Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,10,8,0,-2,0,-10,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.90211\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.90211 q^{2} -1.90211 q^{3} +1.61803 q^{4} +3.61803 q^{6} +4.23607 q^{7} +0.726543 q^{8} +0.618034 q^{9} -5.85410 q^{11} -3.07768 q^{12} +3.07768 q^{13} -8.05748 q^{14} -4.61803 q^{16} -5.23607 q^{17} -1.17557 q^{18} -8.05748 q^{21} +11.1352 q^{22} -4.09017 q^{23} -1.38197 q^{24} -5.85410 q^{26} +4.53077 q^{27} +6.85410 q^{28} +2.80017 q^{29} -1.90211 q^{31} +7.33094 q^{32} +11.1352 q^{33} +9.95959 q^{34} +1.00000 q^{36} -2.80017 q^{37} -5.85410 q^{39} +6.88191 q^{41} +15.3262 q^{42} -0.381966 q^{43} -9.47214 q^{44} +7.77997 q^{46} +1.47214 q^{47} +8.78402 q^{48} +10.9443 q^{49} +9.95959 q^{51} +4.97980 q^{52} +11.1352 q^{53} -8.61803 q^{54} +3.07768 q^{56} -5.32624 q^{58} -14.0413 q^{59} +3.94427 q^{61} +3.61803 q^{62} +2.61803 q^{63} -4.70820 q^{64} -21.1803 q^{66} +5.98385 q^{67} -8.47214 q^{68} +7.77997 q^{69} -0.171513 q^{71} +0.449028 q^{72} +1.00000 q^{73} +5.32624 q^{74} -24.7984 q^{77} +11.1352 q^{78} +5.25731 q^{79} -10.4721 q^{81} -13.0902 q^{82} +8.76393 q^{83} -13.0373 q^{84} +0.726543 q^{86} -5.32624 q^{87} -4.25325 q^{88} -7.77997 q^{89} +13.0373 q^{91} -6.61803 q^{92} +3.61803 q^{93} -2.80017 q^{94} -13.9443 q^{96} -2.62866 q^{97} -20.8172 q^{98} -3.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90211 −1.34500 −0.672499 − 0.740098i \(-0.734779\pi\)
−0.672499 + 0.740098i \(0.734779\pi\)
\(3\) −1.90211 −1.09819 −0.549093 − 0.835761i \(-0.685027\pi\)
−0.549093 + 0.835761i \(0.685027\pi\)
\(4\) 1.61803 0.809017
\(5\) 0 0
\(6\) 3.61803 1.47706
\(7\) 4.23607 1.60108 0.800542 − 0.599277i \(-0.204545\pi\)
0.800542 + 0.599277i \(0.204545\pi\)
\(8\) 0.726543 0.256872
\(9\) 0.618034 0.206011
\(10\) 0 0
\(11\) −5.85410 −1.76508 −0.882539 − 0.470239i \(-0.844168\pi\)
−0.882539 + 0.470239i \(0.844168\pi\)
\(12\) −3.07768 −0.888451
\(13\) 3.07768 0.853596 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(14\) −8.05748 −2.15345
\(15\) 0 0
\(16\) −4.61803 −1.15451
\(17\) −5.23607 −1.26993 −0.634967 − 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) −1.17557 −0.277085
\(19\) 0 0
\(20\) 0 0
\(21\) −8.05748 −1.75829
\(22\) 11.1352 2.37402
\(23\) −4.09017 −0.852859 −0.426430 − 0.904521i \(-0.640229\pi\)
−0.426430 + 0.904521i \(0.640229\pi\)
\(24\) −1.38197 −0.282093
\(25\) 0 0
\(26\) −5.85410 −1.14808
\(27\) 4.53077 0.871947
\(28\) 6.85410 1.29530
\(29\) 2.80017 0.519978 0.259989 − 0.965612i \(-0.416281\pi\)
0.259989 + 0.965612i \(0.416281\pi\)
\(30\) 0 0
\(31\) −1.90211 −0.341630 −0.170815 − 0.985303i \(-0.554640\pi\)
−0.170815 + 0.985303i \(0.554640\pi\)
\(32\) 7.33094 1.29594
\(33\) 11.1352 1.93838
\(34\) 9.95959 1.70806
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.80017 −0.460345 −0.230172 − 0.973150i \(-0.573929\pi\)
−0.230172 + 0.973150i \(0.573929\pi\)
\(38\) 0 0
\(39\) −5.85410 −0.937407
\(40\) 0 0
\(41\) 6.88191 1.07477 0.537387 − 0.843336i \(-0.319412\pi\)
0.537387 + 0.843336i \(0.319412\pi\)
\(42\) 15.3262 2.36489
\(43\) −0.381966 −0.0582493 −0.0291246 − 0.999576i \(-0.509272\pi\)
−0.0291246 + 0.999576i \(0.509272\pi\)
\(44\) −9.47214 −1.42798
\(45\) 0 0
\(46\) 7.77997 1.14709
\(47\) 1.47214 0.214733 0.107367 − 0.994220i \(-0.465758\pi\)
0.107367 + 0.994220i \(0.465758\pi\)
\(48\) 8.78402 1.26786
\(49\) 10.9443 1.56347
\(50\) 0 0
\(51\) 9.95959 1.39462
\(52\) 4.97980 0.690574
\(53\) 11.1352 1.52953 0.764766 − 0.644308i \(-0.222854\pi\)
0.764766 + 0.644308i \(0.222854\pi\)
\(54\) −8.61803 −1.17277
\(55\) 0 0
\(56\) 3.07768 0.411273
\(57\) 0 0
\(58\) −5.32624 −0.699369
\(59\) −14.0413 −1.82803 −0.914013 − 0.405685i \(-0.867033\pi\)
−0.914013 + 0.405685i \(0.867033\pi\)
\(60\) 0 0
\(61\) 3.94427 0.505012 0.252506 − 0.967595i \(-0.418745\pi\)
0.252506 + 0.967595i \(0.418745\pi\)
\(62\) 3.61803 0.459491
\(63\) 2.61803 0.329841
\(64\) −4.70820 −0.588525
\(65\) 0 0
\(66\) −21.1803 −2.60712
\(67\) 5.98385 0.731044 0.365522 − 0.930803i \(-0.380890\pi\)
0.365522 + 0.930803i \(0.380890\pi\)
\(68\) −8.47214 −1.02740
\(69\) 7.77997 0.936598
\(70\) 0 0
\(71\) −0.171513 −0.0203549 −0.0101774 − 0.999948i \(-0.503240\pi\)
−0.0101774 + 0.999948i \(0.503240\pi\)
\(72\) 0.449028 0.0529185
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 5.32624 0.619163
\(75\) 0 0
\(76\) 0 0
\(77\) −24.7984 −2.82604
\(78\) 11.1352 1.26081
\(79\) 5.25731 0.591494 0.295747 − 0.955266i \(-0.404432\pi\)
0.295747 + 0.955266i \(0.404432\pi\)
\(80\) 0 0
\(81\) −10.4721 −1.16357
\(82\) −13.0902 −1.44557
\(83\) 8.76393 0.961967 0.480983 − 0.876730i \(-0.340280\pi\)
0.480983 + 0.876730i \(0.340280\pi\)
\(84\) −13.0373 −1.42248
\(85\) 0 0
\(86\) 0.726543 0.0783451
\(87\) −5.32624 −0.571033
\(88\) −4.25325 −0.453398
\(89\) −7.77997 −0.824675 −0.412337 − 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(90\) 0 0
\(91\) 13.0373 1.36668
\(92\) −6.61803 −0.689978
\(93\) 3.61803 0.375173
\(94\) −2.80017 −0.288815
\(95\) 0 0
\(96\) −13.9443 −1.42318
\(97\) −2.62866 −0.266900 −0.133450 − 0.991056i \(-0.542605\pi\)
−0.133450 + 0.991056i \(0.542605\pi\)
\(98\) −20.8172 −2.10286
\(99\) −3.61803 −0.363626
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 9025.2.a.bl.1.1 4
5.4 even 2 1805.2.a.l.1.4 yes 4
19.18 odd 2 inner 9025.2.a.bl.1.4 4
95.94 odd 2 1805.2.a.l.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.l.1.1 ✓ 4 95.94 odd 2
1805.2.a.l.1.4 yes 4 5.4 even 2
9025.2.a.bl.1.1 4 1.1 even 1 trivial
9025.2.a.bl.1.4 4 19.18 odd 2 inner