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Properties

Label 1815.2.a.m.1.1
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [1815,2,Mod(1,1815)] 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("1815.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1815, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1,3,5,3,1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.90321 q^{2} +1.00000 q^{3} +1.62222 q^{4} +1.00000 q^{5} -1.90321 q^{6} +4.42864 q^{7} +0.719004 q^{8} +1.00000 q^{9} -1.90321 q^{10} +1.62222 q^{12} +0.622216 q^{13} -8.42864 q^{14} +1.00000 q^{15} -4.61285 q^{16} +5.18421 q^{17} -1.90321 q^{18} -7.05086 q^{19} +1.62222 q^{20} +4.42864 q^{21} +8.85728 q^{23} +0.719004 q^{24} +1.00000 q^{25} -1.18421 q^{26} +1.00000 q^{27} +7.18421 q^{28} +7.80642 q^{29} -1.90321 q^{30} +2.75557 q^{31} +7.34122 q^{32} -9.86665 q^{34} +4.42864 q^{35} +1.62222 q^{36} -2.00000 q^{37} +13.4193 q^{38} +0.622216 q^{39} +0.719004 q^{40} +0.193576 q^{41} -8.42864 q^{42} -5.67307 q^{43} +1.00000 q^{45} -16.8573 q^{46} -2.75557 q^{47} -4.61285 q^{48} +12.6128 q^{49} -1.90321 q^{50} +5.18421 q^{51} +1.00937 q^{52} -10.8573 q^{53} -1.90321 q^{54} +3.18421 q^{56} -7.05086 q^{57} -14.8573 q^{58} -4.85728 q^{59} +1.62222 q^{60} -6.85728 q^{61} -5.24443 q^{62} +4.42864 q^{63} -4.74620 q^{64} +0.622216 q^{65} -1.24443 q^{67} +8.40990 q^{68} +8.85728 q^{69} -8.42864 q^{70} +2.75557 q^{71} +0.719004 q^{72} -4.23506 q^{73} +3.80642 q^{74} +1.00000 q^{75} -11.4380 q^{76} -1.18421 q^{78} -8.56199 q^{79} -4.61285 q^{80} +1.00000 q^{81} -0.368416 q^{82} -0.133353 q^{83} +7.18421 q^{84} +5.18421 q^{85} +10.7971 q^{86} +7.80642 q^{87} +5.61285 q^{89} -1.90321 q^{90} +2.75557 q^{91} +14.3684 q^{92} +2.75557 q^{93} +5.24443 q^{94} -7.05086 q^{95} +7.34122 q^{96} +7.24443 q^{97} -24.0049 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} + q^{6} + 9 q^{8} + 3 q^{9} + q^{10} + 5 q^{12} + 2 q^{13} - 12 q^{14} + 3 q^{15} + 13 q^{16} + 2 q^{17} + q^{18} - 8 q^{19} + 5 q^{20} + 9 q^{24} + 3 q^{25}+ \cdots - 39 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90321 −1.34577 −0.672887 − 0.739745i \(-0.734946\pi\)
−0.672887 + 0.739745i \(0.734946\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.62222 0.811108
\(5\) 1.00000 0.447214
\(6\) −1.90321 −0.776983
\(7\) 4.42864 1.67387 0.836934 − 0.547304i \(-0.184346\pi\)
0.836934 + 0.547304i \(0.184346\pi\)
\(8\) 0.719004 0.254206
\(9\) 1.00000 0.333333
\(10\) −1.90321 −0.601848
\(11\) 0 0
\(12\) 1.62222 0.468293
\(13\) 0.622216 0.172572 0.0862858 − 0.996270i \(-0.472500\pi\)
0.0862858 + 0.996270i \(0.472500\pi\)
\(14\) −8.42864 −2.25265
\(15\) 1.00000 0.258199
\(16\) −4.61285 −1.15321
\(17\) 5.18421 1.25736 0.628678 − 0.777666i \(-0.283597\pi\)
0.628678 + 0.777666i \(0.283597\pi\)
\(18\) −1.90321 −0.448591
\(19\) −7.05086 −1.61758 −0.808789 − 0.588100i \(-0.799876\pi\)
−0.808789 + 0.588100i \(0.799876\pi\)
\(20\) 1.62222 0.362738
\(21\) 4.42864 0.966408
\(22\) 0 0
\(23\) 8.85728 1.84687 0.923435 − 0.383754i \(-0.125369\pi\)
0.923435 + 0.383754i \(0.125369\pi\)
\(24\) 0.719004 0.146766
\(25\) 1.00000 0.200000
\(26\) −1.18421 −0.232242
\(27\) 1.00000 0.192450
\(28\) 7.18421 1.35769
\(29\) 7.80642 1.44962 0.724808 − 0.688951i \(-0.241928\pi\)
0.724808 + 0.688951i \(0.241928\pi\)
\(30\) −1.90321 −0.347477
\(31\) 2.75557 0.494915 0.247457 − 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) 7.34122 1.29776
\(33\) 0 0
\(34\) −9.86665 −1.69212
\(35\) 4.42864 0.748577
\(36\) 1.62222 0.270369
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 13.4193 2.17689
\(39\) 0.622216 0.0996342
\(40\) 0.719004 0.113684
\(41\) 0.193576 0.0302315 0.0151158 − 0.999886i \(-0.495188\pi\)
0.0151158 + 0.999886i \(0.495188\pi\)
\(42\) −8.42864 −1.30057
\(43\) −5.67307 −0.865135 −0.432568 − 0.901602i \(-0.642392\pi\)
−0.432568 + 0.901602i \(0.642392\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −16.8573 −2.48547
\(47\) −2.75557 −0.401941 −0.200971 − 0.979597i \(-0.564410\pi\)
−0.200971 + 0.979597i \(0.564410\pi\)
\(48\) −4.61285 −0.665807
\(49\) 12.6128 1.80184
\(50\) −1.90321 −0.269155
\(51\) 5.18421 0.725934
\(52\) 1.00937 0.139974
\(53\) −10.8573 −1.49136 −0.745681 − 0.666303i \(-0.767876\pi\)
−0.745681 + 0.666303i \(0.767876\pi\)
\(54\) −1.90321 −0.258994
\(55\) 0 0
\(56\) 3.18421 0.425508
\(57\) −7.05086 −0.933909
\(58\) −14.8573 −1.95086
\(59\) −4.85728 −0.632364 −0.316182 − 0.948699i \(-0.602401\pi\)
−0.316182 + 0.948699i \(0.602401\pi\)
\(60\) 1.62222 0.209427
\(61\) −6.85728 −0.877985 −0.438992 − 0.898491i \(-0.644664\pi\)
−0.438992 + 0.898491i \(0.644664\pi\)
\(62\) −5.24443 −0.666043
\(63\) 4.42864 0.557956
\(64\) −4.74620 −0.593275
\(65\) 0.622216 0.0771764
\(66\) 0 0
\(67\) −1.24443 −0.152031 −0.0760157 − 0.997107i \(-0.524220\pi\)
−0.0760157 + 0.997107i \(0.524220\pi\)
\(68\) 8.40990 1.01985
\(69\) 8.85728 1.06629
\(70\) −8.42864 −1.00742
\(71\) 2.75557 0.327026 0.163513 − 0.986541i \(-0.447717\pi\)
0.163513 + 0.986541i \(0.447717\pi\)
\(72\) 0.719004 0.0847354
\(73\) −4.23506 −0.495677 −0.247838 − 0.968801i \(-0.579720\pi\)
−0.247838 + 0.968801i \(0.579720\pi\)
\(74\) 3.80642 0.442488
\(75\) 1.00000 0.115470
\(76\) −11.4380 −1.31203
\(77\) 0 0
\(78\) −1.18421 −0.134085
\(79\) −8.56199 −0.963299 −0.481650 − 0.876364i \(-0.659962\pi\)
−0.481650 + 0.876364i \(0.659962\pi\)
\(80\) −4.61285 −0.515732
\(81\) 1.00000 0.111111
\(82\) −0.368416 −0.0406848
\(83\) −0.133353 −0.0146374 −0.00731870 − 0.999973i \(-0.502330\pi\)
−0.00731870 + 0.999973i \(0.502330\pi\)
\(84\) 7.18421 0.783861
\(85\) 5.18421 0.562306
\(86\) 10.7971 1.16428
\(87\) 7.80642 0.836936
\(88\) 0 0
\(89\) 5.61285 0.594961 0.297480 − 0.954728i \(-0.403854\pi\)
0.297480 + 0.954728i \(0.403854\pi\)
\(90\) −1.90321 −0.200616
\(91\) 2.75557 0.288862
\(92\) 14.3684 1.49801
\(93\) 2.75557 0.285739
\(94\) 5.24443 0.540922
\(95\) −7.05086 −0.723402
\(96\) 7.34122 0.749260
\(97\) 7.24443 0.735561 0.367780 − 0.929913i \(-0.380118\pi\)
0.367780 + 0.929913i \(0.380118\pi\)
\(98\) −24.0049 −2.42486
\(99\) 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 1815.2.a.m.1.1 3
3.2 odd 2 5445.2.a.z.1.3 3
5.4 even 2 9075.2.a.cf.1.3 3
11.10 odd 2 165.2.a.c.1.3 ✓ 3
33.32 even 2 495.2.a.e.1.1 3
44.43 even 2 2640.2.a.be.1.3 3
55.32 even 4 825.2.c.g.199.5 6
55.43 even 4 825.2.c.g.199.2 6
55.54 odd 2 825.2.a.k.1.1 3
77.76 even 2 8085.2.a.bk.1.3 3
132.131 odd 2 7920.2.a.cj.1.3 3
165.32 odd 4 2475.2.c.r.199.2 6
165.98 odd 4 2475.2.c.r.199.5 6
165.164 even 2 2475.2.a.bb.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 ✓ 3 11.10 odd 2
495.2.a.e.1.1 3 33.32 even 2
825.2.a.k.1.1 3 55.54 odd 2
825.2.c.g.199.2 6 55.43 even 4
825.2.c.g.199.5 6 55.32 even 4
1815.2.a.m.1.1 3 1.1 even 1 trivial
2475.2.a.bb.1.3 3 165.164 even 2
2475.2.c.r.199.2 6 165.32 odd 4
2475.2.c.r.199.5 6 165.98 odd 4
2640.2.a.be.1.3 3 44.43 even 2
5445.2.a.z.1.3 3 3.2 odd 2
7920.2.a.cj.1.3 3 132.131 odd 2
8085.2.a.bk.1.3 3 77.76 even 2
9075.2.a.cf.1.3 3 5.4 even 2