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Properties

Label 2250.2.c.g.1999.3
Level $2250$
Weight $2$
Character 2250.1999
Analytic conductor $17.966$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2250,2,Mod(1999,2250)] 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("2250.1999"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2250, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2250 = 2 \cdot 3^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2250.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,0,0,0,0,10,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.9663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 750)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1999.3
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 2250.1999
Dual form 2250.2.c.g.1999.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +2.38197i q^{7} -1.00000i q^{8} +5.85410 q^{11} -3.38197i q^{13} -2.38197 q^{14} +1.00000 q^{16} -7.70820i q^{17} -4.09017 q^{19} +5.85410i q^{22} -3.09017i q^{23} +3.38197 q^{26} -2.38197i q^{28} +5.70820 q^{29} -6.47214 q^{31} +1.00000i q^{32} +7.70820 q^{34} -1.61803i q^{37} -4.09017i q^{38} -0.381966 q^{41} -7.70820i q^{43} -5.85410 q^{44} +3.09017 q^{46} -8.61803i q^{47} +1.32624 q^{49} +3.38197i q^{52} +0.381966i q^{53} +2.38197 q^{56} +5.70820i q^{58} -10.8541 q^{59} +11.7082 q^{61} -6.47214i q^{62} -1.00000 q^{64} +3.23607i q^{67} +7.70820i q^{68} +4.47214 q^{71} +8.00000i q^{73} +1.61803 q^{74} +4.09017 q^{76} +13.9443i q^{77} -12.4721 q^{79} -0.381966i q^{82} +2.00000i q^{83} +7.70820 q^{86} -5.85410i q^{88} +15.5623 q^{89} +8.05573 q^{91} +3.09017i q^{92} +8.61803 q^{94} -14.1803i q^{97} +1.32624i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 10 q^{11} - 14 q^{14} + 4 q^{16} + 6 q^{19} + 18 q^{26} - 4 q^{29} - 8 q^{31} + 4 q^{34} - 6 q^{41} - 10 q^{44} - 10 q^{46} - 26 q^{49} + 14 q^{56} - 30 q^{59} + 20 q^{61} - 4 q^{64} + 2 q^{74}+ \cdots + 30 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2250\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(1001\)
\(\chi(n)\) \(-1\) \(1\)

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
\(7\) 2.38197i 0.900299i 0.892953 + 0.450149i \(0.148629\pi\)
−0.892953 + 0.450149i \(0.851371\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.85410 1.76508 0.882539 − 0.470239i \(-0.155832\pi\)
0.882539 + 0.470239i \(0.155832\pi\)
\(12\) 0 0
\(13\) − 3.38197i − 0.937989i −0.883201 − 0.468994i \(-0.844616\pi\)
0.883201 − 0.468994i \(-0.155384\pi\)
\(14\) −2.38197 −0.636607
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 7.70820i − 1.86951i −0.355288 − 0.934757i \(-0.615617\pi\)
0.355288 − 0.934757i \(-0.384383\pi\)
\(18\) 0 0
\(19\) −4.09017 −0.938349 −0.469175 − 0.883105i \(-0.655449\pi\)
−0.469175 + 0.883105i \(0.655449\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.85410i 1.24810i
\(23\) − 3.09017i − 0.644345i −0.946681 − 0.322172i \(-0.895587\pi\)
0.946681 − 0.322172i \(-0.104413\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.38197 0.663258
\(27\) 0 0
\(28\) − 2.38197i − 0.450149i
\(29\) 5.70820 1.05999 0.529993 − 0.848002i \(-0.322194\pi\)
0.529993 + 0.848002i \(0.322194\pi\)
\(30\) 0 0
\(31\) −6.47214 −1.16243 −0.581215 − 0.813750i \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 7.70820 1.32195
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.61803i − 0.266003i −0.991116 − 0.133002i \(-0.957538\pi\)
0.991116 − 0.133002i \(-0.0424615\pi\)
\(38\) − 4.09017i − 0.663513i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.381966 −0.0596531 −0.0298265 − 0.999555i \(-0.509495\pi\)
−0.0298265 + 0.999555i \(0.509495\pi\)
\(42\) 0 0
\(43\) − 7.70820i − 1.17549i −0.809046 − 0.587745i \(-0.800016\pi\)
0.809046 − 0.587745i \(-0.199984\pi\)
\(44\) −5.85410 −0.882539
\(45\) 0 0
\(46\) 3.09017 0.455621
\(47\) − 8.61803i − 1.25707i −0.777782 − 0.628535i \(-0.783655\pi\)
0.777782 − 0.628535i \(-0.216345\pi\)
\(48\) 0 0
\(49\) 1.32624 0.189463
\(50\) 0 0
\(51\) 0 0
\(52\) 3.38197i 0.468994i
\(53\) 0.381966i 0.0524671i 0.999656 + 0.0262335i \(0.00835135\pi\)
−0.999656 + 0.0262335i \(0.991649\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.38197 0.318304
\(57\) 0 0
\(58\) 5.70820i 0.749524i
\(59\) −10.8541 −1.41308 −0.706542 − 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(60\) 0 0
\(61\) 11.7082 1.49908 0.749541 − 0.661958i \(-0.230274\pi\)
0.749541 + 0.661958i \(0.230274\pi\)
\(62\) − 6.47214i − 0.821962i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.23607i 0.395349i 0.980268 + 0.197674i \(0.0633388\pi\)
−0.980268 + 0.197674i \(0.936661\pi\)
\(68\) 7.70820i 0.934757i
\(69\) 0 0
\(70\) 0 0
\(71\) 4.47214 0.530745 0.265372 − 0.964146i \(-0.414505\pi\)
0.265372 + 0.964146i \(0.414505\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 1.61803 0.188093
\(75\) 0 0
\(76\) 4.09017 0.469175
\(77\) 13.9443i 1.58910i
\(78\) 0 0
\(79\) −12.4721 −1.40322 −0.701612 − 0.712559i \(-0.747536\pi\)
−0.701612 + 0.712559i \(0.747536\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 0.381966i − 0.0421811i
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.70820 0.831197
\(87\) 0 0
\(88\) − 5.85410i − 0.624049i
\(89\) 15.5623 1.64960 0.824801 − 0.565424i \(-0.191287\pi\)
0.824801 + 0.565424i \(0.191287\pi\)
\(90\) 0 0
\(91\) 8.05573 0.844470
\(92\) 3.09017i 0.322172i
\(93\) 0 0
\(94\) 8.61803 0.888882
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.1803i − 1.43980i −0.694080 − 0.719898i \(-0.744189\pi\)
0.694080 − 0.719898i \(-0.255811\pi\)
\(98\) 1.32624i 0.133970i
\(99\) 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 2250.2.c.g.1999.3 4
3.2 odd 2 750.2.c.a.499.1 4
5.2 odd 4 2250.2.a.a.1.2 2
5.3 odd 4 2250.2.a.p.1.1 2
5.4 even 2 inner 2250.2.c.g.1999.2 4
12.11 even 2 6000.2.f.k.1249.4 4
15.2 even 4 750.2.a.e.1.2 yes 2
15.8 even 4 750.2.a.d.1.1 ✓ 2
15.14 odd 2 750.2.c.a.499.4 4
60.23 odd 4 6000.2.a.a.1.2 2
60.47 odd 4 6000.2.a.bb.1.1 2
60.59 even 2 6000.2.f.k.1249.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
750.2.a.d.1.1 ✓ 2 15.8 even 4
750.2.a.e.1.2 yes 2 15.2 even 4
750.2.c.a.499.1 4 3.2 odd 2
750.2.c.a.499.4 4 15.14 odd 2
2250.2.a.a.1.2 2 5.2 odd 4
2250.2.a.p.1.1 2 5.3 odd 4
2250.2.c.g.1999.2 4 5.4 even 2 inner
2250.2.c.g.1999.3 4 1.1 even 1 trivial
6000.2.a.a.1.2 2 60.23 odd 4
6000.2.a.bb.1.1 2 60.47 odd 4
6000.2.f.k.1249.1 4 60.59 even 2
6000.2.f.k.1249.4 4 12.11 even 2