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Properties

Label 323.2.bb.a.9.11
Level $323$
Weight $2$
Character 323.9
Analytic conductor $2.579$
Analytic rank $0$
Dimension $672$
Inner twists $4$

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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] = [323,2,Mod(9,323)] 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("323.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(323, base_ring=CyclotomicField(72)) chi = DirichletCharacter(H, H._module([9, 32])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 323 = 17 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 323.bb (of order \(72\), degree \(24\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.57916798529\)
Analytic rank: \(0\)
Dimension: \(672\)
Relative dimension: \(28\) over \(\Q(\zeta_{72})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{72}]$

Embedding invariants

Embedding label 9.11
Character \(\chi\) \(=\) 323.9
Dual form 323.2.bb.a.36.11

$q$-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+(-0.510179 - 0.357231i) q^{2} +(2.21119 - 0.697187i) q^{3} +(-0.551372 - 1.51488i) q^{4} +(2.43825 - 2.66089i) q^{5} +(-1.37716 - 0.434218i) q^{6} +(1.48595 + 0.195629i) q^{7} +(-0.582257 + 2.17301i) q^{8} +(1.94586 - 1.36250i) q^{9} +(-2.19450 + 0.486509i) q^{10} +(2.16456 + 1.66093i) q^{11} +(-2.27535 - 2.96529i) q^{12} +(-4.32122 + 5.14983i) q^{13} +(-0.688215 - 0.630633i) q^{14} +(3.53632 - 7.58365i) q^{15} +(-1.39656 + 1.17185i) q^{16} +(-3.73643 - 1.74330i) q^{17} -1.47946 q^{18} +(1.89229 + 3.92674i) q^{19} +(-5.37531 - 2.22653i) q^{20} +(3.42211 - 0.603411i) q^{21} +(-0.510980 - 1.62062i) q^{22} +(-4.18903 + 3.83854i) q^{23} +(0.227513 + 5.21090i) q^{24} +(-0.699459 - 7.99485i) q^{25} +(4.04428 - 1.08366i) q^{26} +(-0.881497 + 1.14879i) q^{27} +(-0.522955 - 2.35890i) q^{28} +(2.24358 + 1.42932i) q^{29} +(-4.51327 + 2.60574i) q^{30} +(5.20755 - 3.99589i) q^{31} +(5.61333 - 0.491103i) q^{32} +(5.94425 + 2.16353i) q^{33} +(1.28349 + 2.22417i) q^{34} +(4.14366 - 3.47695i) q^{35} +(-3.13692 - 2.19649i) q^{36} +(-3.06459 - 1.26940i) q^{37} +(0.437349 - 2.67932i) q^{38} +(-5.96467 + 14.4000i) q^{39} +(4.36245 + 6.84767i) q^{40} +(3.07532 + 1.60091i) q^{41} +(-1.96145 - 0.914638i) q^{42} +(-2.87340 - 6.16203i) q^{43} +(1.32263 - 4.19485i) q^{44} +(1.11902 - 8.49983i) q^{45} +(3.50841 - 0.461890i) q^{46} +(-1.85696 - 0.327433i) q^{47} +(-2.27107 + 3.56486i) q^{48} +(-4.59171 - 1.23034i) q^{49} +(-2.49916 + 4.32868i) q^{50} +(-9.47738 - 1.24979i) q^{51} +(10.1840 + 3.70667i) q^{52} +(0.744860 - 1.59736i) q^{53} +(0.860105 - 0.271190i) q^{54} +(9.69730 - 1.70990i) q^{55} +(-1.29031 + 3.11508i) q^{56} +(6.92188 + 7.36350i) q^{57} +(-0.634031 - 1.53069i) q^{58} +(3.17763 - 4.53812i) q^{59} +(-13.4382 - 1.17569i) q^{60} +(-9.20455 - 10.0450i) q^{61} +(-4.08424 + 0.178322i) q^{62} +(3.15799 - 1.64394i) q^{63} +(0.118425 + 0.0683727i) q^{64} +(3.16688 + 24.0549i) q^{65} +(-2.25975 - 3.22726i) q^{66} +(2.04685 - 11.6083i) q^{67} +(-0.580736 + 6.62145i) q^{68} +(-6.58659 + 11.4083i) q^{69} +(-3.35609 + 0.293620i) q^{70} +(4.18580 + 3.83558i) q^{71} +(1.82775 + 5.02170i) q^{72} +(0.0321163 + 0.101860i) q^{73} +(1.11002 + 1.74239i) q^{74} +(-7.12055 - 17.1905i) q^{75} +(4.90519 - 5.03168i) q^{76} +(2.89151 + 2.89151i) q^{77} +(8.18718 - 5.21581i) q^{78} +(-7.29665 + 14.0167i) q^{79} +(-0.286999 + 6.57337i) q^{80} +(-3.58560 + 9.85135i) q^{81} +(-0.997070 - 1.91535i) q^{82} +(-0.734336 - 2.74058i) q^{83} +(-2.80095 - 4.85139i) q^{84} +(-13.7491 + 5.69160i) q^{85} +(-0.735321 + 4.17021i) q^{86} +(5.95749 + 1.59631i) q^{87} +(-4.86955 + 3.73654i) q^{88} +(-2.61485 + 3.11625i) q^{89} +(-3.60731 + 3.93669i) q^{90} +(-7.42857 + 6.80703i) q^{91} +(8.12465 + 4.22942i) q^{92} +(8.72902 - 12.4663i) q^{93} +(0.830415 + 0.830415i) q^{94} +(15.0625 + 4.53922i) q^{95} +(12.0698 - 4.99947i) q^{96} +(8.54002 + 1.89328i) q^{97} +(1.90308 + 2.26800i) q^{98} +(6.47495 + 0.282702i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 672 q - 24 q^{2} - 24 q^{3} - 24 q^{5} - 48 q^{6} - 12 q^{7} - 12 q^{8} - 60 q^{9} - 24 q^{10} - 12 q^{12} - 24 q^{14} + 12 q^{15} - 48 q^{16} - 72 q^{17} + 48 q^{18} - 24 q^{19} - 48 q^{20} - 24 q^{23}+ \cdots - 156 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/323\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.510179 − 0.357231i −0.360751 − 0.252601i 0.379123 − 0.925346i \(-0.376226\pi\)
−0.739874 + 0.672746i \(0.765115\pi\)
\(3\) 2.21119 − 0.697187i 1.27663 − 0.402521i 0.414893 − 0.909870i \(-0.363819\pi\)
0.861741 + 0.507349i \(0.169375\pi\)
\(4\) −0.551372 − 1.51488i −0.275686 − 0.757441i
\(5\) 2.43825 − 2.66089i 1.09042 − 1.18998i 0.109823 − 0.993951i \(-0.464972\pi\)
0.980597 − 0.196033i \(-0.0628060\pi\)
\(6\) −1.37716 − 0.434218i −0.562224 − 0.177269i
\(7\) 1.48595 + 0.195629i 0.561636 + 0.0739408i 0.405999 − 0.913874i \(-0.366924\pi\)
0.155637 + 0.987814i \(0.450257\pi\)
\(8\) −0.582257 + 2.17301i −0.205859 + 0.768276i
\(9\) 1.94586 − 1.36250i 0.648619 − 0.454168i
\(10\) −2.19450 + 0.486509i −0.693961 + 0.153848i
\(11\) 2.16456 + 1.66093i 0.652641 + 0.500789i 0.881202 − 0.472741i \(-0.156735\pi\)
−0.228561 + 0.973530i \(0.573402\pi\)
\(12\) −2.27535 − 2.96529i −0.656836 − 0.856005i
\(13\) −4.32122 + 5.14983i −1.19849 + 1.42831i −0.322101 + 0.946705i \(0.604389\pi\)
−0.876390 + 0.481602i \(0.840055\pi\)
\(14\) −0.688215 − 0.630633i −0.183933 − 0.168544i
\(15\) 3.53632 − 7.58365i 0.913073 − 1.95809i
\(16\) −1.39656 + 1.17185i −0.349140 + 0.292963i
\(17\) −3.73643 − 1.74330i −0.906217 − 0.422813i
\(18\) −1.47946 −0.348713
\(19\) 1.89229 + 3.92674i 0.434120 + 0.900855i
\(20\) −5.37531 − 2.22653i −1.20196 − 0.497866i
\(21\) 3.42211 − 0.603411i 0.746766 − 0.131675i
\(22\) −0.510980 − 1.62062i −0.108941 − 0.345518i
\(23\) −4.18903 + 3.83854i −0.873474 + 0.800391i −0.981040 − 0.193807i \(-0.937917\pi\)
0.107566 + 0.994198i \(0.465694\pi\)
\(24\) 0.227513 + 5.21090i 0.0464408 + 1.06367i
\(25\) −0.699459 − 7.99485i −0.139892 − 1.59897i
\(26\) 4.04428 − 1.08366i 0.793149 − 0.212524i
\(27\) −0.881497 + 1.14879i −0.169644 + 0.221085i
\(28\) −0.522955 − 2.35890i −0.0988293 − 0.445790i
\(29\) 2.24358 + 1.42932i 0.416622 + 0.265418i 0.728964 − 0.684552i \(-0.240002\pi\)
−0.312342 + 0.949970i \(0.601113\pi\)
\(30\) −4.51327 + 2.60574i −0.824008 + 0.475741i
\(31\) 5.20755 − 3.99589i 0.935304 − 0.717684i −0.0241031 − 0.999709i \(-0.507673\pi\)
0.959407 + 0.282026i \(0.0910063\pi\)
\(32\) 5.61333 − 0.491103i 0.992307 − 0.0868156i
\(33\) 5.94425 + 2.16353i 1.03476 + 0.376622i
\(34\) 1.28349 + 2.22417i 0.220116 + 0.381441i
\(35\) 4.14366 − 3.47695i 0.700407 − 0.587711i
\(36\) −3.13692 − 2.19649i −0.522820 − 0.366082i
\(37\) −3.06459 − 1.26940i −0.503816 − 0.208687i 0.116276 − 0.993217i \(-0.462904\pi\)
−0.620091 + 0.784530i \(0.712904\pi\)
\(38\) 0.437349 − 2.67932i 0.0709473 − 0.434644i
\(39\) −5.96467 + 14.4000i −0.955111 + 2.30584i
\(40\) 4.36245 + 6.84767i 0.689764 + 1.08271i
\(41\) 3.07532 + 1.60091i 0.480285 + 0.250020i 0.684647 − 0.728875i \(-0.259956\pi\)
−0.204362 + 0.978895i \(0.565512\pi\)
\(42\) −1.96145 − 0.914638i −0.302658 − 0.141132i
\(43\) −2.87340 − 6.16203i −0.438190 − 0.939701i −0.994101 − 0.108459i \(-0.965408\pi\)
0.555911 − 0.831242i \(-0.312369\pi\)
\(44\) 1.32263 − 4.19485i 0.199394 − 0.632397i
\(45\) 1.11902 − 8.49983i 0.166814 − 1.26708i
\(46\) 3.50841 − 0.461890i 0.517286 − 0.0681020i
\(47\) −1.85696 − 0.327433i −0.270866 − 0.0477610i 0.0365654 − 0.999331i \(-0.488358\pi\)
−0.307431 + 0.951570i \(0.599469\pi\)
\(48\) −2.27107 + 3.56486i −0.327800 + 0.514543i
\(49\) −4.59171 − 1.23034i −0.655958 − 0.175764i
\(50\) −2.49916 + 4.32868i −0.353435 + 0.612167i
\(51\) −9.47738 − 1.24979i −1.32710 − 0.175006i
\(52\) 10.1840 + 3.70667i 1.41226 + 0.514022i
\(53\) 0.744860 − 1.59736i 0.102314 − 0.219414i −0.848486 − 0.529218i \(-0.822486\pi\)
0.950801 + 0.309804i \(0.100263\pi\)
\(54\) 0.860105 − 0.271190i 0.117046 − 0.0369043i
\(55\) 9.69730 − 1.70990i 1.30758 − 0.230562i
\(56\) −1.29031 + 3.11508i −0.172425 + 0.416270i
\(57\) 6.92188 + 7.36350i 0.916825 + 0.975320i
\(58\) −0.634031 − 1.53069i −0.0832523 − 0.200989i
\(59\) 3.17763 − 4.53812i 0.413692 − 0.590814i −0.557186 − 0.830388i \(-0.688119\pi\)
0.970878 + 0.239574i \(0.0770078\pi\)
\(60\) −13.4382 − 1.17569i −1.73486 − 0.151780i
\(61\) −9.20455 − 10.0450i −1.17852 − 1.28613i −0.949878 − 0.312620i \(-0.898793\pi\)
−0.228644 − 0.973510i \(-0.573429\pi\)
\(62\) −4.08424 + 0.178322i −0.518699 + 0.0226469i
\(63\) 3.15799 − 1.64394i 0.397869 − 0.207117i
\(64\) 0.118425 + 0.0683727i 0.0148031 + 0.00854658i
\(65\) 3.16688 + 24.0549i 0.392804 + 2.98364i
\(66\) −2.25975 − 3.22726i −0.278156 − 0.397249i
\(67\) 2.04685 − 11.6083i 0.250062 − 1.41817i −0.558374 − 0.829590i \(-0.688574\pi\)
0.808436 − 0.588584i \(-0.200314\pi\)
\(68\) −0.580736 + 6.62145i −0.0704246 + 0.802969i
\(69\) −6.58659 + 11.4083i −0.792932 + 1.37340i
\(70\) −3.35609 + 0.293620i −0.401129 + 0.0350942i
\(71\) 4.18580 + 3.83558i 0.496763 + 0.455200i 0.884480 − 0.466579i \(-0.154514\pi\)
−0.387716 + 0.921779i \(0.626736\pi\)
\(72\) 1.82775 + 5.02170i 0.215402 + 0.591812i
\(73\) 0.0321163 + 0.101860i 0.00375892 + 0.0119218i 0.955578 − 0.294739i \(-0.0952327\pi\)
−0.951819 + 0.306661i \(0.900788\pi\)
\(74\) 1.11002 + 1.74239i 0.129038 + 0.202548i
\(75\) −7.12055 − 17.1905i −0.822210 − 1.98499i
\(76\) 4.90519 − 5.03168i 0.562663 − 0.577173i
\(77\) 2.89151 + 2.89151i 0.329518 + 0.329518i
\(78\) 8.18718 − 5.21581i 0.927015 − 0.590574i
\(79\) −7.29665 + 14.0167i −0.820938 + 1.57701i −0.00420458 + 0.999991i \(0.501338\pi\)
−0.816733 + 0.577016i \(0.804217\pi\)
\(80\) −0.286999 + 6.57337i −0.0320875 + 0.734925i
\(81\) −3.58560 + 9.85135i −0.398400 + 1.09459i
\(82\) −0.997070 − 1.91535i −0.110108 − 0.211515i
\(83\) −0.734336 − 2.74058i −0.0806039 − 0.300818i 0.913842 − 0.406071i \(-0.133101\pi\)
−0.994445 + 0.105253i \(0.966435\pi\)
\(84\) −2.80095 − 4.85139i −0.305609 − 0.529330i
\(85\) −13.7491 + 5.69160i −1.49130 + 0.617340i
\(86\) −0.735321 + 4.17021i −0.0792916 + 0.449685i
\(87\) 5.95749 + 1.59631i 0.638710 + 0.171142i
\(88\) −4.86955 + 3.73654i −0.519096 + 0.398316i
\(89\) −2.61485 + 3.11625i −0.277173 + 0.330322i −0.886615 − 0.462509i \(-0.846949\pi\)
0.609441 + 0.792831i \(0.291394\pi\)
\(90\) −3.60731 + 3.93669i −0.380244 + 0.414963i
\(91\) −7.42857 + 6.80703i −0.778726 + 0.713571i
\(92\) 8.12465 + 4.22942i 0.847053 + 0.440948i
\(93\) 8.72902 − 12.4663i 0.905157 − 1.29270i
\(94\) 0.830415 + 0.830415i 0.0856507 + 0.0856507i
\(95\) 15.0625 + 4.53922i 1.54538 + 0.465714i
\(96\) 12.0698 − 4.99947i 1.23187 − 0.510256i
\(97\) 8.54002 + 1.89328i 0.867107 + 0.192233i 0.627481 − 0.778632i \(-0.284086\pi\)
0.239626 + 0.970865i \(0.422975\pi\)
\(98\) 1.90308 + 2.26800i 0.192240 + 0.229102i
\(99\) 6.47495 + 0.282702i 0.650757 + 0.0284127i
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 323.2.bb.a.9.11 ✓ 672
17.2 even 8 inner 323.2.bb.a.104.18 yes 672
19.17 even 9 inner 323.2.bb.a.264.18 yes 672
323.36 even 72 inner 323.2.bb.a.36.11 yes 672
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
323.2.bb.a.9.11 ✓ 672 1.1 even 1 trivial
323.2.bb.a.36.11 yes 672 323.36 even 72 inner
323.2.bb.a.104.18 yes 672 17.2 even 8 inner
323.2.bb.a.264.18 yes 672 19.17 even 9 inner