Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,2,0,0,-4,6,0,0,7,0,0,6,0,-4,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.6956554098\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2325)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 6975.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.21432 q^{2} +2.90321 q^{4} +0.903212 q^{7} +2.00000 q^{8} +4.59210 q^{11} -4.49532 q^{13} +2.00000 q^{14} -1.37778 q^{16} -6.73975 q^{17} -3.62222 q^{19} +10.1684 q^{22} -8.39853 q^{23} -9.95407 q^{26} +2.62222 q^{28} +1.68889 q^{29} -1.00000 q^{31} -7.05086 q^{32} -14.9240 q^{34} -8.68889 q^{37} -8.02074 q^{38} -4.02074 q^{41} -11.4795 q^{43} +13.3319 q^{44} -18.5970 q^{46} -1.18421 q^{47} -6.18421 q^{49} -13.0509 q^{52} +0.873100 q^{53} +1.80642 q^{56} +3.73975 q^{58} +11.2652 q^{59} +12.8573 q^{61} -2.21432 q^{62} -12.8573 q^{64} +10.0049 q^{67} -19.5669 q^{68} +13.0509 q^{71} -8.42864 q^{73} -19.2400 q^{74} -10.5161 q^{76} +4.14764 q^{77} -1.11753 q^{79} -8.90321 q^{82} -3.96989 q^{83} -25.4193 q^{86} +9.18421 q^{88} +9.50961 q^{89} -4.06022 q^{91} -24.3827 q^{92} -2.62222 q^{94} +17.1891 q^{97} -13.6938 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} - 4 q^{7} + 6 q^{8} + 7 q^{11} + 6 q^{14} - 4 q^{16} - 7 q^{17} - 11 q^{19} + 4 q^{22} - 5 q^{23} - 10 q^{26} + 8 q^{28} + 5 q^{29} - 3 q^{31} - 8 q^{32} - 18 q^{34} - 26 q^{37} - 4 q^{38}+ \cdots - 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.21432 1.56576 0.782880 − 0.622172i \(-0.213750\pi\)
0.782880 + 0.622172i \(0.213750\pi\)
\(3\) 0 0
\(4\) 2.90321 1.45161
\(5\) 0 0
\(6\) 0 0
\(7\) 0.903212 0.341382 0.170691 − 0.985325i \(-0.445400\pi\)
0.170691 + 0.985325i \(0.445400\pi\)
\(8\) 2.00000 0.707107
\(9\) 0 0
\(10\) 0 0
\(11\) 4.59210 1.38457 0.692286 − 0.721623i \(-0.256604\pi\)
0.692286 + 0.721623i \(0.256604\pi\)
\(12\) 0 0
\(13\) −4.49532 −1.24678 −0.623388 − 0.781913i \(-0.714244\pi\)
−0.623388 + 0.781913i \(0.714244\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.37778 −0.344446
\(17\) −6.73975 −1.63463 −0.817314 − 0.576192i \(-0.804538\pi\)
−0.817314 + 0.576192i \(0.804538\pi\)
\(18\) 0 0
\(19\) −3.62222 −0.830993 −0.415497 − 0.909595i \(-0.636392\pi\)
−0.415497 + 0.909595i \(0.636392\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 10.1684 2.16791
\(23\) −8.39853 −1.75121 −0.875607 − 0.483024i \(-0.839538\pi\)
−0.875607 + 0.483024i \(0.839538\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −9.95407 −1.95215
\(27\) 0 0
\(28\) 2.62222 0.495552
\(29\) 1.68889 0.313619 0.156810 − 0.987629i \(-0.449879\pi\)
0.156810 + 0.987629i \(0.449879\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −7.05086 −1.24643
\(33\) 0 0
\(34\) −14.9240 −2.55944
\(35\) 0 0
\(36\) 0 0
\(37\) −8.68889 −1.42845 −0.714223 − 0.699919i \(-0.753220\pi\)
−0.714223 + 0.699919i \(0.753220\pi\)
\(38\) −8.02074 −1.30114
\(39\) 0 0
\(40\) 0 0
\(41\) −4.02074 −0.627935 −0.313967 − 0.949434i \(-0.601658\pi\)
−0.313967 + 0.949434i \(0.601658\pi\)
\(42\) 0 0
\(43\) −11.4795 −1.75061 −0.875303 − 0.483574i \(-0.839338\pi\)
−0.875303 + 0.483574i \(0.839338\pi\)
\(44\) 13.3319 2.00985
\(45\) 0 0
\(46\) −18.5970 −2.74198
\(47\) −1.18421 −0.172735 −0.0863673 − 0.996263i \(-0.527526\pi\)
−0.0863673 + 0.996263i \(0.527526\pi\)
\(48\) 0 0
\(49\) −6.18421 −0.883458
\(50\) 0 0
\(51\) 0 0
\(52\) −13.0509 −1.80983
\(53\) 0.873100 0.119930 0.0599648 − 0.998200i \(-0.480901\pi\)
0.0599648 + 0.998200i \(0.480901\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.80642 0.241394
\(57\) 0 0
\(58\) 3.73975 0.491053
\(59\) 11.2652 1.46660 0.733300 − 0.679905i \(-0.237979\pi\)
0.733300 + 0.679905i \(0.237979\pi\)
\(60\) 0 0
\(61\) 12.8573 1.64621 0.823103 − 0.567892i \(-0.192241\pi\)
0.823103 + 0.567892i \(0.192241\pi\)
\(62\) −2.21432 −0.281219
\(63\) 0 0
\(64\) −12.8573 −1.60716
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0049 1.22230 0.611148 − 0.791516i \(-0.290708\pi\)
0.611148 + 0.791516i \(0.290708\pi\)
\(68\) −19.5669 −2.37284
\(69\) 0 0
\(70\) 0 0
\(71\) 13.0509 1.54885 0.774426 − 0.632665i \(-0.218039\pi\)
0.774426 + 0.632665i \(0.218039\pi\)
\(72\) 0 0
\(73\) −8.42864 −0.986498 −0.493249 − 0.869888i \(-0.664191\pi\)
−0.493249 + 0.869888i \(0.664191\pi\)
\(74\) −19.2400 −2.23660
\(75\) 0 0
\(76\) −10.5161 −1.20627
\(77\) 4.14764 0.472668
\(78\) 0 0
\(79\) −1.11753 −0.125732 −0.0628661 − 0.998022i \(-0.520024\pi\)
−0.0628661 + 0.998022i \(0.520024\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −8.90321 −0.983195
\(83\) −3.96989 −0.435752 −0.217876 − 0.975976i \(-0.569913\pi\)
−0.217876 + 0.975976i \(0.569913\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −25.4193 −2.74103
\(87\) 0 0
\(88\) 9.18421 0.979040
\(89\) 9.50961 1.00802 0.504008 − 0.863699i \(-0.331858\pi\)
0.504008 + 0.863699i \(0.331858\pi\)
\(90\) 0 0
\(91\) −4.06022 −0.425627
\(92\) −24.3827 −2.54207
\(93\) 0 0
\(94\) −2.62222 −0.270461
\(95\) 0 0
\(96\) 0 0
\(97\) 17.1891 1.74529 0.872646 − 0.488354i \(-0.162402\pi\)
0.872646 + 0.488354i \(0.162402\pi\)
\(98\) −13.6938 −1.38328
\(99\) 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 6975.2.a.bc.1.3 3
3.2 odd 2 2325.2.a.u.1.1 yes 3
5.4 even 2 6975.2.a.bd.1.1 3
15.2 even 4 2325.2.c.m.1024.1 6
15.8 even 4 2325.2.c.m.1024.6 6
15.14 odd 2 2325.2.a.t.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2325.2.a.t.1.3 ✓ 3 15.14 odd 2
2325.2.a.u.1.1 yes 3 3.2 odd 2
2325.2.c.m.1024.1 6 15.2 even 4
2325.2.c.m.1024.6 6 15.8 even 4
6975.2.a.bc.1.3 3 1.1 even 1 trivial
6975.2.a.bd.1.1 3 5.4 even 2