# Properties

 Label 1200.4.f.m.49.1 Level $1200$ Weight $4$ Character 1200.49 Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1200.f (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$70.8022920069$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.49 Dual form 1200.4.f.m.49.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000i q^{3} -20.0000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{3} -20.0000i q^{7} -9.00000 q^{9} +24.0000 q^{11} -74.0000i q^{13} +54.0000i q^{17} -124.000 q^{19} -60.0000 q^{21} -120.000i q^{23} +27.0000i q^{27} +78.0000 q^{29} -200.000 q^{31} -72.0000i q^{33} -70.0000i q^{37} -222.000 q^{39} +330.000 q^{41} +92.0000i q^{43} +24.0000i q^{47} -57.0000 q^{49} +162.000 q^{51} -450.000i q^{53} +372.000i q^{57} +24.0000 q^{59} -322.000 q^{61} +180.000i q^{63} +196.000i q^{67} -360.000 q^{69} +288.000 q^{71} +430.000i q^{73} -480.000i q^{77} -520.000 q^{79} +81.0000 q^{81} +156.000i q^{83} -234.000i q^{87} -1026.00 q^{89} -1480.00 q^{91} +600.000i q^{93} -286.000i q^{97} -216.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} + 48 q^{11} - 248 q^{19} - 120 q^{21} + 156 q^{29} - 400 q^{31} - 444 q^{39} + 660 q^{41} - 114 q^{49} + 324 q^{51} + 48 q^{59} - 644 q^{61} - 720 q^{69} + 576 q^{71} - 1040 q^{79} + 162 q^{81} - 2052 q^{89} - 2960 q^{91} - 432 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 + 48 * q^11 - 248 * q^19 - 120 * q^21 + 156 * q^29 - 400 * q^31 - 444 * q^39 + 660 * q^41 - 114 * q^49 + 324 * q^51 + 48 * q^59 - 644 * q^61 - 720 * q^69 + 576 * q^71 - 1040 * q^79 + 162 * q^81 - 2052 * q^89 - 2960 * q^91 - 432 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 3.00000i − 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 20.0000i − 1.07990i −0.841698 0.539949i $$-0.818443\pi$$
0.841698 0.539949i $$-0.181557\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 24.0000 0.657843 0.328921 0.944357i $$-0.393315\pi$$
0.328921 + 0.944357i $$0.393315\pi$$
$$12$$ 0 0
$$13$$ − 74.0000i − 1.57876i −0.613904 0.789381i $$-0.710402\pi$$
0.613904 0.789381i $$-0.289598\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 54.0000i 0.770407i 0.922832 + 0.385204i $$0.125869\pi$$
−0.922832 + 0.385204i $$0.874131\pi$$
$$18$$ 0 0
$$19$$ −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
$$22$$ 0 0
$$23$$ − 120.000i − 1.08790i −0.839117 0.543951i $$-0.816928\pi$$
0.839117 0.543951i $$-0.183072\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 27.0000i 0.192450i
$$28$$ 0 0
$$29$$ 78.0000 0.499456 0.249728 0.968316i $$-0.419659\pi$$
0.249728 + 0.968316i $$0.419659\pi$$
$$30$$ 0 0
$$31$$ −200.000 −1.15874 −0.579372 0.815063i $$-0.696702\pi$$
−0.579372 + 0.815063i $$0.696702\pi$$
$$32$$ 0 0
$$33$$ − 72.0000i − 0.379806i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 70.0000i − 0.311025i −0.987834 0.155513i $$-0.950297\pi$$
0.987834 0.155513i $$-0.0497029\pi$$
$$38$$ 0 0
$$39$$ −222.000 −0.911499
$$40$$ 0 0
$$41$$ 330.000 1.25701 0.628504 0.777806i $$-0.283668\pi$$
0.628504 + 0.777806i $$0.283668\pi$$
$$42$$ 0 0
$$43$$ 92.0000i 0.326276i 0.986603 + 0.163138i $$0.0521616\pi$$
−0.986603 + 0.163138i $$0.947838\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 24.0000i 0.0744843i 0.999306 + 0.0372421i $$0.0118573\pi$$
−0.999306 + 0.0372421i $$0.988143\pi$$
$$48$$ 0 0
$$49$$ −57.0000 −0.166181
$$50$$ 0 0
$$51$$ 162.000 0.444795
$$52$$ 0 0
$$53$$ − 450.000i − 1.16627i −0.812376 0.583134i $$-0.801826\pi$$
0.812376 0.583134i $$-0.198174\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 372.000i 0.864432i
$$58$$ 0 0
$$59$$ 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 $$0.609728\pi$$
$$62$$ 0 0
$$63$$ 180.000i 0.359966i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 196.000i 0.357391i 0.983904 + 0.178696i $$0.0571877\pi$$
−0.983904 + 0.178696i $$0.942812\pi$$
$$68$$ 0 0
$$69$$ −360.000 −0.628100
$$70$$ 0 0
$$71$$ 288.000 0.481399 0.240699 0.970600i $$-0.422623\pi$$
0.240699 + 0.970600i $$0.422623\pi$$
$$72$$ 0 0
$$73$$ 430.000i 0.689420i 0.938709 + 0.344710i $$0.112023\pi$$
−0.938709 + 0.344710i $$0.887977\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 480.000i − 0.710404i
$$78$$ 0 0
$$79$$ −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
$$82$$ 0 0
$$83$$ 156.000i 0.206304i 0.994666 + 0.103152i $$0.0328928\pi$$
−0.994666 + 0.103152i $$0.967107\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 234.000i − 0.288361i
$$88$$ 0 0
$$89$$ −1026.00 −1.22198 −0.610988 0.791640i $$-0.709227\pi$$
−0.610988 + 0.791640i $$0.709227\pi$$
$$90$$ 0 0
$$91$$ −1480.00 −1.70490
$$92$$ 0 0
$$93$$ 600.000i 0.669001i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 286.000i − 0.299370i −0.988734 0.149685i $$-0.952174\pi$$
0.988734 0.149685i $$-0.0478260\pi$$
$$98$$ 0 0
$$99$$ −216.000 −0.219281
$$100$$ 0 0
$$101$$ −1734.00 −1.70831 −0.854156 0.520017i $$-0.825925\pi$$
−0.854156 + 0.520017i $$0.825925\pi$$
$$102$$ 0 0
$$103$$ 452.000i 0.432397i 0.976349 + 0.216198i $$0.0693658\pi$$
−0.976349 + 0.216198i $$0.930634\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1404.00i 1.26850i 0.773127 + 0.634251i $$0.218692\pi$$
−0.773127 + 0.634251i $$0.781308\pi$$
$$108$$ 0 0
$$109$$ 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
$$112$$ 0 0
$$113$$ − 1086.00i − 0.904091i −0.891995 0.452046i $$-0.850694\pi$$
0.891995 0.452046i $$-0.149306\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 666.000i 0.526254i
$$118$$ 0 0
$$119$$ 1080.00 0.831962
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 0 0
$$123$$ − 990.000i − 0.725734i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 1244.00i − 0.869190i −0.900626 0.434595i $$-0.856891\pi$$
0.900626 0.434595i $$-0.143109\pi$$
$$128$$ 0 0
$$129$$ 276.000 0.188376
$$130$$ 0 0
$$131$$ −2328.00 −1.55266 −0.776329 0.630327i $$-0.782921\pi$$
−0.776329 + 0.630327i $$0.782921\pi$$
$$132$$ 0 0
$$133$$ 2480.00i 1.61687i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 2118.00i 1.32082i 0.750903 + 0.660412i $$0.229618\pi$$
−0.750903 + 0.660412i $$0.770382\pi$$
$$138$$ 0 0
$$139$$ 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
$$142$$ 0 0
$$143$$ − 1776.00i − 1.03858i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 171.000i 0.0959445i
$$148$$ 0 0
$$149$$ −258.000 −0.141854 −0.0709268 0.997482i $$-0.522596\pi$$
−0.0709268 + 0.997482i $$0.522596\pi$$
$$150$$ 0 0
$$151$$ 808.000 0.435458 0.217729 0.976009i $$-0.430135\pi$$
0.217729 + 0.976009i $$0.430135\pi$$
$$152$$ 0 0
$$153$$ − 486.000i − 0.256802i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2378.00i 1.20882i 0.796673 + 0.604411i $$0.206592\pi$$
−0.796673 + 0.604411i $$0.793408\pi$$
$$158$$ 0 0
$$159$$ −1350.00 −0.673346
$$160$$ 0 0
$$161$$ −2400.00 −1.17482
$$162$$ 0 0
$$163$$ − 52.0000i − 0.0249874i −0.999922 0.0124937i $$-0.996023\pi$$
0.999922 0.0124937i $$-0.00397698\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3720.00i 1.72373i 0.507141 + 0.861863i $$0.330702\pi$$
−0.507141 + 0.861863i $$0.669298\pi$$
$$168$$ 0 0
$$169$$ −3279.00 −1.49249
$$170$$ 0 0
$$171$$ 1116.00 0.499080
$$172$$ 0 0
$$173$$ − 426.000i − 0.187215i −0.995609 0.0936075i $$-0.970160\pi$$
0.995609 0.0936075i $$-0.0298399\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 72.0000i − 0.0305754i
$$178$$ 0 0
$$179$$ −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 $$0.722179\pi$$
$$182$$ 0 0
$$183$$ 966.000i 0.390212i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1296.00i 0.506807i
$$188$$ 0 0
$$189$$ 540.000 0.207827
$$190$$ 0 0
$$191$$ −3576.00 −1.35471 −0.677357 0.735655i $$-0.736875\pi$$
−0.677357 + 0.735655i $$0.736875\pi$$
$$192$$ 0 0
$$193$$ − 2666.00i − 0.994315i −0.867660 0.497158i $$-0.834377\pi$$
0.867660 0.497158i $$-0.165623\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 2718.00i − 0.982992i −0.870880 0.491496i $$-0.836450\pi$$
0.870880 0.491496i $$-0.163550\pi$$
$$198$$ 0 0
$$199$$ −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
$$202$$ 0 0
$$203$$ − 1560.00i − 0.539362i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1080.00i 0.362634i
$$208$$ 0 0
$$209$$ −2976.00 −0.984948
$$210$$ 0 0
$$211$$ −1100.00 −0.358896 −0.179448 0.983767i $$-0.557431\pi$$
−0.179448 + 0.983767i $$0.557431\pi$$
$$212$$ 0 0
$$213$$ − 864.000i − 0.277936i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4000.00i 1.25133i
$$218$$ 0 0
$$219$$ 1290.00 0.398037
$$220$$ 0 0
$$221$$ 3996.00 1.21629
$$222$$ 0 0
$$223$$ 1964.00i 0.589772i 0.955532 + 0.294886i $$0.0952817\pi$$
−0.955532 + 0.294886i $$0.904718\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 660.000i − 0.192977i −0.995334 0.0964884i $$-0.969239\pi$$
0.995334 0.0964884i $$-0.0307611\pi$$
$$228$$ 0 0
$$229$$ 1906.00 0.550009 0.275004 0.961443i $$-0.411321\pi$$
0.275004 + 0.961443i $$0.411321\pi$$
$$230$$ 0 0
$$231$$ −1440.00 −0.410152
$$232$$ 0 0
$$233$$ 1458.00i 0.409943i 0.978768 + 0.204972i $$0.0657102\pi$$
−0.978768 + 0.204972i $$0.934290\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 1560.00i 0.427565i
$$238$$ 0 0
$$239$$ 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 $$0.463078\pi$$
$$242$$ 0 0
$$243$$ − 243.000i − 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 9176.00i 2.36379i
$$248$$ 0 0
$$249$$ 468.000 0.119110
$$250$$ 0 0
$$251$$ −432.000 −0.108636 −0.0543179 0.998524i $$-0.517298\pi$$
−0.0543179 + 0.998524i $$0.517298\pi$$
$$252$$ 0 0
$$253$$ − 2880.00i − 0.715668i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2526.00i 0.613103i 0.951854 + 0.306552i $$0.0991752\pi$$
−0.951854 + 0.306552i $$0.900825\pi$$
$$258$$ 0 0
$$259$$ −1400.00 −0.335876
$$260$$ 0 0
$$261$$ −702.000 −0.166485
$$262$$ 0 0
$$263$$ 5448.00i 1.27733i 0.769484 + 0.638666i $$0.220513\pi$$
−0.769484 + 0.638666i $$0.779487\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 3078.00i 0.705508i
$$268$$ 0 0
$$269$$ 2574.00 0.583418 0.291709 0.956507i $$-0.405776\pi$$
0.291709 + 0.956507i $$0.405776\pi$$
$$270$$ 0 0
$$271$$ 3184.00 0.713706 0.356853 0.934161i $$-0.383850\pi$$
0.356853 + 0.934161i $$0.383850\pi$$
$$272$$ 0 0
$$273$$ 4440.00i 0.984326i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 3962.00i 0.859399i 0.902972 + 0.429699i $$0.141380\pi$$
−0.902972 + 0.429699i $$0.858620\pi$$
$$278$$ 0 0
$$279$$ 1800.00 0.386248
$$280$$ 0 0
$$281$$ −8286.00 −1.75908 −0.879540 0.475825i $$-0.842149\pi$$
−0.879540 + 0.475825i $$0.842149\pi$$
$$282$$ 0 0
$$283$$ − 2716.00i − 0.570493i −0.958454 0.285246i $$-0.907925\pi$$
0.958454 0.285246i $$-0.0920754\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 6600.00i − 1.35744i
$$288$$ 0 0
$$289$$ 1997.00 0.406473
$$290$$ 0 0
$$291$$ −858.000 −0.172841
$$292$$ 0 0
$$293$$ − 6018.00i − 1.19992i −0.800032 0.599958i $$-0.795184\pi$$
0.800032 0.599958i $$-0.204816\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 648.000i 0.126602i
$$298$$ 0 0
$$299$$ −8880.00 −1.71754
$$300$$ 0 0
$$301$$ 1840.00 0.352345
$$302$$ 0 0
$$303$$ 5202.00i 0.986294i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 9236.00i − 1.71702i −0.512793 0.858512i $$-0.671389\pi$$
0.512793 0.858512i $$-0.328611\pi$$
$$308$$ 0 0
$$309$$ 1356.00 0.249644
$$310$$ 0 0
$$311$$ −1536.00 −0.280060 −0.140030 0.990147i $$-0.544720\pi$$
−0.140030 + 0.990147i $$0.544720\pi$$
$$312$$ 0 0
$$313$$ 7342.00i 1.32586i 0.748681 + 0.662930i $$0.230687\pi$$
−0.748681 + 0.662930i $$0.769313\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 3894.00i − 0.689933i −0.938615 0.344967i $$-0.887890\pi$$
0.938615 0.344967i $$-0.112110\pi$$
$$318$$ 0 0
$$319$$ 1872.00 0.328564
$$320$$ 0 0
$$321$$ 4212.00 0.732370
$$322$$ 0 0
$$323$$ − 6696.00i − 1.15348i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 4422.00i − 0.747820i
$$328$$ 0 0
$$329$$ 480.000 0.0804354
$$330$$ 0 0
$$331$$ −3692.00 −0.613084 −0.306542 0.951857i $$-0.599172\pi$$
−0.306542 + 0.951857i $$0.599172\pi$$
$$332$$ 0 0
$$333$$ 630.000i 0.103675i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 8998.00i − 1.45446i −0.686395 0.727229i $$-0.740808\pi$$
0.686395 0.727229i $$-0.259192\pi$$
$$338$$ 0 0
$$339$$ −3258.00 −0.521977
$$340$$ 0 0
$$341$$ −4800.00 −0.762271
$$342$$ 0 0
$$343$$ − 5720.00i − 0.900440i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 5244.00i − 0.811276i −0.914034 0.405638i $$-0.867049\pi$$
0.914034 0.405638i $$-0.132951\pi$$
$$348$$ 0 0
$$349$$ −6302.00 −0.966585 −0.483293 0.875459i $$-0.660559\pi$$
−0.483293 + 0.875459i $$0.660559\pi$$
$$350$$ 0 0
$$351$$ 1998.00 0.303833
$$352$$ 0 0
$$353$$ − 3414.00i − 0.514756i −0.966311 0.257378i $$-0.917141\pi$$
0.966311 0.257378i $$-0.0828586\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 3240.00i − 0.480333i
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ 2265.00i 0.327498i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3508.00i 0.498954i 0.968381 + 0.249477i $$0.0802587\pi$$
−0.968381 + 0.249477i $$0.919741\pi$$
$$368$$ 0 0
$$369$$ −2970.00 −0.419003
$$370$$ 0 0
$$371$$ −9000.00 −1.25945
$$372$$ 0 0
$$373$$ − 10802.0i − 1.49948i −0.661732 0.749740i $$-0.730178\pi$$
0.661732 0.749740i $$-0.269822\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 5772.00i − 0.788523i
$$378$$ 0 0
$$379$$ 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
$$382$$ 0 0
$$383$$ − 4872.00i − 0.649994i −0.945715 0.324997i $$-0.894637\pi$$
0.945715 0.324997i $$-0.105363\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 828.000i − 0.108759i
$$388$$ 0 0
$$389$$ 14046.0 1.83075 0.915373 0.402606i $$-0.131896\pi$$
0.915373 + 0.402606i $$0.131896\pi$$
$$390$$ 0 0
$$391$$ 6480.00 0.838127
$$392$$ 0 0
$$393$$ 6984.00i 0.896428i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 2734.00i − 0.345631i −0.984954 0.172816i $$-0.944714\pi$$
0.984954 0.172816i $$-0.0552864\pi$$
$$398$$ 0 0
$$399$$ 7440.00 0.933498
$$400$$ 0 0
$$401$$ −15942.0 −1.98530 −0.992650 0.121019i $$-0.961384\pi$$
−0.992650 + 0.121019i $$0.961384\pi$$
$$402$$ 0 0
$$403$$ 14800.0i 1.82938i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 1680.00i − 0.204606i
$$408$$ 0 0
$$409$$ −8714.00 −1.05350 −0.526748 0.850022i $$-0.676589\pi$$
−0.526748 + 0.850022i $$0.676589\pi$$
$$410$$ 0 0
$$411$$ 6354.00 0.762578
$$412$$ 0 0
$$413$$ − 480.000i − 0.0571895i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 6972.00i − 0.818754i
$$418$$ 0 0
$$419$$ 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 $$0.279004\pi$$
$$422$$ 0 0
$$423$$ − 216.000i − 0.0248281i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 6440.00i 0.729868i
$$428$$ 0 0
$$429$$ −5328.00 −0.599623
$$430$$ 0 0
$$431$$ −720.000 −0.0804668 −0.0402334 0.999190i $$-0.512810\pi$$
−0.0402334 + 0.999190i $$0.512810\pi$$
$$432$$ 0 0
$$433$$ 15622.0i 1.73382i 0.498462 + 0.866912i $$0.333898\pi$$
−0.498462 + 0.866912i $$0.666102\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 14880.0i 1.62885i
$$438$$ 0 0
$$439$$ −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
$$442$$ 0 0
$$443$$ − 16116.0i − 1.72843i −0.503123 0.864215i $$-0.667816\pi$$
0.503123 0.864215i $$-0.332184\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 774.000i 0.0818992i
$$448$$ 0 0
$$449$$ −9018.00 −0.947852 −0.473926 0.880565i $$-0.657164\pi$$
−0.473926 + 0.880565i $$0.657164\pi$$
$$450$$ 0 0
$$451$$ 7920.00 0.826914
$$452$$ 0 0
$$453$$ − 2424.00i − 0.251412i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 3670.00i − 0.375657i −0.982202 0.187829i $$-0.939855\pi$$
0.982202 0.187829i $$-0.0601450\pi$$
$$458$$ 0 0
$$459$$ −1458.00 −0.148265
$$460$$ 0 0
$$461$$ 17562.0 1.77428 0.887141 0.461499i $$-0.152688\pi$$
0.887141 + 0.461499i $$0.152688\pi$$
$$462$$ 0 0
$$463$$ 1172.00i 0.117640i 0.998269 + 0.0588202i $$0.0187338\pi$$
−0.998269 + 0.0588202i $$0.981266\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 6876.00i − 0.681335i −0.940184 0.340667i $$-0.889347\pi$$
0.940184 0.340667i $$-0.110653\pi$$
$$468$$ 0 0
$$469$$ 3920.00 0.385946
$$470$$ 0 0
$$471$$ 7134.00 0.697914
$$472$$ 0 0
$$473$$ 2208.00i 0.214638i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 4050.00i 0.388756i
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ 7200.00i 0.678284i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3076.00i 0.286215i 0.989707 + 0.143108i $$0.0457095\pi$$
−0.989707 + 0.143108i $$0.954290\pi$$
$$488$$ 0 0
$$489$$ −156.000 −0.0144265
$$490$$ 0 0
$$491$$ 18912.0 1.73826 0.869131 0.494582i $$-0.164679\pi$$
0.869131 + 0.494582i $$0.164679\pi$$
$$492$$ 0 0
$$493$$ 4212.00i 0.384785i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 5760.00i − 0.519862i
$$498$$ 0 0
$$499$$ 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
$$502$$ 0 0
$$503$$ − 10656.0i − 0.944588i −0.881441 0.472294i $$-0.843426\pi$$
0.881441 0.472294i $$-0.156574\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9837.00i 0.861689i
$$508$$ 0 0
$$509$$ 2766.00 0.240866 0.120433 0.992721i $$-0.461572\pi$$
0.120433 + 0.992721i $$0.461572\pi$$
$$510$$ 0 0
$$511$$ 8600.00 0.744504
$$512$$ 0 0
$$513$$ − 3348.00i − 0.288144i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 576.000i 0.0489989i
$$518$$ 0 0
$$519$$ −1278.00 −0.108089
$$520$$ 0 0
$$521$$ 10530.0 0.885466 0.442733 0.896654i $$-0.354009\pi$$
0.442733 + 0.896654i $$0.354009\pi$$
$$522$$ 0 0
$$523$$ 12692.0i 1.06115i 0.847637 + 0.530576i $$0.178024\pi$$
−0.847637 + 0.530576i $$0.821976\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 10800.0i − 0.892705i
$$528$$ 0 0
$$529$$ −2233.00 −0.183529
$$530$$ 0 0
$$531$$ −216.000 −0.0176527
$$532$$ 0 0
$$533$$ − 24420.0i − 1.98452i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 4320.00i 0.347154i
$$538$$ 0 0
$$539$$ −1368.00 −0.109321
$$540$$ 0 0
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ 0 0
$$543$$ 9390.00i 0.742106i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 3620.00i − 0.282962i −0.989941 0.141481i $$-0.954814\pi$$
0.989941 0.141481i $$-0.0451864\pi$$
$$548$$ 0 0
$$549$$ 2898.00 0.225289
$$550$$ 0 0
$$551$$ −9672.00 −0.747806
$$552$$ 0 0
$$553$$ 10400.0i 0.799734i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 14166.0i − 1.07762i −0.842428 0.538809i $$-0.818875\pi$$
0.842428 0.538809i $$-0.181125\pi$$
$$558$$ 0 0
$$559$$ 6808.00 0.515112
$$560$$ 0 0
$$561$$ 3888.00 0.292605
$$562$$ 0 0
$$563$$ − 13404.0i − 1.00339i −0.865043 0.501697i $$-0.832709\pi$$
0.865043 0.501697i $$-0.167291\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 1620.00i − 0.119989i
$$568$$ 0 0
$$569$$ 18654.0 1.37437 0.687185 0.726483i $$-0.258846\pi$$
0.687185 + 0.726483i $$0.258846\pi$$
$$570$$ 0 0
$$571$$ 7684.00 0.563162 0.281581 0.959537i $$-0.409141\pi$$
0.281581 + 0.959537i $$0.409141\pi$$
$$572$$ 0 0
$$573$$ 10728.0i 0.782144i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 1726.00i − 0.124531i −0.998060 0.0622654i $$-0.980167\pi$$
0.998060 0.0622654i $$-0.0198325\pi$$
$$578$$ 0 0
$$579$$ −7998.00 −0.574068
$$580$$ 0 0
$$581$$ 3120.00 0.222787
$$582$$ 0 0
$$583$$ − 10800.0i − 0.767222i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 10596.0i − 0.745049i −0.928022 0.372524i $$-0.878492\pi$$
0.928022 0.372524i $$-0.121508\pi$$
$$588$$ 0 0
$$589$$ 24800.0 1.73492
$$590$$ 0 0
$$591$$ −8154.00 −0.567531
$$592$$ 0 0
$$593$$ − 2862.00i − 0.198193i −0.995078 0.0990963i $$-0.968405\pi$$
0.995078 0.0990963i $$-0.0315952\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11496.0i 0.788107i
$$598$$ 0 0
$$599$$ −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 $$0.605331\pi$$
$$602$$ 0 0
$$603$$ − 1764.00i − 0.119130i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 17444.0i − 1.16644i −0.812314 0.583221i $$-0.801792\pi$$
0.812314 0.583221i $$-0.198208\pi$$
$$608$$ 0 0
$$609$$ −4680.00 −0.311401
$$610$$ 0 0
$$611$$ 1776.00 0.117593
$$612$$ 0 0
$$613$$ 2374.00i 0.156419i 0.996937 + 0.0782096i $$0.0249203\pi$$
−0.996937 + 0.0782096i $$0.975080\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 12162.0i − 0.793555i −0.917915 0.396778i $$-0.870128\pi$$
0.917915 0.396778i $$-0.129872\pi$$
$$618$$ 0 0
$$619$$ 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
$$622$$ 0 0
$$623$$ 20520.0i 1.31961i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 8928.00i 0.568660i
$$628$$ 0 0
$$629$$ 3780.00 0.239616
$$630$$ 0 0
$$631$$ 12688.0 0.800478 0.400239 0.916411i $$-0.368927\pi$$
0.400239 + 0.916411i $$0.368927\pi$$
$$632$$ 0 0
$$633$$ 3300.00i 0.207209i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4218.00i 0.262360i
$$638$$ 0 0
$$639$$ −2592.00 −0.160466
$$640$$ 0 0
$$641$$ −9150.00 −0.563812 −0.281906 0.959442i $$-0.590967\pi$$
−0.281906 + 0.959442i $$0.590967\pi$$
$$642$$ 0 0
$$643$$ 25292.0i 1.55120i 0.631227 + 0.775598i $$0.282552\pi$$
−0.631227 + 0.775598i $$0.717448\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 2736.00i 0.166249i 0.996539 + 0.0831246i $$0.0264900\pi$$
−0.996539 + 0.0831246i $$0.973510\pi$$
$$648$$ 0 0
$$649$$ 576.000 0.0348382
$$650$$ 0 0
$$651$$ 12000.0 0.722453
$$652$$ 0 0
$$653$$ − 22218.0i − 1.33148i −0.746183 0.665741i $$-0.768116\pi$$
0.746183 0.665741i $$-0.231884\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 3870.00i − 0.229807i
$$658$$ 0 0
$$659$$ 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 $$0.601133\pi$$
$$662$$ 0 0
$$663$$ − 11988.0i − 0.702225i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 9360.00i − 0.543359i
$$668$$ 0 0
$$669$$ 5892.00 0.340505
$$670$$ 0 0
$$671$$ −7728.00 −0.444614
$$672$$ 0 0
$$673$$ − 1370.00i − 0.0784690i −0.999230 0.0392345i $$-0.987508\pi$$
0.999230 0.0392345i $$-0.0124919\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 13758.0i − 0.781038i −0.920595 0.390519i $$-0.872296\pi$$
0.920595 0.390519i $$-0.127704\pi$$
$$678$$ 0 0
$$679$$ −5720.00 −0.323289
$$680$$ 0 0
$$681$$ −1980.00 −0.111415
$$682$$ 0 0
$$683$$ 11988.0i 0.671608i 0.941932 + 0.335804i $$0.109008\pi$$
−0.941932 + 0.335804i $$0.890992\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 5718.00i − 0.317548i
$$688$$ 0 0
$$689$$ −33300.0 −1.84126
$$690$$ 0 0
$$691$$ −32996.0 −1.81654 −0.908268 0.418388i $$-0.862595\pi$$
−0.908268 + 0.418388i $$0.862595\pi$$
$$692$$ 0 0
$$693$$ 4320.00i 0.236801i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 17820.0i 0.968408i
$$698$$ 0 0
$$699$$ 4374.00 0.236681
$$700$$ 0 0
$$701$$ −25902.0 −1.39558 −0.697792 0.716300i $$-0.745834\pi$$
−0.697792 + 0.716300i $$0.745834\pi$$
$$702$$ 0 0
$$703$$ 8680.00i 0.465679i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 34680.0i 1.84480i
$$708$$ 0 0
$$709$$ 27394.0 1.45106 0.725531 0.688189i $$-0.241594\pi$$
0.725531 + 0.688189i $$0.241594\pi$$
$$710$$ 0 0
$$711$$ 4680.00 0.246855
$$712$$ 0 0
$$713$$ 24000.0i 1.26060i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 3528.00i − 0.183760i
$$718$$ 0 0
$$719$$ 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
$$722$$ 0 0
$$723$$ − 2598.00i − 0.133639i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 28028.0i − 1.42985i −0.699201 0.714925i $$-0.746461\pi$$
0.699201 0.714925i $$-0.253539\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −4968.00 −0.251365
$$732$$ 0 0
$$733$$ − 18002.0i − 0.907120i −0.891226 0.453560i $$-0.850154\pi$$
0.891226 0.453560i $$-0.149846\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 4704.00i 0.235107i
$$738$$ 0 0
$$739$$ 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
$$742$$ 0 0
$$743$$ − 18768.0i − 0.926691i −0.886178 0.463345i $$-0.846649\pi$$
0.886178 0.463345i $$-0.153351\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 1404.00i − 0.0687680i
$$748$$ 0 0
$$749$$ 28080.0 1.36985
$$750$$ 0 0
$$751$$ −8696.00 −0.422532 −0.211266 0.977429i $$-0.567759\pi$$
−0.211266 + 0.977429i $$0.567759\pi$$
$$752$$ 0 0
$$753$$ 1296.00i 0.0627209i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 38662.0i − 1.85627i −0.372247 0.928134i $$-0.621413\pi$$
0.372247 0.928134i $$-0.378587\pi$$
$$758$$ 0 0
$$759$$ −8640.00 −0.413191
$$760$$ 0 0
$$761$$ 23874.0 1.13723 0.568615 0.822604i $$-0.307479\pi$$
0.568615 + 0.822604i $$0.307479\pi$$
$$762$$ 0 0
$$763$$ − 29480.0i − 1.39875i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 1776.00i − 0.0836084i
$$768$$ 0 0
$$769$$ −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
$$772$$ 0 0
$$773$$ − 11538.0i − 0.536860i −0.963299 0.268430i $$-0.913495\pi$$
0.963299 0.268430i $$-0.0865049\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 4200.00i 0.193918i
$$778$$ 0 0
$$779$$ −40920.0 −1.88204
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ 0 0
$$783$$ 2106.00i 0.0961204i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14884.0i 0.674152i 0.941478 + 0.337076i $$0.109438\pi$$
−0.941478 + 0.337076i $$0.890562\pi$$
$$788$$ 0 0
$$789$$ 16344.0 0.737467
$$790$$ 0 0
$$791$$ −21720.0 −0.976327
$$792$$ 0 0
$$793$$ 23828.0i 1.06703i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 11334.0i − 0.503728i −0.967763 0.251864i $$-0.918957\pi$$
0.967763 0.251864i $$-0.0810435\pi$$
$$798$$ 0 0
$$799$$ −1296.00 −0.0573832
$$800$$ 0 0
$$801$$ 9234.00 0.407325
$$802$$ 0 0
$$803$$ 10320.0i 0.453530i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 7722.00i − 0.336837i
$$808$$ 0 0
$$809$$ −44730.0 −1.94391 −0.971955 0.235167i $$-0.924436\pi$$
−0.971955 + 0.235167i $$0.924436\pi$$
$$810$$ 0 0
$$811$$ 42748.0 1.85091 0.925453 0.378862i $$-0.123684\pi$$
0.925453 + 0.378862i $$0.123684\pi$$
$$812$$ 0 0
$$813$$ − 9552.00i − 0.412058i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 11408.0i − 0.488513i
$$818$$ 0 0
$$819$$ 13320.0 0.568301
$$820$$ 0 0
$$821$$ −31686.0 −1.34695 −0.673477 0.739208i $$-0.735200\pi$$
−0.673477 + 0.739208i $$0.735200\pi$$
$$822$$ 0 0
$$823$$ 11036.0i 0.467425i 0.972306 + 0.233713i $$0.0750875\pi$$
−0.972306 + 0.233713i $$0.924913\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 25884.0i − 1.08836i −0.838968 0.544181i $$-0.816841\pi$$
0.838968 0.544181i $$-0.183159\pi$$
$$828$$ 0 0
$$829$$ −15950.0 −0.668234 −0.334117 0.942532i $$-0.608438\pi$$
−0.334117 + 0.942532i $$0.608438\pi$$
$$830$$ 0 0
$$831$$ 11886.0 0.496174
$$832$$ 0 0
$$833$$ − 3078.00i − 0.128027i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 5400.00i − 0.223000i
$$838$$ 0 0
$$839$$ 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
$$842$$ 0 0
$$843$$ 24858.0i 1.01560i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 15100.0i 0.612565i
$$848$$ 0 0
$$849$$ −8148.00 −0.329374
$$850$$ 0 0
$$851$$ −8400.00 −0.338365
$$852$$ 0 0
$$853$$ 27862.0i 1.11838i 0.829040 + 0.559189i $$0.188887\pi$$
−0.829040 + 0.559189i $$0.811113\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 7314.00i − 0.291530i −0.989319 0.145765i $$-0.953436\pi$$
0.989319 0.145765i $$-0.0465644\pi$$
$$858$$ 0 0
$$859$$ −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
$$862$$ 0 0
$$863$$ − 32688.0i − 1.28935i −0.764455 0.644677i $$-0.776992\pi$$
0.764455 0.644677i $$-0.223008\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 5991.00i − 0.234677i
$$868$$ 0 0
$$869$$ −12480.0 −0.487175
$$870$$ 0 0
$$871$$ 14504.0 0.564236
$$872$$ 0 0
$$873$$ 2574.00i 0.0997900i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 36650.0i 1.41115i 0.708633 + 0.705577i $$0.249312\pi$$
−0.708633 + 0.705577i $$0.750688\pi$$
$$878$$ 0 0
$$879$$ −18054.0 −0.692772
$$880$$ 0 0
$$881$$ −2646.00 −0.101187 −0.0505936 0.998719i $$-0.516111\pi$$
−0.0505936 + 0.998719i $$0.516111\pi$$
$$882$$ 0 0
$$883$$ 10892.0i 0.415113i 0.978223 + 0.207557i $$0.0665511\pi$$
−0.978223 + 0.207557i $$0.933449\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 43464.0i 1.64530i 0.568550 + 0.822648i $$0.307504\pi$$
−0.568550 + 0.822648i $$0.692496\pi$$
$$888$$ 0 0
$$889$$ −24880.0 −0.938637
$$890$$ 0 0
$$891$$ 1944.00 0.0730937
$$892$$ 0 0
$$893$$ − 2976.00i − 0.111521i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 26640.0i 0.991621i
$$898$$ 0 0
$$899$$ −15600.0 −0.578742
$$900$$ 0 0
$$901$$ 24300.0 0.898502
$$902$$ 0 0
$$903$$ − 5520.00i − 0.203426i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 14884.0i 0.544890i 0.962171 + 0.272445i $$0.0878323\pi$$
−0.962171 + 0.272445i $$0.912168\pi$$
$$908$$ 0 0
$$909$$ 15606.0 0.569437
$$910$$ 0 0
$$911$$ 1248.00 0.0453876 0.0226938 0.999742i $$-0.492776\pi$$
0.0226938 + 0.999742i $$0.492776\pi$$
$$912$$ 0 0
$$913$$ 3744.00i 0.135716i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 46560.0i 1.67671i
$$918$$ 0 0
$$919$$ −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
$$922$$ 0 0
$$923$$ − 21312.0i − 0.760014i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 4068.00i − 0.144132i
$$928$$ 0 0
$$929$$ −29946.0 −1.05758 −0.528792 0.848751i $$-0.677355\pi$$
−0.528792 + 0.848751i $$0.677355\pi$$
$$930$$ 0 0
$$931$$ 7068.00 0.248812
$$932$$ 0 0
$$933$$ 4608.00i 0.161693i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 45002.0i 1.56900i 0.620130 + 0.784499i $$0.287080\pi$$
−0.620130 + 0.784499i $$0.712920\pi$$
$$938$$ 0 0
$$939$$ 22026.0 0.765486
$$940$$ 0 0
$$941$$ 6090.00 0.210976 0.105488 0.994421i $$-0.466360\pi$$
0.105488 + 0.994421i $$0.466360\pi$$
$$942$$ 0 0
$$943$$ − 39600.0i − 1.36750i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 56388.0i − 1.93491i −0.253035 0.967457i $$-0.581429\pi$$
0.253035 0.967457i $$-0.418571\pi$$
$$948$$ 0 0
$$949$$ 31820.0 1.08843
$$950$$ 0 0
$$951$$ −11682.0 −0.398333
$$952$$ 0 0
$$953$$ − 10854.0i − 0.368936i −0.982839 0.184468i $$-0.940944\pi$$
0.982839 0.184468i $$-0.0590561\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 5616.00i − 0.189696i
$$958$$ 0 0
$$959$$ 42360.0 1.42636
$$960$$ 0 0
$$961$$ 10209.0 0.342687
$$962$$ 0 0
$$963$$ − 12636.0i − 0.422834i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 42316.0i 1.40723i 0.710582 + 0.703615i $$0.248432\pi$$
−0.710582 + 0.703615i $$0.751568\pi$$
$$968$$ 0 0
$$969$$ −20088.0 −0.665964
$$970$$ 0 0
$$971$$ −24480.0 −0.809063 −0.404532 0.914524i $$-0.632565\pi$$
−0.404532 + 0.914524i $$0.632565\pi$$
$$972$$ 0 0
$$973$$ − 46480.0i − 1.53143i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 6906.00i − 0.226144i −0.993587 0.113072i $$-0.963931\pi$$
0.993587 0.113072i $$-0.0360690\pi$$
$$978$$ 0 0
$$979$$ −24624.0 −0.803868
$$980$$ 0 0
$$981$$ −13266.0 −0.431754
$$982$$ 0 0
$$983$$ 6960.00i 0.225829i 0.993605 + 0.112914i $$0.0360186\pi$$
−0.993605 + 0.112914i $$0.963981\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 1440.00i − 0.0464394i
$$988$$ 0 0
$$989$$ 11040.0 0.354956
$$990$$ 0 0
$$991$$ −47792.0 −1.53195 −0.765975 0.642870i $$-0.777744\pi$$
−0.765975 + 0.642870i $$0.777744\pi$$
$$992$$ 0 0
$$993$$ 11076.0i 0.353964i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 9938.00i 0.315687i 0.987464 + 0.157843i $$0.0504541\pi$$
−0.987464 + 0.157843i $$0.949546\pi$$
$$998$$ 0 0
$$999$$ 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 1200.4.f.m.49.1 2
4.3 odd 2 75.4.b.a.49.2 2
5.2 odd 4 1200.4.a.o.1.1 1
5.3 odd 4 240.4.a.f.1.1 1
5.4 even 2 inner 1200.4.f.m.49.2 2
12.11 even 2 225.4.b.d.199.1 2
15.8 even 4 720.4.a.r.1.1 1
20.3 even 4 15.4.a.b.1.1 1
20.7 even 4 75.4.a.a.1.1 1
20.19 odd 2 75.4.b.a.49.1 2
40.3 even 4 960.4.a.bi.1.1 1
40.13 odd 4 960.4.a.l.1.1 1
60.23 odd 4 45.4.a.b.1.1 1
60.47 odd 4 225.4.a.g.1.1 1
60.59 even 2 225.4.b.d.199.2 2
140.83 odd 4 735.4.a.i.1.1 1
180.23 odd 12 405.4.e.k.136.1 2
180.43 even 12 405.4.e.d.271.1 2
180.83 odd 12 405.4.e.k.271.1 2
180.103 even 12 405.4.e.d.136.1 2
220.43 odd 4 1815.4.a.a.1.1 1
420.83 even 4 2205.4.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.b.1.1 1 20.3 even 4
45.4.a.b.1.1 1 60.23 odd 4
75.4.a.a.1.1 1 20.7 even 4
75.4.b.a.49.1 2 20.19 odd 2
75.4.b.a.49.2 2 4.3 odd 2
225.4.a.g.1.1 1 60.47 odd 4
225.4.b.d.199.1 2 12.11 even 2
225.4.b.d.199.2 2 60.59 even 2
240.4.a.f.1.1 1 5.3 odd 4
405.4.e.d.136.1 2 180.103 even 12
405.4.e.d.271.1 2 180.43 even 12
405.4.e.k.136.1 2 180.23 odd 12
405.4.e.k.271.1 2 180.83 odd 12
720.4.a.r.1.1 1 15.8 even 4
735.4.a.i.1.1 1 140.83 odd 4
960.4.a.l.1.1 1 40.13 odd 4
960.4.a.bi.1.1 1 40.3 even 4
1200.4.a.o.1.1 1 5.2 odd 4
1200.4.f.m.49.1 2 1.1 even 1 trivial
1200.4.f.m.49.2 2 5.4 even 2 inner
1815.4.a.a.1.1 1 220.43 odd 4
2205.4.a.c.1.1 1 420.83 even 4