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Properties

Label 435.2.f.c.289.1
Level $435$
Weight $2$
Character 435.289
Analytic conductor $3.473$
Analytic rank $0$
Dimension $2$
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] = [435,2,Mod(289,435)] 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("435.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(435, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,2] 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: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 435.289
Dual form 435.2.f.c.289.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +(-1.00000 - 2.00000i) q^{5} +1.00000 q^{6} -2.00000i q^{7} -3.00000 q^{8} +1.00000 q^{9} +(-1.00000 - 2.00000i) q^{10} -2.00000i q^{11} -1.00000 q^{12} -4.00000i q^{13} -2.00000i q^{14} +(-1.00000 - 2.00000i) q^{15} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -2.00000i q^{19} +(1.00000 + 2.00000i) q^{20} -2.00000i q^{21} -2.00000i q^{22} +6.00000i q^{23} -3.00000 q^{24} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{26} +1.00000 q^{27} +2.00000i q^{28} +(-5.00000 - 2.00000i) q^{29} +(-1.00000 - 2.00000i) q^{30} +2.00000i q^{31} +5.00000 q^{32} -2.00000i q^{33} +6.00000 q^{34} +(-4.00000 + 2.00000i) q^{35} -1.00000 q^{36} +2.00000 q^{37} -2.00000i q^{38} -4.00000i q^{39} +(3.00000 + 6.00000i) q^{40} -2.00000i q^{42} -4.00000 q^{43} +2.00000i q^{44} +(-1.00000 - 2.00000i) q^{45} +6.00000i q^{46} +8.00000 q^{47} -1.00000 q^{48} +3.00000 q^{49} +(-3.00000 + 4.00000i) q^{50} +6.00000 q^{51} +4.00000i q^{52} +12.0000i q^{53} +1.00000 q^{54} +(-4.00000 + 2.00000i) q^{55} +6.00000i q^{56} -2.00000i q^{57} +(-5.00000 - 2.00000i) q^{58} -4.00000 q^{59} +(1.00000 + 2.00000i) q^{60} -12.0000i q^{61} +2.00000i q^{62} -2.00000i q^{63} +7.00000 q^{64} +(-8.00000 + 4.00000i) q^{65} -2.00000i q^{66} -6.00000i q^{67} -6.00000 q^{68} +6.00000i q^{69} +(-4.00000 + 2.00000i) q^{70} +8.00000 q^{71} -3.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +(-3.00000 + 4.00000i) q^{75} +2.00000i q^{76} -4.00000 q^{77} -4.00000i q^{78} +2.00000i q^{79} +(1.00000 + 2.00000i) q^{80} +1.00000 q^{81} +2.00000i q^{83} +2.00000i q^{84} +(-6.00000 - 12.0000i) q^{85} -4.00000 q^{86} +(-5.00000 - 2.00000i) q^{87} +6.00000i q^{88} +16.0000i q^{89} +(-1.00000 - 2.00000i) q^{90} -8.00000 q^{91} -6.00000i q^{92} +2.00000i q^{93} +8.00000 q^{94} +(-4.00000 + 2.00000i) q^{95} +5.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{6} - 6 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 2 q^{15} - 2 q^{16} + 12 q^{17} + 2 q^{18} + 2 q^{20} - 6 q^{24} - 6 q^{25} + 2 q^{27} - 10 q^{29} - 2 q^{30}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/435\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(146\) \(262\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) −1.00000 − 2.00000i −0.447214 − 0.894427i
\(6\) 1.00000 0.408248
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −1.00000 − 2.00000i −0.316228 − 0.632456i
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) −1.00000 −0.288675
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) − 2.00000i − 0.534522i
\(15\) −1.00000 − 2.00000i −0.258199 − 0.516398i
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 1.00000 + 2.00000i 0.223607 + 0.447214i
\(21\) − 2.00000i − 0.436436i
\(22\) − 2.00000i − 0.426401i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) −3.00000 −0.612372
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) − 4.00000i − 0.784465i
\(27\) 1.00000 0.192450
\(28\) 2.00000i 0.377964i
\(29\) −5.00000 − 2.00000i −0.928477 − 0.371391i
\(30\) −1.00000 − 2.00000i −0.182574 − 0.365148i
\(31\) 2.00000i 0.359211i 0.983739 + 0.179605i \(0.0574821\pi\)
−0.983739 + 0.179605i \(0.942518\pi\)
\(32\) 5.00000 0.883883
\(33\) − 2.00000i − 0.348155i
\(34\) 6.00000 1.02899
\(35\) −4.00000 + 2.00000i −0.676123 + 0.338062i
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) − 4.00000i − 0.640513i
\(40\) 3.00000 + 6.00000i 0.474342 + 0.948683i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000i 0.301511i
\(45\) −1.00000 − 2.00000i −0.149071 − 0.298142i
\(46\) 6.00000i 0.884652i
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) −3.00000 + 4.00000i −0.424264 + 0.565685i
\(51\) 6.00000 0.840168
\(52\) 4.00000i 0.554700i
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 + 2.00000i −0.539360 + 0.269680i
\(56\) 6.00000i 0.801784i
\(57\) − 2.00000i − 0.264906i
\(58\) −5.00000 − 2.00000i −0.656532 − 0.262613i
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 + 2.00000i 0.129099 + 0.258199i
\(61\) − 12.0000i − 1.53644i −0.640184 − 0.768221i \(-0.721142\pi\)
0.640184 − 0.768221i \(-0.278858\pi\)
\(62\) 2.00000i 0.254000i
\(63\) − 2.00000i − 0.251976i
\(64\) 7.00000 0.875000
\(65\) −8.00000 + 4.00000i −0.992278 + 0.496139i
\(66\) − 2.00000i − 0.246183i
\(67\) − 6.00000i − 0.733017i −0.930415 − 0.366508i \(-0.880553\pi\)
0.930415 − 0.366508i \(-0.119447\pi\)
\(68\) −6.00000 −0.727607
\(69\) 6.00000i 0.722315i
\(70\) −4.00000 + 2.00000i −0.478091 + 0.239046i
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −3.00000 −0.353553
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 2.00000 0.232495
\(75\) −3.00000 + 4.00000i −0.346410 + 0.461880i
\(76\) 2.00000i 0.229416i
\(77\) −4.00000 −0.455842
\(78\) − 4.00000i − 0.452911i
\(79\) 2.00000i 0.225018i 0.993651 + 0.112509i \(0.0358886\pi\)
−0.993651 + 0.112509i \(0.964111\pi\)
\(80\) 1.00000 + 2.00000i 0.111803 + 0.223607i
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 2.00000i 0.218218i
\(85\) −6.00000 − 12.0000i −0.650791 − 1.30158i
\(86\) −4.00000 −0.431331
\(87\) −5.00000 − 2.00000i −0.536056 − 0.214423i
\(88\) 6.00000i 0.639602i
\(89\) 16.0000i 1.69600i 0.529999 + 0.847998i \(0.322192\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −1.00000 − 2.00000i −0.105409 − 0.210819i
\(91\) −8.00000 −0.838628
\(92\) − 6.00000i − 0.625543i
\(93\) 2.00000i 0.207390i
\(94\) 8.00000 0.825137
\(95\) −4.00000 + 2.00000i −0.410391 + 0.205196i
\(96\) 5.00000 0.510310
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 3.00000 0.303046
\(99\) − 2.00000i − 0.201008i
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 435.2.f.c.289.1 yes 2
3.2 odd 2 1305.2.f.b.289.2 2
5.2 odd 4 2175.2.d.d.376.2 2
5.3 odd 4 2175.2.d.c.376.1 2
5.4 even 2 435.2.f.b.289.2 yes 2
15.14 odd 2 1305.2.f.c.289.1 2
29.28 even 2 435.2.f.b.289.1 ✓ 2
87.86 odd 2 1305.2.f.c.289.2 2
145.28 odd 4 2175.2.d.c.376.2 2
145.57 odd 4 2175.2.d.d.376.1 2
145.144 even 2 inner 435.2.f.c.289.2 yes 2
435.434 odd 2 1305.2.f.b.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.f.b.289.1 ✓ 2 29.28 even 2
435.2.f.b.289.2 yes 2 5.4 even 2
435.2.f.c.289.1 yes 2 1.1 even 1 trivial
435.2.f.c.289.2 yes 2 145.144 even 2 inner
1305.2.f.b.289.1 2 435.434 odd 2
1305.2.f.b.289.2 2 3.2 odd 2
1305.2.f.c.289.1 2 15.14 odd 2
1305.2.f.c.289.2 2 87.86 odd 2
2175.2.d.c.376.1 2 5.3 odd 4
2175.2.d.c.376.2 2 145.28 odd 4
2175.2.d.d.376.1 2 145.57 odd 4
2175.2.d.d.376.2 2 5.2 odd 4