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Properties

Label 825.2.c.g.199.5
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $6$
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] = [825,2,Mod(199,825)] 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("825.199"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-10,0,-2,0,0,-6,0,6,0,0,24] 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.5
Root \(1.45161 - 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.g.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.90321i q^{2} -1.00000i q^{3} -1.62222 q^{4} +1.90321 q^{6} -4.42864i q^{7} +0.719004i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.62222i q^{12} +0.622216i q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421i q^{17} -1.90321i q^{18} -7.05086 q^{19} -4.42864 q^{21} +1.90321i q^{22} -8.85728i q^{23} +0.719004 q^{24} -1.18421 q^{26} +1.00000i q^{27} +7.18421i q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122i q^{32} -1.00000i q^{33} +9.86665 q^{34} +1.62222 q^{36} -2.00000i q^{37} -13.4193i q^{38} +0.622216 q^{39} -0.193576 q^{41} -8.42864i q^{42} -5.67307i q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557i q^{47} +4.61285i q^{48} -12.6128 q^{49} -5.18421 q^{51} -1.00937i q^{52} +10.8573i q^{53} -1.90321 q^{54} +3.18421 q^{56} +7.05086i q^{57} +14.8573i q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443i q^{62} +4.42864i q^{63} +4.74620 q^{64} +1.90321 q^{66} -1.24443i q^{67} +8.40990i q^{68} -8.85728 q^{69} +2.75557 q^{71} -0.719004i q^{72} -4.23506i q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864i q^{77} +1.18421i q^{78} -8.56199 q^{79} +1.00000 q^{81} -0.368416i q^{82} -0.133353i q^{83} +7.18421 q^{84} +10.7971 q^{86} -7.80642i q^{87} +0.719004i q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684i q^{92} -2.75557i q^{93} +5.24443 q^{94} -7.34122 q^{96} +7.24443i q^{97} -24.0049i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.90321i 1.34577i 0.739745 + 0.672887i \(0.234946\pi\)
−0.739745 + 0.672887i \(0.765054\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −1.62222 −0.811108
\(5\) 0 0
\(6\) 1.90321 0.776983
\(7\) − 4.42864i − 1.67387i −0.547304 − 0.836934i \(-0.684346\pi\)
0.547304 − 0.836934i \(-0.315654\pi\)
\(8\) 0.719004i 0.254206i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 1.62222i 0.468293i
\(13\) 0.622216i 0.172572i 0.996270 + 0.0862858i \(0.0274998\pi\)
−0.996270 + 0.0862858i \(0.972500\pi\)
\(14\) 8.42864 2.25265
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) − 5.18421i − 1.25736i −0.777666 − 0.628678i \(-0.783597\pi\)
0.777666 − 0.628678i \(-0.216403\pi\)
\(18\) − 1.90321i − 0.448591i
\(19\) −7.05086 −1.61758 −0.808789 − 0.588100i \(-0.799876\pi\)
−0.808789 + 0.588100i \(0.799876\pi\)
\(20\) 0 0
\(21\) −4.42864 −0.966408
\(22\) 1.90321i 0.405766i
\(23\) − 8.85728i − 1.84687i −0.383754 − 0.923435i \(-0.625369\pi\)
0.383754 − 0.923435i \(-0.374631\pi\)
\(24\) 0.719004 0.146766
\(25\) 0 0
\(26\) −1.18421 −0.232242
\(27\) 1.00000i 0.192450i
\(28\) 7.18421i 1.35769i
\(29\) 7.80642 1.44962 0.724808 − 0.688951i \(-0.241928\pi\)
0.724808 + 0.688951i \(0.241928\pi\)
\(30\) 0 0
\(31\) 2.75557 0.494915 0.247457 − 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) − 7.34122i − 1.29776i
\(33\) − 1.00000i − 0.174078i
\(34\) 9.86665 1.69212
\(35\) 0 0
\(36\) 1.62222 0.270369
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) − 13.4193i − 2.17689i
\(39\) 0.622216 0.0996342
\(40\) 0 0
\(41\) −0.193576 −0.0302315 −0.0151158 − 0.999886i \(-0.504812\pi\)
−0.0151158 + 0.999886i \(0.504812\pi\)
\(42\) − 8.42864i − 1.30057i
\(43\) − 5.67307i − 0.865135i −0.901602 − 0.432568i \(-0.857608\pi\)
0.901602 − 0.432568i \(-0.142392\pi\)
\(44\) −1.62222 −0.244558
\(45\) 0 0
\(46\) 16.8573 2.48547
\(47\) − 2.75557i − 0.401941i −0.979597 − 0.200971i \(-0.935590\pi\)
0.979597 − 0.200971i \(-0.0644095\pi\)
\(48\) 4.61285i 0.665807i
\(49\) −12.6128 −1.80184
\(50\) 0 0
\(51\) −5.18421 −0.725934
\(52\) − 1.00937i − 0.139974i
\(53\) 10.8573i 1.49136i 0.666303 + 0.745681i \(0.267876\pi\)
−0.666303 + 0.745681i \(0.732124\pi\)
\(54\) −1.90321 −0.258994
\(55\) 0 0
\(56\) 3.18421 0.425508
\(57\) 7.05086i 0.933909i
\(58\) 14.8573i 1.95086i
\(59\) 4.85728 0.632364 0.316182 − 0.948699i \(-0.397599\pi\)
0.316182 + 0.948699i \(0.397599\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 − 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) 5.24443i 0.666043i
\(63\) 4.42864i 0.557956i
\(64\) 4.74620 0.593275
\(65\) 0 0
\(66\) 1.90321 0.234269
\(67\) − 1.24443i − 0.152031i −0.997107 − 0.0760157i \(-0.975780\pi\)
0.997107 − 0.0760157i \(-0.0242199\pi\)
\(68\) 8.40990i 1.01985i
\(69\) −8.85728 −1.06629
\(70\) 0 0
\(71\) 2.75557 0.327026 0.163513 − 0.986541i \(-0.447717\pi\)
0.163513 + 0.986541i \(0.447717\pi\)
\(72\) − 0.719004i − 0.0847354i
\(73\) − 4.23506i − 0.495677i −0.968801 − 0.247838i \(-0.920280\pi\)
0.968801 − 0.247838i \(-0.0797202\pi\)
\(74\) 3.80642 0.442488
\(75\) 0 0
\(76\) 11.4380 1.31203
\(77\) − 4.42864i − 0.504690i
\(78\) 1.18421i 0.134085i
\(79\) −8.56199 −0.963299 −0.481650 − 0.876364i \(-0.659962\pi\)
−0.481650 + 0.876364i \(0.659962\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 0.368416i − 0.0406848i
\(83\) − 0.133353i − 0.0146374i −0.999973 − 0.00731870i \(-0.997670\pi\)
0.999973 − 0.00731870i \(-0.00232964\pi\)
\(84\) 7.18421 0.783861
\(85\) 0 0
\(86\) 10.7971 1.16428
\(87\) − 7.80642i − 0.836936i
\(88\) 0.719004i 0.0766461i
\(89\) −5.61285 −0.594961 −0.297480 − 0.954728i \(-0.596146\pi\)
−0.297480 + 0.954728i \(0.596146\pi\)
\(90\) 0 0
\(91\) 2.75557 0.288862
\(92\) 14.3684i 1.49801i
\(93\) − 2.75557i − 0.285739i
\(94\) 5.24443 0.540922
\(95\) 0 0
\(96\) −7.34122 −0.749260
\(97\) 7.24443i 0.735561i 0.929913 + 0.367780i \(0.119882\pi\)
−0.929913 + 0.367780i \(0.880118\pi\)
\(98\) − 24.0049i − 2.42486i
\(99\) −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.c.g.199.5 6
3.2 odd 2 2475.2.c.r.199.2 6
5.2 odd 4 825.2.a.k.1.1 3
5.3 odd 4 165.2.a.c.1.3 ✓ 3
5.4 even 2 inner 825.2.c.g.199.2 6
15.2 even 4 2475.2.a.bb.1.3 3
15.8 even 4 495.2.a.e.1.1 3
15.14 odd 2 2475.2.c.r.199.5 6
20.3 even 4 2640.2.a.be.1.3 3
35.13 even 4 8085.2.a.bk.1.3 3
55.32 even 4 9075.2.a.cf.1.3 3
55.43 even 4 1815.2.a.m.1.1 3
60.23 odd 4 7920.2.a.cj.1.3 3
165.98 odd 4 5445.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 ✓ 3 5.3 odd 4
495.2.a.e.1.1 3 15.8 even 4
825.2.a.k.1.1 3 5.2 odd 4
825.2.c.g.199.2 6 5.4 even 2 inner
825.2.c.g.199.5 6 1.1 even 1 trivial
1815.2.a.m.1.1 3 55.43 even 4
2475.2.a.bb.1.3 3 15.2 even 4
2475.2.c.r.199.2 6 3.2 odd 2
2475.2.c.r.199.5 6 15.14 odd 2
2640.2.a.be.1.3 3 20.3 even 4
5445.2.a.z.1.3 3 165.98 odd 4
7920.2.a.cj.1.3 3 60.23 odd 4
8085.2.a.bk.1.3 3 35.13 even 4
9075.2.a.cf.1.3 3 55.32 even 4