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Properties

Label 343.4.a.d.1.12
Level $343$
Weight $4$
Character 343.1
Self dual yes
Analytic conductor $20.238$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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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] = [343,4,Mod(1,343)] 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("343.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(343, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 343 = 7^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 343.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,2,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(20.2376551320\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 101 x^{16} + 200 x^{15} + 4071 x^{14} - 7805 x^{13} - 84126 x^{12} + 151605 x^{11} + \cdots + 1016000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 7^{10} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(1.84535\) of defining polynomial
Character \(\chi\) \(=\) 343.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.84535 q^{2} +2.92703 q^{3} -4.59470 q^{4} +6.48305 q^{5} +5.40139 q^{6} -23.2416 q^{8} -18.4325 q^{9} +11.9635 q^{10} -12.0672 q^{11} -13.4488 q^{12} -60.7848 q^{13} +18.9761 q^{15} -6.13120 q^{16} +38.2229 q^{17} -34.0143 q^{18} -140.830 q^{19} -29.7877 q^{20} -22.2682 q^{22} +98.8923 q^{23} -68.0288 q^{24} -82.9700 q^{25} -112.169 q^{26} -132.982 q^{27} +87.8009 q^{29} +35.0175 q^{30} +93.4330 q^{31} +174.618 q^{32} -35.3212 q^{33} +70.5344 q^{34} +84.6917 q^{36} -267.693 q^{37} -259.879 q^{38} -177.919 q^{39} -150.676 q^{40} -48.2771 q^{41} -24.5034 q^{43} +55.4453 q^{44} -119.499 q^{45} +182.491 q^{46} -548.470 q^{47} -17.9462 q^{48} -153.108 q^{50} +111.880 q^{51} +279.288 q^{52} +503.056 q^{53} -245.398 q^{54} -78.2326 q^{55} -412.213 q^{57} +162.023 q^{58} -112.834 q^{59} -87.1894 q^{60} -762.471 q^{61} +172.416 q^{62} +371.281 q^{64} -394.071 q^{65} -65.1799 q^{66} -91.9446 q^{67} -175.622 q^{68} +289.461 q^{69} +147.463 q^{71} +428.400 q^{72} +834.710 q^{73} -493.986 q^{74} -242.856 q^{75} +647.069 q^{76} -328.322 q^{78} -411.428 q^{79} -39.7489 q^{80} +108.434 q^{81} -89.0880 q^{82} +833.158 q^{83} +247.801 q^{85} -45.2172 q^{86} +256.996 q^{87} +280.462 q^{88} +670.346 q^{89} -220.517 q^{90} -454.380 q^{92} +273.482 q^{93} -1012.12 q^{94} -913.006 q^{95} +511.114 q^{96} -1145.11 q^{97} +222.429 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} - 22 q^{3} + 62 q^{4} - 62 q^{5} - 41 q^{6} - 18 q^{8} + 138 q^{9} - 78 q^{10} - 26 q^{11} - 245 q^{12} - 168 q^{13} + 118 q^{15} + 358 q^{16} - 568 q^{17} + 106 q^{18} - 188 q^{19} - 756 q^{20}+ \cdots + 512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.84535 0.652429 0.326214 − 0.945296i \(-0.394227\pi\)
0.326214 + 0.945296i \(0.394227\pi\)
\(3\) 2.92703 0.563308 0.281654 − 0.959516i \(-0.409117\pi\)
0.281654 + 0.959516i \(0.409117\pi\)
\(4\) −4.59470 −0.574337
\(5\) 6.48305 0.579862 0.289931 − 0.957048i \(-0.406368\pi\)
0.289931 + 0.957048i \(0.406368\pi\)
\(6\) 5.40139 0.367518
\(7\) 0 0
\(8\) −23.2416 −1.02714
\(9\) −18.4325 −0.682685
\(10\) 11.9635 0.378318
\(11\) −12.0672 −0.330765 −0.165382 − 0.986230i \(-0.552886\pi\)
−0.165382 + 0.986230i \(0.552886\pi\)
\(12\) −13.4488 −0.323528
\(13\) −60.7848 −1.29682 −0.648411 − 0.761291i \(-0.724566\pi\)
−0.648411 + 0.761291i \(0.724566\pi\)
\(14\) 0 0
\(15\) 18.9761 0.326641
\(16\) −6.13120 −0.0958000
\(17\) 38.2229 0.545318 0.272659 − 0.962111i \(-0.412097\pi\)
0.272659 + 0.962111i \(0.412097\pi\)
\(18\) −34.0143 −0.445403
\(19\) −140.830 −1.70045 −0.850225 − 0.526420i \(-0.823534\pi\)
−0.850225 + 0.526420i \(0.823534\pi\)
\(20\) −29.7877 −0.333036
\(21\) 0 0
\(22\) −22.2682 −0.215800
\(23\) 98.8923 0.896542 0.448271 − 0.893898i \(-0.352040\pi\)
0.448271 + 0.893898i \(0.352040\pi\)
\(24\) −68.0288 −0.578597
\(25\) −82.9700 −0.663760
\(26\) −112.169 −0.846083
\(27\) −132.982 −0.947869
\(28\) 0 0
\(29\) 87.8009 0.562215 0.281107 − 0.959676i \(-0.409298\pi\)
0.281107 + 0.959676i \(0.409298\pi\)
\(30\) 35.0175 0.213110
\(31\) 93.4330 0.541325 0.270662 − 0.962674i \(-0.412757\pi\)
0.270662 + 0.962674i \(0.412757\pi\)
\(32\) 174.618 0.964640
\(33\) −35.3212 −0.186322
\(34\) 70.5344 0.355781
\(35\) 0 0
\(36\) 84.6917 0.392091
\(37\) −267.693 −1.18942 −0.594708 − 0.803942i \(-0.702732\pi\)
−0.594708 + 0.803942i \(0.702732\pi\)
\(38\) −259.879 −1.10942
\(39\) −177.919 −0.730509
\(40\) −150.676 −0.595601
\(41\) −48.2771 −0.183893 −0.0919466 − 0.995764i \(-0.529309\pi\)
−0.0919466 + 0.995764i \(0.529309\pi\)
\(42\) 0 0
\(43\) −24.5034 −0.0869006 −0.0434503 − 0.999056i \(-0.513835\pi\)
−0.0434503 + 0.999056i \(0.513835\pi\)
\(44\) 55.4453 0.189970
\(45\) −119.499 −0.395863
\(46\) 182.491 0.584930
\(47\) −548.470 −1.70218 −0.851091 − 0.525018i \(-0.824058\pi\)
−0.851091 + 0.525018i \(0.824058\pi\)
\(48\) −17.9462 −0.0539649
\(49\) 0 0
\(50\) −153.108 −0.433056
\(51\) 111.880 0.307182
\(52\) 279.288 0.744812
\(53\) 503.056 1.30378 0.651888 − 0.758315i \(-0.273977\pi\)
0.651888 + 0.758315i \(0.273977\pi\)
\(54\) −245.398 −0.618417
\(55\) −78.2326 −0.191798
\(56\) 0 0
\(57\) −412.213 −0.957876
\(58\) 162.023 0.366805
\(59\) −112.834 −0.248979 −0.124489 − 0.992221i \(-0.539729\pi\)
−0.124489 + 0.992221i \(0.539729\pi\)
\(60\) −87.1894 −0.187602
\(61\) −762.471 −1.60040 −0.800200 − 0.599733i \(-0.795273\pi\)
−0.800200 + 0.599733i \(0.795273\pi\)
\(62\) 172.416 0.353176
\(63\) 0 0
\(64\) 371.281 0.725159
\(65\) −394.071 −0.751977
\(66\) −65.1799 −0.121562
\(67\) −91.9446 −0.167654 −0.0838271 − 0.996480i \(-0.526714\pi\)
−0.0838271 + 0.996480i \(0.526714\pi\)
\(68\) −175.622 −0.313196
\(69\) 289.461 0.505029
\(70\) 0 0
\(71\) 147.463 0.246488 0.123244 − 0.992376i \(-0.460670\pi\)
0.123244 + 0.992376i \(0.460670\pi\)
\(72\) 428.400 0.701214
\(73\) 834.710 1.33829 0.669147 − 0.743130i \(-0.266660\pi\)
0.669147 + 0.743130i \(0.266660\pi\)
\(74\) −493.986 −0.776009
\(75\) −242.856 −0.373901
\(76\) 647.069 0.976631
\(77\) 0 0
\(78\) −328.322 −0.476605
\(79\) −411.428 −0.585940 −0.292970 − 0.956122i \(-0.594644\pi\)
−0.292970 + 0.956122i \(0.594644\pi\)
\(80\) −39.7489 −0.0555508
\(81\) 108.434 0.148743
\(82\) −89.0880 −0.119977
\(83\) 833.158 1.10182 0.550910 − 0.834565i \(-0.314281\pi\)
0.550910 + 0.834565i \(0.314281\pi\)
\(84\) 0 0
\(85\) 247.801 0.316209
\(86\) −45.2172 −0.0566964
\(87\) 256.996 0.316700
\(88\) 280.462 0.339742
\(89\) 670.346 0.798388 0.399194 − 0.916866i \(-0.369290\pi\)
0.399194 + 0.916866i \(0.369290\pi\)
\(90\) −220.517 −0.258272
\(91\) 0 0
\(92\) −454.380 −0.514917
\(93\) 273.482 0.304932
\(94\) −1012.12 −1.11055
\(95\) −913.006 −0.986026
\(96\) 511.114 0.543389
\(97\) −1145.11 −1.19865 −0.599323 − 0.800507i \(-0.704563\pi\)
−0.599323 + 0.800507i \(0.704563\pi\)
\(98\) 0 0
\(99\) 222.429 0.225808
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 343.4.a.d.1.12 ✓ 18
7.6 odd 2 343.4.a.e.1.12 yes 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
343.4.a.d.1.12 ✓ 18 1.1 even 1 trivial
343.4.a.e.1.12 yes 18 7.6 odd 2