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Properties

Label 1900.2.i.g.501.5
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 501.5
Root \(0.226426 - 0.392182i\) of defining polynomial
Character \(\chi\) \(=\) 1900.501
Dual form 1900.2.i.g.201.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.226426 - 0.392182i) q^{3} +2.54366 q^{7} +(1.39746 - 2.42048i) q^{9} +2.22377 q^{11} +(3.51096 - 6.08116i) q^{13} +(1.27878 + 2.21492i) q^{17} +(-2.70498 - 3.41805i) q^{19} +(-0.575952 - 0.997578i) q^{21} +(-4.01448 + 6.95328i) q^{23} -2.62425 q^{27} +(0.941734 - 1.63113i) q^{29} +5.98111 q^{31} +(-0.503519 - 0.872121i) q^{33} -2.86105 q^{37} -3.17989 q^{39} +(-3.67524 - 6.36571i) q^{41} +(-1.84706 - 3.19919i) q^{43} +(-2.36448 + 4.09540i) q^{47} -0.529782 q^{49} +(0.579100 - 1.00303i) q^{51} +(-5.14549 + 8.91226i) q^{53} +(-0.728020 + 1.83478i) q^{57} +(3.73666 + 6.47208i) q^{59} +(4.17839 - 7.23719i) q^{61} +(3.55467 - 6.15687i) q^{63} +(6.17997 - 10.7040i) q^{67} +3.63593 q^{69} +(-4.13931 - 7.16950i) q^{71} +(6.32134 + 10.9489i) q^{73} +5.65651 q^{77} +(2.13067 + 3.69043i) q^{79} +(-3.59819 - 6.23225i) q^{81} +14.7613 q^{83} -0.852933 q^{87} +(7.19403 - 12.4604i) q^{89} +(8.93069 - 15.4684i) q^{91} +(-1.35428 - 2.34568i) q^{93} +(2.91488 + 5.04871i) q^{97} +(3.10763 - 5.38258i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) 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
\(3\) −0.226426 − 0.392182i −0.130727 − 0.226426i 0.793230 − 0.608922i \(-0.208398\pi\)
−0.923957 + 0.382496i \(0.875065\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.54366 0.961414 0.480707 − 0.876881i \(-0.340380\pi\)
0.480707 + 0.876881i \(0.340380\pi\)
\(8\) 0 0
\(9\) 1.39746 − 2.42048i 0.465821 − 0.806825i
\(10\) 0 0
\(11\) 2.22377 0.670491 0.335246 − 0.942131i \(-0.391181\pi\)
0.335246 + 0.942131i \(0.391181\pi\)
\(12\) 0 0
\(13\) 3.51096 − 6.08116i 0.973765 − 1.68661i 0.289812 − 0.957084i \(-0.406407\pi\)
0.683953 − 0.729526i \(-0.260259\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.27878 + 2.21492i 0.310151 + 0.537197i 0.978395 − 0.206745i \(-0.0662872\pi\)
−0.668244 + 0.743942i \(0.732954\pi\)
\(18\) 0 0
\(19\) −2.70498 − 3.41805i −0.620565 − 0.784155i
\(20\) 0 0
\(21\) −0.575952 − 0.997578i −0.125683 − 0.217689i
\(22\) 0 0
\(23\) −4.01448 + 6.95328i −0.837076 + 1.44986i 0.0552521 + 0.998472i \(0.482404\pi\)
−0.892329 + 0.451387i \(0.850930\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.62425 −0.505036
\(28\) 0 0
\(29\) 0.941734 − 1.63113i 0.174876 − 0.302893i −0.765243 − 0.643742i \(-0.777381\pi\)
0.940118 + 0.340849i \(0.110714\pi\)
\(30\) 0 0
\(31\) 5.98111 1.07424 0.537120 − 0.843506i \(-0.319512\pi\)
0.537120 + 0.843506i \(0.319512\pi\)
\(32\) 0 0
\(33\) −0.503519 − 0.872121i −0.0876515 − 0.151817i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.86105 −0.470353 −0.235177 − 0.971953i \(-0.575567\pi\)
−0.235177 + 0.971953i \(0.575567\pi\)
\(38\) 0 0
\(39\) −3.17989 −0.509190
\(40\) 0 0
\(41\) −3.67524 − 6.36571i −0.573977 − 0.994157i −0.996152 − 0.0876426i \(-0.972067\pi\)
0.422175 − 0.906514i \(-0.361267\pi\)
\(42\) 0 0
\(43\) −1.84706 − 3.19919i −0.281673 − 0.487873i 0.690124 − 0.723691i \(-0.257556\pi\)
−0.971797 + 0.235819i \(0.924223\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.36448 + 4.09540i −0.344895 + 0.597376i −0.985335 − 0.170633i \(-0.945419\pi\)
0.640440 + 0.768008i \(0.278752\pi\)
\(48\) 0 0
\(49\) −0.529782 −0.0756832
\(50\) 0 0
\(51\) 0.579100 − 1.00303i 0.0810903 − 0.140453i
\(52\) 0 0
\(53\) −5.14549 + 8.91226i −0.706788 + 1.22419i 0.259255 + 0.965809i \(0.416523\pi\)
−0.966042 + 0.258383i \(0.916810\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.728020 + 1.83478i −0.0964286 + 0.243023i
\(58\) 0 0
\(59\) 3.73666 + 6.47208i 0.486472 + 0.842593i 0.999879 − 0.0155515i \(-0.00495040\pi\)
−0.513408 + 0.858145i \(0.671617\pi\)
\(60\) 0 0
\(61\) 4.17839 − 7.23719i 0.534988 − 0.926627i −0.464176 − 0.885743i \(-0.653649\pi\)
0.999164 − 0.0408838i \(-0.0130174\pi\)
\(62\) 0 0
\(63\) 3.55467 − 6.15687i 0.447847 − 0.775693i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.17997 − 10.7040i 0.755004 − 1.30771i −0.190368 − 0.981713i \(-0.560968\pi\)
0.945372 − 0.325993i \(-0.105698\pi\)
\(68\) 0 0
\(69\) 3.63593 0.437715
\(70\) 0 0
\(71\) −4.13931 − 7.16950i −0.491246 − 0.850863i 0.508703 − 0.860942i \(-0.330125\pi\)
−0.999949 + 0.0100790i \(0.996792\pi\)
\(72\) 0 0
\(73\) 6.32134 + 10.9489i 0.739857 + 1.28147i 0.952559 + 0.304352i \(0.0984401\pi\)
−0.212703 + 0.977117i \(0.568227\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.65651 0.644620
\(78\) 0 0
\(79\) 2.13067 + 3.69043i 0.239719 + 0.415206i 0.960634 − 0.277818i \(-0.0896112\pi\)
−0.720914 + 0.693024i \(0.756278\pi\)
\(80\) 0 0
\(81\) −3.59819 − 6.23225i −0.399799 − 0.692472i
\(82\) 0 0
\(83\) 14.7613 1.62026 0.810132 − 0.586248i \(-0.199396\pi\)
0.810132 + 0.586248i \(0.199396\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.852933 −0.0914440
\(88\) 0 0
\(89\) 7.19403 − 12.4604i 0.762566 − 1.32080i −0.178958 − 0.983857i \(-0.557273\pi\)
0.941524 − 0.336946i \(-0.109394\pi\)
\(90\) 0 0
\(91\) 8.93069 − 15.4684i 0.936191 − 1.62153i
\(92\) 0 0
\(93\) −1.35428 − 2.34568i −0.140432 − 0.243236i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.91488 + 5.04871i 0.295961 + 0.512619i 0.975208 − 0.221291i \(-0.0710269\pi\)
−0.679247 + 0.733910i \(0.737694\pi\)
\(98\) 0 0
\(99\) 3.10763 − 5.38258i 0.312329 − 0.540969i
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.501.5 20
5.2 odd 4 380.2.r.a.349.5 yes 20
5.3 odd 4 380.2.r.a.349.6 yes 20
5.4 even 2 inner 1900.2.i.g.501.6 20
15.2 even 4 3420.2.bj.c.2629.3 20
15.8 even 4 3420.2.bj.c.2629.5 20
19.11 even 3 inner 1900.2.i.g.201.5 20
95.49 even 6 inner 1900.2.i.g.201.6 20
95.68 odd 12 380.2.r.a.49.5 ✓ 20
95.87 odd 12 380.2.r.a.49.6 yes 20
285.68 even 12 3420.2.bj.c.1189.3 20
285.182 even 12 3420.2.bj.c.1189.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.5 ✓ 20 95.68 odd 12
380.2.r.a.49.6 yes 20 95.87 odd 12
380.2.r.a.349.5 yes 20 5.2 odd 4
380.2.r.a.349.6 yes 20 5.3 odd 4
1900.2.i.g.201.5 20 19.11 even 3 inner
1900.2.i.g.201.6 20 95.49 even 6 inner
1900.2.i.g.501.5 20 1.1 even 1 trivial
1900.2.i.g.501.6 20 5.4 even 2 inner
3420.2.bj.c.1189.3 20 285.68 even 12
3420.2.bj.c.1189.5 20 285.182 even 12
3420.2.bj.c.2629.3 20 15.2 even 4
3420.2.bj.c.2629.5 20 15.8 even 4