# Properties

 Label 1425.2.c.c.799.2 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.c (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: $$11.3786822880$$ 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 285) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.c.799.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} -6.00000 q^{11} +1.00000i q^{12} -2.00000 q^{14} -1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -1.00000 q^{19} -2.00000 q^{21} -6.00000i q^{22} -8.00000i q^{23} -3.00000 q^{24} -1.00000i q^{27} +2.00000i q^{28} -4.00000 q^{29} +5.00000i q^{32} -6.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} -2.00000i q^{42} -2.00000i q^{43} -6.00000 q^{44} +8.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +3.00000 q^{49} -6.00000 q^{51} +2.00000i q^{53} +1.00000 q^{54} -6.00000 q^{56} -1.00000i q^{57} -4.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} -2.00000i q^{63} -7.00000 q^{64} +6.00000 q^{66} +8.00000i q^{67} +6.00000i q^{68} +8.00000 q^{69} +16.0000 q^{71} -3.00000i q^{72} +14.0000i q^{73} +4.00000 q^{74} -1.00000 q^{76} -12.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} -2.00000 q^{84} +2.00000 q^{86} -4.00000i q^{87} -18.0000i q^{88} -8.00000i q^{92} -8.00000 q^{94} -5.00000 q^{96} +12.0000i q^{97} +3.00000i q^{98} +6.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 12 q^{11} - 4 q^{14} - 2 q^{16} - 2 q^{19} - 4 q^{21} - 6 q^{24} - 8 q^{29} - 12 q^{34} - 2 q^{36} - 12 q^{44} + 16 q^{46} + 6 q^{49} - 12 q^{51} + 2 q^{54} - 12 q^{56} - 24 q^{59} + 4 q^{61} - 14 q^{64} + 12 q^{66} + 16 q^{69} + 32 q^{71} + 8 q^{74} - 2 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 4 q^{86} - 16 q^{94} - 10 q^{96} + 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 12 * q^11 - 4 * q^14 - 2 * q^16 - 2 * q^19 - 4 * q^21 - 6 * q^24 - 8 * q^29 - 12 * q^34 - 2 * q^36 - 12 * q^44 + 16 * q^46 + 6 * q^49 - 12 * q^51 + 2 * q^54 - 12 * q^56 - 24 * q^59 + 4 * q^61 - 14 * q^64 + 12 * q^66 + 16 * q^69 + 32 * q^71 + 8 * q^74 - 2 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 4 * q^86 - 16 * q^94 - 10 * q^96 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 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 $$0.876624\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ − 6.00000i − 1.27920i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 1.00000i − 0.192450i
$$28$$ 2.00000i 0.377964i
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ − 6.00000i − 1.04447i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 4.00000i − 0.657596i −0.944400 0.328798i $$-0.893356\pi$$
0.944400 0.328798i $$-0.106644\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 0 0
$$53$$ 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
$$56$$ −6.00000 −0.801784
$$57$$ − 1.00000i − 0.132453i
$$58$$ − 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
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ − 2.00000i − 0.251976i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ − 12.0000i − 1.36753i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ − 4.00000i − 0.428845i
$$88$$ − 18.0000i − 1.91881i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 8.00000i − 0.834058i
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ 12.0000i 1.21842i 0.793011 + 0.609208i $$0.208512\pi$$
−0.793011 + 0.609208i $$0.791488\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 6.00000 0.603023
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ − 2.00000i − 0.188982i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −4.00000 −0.371391
$$117$$ 0 0
$$118$$ − 12.0000i − 1.10469i
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ −2.00000 −0.174741 −0.0873704 0.996176i $$-0.527846\pi$$
−0.0873704 + 0.996176i $$0.527846\pi$$
$$132$$ − 6.00000i − 0.522233i
$$133$$ − 2.00000i − 0.173422i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 16.0000i 1.34269i
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ 3.00000i 0.247436i
$$148$$ − 4.00000i − 0.328798i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ − 3.00000i − 0.243332i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 12.0000 0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 1.00000i 0.0785674i
$$163$$ − 22.0000i − 1.72317i −0.507611 0.861586i $$-0.669471\pi$$
0.507611 0.861586i $$-0.330529\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 16.0000i − 1.23812i −0.785345 0.619059i $$-0.787514\pi$$
0.785345 0.619059i $$-0.212486\pi$$
$$168$$ − 6.00000i − 0.462910i
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ − 2.00000i − 0.152499i
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ 4.00000 0.303239
$$175$$ 0 0
$$176$$ 6.00000 0.452267
$$177$$ − 12.0000i − 0.901975i
$$178$$ 0 0
$$179$$ 20.0000 1.49487 0.747435 0.664335i $$-0.231285\pi$$
0.747435 + 0.664335i $$0.231285\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ 24.0000 1.76930
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 36.0000i − 2.63258i
$$188$$ 8.00000i 0.583460i
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ − 7.00000i − 0.505181i
$$193$$ 24.0000i 1.72756i 0.503871 + 0.863779i $$0.331909\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −12.0000 −0.861550
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ − 6.00000i − 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 6.00000i 0.426401i
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ − 18.0000i − 1.26648i
$$203$$ − 8.00000i − 0.561490i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 8.00000i 0.556038i
$$208$$ 0 0
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 16.0000i 1.09630i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ 6.00000i 0.406371i
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 4.00000i 0.268462i
$$223$$ 8.00000i 0.535720i 0.963458 + 0.267860i $$0.0863164\pi$$
−0.963458 + 0.267860i $$0.913684\pi$$
$$224$$ −10.0000 −0.668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 28.0000i 1.85843i 0.369546 + 0.929213i $$0.379513\pi$$
−0.369546 + 0.929213i $$0.620487\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 12.0000 0.789542
$$232$$ − 12.0000i − 0.787839i
$$233$$ 22.0000i 1.44127i 0.693316 + 0.720634i $$0.256149\pi$$
−0.693316 + 0.720634i $$0.743851\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ − 8.00000i − 0.519656i
$$238$$ − 12.0000i − 0.777844i
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 25.0000i 1.60706i
$$243$$ 1.00000i 0.0641500i
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6.00000 −0.378717 −0.189358 0.981908i $$-0.560641\pi$$
−0.189358 + 0.981908i $$0.560641\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 48.0000i 3.01773i
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 22.0000i 1.37232i 0.727450 + 0.686161i $$0.240706\pi$$
−0.727450 + 0.686161i $$0.759294\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ − 2.00000i − 0.123560i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 18.0000 1.10782
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ 8.00000i 0.488678i
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ − 6.00000i − 0.363803i
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ 2.00000i 0.120168i 0.998193 + 0.0600842i $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ 16.0000i 0.959616i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ − 14.0000i − 0.832214i −0.909316 0.416107i $$-0.863394\pi$$
0.909316 0.416107i $$-0.136606\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ − 5.00000i − 0.294628i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −12.0000 −0.703452
$$292$$ 14.0000i 0.819288i
$$293$$ 30.0000i 1.75262i 0.481749 + 0.876309i $$0.340002\pi$$
−0.481749 + 0.876309i $$0.659998\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ 12.0000 0.697486
$$297$$ 6.00000i 0.348155i
$$298$$ − 6.00000i − 0.347571i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ − 16.0000i − 0.920697i
$$303$$ − 18.0000i − 1.03407i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 8.00000i 0.456584i 0.973593 + 0.228292i $$0.0733141\pi$$
−0.973593 + 0.228292i $$0.926686\pi$$
$$308$$ − 12.0000i − 0.683763i
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 34.0000 1.92796 0.963982 0.265969i $$-0.0856919\pi$$
0.963982 + 0.265969i $$0.0856919\pi$$
$$312$$ 0 0
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 2.00000i − 0.112154i
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 16.0000i 0.891645i
$$323$$ − 6.00000i − 0.333849i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 22.0000 1.21847
$$327$$ 6.00000i 0.331801i
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ 4.00000i 0.219199i
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 12.0000i 0.653682i 0.945079 + 0.326841i $$0.105984\pi$$
−0.945079 + 0.326841i $$0.894016\pi$$
$$338$$ 13.0000i 0.707107i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 1.00000i 0.0540738i
$$343$$ 20.0000i 1.07990i
$$344$$ 6.00000 0.323498
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ − 8.00000i − 0.429463i −0.976673 0.214731i $$-0.931112\pi$$
0.976673 0.214731i $$-0.0688876\pi$$
$$348$$ − 4.00000i − 0.214423i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 30.0000i − 1.59901i
$$353$$ − 34.0000i − 1.80964i −0.425797 0.904819i $$-0.640006\pi$$
0.425797 0.904819i $$-0.359994\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 12.0000i − 0.635107i
$$358$$ 20.0000i 1.05703i
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 10.0000i 0.525588i
$$363$$ 25.0000i 1.31216i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ 14.0000i 0.730794i 0.930852 + 0.365397i $$0.119067\pi$$
−0.930852 + 0.365397i $$0.880933\pi$$
$$368$$ 8.00000i 0.417029i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 36.0000 1.86152
$$375$$ 0 0
$$376$$ −24.0000 −1.23771
$$377$$ 0 0
$$378$$ 2.00000i 0.102869i
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 10.0000i 0.511645i
$$383$$ 8.00000i 0.408781i 0.978889 + 0.204390i $$0.0655212\pi$$
−0.978889 + 0.204390i $$0.934479\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −24.0000 −1.22157
$$387$$ 2.00000i 0.101666i
$$388$$ 12.0000i 0.609208i
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 48.0000 2.42746
$$392$$ 9.00000i 0.454569i
$$393$$ − 2.00000i − 0.100887i
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 6.00000 0.301511
$$397$$ − 6.00000i − 0.301131i −0.988600 0.150566i $$-0.951890\pi$$
0.988600 0.150566i $$-0.0481095\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ 2.00000 0.100125
$$400$$ 0 0
$$401$$ 4.00000 0.199750 0.0998752 0.995000i $$-0.468156\pi$$
0.0998752 + 0.995000i $$0.468156\pi$$
$$402$$ − 8.00000i − 0.399004i
$$403$$ 0 0
$$404$$ −18.0000 −0.895533
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ 24.0000i 1.18964i
$$408$$ − 18.0000i − 0.891133i
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ − 8.00000i − 0.394132i
$$413$$ − 24.0000i − 1.18096i
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 16.0000i 0.783523i
$$418$$ 6.00000i 0.293470i
$$419$$ 30.0000 1.46560 0.732798 0.680446i $$-0.238214\pi$$
0.732798 + 0.680446i $$0.238214\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ − 8.00000i − 0.388973i
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ −16.0000 −0.775203
$$427$$ 4.00000i 0.193574i
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 16.0000i − 0.768911i −0.923144 0.384455i $$-0.874389\pi$$
0.923144 0.384455i $$-0.125611\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 6.00000 0.287348
$$437$$ 8.00000i 0.382692i
$$438$$ − 14.0000i − 0.668946i
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 4.00000 0.189832
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ − 6.00000i − 0.283790i
$$448$$ − 14.0000i − 0.661438i
$$449$$ 8.00000 0.377543 0.188772 0.982021i $$-0.439549\pi$$
0.188772 + 0.982021i $$0.439549\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ − 16.0000i − 0.751746i
$$454$$ −28.0000 −1.31411
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ 6.00000i 0.280668i 0.990104 + 0.140334i $$0.0448177\pi$$
−0.990104 + 0.140334i $$0.955182\pi$$
$$458$$ − 2.00000i − 0.0934539i
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 12.0000i 0.558291i
$$463$$ − 14.0000i − 0.650635i −0.945605 0.325318i $$-0.894529\pi$$
0.945605 0.325318i $$-0.105471\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ − 36.0000i − 1.65703i
$$473$$ 12.0000i 0.551761i
$$474$$ 8.00000 0.367452
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ − 2.00000i − 0.0915737i
$$478$$ − 6.00000i − 0.274434i
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 18.0000i 0.819878i
$$483$$ 16.0000i 0.728025i
$$484$$ 25.0000 1.13636
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ 6.00000i 0.271607i
$$489$$ 22.0000 0.994874
$$490$$ 0 0
$$491$$ −26.0000 −1.17336 −0.586682 0.809818i $$-0.699566\pi$$
−0.586682 + 0.809818i $$0.699566\pi$$
$$492$$ 0 0
$$493$$ − 24.0000i − 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.0000i 1.43540i
$$498$$ 0 0
$$499$$ 24.0000 1.07439 0.537194 0.843459i $$-0.319484\pi$$
0.537194 + 0.843459i $$0.319484\pi$$
$$500$$ 0 0
$$501$$ 16.0000 0.714827
$$502$$ − 6.00000i − 0.267793i
$$503$$ − 36.0000i − 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ 6.00000 0.267261
$$505$$ 0 0
$$506$$ −48.0000 −2.13386
$$507$$ 13.0000i 0.577350i
$$508$$ − 4.00000i − 0.177471i
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ −28.0000 −1.23865
$$512$$ − 11.0000i − 0.486136i
$$513$$ 1.00000i 0.0441511i
$$514$$ −22.0000 −0.970378
$$515$$ 0 0
$$516$$ 2.00000 0.0880451
$$517$$ − 48.0000i − 2.11104i
$$518$$ 8.00000i 0.351500i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −4.00000 −0.175243 −0.0876216 0.996154i $$-0.527927\pi$$
−0.0876216 + 0.996154i $$0.527927\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ − 4.00000i − 0.174908i −0.996169 0.0874539i $$-0.972127\pi$$
0.996169 0.0874539i $$-0.0278730\pi$$
$$524$$ −2.00000 −0.0873704
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 0 0
$$528$$ 6.00000i 0.261116i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ − 2.00000i − 0.0867110i
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −24.0000 −1.03664
$$537$$ 20.0000i 0.863064i
$$538$$ − 12.0000i − 0.517357i
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ − 20.0000i − 0.859074i
$$543$$ 10.0000i 0.429141i
$$544$$ −30.0000 −1.28624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ 10.0000i 0.427179i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 4.00000 0.170406
$$552$$ 24.0000i 1.02151i
$$553$$ − 16.0000i − 0.680389i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ − 18.0000i − 0.762684i −0.924434 0.381342i $$-0.875462\pi$$
0.924434 0.381342i $$-0.124538\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 36.0000 1.51992
$$562$$ − 4.00000i − 0.168730i
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 14.0000 0.588464
$$567$$ 2.00000i 0.0839921i
$$568$$ 48.0000i 2.01404i
$$569$$ 8.00000 0.335377 0.167689 0.985840i $$-0.446370\pi$$
0.167689 + 0.985840i $$0.446370\pi$$
$$570$$ 0 0
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ 0 0
$$573$$ 10.0000i 0.417756i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ − 2.00000i − 0.0832611i −0.999133 0.0416305i $$-0.986745\pi$$
0.999133 0.0416305i $$-0.0132552\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −24.0000 −0.997406
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 12.0000i − 0.497416i
$$583$$ − 12.0000i − 0.496989i
$$584$$ −42.0000 −1.73797
$$585$$ 0 0
$$586$$ −30.0000 −1.23929
$$587$$ 20.0000i 0.825488i 0.910847 + 0.412744i $$0.135430\pi$$
−0.910847 + 0.412744i $$0.864570\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ 4.00000i 0.164399i
$$593$$ − 10.0000i − 0.410651i −0.978694 0.205325i $$-0.934175\pi$$
0.978694 0.205325i $$-0.0658253\pi$$
$$594$$ −6.00000 −0.246183
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ 20.0000i 0.818546i
$$598$$ 0 0
$$599$$ 20.0000 0.817178 0.408589 0.912719i $$-0.366021\pi$$
0.408589 + 0.912719i $$0.366021\pi$$
$$600$$ 0 0
$$601$$ 22.0000 0.897399 0.448699 0.893683i $$-0.351887\pi$$
0.448699 + 0.893683i $$0.351887\pi$$
$$602$$ 4.00000i 0.163028i
$$603$$ − 8.00000i − 0.325785i
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ 18.0000 0.731200
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ − 5.00000i − 0.202777i
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 6.00000i − 0.242536i
$$613$$ − 2.00000i − 0.0807792i −0.999184 0.0403896i $$-0.987140\pi$$
0.999184 0.0403896i $$-0.0128599\pi$$
$$614$$ −8.00000 −0.322854
$$615$$ 0 0
$$616$$ 36.0000 1.45048
$$617$$ − 34.0000i − 1.36879i −0.729112 0.684394i $$-0.760067\pi$$
0.729112 0.684394i $$-0.239933\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −8.00000 −0.321547 −0.160774 0.986991i $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 34.0000i 1.36328i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 6.00000i 0.239617i
$$628$$ 2.00000i 0.0798087i
$$629$$ 24.0000 0.956943
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ − 24.0000i − 0.954669i
$$633$$ − 20.0000i − 0.794929i
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ 0 0
$$638$$ 24.0000i 0.950169i
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 40.0000 1.57991 0.789953 0.613168i $$-0.210105\pi$$
0.789953 + 0.613168i $$0.210105\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ 2.00000i 0.0788723i 0.999222 + 0.0394362i $$0.0125562\pi$$
−0.999222 + 0.0394362i $$0.987444\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ − 20.0000i − 0.786281i −0.919478 0.393141i $$-0.871389\pi$$
0.919478 0.393141i $$-0.128611\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ 72.0000 2.82625
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 22.0000i − 0.861586i
$$653$$ − 18.0000i − 0.704394i −0.935926 0.352197i $$-0.885435\pi$$
0.935926 0.352197i $$-0.114565\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 14.0000i − 0.546192i
$$658$$ − 16.0000i − 0.623745i
$$659$$ −48.0000 −1.86981 −0.934907 0.354892i $$-0.884518\pi$$
−0.934907 + 0.354892i $$0.884518\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 28.0000i 1.08825i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −4.00000 −0.154997
$$667$$ 32.0000i 1.23904i
$$668$$ − 16.0000i − 0.619059i
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ −12.0000 −0.463255
$$672$$ − 10.0000i − 0.385758i
$$673$$ 36.0000i 1.38770i 0.720121 + 0.693849i $$0.244086\pi$$
−0.720121 + 0.693849i $$0.755914\pi$$
$$674$$ −12.0000 −0.462223
$$675$$ 0 0
$$676$$ 13.0000 0.500000
$$677$$ − 18.0000i − 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 6.00000i 0.230429i
$$679$$ −24.0000 −0.921035
$$680$$ 0 0
$$681$$ −28.0000 −1.07296
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ −20.0000 −0.763604
$$687$$ − 2.00000i − 0.0763048i
$$688$$ 2.00000i 0.0762493i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 32.0000 1.21734 0.608669 0.793424i $$-0.291704\pi$$
0.608669 + 0.793424i $$0.291704\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 12.0000i 0.455842i
$$694$$ 8.00000 0.303676
$$695$$ 0 0
$$696$$ 12.0000 0.454859
$$697$$ 0 0
$$698$$ 2.00000i 0.0757011i
$$699$$ −22.0000 −0.832116
$$700$$ 0 0
$$701$$ −42.0000 −1.58632 −0.793159 0.609015i $$-0.791565\pi$$
−0.793159 + 0.609015i $$0.791565\pi$$
$$702$$ 0 0
$$703$$ 4.00000i 0.150863i
$$704$$ 42.0000 1.58293
$$705$$ 0 0
$$706$$ 34.0000 1.27961
$$707$$ − 36.0000i − 1.35392i
$$708$$ − 12.0000i − 0.450988i
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ 20.0000 0.747435
$$717$$ − 6.00000i − 0.224074i
$$718$$ − 10.0000i − 0.373197i
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 1.00000i 0.0372161i
$$723$$ 18.0000i 0.669427i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ −25.0000 −0.927837
$$727$$ − 26.0000i − 0.964287i −0.876092 0.482143i $$-0.839858\pi$$
0.876092 0.482143i $$-0.160142\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 2.00000i 0.0739221i
$$733$$ − 22.0000i − 0.812589i −0.913742 0.406294i $$-0.866821\pi$$
0.913742 0.406294i $$-0.133179\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 0 0
$$736$$ 40.0000 1.47442
$$737$$ − 48.0000i − 1.76810i
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 4.00000i − 0.146845i
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ − 36.0000i − 1.31629i
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ − 6.00000i − 0.218652i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ 38.0000i 1.38113i 0.723269 + 0.690567i $$0.242639\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ − 20.0000i − 0.726433i
$$759$$ −48.0000 −1.74229
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ 12.0000i 0.434429i
$$764$$ 10.0000 0.361787
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ − 17.0000i − 0.613435i
$$769$$ 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
$$772$$ 24.0000i 0.863779i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ −36.0000 −1.29232
$$777$$ 8.00000i 0.286998i
$$778$$ 14.0000i 0.501924i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −96.0000 −3.43515
$$782$$ 48.0000i 1.71648i
$$783$$ 4.00000i 0.142948i
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 2.00000 0.0713376
$$787$$ 28.0000i 0.998092i 0.866575 + 0.499046i $$0.166316\pi$$
−0.866575 + 0.499046i $$0.833684\pi$$
$$788$$ − 6.00000i − 0.213741i
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 12.0000 0.426671
$$792$$ 18.0000i 0.639602i
$$793$$ 0 0
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 26.0000i 0.920967i 0.887668 + 0.460484i $$0.152324\pi$$
−0.887668 + 0.460484i $$0.847676\pi$$
$$798$$ 2.00000i 0.0707992i
$$799$$ −48.0000 −1.69812
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 4.00000i 0.141245i
$$803$$ − 84.0000i − 2.96430i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 12.0000i − 0.422420i
$$808$$ − 54.0000i − 1.89971i
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ − 8.00000i − 0.280745i
$$813$$ − 20.0000i − 0.701431i
$$814$$ −24.0000 −0.841200
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 2.00000i 0.0699711i
$$818$$ 10.0000i 0.349642i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ − 10.0000i − 0.348790i
$$823$$ − 14.0000i − 0.488009i −0.969774 0.244005i $$-0.921539\pi$$
0.969774 0.244005i $$-0.0784612\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ − 36.0000i − 1.25184i −0.779886 0.625921i $$-0.784723\pi$$
0.779886 0.625921i $$-0.215277\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 18.0000i 0.623663i
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ 6.00000 0.207514
$$837$$ 0 0
$$838$$ 30.0000i 1.03633i
$$839$$ −8.00000 −0.276191 −0.138095 0.990419i $$-0.544098\pi$$
−0.138095 + 0.990419i $$0.544098\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 6.00000i 0.206774i
$$843$$ − 4.00000i − 0.137767i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 50.0000i 1.71802i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 14.0000 0.480479
$$850$$ 0 0
$$851$$ −32.0000 −1.09695
$$852$$ 16.0000i 0.548151i
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ − 46.0000i − 1.57133i −0.618652 0.785665i $$-0.712321\pi$$
0.618652 0.785665i $$-0.287679\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 36.0000i 1.22616i
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 16.0000 0.543702
$$867$$ − 19.0000i − 0.645274i
$$868$$ 0 0
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 18.0000i 0.609557i
$$873$$ − 12.0000i − 0.406138i
$$874$$ −8.00000 −0.270604
$$875$$ 0 0
$$876$$ −14.0000 −0.473016
$$877$$ 24.0000i 0.810422i 0.914223 + 0.405211i $$0.132802\pi$$
−0.914223 + 0.405211i $$0.867198\pi$$
$$878$$ − 24.0000i − 0.809961i
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ −42.0000 −1.41502 −0.707508 0.706705i $$-0.750181\pi$$
−0.707508 + 0.706705i $$0.750181\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 2.00000i 0.0673054i 0.999434 + 0.0336527i $$0.0107140\pi$$
−0.999434 + 0.0336527i $$0.989286\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ 12.0000i 0.402694i
$$889$$ 8.00000 0.268311
$$890$$ 0 0
$$891$$ −6.00000 −0.201008
$$892$$ 8.00000i 0.267860i
$$893$$ − 8.00000i − 0.267710i
$$894$$ 6.00000 0.200670
$$895$$ 0 0
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ 8.00000i 0.266963i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ 4.00000i 0.133112i
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 52.0000i 1.72663i 0.504664 + 0.863316i $$0.331616\pi$$
−0.504664 + 0.863316i $$0.668384\pi$$
$$908$$ 28.0000i 0.929213i
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ 0 0
$$914$$ −6.00000 −0.198462
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ − 4.00000i − 0.132092i
$$918$$ 6.00000i 0.198030i
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −8.00000 −0.263609
$$922$$ 30.0000i 0.987997i
$$923$$ 0 0
$$924$$ 12.0000 0.394771
$$925$$ 0 0
$$926$$ 14.0000 0.460069
$$927$$ 8.00000i 0.262754i
$$928$$ − 20.0000i − 0.656532i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ −3.00000 −0.0983210
$$932$$ 22.0000i 0.720634i
$$933$$ 34.0000i 1.11311i
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 30.0000i − 0.980057i −0.871706 0.490029i $$-0.836986\pi$$
0.871706 0.490029i $$-0.163014\pi$$
$$938$$ − 16.0000i − 0.522419i
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 44.0000 1.43436 0.717180 0.696888i $$-0.245433\pi$$
0.717180 + 0.696888i $$0.245433\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ − 48.0000i − 1.55979i −0.625910 0.779895i $$-0.715272\pi$$
0.625910 0.779895i $$-0.284728\pi$$
$$948$$ − 8.00000i − 0.259828i
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ − 36.0000i − 1.16677i
$$953$$ − 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
$$956$$ −6.00000 −0.194054
$$957$$ 24.0000i 0.775810i
$$958$$ − 18.0000i − 0.581554i
$$959$$ −20.0000 −0.645834
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ − 12.0000i − 0.386695i
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 14.0000i 0.450210i 0.974335 + 0.225105i $$0.0722725\pi$$
−0.974335 + 0.225105i $$0.927728\pi$$
$$968$$ 75.0000i 2.41059i
$$969$$ 6.00000 0.192748
$$970$$ 0 0
$$971$$ −8.00000 −0.256732 −0.128366 0.991727i $$-0.540973\pi$$
−0.128366 + 0.991727i $$0.540973\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 32.0000i 1.02587i
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 22.0000i 0.703482i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ − 26.0000i − 0.829693i
$$983$$ 8.00000i 0.255160i 0.991828 + 0.127580i $$0.0407210\pi$$
−0.991828 + 0.127580i $$0.959279\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 24.0000 0.764316
$$987$$ − 16.0000i − 0.509286i
$$988$$ 0 0
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 28.0000i 0.888553i
$$994$$ −32.0000 −1.01498
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 14.0000i − 0.443384i −0.975117 0.221692i $$-0.928842\pi$$
0.975117 0.221692i $$-0.0711580\pi$$
$$998$$ 24.0000i 0.759707i
$$999$$ −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 1425.2.c.c.799.2 2
5.2 odd 4 285.2.a.a.1.1 1
5.3 odd 4 1425.2.a.g.1.1 1
5.4 even 2 inner 1425.2.c.c.799.1 2
15.2 even 4 855.2.a.c.1.1 1
15.8 even 4 4275.2.a.h.1.1 1
20.7 even 4 4560.2.a.h.1.1 1
95.37 even 4 5415.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.a.1.1 1 5.2 odd 4
855.2.a.c.1.1 1 15.2 even 4
1425.2.a.g.1.1 1 5.3 odd 4
1425.2.c.c.799.1 2 5.4 even 2 inner
1425.2.c.c.799.2 2 1.1 even 1 trivial
4275.2.a.h.1.1 1 15.8 even 4
4560.2.a.h.1.1 1 20.7 even 4
5415.2.a.h.1.1 1 95.37 even 4