# Properties

 Label 2550.2.d.l.2449.2 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ Analytic rank $0$ 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: $$0$$ 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -5.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +3.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -3.00000 q^{21} -5.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +4.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +5.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +9.00000i q^{37} +1.00000i q^{38} +4.00000 q^{39} -10.0000 q^{41} -3.00000i q^{42} +11.0000i q^{43} +5.00000 q^{44} +4.00000 q^{46} -9.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} -4.00000i q^{52} +3.00000i q^{53} -1.00000 q^{54} -3.00000 q^{56} -1.00000i q^{57} +4.00000i q^{58} +8.00000 q^{59} -14.0000 q^{61} -1.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} -5.00000 q^{66} +7.00000i q^{67} -1.00000i q^{68} -4.00000 q^{69} +14.0000 q^{71} +1.00000i q^{72} +2.00000i q^{73} -9.00000 q^{74} -1.00000 q^{76} +15.0000i q^{77} +4.00000i q^{78} +5.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +8.00000i q^{83} +3.00000 q^{84} -11.0000 q^{86} -4.00000i q^{87} +5.00000i q^{88} +2.00000 q^{89} +12.0000 q^{91} +4.00000i q^{92} +1.00000i q^{93} +9.00000 q^{94} +1.00000 q^{96} +14.0000i q^{97} -2.00000i q^{98} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 10 q^{11} + 6 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{21} - 2 q^{24} - 8 q^{26} + 8 q^{29} - 2 q^{31} - 2 q^{34} + 2 q^{36} + 8 q^{39} - 20 q^{41} + 10 q^{44} + 8 q^{46} - 4 q^{49} + 2 q^{51} - 2 q^{54} - 6 q^{56} + 16 q^{59} - 28 q^{61} - 2 q^{64} - 10 q^{66} - 8 q^{69} + 28 q^{71} - 18 q^{74} - 2 q^{76} + 10 q^{79} + 2 q^{81} + 6 q^{84} - 22 q^{86} + 4 q^{89} + 24 q^{91} + 18 q^{94} + 2 q^{96} + 10 q^{99}+O(q^{100})$$ 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 - 8 * q^26 + 8 * q^29 - 2 * q^31 - 2 * q^34 + 2 * q^36 + 8 * q^39 - 20 * q^41 + 10 * q^44 + 8 * q^46 - 4 * q^49 + 2 * q^51 - 2 * q^54 - 6 * q^56 + 16 * q^59 - 28 * q^61 - 2 * q^64 - 10 * q^66 - 8 * q^69 + 28 * q^71 - 18 * q^74 - 2 * q^76 + 10 * q^79 + 2 * q^81 + 6 * q^84 - 22 * q^86 + 4 * q^89 + 24 * q^91 + 18 * 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
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 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 $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ − 5.00000i − 1.06600i
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000i 0.192450i
$$28$$ 3.00000i 0.566947i
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605 −0.0898027 0.995960i $$-0.528624\pi$$
−0.0898027 + 0.995960i $$0.528624\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 5.00000i 0.870388i
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 9.00000i 1.47959i 0.672832 + 0.739795i $$0.265078\pi$$
−0.672832 + 0.739795i $$0.734922\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ − 3.00000i − 0.462910i
$$43$$ 11.0000i 1.67748i 0.544529 + 0.838742i $$0.316708\pi$$
−0.544529 + 0.838742i $$0.683292\pi$$
$$44$$ 5.00000 0.753778
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 9.00000i − 1.31278i −0.754420 0.656392i $$-0.772082\pi$$
0.754420 0.656392i $$-0.227918\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ − 4.00000i − 0.554700i
$$53$$ 3.00000i 0.412082i 0.978543 + 0.206041i $$0.0660580\pi$$
−0.978543 + 0.206041i $$0.933942\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −3.00000 −0.400892
$$57$$ − 1.00000i − 0.132453i
$$58$$ 4.00000i 0.525226i
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −14.0000 −1.79252 −0.896258 0.443533i $$-0.853725\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ 3.00000i 0.377964i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −5.00000 −0.615457
$$67$$ 7.00000i 0.855186i 0.903971 + 0.427593i $$0.140638\pi$$
−0.903971 + 0.427593i $$0.859362\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 14.0000 1.66149 0.830747 0.556650i $$-0.187914\pi$$
0.830747 + 0.556650i $$0.187914\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ −9.00000 −1.04623
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 15.0000i 1.70941i
$$78$$ 4.00000i 0.452911i
$$79$$ 5.00000 0.562544 0.281272 0.959628i $$-0.409244\pi$$
0.281272 + 0.959628i $$0.409244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 0 0
$$86$$ −11.0000 −1.18616
$$87$$ − 4.00000i − 0.428845i
$$88$$ 5.00000i 0.533002i
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 4.00000i 0.417029i
$$93$$ 1.00000i 0.103695i
$$94$$ 9.00000 0.928279
$$95$$ 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
$$101$$ 9.00000 0.895533 0.447767 0.894150i $$-0.352219\pi$$
0.447767 + 0.894150i $$0.352219\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −3.00000 −0.291386
$$107$$ 15.0000i 1.45010i 0.688694 + 0.725052i $$0.258184\pi$$
−0.688694 + 0.725052i $$0.741816\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 0 0
$$111$$ 9.00000 0.854242
$$112$$ − 3.00000i − 0.283473i
$$113$$ − 3.00000i − 0.282216i −0.989994 0.141108i $$-0.954933\pi$$
0.989994 0.141108i $$-0.0450665\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ − 4.00000i − 0.369800i
$$118$$ 8.00000i 0.736460i
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ − 14.0000i − 1.26750i
$$123$$ 10.0000i 0.901670i
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ −3.00000 −0.267261
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 11.0000 0.968496
$$130$$ 0 0
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ − 5.00000i − 0.435194i
$$133$$ − 3.00000i − 0.260133i
$$134$$ −7.00000 −0.604708
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ −9.00000 −0.757937
$$142$$ 14.0000i 1.17485i
$$143$$ − 20.0000i − 1.67248i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 2.00000i 0.164957i
$$148$$ − 9.00000i − 0.739795i
$$149$$ −14.0000 −1.14692 −0.573462 0.819232i $$-0.694400\pi$$
−0.573462 + 0.819232i $$0.694400\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ − 1.00000i − 0.0811107i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ −15.0000 −1.20873
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 24.0000i 1.91541i 0.287754 + 0.957704i $$0.407091\pi$$
−0.287754 + 0.957704i $$0.592909\pi$$
$$158$$ 5.00000i 0.397779i
$$159$$ 3.00000 0.237915
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 1.00000i 0.0785674i
$$163$$ − 18.0000i − 1.40987i −0.709273 0.704934i $$-0.750976\pi$$
0.709273 0.704934i $$-0.249024\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ 3.00000i 0.231455i
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ − 11.0000i − 0.838742i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ −5.00000 −0.376889
$$177$$ − 8.00000i − 0.601317i
$$178$$ 2.00000i 0.149906i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 12.0000i 0.889499i
$$183$$ 14.0000i 1.03491i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ − 5.00000i − 0.365636i
$$188$$ 9.00000i 0.656392i
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 6.00000i 0.431889i 0.976406 + 0.215945i $$0.0692831\pi$$
−0.976406 + 0.215945i $$0.930717\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 5.00000i 0.355335i
$$199$$ 1.00000 0.0708881 0.0354441 0.999372i $$-0.488715\pi$$
0.0354441 + 0.999372i $$0.488715\pi$$
$$200$$ 0 0
$$201$$ 7.00000 0.493742
$$202$$ 9.00000i 0.633238i
$$203$$ − 12.0000i − 0.842235i
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 4.00000i 0.278019i
$$208$$ 4.00000i 0.277350i
$$209$$ −5.00000 −0.345857
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ − 3.00000i − 0.206041i
$$213$$ − 14.0000i − 0.959264i
$$214$$ −15.0000 −1.02538
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 3.00000i 0.203653i
$$218$$ − 1.00000i − 0.0677285i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 9.00000i 0.604040i
$$223$$ 6.00000i 0.401790i 0.979613 + 0.200895i $$0.0643850\pi$$
−0.979613 + 0.200895i $$0.935615\pi$$
$$224$$ 3.00000 0.200446
$$225$$ 0 0
$$226$$ 3.00000 0.199557
$$227$$ 21.0000i 1.39382i 0.717159 + 0.696909i $$0.245442\pi$$
−0.717159 + 0.696909i $$0.754558\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ 12.0000 0.792982 0.396491 0.918039i $$-0.370228\pi$$
0.396491 + 0.918039i $$0.370228\pi$$
$$230$$ 0 0
$$231$$ 15.0000 0.986928
$$232$$ − 4.00000i − 0.262613i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ −8.00000 −0.520756
$$237$$ − 5.00000i − 0.324785i
$$238$$ 3.00000i 0.194461i
$$239$$ −5.00000 −0.323423 −0.161712 0.986838i $$-0.551701\pi$$
−0.161712 + 0.986838i $$0.551701\pi$$
$$240$$ 0 0
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 14.0000i 0.899954i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 14.0000 0.896258
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 4.00000i 0.254514i
$$248$$ 1.00000i 0.0635001i
$$249$$ 8.00000 0.506979
$$250$$ 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
$$253$$ 20.0000i 1.25739i
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 11.0000i 0.684830i
$$259$$ 27.0000 1.67770
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 4.00000i 0.247121i
$$263$$ − 7.00000i − 0.431638i −0.976433 0.215819i $$-0.930758\pi$$
0.976433 0.215819i $$-0.0692422\pi$$
$$264$$ 5.00000 0.307729
$$265$$ 0 0
$$266$$ 3.00000 0.183942
$$267$$ − 2.00000i − 0.122398i
$$268$$ − 7.00000i − 0.427593i
$$269$$ −28.0000 −1.70719 −0.853595 0.520937i $$-0.825583\pi$$
−0.853595 + 0.520937i $$0.825583\pi$$
$$270$$ 0 0
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ − 12.0000i − 0.726273i
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ 13.0000i 0.781094i 0.920583 + 0.390547i $$0.127714\pi$$
−0.920583 + 0.390547i $$0.872286\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 9.00000i − 0.535942i
$$283$$ 22.0000i 1.30776i 0.756596 + 0.653882i $$0.226861\pi$$
−0.756596 + 0.653882i $$0.773139\pi$$
$$284$$ −14.0000 −0.830747
$$285$$ 0 0
$$286$$ 20.0000 1.18262
$$287$$ 30.0000i 1.77084i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ − 2.00000i − 0.117041i
$$293$$ − 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
$$296$$ 9.00000 0.523114
$$297$$ − 5.00000i − 0.290129i
$$298$$ − 14.0000i − 0.810998i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 33.0000 1.90209
$$302$$ − 24.0000i − 1.38104i
$$303$$ − 9.00000i − 0.517036i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ − 8.00000i − 0.456584i −0.973593 0.228292i $$-0.926686\pi$$
0.973593 0.228292i $$-0.0733141\pi$$
$$308$$ − 15.0000i − 0.854704i
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 10.0000 0.567048 0.283524 0.958965i $$-0.408496\pi$$
0.283524 + 0.958965i $$0.408496\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ − 16.0000i − 0.904373i −0.891923 0.452187i $$-0.850644\pi$$
0.891923 0.452187i $$-0.149356\pi$$
$$314$$ −24.0000 −1.35440
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ − 4.00000i − 0.224662i −0.993671 0.112331i $$-0.964168\pi$$
0.993671 0.112331i $$-0.0358318\pi$$
$$318$$ 3.00000i 0.168232i
$$319$$ −20.0000 −1.11979
$$320$$ 0 0
$$321$$ 15.0000 0.837218
$$322$$ − 12.0000i − 0.668734i
$$323$$ 1.00000i 0.0556415i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 18.0000 0.996928
$$327$$ 1.00000i 0.0553001i
$$328$$ 10.0000i 0.552158i
$$329$$ −27.0000 −1.48856
$$330$$ 0 0
$$331$$ 5.00000 0.274825 0.137412 0.990514i $$-0.456121\pi$$
0.137412 + 0.990514i $$0.456121\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ − 9.00000i − 0.493197i
$$334$$ 18.0000 0.984916
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ − 24.0000i − 1.30736i −0.756770 0.653682i $$-0.773224\pi$$
0.756770 0.653682i $$-0.226776\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ −3.00000 −0.162938
$$340$$ 0 0
$$341$$ 5.00000 0.270765
$$342$$ − 1.00000i − 0.0540738i
$$343$$ − 15.0000i − 0.809924i
$$344$$ 11.0000 0.593080
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 11.0000i − 0.590511i −0.955418 0.295255i $$-0.904595\pi$$
0.955418 0.295255i $$-0.0954048\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ − 5.00000i − 0.266501i
$$353$$ 24.0000i 1.27739i 0.769460 + 0.638696i $$0.220526\pi$$
−0.769460 + 0.638696i $$0.779474\pi$$
$$354$$ 8.00000 0.425195
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ − 3.00000i − 0.158777i
$$358$$ 12.0000i 0.634220i
$$359$$ 25.0000 1.31945 0.659725 0.751507i $$-0.270673\pi$$
0.659725 + 0.751507i $$0.270673\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ − 5.00000i − 0.262794i
$$363$$ − 14.0000i − 0.734809i
$$364$$ −12.0000 −0.628971
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ − 25.0000i − 1.30499i −0.757793 0.652495i $$-0.773722\pi$$
0.757793 0.652495i $$-0.226278\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 9.00000 0.467257
$$372$$ − 1.00000i − 0.0518476i
$$373$$ 6.00000i 0.310668i 0.987862 + 0.155334i $$0.0496454\pi$$
−0.987862 + 0.155334i $$0.950355\pi$$
$$374$$ 5.00000 0.258544
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ 16.0000i 0.824042i
$$378$$ 3.00000i 0.154303i
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 2.00000 0.102463
$$382$$ − 9.00000i − 0.460480i
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ − 11.0000i − 0.559161i
$$388$$ − 14.0000i − 0.710742i
$$389$$ −21.0000 −1.06474 −0.532371 0.846511i $$-0.678699\pi$$
−0.532371 + 0.846511i $$0.678699\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ 2.00000i 0.101015i
$$393$$ − 4.00000i − 0.201773i
$$394$$ −12.0000 −0.604551
$$395$$ 0 0
$$396$$ −5.00000 −0.251259
$$397$$ − 15.0000i − 0.752828i −0.926451 0.376414i $$-0.877157\pi$$
0.926451 0.376414i $$-0.122843\pi$$
$$398$$ 1.00000i 0.0501255i
$$399$$ −3.00000 −0.150188
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 7.00000i 0.349128i
$$403$$ − 4.00000i − 0.199254i
$$404$$ −9.00000 −0.447767
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ − 45.0000i − 2.23057i
$$408$$ − 1.00000i − 0.0495074i
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ − 14.0000i − 0.689730i
$$413$$ − 24.0000i − 1.18096i
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 8.00000i 0.391762i
$$418$$ − 5.00000i − 0.244558i
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 24.0000 1.16969 0.584844 0.811146i $$-0.301156\pi$$
0.584844 + 0.811146i $$0.301156\pi$$
$$422$$ − 12.0000i − 0.584151i
$$423$$ 9.00000i 0.437595i
$$424$$ 3.00000 0.145693
$$425$$ 0 0
$$426$$ 14.0000 0.678302
$$427$$ 42.0000i 2.03252i
$$428$$ − 15.0000i − 0.725052i
$$429$$ −20.0000 −0.965609
$$430$$ 0 0
$$431$$ 14.0000 0.674356 0.337178 0.941441i $$-0.390528\pi$$
0.337178 + 0.941441i $$0.390528\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 27.0000i 1.29754i 0.760986 + 0.648769i $$0.224716\pi$$
−0.760986 + 0.648769i $$0.775284\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ 1.00000 0.0478913
$$437$$ − 4.00000i − 0.191346i
$$438$$ 2.00000i 0.0955637i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ − 4.00000i − 0.190261i
$$443$$ − 38.0000i − 1.80543i −0.430234 0.902717i $$-0.641569\pi$$
0.430234 0.902717i $$-0.358431\pi$$
$$444$$ −9.00000 −0.427121
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ 14.0000i 0.662177i
$$448$$ 3.00000i 0.141737i
$$449$$ 41.0000 1.93491 0.967455 0.253044i $$-0.0814317\pi$$
0.967455 + 0.253044i $$0.0814317\pi$$
$$450$$ 0 0
$$451$$ 50.0000 2.35441
$$452$$ 3.00000i 0.141108i
$$453$$ 24.0000i 1.12762i
$$454$$ −21.0000 −0.985579
$$455$$ 0 0
$$456$$ −1.00000 −0.0468293
$$457$$ − 37.0000i − 1.73079i −0.501093 0.865393i $$-0.667069\pi$$
0.501093 0.865393i $$-0.332931\pi$$
$$458$$ 12.0000i 0.560723i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −5.00000 −0.232873 −0.116437 0.993198i $$-0.537147\pi$$
−0.116437 + 0.993198i $$0.537147\pi$$
$$462$$ 15.0000i 0.697863i
$$463$$ 36.0000i 1.67306i 0.547920 + 0.836531i $$0.315420\pi$$
−0.547920 + 0.836531i $$0.684580\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 6.00000i 0.277647i 0.990317 + 0.138823i $$0.0443321\pi$$
−0.990317 + 0.138823i $$0.955668\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ 21.0000 0.969690
$$470$$ 0 0
$$471$$ 24.0000 1.10586
$$472$$ − 8.00000i − 0.368230i
$$473$$ − 55.0000i − 2.52890i
$$474$$ 5.00000 0.229658
$$475$$ 0 0
$$476$$ −3.00000 −0.137505
$$477$$ − 3.00000i − 0.137361i
$$478$$ − 5.00000i − 0.228695i
$$479$$ −30.0000 −1.37073 −0.685367 0.728197i $$-0.740358\pi$$
−0.685367 + 0.728197i $$0.740358\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ − 8.00000i − 0.364390i
$$483$$ 12.0000i 0.546019i
$$484$$ −14.0000 −0.636364
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000i 0.362515i 0.983436 + 0.181257i $$0.0580167\pi$$
−0.983436 + 0.181257i $$0.941983\pi$$
$$488$$ 14.0000i 0.633750i
$$489$$ −18.0000 −0.813988
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ 4.00000i 0.180151i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ − 42.0000i − 1.88396i
$$498$$ 8.00000i 0.358489i
$$499$$ 38.0000 1.70111 0.850557 0.525883i $$-0.176265\pi$$
0.850557 + 0.525883i $$0.176265\pi$$
$$500$$ 0 0
$$501$$ −18.0000 −0.804181
$$502$$ − 18.0000i − 0.803379i
$$503$$ 34.0000i 1.51599i 0.652263 + 0.757993i $$0.273820\pi$$
−0.652263 + 0.757993i $$0.726180\pi$$
$$504$$ 3.00000 0.133631
$$505$$ 0 0
$$506$$ −20.0000 −0.889108
$$507$$ 3.00000i 0.133235i
$$508$$ − 2.00000i − 0.0887357i
$$509$$ 9.00000 0.398918 0.199459 0.979906i $$-0.436082\pi$$
0.199459 + 0.979906i $$0.436082\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ 1.00000i 0.0441942i
$$513$$ 1.00000i 0.0441511i
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ −11.0000 −0.484248
$$517$$ 45.0000i 1.97910i
$$518$$ 27.0000i 1.18631i
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ 15.0000 0.657162 0.328581 0.944476i $$-0.393430\pi$$
0.328581 + 0.944476i $$0.393430\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 7.00000 0.305215
$$527$$ − 1.00000i − 0.0435607i
$$528$$ 5.00000i 0.217597i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 3.00000i 0.130066i
$$533$$ − 40.0000i − 1.73259i
$$534$$ 2.00000 0.0865485
$$535$$ 0 0
$$536$$ 7.00000 0.302354
$$537$$ − 12.0000i − 0.517838i
$$538$$ − 28.0000i − 1.20717i
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ 23.0000 0.988847 0.494424 0.869221i $$-0.335379\pi$$
0.494424 + 0.869221i $$0.335379\pi$$
$$542$$ 4.00000i 0.171815i
$$543$$ 5.00000i 0.214571i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ − 2.00000i − 0.0855138i −0.999086 0.0427569i $$-0.986386\pi$$
0.999086 0.0427569i $$-0.0136141\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 4.00000i 0.170251i
$$553$$ − 15.0000i − 0.637865i
$$554$$ −13.0000 −0.552317
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 9.00000i − 0.381342i −0.981654 0.190671i $$-0.938934\pi$$
0.981654 0.190671i $$-0.0610664\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ −44.0000 −1.86100
$$560$$ 0 0
$$561$$ −5.00000 −0.211100
$$562$$ − 18.0000i − 0.759284i
$$563$$ 16.0000i 0.674320i 0.941447 + 0.337160i $$0.109466\pi$$
−0.941447 + 0.337160i $$0.890534\pi$$
$$564$$ 9.00000 0.378968
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ − 3.00000i − 0.125988i
$$568$$ − 14.0000i − 0.587427i
$$569$$ −14.0000 −0.586911 −0.293455 0.955973i $$-0.594805\pi$$
−0.293455 + 0.955973i $$0.594805\pi$$
$$570$$ 0 0
$$571$$ −26.0000 −1.08807 −0.544033 0.839064i $$-0.683103\pi$$
−0.544033 + 0.839064i $$0.683103\pi$$
$$572$$ 20.0000i 0.836242i
$$573$$ 9.00000i 0.375980i
$$574$$ −30.0000 −1.25218
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 45.0000i 1.87337i 0.350167 + 0.936687i $$0.386125\pi$$
−0.350167 + 0.936687i $$0.613875\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 14.0000i 0.580319i
$$583$$ − 15.0000i − 0.621237i
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ − 6.00000i − 0.247647i −0.992304 0.123823i $$-0.960484\pi$$
0.992304 0.123823i $$-0.0395156\pi$$
$$588$$ − 2.00000i − 0.0824786i
$$589$$ −1.00000 −0.0412043
$$590$$ 0 0
$$591$$ 12.0000 0.493614
$$592$$ 9.00000i 0.369898i
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 5.00000 0.205152
$$595$$ 0 0
$$596$$ 14.0000 0.573462
$$597$$ − 1.00000i − 0.0409273i
$$598$$ 16.0000i 0.654289i
$$599$$ −39.0000 −1.59350 −0.796748 0.604311i $$-0.793448\pi$$
−0.796748 + 0.604311i $$0.793448\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ 33.0000i 1.34498i
$$603$$ − 7.00000i − 0.285062i
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ 9.00000 0.365600
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ 1.00000i 0.0405554i
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ 36.0000 1.45640
$$612$$ 1.00000i 0.0404226i
$$613$$ − 10.0000i − 0.403896i −0.979396 0.201948i $$-0.935273\pi$$
0.979396 0.201948i $$-0.0647272\pi$$
$$614$$ 8.00000 0.322854
$$615$$ 0 0
$$616$$ 15.0000 0.604367
$$617$$ − 19.0000i − 0.764911i −0.923974 0.382456i $$-0.875078\pi$$
0.923974 0.382456i $$-0.124922\pi$$
$$618$$ 14.0000i 0.563163i
$$619$$ 40.0000 1.60774 0.803868 0.594808i $$-0.202772\pi$$
0.803868 + 0.594808i $$0.202772\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 10.0000i 0.400963i
$$623$$ − 6.00000i − 0.240385i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ 16.0000 0.639489
$$627$$ 5.00000i 0.199681i
$$628$$ − 24.0000i − 0.957704i
$$629$$ −9.00000 −0.358854
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ − 5.00000i − 0.198889i
$$633$$ 12.0000i 0.476957i
$$634$$ 4.00000 0.158860
$$635$$ 0 0
$$636$$ −3.00000 −0.118958
$$637$$ − 8.00000i − 0.316972i
$$638$$ − 20.0000i − 0.791808i
$$639$$ −14.0000 −0.553831
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 15.0000i 0.592003i
$$643$$ 44.0000i 1.73519i 0.497271 + 0.867595i $$0.334335\pi$$
−0.497271 + 0.867595i $$0.665665\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ −1.00000 −0.0393445
$$647$$ − 8.00000i − 0.314512i −0.987558 0.157256i $$-0.949735\pi$$
0.987558 0.157256i $$-0.0502649\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −40.0000 −1.57014
$$650$$ 0 0
$$651$$ 3.00000 0.117579
$$652$$ 18.0000i 0.704934i
$$653$$ − 10.0000i − 0.391330i −0.980671 0.195665i $$-0.937313\pi$$
0.980671 0.195665i $$-0.0626866\pi$$
$$654$$ −1.00000 −0.0391031
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ − 2.00000i − 0.0780274i
$$658$$ − 27.0000i − 1.05257i
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 5.00000i 0.194331i
$$663$$ 4.00000i 0.155347i
$$664$$ 8.00000 0.310460
$$665$$ 0 0
$$666$$ 9.00000 0.348743
$$667$$ − 16.0000i − 0.619522i
$$668$$ 18.0000i 0.696441i
$$669$$ 6.00000 0.231973
$$670$$ 0 0
$$671$$ 70.0000 2.70232
$$672$$ − 3.00000i − 0.115728i
$$673$$ 4.00000i 0.154189i 0.997024 + 0.0770943i $$0.0245643\pi$$
−0.997024 + 0.0770943i $$0.975436\pi$$
$$674$$ 24.0000 0.924445
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ − 3.00000i − 0.115214i
$$679$$ 42.0000 1.61181
$$680$$ 0 0
$$681$$ 21.0000 0.804722
$$682$$ 5.00000i 0.191460i
$$683$$ − 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
$$687$$ − 12.0000i − 0.457829i
$$688$$ 11.0000i 0.419371i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 6.00000 0.228251 0.114125 0.993466i $$-0.463593\pi$$
0.114125 + 0.993466i $$0.463593\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ − 15.0000i − 0.569803i
$$694$$ 11.0000 0.417554
$$695$$ 0 0
$$696$$ −4.00000 −0.151620
$$697$$ − 10.0000i − 0.378777i
$$698$$ 10.0000i 0.378506i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ − 4.00000i − 0.150970i
$$703$$ 9.00000i 0.339441i
$$704$$ 5.00000 0.188445
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ − 27.0000i − 1.01544i
$$708$$ 8.00000i 0.300658i
$$709$$ −49.0000 −1.84023 −0.920117 0.391644i $$-0.871906\pi$$
−0.920117 + 0.391644i $$0.871906\pi$$
$$710$$ 0 0
$$711$$ −5.00000 −0.187515
$$712$$ − 2.00000i − 0.0749532i
$$713$$ 4.00000i 0.149801i
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 5.00000i 0.186728i
$$718$$ 25.0000i 0.932992i
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 42.0000 1.56416
$$722$$ − 18.0000i − 0.669891i
$$723$$ 8.00000i 0.297523i
$$724$$ 5.00000 0.185824
$$725$$ 0 0
$$726$$ 14.0000 0.519589
$$727$$ − 32.0000i − 1.18681i −0.804902 0.593407i $$-0.797782\pi$$
0.804902 0.593407i $$-0.202218\pi$$
$$728$$ − 12.0000i − 0.444750i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −11.0000 −0.406850
$$732$$ − 14.0000i − 0.517455i
$$733$$ − 2.00000i − 0.0738717i −0.999318 0.0369358i $$-0.988240\pi$$
0.999318 0.0369358i $$-0.0117597\pi$$
$$734$$ 25.0000 0.922767
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ − 35.0000i − 1.28924i
$$738$$ 10.0000i 0.368105i
$$739$$ 17.0000 0.625355 0.312678 0.949859i $$-0.398774\pi$$
0.312678 + 0.949859i $$0.398774\pi$$
$$740$$ 0 0
$$741$$ 4.00000 0.146944
$$742$$ 9.00000i 0.330400i
$$743$$ 38.0000i 1.39408i 0.717030 + 0.697042i $$0.245501\pi$$
−0.717030 + 0.697042i $$0.754499\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ −6.00000 −0.219676
$$747$$ − 8.00000i − 0.292705i
$$748$$ 5.00000i 0.182818i
$$749$$ 45.0000 1.64426
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ − 9.00000i − 0.328196i
$$753$$ 18.0000i 0.655956i
$$754$$ −16.0000 −0.582686
$$755$$ 0 0
$$756$$ −3.00000 −0.109109
$$757$$ 46.0000i 1.67190i 0.548807 + 0.835949i $$0.315082\pi$$
−0.548807 + 0.835949i $$0.684918\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ 20.0000 0.725954
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 2.00000i 0.0724524i
$$763$$ 3.00000i 0.108607i
$$764$$ 9.00000 0.325609
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 32.0000i 1.15545i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −11.0000 −0.396670 −0.198335 0.980134i $$-0.563553\pi$$
−0.198335 + 0.980134i $$0.563553\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ − 6.00000i − 0.215945i
$$773$$ 22.0000i 0.791285i 0.918405 + 0.395643i $$0.129478\pi$$
−0.918405 + 0.395643i $$0.870522\pi$$
$$774$$ 11.0000 0.395387
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ − 27.0000i − 0.968620i
$$778$$ − 21.0000i − 0.752886i
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ −70.0000 −2.50480
$$782$$ 4.00000i 0.143040i
$$783$$ 4.00000i 0.142948i
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ 4.00000 0.142675
$$787$$ − 50.0000i − 1.78231i −0.453701 0.891154i $$-0.649897\pi$$
0.453701 0.891154i $$-0.350103\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ −7.00000 −0.249207
$$790$$ 0 0
$$791$$ −9.00000 −0.320003
$$792$$ − 5.00000i − 0.177667i
$$793$$ − 56.0000i − 1.98862i
$$794$$ 15.0000 0.532330
$$795$$ 0 0
$$796$$ −1.00000 −0.0354441
$$797$$ 39.0000i 1.38145i 0.723117 + 0.690725i $$0.242709\pi$$
−0.723117 + 0.690725i $$0.757291\pi$$
$$798$$ − 3.00000i − 0.106199i
$$799$$ 9.00000 0.318397
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ − 30.0000i − 1.05934i
$$803$$ − 10.0000i − 0.352892i
$$804$$ −7.00000 −0.246871
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 28.0000i 0.985647i
$$808$$ − 9.00000i − 0.316619i
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ −22.0000 −0.772524 −0.386262 0.922389i $$-0.626234\pi$$
−0.386262 + 0.922389i $$0.626234\pi$$
$$812$$ 12.0000i 0.421117i
$$813$$ − 4.00000i − 0.140286i
$$814$$ 45.0000 1.57725
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ 11.0000i 0.384841i
$$818$$ 26.0000i 0.909069i
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 14.0000 0.487713
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 39.0000i 1.35616i 0.734987 + 0.678081i $$0.237188\pi$$
−0.734987 + 0.678081i $$0.762812\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ 24.0000 0.833554 0.416777 0.909009i $$-0.363160\pi$$
0.416777 + 0.909009i $$0.363160\pi$$
$$830$$ 0 0
$$831$$ 13.0000 0.450965
$$832$$ − 4.00000i − 0.138675i
$$833$$ − 2.00000i − 0.0692959i
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 5.00000 0.172929
$$837$$ − 1.00000i − 0.0345651i
$$838$$ − 28.0000i − 0.967244i
$$839$$ −10.0000 −0.345238 −0.172619 0.984989i $$-0.555223\pi$$
−0.172619 + 0.984989i $$0.555223\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 24.0000i 0.827095i
$$843$$ 18.0000i 0.619953i
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ −9.00000 −0.309426
$$847$$ − 42.0000i − 1.44314i
$$848$$ 3.00000i 0.103020i
$$849$$ 22.0000 0.755038
$$850$$ 0 0
$$851$$ 36.0000 1.23406
$$852$$ 14.0000i 0.479632i
$$853$$ 9.00000i 0.308154i 0.988059 + 0.154077i $$0.0492404\pi$$
−0.988059 + 0.154077i $$0.950760\pi$$
$$854$$ −42.0000 −1.43721
$$855$$ 0 0
$$856$$ 15.0000 0.512689
$$857$$ 5.00000i 0.170797i 0.996347 + 0.0853984i $$0.0272163\pi$$
−0.996347 + 0.0853984i $$0.972784\pi$$
$$858$$ − 20.0000i − 0.682789i
$$859$$ 25.0000 0.852989 0.426494 0.904490i $$-0.359748\pi$$
0.426494 + 0.904490i $$0.359748\pi$$
$$860$$ 0 0
$$861$$ 30.0000 1.02240
$$862$$ 14.0000i 0.476842i
$$863$$ 17.0000i 0.578687i 0.957225 + 0.289343i $$0.0934369\pi$$
−0.957225 + 0.289343i $$0.906563\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −27.0000 −0.917497
$$867$$ 1.00000i 0.0339618i
$$868$$ − 3.00000i − 0.101827i
$$869$$ −25.0000 −0.848067
$$870$$ 0 0
$$871$$ −28.0000 −0.948744
$$872$$ 1.00000i 0.0338643i
$$873$$ − 14.0000i − 0.473828i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ − 46.0000i − 1.55331i −0.629926 0.776655i $$-0.716915\pi$$
0.629926 0.776655i $$-0.283085\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ −35.0000 −1.17918 −0.589590 0.807703i $$-0.700711\pi$$
−0.589590 + 0.807703i $$0.700711\pi$$
$$882$$ 2.00000i 0.0673435i
$$883$$ 48.0000i 1.61533i 0.589643 + 0.807664i $$0.299269\pi$$
−0.589643 + 0.807664i $$0.700731\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 38.0000 1.27663
$$887$$ 12.0000i 0.402921i 0.979497 + 0.201460i $$0.0645687\pi$$
−0.979497 + 0.201460i $$0.935431\pi$$
$$888$$ − 9.00000i − 0.302020i
$$889$$ 6.00000 0.201234
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ − 6.00000i − 0.200895i
$$893$$ − 9.00000i − 0.301174i
$$894$$ −14.0000 −0.468230
$$895$$ 0 0
$$896$$ −3.00000 −0.100223
$$897$$ − 16.0000i − 0.534224i
$$898$$ 41.0000i 1.36819i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −3.00000 −0.0999445
$$902$$ 50.0000i 1.66482i
$$903$$ − 33.0000i − 1.09817i
$$904$$ −3.00000 −0.0997785
$$905$$ 0 0
$$906$$ −24.0000 −0.797347
$$907$$ − 24.0000i − 0.796907i −0.917189 0.398453i $$-0.869547\pi$$
0.917189 0.398453i $$-0.130453\pi$$
$$908$$ − 21.0000i − 0.696909i
$$909$$ −9.00000 −0.298511
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 40.0000i − 1.32381i
$$914$$ 37.0000 1.22385
$$915$$ 0 0
$$916$$ −12.0000 −0.396491
$$917$$ − 12.0000i − 0.396275i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ 14.0000 0.461817 0.230909 0.972975i $$-0.425830\pi$$
0.230909 + 0.972975i $$0.425830\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ − 5.00000i − 0.164666i
$$923$$ 56.0000i 1.84326i
$$924$$ −15.0000 −0.493464
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ − 14.0000i − 0.459820i
$$928$$ 4.00000i 0.131306i
$$929$$ −1.00000 −0.0328089 −0.0164045 0.999865i $$-0.505222\pi$$
−0.0164045 + 0.999865i $$0.505222\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 6.00000i 0.196537i
$$933$$ − 10.0000i − 0.327385i
$$934$$ −6.00000 −0.196326
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ 38.0000i 1.24141i 0.784046 + 0.620703i $$0.213153\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ 21.0000i 0.685674i
$$939$$ −16.0000 −0.522140
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ 24.0000i 0.781962i
$$943$$ 40.0000i 1.30258i
$$944$$ 8.00000 0.260378
$$945$$ 0 0
$$946$$ 55.0000 1.78820
$$947$$ 29.0000i 0.942373i 0.882034 + 0.471187i $$0.156174\pi$$
−0.882034 + 0.471187i $$0.843826\pi$$
$$948$$ 5.00000i 0.162392i
$$949$$ −8.00000 −0.259691
$$950$$ 0 0
$$951$$ −4.00000 −0.129709
$$952$$ − 3.00000i − 0.0972306i
$$953$$ 12.0000i 0.388718i 0.980930 + 0.194359i $$0.0622627\pi$$
−0.980930 + 0.194359i $$0.937737\pi$$
$$954$$ 3.00000 0.0971286
$$955$$ 0 0
$$956$$ 5.00000 0.161712
$$957$$ 20.0000i 0.646508i
$$958$$ − 30.0000i − 0.969256i
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ − 36.0000i − 1.16069i
$$963$$ − 15.0000i − 0.483368i
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ 24.0000i 0.771788i 0.922543 + 0.385894i $$0.126107\pi$$
−0.922543 + 0.385894i $$0.873893\pi$$
$$968$$ − 14.0000i − 0.449977i
$$969$$ 1.00000 0.0321246
$$970$$ 0 0
$$971$$ −10.0000 −0.320915 −0.160458 0.987043i $$-0.551297\pi$$
−0.160458 + 0.987043i $$0.551297\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 24.0000i 0.769405i
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −14.0000 −0.448129
$$977$$ 48.0000i 1.53566i 0.640656 + 0.767828i $$0.278662\pi$$
−0.640656 + 0.767828i $$0.721338\pi$$
$$978$$ − 18.0000i − 0.575577i
$$979$$ −10.0000 −0.319601
$$980$$ 0 0
$$981$$ 1.00000 0.0319275
$$982$$ − 30.0000i − 0.957338i
$$983$$ − 28.0000i − 0.893061i −0.894768 0.446531i $$-0.852659\pi$$
0.894768 0.446531i $$-0.147341\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 27.0000i 0.859419i
$$988$$ − 4.00000i − 0.127257i
$$989$$ 44.0000 1.39912
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ − 5.00000i − 0.158670i
$$994$$ 42.0000 1.33216
$$995$$ 0 0
$$996$$ −8.00000 −0.253490
$$997$$ 25.0000i 0.791758i 0.918303 + 0.395879i $$0.129560\pi$$
−0.918303 + 0.395879i $$0.870440\pi$$
$$998$$ 38.0000i 1.20287i
$$999$$ −9.00000 −0.284747
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.l.2449.2 2
5.2 odd 4 2550.2.a.f.1.1 1
5.3 odd 4 2550.2.a.bb.1.1 yes 1
5.4 even 2 inner 2550.2.d.l.2449.1 2
15.2 even 4 7650.2.a.ch.1.1 1
15.8 even 4 7650.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.f.1.1 1 5.2 odd 4
2550.2.a.bb.1.1 yes 1 5.3 odd 4
2550.2.d.l.2449.1 2 5.4 even 2 inner
2550.2.d.l.2449.2 2 1.1 even 1 trivial
7650.2.a.f.1.1 1 15.8 even 4
7650.2.a.ch.1.1 1 15.2 even 4