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(101,425)] 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("425.101"); 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([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,2,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-37 +3 \sqrt{33}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.3
Root \(-7.36435i\) of defining polynomial
Character \(\chi\) \(=\) 425.101
Dual form 425.4.d.c.101.4

$q$-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)
 
\(f(q)\) \(=\) \(q+3.37228 q^{2} -7.36435i q^{3} +3.37228 q^{4} -24.8347i q^{6} +17.4703i q^{7} -15.6060 q^{8} -27.2337 q^{9} +51.5505i q^{11} -24.8347i q^{12} -75.2119 q^{13} +58.9148i q^{14} -79.6060 q^{16} +(12.2119 + 69.0208i) q^{17} -91.8397 q^{18} -28.0000 q^{19} +128.658 q^{21} +173.843i q^{22} -19.1913i q^{23} +114.928i q^{24} -253.636 q^{26} +1.72096i q^{27} +58.9148i q^{28} -70.7417i q^{29} -41.4445i q^{31} -143.606 q^{32} +379.636 q^{33} +(41.1821 + 232.757i) q^{34} -91.8397 q^{36} +135.460i q^{37} -94.4239 q^{38} +553.887i q^{39} -288.771i q^{41} +433.870 q^{42} -88.2934 q^{43} +173.843i q^{44} -64.7184i q^{46} -157.576 q^{47} +586.246i q^{48} +37.7881 q^{49} +(508.293 - 89.9330i) q^{51} -253.636 q^{52} -120.250 q^{53} +5.80356i q^{54} -272.641i q^{56} +206.202i q^{57} -238.561i q^{58} -696.119 q^{59} +683.544i q^{61} -139.763i q^{62} -475.781i q^{63} +152.568 q^{64} +1280.24 q^{66} -123.826 q^{67} +(41.1821 + 232.757i) q^{68} -141.331 q^{69} -225.393i q^{71} +425.008 q^{72} +919.423i q^{73} +456.810i q^{74} -94.4239 q^{76} -900.603 q^{77} +1867.86i q^{78} +354.830i q^{79} -722.636 q^{81} -973.815i q^{82} +955.272 q^{83} +433.870 q^{84} -297.750 q^{86} -520.967 q^{87} -804.495i q^{88} +617.636 q^{89} -1313.98i q^{91} -64.7184i q^{92} -305.212 q^{93} -531.391 q^{94} +1057.56i q^{96} -428.533i q^{97} +127.432 q^{98} -1403.91i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 18 q^{8} - 40 q^{9} - 140 q^{13} - 238 q^{16} - 112 q^{17} - 218 q^{18} - 112 q^{19} + 124 q^{21} - 532 q^{26} - 494 q^{32} + 1036 q^{33} + 406 q^{34} - 218 q^{36} - 56 q^{38}+ \cdots - 306 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.37228 1.19228 0.596141 − 0.802880i \(-0.296700\pi\)
0.596141 + 0.802880i \(0.296700\pi\)
\(3\) − 7.36435i − 1.41727i −0.705575 − 0.708635i \(-0.749311\pi\)
0.705575 − 0.708635i \(-0.250689\pi\)
\(4\) 3.37228 0.421535
\(5\) 0 0
\(6\) − 24.8347i − 1.68979i
\(7\) 17.4703i 0.943308i 0.881784 + 0.471654i \(0.156343\pi\)
−0.881784 + 0.471654i \(0.843657\pi\)
\(8\) −15.6060 −0.689693
\(9\) −27.2337 −1.00866
\(10\) 0 0
\(11\) 51.5505i 1.41300i 0.707711 + 0.706502i \(0.249728\pi\)
−0.707711 + 0.706502i \(0.750272\pi\)
\(12\) − 24.8347i − 0.597429i
\(13\) −75.2119 −1.60462 −0.802309 − 0.596909i \(-0.796395\pi\)
−0.802309 + 0.596909i \(0.796395\pi\)
\(14\) 58.9148i 1.12469i
\(15\) 0 0
\(16\) −79.6060 −1.24384
\(17\) 12.2119 + 69.0208i 0.174225 + 0.984706i
\(18\) −91.8397 −1.20260
\(19\) −28.0000 −0.338086 −0.169043 − 0.985609i \(-0.554068\pi\)
−0.169043 + 0.985609i \(0.554068\pi\)
\(20\) 0 0
\(21\) 128.658 1.33692
\(22\) 173.843i 1.68470i
\(23\) − 19.1913i − 0.173985i −0.996209 − 0.0869926i \(-0.972274\pi\)
0.996209 − 0.0869926i \(-0.0277256\pi\)
\(24\) 114.928i 0.977481i
\(25\) 0 0
\(26\) −253.636 −1.91316
\(27\) 1.72096i 0.0122666i
\(28\) 58.9148i 0.397638i
\(29\) − 70.7417i − 0.452980i −0.974014 − 0.226490i \(-0.927275\pi\)
0.974014 − 0.226490i \(-0.0727250\pi\)
\(30\) 0 0
\(31\) − 41.4445i − 0.240118i −0.992767 − 0.120059i \(-0.961692\pi\)
0.992767 − 0.120059i \(-0.0383083\pi\)
\(32\) −143.606 −0.793318
\(33\) 379.636 2.00261
\(34\) 41.1821 + 232.757i 0.207726 + 1.17405i
\(35\) 0 0
\(36\) −91.8397 −0.425184
\(37\) 135.460i 0.601879i 0.953643 + 0.300939i \(0.0973002\pi\)
−0.953643 + 0.300939i \(0.902700\pi\)
\(38\) −94.4239 −0.403094
\(39\) 553.887i 2.27418i
\(40\) 0 0
\(41\) − 288.771i − 1.09996i −0.835178 − 0.549980i \(-0.814635\pi\)
0.835178 − 0.549980i \(-0.185365\pi\)
\(42\) 433.870 1.59399
\(43\) −88.2934 −0.313131 −0.156565 − 0.987668i \(-0.550042\pi\)
−0.156565 + 0.987668i \(0.550042\pi\)
\(44\) 173.843i 0.595631i
\(45\) 0 0
\(46\) − 64.7184i − 0.207439i
\(47\) −157.576 −0.489039 −0.244520 − 0.969644i \(-0.578630\pi\)
−0.244520 + 0.969644i \(0.578630\pi\)
\(48\) 586.246i 1.76286i
\(49\) 37.7881 0.110169
\(50\) 0 0
\(51\) 508.293 − 89.9330i 1.39559 − 0.246924i
\(52\) −253.636 −0.676403
\(53\) −120.250 −0.311653 −0.155826 − 0.987784i \(-0.549804\pi\)
−0.155826 + 0.987784i \(0.549804\pi\)
\(54\) 5.80356i 0.0146253i
\(55\) 0 0
\(56\) − 272.641i − 0.650593i
\(57\) 206.202i 0.479160i
\(58\) − 238.561i − 0.540079i
\(59\) −696.119 −1.53605 −0.768026 − 0.640419i \(-0.778761\pi\)
−0.768026 + 0.640419i \(0.778761\pi\)
\(60\) 0 0
\(61\) 683.544i 1.43473i 0.696695 + 0.717367i \(0.254653\pi\)
−0.696695 + 0.717367i \(0.745347\pi\)
\(62\) − 139.763i − 0.286288i
\(63\) − 475.781i − 0.951473i
\(64\) 152.568 0.297984
\(65\) 0 0
\(66\) 1280.24 2.38767
\(67\) −123.826 −0.225787 −0.112894 − 0.993607i \(-0.536012\pi\)
−0.112894 + 0.993607i \(0.536012\pi\)
\(68\) 41.1821 + 232.757i 0.0734421 + 0.415088i
\(69\) −141.331 −0.246584
\(70\) 0 0
\(71\) − 225.393i − 0.376750i −0.982097 − 0.188375i \(-0.939678\pi\)
0.982097 − 0.188375i \(-0.0603220\pi\)
\(72\) 425.008 0.695662
\(73\) 919.423i 1.47411i 0.675831 + 0.737057i \(0.263785\pi\)
−0.675831 + 0.737057i \(0.736215\pi\)
\(74\) 456.810i 0.717609i
\(75\) 0 0
\(76\) −94.4239 −0.142515
\(77\) −900.603 −1.33290
\(78\) 1867.86i 2.71146i
\(79\) 354.830i 0.505335i 0.967553 + 0.252668i \(0.0813079\pi\)
−0.967553 + 0.252668i \(0.918692\pi\)
\(80\) 0 0
\(81\) −722.636 −0.991270
\(82\) − 973.815i − 1.31146i
\(83\) 955.272 1.26331 0.631655 − 0.775250i \(-0.282376\pi\)
0.631655 + 0.775250i \(0.282376\pi\)
\(84\) 433.870 0.563560
\(85\) 0 0
\(86\) −297.750 −0.373340
\(87\) −520.967 −0.641995
\(88\) − 804.495i − 0.974539i
\(89\) 617.636 0.735610 0.367805 − 0.929903i \(-0.380109\pi\)
0.367805 + 0.929903i \(0.380109\pi\)
\(90\) 0 0
\(91\) − 1313.98i − 1.51365i
\(92\) − 64.7184i − 0.0733408i
\(93\) −305.212 −0.340312
\(94\) −531.391 −0.583072
\(95\) 0 0
\(96\) 1057.56i 1.12435i
\(97\) − 428.533i − 0.448566i −0.974524 − 0.224283i \(-0.927996\pi\)
0.974524 − 0.224283i \(-0.0720041\pi\)
\(98\) 127.432 0.131353
\(99\) − 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.d.c.101.3 4
5.2 odd 4 425.4.c.c.424.7 8
5.3 odd 4 425.4.c.c.424.2 8
5.4 even 2 17.4.b.a.16.2 yes 4
15.14 odd 2 153.4.d.b.118.3 4
17.16 even 2 inner 425.4.d.c.101.4 4
20.19 odd 2 272.4.b.d.33.1 4
85.4 even 4 289.4.a.e.1.4 4
85.33 odd 4 425.4.c.c.424.1 8
85.64 even 4 289.4.a.e.1.3 4
85.67 odd 4 425.4.c.c.424.8 8
85.84 even 2 17.4.b.a.16.1 ✓ 4
255.254 odd 2 153.4.d.b.118.4 4
340.339 odd 2 272.4.b.d.33.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.1 ✓ 4 85.84 even 2
17.4.b.a.16.2 yes 4 5.4 even 2
153.4.d.b.118.3 4 15.14 odd 2
153.4.d.b.118.4 4 255.254 odd 2
272.4.b.d.33.1 4 20.19 odd 2
272.4.b.d.33.4 4 340.339 odd 2
289.4.a.e.1.3 4 85.64 even 4
289.4.a.e.1.4 4 85.4 even 4
425.4.c.c.424.1 8 85.33 odd 4
425.4.c.c.424.2 8 5.3 odd 4
425.4.c.c.424.7 8 5.2 odd 4
425.4.c.c.424.8 8 85.67 odd 4
425.4.d.c.101.3 4 1.1 even 1 trivial
425.4.d.c.101.4 4 17.16 even 2 inner