↑

Properties

Label 6975.2.a.bi.1.1
Level $6975$
Weight $2$
Character 6975.1
Self dual yes
Analytic conductor $55.696$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 = [3,3,0,5,0,0,-2,9,0,0,-2,0,6,-6,0,5,4] 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: \(0\)
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 465)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-1.21432 q^{2} -0.525428 q^{4} +1.59210 q^{7} +3.06668 q^{8} -0.622216 q^{11} -0.214320 q^{13} -1.93332 q^{14} -2.67307 q^{16} +3.52543 q^{17} +1.80642 q^{19} +0.755569 q^{22} +6.90321 q^{23} +0.260253 q^{26} -0.836535 q^{28} -9.73975 q^{29} -1.00000 q^{31} -2.88739 q^{32} -4.28100 q^{34} +4.83654 q^{37} -2.19358 q^{38} +7.47949 q^{41} -8.23506 q^{43} +0.326929 q^{44} -8.38271 q^{46} +11.4652 q^{47} -4.46520 q^{49} +0.112610 q^{52} +13.7605 q^{53} +4.88247 q^{56} +11.8272 q^{58} +4.26025 q^{59} +2.85728 q^{61} +1.21432 q^{62} +8.85236 q^{64} +2.08097 q^{67} -1.85236 q^{68} -1.31111 q^{71} +1.65233 q^{73} -5.87310 q^{74} -0.949145 q^{76} -0.990632 q^{77} -5.19850 q^{79} -9.08250 q^{82} -5.65878 q^{83} +10.0000 q^{86} -1.90813 q^{88} +1.93332 q^{89} -0.341219 q^{91} -3.62714 q^{92} -13.9224 q^{94} -6.91750 q^{97} +5.42219 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} - 2 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} + 4 q^{17} - 8 q^{19} + 2 q^{22} + 14 q^{23} + 14 q^{26} + 4 q^{28} - 16 q^{29} - 3 q^{31} + 11 q^{32} - 6 q^{34}+ \cdots + 17 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21432 −0.858654 −0.429327 − 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(3\) 0 0
\(4\) −0.525428 −0.262714
\(5\) 0 0
\(6\) 0 0
\(7\) 1.59210 0.601759 0.300879 − 0.953662i \(-0.402720\pi\)
0.300879 + 0.953662i \(0.402720\pi\)
\(8\) 3.06668 1.08423
\(9\) 0 0
\(10\) 0 0
\(11\) −0.622216 −0.187605 −0.0938025 − 0.995591i \(-0.529902\pi\)
−0.0938025 + 0.995591i \(0.529902\pi\)
\(12\) 0 0
\(13\) −0.214320 −0.0594416 −0.0297208 − 0.999558i \(-0.509462\pi\)
−0.0297208 + 0.999558i \(0.509462\pi\)
\(14\) −1.93332 −0.516702
\(15\) 0 0
\(16\) −2.67307 −0.668268
\(17\) 3.52543 0.855042 0.427521 − 0.904005i \(-0.359387\pi\)
0.427521 + 0.904005i \(0.359387\pi\)
\(18\) 0 0
\(19\) 1.80642 0.414422 0.207211 − 0.978296i \(-0.433561\pi\)
0.207211 + 0.978296i \(0.433561\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.755569 0.161088
\(23\) 6.90321 1.43942 0.719710 − 0.694275i \(-0.244275\pi\)
0.719710 + 0.694275i \(0.244275\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.260253 0.0510398
\(27\) 0 0
\(28\) −0.836535 −0.158090
\(29\) −9.73975 −1.80863 −0.904313 − 0.426870i \(-0.859616\pi\)
−0.904313 + 0.426870i \(0.859616\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −2.88739 −0.510423
\(33\) 0 0
\(34\) −4.28100 −0.734185
\(35\) 0 0
\(36\) 0 0
\(37\) 4.83654 0.795122 0.397561 − 0.917576i \(-0.369857\pi\)
0.397561 + 0.917576i \(0.369857\pi\)
\(38\) −2.19358 −0.355845
\(39\) 0 0
\(40\) 0 0
\(41\) 7.47949 1.16810 0.584050 − 0.811717i \(-0.301467\pi\)
0.584050 + 0.811717i \(0.301467\pi\)
\(42\) 0 0
\(43\) −8.23506 −1.25584 −0.627918 − 0.778280i \(-0.716093\pi\)
−0.627918 + 0.778280i \(0.716093\pi\)
\(44\) 0.326929 0.0492864
\(45\) 0 0
\(46\) −8.38271 −1.23596
\(47\) 11.4652 1.67237 0.836186 − 0.548446i \(-0.184780\pi\)
0.836186 + 0.548446i \(0.184780\pi\)
\(48\) 0 0
\(49\) −4.46520 −0.637886
\(50\) 0 0
\(51\) 0 0
\(52\) 0.112610 0.0156161
\(53\) 13.7605 1.89015 0.945074 − 0.326855i \(-0.105989\pi\)
0.945074 + 0.326855i \(0.105989\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.88247 0.652447
\(57\) 0 0
\(58\) 11.8272 1.55298
\(59\) 4.26025 0.554638 0.277319 − 0.960778i \(-0.410554\pi\)
0.277319 + 0.960778i \(0.410554\pi\)
\(60\) 0 0
\(61\) 2.85728 0.365837 0.182919 − 0.983128i \(-0.441446\pi\)
0.182919 + 0.983128i \(0.441446\pi\)
\(62\) 1.21432 0.154219
\(63\) 0 0
\(64\) 8.85236 1.10654
\(65\) 0 0
\(66\) 0 0
\(67\) 2.08097 0.254231 0.127115 − 0.991888i \(-0.459428\pi\)
0.127115 + 0.991888i \(0.459428\pi\)
\(68\) −1.85236 −0.224631
\(69\) 0 0
\(70\) 0 0
\(71\) −1.31111 −0.155600 −0.0777999 − 0.996969i \(-0.524790\pi\)
−0.0777999 + 0.996969i \(0.524790\pi\)
\(72\) 0 0
\(73\) 1.65233 0.193390 0.0966951 − 0.995314i \(-0.469173\pi\)
0.0966951 + 0.995314i \(0.469173\pi\)
\(74\) −5.87310 −0.682734
\(75\) 0 0
\(76\) −0.949145 −0.108874
\(77\) −0.990632 −0.112893
\(78\) 0 0
\(79\) −5.19850 −0.584877 −0.292438 − 0.956284i \(-0.594467\pi\)
−0.292438 + 0.956284i \(0.594467\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −9.08250 −1.00299
\(83\) −5.65878 −0.621132 −0.310566 − 0.950552i \(-0.600519\pi\)
−0.310566 + 0.950552i \(0.600519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) −1.90813 −0.203408
\(89\) 1.93332 0.204932 0.102466 − 0.994737i \(-0.467327\pi\)
0.102466 + 0.994737i \(0.467327\pi\)
\(90\) 0 0
\(91\) −0.341219 −0.0357695
\(92\) −3.62714 −0.378155
\(93\) 0 0
\(94\) −13.9224 −1.43599
\(95\) 0 0
\(96\) 0 0
\(97\) −6.91750 −0.702366 −0.351183 − 0.936307i \(-0.614220\pi\)
−0.351183 + 0.936307i \(0.614220\pi\)
\(98\) 5.42219 0.547723
\(99\) 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.bi.1.1 3
3.2 odd 2 2325.2.a.p.1.3 3
5.4 even 2 1395.2.a.h.1.3 3
15.2 even 4 2325.2.c.l.1024.4 6
15.8 even 4 2325.2.c.l.1024.3 6
15.14 odd 2 465.2.a.g.1.1 ✓ 3
60.59 even 2 7440.2.a.bm.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.1 ✓ 3 15.14 odd 2
1395.2.a.h.1.3 3 5.4 even 2
2325.2.a.p.1.3 3 3.2 odd 2
2325.2.c.l.1024.3 6 15.8 even 4
2325.2.c.l.1024.4 6 15.2 even 4
6975.2.a.bi.1.1 3 1.1 even 1 trivial
7440.2.a.bm.1.3 3 60.59 even 2