# Properties

 Label 1815.4.a.a.1.1 Level $1815$ Weight $4$ Character 1815.1 Self dual yes Analytic conductor $107.088$ Analytic rank $1$ 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] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.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: $$107.088466660$$ 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 $$\chi$$ $$=$$ 1815.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -5.00000 q^{5} +9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -5.00000 q^{5} +9.00000 q^{6} -20.0000 q^{7} +21.0000 q^{8} +9.00000 q^{9} +15.0000 q^{10} -3.00000 q^{12} -74.0000 q^{13} +60.0000 q^{14} +15.0000 q^{15} -71.0000 q^{16} -54.0000 q^{17} -27.0000 q^{18} +124.000 q^{19} -5.00000 q^{20} +60.0000 q^{21} -120.000 q^{23} -63.0000 q^{24} +25.0000 q^{25} +222.000 q^{26} -27.0000 q^{27} -20.0000 q^{28} +78.0000 q^{29} -45.0000 q^{30} +200.000 q^{31} +45.0000 q^{32} +162.000 q^{34} +100.000 q^{35} +9.00000 q^{36} -70.0000 q^{37} -372.000 q^{38} +222.000 q^{39} -105.000 q^{40} -330.000 q^{41} -180.000 q^{42} -92.0000 q^{43} -45.0000 q^{45} +360.000 q^{46} -24.0000 q^{47} +213.000 q^{48} +57.0000 q^{49} -75.0000 q^{50} +162.000 q^{51} -74.0000 q^{52} +450.000 q^{53} +81.0000 q^{54} -420.000 q^{56} -372.000 q^{57} -234.000 q^{58} +24.0000 q^{59} +15.0000 q^{60} +322.000 q^{61} -600.000 q^{62} -180.000 q^{63} +433.000 q^{64} +370.000 q^{65} -196.000 q^{67} -54.0000 q^{68} +360.000 q^{69} -300.000 q^{70} -288.000 q^{71} +189.000 q^{72} +430.000 q^{73} +210.000 q^{74} -75.0000 q^{75} +124.000 q^{76} -666.000 q^{78} +520.000 q^{79} +355.000 q^{80} +81.0000 q^{81} +990.000 q^{82} -156.000 q^{83} +60.0000 q^{84} +270.000 q^{85} +276.000 q^{86} -234.000 q^{87} +1026.00 q^{89} +135.000 q^{90} +1480.00 q^{91} -120.000 q^{92} -600.000 q^{93} +72.0000 q^{94} -620.000 q^{95} -135.000 q^{96} -286.000 q^{97} -171.000 q^{98} +O(q^{100})$$

## 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
$$5$$ −5.00000 −0.447214
$$6$$ 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
$$10$$ 15.0000 0.474342
$$11$$ 0 0
$$12$$ −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
$$15$$ 15.0000 0.258199
$$16$$ −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 $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ −5.00000 −0.0559017
$$21$$ 60.0000 0.623480
$$22$$ 0 0
$$23$$ −120.000 −1.08790 −0.543951 0.839117i $$-0.683072\pi$$
−0.543951 + 0.839117i $$0.683072\pi$$
$$24$$ −63.0000 −0.535826
$$25$$ 25.0000 0.200000
$$26$$ 222.000 1.67453
$$27$$ −27.0000 −0.192450
$$28$$ −20.0000 −0.134987
$$29$$ 78.0000 0.499456 0.249728 0.968316i $$-0.419659\pi$$
0.249728 + 0.968316i $$0.419659\pi$$
$$30$$ −45.0000 −0.273861
$$31$$ 200.000 1.15874 0.579372 0.815063i $$-0.303298\pi$$
0.579372 + 0.815063i $$0.303298\pi$$
$$32$$ 45.0000 0.248592
$$33$$ 0 0
$$34$$ 162.000 0.817140
$$35$$ 100.000 0.482945
$$36$$ 9.00000 0.0416667
$$37$$ −70.0000 −0.311025 −0.155513 0.987834i $$-0.549703\pi$$
−0.155513 + 0.987834i $$0.549703\pi$$
$$38$$ −372.000 −1.58806
$$39$$ 222.000 0.911499
$$40$$ −105.000 −0.415049
$$41$$ −330.000 −1.25701 −0.628504 0.777806i $$-0.716332\pi$$
−0.628504 + 0.777806i $$0.716332\pi$$
$$42$$ −180.000 −0.661300
$$43$$ −92.0000 −0.326276 −0.163138 0.986603i $$-0.552162\pi$$
−0.163138 + 0.986603i $$0.552162\pi$$
$$44$$ 0 0
$$45$$ −45.0000 −0.149071
$$46$$ 360.000 1.15389
$$47$$ −24.0000 −0.0744843 −0.0372421 0.999306i $$-0.511857\pi$$
−0.0372421 + 0.999306i $$0.511857\pi$$
$$48$$ 213.000 0.640498
$$49$$ 57.0000 0.166181
$$50$$ −75.0000 −0.212132
$$51$$ 162.000 0.444795
$$52$$ −74.0000 −0.197345
$$53$$ 450.000 1.16627 0.583134 0.812376i $$-0.301826\pi$$
0.583134 + 0.812376i $$0.301826\pi$$
$$54$$ 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 $$0.491570\pi$$
$$60$$ 15.0000 0.0322749
$$61$$ 322.000 0.675867 0.337933 0.941170i $$-0.390272\pi$$
0.337933 + 0.941170i $$0.390272\pi$$
$$62$$ −600.000 −1.22903
$$63$$ −180.000 −0.359966
$$64$$ 433.000 0.845703
$$65$$ 370.000 0.706044
$$66$$ 0 0
$$67$$ −196.000 −0.357391 −0.178696 0.983904i $$-0.557188\pi$$
−0.178696 + 0.983904i $$0.557188\pi$$
$$68$$ −54.0000 −0.0963009
$$69$$ 360.000 0.628100
$$70$$ −300.000 −0.512241
$$71$$ −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
$$75$$ −75.0000 −0.115470
$$76$$ 124.000 0.187155
$$77$$ 0 0
$$78$$ −666.000 −0.966790
$$79$$ 520.000 0.740564 0.370282 0.928919i $$-0.379261\pi$$
0.370282 + 0.928919i $$0.379261\pi$$
$$80$$ 355.000 0.496128
$$81$$ 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
$$85$$ 270.000 0.344537
$$86$$ 276.000 0.346068
$$87$$ −234.000 −0.288361
$$88$$ 0 0
$$89$$ 1026.00 1.22198 0.610988 0.791640i $$-0.290773\pi$$
0.610988 + 0.791640i $$0.290773\pi$$
$$90$$ 135.000 0.158114
$$91$$ 1480.00 1.70490
$$92$$ −120.000 −0.135988
$$93$$ −600.000 −0.669001
$$94$$ 72.0000 0.0790025
$$95$$ −620.000 −0.669586
$$96$$ −135.000 −0.143525
$$97$$ −286.000 −0.299370 −0.149685 0.988734i $$-0.547826\pi$$
−0.149685 + 0.988734i $$0.547826\pi$$
$$98$$ −171.000 −0.176261
$$99$$ 0 0
$$100$$ 25.0000 0.0250000
$$101$$ 1734.00 1.70831 0.854156 0.520017i $$-0.174075\pi$$
0.854156 + 0.520017i $$0.174075\pi$$
$$102$$ −486.000 −0.471776
$$103$$ 452.000 0.432397 0.216198 0.976349i $$-0.430634\pi$$
0.216198 + 0.976349i $$0.430634\pi$$
$$104$$ −1554.00 −1.46521
$$105$$ −300.000 −0.278829
$$106$$ −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 $$-0.275760\pi$$
0.647631 + 0.761954i $$0.275760\pi$$
$$110$$ 0 0
$$111$$ 210.000 0.179570
$$112$$ 1420.00 1.19801
$$113$$ 1086.00 0.904091 0.452046 0.891995i $$-0.350694\pi$$
0.452046 + 0.891995i $$0.350694\pi$$
$$114$$ 1116.00 0.916868
$$115$$ 600.000 0.486524
$$116$$ 78.0000 0.0624321
$$117$$ −666.000 −0.526254
$$118$$ −72.0000 −0.0561707
$$119$$ 1080.00 0.831962
$$120$$ 315.000 0.239629
$$121$$ 0 0
$$122$$ −966.000 −0.716865
$$123$$ 990.000 0.725734
$$124$$ 200.000 0.144843
$$125$$ −125.000 −0.0894427
$$126$$ 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
$$130$$ −1110.00 −0.748873
$$131$$ −2328.00 −1.55266 −0.776329 0.630327i $$-0.782921\pi$$
−0.776329 + 0.630327i $$0.782921\pi$$
$$132$$ 0 0
$$133$$ −2480.00 −1.61687
$$134$$ 588.000 0.379071
$$135$$ 135.000 0.0860663
$$136$$ −1134.00 −0.714998
$$137$$ 2118.00 1.32082 0.660412 0.750903i $$-0.270382\pi$$
0.660412 + 0.750903i $$0.270382\pi$$
$$138$$ −1080.00 −0.666201
$$139$$ −2324.00 −1.41812 −0.709062 0.705147i $$-0.750881\pi$$
−0.709062 + 0.705147i $$0.750881\pi$$
$$140$$ 100.000 0.0603682
$$141$$ 72.0000 0.0430035
$$142$$ 864.000 0.510600
$$143$$ 0 0
$$144$$ −639.000 −0.369792
$$145$$ −390.000 −0.223364
$$146$$ −1290.00 −0.731241
$$147$$ −171.000 −0.0959445
$$148$$ −70.0000 −0.0388781
$$149$$ −258.000 −0.141854 −0.0709268 0.997482i $$-0.522596\pi$$
−0.0709268 + 0.997482i $$0.522596\pi$$
$$150$$ 225.000 0.122474
$$151$$ 808.000 0.435458 0.217729 0.976009i $$-0.430135\pi$$
0.217729 + 0.976009i $$0.430135\pi$$
$$152$$ 2604.00 1.38955
$$153$$ −486.000 −0.256802
$$154$$ 0 0
$$155$$ −1000.00 −0.518206
$$156$$ 222.000 0.113937
$$157$$ 2378.00 1.20882 0.604411 0.796673i $$-0.293408\pi$$
0.604411 + 0.796673i $$0.293408\pi$$
$$158$$ −1560.00 −0.785487
$$159$$ −1350.00 −0.673346
$$160$$ −225.000 −0.111174
$$161$$ 2400.00 1.17482
$$162$$ −243.000 −0.117851
$$163$$ −52.0000 −0.0249874 −0.0124937 0.999922i $$-0.503977\pi$$
−0.0124937 + 0.999922i $$0.503977\pi$$
$$164$$ −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
$$170$$ −810.000 −0.365436
$$171$$ 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
$$175$$ −500.000 −0.215980
$$176$$ 0 0
$$177$$ −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 $$0.597202\pi$$
$$180$$ −45.0000 −0.0186339
$$181$$ −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
$$185$$ 350.000 0.139095
$$186$$ 1800.00 0.709583
$$187$$ 0 0
$$188$$ −24.0000 −0.00931053
$$189$$ 540.000 0.207827
$$190$$ 1860.00 0.710203
$$191$$ 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
$$195$$ −1110.00 −0.407635
$$196$$ 57.0000 0.0207726
$$197$$ 2718.00 0.982992 0.491496 0.870880i $$-0.336450\pi$$
0.491496 + 0.870880i $$0.336450\pi$$
$$198$$ 0 0
$$199$$ −3832.00 −1.36504 −0.682521 0.730866i $$-0.739116\pi$$
−0.682521 + 0.730866i $$0.739116\pi$$
$$200$$ 525.000 0.185616
$$201$$ 588.000 0.206340
$$202$$ −5202.00 −1.81194
$$203$$ −1560.00 −0.539362
$$204$$ 162.000 0.0555994
$$205$$ 1650.00 0.562151
$$206$$ −1356.00 −0.458626
$$207$$ −1080.00 −0.362634
$$208$$ 5254.00 1.75144
$$209$$ 0 0
$$210$$ 900.000 0.295742
$$211$$ −1100.00 −0.358896 −0.179448 0.983767i $$-0.557431\pi$$
−0.179448 + 0.983767i $$0.557431\pi$$
$$212$$ 450.000 0.145784
$$213$$ 864.000 0.277936
$$214$$ −4212.00 −1.34545
$$215$$ 460.000 0.145915
$$216$$ −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 $$-0.404718\pi$$
0.294886 + 0.955532i $$0.404718\pi$$
$$224$$ −900.000 −0.268454
$$225$$ 225.000 0.0666667
$$226$$ −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 $$0.588679\pi$$
$$230$$ −1800.00 −0.516037
$$231$$ 0 0
$$232$$ 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
$$235$$ 120.000 0.0333104
$$236$$ 24.0000 0.00661978
$$237$$ −1560.00 −0.427565
$$238$$ −3240.00 −0.882429
$$239$$ −1176.00 −0.318281 −0.159140 0.987256i $$-0.550872\pi$$
−0.159140 + 0.987256i $$0.550872\pi$$
$$240$$ −1065.00 −0.286439
$$241$$ −866.000 −0.231469 −0.115734 0.993280i $$-0.536922\pi$$
−0.115734 + 0.993280i $$0.536922\pi$$
$$242$$ 0 0
$$243$$ −243.000 −0.0641500
$$244$$ 322.000 0.0844834
$$245$$ −285.000 −0.0743183
$$246$$ −2970.00 −0.769757
$$247$$ −9176.00 −2.36379
$$248$$ 4200.00 1.07540
$$249$$ 468.000 0.119110
$$250$$ 375.000 0.0948683
$$251$$ 432.000 0.108636 0.0543179 0.998524i $$-0.482702\pi$$
0.0543179 + 0.998524i $$0.482702\pi$$
$$252$$ −180.000 −0.0449958
$$253$$ 0 0
$$254$$ 3732.00 0.921915
$$255$$ −810.000 −0.198918
$$256$$ 1513.00 0.369385
$$257$$ 2526.00 0.613103 0.306552 0.951854i $$-0.400825\pi$$
0.306552 + 0.951854i $$0.400825\pi$$
$$258$$ −828.000 −0.199802
$$259$$ 1400.00 0.335876
$$260$$ 370.000 0.0882555
$$261$$ 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 $$0.720513\pi$$
$$264$$ 0 0
$$265$$ −2250.00 −0.521571
$$266$$ 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 $$0.594224\pi$$
$$270$$ −405.000 −0.0912871
$$271$$ 3184.00 0.713706 0.356853 0.934161i $$-0.383850\pi$$
0.356853 + 0.934161i $$0.383850\pi$$
$$272$$ 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
$$280$$ 2100.00 0.448211
$$281$$ 8286.00 1.75908 0.879540 0.475825i $$-0.157851\pi$$
0.879540 + 0.475825i $$0.157851\pi$$
$$282$$ −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
$$285$$ 1860.00 0.386586
$$286$$ 0 0
$$287$$ 6600.00 1.35744
$$288$$ 405.000 0.0828641
$$289$$ −1997.00 −0.406473
$$290$$ 1170.00 0.236913
$$291$$ 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
$$295$$ −120.000 −0.0236836
$$296$$ −1470.00 −0.288655
$$297$$ 0 0
$$298$$ 774.000 0.150458
$$299$$ 8880.00 1.71754
$$300$$ −75.0000 −0.0144338
$$301$$ 1840.00 0.352345
$$302$$ −2424.00 −0.461873
$$303$$ −5202.00 −0.986294
$$304$$ −8804.00 −1.66100
$$305$$ −1610.00 −0.302257
$$306$$ 1458.00 0.272380
$$307$$ −9236.00 −1.71702 −0.858512 0.512793i $$-0.828611\pi$$
−0.858512 + 0.512793i $$0.828611\pi$$
$$308$$ 0 0
$$309$$ −1356.00 −0.249644
$$310$$ 3000.00 0.549640
$$311$$ 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 $$-0.730687\pi$$
−0.662930 + 0.748681i $$0.730687\pi$$
$$314$$ −7134.00 −1.28215
$$315$$ 900.000 0.160982
$$316$$ 520.000 0.0925705
$$317$$ −3894.00 −0.689933 −0.344967 0.938615i $$-0.612110\pi$$
−0.344967 + 0.938615i $$0.612110\pi$$
$$318$$ 4050.00 0.714191
$$319$$ 0 0
$$320$$ −2165.00 −0.378210
$$321$$ −4212.00 −0.732370
$$322$$ −7200.00 −1.24609
$$323$$ −6696.00 −1.15348
$$324$$ 81.0000 0.0138889
$$325$$ −1850.00 −0.315752
$$326$$ 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
$$335$$ 980.000 0.159830
$$336$$ −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
$$340$$ 270.000 0.0430671
$$341$$ 0 0
$$342$$ −3348.00 −0.529354
$$343$$ 5720.00 0.900440
$$344$$ −1932.00 −0.302809
$$345$$ −1800.00 −0.280895
$$346$$ 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 $$-0.660559\pi$$
−0.483293 + 0.875459i $$0.660559\pi$$
$$350$$ 1500.00 0.229081
$$351$$ 1998.00 0.303833
$$352$$ 0 0
$$353$$ 3414.00 0.514756 0.257378 0.966311i $$-0.417141\pi$$
0.257378 + 0.966311i $$0.417141\pi$$
$$354$$ 216.000 0.0324301
$$355$$ 1440.00 0.215288
$$356$$ 1026.00 0.152747
$$357$$ −3240.00 −0.480333
$$358$$ 4320.00 0.637763
$$359$$ −4824.00 −0.709195 −0.354597 0.935019i $$-0.615382\pi$$
−0.354597 + 0.935019i $$0.615382\pi$$
$$360$$ −945.000 −0.138350
$$361$$ 8517.00 1.24173
$$362$$ 9390.00 1.36334
$$363$$ 0 0
$$364$$ 1480.00 0.213113
$$365$$ −2150.00 −0.308318
$$366$$ 2898.00 0.413882
$$367$$ −3508.00 −0.498954 −0.249477 0.968381i $$-0.580259\pi$$
−0.249477 + 0.968381i $$0.580259\pi$$
$$368$$ 8520.00 1.20689
$$369$$ −2970.00 −0.419003
$$370$$ −1050.00 −0.147532
$$371$$ −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 $$0.769822\pi$$
$$374$$ 0 0
$$375$$ 375.000 0.0516398
$$376$$ −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 $$0.468455\pi$$
$$380$$ −620.000 −0.0836982
$$381$$ 3732.00 0.501827
$$382$$ −10728.0 −1.43689
$$383$$ −4872.00 −0.649994 −0.324997 0.945715i $$-0.605363\pi$$
−0.324997 + 0.945715i $$0.605363\pi$$
$$384$$ 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 $$0.868104\pi$$
$$390$$ 3330.00 0.432362
$$391$$ 6480.00 0.838127
$$392$$ 1197.00 0.154229
$$393$$ 6984.00 0.896428
$$394$$ −8154.00 −1.04262
$$395$$ −2600.00 −0.331190
$$396$$ 0 0
$$397$$ −2734.00 −0.345631 −0.172816 0.984954i $$-0.555286\pi$$
−0.172816 + 0.984954i $$0.555286\pi$$
$$398$$ 11496.0 1.44785
$$399$$ 7440.00 0.933498
$$400$$ −1775.00 −0.221875
$$401$$ −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
$$405$$ −405.000 −0.0496904
$$406$$ 4680.00 0.572080
$$407$$ 0 0
$$408$$ 3402.00 0.412804
$$409$$ −8714.00 −1.05350 −0.526748 0.850022i $$-0.676589\pi$$
−0.526748 + 0.850022i $$0.676589\pi$$
$$410$$ −4950.00 −0.596251
$$411$$ −6354.00 −0.762578
$$412$$ 452.000 0.0540496
$$413$$ −480.000 −0.0571895
$$414$$ 3240.00 0.384631
$$415$$ 780.000 0.0922619
$$416$$ −3330.00 −0.392468
$$417$$ 6972.00 0.818754
$$418$$ 0 0
$$419$$ 11976.0 1.39634 0.698169 0.715933i $$-0.253998\pi$$
0.698169 + 0.715933i $$0.253998\pi$$
$$420$$ −300.000 −0.0348536
$$421$$ 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
$$425$$ −1350.00 −0.154081
$$426$$ −2592.00 −0.294795
$$427$$ −6440.00 −0.729868
$$428$$ 1404.00 0.158563
$$429$$ 0 0
$$430$$ −1380.00 −0.154766
$$431$$ −720.000 −0.0804668 −0.0402334 0.999190i $$-0.512810\pi$$
−0.0402334 + 0.999190i $$0.512810\pi$$
$$432$$ 1917.00 0.213499
$$433$$ −15622.0 −1.73382 −0.866912 0.498462i $$-0.833898\pi$$
−0.866912 + 0.498462i $$0.833898\pi$$
$$434$$ 12000.0 1.32723
$$435$$ 1170.00 0.128959
$$436$$ 1474.00 0.161908
$$437$$ −14880.0 −1.62885
$$438$$ 3870.00 0.422182
$$439$$ 9880.00 1.07414 0.537069 0.843538i $$-0.319531\pi$$
0.537069 + 0.843538i $$0.319531\pi$$
$$440$$ 0 0
$$441$$ 513.000 0.0553936
$$442$$ −11988.0 −1.29007
$$443$$ −16116.0 −1.72843 −0.864215 0.503123i $$-0.832184\pi$$
−0.864215 + 0.503123i $$0.832184\pi$$
$$444$$ 210.000 0.0224463
$$445$$ −5130.00 −0.546484
$$446$$ −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 $$0.342836\pi$$
$$450$$ −675.000 −0.0707107
$$451$$ 0 0
$$452$$ 1086.00 0.113011
$$453$$ −2424.00 −0.251412
$$454$$ 1980.00 0.204683
$$455$$ −7400.00 −0.762456
$$456$$ −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
$$460$$ 600.000 0.0608155
$$461$$ −17562.0 −1.77428 −0.887141 0.461499i $$-0.847312\pi$$
−0.887141 + 0.461499i $$0.847312\pi$$
$$462$$ 0 0
$$463$$ 1172.00 0.117640 0.0588202 0.998269i $$-0.481266\pi$$
0.0588202 + 0.998269i $$0.481266\pi$$
$$464$$ −5538.00 −0.554084
$$465$$ 3000.00 0.299186
$$466$$ −4374.00 −0.434810
$$467$$ 6876.00 0.681335 0.340667 0.940184i $$-0.389347\pi$$
0.340667 + 0.940184i $$0.389347\pi$$
$$468$$ −666.000 −0.0657818
$$469$$ 3920.00 0.385946
$$470$$ −360.000 −0.0353310
$$471$$ −7134.00 −0.697914
$$472$$ 504.000 0.0491493
$$473$$ 0 0
$$474$$ 4680.00 0.453501
$$475$$ 3100.00 0.299448
$$476$$ 1080.00 0.103995
$$477$$ 4050.00 0.388756
$$478$$ 3528.00 0.337588
$$479$$ −2280.00 −0.217486 −0.108743 0.994070i $$-0.534683\pi$$
−0.108743 + 0.994070i $$0.534683\pi$$
$$480$$ 675.000 0.0641862
$$481$$ 5180.00 0.491035
$$482$$ 2598.00 0.245510
$$483$$ −7200.00 −0.678284
$$484$$ 0 0
$$485$$ 1430.00 0.133882
$$486$$ 729.000 0.0680414
$$487$$ −3076.00 −0.286215 −0.143108 0.989707i $$-0.545710\pi$$
−0.143108 + 0.989707i $$0.545710\pi$$
$$488$$ 6762.00 0.627257
$$489$$ 156.000 0.0144265
$$490$$ 855.000 0.0788265
$$491$$ 18912.0 1.73826 0.869131 0.494582i $$-0.164679\pi$$
0.869131 + 0.494582i $$0.164679\pi$$
$$492$$ 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 $$0.352640\pi$$
$$500$$ −125.000 −0.0111803
$$501$$ −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
$$505$$ −8670.00 −0.763980
$$506$$ 0 0
$$507$$ −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 $$0.538428\pi$$
$$510$$ 2430.00 0.210985
$$511$$ −8600.00 −0.744504
$$512$$ 8733.00 0.753804
$$513$$ −3348.00 −0.288144
$$514$$ −7578.00 −0.650294
$$515$$ −2260.00 −0.193374
$$516$$ 276.000 0.0235469
$$517$$ 0 0
$$518$$ −4200.00 −0.356250
$$519$$ 1278.00 0.108089
$$520$$ 7770.00 0.655264
$$521$$ 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
$$525$$ 1500.00 0.124696
$$526$$ 16344.0 1.35481
$$527$$ −10800.0 −0.892705
$$528$$ 0 0
$$529$$ 2233.00 0.183529
$$530$$ 6750.00 0.553210
$$531$$ 216.000 0.0176527
$$532$$ −2480.00 −0.202108
$$533$$ 24420.0 1.98452
$$534$$ 9234.00 0.748304
$$535$$ −7020.00 −0.567292
$$536$$ −4116.00 −0.331687
$$537$$ 4320.00 0.347154
$$538$$ 7722.00 0.618809
$$539$$ 0 0
$$540$$ 135.000 0.0107583
$$541$$ −18110.0 −1.43920 −0.719602 0.694386i $$-0.755676\pi$$
−0.719602 + 0.694386i $$0.755676\pi$$
$$542$$ −9552.00 −0.756999
$$543$$ 9390.00 0.742106
$$544$$ −2430.00 −0.191517
$$545$$ −7370.00 −0.579259
$$546$$ 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
$$555$$ −1050.00 −0.0803063
$$556$$ −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
$$560$$ −7100.00 −0.535767
$$561$$ 0 0
$$562$$ −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
$$565$$ −5430.00 −0.404322
$$566$$ −8148.00 −0.605099
$$567$$ −1620.00 −0.119989
$$568$$ −6048.00 −0.446775
$$569$$ 18654.0 1.37437 0.687185 0.726483i $$-0.258846\pi$$
0.687185 + 0.726483i $$0.258846\pi$$
$$570$$ −5580.00 −0.410036
$$571$$ 7684.00 0.563162 0.281581 0.959537i $$-0.409141\pi$$
0.281581 + 0.959537i $$0.409141\pi$$
$$572$$ 0 0
$$573$$ −10728.0 −0.782144
$$574$$ −19800.0 −1.43978
$$575$$ −3000.00 −0.217580
$$576$$ 3897.00 0.281901
$$577$$ −1726.00 −0.124531 −0.0622654 0.998060i $$-0.519833\pi$$
−0.0622654 + 0.998060i $$0.519833\pi$$
$$578$$ 5991.00 0.431129
$$579$$ 7998.00 0.574068
$$580$$ −390.000 −0.0279205
$$581$$ 3120.00 0.222787
$$582$$ −2574.00 −0.183326
$$583$$ 0 0
$$584$$ 9030.00 0.639836
$$585$$ 3330.00 0.235348
$$586$$ 18054.0 1.27270
$$587$$ 10596.0 0.745049 0.372524 0.928022i $$-0.378492\pi$$
0.372524 + 0.928022i $$0.378492\pi$$
$$588$$ −171.000 −0.0119931
$$589$$ 24800.0 1.73492
$$590$$ 360.000 0.0251203
$$591$$ −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 $$0.531595\pi$$
$$594$$ 0 0
$$595$$ −5400.00 −0.372065
$$596$$ −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 $$0.797635\pi$$
$$600$$ −1575.00 −0.107165
$$601$$ 9574.00 0.649803 0.324902 0.945748i $$-0.394669\pi$$
0.324902 + 0.945748i $$0.394669\pi$$
$$602$$ −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
$$610$$ 4830.00 0.320592
$$611$$ 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
$$615$$ −4950.00 −0.324558
$$616$$ 0 0
$$617$$ −12162.0 −0.793555 −0.396778 0.917915i $$-0.629872\pi$$
−0.396778 + 0.917915i $$0.629872\pi$$
$$618$$ 4068.00 0.264788
$$619$$ 8804.00 0.571668 0.285834 0.958279i $$-0.407729\pi$$
0.285834 + 0.958279i $$0.407729\pi$$
$$620$$ −1000.00 −0.0647758
$$621$$ 3240.00 0.209367
$$622$$ −4608.00 −0.297048
$$623$$ −20520.0 −1.31961
$$624$$ −15762.0 −1.01119
$$625$$ 625.000 0.0400000
$$626$$ 22026.0 1.40629
$$627$$ 0 0
$$628$$ 2378.00 0.151103
$$629$$ 3780.00 0.239616
$$630$$ −2700.00 −0.170747
$$631$$ −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
$$635$$ 6220.00 0.388714
$$636$$ −1350.00 −0.0841682
$$637$$ −4218.00 −0.262360
$$638$$ 0 0
$$639$$ −2592.00 −0.160466
$$640$$ 8295.00 0.512326
$$641$$ −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 $$-0.217448\pi$$
0.775598 + 0.631227i $$0.217448\pi$$
$$644$$ 2400.00 0.146853
$$645$$ −1380.00 −0.0842441
$$646$$ 20088.0 1.22345
$$647$$ −2736.00 −0.166249 −0.0831246 0.996539i $$-0.526490\pi$$
−0.0831246 + 0.996539i $$0.526490\pi$$
$$648$$ 1701.00 0.103120
$$649$$ 0 0
$$650$$ 5550.00 0.334906
$$651$$ 12000.0 0.722453
$$652$$ −52.0000 −0.00312343
$$653$$ 22218.0 1.33148 0.665741 0.746183i $$-0.268116\pi$$
0.665741 + 0.746183i $$0.268116\pi$$
$$654$$ 13266.0 0.793183
$$655$$ 11640.0 0.694370
$$656$$ 23430.0 1.39449
$$657$$ 3870.00 0.229807
$$658$$ −1440.00 −0.0853147
$$659$$ −14520.0 −0.858299 −0.429149 0.903234i $$-0.641187\pi$$
−0.429149 + 0.903234i $$0.641187\pi$$
$$660$$ 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
$$665$$ 12400.0 0.723085
$$666$$ 1890.00 0.109964
$$667$$ −9360.00 −0.543359
$$668$$ 3720.00 0.215466
$$669$$ −5892.00 −0.340505
$$670$$ −2940.00 −0.169526
$$671$$ 0 0
$$672$$ 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
$$675$$ −675.000 −0.0384900
$$676$$ 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
$$680$$ 5670.00 0.319757
$$681$$ 1980.00 0.111415
$$682$$ 0 0
$$683$$ 11988.0 0.671608 0.335804 0.941932i $$-0.390992\pi$$
0.335804 + 0.941932i $$0.390992\pi$$
$$684$$ 1116.00 0.0623850
$$685$$ −10590.0 −0.590691
$$686$$ −17160.0 −0.955061
$$687$$ 5718.00 0.317548
$$688$$ 6532.00 0.361962
$$689$$ −33300.0 −1.84126
$$690$$ 5400.00 0.297934
$$691$$ 32996.0 1.81654 0.908268 0.418388i $$-0.137405\pi$$
0.908268 + 0.418388i $$0.137405\pi$$
$$692$$ −426.000 −0.0234019
$$693$$ 0 0
$$694$$ 15732.0 0.860488
$$695$$ 11620.0 0.634204
$$696$$ −4914.00 −0.267622
$$697$$ 17820.0 0.968408
$$698$$ 18906.0 1.02522
$$699$$ −4374.00 −0.236681
$$700$$ −500.000 −0.0269975
$$701$$ 25902.0 1.39558 0.697792 0.716300i $$-0.254166\pi$$
0.697792 + 0.716300i $$0.254166\pi$$
$$702$$ −5994.00 −0.322263
$$703$$ −8680.00 −0.465679
$$704$$ 0 0
$$705$$ −360.000 −0.0192318
$$706$$ −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 $$0.758406\pi$$
$$710$$ −4320.00 −0.228347
$$711$$ 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 $$0.140793\pi$$
$$720$$ 3195.00 0.165376
$$721$$ −9040.00 −0.466945
$$722$$ −25551.0 −1.31705
$$723$$ 2598.00 0.133639
$$724$$ −3130.00 −0.160671
$$725$$ 1950.00 0.0998913
$$726$$ 0 0
$$727$$ 28028.0 1.42985 0.714925 0.699201i $$-0.246461\pi$$
0.714925 + 0.699201i $$0.246461\pi$$
$$728$$ 31080.0 1.58228
$$729$$ 729.000 0.0370370
$$730$$ 6450.00 0.327021
$$731$$ 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
$$735$$ 855.000 0.0429077
$$736$$ −5400.00 −0.270444
$$737$$ 0 0
$$738$$ 8910.00 0.444420
$$739$$ −15284.0 −0.760800 −0.380400 0.924822i $$-0.624214\pi$$
−0.380400 + 0.924822i $$0.624214\pi$$
$$740$$ 350.000 0.0173868
$$741$$ 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
$$745$$ 1290.00 0.0634388
$$746$$ 32406.0 1.59044
$$747$$ −1404.00 −0.0687680
$$748$$ 0 0
$$749$$ −28080.0 −1.36985
$$750$$ −1125.00 −0.0547723
$$751$$ 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
$$755$$ −4040.00 −0.194743
$$756$$ 540.000 0.0259783
$$757$$ −38662.0 −1.85627 −0.928134 0.372247i $$-0.878587\pi$$
−0.928134 + 0.372247i $$0.878587\pi$$
$$758$$ −4380.00 −0.209880
$$759$$ 0 0
$$760$$ −13020.0 −0.621428
$$761$$ −23874.0 −1.13723 −0.568615 0.822604i $$-0.692521\pi$$
−0.568615 + 0.822604i $$0.692521\pi$$
$$762$$ −11196.0 −0.532268
$$763$$ −29480.0 −1.39875
$$764$$ 3576.00 0.169339
$$765$$ 2430.00 0.114846
$$766$$ 14616.0 0.689422
$$767$$ −1776.00 −0.0836084
$$768$$ −4539.00 −0.213264
$$769$$ −23618.0 −1.10753 −0.553763 0.832675i $$-0.686808\pi$$
−0.553763 + 0.832675i $$0.686808\pi$$
$$770$$ 0 0
$$771$$ −7578.00 −0.353975
$$772$$ −2666.00 −0.124289
$$773$$ 11538.0 0.536860 0.268430 0.963299i $$-0.413495\pi$$
0.268430 + 0.963299i $$0.413495\pi$$
$$774$$ 2484.00 0.115356
$$775$$ 5000.00 0.231749
$$776$$ −6006.00 −0.277839
$$777$$ −4200.00 −0.193918
$$778$$ 42138.0 1.94180
$$779$$ −40920.0 −1.88204
$$780$$ −1110.00 −0.0509543
$$781$$ 0 0
$$782$$ −19440.0 −0.888968
$$783$$ −2106.00 −0.0961204
$$784$$ −4047.00 −0.184357
$$785$$ −11890.0 −0.540602
$$786$$ −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
$$790$$ 7800.00 0.351280
$$791$$ −21720.0 −0.976327
$$792$$ 0 0
$$793$$ −23828.0 −1.06703
$$794$$ 8202.00 0.366597
$$795$$ 6750.00 0.301129
$$796$$ −3832.00 −0.170630
$$797$$ −11334.0 −0.503728 −0.251864 0.967763i $$-0.581043\pi$$
−0.251864 + 0.967763i $$0.581043\pi$$
$$798$$ −22320.0 −0.990125
$$799$$ 1296.00 0.0573832
$$800$$ 1125.00 0.0497184
$$801$$ 9234.00 0.407325
$$802$$ 47826.0 2.10573
$$803$$ 0 0
$$804$$ 588.000 0.0257925
$$805$$ −12000.0 −0.525397
$$806$$ 44400.0 1.94035
$$807$$ 7722.00 0.336837
$$808$$ 36414.0 1.58545
$$809$$ −44730.0 −1.94391 −0.971955 0.235167i $$-0.924436\pi$$
−0.971955 + 0.235167i $$0.924436\pi$$
$$810$$ 1215.00 0.0527046
$$811$$ 42748.0 1.85091 0.925453 0.378862i $$-0.123684\pi$$
0.925453 + 0.378862i $$0.123684\pi$$
$$812$$ −1560.00 −0.0674203
$$813$$ −9552.00 −0.412058
$$814$$ 0 0
$$815$$ 260.000 0.0111747
$$816$$ −11502.0 −0.493444
$$817$$ −11408.0 −0.488513
$$818$$ 26142.0 1.11740
$$819$$ 13320.0 0.568301
$$820$$ 1650.00 0.0702689
$$821$$ 31686.0 1.34695 0.673477 0.739208i $$-0.264800\pi$$
0.673477 + 0.739208i $$0.264800\pi$$
$$822$$ 19062.0 0.808836
$$823$$ 11036.0 0.467425 0.233713 0.972306i $$-0.424913\pi$$
0.233713 + 0.972306i $$0.424913\pi$$
$$824$$ 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 $$0.391562\pi$$
$$830$$ −2340.00 −0.0978585
$$831$$ 11886.0 0.496174
$$832$$ −32042.0 −1.33516
$$833$$ −3078.00 −0.128027
$$834$$ −20916.0 −0.868419
$$835$$ −18600.0 −0.770874
$$836$$ 0 0
$$837$$ −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 $$0.408363\pi$$
$$840$$ −6300.00 −0.258775
$$841$$ −18305.0 −0.750543
$$842$$ −33162.0 −1.35729
$$843$$ −24858.0 −1.01560
$$844$$ −1100.00 −0.0448620
$$845$$ −16395.0 −0.667462
$$846$$ 648.000 0.0263342
$$847$$ 0 0
$$848$$ −31950.0 −1.29383
$$849$$ −8148.00 −0.329374
$$850$$ 4050.00 0.163428
$$851$$ 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
$$855$$ −5580.00 −0.223195
$$856$$ 29484.0 1.17727
$$857$$ 7314.00 0.291530 0.145765 0.989319i $$-0.453436\pi$$
0.145765 + 0.989319i $$0.453436\pi$$
$$858$$ 0 0
$$859$$ −28780.0 −1.14314 −0.571572 0.820552i $$-0.693666\pi$$
−0.571572 + 0.820552i $$0.693666\pi$$
$$860$$ 460.000 0.0182394
$$861$$ −19800.0 −0.783719
$$862$$ 2160.00 0.0853479
$$863$$ −32688.0 −1.28935 −0.644677 0.764455i $$-0.723008\pi$$
−0.644677 + 0.764455i $$0.723008\pi$$
$$864$$ −1215.00 −0.0478416
$$865$$ 2130.00 0.0837251
$$866$$ 46866.0 1.83900
$$867$$ 5991.00 0.234677
$$868$$ −4000.00 −0.156416
$$869$$ 0 0
$$870$$ −3510.00 −0.136782
$$871$$ 14504.0 0.564236
$$872$$ 30954.0 1.20210
$$873$$ −2574.00 −0.0997900
$$874$$ 44640.0 1.72766
$$875$$ 2500.00 0.0965891
$$876$$ −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 $$-0.433449\pi$$
0.207557 + 0.978223i $$0.433449\pi$$
$$884$$ 3996.00 0.152036
$$885$$ 360.000 0.0136737
$$886$$ 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
$$890$$ 15390.0 0.579634
$$891$$ 0 0
$$892$$ 1964.00 0.0737215
$$893$$ −2976.00 −0.111521
$$894$$ −2322.00 −0.0868672
$$895$$ 7200.00 0.268904
$$896$$ 33180.0 1.23713
$$897$$ −26640.0 −0.991621
$$898$$ −27054.0 −1.00535
$$899$$ 15600.0 0.578742
$$900$$ 225.000 0.00833333
$$901$$ −24300.0 −0.898502
$$902$$ 0 0
$$903$$ −5520.00 −0.203426
$$904$$ 22806.0 0.839067
$$905$$ 15650.0 0.574833
$$906$$ 7272.00 0.266662
$$907$$ −14884.0 −0.544890 −0.272445 0.962171i $$-0.587832\pi$$
−0.272445 + 0.962171i $$0.587832\pi$$
$$908$$ −660.000 −0.0241221
$$909$$ 15606.0 0.569437
$$910$$ 22200.0 0.808706
$$911$$ −1248.00 −0.0453876 −0.0226938 0.999742i $$-0.507224\pi$$
−0.0226938 + 0.999742i $$0.507224\pi$$
$$912$$ 26412.0 0.958979
$$913$$ 0 0
$$914$$ −11010.0 −0.398445
$$915$$ 4830.00 0.174508
$$916$$ −1906.00 −0.0687511
$$917$$ 46560.0 1.67671
$$918$$ −4374.00 −0.157259
$$919$$ 6640.00 0.238339 0.119169 0.992874i $$-0.461977\pi$$
0.119169 + 0.992874i $$0.461977\pi$$
$$920$$ 12600.0 0.451532
$$921$$ 27708.0 0.991324
$$922$$ 52686.0 1.88191
$$923$$ 21312.0 0.760014
$$924$$ 0 0
$$925$$ −1750.00 −0.0622050
$$926$$ −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 $$0.322645\pi$$
$$930$$ −9000.00 −0.317335
$$931$$ 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
$$940$$ 120.000 0.00416380
$$941$$ −6090.00 −0.210976 −0.105488 0.994421i $$-0.533640\pi$$
−0.105488 + 0.994421i $$0.533640\pi$$
$$942$$ 21402.0 0.740249
$$943$$ 39600.0 1.36750
$$944$$ −1704.00 −0.0587505
$$945$$ −2700.00 −0.0929429
$$946$$ 0 0
$$947$$ 56388.0 1.93491 0.967457 0.253035i $$-0.0814288\pi$$
0.967457 + 0.253035i $$0.0814288\pi$$
$$948$$ −1560.00 −0.0534456
$$949$$ −31820.0 −1.08843
$$950$$ −9300.00 −0.317612
$$951$$ 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
$$955$$ −17880.0 −0.605846
$$956$$ −1176.00 −0.0397851
$$957$$ 0 0
$$958$$ 6840.00 0.230679
$$959$$ −42360.0 −1.42636
$$960$$ 6495.00 0.218360
$$961$$ 10209.0 0.342687
$$962$$ −15540.0 −0.520821
$$963$$ 12636.0 0.422834
$$964$$ −866.000 −0.0289336
$$965$$ 13330.0 0.444671
$$966$$ 21600.0 0.719429
$$967$$ 42316.0 1.40723 0.703615 0.710582i $$-0.251568\pi$$
0.703615 + 0.710582i $$0.251568\pi$$
$$968$$ 0 0
$$969$$ 20088.0 0.665964
$$970$$ −4290.00 −0.142004
$$971$$ 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
$$975$$ 5550.00 0.182300
$$976$$ −22862.0 −0.749790
$$977$$ −6906.00 −0.226144 −0.113072 0.993587i $$-0.536069\pi$$
−0.113072 + 0.993587i $$0.536069\pi$$
$$978$$ −468.000 −0.0153016
$$979$$ 0 0
$$980$$ −285.000 −0.00928979
$$981$$ 13266.0 0.431754
$$982$$ −56736.0 −1.84371
$$983$$ 6960.00 0.225829 0.112914 0.993605i $$-0.463981\pi$$
0.112914 + 0.993605i $$0.463981\pi$$
$$984$$ 20790.0 0.673538
$$985$$ −13590.0 −0.439608
$$986$$ 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
$$995$$ 19160.0 0.610465
$$996$$ 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 1815.4.a.a.1.1 1
11.10 odd 2 15.4.a.b.1.1 1
33.32 even 2 45.4.a.b.1.1 1
44.43 even 2 240.4.a.f.1.1 1
55.32 even 4 75.4.b.a.49.2 2
55.43 even 4 75.4.b.a.49.1 2
55.54 odd 2 75.4.a.a.1.1 1
77.76 even 2 735.4.a.i.1.1 1
88.21 odd 2 960.4.a.bi.1.1 1
88.43 even 2 960.4.a.l.1.1 1
99.32 even 6 405.4.e.k.136.1 2
99.43 odd 6 405.4.e.d.271.1 2
99.65 even 6 405.4.e.k.271.1 2
99.76 odd 6 405.4.e.d.136.1 2
132.131 odd 2 720.4.a.r.1.1 1
165.32 odd 4 225.4.b.d.199.1 2
165.98 odd 4 225.4.b.d.199.2 2
165.164 even 2 225.4.a.g.1.1 1
220.43 odd 4 1200.4.f.m.49.2 2
220.87 odd 4 1200.4.f.m.49.1 2
220.219 even 2 1200.4.a.o.1.1 1
231.230 odd 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 11.10 odd 2
45.4.a.b.1.1 1 33.32 even 2
75.4.a.a.1.1 1 55.54 odd 2
75.4.b.a.49.1 2 55.43 even 4
75.4.b.a.49.2 2 55.32 even 4
225.4.a.g.1.1 1 165.164 even 2
225.4.b.d.199.1 2 165.32 odd 4
225.4.b.d.199.2 2 165.98 odd 4
240.4.a.f.1.1 1 44.43 even 2
405.4.e.d.136.1 2 99.76 odd 6
405.4.e.d.271.1 2 99.43 odd 6
405.4.e.k.136.1 2 99.32 even 6
405.4.e.k.271.1 2 99.65 even 6
720.4.a.r.1.1 1 132.131 odd 2
735.4.a.i.1.1 1 77.76 even 2
960.4.a.l.1.1 1 88.43 even 2
960.4.a.bi.1.1 1 88.21 odd 2
1200.4.a.o.1.1 1 220.219 even 2
1200.4.f.m.49.1 2 220.87 odd 4
1200.4.f.m.49.2 2 220.43 odd 4
1815.4.a.a.1.1 1 1.1 even 1 trivial
2205.4.a.c.1.1 1 231.230 odd 2