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Properties

Label 825.2.a.k.1.1
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
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] = [825,2,Mod(1,825)] 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("825.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1,-3,5,0,-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: \(6.58765816676\)
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 \(\chi\) \(=\) 825.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.90321 q^{2} -1.00000 q^{3} +1.62222 q^{4} +1.90321 q^{6} +4.42864 q^{7} +0.719004 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.62222 q^{12} +0.622216 q^{13} -8.42864 q^{14} -4.61285 q^{16} +5.18421 q^{17} -1.90321 q^{18} +7.05086 q^{19} -4.42864 q^{21} -1.90321 q^{22} -8.85728 q^{23} -0.719004 q^{24} -1.18421 q^{26} -1.00000 q^{27} +7.18421 q^{28} -7.80642 q^{29} +2.75557 q^{31} +7.34122 q^{32} -1.00000 q^{33} -9.86665 q^{34} +1.62222 q^{36} +2.00000 q^{37} -13.4193 q^{38} -0.622216 q^{39} -0.193576 q^{41} +8.42864 q^{42} -5.67307 q^{43} +1.62222 q^{44} +16.8573 q^{46} +2.75557 q^{47} +4.61285 q^{48} +12.6128 q^{49} -5.18421 q^{51} +1.00937 q^{52} +10.8573 q^{53} +1.90321 q^{54} +3.18421 q^{56} -7.05086 q^{57} +14.8573 q^{58} -4.85728 q^{59} +6.85728 q^{61} -5.24443 q^{62} +4.42864 q^{63} -4.74620 q^{64} +1.90321 q^{66} +1.24443 q^{67} +8.40990 q^{68} +8.85728 q^{69} +2.75557 q^{71} +0.719004 q^{72} -4.23506 q^{73} -3.80642 q^{74} +11.4380 q^{76} +4.42864 q^{77} +1.18421 q^{78} +8.56199 q^{79} +1.00000 q^{81} +0.368416 q^{82} -0.133353 q^{83} -7.18421 q^{84} +10.7971 q^{86} +7.80642 q^{87} +0.719004 q^{88} +5.61285 q^{89} +2.75557 q^{91} -14.3684 q^{92} -2.75557 q^{93} -5.24443 q^{94} -7.34122 q^{96} -7.24443 q^{97} -24.0049 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 3 q^{3} + 5 q^{4} - q^{6} + 9 q^{8} + 3 q^{9} + 3 q^{11} - 5 q^{12} + 2 q^{13} - 12 q^{14} + 13 q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + q^{22} - 9 q^{24} + 10 q^{26} - 3 q^{27} + 8 q^{28}+ \cdots + 3 q^{99}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 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
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −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
\(15\) 0 0
\(16\) −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 \(-0.200124\pi\)
0.808789 + 0.588100i \(0.200124\pi\)
\(20\) 0 0
\(21\) −4.42864 −0.966408
\(22\) −1.90321 −0.405766
\(23\) −8.85728 −1.84687 −0.923435 − 0.383754i \(-0.874631\pi\)
−0.923435 + 0.383754i \(0.874631\pi\)
\(24\) −0.719004 −0.146766
\(25\) 0 0
\(26\) −1.18421 −0.232242
\(27\) −1.00000 −0.192450
\(28\) 7.18421 1.35769
\(29\) −7.80642 −1.44962 −0.724808 − 0.688951i \(-0.758072\pi\)
−0.724808 + 0.688951i \(0.758072\pi\)
\(30\) 0 0
\(31\) 2.75557 0.494915 0.247457 − 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) 7.34122 1.29776
\(33\) −1.00000 −0.174078
\(34\) −9.86665 −1.69212
\(35\) 0 0
\(36\) 1.62222 0.270369
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −13.4193 −2.17689
\(39\) −0.622216 −0.0996342
\(40\) 0 0
\(41\) −0.193576 −0.0302315 −0.0151158 − 0.999886i \(-0.504812\pi\)
−0.0151158 + 0.999886i \(0.504812\pi\)
\(42\) 8.42864 1.30057
\(43\) −5.67307 −0.865135 −0.432568 − 0.901602i \(-0.642392\pi\)
−0.432568 + 0.901602i \(0.642392\pi\)
\(44\) 1.62222 0.244558
\(45\) 0 0
\(46\) 16.8573 2.48547
\(47\) 2.75557 0.401941 0.200971 − 0.979597i \(-0.435590\pi\)
0.200971 + 0.979597i \(0.435590\pi\)
\(48\) 4.61285 0.665807
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) −5.18421 −0.725934
\(52\) 1.00937 0.139974
\(53\) 10.8573 1.49136 0.745681 − 0.666303i \(-0.232124\pi\)
0.745681 + 0.666303i \(0.232124\pi\)
\(54\) 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 \(0.602401\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 − 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) −5.24443 −0.666043
\(63\) 4.42864 0.557956
\(64\) −4.74620 −0.593275
\(65\) 0 0
\(66\) 1.90321 0.234269
\(67\) 1.24443 0.152031 0.0760157 − 0.997107i \(-0.475780\pi\)
0.0760157 + 0.997107i \(0.475780\pi\)
\(68\) 8.40990 1.01985
\(69\) 8.85728 1.06629
\(70\) 0 0
\(71\) 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
\(75\) 0 0
\(76\) 11.4380 1.31203
\(77\) 4.42864 0.504690
\(78\) 1.18421 0.134085
\(79\) 8.56199 0.963299 0.481650 − 0.876364i \(-0.340038\pi\)
0.481650 + 0.876364i \(0.340038\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) 10.7971 1.16428
\(87\) 7.80642 0.836936
\(88\) 0.719004 0.0766461
\(89\) 5.61285 0.594961 0.297480 − 0.954728i \(-0.403854\pi\)
0.297480 + 0.954728i \(0.403854\pi\)
\(90\) 0 0
\(91\) 2.75557 0.288862
\(92\) −14.3684 −1.49801
\(93\) −2.75557 −0.285739
\(94\) −5.24443 −0.540922
\(95\) 0 0
\(96\) −7.34122 −0.749260
\(97\) −7.24443 −0.735561 −0.367780 − 0.929913i \(-0.619882\pi\)
−0.367780 + 0.929913i \(0.619882\pi\)
\(98\) −24.0049 −2.42486
\(99\) 1.00000 0.100504
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 825.2.a.k.1.1 3
3.2 odd 2 2475.2.a.bb.1.3 3
5.2 odd 4 825.2.c.g.199.2 6
5.3 odd 4 825.2.c.g.199.5 6
5.4 even 2 165.2.a.c.1.3 ✓ 3
11.10 odd 2 9075.2.a.cf.1.3 3
15.2 even 4 2475.2.c.r.199.5 6
15.8 even 4 2475.2.c.r.199.2 6
15.14 odd 2 495.2.a.e.1.1 3
20.19 odd 2 2640.2.a.be.1.3 3
35.34 odd 2 8085.2.a.bk.1.3 3
55.54 odd 2 1815.2.a.m.1.1 3
60.59 even 2 7920.2.a.cj.1.3 3
165.164 even 2 5445.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 ✓ 3 5.4 even 2
495.2.a.e.1.1 3 15.14 odd 2
825.2.a.k.1.1 3 1.1 even 1 trivial
825.2.c.g.199.2 6 5.2 odd 4
825.2.c.g.199.5 6 5.3 odd 4
1815.2.a.m.1.1 3 55.54 odd 2
2475.2.a.bb.1.3 3 3.2 odd 2
2475.2.c.r.199.2 6 15.8 even 4
2475.2.c.r.199.5 6 15.2 even 4
2640.2.a.be.1.3 3 20.19 odd 2
5445.2.a.z.1.3 3 165.164 even 2
7920.2.a.cj.1.3 3 60.59 even 2
8085.2.a.bk.1.3 3 35.34 odd 2
9075.2.a.cf.1.3 3 11.10 odd 2