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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [435,2,Mod(349,435)] 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("435.349"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 435.349
Dual form 435.2.c.b.349.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +1.00000 q^{6} -2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +(2.00000 + 1.00000i) q^{10} -1.00000i q^{12} -4.00000i q^{13} +2.00000 q^{14} +(-2.00000 - 1.00000i) q^{15} -1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} +(1.00000 - 2.00000i) q^{20} -2.00000 q^{21} -2.00000i q^{23} +3.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} -1.00000 q^{29} +(1.00000 - 2.00000i) q^{30} +4.00000 q^{31} +5.00000i q^{32} -2.00000 q^{34} +(-4.00000 - 2.00000i) q^{35} -1.00000 q^{36} +2.00000i q^{37} -4.00000 q^{39} +(6.00000 + 3.00000i) q^{40} +10.0000 q^{41} -2.00000i q^{42} +(-1.00000 + 2.00000i) q^{45} +2.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +3.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +2.00000 q^{51} -4.00000i q^{52} +12.0000i q^{53} -1.00000 q^{54} +6.00000 q^{56} -1.00000i q^{58} -4.00000 q^{59} +(-2.00000 - 1.00000i) q^{60} +2.00000 q^{61} +4.00000i q^{62} +2.00000i q^{63} -7.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} +2.00000i q^{67} +2.00000i q^{68} -2.00000 q^{69} +(2.00000 - 4.00000i) q^{70} -8.00000 q^{71} -3.00000i q^{72} -14.0000i q^{73} -2.00000 q^{74} +(-4.00000 + 3.00000i) q^{75} -4.00000i q^{78} -8.00000 q^{79} +(-1.00000 + 2.00000i) q^{80} +1.00000 q^{81} +10.0000i q^{82} +6.00000i q^{83} -2.00000 q^{84} +(4.00000 + 2.00000i) q^{85} +1.00000i q^{87} -10.0000 q^{89} +(-2.00000 - 1.00000i) q^{90} -8.00000 q^{91} -2.00000i q^{92} -4.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} +10.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{9} + 4 q^{10} + 4 q^{14} - 4 q^{15} - 2 q^{16} + 2 q^{20} - 4 q^{21} + 6 q^{24} - 6 q^{25} + 8 q^{26} - 2 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{34} - 8 q^{35} - 2 q^{36}+ \cdots + 10 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 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 \(-0.123376\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 2.00000 + 1.00000i 0.632456 + 0.316228i
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 2.00000 0.534522
\(15\) −2.00000 − 1.00000i −0.516398 − 0.258199i
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 − 2.00000i 0.223607 − 0.447214i
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) − 2.00000i − 0.417029i −0.978019 − 0.208514i \(-0.933137\pi\)
0.978019 − 0.208514i \(-0.0668628\pi\)
\(24\) 3.00000 0.612372
\(25\) −3.00000 − 4.00000i −0.600000 − 0.800000i
\(26\) 4.00000 0.784465
\(27\) 1.00000i 0.192450i
\(28\) − 2.00000i − 0.377964i
\(29\) −1.00000 −0.185695
\(30\) 1.00000 − 2.00000i 0.182574 − 0.365148i
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −4.00000 − 2.00000i −0.676123 − 0.338062i
\(36\) −1.00000 −0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 6.00000 + 3.00000i 0.948683 + 0.474342i
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −1.00000 + 2.00000i −0.149071 + 0.298142i
\(46\) 2.00000 0.294884
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 3.00000 0.428571
\(50\) 4.00000 − 3.00000i 0.565685 − 0.424264i
\(51\) 2.00000 0.280056
\(52\) − 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
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) 0 0
\(58\) − 1.00000i − 0.131306i
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −2.00000 − 1.00000i −0.258199 − 0.129099i
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 2.00000i 0.251976i
\(64\) −7.00000 −0.875000
\(65\) −8.00000 − 4.00000i −0.992278 − 0.496139i
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −2.00000 −0.240772
\(70\) 2.00000 − 4.00000i 0.239046 − 0.478091i
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 14.0000i − 1.63858i −0.573382 − 0.819288i \(-0.694369\pi\)
0.573382 − 0.819288i \(-0.305631\pi\)
\(74\) −2.00000 −0.232495
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 0 0
\(77\) 0 0
\(78\) − 4.00000i − 0.452911i
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 + 2.00000i −0.111803 + 0.223607i
\(81\) 1.00000 0.111111
\(82\) 10.0000i 1.10432i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) −2.00000 −0.218218
\(85\) 4.00000 + 2.00000i 0.433861 + 0.216930i
\(86\) 0 0
\(87\) 1.00000i 0.107211i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 − 1.00000i −0.210819 − 0.105409i
\(91\) −8.00000 −0.838628
\(92\) − 2.00000i − 0.208514i
\(93\) − 4.00000i − 0.414781i
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 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 435.2.c.b.349.2 yes 2
3.2 odd 2 1305.2.c.b.784.1 2
5.2 odd 4 2175.2.a.c.1.1 1
5.3 odd 4 2175.2.a.i.1.1 1
5.4 even 2 inner 435.2.c.b.349.1 ✓ 2
15.2 even 4 6525.2.a.k.1.1 1
15.8 even 4 6525.2.a.c.1.1 1
15.14 odd 2 1305.2.c.b.784.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.b.349.1 ✓ 2 5.4 even 2 inner
435.2.c.b.349.2 yes 2 1.1 even 1 trivial
1305.2.c.b.784.1 2 3.2 odd 2
1305.2.c.b.784.2 2 15.14 odd 2
2175.2.a.c.1.1 1 5.2 odd 4
2175.2.a.i.1.1 1 5.3 odd 4
6525.2.a.c.1.1 1 15.8 even 4
6525.2.a.k.1.1 1 15.2 even 4