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Properties

Label 261.3.f.b.244.2
Level $261$
Weight $3$
Character 261.244
Analytic conductor $7.112$
Analytic rank $0$
Dimension $8$
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] = [261,3,Mod(46,261)] 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("261.46"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(261, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 210x^{6} + 644x^{5} + 17515x^{4} - 36108x^{3} - 683586x^{2} + 701748x + 10556010 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 244.2
Root \(-6.88412 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 261.244
Dual form 261.3.f.b.46.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.224745 - 0.224745i) q^{2} -3.89898i q^{4} +9.10887i q^{5} -4.65938 q^{7} +(-1.77526 + 1.77526i) q^{8} +(2.04717 - 2.04717i) q^{10} +(-13.5584 - 13.5584i) q^{11} +1.10102i q^{13} +(1.04717 + 1.04717i) q^{14} -14.7980 q^{16} +(-9.49666 - 9.49666i) q^{17} +(-1.19243 - 1.19243i) q^{19} +35.5153 q^{20} +6.09434i q^{22} -23.0090 q^{23} -57.9715 q^{25} +(0.247449 - 0.247449i) q^{26} +18.1668i q^{28} +(-8.79271 + 27.6349i) q^{29} +(36.8783 + 36.8783i) q^{31} +(10.4268 + 10.4268i) q^{32} +4.26865i q^{34} -42.4416i q^{35} +(-9.49666 + 9.49666i) q^{37} +0.535986i q^{38} +(-16.1706 - 16.1706i) q^{40} +(1.67948 - 1.67948i) q^{41} +(-46.0343 - 46.0343i) q^{43} +(-52.8638 + 52.8638i) q^{44} +(5.17116 + 5.17116i) q^{46} +(31.9889 - 31.9889i) q^{47} -27.2902 q^{49} +(13.0288 + 13.0288i) q^{50} +4.29286 q^{52} -12.3575 q^{53} +(123.501 - 123.501i) q^{55} +(8.27158 - 8.27158i) q^{56} +(8.18692 - 4.23469i) q^{58} -12.4231 q^{59} +(53.4742 + 53.4742i) q^{61} -16.5764i q^{62} +54.5051i q^{64} -10.0290 q^{65} -50.5589i q^{67} +(-37.0273 + 37.0273i) q^{68} +(-9.53854 + 9.53854i) q^{70} -17.4800i q^{71} +(-0.00901400 + 0.00901400i) q^{73} +4.26865 q^{74} +(-4.64927 + 4.64927i) q^{76} +(63.1735 + 63.1735i) q^{77} +(69.5161 + 69.5161i) q^{79} -134.793i q^{80} -0.754910 q^{82} +117.205 q^{83} +(86.5038 - 86.5038i) q^{85} +20.6920i q^{86} +48.1391 q^{88} +(27.7333 + 27.7333i) q^{89} -5.13007i q^{91} +89.7117i q^{92} -14.3787 q^{94} +(10.8617 - 10.8617i) q^{95} +(-83.4159 + 83.4159i) q^{97} +(6.13333 + 6.13333i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 12 q^{7} - 24 q^{8} + 8 q^{10} - 20 q^{11} - 40 q^{16} - 48 q^{17} - 28 q^{19} + 44 q^{20} + 88 q^{23} - 260 q^{25} - 96 q^{26} - 32 q^{29} - 56 q^{31} + 152 q^{32} - 48 q^{37} - 56 q^{41}+ \cdots + 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.224745 − 0.224745i −0.112372 − 0.112372i 0.648685 − 0.761057i \(-0.275319\pi\)
−0.761057 + 0.648685i \(0.775319\pi\)
\(3\) 0 0
\(4\) − 3.89898i − 0.974745i
\(5\) 9.10887i 1.82177i 0.412657 + 0.910887i \(0.364601\pi\)
−0.412657 + 0.910887i \(0.635399\pi\)
\(6\) 0 0
\(7\) −4.65938 −0.665625 −0.332813 − 0.942993i \(-0.607998\pi\)
−0.332813 + 0.942993i \(0.607998\pi\)
\(8\) −1.77526 + 1.77526i −0.221907 + 0.221907i
\(9\) 0 0
\(10\) 2.04717 − 2.04717i 0.204717 − 0.204717i
\(11\) −13.5584 − 13.5584i −1.23258 − 1.23258i −0.962970 − 0.269608i \(-0.913106\pi\)
−0.269608 − 0.962970i \(-0.586894\pi\)
\(12\) 0 0
\(13\) 1.10102i 0.0846939i 0.999103 + 0.0423469i \(0.0134835\pi\)
−0.999103 + 0.0423469i \(0.986517\pi\)
\(14\) 1.04717 + 1.04717i 0.0747979 + 0.0747979i
\(15\) 0 0
\(16\) −14.7980 −0.924872
\(17\) −9.49666 − 9.49666i −0.558627 − 0.558627i 0.370289 − 0.928916i \(-0.379259\pi\)
−0.928916 + 0.370289i \(0.879259\pi\)
\(18\) 0 0
\(19\) −1.19243 − 1.19243i −0.0627596 − 0.0627596i 0.675030 − 0.737790i \(-0.264131\pi\)
−0.737790 + 0.675030i \(0.764131\pi\)
\(20\) 35.5153 1.77576
\(21\) 0 0
\(22\) 6.09434i 0.277016i
\(23\) −23.0090 −1.00039 −0.500196 − 0.865912i \(-0.666739\pi\)
−0.500196 + 0.865912i \(0.666739\pi\)
\(24\) 0 0
\(25\) −57.9715 −2.31886
\(26\) 0.247449 − 0.247449i 0.00951726 − 0.00951726i
\(27\) 0 0
\(28\) 18.1668i 0.648815i
\(29\) −8.79271 + 27.6349i −0.303197 + 0.952928i
\(30\) 0 0
\(31\) 36.8783 + 36.8783i 1.18962 + 1.18962i 0.977172 + 0.212450i \(0.0681442\pi\)
0.212450 + 0.977172i \(0.431856\pi\)
\(32\) 10.4268 + 10.4268i 0.325837 + 0.325837i
\(33\) 0 0
\(34\) 4.26865i 0.125549i
\(35\) − 42.4416i − 1.21262i
\(36\) 0 0
\(37\) −9.49666 + 9.49666i −0.256667 + 0.256667i −0.823697 − 0.567030i \(-0.808092\pi\)
0.567030 + 0.823697i \(0.308092\pi\)
\(38\) 0.535986i 0.0141049i
\(39\) 0 0
\(40\) −16.1706 − 16.1706i −0.404264 − 0.404264i
\(41\) 1.67948 − 1.67948i 0.0409630 − 0.0409630i −0.686329 − 0.727292i \(-0.740779\pi\)
0.727292 + 0.686329i \(0.240779\pi\)
\(42\) 0 0
\(43\) −46.0343 − 46.0343i −1.07057 − 1.07057i −0.997313 − 0.0732520i \(-0.976662\pi\)
−0.0732520 − 0.997313i \(-0.523338\pi\)
\(44\) −52.8638 + 52.8638i −1.20145 + 1.20145i
\(45\) 0 0
\(46\) 5.17116 + 5.17116i 0.112416 + 0.112416i
\(47\) 31.9889 − 31.9889i 0.680615 − 0.680615i −0.279524 − 0.960139i \(-0.590177\pi\)
0.960139 + 0.279524i \(0.0901766\pi\)
\(48\) 0 0
\(49\) −27.2902 −0.556943
\(50\) 13.0288 + 13.0288i 0.260576 + 0.260576i
\(51\) 0 0
\(52\) 4.29286 0.0825549
\(53\) −12.3575 −0.233160 −0.116580 − 0.993181i \(-0.537193\pi\)
−0.116580 + 0.993181i \(0.537193\pi\)
\(54\) 0 0
\(55\) 123.501 − 123.501i 2.24548 − 2.24548i
\(56\) 8.27158 − 8.27158i 0.147707 − 0.147707i
\(57\) 0 0
\(58\) 8.18692 − 4.23469i 0.141154 − 0.0730119i
\(59\) −12.4231 −0.210561 −0.105280 − 0.994443i \(-0.533574\pi\)
−0.105280 + 0.994443i \(0.533574\pi\)
\(60\) 0 0
\(61\) 53.4742 + 53.4742i 0.876626 + 0.876626i 0.993184 − 0.116558i \(-0.0371860\pi\)
−0.116558 + 0.993184i \(0.537186\pi\)
\(62\) − 16.5764i − 0.267361i
\(63\) 0 0
\(64\) 54.5051i 0.851642i
\(65\) −10.0290 −0.154293
\(66\) 0 0
\(67\) − 50.5589i − 0.754610i −0.926089 − 0.377305i \(-0.876851\pi\)
0.926089 − 0.377305i \(-0.123149\pi\)
\(68\) −37.0273 + 37.0273i −0.544519 + 0.544519i
\(69\) 0 0
\(70\) −9.53854 + 9.53854i −0.136265 + 0.136265i
\(71\) − 17.4800i − 0.246197i −0.992394 − 0.123099i \(-0.960717\pi\)
0.992394 − 0.123099i \(-0.0392832\pi\)
\(72\) 0 0
\(73\) −0.00901400 + 0.00901400i −0.000123479 + 0.000123479i −0.707169 − 0.707045i \(-0.750028\pi\)
0.707045 + 0.707169i \(0.250028\pi\)
\(74\) 4.26865 0.0576845
\(75\) 0 0
\(76\) −4.64927 + 4.64927i −0.0611745 + 0.0611745i
\(77\) 63.1735 + 63.1735i 0.820435 + 0.820435i
\(78\) 0 0
\(79\) 69.5161 + 69.5161i 0.879951 + 0.879951i 0.993529 − 0.113578i \(-0.0362313\pi\)
−0.113578 + 0.993529i \(0.536231\pi\)
\(80\) − 134.793i − 1.68491i
\(81\) 0 0
\(82\) −0.754910 −0.00920622
\(83\) 117.205 1.41210 0.706052 − 0.708160i \(-0.250474\pi\)
0.706052 + 0.708160i \(0.250474\pi\)
\(84\) 0 0
\(85\) 86.5038 − 86.5038i 1.01769 − 1.01769i
\(86\) 20.6920i 0.240604i
\(87\) 0 0
\(88\) 48.1391 0.547035
\(89\) 27.7333 + 27.7333i 0.311610 + 0.311610i 0.845533 − 0.533923i \(-0.179283\pi\)
−0.533923 + 0.845533i \(0.679283\pi\)
\(90\) 0 0
\(91\) − 5.13007i − 0.0563744i
\(92\) 89.7117i 0.975127i
\(93\) 0 0
\(94\) −14.3787 −0.152965
\(95\) 10.8617 − 10.8617i 0.114334 − 0.114334i
\(96\) 0 0
\(97\) −83.4159 + 83.4159i −0.859958 + 0.859958i −0.991333 − 0.131375i \(-0.958061\pi\)
0.131375 + 0.991333i \(0.458061\pi\)
\(98\) 6.13333 + 6.13333i 0.0625850 + 0.0625850i
\(99\) 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 261.3.f.b.244.2 8
3.2 odd 2 87.3.e.a.70.3 yes 8
29.17 odd 4 inner 261.3.f.b.46.1 8
87.17 even 4 87.3.e.a.46.4 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.3.e.a.46.4 ✓ 8 87.17 even 4
87.3.e.a.70.3 yes 8 3.2 odd 2
261.3.f.b.46.1 8 29.17 odd 4 inner
261.3.f.b.244.2 8 1.1 even 1 trivial