# Properties

 Label 45.3.c.a.26.1 Level $45$ Weight $3$ Character 45.26 Analytic conductor $1.226$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,3,Mod(26,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("45.26");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 45.c (of order $$2$$, degree $$1$$, 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: $$1.22616118962$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 26.1 Root $$1.58114 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 45.26 Dual form 45.3.c.a.26.4

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.65028i q^{2} -9.32456 q^{4} -2.23607i q^{5} +7.16228 q^{7} +19.4361i q^{8} +O(q^{10})$$ $$q-3.65028i q^{2} -9.32456 q^{4} -2.23607i q^{5} +7.16228 q^{7} +19.4361i q^{8} -8.16228 q^{10} -5.42736i q^{11} +9.81139 q^{13} -26.1443i q^{14} +33.6491 q^{16} +12.2317i q^{17} +6.32456 q^{19} +20.8503i q^{20} -19.8114 q^{22} +12.0394i q^{23} -5.00000 q^{25} -35.8143i q^{26} -66.7851 q^{28} +44.9881i q^{29} -58.2719 q^{31} -45.0842i q^{32} +44.6491 q^{34} -16.0153i q^{35} +66.4605 q^{37} -23.0864i q^{38} +43.4605 q^{40} -16.4743i q^{41} -43.6228 q^{43} +50.6077i q^{44} +43.9473 q^{46} -40.0570i q^{47} +2.29822 q^{49} +18.2514i q^{50} -91.4868 q^{52} -13.2242i q^{53} -12.1359 q^{55} +139.207i q^{56} +164.219 q^{58} -25.1519i q^{59} -35.6754 q^{61} +212.709i q^{62} -29.9737 q^{64} -21.9389i q^{65} +26.7018 q^{67} -114.055i q^{68} -58.4605 q^{70} -92.7301i q^{71} +60.3246 q^{73} -242.600i q^{74} -58.9737 q^{76} -38.8723i q^{77} -96.2192 q^{79} -75.2417i q^{80} -60.1359 q^{82} +79.1215i q^{83} +27.3509 q^{85} +159.235i q^{86} +105.487 q^{88} +107.443i q^{89} +70.2719 q^{91} -112.262i q^{92} -146.219 q^{94} -14.1421i q^{95} -1.07900 q^{97} -8.38915i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{4} + 16 q^{7}+O(q^{10})$$ 4 * q - 12 * q^4 + 16 * q^7 $$4 q - 12 q^{4} + 16 q^{7} - 20 q^{10} - 24 q^{13} + 84 q^{16} - 16 q^{22} - 20 q^{25} - 128 q^{28} - 56 q^{31} + 128 q^{34} + 152 q^{37} + 60 q^{40} - 48 q^{43} + 24 q^{46} - 92 q^{49} - 328 q^{52} + 40 q^{55} + 328 q^{58} - 168 q^{61} - 44 q^{64} + 208 q^{67} - 120 q^{70} + 216 q^{73} - 160 q^{76} - 56 q^{79} - 152 q^{82} + 160 q^{85} + 384 q^{88} + 104 q^{91} - 256 q^{94} - 232 q^{97}+O(q^{100})$$ 4 * q - 12 * q^4 + 16 * q^7 - 20 * q^10 - 24 * q^13 + 84 * q^16 - 16 * q^22 - 20 * q^25 - 128 * q^28 - 56 * q^31 + 128 * q^34 + 152 * q^37 + 60 * q^40 - 48 * q^43 + 24 * q^46 - 92 * q^49 - 328 * q^52 + 40 * q^55 + 328 * q^58 - 168 * q^61 - 44 * q^64 + 208 * q^67 - 120 * q^70 + 216 * q^73 - 160 * q^76 - 56 * q^79 - 152 * q^82 + 160 * q^85 + 384 * q^88 + 104 * q^91 - 256 * q^94 - 232 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 3.65028i − 1.82514i −0.408920 0.912570i $$-0.634094\pi$$
0.408920 0.912570i $$-0.365906\pi$$
$$3$$ 0 0
$$4$$ −9.32456 −2.33114
$$5$$ − 2.23607i − 0.447214i
$$6$$ 0 0
$$7$$ 7.16228 1.02318 0.511591 0.859229i $$-0.329056\pi$$
0.511591 + 0.859229i $$0.329056\pi$$
$$8$$ 19.4361i 2.42952i
$$9$$ 0 0
$$10$$ −8.16228 −0.816228
$$11$$ − 5.42736i − 0.493396i −0.969092 0.246698i $$-0.920654\pi$$
0.969092 0.246698i $$-0.0793456\pi$$
$$12$$ 0 0
$$13$$ 9.81139 0.754722 0.377361 0.926066i $$-0.376832\pi$$
0.377361 + 0.926066i $$0.376832\pi$$
$$14$$ − 26.1443i − 1.86745i
$$15$$ 0 0
$$16$$ 33.6491 2.10307
$$17$$ 12.2317i 0.719511i 0.933047 + 0.359756i $$0.117140\pi$$
−0.933047 + 0.359756i $$0.882860\pi$$
$$18$$ 0 0
$$19$$ 6.32456 0.332871 0.166436 0.986052i $$-0.446774\pi$$
0.166436 + 0.986052i $$0.446774\pi$$
$$20$$ 20.8503i 1.04252i
$$21$$ 0 0
$$22$$ −19.8114 −0.900518
$$23$$ 12.0394i 0.523454i 0.965142 + 0.261727i $$0.0842920\pi$$
−0.965142 + 0.261727i $$0.915708\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −0.200000
$$26$$ − 35.8143i − 1.37747i
$$27$$ 0 0
$$28$$ −66.7851 −2.38518
$$29$$ 44.9881i 1.55131i 0.631155 + 0.775657i $$0.282581\pi$$
−0.631155 + 0.775657i $$0.717419\pi$$
$$30$$ 0 0
$$31$$ −58.2719 −1.87974 −0.939869 0.341535i $$-0.889053\pi$$
−0.939869 + 0.341535i $$0.889053\pi$$
$$32$$ − 45.0842i − 1.40888i
$$33$$ 0 0
$$34$$ 44.6491 1.31321
$$35$$ − 16.0153i − 0.457581i
$$36$$ 0 0
$$37$$ 66.4605 1.79623 0.898115 0.439761i $$-0.144937\pi$$
0.898115 + 0.439761i $$0.144937\pi$$
$$38$$ − 23.0864i − 0.607537i
$$39$$ 0 0
$$40$$ 43.4605 1.08651
$$41$$ − 16.4743i − 0.401813i −0.979610 0.200906i $$-0.935611\pi$$
0.979610 0.200906i $$-0.0643887\pi$$
$$42$$ 0 0
$$43$$ −43.6228 −1.01448 −0.507242 0.861804i $$-0.669335\pi$$
−0.507242 + 0.861804i $$0.669335\pi$$
$$44$$ 50.6077i 1.15018i
$$45$$ 0 0
$$46$$ 43.9473 0.955377
$$47$$ − 40.0570i − 0.852276i −0.904658 0.426138i $$-0.859874\pi$$
0.904658 0.426138i $$-0.140126\pi$$
$$48$$ 0 0
$$49$$ 2.29822 0.0469025
$$50$$ 18.2514i 0.365028i
$$51$$ 0 0
$$52$$ −91.4868 −1.75936
$$53$$ − 13.2242i − 0.249512i −0.992187 0.124756i $$-0.960185\pi$$
0.992187 0.124756i $$-0.0398149\pi$$
$$54$$ 0 0
$$55$$ −12.1359 −0.220654
$$56$$ 139.207i 2.48584i
$$57$$ 0 0
$$58$$ 164.219 2.83137
$$59$$ − 25.1519i − 0.426303i −0.977019 0.213151i $$-0.931627\pi$$
0.977019 0.213151i $$-0.0683728\pi$$
$$60$$ 0 0
$$61$$ −35.6754 −0.584843 −0.292422 0.956289i $$-0.594461\pi$$
−0.292422 + 0.956289i $$0.594461\pi$$
$$62$$ 212.709i 3.43079i
$$63$$ 0 0
$$64$$ −29.9737 −0.468339
$$65$$ − 21.9389i − 0.337522i
$$66$$ 0 0
$$67$$ 26.7018 0.398534 0.199267 0.979945i $$-0.436144\pi$$
0.199267 + 0.979945i $$0.436144\pi$$
$$68$$ − 114.055i − 1.67728i
$$69$$ 0 0
$$70$$ −58.4605 −0.835150
$$71$$ − 92.7301i − 1.30606i −0.757333 0.653029i $$-0.773498\pi$$
0.757333 0.653029i $$-0.226502\pi$$
$$72$$ 0 0
$$73$$ 60.3246 0.826364 0.413182 0.910649i $$-0.364417\pi$$
0.413182 + 0.910649i $$0.364417\pi$$
$$74$$ − 242.600i − 3.27837i
$$75$$ 0 0
$$76$$ −58.9737 −0.775969
$$77$$ − 38.8723i − 0.504834i
$$78$$ 0 0
$$79$$ −96.2192 −1.21796 −0.608982 0.793184i $$-0.708422\pi$$
−0.608982 + 0.793184i $$0.708422\pi$$
$$80$$ − 75.2417i − 0.940521i
$$81$$ 0 0
$$82$$ −60.1359 −0.733365
$$83$$ 79.1215i 0.953271i 0.879101 + 0.476635i $$0.158144\pi$$
−0.879101 + 0.476635i $$0.841856\pi$$
$$84$$ 0 0
$$85$$ 27.3509 0.321775
$$86$$ 159.235i 1.85157i
$$87$$ 0 0
$$88$$ 105.487 1.19871
$$89$$ 107.443i 1.20722i 0.797278 + 0.603612i $$0.206272\pi$$
−0.797278 + 0.603612i $$0.793728\pi$$
$$90$$ 0 0
$$91$$ 70.2719 0.772219
$$92$$ − 112.262i − 1.22024i
$$93$$ 0 0
$$94$$ −146.219 −1.55552
$$95$$ − 14.1421i − 0.148865i
$$96$$ 0 0
$$97$$ −1.07900 −0.0111237 −0.00556187 0.999985i $$-0.501770\pi$$
−0.00556187 + 0.999985i $$0.501770\pi$$
$$98$$ − 8.38915i − 0.0856036i
$$99$$ 0 0
$$100$$ 46.6228 0.466228
$$101$$ 170.282i 1.68596i 0.537942 + 0.842982i $$0.319202\pi$$
−0.537942 + 0.842982i $$0.680798\pi$$
$$102$$ 0 0
$$103$$ −128.460 −1.24719 −0.623595 0.781748i $$-0.714328\pi$$
−0.623595 + 0.781748i $$0.714328\pi$$
$$104$$ 190.695i 1.83361i
$$105$$ 0 0
$$106$$ −48.2719 −0.455395
$$107$$ − 76.3675i − 0.713715i −0.934159 0.356858i $$-0.883848\pi$$
0.934159 0.356858i $$-0.116152\pi$$
$$108$$ 0 0
$$109$$ −13.0790 −0.119991 −0.0599954 0.998199i $$-0.519109\pi$$
−0.0599954 + 0.998199i $$0.519109\pi$$
$$110$$ 44.2996i 0.402724i
$$111$$ 0 0
$$112$$ 241.004 2.15182
$$113$$ 20.7170i 0.183336i 0.995790 + 0.0916680i $$0.0292199\pi$$
−0.995790 + 0.0916680i $$0.970780\pi$$
$$114$$ 0 0
$$115$$ 26.9210 0.234096
$$116$$ − 419.494i − 3.61633i
$$117$$ 0 0
$$118$$ −91.8114 −0.778063
$$119$$ 87.6068i 0.736191i
$$120$$ 0 0
$$121$$ 91.5438 0.756560
$$122$$ 130.225i 1.06742i
$$123$$ 0 0
$$124$$ 543.359 4.38193
$$125$$ 11.1803i 0.0894427i
$$126$$ 0 0
$$127$$ −38.0306 −0.299454 −0.149727 0.988727i $$-0.547839\pi$$
−0.149727 + 0.988727i $$0.547839\pi$$
$$128$$ − 70.9246i − 0.554098i
$$129$$ 0 0
$$130$$ −80.0833 −0.616025
$$131$$ 83.9409i 0.640770i 0.947287 + 0.320385i $$0.103812\pi$$
−0.947287 + 0.320385i $$0.896188\pi$$
$$132$$ 0 0
$$133$$ 45.2982 0.340588
$$134$$ − 97.4690i − 0.727381i
$$135$$ 0 0
$$136$$ −237.737 −1.74806
$$137$$ − 15.5936i − 0.113822i −0.998379 0.0569109i $$-0.981875\pi$$
0.998379 0.0569109i $$-0.0181251\pi$$
$$138$$ 0 0
$$139$$ 67.8420 0.488072 0.244036 0.969766i $$-0.421529\pi$$
0.244036 + 0.969766i $$0.421529\pi$$
$$140$$ 149.336i 1.06669i
$$141$$ 0 0
$$142$$ −338.491 −2.38374
$$143$$ − 53.2499i − 0.372377i
$$144$$ 0 0
$$145$$ 100.596 0.693769
$$146$$ − 220.202i − 1.50823i
$$147$$ 0 0
$$148$$ −619.715 −4.18726
$$149$$ − 233.426i − 1.56662i −0.621634 0.783308i $$-0.713531\pi$$
0.621634 0.783308i $$-0.286469\pi$$
$$150$$ 0 0
$$151$$ 185.351 1.22749 0.613745 0.789505i $$-0.289662\pi$$
0.613745 + 0.789505i $$0.289662\pi$$
$$152$$ 122.925i 0.808716i
$$153$$ 0 0
$$154$$ −141.895 −0.921394
$$155$$ 130.300i 0.840645i
$$156$$ 0 0
$$157$$ −111.276 −0.708765 −0.354383 0.935100i $$-0.615309\pi$$
−0.354383 + 0.935100i $$0.615309\pi$$
$$158$$ 351.227i 2.22296i
$$159$$ 0 0
$$160$$ −100.811 −0.630071
$$161$$ 86.2298i 0.535589i
$$162$$ 0 0
$$163$$ 118.763 0.728607 0.364304 0.931280i $$-0.381307\pi$$
0.364304 + 0.931280i $$0.381307\pi$$
$$164$$ 153.616i 0.936682i
$$165$$ 0 0
$$166$$ 288.816 1.73985
$$167$$ − 221.194i − 1.32452i −0.749276 0.662258i $$-0.769598\pi$$
0.749276 0.662258i $$-0.230402\pi$$
$$168$$ 0 0
$$169$$ −72.7367 −0.430394
$$170$$ − 99.8384i − 0.587285i
$$171$$ 0 0
$$172$$ 406.763 2.36490
$$173$$ − 190.807i − 1.10293i −0.834198 0.551466i $$-0.814069\pi$$
0.834198 0.551466i $$-0.185931\pi$$
$$174$$ 0 0
$$175$$ −35.8114 −0.204637
$$176$$ − 182.626i − 1.03765i
$$177$$ 0 0
$$178$$ 392.197 2.20335
$$179$$ 58.1005i 0.324584i 0.986743 + 0.162292i $$0.0518886\pi$$
−0.986743 + 0.162292i $$0.948111\pi$$
$$180$$ 0 0
$$181$$ −162.921 −0.900116 −0.450058 0.892999i $$-0.648597\pi$$
−0.450058 + 0.892999i $$0.648597\pi$$
$$182$$ − 256.512i − 1.40941i
$$183$$ 0 0
$$184$$ −234.000 −1.27174
$$185$$ − 148.610i − 0.803298i
$$186$$ 0 0
$$187$$ 66.3858 0.355004
$$188$$ 373.513i 1.98677i
$$189$$ 0 0
$$190$$ −51.6228 −0.271699
$$191$$ − 100.062i − 0.523884i −0.965084 0.261942i $$-0.915637\pi$$
0.965084 0.261942i $$-0.0843630\pi$$
$$192$$ 0 0
$$193$$ −61.8947 −0.320698 −0.160349 0.987060i $$-0.551262\pi$$
−0.160349 + 0.987060i $$0.551262\pi$$
$$194$$ 3.93866i 0.0203024i
$$195$$ 0 0
$$196$$ −21.4299 −0.109336
$$197$$ 24.8791i 0.126290i 0.998004 + 0.0631449i $$0.0201130\pi$$
−0.998004 + 0.0631449i $$0.979887\pi$$
$$198$$ 0 0
$$199$$ −156.491 −0.786387 −0.393194 0.919456i $$-0.628630\pi$$
−0.393194 + 0.919456i $$0.628630\pi$$
$$200$$ − 97.1806i − 0.485903i
$$201$$ 0 0
$$202$$ 621.579 3.07712
$$203$$ 322.217i 1.58728i
$$204$$ 0 0
$$205$$ −36.8377 −0.179696
$$206$$ 468.917i 2.27630i
$$207$$ 0 0
$$208$$ 330.144 1.58723
$$209$$ − 34.3256i − 0.164237i
$$210$$ 0 0
$$211$$ 237.789 1.12696 0.563482 0.826128i $$-0.309461\pi$$
0.563482 + 0.826128i $$0.309461\pi$$
$$212$$ 123.309i 0.581648i
$$213$$ 0 0
$$214$$ −278.763 −1.30263
$$215$$ 97.5435i 0.453691i
$$216$$ 0 0
$$217$$ −417.359 −1.92332
$$218$$ 47.7420i 0.219000i
$$219$$ 0 0
$$220$$ 113.162 0.514374
$$221$$ 120.010i 0.543031i
$$222$$ 0 0
$$223$$ −182.302 −0.817500 −0.408750 0.912646i $$-0.634035\pi$$
−0.408750 + 0.912646i $$0.634035\pi$$
$$224$$ − 322.906i − 1.44154i
$$225$$ 0 0
$$226$$ 75.6228 0.334614
$$227$$ 406.078i 1.78889i 0.447180 + 0.894444i $$0.352428\pi$$
−0.447180 + 0.894444i $$0.647572\pi$$
$$228$$ 0 0
$$229$$ −27.2982 −0.119206 −0.0596031 0.998222i $$-0.518984\pi$$
−0.0596031 + 0.998222i $$0.518984\pi$$
$$230$$ − 98.2692i − 0.427257i
$$231$$ 0 0
$$232$$ −874.394 −3.76894
$$233$$ − 356.382i − 1.52954i −0.644306 0.764768i $$-0.722854\pi$$
0.644306 0.764768i $$-0.277146\pi$$
$$234$$ 0 0
$$235$$ −89.5701 −0.381149
$$236$$ 234.530i 0.993771i
$$237$$ 0 0
$$238$$ 319.789 1.34365
$$239$$ − 271.690i − 1.13678i −0.822760 0.568389i $$-0.807567\pi$$
0.822760 0.568389i $$-0.192433\pi$$
$$240$$ 0 0
$$241$$ −224.438 −0.931280 −0.465640 0.884974i $$-0.654176\pi$$
−0.465640 + 0.884974i $$0.654176\pi$$
$$242$$ − 334.161i − 1.38083i
$$243$$ 0 0
$$244$$ 332.658 1.36335
$$245$$ − 5.13898i − 0.0209754i
$$246$$ 0 0
$$247$$ 62.0527 0.251225
$$248$$ − 1132.58i − 4.56685i
$$249$$ 0 0
$$250$$ 40.8114 0.163246
$$251$$ − 318.775i − 1.27002i −0.772504 0.635010i $$-0.780996\pi$$
0.772504 0.635010i $$-0.219004\pi$$
$$252$$ 0 0
$$253$$ 65.3423 0.258270
$$254$$ 138.822i 0.546545i
$$255$$ 0 0
$$256$$ −378.789 −1.47965
$$257$$ 371.975i 1.44738i 0.690128 + 0.723688i $$0.257554\pi$$
−0.690128 + 0.723688i $$0.742446\pi$$
$$258$$ 0 0
$$259$$ 476.009 1.83787
$$260$$ 204.571i 0.786811i
$$261$$ 0 0
$$262$$ 306.408 1.16950
$$263$$ 238.549i 0.907031i 0.891248 + 0.453515i $$0.149830\pi$$
−0.891248 + 0.453515i $$0.850170\pi$$
$$264$$ 0 0
$$265$$ −29.5701 −0.111585
$$266$$ − 165.351i − 0.621621i
$$267$$ 0 0
$$268$$ −248.982 −0.929038
$$269$$ 125.871i 0.467922i 0.972246 + 0.233961i $$0.0751688\pi$$
−0.972246 + 0.233961i $$0.924831\pi$$
$$270$$ 0 0
$$271$$ −258.649 −0.954425 −0.477212 0.878788i $$-0.658353\pi$$
−0.477212 + 0.878788i $$0.658353\pi$$
$$272$$ 411.585i 1.51318i
$$273$$ 0 0
$$274$$ −56.9210 −0.207741
$$275$$ 27.1368i 0.0986793i
$$276$$ 0 0
$$277$$ 227.715 0.822074 0.411037 0.911619i $$-0.365167\pi$$
0.411037 + 0.911619i $$0.365167\pi$$
$$278$$ − 247.642i − 0.890800i
$$279$$ 0 0
$$280$$ 311.276 1.11170
$$281$$ − 241.384i − 0.859016i −0.903063 0.429508i $$-0.858687\pi$$
0.903063 0.429508i $$-0.141313\pi$$
$$282$$ 0 0
$$283$$ 208.333 0.736159 0.368080 0.929794i $$-0.380015\pi$$
0.368080 + 0.929794i $$0.380015\pi$$
$$284$$ 864.667i 3.04460i
$$285$$ 0 0
$$286$$ −194.377 −0.679641
$$287$$ − 117.994i − 0.411128i
$$288$$ 0 0
$$289$$ 139.386 0.482304
$$290$$ − 367.205i − 1.26623i
$$291$$ 0 0
$$292$$ −562.500 −1.92637
$$293$$ − 201.693i − 0.688372i −0.938901 0.344186i $$-0.888155\pi$$
0.938901 0.344186i $$-0.111845\pi$$
$$294$$ 0 0
$$295$$ −56.2413 −0.190648
$$296$$ 1291.73i 4.36397i
$$297$$ 0 0
$$298$$ −852.070 −2.85929
$$299$$ 118.124i 0.395062i
$$300$$ 0 0
$$301$$ −312.438 −1.03800
$$302$$ − 676.583i − 2.24034i
$$303$$ 0 0
$$304$$ 212.816 0.700052
$$305$$ 79.7727i 0.261550i
$$306$$ 0 0
$$307$$ 342.824 1.11669 0.558346 0.829608i $$-0.311436\pi$$
0.558346 + 0.829608i $$0.311436\pi$$
$$308$$ 362.466i 1.17684i
$$309$$ 0 0
$$310$$ 475.631 1.53429
$$311$$ − 217.640i − 0.699807i −0.936786 0.349903i $$-0.886214\pi$$
0.936786 0.349903i $$-0.113786\pi$$
$$312$$ 0 0
$$313$$ 281.895 0.900622 0.450311 0.892872i $$-0.351313\pi$$
0.450311 + 0.892872i $$0.351313\pi$$
$$314$$ 406.189i 1.29360i
$$315$$ 0 0
$$316$$ 897.201 2.83925
$$317$$ 15.4013i 0.0485847i 0.999705 + 0.0242923i $$0.00773325\pi$$
−0.999705 + 0.0242923i $$0.992267\pi$$
$$318$$ 0 0
$$319$$ 244.167 0.765412
$$320$$ 67.0232i 0.209447i
$$321$$ 0 0
$$322$$ 314.763 0.977525
$$323$$ 77.3600i 0.239505i
$$324$$ 0 0
$$325$$ −49.0569 −0.150944
$$326$$ − 433.518i − 1.32981i
$$327$$ 0 0
$$328$$ 320.197 0.976211
$$329$$ − 286.899i − 0.872034i
$$330$$ 0 0
$$331$$ 375.517 1.13449 0.567247 0.823548i $$-0.308009\pi$$
0.567247 + 0.823548i $$0.308009\pi$$
$$332$$ − 737.773i − 2.22221i
$$333$$ 0 0
$$334$$ −807.421 −2.41743
$$335$$ − 59.7070i − 0.178230i
$$336$$ 0 0
$$337$$ 188.114 0.558201 0.279101 0.960262i $$-0.409964\pi$$
0.279101 + 0.960262i $$0.409964\pi$$
$$338$$ 265.509i 0.785530i
$$339$$ 0 0
$$340$$ −255.035 −0.750103
$$341$$ 316.262i 0.927456i
$$342$$ 0 0
$$343$$ −334.491 −0.975193
$$344$$ − 847.858i − 2.46470i
$$345$$ 0 0
$$346$$ −696.500 −2.01300
$$347$$ 513.793i 1.48067i 0.672237 + 0.740336i $$0.265334\pi$$
−0.672237 + 0.740336i $$0.734666\pi$$
$$348$$ 0 0
$$349$$ 112.535 0.322451 0.161225 0.986918i $$-0.448455\pi$$
0.161225 + 0.986918i $$0.448455\pi$$
$$350$$ 130.722i 0.373490i
$$351$$ 0 0
$$352$$ −244.688 −0.695137
$$353$$ − 428.172i − 1.21295i −0.795102 0.606475i $$-0.792583\pi$$
0.795102 0.606475i $$-0.207417\pi$$
$$354$$ 0 0
$$355$$ −207.351 −0.584087
$$356$$ − 1001.86i − 2.81421i
$$357$$ 0 0
$$358$$ 212.083 0.592411
$$359$$ − 56.1961i − 0.156535i −0.996932 0.0782676i $$-0.975061\pi$$
0.996932 0.0782676i $$-0.0249389\pi$$
$$360$$ 0 0
$$361$$ −321.000 −0.889197
$$362$$ 594.708i 1.64284i
$$363$$ 0 0
$$364$$ −655.254 −1.80015
$$365$$ − 134.890i − 0.369561i
$$366$$ 0 0
$$367$$ −154.364 −0.420610 −0.210305 0.977636i $$-0.567446\pi$$
−0.210305 + 0.977636i $$0.567446\pi$$
$$368$$ 405.116i 1.10086i
$$369$$ 0 0
$$370$$ −542.469 −1.46613
$$371$$ − 94.7151i − 0.255297i
$$372$$ 0 0
$$373$$ 557.285 1.49406 0.747030 0.664790i $$-0.231479\pi$$
0.747030 + 0.664790i $$0.231479\pi$$
$$374$$ − 242.327i − 0.647932i
$$375$$ 0 0
$$376$$ 778.552 2.07062
$$377$$ 441.396i 1.17081i
$$378$$ 0 0
$$379$$ 147.404 0.388928 0.194464 0.980910i $$-0.437703\pi$$
0.194464 + 0.980910i $$0.437703\pi$$
$$380$$ 131.869i 0.347024i
$$381$$ 0 0
$$382$$ −365.254 −0.956163
$$383$$ 736.619i 1.92329i 0.274302 + 0.961644i $$0.411553\pi$$
−0.274302 + 0.961644i $$0.588447\pi$$
$$384$$ 0 0
$$385$$ −86.9210 −0.225769
$$386$$ 225.933i 0.585319i
$$387$$ 0 0
$$388$$ 10.0612 0.0259310
$$389$$ 296.408i 0.761975i 0.924580 + 0.380987i $$0.124416\pi$$
−0.924580 + 0.380987i $$0.875584\pi$$
$$390$$ 0 0
$$391$$ −147.263 −0.376631
$$392$$ 44.6685i 0.113950i
$$393$$ 0 0
$$394$$ 90.8157 0.230497
$$395$$ 215.153i 0.544690i
$$396$$ 0 0
$$397$$ −457.057 −1.15128 −0.575638 0.817704i $$-0.695246\pi$$
−0.575638 + 0.817704i $$0.695246\pi$$
$$398$$ 571.237i 1.43527i
$$399$$ 0 0
$$400$$ −168.246 −0.420614
$$401$$ 391.141i 0.975415i 0.873007 + 0.487707i $$0.162167\pi$$
−0.873007 + 0.487707i $$0.837833\pi$$
$$402$$ 0 0
$$403$$ −571.728 −1.41868
$$404$$ − 1587.81i − 3.93022i
$$405$$ 0 0
$$406$$ 1176.18 2.89700
$$407$$ − 360.705i − 0.886253i
$$408$$ 0 0
$$409$$ −411.842 −1.00695 −0.503474 0.864010i $$-0.667945\pi$$
−0.503474 + 0.864010i $$0.667945\pi$$
$$410$$ 134.468i 0.327971i
$$411$$ 0 0
$$412$$ 1197.84 2.90737
$$413$$ − 180.145i − 0.436186i
$$414$$ 0 0
$$415$$ 176.921 0.426316
$$416$$ − 442.339i − 1.06331i
$$417$$ 0 0
$$418$$ −125.298 −0.299757
$$419$$ − 653.447i − 1.55954i −0.626066 0.779770i $$-0.715336\pi$$
0.626066 0.779770i $$-0.284664\pi$$
$$420$$ 0 0
$$421$$ 125.035 0.296995 0.148497 0.988913i $$-0.452556\pi$$
0.148497 + 0.988913i $$0.452556\pi$$
$$422$$ − 867.998i − 2.05687i
$$423$$ 0 0
$$424$$ 257.026 0.606194
$$425$$ − 61.1584i − 0.143902i
$$426$$ 0 0
$$427$$ −255.517 −0.598401
$$428$$ 712.093i 1.66377i
$$429$$ 0 0
$$430$$ 356.061 0.828049
$$431$$ − 397.208i − 0.921596i −0.887505 0.460798i $$-0.847563\pi$$
0.887505 0.460798i $$-0.152437\pi$$
$$432$$ 0 0
$$433$$ 560.114 1.29357 0.646783 0.762674i $$-0.276114\pi$$
0.646783 + 0.762674i $$0.276114\pi$$
$$434$$ 1523.48i 3.51032i
$$435$$ 0 0
$$436$$ 121.956 0.279715
$$437$$ 76.1441i 0.174243i
$$438$$ 0 0
$$439$$ −664.386 −1.51341 −0.756704 0.653758i $$-0.773191\pi$$
−0.756704 + 0.653758i $$0.773191\pi$$
$$440$$ − 235.876i − 0.536081i
$$441$$ 0 0
$$442$$ 438.070 0.991108
$$443$$ − 371.305i − 0.838160i −0.907949 0.419080i $$-0.862353\pi$$
0.907949 0.419080i $$-0.137647\pi$$
$$444$$ 0 0
$$445$$ 240.250 0.539887
$$446$$ 665.455i 1.49205i
$$447$$ 0 0
$$448$$ −214.680 −0.479196
$$449$$ 585.471i 1.30395i 0.758243 + 0.651973i $$0.226058\pi$$
−0.758243 + 0.651973i $$0.773942\pi$$
$$450$$ 0 0
$$451$$ −89.4121 −0.198253
$$452$$ − 193.177i − 0.427382i
$$453$$ 0 0
$$454$$ 1482.30 3.26497
$$455$$ − 157.133i − 0.345347i
$$456$$ 0 0
$$457$$ 168.641 0.369017 0.184508 0.982831i $$-0.440931\pi$$
0.184508 + 0.982831i $$0.440931\pi$$
$$458$$ 99.6462i 0.217568i
$$459$$ 0 0
$$460$$ −251.026 −0.545709
$$461$$ − 298.492i − 0.647487i −0.946145 0.323744i $$-0.895058\pi$$
0.946145 0.323744i $$-0.104942\pi$$
$$462$$ 0 0
$$463$$ −595.285 −1.28571 −0.642856 0.765987i $$-0.722251\pi$$
−0.642856 + 0.765987i $$0.722251\pi$$
$$464$$ 1513.81i 3.26252i
$$465$$ 0 0
$$466$$ −1300.89 −2.79162
$$467$$ − 623.655i − 1.33545i −0.744408 0.667725i $$-0.767268\pi$$
0.744408 0.667725i $$-0.232732\pi$$
$$468$$ 0 0
$$469$$ 191.246 0.407773
$$470$$ 326.956i 0.695651i
$$471$$ 0 0
$$472$$ 488.855 1.03571
$$473$$ 236.756i 0.500542i
$$474$$ 0 0
$$475$$ −31.6228 −0.0665743
$$476$$ − 816.894i − 1.71616i
$$477$$ 0 0
$$478$$ −991.745 −2.07478
$$479$$ 131.857i 0.275276i 0.990483 + 0.137638i $$0.0439510\pi$$
−0.990483 + 0.137638i $$0.956049\pi$$
$$480$$ 0 0
$$481$$ 652.070 1.35565
$$482$$ 819.263i 1.69972i
$$483$$ 0 0
$$484$$ −853.605 −1.76365
$$485$$ 2.41272i 0.00497468i
$$486$$ 0 0
$$487$$ 41.8028 0.0858375 0.0429187 0.999079i $$-0.486334\pi$$
0.0429187 + 0.999079i $$0.486334\pi$$
$$488$$ − 693.392i − 1.42089i
$$489$$ 0 0
$$490$$ −18.7587 −0.0382831
$$491$$ 178.817i 0.364189i 0.983281 + 0.182095i $$0.0582877\pi$$
−0.983281 + 0.182095i $$0.941712\pi$$
$$492$$ 0 0
$$493$$ −550.280 −1.11619
$$494$$ − 226.510i − 0.458522i
$$495$$ 0 0
$$496$$ −1960.80 −3.95322
$$497$$ − 664.159i − 1.33634i
$$498$$ 0 0
$$499$$ −39.0961 −0.0783489 −0.0391744 0.999232i $$-0.512473\pi$$
−0.0391744 + 0.999232i $$0.512473\pi$$
$$500$$ − 104.252i − 0.208503i
$$501$$ 0 0
$$502$$ −1163.62 −2.31796
$$503$$ − 578.698i − 1.15049i −0.817980 0.575247i $$-0.804906\pi$$
0.817980 0.575247i $$-0.195094\pi$$
$$504$$ 0 0
$$505$$ 380.763 0.753986
$$506$$ − 238.518i − 0.471379i
$$507$$ 0 0
$$508$$ 354.619 0.698068
$$509$$ − 355.743i − 0.698905i −0.936954 0.349453i $$-0.886368\pi$$
0.936954 0.349453i $$-0.113632\pi$$
$$510$$ 0 0
$$511$$ 432.061 0.845521
$$512$$ 1098.99i 2.14646i
$$513$$ 0 0
$$514$$ 1357.81 2.64166
$$515$$ 287.246i 0.557760i
$$516$$ 0 0
$$517$$ −217.404 −0.420510
$$518$$ − 1737.57i − 3.35437i
$$519$$ 0 0
$$520$$ 426.408 0.820015
$$521$$ 810.952i 1.55653i 0.627936 + 0.778265i $$0.283900\pi$$
−0.627936 + 0.778265i $$0.716100\pi$$
$$522$$ 0 0
$$523$$ 720.483 1.37760 0.688798 0.724953i $$-0.258139\pi$$
0.688798 + 0.724953i $$0.258139\pi$$
$$524$$ − 782.711i − 1.49372i
$$525$$ 0 0
$$526$$ 870.772 1.65546
$$527$$ − 712.764i − 1.35249i
$$528$$ 0 0
$$529$$ 384.052 0.725996
$$530$$ 107.939i 0.203659i
$$531$$ 0 0
$$532$$ −422.386 −0.793958
$$533$$ − 161.636i − 0.303257i
$$534$$ 0 0
$$535$$ −170.763 −0.319183
$$536$$ 518.979i 0.968245i
$$537$$ 0 0
$$538$$ 459.465 0.854024
$$539$$ − 12.4733i − 0.0231415i
$$540$$ 0 0
$$541$$ 347.149 0.641680 0.320840 0.947133i $$-0.396035\pi$$
0.320840 + 0.947133i $$0.396035\pi$$
$$542$$ 944.142i 1.74196i
$$543$$ 0 0
$$544$$ 551.456 1.01371
$$545$$ 29.2455i 0.0536615i
$$546$$ 0 0
$$547$$ −720.833 −1.31779 −0.658896 0.752234i $$-0.728976\pi$$
−0.658896 + 0.752234i $$0.728976\pi$$
$$548$$ 145.403i 0.265334i
$$549$$ 0 0
$$550$$ 99.0569 0.180104
$$551$$ 284.530i 0.516388i
$$552$$ 0 0
$$553$$ −689.149 −1.24620
$$554$$ − 831.222i − 1.50040i
$$555$$ 0 0
$$556$$ −632.596 −1.13776
$$557$$ 429.102i 0.770380i 0.922837 + 0.385190i $$0.125864\pi$$
−0.922837 + 0.385190i $$0.874136\pi$$
$$558$$ 0 0
$$559$$ −428.000 −0.765653
$$560$$ − 538.902i − 0.962325i
$$561$$ 0 0
$$562$$ −881.118 −1.56783
$$563$$ 670.820i 1.19151i 0.803166 + 0.595755i $$0.203147\pi$$
−0.803166 + 0.595755i $$0.796853\pi$$
$$564$$ 0 0
$$565$$ 46.3246 0.0819904
$$566$$ − 760.474i − 1.34359i
$$567$$ 0 0
$$568$$ 1802.31 3.17309
$$569$$ − 368.663i − 0.647914i −0.946072 0.323957i $$-0.894987\pi$$
0.946072 0.323957i $$-0.105013\pi$$
$$570$$ 0 0
$$571$$ 124.289 0.217669 0.108834 0.994060i $$-0.465288\pi$$
0.108834 + 0.994060i $$0.465288\pi$$
$$572$$ 496.532i 0.868063i
$$573$$ 0 0
$$574$$ −430.710 −0.750366
$$575$$ − 60.1972i − 0.104691i
$$576$$ 0 0
$$577$$ 504.236 0.873893 0.436947 0.899487i $$-0.356060\pi$$
0.436947 + 0.899487i $$0.356060\pi$$
$$578$$ − 508.797i − 0.880272i
$$579$$ 0 0
$$580$$ −938.017 −1.61727
$$581$$ 566.690i 0.975370i
$$582$$ 0 0
$$583$$ −71.7722 −0.123108
$$584$$ 1172.48i 2.00766i
$$585$$ 0 0
$$586$$ −736.236 −1.25638
$$587$$ − 39.2256i − 0.0668238i −0.999442 0.0334119i $$-0.989363\pi$$
0.999442 0.0334119i $$-0.0106373\pi$$
$$588$$ 0 0
$$589$$ −368.544 −0.625711
$$590$$ 205.297i 0.347960i
$$591$$ 0 0
$$592$$ 2236.34 3.77760
$$593$$ 621.670i 1.04835i 0.851611 + 0.524174i $$0.175626\pi$$
−0.851611 + 0.524174i $$0.824374\pi$$
$$594$$ 0 0
$$595$$ 195.895 0.329235
$$596$$ 2176.59i 3.65200i
$$597$$ 0 0
$$598$$ 431.184 0.721044
$$599$$ 1119.77i 1.86940i 0.355436 + 0.934701i $$0.384332\pi$$
−0.355436 + 0.934701i $$0.615668\pi$$
$$600$$ 0 0
$$601$$ −323.789 −0.538751 −0.269375 0.963035i $$-0.586817\pi$$
−0.269375 + 0.963035i $$0.586817\pi$$
$$602$$ 1140.49i 1.89450i
$$603$$ 0 0
$$604$$ −1728.31 −2.86145
$$605$$ − 204.698i − 0.338344i
$$606$$ 0 0
$$607$$ 1025.63 1.68966 0.844832 0.535031i $$-0.179700\pi$$
0.844832 + 0.535031i $$0.179700\pi$$
$$608$$ − 285.138i − 0.468976i
$$609$$ 0 0
$$610$$ 291.193 0.477365
$$611$$ − 393.014i − 0.643232i
$$612$$ 0 0
$$613$$ 904.153 1.47496 0.737482 0.675367i $$-0.236015\pi$$
0.737482 + 0.675367i $$0.236015\pi$$
$$614$$ − 1251.40i − 2.03812i
$$615$$ 0 0
$$616$$ 755.526 1.22650
$$617$$ − 710.716i − 1.15189i −0.817488 0.575945i $$-0.804634\pi$$
0.817488 0.575945i $$-0.195366\pi$$
$$618$$ 0 0
$$619$$ −583.737 −0.943032 −0.471516 0.881858i $$-0.656293\pi$$
−0.471516 + 0.881858i $$0.656293\pi$$
$$620$$ − 1214.99i − 1.95966i
$$621$$ 0 0
$$622$$ −794.447 −1.27725
$$623$$ 769.537i 1.23521i
$$624$$ 0 0
$$625$$ 25.0000 0.0400000
$$626$$ − 1028.99i − 1.64376i
$$627$$ 0 0
$$628$$ 1037.60 1.65223
$$629$$ 812.924i 1.29241i
$$630$$ 0 0
$$631$$ −20.0968 −0.0318491 −0.0159246 0.999873i $$-0.505069\pi$$
−0.0159246 + 0.999873i $$0.505069\pi$$
$$632$$ − 1870.13i − 2.95906i
$$633$$ 0 0
$$634$$ 56.2192 0.0886738
$$635$$ 85.0390i 0.133920i
$$636$$ 0 0
$$637$$ 22.5487 0.0353983
$$638$$ − 891.277i − 1.39699i
$$639$$ 0 0
$$640$$ −158.592 −0.247800
$$641$$ 341.607i 0.532928i 0.963845 + 0.266464i $$0.0858553\pi$$
−0.963845 + 0.266464i $$0.914145\pi$$
$$642$$ 0 0
$$643$$ −469.693 −0.730471 −0.365235 0.930915i $$-0.619011\pi$$
−0.365235 + 0.930915i $$0.619011\pi$$
$$644$$ − 804.054i − 1.24853i
$$645$$ 0 0
$$646$$ 282.386 0.437130
$$647$$ − 572.099i − 0.884234i −0.896957 0.442117i $$-0.854228\pi$$
0.896957 0.442117i $$-0.145772\pi$$
$$648$$ 0 0
$$649$$ −136.508 −0.210336
$$650$$ 179.072i 0.275495i
$$651$$ 0 0
$$652$$ −1107.41 −1.69848
$$653$$ 213.540i 0.327014i 0.986542 + 0.163507i $$0.0522806\pi$$
−0.986542 + 0.163507i $$0.947719\pi$$
$$654$$ 0 0
$$655$$ 187.698 0.286561
$$656$$ − 554.347i − 0.845040i
$$657$$ 0 0
$$658$$ −1047.26 −1.59158
$$659$$ − 420.983i − 0.638820i −0.947617 0.319410i $$-0.896515\pi$$
0.947617 0.319410i $$-0.103485\pi$$
$$660$$ 0 0
$$661$$ 434.272 0.656992 0.328496 0.944505i $$-0.393458\pi$$
0.328496 + 0.944505i $$0.393458\pi$$
$$662$$ − 1370.74i − 2.07061i
$$663$$ 0 0
$$664$$ −1537.81 −2.31599
$$665$$ − 101.290i − 0.152316i
$$666$$ 0 0
$$667$$ −541.631 −0.812041
$$668$$ 2062.54i 3.08763i
$$669$$ 0 0
$$670$$ −217.947 −0.325295
$$671$$ 193.623i 0.288560i
$$672$$ 0 0
$$673$$ 72.7801 0.108143 0.0540714 0.998537i $$-0.482780\pi$$
0.0540714 + 0.998537i $$0.482780\pi$$
$$674$$ − 686.669i − 1.01880i
$$675$$ 0 0
$$676$$ 678.237 1.00331
$$677$$ 172.106i 0.254219i 0.991889 + 0.127109i $$0.0405700\pi$$
−0.991889 + 0.127109i $$0.959430\pi$$
$$678$$ 0 0
$$679$$ −7.72811 −0.0113816
$$680$$ 531.595i 0.781758i
$$681$$ 0 0
$$682$$ 1154.45 1.69274
$$683$$ − 792.592i − 1.16046i −0.814454 0.580228i $$-0.802963\pi$$
0.814454 0.580228i $$-0.197037\pi$$
$$684$$ 0 0
$$685$$ −34.8683 −0.0509027
$$686$$ 1220.99i 1.77986i
$$687$$ 0 0
$$688$$ −1467.87 −2.13353
$$689$$ − 129.747i − 0.188313i
$$690$$ 0 0
$$691$$ 154.851 0.224097 0.112049 0.993703i $$-0.464259\pi$$
0.112049 + 0.993703i $$0.464259\pi$$
$$692$$ 1779.19i 2.57109i
$$693$$ 0 0
$$694$$ 1875.49 2.70244
$$695$$ − 151.699i − 0.218272i
$$696$$ 0 0
$$697$$ 201.509 0.289109
$$698$$ − 410.785i − 0.588518i
$$699$$ 0 0
$$700$$ 333.925 0.477036
$$701$$ − 950.544i − 1.35598i −0.735070 0.677991i $$-0.762851\pi$$
0.735070 0.677991i $$-0.237149\pi$$
$$702$$ 0 0
$$703$$ 420.333 0.597913
$$704$$ 162.678i 0.231076i
$$705$$ 0 0
$$706$$ −1562.95 −2.21381
$$707$$ 1219.61i 1.72505i
$$708$$ 0 0
$$709$$ 390.350 0.550564 0.275282 0.961363i $$-0.411229\pi$$
0.275282 + 0.961363i $$0.411229\pi$$
$$710$$ 756.889i 1.06604i
$$711$$ 0 0
$$712$$ −2088.28 −2.93297
$$713$$ − 701.561i − 0.983956i
$$714$$ 0 0
$$715$$ −119.070 −0.166532
$$716$$ − 541.762i − 0.756650i
$$717$$ 0 0
$$718$$ −205.132 −0.285699
$$719$$ 655.227i 0.911303i 0.890158 + 0.455651i $$0.150594\pi$$
−0.890158 + 0.455651i $$0.849406\pi$$
$$720$$ 0 0
$$721$$ −920.070 −1.27610
$$722$$ 1171.74i 1.62291i
$$723$$ 0 0
$$724$$ 1519.17 2.09830
$$725$$ − 224.940i − 0.310263i
$$726$$ 0 0
$$727$$ 1424.25 1.95908 0.979539 0.201254i $$-0.0645017\pi$$
0.979539 + 0.201254i $$0.0645017\pi$$
$$728$$ 1365.81i 1.87612i
$$729$$ 0 0
$$730$$ −492.386 −0.674501
$$731$$ − 533.580i − 0.729932i
$$732$$ 0 0
$$733$$ −946.749 −1.29161 −0.645805 0.763503i $$-0.723478\pi$$
−0.645805 + 0.763503i $$0.723478\pi$$
$$734$$ 563.471i 0.767672i
$$735$$ 0 0
$$736$$ 542.789 0.737485
$$737$$ − 144.920i − 0.196635i
$$738$$ 0 0
$$739$$ 591.429 0.800310 0.400155 0.916447i $$-0.368956\pi$$
0.400155 + 0.916447i $$0.368956\pi$$
$$740$$ 1385.72i 1.87260i
$$741$$ 0 0
$$742$$ −345.737 −0.465952
$$743$$ − 732.202i − 0.985467i −0.870180 0.492734i $$-0.835998\pi$$
0.870180 0.492734i $$-0.164002\pi$$
$$744$$ 0 0
$$745$$ −521.956 −0.700612
$$746$$ − 2034.25i − 2.72687i
$$747$$ 0 0
$$748$$ −619.018 −0.827564
$$749$$ − 546.965i − 0.730261i
$$750$$ 0 0
$$751$$ −215.359 −0.286764 −0.143382 0.989667i $$-0.545798\pi$$
−0.143382 + 0.989667i $$0.545798\pi$$
$$752$$ − 1347.88i − 1.79240i
$$753$$ 0 0
$$754$$ 1611.22 2.13689
$$755$$ − 414.457i − 0.548950i
$$756$$ 0 0
$$757$$ 276.258 0.364938 0.182469 0.983212i $$-0.441591\pi$$
0.182469 + 0.983212i $$0.441591\pi$$
$$758$$ − 538.064i − 0.709848i
$$759$$ 0 0
$$760$$ 274.868 0.361669
$$761$$ − 893.373i − 1.17395i −0.809606 0.586973i $$-0.800319\pi$$
0.809606 0.586973i $$-0.199681\pi$$
$$762$$ 0 0
$$763$$ −93.6754 −0.122773
$$764$$ 933.033i 1.22125i
$$765$$ 0 0
$$766$$ 2688.87 3.51027
$$767$$ − 246.775i − 0.321740i
$$768$$ 0 0
$$769$$ 284.316 0.369722 0.184861 0.982765i $$-0.440817\pi$$
0.184861 + 0.982765i $$0.440817\pi$$
$$770$$ 317.286i 0.412060i
$$771$$ 0 0
$$772$$ 577.140 0.747591
$$773$$ 1059.64i 1.37081i 0.728162 + 0.685405i $$0.240375\pi$$
−0.728162 + 0.685405i $$0.759625\pi$$
$$774$$ 0 0
$$775$$ 291.359 0.375948
$$776$$ − 20.9716i − 0.0270253i
$$777$$ 0 0
$$778$$ 1081.97 1.39071
$$779$$ − 104.193i − 0.133752i
$$780$$ 0 0
$$781$$ −503.280 −0.644404
$$782$$ 537.550i 0.687404i
$$783$$ 0 0
$$784$$ 77.3331 0.0986392
$$785$$ 248.821i 0.316970i
$$786$$ 0 0
$$787$$ −875.517 −1.11247 −0.556237 0.831024i $$-0.687755\pi$$
−0.556237 + 0.831024i $$0.687755\pi$$
$$788$$ − 231.986i − 0.294399i
$$789$$ 0 0
$$790$$ 785.368 0.994137
$$791$$ 148.381i 0.187586i
$$792$$ 0 0
$$793$$ −350.026 −0.441394
$$794$$ 1668.39i 2.10124i
$$795$$ 0 0
$$796$$ 1459.21 1.83318
$$797$$ − 742.449i − 0.931555i −0.884902 0.465777i $$-0.845775\pi$$
0.884902 0.465777i $$-0.154225\pi$$
$$798$$ 0 0
$$799$$ 489.964 0.613222
$$800$$ 225.421i 0.281776i
$$801$$ 0 0
$$802$$ 1427.78 1.78027
$$803$$ − 327.403i − 0.407725i
$$804$$ 0 0
$$805$$ 192.816 0.239523
$$806$$ 2086.97i 2.58929i
$$807$$ 0 0
$$808$$ −3309.63 −4.09608
$$809$$ − 113.720i − 0.140568i −0.997527 0.0702842i $$-0.977609\pi$$
0.997527 0.0702842i $$-0.0223906\pi$$
$$810$$ 0 0
$$811$$ −1466.03 −1.80769 −0.903844 0.427863i $$-0.859267\pi$$
−0.903844 + 0.427863i $$0.859267\pi$$
$$812$$ − 3004.53i − 3.70016i
$$813$$ 0 0
$$814$$ −1316.67 −1.61754
$$815$$ − 265.562i − 0.325843i
$$816$$ 0 0
$$817$$ −275.895 −0.337692
$$818$$ 1503.34i 1.83782i
$$819$$ 0 0
$$820$$ 343.495 0.418897
$$821$$ − 550.073i − 0.670003i −0.942218 0.335002i $$-0.891263\pi$$
0.942218 0.335002i $$-0.108737\pi$$
$$822$$ 0 0
$$823$$ 1392.51 1.69199 0.845997 0.533187i $$-0.179006\pi$$
0.845997 + 0.533187i $$0.179006\pi$$
$$824$$ − 2496.77i − 3.03007i
$$825$$ 0 0
$$826$$ −657.579 −0.796100
$$827$$ 955.922i 1.15589i 0.816075 + 0.577945i $$0.196145\pi$$
−0.816075 + 0.577945i $$0.803855\pi$$
$$828$$ 0 0
$$829$$ −1652.69 −1.99360 −0.996798 0.0799552i $$-0.974522\pi$$
−0.996798 + 0.0799552i $$0.974522\pi$$
$$830$$ − 645.811i − 0.778086i
$$831$$ 0 0
$$832$$ −294.083 −0.353465
$$833$$ 28.1111i 0.0337469i
$$834$$ 0 0
$$835$$ −494.605 −0.592341
$$836$$ 320.071i 0.382860i
$$837$$ 0 0
$$838$$ −2385.27 −2.84638
$$839$$ 1568.11i 1.86903i 0.355927 + 0.934514i $$0.384165\pi$$
−0.355927 + 0.934514i $$0.615835\pi$$
$$840$$ 0 0
$$841$$ −1182.93 −1.40657
$$842$$ − 456.413i − 0.542058i
$$843$$ 0 0
$$844$$ −2217.28 −2.62711
$$845$$ 162.644i 0.192478i
$$846$$ 0 0
$$847$$ 655.662 0.774099
$$848$$ − 444.981i − 0.524742i
$$849$$ 0 0
$$850$$ −223.246 −0.262642
$$851$$ 800.147i 0.940243i
$$852$$ 0 0
$$853$$ −651.232 −0.763461 −0.381730 0.924274i $$-0.624672\pi$$
−0.381730 + 0.924274i $$0.624672\pi$$
$$854$$ 932.711i 1.09217i
$$855$$ 0 0
$$856$$ 1484.29 1.73398
$$857$$ 299.131i 0.349044i 0.984653 + 0.174522i $$0.0558380\pi$$
−0.984653 + 0.174522i $$0.944162\pi$$
$$858$$ 0 0
$$859$$ 1095.72 1.27558 0.637788 0.770212i $$-0.279850\pi$$
0.637788 + 0.770212i $$0.279850\pi$$
$$860$$ − 909.550i − 1.05762i
$$861$$ 0 0
$$862$$ −1449.92 −1.68204
$$863$$ 1221.95i 1.41593i 0.706247 + 0.707965i $$0.250387\pi$$
−0.706247 + 0.707965i $$0.749613\pi$$
$$864$$ 0 0
$$865$$ −426.658 −0.493246
$$866$$ − 2044.57i − 2.36094i
$$867$$ 0 0
$$868$$ 3891.69 4.48352
$$869$$ 522.216i 0.600939i
$$870$$ 0 0
$$871$$ 261.982 0.300782
$$872$$ − 254.205i − 0.291520i
$$873$$ 0 0
$$874$$ 277.947 0.318018
$$875$$ 80.0767i 0.0915162i
$$876$$ 0 0
$$877$$ 766.399 0.873887 0.436944 0.899489i $$-0.356061\pi$$
0.436944 + 0.899489i $$0.356061\pi$$
$$878$$ 2425.20i 2.76218i
$$879$$ 0 0
$$880$$ −408.364 −0.464050
$$881$$ − 310.097i − 0.351983i −0.984392 0.175992i $$-0.943687\pi$$
0.984392 0.175992i $$-0.0563132\pi$$
$$882$$ 0 0
$$883$$ −122.236 −0.138433 −0.0692165 0.997602i $$-0.522050\pi$$
−0.0692165 + 0.997602i $$0.522050\pi$$
$$884$$ − 1119.04i − 1.26588i
$$885$$ 0 0
$$886$$ −1355.37 −1.52976
$$887$$ 265.444i 0.299261i 0.988742 + 0.149630i $$0.0478084\pi$$
−0.988742 + 0.149630i $$0.952192\pi$$
$$888$$ 0 0
$$889$$ −272.386 −0.306396
$$890$$ − 876.980i − 0.985370i
$$891$$ 0 0
$$892$$ 1699.89 1.90571
$$893$$ − 253.343i − 0.283698i
$$894$$ 0 0
$$895$$ 129.917 0.145158
$$896$$ − 507.981i − 0.566944i
$$897$$ 0 0
$$898$$ 2137.14 2.37988
$$899$$ − 2621.54i − 2.91606i
$$900$$ 0 0
$$901$$ 161.754 0.179527
$$902$$ 326.379i 0.361840i
$$903$$ 0 0
$$904$$ −402.658 −0.445418
$$905$$ 364.302i 0.402544i
$$906$$ 0 0
$$907$$ 672.622 0.741590 0.370795 0.928715i $$-0.379085\pi$$
0.370795 + 0.928715i $$0.379085\pi$$
$$908$$ − 3786.49i − 4.17015i
$$909$$ 0 0
$$910$$ −573.579 −0.630306
$$911$$ 1402.48i 1.53949i 0.638350 + 0.769746i $$0.279617\pi$$
−0.638350 + 0.769746i $$0.720383\pi$$
$$912$$ 0 0
$$913$$ 429.421 0.470340
$$914$$ − 615.586i − 0.673507i
$$915$$ 0 0
$$916$$ 254.544 0.277886
$$917$$ 601.208i 0.655625i
$$918$$ 0 0
$$919$$ −338.255 −0.368068 −0.184034 0.982920i $$-0.558916\pi$$
−0.184034 + 0.982920i $$0.558916\pi$$
$$920$$ 523.240i 0.568739i
$$921$$ 0 0
$$922$$ −1089.58 −1.18176
$$923$$ − 909.811i − 0.985711i
$$924$$ 0 0
$$925$$ −332.302 −0.359246
$$926$$ 2172.96i 2.34661i
$$927$$ 0 0
$$928$$ 2028.25 2.18562
$$929$$ 148.207i 0.159533i 0.996814 + 0.0797667i $$0.0254175\pi$$
−0.996814 + 0.0797667i $$0.974582\pi$$
$$930$$ 0 0
$$931$$ 14.5352 0.0156125
$$932$$ 3323.10i 3.56556i
$$933$$ 0 0
$$934$$ −2276.52 −2.43738
$$935$$ − 148.443i − 0.158763i
$$936$$ 0 0
$$937$$ −1416.72 −1.51197 −0.755987 0.654587i $$-0.772843\pi$$
−0.755987 + 0.654587i $$0.772843\pi$$
$$938$$ − 698.100i − 0.744243i
$$939$$ 0 0
$$940$$ 835.201 0.888512
$$941$$ − 1398.92i − 1.48663i −0.668939 0.743317i $$-0.733251\pi$$
0.668939 0.743317i $$-0.266749\pi$$
$$942$$ 0 0
$$943$$ 198.342 0.210330
$$944$$ − 846.338i − 0.896544i
$$945$$ 0 0
$$946$$ 864.228 0.913560
$$947$$ 1050.57i 1.10937i 0.832060 + 0.554686i $$0.187161\pi$$
−0.832060 + 0.554686i $$0.812839\pi$$
$$948$$ 0 0
$$949$$ 591.868 0.623675
$$950$$ 115.432i 0.121507i
$$951$$ 0 0
$$952$$ −1702.74 −1.78859
$$953$$ − 551.928i − 0.579148i −0.957156 0.289574i $$-0.906486\pi$$
0.957156 0.289574i $$-0.0935136\pi$$
$$954$$ 0 0
$$955$$ −223.745 −0.234288
$$956$$ 2533.39i 2.64999i
$$957$$ 0 0
$$958$$ 481.315 0.502417
$$959$$ − 111.686i − 0.116461i
$$960$$ 0 0
$$961$$ 2434.61 2.53342
$$962$$ − 2380.24i − 2.47426i
$$963$$ 0 0
$$964$$ 2092.79 2.17094
$$965$$ 138.401i 0.143420i
$$966$$ 0 0
$$967$$ −357.093 −0.369279 −0.184639 0.982806i $$-0.559112\pi$$
−0.184639 + 0.982806i $$0.559112\pi$$
$$968$$ 1779.26i 1.83807i
$$969$$ 0 0
$$970$$ 8.80711 0.00907950
$$971$$ 308.206i 0.317411i 0.987326 + 0.158705i $$0.0507320\pi$$
−0.987326 + 0.158705i $$0.949268\pi$$
$$972$$ 0 0
$$973$$ 485.903 0.499387
$$974$$ − 152.592i − 0.156665i
$$975$$ 0 0
$$976$$ −1200.45 −1.22997
$$977$$ 253.280i 0.259243i 0.991564 + 0.129621i $$0.0413762\pi$$
−0.991564 + 0.129621i $$0.958624\pi$$
$$978$$ 0 0
$$979$$ 583.132 0.595640
$$980$$ 47.9187i 0.0488966i
$$981$$ 0 0
$$982$$ 652.732 0.664697
$$983$$ 1068.73i 1.08722i 0.839339 + 0.543609i $$0.182942\pi$$
−0.839339 + 0.543609i $$0.817058\pi$$
$$984$$ 0 0
$$985$$ 55.6313 0.0564785
$$986$$ 2008.68i 2.03720i
$$987$$ 0 0
$$988$$ −578.614 −0.585641
$$989$$ − 525.194i − 0.531035i
$$990$$ 0 0
$$991$$ 280.631 0.283179 0.141590 0.989925i $$-0.454779\pi$$
0.141590 + 0.989925i $$0.454779\pi$$
$$992$$ 2627.14i 2.64833i
$$993$$ 0 0
$$994$$ −2424.37 −2.43900
$$995$$ 349.925i 0.351683i
$$996$$ 0 0
$$997$$ −356.574 −0.357647 −0.178824 0.983881i $$-0.557229\pi$$
−0.178824 + 0.983881i $$0.557229\pi$$
$$998$$ 142.712i 0.142998i
$$999$$ 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 45.3.c.a.26.1 4
3.2 odd 2 inner 45.3.c.a.26.4 yes 4
4.3 odd 2 720.3.l.a.161.1 4
5.2 odd 4 225.3.d.b.224.8 8
5.3 odd 4 225.3.d.b.224.1 8
5.4 even 2 225.3.c.c.26.4 4
8.3 odd 2 2880.3.l.c.1601.3 4
8.5 even 2 2880.3.l.g.1601.4 4
9.2 odd 6 405.3.i.d.296.4 8
9.4 even 3 405.3.i.d.26.4 8
9.5 odd 6 405.3.i.d.26.1 8
9.7 even 3 405.3.i.d.296.1 8
12.11 even 2 720.3.l.a.161.3 4
15.2 even 4 225.3.d.b.224.2 8
15.8 even 4 225.3.d.b.224.7 8
15.14 odd 2 225.3.c.c.26.1 4
20.3 even 4 3600.3.c.i.449.8 8
20.7 even 4 3600.3.c.i.449.2 8
20.19 odd 2 3600.3.l.v.1601.4 4
24.5 odd 2 2880.3.l.g.1601.2 4
24.11 even 2 2880.3.l.c.1601.1 4
60.23 odd 4 3600.3.c.i.449.7 8
60.47 odd 4 3600.3.c.i.449.1 8
60.59 even 2 3600.3.l.v.1601.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
45.3.c.a.26.1 4 1.1 even 1 trivial
45.3.c.a.26.4 yes 4 3.2 odd 2 inner
225.3.c.c.26.1 4 15.14 odd 2
225.3.c.c.26.4 4 5.4 even 2
225.3.d.b.224.1 8 5.3 odd 4
225.3.d.b.224.2 8 15.2 even 4
225.3.d.b.224.7 8 15.8 even 4
225.3.d.b.224.8 8 5.2 odd 4
405.3.i.d.26.1 8 9.5 odd 6
405.3.i.d.26.4 8 9.4 even 3
405.3.i.d.296.1 8 9.7 even 3
405.3.i.d.296.4 8 9.2 odd 6
720.3.l.a.161.1 4 4.3 odd 2
720.3.l.a.161.3 4 12.11 even 2
2880.3.l.c.1601.1 4 24.11 even 2
2880.3.l.c.1601.3 4 8.3 odd 2
2880.3.l.g.1601.2 4 24.5 odd 2
2880.3.l.g.1601.4 4 8.5 even 2
3600.3.c.i.449.1 8 60.47 odd 4
3600.3.c.i.449.2 8 20.7 even 4
3600.3.c.i.449.7 8 60.23 odd 4
3600.3.c.i.449.8 8 20.3 even 4
3600.3.l.v.1601.3 4 60.59 even 2
3600.3.l.v.1601.4 4 20.19 odd 2