# Properties

 Label 3330.2.a.bb.1.2 Level $3330$ Weight $2$ Character 3330.1 Self dual yes Analytic conductor $26.590$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3330.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.5901838731$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 3330.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.37228 q^{11} -4.74456 q^{13} -1.37228 q^{14} +1.00000 q^{16} +5.37228 q^{17} -2.00000 q^{19} -1.00000 q^{20} -3.37228 q^{22} -6.74456 q^{23} +1.00000 q^{25} +4.74456 q^{26} +1.37228 q^{28} -8.11684 q^{29} -2.62772 q^{31} -1.00000 q^{32} -5.37228 q^{34} -1.37228 q^{35} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} -5.37228 q^{41} +7.37228 q^{43} +3.37228 q^{44} +6.74456 q^{46} +8.74456 q^{47} -5.11684 q^{49} -1.00000 q^{50} -4.74456 q^{52} -1.37228 q^{53} -3.37228 q^{55} -1.37228 q^{56} +8.11684 q^{58} -12.7446 q^{59} -5.37228 q^{61} +2.62772 q^{62} +1.00000 q^{64} +4.74456 q^{65} +4.74456 q^{67} +5.37228 q^{68} +1.37228 q^{70} +6.74456 q^{71} +8.74456 q^{73} -1.00000 q^{74} -2.00000 q^{76} +4.62772 q^{77} -4.74456 q^{79} -1.00000 q^{80} +5.37228 q^{82} +0.744563 q^{83} -5.37228 q^{85} -7.37228 q^{86} -3.37228 q^{88} -10.0000 q^{89} -6.51087 q^{91} -6.74456 q^{92} -8.74456 q^{94} +2.00000 q^{95} -0.116844 q^{97} +5.11684 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 3 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8} + 2 q^{10} + q^{11} + 2 q^{13} + 3 q^{14} + 2 q^{16} + 5 q^{17} - 4 q^{19} - 2 q^{20} - q^{22} - 2 q^{23} + 2 q^{25} - 2 q^{26} - 3 q^{28} + q^{29} - 11 q^{31} - 2 q^{32} - 5 q^{34} + 3 q^{35} + 2 q^{37} + 4 q^{38} + 2 q^{40} - 5 q^{41} + 9 q^{43} + q^{44} + 2 q^{46} + 6 q^{47} + 7 q^{49} - 2 q^{50} + 2 q^{52} + 3 q^{53} - q^{55} + 3 q^{56} - q^{58} - 14 q^{59} - 5 q^{61} + 11 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{67} + 5 q^{68} - 3 q^{70} + 2 q^{71} + 6 q^{73} - 2 q^{74} - 4 q^{76} + 15 q^{77} + 2 q^{79} - 2 q^{80} + 5 q^{82} - 10 q^{83} - 5 q^{85} - 9 q^{86} - q^{88} - 20 q^{89} - 36 q^{91} - 2 q^{92} - 6 q^{94} + 4 q^{95} + 17 q^{97} - 7 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 3 * q^7 - 2 * q^8 + 2 * q^10 + q^11 + 2 * q^13 + 3 * q^14 + 2 * q^16 + 5 * q^17 - 4 * q^19 - 2 * q^20 - q^22 - 2 * q^23 + 2 * q^25 - 2 * q^26 - 3 * q^28 + q^29 - 11 * q^31 - 2 * q^32 - 5 * q^34 + 3 * q^35 + 2 * q^37 + 4 * q^38 + 2 * q^40 - 5 * q^41 + 9 * q^43 + q^44 + 2 * q^46 + 6 * q^47 + 7 * q^49 - 2 * q^50 + 2 * q^52 + 3 * q^53 - q^55 + 3 * q^56 - q^58 - 14 * q^59 - 5 * q^61 + 11 * q^62 + 2 * q^64 - 2 * q^65 - 2 * q^67 + 5 * q^68 - 3 * q^70 + 2 * q^71 + 6 * q^73 - 2 * q^74 - 4 * q^76 + 15 * q^77 + 2 * q^79 - 2 * q^80 + 5 * q^82 - 10 * q^83 - 5 * q^85 - 9 * q^86 - q^88 - 20 * q^89 - 36 * q^91 - 2 * q^92 - 6 * q^94 + 4 * q^95 + 17 * q^97 - 7 * 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
$$7$$ 1.37228 0.518674 0.259337 0.965787i $$-0.416496\pi$$
0.259337 + 0.965787i $$0.416496\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ 3.37228 1.01678 0.508391 0.861127i $$-0.330241\pi$$
0.508391 + 0.861127i $$0.330241\pi$$
$$12$$ 0 0
$$13$$ −4.74456 −1.31590 −0.657952 0.753059i $$-0.728577\pi$$
−0.657952 + 0.753059i $$0.728577\pi$$
$$14$$ −1.37228 −0.366758
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 5.37228 1.30297 0.651485 0.758662i $$-0.274146\pi$$
0.651485 + 0.758662i $$0.274146\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −3.37228 −0.718973
$$23$$ −6.74456 −1.40634 −0.703169 0.711022i $$-0.748232\pi$$
−0.703169 + 0.711022i $$0.748232\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 4.74456 0.930485
$$27$$ 0 0
$$28$$ 1.37228 0.259337
$$29$$ −8.11684 −1.50726 −0.753630 0.657299i $$-0.771699\pi$$
−0.753630 + 0.657299i $$0.771699\pi$$
$$30$$ 0 0
$$31$$ −2.62772 −0.471952 −0.235976 0.971759i $$-0.575829\pi$$
−0.235976 + 0.971759i $$0.575829\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −5.37228 −0.921339
$$35$$ −1.37228 −0.231958
$$36$$ 0 0
$$37$$ 1.00000 0.164399
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ −5.37228 −0.839009 −0.419505 0.907753i $$-0.637796\pi$$
−0.419505 + 0.907753i $$0.637796\pi$$
$$42$$ 0 0
$$43$$ 7.37228 1.12426 0.562131 0.827048i $$-0.309982\pi$$
0.562131 + 0.827048i $$0.309982\pi$$
$$44$$ 3.37228 0.508391
$$45$$ 0 0
$$46$$ 6.74456 0.994432
$$47$$ 8.74456 1.27553 0.637763 0.770233i $$-0.279860\pi$$
0.637763 + 0.770233i $$0.279860\pi$$
$$48$$ 0 0
$$49$$ −5.11684 −0.730978
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −4.74456 −0.657952
$$53$$ −1.37228 −0.188497 −0.0942487 0.995549i $$-0.530045\pi$$
−0.0942487 + 0.995549i $$0.530045\pi$$
$$54$$ 0 0
$$55$$ −3.37228 −0.454718
$$56$$ −1.37228 −0.183379
$$57$$ 0 0
$$58$$ 8.11684 1.06579
$$59$$ −12.7446 −1.65920 −0.829600 0.558358i $$-0.811432\pi$$
−0.829600 + 0.558358i $$0.811432\pi$$
$$60$$ 0 0
$$61$$ −5.37228 −0.687850 −0.343925 0.938997i $$-0.611757\pi$$
−0.343925 + 0.938997i $$0.611757\pi$$
$$62$$ 2.62772 0.333721
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 4.74456 0.588491
$$66$$ 0 0
$$67$$ 4.74456 0.579641 0.289820 0.957081i $$-0.406404\pi$$
0.289820 + 0.957081i $$0.406404\pi$$
$$68$$ 5.37228 0.651485
$$69$$ 0 0
$$70$$ 1.37228 0.164019
$$71$$ 6.74456 0.800432 0.400216 0.916421i $$-0.368935\pi$$
0.400216 + 0.916421i $$0.368935\pi$$
$$72$$ 0 0
$$73$$ 8.74456 1.02347 0.511737 0.859142i $$-0.329002\pi$$
0.511737 + 0.859142i $$0.329002\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 4.62772 0.527377
$$78$$ 0 0
$$79$$ −4.74456 −0.533805 −0.266903 0.963724i $$-0.586000\pi$$
−0.266903 + 0.963724i $$0.586000\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 5.37228 0.593269
$$83$$ 0.744563 0.0817264 0.0408632 0.999165i $$-0.486989\pi$$
0.0408632 + 0.999165i $$0.486989\pi$$
$$84$$ 0 0
$$85$$ −5.37228 −0.582706
$$86$$ −7.37228 −0.794974
$$87$$ 0 0
$$88$$ −3.37228 −0.359486
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −6.51087 −0.682525
$$92$$ −6.74456 −0.703169
$$93$$ 0 0
$$94$$ −8.74456 −0.901933
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ −0.116844 −0.0118637 −0.00593185 0.999982i $$-0.501888\pi$$
−0.00593185 + 0.999982i $$0.501888\pi$$
$$98$$ 5.11684 0.516879
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 11.4891 1.14321 0.571605 0.820529i $$-0.306321\pi$$
0.571605 + 0.820529i $$0.306321\pi$$
$$102$$ 0 0
$$103$$ −9.48913 −0.934991 −0.467496 0.883995i $$-0.654844\pi$$
−0.467496 + 0.883995i $$0.654844\pi$$
$$104$$ 4.74456 0.465243
$$105$$ 0 0
$$106$$ 1.37228 0.133288
$$107$$ 3.48913 0.337306 0.168653 0.985675i $$-0.446058\pi$$
0.168653 + 0.985675i $$0.446058\pi$$
$$108$$ 0 0
$$109$$ 0.116844 0.0111916 0.00559581 0.999984i $$-0.498219\pi$$
0.00559581 + 0.999984i $$0.498219\pi$$
$$110$$ 3.37228 0.321534
$$111$$ 0 0
$$112$$ 1.37228 0.129668
$$113$$ −17.3723 −1.63425 −0.817123 0.576463i $$-0.804433\pi$$
−0.817123 + 0.576463i $$0.804433\pi$$
$$114$$ 0 0
$$115$$ 6.74456 0.628934
$$116$$ −8.11684 −0.753630
$$117$$ 0 0
$$118$$ 12.7446 1.17323
$$119$$ 7.37228 0.675816
$$120$$ 0 0
$$121$$ 0.372281 0.0338438
$$122$$ 5.37228 0.486383
$$123$$ 0 0
$$124$$ −2.62772 −0.235976
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −16.7446 −1.48584 −0.742920 0.669380i $$-0.766560\pi$$
−0.742920 + 0.669380i $$0.766560\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −4.74456 −0.416126
$$131$$ 3.25544 0.284429 0.142214 0.989836i $$-0.454578\pi$$
0.142214 + 0.989836i $$0.454578\pi$$
$$132$$ 0 0
$$133$$ −2.74456 −0.237984
$$134$$ −4.74456 −0.409868
$$135$$ 0 0
$$136$$ −5.37228 −0.460669
$$137$$ −16.7446 −1.43058 −0.715292 0.698825i $$-0.753706\pi$$
−0.715292 + 0.698825i $$0.753706\pi$$
$$138$$ 0 0
$$139$$ −1.88316 −0.159727 −0.0798636 0.996806i $$-0.525448\pi$$
−0.0798636 + 0.996806i $$0.525448\pi$$
$$140$$ −1.37228 −0.115979
$$141$$ 0 0
$$142$$ −6.74456 −0.565991
$$143$$ −16.0000 −1.33799
$$144$$ 0 0
$$145$$ 8.11684 0.674067
$$146$$ −8.74456 −0.723705
$$147$$ 0 0
$$148$$ 1.00000 0.0821995
$$149$$ 11.4891 0.941226 0.470613 0.882340i $$-0.344033\pi$$
0.470613 + 0.882340i $$0.344033\pi$$
$$150$$ 0 0
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ −4.62772 −0.372912
$$155$$ 2.62772 0.211063
$$156$$ 0 0
$$157$$ 17.3723 1.38646 0.693229 0.720717i $$-0.256187\pi$$
0.693229 + 0.720717i $$0.256187\pi$$
$$158$$ 4.74456 0.377457
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ −9.25544 −0.729431
$$162$$ 0 0
$$163$$ 19.3723 1.51735 0.758677 0.651467i $$-0.225846\pi$$
0.758677 + 0.651467i $$0.225846\pi$$
$$164$$ −5.37228 −0.419505
$$165$$ 0 0
$$166$$ −0.744563 −0.0577893
$$167$$ −1.48913 −0.115232 −0.0576160 0.998339i $$-0.518350\pi$$
−0.0576160 + 0.998339i $$0.518350\pi$$
$$168$$ 0 0
$$169$$ 9.51087 0.731606
$$170$$ 5.37228 0.412035
$$171$$ 0 0
$$172$$ 7.37228 0.562131
$$173$$ −22.8614 −1.73812 −0.869060 0.494706i $$-0.835276\pi$$
−0.869060 + 0.494706i $$0.835276\pi$$
$$174$$ 0 0
$$175$$ 1.37228 0.103735
$$176$$ 3.37228 0.254195
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ 26.2337 1.96080 0.980399 0.197023i $$-0.0631272\pi$$
0.980399 + 0.197023i $$0.0631272\pi$$
$$180$$ 0 0
$$181$$ 7.48913 0.556662 0.278331 0.960485i $$-0.410219\pi$$
0.278331 + 0.960485i $$0.410219\pi$$
$$182$$ 6.51087 0.482618
$$183$$ 0 0
$$184$$ 6.74456 0.497216
$$185$$ −1.00000 −0.0735215
$$186$$ 0 0
$$187$$ 18.1168 1.32483
$$188$$ 8.74456 0.637763
$$189$$ 0 0
$$190$$ −2.00000 −0.145095
$$191$$ −18.6277 −1.34785 −0.673927 0.738798i $$-0.735394\pi$$
−0.673927 + 0.738798i $$0.735394\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0.116844 0.00838891
$$195$$ 0 0
$$196$$ −5.11684 −0.365489
$$197$$ −27.4891 −1.95852 −0.979260 0.202610i $$-0.935058\pi$$
−0.979260 + 0.202610i $$0.935058\pi$$
$$198$$ 0 0
$$199$$ −11.2554 −0.797877 −0.398938 0.916978i $$-0.630621\pi$$
−0.398938 + 0.916978i $$0.630621\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ −11.4891 −0.808372
$$203$$ −11.1386 −0.781776
$$204$$ 0 0
$$205$$ 5.37228 0.375216
$$206$$ 9.48913 0.661139
$$207$$ 0 0
$$208$$ −4.74456 −0.328976
$$209$$ −6.74456 −0.466531
$$210$$ 0 0
$$211$$ −14.1168 −0.971844 −0.485922 0.874002i $$-0.661516\pi$$
−0.485922 + 0.874002i $$0.661516\pi$$
$$212$$ −1.37228 −0.0942487
$$213$$ 0 0
$$214$$ −3.48913 −0.238512
$$215$$ −7.37228 −0.502785
$$216$$ 0 0
$$217$$ −3.60597 −0.244789
$$218$$ −0.116844 −0.00791367
$$219$$ 0 0
$$220$$ −3.37228 −0.227359
$$221$$ −25.4891 −1.71458
$$222$$ 0 0
$$223$$ −1.37228 −0.0918948 −0.0459474 0.998944i $$-0.514631\pi$$
−0.0459474 + 0.998944i $$0.514631\pi$$
$$224$$ −1.37228 −0.0916894
$$225$$ 0 0
$$226$$ 17.3723 1.15559
$$227$$ 11.3723 0.754805 0.377402 0.926049i $$-0.376817\pi$$
0.377402 + 0.926049i $$0.376817\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ −6.74456 −0.444723
$$231$$ 0 0
$$232$$ 8.11684 0.532897
$$233$$ −6.23369 −0.408382 −0.204191 0.978931i $$-0.565456\pi$$
−0.204191 + 0.978931i $$0.565456\pi$$
$$234$$ 0 0
$$235$$ −8.74456 −0.570432
$$236$$ −12.7446 −0.829600
$$237$$ 0 0
$$238$$ −7.37228 −0.477874
$$239$$ −17.6060 −1.13884 −0.569418 0.822048i $$-0.692831\pi$$
−0.569418 + 0.822048i $$0.692831\pi$$
$$240$$ 0 0
$$241$$ −10.2337 −0.659210 −0.329605 0.944119i $$-0.606916\pi$$
−0.329605 + 0.944119i $$0.606916\pi$$
$$242$$ −0.372281 −0.0239311
$$243$$ 0 0
$$244$$ −5.37228 −0.343925
$$245$$ 5.11684 0.326903
$$246$$ 0 0
$$247$$ 9.48913 0.603779
$$248$$ 2.62772 0.166860
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ −11.4891 −0.725187 −0.362594 0.931947i $$-0.618109\pi$$
−0.362594 + 0.931947i $$0.618109\pi$$
$$252$$ 0 0
$$253$$ −22.7446 −1.42994
$$254$$ 16.7446 1.05065
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 20.9783 1.30859 0.654294 0.756241i $$-0.272966\pi$$
0.654294 + 0.756241i $$0.272966\pi$$
$$258$$ 0 0
$$259$$ 1.37228 0.0852694
$$260$$ 4.74456 0.294245
$$261$$ 0 0
$$262$$ −3.25544 −0.201122
$$263$$ −0.116844 −0.00720491 −0.00360245 0.999994i $$-0.501147\pi$$
−0.00360245 + 0.999994i $$0.501147\pi$$
$$264$$ 0 0
$$265$$ 1.37228 0.0842986
$$266$$ 2.74456 0.168280
$$267$$ 0 0
$$268$$ 4.74456 0.289820
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ −6.74456 −0.409703 −0.204852 0.978793i $$-0.565671\pi$$
−0.204852 + 0.978793i $$0.565671\pi$$
$$272$$ 5.37228 0.325742
$$273$$ 0 0
$$274$$ 16.7446 1.01158
$$275$$ 3.37228 0.203356
$$276$$ 0 0
$$277$$ 0.744563 0.0447364 0.0223682 0.999750i $$-0.492879\pi$$
0.0223682 + 0.999750i $$0.492879\pi$$
$$278$$ 1.88316 0.112944
$$279$$ 0 0
$$280$$ 1.37228 0.0820095
$$281$$ −24.7446 −1.47614 −0.738068 0.674726i $$-0.764262\pi$$
−0.738068 + 0.674726i $$0.764262\pi$$
$$282$$ 0 0
$$283$$ 5.48913 0.326295 0.163147 0.986602i $$-0.447835\pi$$
0.163147 + 0.986602i $$0.447835\pi$$
$$284$$ 6.74456 0.400216
$$285$$ 0 0
$$286$$ 16.0000 0.946100
$$287$$ −7.37228 −0.435172
$$288$$ 0 0
$$289$$ 11.8614 0.697730
$$290$$ −8.11684 −0.476637
$$291$$ 0 0
$$292$$ 8.74456 0.511737
$$293$$ 18.8614 1.10190 0.550948 0.834540i $$-0.314266\pi$$
0.550948 + 0.834540i $$0.314266\pi$$
$$294$$ 0 0
$$295$$ 12.7446 0.742017
$$296$$ −1.00000 −0.0581238
$$297$$ 0 0
$$298$$ −11.4891 −0.665547
$$299$$ 32.0000 1.85061
$$300$$ 0 0
$$301$$ 10.1168 0.583125
$$302$$ 20.0000 1.15087
$$303$$ 0 0
$$304$$ −2.00000 −0.114708
$$305$$ 5.37228 0.307616
$$306$$ 0 0
$$307$$ −23.4891 −1.34060 −0.670298 0.742092i $$-0.733834\pi$$
−0.670298 + 0.742092i $$0.733834\pi$$
$$308$$ 4.62772 0.263689
$$309$$ 0 0
$$310$$ −2.62772 −0.149244
$$311$$ 1.37228 0.0778149 0.0389075 0.999243i $$-0.487612\pi$$
0.0389075 + 0.999243i $$0.487612\pi$$
$$312$$ 0 0
$$313$$ −3.48913 −0.197217 −0.0986085 0.995126i $$-0.531439\pi$$
−0.0986085 + 0.995126i $$0.531439\pi$$
$$314$$ −17.3723 −0.980375
$$315$$ 0 0
$$316$$ −4.74456 −0.266903
$$317$$ −25.3723 −1.42505 −0.712525 0.701647i $$-0.752448\pi$$
−0.712525 + 0.701647i $$0.752448\pi$$
$$318$$ 0 0
$$319$$ −27.3723 −1.53255
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 9.25544 0.515785
$$323$$ −10.7446 −0.597843
$$324$$ 0 0
$$325$$ −4.74456 −0.263181
$$326$$ −19.3723 −1.07293
$$327$$ 0 0
$$328$$ 5.37228 0.296635
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 23.4891 1.29108 0.645540 0.763727i $$-0.276633\pi$$
0.645540 + 0.763727i $$0.276633\pi$$
$$332$$ 0.744563 0.0408632
$$333$$ 0 0
$$334$$ 1.48913 0.0814813
$$335$$ −4.74456 −0.259223
$$336$$ 0 0
$$337$$ −7.25544 −0.395229 −0.197614 0.980280i $$-0.563319\pi$$
−0.197614 + 0.980280i $$0.563319\pi$$
$$338$$ −9.51087 −0.517323
$$339$$ 0 0
$$340$$ −5.37228 −0.291353
$$341$$ −8.86141 −0.479872
$$342$$ 0 0
$$343$$ −16.6277 −0.897812
$$344$$ −7.37228 −0.397487
$$345$$ 0 0
$$346$$ 22.8614 1.22904
$$347$$ 22.9783 1.23354 0.616769 0.787145i $$-0.288441\pi$$
0.616769 + 0.787145i $$0.288441\pi$$
$$348$$ 0 0
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ −1.37228 −0.0733515
$$351$$ 0 0
$$352$$ −3.37228 −0.179743
$$353$$ 22.8614 1.21679 0.608395 0.793634i $$-0.291814\pi$$
0.608395 + 0.793634i $$0.291814\pi$$
$$354$$ 0 0
$$355$$ −6.74456 −0.357964
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ −26.2337 −1.38649
$$359$$ 30.9783 1.63497 0.817485 0.575950i $$-0.195368\pi$$
0.817485 + 0.575950i $$0.195368\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −7.48913 −0.393620
$$363$$ 0 0
$$364$$ −6.51087 −0.341263
$$365$$ −8.74456 −0.457711
$$366$$ 0 0
$$367$$ 3.88316 0.202699 0.101350 0.994851i $$-0.467684\pi$$
0.101350 + 0.994851i $$0.467684\pi$$
$$368$$ −6.74456 −0.351585
$$369$$ 0 0
$$370$$ 1.00000 0.0519875
$$371$$ −1.88316 −0.0977686
$$372$$ 0 0
$$373$$ −31.4891 −1.63045 −0.815223 0.579148i $$-0.803385\pi$$
−0.815223 + 0.579148i $$0.803385\pi$$
$$374$$ −18.1168 −0.936800
$$375$$ 0 0
$$376$$ −8.74456 −0.450966
$$377$$ 38.5109 1.98341
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 2.00000 0.102598
$$381$$ 0 0
$$382$$ 18.6277 0.953077
$$383$$ 13.4891 0.689262 0.344631 0.938738i $$-0.388004\pi$$
0.344631 + 0.938738i $$0.388004\pi$$
$$384$$ 0 0
$$385$$ −4.62772 −0.235850
$$386$$ −2.00000 −0.101797
$$387$$ 0 0
$$388$$ −0.116844 −0.00593185
$$389$$ −14.8614 −0.753503 −0.376752 0.926314i $$-0.622959\pi$$
−0.376752 + 0.926314i $$0.622959\pi$$
$$390$$ 0 0
$$391$$ −36.2337 −1.83242
$$392$$ 5.11684 0.258440
$$393$$ 0 0
$$394$$ 27.4891 1.38488
$$395$$ 4.74456 0.238725
$$396$$ 0 0
$$397$$ 24.9783 1.25362 0.626811 0.779171i $$-0.284360\pi$$
0.626811 + 0.779171i $$0.284360\pi$$
$$398$$ 11.2554 0.564184
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −10.0000 −0.499376 −0.249688 0.968326i $$-0.580328\pi$$
−0.249688 + 0.968326i $$0.580328\pi$$
$$402$$ 0 0
$$403$$ 12.4674 0.621044
$$404$$ 11.4891 0.571605
$$405$$ 0 0
$$406$$ 11.1386 0.552799
$$407$$ 3.37228 0.167158
$$408$$ 0 0
$$409$$ 6.23369 0.308236 0.154118 0.988052i $$-0.450746\pi$$
0.154118 + 0.988052i $$0.450746\pi$$
$$410$$ −5.37228 −0.265318
$$411$$ 0 0
$$412$$ −9.48913 −0.467496
$$413$$ −17.4891 −0.860584
$$414$$ 0 0
$$415$$ −0.744563 −0.0365491
$$416$$ 4.74456 0.232621
$$417$$ 0 0
$$418$$ 6.74456 0.329887
$$419$$ −13.4891 −0.658987 −0.329493 0.944158i $$-0.606878\pi$$
−0.329493 + 0.944158i $$0.606878\pi$$
$$420$$ 0 0
$$421$$ −7.48913 −0.364998 −0.182499 0.983206i $$-0.558419\pi$$
−0.182499 + 0.983206i $$0.558419\pi$$
$$422$$ 14.1168 0.687197
$$423$$ 0 0
$$424$$ 1.37228 0.0666439
$$425$$ 5.37228 0.260594
$$426$$ 0 0
$$427$$ −7.37228 −0.356770
$$428$$ 3.48913 0.168653
$$429$$ 0 0
$$430$$ 7.37228 0.355523
$$431$$ 1.37228 0.0661005 0.0330502 0.999454i $$-0.489478\pi$$
0.0330502 + 0.999454i $$0.489478\pi$$
$$432$$ 0 0
$$433$$ 12.9783 0.623695 0.311847 0.950132i $$-0.399052\pi$$
0.311847 + 0.950132i $$0.399052\pi$$
$$434$$ 3.60597 0.173092
$$435$$ 0 0
$$436$$ 0.116844 0.00559581
$$437$$ 13.4891 0.645272
$$438$$ 0 0
$$439$$ −13.3723 −0.638224 −0.319112 0.947717i $$-0.603385\pi$$
−0.319112 + 0.947717i $$0.603385\pi$$
$$440$$ 3.37228 0.160767
$$441$$ 0 0
$$442$$ 25.4891 1.21239
$$443$$ 16.9783 0.806661 0.403331 0.915054i $$-0.367852\pi$$
0.403331 + 0.915054i $$0.367852\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ 1.37228 0.0649794
$$447$$ 0 0
$$448$$ 1.37228 0.0648342
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ −18.1168 −0.853089
$$452$$ −17.3723 −0.817123
$$453$$ 0 0
$$454$$ −11.3723 −0.533728
$$455$$ 6.51087 0.305235
$$456$$ 0 0
$$457$$ 18.8614 0.882299 0.441150 0.897434i $$-0.354571\pi$$
0.441150 + 0.897434i $$0.354571\pi$$
$$458$$ 10.0000 0.467269
$$459$$ 0 0
$$460$$ 6.74456 0.314467
$$461$$ 39.0951 1.82084 0.910420 0.413685i $$-0.135759\pi$$
0.910420 + 0.413685i $$0.135759\pi$$
$$462$$ 0 0
$$463$$ −5.48913 −0.255101 −0.127551 0.991832i $$-0.540712\pi$$
−0.127551 + 0.991832i $$0.540712\pi$$
$$464$$ −8.11684 −0.376815
$$465$$ 0 0
$$466$$ 6.23369 0.288770
$$467$$ −30.3505 −1.40446 −0.702228 0.711953i $$-0.747811\pi$$
−0.702228 + 0.711953i $$0.747811\pi$$
$$468$$ 0 0
$$469$$ 6.51087 0.300644
$$470$$ 8.74456 0.403357
$$471$$ 0 0
$$472$$ 12.7446 0.586616
$$473$$ 24.8614 1.14313
$$474$$ 0 0
$$475$$ −2.00000 −0.0917663
$$476$$ 7.37228 0.337908
$$477$$ 0 0
$$478$$ 17.6060 0.805278
$$479$$ −3.25544 −0.148745 −0.0743724 0.997231i $$-0.523695\pi$$
−0.0743724 + 0.997231i $$0.523695\pi$$
$$480$$ 0 0
$$481$$ −4.74456 −0.216333
$$482$$ 10.2337 0.466132
$$483$$ 0 0
$$484$$ 0.372281 0.0169219
$$485$$ 0.116844 0.00530561
$$486$$ 0 0
$$487$$ −37.7228 −1.70938 −0.854692 0.519136i $$-0.826254\pi$$
−0.854692 + 0.519136i $$0.826254\pi$$
$$488$$ 5.37228 0.243192
$$489$$ 0 0
$$490$$ −5.11684 −0.231155
$$491$$ −14.9783 −0.675959 −0.337979 0.941153i $$-0.609743\pi$$
−0.337979 + 0.941153i $$0.609743\pi$$
$$492$$ 0 0
$$493$$ −43.6060 −1.96391
$$494$$ −9.48913 −0.426936
$$495$$ 0 0
$$496$$ −2.62772 −0.117988
$$497$$ 9.25544 0.415163
$$498$$ 0 0
$$499$$ −24.9783 −1.11818 −0.559090 0.829107i $$-0.688849\pi$$
−0.559090 + 0.829107i $$0.688849\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 11.4891 0.512785
$$503$$ −29.4891 −1.31486 −0.657428 0.753518i $$-0.728355\pi$$
−0.657428 + 0.753518i $$0.728355\pi$$
$$504$$ 0 0
$$505$$ −11.4891 −0.511259
$$506$$ 22.7446 1.01112
$$507$$ 0 0
$$508$$ −16.7446 −0.742920
$$509$$ 11.2554 0.498888 0.249444 0.968389i $$-0.419752\pi$$
0.249444 + 0.968389i $$0.419752\pi$$
$$510$$ 0 0
$$511$$ 12.0000 0.530849
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −20.9783 −0.925311
$$515$$ 9.48913 0.418141
$$516$$ 0 0
$$517$$ 29.4891 1.29693
$$518$$ −1.37228 −0.0602946
$$519$$ 0 0
$$520$$ −4.74456 −0.208063
$$521$$ −27.0951 −1.18706 −0.593529 0.804813i $$-0.702266\pi$$
−0.593529 + 0.804813i $$0.702266\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 3.25544 0.142214
$$525$$ 0 0
$$526$$ 0.116844 0.00509464
$$527$$ −14.1168 −0.614939
$$528$$ 0 0
$$529$$ 22.4891 0.977788
$$530$$ −1.37228 −0.0596081
$$531$$ 0 0
$$532$$ −2.74456 −0.118992
$$533$$ 25.4891 1.10406
$$534$$ 0 0
$$535$$ −3.48913 −0.150848
$$536$$ −4.74456 −0.204934
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ −17.2554 −0.743244
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 6.74456 0.289704
$$543$$ 0 0
$$544$$ −5.37228 −0.230335
$$545$$ −0.116844 −0.00500505
$$546$$ 0 0
$$547$$ 39.3723 1.68344 0.841719 0.539916i $$-0.181544\pi$$
0.841719 + 0.539916i $$0.181544\pi$$
$$548$$ −16.7446 −0.715292
$$549$$ 0 0
$$550$$ −3.37228 −0.143795
$$551$$ 16.2337 0.691578
$$552$$ 0 0
$$553$$ −6.51087 −0.276871
$$554$$ −0.744563 −0.0316334
$$555$$ 0 0
$$556$$ −1.88316 −0.0798636
$$557$$ −8.74456 −0.370519 −0.185260 0.982690i $$-0.559313\pi$$
−0.185260 + 0.982690i $$0.559313\pi$$
$$558$$ 0 0
$$559$$ −34.9783 −1.47942
$$560$$ −1.37228 −0.0579895
$$561$$ 0 0
$$562$$ 24.7446 1.04379
$$563$$ −33.0951 −1.39479 −0.697396 0.716686i $$-0.745658\pi$$
−0.697396 + 0.716686i $$0.745658\pi$$
$$564$$ 0 0
$$565$$ 17.3723 0.730857
$$566$$ −5.48913 −0.230725
$$567$$ 0 0
$$568$$ −6.74456 −0.282996
$$569$$ 32.9783 1.38252 0.691260 0.722606i $$-0.257056\pi$$
0.691260 + 0.722606i $$0.257056\pi$$
$$570$$ 0 0
$$571$$ −27.6060 −1.15527 −0.577637 0.816294i $$-0.696025\pi$$
−0.577637 + 0.816294i $$0.696025\pi$$
$$572$$ −16.0000 −0.668994
$$573$$ 0 0
$$574$$ 7.37228 0.307713
$$575$$ −6.74456 −0.281268
$$576$$ 0 0
$$577$$ −7.48913 −0.311776 −0.155888 0.987775i $$-0.549824\pi$$
−0.155888 + 0.987775i $$0.549824\pi$$
$$578$$ −11.8614 −0.493369
$$579$$ 0 0
$$580$$ 8.11684 0.337034
$$581$$ 1.02175 0.0423893
$$582$$ 0 0
$$583$$ −4.62772 −0.191661
$$584$$ −8.74456 −0.361853
$$585$$ 0 0
$$586$$ −18.8614 −0.779158
$$587$$ −47.8397 −1.97455 −0.987277 0.159010i $$-0.949170\pi$$
−0.987277 + 0.159010i $$0.949170\pi$$
$$588$$ 0 0
$$589$$ 5.25544 0.216547
$$590$$ −12.7446 −0.524685
$$591$$ 0 0
$$592$$ 1.00000 0.0410997
$$593$$ 10.2337 0.420247 0.210124 0.977675i $$-0.432613\pi$$
0.210124 + 0.977675i $$0.432613\pi$$
$$594$$ 0 0
$$595$$ −7.37228 −0.302234
$$596$$ 11.4891 0.470613
$$597$$ 0 0
$$598$$ −32.0000 −1.30858
$$599$$ −17.4891 −0.714586 −0.357293 0.933992i $$-0.616300\pi$$
−0.357293 + 0.933992i $$0.616300\pi$$
$$600$$ 0 0
$$601$$ −5.37228 −0.219140 −0.109570 0.993979i $$-0.534947\pi$$
−0.109570 + 0.993979i $$0.534947\pi$$
$$602$$ −10.1168 −0.412332
$$603$$ 0 0
$$604$$ −20.0000 −0.813788
$$605$$ −0.372281 −0.0151354
$$606$$ 0 0
$$607$$ 17.7228 0.719347 0.359673 0.933078i $$-0.382888\pi$$
0.359673 + 0.933078i $$0.382888\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ −5.37228 −0.217517
$$611$$ −41.4891 −1.67847
$$612$$ 0 0
$$613$$ 43.0951 1.74059 0.870297 0.492527i $$-0.163927\pi$$
0.870297 + 0.492527i $$0.163927\pi$$
$$614$$ 23.4891 0.947944
$$615$$ 0 0
$$616$$ −4.62772 −0.186456
$$617$$ 31.7228 1.27711 0.638556 0.769575i $$-0.279532\pi$$
0.638556 + 0.769575i $$0.279532\pi$$
$$618$$ 0 0
$$619$$ 30.1168 1.21050 0.605249 0.796036i $$-0.293074\pi$$
0.605249 + 0.796036i $$0.293074\pi$$
$$620$$ 2.62772 0.105532
$$621$$ 0 0
$$622$$ −1.37228 −0.0550235
$$623$$ −13.7228 −0.549793
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 3.48913 0.139453
$$627$$ 0 0
$$628$$ 17.3723 0.693229
$$629$$ 5.37228 0.214207
$$630$$ 0 0
$$631$$ −19.0951 −0.760164 −0.380082 0.924953i $$-0.624104\pi$$
−0.380082 + 0.924953i $$0.624104\pi$$
$$632$$ 4.74456 0.188729
$$633$$ 0 0
$$634$$ 25.3723 1.00766
$$635$$ 16.7446 0.664488
$$636$$ 0 0
$$637$$ 24.2772 0.961897
$$638$$ 27.3723 1.08368
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ 49.6060 1.95932 0.979659 0.200670i $$-0.0643117\pi$$
0.979659 + 0.200670i $$0.0643117\pi$$
$$642$$ 0 0
$$643$$ 3.37228 0.132990 0.0664949 0.997787i $$-0.478818\pi$$
0.0664949 + 0.997787i $$0.478818\pi$$
$$644$$ −9.25544 −0.364715
$$645$$ 0 0
$$646$$ 10.7446 0.422739
$$647$$ 13.4891 0.530312 0.265156 0.964205i $$-0.414577\pi$$
0.265156 + 0.964205i $$0.414577\pi$$
$$648$$ 0 0
$$649$$ −42.9783 −1.68704
$$650$$ 4.74456 0.186097
$$651$$ 0 0
$$652$$ 19.3723 0.758677
$$653$$ −0.510875 −0.0199921 −0.00999604 0.999950i $$-0.503182\pi$$
−0.00999604 + 0.999950i $$0.503182\pi$$
$$654$$ 0 0
$$655$$ −3.25544 −0.127200
$$656$$ −5.37228 −0.209752
$$657$$ 0 0
$$658$$ −12.0000 −0.467809
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 4.35053 0.169216 0.0846080 0.996414i $$-0.473036\pi$$
0.0846080 + 0.996414i $$0.473036\pi$$
$$662$$ −23.4891 −0.912931
$$663$$ 0 0
$$664$$ −0.744563 −0.0288946
$$665$$ 2.74456 0.106430
$$666$$ 0 0
$$667$$ 54.7446 2.11972
$$668$$ −1.48913 −0.0576160
$$669$$ 0 0
$$670$$ 4.74456 0.183298
$$671$$ −18.1168 −0.699393
$$672$$ 0 0
$$673$$ −8.51087 −0.328070 −0.164035 0.986455i $$-0.552451\pi$$
−0.164035 + 0.986455i $$0.552451\pi$$
$$674$$ 7.25544 0.279469
$$675$$ 0 0
$$676$$ 9.51087 0.365803
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ 0 0
$$679$$ −0.160343 −0.00615339
$$680$$ 5.37228 0.206018
$$681$$ 0 0
$$682$$ 8.86141 0.339321
$$683$$ −8.62772 −0.330130 −0.165065 0.986283i $$-0.552783\pi$$
−0.165065 + 0.986283i $$0.552783\pi$$
$$684$$ 0 0
$$685$$ 16.7446 0.639777
$$686$$ 16.6277 0.634849
$$687$$ 0 0
$$688$$ 7.37228 0.281066
$$689$$ 6.51087 0.248045
$$690$$ 0 0
$$691$$ 22.3505 0.850254 0.425127 0.905134i $$-0.360229\pi$$
0.425127 + 0.905134i $$0.360229\pi$$
$$692$$ −22.8614 −0.869060
$$693$$ 0 0
$$694$$ −22.9783 −0.872242
$$695$$ 1.88316 0.0714322
$$696$$ 0 0
$$697$$ −28.8614 −1.09320
$$698$$ 22.0000 0.832712
$$699$$ 0 0
$$700$$ 1.37228 0.0518674
$$701$$ 26.4674 0.999659 0.499829 0.866124i $$-0.333396\pi$$
0.499829 + 0.866124i $$0.333396\pi$$
$$702$$ 0 0
$$703$$ −2.00000 −0.0754314
$$704$$ 3.37228 0.127098
$$705$$ 0 0
$$706$$ −22.8614 −0.860400
$$707$$ 15.7663 0.592953
$$708$$ 0 0
$$709$$ 40.1168 1.50662 0.753310 0.657666i $$-0.228456\pi$$
0.753310 + 0.657666i $$0.228456\pi$$
$$710$$ 6.74456 0.253119
$$711$$ 0 0
$$712$$ 10.0000 0.374766
$$713$$ 17.7228 0.663725
$$714$$ 0 0
$$715$$ 16.0000 0.598366
$$716$$ 26.2337 0.980399
$$717$$ 0 0
$$718$$ −30.9783 −1.15610
$$719$$ 34.7446 1.29575 0.647877 0.761745i $$-0.275657\pi$$
0.647877 + 0.761745i $$0.275657\pi$$
$$720$$ 0 0
$$721$$ −13.0217 −0.484955
$$722$$ 15.0000 0.558242
$$723$$ 0 0
$$724$$ 7.48913 0.278331
$$725$$ −8.11684 −0.301452
$$726$$ 0 0
$$727$$ −48.0000 −1.78022 −0.890111 0.455744i $$-0.849373\pi$$
−0.890111 + 0.455744i $$0.849373\pi$$
$$728$$ 6.51087 0.241309
$$729$$ 0 0
$$730$$ 8.74456 0.323651
$$731$$ 39.6060 1.46488
$$732$$ 0 0
$$733$$ −29.3723 −1.08489 −0.542445 0.840091i $$-0.682501\pi$$
−0.542445 + 0.840091i $$0.682501\pi$$
$$734$$ −3.88316 −0.143330
$$735$$ 0 0
$$736$$ 6.74456 0.248608
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ −36.8614 −1.35597 −0.677984 0.735076i $$-0.737146\pi$$
−0.677984 + 0.735076i $$0.737146\pi$$
$$740$$ −1.00000 −0.0367607
$$741$$ 0 0
$$742$$ 1.88316 0.0691328
$$743$$ 5.37228 0.197090 0.0985449 0.995133i $$-0.468581\pi$$
0.0985449 + 0.995133i $$0.468581\pi$$
$$744$$ 0 0
$$745$$ −11.4891 −0.420929
$$746$$ 31.4891 1.15290
$$747$$ 0 0
$$748$$ 18.1168 0.662417
$$749$$ 4.78806 0.174952
$$750$$ 0 0
$$751$$ 10.7446 0.392075 0.196037 0.980596i $$-0.437193\pi$$
0.196037 + 0.980596i $$0.437193\pi$$
$$752$$ 8.74456 0.318881
$$753$$ 0 0
$$754$$ −38.5109 −1.40248
$$755$$ 20.0000 0.727875
$$756$$ 0 0
$$757$$ 43.9565 1.59763 0.798813 0.601579i $$-0.205462\pi$$
0.798813 + 0.601579i $$0.205462\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 0 0
$$760$$ −2.00000 −0.0725476
$$761$$ −5.37228 −0.194745 −0.0973725 0.995248i $$-0.531044\pi$$
−0.0973725 + 0.995248i $$0.531044\pi$$
$$762$$ 0 0
$$763$$ 0.160343 0.00580480
$$764$$ −18.6277 −0.673927
$$765$$ 0 0
$$766$$ −13.4891 −0.487382
$$767$$ 60.4674 2.18335
$$768$$ 0 0
$$769$$ 46.2337 1.66723 0.833615 0.552346i $$-0.186267\pi$$
0.833615 + 0.552346i $$0.186267\pi$$
$$770$$ 4.62772 0.166771
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ 24.3505 0.875828 0.437914 0.899017i $$-0.355717\pi$$
0.437914 + 0.899017i $$0.355717\pi$$
$$774$$ 0 0
$$775$$ −2.62772 −0.0943904
$$776$$ 0.116844 0.00419445
$$777$$ 0 0
$$778$$ 14.8614 0.532807
$$779$$ 10.7446 0.384964
$$780$$ 0 0
$$781$$ 22.7446 0.813864
$$782$$ 36.2337 1.29571
$$783$$ 0 0
$$784$$ −5.11684 −0.182744
$$785$$ −17.3723 −0.620043
$$786$$ 0 0
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ −27.4891 −0.979260
$$789$$ 0 0
$$790$$ −4.74456 −0.168804
$$791$$ −23.8397 −0.847641
$$792$$ 0 0
$$793$$ 25.4891 0.905145
$$794$$ −24.9783 −0.886445
$$795$$ 0 0
$$796$$ −11.2554 −0.398938
$$797$$ 27.4891 0.973715 0.486857 0.873481i $$-0.338143\pi$$
0.486857 + 0.873481i $$0.338143\pi$$
$$798$$ 0 0
$$799$$ 46.9783 1.66197
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 10.0000 0.353112
$$803$$ 29.4891 1.04065
$$804$$ 0 0
$$805$$ 9.25544 0.326211
$$806$$ −12.4674 −0.439145
$$807$$ 0 0
$$808$$ −11.4891 −0.404186
$$809$$ 51.4891 1.81026 0.905131 0.425134i $$-0.139773\pi$$
0.905131 + 0.425134i $$0.139773\pi$$
$$810$$ 0 0
$$811$$ 49.4891 1.73780 0.868899 0.494989i $$-0.164828\pi$$
0.868899 + 0.494989i $$0.164828\pi$$
$$812$$ −11.1386 −0.390888
$$813$$ 0 0
$$814$$ −3.37228 −0.118198
$$815$$ −19.3723 −0.678581
$$816$$ 0 0
$$817$$ −14.7446 −0.515847
$$818$$ −6.23369 −0.217956
$$819$$ 0 0
$$820$$ 5.37228 0.187608
$$821$$ 32.7446 1.14279 0.571397 0.820674i $$-0.306402\pi$$
0.571397 + 0.820674i $$0.306402\pi$$
$$822$$ 0 0
$$823$$ −39.7228 −1.38465 −0.692325 0.721586i $$-0.743414\pi$$
−0.692325 + 0.721586i $$0.743414\pi$$
$$824$$ 9.48913 0.330569
$$825$$ 0 0
$$826$$ 17.4891 0.608524
$$827$$ 54.3505 1.88995 0.944977 0.327138i $$-0.106084\pi$$
0.944977 + 0.327138i $$0.106084\pi$$
$$828$$ 0 0
$$829$$ −40.3505 −1.40143 −0.700716 0.713440i $$-0.747136\pi$$
−0.700716 + 0.713440i $$0.747136\pi$$
$$830$$ 0.744563 0.0258441
$$831$$ 0 0
$$832$$ −4.74456 −0.164488
$$833$$ −27.4891 −0.952442
$$834$$ 0 0
$$835$$ 1.48913 0.0515333
$$836$$ −6.74456 −0.233266
$$837$$ 0 0
$$838$$ 13.4891 0.465974
$$839$$ 29.4891 1.01808 0.509039 0.860744i $$-0.330001\pi$$
0.509039 + 0.860744i $$0.330001\pi$$
$$840$$ 0 0
$$841$$ 36.8832 1.27183
$$842$$ 7.48913 0.258092
$$843$$ 0 0
$$844$$ −14.1168 −0.485922
$$845$$ −9.51087 −0.327184
$$846$$ 0 0
$$847$$ 0.510875 0.0175539
$$848$$ −1.37228 −0.0471243
$$849$$ 0 0
$$850$$ −5.37228 −0.184268
$$851$$ −6.74456 −0.231201
$$852$$ 0 0
$$853$$ 35.4891 1.21512 0.607562 0.794272i $$-0.292148\pi$$
0.607562 + 0.794272i $$0.292148\pi$$
$$854$$ 7.37228 0.252274
$$855$$ 0 0
$$856$$ −3.48913 −0.119256
$$857$$ 29.1386 0.995355 0.497678 0.867362i $$-0.334186\pi$$
0.497678 + 0.867362i $$0.334186\pi$$
$$858$$ 0 0
$$859$$ −19.7228 −0.672934 −0.336467 0.941695i $$-0.609232\pi$$
−0.336467 + 0.941695i $$0.609232\pi$$
$$860$$ −7.37228 −0.251393
$$861$$ 0 0
$$862$$ −1.37228 −0.0467401
$$863$$ −53.8397 −1.83272 −0.916362 0.400352i $$-0.868888\pi$$
−0.916362 + 0.400352i $$0.868888\pi$$
$$864$$ 0 0
$$865$$ 22.8614 0.777311
$$866$$ −12.9783 −0.441019
$$867$$ 0 0
$$868$$ −3.60597 −0.122395
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −22.5109 −0.762752
$$872$$ −0.116844 −0.00395684
$$873$$ 0 0
$$874$$ −13.4891 −0.456276
$$875$$ −1.37228 −0.0463916
$$876$$ 0 0
$$877$$ 49.3723 1.66718 0.833592 0.552381i $$-0.186281\pi$$
0.833592 + 0.552381i $$0.186281\pi$$
$$878$$ 13.3723 0.451293
$$879$$ 0 0
$$880$$ −3.37228 −0.113680
$$881$$ −1.37228 −0.0462333 −0.0231167 0.999733i $$-0.507359\pi$$
−0.0231167 + 0.999733i $$0.507359\pi$$
$$882$$ 0 0
$$883$$ −11.3723 −0.382708 −0.191354 0.981521i $$-0.561288\pi$$
−0.191354 + 0.981521i $$0.561288\pi$$
$$884$$ −25.4891 −0.857292
$$885$$ 0 0
$$886$$ −16.9783 −0.570395
$$887$$ 25.3723 0.851918 0.425959 0.904743i $$-0.359937\pi$$
0.425959 + 0.904743i $$0.359937\pi$$
$$888$$ 0 0
$$889$$ −22.9783 −0.770666
$$890$$ −10.0000 −0.335201
$$891$$ 0 0
$$892$$ −1.37228 −0.0459474
$$893$$ −17.4891 −0.585251
$$894$$ 0 0
$$895$$ −26.2337 −0.876895
$$896$$ −1.37228 −0.0458447
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ 21.3288 0.711355
$$900$$ 0 0
$$901$$ −7.37228 −0.245606
$$902$$ 18.1168 0.603225
$$903$$ 0 0
$$904$$ 17.3723 0.577793
$$905$$ −7.48913 −0.248947
$$906$$ 0 0
$$907$$ 40.4674 1.34370 0.671849 0.740689i $$-0.265501\pi$$
0.671849 + 0.740689i $$0.265501\pi$$
$$908$$ 11.3723 0.377402
$$909$$ 0 0
$$910$$ −6.51087 −0.215833
$$911$$ −17.7663 −0.588624 −0.294312 0.955709i $$-0.595091\pi$$
−0.294312 + 0.955709i $$0.595091\pi$$
$$912$$ 0 0
$$913$$ 2.51087 0.0830978
$$914$$ −18.8614 −0.623880
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ 4.46738 0.147526
$$918$$ 0 0
$$919$$ 7.72281 0.254752 0.127376 0.991854i $$-0.459344\pi$$
0.127376 + 0.991854i $$0.459344\pi$$
$$920$$ −6.74456 −0.222362
$$921$$ 0 0
$$922$$ −39.0951 −1.28753
$$923$$ −32.0000 −1.05329
$$924$$ 0 0
$$925$$ 1.00000 0.0328798
$$926$$ 5.48913 0.180384
$$927$$ 0 0
$$928$$ 8.11684 0.266448
$$929$$ −31.0951 −1.02020 −0.510098 0.860116i $$-0.670391\pi$$
−0.510098 + 0.860116i $$0.670391\pi$$
$$930$$ 0 0
$$931$$ 10.2337 0.335396
$$932$$ −6.23369 −0.204191
$$933$$ 0 0
$$934$$ 30.3505 0.993100
$$935$$ −18.1168 −0.592484
$$936$$ 0 0
$$937$$ 36.9783 1.20803 0.604013 0.796974i $$-0.293567\pi$$
0.604013 + 0.796974i $$0.293567\pi$$
$$938$$ −6.51087 −0.212588
$$939$$ 0 0
$$940$$ −8.74456 −0.285216
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ 36.2337 1.17993
$$944$$ −12.7446 −0.414800
$$945$$ 0 0
$$946$$ −24.8614 −0.808314
$$947$$ −37.0951 −1.20543 −0.602714 0.797957i $$-0.705914\pi$$
−0.602714 + 0.797957i $$0.705914\pi$$
$$948$$ 0 0
$$949$$ −41.4891 −1.34679
$$950$$ 2.00000 0.0648886
$$951$$ 0 0
$$952$$ −7.37228 −0.238937
$$953$$ −46.2337 −1.49766 −0.748828 0.662764i $$-0.769383\pi$$
−0.748828 + 0.662764i $$0.769383\pi$$
$$954$$ 0 0
$$955$$ 18.6277 0.602779
$$956$$ −17.6060 −0.569418
$$957$$ 0 0
$$958$$ 3.25544 0.105178
$$959$$ −22.9783 −0.742006
$$960$$ 0 0
$$961$$ −24.0951 −0.777261
$$962$$ 4.74456 0.152971
$$963$$ 0 0
$$964$$ −10.2337 −0.329605
$$965$$ −2.00000 −0.0643823
$$966$$ 0 0
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ −0.372281 −0.0119656
$$969$$ 0 0
$$970$$ −0.116844 −0.00375163
$$971$$ 19.6060 0.629185 0.314593 0.949227i $$-0.398132\pi$$
0.314593 + 0.949227i $$0.398132\pi$$
$$972$$ 0 0
$$973$$ −2.58422 −0.0828463
$$974$$ 37.7228 1.20872
$$975$$ 0 0
$$976$$ −5.37228 −0.171963
$$977$$ 11.8832 0.380176 0.190088 0.981767i $$-0.439123\pi$$
0.190088 + 0.981767i $$0.439123\pi$$
$$978$$ 0 0
$$979$$ −33.7228 −1.07779
$$980$$ 5.11684 0.163452
$$981$$ 0 0
$$982$$ 14.9783 0.477975
$$983$$ −30.8614 −0.984326 −0.492163 0.870503i $$-0.663794\pi$$
−0.492163 + 0.870503i $$0.663794\pi$$
$$984$$ 0 0
$$985$$ 27.4891 0.875876
$$986$$ 43.6060 1.38870
$$987$$ 0 0
$$988$$ 9.48913 0.301889
$$989$$ −49.7228 −1.58109
$$990$$ 0 0
$$991$$ 25.6060 0.813400 0.406700 0.913562i $$-0.366679\pi$$
0.406700 + 0.913562i $$0.366679\pi$$
$$992$$ 2.62772 0.0834302
$$993$$ 0 0
$$994$$ −9.25544 −0.293565
$$995$$ 11.2554 0.356821
$$996$$ 0 0
$$997$$ 26.2337 0.830829 0.415415 0.909632i $$-0.363637\pi$$
0.415415 + 0.909632i $$0.363637\pi$$
$$998$$ 24.9783 0.790673
$$999$$ 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 3330.2.a.bb.1.2 2
3.2 odd 2 370.2.a.f.1.2 2
12.11 even 2 2960.2.a.o.1.1 2
15.2 even 4 1850.2.b.m.149.4 4
15.8 even 4 1850.2.b.m.149.1 4
15.14 odd 2 1850.2.a.q.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 3.2 odd 2
1850.2.a.q.1.1 2 15.14 odd 2
1850.2.b.m.149.1 4 15.8 even 4
1850.2.b.m.149.4 4 15.2 even 4
2960.2.a.o.1.1 2 12.11 even 2
3330.2.a.bb.1.2 2 1.1 even 1 trivial