Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [98,8,Mod(67,98)] 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("98.67"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 98.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,8,12,-64,-210,192,0,-1024,2043] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.6137324974\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 98.67
Dual form 98.8.c.e.79.1

$q$-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+(4.00000 + 6.92820i) q^{2} +(6.00000 - 10.3923i) q^{3} +(-32.0000 + 55.4256i) q^{4} +(-105.000 - 181.865i) q^{5} +96.0000 q^{6} -512.000 q^{8} +(1021.50 + 1769.29i) q^{9} +(840.000 - 1454.92i) q^{10} +(-546.000 + 945.700i) q^{11} +(384.000 + 665.108i) q^{12} -1382.00 q^{13} -2520.00 q^{15} +(-2048.00 - 3547.24i) q^{16} +(7353.00 - 12735.8i) q^{17} +(-8172.00 + 14154.3i) q^{18} +(-19970.0 - 34589.1i) q^{19} +13440.0 q^{20} -8736.00 q^{22} +(-34356.0 - 59506.3i) q^{23} +(-3072.00 + 5320.86i) q^{24} +(17012.5 - 29466.5i) q^{25} +(-5528.00 - 9574.78i) q^{26} +50760.0 q^{27} -102570. q^{29} +(-10080.0 - 17459.1i) q^{30} +(113776. - 197066. i) q^{31} +(16384.0 - 28377.9i) q^{32} +(6552.00 + 11348.4i) q^{33} +117648. q^{34} -130752. q^{36} +(-80263.0 - 139020. i) q^{37} +(159760. - 276712. i) q^{38} +(-8292.00 + 14362.2i) q^{39} +(53760.0 + 93115.1i) q^{40} -10842.0 q^{41} -630748. q^{43} +(-34944.0 - 60524.8i) q^{44} +(214515. - 371551. i) q^{45} +(274848. - 476051. i) q^{46} +(236328. + 409332. i) q^{47} -49152.0 q^{48} +272200. q^{50} +(-88236.0 - 152829. i) q^{51} +(44224.0 - 76598.2i) q^{52} +(747009. - 1.29386e6i) q^{53} +(203040. + 351676. i) q^{54} +229320. q^{55} -479280. q^{57} +(-410280. - 710626. i) q^{58} +(1.32033e6 - 2.28688e6i) q^{59} +(80640.0 - 139673. i) q^{60} +(413851. + 716811. i) q^{61} +1.82042e6 q^{62} +262144. q^{64} +(145110. + 251338. i) q^{65} +(-52416.0 + 90787.2i) q^{66} +(63002.0 - 109123. i) q^{67} +(470592. + 815089. i) q^{68} -824544. q^{69} -1.41473e6 q^{71} +(-523008. - 905876. i) q^{72} +(490141. - 848949. i) q^{73} +(642104. - 1.11216e6i) q^{74} +(-204150. - 353598. i) q^{75} +2.55616e6 q^{76} -132672. q^{78} +(1.78340e6 + 3.08894e6i) q^{79} +(-430080. + 744920. i) q^{80} +(-1.92946e6 + 3.34192e6i) q^{81} +(-43368.0 - 75115.6i) q^{82} -5.67289e6 q^{83} -3.08826e6 q^{85} +(-2.52299e6 - 4.36995e6i) q^{86} +(-615420. + 1.06594e6i) q^{87} +(279552. - 484198. i) q^{88} +(-5.97560e6 - 1.03500e7i) q^{89} +3.43224e6 q^{90} +4.39757e6 q^{92} +(-1.36531e6 - 2.36479e6i) q^{93} +(-1.89062e6 + 3.27466e6i) q^{94} +(-4.19370e6 + 7.26370e6i) q^{95} +(-196608. - 340535. i) q^{96} -8.68215e6 q^{97} -2.23096e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 12 q^{3} - 64 q^{4} - 210 q^{5} + 192 q^{6} - 1024 q^{8} + 2043 q^{9} + 1680 q^{10} - 1092 q^{11} + 768 q^{12} - 2764 q^{13} - 5040 q^{15} - 4096 q^{16} + 14706 q^{17} - 16344 q^{18} - 39940 q^{19}+ \cdots - 4461912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 + 6.92820i 0.353553 + 0.612372i
\(3\) 6.00000 − 10.3923i 0.128300 − 0.222222i −0.794718 − 0.606979i \(-0.792381\pi\)
0.923018 + 0.384757i \(0.125715\pi\)
\(4\) −32.0000 + 55.4256i −0.250000 + 0.433013i
\(5\) −105.000 − 181.865i −0.375659 − 0.650661i 0.614766 − 0.788709i \(-0.289250\pi\)
−0.990425 + 0.138048i \(0.955917\pi\)
\(6\) 96.0000 0.181444
\(7\) 0 0
\(8\) −512.000 −0.353553
\(9\) 1021.50 + 1769.29i 0.467078 + 0.809003i
\(10\) 840.000 − 1454.92i 0.265631 − 0.460087i
\(11\) −546.000 + 945.700i −0.123685 + 0.214229i −0.921218 − 0.389046i \(-0.872805\pi\)
0.797533 + 0.603275i \(0.206138\pi\)
\(12\) 384.000 + 665.108i 0.0641500 + 0.111111i
\(13\) −1382.00 −0.174464 −0.0872321 − 0.996188i \(-0.527802\pi\)
−0.0872321 + 0.996188i \(0.527802\pi\)
\(14\) 0 0
\(15\) −2520.00 −0.192789
\(16\) −2048.00 − 3547.24i −0.125000 − 0.216506i
\(17\) 7353.00 − 12735.8i 0.362989 − 0.628715i −0.625462 − 0.780254i \(-0.715090\pi\)
0.988451 + 0.151539i \(0.0484230\pi\)
\(18\) −8172.00 + 14154.3i −0.330274 + 0.572052i
\(19\) −19970.0 − 34589.1i −0.667945 − 1.15691i −0.978478 − 0.206352i \(-0.933841\pi\)
0.310533 − 0.950563i \(-0.399492\pi\)
\(20\) 13440.0 0.375659
\(21\) 0 0
\(22\) −8736.00 −0.174917
\(23\) −34356.0 − 59506.3i −0.588783 − 1.01980i −0.994392 − 0.105755i \(-0.966274\pi\)
0.405609 − 0.914047i \(-0.367059\pi\)
\(24\) −3072.00 + 5320.86i −0.0453609 + 0.0785674i
\(25\) 17012.5 − 29466.5i 0.217760 − 0.377171i
\(26\) −5528.00 − 9574.78i −0.0616824 − 0.106837i
\(27\) 50760.0 0.496305
\(28\) 0 0
\(29\) −102570. −0.780957 −0.390479 − 0.920612i \(-0.627690\pi\)
−0.390479 + 0.920612i \(0.627690\pi\)
\(30\) −10080.0 − 17459.1i −0.0681610 − 0.118058i
\(31\) 113776. − 197066.i 0.685938 − 1.18808i −0.287203 − 0.957870i \(-0.592726\pi\)
0.973141 − 0.230209i \(-0.0739411\pi\)
\(32\) 16384.0 − 28377.9i 0.0883883 − 0.153093i
\(33\) 6552.00 + 11348.4i 0.0317377 + 0.0549713i
\(34\) 117648. 0.513344
\(35\) 0 0
\(36\) −130752. −0.467078
\(37\) −80263.0 − 139020.i −0.260501 − 0.451201i 0.705874 − 0.708337i \(-0.250554\pi\)
−0.966375 + 0.257136i \(0.917221\pi\)
\(38\) 159760. − 276712.i 0.472308 − 0.818062i
\(39\) −8292.00 + 14362.2i −0.0223838 + 0.0387698i
\(40\) 53760.0 + 93115.1i 0.132816 + 0.230043i
\(41\) −10842.0 −0.0245678 −0.0122839 − 0.999925i \(-0.503910\pi\)
−0.0122839 + 0.999925i \(0.503910\pi\)
\(42\) 0 0
\(43\) −630748. −1.20981 −0.604904 − 0.796299i \(-0.706788\pi\)
−0.604904 + 0.796299i \(0.706788\pi\)
\(44\) −34944.0 − 60524.8i −0.0618427 − 0.107115i
\(45\) 214515. − 371551.i 0.350925 − 0.607819i
\(46\) 274848. − 476051.i 0.416332 − 0.721109i
\(47\) 236328. + 409332.i 0.332026 + 0.575087i 0.982909 − 0.184092i \(-0.0589343\pi\)
−0.650883 + 0.759178i \(0.725601\pi\)
\(48\) −49152.0 −0.0641500
\(49\) 0 0
\(50\) 272200. 0.307959
\(51\) −88236.0 − 152829.i −0.0931430 − 0.161328i
\(52\) 44224.0 − 76598.2i 0.0436160 − 0.0755452i
\(53\) 747009. − 1.29386e6i 0.689224 − 1.19377i −0.282865 − 0.959160i \(-0.591285\pi\)
0.972089 − 0.234611i \(-0.0753817\pi\)
\(54\) 203040. + 351676.i 0.175470 + 0.303923i
\(55\) 229320. 0.185854
\(56\) 0 0
\(57\) −479280. −0.342789
\(58\) −410280. − 710626.i −0.276110 − 0.478237i
\(59\) 1.32033e6 − 2.28688e6i 0.836952 − 1.44964i −0.0554795 − 0.998460i \(-0.517669\pi\)
0.892431 − 0.451183i \(-0.148998\pi\)
\(60\) 80640.0 − 139673.i 0.0481971 − 0.0834799i
\(61\) 413851. + 716811.i 0.233448 + 0.404343i 0.958820 − 0.284013i \(-0.0916659\pi\)
−0.725373 + 0.688356i \(0.758333\pi\)
\(62\) 1.82042e6 0.970063
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 145110. + 251338.i 0.0655391 + 0.113517i
\(66\) −52416.0 + 90787.2i −0.0224419 + 0.0388706i
\(67\) 63002.0 − 109123.i 0.0255913 − 0.0443255i −0.852946 − 0.521999i \(-0.825186\pi\)
0.878537 + 0.477674i \(0.158520\pi\)
\(68\) 470592. + 815089.i 0.181494 + 0.314358i
\(69\) −824544. −0.302164
\(70\) 0 0
\(71\) −1.41473e6 −0.469104 −0.234552 − 0.972104i \(-0.575362\pi\)
−0.234552 + 0.972104i \(0.575362\pi\)
\(72\) −523008. − 905876.i −0.165137 − 0.286026i
\(73\) 490141. − 848949.i 0.147466 − 0.255418i −0.782824 − 0.622243i \(-0.786222\pi\)
0.930290 + 0.366825i \(0.119555\pi\)
\(74\) 642104. − 1.11216e6i 0.184202 − 0.319047i
\(75\) −204150. − 353598.i −0.0558772 − 0.0967822i
\(76\) 2.55616e6 0.667945
\(77\) 0 0
\(78\) −132672. −0.0316554
\(79\) 1.78340e6 + 3.08894e6i 0.406962 + 0.704879i 0.994548 − 0.104283i \(-0.0332548\pi\)
−0.587586 + 0.809162i \(0.699921\pi\)
\(80\) −430080. + 744920.i −0.0939149 + 0.162665i
\(81\) −1.92946e6 + 3.34192e6i −0.403402 + 0.698713i
\(82\) −43368.0 − 75115.6i −0.00868602 − 0.0150446i
\(83\) −5.67289e6 −1.08901 −0.544504 − 0.838758i \(-0.683282\pi\)
−0.544504 + 0.838758i \(0.683282\pi\)
\(84\) 0 0
\(85\) −3.08826e6 −0.545441
\(86\) −2.52299e6 − 4.36995e6i −0.427732 − 0.740853i
\(87\) −615420. + 1.06594e6i −0.100197 + 0.173546i
\(88\) 279552. − 484198.i 0.0437294 − 0.0757415i
\(89\) −5.97560e6 − 1.03500e7i −0.898496 − 1.55624i −0.829417 − 0.558629i \(-0.811327\pi\)
−0.0690786 − 0.997611i \(-0.522006\pi\)
\(90\) 3.43224e6 0.496282
\(91\) 0 0
\(92\) 4.39757e6 0.588783
\(93\) −1.36531e6 − 2.36479e6i −0.176012 − 0.304861i
\(94\) −1.89062e6 + 3.27466e6i −0.234778 + 0.406648i
\(95\) −4.19370e6 + 7.26370e6i −0.501839 + 0.869211i
\(96\) −196608. − 340535.i −0.0226805 − 0.0392837i
\(97\) −8.68215e6 −0.965886 −0.482943 − 0.875652i \(-0.660432\pi\)
−0.482943 + 0.875652i \(0.660432\pi\)
\(98\) 0 0
\(99\) −2.23096e6 −0.231083
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 98.8.c.e.67.1 2
7.2 even 3 inner 98.8.c.e.79.1 2
7.3 odd 6 2.8.a.a.1.1 ✓ 1
7.4 even 3 98.8.a.a.1.1 1
7.5 odd 6 98.8.c.d.79.1 2
7.6 odd 2 98.8.c.d.67.1 2
21.17 even 6 18.8.a.b.1.1 1
28.3 even 6 16.8.a.b.1.1 1
35.3 even 12 50.8.b.c.49.2 2
35.17 even 12 50.8.b.c.49.1 2
35.24 odd 6 50.8.a.g.1.1 1
56.3 even 6 64.8.a.e.1.1 1
56.45 odd 6 64.8.a.c.1.1 1
63.31 odd 6 162.8.c.l.55.1 2
63.38 even 6 162.8.c.a.109.1 2
63.52 odd 6 162.8.c.l.109.1 2
63.59 even 6 162.8.c.a.55.1 2
77.10 even 6 242.8.a.e.1.1 1
84.59 odd 6 144.8.a.i.1.1 1
91.31 even 12 338.8.b.d.337.2 2
91.38 odd 6 338.8.a.d.1.1 1
91.73 even 12 338.8.b.d.337.1 2
105.17 odd 12 450.8.c.g.199.2 2
105.38 odd 12 450.8.c.g.199.1 2
105.59 even 6 450.8.a.c.1.1 1
112.3 even 12 256.8.b.f.129.1 2
112.45 odd 12 256.8.b.b.129.2 2
112.59 even 12 256.8.b.f.129.2 2
112.101 odd 12 256.8.b.b.129.1 2
119.101 odd 6 578.8.a.b.1.1 1
140.3 odd 12 400.8.c.j.49.1 2
140.59 even 6 400.8.a.l.1.1 1
140.87 odd 12 400.8.c.j.49.2 2
168.59 odd 6 576.8.a.f.1.1 1
168.101 even 6 576.8.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 7.3 odd 6
16.8.a.b.1.1 1 28.3 even 6
18.8.a.b.1.1 1 21.17 even 6
50.8.a.g.1.1 1 35.24 odd 6
50.8.b.c.49.1 2 35.17 even 12
50.8.b.c.49.2 2 35.3 even 12
64.8.a.c.1.1 1 56.45 odd 6
64.8.a.e.1.1 1 56.3 even 6
98.8.a.a.1.1 1 7.4 even 3
98.8.c.d.67.1 2 7.6 odd 2
98.8.c.d.79.1 2 7.5 odd 6
98.8.c.e.67.1 2 1.1 even 1 trivial
98.8.c.e.79.1 2 7.2 even 3 inner
144.8.a.i.1.1 1 84.59 odd 6
162.8.c.a.55.1 2 63.59 even 6
162.8.c.a.109.1 2 63.38 even 6
162.8.c.l.55.1 2 63.31 odd 6
162.8.c.l.109.1 2 63.52 odd 6
242.8.a.e.1.1 1 77.10 even 6
256.8.b.b.129.1 2 112.101 odd 12
256.8.b.b.129.2 2 112.45 odd 12
256.8.b.f.129.1 2 112.3 even 12
256.8.b.f.129.2 2 112.59 even 12
338.8.a.d.1.1 1 91.38 odd 6
338.8.b.d.337.1 2 91.73 even 12
338.8.b.d.337.2 2 91.31 even 12
400.8.a.l.1.1 1 140.59 even 6
400.8.c.j.49.1 2 140.3 odd 12
400.8.c.j.49.2 2 140.87 odd 12
450.8.a.c.1.1 1 105.59 even 6
450.8.c.g.199.1 2 105.38 odd 12
450.8.c.g.199.2 2 105.17 odd 12
576.8.a.f.1.1 1 168.59 odd 6
576.8.a.g.1.1 1 168.101 even 6
578.8.a.b.1.1 1 119.101 odd 6