Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4598,2,Mod(1,4598)] 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("4598.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4598, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,0,3,-3,0,6,3,3,-3,0,0,0] 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: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.66908 q^{3} +1.00000 q^{4} -4.12398 q^{5} +2.66908 q^{6} +4.21417 q^{7} +1.00000 q^{8} +4.12398 q^{9} -4.12398 q^{10} +2.66908 q^{12} +2.21417 q^{13} +4.21417 q^{14} -11.0072 q^{15} +1.00000 q^{16} +3.45490 q^{17} +4.12398 q^{18} +1.00000 q^{19} -4.12398 q^{20} +11.2480 q^{21} -5.45490 q^{23} +2.66908 q^{24} +12.0072 q^{25} +2.21417 q^{26} +3.00000 q^{27} +4.21417 q^{28} -5.57889 q^{29} -11.0072 q^{30} +7.00724 q^{31} +1.00000 q^{32} +3.45490 q^{34} -17.3792 q^{35} +4.12398 q^{36} -2.90981 q^{37} +1.00000 q^{38} +5.90981 q^{39} -4.12398 q^{40} +11.9170 q^{41} +11.2480 q^{42} -1.46214 q^{43} -17.0072 q^{45} -5.45490 q^{46} +7.58612 q^{47} +2.66908 q^{48} +10.7593 q^{49} +12.0072 q^{50} +9.22141 q^{51} +2.21417 q^{52} -13.2214 q^{53} +3.00000 q^{54} +4.21417 q^{56} +2.66908 q^{57} -5.57889 q^{58} +4.79306 q^{59} -11.0072 q^{60} -8.90981 q^{61} +7.00724 q^{62} +17.3792 q^{63} +1.00000 q^{64} -9.13122 q^{65} +1.30437 q^{67} +3.45490 q^{68} -14.5596 q^{69} -17.3792 q^{70} -6.80030 q^{71} +4.12398 q^{72} +1.45490 q^{73} -2.90981 q^{74} +32.0483 q^{75} +1.00000 q^{76} +5.90981 q^{78} +9.15777 q^{79} -4.12398 q^{80} -4.36471 q^{81} +11.9170 q^{82} +13.2142 q^{83} +11.2480 q^{84} -14.2480 q^{85} -1.46214 q^{86} -14.8905 q^{87} +8.24797 q^{89} -17.0072 q^{90} +9.33092 q^{91} -5.45490 q^{92} +18.7029 q^{93} +7.58612 q^{94} -4.12398 q^{95} +2.66908 q^{96} +2.18038 q^{97} +10.7593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{14} - 9 q^{15} + 3 q^{16} + 9 q^{17} + 3 q^{18} + 3 q^{19} - 3 q^{20} + 15 q^{21} - 15 q^{23} + 12 q^{25} + 9 q^{27} + 6 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.66908 1.54099 0.770497 − 0.637444i \(-0.220008\pi\)
0.770497 + 0.637444i \(0.220008\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.12398 −1.84430 −0.922151 − 0.386831i \(-0.873570\pi\)
−0.922151 + 0.386831i \(0.873570\pi\)
\(6\) 2.66908 1.08965
\(7\) 4.21417 1.59281 0.796404 − 0.604765i \(-0.206733\pi\)
0.796404 + 0.604765i \(0.206733\pi\)
\(8\) 1.00000 0.353553
\(9\) 4.12398 1.37466
\(10\) −4.12398 −1.30412
\(11\) 0 0
\(12\) 2.66908 0.770497
\(13\) 2.21417 0.614102 0.307051 − 0.951693i \(-0.400658\pi\)
0.307051 + 0.951693i \(0.400658\pi\)
\(14\) 4.21417 1.12629
\(15\) −11.0072 −2.84206
\(16\) 1.00000 0.250000
\(17\) 3.45490 0.837937 0.418969 − 0.908001i \(-0.362392\pi\)
0.418969 + 0.908001i \(0.362392\pi\)
\(18\) 4.12398 0.972032
\(19\) 1.00000 0.229416
\(20\) −4.12398 −0.922151
\(21\) 11.2480 2.45451
\(22\) 0 0
\(23\) −5.45490 −1.13743 −0.568713 − 0.822536i \(-0.692559\pi\)
−0.568713 + 0.822536i \(0.692559\pi\)
\(24\) 2.66908 0.544823
\(25\) 12.0072 2.40145
\(26\) 2.21417 0.434235
\(27\) 3.00000 0.577350
\(28\) 4.21417 0.796404
\(29\) −5.57889 −1.03597 −0.517987 − 0.855389i \(-0.673318\pi\)
−0.517987 + 0.855389i \(0.673318\pi\)
\(30\) −11.0072 −2.00964
\(31\) 7.00724 1.25854 0.629268 − 0.777188i \(-0.283355\pi\)
0.629268 + 0.777188i \(0.283355\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.45490 0.592511
\(35\) −17.3792 −2.93762
\(36\) 4.12398 0.687331
\(37\) −2.90981 −0.478370 −0.239185 − 0.970974i \(-0.576880\pi\)
−0.239185 + 0.970974i \(0.576880\pi\)
\(38\) 1.00000 0.162221
\(39\) 5.90981 0.946327
\(40\) −4.12398 −0.652059
\(41\) 11.9170 1.86113 0.930565 − 0.366127i \(-0.119316\pi\)
0.930565 + 0.366127i \(0.119316\pi\)
\(42\) 11.2480 1.73560
\(43\) −1.46214 −0.222974 −0.111487 − 0.993766i \(-0.535561\pi\)
−0.111487 + 0.993766i \(0.535561\pi\)
\(44\) 0 0
\(45\) −17.0072 −2.53529
\(46\) −5.45490 −0.804282
\(47\) 7.58612 1.10655 0.553275 − 0.832999i \(-0.313378\pi\)
0.553275 + 0.832999i \(0.313378\pi\)
\(48\) 2.66908 0.385248
\(49\) 10.7593 1.53704
\(50\) 12.0072 1.69808
\(51\) 9.22141 1.29126
\(52\) 2.21417 0.307051
\(53\) −13.2214 −1.81610 −0.908050 − 0.418861i \(-0.862429\pi\)
−0.908050 + 0.418861i \(0.862429\pi\)
\(54\) 3.00000 0.408248
\(55\) 0 0
\(56\) 4.21417 0.563143
\(57\) 2.66908 0.353528
\(58\) −5.57889 −0.732544
\(59\) 4.79306 0.624004 0.312002 − 0.950082i \(-0.399001\pi\)
0.312002 + 0.950082i \(0.399001\pi\)
\(60\) −11.0072 −1.42103
\(61\) −8.90981 −1.14078 −0.570392 − 0.821373i \(-0.693209\pi\)
−0.570392 + 0.821373i \(0.693209\pi\)
\(62\) 7.00724 0.889920
\(63\) 17.3792 2.18957
\(64\) 1.00000 0.125000
\(65\) −9.13122 −1.13259
\(66\) 0 0
\(67\) 1.30437 0.159354 0.0796769 − 0.996821i \(-0.474611\pi\)
0.0796769 + 0.996821i \(0.474611\pi\)
\(68\) 3.45490 0.418969
\(69\) −14.5596 −1.75277
\(70\) −17.3792 −2.07721
\(71\) −6.80030 −0.807047 −0.403524 − 0.914969i \(-0.632215\pi\)
−0.403524 + 0.914969i \(0.632215\pi\)
\(72\) 4.12398 0.486016
\(73\) 1.45490 0.170284 0.0851418 − 0.996369i \(-0.472866\pi\)
0.0851418 + 0.996369i \(0.472866\pi\)
\(74\) −2.90981 −0.338258
\(75\) 32.0483 3.70061
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 5.90981 0.669154
\(79\) 9.15777 1.03033 0.515165 − 0.857091i \(-0.327731\pi\)
0.515165 + 0.857091i \(0.327731\pi\)
\(80\) −4.12398 −0.461075
\(81\) −4.36471 −0.484968
\(82\) 11.9170 1.31602
\(83\) 13.2142 1.45044 0.725222 − 0.688515i \(-0.241737\pi\)
0.725222 + 0.688515i \(0.241737\pi\)
\(84\) 11.2480 1.22725
\(85\) −14.2480 −1.54541
\(86\) −1.46214 −0.157667
\(87\) −14.8905 −1.59643
\(88\) 0 0
\(89\) 8.24797 0.874283 0.437141 − 0.899393i \(-0.355991\pi\)
0.437141 + 0.899393i \(0.355991\pi\)
\(90\) −17.0072 −1.79272
\(91\) 9.33092 0.978146
\(92\) −5.45490 −0.568713
\(93\) 18.7029 1.93940
\(94\) 7.58612 0.782449
\(95\) −4.12398 −0.423112
\(96\) 2.66908 0.272412
\(97\) 2.18038 0.221384 0.110692 − 0.993855i \(-0.464693\pi\)
0.110692 + 0.993855i \(0.464693\pi\)
\(98\) 10.7593 1.08685
\(99\) 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 4598.2.a.bo.1.3 3
11.10 odd 2 418.2.a.g.1.3 ✓ 3
33.32 even 2 3762.2.a.bg.1.3 3
44.43 even 2 3344.2.a.q.1.1 3
209.208 even 2 7942.2.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.g.1.3 ✓ 3 11.10 odd 2
3344.2.a.q.1.1 3 44.43 even 2
3762.2.a.bg.1.3 3 33.32 even 2
4598.2.a.bo.1.3 3 1.1 even 1 trivial
7942.2.a.bi.1.1 3 209.208 even 2