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Properties

Label 323.2.bb.a.9.14
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 9.14
Character \(\chi\) \(=\) 323.9
Dual form 323.2.bb.a.36.14

$q$-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+(-0.0361768 - 0.0253313i) q^{2} +(-1.65742 + 0.522584i) q^{3} +(-0.683373 - 1.87755i) q^{4} +(-0.496668 + 0.542018i) q^{5} +(0.0731980 + 0.0230792i) q^{6} +(0.0705367 + 0.00928633i) q^{7} +(-0.0456993 + 0.170552i) q^{8} +(0.0165057 - 0.0115574i) q^{9} +(0.0316978 - 0.00702724i) q^{10} +(3.37080 + 2.58651i) q^{11} +(2.11382 + 2.75478i) q^{12} +(-2.12630 + 2.53403i) q^{13} +(-0.00231656 - 0.00212273i) q^{14} +(0.539940 - 1.15790i) q^{15} +(-3.05522 + 2.56363i) q^{16} +(4.08877 + 0.530973i) q^{17} -0.000889884 q^{18} +(-0.263992 + 4.35090i) q^{19} +(1.35708 + 0.562120i) q^{20} +(-0.121762 + 0.0214700i) q^{21} +(-0.0564253 - 0.178958i) q^{22} +(-3.49241 + 3.20021i) q^{23} +(-0.0133847 - 0.306559i) q^{24} +(0.388674 + 4.44257i) q^{25} +(0.141113 - 0.0378111i) q^{26} +(3.15251 - 4.10842i) q^{27} +(-0.0307673 - 0.138782i) q^{28} +(-3.20632 - 2.04265i) q^{29} +(-0.0488645 + 0.0282119i) q^{30} +(-1.17891 + 0.904609i) q^{31} +(0.527261 - 0.0461294i) q^{32} +(-6.93852 - 2.52541i) q^{33} +(-0.134468 - 0.122783i) q^{34} +(-0.0400667 + 0.0336199i) q^{35} +(-0.0329791 - 0.0230922i) q^{36} +(6.66847 + 2.76217i) q^{37} +(0.119764 - 0.150714i) q^{38} +(2.19994 - 5.31113i) q^{39} +(-0.0697450 - 0.109478i) q^{40} +(-5.66514 - 2.94909i) q^{41} +(0.00494882 + 0.00230767i) q^{42} +(4.36596 + 9.36283i) q^{43} +(2.55279 - 8.09641i) q^{44} +(-0.00193352 + 0.0146865i) q^{45} +(0.207410 - 0.0273060i) q^{46} +(-3.04993 - 0.537785i) q^{47} +(3.72408 - 5.84563i) q^{48} +(-6.75659 - 1.81042i) q^{49} +(0.0984748 - 0.170563i) q^{50} +(-7.05431 + 1.25668i) q^{51} +(6.21083 + 2.26056i) q^{52} +(4.41946 - 9.47755i) q^{53} +(-0.218119 + 0.0687727i) q^{54} +(-3.07610 + 0.542400i) q^{55} +(-0.00480728 + 0.0116058i) q^{56} +(-1.83616 - 7.34924i) q^{57} +(0.0642514 + 0.155116i) q^{58} +(-1.71551 + 2.45000i) q^{59} +(-2.54301 - 0.222484i) q^{60} +(-4.81192 - 5.25129i) q^{61} +(0.0655640 - 0.00286259i) q^{62} +(0.00127158 - 0.000661943i) q^{63} +(6.88770 + 3.97661i) q^{64} +(-0.317423 - 2.41106i) q^{65} +(0.187041 + 0.267123i) q^{66} +(1.18423 - 6.71609i) q^{67} +(-1.79723 - 8.03974i) q^{68} +(4.11603 - 7.12918i) q^{69} +(0.00230112 - 0.000201322i) q^{70} +(1.79095 + 1.64111i) q^{71} +(0.00121684 + 0.00334324i) q^{72} +(2.38613 + 7.56782i) q^{73} +(-0.171275 - 0.268847i) q^{74} +(-2.96581 - 7.16011i) q^{75} +(8.34944 - 2.47763i) q^{76} +(0.213746 + 0.213746i) q^{77} +(-0.214124 + 0.136412i) q^{78} +(4.56773 - 8.77454i) q^{79} +(0.127894 - 2.92926i) q^{80} +(-3.09872 + 8.51366i) q^{81} +(0.130243 + 0.250194i) q^{82} +(3.97159 + 14.8222i) q^{83} +(0.123520 + 0.213943i) q^{84} +(-2.31856 + 1.95247i) q^{85} +(0.0792259 - 0.449312i) q^{86} +(6.38168 + 1.70997i) q^{87} +(-0.595178 + 0.456696i) q^{88} +(-6.53860 + 7.79240i) q^{89} +(0.000441977 - 0.000482333i) q^{90} +(-0.173514 + 0.158996i) q^{91} +(8.39518 + 4.37025i) q^{92} +(1.48122 - 2.11540i) q^{93} +(0.0967139 + 0.0967139i) q^{94} +(-2.22715 - 2.30404i) q^{95} +(-0.849789 + 0.351994i) q^{96} +(-11.7039 - 2.59468i) q^{97} +(0.198571 + 0.236648i) q^{98} +(0.0855306 + 0.00373435i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0361768 − 0.0253313i −0.0255808 − 0.0179119i 0.560716 − 0.828008i \(-0.310526\pi\)
−0.586297 + 0.810096i \(0.699415\pi\)
\(3\) −1.65742 + 0.522584i −0.956914 + 0.301714i −0.738584 − 0.674161i \(-0.764505\pi\)
−0.218330 + 0.975875i \(0.570061\pi\)
\(4\) −0.683373 − 1.87755i −0.341687 − 0.938776i
\(5\) −0.496668 + 0.542018i −0.222117 + 0.242398i −0.838306 − 0.545201i \(-0.816454\pi\)
0.616189 + 0.787598i \(0.288676\pi\)
\(6\) 0.0731980 + 0.0230792i 0.0298830 + 0.00942206i
\(7\) 0.0705367 + 0.00928633i 0.0266604 + 0.00350990i 0.143845 − 0.989600i \(-0.454053\pi\)
−0.117184 + 0.993110i \(0.537387\pi\)
\(8\) −0.0456993 + 0.170552i −0.0161572 + 0.0602993i
\(9\) 0.0165057 − 0.0115574i 0.00550188 − 0.00385246i
\(10\) 0.0316978 − 0.00702724i 0.0100237 − 0.00222221i
\(11\) 3.37080 + 2.58651i 1.01634 + 0.779862i 0.975655 − 0.219311i \(-0.0703808\pi\)
0.0406804 + 0.999172i \(0.487047\pi\)
\(12\) 2.11382 + 2.75478i 0.610207 + 0.795237i
\(13\) −2.12630 + 2.53403i −0.589730 + 0.702813i −0.975554 − 0.219760i \(-0.929473\pi\)
0.385824 + 0.922572i \(0.373917\pi\)
\(14\) −0.00231656 − 0.00212273i −0.000619126 − 0.000567324i
\(15\) 0.539940 − 1.15790i 0.139412 − 0.298970i
\(16\) −3.05522 + 2.56363i −0.763804 + 0.640908i
\(17\) 4.08877 + 0.530973i 0.991673 + 0.128780i
\(18\) −0.000889884 0 −0.000209748 0
\(19\) −0.263992 + 4.35090i −0.0605639 + 0.998164i
\(20\) 1.35708 + 0.562120i 0.303452 + 0.125694i
\(21\) −0.121762 + 0.0214700i −0.0265707 + 0.00468513i
\(22\) −0.0564253 − 0.178958i −0.0120299 − 0.0381540i
\(23\) −3.49241 + 3.20021i −0.728218 + 0.667289i −0.951564 − 0.307450i \(-0.900524\pi\)
0.223346 + 0.974739i \(0.428302\pi\)
\(24\) −0.0133847 − 0.306559i −0.00273213 − 0.0625761i
\(25\) 0.388674 + 4.44257i 0.0777349 + 0.888514i
\(26\) 0.141113 − 0.0378111i 0.0276745 − 0.00741536i
\(27\) 3.15251 − 4.10842i 0.606700 − 0.790667i
\(28\) −0.0307673 − 0.138782i −0.00581448 − 0.0262274i
\(29\) −3.20632 − 2.04265i −0.595398 − 0.379310i 0.205015 − 0.978759i \(-0.434276\pi\)
−0.800413 + 0.599448i \(0.795387\pi\)
\(30\) −0.0488645 + 0.0282119i −0.00892139 + 0.00515077i
\(31\) −1.17891 + 0.904609i −0.211738 + 0.162473i −0.709186 − 0.705021i \(-0.750937\pi\)
0.497448 + 0.867494i \(0.334271\pi\)
\(32\) 0.527261 − 0.0461294i 0.0932075 − 0.00815460i
\(33\) −6.93852 − 2.52541i −1.20784 − 0.439618i
\(34\) −0.134468 − 0.122783i −0.0230611 − 0.0210571i
\(35\) −0.0400667 + 0.0336199i −0.00677251 + 0.00568281i
\(36\) −0.0329791 − 0.0230922i −0.00549652 − 0.00384870i
\(37\) 6.66847 + 2.76217i 1.09629 + 0.454098i 0.856196 − 0.516652i \(-0.172822\pi\)
0.240094 + 0.970750i \(0.422822\pi\)
\(38\) 0.119764 − 0.150714i 0.0194283 − 0.0244491i
\(39\) 2.19994 − 5.31113i 0.352273 − 0.850461i
\(40\) −0.0697450 − 0.109478i −0.0110276 − 0.0173099i
\(41\) −5.66514 − 2.94909i −0.884747 − 0.460570i −0.0421475 − 0.999111i \(-0.513420\pi\)
−0.842599 + 0.538542i \(0.818976\pi\)
\(42\) 0.00494882 + 0.00230767i 0.000763620 + 0.000356082i
\(43\) 4.36596 + 9.36283i 0.665802 + 1.42782i 0.891227 + 0.453557i \(0.149845\pi\)
−0.225425 + 0.974261i \(0.572377\pi\)
\(44\) 2.55279 − 8.09641i 0.384847 − 1.22058i
\(45\) −0.00193352 + 0.0146865i −0.000288232 + 0.00218934i
\(46\) 0.207410 − 0.0273060i 0.0305809 − 0.00402605i
\(47\) −3.04993 − 0.537785i −0.444878 − 0.0784440i −0.0532784 − 0.998580i \(-0.516967\pi\)
−0.391600 + 0.920136i \(0.628078\pi\)
\(48\) 3.72408 − 5.84563i 0.537524 − 0.843744i
\(49\) −6.75659 − 1.81042i −0.965227 − 0.258632i
\(50\) 0.0984748 − 0.170563i 0.0139264 − 0.0241213i
\(51\) −7.05431 + 1.25668i −0.987801 + 0.175970i
\(52\) 6.21083 + 2.26056i 0.861287 + 0.313483i
\(53\) 4.41946 − 9.47755i 0.607059 − 1.30184i −0.327102 − 0.944989i \(-0.606072\pi\)
0.934161 − 0.356853i \(-0.116150\pi\)
\(54\) −0.218119 + 0.0687727i −0.0296822 + 0.00935878i
\(55\) −3.07610 + 0.542400i −0.414782 + 0.0731372i
\(56\) −0.00480728 + 0.0116058i −0.000642400 + 0.00155089i
\(57\) −1.83616 − 7.34924i −0.243206 − 0.973431i
\(58\) 0.0642514 + 0.155116i 0.00843662 + 0.0203678i
\(59\) −1.71551 + 2.45000i −0.223341 + 0.318963i −0.915147 − 0.403119i \(-0.867926\pi\)
0.691807 + 0.722082i \(0.256815\pi\)
\(60\) −2.54301 − 0.222484i −0.328301 − 0.0287226i
\(61\) −4.81192 − 5.25129i −0.616103 − 0.672358i 0.348120 − 0.937450i \(-0.386820\pi\)
−0.964223 + 0.265092i \(0.914598\pi\)
\(62\) 0.0655640 − 0.00286259i 0.00832664 − 0.000363549i
\(63\) 0.00127158 0.000661943i 0.000160204 8.33969e-5i
\(64\) 6.88770 + 3.97661i 0.860962 + 0.497077i
\(65\) −0.317423 − 2.41106i −0.0393714 − 0.299056i
\(66\) 0.187041 + 0.267123i 0.0230232 + 0.0328805i
\(67\) 1.18423 − 6.71609i 0.144676 − 0.820500i −0.822950 − 0.568114i \(-0.807673\pi\)
0.967626 − 0.252387i \(-0.0812155\pi\)
\(68\) −1.79723 − 8.03974i −0.217946 − 0.974962i
\(69\) 4.11603 − 7.12918i 0.495512 − 0.858252i
\(70\) 0.00230112 0.000201322i 0.000275036 2.40626e-5i
\(71\) 1.79095 + 1.64111i 0.212547 + 0.194764i 0.774809 − 0.632196i \(-0.217846\pi\)
−0.562261 + 0.826960i \(0.690068\pi\)
\(72\) 0.00121684 + 0.00334324i 0.000143406 + 0.000394005i
\(73\) 2.38613 + 7.56782i 0.279275 + 0.885747i 0.984291 + 0.176555i \(0.0564954\pi\)
−0.705016 + 0.709191i \(0.749060\pi\)
\(74\) −0.171275 − 0.268847i −0.0199103 − 0.0312529i
\(75\) −2.96581 − 7.16011i −0.342463 − 0.826778i
\(76\) 8.34944 − 2.47763i 0.957747 − 0.284203i
\(77\) 0.213746 + 0.213746i 0.0243586 + 0.0243586i
\(78\) −0.214124 + 0.136412i −0.0242448 + 0.0154457i
\(79\) 4.56773 − 8.77454i 0.513910 − 0.987213i −0.480026 − 0.877254i \(-0.659373\pi\)
0.993936 − 0.109958i \(-0.0350717\pi\)
\(80\) 0.127894 − 2.92926i 0.0142990 − 0.327501i
\(81\) −3.09872 + 8.51366i −0.344302 + 0.945962i
\(82\) 0.130243 + 0.250194i 0.0143829 + 0.0276293i
\(83\) 3.97159 + 14.8222i 0.435938 + 1.62694i 0.738811 + 0.673912i \(0.235387\pi\)
−0.302873 + 0.953031i \(0.597946\pi\)
\(84\) 0.123520 + 0.213943i 0.0134771 + 0.0233431i
\(85\) −2.31856 + 1.95247i −0.251483 + 0.211775i
\(86\) 0.0792259 − 0.449312i 0.00854314 − 0.0484506i
\(87\) 6.38168 + 1.70997i 0.684188 + 0.183328i
\(88\) −0.595178 + 0.456696i −0.0634462 + 0.0486840i
\(89\) −6.53860 + 7.79240i −0.693090 + 0.825993i −0.991726 − 0.128374i \(-0.959024\pi\)
0.298635 + 0.954367i \(0.403469\pi\)
\(90\) 0.000441977 0 0.000482333i 4.65885e−5 0 5.08424e-5i
\(91\) −0.173514 + 0.158996i −0.0181892 + 0.0166674i
\(92\) 8.39518 + 4.37025i 0.875258 + 0.455630i
\(93\) 1.48122 − 2.11540i 0.153595 − 0.219357i
\(94\) 0.0967139 + 0.0967139i 0.00997528 + 0.00997528i
\(95\) −2.22715 − 2.30404i −0.228501 − 0.236389i
\(96\) −0.849789 + 0.351994i −0.0867313 + 0.0359253i
\(97\) −11.7039 − 2.59468i −1.18835 − 0.263450i −0.422434 − 0.906394i \(-0.638824\pi\)
−0.765914 + 0.642943i \(0.777713\pi\)
\(98\) 0.198571 + 0.236648i 0.0200587 + 0.0239051i
\(99\) 0.0855306 + 0.00373435i 0.00859615 + 0.000375316i
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.9.14 ✓ 672
17.2 even 8 inner 323.2.bb.a.104.15 yes 672
19.17 even 9 inner 323.2.bb.a.264.15 yes 672
323.36 even 72 inner 323.2.bb.a.36.14 yes 672
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
323.2.bb.a.9.14 ✓ 672 1.1 even 1 trivial
323.2.bb.a.36.14 yes 672 323.36 even 72 inner
323.2.bb.a.104.15 yes 672 17.2 even 8 inner
323.2.bb.a.264.15 yes 672 19.17 even 9 inner