58.9148i − 1.12469i
\(15\) 0 0
\(16\) −79.6060 −1.24384
\(17\) 69.0208 − 12.2119i 0.984706 − 0.174225i
\(18\) − 91.8397i − 1.20260i
\(19\) 28.0000 0.338086 0.169043 − 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 0 0
\(21\) 128.658 1.33692
\(22\) 173.843 1.68470
\(23\) 19.1913 0.173985 0.0869926 − 0.996209i \(-0.472274\pi\)
0.0869926 + 0.996209i \(0.472274\pi\)
\(24\) − 114.928i − 0.977481i
\(25\) 0 0
\(26\) −253.636 −1.91316
\(27\) 1.72096 0.0122666
\(28\) −58.9148 −0.397638
\(29\) 70.7417i 0.452980i 0.974014 + 0.226490i \(0.0727250\pi\)
−0.974014 + 0.226490i \(0.927275\pi\)
\(30\) 0 0
\(31\) − 41.4445i − 0.240118i −0.992767 − 0.120059i \(-0.961692\pi\)
0.992767 − 0.120059i \(-0.0383083\pi\)
\(32\) 143.606i 0.793318i
\(33\) 379.636i 2.00261i
\(34\) −41.1821 − 232.757i −0.207726 − 1.17405i
\(35\) 0 0
\(36\) −91.8397 −0.425184
\(37\) 135.460 0.601879 0.300939 − 0.953643i \(-0.402700\pi\)
0.300939 + 0.953643i \(0.402700\pi\)
\(38\) − 94.4239i − 0.403094i
\(39\) − 553.887i − 2.27418i
\(40\) 0 0
\(41\) − 288.771i − 1.09996i −0.835178 − 0.549980i \(-0.814635\pi\)
0.835178 − 0.549980i \(-0.185365\pi\)
\(42\) − 433.870i − 1.59399i
\(43\) − 88.2934i − 0.313131i −0.987668 − 0.156565i \(-0.949958\pi\)
0.987668 − 0.156565i \(-0.0500422\pi\)
\(44\) − 173.843i − 0.595631i
\(45\) 0 0
\(46\) − 64.7184i − 0.207439i
\(47\) 157.576i 0.489039i 0.969644 + 0.244520i \(0.0786302\pi\)
−0.969644 + 0.244520i \(0.921370\pi\)
\(48\) −586.246 −1.76286
\(49\) −37.7881 −0.110169
\(50\) 0 0
\(51\) 508.293 − 89.9330i 1.39559 − 0.246924i
\(52\) 253.636i 0.676403i
\(53\) − 120.250i − 0.311653i −0.987784 − 0.155826i \(-0.950196\pi\)
0.987784 − 0.155826i \(-0.0498040\pi\)
\(54\) − 5.80356i − 0.0146253i
\(55\) 0 0
\(56\) − 272.641i − 0.650593i
\(57\) 206.202 0.479160
\(58\) 238.561 0.540079
\(59\) 696.119 1.53605 0.768026 − 0.640419i \(-0.221239\pi\)
0.768026 + 0.640419i \(0.221239\pi\)
\(60\) 0 0
\(61\) 683.544i 1.43473i 0.696695 + 0.717367i \(0.254653\pi\)
−0.696695 + 0.717367i \(0.745347\pi\)
\(62\) −139.763 −0.286288
\(63\) 475.781 0.951473
\(64\) −152.568 −0.297984
\(65\) 0 0
\(66\) 1280.24 2.38767
\(67\) 123.826i 0.225787i 0.993607 + 0.112894i \(0.0360120\pi\)
−0.993607 + 0.112894i \(0.963988\pi\)
\(68\) −232.757 + 41.1821i −0.415088 + 0.0734421i
\(69\) 141.331 0.246584
\(70\) 0 0
\(71\) − 225.393i − 0.376750i −0.982097 − 0.188375i \(-0.939678\pi\)
0.982097 − 0.188375i \(-0.0603220\pi\)
\(72\) − 425.008i − 0.695662i
\(73\) −919.423 −1.47411 −0.737057 − 0.675831i \(-0.763785\pi\)
−0.737057 + 0.675831i \(0.763785\pi\)
\(74\) − 456.810i − 0.717609i
\(75\) 0 0
\(76\) −94.4239 −0.142515
\(77\) 900.603i 1.33290i
\(78\) −1867.86 −2.71146
\(79\) − 354.830i − 0.505335i −0.967553 − 0.252668i \(-0.918692\pi\)
0.967553 − 0.252668i \(-0.0813079\pi\)
\(80\) 0 0
\(81\) −722.636 −0.991270
\(82\) −973.815 −1.31146
\(83\) 955.272i 1.26331i 0.775250 + 0.631655i \(0.217624\pi\)
−0.775250 + 0.631655i \(0.782376\pi\)
\(84\) −433.870 −0.563560
\(85\) 0 0
\(86\) −297.750 −0.373340
\(87\) 520.967i 0.641995i
\(88\) 804.495 0.974539
\(89\) −617.636 −0.735610 −0.367805 − 0.929903i \(-0.619891\pi\)
−0.367805 + 0.929903i \(0.619891\pi\)
\(90\) 0 0
\(91\) − 1313.98i − 1.51365i
\(92\) −64.7184 −0.0733408
\(93\) − 305.212i − 0.340312i
\(94\) 531.391 0.583072
\(95\) 0 0
\(96\) 1057.56i 1.12435i
\(97\) −428.533 −0.448566 −0.224283 − 0.974524i \(-0.572004\pi\)
−0.224283 + 0.974524i \(0.572004\pi\)
\(98\) 127.432i 0.131353i
\(99\) 1403.91i 1.42523i
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.c.c.424.2 8
5.2 odd 4 425.4.d.c.101.3 4
5.3 odd 4 17.4.b.a.16.2 yes 4
5.4 even 2 inner 425.4.c.c.424.7 8
15.8 even 4 153.4.d.b.118.3 4
17.16 even 2 inner 425.4.c.c.424.1 8
20.3 even 4 272.4.b.d.33.1 4
85.13 odd 4 289.4.a.e.1.3 4
85.33 odd 4 17.4.b.a.16.1 ✓ 4
85.38 odd 4 289.4.a.e.1.4 4
85.67 odd 4 425.4.d.c.101.4 4
85.84 even 2 inner 425.4.c.c.424.8 8
255.203 even 4 153.4.d.b.118.4 4
340.203 even 4 272.4.b.d.33.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.1 ✓ 4 85.33 odd 4
17.4.b.a.16.2 yes 4 5.3 odd 4
153.4.d.b.118.3 4 15.8 even 4
153.4.d.b.118.4 4 255.203 even 4
272.4.b.d.33.1 4 20.3 even 4
272.4.b.d.33.4 4 340.203 even 4
289.4.a.e.1.3 4 85.13 odd 4
289.4.a.e.1.4 4 85.38 odd 4
425.4.c.c.424.1 8 17.16 even 2 inner
425.4.c.c.424.2 8 1.1 even 1 trivial
425.4.c.c.424.7 8 5.4 even 2 inner
425.4.c.c.424.8 8 85.84 even 2 inner
425.4.d.c.101.3 4 5.2 odd 4
425.4.d.c.101.4 4 85.67 odd 4