# Properties

 Label 1521.2.b.l.1351.4 Level $1521$ Weight $2$ Character 1521.1351 Analytic conductor $12.145$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.b (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: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 169) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.4 Root $$1.80194i$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1351 Dual form 1521.2.b.l.1351.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.554958i q^{2} +1.69202 q^{4} +2.80194i q^{5} +2.69202i q^{7} +2.04892i q^{8} +O(q^{10})$$ $$q+0.554958i q^{2} +1.69202 q^{4} +2.80194i q^{5} +2.69202i q^{7} +2.04892i q^{8} -1.55496 q^{10} -1.19806i q^{11} -1.49396 q^{14} +2.24698 q^{16} +1.13706 q^{17} +1.93900i q^{19} +4.74094i q^{20} +0.664874 q^{22} -4.60388 q^{23} -2.85086 q^{25} +4.55496i q^{28} +7.89977 q^{29} +5.89977i q^{31} +5.34481i q^{32} +0.631023i q^{34} -7.54288 q^{35} -0.951083i q^{37} -1.07606 q^{38} -5.74094 q^{40} -3.31767i q^{41} -7.15883 q^{43} -2.02715i q^{44} -2.55496i q^{46} -7.69202i q^{47} -0.246980 q^{49} -1.58211i q^{50} -5.87263 q^{53} +3.35690 q^{55} -5.51573 q^{56} +4.38404i q^{58} -0.0120816i q^{59} -8.03684 q^{61} -3.27413 q^{62} +1.52781 q^{64} -9.25667i q^{67} +1.92394 q^{68} -4.18598i q^{70} +13.7409i q^{71} -12.8170i q^{73} +0.527811 q^{74} +3.28083i q^{76} +3.22521 q^{77} +0.807315 q^{79} +6.29590i q^{80} +1.84117 q^{82} +16.3327i q^{83} +3.18598i q^{85} -3.97285i q^{86} +2.45473 q^{88} -14.7289i q^{89} -7.78986 q^{92} +4.26875 q^{94} -5.43296 q^{95} +3.13169i q^{97} -0.137063i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q - 10 q^{10} + 10 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{22} - 10 q^{23} + 10 q^{25} + 2 q^{29} - 8 q^{35} + 24 q^{38} - 6 q^{40} - 26 q^{43} + 8 q^{49} - 2 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{61} + 2 q^{62} + 22 q^{64} + 42 q^{68} + 16 q^{74} + 16 q^{77} - 10 q^{79} + 28 q^{82} - 30 q^{88} + 10 q^{94} + 6 q^{95}+O(q^{100})$$ 6 * q - 10 * q^10 + 10 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^22 - 10 * q^23 + 10 * q^25 + 2 * q^29 - 8 * q^35 + 24 * q^38 - 6 * q^40 - 26 * q^43 + 8 * q^49 - 2 * q^53 + 12 * q^55 - 8 * q^56 + 8 * q^61 + 2 * q^62 + 22 * q^64 + 42 * q^68 + 16 * q^74 + 16 * q^77 - 10 * q^79 + 28 * q^82 - 30 * q^88 + 10 * q^94 + 6 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.554958i 0.392415i 0.980562 + 0.196207i $$0.0628626\pi$$
−0.980562 + 0.196207i $$0.937137\pi$$
$$3$$ 0 0
$$4$$ 1.69202 0.846011
$$5$$ 2.80194i 1.25306i 0.779395 + 0.626532i $$0.215526\pi$$
−0.779395 + 0.626532i $$0.784474\pi$$
$$6$$ 0 0
$$7$$ 2.69202i 1.01749i 0.860918 + 0.508744i $$0.169890\pi$$
−0.860918 + 0.508744i $$0.830110\pi$$
$$8$$ 2.04892i 0.724402i
$$9$$ 0 0
$$10$$ −1.55496 −0.491721
$$11$$ − 1.19806i − 0.361229i −0.983554 0.180615i $$-0.942191\pi$$
0.983554 0.180615i $$-0.0578087\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −1.49396 −0.399277
$$15$$ 0 0
$$16$$ 2.24698 0.561745
$$17$$ 1.13706 0.275778 0.137889 0.990448i $$-0.455968\pi$$
0.137889 + 0.990448i $$0.455968\pi$$
$$18$$ 0 0
$$19$$ 1.93900i 0.444837i 0.974951 + 0.222419i $$0.0713952\pi$$
−0.974951 + 0.222419i $$0.928605\pi$$
$$20$$ 4.74094i 1.06011i
$$21$$ 0 0
$$22$$ 0.664874 0.141752
$$23$$ −4.60388 −0.959974 −0.479987 0.877275i $$-0.659359\pi$$
−0.479987 + 0.877275i $$0.659359\pi$$
$$24$$ 0 0
$$25$$ −2.85086 −0.570171
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 4.55496i 0.860806i
$$29$$ 7.89977 1.46695 0.733475 0.679716i $$-0.237897\pi$$
0.733475 + 0.679716i $$0.237897\pi$$
$$30$$ 0 0
$$31$$ 5.89977i 1.05963i 0.848113 + 0.529815i $$0.177739\pi$$
−0.848113 + 0.529815i $$0.822261\pi$$
$$32$$ 5.34481i 0.944839i
$$33$$ 0 0
$$34$$ 0.631023i 0.108219i
$$35$$ −7.54288 −1.27498
$$36$$ 0 0
$$37$$ − 0.951083i − 0.156357i −0.996939 0.0781785i $$-0.975090\pi$$
0.996939 0.0781785i $$-0.0249104\pi$$
$$38$$ −1.07606 −0.174561
$$39$$ 0 0
$$40$$ −5.74094 −0.907722
$$41$$ − 3.31767i − 0.518133i −0.965860 0.259066i $$-0.916585\pi$$
0.965860 0.259066i $$-0.0834148\pi$$
$$42$$ 0 0
$$43$$ −7.15883 −1.09171 −0.545856 0.837879i $$-0.683795\pi$$
−0.545856 + 0.837879i $$0.683795\pi$$
$$44$$ − 2.02715i − 0.305604i
$$45$$ 0 0
$$46$$ − 2.55496i − 0.376708i
$$47$$ − 7.69202i − 1.12200i −0.827817 0.560998i $$-0.810417\pi$$
0.827817 0.560998i $$-0.189583\pi$$
$$48$$ 0 0
$$49$$ −0.246980 −0.0352828
$$50$$ − 1.58211i − 0.223743i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −5.87263 −0.806667 −0.403334 0.915053i $$-0.632149\pi$$
−0.403334 + 0.915053i $$0.632149\pi$$
$$54$$ 0 0
$$55$$ 3.35690 0.452644
$$56$$ −5.51573 −0.737070
$$57$$ 0 0
$$58$$ 4.38404i 0.575653i
$$59$$ − 0.0120816i − 0.00157289i −1.00000 0.000786444i $$-0.999750\pi$$
1.00000 0.000786444i $$-0.000250333\pi$$
$$60$$ 0 0
$$61$$ −8.03684 −1.02901 −0.514506 0.857487i $$-0.672025\pi$$
−0.514506 + 0.857487i $$0.672025\pi$$
$$62$$ −3.27413 −0.415815
$$63$$ 0 0
$$64$$ 1.52781 0.190976
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 9.25667i − 1.13088i −0.824789 0.565441i $$-0.808706\pi$$
0.824789 0.565441i $$-0.191294\pi$$
$$68$$ 1.92394 0.233311
$$69$$ 0 0
$$70$$ − 4.18598i − 0.500320i
$$71$$ 13.7409i 1.63075i 0.578934 + 0.815375i $$0.303469\pi$$
−0.578934 + 0.815375i $$0.696531\pi$$
$$72$$ 0 0
$$73$$ − 12.8170i − 1.50012i −0.661372 0.750058i $$-0.730025\pi$$
0.661372 0.750058i $$-0.269975\pi$$
$$74$$ 0.527811 0.0613568
$$75$$ 0 0
$$76$$ 3.28083i 0.376337i
$$77$$ 3.22521 0.367547
$$78$$ 0 0
$$79$$ 0.807315 0.0908300 0.0454150 0.998968i $$-0.485539\pi$$
0.0454150 + 0.998968i $$0.485539\pi$$
$$80$$ 6.29590i 0.703903i
$$81$$ 0 0
$$82$$ 1.84117 0.203323
$$83$$ 16.3327i 1.79275i 0.443296 + 0.896375i $$0.353809\pi$$
−0.443296 + 0.896375i $$0.646191\pi$$
$$84$$ 0 0
$$85$$ 3.18598i 0.345568i
$$86$$ − 3.97285i − 0.428404i
$$87$$ 0 0
$$88$$ 2.45473 0.261675
$$89$$ − 14.7289i − 1.56126i −0.624996 0.780628i $$-0.714899\pi$$
0.624996 0.780628i $$-0.285101\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −7.78986 −0.812149
$$93$$ 0 0
$$94$$ 4.26875 0.440288
$$95$$ −5.43296 −0.557410
$$96$$ 0 0
$$97$$ 3.13169i 0.317975i 0.987281 + 0.158987i $$0.0508229\pi$$
−0.987281 + 0.158987i $$0.949177\pi$$
$$98$$ − 0.137063i − 0.0138455i
$$99$$ 0 0
$$100$$ −4.82371 −0.482371
$$101$$ 5.29052 0.526426 0.263213 0.964738i $$-0.415218\pi$$
0.263213 + 0.964738i $$0.415218\pi$$
$$102$$ 0 0
$$103$$ 13.5308 1.33323 0.666614 0.745403i $$-0.267743\pi$$
0.666614 + 0.745403i $$0.267743\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ − 3.25906i − 0.316548i
$$107$$ −5.63102 −0.544371 −0.272186 0.962245i $$-0.587747\pi$$
−0.272186 + 0.962245i $$0.587747\pi$$
$$108$$ 0 0
$$109$$ 4.17629i 0.400016i 0.979794 + 0.200008i $$0.0640969\pi$$
−0.979794 + 0.200008i $$0.935903\pi$$
$$110$$ 1.86294i 0.177624i
$$111$$ 0 0
$$112$$ 6.04892i 0.571569i
$$113$$ −7.64310 −0.719003 −0.359501 0.933145i $$-0.617053\pi$$
−0.359501 + 0.933145i $$0.617053\pi$$
$$114$$ 0 0
$$115$$ − 12.8998i − 1.20291i
$$116$$ 13.3666 1.24106
$$117$$ 0 0
$$118$$ 0.00670477 0.000617224 0
$$119$$ 3.06100i 0.280601i
$$120$$ 0 0
$$121$$ 9.56465 0.869513
$$122$$ − 4.46011i − 0.403799i
$$123$$ 0 0
$$124$$ 9.98254i 0.896459i
$$125$$ 6.02177i 0.538604i
$$126$$ 0 0
$$127$$ −6.77777 −0.601430 −0.300715 0.953714i $$-0.597225\pi$$
−0.300715 + 0.953714i $$0.597225\pi$$
$$128$$ 11.5375i 1.01978i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 13.6799 1.19522 0.597611 0.801786i $$-0.296117\pi$$
0.597611 + 0.801786i $$0.296117\pi$$
$$132$$ 0 0
$$133$$ −5.21983 −0.452617
$$134$$ 5.13706 0.443775
$$135$$ 0 0
$$136$$ 2.32975i 0.199774i
$$137$$ 12.9879i 1.10963i 0.831973 + 0.554816i $$0.187211\pi$$
−0.831973 + 0.554816i $$0.812789\pi$$
$$138$$ 0 0
$$139$$ 12.0465 1.02177 0.510886 0.859648i $$-0.329317\pi$$
0.510886 + 0.859648i $$0.329317\pi$$
$$140$$ −12.7627 −1.07865
$$141$$ 0 0
$$142$$ −7.62565 −0.639930
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 22.1347i 1.83818i
$$146$$ 7.11290 0.588668
$$147$$ 0 0
$$148$$ − 1.60925i − 0.132280i
$$149$$ 0.740939i 0.0607001i 0.999539 + 0.0303500i $$0.00966220\pi$$
−0.999539 + 0.0303500i $$0.990338\pi$$
$$150$$ 0 0
$$151$$ 19.0737i 1.55219i 0.630614 + 0.776097i $$0.282803\pi$$
−0.630614 + 0.776097i $$0.717197\pi$$
$$152$$ −3.97285 −0.322241
$$153$$ 0 0
$$154$$ 1.78986i 0.144231i
$$155$$ −16.5308 −1.32779
$$156$$ 0 0
$$157$$ −4.02177 −0.320972 −0.160486 0.987038i $$-0.551306\pi$$
−0.160486 + 0.987038i $$0.551306\pi$$
$$158$$ 0.448026i 0.0356430i
$$159$$ 0 0
$$160$$ −14.9758 −1.18394
$$161$$ − 12.3937i − 0.976763i
$$162$$ 0 0
$$163$$ − 15.1371i − 1.18563i −0.805340 0.592813i $$-0.798017\pi$$
0.805340 0.592813i $$-0.201983\pi$$
$$164$$ − 5.61356i − 0.438346i
$$165$$ 0 0
$$166$$ −9.06398 −0.703502
$$167$$ − 6.26337i − 0.484674i −0.970192 0.242337i $$-0.922086\pi$$
0.970192 0.242337i $$-0.0779140\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −1.76809 −0.135606
$$171$$ 0 0
$$172$$ −12.1129 −0.923600
$$173$$ 16.3913 1.24621 0.623105 0.782138i $$-0.285871\pi$$
0.623105 + 0.782138i $$0.285871\pi$$
$$174$$ 0 0
$$175$$ − 7.67456i − 0.580142i
$$176$$ − 2.69202i − 0.202919i
$$177$$ 0 0
$$178$$ 8.17390 0.612660
$$179$$ −2.45473 −0.183475 −0.0917376 0.995783i $$-0.529242\pi$$
−0.0917376 + 0.995783i $$0.529242\pi$$
$$180$$ 0 0
$$181$$ −11.8073 −0.877631 −0.438815 0.898577i $$-0.644602\pi$$
−0.438815 + 0.898577i $$0.644602\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ − 9.43296i − 0.695407i
$$185$$ 2.66487 0.195925
$$186$$ 0 0
$$187$$ − 1.36227i − 0.0996192i
$$188$$ − 13.0151i − 0.949221i
$$189$$ 0 0
$$190$$ − 3.01507i − 0.218736i
$$191$$ 8.99330 0.650732 0.325366 0.945588i $$-0.394512\pi$$
0.325366 + 0.945588i $$0.394512\pi$$
$$192$$ 0 0
$$193$$ − 13.5254i − 0.973581i −0.873519 0.486790i $$-0.838168\pi$$
0.873519 0.486790i $$-0.161832\pi$$
$$194$$ −1.73795 −0.124778
$$195$$ 0 0
$$196$$ −0.417895 −0.0298496
$$197$$ − 12.9758i − 0.924490i −0.886752 0.462245i $$-0.847044\pi$$
0.886752 0.462245i $$-0.152956\pi$$
$$198$$ 0 0
$$199$$ 13.5864 0.963116 0.481558 0.876414i $$-0.340071\pi$$
0.481558 + 0.876414i $$0.340071\pi$$
$$200$$ − 5.84117i − 0.413033i
$$201$$ 0 0
$$202$$ 2.93602i 0.206577i
$$203$$ 21.2664i 1.49261i
$$204$$ 0 0
$$205$$ 9.29590 0.649254
$$206$$ 7.50902i 0.523179i
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2.32304 0.160688
$$210$$ 0 0
$$211$$ 10.4601 0.720103 0.360052 0.932932i $$-0.382759\pi$$
0.360052 + 0.932932i $$0.382759\pi$$
$$212$$ −9.93661 −0.682449
$$213$$ 0 0
$$214$$ − 3.12498i − 0.213619i
$$215$$ − 20.0586i − 1.36799i
$$216$$ 0 0
$$217$$ −15.8823 −1.07816
$$218$$ −2.31767 −0.156972
$$219$$ 0 0
$$220$$ 5.67994 0.382941
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 11.4058i − 0.763790i −0.924206 0.381895i $$-0.875272\pi$$
0.924206 0.381895i $$-0.124728\pi$$
$$224$$ −14.3884 −0.961362
$$225$$ 0 0
$$226$$ − 4.24160i − 0.282147i
$$227$$ − 10.6407i − 0.706249i −0.935576 0.353124i $$-0.885119\pi$$
0.935576 0.353124i $$-0.114881\pi$$
$$228$$ 0 0
$$229$$ 1.13946i 0.0752974i 0.999291 + 0.0376487i $$0.0119868\pi$$
−0.999291 + 0.0376487i $$0.988013\pi$$
$$230$$ 7.15883 0.472040
$$231$$ 0 0
$$232$$ 16.1860i 1.06266i
$$233$$ −10.8509 −0.710863 −0.355432 0.934702i $$-0.615666\pi$$
−0.355432 + 0.934702i $$0.615666\pi$$
$$234$$ 0 0
$$235$$ 21.5526 1.40593
$$236$$ − 0.0204423i − 0.00133068i
$$237$$ 0 0
$$238$$ −1.69873 −0.110112
$$239$$ 11.9293i 0.771643i 0.922573 + 0.385822i $$0.126082\pi$$
−0.922573 + 0.385822i $$0.873918\pi$$
$$240$$ 0 0
$$241$$ 3.64848i 0.235019i 0.993072 + 0.117510i $$0.0374911\pi$$
−0.993072 + 0.117510i $$0.962509\pi$$
$$242$$ 5.30798i 0.341210i
$$243$$ 0 0
$$244$$ −13.5985 −0.870555
$$245$$ − 0.692021i − 0.0442116i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −12.0881 −0.767598
$$249$$ 0 0
$$250$$ −3.34183 −0.211356
$$251$$ 1.37329 0.0866813 0.0433406 0.999060i $$-0.486200\pi$$
0.0433406 + 0.999060i $$0.486200\pi$$
$$252$$ 0 0
$$253$$ 5.51573i 0.346771i
$$254$$ − 3.76138i − 0.236010i
$$255$$ 0 0
$$256$$ −3.34721 −0.209200
$$257$$ 29.4359 1.83616 0.918082 0.396391i $$-0.129737\pi$$
0.918082 + 0.396391i $$0.129737\pi$$
$$258$$ 0 0
$$259$$ 2.56033 0.159091
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 7.59179i 0.469023i
$$263$$ −10.6963 −0.659564 −0.329782 0.944057i $$-0.606975\pi$$
−0.329782 + 0.944057i $$0.606975\pi$$
$$264$$ 0 0
$$265$$ − 16.4547i − 1.01081i
$$266$$ − 2.89679i − 0.177613i
$$267$$ 0 0
$$268$$ − 15.6625i − 0.956738i
$$269$$ 10.1860 0.621050 0.310525 0.950565i $$-0.399495\pi$$
0.310525 + 0.950565i $$0.399495\pi$$
$$270$$ 0 0
$$271$$ 29.4523i 1.78910i 0.446966 + 0.894551i $$0.352505\pi$$
−0.446966 + 0.894551i $$0.647495\pi$$
$$272$$ 2.55496 0.154917
$$273$$ 0 0
$$274$$ −7.20775 −0.435436
$$275$$ 3.41550i 0.205963i
$$276$$ 0 0
$$277$$ 10.2446 0.615538 0.307769 0.951461i $$-0.400418\pi$$
0.307769 + 0.951461i $$0.400418\pi$$
$$278$$ 6.68532i 0.400959i
$$279$$ 0 0
$$280$$ − 15.4547i − 0.923597i
$$281$$ − 11.5646i − 0.689889i −0.938623 0.344944i $$-0.887898\pi$$
0.938623 0.344944i $$-0.112102\pi$$
$$282$$ 0 0
$$283$$ 30.7090 1.82546 0.912730 0.408562i $$-0.133970\pi$$
0.912730 + 0.408562i $$0.133970\pi$$
$$284$$ 23.2500i 1.37963i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.93123 0.527194
$$288$$ 0 0
$$289$$ −15.7071 −0.923946
$$290$$ −12.2838 −0.721330
$$291$$ 0 0
$$292$$ − 21.6866i − 1.26911i
$$293$$ − 18.6082i − 1.08710i −0.839376 0.543551i $$-0.817079\pi$$
0.839376 0.543551i $$-0.182921\pi$$
$$294$$ 0 0
$$295$$ 0.0338518 0.00197093
$$296$$ 1.94869 0.113265
$$297$$ 0 0
$$298$$ −0.411190 −0.0238196
$$299$$ 0 0
$$300$$ 0 0
$$301$$ − 19.2717i − 1.11080i
$$302$$ −10.5851 −0.609103
$$303$$ 0 0
$$304$$ 4.35690i 0.249885i
$$305$$ − 22.5187i − 1.28942i
$$306$$ 0 0
$$307$$ − 8.94438i − 0.510483i −0.966877 0.255241i $$-0.917845\pi$$
0.966877 0.255241i $$-0.0821549\pi$$
$$308$$ 5.45712 0.310948
$$309$$ 0 0
$$310$$ − 9.17390i − 0.521042i
$$311$$ 21.0398 1.19306 0.596529 0.802591i $$-0.296546\pi$$
0.596529 + 0.802591i $$0.296546\pi$$
$$312$$ 0 0
$$313$$ −7.12737 −0.402863 −0.201432 0.979503i $$-0.564559\pi$$
−0.201432 + 0.979503i $$0.564559\pi$$
$$314$$ − 2.23191i − 0.125954i
$$315$$ 0 0
$$316$$ 1.36599 0.0768431
$$317$$ − 23.9651i − 1.34601i −0.739636 0.673007i $$-0.765003\pi$$
0.739636 0.673007i $$-0.234997\pi$$
$$318$$ 0 0
$$319$$ − 9.46442i − 0.529906i
$$320$$ 4.28083i 0.239306i
$$321$$ 0 0
$$322$$ 6.87800 0.383296
$$323$$ 2.20477i 0.122677i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 8.40044 0.465257
$$327$$ 0 0
$$328$$ 6.79763 0.375336
$$329$$ 20.7071 1.14162
$$330$$ 0 0
$$331$$ 2.89546i 0.159149i 0.996829 + 0.0795745i $$0.0253561\pi$$
−0.996829 + 0.0795745i $$0.974644\pi$$
$$332$$ 27.6353i 1.51669i
$$333$$ 0 0
$$334$$ 3.47591 0.190193
$$335$$ 25.9366 1.41707
$$336$$ 0 0
$$337$$ 3.10560 0.169173 0.0845865 0.996416i $$-0.473043\pi$$
0.0845865 + 0.996416i $$0.473043\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 5.39075i 0.292354i
$$341$$ 7.06829 0.382770
$$342$$ 0 0
$$343$$ 18.1793i 0.981589i
$$344$$ − 14.6679i − 0.790838i
$$345$$ 0 0
$$346$$ 9.09651i 0.489031i
$$347$$ 11.3787 0.610839 0.305419 0.952218i $$-0.401203\pi$$
0.305419 + 0.952218i $$0.401203\pi$$
$$348$$ 0 0
$$349$$ 3.34721i 0.179172i 0.995979 + 0.0895859i $$0.0285544\pi$$
−0.995979 + 0.0895859i $$0.971446\pi$$
$$350$$ 4.25906 0.227656
$$351$$ 0 0
$$352$$ 6.40342 0.341303
$$353$$ − 0.637727i − 0.0339428i −0.999856 0.0169714i $$-0.994598\pi$$
0.999856 0.0169714i $$-0.00540242\pi$$
$$354$$ 0 0
$$355$$ −38.5013 −2.04343
$$356$$ − 24.9215i − 1.32084i
$$357$$ 0 0
$$358$$ − 1.36227i − 0.0719983i
$$359$$ − 21.4590i − 1.13256i −0.824211 0.566282i $$-0.808381\pi$$
0.824211 0.566282i $$-0.191619\pi$$
$$360$$ 0 0
$$361$$ 15.2403 0.802120
$$362$$ − 6.55257i − 0.344395i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 35.9124 1.87974
$$366$$ 0 0
$$367$$ −9.38703 −0.489999 −0.244999 0.969523i $$-0.578788\pi$$
−0.244999 + 0.969523i $$0.578788\pi$$
$$368$$ −10.3448 −0.539261
$$369$$ 0 0
$$370$$ 1.47889i 0.0768840i
$$371$$ − 15.8092i − 0.820775i
$$372$$ 0 0
$$373$$ 27.7265 1.43562 0.717811 0.696238i $$-0.245144\pi$$
0.717811 + 0.696238i $$0.245144\pi$$
$$374$$ 0.756004 0.0390921
$$375$$ 0 0
$$376$$ 15.7603 0.812776
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 35.8702i 1.84253i 0.388935 + 0.921265i $$0.372843\pi$$
−0.388935 + 0.921265i $$0.627157\pi$$
$$380$$ −9.19269 −0.471575
$$381$$ 0 0
$$382$$ 4.99090i 0.255357i
$$383$$ − 4.85517i − 0.248087i −0.992277 0.124044i $$-0.960414\pi$$
0.992277 0.124044i $$-0.0395863\pi$$
$$384$$ 0 0
$$385$$ 9.03684i 0.460560i
$$386$$ 7.50604 0.382047
$$387$$ 0 0
$$388$$ 5.29888i 0.269010i
$$389$$ 2.38537 0.120943 0.0604716 0.998170i $$-0.480740\pi$$
0.0604716 + 0.998170i $$0.480740\pi$$
$$390$$ 0 0
$$391$$ −5.23490 −0.264740
$$392$$ − 0.506041i − 0.0255589i
$$393$$ 0 0
$$394$$ 7.20105 0.362783
$$395$$ 2.26205i 0.113816i
$$396$$ 0 0
$$397$$ 15.2664i 0.766196i 0.923708 + 0.383098i $$0.125143\pi$$
−0.923708 + 0.383098i $$0.874857\pi$$
$$398$$ 7.53989i 0.377941i
$$399$$ 0 0
$$400$$ −6.40581 −0.320291
$$401$$ 12.7584i 0.637124i 0.947902 + 0.318562i $$0.103200\pi$$
−0.947902 + 0.318562i $$0.896800\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 8.95167 0.445362
$$405$$ 0 0
$$406$$ −11.8019 −0.585720
$$407$$ −1.13946 −0.0564807
$$408$$ 0 0
$$409$$ − 25.3588i − 1.25391i −0.779054 0.626956i $$-0.784300\pi$$
0.779054 0.626956i $$-0.215700\pi$$
$$410$$ 5.15883i 0.254777i
$$411$$ 0 0
$$412$$ 22.8944 1.12793
$$413$$ 0.0325239 0.00160040
$$414$$ 0 0
$$415$$ −45.7633 −2.24643
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 1.28919i 0.0630565i
$$419$$ 11.6673 0.569983 0.284992 0.958530i $$-0.408009\pi$$
0.284992 + 0.958530i $$0.408009\pi$$
$$420$$ 0 0
$$421$$ − 8.29291i − 0.404172i −0.979368 0.202086i $$-0.935228\pi$$
0.979368 0.202086i $$-0.0647720\pi$$
$$422$$ 5.80492i 0.282579i
$$423$$ 0 0
$$424$$ − 12.0325i − 0.584351i
$$425$$ −3.24160 −0.157241
$$426$$ 0 0
$$427$$ − 21.6353i − 1.04701i
$$428$$ −9.52781 −0.460544
$$429$$ 0 0
$$430$$ 11.1317 0.536818
$$431$$ − 0.932296i − 0.0449071i −0.999748 0.0224536i $$-0.992852\pi$$
0.999748 0.0224536i $$-0.00714779\pi$$
$$432$$ 0 0
$$433$$ 13.3502 0.641569 0.320785 0.947152i $$-0.396053\pi$$
0.320785 + 0.947152i $$0.396053\pi$$
$$434$$ − 8.81402i − 0.423086i
$$435$$ 0 0
$$436$$ 7.06638i 0.338418i
$$437$$ − 8.92692i − 0.427032i
$$438$$ 0 0
$$439$$ −13.9922 −0.667813 −0.333906 0.942606i $$-0.608367\pi$$
−0.333906 + 0.942606i $$0.608367\pi$$
$$440$$ 6.87800i 0.327896i
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −23.7017 −1.12610 −0.563051 0.826422i $$-0.690373\pi$$
−0.563051 + 0.826422i $$0.690373\pi$$
$$444$$ 0 0
$$445$$ 41.2693 1.95635
$$446$$ 6.32975 0.299722
$$447$$ 0 0
$$448$$ 4.11290i 0.194316i
$$449$$ − 12.5864i − 0.593990i −0.954879 0.296995i $$-0.904016\pi$$
0.954879 0.296995i $$-0.0959844\pi$$
$$450$$ 0 0
$$451$$ −3.97477 −0.187165
$$452$$ −12.9323 −0.608284
$$453$$ 0 0
$$454$$ 5.90515 0.277142
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 33.6383i − 1.57353i −0.617250 0.786767i $$-0.711753\pi$$
0.617250 0.786767i $$-0.288247\pi$$
$$458$$ −0.632351 −0.0295478
$$459$$ 0 0
$$460$$ − 21.8267i − 1.01767i
$$461$$ 1.40283i 0.0653363i 0.999466 + 0.0326681i $$0.0104004\pi$$
−0.999466 + 0.0326681i $$0.989600\pi$$
$$462$$ 0 0
$$463$$ 15.2010i 0.706453i 0.935538 + 0.353226i $$0.114915\pi$$
−0.935538 + 0.353226i $$0.885085\pi$$
$$464$$ 17.7506 0.824052
$$465$$ 0 0
$$466$$ − 6.02177i − 0.278953i
$$467$$ −39.3414 −1.82050 −0.910250 0.414058i $$-0.864111\pi$$
−0.910250 + 0.414058i $$0.864111\pi$$
$$468$$ 0 0
$$469$$ 24.9191 1.15066
$$470$$ 11.9608i 0.551709i
$$471$$ 0 0
$$472$$ 0.0247542 0.00113940
$$473$$ 8.57673i 0.394358i
$$474$$ 0 0
$$475$$ − 5.52781i − 0.253633i
$$476$$ 5.17928i 0.237392i
$$477$$ 0 0
$$478$$ −6.62027 −0.302804
$$479$$ − 22.3690i − 1.02206i −0.859561 0.511032i $$-0.829263\pi$$
0.859561 0.511032i $$-0.170737\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −2.02475 −0.0922250
$$483$$ 0 0
$$484$$ 16.1836 0.735618
$$485$$ −8.77479 −0.398443
$$486$$ 0 0
$$487$$ 22.9205i 1.03863i 0.854584 + 0.519313i $$0.173812\pi$$
−0.854584 + 0.519313i $$0.826188\pi$$
$$488$$ − 16.4668i − 0.745418i
$$489$$ 0 0
$$490$$ 0.384043 0.0173493
$$491$$ 1.84356 0.0831987 0.0415993 0.999134i $$-0.486755\pi$$
0.0415993 + 0.999134i $$0.486755\pi$$
$$492$$ 0 0
$$493$$ 8.98254 0.404553
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 13.2567i 0.595242i
$$497$$ −36.9909 −1.65927
$$498$$ 0 0
$$499$$ − 12.0344i − 0.538736i −0.963037 0.269368i $$-0.913185\pi$$
0.963037 0.269368i $$-0.0868147\pi$$
$$500$$ 10.1890i 0.455664i
$$501$$ 0 0
$$502$$ 0.762118i 0.0340150i
$$503$$ 30.5056 1.36018 0.680088 0.733130i $$-0.261942\pi$$
0.680088 + 0.733130i $$0.261942\pi$$
$$504$$ 0 0
$$505$$ 14.8237i 0.659646i
$$506$$ −3.06100 −0.136078
$$507$$ 0 0
$$508$$ −11.4681 −0.508816
$$509$$ 1.51142i 0.0669924i 0.999439 + 0.0334962i $$0.0106642\pi$$
−0.999439 + 0.0334962i $$0.989336\pi$$
$$510$$ 0 0
$$511$$ 34.5036 1.52635
$$512$$ 21.2174i 0.937687i
$$513$$ 0 0
$$514$$ 16.3357i 0.720538i
$$515$$ 37.9124i 1.67062i
$$516$$ 0 0
$$517$$ −9.21552 −0.405298
$$518$$ 1.42088i 0.0624298i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 5.64012 0.247098 0.123549 0.992338i $$-0.460572\pi$$
0.123549 + 0.992338i $$0.460572\pi$$
$$522$$ 0 0
$$523$$ −31.7506 −1.38836 −0.694179 0.719802i $$-0.744232\pi$$
−0.694179 + 0.719802i $$0.744232\pi$$
$$524$$ 23.1468 1.01117
$$525$$ 0 0
$$526$$ − 5.93602i − 0.258823i
$$527$$ 6.70841i 0.292223i
$$528$$ 0 0
$$529$$ −1.80433 −0.0784492
$$530$$ 9.13169 0.396655
$$531$$ 0 0
$$532$$ −8.83207 −0.382919
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 15.7778i − 0.682133i
$$536$$ 18.9661 0.819213
$$537$$ 0 0
$$538$$ 5.65279i 0.243709i
$$539$$ 0.295897i 0.0127452i
$$540$$ 0 0
$$541$$ − 24.3297i − 1.04602i −0.852327 0.523009i $$-0.824809\pi$$
0.852327 0.523009i $$-0.175191\pi$$
$$542$$ −16.3448 −0.702070
$$543$$ 0 0
$$544$$ 6.07739i 0.260566i
$$545$$ −11.7017 −0.501246
$$546$$ 0 0
$$547$$ −8.18896 −0.350135 −0.175067 0.984556i $$-0.556014\pi$$
−0.175067 + 0.984556i $$0.556014\pi$$
$$548$$ 21.9758i 0.938761i
$$549$$ 0 0
$$550$$ −1.89546 −0.0808227
$$551$$ 15.3177i 0.652555i
$$552$$ 0 0
$$553$$ 2.17331i 0.0924185i
$$554$$ 5.68532i 0.241546i
$$555$$ 0 0
$$556$$ 20.3830 0.864431
$$557$$ 25.3327i 1.07338i 0.843779 + 0.536691i $$0.180326\pi$$
−0.843779 + 0.536691i $$0.819674\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ −16.9487 −0.716213
$$561$$ 0 0
$$562$$ 6.41789 0.270723
$$563$$ −25.3937 −1.07022 −0.535109 0.844783i $$-0.679730\pi$$
−0.535109 + 0.844783i $$0.679730\pi$$
$$564$$ 0 0
$$565$$ − 21.4155i − 0.900957i
$$566$$ 17.0422i 0.716338i
$$567$$ 0 0
$$568$$ −28.1540 −1.18132
$$569$$ −31.1347 −1.30523 −0.652617 0.757688i $$-0.726329\pi$$
−0.652617 + 0.757688i $$0.726329\pi$$
$$570$$ 0 0
$$571$$ 20.5090 0.858276 0.429138 0.903239i $$-0.358817\pi$$
0.429138 + 0.903239i $$0.358817\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 4.95646i 0.206879i
$$575$$ 13.1250 0.547350
$$576$$ 0 0
$$577$$ 15.6890i 0.653143i 0.945172 + 0.326572i $$0.105893\pi$$
−0.945172 + 0.326572i $$0.894107\pi$$
$$578$$ − 8.71678i − 0.362570i
$$579$$ 0 0
$$580$$ 37.4523i 1.55512i
$$581$$ −43.9681 −1.82410
$$582$$ 0 0
$$583$$ 7.03577i 0.291392i
$$584$$ 26.2610 1.08669
$$585$$ 0 0
$$586$$ 10.3268 0.426595
$$587$$ − 30.5687i − 1.26171i −0.775903 0.630853i $$-0.782705\pi$$
0.775903 0.630853i $$-0.217295\pi$$
$$588$$ 0 0
$$589$$ −11.4397 −0.471363
$$590$$ 0.0187864i 0 0.000773422i
$$591$$ 0 0
$$592$$ − 2.13706i − 0.0878328i
$$593$$ − 29.6883i − 1.21915i −0.792727 0.609576i $$-0.791340\pi$$
0.792727 0.609576i $$-0.208660\pi$$
$$594$$ 0 0
$$595$$ −8.57673 −0.351612
$$596$$ 1.25368i 0.0513529i
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −24.2325 −0.990113 −0.495057 0.868861i $$-0.664853\pi$$
−0.495057 + 0.868861i $$0.664853\pi$$
$$600$$ 0 0
$$601$$ 16.4819 0.672310 0.336155 0.941807i $$-0.390873\pi$$
0.336155 + 0.941807i $$0.390873\pi$$
$$602$$ 10.6950 0.435896
$$603$$ 0 0
$$604$$ 32.2731i 1.31317i
$$605$$ 26.7995i 1.08956i
$$606$$ 0 0
$$607$$ 1.43190 0.0581188 0.0290594 0.999578i $$-0.490749\pi$$
0.0290594 + 0.999578i $$0.490749\pi$$
$$608$$ −10.3636 −0.420300
$$609$$ 0 0
$$610$$ 12.4969 0.505986
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 3.84846i 0.155438i 0.996975 + 0.0777190i $$0.0247637\pi$$
−0.996975 + 0.0777190i $$0.975236\pi$$
$$614$$ 4.96376 0.200321
$$615$$ 0 0
$$616$$ 6.60819i 0.266251i
$$617$$ 15.0388i 0.605437i 0.953080 + 0.302719i $$0.0978943\pi$$
−0.953080 + 0.302719i $$0.902106\pi$$
$$618$$ 0 0
$$619$$ − 12.8170i − 0.515159i −0.966257 0.257579i $$-0.917075\pi$$
0.966257 0.257579i $$-0.0829249\pi$$
$$620$$ −27.9705 −1.12332
$$621$$ 0 0
$$622$$ 11.6762i 0.468174i
$$623$$ 39.6504 1.58856
$$624$$ 0 0
$$625$$ −31.1269 −1.24508
$$626$$ − 3.95539i − 0.158089i
$$627$$ 0 0
$$628$$ −6.80492 −0.271546
$$629$$ − 1.08144i − 0.0431199i
$$630$$ 0 0
$$631$$ − 25.7517i − 1.02516i −0.858640 0.512579i $$-0.828690\pi$$
0.858640 0.512579i $$-0.171310\pi$$
$$632$$ 1.65412i 0.0657974i
$$633$$ 0 0
$$634$$ 13.2996 0.528195
$$635$$ − 18.9909i − 0.753631i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 5.25236 0.207943
$$639$$ 0 0
$$640$$ −32.3274 −1.27785
$$641$$ 24.4571 0.965998 0.482999 0.875621i $$-0.339547\pi$$
0.482999 + 0.875621i $$0.339547\pi$$
$$642$$ 0 0
$$643$$ − 9.97344i − 0.393314i −0.980472 0.196657i $$-0.936991\pi$$
0.980472 0.196657i $$-0.0630086\pi$$
$$644$$ − 20.9705i − 0.826352i
$$645$$ 0 0
$$646$$ −1.22355 −0.0481401
$$647$$ 11.8431 0.465600 0.232800 0.972525i $$-0.425211\pi$$
0.232800 + 0.972525i $$0.425211\pi$$
$$648$$ 0 0
$$649$$ −0.0144745 −0.000568173 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 25.6122i − 1.00305i
$$653$$ 7.47411 0.292484 0.146242 0.989249i $$-0.453282\pi$$
0.146242 + 0.989249i $$0.453282\pi$$
$$654$$ 0 0
$$655$$ 38.3303i 1.49769i
$$656$$ − 7.45473i − 0.291058i
$$657$$ 0 0
$$658$$ 11.4916i 0.447988i
$$659$$ −34.1739 −1.33123 −0.665613 0.746297i $$-0.731830\pi$$
−0.665613 + 0.746297i $$0.731830\pi$$
$$660$$ 0 0
$$661$$ − 33.6088i − 1.30723i −0.756827 0.653615i $$-0.773252\pi$$
0.756827 0.653615i $$-0.226748\pi$$
$$662$$ −1.60686 −0.0624524
$$663$$ 0 0
$$664$$ −33.4644 −1.29867
$$665$$ − 14.6256i − 0.567158i
$$666$$ 0 0
$$667$$ −36.3696 −1.40824
$$668$$ − 10.5978i − 0.410040i
$$669$$ 0 0
$$670$$ 14.3937i 0.556078i
$$671$$ 9.62863i 0.371709i
$$672$$ 0 0
$$673$$ −48.0320 −1.85150 −0.925750 0.378137i $$-0.876565\pi$$
−0.925750 + 0.378137i $$0.876565\pi$$
$$674$$ 1.72348i 0.0663860i
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 33.6582 1.29359 0.646794 0.762665i $$-0.276109\pi$$
0.646794 + 0.762665i $$0.276109\pi$$
$$678$$ 0 0
$$679$$ −8.43057 −0.323535
$$680$$ −6.52781 −0.250330
$$681$$ 0 0
$$682$$ 3.92261i 0.150204i
$$683$$ 15.9041i 0.608553i 0.952584 + 0.304276i $$0.0984147\pi$$
−0.952584 + 0.304276i $$0.901585\pi$$
$$684$$ 0 0
$$685$$ −36.3913 −1.39044
$$686$$ −10.0887 −0.385190
$$687$$ 0 0
$$688$$ −16.0858 −0.613264
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 33.1903i 1.26262i 0.775531 + 0.631309i $$0.217482\pi$$
−0.775531 + 0.631309i $$0.782518\pi$$
$$692$$ 27.7345 1.05431
$$693$$ 0 0
$$694$$ 6.31468i 0.239702i
$$695$$ 33.7536i 1.28035i
$$696$$ 0 0
$$697$$ − 3.77240i − 0.142890i
$$698$$ −1.85756 −0.0703097
$$699$$ 0 0
$$700$$ − 12.9855i − 0.490807i
$$701$$ −14.9129 −0.563253 −0.281627 0.959524i $$-0.590874\pi$$
−0.281627 + 0.959524i $$0.590874\pi$$
$$702$$ 0 0
$$703$$ 1.84415 0.0695534
$$704$$ − 1.83041i − 0.0689863i
$$705$$ 0 0
$$706$$ 0.353912 0.0133197
$$707$$ 14.2422i 0.535633i
$$708$$ 0 0
$$709$$ − 38.4312i − 1.44331i −0.692252 0.721656i $$-0.743381\pi$$
0.692252 0.721656i $$-0.256619\pi$$
$$710$$ − 21.3666i − 0.801874i
$$711$$ 0 0
$$712$$ 30.1782 1.13098
$$713$$ − 27.1618i − 1.01722i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −4.15346 −0.155222
$$717$$ 0 0
$$718$$ 11.9089 0.444435
$$719$$ −11.4373 −0.426538 −0.213269 0.976993i $$-0.568411\pi$$
−0.213269 + 0.976993i $$0.568411\pi$$
$$720$$ 0 0
$$721$$ 36.4252i 1.35654i
$$722$$ 8.45771i 0.314764i
$$723$$ 0 0
$$724$$ −19.9782 −0.742485
$$725$$ −22.5211 −0.836413
$$726$$ 0 0
$$727$$ −3.63640 −0.134867 −0.0674333 0.997724i $$-0.521481\pi$$
−0.0674333 + 0.997724i $$0.521481\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 19.9299i 0.737639i
$$731$$ −8.14005 −0.301071
$$732$$ 0 0
$$733$$ − 3.52217i − 0.130094i −0.997882 0.0650472i $$-0.979280\pi$$
0.997882 0.0650472i $$-0.0207198\pi$$
$$734$$ − 5.20941i − 0.192283i
$$735$$ 0 0
$$736$$ − 24.6069i − 0.907021i
$$737$$ −11.0901 −0.408508
$$738$$ 0 0
$$739$$ − 0.420288i − 0.0154605i −0.999970 0.00773027i $$-0.997539\pi$$
0.999970 0.00773027i $$-0.00246064\pi$$
$$740$$ 4.50902 0.165755
$$741$$ 0 0
$$742$$ 8.77346 0.322084
$$743$$ 25.3623i 0.930452i 0.885192 + 0.465226i $$0.154027\pi$$
−0.885192 + 0.465226i $$0.845973\pi$$
$$744$$ 0 0
$$745$$ −2.07606 −0.0760611
$$746$$ 15.3870i 0.563359i
$$747$$ 0 0
$$748$$ − 2.30499i − 0.0842790i
$$749$$ − 15.1588i − 0.553892i
$$750$$ 0 0
$$751$$ −0.650874 −0.0237507 −0.0118754 0.999929i $$-0.503780\pi$$
−0.0118754 + 0.999929i $$0.503780\pi$$
$$752$$ − 17.2838i − 0.630276i
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −53.4432 −1.94500
$$756$$ 0 0
$$757$$ −16.7909 −0.610276 −0.305138 0.952308i $$-0.598703\pi$$
−0.305138 + 0.952308i $$0.598703\pi$$
$$758$$ −19.9065 −0.723036
$$759$$ 0 0
$$760$$ − 11.1317i − 0.403789i
$$761$$ 30.9221i 1.12093i 0.828179 + 0.560463i $$0.189377\pi$$
−0.828179 + 0.560463i $$0.810623\pi$$
$$762$$ 0 0
$$763$$ −11.2427 −0.407012
$$764$$ 15.2168 0.550526
$$765$$ 0 0
$$766$$ 2.69441 0.0973531
$$767$$ 0 0
$$768$$ 0 0
$$769$$ − 43.7689i − 1.57835i −0.614169 0.789174i $$-0.710509\pi$$
0.614169 0.789174i $$-0.289491\pi$$
$$770$$ −5.01507 −0.180730
$$771$$ 0 0
$$772$$ − 22.8853i − 0.823660i
$$773$$ 42.4209i 1.52577i 0.646532 + 0.762886i $$0.276218\pi$$
−0.646532 + 0.762886i $$0.723782\pi$$
$$774$$ 0 0
$$775$$ − 16.8194i − 0.604171i
$$776$$ −6.41657 −0.230341
$$777$$ 0 0
$$778$$ 1.32378i 0.0474598i
$$779$$ 6.43296 0.230485
$$780$$ 0 0
$$781$$ 16.4625 0.589075
$$782$$ − 2.90515i − 0.103888i
$$783$$ 0 0
$$784$$ −0.554958 −0.0198199
$$785$$ − 11.2687i − 0.402199i
$$786$$ 0 0
$$787$$ 36.0116i 1.28368i 0.766841 + 0.641838i $$0.221828\pi$$
−0.766841 + 0.641838i $$0.778172\pi$$
$$788$$ − 21.9554i − 0.782129i
$$789$$ 0 0
$$790$$ −1.25534 −0.0446630
$$791$$ − 20.5754i − 0.731577i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −8.47219 −0.300667
$$795$$ 0 0
$$796$$ 22.9885 0.814806
$$797$$ −31.7101 −1.12323 −0.561614 0.827399i $$-0.689819\pi$$
−0.561614 + 0.827399i $$0.689819\pi$$
$$798$$ 0 0
$$799$$ − 8.74632i − 0.309422i
$$800$$ − 15.2373i − 0.538720i
$$801$$ 0 0
$$802$$ −7.08038 −0.250017
$$803$$ −15.3556 −0.541886
$$804$$ 0 0
$$805$$ 34.7265 1.22395
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 10.8398i 0.381344i
$$809$$ 45.2814 1.59201 0.796005 0.605290i $$-0.206943\pi$$
0.796005 + 0.605290i $$0.206943\pi$$
$$810$$ 0 0
$$811$$ − 42.8635i − 1.50514i −0.658511 0.752571i $$-0.728813\pi$$
0.658511 0.752571i $$-0.271187\pi$$
$$812$$ 35.9831i 1.26276i
$$813$$ 0 0
$$814$$ − 0.632351i − 0.0221639i
$$815$$ 42.4131 1.48567
$$816$$ 0 0
$$817$$ − 13.8810i − 0.485634i
$$818$$ 14.0731 0.492054
$$819$$ 0 0
$$820$$ 15.7289 0.549276
$$821$$ 7.82776i 0.273191i 0.990627 + 0.136595i $$0.0436160\pi$$
−0.990627 + 0.136595i $$0.956384\pi$$
$$822$$ 0 0
$$823$$ 36.7754 1.28191 0.640955 0.767579i $$-0.278539\pi$$
0.640955 + 0.767579i $$0.278539\pi$$
$$824$$ 27.7235i 0.965793i
$$825$$ 0 0
$$826$$ 0.0180494i 0 0.000628019i
$$827$$ 47.3293i 1.64580i 0.568186 + 0.822900i $$0.307645\pi$$
−0.568186 + 0.822900i $$0.692355\pi$$
$$828$$ 0 0
$$829$$ −25.2687 −0.877620 −0.438810 0.898580i $$-0.644600\pi$$
−0.438810 + 0.898580i $$0.644600\pi$$
$$830$$ − 25.3967i − 0.881533i
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −0.280831 −0.00973023
$$834$$ 0 0
$$835$$ 17.5496 0.607328
$$836$$ 3.93064 0.135944
$$837$$ 0 0
$$838$$ 6.47484i 0.223670i
$$839$$ − 37.6883i − 1.30114i −0.759444 0.650572i $$-0.774529\pi$$
0.759444 0.650572i $$-0.225471\pi$$
$$840$$ 0 0
$$841$$ 33.4064 1.15194
$$842$$ 4.60222 0.158603
$$843$$ 0 0
$$844$$ 17.6987 0.609215
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.7482i 0.884720i
$$848$$ −13.1957 −0.453141
$$849$$ 0 0
$$850$$ − 1.79895i − 0.0617036i
$$851$$ 4.37867i 0.150099i
$$852$$ 0 0
$$853$$ 31.0121i 1.06183i 0.847424 + 0.530917i $$0.178152\pi$$
−0.847424 + 0.530917i $$0.821848\pi$$
$$854$$ 12.0067 0.410861
$$855$$ 0 0
$$856$$ − 11.5375i − 0.394344i
$$857$$ 12.4692 0.425940 0.212970 0.977059i $$-0.431686\pi$$
0.212970 + 0.977059i $$0.431686\pi$$
$$858$$ 0 0
$$859$$ −17.3163 −0.590826 −0.295413 0.955370i $$-0.595457\pi$$
−0.295413 + 0.955370i $$0.595457\pi$$
$$860$$ − 33.9396i − 1.15733i
$$861$$ 0 0
$$862$$ 0.517385 0.0176222
$$863$$ 3.46383i 0.117910i 0.998261 + 0.0589550i $$0.0187769\pi$$
−0.998261 + 0.0589550i $$0.981223\pi$$
$$864$$ 0 0
$$865$$ 45.9275i 1.56158i
$$866$$ 7.40880i 0.251761i
$$867$$ 0 0
$$868$$ −26.8732 −0.912136
$$869$$ − 0.967213i − 0.0328105i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −8.55688 −0.289772
$$873$$ 0 0
$$874$$ 4.95407 0.167574
$$875$$ −16.2107 −0.548023
$$876$$ 0 0
$$877$$ 57.2549i 1.93336i 0.255989 + 0.966680i $$0.417599\pi$$
−0.255989 + 0.966680i $$0.582401\pi$$
$$878$$ − 7.76510i − 0.262059i
$$879$$ 0 0
$$880$$ 7.54288 0.254270
$$881$$ −43.1782 −1.45471 −0.727355 0.686261i $$-0.759251\pi$$
−0.727355 + 0.686261i $$0.759251\pi$$
$$882$$ 0 0
$$883$$ −49.9560 −1.68115 −0.840576 0.541693i $$-0.817784\pi$$
−0.840576 + 0.541693i $$0.817784\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ − 13.1535i − 0.441899i
$$887$$ −17.6746 −0.593454 −0.296727 0.954962i $$-0.595895\pi$$
−0.296727 + 0.954962i $$0.595895\pi$$
$$888$$ 0 0
$$889$$ − 18.2459i − 0.611948i
$$890$$ 22.9028i 0.767702i
$$891$$ 0 0
$$892$$ − 19.2989i − 0.646174i
$$893$$ 14.9148 0.499106
$$894$$ 0 0
$$895$$ − 6.87800i − 0.229906i
$$896$$ −31.0592 −1.03761
$$897$$ 0 0
$$898$$ 6.98493 0.233090
$$899$$ 46.6069i 1.55443i
$$900$$ 0 0
$$901$$ −6.67755 −0.222461
$$902$$ − 2.20583i − 0.0734462i
$$903$$ 0 0
$$904$$ − 15.6601i − 0.520847i
$$905$$ − 33.0834i − 1.09973i
$$906$$ 0 0
$$907$$ −7.73423 −0.256811 −0.128406 0.991722i $$-0.540986\pi$$
−0.128406 + 0.991722i $$0.540986\pi$$
$$908$$ − 18.0043i − 0.597494i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −39.6179 −1.31260 −0.656299 0.754501i $$-0.727879\pi$$
−0.656299 + 0.754501i $$0.727879\pi$$
$$912$$ 0 0
$$913$$ 19.5676 0.647594
$$914$$ 18.6679 0.617478
$$915$$ 0 0
$$916$$ 1.92798i 0.0637024i
$$917$$ 36.8267i 1.21612i
$$918$$ 0 0
$$919$$ 14.6213 0.482313 0.241157 0.970486i $$-0.422473\pi$$
0.241157 + 0.970486i $$0.422473\pi$$
$$920$$ 26.4306 0.871390
$$921$$ 0 0
$$922$$ −0.778512 −0.0256389
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 2.71140i 0.0891502i
$$926$$ −8.43594 −0.277222
$$927$$ 0 0
$$928$$ 42.2228i 1.38603i
$$929$$ − 3.55735i − 0.116713i −0.998296 0.0583565i $$-0.981414\pi$$
0.998296 0.0583565i $$-0.0185860\pi$$
$$930$$ 0 0
$$931$$ − 0.478894i − 0.0156951i
$$932$$ −18.3599 −0.601398
$$933$$ 0 0
$$934$$ − 21.8328i − 0.714391i
$$935$$ 3.81700 0.124829
$$936$$ 0 0
$$937$$ 34.5526 1.12878 0.564392 0.825507i $$-0.309111\pi$$
0.564392 + 0.825507i $$0.309111\pi$$
$$938$$ 13.8291i 0.451536i
$$939$$ 0 0
$$940$$ 36.4674 1.18944
$$941$$ − 20.6233i − 0.672299i −0.941809 0.336149i $$-0.890875\pi$$
0.941809 0.336149i $$-0.109125\pi$$
$$942$$ 0 0
$$943$$ 15.2741i 0.497394i
$$944$$ − 0.0271471i 0 0.000883562i
$$945$$ 0 0
$$946$$ −4.75973 −0.154752
$$947$$ 29.4999i 0.958619i 0.877646 + 0.479309i $$0.159113\pi$$
−0.877646 + 0.479309i $$0.840887\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 3.06770 0.0995295
$$951$$ 0 0
$$952$$ −6.27173 −0.203268
$$953$$ 26.2389 0.849963 0.424981 0.905202i $$-0.360281\pi$$
0.424981 + 0.905202i $$0.360281\pi$$
$$954$$ 0 0
$$955$$ 25.1987i 0.815409i
$$956$$ 20.1847i 0.652818i
$$957$$ 0 0
$$958$$ 12.4138 0.401073
$$959$$ −34.9638 −1.12904
$$960$$ 0 0
$$961$$ −3.80731 −0.122817
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 6.17331i 0.198829i
$$965$$ 37.8974 1.21996
$$966$$ 0 0
$$967$$ 17.5176i 0.563330i 0.959513 + 0.281665i $$0.0908866\pi$$
−0.959513 + 0.281665i $$0.909113\pi$$
$$968$$ 19.5972i 0.629877i
$$969$$ 0 0
$$970$$ − 4.86964i − 0.156355i
$$971$$ −20.5120 −0.658262 −0.329131 0.944284i $$-0.606756\pi$$
−0.329131 + 0.944284i $$0.606756\pi$$
$$972$$ 0 0
$$973$$ 32.4295i 1.03964i
$$974$$ −12.7199 −0.407572
$$975$$ 0 0
$$976$$ −18.0586 −0.578042
$$977$$ − 25.4450i − 0.814059i −0.913415 0.407030i $$-0.866565\pi$$
0.913415 0.407030i $$-0.133435\pi$$
$$978$$ 0 0
$$979$$ −17.6461 −0.563971
$$980$$ − 1.17092i − 0.0374035i
$$981$$ 0 0
$$982$$ 1.02310i 0.0326484i
$$983$$ − 39.5244i − 1.26063i −0.776339 0.630316i $$-0.782926\pi$$
0.776339 0.630316i $$-0.217074\pi$$
$$984$$ 0 0
$$985$$ 36.3575 1.15845
$$986$$ 4.98493i 0.158753i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 32.9584 1.04802
$$990$$ 0 0
$$991$$ −29.8377 −0.947826 −0.473913 0.880572i $$-0.657159\pi$$
−0.473913 + 0.880572i $$0.657159\pi$$
$$992$$ −31.5332 −1.00118
$$993$$ 0 0
$$994$$ − 20.5284i − 0.651121i
$$995$$ 38.0683i 1.20685i
$$996$$ 0 0
$$997$$ −4.93123 −0.156174 −0.0780868 0.996947i $$-0.524881\pi$$
−0.0780868 + 0.996947i $$0.524881\pi$$
$$998$$ 6.67861 0.211408
$$999$$ 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 1521.2.b.l.1351.4 6
3.2 odd 2 169.2.b.b.168.3 6
12.11 even 2 2704.2.f.o.337.1 6
13.5 odd 4 1521.2.a.r.1.2 3
13.8 odd 4 1521.2.a.o.1.2 3
13.12 even 2 inner 1521.2.b.l.1351.3 6
39.2 even 12 169.2.c.c.22.2 6
39.5 even 4 169.2.a.b.1.2 3
39.8 even 4 169.2.a.c.1.2 yes 3
39.11 even 12 169.2.c.b.22.2 6
39.17 odd 6 169.2.e.b.23.4 12
39.20 even 12 169.2.c.b.146.2 6
39.23 odd 6 169.2.e.b.147.3 12
39.29 odd 6 169.2.e.b.147.4 12
39.32 even 12 169.2.c.c.146.2 6
39.35 odd 6 169.2.e.b.23.3 12
39.38 odd 2 169.2.b.b.168.4 6
156.47 odd 4 2704.2.a.ba.1.1 3
156.83 odd 4 2704.2.a.z.1.1 3
156.155 even 2 2704.2.f.o.337.2 6
195.44 even 4 4225.2.a.bg.1.2 3
195.164 even 4 4225.2.a.bb.1.2 3
273.83 odd 4 8281.2.a.bf.1.2 3
273.125 odd 4 8281.2.a.bj.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 39.5 even 4
169.2.a.c.1.2 yes 3 39.8 even 4
169.2.b.b.168.3 6 3.2 odd 2
169.2.b.b.168.4 6 39.38 odd 2
169.2.c.b.22.2 6 39.11 even 12
169.2.c.b.146.2 6 39.20 even 12
169.2.c.c.22.2 6 39.2 even 12
169.2.c.c.146.2 6 39.32 even 12
169.2.e.b.23.3 12 39.35 odd 6
169.2.e.b.23.4 12 39.17 odd 6
169.2.e.b.147.3 12 39.23 odd 6
169.2.e.b.147.4 12 39.29 odd 6
1521.2.a.o.1.2 3 13.8 odd 4
1521.2.a.r.1.2 3 13.5 odd 4
1521.2.b.l.1351.3 6 13.12 even 2 inner
1521.2.b.l.1351.4 6 1.1 even 1 trivial
2704.2.a.z.1.1 3 156.83 odd 4
2704.2.a.ba.1.1 3 156.47 odd 4
2704.2.f.o.337.1 6 12.11 even 2
2704.2.f.o.337.2 6 156.155 even 2
4225.2.a.bb.1.2 3 195.164 even 4
4225.2.a.bg.1.2 3 195.44 even 4
8281.2.a.bf.1.2 3 273.83 odd 4
8281.2.a.bj.1.2 3 273.125 odd 4