# Properties

 Label 117.4.a.e.1.2 Level $117$ Weight $4$ Character 117.1 Self dual yes Analytic conductor $6.903$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("117.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.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: $$6.90322347067$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.64575$$ of defining polynomial Character $$\chi$$ $$=$$ 117.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} +O(q^{10})$$ $$q+2.64575 q^{2} -1.00000 q^{4} -10.5830 q^{5} -22.0000 q^{7} -23.8118 q^{8} -28.0000 q^{10} -5.29150 q^{11} +13.0000 q^{13} -58.2065 q^{14} -55.0000 q^{16} +116.413 q^{17} -126.000 q^{19} +10.5830 q^{20} -14.0000 q^{22} +31.7490 q^{23} -13.0000 q^{25} +34.3948 q^{26} +22.0000 q^{28} +52.9150 q^{29} -182.000 q^{31} +44.9778 q^{32} +308.000 q^{34} +232.826 q^{35} -86.0000 q^{37} -333.365 q^{38} +252.000 q^{40} -444.486 q^{41} +96.0000 q^{43} +5.29150 q^{44} +84.0000 q^{46} +365.114 q^{47} +141.000 q^{49} -34.3948 q^{50} -13.0000 q^{52} -190.494 q^{53} +56.0000 q^{55} +523.859 q^{56} +140.000 q^{58} -587.357 q^{59} +574.000 q^{61} -481.527 q^{62} +559.000 q^{64} -137.579 q^{65} -530.000 q^{67} -116.413 q^{68} +616.000 q^{70} +809.600 q^{71} -154.000 q^{73} -227.535 q^{74} +126.000 q^{76} +116.413 q^{77} -460.000 q^{79} +582.065 q^{80} -1176.00 q^{82} -322.782 q^{83} -1232.00 q^{85} +253.992 q^{86} +126.000 q^{88} +1439.29 q^{89} -286.000 q^{91} -31.7490 q^{92} +966.000 q^{94} +1333.46 q^{95} +70.0000 q^{97} +373.051 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 44 q^{7}+O(q^{10})$$ 2 * q - 2 * q^4 - 44 * q^7 $$2 q - 2 q^{4} - 44 q^{7} - 56 q^{10} + 26 q^{13} - 110 q^{16} - 252 q^{19} - 28 q^{22} - 26 q^{25} + 44 q^{28} - 364 q^{31} + 616 q^{34} - 172 q^{37} + 504 q^{40} + 192 q^{43} + 168 q^{46} + 282 q^{49} - 26 q^{52} + 112 q^{55} + 280 q^{58} + 1148 q^{61} + 1118 q^{64} - 1060 q^{67} + 1232 q^{70} - 308 q^{73} + 252 q^{76} - 920 q^{79} - 2352 q^{82} - 2464 q^{85} + 252 q^{88} - 572 q^{91} + 1932 q^{94} + 140 q^{97}+O(q^{100})$$ 2 * q - 2 * q^4 - 44 * q^7 - 56 * q^10 + 26 * q^13 - 110 * q^16 - 252 * q^19 - 28 * q^22 - 26 * q^25 + 44 * q^28 - 364 * q^31 + 616 * q^34 - 172 * q^37 + 504 * q^40 + 192 * q^43 + 168 * q^46 + 282 * q^49 - 26 * q^52 + 112 * q^55 + 280 * q^58 + 1148 * q^61 + 1118 * q^64 - 1060 * q^67 + 1232 * q^70 - 308 * q^73 + 252 * q^76 - 920 * q^79 - 2352 * q^82 - 2464 * q^85 + 252 * q^88 - 572 * q^91 + 1932 * q^94 + 140 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.64575 0.935414 0.467707 0.883883i $$-0.345080\pi$$
0.467707 + 0.883883i $$0.345080\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.125000
$$5$$ −10.5830 −0.946573 −0.473286 0.880909i $$-0.656932\pi$$
−0.473286 + 0.880909i $$0.656932\pi$$
$$6$$ 0 0
$$7$$ −22.0000 −1.18789 −0.593944 0.804506i $$-0.702430\pi$$
−0.593944 + 0.804506i $$0.702430\pi$$
$$8$$ −23.8118 −1.05234
$$9$$ 0 0
$$10$$ −28.0000 −0.885438
$$11$$ −5.29150 −0.145041 −0.0725204 0.997367i $$-0.523104\pi$$
−0.0725204 + 0.997367i $$0.523104\pi$$
$$12$$ 0 0
$$13$$ 13.0000 0.277350
$$14$$ −58.2065 −1.11117
$$15$$ 0 0
$$16$$ −55.0000 −0.859375
$$17$$ 116.413 1.66084 0.830421 0.557136i $$-0.188100\pi$$
0.830421 + 0.557136i $$0.188100\pi$$
$$18$$ 0 0
$$19$$ −126.000 −1.52139 −0.760694 0.649110i $$-0.775141\pi$$
−0.760694 + 0.649110i $$0.775141\pi$$
$$20$$ 10.5830 0.118322
$$21$$ 0 0
$$22$$ −14.0000 −0.135673
$$23$$ 31.7490 0.287832 0.143916 0.989590i $$-0.454031\pi$$
0.143916 + 0.989590i $$0.454031\pi$$
$$24$$ 0 0
$$25$$ −13.0000 −0.104000
$$26$$ 34.3948 0.259437
$$27$$ 0 0
$$28$$ 22.0000 0.148486
$$29$$ 52.9150 0.338830 0.169415 0.985545i $$-0.445812\pi$$
0.169415 + 0.985545i $$0.445812\pi$$
$$30$$ 0 0
$$31$$ −182.000 −1.05446 −0.527228 0.849724i $$-0.676769\pi$$
−0.527228 + 0.849724i $$0.676769\pi$$
$$32$$ 44.9778 0.248469
$$33$$ 0 0
$$34$$ 308.000 1.55358
$$35$$ 232.826 1.12442
$$36$$ 0 0
$$37$$ −86.0000 −0.382117 −0.191058 0.981579i $$-0.561192\pi$$
−0.191058 + 0.981579i $$0.561192\pi$$
$$38$$ −333.365 −1.42313
$$39$$ 0 0
$$40$$ 252.000 0.996117
$$41$$ −444.486 −1.69310 −0.846550 0.532310i $$-0.821324\pi$$
−0.846550 + 0.532310i $$0.821324\pi$$
$$42$$ 0 0
$$43$$ 96.0000 0.340462 0.170231 0.985404i $$-0.445549\pi$$
0.170231 + 0.985404i $$0.445549\pi$$
$$44$$ 5.29150 0.0181301
$$45$$ 0 0
$$46$$ 84.0000 0.269242
$$47$$ 365.114 1.13313 0.566567 0.824016i $$-0.308271\pi$$
0.566567 + 0.824016i $$0.308271\pi$$
$$48$$ 0 0
$$49$$ 141.000 0.411079
$$50$$ −34.3948 −0.0972831
$$51$$ 0 0
$$52$$ −13.0000 −0.0346688
$$53$$ −190.494 −0.493705 −0.246853 0.969053i $$-0.579396\pi$$
−0.246853 + 0.969053i $$0.579396\pi$$
$$54$$ 0 0
$$55$$ 56.0000 0.137292
$$56$$ 523.859 1.25006
$$57$$ 0 0
$$58$$ 140.000 0.316947
$$59$$ −587.357 −1.29606 −0.648028 0.761616i $$-0.724406\pi$$
−0.648028 + 0.761616i $$0.724406\pi$$
$$60$$ 0 0
$$61$$ 574.000 1.20481 0.602403 0.798192i $$-0.294210\pi$$
0.602403 + 0.798192i $$0.294210\pi$$
$$62$$ −481.527 −0.986354
$$63$$ 0 0
$$64$$ 559.000 1.09180
$$65$$ −137.579 −0.262532
$$66$$ 0 0
$$67$$ −530.000 −0.966415 −0.483208 0.875506i $$-0.660528\pi$$
−0.483208 + 0.875506i $$0.660528\pi$$
$$68$$ −116.413 −0.207605
$$69$$ 0 0
$$70$$ 616.000 1.05180
$$71$$ 809.600 1.35327 0.676633 0.736321i $$-0.263439\pi$$
0.676633 + 0.736321i $$0.263439\pi$$
$$72$$ 0 0
$$73$$ −154.000 −0.246909 −0.123454 0.992350i $$-0.539397\pi$$
−0.123454 + 0.992350i $$0.539397\pi$$
$$74$$ −227.535 −0.357437
$$75$$ 0 0
$$76$$ 126.000 0.190174
$$77$$ 116.413 0.172292
$$78$$ 0 0
$$79$$ −460.000 −0.655114 −0.327557 0.944831i $$-0.606225\pi$$
−0.327557 + 0.944831i $$0.606225\pi$$
$$80$$ 582.065 0.813461
$$81$$ 0 0
$$82$$ −1176.00 −1.58375
$$83$$ −322.782 −0.426866 −0.213433 0.976958i $$-0.568465\pi$$
−0.213433 + 0.976958i $$0.568465\pi$$
$$84$$ 0 0
$$85$$ −1232.00 −1.57211
$$86$$ 253.992 0.318473
$$87$$ 0 0
$$88$$ 126.000 0.152632
$$89$$ 1439.29 1.71421 0.857103 0.515145i $$-0.172262\pi$$
0.857103 + 0.515145i $$0.172262\pi$$
$$90$$ 0 0
$$91$$ −286.000 −0.329461
$$92$$ −31.7490 −0.0359790
$$93$$ 0 0
$$94$$ 966.000 1.05995
$$95$$ 1333.46 1.44010
$$96$$ 0 0
$$97$$ 70.0000 0.0732724 0.0366362 0.999329i $$-0.488336\pi$$
0.0366362 + 0.999329i $$0.488336\pi$$
$$98$$ 373.051 0.384529
$$99$$ 0 0
$$100$$ 13.0000 0.0130000
$$101$$ −1460.45 −1.43882 −0.719409 0.694586i $$-0.755587\pi$$
−0.719409 + 0.694586i $$0.755587\pi$$
$$102$$ 0 0
$$103$$ −1428.00 −1.36607 −0.683034 0.730387i $$-0.739340\pi$$
−0.683034 + 0.730387i $$0.739340\pi$$
$$104$$ −309.553 −0.291867
$$105$$ 0 0
$$106$$ −504.000 −0.461819
$$107$$ −1619.20 −1.46293 −0.731467 0.681877i $$-0.761164\pi$$
−0.731467 + 0.681877i $$0.761164\pi$$
$$108$$ 0 0
$$109$$ −338.000 −0.297014 −0.148507 0.988911i $$-0.547447\pi$$
−0.148507 + 0.988911i $$0.547447\pi$$
$$110$$ 148.162 0.128425
$$111$$ 0 0
$$112$$ 1210.00 1.02084
$$113$$ −1682.70 −1.40084 −0.700420 0.713731i $$-0.747004\pi$$
−0.700420 + 0.713731i $$0.747004\pi$$
$$114$$ 0 0
$$115$$ −336.000 −0.272454
$$116$$ −52.9150 −0.0423538
$$117$$ 0 0
$$118$$ −1554.00 −1.21235
$$119$$ −2561.09 −1.97289
$$120$$ 0 0
$$121$$ −1303.00 −0.978963
$$122$$ 1518.66 1.12699
$$123$$ 0 0
$$124$$ 182.000 0.131807
$$125$$ 1460.45 1.04502
$$126$$ 0 0
$$127$$ −376.000 −0.262713 −0.131357 0.991335i $$-0.541933\pi$$
−0.131357 + 0.991335i $$0.541933\pi$$
$$128$$ 1119.15 0.772813
$$129$$ 0 0
$$130$$ −364.000 −0.245576
$$131$$ −687.895 −0.458792 −0.229396 0.973333i $$-0.573675\pi$$
−0.229396 + 0.973333i $$0.573675\pi$$
$$132$$ 0 0
$$133$$ 2772.00 1.80724
$$134$$ −1402.25 −0.903998
$$135$$ 0 0
$$136$$ −2772.00 −1.74777
$$137$$ 1396.96 0.871168 0.435584 0.900148i $$-0.356542\pi$$
0.435584 + 0.900148i $$0.356542\pi$$
$$138$$ 0 0
$$139$$ 2100.00 1.28144 0.640718 0.767776i $$-0.278637\pi$$
0.640718 + 0.767776i $$0.278637\pi$$
$$140$$ −232.826 −0.140553
$$141$$ 0 0
$$142$$ 2142.00 1.26586
$$143$$ −68.7895 −0.0402271
$$144$$ 0 0
$$145$$ −560.000 −0.320727
$$146$$ −407.446 −0.230962
$$147$$ 0 0
$$148$$ 86.0000 0.0477646
$$149$$ −2000.19 −1.09974 −0.549872 0.835249i $$-0.685323\pi$$
−0.549872 + 0.835249i $$0.685323\pi$$
$$150$$ 0 0
$$151$$ 3526.00 1.90028 0.950138 0.311828i $$-0.100941\pi$$
0.950138 + 0.311828i $$0.100941\pi$$
$$152$$ 3000.28 1.60102
$$153$$ 0 0
$$154$$ 308.000 0.161165
$$155$$ 1926.11 0.998120
$$156$$ 0 0
$$157$$ 3066.00 1.55856 0.779278 0.626678i $$-0.215586\pi$$
0.779278 + 0.626678i $$0.215586\pi$$
$$158$$ −1217.05 −0.612803
$$159$$ 0 0
$$160$$ −476.000 −0.235194
$$161$$ −698.478 −0.341912
$$162$$ 0 0
$$163$$ −3442.00 −1.65398 −0.826988 0.562219i $$-0.809948\pi$$
−0.826988 + 0.562219i $$0.809948\pi$$
$$164$$ 444.486 0.211637
$$165$$ 0 0
$$166$$ −854.000 −0.399297
$$167$$ 2693.37 1.24802 0.624011 0.781416i $$-0.285502\pi$$
0.624011 + 0.781416i $$0.285502\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ −3259.57 −1.47057
$$171$$ 0 0
$$172$$ −96.0000 −0.0425577
$$173$$ 3492.39 1.53481 0.767404 0.641164i $$-0.221548\pi$$
0.767404 + 0.641164i $$0.221548\pi$$
$$174$$ 0 0
$$175$$ 286.000 0.123540
$$176$$ 291.033 0.124644
$$177$$ 0 0
$$178$$ 3808.00 1.60349
$$179$$ 169.328 0.0707049 0.0353524 0.999375i $$-0.488745\pi$$
0.0353524 + 0.999375i $$0.488745\pi$$
$$180$$ 0 0
$$181$$ 3374.00 1.38557 0.692783 0.721146i $$-0.256384\pi$$
0.692783 + 0.721146i $$0.256384\pi$$
$$182$$ −756.685 −0.308182
$$183$$ 0 0
$$184$$ −756.000 −0.302897
$$185$$ 910.138 0.361701
$$186$$ 0 0
$$187$$ −616.000 −0.240890
$$188$$ −365.114 −0.141642
$$189$$ 0 0
$$190$$ 3528.00 1.34709
$$191$$ −1185.30 −0.449032 −0.224516 0.974470i $$-0.572080\pi$$
−0.224516 + 0.974470i $$0.572080\pi$$
$$192$$ 0 0
$$193$$ −1542.00 −0.575107 −0.287553 0.957765i $$-0.592842\pi$$
−0.287553 + 0.957765i $$0.592842\pi$$
$$194$$ 185.203 0.0685401
$$195$$ 0 0
$$196$$ −141.000 −0.0513848
$$197$$ 2127.18 0.769318 0.384659 0.923059i $$-0.374319\pi$$
0.384659 + 0.923059i $$0.374319\pi$$
$$198$$ 0 0
$$199$$ −952.000 −0.339123 −0.169562 0.985520i $$-0.554235\pi$$
−0.169562 + 0.985520i $$0.554235\pi$$
$$200$$ 309.553 0.109443
$$201$$ 0 0
$$202$$ −3864.00 −1.34589
$$203$$ −1164.13 −0.402492
$$204$$ 0 0
$$205$$ 4704.00 1.60264
$$206$$ −3778.13 −1.27784
$$207$$ 0 0
$$208$$ −715.000 −0.238348
$$209$$ 666.729 0.220663
$$210$$ 0 0
$$211$$ −1640.00 −0.535082 −0.267541 0.963547i $$-0.586211\pi$$
−0.267541 + 0.963547i $$0.586211\pi$$
$$212$$ 190.494 0.0617132
$$213$$ 0 0
$$214$$ −4284.00 −1.36845
$$215$$ −1015.97 −0.322272
$$216$$ 0 0
$$217$$ 4004.00 1.25258
$$218$$ −894.264 −0.277831
$$219$$ 0 0
$$220$$ −56.0000 −0.0171615
$$221$$ 1513.37 0.460635
$$222$$ 0 0
$$223$$ −4886.00 −1.46722 −0.733612 0.679569i $$-0.762167\pi$$
−0.733612 + 0.679569i $$0.762167\pi$$
$$224$$ −989.511 −0.295154
$$225$$ 0 0
$$226$$ −4452.00 −1.31037
$$227$$ 1867.90 0.546154 0.273077 0.961992i $$-0.411959\pi$$
0.273077 + 0.961992i $$0.411959\pi$$
$$228$$ 0 0
$$229$$ 5558.00 1.60386 0.801928 0.597421i $$-0.203808\pi$$
0.801928 + 0.597421i $$0.203808\pi$$
$$230$$ −888.972 −0.254857
$$231$$ 0 0
$$232$$ −1260.00 −0.356565
$$233$$ −3577.06 −1.00575 −0.502877 0.864358i $$-0.667725\pi$$
−0.502877 + 0.864358i $$0.667725\pi$$
$$234$$ 0 0
$$235$$ −3864.00 −1.07259
$$236$$ 587.357 0.162007
$$237$$ 0 0
$$238$$ −6776.00 −1.84547
$$239$$ −2936.78 −0.794832 −0.397416 0.917639i $$-0.630093\pi$$
−0.397416 + 0.917639i $$0.630093\pi$$
$$240$$ 0 0
$$241$$ −602.000 −0.160906 −0.0804528 0.996758i $$-0.525637\pi$$
−0.0804528 + 0.996758i $$0.525637\pi$$
$$242$$ −3447.41 −0.915736
$$243$$ 0 0
$$244$$ −574.000 −0.150601
$$245$$ −1492.20 −0.389116
$$246$$ 0 0
$$247$$ −1638.00 −0.421957
$$248$$ 4333.74 1.10965
$$249$$ 0 0
$$250$$ 3864.00 0.977523
$$251$$ −3524.14 −0.886222 −0.443111 0.896467i $$-0.646125\pi$$
−0.443111 + 0.896467i $$0.646125\pi$$
$$252$$ 0 0
$$253$$ −168.000 −0.0417473
$$254$$ −994.802 −0.245746
$$255$$ 0 0
$$256$$ −1511.00 −0.368896
$$257$$ 2942.08 0.714092 0.357046 0.934087i $$-0.383784\pi$$
0.357046 + 0.934087i $$0.383784\pi$$
$$258$$ 0 0
$$259$$ 1892.00 0.453912
$$260$$ 137.579 0.0328165
$$261$$ 0 0
$$262$$ −1820.00 −0.429160
$$263$$ 857.223 0.200984 0.100492 0.994938i $$-0.467958\pi$$
0.100492 + 0.994938i $$0.467958\pi$$
$$264$$ 0 0
$$265$$ 2016.00 0.467328
$$266$$ 7334.02 1.69052
$$267$$ 0 0
$$268$$ 530.000 0.120802
$$269$$ 328.073 0.0743605 0.0371802 0.999309i $$-0.488162\pi$$
0.0371802 + 0.999309i $$0.488162\pi$$
$$270$$ 0 0
$$271$$ −2814.00 −0.630769 −0.315384 0.948964i $$-0.602133\pi$$
−0.315384 + 0.948964i $$0.602133\pi$$
$$272$$ −6402.72 −1.42729
$$273$$ 0 0
$$274$$ 3696.00 0.814903
$$275$$ 68.7895 0.0150842
$$276$$ 0 0
$$277$$ −3190.00 −0.691944 −0.345972 0.938245i $$-0.612451\pi$$
−0.345972 + 0.938245i $$0.612451\pi$$
$$278$$ 5556.08 1.19867
$$279$$ 0 0
$$280$$ −5544.00 −1.18328
$$281$$ 6116.98 1.29861 0.649303 0.760530i $$-0.275061\pi$$
0.649303 + 0.760530i $$0.275061\pi$$
$$282$$ 0 0
$$283$$ −4788.00 −1.00571 −0.502857 0.864370i $$-0.667718\pi$$
−0.502857 + 0.864370i $$0.667718\pi$$
$$284$$ −809.600 −0.169158
$$285$$ 0 0
$$286$$ −182.000 −0.0376290
$$287$$ 9778.70 2.01121
$$288$$ 0 0
$$289$$ 8639.00 1.75840
$$290$$ −1481.62 −0.300013
$$291$$ 0 0
$$292$$ 154.000 0.0308636
$$293$$ −6699.04 −1.33571 −0.667854 0.744293i $$-0.732787\pi$$
−0.667854 + 0.744293i $$0.732787\pi$$
$$294$$ 0 0
$$295$$ 6216.00 1.22681
$$296$$ 2047.81 0.402117
$$297$$ 0 0
$$298$$ −5292.00 −1.02872
$$299$$ 412.737 0.0798301
$$300$$ 0 0
$$301$$ −2112.00 −0.404431
$$302$$ 9328.92 1.77755
$$303$$ 0 0
$$304$$ 6930.00 1.30744
$$305$$ −6074.65 −1.14044
$$306$$ 0 0
$$307$$ −406.000 −0.0754777 −0.0377388 0.999288i $$-0.512015\pi$$
−0.0377388 + 0.999288i $$0.512015\pi$$
$$308$$ −116.413 −0.0215365
$$309$$ 0 0
$$310$$ 5096.00 0.933656
$$311$$ 8286.49 1.51088 0.755440 0.655217i $$-0.227423\pi$$
0.755440 + 0.655217i $$0.227423\pi$$
$$312$$ 0 0
$$313$$ −5586.00 −1.00875 −0.504376 0.863484i $$-0.668277\pi$$
−0.504376 + 0.863484i $$0.668277\pi$$
$$314$$ 8111.87 1.45790
$$315$$ 0 0
$$316$$ 460.000 0.0818893
$$317$$ −8392.32 −1.48694 −0.743470 0.668770i $$-0.766821\pi$$
−0.743470 + 0.668770i $$0.766821\pi$$
$$318$$ 0 0
$$319$$ −280.000 −0.0491442
$$320$$ −5915.90 −1.03347
$$321$$ 0 0
$$322$$ −1848.00 −0.319829
$$323$$ −14668.0 −2.52679
$$324$$ 0 0
$$325$$ −169.000 −0.0288444
$$326$$ −9106.68 −1.54715
$$327$$ 0 0
$$328$$ 10584.0 1.78172
$$329$$ −8032.50 −1.34604
$$330$$ 0 0
$$331$$ −4426.00 −0.734970 −0.367485 0.930030i $$-0.619781\pi$$
−0.367485 + 0.930030i $$0.619781\pi$$
$$332$$ 322.782 0.0533583
$$333$$ 0 0
$$334$$ 7126.00 1.16742
$$335$$ 5608.99 0.914782
$$336$$ 0 0
$$337$$ 8370.00 1.35295 0.676473 0.736467i $$-0.263507\pi$$
0.676473 + 0.736467i $$0.263507\pi$$
$$338$$ 447.132 0.0719549
$$339$$ 0 0
$$340$$ 1232.00 0.196513
$$341$$ 963.053 0.152939
$$342$$ 0 0
$$343$$ 4444.00 0.699573
$$344$$ −2285.93 −0.358282
$$345$$ 0 0
$$346$$ 9240.00 1.43568
$$347$$ 6095.81 0.943056 0.471528 0.881851i $$-0.343703\pi$$
0.471528 + 0.881851i $$0.343703\pi$$
$$348$$ 0 0
$$349$$ 4354.00 0.667806 0.333903 0.942607i $$-0.391634\pi$$
0.333903 + 0.942607i $$0.391634\pi$$
$$350$$ 756.685 0.115561
$$351$$ 0 0
$$352$$ −238.000 −0.0360382
$$353$$ 3407.73 0.513810 0.256905 0.966437i $$-0.417297\pi$$
0.256905 + 0.966437i $$0.417297\pi$$
$$354$$ 0 0
$$355$$ −8568.00 −1.28096
$$356$$ −1439.29 −0.214276
$$357$$ 0 0
$$358$$ 448.000 0.0661384
$$359$$ −7762.63 −1.14121 −0.570607 0.821223i $$-0.693292\pi$$
−0.570607 + 0.821223i $$0.693292\pi$$
$$360$$ 0 0
$$361$$ 9017.00 1.31462
$$362$$ 8926.76 1.29608
$$363$$ 0 0
$$364$$ 286.000 0.0411826
$$365$$ 1629.78 0.233717
$$366$$ 0 0
$$367$$ −7784.00 −1.10714 −0.553572 0.832802i $$-0.686735\pi$$
−0.553572 + 0.832802i $$0.686735\pi$$
$$368$$ −1746.20 −0.247355
$$369$$ 0 0
$$370$$ 2408.00 0.338340
$$371$$ 4190.87 0.586467
$$372$$ 0 0
$$373$$ −8510.00 −1.18132 −0.590658 0.806922i $$-0.701132\pi$$
−0.590658 + 0.806922i $$0.701132\pi$$
$$374$$ −1629.78 −0.225332
$$375$$ 0 0
$$376$$ −8694.00 −1.19244
$$377$$ 687.895 0.0939746
$$378$$ 0 0
$$379$$ 1650.00 0.223627 0.111814 0.993729i $$-0.464334\pi$$
0.111814 + 0.993729i $$0.464334\pi$$
$$380$$ −1333.46 −0.180013
$$381$$ 0 0
$$382$$ −3136.00 −0.420031
$$383$$ −8662.19 −1.15566 −0.577829 0.816158i $$-0.696100\pi$$
−0.577829 + 0.816158i $$0.696100\pi$$
$$384$$ 0 0
$$385$$ −1232.00 −0.163087
$$386$$ −4079.75 −0.537963
$$387$$ 0 0
$$388$$ −70.0000 −0.00915905
$$389$$ 2423.51 0.315879 0.157939 0.987449i $$-0.449515\pi$$
0.157939 + 0.987449i $$0.449515\pi$$
$$390$$ 0 0
$$391$$ 3696.00 0.478043
$$392$$ −3357.46 −0.432595
$$393$$ 0 0
$$394$$ 5628.00 0.719631
$$395$$ 4868.18 0.620114
$$396$$ 0 0
$$397$$ −1414.00 −0.178757 −0.0893786 0.995998i $$-0.528488\pi$$
−0.0893786 + 0.995998i $$0.528488\pi$$
$$398$$ −2518.76 −0.317221
$$399$$ 0 0
$$400$$ 715.000 0.0893750
$$401$$ −5228.00 −0.651058 −0.325529 0.945532i $$-0.605542\pi$$
−0.325529 + 0.945532i $$0.605542\pi$$
$$402$$ 0 0
$$403$$ −2366.00 −0.292454
$$404$$ 1460.45 0.179852
$$405$$ 0 0
$$406$$ −3080.00 −0.376497
$$407$$ 455.069 0.0554225
$$408$$ 0 0
$$409$$ 5782.00 0.699026 0.349513 0.936932i $$-0.386347\pi$$
0.349513 + 0.936932i $$0.386347\pi$$
$$410$$ 12445.6 1.49913
$$411$$ 0 0
$$412$$ 1428.00 0.170759
$$413$$ 12921.8 1.53957
$$414$$ 0 0
$$415$$ 3416.00 0.404060
$$416$$ 584.711 0.0689130
$$417$$ 0 0
$$418$$ 1764.00 0.206412
$$419$$ −11482.6 −1.33881 −0.669403 0.742899i $$-0.733450\pi$$
−0.669403 + 0.742899i $$0.733450\pi$$
$$420$$ 0 0
$$421$$ −14194.0 −1.64317 −0.821583 0.570088i $$-0.806909\pi$$
−0.821583 + 0.570088i $$0.806909\pi$$
$$422$$ −4339.03 −0.500523
$$423$$ 0 0
$$424$$ 4536.00 0.519546
$$425$$ −1513.37 −0.172728
$$426$$ 0 0
$$427$$ −12628.0 −1.43118
$$428$$ 1619.20 0.182867
$$429$$ 0 0
$$430$$ −2688.00 −0.301458
$$431$$ 5222.71 0.583687 0.291844 0.956466i $$-0.405731\pi$$
0.291844 + 0.956466i $$0.405731\pi$$
$$432$$ 0 0
$$433$$ −686.000 −0.0761364 −0.0380682 0.999275i $$-0.512120\pi$$
−0.0380682 + 0.999275i $$0.512120\pi$$
$$434$$ 10593.6 1.17168
$$435$$ 0 0
$$436$$ 338.000 0.0371268
$$437$$ −4000.38 −0.437904
$$438$$ 0 0
$$439$$ −1372.00 −0.149162 −0.0745809 0.997215i $$-0.523762\pi$$
−0.0745809 + 0.997215i $$0.523762\pi$$
$$440$$ −1333.46 −0.144478
$$441$$ 0 0
$$442$$ 4004.00 0.430884
$$443$$ 2338.84 0.250839 0.125420 0.992104i $$-0.459972\pi$$
0.125420 + 0.992104i $$0.459972\pi$$
$$444$$ 0 0
$$445$$ −15232.0 −1.62262
$$446$$ −12927.1 −1.37246
$$447$$ 0 0
$$448$$ −12298.0 −1.29693
$$449$$ 17250.3 1.81312 0.906561 0.422074i $$-0.138698\pi$$
0.906561 + 0.422074i $$0.138698\pi$$
$$450$$ 0 0
$$451$$ 2352.00 0.245568
$$452$$ 1682.70 0.175105
$$453$$ 0 0
$$454$$ 4942.00 0.510880
$$455$$ 3026.74 0.311859
$$456$$ 0 0
$$457$$ −17866.0 −1.82874 −0.914372 0.404875i $$-0.867315\pi$$
−0.914372 + 0.404875i $$0.867315\pi$$
$$458$$ 14705.1 1.50027
$$459$$ 0 0
$$460$$ 336.000 0.0340567
$$461$$ 2106.02 0.212770 0.106385 0.994325i $$-0.466072\pi$$
0.106385 + 0.994325i $$0.466072\pi$$
$$462$$ 0 0
$$463$$ −13718.0 −1.37695 −0.688477 0.725258i $$-0.741720\pi$$
−0.688477 + 0.725258i $$0.741720\pi$$
$$464$$ −2910.33 −0.291182
$$465$$ 0 0
$$466$$ −9464.00 −0.940797
$$467$$ 4095.62 0.405830 0.202915 0.979196i $$-0.434958\pi$$
0.202915 + 0.979196i $$0.434958\pi$$
$$468$$ 0 0
$$469$$ 11660.0 1.14799
$$470$$ −10223.2 −1.00332
$$471$$ 0 0
$$472$$ 13986.0 1.36389
$$473$$ −507.984 −0.0493808
$$474$$ 0 0
$$475$$ 1638.00 0.158224
$$476$$ 2561.09 0.246612
$$477$$ 0 0
$$478$$ −7770.00 −0.743497
$$479$$ 8715.10 0.831322 0.415661 0.909520i $$-0.363550\pi$$
0.415661 + 0.909520i $$0.363550\pi$$
$$480$$ 0 0
$$481$$ −1118.00 −0.105980
$$482$$ −1592.74 −0.150513
$$483$$ 0 0
$$484$$ 1303.00 0.122370
$$485$$ −740.810 −0.0693577
$$486$$ 0 0
$$487$$ 4506.00 0.419274 0.209637 0.977779i $$-0.432772\pi$$
0.209637 + 0.977779i $$0.432772\pi$$
$$488$$ −13668.0 −1.26787
$$489$$ 0 0
$$490$$ −3948.00 −0.363985
$$491$$ −21621.1 −1.98726 −0.993631 0.112683i $$-0.964056\pi$$
−0.993631 + 0.112683i $$0.964056\pi$$
$$492$$ 0 0
$$493$$ 6160.00 0.562743
$$494$$ −4333.74 −0.394705
$$495$$ 0 0
$$496$$ 10010.0 0.906174
$$497$$ −17811.2 −1.60753
$$498$$ 0 0
$$499$$ −786.000 −0.0705134 −0.0352567 0.999378i $$-0.511225\pi$$
−0.0352567 + 0.999378i $$0.511225\pi$$
$$500$$ −1460.45 −0.130627
$$501$$ 0 0
$$502$$ −9324.00 −0.828985
$$503$$ 2106.02 0.186685 0.0933426 0.995634i $$-0.470245\pi$$
0.0933426 + 0.995634i $$0.470245\pi$$
$$504$$ 0 0
$$505$$ 15456.0 1.36195
$$506$$ −444.486 −0.0390510
$$507$$ 0 0
$$508$$ 376.000 0.0328392
$$509$$ 8392.32 0.730812 0.365406 0.930848i $$-0.380930\pi$$
0.365406 + 0.930848i $$0.380930\pi$$
$$510$$ 0 0
$$511$$ 3388.00 0.293300
$$512$$ −12951.0 −1.11788
$$513$$ 0 0
$$514$$ 7784.00 0.667972
$$515$$ 15112.5 1.29308
$$516$$ 0 0
$$517$$ −1932.00 −0.164351
$$518$$ 5005.76 0.424596
$$519$$ 0 0
$$520$$ 3276.00 0.276273
$$521$$ 3905.13 0.328382 0.164191 0.986429i $$-0.447499\pi$$
0.164191 + 0.986429i $$0.447499\pi$$
$$522$$ 0 0
$$523$$ −17668.0 −1.47718 −0.738592 0.674152i $$-0.764509\pi$$
−0.738592 + 0.674152i $$0.764509\pi$$
$$524$$ 687.895 0.0573489
$$525$$ 0 0
$$526$$ 2268.00 0.188003
$$527$$ −21187.2 −1.75129
$$528$$ 0 0
$$529$$ −11159.0 −0.917153
$$530$$ 5333.83 0.437145
$$531$$ 0 0
$$532$$ −2772.00 −0.225905
$$533$$ −5778.32 −0.469581
$$534$$ 0 0
$$535$$ 17136.0 1.38477
$$536$$ 12620.2 1.01700
$$537$$ 0 0
$$538$$ 868.000 0.0695579
$$539$$ −746.102 −0.0596232
$$540$$ 0 0
$$541$$ −1650.00 −0.131126 −0.0655629 0.997848i $$-0.520884\pi$$
−0.0655629 + 0.997848i $$0.520884\pi$$
$$542$$ −7445.14 −0.590030
$$543$$ 0 0
$$544$$ 5236.00 0.412668
$$545$$ 3577.06 0.281145
$$546$$ 0 0
$$547$$ 3796.00 0.296719 0.148359 0.988934i $$-0.452601\pi$$
0.148359 + 0.988934i $$0.452601\pi$$
$$548$$ −1396.96 −0.108896
$$549$$ 0 0
$$550$$ 182.000 0.0141100
$$551$$ −6667.29 −0.515492
$$552$$ 0 0
$$553$$ 10120.0 0.778203
$$554$$ −8439.95 −0.647254
$$555$$ 0 0
$$556$$ −2100.00 −0.160180
$$557$$ −2063.69 −0.156986 −0.0784930 0.996915i $$-0.525011\pi$$
−0.0784930 + 0.996915i $$0.525011\pi$$
$$558$$ 0 0
$$559$$ 1248.00 0.0944271
$$560$$ −12805.4 −0.966301
$$561$$ 0 0
$$562$$ 16184.0 1.21473
$$563$$ 26034.2 1.94886 0.974432 0.224683i $$-0.0721346\pi$$
0.974432 + 0.224683i $$0.0721346\pi$$
$$564$$ 0 0
$$565$$ 17808.0 1.32600
$$566$$ −12667.9 −0.940759
$$567$$ 0 0
$$568$$ −19278.0 −1.42410
$$569$$ −3640.55 −0.268225 −0.134112 0.990966i $$-0.542818\pi$$
−0.134112 + 0.990966i $$0.542818\pi$$
$$570$$ 0 0
$$571$$ −19612.0 −1.43737 −0.718684 0.695337i $$-0.755255\pi$$
−0.718684 + 0.695337i $$0.755255\pi$$
$$572$$ 68.7895 0.00502838
$$573$$ 0 0
$$574$$ 25872.0 1.88132
$$575$$ −412.737 −0.0299345
$$576$$ 0 0
$$577$$ 15722.0 1.13434 0.567171 0.823600i $$-0.308038\pi$$
0.567171 + 0.823600i $$0.308038\pi$$
$$578$$ 22856.6 1.64483
$$579$$ 0 0
$$580$$ 560.000 0.0400909
$$581$$ 7101.20 0.507069
$$582$$ 0 0
$$583$$ 1008.00 0.0716074
$$584$$ 3667.01 0.259832
$$585$$ 0 0
$$586$$ −17724.0 −1.24944
$$587$$ 2725.12 0.191615 0.0958074 0.995400i $$-0.469457\pi$$
0.0958074 + 0.995400i $$0.469457\pi$$
$$588$$ 0 0
$$589$$ 22932.0 1.60424
$$590$$ 16446.0 1.14758
$$591$$ 0 0
$$592$$ 4730.00 0.328381
$$593$$ 18012.3 1.24734 0.623672 0.781686i $$-0.285640\pi$$
0.623672 + 0.781686i $$0.285640\pi$$
$$594$$ 0 0
$$595$$ 27104.0 1.86749
$$596$$ 2000.19 0.137468
$$597$$ 0 0
$$598$$ 1092.00 0.0746742
$$599$$ 12181.0 0.830891 0.415446 0.909618i $$-0.363626\pi$$
0.415446 + 0.909618i $$0.363626\pi$$
$$600$$ 0 0
$$601$$ 5950.00 0.403836 0.201918 0.979402i $$-0.435283\pi$$
0.201918 + 0.979402i $$0.435283\pi$$
$$602$$ −5587.83 −0.378310
$$603$$ 0 0
$$604$$ −3526.00 −0.237535
$$605$$ 13789.7 0.926660
$$606$$ 0 0
$$607$$ 14168.0 0.947383 0.473691 0.880691i $$-0.342921\pi$$
0.473691 + 0.880691i $$0.342921\pi$$
$$608$$ −5667.20 −0.378019
$$609$$ 0 0
$$610$$ −16072.0 −1.06678
$$611$$ 4746.48 0.314275
$$612$$ 0 0
$$613$$ −6326.00 −0.416810 −0.208405 0.978043i $$-0.566827\pi$$
−0.208405 + 0.978043i $$0.566827\pi$$
$$614$$ −1074.18 −0.0706029
$$615$$ 0 0
$$616$$ −2772.00 −0.181310
$$617$$ 8805.06 0.574519 0.287260 0.957853i $$-0.407256\pi$$
0.287260 + 0.957853i $$0.407256\pi$$
$$618$$ 0 0
$$619$$ −24486.0 −1.58994 −0.794972 0.606646i $$-0.792515\pi$$
−0.794972 + 0.606646i $$0.792515\pi$$
$$620$$ −1926.11 −0.124765
$$621$$ 0 0
$$622$$ 21924.0 1.41330
$$623$$ −31664.4 −2.03628
$$624$$ 0 0
$$625$$ −13831.0 −0.885184
$$626$$ −14779.2 −0.943601
$$627$$ 0 0
$$628$$ −3066.00 −0.194820
$$629$$ −10011.5 −0.634635
$$630$$ 0 0
$$631$$ 22430.0 1.41509 0.707547 0.706666i $$-0.249802\pi$$
0.707547 + 0.706666i $$0.249802\pi$$
$$632$$ 10953.4 0.689404
$$633$$ 0 0
$$634$$ −22204.0 −1.39090
$$635$$ 3979.21 0.248677
$$636$$ 0 0
$$637$$ 1833.00 0.114013
$$638$$ −740.810 −0.0459702
$$639$$ 0 0
$$640$$ −11844.0 −0.731524
$$641$$ 27484.1 1.69353 0.846767 0.531964i $$-0.178546\pi$$
0.846767 + 0.531964i $$0.178546\pi$$
$$642$$ 0 0
$$643$$ 16478.0 1.01062 0.505310 0.862938i $$-0.331378\pi$$
0.505310 + 0.862938i $$0.331378\pi$$
$$644$$ 698.478 0.0427390
$$645$$ 0 0
$$646$$ −38808.0 −2.36359
$$647$$ −26563.3 −1.61408 −0.807042 0.590494i $$-0.798933\pi$$
−0.807042 + 0.590494i $$0.798933\pi$$
$$648$$ 0 0
$$649$$ 3108.00 0.187981
$$650$$ −447.132 −0.0269815
$$651$$ 0 0
$$652$$ 3442.00 0.206747
$$653$$ −20287.6 −1.21580 −0.607899 0.794014i $$-0.707987\pi$$
−0.607899 + 0.794014i $$0.707987\pi$$
$$654$$ 0 0
$$655$$ 7280.00 0.434280
$$656$$ 24446.7 1.45501
$$657$$ 0 0
$$658$$ −21252.0 −1.25910
$$659$$ −656.146 −0.0387858 −0.0193929 0.999812i $$-0.506173\pi$$
−0.0193929 + 0.999812i $$0.506173\pi$$
$$660$$ 0 0
$$661$$ −14238.0 −0.837812 −0.418906 0.908030i $$-0.637586\pi$$
−0.418906 + 0.908030i $$0.637586\pi$$
$$662$$ −11710.1 −0.687501
$$663$$ 0 0
$$664$$ 7686.00 0.449209
$$665$$ −29336.1 −1.71068
$$666$$ 0 0
$$667$$ 1680.00 0.0975260
$$668$$ −2693.37 −0.156003
$$669$$ 0 0
$$670$$ 14840.0 0.855700
$$671$$ −3037.32 −0.174746
$$672$$ 0 0
$$673$$ −4874.00 −0.279166 −0.139583 0.990210i $$-0.544576\pi$$
−0.139583 + 0.990210i $$0.544576\pi$$
$$674$$ 22144.9 1.26557
$$675$$ 0 0
$$676$$ −169.000 −0.00961538
$$677$$ −21801.0 −1.23764 −0.618818 0.785534i $$-0.712388\pi$$
−0.618818 + 0.785534i $$0.712388\pi$$
$$678$$ 0 0
$$679$$ −1540.00 −0.0870394
$$680$$ 29336.1 1.65439
$$681$$ 0 0
$$682$$ 2548.00 0.143062
$$683$$ −8746.85 −0.490028 −0.245014 0.969520i $$-0.578793\pi$$
−0.245014 + 0.969520i $$0.578793\pi$$
$$684$$ 0 0
$$685$$ −14784.0 −0.824624
$$686$$ 11757.7 0.654390
$$687$$ 0 0
$$688$$ −5280.00 −0.292584
$$689$$ −2476.42 −0.136929
$$690$$ 0 0
$$691$$ 294.000 0.0161857 0.00809283 0.999967i $$-0.497424\pi$$
0.00809283 + 0.999967i $$0.497424\pi$$
$$692$$ −3492.39 −0.191851
$$693$$ 0 0
$$694$$ 16128.0 0.882148
$$695$$ −22224.3 −1.21297
$$696$$ 0 0
$$697$$ −51744.0 −2.81197
$$698$$ 11519.6 0.624675
$$699$$ 0 0
$$700$$ −286.000 −0.0154425
$$701$$ −15758.1 −0.849037 −0.424519 0.905419i $$-0.639557\pi$$
−0.424519 + 0.905419i $$0.639557\pi$$
$$702$$ 0 0
$$703$$ 10836.0 0.581348
$$704$$ −2957.95 −0.158355
$$705$$ 0 0
$$706$$ 9016.00 0.480626
$$707$$ 32130.0 1.70916
$$708$$ 0 0
$$709$$ −6722.00 −0.356065 −0.178032 0.984025i $$-0.556973\pi$$
−0.178032 + 0.984025i $$0.556973\pi$$
$$710$$ −22668.8 −1.19823
$$711$$ 0 0
$$712$$ −34272.0 −1.80393
$$713$$ −5778.32 −0.303506
$$714$$ 0 0
$$715$$ 728.000 0.0380778
$$716$$ −169.328 −0.00883811
$$717$$ 0 0
$$718$$ −20538.0 −1.06751
$$719$$ 31.7490 0.00164679 0.000823393 1.00000i $$-0.499738\pi$$
0.000823393 1.00000i $$0.499738\pi$$
$$720$$ 0 0
$$721$$ 31416.0 1.62274
$$722$$ 23856.7 1.22972
$$723$$ 0 0
$$724$$ −3374.00 −0.173196
$$725$$ −687.895 −0.0352383
$$726$$ 0 0
$$727$$ 12824.0 0.654217 0.327109 0.944987i $$-0.393926\pi$$
0.327109 + 0.944987i $$0.393926\pi$$
$$728$$ 6810.16 0.346705
$$729$$ 0 0
$$730$$ 4312.00 0.218622
$$731$$ 11175.7 0.565453
$$732$$ 0 0
$$733$$ −29610.0 −1.49205 −0.746023 0.665920i $$-0.768039\pi$$
−0.746023 + 0.665920i $$0.768039\pi$$
$$734$$ −20594.5 −1.03564
$$735$$ 0 0
$$736$$ 1428.00 0.0715174
$$737$$ 2804.50 0.140170
$$738$$ 0 0
$$739$$ −15622.0 −0.777625 −0.388812 0.921317i $$-0.627115\pi$$
−0.388812 + 0.921317i $$0.627115\pi$$
$$740$$ −910.138 −0.0452126
$$741$$ 0 0
$$742$$ 11088.0 0.548589
$$743$$ −8588.11 −0.424047 −0.212024 0.977265i $$-0.568005\pi$$
−0.212024 + 0.977265i $$0.568005\pi$$
$$744$$ 0 0
$$745$$ 21168.0 1.04099
$$746$$ −22515.3 −1.10502
$$747$$ 0 0
$$748$$ 616.000 0.0301112
$$749$$ 35622.4 1.73780
$$750$$ 0 0
$$751$$ 29468.0 1.43183 0.715914 0.698189i $$-0.246010\pi$$
0.715914 + 0.698189i $$0.246010\pi$$
$$752$$ −20081.3 −0.973787
$$753$$ 0 0
$$754$$ 1820.00 0.0879052
$$755$$ −37315.7 −1.79875
$$756$$ 0 0
$$757$$ −35030.0 −1.68189 −0.840943 0.541124i $$-0.817999\pi$$
−0.840943 + 0.541124i $$0.817999\pi$$
$$758$$ 4365.49 0.209184
$$759$$ 0 0
$$760$$ −31752.0 −1.51548
$$761$$ −22330.1 −1.06369 −0.531844 0.846842i $$-0.678501\pi$$
−0.531844 + 0.846842i $$0.678501\pi$$
$$762$$ 0 0
$$763$$ 7436.00 0.352819
$$764$$ 1185.30 0.0561290
$$765$$ 0 0
$$766$$ −22918.0 −1.08102
$$767$$ −7635.64 −0.359461
$$768$$ 0 0
$$769$$ −27342.0 −1.28216 −0.641078 0.767476i $$-0.721512\pi$$
−0.641078 + 0.767476i $$0.721512\pi$$
$$770$$ −3259.57 −0.152554
$$771$$ 0 0
$$772$$ 1542.00 0.0718883
$$773$$ 11884.7 0.552993 0.276496 0.961015i $$-0.410827\pi$$
0.276496 + 0.961015i $$0.410827\pi$$
$$774$$ 0 0
$$775$$ 2366.00 0.109664
$$776$$ −1666.82 −0.0771076
$$777$$ 0 0
$$778$$ 6412.00 0.295477
$$779$$ 56005.3 2.57586
$$780$$ 0 0
$$781$$ −4284.00 −0.196279
$$782$$ 9778.70 0.447168
$$783$$ 0 0
$$784$$ −7755.00 −0.353271
$$785$$ −32447.5 −1.47529
$$786$$ 0 0
$$787$$ 22666.0 1.02663 0.513314 0.858201i $$-0.328418\pi$$
0.513314 + 0.858201i $$0.328418\pi$$
$$788$$ −2127.18 −0.0961647
$$789$$ 0 0
$$790$$ 12880.0 0.580063
$$791$$ 37019.4 1.66404
$$792$$ 0 0
$$793$$ 7462.00 0.334153
$$794$$ −3741.09 −0.167212
$$795$$ 0 0
$$796$$ 952.000 0.0423904
$$797$$ 582.065 0.0258693 0.0129346 0.999916i $$-0.495883\pi$$
0.0129346 + 0.999916i $$0.495883\pi$$
$$798$$ 0 0
$$799$$ 42504.0 1.88196
$$800$$ −584.711 −0.0258408
$$801$$ 0 0
$$802$$ −13832.0 −0.609009
$$803$$ 814.891 0.0358118
$$804$$ 0 0
$$805$$ 7392.00 0.323644
$$806$$ −6259.85 −0.273565
$$807$$ 0 0
$$808$$ 34776.0 1.51413
$$809$$ 793.725 0.0344943 0.0172472 0.999851i $$-0.494510\pi$$
0.0172472 + 0.999851i $$0.494510\pi$$
$$810$$ 0 0
$$811$$ −9478.00 −0.410379 −0.205190 0.978722i $$-0.565781\pi$$
−0.205190 + 0.978722i $$0.565781\pi$$
$$812$$ 1164.13 0.0503115
$$813$$ 0 0
$$814$$ 1204.00 0.0518430
$$815$$ 36426.7 1.56561
$$816$$ 0 0
$$817$$ −12096.0 −0.517975
$$818$$ 15297.7 0.653879
$$819$$ 0 0
$$820$$ −4704.00 −0.200330
$$821$$ 3227.82 0.137213 0.0686063 0.997644i $$-0.478145\pi$$
0.0686063 + 0.997644i $$0.478145\pi$$
$$822$$ 0 0
$$823$$ 40476.0 1.71434 0.857172 0.515031i $$-0.172219\pi$$
0.857172 + 0.515031i $$0.172219\pi$$
$$824$$ 34003.2 1.43757
$$825$$ 0 0
$$826$$ 34188.0 1.44014
$$827$$ 7169.99 0.301481 0.150741 0.988573i $$-0.451834\pi$$
0.150741 + 0.988573i $$0.451834\pi$$
$$828$$ 0 0
$$829$$ 27482.0 1.15137 0.575687 0.817670i $$-0.304735\pi$$
0.575687 + 0.817670i $$0.304735\pi$$
$$830$$ 9037.89 0.377963
$$831$$ 0 0
$$832$$ 7267.00 0.302810
$$833$$ 16414.2 0.682737
$$834$$ 0 0
$$835$$ −28504.0 −1.18134
$$836$$ −666.729 −0.0275829
$$837$$ 0 0
$$838$$ −30380.0 −1.25234
$$839$$ 19128.8 0.787126 0.393563 0.919298i $$-0.371242\pi$$
0.393563 + 0.919298i $$0.371242\pi$$
$$840$$ 0 0
$$841$$ −21589.0 −0.885194
$$842$$ −37553.8 −1.53704
$$843$$ 0 0
$$844$$ 1640.00 0.0668852
$$845$$ −1788.53 −0.0728133
$$846$$ 0 0
$$847$$ 28666.0 1.16290
$$848$$ 10477.2 0.424278
$$849$$ 0 0
$$850$$ −4004.00 −0.161572
$$851$$ −2730.42 −0.109985
$$852$$ 0 0
$$853$$ 31962.0 1.28295 0.641476 0.767143i $$-0.278322\pi$$
0.641476 + 0.767143i $$0.278322\pi$$
$$854$$ −33410.5 −1.33874
$$855$$ 0 0
$$856$$ 38556.0 1.53951
$$857$$ −4931.68 −0.196573 −0.0982865 0.995158i $$-0.531336\pi$$
−0.0982865 + 0.995158i $$0.531336\pi$$
$$858$$ 0 0
$$859$$ 11704.0 0.464884 0.232442 0.972610i $$-0.425328\pi$$
0.232442 + 0.972610i $$0.425328\pi$$
$$860$$ 1015.97 0.0402840
$$861$$ 0 0
$$862$$ 13818.0 0.545989
$$863$$ 2280.64 0.0899581 0.0449790 0.998988i $$-0.485678\pi$$
0.0449790 + 0.998988i $$0.485678\pi$$
$$864$$ 0 0
$$865$$ −36960.0 −1.45281
$$866$$ −1814.99 −0.0712191
$$867$$ 0 0
$$868$$ −4004.00 −0.156572
$$869$$ 2434.09 0.0950183
$$870$$ 0 0
$$871$$ −6890.00 −0.268035
$$872$$ 8048.38 0.312560
$$873$$ 0 0
$$874$$ −10584.0 −0.409621
$$875$$ −32130.0 −1.24136
$$876$$ 0 0
$$877$$ −1006.00 −0.0387346 −0.0193673 0.999812i $$-0.506165\pi$$
−0.0193673 + 0.999812i $$0.506165\pi$$
$$878$$ −3629.97 −0.139528
$$879$$ 0 0
$$880$$ −3080.00 −0.117985
$$881$$ 40681.1 1.55571 0.777855 0.628444i $$-0.216308\pi$$
0.777855 + 0.628444i $$0.216308\pi$$
$$882$$ 0 0
$$883$$ 124.000 0.00472586 0.00236293 0.999997i $$-0.499248\pi$$
0.00236293 + 0.999997i $$0.499248\pi$$
$$884$$ −1513.37 −0.0575793
$$885$$ 0 0
$$886$$ 6188.00 0.234639
$$887$$ −16573.0 −0.627358 −0.313679 0.949529i $$-0.601562\pi$$
−0.313679 + 0.949529i $$0.601562\pi$$
$$888$$ 0 0
$$889$$ 8272.00 0.312074
$$890$$ −40300.1 −1.51782
$$891$$ 0 0
$$892$$ 4886.00 0.183403
$$893$$ −46004.3 −1.72394
$$894$$ 0 0
$$895$$ −1792.00 −0.0669273
$$896$$ −24621.4 −0.918016
$$897$$ 0 0
$$898$$ 45640.0 1.69602
$$899$$ −9630.53 −0.357282
$$900$$ 0 0
$$901$$ −22176.0 −0.819966
$$902$$ 6222.81 0.229708
$$903$$ 0 0
$$904$$ 40068.0 1.47416
$$905$$ −35707.1 −1.31154
$$906$$ 0 0
$$907$$ −42996.0 −1.57404 −0.787022 0.616924i $$-0.788378\pi$$
−0.787022 + 0.616924i $$0.788378\pi$$
$$908$$ −1867.90 −0.0682692
$$909$$ 0 0
$$910$$ 8008.00 0.291717
$$911$$ 53169.0 1.93366 0.966832 0.255413i $$-0.0822113\pi$$
0.966832 + 0.255413i $$0.0822113\pi$$
$$912$$ 0 0
$$913$$ 1708.00 0.0619130
$$914$$ −47269.0 −1.71063
$$915$$ 0 0
$$916$$ −5558.00 −0.200482
$$917$$ 15133.7 0.544993
$$918$$ 0 0
$$919$$ 8244.00 0.295913 0.147957 0.988994i $$-0.452730\pi$$
0.147957 + 0.988994i $$0.452730\pi$$
$$920$$ 8000.75 0.286714
$$921$$ 0 0
$$922$$ 5572.00 0.199028
$$923$$ 10524.8 0.375328
$$924$$ 0 0
$$925$$ 1118.00 0.0397401
$$926$$ −36294.4 −1.28802
$$927$$ 0 0
$$928$$ 2380.00 0.0841889
$$929$$ 16192.0 0.571843 0.285922 0.958253i $$-0.407700\pi$$
0.285922 + 0.958253i $$0.407700\pi$$
$$930$$ 0 0
$$931$$ −17766.0 −0.625410
$$932$$ 3577.06 0.125719
$$933$$ 0 0
$$934$$ 10836.0 0.379620
$$935$$ 6519.13 0.228020
$$936$$ 0 0
$$937$$ −18214.0 −0.635032 −0.317516 0.948253i $$-0.602849\pi$$
−0.317516 + 0.948253i $$0.602849\pi$$
$$938$$ 30849.5 1.07385
$$939$$ 0 0
$$940$$ 3864.00 0.134074
$$941$$ 13916.7 0.482115 0.241057 0.970511i $$-0.422506\pi$$
0.241057 + 0.970511i $$0.422506\pi$$
$$942$$ 0 0
$$943$$ −14112.0 −0.487328
$$944$$ 32304.6 1.11380
$$945$$ 0 0
$$946$$ −1344.00 −0.0461916
$$947$$ −40400.6 −1.38632 −0.693159 0.720784i $$-0.743782\pi$$
−0.693159 + 0.720784i $$0.743782\pi$$
$$948$$ 0 0
$$949$$ −2002.00 −0.0684802
$$950$$ 4333.74 0.148005
$$951$$ 0 0
$$952$$ 60984.0 2.07616
$$953$$ −21271.8 −0.723046 −0.361523 0.932363i $$-0.617743\pi$$
−0.361523 + 0.932363i $$0.617743\pi$$
$$954$$ 0 0
$$955$$ 12544.0 0.425041
$$956$$ 2936.78 0.0993540
$$957$$ 0 0
$$958$$ 23058.0 0.777631
$$959$$ −30733.0 −1.03485
$$960$$ 0 0
$$961$$ 3333.00 0.111879
$$962$$ −2957.95 −0.0991353
$$963$$ 0 0
$$964$$ 602.000 0.0201132
$$965$$ 16319.0 0.544380
$$966$$ 0 0
$$967$$ 9578.00 0.318519 0.159259 0.987237i $$-0.449089\pi$$
0.159259 + 0.987237i $$0.449089\pi$$
$$968$$ 31026.7 1.03020
$$969$$ 0 0
$$970$$ −1960.00 −0.0648782
$$971$$ −44956.6 −1.48581 −0.742907 0.669394i $$-0.766554\pi$$
−0.742907 + 0.669394i $$0.766554\pi$$
$$972$$ 0 0
$$973$$ −46200.0 −1.52220
$$974$$ 11921.8 0.392195
$$975$$ 0 0
$$976$$ −31570.0 −1.03538
$$977$$ 32765.0 1.07292 0.536461 0.843925i $$-0.319761\pi$$
0.536461 + 0.843925i $$0.319761\pi$$
$$978$$ 0 0
$$979$$ −7616.00 −0.248630
$$980$$ 1492.20 0.0486395
$$981$$ 0 0
$$982$$ −57204.0 −1.85891
$$983$$ −47438.3 −1.53921 −0.769607 0.638518i $$-0.779548\pi$$
−0.769607 + 0.638518i $$0.779548\pi$$
$$984$$ 0 0
$$985$$ −22512.0 −0.728215
$$986$$ 16297.8 0.526398
$$987$$ 0 0
$$988$$ 1638.00 0.0527447
$$989$$ 3047.91 0.0979957
$$990$$ 0 0
$$991$$ −6848.00 −0.219509 −0.109755 0.993959i $$-0.535007\pi$$
−0.109755 + 0.993959i $$0.535007\pi$$
$$992$$ −8185.95 −0.262000
$$993$$ 0 0
$$994$$ −47124.0 −1.50370
$$995$$ 10075.0 0.321005
$$996$$ 0 0
$$997$$ −5810.00 −0.184558 −0.0922791 0.995733i $$-0.529415\pi$$
−0.0922791 + 0.995733i $$0.529415\pi$$
$$998$$ −2079.56 −0.0659593
$$999$$ 0 0
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 117.4.a.e.1.2 yes 2
3.2 odd 2 inner 117.4.a.e.1.1 2
4.3 odd 2 1872.4.a.ba.1.1 2
12.11 even 2 1872.4.a.ba.1.2 2
13.12 even 2 1521.4.a.p.1.1 2
39.38 odd 2 1521.4.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
117.4.a.e.1.1 2 3.2 odd 2 inner
117.4.a.e.1.2 yes 2 1.1 even 1 trivial
1521.4.a.p.1.1 2 13.12 even 2
1521.4.a.p.1.2 2 39.38 odd 2
1872.4.a.ba.1.1 2 4.3 odd 2
1872.4.a.ba.1.2 2 12.11 even 2