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 = [7,1,0,7,0,0,0,0,0,0,-16,0,1] 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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 23x^{2} - 9x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 925)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.13289\) 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)
 
\(f(q)\) \(=\) \(q-2.13289 q^{2} +2.54923 q^{4} -3.01925 q^{7} -1.17144 q^{8} -5.51375 q^{11} -0.501630 q^{13} +6.43973 q^{14} -2.59990 q^{16} +6.61915 q^{17} +4.42510 q^{19} +11.7602 q^{22} +1.67307 q^{23} +1.06992 q^{26} -7.69675 q^{28} +1.51711 q^{29} -9.00821 q^{31} +7.88818 q^{32} -14.1179 q^{34} -1.00000 q^{37} -9.43826 q^{38} -3.32768 q^{41} +3.05695 q^{43} -14.0558 q^{44} -3.56848 q^{46} +3.80128 q^{47} +2.11587 q^{49} -1.27877 q^{52} -6.03086 q^{53} +3.53687 q^{56} -3.23583 q^{58} -13.4763 q^{59} +11.9483 q^{61} +19.2135 q^{62} -11.6248 q^{64} -10.8898 q^{67} +16.8737 q^{68} +2.46808 q^{71} +13.2652 q^{73} +2.13289 q^{74} +11.2806 q^{76} +16.6474 q^{77} -11.4110 q^{79} +7.09758 q^{82} +8.36622 q^{83} -6.52014 q^{86} +6.45903 q^{88} +7.41023 q^{89} +1.51454 q^{91} +4.26503 q^{92} -8.10773 q^{94} +5.75342 q^{97} -4.51291 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 7 q^{4} - 16 q^{11} + q^{13} + 3 q^{14} + 3 q^{16} + 4 q^{17} + 9 q^{19} - q^{22} - q^{23} - 24 q^{26} - 7 q^{28} - 3 q^{29} + 11 q^{31} + 21 q^{32} - 23 q^{34} - 7 q^{37} - 32 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.13289 −1.50818 −0.754091 − 0.656770i \(-0.771922\pi\)
−0.754091 + 0.656770i \(0.771922\pi\)
\(3\) 0 0
\(4\) 2.54923 1.27461
\(5\) 0 0
\(6\) 0 0
\(7\) −3.01925 −1.14117 −0.570585 − 0.821239i \(-0.693283\pi\)
−0.570585 + 0.821239i \(0.693283\pi\)
\(8\) −1.17144 −0.414167
\(9\) 0 0
\(10\) 0 0
\(11\) −5.51375 −1.66246 −0.831230 − 0.555929i \(-0.812363\pi\)
−0.831230 + 0.555929i \(0.812363\pi\)
\(12\) 0 0
\(13\) −0.501630 −0.139127 −0.0695635 − 0.997578i \(-0.522161\pi\)
−0.0695635 + 0.997578i \(0.522161\pi\)
\(14\) 6.43973 1.72109
\(15\) 0 0
\(16\) −2.59990 −0.649975
\(17\) 6.61915 1.60538 0.802690 − 0.596397i \(-0.203402\pi\)
0.802690 + 0.596397i \(0.203402\pi\)
\(18\) 0 0
\(19\) 4.42510 1.01519 0.507594 − 0.861597i \(-0.330535\pi\)
0.507594 + 0.861597i \(0.330535\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 11.7602 2.50729
\(23\) 1.67307 0.348859 0.174430 − 0.984670i \(-0.444192\pi\)
0.174430 + 0.984670i \(0.444192\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.06992 0.209829
\(27\) 0 0
\(28\) −7.69675 −1.45455
\(29\) 1.51711 0.281721 0.140860 − 0.990029i \(-0.455013\pi\)
0.140860 + 0.990029i \(0.455013\pi\)
\(30\) 0 0
\(31\) −9.00821 −1.61792 −0.808961 − 0.587862i \(-0.799970\pi\)
−0.808961 + 0.587862i \(0.799970\pi\)
\(32\) 7.88818 1.39445
\(33\) 0 0
\(34\) −14.1179 −2.42120
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −9.43826 −1.53109
\(39\) 0 0
\(40\) 0 0
\(41\) −3.32768 −0.519696 −0.259848 − 0.965649i \(-0.583673\pi\)
−0.259848 + 0.965649i \(0.583673\pi\)
\(42\) 0 0
\(43\) 3.05695 0.466180 0.233090 − 0.972455i \(-0.425116\pi\)
0.233090 + 0.972455i \(0.425116\pi\)
\(44\) −14.0558 −2.11899
\(45\) 0 0
\(46\) −3.56848 −0.526143
\(47\) 3.80128 0.554474 0.277237 − 0.960802i \(-0.410581\pi\)
0.277237 + 0.960802i \(0.410581\pi\)
\(48\) 0 0
\(49\) 2.11587 0.302267
\(50\) 0 0
\(51\) 0 0
\(52\) −1.27877 −0.177333
\(53\) −6.03086 −0.828403 −0.414202 − 0.910185i \(-0.635939\pi\)
−0.414202 + 0.910185i \(0.635939\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.53687 0.472634
\(57\) 0 0
\(58\) −3.23583 −0.424886
\(59\) −13.4763 −1.75446 −0.877231 − 0.480069i \(-0.840612\pi\)
−0.877231 + 0.480069i \(0.840612\pi\)
\(60\) 0 0
\(61\) 11.9483 1.52982 0.764909 − 0.644139i \(-0.222784\pi\)
0.764909 + 0.644139i \(0.222784\pi\)
\(62\) 19.2135 2.44012
\(63\) 0 0
\(64\) −11.6248 −1.45310
\(65\) 0 0
\(66\) 0 0
\(67\) −10.8898 −1.33040 −0.665198 − 0.746667i \(-0.731653\pi\)
−0.665198 + 0.746667i \(0.731653\pi\)
\(68\) 16.8737 2.04624
\(69\) 0 0
\(70\) 0 0
\(71\) 2.46808 0.292907 0.146454 − 0.989218i \(-0.453214\pi\)
0.146454 + 0.989218i \(0.453214\pi\)
\(72\) 0 0
\(73\) 13.2652 1.55258 0.776289 − 0.630378i \(-0.217100\pi\)
0.776289 + 0.630378i \(0.217100\pi\)
\(74\) 2.13289 0.247944
\(75\) 0 0
\(76\) 11.2806 1.29397
\(77\) 16.6474 1.89715
\(78\) 0 0
\(79\) −11.4110 −1.28384 −0.641918 − 0.766773i \(-0.721861\pi\)
−0.641918 + 0.766773i \(0.721861\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 7.09758 0.783797
\(83\) 8.36622 0.918312 0.459156 − 0.888356i \(-0.348152\pi\)
0.459156 + 0.888356i \(0.348152\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.52014 −0.703085
\(87\) 0 0
\(88\) 6.45903 0.688535
\(89\) 7.41023 0.785483 0.392741 − 0.919649i \(-0.371527\pi\)
0.392741 + 0.919649i \(0.371527\pi\)
\(90\) 0 0
\(91\) 1.51454 0.158767
\(92\) 4.26503 0.444660
\(93\) 0 0
\(94\) −8.10773 −0.836248
\(95\) 0 0
\(96\) 0 0
\(97\) 5.75342 0.584171 0.292085 − 0.956392i \(-0.405651\pi\)
0.292085 + 0.956392i \(0.405651\pi\)
\(98\) −4.51291 −0.455873
\(99\) 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.cn.1.2 7
3.2 odd 2 925.2.a.j.1.6 ✓ 7
5.4 even 2 8325.2.a.cm.1.6 7
15.2 even 4 925.2.b.i.149.12 14
15.8 even 4 925.2.b.i.149.3 14
15.14 odd 2 925.2.a.k.1.2 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.j.1.6 ✓ 7 3.2 odd 2
925.2.a.k.1.2 yes 7 15.14 odd 2
925.2.b.i.149.3 14 15.8 even 4
925.2.b.i.149.12 14 15.2 even 4
8325.2.a.cm.1.6 7 5.4 even 2
8325.2.a.cn.1.2 7 1.1 even 1 trivial