Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7098,2,Mod(1,7098)] 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("7098.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7098, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,2,2,1,2,2,2,2,1,-8,2,0,2,1,2,6] 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: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.27492 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.27492 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} -3.27492 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -8.54983 q^{19} -3.27492 q^{20} +1.00000 q^{21} -4.00000 q^{22} +2.27492 q^{23} +1.00000 q^{24} +5.72508 q^{25} +1.00000 q^{27} +1.00000 q^{28} +0.725083 q^{29} -3.27492 q^{30} +6.27492 q^{31} +1.00000 q^{32} -4.00000 q^{33} +3.00000 q^{34} -3.27492 q^{35} +1.00000 q^{36} -7.27492 q^{37} -8.54983 q^{38} -3.27492 q^{40} +0.725083 q^{41} +1.00000 q^{42} +10.8248 q^{43} -4.00000 q^{44} -3.27492 q^{45} +2.27492 q^{46} +8.54983 q^{47} +1.00000 q^{48} +1.00000 q^{49} +5.72508 q^{50} +3.00000 q^{51} +11.5498 q^{53} +1.00000 q^{54} +13.0997 q^{55} +1.00000 q^{56} -8.54983 q^{57} +0.725083 q^{58} +10.8248 q^{59} -3.27492 q^{60} -9.00000 q^{61} +6.27492 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -10.2749 q^{67} +3.00000 q^{68} +2.27492 q^{69} -3.27492 q^{70} -2.27492 q^{71} +1.00000 q^{72} +13.2749 q^{73} -7.27492 q^{74} +5.72508 q^{75} -8.54983 q^{76} -4.00000 q^{77} -8.00000 q^{79} -3.27492 q^{80} +1.00000 q^{81} +0.725083 q^{82} +10.8248 q^{83} +1.00000 q^{84} -9.82475 q^{85} +10.8248 q^{86} +0.725083 q^{87} -4.00000 q^{88} +8.27492 q^{89} -3.27492 q^{90} +2.27492 q^{92} +6.27492 q^{93} +8.54983 q^{94} +28.0000 q^{95} +1.00000 q^{96} +14.5498 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + q^{10} - 8 q^{11} + 2 q^{12} + 2 q^{14} + q^{15} + 2 q^{16} + 6 q^{17} + 2 q^{18} - 2 q^{19} + q^{20} + 2 q^{21} - 8 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.27492 −1.46459 −0.732294 − 0.680989i \(-0.761550\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.27492 −1.03562
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) −3.27492 −0.845580
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.54983 −1.96147 −0.980733 − 0.195352i \(-0.937415\pi\)
−0.980733 + 0.195352i \(0.937415\pi\)
\(20\) −3.27492 −0.732294
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 2.27492 0.474353 0.237177 − 0.971467i \(-0.423778\pi\)
0.237177 + 0.971467i \(0.423778\pi\)
\(24\) 1.00000 0.204124
\(25\) 5.72508 1.14502
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 0.725083 0.134644 0.0673222 − 0.997731i \(-0.478554\pi\)
0.0673222 + 0.997731i \(0.478554\pi\)
\(30\) −3.27492 −0.597915
\(31\) 6.27492 1.12701 0.563504 − 0.826113i \(-0.309453\pi\)
0.563504 + 0.826113i \(0.309453\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 3.00000 0.514496
\(35\) −3.27492 −0.553562
\(36\) 1.00000 0.166667
\(37\) −7.27492 −1.19599 −0.597995 − 0.801500i \(-0.704036\pi\)
−0.597995 + 0.801500i \(0.704036\pi\)
\(38\) −8.54983 −1.38697
\(39\) 0 0
\(40\) −3.27492 −0.517810
\(41\) 0.725083 0.113239 0.0566195 − 0.998396i \(-0.481968\pi\)
0.0566195 + 0.998396i \(0.481968\pi\)
\(42\) 1.00000 0.154303
\(43\) 10.8248 1.65076 0.825380 − 0.564578i \(-0.190961\pi\)
0.825380 + 0.564578i \(0.190961\pi\)
\(44\) −4.00000 −0.603023
\(45\) −3.27492 −0.488196
\(46\) 2.27492 0.335418
\(47\) 8.54983 1.24712 0.623561 − 0.781775i \(-0.285685\pi\)
0.623561 + 0.781775i \(0.285685\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 5.72508 0.809649
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 11.5498 1.58649 0.793246 − 0.608901i \(-0.208390\pi\)
0.793246 + 0.608901i \(0.208390\pi\)
\(54\) 1.00000 0.136083
\(55\) 13.0997 1.76636
\(56\) 1.00000 0.133631
\(57\) −8.54983 −1.13245
\(58\) 0.725083 0.0952080
\(59\) 10.8248 1.40926 0.704631 − 0.709574i \(-0.251112\pi\)
0.704631 + 0.709574i \(0.251112\pi\)
\(60\) −3.27492 −0.422790
\(61\) −9.00000 −1.15233 −0.576166 − 0.817333i \(-0.695452\pi\)
−0.576166 + 0.817333i \(0.695452\pi\)
\(62\) 6.27492 0.796915
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −10.2749 −1.25528 −0.627640 − 0.778503i \(-0.715979\pi\)
−0.627640 + 0.778503i \(0.715979\pi\)
\(68\) 3.00000 0.363803
\(69\) 2.27492 0.273868
\(70\) −3.27492 −0.391427
\(71\) −2.27492 −0.269983 −0.134992 − 0.990847i \(-0.543101\pi\)
−0.134992 + 0.990847i \(0.543101\pi\)
\(72\) 1.00000 0.117851
\(73\) 13.2749 1.55371 0.776856 − 0.629679i \(-0.216813\pi\)
0.776856 + 0.629679i \(0.216813\pi\)
\(74\) −7.27492 −0.845692
\(75\) 5.72508 0.661076
\(76\) −8.54983 −0.980733
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −3.27492 −0.366147
\(81\) 1.00000 0.111111
\(82\) 0.725083 0.0800720
\(83\) 10.8248 1.18817 0.594085 − 0.804402i \(-0.297514\pi\)
0.594085 + 0.804402i \(0.297514\pi\)
\(84\) 1.00000 0.109109
\(85\) −9.82475 −1.06564
\(86\) 10.8248 1.16726
\(87\) 0.725083 0.0777370
\(88\) −4.00000 −0.426401
\(89\) 8.27492 0.877139 0.438570 − 0.898697i \(-0.355485\pi\)
0.438570 + 0.898697i \(0.355485\pi\)
\(90\) −3.27492 −0.345207
\(91\) 0 0
\(92\) 2.27492 0.237177
\(93\) 6.27492 0.650679
\(94\) 8.54983 0.881848
\(95\) 28.0000 2.87274
\(96\) 1.00000 0.102062
\(97\) 14.5498 1.47731 0.738656 − 0.674083i \(-0.235461\pi\)
0.738656 + 0.674083i \(0.235461\pi\)
\(98\) 1.00000 0.101015
\(99\) −4.00000 −0.402015
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 7098.2.a.ca.1.1 2
13.3 even 3 546.2.l.j.295.1 yes 4
13.9 even 3 546.2.l.j.211.1 ✓ 4
13.12 even 2 7098.2.a.bm.1.2 2
39.29 odd 6 1638.2.r.x.1387.2 4
39.35 odd 6 1638.2.r.x.757.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.j.211.1 ✓ 4 13.9 even 3
546.2.l.j.295.1 yes 4 13.3 even 3
1638.2.r.x.757.2 4 39.35 odd 6
1638.2.r.x.1387.2 4 39.29 odd 6
7098.2.a.bm.1.2 2 13.12 even 2
7098.2.a.ca.1.1 2 1.1 even 1 trivial