Properties

 Label 78.4.a.c.1.1 Level $78$ Weight $4$ Character 78.1 Self dual yes Analytic conductor $4.602$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,4,Mod(1,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, 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("78.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 78.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: $$4.60214898045$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 78.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +10.0000 q^{5} -6.00000 q^{6} -8.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +10.0000 q^{5} -6.00000 q^{6} -8.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -20.0000 q^{10} +40.0000 q^{11} +12.0000 q^{12} +13.0000 q^{13} +16.0000 q^{14} +30.0000 q^{15} +16.0000 q^{16} +130.000 q^{17} -18.0000 q^{18} -20.0000 q^{19} +40.0000 q^{20} -24.0000 q^{21} -80.0000 q^{22} -24.0000 q^{24} -25.0000 q^{25} -26.0000 q^{26} +27.0000 q^{27} -32.0000 q^{28} -18.0000 q^{29} -60.0000 q^{30} -184.000 q^{31} -32.0000 q^{32} +120.000 q^{33} -260.000 q^{34} -80.0000 q^{35} +36.0000 q^{36} -74.0000 q^{37} +40.0000 q^{38} +39.0000 q^{39} -80.0000 q^{40} -362.000 q^{41} +48.0000 q^{42} +76.0000 q^{43} +160.000 q^{44} +90.0000 q^{45} -452.000 q^{47} +48.0000 q^{48} -279.000 q^{49} +50.0000 q^{50} +390.000 q^{51} +52.0000 q^{52} +382.000 q^{53} -54.0000 q^{54} +400.000 q^{55} +64.0000 q^{56} -60.0000 q^{57} +36.0000 q^{58} +464.000 q^{59} +120.000 q^{60} +358.000 q^{61} +368.000 q^{62} -72.0000 q^{63} +64.0000 q^{64} +130.000 q^{65} -240.000 q^{66} -700.000 q^{67} +520.000 q^{68} +160.000 q^{70} -748.000 q^{71} -72.0000 q^{72} +1058.00 q^{73} +148.000 q^{74} -75.0000 q^{75} -80.0000 q^{76} -320.000 q^{77} -78.0000 q^{78} -976.000 q^{79} +160.000 q^{80} +81.0000 q^{81} +724.000 q^{82} -1008.00 q^{83} -96.0000 q^{84} +1300.00 q^{85} -152.000 q^{86} -54.0000 q^{87} -320.000 q^{88} -386.000 q^{89} -180.000 q^{90} -104.000 q^{91} -552.000 q^{93} +904.000 q^{94} -200.000 q^{95} -96.0000 q^{96} -614.000 q^{97} +558.000 q^{98} +360.000 q^{99} +O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ 3.00000 0.577350
$$4$$ 4.00000 0.500000
$$5$$ 10.0000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ −6.00000 −0.408248
$$7$$ −8.00000 −0.431959 −0.215980 0.976398i $$-0.569295\pi$$
−0.215980 + 0.976398i $$0.569295\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 9.00000 0.333333
$$10$$ −20.0000 −0.632456
$$11$$ 40.0000 1.09640 0.548202 0.836346i $$-0.315312\pi$$
0.548202 + 0.836346i $$0.315312\pi$$
$$12$$ 12.0000 0.288675
$$13$$ 13.0000 0.277350
$$14$$ 16.0000 0.305441
$$15$$ 30.0000 0.516398
$$16$$ 16.0000 0.250000
$$17$$ 130.000 1.85468 0.927342 0.374215i $$-0.122088\pi$$
0.927342 + 0.374215i $$0.122088\pi$$
$$18$$ −18.0000 −0.235702
$$19$$ −20.0000 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ 40.0000 0.447214
$$21$$ −24.0000 −0.249392
$$22$$ −80.0000 −0.775275
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −24.0000 −0.204124
$$25$$ −25.0000 −0.200000
$$26$$ −26.0000 −0.196116
$$27$$ 27.0000 0.192450
$$28$$ −32.0000 −0.215980
$$29$$ −18.0000 −0.115259 −0.0576296 0.998338i $$-0.518354\pi$$
−0.0576296 + 0.998338i $$0.518354\pi$$
$$30$$ −60.0000 −0.365148
$$31$$ −184.000 −1.06604 −0.533022 0.846101i $$-0.678944\pi$$
−0.533022 + 0.846101i $$0.678944\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 120.000 0.633010
$$34$$ −260.000 −1.31146
$$35$$ −80.0000 −0.386356
$$36$$ 36.0000 0.166667
$$37$$ −74.0000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 40.0000 0.170759
$$39$$ 39.0000 0.160128
$$40$$ −80.0000 −0.316228
$$41$$ −362.000 −1.37890 −0.689450 0.724333i $$-0.742148\pi$$
−0.689450 + 0.724333i $$0.742148\pi$$
$$42$$ 48.0000 0.176347
$$43$$ 76.0000 0.269532 0.134766 0.990877i $$-0.456972\pi$$
0.134766 + 0.990877i $$0.456972\pi$$
$$44$$ 160.000 0.548202
$$45$$ 90.0000 0.298142
$$46$$ 0 0
$$47$$ −452.000 −1.40279 −0.701393 0.712774i $$-0.747438\pi$$
−0.701393 + 0.712774i $$0.747438\pi$$
$$48$$ 48.0000 0.144338
$$49$$ −279.000 −0.813411
$$50$$ 50.0000 0.141421
$$51$$ 390.000 1.07080
$$52$$ 52.0000 0.138675
$$53$$ 382.000 0.990033 0.495016 0.868884i $$-0.335162\pi$$
0.495016 + 0.868884i $$0.335162\pi$$
$$54$$ −54.0000 −0.136083
$$55$$ 400.000 0.980654
$$56$$ 64.0000 0.152721
$$57$$ −60.0000 −0.139424
$$58$$ 36.0000 0.0815005
$$59$$ 464.000 1.02386 0.511929 0.859028i $$-0.328931\pi$$
0.511929 + 0.859028i $$0.328931\pi$$
$$60$$ 120.000 0.258199
$$61$$ 358.000 0.751430 0.375715 0.926735i $$-0.377397\pi$$
0.375715 + 0.926735i $$0.377397\pi$$
$$62$$ 368.000 0.753807
$$63$$ −72.0000 −0.143986
$$64$$ 64.0000 0.125000
$$65$$ 130.000 0.248069
$$66$$ −240.000 −0.447605
$$67$$ −700.000 −1.27640 −0.638199 0.769872i $$-0.720320\pi$$
−0.638199 + 0.769872i $$0.720320\pi$$
$$68$$ 520.000 0.927342
$$69$$ 0 0
$$70$$ 160.000 0.273195
$$71$$ −748.000 −1.25030 −0.625150 0.780505i $$-0.714962\pi$$
−0.625150 + 0.780505i $$0.714962\pi$$
$$72$$ −72.0000 −0.117851
$$73$$ 1058.00 1.69629 0.848147 0.529760i $$-0.177718\pi$$
0.848147 + 0.529760i $$0.177718\pi$$
$$74$$ 148.000 0.232495
$$75$$ −75.0000 −0.115470
$$76$$ −80.0000 −0.120745
$$77$$ −320.000 −0.473602
$$78$$ −78.0000 −0.113228
$$79$$ −976.000 −1.38998 −0.694991 0.719018i $$-0.744592\pi$$
−0.694991 + 0.719018i $$0.744592\pi$$
$$80$$ 160.000 0.223607
$$81$$ 81.0000 0.111111
$$82$$ 724.000 0.975030
$$83$$ −1008.00 −1.33304 −0.666520 0.745487i $$-0.732217\pi$$
−0.666520 + 0.745487i $$0.732217\pi$$
$$84$$ −96.0000 −0.124696
$$85$$ 1300.00 1.65888
$$86$$ −152.000 −0.190588
$$87$$ −54.0000 −0.0665449
$$88$$ −320.000 −0.387638
$$89$$ −386.000 −0.459729 −0.229865 0.973223i $$-0.573828\pi$$
−0.229865 + 0.973223i $$0.573828\pi$$
$$90$$ −180.000 −0.210819
$$91$$ −104.000 −0.119804
$$92$$ 0 0
$$93$$ −552.000 −0.615481
$$94$$ 904.000 0.991920
$$95$$ −200.000 −0.215995
$$96$$ −96.0000 −0.102062
$$97$$ −614.000 −0.642704 −0.321352 0.946960i $$-0.604137\pi$$
−0.321352 + 0.946960i $$0.604137\pi$$
$$98$$ 558.000 0.575168
$$99$$ 360.000 0.365468
$$100$$ −100.000 −0.100000
$$101$$ 518.000 0.510326 0.255163 0.966898i $$-0.417871\pi$$
0.255163 + 0.966898i $$0.417871\pi$$
$$102$$ −780.000 −0.757172
$$103$$ 112.000 0.107143 0.0535713 0.998564i $$-0.482940\pi$$
0.0535713 + 0.998564i $$0.482940\pi$$
$$104$$ −104.000 −0.0980581
$$105$$ −240.000 −0.223063
$$106$$ −764.000 −0.700059
$$107$$ −372.000 −0.336099 −0.168050 0.985779i $$-0.553747\pi$$
−0.168050 + 0.985779i $$0.553747\pi$$
$$108$$ 108.000 0.0962250
$$109$$ 934.000 0.820743 0.410371 0.911918i $$-0.365399\pi$$
0.410371 + 0.911918i $$0.365399\pi$$
$$110$$ −800.000 −0.693427
$$111$$ −222.000 −0.189832
$$112$$ −128.000 −0.107990
$$113$$ 1914.00 1.59340 0.796699 0.604376i $$-0.206578\pi$$
0.796699 + 0.604376i $$0.206578\pi$$
$$114$$ 120.000 0.0985880
$$115$$ 0 0
$$116$$ −72.0000 −0.0576296
$$117$$ 117.000 0.0924500
$$118$$ −928.000 −0.723977
$$119$$ −1040.00 −0.801148
$$120$$ −240.000 −0.182574
$$121$$ 269.000 0.202104
$$122$$ −716.000 −0.531341
$$123$$ −1086.00 −0.796108
$$124$$ −736.000 −0.533022
$$125$$ −1500.00 −1.07331
$$126$$ 144.000 0.101814
$$127$$ 1296.00 0.905523 0.452761 0.891632i $$-0.350439\pi$$
0.452761 + 0.891632i $$0.350439\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 228.000 0.155615
$$130$$ −260.000 −0.175412
$$131$$ −892.000 −0.594919 −0.297460 0.954734i $$-0.596139\pi$$
−0.297460 + 0.954734i $$0.596139\pi$$
$$132$$ 480.000 0.316505
$$133$$ 160.000 0.104314
$$134$$ 1400.00 0.902549
$$135$$ 270.000 0.172133
$$136$$ −1040.00 −0.655730
$$137$$ 2326.00 1.45054 0.725269 0.688466i $$-0.241716\pi$$
0.725269 + 0.688466i $$0.241716\pi$$
$$138$$ 0 0
$$139$$ 1932.00 1.17892 0.589461 0.807797i $$-0.299340\pi$$
0.589461 + 0.807797i $$0.299340\pi$$
$$140$$ −320.000 −0.193178
$$141$$ −1356.00 −0.809899
$$142$$ 1496.00 0.884095
$$143$$ 520.000 0.304088
$$144$$ 144.000 0.0833333
$$145$$ −180.000 −0.103091
$$146$$ −2116.00 −1.19946
$$147$$ −837.000 −0.469623
$$148$$ −296.000 −0.164399
$$149$$ 882.000 0.484941 0.242471 0.970159i $$-0.422042\pi$$
0.242471 + 0.970159i $$0.422042\pi$$
$$150$$ 150.000 0.0816497
$$151$$ −1776.00 −0.957145 −0.478572 0.878048i $$-0.658846\pi$$
−0.478572 + 0.878048i $$0.658846\pi$$
$$152$$ 160.000 0.0853797
$$153$$ 1170.00 0.618228
$$154$$ 640.000 0.334887
$$155$$ −1840.00 −0.953499
$$156$$ 156.000 0.0800641
$$157$$ −2410.00 −1.22509 −0.612544 0.790436i $$-0.709854\pi$$
−0.612544 + 0.790436i $$0.709854\pi$$
$$158$$ 1952.00 0.982866
$$159$$ 1146.00 0.571596
$$160$$ −320.000 −0.158114
$$161$$ 0 0
$$162$$ −162.000 −0.0785674
$$163$$ 3212.00 1.54346 0.771728 0.635953i $$-0.219393\pi$$
0.771728 + 0.635953i $$0.219393\pi$$
$$164$$ −1448.00 −0.689450
$$165$$ 1200.00 0.566181
$$166$$ 2016.00 0.942602
$$167$$ 1668.00 0.772896 0.386448 0.922311i $$-0.373702\pi$$
0.386448 + 0.922311i $$0.373702\pi$$
$$168$$ 192.000 0.0881733
$$169$$ 169.000 0.0769231
$$170$$ −2600.00 −1.17301
$$171$$ −180.000 −0.0804967
$$172$$ 304.000 0.134766
$$173$$ 3598.00 1.58122 0.790609 0.612321i $$-0.209764\pi$$
0.790609 + 0.612321i $$0.209764\pi$$
$$174$$ 108.000 0.0470544
$$175$$ 200.000 0.0863919
$$176$$ 640.000 0.274101
$$177$$ 1392.00 0.591125
$$178$$ 772.000 0.325078
$$179$$ 1068.00 0.445956 0.222978 0.974824i $$-0.428422\pi$$
0.222978 + 0.974824i $$0.428422\pi$$
$$180$$ 360.000 0.149071
$$181$$ −4786.00 −1.96542 −0.982709 0.185158i $$-0.940720\pi$$
−0.982709 + 0.185158i $$0.940720\pi$$
$$182$$ 208.000 0.0847142
$$183$$ 1074.00 0.433838
$$184$$ 0 0
$$185$$ −740.000 −0.294086
$$186$$ 1104.00 0.435211
$$187$$ 5200.00 2.03348
$$188$$ −1808.00 −0.701393
$$189$$ −216.000 −0.0831306
$$190$$ 400.000 0.152732
$$191$$ −1312.00 −0.497031 −0.248516 0.968628i $$-0.579943\pi$$
−0.248516 + 0.968628i $$0.579943\pi$$
$$192$$ 192.000 0.0721688
$$193$$ −350.000 −0.130537 −0.0652683 0.997868i $$-0.520790\pi$$
−0.0652683 + 0.997868i $$0.520790\pi$$
$$194$$ 1228.00 0.454460
$$195$$ 390.000 0.143223
$$196$$ −1116.00 −0.406706
$$197$$ −342.000 −0.123688 −0.0618439 0.998086i $$-0.519698\pi$$
−0.0618439 + 0.998086i $$0.519698\pi$$
$$198$$ −720.000 −0.258425
$$199$$ −3368.00 −1.19975 −0.599877 0.800092i $$-0.704784\pi$$
−0.599877 + 0.800092i $$0.704784\pi$$
$$200$$ 200.000 0.0707107
$$201$$ −2100.00 −0.736928
$$202$$ −1036.00 −0.360855
$$203$$ 144.000 0.0497873
$$204$$ 1560.00 0.535401
$$205$$ −3620.00 −1.23333
$$206$$ −224.000 −0.0757613
$$207$$ 0 0
$$208$$ 208.000 0.0693375
$$209$$ −800.000 −0.264771
$$210$$ 480.000 0.157729
$$211$$ −2004.00 −0.653844 −0.326922 0.945051i $$-0.606011\pi$$
−0.326922 + 0.945051i $$0.606011\pi$$
$$212$$ 1528.00 0.495016
$$213$$ −2244.00 −0.721861
$$214$$ 744.000 0.237658
$$215$$ 760.000 0.241077
$$216$$ −216.000 −0.0680414
$$217$$ 1472.00 0.460488
$$218$$ −1868.00 −0.580353
$$219$$ 3174.00 0.979356
$$220$$ 1600.00 0.490327
$$221$$ 1690.00 0.514397
$$222$$ 444.000 0.134231
$$223$$ −5608.00 −1.68403 −0.842017 0.539451i $$-0.818632\pi$$
−0.842017 + 0.539451i $$0.818632\pi$$
$$224$$ 256.000 0.0763604
$$225$$ −225.000 −0.0666667
$$226$$ −3828.00 −1.12670
$$227$$ −1928.00 −0.563726 −0.281863 0.959455i $$-0.590952\pi$$
−0.281863 + 0.959455i $$0.590952\pi$$
$$228$$ −240.000 −0.0697122
$$229$$ −3938.00 −1.13638 −0.568189 0.822898i $$-0.692356\pi$$
−0.568189 + 0.822898i $$0.692356\pi$$
$$230$$ 0 0
$$231$$ −960.000 −0.273434
$$232$$ 144.000 0.0407503
$$233$$ 2562.00 0.720353 0.360176 0.932884i $$-0.382717\pi$$
0.360176 + 0.932884i $$0.382717\pi$$
$$234$$ −234.000 −0.0653720
$$235$$ −4520.00 −1.25469
$$236$$ 1856.00 0.511929
$$237$$ −2928.00 −0.802506
$$238$$ 2080.00 0.566497
$$239$$ 7164.00 1.93891 0.969457 0.245260i $$-0.0788733\pi$$
0.969457 + 0.245260i $$0.0788733\pi$$
$$240$$ 480.000 0.129099
$$241$$ −6182.00 −1.65236 −0.826178 0.563410i $$-0.809489\pi$$
−0.826178 + 0.563410i $$0.809489\pi$$
$$242$$ −538.000 −0.142909
$$243$$ 243.000 0.0641500
$$244$$ 1432.00 0.375715
$$245$$ −2790.00 −0.727537
$$246$$ 2172.00 0.562934
$$247$$ −260.000 −0.0669773
$$248$$ 1472.00 0.376904
$$249$$ −3024.00 −0.769631
$$250$$ 3000.00 0.758947
$$251$$ −1396.00 −0.351055 −0.175527 0.984475i $$-0.556163\pi$$
−0.175527 + 0.984475i $$0.556163\pi$$
$$252$$ −288.000 −0.0719932
$$253$$ 0 0
$$254$$ −2592.00 −0.640301
$$255$$ 3900.00 0.957755
$$256$$ 256.000 0.0625000
$$257$$ 6906.00 1.67620 0.838102 0.545514i $$-0.183665\pi$$
0.838102 + 0.545514i $$0.183665\pi$$
$$258$$ −456.000 −0.110036
$$259$$ 592.000 0.142027
$$260$$ 520.000 0.124035
$$261$$ −162.000 −0.0384197
$$262$$ 1784.00 0.420671
$$263$$ −6848.00 −1.60557 −0.802787 0.596266i $$-0.796650\pi$$
−0.802787 + 0.596266i $$0.796650\pi$$
$$264$$ −960.000 −0.223803
$$265$$ 3820.00 0.885512
$$266$$ −320.000 −0.0737611
$$267$$ −1158.00 −0.265425
$$268$$ −2800.00 −0.638199
$$269$$ −6034.00 −1.36766 −0.683828 0.729643i $$-0.739686\pi$$
−0.683828 + 0.729643i $$0.739686\pi$$
$$270$$ −540.000 −0.121716
$$271$$ 4832.00 1.08311 0.541556 0.840665i $$-0.317836\pi$$
0.541556 + 0.840665i $$0.317836\pi$$
$$272$$ 2080.00 0.463671
$$273$$ −312.000 −0.0691689
$$274$$ −4652.00 −1.02568
$$275$$ −1000.00 −0.219281
$$276$$ 0 0
$$277$$ −4082.00 −0.885428 −0.442714 0.896663i $$-0.645984\pi$$
−0.442714 + 0.896663i $$0.645984\pi$$
$$278$$ −3864.00 −0.833623
$$279$$ −1656.00 −0.355348
$$280$$ 640.000 0.136598
$$281$$ 3350.00 0.711189 0.355595 0.934640i $$-0.384278\pi$$
0.355595 + 0.934640i $$0.384278\pi$$
$$282$$ 2712.00 0.572685
$$283$$ 7796.00 1.63754 0.818770 0.574121i $$-0.194656\pi$$
0.818770 + 0.574121i $$0.194656\pi$$
$$284$$ −2992.00 −0.625150
$$285$$ −600.000 −0.124705
$$286$$ −1040.00 −0.215023
$$287$$ 2896.00 0.595629
$$288$$ −288.000 −0.0589256
$$289$$ 11987.0 2.43985
$$290$$ 360.000 0.0728963
$$291$$ −1842.00 −0.371065
$$292$$ 4232.00 0.848147
$$293$$ 3922.00 0.781999 0.390999 0.920391i $$-0.372129\pi$$
0.390999 + 0.920391i $$0.372129\pi$$
$$294$$ 1674.00 0.332074
$$295$$ 4640.00 0.915767
$$296$$ 592.000 0.116248
$$297$$ 1080.00 0.211003
$$298$$ −1764.00 −0.342905
$$299$$ 0 0
$$300$$ −300.000 −0.0577350
$$301$$ −608.000 −0.116427
$$302$$ 3552.00 0.676803
$$303$$ 1554.00 0.294637
$$304$$ −320.000 −0.0603726
$$305$$ 3580.00 0.672099
$$306$$ −2340.00 −0.437153
$$307$$ 5956.00 1.10725 0.553627 0.832765i $$-0.313243\pi$$
0.553627 + 0.832765i $$0.313243\pi$$
$$308$$ −1280.00 −0.236801
$$309$$ 336.000 0.0618588
$$310$$ 3680.00 0.674226
$$311$$ 2352.00 0.428841 0.214421 0.976741i $$-0.431214\pi$$
0.214421 + 0.976741i $$0.431214\pi$$
$$312$$ −312.000 −0.0566139
$$313$$ 8442.00 1.52450 0.762252 0.647280i $$-0.224093\pi$$
0.762252 + 0.647280i $$0.224093\pi$$
$$314$$ 4820.00 0.866269
$$315$$ −720.000 −0.128785
$$316$$ −3904.00 −0.694991
$$317$$ −5550.00 −0.983341 −0.491670 0.870781i $$-0.663614\pi$$
−0.491670 + 0.870781i $$0.663614\pi$$
$$318$$ −2292.00 −0.404179
$$319$$ −720.000 −0.126371
$$320$$ 640.000 0.111803
$$321$$ −1116.00 −0.194047
$$322$$ 0 0
$$323$$ −2600.00 −0.447888
$$324$$ 324.000 0.0555556
$$325$$ −325.000 −0.0554700
$$326$$ −6424.00 −1.09139
$$327$$ 2802.00 0.473856
$$328$$ 2896.00 0.487515
$$329$$ 3616.00 0.605947
$$330$$ −2400.00 −0.400350
$$331$$ 140.000 0.0232480 0.0116240 0.999932i $$-0.496300\pi$$
0.0116240 + 0.999932i $$0.496300\pi$$
$$332$$ −4032.00 −0.666520
$$333$$ −666.000 −0.109599
$$334$$ −3336.00 −0.546520
$$335$$ −7000.00 −1.14164
$$336$$ −384.000 −0.0623480
$$337$$ −6174.00 −0.997980 −0.498990 0.866608i $$-0.666296\pi$$
−0.498990 + 0.866608i $$0.666296\pi$$
$$338$$ −338.000 −0.0543928
$$339$$ 5742.00 0.919949
$$340$$ 5200.00 0.829440
$$341$$ −7360.00 −1.16882
$$342$$ 360.000 0.0569198
$$343$$ 4976.00 0.783320
$$344$$ −608.000 −0.0952941
$$345$$ 0 0
$$346$$ −7196.00 −1.11809
$$347$$ −2988.00 −0.462260 −0.231130 0.972923i $$-0.574242\pi$$
−0.231130 + 0.972923i $$0.574242\pi$$
$$348$$ −216.000 −0.0332725
$$349$$ −162.000 −0.0248472 −0.0124236 0.999923i $$-0.503955\pi$$
−0.0124236 + 0.999923i $$0.503955\pi$$
$$350$$ −400.000 −0.0610883
$$351$$ 351.000 0.0533761
$$352$$ −1280.00 −0.193819
$$353$$ −10754.0 −1.62147 −0.810733 0.585416i $$-0.800931\pi$$
−0.810733 + 0.585416i $$0.800931\pi$$
$$354$$ −2784.00 −0.417989
$$355$$ −7480.00 −1.11830
$$356$$ −1544.00 −0.229865
$$357$$ −3120.00 −0.462543
$$358$$ −2136.00 −0.315338
$$359$$ 3588.00 0.527486 0.263743 0.964593i $$-0.415043\pi$$
0.263743 + 0.964593i $$0.415043\pi$$
$$360$$ −720.000 −0.105409
$$361$$ −6459.00 −0.941682
$$362$$ 9572.00 1.38976
$$363$$ 807.000 0.116685
$$364$$ −416.000 −0.0599020
$$365$$ 10580.0 1.51721
$$366$$ −2148.00 −0.306770
$$367$$ 11272.0 1.60325 0.801626 0.597826i $$-0.203968\pi$$
0.801626 + 0.597826i $$0.203968\pi$$
$$368$$ 0 0
$$369$$ −3258.00 −0.459633
$$370$$ 1480.00 0.207950
$$371$$ −3056.00 −0.427654
$$372$$ −2208.00 −0.307741
$$373$$ −10914.0 −1.51503 −0.757514 0.652819i $$-0.773586\pi$$
−0.757514 + 0.652819i $$0.773586\pi$$
$$374$$ −10400.0 −1.43789
$$375$$ −4500.00 −0.619677
$$376$$ 3616.00 0.495960
$$377$$ −234.000 −0.0319671
$$378$$ 432.000 0.0587822
$$379$$ 8100.00 1.09781 0.548904 0.835886i $$-0.315045\pi$$
0.548904 + 0.835886i $$0.315045\pi$$
$$380$$ −800.000 −0.107998
$$381$$ 3888.00 0.522804
$$382$$ 2624.00 0.351454
$$383$$ 6180.00 0.824499 0.412250 0.911071i $$-0.364743\pi$$
0.412250 + 0.911071i $$0.364743\pi$$
$$384$$ −384.000 −0.0510310
$$385$$ −3200.00 −0.423603
$$386$$ 700.000 0.0923033
$$387$$ 684.000 0.0898441
$$388$$ −2456.00 −0.321352
$$389$$ −7522.00 −0.980413 −0.490206 0.871606i $$-0.663079\pi$$
−0.490206 + 0.871606i $$0.663079\pi$$
$$390$$ −780.000 −0.101274
$$391$$ 0 0
$$392$$ 2232.00 0.287584
$$393$$ −2676.00 −0.343477
$$394$$ 684.000 0.0874605
$$395$$ −9760.00 −1.24324
$$396$$ 1440.00 0.182734
$$397$$ 6078.00 0.768378 0.384189 0.923254i $$-0.374481\pi$$
0.384189 + 0.923254i $$0.374481\pi$$
$$398$$ 6736.00 0.848355
$$399$$ 480.000 0.0602257
$$400$$ −400.000 −0.0500000
$$401$$ 1830.00 0.227895 0.113947 0.993487i $$-0.463650\pi$$
0.113947 + 0.993487i $$0.463650\pi$$
$$402$$ 4200.00 0.521087
$$403$$ −2392.00 −0.295668
$$404$$ 2072.00 0.255163
$$405$$ 810.000 0.0993808
$$406$$ −288.000 −0.0352049
$$407$$ −2960.00 −0.360496
$$408$$ −3120.00 −0.378586
$$409$$ 12434.0 1.50323 0.751616 0.659601i $$-0.229275\pi$$
0.751616 + 0.659601i $$0.229275\pi$$
$$410$$ 7240.00 0.872093
$$411$$ 6978.00 0.837468
$$412$$ 448.000 0.0535713
$$413$$ −3712.00 −0.442265
$$414$$ 0 0
$$415$$ −10080.0 −1.19231
$$416$$ −416.000 −0.0490290
$$417$$ 5796.00 0.680651
$$418$$ 1600.00 0.187221
$$419$$ −14188.0 −1.65425 −0.827123 0.562021i $$-0.810024\pi$$
−0.827123 + 0.562021i $$0.810024\pi$$
$$420$$ −960.000 −0.111531
$$421$$ 8638.00 0.999977 0.499989 0.866032i $$-0.333338\pi$$
0.499989 + 0.866032i $$0.333338\pi$$
$$422$$ 4008.00 0.462337
$$423$$ −4068.00 −0.467596
$$424$$ −3056.00 −0.350029
$$425$$ −3250.00 −0.370937
$$426$$ 4488.00 0.510433
$$427$$ −2864.00 −0.324587
$$428$$ −1488.00 −0.168050
$$429$$ 1560.00 0.175565
$$430$$ −1520.00 −0.170467
$$431$$ 4292.00 0.479671 0.239836 0.970813i $$-0.422906\pi$$
0.239836 + 0.970813i $$0.422906\pi$$
$$432$$ 432.000 0.0481125
$$433$$ −5982.00 −0.663918 −0.331959 0.943294i $$-0.607710\pi$$
−0.331959 + 0.943294i $$0.607710\pi$$
$$434$$ −2944.00 −0.325614
$$435$$ −540.000 −0.0595196
$$436$$ 3736.00 0.410371
$$437$$ 0 0
$$438$$ −6348.00 −0.692510
$$439$$ 256.000 0.0278319 0.0139160 0.999903i $$-0.495570\pi$$
0.0139160 + 0.999903i $$0.495570\pi$$
$$440$$ −3200.00 −0.346714
$$441$$ −2511.00 −0.271137
$$442$$ −3380.00 −0.363733
$$443$$ 12556.0 1.34662 0.673311 0.739359i $$-0.264872\pi$$
0.673311 + 0.739359i $$0.264872\pi$$
$$444$$ −888.000 −0.0949158
$$445$$ −3860.00 −0.411194
$$446$$ 11216.0 1.19079
$$447$$ 2646.00 0.279981
$$448$$ −512.000 −0.0539949
$$449$$ 5574.00 0.585865 0.292932 0.956133i $$-0.405369\pi$$
0.292932 + 0.956133i $$0.405369\pi$$
$$450$$ 450.000 0.0471405
$$451$$ −14480.0 −1.51183
$$452$$ 7656.00 0.796699
$$453$$ −5328.00 −0.552608
$$454$$ 3856.00 0.398615
$$455$$ −1040.00 −0.107156
$$456$$ 480.000 0.0492940
$$457$$ 1266.00 0.129586 0.0647932 0.997899i $$-0.479361\pi$$
0.0647932 + 0.997899i $$0.479361\pi$$
$$458$$ 7876.00 0.803540
$$459$$ 3510.00 0.356934
$$460$$ 0 0
$$461$$ 7554.00 0.763178 0.381589 0.924332i $$-0.375377\pi$$
0.381589 + 0.924332i $$0.375377\pi$$
$$462$$ 1920.00 0.193347
$$463$$ −6752.00 −0.677737 −0.338868 0.940834i $$-0.610044\pi$$
−0.338868 + 0.940834i $$0.610044\pi$$
$$464$$ −288.000 −0.0288148
$$465$$ −5520.00 −0.550503
$$466$$ −5124.00 −0.509366
$$467$$ 7924.00 0.785180 0.392590 0.919714i $$-0.371579\pi$$
0.392590 + 0.919714i $$0.371579\pi$$
$$468$$ 468.000 0.0462250
$$469$$ 5600.00 0.551352
$$470$$ 9040.00 0.887200
$$471$$ −7230.00 −0.707305
$$472$$ −3712.00 −0.361989
$$473$$ 3040.00 0.295517
$$474$$ 5856.00 0.567458
$$475$$ 500.000 0.0482980
$$476$$ −4160.00 −0.400574
$$477$$ 3438.00 0.330011
$$478$$ −14328.0 −1.37102
$$479$$ −11084.0 −1.05729 −0.528644 0.848844i $$-0.677299\pi$$
−0.528644 + 0.848844i $$0.677299\pi$$
$$480$$ −960.000 −0.0912871
$$481$$ −962.000 −0.0911922
$$482$$ 12364.0 1.16839
$$483$$ 0 0
$$484$$ 1076.00 0.101052
$$485$$ −6140.00 −0.574852
$$486$$ −486.000 −0.0453609
$$487$$ 4432.00 0.412388 0.206194 0.978511i $$-0.433892\pi$$
0.206194 + 0.978511i $$0.433892\pi$$
$$488$$ −2864.00 −0.265670
$$489$$ 9636.00 0.891114
$$490$$ 5580.00 0.514446
$$491$$ −1140.00 −0.104781 −0.0523905 0.998627i $$-0.516684\pi$$
−0.0523905 + 0.998627i $$0.516684\pi$$
$$492$$ −4344.00 −0.398054
$$493$$ −2340.00 −0.213769
$$494$$ 520.000 0.0473601
$$495$$ 3600.00 0.326885
$$496$$ −2944.00 −0.266511
$$497$$ 5984.00 0.540079
$$498$$ 6048.00 0.544212
$$499$$ 1764.00 0.158251 0.0791257 0.996865i $$-0.474787\pi$$
0.0791257 + 0.996865i $$0.474787\pi$$
$$500$$ −6000.00 −0.536656
$$501$$ 5004.00 0.446232
$$502$$ 2792.00 0.248233
$$503$$ 16976.0 1.50482 0.752408 0.658697i $$-0.228892\pi$$
0.752408 + 0.658697i $$0.228892\pi$$
$$504$$ 576.000 0.0509069
$$505$$ 5180.00 0.456449
$$506$$ 0 0
$$507$$ 507.000 0.0444116
$$508$$ 5184.00 0.452761
$$509$$ 9474.00 0.825005 0.412503 0.910956i $$-0.364655\pi$$
0.412503 + 0.910956i $$0.364655\pi$$
$$510$$ −7800.00 −0.677235
$$511$$ −8464.00 −0.732731
$$512$$ −512.000 −0.0441942
$$513$$ −540.000 −0.0464748
$$514$$ −13812.0 −1.18526
$$515$$ 1120.00 0.0958313
$$516$$ 912.000 0.0778073
$$517$$ −18080.0 −1.53802
$$518$$ −1184.00 −0.100429
$$519$$ 10794.0 0.912917
$$520$$ −1040.00 −0.0877058
$$521$$ 14114.0 1.18684 0.593422 0.804892i $$-0.297777\pi$$
0.593422 + 0.804892i $$0.297777\pi$$
$$522$$ 324.000 0.0271668
$$523$$ 20284.0 1.69590 0.847952 0.530074i $$-0.177836\pi$$
0.847952 + 0.530074i $$0.177836\pi$$
$$524$$ −3568.00 −0.297460
$$525$$ 600.000 0.0498784
$$526$$ 13696.0 1.13531
$$527$$ −23920.0 −1.97718
$$528$$ 1920.00 0.158252
$$529$$ −12167.0 −1.00000
$$530$$ −7640.00 −0.626152
$$531$$ 4176.00 0.341286
$$532$$ 640.000 0.0521570
$$533$$ −4706.00 −0.382438
$$534$$ 2316.00 0.187684
$$535$$ −3720.00 −0.300616
$$536$$ 5600.00 0.451275
$$537$$ 3204.00 0.257473
$$538$$ 12068.0 0.967079
$$539$$ −11160.0 −0.891828
$$540$$ 1080.00 0.0860663
$$541$$ −14362.0 −1.14135 −0.570675 0.821176i $$-0.693318\pi$$
−0.570675 + 0.821176i $$0.693318\pi$$
$$542$$ −9664.00 −0.765875
$$543$$ −14358.0 −1.13473
$$544$$ −4160.00 −0.327865
$$545$$ 9340.00 0.734095
$$546$$ 624.000 0.0489098
$$547$$ −20956.0 −1.63805 −0.819025 0.573757i $$-0.805485\pi$$
−0.819025 + 0.573757i $$0.805485\pi$$
$$548$$ 9304.00 0.725269
$$549$$ 3222.00 0.250477
$$550$$ 2000.00 0.155055
$$551$$ 360.000 0.0278340
$$552$$ 0 0
$$553$$ 7808.00 0.600416
$$554$$ 8164.00 0.626092
$$555$$ −2220.00 −0.169791
$$556$$ 7728.00 0.589461
$$557$$ −4134.00 −0.314476 −0.157238 0.987561i $$-0.550259\pi$$
−0.157238 + 0.987561i $$0.550259\pi$$
$$558$$ 3312.00 0.251269
$$559$$ 988.000 0.0747548
$$560$$ −1280.00 −0.0965891
$$561$$ 15600.0 1.17403
$$562$$ −6700.00 −0.502887
$$563$$ −16228.0 −1.21479 −0.607397 0.794399i $$-0.707786\pi$$
−0.607397 + 0.794399i $$0.707786\pi$$
$$564$$ −5424.00 −0.404950
$$565$$ 19140.0 1.42518
$$566$$ −15592.0 −1.15792
$$567$$ −648.000 −0.0479955
$$568$$ 5984.00 0.442048
$$569$$ 2514.00 0.185224 0.0926119 0.995702i $$-0.470478\pi$$
0.0926119 + 0.995702i $$0.470478\pi$$
$$570$$ 1200.00 0.0881798
$$571$$ −11612.0 −0.851046 −0.425523 0.904948i $$-0.639910\pi$$
−0.425523 + 0.904948i $$0.639910\pi$$
$$572$$ 2080.00 0.152044
$$573$$ −3936.00 −0.286961
$$574$$ −5792.00 −0.421173
$$575$$ 0 0
$$576$$ 576.000 0.0416667
$$577$$ 6354.00 0.458441 0.229221 0.973375i $$-0.426382\pi$$
0.229221 + 0.973375i $$0.426382\pi$$
$$578$$ −23974.0 −1.72524
$$579$$ −1050.00 −0.0753653
$$580$$ −720.000 −0.0515455
$$581$$ 8064.00 0.575819
$$582$$ 3684.00 0.262383
$$583$$ 15280.0 1.08548
$$584$$ −8464.00 −0.599731
$$585$$ 1170.00 0.0826898
$$586$$ −7844.00 −0.552957
$$587$$ −13240.0 −0.930960 −0.465480 0.885059i $$-0.654118\pi$$
−0.465480 + 0.885059i $$0.654118\pi$$
$$588$$ −3348.00 −0.234812
$$589$$ 3680.00 0.257439
$$590$$ −9280.00 −0.647545
$$591$$ −1026.00 −0.0714112
$$592$$ −1184.00 −0.0821995
$$593$$ −1146.00 −0.0793602 −0.0396801 0.999212i $$-0.512634\pi$$
−0.0396801 + 0.999212i $$0.512634\pi$$
$$594$$ −2160.00 −0.149202
$$595$$ −10400.0 −0.716569
$$596$$ 3528.00 0.242471
$$597$$ −10104.0 −0.692679
$$598$$ 0 0
$$599$$ 10464.0 0.713769 0.356884 0.934149i $$-0.383839\pi$$
0.356884 + 0.934149i $$0.383839\pi$$
$$600$$ 600.000 0.0408248
$$601$$ 6650.00 0.451346 0.225673 0.974203i $$-0.427542\pi$$
0.225673 + 0.974203i $$0.427542\pi$$
$$602$$ 1216.00 0.0823263
$$603$$ −6300.00 −0.425466
$$604$$ −7104.00 −0.478572
$$605$$ 2690.00 0.180767
$$606$$ −3108.00 −0.208340
$$607$$ −6664.00 −0.445607 −0.222803 0.974863i $$-0.571521\pi$$
−0.222803 + 0.974863i $$0.571521\pi$$
$$608$$ 640.000 0.0426898
$$609$$ 432.000 0.0287447
$$610$$ −7160.00 −0.475246
$$611$$ −5876.00 −0.389063
$$612$$ 4680.00 0.309114
$$613$$ 2134.00 0.140606 0.0703030 0.997526i $$-0.477603\pi$$
0.0703030 + 0.997526i $$0.477603\pi$$
$$614$$ −11912.0 −0.782947
$$615$$ −10860.0 −0.712061
$$616$$ 2560.00 0.167444
$$617$$ −714.000 −0.0465876 −0.0232938 0.999729i $$-0.507415\pi$$
−0.0232938 + 0.999729i $$0.507415\pi$$
$$618$$ −672.000 −0.0437408
$$619$$ 29228.0 1.89786 0.948928 0.315494i $$-0.102170\pi$$
0.948928 + 0.315494i $$0.102170\pi$$
$$620$$ −7360.00 −0.476750
$$621$$ 0 0
$$622$$ −4704.00 −0.303237
$$623$$ 3088.00 0.198584
$$624$$ 624.000 0.0400320
$$625$$ −11875.0 −0.760000
$$626$$ −16884.0 −1.07799
$$627$$ −2400.00 −0.152866
$$628$$ −9640.00 −0.612544
$$629$$ −9620.00 −0.609816
$$630$$ 1440.00 0.0910650
$$631$$ −13536.0 −0.853977 −0.426989 0.904257i $$-0.640426\pi$$
−0.426989 + 0.904257i $$0.640426\pi$$
$$632$$ 7808.00 0.491433
$$633$$ −6012.00 −0.377497
$$634$$ 11100.0 0.695327
$$635$$ 12960.0 0.809924
$$636$$ 4584.00 0.285798
$$637$$ −3627.00 −0.225600
$$638$$ 1440.00 0.0893576
$$639$$ −6732.00 −0.416767
$$640$$ −1280.00 −0.0790569
$$641$$ 17218.0 1.06095 0.530476 0.847700i $$-0.322013\pi$$
0.530476 + 0.847700i $$0.322013\pi$$
$$642$$ 2232.00 0.137212
$$643$$ 15044.0 0.922671 0.461335 0.887226i $$-0.347370\pi$$
0.461335 + 0.887226i $$0.347370\pi$$
$$644$$ 0 0
$$645$$ 2280.00 0.139186
$$646$$ 5200.00 0.316705
$$647$$ 25176.0 1.52978 0.764892 0.644158i $$-0.222792\pi$$
0.764892 + 0.644158i $$0.222792\pi$$
$$648$$ −648.000 −0.0392837
$$649$$ 18560.0 1.12256
$$650$$ 650.000 0.0392232
$$651$$ 4416.00 0.265863
$$652$$ 12848.0 0.771728
$$653$$ −16034.0 −0.960887 −0.480443 0.877026i $$-0.659524\pi$$
−0.480443 + 0.877026i $$0.659524\pi$$
$$654$$ −5604.00 −0.335067
$$655$$ −8920.00 −0.532112
$$656$$ −5792.00 −0.344725
$$657$$ 9522.00 0.565432
$$658$$ −7232.00 −0.428469
$$659$$ 25356.0 1.49883 0.749415 0.662100i $$-0.230335\pi$$
0.749415 + 0.662100i $$0.230335\pi$$
$$660$$ 4800.00 0.283091
$$661$$ 18310.0 1.07742 0.538711 0.842490i $$-0.318911\pi$$
0.538711 + 0.842490i $$0.318911\pi$$
$$662$$ −280.000 −0.0164388
$$663$$ 5070.00 0.296987
$$664$$ 8064.00 0.471301
$$665$$ 1600.00 0.0933013
$$666$$ 1332.00 0.0774984
$$667$$ 0 0
$$668$$ 6672.00 0.386448
$$669$$ −16824.0 −0.972277
$$670$$ 14000.0 0.807264
$$671$$ 14320.0 0.823871
$$672$$ 768.000 0.0440867
$$673$$ 24802.0 1.42057 0.710287 0.703912i $$-0.248565\pi$$
0.710287 + 0.703912i $$0.248565\pi$$
$$674$$ 12348.0 0.705678
$$675$$ −675.000 −0.0384900
$$676$$ 676.000 0.0384615
$$677$$ −22706.0 −1.28901 −0.644507 0.764598i $$-0.722937\pi$$
−0.644507 + 0.764598i $$0.722937\pi$$
$$678$$ −11484.0 −0.650502
$$679$$ 4912.00 0.277622
$$680$$ −10400.0 −0.586503
$$681$$ −5784.00 −0.325467
$$682$$ 14720.0 0.826478
$$683$$ −14792.0 −0.828697 −0.414349 0.910118i $$-0.635991\pi$$
−0.414349 + 0.910118i $$0.635991\pi$$
$$684$$ −720.000 −0.0402484
$$685$$ 23260.0 1.29740
$$686$$ −9952.00 −0.553891
$$687$$ −11814.0 −0.656088
$$688$$ 1216.00 0.0673831
$$689$$ 4966.00 0.274586
$$690$$ 0 0
$$691$$ −1148.00 −0.0632011 −0.0316006 0.999501i $$-0.510060\pi$$
−0.0316006 + 0.999501i $$0.510060\pi$$
$$692$$ 14392.0 0.790609
$$693$$ −2880.00 −0.157867
$$694$$ 5976.00 0.326867
$$695$$ 19320.0 1.05446
$$696$$ 432.000 0.0235272
$$697$$ −47060.0 −2.55742
$$698$$ 324.000 0.0175696
$$699$$ 7686.00 0.415896
$$700$$ 800.000 0.0431959
$$701$$ 14870.0 0.801187 0.400594 0.916256i $$-0.368804\pi$$
0.400594 + 0.916256i $$0.368804\pi$$
$$702$$ −702.000 −0.0377426
$$703$$ 1480.00 0.0794015
$$704$$ 2560.00 0.137051
$$705$$ −13560.0 −0.724396
$$706$$ 21508.0 1.14655
$$707$$ −4144.00 −0.220440
$$708$$ 5568.00 0.295563
$$709$$ −6354.00 −0.336572 −0.168286 0.985738i $$-0.553823\pi$$
−0.168286 + 0.985738i $$0.553823\pi$$
$$710$$ 14960.0 0.790759
$$711$$ −8784.00 −0.463327
$$712$$ 3088.00 0.162539
$$713$$ 0 0
$$714$$ 6240.00 0.327067
$$715$$ 5200.00 0.271985
$$716$$ 4272.00 0.222978
$$717$$ 21492.0 1.11943
$$718$$ −7176.00 −0.372989
$$719$$ 9288.00 0.481758 0.240879 0.970555i $$-0.422564\pi$$
0.240879 + 0.970555i $$0.422564\pi$$
$$720$$ 1440.00 0.0745356
$$721$$ −896.000 −0.0462813
$$722$$ 12918.0 0.665870
$$723$$ −18546.0 −0.953988
$$724$$ −19144.0 −0.982709
$$725$$ 450.000 0.0230518
$$726$$ −1614.00 −0.0825085
$$727$$ −21544.0 −1.09907 −0.549534 0.835471i $$-0.685195\pi$$
−0.549534 + 0.835471i $$0.685195\pi$$
$$728$$ 832.000 0.0423571
$$729$$ 729.000 0.0370370
$$730$$ −21160.0 −1.07283
$$731$$ 9880.00 0.499897
$$732$$ 4296.00 0.216919
$$733$$ 19990.0 1.00730 0.503648 0.863909i $$-0.331991\pi$$
0.503648 + 0.863909i $$0.331991\pi$$
$$734$$ −22544.0 −1.13367
$$735$$ −8370.00 −0.420044
$$736$$ 0 0
$$737$$ −28000.0 −1.39945
$$738$$ 6516.00 0.325010
$$739$$ 532.000 0.0264816 0.0132408 0.999912i $$-0.495785\pi$$
0.0132408 + 0.999912i $$0.495785\pi$$
$$740$$ −2960.00 −0.147043
$$741$$ −780.000 −0.0386694
$$742$$ 6112.00 0.302397
$$743$$ −25452.0 −1.25672 −0.628360 0.777922i $$-0.716274\pi$$
−0.628360 + 0.777922i $$0.716274\pi$$
$$744$$ 4416.00 0.217605
$$745$$ 8820.00 0.433745
$$746$$ 21828.0 1.07129
$$747$$ −9072.00 −0.444347
$$748$$ 20800.0 1.01674
$$749$$ 2976.00 0.145181
$$750$$ 9000.00 0.438178
$$751$$ 6440.00 0.312915 0.156457 0.987685i $$-0.449993\pi$$
0.156457 + 0.987685i $$0.449993\pi$$
$$752$$ −7232.00 −0.350697
$$753$$ −4188.00 −0.202682
$$754$$ 468.000 0.0226042
$$755$$ −17760.0 −0.856096
$$756$$ −864.000 −0.0415653
$$757$$ −786.000 −0.0377380 −0.0188690 0.999822i $$-0.506007\pi$$
−0.0188690 + 0.999822i $$0.506007\pi$$
$$758$$ −16200.0 −0.776267
$$759$$ 0 0
$$760$$ 1600.00 0.0763659
$$761$$ −1498.00 −0.0713567 −0.0356784 0.999363i $$-0.511359\pi$$
−0.0356784 + 0.999363i $$0.511359\pi$$
$$762$$ −7776.00 −0.369678
$$763$$ −7472.00 −0.354528
$$764$$ −5248.00 −0.248516
$$765$$ 11700.0 0.552960
$$766$$ −12360.0 −0.583009
$$767$$ 6032.00 0.283967
$$768$$ 768.000 0.0360844
$$769$$ 14738.0 0.691113 0.345556 0.938398i $$-0.387690\pi$$
0.345556 + 0.938398i $$0.387690\pi$$
$$770$$ 6400.00 0.299532
$$771$$ 20718.0 0.967757
$$772$$ −1400.00 −0.0652683
$$773$$ −3822.00 −0.177837 −0.0889184 0.996039i $$-0.528341\pi$$
−0.0889184 + 0.996039i $$0.528341\pi$$
$$774$$ −1368.00 −0.0635294
$$775$$ 4600.00 0.213209
$$776$$ 4912.00 0.227230
$$777$$ 1776.00 0.0819995
$$778$$ 15044.0 0.693256
$$779$$ 7240.00 0.332991
$$780$$ 1560.00 0.0716115
$$781$$ −29920.0 −1.37083
$$782$$ 0 0
$$783$$ −486.000 −0.0221816
$$784$$ −4464.00 −0.203353
$$785$$ −24100.0 −1.09575
$$786$$ 5352.00 0.242875
$$787$$ −11900.0 −0.538995 −0.269498 0.963001i $$-0.586858\pi$$
−0.269498 + 0.963001i $$0.586858\pi$$
$$788$$ −1368.00 −0.0618439
$$789$$ −20544.0 −0.926978
$$790$$ 19520.0 0.879102
$$791$$ −15312.0 −0.688283
$$792$$ −2880.00 −0.129213
$$793$$ 4654.00 0.208409
$$794$$ −12156.0 −0.543325
$$795$$ 11460.0 0.511251
$$796$$ −13472.0 −0.599877
$$797$$ −21274.0 −0.945500 −0.472750 0.881197i $$-0.656739\pi$$
−0.472750 + 0.881197i $$0.656739\pi$$
$$798$$ −960.000 −0.0425860
$$799$$ −58760.0 −2.60173
$$800$$ 800.000 0.0353553
$$801$$ −3474.00 −0.153243
$$802$$ −3660.00 −0.161146
$$803$$ 42320.0 1.85983
$$804$$ −8400.00 −0.368464
$$805$$ 0 0
$$806$$ 4784.00 0.209069
$$807$$ −18102.0 −0.789617
$$808$$ −4144.00 −0.180427
$$809$$ −27566.0 −1.19798 −0.598992 0.800755i $$-0.704432\pi$$
−0.598992 + 0.800755i $$0.704432\pi$$
$$810$$ −1620.00 −0.0702728
$$811$$ −11244.0 −0.486844 −0.243422 0.969921i $$-0.578270\pi$$
−0.243422 + 0.969921i $$0.578270\pi$$
$$812$$ 576.000 0.0248936
$$813$$ 14496.0 0.625334
$$814$$ 5920.00 0.254909
$$815$$ 32120.0 1.38051
$$816$$ 6240.00 0.267701
$$817$$ −1520.00 −0.0650894
$$818$$ −24868.0 −1.06295
$$819$$ −936.000 −0.0399347
$$820$$ −14480.0 −0.616663
$$821$$ 13554.0 0.576173 0.288086 0.957604i $$-0.406981\pi$$
0.288086 + 0.957604i $$0.406981\pi$$
$$822$$ −13956.0 −0.592179
$$823$$ 14384.0 0.609228 0.304614 0.952476i $$-0.401473\pi$$
0.304614 + 0.952476i $$0.401473\pi$$
$$824$$ −896.000 −0.0378806
$$825$$ −3000.00 −0.126602
$$826$$ 7424.00 0.312729
$$827$$ −2488.00 −0.104615 −0.0523073 0.998631i $$-0.516658\pi$$
−0.0523073 + 0.998631i $$0.516658\pi$$
$$828$$ 0 0
$$829$$ −20858.0 −0.873858 −0.436929 0.899496i $$-0.643934\pi$$
−0.436929 + 0.899496i $$0.643934\pi$$
$$830$$ 20160.0 0.843089
$$831$$ −12246.0 −0.511202
$$832$$ 832.000 0.0346688
$$833$$ −36270.0 −1.50862
$$834$$ −11592.0 −0.481293
$$835$$ 16680.0 0.691300
$$836$$ −3200.00 −0.132386
$$837$$ −4968.00 −0.205160
$$838$$ 28376.0 1.16973
$$839$$ 23116.0 0.951195 0.475598 0.879663i $$-0.342232\pi$$
0.475598 + 0.879663i $$0.342232\pi$$
$$840$$ 1920.00 0.0788646
$$841$$ −24065.0 −0.986715
$$842$$ −17276.0 −0.707091
$$843$$ 10050.0 0.410605
$$844$$ −8016.00 −0.326922
$$845$$ 1690.00 0.0688021
$$846$$ 8136.00 0.330640
$$847$$ −2152.00 −0.0873006
$$848$$ 6112.00 0.247508
$$849$$ 23388.0 0.945435
$$850$$ 6500.00 0.262292
$$851$$ 0 0
$$852$$ −8976.00 −0.360930
$$853$$ 934.000 0.0374907 0.0187453 0.999824i $$-0.494033\pi$$
0.0187453 + 0.999824i $$0.494033\pi$$
$$854$$ 5728.00 0.229518
$$855$$ −1800.00 −0.0719985
$$856$$ 2976.00 0.118829
$$857$$ 12642.0 0.503900 0.251950 0.967740i $$-0.418928\pi$$
0.251950 + 0.967740i $$0.418928\pi$$
$$858$$ −3120.00 −0.124143
$$859$$ −22796.0 −0.905459 −0.452730 0.891648i $$-0.649550\pi$$
−0.452730 + 0.891648i $$0.649550\pi$$
$$860$$ 3040.00 0.120539
$$861$$ 8688.00 0.343886
$$862$$ −8584.00 −0.339179
$$863$$ −76.0000 −0.00299776 −0.00149888 0.999999i $$-0.500477\pi$$
−0.00149888 + 0.999999i $$0.500477\pi$$
$$864$$ −864.000 −0.0340207
$$865$$ 35980.0 1.41429
$$866$$ 11964.0 0.469461
$$867$$ 35961.0 1.40865
$$868$$ 5888.00 0.230244
$$869$$ −39040.0 −1.52398
$$870$$ 1080.00 0.0420867
$$871$$ −9100.00 −0.354009
$$872$$ −7472.00 −0.290176
$$873$$ −5526.00 −0.214235
$$874$$ 0 0
$$875$$ 12000.0 0.463627
$$876$$ 12696.0 0.489678
$$877$$ −46130.0 −1.77617 −0.888084 0.459681i $$-0.847964\pi$$
−0.888084 + 0.459681i $$0.847964\pi$$
$$878$$ −512.000 −0.0196801
$$879$$ 11766.0 0.451487
$$880$$ 6400.00 0.245164
$$881$$ 6682.00 0.255530 0.127765 0.991804i $$-0.459220\pi$$
0.127765 + 0.991804i $$0.459220\pi$$
$$882$$ 5022.00 0.191723
$$883$$ 47404.0 1.80665 0.903325 0.428957i $$-0.141119\pi$$
0.903325 + 0.428957i $$0.141119\pi$$
$$884$$ 6760.00 0.257198
$$885$$ 13920.0 0.528718
$$886$$ −25112.0 −0.952206
$$887$$ 33672.0 1.27463 0.637314 0.770604i $$-0.280045\pi$$
0.637314 + 0.770604i $$0.280045\pi$$
$$888$$ 1776.00 0.0671156
$$889$$ −10368.0 −0.391149
$$890$$ 7720.00 0.290758
$$891$$ 3240.00 0.121823
$$892$$ −22432.0 −0.842017
$$893$$ 9040.00 0.338759
$$894$$ −5292.00 −0.197976
$$895$$ 10680.0 0.398875
$$896$$ 1024.00 0.0381802
$$897$$ 0 0
$$898$$ −11148.0 −0.414269
$$899$$ 3312.00 0.122871
$$900$$ −900.000 −0.0333333
$$901$$ 49660.0 1.83620
$$902$$ 28960.0 1.06903
$$903$$ −1824.00 −0.0672192
$$904$$ −15312.0 −0.563351
$$905$$ −47860.0 −1.75792
$$906$$ 10656.0 0.390753
$$907$$ −14540.0 −0.532296 −0.266148 0.963932i $$-0.585751\pi$$
−0.266148 + 0.963932i $$0.585751\pi$$
$$908$$ −7712.00 −0.281863
$$909$$ 4662.00 0.170109
$$910$$ 2080.00 0.0757707
$$911$$ −7840.00 −0.285127 −0.142564 0.989786i $$-0.545535\pi$$
−0.142564 + 0.989786i $$0.545535\pi$$
$$912$$ −960.000 −0.0348561
$$913$$ −40320.0 −1.46155
$$914$$ −2532.00 −0.0916314
$$915$$ 10740.0 0.388037
$$916$$ −15752.0 −0.568189
$$917$$ 7136.00 0.256981
$$918$$ −7020.00 −0.252391
$$919$$ 47720.0 1.71288 0.856440 0.516246i $$-0.172671\pi$$
0.856440 + 0.516246i $$0.172671\pi$$
$$920$$ 0 0
$$921$$ 17868.0 0.639273
$$922$$ −15108.0 −0.539648
$$923$$ −9724.00 −0.346771
$$924$$ −3840.00 −0.136717
$$925$$ 1850.00 0.0657596
$$926$$ 13504.0 0.479232
$$927$$ 1008.00 0.0357142
$$928$$ 576.000 0.0203751
$$929$$ 7502.00 0.264944 0.132472 0.991187i $$-0.457709\pi$$
0.132472 + 0.991187i $$0.457709\pi$$
$$930$$ 11040.0 0.389264
$$931$$ 5580.00 0.196431
$$932$$ 10248.0 0.360176
$$933$$ 7056.00 0.247592
$$934$$ −15848.0 −0.555206
$$935$$ 52000.0 1.81880
$$936$$ −936.000 −0.0326860
$$937$$ 22058.0 0.769054 0.384527 0.923114i $$-0.374365\pi$$
0.384527 + 0.923114i $$0.374365\pi$$
$$938$$ −11200.0 −0.389865
$$939$$ 25326.0 0.880173
$$940$$ −18080.0 −0.627345
$$941$$ 23338.0 0.808498 0.404249 0.914649i $$-0.367533\pi$$
0.404249 + 0.914649i $$0.367533\pi$$
$$942$$ 14460.0 0.500140
$$943$$ 0 0
$$944$$ 7424.00 0.255965
$$945$$ −2160.00 −0.0743543
$$946$$ −6080.00 −0.208962
$$947$$ −30488.0 −1.04617 −0.523087 0.852279i $$-0.675220\pi$$
−0.523087 + 0.852279i $$0.675220\pi$$
$$948$$ −11712.0 −0.401253
$$949$$ 13754.0 0.470468
$$950$$ −1000.00 −0.0341519
$$951$$ −16650.0 −0.567732
$$952$$ 8320.00 0.283249
$$953$$ 9522.00 0.323660 0.161830 0.986819i $$-0.448260\pi$$
0.161830 + 0.986819i $$0.448260\pi$$
$$954$$ −6876.00 −0.233353
$$955$$ −13120.0 −0.444558
$$956$$ 28656.0 0.969457
$$957$$ −2160.00 −0.0729602
$$958$$ 22168.0 0.747615
$$959$$ −18608.0 −0.626573
$$960$$ 1920.00 0.0645497
$$961$$ 4065.00 0.136451
$$962$$ 1924.00 0.0644826
$$963$$ −3348.00 −0.112033
$$964$$ −24728.0 −0.826178
$$965$$ −3500.00 −0.116755
$$966$$ 0 0
$$967$$ −7616.00 −0.253272 −0.126636 0.991949i $$-0.540418\pi$$
−0.126636 + 0.991949i $$0.540418\pi$$
$$968$$ −2152.00 −0.0714544
$$969$$ −7800.00 −0.258588
$$970$$ 12280.0 0.406481
$$971$$ 51316.0 1.69599 0.847996 0.530002i $$-0.177809\pi$$
0.847996 + 0.530002i $$0.177809\pi$$
$$972$$ 972.000 0.0320750
$$973$$ −15456.0 −0.509246
$$974$$ −8864.00 −0.291603
$$975$$ −975.000 −0.0320256
$$976$$ 5728.00 0.187857
$$977$$ −48666.0 −1.59362 −0.796808 0.604232i $$-0.793480\pi$$
−0.796808 + 0.604232i $$0.793480\pi$$
$$978$$ −19272.0 −0.630113
$$979$$ −15440.0 −0.504050
$$980$$ −11160.0 −0.363768
$$981$$ 8406.00 0.273581
$$982$$ 2280.00 0.0740914
$$983$$ 17388.0 0.564182 0.282091 0.959388i $$-0.408972\pi$$
0.282091 + 0.959388i $$0.408972\pi$$
$$984$$ 8688.00 0.281467
$$985$$ −3420.00 −0.110630
$$986$$ 4680.00 0.151158
$$987$$ 10848.0 0.349844
$$988$$ −1040.00 −0.0334887
$$989$$ 0 0
$$990$$ −7200.00 −0.231142
$$991$$ 11496.0 0.368499 0.184249 0.982880i $$-0.441015\pi$$
0.184249 + 0.982880i $$0.441015\pi$$
$$992$$ 5888.00 0.188452
$$993$$ 420.000 0.0134223
$$994$$ −11968.0 −0.381893
$$995$$ −33680.0 −1.07309
$$996$$ −12096.0 −0.384816
$$997$$ 48862.0 1.55213 0.776066 0.630652i $$-0.217212\pi$$
0.776066 + 0.630652i $$0.217212\pi$$
$$998$$ −3528.00 −0.111901
$$999$$ −1998.00 −0.0632772
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 78.4.a.c.1.1 1
3.2 odd 2 234.4.a.h.1.1 1
4.3 odd 2 624.4.a.d.1.1 1
5.4 even 2 1950.4.a.l.1.1 1
8.3 odd 2 2496.4.a.j.1.1 1
8.5 even 2 2496.4.a.a.1.1 1
12.11 even 2 1872.4.a.d.1.1 1
13.5 odd 4 1014.4.b.h.337.2 2
13.8 odd 4 1014.4.b.h.337.1 2
13.12 even 2 1014.4.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.c.1.1 1 1.1 even 1 trivial
234.4.a.h.1.1 1 3.2 odd 2
624.4.a.d.1.1 1 4.3 odd 2
1014.4.a.j.1.1 1 13.12 even 2
1014.4.b.h.337.1 2 13.8 odd 4
1014.4.b.h.337.2 2 13.5 odd 4
1872.4.a.d.1.1 1 12.11 even 2
1950.4.a.l.1.1 1 5.4 even 2
2496.4.a.a.1.1 1 8.5 even 2
2496.4.a.j.1.1 1 8.3 odd 2