# Properties

 Label 1680.4.a.y.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: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 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 $$3.70156$$ 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)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -37.4031 q^{11} +29.0156 q^{13} +15.0000 q^{15} +58.4187 q^{17} +54.5969 q^{19} +21.0000 q^{21} -161.675 q^{23} +25.0000 q^{25} -27.0000 q^{27} +137.581 q^{29} -154.659 q^{31} +112.209 q^{33} +35.0000 q^{35} -350.125 q^{37} -87.0469 q^{39} +353.769 q^{41} +518.156 q^{43} -45.0000 q^{45} +542.219 q^{47} +49.0000 q^{49} -175.256 q^{51} +305.309 q^{53} +187.016 q^{55} -163.791 q^{57} -14.6813 q^{59} -171.069 q^{61} -63.0000 q^{63} -145.078 q^{65} -551.956 q^{67} +485.025 q^{69} +120.334 q^{71} +284.659 q^{73} -75.0000 q^{75} +261.822 q^{77} -941.612 q^{79} +81.0000 q^{81} -377.150 q^{83} -292.094 q^{85} -412.744 q^{87} -677.725 q^{89} -203.109 q^{91} +463.978 q^{93} -272.984 q^{95} -1225.03 q^{97} -336.628 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 $$2 q - 6 q^{3} - 10 q^{5} - 14 q^{7} + 18 q^{9} - 62 q^{11} - 6 q^{13} + 30 q^{15} + 40 q^{17} + 122 q^{19} + 42 q^{21} - 16 q^{23} + 50 q^{25} - 54 q^{27} + 352 q^{29} - 66 q^{31} + 186 q^{33} + 70 q^{35} - 188 q^{37} + 18 q^{39} + 16 q^{41} + 396 q^{43} - 90 q^{45} + 188 q^{47} + 98 q^{49} - 120 q^{51} + 982 q^{53} + 310 q^{55} - 366 q^{57} - 516 q^{59} - 880 q^{61} - 126 q^{63} + 30 q^{65} + 356 q^{67} + 48 q^{69} - 310 q^{71} + 326 q^{73} - 150 q^{75} + 434 q^{77} - 1832 q^{79} + 162 q^{81} + 680 q^{83} - 200 q^{85} - 1056 q^{87} + 796 q^{89} + 42 q^{91} + 198 q^{93} - 610 q^{95} - 670 q^{97} - 558 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 10 * q^5 - 14 * q^7 + 18 * q^9 - 62 * q^11 - 6 * q^13 + 30 * q^15 + 40 * q^17 + 122 * q^19 + 42 * q^21 - 16 * q^23 + 50 * q^25 - 54 * q^27 + 352 * q^29 - 66 * q^31 + 186 * q^33 + 70 * q^35 - 188 * q^37 + 18 * q^39 + 16 * q^41 + 396 * q^43 - 90 * q^45 + 188 * q^47 + 98 * q^49 - 120 * q^51 + 982 * q^53 + 310 * q^55 - 366 * q^57 - 516 * q^59 - 880 * q^61 - 126 * q^63 + 30 * q^65 + 356 * q^67 + 48 * q^69 - 310 * q^71 + 326 * q^73 - 150 * q^75 + 434 * q^77 - 1832 * q^79 + 162 * q^81 + 680 * q^83 - 200 * q^85 - 1056 * q^87 + 796 * q^89 + 42 * q^91 + 198 * q^93 - 610 * q^95 - 670 * q^97 - 558 * 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
$$11$$ −37.4031 −1.02522 −0.512612 0.858620i $$-0.671322\pi$$
−0.512612 + 0.858620i $$0.671322\pi$$
$$12$$ 0 0
$$13$$ 29.0156 0.619037 0.309519 0.950893i $$-0.399832\pi$$
0.309519 + 0.950893i $$0.399832\pi$$
$$14$$ 0 0
$$15$$ 15.0000 0.258199
$$16$$ 0 0
$$17$$ 58.4187 0.833449 0.416724 0.909033i $$-0.363178\pi$$
0.416724 + 0.909033i $$0.363178\pi$$
$$18$$ 0 0
$$19$$ 54.5969 0.659231 0.329615 0.944115i $$-0.393081\pi$$
0.329615 + 0.944115i $$0.393081\pi$$
$$20$$ 0 0
$$21$$ 21.0000 0.218218
$$22$$ 0 0
$$23$$ −161.675 −1.46572 −0.732860 0.680379i $$-0.761815\pi$$
−0.732860 + 0.680379i $$0.761815\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 0 0
$$29$$ 137.581 0.880972 0.440486 0.897759i $$-0.354806\pi$$
0.440486 + 0.897759i $$0.354806\pi$$
$$30$$ 0 0
$$31$$ −154.659 −0.896053 −0.448026 0.894020i $$-0.647873\pi$$
−0.448026 + 0.894020i $$0.647873\pi$$
$$32$$ 0 0
$$33$$ 112.209 0.591913
$$34$$ 0 0
$$35$$ 35.0000 0.169031
$$36$$ 0 0
$$37$$ −350.125 −1.55568 −0.777840 0.628462i $$-0.783685\pi$$
−0.777840 + 0.628462i $$0.783685\pi$$
$$38$$ 0 0
$$39$$ −87.0469 −0.357401
$$40$$ 0 0
$$41$$ 353.769 1.34755 0.673773 0.738938i $$-0.264673\pi$$
0.673773 + 0.738938i $$0.264673\pi$$
$$42$$ 0 0
$$43$$ 518.156 1.83763 0.918815 0.394689i $$-0.129148\pi$$
0.918815 + 0.394689i $$0.129148\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 0 0
$$47$$ 542.219 1.68278 0.841391 0.540427i $$-0.181737\pi$$
0.841391 + 0.540427i $$0.181737\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ 0 0
$$51$$ −175.256 −0.481192
$$52$$ 0 0
$$53$$ 305.309 0.791273 0.395637 0.918407i $$-0.370524\pi$$
0.395637 + 0.918407i $$0.370524\pi$$
$$54$$ 0 0
$$55$$ 187.016 0.458494
$$56$$ 0 0
$$57$$ −163.791 −0.380607
$$58$$ 0 0
$$59$$ −14.6813 −0.0323956 −0.0161978 0.999869i $$-0.505156\pi$$
−0.0161978 + 0.999869i $$0.505156\pi$$
$$60$$ 0 0
$$61$$ −171.069 −0.359067 −0.179534 0.983752i $$-0.557459\pi$$
−0.179534 + 0.983752i $$0.557459\pi$$
$$62$$ 0 0
$$63$$ −63.0000 −0.125988
$$64$$ 0 0
$$65$$ −145.078 −0.276842
$$66$$ 0 0
$$67$$ −551.956 −1.00645 −0.503225 0.864155i $$-0.667853\pi$$
−0.503225 + 0.864155i $$0.667853\pi$$
$$68$$ 0 0
$$69$$ 485.025 0.846234
$$70$$ 0 0
$$71$$ 120.334 0.201142 0.100571 0.994930i $$-0.467933\pi$$
0.100571 + 0.994930i $$0.467933\pi$$
$$72$$ 0 0
$$73$$ 284.659 0.456395 0.228198 0.973615i $$-0.426717\pi$$
0.228198 + 0.973615i $$0.426717\pi$$
$$74$$ 0 0
$$75$$ −75.0000 −0.115470
$$76$$ 0 0
$$77$$ 261.822 0.387498
$$78$$ 0 0
$$79$$ −941.612 −1.34101 −0.670504 0.741906i $$-0.733922\pi$$
−0.670504 + 0.741906i $$0.733922\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −377.150 −0.498766 −0.249383 0.968405i $$-0.580228\pi$$
−0.249383 + 0.968405i $$0.580228\pi$$
$$84$$ 0 0
$$85$$ −292.094 −0.372730
$$86$$ 0 0
$$87$$ −412.744 −0.508630
$$88$$ 0 0
$$89$$ −677.725 −0.807176 −0.403588 0.914941i $$-0.632237\pi$$
−0.403588 + 0.914941i $$0.632237\pi$$
$$90$$ 0 0
$$91$$ −203.109 −0.233974
$$92$$ 0 0
$$93$$ 463.978 0.517336
$$94$$ 0 0
$$95$$ −272.984 −0.294817
$$96$$ 0 0
$$97$$ −1225.03 −1.28230 −0.641151 0.767414i $$-0.721543\pi$$
−0.641151 + 0.767414i $$0.721543\pi$$
$$98$$ 0 0
$$99$$ −336.628 −0.341741
$$100$$ 0 0
$$101$$ −338.144 −0.333134 −0.166567 0.986030i $$-0.553268\pi$$
−0.166567 + 0.986030i $$0.553268\pi$$
$$102$$ 0 0
$$103$$ 566.700 0.542122 0.271061 0.962562i $$-0.412625\pi$$
0.271061 + 0.962562i $$0.412625\pi$$
$$104$$ 0 0
$$105$$ −105.000 −0.0975900
$$106$$ 0 0
$$107$$ 562.531 0.508242 0.254121 0.967172i $$-0.418214\pi$$
0.254121 + 0.967172i $$0.418214\pi$$
$$108$$ 0 0
$$109$$ 1830.79 1.60879 0.804396 0.594094i $$-0.202489\pi$$
0.804396 + 0.594094i $$0.202489\pi$$
$$110$$ 0 0
$$111$$ 1050.37 0.898173
$$112$$ 0 0
$$113$$ −31.8032 −0.0264761 −0.0132380 0.999912i $$-0.504214\pi$$
−0.0132380 + 0.999912i $$0.504214\pi$$
$$114$$ 0 0
$$115$$ 808.375 0.655490
$$116$$ 0 0
$$117$$ 261.141 0.206346
$$118$$ 0 0
$$119$$ −408.931 −0.315014
$$120$$ 0 0
$$121$$ 67.9937 0.0510847
$$122$$ 0 0
$$123$$ −1061.31 −0.778006
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −2220.81 −1.55169 −0.775847 0.630921i $$-0.782677\pi$$
−0.775847 + 0.630921i $$0.782677\pi$$
$$128$$ 0 0
$$129$$ −1554.47 −1.06096
$$130$$ 0 0
$$131$$ −646.512 −0.431191 −0.215596 0.976483i $$-0.569169\pi$$
−0.215596 + 0.976483i $$0.569169\pi$$
$$132$$ 0 0
$$133$$ −382.178 −0.249166
$$134$$ 0 0
$$135$$ 135.000 0.0860663
$$136$$ 0 0
$$137$$ −896.009 −0.558768 −0.279384 0.960179i $$-0.590130\pi$$
−0.279384 + 0.960179i $$0.590130\pi$$
$$138$$ 0 0
$$139$$ 2313.61 1.41178 0.705891 0.708320i $$-0.250547\pi$$
0.705891 + 0.708320i $$0.250547\pi$$
$$140$$ 0 0
$$141$$ −1626.66 −0.971554
$$142$$ 0 0
$$143$$ −1085.27 −0.634652
$$144$$ 0 0
$$145$$ −687.906 −0.393983
$$146$$ 0 0
$$147$$ −147.000 −0.0824786
$$148$$ 0 0
$$149$$ 819.337 0.450488 0.225244 0.974302i $$-0.427682\pi$$
0.225244 + 0.974302i $$0.427682\pi$$
$$150$$ 0 0
$$151$$ −534.744 −0.288191 −0.144095 0.989564i $$-0.546027\pi$$
−0.144095 + 0.989564i $$0.546027\pi$$
$$152$$ 0 0
$$153$$ 525.769 0.277816
$$154$$ 0 0
$$155$$ 773.297 0.400727
$$156$$ 0 0
$$157$$ 1564.76 0.795423 0.397711 0.917511i $$-0.369805\pi$$
0.397711 + 0.917511i $$0.369805\pi$$
$$158$$ 0 0
$$159$$ −915.928 −0.456842
$$160$$ 0 0
$$161$$ 1131.72 0.553990
$$162$$ 0 0
$$163$$ 1114.31 0.535455 0.267728 0.963495i $$-0.413727\pi$$
0.267728 + 0.963495i $$0.413727\pi$$
$$164$$ 0 0
$$165$$ −561.047 −0.264712
$$166$$ 0 0
$$167$$ 1774.47 0.822231 0.411115 0.911583i $$-0.365139\pi$$
0.411115 + 0.911583i $$0.365139\pi$$
$$168$$ 0 0
$$169$$ −1355.09 −0.616793
$$170$$ 0 0
$$171$$ 491.372 0.219744
$$172$$ 0 0
$$173$$ −4215.88 −1.85276 −0.926380 0.376590i $$-0.877097\pi$$
−0.926380 + 0.376590i $$0.877097\pi$$
$$174$$ 0 0
$$175$$ −175.000 −0.0755929
$$176$$ 0 0
$$177$$ 44.0438 0.0187036
$$178$$ 0 0
$$179$$ 2430.70 1.01497 0.507483 0.861662i $$-0.330576\pi$$
0.507483 + 0.861662i $$0.330576\pi$$
$$180$$ 0 0
$$181$$ −2700.91 −1.10916 −0.554578 0.832132i $$-0.687120\pi$$
−0.554578 + 0.832132i $$0.687120\pi$$
$$182$$ 0 0
$$183$$ 513.206 0.207308
$$184$$ 0 0
$$185$$ 1750.62 0.695722
$$186$$ 0 0
$$187$$ −2185.04 −0.854472
$$188$$ 0 0
$$189$$ 189.000 0.0727393
$$190$$ 0 0
$$191$$ −3611.10 −1.36801 −0.684005 0.729478i $$-0.739763\pi$$
−0.684005 + 0.729478i $$0.739763\pi$$
$$192$$ 0 0
$$193$$ −4468.33 −1.66651 −0.833257 0.552886i $$-0.813526\pi$$
−0.833257 + 0.552886i $$0.813526\pi$$
$$194$$ 0 0
$$195$$ 435.234 0.159835
$$196$$ 0 0
$$197$$ −434.422 −0.157113 −0.0785566 0.996910i $$-0.525031\pi$$
−0.0785566 + 0.996910i $$0.525031\pi$$
$$198$$ 0 0
$$199$$ 468.915 0.167038 0.0835189 0.996506i $$-0.473384\pi$$
0.0835189 + 0.996506i $$0.473384\pi$$
$$200$$ 0 0
$$201$$ 1655.87 0.581074
$$202$$ 0 0
$$203$$ −963.069 −0.332976
$$204$$ 0 0
$$205$$ −1768.84 −0.602641
$$206$$ 0 0
$$207$$ −1455.07 −0.488573
$$208$$ 0 0
$$209$$ −2042.09 −0.675859
$$210$$ 0 0
$$211$$ −3735.51 −1.21878 −0.609392 0.792869i $$-0.708586\pi$$
−0.609392 + 0.792869i $$0.708586\pi$$
$$212$$ 0 0
$$213$$ −361.003 −0.116129
$$214$$ 0 0
$$215$$ −2590.78 −0.821813
$$216$$ 0 0
$$217$$ 1082.62 0.338676
$$218$$ 0 0
$$219$$ −853.978 −0.263500
$$220$$ 0 0
$$221$$ 1695.06 0.515936
$$222$$ 0 0
$$223$$ −842.806 −0.253087 −0.126544 0.991961i $$-0.540388\pi$$
−0.126544 + 0.991961i $$0.540388\pi$$
$$224$$ 0 0
$$225$$ 225.000 0.0666667
$$226$$ 0 0
$$227$$ −992.150 −0.290094 −0.145047 0.989425i $$-0.546333\pi$$
−0.145047 + 0.989425i $$0.546333\pi$$
$$228$$ 0 0
$$229$$ −6411.39 −1.85012 −0.925059 0.379825i $$-0.875984\pi$$
−0.925059 + 0.379825i $$0.875984\pi$$
$$230$$ 0 0
$$231$$ −785.466 −0.223722
$$232$$ 0 0
$$233$$ −2274.35 −0.639476 −0.319738 0.947506i $$-0.603595\pi$$
−0.319738 + 0.947506i $$0.603595\pi$$
$$234$$ 0 0
$$235$$ −2711.09 −0.752563
$$236$$ 0 0
$$237$$ 2824.84 0.774232
$$238$$ 0 0
$$239$$ −2863.12 −0.774893 −0.387447 0.921892i $$-0.626643\pi$$
−0.387447 + 0.921892i $$0.626643\pi$$
$$240$$ 0 0
$$241$$ −5364.23 −1.43378 −0.716889 0.697187i $$-0.754435\pi$$
−0.716889 + 0.697187i $$0.754435\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 0 0
$$245$$ −245.000 −0.0638877
$$246$$ 0 0
$$247$$ 1584.16 0.408088
$$248$$ 0 0
$$249$$ 1131.45 0.287963
$$250$$ 0 0
$$251$$ −5569.81 −1.40065 −0.700325 0.713824i $$-0.746961\pi$$
−0.700325 + 0.713824i $$0.746961\pi$$
$$252$$ 0 0
$$253$$ 6047.15 1.50269
$$254$$ 0 0
$$255$$ 876.281 0.215196
$$256$$ 0 0
$$257$$ 2095.36 0.508580 0.254290 0.967128i $$-0.418158\pi$$
0.254290 + 0.967128i $$0.418158\pi$$
$$258$$ 0 0
$$259$$ 2450.87 0.587992
$$260$$ 0 0
$$261$$ 1238.23 0.293657
$$262$$ 0 0
$$263$$ 7465.88 1.75044 0.875220 0.483724i $$-0.160716\pi$$
0.875220 + 0.483724i $$0.160716\pi$$
$$264$$ 0 0
$$265$$ −1526.55 −0.353868
$$266$$ 0 0
$$267$$ 2033.17 0.466023
$$268$$ 0 0
$$269$$ −6521.38 −1.47812 −0.739062 0.673637i $$-0.764731\pi$$
−0.739062 + 0.673637i $$0.764731\pi$$
$$270$$ 0 0
$$271$$ −2409.70 −0.540144 −0.270072 0.962840i $$-0.587048\pi$$
−0.270072 + 0.962840i $$0.587048\pi$$
$$272$$ 0 0
$$273$$ 609.328 0.135085
$$274$$ 0 0
$$275$$ −935.078 −0.205045
$$276$$ 0 0
$$277$$ −2219.83 −0.481503 −0.240752 0.970587i $$-0.577394\pi$$
−0.240752 + 0.970587i $$0.577394\pi$$
$$278$$ 0 0
$$279$$ −1391.93 −0.298684
$$280$$ 0 0
$$281$$ 5838.56 1.23950 0.619749 0.784800i $$-0.287234\pi$$
0.619749 + 0.784800i $$0.287234\pi$$
$$282$$ 0 0
$$283$$ 3645.04 0.765636 0.382818 0.923824i $$-0.374954\pi$$
0.382818 + 0.923824i $$0.374954\pi$$
$$284$$ 0 0
$$285$$ 818.953 0.170213
$$286$$ 0 0
$$287$$ −2476.38 −0.509325
$$288$$ 0 0
$$289$$ −1500.25 −0.305363
$$290$$ 0 0
$$291$$ 3675.10 0.740338
$$292$$ 0 0
$$293$$ 3777.91 0.753268 0.376634 0.926362i $$-0.377081\pi$$
0.376634 + 0.926362i $$0.377081\pi$$
$$294$$ 0 0
$$295$$ 73.4064 0.0144877
$$296$$ 0 0
$$297$$ 1009.88 0.197304
$$298$$ 0 0
$$299$$ −4691.10 −0.907336
$$300$$ 0 0
$$301$$ −3627.09 −0.694559
$$302$$ 0 0
$$303$$ 1014.43 0.192335
$$304$$ 0 0
$$305$$ 855.344 0.160580
$$306$$ 0 0
$$307$$ 4799.64 0.892281 0.446140 0.894963i $$-0.352798\pi$$
0.446140 + 0.894963i $$0.352798\pi$$
$$308$$ 0 0
$$309$$ −1700.10 −0.312994
$$310$$ 0 0
$$311$$ −580.113 −0.105772 −0.0528861 0.998601i $$-0.516842\pi$$
−0.0528861 + 0.998601i $$0.516842\pi$$
$$312$$ 0 0
$$313$$ −6114.78 −1.10424 −0.552121 0.833764i $$-0.686182\pi$$
−0.552121 + 0.833764i $$0.686182\pi$$
$$314$$ 0 0
$$315$$ 315.000 0.0563436
$$316$$ 0 0
$$317$$ −4300.63 −0.761979 −0.380989 0.924579i $$-0.624417\pi$$
−0.380989 + 0.924579i $$0.624417\pi$$
$$318$$ 0 0
$$319$$ −5145.97 −0.903194
$$320$$ 0 0
$$321$$ −1687.59 −0.293434
$$322$$ 0 0
$$323$$ 3189.48 0.549435
$$324$$ 0 0
$$325$$ 725.391 0.123807
$$326$$ 0 0
$$327$$ −5492.38 −0.928836
$$328$$ 0 0
$$329$$ −3795.53 −0.636032
$$330$$ 0 0
$$331$$ 6687.54 1.11051 0.555257 0.831679i $$-0.312620\pi$$
0.555257 + 0.831679i $$0.312620\pi$$
$$332$$ 0 0
$$333$$ −3151.12 −0.518560
$$334$$ 0 0
$$335$$ 2759.78 0.450098
$$336$$ 0 0
$$337$$ 5869.28 0.948723 0.474362 0.880330i $$-0.342679\pi$$
0.474362 + 0.880330i $$0.342679\pi$$
$$338$$ 0 0
$$339$$ 95.4097 0.0152860
$$340$$ 0 0
$$341$$ 5784.74 0.918655
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 0 0
$$345$$ −2425.12 −0.378447
$$346$$ 0 0
$$347$$ −1937.22 −0.299699 −0.149850 0.988709i $$-0.547879\pi$$
−0.149850 + 0.988709i $$0.547879\pi$$
$$348$$ 0 0
$$349$$ −9748.82 −1.49525 −0.747625 0.664121i $$-0.768806\pi$$
−0.747625 + 0.664121i $$0.768806\pi$$
$$350$$ 0 0
$$351$$ −783.422 −0.119134
$$352$$ 0 0
$$353$$ −4576.61 −0.690052 −0.345026 0.938593i $$-0.612130\pi$$
−0.345026 + 0.938593i $$0.612130\pi$$
$$354$$ 0 0
$$355$$ −601.672 −0.0899533
$$356$$ 0 0
$$357$$ 1226.79 0.181873
$$358$$ 0 0
$$359$$ −10849.9 −1.59509 −0.797546 0.603258i $$-0.793869\pi$$
−0.797546 + 0.603258i $$0.793869\pi$$
$$360$$ 0 0
$$361$$ −3878.18 −0.565415
$$362$$ 0 0
$$363$$ −203.981 −0.0294938
$$364$$ 0 0
$$365$$ −1423.30 −0.204106
$$366$$ 0 0
$$367$$ −11467.7 −1.63108 −0.815541 0.578699i $$-0.803561\pi$$
−0.815541 + 0.578699i $$0.803561\pi$$
$$368$$ 0 0
$$369$$ 3183.92 0.449182
$$370$$ 0 0
$$371$$ −2137.17 −0.299073
$$372$$ 0 0
$$373$$ 539.982 0.0749576 0.0374788 0.999297i $$-0.488067\pi$$
0.0374788 + 0.999297i $$0.488067\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ 0 0
$$377$$ 3992.01 0.545355
$$378$$ 0 0
$$379$$ −8577.57 −1.16253 −0.581267 0.813713i $$-0.697443\pi$$
−0.581267 + 0.813713i $$0.697443\pi$$
$$380$$ 0 0
$$381$$ 6662.44 0.895871
$$382$$ 0 0
$$383$$ −8627.96 −1.15109 −0.575546 0.817770i $$-0.695210\pi$$
−0.575546 + 0.817770i $$0.695210\pi$$
$$384$$ 0 0
$$385$$ −1309.11 −0.173295
$$386$$ 0 0
$$387$$ 4663.41 0.612543
$$388$$ 0 0
$$389$$ 9234.06 1.20356 0.601781 0.798661i $$-0.294458\pi$$
0.601781 + 0.798661i $$0.294458\pi$$
$$390$$ 0 0
$$391$$ −9444.85 −1.22160
$$392$$ 0 0
$$393$$ 1939.54 0.248948
$$394$$ 0 0
$$395$$ 4708.06 0.599717
$$396$$ 0 0
$$397$$ 11618.0 1.46874 0.734372 0.678747i $$-0.237477\pi$$
0.734372 + 0.678747i $$0.237477\pi$$
$$398$$ 0 0
$$399$$ 1146.53 0.143856
$$400$$ 0 0
$$401$$ 11157.1 1.38942 0.694711 0.719289i $$-0.255532\pi$$
0.694711 + 0.719289i $$0.255532\pi$$
$$402$$ 0 0
$$403$$ −4487.54 −0.554690
$$404$$ 0 0
$$405$$ −405.000 −0.0496904
$$406$$ 0 0
$$407$$ 13095.8 1.59492
$$408$$ 0 0
$$409$$ −7428.08 −0.898031 −0.449015 0.893524i $$-0.648225\pi$$
−0.449015 + 0.893524i $$0.648225\pi$$
$$410$$ 0 0
$$411$$ 2688.03 0.322605
$$412$$ 0 0
$$413$$ 102.769 0.0122444
$$414$$ 0 0
$$415$$ 1885.75 0.223055
$$416$$ 0 0
$$417$$ −6940.83 −0.815093
$$418$$ 0 0
$$419$$ −9644.74 −1.12453 −0.562263 0.826959i $$-0.690069\pi$$
−0.562263 + 0.826959i $$0.690069\pi$$
$$420$$ 0 0
$$421$$ 9918.18 1.14818 0.574088 0.818793i $$-0.305357\pi$$
0.574088 + 0.818793i $$0.305357\pi$$
$$422$$ 0 0
$$423$$ 4879.97 0.560927
$$424$$ 0 0
$$425$$ 1460.47 0.166690
$$426$$ 0 0
$$427$$ 1197.48 0.135715
$$428$$ 0 0
$$429$$ 3255.82 0.366417
$$430$$ 0 0
$$431$$ 16324.1 1.82437 0.912185 0.409779i $$-0.134394\pi$$
0.912185 + 0.409779i $$0.134394\pi$$
$$432$$ 0 0
$$433$$ −5168.75 −0.573659 −0.286829 0.957982i $$-0.592601\pi$$
−0.286829 + 0.957982i $$0.592601\pi$$
$$434$$ 0 0
$$435$$ 2063.72 0.227466
$$436$$ 0 0
$$437$$ −8826.95 −0.966248
$$438$$ 0 0
$$439$$ −18339.0 −1.99378 −0.996892 0.0787782i $$-0.974898\pi$$
−0.996892 + 0.0787782i $$0.974898\pi$$
$$440$$ 0 0
$$441$$ 441.000 0.0476190
$$442$$ 0 0
$$443$$ 1613.28 0.173023 0.0865113 0.996251i $$-0.472428\pi$$
0.0865113 + 0.996251i $$0.472428\pi$$
$$444$$ 0 0
$$445$$ 3388.62 0.360980
$$446$$ 0 0
$$447$$ −2458.01 −0.260089
$$448$$ 0 0
$$449$$ −886.750 −0.0932034 −0.0466017 0.998914i $$-0.514839\pi$$
−0.0466017 + 0.998914i $$0.514839\pi$$
$$450$$ 0 0
$$451$$ −13232.1 −1.38154
$$452$$ 0 0
$$453$$ 1604.23 0.166387
$$454$$ 0 0
$$455$$ 1015.55 0.104636
$$456$$ 0 0
$$457$$ 7391.22 0.756557 0.378279 0.925692i $$-0.376516\pi$$
0.378279 + 0.925692i $$0.376516\pi$$
$$458$$ 0 0
$$459$$ −1577.31 −0.160397
$$460$$ 0 0
$$461$$ −7133.35 −0.720679 −0.360340 0.932821i $$-0.617339\pi$$
−0.360340 + 0.932821i $$0.617339\pi$$
$$462$$ 0 0
$$463$$ −14461.8 −1.45162 −0.725808 0.687897i $$-0.758534\pi$$
−0.725808 + 0.687897i $$0.758534\pi$$
$$464$$ 0 0
$$465$$ −2319.89 −0.231360
$$466$$ 0 0
$$467$$ 16393.5 1.62441 0.812206 0.583370i $$-0.198266\pi$$
0.812206 + 0.583370i $$0.198266\pi$$
$$468$$ 0 0
$$469$$ 3863.69 0.380403
$$470$$ 0 0
$$471$$ −4694.28 −0.459238
$$472$$ 0 0
$$473$$ −19380.7 −1.88398
$$474$$ 0 0
$$475$$ 1364.92 0.131846
$$476$$ 0 0
$$477$$ 2747.78 0.263758
$$478$$ 0 0
$$479$$ 12991.4 1.23923 0.619617 0.784904i $$-0.287288\pi$$
0.619617 + 0.784904i $$0.287288\pi$$
$$480$$ 0 0
$$481$$ −10159.1 −0.963025
$$482$$ 0 0
$$483$$ −3395.17 −0.319846
$$484$$ 0 0
$$485$$ 6125.17 0.573463
$$486$$ 0 0
$$487$$ 12863.5 1.19692 0.598461 0.801152i $$-0.295779\pi$$
0.598461 + 0.801152i $$0.295779\pi$$
$$488$$ 0 0
$$489$$ −3342.92 −0.309145
$$490$$ 0 0
$$491$$ 4898.10 0.450200 0.225100 0.974336i $$-0.427729\pi$$
0.225100 + 0.974336i $$0.427729\pi$$
$$492$$ 0 0
$$493$$ 8037.32 0.734245
$$494$$ 0 0
$$495$$ 1683.14 0.152831
$$496$$ 0 0
$$497$$ −842.340 −0.0760244
$$498$$ 0 0
$$499$$ −10308.0 −0.924746 −0.462373 0.886686i $$-0.653002\pi$$
−0.462373 + 0.886686i $$0.653002\pi$$
$$500$$ 0 0
$$501$$ −5323.41 −0.474715
$$502$$ 0 0
$$503$$ 15119.6 1.34026 0.670130 0.742244i $$-0.266238\pi$$
0.670130 + 0.742244i $$0.266238\pi$$
$$504$$ 0 0
$$505$$ 1690.72 0.148982
$$506$$ 0 0
$$507$$ 4065.28 0.356105
$$508$$ 0 0
$$509$$ 14183.8 1.23514 0.617571 0.786515i $$-0.288117\pi$$
0.617571 + 0.786515i $$0.288117\pi$$
$$510$$ 0 0
$$511$$ −1992.62 −0.172501
$$512$$ 0 0
$$513$$ −1474.12 −0.126869
$$514$$ 0 0
$$515$$ −2833.50 −0.242444
$$516$$ 0 0
$$517$$ −20280.7 −1.72523
$$518$$ 0 0
$$519$$ 12647.6 1.06969
$$520$$ 0 0
$$521$$ −7464.08 −0.627653 −0.313827 0.949480i $$-0.601611\pi$$
−0.313827 + 0.949480i $$0.601611\pi$$
$$522$$ 0 0
$$523$$ 16642.9 1.39148 0.695739 0.718295i $$-0.255077\pi$$
0.695739 + 0.718295i $$0.255077\pi$$
$$524$$ 0 0
$$525$$ 525.000 0.0436436
$$526$$ 0 0
$$527$$ −9035.01 −0.746814
$$528$$ 0 0
$$529$$ 13971.8 1.14834
$$530$$ 0 0
$$531$$ −132.132 −0.0107985
$$532$$ 0 0
$$533$$ 10264.8 0.834181
$$534$$ 0 0
$$535$$ −2812.66 −0.227293
$$536$$ 0 0
$$537$$ −7292.09 −0.585991
$$538$$ 0 0
$$539$$ −1832.75 −0.146461
$$540$$ 0 0
$$541$$ 67.8755 0.00539408 0.00269704 0.999996i $$-0.499142\pi$$
0.00269704 + 0.999996i $$0.499142\pi$$
$$542$$ 0 0
$$543$$ 8102.74 0.640372
$$544$$ 0 0
$$545$$ −9153.97 −0.719473
$$546$$ 0 0
$$547$$ 9212.91 0.720138 0.360069 0.932926i $$-0.382753\pi$$
0.360069 + 0.932926i $$0.382753\pi$$
$$548$$ 0 0
$$549$$ −1539.62 −0.119689
$$550$$ 0 0
$$551$$ 7511.51 0.580764
$$552$$ 0 0
$$553$$ 6591.29 0.506854
$$554$$ 0 0
$$555$$ −5251.87 −0.401675
$$556$$ 0 0
$$557$$ 16699.6 1.27035 0.635173 0.772370i $$-0.280929\pi$$
0.635173 + 0.772370i $$0.280929\pi$$
$$558$$ 0 0
$$559$$ 15034.6 1.13756
$$560$$ 0 0
$$561$$ 6555.13 0.493329
$$562$$ 0 0
$$563$$ −14772.8 −1.10586 −0.552931 0.833227i $$-0.686491\pi$$
−0.552931 + 0.833227i $$0.686491\pi$$
$$564$$ 0 0
$$565$$ 159.016 0.0118405
$$566$$ 0 0
$$567$$ −567.000 −0.0419961
$$568$$ 0 0
$$569$$ 5663.76 0.417289 0.208644 0.977992i $$-0.433095\pi$$
0.208644 + 0.977992i $$0.433095\pi$$
$$570$$ 0 0
$$571$$ −5579.58 −0.408929 −0.204464 0.978874i $$-0.565545\pi$$
−0.204464 + 0.978874i $$0.565545\pi$$
$$572$$ 0 0
$$573$$ 10833.3 0.789821
$$574$$ 0 0
$$575$$ −4041.87 −0.293144
$$576$$ 0 0
$$577$$ −2301.23 −0.166034 −0.0830170 0.996548i $$-0.526456\pi$$
−0.0830170 + 0.996548i $$0.526456\pi$$
$$578$$ 0 0
$$579$$ 13405.0 0.962162
$$580$$ 0 0
$$581$$ 2640.05 0.188516
$$582$$ 0 0
$$583$$ −11419.5 −0.811232
$$584$$ 0 0
$$585$$ −1305.70 −0.0922806
$$586$$ 0 0
$$587$$ 16470.4 1.15810 0.579052 0.815291i $$-0.303423\pi$$
0.579052 + 0.815291i $$0.303423\pi$$
$$588$$ 0 0
$$589$$ −8443.92 −0.590706
$$590$$ 0 0
$$591$$ 1303.27 0.0907093
$$592$$ 0 0
$$593$$ −13570.0 −0.939715 −0.469858 0.882742i $$-0.655695\pi$$
−0.469858 + 0.882742i $$0.655695\pi$$
$$594$$ 0 0
$$595$$ 2044.66 0.140879
$$596$$ 0 0
$$597$$ −1406.75 −0.0964393
$$598$$ 0 0
$$599$$ −27814.1 −1.89725 −0.948625 0.316403i $$-0.897525\pi$$
−0.948625 + 0.316403i $$0.897525\pi$$
$$600$$ 0 0
$$601$$ 20646.1 1.40128 0.700641 0.713514i $$-0.252898\pi$$
0.700641 + 0.713514i $$0.252898\pi$$
$$602$$ 0 0
$$603$$ −4967.61 −0.335483
$$604$$ 0 0
$$605$$ −339.969 −0.0228458
$$606$$ 0 0
$$607$$ −3315.28 −0.221686 −0.110843 0.993838i $$-0.535355\pi$$
−0.110843 + 0.993838i $$0.535355\pi$$
$$608$$ 0 0
$$609$$ 2889.21 0.192244
$$610$$ 0 0
$$611$$ 15732.8 1.04170
$$612$$ 0 0
$$613$$ −11113.9 −0.732278 −0.366139 0.930560i $$-0.619320\pi$$
−0.366139 + 0.930560i $$0.619320\pi$$
$$614$$ 0 0
$$615$$ 5306.53 0.347935
$$616$$ 0 0
$$617$$ 7871.34 0.513595 0.256797 0.966465i $$-0.417333\pi$$
0.256797 + 0.966465i $$0.417333\pi$$
$$618$$ 0 0
$$619$$ 19107.1 1.24068 0.620339 0.784334i $$-0.286995\pi$$
0.620339 + 0.784334i $$0.286995\pi$$
$$620$$ 0 0
$$621$$ 4365.22 0.282078
$$622$$ 0 0
$$623$$ 4744.07 0.305084
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 6126.28 0.390208
$$628$$ 0 0
$$629$$ −20453.9 −1.29658
$$630$$ 0 0
$$631$$ 25769.9 1.62580 0.812902 0.582401i $$-0.197887\pi$$
0.812902 + 0.582401i $$0.197887\pi$$
$$632$$ 0 0
$$633$$ 11206.5 0.703665
$$634$$ 0 0
$$635$$ 11104.1 0.693939
$$636$$ 0 0
$$637$$ 1421.77 0.0884339
$$638$$ 0 0
$$639$$ 1083.01 0.0670472
$$640$$ 0 0
$$641$$ −1954.61 −0.120440 −0.0602202 0.998185i $$-0.519180\pi$$
−0.0602202 + 0.998185i $$0.519180\pi$$
$$642$$ 0 0
$$643$$ 19396.5 1.18961 0.594807 0.803868i $$-0.297228\pi$$
0.594807 + 0.803868i $$0.297228\pi$$
$$644$$ 0 0
$$645$$ 7772.34 0.474474
$$646$$ 0 0
$$647$$ −31264.3 −1.89973 −0.949865 0.312661i $$-0.898780\pi$$
−0.949865 + 0.312661i $$0.898780\pi$$
$$648$$ 0 0
$$649$$ 549.126 0.0332127
$$650$$ 0 0
$$651$$ −3247.85 −0.195535
$$652$$ 0 0
$$653$$ 6442.75 0.386102 0.193051 0.981189i $$-0.438162\pi$$
0.193051 + 0.981189i $$0.438162\pi$$
$$654$$ 0 0
$$655$$ 3232.56 0.192835
$$656$$ 0 0
$$657$$ 2561.93 0.152132
$$658$$ 0 0
$$659$$ 3584.70 0.211897 0.105949 0.994372i $$-0.466212\pi$$
0.105949 + 0.994372i $$0.466212\pi$$
$$660$$ 0 0
$$661$$ 6294.43 0.370386 0.185193 0.982702i $$-0.440709\pi$$
0.185193 + 0.982702i $$0.440709\pi$$
$$662$$ 0 0
$$663$$ −5085.17 −0.297876
$$664$$ 0 0
$$665$$ 1910.89 0.111430
$$666$$ 0 0
$$667$$ −22243.4 −1.29126
$$668$$ 0 0
$$669$$ 2528.42 0.146120
$$670$$ 0 0
$$671$$ 6398.51 0.368125
$$672$$ 0 0
$$673$$ −10233.9 −0.586162 −0.293081 0.956088i $$-0.594681\pi$$
−0.293081 + 0.956088i $$0.594681\pi$$
$$674$$ 0 0
$$675$$ −675.000 −0.0384900
$$676$$ 0 0
$$677$$ −7100.75 −0.403108 −0.201554 0.979477i $$-0.564599\pi$$
−0.201554 + 0.979477i $$0.564599\pi$$
$$678$$ 0 0
$$679$$ 8575.24 0.484665
$$680$$ 0 0
$$681$$ 2976.45 0.167486
$$682$$ 0 0
$$683$$ 35274.6 1.97620 0.988100 0.153813i $$-0.0491554\pi$$
0.988100 + 0.153813i $$0.0491554\pi$$
$$684$$ 0 0
$$685$$ 4480.05 0.249889
$$686$$ 0 0
$$687$$ 19234.2 1.06817
$$688$$ 0 0
$$689$$ 8858.74 0.489828
$$690$$ 0 0
$$691$$ −4945.12 −0.272245 −0.136122 0.990692i $$-0.543464\pi$$
−0.136122 + 0.990692i $$0.543464\pi$$
$$692$$ 0 0
$$693$$ 2356.40 0.129166
$$694$$ 0 0
$$695$$ −11568.0 −0.631368
$$696$$ 0 0
$$697$$ 20666.7 1.12311
$$698$$ 0 0
$$699$$ 6823.06 0.369201
$$700$$ 0 0
$$701$$ −15300.4 −0.824379 −0.412190 0.911098i $$-0.635236\pi$$
−0.412190 + 0.911098i $$0.635236\pi$$
$$702$$ 0 0
$$703$$ −19115.7 −1.02555
$$704$$ 0 0
$$705$$ 8133.28 0.434492
$$706$$ 0 0
$$707$$ 2367.01 0.125913
$$708$$ 0 0
$$709$$ −28297.4 −1.49892 −0.749458 0.662052i $$-0.769686\pi$$
−0.749458 + 0.662052i $$0.769686\pi$$
$$710$$ 0 0
$$711$$ −8474.51 −0.447003
$$712$$ 0 0
$$713$$ 25004.5 1.31336
$$714$$ 0 0
$$715$$ 5426.37 0.283825
$$716$$ 0 0
$$717$$ 8589.35 0.447385
$$718$$ 0 0
$$719$$ −8548.96 −0.443425 −0.221712 0.975112i $$-0.571165\pi$$
−0.221712 + 0.975112i $$0.571165\pi$$
$$720$$ 0 0
$$721$$ −3966.90 −0.204903
$$722$$ 0 0
$$723$$ 16092.7 0.827792
$$724$$ 0 0
$$725$$ 3439.53 0.176194
$$726$$ 0 0
$$727$$ −14345.3 −0.731827 −0.365913 0.930649i $$-0.619243\pi$$
−0.365913 + 0.930649i $$0.619243\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 30270.0 1.53157
$$732$$ 0 0
$$733$$ −22624.0 −1.14002 −0.570012 0.821637i $$-0.693061\pi$$
−0.570012 + 0.821637i $$0.693061\pi$$
$$734$$ 0 0
$$735$$ 735.000 0.0368856
$$736$$ 0 0
$$737$$ 20644.9 1.03184
$$738$$ 0 0
$$739$$ −14837.3 −0.738566 −0.369283 0.929317i $$-0.620397\pi$$
−0.369283 + 0.929317i $$0.620397\pi$$
$$740$$ 0 0
$$741$$ −4752.49 −0.235610
$$742$$ 0 0
$$743$$ −13073.1 −0.645497 −0.322749 0.946485i $$-0.604607\pi$$
−0.322749 + 0.946485i $$0.604607\pi$$
$$744$$ 0 0
$$745$$ −4096.69 −0.201464
$$746$$ 0 0
$$747$$ −3394.35 −0.166255
$$748$$ 0 0
$$749$$ −3937.72 −0.192098
$$750$$ 0 0
$$751$$ −16213.3 −0.787791 −0.393895 0.919155i $$-0.628873\pi$$
−0.393895 + 0.919155i $$0.628873\pi$$
$$752$$ 0 0
$$753$$ 16709.4 0.808665
$$754$$ 0 0
$$755$$ 2673.72 0.128883
$$756$$ 0 0
$$757$$ 19903.9 0.955642 0.477821 0.878457i $$-0.341427\pi$$
0.477821 + 0.878457i $$0.341427\pi$$
$$758$$ 0 0
$$759$$ −18141.4 −0.867580
$$760$$ 0 0
$$761$$ −30125.5 −1.43502 −0.717509 0.696549i $$-0.754718\pi$$
−0.717509 + 0.696549i $$0.754718\pi$$
$$762$$ 0 0
$$763$$ −12815.6 −0.608066
$$764$$ 0 0
$$765$$ −2628.84 −0.124243
$$766$$ 0 0
$$767$$ −425.986 −0.0200541
$$768$$ 0 0
$$769$$ −36049.1 −1.69046 −0.845230 0.534402i $$-0.820537\pi$$
−0.845230 + 0.534402i $$0.820537\pi$$
$$770$$ 0 0
$$771$$ −6286.09 −0.293629
$$772$$ 0 0
$$773$$ −9644.77 −0.448769 −0.224384 0.974501i $$-0.572037\pi$$
−0.224384 + 0.974501i $$0.572037\pi$$
$$774$$ 0 0
$$775$$ −3866.48 −0.179211
$$776$$ 0 0
$$777$$ −7352.62 −0.339477
$$778$$ 0 0
$$779$$ 19314.7 0.888344
$$780$$ 0 0
$$781$$ −4500.88 −0.206215
$$782$$ 0 0
$$783$$ −3714.69 −0.169543
$$784$$ 0 0
$$785$$ −7823.80 −0.355724
$$786$$ 0 0
$$787$$ 25218.6 1.14224 0.571122 0.820865i $$-0.306508\pi$$
0.571122 + 0.820865i $$0.306508\pi$$
$$788$$ 0 0
$$789$$ −22397.6 −1.01062
$$790$$ 0 0
$$791$$ 222.623 0.0100070
$$792$$ 0 0
$$793$$ −4963.67 −0.222276
$$794$$ 0 0
$$795$$ 4579.64 0.204306
$$796$$ 0 0
$$797$$ 32042.3 1.42409 0.712044 0.702135i $$-0.247770\pi$$
0.712044 + 0.702135i $$0.247770\pi$$
$$798$$ 0 0
$$799$$ 31675.7 1.40251
$$800$$ 0 0
$$801$$ −6099.52 −0.269059
$$802$$ 0 0
$$803$$ −10647.1 −0.467908
$$804$$ 0 0
$$805$$ −5658.62 −0.247752
$$806$$ 0 0
$$807$$ 19564.1 0.853396
$$808$$ 0 0
$$809$$ 3427.90 0.148972 0.0744860 0.997222i $$-0.476268\pi$$
0.0744860 + 0.997222i $$0.476268\pi$$
$$810$$ 0 0
$$811$$ −23094.4 −0.999943 −0.499972 0.866042i $$-0.666656\pi$$
−0.499972 + 0.866042i $$0.666656\pi$$
$$812$$ 0 0
$$813$$ 7229.11 0.311852
$$814$$ 0 0
$$815$$ −5571.53 −0.239463
$$816$$ 0 0
$$817$$ 28289.7 1.21142
$$818$$ 0 0
$$819$$ −1827.98 −0.0779914
$$820$$ 0 0
$$821$$ 474.741 0.0201810 0.0100905 0.999949i $$-0.496788\pi$$
0.0100905 + 0.999949i $$0.496788\pi$$
$$822$$ 0 0
$$823$$ −24159.8 −1.02328 −0.511638 0.859201i $$-0.670961\pi$$
−0.511638 + 0.859201i $$0.670961\pi$$
$$824$$ 0 0
$$825$$ 2805.23 0.118383
$$826$$ 0 0
$$827$$ 7566.35 0.318147 0.159074 0.987267i $$-0.449149\pi$$
0.159074 + 0.987267i $$0.449149\pi$$
$$828$$ 0 0
$$829$$ −10580.9 −0.443295 −0.221648 0.975127i $$-0.571143\pi$$
−0.221648 + 0.975127i $$0.571143\pi$$
$$830$$ 0 0
$$831$$ 6659.48 0.277996
$$832$$ 0 0
$$833$$ 2862.52 0.119064
$$834$$ 0 0
$$835$$ −8872.34 −0.367713
$$836$$ 0 0
$$837$$ 4175.80 0.172445
$$838$$ 0 0
$$839$$ −15315.6 −0.630218 −0.315109 0.949055i $$-0.602041\pi$$
−0.315109 + 0.949055i $$0.602041\pi$$
$$840$$ 0 0
$$841$$ −5460.40 −0.223888
$$842$$ 0 0
$$843$$ −17515.7 −0.715625
$$844$$ 0 0
$$845$$ 6775.47 0.275838
$$846$$ 0 0
$$847$$ −475.956 −0.0193082
$$848$$ 0 0
$$849$$ −10935.1 −0.442040
$$850$$ 0 0
$$851$$ 56606.4 2.28019
$$852$$ 0 0
$$853$$ 18598.2 0.746528 0.373264 0.927725i $$-0.378239\pi$$
0.373264 + 0.927725i $$0.378239\pi$$
$$854$$ 0 0
$$855$$ −2456.86 −0.0982723
$$856$$ 0 0
$$857$$ 41775.3 1.66513 0.832566 0.553926i $$-0.186871\pi$$
0.832566 + 0.553926i $$0.186871\pi$$
$$858$$ 0 0
$$859$$ −32414.7 −1.28752 −0.643758 0.765229i $$-0.722626\pi$$
−0.643758 + 0.765229i $$0.722626\pi$$
$$860$$ 0 0
$$861$$ 7429.14 0.294059
$$862$$ 0 0
$$863$$ 31299.7 1.23459 0.617297 0.786730i $$-0.288228\pi$$
0.617297 + 0.786730i $$0.288228\pi$$
$$864$$ 0 0
$$865$$ 21079.4 0.828580
$$866$$ 0 0
$$867$$ 4500.75 0.176302
$$868$$ 0 0
$$869$$ 35219.2 1.37483
$$870$$ 0 0
$$871$$ −16015.4 −0.623030
$$872$$ 0 0
$$873$$ −11025.3 −0.427434
$$874$$ 0 0
$$875$$ 875.000 0.0338062
$$876$$ 0 0
$$877$$ −19973.0 −0.769031 −0.384515 0.923119i $$-0.625631\pi$$
−0.384515 + 0.923119i $$0.625631\pi$$
$$878$$ 0 0
$$879$$ −11333.7 −0.434900
$$880$$ 0 0
$$881$$ 17367.9 0.664176 0.332088 0.943248i $$-0.392247\pi$$
0.332088 + 0.943248i $$0.392247\pi$$
$$882$$ 0 0
$$883$$ 14364.2 0.547446 0.273723 0.961809i $$-0.411745\pi$$
0.273723 + 0.961809i $$0.411745\pi$$
$$884$$ 0 0
$$885$$ −220.219 −0.00836450
$$886$$ 0 0
$$887$$ 33738.1 1.27713 0.638564 0.769568i $$-0.279529\pi$$
0.638564 + 0.769568i $$0.279529\pi$$
$$888$$ 0 0
$$889$$ 15545.7 0.586485
$$890$$ 0 0
$$891$$ −3029.65 −0.113914
$$892$$ 0 0
$$893$$ 29603.4 1.10934
$$894$$ 0 0
$$895$$ −12153.5 −0.453906
$$896$$ 0 0
$$897$$ 14073.3 0.523850
$$898$$ 0 0
$$899$$ −21278.2 −0.789398
$$900$$ 0 0
$$901$$ 17835.8 0.659485
$$902$$ 0 0
$$903$$ 10881.3 0.401004
$$904$$ 0 0
$$905$$ 13504.6 0.496030
$$906$$ 0 0
$$907$$ 32998.6 1.20805 0.604024 0.796966i $$-0.293563\pi$$
0.604024 + 0.796966i $$0.293563\pi$$
$$908$$ 0 0
$$909$$ −3043.29 −0.111045
$$910$$ 0 0
$$911$$ −33446.3 −1.21638 −0.608192 0.793790i $$-0.708105\pi$$
−0.608192 + 0.793790i $$0.708105\pi$$
$$912$$ 0 0
$$913$$ 14106.6 0.511347
$$914$$ 0 0
$$915$$ −2566.03 −0.0927108
$$916$$ 0 0
$$917$$ 4525.59 0.162975
$$918$$ 0 0
$$919$$ −41708.7 −1.49711 −0.748554 0.663074i $$-0.769252\pi$$
−0.748554 + 0.663074i $$0.769252\pi$$
$$920$$ 0 0
$$921$$ −14398.9 −0.515158
$$922$$ 0 0
$$923$$ 3491.58 0.124514
$$924$$ 0 0
$$925$$ −8753.12 −0.311136
$$926$$ 0 0
$$927$$ 5100.30 0.180707
$$928$$ 0 0
$$929$$ 49024.6 1.73137 0.865686 0.500587i $$-0.166883\pi$$
0.865686 + 0.500587i $$0.166883\pi$$
$$930$$ 0 0
$$931$$ 2675.25 0.0941758
$$932$$ 0 0
$$933$$ 1740.34 0.0610677
$$934$$ 0 0
$$935$$ 10925.2 0.382131
$$936$$ 0 0
$$937$$ −5447.58 −0.189930 −0.0949651 0.995481i $$-0.530274\pi$$
−0.0949651 + 0.995481i $$0.530274\pi$$
$$938$$ 0 0
$$939$$ 18344.4 0.637535
$$940$$ 0 0
$$941$$ 4125.75 0.142928 0.0714642 0.997443i $$-0.477233\pi$$
0.0714642 + 0.997443i $$0.477233\pi$$
$$942$$ 0 0
$$943$$ −57195.5 −1.97513
$$944$$ 0 0
$$945$$ −945.000 −0.0325300
$$946$$ 0 0
$$947$$ −17332.3 −0.594747 −0.297374 0.954761i $$-0.596111\pi$$
−0.297374 + 0.954761i $$0.596111\pi$$
$$948$$ 0 0
$$949$$ 8259.57 0.282526
$$950$$ 0 0
$$951$$ 12901.9 0.439929
$$952$$ 0 0
$$953$$ 56839.4 1.93201 0.966007 0.258517i $$-0.0832337\pi$$
0.966007 + 0.258517i $$0.0832337\pi$$
$$954$$ 0 0
$$955$$ 18055.5 0.611792
$$956$$ 0 0
$$957$$ 15437.9 0.521459
$$958$$ 0 0
$$959$$ 6272.07 0.211195
$$960$$ 0 0
$$961$$ −5871.48 −0.197089
$$962$$ 0 0
$$963$$ 5062.78 0.169414
$$964$$ 0 0
$$965$$ 22341.6 0.745287
$$966$$ 0 0
$$967$$ 13284.2 0.441769 0.220884 0.975300i $$-0.429106\pi$$
0.220884 + 0.975300i $$0.429106\pi$$
$$968$$ 0 0
$$969$$ −9568.44 −0.317216
$$970$$ 0 0
$$971$$ −12153.0 −0.401658 −0.200829 0.979626i $$-0.564364\pi$$
−0.200829 + 0.979626i $$0.564364\pi$$
$$972$$ 0 0
$$973$$ −16195.3 −0.533604
$$974$$ 0 0
$$975$$ −2176.17 −0.0714803
$$976$$ 0 0
$$977$$ −37999.3 −1.24433 −0.622163 0.782888i $$-0.713746\pi$$
−0.622163 + 0.782888i $$0.713746\pi$$
$$978$$ 0 0
$$979$$ 25349.0 0.827537
$$980$$ 0 0
$$981$$ 16477.1 0.536264
$$982$$ 0 0
$$983$$ −22375.0 −0.725993 −0.362996 0.931791i $$-0.618246\pi$$
−0.362996 + 0.931791i $$0.618246\pi$$
$$984$$ 0 0
$$985$$ 2172.11 0.0702631
$$986$$ 0 0
$$987$$ 11386.6 0.367213
$$988$$ 0 0
$$989$$ −83772.9 −2.69345
$$990$$ 0 0
$$991$$ −18985.3 −0.608564 −0.304282 0.952582i $$-0.598417\pi$$
−0.304282 + 0.952582i $$0.598417\pi$$
$$992$$ 0 0
$$993$$ −20062.6 −0.641156
$$994$$ 0 0
$$995$$ −2344.58 −0.0747016
$$996$$ 0 0
$$997$$ −56476.5 −1.79401 −0.897006 0.442019i $$-0.854262\pi$$
−0.897006 + 0.442019i $$0.854262\pi$$
$$998$$ 0 0
$$999$$ 9453.37 0.299391
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.y.1.1 2
4.3 odd 2 105.4.a.g.1.1 2
12.11 even 2 315.4.a.g.1.2 2
20.3 even 4 525.4.d.j.274.3 4
20.7 even 4 525.4.d.j.274.2 4
20.19 odd 2 525.4.a.i.1.2 2
28.27 even 2 735.4.a.q.1.1 2
60.59 even 2 1575.4.a.y.1.1 2
84.83 odd 2 2205.4.a.v.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.g.1.1 2 4.3 odd 2
315.4.a.g.1.2 2 12.11 even 2
525.4.a.i.1.2 2 20.19 odd 2
525.4.d.j.274.2 4 20.7 even 4
525.4.d.j.274.3 4 20.3 even 4
735.4.a.q.1.1 2 28.27 even 2
1575.4.a.y.1.1 2 60.59 even 2
1680.4.a.y.1.1 2 1.1 even 1 trivial
2205.4.a.v.1.2 2 84.83 odd 2