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Properties

Label 35.5.l.a.18.6
Level $35$
Weight $5$
Character 35.18
Analytic conductor $3.618$
Analytic rank $0$
Dimension $56$
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] = [35,5,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), 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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 18.6
Character \(\chi\) \(=\) 35.18
Dual form 35.5.l.a.2.6

$q$-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+(-1.93554 - 0.518627i) q^{2} +(0.952096 - 0.255113i) q^{3} +(-10.3791 - 5.99235i) q^{4} +(3.12714 + 24.8036i) q^{5} -1.97513 q^{6} +(-47.7288 + 11.0890i) q^{7} +(39.6520 + 39.6520i) q^{8} +(-69.3067 + 40.0142i) q^{9} +(6.81113 - 49.6304i) q^{10} +(-46.1603 + 79.9519i) q^{11} +(-11.4106 - 3.05746i) q^{12} +(-64.8218 - 64.8218i) q^{13} +(98.1321 + 3.29027i) q^{14} +(9.30508 + 22.8177i) q^{15} +(39.6941 + 68.7522i) q^{16} +(-17.2363 - 64.3266i) q^{17} +(154.898 - 41.5049i) q^{18} +(270.008 - 155.889i) q^{19} +(116.175 - 276.177i) q^{20} +(-42.6134 + 22.7340i) q^{21} +(130.810 - 130.810i) q^{22} +(225.082 - 840.019i) q^{23} +(47.8682 + 27.6367i) q^{24} +(-605.442 + 155.129i) q^{25} +(91.8471 + 159.084i) q^{26} +(-112.234 + 112.234i) q^{27} +(561.828 + 170.915i) q^{28} +1133.18i q^{29} +(-6.17651 - 48.9905i) q^{30} +(-858.076 + 1486.23i) q^{31} +(-273.391 - 1020.31i) q^{32} +(-23.5522 + 87.8980i) q^{33} +133.446i q^{34} +(-424.301 - 1149.17i) q^{35} +959.117 q^{36} +(-370.889 - 99.3793i) q^{37} +(-603.460 + 161.697i) q^{38} +(-78.2535 - 45.1797i) q^{39} +(-859.516 + 1107.51i) q^{40} -1219.29 q^{41} +(94.2706 - 21.9022i) q^{42} +(1625.68 + 1625.68i) q^{43} +(958.200 - 553.217i) q^{44} +(-1209.23 - 1593.93i) q^{45} +(-871.313 + 1509.16i) q^{46} +(-272.196 - 72.9348i) q^{47} +(55.3322 + 55.3322i) q^{48} +(2155.07 - 1058.52i) q^{49} +(1252.31 + 13.7398i) q^{50} +(-32.8211 - 56.8479i) q^{51} +(284.354 + 1061.22i) q^{52} +(812.315 - 217.659i) q^{53} +(275.441 - 159.026i) q^{54} +(-2127.45 - 894.922i) q^{55} +(-2332.24 - 1452.84i) q^{56} +(217.304 - 217.304i) q^{57} +(587.697 - 2193.31i) q^{58} +(77.1325 + 44.5324i) q^{59} +(40.1536 - 292.585i) q^{60} +(1434.14 + 2484.01i) q^{61} +(2431.64 - 2431.64i) q^{62} +(2864.20 - 2678.37i) q^{63} +846.429i q^{64} +(1405.11 - 1810.52i) q^{65} +(91.1726 - 157.916i) q^{66} +(946.688 + 3533.09i) q^{67} +(-206.571 + 770.935i) q^{68} -857.200i q^{69} +(225.262 + 2444.32i) q^{70} +5389.96 q^{71} +(-4334.79 - 1161.50i) q^{72} +(-9116.45 + 2442.75i) q^{73} +(666.330 + 384.706i) q^{74} +(-536.863 + 302.154i) q^{75} -3736.57 q^{76} +(1316.59 - 4327.88i) q^{77} +(128.032 + 128.032i) q^{78} +(-2335.70 + 1348.52i) q^{79} +(-1581.18 + 1199.56i) q^{80} +(3162.93 - 5478.35i) q^{81} +(2359.99 + 632.357i) q^{82} +(4334.36 + 4334.36i) q^{83} +(578.517 + 19.3971i) q^{84} +(1541.63 - 628.680i) q^{85} +(-2303.45 - 3989.69i) q^{86} +(289.089 + 1078.89i) q^{87} +(-5000.60 + 1339.91i) q^{88} +(-12188.7 + 7037.13i) q^{89} +(1513.86 + 3712.26i) q^{90} +(3812.67 + 2375.06i) q^{91} +(-7369.83 + 7369.83i) q^{92} +(-437.813 + 1633.94i) q^{93} +(489.022 + 282.337i) q^{94} +(4710.97 + 6209.69i) q^{95} +(-520.589 - 901.686i) q^{96} +(-2885.72 + 2885.72i) q^{97} +(-4720.21 + 931.143i) q^{98} -7388.27i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22}+ \cdots - 78606 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93554 − 0.518627i −0.483886 − 0.129657i 0.00862561 − 0.999963i \(-0.497254\pi\)
−0.492511 + 0.870306i \(0.663921\pi\)
\(3\) 0.952096 − 0.255113i 0.105788 − 0.0283459i −0.205536 − 0.978649i \(-0.565894\pi\)
0.311325 + 0.950304i \(0.399227\pi\)
\(4\) −10.3791 − 5.99235i −0.648691 − 0.374522i
\(5\) 3.12714 + 24.8036i 0.125086 + 0.992146i
\(6\) −1.97513 −0.0548648
\(7\) −47.7288 + 11.0890i −0.974056 + 0.226305i
\(8\) 39.6520 + 39.6520i 0.619562 + 0.619562i
\(9\) −69.3067 + 40.0142i −0.855638 + 0.494003i
\(10\) 6.81113 − 49.6304i 0.0681113 − 0.496304i
\(11\) −46.1603 + 79.9519i −0.381490 + 0.660760i −0.991275 − 0.131807i \(-0.957922\pi\)
0.609786 + 0.792566i \(0.291256\pi\)
\(12\) −11.4106 − 3.05746i −0.0792401 − 0.0212323i
\(13\) −64.8218 − 64.8218i −0.383561 − 0.383561i 0.488822 − 0.872383i \(-0.337427\pi\)
−0.872383 + 0.488822i \(0.837427\pi\)
\(14\) 98.1321 + 3.29027i 0.500674 + 0.0167871i
\(15\) 9.30508 + 22.8177i 0.0413559 + 0.101412i
\(16\) 39.6941 + 68.7522i 0.155055 + 0.268563i
\(17\) −17.2363 − 64.3266i −0.0596410 − 0.222583i 0.929673 − 0.368387i \(-0.120090\pi\)
−0.989314 + 0.145804i \(0.953423\pi\)
\(18\) 154.898 − 41.5049i 0.478082 − 0.128102i
\(19\) 270.008 − 155.889i 0.747944 − 0.431826i −0.0770064 − 0.997031i \(-0.524536\pi\)
0.824951 + 0.565205i \(0.191203\pi\)
\(20\) 116.175 − 276.177i 0.290438 − 0.690443i
\(21\) −42.6134 + 22.7340i −0.0966291 + 0.0515510i
\(22\) 130.810 − 130.810i 0.270270 − 0.270270i
\(23\) 225.082 − 840.019i 0.425486 − 1.58794i −0.337372 − 0.941371i \(-0.609538\pi\)
0.762859 − 0.646565i \(-0.223795\pi\)
\(24\) 47.8682 + 27.6367i 0.0831045 + 0.0479804i
\(25\) −605.442 + 155.129i −0.968707 + 0.248206i
\(26\) 91.8471 + 159.084i 0.135868 + 0.235331i
\(27\) −112.234 + 112.234i −0.153956 + 0.153956i
\(28\) 561.828 + 170.915i 0.716618 + 0.218003i
\(29\) 1133.18i 1.34742i 0.738997 + 0.673708i \(0.235300\pi\)
−0.738997 + 0.673708i \(0.764700\pi\)
\(30\) −6.17651 − 48.9905i −0.00686279 − 0.0544338i
\(31\) −858.076 + 1486.23i −0.892899 + 1.54655i −0.0565160 + 0.998402i \(0.517999\pi\)
−0.836383 + 0.548145i \(0.815334\pi\)
\(32\) −273.391 − 1020.31i −0.266983 − 0.996395i
\(33\) −23.5522 + 87.8980i −0.0216274 + 0.0807144i
\(34\) 133.446i 0.115438i
\(35\) −424.301 − 1149.17i −0.346368 − 0.938099i
\(36\) 959.117 0.740059
\(37\) −370.889 − 99.3793i −0.270919 − 0.0725926i 0.120802 − 0.992677i \(-0.461453\pi\)
−0.391721 + 0.920084i \(0.628120\pi\)
\(38\) −603.460 + 161.697i −0.417909 + 0.111978i
\(39\) −78.2535 − 45.1797i −0.0514487 − 0.0297039i
\(40\) −859.516 + 1107.51i −0.537198 + 0.692194i
\(41\) −1219.29 −0.725336 −0.362668 − 0.931918i \(-0.618134\pi\)
−0.362668 + 0.931918i \(0.618134\pi\)
\(42\) 94.2706 − 21.9022i 0.0534414 − 0.0124162i
\(43\) 1625.68 + 1625.68i 0.879221 + 0.879221i 0.993454 − 0.114233i \(-0.0364411\pi\)
−0.114233 + 0.993454i \(0.536441\pi\)
\(44\) 958.200 − 553.217i 0.494938 − 0.285753i
\(45\) −1209.23 − 1593.93i −0.597151 − 0.787125i
\(46\) −871.313 + 1509.16i −0.411774 + 0.713213i
\(47\) −272.196 − 72.9348i −0.123222 − 0.0330171i 0.196681 − 0.980467i \(-0.436984\pi\)
−0.319903 + 0.947450i \(0.603650\pi\)
\(48\) 55.3322 + 55.3322i 0.0240157 + 0.0240157i
\(49\) 2155.07 − 1058.52i 0.897572 − 0.440868i
\(50\) 1252.31 + 13.7398i 0.500925 + 0.00549590i
\(51\) −32.8211 − 56.8479i −0.0126187 − 0.0218562i
\(52\) 284.354 + 1061.22i 0.105161 + 0.392465i
\(53\) 812.315 − 217.659i 0.289183 − 0.0774863i −0.111311 − 0.993786i \(-0.535505\pi\)
0.400494 + 0.916299i \(0.368838\pi\)
\(54\) 275.441 − 159.026i 0.0944586 − 0.0545357i
\(55\) −2127.45 − 894.922i −0.703289 − 0.295842i
\(56\) −2332.24 − 1452.84i −0.743698 − 0.463278i
\(57\) 217.304 − 217.304i 0.0668833 − 0.0668833i
\(58\) 587.697 − 2193.31i 0.174702 − 0.651996i
\(59\) 77.1325 + 44.5324i 0.0221581 + 0.0127930i 0.511038 − 0.859558i \(-0.329261\pi\)
−0.488880 + 0.872351i \(0.662594\pi\)
\(60\) 40.1536 − 292.585i 0.0111538 − 0.0812736i
\(61\) 1434.14 + 2484.01i 0.385419 + 0.667565i 0.991827 − 0.127588i \(-0.0407236\pi\)
−0.606408 + 0.795153i \(0.707390\pi\)
\(62\) 2431.64 − 2431.64i 0.632582 − 0.632582i
\(63\) 2864.20 − 2678.37i 0.721644 − 0.674822i
\(64\) 846.429i 0.206648i
\(65\) 1405.11 − 1810.52i 0.332571 − 0.428527i
\(66\) 91.1726 − 157.916i 0.0209303 − 0.0362524i
\(67\) 946.688 + 3533.09i 0.210891 + 0.787054i 0.987573 + 0.157162i \(0.0502344\pi\)
−0.776682 + 0.629892i \(0.783099\pi\)
\(68\) −206.571 + 770.935i −0.0446737 + 0.166725i
\(69\) − 857.200i − 0.180046i
\(70\) 225.262 + 2444.32i 0.0459719 + 0.498842i
\(71\) 5389.96 1.06922 0.534612 − 0.845098i \(-0.320458\pi\)
0.534612 + 0.845098i \(0.320458\pi\)
\(72\) −4334.79 − 1161.50i −0.836186 − 0.224055i
\(73\) −9116.45 + 2442.75i −1.71072 + 0.458387i −0.975602 − 0.219547i \(-0.929542\pi\)
−0.735123 + 0.677934i \(0.762875\pi\)
\(74\) 666.330 + 384.706i 0.121682 + 0.0702531i
\(75\) −536.863 + 302.154i −0.0954424 + 0.0537163i
\(76\) −3736.57 −0.646913
\(77\) 1316.59 − 4327.88i 0.222059 − 0.729950i
\(78\) 128.032 + 128.032i 0.0210440 + 0.0210440i
\(79\) −2335.70 + 1348.52i −0.374251 + 0.216074i −0.675314 − 0.737530i \(-0.735992\pi\)
0.301063 + 0.953604i \(0.402659\pi\)
\(80\) −1581.18 + 1199.56i −0.247059 + 0.187431i
\(81\) 3162.93 − 5478.35i 0.482080 − 0.834987i
\(82\) 2359.99 + 632.357i 0.350980 + 0.0940447i
\(83\) 4334.36 + 4334.36i 0.629171 + 0.629171i 0.947859 − 0.318689i \(-0.103242\pi\)
−0.318689 + 0.947859i \(0.603242\pi\)
\(84\) 578.517 + 19.3971i 0.0819894 + 0.00274902i
\(85\) 1541.63 − 628.680i 0.213375 − 0.0870146i
\(86\) −2303.45 − 3989.69i −0.311445 − 0.539439i
\(87\) 289.089 + 1078.89i 0.0381938 + 0.142541i
\(88\) −5000.60 + 1339.91i −0.645738 + 0.173025i
\(89\) −12188.7 + 7037.13i −1.53878 + 0.888414i −0.539868 + 0.841749i \(0.681526\pi\)
−0.998911 + 0.0466649i \(0.985141\pi\)
\(90\) 1513.86 + 3712.26i 0.186897 + 0.458303i
\(91\) 3812.67 + 2375.06i 0.460412 + 0.286808i
\(92\) −7369.83 + 7369.83i −0.870726 + 0.870726i
\(93\) −437.813 + 1633.94i −0.0506201 + 0.188917i
\(94\) 489.022 + 282.337i 0.0553443 + 0.0319530i
\(95\) 4710.97 + 6209.69i 0.521991 + 0.688055i
\(96\) −520.589 − 901.686i −0.0564875 − 0.0978392i
\(97\) −2885.72 + 2885.72i −0.306698 + 0.306698i −0.843627 − 0.536929i \(-0.819584\pi\)
0.536929 + 0.843627i \(0.319584\pi\)
\(98\) −4720.21 + 931.143i −0.491484 + 0.0969536i
\(99\) − 7388.27i − 0.753828i
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 35.5.l.a.18.6 yes 56
5.2 odd 4 inner 35.5.l.a.32.9 yes 56
7.2 even 3 inner 35.5.l.a.23.9 yes 56
35.2 odd 12 inner 35.5.l.a.2.6 ✓ 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.6 ✓ 56 35.2 odd 12 inner
35.5.l.a.18.6 yes 56 1.1 even 1 trivial
35.5.l.a.23.9 yes 56 7.2 even 3 inner
35.5.l.a.32.9 yes 56 5.2 odd 4 inner