# Properties

 Label 75.4.a.a.1.1 Level $75$ Weight $4$ Character 75.1 Self dual yes Analytic conductor $4.425$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.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: $$4.42514325043$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 75.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} -24.0000 q^{11} +3.00000 q^{12} -74.0000 q^{13} +60.0000 q^{14} -71.0000 q^{16} -54.0000 q^{17} -27.0000 q^{18} -124.000 q^{19} -60.0000 q^{21} +72.0000 q^{22} +120.000 q^{23} +63.0000 q^{24} +222.000 q^{26} +27.0000 q^{27} -20.0000 q^{28} -78.0000 q^{29} +200.000 q^{31} +45.0000 q^{32} -72.0000 q^{33} +162.000 q^{34} +9.00000 q^{36} +70.0000 q^{37} +372.000 q^{38} -222.000 q^{39} +330.000 q^{41} +180.000 q^{42} -92.0000 q^{43} -24.0000 q^{44} -360.000 q^{46} +24.0000 q^{47} -213.000 q^{48} +57.0000 q^{49} -162.000 q^{51} -74.0000 q^{52} -450.000 q^{53} -81.0000 q^{54} -420.000 q^{56} -372.000 q^{57} +234.000 q^{58} +24.0000 q^{59} -322.000 q^{61} -600.000 q^{62} -180.000 q^{63} +433.000 q^{64} +216.000 q^{66} +196.000 q^{67} -54.0000 q^{68} +360.000 q^{69} -288.000 q^{71} +189.000 q^{72} +430.000 q^{73} -210.000 q^{74} -124.000 q^{76} +480.000 q^{77} +666.000 q^{78} -520.000 q^{79} +81.0000 q^{81} -990.000 q^{82} -156.000 q^{83} -60.0000 q^{84} +276.000 q^{86} -234.000 q^{87} -504.000 q^{88} +1026.00 q^{89} +1480.00 q^{91} +120.000 q^{92} +600.000 q^{93} -72.0000 q^{94} +135.000 q^{96} +286.000 q^{97} -171.000 q^{98} -216.000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.00000 −1.06066 −0.530330 0.847791i $$-0.677932\pi$$
−0.530330 + 0.847791i $$0.677932\pi$$
$$3$$ 3.00000 0.577350
$$4$$ 1.00000 0.125000
$$5$$ 0 0
$$6$$ −9.00000 −0.612372
$$7$$ −20.0000 −1.07990 −0.539949 0.841698i $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ 21.0000 0.928078
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 3.00000 0.0721688
$$13$$ −74.0000 −1.57876 −0.789381 0.613904i $$-0.789598\pi$$
−0.789381 + 0.613904i $$0.789598\pi$$
$$14$$ 60.0000 1.14541
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ −54.0000 −0.770407 −0.385204 0.922832i $$-0.625869\pi$$
−0.385204 + 0.922832i $$0.625869\pi$$
$$18$$ −27.0000 −0.353553
$$19$$ −124.000 −1.49724 −0.748620 0.663000i $$-0.769283\pi$$
−0.748620 + 0.663000i $$0.769283\pi$$
$$20$$ 0 0
$$21$$ −60.0000 −0.623480
$$22$$ 72.0000 0.697748
$$23$$ 120.000 1.08790 0.543951 0.839117i $$-0.316928\pi$$
0.543951 + 0.839117i $$0.316928\pi$$
$$24$$ 63.0000 0.535826
$$25$$ 0 0
$$26$$ 222.000 1.67453
$$27$$ 27.0000 0.192450
$$28$$ −20.0000 −0.134987
$$29$$ −78.0000 −0.499456 −0.249728 0.968316i $$-0.580341\pi$$
−0.249728 + 0.968316i $$0.580341\pi$$
$$30$$ 0 0
$$31$$ 200.000 1.15874 0.579372 0.815063i $$-0.303298\pi$$
0.579372 + 0.815063i $$0.303298\pi$$
$$32$$ 45.0000 0.248592
$$33$$ −72.0000 −0.379806
$$34$$ 162.000 0.817140
$$35$$ 0 0
$$36$$ 9.00000 0.0416667
$$37$$ 70.0000 0.311025 0.155513 0.987834i $$-0.450297\pi$$
0.155513 + 0.987834i $$0.450297\pi$$
$$38$$ 372.000 1.58806
$$39$$ −222.000 −0.911499
$$40$$ 0 0
$$41$$ 330.000 1.25701 0.628504 0.777806i $$-0.283668\pi$$
0.628504 + 0.777806i $$0.283668\pi$$
$$42$$ 180.000 0.661300
$$43$$ −92.0000 −0.326276 −0.163138 0.986603i $$-0.552162\pi$$
−0.163138 + 0.986603i $$0.552162\pi$$
$$44$$ −24.0000 −0.0822304
$$45$$ 0 0
$$46$$ −360.000 −1.15389
$$47$$ 24.0000 0.0744843 0.0372421 0.999306i $$-0.488143\pi$$
0.0372421 + 0.999306i $$0.488143\pi$$
$$48$$ −213.000 −0.640498
$$49$$ 57.0000 0.166181
$$50$$ 0 0
$$51$$ −162.000 −0.444795
$$52$$ −74.0000 −0.197345
$$53$$ −450.000 −1.16627 −0.583134 0.812376i $$-0.698174\pi$$
−0.583134 + 0.812376i $$0.698174\pi$$
$$54$$ −81.0000 −0.204124
$$55$$ 0 0
$$56$$ −420.000 −1.00223
$$57$$ −372.000 −0.864432
$$58$$ 234.000 0.529754
$$59$$ 24.0000 0.0529582 0.0264791 0.999649i $$-0.491570\pi$$
0.0264791 + 0.999649i $$0.491570\pi$$
$$60$$ 0 0
$$61$$ −322.000 −0.675867 −0.337933 0.941170i $$-0.609728\pi$$
−0.337933 + 0.941170i $$0.609728\pi$$
$$62$$ −600.000 −1.22903
$$63$$ −180.000 −0.359966
$$64$$ 433.000 0.845703
$$65$$ 0 0
$$66$$ 216.000 0.402845
$$67$$ 196.000 0.357391 0.178696 0.983904i $$-0.442812\pi$$
0.178696 + 0.983904i $$0.442812\pi$$
$$68$$ −54.0000 −0.0963009
$$69$$ 360.000 0.628100
$$70$$ 0 0
$$71$$ −288.000 −0.481399 −0.240699 0.970600i $$-0.577377\pi$$
−0.240699 + 0.970600i $$0.577377\pi$$
$$72$$ 189.000 0.309359
$$73$$ 430.000 0.689420 0.344710 0.938709i $$-0.387977\pi$$
0.344710 + 0.938709i $$0.387977\pi$$
$$74$$ −210.000 −0.329892
$$75$$ 0 0
$$76$$ −124.000 −0.187155
$$77$$ 480.000 0.710404
$$78$$ 666.000 0.966790
$$79$$ −520.000 −0.740564 −0.370282 0.928919i $$-0.620739\pi$$
−0.370282 + 0.928919i $$0.620739\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ −990.000 −1.33326
$$83$$ −156.000 −0.206304 −0.103152 0.994666i $$-0.532893\pi$$
−0.103152 + 0.994666i $$0.532893\pi$$
$$84$$ −60.0000 −0.0779350
$$85$$ 0 0
$$86$$ 276.000 0.346068
$$87$$ −234.000 −0.288361
$$88$$ −504.000 −0.610529
$$89$$ 1026.00 1.22198 0.610988 0.791640i $$-0.290773\pi$$
0.610988 + 0.791640i $$0.290773\pi$$
$$90$$ 0 0
$$91$$ 1480.00 1.70490
$$92$$ 120.000 0.135988
$$93$$ 600.000 0.669001
$$94$$ −72.0000 −0.0790025
$$95$$ 0 0
$$96$$ 135.000 0.143525
$$97$$ 286.000 0.299370 0.149685 0.988734i $$-0.452174\pi$$
0.149685 + 0.988734i $$0.452174\pi$$
$$98$$ −171.000 −0.176261
$$99$$ −216.000 −0.219281
$$100$$ 0 0
$$101$$ −1734.00 −1.70831 −0.854156 0.520017i $$-0.825925\pi$$
−0.854156 + 0.520017i $$0.825925\pi$$
$$102$$ 486.000 0.471776
$$103$$ −452.000 −0.432397 −0.216198 0.976349i $$-0.569366\pi$$
−0.216198 + 0.976349i $$0.569366\pi$$
$$104$$ −1554.00 −1.46521
$$105$$ 0 0
$$106$$ 1350.00 1.23702
$$107$$ 1404.00 1.26850 0.634251 0.773127i $$-0.281308\pi$$
0.634251 + 0.773127i $$0.281308\pi$$
$$108$$ 27.0000 0.0240563
$$109$$ −1474.00 −1.29526 −0.647631 0.761954i $$-0.724240\pi$$
−0.647631 + 0.761954i $$0.724240\pi$$
$$110$$ 0 0
$$111$$ 210.000 0.179570
$$112$$ 1420.00 1.19801
$$113$$ −1086.00 −0.904091 −0.452046 0.891995i $$-0.649306\pi$$
−0.452046 + 0.891995i $$0.649306\pi$$
$$114$$ 1116.00 0.916868
$$115$$ 0 0
$$116$$ −78.0000 −0.0624321
$$117$$ −666.000 −0.526254
$$118$$ −72.0000 −0.0561707
$$119$$ 1080.00 0.831962
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 966.000 0.716865
$$123$$ 990.000 0.725734
$$124$$ 200.000 0.144843
$$125$$ 0 0
$$126$$ 540.000 0.381802
$$127$$ −1244.00 −0.869190 −0.434595 0.900626i $$-0.643109\pi$$
−0.434595 + 0.900626i $$0.643109\pi$$
$$128$$ −1659.00 −1.14560
$$129$$ −276.000 −0.188376
$$130$$ 0 0
$$131$$ 2328.00 1.55266 0.776329 0.630327i $$-0.217079\pi$$
0.776329 + 0.630327i $$0.217079\pi$$
$$132$$ −72.0000 −0.0474757
$$133$$ 2480.00 1.61687
$$134$$ −588.000 −0.379071
$$135$$ 0 0
$$136$$ −1134.00 −0.714998
$$137$$ −2118.00 −1.32082 −0.660412 0.750903i $$-0.729618\pi$$
−0.660412 + 0.750903i $$0.729618\pi$$
$$138$$ −1080.00 −0.666201
$$139$$ 2324.00 1.41812 0.709062 0.705147i $$-0.249119\pi$$
0.709062 + 0.705147i $$0.249119\pi$$
$$140$$ 0 0
$$141$$ 72.0000 0.0430035
$$142$$ 864.000 0.510600
$$143$$ 1776.00 1.03858
$$144$$ −639.000 −0.369792
$$145$$ 0 0
$$146$$ −1290.00 −0.731241
$$147$$ 171.000 0.0959445
$$148$$ 70.0000 0.0388781
$$149$$ 258.000 0.141854 0.0709268 0.997482i $$-0.477404\pi$$
0.0709268 + 0.997482i $$0.477404\pi$$
$$150$$ 0 0
$$151$$ −808.000 −0.435458 −0.217729 0.976009i $$-0.569865\pi$$
−0.217729 + 0.976009i $$0.569865\pi$$
$$152$$ −2604.00 −1.38955
$$153$$ −486.000 −0.256802
$$154$$ −1440.00 −0.753497
$$155$$ 0 0
$$156$$ −222.000 −0.113937
$$157$$ −2378.00 −1.20882 −0.604411 0.796673i $$-0.706592\pi$$
−0.604411 + 0.796673i $$0.706592\pi$$
$$158$$ 1560.00 0.785487
$$159$$ −1350.00 −0.673346
$$160$$ 0 0
$$161$$ −2400.00 −1.17482
$$162$$ −243.000 −0.117851
$$163$$ 52.0000 0.0249874 0.0124937 0.999922i $$-0.496023\pi$$
0.0124937 + 0.999922i $$0.496023\pi$$
$$164$$ 330.000 0.157126
$$165$$ 0 0
$$166$$ 468.000 0.218818
$$167$$ 3720.00 1.72373 0.861863 0.507141i $$-0.169298\pi$$
0.861863 + 0.507141i $$0.169298\pi$$
$$168$$ −1260.00 −0.578638
$$169$$ 3279.00 1.49249
$$170$$ 0 0
$$171$$ −1116.00 −0.499080
$$172$$ −92.0000 −0.0407845
$$173$$ −426.000 −0.187215 −0.0936075 0.995609i $$-0.529840\pi$$
−0.0936075 + 0.995609i $$0.529840\pi$$
$$174$$ 702.000 0.305853
$$175$$ 0 0
$$176$$ 1704.00 0.729795
$$177$$ 72.0000 0.0305754
$$178$$ −3078.00 −1.29610
$$179$$ −1440.00 −0.601289 −0.300644 0.953736i $$-0.597202\pi$$
−0.300644 + 0.953736i $$0.597202\pi$$
$$180$$ 0 0
$$181$$ −3130.00 −1.28537 −0.642683 0.766133i $$-0.722179\pi$$
−0.642683 + 0.766133i $$0.722179\pi$$
$$182$$ −4440.00 −1.80832
$$183$$ −966.000 −0.390212
$$184$$ 2520.00 1.00966
$$185$$ 0 0
$$186$$ −1800.00 −0.709583
$$187$$ 1296.00 0.506807
$$188$$ 24.0000 0.00931053
$$189$$ −540.000 −0.207827
$$190$$ 0 0
$$191$$ 3576.00 1.35471 0.677357 0.735655i $$-0.263125\pi$$
0.677357 + 0.735655i $$0.263125\pi$$
$$192$$ 1299.00 0.488267
$$193$$ −2666.00 −0.994315 −0.497158 0.867660i $$-0.665623\pi$$
−0.497158 + 0.867660i $$0.665623\pi$$
$$194$$ −858.000 −0.317530
$$195$$ 0 0
$$196$$ 57.0000 0.0207726
$$197$$ 2718.00 0.982992 0.491496 0.870880i $$-0.336450\pi$$
0.491496 + 0.870880i $$0.336450\pi$$
$$198$$ 648.000 0.232583
$$199$$ −3832.00 −1.36504 −0.682521 0.730866i $$-0.739116\pi$$
−0.682521 + 0.730866i $$0.739116\pi$$
$$200$$ 0 0
$$201$$ 588.000 0.206340
$$202$$ 5202.00 1.81194
$$203$$ 1560.00 0.539362
$$204$$ −162.000 −0.0555994
$$205$$ 0 0
$$206$$ 1356.00 0.458626
$$207$$ 1080.00 0.362634
$$208$$ 5254.00 1.75144
$$209$$ 2976.00 0.984948
$$210$$ 0 0
$$211$$ 1100.00 0.358896 0.179448 0.983767i $$-0.442569\pi$$
0.179448 + 0.983767i $$0.442569\pi$$
$$212$$ −450.000 −0.145784
$$213$$ −864.000 −0.277936
$$214$$ −4212.00 −1.34545
$$215$$ 0 0
$$216$$ 567.000 0.178609
$$217$$ −4000.00 −1.25133
$$218$$ 4422.00 1.37383
$$219$$ 1290.00 0.398037
$$220$$ 0 0
$$221$$ 3996.00 1.21629
$$222$$ −630.000 −0.190463
$$223$$ −1964.00 −0.589772 −0.294886 0.955532i $$-0.595282\pi$$
−0.294886 + 0.955532i $$0.595282\pi$$
$$224$$ −900.000 −0.268454
$$225$$ 0 0
$$226$$ 3258.00 0.958933
$$227$$ −660.000 −0.192977 −0.0964884 0.995334i $$-0.530761\pi$$
−0.0964884 + 0.995334i $$0.530761\pi$$
$$228$$ −372.000 −0.108054
$$229$$ −1906.00 −0.550009 −0.275004 0.961443i $$-0.588679\pi$$
−0.275004 + 0.961443i $$0.588679\pi$$
$$230$$ 0 0
$$231$$ 1440.00 0.410152
$$232$$ −1638.00 −0.463534
$$233$$ 1458.00 0.409943 0.204972 0.978768i $$-0.434290\pi$$
0.204972 + 0.978768i $$0.434290\pi$$
$$234$$ 1998.00 0.558177
$$235$$ 0 0
$$236$$ 24.0000 0.00661978
$$237$$ −1560.00 −0.427565
$$238$$ −3240.00 −0.882429
$$239$$ 1176.00 0.318281 0.159140 0.987256i $$-0.449128\pi$$
0.159140 + 0.987256i $$0.449128\pi$$
$$240$$ 0 0
$$241$$ 866.000 0.231469 0.115734 0.993280i $$-0.463078\pi$$
0.115734 + 0.993280i $$0.463078\pi$$
$$242$$ 2265.00 0.601652
$$243$$ 243.000 0.0641500
$$244$$ −322.000 −0.0844834
$$245$$ 0 0
$$246$$ −2970.00 −0.769757
$$247$$ 9176.00 2.36379
$$248$$ 4200.00 1.07540
$$249$$ −468.000 −0.119110
$$250$$ 0 0
$$251$$ 432.000 0.108636 0.0543179 0.998524i $$-0.482702\pi$$
0.0543179 + 0.998524i $$0.482702\pi$$
$$252$$ −180.000 −0.0449958
$$253$$ −2880.00 −0.715668
$$254$$ 3732.00 0.921915
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ −2526.00 −0.613103 −0.306552 0.951854i $$-0.599175\pi$$
−0.306552 + 0.951854i $$0.599175\pi$$
$$258$$ 828.000 0.199802
$$259$$ −1400.00 −0.335876
$$260$$ 0 0
$$261$$ −702.000 −0.166485
$$262$$ −6984.00 −1.64684
$$263$$ −5448.00 −1.27733 −0.638666 0.769484i $$-0.720513\pi$$
−0.638666 + 0.769484i $$0.720513\pi$$
$$264$$ −1512.00 −0.352489
$$265$$ 0 0
$$266$$ −7440.00 −1.71495
$$267$$ 3078.00 0.705508
$$268$$ 196.000 0.0446739
$$269$$ −2574.00 −0.583418 −0.291709 0.956507i $$-0.594224\pi$$
−0.291709 + 0.956507i $$0.594224\pi$$
$$270$$ 0 0
$$271$$ −3184.00 −0.713706 −0.356853 0.934161i $$-0.616150\pi$$
−0.356853 + 0.934161i $$0.616150\pi$$
$$272$$ 3834.00 0.854671
$$273$$ 4440.00 0.984326
$$274$$ 6354.00 1.40095
$$275$$ 0 0
$$276$$ 360.000 0.0785125
$$277$$ −3962.00 −0.859399 −0.429699 0.902972i $$-0.641380\pi$$
−0.429699 + 0.902972i $$0.641380\pi$$
$$278$$ −6972.00 −1.50415
$$279$$ 1800.00 0.386248
$$280$$ 0 0
$$281$$ −8286.00 −1.75908 −0.879540 0.475825i $$-0.842149\pi$$
−0.879540 + 0.475825i $$0.842149\pi$$
$$282$$ −216.000 −0.0456121
$$283$$ 2716.00 0.570493 0.285246 0.958454i $$-0.407925\pi$$
0.285246 + 0.958454i $$0.407925\pi$$
$$284$$ −288.000 −0.0601748
$$285$$ 0 0
$$286$$ −5328.00 −1.10158
$$287$$ −6600.00 −1.35744
$$288$$ 405.000 0.0828641
$$289$$ −1997.00 −0.406473
$$290$$ 0 0
$$291$$ 858.000 0.172841
$$292$$ 430.000 0.0861776
$$293$$ −6018.00 −1.19992 −0.599958 0.800032i $$-0.704816\pi$$
−0.599958 + 0.800032i $$0.704816\pi$$
$$294$$ −513.000 −0.101765
$$295$$ 0 0
$$296$$ 1470.00 0.288655
$$297$$ −648.000 −0.126602
$$298$$ −774.000 −0.150458
$$299$$ −8880.00 −1.71754
$$300$$ 0 0
$$301$$ 1840.00 0.352345
$$302$$ 2424.00 0.461873
$$303$$ −5202.00 −0.986294
$$304$$ 8804.00 1.66100
$$305$$ 0 0
$$306$$ 1458.00 0.272380
$$307$$ −9236.00 −1.71702 −0.858512 0.512793i $$-0.828611\pi$$
−0.858512 + 0.512793i $$0.828611\pi$$
$$308$$ 480.000 0.0888004
$$309$$ −1356.00 −0.249644
$$310$$ 0 0
$$311$$ 1536.00 0.280060 0.140030 0.990147i $$-0.455280\pi$$
0.140030 + 0.990147i $$0.455280\pi$$
$$312$$ −4662.00 −0.845942
$$313$$ 7342.00 1.32586 0.662930 0.748681i $$-0.269313\pi$$
0.662930 + 0.748681i $$0.269313\pi$$
$$314$$ 7134.00 1.28215
$$315$$ 0 0
$$316$$ −520.000 −0.0925705
$$317$$ 3894.00 0.689933 0.344967 0.938615i $$-0.387890\pi$$
0.344967 + 0.938615i $$0.387890\pi$$
$$318$$ 4050.00 0.714191
$$319$$ 1872.00 0.328564
$$320$$ 0 0
$$321$$ 4212.00 0.732370
$$322$$ 7200.00 1.24609
$$323$$ 6696.00 1.15348
$$324$$ 81.0000 0.0138889
$$325$$ 0 0
$$326$$ −156.000 −0.0265032
$$327$$ −4422.00 −0.747820
$$328$$ 6930.00 1.16660
$$329$$ −480.000 −0.0804354
$$330$$ 0 0
$$331$$ 3692.00 0.613084 0.306542 0.951857i $$-0.400828\pi$$
0.306542 + 0.951857i $$0.400828\pi$$
$$332$$ −156.000 −0.0257880
$$333$$ 630.000 0.103675
$$334$$ −11160.0 −1.82829
$$335$$ 0 0
$$336$$ 4260.00 0.691673
$$337$$ 8998.00 1.45446 0.727229 0.686395i $$-0.240808\pi$$
0.727229 + 0.686395i $$0.240808\pi$$
$$338$$ −9837.00 −1.58302
$$339$$ −3258.00 −0.521977
$$340$$ 0 0
$$341$$ −4800.00 −0.762271
$$342$$ 3348.00 0.529354
$$343$$ 5720.00 0.900440
$$344$$ −1932.00 −0.302809
$$345$$ 0 0
$$346$$ 1278.00 0.198571
$$347$$ −5244.00 −0.811276 −0.405638 0.914034i $$-0.632951\pi$$
−0.405638 + 0.914034i $$0.632951\pi$$
$$348$$ −234.000 −0.0360452
$$349$$ 6302.00 0.966585 0.483293 0.875459i $$-0.339441\pi$$
0.483293 + 0.875459i $$0.339441\pi$$
$$350$$ 0 0
$$351$$ −1998.00 −0.303833
$$352$$ −1080.00 −0.163535
$$353$$ −3414.00 −0.514756 −0.257378 0.966311i $$-0.582859\pi$$
−0.257378 + 0.966311i $$0.582859\pi$$
$$354$$ −216.000 −0.0324301
$$355$$ 0 0
$$356$$ 1026.00 0.152747
$$357$$ 3240.00 0.480333
$$358$$ 4320.00 0.637763
$$359$$ 4824.00 0.709195 0.354597 0.935019i $$-0.384618\pi$$
0.354597 + 0.935019i $$0.384618\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ 9390.00 1.36334
$$363$$ −2265.00 −0.327498
$$364$$ 1480.00 0.213113
$$365$$ 0 0
$$366$$ 2898.00 0.413882
$$367$$ 3508.00 0.498954 0.249477 0.968381i $$-0.419741\pi$$
0.249477 + 0.968381i $$0.419741\pi$$
$$368$$ −8520.00 −1.20689
$$369$$ 2970.00 0.419003
$$370$$ 0 0
$$371$$ 9000.00 1.25945
$$372$$ 600.000 0.0836251
$$373$$ −10802.0 −1.49948 −0.749740 0.661732i $$-0.769822\pi$$
−0.749740 + 0.661732i $$0.769822\pi$$
$$374$$ −3888.00 −0.537550
$$375$$ 0 0
$$376$$ 504.000 0.0691272
$$377$$ 5772.00 0.788523
$$378$$ 1620.00 0.220433
$$379$$ 1460.00 0.197876 0.0989382 0.995094i $$-0.468455\pi$$
0.0989382 + 0.995094i $$0.468455\pi$$
$$380$$ 0 0
$$381$$ −3732.00 −0.501827
$$382$$ −10728.0 −1.43689
$$383$$ 4872.00 0.649994 0.324997 0.945715i $$-0.394637\pi$$
0.324997 + 0.945715i $$0.394637\pi$$
$$384$$ −4977.00 −0.661410
$$385$$ 0 0
$$386$$ 7998.00 1.05463
$$387$$ −828.000 −0.108759
$$388$$ 286.000 0.0374213
$$389$$ −14046.0 −1.83075 −0.915373 0.402606i $$-0.868104\pi$$
−0.915373 + 0.402606i $$0.868104\pi$$
$$390$$ 0 0
$$391$$ −6480.00 −0.838127
$$392$$ 1197.00 0.154229
$$393$$ 6984.00 0.896428
$$394$$ −8154.00 −1.04262
$$395$$ 0 0
$$396$$ −216.000 −0.0274101
$$397$$ 2734.00 0.345631 0.172816 0.984954i $$-0.444714\pi$$
0.172816 + 0.984954i $$0.444714\pi$$
$$398$$ 11496.0 1.44785
$$399$$ 7440.00 0.933498
$$400$$ 0 0
$$401$$ −15942.0 −1.98530 −0.992650 0.121019i $$-0.961384\pi$$
−0.992650 + 0.121019i $$0.961384\pi$$
$$402$$ −1764.00 −0.218857
$$403$$ −14800.0 −1.82938
$$404$$ −1734.00 −0.213539
$$405$$ 0 0
$$406$$ −4680.00 −0.572080
$$407$$ −1680.00 −0.204606
$$408$$ −3402.00 −0.412804
$$409$$ 8714.00 1.05350 0.526748 0.850022i $$-0.323411\pi$$
0.526748 + 0.850022i $$0.323411\pi$$
$$410$$ 0 0
$$411$$ −6354.00 −0.762578
$$412$$ −452.000 −0.0540496
$$413$$ −480.000 −0.0571895
$$414$$ −3240.00 −0.384631
$$415$$ 0 0
$$416$$ −3330.00 −0.392468
$$417$$ 6972.00 0.818754
$$418$$ −8928.00 −1.04470
$$419$$ 11976.0 1.39634 0.698169 0.715933i $$-0.253998\pi$$
0.698169 + 0.715933i $$0.253998\pi$$
$$420$$ 0 0
$$421$$ 11054.0 1.27967 0.639833 0.768514i $$-0.279004\pi$$
0.639833 + 0.768514i $$0.279004\pi$$
$$422$$ −3300.00 −0.380667
$$423$$ 216.000 0.0248281
$$424$$ −9450.00 −1.08239
$$425$$ 0 0
$$426$$ 2592.00 0.294795
$$427$$ 6440.00 0.729868
$$428$$ 1404.00 0.158563
$$429$$ 5328.00 0.599623
$$430$$ 0 0
$$431$$ 720.000 0.0804668 0.0402334 0.999190i $$-0.487190\pi$$
0.0402334 + 0.999190i $$0.487190\pi$$
$$432$$ −1917.00 −0.213499
$$433$$ 15622.0 1.73382 0.866912 0.498462i $$-0.166102\pi$$
0.866912 + 0.498462i $$0.166102\pi$$
$$434$$ 12000.0 1.32723
$$435$$ 0 0
$$436$$ −1474.00 −0.161908
$$437$$ −14880.0 −1.62885
$$438$$ −3870.00 −0.422182
$$439$$ −9880.00 −1.07414 −0.537069 0.843538i $$-0.680469\pi$$
−0.537069 + 0.843538i $$0.680469\pi$$
$$440$$ 0 0
$$441$$ 513.000 0.0553936
$$442$$ −11988.0 −1.29007
$$443$$ 16116.0 1.72843 0.864215 0.503123i $$-0.167816\pi$$
0.864215 + 0.503123i $$0.167816\pi$$
$$444$$ 210.000 0.0224463
$$445$$ 0 0
$$446$$ 5892.00 0.625548
$$447$$ 774.000 0.0818992
$$448$$ −8660.00 −0.913274
$$449$$ 9018.00 0.947852 0.473926 0.880565i $$-0.342836\pi$$
0.473926 + 0.880565i $$0.342836\pi$$
$$450$$ 0 0
$$451$$ −7920.00 −0.826914
$$452$$ −1086.00 −0.113011
$$453$$ −2424.00 −0.251412
$$454$$ 1980.00 0.204683
$$455$$ 0 0
$$456$$ −7812.00 −0.802260
$$457$$ 3670.00 0.375657 0.187829 0.982202i $$-0.439855\pi$$
0.187829 + 0.982202i $$0.439855\pi$$
$$458$$ 5718.00 0.583372
$$459$$ −1458.00 −0.148265
$$460$$ 0 0
$$461$$ 17562.0 1.77428 0.887141 0.461499i $$-0.152688\pi$$
0.887141 + 0.461499i $$0.152688\pi$$
$$462$$ −4320.00 −0.435032
$$463$$ −1172.00 −0.117640 −0.0588202 0.998269i $$-0.518734\pi$$
−0.0588202 + 0.998269i $$0.518734\pi$$
$$464$$ 5538.00 0.554084
$$465$$ 0 0
$$466$$ −4374.00 −0.434810
$$467$$ −6876.00 −0.681335 −0.340667 0.940184i $$-0.610653\pi$$
−0.340667 + 0.940184i $$0.610653\pi$$
$$468$$ −666.000 −0.0657818
$$469$$ −3920.00 −0.385946
$$470$$ 0 0
$$471$$ −7134.00 −0.697914
$$472$$ 504.000 0.0491493
$$473$$ 2208.00 0.214638
$$474$$ 4680.00 0.453501
$$475$$ 0 0
$$476$$ 1080.00 0.103995
$$477$$ −4050.00 −0.388756
$$478$$ −3528.00 −0.337588
$$479$$ 2280.00 0.217486 0.108743 0.994070i $$-0.465317\pi$$
0.108743 + 0.994070i $$0.465317\pi$$
$$480$$ 0 0
$$481$$ −5180.00 −0.491035
$$482$$ −2598.00 −0.245510
$$483$$ −7200.00 −0.678284
$$484$$ −755.000 −0.0709053
$$485$$ 0 0
$$486$$ −729.000 −0.0680414
$$487$$ 3076.00 0.286215 0.143108 0.989707i $$-0.454290\pi$$
0.143108 + 0.989707i $$0.454290\pi$$
$$488$$ −6762.00 −0.627257
$$489$$ 156.000 0.0144265
$$490$$ 0 0
$$491$$ −18912.0 −1.73826 −0.869131 0.494582i $$-0.835321\pi$$
−0.869131 + 0.494582i $$0.835321\pi$$
$$492$$ 990.000 0.0907168
$$493$$ 4212.00 0.384785
$$494$$ −27528.0 −2.50717
$$495$$ 0 0
$$496$$ −14200.0 −1.28548
$$497$$ 5760.00 0.519862
$$498$$ 1404.00 0.126335
$$499$$ 9956.00 0.893170 0.446585 0.894741i $$-0.352640\pi$$
0.446585 + 0.894741i $$0.352640\pi$$
$$500$$ 0 0
$$501$$ 11160.0 0.995194
$$502$$ −1296.00 −0.115226
$$503$$ 10656.0 0.944588 0.472294 0.881441i $$-0.343426\pi$$
0.472294 + 0.881441i $$0.343426\pi$$
$$504$$ −3780.00 −0.334077
$$505$$ 0 0
$$506$$ 8640.00 0.759081
$$507$$ 9837.00 0.861689
$$508$$ −1244.00 −0.108649
$$509$$ −2766.00 −0.240866 −0.120433 0.992721i $$-0.538428\pi$$
−0.120433 + 0.992721i $$0.538428\pi$$
$$510$$ 0 0
$$511$$ −8600.00 −0.744504
$$512$$ 8733.00 0.753804
$$513$$ −3348.00 −0.288144
$$514$$ 7578.00 0.650294
$$515$$ 0 0
$$516$$ −276.000 −0.0235469
$$517$$ −576.000 −0.0489989
$$518$$ 4200.00 0.356250
$$519$$ −1278.00 −0.108089
$$520$$ 0 0
$$521$$ 10530.0 0.885466 0.442733 0.896654i $$-0.354009\pi$$
0.442733 + 0.896654i $$0.354009\pi$$
$$522$$ 2106.00 0.176585
$$523$$ −12692.0 −1.06115 −0.530576 0.847637i $$-0.678024\pi$$
−0.530576 + 0.847637i $$0.678024\pi$$
$$524$$ 2328.00 0.194082
$$525$$ 0 0
$$526$$ 16344.0 1.35481
$$527$$ −10800.0 −0.892705
$$528$$ 5112.00 0.421347
$$529$$ 2233.00 0.183529
$$530$$ 0 0
$$531$$ 216.000 0.0176527
$$532$$ 2480.00 0.202108
$$533$$ −24420.0 −1.98452
$$534$$ −9234.00 −0.748304
$$535$$ 0 0
$$536$$ 4116.00 0.331687
$$537$$ −4320.00 −0.347154
$$538$$ 7722.00 0.618809
$$539$$ −1368.00 −0.109321
$$540$$ 0 0
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ 9552.00 0.756999
$$543$$ −9390.00 −0.742106
$$544$$ −2430.00 −0.191517
$$545$$ 0 0
$$546$$ −13320.0 −1.04404
$$547$$ −3620.00 −0.282962 −0.141481 0.989941i $$-0.545186\pi$$
−0.141481 + 0.989941i $$0.545186\pi$$
$$548$$ −2118.00 −0.165103
$$549$$ −2898.00 −0.225289
$$550$$ 0 0
$$551$$ 9672.00 0.747806
$$552$$ 7560.00 0.582926
$$553$$ 10400.0 0.799734
$$554$$ 11886.0 0.911530
$$555$$ 0 0
$$556$$ 2324.00 0.177265
$$557$$ 14166.0 1.07762 0.538809 0.842428i $$-0.318875\pi$$
0.538809 + 0.842428i $$0.318875\pi$$
$$558$$ −5400.00 −0.409678
$$559$$ 6808.00 0.515112
$$560$$ 0 0
$$561$$ 3888.00 0.292605
$$562$$ 24858.0 1.86579
$$563$$ 13404.0 1.00339 0.501697 0.865043i $$-0.332709\pi$$
0.501697 + 0.865043i $$0.332709\pi$$
$$564$$ 72.0000 0.00537544
$$565$$ 0 0
$$566$$ −8148.00 −0.605099
$$567$$ −1620.00 −0.119989
$$568$$ −6048.00 −0.446775
$$569$$ −18654.0 −1.37437 −0.687185 0.726483i $$-0.741154\pi$$
−0.687185 + 0.726483i $$0.741154\pi$$
$$570$$ 0 0
$$571$$ −7684.00 −0.563162 −0.281581 0.959537i $$-0.590859\pi$$
−0.281581 + 0.959537i $$0.590859\pi$$
$$572$$ 1776.00 0.129822
$$573$$ 10728.0 0.782144
$$574$$ 19800.0 1.43978
$$575$$ 0 0
$$576$$ 3897.00 0.281901
$$577$$ 1726.00 0.124531 0.0622654 0.998060i $$-0.480167\pi$$
0.0622654 + 0.998060i $$0.480167\pi$$
$$578$$ 5991.00 0.431129
$$579$$ −7998.00 −0.574068
$$580$$ 0 0
$$581$$ 3120.00 0.222787
$$582$$ −2574.00 −0.183326
$$583$$ 10800.0 0.767222
$$584$$ 9030.00 0.639836
$$585$$ 0 0
$$586$$ 18054.0 1.27270
$$587$$ −10596.0 −0.745049 −0.372524 0.928022i $$-0.621508\pi$$
−0.372524 + 0.928022i $$0.621508\pi$$
$$588$$ 171.000 0.0119931
$$589$$ −24800.0 −1.73492
$$590$$ 0 0
$$591$$ 8154.00 0.567531
$$592$$ −4970.00 −0.345043
$$593$$ −2862.00 −0.198193 −0.0990963 0.995078i $$-0.531595\pi$$
−0.0990963 + 0.995078i $$0.531595\pi$$
$$594$$ 1944.00 0.134282
$$595$$ 0 0
$$596$$ 258.000 0.0177317
$$597$$ −11496.0 −0.788107
$$598$$ 26640.0 1.82172
$$599$$ −23592.0 −1.60925 −0.804627 0.593781i $$-0.797635\pi$$
−0.804627 + 0.593781i $$0.797635\pi$$
$$600$$ 0 0
$$601$$ −9574.00 −0.649803 −0.324902 0.945748i $$-0.605331\pi$$
−0.324902 + 0.945748i $$0.605331\pi$$
$$602$$ −5520.00 −0.373718
$$603$$ 1764.00 0.119130
$$604$$ −808.000 −0.0544322
$$605$$ 0 0
$$606$$ 15606.0 1.04612
$$607$$ −17444.0 −1.16644 −0.583221 0.812314i $$-0.698208\pi$$
−0.583221 + 0.812314i $$0.698208\pi$$
$$608$$ −5580.00 −0.372202
$$609$$ 4680.00 0.311401
$$610$$ 0 0
$$611$$ −1776.00 −0.117593
$$612$$ −486.000 −0.0321003
$$613$$ 2374.00 0.156419 0.0782096 0.996937i $$-0.475080\pi$$
0.0782096 + 0.996937i $$0.475080\pi$$
$$614$$ 27708.0 1.82118
$$615$$ 0 0
$$616$$ 10080.0 0.659310
$$617$$ 12162.0 0.793555 0.396778 0.917915i $$-0.370128\pi$$
0.396778 + 0.917915i $$0.370128\pi$$
$$618$$ 4068.00 0.264788
$$619$$ 8804.00 0.571668 0.285834 0.958279i $$-0.407729\pi$$
0.285834 + 0.958279i $$0.407729\pi$$
$$620$$ 0 0
$$621$$ 3240.00 0.209367
$$622$$ −4608.00 −0.297048
$$623$$ −20520.0 −1.31961
$$624$$ 15762.0 1.01119
$$625$$ 0 0
$$626$$ −22026.0 −1.40629
$$627$$ 8928.00 0.568660
$$628$$ −2378.00 −0.151103
$$629$$ −3780.00 −0.239616
$$630$$ 0 0
$$631$$ −12688.0 −0.800478 −0.400239 0.916411i $$-0.631073\pi$$
−0.400239 + 0.916411i $$0.631073\pi$$
$$632$$ −10920.0 −0.687301
$$633$$ 3300.00 0.207209
$$634$$ −11682.0 −0.731785
$$635$$ 0 0
$$636$$ −1350.00 −0.0841682
$$637$$ −4218.00 −0.262360
$$638$$ −5616.00 −0.348495
$$639$$ −2592.00 −0.160466
$$640$$ 0 0
$$641$$ −9150.00 −0.563812 −0.281906 0.959442i $$-0.590967\pi$$
−0.281906 + 0.959442i $$0.590967\pi$$
$$642$$ −12636.0 −0.776796
$$643$$ −25292.0 −1.55120 −0.775598 0.631227i $$-0.782552\pi$$
−0.775598 + 0.631227i $$0.782552\pi$$
$$644$$ −2400.00 −0.146853
$$645$$ 0 0
$$646$$ −20088.0 −1.22345
$$647$$ 2736.00 0.166249 0.0831246 0.996539i $$-0.473510\pi$$
0.0831246 + 0.996539i $$0.473510\pi$$
$$648$$ 1701.00 0.103120
$$649$$ −576.000 −0.0348382
$$650$$ 0 0
$$651$$ −12000.0 −0.722453
$$652$$ 52.0000 0.00312343
$$653$$ −22218.0 −1.33148 −0.665741 0.746183i $$-0.731884\pi$$
−0.665741 + 0.746183i $$0.731884\pi$$
$$654$$ 13266.0 0.793183
$$655$$ 0 0
$$656$$ −23430.0 −1.39449
$$657$$ 3870.00 0.229807
$$658$$ 1440.00 0.0853147
$$659$$ 14520.0 0.858299 0.429149 0.903234i $$-0.358813\pi$$
0.429149 + 0.903234i $$0.358813\pi$$
$$660$$ 0 0
$$661$$ −10618.0 −0.624799 −0.312400 0.949951i $$-0.601133\pi$$
−0.312400 + 0.949951i $$0.601133\pi$$
$$662$$ −11076.0 −0.650273
$$663$$ 11988.0 0.702225
$$664$$ −3276.00 −0.191466
$$665$$ 0 0
$$666$$ −1890.00 −0.109964
$$667$$ −9360.00 −0.543359
$$668$$ 3720.00 0.215466
$$669$$ −5892.00 −0.340505
$$670$$ 0 0
$$671$$ 7728.00 0.444614
$$672$$ −2700.00 −0.154992
$$673$$ −1370.00 −0.0784690 −0.0392345 0.999230i $$-0.512492\pi$$
−0.0392345 + 0.999230i $$0.512492\pi$$
$$674$$ −26994.0 −1.54269
$$675$$ 0 0
$$676$$ 3279.00 0.186561
$$677$$ 13758.0 0.781038 0.390519 0.920595i $$-0.372296\pi$$
0.390519 + 0.920595i $$0.372296\pi$$
$$678$$ 9774.00 0.553640
$$679$$ −5720.00 −0.323289
$$680$$ 0 0
$$681$$ −1980.00 −0.111415
$$682$$ 14400.0 0.808511
$$683$$ −11988.0 −0.671608 −0.335804 0.941932i $$-0.609008\pi$$
−0.335804 + 0.941932i $$0.609008\pi$$
$$684$$ −1116.00 −0.0623850
$$685$$ 0 0
$$686$$ −17160.0 −0.955061
$$687$$ −5718.00 −0.317548
$$688$$ 6532.00 0.361962
$$689$$ 33300.0 1.84126
$$690$$ 0 0
$$691$$ 32996.0 1.81654 0.908268 0.418388i $$-0.137405\pi$$
0.908268 + 0.418388i $$0.137405\pi$$
$$692$$ −426.000 −0.0234019
$$693$$ 4320.00 0.236801
$$694$$ 15732.0 0.860488
$$695$$ 0 0
$$696$$ −4914.00 −0.267622
$$697$$ −17820.0 −0.968408
$$698$$ −18906.0 −1.02522
$$699$$ 4374.00 0.236681
$$700$$ 0 0
$$701$$ −25902.0 −1.39558 −0.697792 0.716300i $$-0.745834\pi$$
−0.697792 + 0.716300i $$0.745834\pi$$
$$702$$ 5994.00 0.322263
$$703$$ −8680.00 −0.465679
$$704$$ −10392.0 −0.556340
$$705$$ 0 0
$$706$$ 10242.0 0.545981
$$707$$ 34680.0 1.84480
$$708$$ 72.0000 0.00382193
$$709$$ −27394.0 −1.45106 −0.725531 0.688189i $$-0.758406\pi$$
−0.725531 + 0.688189i $$0.758406\pi$$
$$710$$ 0 0
$$711$$ −4680.00 −0.246855
$$712$$ 21546.0 1.13409
$$713$$ 24000.0 1.26060
$$714$$ −9720.00 −0.509470
$$715$$ 0 0
$$716$$ −1440.00 −0.0751611
$$717$$ 3528.00 0.183760
$$718$$ −14472.0 −0.752215
$$719$$ 34848.0 1.80753 0.903763 0.428033i $$-0.140793\pi$$
0.903763 + 0.428033i $$0.140793\pi$$
$$720$$ 0 0
$$721$$ 9040.00 0.466945
$$722$$ −25551.0 −1.31705
$$723$$ 2598.00 0.133639
$$724$$ −3130.00 −0.160671
$$725$$ 0 0
$$726$$ 6795.00 0.347364
$$727$$ −28028.0 −1.42985 −0.714925 0.699201i $$-0.753539\pi$$
−0.714925 + 0.699201i $$0.753539\pi$$
$$728$$ 31080.0 1.58228
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 4968.00 0.251365
$$732$$ −966.000 −0.0487765
$$733$$ −18002.0 −0.907120 −0.453560 0.891226i $$-0.649846\pi$$
−0.453560 + 0.891226i $$0.649846\pi$$
$$734$$ −10524.0 −0.529221
$$735$$ 0 0
$$736$$ 5400.00 0.270444
$$737$$ −4704.00 −0.235107
$$738$$ −8910.00 −0.444420
$$739$$ 15284.0 0.760800 0.380400 0.924822i $$-0.375786\pi$$
0.380400 + 0.924822i $$0.375786\pi$$
$$740$$ 0 0
$$741$$ 27528.0 1.36473
$$742$$ −27000.0 −1.33585
$$743$$ 18768.0 0.926691 0.463345 0.886178i $$-0.346649\pi$$
0.463345 + 0.886178i $$0.346649\pi$$
$$744$$ 12600.0 0.620885
$$745$$ 0 0
$$746$$ 32406.0 1.59044
$$747$$ −1404.00 −0.0687680
$$748$$ 1296.00 0.0633509
$$749$$ −28080.0 −1.36985
$$750$$ 0 0
$$751$$ 8696.00 0.422532 0.211266 0.977429i $$-0.432241\pi$$
0.211266 + 0.977429i $$0.432241\pi$$
$$752$$ −1704.00 −0.0826310
$$753$$ 1296.00 0.0627209
$$754$$ −17316.0 −0.836355
$$755$$ 0 0
$$756$$ −540.000 −0.0259783
$$757$$ 38662.0 1.85627 0.928134 0.372247i $$-0.121413\pi$$
0.928134 + 0.372247i $$0.121413\pi$$
$$758$$ −4380.00 −0.209880
$$759$$ −8640.00 −0.413191
$$760$$ 0 0
$$761$$ 23874.0 1.13723 0.568615 0.822604i $$-0.307479\pi$$
0.568615 + 0.822604i $$0.307479\pi$$
$$762$$ 11196.0 0.532268
$$763$$ 29480.0 1.39875
$$764$$ 3576.00 0.169339
$$765$$ 0 0
$$766$$ −14616.0 −0.689422
$$767$$ −1776.00 −0.0836084
$$768$$ 4539.00 0.213264
$$769$$ 23618.0 1.10753 0.553763 0.832675i $$-0.313192\pi$$
0.553763 + 0.832675i $$0.313192\pi$$
$$770$$ 0 0
$$771$$ −7578.00 −0.353975
$$772$$ −2666.00 −0.124289
$$773$$ −11538.0 −0.536860 −0.268430 0.963299i $$-0.586505\pi$$
−0.268430 + 0.963299i $$0.586505\pi$$
$$774$$ 2484.00 0.115356
$$775$$ 0 0
$$776$$ 6006.00 0.277839
$$777$$ −4200.00 −0.193918
$$778$$ 42138.0 1.94180
$$779$$ −40920.0 −1.88204
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ 19440.0 0.888968
$$783$$ −2106.00 −0.0961204
$$784$$ −4047.00 −0.184357
$$785$$ 0 0
$$786$$ −20952.0 −0.950805
$$787$$ 14884.0 0.674152 0.337076 0.941478i $$-0.390562\pi$$
0.337076 + 0.941478i $$0.390562\pi$$
$$788$$ 2718.00 0.122874
$$789$$ −16344.0 −0.737467
$$790$$ 0 0
$$791$$ 21720.0 0.976327
$$792$$ −4536.00 −0.203510
$$793$$ 23828.0 1.06703
$$794$$ −8202.00 −0.366597
$$795$$ 0 0
$$796$$ −3832.00 −0.170630
$$797$$ 11334.0 0.503728 0.251864 0.967763i $$-0.418957\pi$$
0.251864 + 0.967763i $$0.418957\pi$$
$$798$$ −22320.0 −0.990125
$$799$$ −1296.00 −0.0573832
$$800$$ 0 0
$$801$$ 9234.00 0.407325
$$802$$ 47826.0 2.10573
$$803$$ −10320.0 −0.453530
$$804$$ 588.000 0.0257925
$$805$$ 0 0
$$806$$ 44400.0 1.94035
$$807$$ −7722.00 −0.336837
$$808$$ −36414.0 −1.58545
$$809$$ 44730.0 1.94391 0.971955 0.235167i $$-0.0755638\pi$$
0.971955 + 0.235167i $$0.0755638\pi$$
$$810$$ 0 0
$$811$$ −42748.0 −1.85091 −0.925453 0.378862i $$-0.876316\pi$$
−0.925453 + 0.378862i $$0.876316\pi$$
$$812$$ 1560.00 0.0674203
$$813$$ −9552.00 −0.412058
$$814$$ 5040.00 0.217017
$$815$$ 0 0
$$816$$ 11502.0 0.493444
$$817$$ 11408.0 0.488513
$$818$$ −26142.0 −1.11740
$$819$$ 13320.0 0.568301
$$820$$ 0 0
$$821$$ −31686.0 −1.34695 −0.673477 0.739208i $$-0.735200\pi$$
−0.673477 + 0.739208i $$0.735200\pi$$
$$822$$ 19062.0 0.808836
$$823$$ −11036.0 −0.467425 −0.233713 0.972306i $$-0.575087\pi$$
−0.233713 + 0.972306i $$0.575087\pi$$
$$824$$ −9492.00 −0.401298
$$825$$ 0 0
$$826$$ 1440.00 0.0606586
$$827$$ −25884.0 −1.08836 −0.544181 0.838968i $$-0.683159\pi$$
−0.544181 + 0.838968i $$0.683159\pi$$
$$828$$ 1080.00 0.0453292
$$829$$ 15950.0 0.668234 0.334117 0.942532i $$-0.391562\pi$$
0.334117 + 0.942532i $$0.391562\pi$$
$$830$$ 0 0
$$831$$ −11886.0 −0.496174
$$832$$ −32042.0 −1.33516
$$833$$ −3078.00 −0.128027
$$834$$ −20916.0 −0.868419
$$835$$ 0 0
$$836$$ 2976.00 0.123119
$$837$$ 5400.00 0.223000
$$838$$ −35928.0 −1.48104
$$839$$ 13800.0 0.567853 0.283927 0.958846i $$-0.408363\pi$$
0.283927 + 0.958846i $$0.408363\pi$$
$$840$$ 0 0
$$841$$ −18305.0 −0.750543
$$842$$ −33162.0 −1.35729
$$843$$ −24858.0 −1.01560
$$844$$ 1100.00 0.0448620
$$845$$ 0 0
$$846$$ −648.000 −0.0263342
$$847$$ 15100.0 0.612565
$$848$$ 31950.0 1.29383
$$849$$ 8148.00 0.329374
$$850$$ 0 0
$$851$$ 8400.00 0.338365
$$852$$ −864.000 −0.0347420
$$853$$ 27862.0 1.11838 0.559189 0.829040i $$-0.311113\pi$$
0.559189 + 0.829040i $$0.311113\pi$$
$$854$$ −19320.0 −0.774141
$$855$$ 0 0
$$856$$ 29484.0 1.17727
$$857$$ 7314.00 0.291530 0.145765 0.989319i $$-0.453436\pi$$
0.145765 + 0.989319i $$0.453436\pi$$
$$858$$ −15984.0 −0.635996
$$859$$ −28780.0 −1.14314 −0.571572 0.820552i $$-0.693666\pi$$
−0.571572 + 0.820552i $$0.693666\pi$$
$$860$$ 0 0
$$861$$ −19800.0 −0.783719
$$862$$ −2160.00 −0.0853479
$$863$$ 32688.0 1.28935 0.644677 0.764455i $$-0.276992\pi$$
0.644677 + 0.764455i $$0.276992\pi$$
$$864$$ 1215.00 0.0478416
$$865$$ 0 0
$$866$$ −46866.0 −1.83900
$$867$$ −5991.00 −0.234677
$$868$$ −4000.00 −0.156416
$$869$$ 12480.0 0.487175
$$870$$ 0 0
$$871$$ −14504.0 −0.564236
$$872$$ −30954.0 −1.20210
$$873$$ 2574.00 0.0997900
$$874$$ 44640.0 1.72766
$$875$$ 0 0
$$876$$ 1290.00 0.0497546
$$877$$ −36650.0 −1.41115 −0.705577 0.708633i $$-0.749312\pi$$
−0.705577 + 0.708633i $$0.749312\pi$$
$$878$$ 29640.0 1.13930
$$879$$ −18054.0 −0.692772
$$880$$ 0 0
$$881$$ −2646.00 −0.101187 −0.0505936 0.998719i $$-0.516111\pi$$
−0.0505936 + 0.998719i $$0.516111\pi$$
$$882$$ −1539.00 −0.0587538
$$883$$ −10892.0 −0.415113 −0.207557 0.978223i $$-0.566551\pi$$
−0.207557 + 0.978223i $$0.566551\pi$$
$$884$$ 3996.00 0.152036
$$885$$ 0 0
$$886$$ −48348.0 −1.83328
$$887$$ 43464.0 1.64530 0.822648 0.568550i $$-0.192496\pi$$
0.822648 + 0.568550i $$0.192496\pi$$
$$888$$ 4410.00 0.166655
$$889$$ 24880.0 0.938637
$$890$$ 0 0
$$891$$ −1944.00 −0.0730937
$$892$$ −1964.00 −0.0737215
$$893$$ −2976.00 −0.111521
$$894$$ −2322.00 −0.0868672
$$895$$ 0 0
$$896$$ 33180.0 1.23713
$$897$$ −26640.0 −0.991621
$$898$$ −27054.0 −1.00535
$$899$$ −15600.0 −0.578742
$$900$$ 0 0
$$901$$ 24300.0 0.898502
$$902$$ 23760.0 0.877075
$$903$$ 5520.00 0.203426
$$904$$ −22806.0 −0.839067
$$905$$ 0 0
$$906$$ 7272.00 0.266662
$$907$$ 14884.0 0.544890 0.272445 0.962171i $$-0.412168\pi$$
0.272445 + 0.962171i $$0.412168\pi$$
$$908$$ −660.000 −0.0241221
$$909$$ −15606.0 −0.569437
$$910$$ 0 0
$$911$$ −1248.00 −0.0453876 −0.0226938 0.999742i $$-0.507224\pi$$
−0.0226938 + 0.999742i $$0.507224\pi$$
$$912$$ 26412.0 0.958979
$$913$$ 3744.00 0.135716
$$914$$ −11010.0 −0.398445
$$915$$ 0 0
$$916$$ −1906.00 −0.0687511
$$917$$ −46560.0 −1.67671
$$918$$ 4374.00 0.157259
$$919$$ −6640.00 −0.238339 −0.119169 0.992874i $$-0.538023\pi$$
−0.119169 + 0.992874i $$0.538023\pi$$
$$920$$ 0 0
$$921$$ −27708.0 −0.991324
$$922$$ −52686.0 −1.88191
$$923$$ 21312.0 0.760014
$$924$$ 1440.00 0.0512690
$$925$$ 0 0
$$926$$ 3516.00 0.124776
$$927$$ −4068.00 −0.144132
$$928$$ −3510.00 −0.124161
$$929$$ 29946.0 1.05758 0.528792 0.848751i $$-0.322645\pi$$
0.528792 + 0.848751i $$0.322645\pi$$
$$930$$ 0 0
$$931$$ −7068.00 −0.248812
$$932$$ 1458.00 0.0512429
$$933$$ 4608.00 0.161693
$$934$$ 20628.0 0.722665
$$935$$ 0 0
$$936$$ −13986.0 −0.488405
$$937$$ −45002.0 −1.56900 −0.784499 0.620130i $$-0.787080\pi$$
−0.784499 + 0.620130i $$0.787080\pi$$
$$938$$ 11760.0 0.409358
$$939$$ 22026.0 0.765486
$$940$$ 0 0
$$941$$ 6090.00 0.210976 0.105488 0.994421i $$-0.466360\pi$$
0.105488 + 0.994421i $$0.466360\pi$$
$$942$$ 21402.0 0.740249
$$943$$ 39600.0 1.36750
$$944$$ −1704.00 −0.0587505
$$945$$ 0 0
$$946$$ −6624.00 −0.227658
$$947$$ −56388.0 −1.93491 −0.967457 0.253035i $$-0.918571\pi$$
−0.967457 + 0.253035i $$0.918571\pi$$
$$948$$ −1560.00 −0.0534456
$$949$$ −31820.0 −1.08843
$$950$$ 0 0
$$951$$ 11682.0 0.398333
$$952$$ 22680.0 0.772125
$$953$$ −10854.0 −0.368936 −0.184468 0.982839i $$-0.559056\pi$$
−0.184468 + 0.982839i $$0.559056\pi$$
$$954$$ 12150.0 0.412338
$$955$$ 0 0
$$956$$ 1176.00 0.0397851
$$957$$ 5616.00 0.189696
$$958$$ −6840.00 −0.230679
$$959$$ 42360.0 1.42636
$$960$$ 0 0
$$961$$ 10209.0 0.342687
$$962$$ 15540.0 0.520821
$$963$$ 12636.0 0.422834
$$964$$ 866.000 0.0289336
$$965$$ 0 0
$$966$$ 21600.0 0.719429
$$967$$ 42316.0 1.40723 0.703615 0.710582i $$-0.251568\pi$$
0.703615 + 0.710582i $$0.251568\pi$$
$$968$$ −15855.0 −0.526445
$$969$$ 20088.0 0.665964
$$970$$ 0 0
$$971$$ 24480.0 0.809063 0.404532 0.914524i $$-0.367435\pi$$
0.404532 + 0.914524i $$0.367435\pi$$
$$972$$ 243.000 0.00801875
$$973$$ −46480.0 −1.53143
$$974$$ −9228.00 −0.303577
$$975$$ 0 0
$$976$$ 22862.0 0.749790
$$977$$ 6906.00 0.226144 0.113072 0.993587i $$-0.463931\pi$$
0.113072 + 0.993587i $$0.463931\pi$$
$$978$$ −468.000 −0.0153016
$$979$$ −24624.0 −0.803868
$$980$$ 0 0
$$981$$ −13266.0 −0.431754
$$982$$ 56736.0 1.84371
$$983$$ −6960.00 −0.225829 −0.112914 0.993605i $$-0.536019\pi$$
−0.112914 + 0.993605i $$0.536019\pi$$
$$984$$ 20790.0 0.673538
$$985$$ 0 0
$$986$$ −12636.0 −0.408126
$$987$$ −1440.00 −0.0464394
$$988$$ 9176.00 0.295473
$$989$$ −11040.0 −0.354956
$$990$$ 0 0
$$991$$ 47792.0 1.53195 0.765975 0.642870i $$-0.222256\pi$$
0.765975 + 0.642870i $$0.222256\pi$$
$$992$$ 9000.00 0.288055
$$993$$ 11076.0 0.353964
$$994$$ −17280.0 −0.551397
$$995$$ 0 0
$$996$$ −468.000 −0.0148887
$$997$$ −9938.00 −0.315687 −0.157843 0.987464i $$-0.550454\pi$$
−0.157843 + 0.987464i $$0.550454\pi$$
$$998$$ −29868.0 −0.947350
$$999$$ 1890.00 0.0598568
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 75.4.a.a.1.1 1
3.2 odd 2 225.4.a.g.1.1 1
4.3 odd 2 1200.4.a.o.1.1 1
5.2 odd 4 75.4.b.a.49.1 2
5.3 odd 4 75.4.b.a.49.2 2
5.4 even 2 15.4.a.b.1.1 1
15.2 even 4 225.4.b.d.199.2 2
15.8 even 4 225.4.b.d.199.1 2
15.14 odd 2 45.4.a.b.1.1 1
20.3 even 4 1200.4.f.m.49.1 2
20.7 even 4 1200.4.f.m.49.2 2
20.19 odd 2 240.4.a.f.1.1 1
35.34 odd 2 735.4.a.i.1.1 1
40.19 odd 2 960.4.a.l.1.1 1
40.29 even 2 960.4.a.bi.1.1 1
45.4 even 6 405.4.e.d.136.1 2
45.14 odd 6 405.4.e.k.136.1 2
45.29 odd 6 405.4.e.k.271.1 2
45.34 even 6 405.4.e.d.271.1 2
55.54 odd 2 1815.4.a.a.1.1 1
60.59 even 2 720.4.a.r.1.1 1
105.104 even 2 2205.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 5.4 even 2
45.4.a.b.1.1 1 15.14 odd 2
75.4.a.a.1.1 1 1.1 even 1 trivial
75.4.b.a.49.1 2 5.2 odd 4
75.4.b.a.49.2 2 5.3 odd 4
225.4.a.g.1.1 1 3.2 odd 2
225.4.b.d.199.1 2 15.8 even 4
225.4.b.d.199.2 2 15.2 even 4
240.4.a.f.1.1 1 20.19 odd 2
405.4.e.d.136.1 2 45.4 even 6
405.4.e.d.271.1 2 45.34 even 6
405.4.e.k.136.1 2 45.14 odd 6
405.4.e.k.271.1 2 45.29 odd 6
720.4.a.r.1.1 1 60.59 even 2
735.4.a.i.1.1 1 35.34 odd 2
960.4.a.l.1.1 1 40.19 odd 2
960.4.a.bi.1.1 1 40.29 even 2
1200.4.a.o.1.1 1 4.3 odd 2
1200.4.f.m.49.1 2 20.3 even 4
1200.4.f.m.49.2 2 20.7 even 4
1815.4.a.a.1.1 1 55.54 odd 2
2205.4.a.c.1.1 1 105.104 even 2