Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [637,2,Mod(1,637)] 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("637.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-2,4,6,5,2,0,-6,11,-14,-4,18,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) 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 1.1
Root \(-1.65544\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.65544 q^{2} +2.39593 q^{3} +5.05137 q^{4} +3.65544 q^{5} -6.36226 q^{6} -8.10275 q^{8} +2.74049 q^{9} -9.70682 q^{10} +0.655442 q^{11} +12.1027 q^{12} -1.00000 q^{13} +8.75819 q^{15} +11.4136 q^{16} +2.39593 q^{17} -7.27721 q^{18} -2.70682 q^{19} +18.4650 q^{20} -1.74049 q^{22} +7.36226 q^{23} -19.4136 q^{24} +8.36226 q^{25} +2.65544 q^{26} -0.621770 q^{27} -0.208136 q^{29} -23.2569 q^{30} -1.13642 q^{31} -14.1027 q^{32} +1.57040 q^{33} -6.36226 q^{34} +13.8432 q^{36} -7.44731 q^{37} +7.18780 q^{38} -2.39593 q^{39} -29.6191 q^{40} -10.2055 q^{41} -3.10275 q^{43} +3.31088 q^{44} +10.0177 q^{45} -19.5501 q^{46} +4.60407 q^{47} +27.3463 q^{48} -22.2055 q^{50} +5.74049 q^{51} -5.05137 q^{52} +5.25951 q^{53} +1.65107 q^{54} +2.39593 q^{55} -6.48535 q^{57} +0.552694 q^{58} +8.25951 q^{59} +44.2409 q^{60} -1.89725 q^{61} +3.01770 q^{62} +14.6218 q^{64} -3.65544 q^{65} -4.17009 q^{66} -12.8946 q^{67} +12.1027 q^{68} +17.6395 q^{69} -6.75819 q^{71} -22.2055 q^{72} +12.5367 q^{73} +19.7759 q^{74} +20.0354 q^{75} -13.6731 q^{76} +6.36226 q^{78} -1.51902 q^{79} +41.7219 q^{80} -9.71119 q^{81} +27.1001 q^{82} +15.7582 q^{83} +8.75819 q^{85} +8.23917 q^{86} -0.498680 q^{87} -5.31088 q^{88} -14.8096 q^{89} -26.6014 q^{90} +37.1895 q^{92} -2.72279 q^{93} -12.2258 q^{94} -9.89461 q^{95} -33.7892 q^{96} -10.0177 q^{97} +1.79623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 4 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} - 6 q^{8} + 11 q^{9} - 14 q^{10} - 4 q^{11} + 18 q^{12} - 3 q^{13} + 2 q^{15} + 4 q^{16} + 4 q^{17} + 8 q^{18} + 7 q^{19} + 16 q^{20} - 8 q^{22} + q^{23}+ \cdots - 30 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.65544 −1.87768 −0.938841 − 0.344352i \(-0.888099\pi\)
−0.938841 + 0.344352i \(0.888099\pi\)
\(3\) 2.39593 1.38329 0.691646 − 0.722237i \(-0.256886\pi\)
0.691646 + 0.722237i \(0.256886\pi\)
\(4\) 5.05137 2.52569
\(5\) 3.65544 1.63476 0.817382 − 0.576096i \(-0.195425\pi\)
0.817382 + 0.576096i \(0.195425\pi\)
\(6\) −6.36226 −2.59738
\(7\) 0 0
\(8\) −8.10275 −2.86475
\(9\) 2.74049 0.913496
\(10\) −9.70682 −3.06956
\(11\) 0.655442 0.197623 0.0988117 − 0.995106i \(-0.468496\pi\)
0.0988117 + 0.995106i \(0.468496\pi\)
\(12\) 12.1027 3.49376
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 8.75819 2.26136
\(16\) 11.4136 2.85341
\(17\) 2.39593 0.581099 0.290549 − 0.956860i \(-0.406162\pi\)
0.290549 + 0.956860i \(0.406162\pi\)
\(18\) −7.27721 −1.71526
\(19\) −2.70682 −0.620986 −0.310493 − 0.950576i \(-0.600494\pi\)
−0.310493 + 0.950576i \(0.600494\pi\)
\(20\) 18.4650 4.12890
\(21\) 0 0
\(22\) −1.74049 −0.371074
\(23\) 7.36226 1.53514 0.767569 − 0.640967i \(-0.221466\pi\)
0.767569 + 0.640967i \(0.221466\pi\)
\(24\) −19.4136 −3.96279
\(25\) 8.36226 1.67245
\(26\) 2.65544 0.520775
\(27\) −0.621770 −0.119660
\(28\) 0 0
\(29\) −0.208136 −0.0386499 −0.0193250 − 0.999813i \(-0.506152\pi\)
−0.0193250 + 0.999813i \(0.506152\pi\)
\(30\) −23.2569 −4.24610
\(31\) −1.13642 −0.204107 −0.102054 − 0.994779i \(-0.532541\pi\)
−0.102054 + 0.994779i \(0.532541\pi\)
\(32\) −14.1027 −2.49304
\(33\) 1.57040 0.273371
\(34\) −6.36226 −1.09112
\(35\) 0 0
\(36\) 13.8432 2.30721
\(37\) −7.44731 −1.22433 −0.612165 − 0.790730i \(-0.709701\pi\)
−0.612165 + 0.790730i \(0.709701\pi\)
\(38\) 7.18780 1.16601
\(39\) −2.39593 −0.383656
\(40\) −29.6191 −4.68320
\(41\) −10.2055 −1.59383 −0.796915 − 0.604091i \(-0.793536\pi\)
−0.796915 + 0.604091i \(0.793536\pi\)
\(42\) 0 0
\(43\) −3.10275 −0.473165 −0.236582 − 0.971611i \(-0.576027\pi\)
−0.236582 + 0.971611i \(0.576027\pi\)
\(44\) 3.31088 0.499135
\(45\) 10.0177 1.49335
\(46\) −19.5501 −2.88250
\(47\) 4.60407 0.671572 0.335786 − 0.941938i \(-0.390998\pi\)
0.335786 + 0.941938i \(0.390998\pi\)
\(48\) 27.3463 3.94710
\(49\) 0 0
\(50\) −22.2055 −3.14033
\(51\) 5.74049 0.803829
\(52\) −5.05137 −0.700500
\(53\) 5.25951 0.722449 0.361225 − 0.932479i \(-0.382359\pi\)
0.361225 + 0.932479i \(0.382359\pi\)
\(54\) 1.65107 0.224683
\(55\) 2.39593 0.323067
\(56\) 0 0
\(57\) −6.48535 −0.859005
\(58\) 0.552694 0.0725723
\(59\) 8.25951 1.07530 0.537648 − 0.843169i \(-0.319313\pi\)
0.537648 + 0.843169i \(0.319313\pi\)
\(60\) 44.2409 5.71148
\(61\) −1.89725 −0.242918 −0.121459 − 0.992596i \(-0.538757\pi\)
−0.121459 + 0.992596i \(0.538757\pi\)
\(62\) 3.01770 0.383248
\(63\) 0 0
\(64\) 14.6218 1.82772
\(65\) −3.65544 −0.453402
\(66\) −4.17009 −0.513303
\(67\) −12.8946 −1.57533 −0.787664 − 0.616105i \(-0.788710\pi\)
−0.787664 + 0.616105i \(0.788710\pi\)
\(68\) 12.1027 1.46767
\(69\) 17.6395 2.12354
\(70\) 0 0
\(71\) −6.75819 −0.802050 −0.401025 − 0.916067i \(-0.631346\pi\)
−0.401025 + 0.916067i \(0.631346\pi\)
\(72\) −22.2055 −2.61694
\(73\) 12.5367 1.46731 0.733656 − 0.679521i \(-0.237812\pi\)
0.733656 + 0.679521i \(0.237812\pi\)
\(74\) 19.7759 2.29890
\(75\) 20.0354 2.31349
\(76\) −13.6731 −1.56842
\(77\) 0 0
\(78\) 6.36226 0.720384
\(79\) −1.51902 −0.170903 −0.0854516 − 0.996342i \(-0.527233\pi\)
−0.0854516 + 0.996342i \(0.527233\pi\)
\(80\) 41.7219 4.66465
\(81\) −9.71119 −1.07902
\(82\) 27.1001 2.99271
\(83\) 15.7582 1.72969 0.864843 − 0.502042i \(-0.167418\pi\)
0.864843 + 0.502042i \(0.167418\pi\)
\(84\) 0 0
\(85\) 8.75819 0.949959
\(86\) 8.23917 0.888453
\(87\) −0.498680 −0.0534641
\(88\) −5.31088 −0.566142
\(89\) −14.8096 −1.56981 −0.784905 − 0.619616i \(-0.787288\pi\)
−0.784905 + 0.619616i \(0.787288\pi\)
\(90\) −26.6014 −2.80404
\(91\) 0 0
\(92\) 37.1895 3.87728
\(93\) −2.72279 −0.282340
\(94\) −12.2258 −1.26100
\(95\) −9.89461 −1.01517
\(96\) −33.7892 −3.44860
\(97\) −10.0177 −1.01714 −0.508572 − 0.861020i \(-0.669826\pi\)
−0.508572 + 0.861020i \(0.669826\pi\)
\(98\) 0 0
\(99\) 1.79623 0.180528
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 637.2.a.i.1.1 yes 3
3.2 odd 2 5733.2.a.bd.1.3 3
7.2 even 3 637.2.e.k.508.3 6
7.3 odd 6 637.2.e.l.79.3 6
7.4 even 3 637.2.e.k.79.3 6
7.5 odd 6 637.2.e.l.508.3 6
7.6 odd 2 637.2.a.h.1.1 ✓ 3
13.12 even 2 8281.2.a.bk.1.3 3
21.20 even 2 5733.2.a.be.1.3 3
91.90 odd 2 8281.2.a.bh.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.h.1.1 ✓ 3 7.6 odd 2
637.2.a.i.1.1 yes 3 1.1 even 1 trivial
637.2.e.k.79.3 6 7.4 even 3
637.2.e.k.508.3 6 7.2 even 3
637.2.e.l.79.3 6 7.3 odd 6
637.2.e.l.508.3 6 7.5 odd 6
5733.2.a.bd.1.3 3 3.2 odd 2
5733.2.a.be.1.3 3 21.20 even 2
8281.2.a.bh.1.3 3 91.90 odd 2
8281.2.a.bk.1.3 3 13.12 even 2