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Properties

Label 6975.2.a.bi.1.2
Level $6975$
Weight $2$
Character 6975.1
Self dual yes
Analytic conductor $55.696$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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 1.2
Root \(2.17009\) 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)
 
\(f(q)\) \(=\) \(q+1.53919 q^{2} +0.369102 q^{4} -4.87936 q^{7} -2.51026 q^{8} -4.34017 q^{11} +2.53919 q^{13} -7.51026 q^{14} -4.60197 q^{16} +2.63090 q^{17} -7.41855 q^{19} -6.68035 q^{22} +2.29072 q^{23} +3.90829 q^{26} -1.80098 q^{28} -6.09171 q^{29} -1.00000 q^{31} -2.06278 q^{32} +4.04945 q^{34} +5.80098 q^{37} -11.4186 q^{38} +0.183417 q^{41} +6.49693 q^{43} -1.60197 q^{44} +3.52586 q^{46} -9.80817 q^{47} +16.8082 q^{49} +0.937221 q^{52} -1.86603 q^{53} +12.2485 q^{56} -9.37629 q^{58} +7.90829 q^{59} -8.15676 q^{61} -1.53919 q^{62} +6.02893 q^{64} +10.4813 q^{67} +0.971071 q^{68} -3.17009 q^{71} +15.5597 q^{73} +8.92881 q^{74} -2.73820 q^{76} +21.1773 q^{77} -6.23287 q^{79} +0.282314 q^{82} +6.38962 q^{83} +10.0000 q^{86} +10.8950 q^{88} +7.51026 q^{89} -12.3896 q^{91} +0.845512 q^{92} -15.0966 q^{94} -16.2823 q^{97} +25.8710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} - 2 q^{11} + 6 q^{13} - 6 q^{14} + 5 q^{16} + 4 q^{17} - 8 q^{19} + 2 q^{22} + 14 q^{23} + 14 q^{26} + 4 q^{28} - 16 q^{29} - 3 q^{31} + 11 q^{32} - 6 q^{34}+ \cdots + 17 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53919 1.08837 0.544185 − 0.838965i \(-0.316839\pi\)
0.544185 + 0.838965i \(0.316839\pi\)
\(3\) 0 0
\(4\) 0.369102 0.184551
\(5\) 0 0
\(6\) 0 0
\(7\) −4.87936 −1.84423 −0.922113 − 0.386921i \(-0.873538\pi\)
−0.922113 + 0.386921i \(0.873538\pi\)
\(8\) −2.51026 −0.887511
\(9\) 0 0
\(10\) 0 0
\(11\) −4.34017 −1.30861 −0.654306 − 0.756230i \(-0.727039\pi\)
−0.654306 + 0.756230i \(0.727039\pi\)
\(12\) 0 0
\(13\) 2.53919 0.704244 0.352122 − 0.935954i \(-0.385460\pi\)
0.352122 + 0.935954i \(0.385460\pi\)
\(14\) −7.51026 −2.00720
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) 2.63090 0.638086 0.319043 − 0.947740i \(-0.396638\pi\)
0.319043 + 0.947740i \(0.396638\pi\)
\(18\) 0 0
\(19\) −7.41855 −1.70193 −0.850966 − 0.525221i \(-0.823983\pi\)
−0.850966 + 0.525221i \(0.823983\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.68035 −1.42425
\(23\) 2.29072 0.477649 0.238825 − 0.971063i \(-0.423238\pi\)
0.238825 + 0.971063i \(0.423238\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.90829 0.766479
\(27\) 0 0
\(28\) −1.80098 −0.340354
\(29\) −6.09171 −1.13120 −0.565601 − 0.824679i \(-0.691356\pi\)
−0.565601 + 0.824679i \(0.691356\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −2.06278 −0.364651
\(33\) 0 0
\(34\) 4.04945 0.694475
\(35\) 0 0
\(36\) 0 0
\(37\) 5.80098 0.953676 0.476838 − 0.878991i \(-0.341783\pi\)
0.476838 + 0.878991i \(0.341783\pi\)
\(38\) −11.4186 −1.85233
\(39\) 0 0
\(40\) 0 0
\(41\) 0.183417 0.0286450 0.0143225 − 0.999897i \(-0.495441\pi\)
0.0143225 + 0.999897i \(0.495441\pi\)
\(42\) 0 0
\(43\) 6.49693 0.990772 0.495386 − 0.868673i \(-0.335027\pi\)
0.495386 + 0.868673i \(0.335027\pi\)
\(44\) −1.60197 −0.241506
\(45\) 0 0
\(46\) 3.52586 0.519859
\(47\) −9.80817 −1.43067 −0.715334 − 0.698782i \(-0.753726\pi\)
−0.715334 + 0.698782i \(0.753726\pi\)
\(48\) 0 0
\(49\) 16.8082 2.40117
\(50\) 0 0
\(51\) 0 0
\(52\) 0.937221 0.129969
\(53\) −1.86603 −0.256319 −0.128160 − 0.991754i \(-0.540907\pi\)
−0.128160 + 0.991754i \(0.540907\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 12.2485 1.63677
\(57\) 0 0
\(58\) −9.37629 −1.23117
\(59\) 7.90829 1.02957 0.514786 − 0.857319i \(-0.327871\pi\)
0.514786 + 0.857319i \(0.327871\pi\)
\(60\) 0 0
\(61\) −8.15676 −1.04437 −0.522183 − 0.852834i \(-0.674882\pi\)
−0.522183 + 0.852834i \(0.674882\pi\)
\(62\) −1.53919 −0.195477
\(63\) 0 0
\(64\) 6.02893 0.753616
\(65\) 0 0
\(66\) 0 0
\(67\) 10.4813 1.28050 0.640249 − 0.768167i \(-0.278831\pi\)
0.640249 + 0.768167i \(0.278831\pi\)
\(68\) 0.971071 0.117760
\(69\) 0 0
\(70\) 0 0
\(71\) −3.17009 −0.376220 −0.188110 − 0.982148i \(-0.560236\pi\)
−0.188110 + 0.982148i \(0.560236\pi\)
\(72\) 0 0
\(73\) 15.5597 1.82113 0.910563 − 0.413370i \(-0.135648\pi\)
0.910563 + 0.413370i \(0.135648\pi\)
\(74\) 8.92881 1.03795
\(75\) 0 0
\(76\) −2.73820 −0.314094
\(77\) 21.1773 2.41337
\(78\) 0 0
\(79\) −6.23287 −0.701252 −0.350626 − 0.936516i \(-0.614031\pi\)
−0.350626 + 0.936516i \(0.614031\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.282314 0.0311764
\(83\) 6.38962 0.701352 0.350676 − 0.936497i \(-0.385952\pi\)
0.350676 + 0.936497i \(0.385952\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) 10.8950 1.16141
\(89\) 7.51026 0.796086 0.398043 − 0.917367i \(-0.369689\pi\)
0.398043 + 0.917367i \(0.369689\pi\)
\(90\) 0 0
\(91\) −12.3896 −1.29879
\(92\) 0.845512 0.0881507
\(93\) 0 0
\(94\) −15.0966 −1.55710
\(95\) 0 0
\(96\) 0 0
\(97\) −16.2823 −1.65322 −0.826609 − 0.562776i \(-0.809733\pi\)
−0.826609 + 0.562776i \(0.809733\pi\)
\(98\) 25.8710 2.61336
\(99\) 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.1.2 3
3.2 odd 2 2325.2.a.p.1.2 3
5.4 even 2 1395.2.a.h.1.2 3
15.2 even 4 2325.2.c.l.1024.2 6
15.8 even 4 2325.2.c.l.1024.5 6
15.14 odd 2 465.2.a.g.1.2 ✓ 3
60.59 even 2 7440.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.2 ✓ 3 15.14 odd 2
1395.2.a.h.1.2 3 5.4 even 2
2325.2.a.p.1.2 3 3.2 odd 2
2325.2.c.l.1024.2 6 15.2 even 4
2325.2.c.l.1024.5 6 15.8 even 4
6975.2.a.bi.1.2 3 1.1 even 1 trivial
7440.2.a.bm.1.1 3 60.59 even 2