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Properties

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

$q$-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 \(-0.765054\pi\)
0.739745 − 0.672887i \(-0.234946\pi\)
\(3\) 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 \(0.315654\pi\)
−0.547304 + 0.836934i \(0.684346\pi\)
\(8\) − 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 \(-0.972500\pi\)
0.996270 − 0.0862858i \(-0.0274998\pi\)
\(14\) 8.42864 2.25265
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) 5.18421i 1.25736i 0.777666 + 0.628678i \(0.216403\pi\)
−0.777666 + 0.628678i \(0.783597\pi\)
\(18\) 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 \(0.374631\pi\)
−0.383754 + 0.923435i \(0.625369\pi\)
\(24\) 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 \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 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 \(0.142392\pi\)
−0.901602 + 0.432568i \(0.857608\pi\)
\(44\) −1.62222 −0.244558
\(45\) 0 0
\(46\) 16.8573 2.48547
\(47\) 2.75557i 0.401941i 0.979597 + 0.200971i \(0.0644095\pi\)
−0.979597 + 0.200971i \(0.935590\pi\)
\(48\) − 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 \(-0.732124\pi\)
0.666303 − 0.745681i \(-0.267876\pi\)
\(54\) −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 \(0.0242199\pi\)
−0.997107 + 0.0760157i \(0.975780\pi\)
\(68\) − 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 \(0.0797202\pi\)
−0.968801 + 0.247838i \(0.920280\pi\)
\(74\) 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 \(0.00232964\pi\)
−0.999973 + 0.00731870i \(0.997670\pi\)
\(84\) 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 \(-0.880118\pi\)
0.929913 − 0.367780i \(-0.119882\pi\)
\(98\) 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.199.2 6
3.2 odd 2 2475.2.c.r.199.5 6
5.2 odd 4 165.2.a.c.1.3 ✓ 3
5.3 odd 4 825.2.a.k.1.1 3
5.4 even 2 inner 825.2.c.g.199.5 6
15.2 even 4 495.2.a.e.1.1 3
15.8 even 4 2475.2.a.bb.1.3 3
15.14 odd 2 2475.2.c.r.199.2 6
20.7 even 4 2640.2.a.be.1.3 3
35.27 even 4 8085.2.a.bk.1.3 3
55.32 even 4 1815.2.a.m.1.1 3
55.43 even 4 9075.2.a.cf.1.3 3
60.47 odd 4 7920.2.a.cj.1.3 3
165.32 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 5.2 odd 4
495.2.a.e.1.1 3 15.2 even 4
825.2.a.k.1.1 3 5.3 odd 4
825.2.c.g.199.2 6 1.1 even 1 trivial
825.2.c.g.199.5 6 5.4 even 2 inner
1815.2.a.m.1.1 3 55.32 even 4
2475.2.a.bb.1.3 3 15.8 even 4
2475.2.c.r.199.2 6 15.14 odd 2
2475.2.c.r.199.5 6 3.2 odd 2
2640.2.a.be.1.3 3 20.7 even 4
5445.2.a.z.1.3 3 165.32 odd 4
7920.2.a.cj.1.3 3 60.47 odd 4
8085.2.a.bk.1.3 3 35.27 even 4
9075.2.a.cf.1.3 3 55.43 even 4