# Properties

 Label 1680.4.a.bg.1.1 Level $1680$ Weight $4$ Character 1680.1 Self dual yes Analytic conductor $99.123$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1680.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$99.1232088096$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.53113$$ of defining polynomial Character $$\chi$$ $$=$$ 1680.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +2.93774 q^{11} -19.0623 q^{13} -15.0000 q^{15} +122.498 q^{17} -107.436 q^{19} -21.0000 q^{21} -210.623 q^{23} +25.0000 q^{25} -27.0000 q^{27} +95.4942 q^{29} +94.3074 q^{31} -8.81323 q^{33} +35.0000 q^{35} +97.1206 q^{37} +57.1868 q^{39} -491.113 q^{41} +43.0039 q^{43} +45.0000 q^{45} -473.494 q^{47} +49.0000 q^{49} -367.494 q^{51} -183.677 q^{53} +14.6887 q^{55} +322.307 q^{57} +760.615 q^{59} -198.747 q^{61} +63.0000 q^{63} -95.3113 q^{65} +309.992 q^{67} +631.868 q^{69} -665.693 q^{71} +621.288 q^{73} -75.0000 q^{75} +20.5642 q^{77} +24.7626 q^{79} +81.0000 q^{81} +406.724 q^{83} +612.490 q^{85} -286.483 q^{87} +261.751 q^{89} -133.436 q^{91} -282.922 q^{93} -537.179 q^{95} -1004.77 q^{97} +26.4397 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 $$2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 - 22 * q^13 - 30 * q^15 + 116 * q^17 - 102 * q^19 - 42 * q^21 - 260 * q^23 + 50 * q^25 - 54 * q^27 - 196 * q^29 - 150 * q^31 - 66 * q^33 + 70 * q^35 - 96 * q^37 + 66 * q^39 - 176 * q^41 + 344 * q^43 + 90 * q^45 - 560 * q^47 + 98 * q^49 - 348 * q^51 + 326 * q^53 + 110 * q^55 + 306 * q^57 + 844 * q^59 - 204 * q^61 + 126 * q^63 - 110 * q^65 + 104 * q^67 + 780 * q^69 - 1670 * q^71 - 386 * q^73 - 150 * q^75 + 154 * q^77 + 888 * q^79 + 162 * q^81 - 928 * q^83 + 580 * q^85 + 588 * q^87 + 588 * q^89 - 154 * q^91 + 450 * q^93 - 510 * q^95 + 522 * q^97 + 198 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −3.00000 −0.577350
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 7.00000 0.377964
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 2.93774 0.0805239 0.0402619 0.999189i $$-0.487181\pi$$
0.0402619 + 0.999189i $$0.487181\pi$$
$$12$$ 0 0
$$13$$ −19.0623 −0.406686 −0.203343 0.979108i $$-0.565181\pi$$
−0.203343 + 0.979108i $$0.565181\pi$$
$$14$$ 0 0
$$15$$ −15.0000 −0.258199
$$16$$ 0 0
$$17$$ 122.498 1.74766 0.873828 0.486236i $$-0.161630\pi$$
0.873828 + 0.486236i $$0.161630\pi$$
$$18$$ 0 0
$$19$$ −107.436 −1.29723 −0.648617 0.761115i $$-0.724652\pi$$
−0.648617 + 0.761115i $$0.724652\pi$$
$$20$$ 0 0
$$21$$ −21.0000 −0.218218
$$22$$ 0 0
$$23$$ −210.623 −1.90947 −0.954736 0.297455i $$-0.903862\pi$$
−0.954736 + 0.297455i $$0.903862\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 95.4942 0.611477 0.305738 0.952116i $$-0.401097\pi$$
0.305738 + 0.952116i $$0.401097\pi$$
$$30$$ 0 0
$$31$$ 94.3074 0.546391 0.273195 0.961959i $$-0.411919\pi$$
0.273195 + 0.961959i $$0.411919\pi$$
$$32$$ 0 0
$$33$$ −8.81323 −0.0464905
$$34$$ 0 0
$$35$$ 35.0000 0.169031
$$36$$ 0 0
$$37$$ 97.1206 0.431528 0.215764 0.976446i $$-0.430776\pi$$
0.215764 + 0.976446i $$0.430776\pi$$
$$38$$ 0 0
$$39$$ 57.1868 0.234800
$$40$$ 0 0
$$41$$ −491.113 −1.87071 −0.935353 0.353716i $$-0.884918\pi$$
−0.935353 + 0.353716i $$0.884918\pi$$
$$42$$ 0 0
$$43$$ 43.0039 0.152512 0.0762562 0.997088i $$-0.475703\pi$$
0.0762562 + 0.997088i $$0.475703\pi$$
$$44$$ 0 0
$$45$$ 45.0000 0.149071
$$46$$ 0 0
$$47$$ −473.494 −1.46949 −0.734747 0.678341i $$-0.762699\pi$$
−0.734747 + 0.678341i $$0.762699\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −367.494 −1.00901
$$52$$ 0 0
$$53$$ −183.677 −0.476038 −0.238019 0.971261i $$-0.576498\pi$$
−0.238019 + 0.971261i $$0.576498\pi$$
$$54$$ 0 0
$$55$$ 14.6887 0.0360114
$$56$$ 0 0
$$57$$ 322.307 0.748959
$$58$$ 0 0
$$59$$ 760.615 1.67837 0.839183 0.543849i $$-0.183034\pi$$
0.839183 + 0.543849i $$0.183034\pi$$
$$60$$ 0 0
$$61$$ −198.747 −0.417163 −0.208582 0.978005i $$-0.566885\pi$$
−0.208582 + 0.978005i $$0.566885\pi$$
$$62$$ 0 0
$$63$$ 63.0000 0.125988
$$64$$ 0 0
$$65$$ −95.3113 −0.181876
$$66$$ 0 0
$$67$$ 309.992 0.565247 0.282624 0.959231i $$-0.408795\pi$$
0.282624 + 0.959231i $$0.408795\pi$$
$$68$$ 0 0
$$69$$ 631.868 1.10243
$$70$$ 0 0
$$71$$ −665.693 −1.11272 −0.556360 0.830941i $$-0.687803\pi$$
−0.556360 + 0.830941i $$0.687803\pi$$
$$72$$ 0 0
$$73$$ 621.288 0.996113 0.498057 0.867145i $$-0.334047\pi$$
0.498057 + 0.867145i $$0.334047\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ 20.5642 0.0304352
$$78$$ 0 0
$$79$$ 24.7626 0.0352659 0.0176330 0.999845i $$-0.494387\pi$$
0.0176330 + 0.999845i $$0.494387\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 406.724 0.537876 0.268938 0.963157i $$-0.413327\pi$$
0.268938 + 0.963157i $$0.413327\pi$$
$$84$$ 0 0
$$85$$ 612.490 0.781575
$$86$$ 0 0
$$87$$ −286.483 −0.353036
$$88$$ 0 0
$$89$$ 261.751 0.311748 0.155874 0.987777i $$-0.450181\pi$$
0.155874 + 0.987777i $$0.450181\pi$$
$$90$$ 0 0
$$91$$ −133.436 −0.153713
$$92$$ 0 0
$$93$$ −282.922 −0.315459
$$94$$ 0 0
$$95$$ −537.179 −0.580141
$$96$$ 0 0
$$97$$ −1004.77 −1.05175 −0.525873 0.850563i $$-0.676261\pi$$
−0.525873 + 0.850563i $$0.676261\pi$$
$$98$$ 0 0
$$99$$ 26.4397 0.0268413
$$100$$ 0 0
$$101$$ −128.872 −0.126962 −0.0634812 0.997983i $$-0.520220\pi$$
−0.0634812 + 0.997983i $$0.520220\pi$$
$$102$$ 0 0
$$103$$ −806.008 −0.771051 −0.385526 0.922697i $$-0.625980\pi$$
−0.385526 + 0.922697i $$0.625980\pi$$
$$104$$ 0 0
$$105$$ −105.000 −0.0975900
$$106$$ 0 0
$$107$$ 769.712 0.695429 0.347714 0.937600i $$-0.386958\pi$$
0.347714 + 0.937600i $$0.386958\pi$$
$$108$$ 0 0
$$109$$ −780.856 −0.686169 −0.343085 0.939304i $$-0.611472\pi$$
−0.343085 + 0.939304i $$0.611472\pi$$
$$110$$ 0 0
$$111$$ −291.362 −0.249143
$$112$$ 0 0
$$113$$ −1115.65 −0.928771 −0.464386 0.885633i $$-0.653725\pi$$
−0.464386 + 0.885633i $$0.653725\pi$$
$$114$$ 0 0
$$115$$ −1053.11 −0.853942
$$116$$ 0 0
$$117$$ −171.560 −0.135562
$$118$$ 0 0
$$119$$ 857.486 0.660552
$$120$$ 0 0
$$121$$ −1322.37 −0.993516
$$122$$ 0 0
$$123$$ 1473.34 1.08005
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ 1875.98 1.31076 0.655381 0.755299i $$-0.272508\pi$$
0.655381 + 0.755299i $$0.272508\pi$$
$$128$$ 0 0
$$129$$ −129.012 −0.0880530
$$130$$ 0 0
$$131$$ −364.203 −0.242905 −0.121452 0.992597i $$-0.538755\pi$$
−0.121452 + 0.992597i $$0.538755\pi$$
$$132$$ 0 0
$$133$$ −752.051 −0.490309
$$134$$ 0 0
$$135$$ −135.000 −0.0860663
$$136$$ 0 0
$$137$$ 1603.13 0.999743 0.499872 0.866099i $$-0.333380\pi$$
0.499872 + 0.866099i $$0.333380\pi$$
$$138$$ 0 0
$$139$$ −2431.12 −1.48349 −0.741746 0.670681i $$-0.766002\pi$$
−0.741746 + 0.670681i $$0.766002\pi$$
$$140$$ 0 0
$$141$$ 1420.48 0.848413
$$142$$ 0 0
$$143$$ −56.0000 −0.0327479
$$144$$ 0 0
$$145$$ 477.471 0.273461
$$146$$ 0 0
$$147$$ −147.000 −0.0824786
$$148$$ 0 0
$$149$$ 2341.57 1.28744 0.643722 0.765260i $$-0.277389\pi$$
0.643722 + 0.765260i $$0.277389\pi$$
$$150$$ 0 0
$$151$$ 2104.07 1.13395 0.566976 0.823734i $$-0.308113\pi$$
0.566976 + 0.823734i $$0.308113\pi$$
$$152$$ 0 0
$$153$$ 1102.48 0.582552
$$154$$ 0 0
$$155$$ 471.537 0.244353
$$156$$ 0 0
$$157$$ −593.467 −0.301680 −0.150840 0.988558i $$-0.548198\pi$$
−0.150840 + 0.988558i $$0.548198\pi$$
$$158$$ 0 0
$$159$$ 551.031 0.274840
$$160$$ 0 0
$$161$$ −1474.36 −0.721712
$$162$$ 0 0
$$163$$ −2178.71 −1.04693 −0.523465 0.852047i $$-0.675361\pi$$
−0.523465 + 0.852047i $$0.675361\pi$$
$$164$$ 0 0
$$165$$ −44.0661 −0.0207912
$$166$$ 0 0
$$167$$ −799.502 −0.370463 −0.185231 0.982695i $$-0.559303\pi$$
−0.185231 + 0.982695i $$0.559303\pi$$
$$168$$ 0 0
$$169$$ −1833.63 −0.834606
$$170$$ 0 0
$$171$$ −966.922 −0.432412
$$172$$ 0 0
$$173$$ −1444.36 −0.634754 −0.317377 0.948299i $$-0.602802\pi$$
−0.317377 + 0.948299i $$0.602802\pi$$
$$174$$ 0 0
$$175$$ 175.000 0.0755929
$$176$$ 0 0
$$177$$ −2281.84 −0.969005
$$178$$ 0 0
$$179$$ −3343.49 −1.39611 −0.698056 0.716043i $$-0.745952\pi$$
−0.698056 + 0.716043i $$0.745952\pi$$
$$180$$ 0 0
$$181$$ 2251.81 0.924729 0.462365 0.886690i $$-0.347001\pi$$
0.462365 + 0.886690i $$0.347001\pi$$
$$182$$ 0 0
$$183$$ 596.241 0.240849
$$184$$ 0 0
$$185$$ 485.603 0.192985
$$186$$ 0 0
$$187$$ 359.868 0.140728
$$188$$ 0 0
$$189$$ −189.000 −0.0727393
$$190$$ 0 0
$$191$$ 1001.93 0.379565 0.189782 0.981826i $$-0.439222\pi$$
0.189782 + 0.981826i $$0.439222\pi$$
$$192$$ 0 0
$$193$$ −4054.97 −1.51235 −0.756173 0.654372i $$-0.772933\pi$$
−0.756173 + 0.654372i $$0.772933\pi$$
$$194$$ 0 0
$$195$$ 285.934 0.105006
$$196$$ 0 0
$$197$$ −5140.23 −1.85902 −0.929508 0.368802i $$-0.879768\pi$$
−0.929508 + 0.368802i $$0.879768\pi$$
$$198$$ 0 0
$$199$$ −585.631 −0.208614 −0.104307 0.994545i $$-0.533263\pi$$
−0.104307 + 0.994545i $$0.533263\pi$$
$$200$$ 0 0
$$201$$ −929.977 −0.326346
$$202$$ 0 0
$$203$$ 668.459 0.231116
$$204$$ 0 0
$$205$$ −2455.56 −0.836605
$$206$$ 0 0
$$207$$ −1895.60 −0.636490
$$208$$ 0 0
$$209$$ −315.619 −0.104458
$$210$$ 0 0
$$211$$ 1055.16 0.344266 0.172133 0.985074i $$-0.444934\pi$$
0.172133 + 0.985074i $$0.444934\pi$$
$$212$$ 0 0
$$213$$ 1997.08 0.642430
$$214$$ 0 0
$$215$$ 215.019 0.0682056
$$216$$ 0 0
$$217$$ 660.152 0.206516
$$218$$ 0 0
$$219$$ −1863.86 −0.575106
$$220$$ 0 0
$$221$$ −2335.09 −0.710747
$$222$$ 0 0
$$223$$ −4675.85 −1.40412 −0.702059 0.712119i $$-0.747736\pi$$
−0.702059 + 0.712119i $$0.747736\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −5443.11 −1.59151 −0.795754 0.605621i $$-0.792925\pi$$
−0.795754 + 0.605621i $$0.792925\pi$$
$$228$$ 0 0
$$229$$ −536.303 −0.154759 −0.0773797 0.997002i $$-0.524655\pi$$
−0.0773797 + 0.997002i $$0.524655\pi$$
$$230$$ 0 0
$$231$$ −61.6926 −0.0175717
$$232$$ 0 0
$$233$$ −183.490 −0.0515916 −0.0257958 0.999667i $$-0.508212\pi$$
−0.0257958 + 0.999667i $$0.508212\pi$$
$$234$$ 0 0
$$235$$ −2367.47 −0.657178
$$236$$ 0 0
$$237$$ −74.2878 −0.0203608
$$238$$ 0 0
$$239$$ −643.218 −0.174085 −0.0870425 0.996205i $$-0.527742\pi$$
−0.0870425 + 0.996205i $$0.527742\pi$$
$$240$$ 0 0
$$241$$ −5755.61 −1.53839 −0.769194 0.639015i $$-0.779342\pi$$
−0.769194 + 0.639015i $$0.779342\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ 245.000 0.0638877
$$246$$ 0 0
$$247$$ 2047.97 0.527567
$$248$$ 0 0
$$249$$ −1220.17 −0.310543
$$250$$ 0 0
$$251$$ 5132.27 1.29062 0.645311 0.763920i $$-0.276728\pi$$
0.645311 + 0.763920i $$0.276728\pi$$
$$252$$ 0 0
$$253$$ −618.755 −0.153758
$$254$$ 0 0
$$255$$ −1837.47 −0.451243
$$256$$ 0 0
$$257$$ 5041.74 1.22372 0.611859 0.790967i $$-0.290422\pi$$
0.611859 + 0.790967i $$0.290422\pi$$
$$258$$ 0 0
$$259$$ 679.844 0.163102
$$260$$ 0 0
$$261$$ 859.448 0.203826
$$262$$ 0 0
$$263$$ −7577.00 −1.77649 −0.888246 0.459367i $$-0.848076\pi$$
−0.888246 + 0.459367i $$0.848076\pi$$
$$264$$ 0 0
$$265$$ −918.385 −0.212890
$$266$$ 0 0
$$267$$ −785.253 −0.179988
$$268$$ 0 0
$$269$$ 1023.10 0.231893 0.115947 0.993255i $$-0.463010\pi$$
0.115947 + 0.993255i $$0.463010\pi$$
$$270$$ 0 0
$$271$$ 2251.98 0.504790 0.252395 0.967624i $$-0.418782\pi$$
0.252395 + 0.967624i $$0.418782\pi$$
$$272$$ 0 0
$$273$$ 400.307 0.0887462
$$274$$ 0 0
$$275$$ 73.4436 0.0161048
$$276$$ 0 0
$$277$$ −8630.72 −1.87209 −0.936047 0.351875i $$-0.885544\pi$$
−0.936047 + 0.351875i $$0.885544\pi$$
$$278$$ 0 0
$$279$$ 848.767 0.182130
$$280$$ 0 0
$$281$$ −7521.62 −1.59680 −0.798402 0.602124i $$-0.794321\pi$$
−0.798402 + 0.602124i $$0.794321\pi$$
$$282$$ 0 0
$$283$$ −14.8169 −0.00311226 −0.00155613 0.999999i $$-0.500495\pi$$
−0.00155613 + 0.999999i $$0.500495\pi$$
$$284$$ 0 0
$$285$$ 1611.54 0.334945
$$286$$ 0 0
$$287$$ −3437.79 −0.707060
$$288$$ 0 0
$$289$$ 10092.8 2.05430
$$290$$ 0 0
$$291$$ 3014.32 0.607226
$$292$$ 0 0
$$293$$ 6913.39 1.37844 0.689222 0.724550i $$-0.257952\pi$$
0.689222 + 0.724550i $$0.257952\pi$$
$$294$$ 0 0
$$295$$ 3803.07 0.750588
$$296$$ 0 0
$$297$$ −79.3190 −0.0154968
$$298$$ 0 0
$$299$$ 4014.94 0.776555
$$300$$ 0 0
$$301$$ 301.027 0.0576442
$$302$$ 0 0
$$303$$ 386.615 0.0733018
$$304$$ 0 0
$$305$$ −993.735 −0.186561
$$306$$ 0 0
$$307$$ 7644.12 1.42108 0.710542 0.703655i $$-0.248450\pi$$
0.710542 + 0.703655i $$0.248450\pi$$
$$308$$ 0 0
$$309$$ 2418.02 0.445167
$$310$$ 0 0
$$311$$ −7593.99 −1.38462 −0.692308 0.721602i $$-0.743406\pi$$
−0.692308 + 0.721602i $$0.743406\pi$$
$$312$$ 0 0
$$313$$ −9127.84 −1.64836 −0.824179 0.566329i $$-0.808363\pi$$
−0.824179 + 0.566329i $$0.808363\pi$$
$$314$$ 0 0
$$315$$ 315.000 0.0563436
$$316$$ 0 0
$$317$$ −4929.81 −0.873456 −0.436728 0.899593i $$-0.643863\pi$$
−0.436728 + 0.899593i $$0.643863\pi$$
$$318$$ 0 0
$$319$$ 280.537 0.0492385
$$320$$ 0 0
$$321$$ −2309.14 −0.401506
$$322$$ 0 0
$$323$$ −13160.7 −2.26712
$$324$$ 0 0
$$325$$ −476.556 −0.0813372
$$326$$ 0 0
$$327$$ 2342.57 0.396160
$$328$$ 0 0
$$329$$ −3314.46 −0.555417
$$330$$ 0 0
$$331$$ −1221.67 −0.202867 −0.101433 0.994842i $$-0.532343\pi$$
−0.101433 + 0.994842i $$0.532343\pi$$
$$332$$ 0 0
$$333$$ 874.086 0.143843
$$334$$ 0 0
$$335$$ 1549.96 0.252786
$$336$$ 0 0
$$337$$ −8744.83 −1.41354 −0.706768 0.707446i $$-0.749847\pi$$
−0.706768 + 0.707446i $$0.749847\pi$$
$$338$$ 0 0
$$339$$ 3346.94 0.536226
$$340$$ 0 0
$$341$$ 277.051 0.0439975
$$342$$ 0 0
$$343$$ 343.000 0.0539949
$$344$$ 0 0
$$345$$ 3159.34 0.493023
$$346$$ 0 0
$$347$$ −4589.56 −0.710031 −0.355015 0.934860i $$-0.615524\pi$$
−0.355015 + 0.934860i $$0.615524\pi$$
$$348$$ 0 0
$$349$$ −3989.89 −0.611960 −0.305980 0.952038i $$-0.598984\pi$$
−0.305980 + 0.952038i $$0.598984\pi$$
$$350$$ 0 0
$$351$$ 514.681 0.0782668
$$352$$ 0 0
$$353$$ 2416.35 0.364333 0.182166 0.983268i $$-0.441689\pi$$
0.182166 + 0.983268i $$0.441689\pi$$
$$354$$ 0 0
$$355$$ −3328.46 −0.497624
$$356$$ 0 0
$$357$$ −2572.46 −0.381370
$$358$$ 0 0
$$359$$ 2756.24 0.405206 0.202603 0.979261i $$-0.435060\pi$$
0.202603 + 0.979261i $$0.435060\pi$$
$$360$$ 0 0
$$361$$ 4683.45 0.682818
$$362$$ 0 0
$$363$$ 3967.11 0.573607
$$364$$ 0 0
$$365$$ 3106.44 0.445475
$$366$$ 0 0
$$367$$ −11112.8 −1.58061 −0.790307 0.612711i $$-0.790079\pi$$
−0.790307 + 0.612711i $$0.790079\pi$$
$$368$$ 0 0
$$369$$ −4420.02 −0.623569
$$370$$ 0 0
$$371$$ −1285.74 −0.179925
$$372$$ 0 0
$$373$$ 6091.09 0.845535 0.422768 0.906238i $$-0.361059\pi$$
0.422768 + 0.906238i $$0.361059\pi$$
$$374$$ 0 0
$$375$$ −375.000 −0.0516398
$$376$$ 0 0
$$377$$ −1820.33 −0.248679
$$378$$ 0 0
$$379$$ −3984.29 −0.539998 −0.269999 0.962861i $$-0.587023\pi$$
−0.269999 + 0.962861i $$0.587023\pi$$
$$380$$ 0 0
$$381$$ −5627.95 −0.756768
$$382$$ 0 0
$$383$$ −318.475 −0.0424890 −0.0212445 0.999774i $$-0.506763\pi$$
−0.0212445 + 0.999774i $$0.506763\pi$$
$$384$$ 0 0
$$385$$ 102.821 0.0136110
$$386$$ 0 0
$$387$$ 387.035 0.0508374
$$388$$ 0 0
$$389$$ 3885.46 0.506429 0.253214 0.967410i $$-0.418512\pi$$
0.253214 + 0.967410i $$0.418512\pi$$
$$390$$ 0 0
$$391$$ −25800.9 −3.33710
$$392$$ 0 0
$$393$$ 1092.61 0.140241
$$394$$ 0 0
$$395$$ 123.813 0.0157714
$$396$$ 0 0
$$397$$ 4806.04 0.607578 0.303789 0.952739i $$-0.401748\pi$$
0.303789 + 0.952739i $$0.401748\pi$$
$$398$$ 0 0
$$399$$ 2256.15 0.283080
$$400$$ 0 0
$$401$$ 3618.59 0.450633 0.225316 0.974286i $$-0.427658\pi$$
0.225316 + 0.974286i $$0.427658\pi$$
$$402$$ 0 0
$$403$$ −1797.71 −0.222209
$$404$$ 0 0
$$405$$ 405.000 0.0496904
$$406$$ 0 0
$$407$$ 285.315 0.0347483
$$408$$ 0 0
$$409$$ −2109.05 −0.254978 −0.127489 0.991840i $$-0.540692\pi$$
−0.127489 + 0.991840i $$0.540692\pi$$
$$410$$ 0 0
$$411$$ −4809.40 −0.577202
$$412$$ 0 0
$$413$$ 5324.30 0.634363
$$414$$ 0 0
$$415$$ 2033.62 0.240546
$$416$$ 0 0
$$417$$ 7293.37 0.856494
$$418$$ 0 0
$$419$$ 6905.91 0.805193 0.402597 0.915377i $$-0.368108\pi$$
0.402597 + 0.915377i $$0.368108\pi$$
$$420$$ 0 0
$$421$$ −9647.54 −1.11685 −0.558423 0.829556i $$-0.688593\pi$$
−0.558423 + 0.829556i $$0.688593\pi$$
$$422$$ 0 0
$$423$$ −4261.45 −0.489831
$$424$$ 0 0
$$425$$ 3062.45 0.349531
$$426$$ 0 0
$$427$$ −1391.23 −0.157673
$$428$$ 0 0
$$429$$ 168.000 0.0189070
$$430$$ 0 0
$$431$$ 13002.7 1.45318 0.726589 0.687073i $$-0.241105\pi$$
0.726589 + 0.687073i $$0.241105\pi$$
$$432$$ 0 0
$$433$$ 7356.07 0.816420 0.408210 0.912888i $$-0.366153\pi$$
0.408210 + 0.912888i $$0.366153\pi$$
$$434$$ 0 0
$$435$$ −1432.41 −0.157883
$$436$$ 0 0
$$437$$ 22628.4 2.47703
$$438$$ 0 0
$$439$$ −6909.21 −0.751159 −0.375579 0.926790i $$-0.622556\pi$$
−0.375579 + 0.926790i $$0.622556\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ 14812.6 1.58864 0.794318 0.607502i $$-0.207828\pi$$
0.794318 + 0.607502i $$0.207828\pi$$
$$444$$ 0 0
$$445$$ 1308.75 0.139418
$$446$$ 0 0
$$447$$ −7024.72 −0.743306
$$448$$ 0 0
$$449$$ −10654.5 −1.11986 −0.559932 0.828538i $$-0.689173\pi$$
−0.559932 + 0.828538i $$0.689173\pi$$
$$450$$ 0 0
$$451$$ −1442.76 −0.150636
$$452$$ 0 0
$$453$$ −6312.21 −0.654688
$$454$$ 0 0
$$455$$ −667.179 −0.0687425
$$456$$ 0 0
$$457$$ −5855.16 −0.599328 −0.299664 0.954045i $$-0.596875\pi$$
−0.299664 + 0.954045i $$0.596875\pi$$
$$458$$ 0 0
$$459$$ −3307.45 −0.336336
$$460$$ 0 0
$$461$$ 3204.74 0.323774 0.161887 0.986809i $$-0.448242\pi$$
0.161887 + 0.986809i $$0.448242\pi$$
$$462$$ 0 0
$$463$$ −371.658 −0.0373054 −0.0186527 0.999826i $$-0.505938\pi$$
−0.0186527 + 0.999826i $$0.505938\pi$$
$$464$$ 0 0
$$465$$ −1414.61 −0.141077
$$466$$ 0 0
$$467$$ 19752.3 1.95723 0.978614 0.205703i $$-0.0659482\pi$$
0.978614 + 0.205703i $$0.0659482\pi$$
$$468$$ 0 0
$$469$$ 2169.95 0.213643
$$470$$ 0 0
$$471$$ 1780.40 0.174175
$$472$$ 0 0
$$473$$ 126.334 0.0122809
$$474$$ 0 0
$$475$$ −2685.90 −0.259447
$$476$$ 0 0
$$477$$ −1653.09 −0.158679
$$478$$ 0 0
$$479$$ 20762.0 1.98046 0.990232 0.139433i $$-0.0445279\pi$$
0.990232 + 0.139433i $$0.0445279\pi$$
$$480$$ 0 0
$$481$$ −1851.34 −0.175496
$$482$$ 0 0
$$483$$ 4423.07 0.416681
$$484$$ 0 0
$$485$$ −5023.87 −0.470355
$$486$$ 0 0
$$487$$ −17647.6 −1.64207 −0.821035 0.570878i $$-0.806603\pi$$
−0.821035 + 0.570878i $$0.806603\pi$$
$$488$$ 0 0
$$489$$ 6536.12 0.604445
$$490$$ 0 0
$$491$$ 5637.46 0.518157 0.259078 0.965856i $$-0.416581\pi$$
0.259078 + 0.965856i $$0.416581\pi$$
$$492$$ 0 0
$$493$$ 11697.9 1.06865
$$494$$ 0 0
$$495$$ 132.198 0.0120038
$$496$$ 0 0
$$497$$ −4659.85 −0.420569
$$498$$ 0 0
$$499$$ 17474.1 1.56764 0.783818 0.620991i $$-0.213270\pi$$
0.783818 + 0.620991i $$0.213270\pi$$
$$500$$ 0 0
$$501$$ 2398.51 0.213887
$$502$$ 0 0
$$503$$ −7444.81 −0.659936 −0.329968 0.943992i $$-0.607038\pi$$
−0.329968 + 0.943992i $$0.607038\pi$$
$$504$$ 0 0
$$505$$ −644.358 −0.0567793
$$506$$ 0 0
$$507$$ 5500.89 0.481860
$$508$$ 0 0
$$509$$ −3384.48 −0.294724 −0.147362 0.989083i $$-0.547078\pi$$
−0.147362 + 0.989083i $$0.547078\pi$$
$$510$$ 0 0
$$511$$ 4349.02 0.376495
$$512$$ 0 0
$$513$$ 2900.77 0.249653
$$514$$ 0 0
$$515$$ −4030.04 −0.344825
$$516$$ 0 0
$$517$$ −1391.00 −0.118329
$$518$$ 0 0
$$519$$ 4333.07 0.366476
$$520$$ 0 0
$$521$$ 2973.12 0.250009 0.125005 0.992156i $$-0.460105\pi$$
0.125005 + 0.992156i $$0.460105\pi$$
$$522$$ 0 0
$$523$$ −2689.02 −0.224823 −0.112412 0.993662i $$-0.535858\pi$$
−0.112412 + 0.993662i $$0.535858\pi$$
$$524$$ 0 0
$$525$$ −525.000 −0.0436436
$$526$$ 0 0
$$527$$ 11552.5 0.954903
$$528$$ 0 0
$$529$$ 32194.9 2.64608
$$530$$ 0 0
$$531$$ 6845.53 0.559455
$$532$$ 0 0
$$533$$ 9361.72 0.760790
$$534$$ 0 0
$$535$$ 3848.56 0.311005
$$536$$ 0 0
$$537$$ 10030.5 0.806046
$$538$$ 0 0
$$539$$ 143.949 0.0115034
$$540$$ 0 0
$$541$$ −14429.5 −1.14671 −0.573356 0.819306i $$-0.694359\pi$$
−0.573356 + 0.819306i $$0.694359\pi$$
$$542$$ 0 0
$$543$$ −6755.44 −0.533893
$$544$$ 0 0
$$545$$ −3904.28 −0.306864
$$546$$ 0 0
$$547$$ −13811.2 −1.07957 −0.539784 0.841804i $$-0.681494\pi$$
−0.539784 + 0.841804i $$0.681494\pi$$
$$548$$ 0 0
$$549$$ −1788.72 −0.139054
$$550$$ 0 0
$$551$$ −10259.5 −0.793229
$$552$$ 0 0
$$553$$ 173.338 0.0133293
$$554$$ 0 0
$$555$$ −1456.81 −0.111420
$$556$$ 0 0
$$557$$ −6033.26 −0.458954 −0.229477 0.973314i $$-0.573702\pi$$
−0.229477 + 0.973314i $$0.573702\pi$$
$$558$$ 0 0
$$559$$ −819.751 −0.0620246
$$560$$ 0 0
$$561$$ −1079.60 −0.0812493
$$562$$ 0 0
$$563$$ 6958.47 0.520896 0.260448 0.965488i $$-0.416130\pi$$
0.260448 + 0.965488i $$0.416130\pi$$
$$564$$ 0 0
$$565$$ −5578.23 −0.415359
$$566$$ 0 0
$$567$$ 567.000 0.0419961
$$568$$ 0 0
$$569$$ −13396.4 −0.987009 −0.493505 0.869743i $$-0.664284\pi$$
−0.493505 + 0.869743i $$0.664284\pi$$
$$570$$ 0 0
$$571$$ 8055.84 0.590414 0.295207 0.955433i $$-0.404611\pi$$
0.295207 + 0.955433i $$0.404611\pi$$
$$572$$ 0 0
$$573$$ −3005.78 −0.219142
$$574$$ 0 0
$$575$$ −5265.56 −0.381894
$$576$$ 0 0
$$577$$ −21456.9 −1.54812 −0.774059 0.633114i $$-0.781777\pi$$
−0.774059 + 0.633114i $$0.781777\pi$$
$$578$$ 0 0
$$579$$ 12164.9 0.873153
$$580$$ 0 0
$$581$$ 2847.07 0.203298
$$582$$ 0 0
$$583$$ −539.596 −0.0383324
$$584$$ 0 0
$$585$$ −857.802 −0.0606252
$$586$$ 0 0
$$587$$ −20156.3 −1.41728 −0.708638 0.705572i $$-0.750690\pi$$
−0.708638 + 0.705572i $$0.750690\pi$$
$$588$$ 0 0
$$589$$ −10132.0 −0.708797
$$590$$ 0 0
$$591$$ 15420.7 1.07330
$$592$$ 0 0
$$593$$ 599.307 0.0415018 0.0207509 0.999785i $$-0.493394\pi$$
0.0207509 + 0.999785i $$0.493394\pi$$
$$594$$ 0 0
$$595$$ 4287.43 0.295408
$$596$$ 0 0
$$597$$ 1756.89 0.120444
$$598$$ 0 0
$$599$$ 5493.05 0.374691 0.187346 0.982294i $$-0.440012\pi$$
0.187346 + 0.982294i $$0.440012\pi$$
$$600$$ 0 0
$$601$$ 24292.8 1.64879 0.824396 0.566014i $$-0.191515\pi$$
0.824396 + 0.566014i $$0.191515\pi$$
$$602$$ 0 0
$$603$$ 2789.93 0.188416
$$604$$ 0 0
$$605$$ −6611.85 −0.444314
$$606$$ 0 0
$$607$$ −3029.50 −0.202576 −0.101288 0.994857i $$-0.532296\pi$$
−0.101288 + 0.994857i $$0.532296\pi$$
$$608$$ 0 0
$$609$$ −2005.38 −0.133435
$$610$$ 0 0
$$611$$ 9025.87 0.597623
$$612$$ 0 0
$$613$$ −19339.6 −1.27426 −0.637129 0.770757i $$-0.719878\pi$$
−0.637129 + 0.770757i $$0.719878\pi$$
$$614$$ 0 0
$$615$$ 7366.69 0.483014
$$616$$ 0 0
$$617$$ −5743.91 −0.374783 −0.187391 0.982285i $$-0.560003\pi$$
−0.187391 + 0.982285i $$0.560003\pi$$
$$618$$ 0 0
$$619$$ 8243.35 0.535264 0.267632 0.963521i $$-0.413759\pi$$
0.267632 + 0.963521i $$0.413759\pi$$
$$620$$ 0 0
$$621$$ 5686.81 0.367478
$$622$$ 0 0
$$623$$ 1832.26 0.117830
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 946.856 0.0603091
$$628$$ 0 0
$$629$$ 11897.1 0.754162
$$630$$ 0 0
$$631$$ 4376.56 0.276114 0.138057 0.990424i $$-0.455914\pi$$
0.138057 + 0.990424i $$0.455914\pi$$
$$632$$ 0 0
$$633$$ −3165.48 −0.198762
$$634$$ 0 0
$$635$$ 9379.92 0.586190
$$636$$ 0 0
$$637$$ −934.051 −0.0580980
$$638$$ 0 0
$$639$$ −5991.23 −0.370907
$$640$$ 0 0
$$641$$ 11836.6 0.729357 0.364678 0.931133i $$-0.381179\pi$$
0.364678 + 0.931133i $$0.381179\pi$$
$$642$$ 0 0
$$643$$ −1448.21 −0.0888209 −0.0444104 0.999013i $$-0.514141\pi$$
−0.0444104 + 0.999013i $$0.514141\pi$$
$$644$$ 0 0
$$645$$ −645.058 −0.0393785
$$646$$ 0 0
$$647$$ 8732.95 0.530646 0.265323 0.964160i $$-0.414521\pi$$
0.265323 + 0.964160i $$0.414521\pi$$
$$648$$ 0 0
$$649$$ 2234.49 0.135149
$$650$$ 0 0
$$651$$ −1980.46 −0.119232
$$652$$ 0 0
$$653$$ −21978.4 −1.31712 −0.658562 0.752527i $$-0.728835\pi$$
−0.658562 + 0.752527i $$0.728835\pi$$
$$654$$ 0 0
$$655$$ −1821.01 −0.108630
$$656$$ 0 0
$$657$$ 5591.59 0.332038
$$658$$ 0 0
$$659$$ 27761.7 1.64103 0.820516 0.571623i $$-0.193686\pi$$
0.820516 + 0.571623i $$0.193686\pi$$
$$660$$ 0 0
$$661$$ −8573.72 −0.504507 −0.252254 0.967661i $$-0.581172\pi$$
−0.252254 + 0.967661i $$0.581172\pi$$
$$662$$ 0 0
$$663$$ 7005.27 0.410350
$$664$$ 0 0
$$665$$ −3760.25 −0.219273
$$666$$ 0 0
$$667$$ −20113.2 −1.16760
$$668$$ 0 0
$$669$$ 14027.6 0.810668
$$670$$ 0 0
$$671$$ −583.868 −0.0335916
$$672$$ 0 0
$$673$$ 27159.2 1.55559 0.777795 0.628518i $$-0.216338\pi$$
0.777795 + 0.628518i $$0.216338\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −1392.30 −0.0790404 −0.0395202 0.999219i $$-0.512583\pi$$
−0.0395202 + 0.999219i $$0.512583\pi$$
$$678$$ 0 0
$$679$$ −7033.42 −0.397523
$$680$$ 0 0
$$681$$ 16329.3 0.918857
$$682$$ 0 0
$$683$$ 8675.09 0.486007 0.243004 0.970025i $$-0.421867\pi$$
0.243004 + 0.970025i $$0.421867\pi$$
$$684$$ 0 0
$$685$$ 8015.66 0.447099
$$686$$ 0 0
$$687$$ 1608.91 0.0893504
$$688$$ 0 0
$$689$$ 3501.30 0.193598
$$690$$ 0 0
$$691$$ 21426.0 1.17957 0.589785 0.807561i $$-0.299213\pi$$
0.589785 + 0.807561i $$0.299213\pi$$
$$692$$ 0 0
$$693$$ 185.078 0.0101451
$$694$$ 0 0
$$695$$ −12155.6 −0.663438
$$696$$ 0 0
$$697$$ −60160.4 −3.26935
$$698$$ 0 0
$$699$$ 550.470 0.0297864
$$700$$ 0 0
$$701$$ 24840.5 1.33839 0.669197 0.743085i $$-0.266638\pi$$
0.669197 + 0.743085i $$0.266638\pi$$
$$702$$ 0 0
$$703$$ −10434.2 −0.559793
$$704$$ 0 0
$$705$$ 7102.41 0.379422
$$706$$ 0 0
$$707$$ −902.101 −0.0479873
$$708$$ 0 0
$$709$$ 12525.0 0.663450 0.331725 0.943376i $$-0.392369\pi$$
0.331725 + 0.943376i $$0.392369\pi$$
$$710$$ 0 0
$$711$$ 222.863 0.0117553
$$712$$ 0 0
$$713$$ −19863.3 −1.04332
$$714$$ 0 0
$$715$$ −280.000 −0.0146453
$$716$$ 0 0
$$717$$ 1929.65 0.100508
$$718$$ 0 0
$$719$$ −28085.0 −1.45674 −0.728369 0.685185i $$-0.759721\pi$$
−0.728369 + 0.685185i $$0.759721\pi$$
$$720$$ 0 0
$$721$$ −5642.05 −0.291430
$$722$$ 0 0
$$723$$ 17266.8 0.888189
$$724$$ 0 0
$$725$$ 2387.35 0.122295
$$726$$ 0 0
$$727$$ 14326.2 0.730851 0.365426 0.930841i $$-0.380923\pi$$
0.365426 + 0.930841i $$0.380923\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 5267.89 0.266539
$$732$$ 0 0
$$733$$ 6727.85 0.339016 0.169508 0.985529i $$-0.445782\pi$$
0.169508 + 0.985529i $$0.445782\pi$$
$$734$$ 0 0
$$735$$ −735.000 −0.0368856
$$736$$ 0 0
$$737$$ 910.677 0.0455159
$$738$$ 0 0
$$739$$ 3418.51 0.170165 0.0850826 0.996374i $$-0.472885\pi$$
0.0850826 + 0.996374i $$0.472885\pi$$
$$740$$ 0 0
$$741$$ −6143.91 −0.304591
$$742$$ 0 0
$$743$$ −8095.50 −0.399724 −0.199862 0.979824i $$-0.564049\pi$$
−0.199862 + 0.979824i $$0.564049\pi$$
$$744$$ 0 0
$$745$$ 11707.9 0.575762
$$746$$ 0 0
$$747$$ 3660.51 0.179292
$$748$$ 0 0
$$749$$ 5387.99 0.262847
$$750$$ 0 0
$$751$$ −13446.8 −0.653371 −0.326686 0.945133i $$-0.605932\pi$$
−0.326686 + 0.945133i $$0.605932\pi$$
$$752$$ 0 0
$$753$$ −15396.8 −0.745141
$$754$$ 0 0
$$755$$ 10520.4 0.507119
$$756$$ 0 0
$$757$$ −2593.24 −0.124508 −0.0622541 0.998060i $$-0.519829\pi$$
−0.0622541 + 0.998060i $$0.519829\pi$$
$$758$$ 0 0
$$759$$ 1856.26 0.0887722
$$760$$ 0 0
$$761$$ 27079.4 1.28992 0.644959 0.764217i $$-0.276875\pi$$
0.644959 + 0.764217i $$0.276875\pi$$
$$762$$ 0 0
$$763$$ −5465.99 −0.259348
$$764$$ 0 0
$$765$$ 5512.41 0.260525
$$766$$ 0 0
$$767$$ −14499.0 −0.682568
$$768$$ 0 0
$$769$$ 2138.72 0.100292 0.0501458 0.998742i $$-0.484031\pi$$
0.0501458 + 0.998742i $$0.484031\pi$$
$$770$$ 0 0
$$771$$ −15125.2 −0.706513
$$772$$ 0 0
$$773$$ 25864.0 1.20345 0.601724 0.798704i $$-0.294481\pi$$
0.601724 + 0.798704i $$0.294481\pi$$
$$774$$ 0 0
$$775$$ 2357.69 0.109278
$$776$$ 0 0
$$777$$ −2039.53 −0.0941671
$$778$$ 0 0
$$779$$ 52763.1 2.42675
$$780$$ 0 0
$$781$$ −1955.63 −0.0896006
$$782$$ 0 0
$$783$$ −2578.34 −0.117679
$$784$$ 0 0
$$785$$ −2967.33 −0.134916
$$786$$ 0 0
$$787$$ 32371.3 1.46621 0.733107 0.680113i $$-0.238069\pi$$
0.733107 + 0.680113i $$0.238069\pi$$
$$788$$ 0 0
$$789$$ 22731.0 1.02566
$$790$$ 0 0
$$791$$ −7809.52 −0.351043
$$792$$ 0 0
$$793$$ 3788.57 0.169654
$$794$$ 0 0
$$795$$ 2755.16 0.122912
$$796$$ 0 0
$$797$$ 2024.33 0.0899691 0.0449845 0.998988i $$-0.485676\pi$$
0.0449845 + 0.998988i $$0.485676\pi$$
$$798$$ 0 0
$$799$$ −58002.1 −2.56817
$$800$$ 0 0
$$801$$ 2355.76 0.103916
$$802$$ 0 0
$$803$$ 1825.18 0.0802109
$$804$$ 0 0
$$805$$ −7371.79 −0.322760
$$806$$ 0 0
$$807$$ −3069.29 −0.133884
$$808$$ 0 0
$$809$$ 12391.7 0.538526 0.269263 0.963067i $$-0.413220\pi$$
0.269263 + 0.963067i $$0.413220\pi$$
$$810$$ 0 0
$$811$$ −14654.5 −0.634511 −0.317256 0.948340i $$-0.602761\pi$$
−0.317256 + 0.948340i $$0.602761\pi$$
$$812$$ 0 0
$$813$$ −6755.94 −0.291441
$$814$$ 0 0
$$815$$ −10893.5 −0.468201
$$816$$ 0 0
$$817$$ −4620.16 −0.197844
$$818$$ 0 0
$$819$$ −1200.92 −0.0512376
$$820$$ 0 0
$$821$$ 23887.9 1.01546 0.507731 0.861516i $$-0.330485\pi$$
0.507731 + 0.861516i $$0.330485\pi$$
$$822$$ 0 0
$$823$$ 4008.41 0.169774 0.0848871 0.996391i $$-0.472947\pi$$
0.0848871 + 0.996391i $$0.472947\pi$$
$$824$$ 0 0
$$825$$ −220.331 −0.00929810
$$826$$ 0 0
$$827$$ 45110.4 1.89679 0.948394 0.317096i $$-0.102708\pi$$
0.948394 + 0.317096i $$0.102708\pi$$
$$828$$ 0 0
$$829$$ 16165.4 0.677260 0.338630 0.940920i $$-0.390036\pi$$
0.338630 + 0.940920i $$0.390036\pi$$
$$830$$ 0 0
$$831$$ 25892.2 1.08085
$$832$$ 0 0
$$833$$ 6002.41 0.249665
$$834$$ 0 0
$$835$$ −3997.51 −0.165676
$$836$$ 0 0
$$837$$ −2546.30 −0.105153
$$838$$ 0 0
$$839$$ 25244.4 1.03878 0.519388 0.854538i $$-0.326160\pi$$
0.519388 + 0.854538i $$0.326160\pi$$
$$840$$ 0 0
$$841$$ −15269.9 −0.626096
$$842$$ 0 0
$$843$$ 22564.9 0.921916
$$844$$ 0 0
$$845$$ −9168.15 −0.373247
$$846$$ 0 0
$$847$$ −9256.59 −0.375514
$$848$$ 0 0
$$849$$ 44.4506 0.00179687
$$850$$ 0 0
$$851$$ −20455.8 −0.823990
$$852$$ 0 0
$$853$$ −30168.1 −1.21094 −0.605472 0.795867i $$-0.707016\pi$$
−0.605472 + 0.795867i $$0.707016\pi$$
$$854$$ 0 0
$$855$$ −4834.61 −0.193380
$$856$$ 0 0
$$857$$ −13393.6 −0.533857 −0.266929 0.963716i $$-0.586009\pi$$
−0.266929 + 0.963716i $$0.586009\pi$$
$$858$$ 0 0
$$859$$ −19060.4 −0.757081 −0.378541 0.925585i $$-0.623574\pi$$
−0.378541 + 0.925585i $$0.623574\pi$$
$$860$$ 0 0
$$861$$ 10313.4 0.408222
$$862$$ 0 0
$$863$$ 9466.86 0.373413 0.186707 0.982416i $$-0.440219\pi$$
0.186707 + 0.982416i $$0.440219\pi$$
$$864$$ 0 0
$$865$$ −7221.79 −0.283871
$$866$$ 0 0
$$867$$ −30278.3 −1.18605
$$868$$ 0 0
$$869$$ 72.7461 0.00283975
$$870$$ 0 0
$$871$$ −5909.15 −0.229878
$$872$$ 0 0
$$873$$ −9042.97 −0.350582
$$874$$ 0 0
$$875$$ 875.000 0.0338062
$$876$$ 0 0
$$877$$ 37740.6 1.45315 0.726573 0.687090i $$-0.241112\pi$$
0.726573 + 0.687090i $$0.241112\pi$$
$$878$$ 0 0
$$879$$ −20740.2 −0.795845
$$880$$ 0 0
$$881$$ 25991.5 0.993957 0.496979 0.867763i $$-0.334443\pi$$
0.496979 + 0.867763i $$0.334443\pi$$
$$882$$ 0 0
$$883$$ −39420.3 −1.50238 −0.751189 0.660087i $$-0.770519\pi$$
−0.751189 + 0.660087i $$0.770519\pi$$
$$884$$ 0 0
$$885$$ −11409.2 −0.433352
$$886$$ 0 0
$$887$$ −46005.2 −1.74149 −0.870745 0.491735i $$-0.836363\pi$$
−0.870745 + 0.491735i $$0.836363\pi$$
$$888$$ 0 0
$$889$$ 13131.9 0.495421
$$890$$ 0 0
$$891$$ 237.957 0.00894710
$$892$$ 0 0
$$893$$ 50870.2 1.90628
$$894$$ 0 0
$$895$$ −16717.5 −0.624361
$$896$$ 0 0
$$897$$ −12044.8 −0.448345
$$898$$ 0 0
$$899$$ 9005.81 0.334105
$$900$$ 0 0
$$901$$ −22500.1 −0.831950
$$902$$ 0 0
$$903$$ −903.081 −0.0332809
$$904$$ 0 0
$$905$$ 11259.1 0.413551
$$906$$ 0 0
$$907$$ 2838.97 0.103932 0.0519661 0.998649i $$-0.483451\pi$$
0.0519661 + 0.998649i $$0.483451\pi$$
$$908$$ 0 0
$$909$$ −1159.84 −0.0423208
$$910$$ 0 0
$$911$$ −39890.9 −1.45076 −0.725382 0.688347i $$-0.758337\pi$$
−0.725382 + 0.688347i $$0.758337\pi$$
$$912$$ 0 0
$$913$$ 1194.85 0.0433119
$$914$$ 0 0
$$915$$ 2981.21 0.107711
$$916$$ 0 0
$$917$$ −2549.42 −0.0918094
$$918$$ 0 0
$$919$$ 646.475 0.0232048 0.0116024 0.999933i $$-0.496307\pi$$
0.0116024 + 0.999933i $$0.496307\pi$$
$$920$$ 0 0
$$921$$ −22932.4 −0.820463
$$922$$ 0 0
$$923$$ 12689.6 0.452528
$$924$$ 0 0
$$925$$ 2428.02 0.0863056
$$926$$ 0 0
$$927$$ −7254.07 −0.257017
$$928$$ 0 0
$$929$$ 51188.2 1.80778 0.903892 0.427760i $$-0.140697\pi$$
0.903892 + 0.427760i $$0.140697\pi$$
$$930$$ 0 0
$$931$$ −5264.35 −0.185319
$$932$$ 0 0
$$933$$ 22782.0 0.799409
$$934$$ 0 0
$$935$$ 1799.34 0.0629355
$$936$$ 0 0
$$937$$ 29786.1 1.03849 0.519247 0.854624i $$-0.326212\pi$$
0.519247 + 0.854624i $$0.326212\pi$$
$$938$$ 0 0
$$939$$ 27383.5 0.951680
$$940$$ 0 0
$$941$$ −44817.4 −1.55261 −0.776304 0.630358i $$-0.782908\pi$$
−0.776304 + 0.630358i $$0.782908\pi$$
$$942$$ 0 0
$$943$$ 103439. 3.57206
$$944$$ 0 0
$$945$$ −945.000 −0.0325300
$$946$$ 0 0
$$947$$ −54697.1 −1.87689 −0.938446 0.345425i $$-0.887735\pi$$
−0.938446 + 0.345425i $$0.887735\pi$$
$$948$$ 0 0
$$949$$ −11843.2 −0.405105
$$950$$ 0 0
$$951$$ 14789.4 0.504290
$$952$$ 0 0
$$953$$ −7577.51 −0.257565 −0.128783 0.991673i $$-0.541107\pi$$
−0.128783 + 0.991673i $$0.541107\pi$$
$$954$$ 0 0
$$955$$ 5009.63 0.169746
$$956$$ 0 0
$$957$$ −841.612 −0.0284278
$$958$$ 0 0
$$959$$ 11221.9 0.377868
$$960$$ 0 0
$$961$$ −20897.1 −0.701457
$$962$$ 0 0
$$963$$ 6927.41 0.231810
$$964$$ 0 0
$$965$$ −20274.8 −0.676342
$$966$$ 0 0
$$967$$ −50779.0 −1.68867 −0.844334 0.535817i $$-0.820004\pi$$
−0.844334 + 0.535817i $$0.820004\pi$$
$$968$$ 0 0
$$969$$ 39482.0 1.30892
$$970$$ 0 0
$$971$$ 15313.2 0.506102 0.253051 0.967453i $$-0.418566\pi$$
0.253051 + 0.967453i $$0.418566\pi$$
$$972$$ 0 0
$$973$$ −17017.9 −0.560707
$$974$$ 0 0
$$975$$ 1429.67 0.0469601
$$976$$ 0 0
$$977$$ 46620.4 1.52663 0.763316 0.646025i $$-0.223570\pi$$
0.763316 + 0.646025i $$0.223570\pi$$
$$978$$ 0 0
$$979$$ 768.957 0.0251031
$$980$$ 0 0
$$981$$ −7027.70 −0.228723
$$982$$ 0 0
$$983$$ 2824.37 0.0916414 0.0458207 0.998950i $$-0.485410\pi$$
0.0458207 + 0.998950i $$0.485410\pi$$
$$984$$ 0 0
$$985$$ −25701.1 −0.831377
$$986$$ 0 0
$$987$$ 9943.38 0.320670
$$988$$ 0 0
$$989$$ −9057.59 −0.291218
$$990$$ 0 0
$$991$$ −16951.4 −0.543370 −0.271685 0.962386i $$-0.587581\pi$$
−0.271685 + 0.962386i $$0.587581\pi$$
$$992$$ 0 0
$$993$$ 3665.00 0.117125
$$994$$ 0 0
$$995$$ −2928.15 −0.0932952
$$996$$ 0 0
$$997$$ 23847.8 0.757540 0.378770 0.925491i $$-0.376347\pi$$
0.378770 + 0.925491i $$0.376347\pi$$
$$998$$ 0 0
$$999$$ −2622.26 −0.0830476
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1680.4.a.bg.1.1 2
4.3 odd 2 105.4.a.f.1.1 2
12.11 even 2 315.4.a.i.1.2 2
20.3 even 4 525.4.d.h.274.3 4
20.7 even 4 525.4.d.h.274.2 4
20.19 odd 2 525.4.a.k.1.2 2
28.27 even 2 735.4.a.p.1.1 2
60.59 even 2 1575.4.a.w.1.1 2
84.83 odd 2 2205.4.a.z.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.1 2 4.3 odd 2
315.4.a.i.1.2 2 12.11 even 2
525.4.a.k.1.2 2 20.19 odd 2
525.4.d.h.274.2 4 20.7 even 4
525.4.d.h.274.3 4 20.3 even 4
735.4.a.p.1.1 2 28.27 even 2
1575.4.a.w.1.1 2 60.59 even 2
1680.4.a.bg.1.1 2 1.1 even 1 trivial
2205.4.a.z.1.2 2 84.83 odd 2