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Properties

Label 35.5.l.a.2.2
Level $35$
Weight $5$
Character 35.2
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 2.2
Character \(\chi\) \(=\) 35.2
Dual form 35.5.l.a.18.2

$q$-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+(-5.71329 + 1.53087i) q^{2} +(-15.4449 - 4.13845i) q^{3} +(16.4417 - 9.49261i) q^{4} +(-22.7561 - 10.3519i) q^{5} +94.5767 q^{6} +(-15.2970 + 46.5511i) q^{7} +(-12.4854 + 12.4854i) q^{8} +(151.270 + 87.3360i) q^{9} +(145.859 + 24.3067i) q^{10} +(-42.5401 - 73.6816i) q^{11} +(-293.225 + 78.5694i) q^{12} +(-10.7310 + 10.7310i) q^{13} +(16.1328 - 289.377i) q^{14} +(308.625 + 254.059i) q^{15} +(-99.6625 + 172.620i) q^{16} +(93.8444 - 350.232i) q^{17} +(-997.952 - 267.400i) q^{18} +(-81.5338 - 47.0736i) q^{19} +(-472.414 + 45.8122i) q^{20} +(428.911 - 655.671i) q^{21} +(355.841 + 355.841i) q^{22} +(143.194 + 534.406i) q^{23} +(244.507 - 141.166i) q^{24} +(410.677 + 471.136i) q^{25} +(44.8815 - 77.7370i) q^{26} +(-1059.10 - 1059.10i) q^{27} +(190.382 + 910.587i) q^{28} +228.863i q^{29} +(-2152.19 - 979.046i) q^{30} +(408.154 + 706.943i) q^{31} +(378.261 - 1411.69i) q^{32} +(352.100 + 1314.06i) q^{33} +2144.64i q^{34} +(829.991 - 900.966i) q^{35} +3316.19 q^{36} +(1937.56 - 519.167i) q^{37} +(537.890 + 144.127i) q^{38} +(210.149 - 121.330i) q^{39} +(413.367 - 154.872i) q^{40} -151.398 q^{41} +(-1446.74 + 4402.64i) q^{42} +(209.207 - 209.207i) q^{43} +(-1398.86 - 807.633i) q^{44} +(-2538.23 - 3553.36i) q^{45} +(-1636.21 - 2834.01i) q^{46} +(-1306.86 + 350.173i) q^{47} +(2253.66 - 2253.66i) q^{48} +(-1933.00 - 1424.19i) q^{49} +(-3067.57 - 2063.04i) q^{50} +(-2898.84 + 5020.93i) q^{51} +(-74.5705 + 278.301i) q^{52} +(1024.67 + 274.560i) q^{53} +(7672.29 + 4429.60i) q^{54} +(205.302 + 2117.07i) q^{55} +(-390.220 - 772.201i) q^{56} +(1064.47 + 1064.47i) q^{57} +(-350.359 - 1307.56i) q^{58} +(-1517.42 + 876.081i) q^{59} +(7485.99 + 1247.50i) q^{60} +(3536.71 - 6125.76i) q^{61} +(-3414.14 - 3414.14i) q^{62} +(-6379.58 + 5705.82i) q^{63} +5455.25i q^{64} +(355.281 - 133.109i) q^{65} +(-4023.30 - 6968.56i) q^{66} +(-786.257 + 2934.35i) q^{67} +(-1781.66 - 6649.24i) q^{68} -8846.46i q^{69} +(-3362.72 + 6418.09i) q^{70} +4893.58 q^{71} +(-2979.11 + 798.249i) q^{72} +(7641.14 + 2047.44i) q^{73} +(-10275.0 + 5932.30i) q^{74} +(-4393.10 - 8976.22i) q^{75} -1787.40 q^{76} +(4080.69 - 853.175i) q^{77} +(-1014.90 + 1014.90i) q^{78} +(4206.19 + 2428.45i) q^{79} +(4054.87 - 2896.47i) q^{80} +(4900.45 + 8487.83i) q^{81} +(864.978 - 231.770i) q^{82} +(-3185.61 + 3185.61i) q^{83} +(827.988 - 14851.8i) q^{84} +(-5761.09 + 6998.44i) q^{85} +(-874.990 + 1515.53i) q^{86} +(947.137 - 3534.76i) q^{87} +(1451.08 + 388.815i) q^{88} +(-1185.37 - 684.375i) q^{89} +(19941.4 + 16415.7i) q^{90} +(-335.387 - 663.692i) q^{91} +(7427.26 + 7427.26i) q^{92} +(-3378.25 - 12607.8i) q^{93} +(6930.42 - 4001.28i) q^{94} +(1368.09 + 1915.24i) q^{95} +(-11684.4 + 20238.0i) q^{96} +(3098.05 + 3098.05i) q^{97} +(13224.0 + 5177.62i) q^{98} -14861.1i 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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.71329 + 1.53087i −1.42832 + 0.382718i −0.888428 − 0.459016i \(-0.848202\pi\)
−0.539894 + 0.841733i \(0.681536\pi\)
\(3\) −15.4449 − 4.13845i −1.71610 − 0.459828i −0.739194 − 0.673492i \(-0.764793\pi\)
−0.976907 + 0.213664i \(0.931460\pi\)
\(4\) 16.4417 − 9.49261i 1.02761 − 0.593288i
\(5\) −22.7561 − 10.3519i −0.910243 − 0.414075i
\(6\) 94.5767 2.62713
\(7\) −15.2970 + 46.5511i −0.312185 + 0.950021i
\(8\) −12.4854 + 12.4854i −0.195085 + 0.195085i
\(9\) 151.270 + 87.3360i 1.86754 + 1.07822i
\(10\) 145.859 + 24.3067i 1.45859 + 0.243067i
\(11\) −42.5401 − 73.6816i −0.351571 − 0.608939i 0.634954 − 0.772550i \(-0.281019\pi\)
−0.986525 + 0.163611i \(0.947686\pi\)
\(12\) −293.225 + 78.5694i −2.03629 + 0.545621i
\(13\) −10.7310 + 10.7310i −0.0634970 + 0.0634970i −0.738142 − 0.674645i \(-0.764297\pi\)
0.674645 + 0.738142i \(0.264297\pi\)
\(14\) 16.1328 − 289.377i 0.0823101 − 1.47642i
\(15\) 308.625 + 254.059i 1.37167 + 1.12915i
\(16\) −99.6625 + 172.620i −0.389306 + 0.674299i
\(17\) 93.8444 − 350.232i 0.324721 − 1.21188i −0.589871 − 0.807498i \(-0.700821\pi\)
0.914592 − 0.404378i \(-0.132512\pi\)
\(18\) −997.952 − 267.400i −3.08010 − 0.825310i
\(19\) −81.5338 − 47.0736i −0.225855 − 0.130398i 0.382803 − 0.923830i \(-0.374959\pi\)
−0.608659 + 0.793432i \(0.708292\pi\)
\(20\) −472.414 + 45.8122i −1.18104 + 0.114530i
\(21\) 428.911 − 655.671i 0.972587 − 1.48678i
\(22\) 355.841 + 355.841i 0.735208 + 0.735208i
\(23\) 143.194 + 534.406i 0.270688 + 1.01022i 0.958676 + 0.284499i \(0.0918271\pi\)
−0.687989 + 0.725721i \(0.741506\pi\)
\(24\) 244.507 − 141.166i 0.424491 − 0.245080i
\(25\) 410.677 + 471.136i 0.657084 + 0.753818i
\(26\) 44.8815 − 77.7370i 0.0663927 − 0.114996i
\(27\) −1059.10 − 1059.10i −1.45281 − 1.45281i
\(28\) 190.382 + 910.587i 0.242834 + 1.16146i
\(29\) 228.863i 0.272131i 0.990700 + 0.136066i \(0.0434458\pi\)
−0.990700 + 0.136066i \(0.956554\pi\)
\(30\) −2152.19 − 979.046i −2.39133 − 1.08783i
\(31\) 408.154 + 706.943i 0.424718 + 0.735633i 0.996394 − 0.0848463i \(-0.0270399\pi\)
−0.571676 + 0.820479i \(0.693707\pi\)
\(32\) 378.261 − 1411.69i 0.369395 − 1.37860i
\(33\) 352.100 + 1314.06i 0.323324 + 1.20666i
\(34\) 2144.64i 1.85523i
\(35\) 829.991 − 900.966i 0.677544 − 0.735482i
\(36\) 3316.19 2.55879
\(37\) 1937.56 − 519.167i 1.41531 − 0.379231i 0.531492 − 0.847063i \(-0.321632\pi\)
0.883817 + 0.467833i \(0.154965\pi\)
\(38\) 537.890 + 144.127i 0.372500 + 0.0998110i
\(39\) 210.149 − 121.330i 0.138165 − 0.0797696i
\(40\) 413.367 − 154.872i 0.258355 − 0.0967949i
\(41\) −151.398 −0.0900641 −0.0450320 − 0.998986i \(-0.514339\pi\)
−0.0450320 + 0.998986i \(0.514339\pi\)
\(42\) −1446.74 + 4402.64i −0.820149 + 2.49583i
\(43\) 209.207 − 209.207i 0.113146 − 0.113146i −0.648267 − 0.761413i \(-0.724506\pi\)
0.761413 + 0.648267i \(0.224506\pi\)
\(44\) −1398.86 − 807.633i −0.722552 − 0.417166i
\(45\) −2538.23 − 3553.36i −1.25345 − 1.75474i
\(46\) −1636.21 − 2834.01i −0.773258 − 1.33932i
\(47\) −1306.86 + 350.173i −0.591609 + 0.158521i −0.542188 − 0.840257i \(-0.682404\pi\)
−0.0494204 + 0.998778i \(0.515737\pi\)
\(48\) 2253.66 − 2253.66i 0.978151 − 0.978151i
\(49\) −1933.00 − 1424.19i −0.805082 − 0.593164i
\(50\) −3067.57 − 2063.04i −1.22703 − 0.825217i
\(51\) −2898.84 + 5020.93i −1.11451 + 1.93039i
\(52\) −74.5705 + 278.301i −0.0275778 + 0.102922i
\(53\) 1024.67 + 274.560i 0.364781 + 0.0977428i 0.436554 − 0.899678i \(-0.356199\pi\)
−0.0717725 + 0.997421i \(0.522866\pi\)
\(54\) 7672.29 + 4429.60i 2.63110 + 1.51907i
\(55\) 205.302 + 2117.07i 0.0678685 + 0.699859i
\(56\) −390.220 − 772.201i −0.124432 − 0.246237i
\(57\) 1064.47 + 1064.47i 0.327630 + 0.327630i
\(58\) −350.359 − 1307.56i −0.104150 − 0.388691i
\(59\) −1517.42 + 876.081i −0.435914 + 0.251675i −0.701863 − 0.712312i \(-0.747648\pi\)
0.265949 + 0.963987i \(0.414315\pi\)
\(60\) 7485.99 + 1247.50i 2.07944 + 0.346527i
\(61\) 3536.71 − 6125.76i 0.950472 − 1.64627i 0.206068 − 0.978538i \(-0.433933\pi\)
0.744405 − 0.667729i \(-0.232733\pi\)
\(62\) −3414.14 − 3414.14i −0.888174 − 0.888174i
\(63\) −6379.58 + 5705.82i −1.60735 + 1.43760i
\(64\) 5455.25i 1.33185i
\(65\) 355.281 − 133.109i 0.0840902 − 0.0315052i
\(66\) −4023.30 − 6968.56i −0.923622 − 1.59976i
\(67\) −786.257 + 2934.35i −0.175152 + 0.653676i 0.821374 + 0.570390i \(0.193208\pi\)
−0.996526 + 0.0832854i \(0.973459\pi\)
\(68\) −1781.66 − 6649.24i −0.385306 − 1.43798i
\(69\) − 8846.46i − 1.85811i
\(70\) −3362.72 + 6418.09i −0.686269 + 1.30981i
\(71\) 4893.58 0.970756 0.485378 − 0.874304i \(-0.338682\pi\)
0.485378 + 0.874304i \(0.338682\pi\)
\(72\) −2979.11 + 798.249i −0.574673 + 0.153983i
\(73\) 7641.14 + 2047.44i 1.43388 + 0.384207i 0.890386 − 0.455206i \(-0.150435\pi\)
0.543494 + 0.839413i \(0.317101\pi\)
\(74\) −10275.0 + 5932.30i −1.87638 + 1.08333i
\(75\) −4393.10 − 8976.22i −0.780996 − 1.59577i
\(76\) −1787.40 −0.309454
\(77\) 4080.69 − 853.175i 0.688260 − 0.143899i
\(78\) −1014.90 + 1014.90i −0.166815 + 0.166815i
\(79\) 4206.19 + 2428.45i 0.673961 + 0.389112i 0.797576 − 0.603219i \(-0.206115\pi\)
−0.123615 + 0.992330i \(0.539449\pi\)
\(80\) 4054.87 − 2896.47i 0.633574 − 0.452573i
\(81\) 4900.45 + 8487.83i 0.746906 + 1.29368i
\(82\) 864.978 − 231.770i 0.128640 − 0.0344691i
\(83\) −3185.61 + 3185.61i −0.462420 + 0.462420i −0.899448 − 0.437028i \(-0.856031\pi\)
0.437028 + 0.899448i \(0.356031\pi\)
\(84\) 827.988 − 14851.8i 0.117345 − 2.10485i
\(85\) −5761.09 + 6998.44i −0.797383 + 0.968642i
\(86\) −874.990 + 1515.53i −0.118306 + 0.204912i
\(87\) 947.137 − 3534.76i 0.125134 − 0.467005i
\(88\) 1451.08 + 388.815i 0.187381 + 0.0502086i
\(89\) −1185.37 − 684.375i −0.149649 − 0.0864001i 0.423306 − 0.905987i \(-0.360870\pi\)
−0.572955 + 0.819587i \(0.694203\pi\)
\(90\) 19941.4 + 16415.7i 2.46190 + 2.02662i
\(91\) −335.387 − 663.692i −0.0405007 − 0.0801463i
\(92\) 7427.26 + 7427.26i 0.877511 + 0.877511i
\(93\) −3378.25 − 12607.8i −0.390594 − 1.45772i
\(94\) 6930.42 − 4001.28i 0.784339 − 0.452838i
\(95\) 1368.09 + 1915.24i 0.151589 + 0.212215i
\(96\) −11684.4 + 20238.0i −1.26784 + 2.19596i
\(97\) 3098.05 + 3098.05i 0.329264 + 0.329264i 0.852307 − 0.523042i \(-0.175203\pi\)
−0.523042 + 0.852307i \(0.675203\pi\)
\(98\) 13224.0 + 5177.62i 1.37693 + 0.539110i
\(99\) − 14861.1i − 1.51629i
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.2.2 ✓ 56
5.3 odd 4 inner 35.5.l.a.23.13 yes 56
7.4 even 3 inner 35.5.l.a.32.13 yes 56
35.18 odd 12 inner 35.5.l.a.18.2 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.5.l.a.2.2 ✓ 56 1.1 even 1 trivial
35.5.l.a.18.2 yes 56 35.18 odd 12 inner
35.5.l.a.23.13 yes 56 5.3 odd 4 inner
35.5.l.a.32.13 yes 56 7.4 even 3 inner