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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4950,2,Mod(199,4950)] 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("4950.199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4950.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,0,0,0,0,4,0,0,-2,0,4,0,0,-14,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.5259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(3.37228i\) of defining polynomial
Character \(\chi\) \(=\) 4950.199
Dual form 4950.2.c.bc.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +3.37228i q^{7} -1.00000i q^{8} +1.00000 q^{11} -2.00000i q^{13} -3.37228 q^{14} +1.00000 q^{16} -1.37228i q^{17} -0.627719 q^{19} +1.00000i q^{22} +2.74456i q^{23} +2.00000 q^{26} -3.37228i q^{28} +1.37228 q^{29} +3.37228 q^{31} +1.00000i q^{32} +1.37228 q^{34} +9.37228i q^{37} -0.627719i q^{38} +11.4891 q^{41} +4.00000i q^{43} -1.00000 q^{44} -2.74456 q^{46} -2.74456i q^{47} -4.37228 q^{49} +2.00000i q^{52} -4.11684i q^{53} +3.37228 q^{56} +1.37228i q^{58} -2.74456 q^{59} -5.37228 q^{61} +3.37228i q^{62} -1.00000 q^{64} +8.00000i q^{67} +1.37228i q^{68} -10.1168 q^{71} +15.4891i q^{73} -9.37228 q^{74} +0.627719 q^{76} +3.37228i q^{77} +1.25544 q^{79} +11.4891i q^{82} -2.74456i q^{83} -4.00000 q^{86} -1.00000i q^{88} -1.37228 q^{89} +6.74456 q^{91} -2.74456i q^{92} +2.74456 q^{94} -12.7446i q^{97} -4.37228i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{11} - 2 q^{14} + 4 q^{16} - 14 q^{19} + 8 q^{26} - 6 q^{29} + 2 q^{31} - 6 q^{34} - 4 q^{44} + 12 q^{46} - 6 q^{49} + 2 q^{56} + 12 q^{59} - 10 q^{61} - 4 q^{64} - 6 q^{71} - 26 q^{74}+ \cdots - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2377\) \(4501\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.37228i 1.27460i 0.770615 + 0.637301i \(0.219949\pi\)
−0.770615 + 0.637301i \(0.780051\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) −3.37228 −0.901280
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.37228i − 0.332827i −0.986056 − 0.166414i \(-0.946781\pi\)
0.986056 − 0.166414i \(-0.0532187\pi\)
\(18\) 0 0
\(19\) −0.627719 −0.144009 −0.0720043 − 0.997404i \(-0.522940\pi\)
−0.0720043 + 0.997404i \(0.522940\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000i 0.213201i
\(23\) 2.74456i 0.572281i 0.958188 + 0.286140i \(0.0923724\pi\)
−0.958188 + 0.286140i \(0.907628\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) − 3.37228i − 0.637301i
\(29\) 1.37228 0.254826 0.127413 − 0.991850i \(-0.459333\pi\)
0.127413 + 0.991850i \(0.459333\pi\)
\(30\) 0 0
\(31\) 3.37228 0.605680 0.302840 − 0.953041i \(-0.402065\pi\)
0.302840 + 0.953041i \(0.402065\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 1.37228 0.235344
\(35\) 0 0
\(36\) 0 0
\(37\) 9.37228i 1.54079i 0.637565 + 0.770397i \(0.279942\pi\)
−0.637565 + 0.770397i \(0.720058\pi\)
\(38\) − 0.627719i − 0.101829i
\(39\) 0 0
\(40\) 0 0
\(41\) 11.4891 1.79430 0.897150 − 0.441726i \(-0.145634\pi\)
0.897150 + 0.441726i \(0.145634\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −2.74456 −0.404664
\(47\) − 2.74456i − 0.400336i −0.979762 − 0.200168i \(-0.935851\pi\)
0.979762 − 0.200168i \(-0.0641487\pi\)
\(48\) 0 0
\(49\) −4.37228 −0.624612
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) − 4.11684i − 0.565492i −0.959195 − 0.282746i \(-0.908755\pi\)
0.959195 − 0.282746i \(-0.0912454\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.37228 0.450640
\(57\) 0 0
\(58\) 1.37228i 0.180189i
\(59\) −2.74456 −0.357312 −0.178656 − 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 − 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) 3.37228i 0.428280i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 1.37228i 0.166414i
\(69\) 0 0
\(70\) 0 0
\(71\) −10.1168 −1.20065 −0.600324 − 0.799757i \(-0.704962\pi\)
−0.600324 + 0.799757i \(0.704962\pi\)
\(72\) 0 0
\(73\) 15.4891i 1.81286i 0.422351 + 0.906432i \(0.361205\pi\)
−0.422351 + 0.906432i \(0.638795\pi\)
\(74\) −9.37228 −1.08951
\(75\) 0 0
\(76\) 0.627719 0.0720043
\(77\) 3.37228i 0.384307i
\(78\) 0 0
\(79\) 1.25544 0.141248 0.0706239 − 0.997503i \(-0.477501\pi\)
0.0706239 + 0.997503i \(0.477501\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 11.4891i 1.26876i
\(83\) − 2.74456i − 0.301255i −0.988591 − 0.150627i \(-0.951871\pi\)
0.988591 − 0.150627i \(-0.0481294\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) − 1.00000i − 0.106600i
\(89\) −1.37228 −0.145462 −0.0727308 − 0.997352i \(-0.523171\pi\)
−0.0727308 + 0.997352i \(0.523171\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) − 2.74456i − 0.286140i
\(93\) 0 0
\(94\) 2.74456 0.283080
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.7446i − 1.29401i −0.762484 − 0.647007i \(-0.776020\pi\)
0.762484 − 0.647007i \(-0.223980\pi\)
\(98\) − 4.37228i − 0.441667i
\(99\) 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 4950.2.c.bc.199.4 4
3.2 odd 2 550.2.b.f.199.2 4
5.2 odd 4 4950.2.a.bw.1.1 2
5.3 odd 4 990.2.a.m.1.2 2
5.4 even 2 inner 4950.2.c.bc.199.1 4
12.11 even 2 4400.2.b.p.4049.1 4
15.2 even 4 550.2.a.n.1.2 2
15.8 even 4 110.2.a.d.1.1 ✓ 2
15.14 odd 2 550.2.b.f.199.3 4
20.3 even 4 7920.2.a.bq.1.1 2
60.23 odd 4 880.2.a.n.1.2 2
60.47 odd 4 4400.2.a.bl.1.1 2
60.59 even 2 4400.2.b.p.4049.4 4
105.83 odd 4 5390.2.a.bp.1.2 2
120.53 even 4 3520.2.a.bq.1.2 2
120.83 odd 4 3520.2.a.bj.1.1 2
165.32 odd 4 6050.2.a.cb.1.2 2
165.98 odd 4 1210.2.a.r.1.1 2
660.263 even 4 9680.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 ✓ 2 15.8 even 4
550.2.a.n.1.2 2 15.2 even 4
550.2.b.f.199.2 4 3.2 odd 2
550.2.b.f.199.3 4 15.14 odd 2
880.2.a.n.1.2 2 60.23 odd 4
990.2.a.m.1.2 2 5.3 odd 4
1210.2.a.r.1.1 2 165.98 odd 4
3520.2.a.bj.1.1 2 120.83 odd 4
3520.2.a.bq.1.2 2 120.53 even 4
4400.2.a.bl.1.1 2 60.47 odd 4
4400.2.b.p.4049.1 4 12.11 even 2
4400.2.b.p.4049.4 4 60.59 even 2
4950.2.a.bw.1.1 2 5.2 odd 4
4950.2.c.bc.199.1 4 5.4 even 2 inner
4950.2.c.bc.199.4 4 1.1 even 1 trivial
5390.2.a.bp.1.2 2 105.83 odd 4
6050.2.a.cb.1.2 2 165.32 odd 4
7920.2.a.bq.1.1 2 20.3 even 4
9680.2.a.bt.1.2 2 660.263 even 4