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,-2,0,2,0,0,8,-6,0,0,-3,0,4,-10,0,4,-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: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
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)
 
\(f(q)\) \(=\) \(q-0.688892 q^{2} -1.52543 q^{4} +4.90321 q^{7} +2.42864 q^{8} -3.21432 q^{11} -0.836535 q^{13} -3.37778 q^{14} +1.37778 q^{16} -6.73975 q^{17} +3.00000 q^{19} +2.21432 q^{22} -5.21432 q^{23} +0.576283 q^{26} -7.47949 q^{28} +1.49532 q^{29} +1.00000 q^{31} -5.80642 q^{32} +4.64296 q^{34} +0.407896 q^{37} -2.06668 q^{38} -9.54617 q^{41} -7.61285 q^{43} +4.90321 q^{44} +3.59210 q^{46} +6.62222 q^{47} +17.0415 q^{49} +1.27607 q^{52} +12.8731 q^{53} +11.9081 q^{56} -1.03011 q^{58} +8.16839 q^{59} +0.755569 q^{61} -0.688892 q^{62} +1.24443 q^{64} +2.85236 q^{67} +10.2810 q^{68} +8.29529 q^{71} +14.0415 q^{73} -0.280996 q^{74} -4.57628 q^{76} -15.7605 q^{77} -2.77631 q^{79} +6.57628 q^{82} -10.8874 q^{83} +5.24443 q^{86} -7.80642 q^{88} +2.78568 q^{89} -4.10171 q^{91} +7.95407 q^{92} -4.56199 q^{94} +8.95407 q^{97} -11.7397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 6 q^{8} - 3 q^{11} + 4 q^{13} - 10 q^{14} + 4 q^{16} - 7 q^{17} + 9 q^{19} - 9 q^{23} - 18 q^{26} + 4 q^{28} - 9 q^{29} + 3 q^{31} - 4 q^{32} - 6 q^{34} + 8 q^{37} - 6 q^{38}+ \cdots - 22 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.688892 −0.487120 −0.243560 − 0.969886i \(-0.578315\pi\)
−0.243560 + 0.969886i \(0.578315\pi\)
\(3\) 0 0
\(4\) −1.52543 −0.762714
\(5\) 0 0
\(6\) 0 0
\(7\) 4.90321 1.85324 0.926620 − 0.375999i \(-0.122700\pi\)
0.926620 + 0.375999i \(0.122700\pi\)
\(8\) 2.42864 0.858654
\(9\) 0 0
\(10\) 0 0
\(11\) −3.21432 −0.969154 −0.484577 − 0.874749i \(-0.661026\pi\)
−0.484577 + 0.874749i \(0.661026\pi\)
\(12\) 0 0
\(13\) −0.836535 −0.232013 −0.116007 − 0.993248i \(-0.537009\pi\)
−0.116007 + 0.993248i \(0.537009\pi\)
\(14\) −3.37778 −0.902751
\(15\) 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
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.21432 0.472095
\(23\) −5.21432 −1.08726 −0.543630 − 0.839325i \(-0.682951\pi\)
−0.543630 + 0.839325i \(0.682951\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.576283 0.113018
\(27\) 0 0
\(28\) −7.47949 −1.41349
\(29\) 1.49532 0.277673 0.138837 − 0.990315i \(-0.455664\pi\)
0.138837 + 0.990315i \(0.455664\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −5.80642 −1.02644
\(33\) 0 0
\(34\) 4.64296 0.796261
\(35\) 0 0
\(36\) 0 0
\(37\) 0.407896 0.0670577 0.0335288 − 0.999438i \(-0.489325\pi\)
0.0335288 + 0.999438i \(0.489325\pi\)
\(38\) −2.06668 −0.335259
\(39\) 0 0
\(40\) 0 0
\(41\) −9.54617 −1.49086 −0.745431 − 0.666583i \(-0.767756\pi\)
−0.745431 + 0.666583i \(0.767756\pi\)
\(42\) 0 0
\(43\) −7.61285 −1.16095 −0.580474 − 0.814279i \(-0.697133\pi\)
−0.580474 + 0.814279i \(0.697133\pi\)
\(44\) 4.90321 0.739187
\(45\) 0 0
\(46\) 3.59210 0.529627
\(47\) 6.62222 0.965949 0.482975 − 0.875634i \(-0.339556\pi\)
0.482975 + 0.875634i \(0.339556\pi\)
\(48\) 0 0
\(49\) 17.0415 2.43450
\(50\) 0 0
\(51\) 0 0
\(52\) 1.27607 0.176960
\(53\) 12.8731 1.76826 0.884128 − 0.467244i \(-0.154753\pi\)
0.884128 + 0.467244i \(0.154753\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 11.9081 1.59129
\(57\) 0 0
\(58\) −1.03011 −0.135260
\(59\) 8.16839 1.06343 0.531717 − 0.846922i \(-0.321547\pi\)
0.531717 + 0.846922i \(0.321547\pi\)
\(60\) 0 0
\(61\) 0.755569 0.0967407 0.0483703 − 0.998829i \(-0.484597\pi\)
0.0483703 + 0.998829i \(0.484597\pi\)
\(62\) −0.688892 −0.0874894
\(63\) 0 0
\(64\) 1.24443 0.155554
\(65\) 0 0
\(66\) 0 0
\(67\) 2.85236 0.348471 0.174235 − 0.984704i \(-0.444255\pi\)
0.174235 + 0.984704i \(0.444255\pi\)
\(68\) 10.2810 1.24675
\(69\) 0 0
\(70\) 0 0
\(71\) 8.29529 0.984469 0.492235 − 0.870463i \(-0.336180\pi\)
0.492235 + 0.870463i \(0.336180\pi\)
\(72\) 0 0
\(73\) 14.0415 1.64343 0.821716 − 0.569897i \(-0.193017\pi\)
0.821716 + 0.569897i \(0.193017\pi\)
\(74\) −0.280996 −0.0326652
\(75\) 0 0
\(76\) −4.57628 −0.524936
\(77\) −15.7605 −1.79607
\(78\) 0 0
\(79\) −2.77631 −0.312360 −0.156180 − 0.987729i \(-0.549918\pi\)
−0.156180 + 0.987729i \(0.549918\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.57628 0.726229
\(83\) −10.8874 −1.19505 −0.597523 − 0.801852i \(-0.703848\pi\)
−0.597523 + 0.801852i \(0.703848\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.24443 0.565522
\(87\) 0 0
\(88\) −7.80642 −0.832168
\(89\) 2.78568 0.295282 0.147641 − 0.989041i \(-0.452832\pi\)
0.147641 + 0.989041i \(0.452832\pi\)
\(90\) 0 0
\(91\) −4.10171 −0.429976
\(92\) 7.95407 0.829269
\(93\) 0 0
\(94\) −4.56199 −0.470534
\(95\) 0 0
\(96\) 0 0
\(97\) 8.95407 0.909148 0.454574 − 0.890709i \(-0.349792\pi\)
0.454574 + 0.890709i \(0.349792\pi\)
\(98\) −11.7397 −1.18589
\(99\) 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.ba.1.2 yes 3
3.2 odd 2 6975.2.a.bh.1.2 yes 3
5.4 even 2 6975.2.a.bg.1.2 yes 3
15.14 odd 2 6975.2.a.z.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6975.2.a.z.1.2 ✓ 3 15.14 odd 2
6975.2.a.ba.1.2 yes 3 1.1 even 1 trivial
6975.2.a.bg.1.2 yes 3 5.4 even 2
6975.2.a.bh.1.2 yes 3 3.2 odd 2