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Properties

Label 1900.2.i.g.501.10
Level $1900$
Weight $2$
Character 1900.501
Analytic conductor $15.172$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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] = [1900,2,Mod(201,1900)] 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("1900.201"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 20 x^{18} + 261 x^{16} + 1994 x^{14} + 11074 x^{12} + 39211 x^{10} + 99376 x^{8} + 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 501.10
Root \(-1.43632 + 2.48777i\) of defining polynomial
Character \(\chi\) \(=\) 1900.501
Dual form 1900.2.i.g.201.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.43632 + 2.48777i) q^{3} -3.54568 q^{7} +(-2.62601 + 4.54838i) q^{9} -1.81575 q^{11} +(1.60681 - 2.78308i) q^{13} +(-3.99638 - 6.92193i) q^{17} +(-0.863760 - 4.27246i) q^{19} +(-5.09271 - 8.82084i) q^{21} +(-4.21480 + 7.30026i) q^{23} -6.46921 q^{27} +(4.29124 - 7.43265i) q^{29} -1.70874 q^{31} +(-2.60799 - 4.51718i) q^{33} -5.50608 q^{37} +9.23155 q^{39} +(4.05694 + 7.02683i) q^{41} +(-2.51363 - 4.35373i) q^{43} +(-0.674543 + 1.16834i) q^{47} +5.57183 q^{49} +(11.4801 - 19.8842i) q^{51} +(-1.10967 + 1.92201i) q^{53} +(9.38828 - 8.28544i) q^{57} +(-0.960774 - 1.66411i) q^{59} +(2.83047 - 4.90251i) q^{61} +(9.31098 - 16.1271i) q^{63} +(-4.64397 + 8.04360i) q^{67} -24.2152 q^{69} +(-2.94365 - 5.09854i) q^{71} +(1.63226 + 2.82716i) q^{73} +6.43807 q^{77} +(-2.08739 - 3.61546i) q^{79} +(-1.41381 - 2.44879i) q^{81} -6.30268 q^{83} +24.6543 q^{87} +(-2.73646 + 4.73968i) q^{89} +(-5.69723 + 9.86789i) q^{91} +(-2.45429 - 4.25096i) q^{93} +(3.99096 + 6.91255i) q^{97} +(4.76818 - 8.25873i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.43632 + 2.48777i 0.829257 + 1.43632i 0.898622 + 0.438725i \(0.144570\pi\)
−0.0693641 + 0.997591i \(0.522097\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.54568 −1.34014 −0.670070 − 0.742298i \(-0.733736\pi\)
−0.670070 + 0.742298i \(0.733736\pi\)
\(8\) 0 0
\(9\) −2.62601 + 4.54838i −0.875336 + 1.51613i
\(10\) 0 0
\(11\) −1.81575 −0.547470 −0.273735 − 0.961805i \(-0.588259\pi\)
−0.273735 + 0.961805i \(0.588259\pi\)
\(12\) 0 0
\(13\) 1.60681 − 2.78308i 0.445649 − 0.771887i −0.552448 − 0.833547i \(-0.686306\pi\)
0.998097 + 0.0616606i \(0.0196396\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.99638 − 6.92193i −0.969264 − 1.67881i −0.697694 − 0.716396i \(-0.745790\pi\)
−0.271570 − 0.962419i \(-0.587543\pi\)
\(18\) 0 0
\(19\) −0.863760 − 4.27246i −0.198160 − 0.980170i
\(20\) 0 0
\(21\) −5.09271 − 8.82084i −1.11132 − 1.92486i
\(22\) 0 0
\(23\) −4.21480 + 7.30026i −0.878848 + 1.52221i −0.0262406 + 0.999656i \(0.508354\pi\)
−0.852607 + 0.522553i \(0.824980\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −6.46921 −1.24500
\(28\) 0 0
\(29\) 4.29124 − 7.43265i 0.796863 − 1.38021i −0.124786 − 0.992184i \(-0.539824\pi\)
0.921649 − 0.388024i \(-0.126842\pi\)
\(30\) 0 0
\(31\) −1.70874 −0.306899 −0.153450 − 0.988156i \(-0.549038\pi\)
−0.153450 + 0.988156i \(0.549038\pi\)
\(32\) 0 0
\(33\) −2.60799 − 4.51718i −0.453994 − 0.786340i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.50608 −0.905193 −0.452597 − 0.891715i \(-0.649502\pi\)
−0.452597 + 0.891715i \(0.649502\pi\)
\(38\) 0 0
\(39\) 9.23155 1.47823
\(40\) 0 0
\(41\) 4.05694 + 7.02683i 0.633588 + 1.09741i 0.986812 + 0.161868i \(0.0517520\pi\)
−0.353224 + 0.935539i \(0.614915\pi\)
\(42\) 0 0
\(43\) −2.51363 − 4.35373i −0.383325 − 0.663938i 0.608211 − 0.793776i \(-0.291888\pi\)
−0.991535 + 0.129838i \(0.958554\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.674543 + 1.16834i −0.0983922 + 0.170420i −0.911019 − 0.412364i \(-0.864703\pi\)
0.812627 + 0.582784i \(0.198037\pi\)
\(48\) 0 0
\(49\) 5.57183 0.795976
\(50\) 0 0
\(51\) 11.4801 − 19.8842i 1.60754 − 2.78434i
\(52\) 0 0
\(53\) −1.10967 + 1.92201i −0.152426 + 0.264009i −0.932119 − 0.362153i \(-0.882042\pi\)
0.779693 + 0.626162i \(0.215375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.38828 − 8.28544i 1.24351 − 1.09743i
\(58\) 0 0
\(59\) −0.960774 − 1.66411i −0.125082 − 0.216649i 0.796683 − 0.604398i \(-0.206586\pi\)
−0.921765 + 0.387749i \(0.873253\pi\)
\(60\) 0 0
\(61\) 2.83047 − 4.90251i 0.362404 − 0.627702i −0.625952 − 0.779862i \(-0.715289\pi\)
0.988356 + 0.152159i \(0.0486227\pi\)
\(62\) 0 0
\(63\) 9.31098 − 16.1271i 1.17307 − 2.03182i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.64397 + 8.04360i −0.567352 + 0.982682i 0.429475 + 0.903079i \(0.358699\pi\)
−0.996827 + 0.0796032i \(0.974635\pi\)
\(68\) 0 0
\(69\) −24.2152 −2.91516
\(70\) 0 0
\(71\) −2.94365 − 5.09854i −0.349346 − 0.605086i 0.636787 − 0.771040i \(-0.280263\pi\)
−0.986134 + 0.165954i \(0.946930\pi\)
\(72\) 0 0
\(73\) 1.63226 + 2.82716i 0.191042 + 0.330894i 0.945596 − 0.325344i \(-0.105480\pi\)
−0.754554 + 0.656238i \(0.772147\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.43807 0.733687
\(78\) 0 0
\(79\) −2.08739 − 3.61546i −0.234850 − 0.406771i 0.724379 − 0.689402i \(-0.242126\pi\)
−0.959229 + 0.282630i \(0.908793\pi\)
\(80\) 0 0
\(81\) −1.41381 − 2.44879i −0.157090 − 0.272087i
\(82\) 0 0
\(83\) −6.30268 −0.691809 −0.345905 − 0.938270i \(-0.612428\pi\)
−0.345905 + 0.938270i \(0.612428\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 24.6543 2.64322
\(88\) 0 0
\(89\) −2.73646 + 4.73968i −0.290064 + 0.502405i −0.973825 − 0.227301i \(-0.927010\pi\)
0.683761 + 0.729706i \(0.260343\pi\)
\(90\) 0 0
\(91\) −5.69723 + 9.86789i −0.597232 + 1.03444i
\(92\) 0 0
\(93\) −2.45429 − 4.25096i −0.254499 − 0.440804i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.99096 + 6.91255i 0.405221 + 0.701863i 0.994347 − 0.106178i \(-0.0338613\pi\)
−0.589126 + 0.808041i \(0.700528\pi\)
\(98\) 0 0
\(99\) 4.76818 − 8.25873i 0.479220 − 0.830034i
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 1900.2.i.g.501.10 20
5.2 odd 4 380.2.r.a.349.10 yes 20
5.3 odd 4 380.2.r.a.349.1 yes 20
5.4 even 2 inner 1900.2.i.g.501.1 20
15.2 even 4 3420.2.bj.c.2629.10 20
15.8 even 4 3420.2.bj.c.2629.4 20
19.11 even 3 inner 1900.2.i.g.201.10 20
95.49 even 6 inner 1900.2.i.g.201.1 20
95.68 odd 12 380.2.r.a.49.10 yes 20
95.87 odd 12 380.2.r.a.49.1 ✓ 20
285.68 even 12 3420.2.bj.c.1189.10 20
285.182 even 12 3420.2.bj.c.1189.4 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.1 ✓ 20 95.87 odd 12
380.2.r.a.49.10 yes 20 95.68 odd 12
380.2.r.a.349.1 yes 20 5.3 odd 4
380.2.r.a.349.10 yes 20 5.2 odd 4
1900.2.i.g.201.1 20 95.49 even 6 inner
1900.2.i.g.201.10 20 19.11 even 3 inner
1900.2.i.g.501.1 20 5.4 even 2 inner
1900.2.i.g.501.10 20 1.1 even 1 trivial
3420.2.bj.c.1189.4 20 285.182 even 12
3420.2.bj.c.1189.10 20 285.68 even 12
3420.2.bj.c.2629.4 20 15.8 even 4
3420.2.bj.c.2629.10 20 15.2 even 4