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Properties

Label 3330.2.d.p.1999.5
Level $3330$
Weight $2$
Character 3330.1999
Analytic conductor $26.590$
Analytic rank $0$
Dimension $10$
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] = [3330,2,Mod(1999,3330)] 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("3330.1999"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3330, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-10,-6,0,0,0,0,2,-6,0,0,-2] 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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.12837029094400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 51x^{6} - 124x^{5} + 154x^{4} - 46x^{3} + x^{2} + 4x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1999.5
Root \(1.51933 + 1.51933i\) of defining polynomial
Character \(\chi\) \(=\) 3330.1999
Dual form 3330.2.d.p.1999.10

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(1.42149 - 1.72608i) q^{5} -4.14336i q^{7} +1.00000i q^{8} +(-1.72608 - 1.42149i) q^{10} +4.76853 q^{11} +3.91744i q^{13} -4.14336 q^{14} +1.00000 q^{16} -3.31637i q^{17} +1.85109 q^{19} +(-1.42149 + 1.72608i) q^{20} -4.76853i q^{22} -1.54229i q^{23} +(-0.958719 - 4.90723i) q^{25} +3.91744 q^{26} +4.14336i q^{28} +8.87323 q^{29} +9.75286 q^{31} -1.00000i q^{32} -3.31637 q^{34} +(-7.15179 - 5.88976i) q^{35} +1.00000i q^{37} -1.85109i q^{38} +(1.72608 + 1.42149i) q^{40} +5.06977 q^{41} +9.99446i q^{43} -4.76853 q^{44} -1.54229 q^{46} -4.82700i q^{47} -10.1675 q^{49} +(-4.90723 + 0.958719i) q^{50} -3.91744i q^{52} -5.13611i q^{53} +(6.77843 - 8.23088i) q^{55} +4.14336 q^{56} -8.87323i q^{58} -1.05324 q^{59} -4.14422 q^{61} -9.75286i q^{62} -1.00000 q^{64} +(6.76182 + 5.56861i) q^{65} +1.00811i q^{67} +3.31637i q^{68} +(-5.88976 + 7.15179i) q^{70} -6.45248 q^{71} -10.7714i q^{73} +1.00000 q^{74} -1.85109 q^{76} -19.7578i q^{77} +1.19856 q^{79} +(1.42149 - 1.72608i) q^{80} -5.06977i q^{82} +10.6932i q^{83} +(-5.72432 - 4.71419i) q^{85} +9.99446 q^{86} +4.76853i q^{88} +7.29569 q^{89} +16.2314 q^{91} +1.54229i q^{92} -4.82700 q^{94} +(2.63131 - 3.19514i) q^{95} +14.8988i q^{97} +10.1675i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} - 6 q^{5} + 2 q^{10} - 6 q^{11} - 2 q^{14} + 10 q^{16} - 8 q^{19} + 6 q^{20} + 4 q^{25} + 12 q^{26} + 22 q^{29} + 46 q^{31} - 18 q^{34} - 32 q^{35} - 2 q^{40} + 14 q^{41} + 6 q^{44} + 12 q^{46}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) 1.42149 − 1.72608i 0.635711 − 0.771927i
\(6\) 0 0
\(7\) − 4.14336i − 1.56604i −0.621994 − 0.783022i \(-0.713677\pi\)
0.621994 − 0.783022i \(-0.286323\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −1.72608 − 1.42149i −0.545835 − 0.449515i
\(11\) 4.76853 1.43777 0.718883 − 0.695131i \(-0.244654\pi\)
0.718883 + 0.695131i \(0.244654\pi\)
\(12\) 0 0
\(13\) 3.91744i 1.08650i 0.839570 + 0.543251i \(0.182807\pi\)
−0.839570 + 0.543251i \(0.817193\pi\)
\(14\) −4.14336 −1.10736
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 3.31637i − 0.804337i −0.915566 − 0.402169i \(-0.868257\pi\)
0.915566 − 0.402169i \(-0.131743\pi\)
\(18\) 0 0
\(19\) 1.85109 0.424670 0.212335 − 0.977197i \(-0.431893\pi\)
0.212335 + 0.977197i \(0.431893\pi\)
\(20\) −1.42149 + 1.72608i −0.317855 + 0.385964i
\(21\) 0 0
\(22\) − 4.76853i − 1.01665i
\(23\) − 1.54229i − 0.321590i −0.986988 − 0.160795i \(-0.948594\pi\)
0.986988 − 0.160795i \(-0.0514058\pi\)
\(24\) 0 0
\(25\) −0.958719 − 4.90723i −0.191744 − 0.981445i
\(26\) 3.91744 0.768273
\(27\) 0 0
\(28\) 4.14336i 0.783022i
\(29\) 8.87323 1.64772 0.823859 − 0.566795i \(-0.191817\pi\)
0.823859 + 0.566795i \(0.191817\pi\)
\(30\) 0 0
\(31\) 9.75286 1.75166 0.875832 − 0.482615i \(-0.160313\pi\)
0.875832 + 0.482615i \(0.160313\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −3.31637 −0.568752
\(35\) −7.15179 − 5.88976i −1.20887 − 0.995551i
\(36\) 0 0
\(37\) 1.00000i 0.164399i
\(38\) − 1.85109i − 0.300287i
\(39\) 0 0
\(40\) 1.72608 + 1.42149i 0.272918 + 0.224758i
\(41\) 5.06977 0.791764 0.395882 − 0.918301i \(-0.370439\pi\)
0.395882 + 0.918301i \(0.370439\pi\)
\(42\) 0 0
\(43\) 9.99446i 1.52414i 0.647494 + 0.762070i \(0.275817\pi\)
−0.647494 + 0.762070i \(0.724183\pi\)
\(44\) −4.76853 −0.718883
\(45\) 0 0
\(46\) −1.54229 −0.227399
\(47\) − 4.82700i − 0.704090i −0.935983 − 0.352045i \(-0.885486\pi\)
0.935983 − 0.352045i \(-0.114514\pi\)
\(48\) 0 0
\(49\) −10.1675 −1.45249
\(50\) −4.90723 + 0.958719i −0.693986 + 0.135583i
\(51\) 0 0
\(52\) − 3.91744i − 0.543251i
\(53\) − 5.13611i − 0.705499i −0.935718 − 0.352750i \(-0.885247\pi\)
0.935718 − 0.352750i \(-0.114753\pi\)
\(54\) 0 0
\(55\) 6.77843 − 8.23088i 0.914003 − 1.10985i
\(56\) 4.14336 0.553680
\(57\) 0 0
\(58\) − 8.87323i − 1.16511i
\(59\) −1.05324 −0.137120 −0.0685598 − 0.997647i \(-0.521840\pi\)
−0.0685598 + 0.997647i \(0.521840\pi\)
\(60\) 0 0
\(61\) −4.14422 −0.530613 −0.265306 − 0.964164i \(-0.585473\pi\)
−0.265306 + 0.964164i \(0.585473\pi\)
\(62\) − 9.75286i − 1.23861i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 6.76182 + 5.56861i 0.838701 + 0.690701i
\(66\) 0 0
\(67\) 1.00811i 0.123160i 0.998102 + 0.0615800i \(0.0196139\pi\)
−0.998102 + 0.0615800i \(0.980386\pi\)
\(68\) 3.31637i 0.402169i
\(69\) 0 0
\(70\) −5.88976 + 7.15179i −0.703961 + 0.854802i
\(71\) −6.45248 −0.765768 −0.382884 − 0.923796i \(-0.625069\pi\)
−0.382884 + 0.923796i \(0.625069\pi\)
\(72\) 0 0
\(73\) − 10.7714i − 1.26070i −0.776312 − 0.630349i \(-0.782912\pi\)
0.776312 − 0.630349i \(-0.217088\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −1.85109 −0.212335
\(77\) − 19.7578i − 2.25161i
\(78\) 0 0
\(79\) 1.19856 0.134849 0.0674243 − 0.997724i \(-0.478522\pi\)
0.0674243 + 0.997724i \(0.478522\pi\)
\(80\) 1.42149 − 1.72608i 0.158928 − 0.192982i
\(81\) 0 0
\(82\) − 5.06977i − 0.559862i
\(83\) 10.6932i 1.17373i 0.809683 + 0.586867i \(0.199639\pi\)
−0.809683 + 0.586867i \(0.800361\pi\)
\(84\) 0 0
\(85\) −5.72432 − 4.71419i −0.620890 − 0.511326i
\(86\) 9.99446 1.07773
\(87\) 0 0
\(88\) 4.76853i 0.508327i
\(89\) 7.29569 0.773342 0.386671 − 0.922218i \(-0.373625\pi\)
0.386671 + 0.922218i \(0.373625\pi\)
\(90\) 0 0
\(91\) 16.2314 1.70151
\(92\) 1.54229i 0.160795i
\(93\) 0 0
\(94\) −4.82700 −0.497867
\(95\) 2.63131 − 3.19514i 0.269967 − 0.327814i
\(96\) 0 0
\(97\) 14.8988i 1.51274i 0.654143 + 0.756371i \(0.273030\pi\)
−0.654143 + 0.756371i \(0.726970\pi\)
\(98\) 10.1675i 1.02707i
\(99\) 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 3330.2.d.p.1999.5 10
3.2 odd 2 370.2.b.d.149.10 yes 10
5.4 even 2 inner 3330.2.d.p.1999.10 10
15.2 even 4 1850.2.a.bd.1.5 5
15.8 even 4 1850.2.a.be.1.1 5
15.14 odd 2 370.2.b.d.149.1 ✓ 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.1 ✓ 10 15.14 odd 2
370.2.b.d.149.10 yes 10 3.2 odd 2
1850.2.a.bd.1.5 5 15.2 even 4
1850.2.a.be.1.1 5 15.8 even 4
3330.2.d.p.1999.5 10 1.1 even 1 trivial
3330.2.d.p.1999.10 10 5.4 even 2 inner