# Properties

 Label 1014.4.b.h.337.1 Level $1014$ Weight $4$ Character 1014.337 Analytic conductor $59.828$ 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] = [1014,4,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, 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("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1014.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: $$59.8279367458$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.337 Dual form 1014.4.b.h.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +10.0000i q^{5} -6.00000i q^{6} +8.00000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} +10.0000i q^{5} -6.00000i q^{6} +8.00000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +20.0000 q^{10} -40.0000i q^{11} -12.0000 q^{12} +16.0000 q^{14} +30.0000i q^{15} +16.0000 q^{16} -130.000 q^{17} -18.0000i q^{18} -20.0000i q^{19} -40.0000i q^{20} +24.0000i q^{21} -80.0000 q^{22} +24.0000i q^{24} +25.0000 q^{25} +27.0000 q^{27} -32.0000i q^{28} -18.0000 q^{29} +60.0000 q^{30} -184.000i q^{31} -32.0000i q^{32} -120.000i q^{33} +260.000i q^{34} -80.0000 q^{35} -36.0000 q^{36} +74.0000i q^{37} -40.0000 q^{38} -80.0000 q^{40} -362.000i q^{41} +48.0000 q^{42} -76.0000 q^{43} +160.000i q^{44} +90.0000i q^{45} +452.000i q^{47} +48.0000 q^{48} +279.000 q^{49} -50.0000i q^{50} -390.000 q^{51} +382.000 q^{53} -54.0000i q^{54} +400.000 q^{55} -64.0000 q^{56} -60.0000i q^{57} +36.0000i q^{58} -464.000i q^{59} -120.000i q^{60} +358.000 q^{61} -368.000 q^{62} +72.0000i q^{63} -64.0000 q^{64} -240.000 q^{66} -700.000i q^{67} +520.000 q^{68} +160.000i q^{70} -748.000i q^{71} +72.0000i q^{72} -1058.00i q^{73} +148.000 q^{74} +75.0000 q^{75} +80.0000i q^{76} +320.000 q^{77} -976.000 q^{79} +160.000i q^{80} +81.0000 q^{81} -724.000 q^{82} -1008.00i q^{83} -96.0000i q^{84} -1300.00i q^{85} +152.000i q^{86} -54.0000 q^{87} +320.000 q^{88} +386.000i q^{89} +180.000 q^{90} -552.000i q^{93} +904.000 q^{94} +200.000 q^{95} -96.0000i q^{96} -614.000i q^{97} -558.000i q^{98} -360.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 8 * q^4 + 18 * q^9 $$2 q + 6 q^{3} - 8 q^{4} + 18 q^{9} + 40 q^{10} - 24 q^{12} + 32 q^{14} + 32 q^{16} - 260 q^{17} - 160 q^{22} + 50 q^{25} + 54 q^{27} - 36 q^{29} + 120 q^{30} - 160 q^{35} - 72 q^{36} - 80 q^{38} - 160 q^{40} + 96 q^{42} - 152 q^{43} + 96 q^{48} + 558 q^{49} - 780 q^{51} + 764 q^{53} + 800 q^{55} - 128 q^{56} + 716 q^{61} - 736 q^{62} - 128 q^{64} - 480 q^{66} + 1040 q^{68} + 296 q^{74} + 150 q^{75} + 640 q^{77} - 1952 q^{79} + 162 q^{81} - 1448 q^{82} - 108 q^{87} + 640 q^{88} + 360 q^{90} + 1808 q^{94} + 400 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 - 8 * q^4 + 18 * q^9 + 40 * q^10 - 24 * q^12 + 32 * q^14 + 32 * q^16 - 260 * q^17 - 160 * q^22 + 50 * q^25 + 54 * q^27 - 36 * q^29 + 120 * q^30 - 160 * q^35 - 72 * q^36 - 80 * q^38 - 160 * q^40 + 96 * q^42 - 152 * q^43 + 96 * q^48 + 558 * q^49 - 780 * q^51 + 764 * q^53 + 800 * q^55 - 128 * q^56 + 716 * q^61 - 736 * q^62 - 128 * q^64 - 480 * q^66 + 1040 * q^68 + 296 * q^74 + 150 * q^75 + 640 * q^77 - 1952 * q^79 + 162 * q^81 - 1448 * q^82 - 108 * q^87 + 640 * q^88 + 360 * q^90 + 1808 * q^94 + 400 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 0.707107i
$$3$$ 3.00000 0.577350
$$4$$ −4.00000 −0.500000
$$5$$ 10.0000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ − 6.00000i − 0.408248i
$$7$$ 8.00000i 0.431959i 0.976398 + 0.215980i $$0.0692945\pi$$
−0.976398 + 0.215980i $$0.930705\pi$$
$$8$$ 8.00000i 0.353553i
$$9$$ 9.00000 0.333333
$$10$$ 20.0000 0.632456
$$11$$ − 40.0000i − 1.09640i −0.836346 0.548202i $$-0.815312\pi$$
0.836346 0.548202i $$-0.184688\pi$$
$$12$$ −12.0000 −0.288675
$$13$$ 0 0
$$14$$ 16.0000 0.305441
$$15$$ 30.0000i 0.516398i
$$16$$ 16.0000 0.250000
$$17$$ −130.000 −1.85468 −0.927342 0.374215i $$-0.877912\pi$$
−0.927342 + 0.374215i $$0.877912\pi$$
$$18$$ − 18.0000i − 0.235702i
$$19$$ − 20.0000i − 0.241490i −0.992684 0.120745i $$-0.961472\pi$$
0.992684 0.120745i $$-0.0385284\pi$$
$$20$$ − 40.0000i − 0.447214i
$$21$$ 24.0000i 0.249392i
$$22$$ −80.0000 −0.775275
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 24.0000i 0.204124i
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ − 32.0000i − 0.215980i
$$29$$ −18.0000 −0.115259 −0.0576296 0.998338i $$-0.518354\pi$$
−0.0576296 + 0.998338i $$0.518354\pi$$
$$30$$ 60.0000 0.365148
$$31$$ − 184.000i − 1.06604i −0.846101 0.533022i $$-0.821056\pi$$
0.846101 0.533022i $$-0.178944\pi$$
$$32$$ − 32.0000i − 0.176777i
$$33$$ − 120.000i − 0.633010i
$$34$$ 260.000i 1.31146i
$$35$$ −80.0000 −0.386356
$$36$$ −36.0000 −0.166667
$$37$$ 74.0000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ −40.0000 −0.170759
$$39$$ 0 0
$$40$$ −80.0000 −0.316228
$$41$$ − 362.000i − 1.37890i −0.724333 0.689450i $$-0.757852\pi$$
0.724333 0.689450i $$-0.242148\pi$$
$$42$$ 48.0000 0.176347
$$43$$ −76.0000 −0.269532 −0.134766 0.990877i $$-0.543028\pi$$
−0.134766 + 0.990877i $$0.543028\pi$$
$$44$$ 160.000i 0.548202i
$$45$$ 90.0000i 0.298142i
$$46$$ 0 0
$$47$$ 452.000i 1.40279i 0.712774 + 0.701393i $$0.247438\pi$$
−0.712774 + 0.701393i $$0.752562\pi$$
$$48$$ 48.0000 0.144338
$$49$$ 279.000 0.813411
$$50$$ − 50.0000i − 0.141421i
$$51$$ −390.000 −1.07080
$$52$$ 0 0
$$53$$ 382.000 0.990033 0.495016 0.868884i $$-0.335162\pi$$
0.495016 + 0.868884i $$0.335162\pi$$
$$54$$ − 54.0000i − 0.136083i
$$55$$ 400.000 0.980654
$$56$$ −64.0000 −0.152721
$$57$$ − 60.0000i − 0.139424i
$$58$$ 36.0000i 0.0815005i
$$59$$ − 464.000i − 1.02386i −0.859028 0.511929i $$-0.828931\pi$$
0.859028 0.511929i $$-0.171069\pi$$
$$60$$ − 120.000i − 0.258199i
$$61$$ 358.000 0.751430 0.375715 0.926735i $$-0.377397\pi$$
0.375715 + 0.926735i $$0.377397\pi$$
$$62$$ −368.000 −0.753807
$$63$$ 72.0000i 0.143986i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ −240.000 −0.447605
$$67$$ − 700.000i − 1.27640i −0.769872 0.638199i $$-0.779680\pi$$
0.769872 0.638199i $$-0.220320\pi$$
$$68$$ 520.000 0.927342
$$69$$ 0 0
$$70$$ 160.000i 0.273195i
$$71$$ − 748.000i − 1.25030i −0.780505 0.625150i $$-0.785038\pi$$
0.780505 0.625150i $$-0.214962\pi$$
$$72$$ 72.0000i 0.117851i
$$73$$ − 1058.00i − 1.69629i −0.529760 0.848147i $$-0.677718\pi$$
0.529760 0.848147i $$-0.322282\pi$$
$$74$$ 148.000 0.232495
$$75$$ 75.0000 0.115470
$$76$$ 80.0000i 0.120745i
$$77$$ 320.000 0.473602
$$78$$ 0 0
$$79$$ −976.000 −1.38998 −0.694991 0.719018i $$-0.744592\pi$$
−0.694991 + 0.719018i $$0.744592\pi$$
$$80$$ 160.000i 0.223607i
$$81$$ 81.0000 0.111111
$$82$$ −724.000 −0.975030
$$83$$ − 1008.00i − 1.33304i −0.745487 0.666520i $$-0.767783\pi$$
0.745487 0.666520i $$-0.232217\pi$$
$$84$$ − 96.0000i − 0.124696i
$$85$$ − 1300.00i − 1.65888i
$$86$$ 152.000i 0.190588i
$$87$$ −54.0000 −0.0665449
$$88$$ 320.000 0.387638
$$89$$ 386.000i 0.459729i 0.973223 + 0.229865i $$0.0738284\pi$$
−0.973223 + 0.229865i $$0.926172\pi$$
$$90$$ 180.000 0.210819
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 552.000i − 0.615481i
$$94$$ 904.000 0.991920
$$95$$ 200.000 0.215995
$$96$$ − 96.0000i − 0.102062i
$$97$$ − 614.000i − 0.642704i −0.946960 0.321352i $$-0.895863\pi$$
0.946960 0.321352i $$-0.104137\pi$$
$$98$$ − 558.000i − 0.575168i
$$99$$ − 360.000i − 0.365468i
$$100$$ −100.000 −0.100000
$$101$$ −518.000 −0.510326 −0.255163 0.966898i $$-0.582129\pi$$
−0.255163 + 0.966898i $$0.582129\pi$$
$$102$$ 780.000i 0.757172i
$$103$$ −112.000 −0.107143 −0.0535713 0.998564i $$-0.517060\pi$$
−0.0535713 + 0.998564i $$0.517060\pi$$
$$104$$ 0 0
$$105$$ −240.000 −0.223063
$$106$$ − 764.000i − 0.700059i
$$107$$ −372.000 −0.336099 −0.168050 0.985779i $$-0.553747\pi$$
−0.168050 + 0.985779i $$0.553747\pi$$
$$108$$ −108.000 −0.0962250
$$109$$ 934.000i 0.820743i 0.911918 + 0.410371i $$0.134601\pi$$
−0.911918 + 0.410371i $$0.865399\pi$$
$$110$$ − 800.000i − 0.693427i
$$111$$ 222.000i 0.189832i
$$112$$ 128.000i 0.107990i
$$113$$ 1914.00 1.59340 0.796699 0.604376i $$-0.206578\pi$$
0.796699 + 0.604376i $$0.206578\pi$$
$$114$$ −120.000 −0.0985880
$$115$$ 0 0
$$116$$ 72.0000 0.0576296
$$117$$ 0 0
$$118$$ −928.000 −0.723977
$$119$$ − 1040.00i − 0.801148i
$$120$$ −240.000 −0.182574
$$121$$ −269.000 −0.202104
$$122$$ − 716.000i − 0.531341i
$$123$$ − 1086.00i − 0.796108i
$$124$$ 736.000i 0.533022i
$$125$$ 1500.00i 1.07331i
$$126$$ 144.000 0.101814
$$127$$ −1296.00 −0.905523 −0.452761 0.891632i $$-0.649561\pi$$
−0.452761 + 0.891632i $$0.649561\pi$$
$$128$$ 128.000i 0.0883883i
$$129$$ −228.000 −0.155615
$$130$$ 0 0
$$131$$ −892.000 −0.594919 −0.297460 0.954734i $$-0.596139\pi$$
−0.297460 + 0.954734i $$0.596139\pi$$
$$132$$ 480.000i 0.316505i
$$133$$ 160.000 0.104314
$$134$$ −1400.00 −0.902549
$$135$$ 270.000i 0.172133i
$$136$$ − 1040.00i − 0.655730i
$$137$$ − 2326.00i − 1.45054i −0.688466 0.725269i $$-0.741716\pi$$
0.688466 0.725269i $$-0.258284\pi$$
$$138$$ 0 0
$$139$$ 1932.00 1.17892 0.589461 0.807797i $$-0.299340\pi$$
0.589461 + 0.807797i $$0.299340\pi$$
$$140$$ 320.000 0.193178
$$141$$ 1356.00i 0.809899i
$$142$$ −1496.00 −0.884095
$$143$$ 0 0
$$144$$ 144.000 0.0833333
$$145$$ − 180.000i − 0.103091i
$$146$$ −2116.00 −1.19946
$$147$$ 837.000 0.469623
$$148$$ − 296.000i − 0.164399i
$$149$$ 882.000i 0.484941i 0.970159 + 0.242471i $$0.0779578\pi$$
−0.970159 + 0.242471i $$0.922042\pi$$
$$150$$ − 150.000i − 0.0816497i
$$151$$ 1776.00i 0.957145i 0.878048 + 0.478572i $$0.158846\pi$$
−0.878048 + 0.478572i $$0.841154\pi$$
$$152$$ 160.000 0.0853797
$$153$$ −1170.00 −0.618228
$$154$$ − 640.000i − 0.334887i
$$155$$ 1840.00 0.953499
$$156$$ 0 0
$$157$$ −2410.00 −1.22509 −0.612544 0.790436i $$-0.709854\pi$$
−0.612544 + 0.790436i $$0.709854\pi$$
$$158$$ 1952.00i 0.982866i
$$159$$ 1146.00 0.571596
$$160$$ 320.000 0.158114
$$161$$ 0 0
$$162$$ − 162.000i − 0.0785674i
$$163$$ − 3212.00i − 1.54346i −0.635953 0.771728i $$-0.719393\pi$$
0.635953 0.771728i $$-0.280607\pi$$
$$164$$ 1448.00i 0.689450i
$$165$$ 1200.00 0.566181
$$166$$ −2016.00 −0.942602
$$167$$ − 1668.00i − 0.772896i −0.922311 0.386448i $$-0.873702\pi$$
0.922311 0.386448i $$-0.126298\pi$$
$$168$$ −192.000 −0.0881733
$$169$$ 0 0
$$170$$ −2600.00 −1.17301
$$171$$ − 180.000i − 0.0804967i
$$172$$ 304.000 0.134766
$$173$$ −3598.00 −1.58122 −0.790609 0.612321i $$-0.790236\pi$$
−0.790609 + 0.612321i $$0.790236\pi$$
$$174$$ 108.000i 0.0470544i
$$175$$ 200.000i 0.0863919i
$$176$$ − 640.000i − 0.274101i
$$177$$ − 1392.00i − 0.591125i
$$178$$ 772.000 0.325078
$$179$$ −1068.00 −0.445956 −0.222978 0.974824i $$-0.571578\pi$$
−0.222978 + 0.974824i $$0.571578\pi$$
$$180$$ − 360.000i − 0.149071i
$$181$$ 4786.00 1.96542 0.982709 0.185158i $$-0.0592797\pi$$
0.982709 + 0.185158i $$0.0592797\pi$$
$$182$$ 0 0
$$183$$ 1074.00 0.433838
$$184$$ 0 0
$$185$$ −740.000 −0.294086
$$186$$ −1104.00 −0.435211
$$187$$ 5200.00i 2.03348i
$$188$$ − 1808.00i − 0.701393i
$$189$$ 216.000i 0.0831306i
$$190$$ − 400.000i − 0.152732i
$$191$$ −1312.00 −0.497031 −0.248516 0.968628i $$-0.579943\pi$$
−0.248516 + 0.968628i $$0.579943\pi$$
$$192$$ −192.000 −0.0721688
$$193$$ 350.000i 0.130537i 0.997868 + 0.0652683i $$0.0207903\pi$$
−0.997868 + 0.0652683i $$0.979210\pi$$
$$194$$ −1228.00 −0.454460
$$195$$ 0 0
$$196$$ −1116.00 −0.406706
$$197$$ − 342.000i − 0.123688i −0.998086 0.0618439i $$-0.980302\pi$$
0.998086 0.0618439i $$-0.0196981\pi$$
$$198$$ −720.000 −0.258425
$$199$$ 3368.00 1.19975 0.599877 0.800092i $$-0.295216\pi$$
0.599877 + 0.800092i $$0.295216\pi$$
$$200$$ 200.000i 0.0707107i
$$201$$ − 2100.00i − 0.736928i
$$202$$ 1036.00i 0.360855i
$$203$$ − 144.000i − 0.0497873i
$$204$$ 1560.00 0.535401
$$205$$ 3620.00 1.23333
$$206$$ 224.000i 0.0757613i
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −800.000 −0.264771
$$210$$ 480.000i 0.157729i
$$211$$ −2004.00 −0.653844 −0.326922 0.945051i $$-0.606011\pi$$
−0.326922 + 0.945051i $$0.606011\pi$$
$$212$$ −1528.00 −0.495016
$$213$$ − 2244.00i − 0.721861i
$$214$$ 744.000i 0.237658i
$$215$$ − 760.000i − 0.241077i
$$216$$ 216.000i 0.0680414i
$$217$$ 1472.00 0.460488
$$218$$ 1868.00 0.580353
$$219$$ − 3174.00i − 0.979356i
$$220$$ −1600.00 −0.490327
$$221$$ 0 0
$$222$$ 444.000 0.134231
$$223$$ − 5608.00i − 1.68403i −0.539451 0.842017i $$-0.681368\pi$$
0.539451 0.842017i $$-0.318632\pi$$
$$224$$ 256.000 0.0763604
$$225$$ 225.000 0.0666667
$$226$$ − 3828.00i − 1.12670i
$$227$$ − 1928.00i − 0.563726i −0.959455 0.281863i $$-0.909048\pi$$
0.959455 0.281863i $$-0.0909524\pi$$
$$228$$ 240.000i 0.0697122i
$$229$$ 3938.00i 1.13638i 0.822898 + 0.568189i $$0.192356\pi$$
−0.822898 + 0.568189i $$0.807644\pi$$
$$230$$ 0 0
$$231$$ 960.000 0.273434
$$232$$ − 144.000i − 0.0407503i
$$233$$ −2562.00 −0.720353 −0.360176 0.932884i $$-0.617283\pi$$
−0.360176 + 0.932884i $$0.617283\pi$$
$$234$$ 0 0
$$235$$ −4520.00 −1.25469
$$236$$ 1856.00i 0.511929i
$$237$$ −2928.00 −0.802506
$$238$$ −2080.00 −0.566497
$$239$$ 7164.00i 1.93891i 0.245260 + 0.969457i $$0.421127\pi$$
−0.245260 + 0.969457i $$0.578873\pi$$
$$240$$ 480.000i 0.129099i
$$241$$ 6182.00i 1.65236i 0.563410 + 0.826178i $$0.309489\pi$$
−0.563410 + 0.826178i $$0.690511\pi$$
$$242$$ 538.000i 0.142909i
$$243$$ 243.000 0.0641500
$$244$$ −1432.00 −0.375715
$$245$$ 2790.00i 0.727537i
$$246$$ −2172.00 −0.562934
$$247$$ 0 0
$$248$$ 1472.00 0.376904
$$249$$ − 3024.00i − 0.769631i
$$250$$ 3000.00 0.758947
$$251$$ 1396.00 0.351055 0.175527 0.984475i $$-0.443837\pi$$
0.175527 + 0.984475i $$0.443837\pi$$
$$252$$ − 288.000i − 0.0719932i
$$253$$ 0 0
$$254$$ 2592.00i 0.640301i
$$255$$ − 3900.00i − 0.957755i
$$256$$ 256.000 0.0625000
$$257$$ −6906.00 −1.67620 −0.838102 0.545514i $$-0.816335\pi$$
−0.838102 + 0.545514i $$0.816335\pi$$
$$258$$ 456.000i 0.110036i
$$259$$ −592.000 −0.142027
$$260$$ 0 0
$$261$$ −162.000 −0.0384197
$$262$$ 1784.00i 0.420671i
$$263$$ −6848.00 −1.60557 −0.802787 0.596266i $$-0.796650\pi$$
−0.802787 + 0.596266i $$0.796650\pi$$
$$264$$ 960.000 0.223803
$$265$$ 3820.00i 0.885512i
$$266$$ − 320.000i − 0.0737611i
$$267$$ 1158.00i 0.265425i
$$268$$ 2800.00i 0.638199i
$$269$$ −6034.00 −1.36766 −0.683828 0.729643i $$-0.739686\pi$$
−0.683828 + 0.729643i $$0.739686\pi$$
$$270$$ 540.000 0.121716
$$271$$ − 4832.00i − 1.08311i −0.840665 0.541556i $$-0.817836\pi$$
0.840665 0.541556i $$-0.182164\pi$$
$$272$$ −2080.00 −0.463671
$$273$$ 0 0
$$274$$ −4652.00 −1.02568
$$275$$ − 1000.00i − 0.219281i
$$276$$ 0 0
$$277$$ 4082.00 0.885428 0.442714 0.896663i $$-0.354016\pi$$
0.442714 + 0.896663i $$0.354016\pi$$
$$278$$ − 3864.00i − 0.833623i
$$279$$ − 1656.00i − 0.355348i
$$280$$ − 640.000i − 0.136598i
$$281$$ − 3350.00i − 0.711189i −0.934640 0.355595i $$-0.884278\pi$$
0.934640 0.355595i $$-0.115722\pi$$
$$282$$ 2712.00 0.572685
$$283$$ −7796.00 −1.63754 −0.818770 0.574121i $$-0.805344\pi$$
−0.818770 + 0.574121i $$0.805344\pi$$
$$284$$ 2992.00i 0.625150i
$$285$$ 600.000 0.124705
$$286$$ 0 0
$$287$$ 2896.00 0.595629
$$288$$ − 288.000i − 0.0589256i
$$289$$ 11987.0 2.43985
$$290$$ −360.000 −0.0728963
$$291$$ − 1842.00i − 0.371065i
$$292$$ 4232.00i 0.848147i
$$293$$ − 3922.00i − 0.781999i −0.920391 0.390999i $$-0.872129\pi$$
0.920391 0.390999i $$-0.127871\pi$$
$$294$$ − 1674.00i − 0.332074i
$$295$$ 4640.00 0.915767
$$296$$ −592.000 −0.116248
$$297$$ − 1080.00i − 0.211003i
$$298$$ 1764.00 0.342905
$$299$$ 0 0
$$300$$ −300.000 −0.0577350
$$301$$ − 608.000i − 0.116427i
$$302$$ 3552.00 0.676803
$$303$$ −1554.00 −0.294637
$$304$$ − 320.000i − 0.0603726i
$$305$$ 3580.00i 0.672099i
$$306$$ 2340.00i 0.437153i
$$307$$ − 5956.00i − 1.10725i −0.832765 0.553627i $$-0.813243\pi$$
0.832765 0.553627i $$-0.186757\pi$$
$$308$$ −1280.00 −0.236801
$$309$$ −336.000 −0.0618588
$$310$$ − 3680.00i − 0.674226i
$$311$$ −2352.00 −0.428841 −0.214421 0.976741i $$-0.568786\pi$$
−0.214421 + 0.976741i $$0.568786\pi$$
$$312$$ 0 0
$$313$$ 8442.00 1.52450 0.762252 0.647280i $$-0.224093\pi$$
0.762252 + 0.647280i $$0.224093\pi$$
$$314$$ 4820.00i 0.866269i
$$315$$ −720.000 −0.128785
$$316$$ 3904.00 0.694991
$$317$$ − 5550.00i − 0.983341i −0.870781 0.491670i $$-0.836386\pi$$
0.870781 0.491670i $$-0.163614\pi$$
$$318$$ − 2292.00i − 0.404179i
$$319$$ 720.000i 0.126371i
$$320$$ − 640.000i − 0.111803i
$$321$$ −1116.00 −0.194047
$$322$$ 0 0
$$323$$ 2600.00i 0.447888i
$$324$$ −324.000 −0.0555556
$$325$$ 0 0
$$326$$ −6424.00 −1.09139
$$327$$ 2802.00i 0.473856i
$$328$$ 2896.00 0.487515
$$329$$ −3616.00 −0.605947
$$330$$ − 2400.00i − 0.400350i
$$331$$ 140.000i 0.0232480i 0.999932 + 0.0116240i $$0.00370012\pi$$
−0.999932 + 0.0116240i $$0.996300\pi$$
$$332$$ 4032.00i 0.666520i
$$333$$ 666.000i 0.109599i
$$334$$ −3336.00 −0.546520
$$335$$ 7000.00 1.14164
$$336$$ 384.000i 0.0623480i
$$337$$ 6174.00 0.997980 0.498990 0.866608i $$-0.333704\pi$$
0.498990 + 0.866608i $$0.333704\pi$$
$$338$$ 0 0
$$339$$ 5742.00 0.919949
$$340$$ 5200.00i 0.829440i
$$341$$ −7360.00 −1.16882
$$342$$ −360.000 −0.0569198
$$343$$ 4976.00i 0.783320i
$$344$$ − 608.000i − 0.0952941i
$$345$$ 0 0
$$346$$ 7196.00i 1.11809i
$$347$$ −2988.00 −0.462260 −0.231130 0.972923i $$-0.574242\pi$$
−0.231130 + 0.972923i $$0.574242\pi$$
$$348$$ 216.000 0.0332725
$$349$$ 162.000i 0.0248472i 0.999923 + 0.0124236i $$0.00395465\pi$$
−0.999923 + 0.0124236i $$0.996045\pi$$
$$350$$ 400.000 0.0610883
$$351$$ 0 0
$$352$$ −1280.00 −0.193819
$$353$$ − 10754.0i − 1.62147i −0.585416 0.810733i $$-0.699069\pi$$
0.585416 0.810733i $$-0.300931\pi$$
$$354$$ −2784.00 −0.417989
$$355$$ 7480.00 1.11830
$$356$$ − 1544.00i − 0.229865i
$$357$$ − 3120.00i − 0.462543i
$$358$$ 2136.00i 0.315338i
$$359$$ − 3588.00i − 0.527486i −0.964593 0.263743i $$-0.915043\pi$$
0.964593 0.263743i $$-0.0849570\pi$$
$$360$$ −720.000 −0.105409
$$361$$ 6459.00 0.941682
$$362$$ − 9572.00i − 1.38976i
$$363$$ −807.000 −0.116685
$$364$$ 0 0
$$365$$ 10580.0 1.51721
$$366$$ − 2148.00i − 0.306770i
$$367$$ 11272.0 1.60325 0.801626 0.597826i $$-0.203968\pi$$
0.801626 + 0.597826i $$0.203968\pi$$
$$368$$ 0 0
$$369$$ − 3258.00i − 0.459633i
$$370$$ 1480.00i 0.207950i
$$371$$ 3056.00i 0.427654i
$$372$$ 2208.00i 0.307741i
$$373$$ −10914.0 −1.51503 −0.757514 0.652819i $$-0.773586\pi$$
−0.757514 + 0.652819i $$0.773586\pi$$
$$374$$ 10400.0 1.43789
$$375$$ 4500.00i 0.619677i
$$376$$ −3616.00 −0.495960
$$377$$ 0 0
$$378$$ 432.000 0.0587822
$$379$$ 8100.00i 1.09781i 0.835886 + 0.548904i $$0.184955\pi$$
−0.835886 + 0.548904i $$0.815045\pi$$
$$380$$ −800.000 −0.107998
$$381$$ −3888.00 −0.522804
$$382$$ 2624.00i 0.351454i
$$383$$ 6180.00i 0.824499i 0.911071 + 0.412250i $$0.135257\pi$$
−0.911071 + 0.412250i $$0.864743\pi$$
$$384$$ 384.000i 0.0510310i
$$385$$ 3200.00i 0.423603i
$$386$$ 700.000 0.0923033
$$387$$ −684.000 −0.0898441
$$388$$ 2456.00i 0.321352i
$$389$$ 7522.00 0.980413 0.490206 0.871606i $$-0.336921\pi$$
0.490206 + 0.871606i $$0.336921\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 2232.00i 0.287584i
$$393$$ −2676.00 −0.343477
$$394$$ −684.000 −0.0874605
$$395$$ − 9760.00i − 1.24324i
$$396$$ 1440.00i 0.182734i
$$397$$ − 6078.00i − 0.768378i −0.923254 0.384189i $$-0.874481\pi$$
0.923254 0.384189i $$-0.125519\pi$$
$$398$$ − 6736.00i − 0.848355i
$$399$$ 480.000 0.0602257
$$400$$ 400.000 0.0500000
$$401$$ − 1830.00i − 0.227895i −0.993487 0.113947i $$-0.963650\pi$$
0.993487 0.113947i $$-0.0363495\pi$$
$$402$$ −4200.00 −0.521087
$$403$$ 0 0
$$404$$ 2072.00 0.255163
$$405$$ 810.000i 0.0993808i
$$406$$ −288.000 −0.0352049
$$407$$ 2960.00 0.360496
$$408$$ − 3120.00i − 0.378586i
$$409$$ 12434.0i 1.50323i 0.659601 + 0.751616i $$0.270725\pi$$
−0.659601 + 0.751616i $$0.729275\pi$$
$$410$$ − 7240.00i − 0.872093i
$$411$$ − 6978.00i − 0.837468i
$$412$$ 448.000 0.0535713
$$413$$ 3712.00 0.442265
$$414$$ 0 0
$$415$$ 10080.0 1.19231
$$416$$ 0 0
$$417$$ 5796.00 0.680651
$$418$$ 1600.00i 0.187221i
$$419$$ −14188.0 −1.65425 −0.827123 0.562021i $$-0.810024\pi$$
−0.827123 + 0.562021i $$0.810024\pi$$
$$420$$ 960.000 0.111531
$$421$$ 8638.00i 0.999977i 0.866032 + 0.499989i $$0.166662\pi$$
−0.866032 + 0.499989i $$0.833338\pi$$
$$422$$ 4008.00i 0.462337i
$$423$$ 4068.00i 0.467596i
$$424$$ 3056.00i 0.350029i
$$425$$ −3250.00 −0.370937
$$426$$ −4488.00 −0.510433
$$427$$ 2864.00i 0.324587i
$$428$$ 1488.00 0.168050
$$429$$ 0 0
$$430$$ −1520.00 −0.170467
$$431$$ 4292.00i 0.479671i 0.970813 + 0.239836i $$0.0770936\pi$$
−0.970813 + 0.239836i $$0.922906\pi$$
$$432$$ 432.000 0.0481125
$$433$$ 5982.00 0.663918 0.331959 0.943294i $$-0.392290\pi$$
0.331959 + 0.943294i $$0.392290\pi$$
$$434$$ − 2944.00i − 0.325614i
$$435$$ − 540.000i − 0.0595196i
$$436$$ − 3736.00i − 0.410371i
$$437$$ 0 0
$$438$$ −6348.00 −0.692510
$$439$$ −256.000 −0.0278319 −0.0139160 0.999903i $$-0.504430\pi$$
−0.0139160 + 0.999903i $$0.504430\pi$$
$$440$$ 3200.00i 0.346714i
$$441$$ 2511.00 0.271137
$$442$$ 0 0
$$443$$ 12556.0 1.34662 0.673311 0.739359i $$-0.264872\pi$$
0.673311 + 0.739359i $$0.264872\pi$$
$$444$$ − 888.000i − 0.0949158i
$$445$$ −3860.00 −0.411194
$$446$$ −11216.0 −1.19079
$$447$$ 2646.00i 0.279981i
$$448$$ − 512.000i − 0.0539949i
$$449$$ − 5574.00i − 0.585865i −0.956133 0.292932i $$-0.905369\pi$$
0.956133 0.292932i $$-0.0946311\pi$$
$$450$$ − 450.000i − 0.0471405i
$$451$$ −14480.0 −1.51183
$$452$$ −7656.00 −0.796699
$$453$$ 5328.00i 0.552608i
$$454$$ −3856.00 −0.398615
$$455$$ 0 0
$$456$$ 480.000 0.0492940
$$457$$ 1266.00i 0.129586i 0.997899 + 0.0647932i $$0.0206388\pi$$
−0.997899 + 0.0647932i $$0.979361\pi$$
$$458$$ 7876.00 0.803540
$$459$$ −3510.00 −0.356934
$$460$$ 0 0
$$461$$ 7554.00i 0.763178i 0.924332 + 0.381589i $$0.124623\pi$$
−0.924332 + 0.381589i $$0.875377\pi$$
$$462$$ − 1920.00i − 0.193347i
$$463$$ 6752.00i 0.677737i 0.940834 + 0.338868i $$0.110044\pi$$
−0.940834 + 0.338868i $$0.889956\pi$$
$$464$$ −288.000 −0.0288148
$$465$$ 5520.00 0.550503
$$466$$ 5124.00i 0.509366i
$$467$$ −7924.00 −0.785180 −0.392590 0.919714i $$-0.628421\pi$$
−0.392590 + 0.919714i $$0.628421\pi$$
$$468$$ 0 0
$$469$$ 5600.00 0.551352
$$470$$ 9040.00i 0.887200i
$$471$$ −7230.00 −0.707305
$$472$$ 3712.00 0.361989
$$473$$ 3040.00i 0.295517i
$$474$$ 5856.00i 0.567458i
$$475$$ − 500.000i − 0.0482980i
$$476$$ 4160.00i 0.400574i
$$477$$ 3438.00 0.330011
$$478$$ 14328.0 1.37102
$$479$$ 11084.0i 1.05729i 0.848844 + 0.528644i $$0.177299\pi$$
−0.848844 + 0.528644i $$0.822701\pi$$
$$480$$ 960.000 0.0912871
$$481$$ 0 0
$$482$$ 12364.0 1.16839
$$483$$ 0 0
$$484$$ 1076.00 0.101052
$$485$$ 6140.00 0.574852
$$486$$ − 486.000i − 0.0453609i
$$487$$ 4432.00i 0.412388i 0.978511 + 0.206194i $$0.0661078\pi$$
−0.978511 + 0.206194i $$0.933892\pi$$
$$488$$ 2864.00i 0.265670i
$$489$$ − 9636.00i − 0.891114i
$$490$$ 5580.00 0.514446
$$491$$ 1140.00 0.104781 0.0523905 0.998627i $$-0.483316\pi$$
0.0523905 + 0.998627i $$0.483316\pi$$
$$492$$ 4344.00i 0.398054i
$$493$$ 2340.00 0.213769
$$494$$ 0 0
$$495$$ 3600.00 0.326885
$$496$$ − 2944.00i − 0.266511i
$$497$$ 5984.00 0.540079
$$498$$ −6048.00 −0.544212
$$499$$ 1764.00i 0.158251i 0.996865 + 0.0791257i $$0.0252129\pi$$
−0.996865 + 0.0791257i $$0.974787\pi$$
$$500$$ − 6000.00i − 0.536656i
$$501$$ − 5004.00i − 0.446232i
$$502$$ − 2792.00i − 0.248233i
$$503$$ 16976.0 1.50482 0.752408 0.658697i $$-0.228892\pi$$
0.752408 + 0.658697i $$0.228892\pi$$
$$504$$ −576.000 −0.0509069
$$505$$ − 5180.00i − 0.456449i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 5184.00 0.452761
$$509$$ 9474.00i 0.825005i 0.910956 + 0.412503i $$0.135345\pi$$
−0.910956 + 0.412503i $$0.864655\pi$$
$$510$$ −7800.00 −0.677235
$$511$$ 8464.00 0.732731
$$512$$ − 512.000i − 0.0441942i
$$513$$ − 540.000i − 0.0464748i
$$514$$ 13812.0i 1.18526i
$$515$$ − 1120.00i − 0.0958313i
$$516$$ 912.000 0.0778073
$$517$$ 18080.0 1.53802
$$518$$ 1184.00i 0.100429i
$$519$$ −10794.0 −0.912917
$$520$$ 0 0
$$521$$ 14114.0 1.18684 0.593422 0.804892i $$-0.297777\pi$$
0.593422 + 0.804892i $$0.297777\pi$$
$$522$$ 324.000i 0.0271668i
$$523$$ 20284.0 1.69590 0.847952 0.530074i $$-0.177836\pi$$
0.847952 + 0.530074i $$0.177836\pi$$
$$524$$ 3568.00 0.297460
$$525$$ 600.000i 0.0498784i
$$526$$ 13696.0i 1.13531i
$$527$$ 23920.0i 1.97718i
$$528$$ − 1920.00i − 0.158252i
$$529$$ −12167.0 −1.00000
$$530$$ 7640.00 0.626152
$$531$$ − 4176.00i − 0.341286i
$$532$$ −640.000 −0.0521570
$$533$$ 0 0
$$534$$ 2316.00 0.187684
$$535$$ − 3720.00i − 0.300616i
$$536$$ 5600.00 0.451275
$$537$$ −3204.00 −0.257473
$$538$$ 12068.0i 0.967079i
$$539$$ − 11160.0i − 0.891828i
$$540$$ − 1080.00i − 0.0860663i
$$541$$ 14362.0i 1.14135i 0.821176 + 0.570675i $$0.193318\pi$$
−0.821176 + 0.570675i $$0.806682\pi$$
$$542$$ −9664.00 −0.765875
$$543$$ 14358.0 1.13473
$$544$$ 4160.00i 0.327865i
$$545$$ −9340.00 −0.734095
$$546$$ 0 0
$$547$$ −20956.0 −1.63805 −0.819025 0.573757i $$-0.805485\pi$$
−0.819025 + 0.573757i $$0.805485\pi$$
$$548$$ 9304.00i 0.725269i
$$549$$ 3222.00 0.250477
$$550$$ −2000.00 −0.155055
$$551$$ 360.000i 0.0278340i
$$552$$ 0 0
$$553$$ − 7808.00i − 0.600416i
$$554$$ − 8164.00i − 0.626092i
$$555$$ −2220.00 −0.169791
$$556$$ −7728.00 −0.589461
$$557$$ 4134.00i 0.314476i 0.987561 + 0.157238i $$0.0502590\pi$$
−0.987561 + 0.157238i $$0.949741\pi$$
$$558$$ −3312.00 −0.251269
$$559$$ 0 0
$$560$$ −1280.00 −0.0965891
$$561$$ 15600.0i 1.17403i
$$562$$ −6700.00 −0.502887
$$563$$ 16228.0 1.21479 0.607397 0.794399i $$-0.292214\pi$$
0.607397 + 0.794399i $$0.292214\pi$$
$$564$$ − 5424.00i − 0.404950i
$$565$$ 19140.0i 1.42518i
$$566$$ 15592.0i 1.15792i
$$567$$ 648.000i 0.0479955i
$$568$$ 5984.00 0.442048
$$569$$ −2514.00 −0.185224 −0.0926119 0.995702i $$-0.529522\pi$$
−0.0926119 + 0.995702i $$0.529522\pi$$
$$570$$ − 1200.00i − 0.0881798i
$$571$$ 11612.0 0.851046 0.425523 0.904948i $$-0.360090\pi$$
0.425523 + 0.904948i $$0.360090\pi$$
$$572$$ 0 0
$$573$$ −3936.00 −0.286961
$$574$$ − 5792.00i − 0.421173i
$$575$$ 0 0
$$576$$ −576.000 −0.0416667
$$577$$ 6354.00i 0.458441i 0.973375 + 0.229221i $$0.0736177\pi$$
−0.973375 + 0.229221i $$0.926382\pi$$
$$578$$ − 23974.0i − 1.72524i
$$579$$ 1050.00i 0.0753653i
$$580$$ 720.000i 0.0515455i
$$581$$ 8064.00 0.575819
$$582$$ −3684.00 −0.262383
$$583$$ − 15280.0i − 1.08548i
$$584$$ 8464.00 0.599731
$$585$$ 0 0
$$586$$ −7844.00 −0.552957
$$587$$ − 13240.0i − 0.930960i −0.885059 0.465480i $$-0.845882\pi$$
0.885059 0.465480i $$-0.154118\pi$$
$$588$$ −3348.00 −0.234812
$$589$$ −3680.00 −0.257439
$$590$$ − 9280.00i − 0.647545i
$$591$$ − 1026.00i − 0.0714112i
$$592$$ 1184.00i 0.0821995i
$$593$$ 1146.00i 0.0793602i 0.999212 + 0.0396801i $$0.0126339\pi$$
−0.999212 + 0.0396801i $$0.987366\pi$$
$$594$$ −2160.00 −0.149202
$$595$$ 10400.0 0.716569
$$596$$ − 3528.00i − 0.242471i
$$597$$ 10104.0 0.692679
$$598$$ 0 0
$$599$$ 10464.0 0.713769 0.356884 0.934149i $$-0.383839\pi$$
0.356884 + 0.934149i $$0.383839\pi$$
$$600$$ 600.000i 0.0408248i
$$601$$ 6650.00 0.451346 0.225673 0.974203i $$-0.427542\pi$$
0.225673 + 0.974203i $$0.427542\pi$$
$$602$$ −1216.00 −0.0823263
$$603$$ − 6300.00i − 0.425466i
$$604$$ − 7104.00i − 0.478572i
$$605$$ − 2690.00i − 0.180767i
$$606$$ 3108.00i 0.208340i
$$607$$ −6664.00 −0.445607 −0.222803 0.974863i $$-0.571521\pi$$
−0.222803 + 0.974863i $$0.571521\pi$$
$$608$$ −640.000 −0.0426898
$$609$$ − 432.000i − 0.0287447i
$$610$$ 7160.00 0.475246
$$611$$ 0 0
$$612$$ 4680.00 0.309114
$$613$$ 2134.00i 0.140606i 0.997526 + 0.0703030i $$0.0223966\pi$$
−0.997526 + 0.0703030i $$0.977603\pi$$
$$614$$ −11912.0 −0.782947
$$615$$ 10860.0 0.712061
$$616$$ 2560.00i 0.167444i
$$617$$ − 714.000i − 0.0465876i −0.999729 0.0232938i $$-0.992585\pi$$
0.999729 0.0232938i $$-0.00741532\pi$$
$$618$$ 672.000i 0.0437408i
$$619$$ − 29228.0i − 1.89786i −0.315494 0.948928i $$-0.602170\pi$$
0.315494 0.948928i $$-0.397830\pi$$
$$620$$ −7360.00 −0.476750
$$621$$ 0 0
$$622$$ 4704.00i 0.303237i
$$623$$ −3088.00 −0.198584
$$624$$ 0 0
$$625$$ −11875.0 −0.760000
$$626$$ − 16884.0i − 1.07799i
$$627$$ −2400.00 −0.152866
$$628$$ 9640.00 0.612544
$$629$$ − 9620.00i − 0.609816i
$$630$$ 1440.00i 0.0910650i
$$631$$ 13536.0i 0.853977i 0.904257 + 0.426989i $$0.140426\pi$$
−0.904257 + 0.426989i $$0.859574\pi$$
$$632$$ − 7808.00i − 0.491433i
$$633$$ −6012.00 −0.377497
$$634$$ −11100.0 −0.695327
$$635$$ − 12960.0i − 0.809924i
$$636$$ −4584.00 −0.285798
$$637$$ 0 0
$$638$$ 1440.00 0.0893576
$$639$$ − 6732.00i − 0.416767i
$$640$$ −1280.00 −0.0790569
$$641$$ −17218.0 −1.06095 −0.530476 0.847700i $$-0.677987\pi$$
−0.530476 + 0.847700i $$0.677987\pi$$
$$642$$ 2232.00i 0.137212i
$$643$$ 15044.0i 0.922671i 0.887226 + 0.461335i $$0.152630\pi$$
−0.887226 + 0.461335i $$0.847370\pi$$
$$644$$ 0 0
$$645$$ − 2280.00i − 0.139186i
$$646$$ 5200.00 0.316705
$$647$$ −25176.0 −1.52978 −0.764892 0.644158i $$-0.777208\pi$$
−0.764892 + 0.644158i $$0.777208\pi$$
$$648$$ 648.000i 0.0392837i
$$649$$ −18560.0 −1.12256
$$650$$ 0 0
$$651$$ 4416.00 0.265863
$$652$$ 12848.0i 0.771728i
$$653$$ −16034.0 −0.960887 −0.480443 0.877026i $$-0.659524\pi$$
−0.480443 + 0.877026i $$0.659524\pi$$
$$654$$ 5604.00 0.335067
$$655$$ − 8920.00i − 0.532112i
$$656$$ − 5792.00i − 0.344725i
$$657$$ − 9522.00i − 0.565432i
$$658$$ 7232.00i 0.428469i
$$659$$ 25356.0 1.49883 0.749415 0.662100i $$-0.230335\pi$$
0.749415 + 0.662100i $$0.230335\pi$$
$$660$$ −4800.00 −0.283091
$$661$$ − 18310.0i − 1.07742i −0.842490 0.538711i $$-0.818911\pi$$
0.842490 0.538711i $$-0.181089\pi$$
$$662$$ 280.000 0.0164388
$$663$$ 0 0
$$664$$ 8064.00 0.471301
$$665$$ 1600.00i 0.0933013i
$$666$$ 1332.00 0.0774984
$$667$$ 0 0
$$668$$ 6672.00i 0.386448i
$$669$$ − 16824.0i − 0.972277i
$$670$$ − 14000.0i − 0.807264i
$$671$$ − 14320.0i − 0.823871i
$$672$$ 768.000 0.0440867
$$673$$ −24802.0 −1.42057 −0.710287 0.703912i $$-0.751435\pi$$
−0.710287 + 0.703912i $$0.751435\pi$$
$$674$$ − 12348.0i − 0.705678i
$$675$$ 675.000 0.0384900
$$676$$ 0 0
$$677$$ −22706.0 −1.28901 −0.644507 0.764598i $$-0.722937\pi$$
−0.644507 + 0.764598i $$0.722937\pi$$
$$678$$ − 11484.0i − 0.650502i
$$679$$ 4912.00 0.277622
$$680$$ 10400.0 0.586503
$$681$$ − 5784.00i − 0.325467i
$$682$$ 14720.0i 0.826478i
$$683$$ 14792.0i 0.828697i 0.910118 + 0.414349i $$0.135991\pi$$
−0.910118 + 0.414349i $$0.864009\pi$$
$$684$$ 720.000i 0.0402484i
$$685$$ 23260.0 1.29740
$$686$$ 9952.00 0.553891
$$687$$ 11814.0i 0.656088i
$$688$$ −1216.00 −0.0673831
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 1148.00i − 0.0632011i −0.999501 0.0316006i $$-0.989940\pi$$
0.999501 0.0316006i $$-0.0100604\pi$$
$$692$$ 14392.0 0.790609
$$693$$ 2880.00 0.157867
$$694$$ 5976.00i 0.326867i
$$695$$ 19320.0i 1.05446i
$$696$$ − 432.000i − 0.0235272i
$$697$$ 47060.0i 2.55742i
$$698$$ 324.000 0.0175696
$$699$$ −7686.00 −0.415896
$$700$$ − 800.000i − 0.0431959i
$$701$$ −14870.0 −0.801187 −0.400594 0.916256i $$-0.631196\pi$$
−0.400594 + 0.916256i $$0.631196\pi$$
$$702$$ 0 0
$$703$$ 1480.00 0.0794015
$$704$$ 2560.00i 0.137051i
$$705$$ −13560.0 −0.724396
$$706$$ −21508.0 −1.14655
$$707$$ − 4144.00i − 0.220440i
$$708$$ 5568.00i 0.295563i
$$709$$ 6354.00i 0.336572i 0.985738 + 0.168286i $$0.0538232\pi$$
−0.985738 + 0.168286i $$0.946177\pi$$
$$710$$ − 14960.0i − 0.790759i
$$711$$ −8784.00 −0.463327
$$712$$ −3088.00 −0.162539
$$713$$ 0 0
$$714$$ −6240.00 −0.327067
$$715$$ 0 0
$$716$$ 4272.00 0.222978
$$717$$ 21492.0i 1.11943i
$$718$$ −7176.00 −0.372989
$$719$$ −9288.00 −0.481758 −0.240879 0.970555i $$-0.577436\pi$$
−0.240879 + 0.970555i $$0.577436\pi$$
$$720$$ 1440.00i 0.0745356i
$$721$$ − 896.000i − 0.0462813i
$$722$$ − 12918.0i − 0.665870i
$$723$$ 18546.0i 0.953988i
$$724$$ −19144.0 −0.982709
$$725$$ −450.000 −0.0230518
$$726$$ 1614.00i 0.0825085i
$$727$$ 21544.0 1.09907 0.549534 0.835471i $$-0.314805\pi$$
0.549534 + 0.835471i $$0.314805\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ − 21160.0i − 1.07283i
$$731$$ 9880.00 0.499897
$$732$$ −4296.00 −0.216919
$$733$$ 19990.0i 1.00730i 0.863909 + 0.503648i $$0.168009\pi$$
−0.863909 + 0.503648i $$0.831991\pi$$
$$734$$ − 22544.0i − 1.13367i
$$735$$ 8370.00i 0.420044i
$$736$$ 0 0
$$737$$ −28000.0 −1.39945
$$738$$ −6516.00 −0.325010
$$739$$ − 532.000i − 0.0264816i −0.999912 0.0132408i $$-0.995785\pi$$
0.999912 0.0132408i $$-0.00421481\pi$$
$$740$$ 2960.00 0.147043
$$741$$ 0 0
$$742$$ 6112.00 0.302397
$$743$$ − 25452.0i − 1.25672i −0.777922 0.628360i $$-0.783726\pi$$
0.777922 0.628360i $$-0.216274\pi$$
$$744$$ 4416.00 0.217605
$$745$$ −8820.00 −0.433745
$$746$$ 21828.0i 1.07129i
$$747$$ − 9072.00i − 0.444347i
$$748$$ − 20800.0i − 1.01674i
$$749$$ − 2976.00i − 0.145181i
$$750$$ 9000.00 0.438178
$$751$$ −6440.00 −0.312915 −0.156457 0.987685i $$-0.550007\pi$$
−0.156457 + 0.987685i $$0.550007\pi$$
$$752$$ 7232.00i 0.350697i
$$753$$ 4188.00 0.202682
$$754$$ 0 0
$$755$$ −17760.0 −0.856096
$$756$$ − 864.000i − 0.0415653i
$$757$$ −786.000 −0.0377380 −0.0188690 0.999822i $$-0.506007\pi$$
−0.0188690 + 0.999822i $$0.506007\pi$$
$$758$$ 16200.0 0.776267
$$759$$ 0 0
$$760$$ 1600.00i 0.0763659i
$$761$$ 1498.00i 0.0713567i 0.999363 + 0.0356784i $$0.0113592\pi$$
−0.999363 + 0.0356784i $$0.988641\pi$$
$$762$$ 7776.00i 0.369678i
$$763$$ −7472.00 −0.354528
$$764$$ 5248.00 0.248516
$$765$$ − 11700.0i − 0.552960i
$$766$$ 12360.0 0.583009
$$767$$ 0 0
$$768$$ 768.000 0.0360844
$$769$$ 14738.0i 0.691113i 0.938398 + 0.345556i $$0.112310\pi$$
−0.938398 + 0.345556i $$0.887690\pi$$
$$770$$ 6400.00 0.299532
$$771$$ −20718.0 −0.967757
$$772$$ − 1400.00i − 0.0652683i
$$773$$ − 3822.00i − 0.177837i −0.996039 0.0889184i $$-0.971659\pi$$
0.996039 0.0889184i $$-0.0283410\pi$$
$$774$$ 1368.00i 0.0635294i
$$775$$ − 4600.00i − 0.213209i
$$776$$ 4912.00 0.227230
$$777$$ −1776.00 −0.0819995
$$778$$ − 15044.0i − 0.693256i
$$779$$ −7240.00 −0.332991
$$780$$ 0 0
$$781$$ −29920.0 −1.37083
$$782$$ 0 0
$$783$$ −486.000 −0.0221816
$$784$$ 4464.00 0.203353
$$785$$ − 24100.0i − 1.09575i
$$786$$ 5352.00i 0.242875i
$$787$$ 11900.0i 0.538995i 0.963001 + 0.269498i $$0.0868576\pi$$
−0.963001 + 0.269498i $$0.913142\pi$$
$$788$$ 1368.00i 0.0618439i
$$789$$ −20544.0 −0.926978
$$790$$ −19520.0 −0.879102
$$791$$ 15312.0i 0.688283i
$$792$$ 2880.00 0.129213
$$793$$ 0 0
$$794$$ −12156.0 −0.543325
$$795$$ 11460.0i 0.511251i
$$796$$ −13472.0 −0.599877
$$797$$ 21274.0 0.945500 0.472750 0.881197i $$-0.343261\pi$$
0.472750 + 0.881197i $$0.343261\pi$$
$$798$$ − 960.000i − 0.0425860i
$$799$$ − 58760.0i − 2.60173i
$$800$$ − 800.000i − 0.0353553i
$$801$$ 3474.00i 0.153243i
$$802$$ −3660.00 −0.161146
$$803$$ −42320.0 −1.85983
$$804$$ 8400.00i 0.368464i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −18102.0 −0.789617
$$808$$ − 4144.00i − 0.180427i
$$809$$ −27566.0 −1.19798 −0.598992 0.800755i $$-0.704432\pi$$
−0.598992 + 0.800755i $$0.704432\pi$$
$$810$$ 1620.00 0.0702728
$$811$$ − 11244.0i − 0.486844i −0.969921 0.243422i $$-0.921730\pi$$
0.969921 0.243422i $$-0.0782699\pi$$
$$812$$ 576.000i 0.0248936i
$$813$$ − 14496.0i − 0.625334i
$$814$$ − 5920.00i − 0.254909i
$$815$$ 32120.0 1.38051
$$816$$ −6240.00 −0.267701
$$817$$ 1520.00i 0.0650894i
$$818$$ 24868.0 1.06295
$$819$$ 0 0
$$820$$ −14480.0 −0.616663
$$821$$ 13554.0i 0.576173i 0.957604 + 0.288086i $$0.0930190\pi$$
−0.957604 + 0.288086i $$0.906981\pi$$
$$822$$ −13956.0 −0.592179
$$823$$ −14384.0 −0.609228 −0.304614 0.952476i $$-0.598527\pi$$
−0.304614 + 0.952476i $$0.598527\pi$$
$$824$$ − 896.000i − 0.0378806i
$$825$$ − 3000.00i − 0.126602i
$$826$$ − 7424.00i − 0.312729i
$$827$$ 2488.00i 0.104615i 0.998631 + 0.0523073i $$0.0166575\pi$$
−0.998631 + 0.0523073i $$0.983342\pi$$
$$828$$ 0 0
$$829$$ 20858.0 0.873858 0.436929 0.899496i $$-0.356066\pi$$
0.436929 + 0.899496i $$0.356066\pi$$
$$830$$ − 20160.0i − 0.843089i
$$831$$ 12246.0 0.511202
$$832$$ 0 0
$$833$$ −36270.0 −1.50862
$$834$$ − 11592.0i − 0.481293i
$$835$$ 16680.0 0.691300
$$836$$ 3200.00 0.132386
$$837$$ − 4968.00i − 0.205160i
$$838$$ 28376.0i 1.16973i
$$839$$ − 23116.0i − 0.951195i −0.879663 0.475598i $$-0.842232\pi$$
0.879663 0.475598i $$-0.157768\pi$$
$$840$$ − 1920.00i − 0.0788646i
$$841$$ −24065.0 −0.986715
$$842$$ 17276.0 0.707091
$$843$$ − 10050.0i − 0.410605i
$$844$$ 8016.00 0.326922
$$845$$ 0 0
$$846$$ 8136.00 0.330640
$$847$$ − 2152.00i − 0.0873006i
$$848$$ 6112.00 0.247508
$$849$$ −23388.0 −0.945435
$$850$$ 6500.00i 0.262292i
$$851$$ 0 0
$$852$$ 8976.00i 0.360930i
$$853$$ − 934.000i − 0.0374907i −0.999824 0.0187453i $$-0.994033\pi$$
0.999824 0.0187453i $$-0.00596718\pi$$
$$854$$ 5728.00 0.229518
$$855$$ 1800.00 0.0719985
$$856$$ − 2976.00i − 0.118829i
$$857$$ −12642.0 −0.503900 −0.251950 0.967740i $$-0.581072\pi$$
−0.251950 + 0.967740i $$0.581072\pi$$
$$858$$ 0 0
$$859$$ −22796.0 −0.905459 −0.452730 0.891648i $$-0.649550\pi$$
−0.452730 + 0.891648i $$0.649550\pi$$
$$860$$ 3040.00i 0.120539i
$$861$$ 8688.00 0.343886
$$862$$ 8584.00 0.339179
$$863$$ − 76.0000i − 0.00299776i −0.999999 0.00149888i $$-0.999523\pi$$
0.999999 0.00149888i $$-0.000477109\pi$$
$$864$$ − 864.000i − 0.0340207i
$$865$$ − 35980.0i − 1.41429i
$$866$$ − 11964.0i − 0.469461i
$$867$$ 35961.0 1.40865
$$868$$ −5888.00 −0.230244
$$869$$ 39040.0i 1.52398i
$$870$$ −1080.00 −0.0420867
$$871$$ 0 0
$$872$$ −7472.00 −0.290176
$$873$$ − 5526.00i − 0.214235i
$$874$$ 0 0
$$875$$ −12000.0 −0.463627
$$876$$ 12696.0i 0.489678i
$$877$$ − 46130.0i − 1.77617i −0.459681 0.888084i $$-0.652036\pi$$
0.459681 0.888084i $$-0.347964\pi$$
$$878$$ 512.000i 0.0196801i
$$879$$ − 11766.0i − 0.451487i
$$880$$ 6400.00 0.245164
$$881$$ −6682.00 −0.255530 −0.127765 0.991804i $$-0.540780\pi$$
−0.127765 + 0.991804i $$0.540780\pi$$
$$882$$ − 5022.00i − 0.191723i
$$883$$ −47404.0 −1.80665 −0.903325 0.428957i $$-0.858881\pi$$
−0.903325 + 0.428957i $$0.858881\pi$$
$$884$$ 0 0
$$885$$ 13920.0 0.528718
$$886$$ − 25112.0i − 0.952206i
$$887$$ 33672.0 1.27463 0.637314 0.770604i $$-0.280045\pi$$
0.637314 + 0.770604i $$0.280045\pi$$
$$888$$ −1776.00 −0.0671156
$$889$$ − 10368.0i − 0.391149i
$$890$$ 7720.00i 0.290758i
$$891$$ − 3240.00i − 0.121823i
$$892$$ 22432.0i 0.842017i
$$893$$ 9040.00 0.338759
$$894$$ 5292.00 0.197976
$$895$$ − 10680.0i − 0.398875i
$$896$$ −1024.00 −0.0381802
$$897$$ 0 0
$$898$$ −11148.0 −0.414269
$$899$$ 3312.00i 0.122871i
$$900$$ −900.000 −0.0333333
$$901$$ −49660.0 −1.83620
$$902$$ 28960.0i 1.06903i
$$903$$ − 1824.00i − 0.0672192i
$$904$$ 15312.0i 0.563351i
$$905$$ 47860.0i 1.75792i
$$906$$ 10656.0 0.390753
$$907$$ 14540.0 0.532296 0.266148 0.963932i $$-0.414249\pi$$
0.266148 + 0.963932i $$0.414249\pi$$
$$908$$ 7712.00i 0.281863i
$$909$$ −4662.00 −0.170109
$$910$$ 0 0
$$911$$ −7840.00 −0.285127 −0.142564 0.989786i $$-0.545535\pi$$
−0.142564 + 0.989786i $$0.545535\pi$$
$$912$$ − 960.000i − 0.0348561i
$$913$$ −40320.0 −1.46155
$$914$$ 2532.00 0.0916314
$$915$$ 10740.0i 0.388037i
$$916$$ − 15752.0i − 0.568189i
$$917$$ − 7136.00i − 0.256981i
$$918$$ 7020.00i 0.252391i
$$919$$ 47720.0 1.71288 0.856440 0.516246i $$-0.172671\pi$$
0.856440 + 0.516246i $$0.172671\pi$$
$$920$$ 0 0
$$921$$ − 17868.0i − 0.639273i
$$922$$ 15108.0 0.539648
$$923$$ 0 0
$$924$$ −3840.00 −0.136717
$$925$$ 1850.00i 0.0657596i
$$926$$ 13504.0 0.479232
$$927$$ −1008.00 −0.0357142
$$928$$ 576.000i 0.0203751i
$$929$$ 7502.00i 0.264944i 0.991187 + 0.132472i $$0.0422914\pi$$
−0.991187 + 0.132472i $$0.957709\pi$$
$$930$$ − 11040.0i − 0.389264i
$$931$$ − 5580.00i − 0.196431i
$$932$$ 10248.0 0.360176
$$933$$ −7056.00 −0.247592
$$934$$ 15848.0i 0.555206i
$$935$$ −52000.0 −1.81880
$$936$$ 0 0
$$937$$ 22058.0 0.769054 0.384527 0.923114i $$-0.374365\pi$$
0.384527 + 0.923114i $$0.374365\pi$$
$$938$$ − 11200.0i − 0.389865i
$$939$$ 25326.0 0.880173
$$940$$ 18080.0 0.627345
$$941$$ 23338.0i 0.808498i 0.914649 + 0.404249i $$0.132467\pi$$
−0.914649 + 0.404249i $$0.867533\pi$$
$$942$$ 14460.0i 0.500140i
$$943$$ 0 0
$$944$$ − 7424.00i − 0.255965i
$$945$$ −2160.00 −0.0743543
$$946$$ 6080.00 0.208962
$$947$$ 30488.0i 1.04617i 0.852279 + 0.523087i $$0.175220\pi$$
−0.852279 + 0.523087i $$0.824780\pi$$
$$948$$ 11712.0 0.401253
$$949$$ 0 0
$$950$$ −1000.00 −0.0341519
$$951$$ − 16650.0i − 0.567732i
$$952$$ 8320.00 0.283249
$$953$$ −9522.00 −0.323660 −0.161830 0.986819i $$-0.551740\pi$$
−0.161830 + 0.986819i $$0.551740\pi$$
$$954$$ − 6876.00i − 0.233353i
$$955$$ − 13120.0i − 0.444558i
$$956$$ − 28656.0i − 0.969457i
$$957$$ 2160.00i 0.0729602i
$$958$$ 22168.0 0.747615
$$959$$ 18608.0 0.626573
$$960$$ − 1920.00i − 0.0645497i
$$961$$ −4065.00 −0.136451
$$962$$ 0 0
$$963$$ −3348.00 −0.112033
$$964$$ − 24728.0i − 0.826178i
$$965$$ −3500.00 −0.116755
$$966$$ 0 0
$$967$$ − 7616.00i − 0.253272i −0.991949 0.126636i $$-0.959582\pi$$
0.991949 0.126636i $$-0.0404180\pi$$
$$968$$ − 2152.00i − 0.0714544i
$$969$$ 7800.00i 0.258588i
$$970$$ − 12280.0i − 0.406481i
$$971$$ 51316.0 1.69599 0.847996 0.530002i $$-0.177809\pi$$
0.847996 + 0.530002i $$0.177809\pi$$
$$972$$ −972.000 −0.0320750
$$973$$ 15456.0i 0.509246i
$$974$$ 8864.00 0.291603
$$975$$ 0 0
$$976$$ 5728.00 0.187857
$$977$$ − 48666.0i − 1.59362i −0.604232 0.796808i $$-0.706520\pi$$
0.604232 0.796808i $$-0.293480\pi$$
$$978$$ −19272.0 −0.630113
$$979$$ 15440.0 0.504050
$$980$$ − 11160.0i − 0.363768i
$$981$$ 8406.00i 0.273581i
$$982$$ − 2280.00i − 0.0740914i
$$983$$ − 17388.0i − 0.564182i −0.959388 0.282091i $$-0.908972\pi$$
0.959388 0.282091i $$-0.0910280\pi$$
$$984$$ 8688.00 0.281467
$$985$$ 3420.00 0.110630
$$986$$ − 4680.00i − 0.151158i
$$987$$ −10848.0 −0.349844
$$988$$ 0 0
$$989$$ 0 0
$$990$$ − 7200.00i − 0.231142i
$$991$$ 11496.0 0.368499 0.184249 0.982880i $$-0.441015\pi$$
0.184249 + 0.982880i $$0.441015\pi$$
$$992$$ −5888.00 −0.188452
$$993$$ 420.000i 0.0134223i
$$994$$ − 11968.0i − 0.381893i
$$995$$ 33680.0i 1.07309i
$$996$$ 12096.0i 0.384816i
$$997$$ 48862.0 1.55213 0.776066 0.630652i $$-0.217212\pi$$
0.776066 + 0.630652i $$0.217212\pi$$
$$998$$ 3528.00 0.111901
$$999$$ 1998.00i 0.0632772i
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 1014.4.b.h.337.1 2
13.5 odd 4 78.4.a.c.1.1 1
13.8 odd 4 1014.4.a.j.1.1 1
13.12 even 2 inner 1014.4.b.h.337.2 2
39.5 even 4 234.4.a.h.1.1 1
52.31 even 4 624.4.a.d.1.1 1
65.44 odd 4 1950.4.a.l.1.1 1
104.5 odd 4 2496.4.a.a.1.1 1
104.83 even 4 2496.4.a.j.1.1 1
156.83 odd 4 1872.4.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.c.1.1 1 13.5 odd 4
234.4.a.h.1.1 1 39.5 even 4
624.4.a.d.1.1 1 52.31 even 4
1014.4.a.j.1.1 1 13.8 odd 4
1014.4.b.h.337.1 2 1.1 even 1 trivial
1014.4.b.h.337.2 2 13.12 even 2 inner
1872.4.a.d.1.1 1 156.83 odd 4
1950.4.a.l.1.1 1 65.44 odd 4
2496.4.a.a.1.1 1 104.5 odd 4
2496.4.a.j.1.1 1 104.83 even 4