# Properties

 Label 2550.2.d.a.2449.1 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.d (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: $$20.3618525154$$ Analytic rank: $$1$$ 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2449.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2550.2449 Dual form 2550.2.d.a.2449.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -5.00000 q^{11} +1.00000i q^{12} -2.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -3.00000 q^{21} +5.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} +3.00000i q^{28} -10.0000 q^{29} +5.00000 q^{31} -1.00000i q^{32} +5.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000i q^{37} +1.00000i q^{38} -2.00000 q^{39} +6.00000 q^{41} +3.00000i q^{42} -1.00000i q^{43} +5.00000 q^{44} +6.00000 q^{46} -3.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -1.00000 q^{51} +2.00000i q^{52} -1.00000i q^{53} +1.00000 q^{54} +3.00000 q^{56} +1.00000i q^{57} +10.0000i q^{58} -8.00000 q^{59} -2.00000 q^{61} -5.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} +5.00000 q^{66} +11.0000i q^{67} +1.00000i q^{68} +6.00000 q^{69} +6.00000 q^{71} -1.00000i q^{72} -12.0000i q^{73} +3.00000 q^{74} +1.00000 q^{76} +15.0000i q^{77} +2.00000i q^{78} -5.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +18.0000i q^{83} +3.00000 q^{84} -1.00000 q^{86} +10.0000i q^{87} -5.00000i q^{88} -12.0000 q^{89} -6.00000 q^{91} -6.00000i q^{92} -5.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} +14.0000i q^{97} +2.00000i q^{98} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 10 q^{11} - 6 q^{14} + 2 q^{16} - 2 q^{19} - 6 q^{21} + 2 q^{24} - 4 q^{26} - 20 q^{29} + 10 q^{31} - 2 q^{34} + 2 q^{36} - 4 q^{39} + 12 q^{41} + 10 q^{44} + 12 q^{46} - 4 q^{49} - 2 q^{51} + 2 q^{54} + 6 q^{56} - 16 q^{59} - 4 q^{61} - 2 q^{64} + 10 q^{66} + 12 q^{69} + 12 q^{71} + 6 q^{74} + 2 q^{76} - 10 q^{79} + 2 q^{81} + 6 q^{84} - 2 q^{86} - 24 q^{89} - 12 q^{91} - 6 q^{94} - 2 q^{96} + 10 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 10 * q^11 - 6 * q^14 + 2 * q^16 - 2 * q^19 - 6 * q^21 + 2 * q^24 - 4 * q^26 - 20 * q^29 + 10 * q^31 - 2 * q^34 + 2 * q^36 - 4 * q^39 + 12 * q^41 + 10 * q^44 + 12 * q^46 - 4 * q^49 - 2 * q^51 + 2 * q^54 + 6 * q^56 - 16 * q^59 - 4 * q^61 - 2 * q^64 + 10 * q^66 + 12 * q^69 + 12 * q^71 + 6 * q^74 + 2 * q^76 - 10 * q^79 + 2 * q^81 + 6 * q^84 - 2 * q^86 - 24 * q^89 - 12 * q^91 - 6 * q^94 - 2 * q^96 + 10 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.00000i − 0.707107i
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 1.00000i − 0.242536i
$$18$$ 1.00000i 0.235702i
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 5.00000i 1.06600i
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000i 0.192450i
$$28$$ 3.00000i 0.566947i
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 5.00000i 0.870388i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 3.00000i 0.462910i
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 5.00000 0.753778
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ − 3.00000i − 0.437595i −0.975770 0.218797i $$-0.929787\pi$$
0.975770 0.218797i $$-0.0702134\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000i 0.277350i
$$53$$ − 1.00000i − 0.137361i −0.997639 0.0686803i $$-0.978121\pi$$
0.997639 0.0686803i $$-0.0218788\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 1.00000i 0.132453i
$$58$$ 10.0000i 1.31306i
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ − 5.00000i − 0.635001i
$$63$$ 3.00000i 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 5.00000 0.615457
$$67$$ 11.0000i 1.34386i 0.740613 + 0.671932i $$0.234535\pi$$
−0.740613 + 0.671932i $$0.765465\pi$$
$$68$$ 1.00000i 0.121268i
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 12.0000i − 1.40449i −0.711934 0.702247i $$-0.752180\pi$$
0.711934 0.702247i $$-0.247820\pi$$
$$74$$ 3.00000 0.348743
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 15.0000i 1.70941i
$$78$$ 2.00000i 0.226455i
$$79$$ −5.00000 −0.562544 −0.281272 0.959628i $$-0.590756\pi$$
−0.281272 + 0.959628i $$0.590756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ 18.0000i 1.97576i 0.155230 + 0.987878i $$0.450388\pi$$
−0.155230 + 0.987878i $$0.549612\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 0 0
$$86$$ −1.00000 −0.107833
$$87$$ 10.0000i 1.07211i
$$88$$ − 5.00000i − 0.533002i
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ − 6.00000i − 0.625543i
$$93$$ − 5.00000i − 0.518476i
$$94$$ −3.00000 −0.309426
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 2.00000i 0.202031i
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ 11.0000 1.09454 0.547270 0.836956i $$-0.315667\pi$$
0.547270 + 0.836956i $$0.315667\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ 5.00000i 0.483368i 0.970355 + 0.241684i $$0.0776998\pi$$
−0.970355 + 0.241684i $$0.922300\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ − 3.00000i − 0.283473i
$$113$$ 1.00000i 0.0940721i 0.998893 + 0.0470360i $$0.0149776\pi$$
−0.998893 + 0.0470360i $$0.985022\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ 10.0000 0.928477
$$117$$ 2.00000i 0.184900i
$$118$$ 8.00000i 0.736460i
$$119$$ −3.00000 −0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 2.00000i 0.181071i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −5.00000 −0.449013
$$125$$ 0 0
$$126$$ 3.00000 0.267261
$$127$$ − 12.0000i − 1.06483i −0.846484 0.532414i $$-0.821285\pi$$
0.846484 0.532414i $$-0.178715\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −1.00000 −0.0880451
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ − 5.00000i − 0.435194i
$$133$$ 3.00000i 0.260133i
$$134$$ 11.0000 0.950255
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ − 16.0000i − 1.36697i −0.729964 0.683486i $$-0.760463\pi$$
0.729964 0.683486i $$-0.239537\pi$$
$$138$$ − 6.00000i − 0.510754i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ −3.00000 −0.252646
$$142$$ − 6.00000i − 0.503509i
$$143$$ 10.0000i 0.836242i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ 2.00000i 0.164957i
$$148$$ − 3.00000i − 0.246598i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ 1.00000i 0.0808452i
$$154$$ 15.0000 1.20873
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 5.00000i 0.397779i
$$159$$ −1.00000 −0.0793052
$$160$$ 0 0
$$161$$ 18.0000 1.41860
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 18.0000 1.39707
$$167$$ 18.0000i 1.39288i 0.717614 + 0.696441i $$0.245234\pi$$
−0.717614 + 0.696441i $$0.754766\pi$$
$$168$$ − 3.00000i − 0.231455i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 1.00000i 0.0762493i
$$173$$ 20.0000i 1.52057i 0.649589 + 0.760286i $$0.274941\pi$$
−0.649589 + 0.760286i $$0.725059\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 0 0
$$176$$ −5.00000 −0.376889
$$177$$ 8.00000i 0.601317i
$$178$$ 12.0000i 0.899438i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −17.0000 −1.26360 −0.631800 0.775131i $$-0.717684\pi$$
−0.631800 + 0.775131i $$0.717684\pi$$
$$182$$ 6.00000i 0.444750i
$$183$$ 2.00000i 0.147844i
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −5.00000 −0.366618
$$187$$ 5.00000i 0.365636i
$$188$$ 3.00000i 0.218797i
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ 3.00000 0.217072 0.108536 0.994092i $$-0.465384\pi$$
0.108536 + 0.994092i $$0.465384\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 24.0000i − 1.72756i −0.503871 0.863779i $$-0.668091\pi$$
0.503871 0.863779i $$-0.331909\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ − 5.00000i − 0.355335i
$$199$$ −17.0000 −1.20510 −0.602549 0.798082i $$-0.705848\pi$$
−0.602549 + 0.798082i $$0.705848\pi$$
$$200$$ 0 0
$$201$$ 11.0000 0.775880
$$202$$ − 11.0000i − 0.773957i
$$203$$ 30.0000i 2.10559i
$$204$$ 1.00000 0.0700140
$$205$$ 0 0
$$206$$ 4.00000 0.278693
$$207$$ − 6.00000i − 0.417029i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 5.00000 0.345857
$$210$$ 0 0
$$211$$ 18.0000 1.23917 0.619586 0.784929i $$-0.287301\pi$$
0.619586 + 0.784929i $$0.287301\pi$$
$$212$$ 1.00000i 0.0686803i
$$213$$ − 6.00000i − 0.411113i
$$214$$ 5.00000 0.341793
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ − 15.0000i − 1.01827i
$$218$$ 5.00000i 0.338643i
$$219$$ −12.0000 −0.810885
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ − 3.00000i − 0.201347i
$$223$$ 18.0000i 1.20537i 0.797980 + 0.602685i $$0.205902\pi$$
−0.797980 + 0.602685i $$0.794098\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ 1.00000 0.0665190
$$227$$ 27.0000i 1.79205i 0.444001 + 0.896026i $$0.353559\pi$$
−0.444001 + 0.896026i $$0.646441\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −24.0000 −1.58596 −0.792982 0.609245i $$-0.791473\pi$$
−0.792982 + 0.609245i $$0.791473\pi$$
$$230$$ 0 0
$$231$$ 15.0000 0.986928
$$232$$ − 10.0000i − 0.656532i
$$233$$ − 10.0000i − 0.655122i −0.944830 0.327561i $$-0.893773\pi$$
0.944830 0.327561i $$-0.106227\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 8.00000 0.520756
$$237$$ 5.00000i 0.324785i
$$238$$ 3.00000i 0.194461i
$$239$$ −13.0000 −0.840900 −0.420450 0.907316i $$-0.638128\pi$$
−0.420450 + 0.907316i $$0.638128\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ − 14.0000i − 0.899954i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 2.00000i 0.127257i
$$248$$ 5.00000i 0.317500i
$$249$$ 18.0000 1.14070
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ − 3.00000i − 0.188982i
$$253$$ − 30.0000i − 1.88608i
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 2.00000i − 0.124757i −0.998053 0.0623783i $$-0.980131\pi$$
0.998053 0.0623783i $$-0.0198685\pi$$
$$258$$ 1.00000i 0.0622573i
$$259$$ 9.00000 0.559233
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ 20.0000i 1.23560i
$$263$$ 3.00000i 0.184988i 0.995713 + 0.0924940i $$0.0294839\pi$$
−0.995713 + 0.0924940i $$0.970516\pi$$
$$264$$ −5.00000 −0.307729
$$265$$ 0 0
$$266$$ 3.00000 0.183942
$$267$$ 12.0000i 0.734388i
$$268$$ − 11.0000i − 0.671932i
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 14.0000 0.850439 0.425220 0.905090i $$-0.360197\pi$$
0.425220 + 0.905090i $$0.360197\pi$$
$$272$$ − 1.00000i − 0.0606339i
$$273$$ 6.00000i 0.363137i
$$274$$ −16.0000 −0.966595
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 31.0000i 1.86261i 0.364241 + 0.931305i $$0.381328\pi$$
−0.364241 + 0.931305i $$0.618672\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ −5.00000 −0.299342
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 3.00000i 0.178647i
$$283$$ − 14.0000i − 0.832214i −0.909316 0.416107i $$-0.863394\pi$$
0.909316 0.416107i $$-0.136606\pi$$
$$284$$ −6.00000 −0.356034
$$285$$ 0 0
$$286$$ 10.0000 0.591312
$$287$$ − 18.0000i − 1.06251i
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 12.0000i 0.702247i
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ −3.00000 −0.174371
$$297$$ − 5.00000i − 0.290129i
$$298$$ − 6.00000i − 0.347571i
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ − 8.00000i − 0.460348i
$$303$$ − 11.0000i − 0.631933i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ − 15.0000i − 0.854704i
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ − 2.00000i − 0.113228i
$$313$$ − 34.0000i − 1.92179i −0.276907 0.960897i $$-0.589309\pi$$
0.276907 0.960897i $$-0.410691\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 5.00000 0.281272
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 1.00000i 0.0560772i
$$319$$ 50.0000 2.79946
$$320$$ 0 0
$$321$$ 5.00000 0.279073
$$322$$ − 18.0000i − 1.00310i
$$323$$ 1.00000i 0.0556415i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 5.00000i 0.276501i
$$328$$ 6.00000i 0.331295i
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ − 18.0000i − 0.987878i
$$333$$ − 3.00000i − 0.164399i
$$334$$ 18.0000 0.984916
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ − 12.0000i − 0.653682i −0.945079 0.326841i $$-0.894016\pi$$
0.945079 0.326841i $$-0.105984\pi$$
$$338$$ − 9.00000i − 0.489535i
$$339$$ 1.00000 0.0543125
$$340$$ 0 0
$$341$$ −25.0000 −1.35383
$$342$$ − 1.00000i − 0.0540738i
$$343$$ − 15.0000i − 0.809924i
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ 20.0000 1.07521
$$347$$ 23.0000i 1.23470i 0.786687 + 0.617352i $$0.211795\pi$$
−0.786687 + 0.617352i $$0.788205\pi$$
$$348$$ − 10.0000i − 0.536056i
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 5.00000i 0.266501i
$$353$$ − 2.00000i − 0.106449i −0.998583 0.0532246i $$-0.983050\pi$$
0.998583 0.0532246i $$-0.0169499\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ 12.0000 0.635999
$$357$$ 3.00000i 0.158777i
$$358$$ − 12.0000i − 0.634220i
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 17.0000i 0.893500i
$$363$$ − 14.0000i − 0.734809i
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ − 5.00000i − 0.260998i −0.991448 0.130499i $$-0.958342\pi$$
0.991448 0.130499i $$-0.0416579\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ 5.00000i 0.259238i
$$373$$ − 18.0000i − 0.932005i −0.884783 0.466002i $$-0.845694\pi$$
0.884783 0.466002i $$-0.154306\pi$$
$$374$$ 5.00000 0.258544
$$375$$ 0 0
$$376$$ 3.00000 0.154713
$$377$$ 20.0000i 1.03005i
$$378$$ − 3.00000i − 0.154303i
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ − 3.00000i − 0.153493i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −24.0000 −1.22157
$$387$$ 1.00000i 0.0508329i
$$388$$ − 14.0000i − 0.710742i
$$389$$ 5.00000 0.253510 0.126755 0.991934i $$-0.459544\pi$$
0.126755 + 0.991934i $$0.459544\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ − 2.00000i − 0.101015i
$$393$$ 20.0000i 1.00887i
$$394$$ −12.0000 −0.604551
$$395$$ 0 0
$$396$$ −5.00000 −0.251259
$$397$$ − 21.0000i − 1.05396i −0.849878 0.526980i $$-0.823324\pi$$
0.849878 0.526980i $$-0.176676\pi$$
$$398$$ 17.0000i 0.852133i
$$399$$ 3.00000 0.150188
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ − 11.0000i − 0.548630i
$$403$$ − 10.0000i − 0.498135i
$$404$$ −11.0000 −0.547270
$$405$$ 0 0
$$406$$ 30.0000 1.48888
$$407$$ − 15.0000i − 0.743522i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ −16.0000 −0.789222
$$412$$ − 4.00000i − 0.197066i
$$413$$ 24.0000i 1.18096i
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 8.00000i 0.391762i
$$418$$ − 5.00000i − 0.244558i
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ 18.0000 0.877266 0.438633 0.898666i $$-0.355463\pi$$
0.438633 + 0.898666i $$0.355463\pi$$
$$422$$ − 18.0000i − 0.876226i
$$423$$ 3.00000i 0.145865i
$$424$$ 1.00000 0.0485643
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ 6.00000i 0.290360i
$$428$$ − 5.00000i − 0.241684i
$$429$$ 10.0000 0.482805
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 21.0000i 1.00920i 0.863355 + 0.504598i $$0.168359\pi$$
−0.863355 + 0.504598i $$0.831641\pi$$
$$434$$ −15.0000 −0.720023
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ − 6.00000i − 0.287019i
$$438$$ 12.0000i 0.573382i
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 2.00000i 0.0951303i
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ −3.00000 −0.142374
$$445$$ 0 0
$$446$$ 18.0000 0.852325
$$447$$ − 6.00000i − 0.283790i
$$448$$ 3.00000i 0.141737i
$$449$$ −21.0000 −0.991051 −0.495526 0.868593i $$-0.665025\pi$$
−0.495526 + 0.868593i $$0.665025\pi$$
$$450$$ 0 0
$$451$$ −30.0000 −1.41264
$$452$$ − 1.00000i − 0.0470360i
$$453$$ − 8.00000i − 0.375873i
$$454$$ 27.0000 1.26717
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ 25.0000i 1.16945i 0.811231 + 0.584725i $$0.198798\pi$$
−0.811231 + 0.584725i $$0.801202\pi$$
$$458$$ 24.0000i 1.12145i
$$459$$ 1.00000 0.0466760
$$460$$ 0 0
$$461$$ −15.0000 −0.698620 −0.349310 0.937007i $$-0.613584\pi$$
−0.349310 + 0.937007i $$0.613584\pi$$
$$462$$ − 15.0000i − 0.697863i
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ −10.0000 −0.464238
$$465$$ 0 0
$$466$$ −10.0000 −0.463241
$$467$$ − 18.0000i − 0.832941i −0.909149 0.416470i $$-0.863267\pi$$
0.909149 0.416470i $$-0.136733\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 33.0000 1.52380
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 8.00000i − 0.368230i
$$473$$ 5.00000i 0.229900i
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ 3.00000 0.137505
$$477$$ 1.00000i 0.0457869i
$$478$$ 13.0000i 0.594606i
$$479$$ −26.0000 −1.18797 −0.593985 0.804476i $$-0.702446\pi$$
−0.593985 + 0.804476i $$0.702446\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 0 0
$$483$$ − 18.0000i − 0.819028i
$$484$$ −14.0000 −0.636364
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 40.0000i 1.81257i 0.422664 + 0.906287i $$0.361095\pi$$
−0.422664 + 0.906287i $$0.638905\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 10.0000i 0.450377i
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 5.00000 0.224507
$$497$$ − 18.0000i − 0.807410i
$$498$$ − 18.0000i − 0.806599i
$$499$$ −22.0000 −0.984855 −0.492428 0.870353i $$-0.663890\pi$$
−0.492428 + 0.870353i $$0.663890\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ 18.0000i 0.803379i
$$503$$ 6.00000i 0.267527i 0.991013 + 0.133763i $$0.0427062\pi$$
−0.991013 + 0.133763i $$0.957294\pi$$
$$504$$ −3.00000 −0.133631
$$505$$ 0 0
$$506$$ −30.0000 −1.33366
$$507$$ − 9.00000i − 0.399704i
$$508$$ 12.0000i 0.532414i
$$509$$ −9.00000 −0.398918 −0.199459 0.979906i $$-0.563918\pi$$
−0.199459 + 0.979906i $$0.563918\pi$$
$$510$$ 0 0
$$511$$ −36.0000 −1.59255
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ −2.00000 −0.0882162
$$515$$ 0 0
$$516$$ 1.00000 0.0440225
$$517$$ 15.0000i 0.659699i
$$518$$ − 9.00000i − 0.395437i
$$519$$ 20.0000 0.877903
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ − 10.0000i − 0.437688i
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 20.0000 0.873704
$$525$$ 0 0
$$526$$ 3.00000 0.130806
$$527$$ − 5.00000i − 0.217803i
$$528$$ 5.00000i 0.217597i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 8.00000 0.347170
$$532$$ − 3.00000i − 0.130066i
$$533$$ − 12.0000i − 0.519778i
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ −11.0000 −0.475128
$$537$$ − 12.0000i − 0.517838i
$$538$$ − 6.00000i − 0.258678i
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ −41.0000 −1.76273 −0.881364 0.472438i $$-0.843374\pi$$
−0.881364 + 0.472438i $$0.843374\pi$$
$$542$$ − 14.0000i − 0.601351i
$$543$$ 17.0000i 0.729540i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ − 22.0000i − 0.940652i −0.882493 0.470326i $$-0.844136\pi$$
0.882493 0.470326i $$-0.155864\pi$$
$$548$$ 16.0000i 0.683486i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 10.0000 0.426014
$$552$$ 6.00000i 0.255377i
$$553$$ 15.0000i 0.637865i
$$554$$ 31.0000 1.31706
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 17.0000i − 0.720313i −0.932892 0.360157i $$-0.882723\pi$$
0.932892 0.360157i $$-0.117277\pi$$
$$558$$ 5.00000i 0.211667i
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 5.00000 0.211100
$$562$$ 30.0000i 1.26547i
$$563$$ − 8.00000i − 0.337160i −0.985688 0.168580i $$-0.946082\pi$$
0.985688 0.168580i $$-0.0539181\pi$$
$$564$$ 3.00000 0.126323
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ − 3.00000i − 0.125988i
$$568$$ 6.00000i 0.251754i
$$569$$ 36.0000 1.50920 0.754599 0.656186i $$-0.227831\pi$$
0.754599 + 0.656186i $$0.227831\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ − 10.0000i − 0.418121i
$$573$$ − 3.00000i − 0.125327i
$$574$$ −18.0000 −0.751305
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 33.0000i − 1.37381i −0.726748 0.686904i $$-0.758969\pi$$
0.726748 0.686904i $$-0.241031\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ 54.0000 2.24030
$$582$$ − 14.0000i − 0.580319i
$$583$$ 5.00000i 0.207079i
$$584$$ 12.0000 0.496564
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ − 2.00000i − 0.0824786i
$$589$$ −5.00000 −0.206021
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 3.00000i 0.123299i
$$593$$ 12.0000i 0.492781i 0.969171 + 0.246390i $$0.0792446\pi$$
−0.969171 + 0.246390i $$0.920755\pi$$
$$594$$ −5.00000 −0.205152
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 17.0000i 0.695764i
$$598$$ − 12.0000i − 0.490716i
$$599$$ −35.0000 −1.43006 −0.715031 0.699093i $$-0.753587\pi$$
−0.715031 + 0.699093i $$0.753587\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 3.00000i 0.122271i
$$603$$ − 11.0000i − 0.447955i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −11.0000 −0.446844
$$607$$ − 20.0000i − 0.811775i −0.913923 0.405887i $$-0.866962\pi$$
0.913923 0.405887i $$-0.133038\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ − 1.00000i − 0.0404226i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ −15.0000 −0.604367
$$617$$ 13.0000i 0.523360i 0.965155 + 0.261680i $$0.0842766\pi$$
−0.965155 + 0.261680i $$0.915723\pi$$
$$618$$ − 4.00000i − 0.160904i
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 2.00000i 0.0801927i
$$623$$ 36.0000i 1.44231i
$$624$$ −2.00000 −0.0800641
$$625$$ 0 0
$$626$$ −34.0000 −1.35891
$$627$$ − 5.00000i − 0.199681i
$$628$$ 0 0
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ −14.0000 −0.557331 −0.278666 0.960388i $$-0.589892\pi$$
−0.278666 + 0.960388i $$0.589892\pi$$
$$632$$ − 5.00000i − 0.198889i
$$633$$ − 18.0000i − 0.715436i
$$634$$ 12.0000 0.476581
$$635$$ 0 0
$$636$$ 1.00000 0.0396526
$$637$$ 4.00000i 0.158486i
$$638$$ − 50.0000i − 1.97952i
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ − 5.00000i − 0.197334i
$$643$$ 34.0000i 1.34083i 0.741987 + 0.670415i $$0.233884\pi$$
−0.741987 + 0.670415i $$0.766116\pi$$
$$644$$ −18.0000 −0.709299
$$645$$ 0 0
$$646$$ 1.00000 0.0393445
$$647$$ − 24.0000i − 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 40.0000 1.57014
$$650$$ 0 0
$$651$$ −15.0000 −0.587896
$$652$$ 0 0
$$653$$ − 16.0000i − 0.626128i −0.949732 0.313064i $$-0.898644\pi$$
0.949732 0.313064i $$-0.101356\pi$$
$$654$$ 5.00000 0.195515
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 12.0000i 0.468165i
$$658$$ 9.00000i 0.350857i
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ 8.00000 0.311164 0.155582 0.987823i $$-0.450275\pi$$
0.155582 + 0.987823i $$0.450275\pi$$
$$662$$ 25.0000i 0.971653i
$$663$$ 2.00000i 0.0776736i
$$664$$ −18.0000 −0.698535
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ − 60.0000i − 2.32321i
$$668$$ − 18.0000i − 0.696441i
$$669$$ 18.0000 0.695920
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ 3.00000i 0.115728i
$$673$$ 14.0000i 0.539660i 0.962908 + 0.269830i $$0.0869676\pi$$
−0.962908 + 0.269830i $$0.913032\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 12.0000i − 0.461197i −0.973049 0.230599i $$-0.925932\pi$$
0.973049 0.230599i $$-0.0740685\pi$$
$$678$$ − 1.00000i − 0.0384048i
$$679$$ 42.0000 1.61181
$$680$$ 0 0
$$681$$ 27.0000 1.03464
$$682$$ 25.0000i 0.957299i
$$683$$ − 8.00000i − 0.306111i −0.988218 0.153056i $$-0.951089\pi$$
0.988218 0.153056i $$-0.0489114\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 24.0000i 0.915657i
$$688$$ − 1.00000i − 0.0381246i
$$689$$ −2.00000 −0.0761939
$$690$$ 0 0
$$691$$ −48.0000 −1.82601 −0.913003 0.407953i $$-0.866243\pi$$
−0.913003 + 0.407953i $$0.866243\pi$$
$$692$$ − 20.0000i − 0.760286i
$$693$$ − 15.0000i − 0.569803i
$$694$$ 23.0000 0.873068
$$695$$ 0 0
$$696$$ −10.0000 −0.379049
$$697$$ − 6.00000i − 0.227266i
$$698$$ − 14.0000i − 0.529908i
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ − 3.00000i − 0.113147i
$$704$$ 5.00000 0.188445
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ − 33.0000i − 1.24109i
$$708$$ − 8.00000i − 0.300658i
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ 0 0
$$711$$ 5.00000 0.187515
$$712$$ − 12.0000i − 0.449719i
$$713$$ 30.0000i 1.12351i
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 13.0000i 0.485494i
$$718$$ 15.0000i 0.559795i
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 0 0
$$721$$ 12.0000 0.446903
$$722$$ 18.0000i 0.669891i
$$723$$ 0 0
$$724$$ 17.0000 0.631800
$$725$$ 0 0
$$726$$ −14.0000 −0.519589
$$727$$ − 2.00000i − 0.0741759i −0.999312 0.0370879i $$-0.988192\pi$$
0.999312 0.0370879i $$-0.0118082\pi$$
$$728$$ − 6.00000i − 0.222375i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −1.00000 −0.0369863
$$732$$ − 2.00000i − 0.0739221i
$$733$$ − 16.0000i − 0.590973i −0.955347 0.295487i $$-0.904518\pi$$
0.955347 0.295487i $$-0.0954818\pi$$
$$734$$ −5.00000 −0.184553
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ − 55.0000i − 2.02595i
$$738$$ 6.00000i 0.220863i
$$739$$ 19.0000 0.698926 0.349463 0.936950i $$-0.386364\pi$$
0.349463 + 0.936950i $$0.386364\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 3.00000i 0.110133i
$$743$$ 18.0000i 0.660356i 0.943919 + 0.330178i $$0.107109\pi$$
−0.943919 + 0.330178i $$0.892891\pi$$
$$744$$ 5.00000 0.183309
$$745$$ 0 0
$$746$$ −18.0000 −0.659027
$$747$$ − 18.0000i − 0.658586i
$$748$$ − 5.00000i − 0.182818i
$$749$$ 15.0000 0.548088
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ − 3.00000i − 0.109399i
$$753$$ 18.0000i 0.655956i
$$754$$ 20.0000 0.728357
$$755$$ 0 0
$$756$$ −3.00000 −0.109109
$$757$$ − 14.0000i − 0.508839i −0.967094 0.254419i $$-0.918116\pi$$
0.967094 0.254419i $$-0.0818843\pi$$
$$758$$ − 8.00000i − 0.290573i
$$759$$ −30.0000 −1.08893
$$760$$ 0 0
$$761$$ 36.0000 1.30500 0.652499 0.757789i $$-0.273720\pi$$
0.652499 + 0.757789i $$0.273720\pi$$
$$762$$ 12.0000i 0.434714i
$$763$$ 15.0000i 0.543036i
$$764$$ −3.00000 −0.108536
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 16.0000i 0.577727i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 37.0000 1.33425 0.667127 0.744944i $$-0.267524\pi$$
0.667127 + 0.744944i $$0.267524\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ 24.0000i 0.863779i
$$773$$ − 34.0000i − 1.22290i −0.791285 0.611448i $$-0.790588\pi$$
0.791285 0.611448i $$-0.209412\pi$$
$$774$$ 1.00000 0.0359443
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ − 9.00000i − 0.322873i
$$778$$ − 5.00000i − 0.179259i
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ −30.0000 −1.07348
$$782$$ − 6.00000i − 0.214560i
$$783$$ − 10.0000i − 0.357371i
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ − 18.0000i − 0.641631i −0.947142 0.320815i $$-0.896043\pi$$
0.947142 0.320815i $$-0.103957\pi$$
$$788$$ 12.0000i 0.427482i
$$789$$ 3.00000 0.106803
$$790$$ 0 0
$$791$$ 3.00000 0.106668
$$792$$ 5.00000i 0.177667i
$$793$$ 4.00000i 0.142044i
$$794$$ −21.0000 −0.745262
$$795$$ 0 0
$$796$$ 17.0000 0.602549
$$797$$ 11.0000i 0.389640i 0.980839 + 0.194820i $$0.0624123\pi$$
−0.980839 + 0.194820i $$0.937588\pi$$
$$798$$ − 3.00000i − 0.106199i
$$799$$ −3.00000 −0.106132
$$800$$ 0 0
$$801$$ 12.0000 0.423999
$$802$$ 18.0000i 0.635602i
$$803$$ 60.0000i 2.11735i
$$804$$ −11.0000 −0.387940
$$805$$ 0 0
$$806$$ −10.0000 −0.352235
$$807$$ − 6.00000i − 0.211210i
$$808$$ 11.0000i 0.386979i
$$809$$ 39.0000 1.37117 0.685583 0.727994i $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 0 0
$$811$$ 56.0000 1.96643 0.983213 0.182462i $$-0.0584065\pi$$
0.983213 + 0.182462i $$0.0584065\pi$$
$$812$$ − 30.0000i − 1.05279i
$$813$$ − 14.0000i − 0.491001i
$$814$$ −15.0000 −0.525750
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 1.00000i 0.0349856i
$$818$$ − 2.00000i − 0.0699284i
$$819$$ 6.00000 0.209657
$$820$$ 0 0
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 16.0000i 0.558064i
$$823$$ 4.00000i 0.139431i 0.997567 + 0.0697156i $$0.0222092\pi$$
−0.997567 + 0.0697156i $$0.977791\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 53.0000i 1.84299i 0.388390 + 0.921495i $$0.373032\pi$$
−0.388390 + 0.921495i $$0.626968\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 31.0000 1.07538
$$832$$ 2.00000i 0.0693375i
$$833$$ 2.00000i 0.0692959i
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ −5.00000 −0.172929
$$837$$ 5.00000i 0.172825i
$$838$$ 24.0000i 0.829066i
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ − 18.0000i − 0.620321i
$$843$$ 30.0000i 1.03325i
$$844$$ −18.0000 −0.619586
$$845$$ 0 0
$$846$$ 3.00000 0.103142
$$847$$ − 42.0000i − 1.44314i
$$848$$ − 1.00000i − 0.0343401i
$$849$$ −14.0000 −0.480479
$$850$$ 0 0
$$851$$ −18.0000 −0.617032
$$852$$ 6.00000i 0.205557i
$$853$$ − 37.0000i − 1.26686i −0.773802 0.633428i $$-0.781647\pi$$
0.773802 0.633428i $$-0.218353\pi$$
$$854$$ 6.00000 0.205316
$$855$$ 0 0
$$856$$ −5.00000 −0.170896
$$857$$ 21.0000i 0.717346i 0.933463 + 0.358673i $$0.116771\pi$$
−0.933463 + 0.358673i $$0.883229\pi$$
$$858$$ − 10.0000i − 0.341394i
$$859$$ 23.0000 0.784750 0.392375 0.919805i $$-0.371654\pi$$
0.392375 + 0.919805i $$0.371654\pi$$
$$860$$ 0 0
$$861$$ −18.0000 −0.613438
$$862$$ 0 0
$$863$$ − 33.0000i − 1.12333i −0.827364 0.561667i $$-0.810160\pi$$
0.827364 0.561667i $$-0.189840\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 21.0000 0.713609
$$867$$ 1.00000i 0.0339618i
$$868$$ 15.0000i 0.509133i
$$869$$ 25.0000 0.848067
$$870$$ 0 0
$$871$$ 22.0000 0.745442
$$872$$ − 5.00000i − 0.169321i
$$873$$ − 14.0000i − 0.473828i
$$874$$ −6.00000 −0.202953
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ 16.0000i 0.539974i
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ −9.00000 −0.303218 −0.151609 0.988441i $$-0.548445\pi$$
−0.151609 + 0.988441i $$0.548445\pi$$
$$882$$ − 2.00000i − 0.0673435i
$$883$$ − 20.0000i − 0.673054i −0.941674 0.336527i $$-0.890748\pi$$
0.941674 0.336527i $$-0.109252\pi$$
$$884$$ 2.00000 0.0672673
$$885$$ 0 0
$$886$$ −24.0000 −0.806296
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 3.00000i 0.100673i
$$889$$ −36.0000 −1.20740
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ − 18.0000i − 0.602685i
$$893$$ 3.00000i 0.100391i
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ − 12.0000i − 0.400668i
$$898$$ 21.0000i 0.700779i
$$899$$ −50.0000 −1.66759
$$900$$ 0 0
$$901$$ −1.00000 −0.0333148
$$902$$ 30.0000i 0.998891i
$$903$$ 3.00000i 0.0998337i
$$904$$ −1.00000 −0.0332595
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 30.0000i 0.996134i 0.867139 + 0.498067i $$0.165957\pi$$
−0.867139 + 0.498067i $$0.834043\pi$$
$$908$$ − 27.0000i − 0.896026i
$$909$$ −11.0000 −0.364847
$$910$$ 0 0
$$911$$ 56.0000 1.85536 0.927681 0.373373i $$-0.121799\pi$$
0.927681 + 0.373373i $$0.121799\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ − 90.0000i − 2.97857i
$$914$$ 25.0000 0.826927
$$915$$ 0 0
$$916$$ 24.0000 0.792982
$$917$$ 60.0000i 1.98137i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −44.0000 −1.45143 −0.725713 0.687998i $$-0.758490\pi$$
−0.725713 + 0.687998i $$0.758490\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 15.0000i 0.493999i
$$923$$ − 12.0000i − 0.394985i
$$924$$ −15.0000 −0.493464
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 4.00000i − 0.131377i
$$928$$ 10.0000i 0.328266i
$$929$$ −27.0000 −0.885841 −0.442921 0.896561i $$-0.646058\pi$$
−0.442921 + 0.896561i $$0.646058\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 10.0000i 0.327561i
$$933$$ 2.00000i 0.0654771i
$$934$$ −18.0000 −0.588978
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 10.0000i − 0.326686i −0.986569 0.163343i $$-0.947772\pi$$
0.986569 0.163343i $$-0.0522277\pi$$
$$938$$ − 33.0000i − 1.07749i
$$939$$ −34.0000 −1.10955
$$940$$ 0 0
$$941$$ 22.0000 0.717180 0.358590 0.933495i $$-0.383258\pi$$
0.358590 + 0.933495i $$0.383258\pi$$
$$942$$ 0 0
$$943$$ 36.0000i 1.17232i
$$944$$ −8.00000 −0.260378
$$945$$ 0 0
$$946$$ 5.00000 0.162564
$$947$$ − 61.0000i − 1.98223i −0.132994 0.991117i $$-0.542459\pi$$
0.132994 0.991117i $$-0.457541\pi$$
$$948$$ − 5.00000i − 0.162392i
$$949$$ −24.0000 −0.779073
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ − 3.00000i − 0.0972306i
$$953$$ − 14.0000i − 0.453504i −0.973952 0.226752i $$-0.927189\pi$$
0.973952 0.226752i $$-0.0728108\pi$$
$$954$$ 1.00000 0.0323762
$$955$$ 0 0
$$956$$ 13.0000 0.420450
$$957$$ − 50.0000i − 1.61627i
$$958$$ 26.0000i 0.840022i
$$959$$ −48.0000 −1.55000
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ − 6.00000i − 0.193448i
$$963$$ − 5.00000i − 0.161123i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ −18.0000 −0.579141
$$967$$ 36.0000i 1.15768i 0.815440 + 0.578841i $$0.196495\pi$$
−0.815440 + 0.578841i $$0.803505\pi$$
$$968$$ 14.0000i 0.449977i
$$969$$ 1.00000 0.0321246
$$970$$ 0 0
$$971$$ −38.0000 −1.21948 −0.609739 0.792602i $$-0.708726\pi$$
−0.609739 + 0.792602i $$0.708726\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 24.0000i 0.769405i
$$974$$ 40.0000 1.28168
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 10.0000i 0.319928i 0.987123 + 0.159964i $$0.0511379\pi$$
−0.987123 + 0.159964i $$0.948862\pi$$
$$978$$ 0 0
$$979$$ 60.0000 1.91761
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ − 4.00000i − 0.127645i
$$983$$ − 6.00000i − 0.191370i −0.995412 0.0956851i $$-0.969496\pi$$
0.995412 0.0956851i $$-0.0305042\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 10.0000 0.318465
$$987$$ 9.00000i 0.286473i
$$988$$ − 2.00000i − 0.0636285i
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ − 5.00000i − 0.158750i
$$993$$ 25.0000i 0.793351i
$$994$$ −18.0000 −0.570925
$$995$$ 0 0
$$996$$ −18.0000 −0.570352
$$997$$ 43.0000i 1.36182i 0.732365 + 0.680912i $$0.238416\pi$$
−0.732365 + 0.680912i $$0.761584\pi$$
$$998$$ 22.0000i 0.696398i
$$999$$ −3.00000 −0.0949158
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 2550.2.d.a.2449.1 2
5.2 odd 4 2550.2.a.z.1.1 yes 1
5.3 odd 4 2550.2.a.h.1.1 1
5.4 even 2 inner 2550.2.d.a.2449.2 2
15.2 even 4 7650.2.a.be.1.1 1
15.8 even 4 7650.2.a.bl.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.h.1.1 1 5.3 odd 4
2550.2.a.z.1.1 yes 1 5.2 odd 4
2550.2.d.a.2449.1 2 1.1 even 1 trivial
2550.2.d.a.2449.2 2 5.4 even 2 inner
7650.2.a.be.1.1 1 15.2 even 4
7650.2.a.bl.1.1 1 15.8 even 4