# Properties

 Label 4650.2.d.o.3349.2 Level $4650$ Weight $2$ Character 4650.3349 Analytic conductor $37.130$ 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] = [4650,2,Mod(3349,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.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: $$37.1304369399$$ 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: no (minimal twist has level 930) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3349.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 4650.3349 Dual form 4650.2.d.o.3349.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -16.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +18.0000 q^{89} +4.00000i q^{92} +1.00000i q^{93} +1.00000 q^{96} +14.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} - 12 q^{26} - 4 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{59} - 12 q^{61} - 2 q^{64} - 8 q^{66} - 8 q^{69} - 24 q^{71} - 4 q^{74} - 8 q^{76} + 32 q^{79} + 2 q^{81} + 8 q^{86} + 36 q^{89} + 2 q^{96} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 8 * q^11 + 2 * q^16 + 8 * q^19 - 2 * q^24 - 12 * q^26 - 4 * q^29 - 2 * q^31 + 4 * q^34 + 2 * q^36 + 12 * q^39 - 12 * q^41 + 8 * q^44 + 8 * q^46 + 14 * q^49 - 4 * q^51 - 2 * q^54 + 8 * q^59 - 12 * q^61 - 2 * q^64 - 8 * q^66 - 8 * q^69 - 24 * q^71 - 4 * q^74 - 8 * q^76 + 32 * q^79 + 2 * q^81 + 8 * q^86 + 36 * q^89 + 2 * q^96 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 6.00000i 1.66410i 0.554700 + 0.832050i $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ − 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
$$26$$ −6.00000 −1.17670
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605
$$32$$ 1.00000i 0.176777i
$$33$$ 4.00000i 0.696311i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ − 6.00000i − 0.832050i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 4.00000i − 0.529813i
$$58$$ − 2.00000i − 0.262613i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ − 1.00000i − 0.127000i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ − 16.0000i − 1.95471i −0.211604 0.977356i $$-0.567869\pi$$
0.211604 0.977356i $$-0.432131\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 6.00000i 0.679366i
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 2.00000i 0.214423i
$$88$$ 4.00000i 0.426401i
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 4.00000i 0.417029i
$$93$$ 1.00000i 0.103695i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 14.0000i 1.42148i 0.703452 + 0.710742i $$0.251641\pi$$
−0.703452 + 0.710742i $$0.748359\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ − 10.0000i − 0.940721i −0.882474 0.470360i $$-0.844124\pi$$
0.882474 0.470360i $$-0.155876\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 2.00000 0.185695
$$117$$ − 6.00000i − 0.554700i
$$118$$ 4.00000i 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ − 6.00000i − 0.543214i
$$123$$ 6.00000i 0.541002i
$$124$$ 1.00000 0.0898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ − 4.00000i − 0.348155i
$$133$$ 0 0
$$134$$ 16.0000 1.38219
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ − 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
$$141$$ 0 0
$$142$$ − 12.0000i − 1.00702i
$$143$$ − 24.0000i − 2.00698i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ − 7.00000i − 0.577350i
$$148$$ − 2.00000i − 0.164399i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 2.00000i 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ − 18.0000i − 1.43656i −0.695756 0.718278i $$-0.744931\pi$$
0.695756 0.718278i $$-0.255069\pi$$
$$158$$ 16.0000i 1.27289i
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ − 24.0000i − 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000i 0.304997i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ − 4.00000i − 0.300658i
$$178$$ 18.0000i 1.34916i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 6.00000i 0.443533i
$$184$$ −4.00000 −0.294884
$$185$$ 0 0
$$186$$ −1.00000 −0.0733236
$$187$$ 8.00000i 0.585018i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 22.0000i − 1.58359i −0.610784 0.791797i $$-0.709146\pi$$
0.610784 0.791797i $$-0.290854\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 4.00000i 0.284268i
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 2.00000i 0.140720i
$$203$$ 0 0
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 4.00000i 0.278019i
$$208$$ 6.00000i 0.416025i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ − 2.00000i − 0.137361i
$$213$$ 12.0000i 0.822226i
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ − 6.00000i − 0.406371i
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 2.00000i 0.134231i
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 10.0000 0.665190
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000i 0.131306i
$$233$$ 22.0000i 1.44127i 0.693316 + 0.720634i $$0.256149\pi$$
−0.693316 + 0.720634i $$0.743851\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ − 16.0000i − 1.03931i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 24.0000i 1.52708i
$$248$$ 1.00000i 0.0635001i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 16.0000i 1.00591i
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 22.0000i − 1.37232i −0.727450 0.686161i $$-0.759294\pi$$
0.727450 0.686161i $$-0.240706\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ − 4.00000i − 0.247121i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 18.0000i − 1.10158i
$$268$$ 16.0000i 0.977356i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 6.00000i − 0.360505i −0.983620 0.180253i $$-0.942309\pi$$
0.983620 0.180253i $$-0.0576915\pi$$
$$278$$ − 8.00000i − 0.479808i
$$279$$ 1.00000 0.0598684
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 24.0000 1.41915
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 6.00000i 0.351123i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 10.0000i − 0.579284i
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000i 0.920697i
$$303$$ − 2.00000i − 0.114897i
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 16.0000i 0.913168i 0.889680 + 0.456584i $$0.150927\pi$$
−0.889680 + 0.456584i $$0.849073\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −28.0000 −1.58773 −0.793867 0.608091i $$-0.791935\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ − 6.00000i − 0.339683i
$$313$$ 26.0000i 1.46961i 0.678280 + 0.734803i $$0.262726\pi$$
−0.678280 + 0.734803i $$0.737274\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 2.00000i 0.112154i
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ − 8.00000i − 0.445132i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ 6.00000i 0.331801i
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 24.0000 1.31916 0.659580 0.751635i $$-0.270734\pi$$
0.659580 + 0.751635i $$0.270734\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ − 2.00000i − 0.109599i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 2.00000i − 0.108947i −0.998515 0.0544735i $$-0.982652\pi$$
0.998515 0.0544735i $$-0.0173480\pi$$
$$338$$ − 23.0000i − 1.25104i
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ 4.00000 0.216612
$$342$$ − 4.00000i − 0.216295i
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ − 2.00000i − 0.107211i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ − 4.00000i − 0.213201i
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ 0 0
$$358$$ 4.00000i 0.211407i
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 6.00000i − 0.315353i
$$363$$ − 5.00000i − 0.262432i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ − 16.0000i − 0.835193i −0.908633 0.417597i $$-0.862873\pi$$
0.908633 0.417597i $$-0.137127\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 1.00000i − 0.0518476i
$$373$$ − 38.0000i − 1.96757i −0.179364 0.983783i $$-0.557404\pi$$
0.179364 0.983783i $$-0.442596\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 12.0000i − 0.618031i
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 8.00000 0.409852
$$382$$ 4.00000i 0.204658i
$$383$$ 4.00000i 0.204390i 0.994764 + 0.102195i $$0.0325866\pi$$
−0.994764 + 0.102195i $$0.967413\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ 4.00000i 0.203331i
$$388$$ − 14.0000i − 0.710742i
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ − 7.00000i − 0.353553i
$$393$$ 4.00000i 0.201773i
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ 14.0000i 0.702640i 0.936255 + 0.351320i $$0.114267\pi$$
−0.936255 + 0.351320i $$0.885733\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 14.0000 0.699127 0.349563 0.936913i $$-0.386330\pi$$
0.349563 + 0.936913i $$0.386330\pi$$
$$402$$ − 16.0000i − 0.798007i
$$403$$ − 6.00000i − 0.298881i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.00000i − 0.396545i
$$408$$ 2.00000i 0.0990148i
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ − 8.00000i − 0.394132i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −6.00000 −0.294174
$$417$$ 8.00000i 0.391762i
$$418$$ − 16.0000i − 0.782586i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 0 0
$$424$$ 2.00000 0.0971286
$$425$$ 0 0
$$426$$ −12.0000 −0.581402
$$427$$ 0 0
$$428$$ 4.00000i 0.193347i
$$429$$ −24.0000 −1.15873
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ 2.00000i 0.0961139i 0.998845 + 0.0480569i $$0.0153029\pi$$
−0.998845 + 0.0480569i $$0.984697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ − 16.0000i − 0.765384i
$$438$$ − 6.00000i − 0.286691i
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 12.0000i 0.570782i
$$443$$ − 4.00000i − 0.190046i −0.995475 0.0950229i $$-0.969708\pi$$
0.995475 0.0950229i $$-0.0302924\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 10.0000i 0.472984i
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 10.0000i 0.470360i
$$453$$ − 16.0000i − 0.751746i
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 22.0000i 1.02912i 0.857455 + 0.514558i $$0.172044\pi$$
−0.857455 + 0.514558i $$0.827956\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ − 24.0000i − 1.11537i −0.830051 0.557687i $$-0.811689\pi$$
0.830051 0.557687i $$-0.188311\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ 4.00000i 0.185098i 0.995708 + 0.0925490i $$0.0295015\pi$$
−0.995708 + 0.0925490i $$0.970499\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ − 4.00000i − 0.184115i
$$473$$ 16.0000i 0.735681i
$$474$$ 16.0000 0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 0 0
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ − 14.0000i − 0.637683i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 40.0000i 1.81257i 0.422664 + 0.906287i $$0.361095\pi$$
−0.422664 + 0.906287i $$0.638905\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ −24.0000 −1.08532
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ − 6.00000i − 0.270501i
$$493$$ 4.00000i 0.180151i
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ −1.00000 −0.0449013
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 20.0000i 0.892644i
$$503$$ 32.0000i 1.42681i 0.700752 + 0.713405i $$0.252848\pi$$
−0.700752 + 0.713405i $$0.747152\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −16.0000 −0.711287
$$507$$ 23.0000i 1.02147i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 4.00000i 0.176604i
$$514$$ 22.0000 0.970378
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ 20.0000i 0.874539i 0.899331 + 0.437269i $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 2.00000i 0.0871214i
$$528$$ 4.00000i 0.174078i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ − 36.0000i − 1.55933i
$$534$$ 18.0000 0.778936
$$535$$ 0 0
$$536$$ −16.0000 −0.691095
$$537$$ − 4.00000i − 0.172613i
$$538$$ − 18.0000i − 0.776035i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ 6.00000i 0.257485i
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 16.0000i − 0.684111i −0.939680 0.342055i $$-0.888877\pi$$
0.939680 0.342055i $$-0.111123\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 4.00000i 0.170251i
$$553$$ 0 0
$$554$$ 6.00000 0.254916
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 1.00000i 0.0423334i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 10.0000i 0.421825i
$$563$$ − 12.0000i − 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 12.0000i 0.503509i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 24.0000i 1.00349i
$$573$$ − 4.00000i − 0.167102i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000i 1.24892i 0.781058 + 0.624458i $$0.214680\pi$$
−0.781058 + 0.624458i $$0.785320\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ −22.0000 −0.914289
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 14.0000i 0.580319i
$$583$$ − 8.00000i − 0.331326i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 4.00000i − 0.165098i −0.996587 0.0825488i $$-0.973694\pi$$
0.996587 0.0825488i $$-0.0263060\pi$$
$$588$$ 7.00000i 0.288675i
$$589$$ −4.00000 −0.164817
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 2.00000i 0.0821995i
$$593$$ 14.0000i 0.574911i 0.957794 + 0.287456i $$0.0928094\pi$$
−0.957794 + 0.287456i $$0.907191\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 16.0000i 0.654836i
$$598$$ 24.0000i 0.981433i
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 16.0000i 0.651570i
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 2.00000i − 0.0808452i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 46.0000i − 1.85189i −0.377658 0.925945i $$-0.623271\pi$$
0.377658 0.925945i $$-0.376729\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −32.0000 −1.28619 −0.643094 0.765787i $$-0.722350\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ − 28.0000i − 1.12270i
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −26.0000 −1.03917
$$627$$ 16.0000i 0.638978i
$$628$$ 18.0000i 0.718278i
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ − 16.0000i − 0.636446i
$$633$$ 4.00000i 0.158986i
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ 42.0000i 1.66410i
$$638$$ 8.00000i 0.316723i
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ − 20.0000i − 0.788723i −0.918955 0.394362i $$-0.870966\pi$$
0.918955 0.394362i $$-0.129034\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 8.00000 0.314756
$$647$$ 12.0000i 0.471769i 0.971781 + 0.235884i $$0.0757987\pi$$
−0.971781 + 0.235884i $$0.924201\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 24.0000i 0.939913i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000i 0.234082i
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ −34.0000 −1.32245 −0.661223 0.750189i $$-0.729962\pi$$
−0.661223 + 0.750189i $$0.729962\pi$$
$$662$$ 24.0000i 0.932786i
$$663$$ − 12.0000i − 0.466041i
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 2.00000 0.0774984
$$667$$ 8.00000i 0.309761i
$$668$$ 12.0000i 0.464294i
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 2.00000 0.0770371
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ − 10.0000i − 0.384048i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ 4.00000i 0.153168i
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000i 0.381524i
$$688$$ − 4.00000i − 0.152499i
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ 12.0000i 0.454532i
$$698$$ 2.00000i 0.0757011i
$$699$$ 22.0000 0.832116
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ − 6.00000i − 0.226455i
$$703$$ 8.00000i 0.301726i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 0 0
$$708$$ 4.00000i 0.150329i
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ − 18.0000i − 0.674579i
$$713$$ 4.00000i 0.149801i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ 20.0000i 0.746393i
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ 14.0000i 0.520666i
$$724$$ 6.00000 0.222988
$$725$$ 0 0
$$726$$ 5.00000 0.185567
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ − 6.00000i − 0.221766i
$$733$$ 2.00000i 0.0738717i 0.999318 + 0.0369358i $$0.0117597\pi$$
−0.999318 + 0.0369358i $$0.988240\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 4.00000 0.147442
$$737$$ 64.0000i 2.35747i
$$738$$ 6.00000i 0.220863i
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ 0 0
$$743$$ 12.0000i 0.440237i 0.975473 + 0.220119i $$0.0706445\pi$$
−0.975473 + 0.220119i $$0.929356\pi$$
$$744$$ 1.00000 0.0366618
$$745$$ 0 0
$$746$$ 38.0000 1.39128
$$747$$ 12.0000i 0.439057i
$$748$$ − 8.00000i − 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −24.0000 −0.875772 −0.437886 0.899030i $$-0.644273\pi$$
−0.437886 + 0.899030i $$0.644273\pi$$
$$752$$ 0 0
$$753$$ − 20.0000i − 0.728841i
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 18.0000i 0.654221i 0.944986 + 0.327111i $$0.106075\pi$$
−0.944986 + 0.327111i $$0.893925\pi$$
$$758$$ 20.0000i 0.726433i
$$759$$ 16.0000 0.580763
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ 0 0
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ −4.00000 −0.144526
$$767$$ 24.0000i 0.866590i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −22.0000 −0.792311
$$772$$ 22.0000i 0.791797i
$$773$$ 50.0000i 1.79838i 0.437564 + 0.899188i $$0.355842\pi$$
−0.437564 + 0.899188i $$0.644158\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ 6.00000i 0.215110i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ − 8.00000i − 0.286079i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 2.00000i 0.0712470i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 0 0
$$792$$ − 4.00000i − 0.142134i
$$793$$ − 36.0000i − 1.27840i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ − 18.0000i − 0.637593i −0.947823 0.318796i $$-0.896721\pi$$
0.947823 0.318796i $$-0.103279\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ 14.0000i 0.494357i
$$803$$ 24.0000i 0.846942i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 6.00000 0.211341
$$807$$ 18.0000i 0.633630i
$$808$$ − 2.00000i − 0.0703598i
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 8.00000i 0.280572i
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ − 16.0000i − 0.559769i
$$818$$ 6.00000i 0.209785i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ − 18.0000i − 0.627822i
$$823$$ 40.0000i 1.39431i 0.716919 + 0.697156i $$0.245552\pi$$
−0.716919 + 0.697156i $$0.754448\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 52.0000i − 1.80822i −0.427303 0.904109i $$-0.640536\pi$$
0.427303 0.904109i $$-0.359464\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ −50.0000 −1.73657 −0.868286 0.496064i $$-0.834778\pi$$
−0.868286 + 0.496064i $$0.834778\pi$$
$$830$$ 0 0
$$831$$ −6.00000 −0.208138
$$832$$ − 6.00000i − 0.208013i
$$833$$ − 14.0000i − 0.485071i
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ − 1.00000i − 0.0345651i
$$838$$ 12.0000i 0.414533i
$$839$$ −28.0000 −0.966667 −0.483334 0.875436i $$-0.660574\pi$$
−0.483334 + 0.875436i $$0.660574\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 14.0000i 0.482472i
$$843$$ − 10.0000i − 0.344418i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 2.00000i 0.0686803i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 8.00000 0.274236
$$852$$ − 12.0000i − 0.411113i
$$853$$ 58.0000i 1.98588i 0.118609 + 0.992941i $$0.462157\pi$$
−0.118609 + 0.992941i $$0.537843\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −4.00000 −0.136717
$$857$$ − 54.0000i − 1.84460i −0.386469 0.922302i $$-0.626305\pi$$
0.386469 0.922302i $$-0.373695\pi$$
$$858$$ − 24.0000i − 0.819346i
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 12.0000i − 0.408722i
$$863$$ 36.0000i 1.22545i 0.790295 + 0.612727i $$0.209928\pi$$
−0.790295 + 0.612727i $$0.790072\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ − 13.0000i − 0.441503i
$$868$$ 0 0
$$869$$ −64.0000 −2.17105
$$870$$ 0 0
$$871$$ 96.0000 3.25284
$$872$$ 6.00000i 0.203186i
$$873$$ − 14.0000i − 0.473828i
$$874$$ 16.0000 0.541208
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ 38.0000i 1.28317i 0.767052 + 0.641584i $$0.221723\pi$$
−0.767052 + 0.641584i $$0.778277\pi$$
$$878$$ 32.0000i 1.07995i
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 52.0000i 1.74994i 0.484178 + 0.874970i $$0.339119\pi$$
−0.484178 + 0.874970i $$0.660881\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 4.00000 0.134383
$$887$$ 24.0000i 0.805841i 0.915235 + 0.402921i $$0.132005\pi$$
−0.915235 + 0.402921i $$0.867995\pi$$
$$888$$ − 2.00000i − 0.0671156i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 8.00000i 0.267860i
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 24.0000i − 0.801337i
$$898$$ 18.0000i 0.600668i
$$899$$ 2.00000 0.0667037
$$900$$ 0 0
$$901$$ 4.00000 0.133259
$$902$$ 24.0000i 0.799113i
$$903$$ 0 0
$$904$$ −10.0000 −0.332595
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 56.0000i 1.85945i 0.368255 + 0.929725i $$0.379955\pi$$
−0.368255 + 0.929725i $$0.620045\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 48.0000i 1.58857i
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 0 0
$$918$$ 2.00000i 0.0660098i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 16.0000 0.527218
$$922$$ − 14.0000i − 0.461065i
$$923$$ − 72.0000i − 2.36991i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 24.0000 0.788689
$$927$$ − 8.00000i − 0.262754i
$$928$$ − 2.00000i − 0.0656532i
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ − 22.0000i − 0.720634i
$$933$$ 28.0000i 0.916679i
$$934$$ −4.00000 −0.130884
$$935$$ 0 0
$$936$$ −6.00000 −0.196116
$$937$$ − 10.0000i − 0.326686i −0.986569 0.163343i $$-0.947772\pi$$
0.986569 0.163343i $$-0.0522277\pi$$
$$938$$ 0 0
$$939$$ 26.0000 0.848478
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ − 18.0000i − 0.586472i
$$943$$ 24.0000i 0.781548i
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 44.0000i 1.42981i 0.699223 + 0.714904i $$0.253530\pi$$
−0.699223 + 0.714904i $$0.746470\pi$$
$$948$$ 16.0000i 0.519656i
$$949$$ 36.0000 1.16861
$$950$$ 0 0
$$951$$ −2.00000 −0.0648544
$$952$$ 0 0
$$953$$ − 30.0000i − 0.971795i −0.874016 0.485898i $$-0.838493\pi$$
0.874016 0.485898i $$-0.161507\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 8.00000i − 0.258603i
$$958$$ − 36.0000i − 1.16311i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ − 12.0000i − 0.386896i
$$963$$ 4.00000i 0.128898i
$$964$$ 14.0000 0.450910
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 0 0
$$974$$ −40.0000 −1.28168
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ − 22.0000i − 0.703842i −0.936030 0.351921i $$-0.885529\pi$$
0.936030 0.351921i $$-0.114471\pi$$
$$978$$ − 24.0000i − 0.767435i
$$979$$ −72.0000 −2.30113
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ 20.0000i 0.638226i
$$983$$ − 4.00000i − 0.127580i −0.997963 0.0637901i $$-0.979681\pi$$
0.997963 0.0637901i $$-0.0203188\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ − 24.0000i − 0.763542i
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ − 1.00000i − 0.0317500i
$$993$$ − 24.0000i − 0.761617i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ − 26.0000i − 0.823428i −0.911313 0.411714i $$-0.864930\pi$$
0.911313 0.411714i $$-0.135070\pi$$
$$998$$ − 16.0000i − 0.506471i
$$999$$ −2.00000 −0.0632772
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 4650.2.d.o.3349.2 2
5.2 odd 4 930.2.a.b.1.1 1
5.3 odd 4 4650.2.a.bp.1.1 1
5.4 even 2 inner 4650.2.d.o.3349.1 2
15.2 even 4 2790.2.a.ba.1.1 1
20.7 even 4 7440.2.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.b.1.1 1 5.2 odd 4
2790.2.a.ba.1.1 1 15.2 even 4
4650.2.a.bp.1.1 1 5.3 odd 4
4650.2.d.o.3349.1 2 5.4 even 2 inner
4650.2.d.o.3349.2 2 1.1 even 1 trivial
7440.2.a.q.1.1 1 20.7 even 4