Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,0,0,8,0,0,0,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{14 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.14896\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} +3.32813 q^{7} -2.23607 q^{8} -1.14896 q^{11} -1.85906 q^{13} +5.38503 q^{14} -4.85410 q^{16} -2.67989 q^{17} -1.82083 q^{19} -1.85906 q^{22} +0.420194 q^{23} -3.00802 q^{26} +2.05690 q^{28} +10.0413 q^{29} -6.00306 q^{31} -3.38197 q^{32} -4.33615 q^{34} +1.00000 q^{37} -2.94617 q^{38} +4.14094 q^{41} +8.24409 q^{43} -0.710097 q^{44} +0.679888 q^{46} +11.7212 q^{47} +4.07646 q^{49} -1.14896 q^{52} +1.64166 q^{53} -7.44193 q^{56} +16.2472 q^{58} +7.07150 q^{59} -0.294859 q^{61} -9.71316 q^{62} +4.23607 q^{64} -4.44688 q^{67} -1.65626 q^{68} +10.3518 q^{71} +9.07150 q^{73} +1.61803 q^{74} -1.12533 q^{76} -3.82389 q^{77} +9.63977 q^{79} +6.70018 q^{82} -0.115689 q^{83} +13.3392 q^{86} +2.56916 q^{88} +5.51226 q^{89} -6.18719 q^{91} +0.259694 q^{92} +18.9653 q^{94} -1.38692 q^{97} +6.59584 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 8 q^{7} + 4 q^{11} + 2 q^{13} + 4 q^{14} - 6 q^{16} + 2 q^{17} - 4 q^{19} + 2 q^{22} + 6 q^{26} - 4 q^{28} + 12 q^{29} - 2 q^{31} - 18 q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 26 q^{41}+ \cdots - 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 − 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 0 0
\(7\) 3.32813 1.25792 0.628958 − 0.777440i \(-0.283482\pi\)
0.628958 + 0.777440i \(0.283482\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) −1.14896 −0.346425 −0.173212 − 0.984884i \(-0.555415\pi\)
−0.173212 + 0.984884i \(0.555415\pi\)
\(12\) 0 0
\(13\) −1.85906 −0.515610 −0.257805 − 0.966197i \(-0.582999\pi\)
−0.257805 + 0.966197i \(0.582999\pi\)
\(14\) 5.38503 1.43921
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −2.67989 −0.649968 −0.324984 − 0.945719i \(-0.605359\pi\)
−0.324984 + 0.945719i \(0.605359\pi\)
\(18\) 0 0
\(19\) −1.82083 −0.417727 −0.208864 − 0.977945i \(-0.566976\pi\)
−0.208864 + 0.977945i \(0.566976\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.85906 −0.396353
\(23\) 0.420194 0.0876165 0.0438083 − 0.999040i \(-0.486051\pi\)
0.0438083 + 0.999040i \(0.486051\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.00802 −0.589921
\(27\) 0 0
\(28\) 2.05690 0.388717
\(29\) 10.0413 1.86462 0.932310 − 0.361659i \(-0.117790\pi\)
0.932310 + 0.361659i \(0.117790\pi\)
\(30\) 0 0
\(31\) −6.00306 −1.07818 −0.539091 − 0.842248i \(-0.681232\pi\)
−0.539091 + 0.842248i \(0.681232\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) −4.33615 −0.743644
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −2.94617 −0.477931
\(39\) 0 0
\(40\) 0 0
\(41\) 4.14094 0.646706 0.323353 − 0.946278i \(-0.395190\pi\)
0.323353 + 0.946278i \(0.395190\pi\)
\(42\) 0 0
\(43\) 8.24409 1.25721 0.628606 − 0.777724i \(-0.283626\pi\)
0.628606 + 0.777724i \(0.283626\pi\)
\(44\) −0.710097 −0.107051
\(45\) 0 0
\(46\) 0.679888 0.100244
\(47\) 11.7212 1.70971 0.854855 − 0.518867i \(-0.173646\pi\)
0.854855 + 0.518867i \(0.173646\pi\)
\(48\) 0 0
\(49\) 4.07646 0.582351
\(50\) 0 0
\(51\) 0 0
\(52\) −1.14896 −0.159332
\(53\) 1.64166 0.225499 0.112750 − 0.993623i \(-0.464034\pi\)
0.112750 + 0.993623i \(0.464034\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.44193 −0.994469
\(57\) 0 0
\(58\) 16.2472 2.13336
\(59\) 7.07150 0.920631 0.460315 − 0.887755i \(-0.347736\pi\)
0.460315 + 0.887755i \(0.347736\pi\)
\(60\) 0 0
\(61\) −0.294859 −0.0377528 −0.0188764 − 0.999822i \(-0.506009\pi\)
−0.0188764 + 0.999822i \(0.506009\pi\)
\(62\) −9.71316 −1.23357
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 0 0
\(67\) −4.44688 −0.543273 −0.271637 − 0.962400i \(-0.587565\pi\)
−0.271637 + 0.962400i \(0.587565\pi\)
\(68\) −1.65626 −0.200851
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3518 1.22853 0.614264 − 0.789101i \(-0.289453\pi\)
0.614264 + 0.789101i \(0.289453\pi\)
\(72\) 0 0
\(73\) 9.07150 1.06174 0.530869 − 0.847454i \(-0.321865\pi\)
0.530869 + 0.847454i \(0.321865\pi\)
\(74\) 1.61803 0.188093
\(75\) 0 0
\(76\) −1.12533 −0.129085
\(77\) −3.82389 −0.435773
\(78\) 0 0
\(79\) 9.63977 1.08456 0.542279 − 0.840198i \(-0.317561\pi\)
0.542279 + 0.840198i \(0.317561\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.70018 0.739912
\(83\) −0.115689 −0.0126985 −0.00634927 − 0.999980i \(-0.502021\pi\)
−0.00634927 + 0.999980i \(0.502021\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 13.3392 1.43840
\(87\) 0 0
\(88\) 2.56916 0.273873
\(89\) 5.51226 0.584298 0.292149 − 0.956373i \(-0.405630\pi\)
0.292149 + 0.956373i \(0.405630\pi\)
\(90\) 0 0
\(91\) −6.18719 −0.648594
\(92\) 0.259694 0.0270750
\(93\) 0 0
\(94\) 18.9653 1.95612
\(95\) 0 0
\(96\) 0 0
\(97\) −1.38692 −0.140821 −0.0704103 − 0.997518i \(-0.522431\pi\)
−0.0704103 + 0.997518i \(0.522431\pi\)
\(98\) 6.59584 0.666281
\(99\) 0 0
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 8325.2.a.bw.1.4 4
3.2 odd 2 2775.2.a.w.1.2 4
5.2 odd 4 1665.2.c.d.334.7 8
5.3 odd 4 1665.2.c.d.334.1 8
5.4 even 2 8325.2.a.bt.1.1 4
15.2 even 4 555.2.c.b.334.2 ✓ 8
15.8 even 4 555.2.c.b.334.8 yes 8
15.14 odd 2 2775.2.a.y.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.b.334.2 ✓ 8 15.2 even 4
555.2.c.b.334.8 yes 8 15.8 even 4
1665.2.c.d.334.1 8 5.3 odd 4
1665.2.c.d.334.7 8 5.2 odd 4
2775.2.a.w.1.2 4 3.2 odd 2
2775.2.a.y.1.3 4 15.14 odd 2
8325.2.a.bt.1.1 4 5.4 even 2
8325.2.a.bw.1.4 4 1.1 even 1 trivial