# Properties

 Label 75.4.b.a.49.1 Level $75$ Weight $4$ Character 75.49 Analytic conductor $4.425$ 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] = [75,4,Mod(49,75)]

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, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (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: $$4.42514325043$$ 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 $$\chi$$ $$=$$ 75.49 Dual form 75.4.b.a.49.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} -20.0000i q^{7} -21.0000i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q-3.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -9.00000 q^{6} -20.0000i q^{7} -21.0000i q^{8} -9.00000 q^{9} -24.0000 q^{11} +3.00000i q^{12} +74.0000i q^{13} -60.0000 q^{14} -71.0000 q^{16} -54.0000i q^{17} +27.0000i q^{18} +124.000 q^{19} -60.0000 q^{21} +72.0000i q^{22} -120.000i q^{23} -63.0000 q^{24} +222.000 q^{26} +27.0000i q^{27} +20.0000i q^{28} +78.0000 q^{29} +200.000 q^{31} +45.0000i q^{32} +72.0000i q^{33} -162.000 q^{34} +9.00000 q^{36} +70.0000i q^{37} -372.000i q^{38} +222.000 q^{39} +330.000 q^{41} +180.000i q^{42} +92.0000i q^{43} +24.0000 q^{44} -360.000 q^{46} +24.0000i q^{47} +213.000i q^{48} -57.0000 q^{49} -162.000 q^{51} -74.0000i q^{52} +450.000i q^{53} +81.0000 q^{54} -420.000 q^{56} -372.000i q^{57} -234.000i q^{58} -24.0000 q^{59} -322.000 q^{61} -600.000i q^{62} +180.000i q^{63} -433.000 q^{64} +216.000 q^{66} +196.000i q^{67} +54.0000i q^{68} -360.000 q^{69} -288.000 q^{71} +189.000i q^{72} -430.000i q^{73} +210.000 q^{74} -124.000 q^{76} +480.000i q^{77} -666.000i q^{78} +520.000 q^{79} +81.0000 q^{81} -990.000i q^{82} +156.000i q^{83} +60.0000 q^{84} +276.000 q^{86} -234.000i q^{87} +504.000i q^{88} -1026.00 q^{89} +1480.00 q^{91} +120.000i q^{92} -600.000i q^{93} +72.0000 q^{94} +135.000 q^{96} +286.000i q^{97} +171.000i q^{98} +216.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 18 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 18 * q^6 - 18 * q^9 $$2 q - 2 q^{4} - 18 q^{6} - 18 q^{9} - 48 q^{11} - 120 q^{14} - 142 q^{16} + 248 q^{19} - 120 q^{21} - 126 q^{24} + 444 q^{26} + 156 q^{29} + 400 q^{31} - 324 q^{34} + 18 q^{36} + 444 q^{39} + 660 q^{41} + 48 q^{44} - 720 q^{46} - 114 q^{49} - 324 q^{51} + 162 q^{54} - 840 q^{56} - 48 q^{59} - 644 q^{61} - 866 q^{64} + 432 q^{66} - 720 q^{69} - 576 q^{71} + 420 q^{74} - 248 q^{76} + 1040 q^{79} + 162 q^{81} + 120 q^{84} + 552 q^{86} - 2052 q^{89} + 2960 q^{91} + 144 q^{94} + 270 q^{96} + 432 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 18 * q^6 - 18 * q^9 - 48 * q^11 - 120 * q^14 - 142 * q^16 + 248 * q^19 - 120 * q^21 - 126 * q^24 + 444 * q^26 + 156 * q^29 + 400 * q^31 - 324 * q^34 + 18 * q^36 + 444 * q^39 + 660 * q^41 + 48 * q^44 - 720 * q^46 - 114 * q^49 - 324 * q^51 + 162 * q^54 - 840 * q^56 - 48 * q^59 - 644 * q^61 - 866 * q^64 + 432 * q^66 - 720 * q^69 - 576 * q^71 + 420 * q^74 - 248 * q^76 + 1040 * q^79 + 162 * q^81 + 120 * q^84 + 552 * q^86 - 2052 * q^89 + 2960 * q^91 + 144 * q^94 + 270 * q^96 + 432 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 3.00000i − 1.06066i −0.847791 0.530330i $$-0.822068\pi$$
0.847791 0.530330i $$-0.177932\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ −1.00000 −0.125000
$$5$$ 0 0
$$6$$ −9.00000 −0.612372
$$7$$ − 20.0000i − 1.07990i −0.841698 0.539949i $$-0.818443\pi$$
0.841698 0.539949i $$-0.181557\pi$$
$$8$$ − 21.0000i − 0.928078i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −24.0000 −0.657843 −0.328921 0.944357i $$-0.606685\pi$$
−0.328921 + 0.944357i $$0.606685\pi$$
$$12$$ 3.00000i 0.0721688i
$$13$$ 74.0000i 1.57876i 0.613904 + 0.789381i $$0.289598\pi$$
−0.613904 + 0.789381i $$0.710402\pi$$
$$14$$ −60.0000 −1.14541
$$15$$ 0 0
$$16$$ −71.0000 −1.10938
$$17$$ − 54.0000i − 0.770407i −0.922832 0.385204i $$-0.874131\pi$$
0.922832 0.385204i $$-0.125869\pi$$
$$18$$ 27.0000i 0.353553i
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 0 0
$$21$$ −60.0000 −0.623480
$$22$$ 72.0000i 0.697748i
$$23$$ − 120.000i − 1.08790i −0.839117 0.543951i $$-0.816928\pi$$
0.839117 0.543951i $$-0.183072\pi$$
$$24$$ −63.0000 −0.535826
$$25$$ 0 0
$$26$$ 222.000 1.67453
$$27$$ 27.0000i 0.192450i
$$28$$ 20.0000i 0.134987i
$$29$$ 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 $$-0.303298\pi$$
0.579372 + 0.815063i $$0.303298\pi$$
$$32$$ 45.0000i 0.248592i
$$33$$ 72.0000i 0.379806i
$$34$$ −162.000 −0.817140
$$35$$ 0 0
$$36$$ 9.00000 0.0416667
$$37$$ 70.0000i 0.311025i 0.987834 + 0.155513i $$0.0497029\pi$$
−0.987834 + 0.155513i $$0.950297\pi$$
$$38$$ − 372.000i − 1.58806i
$$39$$ 222.000 0.911499
$$40$$ 0 0
$$41$$ 330.000 1.25701 0.628504 0.777806i $$-0.283668\pi$$
0.628504 + 0.777806i $$0.283668\pi$$
$$42$$ 180.000i 0.661300i
$$43$$ 92.0000i 0.326276i 0.986603 + 0.163138i $$0.0521616\pi$$
−0.986603 + 0.163138i $$0.947838\pi$$
$$44$$ 24.0000 0.0822304
$$45$$ 0 0
$$46$$ −360.000 −1.15389
$$47$$ 24.0000i 0.0744843i 0.999306 + 0.0372421i $$0.0118573\pi$$
−0.999306 + 0.0372421i $$0.988143\pi$$
$$48$$ 213.000i 0.640498i
$$49$$ −57.0000 −0.166181
$$50$$ 0 0
$$51$$ −162.000 −0.444795
$$52$$ − 74.0000i − 0.197345i
$$53$$ 450.000i 1.16627i 0.812376 + 0.583134i $$0.198174\pi$$
−0.812376 + 0.583134i $$0.801826\pi$$
$$54$$ 81.0000 0.204124
$$55$$ 0 0
$$56$$ −420.000 −1.00223
$$57$$ − 372.000i − 0.864432i
$$58$$ − 234.000i − 0.529754i
$$59$$ −24.0000 −0.0529582 −0.0264791 0.999649i $$-0.508430\pi$$
−0.0264791 + 0.999649i $$0.508430\pi$$
$$60$$ 0 0
$$61$$ −322.000 −0.675867 −0.337933 0.941170i $$-0.609728\pi$$
−0.337933 + 0.941170i $$0.609728\pi$$
$$62$$ − 600.000i − 1.22903i
$$63$$ 180.000i 0.359966i
$$64$$ −433.000 −0.845703
$$65$$ 0 0
$$66$$ 216.000 0.402845
$$67$$ 196.000i 0.357391i 0.983904 + 0.178696i $$0.0571877\pi$$
−0.983904 + 0.178696i $$0.942812\pi$$
$$68$$ 54.0000i 0.0963009i
$$69$$ −360.000 −0.628100
$$70$$ 0 0
$$71$$ −288.000 −0.481399 −0.240699 0.970600i $$-0.577377\pi$$
−0.240699 + 0.970600i $$0.577377\pi$$
$$72$$ 189.000i 0.309359i
$$73$$ − 430.000i − 0.689420i −0.938709 0.344710i $$-0.887977\pi$$
0.938709 0.344710i $$-0.112023\pi$$
$$74$$ 210.000 0.329892
$$75$$ 0 0
$$76$$ −124.000 −0.187155
$$77$$ 480.000i 0.710404i
$$78$$ − 666.000i − 0.966790i
$$79$$ 520.000 0.740564 0.370282 0.928919i $$-0.379261\pi$$
0.370282 + 0.928919i $$0.379261\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ − 990.000i − 1.33326i
$$83$$ 156.000i 0.206304i 0.994666 + 0.103152i $$0.0328928\pi$$
−0.994666 + 0.103152i $$0.967107\pi$$
$$84$$ 60.0000 0.0779350
$$85$$ 0 0
$$86$$ 276.000 0.346068
$$87$$ − 234.000i − 0.288361i
$$88$$ 504.000i 0.610529i
$$89$$ −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
$$92$$ 120.000i 0.135988i
$$93$$ − 600.000i − 0.669001i
$$94$$ 72.0000 0.0790025
$$95$$ 0 0
$$96$$ 135.000 0.143525
$$97$$ 286.000i 0.299370i 0.988734 + 0.149685i $$0.0478260\pi$$
−0.988734 + 0.149685i $$0.952174\pi$$
$$98$$ 171.000i 0.176261i
$$99$$ 216.000 0.219281
$$100$$ 0 0
$$101$$ −1734.00 −1.70831 −0.854156 0.520017i $$-0.825925\pi$$
−0.854156 + 0.520017i $$0.825925\pi$$
$$102$$ 486.000i 0.471776i
$$103$$ 452.000i 0.432397i 0.976349 + 0.216198i $$0.0693658\pi$$
−0.976349 + 0.216198i $$0.930634\pi$$
$$104$$ 1554.00 1.46521
$$105$$ 0 0
$$106$$ 1350.00 1.23702
$$107$$ 1404.00i 1.26850i 0.773127 + 0.634251i $$0.218692\pi$$
−0.773127 + 0.634251i $$0.781308\pi$$
$$108$$ − 27.0000i − 0.0240563i
$$109$$ 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
$$112$$ 1420.00i 1.19801i
$$113$$ 1086.00i 0.904091i 0.891995 + 0.452046i $$0.149306\pi$$
−0.891995 + 0.452046i $$0.850694\pi$$
$$114$$ −1116.00 −0.916868
$$115$$ 0 0
$$116$$ −78.0000 −0.0624321
$$117$$ − 666.000i − 0.526254i
$$118$$ 72.0000i 0.0561707i
$$119$$ −1080.00 −0.831962
$$120$$ 0 0
$$121$$ −755.000 −0.567243
$$122$$ 966.000i 0.716865i
$$123$$ − 990.000i − 0.725734i
$$124$$ −200.000 −0.144843
$$125$$ 0 0
$$126$$ 540.000 0.381802
$$127$$ − 1244.00i − 0.869190i −0.900626 0.434595i $$-0.856891\pi$$
0.900626 0.434595i $$-0.143109\pi$$
$$128$$ 1659.00i 1.14560i
$$129$$ 276.000 0.188376
$$130$$ 0 0
$$131$$ 2328.00 1.55266 0.776329 0.630327i $$-0.217079\pi$$
0.776329 + 0.630327i $$0.217079\pi$$
$$132$$ − 72.0000i − 0.0474757i
$$133$$ − 2480.00i − 1.61687i
$$134$$ 588.000 0.379071
$$135$$ 0 0
$$136$$ −1134.00 −0.714998
$$137$$ − 2118.00i − 1.32082i −0.750903 0.660412i $$-0.770382\pi$$
0.750903 0.660412i $$-0.229618\pi$$
$$138$$ 1080.00i 0.666201i
$$139$$ −2324.00 −1.41812 −0.709062 0.705147i $$-0.750881\pi$$
−0.709062 + 0.705147i $$0.750881\pi$$
$$140$$ 0 0
$$141$$ 72.0000 0.0430035
$$142$$ 864.000i 0.510600i
$$143$$ − 1776.00i − 1.03858i
$$144$$ 639.000 0.369792
$$145$$ 0 0
$$146$$ −1290.00 −0.731241
$$147$$ 171.000i 0.0959445i
$$148$$ − 70.0000i − 0.0388781i
$$149$$ −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 $$-0.569865\pi$$
−0.217729 + 0.976009i $$0.569865\pi$$
$$152$$ − 2604.00i − 1.38955i
$$153$$ 486.000i 0.256802i
$$154$$ 1440.00 0.753497
$$155$$ 0 0
$$156$$ −222.000 −0.113937
$$157$$ − 2378.00i − 1.20882i −0.796673 0.604411i $$-0.793408\pi$$
0.796673 0.604411i $$-0.206592\pi$$
$$158$$ − 1560.00i − 0.785487i
$$159$$ 1350.00 0.673346
$$160$$ 0 0
$$161$$ −2400.00 −1.17482
$$162$$ − 243.000i − 0.117851i
$$163$$ − 52.0000i − 0.0249874i −0.999922 0.0124937i $$-0.996023\pi$$
0.999922 0.0124937i $$-0.00397698\pi$$
$$164$$ −330.000 −0.157126
$$165$$ 0 0
$$166$$ 468.000 0.218818
$$167$$ 3720.00i 1.72373i 0.507141 + 0.861863i $$0.330702\pi$$
−0.507141 + 0.861863i $$0.669298\pi$$
$$168$$ 1260.00i 0.578638i
$$169$$ −3279.00 −1.49249
$$170$$ 0 0
$$171$$ −1116.00 −0.499080
$$172$$ − 92.0000i − 0.0407845i
$$173$$ 426.000i 0.187215i 0.995609 + 0.0936075i $$0.0298399\pi$$
−0.995609 + 0.0936075i $$0.970160\pi$$
$$174$$ −702.000 −0.305853
$$175$$ 0 0
$$176$$ 1704.00 0.729795
$$177$$ 72.0000i 0.0305754i
$$178$$ 3078.00i 1.29610i
$$179$$ 1440.00 0.601289 0.300644 0.953736i $$-0.402798\pi$$
0.300644 + 0.953736i $$0.402798\pi$$
$$180$$ 0 0
$$181$$ −3130.00 −1.28537 −0.642683 0.766133i $$-0.722179\pi$$
−0.642683 + 0.766133i $$0.722179\pi$$
$$182$$ − 4440.00i − 1.80832i
$$183$$ 966.000i 0.390212i
$$184$$ −2520.00 −1.00966
$$185$$ 0 0
$$186$$ −1800.00 −0.709583
$$187$$ 1296.00i 0.506807i
$$188$$ − 24.0000i − 0.00931053i
$$189$$ 540.000 0.207827
$$190$$ 0 0
$$191$$ 3576.00 1.35471 0.677357 0.735655i $$-0.263125\pi$$
0.677357 + 0.735655i $$0.263125\pi$$
$$192$$ 1299.00i 0.488267i
$$193$$ 2666.00i 0.994315i 0.867660 + 0.497158i $$0.165623\pi$$
−0.867660 + 0.497158i $$0.834377\pi$$
$$194$$ 858.000 0.317530
$$195$$ 0 0
$$196$$ 57.0000 0.0207726
$$197$$ 2718.00i 0.982992i 0.870880 + 0.491496i $$0.163550\pi$$
−0.870880 + 0.491496i $$0.836450\pi$$
$$198$$ − 648.000i − 0.232583i
$$199$$ 3832.00 1.36504 0.682521 0.730866i $$-0.260884\pi$$
0.682521 + 0.730866i $$0.260884\pi$$
$$200$$ 0 0
$$201$$ 588.000 0.206340
$$202$$ 5202.00i 1.81194i
$$203$$ − 1560.00i − 0.539362i
$$204$$ 162.000 0.0555994
$$205$$ 0 0
$$206$$ 1356.00 0.458626
$$207$$ 1080.00i 0.362634i
$$208$$ − 5254.00i − 1.75144i
$$209$$ −2976.00 −0.984948
$$210$$ 0 0
$$211$$ 1100.00 0.358896 0.179448 0.983767i $$-0.442569\pi$$
0.179448 + 0.983767i $$0.442569\pi$$
$$212$$ − 450.000i − 0.145784i
$$213$$ 864.000i 0.277936i
$$214$$ 4212.00 1.34545
$$215$$ 0 0
$$216$$ 567.000 0.178609
$$217$$ − 4000.00i − 1.25133i
$$218$$ − 4422.00i − 1.37383i
$$219$$ −1290.00 −0.398037
$$220$$ 0 0
$$221$$ 3996.00 1.21629
$$222$$ − 630.000i − 0.190463i
$$223$$ 1964.00i 0.589772i 0.955532 + 0.294886i $$0.0952817\pi$$
−0.955532 + 0.294886i $$0.904718\pi$$
$$224$$ 900.000 0.268454
$$225$$ 0 0
$$226$$ 3258.00 0.958933
$$227$$ − 660.000i − 0.192977i −0.995334 0.0964884i $$-0.969239\pi$$
0.995334 0.0964884i $$-0.0307611\pi$$
$$228$$ 372.000i 0.108054i
$$229$$ 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
$$232$$ − 1638.00i − 0.463534i
$$233$$ − 1458.00i − 0.409943i −0.978768 0.204972i $$-0.934290\pi$$
0.978768 0.204972i $$-0.0657102\pi$$
$$234$$ −1998.00 −0.558177
$$235$$ 0 0
$$236$$ 24.0000 0.00661978
$$237$$ − 1560.00i − 0.427565i
$$238$$ 3240.00i 0.882429i
$$239$$ −1176.00 −0.318281 −0.159140 0.987256i $$-0.550872\pi$$
−0.159140 + 0.987256i $$0.550872\pi$$
$$240$$ 0 0
$$241$$ 866.000 0.231469 0.115734 0.993280i $$-0.463078\pi$$
0.115734 + 0.993280i $$0.463078\pi$$
$$242$$ 2265.00i 0.601652i
$$243$$ − 243.000i − 0.0641500i
$$244$$ 322.000 0.0844834
$$245$$ 0 0
$$246$$ −2970.00 −0.769757
$$247$$ 9176.00i 2.36379i
$$248$$ − 4200.00i − 1.07540i
$$249$$ 468.000 0.119110
$$250$$ 0 0
$$251$$ 432.000 0.108636 0.0543179 0.998524i $$-0.482702\pi$$
0.0543179 + 0.998524i $$0.482702\pi$$
$$252$$ − 180.000i − 0.0449958i
$$253$$ 2880.00i 0.715668i
$$254$$ −3732.00 −0.921915
$$255$$ 0 0
$$256$$ 1513.00 0.369385
$$257$$ − 2526.00i − 0.613103i −0.951854 0.306552i $$-0.900825\pi$$
0.951854 0.306552i $$-0.0991752\pi$$
$$258$$ − 828.000i − 0.199802i
$$259$$ 1400.00 0.335876
$$260$$ 0 0
$$261$$ −702.000 −0.166485
$$262$$ − 6984.00i − 1.64684i
$$263$$ 5448.00i 1.27733i 0.769484 + 0.638666i $$0.220513\pi$$
−0.769484 + 0.638666i $$0.779487\pi$$
$$264$$ 1512.00 0.352489
$$265$$ 0 0
$$266$$ −7440.00 −1.71495
$$267$$ 3078.00i 0.705508i
$$268$$ − 196.000i − 0.0446739i
$$269$$ 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 $$-0.616150\pi$$
−0.356853 + 0.934161i $$0.616150\pi$$
$$272$$ 3834.00i 0.854671i
$$273$$ − 4440.00i − 0.984326i
$$274$$ −6354.00 −1.40095
$$275$$ 0 0
$$276$$ 360.000 0.0785125
$$277$$ − 3962.00i − 0.859399i −0.902972 0.429699i $$-0.858620\pi$$
0.902972 0.429699i $$-0.141380\pi$$
$$278$$ 6972.00i 1.50415i
$$279$$ −1800.00 −0.386248
$$280$$ 0 0
$$281$$ −8286.00 −1.75908 −0.879540 0.475825i $$-0.842149\pi$$
−0.879540 + 0.475825i $$0.842149\pi$$
$$282$$ − 216.000i − 0.0456121i
$$283$$ − 2716.00i − 0.570493i −0.958454 0.285246i $$-0.907925\pi$$
0.958454 0.285246i $$-0.0920754\pi$$
$$284$$ 288.000 0.0601748
$$285$$ 0 0
$$286$$ −5328.00 −1.10158
$$287$$ − 6600.00i − 1.35744i
$$288$$ − 405.000i − 0.0828641i
$$289$$ 1997.00 0.406473
$$290$$ 0 0
$$291$$ 858.000 0.172841
$$292$$ 430.000i 0.0861776i
$$293$$ 6018.00i 1.19992i 0.800032 + 0.599958i $$0.204816\pi$$
−0.800032 + 0.599958i $$0.795184\pi$$
$$294$$ 513.000 0.101765
$$295$$ 0 0
$$296$$ 1470.00 0.288655
$$297$$ − 648.000i − 0.126602i
$$298$$ 774.000i 0.150458i
$$299$$ 8880.00 1.71754
$$300$$ 0 0
$$301$$ 1840.00 0.352345
$$302$$ 2424.00i 0.461873i
$$303$$ 5202.00i 0.986294i
$$304$$ −8804.00 −1.66100
$$305$$ 0 0
$$306$$ 1458.00 0.272380
$$307$$ − 9236.00i − 1.71702i −0.512793 0.858512i $$-0.671389\pi$$
0.512793 0.858512i $$-0.328611\pi$$
$$308$$ − 480.000i − 0.0888004i
$$309$$ 1356.00 0.249644
$$310$$ 0 0
$$311$$ 1536.00 0.280060 0.140030 0.990147i $$-0.455280\pi$$
0.140030 + 0.990147i $$0.455280\pi$$
$$312$$ − 4662.00i − 0.845942i
$$313$$ − 7342.00i − 1.32586i −0.748681 0.662930i $$-0.769313\pi$$
0.748681 0.662930i $$-0.230687\pi$$
$$314$$ −7134.00 −1.28215
$$315$$ 0 0
$$316$$ −520.000 −0.0925705
$$317$$ 3894.00i 0.689933i 0.938615 + 0.344967i $$0.112110\pi$$
−0.938615 + 0.344967i $$0.887890\pi$$
$$318$$ − 4050.00i − 0.714191i
$$319$$ −1872.00 −0.328564
$$320$$ 0 0
$$321$$ 4212.00 0.732370
$$322$$ 7200.00i 1.24609i
$$323$$ − 6696.00i − 1.15348i
$$324$$ −81.0000 −0.0138889
$$325$$ 0 0
$$326$$ −156.000 −0.0265032
$$327$$ − 4422.00i − 0.747820i
$$328$$ − 6930.00i − 1.16660i
$$329$$ 480.000 0.0804354
$$330$$ 0 0
$$331$$ 3692.00 0.613084 0.306542 0.951857i $$-0.400828\pi$$
0.306542 + 0.951857i $$0.400828\pi$$
$$332$$ − 156.000i − 0.0257880i
$$333$$ − 630.000i − 0.103675i
$$334$$ 11160.0 1.82829
$$335$$ 0 0
$$336$$ 4260.00 0.691673
$$337$$ 8998.00i 1.45446i 0.686395 + 0.727229i $$0.259192\pi$$
−0.686395 + 0.727229i $$0.740808\pi$$
$$338$$ 9837.00i 1.58302i
$$339$$ 3258.00 0.521977
$$340$$ 0 0
$$341$$ −4800.00 −0.762271
$$342$$ 3348.00i 0.529354i
$$343$$ − 5720.00i − 0.900440i
$$344$$ 1932.00 0.302809
$$345$$ 0 0
$$346$$ 1278.00 0.198571
$$347$$ − 5244.00i − 0.811276i −0.914034 0.405638i $$-0.867049\pi$$
0.914034 0.405638i $$-0.132951\pi$$
$$348$$ 234.000i 0.0360452i
$$349$$ −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
$$352$$ − 1080.00i − 0.163535i
$$353$$ 3414.00i 0.514756i 0.966311 + 0.257378i $$0.0828586\pi$$
−0.966311 + 0.257378i $$0.917141\pi$$
$$354$$ 216.000 0.0324301
$$355$$ 0 0
$$356$$ 1026.00 0.152747
$$357$$ 3240.00i 0.480333i
$$358$$ − 4320.00i − 0.637763i
$$359$$ −4824.00 −0.709195 −0.354597 0.935019i $$-0.615382\pi$$
−0.354597 + 0.935019i $$0.615382\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ 9390.00i 1.36334i
$$363$$ 2265.00i 0.327498i
$$364$$ −1480.00 −0.213113
$$365$$ 0 0
$$366$$ 2898.00 0.413882
$$367$$ 3508.00i 0.498954i 0.968381 + 0.249477i $$0.0802587\pi$$
−0.968381 + 0.249477i $$0.919741\pi$$
$$368$$ 8520.00i 1.20689i
$$369$$ −2970.00 −0.419003
$$370$$ 0 0
$$371$$ 9000.00 1.25945
$$372$$ 600.000i 0.0836251i
$$373$$ 10802.0i 1.49948i 0.661732 + 0.749740i $$0.269822\pi$$
−0.661732 + 0.749740i $$0.730178\pi$$
$$374$$ 3888.00 0.537550
$$375$$ 0 0
$$376$$ 504.000 0.0691272
$$377$$ 5772.00i 0.788523i
$$378$$ − 1620.00i − 0.220433i
$$379$$ −1460.00 −0.197876 −0.0989382 0.995094i $$-0.531545\pi$$
−0.0989382 + 0.995094i $$0.531545\pi$$
$$380$$ 0 0
$$381$$ −3732.00 −0.501827
$$382$$ − 10728.0i − 1.43689i
$$383$$ − 4872.00i − 0.649994i −0.945715 0.324997i $$-0.894637\pi$$
0.945715 0.324997i $$-0.105363\pi$$
$$384$$ 4977.00 0.661410
$$385$$ 0 0
$$386$$ 7998.00 1.05463
$$387$$ − 828.000i − 0.108759i
$$388$$ − 286.000i − 0.0374213i
$$389$$ 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
$$392$$ 1197.00i 0.154229i
$$393$$ − 6984.00i − 0.896428i
$$394$$ 8154.00 1.04262
$$395$$ 0 0
$$396$$ −216.000 −0.0274101
$$397$$ 2734.00i 0.345631i 0.984954 + 0.172816i $$0.0552864\pi$$
−0.984954 + 0.172816i $$0.944714\pi$$
$$398$$ − 11496.0i − 1.44785i
$$399$$ −7440.00 −0.933498
$$400$$ 0 0
$$401$$ −15942.0 −1.98530 −0.992650 0.121019i $$-0.961384\pi$$
−0.992650 + 0.121019i $$0.961384\pi$$
$$402$$ − 1764.00i − 0.218857i
$$403$$ 14800.0i 1.82938i
$$404$$ 1734.00 0.213539
$$405$$ 0 0
$$406$$ −4680.00 −0.572080
$$407$$ − 1680.00i − 0.204606i
$$408$$ 3402.00i 0.412804i
$$409$$ −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
$$412$$ − 452.000i − 0.0540496i
$$413$$ 480.000i 0.0571895i
$$414$$ 3240.00 0.384631
$$415$$ 0 0
$$416$$ −3330.00 −0.392468
$$417$$ 6972.00i 0.818754i
$$418$$ 8928.00i 1.04470i
$$419$$ −11976.0 −1.39634 −0.698169 0.715933i $$-0.746002\pi$$
−0.698169 + 0.715933i $$0.746002\pi$$
$$420$$ 0 0
$$421$$ 11054.0 1.27967 0.639833 0.768514i $$-0.279004\pi$$
0.639833 + 0.768514i $$0.279004\pi$$
$$422$$ − 3300.00i − 0.380667i
$$423$$ − 216.000i − 0.0248281i
$$424$$ 9450.00 1.08239
$$425$$ 0 0
$$426$$ 2592.00 0.294795
$$427$$ 6440.00i 0.729868i
$$428$$ − 1404.00i − 0.158563i
$$429$$ −5328.00 −0.599623
$$430$$ 0 0
$$431$$ 720.000 0.0804668 0.0402334 0.999190i $$-0.487190\pi$$
0.0402334 + 0.999190i $$0.487190\pi$$
$$432$$ − 1917.00i − 0.213499i
$$433$$ − 15622.0i − 1.73382i −0.498462 0.866912i $$-0.666102\pi$$
0.498462 0.866912i $$-0.333898\pi$$
$$434$$ −12000.0 −1.32723
$$435$$ 0 0
$$436$$ −1474.00 −0.161908
$$437$$ − 14880.0i − 1.62885i
$$438$$ 3870.00i 0.422182i
$$439$$ 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
$$442$$ − 11988.0i − 1.29007i
$$443$$ − 16116.0i − 1.72843i −0.503123 0.864215i $$-0.667816\pi$$
0.503123 0.864215i $$-0.332184\pi$$
$$444$$ −210.000 −0.0224463
$$445$$ 0 0
$$446$$ 5892.00 0.625548
$$447$$ 774.000i 0.0818992i
$$448$$ 8660.00i 0.913274i
$$449$$ −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
$$452$$ − 1086.00i − 0.113011i
$$453$$ 2424.00i 0.251412i
$$454$$ −1980.00 −0.204683
$$455$$ 0 0
$$456$$ −7812.00 −0.802260
$$457$$ 3670.00i 0.375657i 0.982202 + 0.187829i $$0.0601450\pi$$
−0.982202 + 0.187829i $$0.939855\pi$$
$$458$$ − 5718.00i − 0.583372i
$$459$$ 1458.00 0.148265
$$460$$ 0 0
$$461$$ 17562.0 1.77428 0.887141 0.461499i $$-0.152688\pi$$
0.887141 + 0.461499i $$0.152688\pi$$
$$462$$ − 4320.00i − 0.435032i
$$463$$ 1172.00i 0.117640i 0.998269 + 0.0588202i $$0.0187338\pi$$
−0.998269 + 0.0588202i $$0.981266\pi$$
$$464$$ −5538.00 −0.554084
$$465$$ 0 0
$$466$$ −4374.00 −0.434810
$$467$$ − 6876.00i − 0.681335i −0.940184 0.340667i $$-0.889347\pi$$
0.940184 0.340667i $$-0.110653\pi$$
$$468$$ 666.000i 0.0657818i
$$469$$ 3920.00 0.385946
$$470$$ 0 0
$$471$$ −7134.00 −0.697914
$$472$$ 504.000i 0.0491493i
$$473$$ − 2208.00i − 0.214638i
$$474$$ −4680.00 −0.453501
$$475$$ 0 0
$$476$$ 1080.00 0.103995
$$477$$ − 4050.00i − 0.388756i
$$478$$ 3528.00i 0.337588i
$$479$$ −2280.00 −0.217486 −0.108743 0.994070i $$-0.534683\pi$$
−0.108743 + 0.994070i $$0.534683\pi$$
$$480$$ 0 0
$$481$$ −5180.00 −0.491035
$$482$$ − 2598.00i − 0.245510i
$$483$$ 7200.00i 0.678284i
$$484$$ 755.000 0.0709053
$$485$$ 0 0
$$486$$ −729.000 −0.0680414
$$487$$ 3076.00i 0.286215i 0.989707 + 0.143108i $$0.0457095\pi$$
−0.989707 + 0.143108i $$0.954290\pi$$
$$488$$ 6762.00i 0.627257i
$$489$$ −156.000 −0.0144265
$$490$$ 0 0
$$491$$ −18912.0 −1.73826 −0.869131 0.494582i $$-0.835321\pi$$
−0.869131 + 0.494582i $$0.835321\pi$$
$$492$$ 990.000i 0.0907168i
$$493$$ − 4212.00i − 0.384785i
$$494$$ 27528.0 2.50717
$$495$$ 0 0
$$496$$ −14200.0 −1.28548
$$497$$ 5760.00i 0.519862i
$$498$$ − 1404.00i − 0.126335i
$$499$$ −9956.00 −0.893170 −0.446585 0.894741i $$-0.647360\pi$$
−0.446585 + 0.894741i $$0.647360\pi$$
$$500$$ 0 0
$$501$$ 11160.0 0.995194
$$502$$ − 1296.00i − 0.115226i
$$503$$ − 10656.0i − 0.944588i −0.881441 0.472294i $$-0.843426\pi$$
0.881441 0.472294i $$-0.156574\pi$$
$$504$$ 3780.00 0.334077
$$505$$ 0 0
$$506$$ 8640.00 0.759081
$$507$$ 9837.00i 0.861689i
$$508$$ 1244.00i 0.108649i
$$509$$ 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
$$512$$ 8733.00i 0.753804i
$$513$$ 3348.00i 0.288144i
$$514$$ −7578.00 −0.650294
$$515$$ 0 0
$$516$$ −276.000 −0.0235469
$$517$$ − 576.000i − 0.0489989i
$$518$$ − 4200.00i − 0.356250i
$$519$$ 1278.00 0.108089
$$520$$ 0 0
$$521$$ 10530.0 0.885466 0.442733 0.896654i $$-0.354009\pi$$
0.442733 + 0.896654i $$0.354009\pi$$
$$522$$ 2106.00i 0.176585i
$$523$$ 12692.0i 1.06115i 0.847637 + 0.530576i $$0.178024\pi$$
−0.847637 + 0.530576i $$0.821976\pi$$
$$524$$ −2328.00 −0.194082
$$525$$ 0 0
$$526$$ 16344.0 1.35481
$$527$$ − 10800.0i − 0.892705i
$$528$$ − 5112.00i − 0.421347i
$$529$$ −2233.00 −0.183529
$$530$$ 0 0
$$531$$ 216.000 0.0176527
$$532$$ 2480.00i 0.202108i
$$533$$ 24420.0i 1.98452i
$$534$$ 9234.00 0.748304
$$535$$ 0 0
$$536$$ 4116.00 0.331687
$$537$$ − 4320.00i − 0.347154i
$$538$$ − 7722.00i − 0.618809i
$$539$$ 1368.00 0.109321
$$540$$ 0 0
$$541$$ 18110.0 1.43920 0.719602 0.694386i $$-0.244324\pi$$
0.719602 + 0.694386i $$0.244324\pi$$
$$542$$ 9552.00i 0.756999i
$$543$$ 9390.00i 0.742106i
$$544$$ 2430.00 0.191517
$$545$$ 0 0
$$546$$ −13320.0 −1.04404
$$547$$ − 3620.00i − 0.282962i −0.989941 0.141481i $$-0.954814\pi$$
0.989941 0.141481i $$-0.0451864\pi$$
$$548$$ 2118.00i 0.165103i
$$549$$ 2898.00 0.225289
$$550$$ 0 0
$$551$$ 9672.00 0.747806
$$552$$ 7560.00i 0.582926i
$$553$$ − 10400.0i − 0.799734i
$$554$$ −11886.0 −0.911530
$$555$$ 0 0
$$556$$ 2324.00 0.177265
$$557$$ 14166.0i 1.07762i 0.842428 + 0.538809i $$0.181125\pi$$
−0.842428 + 0.538809i $$0.818875\pi$$
$$558$$ 5400.00i 0.409678i
$$559$$ −6808.00 −0.515112
$$560$$ 0 0
$$561$$ 3888.00 0.292605
$$562$$ 24858.0i 1.86579i
$$563$$ − 13404.0i − 1.00339i −0.865043 0.501697i $$-0.832709\pi$$
0.865043 0.501697i $$-0.167291\pi$$
$$564$$ −72.0000 −0.00537544
$$565$$ 0 0
$$566$$ −8148.00 −0.605099
$$567$$ − 1620.00i − 0.119989i
$$568$$ 6048.00i 0.446775i
$$569$$ 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 $$-0.590859\pi$$
−0.281581 + 0.959537i $$0.590859\pi$$
$$572$$ 1776.00i 0.129822i
$$573$$ − 10728.0i − 0.782144i
$$574$$ −19800.0 −1.43978
$$575$$ 0 0
$$576$$ 3897.00 0.281901
$$577$$ 1726.00i 0.124531i 0.998060 + 0.0622654i $$0.0198325\pi$$
−0.998060 + 0.0622654i $$0.980167\pi$$
$$578$$ − 5991.00i − 0.431129i
$$579$$ 7998.00 0.574068
$$580$$ 0 0
$$581$$ 3120.00 0.222787
$$582$$ − 2574.00i − 0.183326i
$$583$$ − 10800.0i − 0.767222i
$$584$$ −9030.00 −0.639836
$$585$$ 0 0
$$586$$ 18054.0 1.27270
$$587$$ − 10596.0i − 0.745049i −0.928022 0.372524i $$-0.878492\pi$$
0.928022 0.372524i $$-0.121508\pi$$
$$588$$ − 171.000i − 0.0119931i
$$589$$ 24800.0 1.73492
$$590$$ 0 0
$$591$$ 8154.00 0.567531
$$592$$ − 4970.00i − 0.345043i
$$593$$ 2862.00i 0.198193i 0.995078 + 0.0990963i $$0.0315952\pi$$
−0.995078 + 0.0990963i $$0.968405\pi$$
$$594$$ −1944.00 −0.134282
$$595$$ 0 0
$$596$$ 258.000 0.0177317
$$597$$ − 11496.0i − 0.788107i
$$598$$ − 26640.0i − 1.82172i
$$599$$ 23592.0 1.60925 0.804627 0.593781i $$-0.202365\pi$$
0.804627 + 0.593781i $$0.202365\pi$$
$$600$$ 0 0
$$601$$ −9574.00 −0.649803 −0.324902 0.945748i $$-0.605331\pi$$
−0.324902 + 0.945748i $$0.605331\pi$$
$$602$$ − 5520.00i − 0.373718i
$$603$$ − 1764.00i − 0.119130i
$$604$$ 808.000 0.0544322
$$605$$ 0 0
$$606$$ 15606.0 1.04612
$$607$$ − 17444.0i − 1.16644i −0.812314 0.583221i $$-0.801792\pi$$
0.812314 0.583221i $$-0.198208\pi$$
$$608$$ 5580.00i 0.372202i
$$609$$ −4680.00 −0.311401
$$610$$ 0 0
$$611$$ −1776.00 −0.117593
$$612$$ − 486.000i − 0.0321003i
$$613$$ − 2374.00i − 0.156419i −0.996937 0.0782096i $$-0.975080\pi$$
0.996937 0.0782096i $$-0.0249203\pi$$
$$614$$ −27708.0 −1.82118
$$615$$ 0 0
$$616$$ 10080.0 0.659310
$$617$$ 12162.0i 0.793555i 0.917915 + 0.396778i $$0.129872\pi$$
−0.917915 + 0.396778i $$0.870128\pi$$
$$618$$ − 4068.00i − 0.264788i
$$619$$ −8804.00 −0.571668 −0.285834 0.958279i $$-0.592271\pi$$
−0.285834 + 0.958279i $$0.592271\pi$$
$$620$$ 0 0
$$621$$ 3240.00 0.209367
$$622$$ − 4608.00i − 0.297048i
$$623$$ 20520.0i 1.31961i
$$624$$ −15762.0 −1.01119
$$625$$ 0 0
$$626$$ −22026.0 −1.40629
$$627$$ 8928.00i 0.568660i
$$628$$ 2378.00i 0.151103i
$$629$$ 3780.00 0.239616
$$630$$ 0 0
$$631$$ −12688.0 −0.800478 −0.400239 0.916411i $$-0.631073\pi$$
−0.400239 + 0.916411i $$0.631073\pi$$
$$632$$ − 10920.0i − 0.687301i
$$633$$ − 3300.00i − 0.207209i
$$634$$ 11682.0 0.731785
$$635$$ 0 0
$$636$$ −1350.00 −0.0841682
$$637$$ − 4218.00i − 0.262360i
$$638$$ 5616.00i 0.348495i
$$639$$ 2592.00 0.160466
$$640$$ 0 0
$$641$$ −9150.00 −0.563812 −0.281906 0.959442i $$-0.590967\pi$$
−0.281906 + 0.959442i $$0.590967\pi$$
$$642$$ − 12636.0i − 0.776796i
$$643$$ 25292.0i 1.55120i 0.631227 + 0.775598i $$0.282552\pi$$
−0.631227 + 0.775598i $$0.717448\pi$$
$$644$$ 2400.00 0.146853
$$645$$ 0 0
$$646$$ −20088.0 −1.22345
$$647$$ 2736.00i 0.166249i 0.996539 + 0.0831246i $$0.0264900\pi$$
−0.996539 + 0.0831246i $$0.973510\pi$$
$$648$$ − 1701.00i − 0.103120i
$$649$$ 576.000 0.0348382
$$650$$ 0 0
$$651$$ −12000.0 −0.722453
$$652$$ 52.0000i 0.00312343i
$$653$$ 22218.0i 1.33148i 0.746183 + 0.665741i $$0.231884\pi$$
−0.746183 + 0.665741i $$0.768116\pi$$
$$654$$ −13266.0 −0.793183
$$655$$ 0 0
$$656$$ −23430.0 −1.39449
$$657$$ 3870.00i 0.229807i
$$658$$ − 1440.00i − 0.0853147i
$$659$$ −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 $$0.601133\pi$$
$$662$$ − 11076.0i − 0.650273i
$$663$$ − 11988.0i − 0.702225i
$$664$$ 3276.00 0.191466
$$665$$ 0 0
$$666$$ −1890.00 −0.109964
$$667$$ − 9360.00i − 0.543359i
$$668$$ − 3720.00i − 0.215466i
$$669$$ 5892.00 0.340505
$$670$$ 0 0
$$671$$ 7728.00 0.444614
$$672$$ − 2700.00i − 0.154992i
$$673$$ 1370.00i 0.0784690i 0.999230 + 0.0392345i $$0.0124919\pi$$
−0.999230 + 0.0392345i $$0.987508\pi$$
$$674$$ 26994.0 1.54269
$$675$$ 0 0
$$676$$ 3279.00 0.186561
$$677$$ 13758.0i 0.781038i 0.920595 + 0.390519i $$0.127704\pi$$
−0.920595 + 0.390519i $$0.872296\pi$$
$$678$$ − 9774.00i − 0.553640i
$$679$$ 5720.00 0.323289
$$680$$ 0 0
$$681$$ −1980.00 −0.111415
$$682$$ 14400.0i 0.808511i
$$683$$ 11988.0i 0.671608i 0.941932 + 0.335804i $$0.109008\pi$$
−0.941932 + 0.335804i $$0.890992\pi$$
$$684$$ 1116.00 0.0623850
$$685$$ 0 0
$$686$$ −17160.0 −0.955061
$$687$$ − 5718.00i − 0.317548i
$$688$$ − 6532.00i − 0.361962i
$$689$$ −33300.0 −1.84126
$$690$$ 0 0
$$691$$ 32996.0 1.81654 0.908268 0.418388i $$-0.137405\pi$$
0.908268 + 0.418388i $$0.137405\pi$$
$$692$$ − 426.000i − 0.0234019i
$$693$$ − 4320.00i − 0.236801i
$$694$$ −15732.0 −0.860488
$$695$$ 0 0
$$696$$ −4914.00 −0.267622
$$697$$ − 17820.0i − 0.968408i
$$698$$ 18906.0i 1.02522i
$$699$$ −4374.00 −0.236681
$$700$$ 0 0
$$701$$ −25902.0 −1.39558 −0.697792 0.716300i $$-0.745834\pi$$
−0.697792 + 0.716300i $$0.745834\pi$$
$$702$$ 5994.00i 0.322263i
$$703$$ 8680.00i 0.465679i
$$704$$ 10392.0 0.556340
$$705$$ 0 0
$$706$$ 10242.0 0.545981
$$707$$ 34680.0i 1.84480i
$$708$$ − 72.0000i − 0.00382193i
$$709$$ 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
$$712$$ 21546.0i 1.13409i
$$713$$ − 24000.0i − 1.26060i
$$714$$ 9720.00 0.509470
$$715$$ 0 0
$$716$$ −1440.00 −0.0751611
$$717$$ 3528.00i 0.183760i
$$718$$ 14472.0i 0.752215i
$$719$$ −34848.0 −1.80753 −0.903763 0.428033i $$-0.859207\pi$$
−0.903763 + 0.428033i $$0.859207\pi$$
$$720$$ 0 0
$$721$$ 9040.00 0.466945
$$722$$ − 25551.0i − 1.31705i
$$723$$ − 2598.00i − 0.133639i
$$724$$ 3130.00 0.160671
$$725$$ 0 0
$$726$$ 6795.00 0.347364
$$727$$ − 28028.0i − 1.42985i −0.699201 0.714925i $$-0.746461\pi$$
0.699201 0.714925i $$-0.253539\pi$$
$$728$$ − 31080.0i − 1.58228i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ 4968.00 0.251365
$$732$$ − 966.000i − 0.0487765i
$$733$$ 18002.0i 0.907120i 0.891226 + 0.453560i $$0.149846\pi$$
−0.891226 + 0.453560i $$0.850154\pi$$
$$734$$ 10524.0 0.529221
$$735$$ 0 0
$$736$$ 5400.00 0.270444
$$737$$ − 4704.00i − 0.235107i
$$738$$ 8910.00i 0.444420i
$$739$$ −15284.0 −0.760800 −0.380400 0.924822i $$-0.624214\pi$$
−0.380400 + 0.924822i $$0.624214\pi$$
$$740$$ 0 0
$$741$$ 27528.0 1.36473
$$742$$ − 27000.0i − 1.33585i
$$743$$ − 18768.0i − 0.926691i −0.886178 0.463345i $$-0.846649\pi$$
0.886178 0.463345i $$-0.153351\pi$$
$$744$$ −12600.0 −0.620885
$$745$$ 0 0
$$746$$ 32406.0 1.59044
$$747$$ − 1404.00i − 0.0687680i
$$748$$ − 1296.00i − 0.0633509i
$$749$$ 28080.0 1.36985
$$750$$ 0 0
$$751$$ 8696.00 0.422532 0.211266 0.977429i $$-0.432241\pi$$
0.211266 + 0.977429i $$0.432241\pi$$
$$752$$ − 1704.00i − 0.0826310i
$$753$$ − 1296.00i − 0.0627209i
$$754$$ 17316.0 0.836355
$$755$$ 0 0
$$756$$ −540.000 −0.0259783
$$757$$ 38662.0i 1.85627i 0.372247 + 0.928134i $$0.378587\pi$$
−0.372247 + 0.928134i $$0.621413\pi$$
$$758$$ 4380.00i 0.209880i
$$759$$ 8640.00 0.413191
$$760$$ 0 0
$$761$$ 23874.0 1.13723 0.568615 0.822604i $$-0.307479\pi$$
0.568615 + 0.822604i $$0.307479\pi$$
$$762$$ 11196.0i 0.532268i
$$763$$ − 29480.0i − 1.39875i
$$764$$ −3576.00 −0.169339
$$765$$ 0 0
$$766$$ −14616.0 −0.689422
$$767$$ − 1776.00i − 0.0836084i
$$768$$ − 4539.00i − 0.213264i
$$769$$ −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
$$772$$ − 2666.00i − 0.124289i
$$773$$ 11538.0i 0.536860i 0.963299 + 0.268430i $$0.0865049\pi$$
−0.963299 + 0.268430i $$0.913495\pi$$
$$774$$ −2484.00 −0.115356
$$775$$ 0 0
$$776$$ 6006.00 0.277839
$$777$$ − 4200.00i − 0.193918i
$$778$$ − 42138.0i − 1.94180i
$$779$$ 40920.0 1.88204
$$780$$ 0 0
$$781$$ 6912.00 0.316685
$$782$$ 19440.0i 0.888968i
$$783$$ 2106.00i 0.0961204i
$$784$$ 4047.00 0.184357
$$785$$ 0 0
$$786$$ −20952.0 −0.950805
$$787$$ 14884.0i 0.674152i 0.941478 + 0.337076i $$0.109438\pi$$
−0.941478 + 0.337076i $$0.890562\pi$$
$$788$$ − 2718.00i − 0.122874i
$$789$$ 16344.0 0.737467
$$790$$ 0 0
$$791$$ 21720.0 0.976327
$$792$$ − 4536.00i − 0.203510i
$$793$$ − 23828.0i − 1.06703i
$$794$$ 8202.00 0.366597
$$795$$ 0 0
$$796$$ −3832.00 −0.170630
$$797$$ 11334.0i 0.503728i 0.967763 + 0.251864i $$0.0810435\pi$$
−0.967763 + 0.251864i $$0.918957\pi$$
$$798$$ 22320.0i 0.990125i
$$799$$ 1296.00 0.0573832
$$800$$ 0 0
$$801$$ 9234.00 0.407325
$$802$$ 47826.0i 2.10573i
$$803$$ 10320.0i 0.453530i
$$804$$ −588.000 −0.0257925
$$805$$ 0 0
$$806$$ 44400.0 1.94035
$$807$$ − 7722.00i − 0.336837i
$$808$$ 36414.0i 1.58545i
$$809$$ −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 $$-0.876316\pi$$
−0.925453 + 0.378862i $$0.876316\pi$$
$$812$$ 1560.00i 0.0674203i
$$813$$ 9552.00i 0.412058i
$$814$$ −5040.00 −0.217017
$$815$$ 0 0
$$816$$ 11502.0 0.493444
$$817$$ 11408.0i 0.488513i
$$818$$ 26142.0i 1.11740i
$$819$$ −13320.0 −0.568301
$$820$$ 0 0
$$821$$ −31686.0 −1.34695 −0.673477 0.739208i $$-0.735200\pi$$
−0.673477 + 0.739208i $$0.735200\pi$$
$$822$$ 19062.0i 0.808836i
$$823$$ 11036.0i 0.467425i 0.972306 + 0.233713i $$0.0750875\pi$$
−0.972306 + 0.233713i $$0.924913\pi$$
$$824$$ 9492.00 0.401298
$$825$$ 0 0
$$826$$ 1440.00 0.0606586
$$827$$ − 25884.0i − 1.08836i −0.838968 0.544181i $$-0.816841\pi$$
0.838968 0.544181i $$-0.183159\pi$$
$$828$$ − 1080.00i − 0.0453292i
$$829$$ −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
$$832$$ − 32042.0i − 1.33516i
$$833$$ 3078.00i 0.128027i
$$834$$ 20916.0 0.868419
$$835$$ 0 0
$$836$$ 2976.00 0.123119
$$837$$ 5400.00i 0.223000i
$$838$$ 35928.0i 1.48104i
$$839$$ −13800.0 −0.567853 −0.283927 0.958846i $$-0.591637\pi$$
−0.283927 + 0.958846i $$0.591637\pi$$
$$840$$ 0 0
$$841$$ −18305.0 −0.750543
$$842$$ − 33162.0i − 1.35729i
$$843$$ 24858.0i 1.01560i
$$844$$ −1100.00 −0.0448620
$$845$$ 0 0
$$846$$ −648.000 −0.0263342
$$847$$ 15100.0i 0.612565i
$$848$$ − 31950.0i − 1.29383i
$$849$$ −8148.00 −0.329374
$$850$$ 0 0
$$851$$ 8400.00 0.338365
$$852$$ − 864.000i − 0.0347420i
$$853$$ − 27862.0i − 1.11838i −0.829040 0.559189i $$-0.811113\pi$$
0.829040 0.559189i $$-0.188887\pi$$
$$854$$ 19320.0 0.774141
$$855$$ 0 0
$$856$$ 29484.0 1.17727
$$857$$ 7314.00i 0.291530i 0.989319 + 0.145765i $$0.0465644\pi$$
−0.989319 + 0.145765i $$0.953436\pi$$
$$858$$ 15984.0i 0.635996i
$$859$$ 28780.0 1.14314 0.571572 0.820552i $$-0.306334\pi$$
0.571572 + 0.820552i $$0.306334\pi$$
$$860$$ 0 0
$$861$$ −19800.0 −0.783719
$$862$$ − 2160.00i − 0.0853479i
$$863$$ − 32688.0i − 1.28935i −0.764455 0.644677i $$-0.776992\pi$$
0.764455 0.644677i $$-0.223008\pi$$
$$864$$ −1215.00 −0.0478416
$$865$$ 0 0
$$866$$ −46866.0 −1.83900
$$867$$ − 5991.00i − 0.234677i
$$868$$ 4000.00i 0.156416i
$$869$$ −12480.0 −0.487175
$$870$$ 0 0
$$871$$ −14504.0 −0.564236
$$872$$ − 30954.0i − 1.20210i
$$873$$ − 2574.00i − 0.0997900i
$$874$$ −44640.0 −1.72766
$$875$$ 0 0
$$876$$ 1290.00 0.0497546
$$877$$ − 36650.0i − 1.41115i −0.708633 0.705577i $$-0.750688\pi$$
0.708633 0.705577i $$-0.249312\pi$$
$$878$$ − 29640.0i − 1.13930i
$$879$$ 18054.0 0.692772
$$880$$ 0 0
$$881$$ −2646.00 −0.101187 −0.0505936 0.998719i $$-0.516111\pi$$
−0.0505936 + 0.998719i $$0.516111\pi$$
$$882$$ − 1539.00i − 0.0587538i
$$883$$ 10892.0i 0.415113i 0.978223 + 0.207557i $$0.0665511\pi$$
−0.978223 + 0.207557i $$0.933449\pi$$
$$884$$ −3996.00 −0.152036
$$885$$ 0 0
$$886$$ −48348.0 −1.83328
$$887$$ 43464.0i 1.64530i 0.568550 + 0.822648i $$0.307504\pi$$
−0.568550 + 0.822648i $$0.692496\pi$$
$$888$$ − 4410.00i − 0.166655i
$$889$$ −24880.0 −0.938637
$$890$$ 0 0
$$891$$ −1944.00 −0.0730937
$$892$$ − 1964.00i − 0.0737215i
$$893$$ 2976.00i 0.111521i
$$894$$ 2322.00 0.0868672
$$895$$ 0 0
$$896$$ 33180.0 1.23713
$$897$$ − 26640.0i − 0.991621i
$$898$$ 27054.0i 1.00535i
$$899$$ 15600.0 0.578742
$$900$$ 0 0
$$901$$ 24300.0 0.898502
$$902$$ 23760.0i 0.877075i
$$903$$ − 5520.00i − 0.203426i
$$904$$ 22806.0 0.839067
$$905$$ 0 0
$$906$$ 7272.00 0.266662
$$907$$ 14884.0i 0.544890i 0.962171 + 0.272445i $$0.0878323\pi$$
−0.962171 + 0.272445i $$0.912168\pi$$
$$908$$ 660.000i 0.0241221i
$$909$$ 15606.0 0.569437
$$910$$ 0 0
$$911$$ −1248.00 −0.0453876 −0.0226938 0.999742i $$-0.507224\pi$$
−0.0226938 + 0.999742i $$0.507224\pi$$
$$912$$ 26412.0i 0.958979i
$$913$$ − 3744.00i − 0.135716i
$$914$$ 11010.0 0.398445
$$915$$ 0 0
$$916$$ −1906.00 −0.0687511
$$917$$ − 46560.0i − 1.67671i
$$918$$ − 4374.00i − 0.157259i
$$919$$ 6640.00 0.238339 0.119169 0.992874i $$-0.461977\pi$$
0.119169 + 0.992874i $$0.461977\pi$$
$$920$$ 0 0
$$921$$ −27708.0 −0.991324
$$922$$ − 52686.0i − 1.88191i
$$923$$ − 21312.0i − 0.760014i
$$924$$ −1440.00 −0.0512690
$$925$$ 0 0
$$926$$ 3516.00 0.124776
$$927$$ − 4068.00i − 0.144132i
$$928$$ 3510.00i 0.124161i
$$929$$ −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
$$932$$ 1458.00i 0.0512429i
$$933$$ − 4608.00i − 0.161693i
$$934$$ −20628.0 −0.722665
$$935$$ 0 0
$$936$$ −13986.0 −0.488405
$$937$$ − 45002.0i − 1.56900i −0.620130 0.784499i $$-0.712920\pi$$
0.620130 0.784499i $$-0.287080\pi$$
$$938$$ − 11760.0i − 0.409358i
$$939$$ −22026.0 −0.765486
$$940$$ 0 0
$$941$$ 6090.00 0.210976 0.105488 0.994421i $$-0.466360\pi$$
0.105488 + 0.994421i $$0.466360\pi$$
$$942$$ 21402.0i 0.740249i
$$943$$ − 39600.0i − 1.36750i
$$944$$ 1704.00 0.0587505
$$945$$ 0 0
$$946$$ −6624.00 −0.227658
$$947$$ − 56388.0i − 1.93491i −0.253035 0.967457i $$-0.581429\pi$$
0.253035 0.967457i $$-0.418571\pi$$
$$948$$ 1560.00i 0.0534456i
$$949$$ 31820.0 1.08843
$$950$$ 0 0
$$951$$ 11682.0 0.398333
$$952$$ 22680.0i 0.772125i
$$953$$ 10854.0i 0.368936i 0.982839 + 0.184468i $$0.0590561\pi$$
−0.982839 + 0.184468i $$0.940944\pi$$
$$954$$ −12150.0 −0.412338
$$955$$ 0 0
$$956$$ 1176.00 0.0397851
$$957$$ 5616.00i 0.189696i
$$958$$ 6840.00i 0.230679i
$$959$$ −42360.0 −1.42636
$$960$$ 0 0
$$961$$ 10209.0 0.342687
$$962$$ 15540.0i 0.520821i
$$963$$ − 12636.0i − 0.422834i
$$964$$ −866.000 −0.0289336
$$965$$ 0 0
$$966$$ 21600.0 0.719429
$$967$$ 42316.0i 1.40723i 0.710582 + 0.703615i $$0.248432\pi$$
−0.710582 + 0.703615i $$0.751568\pi$$
$$968$$ 15855.0i 0.526445i
$$969$$ −20088.0 −0.665964
$$970$$ 0 0
$$971$$ 24480.0 0.809063 0.404532 0.914524i $$-0.367435\pi$$
0.404532 + 0.914524i $$0.367435\pi$$
$$972$$ 243.000i 0.00801875i
$$973$$ 46480.0i 1.53143i
$$974$$ 9228.00 0.303577
$$975$$ 0 0
$$976$$ 22862.0 0.749790
$$977$$ 6906.00i 0.226144i 0.993587 + 0.113072i $$0.0360690\pi$$
−0.993587 + 0.113072i $$0.963931\pi$$
$$978$$ 468.000i 0.0153016i
$$979$$ 24624.0 0.803868
$$980$$ 0 0
$$981$$ −13266.0 −0.431754
$$982$$ 56736.0i 1.84371i
$$983$$ 6960.00i 0.225829i 0.993605 + 0.112914i $$0.0360186\pi$$
−0.993605 + 0.112914i $$0.963981\pi$$
$$984$$ −20790.0 −0.673538
$$985$$ 0 0
$$986$$ −12636.0 −0.408126
$$987$$ − 1440.00i − 0.0464394i
$$988$$ − 9176.00i − 0.295473i
$$989$$ 11040.0 0.354956
$$990$$ 0 0
$$991$$ 47792.0 1.53195 0.765975 0.642870i $$-0.222256\pi$$
0.765975 + 0.642870i $$0.222256\pi$$
$$992$$ 9000.00i 0.288055i
$$993$$ − 11076.0i − 0.353964i
$$994$$ 17280.0 0.551397
$$995$$ 0 0
$$996$$ −468.000 −0.0148887
$$997$$ − 9938.00i − 0.315687i −0.987464 0.157843i $$-0.949546\pi$$
0.987464 0.157843i $$-0.0504541\pi$$
$$998$$ 29868.0i 0.947350i
$$999$$ −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.b.a.49.1 2
3.2 odd 2 225.4.b.d.199.2 2
4.3 odd 2 1200.4.f.m.49.2 2
5.2 odd 4 15.4.a.b.1.1 1
5.3 odd 4 75.4.a.a.1.1 1
5.4 even 2 inner 75.4.b.a.49.2 2
15.2 even 4 45.4.a.b.1.1 1
15.8 even 4 225.4.a.g.1.1 1
15.14 odd 2 225.4.b.d.199.1 2
20.3 even 4 1200.4.a.o.1.1 1
20.7 even 4 240.4.a.f.1.1 1
20.19 odd 2 1200.4.f.m.49.1 2
35.27 even 4 735.4.a.i.1.1 1
40.27 even 4 960.4.a.l.1.1 1
40.37 odd 4 960.4.a.bi.1.1 1
45.2 even 12 405.4.e.k.271.1 2
45.7 odd 12 405.4.e.d.271.1 2
45.22 odd 12 405.4.e.d.136.1 2
45.32 even 12 405.4.e.k.136.1 2
55.32 even 4 1815.4.a.a.1.1 1
60.47 odd 4 720.4.a.r.1.1 1
105.62 odd 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 5.2 odd 4
45.4.a.b.1.1 1 15.2 even 4
75.4.a.a.1.1 1 5.3 odd 4
75.4.b.a.49.1 2 1.1 even 1 trivial
75.4.b.a.49.2 2 5.4 even 2 inner
225.4.a.g.1.1 1 15.8 even 4
225.4.b.d.199.1 2 15.14 odd 2
225.4.b.d.199.2 2 3.2 odd 2
240.4.a.f.1.1 1 20.7 even 4
405.4.e.d.136.1 2 45.22 odd 12
405.4.e.d.271.1 2 45.7 odd 12
405.4.e.k.136.1 2 45.32 even 12
405.4.e.k.271.1 2 45.2 even 12
720.4.a.r.1.1 1 60.47 odd 4
735.4.a.i.1.1 1 35.27 even 4
960.4.a.l.1.1 1 40.27 even 4
960.4.a.bi.1.1 1 40.37 odd 4
1200.4.a.o.1.1 1 20.3 even 4
1200.4.f.m.49.1 2 20.19 odd 2
1200.4.f.m.49.2 2 4.3 odd 2
1815.4.a.a.1.1 1 55.32 even 4
2205.4.a.c.1.1 1 105.62 odd 4