# Properties

 Label 2550.2.d.q.2449.2 Level $2550$ Weight $2$ Character 2550.2449 Analytic conductor $20.362$ Analytic rank $1$ Dimension $2$ 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: no (minimal twist has level 102) 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.q.2449.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -6.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +2.00000 q^{21} +6.00000i q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +4.00000 q^{29} -6.00000 q^{31} +1.00000i q^{32} -1.00000 q^{34} +1.00000 q^{36} +4.00000i q^{37} -4.00000i q^{38} -6.00000 q^{39} -10.0000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -6.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} +1.00000 q^{51} +6.00000i q^{52} -2.00000i q^{53} -1.00000 q^{54} +2.00000 q^{56} +4.00000i q^{57} +4.00000i q^{58} -12.0000 q^{59} -4.00000 q^{61} -6.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} +12.0000i q^{67} -1.00000i q^{68} +6.00000 q^{69} -6.00000 q^{71} +1.00000i q^{72} +2.00000i q^{73} -4.00000 q^{74} +4.00000 q^{76} -6.00000i q^{78} -10.0000 q^{79} +1.00000 q^{81} -10.0000i q^{82} -12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} -4.00000i q^{87} +2.00000 q^{89} +12.0000 q^{91} -6.00000i q^{92} +6.00000i q^{93} +4.00000 q^{94} +1.00000 q^{96} -6.00000i q^{97} +3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 8 q^{19} + 4 q^{21} - 2 q^{24} + 12 q^{26} + 8 q^{29} - 12 q^{31} - 2 q^{34} + 2 q^{36} - 12 q^{39} - 20 q^{41} - 12 q^{46} + 6 q^{49} + 2 q^{51} - 2 q^{54} + 4 q^{56} - 24 q^{59} - 8 q^{61} - 2 q^{64} + 12 q^{69} - 12 q^{71} - 8 q^{74} + 8 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 4 q^{89} + 24 q^{91} + 8 q^{94} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 8 * q^19 + 4 * q^21 - 2 * q^24 + 12 * q^26 + 8 * q^29 - 12 * q^31 - 2 * q^34 + 2 * q^36 - 12 * q^39 - 20 * q^41 - 12 * q^46 + 6 * q^49 + 2 * q^51 - 2 * q^54 + 4 * q^56 - 24 * q^59 - 8 * q^61 - 2 * q^64 + 12 * q^69 - 12 * q^71 - 8 * q^74 + 8 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 + 8 * q^86 + 4 * q^89 + 24 * q^91 + 8 * q^94 + 2 * q^96

## 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
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 1.00000i 0.242536i
$$18$$ − 1.00000i − 0.235702i
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 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
$$26$$ 6.00000 1.17670
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 0 0
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 2.00000i 0.308607i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ − 4.00000i − 0.583460i −0.956501 0.291730i $$-0.905769\pi$$
0.956501 0.291730i $$-0.0942309\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 1.00000 0.140028
$$52$$ 6.00000i 0.832050i
$$53$$ − 2.00000i − 0.274721i −0.990521 0.137361i $$-0.956138\pi$$
0.990521 0.137361i $$-0.0438619\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ 4.00000i 0.529813i
$$58$$ 4.00000i 0.525226i
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ − 6.00000i − 0.762001i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000i 1.46603i 0.680211 + 0.733017i $$0.261888\pi$$
−0.680211 + 0.733017i $$0.738112\pi$$
$$68$$ − 1.00000i − 0.121268i
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ 2.00000i 0.234082i 0.993127 + 0.117041i $$0.0373409\pi$$
−0.993127 + 0.117041i $$0.962659\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ − 6.00000i − 0.679366i
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 10.0000i − 1.10432i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 4.00000i − 0.428845i
$$88$$ 0 0
$$89$$ 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
$$92$$ − 6.00000i − 0.625543i
$$93$$ 6.00000i 0.622171i
$$94$$ 4.00000 0.412568
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 1.00000i 0.0990148i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 2.00000 0.194257
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 2.00000i 0.188982i
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 6.00000i 0.554700i
$$118$$ − 12.0000i − 1.10469i
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 4.00000i − 0.362143i
$$123$$ 10.0000i 0.901670i
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ − 8.00000i − 0.693688i
$$134$$ −12.0000 −1.03664
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 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
$$141$$ −4.00000 −0.336861
$$142$$ − 6.00000i − 0.503509i
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 4.00000i − 0.328798i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ 4.00000i 0.324443i
$$153$$ − 1.00000i − 0.0808452i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ − 6.00000i − 0.478852i −0.970915 0.239426i $$-0.923041\pi$$
0.970915 0.239426i $$-0.0769593\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 1.00000i 0.0785674i
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 2.00000i 0.154765i 0.997001 + 0.0773823i $$0.0246562\pi$$
−0.997001 + 0.0773823i $$0.975344\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ − 4.00000i − 0.304114i −0.988372 0.152057i $$-0.951410\pi$$
0.988372 0.152057i $$-0.0485898\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 12.0000i 0.901975i
$$178$$ 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
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 12.0000i 0.889499i
$$183$$ 4.00000i 0.295689i
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ −6.00000 −0.439941
$$187$$ 0 0
$$188$$ 4.00000i 0.291730i
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ 6.00000i 0.431889i 0.976406 + 0.215945i $$0.0692831\pi$$
−0.976406 + 0.215945i $$0.930717\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 8.00000i − 0.569976i −0.958531 0.284988i $$-0.908010\pi$$
0.958531 0.284988i $$-0.0919897\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 14.0000i 0.985037i
$$203$$ 8.00000i 0.561490i
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ − 6.00000i − 0.417029i
$$208$$ − 6.00000i − 0.416025i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 6.00000i 0.411113i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ − 12.0000i − 0.814613i
$$218$$ − 16.0000i − 1.08366i
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 4.00000i 0.268462i
$$223$$ − 4.00000i − 0.267860i −0.990991 0.133930i $$-0.957240\pi$$
0.990991 0.133930i $$-0.0427597\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 4.00000i − 0.262613i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 10.0000i 0.649570i
$$238$$ − 2.00000i − 0.129641i
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 4.00000 0.256074
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 24.0000i 1.52708i
$$248$$ 6.00000i 0.381000i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ − 16.0000i − 0.988483i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 8.00000 0.490511
$$267$$ − 2.00000i − 0.122398i
$$268$$ − 12.0000i − 0.733017i
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 1.00000i 0.0606339i
$$273$$ − 12.0000i − 0.726273i
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 8.00000i 0.480673i 0.970690 + 0.240337i $$0.0772579\pi$$
−0.970690 + 0.240337i $$0.922742\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 4.00000i − 0.238197i
$$283$$ 32.0000i 1.90220i 0.308879 + 0.951101i $$0.400046\pi$$
−0.308879 + 0.951101i $$0.599954\pi$$
$$284$$ 6.00000 0.356034
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 20.0000i − 1.18056i
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −6.00000 −0.351726
$$292$$ − 2.00000i − 0.117041i
$$293$$ 2.00000i 0.116841i 0.998292 + 0.0584206i $$0.0186065\pi$$
−0.998292 + 0.0584206i $$0.981394\pi$$
$$294$$ 3.00000 0.174964
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ 0 0
$$298$$ 6.00000i 0.347571i
$$299$$ 36.0000 2.08193
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ − 24.0000i − 1.38104i
$$303$$ − 14.0000i − 0.804279i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 6.00000i 0.339683i
$$313$$ − 26.0000i − 1.46961i −0.678280 0.734803i $$-0.737274\pi$$
0.678280 0.734803i $$-0.262726\pi$$
$$314$$ 6.00000 0.338600
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 16.0000i 0.898650i 0.893368 + 0.449325i $$0.148335\pi$$
−0.893368 + 0.449325i $$0.851665\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 12.0000i − 0.668734i
$$323$$ − 4.00000i − 0.222566i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 16.0000i 0.884802i
$$328$$ 10.0000i 0.552158i
$$329$$ 8.00000 0.441054
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ − 4.00000i − 0.219199i
$$334$$ −2.00000 −0.109435
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 6.00000i 0.326841i 0.986557 + 0.163420i $$0.0522527\pi$$
−0.986557 + 0.163420i $$0.947747\pi$$
$$338$$ − 23.0000i − 1.25104i
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000i 0.216295i
$$343$$ 20.0000i 1.07990i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 4.00000 0.215041
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 4.00000i 0.214423i
$$349$$ 30.0000 1.60586 0.802932 0.596071i $$-0.203272\pi$$
0.802932 + 0.596071i $$0.203272\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ 14.0000i 0.745145i 0.928003 + 0.372572i $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 2.00000i 0.105851i
$$358$$ 12.0000i 0.634220i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 20.0000i − 1.05118i
$$363$$ 11.0000i 0.577350i
$$364$$ −12.0000 −0.628971
$$365$$ 0 0
$$366$$ −4.00000 −0.209083
$$367$$ − 10.0000i − 0.521996i −0.965339 0.260998i $$-0.915948\pi$$
0.965339 0.260998i $$-0.0840516\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 4.00000 0.207670
$$372$$ − 6.00000i − 0.311086i
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −4.00000 −0.206284
$$377$$ − 24.0000i − 1.23606i
$$378$$ − 2.00000i − 0.102869i
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ − 4.00000i − 0.204658i
$$383$$ 28.0000i 1.43073i 0.698749 + 0.715367i $$0.253740\pi$$
−0.698749 + 0.715367i $$0.746260\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 4.00000i 0.203331i
$$388$$ 6.00000i 0.304604i
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ − 3.00000i − 0.151523i
$$393$$ 16.0000i 0.807093i
$$394$$ 8.00000 0.403034
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 20.0000i − 1.00377i −0.864934 0.501886i $$-0.832640\pi$$
0.864934 0.501886i $$-0.167360\pi$$
$$398$$ − 14.0000i − 0.701757i
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 12.0000i 0.598506i
$$403$$ 36.0000i 1.79329i
$$404$$ −14.0000 −0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ − 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
$$412$$ − 4.00000i − 0.197066i
$$413$$ − 24.0000i − 1.18096i
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 8.00000i 0.391762i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ 4.00000i 0.194487i
$$424$$ −2.00000 −0.0971286
$$425$$ 0 0
$$426$$ −6.00000 −0.290701
$$427$$ − 8.00000i − 0.387147i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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
$$433$$ − 18.0000i − 0.865025i −0.901628 0.432512i $$-0.857627\pi$$
0.901628 0.432512i $$-0.142373\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 0 0
$$436$$ 16.0000 0.766261
$$437$$ − 24.0000i − 1.14808i
$$438$$ 2.00000i 0.0955637i
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 6.00000i 0.285391i
$$443$$ 12.0000i 0.570137i 0.958507 + 0.285069i $$0.0920164\pi$$
−0.958507 + 0.285069i $$0.907984\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ 4.00000 0.189405
$$447$$ − 6.00000i − 0.283790i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 2.00000i − 0.0940721i
$$453$$ 24.0000i 1.12762i
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 2.00000i 0.0934539i
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 0 0
$$463$$ − 4.00000i − 0.185896i −0.995671 0.0929479i $$-0.970371\pi$$
0.995671 0.0929479i $$-0.0296290\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ − 6.00000i − 0.277350i
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ −6.00000 −0.276465
$$472$$ 12.0000i 0.552345i
$$473$$ 0 0
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ 2.00000i 0.0915737i
$$478$$ 20.0000i 0.914779i
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ − 18.0000i − 0.819878i
$$483$$ 12.0000i 0.546019i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 38.0000i 1.72194i 0.508652 + 0.860972i $$0.330144\pi$$
−0.508652 + 0.860972i $$0.669856\pi$$
$$488$$ 4.00000i 0.181071i
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ 4.00000i 0.180151i
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ −6.00000 −0.269408
$$497$$ − 12.0000i − 0.538274i
$$498$$ − 12.0000i − 0.537733i
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 2.00000 0.0893534
$$502$$ 12.0000i 0.535586i
$$503$$ − 26.0000i − 1.15928i −0.814872 0.579641i $$-0.803193\pi$$
0.814872 0.579641i $$-0.196807\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ 8.00000i 0.354943i
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 1.00000i 0.0441942i
$$513$$ − 4.00000i − 0.176604i
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ − 8.00000i − 0.351500i
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ − 4.00000i − 0.175075i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ 16.0000 0.698963
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ − 6.00000i − 0.261364i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 8.00000i 0.346844i
$$533$$ 60.0000i 2.59889i
$$534$$ 2.00000 0.0865485
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ − 12.0000i − 0.517838i
$$538$$ 12.0000i 0.517357i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −12.0000 −0.515920 −0.257960 0.966156i $$-0.583050\pi$$
−0.257960 + 0.966156i $$0.583050\pi$$
$$542$$ − 16.0000i − 0.687259i
$$543$$ 20.0000i 0.858282i
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 4.00000 0.170716
$$550$$ 0 0
$$551$$ −16.0000 −0.681623
$$552$$ − 6.00000i − 0.255377i
$$553$$ − 20.0000i − 0.850487i
$$554$$ −8.00000 −0.339887
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 14.0000i − 0.593199i −0.955002 0.296600i $$-0.904147\pi$$
0.955002 0.296600i $$-0.0958526\pi$$
$$558$$ 6.00000i 0.254000i
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 18.0000i − 0.759284i
$$563$$ − 4.00000i − 0.168580i −0.996441 0.0842900i $$-0.973138\pi$$
0.996441 0.0842900i $$-0.0268622\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 0 0
$$566$$ −32.0000 −1.34506
$$567$$ 2.00000i 0.0839921i
$$568$$ 6.00000i 0.251754i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ 4.00000i 0.167102i
$$574$$ 20.0000 0.834784
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000i 1.24892i 0.781058 + 0.624458i $$0.214680\pi$$
−0.781058 + 0.624458i $$0.785320\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ − 6.00000i − 0.248708i
$$583$$ 0 0
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −2.00000 −0.0826192
$$587$$ − 36.0000i − 1.48588i −0.669359 0.742940i $$-0.733431\pi$$
0.669359 0.742940i $$-0.266569\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 24.0000 0.988903
$$590$$ 0 0
$$591$$ −8.00000 −0.329076
$$592$$ 4.00000i 0.164399i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 14.0000i 0.572982i
$$598$$ 36.0000i 1.47215i
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ − 12.0000i − 0.488678i
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ 14.0000 0.568711
$$607$$ 38.0000i 1.54237i 0.636610 + 0.771186i $$0.280336\pi$$
−0.636610 + 0.771186i $$0.719664\pi$$
$$608$$ − 4.00000i − 0.162221i
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 1.00000i 0.0404226i
$$613$$ 30.0000i 1.21169i 0.795583 + 0.605844i $$0.207165\pi$$
−0.795583 + 0.605844i $$0.792835\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 30.0000i 1.20289i
$$623$$ 4.00000i 0.160257i
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 6.00000i 0.239426i
$$629$$ −4.00000 −0.159490
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 10.0000i 0.397779i
$$633$$ − 8.00000i − 0.317971i
$$634$$ −16.0000 −0.635441
$$635$$ 0 0
$$636$$ 2.00000 0.0793052
$$637$$ − 18.0000i − 0.713186i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ − 28.0000i − 1.10079i −0.834903 0.550397i $$-0.814476\pi$$
0.834903 0.550397i $$-0.185524\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −12.0000 −0.470317
$$652$$ − 12.0000i − 0.469956i
$$653$$ 20.0000i 0.782660i 0.920250 + 0.391330i $$0.127985\pi$$
−0.920250 + 0.391330i $$0.872015\pi$$
$$654$$ −16.0000 −0.625650
$$655$$ 0 0
$$656$$ −10.0000 −0.390434
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 8.00000i 0.311872i
$$659$$ −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 $$0.587774\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ − 6.00000i − 0.233021i
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ 24.0000i 0.929284i
$$668$$ − 2.00000i − 0.0773823i
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 2.00000i 0.0771517i
$$673$$ − 26.0000i − 1.00223i −0.865382 0.501113i $$-0.832924\pi$$
0.865382 0.501113i $$-0.167076\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ − 8.00000i − 0.307465i −0.988113 0.153732i $$-0.950871\pi$$
0.988113 0.153732i $$-0.0491294\pi$$
$$678$$ 2.00000i 0.0768095i
$$679$$ 12.0000 0.460518
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i 0.973289 + 0.229584i $$0.0737364\pi$$
−0.973289 + 0.229584i $$0.926264\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ − 2.00000i − 0.0763048i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 16.0000 0.608669 0.304334 0.952565i $$-0.401566\pi$$
0.304334 + 0.952565i $$0.401566\pi$$
$$692$$ 4.00000i 0.152057i
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ −4.00000 −0.151620
$$697$$ − 10.0000i − 0.378777i
$$698$$ 30.0000i 1.13552i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ − 16.0000i − 0.603451i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ 28.0000i 1.05305i
$$708$$ − 12.0000i − 0.450988i
$$709$$ 16.0000 0.600893 0.300446 0.953799i $$-0.402864\pi$$
0.300446 + 0.953799i $$0.402864\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ − 2.00000i − 0.0749532i
$$713$$ − 36.0000i − 1.34821i
$$714$$ −2.00000 −0.0748481
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 20.0000i − 0.746914i
$$718$$ 0 0
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ − 3.00000i − 0.111648i
$$723$$ 18.0000i 0.669427i
$$724$$ 20.0000 0.743294
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ − 12.0000i − 0.444750i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ − 4.00000i − 0.147844i
$$733$$ 18.0000i 0.664845i 0.943131 + 0.332423i $$0.107866\pi$$
−0.943131 + 0.332423i $$0.892134\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 0 0
$$738$$ 10.0000i 0.368105i
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ 4.00000i 0.146845i
$$743$$ 38.0000i 1.39408i 0.717030 + 0.697042i $$0.245501\pi$$
−0.717030 + 0.697042i $$0.754499\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −34.0000 −1.24068 −0.620339 0.784334i $$-0.713005\pi$$
−0.620339 + 0.784334i $$0.713005\pi$$
$$752$$ − 4.00000i − 0.145865i
$$753$$ − 12.0000i − 0.437304i
$$754$$ 24.0000 0.874028
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ − 14.0000i − 0.508839i −0.967094 0.254419i $$-0.918116\pi$$
0.967094 0.254419i $$-0.0818843\pi$$
$$758$$ 4.00000i 0.145287i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ − 8.00000i − 0.289809i
$$763$$ − 32.0000i − 1.15848i
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ 72.0000i 2.59977i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ − 6.00000i − 0.215945i
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 8.00000i 0.286998i
$$778$$ 14.0000i 0.501924i
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 6.00000i − 0.214560i
$$783$$ 4.00000i 0.142948i
$$784$$ 3.00000 0.107143
$$785$$ 0 0
$$786$$ −16.0000 −0.570701
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 8.00000i 0.284988i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 0 0
$$793$$ 24.0000i 0.852265i
$$794$$ 20.0000 0.709773
$$795$$ 0 0
$$796$$ 14.0000 0.496217
$$797$$ − 46.0000i − 1.62940i −0.579880 0.814702i $$-0.696901\pi$$
0.579880 0.814702i $$-0.303099\pi$$
$$798$$ − 8.00000i − 0.283197i
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 30.0000i 1.05934i
$$803$$ 0 0
$$804$$ −12.0000 −0.423207
$$805$$ 0 0
$$806$$ −36.0000 −1.26805
$$807$$ − 12.0000i − 0.422420i
$$808$$ − 14.0000i − 0.492518i
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ − 8.00000i − 0.280745i
$$813$$ 16.0000i 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 1.00000 0.0350070
$$817$$ 16.0000i 0.559769i
$$818$$ 26.0000i 0.909069i
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 18.0000i 0.627441i 0.949515 + 0.313720i $$0.101575\pi$$
−0.949515 + 0.313720i $$0.898425\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ − 16.0000i − 0.556375i −0.960527 0.278187i $$-0.910266\pi$$
0.960527 0.278187i $$-0.0897336\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ −6.00000 −0.208389 −0.104194 0.994557i $$-0.533226\pi$$
−0.104194 + 0.994557i $$0.533226\pi$$
$$830$$ 0 0
$$831$$ 8.00000 0.277517
$$832$$ 6.00000i 0.208013i
$$833$$ 3.00000i 0.103944i
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 6.00000i − 0.207390i
$$838$$ 12.0000i 0.414533i
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 34.0000i 1.17172i
$$843$$ 18.0000i 0.619953i
$$844$$ −8.00000 −0.275371
$$845$$ 0 0
$$846$$ −4.00000 −0.137523
$$847$$ − 22.0000i − 0.755929i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 32.0000 1.09824
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ − 6.00000i − 0.205557i
$$853$$ − 16.0000i − 0.547830i −0.961754 0.273915i $$-0.911681\pi$$
0.961754 0.273915i $$-0.0883186\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 10.0000i 0.341593i 0.985306 + 0.170797i $$0.0546341\pi$$
−0.985306 + 0.170797i $$0.945366\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ −20.0000 −0.681598
$$862$$ 14.0000i 0.476842i
$$863$$ − 48.0000i − 1.63394i −0.576681 0.816970i $$-0.695652\pi$$
0.576681 0.816970i $$-0.304348\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 18.0000 0.611665
$$867$$ 1.00000i 0.0339618i
$$868$$ 12.0000i 0.407307i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 72.0000 2.43963
$$872$$ 16.0000i 0.541828i
$$873$$ 6.00000i 0.203069i
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ −2.00000 −0.0675737
$$877$$ 24.0000i 0.810422i 0.914223 + 0.405211i $$0.132802\pi$$
−0.914223 + 0.405211i $$0.867198\pi$$
$$878$$ 10.0000i 0.337484i
$$879$$ 2.00000 0.0674583
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ − 52.0000i − 1.74994i −0.484178 0.874970i $$-0.660881\pi$$
0.484178 0.874970i $$-0.339119\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ − 18.0000i − 0.604381i −0.953248 0.302190i $$-0.902282\pi$$
0.953248 0.302190i $$-0.0977178\pi$$
$$888$$ − 4.00000i − 0.134231i
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 4.00000i 0.133930i
$$893$$ 16.0000i 0.535420i
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ 2.00000 0.0668153
$$897$$ − 36.0000i − 1.20201i
$$898$$ 26.0000i 0.867631i
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 2.00000 0.0666297
$$902$$ 0 0
$$903$$ − 8.00000i − 0.266223i
$$904$$ 2.00000 0.0665190
$$905$$ 0 0
$$906$$ −24.0000 −0.797347
$$907$$ − 24.0000i − 0.796907i −0.917189 0.398453i $$-0.869547\pi$$
0.917189 0.398453i $$-0.130453\pi$$
$$908$$ 4.00000i 0.132745i
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ 26.0000 0.861418 0.430709 0.902491i $$-0.358263\pi$$
0.430709 + 0.902491i $$0.358263\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 0 0
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ − 32.0000i − 1.05673i
$$918$$ − 1.00000i − 0.0330049i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ − 10.0000i − 0.329332i
$$923$$ 36.0000i 1.18495i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ − 4.00000i − 0.131377i
$$928$$ 4.00000i 0.131306i
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 6.00000i 0.196537i
$$933$$ − 30.0000i − 0.982156i
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ − 22.0000i − 0.718709i −0.933201 0.359354i $$-0.882997\pi$$
0.933201 0.359354i $$-0.117003\pi$$
$$938$$ − 24.0000i − 0.783628i
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ − 6.00000i − 0.195491i
$$943$$ − 60.0000i − 1.95387i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 24.0000i 0.779895i 0.920837 + 0.389948i $$0.127507\pi$$
−0.920837 + 0.389948i $$0.872493\pi$$
$$948$$ − 10.0000i − 0.324785i
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 16.0000 0.518836
$$952$$ 2.00000i 0.0648204i
$$953$$ 22.0000i 0.712650i 0.934362 + 0.356325i $$0.115970\pi$$
−0.934362 + 0.356325i $$0.884030\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ −20.0000 −0.646846
$$957$$ 0 0
$$958$$ − 10.0000i − 0.323085i
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 24.0000i 0.773791i
$$963$$ 0 0
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ 44.0000i 1.41494i 0.706741 + 0.707472i $$0.250165\pi$$
−0.706741 + 0.707472i $$0.749835\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 16.0000i − 0.512936i
$$974$$ −38.0000 −1.21760
$$975$$ 0 0
$$976$$ −4.00000 −0.128037
$$977$$ − 2.00000i − 0.0639857i −0.999488 0.0319928i $$-0.989815\pi$$
0.999488 0.0319928i $$-0.0101854\pi$$
$$978$$ 12.0000i 0.383718i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 16.0000 0.510841
$$982$$ − 20.0000i − 0.638226i
$$983$$ − 38.0000i − 1.21201i −0.795460 0.606006i $$-0.792771\pi$$
0.795460 0.606006i $$-0.207229\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ − 8.00000i − 0.254643i
$$988$$ − 24.0000i − 0.763542i
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ − 6.00000i − 0.190500i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 20.0000i 0.633406i 0.948525 + 0.316703i $$0.102576\pi$$
−0.948525 + 0.316703i $$0.897424\pi$$
$$998$$ − 32.0000i − 1.01294i
$$999$$ −4.00000 −0.126554
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.q.2449.2 2
5.2 odd 4 102.2.a.a.1.1 1
5.3 odd 4 2550.2.a.be.1.1 1
5.4 even 2 inner 2550.2.d.q.2449.1 2
15.2 even 4 306.2.a.d.1.1 1
15.8 even 4 7650.2.a.z.1.1 1
20.7 even 4 816.2.a.h.1.1 1
35.27 even 4 4998.2.a.x.1.1 1
40.27 even 4 3264.2.a.p.1.1 1
40.37 odd 4 3264.2.a.bf.1.1 1
60.47 odd 4 2448.2.a.t.1.1 1
85.2 odd 8 1734.2.f.g.1483.1 4
85.32 odd 8 1734.2.f.g.1483.2 4
85.42 odd 8 1734.2.f.g.829.2 4
85.47 odd 4 1734.2.b.d.577.1 2
85.67 odd 4 1734.2.a.h.1.1 1
85.72 odd 4 1734.2.b.d.577.2 2
85.77 odd 8 1734.2.f.g.829.1 4
120.77 even 4 9792.2.a.a.1.1 1
120.107 odd 4 9792.2.a.b.1.1 1
255.152 even 4 5202.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.a.1.1 1 5.2 odd 4
306.2.a.d.1.1 1 15.2 even 4
816.2.a.h.1.1 1 20.7 even 4
1734.2.a.h.1.1 1 85.67 odd 4
1734.2.b.d.577.1 2 85.47 odd 4
1734.2.b.d.577.2 2 85.72 odd 4
1734.2.f.g.829.1 4 85.77 odd 8
1734.2.f.g.829.2 4 85.42 odd 8
1734.2.f.g.1483.1 4 85.2 odd 8
1734.2.f.g.1483.2 4 85.32 odd 8
2448.2.a.t.1.1 1 60.47 odd 4
2550.2.a.be.1.1 1 5.3 odd 4
2550.2.d.q.2449.1 2 5.4 even 2 inner
2550.2.d.q.2449.2 2 1.1 even 1 trivial
3264.2.a.p.1.1 1 40.27 even 4
3264.2.a.bf.1.1 1 40.37 odd 4
4998.2.a.x.1.1 1 35.27 even 4
5202.2.a.g.1.1 1 255.152 even 4
7650.2.a.z.1.1 1 15.8 even 4
9792.2.a.a.1.1 1 120.77 even 4
9792.2.a.b.1.1 1 120.107 odd 4