# Properties

 Label 2790.2.a.bf.1.2 Level $2790$ Weight $2$ Character 2790.1 Self dual yes Analytic conductor $22.278$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(1,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2790.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2790.a (trivial)

## 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: yes Analytic conductor: $$22.2782621639$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 930) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-3.53113$$ of defining polynomial Character $$\chi$$ $$=$$ 2790.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.53113 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +3.53113 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.53113 q^{11} +6.00000 q^{13} -3.53113 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.53113 q^{19} +1.00000 q^{20} -1.53113 q^{22} +1.53113 q^{23} +1.00000 q^{25} -6.00000 q^{26} +3.53113 q^{28} +1.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} +3.53113 q^{35} +9.06226 q^{37} +3.53113 q^{38} -1.00000 q^{40} +9.06226 q^{41} -0.468871 q^{43} +1.53113 q^{44} -1.53113 q^{46} -11.0623 q^{47} +5.46887 q^{49} -1.00000 q^{50} +6.00000 q^{52} -5.53113 q^{53} +1.53113 q^{55} -3.53113 q^{56} -7.06226 q^{59} -11.0623 q^{61} -1.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} -11.0623 q^{67} +4.00000 q^{68} -3.53113 q^{70} +4.46887 q^{71} -0.468871 q^{73} -9.06226 q^{74} -3.53113 q^{76} +5.40661 q^{77} -0.468871 q^{79} +1.00000 q^{80} -9.06226 q^{82} +8.00000 q^{83} +4.00000 q^{85} +0.468871 q^{86} -1.53113 q^{88} -1.53113 q^{89} +21.1868 q^{91} +1.53113 q^{92} +11.0623 q^{94} -3.53113 q^{95} -16.1245 q^{97} -5.46887 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - q^{7} - 2 q^{8} - 2 q^{10} - 5 q^{11} + 12 q^{13} + q^{14} + 2 q^{16} + 8 q^{17} + q^{19} + 2 q^{20} + 5 q^{22} - 5 q^{23} + 2 q^{25} - 12 q^{26} - q^{28} + 2 q^{31} - 2 q^{32} - 8 q^{34} - q^{35} + 2 q^{37} - q^{38} - 2 q^{40} + 2 q^{41} - 9 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 19 q^{49} - 2 q^{50} + 12 q^{52} - 3 q^{53} - 5 q^{55} + q^{56} + 2 q^{59} - 6 q^{61} - 2 q^{62} + 2 q^{64} + 12 q^{65} - 6 q^{67} + 8 q^{68} + q^{70} + 17 q^{71} - 9 q^{73} - 2 q^{74} + q^{76} + 35 q^{77} - 9 q^{79} + 2 q^{80} - 2 q^{82} + 16 q^{83} + 8 q^{85} + 9 q^{86} + 5 q^{88} + 5 q^{89} - 6 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} - 19 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - q^7 - 2 * q^8 - 2 * q^10 - 5 * q^11 + 12 * q^13 + q^14 + 2 * q^16 + 8 * q^17 + q^19 + 2 * q^20 + 5 * q^22 - 5 * q^23 + 2 * q^25 - 12 * q^26 - q^28 + 2 * q^31 - 2 * q^32 - 8 * q^34 - q^35 + 2 * q^37 - q^38 - 2 * q^40 + 2 * q^41 - 9 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 19 * q^49 - 2 * q^50 + 12 * q^52 - 3 * q^53 - 5 * q^55 + q^56 + 2 * q^59 - 6 * q^61 - 2 * q^62 + 2 * q^64 + 12 * q^65 - 6 * q^67 + 8 * q^68 + q^70 + 17 * q^71 - 9 * q^73 - 2 * q^74 + q^76 + 35 * q^77 - 9 * q^79 + 2 * q^80 - 2 * q^82 + 16 * q^83 + 8 * q^85 + 9 * q^86 + 5 * q^88 + 5 * q^89 - 6 * q^91 - 5 * q^92 + 6 * q^94 + q^95 - 19 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.53113 1.33464 0.667321 0.744771i $$-0.267441\pi$$
0.667321 + 0.744771i $$0.267441\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 1.53113 0.461653 0.230826 0.972995i $$-0.425857\pi$$
0.230826 + 0.972995i $$0.425857\pi$$
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ −3.53113 −0.943734
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 0 0
$$19$$ −3.53113 −0.810097 −0.405048 0.914295i $$-0.632745\pi$$
−0.405048 + 0.914295i $$0.632745\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −1.53113 −0.326438
$$23$$ 1.53113 0.319262 0.159631 0.987177i $$-0.448969\pi$$
0.159631 + 0.987177i $$0.448969\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −6.00000 −1.17670
$$27$$ 0 0
$$28$$ 3.53113 0.667321
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 1.00000 0.179605
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 3.53113 0.596870
$$36$$ 0 0
$$37$$ 9.06226 1.48983 0.744913 0.667162i $$-0.232491\pi$$
0.744913 + 0.667162i $$0.232491\pi$$
$$38$$ 3.53113 0.572825
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ 9.06226 1.41529 0.707643 0.706570i $$-0.249758\pi$$
0.707643 + 0.706570i $$0.249758\pi$$
$$42$$ 0 0
$$43$$ −0.468871 −0.0715022 −0.0357511 0.999361i $$-0.511382\pi$$
−0.0357511 + 0.999361i $$0.511382\pi$$
$$44$$ 1.53113 0.230826
$$45$$ 0 0
$$46$$ −1.53113 −0.225753
$$47$$ −11.0623 −1.61360 −0.806798 0.590827i $$-0.798801\pi$$
−0.806798 + 0.590827i $$0.798801\pi$$
$$48$$ 0 0
$$49$$ 5.46887 0.781267
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 6.00000 0.832050
$$53$$ −5.53113 −0.759759 −0.379879 0.925036i $$-0.624035\pi$$
−0.379879 + 0.925036i $$0.624035\pi$$
$$54$$ 0 0
$$55$$ 1.53113 0.206457
$$56$$ −3.53113 −0.471867
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −7.06226 −0.919428 −0.459714 0.888067i $$-0.652048\pi$$
−0.459714 + 0.888067i $$0.652048\pi$$
$$60$$ 0 0
$$61$$ −11.0623 −1.41638 −0.708188 0.706023i $$-0.750487\pi$$
−0.708188 + 0.706023i $$0.750487\pi$$
$$62$$ −1.00000 −0.127000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −11.0623 −1.35147 −0.675735 0.737145i $$-0.736174\pi$$
−0.675735 + 0.737145i $$0.736174\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0 0
$$70$$ −3.53113 −0.422051
$$71$$ 4.46887 0.530357 0.265179 0.964199i $$-0.414569\pi$$
0.265179 + 0.964199i $$0.414569\pi$$
$$72$$ 0 0
$$73$$ −0.468871 −0.0548772 −0.0274386 0.999623i $$-0.508735\pi$$
−0.0274386 + 0.999623i $$0.508735\pi$$
$$74$$ −9.06226 −1.05347
$$75$$ 0 0
$$76$$ −3.53113 −0.405048
$$77$$ 5.40661 0.616141
$$78$$ 0 0
$$79$$ −0.468871 −0.0527521 −0.0263761 0.999652i $$-0.508397\pi$$
−0.0263761 + 0.999652i $$0.508397\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ −9.06226 −1.00076
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0.468871 0.0505597
$$87$$ 0 0
$$88$$ −1.53113 −0.163219
$$89$$ −1.53113 −0.162299 −0.0811497 0.996702i $$-0.525859\pi$$
−0.0811497 + 0.996702i $$0.525859\pi$$
$$90$$ 0 0
$$91$$ 21.1868 2.22098
$$92$$ 1.53113 0.159631
$$93$$ 0 0
$$94$$ 11.0623 1.14098
$$95$$ −3.53113 −0.362286
$$96$$ 0 0
$$97$$ −16.1245 −1.63720 −0.818598 0.574367i $$-0.805248\pi$$
−0.818598 + 0.574367i $$0.805248\pi$$
$$98$$ −5.46887 −0.552439
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 17.5311 1.74441 0.872206 0.489138i $$-0.162689\pi$$
0.872206 + 0.489138i $$0.162689\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 5.53113 0.537231
$$107$$ 14.5934 1.41080 0.705398 0.708811i $$-0.250768\pi$$
0.705398 + 0.708811i $$0.250768\pi$$
$$108$$ 0 0
$$109$$ −1.06226 −0.101746 −0.0508729 0.998705i $$-0.516200\pi$$
−0.0508729 + 0.998705i $$0.516200\pi$$
$$110$$ −1.53113 −0.145987
$$111$$ 0 0
$$112$$ 3.53113 0.333660
$$113$$ −16.5934 −1.56097 −0.780487 0.625172i $$-0.785029\pi$$
−0.780487 + 0.625172i $$0.785029\pi$$
$$114$$ 0 0
$$115$$ 1.53113 0.142779
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 7.06226 0.650134
$$119$$ 14.1245 1.29479
$$120$$ 0 0
$$121$$ −8.65564 −0.786877
$$122$$ 11.0623 1.00153
$$123$$ 0 0
$$124$$ 1.00000 0.0898027
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −9.06226 −0.804145 −0.402073 0.915608i $$-0.631710\pi$$
−0.402073 + 0.915608i $$0.631710\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −6.00000 −0.526235
$$131$$ −7.06226 −0.617032 −0.308516 0.951219i $$-0.599832\pi$$
−0.308516 + 0.951219i $$0.599832\pi$$
$$132$$ 0 0
$$133$$ −12.4689 −1.08119
$$134$$ 11.0623 0.955634
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ −0.937742 −0.0801167 −0.0400584 0.999197i $$-0.512754\pi$$
−0.0400584 + 0.999197i $$0.512754\pi$$
$$138$$ 0 0
$$139$$ 6.00000 0.508913 0.254457 0.967084i $$-0.418103\pi$$
0.254457 + 0.967084i $$0.418103\pi$$
$$140$$ 3.53113 0.298435
$$141$$ 0 0
$$142$$ −4.46887 −0.375019
$$143$$ 9.18677 0.768237
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0.468871 0.0388041
$$147$$ 0 0
$$148$$ 9.06226 0.744913
$$149$$ 10.4689 0.857643 0.428822 0.903389i $$-0.358929\pi$$
0.428822 + 0.903389i $$0.358929\pi$$
$$150$$ 0 0
$$151$$ −18.1245 −1.47495 −0.737476 0.675373i $$-0.763983\pi$$
−0.737476 + 0.675373i $$0.763983\pi$$
$$152$$ 3.53113 0.286412
$$153$$ 0 0
$$154$$ −5.40661 −0.435677
$$155$$ 1.00000 0.0803219
$$156$$ 0 0
$$157$$ −14.4689 −1.15474 −0.577371 0.816482i $$-0.695921\pi$$
−0.577371 + 0.816482i $$0.695921\pi$$
$$158$$ 0.468871 0.0373014
$$159$$ 0 0
$$160$$ −1.00000 −0.0790569
$$161$$ 5.40661 0.426101
$$162$$ 0 0
$$163$$ −11.0623 −0.866463 −0.433231 0.901283i $$-0.642627\pi$$
−0.433231 + 0.901283i $$0.642627\pi$$
$$164$$ 9.06226 0.707643
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ −0.593387 −0.0459176 −0.0229588 0.999736i $$-0.507309\pi$$
−0.0229588 + 0.999736i $$0.507309\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ −4.00000 −0.306786
$$171$$ 0 0
$$172$$ −0.468871 −0.0357511
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 3.53113 0.266928
$$176$$ 1.53113 0.115413
$$177$$ 0 0
$$178$$ 1.53113 0.114763
$$179$$ −13.0623 −0.976319 −0.488159 0.872754i $$-0.662332\pi$$
−0.488159 + 0.872754i $$0.662332\pi$$
$$180$$ 0 0
$$181$$ 26.5934 1.97667 0.988335 0.152293i $$-0.0486657\pi$$
0.988335 + 0.152293i $$0.0486657\pi$$
$$182$$ −21.1868 −1.57047
$$183$$ 0 0
$$184$$ −1.53113 −0.112876
$$185$$ 9.06226 0.666270
$$186$$ 0 0
$$187$$ 6.12452 0.447869
$$188$$ −11.0623 −0.806798
$$189$$ 0 0
$$190$$ 3.53113 0.256175
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 16.1245 1.15767
$$195$$ 0 0
$$196$$ 5.46887 0.390634
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ −0.468871 −0.0332374 −0.0166187 0.999862i $$-0.505290\pi$$
−0.0166187 + 0.999862i $$0.505290\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −17.5311 −1.23349
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 9.06226 0.632936
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 6.00000 0.416025
$$209$$ −5.40661 −0.373983
$$210$$ 0 0
$$211$$ 22.5934 1.55539 0.777696 0.628640i $$-0.216388\pi$$
0.777696 + 0.628640i $$0.216388\pi$$
$$212$$ −5.53113 −0.379879
$$213$$ 0 0
$$214$$ −14.5934 −0.997583
$$215$$ −0.468871 −0.0319767
$$216$$ 0 0
$$217$$ 3.53113 0.239709
$$218$$ 1.06226 0.0719452
$$219$$ 0 0
$$220$$ 1.53113 0.103229
$$221$$ 24.0000 1.61441
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ −3.53113 −0.235933
$$225$$ 0 0
$$226$$ 16.5934 1.10378
$$227$$ 16.4689 1.09308 0.546539 0.837434i $$-0.315945\pi$$
0.546539 + 0.837434i $$0.315945\pi$$
$$228$$ 0 0
$$229$$ 18.5934 1.22869 0.614343 0.789039i $$-0.289421\pi$$
0.614343 + 0.789039i $$0.289421\pi$$
$$230$$ −1.53113 −0.100960
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.53113 −0.624405 −0.312203 0.950016i $$-0.601067\pi$$
−0.312203 + 0.950016i $$0.601067\pi$$
$$234$$ 0 0
$$235$$ −11.0623 −0.721622
$$236$$ −7.06226 −0.459714
$$237$$ 0 0
$$238$$ −14.1245 −0.915556
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 8.65564 0.556406
$$243$$ 0 0
$$244$$ −11.0623 −0.708188
$$245$$ 5.46887 0.349393
$$246$$ 0 0
$$247$$ −21.1868 −1.34808
$$248$$ −1.00000 −0.0635001
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ 13.0623 0.824482 0.412241 0.911075i $$-0.364746\pi$$
0.412241 + 0.911075i $$0.364746\pi$$
$$252$$ 0 0
$$253$$ 2.34436 0.147388
$$254$$ 9.06226 0.568617
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 27.6556 1.72511 0.862556 0.505962i $$-0.168862\pi$$
0.862556 + 0.505962i $$0.168862\pi$$
$$258$$ 0 0
$$259$$ 32.0000 1.98838
$$260$$ 6.00000 0.372104
$$261$$ 0 0
$$262$$ 7.06226 0.436308
$$263$$ 6.93774 0.427800 0.213900 0.976856i $$-0.431383\pi$$
0.213900 + 0.976856i $$0.431383\pi$$
$$264$$ 0 0
$$265$$ −5.53113 −0.339775
$$266$$ 12.4689 0.764516
$$267$$ 0 0
$$268$$ −11.0623 −0.675735
$$269$$ −29.1868 −1.77955 −0.889774 0.456400i $$-0.849138\pi$$
−0.889774 + 0.456400i $$0.849138\pi$$
$$270$$ 0 0
$$271$$ −19.5311 −1.18643 −0.593216 0.805043i $$-0.702142\pi$$
−0.593216 + 0.805043i $$0.702142\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ 0.937742 0.0566511
$$275$$ 1.53113 0.0923305
$$276$$ 0 0
$$277$$ 1.06226 0.0638249 0.0319124 0.999491i $$-0.489840\pi$$
0.0319124 + 0.999491i $$0.489840\pi$$
$$278$$ −6.00000 −0.359856
$$279$$ 0 0
$$280$$ −3.53113 −0.211025
$$281$$ 1.06226 0.0633690 0.0316845 0.999498i $$-0.489913\pi$$
0.0316845 + 0.999498i $$0.489913\pi$$
$$282$$ 0 0
$$283$$ −12.0000 −0.713326 −0.356663 0.934233i $$-0.616086\pi$$
−0.356663 + 0.934233i $$0.616086\pi$$
$$284$$ 4.46887 0.265179
$$285$$ 0 0
$$286$$ −9.18677 −0.543225
$$287$$ 32.0000 1.88890
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −0.468871 −0.0274386
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ −7.06226 −0.411181
$$296$$ −9.06226 −0.526733
$$297$$ 0 0
$$298$$ −10.4689 −0.606445
$$299$$ 9.18677 0.531285
$$300$$ 0 0
$$301$$ −1.65564 −0.0954298
$$302$$ 18.1245 1.04295
$$303$$ 0 0
$$304$$ −3.53113 −0.202524
$$305$$ −11.0623 −0.633423
$$306$$ 0 0
$$307$$ 2.12452 0.121253 0.0606263 0.998161i $$-0.480690\pi$$
0.0606263 + 0.998161i $$0.480690\pi$$
$$308$$ 5.40661 0.308070
$$309$$ 0 0
$$310$$ −1.00000 −0.0567962
$$311$$ 8.00000 0.453638 0.226819 0.973937i $$-0.427167\pi$$
0.226819 + 0.973937i $$0.427167\pi$$
$$312$$ 0 0
$$313$$ 27.0623 1.52965 0.764825 0.644239i $$-0.222826\pi$$
0.764825 + 0.644239i $$0.222826\pi$$
$$314$$ 14.4689 0.816526
$$315$$ 0 0
$$316$$ −0.468871 −0.0263761
$$317$$ −29.0623 −1.63230 −0.816150 0.577841i $$-0.803895\pi$$
−0.816150 + 0.577841i $$0.803895\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −5.40661 −0.301299
$$323$$ −14.1245 −0.785909
$$324$$ 0 0
$$325$$ 6.00000 0.332820
$$326$$ 11.0623 0.612682
$$327$$ 0 0
$$328$$ −9.06226 −0.500379
$$329$$ −39.0623 −2.15357
$$330$$ 0 0
$$331$$ 29.0623 1.59741 0.798703 0.601725i $$-0.205520\pi$$
0.798703 + 0.601725i $$0.205520\pi$$
$$332$$ 8.00000 0.439057
$$333$$ 0 0
$$334$$ 0.593387 0.0324687
$$335$$ −11.0623 −0.604396
$$336$$ 0 0
$$337$$ 29.1868 1.58990 0.794952 0.606672i $$-0.207496\pi$$
0.794952 + 0.606672i $$0.207496\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 0 0
$$340$$ 4.00000 0.216930
$$341$$ 1.53113 0.0829153
$$342$$ 0 0
$$343$$ −5.40661 −0.291930
$$344$$ 0.468871 0.0252798
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ 22.1245 1.18771 0.593853 0.804573i $$-0.297606\pi$$
0.593853 + 0.804573i $$0.297606\pi$$
$$348$$ 0 0
$$349$$ 25.0623 1.34155 0.670776 0.741660i $$-0.265961\pi$$
0.670776 + 0.741660i $$0.265961\pi$$
$$350$$ −3.53113 −0.188747
$$351$$ 0 0
$$352$$ −1.53113 −0.0816094
$$353$$ 23.0623 1.22748 0.613740 0.789508i $$-0.289664\pi$$
0.613740 + 0.789508i $$0.289664\pi$$
$$354$$ 0 0
$$355$$ 4.46887 0.237183
$$356$$ −1.53113 −0.0811497
$$357$$ 0 0
$$358$$ 13.0623 0.690362
$$359$$ −11.5311 −0.608590 −0.304295 0.952578i $$-0.598421\pi$$
−0.304295 + 0.952578i $$0.598421\pi$$
$$360$$ 0 0
$$361$$ −6.53113 −0.343744
$$362$$ −26.5934 −1.39772
$$363$$ 0 0
$$364$$ 21.1868 1.11049
$$365$$ −0.468871 −0.0245418
$$366$$ 0 0
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ 1.53113 0.0798156
$$369$$ 0 0
$$370$$ −9.06226 −0.471124
$$371$$ −19.5311 −1.01401
$$372$$ 0 0
$$373$$ −10.4689 −0.542058 −0.271029 0.962571i $$-0.587364\pi$$
−0.271029 + 0.962571i $$0.587364\pi$$
$$374$$ −6.12452 −0.316691
$$375$$ 0 0
$$376$$ 11.0623 0.570492
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −2.59339 −0.133213 −0.0666067 0.997779i $$-0.521217\pi$$
−0.0666067 + 0.997779i $$0.521217\pi$$
$$380$$ −3.53113 −0.181143
$$381$$ 0 0
$$382$$ −8.00000 −0.409316
$$383$$ −13.0623 −0.667450 −0.333725 0.942670i $$-0.608306\pi$$
−0.333725 + 0.942670i $$0.608306\pi$$
$$384$$ 0 0
$$385$$ 5.40661 0.275547
$$386$$ −14.0000 −0.712581
$$387$$ 0 0
$$388$$ −16.1245 −0.818598
$$389$$ −27.0623 −1.37211 −0.686055 0.727549i $$-0.740659\pi$$
−0.686055 + 0.727549i $$0.740659\pi$$
$$390$$ 0 0
$$391$$ 6.12452 0.309730
$$392$$ −5.46887 −0.276220
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ −0.468871 −0.0235915
$$396$$ 0 0
$$397$$ −18.7179 −0.939425 −0.469712 0.882820i $$-0.655642\pi$$
−0.469712 + 0.882820i $$0.655642\pi$$
$$398$$ 0.468871 0.0235024
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 34.7179 1.73373 0.866865 0.498544i $$-0.166132\pi$$
0.866865 + 0.498544i $$0.166132\pi$$
$$402$$ 0 0
$$403$$ 6.00000 0.298881
$$404$$ 17.5311 0.872206
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 13.8755 0.687782
$$408$$ 0 0
$$409$$ −9.06226 −0.448100 −0.224050 0.974578i $$-0.571928\pi$$
−0.224050 + 0.974578i $$0.571928\pi$$
$$410$$ −9.06226 −0.447553
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −24.9377 −1.22711
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ −6.00000 −0.294174
$$417$$ 0 0
$$418$$ 5.40661 0.264446
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 38.2490 1.86414 0.932072 0.362273i $$-0.117999\pi$$
0.932072 + 0.362273i $$0.117999\pi$$
$$422$$ −22.5934 −1.09983
$$423$$ 0 0
$$424$$ 5.53113 0.268615
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ −39.0623 −1.89036
$$428$$ 14.5934 0.705398
$$429$$ 0 0
$$430$$ 0.468871 0.0226110
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ −1.40661 −0.0675975 −0.0337988 0.999429i $$-0.510761\pi$$
−0.0337988 + 0.999429i $$0.510761\pi$$
$$434$$ −3.53113 −0.169500
$$435$$ 0 0
$$436$$ −1.06226 −0.0508729
$$437$$ −5.40661 −0.258633
$$438$$ 0 0
$$439$$ −8.93774 −0.426575 −0.213288 0.976989i $$-0.568417\pi$$
−0.213288 + 0.976989i $$0.568417\pi$$
$$440$$ −1.53113 −0.0729937
$$441$$ 0 0
$$442$$ −24.0000 −1.14156
$$443$$ −38.5934 −1.83363 −0.916814 0.399316i $$-0.869248\pi$$
−0.916814 + 0.399316i $$0.869248\pi$$
$$444$$ 0 0
$$445$$ −1.53113 −0.0725825
$$446$$ −2.00000 −0.0947027
$$447$$ 0 0
$$448$$ 3.53113 0.166830
$$449$$ −28.1245 −1.32728 −0.663639 0.748053i $$-0.730989\pi$$
−0.663639 + 0.748053i $$0.730989\pi$$
$$450$$ 0 0
$$451$$ 13.8755 0.653371
$$452$$ −16.5934 −0.780487
$$453$$ 0 0
$$454$$ −16.4689 −0.772922
$$455$$ 21.1868 0.993251
$$456$$ 0 0
$$457$$ 31.0623 1.45303 0.726516 0.687150i $$-0.241138\pi$$
0.726516 + 0.687150i $$0.241138\pi$$
$$458$$ −18.5934 −0.868812
$$459$$ 0 0
$$460$$ 1.53113 0.0713893
$$461$$ −24.0000 −1.11779 −0.558896 0.829238i $$-0.688775\pi$$
−0.558896 + 0.829238i $$0.688775\pi$$
$$462$$ 0 0
$$463$$ −35.1868 −1.63527 −0.817634 0.575738i $$-0.804715\pi$$
−0.817634 + 0.575738i $$0.804715\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 9.53113 0.441521
$$467$$ −10.1245 −0.468507 −0.234253 0.972176i $$-0.575265\pi$$
−0.234253 + 0.972176i $$0.575265\pi$$
$$468$$ 0 0
$$469$$ −39.0623 −1.80373
$$470$$ 11.0623 0.510264
$$471$$ 0 0
$$472$$ 7.06226 0.325067
$$473$$ −0.717902 −0.0330092
$$474$$ 0 0
$$475$$ −3.53113 −0.162019
$$476$$ 14.1245 0.647396
$$477$$ 0 0
$$478$$ −8.00000 −0.365911
$$479$$ −41.6556 −1.90329 −0.951647 0.307192i $$-0.900611\pi$$
−0.951647 + 0.307192i $$0.900611\pi$$
$$480$$ 0 0
$$481$$ 54.3735 2.47922
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ −8.65564 −0.393438
$$485$$ −16.1245 −0.732177
$$486$$ 0 0
$$487$$ 6.93774 0.314379 0.157190 0.987568i $$-0.449757\pi$$
0.157190 + 0.987568i $$0.449757\pi$$
$$488$$ 11.0623 0.500765
$$489$$ 0 0
$$490$$ −5.46887 −0.247058
$$491$$ −16.5934 −0.748849 −0.374425 0.927257i $$-0.622160\pi$$
−0.374425 + 0.927257i $$0.622160\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 21.1868 0.953238
$$495$$ 0 0
$$496$$ 1.00000 0.0449013
$$497$$ 15.7802 0.707837
$$498$$ 0 0
$$499$$ 9.06226 0.405682 0.202841 0.979212i $$-0.434982\pi$$
0.202841 + 0.979212i $$0.434982\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −13.0623 −0.582997
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ 17.5311 0.780125
$$506$$ −2.34436 −0.104219
$$507$$ 0 0
$$508$$ −9.06226 −0.402073
$$509$$ −5.87548 −0.260426 −0.130213 0.991486i $$-0.541566\pi$$
−0.130213 + 0.991486i $$0.541566\pi$$
$$510$$ 0 0
$$511$$ −1.65564 −0.0732414
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −27.6556 −1.21984
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −16.9377 −0.744921
$$518$$ −32.0000 −1.40600
$$519$$ 0 0
$$520$$ −6.00000 −0.263117
$$521$$ −20.1245 −0.881671 −0.440836 0.897588i $$-0.645318\pi$$
−0.440836 + 0.897588i $$0.645318\pi$$
$$522$$ 0 0
$$523$$ −14.5934 −0.638124 −0.319062 0.947734i $$-0.603368\pi$$
−0.319062 + 0.947734i $$0.603368\pi$$
$$524$$ −7.06226 −0.308516
$$525$$ 0 0
$$526$$ −6.93774 −0.302500
$$527$$ 4.00000 0.174243
$$528$$ 0 0
$$529$$ −20.6556 −0.898071
$$530$$ 5.53113 0.240257
$$531$$ 0 0
$$532$$ −12.4689 −0.540594
$$533$$ 54.3735 2.35518
$$534$$ 0 0
$$535$$ 14.5934 0.630927
$$536$$ 11.0623 0.477817
$$537$$ 0 0
$$538$$ 29.1868 1.25833
$$539$$ 8.37355 0.360674
$$540$$ 0 0
$$541$$ 28.1245 1.20917 0.604584 0.796542i $$-0.293340\pi$$
0.604584 + 0.796542i $$0.293340\pi$$
$$542$$ 19.5311 0.838934
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ −1.06226 −0.0455021
$$546$$ 0 0
$$547$$ 41.1868 1.76102 0.880510 0.474028i $$-0.157201\pi$$
0.880510 + 0.474028i $$0.157201\pi$$
$$548$$ −0.937742 −0.0400584
$$549$$ 0 0
$$550$$ −1.53113 −0.0652876
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −1.65564 −0.0704052
$$554$$ −1.06226 −0.0451310
$$555$$ 0 0
$$556$$ 6.00000 0.254457
$$557$$ −45.7802 −1.93977 −0.969884 0.243568i $$-0.921682\pi$$
−0.969884 + 0.243568i $$0.921682\pi$$
$$558$$ 0 0
$$559$$ −2.81323 −0.118987
$$560$$ 3.53113 0.149217
$$561$$ 0 0
$$562$$ −1.06226 −0.0448086
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ −16.5934 −0.698089
$$566$$ 12.0000 0.504398
$$567$$ 0 0
$$568$$ −4.46887 −0.187510
$$569$$ −17.5311 −0.734943 −0.367472 0.930035i $$-0.619776\pi$$
−0.367472 + 0.930035i $$0.619776\pi$$
$$570$$ 0 0
$$571$$ 3.87548 0.162184 0.0810920 0.996707i $$-0.474159\pi$$
0.0810920 + 0.996707i $$0.474159\pi$$
$$572$$ 9.18677 0.384118
$$573$$ 0 0
$$574$$ −32.0000 −1.33565
$$575$$ 1.53113 0.0638525
$$576$$ 0 0
$$577$$ −11.1868 −0.465711 −0.232856 0.972511i $$-0.574807\pi$$
−0.232856 + 0.972511i $$0.574807\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 28.2490 1.17197
$$582$$ 0 0
$$583$$ −8.46887 −0.350745
$$584$$ 0.468871 0.0194020
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ −20.2490 −0.835767 −0.417883 0.908501i $$-0.637228\pi$$
−0.417883 + 0.908501i $$0.637228\pi$$
$$588$$ 0 0
$$589$$ −3.53113 −0.145498
$$590$$ 7.06226 0.290749
$$591$$ 0 0
$$592$$ 9.06226 0.372456
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ 14.1245 0.579049
$$596$$ 10.4689 0.428822
$$597$$ 0 0
$$598$$ −9.18677 −0.375675
$$599$$ −10.5934 −0.432834 −0.216417 0.976301i $$-0.569437\pi$$
−0.216417 + 0.976301i $$0.569437\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 1.65564 0.0674790
$$603$$ 0 0
$$604$$ −18.1245 −0.737476
$$605$$ −8.65564 −0.351902
$$606$$ 0 0
$$607$$ −38.5934 −1.56646 −0.783229 0.621734i $$-0.786429\pi$$
−0.783229 + 0.621734i $$0.786429\pi$$
$$608$$ 3.53113 0.143206
$$609$$ 0 0
$$610$$ 11.0623 0.447898
$$611$$ −66.3735 −2.68519
$$612$$ 0 0
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ −2.12452 −0.0857385
$$615$$ 0 0
$$616$$ −5.40661 −0.217839
$$617$$ −20.5934 −0.829059 −0.414529 0.910036i $$-0.636054\pi$$
−0.414529 + 0.910036i $$0.636054\pi$$
$$618$$ 0 0
$$619$$ −34.0000 −1.36658 −0.683288 0.730149i $$-0.739451\pi$$
−0.683288 + 0.730149i $$0.739451\pi$$
$$620$$ 1.00000 0.0401610
$$621$$ 0 0
$$622$$ −8.00000 −0.320771
$$623$$ −5.40661 −0.216611
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −27.0623 −1.08163
$$627$$ 0 0
$$628$$ −14.4689 −0.577371
$$629$$ 36.2490 1.44534
$$630$$ 0 0
$$631$$ −9.65564 −0.384385 −0.192193 0.981357i $$-0.561560\pi$$
−0.192193 + 0.981357i $$0.561560\pi$$
$$632$$ 0.468871 0.0186507
$$633$$ 0 0
$$634$$ 29.0623 1.15421
$$635$$ −9.06226 −0.359625
$$636$$ 0 0
$$637$$ 32.8132 1.30011
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 22.0000 0.868948 0.434474 0.900684i $$-0.356934\pi$$
0.434474 + 0.900684i $$0.356934\pi$$
$$642$$ 0 0
$$643$$ 6.59339 0.260018 0.130009 0.991513i $$-0.458499\pi$$
0.130009 + 0.991513i $$0.458499\pi$$
$$644$$ 5.40661 0.213050
$$645$$ 0 0
$$646$$ 14.1245 0.555722
$$647$$ −1.53113 −0.0601949 −0.0300974 0.999547i $$-0.509582\pi$$
−0.0300974 + 0.999547i $$0.509582\pi$$
$$648$$ 0 0
$$649$$ −10.8132 −0.424456
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ −11.0623 −0.433231
$$653$$ 26.2490 1.02720 0.513602 0.858029i $$-0.328311\pi$$
0.513602 + 0.858029i $$0.328311\pi$$
$$654$$ 0 0
$$655$$ −7.06226 −0.275945
$$656$$ 9.06226 0.353822
$$657$$ 0 0
$$658$$ 39.0623 1.52281
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ −29.0623 −1.12954
$$663$$ 0 0
$$664$$ −8.00000 −0.310460
$$665$$ −12.4689 −0.483522
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −0.593387 −0.0229588
$$669$$ 0 0
$$670$$ 11.0623 0.427372
$$671$$ −16.9377 −0.653874
$$672$$ 0 0
$$673$$ 7.06226 0.272230 0.136115 0.990693i $$-0.456538\pi$$
0.136115 + 0.990693i $$0.456538\pi$$
$$674$$ −29.1868 −1.12423
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ −3.40661 −0.130927 −0.0654634 0.997855i $$-0.520853\pi$$
−0.0654634 + 0.997855i $$0.520853\pi$$
$$678$$ 0 0
$$679$$ −56.9377 −2.18507
$$680$$ −4.00000 −0.153393
$$681$$ 0 0
$$682$$ −1.53113 −0.0586300
$$683$$ −15.5311 −0.594282 −0.297141 0.954834i $$-0.596033\pi$$
−0.297141 + 0.954834i $$0.596033\pi$$
$$684$$ 0 0
$$685$$ −0.937742 −0.0358293
$$686$$ 5.40661 0.206425
$$687$$ 0 0
$$688$$ −0.468871 −0.0178755
$$689$$ −33.1868 −1.26432
$$690$$ 0 0
$$691$$ −7.53113 −0.286498 −0.143249 0.989687i $$-0.545755\pi$$
−0.143249 + 0.989687i $$0.545755\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 0 0
$$694$$ −22.1245 −0.839835
$$695$$ 6.00000 0.227593
$$696$$ 0 0
$$697$$ 36.2490 1.37303
$$698$$ −25.0623 −0.948620
$$699$$ 0 0
$$700$$ 3.53113 0.133464
$$701$$ 21.5311 0.813220 0.406610 0.913602i $$-0.366711\pi$$
0.406610 + 0.913602i $$0.366711\pi$$
$$702$$ 0 0
$$703$$ −32.0000 −1.20690
$$704$$ 1.53113 0.0577066
$$705$$ 0 0
$$706$$ −23.0623 −0.867960
$$707$$ 61.9047 2.32816
$$708$$ 0 0
$$709$$ 25.4066 0.954165 0.477083 0.878858i $$-0.341694\pi$$
0.477083 + 0.878858i $$0.341694\pi$$
$$710$$ −4.46887 −0.167714
$$711$$ 0 0
$$712$$ 1.53113 0.0573815
$$713$$ 1.53113 0.0573412
$$714$$ 0 0
$$715$$ 9.18677 0.343566
$$716$$ −13.0623 −0.488159
$$717$$ 0 0
$$718$$ 11.5311 0.430338
$$719$$ 33.1868 1.23766 0.618829 0.785526i $$-0.287607\pi$$
0.618829 + 0.785526i $$0.287607\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 6.53113 0.243063
$$723$$ 0 0
$$724$$ 26.5934 0.988335
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −50.8424 −1.88564 −0.942820 0.333301i $$-0.891837\pi$$
−0.942820 + 0.333301i $$0.891837\pi$$
$$728$$ −21.1868 −0.785234
$$729$$ 0 0
$$730$$ 0.468871 0.0173537
$$731$$ −1.87548 −0.0693673
$$732$$ 0 0
$$733$$ 4.12452 0.152342 0.0761712 0.997095i $$-0.475730\pi$$
0.0761712 + 0.997095i $$0.475730\pi$$
$$734$$ 10.0000 0.369107
$$735$$ 0 0
$$736$$ −1.53113 −0.0564382
$$737$$ −16.9377 −0.623910
$$738$$ 0 0
$$739$$ 27.1868 1.00008 0.500041 0.866002i $$-0.333318\pi$$
0.500041 + 0.866002i $$0.333318\pi$$
$$740$$ 9.06226 0.333135
$$741$$ 0 0
$$742$$ 19.5311 0.717010
$$743$$ −11.6556 −0.427604 −0.213802 0.976877i $$-0.568585\pi$$
−0.213802 + 0.976877i $$0.568585\pi$$
$$744$$ 0 0
$$745$$ 10.4689 0.383550
$$746$$ 10.4689 0.383293
$$747$$ 0 0
$$748$$ 6.12452 0.223934
$$749$$ 51.5311 1.88291
$$750$$ 0 0
$$751$$ 47.0623 1.71733 0.858663 0.512540i $$-0.171296\pi$$
0.858663 + 0.512540i $$0.171296\pi$$
$$752$$ −11.0623 −0.403399
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −18.1245 −0.659619
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 2.59339 0.0941960
$$759$$ 0 0
$$760$$ 3.53113 0.128088
$$761$$ 12.5934 0.456510 0.228255 0.973601i $$-0.426698\pi$$
0.228255 + 0.973601i $$0.426698\pi$$
$$762$$ 0 0
$$763$$ −3.75097 −0.135794
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 13.0623 0.471959
$$767$$ −42.3735 −1.53002
$$768$$ 0 0
$$769$$ −28.5934 −1.03110 −0.515552 0.856858i $$-0.672413\pi$$
−0.515552 + 0.856858i $$0.672413\pi$$
$$770$$ −5.40661 −0.194841
$$771$$ 0 0
$$772$$ 14.0000 0.503871
$$773$$ −14.4689 −0.520409 −0.260205 0.965554i $$-0.583790\pi$$
−0.260205 + 0.965554i $$0.583790\pi$$
$$774$$ 0 0
$$775$$ 1.00000 0.0359211
$$776$$ 16.1245 0.578836
$$777$$ 0 0
$$778$$ 27.0623 0.970229
$$779$$ −32.0000 −1.14652
$$780$$ 0 0
$$781$$ 6.84242 0.244841
$$782$$ −6.12452 −0.219012
$$783$$ 0 0
$$784$$ 5.46887 0.195317
$$785$$ −14.4689 −0.516416
$$786$$ 0 0
$$787$$ −28.4689 −1.01481 −0.507403 0.861709i $$-0.669394\pi$$
−0.507403 + 0.861709i $$0.669394\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 0 0
$$790$$ 0.468871 0.0166817
$$791$$ −58.5934 −2.08334
$$792$$ 0 0
$$793$$ −66.3735 −2.35699
$$794$$ 18.7179 0.664273
$$795$$ 0 0
$$796$$ −0.468871 −0.0166187
$$797$$ 20.1245 0.712847 0.356423 0.934325i $$-0.383996\pi$$
0.356423 + 0.934325i $$0.383996\pi$$
$$798$$ 0 0
$$799$$ −44.2490 −1.56542
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ −34.7179 −1.22593
$$803$$ −0.717902 −0.0253342
$$804$$ 0 0
$$805$$ 5.40661 0.190558
$$806$$ −6.00000 −0.211341
$$807$$ 0 0
$$808$$ −17.5311 −0.616743
$$809$$ 8.59339 0.302127 0.151064 0.988524i $$-0.451730\pi$$
0.151064 + 0.988524i $$0.451730\pi$$
$$810$$ 0 0
$$811$$ −39.7802 −1.39687 −0.698435 0.715673i $$-0.746120\pi$$
−0.698435 + 0.715673i $$0.746120\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −13.8755 −0.486335
$$815$$ −11.0623 −0.387494
$$816$$ 0 0
$$817$$ 1.65564 0.0579237
$$818$$ 9.06226 0.316854
$$819$$ 0 0
$$820$$ 9.06226 0.316468
$$821$$ −9.87548 −0.344657 −0.172328 0.985040i $$-0.555129\pi$$
−0.172328 + 0.985040i $$0.555129\pi$$
$$822$$ 0 0
$$823$$ 7.87548 0.274522 0.137261 0.990535i $$-0.456170\pi$$
0.137261 + 0.990535i $$0.456170\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 24.9377 0.867695
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ 0 0
$$829$$ −9.65564 −0.335354 −0.167677 0.985842i $$-0.553627\pi$$
−0.167677 + 0.985842i $$0.553627\pi$$
$$830$$ −8.00000 −0.277684
$$831$$ 0 0
$$832$$ 6.00000 0.208013
$$833$$ 21.8755 0.757941
$$834$$ 0 0
$$835$$ −0.593387 −0.0205350
$$836$$ −5.40661 −0.186992
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 54.8424 1.89337 0.946685 0.322160i $$-0.104409\pi$$
0.946685 + 0.322160i $$0.104409\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −38.2490 −1.31815
$$843$$ 0 0
$$844$$ 22.5934 0.777696
$$845$$ 23.0000 0.791224
$$846$$ 0 0
$$847$$ −30.5642 −1.05020
$$848$$ −5.53113 −0.189940
$$849$$ 0 0
$$850$$ −4.00000 −0.137199
$$851$$ 13.8755 0.475645
$$852$$ 0 0
$$853$$ 1.28210 0.0438982 0.0219491 0.999759i $$-0.493013\pi$$
0.0219491 + 0.999759i $$0.493013\pi$$
$$854$$ 39.0623 1.33668
$$855$$ 0 0
$$856$$ −14.5934 −0.498792
$$857$$ −14.0000 −0.478231 −0.239115 0.970991i $$-0.576857\pi$$
−0.239115 + 0.970991i $$0.576857\pi$$
$$858$$ 0 0
$$859$$ −6.93774 −0.236713 −0.118356 0.992971i $$-0.537763\pi$$
−0.118356 + 0.992971i $$0.537763\pi$$
$$860$$ −0.468871 −0.0159884
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ −41.5311 −1.41374 −0.706868 0.707345i $$-0.749893\pi$$
−0.706868 + 0.707345i $$0.749893\pi$$
$$864$$ 0 0
$$865$$ −14.0000 −0.476014
$$866$$ 1.40661 0.0477987
$$867$$ 0 0
$$868$$ 3.53113 0.119854
$$869$$ −0.717902 −0.0243532
$$870$$ 0 0
$$871$$ −66.3735 −2.24898
$$872$$ 1.06226 0.0359726
$$873$$ 0 0
$$874$$ 5.40661 0.182881
$$875$$ 3.53113 0.119374
$$876$$ 0 0
$$877$$ 44.1245 1.48998 0.744990 0.667076i $$-0.232454\pi$$
0.744990 + 0.667076i $$0.232454\pi$$
$$878$$ 8.93774 0.301634
$$879$$ 0 0
$$880$$ 1.53113 0.0516143
$$881$$ −36.1245 −1.21707 −0.608533 0.793529i $$-0.708242\pi$$
−0.608533 + 0.793529i $$0.708242\pi$$
$$882$$ 0 0
$$883$$ −53.6556 −1.80566 −0.902828 0.430002i $$-0.858513\pi$$
−0.902828 + 0.430002i $$0.858513\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 38.5934 1.29657
$$887$$ 25.1868 0.845689 0.422845 0.906202i $$-0.361032\pi$$
0.422845 + 0.906202i $$0.361032\pi$$
$$888$$ 0 0
$$889$$ −32.0000 −1.07325
$$890$$ 1.53113 0.0513236
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ 39.0623 1.30717
$$894$$ 0 0
$$895$$ −13.0623 −0.436623
$$896$$ −3.53113 −0.117967
$$897$$ 0 0
$$898$$ 28.1245 0.938527
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −22.1245 −0.737074
$$902$$ −13.8755 −0.462003
$$903$$ 0 0
$$904$$ 16.5934 0.551888
$$905$$ 26.5934 0.883994
$$906$$ 0 0
$$907$$ −32.2490 −1.07081 −0.535406 0.844595i $$-0.679841\pi$$
−0.535406 + 0.844595i $$0.679841\pi$$
$$908$$ 16.4689 0.546539
$$909$$ 0 0
$$910$$ −21.1868 −0.702335
$$911$$ 16.0000 0.530104 0.265052 0.964234i $$-0.414611\pi$$
0.265052 + 0.964234i $$0.414611\pi$$
$$912$$ 0 0
$$913$$ 12.2490 0.405384
$$914$$ −31.0623 −1.02745
$$915$$ 0 0
$$916$$ 18.5934 0.614343
$$917$$ −24.9377 −0.823517
$$918$$ 0 0
$$919$$ −8.93774 −0.294829 −0.147414 0.989075i $$-0.547095\pi$$
−0.147414 + 0.989075i $$0.547095\pi$$
$$920$$ −1.53113 −0.0504798
$$921$$ 0 0
$$922$$ 24.0000 0.790398
$$923$$ 26.8132 0.882568
$$924$$ 0 0
$$925$$ 9.06226 0.297965
$$926$$ 35.1868 1.15631
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −36.8424 −1.20876 −0.604380 0.796696i $$-0.706579\pi$$
−0.604380 + 0.796696i $$0.706579\pi$$
$$930$$ 0 0
$$931$$ −19.3113 −0.632902
$$932$$ −9.53113 −0.312203
$$933$$ 0 0
$$934$$ 10.1245 0.331284
$$935$$ 6.12452 0.200293
$$936$$ 0 0
$$937$$ −25.0623 −0.818748 −0.409374 0.912367i $$-0.634253\pi$$
−0.409374 + 0.912367i $$0.634253\pi$$
$$938$$ 39.0623 1.27543
$$939$$ 0 0
$$940$$ −11.0623 −0.360811
$$941$$ 21.8755 0.713120 0.356560 0.934272i $$-0.383949\pi$$
0.356560 + 0.934272i $$0.383949\pi$$
$$942$$ 0 0
$$943$$ 13.8755 0.451848
$$944$$ −7.06226 −0.229857
$$945$$ 0 0
$$946$$ 0.717902 0.0233410
$$947$$ −35.0623 −1.13937 −0.569685 0.821863i $$-0.692935\pi$$
−0.569685 + 0.821863i $$0.692935\pi$$
$$948$$ 0 0
$$949$$ −2.81323 −0.0913212
$$950$$ 3.53113 0.114565
$$951$$ 0 0
$$952$$ −14.1245 −0.457778
$$953$$ −54.1245 −1.75327 −0.876633 0.481161i $$-0.840215\pi$$
−0.876633 + 0.481161i $$0.840215\pi$$
$$954$$ 0 0
$$955$$ 8.00000 0.258874
$$956$$ 8.00000 0.258738
$$957$$ 0 0
$$958$$ 41.6556 1.34583
$$959$$ −3.31129 −0.106927
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ −54.3735 −1.75307
$$963$$ 0 0
$$964$$ 2.00000 0.0644157
$$965$$ 14.0000 0.450676
$$966$$ 0 0
$$967$$ −17.0623 −0.548685 −0.274343 0.961632i $$-0.588460\pi$$
−0.274343 + 0.961632i $$0.588460\pi$$
$$968$$ 8.65564 0.278203
$$969$$ 0 0
$$970$$ 16.1245 0.517727
$$971$$ 33.1868 1.06501 0.532507 0.846426i $$-0.321250\pi$$
0.532507 + 0.846426i $$0.321250\pi$$
$$972$$ 0 0
$$973$$ 21.1868 0.679217
$$974$$ −6.93774 −0.222300
$$975$$ 0 0
$$976$$ −11.0623 −0.354094
$$977$$ 38.0000 1.21573 0.607864 0.794041i $$-0.292027\pi$$
0.607864 + 0.794041i $$0.292027\pi$$
$$978$$ 0 0
$$979$$ −2.34436 −0.0749259
$$980$$ 5.46887 0.174697
$$981$$ 0 0
$$982$$ 16.5934 0.529516
$$983$$ −17.0623 −0.544202 −0.272101 0.962269i $$-0.587718\pi$$
−0.272101 + 0.962269i $$0.587718\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −21.1868 −0.674041
$$989$$ −0.717902 −0.0228280
$$990$$ 0 0
$$991$$ −24.4689 −0.777279 −0.388640 0.921390i $$-0.627055\pi$$
−0.388640 + 0.921390i $$0.627055\pi$$
$$992$$ −1.00000 −0.0317500
$$993$$ 0 0
$$994$$ −15.7802 −0.500516
$$995$$ −0.468871 −0.0148642
$$996$$ 0 0
$$997$$ 62.2490 1.97145 0.985723 0.168373i $$-0.0538514\pi$$
0.985723 + 0.168373i $$0.0538514\pi$$
$$998$$ −9.06226 −0.286861
$$999$$ 0 0
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 2790.2.a.bf.1.2 2
3.2 odd 2 930.2.a.q.1.2 2
12.11 even 2 7440.2.a.bd.1.1 2
15.2 even 4 4650.2.d.bg.3349.4 4
15.8 even 4 4650.2.d.bg.3349.1 4
15.14 odd 2 4650.2.a.bz.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.q.1.2 2 3.2 odd 2
2790.2.a.bf.1.2 2 1.1 even 1 trivial
4650.2.a.bz.1.1 2 15.14 odd 2
4650.2.d.bg.3349.1 4 15.8 even 4
4650.2.d.bg.3349.4 4 15.2 even 4
7440.2.a.bd.1.1 2 12.11 even 2