Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4650,2,Mod(3349,4650)] 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("4650.3349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,-4,0,0,-4,0,2,0,0,-2,0,4,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.3
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.bd.3349.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.56155 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.56155 q^{14} +1.00000 q^{16} +5.12311i q^{17} -1.00000i q^{18} -4.68466 q^{19} +1.56155 q^{21} -1.56155i q^{22} -5.56155i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +1.56155i q^{28} +1.12311 q^{29} +1.00000 q^{31} +1.00000i q^{32} -1.56155i q^{33} -5.12311 q^{34} +1.00000 q^{36} +5.12311i q^{37} -4.68466i q^{38} +2.00000 q^{39} -1.12311 q^{41} +1.56155i q^{42} +7.80776i q^{43} +1.56155 q^{44} +5.56155 q^{46} -3.12311i q^{47} +1.00000i q^{48} +4.56155 q^{49} -5.12311 q^{51} +2.00000i q^{52} -11.5616i q^{53} +1.00000 q^{54} -1.56155 q^{56} -4.68466i q^{57} +1.12311i q^{58} +4.87689 q^{59} +6.00000 q^{61} +1.00000i q^{62} +1.56155i q^{63} -1.00000 q^{64} +1.56155 q^{66} +9.36932i q^{67} -5.12311i q^{68} +5.56155 q^{69} +4.68466 q^{71} +1.00000i q^{72} -9.80776i q^{73} -5.12311 q^{74} +4.68466 q^{76} +2.43845i q^{77} +2.00000i q^{78} +16.6847 q^{79} +1.00000 q^{81} -1.12311i q^{82} +2.24621i q^{83} -1.56155 q^{84} -7.80776 q^{86} +1.12311i q^{87} +1.56155i q^{88} +1.31534 q^{89} -3.12311 q^{91} +5.56155i q^{92} +1.00000i q^{93} +3.12311 q^{94} -1.00000 q^{96} -6.00000i q^{97} +4.56155i q^{98} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 2 q^{21} + 4 q^{24} + 8 q^{26} - 12 q^{29} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 12 q^{41} - 2 q^{44} + 14 q^{46}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 1.56155i − 0.590211i −0.955465 − 0.295106i \(-0.904645\pi\)
0.955465 − 0.295106i \(-0.0953549\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 − 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.12311i 1.24254i 0.783598 + 0.621268i \(0.213382\pi\)
−0.783598 + 0.621268i \(0.786618\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −4.68466 −1.07473 −0.537367 − 0.843348i \(-0.680581\pi\)
−0.537367 + 0.843348i \(0.680581\pi\)
\(20\) 0 0
\(21\) 1.56155 0.340759
\(22\) − 1.56155i − 0.332924i
\(23\) − 5.56155i − 1.15966i −0.814736 − 0.579832i \(-0.803118\pi\)
0.814736 − 0.579832i \(-0.196882\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) 1.56155i 0.295106i
\(29\) 1.12311 0.208555 0.104278 − 0.994548i \(-0.466747\pi\)
0.104278 + 0.994548i \(0.466747\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) − 1.56155i − 0.271831i
\(34\) −5.12311 −0.878605
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.12311i 0.842233i 0.907006 + 0.421117i \(0.138362\pi\)
−0.907006 + 0.421117i \(0.861638\pi\)
\(38\) − 4.68466i − 0.759952i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 − 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) 1.56155i 0.240953i
\(43\) 7.80776i 1.19067i 0.803477 + 0.595336i \(0.202981\pi\)
−0.803477 + 0.595336i \(0.797019\pi\)
\(44\) 1.56155 0.235413
\(45\) 0 0
\(46\) 5.56155 0.820006
\(47\) − 3.12311i − 0.455552i −0.973714 − 0.227776i \(-0.926855\pi\)
0.973714 − 0.227776i \(-0.0731454\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 4.56155 0.651650
\(50\) 0 0
\(51\) −5.12311 −0.717378
\(52\) 2.00000i 0.277350i
\(53\) − 11.5616i − 1.58810i −0.607852 − 0.794051i \(-0.707968\pi\)
0.607852 − 0.794051i \(-0.292032\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.56155 −0.208671
\(57\) − 4.68466i − 0.620498i
\(58\) 1.12311i 0.147471i
\(59\) 4.87689 0.634918 0.317459 − 0.948272i \(-0.397170\pi\)
0.317459 + 0.948272i \(0.397170\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 1.56155i 0.196737i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.56155 0.192214
\(67\) 9.36932i 1.14464i 0.820029 + 0.572322i \(0.193957\pi\)
−0.820029 + 0.572322i \(0.806043\pi\)
\(68\) − 5.12311i − 0.621268i
\(69\) 5.56155 0.669532
\(70\) 0 0
\(71\) 4.68466 0.555967 0.277983 − 0.960586i \(-0.410334\pi\)
0.277983 + 0.960586i \(0.410334\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 9.80776i − 1.14791i −0.818886 − 0.573956i \(-0.805408\pi\)
0.818886 − 0.573956i \(-0.194592\pi\)
\(74\) −5.12311 −0.595549
\(75\) 0 0
\(76\) 4.68466 0.537367
\(77\) 2.43845i 0.277887i
\(78\) 2.00000i 0.226455i
\(79\) 16.6847 1.87717 0.938585 − 0.345047i \(-0.112137\pi\)
0.938585 + 0.345047i \(0.112137\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 1.12311i − 0.124026i
\(83\) 2.24621i 0.246554i 0.992372 + 0.123277i \(0.0393403\pi\)
−0.992372 + 0.123277i \(0.960660\pi\)
\(84\) −1.56155 −0.170379
\(85\) 0 0
\(86\) −7.80776 −0.841933
\(87\) 1.12311i 0.120410i
\(88\) 1.56155i 0.166462i
\(89\) 1.31534 0.139426 0.0697130 − 0.997567i \(-0.477792\pi\)
0.0697130 + 0.997567i \(0.477792\pi\)
\(90\) 0 0
\(91\) −3.12311 −0.327390
\(92\) 5.56155i 0.579832i
\(93\) 1.00000i 0.103695i
\(94\) 3.12311 0.322124
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 6.00000i − 0.609208i −0.952479 − 0.304604i \(-0.901476\pi\)
0.952479 − 0.304604i \(-0.0985241\pi\)
\(98\) 4.56155i 0.460786i
\(99\) 1.56155 0.156942
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 4650.2.d.bd.3349.3 4
5.2 odd 4 4650.2.a.ce.1.2 2
5.3 odd 4 930.2.a.p.1.1 ✓ 2
5.4 even 2 inner 4650.2.d.bd.3349.2 4
15.8 even 4 2790.2.a.be.1.1 2
20.3 even 4 7440.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.1 ✓ 2 5.3 odd 4
2790.2.a.be.1.1 2 15.8 even 4
4650.2.a.ce.1.2 2 5.2 odd 4
4650.2.d.bd.3349.2 4 5.4 even 2 inner
4650.2.d.bd.3349.3 4 1.1 even 1 trivial
7440.2.a.bl.1.2 2 20.3 even 4