Properties

 Label 5445.2.a.z.1.3 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.90321 q^{2} +1.62222 q^{4} -1.00000 q^{5} +4.42864 q^{7} -0.719004 q^{8} +O(q^{10})$$ $$q+1.90321 q^{2} +1.62222 q^{4} -1.00000 q^{5} +4.42864 q^{7} -0.719004 q^{8} -1.90321 q^{10} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} -7.05086 q^{19} -1.62222 q^{20} -8.85728 q^{23} +1.00000 q^{25} +1.18421 q^{26} +7.18421 q^{28} -7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} -4.42864 q^{35} -2.00000 q^{37} -13.4193 q^{38} +0.719004 q^{40} -0.193576 q^{41} -5.67307 q^{43} -16.8573 q^{46} +2.75557 q^{47} +12.6128 q^{49} +1.90321 q^{50} +1.00937 q^{52} +10.8573 q^{53} -3.18421 q^{56} -14.8573 q^{58} +4.85728 q^{59} -6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} -0.622216 q^{65} -1.24443 q^{67} -8.40990 q^{68} -8.42864 q^{70} -2.75557 q^{71} -4.23506 q^{73} -3.80642 q^{74} -11.4380 q^{76} -8.56199 q^{79} +4.61285 q^{80} -0.368416 q^{82} +0.133353 q^{83} +5.18421 q^{85} -10.7971 q^{86} -5.61285 q^{89} +2.75557 q^{91} -14.3684 q^{92} +5.24443 q^{94} +7.05086 q^{95} +7.24443 q^{97} +24.0049 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 5 q^{4} - 3 q^{5} - 9 q^{8}+O(q^{10})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 - 9 * q^8 $$3 q - q^{2} + 5 q^{4} - 3 q^{5} - 9 q^{8} + q^{10} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} - 8 q^{19} - 5 q^{20} + 3 q^{25} - 10 q^{26} + 8 q^{28} - 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} - 6 q^{37} + 9 q^{40} - 14 q^{41} - 4 q^{43} - 24 q^{46} + 8 q^{47} + 11 q^{49} - q^{50} + 30 q^{52} + 6 q^{53} + 4 q^{56} - 18 q^{58} - 12 q^{59} + 6 q^{61} + 16 q^{62} + 13 q^{64} - 2 q^{65} - 4 q^{67} + 42 q^{68} - 12 q^{70} - 8 q^{71} + 14 q^{73} + 2 q^{74} - 48 q^{76} - 12 q^{79} - 13 q^{80} + 26 q^{82} + 2 q^{85} + 8 q^{86} + 10 q^{89} + 8 q^{91} - 16 q^{92} + 16 q^{94} + 8 q^{95} + 22 q^{97} + 39 q^{98}+O(q^{100})$$ 3 * q - q^2 + 5 * q^4 - 3 * q^5 - 9 * q^8 + q^10 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 - 8 * q^19 - 5 * q^20 + 3 * q^25 - 10 * q^26 + 8 * q^28 - 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 - 6 * q^37 + 9 * q^40 - 14 * q^41 - 4 * q^43 - 24 * q^46 + 8 * q^47 + 11 * q^49 - q^50 + 30 * q^52 + 6 * q^53 + 4 * q^56 - 18 * q^58 - 12 * q^59 + 6 * q^61 + 16 * q^62 + 13 * q^64 - 2 * q^65 - 4 * q^67 + 42 * q^68 - 12 * q^70 - 8 * q^71 + 14 * q^73 + 2 * q^74 - 48 * q^76 - 12 * q^79 - 13 * q^80 + 26 * q^82 + 2 * q^85 + 8 * q^86 + 10 * q^89 + 8 * q^91 - 16 * q^92 + 16 * q^94 + 8 * q^95 + 22 * q^97 + 39 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.90321 1.34577 0.672887 0.739745i $$-0.265054\pi$$
0.672887 + 0.739745i $$0.265054\pi$$
$$3$$ 0 0
$$4$$ 1.62222 0.811108
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.42864 1.67387 0.836934 0.547304i $$-0.184346\pi$$
0.836934 + 0.547304i $$0.184346\pi$$
$$8$$ −0.719004 −0.254206
$$9$$ 0 0
$$10$$ −1.90321 −0.601848
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 0.622216 0.172572 0.0862858 0.996270i $$-0.472500\pi$$
0.0862858 + 0.996270i $$0.472500\pi$$
$$14$$ 8.42864 2.25265
$$15$$ 0 0
$$16$$ −4.61285 −1.15321
$$17$$ −5.18421 −1.25736 −0.628678 0.777666i $$-0.716403\pi$$
−0.628678 + 0.777666i $$0.716403\pi$$
$$18$$ 0 0
$$19$$ −7.05086 −1.61758 −0.808789 0.588100i $$-0.799876\pi$$
−0.808789 + 0.588100i $$0.799876\pi$$
$$20$$ −1.62222 −0.362738
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −8.85728 −1.84687 −0.923435 0.383754i $$-0.874631\pi$$
−0.923435 + 0.383754i $$0.874631\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.18421 0.232242
$$27$$ 0 0
$$28$$ 7.18421 1.35769
$$29$$ −7.80642 −1.44962 −0.724808 0.688951i $$-0.758072\pi$$
−0.724808 + 0.688951i $$0.758072\pi$$
$$30$$ 0 0
$$31$$ 2.75557 0.494915 0.247457 0.968899i $$-0.420405\pi$$
0.247457 + 0.968899i $$0.420405\pi$$
$$32$$ −7.34122 −1.29776
$$33$$ 0 0
$$34$$ −9.86665 −1.69212
$$35$$ −4.42864 −0.748577
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −13.4193 −2.17689
$$39$$ 0 0
$$40$$ 0.719004 0.113684
$$41$$ −0.193576 −0.0302315 −0.0151158 0.999886i $$-0.504812\pi$$
−0.0151158 + 0.999886i $$0.504812\pi$$
$$42$$ 0 0
$$43$$ −5.67307 −0.865135 −0.432568 0.901602i $$-0.642392\pi$$
−0.432568 + 0.901602i $$0.642392\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −16.8573 −2.48547
$$47$$ 2.75557 0.401941 0.200971 0.979597i $$-0.435590\pi$$
0.200971 + 0.979597i $$0.435590\pi$$
$$48$$ 0 0
$$49$$ 12.6128 1.80184
$$50$$ 1.90321 0.269155
$$51$$ 0 0
$$52$$ 1.00937 0.139974
$$53$$ 10.8573 1.49136 0.745681 0.666303i $$-0.232124\pi$$
0.745681 + 0.666303i $$0.232124\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.18421 −0.425508
$$57$$ 0 0
$$58$$ −14.8573 −1.95086
$$59$$ 4.85728 0.632364 0.316182 0.948699i $$-0.397599\pi$$
0.316182 + 0.948699i $$0.397599\pi$$
$$60$$ 0 0
$$61$$ −6.85728 −0.877985 −0.438992 0.898491i $$-0.644664\pi$$
−0.438992 + 0.898491i $$0.644664\pi$$
$$62$$ 5.24443 0.666043
$$63$$ 0 0
$$64$$ −4.74620 −0.593275
$$65$$ −0.622216 −0.0771764
$$66$$ 0 0
$$67$$ −1.24443 −0.152031 −0.0760157 0.997107i $$-0.524220\pi$$
−0.0760157 + 0.997107i $$0.524220\pi$$
$$68$$ −8.40990 −1.01985
$$69$$ 0 0
$$70$$ −8.42864 −1.00742
$$71$$ −2.75557 −0.327026 −0.163513 0.986541i $$-0.552283\pi$$
−0.163513 + 0.986541i $$0.552283\pi$$
$$72$$ 0 0
$$73$$ −4.23506 −0.495677 −0.247838 0.968801i $$-0.579720\pi$$
−0.247838 + 0.968801i $$0.579720\pi$$
$$74$$ −3.80642 −0.442488
$$75$$ 0 0
$$76$$ −11.4380 −1.31203
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −8.56199 −0.963299 −0.481650 0.876364i $$-0.659962\pi$$
−0.481650 + 0.876364i $$0.659962\pi$$
$$80$$ 4.61285 0.515732
$$81$$ 0 0
$$82$$ −0.368416 −0.0406848
$$83$$ 0.133353 0.0146374 0.00731870 0.999973i $$-0.497670\pi$$
0.00731870 + 0.999973i $$0.497670\pi$$
$$84$$ 0 0
$$85$$ 5.18421 0.562306
$$86$$ −10.7971 −1.16428
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −5.61285 −0.594961 −0.297480 0.954728i $$-0.596146\pi$$
−0.297480 + 0.954728i $$0.596146\pi$$
$$90$$ 0 0
$$91$$ 2.75557 0.288862
$$92$$ −14.3684 −1.49801
$$93$$ 0 0
$$94$$ 5.24443 0.540922
$$95$$ 7.05086 0.723402
$$96$$ 0 0
$$97$$ 7.24443 0.735561 0.367780 0.929913i $$-0.380118\pi$$
0.367780 + 0.929913i $$0.380118\pi$$
$$98$$ 24.0049 2.42486
$$99$$ 0 0
$$100$$ 1.62222 0.162222
$$101$$ 4.66370 0.464056 0.232028 0.972709i $$-0.425464\pi$$
0.232028 + 0.972709i $$0.425464\pi$$
$$102$$ 0 0
$$103$$ −11.6128 −1.14425 −0.572124 0.820167i $$-0.693880\pi$$
−0.572124 + 0.820167i $$0.693880\pi$$
$$104$$ −0.447375 −0.0438688
$$105$$ 0 0
$$106$$ 20.6637 2.00704
$$107$$ 2.62222 0.253499 0.126750 0.991935i $$-0.459546\pi$$
0.126750 + 0.991935i $$0.459546\pi$$
$$108$$ 0 0
$$109$$ 19.7146 1.88831 0.944156 0.329499i $$-0.106880\pi$$
0.944156 + 0.329499i $$0.106880\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −20.4286 −1.93032
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 8.85728 0.825946
$$116$$ −12.6637 −1.17580
$$117$$ 0 0
$$118$$ 9.24443 0.851019
$$119$$ −22.9590 −2.10465
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −13.0509 −1.18157
$$123$$ 0 0
$$124$$ 4.47013 0.401429
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 15.1842 1.34738 0.673690 0.739014i $$-0.264708\pi$$
0.673690 + 0.739014i $$0.264708\pi$$
$$128$$ 5.64941 0.499342
$$129$$ 0 0
$$130$$ −1.18421 −0.103862
$$131$$ 1.24443 0.108726 0.0543632 0.998521i $$-0.482687\pi$$
0.0543632 + 0.998521i $$0.482687\pi$$
$$132$$ 0 0
$$133$$ −31.2257 −2.70761
$$134$$ −2.36842 −0.204600
$$135$$ 0 0
$$136$$ 3.72746 0.319627
$$137$$ 0.488863 0.0417663 0.0208832 0.999782i $$-0.493352\pi$$
0.0208832 + 0.999782i $$0.493352\pi$$
$$138$$ 0 0
$$139$$ −17.8064 −1.51032 −0.755161 0.655540i $$-0.772441\pi$$
−0.755161 + 0.655540i $$0.772441\pi$$
$$140$$ −7.18421 −0.607176
$$141$$ 0 0
$$142$$ −5.24443 −0.440103
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 7.80642 0.648288
$$146$$ −8.06022 −0.667069
$$147$$ 0 0
$$148$$ −3.24443 −0.266691
$$149$$ −1.43801 −0.117806 −0.0589031 0.998264i $$-0.518760\pi$$
−0.0589031 + 0.998264i $$0.518760\pi$$
$$150$$ 0 0
$$151$$ 12.1748 0.990774 0.495387 0.868672i $$-0.335026\pi$$
0.495387 + 0.868672i $$0.335026\pi$$
$$152$$ 5.06959 0.411198
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2.75557 −0.221333
$$156$$ 0 0
$$157$$ 18.4701 1.47408 0.737038 0.675851i $$-0.236224\pi$$
0.737038 + 0.675851i $$0.236224\pi$$
$$158$$ −16.2953 −1.29638
$$159$$ 0 0
$$160$$ 7.34122 0.580374
$$161$$ −39.2257 −3.09142
$$162$$ 0 0
$$163$$ −10.1017 −0.791227 −0.395614 0.918417i $$-0.629468\pi$$
−0.395614 + 0.918417i $$0.629468\pi$$
$$164$$ −0.314022 −0.0245210
$$165$$ 0 0
$$166$$ 0.253799 0.0196986
$$167$$ −16.3368 −1.26418 −0.632089 0.774896i $$-0.717802\pi$$
−0.632089 + 0.774896i $$0.717802\pi$$
$$168$$ 0 0
$$169$$ −12.6128 −0.970219
$$170$$ 9.86665 0.756737
$$171$$ 0 0
$$172$$ −9.20294 −0.701718
$$173$$ −9.18421 −0.698262 −0.349131 0.937074i $$-0.613523\pi$$
−0.349131 + 0.937074i $$0.613523\pi$$
$$174$$ 0 0
$$175$$ 4.42864 0.334774
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −10.6824 −0.800683
$$179$$ 25.3274 1.89306 0.946530 0.322617i $$-0.104563\pi$$
0.946530 + 0.322617i $$0.104563\pi$$
$$180$$ 0 0
$$181$$ −13.6128 −1.01184 −0.505918 0.862582i $$-0.668846\pi$$
−0.505918 + 0.862582i $$0.668846\pi$$
$$182$$ 5.24443 0.388743
$$183$$ 0 0
$$184$$ 6.36842 0.469486
$$185$$ 2.00000 0.147043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 4.47013 0.326017
$$189$$ 0 0
$$190$$ 13.4193 0.973536
$$191$$ −6.10171 −0.441504 −0.220752 0.975330i $$-0.570851\pi$$
−0.220752 + 0.975330i $$0.570851\pi$$
$$192$$ 0 0
$$193$$ −18.3368 −1.31991 −0.659955 0.751305i $$-0.729425\pi$$
−0.659955 + 0.751305i $$0.729425\pi$$
$$194$$ 13.7877 0.989898
$$195$$ 0 0
$$196$$ 20.4608 1.46148
$$197$$ −6.69535 −0.477024 −0.238512 0.971140i $$-0.576660\pi$$
−0.238512 + 0.971140i $$0.576660\pi$$
$$198$$ 0 0
$$199$$ 14.1017 0.999644 0.499822 0.866128i $$-0.333399\pi$$
0.499822 + 0.866128i $$0.333399\pi$$
$$200$$ −0.719004 −0.0508412
$$201$$ 0 0
$$202$$ 8.87601 0.624514
$$203$$ −34.5718 −2.42647
$$204$$ 0 0
$$205$$ 0.193576 0.0135199
$$206$$ −22.1017 −1.53990
$$207$$ 0 0
$$208$$ −2.87019 −0.199012
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 10.6637 0.734120 0.367060 0.930197i $$-0.380364\pi$$
0.367060 + 0.930197i $$0.380364\pi$$
$$212$$ 17.6128 1.20966
$$213$$ 0 0
$$214$$ 4.99063 0.341153
$$215$$ 5.67307 0.386900
$$216$$ 0 0
$$217$$ 12.2034 0.828422
$$218$$ 37.5210 2.54124
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3.22570 −0.216984
$$222$$ 0 0
$$223$$ 8.85728 0.593127 0.296564 0.955013i $$-0.404159\pi$$
0.296564 + 0.955013i $$0.404159\pi$$
$$224$$ −32.5116 −2.17227
$$225$$ 0 0
$$226$$ 11.4193 0.759599
$$227$$ 13.3778 0.887915 0.443957 0.896048i $$-0.353574\pi$$
0.443957 + 0.896048i $$0.353574\pi$$
$$228$$ 0 0
$$229$$ 11.5111 0.760677 0.380339 0.924847i $$-0.375807\pi$$
0.380339 + 0.924847i $$0.375807\pi$$
$$230$$ 16.8573 1.11154
$$231$$ 0 0
$$232$$ 5.61285 0.368502
$$233$$ −4.32693 −0.283467 −0.141733 0.989905i $$-0.545268\pi$$
−0.141733 + 0.989905i $$0.545268\pi$$
$$234$$ 0 0
$$235$$ −2.75557 −0.179753
$$236$$ 7.87955 0.512915
$$237$$ 0 0
$$238$$ −43.6958 −2.83238
$$239$$ −3.34614 −0.216444 −0.108222 0.994127i $$-0.534516\pi$$
−0.108222 + 0.994127i $$0.534516\pi$$
$$240$$ 0 0
$$241$$ 1.34614 0.0867126 0.0433563 0.999060i $$-0.486195\pi$$
0.0433563 + 0.999060i $$0.486195\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −11.1240 −0.712140
$$245$$ −12.6128 −0.805805
$$246$$ 0 0
$$247$$ −4.38715 −0.279148
$$248$$ −1.98126 −0.125810
$$249$$ 0 0
$$250$$ −1.90321 −0.120370
$$251$$ −22.7556 −1.43632 −0.718159 0.695879i $$-0.755015\pi$$
−0.718159 + 0.695879i $$0.755015\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 28.8988 1.81327
$$255$$ 0 0
$$256$$ 20.2444 1.26528
$$257$$ 6.85728 0.427745 0.213873 0.976862i $$-0.431392\pi$$
0.213873 + 0.976862i $$0.431392\pi$$
$$258$$ 0 0
$$259$$ −8.85728 −0.550365
$$260$$ −1.00937 −0.0625983
$$261$$ 0 0
$$262$$ 2.36842 0.146321
$$263$$ −29.5812 −1.82406 −0.912028 0.410129i $$-0.865484\pi$$
−0.912028 + 0.410129i $$0.865484\pi$$
$$264$$ 0 0
$$265$$ −10.8573 −0.666957
$$266$$ −59.4291 −3.64383
$$267$$ 0 0
$$268$$ −2.01874 −0.123314
$$269$$ −8.48886 −0.517575 −0.258788 0.965934i $$-0.583323\pi$$
−0.258788 + 0.965934i $$0.583323\pi$$
$$270$$ 0 0
$$271$$ −14.6637 −0.890757 −0.445378 0.895343i $$-0.646931\pi$$
−0.445378 + 0.895343i $$0.646931\pi$$
$$272$$ 23.9140 1.45000
$$273$$ 0 0
$$274$$ 0.930409 0.0562081
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.6035 −0.877438 −0.438719 0.898624i $$-0.644568\pi$$
−0.438719 + 0.898624i $$0.644568\pi$$
$$278$$ −33.8894 −2.03255
$$279$$ 0 0
$$280$$ 3.18421 0.190293
$$281$$ −0.193576 −0.0115478 −0.00577389 0.999983i $$-0.501838\pi$$
−0.00577389 + 0.999983i $$0.501838\pi$$
$$282$$ 0 0
$$283$$ −27.1842 −1.61593 −0.807967 0.589228i $$-0.799432\pi$$
−0.807967 + 0.589228i $$0.799432\pi$$
$$284$$ −4.47013 −0.265253
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.857279 −0.0506036
$$288$$ 0 0
$$289$$ 9.87601 0.580942
$$290$$ 14.8573 0.872449
$$291$$ 0 0
$$292$$ −6.87019 −0.402047
$$293$$ −2.81579 −0.164500 −0.0822502 0.996612i $$-0.526211\pi$$
−0.0822502 + 0.996612i $$0.526211\pi$$
$$294$$ 0 0
$$295$$ −4.85728 −0.282802
$$296$$ 1.43801 0.0835825
$$297$$ 0 0
$$298$$ −2.73683 −0.158540
$$299$$ −5.51114 −0.318717
$$300$$ 0 0
$$301$$ −25.1240 −1.44812
$$302$$ 23.1713 1.33336
$$303$$ 0 0
$$304$$ 32.5245 1.86541
$$305$$ 6.85728 0.392647
$$306$$ 0 0
$$307$$ 24.4286 1.39422 0.697108 0.716966i $$-0.254470\pi$$
0.697108 + 0.716966i $$0.254470\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −5.24443 −0.297864
$$311$$ −19.8796 −1.12727 −0.563633 0.826025i $$-0.690597\pi$$
−0.563633 + 0.826025i $$0.690597\pi$$
$$312$$ 0 0
$$313$$ −15.7146 −0.888239 −0.444120 0.895967i $$-0.646483\pi$$
−0.444120 + 0.895967i $$0.646483\pi$$
$$314$$ 35.1526 1.98377
$$315$$ 0 0
$$316$$ −13.8894 −0.781340
$$317$$ −16.4889 −0.926107 −0.463053 0.886330i $$-0.653246\pi$$
−0.463053 + 0.886330i $$0.653246\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 4.74620 0.265321
$$321$$ 0 0
$$322$$ −74.6548 −4.16035
$$323$$ 36.5531 2.03387
$$324$$ 0 0
$$325$$ 0.622216 0.0345143
$$326$$ −19.2257 −1.06481
$$327$$ 0 0
$$328$$ 0.139182 0.00768504
$$329$$ 12.2034 0.672796
$$330$$ 0 0
$$331$$ 15.3461 0.843500 0.421750 0.906712i $$-0.361416\pi$$
0.421750 + 0.906712i $$0.361416\pi$$
$$332$$ 0.216327 0.0118725
$$333$$ 0 0
$$334$$ −31.0923 −1.70130
$$335$$ 1.24443 0.0679905
$$336$$ 0 0
$$337$$ −28.2351 −1.53806 −0.769031 0.639212i $$-0.779261\pi$$
−0.769031 + 0.639212i $$0.779261\pi$$
$$338$$ −24.0049 −1.30570
$$339$$ 0 0
$$340$$ 8.40990 0.456091
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 24.8573 1.34217
$$344$$ 4.07896 0.219923
$$345$$ 0 0
$$346$$ −17.4795 −0.939703
$$347$$ 2.62222 0.140768 0.0703840 0.997520i $$-0.477578\pi$$
0.0703840 + 0.997520i $$0.477578\pi$$
$$348$$ 0 0
$$349$$ −5.14272 −0.275284 −0.137642 0.990482i $$-0.543952\pi$$
−0.137642 + 0.990482i $$0.543952\pi$$
$$350$$ 8.42864 0.450530
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 9.34614 0.497445 0.248722 0.968575i $$-0.419989\pi$$
0.248722 + 0.968575i $$0.419989\pi$$
$$354$$ 0 0
$$355$$ 2.75557 0.146250
$$356$$ −9.10525 −0.482577
$$357$$ 0 0
$$358$$ 48.2034 2.54763
$$359$$ 10.7556 0.567657 0.283829 0.958875i $$-0.408395\pi$$
0.283829 + 0.958875i $$0.408395\pi$$
$$360$$ 0 0
$$361$$ 30.7146 1.61656
$$362$$ −25.9081 −1.36170
$$363$$ 0 0
$$364$$ 4.47013 0.234298
$$365$$ 4.23506 0.221673
$$366$$ 0 0
$$367$$ 33.7975 1.76422 0.882108 0.471046i $$-0.156124\pi$$
0.882108 + 0.471046i $$0.156124\pi$$
$$368$$ 40.8573 2.12983
$$369$$ 0 0
$$370$$ 3.80642 0.197887
$$371$$ 48.0830 2.49634
$$372$$ 0 0
$$373$$ −33.9496 −1.75784 −0.878922 0.476965i $$-0.841737\pi$$
−0.878922 + 0.476965i $$0.841737\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −1.98126 −0.102176
$$377$$ −4.85728 −0.250163
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 11.4380 0.586757
$$381$$ 0 0
$$382$$ −11.6128 −0.594165
$$383$$ 14.6351 0.747820 0.373910 0.927465i $$-0.378017\pi$$
0.373910 + 0.927465i $$0.378017\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −34.8988 −1.77630
$$387$$ 0 0
$$388$$ 11.7520 0.596619
$$389$$ 5.61285 0.284583 0.142291 0.989825i $$-0.454553\pi$$
0.142291 + 0.989825i $$0.454553\pi$$
$$390$$ 0 0
$$391$$ 45.9180 2.32217
$$392$$ −9.06868 −0.458038
$$393$$ 0 0
$$394$$ −12.7427 −0.641966
$$395$$ 8.56199 0.430801
$$396$$ 0 0
$$397$$ −12.7556 −0.640184 −0.320092 0.947387i $$-0.603714\pi$$
−0.320092 + 0.947387i $$0.603714\pi$$
$$398$$ 26.8385 1.34529
$$399$$ 0 0
$$400$$ −4.61285 −0.230642
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ 1.71456 0.0854082
$$404$$ 7.56553 0.376399
$$405$$ 0 0
$$406$$ −65.7975 −3.26548
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 7.12399 0.352258 0.176129 0.984367i $$-0.443642\pi$$
0.176129 + 0.984367i $$0.443642\pi$$
$$410$$ 0.368416 0.0181948
$$411$$ 0 0
$$412$$ −18.8385 −0.928108
$$413$$ 21.5111 1.05849
$$414$$ 0 0
$$415$$ −0.133353 −0.00654605
$$416$$ −4.56782 −0.223956
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −15.6128 −0.762738 −0.381369 0.924423i $$-0.624547\pi$$
−0.381369 + 0.924423i $$0.624547\pi$$
$$420$$ 0 0
$$421$$ 7.89829 0.384939 0.192470 0.981303i $$-0.438350\pi$$
0.192470 + 0.981303i $$0.438350\pi$$
$$422$$ 20.2953 0.987959
$$423$$ 0 0
$$424$$ −7.80642 −0.379113
$$425$$ −5.18421 −0.251471
$$426$$ 0 0
$$427$$ −30.3684 −1.46963
$$428$$ 4.25380 0.205615
$$429$$ 0 0
$$430$$ 10.7971 0.520680
$$431$$ 34.3051 1.65242 0.826210 0.563362i $$-0.190492\pi$$
0.826210 + 0.563362i $$0.190492\pi$$
$$432$$ 0 0
$$433$$ 14.4701 0.695390 0.347695 0.937608i $$-0.386964\pi$$
0.347695 + 0.937608i $$0.386964\pi$$
$$434$$ 23.2257 1.11487
$$435$$ 0 0
$$436$$ 31.9813 1.53162
$$437$$ 62.4514 2.98746
$$438$$ 0 0
$$439$$ 19.3176 0.921977 0.460988 0.887406i $$-0.347495\pi$$
0.460988 + 0.887406i $$0.347495\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −6.13918 −0.292011
$$443$$ 13.1240 0.623539 0.311770 0.950158i $$-0.399078\pi$$
0.311770 + 0.950158i $$0.399078\pi$$
$$444$$ 0 0
$$445$$ 5.61285 0.266074
$$446$$ 16.8573 0.798215
$$447$$ 0 0
$$448$$ −21.0192 −0.993064
$$449$$ 32.3051 1.52457 0.762287 0.647240i $$-0.224077\pi$$
0.762287 + 0.647240i $$0.224077\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 9.73329 0.457816
$$453$$ 0 0
$$454$$ 25.4608 1.19493
$$455$$ −2.75557 −0.129183
$$456$$ 0 0
$$457$$ 23.4608 1.09745 0.548724 0.836004i $$-0.315114\pi$$
0.548724 + 0.836004i $$0.315114\pi$$
$$458$$ 21.9081 1.02370
$$459$$ 0 0
$$460$$ 14.3684 0.669931
$$461$$ −28.8671 −1.34448 −0.672238 0.740335i $$-0.734667\pi$$
−0.672238 + 0.740335i $$0.734667\pi$$
$$462$$ 0 0
$$463$$ −19.3461 −0.899091 −0.449546 0.893257i $$-0.648414\pi$$
−0.449546 + 0.893257i $$0.648414\pi$$
$$464$$ 36.0098 1.67172
$$465$$ 0 0
$$466$$ −8.23506 −0.381482
$$467$$ −3.14272 −0.145428 −0.0727139 0.997353i $$-0.523166\pi$$
−0.0727139 + 0.997353i $$0.523166\pi$$
$$468$$ 0 0
$$469$$ −5.51114 −0.254481
$$470$$ −5.24443 −0.241908
$$471$$ 0 0
$$472$$ −3.49240 −0.160751
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −7.05086 −0.323515
$$476$$ −37.2444 −1.70710
$$477$$ 0 0
$$478$$ −6.36842 −0.291285
$$479$$ −24.8573 −1.13576 −0.567879 0.823112i $$-0.692236\pi$$
−0.567879 + 0.823112i $$0.692236\pi$$
$$480$$ 0 0
$$481$$ −1.24443 −0.0567412
$$482$$ 2.56199 0.116696
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −7.24443 −0.328953
$$486$$ 0 0
$$487$$ −11.5299 −0.522468 −0.261234 0.965275i $$-0.584129\pi$$
−0.261234 + 0.965275i $$0.584129\pi$$
$$488$$ 4.93041 0.223189
$$489$$ 0 0
$$490$$ −24.0049 −1.08443
$$491$$ 16.3872 0.739542 0.369771 0.929123i $$-0.379436\pi$$
0.369771 + 0.929123i $$0.379436\pi$$
$$492$$ 0 0
$$493$$ 40.4701 1.82268
$$494$$ −8.34968 −0.375670
$$495$$ 0 0
$$496$$ −12.7110 −0.570742
$$497$$ −12.2034 −0.547398
$$498$$ 0 0
$$499$$ −25.3274 −1.13381 −0.566905 0.823783i $$-0.691859\pi$$
−0.566905 + 0.823783i $$0.691859\pi$$
$$500$$ −1.62222 −0.0725477
$$501$$ 0 0
$$502$$ −43.3087 −1.93296
$$503$$ 19.0923 0.851285 0.425643 0.904891i $$-0.360048\pi$$
0.425643 + 0.904891i $$0.360048\pi$$
$$504$$ 0 0
$$505$$ −4.66370 −0.207532
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 24.6321 1.09287
$$509$$ 32.4514 1.43838 0.719191 0.694812i $$-0.244512\pi$$
0.719191 + 0.694812i $$0.244512\pi$$
$$510$$ 0 0
$$511$$ −18.7556 −0.829698
$$512$$ 27.2306 1.20343
$$513$$ 0 0
$$514$$ 13.0509 0.575649
$$515$$ 11.6128 0.511723
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −16.8573 −0.740666
$$519$$ 0 0
$$520$$ 0.447375 0.0196187
$$521$$ 29.2257 1.28040 0.640200 0.768208i $$-0.278851\pi$$
0.640200 + 0.768208i $$0.278851\pi$$
$$522$$ 0 0
$$523$$ −6.71408 −0.293586 −0.146793 0.989167i $$-0.546895\pi$$
−0.146793 + 0.989167i $$0.546895\pi$$
$$524$$ 2.01874 0.0881889
$$525$$ 0 0
$$526$$ −56.2993 −2.45477
$$527$$ −14.2854 −0.622284
$$528$$ 0 0
$$529$$ 55.4514 2.41093
$$530$$ −20.6637 −0.897574
$$531$$ 0 0
$$532$$ −50.6548 −2.19616
$$533$$ −0.120446 −0.00521710
$$534$$ 0 0
$$535$$ −2.62222 −0.113368
$$536$$ 0.894751 0.0386473
$$537$$ 0 0
$$538$$ −16.1561 −0.696539
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ −27.9081 −1.19876
$$543$$ 0 0
$$544$$ 38.0584 1.63174
$$545$$ −19.7146 −0.844479
$$546$$ 0 0
$$547$$ −41.3689 −1.76881 −0.884403 0.466724i $$-0.845434\pi$$
−0.884403 + 0.466724i $$0.845434\pi$$
$$548$$ 0.793040 0.0338770
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 55.0420 2.34487
$$552$$ 0 0
$$553$$ −37.9180 −1.61244
$$554$$ −27.7935 −1.18083
$$555$$ 0 0
$$556$$ −28.8859 −1.22503
$$557$$ −20.7971 −0.881200 −0.440600 0.897704i $$-0.645234\pi$$
−0.440600 + 0.897704i $$0.645234\pi$$
$$558$$ 0 0
$$559$$ −3.52987 −0.149298
$$560$$ 20.4286 0.863268
$$561$$ 0 0
$$562$$ −0.368416 −0.0155407
$$563$$ 37.7275 1.59002 0.795012 0.606594i $$-0.207465\pi$$
0.795012 + 0.606594i $$0.207465\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ −51.7373 −2.17468
$$567$$ 0 0
$$568$$ 1.98126 0.0831320
$$569$$ −7.33630 −0.307554 −0.153777 0.988106i $$-0.549144\pi$$
−0.153777 + 0.988106i $$0.549144\pi$$
$$570$$ 0 0
$$571$$ −36.6450 −1.53354 −0.766772 0.641919i $$-0.778138\pi$$
−0.766772 + 0.641919i $$0.778138\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −1.63158 −0.0681010
$$575$$ −8.85728 −0.369374
$$576$$ 0 0
$$577$$ 4.22216 0.175771 0.0878853 0.996131i $$-0.471989\pi$$
0.0878853 + 0.996131i $$0.471989\pi$$
$$578$$ 18.7961 0.781817
$$579$$ 0 0
$$580$$ 12.6637 0.525832
$$581$$ 0.590573 0.0245011
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 3.04503 0.126004
$$585$$ 0 0
$$586$$ −5.35905 −0.221380
$$587$$ −34.3684 −1.41854 −0.709268 0.704939i $$-0.750974\pi$$
−0.709268 + 0.704939i $$0.750974\pi$$
$$588$$ 0 0
$$589$$ −19.4291 −0.800563
$$590$$ −9.24443 −0.380587
$$591$$ 0 0
$$592$$ 9.22570 0.379174
$$593$$ 27.9398 1.14735 0.573675 0.819083i $$-0.305517\pi$$
0.573675 + 0.819083i $$0.305517\pi$$
$$594$$ 0 0
$$595$$ 22.9590 0.941227
$$596$$ −2.33276 −0.0955535
$$597$$ 0 0
$$598$$ −10.4889 −0.428921
$$599$$ 31.2257 1.27585 0.637924 0.770100i $$-0.279794\pi$$
0.637924 + 0.770100i $$0.279794\pi$$
$$600$$ 0 0
$$601$$ 8.75557 0.357147 0.178574 0.983927i $$-0.442852\pi$$
0.178574 + 0.983927i $$0.442852\pi$$
$$602$$ −47.8163 −1.94885
$$603$$ 0 0
$$604$$ 19.7502 0.803625
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 15.1842 0.616308 0.308154 0.951336i $$-0.400289\pi$$
0.308154 + 0.951336i $$0.400289\pi$$
$$608$$ 51.7619 2.09922
$$609$$ 0 0
$$610$$ 13.0509 0.528414
$$611$$ 1.71456 0.0693636
$$612$$ 0 0
$$613$$ −42.7239 −1.72560 −0.862802 0.505543i $$-0.831292\pi$$
−0.862802 + 0.505543i $$0.831292\pi$$
$$614$$ 46.4929 1.87630
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 3.51114 0.141353 0.0706765 0.997499i $$-0.477484\pi$$
0.0706765 + 0.997499i $$0.477484\pi$$
$$618$$ 0 0
$$619$$ 17.5941 0.707167 0.353584 0.935403i $$-0.384963\pi$$
0.353584 + 0.935403i $$0.384963\pi$$
$$620$$ −4.47013 −0.179525
$$621$$ 0 0
$$622$$ −37.8350 −1.51705
$$623$$ −24.8573 −0.995886
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −29.9081 −1.19537
$$627$$ 0 0
$$628$$ 29.9625 1.19564
$$629$$ 10.3684 0.413416
$$630$$ 0 0
$$631$$ 15.8163 0.629636 0.314818 0.949152i $$-0.398057\pi$$
0.314818 + 0.949152i $$0.398057\pi$$
$$632$$ 6.15610 0.244877
$$633$$ 0 0
$$634$$ −31.3818 −1.24633
$$635$$ −15.1842 −0.602567
$$636$$ 0 0
$$637$$ 7.84791 0.310946
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −5.64941 −0.223313
$$641$$ −25.8163 −1.01968 −0.509841 0.860269i $$-0.670296\pi$$
−0.509841 + 0.860269i $$0.670296\pi$$
$$642$$ 0 0
$$643$$ 18.1017 0.713862 0.356931 0.934131i $$-0.383823\pi$$
0.356931 + 0.934131i $$0.383823\pi$$
$$644$$ −63.6325 −2.50747
$$645$$ 0 0
$$646$$ 69.5683 2.73713
$$647$$ 47.0420 1.84941 0.924705 0.380684i $$-0.124311\pi$$
0.924705 + 0.380684i $$0.124311\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 1.18421 0.0464485
$$651$$ 0 0
$$652$$ −16.3872 −0.641770
$$653$$ −30.0830 −1.17724 −0.588619 0.808411i $$-0.700328\pi$$
−0.588619 + 0.808411i $$0.700328\pi$$
$$654$$ 0 0
$$655$$ −1.24443 −0.0486240
$$656$$ 0.892937 0.0348633
$$657$$ 0 0
$$658$$ 23.2257 0.905432
$$659$$ −10.2854 −0.400664 −0.200332 0.979728i $$-0.564202\pi$$
−0.200332 + 0.979728i $$0.564202\pi$$
$$660$$ 0 0
$$661$$ −27.7146 −1.07797 −0.538986 0.842315i $$-0.681192\pi$$
−0.538986 + 0.842315i $$0.681192\pi$$
$$662$$ 29.2070 1.13516
$$663$$ 0 0
$$664$$ −0.0958814 −0.00372092
$$665$$ 31.2257 1.21088
$$666$$ 0 0
$$667$$ 69.1437 2.67725
$$668$$ −26.5018 −1.02538
$$669$$ 0 0
$$670$$ 2.36842 0.0914999
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 9.86665 0.380331 0.190166 0.981752i $$-0.439097\pi$$
0.190166 + 0.981752i $$0.439097\pi$$
$$674$$ −53.7373 −2.06988
$$675$$ 0 0
$$676$$ −20.4608 −0.786952
$$677$$ −5.65433 −0.217314 −0.108657 0.994079i $$-0.534655\pi$$
−0.108657 + 0.994079i $$0.534655\pi$$
$$678$$ 0 0
$$679$$ 32.0830 1.23123
$$680$$ −3.72746 −0.142942
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −34.1847 −1.30804 −0.654020 0.756477i $$-0.726919\pi$$
−0.654020 + 0.756477i $$0.726919\pi$$
$$684$$ 0 0
$$685$$ −0.488863 −0.0186785
$$686$$ 47.3087 1.80625
$$687$$ 0 0
$$688$$ 26.1690 0.997684
$$689$$ 6.75557 0.257367
$$690$$ 0 0
$$691$$ −19.2257 −0.731380 −0.365690 0.930737i $$-0.619167\pi$$
−0.365690 + 0.930737i $$0.619167\pi$$
$$692$$ −14.8988 −0.566366
$$693$$ 0 0
$$694$$ 4.99063 0.189442
$$695$$ 17.8064 0.675436
$$696$$ 0 0
$$697$$ 1.00354 0.0380118
$$698$$ −9.78769 −0.370469
$$699$$ 0 0
$$700$$ 7.18421 0.271538
$$701$$ −29.9081 −1.12961 −0.564807 0.825223i $$-0.691050\pi$$
−0.564807 + 0.825223i $$0.691050\pi$$
$$702$$ 0 0
$$703$$ 14.1017 0.531856
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 17.7877 0.669448
$$707$$ 20.6539 0.776768
$$708$$ 0 0
$$709$$ −15.3274 −0.575633 −0.287816 0.957686i $$-0.592929\pi$$
−0.287816 + 0.957686i $$0.592929\pi$$
$$710$$ 5.24443 0.196820
$$711$$ 0 0
$$712$$ 4.03566 0.151243
$$713$$ −24.4068 −0.914043
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 41.0865 1.53548
$$717$$ 0 0
$$718$$ 20.4701 0.763938
$$719$$ −23.8163 −0.888197 −0.444098 0.895978i $$-0.646476\pi$$
−0.444098 + 0.895978i $$0.646476\pi$$
$$720$$ 0 0
$$721$$ −51.4291 −1.91532
$$722$$ 58.4563 2.17552
$$723$$ 0 0
$$724$$ −22.0830 −0.820707
$$725$$ −7.80642 −0.289923
$$726$$ 0 0
$$727$$ −32.9403 −1.22169 −0.610843 0.791752i $$-0.709169\pi$$
−0.610843 + 0.791752i $$0.709169\pi$$
$$728$$ −1.98126 −0.0734305
$$729$$ 0 0
$$730$$ 8.06022 0.298322
$$731$$ 29.4104 1.08778
$$732$$ 0 0
$$733$$ 29.8666 1.10315 0.551575 0.834125i $$-0.314027\pi$$
0.551575 + 0.834125i $$0.314027\pi$$
$$734$$ 64.3239 2.37424
$$735$$ 0 0
$$736$$ 65.0232 2.39679
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 5.06959 0.186488 0.0932440 0.995643i $$-0.470276\pi$$
0.0932440 + 0.995643i $$0.470276\pi$$
$$740$$ 3.24443 0.119268
$$741$$ 0 0
$$742$$ 91.5121 3.35951
$$743$$ 22.4385 0.823188 0.411594 0.911367i $$-0.364972\pi$$
0.411594 + 0.911367i $$0.364972\pi$$
$$744$$ 0 0
$$745$$ 1.43801 0.0526845
$$746$$ −64.6133 −2.36566
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 11.6128 0.424324
$$750$$ 0 0
$$751$$ −6.63512 −0.242119 −0.121060 0.992645i $$-0.538629\pi$$
−0.121060 + 0.992645i $$0.538629\pi$$
$$752$$ −12.7110 −0.463523
$$753$$ 0 0
$$754$$ −9.24443 −0.336662
$$755$$ −12.1748 −0.443088
$$756$$ 0 0
$$757$$ 8.75557 0.318227 0.159113 0.987260i $$-0.449136\pi$$
0.159113 + 0.987260i $$0.449136\pi$$
$$758$$ −38.0642 −1.38256
$$759$$ 0 0
$$760$$ −5.06959 −0.183893
$$761$$ 3.15257 0.114280 0.0571402 0.998366i $$-0.481802\pi$$
0.0571402 + 0.998366i $$0.481802\pi$$
$$762$$ 0 0
$$763$$ 87.3087 3.16079
$$764$$ −9.89829 −0.358108
$$765$$ 0 0
$$766$$ 27.8537 1.00640
$$767$$ 3.02227 0.109128
$$768$$ 0 0
$$769$$ 28.9590 1.04429 0.522144 0.852857i $$-0.325132\pi$$
0.522144 + 0.852857i $$0.325132\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −29.7462 −1.07059
$$773$$ −29.1427 −1.04819 −0.524095 0.851660i $$-0.675596\pi$$
−0.524095 + 0.851660i $$0.675596\pi$$
$$774$$ 0 0
$$775$$ 2.75557 0.0989830
$$776$$ −5.20877 −0.186984
$$777$$ 0 0
$$778$$ 10.6824 0.382984
$$779$$ 1.36488 0.0489018
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 87.3916 3.12512
$$783$$ 0 0
$$784$$ −58.1811 −2.07790
$$785$$ −18.4701 −0.659227
$$786$$ 0 0
$$787$$ 11.2672 0.401632 0.200816 0.979629i $$-0.435641\pi$$
0.200816 + 0.979629i $$0.435641\pi$$
$$788$$ −10.8613 −0.386918
$$789$$ 0 0
$$790$$ 16.2953 0.579760
$$791$$ 26.5718 0.944786
$$792$$ 0 0
$$793$$ −4.26671 −0.151515
$$794$$ −24.2766 −0.861543
$$795$$ 0 0
$$796$$ 22.8760 0.810819
$$797$$ −41.9625 −1.48639 −0.743195 0.669075i $$-0.766690\pi$$
−0.743195 + 0.669075i $$0.766690\pi$$
$$798$$ 0 0
$$799$$ −14.2854 −0.505383
$$800$$ −7.34122 −0.259551
$$801$$ 0 0
$$802$$ −3.80642 −0.134409
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 39.2257 1.38252
$$806$$ 3.26317 0.114940
$$807$$ 0 0
$$808$$ −3.35322 −0.117966
$$809$$ −27.8064 −0.977622 −0.488811 0.872390i $$-0.662569\pi$$
−0.488811 + 0.872390i $$0.662569\pi$$
$$810$$ 0 0
$$811$$ −6.78415 −0.238224 −0.119112 0.992881i $$-0.538005\pi$$
−0.119112 + 0.992881i $$0.538005\pi$$
$$812$$ −56.0830 −1.96813
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 10.1017 0.353847
$$816$$ 0 0
$$817$$ 40.0000 1.39942
$$818$$ 13.5585 0.474060
$$819$$ 0 0
$$820$$ 0.314022 0.0109661
$$821$$ 3.62269 0.126433 0.0632164 0.998000i $$-0.479864\pi$$
0.0632164 + 0.998000i $$0.479864\pi$$
$$822$$ 0 0
$$823$$ 42.0642 1.46627 0.733134 0.680085i $$-0.238057\pi$$
0.733134 + 0.680085i $$0.238057\pi$$
$$824$$ 8.34968 0.290875
$$825$$ 0 0
$$826$$ 40.9403 1.42449
$$827$$ 30.8256 1.07191 0.535956 0.844246i $$-0.319951\pi$$
0.535956 + 0.844246i $$0.319951\pi$$
$$828$$ 0 0
$$829$$ 7.12399 0.247426 0.123713 0.992318i $$-0.460520\pi$$
0.123713 + 0.992318i $$0.460520\pi$$
$$830$$ −0.253799 −0.00880950
$$831$$ 0 0
$$832$$ −2.95316 −0.102382
$$833$$ −65.3876 −2.26555
$$834$$ 0 0
$$835$$ 16.3368 0.565357
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −29.7146 −1.02647
$$839$$ −3.34614 −0.115522 −0.0577608 0.998330i $$-0.518396\pi$$
−0.0577608 + 0.998330i $$0.518396\pi$$
$$840$$ 0 0
$$841$$ 31.9403 1.10139
$$842$$ 15.0321 0.518041
$$843$$ 0 0
$$844$$ 17.2988 0.595450
$$845$$ 12.6128 0.433895
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −50.0830 −1.71986
$$849$$ 0 0
$$850$$ −9.86665 −0.338423
$$851$$ 17.7146 0.607247
$$852$$ 0 0
$$853$$ 26.4197 0.904595 0.452297 0.891867i $$-0.350605\pi$$
0.452297 + 0.891867i $$0.350605\pi$$
$$854$$ −57.7975 −1.97779
$$855$$ 0 0
$$856$$ −1.88538 −0.0644411
$$857$$ 38.7783 1.32464 0.662321 0.749220i $$-0.269571\pi$$
0.662321 + 0.749220i $$0.269571\pi$$
$$858$$ 0 0
$$859$$ −27.3087 −0.931760 −0.465880 0.884848i $$-0.654262\pi$$
−0.465880 + 0.884848i $$0.654262\pi$$
$$860$$ 9.20294 0.313818
$$861$$ 0 0
$$862$$ 65.2899 2.22378
$$863$$ 49.5308 1.68605 0.843024 0.537875i $$-0.180773\pi$$
0.843024 + 0.537875i $$0.180773\pi$$
$$864$$ 0 0
$$865$$ 9.18421 0.312272
$$866$$ 27.5397 0.935838
$$867$$ 0 0
$$868$$ 19.7966 0.671940
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −0.774305 −0.0262363
$$872$$ −14.1748 −0.480021
$$873$$ 0 0
$$874$$ 118.858 4.02044
$$875$$ −4.42864 −0.149715
$$876$$ 0 0
$$877$$ 4.50177 0.152014 0.0760070 0.997107i $$-0.475783\pi$$
0.0760070 + 0.997107i $$0.475783\pi$$
$$878$$ 36.7654 1.24077
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 15.1240 0.509540 0.254770 0.967002i $$-0.418000\pi$$
0.254770 + 0.967002i $$0.418000\pi$$
$$882$$ 0 0
$$883$$ −30.2480 −1.01793 −0.508963 0.860789i $$-0.669971\pi$$
−0.508963 + 0.860789i $$0.669971\pi$$
$$884$$ −5.23277 −0.175997
$$885$$ 0 0
$$886$$ 24.9777 0.839143
$$887$$ 57.1941 1.92039 0.960194 0.279333i $$-0.0901135\pi$$
0.960194 + 0.279333i $$0.0901135\pi$$
$$888$$ 0 0
$$889$$ 67.2454 2.25534
$$890$$ 10.6824 0.358076
$$891$$ 0 0
$$892$$ 14.3684 0.481090
$$893$$ −19.4291 −0.650171
$$894$$ 0 0
$$895$$ −25.3274 −0.846602
$$896$$ 25.0192 0.835833
$$897$$ 0 0
$$898$$ 61.4835 2.05173
$$899$$ −21.5111 −0.717437
$$900$$ 0 0
$$901$$ −56.2864 −1.87517
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −4.31402 −0.143482
$$905$$ 13.6128 0.452506
$$906$$ 0 0
$$907$$ −53.2641 −1.76861 −0.884303 0.466913i $$-0.845366\pi$$
−0.884303 + 0.466913i $$0.845366\pi$$
$$908$$ 21.7017 0.720195
$$909$$ 0 0
$$910$$ −5.24443 −0.173851
$$911$$ 0.590573 0.0195665 0.00978327 0.999952i $$-0.496886\pi$$
0.00978327 + 0.999952i $$0.496886\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 44.6508 1.47692
$$915$$ 0 0
$$916$$ 18.6735 0.616991
$$917$$ 5.51114 0.181994
$$918$$ 0 0
$$919$$ −55.8707 −1.84300 −0.921502 0.388375i $$-0.873037\pi$$
−0.921502 + 0.388375i $$0.873037\pi$$
$$920$$ −6.36842 −0.209960
$$921$$ 0 0
$$922$$ −54.9403 −1.80936
$$923$$ −1.71456 −0.0564354
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −36.8198 −1.20997
$$927$$ 0 0
$$928$$ 57.3087 1.88125
$$929$$ −15.3274 −0.502876 −0.251438 0.967873i $$-0.580903\pi$$
−0.251438 + 0.967873i $$0.580903\pi$$
$$930$$ 0 0
$$931$$ −88.9314 −2.91461
$$932$$ −7.01921 −0.229922
$$933$$ 0 0
$$934$$ −5.98126 −0.195713
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 27.8479 0.909752 0.454876 0.890555i $$-0.349684\pi$$
0.454876 + 0.890555i $$0.349684\pi$$
$$938$$ −10.4889 −0.342474
$$939$$ 0 0
$$940$$ −4.47013 −0.145799
$$941$$ 10.4157 0.339543 0.169772 0.985483i $$-0.445697\pi$$
0.169772 + 0.985483i $$0.445697\pi$$
$$942$$ 0 0
$$943$$ 1.71456 0.0558337
$$944$$ −22.4059 −0.729250
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8.47013 −0.275242 −0.137621 0.990485i $$-0.543946\pi$$
−0.137621 + 0.990485i $$0.543946\pi$$
$$948$$ 0 0
$$949$$ −2.63512 −0.0855397
$$950$$ −13.4193 −0.435379
$$951$$ 0 0
$$952$$ 16.5076 0.535014
$$953$$ −8.71408 −0.282277 −0.141138 0.989990i $$-0.545076\pi$$
−0.141138 + 0.989990i $$0.545076\pi$$
$$954$$ 0 0
$$955$$ 6.10171 0.197447
$$956$$ −5.42816 −0.175559
$$957$$ 0 0
$$958$$ −47.3087 −1.52847
$$959$$ 2.16500 0.0699114
$$960$$ 0 0
$$961$$ −23.4068 −0.755059
$$962$$ −2.36842 −0.0763608
$$963$$ 0 0
$$964$$ 2.18373 0.0703333
$$965$$ 18.3368 0.590282
$$966$$ 0 0
$$967$$ −44.2449 −1.42282 −0.711410 0.702777i $$-0.751943\pi$$
−0.711410 + 0.702777i $$0.751943\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −13.7877 −0.442696
$$971$$ 57.1437 1.83383 0.916914 0.399085i $$-0.130672\pi$$
0.916914 + 0.399085i $$0.130672\pi$$
$$972$$ 0 0
$$973$$ −78.8582 −2.52808
$$974$$ −21.9438 −0.703124
$$975$$ 0 0
$$976$$ 31.6316 1.01250
$$977$$ 16.2480 0.519819 0.259909 0.965633i $$-0.416307\pi$$
0.259909 + 0.965633i $$0.416307\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −20.4608 −0.653595
$$981$$ 0 0
$$982$$ 31.1882 0.995256
$$983$$ −1.12399 −0.0358496 −0.0179248 0.999839i $$-0.505706\pi$$
−0.0179248 + 0.999839i $$0.505706\pi$$
$$984$$ 0 0
$$985$$ 6.69535 0.213331
$$986$$ 77.0232 2.45292
$$987$$ 0 0
$$988$$ −7.11691 −0.226419
$$989$$ 50.2480 1.59779
$$990$$ 0 0
$$991$$ −53.6513 −1.70429 −0.852144 0.523307i $$-0.824698\pi$$
−0.852144 + 0.523307i $$0.824698\pi$$
$$992$$ −20.2292 −0.642279
$$993$$ 0 0
$$994$$ −23.2257 −0.736674
$$995$$ −14.1017 −0.447054
$$996$$ 0 0
$$997$$ 35.7275 1.13150 0.565750 0.824577i $$-0.308587\pi$$
0.565750 + 0.824577i $$0.308587\pi$$
$$998$$ −48.2034 −1.52585
$$999$$ 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 5445.2.a.z.1.3 3
3.2 odd 2 1815.2.a.m.1.1 3
11.10 odd 2 495.2.a.e.1.1 3
15.14 odd 2 9075.2.a.cf.1.3 3
33.32 even 2 165.2.a.c.1.3 3
44.43 even 2 7920.2.a.cj.1.3 3
55.32 even 4 2475.2.c.r.199.2 6
55.43 even 4 2475.2.c.r.199.5 6
55.54 odd 2 2475.2.a.bb.1.3 3
132.131 odd 2 2640.2.a.be.1.3 3
165.32 odd 4 825.2.c.g.199.5 6
165.98 odd 4 825.2.c.g.199.2 6
165.164 even 2 825.2.a.k.1.1 3
231.230 odd 2 8085.2.a.bk.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 33.32 even 2
495.2.a.e.1.1 3 11.10 odd 2
825.2.a.k.1.1 3 165.164 even 2
825.2.c.g.199.2 6 165.98 odd 4
825.2.c.g.199.5 6 165.32 odd 4
1815.2.a.m.1.1 3 3.2 odd 2
2475.2.a.bb.1.3 3 55.54 odd 2
2475.2.c.r.199.2 6 55.32 even 4
2475.2.c.r.199.5 6 55.43 even 4
2640.2.a.be.1.3 3 132.131 odd 2
5445.2.a.z.1.3 3 1.1 even 1 trivial
7920.2.a.cj.1.3 3 44.43 even 2
8085.2.a.bk.1.3 3 231.230 odd 2
9075.2.a.cf.1.3 3 15.14 odd 2